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1. Exponential Smoothing File name SIMXPO XLSX SMOOTH XLSX TYPE NONLINEAR OPTIMIZATION Application Profile Exponential smoothing is one technique that you can use to predict events in the future by studying events in the past By employing weighted averages to smooth past values it lets you forecast the value in the next period The basic model for exponential smoothing is Ba Ror P where SN predicted value at period 4 1 P predicted value in period Y actual value at period amp alpha the smoothing constant If Alpha is set to 1 the forecast for the next period is based entirely on the actual value from the last period If Alpha is set to 0 the actual value from the last period is completely ignored Since neither of these cases will provide much insight into future data we ll constrain Alpha to be between 01 and 99 The Problem in Words In order to minimize costly overstocking and inventory holding your retail outlet needs useful predictions of future sales Background For this simple exponential smoothing problem you have sales data in 1 000 s for eight months You need to find Alpha the smoothing constant that minimizes the sum of the error which in this case is the difference between the actual and predicted sales for each period Objective of Optimization The objective is to determine projected sales and the Alpha smoothing constant while minimizi
2. 9 500 3500 lt 11 500 4000 lt 9 750 3500 lt Total Value Total Weight Maximum of Load of Load Load Weight 10000 10000 WB Status TRUCK 21 You ll notice that the What sBest solution recommends loading 33 of item 1 and all 100 or 1 of item 2 While this is the best solution if fractions of items can be loaded it may be difficult to load or sell one third of a piano To eliminate the fractions you could do some What If trials and round them in the direction that doesn t violate the maximum load constraint If you round the proportion up to 1 00 the weight of the load increases to 15 000 pounds This violates the 10 000 pounds maximum weight constraint Instead round the fraction of item 1 down to zero This drops the value of the load to 7 500 pounds leaving 2 500 pounds of load capacity unused No other item weighs less than 2 500 pounds so the truck can t be filled up further 370 CHAPTER7 Rounding fractional weights as in the conventional method the truck would carry only item 2 for a load weight of 7 500 pounds and a load value of 24 000 The Binary Integer Method This problem can be converted very simply to give an integer solution one with no fractional weights for an item Use the binary nteger command to specify that the adjustable cells in D6 D11 cannot be anything other than zero or one You could also eliminate the constraints in E6 E11 and the constants in
3. The cash required to buy any asset must come from selling some asset The Proceeds From Sales is calculated in E17 by the formula E7 1 C 7 E8 1 C 8 E9 1 C 9 The constraint in cell F17 WB E17 gt G17 requires the Proceeds from Sales to be at least as great as the Cost of Buys in G17 calculated by the formula F7 1 C 7 F8 14 C 8 F9 1 C 9 268 CHAPTER7 Now let s solve the model After solving the WB Status worksheet will open in order to show you the Nonlinearity present warning This warning can be shut off from the General Options dialog box Your solved model now appears as follows The PORTCOST Model After Solving PORTFOLIO WITH TRANSACTION COSTS e D E E G H l Ca Transaction Cost Return Rate Begin Sold ht End Asset1 9 0 1 0 50 0 0 0 28 1 78 1 Asset2 13 5 1 5 30 0 17 6 0 0 124 Asset3 18 0 2 0 20 0 11 3 00 8 7 Covariance Matrix End Value Desired Asset 1 Asset 2 Asset3 Port Return End Value 109 5 gt 109 5 Asset 1 25 0 2 5 0 0 Asset 2 25 150 0 40 0 Proceeds Cost of Asset 3 0 0 40 0 256 0 From Sales Buys 28 4 gt 28 4 Variance 20 88 In the optimal solution you are holding 78 1 of Asset 1 12 4 of Asset 2 and 8 7 of Asset 3 These amounts are percentages of the value of your original portfolio You may notice that the percentages add to only 99 2 rather than 100 This is because 8 of the value of the original portfolio has g
4. IntPreSolvOpt_BadGUBCutsArg Bad GUB Cuts argument IntPreSolvOpt_BadKnapsackCoverCutsArg Bad Knapsack Cover Cuts argument IntPreSolvOpt_BadLatticeCutsArg Bad Lattice Cuts argument IntPreSolvOpt_BadLiftingCutsArg Bad Lifting Cuts argument IntPreSolvOpt_BadPlantLocationCutsArg Bad Plant Location Cuts argument IntPreSolvOpt_BadObjIntegralityCutsArg Bad Objective Integrality Cuts argument IntPreSolvOpt_BadBasicCutsArg Bad Basic Cuts argument IntPreSolvOpt_BadCardinalityCutsArg Bad Cardinality Cuts argument Example If one wished to set all the options the following would work wbSetIntegerPreSolverOptions 2 RRE EC D Dy False False True True False True True False False True True False True To set individual options here setting the ProbingLevel to 4 named arguments should be used wbSetIntegerPreSolverOptions ProbingLevel 4 172 CHAPTER 4 wbSetLinearOptions This routine is used to set the What sBest linear solver options which can be seen in the Linear Solver Options dialog box All arguments are optional For additional discussion of the options available through this routine see the section entitled Options Linear Solver Syntax wbSetLinearOptions LinearSolverMethod ScaleModel Reduction PrimalPricing DualPricing Argument Required Default Description LinearSolverMethod No 0 Solver Decides
5. E9 forces the Footprint in C9 to be no greater than the upper limit of 252 square inches in E9 The Footprint is calculated by the formula Width Length In D11 the formula WB C11 gt E11 forces the Volume in C11 calculated by the formula Length Width Height to be at least as great as the lower limit of 1512 cubic inches required to fit the electronic equipment in the cabinet Finally in D13 and D15 the formulas WB C13 lt E13 and WB C15 gt E15 force the ratio of Height to Width to fall between 718 and 518 where C13 and C15 both calculate Width Height 242 CHAPTER7 Now you re ready to solve the model and find the optimum design The BOX Worksheet After Optimization CABINET DESIGN ACTUAL LIMIT Surface Area 888 000 Length Width Height 23 03 6 87 9 56 Footprint 158 123 Volume 1512 000 UNIT COST 50 97 Width Height 0 718 lt 0 718 Width Height 0 718 gt 0 518 The resulting 23 03 x 6 87 x 9 56 inch box is the lowest cost design 50 97 per unit that meets all the requirements of your various departments SAMPLE MODELS 243 Flow Network Modeling File name FLOWNET XLSX TYPE NONLINEAR The Problem in Words You are the manager of a network of pipelines and you need to determine a balance of flows along each arc of the network while satisfying demand at the nodes Background In the FLOWNET model you have two supply nodes with connections to a
6. In many multi period modeling problems liquid or cash like assets are treated like other commodities so holding cash is just like keeping inventory The following illustrates the major features of multi period models in a financial context The Problem in Words You re a financial advisor for an investment firm Your client needs enough cash to cover commitments for the next five years You ve concluded that to meet these financial obligations he should invest in some very low risk securities such as government bonds You must recommend bonds to buy to minimize his total cost now yet cover his cash flow requirements This is also an example of how to defease debt how to wipe it off the books Background Your client s specific cash flow needs are as follows Year 0 1 2 3 4 Cash Flow Need M 17 0 62 0 23 0 35 0 62 0 In graph form the data look like this 70 00 60 00 50 00 40 00 30 00 20 00 10 00 S 0 00 Year 0 Year 1 Year 2 Year 3 Year 4 252 CHAPTER7 You ve counseled your client to buy government bonds not only because of their low risk but also because a bond owner receives a fixed set of cash payouts For each year until the bond matures your client receives an interest payment and at maturity He also gets back the face value or principal of the bond Generally a broad spectrum of such investments are available on a
7. KY E Best Je A Constraints lt gt After you have created your model by setting up the ABC s you can Solve your worksheet model and find the best answer to your problem 19 20 CHAPTER2 Adjustable The dialog box posted by the Adjustable command appears as follows Adjustable m Make Adjustable Refers to SAS E This command allows you to set properties of selected cells in your workbook Specifically it is used to identify to What sBest the cells that may be adjusted by the solver in it s search for a solution that optimizes the objective cell while satisfying all constraints Enter the range of cells whose properties you wish to set in the Refers To field the current selection of cells is already entered Next select the property you wish to apply to the cells from the drop down list at the top of the dialog box You have the following options 1 Make Adjustable 2 Remove Adjustable 3 Make Adjustable amp Free or Remove Free See below for more detail on these options Clicking OK will apply the change in properties Make Adjustable Select Make Adjustable to specify the range indicated in the Refers To text box as adjustable cells Before What sBest can solve a problem it must know what cells it can change in its search for the optimal solution These are the adjustable cells By default What sBest assumes the adjustable cells can be set to any non
8. Level 4 elimination Level 5 dual reductions Level 6 use dual information Level 7 binary row presolving Level 8 row aggregation Level 9 maximum pass fee OOH e Ca The default setting for Probing Levels is Solver decides which results in the solver setting probing to a level it believes will offer the best result ADDITIONAL COMMANDS 69 Constraint Cuts The options contained in the Constraint Cuts box on the Integer Pre Solver dialog box Constraints Cuts Max Passes Relative Limits Types Iw Coefficent Reduction Iw Lattice Iw Disaggregation M Lifting V Flow Cover Iw Plant Location Iw GCD Obj Integrality V Gomory V Basic V GUB Cardinality Iw Knapsack Cover can be used to influence the solver s cut generation phase on linear models What sBest s integer programming pre solver performs extensive evaluation of your model in order to add constraint cuts Constraint cuts are used to cut away sections of the feasible region of the continuous model i e the model with integer restrictions dropped that are not contained in the feasible region to the integer model On most integer models this will accomplish two things First solutions to the continuous problem will tend to be more naturally integer Thus the branch and bound solver will have to branch on fewer variables Secondly the bounds derived from intermediate solutions will tend to be tighter allowing the solver to
9. Maximize To reset cell F17 to None enter wbBest F17 j None UI WBBIN For additional discussion of the functionality provided see the section entitled Integer which refers to the Binary option button that calls this procedure This routine is used to build binary integer ranges named WBBIN The adjustable cells contained in these ranges will be set to either 0 or 1 by the What sBest solver Syntax WBBIN IntName Refers_to Argument Required Description IntName Yes IntName is a string that refers to the range being made integer Refers_to No Refers_to is the range or address of the cells to be specified as integer If omitted the default is the current selection Error Codes Description Int_BadIntNameArg Bad JntName argument Int_BadRefers_toArg Bad Refers_to argument Int_ProtectedShe etError Unable to set integer cells on a protected sheet Int_CreateError Error setting integer cells Remarks Integer variables can dramatically increase total solution times 142 CHAPTER 4 Example To constrain cell F7 to be a binary integer using the range name OnOff enter WBBIN OnOff UI F7 wbConstraint For additional discussion of the functionality provided see the section entitled Constraints which refers to the dialog box that calls this procedure This routine is also called with appropriate arguments by the three too
10. will delete the existing license key HINT Control Panel of Windows shows all What sBest to remove Cancel Simply click on the Remove Only button Then a dialog box will appear to confirm your choice By clicking Yes your What sBest add in files will be completely removed Selecting No and then Cancel will close the uninstall program After uninstalling the software you might want to verify that Excel has cleaned up its own lists e g Toolbar and Add ins available To do this simply open Excel and choose the Tools Add ins command or the Office Button then Options Add ins If the What sBest add in appears in the list of Add Ins available click on the check box before it An error message should appear informing you that the WBA XLA or WBA XLAM file can not be found and prompt you to delete it Thus click Yes and the entry will be removed from the list Similarly to clean up the Excel list of toolbars choose the View Toolbars Customize command or right click on the Ribbon bar If the What sBest toolbar appears in the list of Toolbars select the What sBest toolbar click the Delete button and click the OK button to delete it from the list INSTALLATION DETAILS 447 The What sBest add in can always be reinstalled by launching the installation program again Appendix B Contacting LINDO Systems If you have questions or problems running the software or you have a suggestion regar
11. In the Preview window we see that the solver was able to find 5 feasible next best solutions to the model The solutions are ranked in order by their objective values There is also a column labeled Trade off Cells which lists the value in each solution of a designated trade off variable Any scalar variable in a model can be selected as the tradeoff variable In this example there are 9 variables and the objective cell WBMAX is automatically selected as a tradeoff variable The idea behind the trade off variable is that it allows you to weigh the trade offs in a model s objective value with a secondary goal In this case our secondary goal is the number of our favorite items in the picnic basket In particular we see that there are three solutions with slightly worse objective values 23 vs 25 that include one of our favorite items For example if we selected the solution Run 2 and pressed the This Selection button we d see the following solution containing one of our favorite items Brownies cell F4 The Take Default button selects the Run 1 which is the solution of the objective cell These buttons allow you to display selected solutions returned by the K Best solver Once a final solution is selected all subsequent status and solution reports will be based on that particular solution 82 CHAPTER3 Options Stochastic Solver The Stochastic Solver option allows you to solve models in which some cells are random var
12. NEWSVENDOR xls gt oer a Se Optimization of eegenen Normal P Stage 1 in the beginning unknown demand is revealed to us 5 Stage 1 atthe end we compute our sales and the resulting profit Step 1 Core model lt lt Stage 0 decis lt lt Stage 1 random demand lt lt Stage 1 recourse decision lt lt Stage 1 constraint lt lt Stage 1 decision and constraint lt lt Stage 1 non negativity constraint lt lt Stage 0 cost computation lt lt Stage 1 holding cost computation lt lt Stage 1 shortage cost computation lt lt Stage 1 revenue computation lt lt Stage 1 expected value maximize MATHEMATICAL MODELING 213 The scenario by scenario report is generated on the WB Stochastic tab A portion of it appears below ed 3 GEIS NEWSVENDOR xlsx Microsoft Excel bala File Home Insert Page La Formul Data Review View Develop Add Ins e gi X DATE GENERATED Apr 05 2011 08 19 AM STOCHASTIC INFORMATION RANDOMS STAGES NODES SCENARIOS 200 Expected Value EV 2 11E 03 Expected Value of Wait and See Model s Objective EVWS 2 80E 03 Expected Value of Perfect Information EVWS EV 6 88E 02 Expected Value of Modeling Uncertainty EV EVEM 2 78E 01 REPORTING CELLS 0 SCENARIO PROBABILITY Model B11 Model B12fodel B13 DLOSTSALES STAGE 0 STAGE 1 STAGE 1 85 752294 104 62219 18 86989 85 752294 82 987394 D
13. Refers_to IntChoice Argument Required Description IntName No IntName is a string that refers to the range being made integer Refers_to No Refers_to is the range or address of the cell s to be specified as integer If omitted the default value is the current selection MACROS THE VBA INTERFACE 155 IntChoice No IntChoice is a string to indicate if the cells should be binary or general integer IntChoice accepts BIN or BINARY for binary 0 1 and INT or GENERAL for general integer If omitted the default is binary For backwards compatibility with earlier versions of What sBest IntChoice also accepts the numbers 1 for binary and 2 for general Error Codes Description Int_BadIntNameArg Bad JntName argument Int_BadRefers_toArg Bad Refers_to argument Int_BadIntChoiceArg Bad JntChoice argument Int_ProtectedSheetError Unable to set integer cells on a protected sheet Int_CreateError Error setting integer cells Remarks Integer variables can dramatically increase the solution time Example To constrain cell F6 to be a general integer using the range name staff enter wbInteger staff F6 General To constrain cell F7 to be a binary integer using the range name OnOff enter wbInteger OnOff F7 Binary Note The second of these two examples could be done without the JntChoice argument of Binary which is the default 156 CHAPTER 4 wbintegerCard Fo
14. Stage 1 shortage cost computation lt lt Stage 1 revenue computation lt lt Stage 1 expected value maximize 210 CHAPTER 6 The NEWSVENDOR Worksheet before Optimization part 2 _ LAJE EWSVENDOR xlsx Microsoft Excel el eat Al A Stochastic Scenario Optimization of Newsvendor Normal Fa T ai G D E F G H l J K b Mle Stochastic data 2d to us ulting profit Step 2a Stage information Core model WESP_VAR Q TC stage 0 WESPLRAND WESP_VAR lt Stage 0 decision lt Stage 1 random demand Step 2b Distribution information lt lt Stage 1 recourse decision WESP_DIST_NORMAL 14 lt lt Stage 1 constraint lt Stage 1 decision and constraint Mean Demand lt Stage 1 non negativity constraint Standard Deviation lt Stage 0 cost computation lt Stage 1 holding cost computation Step 3 Sample size lt lt Stage 1 shortage cost computation WBSP_STSC 20 lt lt Stage 1 revenue computation Stage Scenarios Ke lt Stage 1 expected value maximize Step 4 Reporting cells For this specific problem the five steps mentioned earlier are Step 1 The core model is described numerically in column B and in words in columns A and C Step 2a We specify cell B11 as a stage 0 decision variable by inserting the expression WBSP_VAR 0 B11 into cell I6 We can enter this expression directly or using the Stochastic Support dialog bo
15. Suggestions In rare instances if a formula becomes too complex What sBest may not be able to successfully parse it Things to check for are nested functions that are many layers deep in which case you may want to break the formula up into multiple cells Another possibility is that you have used a text string as part of the formula Finally some Excel functions must follow a specific format in order to be parsed by WB You may need to contact LINDO Systems for assistance FORMULA2 Unsupported Functions What sBest does not directly support all the functions available in Excel If any unsupported functions references are found in any of the cell formulas the following warning message will be displayed on the WB Status tab Warning Message WARNING Unsupported Functions Help Reference FORMULA2 The cells listed contain spreadsheet functions that are not defined in What sBest The numeric values for these cells are taken from the spreadsheet directly without recalculation cell addresses listed at bottom of tab Suggestions Cells in your model can reference Excel functions that are not supported by What sBest However these cells values will be fixed at their calculated values at the start of the solve command The list of cells referencing unsupported functions may be found at the end of the report Supported functions and operators are listed in the Supported Functions and Operations section If th
16. The best solution to this problem meets nutritional needs at the lowest cost Total Cost J16 contains the formula SUMPRODUCT C16 F16 C13 F13 That s the Percentage of Blend times the Cost Per Bushel for each grain SAMPLE MODELS 233 C Specify Constraints There are two limitations on the solution to this problem First the final mix must contain the minimum required levels of nutrients This is enforced by a constraint for each nutrient H7 H10 Each constraint will return the Not gt indicator until the minimum requirement 17 110 for that nutrient is met in the Nutrients Supplied column G7 G10 Second the constraint in G16 forces the percentages in C16 F16 to sum to 100 The Nutrients Supplied cell for Nutrient A G7 contains SUMPRODUCT C7 F7 C 16 F 16 which sums the product of the Percentage of Blend and the amount of Nutrient A for each grain Each grain contains different combinations of each nutrient C7 F10 Now let s solve the model with What sBest to find the optimal solution The HOGFEED Worksheet After Optimization Nutrients Per Unit Weight of Grain Nutrients Minimum Dual ltem 1 2 3 4 Supplied Reqd Value Nutrient A 2 2 3 4 7 2 1 5 3 77 gt 24 0 00 Nutrient B 1 4 14 0 0 0 8 1 0 gt 0 7 0 00 Nutrient C 23 56 1114 13 50 gt 50 4 55 Nutrient D 120 1149 418 521 210 gt 210 0 17 3 Cost Bushel 35 00 50 00 80 00 95 00 Percentage Unity kF of Blend 68 5 13 302
17. What sBest makes use of a number of temporary files when solving your model If What sBest encounters a problem managing its temporary files the following error message will be displayed on the WB Status tab Error Message KERR OR Error Managing Temporary Files Help Reference TEMPFILE An error occurred while attempting to access temporary work files Please delete the following files from the folder containing your workbook LINDOWBS XLS LINDOWBX XLS LINDOWBS _XLSB LINDOWBX LINDOWBRC TXT LINDOWBSOLN TXT LINDOWBSTATUS PRN LINDOWBSOLN PRN LINDOWBSTOC PRN LINDOWBSTOH PRN Also verify the access rights on this folder to create files The folder path name should be shorter and may not contain any symbols Suggestions This error occurs when temporary work files used by What sBest could not be accessed Go to the folder where your model workbook is located and remove the files listed above You should then be able to solve your model UNBOUNDED Unbounded Model If without violating any constraints the value of the cell to be maximized can be increased to infinity or the value of the minimized best cell can be decreased to negative infinity then the problem is said to be unbounded In this case the following error message will be displayed on the WB Status tab Error Message Unbounded Model Help Reference UNBOUNDED The value of the cell to be maximized minimized can be increased decreased without li
18. What sBest Version 11 0 Users Manual Taking your spreadsheet beyond What If 1415 North Dayton Street Chicago Illinois 60642 USA Phone 312 988 7422 Fax 312 988 9065 info lindo com www lindo com COPYRIGHT The What sBest software and its related documentation are copyrighted You may not copy the What sBest software or related documentation except in the manner authorized in the related documentation or with the written permission of LINDO Systems Inc TRADEMARKS What sBest and LINDO are registered trademarks of LINDO Systems Inc Other product and company names mentioned herein are the property of their respective owners DISCLAIMER LINDO Systems Inc warrants that on the date of receipt of your payment the enclosed computer media contains an accurate reproduction of the What sBest software and that the copy of the related documentation is accurately reproduced Due to the inherent complexity of computer programs and computer models the What sBest program may not be completely free of errors You are advised to verify your answers before basing decisions on them NEITHER LINDO SYSTEMS INC NOR ANYONE ELSE ASSOCIATED IN THE CREATION PRODUCTION OR DISTRIBUTION OF THE WHAT SBEST SOFTWARE MAKES ANY OTHER EXPRESSED WARRANTIES REGARDING THE DISKS OR DOCUMENTATION AND MAKES NO WARRANTIES AT ALL EITHER EXPRESSED OR IMPLIED REGARDING THE WHAT SBEST SOFTWARE INCLUDING THE IMPLIED WARRA
19. cceccccceeecee cece eeeeeeaeaeceeeeeeeeeaaaeaeceeeeeeenseeeaeaeeeeeeeeeeennneeees 440 Location of the Add IN Files cceccccecssccecessneeeeeesneeeeesesaeeeeeseaeeeeeseeeeeesseeeeseseeeeenees 440 Installation Related Problems ssnnssssnneeennnesenennnsetrnnsttnnnnennnnntnnnnnttnnnnstnnnnnnnn nesena nennen 440 Settings for Microsoft Excel 2007 2010 440 Settings for Microsoft Excel 1O0 2007 441 ee EE 443 Location of the Add In Files and Update Links 444 Uninstall UE 446 APPENDIX B CONTACTING LINDO SYSTEMS ssccccseeeseseeesseeenseeeeeeeeeeseaesnseeeeeeeeeas 449 TE ee ee EEN 449 xii PREFACE PREFACE xiii Preface What sBest 11 0 makes available to your Microsoft Excel spreadsheet program a highly developed solver capable of performing linear integer quadratic general nonlinear and stochastic optimization on the most difficult of problems What sBest gives you access to this solver from within Excel and may either be run directly or called from within Visual Basic The earliest version of What sBest became available in 1984 for VisiCalc and What sBest has been under continuous development since then Models have been built for What sBest that solve problems in virtually every field of professional endeavor The Sample Models are provided to demonstrate the range of applications for What sBest What sBest 11 0 includes major solver enhancements
20. s solve the model The Shipping Costs Worksheet After Optimization PLANTLOC xlsx Microsoft Exce ei Security Warning Automatic update of links has been disabled WE WEE SS a a a PLANT LOCATION MODEL Shipping Costs Per Ton Month Operating Cost 15 250 4 130 13 200 Trans Costs 31 588 TOTAL OPERATING COST 46 838 332 CHAPTER7 The Amount Shipped Worksheet After Optimization PLANTLOC xls e eier kees Automatic update of links has been disabled Amount Shipped Demand City Supply Supply Capacity Atlanta Boston Chicago Denver Omaha Portland imitation in Tons et Demand Demand Plants are open in Baltimore Cheyenne and Memphis 19 110 and I12 of the Shipping Costs worksheet Total Operating Cost in J18 of the Shipping Costs worksheet has been minimized to the optimally low figure of 46 838 Since the constraint in H9 of the Amount Shipped worksheet is not binding you know that additional sales out of Baltimore are possible SAMPLE MODELS 333 Staff Scheduling File name STAFF XLSX TYPE LINEAR OPTIMIZATION Application Profile In Staff Scheduling problems the goal is to meet specified manpower requirements at minimum cost In general the schedule must meet certain conditions such as those imposed by regulations or union contracts including minimum shift length number of work breaks or maximum overtime hours These models have applications in the scheduling of a
21. sBest the solver automatically converts this nonlinear model to an equivalent linear model The Problem in Words You are a contractor specializing in home construction You have the opportunity to bid on a job that involves building a number of homes in a new housing development In order to place an intelligent bid you need to calculate your expected cost on a per home basis Based on past experience you know the following three factors primarily determine the construction cost of a house total square footage number of bedrooms and number of baths You believe there is a linear relationship between these three factors and total cost Specifically you have come up with the following equation Total Cost B B Sq Feet B Bedrooms B Bathrooms i You pull up the following historical data from projects your company has completed in the past Sq Feet Beds Baths ___Cost 000 1500 1 1 78 5 1600 1 1 105 8 2200 2 1 149 1 2600 3 2 173 8 3000 3 2 224 4 3500 5 2 267 0 4000 4 4 302 7 5200 2 3 302 0 8200 4 4 438 7 8700 6 4 490 4 Objective of Optimization You will use your historical data to estimate values for the beta coefficients 2 2 2 and Bim formula i You want your estimation function to avoid the case where it might be a poor predictor on one type of home Thus you have decided you want your model s objective to minimize the maximum error in your cost estimations Note that this process of fitting a linear f
22. takes a random decision oz leading to realizations of all random events in stage T and T 1 at the end of stage 7 having seen all of nature s T previous decisions as well as all our previous decisions we make the final recourse decision vz 1 Xp 7 This relationship between the decision variables and realizations of random data can be illustrated as follows Decision Xe Decision Zi Stage t 1 Stage t Each decision represented with a rectangle corresponds to an uninterrupted sequence of decisions until the next random event And each random observation corresponds to an uninterrupted sequence of random events until the next decision point Note A stage is the pair random event followed by a decision The initial stage 0 is special in that it consists only of our first decision The final stage may be special in that it may consist of just a random event with no explicit final decision MATHEMATICAL MODELING 207 Recourse Models The decision taken in stage 0 is called the initial decision whereas decisions taken in succeeding stages are called recourse decisions Recourse decisions are interpreted as corrective actions that are based on the actual values the random parameters realized so far as well as the past decisions taken thus far Recourse decisions provide latitude for obtaining improved overall solutions by realigning the initial decision with possible realizations o
23. 00 Dual Value 0 00 0 00 0 00 57 88 4 4 gt WB Status HOGFEED 2 The What sBest solution has a total cost of 48 78 per bushel of the optimal blend Grain 4 which is not a bargain at 95 per bushel is not purchased 234 CHAPTER7 D Dual Values Dual values have been included in two locations in this worksheet C18 F18 and J7 J10 The dual value entries in cells J7 through J10 show the reward i e how much money could be saved if each respective constraint H7 H10 were relaxed by one unit In practical terms this means that you could save 4 55 per bushel if the requirement for Nutrient C were reduced to 4 If slightly less sprightly hogs are tolerable in return for such a savings relaxing this nutritional requirement would be a good decision Likewise if the requirement for Nutrient D were reduced to 20 the cost bushel would go down by 17 Note that the dual values are zero for Nutrients A and B Lowering the requirement by one unit offers no savings because there is already an excess of Nutrients A and B in your optimum blend since constraints H7 and H8 are not tightly satisfied The other dual values in the Swine amp Roses model C18 F18 show the amount by which the price of an unused grain would have to be reduced before it would be cost effective to use it in the minimum cost blend For example the dual value for Grain 4 is 57 88 F18 This tells you how much the price of Grain 4 would have
24. 15 020 Note how effectively What sBest used the available raw materials and how quickly this optimal solution was calculated Note also how many of each product What sBest tells you to produce and that the solution manufactures none of the product with the highest profit contribution This nonintuitive result which you d have difficulty finding through what if is a good indication of the power of What sBest 314 CHAPTER7 The Building Block Method File name BLOCK XLSX TYPE LINEAR OPTIMIZATION Application Profile This example uses the full power of your spreadsheet software to develop larger applications using smaller ones as building blocks In this way comprehensive models can be created that can simulta neously cover many different aspects of a business situation The Problem in Words You run three plants Each plant is capable of producing six products from six raw materials You also operate two steel mills that can ship steel as a raw material to any of your plants You want to know the amount of steel each mill should ship to each plant and the number of products each plant should produce to maximize your total profit Background The PRODMIX shown earlier and SHIPPING shown later worksheets are combined in the BLOCK problem which involves maximizing the profit from several manufacturing plants while minimizing the cost of shipping raw materials from multiple sites The cost of shipping one unit
25. 230 Hold Interrupt 406 Hurdle 75 IF function 187 188 IKBREP 404 INDICMOD 404 Industrial Version 120 Infeasibility 32 37 INFEASIBLE 201 405 INFLARG 406 Ingredients 230 Initial Values 203 Inner Sum Product 192 Installation 4 439 47 INT 187 Integer command refers to 42 Integer names in workbook 42 Integer Pre Solver Options Dialog Box 66 Integer problems 204 binary 2 42 general 2 42 Solution method in 71 Integer Solver Options Dialog Box 71 Integers 120 Integers Bin 33 Integers Integer Binary 41 Integers Semi continuous 46 Integers Special Ordered Set 43 Integrality 72 Interactive environment 7 INTERRUPT 406 Invalid Outside Procedure 397 INVENT XLSX 306 Inventory 284 306 310 Inverse exponential cumulative distribution 190 Inverse multinomial cumulative distribution 191 Inverse triangular cumulative distribution 189 Inverse uniform cumulative distribution 190 Invest 265 269 Investment 273 INVESTMENTCOLLEGE XLSX 215 INVMOD 408 IRRECONST 408 Iteration Limit for Slow Progress 62 INDEX 455 Iterations 32 ITRLIM 409 Joint distribution 222 K K Best Solutions 77 Kendall correlation 222 Knapsack 367 Knapsack Cover 70 Kuhn Tucker 200 Language 127 Latin hypercube 222 Lattice 70 Left hand side 25 LHS 25 LICCAP1 409 LICCAP2 410 LICCAP3 411 LICCAP4 411 LICCAPS 412 License key 4 LICKEY 1 413 LICKEY 2 413 LICKEY3 414 LICK
26. 250 Not lt 300 Not lt 260 Not lt 250 Not lt 240 Not lt 210 Not lt 140 Not lt mo AO bois EI KKK KKK KKK KKK KKK KKK s Source Capacity Cost Unit Source Holding Cost Hd Big WI A Determine Adjustable Cells The adjustable cells D6 F18 are the amounts purchased for each time period from each of the three supply sources B Define Best The best solution to this inventory problem consists of the combination of purchases from each source in each time period that results in the minimum Total Cost H21 The formula in this cell SUM H6 H18 is the sum of period costs Each period cost is composed of the sum of the purchasing and inventory holding costs for that period 308 CHAPTER 7 C Specify Constraints The constraints are that there must be sufficient supply for each time period to satisfy your demand and the capacity of the three suppliers must not be exceeded The first constraints are found in C6 C18 Look at the formula in C6 WB B6 lt G5 SUM D6 F6 The amount held over from the period before G5 is added to the sum of purchases from the three sources and required to be greater than or equal to total demand for period 1 B6 The second group of constraints D27 F39 is that supply sources not be used beyond their capacity is in cells D27 F39 In cell D27 for instance is the formula WB D6 lt D 20 D6 the amount purchased from Source 1 in Period
27. 4 each for Schedules 2 and 8 and 2 each for schedules 3 and 5 Each person s preference score is multiplied by his or her seniority index to get a weighted preference score Employees are assigned to work patterns according to these weighted scores with the highest score employees assigned first The only exception to this rule occurs when assigning an employee with a high score to a first choice It may be necessary to assign employees with low weighted scores to their lowest preferences In such instances the program would assign the senior employee to the second or third choice so as to benefit the junior staff members This is in keeping with the objective of maximizing group preference rather than individual preferences To follow strict seniority in staff assignment just use a different scale for seniority performance scores For example vary the scores between zero and 100 rather than 10 and 20 The first scoring scheme would place a much higher emphasis on senior employees than the latter one 354 CHAPTER7 You can tailor the model to meet specific needs with only slight modifications in the structure of input data The same is true for the stage 1 model A Determine Adjustable Cells The adjustable cells in US5 AF16 of FIXED show whether a person is assigned to a work pattern There are 12 adjustable cells corresponding to each individual being assigned indicating what one of the 12 possible work patterns that person is a
28. 85 752294 57 430381 7 3 85 752294 72 294769 85 Any of the standard statistical tools in Excel can be used to analyze the scenario data 214 CHAPTER 6 The histogram generated on the WB Histogram tab is CJ td 9 gt Te MEIER Microsoft Excel Lei bt File Home Insert Page La Formul Data Review View Develop Add Ins Y ei o ep ZS DATE GENERATED Apr 05 2011 08 19 AN DATA HISTOGRAM Model B22 TOT_PROF Bin Lows Bin Highs Mid Points Bin Probabilitie 1036 132393 805 12242 920 62741 472 580823 217 41946 345 00014 124 706332 272 176355 73 735011 356 74066 745 313288 551 026974 793 259154 1204 70012 998 979634 1233 054598 1635 41617 1434 23538 1662 79893 2106 16119 1884 48006 2128 178245 2548 64441 2338 41133 2556 527149 3001 33029 2778 92872 Histogram Notice that even though the driving random variable Demand has a Normal distribution the total profit has a highly skewed decidedly non Normal distribution MATHEMATICAL MODELING 215 The histogram graph is 0 6 0 5 3 0 4 s 03 g E 02 ot o T T T T Rn T el T ES T T T 1 9 D Ki bh Vv e 5 o i 5 S D Ny Wi ka Wi A Ai D A A 9 E o V E g SA i ff SI e A by Ki Y A TOT_PROF Defining an SP Model in What sBest Multi Stage Example Next we give a slightly more complex example that illustrates ho
29. A stochastic model to maximize revenue by allocating crops to grow on a specified land area over 2 stages and 1 random parameter for the weather condition CROPALLOC XLSX Put Option A stochastic model to maximize wealth by selling an option at the right time over a multi stage structure and random parameters for the stock returns PUTOPTION XLSX 230 CHAPTER7 Blending File name HOGFEED XLSX TYPE LINEAR OPTIMIZATION Application Profile The manufacture of animal feed is an example of a blending problem The question is how do you mix a number of raw components into a minimum cost final blend that has specified amounts of certain ingredients In the case of hogfeed the issue is vital How can raw feedstocks be combined to produce a high nutrition hogfeed at lowest cost For someone with several thousand hungry hogs the quality of this answer can easily spell the difference between profit and loss Since feedstock prices are typically volatile the complexity of the question and the daily importance of reliable solutions is even more apparent Blending problems are also of interest well beyond the field of animal husbandry Examples include finding the least costly mix of ores that will produce an alloy with specified characteristics or deciding on the most efficient combination of advertising media purchases to reach target audience percentages in multiple demographic groups The Problem in Words You are the manag
30. CellRangel CellRange2 CellRange3 Refers_to Place_in_cell NoErrDialog FunctionName Yes WBSP FunctionName is a string argument indicating the name of the function to implement Stage Yes Stage is an integer between and 255 indicating the stage the cell belongs to CellRangel CellRangel is a cell reference for indicating the value of the first CellRange2 argument used for distributions CellRange2 is a cell reference for indicating the value of the first argument used for distributions CellRange3 is a cell reference for indicating the value of the first argument used for distributions gf Wd Wgd Refers_to Yes Refers_to is the range or address of the cell to copy the stochastic function PF ia Place m cell is a cell reference to specify the cell in which to write the WBSP_ function Place_in_cell NoErrDialog Any argument passed here causes all What sBest error dialog boxes from the wbStochasticFunction routine to be suppressed Instead the error number is delivered in this return argument Pass an integer to use any possible returned What sBest error number Useful MACROS THE VBAINTERFACE _ 181 in an embedded application of What sBest Sto_BadFunctionNameArg Bad FunctionName argument Sto_BadStageArg Bad Stage argument Sto_BadCellRange1l Arg Bad CellRangel argument Sto_BadCellRange2Arg Bad CellRange2 argument eas Conran aromen Sto_BadCellRange1CellRange2Cel
31. Correct the formula and the validity of the arguments in the cell below The format should be WBSP_DIST arg1 arg2 cells where argl and arg2 are references and cells must refer to random cells cell address Suggestions There is an incorrectly formatted distribution function cell in the model A distribution function cell must use the format WBSP_DIST arg1 arg2 cells where arg and arg2 are cell references and cells is a reference to the cells you want to apply this two argument distribution The cell reference should be to a random cell For more information on using the stochastic feature refer to section Advanced Stochastic Support SPDISTS3 Stochastic WBSP_DIST 3 Format Using the stochastic feature the user has to associate a distribution to a random cell either discrete or continuous The WBSP_DIST function is used to request a three argument distribution If the arguments to this function are not correctly specified the following error message will be displayed on the WB Status tab TROUBLESHOOTING 427 Error Message eK ERROR Stochastic WBSP_DIST 3 Format Help Reference SPDIST3 A WBSP_DIST cell with 3 arguments are incorrectly formatted Correct the formula and the validity of the arguments in the cell below The format should be WBSP_DIST argl arg2 arg3 cells where r argl arg2 and arg3 are references and cells must refer to random
32. G 6 the current selection The Objective Best Cell in XYZ Z xlsx Microsoft Excel HEH alas Data Review View Developer Add Ins X YRPORATION PRODUCTION PLAN Deluxe PROFIT 0 ANN 14 CHAPTER 1 C Specify Constraints The constraints for this problem are that the Total Usage of components E15 E17 must be less than or equal to the number in stock G115 G17 The formula for Total Usage of Standard tower components is C5 C15 D5 D15 Cells E16 and E17 have similar formulas for the Deluxe tower and hard drives To specify the constraints highlight the range F15 F17 then choose Constraints from the WB menu and click OK Note that the Constraints dialog box has E15 E17 entered as Left hand side LHS G15 G17 entered as Right Hand Side RHS F 15 F 17 entered in Store in and lt less than or equal to being the default is entered as the constraint type Alternately you may use the lt toolbar button after selecting the range to store the constraints See The ABC s for more information on these defaults Automatic Generation of a Constraint Home Insert Page Layout Formulas Data Review View Developer Add Ins EI D ZE LFS Kescht WB E15 lt 615 COMPUTER CORPORATION PRODUCTION PLAN Product Standard Deluxe PROFIT Quantity to Produce 0 0 Profit per Unit 300 KA Components Quantity Required Total Number Standard Delu
33. Licensed If you attempt to use the mixed integer programming solver option without a license the following error message will be displayed on the WB Status tab Error Message ERROR Mixed Integer Solver Not Licensed Help Reference LICOPT4 The license for your installation does not allow for use of the mixed integer solver option TROUBLESHOOTING 417 Suggestions The mixed integer programming solver is an optional feature You ve attempted to use the mixed integer solver with a copy of What sBest that does not have an appropriate license You will need to either disable any integer options WB Options Integer Pre Solver WB Options Integer Solver or contact LINDO Systems regarding a license upgrade that includes the mixed integer option LICOPT5 Stochastic Solver Not Licensed If you attempt to use the stochastic solver option without a license the following error message will be displayed on the WB Status tab Error Message Stochastic Solver Not Licensed Help Reference LICOPT5 The license for your installation does not allow for use of the stochastic solver option Suggestions The stochastic programming solver is an optional feature You have attempted to set up a stochastic model with a copy of What sBest that does not have an appropriate license You will need to either disable any stochastic featuress WB Options Stochastic Solver WB Advanced Stochastic Support or contact LINDO System
34. TRIAINV This function returns the inverse of a triangular cumulative distribution for supplied low mode and high values The syntax is WBTRIAINV Prob Low Mode High Prob The probability corresponding to the triangular distribution The probability must be greater than or equal to 0 and less than or equal to 1 Low The lower limit of the triangular distribution Mode The mode peak value of the distribution Mode must be between the Low and High values High The upper limit of the distribution High must be greater than or equal to Mode a E T For example WBTRIAINV 7 1 2 3 returns 2 2254 This means that for a triangular distribution over the range 1 3 with mode 2 70 percent of the outcomes are less than or equal to 2 2254 190 CHAPTER 5 Inverse Exponential Cumulative Distribution EXPOINV This function returns the inverse of an exponential cumulative distribution for a supplied mean and standard deviation The syntax is WBEXPOINV Prob Mean Prob The probability corresponding to the exponential distribution The probability must be greater than or equal to 0 and less than or equal to 1 Mean The arithmetic mean of the distribution The mean must be greater than or equal to zero For example WBEXPOINV 5 1 returns 0 69315 This means that for an exponential distribution with mean 1 0 50 percent of the outcomes are less than or equal to 0 69315 Inverse Uniform Cumulative Distribution UN
35. WBSP_RAND WBSP_RAND WBSP_RAND Reporting Cells WBSP_REP WBSP_HIST possible returns on stock table WBSP_DIST_DISCRETE_SV WBSP_DIST_DISCRETE_SV WBSP_DIST_DISCRETE_SV WBSP_DIST_DISCRETE_SV WBSP_DIST_DISCRETE_SV Generate a histogram of the PV Scenario Tree SAMPLE MODELS 385 The PUTOPTION Worksheet After Optimization Gaa GF YTS PUTOPTION s Microsoft Excel ue el File Home Insert Pagel Formt Data Beien View Devel Add I Risk Su Y ai o El X The holder of the option has the on to sella porkan stock t at any time the American feature between now and a specified 5 expiration date at a specified strike price The holder makes a profit in the period of exercise if the E strike price exceeds the market price of the stock at the E time of sale Money is borrowed invested at the risk free rate Step 1 i Initial Price 100 Strike price 99 Risk free rate 0 03 Stock Price of return this stock this 0 100 000 i WBSP_VAR 0 08 92 000 j WBSP_VAR 0 01 91 080 WBSP_VAR 0 01 90 169 WBSP_VAR 0 01 89 268 WBSP_VAR 0 03 91 946 i WBSP_VAR Number times sold Step 3 Sample s Can sell at most once WBSP_STSC Stage Scenario PV of final wealth 7 4654 lt Maximize The solver writes a series of solution in the created tab WB _Stochastic with the expected value and displays the first scenario on the spreadsheet 8 Troubleshooting Troublesho
36. and the objective function for the problem This is similar to the section that we create for solving a deterministic optimization problem 2 A section where we enter the stochastic information Information can be entered directly onto the spreadsheet or via the set of dialog boxes Step 1 core model 1 Data and Formulas Specify the quantity required plantation cost selling price and cost price for all the four crops The yield information will be entered at a later stage Also specify the total area of the farm 2 Adjustable Cells For each crop the adjustable cells correspond to the following a Area allocated b Quantity produced c Quantity sold d Quantity purchased 3 Objective function The objective function is the profit given by profit amount obtained from selling the excess crop plantation cost purchase cost 4 Constraint Cells a For each crop c the quantity of crop produced is equal to the product of the yield of that crop and the area allocated to that crop i e quantity c yield c area c b The sum of the areas allocated to all the crops should be less than or equal to the total area of the farm i e sum area c lt total area c For each crop c quantity required c quantity produced c quantity sold c quantity purchased c 5 Random Parameters Finally the Yield cells will be read as random parameters 374 CHARTER Step 2 stochastic informa
37. appropriate and clicking OK Note The default value entered in the For Cell Range is G15 G17 which is not the range of the constraint cells You must make the correction to F15 F17 in order to obtain the true range values In the illustration below we used cells 115 117 and J15 J17 for the upper and lower ranges respectively and then re solved Again we could have placed the dual value cells in any convenient location in the workbook wavua e K D SG EE XYZ COMPUTER CORPORATION PRODUCTION PLAN Product Standard Deluxe Quantity to Produce 60 30 _ Profit per Unit 300 500 Product Component Requirements 2 Components Quantity Required Total Number Dual Upper Lower Standard Deluxe Usage In Stock Value Range Range P 1630 20 2507 40 sol Deluxe Tower 1 30 lt Hard Drive 2 120 lt Standard Tower 0 60 lt 50 60 20 0 0 120 TUTORIAL SAMPLE 96 CHAPTER3 The ranges for the Standard computer tower stock constraint are 60 upper and 40 lower That is the dual value will stay at 50 as the number in stock varies from 20 60 40 through 120 60 60 If the right hand side of the equation is increased to the limit of 120 and the model is re solved the profit increases to 36 000 33 000 50 60 as shown in the following dea mt La _ dy ADDITIONAL COMMANDS 97 Increasing the right hand side of this constraint beyond 120 exceeds the upper r
38. difficult to predict whether a model will solve better or worse with reduction so the user is advised to try both settings to determine which setting offers better performance The default is to enable reduction Solver Method Indicate the method you would like the linear solver in What sBest to employ using the Solver Method drop down box Possible choices include Solver Decides Primal Simplex Dual Simplex and Barrier The simplex method moves along the surface of the feasible region toward the optimal solution while the barrier method moves through the interior region The Solver Decides setting currently defaults to the primal simplex method As a rough guideline primal simplex tends to do better on sparse models with fewer rows than columns Dual simplex tends to do well on sparse models with fewer columns than rows The barrier method is most effective on densely structured or very large models The barrier method is an optional feature in What sBest However most versions ship with a 30 day trial license for the barrier solver If your version of What sBest doesn t currently include the barrier option you may add it at any time Please contact LINDO Systems for information If the barrier solver is chosen without a license What sBest will default to using the primal simplex method The default setting for Solver Method is Solver Decides Pricing Select the type of pricing for the dual simplex and primal simplex algorithm
39. lt lt lt lt Wed Thu lt g lt lt ba Wed Thu lt lt lt lt lt lt lt dE iit i Thu ba lt lt lt dE aTi 2 3 Eg KS 6 8 9 10 EL 12 13 14 15 16 18 19 Ei 21 23 E 25 26 22 OH Diego SAMPLE MODELS 343 The Constraints worksheet contains constraint formulas that ensure each employee works no more than one shift in any 24 hour period A Determine Adjustable Cells The adjustable cells in this model are cells D3 J5 D8 J10 D13 J15 and D18 J20 of the Assignments worksheet These are the schedules for the individual staff members B Define Best The best solution maximizes the total staff preference E4 of the Model worksheet which is the sum of the individual preference totals Individual totals are the product of the shift assignment variables 0 or 1 and the preference values assigned to each shift C Specify Constraints The constraints in this problem are Staffing requirements for each shift must be met e Each staff member must work exactly the prescribed number of shifts during the week no more and no less After working one shift a staff member can t be assigned to the next two consecutive shifts or work more than one shift in a single day To assure that the staffing requirements for each shift are met the constraint formulas in D8 J8 of the Model worksheet re
40. value for the Big M Coefficient may take some experimenting The default value for Big M is 100 000 Status Report Use the Status Report option to specify under what conditions if any you would like a status report created when a model is solved The available options here are Always Created Never Created Only on Error Warning If you select Always Created What sBest will insert a status report worksheet entitled WB Status into the current workbook every time you solve The Never Created selection means that What sBest will not insert a status report By selecting Only on Error Warning What sBest will insert a status report worksheet entitled WB Status only if an error or warning from calculation is encountered If there are no errors or warnings then no status report is inserted ADDITIONAL COMMANDS 53 The status report contains the model s classification statistics options selected and any error messages or warnings generated during the last attempt to solve the model To view the status report just click on the WB Status tab of your workbook The information presented in the status report is discussed in the Solve section or the Solution Outcomes section in Overview of Mathematical Modeling The What sBest default is Always Created meaning that a status report is created and inserted into the workbook upon each solve Solution Report Select the Solution Report checkbox to tell What sBest you would like a solution
41. 0 Solver Decides 1 Depth First 2 Worst Bound 3 Best Bound GlAlgebraicReformulation No 0 Solver Decides GlAlgebraocReformulation indicates the algebraic reformulation 0 Solver Decides 1 None 2 Minimum 3 Medium 4 Maximum Error Codes Description GlobalOpt_BadGiStratGlobalArg Bad Global Solver argument GlobalOpt_BadGIMultistartAttemptsArg Bad Multistart Attempts argument GlobalOpt_BadGlOptimalityToleranceArg Bad Optimality Tolerance argument GlobalOpt_BadGIDeltaToleranceArg Bad Delta Tolerance argument GlobalOpt_BadGlVariableBoundLimitArg Bad Variable Bound Limit argument GlobalOpt_BadGlUseBoundLimitArg Bad Use of Bound Limit argument GlobalOpt_BadGIBranchingDirectionArg Bad Branching Direction argument GlobalOpt_BadGIBoxSelectionArg Bad Box Selection argument GlobalOpt_BadGIAlgebraicReformulationArg Bad Algebraic Reformulation argument 166 CHAPTER 4 Example If one wished to set all the options the following would work wbSetGlobalOptions True 4 0 0001 0 000001 100000000 1 2 1 2 To set individual options here setting Multistart Attempts to 4 named arguments should be used wbSetGlobalOptions GlMultistartAttempts 4 Note The global solver needs the nonlinear option and the mixed integer option to operate wbSetiIntegerOptions This routine is used to set the
42. 0 0 5000 Open 0 0 CostiMonth 1300 975 1000 1100 2 000 0 Fixed Costs 0 0 0 0 0 0 0 Variable Costs 0 0 0 0 0 0 0 Daily Cost of Capital 0 027 TOTAL OPERATING COST 0 AVERAGE MAIL DELAY BETWEEN LOCATIONS Proposed Lockbox Locations Customer Home Locations New York Atlanta Cincinnal Denver Seattle Office Seattle Locations Yew York Atlanta Cincinnal Denver Seattle Seattle Los Angeles Houston Philadelphia Miami ee vvvvw nnnn n nn nn nu vvvvyv ono one nn vvvvw noone own nn vvvvy ono en In cell C18 the Daily Cost of Capital is entered In this example the cost was calculated based on a hypothetical 10 annual rate for a 360 day year prorated to 027 daily This figure is important in calculating the expense due to float time caused by mail delay In H7 H11 the Monthly Cash Flow from each customer is shown Cells B27 F31 display the average mail delay in days between customer sites and proposed lockbox locations In G27 G31 the average delay from customer locations to the home office is found Finally in B14 G14 the fixed operating Cost Month associated with each potential lockbox site is entered Note that no fixed cost specific to the crediting of deposits is associated with the home office cell G14 since this is part of the assumed cost of doing business SAMPLE MODELS 259 A Determine Adjustable Cells The adjustable cells indicate whether or not cash is to be ro
43. 1 is required to be less than or equal to D20 the supply capacity per period Now let s solve the model The Inventory Worksheet After Solving ik la 8 jE Purchase From Source Ending 1 3 Inventory 0 100 lt 140 180 lt 180 220 lt 180 150 lt 180 100 lt 180 200 lt 180 lt 180 lt 180 lt 180 lt 180 240 lt 180 210 lt 180 140 lt 140 Source Capacity 180 TOTAL COST Cost Unit Source 100 Holding Cost pa RS 21 210 14 000 enee ee em What sBest has arrived at a minimized Total Cost of 263 836 H21 SAMPLE MODELS 309 This simple model is intended to demonstrate the basic principles in modeling inventory Add fixed ordering and shipping costs for each source etc for a model more reflective of the real world 310 CHAPTER7 Product Mix File name PRODMIX XLSX TYPE LINEAR OPTIMIZATION PMIXMAC XLSM Application Profile In product mix problems the objective is to find the most profitable allocation of a set of limited resources over a set of desired products or activities Virtually every manufacturer faces this situation in some form or another This example involves the conversion of raw materials into manufactured goods An agricultural example might involve the use of land seed water labor and fertilizer so as to maximize farm output Product mix problems often serve as the building blocks for more comple
44. 125 lie EE 127 4 WHAT SBEST MACROS THE VBA INTERFACE ccccsccsseesseeeeseseeesseesneneeeeeeeeneas 129 Usage Guidelines for Macros in VBAL 129 Introduction to the VBA Interface of What ebesi 129 Object Browser chen hi Anish eat iA ee de aie 129 Macro Recorder EE 130 Calling rel EI 130 Error messages you might encounter learning VBA sssssssssserssssrrssssrrrssrrrrssrrrnssssens 131 Se Ne EE 131 Embedding What sBest in a Visual Basic Droe 131 Concealing and Protecting a Model from the User 132 Tutorial on the VBA Interface AA 133 Building and Solving a Basic Model 133 Solving Multiple Problems with a Looping Mac 136 VBA Interface Procedures cccecceceeececeeeeeeeeeeceaeeeeaaeeeeaeeseeeeecaaeeeseaeeseeeeesaeeesaeeseneeseaees 137 WbAddAdjustableStyle ececccceeceeeceeceeeeeeaeeeeeeeseaeeeeeaeeeeaeeseceeeesaaeeseaaeseeeeessaeeesaeeeenes 138 Viele EE 138 Viet ee EE TEE 138 Ve tel EE 139 WO BOSE EE 140 WBBIN EE 141 WD CONSIVAIN TEE 142 WbDeleteReports eeccccccesecceceeeeeceeeeneeceeeeneeceeesnseceeeenseaeeeenneeeesnsneeeeeneneeeeensneeesensenees 143 wbDeleteWBMenuU ccceceeeceenececeeeeeeeeaeeaeceeeee eee eaaeae cessed sgeeaaaaeaeeeeeeeeeeeeaeaeeeeeeeeeeeea 143 WO KIELEN 144 WDE rror and Error CodeS ccccccccccccccccccecccecccecceceeeeeseeeceseeeaeeeaeeeeeaesesenesananasesanenenaeaeegs 145 VIE TEE 153 PREFACE vii 5 6 WING GOR icin ti ccc sati iineoa aain aa iea
45. 129 64 bit DATE GENERATED Apr 05 2011 08 19 AM STOCHASTIC INFORMATION RANDOMS STAGES NODES SCENARIOS 200 Expected Value EV 2 11E 03 Expected Value of Wait and See Model s Objective EVWS 2 80E 03 Expected Value of Perfect Information EVWS EV 6 88E 02 Expected Value of Modeling Uncertainty EV EVEM 2 78E 01 REPORTING CELLS SCENARIO PROBABILITY Model B11 Model B12fodel B13 DLOSTSALES 23 STAGE 0 STAGE 1 STAGE 1 24 ER 752294 104 62219 18 86989 26 S 752294 82 987394 0 3 27 i 752294 57 430381 D 28 e 752294 72 294769 0 29 005 752294 75 072594 0 iv Z Model Scenarili 4 u U Ge d E 3 4 5 6 SE 8 9 10 11 2 13 14 15 16 17 18 19 20 21 22 Create Histogram Using the Function WBSP_HIST The general format is WBSP_HIST mumber_of_bins cells_to_be_reported e g WBSP_HIST 0 Sheet1 B 1 meaning the solver decides of the number of bins and cell B1 Sheet will appear in the Stochastic Histogram Report After solution a new tab WB _ Histogram will be created It will have four columns reporting the Bin Lows Bin Highs Mid Points and Bin Counts 114 CHAPTER3 Select the checkbox to tell What sBest you would like a stochastic histogram created when the model is solved The next time you solve What sBest will generate a worksheet called WB Histogram containing a listing of the data and the histogram graph for the Bin Counts against the
46. 230 Application re EE aaie aait 230 Chance Constrained BIOnding cccccccceeeceeneeteeeeeeeaeeeeaaeeeeneeceaaeeesaaeseeaeeseeeessaeeeeaeeeeaes 235 Application Profile sas 24 jedan acticin E 235 SERA oe 239 Flow Network Modeling ccesccseeceseeeceeeseeneeeeeceeeeeeeseaeeeseeaaeeeseeaaesesesaeeessseeeeeeeeeenseeees 243 Bond Portfolio OptimiZation cccceeccceceeeeeeeeeceeeee eae ria aar aan ari iaoa Ed ai ia ae Eaa ea 251 Application Profe r tanh E tain 251 ele siet EE 257 Markowitz Portfolio Problem 261 Application Profile sasis an ea a tie a chen EE 261 Portfolio with Transaction Costs c ccccccceceeeceeeeeeeeeeeceeeeeeeeeeeaeeesaaeeseaeeseeeessaeessaeeeeeeeeaas 265 Application Profile iis 3 4a v EE E 265 Portfolio Minimizing Downside Risk sssssssssesesrnssrnssrrssrnsrnsrnssrnssrnssrnssrnnnsrnnnrnnrnsnnt 269 Application Profilert eth lee E EE 269 Portfolio Scenario Model 273 Seasonal Sales Fachoring nt 279 Exponential Gmoothinmg scenes eaaeeeeaaeseceeeeeeaeeeeaaeseeeeeseaeeesaeeseneeee 284 Application de 284 Linearization Option Construction Cost Estimation 0 ccccceccccceeeeseeeeeeeeeesaeeeeeeteeeees 289 Application Profile c ices ceegieveteghen deene dE ee ANERE eee 289 Stratified Gamplmg AA 294 Application Profiles EE 294 E edel DEE EE 298 vedia BUYING EE 302 Application lte TEE EE 302 Multi Period Inventory Management 306 Applications re 306 PLOGUCU Gul
47. 3 Sheet1 B 1 Sheet1 C 1 C 3 Sheet1 A 1 The cell Al then contains the function WBCARD 3 Sheet1 B 1 Sheet1 C 1 C 3 wbIntegerSemic For additional discussion of the functionality provided see the section entitled nteger Semi continuous which refers to the Semi continuous dialog box that calls this procedure This routine is used to build the Semi continuous ranges WBSEMIC Adjustable cells contained in these ranges will be restricted to integral values by the What sBest solver If you wish to delete a SEMIC function using VBA instead of deleting it on the spreadsheet you can use Selection Clear This statement clears the selected cell from any of its content Syntax wbIntegerSemic LowerBound UpperBound ArgList Refers_to NoErrDialog LowerBound Yes LowerBound is a number indicating the lower bound in the range UppererBound UpperBound is a number indicating the upper bound in the range ArgList ArgList is a string list of range or cells reference separated by a comma Refers_to Yes Refers_to is the range or address of the cell s to be specified as SEMIC NoErrDialog No Any argument passed here causes all What sBest error dialog boxes from the wbIntegerSemic routine to be suppressed Instead the error number is delivered in this return argument Pass an integer to use any possible returned What sBest error number Useful in an embedded application of What sBest IntSemic_BadLowerBo
48. 5 102 1 104 7 0 0 10 3 108 3 139 0 113 1 116 0 1 0 0 0 103 5 92 8 100 6 99 8 0 0 15 2 117 6 171 5 190 8 183 6 686 0 0 Expected Return 122 2 Target Return 115 0 Variance 0 0745 Return gt Target gt Semi Variance 0 0103 Invest Total 100 Downside Risk 7 0 0559 PORTSCEN 22 Ray fa Minimized downside risk comes to 0559 Minimization of the three measures of risk yield the following portfolios Asset 1 Asset 2 Asset 3 Minimize Variance 52 5 33 9 13 6 Minimize Semi Variance 46 8 1 53 1 Minimize Downside Risk 6 6 12 4 81 0 It is interesting to observe that although minimizing all three measures of risk Variance Semi variance and Downside Risk individually produces a similar level of risk it also produces rather different portfolio allocation and expected returns This illustrates why it is important to consider the measure of risk that is most appropriate for your investment goals SAMPLE MODELS 279 Seasonal Sales Factoring File name SEASON XLSX TYPE NONLINEAR OPTIMIZATION The Problem in Words You sell a seasonal product and you would like to find out how the seasons affect sales and your sales growth This information would help you in forecasting sales and making future ordering decisions Background This model calculates seasonal factors that measure the effect each season has on sales and fits a straight line to the normalized data by minimizing the sq
49. A tough model may take more time to solve than you are willing to wait For more information on what makes a model tough and techniques for making models easier to solve refer to the section Overview of Mathematical Modeling How do fix the Error Opening File error message This error message results from temporary files that couldn t be closed from the last run of the What sBest Solver Go to the folder where your model file is located and remove any What sBest temporary files These files are named LINDOWBS XLS LINDOWBX XLS LINDOWBS XLSB LINDOWBX LINDOWBRC TXT LINDOWBSOLN TXT LINDOWBSTATUS PRN LINDOWBSOLN PRN LINDOWBSTOC PRN LINDOWBSTOH PRN 394 CHAPTER 8 How do fix the Error in Auto_add 5 Invalid procedure or call argument error message This error will appear when an Excel function or VBA statement cannot be executed properly In the case of What sBest this is typically due to missing program components Attempting to attach the What sBest add ins over a network can lead to this problem in that not all of the What sBest add in files will get copied to the local machine For this reason we recommend installing What sBest on every platform you intend to use it on Running a complete install on each machine will guarantee that all the correct add in files are copied to the correct folder How do fix the Error in returning solution in cell error message Usually this error a
50. A2 in Sheet is in Stage 0 The entry Place function in cell allows you to place this information in a any cell in the workbook Typically you place this information near the decision cell so that you can quickly observe the staging information of a cell without calling up any special menus Randoms WBSP_RAND Use this function to associate a Stage to a Random cell The format is WBSP_RAND stage random_cells_with_this_stage e g WBSP_RAND 1 Sheet1 A 2 meaning cell A2 in Sheet will be at Stage 1 Distributions In Step 2b you specify distribution information about the random cells The format is WBSP_DIST_distribution parameter_cells random_cells e g WBSP_DIST LOGNORMAL C2 D2 Sheet1 A 2 meaning the random cell A2 in Sheet will follow a Lognormal distribution with the value in C2 for the Mean argument 1 and the value in D2 for the Standard Deviation sigma argument 2 As before the entry Place function in cell allows you to place this information in a any cell in the workbook Typically you place this information near the random cell s so that you can quickly observe the distribution information of a cell without calling up any special menus Distributions with one argument WBSP_DIST_ DISCRETE SH argument 1 Discrete values with scenarios read horizontally WBSP_DIST_ DISCRETE SV argument 1 Discrete values with scenarios read vertic
51. Consulting Manufacturing Accounting Government Agricultural Medical C Financial Marketing C Other Telecommunications Insurance What other optimization package have you used eet What will be your primary application of this product 3 ADDITIONAL COMMANDS 125 Use the Register command to register your version of What sBest online You will need a connection to the Internet for this command to work Enter the personal information you deem relevant and select the Register button Your information will be sent directly to LINDO Systems via the Internet Once your registration is complete the following dialog box will appear on your screen Lindo Systems Product Registration 64 bit xs e Your registration has been successfully submitted Thank you Select the OK button to be returned to the main Excel environment LINDO Systems is constantly working to make our products faster and easier to use Registering your software ensures that you will be kept up to date on the latest enhancements and other product news You can also register through the mail or by fax using the registration card included with your software package CheckUpdate Turn the CheckUpdate command on to have What sBest automatically check every time you start the software if there is a more recent version of What sBest available for download on the LINDO Systems website You will need an Internet connectio
52. E CR EE E EE E EE ET i sk Model Stock1 Stock Stock3 38 1 32 4 29 5 Scenario Downside Forcing Return Risk Constraints 7 1 14 4 16 9 7 0 4 0 gt 5 6 10 7 3 5 4 6 6 4 gt 3 8 32 1 13 3 15 8 0 0 gt 8 9 30 5 73 2 34 9 0 0 9 0 19 5 2 1 10 4 0 6 8 3 39 0 13 1 19 7 0 0 3 5 7 2 0 6 0 8 11 8 Desired Threshold Budget Investment Return Return Equals 100 Percent gt 13 0 11 0 ared Downside Risk The Average Squared Downside Risk in cell D19 has been minimized to 2828 SAMPLE MODELS 273 Portfolio Scenario Model File name PORTSCEN XLSX TYPE NONLINEAR OPTIMIZATION The Problem in Words There are a variety of ways to measure the risk of a portfolio In this model you need to observe the changes in the optimal investment allocations resulting from the minimization of risk as measured in three different ways and determine the best measure of risk for your lifestyle Background Once again there are three assets this time with twelve equally likely expected return scenarios Scenarios are outcomes of events with an influence on your analysis Fed raises interest rates by 0 5 point Fed lowers interest rates by 0 25 point housing starts up etc This model calculates three different measures of risk variance semi variance and downside risk You can minimize each of the three and get different levels of each asset that achieve your target return of 15 each of
53. F6 F11 which are now superfluous The re optimized solution should look like the following The TRUCK Worksheet in Integer form After SEIN TRUCK LOADING Maximum Proportion Proportion Value Weight Loaded Loaded 22 500 7500 24 000 7500 8 000 3000 9 500 3500 11 500 4000 9 750 3500 Total Value Total Weight Maximum of Load of Load Load Weight 10000 lt 10000 Note that What sBest has returned a value of 1 in the Proportions Loaded cells of Items 3 4 and 6 and a 0 for the remaining items This results in a total load weight of 10 000 pounds and a total load value of 27 250 This total load value is 3 250 greater than that achieved by rounding fractional answers SAMPLE MODELS 371 Rounding versus Integer Solutions The important things to notice about the integer solution are that the truck s capacity is more fully utilized and more importantly that the value of the load is 3 250 higher As a rule rounding of fractional answers is effective when it results in large integers When the integers are small as in this problem rounding is less effective 372 CHAPTER7 Crop Allocation Under Uncertainty File name CROPALLOC XLSX TYPE LINEAR OPTIMIZATION STOCHASTIC Application Profile The stochastic farm allocation model is an example of a single stage stochastic optimization problem In this problem a random event occurs at stage 1 of the process To handle this random
54. LinearSolverMethod is an integer indicating the linear solver method to employ 0 Solver Decides 1 Primal Simplex 2 Dual Simplex ScaleModel No 1 True ScaleModel is a True False flag indicating whether to scale model Reduction No 1 True Reduction is a True False flag indicating whether to use model reduction PrimalPricing No 0 Solver Decides PrimalPricing is an integer indicating what primal pricing method to use 0 Solver Decides 1 Devex 2 Partial DualPricing No 0 Solver Decides DualPricing is an integer indicating what dual pricing method to use 0 Solver Decides 1 Partial 2 Steepest Edge Error Codes Description LinOpt_BadScaleModelArg Bad ScaleModel argument LinOpt_BadReductionArg Bad Reduction argument LinOpt_BadLinearSolverMethodArg Bad LinearSolverMethod argument LinOpt_BadDualPricingArg Bad DualPricing argument LinOpt_BadPrimaltPricingArg Bad PrimalPricing argument MACROS THE VBA INTERFACE 173 Example If one wished to set all the linear solver options the simplest syntax would be wbSetLinearOptions 3 False True 2 2 To set one or more options named arguments should be used wbSetLinearOptions Reduction True wbSetNonlinearOptions This routine is used to set the What sBest nonlinear solver options seen in the Nonlinear Solver Options dialog box All arguments are optiona
55. Mid Points values Click on the WB Histogram tab to see the stochastic histogram report Select button This button will open a new window To create the list of reporting cells specify the cell or cell range in the Select any cell to report field and the number of bins Once this is done pressing the Add button causes the selected cells to appear in the List of selected cells box You can decide to remove any selected cells by simply going to the list box select the cell reference to remove from the list of names and click on the Remove button The WBSP_HIST function will be placed in the current selection of the field Place the function WBSP_HIST in cell Histogram Chart s Create the histogram for any cells displaying the number of bins and the mid points using the function WBSP_HISTQ Number of bins 0 Select any cells to report multiple selection holding the Ctrl key a List of selected cells select and change by aaa Add or Remove wm Remove Place the function WBSP_HIST in cell sasi zj Help ADDITIONAL COMMANDS 115 Here s an example of the histogram report for the NEWSVENDOR sample model da 3 FIR NEWSVENDOR xlsx Microsoft Excel ec e File Home Insert Page La Formulz Data Review View Develop Add Ins Y Q o X What sBest 1110 0 2 Mar 31 2011 Library 7 0 1 129 64 bit DATE GENERATED Apr 05 2011 08 19 AM 6 DATA HISTOGRAM Mo
56. Staff Needs for some days you begin to violate the constraints for others Adjust your entries for Number Starting This Day to satisfy all the constraints but be sure to meet all staff requirements After each estimate judge your solutions by minimizing Total Cost G18 When you find a solution you might be satisfied with note your final figure for Total Cost 336 CHAPTER7 Now that you have arrived at a What If spreadsheet solution you re ready to solve the model The STAFF Worksheet After Optimization Home Insert Page Layout Formulas Data Review View Developer Add Ins 7 X STAFF SCHEDULING Staff Staff Starting S Needs This Da 180 160 150 160 190 140 110 Total Employees 200 TOTAL COST 44 000 The best possible solution to this problem is a Total Cost of 44 000 Note that the constraints in D7 D13 are all tightly satisfied That is there is no overstaffing at all D Dual Values The best solution to this problem recommends starting some people on each day of the week except Sunday If someone on your staff can only work Sunday through Thursday you can analyze the model to find what the cost would be of starting this person on Sunday instead of some other day Do this by using the Advanced Dual command to display in cell H13 the dual value of cell G13 the Number Starting on Sunday Then solve the model again Refer to the section About Dual Values in Chapter 3 Additional Comman
57. The Worksheet Let s open the BOX sample file Let s look at how the requirements are translated to the spreadsheet format The BOX Worksheet Before Optimization CABINET DESIGN ACTUAL LIMIT Surface Area 6 000 Not gt 888 Length Width Height 1 00 1 00 1 00 Footprint 1 000 lt 252 Volume 1 000 Not gt 1512 UNIT COST Width Height 1 000 Not lt 0 718 Width Height 1 000 gt 0 518 A Determine Adjustable Cells The adjustable cells in the BOX model are the Length Width and Height in H8 J8 These cells are further identified as ranges L W and H respectively SAMPLE MODELS 24 B Define Best The best solution is the minimum cost cabinet that satisfies all the constraints This objective is calculated in cell 113 by means of the following formula 2 0 05 Length Width Length Height 0 1 Height Width The interpretation of which is The cost of the side top bottom panels the sum of the areas of one side and one top bottom panel the cost of front back panels the area of a front back panel all multiplied by 2 C Specify Constraints The constraints are enforced by means of the formulas in D7 D9 D11 D13 and D15 In D7 the formula WB C7 gt E7 forces the total Surface Area of the cabinet in C7 calculated by the formula 2 Length Width Length Height Width Height to be at least as great as the lower limit of 888 square inches in E7 In D9 the formula WB C9 lt
58. The Worksheet Let s look at the sample file called PRICING The PRICING Worksheet Before Solving Miles per Cost Unit H Gallon Rangler 3 i 29 00 Mercurial d Z 4 23 00 Cadimac d 15 5 19 00 Production Rangler l Mercurial K 1 00 Cadimac Capacity year 500 00 100 00 200 00 Constraint lt S lt Price Produced Constraint Rangler 1 00 2 00 lt Mercurial 1 00 3 00 lt Cadimac 1 00 2 00 lt 7 00 gt Produced Requirement Miles Gallon 23 57143 Not gt 24 A Determine Adjustable Cells The adjustable cells in the model are the quantity of each automobile to produce at each plant B8 F10 and the Price in thousands to charge for each unit B15 B17 B Define Best The best cell is the maximized Total Profit in G16 The formula there is SUMPRODUCT B15 B17 C15 C17 B4 B8 C4 C8 C5 C9 D5 D9 E5 E9 E6 E10 F6 F10 This is profit minus production costs C Specify Constraints 300 CHARTER The constraints in the model enforce the three requirements In B12 F12 yearly production at each plant the sum of car models produced at a given plant is required to be no greater than that plant s capacity For instance the constraint for Plant D B12 is WB SUM B8 B10 lt B1 1 In D15 D17 the total quantity Produced of each auto C15 C17 is forced to be no greater than the quantity required according to that model s demand curve formula E15 E17 The estimates for these demand curves h
59. This overview very briefly explains some of the different types of expressions entailed and explores how the expression type affects the solution search An understanding of these basic principles is not required to use What sBest but it can go a long way in helping you to use the software more effectively In the section entitled Linear vs Nonlinear Expressions and Linearization we consider the two expression types linear and nonlinear and how they influence the solution process The section entitled The Solution Process Determining optima introduces some of the issues in finding optima These issues include local versus global optima and the distinction between smooth continuous and non smooth non continuous functions The type of outcomes generated by What sBest is discussed in the section entitled Solution Outcomes Finally some suggestions for modeling are offered in the last section Guidelines for Modeling with What sBest and Guidelines for Stochastic Modeling Linear vs Nonlinear Expressions and Linearization Mathematical expressions can be classified according to their characteristics The broadest and most common distinction is between linear and nonlinear expressions What sBest analyzes your model and describes it as linear or nonlinear in the status report under Model Type Linear Expressions If all the terms of an equation are of the first order the expression is said to be linear This means the expression doesn t co
60. WBSP_ DIST PARETO argument 1 Scale argument 2 Shape WBSP_ DIST UNIFORM argument 1 Upper limit argument 2 Upper limit WBSP_DIST WEIBULL argument 1 Scale argument 2 Shape Distributions with three arguments WBSP_DIST HYPERGEOMETRIC argument 1 Population argument 2 Defective argument 3 Size WBSP_DIST_ TRIANGULAR argument Lower argument 2 Mode argument 3 Upper ADDITIONAL COMMANDS 109 Correlations In Step 2b you can also specify the correlation information about the random cells Correlation is optional The format is WBSP_CORR_correlation matrix random_cells e g WBSP_CORR_SPEARMAN M19 N20 Sheet1 B 13 B 14 meaning the random cells B13 and B14 in Sheet will follow a spearman correlation with the matrix defined in the range M19 N20 This is a 2 column and 2 row matrix for the 2 random cells to correlate As before the entry Place function in cell allows you to place this information in a any cell in the workbook Typically you place this information near the random cell s so that you can quickly observe the correlation information of a cell without calling up any special menus WBSP_CORR_ KENDALL argument 1 matrix WBSP_CORR_ PEARSON argument matrix WBSP_CORR_SPEARMAN argument matrix The NONE choice will actually delete the selected cell Refers To Specify the random cell on which to apply the
61. What sBest integer options displayed in the Integer Solver Options dialog box All arguments are optional For additional discussion of the options available through this routine see the section entitled Options Integer Solver Syntax wbSetIntegerOptions BranchingDirection AbsoluteIntegrality RelativeIntegrality WarmStartLP ColdStartLP AbsoluteOptimality RelativeOptimality TimeToRelativeOptimality HurdleTolerance NodeSelectionTolerance StrongBranchTolerance Argument Required Default Description BranchingDirection No 0 Both BranchingDirection is an integer indicating the preferred direction of branching 0 Both 1 Up 2 Down Absolutelntegrality No 0 000001 AbsoluteIntegrality is a fractional value indicating the absolute amount of violation from integrality that is acceptable for the integer variables Relativelntegrality No 0 000008 Relativelntegrality is a fractional value indicating the relative amount of violation from integrality that is acceptable for the integer variables WarmStartLP No 0 Solver WarmStartLP is an integer value Decides indicating the linear solver to use during branch and bound when a starting basis is present 0 Solver Decides 1 Barrier 2 Primal 3 Dual ColdStartLP No 0 Solver ColdStartLP is an integer value MACROS THE VBA INTERFACE 167 Decides indicating the linear solver t
62. a maximum size of 16384 columns and 1048576 rows per sheet You can save your file with any of these formats TROUBLESHOOTING 395 Error Messages amp Warnings If What sBest encountered an error during the solution process then it opens the status report worksheet tab entitled WB Status as opposed to the worksheet that was open before the run Based upon the error message returned you can troubleshoot your model The default setting for the General Options dialog box Options General is to produce a status report following each solve command Most of the following errors will be found in the status report worksheet after the model is solved Errors from VBA Code Excel generates these error messages when the WBA XLA file has not been checked off as a reference add in for your calls to the What sBest VBA interface To ensure the What sBest procedures are available to your Visual Basic code choose Tools References from the Visual Basic Editor and make sure WBA XLA is checked Note The first step in running What sBest from VBA is to create a reference to What sBest This reference is made by checking the WBA XLA box under Tools References from within the Visual Basic Editor The Visual Basic Editor is called via Tools Macros from the main Excel menu bar If you do not create this reference then any attempt to use the What sBest attributes or procedures will produce the error message Sub or Function n
63. amp d D D Ww oooo oo coo Knapsack Capacity Favorites in Knapsack ro This model uses the K Best feature via the Integer Solver 78 CHAPTER3 Where Cell F12 has the formula SUMPRODUCT C3 C10 WBBINf where WBBINf is the range for the adjustable binary cells in column F Cell E13 has the formula SUMPRODUCT D3 D10 F3 F10 objective cell to maximize Cell G12 is the capacity constraint WB F12 lt H12 Say our friend s ratings of the candidate picnic items is given in the data section above and the rating is different than yours D 10 2 TERE 6 a Waterton 1 ADDITIONAL COMMANDS 79 If we solve the model as is thus solely maximizing our preferences we get the following solution The KNAPSACK_KBEST Worksheet After Solving Al ege E E O a T R Weight Rating tls le e ee fo fe fe fon a feo e f Brats Brownies Beer Ant Repel Blanket Frisbee Salad Watermelon WBIKB_REP NUP b Am w rik bio C10 Co Knapsack Capacity 15 w Favorites in Knapsack 25 This model uses the K Best feature via the Integer Solver 80 CHAPTER3 As indicated by the adjustable cells a few of our favorite items are included in the optimal basket However we are wondering if there isn t another combination of items that our friend might like almost as much that includes most of our favorite items To investigate this question we set the Desired Number paramete
64. and lower ranges for cell E5 displayed in cells F6 and F7 respectively enter wbDualValue E5 F6 Upper Range wbDualValue E5 F7 Lower Range MACROS THE VBA INTERFACE 145 wbError and Error Codes It is very important to handle any possible What sBest errors using some form of the On Error statement as shown in the code example below when error handling code is in place If a What sBest VBA procedure encounters an error then your code handles the particular error For example you can obtain a description of the error and provide it to the user or perhaps provide the user with further information as the code example below demonstrates However if your code that calls the What sBest procedure does not have error handling then Excel handles any What sBest error with a run time error message that can sometimes disguise the error For details see the section entitled Run time Errors Code Sample One below sets the selected cell s as adjustable and solves the model while providing the user with information in case there is an error setting adjustable cells on a protected sheet The code also displays a description of any other error NOTE ON CODE SAMPLE ONE What sBest has defined error numbers beginning at 30001 These error numbers should not be used in your code since they are subject to change Instead use the error codes stored in the wbErr structure as shown in the example below What sBe
65. another cell in the worksheet is deleted or replaced by moving another cell over it This error can also occur when a file is opened that contains functions created on a system in which the What sBest program files were in a different location The latter problem can be corrected using the Update Links button on the General Options dialog box posted by the Options General command If this is not the result of links needing to be updated the references in the specified cells need to be corrected to eliminate all REF errors TROUBLESHOOTING 435 WBCARDFORM Cardinality Cell Format If requested What sBest can specify Cardinality sets Range values tell you over what range a particular set is valid The WBCARD function is used to request Cardinality ranges If the arguments to this function are not correctly specified the following error message will be displayed on the WB Status tab Error Message eKERROR Cardinality Cell Format Help Reference WBCARDFORM A Cardinality cell is incorrectly formatted Correct the formula and the validity of the arguments in the cell below The format should be WBCARD number cells where number is numeric and cells must refer to adjustable cells cell address Suggestions There is an incorrectly formatted WBCARD function in the model A WBCARD function cell must use the format WBCARD number cells where number is an integer and cells is a reference to the ce
66. are three obsolete Deluxe models and they are taken apart for the common parts used in the Standard model gt end Product Standard Deluxe Dual Val Quantity to Produce sel al Low Rng Upp Rng Profit per Unit 300 500 Product Component Requi Components Quantity Required Number Standard Deluxe Usage In Stock Standard Tower 1 0 56 lt Deluxe Tower 0 1 a 50 Hard Drive 1 2 50 lt 50 TUTORIAL SAMPLE In this problem if the original lower bound is exceeded by one 6 the dual value is reduced to zero If the original upper bound is exceeded 26 or more the number of hard drives in stock is insufficient and the model is infeasible Either way the dual value changes when the range is exceeded In some problems the range of a dual value may be very small or zero When this occurs you may find that different starting solutions result in different dual values This may also indicate that more than one combination of the adjustable cells will result in the optimal value of the best cell In this case your model is said to have multiple optima In any event before making use of dual values in decisions of economic importance you should investigate the range over which they re valid ADDITIONAL COMMANDS 101 To retrieve dual values for all adjustable and constraint cells without entering dual value formulas in the worksheet you may request a solution report with the Options General
67. assign individual employees to the work patterns selected in this stage by preference Stage 2 Preference Maximization The Problem in Words The second stage model assigns existing staff members to the work patterns selected in stage 1 You know the preferences of each staff member from the ranks of all work patterns selected in stage 1 and the model assigns individuals to work patterns so as to maximize cumulative preference of the group This model has provisions for specifying employee preference scores for work patterns and individual seniority levels performance indices or a composite of the two Objective of Optimization The objective of this model is to maximize the cumulative preference of the group while assigning staff members to the minimum cost work patterns selected in stage 1 The Worksheet This model is also in 2 screens of the worksheet FIXED2 Screen 1 contains space for up to 12 individual employee names B5 B16 the employee seniority performance index D5 D16 and space for work pattern preferences F5 Q16 352 CHAPTER7 Screen 2 contains the adjustable cells employee names and their assignments and a preference subtotal for each employee The FIXED2 Worksheet screen 2 Before Optimization Ar C CC _ O Esi tT U VW IN ZA ARA AD AE AE AG AH AU e 2 EMPLOYEE ASSIGNMENTS TO SCHEDULES 3 Schedule 10 Staffed LLOYD DIANE TOM JOANNE DAN JIM SUSAN JOY JOHN PAM SHAOIB BEA Met 12 Si
68. back on A checkmark will appear next to the ToolBar command when the toolbar is on The What sBest toolbar offers rapid one click access to eight of the most commonly used operations The What sBest toolbar also makes tool tips available To learn the function of a particular toolbar button just move the cursor over the button for a second and a statement of the button s function will appear Minimize Gt GI Wa G GSitls Bookl Microsoft Excel Home Insert Page La Formule Data Review View Develop wey KS Ace es E Menu Commands Custom Toolbars Al e fe What sBest 11 0 Solve Should you wish to remove the What sBest toolbar you would use View Toolbars Customize command select the What sBest toolbar and press the Delete button You would then have to reinstall What sBest in order to restore the toolbar 122 CHAPTER3 Upgrading via a New License Key The dialog box posted by the Upgrade command appears as follows What sBest R License Key Entry sl If you have a What sBest R license key please paste Crt V or type itin below If you enter the key by typing please be sure to indude all hyphens If you don t have a What sBest R license key you can press the Trial button to run the trial version of What sBest R The Trial version is fully functional but has limited capacity with respect to model size Floating Network License User Infor
69. be the best cell If omitted the default value for BestCell is the current cell BestChoice No This is a string to indicate whether the best cell should be minimized or maximized BestChoice accepts MIN or MINIMIZE upper or lower case for minimize MAX or MAXIMIZE upper or lower case for maximize and NONE upper or lower case to remove any previous best cell If omitted the default value for BestChoice is minimize For backwards compatibility with earlier versions of What sBest BestChoice also accepts the numbers 1 for Minimize 2 for Maximize and 3 for None NoErrDialog No Any argument passed here causes all What sBest error dialog boxes from the wbBest routine to be suppressed Instead the error number is delivered in this return argument Pass an integer to get any possible returned What sBest error number Useful in an embedded application of What sBest MACROS THE VBA INTERFACE _ 141 Error Codes Description Best_BadBestCellArg Bad BestCell argument Best_BadBestChoiceArg Bad BestChoice argument Best_ProtectedSheetError Unable to set the best cell in a protected sheet Best_CreateStyleError Error creating a Best style Remarks Only one cell can be specified as a best cell Example To set I17 as your objective to minimize enter wbBest T17 Minimize To set F17 as the objective to maximize enter wbBest F17
70. breaks Non smooth expressions include nondifferentiable and discontinuous functions Expressions having one or more points for which the first derivative is not defined are called nondifferentiable Graphs of nondifferentiable expressions may have abrupt bends at such points Taking the absolute value of an adjustable cell ABS A1 is an example of a nondifferentiable expression This is illustrated in the following graph Graph of ABS A1 MATHEMATICAL MODELING 199 100 90 80 70 60 50 40 30 20 10 0 100 50 50 100 Here there is an abrupt bend in the graph at the point of zero This can dramatically increase solution times Additional non smooth functions are MAX and MIN Discontinuous functions are even more challenging to a nonlinear solver Discontinuous functions are expressions whose graphs contain breaks The spreadsheet ZF function with adjustable cells as arguments is commonly used to express a discontinuous expression as in the following Graph of IF A1 gt 0 A12 A12 100 200 180 160 140 120 80 60 40 20 N Oo st O O KwK DO MH 200 CHAPTER 6 In simplified terms What sBest searches along the plot of the expression to find a maximum or minimum point representing an optimal solution Breaks and sharp bends on the plots of non smooth functions pose challenges to the solver not posed by smooth continuous functions It is a good practice to avoid formulating your problem with
71. constraints x2 2 22 0 z gt 0 Geometrically the feasible region defined by these two constraints is an ice cream cone with the point of the cone at 0 0 0 The feasible region for the constraint x2 y2 z 2 lt 0 by itself is not convex The feasible region consists of two ice cream cones one right side up the other upside down and with their pointy ends touching The constraint z gt 0 eliminates the upside down cone and leaves the quadratic cone illustrated in Figure 1 Second order cone problems are essentially a generalization of linear models defined over polyhedral cones to ones defined over quadratic cones MATHEMATICAL MODELING 225 Quadratic Come 2 x y z gt 0 More generally in n dimensions a simple quadratic cone ice cream cone constraint is of the form x0 2 x142 x2 2 xn 2 lt 0 x0 gt 0 Second order cone constraints are more general than they might at first appear For another conic form consider the constraints uv x2 lt 0 u v gt 0 The first constraint by itself describes a nonconvex feasible region colored blue and green illustrated in Figure 2 The three constraints together however describe a convex feasible region colored green only called the rotated quadratic cone 226 CHAPTER 6 Rotated Quadratic Cme uv gt x uv gt 0 Figure 2 Rotated Quadratic Cone More generally in n dimensions the rotated quadratic cone constraint in s
72. created tab WB _Stochastic with the expected value and displays the first scenario on the spreadsheet SAMPLE MODELS 379 Put Option File name PUTOPTION XLSX TYPE NONLINEAR OPTIMIZATION STOCHASTIC Application Profile The holder of an American put option has the right to sell a specified stock at any time the American feature between now and a specified expiration date at a specified strike price The holder makes a profit in the period of exercise if the strike price exceeds the market price of the stock at the time of sale Wealth is invested at the risk free rate and the stock return for a specific period is the uncertain parameter This is a multi period stochastic problem under the uncertainty of a stock return The Problem in Words A trader of an American style option has the right to sell a stock There is an initial price of the stock of 100 a strike price of 99 a risk free rate of 4 The trader can sell this stock over a 5 time period Given all prices at the beginning of stage n the market makes a random outcome At the end of stage n having seen all of market s n previous outcomes as well as all the previous decisions the trader makes the next decision The trader has to decide when to decide to sell it once so to generate a profit The Worksheet The worksheet has two sections 1 A section where we enter the data the adjustables the randoms the constraint and the objective functio
73. default setting for Crash Initial Solution is off 60 CHAPTER3 Presolve Check the Presolve checkbox to have What sBest s nonlinear solver identifiy and remove extraneous variables and constraints from the formulation before solving In some cases this greatly reduces the size of the model thus reducing the solving time The default setting for Presolve is on Quadratic Recognition Check the Quadratic Recognition checkbox to have What sBest s nonlinear solver use algebraic preprocessing to determine if an arbitrary nonlinear model is actually a quadratic programming QP model If a model is found to be a QP model then it can be passed to the faster barrier or conic solver Note that the QP solver is not included with the standard version of What sBest but comes as part of the barrier option The default setting for Quadratic Recognition is off Selective Constraint Evaluation Check the Selective Constraint Evaluation checkbox to have What sBest s nonlinear solver evaluate constraints only on an as needed basis Thus not every constraint will be evaluated each iteration This generally leads to faster solution times but can also lead to problems in models that have functions that are undefined in certain regions For example What sBest may not evaluate a constraint for many iterations only to find that the solver has moved to a point in a region where the constraint is no longer defined In this case there may n
74. dialog box provides information about which version of What sBest you are running and its capacity according to the license key you possess If you do not have a license key you can run the Trial version of What sBest In addition the box at the bottom of the dialog box supplies the location of the What sBest add in file WBA XLA and the necessary accompanying files to run What sBest The capacity constraints for the different versions of What sBest are as follows Name Constraints Variables Integers Nonlinear Global Trial Version 150 300 30 30 5 Personal Version 250 500 50 50 5 Commercial Version 1000 2000 200 200 10 Professional Version 4000 8000 800 800 20 Industrial Version 16000 32000 3200 3200 50 Extended Version Unlimited Unlimited Unlimited Unlimited Unlimited ADDITIONAL COMMANDS 121 ToolBar Use the ToolBar command to toggle the What seet toolbar on and off When What sBest is installed the toolbar appears in floating undocked mode as follows What sBest zx Jk L SS k a Xl I e gU Bookl Microsoft Excel Home Insert Page La Formule Data Review View Develop Add Ins A ei o Eil es ap KEKK I Aces Menu Commands Custom Toolbars The user may then reposition the What sBest toolbar to a preferred part of the Excel window Select the ToolBar command to turn the toolbar off Select it again to turn the toolbar
75. disadvantages with respect to finding good quality solutions and speed of convergence The user is advised to try both settings to determine which version offers better performance The default setting for the Solver Version is Solver Decides which merely selects Ver 3 0 ADDITIONAL COMMANDS 63 Options Global Solver The dialog box posted by the Options Global Solver command appears as follows Global Solver Options Xe Strategy be Global Solver Multistart Attempts Solver Decides geckeg m Tolerance 0 000001 0 0000001 Ene int Variable Bound 10000000000 Use of Bound Solver Decides Branching Direction Solver Decides Box Selection Solver Decides Algebraic Reformulation Solver Decides e a ox This command allows you to set a number of options controlling the function of the global solver 64 CHAPTER3 Global Solver Check the Global Solver checkbox to have the What sBest s global solver partition the original nonconvex nonlinear problem into several convex linear subproblems The global solver will then use a branch and bound technique to exhaustively search over these subproblems for the globally optimal solution The global solver will rigorously find a global optimum if allowed to run to completion But be aware that runtimes may increase dramatically particularly on large models when selecting the global option For more information on th
76. distribution Alternately you can accept the currently selected cells which What Beet has automatically placed in this box You can also type the correct cell range directly into the text box Place function in cell Specify the cell in which to write the WBSP_ function Alternately you can accept the currently selected cells which What sBest has automatically placed in this box You can also type the correct cell range directly into the text box 110 CHAPTER 3 Page Step 3 scenario sampling Stochastic Support sl Iw Use Stochastic Modeling Support Step 1 Step2 Step 3 step 4 Enter the scenario sampling information by using the function WBSP_STSC Select range with Column1 for Stages Column2 for Scenarios al Place function in cell S Set N e et Las This command allows you to specify the Scenario Sampling information for the stochastic part The general format is WBSP_STSC stages_and_sample_sizes e g WBSP_STC B1 C3 means range B1 B3 lists the Stages and range C1 C3 lists sample size for each stage Sampling with WBSP_STSC has to be specified if the model contains a continuous distribution With a discrete distribution there is no necessary call for sampling because it takes the discrete values directly from the discrete table Otherwise sampling will be applied on the discrete distribution too In case of a model with both distributions continuous and discrete so the user
77. event a decision has to be made at stage 0 This decision tries to optimize an objective function over all the possible realizations of the random event The Problem in Words A farmer can grow wheat corn or beans on his 500 acres of farm land He requires 200 tons of wheat and 240 tons of corn to feed his cattle For domestic use he requires 100 tons of beans He can either grow these crops or buy them from the market The plantation cost for the crops is given by 150 acre wheat 230 acre corn and 260 acre beans The crops can be sold in the market at 170 ton wheat 150 ton corn and 200 ton beans Any shortfall can be purchased from the market at 238 ton wheat and 210 ton corn and 270 ton beans The yields in tons acre of the crops depend on the weather The farmer believes that the weather can be good fair bad or very bad with equal probability If the yield of a crop is y in fair weather then the variation of the yield with weather is given by Good 1 2y Fair y Bad 0 8y Very bad 0 6y The yields in tons acre of the various crops under the various weather scenarios are Wheat Com Beans Good 3 3 6 24 Fair 2 5 3 20 Bad 2 2 4 16 Very bad 1 5 1 8 12 The farmer has to decide how much land to allocate to each crop so as to maximize his profit The Worksheet The worksheet has two sections SAMPLE MODELS 373 1 A section where we enter the data the adjustables constraints
78. fourth argument is an optional TRUE FALSE value cell address Suggestions The VLOOKUP function requires three arguments as follows VLOOKUP lookup_value table_array col_index_num range _lookup What sBest places the following restrictions on these arguments the col_index_num argument must be a value not a formula or a cell reference e all arguments must evaluate to being numeric rather than text and range_lookup is an optional TRUE FALSE argument Refer to the What sBest sample file VLOOKUPWBSOLVER XLS for an example of valid VLOOKUP usage MEMORY Insufficient Memory If there is not enough random access memory on your machine the following error message will be displayed on the WB Status tab Error Message Insufficient Memory Help Reference MEMORY There is not enough random access memory on the machine to solve the mode to completion Try closing all other applications or adding memory additional message Suggestions The solver was not able to allocate sufficient memory Try shutting down any other applications that are open to free up memory Memory requirements grow along with your model so reducing the model s size may help too Another option is to try and increase the amount of virtual memory available to your computer Unfortunately relying on virtual memory rather than true RAM memory will slow the solver down substantially TROUBLESHOOTING 419 If all else fails you s
79. g an integer passed causes all What sBest error dialog boxes from the wbSolve routine to be suppressed Instead the error number is delivered in this return argument Pass an integer to use any possible returned What sBest error number This is useful in an embedded application of What sBest Error Codes Description Solve_UnmappedDrive Error on drive Solve_NoDLL Error on DLL to access Solve_BadSolutionStatusArg Bad SolutionStatus argument Solve_BadExcelOpenArg Bad ExcelOpen argument Solve_MultipleMaxMinError More than one cell is specified as the objective cell Solve_ProtectedWorkbookError Protection can not be on the workbook only on worksheets Solve_ReadLINDOWBRCError Error reading temporary file LLNDOWBRC TXT Solve_SaveTempFileError Error saving a copy of the active workbook Solve_SymbolInSheetNameError Error on sheet name Solve_MinimumFileFormatError The active workbook must be an Excel 8 0 BIFF8 or Excel 97 format or higher Solve_UnableToLoadSolverError There was a problem loading the What sBest solver Solve_ErrorRunningSolver There was a problem running the What sBest solver Return Value The return value is the same of the Solution Status argument Example Sub JustSolve On Error goto errorhandler Dim lngSolutionStatus as Long Solve the current model wbSolve IngSolutionStatus Display the
80. general integer a range name WBINTQuantity would be assigned to the selected cells and appear in the list on the Integer dialog box Refers to Specify the range of cells that are to be made integer in this text box What sBest will automatically fill the Refers to text box in with the range of the currently selected cells If this is not the range of cells you want you can type the correct range in or select the button on the right edge of the text box to bring up a cursor for cell selection Binary WBBIN Select the Binary radio button in the Integer Type box to cause What sBest to constrain the current range of cells specified in the Refers to text box to have a value of zero or one Binary integer variables 0 1 are useful in making Yes No Open Close or Buy Sell decisions and formulating piecewise linear functions If a cell is specified as binary What sBest will find the best solution that returns a 0 or 1 in that cell General WBINT Select the General radio button in the Integer Type box to cause What sBest to constrain the range of cells specified in the Refers To text box to be a positive whole number General integer variables 0 1 2 can be useful when answers with fractions are of limited or no value Some examples are personnel scheduling buy sell decisions involving round lots and discrete loading models If a cell is specified as general integer What sBest will find the best solution that returns a whole number
81. given day To keep it simple let s assume you come across the following quotations for nine U S Treasury bonds Year to Asking Price Maturit Rate 1000 1 Year 0 996 0 997 2 Year 0 923 0 987 3 Year 0 993 1 061 0 883 4 Year 1 102 0 889 Objective of Optimization Interest Rate 10 5 10 5 10 75 11 2 11 8 12 0 10 0 12 6 10 2 You want to minimize your client s initial investment while covering his cash flow needs over the next five years In other words you wish to know the minimum amount of cash that should be allocated among all the available bonds which will maintain the required cash inflow over the next five years SAMPLE MODELS The Worksheet To get started let s examine the BONDS worksheet Look it over for a minute to see how it s designed D The BONDS Worksheet Part 1 Before Optimization Asking 1 0 996 0 997 2 0 923 0 987 0 993 1 061 0 883 1 102 0 889 Interest 10 5 10 5 10 75 11 2 11 6 12 10 12 6 10 2 Units 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Investment BOND OPTIONS Maturity Price Rate Purchased Bond Issue 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 TOTAL COST S000 Year 0 Year 1 Year 2 Year 3 Year 4 Amount Covered 0 00 0 00 0 00 0 00 0 00 Not gt T Notze T Not gt Not gt Not ze Cash Flow Need 17 00 62 00 23 00 35 00 62 00 A Determine Adjustable Cells Adjustable cells a
82. has been specified to be minimized and contains the equation B3 S N 3 1416 B3 and B5 contains the constraint B4 lt 6 The following graph shows a plot of B4 the expression to be minimized for values of B3 between 0 and 6 You can see that if you re searching for a minimum there are local optima at B3 values of 0 1 564 3 529 and 5 518 in the valleys If you use the global solver it will find a global optimum for this problem at B3 5 518 since that is the last valley before 6 Graph of B3 SIN 3 1416 B3 Imagine the graph as a series of hills You re searching in the dark for the minimum or lowest elevation If you start your search at B3 3 every step to the left takes you uphill and every step right takes you downhill Therefore you move to the right in your search for the lowest point You ll continue to move right as long as this direction leads to lower ground When you reach B3 3 529 you will notice a small flat area slope is equal to zero Continuing to the right begins to lead uphill and retreating to the left leads up the hill you just came down You are ina valley the lowest point in the immediate neighborhood a local minimum but is it the lowest possible point You can t determine that given the information available to you at that particular point This is an extreme example and not typical of well formulated nonlinear problems Nonlinear solvers do their best to tackle such nonlinear models b
83. helpful in building better nonlinear models Operational Problems The following problems generally occur due to an error in operation within Excel Symptom An Excel error appears claiming that a workbook cannot be opened under High Security Level Problem This error occurs when Excel tries to open the What sBest add in What sBest makes use of Excel macros to implement the WB constraint function Suggestions The safest option is to put Excel into Medium security mode To do this run the Tools Macro Security command in Excel and set the Security Level to Medium The problem with this approach is that you will be prompted by Excel each time it loads as to whether or not you wish to allow the What sBest add in to be loaded An alternative is to run the Tools Macro Security command select the Trusted Sources tab then check the Trust all installed add ins and templates button Symptom An Excel error appears referring to multiple copies of WBA XLA WBA XLAM Problem You have a toolbar left from a previous version of What sBest which must be deleted before you reinstall What sBest Suggestions First uninstall What sBest from your machine with the Uninstall program You can search for the word uninstall to find a shortcut entitled uninstall What sBest and double click it Otherwise you can manually remove What sBest related files from your machine The locati
84. in that cell Runtime Concerns in Integer Problems The use of integer variables can considerably increase the time required to find optimal solutions In most nonlinear models that include integer variables your chances of reaching an optimal solution in a reasonable amount of time are diminished further unless the problem is extremely rudimentary or ADDITIONAL COMMANDS 43 constrained in very specific ways To improve performance you may wish to use options such as Tolerance and or Hurdle Known IP Such options can significantly decrease the solution time on some integer problems For details on these options see the section entitled Options Integer Solver Integers Special Ordered Set The Integers Special Ordered Set command allows you to support Special Ordered Sets SOS variables of type 1 2 and 3 and Cardinality sets of variable via the following dialog box Special Ordered Set Cardinality asm Special Ordered Set Cardinality e Type 1 WBSOS1 WBCARD Type 2 WBSOS2 Type 3 WBSOS3 Select any cells multiple selection holding ae Lx asi E List of selected cells select and change by Add or Remove Add Remove Place the function WBSOS 1 in cell KGR ee Special Ordered Set The properties of the three SOS types are via the WBSOSx function WBSOS1 At most one variable belonging to an SOS1 set will be gt 0 44 CHAPTER3 At most two variables in a
85. is a True False flag indicating whether to provide warning of irreconciliable constraints or not Opt_BadFeasibilityToleranceArg Bad Feasibility Tolerance argument Opt_BadlIterLimitArg Bad Iteration Limit argument Opt_BadRuntimeLimitArg Bad Runtime Limit argument Opt_BadSelectRefersToArg Opt_BadMinimizeExcelArg Opt_BadHideStatusArg Opt_BadConstraintIndSlackArg Opt_BadLinearizationDegArg Opt_BadStatusReportArg Opt_BadReportsLocationArg Opt_BadSolutionReportArg Opt_BadNonlinearityArg Opt_BadNoObjectiveArg Opt_BadBlankCellsArg Opt_BadUnsupportedFunctionArg Opt_BadStringArgumentArg Opt_BadIrreConstraintArg Opt_BadInfeasConstraintArg Opt_BadUnboundVarArg Opt_BadSupLookupFctArg 164 CHAPTER 4 Example Arguments should be provided as follows If one wished to set all the options the simplest syntax would be wbSetGeneralOptions 0 0000002 77 88 2 True False False _ 1 0 2 222222 1 1 1 False True False False _ False True False False True To set one or more options named arguments should be used wbSetGeneralOptions goSolutionReport 1 goWrmBlank 1 wbSetGlobalOptions This routine is used to set the What sBest global solver options seen in the Global Solver Options dialog box For additional discussion of the options available through this routine see the section entitled Options Global Solver All arguments are optional Syntax wbSetGlobalOptions GlStratGlobal
86. is allowed by your installation You will need to either reduce the size of your model or upgrade to a larger version Suggestions The model exceeds the integer cell limit for your installation The capacity limits of your version may be found by running the WB About What sBest command One option is to remove integer cells from the model until it satisfies the limit You should also check that there aren t any unintended ranges of integer cells in the model Integer cells are flagged by What sBest using range names that begin with WBINT and WBGIN Review all such range names to be sure that there are no unintended integer cell ranges Some classes of models will return naturally integer values If your model falls in this category you can remove the integer formatting and let What sBest solve the model as a continuous problem You may also want to review your demands for an integer solution For instance in some models you can simply round the solution to the continuous model with little or no ill effects For instance if a model calls for producing 1 247 288 33 pencils you could safely round the number up or down without significant impact On the other hand if the model calls for building 6 space stations rounding up or down will have a huge impact on the objective and feasibility of the constraints In which case rounding is not an option Check to be sure that all adjustable cells in the model are still requ
87. is only free to change the purchase quantities in C18 F18 SAMPLE MODELS 237 C Specify Constraints The limitation to this problem is that the final mix must contain the minimum required levels of nutrients for Swine amp Rose s hogs This limitation is enforced by creating a constraint cell for each nutrient H7 H9 H11 Each constraint will return the Not gt indicator until the Minimum Required 17 19 111 for that nutrient is met in the Nutrients Supplied column G7 G9 G11 Now let s solve the problem with What sBest The HOGCHANC Worksheet After Solving HOGCHANC xIsx Microsoft Excel SWINE amp ROSES Hog Farm Nutrients Per Unit Weight of Grain Nutrients Minimum Dual Item 1 2 3 4 Supplied Reqd Value Nutrient A 22 3 4 7 2 3 7 gt 24 0 00 Nutrient B 14 11 0 0 1 0 9 gt 07 0 00 Nutrient C 2 3 5 6 11 1 5 0 gt 5 0 0 97 Nutrient D Mean Value 12 0 11 9 41 8 52 1 21 0 gt 21 0 1 59 Variance 0 3 2 2 20 5 33 2 Cost Weight 35 00 50 00 80 00 95 00 Weight Units Unity VC wm to Purchase 63 4 0 0 31 3 53 19 Dual Value 0 00 11 62 0 00 0 00 E 22 23 24 4 4 gt M WB Status HOGCHANC AFJ Ready E This What sBest solution costs 52 26 per bushel It s more expensive than the HOGFEED solution in the previous example because you have compensated here for the variability in the amount of Nutrient D in the various component grains It s no
88. is set to Up Setting the Application of the Constraint Cuts to All Nodes may help solve some problems Setting the Probing Levels may also help shorten solution time We recommend setting to a level of three or lower Nonlinear problems are generally difficult to solve Setting a Feasibility Tolerance or Optimality Tolerance may help you to reach a solution Trying different Strategies may also be useful to reaching a solution Many nonlinear problems are made unnecessarily difficult or slow to solve by the presence of nonlinear expressions that could be rewritten in linear form The Linearization option is a tool for automatically rewriting these nonlinear expressions in linear form Linearization can be a great performance boost if all nonlinear formulas in a model can be linearized such that the final model becomes totally linear If linearization can only be applied to a subset of the nonlinear formulas then performance may erode The options described above for solving nonlinear problems are important aids to solving nonlinear problems However if the problem is poorly formulated then the options can only help alleviate the effects of poor formulation The following section provides some suggestions for good formulation of nonlinear problems MATHEMATICAL MODELING 203 Nonlinear Modeling Guideline Nonlinear models can be extremely complex to solve It can pay off in terms of solution speed and reliability to spend a little extra
89. of interest as calculated in cell 110 E10 D10 100 Next year and the year after he will continue to receive interest payments of 1 00 J10 and K10 In three years when the bond matures he will receive another interest payment of 1 00 and additionally a repayment of the principal of 10 units for a total of 11 00 as calculated in cell L10 E10 1 D10 100 You re now ready to solve the model and find the best possible solution to this problem The BONDS Worksheet Part 1 After Optimization d SCH E BOND OPTIONS Maturity Price Rate Purchased Bond Issue 1 0 996 10 5 45 0 44 82 0 997 10 5 0 0 0 00 2 0 923 10 75 10 7 9 90 0 987 11 2 0 0 0 00 0 993 11 8 45 6 45 30 1 061 12 0 0 0 00 0 883 10 0 0 0 00 1 102 12 6 0 0 0 00 0 889 10 2 56 3 50 02 TOTAL COST 5100 Year 0 Year 1 Year 2 Year 3 Year 4 Amount Covered 17 00 62 00 23 00 56 74 62 00 7 7 7 S gt P Cash Flow Need 17 00 62 00 23 00 35 00 62 00 256 CHAPTER7 What sBest returns a total cost of 150 04 million with only one cash surplus in Year 3 of 21 74 million The Income Stream for each bond also appears in its optimized form The BONDS Worksheet Part 2 After Optimization A fel AMALARE x Ric np i Js k Li m n jg Income Stream Year 0 Year 1 Year 2 Year 3 4 7250 49 7250 0 0000 0 0000 1 1529 1 1529 11 8779 0 0000 0 0000 0 0000 5 3834 5 3834 5 3834 51 0055
90. on the two measured features are bounded on the upper end so as to force reliable estimates The variances are computed in B18 and B19 For instance the variance computed for Question 1 in B18 is found by means of the formula B 10 2 B12 2 B 5 C 10 2 C12 2 C 5 D 10 2 D12 2 D 5 E 10 2 E12 2 E 5 B 10 B12 2 B 8 C 10 C12 2 C 8 D 10 D12 2 D 8 E 10 E12 2 E 8 That is for each Stratum its Weigh times Variance is divided by its sample size The resulting figures are summed For each stratum from that result is subtracted the Weight times Variance divided by Population size This formulation is from Sampling Techniques W G Cochran 2nd Ed Wiley New York 1963 to which interested readers may refer SAMPLE MODELS 297 Now you are ready to solve the model After solving the WB Status worksheet will open in order to show you the Nonlinearity present warning This warning can be shut off from the General Options dialog box Your solved model now appears as follows The SAMPLEWB Worksheet After Solving i 4 Sample Size j i 101 23 Lower Bound gt Upper Bound lt Population 200000 100000 Cost 1 1 00 1 00 Weight i i 20 0 10 0 Stratum Variance Question 1 5 5 Question 2 4 8 Fixed Cost 1 00 Total Cost 605 74 Maximum Variance Question 1 0 043 lt 0 043 Question 2 0 014 0 014 The solution tells you the minimal sample sizes that yield reliable results within the toleran
91. one of six categories your model falls into Integer Quadratic Integer Nonlinear Integer Linear Quadratic Nonlinear Linear For further information on these categories see the section entitled Linear vs Nonlinear Expressions and Linearization in Overview of Mathematical Modeling chapter State This shows the state of the current solution Possible states include Globally Optimal Infeasible Unbounded Feasible Infeasible or Unbounded Near Optimal Locally Optimal Locally Infeasible Cutoff Numerical Error Unknown Unloaded Loaded Unknown Error See Overview of Mathematical Modeling for more information on these states Tries This shows the current number of tries or iterations used in solving the model The value is updated periodically in the course of solving Infeasibility This shows the total amount by which all constraints are violated When this value is reported as zero all constraints are satisfied However on integer models all integer restrictions may not be satisfied Objective This shows the current value of the cell to be maximized or minimized Solver Type This shows the type of specialized solver in use if any and will be either Global Branch and Bound or Multistart Best Obj This shows the objective value of best feasible solution found Obj Bound This shows the theoretical bound on the objective for integer or nonlinear programming models What sBest initially determines this value by solving the pro
92. or the barrier method if the barrier option is purchased What sBest options apply only to the current model That means options are defined as Excel names in the workbook and are not saved for other workbooks The What sBest option names usually start with WBxxx and thus are separated alphabetically from the user defined names The Reset to Default command resets the workbook options but not options defined in the Advanced Parameters window The workbook options are General Options Linear Solver Options Nonlinear Solver Options Global Solver Options Integer Pre Solver Options Integer Solver Options and Stochastic Solver 39 40 CHAPTER3 The algorithms used by the solvers are as follows The linear solver in What sBest uses either a primal simplex dual simplex or barrier method The So ver Method default setting of Solver decides selects the primal simplex method The What sBest nonlinear solver uses a generalized reduced gradient GRG algorithm The What sBest global solver combines a series of range bounding techniques within a brand and bound framework to find global solutions to non convex NLPs Finally the integer solver provided in What sBest uses a branch and bound algorithm also referred to as the branch and bound manager ADDITIONAL COMMANDS 4 Integers lInteger Binary The Jntegers Integer Binary command allows you to restrict your adjustable cells to being integers What sBest allow
93. over which the dual value of a constraint or adjustable cell stays the same Note Calculating upper and lower ranges may significantly increase solution times so it is good to weigh the benefits of the additional information against the amount of extra time it will take to get that information This runtime consideration is particularly relevant for large models Report Information in Specify the host range of cells you would like to see the dual information placed in the Report Information in text box What sBest will automatically fill this box with the current selection of cells ADDITIONAL COMMANDS 89 Usage Guidelines for Dual Values Dual values can provide valuable sensitivity information They can give you a feel for how sensitive a solution is to a particular constraint cell or adjustable cell What sBest can provide dual information on any adjustable cell constraint cell or any cell that is a function of adjustable cells Dual Value of a Constraint Cell Shadow Price The dual value of a constraint is the rate of improvement in the best cell as the right hand side of the constraint is increased The dual value of a constraint is often referred to as the shadow price because it foretells the price you should be willing to pay for that item For example in the XYZ sample model referred to earlier in Getting Started dual values can be specified for the constraints in cells F15 F17 These could be put in any convenient pl
94. principles to analyze a much larger set of data covering 2 years The SMOOTH Worksheet Before Solving Squared Squared Sales Predicted Error Error 1 10 1 10 1 52 0 00 Not 1 10 0187 40849 Upper amp Lower 1 76 0 00 0 00 3 10 Bounds 1 86 0 00 0 00 3 46 Alpha 0 00 Not gt lt 1 57 0 00 0 00 2 46 1 64 0 00 0 00 2 69 1 64 0 00 0 00 2 69 2 12 0 00 0 00 449 1 39 0 00 0 00 1 93 2 29 0 00 0 00 5 24 1 67 0 00 0 00 2 79 2 42 0 00 0 00 586 1 84 0 00 0 00 3 39 1 65 0 00 0 00 2 72 1 81 0 00 0 00 3 28 2 13 0 00 nn 4454 emia Leo feat 9 eg ZE 288 CHAPTER7 The solution is shown below The SMOOTH Worksheet After Solving SMOOTH ales Microsoft Exce E Exponential Smoothing Model EI KEE ee Squared Ben Error Error 0 18 F581 Upper amp Lower 0 18 Bounds 0 08 Alpha 056 gt lt 0 03 0 00 0 00 0 23 0 27 0 45 0 11 0 37 0 10 0 11 0 00 014 SAMPLE MODELS 289 Linearization Option Construction Cost Estimation File name LINEARZ XLSX TYPE NONLINEAR LINEAR OPTIMIZATION Application Profile This example demonstrates how one might develop a model for estimating construction costs as well as performance gains resulting from What sBest s exclusive linearization option Note we classify this model as both nonlinear and linear As you will see the model as formulated is nonlinear However after invoking the linearization option in What
95. problem cost is equal to the number of employees times their weekly salary This is shown in cell G18 The formula for Total Employees G15 is the sum of employees starting each day The weekly salary per employee has been entered in cell G16 This means that the formula for Total Cost in cell G18 is G15 G16 SAMPLE MODELS 335 C Specify Constraints The requirement of any solution to this problem is that Staff Size is constrained to be greater than or equal to Staff Needs Without such a constraint the job for What sBest would be easy The least expensive solution would always be to hire nobody Take a look at the Staff Size column C7 C13 The formula for Staff Size equals the Number Starting This Day plus the total of the number starting for the preceding four days Remember each employee has shifts on five consecutive days so any given day s Staff Size equals the five day total For example the formula for the Staff Size on Monday C7 reads G7 G10 G11 G12 G13 The value in C7 must always be greater than or equal to the Staff Needs for Monday To ensure this result a greater than gt constraint was entered in cells D7 D13 with C7 C13 on the left hand side and E7 E13 on the right hand side of the equation What If vs What sBest Try a What If solution to this problem by entering initial estimates of the numbers of employees who will start their five day workweek on each day G7 G13 As you meet
96. program refers to a multistage stochastic model with recourse The term stage time and period are used interchangeably unless otherwise is stated The terms random parameter and stochastic parameter are also used interchangeably Multistage decision making under uncertainty involves making optimal decisions for a T stage horizon before uncertain events random parameters are revealed while trying to protect against unfavorable outcomes that could be observed in the future In its most general form a multistage decision process with T 1 stages follows an alternating sequence of decisions and random events decisions events decisions events decisions 0 1 in stage 0 we make a decision xo taking into account that 1 0 at the beginning of stage 1 Nature takes a set of random decisions o leading to realizations of all random events in stage 1 and 206 CHAPTER 6 1 1 at the end of stage 1 having seen nature s decision as well as our previous decisions we make a recourse decision x xo taking into account that 2 0 at the beginning of stage 2 Nature takes a set of random decisions oz leading to realizations of all random events in stage 2 and 2 1 at the end of stage 2 having seen nature s decision as well as our previous decisions we make a recourse decision x2 Xo 1 X 2 taking into account that T 0 At the beginning of stage T Nature
97. report created when the model is solved The next time you solve What sBest will generate a worksheet called WB Solution containing a listing of the ABC cells and their locations types values and formulas as well as sensitivity analysis dual values for the adjustable and constraint cells or the detailed report with the objective coefficients ranges Click on the WB Solution tab to see the solution report The default is not to create a solution report 54 CHAPTER3 Here s an example of the solution report WB Solution for the XYZ sample model covered earlier Glace Saa Microsoft Excel eel Sal Home Insert Page La Formul Data Review View Develop Add Ins Y Q o ER ZS fe What sBest 11 0 0 2 Mar 31 2011 Library 7 0 1 129 E C D 4 2 Mar 31 2011 Library 7 0 1 129 64 bit DATE GENERATED Apr 05 2011 08 01 AM OBJECTIVE INITIAL XY Z G6 3 300000e 004 3 300000e 004 MAXIMIZE COEFFICIENTS XYZIC5 3 000000e 002 EYZIDS 5 000000e 002 ADJUSTABLE INITIAL TYPE REDUCED COS H Z ICS 6 000000e 001 6 000000e 001 C 0 000000e 0C XYZ D5 3 000000e 001 3 000000e 001 C 0 000000e 0C B Binary C Continuous F Free I Integer N Free Integer CONSTRAINT CELLS DUAL VALUE SLACKS TYPE DECREASE Note that creating solution reports may dramatically increase runtimes on large models Beginning or End Select the Beginning or End radio button to tell What sBest w
98. results Select Case lngSolutionStatus Case 1 MsgBox The model is Globally Optimal Case 2 MACROS THE VBA INTERFACE 179 MsgBox Case 3 MsgBox Case 4 MsgBox Case 5 MsgBox Case 6 MsgBox Case 7 MsgBox Case 8 MsgBox Case 9 MsgBox Case 10 MsgBox Case 11 MsgBox Case 12 MsgBox Case 13 MsgBox Case 14 MsgBox Case Else MsgBox End Select Exit Sub Errorhandler The model is Globally Optimal amp _ Range WBMAX The The The The The The The The The The The The model model model model model model model model model model model model is Infeasible is Unbounded is Feasible is Infeasible or Unbounded is Near Optimal is Locally Optimal is Locally Infeasible is Cutoff is Numerical Error is Unknown is Unloaded is Loaded Solution status unknown Msgbox Error number is wbError Err End Sub amp Err amp Description 180 CHAPTER 4 wbStochasticFunction This routine can be used to set the stochastic support functions seen in the Stochastic Support dialog box All arguments are required except for the error code but a 0 value can be placed for unnecessary arguments For additional discussion of the options available through this routine see the section entitled Advanced Stochastic Support Syntax wbStochasticFunction FunctionName Stage
99. satisfied and try solving again from these starting values 30 CHAPTER2 Explicitly Specifying Convexity to the Solver One of the important considerations for the solver in finding a global optimum is the exploitation of convexity In layman s terms if an optimization problem is convex then a carefully implemented hill climbing search algorithm or valley finding for a minimization problem will find a global optimum More generally if the feasible region is convex then any strict local optimum can justifiably be declared a global optimum Our solver is fairly sophisticated in its ability to identify convex expressions Nevertheless there may be some constraints that you know have convex feasible regions but you suspect might not be identified as convex by our solver What sBest allows special constraint types lt c gt c and c to allow users to identify constraints that have convex feasible regions For example suppose we have the constraints x43 2 x y y 3 lt 6 x gt 5 y gt 5 The feasible region of this set of constraints happens to be convex The user is justified in writing x 3 2 x y y 3 lt c 6 More generally if f x is any function on the left hand side and the user writes fix lt c b The solver will assume that the feasible set for this constraint is convex This is true for example if fx is a convex or quasi convex function Loosely speaking a quasi convex functio
100. select the highest number of exposures over 100 whose summed cost is less than 10 000 Alternatively you could do away with cell D3 and use the formula WB B3 lt 10000 The Constraints command lets you enter single cell references in the left hand side and single cell references single cell range names or values in the right hand side of your formula When entering a constraint formula manually you have the additional option of entering an expression which evaluates to a number not an array on either side of the Indicator such as WB B3 B11 lt D3 D11 Constraint Display Using the Constraint option in the Display box under the Options General command you can set the method of constraint display to either Indicator or Slack If you set constraint display Indicator mode the default then the constraint is shown in the cell by one of the following eight visual indicators lt gt lt gt Not lt Not gt Not or If you set What sBest to display the constraint in Slack mode then a numeric value known as the slack is returned The slack is the amount by which a constraint is not violated For example What sBest would display constraint values based upon the following table Constraint Slack Mode Indicator Mode Tightly satisfied 0 lt Satisfied positive number lt Not satisfied negative number Not lt ABCs 29 Constraint related Problems Constraints m
101. setting currently defaults to the free method The default setting for Solver Method is Solver Decides ADDITIONAL COMMANDS 83 Seed for Random Number Generator This is the Seed to initialize the random number generator Possible values are positive integers and can be set via a single cell reference or a number The default setting for Seed is 1031 Common Size per Stage This is the default number of scenarios to generate per stage Possible values are positive integers The logical states will behave as follow Sampling on Continuous Only Sampling on Checked Continuous Only Not Checked Function Use of Common Size per Stage for Continuous and the Use of Common Size WBSP_STSC Q Defined Table Size for Discrete Distributions per Stage Not Defined Function Use of Function Argument for Continuous and the Use of Function WBSP_STSC Defined Table Size for Discrete Distributions Argument Defined The default setting for Common Sample Size is 2 Sampling on Continuous Distribution Only In case of a model with both distributions continuous and discrete this flag enables the sampling technique only on the continuous distribution leaving the discrete distribution with its own set of data Sampling can be applied with the spreadsheet function WBSP_STSC The default setting is on meaning sampling is applied on continuous distributions 84 CHAPTER3 Expected Value of Objective The Expected Value of Obje
102. such as the Object Browser how to embed your What sBest model in a Visual Basic project as well as how to hide the model The procedures or functions such as Adjust Best and Constraint provided by What sBest are discussed in VBA Interface Procedures The section entitled Tutorial on the VBA Interface is provided to help you get started writing VBA The rest of the section describes the syntax arguments and error codes of particular procedures Usage Guidelines for Macros in VBA Introduction to the VBA Interface of What sBest The very first step in creating a macro for What sBest is to create a reference to the What sBest add in so you can use the add in s attributes and procedures This reference to What sBest is made by checking the WBA XLA or WBA _XLAM box under Tools References from within the Visual Basic Editor The Visual Basic Editor is called via Tools Macros from the main Excel menu bar If you do not create this reference then when your code attempts to call the What sBest attributes or procedures Excel returns the error message Sub or Function not defined Object Browser Excel s Object Browser is a useful tool for developing macros that run What sBest The Object Browser provides for display of all publicly declared objects within a given unit of code such as a module or the What sBest add in WBA XLA or WBA XLAM Once you ve selected such an object the Object Browser provides a v
103. the Add In Files This provides you with the choice of where the add in files are to be located The recommended choice of the Library subdirectory is already selected See the section entitled Location of the Add in Files and Update Links for details Final Instructions This gives some important instructions to complete the installation of What sBest When What sBest setup is almost complete click Finish and Excel will open You may receive a message that WBINTR XLS contains macros You must then select the ENABLE MACROS button in order to finish installation Also you can choose to view the README file Add ins library and executable files from LINDO Systems are digitally signed Installation Related Problems Most errors during installation result in clear on screen messages and instructions for remedying the errors If for instance you specify installation to a non existent directory the setup program asks you if you want to create such a directory now or quit installation Missing spreadsheet software found configuration files and incorrect spreadsheet releases are all handled similarly If after following all on screen instructions and referring to Troubleshooting you still have trouble installing What sBest please call LINDO Systems Settings for Microsofte Excele 2007 2010 Via the Ribbon select File Office Button Excel options INSTALLATION DETAILS 441 Popular Show Developer tab in the Ribbo
104. the dual solver will be used Primal The primal solver will be used exclusively e Dual The dual solver will be used exclusively 74 CHAPTER3 In general Solver Decides will yield the best results However experimentation with the other options may be fruitful Optimality The Optimality box contains three tolerances Absolute Relative and Time to Relative These tolerances allow you to tradeoff solution time vs solution quality Ideally we d always want the solver to find the best solution to a model Unfortunately integer programming problems are very complex and the extra computation required to find an optimum solution can be prohibitive On larger integer models the alternative of getting a solution within a few percentage points of the true optimum after several minutes of runtime as opposed to the true optimum after several days makes the use of these tolerances quite attractive Absolute Use the Absolute text box to set the absolute optimality tolerance This is a positive value r indicating to the branch and bound solver that it should only search for integer solutions with objective values at least r units better than the best integer solution found so far In many integer programming models there are huge numbers of branches with roughly equivalent potential This tolerance helps keep the branch and bound solver from being distracted by branches that can t offer a solution significantly better than the inc
105. the functionality provided see the section entitled Advanced Dual which refers to the dialog box interface that calls this procedure This routine is used to build the dual formulas All arguments are optional Syntax wbDualValue DualCells DualLoc DualChoice Argument Required Description DualCells No DualCells is the range or cell that dual value information is desired for If omitted the default is the offset of one cell to the left of the current selection DualLoc No DualLoc is the range or cell where the information is to be displayed If omitted the default is the current selection DualChoice No DualChoice is a string indicating the type of information desired Dual or Dual Value for dual value Upper or Upper Range for upper range and Lower or Lower Range for lower range If omitted the default is Dual For backwards compatibility with earlier versions of What sBest DualChoice also accepts the numbers 1 for Dual 2 for Lower and 3 for Upper Error Codes Description Dual_BadDualsCellArg Bad DualCells argument Dual_BadDualLocArg Bad DualLoc argument Dual_BadDualChoice Bad DualChoice argument Dual_ProtectedSheetError Unable to enter dual formulas on a protected sheet Remarks On integer models dual value information is only of limited use Example To create the dual value for cell E5 in cell F5 enter wbDualValue E5 F5 Dual Value To create the upper
106. the number of nodes to be created in that stage By default stage 0 will always one node thus the Oth index in the array will be one Other positions in the array corresponding to the number of nodes in stages 1 2 7 1 may take any positive integer values In this framework each node represents a block realization of all the stochastic parameters in that stage and will have a conditional probability of 1 N where Nt represents the number of nodes in stage t Specify the sample size per stochastic parameter In the method the user should provide an integer array of length S the number stochastic parameters in the model and give in each position the sample size for that stochastic parameter In either case What sBest will automatically construct a finite scenario tree with specified dimensions In a case where both stochastic parameters are normally distributed each belonging to a different stage Therefore creating N nodes per stage has the same effect as creating N samples per stochastic parameter whenever there is a single stochastic parameter per stage MATHEMATICAL MODELING 223 Note Sampling a scenario tree is not limited to stochastic parameters that follow parametric distributions It is also possible to use sampling for models which already have a finite scenario tree This is especially useful when the original tree is finite but still too big to handle computationally For instance a 2 stage model may have 30 sto
107. the scenarios set vertically ii the cell address of the random parameter The random demand in cell B19 will take the values entered in the range 19 S22 with equal probabilities WBSP DIST DISCRETE SV S 19 S 22 B19 Step 3 scenario information Specify the number of scenarios per stage by using the function WBSP_STSC This function requires a two column table as an argument 1 Column 1 for the number of stages in ascending order 2 Column 2 for the respective number of scenarios In this problem there are 5 stages with 2 scenarios each set in the range J28 K32 WBSP_STSC J28 K32 It happens the discrete table for the distribution has only 4 possible values so the random parameters will select 2 among the 4 outcomes with replacement The option Sampling on Continuous Only needs to be turned off to take advantage of the sampling feature Step 4 reporting cells Use the function WBSP_REP to specify the cells that should appear in the Stochastic report This function takes a series of arguments that are the address of the cells that should be reported in the final solution WBSP_REP C19 C23 D18 D23 WIth0 Wlth1 Wlth2 Wlth3 Wlth4 WIth5 Here the stochastic report will display the outcomes of scenarios for all the random cells the price of stocks for any periods the decision to sell and the ending wealth to maximize Use the function WBSP_HIST to generate a histogram of the Present Value in a 8 bin graph WBSP
108. to be reduced in order for it to appear in the optimal solution of HOGFEED This is a good number to have handy when negotiating to get a lower price For more information on dual values see the section on Dual Values in Chapter 3 Additional Commands SAMPLE MODELS 235 Chance Constrained Blending File name HOGCHANC XLSX TYPE NONLINEAR OPTIMIZATION Application Profile In the previous HOGFEED blending example we showed you a deterministic model the nutrient content of the different grains to be incorporated in the finished feed were known and constant However what if the nutrient content is not so reliable What if the content varies at random This model is nonlinear and perhaps more likely to occur in the real world The Problem In Words You must determine how much of four available grains to include in blended feed in circumstances under which the content of one of the nutrients varies at random to meet nutrient requirements with some degree of certainty Background You ve had several samples of the four grains tested and you find that their content of Nutrient D varies randomly and is normally distributed You want to find a new blend that ensures the minimum requirements for Nutrient D are met at least 95 of the time You ve calculated the mean and variance for each grain the content means are the same as in the original HOGFEED model The equation that calculates Nutrient D now looks like this Mean Nut
109. to mark cells as being adjustable Adjustable cells are easy to find in that they are displayed in a blue font and have the Adjustable style applied to them When identifying your adjustable cells be sure to place a numeric value any will do in each adjustable cell If an adjustable cell does not contain a numeric value What sBest will not attempt to change its value TROUBLESHOOTING 421 NOBEST No Best Cell Most What sBest models will have an optimization objective i e a best cell It could be profit maximization cost minimization or some other criterion that is a function of the adjustable cells So when a model does not contain a best cell the following warning will be displayed on the WB Status tab Warning Message WARNING No Best Cell Help Reference NOBEST Either no cell has been specified to be maximized or minimized or the cell that is marked is not a function of any adjustable cells If this is an optimization model use the Best command or the Minimize or Maximize toolbar button to specify a best cell as the objective of the optimization This warning can be turned off via the WB Options General menu Suggestions If you failed to indicate a best cell before solving specify a cell to be maximized or minimized and re solve Refer to section The ABC s Three Steps to What sBest for information on how to identify a best cell Occasionally you may have models that don t require a best cell w
110. variance by question for each group Objective of Optimization The objective is to determine the best sample size for each stratum while minimizing the cost to obtain the sample SAMPLE MODELS 295 The Worksheet Let s look at the sample file called SAMPLEWB The SAMPLEWB Worksheet Before Solving Home Insert Page Layout Formulas Data Review View Developer Addins 7 X A Q A Stratified Sampling Plan Design e A n B C D E F G H Je Se Stratum 3 Le Sample Size i i 1 00 Lower Bound gt gt Upper Bound lt g Population 300000 200000 100000 Cost 1 00 1 00 1 00 Weight 30 0 20 0 10 0 Stratum Variance Question 1 5 5 5 Question 2 2 4 8 Fixed Cost 1 00 Total Cost Maximum Variance Question 1 7 4999 0 043 Question2 1 799915 0 014 A Determine Adjustable Cells The adjustable cells are B5 E5 the Sample Sizes for each stratum B Define Best The best cell is the minimized cost of sampling in cell B16 The formula here adds a Fixed Cost a nominal 1 00 in this example to the sum of sample sizes times the cost per unit sampled C Specify Constraints There are four types of constraints in the model In B6 E6 a Lower Bound of 001 is placed on the sample sizes This ensures that optimizer errors caused by division by zero will not occur 296 CHAPTER7 In B7 E7 the sample size is required to be no larger than the population from which it is taken In C18 and C19 the variances
111. where n is the option s setting During these initial levels the solver picks a subset of the fractional variables as branching candidates performs a tentative branch on each of 168 CHAPTER 4 the variables in the subset and selects as the final candidate the variable that offers the greatest improvement in the bound on the objective Error Codes Description IntOpt_BadBranchingDirectionArg Bad BranchingDirection argument IntOpt_BadAbsolutelntegralityArg Bad Absolutelntegrality argument IntOpt_BadRelativelntegralityArg Bad Relativelntegrality argument IntOpt_BadWarmStartArg Bad WarmStart argument IntOpt_BadColdStartArg Bad ColdStart argument IntOpt_BadAbsoluteOptimalityArg Bad AbsoluteOptimality argument IntOpt_BadRelativeOptimalityArg Bad RelativeOptimality argument IntOpt_BadTimeToRelative Bad TimeToRelative argument IntOpt_BadHurdleArg Bad Hurdle argument IntOpt_BadNodeSelectionArg Bad NodeSelection argument IntOpt_BadStrongBranchArg Bad StrongBranch argument Example If one wished to set all the options the following would work wbSetIntegerOptions 1 01 120 O 1 20 000001 00001 3 1 00008 To set individual options here setting the RelativeOptimality tolerance to 10 named arguments should be used wbSetIntegerOptions RelativeOptimality 1 MACROS THE VBA INTER
112. 0 lt 100 120 lt 150 Costs 35 000 54 000 Total Cost 89 000 31 Status SHIPPING AF l l eng Zi E T The least cost solution results in shipping costs of 89 000 SAMPLE MODELS 363 D Dual Values The solution recommends shipping nothing from Mill 2 to Plant A To find out how much your total cost will increase if you transport anything between these two points use the Advanced Dual command to display in an empty cell the dual value for cell E6 Then optimize again and you ll find a dual value of 200 This means that your Total Cost would increase by 200 if you were to ship one unit along that arc If you perform the same operation for cell B7 the amount shipped from Mill 1 to Plant B you ll find a dual value of zero Dual values are generally positive for adjustable cells that have a zero value in the solution An exception to this occurs in cases where there are multiple solutions as in the current problem Because no penalty is incurred in moving from one optimal solution to another such a problem is said to have multiple optima there is more than one combination of adjustable cells that give the same minimum cost 364 CHAPTER 7 Traffic Congestion Cost Minimization File name TRAFFIC XLSX TYPE NONLINEAR OPTIMIZATION Application Profile In the earlier SHIPPING network problem the cost to transport your product from a supply point to a demand point was fixed However in some network proble
113. 0 0000 0 0000 0 0000 0 0000 0 0000 0 0000 0 0000 0 0000 0 0000 0 0000 0 0000 0 0000 5 7387 5 7387 5 7387 5 7387 SAMPLE MODELS 257 Lockbox Location File name LOCKBOX XLSX TYPE LINEAR OPTIMIZATION The Problem in Words As a corporate cash manager or officer of a bank whose clients must deal with the problem of optimizing collections from customers located at large distances from your home office you must decide where to locate postal lockboxes so that customers deposits can be credited with a minimum of expense caused by mail delay or float time while minimizing operating costs associated with the locations In addition each customer must be assigned to exactly one location Background Your customers located in Seattle Los Angeles Houston Philadelphia and Miami whose deposits must be credited with a minimum of float time may be assigned to potential lockbox locations in New York Atlanta Cincinnati Denver or St Louis It may also be more cost efficient to assign them to the home office Objective of Optimization Each customer must be assigned to a postal lockbox so as to minimize the sum of float time expense and monthly fixed operating costs 258 CHAPTER7 The Worksheet Let s open the LOCKBOX sample file and look at its design and formulas Monthly Proposed Lockbox Locations Cash Home Flow Seattle 0 0 5000 Los Angeles 0 0 5000 Houston 0 0 5000 Philadelphia 0 0 5000 Miami
114. 0 0000001 ADDITIONAL COMMANDS 65 Variable Bound Specify the maximum magnitude of variable bounds used in the global solver s convexification process in the Variable Bound text box in the Limit box Any variable bound with a magnitude in excess of this value will be truncated to this setting Setting this parameter appropriately helps the global solver focus on more productive domains The default setting for Variable Bound is 10000000000 Use of Bound Specify how to impose the above variable bound limit on the model variables in the Use of Bound drop down box in the Limit box Possible values are Solver Decides None All Variables or Selected Variables The default setting for Use of Bound is 0 or Solver Decides Branching Direction Specify the direction to branch first when branching on a variable with the Branching Direction drop down box The branch variable is selected as the one that holds the largest magnitude in the measure Possible values are Solver Decides Absolute Width Local Width Global Width Global Distance Absolute Violation or Relative Violation The default setting for Branching Direction is 0 or Solver Decides Box Selection Specify the node selection rule for choosing between all active nodes in the global solver s branch and bound tree in the Box Selection drop down box Possible values are Solver Decides Depth First Worst Bound or Best Bound The default setting for Box Selection
115. 00 Warehouse 1 Warehouse 2 Warehouse 3 Unit 2 100 0 100 0 100 0 300 Not 1200 Unit 3 100 0 100 0 100 0 300 PT Not 600 Unit 4 100 0 100 0 100 0 300 PT Not 400 Total Supply 400 Not 1100 400 Not 700 400 Not 1300 RATE Warehouse 1 39 14 11 14 Warehouse 2 27 9 12 9 Warehouse 3 24 14 17 13 LIMIT 1000 1000 Warehouse 3 lt lt Warehouse 1 500 1000 TIME PENALTY Warehouse 1 Warehouse 2 Warehouse 3 Epsilon Unit 1 4875 3375 2743 0 01 Unit 2 1556 1029 1680 Unit 3 1222 1371 2040 Unit 4 1556 1029 1560 Total Cost A Determine Adjustable Cells The adjustable cells are the amounts shipped along each arc from Warehouse to Intake Center in BS E7 B Define Best The best solution in C36 is the minimized sum of all time penalties C Specify Constraints There are three groups of constraints in the model In G5 G7 the Total shipped from each Warehouse F5 F7 is forced to be equal to its total Supply H5 H7 In B9 E9 the amount shipped to each intake center B8 E8 is forced to be equal to its demand B10 E10 In B23 E25 the amount shipped along 366 CHAPTER7 each route B5 E7 is required to be less than or equal to that route s maximum capacity minus Epsilon 01 in cell B33 see note below Note As traffic Flow approaches the Limit the Time Penalty goes to infinity Observe for instance the formula for
116. 1 on KYZ G6 24 Minimum coefficient in formula HAH WB Status WB Solution XYZ 7 The status report is organized as follows First final values for the fields appearing in the solver status window are presented At its conclusion the status report appends critically important information such as the solution status The fields in the report and the significance of the returned values are as follows Classification Data Total Cells This shows the total number of numeric cells in the model These are the cells useful to the model containing the Numerics Strings and the Constraints Numerics This shows the number of cells displaying a numeric value These are Adjustable Constant and Formula cells Adjustables This shows the total number of adjustable cells in the model Continuous This shows the number of Adjustable cells with a continuous range 0 infinity 36 CHAPTER 2 Free This shows the number of Adjustable cells with a free range infinity infinity Integers This shows the number of Adjustable cells with integer restrictions on a range 0 infinity Binaries This shows the number of Adjustable cells with binary restrictions 0 1 Constants This shows the number of cells containing a constant number or an unsupported function converted into its constant value This does not count the constant arguments within a formula Formulas This shows the number of cells containing a usefull form
117. 2 1 113 3 i 0 0 0 0 0 0 Not 108 9 130 5 173 2 5 0 0 0 0 0 0 Not 109 0 119 5 102 1 0 0 0 0 0 0 Not 108 3 139 0 113 1 0 0 0 0 0 0 Not 103 5 92 8 100 6 3 0 0 0 0 0 0 Not 117 6 171 5 190 8 S 0 0 0 0 0 0 Expected Return 0 0 Target Return 115 0 Variance Return gt Target Not gt Semi Variance 0 0000 Invest Total 100 Not Downside Risk 0 0000 A Determine Adjustable Cells The adjustable cells in the model are the percentages of capital invested in each Asset B3 D3 and the differences Over and Under in G4 H15 The Overs are forced to be the amount by which a Return is greater than the Expected Return The Unders similarly are the amount by which a Return is less than the Expected Return These adjustable cells are forced to take these values by the constraints in 14 115 which we explain in detail under C Specify Constraints below B Define Best You can choose one of three best cells in this model Well show solutions for all three minimization objectives Variance in H18 SAMPLE MODELS 275 Semi Variance in H19 and e Downside Risk in H20 The Variance is calculated in H18 with the formula E4 G4 H4 2 E5 G5 H5 2 E6 G6 H6 2 E7 G7 H7 2 E8 G8 H8 2 E9 G9 H9 2 E10 G10 H10 42 E11 G11 H11 2 E12 G12 H12 2 E13 G13 H13 2 E14 G14 H14 2 E15 G15 H15 2 This is equivalent to the sumproduct of the probabilities and the squared values
118. 83 Optimization The PUTOPTION Worksheet Before Optimization in g IST PUTOP TIONS Microsoft Exel ma File Home Insert Page L Formu Data Reviev View Devel Addi Risks e Qo gi X Stochastic Programming Version of an American Put Option The holder of the option has the right to sell a specified stock at any time the American feature between now and a specified expiration date at a specified strike price The holder makes a profit in the period of exercise if the strike price exceeds the market price of the stock at the time of sale Money is borrowed invested at the risk free rate _10 Step 1 Core one scenario model aif Initial Price 100 Strike price 99 Risk free rate 0 03 Stock Price of Wealth return this stock this this period period Sell 1 period 0 100 000 0 0 000 WBSP_VAR 0 08 92 000 0 0 000 WBSP_VAR 0 01 91 080 0 0 000 WBSP_VAR 0 01 90 169 0 0 000 WBSP_VAR 0 01 89 268 0 0 000 WBSP_VAR 0 03 91 946 0 0 000 WBSP_VAR 0 lt Number times sold Can sell at most once 1 PV of final wealth 0 lt Maximize et 4 4 gt H Model Scenario Tree a tT pOr SON Home Insert Pagel Form Data Reviev View Dee Add Ii aka e o SS zA WBSP_VAR WBSP_VAR WBSP_VAR WBSP_VAR Specify stage of Sell decisions Mark Stock returns as random parameters Step 2b Distribution of Distribution and give stage WBSP_RAND WBSP_RAND
119. 852 for H O and between 1 8 and 2 for other fluids 248 CHAPTER7 Now you re ready to solve the model and determine the balance of flows After solving the WB Status worksheet will open in order to show you the Nonlinearity present warning and No Best Cell warnings These warnings may be turned off from the General Options dialog box The three model worksheets now show the flows as shown below The FLOWNET Model After Solving FLOWNET xlsx Microsoft Excel ARC RESISTANCE Destination Cc D E Zomm pcomb Node Demand 1 2 4 6 Node Pressure 203 204 206 227 232 233 240 240 SAMPLE MODELS 249 FLOWNET xls 0 25 0 13 0 10 12 63 0 23 8 15 6 99 A B cC D E F G H 250 CHAPTER7 A B C D E F G H essure Parameter 1 Flow Parameter 1 Note The pressure parameter equals 1 for incompressible fluids and electricity and 2 for gases The flow parameter equals 1 for electricity 1 852 for H2O and between 1 8 and 2 for other fluids SAMPLE MODELS 251 Bond Portfolio Optimization File name BONDS XLSX TYPE LINEAR OPTIMIZATION FLOWNET Application Profile This multi period model is an example of dynamic modeling Dynamic means that decisions made in this period affect not only this period s returns or costs but future decisions and returns as well For this reason multi period problems cannot be treated as if they were merely a collection of single period problems
120. AL FTEs AND POOL DAYS Recommended FTEs Scheduled FTEs Staff FTEs Pool FTEs moo CO CO CO On Total Allocated FTE s COSTS PER DAY Staff FTEs 120 Pool Days 160 TOTAL COSTS Esseg 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Oo o oO CO On Lee reg Dn de L L d 348 CHAPTER7 The second screen P1 AG20 consists of possible individual schedules R5 AE16 work patterns and the adjustable cells P5 P 16 The FIXED1 Worksheet screen 2 Before Solving Number Schedule Available Schedule Patterns Assigned Number Su Mo Tu We Th Fr Sa Su Mo Tu Weih Fr Ga Days 10 10 10 10 10 10 10 10 10 10 0 0 CO COOC EH ECH wh sch E EH EH EH a sch sch sch O EH wh EN web sch at at sch wh OO O O wh sch sch wh sch sch EH ECH E E E ababab wh ad adb ad adb ab EN EH ch och E web ER web sch sch sch sch ooe sch sch E sch EN sch sch sch sch CO OO OO EN Ah ch ch ch E E EA CH oooc s ss EA EA EA CH E E wh wh BOC sch E sch sch EH EH ch och E EN wh och sch ab ech ab EH CH ch EH sch sch sch sch sch sch sch sch EN EH ch wh sch sch sch sch Oe E a EH OO sch sch sch sch sch EH sch EH CO OH A Ah E E E EN ch ch ch 3 22 3 SOS D S 20 9 CH o Pool Days Screen 2 depicts 10 work patterns that incorporate the assumptions and work rules outlined earlier Each work pattern represents an employee working 5 days a week 10 days per two week time period Employe
121. Custom Toolbars Al v fe What sBest 11 0 Solve Minimize When What sBest is first installed the following events should occur 1 the What sBest add in is loaded 2 the What sBest menu is inserted in the Excel menu bar and 3 the toolbar appears in floating undocked mode The user may then reposition the What sBest toolbar to a preferred part of the Excel window To remove the toolbar from view either go to WB Toolbar or to View Toolbars on the main Excel menu bar and uncheck the What sBest toolbar Reverse the process to return the toolbar to view Should you wish to remove the What sBest toolbar you would use View Toolbars Customize command select the What sBest toolbar and press the Delete button You would then have to reinstall What sBest in order to restore the toolbar What sBest advises the user as to its progress via the Excel status bar displayed at the bottom of the Excel window This bar normally displays Ready but displays other messages when either Excel or What sBest is performing some activity that may be prolonged For further information on the What sBest add in see the Add ins section GETTINGSTARTED 9 Developing a Model in What sBest The ABC s Three Steps to What sBest There are three steps to setting up a model to be solved by What sBest Throughout this help file you ll find them referred to again and again We call th
122. E18 The equations to calculate the predicted points D5 D12 are of the form Predicted Seasonal Factor Base Point Period Trend 282 CHAPTER7 The Error terms are the Predicted points minus the Sales data points For example the Error for Spring of Period I in ES is found with D5 CS Now let s solve the model After solving the WB Status worksheet will open in order to show you the Nonlinearity present warning This warning can be disabled from the General Options dialog box Your solved model now appears as follows The SEASON Worksheet After Solving Sales Season Period 1000 s Predicted Error Spring Summer Fall Winter Spring Summer Fall Winter Trend Base Sum of Squared CO JO On P Wh A 1 55 9 72 10 00 14 00 12 00 19 00 14 00 21 00 19 00 26 00 9 31 14 10 12 85 18 81 14 44 20 93 18 40 26 14 0 69 0 10 0 85 0 19 0 44 0 07 0 60 0 14 Seasonal Factors Spring Summer Fall Winter 0 83 1 10 0 89 1 18 1 00 1 00 Error 1 823 Average The spring seasonal factor is 83 In other words spring sales are 83 of the average season The trend of the solved model is 1 55 This means that after the effects of season are taken into account sales are increasing at an average rate of 1 550 per season SAMPLE MODELS 283 Actual sales compared to predicted sales are shown in the graph below
123. EE 310 PREFACE ix Application dree 310 The Building Block Meibod 314 Application de EE 314 Waste Minimization in Stock Cutting 321 Application Rone EE 321 Plan HOCAtOn EE 327 Staff Scheduling teen Eege cee tn need Leeda 333 Application Protine Aa aaa a a eege a 333 Staff Scheduling Preferred Assignment sssessseeseesissresisrissrrsiisrinrinsrnrinsrustnnrnstnntnnsnntnnnt 338 Application re EE 338 Staff Scheduling Two Stage Fixed hf 345 elle te re EE 345 Stage 1 Cost Minimization cececceceeceeeeeeeeeee eee eeeeaeeeeaeeeseaeeeseaeeeeaaeseeeeeseaeeesiaeenennees 345 Stage 2 Preference Maximization ccccccececceceeeeeeeeceeeeeeeeeesaeeeeaeeseeeeescaeesteaeeseeeeess 351 Pipeline Cp ao anal al Bh ea aan 356 Application Profile c c a si annua Weenies nel nn Se Bae 356 Shipping Cost REGUCtION innia eaii aaa daidai ia adaa ia doddi aaa aidian 360 Application re EE 360 Traffic Congestion Cost Minimization 0 cccceesceeeeeeeeceeeeeeeseeeeeseaeeeeeeeseeeeeseaeeseaeeeeaes 364 Application Profile EE 364 Truck LOACING DEET EE 367 Application Profile iszen Hee E E 367 Crop Allocation Under Uncertainty 0 0 ccccecsceececeececeeeeeee scenes ceaeeesaaeeseeeeseeeeessaeeeenaeeneaes 372 Application lte EE then dha 372 el rd e en EE ae ee tae Re ue el 379 Application Profile ticnatcc5 E 379 8 TROUBLESHOOTING eege ee eege Se Eed EE 387 General amp Operational Problems 387 Content iach t
124. EMIC declaration is as follows written in the selected cell of the spreadsheet WBSEMIC lower bound upper bound range references This will restrict the variable in the range references to be either 0 or to lie within the range lower_bound upper_bound Note Each semi continuous variable will be counted against any integer variable limit for your installation ADDITIONAL COMMANDS 47 Select Any Cells This reference field allows you to select any of the Adjustable cells that will be part of the set You can select multiple ranges by holding the Control Ctrl key Then click on Add to store them in the List of Selected Cells field List of Selected Cells This field will list all the cells that are part of a Semi continuous set You can modify the list by clicking on Add or Remove buttons Place the Function in Cells Specify the host range of cell you would like to see the SEMIC function to be placed in the spreadsheet What sBest automatically fills this box with the current cell selection If the selection already contains a WBSEMIC function then What sBest will display the cells argument into the List of Selected Cells 48 CHAPTER3 Options The Options command allows you to customize the operating parameters of the What sBest solvers the user interface how What sBest runs and how your model is displayed There are several sub commands under the Options command on the WB menu They are
125. EY4 414 LICKEYS 415 LICOPT1 415 LICOPT2 416 LICOPT3 416 LICOPT4 416 LICOPTS 417 LICOPT6 417 Lifting 70 Lindo Systems Inc 449 Linear expressions 193 Formulas 193 regression 289 Linear Solver Options Dialog Box 56 Linear vs Nonlinear Expressions and Linearization 193 linearization 51 Linearization 188 194 289 LINEARZ XLSX 289 Linus Schrage 2 LN 187 Load 367 Local Optima 195 Local Width 65 Locally Optimal 200 Locate 118 Location of the Add in files and Update files 444 LOCKBOX XLSX 257 Locked cell 21 132 LOG 187 Logical operators 187 LOOKUP 418 Looping Macro 136 Louveaux F 215 Lower Range 87 94 LP Solver 73 Macros 133 Macros in VBA 129 Make Adjustable 20 Make Adjustable amp Free or Remove Free 21 22 Markowitz Portfolio Model 261 MARKOWITZ XLSX 261 Mathematical functions 187 MAX function 187 188 Max Passes 69 Maximize Best 23 Media Buying 302 MEDIA XLSX 302 Menu bar 7 Menu commands 7 Microsoft Windows 3 MIN function 187 Minimize Best 23 Minimum and Maximum coefficients 36 MMULT 187 MOD 187 Model Reduction 57 Model Type 32 36 Models Non optimization 201 with Integer Restrictions 2 MODERR 419 Multi Period Inventory Management 306 Multi period Model 251 Multiple Optima 94 100 363 Multistart Attempts 64 456 INDEX N NAME 420 Network installation 4 errors 360 Network models 243 NEWSVE
126. Excel 2010 64 bit on Windows Vista or 7 64 bit This will allow you to solve models with larger sets of data 0 Support of Additional Excel s Default Built in Functions What sBest 11 supports additional Cumulative Distribution and Probability Density functions A complete list is given in the Supported Functions and Operators section PREFACE xv O Support of Excel version 2007 file format Windows Vista 7 The solver module has been redesigned to support up to date platforms and operating systems Windows XP Vista and 7 with Microsoft Excel 2002 or Excel 2007 2010 What sBest can run models saved with the new workbook format XLSX XLSM or XLSB with the maximum size of 16384 columns and 1048576 rows 0 Select Display Language to be English Chinese French or Japanese This feature allows the user to select the language of his choice to be either English Chinese French or Japanese without reinstalling the add in The language selection makes easier the project for companies working in a multi cultural environment The What sBest development team wishes you the Best in all your optimization endeavors Copyright 2011 LINDO Systems Inc xvi PREFACE 1 Getting Started with What s Best What is What sBest What sBest makes available to your Excel spreadsheet program a highly developed solver capable of performing linear and nonlinear optimization on the most diffic
127. FACE 169 wbSetintegerPreSolverOptions This routine is used to set the What sBest integer pre solver options seen in the Integer Pre Solver Options dialog box For additional discussion of the options available through this routine see the section entitled Options Integer Pre Solver All arguments are optional Syntax wbSetIntegerPreSolverOptions HeuristicLevel ProbingLevel MaxCutsPasses RelativeCutsLimit CoefficientReduction Disaggregation FlowCover GCD Gomory GUB KnapsackCover Lattice Lifting PlantLocation ObjIntegrality Basic Cardinality Argument HeuristicLevel Required No Default 3 High Description HeuristicLevel controls the level of integer programming heuristics used by the integer solver These heuristics use the continuous solution at each node in the branch and bound tree to attempt to quickly find a good integer solution 0 None 1 Low 2 Medium 3 High Cutoff Criterion ProbingLevel 0 Solver Decides 0 Solver Decides The Cutoff Criterion is used to control the criterion for terminating heuristics 0 Solver Decides 1 Time 2 Iterations ProbingLevel controls the amount of probing that occurs in integer models Probing involves taking a close look at the integer variables in a model and deducing tighter variable bounds and right hand side values In many cases probing can tighten an integer model sufficientl
128. Feet E End Waste rs t Note that the optimized cutting plan includes no footage cut to Pattern B in cell G5 Also end waste occurs only in the 35 width E13 SAMPLE MODELS 327 Plant Location File name PLANTLOC XLSX TYPE LINEAR OPTIMIZATION This problem is related to the class of network or routing problems In such problems the variables usually include multiple choices of transportation routes to and from a number of points of origin and destination with differing shipping and operating costs associated with each possible route The Plant Location problem is similar to the SHIPPING sample model but allows greater latitude of decision making in that the points of origin plant locations are variable Manufacturers and wholesale businesses are likely to encounter problems of this sort in matching existing customer demand to product availability and minimal transportation costs Background Your firm has a choice of five locations in which to operate a manufacturing facility Six markets exist with a demand for your product Each potential plant has an associated monthly operating cost and shipping routes to the demand cities have varying costs In addition each potential plant will have a production capacity that must not be exceeded Objective of Optimization The objective is to locate plants in such a way that demand is satisfied in each target city potential plant capacity is not exceeded and overall oper
129. G ed NA a 72 BE EE 73 E Oto aleet EE Saeed ete ie ee eel maha 74 elle 75 Usage Guidelines for K Best Solutions 0 0 2 ceeceeeseceeeeeeeeeeeeeeseeeeeeeaaeeeeaeeseeeeseaeeseaeenenes 77 Options Stochastic Solver ees dees SEN O aaa aa a e ieee each 82 Stochastic Modeling Support c ccceeceeseeceececeeeeeeaeeeeeeeeceeeeeeaeseeaaesseeeeseeeeseaeeseneeee 82 Optimization Method e serais an th hn didn inti die th eed 82 Seed for Random Number Generator 83 Common size per e TE 83 Sampling on Continuous Distribution Only 83 Expected Value of Obiechhye A 84 Calculation for Expected Value of Wait and See Model s Objective esseesseeesseeenee 84 Calculation for Expected Value of Policy Based On Mean Outcome ssssnsssenneneneneeee 84 Calculation for Expected Value of Perfect Information ssnnsnssnnssssnnnsrnnnnsennnnnnrnnnnnena 84 Calculation for Expected Value of Modeling Uncertainty c ccssceeeeeeeseeeeeeteeeeeeeeees 84 Print Scenarios Horizontally in Heport 85 Options Reset to Defa lt ee g et ee A a eegene ede 86 ee De TE E techy 87 RETTEN 88 PROPOIT on RE EEN 88 BUER 88 Upper Range and Lower Range ssssessrseessrnesrrrnessernesrrnnnentinnnsnnnnnnnnnnnnntnnnntnnnnnnnnnnnnnnn nne 88 Report Information in c ecccecccceeeeceeeeececeeeeeeaaeeeeeeeceaeeeeaaeseaeeceaeeesaaeeseaaesseneeesaeeseaaeesnees 88 Usage Guidelines for Dual Values 89 Dual Value of a Constraint Cell Shadow Pri
130. GlMultistartAttempts GlOptimalityTolerance GlDeltaTolerance GlVariableBoundLimit GlUseBoundLimit GlBranchingDirection GIBoxSelection GlAlgebraicReformulation Argument Required Default Description GlStratGlobal No 0 False GlStratGlobal is a True False flag indicating whether or not to use this strategy GlMultistartAttempts No 0 Solver GlMultistartAttempts is a positive Decides number indicating the multistart attempts 0 Solver Decides 1 OF 2 or more number of attempts GlOptimalityTolerance No 0 000001 GlOptimalityTolerance is a number between 0 and 1 indicating the optimality tolerance GlDeltaTolerance No 0 0000001 GlDeltaTolerance is a number between 0 and 1 indicating the delta tolerance GlVariableBoundLimit No 1000000000 GlVariableBoundLimit is a positive integer indicating the variable bound limit GlUseBoundLimit No 0 Solver GlUseBoundLimit indicates the use Decides of bound limit MACROS THE VBA INTERFACE 165 0 Solver Decides 1 None 2 All Variables 3 Selected Variables GlBranchingDirection No 0 Solver Decides GlBranchingDirection indicates the branching direction 0 Solver Decides 1 Absolute Width 2 Local Width 3 Global Width 4 Global Distance 5 Absolute Violation 6 Relative Violation GIBoxSelection 0 Solver Decides GlBoxSelection indicates the box selection
131. H 187 SLP Direction 60 Smooth vs Nonsmooth Expressions 198 SMOOTH XLSX 284 SOCP xiv 224 Solution Outcomes 200 Solution Status 36 200 INFEASIBLE 201 Locally Optimal 200 NUMERICAL ERROR 201 Optimal 195 UNBOUNDED 201 Solution Time 37 Solve command 31 Solver decides 76 Solver Method 57 Solver Type 32 37 Solver Version 62 SOLVERR 424 Solving 31 33 Interrupting 200 no feasible solution found 201 optimality conditions 200 runtime 33 Unbounded 201 Solving Second Order Cone Programs 224 SPCORR 425 SPDIST1 425 SPDIST2 426 SPDIST3 426 Spearman correlation 222 Specified 4 Specify Constraints 15 SPERR 427 SPHIST 428 SPRAND 428 SPREP 429 SPSTSC 430 SPVAR 430 SQRT 187 Staff Scheduling 333 Staff Scheduling Preferred Assignment 338 Staff Scheduling Two Stage Fixed Shift 345 STAFF XLSX 333 Stage 205 458 INDEX Standard Deviation 261 Standard normal linear loss function 191 Starting What sBest 7 State of the model 32 Statistical functions 189 91 Steepest Edge 58 61 Steps 32 Stochastic 82 205 Stochastic Monte Carlo Sampling 221 STRARG 431 Stratified Sampling 294 Strictly convex concave 198 String Support Maximum String Length 104 STRLIST 431 Strong Branch 76 STRRES 432 Sub or Function not defined 133 395 SUM 187 SUMIF 188 432 SUMIFWBSOLVER XLS 433 SUMPRODUCT 187 Supported Functions and Operators 187 System R
132. IFINV This function returns the inverse of a uniform cumulative distribution for supplied low and high values The syntax is WBUNIFINV Prob Low High Prob The probability corresponding to the uniform distribution The probability must be greater than or equal to 0 and less than or equal to 1 Low The lower limit of the uniform distribution High The upper limit of the uniform distribution High must be greater than or equal to Low Probability High 5 For example UNIFINV 4 1 2 returns 1 4 This means that for a uniform distribution on the interval 1 2 40 percent of the outcomes are less than or equal to 1 4 FUNCTIONS AND OPERATORS 191 Inverse Multinomial Cumulative Distribution MULTINV This function returns the inverse of a multinomial cumulative distribution for a supplied set of probabilities and corresponding values The syntax is WBMULTINV Prob ProbRange ValueRange Prob The probability corresponding to the multinomial distribution The probability must be greater than or equal to 0 and less than or equal to 1 ProbRange The cell range containing the probabilities ProbRange must be a one dimensional range 1 e a range of cells in either a single row or a single column ValueRange The cell range containing the values that correspond with each probability The ValueRange must be the same size and shape as the ProbRange For example given the following id q s Book2 Microsoft Ex
133. ING COST AVERAGE MAIL DELAY BETWEEN LOCATIONS Proposed Lockbox Locations Customer Home 25 Locations New York Atlanta Cincinnal Denver Seattle Office Seattle 4 4 Di O E Locations Yew York Atlanta Cincinnal Denver Seattle A0 Seattle Los Angeles Houston Philadelphia Miami nn nu vyv nnn E www nn mn v D vvvY oon vvvvw nnnn nu nn nn nu vvvvv nnnn nu vv nn Ki D nu v D WB Status SO DH After optimization What sBest has returned a minimized cost of 11 475 in cell H19 and the optimal solution recommends that lockboxes be opened in New York Atlanta and Denver Note This model is an example of a class of models that returns naturally integer answers even though the adjustable cells have not been specified as integer Since the use of 0 1 integer adjustable cells can cause dramatic increases in computation time it is best to use naturally integer models whenever possible Space here prohibits a detailed explanation of this phenomenon Interested readers may refer to Chapter 8 Introductions to Operations Research by Hillier amp Lieberman 7th ed from McGraw Hill and to Chapter 14 Optimization Modeling with LINGO by Linus Schrage from Duxbury Press The latter text is available from LINDO Systems Call 312 988 7422 for more information SAMPLE MODELS 261 Markowitz Portfolio Problem File name MARKOWIT XLSX TYPE NONLINEAR OPTIMIZATION Application Prof
134. LESHOOTING 407 Suggestions When a model is interrupted there are certain instances when What sBest will be able to return the best solution found so far The best way to determine if an incumbent solution exists is by checking the Best Obj field in the What sBest Solver Status window If this field contains a numeric value then there is an incumbent solution that can be returned If this field does not contain a numeric value then a valid solution does not exist In which case the values returned by the solver will not be valid Keep in mind that at best the returned solution is most likely sub optimal Furthermore whenever you interrupt the solver you should check the final infeasibility value on the WB Status tab of the returned workbook An infeasibility value significantly different from zero means that some constraints are violated in the returned answer and the solution is not valid Note Ifa feasible integer solution is found for an integer optimization model restoring the best integer solution may take some time after the initial interrupt So don t be concerned if the solver doesn t interrupt immediately when the model contains integer adjustable cells The solver is just busy returning to the incumbent solution 408 CHAPTER 8 INVMOD Invalid Model If you attempt to solve a model that is clearly not formatted for What sBest the following error message will be displayed on the WB Status tab Error Me
135. LOC XLSX 327 PMIXMAC XLS 136 PMIXMAC XLSM 310 PORTCOST XLSX 265 Portfolio Minimizing Downside Risk 269 Scenario Model 273 Selection 261 With Transaction Cost 265 PORTSCEN XLSX 273 pre sampling 221 Presolve 60 Pricing 57 PRICING XLSX 298 Primal 73 Primal Pricing 58 Probing Level 67 PRODMIX XLSX 310 Product Mix 310 Professional Version 120 Profit 269 298 314 Protect Sheet 21 26 132 INDEX 457 Put Option 379 PUTOPTION XLSX 379 Q quadratic cone 225 Quadratic Recognition 60 QUAPREC 422 R RAM 3 Random number generator 83 Random number seed 83 Random variable 82 Ranges 94 Reading file 33 Reasonable Bounding Constraints 203 Reduced Cost 91 Refers to 42 Register 124 Relationships 204 Relative 72 Relative Limit 70 Relative Violation 65 Remove adjustable 21 Restrictions 204 Return 251 265 RHS 25 Right Hand Side 25 Risk 265 269 273 Runtime 33 Concerns 42 Errors 396 RUNTIME 423 runtime limit 50 Sales 279 sample size 222 SAMPLEWB XLSX 294 Sampling 294 Savage S 2 Scale Model 57 Scaling the Model 203 Scenario tree 222 Scheduling Model 333 338 345 SEASON XLSX 279 Seasonal Sales Factoring 279 second order cone xiv Selective Constraint Evaluation 60 Semi variance 273 Set Adjustable cells 15 Setting the Model 33 Shadow Price 89 Shipping Cost Reduction 360 SHIPPING XLSX 360 SIMXPO XLSX 284 SIN 187 SIN
136. NDOR XLSX 208 No best cell warning 201 No Feasible Solution Found 201 NOADJ 403 NOBEST 421 NOCONST 421 Node 243 Node Selection 76 None Best 23 Nonlinear 120 Expressions 194 Initial Values 203 Magnitude of Model 203 Modeling Guidelines 203 Models 61 200 203 Models Starting Point 61 Optimization 195 Scaling the model 203 Solver Options Dialog Box 59 Undefined Regions 203 Nonlinears 33 Non optimization Models 2 201 NORMINV 187 NORMSDIST 187 NOT function 188 NPV 187 Numeric cells 32 NUMERICAL ERROR 201 O Obj Bound 32 Obj Direction 33 Objective 13 23 32 Objective Value 37 Omit Cells 160 Omit command what not to omit 102 what to omit 102 OMIT names in workbook 101 OMITTED 422 Operators 187 91 Optimality 64 74 Optimality conditions 200 Optimality Tolerance 61 Optimizables 33 Optimization 2 Models 2 Nonlinear 195 optima 200 Optimum Global 195 Local 195 Locally 200 Options and Solvers 39 Options command 48 Options General 49 Options Global Solver 63 Options Integer Pre Solver 66 Options Integer Solver 71 Options Linear Solver 56 Options Nonlinear Solver 59 Options Reset to Default 86 Options Stochastic Solver 82 OR function 188 P Parameters Dialog Box 117 Partial 58 Pearson correlation 222 Personal Version 120 PI 187 Pipeline Optimization 356 PIPELINE XLSX 356 Plant Location 70 327 PLANT
137. NDS 103 Advanced Function Support This feature allows you to build models using not only Microsoft Excel built in functions but also user defined functions The user defined functions can be VBA macros add in file XLA macros calling a user add in file XLA XLAM or a library file XLL This feature is useful if the user has to develop his own function and wants these functions to be integrated in the optimization model The function support and add in names are entered using the following dialog box Function Support kl Function Support By checking this option every function will be supported in the model Otherwise What sBest will treat user written functions as unsupported functions and will read the cell as a constant equal to its numeric value at the start of optimization This support of user functions will make the model bigger more complex and so may make solution time longer In Microsoft Excel 2002 you may need to adjust a setting from the menu Tools Options Security MacroSecurity TrustedSources so as to grant any project access Thus grant any project access Check the box Trust Access to Visual Basic Project save your model and reopen Excel In Microsoft Excel 2007 adjust the security setting from the Excel Options then click on Advanced TrustCenter TrustCenterSettings Enable macros and grant the same access in the Macro Settings tab In case of your add in has been dig
138. NTIES OF MERCHANTABILITY FITNESS FOR A PARTICULAR PURPOSE OR OTHERWISE Further LINDO Systems Inc reserves the right to revise this software and related documentation and make changes to the content hereof without obligation to notify any person of such revisions or changes ACKNOWLEDGEMENTS We gratefully acknowledge Professor Sam Savage for his contributions to early releases of What sBest Copyright 2011 LINDO Systems Inc Published by 1415 North Dayton Street Chicago Illinois 60642 USA Sales amp Information 312 988 7422 info lindo com Technical Support 312 988 9421 tech lindo com www lindo com Contents CONTENT Siscecicietec enn EE Ee EE lll BREEAGE ees Seege WESSEN EE Ee XIII 1 GETTING STARTED WITH WHAT SBEST cescccsseeceseeeeeseeeeseeeenseeeeeeeeeseeeseseenenseeeeeeees 1 Whats Wat ebeetl a a a aa aaa aa RAAEN 1 Optimization Model 2 Non Optimization Modele 2 Models with Integer Restrictions AA 2 System Heouirements A 3 Installation QOVOrVieW siidist iiidid tanie i a aiat ia diada 4 The What sBest Interactive Environment ssssssssssssissssrrssrirrssinnnsstinnnstinnnntinnnntnnnnstnnnnnnnt 7 Developing a Model in Wat sfiest t 9 The ABC s Three Steps to What sBest eccceccceeeeeeeceeeeeeeceeeeeeaeeeeeeeseaeeesaeeesaaeeeeeeeeaas 9 TUTORIAL EE 10 The XYZ Production Problem AAA 10 The ABC S ue eege ee EEN AEN Need Deeded 12 What s Best lfisin2cticcuh ie ates e M
139. Node Selection and Strong Branch We discuss each of these in the sections that follow Hurdle Enter a value in the Hurdle text box if you know the objective value of a solution to a model This value is used in the branch and bound manager to narrow the search for the optimum More specifically What sBest will only search for integer solutions in which the objective is better than the Hurdle value Any user supplied Hurdle value comes into play when What sBest is searching for an initial integer solution At this point the solver can ignore branches in the search tree with objective values worse than the Hurdle value because a better solution exists i e the Hurdle on some alternate branch Depending on the problem a good Hurdle value can greatly reduce solution time However if you set a Hurdle value better than the optimal integer solution What sBes t will return an infeasible error message because no feasible answer can satisfy the bound you have set Once What sBest finds an initial integer solution the Hurdle tolerance no longer has an effect At this point the relative optimality tolerance comes into play Hurdle differs from the relative optimality tolerance If you use Hurdle by itself and an integer answer is found it will yield the true optimal integer solution For example if you re confident the best integer answer is at least 95 for a maximization problem just enter 95 as the value in the Hurdle field What sBest
140. OPT5 Stochastic Solver Not Licensed cccccceseeecesseeseeeceeeeececsesseaeseeeeseeeseesees 417 LICOPTE Conic Solver Not Uicensed 417 LOOKUP Unsupported Lookup Usage seer eeeeeaeeeeneeseaeeesaaeeeeaeeteeeees 418 MEMORY Insufficient Memory cccccceeeeeeeeeeee cee eeeeaaeeeeeeeceaeeesaaeeseaeeseeeeseaaeseeaeeeenees 418 MODERR Parsing Cell Formula 419 NAME Unsupported Name sinisesse a ienee E NRA e aaaea a E iiS 420 NOADJ No Adjustable Cells AA 420 NOBES F No Best Gelli deet e Ee EE tha EEE A a thea 421 NOCONST No Constraint Cells ccccccccccceeeeesesssseceeeeeceesseaeeeceeeescesceeaeaeseeeeseneseaaes 421 OMITTED Omitted Cell Heterence 422 QUAPREC Quadratic Recognition c cccececeseceeeeeeceeeceaecaecaeceaeceaeseaeseeeetereseneeeaeeeaes 422 RUNTIME Runtime Limit Heached 423 SOLVERR Solver Cor 424 SPCORR Stochastic WBSP_CORR Fommat 425 SPDIST1 Stochastic WBSP_DIST 1 Format ccccccccccececececsesseaeceseeeeeessstsaeeeeeeeeens 425 SPDIST2 Stochastic WBSP_DIST 2 Format ccccccscccesceececsesseaeseceeeeeesesseaeeeeeeeeess 426 SPDISTS Stochastic WBSP_DIST 3 Format ccccccscccecceccecsesseaeseeeeeceesesseaeeeeeeeeees 426 SPERR Stochastic Error 427 SPHIST Stochastic WBSP_HIST Format ccccccccssssssseceseescecsesseaeseseeseeeseseeseeeeeeeens 428 SPRAND Stochastic WBSP_RAND Format ccccccsseceeceeececsesseaececeeseee
141. OS 187 ACOSH 187 Active 32 37 ADDINLINK 397 Additional commands 39 Adjustable button 20 Adjustable cell 9 12 Dual Value for 91 Adjustable command 20 refers to 21 Adjustable dialog box 20 22 Adjustable Best Constraints Steps 9 Adjustables 32 Advanced Parameters 103 117 Advanced Dual 87 Advanced Dual 87 Advanced Function Support 103 Advanced Om 101 Advanced Omit 101 Advanced Parameters 117 Advanced Stochastic Support 115 Advanced String Support 104 Advertising 302 Algebraic Reformulation 65 Algorithms 40 Alpha 284 Index American option 379 AND function 187 188 Antithetic variate 222 Arc 243 ARITHERR 398 ASIN 187 ASINH 187 Asset 265 273 ASSIGN XLSX 338 ATAN 187 ATAN2 187 ATANH 187 Audience 302 AVERAGE 187 B Backsolving 201 Backward analytical 62 Barrier 57 73 Base 279 Best bound 76 Best Button 23 Best cell 9 12 23 none 24 Best command 23 Best dialog box 23 Best Obj 32 Best Objective Bound 37 Beyond VBA 131 Binary integer variables 2 42 Birge J 215 Blending Models 230 BLKCELL 398 BLOCK XLSX 314 Bond portfolio optimization 251 BONDS XLSX 251 Bounding constraints 203 Box Design 239 Box Selection 65 BOX XLSX 239 Branching Direction 65 72 Budget 302 C CALL 397 451 452 INDEX Capacity 120 298 327 356 Car pricing 298 Cash flow 251 Cell best command 24 dual value
142. Obj Direction This shows the direction of the best cell Event This shows the stage of the solution process the solver is currently performing 1 NAARWH Extracting Data Check license copy file Extracting Data Reading file Extracting Data Storing relevant formulas Extracting Data Creating instruction list Building the Model Solving Writing Solution During the last stage What sBest writes the solved values of the adjustable cells directly into your spreadsheet and writes any requested reports Elapsed Runtime This shows the length of time the solver has been running in hours minutes and seconds Note The terms Variables or Optimizables are used interchangebly and refer to the total number of adjustable cells constraint cells and cells containing formulas dependent on adjustable cells that influence the objective function 34 CHAPTER 2 Getting the Best Results You can obtain the results of your call to solve the model by either examining the values displayed in the spreadsheet or by looking at the status report WB Status and or the solution report WB Solution In some cases such as when you have a pretty good estimate of what the objective should be you might feel satisfied after inspecting the best cell However even under such ideal conditions it is a good idea to confirm that your model solved successfully by examining the status report The status report provides detaile
143. Omit You may use the Advanced Omit command to remove parts of your workbook from What sBest s consideration thereby decreasing the time to reach a solution If you can identify cells with a numeric value or equations that are extraneous to the problem being solved then you can place them in an omit range For example some cells in the spreadsheet may be used for evaluation and reporting and are not used in finding the solution If these cells contain unsupported spreadsheet functions placing them in an omit range will eliminate the error message caused by the equations If they contain equations that depend upon adjustable cells placing them in an omit range can shorten the time required for solving What Not To Omit What sBest will ignore the values and equations of every cell within an omit range Therefore it is very important that information pertinent to the problem is not included in an omit range For example you should be especially careful not to include Adjustable cells Any adjustable cell within an omit range will not be adjusted during the solution process Constraint cells Any constraint cells included in an omit range will be ignored during the solution process Cell Precedent to Equations Outside Omit Range An equation outside of any omit range that references a cell within an omit range will cause the solution to be aborted with an error message and the unsolved model will be returned ADDITIONAL COMMA
144. Optima Dual values are generally positive for adjustable cells that have a zero value in the optimal solution An exception to this occurs when there are multiple optima i e more than one combination of adjustable cell values that yield the same optimal objective value because no penalty is incurred in moving from one optimal solution to an alternative one If your model yields zero dual values for any of the adjustable cells it s likely that there are multiple optima for your model Dual Values in Nonlinear Problems Dual values in a nonlinear problem can be interpreted usefully only for small changes in the right hand side It may not be possible to make assumptions about the effect of large changes in a nonlinear model You may want to investigate the effect of such changes manually by inserting proposed changes and re solving Users with knowledge of calculus can think of the dual value as a derivative the rate of change of the best cell value with respect to changes in either the right hand side of a constraint or an adjustable cell value As with the derivative of a linear function a dual value in a linear model has a constant value over a range In nonlinear models the dual value is similar to the derivative of a nonlinear function it is valid at the point where it s evaluated but it may change immediately as you move from that point Dual Values in Integer Problems Dual values for problems with integer variables cannot be u
145. Over qoal 807 Not gt 0 Net utility 80 To be maximized MATHEMATICAL MODELING 217 The INVESTMENTCOLLEGE Worksheet Before Optimization part 2 3 ESTMENTCOLLEGE xlsx Microsoft Excel N Bas EE Home Insert Page Layout Formulas Data Review View Developer Addins X Investment Planning for Going to College After 3 Periods sl 4 H I J K L M N O a SE ze e el x Step 2a Time stage specifications for Random variables Decision variables Step 2b Distribution specifications Growth factor distribution equally likely Scenario Stocks Bonds re 3 sre e by stage Stage is Se EE A Stepping through the five steps again for this multistage example Step 1 The core model is described in the range A3 H16 The growth factors for Stocks and Bonds are in columns B and C They will be declared random in step 2 below The beginning wealth before each allocation decision is given in column D At the beginning it is a constant 55 In subsequent periods the wealth available comes previous period investments and is given by the formulae D10 SUMPRODUCT G9 H9 B10 C10 D11 SUMPRODUCT G10 H10 B11 C11 D12 SUMPRODUCT G11 H11 B12 C12 218 CHAPTER 6 The decision variables of how much to invest in Stocks and Bonds each period is in columns G and H The total amount invested is computed in column F and is given by F9 SUM G9 H9 F10 SUM G10 H10 F12 SUM G11 H11 We require the amount in
146. R arg cells where arg is a numeric value or a cell reference to a numeric value and cells is a reference to the cells you want to apply this timing information The cell reference TROUBLESHOOTING 431 should be to a variable cell For more information on using the stochastic feature refer to section Advanced Stochastic Support STRARG String Arguments Found In general What sBest expects all function arguments to be numeric If any unexpected string arguments were found while parsing the workbook What sBest will display the following warning message on the WB Status tab Warning Message VA RNING String Arguments Found Help Reference STRARG Text arguments have been found in formulas Their values have been taken to be zero This can lead to infeasible or sub optimal solutions Please check the returned solution carefully This warning can be turned off via the WB Options General menu See list of cells below Suggestions In many instances text arguments can be replaced by numeric arguments with the same end effect See if this is possible in the cell formulas listed as part of this error message STRLIST String List Error What sBest supports string cells and arguments under certain conditions If your string listing becomes unsupported the following error message will be displayed on the WB Status tab Error Message RROR String List Error Help Reference STRLIST What sBest enco
147. Report drop down boxes You might want to use the Delete Reports button if you have already solved the model and want to save it without the What sBest reports especially when the reports are at the end of many worksheets Options Linear Solver The dialog box posted by the Options Linear Solver command appears as follows Linear Solver Options ax Iw Scale Model Iw Model Reduction Solver Method Solver Decides D Pricing Dual Pricing Solver Decides e Primal Pricing Solver Decides v Help ADDITIONAL COMMANDS 57 This command allows you to set a number of options controlling the function of the linear solver Scale Model Select the Scale Model checkbox to rescale the matrix coefficients so the ratio of the largest to the smallest coefficients is reduced Rescaling reduces the likelihood of roundoff error and thereby promotes numerical stability and accuracy The default setting is to enable scaling Model Reduction Enable the Model Reduction checkbox to have the solver identify and remove extraneous variables and constraints from the formulation before solving In some cases this greatly reduces the size of the model to be solved to the user s benefit In other cases reduction only adds to the solution time without significantly decreasing the size of the model Because Model Reduction can offer significant improvement in performance this is one of the more significant options However it is extremely
148. SIGNMENTS D5 ASSIGNMENTS E3 lt 1 which excludes Reagan from working more than one of the three Monday shifts or more than one of three shifts beginning with Monday Evening respectively 344 CHAPTER7 You re now ready to solve the problem Model Worksheet in ASSIGN After Optimization MULTIPLE SHIFT ASSIGNMENT MODEL Preference Total Tag Mon Tue Wed Thu DAYS REQD 2 1 1 3 EE SE gt Mon Wed 1 0 EE Mon Wed NITES REQD 1 0 Model Assignments Preferences Constraints Mil Leo Im Dn 1 a The solution returns a Preference Total of 50 The scheduling requirements are tightly satisfied no overstaffing has occurred and each employee has been assigned to three or more preferred shifts SAMPLE MODELS 345 Staff Scheduling Two Stage Fixed Shift File name FIXED1 XLSX FIXED2 XLSX TYPE LINEAR OPTIMIZATION Application Profile Many organizations staff their personnel according to a number of non overlapping daily shifts Manufacturing plants hospitals and banks usually have 2 or 3 shifts per day with prespecified durations and starting times For example a hospital might have 3 daily shifts each 8 hours in length with 8 00 am 4 00 pm and midnight as the starting times In this model you need to fill these shifts at the lowest cost 1 e with minimal overstaffing Another consideration is that worker preferences for certain shifts should be satisfied as much as possibl
149. SP_DIST_distribution table cell list where distribution specifies the distribution e g NORMAL Step 3 Sample size for each stage is stored in WBSP_STSC table Step 4 Cells to be reported are listed in WBSP_REP cell_list or WBSP_HIST bins cell_ list You may type these functions in directly or you may use the Advanced Stochastic Support drop down menu in What sBest to guide you through the steps We illustrate the formulation of an SP model in What sBest with perhaps the simplest SP model possible the one product newsvendor problem model NEWSVENDOR XLSX This model has just two stages In stage 0 half stage we must decide how much to stock Q of a single product At the beginning of stage 1 demand D is revealed At the end of stage 1 we sell the minimum D Q enjoy our sales revenue and pay a penalty if any for unsatisfied demand or pay a holding cost for leftover inventory MATHEMATICAL MODELING 209 The NEWSVENDOR Worksheet before Optimization part 1 Stage 1 in tre be inning unknown demand is revealed to us 5 Stage 1 at we compute our sales and the resulting profit Core model lt lt Stage 0 decision lt lt Stage 1 random demand lt lt Stage 1 recourse decision lt lt Stage 1 constraint lt Stage 1 decision and constraint lt lt Stage 1 non negativity constraint lt lt Stage 0 cost computation Stage 1 holding cost computation lt lt
150. Scenario Tree When the probability distributions for all random variables are discrete and finite then there are only a finite number of outcomes in each stage With each random parameter fixed to one of its possible outcomes one can create a scenario representing one possible realization of the future Enumeration of all possible combinations of outcomes allows us to represent all scenarios in a tree with each scenario being a path from the root of the tree to one of its leaves The nodes visited by each path correspond to values assumed by random parameters in the model Defining an SP Model in What sBest Simple 2 Stage Example There is a relatively straightforward five step process for setting up an SP model in What seet All the information about the SP features is stored explicitly openly on the spreadsheet The five steps are Step 1 Define a core or deterministic model as a regular deterministic What sBest model You can plug regular numbers in a random cell to check results Step 2a Specify staging information about the sequence of decisions and random events This information is stored directly on the spreadsheet using special What sBest functions These functions and formats are Decision variable and their stages are identified by WBSP_VAR stage cell_list and Random variables and their stage are identified by WBSP_RAND stage cell_list Step 2b Distribution information about random cells is stored in WB
151. SlowProg is an integer greater 174 CHAPTER 4 than 2 indicating the iteration limit for slow progress Derivatives 0 Solver Decides Derivatives can take 5 values 0 for Solver Decides 1 for Backward Analytical 2 for Forward Analytical 3 for Central Differences 4 for Forward Differences SolverVersion 0 Solver Decides SolverVersion can take three values 0 for Solver Decides 1 for Ver 2 0 2 for Ver 3 0 Error Codes Description NonlinOpt_BadStratCrashInitialArg Bad StratCrashInitial argument NonlinOpt_BadStratPresolveArg Bad StratPresolve argument NonlinOpt_BadStratQuadraticRecognitionArg Bad StratQuadraticRecognition argument NonlinOpt_BadStratSelectiveConstraintArg Bad StratSelectiveConstraint argument NonlinOpt_BadStratSLPDirectionArg Bad StratSLPDirection argument NonlinOpt_BadStratSteepestEdgeArg Bad StratSteepestEdge argument NonlinOpt_BadStartingPointArg Bad StartingPoint argument NonlinOpt_BadOptimalityToleranceArg Bad OptimalityTolerance argument NonlinOpt_BadItLimitSlowProgArg Bad tLimitSlowProg argument NonlinOpt_BadDerivativesArg Bad Derivatives argument NonlinOpt_BadSolverVersionArg Bad SolverVersion argument Example If one wished to set all the nonlinear options the simplest syntax would be wbSetNonlinearOptions True False True 0 3 6 1 1 True T
152. TER 8 FUNCADDIN Failed to Access the Add in If the What sBest solver was unable to locate an add in that is needed to calculate the user s defined functions the following error message will be displayed on the pop up window Error Message VA RNING Failed to Access the Add in Help Reference FUNCADDIN The workbook is referring to an add in that is not at the specified location add in location This is needed to run the user s defined functions from Excel Reset any call to this add in then save the model and restart Excel Suggestions Clear any Excel call of this add in via the menu Tools add ins reset the add in by browsing to the right location save the model and restart Excel In some situations you may need to remove the add in from your system to clean up any remaining link in the Excel Add ins list Also you can carefully correct the link by accessing the registry editor from the Windows Start menu then run REGEDIT Usually the links are displayed in the registry folder HKEY CURRENT USER Software Microsoft Office 10 0 Excel Options In Excel 2007 adjust the security setting from the Excel Options then click on Advanced TrustCenter TrustCenterSettings Enable macros and grant the same access in the Macro Settings tab In case your add in has been digitally signed you will need to enable this add in to run your model In some situatio
153. Warehouse 1 Unit 1 in cell B23 B13 B5 1 B5 B18 When Flow B5 Limit B18 the formula to calculate time is undefined because it results in division by zero The user should build models that avoid such mathematically undefined regions Therefore rather than constraining the flow along each route to be less than or equal to its limit we ve constrained it to be less than or equal to the limit minus a small amount we call Epsilon Now let s solve the model The TRAFFIC Worksheet After Solving iON Lee ees Home insert Page Layout Formulas Data Review View Developer Add ins OX TRAFFIC FLOW Unit 1 Unit2 Unit3 Unit4 Warehouse 1 200 8 456 4 3112 131 6 Warehouse2 239 4 409 8 498 1 0 Warehouse3 459 8 333 8 238 9 267 5 Total 900 S 1200 S 600 gt 400 a Demand 900 1200 600 400 RATE Warehouse 1 39 14 11 14 4 Warehouse 2 27 9 12 9 Warehouse 3 24 14 7 13 17 LIMIT 18 Warehouse 1 500 1000 1000 Warehouse 3 lt lt TIME PENALTY Unit 1 Unit2 Unit3 Unit4 Warehouse 1 13090 11753 4971 2121 Warehouse2 12402 7562 637 9 Warehouse3 25946 10534 6750 6274 Epsilon 0 01 Total Cost SAMPLE MODELS _ 367 Truck Loading File name TRUCK XLSX TYPE LINEAR OPTIMIZATION Application Profile This is an example of the Knapsack class of problems in which a number of things boxes books or Bradley Fighting Vehicles must be efficiently or profitably packed in a container truck
154. _ HIST 8 E28 382 CHAPTER7 Scenario Tree The Scenario Tree illustrates the concept of modeling under uncertainty In Stage 0 an initial decision has to be made for selling or keeping the stock In Stage 1 the stock return is revealed depending on the market sell or keep action has to be decided From Stage 2 to 4 the same sequence is applied In Stage 5 the stock return is revealed a recourse decision is ma wealth and as a recourse decision the de as well as the final computation The objective is to maximize the total expected profit at the end of planning horizon Stage 5 la GF LP S PUTOPTIONxisx Microsoft Excel bola Home Insert Pagel Fom Data Reie View Dev fe Scenario Tree eh Add tr Risks e gt P 2 Stock Return 3b Stock Foturn 3a Stock Roturn b Shaaed taad Shaaez initial Eeainnina EBeainnina docizion 2random Zrandom Zrandom enzel outcomor sutcaomor sutcomor onmarketroturnr anmarketroturnr anmarketroturnr recourre recourre recourre docizion docirion docirian onsell ansell anzoll Ma Et H Model Scenario Tree Stock Roturnda UN Stock Roturn db Stock Return da 5 Stock Return db Staat ERT Zrandom random sutcomor sutcomer onmarketroturnr onmarketroturnr Staad taat End End rocourro rocourro docirion docizion ansell ansell and final computation Helm Ready 73 50 CL SAMPLE MODELS 3
155. _RAND Format Using the stochastic feature the user has to associate a stage information to a random cell The WBSP_RAND function is used to specify the timing information If the arguments to this function are not correctly specified the following error message will be displayed on the WB Status tab Error Message eK ERROR Stochastic WBSP_RAND Format Help Reference SPRAND A WBSP_RAND cell is incorrectly formatted Correct the formula and the validity of the arguments in the cell below The format should be WBSP_RAND stage cells where stage is numeric and cells must refer to random cells Also verify the number of stages in the mode cell address additional message TROUBLESHOOTING 429 Suggestions There is an incorrectly formatted random function cell in the model A random cell must use the format WBSP_RAND arg cells where arg is a numeric value or a cell reference to a numeric value and cells is a reference to the cells you want to apply this timing information The cell reference should be to a random cell Also make sure of the staging consistency in the model For more information on using the stochastic feature refer to section Advanced Stochastic Support SPREP Stochastic WBSP_REP Format Using the stochastic feature What sBest can return the solution information on any cells These values tell you how sensitive the solutions are to various stages of the model The WBSP
156. _REP function is used to request solution values on variables and outcomes on random cells If the arguments to this function are not correctly specified the following error message will be displayed on the WB Status tab Error Message KE RROR Stochastic WBSP_REP Format Help Reference SPREP A WBSP_REP cell is incorrectly formatted Correct the formula and the validity of the arguments in the cell below The format should be WBSP_REP cells where elle must refer to random or variable cells cell address Suggestions There is an incorrectly formatted Stochastic Report cell in the model A reporting cell must use the format WBSP_REP cells where cells is a reference to the cells you want to report the solutions The cell reference should be to a random or a variable cell For more information on using Stochastic Solution values refer to section Advanced Stochastic Support 430 CHAPTER 8 SPSTSC Stochastic WBSP_STSC Format Using the stochastic feature the user has to specify a table of scenario per stage The WBSP_STSC function is used to select this two column table where stage is in the first column and scenario on the second column If the argument to this function is not correctly specified the following error message will be displayed on the WB Status tab Error Message Krk d RROR Stochastic WBSP_STSC Format Help Reference SPSTSC A WBSP_STSC cell is incorrectly formatted Correc
157. a 87 wbDualValue 144 WBDUFORM 435 whbError and Error Codes 145 WBEXPOINV function 190 WBFREE 153 WBINT 154 whbinteger 154 whbintegerCard 156 whblintegerSemic 157 wbintegerSos 158 WBKBEST 159 WBLOFORM 436 WBLOWER Formula 87 WBMULITINYV function 191 WBNORMSL function 191 WBOMIT 160 wbResetOptionsToDefault 161 WBSEMICFORM 437 wbSetFunctionSupport 185 wbSetGeneralOptions 161 wbSetGlobalOptions 164 wbSetIntegerOptions 166 wbSetIntegerPreSolverOptions 169 wbSetLinearOptions 172 wbSetNonlinearOptions 173 wbSetStochasticOptions 175 wbSetStochasticSupport 177 wbSetStringSupport 185 wbSolve 177 WBSOSFORM 438 WbStochasticFunction 180 181 wbStochasticReport 182 wbStochasticStageScenario 183 WBTRIAINV function 189 WBUNIFINV function 190 wbUpdateLinks 184 WBUPFORM 438 WBUPPER Formula 87 WBxxx 39 WEIBULL 187 What If projections 11 17 Working while Solving 18 Worst Bound 65 76 Writing Solution 33 X XYZ production problem adjustable cell ranges 12 after optimization 17 best cell 12 what s best if 17 XYZ VBP 131 XYZ XLS 10 XYZVBA XLSM 133
158. a number indicating the Delta coefficient goLinearizationBigM 100000 goLinearizationBigM Coefficient is a number indicating the Big M coefficient goStatusReport 0 Always Created goStatusReport is an integer indicating when to show the status report 0 Always Created 1 Never Created 2 Only on Error Warning goStatusReportBegEnd 0 Beginning goStatusReportBegEnd is a flag indicating the location of the reports in the workbook 0 Beginning 1 End goSolutionReport 1 Never Created goSolutionReport is an integer indicating whether to show the solution report 0 Always Created 1 Never Created goWrnNonlinear 1 True Nonlinearity is a True False flag indicating whether to provide warning of nonlinear formulas or not goWrnBest 1 True goWrnBest is a True False flag indicating whether to provide warning on presence of Best cell goWrnBlank 0 False goWrnBlank is a True False flag indicating whether to provide warning of blank cells referenced in formulas or not goWrnFunction 1 True goWrnFunction is a True False flag indicating whether to provide warning of unsupported functions or not goWrnStringArg 1 True goWrnStringArg is a True False flag indicating whether to provide warning of string text argument in formulas MACROS THE VBA INTERFACE 163 goWrnIrreConst No 1 True goWrnIrreConst
159. able lattice cuts 0 Disable 1 Enable Lifting No 1 True Lifting is used to enable or disable lifting cuts 0 Disable 1 Enable PlantLocation No 1 True PlantLocation is used to enable or disable plant location cuts 0 Disable 1 Enable Objlntegrality No 0 False Objintegrality is used to enable or disable the objective integrality cuts 0 Disable 1 Enable Basic No 1 True Basic is used to enable or disable the basic cuts 0 Disable 1 Enable Cardinality No 0 False Cardinality is used to enable or disable the objective cardinality cuts 0 Disable 1 Enable Error Codes Description IntPreSolvOpt_BadHeuristicLevelArg Bad Heuristic Level argument IntPreSolvOpt_BadCutoffCriterionArg Bad Cutoff Criterion argument IntPreSolvOpt_BadProbingLevelArg Bad Probing Level argument MACROS THE VBA INTERFACE 171 IntPreSolvOpt _BadMaxCutsPassesArg Bad Max Cuts Passes argument IntPreSolvOpt_BadRelativeCutsLimitArg Bad Relative Cuts Limit argument IntPreSolvOpt_BadCoefficientReductionCuts Arg Bad Coefficient Reduction Cuts argument IntPreSolvOpt_BadDisaggregationCutsArg Bad Disaggregation Cuts argument IntPreSolvOpt_BadFlowCoverCutsArg Bad Flow Cover Cuts argument IntPreSolvOpt_BadGCDCuts Arg Bad GCD Cuts argument IntPreSolvOpt_BadGomoryCutsArg Bad Gomory Cuts argument
160. ables are fixed at their optimal values from the previous step b the random variables are freed up c the nonanticipativity constraints are dropped and d this wait and see model is solved EVEM is the objective value from this EVWS model Possible flag values are Disable False or Enable True The default setting is off the EVEM is not calculated Calculation for Expected Value of Perfect Information This Expected Value of Perfect Information EVPI is the absolute value of the difference between EV and EVWS This corresponds to the expected improvement to the objective were we to obtain perfect information about the random outcomes As such this is a expected measure of how much we should be willing to pay to obtain perfect information regarding the outcomes of the random variables Possible flag values are Disable False or Enable True The default setting is to enable calculation Calculation for Expected Value of Modeling Uncertainty ThisExpected Value of Modeling Uncertainty EVMU is the absolute value of the difference EV EVEM It is a measure of what we can expect to gain by taking into account uncertainty in our modeling analysis as opposed to mistakenly assuming that random variables always take on their mean outcomes Possible flag values are Disable False or Enable True The default setting is off the EVMU is not calculated ADDITIONAL COMMANDS 85 Note The above approach for computing EM and EVMU ma
161. ace on the spreadsheet We chose H15 H17 below a a T XYZ COMPUTER CORPORATION PRODUCTION PLAN Standard Deluxe PROFIT 60 30 Profit per Unit 300 500 Product Component Requirements Quantity Required Total Standard Deluxe Usage 0 60 lt 1 30 lt 2 120 lt 90 CHAPTER3 The dual cells are inserted by highlighting the cells H15 H17 clicking on WB Advanced Dual selecting F15 F17 for the For Cell Range box and clicking OK When we re solve the dual values for the stock constraints will appear as in the following Ee lech WBDUAL F15 50 B XYZ COMPUTER CORPORATION PRODUCTION en Product Standard Deluxe PROFIT Quantity to Produce 60 30 Profit per Unit 300 500 Product Component Requi Components Quantity Required Total Standard Deluxe Usage Standard Tower 1 0 60 lt Deluxe Tower 0 1 30 Hard Drive 1 2 120 lt The dual value of the constraint function WB E15 lt G15 is shown in cell H15 as 50 Note You must re solve your model to allow What sBest to update any new dual cells This means the value of the best cell would improve by 50 for each unit increase in the right hand side of the constraint equation In other words changing the cell G15 to 61 and re solving the model would return a new solution with a maximum profit of 33 050 assuming a valid range on the dual value of at least one This is also the maxi
162. adimac 40 92 76 23 lt 740 55 gt Produced Requirement Miles Gallon 26 48315 WE u B 302 CHAPTER7 Media Buying File name MEDIA XLSX TYPE LINEAR OPTIMIZATION Application Profile Media selection or buying is the problem of finding the best way to deliver the desired number of advertising exposures to the target audience by determining the most efficient combination of advertising media purchases Even for small media budgets there can be literally thousands of possible media schedules from which to choose The task is to select from this set an effective schedule to become the media plan What sBest provides a natural format for analyzing a media selection problem You can use it to find the media mix that will maximize the number of effective exposures subject to a set of constraints the ad budget minimum and maximum media availabilities minimum desired exposure rates etc A great advantage of What sBest is that new plans can be generated quickly to show the significance of changes made in problem specifications The Problem in Words You re a media buyer for an advertising agency You need to minimize the cost while satisfying minimum exposure needs Background There are six target markets Groups through 6 and each requires a specified number of media exposures There are five different media sources the Times Mirror Tribune Herald and Post over which the advertising dollars are to be a
163. age 1 376 CHAPTER7 CROPALLOC xIsx ne samoree Scenario 1 Scenario 2 Scenario 3 Scenario 4 Stage 0 Stage J Stage J initial Beginning End decision 4random recourse on area outcomes decision on yields on crops and final computation joa kel de SAMPLE MODELS 377 Optimization The CROPALLOC Worksheet Before Optimization Selling pricefor Cost Quantity Plantatio excess price for required ncost produce shortfall Yield 200 150 170 238 240 230 150 210 100 260 200 270 Total area 500 Declare the area Adjustables Wheat alloca y Quantit y Crops ted produc sold purcha Corn Wheat Beans Corn Beans Declare the yield Maximize profit Amount obtained from selling Plantation cost Purchase cost Plantation cost Purchase cost Amount from selling 378 CHAPTER7 The CROPALLOC Worksheet After Optimization CROPALLOC xlsx GE Selling pricefor Cost Quantity Plantatio excess price for Crops required ncost produce shortfall Yield Wheat 200 150 170 238 Corn 240 230 150 210 Beans 100 260 200 270 Step 2a Total area 500 Declare the area Adjustables Wheat alloca gy Quantit y Crops ted produc ysold purcha Corn Wheat D D Beans Corn 0 0 Beans 500 6000 5900 Declare the yield Maximize profit Amount obtained from selling Plantation cost Purchase cost Plantation cost 130000 Purchase cost Amount from selling The solver writes a series of solution in the
164. age will be displayed on the WB Status tab Error Message Global Adjustable Cell Limit Help Reference LICCAP5 The number of global adjustable cells in this model number of globals exceeds the limit ot global adjustable cell limit The model has more nonlinear adjustable cells than is allowed by your installation of the global solver You will need to either reduce the size of your model or upgrade to a larger version Suggestions TROUBLESHOOTING 413 The model exceeds the nonlinear adjustable cell limit for your installation of the global solver This limit is easy to hit on some of the smaller capacity versions of What sBest given that they allow only a handful of nonlinear adjustable cells when running the global solver The capacity limits of your version may be found by running the WB About What sBest command Refer to the previous error message LICCAP4 for information on the definition of a nonlinear adjustable cell You will need to reduce the number of nonlinear adjustable cells in the model or turn off the global solver and use the standard nonlinear solver If this is not possible you will need to contact LINDO Systems regarding a license upgrade LICKEY1 Failed to Process License Key When your What sBest license key can t be found read or processed the following error message will be displayed on the WB Status tab Error Message Failed to Process License Key Help Reference LICKEY1 Unab
165. aints are that the minimum Requirements be met The Exposures formulas in B17 G17 are the sumproduct of the number of exposures per dollar for a media source and the amount in dollars being allocated to that source The formulas in the row Met B16 G16 force these amounts to be greater than or equal to the required exposures for each targeted group B15 G15 What If vs What sBest Work on a conventional What If spreadsheet solution to this problem by adjusting the Dollars Allocated shown in cells H9 H13 You ve reached a viable solution when cells B16 G16 return gt or gt Judge any solution that meets all the exposure requirements by minimizing Total Cost H18 When you arrive at a reasonable answer jot down your total cost figure Now let s solve the model with What sBest The MEDIA Worksheet After Solving Target Markets Dollars Group Group Group3 Group4 Group5 Group6 Allocated Media Times 0 10 1 93 Mirror 0 10 1 41 Tribune 20 0 1 25 Herald 10 0 0 00 Post 0 5 1 63 Exposures 25 41 538 40 14 382 Met gt P gt TOTAL COST Requirement 25 0 40 0 60 0 1200 400 11 0 SAMPLE MODELS 305 As you can see the What sBest solution has a total cost of 6 220 It results in surplus exposures in only two target markets Group 2 and Group 6 It proposes no media purchases from the Herald which offering only 15 total exposures per dollar is not a bargai
166. al Options dialog or by unchecking the Scale option in the Linear Option dialog box Suggestions What sBest calculates the amount by which all constraints are violated after the solver returns a solution In general this value will be quite small when a feasible or optimal solution is returned In some cases the total violations or infeasibilities may be found to be excessive When the total infeasibilities exceeds 0 01 What sBest will display this error Under these conditions you should check the solution carefully You may be able to resolve this error by decreasing the Feasibility Tolerance in the General Options dialog or by unchecking the Scale option in the Linear Option dialog box INTERRUPT Solver Interrupt If the Hold Interrupt button is pressed before completion of the solution process the user can hold the process and continue it some point later or stop the process via the following dialog box What sBest Solver Interrupt kd BEZ Click on Interrupt to halt the solver iw of Continue to resume solving Interrupt Continue By clicking OK the following warning message is displayed on the WB Status tab in the returned workbook Warning Message KAKA KEKE KEKE KEES 8 INTERRUPTED KEKE KEKE EE EEN WARNING Solver Interrupt Help Reference INTERRUPT The solver was interrupted before finding the final solution Check the solution carefully it may be sub optimal or infeasible TROUB
167. al cost and meeting your customers demand for the product Background You know the unit demand for each of the thirteen time periods For instance the demand for Time Period I must be met by using some combination of sources 1 2 and 3 The ending inventory is equal to starting inventory plus the sum of amounts purchased from each source less the unit demand You must not only satisfy the demand for each period but also minimize the amount of inventory held over Obviously it s in your interest to purchase as much as possible from your least expensive source Source 1 However demand in some periods exceeds Source s capacity Objective of Optimization The objective in the Inventory model is to minimize the total cost and determine how much material to buy from each source in each time period to meet customer demand at lowest purchase and holding costs SAMPLE MODELS 307 The Worksheet Let s look at the INVENT worksheet and note that Source is obviously the best bargain D21 However as demand rises from period to period B6 B18 Source 1 with a capacity of 180 won t suffice to meet demand The INVENT Worksheet Before Solving TO e a Time Purchase From eae Ending Period Demand Met 1 3 Inventory 0 100 280 500 650 750 950 1200 1500 1760 2010 2250 2460 2600 TOTAL COST 100 Not lt 180 Not lt 220 Not lt 150 Not lt 100 Not lt 200 Not lt
168. al shipping costs to the city in question from all plant locations For instance the formula in cell B16 is SUMPRODUCT B9 B13 Amount Shipped B9 B13 C Specify Constraints The two blocks of constraints in the Plant Locating problem ensure that demand is satisfied in the six cities and that production capacity at each of the five potential plants is not exceeded First since demand in B16 G16 of the Amount Shipped worksheet must be met the constraints in B15 G15 of the same worksheet require the sum of amount shipped to each demand city Atlanta Boston etc to be greater than or equal to the demand for that city The second constraint is that the amount shipped from each Supply City Baltimore Cheyenne etc must not exceed its supply capacity To enforce this constraint the constraints in H9 H13 of the Amount Shipped worksheet require that the sum of materials shipped from each Supply City is less than or equal to the Monthly Supply Capacity for that city 19 113 of the Amount Shipped worksheet Let s look at the formula in H9 of the Amount Shipped worksheet WB SUM B9 G9 lt 19 Shipping Costs 19 The total amount shipped from Baltimore SUM B9 G9 can be no more than I9 Shipping Costs 19 Baltimore s capacity Since Shipping Costs I9 is either 1 or 0 depending on whether or not a plant is open in Baltimore its use in the formula returns a nonzero value only if the plant is opened SAMPLE MODELS 331 Now let
169. ally WBSP_DIST_CHISQUARE argument 1 Degrees freedom WBSP_ DIST EXPONENTIAL argument 1 Rate WBSP_DIST_ GEOMETRIC argument 1 Success probabilities WBSP_ DIST LOGARITHMIC argument 1 p Factor WBSP_DIST_ POISSON argument 1 Mean WBSP_DIST STUDENTS_T argument 1 Degrees freedom 108 CHAPTER 3 Distributions with two arguments WBSP DIST DISCRETE SH W argument 1 Discrete values argument 2 Weight probabilities with scenarios read horizontally WBSP DIST DISCRETE SV W argument 1 Discrete values argument 2 Weight probabilities with scenarios read vertically WBSP_DIST_BETA argument 1 Shape 1 argument 2 Shape 2 WBSP_DIST_BINOMIAL argument 1 Sample size argument 2 Success probabilities WBSP_DIST_CAUCHY argument 1 Location argument 2 Scale WBSP_DIST_GAMMA argument 1 Shape argument 2 Scale WBSP_DIST_GUMBEL argument 1 Location argument 2 Scale WBSP_DIST_F_DISTRIBUTION argument 1 Degrees freedom 1 argument 2 Degrees freedom 2 WBSP_DIST_LAPLACE argument 1 Location argument 2 Scale WBSP_DIST_LOGISTIC argument 1 Location argument 2 Scale WBSP_DIST_LOGNORMAL argument 1 Mean argument 2 Sigma WBSP_DIST_NEGATIVEBINOMIAL argument 1 r Factor argument 2 Success probabilities WBSP_DIST NORMAL argument 1 Mean argument 2 Sigma
170. an be found in the Library subdirectory or the WB subdirectory on your C drive or elsewhere if you so requested Details are described in Location of the Add in Files and Update Links Please note that it is not possible to disable the What sBest menu if the add in is installed to the XLSTART subdirectory The What sBest functions are only available when the What sBest add in is loaded If you disable remove the What sBest add in from Excel then you must manually re enable it the next time you wish to build or solve a model Note 1 Starting with What sBest release 6 0 1 the add in name is now WBA XLA instead of the former WB XLA name Starting What sBest release 10 0 0 0 the add in name can be either WBA XLA for Excel 2002 or WBA XLAM for Excel 2007 Note 2 Add ins library and executable files from LINDO Systems are digitally signed 444 APPENDIX A Location of the Add In Files and Update Links If you have already installed What sBest then you can check the location of the add in files WBA XLA or WBA XLAM at the bottom of the About What sBest dialog box The installation program will detect if a previous release of What sBest has been installed Thus it will ask if you wish to remove it first or want to write over it Location of the Add in Files and Update Links Your choice of the Default installation option during the setup program will automatically transfer the program files to the Library subd
171. ange for the original dual value of 50 and a dual value of 0 is now in effect Since the dual value is 0 changing cell G15 to 121 and re solving the model does not increase the profit beyond 36 000 as shown in the following XYZ COMPUTER CORPORATION PRODUCTION PLAN Product Standard Deluxe Quantity to Produce 120 Profit per Unit Components Quantity Required Number Dual Standard Deluxe Usage In Stock Value T 1 120 lt 1210 Standard Tower 205 lt 2 D Deluxe Tower 0 D lt 50 0 Hard Drive 1 120 lt 120 300 TUTORIAL SAMPLE Adjustable Cell Ranges In the XYZ Production example the adjustable cells refer to the production of Standard and Deluxe computer tower models The dual value of each of these adjustable cells is the amount by which the profit of producing the models would have to increase in order to make it profitable to put them into the solution The upper and lower range of the dual of an adjustable cell is the range over which the adjustable cell could change with its dual value remaining the same 98 CHAPTER 3 We will use the example referred to earlier for the adjustable cell dual value where the number of hard drives in stock is reduced to 50 An upper and lower range value can be put in the model by selecting the cell to put the dual information in selecting Advanced Dual selecting the adjustable cells D5 to enter in the For cell range choosing Upper Range or Lowe
172. are kept to a minimum ADDITIONAL COMMANDS 127 Language This feature allows the user to select the language of his choice without reinstalling the add in Da ca Home Insert Page Layout Formulas T eeaempwenlsil 16 X CheckUpdate A language 1 V English ES Ke A lt zs Adjustable Best Constraints Solve Integers gt Options gt Advanced gt Locate Help About What sBest ToolBar Upgrade Register bh 9 Francais AAR w After selecting the language the user can revert to the previous setting because the language words will stay displayed in the native language Make sure that your system has installed the international language capabilities This can be verified via the Control Panel of the Windows operating system 128 CHAPTER3 Note English US Chinese Pinyin French France Japanese Katakana 4 What sBest Macros The VBA Interface This section discusses the VBA Visual Basic for Applications interface of What sBest The VBA interface allows the user to write VBA code or macros that execute What sBest commands By using this interface you can build optimization applications that utilize all the power and functionality of What sBest in Excel or any other application offering VBA control The section entitled Usage Guidelines for Macros in VBA introduces some features of the VBA interface
173. ariations of compare such as lt gt symbol for not equal gt lt gt lt SUMIF and VLOOKUP functions Any other operations will return an error message Also a formula cell should return a numeric number rather than a string value ADDITIONAL COMMANDS 105 Advanced Stochastic Support The Advanced Stochastic Support command allows you to support Stochastic Modeling via the series of dialog boxes Note The user can directly enter or modify the stochastic functions in the spreadsheet without reopening the dialog box Stochastic Modeling Support By checking this option the model will be processed as a stochastic model Otherwise What sBest will treat the core model even if some stochastic functions are set in the spreadsheet The default setting is False unchecked Page Step 1 core model Stochastic Support ks Step 1 Step 2 Step 3 Step 4 Design the model with its Adjustable Best Constraint and Random cells 106 CHAPTER3 This step simply reminds you to first formulate a standard deterministic core model Adjustable Best Constraint A cell that represents a Random quantity is simply filled with an arbitrary number Cells containing equations or strings are inappropriate as random cells What sBest cannot rewrite an equation although the value returned by an equation will change if the equation depends upon any adjustable cells Ther
174. art Use the Warm Start option to control the linear solver that is used by the branch and bound solver at each node of the solution tree when a starting basis is present to use as an initial starting point The Cold Start option discussed below determines the solver to use when a previous solution does not exist The available options are Solver Decides What sBest chooses the most appropriate solver Barrier What sBest uses the barrier method assuming you have purchased a license for the barrier solver Otherwise the dual solver will be used e Primal The primal solver will be used exclusively e Dual The dual solver will be used exclusively In general Solver Decides will yield the best results The barrier solver can t make use of a pre existing solution so Barrier usually won t give the best results In general Dual will be faster than Primal for reoptimization during branch and bound Cold Start Use the Cold Start option to control the linear solver that is used by the branch and bound solver at each node of the solution tree when a previous solution is not present The Warm Start option discussed above determines the solver to use when a previous solution does exist The available options are Solver Decides What sBest chooses the most appropriate solver at each node e Barrier What sBest uses the barrier method assuming you have purchased a license for the barrier solver Otherwise
175. as are WBUPPER cell number and WBLOWER cell number The study of these values and their effect on the solution of the model is called sensitivity analysis Please refer to the section Usage Guidelines for Dual Values for a tutorial and information on how to use this option effectively 88 CHAPTER3 For Cell Range This is the target range of cells you wish dual reporting on What sBest will automatically fill this box with the cells directly to the left of the current selection of cells or above them if the entire row is selected If this is not the range of cells you want you can type the correct range in or select the button on the right edge of the text box to bring up a cursor for cell selection Report on Type Dual Value Select Dual Value to prompt What sBest to return the dual value of the adjustable or constraint cells specified in For Cell Range To enter a dual value function put the cursor in the cell for which you want to find the dual value and select Dual Value Then specify the cell in which you want the dual information displayed in the Report Information in box Note When taking the dual value of a constraint be sure to specify the cell containing the constraint function not the right hand side of the constraint Upper Range and Lower Range Select Upper Range or Lower Range to cause What sBest to return the upper and lower valid ranges for a dual value That is the upper and lower bounds of the range
176. ating costs are minimized 328 CHAPTER7 The Worksheet Let s open the PLANTLOC sample and examine its layout and formulas The Shipping Costs Worksheet Before Optimization cE PLANT LOCATION MODEL Shipping Costs Per Ton Month SAMPLE MODELS 329 The Amount Shipped Worksheet Before Optimization Amount Shipped Monthly Proposed Demand City Supply Sup Supply Capacity City Atlanta Boston Chicago Denver Omaha Portland imitation in Tons eet Demand Demand A Determine Adjustable Cells The adjustable cells in PLANTLOC are the quantities shipped from each potential plant to each demand city B9 G13 on the Amount Shipped worksheet and the number of plants open at each location 19 113 of the Shipping Costs worksheet In addition since either a whole plant or none at all must be erected at any location 19 113 of the Shipping Costs worksheet are specified as binary integers What sBest will return only a zero or one as values in these cells 330 CHAPTER7 B Define Best The best solution consists of minimizing Total Operating Cost in cell J18 of the Shipping Costs worksheet This cell contains the sum of the two subtotals for Operating Costs and Transportation costs in cells J15 and J16 of the Shipping Costs worksheet respectively The Transportation cost subtotal in turn is the sum of the values in cells B16 G16 of the Shipping Costs worksheet The formulas in these cells represent the tot
177. ative values would be appropriate For example if you have the ability to either buy or sell a stock you could create a single adjustable cell representing your decision to buy sell shares A positive value would represent the amount to buy and a negative value the amount to sell For more information see the Adjustable command above ABCs 23 Best The dialog box posted by the Best command appears as follows This command is used to specify the cell in your model to maximize or minimize Select the objective sense you wish to apply from the drop down box on the left of the dialog box Possible options are 1 Minimize 2 Maximize 3 None Then enter the cell you wish to set in the text box on the right of the dialog box What sBest defaults to entering the current selection Clicking OK will define the best cell Objective Minimize or Maximize JK L Select Minimize or Maximize to specify whether your objective is to determine the smallest or largest value for the best cell respectively These choices may also be made using the Minimize Je and Maximize 7 toolbar buttons For optimization models you must define a best cell equation solving and goal seeking models do not require a best cell as discussed below The best cell sometimes called the objective of optimization is the cell whose value is to be minimized or maximized during optimization The best cell must be an adjustable cell or a for
178. ave been supplied by your econometrics department For example the demand curve formula for the Mercurial E16 is 495 13 B16 3 B15 B17 Note that the demand curves for the three makes of auto are interdependent the sales of one influence demand for the other two Finally cell C20 contains the constraint WB B20 gt D20 which forces the average Miles Gallon across the produced quantity of all three makes B20 to be at least 24 The formula in B20 is SUMPRODUCT G4 G6 C15 C17 SUM C15 C17 Note The federal guideline is expressed in miles per gallon The constraint could more properly be expressed as gallons per mile as if each car sold were driven one mile a measure taken of total gallons used and the total cars sold divided by total gallons used This is because using miles per gallon assumes each car running until the fuel tank is empty Obviously the higher MPG car would go further on a tank of fuel skewing the overall production in its favor SAMPLE MODELS 301 Now you are ready to solve the model After solving the WB Status worksheet will open in order to show you the Nonlinearity present warning This warning can be shut off from the General Options dialog box Your solved model appears as follows The PRICING Worksheet After Solving i Miles per Gallon 29 00 23 00 19 00 0 00 500 00 100 00 200 00 lt lt Price Produced Constraint 12 39 480 73 S Mercurial 29 96 183 59 lt C
179. ays of doing this in What sBest You can specify an explicit arbitrary discrete joint distribution table form any of WBSP_DIST_ DISCRETE SV table rand_vars Equi probable sequenced vertically WBSP_DIST_ DISCRETE SH table rand_vars Equi probable sequenced horizontally WBSP_DIST_DISCRETE_SV_W table prob rand_vars Weighted prob sequenced vertically WBSP_DIST_DISCRETE_SH_W table prob rand_vars Weighted prob sequenced horizontally Another common way to characterize dependencies among variables drawn from standard univariate distributions such as the Normal or Poisson is by standard correlation measures What sBest supports three correlation types WBSP_CORR_PEARSON matrix rand_vars Pearson s linear correlation WBSP_CORR_SPEARMAN matrix rand_vars Spearman s rank correlation WBSP_CORR_KENDALL matrix rand_vars Kendall s rank correlation Sampling a Scenario Tree For stochastic programs with infinite event space What sBest offers an easy to use function to create finite scenario trees implicitly with user specified dimensions This is especially handy when there are several stochastic parameters and the task of explicit sampling becomes tedious In this context the user can specify the dimensions of a scenario tree by either of the following methods Specify the number of nodes per stage In this method the user should provide an integer array of length 7 number of stages in the model and give in each position
180. bar and the toolbar are integrated in the Ribbon layout xH a e gU Bookl Microsoft Excel Home Insert Page La Formule Data Review View Develop Add Ins A e o pp es WB v KS KS bf 7 lt 2 2s Menu Commands Custom Toolbars In Excel version 2002 1 The What sBest menu WB is embedded in the main Excel menu bar and appears as follows File Edit view Insert Format Tools Data Window Help The WB menu contains all the commands discussed in ARC e Basic Functions and Additional Commands 2 The What sBest toolbar can float on your screen or be repositioned to a preferred part of the Excel window and appears as follows The buttons on the toolbar correspond to the menu commands Adjustable Best Constraints and Solve which are discussed in ABC e Basic Functions 8 CHAPTER 1 Of these two access methods the What sBest menu provides for full interactive use of the entire range of What sBest commands For faster access the What sBest toolbar offers rapid one click access to the most commonly used operations The What sBest toolbar also makes tool tips available To learn the function of a particular toolbar button just move the cursor over the button for a second and a statement of the button s function will appear as follows x el gt UIS Bookl Microsoft Excel Home Insert Page La Formule Data Review View Develop wey RS BK e A lt s Menu Commands
181. be valid over only an infinitesimal range See Additional Commands for a discussion of valid ranges for dual values 338 CHAPTER7 Staff Scheduling Preferred Assignment File name ASSIGN XLSX TYPE LINEAR OPTIMIZATION Application Profile Staff scheduling is usually a very costly and laborious job In institutions where there are several daily shifts and the staff can rotate between shifts during the same period the task is even more complicated The scheduler must match the organization s requirements with the preferences of individual employees Larger numbers of employees and longer time periods make the task more difficult This problem involves scheduling several staff members in an organization that operates in multiple time shifts each day seven days a week The Problem in Words Your business works in three shifts day evening and night For each shift you must meet certain minimum staffing levels Each staff member must be assigned a specific number of shifts and has individual preferences for working the various shifts Moreover because of the nature of the work each employee must have an adequate rest period between shifts in order to perform the job effectively This model demonstrates how you can meet your staffing requirements while satisfying your employees scheduling preferences to the greatest possible degree Background Your specific staffing needs for a particular week are as follows Staff Me
182. blem without integer restrictions It then tightens this bound as it searches more branches of the solution tree The objective bound is useful information in that it is a limit on how good a value the objective can attain On long running models you might want to interrupt the solver once the best solution is sufficiently close to the theoretical limit Steps This shows the number of solver iterations Active This shows the number of pending subproblems to be solved Numerics This shows the total number of numeric cells in the model Adjustables This shows the total number of adjustable cells in the model ABCs 33 Integers Bin This shows the number of adjustables cells with integer binary restrictions Formulas This shows the number of formula cells Constraints This shows the total number of constraint cells in the model Nonlinears This shows the number of adjustable cells that appear nonlinearly in the model We say that a variable appears nonlinearly in a formula when that formula is nonlinear with respect to changes in the variable For instance consider the nonlinear expression X Y 2 This expression is nonlinear with respect to Y but it is linear with respect to X Thus Y would be included in the nonlinear count while X would not Coefficients This shows the total number of coefficients in the best cell and all formulas dependent on adjustable cells Mathematicians refer to this as the number of nonzero elements
183. can finish formulating the model in this tutorial The other sample models have been completely formulated for you and are ready to solve Please note that the following discussion uses Excel naming conventions such as C 5 D4 throughout The XYZ Production Problem XYZ Corporation builds two computer models the Standard and the Deluxe The Standard has a profit per unit of 300 and the Deluxe has a profit per unit of 500 The two models are produced from three components the Standard computer tower the Deluxe computer tower and a hard drive Standard Model Deluxe Model Profit per Unit 300 500 Problem What combination of Standard and Deluxe computer towers will maximize XYZ s profit from the components currently in stock Unless specified otherwise during installation the XYZ XLS file was copied to the WB subdirectory If you haven t already done so open the XYZ XLS sample file GETTING STARTED 11 The XYZ Worksheet Before Optimization IXYZ COMPUTER CORPORATION PRODUCTION PLAN Product Standard Deluxe PROFIT Quantity to Produce 0 0 0 Profit per Unit 300 500 Product Component Requi Components Quantity Required Total Number Standard Deluxe Usage In Stock Standard Tower 0 Deluxe Tower Hard Drive Examine the layout and logic of the model You might want to experiment with various What If projections For example try to adjust the Quantity to Produce in both c
184. ce cccccceceeeeeeceeeeeeeeeeeeeeeseaeeeeeeseeeees 89 Dual Value of an Adjustable Cell Reduced Coste 91 Dual Value of Zero and Multiple Optima 94 Dual Values in Nonlinear Problems ssseesseesseessnesresssnsssrrsennstnnstnnssrnnstnsrnssrnssrnnsrnnnnnt 94 Dual Values in Integer Problems ccccceeeeeeeeeeeeeeeeeaeeeeeee certs caaeeeseaeeseeeesaeeseeeseeneee 94 Valid Ranges for Dual Values eccceccceceeeeeeeeeeeeeeeeeeeaeeeeaeeseceeeeseaeeeeaaeseeeeeseeeesaeeeeeneeeed 94 Constraint E GER 95 Adjustable Cell Ranges AA 97 vi PREFACE Advanceds 2 Om its ege ee Ee 101 OMIT Names in Workbook and Refers fo 101 Usage Guidelines for Om 102 What Ke Eu 102 What otemt enee eegen eeh Eegen Ee 102 Advanced FUNCtION Support ceeeeeeeceeeeeeeeeeaeeeeeee tenes caaeeeeaaeseaeeesaaeeseaaeseeeeesnaeeesaeeeenes 103 FUNCHION SUPPOMts enee cheer ed Ae eee a Aiea 103 Advanced String Gupport 104 String SUPPOMt i eet aie ede tel eed ee eT ee ae 104 Maximum String Length 104 Advanced Stochastic Gupport 105 Advanced Convert Model Format 116 Advanced Parameters oies ierse ia via aea a EANA VAA Eaa EANAN 117 elei EE E As ee ee eed E ee 118 Help amp About What sBest 0 ee eee cece ceeeeeeeeeeecaeecaeecaaesaaeseaeseaeesaeseaeseaeeseeeseeeseneeeneseaes 119 ToolBa EE Alia Ree A ad We 121 Upgrading via a New License key 122 E UE 124 GheckUpdatevs aitiactwetieeei aaa da eee adit ee en Ae
185. cel Home Insert Page Layout Formulas Data Review View WBMULTINV 0 6 A3 A5 B3 B5 WBMULTINV 2 3 0 2 2 4 0 3 1 5 0 5 3 f then WBMULTINV 0 6 A3 A5 B3 B5 returns 3 This means that the minimum outcome X such that 60 percent of the outcomes are less than or equal to_X is X 3 Standard Normal Linear Loss Function NORMSL This function returns the expected value of max 0 Z X where Z is a standard normal random variable In inventory modeling WBNORMSL X is the expected amount that demand exceeds a level X if demand has a normal distribution The syntax is WBNORMSL Value Value Value must be a real number 192 CHAPTERS Inner Sum Product WBINNERPRODUCT This function is the equivalent of SUMPRODUCT but instead of multiplying terms row by row the terms are multiplied row by column and then summed The syntax is WBINNERPRODUCT Rangel Range2 Rangel or The cell range containing the column or row to multiply and sum Rangel and Range2 Range2 can only be one dimensional ranges and the ranges must be the same size and shape Note Nesting of the VBINNERPRODUCT function is not recommended In other words the function WB A1 lt WBINNERPRODUCT B2 B4 A3 A5 should be avoided 6 Overview of Mathematical Modeling Introduction The relationships in your model influence the computation time the solution methods used by What sBest and the type of solution returned
186. cells cell address Suggestions There is an incorrectly formatted distribution function cell in the model A distribution function cell must use the format WBSP_DIST arg1 arg2 arg3 cells where arg1 arg2 and arg3 are cell references and cells is a reference to the cells you want to apply this three argument distribution The cell reference should be to a random cell For more information on using the stochastic feature refer to section Advanced Stochastic Support SPERR Stochastic Error Using the stochastic feature the user has to associate the stochastic information to a core model The set of WBSP_ function is used to set the information If the arguments of these functions are not correctly specified or if any piece of information can not be loaded the following error message will be displayed on the WB Status tab Error Message KE RROR Stochastic Error Help Reference SPERR Your model must have at least one formula that is dependent upon an adjustable cell and a random cell Refer to the Getting Started section of the online help to learn how to format a stochastic model for What sBest cell address additional message Suggestions There is an incorrectly formatted stochastic function cell in the model or the core model is missing the Adjustable Best Constraint cells A stochastic function cell must use the format WBSP _ cells where cells is a reference to the cells having the a
187. ces you ve set This data could be used to calculate the probable success of a marketing plan or in forecasting demand for next year s production 298 CHAPTER7 Car Pricing File name PRICING XLSX TYPE NONLINEAR OPTIMIZATION The Problem in Words You manufacture three automobile models each with its own price structure and miles per gallon rating at your five plants An increase in sales of any one model results in a small decrease in sales of the two others The models compete for production capacity in your five plants Also the federal government has imposed an lower limit on the average fleet mileage of your entire production which makes sales of low mileage cars more desirable You need to set the prices of the three models of automobiles so profit is maximized at the five plants while staying within the federal fleet gas mileage requirement of 24 miles per gallon Background Your known values are the cost of manufacturing each model at each plant the miles per gallon for each model and the annual production capacity at each plant You also know price quantity relationships among the models called demand curves by economists These demands are dependent upon the price of each car related to the prices of the other cars Objective of Optimization The objective is to determine the best price for each car to maximize profit while conforming to all regulations and constraints SAMPLE MODELS 299
188. ch and bound tree where n is the option s setting During these initial levels What sBest picks a subset of the fractional variables as branching candidates What sBest then performs a tentative branch on each of the variables in the subset selecting as the final candidate the variable that offers the greatest improvement in the bound on the objective Although strong branching is useful in tightening the bound quickly it does take additional computation time So you may want to try different settings to determine what works best for your model The default Strong Branch setting is 10 levels ADDITIONAL COMMANDS 77 Usage Guidelines for K Best Solutions As an example we are packing a basket for a picnic we will be taking with a friend You ve constructed a list of items you would like to carry with you on the picnic Each item has a weight associated with it and your knapsack is limited to carrying no more than 15 pounds You have also come up with a to 10 rating for each item which indicates how strongly your friend wants to include the particular item in the knapsack for the picnic The information are listed in the model below KNAPSACK_ KBEST xlsx The KNAPSACK_KBEST Worksheet Before Solving fi E AutoSum gt E Logical E Recently Used A Text Insert Function Financial ES Date amp Time f Weight Rating Brats Brownies Beer Ant Repel Blanket Frisbee Salad Watermelon WBIKB_REP zl ue
189. chastic parameters with two outcomes each This will correspond to a scenario tree with 2 1 0737e 009 scenarios Sampling is essential for models with scenario trees this big 224 CHAPTER 6 Solving Second Order Cone Programs The optimization capabilities of What sBest extend to the solution of second order cone problems SOCP of the following form Optimize AO x b0 c0 x subject to Aix bi cix di 0 for i 0 1 m 1 Lj lt xj lt Uj forj 0 1 n 1 xj is integer for j in a specified J C 0 n 1 where Optimize is either minimize or maximize Ai are matrices of appropriate dimensions i 0 m 1 bi and ci are vectors of constants di are constants x x0 x2 xn 1 is an n vector of decision variables 2 is one of the relational operators lt or gt Without the integrality restrictions SOCPs are nonlinear convex problems that include linear and convex quadratically constrained quadratic programs as special cases Several decision problems in engineering design and control can be formulated as SOCP What sBest solves this class of problems using the so called conic optimizer which uses an interior point algorithm To solve a convex problem using What sBest it may be advantageous to cast the problem e g a QCQP as a SOCP and use the conic optimizer To motivate the second order cone problems and common forms of quadratic cones consider the following two
190. command Advanced Omit The dialog box posted by the Advanced Omit command appears as follows i Omit Omit Names in Workbook m The Omit command allows you to specify a range of cells on the workbook to be ignored by the What sBest solver while it is seeking a solution These omitted areas do not contribute toward any limits What sBest imposes in restricted size versions If used properly omit ranges can decrease the solution time However care should be exercised in applying omit ranges Please refer to the section entitled Usage Guidelines for Omit OMIT Names in Workbook and Refers to What sBest uses range names to specify the cell range to be omitted The target cell range is indicated in the Refers to box What sBest automatically fills the Refers to box with the range of the currently selected cells If this is not the range of cells you want omitted you can type the correct range in or select the button on the right edge of the text box to bring up a cursor for cell selection Next enter any word of your choice in the OMIT Names in Workbook box and click on the Add button The name you entered should then appear in the box below with a WBOMIT prefix If you decide later you would like to remove the omit restriction simply go to the Omit dialog box select the name of the omit range to remove from the list of names and click on the Delete button 102 CHAPTER3 Usage Guidelines for Omit What To
191. constraint formulas require that the sum of amounts pumped from each Well is less than or equal to the Capacities B11 and C11 Second the amount received at each Pump must equal the amount pumped out Such constraints are known as conservation constraints To achieve this cells D7 D9 contain constraint formulas forcing the sum of amounts pumped from the Wells to each Pump to equal the sum of amounts pumped from that Pump to the Refineries Third demand at the Refineries must be satisfied These constraints are in cells E12 H12 and require that the sum of amounts pumped from the Pumps to each Refinery is greater than or equal to the demands in E11 H11 Finally the capacities of the pipelines cannot be exceeded The constraint formulas in B29 C31 and E29 H31 one for each arc accomplish this result SAMPLE MODELS 359 Now let s solve the model The PIPELINE Worksheet After Optimization a Gf PIPELINE NETWORK PROBLEM o K LE A O 0 a R PROBLEM MONTHLY PUMPING VOLUME Pumped From Inflow Well Pumping Costs 338 40 Refinery Pumping Costs 1 31 092 65 _ Total Pumping Costs WO What sBest has returned a minimized Total Pumping Cost of 1 431 05 Note that all Pumps are in use 360 CHAPTER7 Shipping Cost Reduction File name SHIPPING XLSX TYPE LINEAR OPTIMIZATION Application Profile This problem is typical of the class of network problems that generally involve the shipping of goo
192. ctive EV is the expected value for the model s objective over all the scenarios and is the same as the reported objective value for the model This value is provided by default in the report Calculation for Expected Value of Wait and See Model s Objective The Expected Value of Wait and See Model s Objective EVWS reports the expected value of the objective if we could wait and see the outcomes of all the random variables before making our decisions Such a policy would allow us to always make the best decision based on the outcomes for the random variables and of course is not possible in practice For a minimization it s true that EVWS lt EV with the converse holding for a maximization Technically speaking EVWS is a relaxation of the true Stochastic Programming model obtained by dropping the nonanticipativity constraints Possible flag values are Disable False or Enable True The default setting is off the EVWS is not calculated Calculation for Expected Value of Policy Based On Mean Outcome This Expected Value of Policy Based On Mean Outcome EVEM is the expected true objective value if we mistakenly assume that all random variables will always take on exactly their mean values EVEM is computed using a two step process First the values of all random variables are fixed at their means and the resulting deterministic model is solved to yield the optimal values for the stage 0 decision variables Next a the stage 0 vari
193. d quality requirements HOGFEED XLSX Chance constrained Blending As above but quality content in the constituent elements varies at random making the model nonlinear HOGCHANC XLSX Engineering Models Box Design A simple nonlinear model that finds dimensions for a cabinet to meet various design requirements BOX XLSX Flow Network Modeling Calculating flow and pressure across a complex network FLOWNET XLSX Financial Models Bond Portfolio Optimization A multi period model that recommends bond purchases to minimize costs while providing a specified cash flow BONDS XLSX Lockbox Location Locating postal lockboxes to minimize float while still serving all customers LOCKBOX XLSX Markowitz Portfolio Problem Selecting assets to meet a desired return at minimum variance MARKOWIT XLSX Portfolio with Transaction Costs Adjusting a portfolio of assets in such a way that desired return after broker s fees and other transaction costs is met with minimum variance PORTCOST XLSX 227 228 CHAPTER7 Minimizing Downside Risk Purchasing and maintaining a portfolio of assets so as to minimize the risk of losing value DNRISK XLSX Portolio Scenario Model Demonstrating the differences among minimizing three different measures of risk PORTSCEN XLSX Forecasting Models Seasonal Sales Factoring Determining the effect of seasonal factors on historical sales to improve foreca
194. d feedback about your model and the solution outcome If you wish a more detailed view of your model you may request the solution report as well The solution report provides a detailed analysis of the adjustable best and constraint cells showing the initial and final values as well as the locations of the cells The generation of the status report and solution report is controlled by settings at the bottom of the General Options dialog box If an error or warning situation was encountered that prevented the solver from running to completion then by default the solver opens up the status report to give you the error or warning message This message describes the problem the solver encountered and offers suggestions for correcting it The user may change the error and warning situations under which the status report is automatically opened at the bottom of the General Options dialog box ABCs 35 The status report generated by solving the completed sample model XYZ_XLS is shown below EIER Kaka XYZ xlsx Microsoft Excel oid Home Insert Page La Formul Data Review View Develop Add Ins Y Q o ER ZS D 7 Mar 31 2011 Library 7 0 1 129 64 bit Apr 05 2011 08 01 AM Current Capacity Limits Total Cells Numerics Adjustables Unlimited Continuous Free Integers Binaries Unlimited Constants Formulas Strings Constraints Unlimited Nonlinears Unlimited Coefficients 16 Minimum coefficient value
195. dd ins If What sBest is not in the list press the Browse button to locate the What sBest add in file WBA XLA or WBA XLAM so it can be loaded The default location for it is the Library subdirectory below the Excel directory If it is not there it will probably be found in the WB subdirectory If you have a laptop machine and are running Excel it is possible that the files required to support add ins have not been installed If the WB menu item does not appear on the menu after trying to set the What sBest add in to load run the installation program for Excel and choose the Typical installation option when prompted Symptom There are Excel error codes of REF in What sBest cells Problem The What sBest add in provides special functions to Excel giving it the ability to express constraints and dual cells As it does with all add in functions Excel internally stores the location of the add in program files whenever you use one of the What sBest add in functions in a model As a result if you create a model with What sBest add in functions and later open it using a copy of Excel that has the What sBest program files installed in a different location then Excel will not properly handle the What sBest functions and displays the REF error code Suggestions Close the workbook and then reopen it using the following procedure When Excel opens the workbook and detects formulas with inco
196. de simply obtain a new license key from LINDO Systems and enter it here The Floating Network License part is useful for a license that needs to be shared among different users The user Activate the key or automatically requests the key at opening of Excel at Solving and during solving iterations The user can Release the key or automatically lose it on closing Excel on removing the add in or changing the license key By default the key handler is valid for 60 minutes The User Information File is created upon request of Lindo Systems It contains the Machine Name the User Name and the Disk Identification Number This file USERINFO TXT can then be sent to Lindo Systems so to create the full license key By default this file will be created in the What sBest add in folder but you can select a different folder usually in the local disk drive Also you may need to verify where your license key is stored The license file LVDWBxxx LIC is located in the same directory as the What sBest add in files If you have lost your license key please contact LINDO Systems 124 CHAPTER3 Register The dialog box posted by the Register command appears as follows Lindo Systems Product Registration 64 bit Serial Number Lindo Internal Use Only Standard License Linear Barrier Conic Nonlinear Global Integer Stochastic Extended Phone Email What is your company s primary business Education
197. del B22 TOT_PROF 1036 132393 805 12242 920 62741 472 580823 124 706332 356 74066 793 259154 1233 054598 1662 79893 2128 178245 2556 527149 217 41946 272 176355 745 313288 1204 70012 1635 41617 2106 16119 254864441 3001 33029 345 00014 73 735011 551 026974 998 979634 1434 23538 1884 48006 2338 41133 2778 92872 Histogram We Stochastic Please refer to the section Guidelines for Stochastic Modeling for a tutorial and information on how to use Stochastic Support effectively 116 CHAPTER3 Advanced JConvert Model Format This feature allows you to convert a non What sBest format spreadsheet optimization model to What sBest format It will try to extract any existing data or information to set the Adjustable Best cells and it will create a new tab W I Constraint_Sheet1 at the end of the workbook for inserting the constraint functions Convert Model Format 5l This feature will try to convert the model from any other formats into a What sBest format for setting the Adjustables Best and to create a WB _Constraint_ worksheet for the Constraints What sBest will have only one model per workbook The sequence of worksheet tabs may change Current selected worksheet to convert Current Selected Worksheet to Convert What sBest can have one model per workbook but other spreadsheet modeling interfaces can have one model per workshe
198. del beggen serete eesin rn this acc titi ei deed Ed ec 57 solver Method Age ve entail e Dee teeth eho Neate Zeen 57 Ne Le TEE 57 Options Nonlinear SOW ANERER 59 Grashi Inte Solutio Nis nn initiat nae aeea a e aae iaa aae aa Aae a ai a 59 Presoen EE 60 Quadratic Recognition 2 ceccceecceeceeeeeeeeeeeeeeeeeeeeeeeeeeeaeeeaeesaeecaaecaaesaaeeeaeeeeeeeaeseaeeeeeenees 60 Selective Constraint Evaluation esseseeeseesresrssiresirssrissirsssrrssirssrnssrnnstnnsrnnsnnnttnntrnnnt 60 el SAIT EE 60 Steepest Edges EE 61 Starting PON TEE 61 Optimality Tolerance A 61 Iteration Limit for Slow Progress esssessseessessseessesssrrssrnestntstnnntnnstnnstnnnstnsrnstnnsennnnnnnnt 62 Ai VE 62 en EE 62 Options GloDal Goler 63 Eeer 64 Mustat Attempts iis iitoc aaar EEE AEE EE AAEE EEEE A dee 64 PREFACE v CUI A asa Fel E ee eeh 64 Deltgen EE ek 64 varane BOUNO acara a A T A he nant alate Maede raat 65 WS6 0f BOUNG E 65 Branching Direc igisi sdeeed aaaea eat AERE 65 Box SCC CUON EE 65 Algebraic Heiormulatton AA 65 Options Integer Pre Solver AA 66 Heuristics Levels nern uani arai iai eaa easda Pea a eee ees 67 Heuristics Cutoff Criterion ccecsceeeeeeeceeeeeeeeeeceeeeeseaeeeeaaeseeeeseaeeesaaaeseeeeeseaeeesaeeseneeenaees 67 SN Gleis WEE 67 Constat DUSEL eee 69 Options Integer SoN eicinaite aiana Ed ented 71 Branching Et eheont edd EN tapaa AN AEE AAE A cael 72 Ittre 2 g ie gege Neth e EEGENEN e A
199. ders may refer to any good text on integer programming What sBest defaults to enabling all cut generation strategies Coefficient Reduction Disaggregation Flow Cover GCD Gomory GUB Knapsack Cover Lattice Lifting Plant Location Objective Integrality Basic Cardinality ADDITIONAL COMMANDS 71 Options lInteger Solver The dialog box posted by the Options Integer Solver command appears as follows Integer Solver Options oo l Branching m Integrality Absolute 0 000001 LP Solver Warm Start Solver Decides v This command allows you to set a number of options controlling the function of the integer solver To understand the operation of the integer options it is useful to understand how integer problems are normally solved With the default option settings What sBest solves integer problems using the following steps 72 CHAPTER3 1 What sBest begins by solving the continuous relaxation i e the original model with integer restrictions removed This gives an optimistic bound on the objective of the true integer model because the objective of the integer restricted model could never be better than the objective of the continuous approximation 2 After solving the continuous relaxation What sBest uses a process called branch and bound to find the optimal integer solution Branch and bound implicitly enumerates all possible intege
200. dicator i e lt If an indicator is not satisfied at all What sBest will return a Not before the indicator i e Not Slack mode displays the slack value of each constraint The slack value of a constraint tells you how close you are to satisfying the constraint A positive slack is the amount by which the constraint is overly satisfied A zero slack value means the constraint is exactly satisfied A negative slack is the amount by which the constraint is violated Computing the slack of a constraint involves simply subtracting one side of the constraint from the other side The default setting is indicator mode Auto Select Free Int Omit Ranges Check this option if you want the Free Integer or Omit ranges selected as you scroll through any list of What sBest defined named range names these range names begin with WB With this option checked on the range corresponding to a particular name is selected when you click on that name on the list This enables you to easily identify the range corresponding to a particular name The default setting is on Minimize Excele on Solve Check this option if you would like Excel to be minimized while the What sBest solver is running The default setting is off Hide Status Window on Solve Check this option if you would like to hide the Status Window while the What sBest solver is running The default setting is off Linearization The linearization options offer
201. ding the software or documentation you can contact LINDO Systems by e mail phone FAX or mail E mail is often the best means of submitting questions regarding the operation of the software because it allows us to research the questions and provide detailed answers To resolve a problem running a specific model we may require that you attach a copy of the model in your email All models sent to LINDO Systems are treated as strictly confidential Sales and Marketing can be reached at e mail sales lindo com phone 312 988 7422 FAX 312 988 9065 Mail Courier LINDO Systems Inc 1415 North Dayton Street Chicago IL 60642 USA Tech support is available during business hours 9 00 AM to 5 00 PM Central Time e mail tech lindo com Phone 312 988 9421 General information e mail info lindo com phone 800 441 BEST 800 441 2378 www lindo com From our website www lindo com you can download trial versions of any of our optimization products You can also find a database of sample models and past newsletters to assist you in your modeling efforts Our website also accommodates online ordering and upgrading information 449 REF 434 0 0 1 integer variables 2 42 A ABC steps 9 12 About What sBest 119 About What sBest dialog box 120 ABS 187 Absolute 72 Absolute value function ABS 188 Absolute Violation 65 165 Absolute Width 65 165 Access 131 137 Acknowledgements 2 AC
202. djust For additional discussion of the functionality provided see the section entitled Adjustable which refers to the dialog box interface that calls this procedure This routine is also called with appropriate arguments by two of the toolbar buttons This routine is used to either set cells as adjustable or to remove the adjustable attribute All arguments are optional Syntax wbAdjust AdjRange AdjChoice NoErrDialog Argument Required Description AdjRange No This is the range or address of the cells to be rendered adjustable If omitted the default value for AdjRange is the current selection AdjChoice No This is a string to indicate Adjustable or Reset AdjChoice accepts the text Adjust to set the AdjRange as Adjustable or Reset to set the AdjRange to no longer be adjustable If omitted the default value for AdjChoice is Adjust For backwards compatibility with earlier versions of What sBest AdjChoice also accepts the numbers for Adjustable and 2 for Reset NoErrDialog No Any argument passed here causes all What sBest error dialog boxes from the wbAdjust routine to be suppressed Instead the error number is delivered in this return argument Pass an integer to use any possible returned What sBest error number Useful in an embedded application of What sBest Error Codes Description Adj_BadAdjRangeArg Bad AdjRange argument Adj_BadAdjChoiceArg Bad AdjCh
203. ds for more information on dual values SAMPLE MODELS 337 After solving a dual value of 67 0 666 appears in cell H13 This indicates that the total cost would increase by 67 if one person were to start a workweek on Sunday You can confirm this by placing a 1 in cell G13 and using the Remove Adjustable command to lock the value in place Then solve once again The total cost of this solution has increased to 44 067 an increase of 67 from the previous best solution exactly as predicted by the dual value in cell H13 However you get fractional numbers of employees To get integer answers you could round the fractional cell values One possibility is as follows Round From To Cell G8 Number Starting on Monday 79 33333 80 Cell G10 Number Starting on Wednesday 19 33333 20 Cell G11 Number Starting on Thursday 39 33333 39 Cell G13 Number Starting on Saturday 29 33333 29 After entering these new integer values in the adjustable cells you ll find a new solution with a total of 221 employees and 44 200 as the minimum cost Therefore dual values in this case provide us with the theoretical since fractional people are imaginary minimum increase in total cost 67 instead of the actual increase in total cost 200 for a feasible solution Note Before using any dual values in a pricing hiring or purchasing decision be sure to investigate the ranges over which they are valid In some problems dual values may
204. ds through transportation networks or of oil or gas through systems of pipelines The particular shipping example illustrates the use of optimization to provide the least cost distribution for a business that produces a product at many locations transports it to intermediate warehouses and then to final consumption or sales locations Manufacturers utilities retail chains and other distributors are all likely users of shipping optimization systems modeled along similar lines The Problem in Words You work in the shipping department of a manufacturing business and need to minimize shipping costs while meeting demand without exceeding capacity Background In this case two steel mills produce the same product This product must be sent out to three plant locations Each location has a demand for the product that must be met and each steel mill has limited manufacturing capacity Finally shipping costs vary from each steel mill to each plant Shipping Costs Per Unit of Steel From Steel Steel To Mill 1 Mill 2 Plant A 200 500 Plant B 300 400 Plant C 500 600 Objective of Optimization The objective is to meet all plant demand without exceeding steel mill capacity at minimum shipping cost SAMPLE MODELS _ 361 The Worksheet Let s look at the SHIPPING sample file The SHIPPING Worksheet Before Optimization Total Units From From Demand Demand Shipped Steel Mill 1 Cost Steel Mill 2 Cost Constraint To Pla
205. e In fact these may be not only preferences but requirements under union work agreements Here s a two stage model for this specific situation Stage 1 Cost Minimization The Problem in Words The first stage is to determine the lowest cost schedule that meets all staffing requirements You can determine this schedule independent of specific personnel availability or characteristics You know staffing requirements for each day of the week It s not important how these daily requirement figures are obtained as long as they accurately reflect staffing needs They could be based on the output of another model such as a queuing model or an extrapolation of historical figures We also make the following assumptions in this model e There is only one skill level of staff employees are interchangeable Staffing requirements or Full Time Equivalent requirements FTEs can be met with either full time employees working five days per week or part time or pool employees that can be hired on a daily basis e Each full time employee cannot work more than 4 days in a row You could incorporate other assumptions with minor changes to the model Background An employee s schedule for one time period is called a work pattern Each employee can be scheduled under one of several work patterns For example a full time employee working five days a week can either work Monday through Friday or Tuesday through Saturday These work patterns can b
206. e integral values by the What sBest solver Syntax WBINT IntName Refers_to Argument Required Description IntName Yes IntName is a string that refers to the range being made integer Refers_to No Refers_to is the range or address of the cell s to be specified as integer If omitted the default is the current selection Error Codes Description Int_BadIntNameArg Bad IntName argument Int_BadRefers_toArg Bad Refers_to argument Int_ProtectedSheetError Unable to set integer cells on a protected sheet Int_CreateError Error setting integer cells Remarks The use of integer variables can dramatically increase total solution time Example To constrain cell F6 to be a general integer using the range name staff enter WBINT staff F6 wbinteger For additional discussion of the functionality provided see the section entitled Integer which refers to the Integer dialog box that calls this procedure This routine is used to build the integer ranges WBBIN and WBINT Adjustable cells contained in these ranges will be restricted to integral values by the What sBest solver If you wish to delete an integer range name using VBA instead of the Delete button on the Integer dialog box you can use Active Workbook Names WBBINmyRange Delete This statement removes the integer range name thereby returning the adjustable cells contained in the former range to normal status Syntax wbInteger IntName
207. e represented as follows Pattern Mon Tue Wed Thu Fri Sat Sun 1 1 1 1 1 1 0 0 2 0 1 1 1 1 1 0 A one indicates that the employee is working and a zero that he is off 346 CHAPTER7 There may be many patterns consistent with an organization s work rules The first step in scheduling is to find a representative sample of patterns that correspond with organization rules or policies and meet other necessary criteria The example shown here displays 10 patterns Patterns 11 and 12 may be written in or any of the supplied patterns may be written over This process need not be repeated unless there is a change in the policies or rules Objective of Optimization The objective is to minimize staffing costs for a single shift The model creates the minimum cost schedule based on daily requirement figures average cost per employee category and the work patterns It chooses the best out of all work patterns that meet the staffing requirements It also specifies how many people to schedule according to each work pattern and if desired how many pool people to hire on a temporary basis to satisfy requirements SAMPLE MODELS 347 The Worksheet The model is formulated on the left and far right of the worksheet FIXED1 The first screen A1 N20 contains the staffing and full time equivalent FTE requirements and coverage and cost information The FIXED1 Worksheet screen 1 Before Solving STAFF SCHEDULING Fixed Shift TOT
208. e Workbook Names WBKBESTmyRangeName Delete This statement removes the WBKBEST range name Syntax WBBEST KBestName Refers_to Refers_to is the range or address of the cell s to be read If omitted the default is the current selection KBestName Yes KBestName is a string that refers to the range that is to be read as a trade off 160 CHAPTER 4 KBest_BadRefers_toArg Bad Refers_to argument KBest_ProtectedSheetError Unable to set KBest cells on a protected sheet Example To define an KBest trade off range D10 H20 using the range name Tradeoff enter WBKBEST Tradeoff D10 H20 WBOMIT For additional discussion of the functionality provided see the section entitled Advanced Omit which refers to the dialog box that calls this procedure This routine is used to build WBOMIT ranges which are areas of the worksheet that What sBest will ignore while solving If you wish to delete a WBOMIT range name using VBA instead of the Delete button on the Omit dialog box you can use Active Workbook Names WBOMITmyRangeName Delete This statement removes the VWBOMIT range name Syntax WBOMIT OmitName Refers_to Argument Required Description OmitName Yes OmitName is a string that refers to the range that is to be ignored Refers_to No Refers_to is the range or address of the cell s to be ignored If omitted the default is the current selection Error Codes Descr
209. e concept of global versus local optimality refer to section Local Optima vs Global Optima The default setting for Global Solver is off Multistart Attempts Use the Multistart Attempts text box to have the What sBest s nonlinear solver invoke a heuristic for generating multiple good starting points The multistart capability is useful for quickly finding better solutions when a non convex model has several local optima Note The multistart capability is an optional component of the What sBest solver and is packaged as part of the global solver option Your particular installation may not have the global option enabled in which case enabling Multistart Attempts will cause a warning to be generated when solving If you are interested in adding the multistart capability please contact LINDO Systems The default setting for Multistart Attempts is Solver Decides Optimality Specify the global solver s optimality tolerance in the Optimality text box in the Tolerance box This tolerance indicates that candidate solutions must beat the incumbent solution by at least this amount to become the new best solution The default setting for Optimality is 0 000001 Delta Specify the delta tolerance the global solver s convexification process in the Delta text box in the Tolerance box This tolerance is a measure of how closely the additional constraints added as part of convexification should be satisfied The default setting for Delta is
210. e detail on what to do each period For example at the beginning we should split our 55 by investing 41 479 into Stocks and 13 521 in Bonds Notice the interesting policy in period stage 2 If our wealth is well below 80 i e 64 or if our wealth is above 80 then we invest all of our portfolio into Stocks If our wealth is intermediate i e 71 428571 then we invest all portfolio into bonds The reasoning is that if we know we are going to fall short of our target or if we have already achieved our target then we might as well invest in the instrument with highest expected returns Stocks On the other hand if our beginning wealth stage 2 is 71 428571 then if we put all our money into bonds then we are guaranteed to just achieve our target of 80 MATHEMATICAL MODELING 221 Monte Carlo Sampling In stochastic programming where one or more stochastic parameters have continuous or discrete but infinite event space there are an infinite number of possible scenarios making the model computationally intractable For such cases Monte Carlo sampling also called pre sampling can be used to approximate the problem to work with a finite scenario tree As illustrated in the figure below if the model has a single stochastic parameter with a continuous distribution such as the Normal Distribution one can discretize the event space simply by generating N sample points and construct a finite and tractable scenario tree This is also true fo
211. e following error message on the WB Status tab Error Message Solver Error Help Reference SOLVERR The solver encountered an error from which it was unable to recover A brief description of the error follows error description Suggestions Solver errors should be an uncommon occurrence and you may need to contact LINDO Systems to resolve the error Short of that here are some suggestions Ifthe error appears to be numerical in nature and your model is nonlinear you may want to try restarting from a different point This will tend to force the solver to take a different path to the solution and hopefully avoid the error condition Rescale the model so that the ratio of largest to smallest coefficients is reduced to improve numerical stability The What sBest status tab displays the value of these coefficients and the cells in which they appear Often simply changing units of measure e g dollars to thousands of dollars will be enough to bring this ratio into line Experimenting with some of the solver parameters via the WB Options command may help solve the problem However should this not solve the problem be sure to restore all options back to their default settings You may be able to choose an alternate solver For example with linear models you can choose the primal dual or barrier solver For nonlinear models there are two versions of the standard nonlinear solver a quadratic solver and a global solv
212. e function in the teration Limit for Slow Progress text box In other words if the value has not improved by the Nth iteration N having been set as the teration Limit for Slow Progress then the solver terminates the solution process In cases when the model has a flat objective function around the optimal solution a large iteration limit may be required to allow for sufficient iterations to move to the optimal solution The default setting for the Iteration Limit for Slow Progress is 100 Derivatives Use the drop down menu to control the derivative method in the nonlinear solver There are two different methods to use Numerical or Analytical derivatives Numerical derivatives are computed using finite differences There are two choices available Central Differences and Forward Differences Analytical derivatives are computed directly by symbolically analyzing the arithmetic operations in a constraint Again the two choices are Backward Analytical and Forward Analytical By default What sBest uses backward analytical derivatives However different derivatives can improve speed and precision on many nonlinear models Experimenting with the various derivative options may be required to determine which method works the best for a particular model The default setting for the Derivatives is 0 or Solver Decides Solver Version The nonlinear solver is provided in two versions Ver 2 0 and Ver 3 0 Each version has its advantages and
213. e is no solution that satisfies all of the constraints and any integrality conditions in the model Check to make sure all the constraints are properly formulated Consider easing constraints in the returned model that are either not satisfied or tight NOTE The answer returned is not feasible therefore the value of the objective function is NOT optimal The solution returned is only for the purpose of illustrating a scenario with violated constraints so the model can be corrected See the list of cells below that may be contributing to the infeasibilities Suggestions A linear optimization model containing constraints that cannot be satisfied simultaneously by What sBest is said to be infeasible With nonlinear models it s possible that a feasible solution exists but What sBest was unable to identify that solution from the starting adjustable cell values you specified In either case the solver tries to find a solution to the problem that satisfies as many constraints as possible and returns the appropriate solution status in the status report worksheet entitled WB Status If a linear or nonlinear problem is truly infeasible correcting the model can be a very complicated task The first step is to investigate those constraints contributing to the infeasibility These cells have been identified for you by What sBest at the end of the WB Status tab If you have selected the constraint display type in General Options as Indicator then t
214. e model is laid out in a 3D workbook on four separate worksheets a shipping sheet and three separate manufacturing sheets 316 CHAPTER7 The first worksheet is Shipping built by copying the original Shipping Cost Reduction problem into cell Al of anew workbook This sheet gives information on the amount of steel that has been shipped from each steel mill to each manufacturing plant a breakdown of shipping costs and total costs The Shipping Worksheet Before Optimization Total Units From Steel Mill 1 Cost Shipped Steel Mill2 Cost Constraint To 2 Plant A 5 P Mot zs 3 PlantB 4 Not gt 5 Plant C Sp P Not gt P e gg Capacity 0 PROFIT From ALL PLANTS less SHIPPING COSTS The remaining three worksheets Plant A Plant B and Plant C are the manufacturing plant screens They were built by placing the PRODMIX model into each of the three tabbed worksheets Each sheet represents one of the three manufacturing plants SAMPLE MODELS _ 317 With the exception of steel the starting inventories of all the raw materials are the same as in the original problem The starting inventory of steel for each plant now becomes the amounts shipped from the steel mills to that plant For example the formula in cell J13 of Plant A worksheet Starting Steel Inventory for steel at Plant A is Shipping B6 Shipping F6 Each plant s sheet tells the quantity of each item produced inventory total usage amounts and the pr
215. e model without pool employees as part of the solution you can also use it if you have access to part time or pool employees and need not rely only on full timers to meet staffing needs Just make the cells representing pool R18 AE18 adjustable before solving This will result in a 1 760 saving in this case since with pool cells adjustable there s much more flexibility in matching the Scheduled personnel with Recommended levels Sometimes the lowest cost solution results in fractional values for the adjustable cells Although these fractional numbers provide the lowest cost they are not practical for implementation purposes since it s difficult to locate 2 3 of a person who is in any condition to work SAMPLE MODELS 351 Fortunately though realistic whole number solutions can be obtained by rounding the fractional answers without sacrificing much in total cost If the program returns non integer values in cells P5 P16 or in R18 AE18 when the Pool is used you can manually round these values to whole numbers You could also use the Jnteger command to force whole number answers but this can significantly increase the solution time The objective in rounding these cells should be to make sure that the Scheduled gt Recommended constraints in D5 D11 and D13 D19 aren t violated and to have a final cost as close as possible to the one before you started rounding Now that the lowest cost schedule has been found the next step is to
216. e numbers to 10 to the 4 1 100th compared to 10 Simplify Relationships Whenever possible use linear rather than nonlinear relationships Some nonlinear expressions can be reformulated in a linear manner A simple example is a constraint on the ratio of two adjustable cells Consider the constraint WB A1 B1 lt 25 where Al and B1 are adjustable cells that each must be greater than zero The relationship A1 B1 is nonlinear If you multiply both sides of the constraint by B1 the constraint becomes WB A1 lt 25 B1 This is mathematically equivalent to the original constraint but it is now a linear rather than nonlinear constraint As much as possible avoid non smooth expressions notably spreadsheet functions like JF MAX MIN ABS and table lookup functions Models with non smooth relationships are generally more difficult to solve If you can approximate the non smooth relationship with a smooth expression Reduce Integer Restrictions Minimizing the use of integer restrictions can drastically reduce the solution time In instances involving larger numbers you may find that solving the model without integer restrictions and then rounding yields acceptable answers in a fraction of the time the integer model requires MATHEMATICAL MODELING 205 Guidelines for Stochastic Modeling So far we worked with deterministic mathematical programs where model parameters e g coefficients bounds etc are known constants A
217. e unsupported cells are used for reporting only they can be included in a WBOMIT range see the section entitled Advanced Omit If they are necessary to the actual optimization model they must be expressed or approximated in terms of equivalent operations supported by What sBest If you cannot approximate the unsupported functions reformulate the problem to eliminate all such functions TROUBLESHOOTING 401 This warning may be disabled with the Unsupported Function warning checkbox in the General Options dialog box posted by the Options General command Using linked worksheets may also cause this message For more on this subject see the Solve topic FORMULAS External Reference If there are external links to other workbooks that can t be resolved the following error message will be displayed on the WB Status tab Error Message External Reference Help Reference FORMULA3 A link to an external workbook in the cell below could not be resolved Check the validity of the link or remove the external link cell address Suggestions An external reference was found that could not be imported You should check to see if the link is valid and if it can be resolved Array formulas can also generate this error in that they may only reference native data contained in the primary workbook If any of your array formulas have external links you will need to import the external data into the native workbook 402 CHAP
218. e write privileges Users who wish to accomplish the following Installing What sBest on a network Installing on systems with multiple versions of Excel or Operating a non English version of Windows will need to select a Specified installation 439 440 APPENDIX A If Excel is installed on a network server the What sBest add ins files should be installed and run locally If you have a network license or site license then you can follow the instructions in this section to install What sBest to the server and set up directories for each node If you have questions regarding installation of What sBest the type of license you have or the number of copies licensed to you please call LINDO Systems It is important that What sBest users avoid sharing a directory for spreadsheet data files When solving a model What sBest saves a copy of the model and creates a copy of the solved model under the same file names each time If What sBest users share a common data directory over the network the data files of one user could be inadvertently overwritten by the data files of another user What sBest Add in Files The setup program will attempt to locate Excel and find a Library folder in the dialog box but it is possible it may fail or will find a version of Excel different from the desired one Therefore it is important that you confirm that the supplied path is correct and otherwise correct it Location of
219. ecify Constraints The constraints are of two types First cells B40 G44 prevent a customer from being assigned to a location where no lockbox has been opened For instance move the cursor to cell B40 and observe the formula there WB B 13 gt B7 It requires that B13 a 1 or 0 depending on whether or not the New York lockbox is opened be greater than or equal to B7 a 1 or 0 depending on whether or not the Seattle customer is assigned to the New York Lockbox The value in B7 cannot be greater than that in B13 If the Seattle customer is assigned to a lockbox located in New York the location must be open or else the constraint in cell B40 will be unsatisfied The constraints in cells H40 H44 ensure that each customer is assigned to a lockbox by requiring that the sum of lockbox assignments for a given customer be at least 1 260 CHAPTER7 With all constraints in place we re ready to solve the model The LOCKBOX Worksheet After Optimization Vid _LOCKBOX xlsx Microsoft Excel Ejeet Monthly Proposed Lockbox Locations Cash A Customer Home Flow Le Locations Mew York Atlanta Cincinnal Denver Seattle Office 000 Seattle 5000 Los Angeles 5000 Houston 5000 Philadelphia 5000 Miami 5000 1 1 1 0 WIR CostiMonth 1300 975 1000 1100 2 000 0 15 Fixed Costs 1300 975 0 1 100 0 0 17 Variable Costs 1350 1 350 0 5 400 0 0 18 Daily Cost of Capital 0 027 f TOTAL OPERAT
220. ed for arcs where there is no flow This increases the number of numeric cells in the model In your own models you may want to minimize the number of numeric cells B Define Best No best cell is defined in this example since our objective is simply to balance the flows along the arcs and satisfy demand at the nodes SAMPLE MODELS 247 C Specify Constraints The constraints in this model are twofold First conservation of flow requires that the amount entering an arc must equal the amount exiting that arc Arc Flow C13 H13 As an example here s the formula in C13 WB SUM C4 C11 ARC RESISTANCE C13 SUM C4 H4 Second you must account for pressure lost along each arc due to resistance These constraints are found in cells C5 H11 of the Pressure Balance worksheet The formula in C5 for instance requires that pressure applied equals pressure lost on the B gt A arc WB ARC RESISTANCE D14 C 13 ARC RESISTANCE C14 C 13 ARC RESISTANCE C5 ARC FLOW C5 C 14 Disregarding the exponents which in this case are 1 the formula is WB ARC RESISTANCE D14 ARC RESISTANCE C14 ARC RESISTANCE C5 ARC FLOW C5 Thus loss in pressure along arc B gt A pressure at B minus pressure at A must equal resistance along arc B gt A times flow along arc B gt A Note The Pressure parameter is 1 for incompressible fluids and electricity and 2 for gases The Flow parameter is 1 for electricity 1
221. eding graph a strictly convex function with no constraints has a single minimum That is water drizzled on it will collect in a single puddle at the global minimum Therefore minimizing a smooth convex function with no constraints will yield the global optimum regardless of the initial value of the adjustable cells However if the function is not convex there may be more than one local optimum In this case the returned answer may be locally optimal but not globally optimal 198 CHAPTER 6 Determining the convexity of a multiple variable problem is no easy task Mathematicians call a function convex if the matrix of second derivatives is positive definite or has all positive Eigen values A function is called concave if the matrix of second derivatives is negative definite A function can be concave in one section and convex in another The following function is strictly concave A Strictly Concave Function Graph of A12 D 300 250 200 fe 1 150 o f 2 LO oS Oo CH 1 Oo re _ NS 250 300 1 10000 20000 30000 40000 50000 60000 70000 80000 90000 For a mixed function look back at the graph of X SIN 3 1416 B3 in the preceding section Smooth vs Non smooth Expressions Smooth expressions have a defined first derivative slope or gradient at every point Graphically a smooth function of a single variable can be plotted as a single continuous line with no abrupt bends or
222. ee EE 189 Inverse Triangular Cumulative Distribution TRIAINV ccceeeseeeeeeeeeseeeeeneeeeeeeeeees 189 Inverse Exponential Cumulative Distribution EXPOINV c ccccceeeseeeeeeeeseeeeeeeneeeeees 190 Inverse Uniform Cumulative Distribution UNIFINV ccccceeeeeeeeeeeeeeeeeeeeeeeeeeeetees 190 Inverse Multinomial Cumulative Distribution MULTINN 191 Standard Normal Linear Loss Function NODMSL 191 OVERVIEW OF MATHEMATICAL MODELING cccseeecsseecesseeeeeeeenseeeseeesenseeenseees 193 INMCOGUCUION EE 193 Linear vs Nonlinear Expressions and Linearization cc ceeeeeeeeeeeeeeeeeeeeeeseeeeeeeeeeeees 193 Lim ar Expressions een oiea e e e ai a aea e aia a i aadi 193 Nonlinear Expressions 0 ccccccecescceceeeeeeeeeeeeneeceeeeceaeeeeaaeeseaeeecaaeessaaesseneeseeeesaeesseneeees 194 LNG ANZ ATOM saree e E i e E e a e ea A 194 The Solution Process Determining Optima sssesseseeesenesinrernssrnssrnssrnssrnssrnssrnnsrnnssrnsnn 195 Local Optima vs Global Optima ccceceeeeeeceeeeeceeeeeeeaeeeeeeeceeaeeesaaeseeeeeseeeesaeeeeaaeeee 195 GONVEXILY cect cts naea an E e A A a dE ee dee eet aes 197 Smooth vs Non smooth Expression 198 SOMMMOM OUTCOMES E 200 Case 1 Optimization Models cccccccecsecceceeseceeecseeeeeeseeeaeeeeseeeeeeeseneseessseeaeesseeeaaeess 200 viii PREFACE Case 2 Non optimization Models esseeeseesseeesinesrnreen rest nesrnssrnsttnstrnssrnssnnnnnnnsnn
223. eeceeececseseaeceeeesesesseseeaeees 403 IKBREP K Best WBIKB_REP Fommat 404 INDICMOD Indicator Model Cell Reference ccccccccccecsesssseceseeeceeseseaeseseeeseeesesseaeees 404 INFEASIBLE No Feasible Solution Found 405 INFLARG Large Infeasibility cccceccececeeeeeeeeeeceeeeeeeaeeeeaeeseeeeeceaeeeseaeeeeaeeseeeesaaeeeeneeee 406 INTERRUPT Solver Interrupt ccceecceceeeeeeeeeeeeeeeceaeeeseaeeseeeeecaaeeeseaeeseaeeseeeesaeeeteneeee 406 INVMOD Invalid Model 408 IRRECONST Irreconcilable Consiraimts 408 ITRLIM Iteration Limit Beached 409 LIGGAP1 Constraint Bn TIET 409 LICCAP2 Adjustable Cell mp 410 LICCAPS Integer Cell Limit 00 ccc ceeececececeeeeseceeeeeceeeeeeeaeeseaaeecaeeesaaesseaaeseeeeseaeeseaaaeennees 411 LICCAP4 Nonlinear Adjustable Cell Limit 0 ccccecseeeeeeeeeene esses eeeeeeseeeesaeeseeeseneees 411 LICCAPS5 Global Adjustable Cell mit 412 LICKEY1 Failed to Process License key 413 LICKEY2 Pending License Expiration nnns 413 LICKEY3 Expired License key 414 LICKEY4 Pending Expiration of Option Tal 414 LICKEY5 License Key Dongle Heouired 415 LICOPT1 Global Solver Not Uicensed 415 LICOPT2 Nonlinear Solver Not LUicensed 416 LICOPTS Barrier Solver Not Licensed ccccceesssseceeeeeceesesaeceeeeeeeceeesesasaeceeeesesenseaees 416 LICOPT4 Mixed Integer Solver Not Licensed 0 ccccceeeeeeeeeeeeeeeeeeeee teense seaaeeeeeeteeeees 416 LIC
224. eee dia ee deel deel 17 Working While Gong 18 The NextStep insta oad ial iene bien te i eh i eee eta ale bay 18 2 ABC S THE BASICS OF WHAT SBEST ccccceceseessteeenseeeeeeeeestneeenseeeseneeessaeenseeeeneneess 19 Adjustable EE 20 Make Adjustable wai cinta devel Seege eet lieth idan dda 20 Remove Adjustable EE 21 Make Adjustable amp Free or REMOVE Free 00 0 eeeeeceeenneceeceeneeeeeeaeeeeeeaaeeeeeeaaeeeeeeeaeeeeneaas 21 tee stave tai ergeet Egeter eege a eines 22 BeStiatci stig din ee ate ei ate ada ede cited 23 ODJECE eeh e ead eeh te Slater eege i bi Ny Aenea ad 23 Celanese tee ate i te Atte hated a AA ee i ee ofa 24 Constraintss Ofte cated whet ge deet aie 25 Left Hand Side LHS Right Hand Side RHS and Stored Im 25 Usage Guidelines for Constraints cccccccccceeececeeeeeeceaeeeeaaeceeeeeceaeeesaaeeseaeeseeeeeseaeessaaeeeaes 27 Constraint related Problems sssseeseeseesrssiesissiesinssiitsrrsstissrinstnnstnnsnnnntnnntnnnnnnnnnnn ne 29 Explicitly Specifying Convexity to the Solver ssssssssssesssseessrrssrrsrnsrnssrnssrnssrnssrnssrnssens 30 GEREENT Eed heet ee Ee En 31 Getting the Best Hesuhte 34 3 ADDITIONAL COMMANDS eecccceeecesteeeseeeeneeeeeeeneescaeensneeenseeesnaesaseeeeneeeeseeeseseaeenseeees 39 Options and SolWers irinna tarea adaa aia edee adaa aaao aad eerie feeeeteap athe 39 Integers lnteger Binary ccccescccceceecceeeeeeeceeeeeeeceeeeaaeeeeeeaeeeeeaaeaeseeceee
225. een Assets i and j y for i j is the variance of Asset i This is computed by means of the following formula in cell B16 G6 B6 2 G7 B6 B7 G8 B6 B8 HH6 B7 B6 H7 B7 2 H8 B7 B8 16 B6 B8 17 B7 B8 18 B8 2 C Specify Constraints There are three types of constraints in the Markowitz model The first in C6 C8 requires that no percentage invested in any single stock exceed the Upper Bound for each stock of 75 D6 D8 For example C6 has WB B6 lt D6 The second in C15 is WB B15 gt D15 This requires that the combined return on the three Assets B15 be greater than or equal to the desired return in cell D15 15 The third in C10 is WB B10 D10 This requires the sum of the percentages invested in each stock B10 add up to 100 D10 264 CHAPTER7 Now let s solve the model After solving the WB Status worksheet will open in order to show you the Nonlinearity present warning This warning can be shut off from the General Options dialog box Your solved model now appears as follows The MARKOWIT Model After Solving Percentage Upper Covariance Matrix Invested Bound Return Asset 1 Asset 2 Asset 3 Asset1 18 3 75 0 30 0 3 0 1 0 0 5 Asset2 24 8 75 0 20 0 1 0 2 0 0 4 Asset3 56 9 750 8 0 0 5 0 4 1 0 Invested 100 0 100 0 Desired Portfolio Return Retum 15 0 gt 15 0 Variance 042 What sBest returns a minimized variance of 0 42 An interestin
226. eeneeeeeneeeeeeeeteaes 41 Integer Names in Workbook AAA 42 Refers Bette hiatal hel itt ale ahaha otk ah tet 42 BinarysWBBIN WEE 42 General WBIN Ta AE 42 Runtime Concerns in Integer Problems s ssssesseesseessrnesrnesrnssrnssrrnssrnsrnssrnssrnssrnssrnssens 42 iv PREFACE Integers Special Ordered Get 43 Special Ordered EEN 43 Gardinallty Ee 44 eler ANY COllS agar ee eebe Ee Ee 45 List ot selected Cells urrina Beate AEG ates AN eet aaa see 45 Place the Function in Cells 0 ccccceeececeeeeeeeeeeeeeeeceeeeeceaaeeeeaeeseaeeesaaeeesaaeseeeeeseaaeeeeaeeseeees 45 Integers S Mi CONLINUOUS EE 46 EE 47 List of Selected CellS 2 ii nariais iniiai aia deer Eege cian 47 Place the Function in Cells A 47 E le IEN 48 Options Generals icss sei AEN Siete Dei aed Ae ee ie H 49 Feasibility Tolerate dead ENEE ENEE Ed Meee At Sue 50 Iteration LIM Ite tities Ai eaten Wie ie Re ee ea 50 Runtime Bu Le 50 Constraint Display ss ctacwcasaieta A fae eine ei pee aad cee 50 Auto Select Free Int Omit Ranges cee eeeaaeeeeeeeeseaeeesaeeeeeeeeeeee 51 Minimize Excel on Solve A 51 Hide Status Window on Gohye 51 EE 51 Delia Coeficiente E dech dees Ze eg ee deed EE een eee canted 52 le Deele EH 52 Status ele TEE 52 Begimning OF End ET 54 WarninGSs tcani nia tane dite dati ate ein ATE E tion aot ene 55 Update ENKS ii eebe ee Geert gereent gehae 55 NC CEET 56 Options Linear ht gert Meuse Eeer ee 56 scade lee E EE 57 Mo
227. efficiently using the other raw materials on hand SAMPLE MODELS 321 Waste Minimization in Stock Cutting File name CUTSTOCK XLSX TYPE LINEAR OPTIMIZATION Application Profile Consider a product produced in rolls or sheets fabric sheet metal or carpeting A manufacturer of this product is constantly faced with a problem His machinery produces rolls of a single standard width but he must sell to an unstandardized marketplace This model applies linear programming to the problem of cut patterns Given the sizes required the amount of each size demanded and the cut patterns available the model minimizes the waste cost associated with meeting demand The Problem in Words You re the production manager in a sheet metal plant You turn out rolls of sheet metal of a particular width Your customers use your product in different ways requiring rolls of different widths To accommodate their needs you can cut a roll of sheet metal into more than one narrower roll Your goal is to waste as little of the sheet metal as possible by choosing patterns that best utilize the width of a roll and closely matching the length requirements of various sizes Each wasted edge strip and each length of metal cut to an unneeded width adds to your total waste cost You must select cut patterns that meet customer needs and minimize total waste costs Background Your factory produces sheet metal in various widths that you cut from 100 inch widths I
228. efore be sure to enter a numeric value in each of your random cells Note The Random cells will be identified during the Step 2 below Page Step 2 sequencing and distributions l Stochastic Support s V Use Stochastic Modeling Support Step1 Step 2 step 3 Step 4 Specify a the stage information for the Variables Adjustables Formulas Constraints cells the Random cells and b associated distributions using the set of WBSP_ functions Refers to Stage WBSP_VAR DI 0 A 1 S Place function in cell al me This step provides two related pieces of information a the sequencing of decision and random events or staging and b the distributions of the random variables The general sequence of events is as follows Stage 0 We make initial decisions Stage 1 beginning First set of random variables are observed Stage 1 end We make additional decision or calculations based on observations so far Stage 2 beginning the second set of random variables are observed etc This staging information is inserted into the sheet with the WBSP_VAR and WBSP_RAND functions as follows In the drop down list you can select a WBSP_ function with its associated arguments ADDITIONAL COMMANDS 107 Variables WBSP_VAR Use this function to associate a Stage to an Adjustable Constraint or formula cell The format is WBSP_VAR stage cells_with_this_stage e g WBSP_VAR 0 Sheet1 A 2 means cell
229. egrality that will be tolerated Absolute Specify the absolute amount of violation from integrality that is acceptable in the Absolute text box Specifically if Z is the closest integer value to X X will be considered an integer if X lt Absolute Integrality Tolerance The default value for this tolerance is 000001 Although one might be tempted to set this tolerance to 0 this may result in feasible models being reported as infeasible Relative Specify the relative amount of violation from integrality that is acceptable in the Relative text box Specifically if Z is the closest integer value to X X will be considered an integer if X I lt Relative Integrality Tolerance xX ADDITIONAL COMMANDS 73 The default value for the relative integrality tolerance is 000008 Although one might be tempted to set this tolerance to 0 this may result in feasible models being reported as infeasible LP Solver When solving a mixed linear integer programming model the branch and bound solver solves a linear programming LP model at each node of the solution tree You may choose between the primal simplex dual simplex or barrier assuming the barrier option was purchased with your license solver for handling these linear programs The two options in the LP Solver box Warm Start and Cold Start control this choice of LP solver based on whether there is a starting basis warm start or no starting basis cold start Warm St
230. egrated in the Ribbon bar in Excel version 2007 OGFEED xIs Adjustable Best Integers Options Advanced loaie Unit v Goen Help About What sBest ToolBar Upgrade Register CheckUpdate Al Language Nutrients Minimum 4 Supplied Regd 0 0 Not gt 24 0 0 Not gt 0 7 0 0 Not gt 5 0 0 0 Not gt 21 0 6 CHAPTER 1 In Excel version 2002 the menu and toolbar show as follow HI Microsoft Excel HOGFEED XLS File Edit View Insert Format Tools Data Window WB LS Geneva 710 Eee U E E Adjustable ISS K kK Ace s BO Best WEMIN fe SUMPRODUCT C16 F16 C13 constransi A B C D E G Solve Integers SWINE amp ROSES Hog Farm Options Nutrients Per Unit Weight of Grain Nutri SES ltem 1 2 3 4 Supp Locate Nutrient A 2 2 ee Nutrient B 1 4 0 0 S Nutrient C 2 3 Ia i About What sBest Nutrient D 12 0 i 41 8 ToolBar Upgrade Cost Bushel 35 00 50 00 30 00 95 00 Register Percentage IEN CheckUpdate of Blend 0 0 00 00 0 0 No D Language Help 1 2 2 A 5 6 fi 8 9 Dual Value 1 00 1 00 1 00 1 00 19 20 21 M 4 gt A HOGFEED Ready GETTING STARTED 7 The What sBest Interactive Environment What sBest offers two interactive methods to access commands the What sBest menu and toolbar This section describes how to effectively use this interface In Excel version 2010 both the menu
231. ells C5 and D5 so as to maximize profit G6 without allowing your Total Usage E15 E17 to exceed the number of components in stock G15 G17 For example a sensible production plan might be to make as many Deluxe models as possible since these yield the highest per unit profit Then with what is left make as many Standard models as possible This production plan would use all 50 Deluxe computer towers E16 20 Standard computer towers E15 and all 120 hard drives E17 currently in stock It would result in a total profit of 31 000 G6 However this solution can be improved by using What sBest 12 CHAPTER 1 The ABC s Now let s apply the ABC s to this spreadsheet to show how What sBest provides you with the best possible answer A Determine Adjustable Cells In this example we first want What sBest to adjust the value for Quantities to Produce for both models of computer in cells C5 D5 What sBest requires that numbers be placed in all adjustable cells You may simply enter zeroes in these cells although any number will suffice Next specify that cells C5 and DS are adjustable by selecting both cells and either 1 choose Adjustable from the WB menu and click OK or Kx KX 2 use the Make Adjustable toolbar button or to Remove Adjustable If you use the WB menu to access the Adjustable dialog box you ll notice that the Adjustable dialog box has C 5 D 5 the current selection entered i
232. els the ranges over which dual values are valid may be very small Before basing a pricing or purchasing decision on a dual price or reduced cost in a nonlinear model you should test the returned value by making the specified change to the variable or constraint using Adjustable Remove Adjustable on the variable and re solving SAMPLE MODELS 239 Box Design File name BOX XLSX TYPE NONLINEAR OPTIMIZATION The Problem in Words As a manufacturer of electronic equipment you must design a cabinet for your new product that meets the specifications of various departments within your organization at minimum cost Background Your engineering department has determined that the equipment requires a volume of at least 1512 cubic inches and that a minimum surface area of 888 square inches will suffice to dissipate heat Marketing will find it easiest to sell the finished device if the footprint of the cabinet is no more than 252 square inches Finally your designers have decreed that for aesthetic reasons the ratio of height to width should be 618 1 That is between 518 and 718 inclusively The sheet metal from which the cabinet will be made costs 05 per square inch Extra labor required on the front and back panels raises the cost for these components to 10 per square inch Objective of Optimization The objective is to determine the specifications of the box with the smallest production cost 240 CHAPTER7
233. em the ABC s Adjustable Best and Constraints A Identify Adjustable Cells The adjustable cells are the cells in the worksheet that What sBest can adjust in its quest for a solution In mathematical programming terms these are called variables These can be defined using either the menu command Adjustable or the toolbar button KY Ex B Define Best The best cell is the goal or objective of your solution Typically this is to maximize or minimize an adjustable cell or some function of the adjustable cells What sBest allows only one best cell in the model No best cell is needed when equation solving or goal seeking The menu command Best or the toolbar button Je L can be used to define the best cell C Specify Constraints The constraint cells identify any limitations in a model For example Raw materials used in production must be less than or equal to raw materials on hand The constraint cells enforce these restrictions They are defined by either the menu command Constraints or the toolbar buttons lt gt Once you ve specified the ABC s you can solve your worksheet model and find the best answer to your problem 10 CHAPTER 1 TUTORIAL This tutorial will introduce you to What sBest by showing you how to solve a simple linear problem called the XYZ Production Problem You should open the sample problem file XYZ XZLS located in the WB directory in your C drive so you
234. equirements 3 T TANH 187 TEMPFILE 433 Textbook 2 The ABC s 19 The Building Block Method 314 Time to Relative 75 Tolerance command 76 Tolerances 75 Toolbar 7 ToolBar command 121 Tools Add ins 4 Tools Macros 133 Tools References 133 Trademarks 2 Traffic Congestion Cost Minimization 364 TRAFFIC XLSX 364 TRANSPOSE 187 Trend 279 Trial Version 120 Tries 32 37 Truck Loading 367 TRUCK XLSX 367 TRUE 187 Tutorial 10 on the VBA Interface 133 Types 70 U JNBOUNDED 201 433 INDEFREF 434 pgrade 122 pgrading via New License Key 122 pper Range 88 sage Guidelines for Dual values 89 for Omit 102 Use of Bound 65 C CGG GGO Vv Valid Ranges for Dual Values 94 Variable Bounds 65 Variables 120 integer 2 42 Variance 273 VBA 131 133 137 153 154 160 VBA Interface Procedures 137 VBA Interface Tutorial 133 Ver 2 0 62 Ver 3 0 62 View Toolbars Customize Menu 8 Visual Basic Editor 133 for applications VBA 129 86 VLOOKUP 188 418 VLOOKUPWBSOLVER XLS 418 W Warm Start 73 Waste Minimization in Stock Cutting 321 WB formula 26 WB Menu 4 7 19 WB Solution 34 54 WB Status 34 53 WB _ Histogram 56 214 WB Stochastic 56 213 220 wbAddAdjustableStyle 138 wbAddBestStyle 138 wbAddWBMenu 138 wbAdjust 139 wbBest 140 WBBIN 141 WBCARDFORM 435 wbConstraint 142 INDEX 459 wbDeleteReports 143 wbDeleteWBMenu 143 WBDUAL Formul
235. er arguments wbSetStringSupport This routine can be used to set the What sBest string support option seen in the String Support dialog box The first argument is required the others are optional For additional discussion of the options available through this routine see the section entitled Advanced String Support Syntax wbSetStringSupport AdvStringSupport AdvStringLength Argument Required Default Description AdvStringSupport Yes 0 False AdvStringSupport is a True False flag indicating whether or not to use this support AdvStringLength No 20 AdvStringLength is an integer between 1 and 255 indicating the maximum string length 186 CHAPTER 4 Error Codes Description AdvStr_BadAdvStringSupportArg Bad AdvStringSupport argument AdvStr_BadAdvStringLengthArg Bad AdvStringLength argument Examples If one wished to set all the string support options the simplest syntax would be wbSetStringSupport True To set one or more options named arguments should be used wbSetStringSupport AdvStringSupport True Note The AdvStringSupport argument should be set to TRUE in order to use the AdvStringLength argument 5 Functions and Operators What sBest supports many mathematical and logical functions of Excel and also adds some functions of its own List of Supported Functions and Operators The following Excel functions and opera
236. er of the Swine amp Roses S amp R Hogfarm and your hogfeed must meet certain minimum requirements for four nutrients A B C and D to assure fast growth and large robust animals SAMPLE MODELS 231 Background The nutrients are found in four different grains Each grain contains a different combination with a price that fluctuates with market conditions The nutrient content by grain is shown in the following graph Nutrient Content By Grain Objective of Optimization The objective is to determine how much of each grain S amp R should buy at today s prices to meet their nutritional requirements at lowest cost This always has a chance of being wrong if price fluctuates unexpectedly 232 CHAPTER7 The Worksheet Let s open the HOGFEED sample file and look over the ABC s The HOGFEED Worksheet Before Optimization SWINE amp ROSES Hog Farm Nutrients Per Unit Weight of Grain Nutrients Minimum ltem 1 2 3 4 Supplied Regd Nutrient A 2 2 3 4 7 2 1 5 0 0 Not gt 24 Nutrient B 1 4 1 1 0 0 0 8 0 0 Not gt 0 7 Nutrient C 2 3 5 6 11 1 1 3 0 0 Not gt 5 0 Nutrient D 12 0 11 9 41 8 52 1 0 0 Not gt 21 0 Cost Bushel 35 00 50 00 80 00 95 00 Percentage Unity ky of Blend 0 0 0 0 00 0 0 Not Dual Value 1 00 A Determine Adjustable Cells The cells What sBest is free to change in this model are the Percentage of Blend contributed by each grain C16 F16 B Define Best
237. er that isn t available through the normal What sBest interface If this is found to be the case you will be provided with a list of settings by a LINDO Systems Technical support representative after investigation of the model Note The Advanced Parameters command should only be used with guidance from LINDO Systems technical support The parameter names and their values are entered using the following dialog box Parameters ox The Advanced Parameters command should ox only be used under the guidance of LINDO Systems technical support Real valued parameters have names beginning with WBLS_DPARAM while integer valued parameters names begin with WBLS_IPARAM Rm The WB Options Reset To Default command will not reset the custom parameters defined in this window Instead you must delete them individually using the Delete button pictured above Users accessing What sBest via VBA macros can set customer parameters by using the standard VBA code for instance ActiveWorkbook Names Add Name WBLS _DPARAM ANYPARM RefersToRIC1 1 23 118 CHAPTER3 Locate The dialog box posted by the Locate command appears as follows Locate sl Locate Adjustable v cells Using Method Identify One by One D Across Entire Workbook DI pe cma Specify the type of cell you would like to find in the Locate drop down box to search the worksheet or workbook for Adjustable Best Constraint Vi
238. er to choose from TROUBLESHOOTING 425 SPCORR Stochastic WBSP_CORR Format Using the stochastic feature the user has to associate a distribution to a random cell either discrete or continuous The WBSP_CORR function is used to request a one argument correlation If the arguments to this function are not correctly specified the following error message will be displayed on the WB Status tab Error Message d t RROR Stochastic WBSP_CORR Format Help Reference SPCORR A WBSP_CORR cell with 1 argument is incorrectly formatted Correct the formula and the validity of the arguments in the cell below The format should be WBSP_CORR arg1 cells where argl is a reference and elle must refer to random cells cell address Suggestions There is an incorrectly formatted distribution function cell in the model A correlation function cell must use the format WBSP_CORR arg1 cells where arg is a cell reference and cells is a reference to the cells you want to apply this correlation type Kendall Pearson or Spearman The cell reference should be to a random cell For more information on using the stochastic feature refer to section Advanced Stochastic Support SPDIST1 Stochastic WBSP_DIST 1 Format Using the stochastic feature the user has to associate a distribution to a random cell either discrete or continuous The WBSP_DIST function is used to request a one argument distribution If t
239. erns is tallied using formulas in B9 E9 B11 E11 contain constraints forcing these requirements to be met The formula for 15 End Waste in B11 for example is WB B9 gt B10 This ensures that actual footage B9 is at least equal to the required amount B10 Now you re ready to solve the model The CUTSTOCK Worksheet After Optimization Cc D Feet Cut Edge Waste In Pattern Ins 1 483 50 4 2 901 2 0 00 0 0 0 447 88 0 0 1 725 00 10 10 875 Cut 1450 3000 1208 5 Need 1450 967 3000 1020 End Waste In Stock Width Inches 0 188 5 Inches 100 0 4 948 End Waste Cost Per Inch Ft 0 75 Total End Waste Cost 4 943 Total Edge Waste Cost 13 776 Edge Waste Cost Total Waste Cost 18 724 Per Inch Ft 1 50 The solution to the CUTSTOCK model is a Total Waste Cost of 18 724 13 776 Edge Waste E18 and 4 948 End Waste E17 Rounding may give a slightly different answer Here s a schematic of the solution 326 CHAPTER 7 Note If multiple raw material types were available this model would be reformulated to minimize total material costs Otherwise very expensive material might be purchased although it would be used efficiently Another model might also take into account the value of End Waste that could be re entered in inventory SS 35 ag A n dg Pattern A Ge 48d feet E a ag ag ag ag Patter C d4 feat 35 a5 15 ria name D Tau Test Qe 129
240. es get one out of two weekends off and nobody works more than 4 days at a stretch Cell L16 contains the average daily cost per full time employee and cell L17 contains the average daily cost per pool employee These estimates should be based on accounting and payroll data and can be updated on an as needed basis A Determine Adjustable Cells The adjustable cells are the number of people scheduled according to each work pattern P5 P16 Pool employees can be added to the model by making cells R18 AE18 adjustable and re solving SAMPLE MODELS 349 B Define Best The best solution is the one that results in the minimum total cost in cell L19 Cell L6 contains the aggregate requirement level L8 and L9 contain the minimum cost staffing levels with breakdown between full time and pool employees Cell L11 contains the total employees scheduled and cell L19 is the objective of optimization the total cost figure C Specify Constraints The constraints in cells D5 D11 and D13 D19 require that the employees scheduled C5 C11 and C13 C19 for each day of the scheduling period be greater than or equal to staffing needs for that day E5 E11 and E13 E19 With the ABC s in place go ahead and solve the model The FIXED1 Worksheet screen 1 After Solving No Pool STAFF SCHEDULING Fixed Shift Scheduled Recommended TOTAL FTEs AND POOL DAYS Sun 6 gt 6 Mon 10 gt 8 Recommended FTEs Tue 8 gt Scheduled FTEs Wed 12 gt S
241. est will not perform any linearization What sBest defaults to using Solver Decides The linearization process may add a considerable number of constraints and variables to the mathematical program generated to optimize your model The number of such constraints and variables added is noted in the WB Status report Delta Coefficient You can specify how closely you want the additional constraints added as part of linearization to be satisfied with the Delta Coefficient text box What sBest defaults to a Delta Coefficient coefficient of 0 000001 Big M Coefficient Specify the Big M value to use during linearization with this text box When What sBest linearizes a model it will add forcing constraints to the mathematical program generated to optimize your model These forcing constraints are of the form f Adjustable Cells M ey where M is the Big M Coefficient and y is a 0 1 variable The idea being that if some activity in the adjustable cells is occurring the forcing constraint will drive y to take on the value of 1 Given this if we set the Big M value to be too small we may end up with an infeasible model Therefore the astute reader might then conclude that it would be smart to make Big M quite large thereby minimizing the chance of an infeasible model Unfortunately setting Big M to a large number can lead to numerical roundoff problems in the solver resulting in infeasible or sub optimal solutions So getting a good
242. et The user needs to select the current worksheet to convert then invoke the feature The feature will apply the Adjustable Style for the decision variable cells the WBMIN or WBMAX Name for the Objective cell and create the WB functions in a separate tab WB Constraint Sheet at the end of the workbook The user may need to reset the tolerances via the numerous Additional Commands dialog box for the type of the model What sBest will then detect automatically the type of the model once the user starts the solver via the Solve button For instance the workbook has several tabs but you want to convert the model in Sheet Select the tab Sheetl go to Advanced and Convert Model Format via the WB menu and press OK You will see the Adjustable cells with a blue like font the Best cell with a blue background and the tab WB Constraint_Sheet1 for the constraints functions such as WB Sheet1 A1 lt Sheet2 C1 Note This feature creates a new model format but does not remove the existing format Any cells difference that could not be transported will be listed in the status report This feature cannot be invoked via the VBA interface ADDITIONAL COMMANDS 117 Advanced Parameters This feature allows you to set arbitrary parameters in the Lindo API which is the solver engine used by What sBest In order to solve your model successfully there may be rare occasions when you will need to set an API paramet
243. f uncertainties in the best possible way Restricting ourselves to linear multistage stochastic programs for illustration we have the following form for a multistage stochastic program with 7 1 stages Minimize or maximize coxo Eileixi Elex Erler Such that Ao0X0 bo A 1 10 0 AC 1X1 bei Ate 2 20X0 dien O2 21 1 dien W2 22X2 b 2 2 A Q Or 7X dien OFX A 1 zl to D Orr Lo lt x0 lt Uo La sum S Ua Lien oclnz XT lt U Or r where en 2 represents random outcomes from event space Q Q up to stage t A Q Oy is the coefficient matrix generated by outcomes up to stage for all p 1 6 t 1 T C t is the objective coefficients generated by outcomes up to stage for all t 1 T b 1 fl is the right hand side values generated by outcomes up to stage t for all t 1 T L q and U a ol are the lower and upper bounds generated by outcomes up to stage t for all t 1 7 is one of the relational operators lt or gt and Xo and x x Wo are the decision variables unknowns for which optimal values are sought The expression being optimized is called the cost due to initial stage plus the expected cost of recourse Note What sBest can solve linear nonlinear and integer multistage stochastic programming problems 208 CHAPTER 6
244. f values may help shorten the solution time For example assume you know that reasonable values for an adjustable cell A1 are between 500 and 1000 Values outside of this range are not mathematically impossible they just don t make sense Add the constraints WB A1 gt 500 and WB A1 lt 1000 This ensures that What sBest won t waste computation time investigating solutions you would view as meaningless Similarly adding bounds to restrict expressions from investigating values at or very near undefined regions i e division by zero may reduce computation time and increase the reliability of the solution Scale the Model to a Reasonable Range of Units Try to model your problem so that the units involved are of similar orders of magnitude Mathematically manipulating numbers that differ in size by a large amount can increase the computation time and may introduce problems due to round off error 204 CHAPTER 6 For example consider a financial problem with equations expressing an interest rate of 8 5 085 and budget constraints of 12 850 000 The difference in magnitude between these numbers is on the order of 10 to the 9 1 100th compared to 10 000 000 A difference of 10 to the 4 or less between the largest and smallest units would be preferable In this case the budget could be expressed in units of millions of dollars That is 12 85 to represent 12 850 000 This lowers the difference in magnitude of the units of thes
245. fathom i e drop from consideration branches higher in the branch and bound tree These improvements should dramatically speed solution times on most integer models Max Passes Set the maximum number of passes allowed by the integer pre solver to any non negative integer value with the Max Passes text box The integer pre solver makes iterative passes through a model determining appropriate constraint cuts to append to the formulation In general the benefit of each successive pass declines At some point additional passes will only add to the total solution time Thus What sBest imposes a limit on the maximum number of passes The default limit is 200 passes 70 CHAPTER3 Relative Limit Set the relative constraint cut limit by changing it to any desired fractional value with the Relative Limits text box Most integer programming models benefit from the addition of some constraint cuts However at some point additional cuts take more time to generate than they save in solution time For this reason What sBest imposes a relative limit on the number of constraint cuts that are generated The default limit is set to 0 5 times the number of true constraints in the original formulation Types Enable or disable any of the different strategies What sBest uses for generating constraint cuts with the checkboxes in the Types box It is beyond the scope of this help file to go into the details of the various strategies Interested rea
246. fers the lowest risk depending on which of the three measures you select Variance is a measure of the fluctuation of the expected return However this measure of risk considers a scenario that returns 5 above the target to be as desirable as a scenario that returns 5 below the target You re probably only worried about the risk that the return will be below your target return Both Semi variance and downside risk look only at the risk that your return will be below the target Downside risk minimizes the difference between the target and returns below the target The semi variance minimizes the difference between the target and the square of the returns below the target Therefore it puts a higher weight on larger differences below the target Objective of Optimization The objective is to determine the percent to invest in each asset while minimizing the risk of the entire portfolio using the risk measure that is most important to you 274 CHAPTER7 The Worksheet Let s look at the supplied PORTSCEN sample file The PORTSCEN Worksheet Before Solving l K tzend Asset 1 Asset 2 Asset 3 Return Difference Forcing 0 0 00 0 0 Over Under Constraints 130 0 122 5 114 9 0 0 0 0 0 0 Not 110 3 129 0 126 0 S 0 0 0 0 0 0 Not 121 6 121 6 141 9 i 0 0 0 0 0 0 Not 954 728 92 2 3 0 0 0 0 0 0 Not 92 9 114 4 116 9 i 0 0 0 0 0 0 Not 105 6 107 0 965 0 0 0 0 0 0 Not 103 8 13
247. fers_to is the range or address of the cell s to be specified as SOS NoErrDialog No Any argument passed here causes all What sBest error dialog boxes from the wbIntegerSos routine to be suppressed Instead the error number is delivered in this return argument Pass an integer to use any possible returned What sBest error number Useful in an embedded application of What sBest MACROS THE VBA INTERFACE 159 IntSos_BadTypeSosArg Bad TypeSos argument IntSos_BadRefersToArg Bad Refers_to argument IntSos_ProtectedSheetError Unable to set integer cells on a protected sheet Remarks IntSos_BadArgListArg Bad ArgList argument Integer variables can dramatically increase the solution time Example To constrain cell F6 to be a general integer using the range name staff enter wbIntegerSos 1 Sheet1 B 1 Sheet1 C 1 C 3 Sheet1 A 1 The cell Al then contains the function WBSOS 1 Sheet 1 B 1 Sheet1 C 1 C 3 WBKBEST For additional discussion of the functionality provided see the section entitled Options Integer Solver and Usage Guidelines for K Best Solutions which refers to the dialog box that calls this procedure This routine is used to build WBKBEST ranges which are areas of the worksheet that What sBest will see the trade off cells while solving If you wish to delete a WBKBEST range name using VBA instead of the Delete button on the K Best Trade off Cells dialog box you can use Activ
248. ff OO General Linear Solver Nonlinear Solver Global Solver Integer Pre Solver Integer Solver Stochastic Solver Reset to Default From the dialog boxes posted by these commands you can set the options by simply selecting the options you would like and clicking the OK button If you wish to restore the options to their default values you should use the WB Options Reset to Default command to return all What sBest options to their default values ADDITIONAL COMMANDS 49 Options General The dialog box posted by the Options General command appears as follows General Options k Solver Feasibility Tolerance Limits Iteration Limit iterations Runtime Limit seconds Display Constraint Indicator Iw Auto Select Free Int Omit Ranges IT Minimize Excel on Solve J Hide Status Window on Solve m Linearization Degree m Reports Location and Warnings Status Report Always Created v Beginning Solution Report Never Created D Warnings JW Nonlinearity Present IV No Best Cell Reference to Blank Cell Je Unsupported Function Iw String Argument Present 50 CHAPTER3 The General Options dialog box contains options that let you control the ways in which What sBest displays processes and saves information All of the options in the General Options dialog are saved with the workbook and are therefore called workbo
249. forced into the solution even though it is not optimal to do so If the adjustable cell is a positive number the dual value will always be 0 The dual value of an adjustable cell is often referred to as the reduced cost because it is the amount by which the cost of producing that adjustable item would have to be reduced in order to make it profitable The dual values for the adjustable cells can be placed in the model by clicking on the cells to put the information in choosing Advanced Dual entering the adjustable cells in the For Cell Range edit box and clicking OK The dual values can be placed in any convenient location in the spreadsheet 92 CHAPTER3 For illustration we continue with the XYZ model changing the available number of hard drives G17 to 50 and re solving causing the number of Deluxes adjustable cell D5 to drop to 0 and the total profit to be reduced to 15 000 We also requested dual values on our two adjustable cells placing the dual values in cells C6 D6 After re solving we get the following results EEN El EE e B oH COMPUTER CORPORATION PRODUCTION n Product Standard Deluxe PROFIT Quantity to Produce T el Profit per Unit 300 500 Product Component Requi Components Quantity Required Total Standard Deluxe Usage Standard Tower 50 Deluxe Tower 0 0 lt Hard Drive 50 lt The dual value of adjustable cell DS is now 100 This means the value of the best ce
250. functions you will see that it contains the path of the What sBest add in file at the time the function was created For example a function of the form 56 CHAPTER3 WB A1 lt B1 has a prepended path and appears something like this C PROGRAM FILES MICROSOFT OFFICE OFFICE10 LIBRARY wba xla WB A1 lt C1 Or in Excel 2007 C PROGRAM FILES MICROSOFT OFFICE OFFICE12 LIBRARY LINDOWB wba xlam WB A1 lt C1 In the case of the sample model files the prepended path may not coincide with the path to your What sBest add in file For this reason Excel prompts you to update links and after declining this you should use the What sBest Update Links button in the General Options dialog box The Update Links command in What sBest finds the path to your What sBest add in file and then returns the equations to the previous form without a prepended path If you leave the paths in their incorrect form then Excel fails to find your What sBest add in file and it inserts an error code of REF into each of the cells having an incorrect path After updating is completed the workbook sheet should be saved Delete Reports Click on the Delete Reports button to remove the WB Status the WB Solution WB _ Stochastic and the WB _ Histogram reports from the workbook you have open The next time you solve the model new reports will be generated according to your specifications in the Status Report and the Solution
251. g box method offers an error checking interface for out of range and other errors The last two methods allow you the flexibility of entering complex expressions as well as cell references To learn how to use VBA to enter constraints see the section titled wbhConstraint Let s consider the following constraints a resource limitation constraint Advertising expenditures must be no more than lt 10 000 and a performance requirement constraint Advertising exposures must be at least gt 100 sew Vi Home Insert Page Layout Formulas Data Review View WB B6 lt D6 A B em o E Advertising Projection EXPENDITURE BUDGET r 0 00 lt 10 000 00 OW FP win ke EXPOSURES REQUIREMENT 6 of lt l 100 28 CHAPTER 2 In your What sBest model if an adjustable cell representing the advertising expenditure was in cell B3 and D3 was a fixed cell with a value of 10 000 you could enter the constraint formula WB B3 lt D3 anywhere in the spreadsheet to enforce the first constraint We ve entered it in cell C3 where it makes visual sense Likewise in cell C6 the formula WB B6 gt D6 enforces the second constraint What sBest would then optimize so as to satisfy these constraints With B3 selected as the objective to minimize the program will select at least 100 exposures that add up to the lowest possible expenditure With B6 selected as the objective to maximize the program will
252. g box that calls this procedure This routine builds WBFREE ranges which are areas of the worksheet where adjustable cells are allowed to be negative If you wish to delete a WBFREE range name using VBA instead of the Delete button on the Free dialog box you can use ActiveWorkbook Names WBFREEmyRangeName Delete This statement removes the WBFREE range name thereby returning the adjustable cells contained in the former range to normal status Syntax WBFREE FreeName Refers_to Argument Required Description FreeName Yes FreeName is a string that refers to the range being allowed to be negative Refers_to No Refers_to is the range or address of the cells to be allowed to be negative If omitted the default is the current selection Error Codes Description Free_BadRefers_toArg Bad Refers_to argument Free_ProtectedSheetError Unable to set free cells on a protected sheet Free_CreateError Error setting free cells Example To allow cell G8 to be positive or negative using the range name BuySell enter WBFREE BuySell G8 154 CHAPTER 4 WBINT For additional discussion of the functionality provided see the section entitled Integer which refers to the Integer dialog box having a General option button that calls this procedure This routine is used to build general integer ranges named WBINT The adjustable cells contained in these ranges will be set to non negativ
253. g exercise is to solve this model for different levels of Desired Return in cell D15 recording the portfolio variance at each step A graph of the resulting data illustrating the tradeoff between risk and return is called the efficient frontier SAMPLE MODELS 265 Portfolio with Transaction Costs File name PORTCOST XLSX TYPE NONLINEAR OPTIMIZATION Application Profile As the expected return and variances of each asset change the optimal portfolio is almost certain to change Adjusting your investments with every change may cheer up your broker but the commissions or transaction costs will slowly erode the value of your portfolio Typically there s a cost associated with each sale or purchase Using this model you need to take these transaction costs into account in calculating how to adjust your portfolio The Problem in Words According to your estimates your current holdings will yield a 12 2 return You are interested in adjusting your portfolio to lower the variance yet achieve a return of at least 9 5 Background Your portfolio currently consists of 50 of Asset 1 30 of Asset 2 and 20 of Asset 3 You pay a transaction fee at the beginning of the period that is a percentage of the amount bought or sold The transaction fee is 1 0 for Asset 1 1 5 for Asset 2 and 2 0 for Asset 3 You have estimated the expected return and the covariance terms for each asset Objective of Optimization The objective is to dete
254. g with LINGO or the LINDO textbook Optimization Modeling with LINDO both by Linus Schrage and available through LINDO Systems Non Optimization Models Non optimization models do not have a cell defining an objective function that is to be maximized or minimized A common purpose of such models is to solve an equation What sBest can find values that satisfy sets of equations or satisfy circular references Similarly What sBest can find values to accomplish a desired result commonly called goal seeking or backsolving The sample model Flow Network Modeling is an example of a non optimization model Models with Integer Restrictions Users frequently need to find answers that consist of whole units e variables expressible as integers In personnel scheduling for example the most efficient use of workers may call for 2 37 persons on a shift but it s difficult to find 37 of a person who is capable of meaningful work Therefore What sBest allows you to restrict values to be whole numbers or general integers Binary integer 0 1 restrictions can also be helpful in yes no or on off decisions GETTINGSTARTED 3 System Requirements To install and run What sBest check that you have the following Software e Microsoft Windows 7 Vista or Windows NT 4 0 Windows XP e Microsoft Excel 32 bit version 2002 or higher version 2007 2010 e or Microsoft Excel 64 bit version 2010 Microsoft NET F
255. ge scenario table seen in the Stochastic Support dialog box All arguments are required except for the error code For additional discussion of the options available through this routine see the section entitled Advanced Stochastic Support Syntax wbStochasticStageScenario CellRangel Place_in_cell RangeChoice NoErrDialog CellRange 1 CellRangel is a cell reference for indicating the range of the table Place_in_ cell Yes Place m cell is a cell reference to specify the cell in which to write the WBSP_STSC function 184 CHAPTER 4 RangeChoice NoErrDialog Error Codes Sto_BadCellRangelArg Sto_BadPlaceInCellArg Sto_BadRangeChoiceArg Examples To set a Stage Scenario table the simplest syntax would be wbStochasticStageScenario Sheet1 B 1 C 3 D1 SET 0 RangeChoice is a string SET to enter the function or NONE to delete the selected cell Any argument passed here causes all What sBest error dialog boxes from the wbStochasticFunction routine to be suppressed Instead the error number is delivered in this return argument Pass an integer to use any possible returned What sBest error number Useful in an embedded application of What sBest Bad CellRangel argument Bad PlaceInCell argument Bad RangeChoice argument wbUpdateLinks For additional discussion of the functionality provided by this routine see the description of Update Links in the section entitled Options Ge
256. ging the linearization option For more information on setting the linearization options see Selecting Options For more information on the linearization process see Overview of Mathematical Modeling For a list of the functions that can be linearized automatically by What sBest see Functions and Operators 294 CHAPTER7 Stratified Sampling File name SAMPLEWB XLSX TYPE NONLINEAR OPTIMIZATION Application Profile This model is useful wherever you need to determine an optimal statistical sampling strategy Opinion polling and market research are two areas in which models similar to this one might be used The Problem in Words In your work in market research for an appliance manufacturer you must determine the consumer s trend toward purchasing your product You need to determine two population means using the smallest least costly sample likely to give reliable results To reduce error you break up the population by income groups You re asking two questions e How much do you plan to spend on major appliances next year e Onascale of 1 to 100 how likely are you to buy our brand Background The population data looks like this Variance in Response Stratum Income Group Population Question 1 Question 2 1 50 001 400 000 5 1 2 35 001 50 000 300 000 5 2 3 22 501 35 000 200 000 5 4 4 under 22 500 100 000 5 8 You have determined your tolerable upper limits of variance for both sample means and you know the
257. han a local drive Your system runs on a non English version of Windows If you are updating from an earlier release the location that you specify for the program files may affect how your existing models are interpreted Carefully read the information displayed during installation as well as the Location of the Add In Files and Update Links section As the last step in installing What sBest Excel will open and you will find a new toolbar What sBest and menu WB specific to the What sBest program These are discussed in the The What sBest Environment Menu and Toolbar section If you do not find the WB menu present please go to Tools Add ins and check the What sBest entry or try reinstalling What sBest You may be prompted for a license key the first time the software runs You will find your What sBest license key enclosed with your CD Please enter the license key exactly as it appears including all hyphens If you have requested and received a license key via e mail you may copy and paste it directly into the space provided Ctrl C to copy Ctrl V to paste If you would like to run a trial version of What sBest 150 constraints 300 variables and 30 integers no license key is required and just click Trial here If you have misplaced your license key you will need to contact LINDO Systems GETTING STARTED 5 You can now start building and solving models using the What sBest menu and toolbar int
258. has already been installed If it has you will be asked you if you would like to remove the previous version or to write over it In the next step you will be presented with the standard license agreement for What sBest which you must agree to What sBest will then confirm that your system requirements are adequate If so you will then choose a destination directory for the What sBest sample files Next you will be given a choice between a Default and a Specified setup for the add in files The Default setup is recommended for English language versions of Windows This will install the What sBest add in files in Excel s Library folder On the other hand should you choose the Specified setup the add in files will be installed in a directory that you specify The Specified option is recommended for non English versions of Windows or network installations At this point What sBest has enough information to begin copying files Once the files are done copying a Finish button should appear After clicking Finish Excel will open You may receive a message that wbintr xls contains macros You must then select the Enable Macros button in order to finish installation 392 CHAPTER 8 Where are the What sBest add in files installed in my computer The Default installation will automatically transfer the program files to the Library subdirectory of your main Excel directory usually C Program Files Microsoft Off
259. he arguments to this function are not correctly specified the following error message will be displayed on the WB Status tab Error Message aE RROR Stochastic WBSP_DIST 1 Format Help Reference SPDIST1 A WBSP_DIST cell with 1 argument is incorrectly formatted Correct the formula and the validity of the arguments in the cell below The format should be WBSP_DIST arg1 cells where argl is a reference and cells must refer to random cells cell address 426 CHAPTER 8 Suggestions There is an incorrectly formatted distribution function cell in the model A distribution function cell must use the format WBSP_DIST argl cells where argl is a cell reference and cells is a reference to the cells you want to apply this one argument distribution The cell reference should be to a random cell For more information on using the stochastic feature refer to section Advanced Stochastic Support SPDIST2 Stochastic WBSP_DIST 2 Format Using the stochastic feature the user has to associate a distribution to a random cell either discrete or continuous The WBSP_DIST function is used to request a two argument distribution If the arguments to this function are not correctly specified the following error message will be displayed on the WB Status tab Error Message KE RROR Stochastic WBSP_DIST 2 Format Help Reference SPDIST2 A WBSP_DIST cell with 2 arguments are incorrectly formatted
260. he cells with unsatisfied constraints will display Not Not lt or Not gt Investigating their relationships to satisfied constraints may help you find the conflicts Next find all the constraints that are contradictory to the constraints identified in the first step Finally eliminate all the contradictory constraints except the ones that most accurately model your business situation If a nonlinear problem returns this error message and you suspect there is a feasible solution change the starting adjustable cell values to a feasible or near feasible solution and solve again Note that a very discontinuous problem may have a feasible solution that can only be found by starting at the solution point itself If you have the global solver option you may be able to use the multistart capability to automate the selection of alternate starting points You may also try invoking the global solver which assuming the model is not too large will always find a feasible point if one exists 406 CHAPTER 8 INFLARG Large Infeasibility If the infeasibility is too large the following warning message will be displayed on the WB Status tab Warning Message VARNING Infeasibility too large for a trusted solution Help Reference INFLARG Constraint violations exceeding tolerances were found Check the solution carefully before proceeding You may be able to resolve this error by decreasing the Feasibility Tolerance in the Gener
261. heet Hide Now the user will only see the Inputs and Outputs sheets A clever user would be able to determine that there is a Model sheet but he or she would not be able to view or modify it In the WB subdirectory there is a file called PRODMIX2 XLS which has the changes described above Note It is not possible to use Excel s Tools Protection Protect Structure command with What sBest This command does not allow the WB Status or WB Solution worksheets to be added to the workbook Tutorial on the VBA Interface This section provides a tutorial on the basics of running What sBest commands using VBA Visual Basic for Applications It includes detailed discussion and step by step examples of a variety of different common applications combining Visual Basic and What sBest Note The first step in running any What sBest from VBA is to create a reference to What sBest This reference is made by checking the WBA XLA box under Tools References from within the Visual Basic Editor The Visual Basic Editor is called via Tools Macros from the main Excel menu bar If you do not create this reference then any attempt to use the What sBest attributes or procedures will produce the error message Sub or Function not defined Building and Solving a Basic Model Let s revisit the XYZ Production problem discussed in the Tutorial of Getting Starting We will look at how the ABC s of a model can be specified using Visual Basic Un
262. here all you want to do is find a set of values for the adjustable cells that satisfy all the constraints In this case you should disable this warning by clearing the No Best Cell checkbox on the General Options dialog box NOCONST No Constraint Cells Most What sBest models will have at least one and most likely many constraints on the adjustable cells Constraints place limitations on the adjustable cells and or functions of the adjustable cells For instance resources will never be unlimited At some usage level resources will be exhausted Another example would be that orders must be filled A cost minimization model with no constraints to fill orders would minimize cost by producing nothing which clearly is not an optimal answer to the real world problem Constraints are used to indicate these types of limitations It would be unusual for a model to have no constraints So when What sBest encounters such a situation it displays the following warning on the WB Status tab Warning Message WARNING No Constraint Cells Help Reference NOCONST The solver recognized no valid constraints The model either contained no constraint functions or only constraint functions that did not depend on any adjustable cells If the model was developed for an earlier version of What sBest the constraints may need to be converted to the current format for constraint functions 422 CHAPTER 8 Suggestions You will need to add at least
263. hether to place any reports status and or solution report at the beginning of the worksheet tabs far left or at the end of all worksheet tabs far right The default is to insert the reports at the Beginning ADDITIONAL COMMANDS 55 Warnings These checkboxes allow you to select the warnings you wish to display in the status report The warnings are Nonlinearity Present warning that nonlinear functions or formulas are present in the model Nonlinear functions can adversely affect the speed and performance of the What sBest solver When possible nonlinearities should be avoided See Overview of Mathematical Modeling for a discussion of nonlinear relationships No Best Cell warning that no objective best cell is specified for the model The omission of a best cell may be intentional e g in a goal seeking model or unintentional e g in an optimization model Reference to Blank Cell warning that blank cells are referenced in formulas The reference of a blank cell in a formula may be intentional e g summing across a large range that contains a few blank cells or unintentional e g accidentally referencing A21 instead of A12 Unsupported Function warning that an unsupported function appears in the model The value of a cell containing an unsupported function will not be recalculated during the solution process String Argument Present warning that a text argument appears in a function that is ex
264. hould consider adding more memory to your machine This assumes that you haven t already added the maximum amount of memory possible to your machine In the case where you have already maxed out your machine you will need to reduce the size of the model MODERR Parsing Cell Formula If What sBest can t parse a particular cell formula the following error message will be displayed on the WB Status tab Error Message Parsing Cell Formula Help Reference MODERR What sBest was not able to parse the formula in the following cell cell address additional message Suggestions If a formula contains an unexpected inappropriate or unrecognized operation where a comma is expected instead of the multiplication operator What sBest may not know how to proceed Here is a list of possible causes The formula contains an unsupported combination of cells and areas e g multiplying a range of cells by a numeric value The formula may reference a cell in an error state ep NUM The cell s formula may exploit some feature unsupported by What sBest Some Excel functions need to meet a particular format to be supported by What sBest If the formula still results in this message and the cell is used only for reporting purposes you may wish to place it ina WBOMIT range to force What sBest to ignore the cell 420 CHAPTER 8 NAME Unsupported Name If What sBest can t parse a particular na
265. iables The dialog box posted by the Options Stochastic Solver command appears as follows Stochastic Solver Options l _poesessscencesesoussecseossecsecuasecssnscuccssvescesneossecseusenssssenssesses aan uc Modeling support Optimization Method Solver Deddes E Seed for Random Generator e Common Size per Stage f 0 V Sampling on Continuous Distribution Only Report Information Expected Value of Wait and See Model s Objective Expected Value of Policy Based On Mean Outcome Iw Expected Value of Perfect Information Expected Value of Modeling Uncertainty Print Scenarios Horizontally in Report e cet x This command allows you to set a number of options controlling the function of the stochastic solver Typically you want to check Use Stochastic and Print Scenarios Stochastic Modeling Support By checking this option the model will be processed as a stochastic model Otherwise What sBest will solve the deterministic core model even if some stochastic functions are set in the spreadsheet The default setting is False unchecked Optimization Method Indicates the method you would like the stochastic solver in What sBest to employ using the Optimization Method drop down box Possible choices include Solver Decides Free Deterministic Equivalent Nested Benders Decomposition Augmented Lagrangian Decomposition and Simple Benders Decomposition The Solver Decides
266. ice Office10 Library LindoWB for Excel 2002 or in C Program Files Microsoft Office Office14 Library LindoWB for Excel 2010 You should find the following files CONOPT3 DLL DFORMD DLL LIBIOMPSMD DLL LINDO7_0 DLL LINDOCU_11 DLL LINDOPR4 DLL LINDOWBEF DLL LINDOWBIL DLL LNDWBxxx LIC MOSEK6_0 DLL MSVCR71 DLL MXST32 LIB README WRI USERINFO TXT WBA XLA Excel 2003 or WBA XLAM Excel 2007 2010 WBINTR XLS WBOPT DLL WBOPTLINK EXE WBUNCHADD EXE The Help file will be stored in the same directory LINDOWBHELP CHM Finally sample workbooks will be installed in C WB The Specified installation differs from the Default installation in that it lets you choose the specific location for the What sBest add in files The 64 bit files are identified with the 64 inside the file name What can I do if I receive an error message What sBest displays two kinds of error messages those generated by Excel and those generated by What sBest itself Errors from Excel e g Illegal Operation or Runtime Error typically result from operational problems in the loading and running of What sBest You should check that your model is pointing to the right What sBest add in file and that there is no loss of links To do this check the Tools Add ins list and Browse to find What sBest If you are developing macros with the Visual Basic Editor and using What sBest functions verify that the Tools Refere
267. id any previously specified best cell If no best cell is specified in a model What sBest will try to find a solution that satisfies all constraints and relationships in the model without trying to maximize or minimize any particular cell This is referred to as goal seeking or backsolving If you solve a model in which no best cell has been specified a No Best Cell Specified warning appears in the status report after the solution process has completed This warning is intended to notify the user that the specification of a best cell may have been inadvertently omitted You may turn this warning on or off by the No Best Cell checkbox in the General Options dialog box posted by Options General command Cell range Specify the cell to be maximized or minimized in this text box by using the button on the right edge to bring up a cursor for cell selection Alternately you may accept the currently selected cell which is automatically selected by What sBest You may also type the cell range to be specified as the best cell directly into the text box ABCs 25 Constraints or 202 The dialog box posted by the Constraints command appears as follows Constraints Ko Left Hand Side LHS Right Hand Side RHS a ed gt al Stored in sei El op ma This command is used to specify the resource constraints in your model To define a constraint you must supply the left hand side of the constraint the right hand
268. ie aaa aa aa Eed bebe dee Ehe Ben 154 wobIntegerGard EE 156 WDINTEGEISEMUC EE 157 WDIMLEQG ESOS EE 158 EE 159 ELE NEE EE 160 wbbieset pttons Tofietauht AA 161 wbSetGeneralOptions sssssssssesnseenneetnsenrsstnsstnsstnsstnsstenstenssensnnenstensnensnensnnnsnnnsnnnnsnnnt 161 wbSetGlobalOptions iiniu iiaiai aia diedan ede 164 WbSetlntegetOptonS spini nitar auas aia naait daaa ia iaaea a doida daai aa 166 wGeilntegerbrefohverptions 169 weil InearCptlons 172 wbSetNonlinearOptions ssseesseeeeseesseeesersstnsstnsstnsstesstesstessrensrensnrensrensnenstnnsnnnnnnnnnnnnt 173 wGetGtochasttet tions 175 wGetGtochasttctGupport 177 WDSOlVe viii ein cele wa aia ee AL ee aa 177 WbStochastic Functio Naa ia aa aa aa a nal Ae ea 180 WDStochastiCHistOQram rissin iina iani da oiana a doidas siida adaa a oriana didaa 181 WbStochasticREpOr sidi ii iarri aa aid a eid a A alee idend diia 182 WbStochasticStageSCenario piieis etant ienaa aie adii aat iaer a ad iiaae aiaa aa 183 Ve ee Tut 184 WbSetFunctionSupportin advee etic diana i den ete 185 WDSeIStringSuppOMt i fat seeciels lett E 185 FUNCTIONS AND OPERATORS esccssseeceseeeeeeeeesneeenseeeeeeessaseaeenseeeseeeseseaesaseeeeeseeeseas 187 List of Supported Functions and Operators ttn nrnnsennetnnennnnnn 187 FRUINCUIONS ee Riet ees ad even Ee eege ee Eech ee ee Dede ech 187 Operators ege ee ates wie cea hia aia tee tert ath Ee 187 LIM ANIZ ATION ET 188 Global SOVET areia na E EE 188 Elle St
269. ield allows you to select any of the Adjustable cells that will be part of the set You can select multiple range by holding the Control Ctrl key Then click on Add to store them in the List of Selected Cells field List of Selected Cells This field will list all the cells that are part of an SOS or CARD set You can modify the list by clicking on Add or Remove buttons Place the Function in Cells Specify the host range of cell you would like to see the SOS or CARD function to be placed in the spreadsheet What sBest automatically fills this box with the current cell selection If the selection already contains a WBSOSx or a WBCARD function then What sBest will display the cells argument into the List of Selected Cells 46 CHAPTER3 Integers Semi continuous The Integers Semi continuous command allows you to support Semi continuous variables via the following dialog box Semi continuous S Lower Bound ooo Upper Bound oOo Select any cells multiple selection holding se Lx _ SAS1 List of selected cells select and change by Add or Remove Place the function WBSEMIC in cell Tan a ef Many models require certain variables to either be 0 or lie within some nonnegative range e g 10 to 20 Variables with this property are said to be semi continuous Modeling semi continuity in What sBest in the past meant having to add an additional 0 1 variable and two additional constraints The syntax for the WBS
270. iew of all objects procedures and attributes that are available to any macros you write for that object For example by selecting the What sBest add in object WBA XLA within the Object Browser you can view the particular attributes and procedures of What sBest that you can employ in a macro 129 130 CHAPTER 4 The Object Browser is a feature of the Visual Basic Editor and is opened from the Excel menu by Tools Macro Visual Basic Editor Having opened the Visual Basic Editor select View Object Browser to start the Object Browser From the Object Browser you designate What sBest as your object by selecting WBA XLA for the Library Workbooks instead of lt All Libraries gt If you do not see WBA XLA in the drop down list then you must check in the What sBest add in via the Excel menu s or Ribbon s Tools Add ins or re install the What sBest add in After you ve selected WBA XLA in the Object Browser you should see WBUsers among the Classes listed This class is your primary source of What sBest procedures As you select this class the right window should display all of the Members procedures available to you from that class Macro Recorder Excel s Macro Recorder can be a very valuable tool for creating macros of tasks in Excel Unfortunately it generally does not work well in conjunction with Excel add ins such as What sBest The Macro Recorder of Excel 97 and later versions is unable t
271. ile In the March 1952 issue of Journal of Finance Harry M Markowitz published an article called Portfolio Selection In it he demonstrated how to reduce the standard deviation of returns on asset portfolios by selecting assets with prices that don t fluctuate in exactly the same ways At the same time he laid down some basic principles for establishing an advantageous relationship between risk and return and his work is still in use forty years later The Problem in Words You re considering investing in three assets and historical data reveal that the return from each asset has fluctuated over time You want to reduce variability or risk by spreading your investment over the three stocks Background From the historical data you have calculated an expected return the variance of the return rate and the covariance of the return between the different assets Variance is a measure of the fluctuation in the return the higher the variance the riskier the investment The covariance is a measure of the correlation of return fluctuations of one stock with the fluctuations of another High covariance indicates that an increase in one stock s return is likely to correspond to an increase in the other A low covariance means the return rates are relatively independent and a negative covariance means that an increase in one stock s return is likely to correspond to a decrease in the other You have a target return of 15 What percentage
272. imize Total Profit A3 by inserting and adjusting the Quantities Produced B8 G8 for each product As you go through these What If projections be sure to use no more of any raw material than you currently have in your Starting Inventory As you near exhaustion of any raw material look at the Product Resource Requirements table B14 G19 for products that require relatively little of that material Also remember to use Profit Unit B6 G6 as a guide to the effect of incremental production of each product Judge each of your hypothetical solutions by the Total Profit A3 generated Once you arrive at a reasonable solution jot down its total profit SAMPLE MODELS 313 Now that you ve experimented on your own you re ready to solve the model The PRODMIX Worksheet After Optimization Home Insert Page Layout Formulas Data Review View Developer Addins 7 X E O pn mel LG Ea B c D E G G H l J K b r PRODUCT MIX Product 2 3 2 eS G Profit Unit 30 45 24 26 24 30 Quantity 120 0 220 160 20 50 Produced Product Resource Requirements Total Start Usage Inv 800 lt 800 1160 lt 1160 Wood Plastic 1780 lt 1780 1050 lt 1050 1240 lt 1360 1240 lt zb MAM E P sch E E eat Om P N ch 0 GA e ANN OOS WN ch Ah A N Fb oe 1240 The best possible solution to this problem is a total profit of
273. ing the returned solution apply as when interactively interrupting the solver For more details refer to section Solver Interrupt above LICCAP1 Constraint Limit If your model has more constraints than is allowed for your installation the following error message will be displayed on the WB Status tab Error Message Constraint Limit Help Reference LICCAP1 The number of constraints in this model number of constraints exceeds the limit ot constraint limit The model has more constraints than is allowed by your installation You will need to either reduce the size of your model or upgrade to a larger version Suggestions The model exceeds the constraint limit for your installation The capacity limits of your version may be found by running the WB About What sBest command One option is to remove constraint cells from the model until it satisfies the constraint limit Keep in mind that What sBest defaults to forcing adjustable cells to be non negative So any constraints that put a lower bound of zero on the adjustable cells are not required and may be removed Also if the workbook contains multiple independent models you can separate them into different workbooks in order to meet the limit 410 CHAPTER 8 If eliminating constraints is not an alternative then you may wish to contact LINDO Systems regarding a license upgrade to handle the additional constraints LICCAP2 Adjustable Cell Limit If your model has
274. ion Status No Feasible Solution Unbounded The message Solution Status UNBOUNDED is returned if without violating any constraints the value of the cell to be minimized can be decreased without limit or the value of the cell to be maximized can be increased without limit For details please see the topic Solution Status UNBOUNDED Numerical Error The message Solution Status NUMERICAL ERROR is returned if there was a serious error For details please see the topic Solution Status NUMERICAL ERROR Case 2 Non optimization Models If no best cell has been specified i e there is no objective function there are only two possible outcomes of an attempt to solve the model What sBest either finds a feasible solution or reports that a feasible solution was not found As with nonlinear optimization models if the model is nonlinear there may be a feasible solution that What sBest was unable to find You may want to consider re solving with different initial values in the adjustable cells 202 CHAPTER 6 Guidelines for Modeling with What sBest General Modeling Guidelines An example showing how to build a simple linear model is provided in the Tutorial in Getting Started In addition the Sample Models demonstrate how linear and nonlinear models can be constructed for a variety of applications As you construct your own model it may be useful to periodically solve the model and check the status report to determine the type
275. iption Omit_BadOmitNameArg Bad OmitName argument Omit_BadRefers_toArg Bad Refers_to argument Omit_ProtectedSheetError Unable to set omit cells on a protected sheet Omit_CreateError Error setting omit cells Remarks Cells that are in an omit range can t be referenced by any cell outside of an omit range Example To define an omit range D10 H20 using the range name Report enter WBOMIT Report D10 H20 MACROS THE VBAINTERFACE _ 161 wbResetOptionsToDefault For further discussion of the functionality provided see the section entitled Options Reset To Default This procedure has no arguments and resets all of the workbook options to their default values Syntax wbResetOptionsToDefault Error Code Description ResetOptionsToDefaultError Error in reseting options to default wbSetGeneralOptions This routine is used to set the What sBest general options These options constitute the majority of the options seen in the General Options dialog box All arguments are optional For additional discussion of the options available through this routine see the section entitled Options General Syntax wbSetGeneralOptions goFeasTol golterLimit goRuntimeLimit goIndSlack goAutoSelectFreeIntOmit goMinimizeExcel goHideStatusWindow goLinearizationDegree goLinearizationDelta goLinearizationBigM goStatusReport goStatusReportBegEnd goSolutionReport goWrmNonlinear goWr
276. irectory of your main Excel directory usually C Program Files Microsoft Office Office 10 Library LindoWB for Excel 2002 or in C Program Files Microsoft Office O ffice 14 Library LindoWB for Excel 2010 You will find the following files CONOPT3 DLL DFORMD DLL LIBIOMPSMD DLL LINDO7_0 DLL LINDOCU_11 DLL LINDOPR4 DLL LINDOWBEF DLL LINDOWBIL DLL LNDWBxxx LIC MOSEK6_0 DLL MSVCR71 DLL MXST32 LIB README WRI USERINFO TXT WBA XLA Excel 2003 or WBA XLAM Excel 2007 2010 WBINTR XLS WBOPT DLL WBOPTLINK EXE WBUNCHADD EXE The Help file will be stored in the same directory LINDOWBHELP CHM Finally sample workbooks will be installed in C WB The 64 bit files are identified with the 64 inside the file name The Specified installation differs from the Default option only with respect to placement of the add in files It first check if a Library subdirectory already exists but then the setup program lets you choose the most convenient place for locating the What sBest add in files As a reminder you might consider the following folders for What sBest add in files INSTALLATION DETAILS 445 LIBRARY the Library subdirectory of your main Excel directory usually a path ending in Program Files Microsoft Office Office14 Library LindoWB This LindoWB subdirectory is the recommended location for installing the What sBest program files XLSTART the XLSTART subdirectory of your main Excel directory Unlike u
277. ired Also if the workbook contains multiple independent models you can separate them into different workbooks in order to meet the limit If eliminating integer cells is not an alternative then you may wish to contact LINDO Systems regarding a license upgrade to handle the additional constraints LICCAP4 Nonlinear Adjustable Cell Limit If your model has more nonlinear cells than is allowed for your installation the following error message will be displayed on the WB Status tab Error Message RROR Nonlinear Adjustable Cell Limit Help Reference LICCAP4 412 CHAPTER 8 The number of nonlinear adjustable cells in this model number of nonlinears exceeds the limit ot nonlinear adjustable cell limit The model has more nonlinear adjustable cells than is allowed by your installation You will need to either reduce the size of your model or upgrade to a larger version Suggestions The model exceeds the nonlinear adjustable cell limit for your installation The capacity limits of your version may be found by running the WB About What sBest command An adjustable cell is considered nonlinear if it appears in a nonlinear manner in one or more cell formulas in your workbook As an example suppose that cells A and B are both adjustable cells Furthermore suppose that we have the following formula somewhere in the workbook SIN A1 3 B1 The SIN function is a nonlinear function so this formula causes A1 to be co
278. irline flight crews hospital and office staffs and restaurant personnel to name just a few The Problem in Words You are running a business with daily staff needs on varying work loads each day of the week Background The daily staff needs range from 110 to 190 people as shown below Staff Scheduling Needs by Day of Week 150 100 50 Tue Wed Thu Fri Sat Pea In addition there s a labor requirement Employees must work a five consecutive day workweek followed by two days off Thus the allowable shifts are Monday through Friday Tuesday through Saturday Wednesday through Sunday etc Each employee earns 200 per week Objective of Optimization The objective is to cover staff needs with five day shifts at minimum weekly payroll cost 334 CHAPTER7 The Worksheet Let s look at the STAFF worksheet included in the sample files Look it over for a minute and examine its layout and formulas to see how it has been designed STAFF SCHEDULING Staff Staff Starting S Needs This Da 180 160 150 160 190 140 110 v H gt gt gt gt vv U Total Employees 200 TOTAL cosT x A Determine Adjustable Cells The adjustable cells in this model contain the number of people whose five day workweeks begin on each day of the week These are shown in the Number Starting This Day column G7 G13 B Define Best The best solution is the one that minimizes total payroll costs In this
279. is Infeasible Case 4 MsgBox The model is Unbounded Case 5 MsgBox The model is Feasible Case 6 MsgBox The model is Infeasible or Unbounded Case 7 MsgBox The model is Near Optimal Case 8 MsgBox The model is Locally Optimal Case 9 MsgBox The model is Locally Infeasible Case 10 MsgBox The model is Cutoff Case 11 MsgBox The model is Numerical Error Case 12 MsgBox The model is Unknown Case 13 MsgBox The model is Unloaded Case 14 MsgBox The model is Loaded Case Else MsgBox Solution status unknown End Select Exit Sub MACROS THE VBA INTERFACE 135 Handler MsgBox The error description is amp wbError Err End Sub In the above example any line starting with a single quotation mark is a comment Note If Excel returns the error message Sub or Function not defined then What sBest needs to be set as an available reference To do this choose Tools References from the Visual Basic Editor and check WBA XLA in the list Now you should be able to run the procedure without error The procedure BuildXYZ shown above is very simple It sets C5 D5 to be adjustable cells makes the Total Profit in G6 the cell to maximize and constrains Total Usage E15 E17 to be less than the number in stock G15 G17 When these steps have been performed the macro code solves the model and displays the results To make this example more readable the ranges passed to the What sBes
280. is 0 or Solver Decides Algebraic Reformulation Control the extent of algebraic reformulation e g whether x y x is replaced by x y x with the Algebraic Reformulation drop down box The algebraic reformulation process is crucial in building a tight convex envelope to enclose the nonlinear nonconvex functions thereby helping to improve overall convergence Possible values are Solver Decides None Minimum Medium or Maximum The default setting for Algebraic Reformulation is Solver Decides Note The global solver needs the nonlinear option and the mixed integer option to operate 66 CHAPTER3 Options lInteger Pre Solver The dialog box posted by the Options Integer Pre Solver command appears as follows Integer Pre Solver Options xs SE Level 3 Cutoff Criterion Solver Decides D Cancel Probing Level Solver Decides gt Deddes Help Constraints Cuts Max Passes Relative Limits 200 0 75 Types This command allows you to set a number of options controlling the function of the integer pre solver Integer programming models are inherently difficult to solve This is due to the many possible combinations of values that the integer variables may assume In fact the possible number of combinations grows exponentially with the number of integer variables Given this rate of growth in complexity it doesn t take a whole lot of integer variables to create a problem that is very tough to so
281. itally signed you will need to enable this add in to run your model In some situations a time delay may occur after enabling the add in The default setting is False unchecked 104 CHAPTER3 Advanced String Support This feature allows you to build models using strings as arguments e g in functions such as IF VLOOKUP and SUMIF String support related settings are entered using the following dialog box String Support xs String Support Maximum StringLength Te Ss e _ or String Support By checking this option string arguments will be supported in the model Otherwise What sBest will treat strings as arguments with a numeric value of zero String support may make the model bigger so it may need more resources to be solved The default setting is False unchecked which means string arguments will be read as a 0 value number Maximum String Length This length can be set between and 255 By making this setting small you reduce the memory required by the string support feature however making the max string length too small may truncate two different strings such as New_York and New_Jersey make them appear to be identical and lead to incorrect results The maximum string length default value is 20 characters Note If the String Support option is set to FALSE then the Maximum String Length setting will be disregarded String operations are allowed only in v
282. kes unambiguous sense only for models with a stage 0 and a stage 1 If there are later random variables in stages 2 3 etc then there are complications For example for decisions in later stages we have seen the outcomes from the random variables in earlier stages so considering these random variables to take on their mean value does not make sense For models with additional stages beyond 0 and 1 EVMU will merely be an approximation of the true expected value of modeling uncertainty Note Computing these expected value statistics can be very time consuming for large models Print Scenarios Horizontally in Report This flag enables to print the scenarios along a row in the WB _Stochastic report so the reporting cells will be displayed along a column It is useful for models reporting numerous cells for a few scenarios The default setting is off meaning the scenarios will appear vertically in the report 86 CHAPTER3 Options Reset to Default This command resets all workbook options to their default values Such options can be seen in the Insert Name Define or Name Manager menu as WBxxx names This command does not restore options defined in the Advanced Parameter window to default values See the section entitled Options and Solvers for more information about workbook options Upon first use the Reset to Default command displays the following dialog box Reset to Default Warning ks goeecessseccssesesossesosossecesoese
283. l For additional discussion of the options available through this routine see the section entitled Options Nonlinear Options Syntax wbSetNonlinearOptions StratCrashInitial StartPresolve StratQuadraticRecognition StratSelectiveConstraint StratSLPDirection StratSteepestEdge StartingPoint OptimalityTolerance ItLimitSlowProg Derivatives SolverVersion Argument Required Default Description StratCrashInitial No 0 False StratCrashInitial is a True False flag indicating whether or not to use this strategy StratPresolve No 1 True StratPresolve is a True False flag indicating whether or not to use this strategy StratOuadraticRecognition No 0 False StratOQuadraticRecognition is a True False flag indicating whether or not to use this strategy StratSelectiveConstraint No 1 True StratSelectiveConstraint is a True False flag indicating whether or not to use this strategy StratSLPDirection No 1 True StratSLPDirection is a True False flag indicating whether or not to use this strategy StratSteepestEdge No 0 False StratSteepestEdge is a True False flag indicating whether or not to use this strategy StartingPoint No 0 False StartingPoint is a True False flag indicating whether or not to use this feature OptimalityTolerance No 0 0000001 OptimalityTolerance is a positive number indicating the optimality tolerance ItLimitSlowProg No 100 ItLimit
284. l Format 435 Error messages amp Warnings Semi continuous Cell Format 437 Error messages amp Warnings Special Ordered Sets Cell Format 438 Estimation Model 289 Event 33 Excel 3 4 131 137 EXLVER 399 EXP 187 Exponential smoothing 284 Exposure 302 Expressions convex 197 linear 193 nonlinear 194 Nonsmooth 198 Smooth 198 Extended Version 120 Extracting Data 33 FALSE 187 FAQs 390 feasibility tolerance 50 FIXED1 XLSX 345 454 INDEX FIXED2 XLSX 345 Flow 243 364 Flow Cover 70 Flow Network Modeling 243 FLOWNET XLSX 243 Forecasting Models 279 284 FORMULA1 400 FORMULA2 400 FORMULA3 401 Forward Analytical 62 Forward Differences 62 Free 22 Deleting designation 22 variables 22 Frequently Asked Questions 390 FUNCADDIN 402 FUNCMACRO 402 FUNCSERVER 403 Function 133 Functions 187 91 convex 197 linear 193 nonlinear 194 Statistical EXPOINYV 190 MULTINV 191 192 TRIAINY 189 UNIFINV 190 WBNORMSL 191 G GCD 70 General and Operational Problems 387 General integer variables 2 42 General Modeling Guidelines 202 General Options Dialog Box 49 Global 120 Global Distance 65 Global optima 200 Global Solver 64 Global Solver Options Dialog Box 63 Global Width 65 Goal seeking 201 Gomory 70 GUB 70 Help 119 Heuristics Level 67 Hidden cell 21 132 Hide model 132 histogram 211 HLOOKUP 187 188 418 HOGCHANC XLSX 235 HOGFEED XLS
285. l Run wbAdjust C5 D5 intError If intError gt 0 Then GoTo Quit End If Omitted code similar to above for wbBest wbConstraint and wbSolve as well as code for displaying the results of solve Exit Sub Quit Dim strError as String gobjExcel Run wbError intError strError MsgBox Error amp intError amp Description amp _ strError Exit Sub End Sub MACROS THE VBA INTERFACE 147 The error numbers and the codes stored in the wbErr structure are as follows wee Ee 0011 Best BasBenGhoneng SSCSCSC 20012 Best oere ege ze feon seansa OOOO 148 CHAPTER 4 EE E ze Jus BadRetesTows O s0064 Jus Eege 0072 in Proeareasneeer MACROS THE VBA INTERFACE 149 Semer Spang SSSSSCSC zeng unOpBecinearSoverNetodag zem eg Setage CS Sat eg BadsratPresonewrg 150 CHAPTER 4 S0155 InPreSovOpt BacDsaqoeoaionGuehy Set apen BacFowCoveGuig za Jaen BadGcDGuemy ze upon BadkrapsackCovercusag MACROS THE VBAINTERFACE _ 151 20199 are BasSechasteSamotConag EEN 20195 Jar BasSiochasceVEVPRg 20198 are BadStoorasteRepscentonizas 152 CHAPTER 4 0224 Jar Beamon SSSCSCS S 20225 ar eegene 20220 Jan opene zen esu MACROS THE VBA INTERFACE 153 WBFREE For additional discussion of the functionality provided see the section entitled Free which refers to the dialo
286. lRange3Arg Bad CellRangel CellRange2CellRange3 argument Examples To set a stage information to a variable cell the simplest syntax would be wbStochasticFunction WBSP_VAR 1 0 0 0 A2 A3 0 To set a distribution to a random cell wbStochasticFunction WBSP_DIST_LOGNORMAL 0 C2 D2 0 A2 A3 0 wbStochasticHistogram This routine can be used to set the stochastic histogram seen in the Stochastic Support dialog box All arguments are required except for the error code For additional discussion of the options available through this routine see the section entitled Advanced Stochastic Support Syntax wbStochasticHistogram NumberBins Arglist Place_in_ cell NoErrDialog NumberBins Yes NumberBins is an integer between 1 and 255 indicating the number of bins for the reporting cell 182 CHAPTER 4 references for indicating the cells to display in the stochastic report Place_in cell Yes Place_in cell is a cell reference to specify the cell in which to write the WBSP_HIST function NoErrDialog No Any argument passed here causes all What sBest error dialog boxes from the wbStochasticHistogram routine to be suppressed Instead the error number is delivered in this return argument Pass an integer to use any possible returned What sBest error number Useful in an embedded application of What sBest EE Examples To set several reporting cells so to display the histogram
287. lbar buttons displayed above This routine inserts a constraint formula within the selected cell Syntax wbConstraint LHS Direction RHS ConLoc NoErrDialog Argument Required Description LHS Yes LHS is the left hand side of the constraint equation Direction Yes This is a string to indicate what type of constraint lt less than or equal to gt greater than or equal to equal to lt c convex gt c concave or c convex For backward compatibility with earlier versions of What sBest Direction also accepts the numbers for lt 2 for and 3 for gt 4 for lt c 5 for c 6 for gt c 7 for None RHS Yes RHS is the right hand side of the constraint equation It can be a range or a value see the Remarks below ConLoc Yes ConLoc is the range to store the constraints NoErrDialog No Any argument passed causes all What sBest error dialog boxes from the wbConstraint routine to be suppressed Instead the error number is delivered in this return argument Pass an integer to get any possible returned What sBest error number Useful in an embedded application of What sBest Error Codes Description Con_BadLHSArg Bad LHS argument Con_BadDirectionArg Bad Direction argument Con_BadRHSArg Bad RHS argument Con_BadConLocArg Bad ConLoc argument Con_RangeSizeError LHS must be same size as ConLoc Con_Protec
288. le to find or read a valid license key Run the WB Upgrade command to install a valid license key Suggestions You may need to reinstall your license key with the WB Upgrade command Also make sure the license file LVDWB80 LIC is installed in the directory along the What sBest add in files typically Program Files Microsoft Office Office10 Library You may need to contact LINDO Systems to obtain a valid license key LICKEY2 Pending License Expiration If your license key will expire in the next day the following warning message will be displayed on the WB Status tab Warning Message WARNING Pending License Expiration Help Reference LICKEY2 Your What sBest license key will expire at the end of the day Suggestions You have a temporary license key that will expire at the end of the day Note that you will still be able to use What sBest after the license expires but it will revert to a demonstration version with limited capacity You may want to contact LINDO Systems to obtain a new license key 414 CHAPTER 8 LICKEY3 Expired License Key When your What sBest license key has expired the following warning message will be displayed on the WB Status tab Warning Message WARNING Expired License Key Help Reference LICKEY3 Your license key has expired You may wish to contact LINDO Systems to obtain a new license key Suggestions Your license key has expired You may continue on
289. lerance quite attractive The default value for the relative optimality tolerance is 0 00001 Note Generally speaking increasing the relative optimality tolerance is the change that will most likely improve runtimes on integer models ADDITIONAL COMMANDS 75 Time to Relative If an integer programming model is relatively easy to solve then we would like to have the solver press on to the true optimal solution without immediately resorting to a relative optimality tolerance discussed above On the other hand if after running for a while it becomes apparent that the optimal solution won t be immediately forthcoming then we might want the solver to switch to using a relative optimality tolerance The Time to Relative tolerance can be used in this manner This tolerance is the number of seconds before the branch and bound solver begins using the relative optimality tolerance So for the first n seconds where n is the value of the Time to Relative tolerance the branch and bound solver will not use the relative optimality tolerance and will attempt to find the true optimal solution to the model Thereafter the solver will use the relative optimality tolerance to confine its search The default value for the Time to Relative tolerance is 100 seconds Tolerances The Tolerances group box contains three miscellaneous tolerances for controlling the branching strategy used by the branch and bound solver The three tolerances are Hurdle
290. less specified otherwise a sample file named XYZVBA XLSM was copied to the WB subdirectory during installation Along with the main XYZ worksheet there are two VBA modules called 4BC s and Dual The ABC s module has three procedures in it BuildXYZ which uses Visual Basic to build and solve the model MaxDeluxe which changes the objective to maximizing the use of Deluxe computer towers and solves the new model and MaxDeluxe2 which uses the same objective as MaxDeluxe but imposes a restriction that profit must be greater than or equal to 32 000 You will notice that the code to build the model is very straightforward For information on the complete syntax for each of the functions refer to the section on Procedures The BuildXYZ procedure is listed below Sub BuildxXyYZ Dim lngSolutionStatus As Long Go to line label Handler for code that handles error On Error GoTo Handler Make Quantities to Produce C5 D5 Adjustable 134 CHAPTER 4 wbAdjust C5 D5 Make Total Profit G6 the best cell to Maximize wbBest G6 Maximize Constrain Total Usage E15 E17 to be less than Number in Stock G15 G17 wbConstraint E15 E17 lt G15 G17 F15 F17 Solve the model wbSolve lIngSolutionStatus Display the results Select Case lngSolutionStatus Case 1 MsgBox The model is Globally Optimal Case 2 MsgBox The model is Globally Optimal amp _ Range WBMAX Case 3 MsgBox The model
291. ll would be penalized in this case decrease by 100 for each unit increase in D5 In other words forcing the number of Deluxe computers from 0 to 1 would return a new solution with a profit of 14 900 15 000 100 assuming a valid range on the dual value of at least one see the Valid Ranges for Dual Values section below ADDITIONAL COMMANDS 93 In the window below we show the re solved model with the Deluxe adjustable cell forced to be 1 by putting the constraint WB DS5 1 in cell D7 As you can see the dual value is now zero because the variable is in the solution The quantity of Deluxe computers to produce could also be forced to one by choosing Remove Adjustable in the Adjustable dialog box or the toolbar button to make D5 into a non adjustable cell and entering 1 as a constant value x XYZ COMPUTER CORPORATION PRODUCTION PLAN Product Standard Deluxe PROFIT Quantity to Produce r Profit per Unit Components Quantity Required Number Standard Deluxe Usage In Stock Standard Tower 48 Deluxe Tower 0 lt 50 17 Hard Drive 50 lt 50 Note Itis important to note that the dual value of a particular constraint or adjustable cell assumes all other information in the model remains unchanged That is given the dual information you cannot predict the effect on the best cell of simultaneously changing two or more constraints 94 CHAPTER3 Dual Value of Zero and Multiple
292. llocated Each of the media sources offers various combinations of exposures per dollar spent for each of the different target groups Objective of Optimization The objective is to determine how much of each of the media sources should be purchased to meet the exposure requirements at lowest cost SAMPLE MODELS 303 The Worksheet Let s open the MEDIA sample file and look at its structure and formulas The MEDIA Worksheet Before Solving m Qu fejoprimatmenieuing _ H r Market and Medium Thousands Target Markets Dollars Group Group Group3 Group4 Group5 Group6 Allocated Media Times 10 Mirror 10 Tribune Herald Post Exposures 0 0 0 Met Not gt Not gt Not gt Not gt Not gt Not ss TOTAL COST Requirement 25 0 40 0 60 0 120 0 400 11 0 The exposures per dollar are listed for each media source and each media source in B9 G13 Notice that each media source reaches different combinations of target groups per dollar spent Cells B15 G15 show the minimum required exposures for each of the target groups A Determine Adjustable Cells The adjustable cells in this model are the Dollars Allocated to each media source cells H9 through H13 B Define Best The best solution is the one that meets Requirements at the lowest cost The Total Cost cell H17 shows that every change in Dollars Allocated will have a direct bearing on cost 204 CHAPTER7 C Specify Constraints The constr
293. lls you want the CARD on The cell reference should be to an adjustable cell For more information on range values refer to section Integer Special Ordered Sets WBDUFORM Dual Cell Format If requested What sBest can return dual solution information Dual values tell you how how sensitive the final solution is to various parts of the model The WBDUAL function is used to request dual values If the arguments to this function are not correctly specified the following error message will be displayed on the WB Status tab Error Message Dual Cell Format Help Reference WBDUFORM A dual cell is incorrectly formatted Correct the formula in the cell below The format should be WBDUAL cell value cell address Suggestions There is an incorrectly formatted dual cell in the model A dual cell must use the format WBDUAL cell value where cell is a reference to the cell you want the dual value on and value is a numeric quantity The cell reference should be to either a constraint cell or an adjustable cell You may use any numeric value for value What sBest will replace value with the actual dual value the next time you run the solver For more information on using dual values refer to section Advanced Dual 436 CHAPTER 8 WBLOFORM Lower Range Cell Format If requested What sBest can return ranging information Range values tell you over what range a particular dual value is valid The WBLOWER functi
294. lth2 G11 H11 D12 D14 D15 D16 into cell J25 MATHEMATICAL MODELING 219 The INVESTMENTCOLLEGE Worksheet After Optimization rn ae estmeint Planning for Going to College After 3 Periods wo investment options each stage Stocks and Bonds Ref Birge amp Louveaux 80 Goal for wealth at beginning of period 4 4 Penalty unit for wealth under goal 1 Utility of wealth unit over goal Step 1 Core model Growth factor Beginning Total Invest in WBS tocks Bonds Wealth invested Stocks Bonds WBS 55 55 0000 41 4793 13 5207 WBS 1 25 1 14 67 26272 67 2627 65 0946 2 1681 1 25 1 14 83 839905 83 8399 83 8399 0 0000 Step 1 25 1 14 104 79988 WBS WBS Under goal 0 WBS Over goal 24 799881 gt o Net utility 24799881 To be maximized 220 CHAPTER 6 The scenario by scenario report is generated on the WB Stochastic tab r Model G10 Model H10fModel D11 Model G11 Model H11 Model D12 Moc STOCKINVEST1BONDINVEST1 WEALTH2 STOCKINVEST2 BONDINVEST2 WEALTH3 UNI STAGE 1 STAGE 2 STAGE 2 STAGE 2 094582 03991 83 839905 094582 83991 83 839905 094582 42857 0 094582 42857 743215 42857 743215 42857 743215 743215 The scenario report we obtain is show above Our expected utility at the end of the three periods is 1 514085 This means that there were some outcomes under which our final wealth fell short of our target wealth of 80 This report gives complet
295. lve The integer pre solver carefully examines integer models using a number of heuristic procedures The goal of these procedures is to reduce the number of integer variable combinations that need to be examined In some cases the pre solver can deduce a model s solution without the need to pass it on to the full blown integer solver ADDITIONAL COMMANDS 67 The integer pre solver options available to the user are listed below Heuristics Level Use the Heuristics Level drop down box to control the level of integer programming heuristics These heuristics use the continuous solution at each node in the branch and bound tree to attempt to quickly find a good integer solution If an integer solution better than the incumbent is found then it is used to fix or tighten global and local variable bounds There are four options available None Low Medium and High The None option disables heuristics The remaining three options allow you to set the level of the amount of heuristics performed to a low medium or high level What sBest defaults to performing a high level of heuristics Heuristics Cutoff Criterion The Heuristic Cutoff Criterion is used to control the criterion for terminating heuristics Choices here are Solver Decides Time and Iterations Under the Time setting What sBest terminates heuristics after a certain amount of elapsed time The Iterations option terminates heuristics after a certain number of iterations I
296. lver in that they aren t continuously differentiable Both functions have kinks in their graphs at various points that the nonlinear solver is unable to see around With a fair amount of effort one could linearize this model by adding a number of additional variables and constraints However the linearization option in What sBest can do this for you automatically SAMPLE MODELS 293 For illustration we ran this model twice once with linearization off and once with it on using the two Linearization Degree settings of None and Maximum The adjustable cells were reset to 0 for each run Here are some comparative results Linearization Off On Adjustables 4 4 Constraints 0 61 Integers Bin 0 20 Variables 25 76 Nonlinears 20 0 Solution Time sec 1 lt 1 As you can see the model is considerably larger with linearization enabled because it has internally increased the number of variables from 25 to 76 integers from 0 to 20 and constraints from 0 to 61 The crucial observation however is that the number of nonlinears goes from 20 to 0 Thus the linearization completely removes all the nonlinearities from the model This allows What sBest to invoke its faster and more accurate linear solver which reduces total solution time down to less than 1 second In conclusion if your model has nonlinearities that can all be handled by the What sBest linearizer solution times can be reduced dramatically by enga
297. m the user It will contain data and formulas that are considered proprietary We now populate the Jnputs and Outputs sheets as follows First we copy all of the data from A5 G6 on the Model sheet to the Inputs sheet We now set the Profit Unit for each of the products to unlocked cells on this Inputs sheet We can then protect the Inputs sheet with the Tools Protect Sheet command and enter a password Turning to the Outputs sheet we copy and change the Quantity to Produce for each of the products to formulas that reference the Model sheet We can then protect the Outputs sheet from the user so nothing can be changed First select the Quantity to Produce cells and choose Format Cells On the Protection tab select Hidden leave the cells Locked so that the user can not see the formulas in these cells Then choose Tools Protection Protect Sheet and enter a password On the Model sheet we change the Profit Unit for each of the products to formulas that reference the Inputs sheet The last thing to do is to hide and protect the Model sheet To do this select the entire worksheet and choose Format Cells On the Protection tab select Hidden Then with the entire spreadsheet still selected choose Format Row Hide and Format Column Hide Next protect the sheet by giving the MACROS THE VBA INTERFACE 133 command Tools Protection Protect Sheet and enter a password The final step is to hide the Model sheet with the command Format S
298. mation File Upon request of Lindo Systems this USERINFO TXT file is created to contain the Machine Name the User Name and the Disk Identification Number This file should then be sent to Lindo Systems to create the full License Key Select Destination Folder Browse Create C Program Files Microsoft Office Office 14 Library LindoWB al wa cos ter Use this command to input your What sBest license key If you would like to run the downloaded trial version of What sBest 150 constraints 300 variables 30 integers 30 nonlinear variables and 5 global variables just click the Trial button or Cancel and you will be issued a 30 day trial license This is a fully functional license but is limited in capacity If you have already purchased a larger version of What sBest you will find your What sBest license key enclosed with your CD Please enter the license key in the text box exactly as it appears including all hyphens and press the OK button If you have requested and received a license key via e mail you may copy and paste it directly into the space provided Ctrl C to copy Ctrl V to paste ADDITIONAL COMMANDS 123 This command can also be used to upgrade What sBest to handle larger problems greater number of constraints variables and integers or to add options to solve nonlinear problems use the barrier solver on linear problems or use the global solver on nonlinear problems To upgra
299. may want to apply sampling only on the continuous with the option Sampling on Continuous Distribution Only via the Stochastic Solver options box Select Range Select the a two column range where the column is for the Stage in ascending order and the second column for the number of scenarios associated to the stage ADDITIONAL COMMANDS 111 Place function in cell Specify the cell in which to write the WBSP_STSC function Set button Place the function in the spreadsheet None button Delete the selected cell Page Step 4 reports l Stochastic Support reno Iw Use Stochastic Modeling Support Step 1 Step 2 Step3 Step 4 Select the cells to appear in the Stochastic Report The solution of an SP model can produce a lot of information You are probably interested in only a small portion of these results These features allow you to specify the cells for which to report the values in the WB Stochastic and the WB _ Histogram sheets Create Report Using the Function WBSP_REP The general format is WBSP_REP cells_to_be_reported e g WBSP_REP Sheet1 B 1 Sheet1 B 2 meaning the cells B1 and B2 of Sheet will appear in the Stochastic Scenario Report After solution a new tab WB Stochastic will be created It will have one column for every cell to be reported and one row for every scenario 112 CHAPTER 3 Select the checkbox to tell What sBest you would like a stochastic report created
300. mbers Required Mon Tue Wed Thu _ Fri Sat Sun Day 2 1 1 3 1 1 1 Eve 1 1 0 0 2 1 1 Nite 1 0 0 0 1 1 1 Your staff members Nixon Ford Bush and Reagan must each be assigned to five shifts for the week and have individually selected their preferred shifts on a descending scale of 5 to 1 These rankings appear on the Preferences worksheet of the model In addition to considering your staff s preferences you must abide by two rules in scheduling the shifts Staff members cannot work more than one shift in a single day After working a shift a staff member can t work the next two shifts SAMPLE MODELS 339 Objective of Optimization The objective is to maximize overall staff preference by assigning employees to the shifts they deem most desirable The Worksheet Let s open the ASSIGN sample file and look it over The model consists of four worksheets in one workbook The Model Worksheet Before Optimization MULTIPLE SHIFT ASSIGNMENT MODEL Preference Total 7 0 Mon Tue Wed Thu Fri Sat Sun 2 1 1 3 1 1 1 Not gt Not gt Not gt Not gt Not gt Not gt Not gt Mon Tue Wed Thu Fri Ga Sun 1 1 0 0 2 1 1 Not gt Not ze gt gt Not gt Not gt Not gt Mon Tue Wed Thu Fri Sat Sun 1 0 0 0 1 1 1 Hot gt gt Not gt Not gt Not gt 340 CHAPTER7 The Model worksheet shows the number of people required for each shift for the week and constraints ensuring the
301. me the following warning message will be displayed on the WB Status tab Warning Message WARNING Unsupported external value or range in the Name list Help Reference NAME Check name value or the validity of the link Name will be taken as a 0 number Name name Suggestions What sBest does not support names containing any of the following links to external workbooks formulas and symbols Verify all names using Insert Name Define remove any unused names or copy the external data directly into the current workbook You may need to call LINDO Systems for assistance NOAID No Adjustable Cells Adjustable cells represent activities that are under the solver s direct control In other words these cells have been identified as ones that are allowed to be changed by the solver during the optimization process If there are no adjustable cells in the model the following error message is displayed on the WB Status tab Error Message No Adjustable Cells Help Reference NOADJ No cells have been specified as adjustable You must use the Adjustable command to specify the cells you want What sBest to adjust to find the solution Adjustable cells must contain numbers Adjustable cells that are blank or contain formulas or text will be ignored by the solver Suggestions You will need to add at least one adjustable cell to the model See section The ABC e Three Steps to What sBest for information on how
302. mit and without violating a constraint The problem may be incorrectly formulated Be sure that the best cell has been correctly specified and all equations and constraints have been correctly formulated Refer to Help at Solution Status Unbounded Suggestions Check the formulation to ensure that the adjustable cells the best cell and the constraint cells have been properly specified and no constraints have been left out The returned worksheet will display a large positive or negative number in the best cell and may have large numbers in one or more adjustable cells This information can give you an indication of where formulation errors lie An incorrectly specified best cell or constraint i e Maximizing a cell that should be minimized or 434 CHAPTER 8 incorrectly specifying a greater than or equal to constraint as a less than or equal to constraint may be causing the problem UNDEFREF Undefined Reference If What sBest encounters formulas with REF type Excel errors the following error message will be displayed on the WB Status tab Error Message Undefined Reference Help Reference UNDEFREF An invalid cell reference or an invalid output value was found in the formula of the cell listed below You must correct any illegal references to continue Running the Update Links command from the WB Options General menu may correct the error cell address Suggestions This error can occur if a cell referenced by
303. more adjustable cells than is allowed for your installation the following error message will be displayed on the WB Status tab Error Message KE RR OR Adjustable Cell Limit Help Reference LICCAP2 The number of adjustable cells in this model number of adjustables exceeds the limit of adjustable limit The model has more adjustable cells than is allowed by your installation You will need to either reduce the size of your model or upgrade to a larger version Suggestions The model exceeds the adjustable cell limit for your installation The capacity limits of your version may be found by running the WB About What sBest command One option is to remove adjustable cells from the model until it satisfies the limit Also if the workbook contains multiple independent models you can separate them into different workbooks in order to meet the limit If eliminating adjustables is not an alternative then you may wish to contact LINDO Systems regarding a license upgrade to handle the additional constraints TROUBLESHOOTING 411 LICCAP3 Integer Cell Limit If your model has more integer cells than is allowed for your installation the following error message will be displayed on the WB Status tab Error Message E ERROR Integer Cell Limit Help Reference LICCAP3 The number of integer cells in this model number of integers exceeds the limit of integer limit The model has more integer cells than
304. ms the costs vary with the amount transported along each arc If you ve ever driven to or from a major city during rush hour you ve experienced this phenomenon As the number of cars on the road increases the cost in terms of time required of getting from point A to point B increases In many cases the cost does not increase linearly For example doubling the traffic on a lightly traveled road may not double the travel time but doubling the traffic again may effectively grind the flow of vehicles nearly to a halt The Problem in Words As Quartermaster of a military base you need to distribute uniforms from three warehouses to four intake centers within the base Background You know that the time required to go from a particular warehouse to a particular unit obeys the following formula Time Rate Flow 1 Flow Limit Where Rate time required to transport one unit if there is no congestion along this route Flow amount of product moving along this route Limit the maximum amount that can be moved along this route You also know the rates and limits for each route or arc in the network Objective of Optimization The objective is to ship all the uniforms to the Jntake Centers at minimum cost while satisfying demand at each center SAMPLE MODELS 365 The Worksheet Let s look at the sample file TRAFFIC The TRAFFIC Worksheet Before Solving TRAFFIC FLOW 100 0 100 0 100 0 300 T Not 9
305. mula depending upon an adjustable cell defining exactly what it is you want to optimize 24 CHAPTER 2 The two most common objectives of optimization models are to minimize cost or maximize profit but you can choose to maximize or minimize any adjustable cell or any equation that depends upon one or more adjustable cells Other frequent targets for minimization are waste conflict time required surplus and risk A maximization cell could be something tangible an equation representing total output or something less concrete such as a calculation to estimate employee job satisfaction or the effectiveness of customer service You may specify no more than one cell to be maximized or minimized Whenever you designate a best cell any previously specified best cell returns to being a non optimized cell In a scenario with multiple goals you should consider maximizing or minimizing one goal and constraining the others or combining goals into a single equation to be maximized or minimized For example if the goals for your factory are to maximize production output and minimize costs there are clearly trade offs between the two goals You can choose to 1 Maximize output while constraining costs to be less than a desired level 2 Minimize costs while constraining output to be greater than a desired level 3 Combine both goals into a single equation of output minus total cost and maximize the single equation None Select None to vo
306. mum price at which you should be willing to purchase an additional Standard computer tower ADDITIONAL COMMANDS 91 Note In What sBest we use the convention of having dual values on constraints return the rate of improvement in the objective for an increase in the constraint s right hand side value A positive dual implies that the objective will improve when increasing the right hand side while a negative implies that the objective will suffer On maximization problems as we have shown above a positive dual value means that increasing the right hand side of the constraint will cause the optimal objective to improve or increase On the other hand for minimization models if the sign of the dual is positive then the optimal objective value will decrease as the right hand side increases Note Ifthe constraint on Deluxe computer tower usage WB E16 lt G16 is not tight i e E16 is not equal to G16 but is less than 50 then the dual value is 0 Increasing the right hand side from 50 to 51 would not affect the optimal solution and consequently would not change the best cell value Note When taking the dual value of a constraint be sure to specify the cell containing the constraint function not the right hand side of the constraint Dual Value of an Adjustable Cell Reduced Cost The dual value of an adjustable cell is the rate at which the best cell would be penalized if an adjustable cell with an optimal value of zero were
307. n D Dual Values Dual values can be applied to cells H9 H13 giving the cost penalty associated with using a media source when the optimizer determines zero dollars should be allocated for that source In this example the optimizer determines that zero dollars should be allocated for the Herald By using the Advanced Dual command you would find that the total cost 6 220 will be increased by 500 for every 1 000 spent on the Herald Dual values for cells B16 G16 would give the dollar cost of requiring a thousand more exposures 306 CHAPTER7 Multi Period Inventory Management File name INVENT XLSX TYPE LINEAR OPTIMIZATION Application Profile What if your product is made from materials that are in limited supply and require special or expensive storage facilities In such a situation you have to keep purchasing and inventory costs down but still keep enough stock to meet demand If you use perishable resources or materials subject to supply limitations you may encounter problems like this one The Problem in Words Your production demands vary over thirteen time periods Raw materials available at varying costs from three different suppliers must occasionally be held in sufficient quantity from time period to time period to meet production needs A holding cost associated with inventory held over further complicates the picture How can you determine how much inventory to hold and which supplier to use while minimizing tot
308. n File Office Button Excel options Manage Excel Add ins Go Check _What sBest or Browse to WBA XLAM usually in C Program Files Microsoft Office O ffice 12 Library LindoWB File Office Button Excel options Trust Center Trust Center Settings Macro Settings Enable all macros possibly Trusted Publishers to Lindo Systems Inc Settings for Microsofte Excele 1997 2003 Via the Menubar select Tools Add ins Check _What sBest or Browse to WBA XLA usually in C Program Files Microsoft Office O ffice 1 1 Library LindoWB Tools Customize Toolbars Check _What sBest Tools Options Security Macro Security Security Level to medium 442 APPENDIX A Trusted Publishers to Lindo Systems Inc Note A previous add in link may be remaining under the name wba that should be deleted INSTALLATION DETAILS 443 Add ins If you wish to disable remove the What sBest add in and menu you may do so by going to the Excel menu s Tools Add ins dialog You can then uncheck the What sBest add in from the list which disables the add in while you are still in Excel When you reopen Excel the WB menu item will not be present The What sBest add in and its menu are reenabled by checking the What sBest add in from the list posted by the Tools Add Ins dialog If What sBest does not appear in the current list of add ins the browse button can be used to locate it The add in file WBA XLA or WBA XLAM c
309. n SOS2 set can be gt 0 If two variables are nonzero then the variables will be adjacent to one another SOS2 sets are particularly useful for implementing piecewise linear functions in models IWBSOS3 Exactly one variable from a given SOS3 set will be equal to 1 All remaining variables will be equal to 0 The syntax for the WBSOSx declarations is as follows written in the selected cell of the spreadsheet WBSOS range reference Note Each variable in an SOS set counts against the integer variable limit imposed in limited versions of What sBest SOS sets are supported for linear models only Cardinality Related to the SOS capability discussed above What sBest also supports cardinality sets of variables via the WBCARD function The cardinality feature allows you to specify a set of variables with a cardinality of N meaning that at most N of the variables in the set will be allowed to be nonzero As with SOS sets cardinality sets help the integer solver branch more efficiently and they reduce the number of variables and constraints in your models The syntax for the WBCARD declarations is as follows written in the selected cell of the spreadsheet WBCARD number range reference Note Each variable in a CARD set counts against the integer variable limit imposed in limited versions of What sBest CARD sets are supported for linear models only ADDITIONAL COMMANDS 45 Select Any Cells This reference f
310. n any order you choose Using named arguments can also make the code easier to read Error messages you might encounter learning VBA When initially learning to run What sBest using VBA code the user may encounter the following error messages which are easily resolved Sub or Function not defined This error message is often the result of not having WBA_XLA checked as an available reference Checking it as a reference makes the What sBest procedures available to your Visual Basic code Instructions for setting this reference are provided at the beginning of this section Invalid outside procedure This message is often the result of using parenthesis without the CALL keyword Either put the keyword CALL in front of procedure calls where you use parenthesis or remove the parenthesis Beyond VBA Model developers may prefer the ability to provide the user with a more elaborate polished interface than can be provided using Excel workbooks running Visual Basic macros Developers can create such applications using the full featured development environments of Visual Basic or Microsoft Access that run What sBest with Excel running in the background The Access or VB application can easily prepare the data for the optimization model insert it into an Excel workbook and run What sBest so that Excel is completely hidden from the user The development of the application can progress in natural stages The optimization m
311. n for the problem This is similar to the section for solving a deterministic optimization problem 2 A section where we enter the stochastic information Information can be entered directly onto the spreadsheet or via the set of dialog boxes Step 1 core model 1 Data and Formulas Specify the initial price 100 the strike price 99 and the risk free rate 5 The price stock of this period price of the previous period 1 the stock return of this period This is written in the cells C18 to C23 The wealth of this period the wealth of the previous period 1 the risk free rate the decision to sell strike price price stock of this period This is written in cells E18 to E23 380 CHARTER 2 Adjustable Cell There are 6 adjustable cells in cells E18 to E23 one for each period The meaning is 1 for selling 0 for holding 3 Objective Function The objective function is to maximize the wealth at the end of period 5 cell E28 4 Constraint Cell There is one constraints in D26 of the model The total number of selling decisions has to be one 5 Random Cell There are 5 random parameters in cells B19 to B23 one at each period expressing the uncertainty of the stock return Step 2 stochastic information for stage and distribution 1 Decision Variables Specify the stage information for the decision to sell or to hold which is a decision variable adjustable This is done by using
312. n general the Time setting results in the fastest performance However due to shifting computational loads on a machine solution paths may change under the Time setting from one solve to the next potentially resulting in non reproducible solutions If reproducibility of a runs is a concern then the Iterations option should be selected Under the Solver Decides setting What sBest chooses the most appropriate strategy What sBest defaults to Solver Decides Probing Level Specify the level to which you would like probing applied to your model with the Probing Level drop down box Level is the lowest level of probing while Level 9 is the highest Setting the Probing Level to None will turn probing off The Probing Level option is applicable to mixed integer linear models Probing examines the integer variables to deduce tighter variable bounds and right hand side values This process is referred to as tightening the formulation and can result in the integer variables taking on values that are closer to being integral when the continuous relaxation of the model is solved Probing can often tighten an integer model enough to dramatically speed solution times In other cases probing cannot adequately tighten a model and the expense of the probing step simply slows down the overall solution time 68 CHAPTER3 Settings are Ca None no probing Level 1 minimum simple presolving Level 2 probing Level 3 coefficient reduction
313. n is uni modal if you pour water into it only one puddle would form A convex function is also quasi convex If you have the constraint f x gt b and you know that f x is a concave function then you are allowed to write Ax gt c b Finally if you have the constraint f x b and you know that f x is a quasi convex function then you are allowed to write Ax c b The interpretation is that we require f x b however the feasible region to the relaxation f x lt b is convex Similarly if you have the constraint f x b and you know that f x is a quasi concave function then you are allowed to write f x c b The interpretation is that we require f x b however the feasible region to the relaxation f x lt b or f x gt b is convex ABCs 31 Solve Once the ABC s Adjustable Best and Constraint cells have been specified you are ready to solve your model Choosing the Solve command prompts the What sBest solver to begin examining and solving your model During the time that the solver is computing it displays the Solver Status window which appears as shown below HE What sBest Solver Status 64 bit Lindo Systems Inc GU Copyright 2011 64 bit What sBest 11 0 0 2 Mar 31 2011 Library 7 0 1 129 Extended License m Solver Status Model Type State Tries Infeasibility Objective Classification Statistic Category Current Numeric
314. n order to meet customer demand you must produce rolls of these widths and footage Width Total Footage 35 1 020 25 3 000 18 967 15 1 450 For each roll of 100 metal you select one of four cut patterns Pattern A B C D Rolls 35 35 25 35 cut into 25 35 29 25 widths 18 15 22 15 of 18 15 25 15 322 CHAPTER7 Schematically the patterns look like this if TD Fees 9 __ EEN 25 15 15 4 Patter A Edge Waste _ 33 33 15 15 Fatter B 25 KEN z5 25 Patter C EN 25 45 45 Pattern D 10 Edge Waste Each length of 100 metal must be cut into one of these patterns but how many feet should be cut using each pattern Objective of Optimization Your objective is to minimize the total waste cost the sum of edge waste and end waste How many feet of each pattern should you cut in order to meet customer demand at the lowest cost SAMPLE MODELS 323 The Worksheet Let s examine the CUTSTOCK sample file and see how the edge waste and end waste are calculated The CUTSTOCK Worksheet Before Optimization oo ans PRS EI ROG PE Pa EY ER TE a v Qu fe WidthsTocutininches sd Tal B G D F G H Edge Waste In Ins 4 0 0 10 Cut 0 0 Need 1450 967 3000 1020 Not gt Not gt Not gt Not gt End Waste In Stock Width Inches 1450 967 3000 1020 Inches 100 16 313 13 055 56 250 26 775 End Waste Cost Per Inch Ft 0 75 Total End Was
315. n the Refers To text box In the drop down box the default setting Make Adjustable is already selected so you need only click OK What sBest identifies an adjustable cell by applying an Adjustable style distinguished by a default font color of blue as a visual reminder and unlocked cell status as follows The Adjustable Cells in XYZ vi ZEN h XYZ xlsx Microsoft Excel A Home Insert Page Layout Formulas Data Review View Develc XYZ COMPUTER CORPORATION PRODUCTION PIL Product Standard Deluxe Quantity to Produce 0 ol 1 2 3 4 Profit per Unit 300 500 LO OO Nio B Define Best XYZ Corporation s goal is to maximize profit which may be expressed as Total Profit Quantity of Standard Models Produced TIMES Profit per Unit of Standard Models PLUS Quantity of Deluxe Models Produced TIMES Profit per Unit of Deluxe Models GETTING STARTED 13 This formula appears in cell G6 as C5 C8 D5 D8 This is the sumproduct of C5 D5 and C8 D8 and could also be entered as SUMPRODUCT CS DS C8 D8 This function can be especially useful when doing similar operations on larger ranges To make G6 the objective best cell move the cursor to that cell and either 1 choose Best from the WB menu select Maximize and click OK or 2 use the Maximize toolbar button LZ If you use the Best dialog box via the menu you II notice that the right text box on the Best dialog box indicates
316. n to the Internet for this command to work When you issue the CheckUpdate command or start a version of What sBest with CheckUpdate enabled What sBest will search the Internet to see if an updated version of the software is available for download If you currently have the most recent version then you will be returned to the main What sBest environment If a newer build of the software is available you will be presented with the following dialog box 126 CHAPTER 3 Lindo Systems Check Update 64 bit x WW There is a new version available QD what sbest 11 0 03 Please go to our Website at www lindo com downloads WB WINDOWS 64x86 11 0 zip to download it Thank you A checkmark next to the command in the WB menu indicates that the command is still enabled You will need to download the new version of the software from the Website at the address indicated You can disable this option by checking off the command item Upon completion of downloading you will be prompted to begin installation of the new update Close Excel and double click on the file to open the install wizard which guides you through the setup process to update What sBest For your convenience your license key will be automatically transferred to the new version during installation Keeping your software up to date helps ensure that you are using the most recent version of the software and that compatibility and operational problems
317. n with the new constraint in place results in the production of 40 units of each model a profit of 32 000 and a surplus of 10 Deluxe towers Working While Solving This simple model solves very quickly While larger models are solving you may want to do other work as you wait for your solution If you want to work in Excel while your model is being solved then just minimize the model being solved and open another copy of Excel or whatever application you wish to you use Simply leave the minimized Excel icon on the taskbar at the bottom of the screen and your model will continue to solve in the background The Next Step The ABC s Basic Functions and Advanced Functions sections explain the What sBest commands in depth To learn more about the principles of optimization modeling see Overview of Mathematical Modeling If you d like to move on to more sophisticated problems see Sample Models 2 ABC s The Basics of What sBest This chapter describes how to use the three basic commands that are used in building almost every model for What sBest We ll refer to them as the ABC s Adjustable Best Constraints These three commands are the first commands at the top of the main WB menu item on the Excel menu In addition to the menu commands you may use the corresponding toolbar buttons on the What sBest toolbar The sections below include in depth discussion of the use of these commands Adjustable
318. nBest goWrnBlank goWrnFunction goWrmStringArg goWrnIrreConst goWrnInfeasConst goWrnUnboundVar goWmSupLookupFct Argument Required Default Description goFeasTol No 0 0000001 goFeasibilityTol is a positive number indicating the amount of violation tolerated in constraints golterLimit No None golterLimit is an integer indicating how many tries to allow in solving the model goRuntimeLimit No None goRuntimeLimit is an integer indicating the maximum number of seconds to be used in solving the model golndSlack No 1 Indicator goIndSlack is an integer indicating what form to display constraints in 1 Indicator 2 Slack goAutoSelectFreeIntOmit No 1 True goAutoSelectFreeInt is a True False flag indicating whether to automatically select Free Integer or Omit range names goMinimizeExcel No 0 False goMinimizeExcel is a True False flag indicating whether to minimize Excel during the solution process goHideStatusWindow No 0 False goHideStatus Window is a True False 162 CHAPTER 4 flag indicating whether to minimize the status window during the solution process goLinearizationDegree 0 Solver Decides goLinearizationDegree is an integer indicating the degree of linearization to use 0 Solver Decides 1 None 2 Mathematical 3 Mathematical Logical goLinearizationDelta 0 000001 goLinearizationDelta is
319. nces command is referring to the correct WBA_XLA file In Excel 2007 go to the OfficeButton ExcelOptions Add Ins Go to browse to the correct location TROUBLESHOOTING 393 What sBest generated errors e g Error building the model are typically encountered when building and solving a model Check to see that all adjustable best and constraint cells are correctly specified Also if you have moved your model from one computer to another try running the General Options command and clicking the Update Links button You may also not nest any What sBest functions in a single expression Can I protect my workbook from viewing You can password protect worksheets data and macros from viewing However you may not protect any of the adjustable dual or range cells What sBest will still be able to solve the model but it won t be able to access these protected cells in order to write the results How large can my model be The size model you can solve will depend primarily on the size license you purchased For a list of the various sizes and their specific limitations see section About What s Beet Other indirect limitations are memory and time Your model may physically fit within the limits of your version of What sBest but particularly for very large models there may not be enough random access memory available to successfully solve your model Also certain classes of models are very difficult to solve
320. nd in the drop down list in the Adjustable dialog box appears as follows l Free axe Free Names in Workbook Cancel Add Delete Refers to eg ee By default adjustable cells are restricted to being greater than or equal to zero However you may override this default lower bound of zero on an adjustable cell by making it free Adjustable cells that may assume negative as well as positive values are referred to as free variables To create an adjustable and free cell s specify the cell or cell range you wish to change in the Refers to field Then enter a name in the FREE Names in Workbook field You can choose any combination of letters for your range name Once this is done pressing the Add button causes What sBest to assign a WBFREE range name to the selected cells which will appear in the list on the Free dialog box For instance if you used the name BuySell in the FREE Names in Workbook text box a range name WBFREEBuySell would be assigned to the selected cells If the cells are already adjustable you can make them adjustable and free by following the steps above Alternately if you are removing a free variable designation for a given cell range simply select the name of the range to remove and press the Delete button A negative adjustable cell representing the number of televisions to produce would not be meaningful However there are scenarios in which allowing an adjustable cell to assume neg
321. negative value Adjustable cells also called decision variables usually represent quantities or activities over which you have direct control Traditionally mathematicians refer to them as variables Some examples might be the number of televisions to produce the amount of money to be spent on advertising the number of shares to purchase of a particular stock ABCs 21 the location of your next service facility Inappropriate adjustable cells would be things you have no control over such as the total demand for televisions the price of advertising to be purchased e the risk of a particular stock the local building restrictions Cells containing equations are also inappropriate as adjustable cells What sBest cannot rewrite an equation although the value returned by an equation will change if the equation depends upon any adjustable cells What sBest will not alter the contents of cells specified as adjustable that contain equations text or blanks In other words the cells must be numeric Therefore be sure to enter a numeric value in each of your adjustable cells Note When the adjustable quality is applied to a cell What sBest assigns a predefined Adjustable style to the cell This style is automatically made available to each of your What sBest models and can be customized e g font color as you wish F x Remove Adjustable Select Remove Adjustable to return an adjustable cell to its fixed
322. neral This routine is used to update the links to the WB constraints and functions Syntax wbUpdateLinks The wbhUpdateLinks procedure has no arguments Error Codes Description UpdateLinksProtectedError Unable to update links on a protected sheet UpdateLinksHiddenError Unable to determine if there are constraints in hidden cells on protected sheets Remarks This procedure is provided because Excel does not properly update links of add in functions MACROS THE VBA INTERFACE 185 wbSetFunctionSupport This routine can be used to set the What sBest function support option seen in the Function Support dialog box There is only one argument required For additional discussion of the options available through this routine see the section entitled Advanced FunctionSupport Syntax wbSetFunctionSupport AdvFunctionSupport Argument Required Default Description AdvFunctionSupport Yes 0 False AdvFunctionSupport is a True False flag indicating whether or not to use this support Error Codes Description AdvFunc_BadAdvFunctionSupportArg Bad AdvFunctionSupport argument Example If one wished to set all the function support options the simplest syntax would be wbSetFunctionSupport True or by the named argument wbSetFunctionSupport AdvFunctionSupport True Note The AdvFunctionSupport argument should be set to TRUE in order to use the oth
323. nformation on using K Best Solution values refer to section Options Integer Solver INDICMOD Indicator Model Cell Reference If your model has references to WB constraint cells without being in a Slack mode the following warning message will be displayed on the WB Status tab Error Message WARNING Indicator Mode Cell Reference Help Reference INDICMOD The following cells directly refer to WB constraint functions being in Indicator mode rather than Slack mode You need to modify the range selection or to specify the Slack mode via the WB Options General menu cell addresses listed at bottom of tab Suggestions You should verify that the cells being referenced were actually intended to be in Slack rather than Indicator mode checking the option via the WB Options General command Otherwise you may need to change the range selection in the cells listed at the bottom of the status report Some other What sBest functions such as WBDUAL WBLOWER O or WBUPPER for the dual values can have references to WB constraint cells without being in the Slack mode TROUBLESHOOTING 405 INFEASIBLE No Feasible Solution Found If the What sBest solver was unable to find a solution that satisfies all the constraints cells and any integer requirements in your model the following error message will be displayed on the WB Status tab Error Message No Feasible Solution Found Help Reference INFEASIBLE Ther
324. ng the squared error SAMPLE MODELS 285 The Worksheet Let s look at the supplied sample file called SIMXPO The SIMXPO Worksheet Before Solving Home Insert Page Layout Formulas Data Review View Developer Addins amp X A e itl Simple Exponential Smoothing e Fa B G D E F G H Sales Predicted 10 14 12 19 14 21 19 26 oinunuunus Alpha Lower bound Upper bound A Determine Adjustable Cells The adjustable cells are the Predicted sales in C5 C11 and the Alpha smoothing term in C14 Note that Period 1 in cell C4 is fixed at 10 since there is no previous period on which to base a prediction B Define Best The best cell is the minimized Sum of Squared Errors differences between Sales and Predicted in cells E4 E11 in E13 C Specify Constraints The constraints in D5 D11 reproduce the basic model for exponential smoothing see preceding page for each Predicted period In C15 C16 the Alpha term is bounded by 001 and 999 286 CHAPTER7 Now you are ready to solve the model After solving the WB Status worksheet will open in order to show you the Nonlinearity present warning This warning can be disabled from the General Options dialog box Your solved model now appears as follows The SIMXPO Worksheet After Solving SIMXPO xIsx Microsoft Excel gett L o fe TR SAMPLE MODELS 287 Following is another sample file called SMOOTH which uses the same
325. ngle Pref 0 Assign Sub lt lt lt lt lt lt lt lt lt lt lt lt am E a E o E o E a E a E o E o E E o E E A KEE KKK oooooococoooooccocow GER H eeeeeeeeeeeehiO KKK E n E E 2 KE H eeeeeeeeeeeeb e oooocooococococcoceoo CO a a E a E a E a a E P oooocoococococoeoo oooococoocoeoocoe oooocoococococo H H Il M H 1i ll S 1i H STAFF NEEDS MET Not gt PREFERENCE TOTAL EZ y Wl ES ef You must now copy the optimal work patterns selected information from cells P5 P16 of the Stage 1 model after optimization and enter into cells U4 AF4 For this example we ve used the optimal scheduling generated without using Pool FTE s SAMPLE MODELS 353 After the data from the first stage is entered the model looks like this 2 m MPLOYEE ASSIGNMENTS TO SCHEDULES 10 12 Single Pref 0 Assign Sub lt 0 lt lt Schedule Staffed LLOYD DIANE TOM JOANNE DAN JIM SUSAN JOY JOHN PAM SHAOIB BEA Met lt lt lt lt lt lt lt 2 0 0 0 0 0 0 0 0 0 0 0 0 0 i A GA eeneneeeeeeeeebh be eeneneoeeeneaeeeene On eeneneaeeenenee enen E zl ZC eeeeeneeeeeecb On eeneneeeeneaeeeneneu eoocooccocecocoos eceoooocococccocd coocooococccd z 2 i amp 1 1 1 z E u 1 z E 1 STAFF NEEDS MET Not gt PREFERENCE TOTAL ES A total of 10 people need to be scheduled
326. ning solution in cell error message When using Function Support the system seems to hang with the message Server Busy What Excel file format should be used ef Oe oo e e FH FH OH o TROUBLESHOOTING 391 What are the system requirements to install What sBest To install and run What sBest check that you have the following Software Microsoft Windows 7 Vista or Windows NT 4 0 Windows XP Microsoft Excel 32 bit version 2002 or higher version 2007 2010 or Microsoft Excel 64 bit version 2010 Microsoft NET Framework 3 5 or higher Hardware Pentium class PC 500 MB of RAM 50 MB of free disk space Make sure you have administrative privileges to install files on your default drive System and Program Files folders An Internet connection is required to download the latest version of What sBest You can also contact LINDO Systems to obtain a copy Additional information can be found via the Help command on the What sBest menu How do l install What sBest add in on my computer If you are installing What sBest from the original CD open the What sBest folder click on setup exe If you downloaded the demonstration version from the website simply run the executable or unzip the zip file and then run the executable Excel should be closed during the install process An installation program will then commence and will verify if What sBest
327. nnnnnnnn 201 Guidelines for Modeling with What SBest 0 ccccceceeeseeeeeeeeeseaeeeeeeeseeeeesaeeesaeeeeneeeeaees 202 General Modeling Guidelines ccccceseseceeeeeceeeeesaeeeeaeeseeeeeeeaaeeeeaaeseeeeeseaeestaeenenaees 202 Nonlinear Modeling Guideline ccccccceeeeeeeeceeeeeceeeeeseeeeseeeeecaeeesaaeeseeeeseeeeseaeeseneeee 203 Scale the Model to a Reasonable Range of Units ccececeeeeeeeeeeeeeeeeeseaeeeeneeeeeneees 203 Simplify Relationships AAA 204 Reduce Integer Hestrtctons 204 Guidelines for Stochastic Modeling cceeececeeeeeeeeeeeeaee teense caaeeeeaeeseeeeesaeeesaeeneneeseaees 205 Multistage Decision Making Under Uncertainty c ccceccceceeeeeeeeeeeeeeeeeneeeeeaeeeeeeeeeas 205 Recourse Models gd ege AER 207 Scenario Tree wisi nee eas ceed ea ee ae ete 208 Defining an SP Model in What sBest Simple 2 Stage Example sssseseeeseeeeseeeseene 208 Defining an SP Model in What sBest Multi Stage Example 215 Monte Carlo Sampling Ave 221 Generating Dependent Samples ccccccceeeeceeeeeeeeeeeeeeeesaeeeeaaeeeeeeeseeeesaeeeeeeseeeeess 222 Sampling a Scenario Tree cccccecceceececeenceceeeeeceaeeeeaaeeceaeeeceaeeseaaesgeneeeeaeeesaeseeneeeneeeess 222 Solving Second Order Cone Programs ssesseeessesssessisssissrissississrnnstnntennntnnntnnntnnsnn te 224 ZC SAMPLE MODEL Eeer sek ameen aeaa a a teanen aana aa Eaei aaoi aane 227 Bletpitmgt eege deed
328. non adjustable state By default a cell is fixed until it has been specified as adjustable When you instruct What sBest to remove a cell s adjustable status the Adjustable style is removed from the cell and you will see the font color revert from blue or the font color you defined for the Adjustable style to the original color After solving a model you may wish use Remove Adjustable to fix some of the adjustable cells at their new values turning them into constants and solve again for the remaining adjustable cells Make Adjustable amp Free or Remove Free Select this to take the referenced cells and either set them as adjustable and free capable of assuming negative as well as positive values or remove their free status Making this selection and clicking OK will post the Free dialog box to facilitate setting or removing the free status For further discussion see the section entitled Free Refers To Specify the range of cells that are to be made adjustable or fixed in this text box by using the button on the right edge to bring up a cursor for cell selection Alternately you can accept the currently selected cells which What sBest has automatically placed in this box You can also type the correct cell range directly into the text box Note Adjustable cells should not be locked or hidden on a protected sheet 22 CHAPTER 2 Free The dialog box posted by selecting the Make Adjustable amp Free or Remove Free comma
329. non smooth functions Solution Outcomes The solution and analysis of the outcome of the solution process is presented in the status report worksheet The status report worksheet is inserted following the completion of the solution process unless you have disabled the status report regarding disabling reports see the Reports Location and Warnings box on the General Options dialog box posted by the Options General command The status report is opened by clicking the WB Status tab among your Excel worksheets Unless you have imposed a limit on the number of iterations in the solution search or have interrupted the solver by pressing the Hold Interrupt button on the What sBest solver status window the message provided in the field Solution Status in the status report should be one of those listed below Case 1 Optimization Models If a cell has been specified to be maximized or minimized there are several possible outcomes of an attempt to solve the model Globally Optimal During the solution search the solver found the best possible solution that satisfies all the constraints The message Solution Status GLOBALLY OPTIMAL will be displayed in the status report upon successful solution of a linear model There is no better answer that satisfies all constraints Locally Optimal A locally optimal outcome applies to a nonlinear model when the solver has found the best solution within a local area that satisfies all of the constrain
330. ns a time delay may occur after enabling the add in Refer to the What sBest sample file Shipping MacroFunctions xls for an example of valid user defined function usage FUNCMACRO Failed to Access the Macro If the What sBest solver is unable to execute a macro that is needed to calculate a user defined function the following error message will be displayed on the pop up window Error Message WARNING Failed to Access the Macro Help Reference FUNCMACRO The solver needs to access the VBA code to run the users defined function from Excel Select Trust Access to Visual Basic Project via the Tools Macro Security TrustedSources menu Then save the model and restart Excel TROUBLESHOOTING 403 Suggestions In Excel 2002 there is an additional security setting that the user needs to set for allowing the server to access a user defined macro Select Trust Access to Visual Basic Project via the Tools Options Security MacroSecurity TrustedSources menu Then save the model and restart Excel FUNCSERVER Failed to Access the Server If the What sBest solver was unable to access Excel to calculate the user s defined functions the following error message will be displayed on the pop up window Error Message VARNING Failed to Access the Server Help Reference FUNCSERVER The solver needs to access Excel to run the users defined functions Make sure only one Excel application i
331. nsidered a nonlinear adjustable cell On the other hand multiplying an adjustable cell by a constant is a linear operation so this formula alone would not make B a nonlinear adjustable cell For more information on identifying and solving linear and nonlinear expressions see Overview of Mathematical Modeling Evaluate the formulas in which the nonlinear adjustable cells appear to determine if these formulas can be rewritten as linear expressions If this is not possible you will need to either reduce the size of your model or upgrade to a larger version of What sBest When possible formulate your problem using linear expressions Problems composed entirely of linear relationships solve faster and more reliably However in many cases linear reformulation may not be possible If your model contains inherently nonlinear expressions you may choose to turn off this nonlinear warning with the Nonlinearity present checkbox in the General Options dialog box All versions of What sBest are supplied with a linear solver The downloadable and Solver Suite versions of What sBest provide a nonlinear solver with capacity for a few nonlinear adjustable cells For the Commercial and more powerful versions of What sBest the nonlinear option may be purchased from LINDO Systems LICCAP5 Global Adjustable Cell Limit If your model has more nonlinear cells than is allowed for your license when you are running the global solver the following error mess
332. nt A 200 0 500 Not gt Plant B 300 0 400 Not gt Plant C 500 0 600 Not gt Output Capacity Output Capacity e 100 0 lt 150 0 0 Total Cost 0 A Determine Adjustable Cells The adjustable cells in this model are the quantities to be shipped from each steel mill to each plant B6 B8 and E6 E8 B Define Best The best solution minimizes Total Cost in cell 112 The Total Cost formula is the sum of the two Cost per Mill cells in B12 and E12 Each of these is the sumproduct of the Units Shipped from each Mill to each Plant B6 B8 and E6 E8 and the cost associated with each shipping arc D6 D8 and G6 G8 C Specify Constraints There are two sets of constraints First each plant location requires certain levels of shipments Second the production of each steel mill must not exceed its capacity 362 CHAPTER7 The constraints in H6 H8 force the total shipped to each plant to be greater than or equal to that plant s demand 16 18 To guard against exceeding the capacity of a Steel Mill the capacity constraints in C11 and F11 require that a Steel Mill s output the sum of amounts shipped from it be less than or equal to that Mill s capacity Now let s solve the model The SHIPPING Worksheet After Optimization Total Units From Demand Demand Shipped Steel Mill 1 Cost Steel Mill 2 Cost Constraint 200 0 500 9 gt 300 90 400 gt 500 30 600 gt Capacity Output Capacity 10
333. ntain a variable squared cubed or taken to any power other than one a term divided by a variable or variables multiplied by each other In other words proportionality exists That is for every unit increase or decrease in a variable the expression increases or decreases by a fixed amount In their simplest form linear formulas are straight line relationships For example suppose you re buying tomatoes for 1 50 per pound The expression or function used to calculate the Cost C in terms of the amount of tomatoes purchased 7 is C 15 T 193 194 CHAPTER6 As you might expect a graph of this expression for cost is a straight line 5 10 15 20 25 30 35 40 Tomatoes Purchased Ibs Linear expressions can have multiple variables For example if you added potatoes P at 0 75 per pound and apples A at 1 25 per pound your cost function would become C 15 7T 0 75 P 1 25 A This new cost expression is linear you could think of it as the sum of three simpler linear expressions Nonlinear Expressions By definition all expressions that are not linear are nonlinear Nonlinear expressions include relationships in which variables are squared cubed or taken to powers other than one variables that are multiplied by each other and many expressions using nonlinear spreadsheet functions such as F MAX and MIN Models with nonlinear expressions are intrinsically much more difficult to solve than linear model
334. ntax wbSetStochasticSupport AdvStochasticSupport AdvStochasticSupport Yes AdvStochasticSupport is a True False flag indicating whether or not to use this support Default 0 False Sto_BadAdvStochasticSupportArg Bad AdvStochasticSupport argument Example If one wished to set all the function support options the simplest syntax would be wbSetStochasticSupport True or by the named argument wbSetStochasticSupport AdvStochasticSupport True wbSolve For additional discussion of the functionality provided by this routine see Solve This routine is used to solve the active model Syntax wbSolve SolutionStatus ExcelOpen NoErrDialog Argument Required Description SolutionStatus No SolutionStatus is a return integer on the condition of the solution Globally Optimal 1 Globally Optimal 2 Infeasible 3 Unbounded 4 Feasible 5 Infeasible or Unbounded 6 Near Optimal 7 Locally Optimal 8 Locally Infeasible 9 Cutoff 10 Numerical Error 11 Unknown 12 Unloaded 13 Loaded 14 Unknown Error other ExcelOpen No ExcelOpen is a True False flag to indicate whether Excel should remain in the open state or be minimized True to leave 178 CHAPTER 4 Excel open False to minimize Excel If this argument is omitted it defaults to the value of the Minimize Excel on Solve setting in the General Options dialog box NoErrDialog No Any argument e
335. o record the What sBest add in commands so we recommend you do not try to use it to generate VBA code You can use Excel s Object Browser described above to more quickly and accurately write your VBA code Calling Procedures Visual Basic for applications is quite liberal in the syntax it allows when calling procedures For the sake of simplicity the calling procedures used for illustration below will stick to one or two conventions depending upon whether or not named arguments are used For example to call the procedure to set cells D8 E10 as adjustable without named arguments would be wbAdjust D8 E10 The same procedure call with named arguments would be wbAdjust AdjRange D8 E10 Note VBA is case insensitive to the procedure name when it is called so the procedure could alternately be called with its name typed entirely in upper case e as WBADJUST instead of wbAdjust MACROS THE VBAINTERFACE _ 131 Named arguments such as shown above are necessary when you are skipping over some arguments With some programming languages it is possible to skip an argument by leaving it blank This is not possible with VBA Named arguments must be used instead INVALID whbSetGeneralOptions False True True VALID wbSetGeneralOptions goStatusReport True goSolutionReport False goWmBlank True When calling procedures with multiple arguments using named arguments allows you to list the arguments i
336. o the table Sheets PRODMIX Select Cells TableRow 12 Range C6 Copy the solution so it can be pasted into the table Range B8 68 Copy Paste the solution to the table We paste in the values only to prevent the cells in the table from being treated as Adjustable Range Cells TableRow 13 Cells TableRow 18 PasteSpecial Paste xlValues Save the Total Profit in the table Cells TableRow 19 Range A3 Next Count End Sub Lines beginning with a single quotation mark are comments MACROS THE VBA INTERFACE 137 The macro shown here performs range analysis by incrementing the profit unit of product 2 from 45 to 64 The For Next form of looping in Visual Basic is used here but other looping forms such as Do Loop and For Each Next forms are available Please consult the Visual Basic or Excel Help for more information After running the macro the adjustable cells and the best cell for each increment are copied to the table in the range L2 S22 The row is incremented each time through the variable Table that is always two greater than the Count the iteration number In this way a convenient report is created providing sensitivity analysis of the profit unit of Product 2 As shown here there are changes in the optimal solution when the profit unit for product 2 is 50 or greater In addition to simply having the macro increment the profit unit for product 2 through a range the p
337. o use during branch and bound when a starting basis is not present 0 Solver Decides 1 Barrier 2 Primal 3 Dual AbsoluteOptimality AbsoluteOptimality is a positive value r indicating to the branch and bound solver that it should only search for integer solutions with objective values at least r units better than the best integer solution found so far RelativeOptimality 0 00001 RelativeOptimality is a value r ranging from 0 to 1 indicating to the branch and bound solver that it should only search for integer solutions with objective values at least 100 7 better than the best integer solution found so far TimeToRelativeOptimality 100 TimeToRelativeOptimality is the number of seconds before the branch and bound solver resorts to using the RelativeOptimality tolerance HurdleTolerance None HurdleTolerance is a value of a known integer solution What sBest will only search for any feasible integer solution if it is at least as good as the bound specified in HurdleTolerance 0 None NodeSelectionTolerance 0 Solver Decides NodeSelectionTolerance controls the order in which the branch and bound solver selects branch nodes in the tree 0 Solver Decides 1 Depth First 2 Worst Bound 3 Best Bound StrongBranchTolerance The StrongBranchTolerance option uses a more intensive branching strategy during the first levels of the branch and bound tree
338. odel can first be prototyped and tested in Excel Once the structure of the model is decided upon Visual Basic code can be written to automate the process of creating the model within Excel Finally the code can be incorporated into an Access or Visual Basic project in which the What sBest model is built based upon a data set provided by the application Embedding What sBest in a Visual Basice Project We again return to the XYZ problem to look at how it could be incorporated into a Visual Basic application Assuming that Visual Basic is installed on the computer start Visual Basic and choose File Open Project and select the XYZ project XYZ VBP in the VB60 subdirectory of your WB directory In the frmMain form there is code generated using the VB Application Wizard for a simple application with a menu and toolbar 132 CHAPTER 4 A WB menu item has been added with three commands Solve Show Excel and Hide Excel Along with the menu items one button has been added to the far right of the toolbar for the So ve command When Solve is selected Excel is loaded if it is not already open the XYZ model is opened the adjustable cells best cell and constraints are added and the model is solved All of this happens with Excel hidden the user only sees the What sBest solver window To see the results choose Show Excel from the WB menu in the VB project The comment Code for WB appears above all code fo
339. of model you have linear nonlinear and its characteristics In general linear models are the fastest group of models to solve and nonlinear models are the most difficult to solve It is not unusual that a nonlinear model will take many times longer to solve than a linear model with the same number of variables If a linear or nonlinear model is close to a solution De constraints are only slightly violated then increasing one of the feasibility tolerances may allow you to reach a solution Tolerances are set in both the Linear Solver Options and Nonlinear Solver Options dialog boxes For a difficult linear model the options provided in the Linear Solver Options dialog box may help you to reach a solution or shorten the solution time One of the first options you might try is turning on or off Model Reduction Some linear problems will solve faster by a different Solver Method The Barrier method may be considerably faster on big models while the two simplex methods will tend to be faster on smaller sparse models Large integer problems can be very difficult to solve and take a great deal of time arriving at a solution To solve an infeasible problem or to shorten the solution time you can try various options on the Integer Solver Options dialog box Setting an Optimality Tolerance can improve run times dramatically if you are happy being within a certain percentage of the optimal solution Some problems solve faster when the Branching Direction
340. of steel from each of the mills to each of the plants is shown in the table below Shipping Costs Per Unit of Steel From Steel Steel To Mill1 Mill 2 Plant A 2 5 Plant B 3 4 Plant C 5 6 Mill I has a maximum total capacity of 1200 units and Mill 2 has a maximum total capacity of 1800 units Each plant has an 800 unit minimum requirement of steel for the period that has been specified by each plant manager The actual starting inventory of steel at each plant is determined by the total number of units shipped to that plant from the two steel mills The starting inventories of the other five raw materials at each plant are listed below Resource Units on Hand Wood 1160 Plastic 1780 Rubber 1050 Glass 1360 Paint 1240 The profit contribution and product resource requirements for each of the six products are the same as in the Product Mix problem SAMPLE MODELS 315 Objective of Optimization The objective of optimization is to maximize total profits The total profit is the sum of individual plant profits minus the sum of the shipping costs from the individual steel mills The Worksheet Let s look at the BLOCK sample file The BLOCK worksheet was built by directly combining the SHIPPING and PRODMIX worksheets After combining one copy of the Shipping Cost Reduction model with three copies of the Product Mix model we performed only slight modifications to link the four individual problems into one comprehensive model Th
341. of the difference between Return and Target Return The Downside Risk is calculated as the sumproduct of the probabilities in Column E and the Unders in Column H The Semi variance is formulated as the sumproduct of the probabilities and the squares of the Unders C Specify Constraints There are three types of constraints in the PORTSCEN model The first requires a bit of thought The Forcing Constraints in 14 115 force the difference between Over G4 G15 and Under H4 H15 to be equal to the difference between the Expected Return F17 and Return F4 F15 For instance in 14 the formula is WB G4 H4 F4 C 17 To find the semi variance and downside risk we need to calculate the amount by which each Scenario Risk SR is below the Target Return TR A shrewd spreadsheet builder will recognize that this calculation can be done using an F function of the form IF SR lt TR TR SR 0 This function returns the difference between the target return and the scenario return if the scenario return is less than the target return Otherwise it returns zero The bend at the origin of this ZF statement makes it a non smooth function A smooth function has no breaks or sharp bends Most ZF functions that depend directly or indirectly on adjustable cells are non smooth and it s wise to avoid non smooth functions since they can be difficult to solve reliably see Chapter 7 Overview of Mathematical Modeling for a discussion of non smooth relati
342. ofit contribution from that plant The Plant A Worksheet Before Optimization gt sx Microsoft Excel ke Se Profit at PLANT A M C eg CH B c D E F G H I J Profit at PLANT A 0 00 Product F 3 Af 5 Profit Unit 45 24 26 24 Quantity 0 0 0 0 Produced 800 Minimum Steel Requirement Product Resource Requirements rivera Shipping Plant A Plant B Plant C 273 Ready 7 With the exception of the best cell the ABC s in the BLOCK model are the same as in its PRODMIX and SHIPPING components A Determine Adjustable Cells The adjustable cells B6 B8 F6 F8 of the Shipping worksheet are the quantities shipped from each mill to each plant and the quantities to produce of each product from each plant B6 G6 of the Plant A B amp C worksheets 318 CHAPTER7 B Define Best The best solution is the shipping and production decision that maximizes profit at all three plants less all shipping costs This formula is in cell A15 of the Shipping worksheet C Specify Constraints The constraints are to ship at least the minimum amount of steel required by each plant 16 18 of the Shipping worksheet to ship no more steel than the capacity of each mill C11 G11 of the Shipping worksheet and to use no more raw material at each plant than has been shipped or is being held in stock 113 118 of the Plant A B amp C worksheets What If vs What sBest If you spend a fe
343. oice argument Adj_ProtectedSheetError Unable to set cells as adjustable on a protected sheet Adj_CreateStyleError Error creating an Adjustable style Adj_CheckStyleUnlockedError Error checking that the Adjustable style is unlocked Adj_CheckStyleProtectionError Error checking that the Adjustable style includes protection Remarks An adjustable cell must contain a number or the cell is ignored by the solver The Adjust procedure formats the cell s with a style called Adjustable which carries a blue font color as a visual reminder All adjustable cells will return non negative values unless they are ina WBFREE range Example To set the cells D8 E10 as adjustable enter the macro wbAdjust D8 E10 or wbAdjust Range Cells 8 4 cells 10 5 140 CHAPTER 4 To reset the cells D8 D10 to no longer be adjustable enter wbAdjust D8 D10 Reset or wbAdjust Range Cells 8 4 cells 10 4 Reset wbBest For additional discussion of the functionality provided see the section entitled Best which refers to the dialog box that calls this procedure This routine is also called with appropriate arguments by the either of the toolbar buttons displayed above This routine is used to create a best cell replacing any previously defined best cell All arguments are optional Syntax wbBest BestCell BestChoice NoErrDialog Argument Required Description BestCell No This is the cell or address of the cell to
344. ok options All workbook options are reset to default settings every time you create a new workbook You may also use the Reset to Default command to return all workbook options in the current model to their default values with the exception of the Advanced Parameters options Discussion of the General Options follows below Feasibility Tolerance You can specify the amount of violation that will be tolerated in a model s constraints with the Feasibility Tolerance text box Increasing the feasibility tolerance can allow you to find a feasible solution to a model that was previously infeasible For example changing a feasibility tolerance would be useful if you have a poorly scaled model with very large and very small coefficients that is nearly feasible You would know that the model is nearly feasible if you found the slack values to be small The best solution would be to rescale the model but setting the feasibility tolerance offers a simpler alternative By entering a larger feasibility tolerance you allow the solver to tolerate slight constraint violation such as a few pennies in a budget of tens of thousands in order to reach a feasible solution The default value for feasibility tolerance is 0 0000001 Iteration Limit You can specify a limit on the number of tries or iterations undertaken during the solution process with the Iteration Limit Iterations textbox If this limit is encountered before a solution has been found then the sol
345. olated or Dual cells If you choose to locate the best cell then What sBest searches the entire workbook for the maximized or minimized cell and selects that cell If you choose to locate Adjustable Constraint Violated or Dual cells then specify whether you wish to simultaneously Select All Such Cells or Identify One by One with the Using Method drop down box If you choose to Identify One by One What sBest proceeds to search by row until it finds the first such cell If there are more cells of the same type in the worksheet or workbook then the following dialog box appears Locate Next S For example if the current cell is Al you are searching for adjustable cells one by one and A2 and B1 are both adjustable cells the next cell selected will be B1 Finally select whether you want What sBest to search across the Entire Workbook or This Worksheet in the Across drop down box If no cells of the type you requested are found in the worksheet or workbook then What sBest lets you know ADDITIONAL COMMANDS 119 Help amp About What sBest The dialog box posted by the About What sBest command appears as follows About What sBest R 64 bit What sBest 11 0 1 0 May 12 2011 x Copyright 2011 Lindo Systems Inc Library 7 0 1 196 Extended License Help Location of the license file LNDWB110 LIC G 046 6 1 46 6 141 120 CHAPTER3 The About What sBest
346. om variables 5 Distribution Information for the Random Parameters Declare a joint discrete distribution for the yields of the various crops This is done by using the What s Beat function WBSP_DIST_ DISCRETE SV This function takes in two parameters i the set of possible values of the random variable set vertically ii the cell address of the random variable SAMPLE MODELS 375 Step 3 scenario information Specify the scenario information by using the function WBSP_STSC This function requires a two column table as an argument 1 Column 1 for the number of stages in ascending order 2 Column 2 for the respective number of scenarios In this problem the number of stages is 1 and the number of scenarios is 4 Step 4 reporting cells Use the function WBSP_REP to specify the cells that you wish to report This function takes in one parameter that is the address of the cell that we wish to report in the final solution Here we are reporting the area allocated to each crop and the yields for the various crops Scenario Tree The Scenario Tree illustrates the concept of modeling under uncertainty In Stage 0 the allocated area has to be decided for each crop to grow In Stage 1 the crop yield is revealed depending on the weather and as a recourse decision the quantity to produce sell and purchase have to be decided The objective is to maximize the total expected profit at the end of planning horizon St
347. omitted cell formulas QUAPREC Quadratic Recognition A quadratic program QP is any model that is linear with the exception of product terms involving two variables e g 3 X Y If you have enabled the Quadratic Recognition option and you try and solve a quadratic programming model that is non convex the following error message will be displayed on the WB Status tab Error Message Quadratic Recognition Help Reference QUAPREC The quadratic solver requires that the model be convex The model does not satisfy this condition You must disable the quadratic solver by turning off quadratic recognition and then re solve This will cause the general purpose nonlinear solver to be invoked TROUBLESHOOTING 423 Suggestions The Quadratic Recognition option is controlled with the WB Options Nonlinear Solver command When this option is enabled the nonlinear solver uses algebraic preprocessing to determine if an arbitrary nonlinear model is actually a quadratic program in which case it can be passed to the faster quadratic solver The quadratic solver assesses the model to determine if it is convex If the model is found to be non convex you will receive this error message At which point you must either reformulate the model so that it becomes convex or turn off quadratic recognition and use the general purpose nonlinear solver Refer to Overview of Mathematical Modeling for a discussion of the concept of convexity in math program
348. on is used to request lower ranges If the arguments to this function are not correctly specified the following error message will be displayed on the WB Status tab Error Message ERROR Lower Range Cell Format Help Reference WBLOFORM A lower range cell is incorrectly formatted Correct the formula in the cell below The format should be WBLOWER cell value cell address Suggestions There is an incorrectly formatted lower range cell in the model A lower range cell must use the format WBLOWER cell value where cell is a reference to the cell you want the range value on and value is anumeric quantity The cell reference should be to either a constraint cell or an adjustable cell You may use any numeric value for value What sBest will replace value with the actual lower range value the next time you run the solver For more information on using range values refer to section Advanced Dual TROUBLESHOOTING 437 WBSEMICFORM Semi continuous Cell Format If requested What sBest can specify Semi continuous sets Range values tell you over what range a particular set is valid The WBSEMIC function is used to request Semi continuous ranges If the arguments to this function are not correctly specified the following error message will be displayed on the WB Status tab Error Message x x ERROR Semi continuous Cell Format Help Reference WBSEMICFORM A Semi continuous cell is incorrectly fo
349. on of the What sBest files is indicated at the bottom of the About What sBest dialog box Then to delete the old toolbar use the View Toolbars Customize Excel 2002 command to bring up a list of the available toolbars or right click on the ribbon to select Delete Excel 2007 Scroll down to the What sBest toolbar which may or may not be checked off and click on highlight the words What sBest Then click the Delete button to remove the old toolbar Now install What sBest and the new toolbar will be installed without conflict with the old toolbar TROUBLESHOOTING 389 Symptom When I try to close Excel an error appears stating This workbook is currently referenced by another workbook and cannot be closed Problem You have set a reference to WBA XLA or WBA XLAM in the VBA Editor that must be removed before the What sBest add in can be removed and Excel can close Suggestions From the Visual Basic Editor Tools Macro Visual Basic Editor go to Tools References to open the list of references Unselect WBA XLA or WBA XLAM and you should be able to close Excel Symptom The What sBest menu does not load Problem The add in is not set to load or there is a conflict such that it is unable to load Suggestions In Excel to set the What sBest add in to load choose Tools Add ins from the Excel menu and select What sBest from the list of a
350. one constraints and thus be solved by the faster conic solver e Improved ability for efficiency handling polynomial terms e Improved bounds for non convex quadratic terms using SDP and eigenvalue reformulations 0 Constraint Convexity One of the important considerations for the solver in finding a global optimum is the exploitation of convexity Our solver is fairly sophisticated in its ability to identify convex expressions Nevertheless there may be some constraints that you know have convex feasible regions What sBest allows special constraint types lt c gt c and c to allow users to identify constraints that have convex feasible regions QO Conic Solver SOCP SOCPs or Second Order Cone Programming SOCP are nonlinear convex problems that include linear and convex quadratically constrained quadratic programs as special cases The Conic Solver is invoked when the model is Convex with Global option set and checked the Quadratic Recognition on and the Conic option set U Convert Model Format What sBest 11 has a command to help you convert a model made via competing spreadsheet modeling interfaces to a What sBest format It will extract any existing data or information to set the Adjustable Best cells and Constraint cells from the default selected spreadsheet tab Then just click on Solve to see the What sBest results 0 64 bit Compatibility Install What sBest 64 bit version if you have Microsoft
351. one constraint cell to the model Constraints can be created with the WB Constraints command You may also want to refer to section The ABC s Three Steps to What sBest for detailed information on how to create a constraint cell On rare occasions it may actually make sense to have an unconstrained model This would be a model where you need to find the extreme point of an objective function dependent upon unconstrained adjustable cells In this case you may want to create a single vacuous constraint cell in order to avoid this error message ep X gt 1 e20 OMITTED Omitted Cell Reference Cells within WBOMIT ranges e cells that have been omitted from the optimization via the WB Advanced Omit command cannot be referenced in cell formulas that lie outside WBOMIT ranges When this situation is encountered What sBest will display the following error message on the WB Status tab Error Message Omitted Cell Reference Help Reference OMITTED The solver has been halted because the following cells are contained in WBOMIT ranges and have been referenced by formulas that are not contained in any WBOMIT range cell addresses Suggestions There are several options here e Remove the omitted cells that are being referenced i e the cells listed in the error message from all WBOMIT ranges e Place the cells referencing the omitted cells in WBOMIT ranges too e Eliminate the references to the omitted cells from the un
352. one to the transaction costs required to shift your investment levels Note also that the variance has been lowered to 20 85 from 35 54 SAMPLE MODELS 269 Portfolio Minimizing Downside Risk File name DNRISK XLSX TYPE NONLINEAR OPTIMIZATION Application Profile An obvious objective and common strategy in investing is to minimize the risk of losing ground This model demonstrates how to accomplish this The Problem in Words You re considering investing in three different assets You need to determine how much to invest in each of three assets to minimize the risk of having to sell your baseball cards Background You have forecasts for the expected return of each asset under seven equally likely scenarios there is a transportation strike interest rates rise or fall crops fail Cubs win the World Series Bulls win NBA finals etc You know that if you invest 100 of your capital in Asset 2 the average return under all possible scenarios your expected return is 19 9 You plan to live on the return from your portfolio After figuring your living expenses you find that a return below 11 will force you to start selling off your beloved baseball card collection to live in the manner to which you ve become accustomed Call that 11 your threshold return The return on Asset 2 is below your threshold return of 11 in a couple of scenarios and under Scenario 7 you actually lose money on your portfolio perhaps forcing y
353. ons You should verify that the blank cells being referenced were actually intended to be blank If the model looks correct then you may choose to disable this warning message using the WB Options General command TROUBLESHOOTING 399 EXLVER Excel File Format If your workbook is not saved in a current workbook file format the following warning message will be displayed on the WB Status tab Warning Message WARNING Excel File Format Help Reference EXLVER The file format you have chosen is from an older release of Excel Please resave your file using the latest Excel file format Suggestions What sBest requires workbooks to be saved in Excel 97 format or higher You will need to work with a more current Excel file format Resave the file using the File Save As command in Excel When presented with the file save dialog box select a more current file format from the Save as type dropdown box 400 CHAPTER 8 FORMULA Formula Parsing If What sBest was unable to parse a cell formula in your workbook the following error message will be displayed on the WB Status tab Error Message Formula Parsing Help Reference FORMULAI An error occurred while attempting to parse the cell formula listed below Check the formula in this cell for potential problems such as complex nested functions or text strings as arguments Some Excel functions also require a specific format cell address
354. onships By adding adjustable cells and carefully formulated constraints it turns out you can avoid the need for most non smooth JF functions You can choose to use ZF functions instead of finding ways around them but you run the risk of suboptimal answers from the solver In this case we ve emulated the F function using two adjustable cells and one constraint For each of the scenarios we ve added an adjustable cell in columns G and H Then we use a constraint to force the adjustable cell in column G to be equal to the amount the scenario return is Over the target return or zero if the return is less than the target The adjustable cell in column H is likewise required to be equal to the amount the scenario return is Under the target return or zero if the return is greater than the target The constraints in cells 14 115 force the difference between the adjustable cells G4 H15 to equal the difference between the return for this scenario F4 F15 and the Target Return C19 276 CHAPTER7 For example for scenario 1 the constraint in cell 14 reads VWB G4 H4 F4 C 18 If Target Return C18 is 115 and the scenario return F4 is 120 the constraint forces G4 H4 to be equal to 5 As adjustable cells G4 and H4 are implicitly nonnegative Since the objective is to minimize a function of these adjustable cells they re forced to the smallest values that satisfy all constraints Thus G4 will be forced to 5 and H4 will be f
355. or occurs the following error message will be displayed on the WB Status tab Error Message RROR Arithmetic Error Help Reference ARITHERR What sBest encountered an undefined arithmetic operation in the cell listed below One example of an undefined arithmetic operation would be division by zero but there are many others Check the cell below to determine the source of the error You must either rewrite the formula or constrain the adjustable cells to avoid the error cell address Suggestions Examples of undefined arithmetic operations are division by zero multiplying by a text string and evaluating to a number larger than 1030 If the cells are unrelated to solving the model include them in a WBOMIT range or delete them Otherwise change the initial values of the adjustable cells to move away from the undefined region Also consider rewriting the model in such a way that all formulas are defined over the entire domain of the adjustable cells BLKCELL Blank Cell Warning If your model has references to blank cells the following warning message will be displayed on the WB Status tab Warning Message WARNING Blank Cell Warning Help Reference BLKCELL Blank cells have been referenced in formulas During the solution process their values have been taken to be zero This warning can be turned off via the WB Options General menu See cell list below cell addresses listed at bottom of tab Suggesti
356. orced to 0 Similarly if the scenario return F4 is 110 then the constraint forces G4 H4 to return 5 G4 is forced to 0 and H4 to 5 If you experiment with the asset allocations in cells G4 H15 you can confirm that the forcing constraints work properly In C19 the constraint formula WB C17 gt C18 forces the Expected Return in C17 to be at least as great as the Target Return in C18 In C20 a constraint requires that 100 percent of the funds are invested That is that B3 D3 sum to 1 Now you re ready to solve the model After solving the WB Status worksheet will open in order to show you the Nonlinearity present warning This warning can be disabled from the General Options dialog box The solutions for all three objectives follow The PORTSCEN Worksheet After Minimizing Variance ORTSCEN xlsx Moot ool Eh Gem El L 14 PORE Wey MODEL Asset 2 Return Difference Forcing 52 5 33 9 i Over Under Constraints 130 0 122 5 i 125 4 104 0 0 110 3 129 0 118 8 3 8 0 0 121 6 121 6 124 4 94 0 0 954 72 8 87 3 0 0 27 7 92 9 114 4 103 4 0 0 116 105 6 107 0 104 8 0 0 10 2 103 8 132 1 K 114 7 0 0 0 3 108 9 130 5 3 2 125 0 10 0 0 0 109 0 119 5 a 111 6 0 0 3 4 108 3 139 0 119 3 4 3 0 0 103 5 92 8 A i 99 5 0 0 155 117 6 171 5 i 145 8 30 8 0 0 COON OME WUN Expected Return 115 0 Target Return 115 0 Variance Return gt Ta
357. osessseog j Delete all options and names in Help workbook Reset the options will remove the WBxxx names in the workbook These names belong to What sBest and were created via the Options dialog boxes Deleting hidden names will only remove names in the workbook that are not visible or listed via the Name Manager of Excel Deleting all names and options will actually remove any names from the workbook These names could have been defined by What sBest the user Excel or other applications saving names in this workbook file Removing these visible or invisible names will clean up the file and the user has to redefine the useful names ADDITIONAL COMMANDS 87 Advanced Dual The dialog box posted by the Advanced Dual command appears as follows l Dual S For Cell Range Report on Type Report Information in B 1 ml el The Dual command allows for reporting of dual values for the chosen cells or range of cells If you are requesting dual values for a range of cells then the range in Report Information in should be identical in dimension to the range in For Cell Range What sBest provides three dual value options in the Report on Type box The first Dual Value lets you define the dual value of a cell or range of cells The formula will be WBDUAL cell number The other two Upper Range and Lower Range let you define the upper and lower ranges for dual values The formul
358. ot be a valid point for the solver to retreat to and the solution process terminates with an error Turning off Selective Constraint Evaluation eliminates these errors The default setting for Selective Constraint Evaluation is on SLP Direction Check the SLP Direction checkbox to have What sBest s nonlinear solver use successive linear programming SLP techniques to compute new search directions This technique uses a linear approximation in search computations in order to speed iteration times In general the number of total iterations will tend to rise when this method is used However runtimes will tend to be shorter The default setting for SLP Direction is on ADDITIONAL COMMANDS _ 61 Steepest Edge When What sBest is not in Steepest Edge mode the nonlinear solver will tend to select variables that offer the highest absolute rate of improvement to the objective regardless of how far other variables may have to move per unit of movement in the newly introduced variable The problem with this strategy is that other variables may quickly hit a bound resulting in little gain to the objective When the steepest edge option is enabled the nonlinear solver spends a little more time in selecting variables by looking at the rate at which the objective will improve relative to movements in the other nonzero variables Therefore on average each iteration will lead to larger gains in the objective In general the Steepest Edge o
359. ot defined 396 CHAPTER 8 Runtime Errors If you did not place error handling into your VBA code then Excel handles a What sBest error by generating a run time error message box In many instances this runtime error message provides you with a What sBest error code a number over 30000 However in some cases you may only receive a Run time error 5 Invalid Procedure Call or Argument message box like the one shown below Whether you receive a What sBest error code or an Excel error code you must take steps to 1 correct the source of this error and 2 insert error handling into your code to anticipate any further errors Error handling is placed in your VBA code by adding a suitable On Error statement before any calls to What sBest routines and inserting some appropriate error handling code Including error handling in your code will prevent Excel runtime error messages like that shown above For an example of such code see the section entitled wbError and Error Codes in chapter VBA Interface If you don t correct the source of the error fail to insert error handling and you again run the routine that produced the error then Excel will generate the following cryptic error message Microsoft Visual Basic s i Run time error 91 Object variable or With block variable not set This message signifies that Excel has an unhandled error left over from the previous run and cannot display the new error tha
360. oting is divided into two major sections General and Operational Problems and Error Messages amp Warnings The General and Operational Problems section lists general problems and symptoms that may occur due to some operational problem especially in loading and running What sBest within Excel You will also find a Frequently Asked Questions section Error Messages amp Warnings is a list of messages that may be encountered in building and solving a model Most of these messages are returned in the WB Status Report If you encounter an error message that is not discussed here please contact LINDO Systems General amp Operational Problems Content The solution returned from What sBest is not optimal i e there is a solution that yields a better value in the best cell and satisfies all constraints An Excel error appears claiming that a workbook cannot be opened under High Security Level An Excel error appears referring to multiple copies of WBA XLA WBA XLAM When I try to close Excel an error appears stating This workbook is currently referenced by another workbook and cannot be closed The What sBest menu does not load There are Excel error codes of REF in What sBest cells General Problems Symptom The solution returned from What sBest is not optimal i e there is a solution that yields a better value in the best cell and satisfies all constraints Problem The model ma
361. ou to sell the entire baseball card collection While the 19 9 return is very appealing the thought of parting with even some of your baseball cards is frightening You ve decided you would like an average or expected return of at least 13 call this your desired return but you want to minimize the likelihood that you ll have to sell off some of those valuable cards Objective of Optimization The objective is to invest for a profit while minimizing the risk of diminishing your baseball card collection 270 CHAPTER7 The Worksheet Let s look at the DNRISK sample file The DNAISK Worksheet Before Solving i isk Stock1 Stock2 Stock3 0 0 0 0 0 0 Scenario Downside Forcing Return Risk Constraints 7 1 14 4 16 9 0 0 0 0 Not gt 5 6 10 7 3 5 0 0 0 0 Not gt 3 8 32 1 13 3 0 0 0 0 Not gt 8 9 30 5 73 2 0 0 0 0 Not gt 9 0 19 5 2 1 0 0 0 0 Not gt 8 3 39 0 13 1 0 0 0 0 Not gt 3 5 7 2 0 6 0 0 0 0 Not gt Average Desired Threshold Budget Investment Return Return Return Equals 100 Percent 0 0 Not gt 13 0 11 0 Not Average WI Downside Risk A Determine Adjustable Cells The adjustable cells are the percentages of your capital to invest in each of the three assets in cells C4 E4 and the percentages of Downside Risk for each Scenario in cells G6 G12 B Define Best The best cell is the average of the sum of the squares of the downside risk pe
362. ould be under the Partial method Devex is useful with degenerate models but it is generally difficult to predict in advance what method to use The default value for Primal Pricing is Solver Decides ADDITIONAL COMMANDS 59 Options Nonlinear Solver This command allows you to set a number of options controlling the nonlinear solver The dialog box posted by the Options Nonlinear Solver command appears as follows Nonlinear Solver Options Strategies Crash Initial Solution JV Presolve IT Quadratic Recog Iw Selective Constraint Evaluation Iw SLP Direction Steepest Edge Gather Information for Starting Point Optimatility Tolerance 0 0000001 Iteration Limit for Slow Progress 100 Derivatives Solver Decides Solver Version Solver Decides The nonlinear solver is an option on larger versions of What sBest You can try the nonlinear solver in the Demo version of What sBest or contact LINDO Systems to purchase a nonlinear solver license for a larger version For more information regarding the difference between linear and nonlinear models see Linear vs Nonlinear Expressions and Linearization Crash Initial Solution Check the Crash Initial Solution checkbox to have What sBest s nonlinear solver invoke a heuristic for generating a good starting point when you solve a model If this initial point is relatively good subsequent solver iterations should be reduced along with overall runtimes The
363. ow the prompts Following is a description of the screens that will be seen in the setup program The third screen entitled Setup Type will give a choice between a Default or Specified installation If for some reason you are prompted with a screen not described here follow the directions on it as it has the latest information available Please close all applications Excel included during the installation Version 32 bit 64 bit What sBest 64 bit should be installed with Excel 64 bit on a 64 bit operating system Otherwise install the 32 bit version of What sBest What sBest Example Files Example files can be installed on a local drive or any drive on the network in which you have write privileges It is important that What sBest users avoid sharing a directory for Excel data files If What sBest users share a common data directory over the network the data files of one user could be inadvertently overwritten by the data files of another user If this installation is for multiple users it is recommended that the example files XLS be copied to a private data directory for each user Setup Type When prompted for setup type of Default or Specified choosing Specified installation will ensure that you are prompted to confirm the location of the version of Excel you are installing with It also prompts you for the location of the What sBest program files which can be on a local drive or any drive on the network in which you hav
364. pecting a numeric argument Unexpected text arguments are treated as if they have a numeric value of zero Irreconcilable Constraint warning that a violated constraint does not depend upon adjustable cells and therefore cannot be reconciled Infeasible Constraint warning that the debugging feature will list the infeasible constraints These constraints are contributing to the infeasibility of the model Unbounded Variable warning that the debugging feature will list the unbounded variables These constraints are contributing to unbound the model Support Lookup Functions warning that the Lookup functions will be read and processed otherwise these functions will be treated as unsupported functions The default is to display all warnings except the Reference to Blank Cell and the Infeasible Constraint debugger warnings Update Links Select the Update Links button to update all What sBest functions in the current workbook to point to the present location of the What sBest add in file WBA XLA When a file is opened that contains What sBest functions created on a system in which the What sBest program files were in a different location Excel displays a message about automatic links and asks Update all Linked Information You may reply No to this dialog box given that the Excel link update won t update the links to the What sBest add in If you now look at a cell containing one of the What sBest
365. ppears when What sBest could not access the worksheet in order to copy the solution back to the adjustable cells or the temporary solution file could not be read First verify that your worksheet is not protected or locked Also make sure Excel is calling the right add in in the Library folder When using Function Support the system seems to hang with the message Server Busy Usually this Windows message appears when a Microsoft Office component could not access the add in in order to read it and to execute it If your add in has been digitally signed you will need to enable this add in to run your model then retry the call In some situations a time delay may occur after enabling the add in Also make sure Excel is calling the add in in the right folder In Excel 2002 there is an additional security setting that the user needs to set for accessing and executing a macro from the Solver Select Trust Access to Visual Basic Project via the Tools Options Security MacroSecurity TrustedSources menu Then save the model and restart Excel In Excel 2007 This feature can be set via the OfficeButton ExcelOptions TrustCenter TrustCenterSettings What Excele file format should be used In Excel 97 2003 the file format is XLS with a maximum size of 256 columns and 65536 rows per sheet In Excel version 2007 the new format is either XLSX XLSB for workbook or XLSM for macro enabled workbook with
366. procedure or remarks The procedures with arguments that require cells can accept either range objects or the address of the cell s Argument with square brackets and around them are optional non bracketed arguments are required 138 CHAPTER 4 wbAddAdjustableStyle This routine is used to add the adjustable style to a workbook that does not have one yet Syntax wbAddAdjustableStyle The wbAddAdjustableStyle procedure has no arguments Error Codes Description Adj_CreateStyleError Error Creating an Adjustable style Remarks Because this procedure is called by the wbAdjust procedure in most cases it should not be necessary for the developer to call it directly wbAdaBestStyle This routine is used to add the best style to a workbook that does not have one yet Syntax wbAddBestStyle The wbAddBestStyle procedure has no arguments Error Codes Description Best_CreateStyleError Error Creating a Best style Remarks Only one cell the objective cell should use this style wbAddWBMenu This routine is used to add the WB menu item to the Excel menu Syntax woAddWBMenu The wbAddWBMenu procedure has no arguments Error Code Description AddWBMenuError Error in adding the WB menu Remarks This procedure and the wbDeleteWBMenu procedure allows a developer to remove and add the WB menu as desired MACROS THE VBA INTERFACE 139 wbA
367. ptimization is to maximize the total value of the items loaded onto the truck without exceeding the truck s weight capacity 368 CHAPTER7 The Worksheet Let s look at the TRUCK sample file The six items scheduled for shipment A6 A11 along with their corresponding dollar Values B6 B11 and Weights C6 C11 appear in the worksheet D TRUCK LOADING Proportion Proportion Weight Loaded Loaded 7500 lt 7500 lt 3000 lt 3500 lt 4000 lt 3500 lt Total Value Total Weight Maximum of Load of Load Load Weight 0 10000 A Determine Adjustable Cells The adjustable cells are the Proportion Loaded of each item in D6 D11 B Define Best The best solution is the one that maximizes the value of the load in cell B15 The formula there SUMPRODUCT B6 B11 D6 D11 multiplies each individual item s value times the Proportion Loaded and sums them SAMPLE MODELS 369 C Specify Constraints There are two constraints The first in D15 specifies that the Total Weight of Load C15 remain less than or equal to the Maximum Load Weight E15 The second in E6 E11 requires that the number of each item loaded onto the truck is less than or equal to 1 Now let s solve the worksheet TRUCK Worksheet After Optimization by Conventional Method lale ez TRUCK LOADING Proportion Proportion Value Weight Loaded Loaded 22 500 7500 0 33333333 lt 24 000 7500 T 8 000 3000 7
368. ption will result in fewer iterations However each iteration will take longer The default setting for Steepest Edge is off Starting Point The Gather Information for Starting Point option may be useful when solving nonlinear models This option computes the slack values for the constraint cells and uses these values internally to compute an improved starting point After solving the spreadsheet goes back to the original Slack Indicator display The default value for Gather Information for Starting Point is off Optimality Tolerance Specify the Optimality Tolerance in this text box The nonlinear solver operates by taking each variable and making small changes to assess the rate of improvement in the objective function The nonlinear solver s Optimality Tolerance is a tolerance placed upon that rate of improvement calculation for the objective function If for a given variable the computed value proves to be less than or equal to the Optimality Tolerance then the solver does not attempt further adjustments to that variable s value Decreasing the Optimality Tolerance toward zero will tend to make the solver run longer but may lead to better solutions to poorly formulated or poorly scaled models The default setting for Optimality Tolerance is 0 0000001 62 CHAPTER3 Iteration Limit for Slow Progress Specify an integer limit upon the number of successive iterations made without a significant improvement in the objectiv
369. quire the number of people assigned to each shift to be greater than or equal to the number required On the Monday Days shift Model D8 for instance the formula is WB ASSIGNMENTS D3 ASSIGNMENTS D8 ASSIGNMENTS D13 ASSIGNMENTS D18 gt D7 This adds the shift assignment variables for the Monday Day shift for all four staff members and forces this sum to be greater than or equal to at least as great as the number of people required for that shift The model uses an equality constraint for each staff member to force the number of shifts assigned each week to equal the number he s expected to work The formula in cell L3 of the Constraints worksheet for example WB ASSIGNMENTS B3 SUM ASSIGNMENTS D3 J3 forces this equality for Reagan Assigning him to more than 5 shifts causes the constraint to return Not Finally the model must assure that after working one shift a staff member isn t assigned to either of the following two shifts and that each staff member is assigned to only one shift per day This is accomplished by summing the assignment variables 0 s and 1 s for each set of three consecutive shifts and forcing that total to be less than or equal to one If for instance a staff member were assigned to two shifts out of a consecutive block of three the constraint would be violated Examine the formula in D3 and D4 of the Constraints worksheet WB SUM ASSIGNMENTS D3 D5 lt 1 and WB ASSIGNMENTS D4 AS
370. r what values are chosen for the adjustable cells 0 will never be greater than 1 What sBest will provide you with a list of the irreconcilable cell formulas You will need to either delete them or correct them so that they are functions of the adjustable cells To correct them check the cell formulas for errors and be sure that all the adjustable cells have been specified TROUBLESHOOTING 409 ITRLIM Iteration Limit Reached By default What sBest does not put a limit on the number of solver iterations However by using the Iteration Limit option on the General Options dialog box posted by the Options General command the user may limit the number of iterations When the program has executed the number of iterations you specified without finding a solution the following warning message will be displayed on the WB Status tab Warning Message KAKA KEE KEKE KEES b INTERRUPTED S EEEEEEEEEEEKEEEEEEEE EE EE EEN WARNING Iteration Limit Reached Help Reference ITRLIM The limit for the maximum number of iterations was reached before the final solution could be found Check the solution carefully it may be sub optimal or infeasible The iteration limit is set via the WB Options General menu Suggestions The iteration limit is set through the WB Options General menu In some cases the solver may be able to return the best solution found so far after hitting an iteration limit However the same precautions regard
371. r Range whichever is appropriate and clicking OK In the box below we used cells D6 and E6 for the lower and upper ranges respectively and then re solved Standard Deluxe Dual Val 100 S 251 Low Rng Upp Rng 300 500 Product Component Requi Quantity Required Total Standard Deluxe Usage 1 0 1 TUTORIAL SAMPLE ADDITIONAL COMMANDS 99 The ranges for the number of Deluxe computer towers to produce are 25 upper and 5 lower That is the dual value will stay at 100 as the quantity of Deluxe computers to produce varies from 5 through 25 If cell DS Quantity to Produce is increased to 15 made non adjustable using the Remove Adjustable command via the Adjustable dialog box or the toolbar button and the model is re solved then profit decreases to 13 500 15 000 100 15 as shown in the following LS XYZ xlsx Microsoft Excel Low Rng Upp Rng 300 500 Product Component Requi Quantity Required Total Standard Deluxe Usage 1 0 20 lt 1 15 lt 1 2 50 lt TUTORIAL SAMPLE 100 CHAPTER 3 Note that since the dual value formulas now refer to a non adjustable cell What sBest ignores these cells and they are not changed If the quantity of Deluxe computers to produce is decreased to 3 and the model is re solved the profit increases to 15 300 15 000 100 3 and the number of Standard to produce increases to 56 as shown in the following This might occur if there
372. r What sBest that has been added to the original project and this includes a few subroutines that have been added to build and solve the XYZ model Please pay particular attention to the code for error handling provided in the subroutine VBABuildXYZ This form of error handling allows your Visual Basic application to handle the most common What sBest errors in a manner appropriate to your application Concealing and Protecting a Model from the User What sBest allows model developers to hide aspects of a model that might be proprietary and protect sections of a model from inadvertent or unauthorized modification while allowing the user to access and modify input data While this does not require the use of Visual Basic these techniques can easily be used in conjunction with VBA Information in a What sBest model that is to be hidden from the user must reside on one or more worksheets that are separate from the worksheet s to which the user has access Similarly information that is to be protected must reside on one or more worksheets that are separate from the rest of the model To demonstrate how to conceal and protect a model from the user we will modify the Product Mix model First we rename the Prodmix sheet as Model Second we will create two new worksheets that have all the information that the user needs one called Inputs and the other called Outputs When we are done the Model sheet will be entirely hidden and protected fro
373. r additional discussion of the functionality provided see the section entitled Integer Special Ordered Set which refers to the Special Ordered Set Cardinality dialog box that calls this procedure This routine is used to build the Cardinality ranges WBCARD Adjustable cells contained in these ranges will be restricted to integral values by the What sBest solver If you wish to delete a CARD function using VBA instead of deleting it on the spreadsheet you can use Selection Clear This statement clears the selected cell from any of its content Syntax wbIntegerCard CardNumber ArgList Refers_to NoErrDialog CardNumber CardNumber i is an integer meaning that at most N of the variables in the set will be allowed to be nonzero ArgList ArgList is a string list of range or cells reference separated by a comma Refers_to Refers_to is the range or address of the cell s to be specified as SOS NoErrDialog Any argument passed here causes all What sBest error dialog boxes from the wbIntegerCard routine to be suppressed Instead the error number is delivered in this return argument Pass an integer to use any possible returned What sBest error number Useful in an embedded application of What sBest MACROS THE VBA INTERFACE 157 Remarks Integer variables can dramatically increase the solution time Example To constrain cell F6 to be a general integer using the range name staff enter wbIntegerCard
374. r an example of a simple nonlinear to linear conversion consider the following equation X Y 10 As written this equation is nonlinear because of the division by Y By simply multiplying both sides of the equation through by Y this can be converted to the equivalent linear equation X 10 By utilizing as many nonlinear to linear conversions as possible the nonlinear model might possibly be reduced to a linear model When a nonlinear model can be fully linearized the payoff can sometimes be enormous e g finding a solution where none could be found before or reducing solution time from days to seconds The sample model Linearization Option and Construction Cost Estimation demonstrates the benefits of successful linearization The What sBest linearization option cannot reduce all nonlinear models to linear models nor does the effort necessarily improve the solution time In some exceptional cases it may be better to run What sBest with linearization disabled For this reason the linearization option is provided with some user adjustable parameters that are set in the General Options dialog box You may wish to try different degrees of linearization to see what works best for your model It is highly recommended that you consider performing linearization on your own If What sBest indicates that your model is nonlinear it can be beneficial to investigate the expressions in the nonlinear cells and determine whether any could be refo
375. r discrete distributions with infinite event space like the Poisson distribution Note Sampling a scenario tree prior to the optimization process is also called pre sampling This is to distinguish this type of sampling from the one that is used during optimization process In What sBest sampling refers to pre sampling unless otherwise is stated SAMPLING N EVENTS FROM Q e gt j ae D 00y o P e UN wo l o Poy AN i OD ee i o l Ee P2 Decision Decision K On O Plon UN i OSS i Ss Ow Onn NS Si Plon VN SS N Se T Ploy 1 N Uncountably many scenarios each with A finite set of sampled scenarios each with a point probability of zero a point probability of 1 N Note Because the point probability of each scenario in the original model is zero it is customary to set the probabilities of sampled scenarios to 1 N However the user can always define customized sampling approaches to work with different scenario probabilities 222 CHAPTER 6 Given the parametric distribution of each stochastic parameter What sBest s sampling routines can be used to generate univariate samples from these distributions efficiently The user has the option to use antitheticvariate s or Latin hypercube sampling to reduce the sample variance Generating Dependent Samples In certain situations the modeler may require some of the random variables to be dependent on each other There are several w
376. r of the K Best Solutions box to 5 via the Options Integer Solver K Best Solutions Desired Number 5 Specify Reporting Cells We also define the cells to look at via the button Specify Reporting Cells here all the Adjustable cells select the range of cells or cell by cell select the cell to refer then click Add You can modify or delete an entry by clicking on the cell reference in the list Cell H3 has the formula WBIKB_REP F3 F10 K Best Trade off Cells a axe Select any cells to report multiple selection OK holding the Ctrl key Knapsack F 3 F 10 List of selected cells select and change by gei Add or Remove Knapsack F 3 F 10 Add Remove ADDITIONAL COMMANDS 81 This means that we would like to generate the 5 best solutions to the model We then click OK and then run the Solve command The solver sees that the K Best feature is being requested and it automatically generates the 5 best solutions to the model At which point we are presented with the following dialog box What sBest K Best Preview for Integer Solver axe Select the Desired Run Yalues for the Trade off Cells KNAPSACK IF10 1 000000 KNAPSACK IF3 0 000000 KNAPSACK IF4 0 000000 KNAPSACK IFS 0 000000 KNAPSACK IFG 0 000000 KNAPSACK IF 0 000000 KNAPSACK FS 1 000000 KNAPSACK IFS 1 000000 WBMAX 25 000000 Report solution into the spreadsheet This Selection Take Default
377. r solutions in an intelligent fashion minimizing the number of solutions that have to be explicitly examined However the number of potential solutions grows exponentially with the number of integer adjustable cells Thus models with a large number of integers may take a very long time to solve Options such as the optimality tolerances set limits upon how exhaustively this branch and bound search will be carried out Optimality Tolerance is particularly useful in some integer problems as a means of significantly decreasing the solution time Branching Direction Use the Direction drop down box in the Branching box to govern the preferred direction of branching Branching occurs when the branch and bound manager forces an integer variable that is currently fractional to an integer value When the branching direction is set to Up the branch and bound manager will branch on a fractional integer variable by first forcing it to the next largest integer The reverse is true when this option is set to Down When the option is set to Both the branch and bound manager makes an educated guess as to the best initial branching direction for each fractional variable The default setting is Both Integrality Due to the potential for round off error on digital computers it is not always possible to find exact integer values for the integer variables The two tolerances in the ntegrality box Absolute and Relative control the amount of deviation from int
378. ramework 3 5 or higher Hardware e Pentium class PC e 500 MB of RAM 50 MB of free disk space An Internet connection is required to download the latest version of What sBest You can also contact LINDO Systems to obtain a copy Add ins library and executable files from LINDO Systems are digitally signed Additional information can be found in the nstallation Overview 4 CHAPTER 1 Installation Overview To get up and running quickly first close all programs and simply run setup exe from the Windows desktop or from the CD and follow the prompts in the dialog boxes that follow The setup program offers Default and Specified installation The Default installation option analyzes your system for the critical information required to successfully install What sBest When you select Default as the installation type in the initial dialog box the add in file is copied into a directory entitled Library within the main Excel directory The Specified installation option is available for situations that require more information from the user and for users who prefer controlling the details of installation See the section entitled Jnstallation for more information Default installation is recommended for most environments However we recommend you select Specified installation under the following situations You have more than one copy release of Excel installed on your system e Excel is installed on a network server rather t
379. range 88 Central Differences 62 Checking license 33 CheckUpdate 125 Classification Data 35 Coefficient Reduction 70 Coefficients 33 Cold Start 73 Commercial Version 120 concave 25 30 Concavity 198 conic programming 60 224 417 Constraint cells 9 automatic generation of 14 creating 25 Constraint Cuts 69 Constraint Ranges 95 Constraint Related Problems 29 Constraints 33 120 Button 25 command 25 dialog box 25 Constraints 25 Contacting Lindo Systems 449 Container 367 convert model format xiv 116 convex 25 30 convex 197 Convexity 197 Copyright 2 COS 187 COSH 187 Cost 230 235 239 257 265 289 294 321 327 333 345 356 364 Covariance matrix 261 Crash Initial Solution 59 Crop Allocation 372 CROPALLOC XLSX 372 CUTSTOCK XLSX 321 D Default 4 Define Best 15 Delta 64 Demand 306 321 364 Dependent random variables 222 Depth First 65 76 Derivatives 62 Design 239 Determine Adjustable cells 9 Determine Best 9 Determine Constraints 9 Devex 58 Direction 33 37 Disaggregation 70 Disclaimer 2 Discrete distribution table 222 Disk space 3 Distribution 189 360 DNRISK XLSX 269 Downside risk 273 Dual 73 Dual dialog box 87 Dual Pricing 58 Dual Value 88 of a Constraint Cell 89 of an Adjustable cell 91 of Zero and Multiple Optima 94 Dual values 89 and Multiple Optima 94 in Integer Problems 94 in Nonlinear problems 94 in Nonlinear Problem
380. rcentages D19 The sum of squares is used to increase the relative penalty on outcomes that are far below the return threshold SAMPLE MODELS 271 C Specify Constraints There are three kinds of constraints in this model The Forcing Constraints in 16 112 force the Downside Risk for each Scenario to be either the Threshold Return minus Scenario Return when the Scenario Return is less than the Threshold Return or zero when the Scenario Return is more than the Threshold Return For example the formula in I6 is WB G6 gt E 16 F6 Since G6 is an adjustable cell which is prohibited from returning a negative value and since this is a minimization problem it will return the smallest possible nonnegative value If E16 Threshold Return is less than F6 Scenario Return E16 F6 is a negative number so the constraint will be satisfied by zero which is greater than any negative number The constraint in H16 forces the sum of percentages invested in each stock to be 100 The constraint in C16 forces the average return for the scenarios to be greater than or equal to the Desired Return 272 CHAPTER7 Now let s solve the model After solving the WB Status worksheet will open in order to show you the Nonlinearity present warning This warning can be shut off from the General Options dialog box Your solved model now appears as follows The DNAISK Worksheet After Solving aa Downside Risk Portfolio Model E WS
381. re the units of bonds purchased for each bond quoted E4 E12 B Define Best The Best possible solution is the particular mix of units of bonds purchased that result in minimum initial investment in cell F14 Examine the formula for the Total Cost cell SUM F4 F12 This formula sums the amount invested in each bond issued for each of the nine bonds offered 254 CHAPTER7 C Specify Constraints The constraints in this problem B18 F18 require that the total cash inflows in B17 F17 Ge from the interest and or principal income each year remain greater than or equal to cash outflows needed per period in B19 F19 The cells B17 F17 Amount Covered contains formulas that add the cash received from bond interest payments as well as the repaid principal when applicable for each year This Income Stream is detailed in cells 14 M12 and appears as follows _ The BONDS Worksheet Part 2 Before Optimization r Income Stream Year of Year 0 Year 2 Year 3 Maturity 0 0000 0 0000 0 0000 0 0000 0 0000 0 0000 0 0000 0 0000 0 0000 For instance suppose that in the beginning of year 0 your client invests in 10 units of a 10 percent 3 year U S Treasury bond see the bond listed in C10 F10 A 10 0 would then be inserted in cell E10 Each unit represents a 1000 face value or principal SAMPLE MODELS 255 Thus in the current year your client will receive an interest payment of 1 00 or 10 units 10 rate
382. re you use parenthesis or remove the parenthesis ADDINLINK Multiple Add in Links In the case of multiple installations of What sBest the solver may be confused on the source link to the add in The following error message will be displayed on the WB Status tab Error Message ERROR Multiple Add in Links Help reference ADDINLINK The workbook contains multiple links to the What sBest add in Only one source should be defined in the workbook Make sure to correct the add in link reference via the menu Edit Links or OfficeButton Prepare EditLinksToFiles and remove the corrupted link Then update the workbook links via the menu WB Options General UpdateLinks save the file and reopen it Suggestions Make sure of the location of the current add in via the Excel menu Tools Add ins or OfficeButton ExcelOptions Add Ins Go You may need to reset the add in by browsing to the installation location Remove any corrupted or wrong add in link from your workbook via the menu Edit Links or OfficeButton Prepare EditLinksToFiles Reset the links on the worksheet to the What sBest functions via the menu WB Options General UpdateLinks Save the file close Excel and reopen it 398 CHAPTER 8 ARITHERR Arithmetic Error Many mathematical operations are not defined e g division by zero What sBest tries to avoid such errors but this is not always possible When an irresolvable numerical err
383. report the simplest syntax would be wbStochasticHistogram 0 Sheet1 C 1 wbStochasticReport This routine can be used to set the stochastic reporting cells seen in the Stochastic Support dialog box All arguments are required except for the error code For additional discussion of the options available through this routine see the section entitled Advanced Stochastic Support Syntax wbStochasticReport ArgList Place_in_cell NoErrDialog ArgList ArgList is a listing of cell MACROS THE VBA INTERFACE 183 references for indicating the cells to display in the stochastic report Place_in_cell Place_in_ cell is a cell reference to specify the cell in which to write the WBSP_REP function NoErrDialog No Any argument passed here causes all What sBest error dialog boxes from the wbStochasticFunction routine to be suppressed Instead the error number is delivered in this return argument Pass an integer to use any possible returned What sBest error number Useful in an embedded application of What sBest Sto_BadArgListArg Bad ArgList argument Sto_BadPlaceInCellArg Bad PlaceInCell argument Sto_ProtectedSheetRepError Bad ProtectedSheetRep argument Examples To set several reporting cells so to display the stochastic report the simplest syntax would be wbStochasticReport Sheet1 B 1 B 3 Sheet1 C 1 A1 0 wbStochasticStageScenario This routine can be used to set the stochastic sta
384. rget gt Semi Variance 0 0105 20 Invest Total 100 Downside Risk 0 0572 SAMPLE MODELS 277 Variance minimizes to 0211 Solutions for downside risk and semi variance appear below The PORTSCEN Worksheet After Minimizing Semi Variance Asset2 Asset3 Prob Return Difference Forcing 0 1 53 0 Over Under Constraints 122 5 114 9 8 3 122 0 7 0 0 0 129 0 126 0 8 3 118 7 3 7 0 0 121 6 141 9 8 3 1324 174 0 0 72 8 922 8 3 93 7 0 0 21 3 114 4 116 9 8 3 105 7 0 0 9 3 107 0 965 8 3 100 8 0 0 14 2 132 1 113 3 8 3 108 9 0 0 6 1 130 5 173 2 8 3 143 0 28 0 0 0 119 5 102 1 8 3 105 4 0 0 9 6 139 0 113 1 8 3 110 9 0 0 41 92 8 100 6 8 3 101 9 0 0 13 1 171 5 190 8 8 3 156 5 415 0 0 116 6 115 0 Variance 0 0328 gt Semi Variance Downside Risk 0 0649 After solving the semi variance is minimized to 0089 278 CHAPTER7 The PORTSCEN Worksheet After Minimizing Downside Risk z PORTSCEN xlsx Microsoft Excel bala MO SCENARIO MODEL Asset 1 Asset 2 Asset 3 Return Difference Forcing 66 124 81 0 Over Under Constraints 130 0 122 5 114 9 116 8 1 8 00 110 3 129 0 126 0 125 3 10 3 0 0 121 6 121 6 141 9 138 0 23 0 0 0 95 4 72 8 92 2 90 0 0 0 25 0 92 9 114 4 116 9 115 0 0 0 0 0 105 6 107 0 965 98 4 0 0 16 6 103 8 132 1 113 3 115 0 0 0 0 0 108 9 130 5 173 2 163 6 48 6 0 0 109 0 119
385. rgument value For more information on using the stochastic feature refer to section Advanced Stochastic Support 428 CHAPTER 8 SPHIST Stochastic WBSP_HIST Format Using the stochastic feature What sBest can return the histogram information on any cells These values tell you how sensitive the solutions are to various stages of the model The WBSP_HIST function is used to request histogram values on any cell If the arguments to this function are not correctly specified the following error message will be displayed on the WB Status tab Error Message 4K ERROR Aa Stochastic WBSP_HIST Format Help Reference SPHIST A WBSP_HIST cell is incorrectly formatted Correct the formula and the validity of the arguments in the cell below The format should be WBSP_HIST number cells where the number states the number of desired bins and cells must refer to any cells cell address Suggestions There is an incorrectly formatted Stochastic Histogram Report cell in the model A reporting cell must use the format WBSP_HIST number cells where number is a numeric number referring to the number of bins and cells is a reference to the cells you want to report the histogram The number can be left to 0 so the solver will decide the number of bins The ce s reference can be any cells For more information on using the Stochastic Histogram values refer to section Advanced Stochastic Support SPRAND Stochastic WBSP
386. rient D in the blend Z standard deviation Nutrient D gt 21 For 95 confidence in the content of Nutrient D set Z 1 645 Refer to any elementary text on statistics for a discussion of Z values Objective of Optimization The objective is to determine how much of each grain you should buy at today s prices to meet their nutritional requirements at lowest cost 236 CHAPTER7 The Worksheet Let s look at the HOGCHANC worksheet as supplied in your sample files r B SWINE amp ROSES Hog Farm Nutrients Per Unit Weight of Grain Nutrients Minimum Dual ltem 1 2 3 4 Supplied Regd Value Nutrient A 22 3 4 T2 d 0 0 Not gt 24 1 00 Nutrient B 14 1 1 0 0 i 0 0 Not gt 07 1 00 Nutrient C 2s 5 6 11 1 0 0 Not gt 50 1 00 Nutrient D Mean Value 12 0 11 9 41 8 52 1 0 0 Not gt 21 0 1 00 Variance 0 3 2 2 20 5 33 2 Cost Weight 35 00 50 00 80 00 95 00 Weight Units Unity to Purchase 0 0 0 0 0 0 0 0 Not Dual Value 1 00 1 00 1 00 1 00 A Determine Adjustable Cells The cells What sBest is free to manipulate in this model are the Weight Unit Percentages to Purchase for each grain C18 F18 B Define Best The optimal solution to this problem is the one that meets needs at lowest cost If you examine the formula in the Total Cost cell J18 SUMPRODUCT C18 F18 C15 F15 you see that every change in price or purchase quantities has a direct bearing on cost What sBest of course
387. rmatted Correct the formula and the validity of the arguments in the cell below The format should be WBSEMIC lower upper cells where min and max are constant numbers and elle must refer to adjustable cells cell address Suggestions There is an incorrectly formatted WBSEMIC function in the model A WBSEMIC function cell must use the format WBSEMIC lower upper cells where lower and upper are numbers and cells is a reference to the cells you want the semic continuous on The cell reference should be to an adjustable cell For more information on range values refer to section Integer Semi continuous 438 CHAPTER 8 WBSOSFORM Special Ordered Sets Cell Format If requested What sBest can specify Special Ordered Sets Range values tell you over what range a particular set is valid The WBSOS1 WBSOS2 or WBSOS3 functions are used to request SOS ranges If the arguments to this function are not correctly specified the following error message will be displayed on the WB Status tab Error Message Krk ERROR Special Ordered Sets Cell Format Help Reference WBSOSFORM A SOS cell is incorrectly formatted Correct the formula and the validity of the arguments in the cell below The format should be WBSOS1 cells where cells must refer to adjustable cells cell address Suggestions There is an incorrectly formatted WBSOSx function in the model A WBSOSx function cell must use
388. rmine the percent to invest in each asset while minimizing the risk of the entire portfolio 266 CHAPTER7 The Worksheet Let s look at the PORTCOST sample file The PORTCOST Model Before Solving e BE EE COSTS Transaction Cost Return Rate Begin Asset1 9 0 1 0 50 0 Asset2 13 5 1 5 30 0 Asset3 18 0 2 0 20 0 Covariance Matrix Asset 1 25 0 25 0 0 Asset 2 25 150 0 40 0 Asset 3 0 0 40 0 256 0 Variance 35 54 A Determine Adjustable Cells Asset 1 Asset 2 Asset 3 Sold ht End 0 0 0 0 50 0 0 0 0 0 30 0 0 0 0 0 20 0 End Value Desired Port Return End Value 112 2 109 5 Proceeds Cost of From Sales Buys 0 0 gt 0 0 The adjustable cells are the percentages of each asset to be Bought or Sold in cells E7 F9 B Define Best The best solution minimizes the variance of the portfolio This is accomplished in cell B18 by means of the following formula explained in the previous sample model G7 2 B14 G7 G8 B15 G7 G9 B16 G8 G7 C14 G8 2 C15 G8 G9 C16 G9 G7 D14 G9 G8 D 15 G9 2 D 16 SAMPLE MODELS 267 C Specify Constraints Cell E13 calculates the value of the ending portfolio plus the return over the period with the formula 1 B7 G7 1 B8 G8 1 B9 G9 The constraint in cell F13 WB E13 gt G13 ensures that the End Value Port Return E13 is greater than or equal to the Desired End Value G13 of 109 5 of the original portfolio value
389. rmulated in a linear manner Note The locations of all nonlinear cells will be displayed in the staus report assuming that the Nonlinearity present warning has been enabled in the General Options dialog box The Solution Process Determining Optima Local Optima vs Global Optima Linear and nonlinear problems are distinguishable by their different kinds of solutions When the What sBest linear solver finds a solution to a linear optimization model that solution is a best solution Such a solution is called a global optimum because there is no better feasible solution By contrast conventional nonlinear solvers are only able to identify local optima i e solutions for which no better feasible solutions can be found nearby What sBest tells you what type of solution you have reached by returning either Globally Optimal or Locally Optimal in the Solution Status field of the status report If a model is nonlinear and the global solver is not used then the status will always be locally optimal if a solution is found It is a property of nonlinear optimization models that they may have several solutions that are locally optimal Such locally optimal solutions to nonlinear optimization models cannot be assumed to be 196 CHAPTER6 globally optimal The following example illustrates this property and explains how you might explore the various local optima Consider the following model Cell B3 has been specified as an adjustable cell B4
390. roduct 2 and so on You want to find a combination of values B8 G8 that will yield the highest Total Profit A3 312 CHAPTER7 Note The SUMPRODUCT method of expression eliminates the need for lengthy formulas while simplifying later addition of other products Inserting rows not appending within the ranges allows you to enlarge them without rewriting the Total Cost formula In writing small models you may find it easier to write such formulas without SUMPRODUCT In large models or ones you intend to enlarge it s probably best to use SUMPRODUCT C Specify Constraints The constraints in this problem 114 119 must reflect the requirement that total usage of raw materials remain less than or equal to the quantities of raw materials in stock This necessitates a separate constraint for each of the six raw materials For instance the constraint in 114 WB H14 lt J14 requires that the Total Usage H14 be less than or equal to the Starting Inventory J14 Total Usage H14 H19 is determined based on the Product Resource Requirements defined within the model B14 G19 For instance the formula in H14 for Total Steel Usage is SUMPRODUCT B14 G14 B 8 G 8 That is interpreted as the Steel requirement per unit of Product 1 Quantity Produced of Product I Steel requirement per unit of Product 2 Quantity Produced of Product 2 and so on What if vs What s Best Spend a few minutes working with the model to try to max
391. rofit unit values could have been read from a database or a text file This model is a looping version of the Product Mix model VBA Interface Procedures What sBest provides a set of Visual Basic Procedures that allow models to be built and automated using Visual Basic for Applications VBA commands These procedures are the same ones that are called when What sBest commands are selected from the WB menu or the What sBest toolbar For example the Solve procedure invoked in a macro is also invoked when you press the button on the What sBest toolbar If you wish to automate the creation of a What sBest model or if you wish to incorporate What sBest as a part of a larger application then you will find these procedures to be invaluable They make it relatively easy to incorporate What sBest into a Microsoft Excel or Access application or any other application offering Visual Basic for Applications All of the following procedures with arguments requiring cells can accept either range objects or the address of the cell s Note VBA is case insensitive to the procedure name when it is called so any procedure could alternately be called with its name typed entirely in lower case Oe as wbaddadjustablestyle instead of wbAddAdjustableStyle All of the What sBest procedures available are listed below with details on their syntax arguments error codes and any return values Some also include examples of using the
392. ror 431 String Result of Formula 432 Unbounded Model 433 Undefined Reference 434 Unsupported Functions 400 Unsupported Lookup Usage 418 Unsupported Name 420 Unsupported Sumif Usage 432 Upper Range Cell Format 438 Error messages amp Warnings Multiple Add in Links 397 Error messages amp Warnings Arithmetic Error 398 Error messages amp Warnings Blank Cell Warning 398 Error messages amp Warnings K Best WBIKB_REP Format 404 Error messages amp Warnings Indicator Model Cell Reference 404 Error messages amp Warnings License key Dongle Required 415 Error messages amp Warnings Mixed Integer Solver Not Licensed 416 Error messages amp Warnings Stochastic Solver Not Licensed 417 Error messages amp Warnings Conic Solver Not Licensed 417 Error messages amp Warnings Stochastic WBSP_CORR Format 425 Error messages amp Warnings Stochastic WBSP_DIST 1 Format 425 Error messages amp Warnings Stochastic WBSP_DIST 2 Format 426 Error messages amp Warnings Stochastic WBSP_DIST 3 Format 426 Error messages amp Warnings Stochastic Error 427 Error messages amp Warnings Stochastic WBSP_HIST Format 428 Error messages amp Warnings Stochastic WBSP_RAND Format 428 Error messages amp Warnings Stochastic WBSP_REP Foramt 429 Error messages amp Warnings Stochastic WBSP STSC Format 430 Error messages amp Warnings Stochastic WBSP_VAR Format 430 Error messages amp Warnings Cardinality Cel
393. rrect paths to your What sBest add in it will present you with a message about automatic links and then ask Do you want to update all linked information Answer Yes and use the Browse button to find the WBA XLA or WBA XLAM file in your LIBRARY subdirectory of your main Excel directory If this does not remove the REF error then use the What sBest Update Links button on the General Options dialog box This should correctly update the path If you fail to update the links with the What sBest Update Links button then Excel will place the error code of REF into the cells with the incorrect path to WBA XLA or WBA XLAM If you frequently exchange models with other What sBest users that have their program files in a different location or you have updated or re installed the What sBest program files in a different location please refer to the discussion of the Update Links button under Options General 390 CHAPTER 8 Frequently Asked Questions Content What are the system requirements to install What sBest How do I install What sBest add in on my computer Where are the What sBest add in files installed on my computer What can I do if I receive an error message Can I protect my workbook from viewing How large can my model be How do I fix the Error Opening File error message How do I fix the Error in Auto_add 5 Invalid procedure or call argument error message How do I fix the Error in retur
394. rue True True _ To set one or more options named arguments should be used wbSetNonlinearOptions OptimalityTolerance 0 00003 MACROS THE VBA INTERFACE 175 wbSetStochasticOptions This routine can be used to set the What sBest stochastic solver options seen in the Stochastic Solver dialog box All arguments are optional For additional discussion of the options available through this routine see the section entitled Options Stochastic Solver Syntax wbSetStochasticOptions StochasticSolverMethod StochasticSeed StochasticSamp Size StochasticSamp Cont StochasticEVWS StochasticEVEM StochasticEVPI StochasticEVMU StochasticRepScenHoriz StochasticSolverMethod StochasticSolverMethod is an i integer indicating the stochastic solver method to employ 0 Solver Decides 1 Free 2 Deterministic Equivalent 3 Nested Benders Decomposition 4 Augmented Lagrangian Decomposition 5 Simple Benders Decomposition StochasticSeed StochasticSeed is a cell reference or an integer greater than 1 indicating the seed value for random generator StochasticSamplSize StochasticSampISize is an integer greater than 1 indicating the default number of scenarios per stage StochasticSamp Cont StochasticSamp Cont is a True False flag indicating whether or not to use this option StochasticEVWS StochasticEVWS is a True False flag indicating whether or not to use this option StochasticE VEM Stocha
395. running a trial version with limited capacity or you can contact LINDO Systems to obtain a new license LICKEY4 Pending Expiration of Option Trial Some versions of What sBest include trial licenses for some of the optional capabilities e g the global solver If the trial license for the options is about to expire the following warning message will be displayed on the WB Status tab Warning Message WARNING Pending Expiration of Option Trial Help Reference LICKEY4 Your trial period for the optional features will expire at the end of the day You can continue to use What sBest but some of the optional features may no longer be available Suggestions Some installations come with a free trial period for the optional solvers Nonlinear Global and Barrier In this case the trial period is about to expire What sBest will continue to operate but you will no longer have access to all the optional solver engines You may want to contact LINDO Systems to order a license upgrade that includes permanent access to one or more of the optional solvers TROUBLESHOOTING 415 LICKEY5 License Key Dongle Required Some versions of What sBest include a dongle key with the license The dongle key is in the form of a key to insert into a USB port of the computer If this dongle key is required and missing the following error message will be displayed on the WB Status tab Warning Message License Key Dongle Req
396. s Adjustables Maximum Integers Bin Formulas Constraints Nonlinears Coefficients Obj Direction r Extended Solver Solver Type Obj Bound Steps Activity Extracting Data Reading File Elapsed Runtime 00 00 00 Best Obj Active Hold Interrupt Help 32 CHAPTER 2 The solver status window remains present for the duration of the solution process and is updated periodically Once the solver completes its run it returns to the workbook to install a new solution and inserts a new worksheet entitled WB Status This new worksheet is referred to as the status report The status report is an easily understood analysis of the model and solution It provides a hard copy of the information appearing in the solver status window The section entitled Getting the Best Results discusses the information contained in the status report in further detail You may use the nterrupt Solver button on the solver status window to interrupt the solver while it is solving If you do you can not expect any useful information to be returned unless you have an integer problem In the case of an integer problem the best solution to the point of interruption is returned For linear problems and non linear problems the returned values will not be useful The definitions of the information in the What sBest solver status box are shown below Model Type This shows
397. s RUNTIME Runtime Limit Reached By default What sBest does not put a limit on the total time that the solver will run However by using the Runtime Limit option on the General Options dialog box posted by the Options General command the user may limit the number of iterations When the program exceeds the specified runtime limit the following warning message will be displayed on the WB Status tab Warning Message RAKE KEE EKER EE EE EEN INTERRUPTED KEKE KEKE KEEEKE WARNING Runtime Limit Reached Help Reference RUNTIME The limit for the maximum runtime was reached before the final solution could be found Check the solution carefully it may be sub optimal or infeasible The runtime limit is set via the WB Options General menu Suggestions The runtime limit is set through the WB Options General menu In some cases the solver may be able to return the best solution found so far after hitting a runtime limit However the same precautions regarding the returned solution apply as when interactively interrupting the solver For more details refer to section Solver Interrupt above 424 CHAPTER 8 SOLVERR Solver Error Mathematical programming problems are inherently difficult to solve and as a result the What sBest solver may encounter an error from which it is unable to recover This should not be a common occurrence but should the solver encounter such an error What sBest will display th
398. s Unlike linear models a badly formulated nonlinear model may prevent What sBest from finding a solution even though one in fact exists Another problem posed by some nonlinear models is that What sBest may detect a locally optimal solution to the model that appears to be the best whereas a better but also locally optimal solution still exists For more on what you can do to help minimize the occurrence of these undesirable results see the section titled Guidelines for Modeling with What sBest Linearization The presence of any nonlinear expressions in the model will cause the nonlinear solver to be invoked and the model to be classified as nonlinear You can determine whether your model is linear or nonlinear by the classification under Model Type in the solver status window or the status report The nonlinear models take much longer to solve than a linear model with the same number of constraints The process of converting a nonlinear expression to a linear expression is called Jinearization What sBest can perform different degrees of linearization in the pre processing stage of the solution MATHEMATICAL MODELING 195 process on the ABS IF MAX and MIN nonlinear spreadsheet functions as well as all logical operators See Chapter 6 Functions and Operators for a comprehensive list Ideally all nonlinear expressions would be converted to linear equivalents so the linear solver can be used instead of the nonlinear solver Fo
399. s 94 Valid ranges for 94 E Elapsed Runtime 33 Embedding in a Visual Basic project 131 Engineering Models 239 243 Error 279 284 Error Codes 145 Error from VBA code 395 Error messages Iteration Limit 409 423 no feasible solution found 201 solution status globally optimal 200 locally optimal 195 Undefined Arithmetic Value 398 Unexpected Operation 419 Error messages amp Warnings 395 Adjustable Cell Limit 410 Barrier Solver Not Licensed 416 Constraint Limit 409 Dual Cell Format 435 Error Managing Temp Files 433 Excel File Format 399 INDEX 453 Expired License Key 414 External Reference 401 Failed to Access the Add in 402 Failed to Access the Macro 402 Failed to Access the Server 403 Failed to Process License Key 413 Formula Parsing 400 Global Adjustable Cell Limit 412 Global Solver Not Licensed 415 Integer Cell Limit 411 Invalid Model 408 Irreconcilable Constraints 408 Iteration Limit Reached 409 Large Infeasibility 406 Lower Range Cell Format 436 No Adjustable Cells 403 No Best Cell 421 No Constraint Cells 421 No Feasible Solution Found 405 Nonlinear Adjustable Cell Limit 411 Nonlinear Solver Not Licensed 416 Omitted Cell Reference 422 Parsing Cell Formula 419 Pending Expiration of Option Trial 414 Pending License Expiration 413 Quadratic Recognition 422 Runtime Limit Reached 423 Solver Error 424 Solver Interrupt 406 String Arguments Found 431 String List Er
400. s This of course is our objective and we tell What sBest to minimize it C Specify Constraints This is an unusual model in that it doesn t make use of any limited resources so no constraints are required You are almost ready to solve the model but first you will need to set a couple of options on the General Options tab First you will need to set the Linerization Degree to Maximum to have the What sBest linearizer automatically convert this otherwise nonlinear model to linearity Next you will need to reduce the Big M Coefficient from its default value of 100 000 to 1 000 Linearized models are very sensitive to this parameter If Big M is too large you can end up with poorly scaled models On the other hand setting this parameter too low will result in infeasible models For this particular model a setting of 1 000 works well After solving you ll see the following result 292 CHAPTER7 The LINEARZ Worksheet After Optimization Allee Coefficients 5 6896 0 0373 19 5877 15 0613 Thus our optimal solution minimizes the maximum error at 17 8 using the following formula Total Cost 5 6896 0 0373 Sq Feet 19 5877 Beds 15 0613 Baths Linearization Note that as formulated this model is nonlinear This is the result of the ABS and MAX functions used in computing the error terms and the objective value respectively Both ABS and MAX can create problems for the nonlinear so
401. s crates or C130 aircraft The objective can be to minimize wasted space in the container to maximize or minimize total load weight or as in this case to maximize the value of the load The Problem in Words You have to decide which items to load on a truck Each item must be shipped in its entirety or not at all e 25 of an item cannot be shipped The Truck Loading problem illustrates a situation in which non integer fractional answers are not acceptable To demonstrate potential problems involved in rounding fractional answers we ll solve the problem using two different methods In the first case wel use What sBest to optimize by the conventional method The solution may suggest fractional answers but we ll just round the fractions to the nearest whole number that does not violate any constraints In the second case well find an integer solution using the binary Integer command Background Items are to be loaded onto a truck with a 10 000 pound capacity However the items currently scheduled for shipment will exceed the capacity of the truck so you have to make a yes no decision as to whether each item gets shipped Each item has an associated dollar value and weight as shown below Dollar Value and Weight for Each Item Item Value Weight 1 22 500 7 500 lbs 2 24 000 7 500 Ibs 3 8 000 3 000 Ibs 4 9 500 3 500 Ibs 5 11 500 4 000 Ibs 6 9 750 3 500 Ibs Objective of Optimization The objective of o
402. s in the Dual Pricing and Primal Pricing drop down boxes respectively Pricing is a way of assessing the relative attractiveness of variables for selection in the simplex algorithm 58 CHAPTER3 Dual Pricing One pricing algorithm for dual simplex is called Partial according to which the dual simplex solver prefers variables offering the highest absolute rate of improvement toward the objective This preference is acted upon regardless of how far other variables have to move in tandem with the chosen variables Since these other variables may quickly hit a bound the resulting gain for the objective may actually be quite small The pricing mode called Steepest Edge takes a different approach of more painstakingly selecting variables according to the actual improvement in the objective Because the objective s gain is higher per iteration with Steepest Edge fewer iterations of somewhat longer duration are required The default value for Dual Pricing is Solver Decides Primal Pricing For the primal simplex algorithm there are two primary methods The Partial method prices a small subset of variables each iteration and intermittently prices out all the variables to determine a new subset of interesting variables Devex prices out all columns during each iteration using a steepest edge approximation Use of the thorough steepest edge algorithm means that Devex results in fewer overall iterations though each iteration is longer than it w
403. s of your funds should you invest in each of the three assets to achieve this target and minimize the variance or risk of the portfolio As an additional safety feature you decide to invest no more than 75 in any single asset Objective of Optimization The objective is to determine the percent to invest in each asset while minimizing risk of the entire portfolio 262 CHAPTER7 The Worksheet Let s look at the VARKOWIT sample file as supplied The MARKOWIT Model Before Solving MARKOWIT xts al ein f STANDARD MARKOWITZ PORTFOLIO PROBLEM Is To AT RE E EC E VW OP MARKOWITZ PORTFOLIO PROBLEM Upper Covariance Matrix Invested Bound Return Asset 1 Asset 2 Asset 3 0 1 lt 75 0 30 0 3 0 1 0 0 5 0 2 lt 75 0 20 0 1 0 2 0 0 4 0 3 lt 75 0 8 0 0 5 0 4 1 0 0 6 Not 100 0 Desired Return 0 1 Not gt 15 0 Variance j Re Wem DL a A Determine Adjustable Cells The adjustable cells are B6 B8 the Percentages Invested in each of the three stocks SAMPLE MODELS 263 B Define Best The best solution is the one that minimizes the risk of the entire portfolio In the Markowitz model a portfolio s variance is used as the measure of risk Using the variance gives a relatively greater penalty to outcomes that are far from the mean The variance of a portfolio is iji Oj where x is the percentage invested in Asset i O for i j is the covariance betw
404. s regarding a license upgrade that includes the stochastic option LICOPT6 Conic Solver Not Licensed If you attempt to use the conic solver option without a license the following error message will be displayed on the WB Status tab Error Message Conic Solver Not Licensed Help Reference LICOPT6 The license for your installation does not allow for use of the conic solver option Suggestions The Conic Solver or Second Order Cone Programming SOCP is invoked when the model is Convex with Global option set and checked the Quadratic Recognition on and the Conic option set The conic programming solver is an optional feature You have attempted to set up a conic model with a copy of What sBest that does not have an appropriate license You will need to contact LINDO Systems regarding a license upgrade that includes the conic option 418 CHAPTER 8 LOOKUP Unsupported Lookup Usage What sBest supports the VLOOKUP and HLOOKUP Excel functions only under certain conditions If you use an unsupported variant of either of these lookup functions the following error message will be displayed on the WB Status tab Error Message Unsupported Lookup Usage Help Reference LOOKUP The solver has been halted because the following cell contains an equation with an unsupported reference to either VLOOKUP or HLOOKUP All arguments must evaluate to being numeric the third argument must be a numeric constant and the
405. s running version 2002 or later via the Task Manager Also verify that the application is not already busy with any macro virus scan Suggestions The solver could not link into Excel to execute the functions e Make sure to run Excel version 2002 or later e Make sure Excel is not already busy with another process e Turn off the virus scan for macro applications In the mean time select Retry from the Server Busy window 404 CHAPTER 8 IKBREP K Best WBIKB_REP Format Using the K Best feature What sBest can return the trade off solution information Trade off values tell you how sensitive the successive solutions are to various parts of the model The WBIKB_REP function is used to request trade off values If the arguments to this function are not correctly specified the following error message will be displayed on the WB Status tab Error Message K ERROR K Best WBIKB_REP Format Help Reference IKBREP A WBIKB_REP cell is incorrectly formatted Correct the formula and the validity of the arguments in the cell below The format should be WBIKB_REP cells where cells must refer to variable cells cell address Suggestions There is an incorrectly formatted K Best Report cell in the model A reporting cell must use the format WBIKB_REP cells where cells is a reference to the cells you want to report the trade off value The cell reference should be to a variable cell For more i
406. s you to define integers in either binary or general format A binary integer will return a zero or one whereas any nonnegative whole number 0 1 2 3 will be returned for a general integer The nteger dialog box appears as follows Integer Binary To create an integer range specify the cell or cell range you wish to make integer in the Refers to text box Then enter a name in the Integer Names in Workbook text box Next select the Binary WBBIN or General WBINT radio button in the Integer Type box Once this is done pressing the Add button causes What sBest to assign a WBBIN or WBINT range name to the selected cells which will appear in the list on the Integer dialog box If you decide later you would like to remove the integer restriction simply select the Integer command click on the name of the integer range to remove and press the Delete button It is important to remember that integer restrictions can increase solution times as discussed in the section entitled Runtime Concerns 42 CHAPTER3 Integer Names in Workbook What sBest uses range names to specify which cells in a model should be integers You can choose any combination of letters for your range name When the range name is created What sBest adds WBBIN or WBINT to the front of the name to represent respectively binary integers and general integers For instance if you used the name Quantity in the Integer Names in Workbook textbox for a
407. se needs are met The best cell also appears in this screen The Assignments Worksheet Before Optimization Weem DH p f SAMPLE MODELS 34 The Assignments worksheet shows individual employees who are assigned to weekly shifts A 1 indicates that a staff member has been assigned to a particular shift A 0 means he isn t working that shift The Work cell by each name specifies the exact number of shifts each one should work during the week Since no employee can work more than 1 shift during any 24 hour period the maximum number of shifts any employee can work during the week is 7 The Preferences Worksheet Before Optimization amp os08 3 noo oso oone n ono S oona Oow oowaza oofrc l i 2 ER Eg 5 LB ER 8 Ma 11 12 13 KI 15 16 18 19 20 23 EI 25 26 22 342 CHAPTER7 Each employee ranks his five most preferred shifts for the week on the Preferences worksheet Employees can specify preferences for more than 5 shifts during the week as long as they follow the same procedure e g each employee can rank his first seven choices from 7 to 1 However each em ployee would still be assigned the exact number of shifts specified in column B of the Assignments worksheet The Constraints Worksheet Before Optimization CONSTRAINTS Mon Tue Wed Thu Fri Sat Sun SHIFTS lt lt lt lt Not z g lt lt
408. se the nonlinear solver option without a license the following error message will be displayed on the WB Status tab Error Message E ERROR Nonlinear Solver Not Licensed Help Reference LICOPT2 The license for your installation does not allow for the solution of nonlinear models Suggestions The nonlinear solver capability is an optional feature You ve attempted to solve a nonlinear model with a copy of What sBest that does not have the option enabled You need to either reformulate the model to eliminate all nonlinear cells or contact LINDO Systems regarding a license upgrade that includes the nolinear solver option For more information on identifying nonlinear expressions see Overview of Mathematical Modeling LICOPTS Barrier Solver Not Licensed If you attempt to use the barrier solver option without a license the following error message will be displayed on the WB Status tab Error Message ERROR Barrier Solver Not Licensed Help Reference LICOPT3 The license for your installation does not allow for use of the global solver option Suggestions The barrier solver is an optional feature You ve attempted to use the barrier solver with a copy of What sBest that does not have an appropriate license You will need to either disable the barrier option WB Options Linear Solver or contact LINDO Systems regarding a license upgrade that includes the barrier option LICOPT4 Mixed Integer Solver Not
409. sefully interpreted because of the way the branch and bound technique solves integer models If a dual value is requested in an integer model What sBest will return a value However this value is of no practical use Valid Ranges for Dual Values You may also wish to determine the upper and lower ranges of any specified dual value Changing the right hand side of a constraint or using more or less of a resource as described above eventually will cause a change in the dual value The valid range for a dual value is the amount of change in the right hand side of a constraint or in resource use in an adjustable cell that is possible before a change in the dual value will occur The range over which the use or availability of a resource or the right hand side of a constraint can decrease before the dual value changes is called the Jower range The range over which the use or availability of a resource or the right hand side of a constraint can increase before the dual value changes is called the upper range ADDITIONAL COMMANDS 95 Constraint Ranges For example building on the XYZ example an upper and lower range can be determined for the dual values of the stock constraints These ranges are put into the model by highlighting the cells you would like the range to appear in selecting Advanced Dual specifically selecting the constraint cells F15 F17 to display in For Cell Range choosing Upper Range or Lower Range whichever is
410. selecting cell B1 and pressing the lt toolbar button or typing the WB A1 lt C1 formula directly into the cell Although it is typically easier to use the What sBest commands constraints can be created in a What sBest model by copying or modifying existing constraints What sBest will treat copied constraint cells the same as any constraint cell created by the menu or toolbar Constraints command Note Itis not recommended to lock or hide constraint cells on a protected sheet ABCs 27 Usage Guidelines for Constraints In the real world decisions are always constrained in some fashion In the steel business for example you work within a budget to acquire finite amounts of ore coke alloy metals and power Orders must be delivered on time and up to specifications or you risk losing business Such limits to doing business are referred to as constraints in the language of optimization modeling With What sBest you can use the operators gt greater than or equal to lt less than or equal to or equal to to express your particular constraints What sBest allows you to enter constraints using the four following methods 1 The Constraints dialog box 2 The three constraint toolbar buttons 3 Manually entering constraint functions as cell formulas 4 Using VBA to automatically enter constraint functions as cell formulas The first two methods only allow the entry of cell references The dialo
411. seseaeeeeeeeeees 428 SPREP Stochastic WBSP_REP Format 429 SPSTSC Stochastic WDGbP GTGCkomat 430 SPVAR Stochastic WBSP_VAR Format ccccccccccsssesssesesecececsesssaeseeeeseeeseeseaeeeseeeens 430 STRARG String Arguments Found 431 PREFACE xi STRLEIST ASING Te Erte na a dt Nee eege SEENEN 431 STRRES String Result of Formula seeeeeeeeeeeeseeeiesresrirssrssrissrissrnssrnnsrnnsrnnsrnnnsrnnt 432 SUMIF Unsupported Sumif Usage cccceeseeceeeeeceeeeeeaeeeeeeeseeeeeeaaeeeeeeeseeeeseaeeeeaeesenes 432 TEMPFILE Error Managing Temp Files 0 eeccceeeeeneeeeeenceeeeeeeceeeeetaeeeeetaeeeeeeaeeeeenena 433 UNBOUNDED Unbounded Model 433 UNDEFREF Undefined Heterence annern anann ennnen 434 WBCARDFORNM Cardinality Cell Fomat cee eee e reas eee ee sense eeeaeennaeeeaes 435 WBDUFORM Dual Cell Format 435 WBLOFORM Lower Range Cell Format 436 WBSEMICFORM Semi continuous Cell Fommat 437 WBSOSFOR M Special Ordered Sets Cell Format cccceeceeceeeeeeeseeeeeeeseeeesaeeseaes 438 WBUPFORM Upper Range Cell Fomat 438 APPENDIX A INSTALLATION DETAILS sccsseeeseeeeeeeeeeeeeeseseeeeenseeeseneeeseaeseseeeenseeeeeas 439 Installation estate ee Aert Age SEENEN ee Weide tial eg ee ee es 439 What sBest Example Files cccccccceeeeeceeeeeceeeeeeaeeeeeaeeeceeeeseaeeesaaeeseaeeseeeeseaeeesnaeeeeaes 439 Setup TPG eebe Lae i a sed eee eee 439 What sBest Add in Files
412. side of the constraint and the host cell where the constraint should appear These go in the Left Hand Side LHS Right Hand Side RHS and Stored in fields respectively Then select lt less than or equal to gt greater than or equal to equal to lt c convex gt c concave c convex in order to specify the direction of the constraint If no type is selected What sBest will default to lt less than or equal to Selecting None will clear the cell Clicking OK will apply the constraints The user can explicitly specify the constraint convexity to the Solver by integrating a c in the direction see the Usage Guidelines for Constraints part for additional information You need be concerned with lt c or gt c only if using the Global solver Note What sBest uses the loose inequality operators rather than the strict inequality operators of gt greater than and lt less than Left Hand Side LHS Right Hand Side RHS and Stored in Specify the cells or ranges to be constrained in these text boxes What sBest attempts to automatically fill in these boxes based upon what the current selection is The cells just to the left of the currently selected cells are entered as the Left Hand Side of the constraint The cells just to the right of the currently selected cells are entered as the Right Hand Side of the constraint The currently selected cells become the Stored in range displayed When a ro
413. significant performance gains to some nonlinear models by replacing nonlinear operations with mathematically equivalent linear operations An example of such performance gains may be seen in the sample model entitled Linearization Option and Construction Cost Estimation Degree Many nonlinear operations such as abs x and min x y can be replaced by linear operations that are mathematically equivalent The ultimate goal is to replace all the nonlinear operations in a model with equivalent linear ones thus allowing you to use the faster and more robust linear solvers in What sBest We refer to this process as linearization 52 CHAPTER3 Specify the extent to which What sBest should attempt to linearize models with the Degree drop down box The available options here are Solver Decides None Mathematical and Mathematical Logical Under the None option no linearization occurs With the Mathematical option What sBest linearizes ABS MAX and MIN function references as well as any product of binary variables and at most one continuous variable The Mathematical Logical option is equivalent to the Mathematical option plus it will linearize AND IF INT MOTO OR SIGN SUMIF VLOOKUP and HLOOKUP functions and all logical operators lt gt and lt gt Under the Solver Decides option What sBest will do maximum linearization if the number of adjustable cells doesn t exceed 12 Otherwise What sB
414. sing other location options if installed to the XLSTART directory the What sBest menu cannot be disabled through the dialog box posted by the Excel menu command Tools Add ins WB the What sBest directory where the sample files are installed Note Ifyou are running Excel from a server verify if you have write access to Excel s Library or XLSTART subdirectories In the Windows Registry What sBest creates two keys in the following locations HKEY CURRENT USER Software Lindo Systems Inc for CheckUpdate Language and Registration Note When you re enable What sBest it appears on the list given by Tools Add ins if the add in is installed to the Library directory If it is not on the list you may use the Browse button on the Add ins dialog box to specify the location of the WBA XLA or WBA XLAM add in file in the WB directory 446 APPENDIX A Uninstall Files There are two ways to uninstall What sBest First you can rerun the executable file used for the initial installation Alternatively you can open the Control Panel of your Windows operating system select the Add Remove Programs icon click on What sBest in the list of programs and click the Add Remove button Both of these ways will redirect you to a common What sBest maintenance window that resembles the following Welcome to the What sBest 64 bit Maintenance Window Please select one of the following EI Install NOTE Remove
415. ssage Invalid Model Help Reference INVMOD Your model must have at least one adjustable cell and one formula that is dependent upon an adjustable cell Refer to the Getting Started section of the online help to learn how to format a model for What seet Suggestions The model you are attempting to solve does not have the appropriate formatting for What sBest You will need to identify the adjustable cells identify the best cell and input your constraint cells Refer to section ABC s Basic Functions for details on how to format a model IRRECONST Irreconcilable Constraints If a constraint is violated and it does not depend upon any adjustable cells What sBest will not be able to satisfy the constraint and it is considered irreconcilable In which case the following error message will be displayed on the WB Status tab Error Message Irreconcilable Constraints Help Reference IRRECONST The cells listed below contain violated constraints that do not depend upon adjustable cells and therefore cannot be reconciled These constraints were ignored during the solution process An example of an irreconcilable constraint would be WB 1 lt 0 Please check to see that these constraints have been entered correctly and that all adjustable cells have been specified cell addresses listed at bottom of tab Suggestions A simple example of an irreconcilable constraint would be WB 0 gt 1 Clearly no matte
416. ssigned to A 1 in an employee s row indicates that he is assigned that column s schedule number B Define Best The best solution is the one that maximizes total preference score for the group Total preference is shown in AH19 This is the sum of the preference subtotals for each employee C Specify Constraints Constraints are of the following three types No more work patterns must be assigned than recommended by the stage 1 optimization X19 Employees must be assigned to only the work patterns selected in stage 1 U17 AF17 and e Fach person should be assigned to only one work pattern AGS AG16 SAMPLE MODELS 355 Now let s solve the model The FIXED2 Worksheet screen 2 After Optimization FIXED2 xIsx RECH D 2 EMPLOYEE ASSIGNMENTS TO SCHEDULES 10 12 Single Pref Staffed LLOYD DIANE TOM JOANNE DAN JIM SUSAN JOY JOHN PAM SHAOIB BEA Met OC ECO EE ECH E sch oooocoocoocoococooc cocoon COOC LG OC EC h EECH COA CONS OCH Ah EECH EN h MO ooooocoococoocococcocoon OCH E zl CO zb E E E EE zb h h ECH P OO CC ch E ch sch E sch E EECH EH CH amp CO CEO CEO CEO CH E KKK STAFF NEEDS MET gt PREFERENCE TOTAL The maximized preference total in cell AH19 is 150 and the staffing needs established by stage 1 are met To verify that assignments have been made according to seniority and preference go back to screen and observe that Tom in row 7 with a seniorit
417. st provides a function to return the error text for these wbErr codes This function is similar to Excel s Error function The wbError function takes an error code or error number as an argument and returns the error text CODE SAMPLE ONE Sub mySub On Error GoTo ErrorHandler Trap any What sBest error wbAdjust Make current cell adjustable wbSolve Solve the model Exit Sub ErrorHandler Process any What sBest error If Err wbErr Adj ProtectedSheetError Then MsgBox To the user Please unprotect all the sheets in your models using the Tools Protect command Else If Err gt 30000 Then MsgBox The error description is amp _ wbError Err Else MsgBox The error description is amp _ Err Description 146 CHAPTER 4 End If End If Exit Sub End Sub If you are running What sBest within a VB project you should use an alternative method for error handling This method will prevent possible What sBest errors from being transformed into unusable Visual Basic runtime errors The following Code Sample Two is excerpted from the VBABuildX YZ subroutine found in the Visual Basic project file XYZ VBP available in the VB60 subdirectory of the WB directory For details see the section entitled Usage Guidelines for Macros in VBA CODE SAMPLE TWO Sub VBABuildxYZ Omitted code that created an Excel object named gobjExcel and opens the problem to be solved Dim intError As Integer gobjExce
418. sticEVEM is a True False flag indicating whether or not to use this option 176 CHAPTER 4 StochasticE VPI StochasticEVPI is a True False flag indicating whether or not to use this option StochasticEVMU StochasticEVMU is a True False flag indicating whether or not to use this option StochasticRepScenHoriz StochasticRepScenHoriz is a True False flag indicating whether or not to use this option StoOpt_BadStochasticSolverMethodArg StoOpt_BadStochasticSeedArg StoOpt_BadStochasticSamp ContArg StoOpt_BadStochasticEVWSArg StoOpt_BadStochasticE VEMArg StoOpt_BadStochasticEVPIArg StoOpt_BadStochasticEVMUArg StoOpt_BadStochasticRepScenHorizArg Bad StochasticRepScenHoriz argument Example If one wished to set all the stochastic support options the simplest syntax would be wbSetStochasticOptions 2 2027 False False False False True False To set one or more options named arguments should be used wbSetStochasticOptions StochasticEVPI False Note The AdvStochasticSupport argument should be set to TRUE via the wbSetStochasticSupport function in order to use these options MACROS THE VBA INTERFACE 177 wbSetStochasticSupport This routine can be used to set the What sBest stochastic support option seen in the Stochastic Support dialog box There is only one argument For additional discussion of the options available through this routine see the section entitled Advanced Stochastic Support Sy
419. sting SEASON XLSX Exponential Smoothing Two models illustrating one technique for using historical data to predict future sales SIMXPO XLSX SMOOTH XLSX Linearization Option and Construction Cost Estimation Models Linearization Option and Construction Cost Estimation This model demonstrates the performance gains resulting from using the What sBest linearization option and illustrates a case of estimating construction costs LINEARZ XLSX Marketing Models Stratified Sampling Determining the least costly polling sample likely to give reliable results SAMPLEWB XLSX Car Pricing A nonlinear pricing model illustrating a case in which sales of products are interdependent PRICING XLSX Media Buying Purchasing advertising media space to meet an exposure target at minimum cost MEDIA XLSX Production Models Multi Period Inventory Management Managing inventory across multiple periods to minimize holding costs while maintaining sufficient stock INVENT XLSX Product Mix Using available resources to manufacture a mix of products that yield the highest profit PRODMIX XLSX PMIXMAC XLSM The Building Block Method Combining production and shipping models into one large model a common approach to problem solving BLOCK XLSX Waste Minimization in Stock Cutting Cutting sheet or coil materials to varying lengths while minimizing waste CUTSTOCK XLSX Plant Locating Demonstrating ho
420. stochastic program SP is a mathematical program linear nonlinear or mixed integer in which some of the model parameters are not known with certainty and the uncertainty can be described with known probability distributions Applications arise in several industries e Multi period financial portfolio planning with uncertain prices interest rates and exchange rates e Exploration planning for petroleum companies e Fuel purchasing when facing uncertain future fuel demand e Fleet assignment vehicle type to route assignment in face of uncertain route demand e Electricity generator unit commitment in face of uncertain demand e Hydro management and flood control in face of uncertain rainfall e Optimal time to exercise for options in face of uncertain prices e Capacity and Production planning in face of uncertain future demands and prices e Foundry metal blending in face of uncertain input scrap qualities e Product planning in face of future technology uncertainty e Revenue management in the hospitality and transport industries Multistage Decision Making Under Uncertainty Stochastic programs fall into two major categories a Multistage Stochastic Programs with Recourse and b Chance Constrained Programs What sBest capabilities are extended to solve models in the first category namely multistage stochastic recourse models Chance constrained models will be supported in future versions In this chapter the term stochastic
421. straints such that the modified model is mathematically equivalent to the original with the non smooth functions or operators eliminated Non smooth functions and operators that may be eliminated through linearization are FUNCTIONS OPERATORS ABS lt AND lt HLOOKUP lt gt IF INT lt MAX gt NOT OR SIGN SUMIF VLOOKUP In addition to the above What sBest can also linearize products of 0 1 and continuous variables For more information on the linearization process refer to the section entitled Options General Global Solver Functions currently supported are 1 Standard smooth functions x y x y x y log x exp x sqrt x sin x cos x 2 Continuous non smooth abs x max x y min x y 3 Smooth not quite continuous x y xy tan x floor x 4 Logical and relational operators if x y z gt lt AND OR NOT 5 Probability distributions psn z Normal Distribution psl z Normal Linear Loss FUNCTIONS AND OPERATORS 189 Statistical Functions What sBest supplies six built in statistical functions These functions can be used within your What sBest models as if they were native supported functions in your spreadsheet e g SUM SUMPRODUCT ABS etc You should precede the spreadsheet function name with For example enter the inverse triangular cumulative distribution distribution as WBTRIAINV 7 1 2 3 Inverse Triangular Cumulative Distribution
422. t procedures are text references Alternately the ranges could also have been passed as objects to allow the developer to work with row and column numbers instead of the text The above code presented with object references would look like this Make Quantities to Produce C5 D5 Adjustable wbAdjust Range Cells 5 3 Cells 5 4 Make Total Profit G6 the best cell to Maximize wbBest Cells 6 7 Maximize Constrain Total Usage E15 E17 to be less than Number in Stock G15 G17 wbConstraint Range Cells 15 5 Cells 17 5 lt _ Range Cells 15 7 Cells 17 7 Range Cells 15 6 Cells 17 6 The two other procedures in the ABC e module MaxDeluxe and MaxDeluxe2 simply change the XYZ model slightly and solve it again The Dual module contains a couple of subroutines that demonstrate putting dual value functions in the model In the examples above none of the arguments used the named argument syntax To do so simply list the argument name followed by a colon and equal sign and then the argument To set the range C5 D5 as adjustable with the named argument syntax enter wbAdjust AdjRange C5 D5 AdjChoice Adjustable Using the named argument syntax can make the code easier to read and when calling procedures with multiple arguments allows you to list the arguments in any order you choose 136 CHAPTER 4 Solving Multiple Problems with a Looping Macro When automating a process using macros you may want to r
423. t a discontinuity or a point where the objective function is non smooth the slope is undefined At a point for which the slope is undefined the optimality conditions cannot be said to be satisfied hence they are determined uncertain Second if in the search for a solution What sBest completes several iterations without making any significant improvement in the objective function it will stop even though the optimality conditions are not satisfied This may be a situation in which the objective function is very nearly flat over an unusually large region Whether the optimality conditions are satisfied or uncertain if the solution is locally optimal then another local optimum may exist that improves on the current solution Even though the mathematical conditions to prove optimality may be inconclusive the value returned in the best cell is locally optimal Therefore you should always consider re solving the model with different values for the adjustable cells to see whether another local optimum improves the solution No Feasible Solution Found The message Solution Status INFEASIBLE is displayed in the status report if What sBest was unable to find a solution that satisfied all the constraints If the model is linear then there is no answer that satisfies all constraints If the model is nonlinear then either no feasible answer exists or a feasible answer exists but What sBest was unable to find it For details please see the topic Solut
424. t the formula and the validity of the arguments in the cell below The format should be WBSP_STSC cells where cells must refer to a two column range Also review the value of the arguments in the distribution function additional message Suggestions There is an incorrectly formatted Stochastic Scenario cell in the model Such cell must use the format WBSP_STSC cells where cells is a reference to a two column range Also review the value of the arguments in the distribution function some of them could be out of range For more information on using Stochastic Solution values refer to section Advanced Stochastic Support SPVAR Stochastic WBSP_VAR Format Using the stochastic feature the user has to associate a stage information to a variable cell The WBSP_VAR function is used to specify the timing information If the arguments to this function are not correctly specified the following error message will be displayed on the WB Status tab Error Message ERROR Stochastic WBSP_VAR Format Help Reference SPVAR A WBSP_VAR cell is incorrectly formatted Correct the formula and the validity of the arguments in the cell below The format should be WBSP_VAR stage cells where stage is numeric and cells must refer to variable cells cell address additional message Suggestions There is an incorrectly formatted variable function cell in the model A variable cell must use the format WBSP_VA
425. t was just generated You must now try to correct the source of the error but this time you do not have the benefit of a What sBest error code If you can t manage to guess what the source of the error is then you must resort to unchecking the What sBest add in from the list generated by the Tools Add ins command This forces the removal of the previous unhandled error Now you can return to the add in list and check the What sBest add in to reinstate it You now have another chance to determine the source of the error and insert error handling code TROUBLESHOOTING 397 Errors from an embedded application In the case of an error running What sBest embedded within a Visual Basic project the Run time error 5 Invalid Procedure Call or Argument message box shown above will be displayed This may be followed by a Run time error 440 Automation Error To correct this problem you will need to terminate the application insert appropriate error handling and run the application again The error handling for an embedded application is distinct from normal error handling for a stand alone Excel application A code sample of this error handling is provided toward the end of the section entitled wbError and Error Codes in chapter VBA Interface Invalid Outside Procedure This is often the result of using parenthesis without the CALL keyword in Visual Basic code Either put the keyword CALL in front of the procedure calls whe
426. taff FTEs Thu 10 Pool FTEs Fri Sat o CO CO Total Allocated FTE s Sun Mon Tue Wed Thu Fri Sat 6 COSTS PER DAY Staff FTEs 120 Pool Days 160 TOTAL COSTS 14 400 Oo o oO an 350 CHAPTER7 The FIXED1 Worksheet screen 2 After Solving No Pool t Microsoft Bxcel Ea Smal TU VW XY Z AA ABJAC AD AE AF Age Number Schedule Available Schedule Patterns Total Assigned Number Su Mo Tu We Th Fr Sa Su Mo Tu Weih Fr Sa Days 10 10 10 10 10 10 10 10 10 10 0 0 OONOAnMFEWNHA CO 10 11 12 2 33 4 33 e DG WK De Ha RE IW 0 2 4 0 2 2 2 0 0 0 0 0 E E sch OOOO a sch sch sch O EH ch EN sch wh sch sch wc sch OO E E b ab ad ab sch sch Oo web OOO ab ata ad ab ab ad ab ad EH CH wh sch E sch E sch sch sch sch sch ooe sch sch E sch E sch sch sch sch CO OO OO EN A ch ch 320000 CO OO OO OO A Ah bh bh EA EA EA CH EN EH ch E sch wh EH EH web sch sch sch E E eh ech EN ER web sch wh och sch sch EH CH ch EH 2 sch sch sch sch sch sch sch ER ER eh wb eh wb eh oh EI sch ER a EH OO ch sch sch sch sch sch EH sch EH CO OO A ch E EA OO OO zb ch zb sch CH CH CH CH CH CH CH CH CH CH CH CH CH CH Pool Days The above screens show the solution when only the cells for full time employees are adjustable The total cost is 14 400 The full time staff needed to cover these requirements is 12 people Note Though we ve illustrated th
427. tandard form is 2x0 x1 x2 2 x3 2 xn 2 lt 0 x0 x1 gt 0 In both simple and rotated quadratic cones a variable can appear in at most one cone constraint If naturally you would like to have a variable say x2 appear in two cone constraints then you must introduce an extra copy of the variable say y2 for the second cone constraint and then connect the two with the linear constraint x2 y2 0 Notice using a standard transformation rotated quadratic cone constraints can be shown to be equivalent to quadratic cone constraints y u vy 2 z u v 2 x2 y2 2 2 lt 0 z gt 0 7 Sample Models Introduction This section provides sample models for problems in manufacturing finance engineering scheduling etc They are real world problems that demonstrate the practical use of the elegant concepts of linear and nonlinear programming A major strength and advantage of the spreadsheet environment compared to the more traditional methods of mathematical modeling is the ease with which relationships among data are visualized We ve assembled a wide range of sample spreadsheet models for you to investigate Some of them are very nearly full fledged applications All of them clearly demonstrate the principles of modeling with What sBest We ve organized them in the following groups Blending Models Blending Blending elements with various qualities into the lowest cost product to meet designate
428. te Cost 112 gei Total Edge Waste Cost Edge Waste Cost Total Waste Cost EE Per Inch Ft 1 50 In B4 B7 the number of cuts of the 15 inch width is entered for each pattern e g pattern has no 15 inch cuts and pattern B has two as shown in the schematic on the previous page The total Cut cells B9 E9 display the footage cut to each width In G4 G7 Edge Waste in inches is shown Edge waste is the total of inches cut per pattern subtracted from the stock width of 100 inches End Waste in inches B13 E13 is the amount cut less the amount needed A Determine Adjustable Cells The adjustable cells in this model are the Feet Cut Pattern F4 F7 They contain the number of feet to be cut into each of the four available patterns 324 CHAPTER7 B Define Best The best solution to this worksheet is the combination of cut patterns that minimizes total Waste Costs while meeting the required footage for each width The value of Total Waste Costs is located in cell E19 and is the sum of the edge waste costs resulting from each cut pattern and the end waste costs of excess footage of any width produced SAMPLE MODELS 325 C Specify Constraints The constraints in this problem are simply that the production of sheet metal in each of the four widths must meet the amount demanded The requirements for 15 18 25 and 35 inch sheet metal appear in B10 E10 The actual footage of each width resulting from the selected Cut patt
429. tedSheetError Unable to enter constraints on a protected sheet Remarks LHS and ConLoc ranges must be the same size RHS can be a range or a value MACROS THE VBA INTERFACE 143 Example To enter the constraint H8 gt 10 into cell I8 enter wbConstraint H8 gt 10 I8 or wbConstraint Cells 8 8 gt 10 Cells 8 9 To constrain H8 H10 to be less than or equal to J8 J10 and store the constraints in 18 110 enter wbConstraint H8 H10 lt J8 J10 I8 1I10 or wbConstraint Range Cells 8 8 Cells 10 8 lt _ Range Cells 8 10 Cells 10 10 Range Cells 8 9 Cells 10 9 wbDeleteReports This routine is called by the Delete Reports button on the General Options dialog box posted by the Options General command This routine is used to delete any What sBest report worksheets WB Status and WB Solution in the current workbook Syntax wbDeleteReports The whDeleteReports procedure has no arguments Error Codes Value Description Rep_DeleteReportsError Error deleting one of the reports wbDeleteWBMenu This routine is used to delete the WB menu item from Excel s menu Syntax whDeleteWBMenu The whDeleteWBMenu procedure has no arguments Remarks This procedure together with the whAddWBMenu procedure allows a developer to remove and add the WB menu as desired 144 CHAPTER 4 wbDualValue For additional discussion of
430. tered the following situations on the string support The return value of a formula cell should be numeric rather than text A numeric result displayed in a text form An illegal string operation An omitted string cells or sheets that are not relevant to the model Refer to the What sBest sample file VehicleRouting xls for an example of valid string argument usage SUMIF Unsupported Sumif Usage What sBest supports the SUMIF Excel function only under certain conditions If you use an unsupported variant of SUMIF the following error message will be displayed on the WB Status tab Error Message AK ERROR Unsupported Sumif Usage Help Reference SUMIF The solver has been halted because the following cell contains an equation with an unsupported reference to SUMIF The first and third arguments to a SUMIF function must be cell ranges with the same shape and size Furthermore all arguments must evaluate to being numeric not text cell address Suggestions The SUMIF function requires three arguments as follows SUMIF range criteria sum_range What sBest places the following restrictions on these arguments the range and sum_range arguments must be cell ranges with the same shape and size all arguments must evaluate to being numeric rather than text TROUBLESHOOTING 433 Refer to the What sBest sample file SUMIFWBSOLVER XLS for an example of valid SUMIF usage TEMPFILE Error Managing Temp Files
431. ters to be produced is greater than 60 What sBest will return the Solution Status INFEASIBLE message in the WB Status worksheet The original constraint requires that no more than 50 Deluxe computer towers be used However this new constraint requires production of at least 60 Deluxe computers exceeding the limitation of the first constraint Anytime What sBest can t find a solution to a linear problem that satisfies all constraints you ll see the INFEASIBLE error message By examining the state of the constraints in the returned infeasible solution you may be able to determine how to reformulate and eliminate the source of infeasibility No Feasible Solution Found Nonlinear In a nonlinear problem the Solution Status INFEASIBLE message may not mean that there is no solution that satisfies all constraints It only indicates that none can be found from the starting values you ve entered in the adjustable cells If you know or suspect there is a feasible answer to the problem enter adjustable cell values that yield a feasible or near feasible solution and try solving from that starting point Alternatively using the Global Solver option may help One other possible course to follow is to view the constraints in slack mode rather than Indicator mode This helps you determine which constraints are furthest from being satisfied Then enter educated guesses for the adjustable cells values that cause the constraints to come closer to being
432. the format WBSOS1 cells where cells is a reference to the cells you want the SOS on The cell reference should be to an adjustable cell For more information on range values refer to section Integer Special Ordered WBUPFORM Upper Range Cell Format If requested What sBest can return ranging information Range values tell you over what range a particular dual value is valid The WBUPPER function is used to request upper ranges If the arguments to this function are not correctly specified the following error message will be displayed on the WB Status tab Error Message RROR Upper Range Cell Format Help Reference WBUPFORM An upperr range cell is incorrectly formatted Correct the formula in the cell below The format should be WBUPPER cell value cell address Suggestions There is an incorrectly formatted upper range cell in the model An upper range cell must use the format WBUPPER cell value where cell is a reference to the cell you want the range value on and value is anumeric quantity The cell reference should be to either a constraint cell or an adjustable cell You may use any numeric value for value What sBest will replace value with the actual lower range value the next time you run the solver For more information on range values refer to section Advanced Dual Appendix A Installation Details Installation To install What sBest run the What sBest setup program and foll
433. the function WBSP_VAR This function requires two parameters i stage of the variable ii cell address of the variable In this problem the decision in cell D18 is a recourse variable that has to be decided at stage 0 WBSP_VAR 0 D18 Then the other adjustables belong to their respective period For instance D23 belongs to the stage 5 2 Random Parameters Specify the stage information for the stock return This is done by using the function WBSP_RAND This function requires two parameters i stage of the variable ii cell address of the parameter In this problem the return in cell B19 occurs randomly at stage 1 WBSP_RAND 1 B19 Then the random cell in B23 occurs at stage 5 3 Other Variables Declare the stage information for each variable dependent on the recourse variables or random parameters The following variables are assigned a stage 1 using the function WBSP_ VAR because they are dependent of the random parameter which is at stage 1 SAMPLE MODELS 381 a Price of stock this period cell C19 b Wealth this period cell E19 For instance these variables take a value only at stage 1 WBSP_VAR C19 D19 E19 The constraint in B26 belongs to stage 5 4 Distribution Type for the Random Parameter Specify a discrete distribution for the stock return parameter This is done by using the function WBSP_DIST_ DISCRETE SV This function takes in two parameters i the set of possible discrete values for
434. time to make sure the model is formulated in a way that is most efficient to solve This section gives some general guidelines to consider when building and solving nonlinear models Supply Good Initial Values The initial adjustable cell values can affect the path What sBest takes to the solution Starting with initial adjustable cell values near the optimal solution can reduce the solution time In many situations you may not know good initial values However when reasonable values are known they should be used If near optimal values are not known simply starting with values that satisfy all constraints may be helpful Starting from any reasonable answer may be better than starting with zeroes in the adjustable cells Always start with adjustable cell values that avoid undefined values in adjustable dependent equations For example if an equation divides by an adjustable cell then a value of zero for that adjustable cell makes the equation take an undefined value Start the solve attempt with a nonzero value for that adjustable cell Consider changing the initial values and re solving the model if 1 you suspect there is an answer better than the answer returned by What sBest 2 you know a feasible solution exists even though What sBest returns the message No Feasible Solution Found Use Reasonable Bounding Constraints Adding constraints that bound the upper and or lower values of adjustable cells to fall within a reasonable range o
435. tion for stage and distribution lInitial Decision Variables Specify the area allocated to each crop as a recourse variable This is done by using the function WBSP_VAR This function requires two parameters i stage of the variable ii cell address of the variable In this problem the area allocated to each crop is a recourse variable that has to be fixed at stage 0 2 Random Parameters Enter the yield information for all the crops under the various weather scenarios Then declare the yields of each crop as a random variable This is done by using the function WBSP_RAND This function requires two parameters i stage of the variable ii cell address of the variable In this problem the yields of the crop are the random variables that are realized at stage 1 3 Stage Information for Recourse Variables Declare the stage information for each variable dependent on the recourse or random variables The following variables adjustables are assigned a stage 1 using the function WBSP_VAR i Quantity produced ii Quantity sold iii Quantity purchased The stage is because these variables depend on the random variables and therefore take a value only at stage 1 4 Stage Information for the Constraints and the Objective Function Assign the objective function and all the constraints a stage 1 by using the function WBSP_VAR The stage is because these variables take on values after a realization of a value by the rand
436. to assist you in tracking them down Note that lt RHS gt signifies the Right Hand Side of a constraint If the location given is this means that no specific cell location is available and that the coefficient appears in a constraint added to the model as part of the linearization option Model Type What sBest examines your model to determine which of six types your model falls under Integer Quadratic Integer Nonlinear Integer Linear Quadratic Nonlinear Linear For further information on these categories see the section entitled Linear vs Nonlinear Expressions and Linearization Solution Status This field gives the status of the final solution The list of possible states is Globally Optimal Infeasible Unbounded Feasible Infeasible or Unbounded Near Optimal Local Optimal Locally Infeasible Cutoff Numerical Error Unknown Unloaded Loaded and Unknown Error See Solution Outcomes in Overview of Mathematical Modeling for more information ABCs 37 Optimality Condition This shows the optimality condition for nonlinear models either Satisfied or Uncertain See Solution Outcomes for more information Objective Value Assuming your model contains a best cell to be maximized or minimized this field displays its final value Direction Assuming your model contains a best cell this field displays its direction The direction will be listed as either Maximize or Minimize Solver Type This shows the t
437. to increase speed and reliability on broad classes of linear integer quadratic and general nonlinear models Other significant enhancements include an improved global solver an updated pre solver and greater user control over the solution process O Stochastic Support Improvements This feature allows the user to build models using not only decision variables but also random outcomes from a distribution Scenario Planning Stochastic Programming SP is an approach for solving problems of multi stage decision making under uncertainty e Improved warm start in solving multistage problems e Improved method to induce correlations among stochastic parameters 0 Mixed Integer Solver Improvements The MIP solver includes a number of new features e Significant improvements in root node heuristics for quickly finding good integer feasible solutions e Improved identification of special structures in certain classes of models as in multi period models and the ability to exploit this structure to achieve significant reductions in solve times xiv PREFACE LU Global Solver Improvements The global solver now includes e Improved heuristics for finding a good feasible solution quickly e Constraints may now be flagged as being convex in cases where the constraint s complexity make it impossible for the global solver to automatically determine convexity e Improved ability to identify constraints that can be reformulated as conic Second Order C
438. tors are supported by What sBest Functions ABS ACOS ACOSH AND ASIN ASINH ATAN ATAN2 ATANH AVERAGE BINOMDIST CHIDIST CHIINV COS COSH EXP EXPONDIST FALSE FDIST FINV GAMMADIST GAMMAINV HLOOKUP HYPGEOMDIST IF INT LN LOG LOGINV LOGNORMDIST MAX MIN MOD MMULT NEGBINOMDIST NORMDIST NORMINV NORMSDIST NORMSINV NOT NPV OR PI POISSON PRODUCT SIGN SIN SINH SQRT SUM SUMIF SUMPRODUCT TAN TANH TRANSPOSE TRUE TRUNC VLOOKUP WBINNERPRODUCT WEIBULL Operators A ok lt gt lt gt Se lt gt The function is non smooth Its use may result in long solution times and if the Global Solver is not used non optimal solutions See the discussion of Smoothness in Overview of Mathematical Modeling These functions are supported only under certain conditions Refer to the sections Function SUMIF or Function VLOOKUP HLOOKUP for specifics WBINNERPRODUCT is a function supplied with What sBest This function is the equivalent of SUMPRODUCT but instead of multiplying terms row by row the terms are multiplied row by column TOG can support one argument assuming base 10 or two arguments 187 188 CHAPTER 5 Linearization Many of the non smooth functions and operators supported by What sBest may be automatically smoothed by the solver through a process referred to as linearization Linearization replaces a non smooth function or operator with a collection of additional linear variables and con
439. total of six other nodes The pressure at the supply nodes G and H is fixed at 240 Ibs square inch PSI Theresa known demand at each of the six other nodes in C13 H13 of the Arc Resistance worksheet You need to calculate the pressure at each node and the flow across each arc The following diagram helps visualize this Supply 240 Ibs Sie si re ae Fa ote A A LU E e ei Flow Direction amp P Rb e Arc Resistance A i RK A A 3 Supply 240 Ib lbs 244 CHAPTER7 The Worksheet The FLOWNET Model Before Solving TOWNE die 30 1 A B cC D E F G H WE We Pressure E SAMPLE MODELS 245 FLOWNET xls O 4 aa a 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 A B Cc D E F G H Flow Conservation Not Not Not Not Not Not 246 CHAPTER7 PRESSURE BALANCE Destination D E Not Not Not A B Cc D E F G H Pressure Parameter 1 Flow Parameter 1 Note The pressure parameter equals 1 for incompressible fluids and electricity and 2 for gases The flow parameter equals 1 for electricity 1 852 for H2O and between 1 8 and 2 for other fluids ape w A Determine Adjustable Cells The adjustable cells are the Pressures at each Node Arc Resistance C15 H15 and the Flow along each arc the cells outlined above in the range C4 J11 of the Arc Flow worksheet To facilitate modeling the problem nonadjustable zeroes have been enter
440. ts In other words no better solution exists within that local area This means it is possible there is a locally optimal solution outside this immediate area that provides a better answer It may be wise to consider re solving the model with different starting values for the adjustable cells in attempting to investigate other local optima Optimality Conditions in Nonlinear Models After successfully solving a nonlinear model in addition to reporting a solution status of locally optimal the status report also indicates whether the optimality conditions are SATISFIED or UNCERTAIN A mathematician might refer to these optimality conditions as the Kuhn Tucker conditions A layman on the other hand might prefer to think of the optimality conditions as the flatness conditions Geometrically for functions with no constraints satisfaction of the optimality conditions means the objective function is flat the slope is zero in the neighborhood of the returned solution e it is at the top of the peak or bottom of the valley Algebraically it means that for each variable the partial MATHEMATICAL MODELING 201 derivative is zero Explaining these conditions on models with constraints lies beyond the scope of this help file It suffices to say that the interpretation can be generalized to constrained models There are two primary reasons for the optimality conditions to be reported as uncertain First if the optimum best cell value lies a
441. ts ete ee Re athe he ee tla a E oll tei dt eS 387 G neral Problems e tegt gehen d EEN nein tae ied 387 Operational Probl ms gege a inviehin tiie ia tan en tee 388 Frequently Asked Questions ssssessseseeennesrnsst nest nernsttn retn nesnnsttneetnntnnsetnnotnnennsennsennenn 390 CONtGh tiie tee EE a heh Scie Eed 390 Error Messages amp Warnings osrin erenneren nsronieuinin enn aae enia Vaaia a aE paa ENANA EEEE a 395 Errors trom VBA Gol Z tteg el ng See EE AER ees 395 dutt 396 Errors from an embedded application 397 Invalid Outside Procedure ccccceceececeeeeeceeeeeeaaeeeeeeeceaeeeseaaeseeeeecaeeesaaeeseaeeseeesseaaeeneneeee 397 ADDINLINK Multiple Add in Links 000 2 cece ceeeeeeeeeeeeeeecaeeeeeaeeeeeeeeaaeeeeaaeseneeeseaeeeeaeeteaes 397 ARITHERR Arithmetic ENO ereere oae i ia d a ii e 398 BLKCELL Blank Cell Warming cee eeeeaeeeceeeeeaeeeeeaaeseeeeeseaeeesaeeeeaeeeeeeees 398 EXLVER Excel File Format 399 FORMULA1 Formula Parsing ccccccccceeeeeseeceeeeeecaeeeeaeeseeeesaeeeseaeeeeeeeseeeeenaeeeeaaeeee 400 FORMULAZ2 Unsupported Functons 400 FORMULAS External Reference ccccccceeeseceeeeeceeeeeaeeseeeeeceaeeeseaeeseneeesaeeesaaeseeneeeaas 401 FUNCADDIN Failed to Access the Add in 00 ecccccceeeeeeceeeeeeeeeeeeeaaeeseeeeesaeeesaeeeeeeeeeas 402 FUNCMACRO Failed to Access the Macro 402 X PREFACE FUNCSERVER Failed to Access the Server cc cccccccccccssssecee
442. uared error The trend or slope of the line is the sales growth per season The base is the sales figure at time zero You have the sales data by season for the last two years rh Summer Fal Whiter g Objective of Optimization The objective is to predict sales for the next two years based on the Base Trend and Seasonal Factors while minimizing the error 280 CHAPTER7 The Worksheet Let s look at the SEASON sample file The SEASON Worksheet Before Solving Season Period 1000 s Predicted Error 10 00 0 00 10 00 14 00 0 00 14 00 12 00 0 00 12 00 19 00 0 00 19 00 14 00 0 00 14 00 21 00 0 00 21 00 19 00 0 00 19 00 26 00 0 00 26 00 Spring Summer Fall Winter Spring Summer Fall Winter ON OnMF WH Seasonal Factors Trend Spring 0 00 Base Summer 0 00 Fall 0 00 Sum of Squared Winter 0 00 Error 2475 000 Average 0 00 Not 1 00 A Determine Adjustable Cells The adjustable cells in the model are the Trend in B15 the Base in B16 and the four Seasonal Factors in E15 E18 B Define Best The best cell is the Sum of Squared Error in B19 The formula there is SUMPRODUCT ES E12 E5 E12 This is the sum of the range of Errors in E5 E12 squared SAMPLE MODELS 281 C Specify Constraints The only constraint F19 is the requirement that the Seasonal Factors average to 1 This is enforced by the formula WB E19 G19 where E19 contains AVERAGE E15
443. uired Help Reference LICKEY5 Unable to find or read a valid license key with dongle Run the WB Upgrade command to install a valid license key Suggestions Some licenses require a dongle key at the time of order to insert into a USB port of the computer If the dongle key is missing corrupted or does not match the license file so you will not be able to run What sBest What sBest can still continue to operate on a demo license capacity You may want to contact LINDO Systems to order a license upgrade that includes permanent access to one or more of the optional solvers including a dongle key LICOPT1 Global Solver Not Licensed If you attempt to use the global solver option without a license the following error message will be displayed on the WB Status tab Error Message 4K ERROR e Global Solver Not Licensed Help Reference LICOPT1 The license for your installation does not allow for use of the global solver option Suggestions The global solver is an optional feature You ve attempted to use the global solver with a copy of What sBest that does not have an appropriate license You will need to either disable the global option WB Options Global Solver or contact LINDO Systems regarding a license upgrade that includes the global option Note The global solver needs the nonlinear option and the mixed integer option to operate 416 CHAPTER 8 LICOPT2 Nonlinear Solver Not Licensed If you attempt to u
444. ula to the model This does not count the constant arguments within a formula Strings This shows the number of cells containing a string of characters or an unsupported function converted into its string value This does not count the string arguments within a formula Constraints This shows the total number of constraint cells in the model Nonlinears This shows the number of adjustable cells that appear nonlinearly in the model We say that a variable appears nonlinearly in a formula when that formula is nonlinear with respect to changes in the variable For instance consider the nonlinear expression X Y 2 This expression is nonlinear with respect to Y but it is linear with respect to X Thus Y would be counted in the nonlinear count while X would not Coefficients This shows the total number of coefficients in the best cell and all formulas dependent on adjustable cells Variables This shows the total number of adjustable cells constraint cells and cells containing formulas dependent on adjustable cells that influence the best cell This information appears after an automatic linearization Minimum and Maximum coefficients These are the largest and smallest absolute values of coefficients appearing in the model When the ratio of these two values becomes very large the model is said to be poorly scaled Poorly scaled models can lead to numerical round off errors in the solver These values are supplied with their location
445. ult of problems What sBest gives you access to this solver from within Excel and may either be run directly or called from within Visual Basic People in business finance science math and many other fields use What sBest every day to model and solve problems in production financial planning personnel scheduling resource allocation portfolio management stock cutting inventory control It s a long list to which you will want to add your own application To demonstrate some of the range of applications for What sBest sample models are provided with the software and many are explained in Chapter 8 Sample models 2 CHAPTER 1 Optimization Models Optimization helps you find the answer that yields the most desirable result the one that attains the highest profit output or happiness or the one that achieves the lowest cost waste or discomfort Often these problems involve making the most efficient use of your resources including money time machinery staff inventory etc Optimization problems are classified as linear or nonlinear depending on whether the relationships in the problem are linear with respect to the variables What sBest can solve both linear and nonlinear problems with optional integer restrictions For more information on optimization and the solution process please refer to Overview of Mathematical Modeling For additional reference we recommend the LINGO textbook Optimization Modelin
446. umbent solution In general you shouldn t have to set this tolerance Occasionally particularly on poorly formulated models you might need to increase this tolerance slightly to improve performance In most cases you should experiment with the relative optimality tolerance discussed below in order to improve performance The default value for the Absolute optimality tolerance is 0 0 Relative Use the Relative text box to set the relative optimality tolerance This is a value r ranging from 0 to 1 indicating to the branch and bound solver that it should only search for integer solutions with objective values at least 100 7 better than the best integer solution found so far The end results of modifying the search procedure in this way are twofold First on the positive side solution times can be improved tremendously Second on the negative side the final solution obtained by What sBest may not be the true optimal solution You will however be guaranteed the solution is within 100 r of the true optimum on linear integer models Typical values for the relative optimality tolerance would be in the range 01 to 05 In other words you would be happy to get a solution within 1 to 5 of the true optimal value On large integer models the alternative of getting a good solution within a few percentage points of the true optimum after several minutes of runtime as opposed to the true optimum after several days makes the use of this to
447. un the same macro many times on different data while storing the optimal solution for each data set For example you may have a database filled with information on the return on a group of investments at a range of interest rates What sBest can be used to determine the optimal portfolio in each situation by using a looping macro Using the PMIXMAC_XLS sample file which was copied to the WB subdirectory during installation let s look at how a looping macro might be created Use the Visual Basic Editor to go to the Looping Macro module in your workbook and your screen should look like the following s Sab Macroloopi This macro will increment the Profit Unit of Product 2 from 45 60 solve the model copy the Total Profit and Quantity Produced of Products 1 6 to a table and repeat Dim Count As Integer Dim TableRow As Integer Initialize the profit of product Z to 45 Sheets PRODMIX Select Range C6 44 Evaluate the effect of changing the profit on product Z by increasing it by one unit and resolving Solve the model save the results in a table and increnent the profit on product 2 l To 20 For Count Copy a new value to the profit of product Z Range C6 Range C6 1 Solve the current model 4pplication Run macro WBUsers whSolve Set TableRow to Count plus Z since the table will start in row 3 TableRow Count Z Copy the profit per unit for product Z t
448. unction to a set of data points is referred to as linear regression 290 CHAPTER7 The Worksheet A picture of the worksheet used to solve this problem is listed below You can find it in your sample models directory under the name LINEARZ sample file The LINEARZ Worksheet Before Optimization r 1500 1600 2200 2600 3000 3500 4000 5200 6200 8700 ABDA eae Coefficients 0 0000 0 0000 0 0000 0 0000 A Determine Adjustable Cells The adjustable cells in the model are the beta coefficients We need What sBest to pick the values for the coefficients that minimize the maximum prediction error There are four beta Coefficients so we added four adjustable cells to represent them in the range B16 E16 B16 is the intercept coefficient C16 is the coefficient for square feet D16 is the coefficient on the number of bedrooms and finally E16 is the coefficient on the number of baths SAMPLE MODELS 291 B Define Best We want to minimize the Max Error which is computed in cell H16 with the formula MAX H4 H13 The range H4 H13 contains the amount of Error in our forecast for each of the historical cost observations For example cell H4 contains the formula ABS F4 G4 In words this is the absolute value of the difference of the Actual Cost F4 minus the Predicted Cost G4 This gives us the amount of forecast error on the first observation Cell H16 simply takes the maximum of all these error term
449. undArg Bad LowerBound argument 158 CHAPTER 4 IntSemic_BadUpperBoundArg Bad UpperBound argument IntSemic_BadArgListArg Bad ArgList argument IntSemic_BadRefersToArg Bad Refers_to argument IntSemic_ProtectedSheetError Unable to set integer cells on a protected sheet Remarks Integer variables can dramatically increase the solution time Example To constrain cell F6 to be a general integer using the range name staff enter whIntegerSemic 10 20 Sheet1 B 1 Sheet1 C 1 C 3 Sheet1 A 1 The cell Al then contains the function WBCARD 10 20 Sheet1 B 1 Sheet1 C 1 C 3 wbintegerSos For additional discussion of the functionality provided see the section entitled Integer Special Ordered Set which refers to the Special Ordered Set Cardinality dialog box that calls this procedure This routine is used to build the SOS ranges WBSOSx Adjustable cells contained in these ranges will be restricted to integral values by the What sBest solver If you wish to delete a SOS function using VBA instead of deleting it on the spreadsheet you can use Se ection Clear This statement clears the selected cell from any of its content Syntax wbIntegerSos TypeSos ArgList Refers_to NoErrDialog TypeSos TypeSos is an integer that refers to the type of the SOS 1 for type 1 2 for type 2 3 for type 3 ArgList ArgList is a string list of range or cells reference separated by a comma Refers_to Yes Re
450. untered an error while loading all the strings You may need to shorten the string length to save memory omit cells containing strings or simplify the format of these strings Suggestions What sBest may have encountered the following restrictions on the string support Too many strings to load to the current allowable memory String containing unsupported characters Illegal string operations The maximum length of the string is too large The return value of a formula cell should be numeric rather than text Omit string cells or sheets that are not relevant to the model Refer to the What sBest sample file VehicleRouting xls for an example of valid string argument usage 432 CHAPTER 8 STRRES String Result of Formula What sBest supports string cells and arguments under certain conditions but the formula should return a numeric value instead of a string The following warning message will be displayed on the WB Status tab Warning Message WARNING String Result of Formula Help Reference STRRES What sBest encountered formulas displaying results in a text form You can use the String Support feature which allows you to use strings as arguments but the formula should return a numeric value instead of a string so their results have been taken to be zero Otherwise you may want to set omitted areas to discard these cells cell addresses listed at bottom of tab Suggestions What sBest may have encoun
451. upply SAMPLE MODELS 357 Objective of Optimization The objective in this model is to minimize the Total Pumping Cost without exceeding either the monthly output capacity of the wells or the capacity of the pipelines while meeting demand at each refinery The Worksheet Open the PIPELINE sample file and let s have a look at it The PIPELINE Worksheet Before Optimization MONTHLY PUMPING VOLUME From Inflow Well2 Outflow Well Pumping Costs 0 00 Refinery Pumping Costs 0 00 Total Pumping Costs 358 CHAPTER7 A Determine Adjustable Cells The adjustable cells are the amounts sent from the two Wells to the three Pumps B7 B8 and C7 C9 and the amounts sent from the three Pumps to the Refineries E7 E8 F7 F9 G8 G9 H8 H9 Cells B9 E9 G7 and H7 are fixed cells with a value of zero representing pipelines that are not open B Define Best The best solution to this problem is the one that minimizes Total Pumping Cost in cell E37 The formula in E37 is the sum of the two subtotals for Well and Refinery pumping costs Each subtotal in turn is the sumproduct of pumping costs along each open pipeline from the Wells to the Pumps in the first instance and from the Pumps to Refineries in the second and amounts carried on that pipeline C Specify Constraints The constraints in the PIPELINE model must perform several functions First supply capacity at the Wells must not be exceeded In cells B12 and C12
452. ust be properly formulated or they can be a source of problems Some of the problems are discussed below Too Few Constraints Unbounded Solutions Let s return for a moment to the XYZ Production model presented in the Tutorial of Getting Started Had you forgotten to include constraints in the model and tried to solve What sBest would have given an error message indicating Solution Status UNBOUNDED in the WB Status worksheet An unbounded solution error occurs when a model is not properly constrained Profit could be increased without limit by producing an infinite number of computers What sBest knows you ve left out one or more constraints and indicates that the problem is unbounded In an actual production situation there may be constraints other than raw material or component availability such as limited plant capacity or labor supply Excluding any of these could lead to an unbounded solution Also What sBest might give you an unbounded error message if you maximize the wrong cell or maximize the objective cell when in fact you really wanted to minimize it Anytime What sBest finds the objective heading toward infinity you ll receive the Solution Status UNBOUNDED error message For details see the topic UNBOUNDED in Troubleshooting No Feasible Solution Found Linear What if you add a constraint that contradicts one or more of the existing constraints In the XYZ problem if you add a requirement that the number of Deluxe compu
453. uted from a given customer to a proposed lockbox location B7 G11 and whether or not a lockbox is opened at a given site B13 G13 These cells display a figure to show if a customer is assigned to a lockbox location 1 or not 0 B Define Best The best cell is the sum of all operating and float costs in H19 A cost summary broken down into subtotals of fixed and variable costs is found in A15 H19 Let s look at the formulas used to determine these costs Considering first the variable costs associated with a lockbox in New York City the formula in cell B17 is B7 B27 H 7 B8 B28 H 8 B9 B29 H 9 B 10 B30 H 10 B11 B31 H 11 1000 C 18 Thus for the NYC lockbox the average mail delay from each customer location B27 B31 is multiplied by the cell indicating whether a lockbox has been assigned B7 B11 a zero or a one then by the amount of cash flow from the given customer location H7 H11 The sum of these products is then multiplied by 1 000 to compensate for the fact that monthly cash flow figures are abbreviated by three digits and by the Daily Cost of Capital in cell C18 Fixed costs are calculated simply by multiplying the cost per month for each proposed lockbox cell B14 in the case of NYC by the cell indicating whether the location has been opened B13 These costs per location are added to produce subtotals in cells H15 and H17 The sum of the subtotals Total Operating Costs appears in H19 the objective cell C Sp
454. ution search is terminated and if the model contains integer variables the best integer solution found up to that point is returned If this limit is encountered and the model does not contain integer variables the returned solution is meaningless Please note that returning to any previously established best solution may take some time The default value is None signifying that no iteration limit is imposed Runtime Limit You can specify the maximum amount of time in seconds the solver should spend searching for a solution with the Runtime Limit seconds text box If this runtime limit is encountered before a solution has been found then the solution search is terminated and if the model contains integer variables the best integer solution found up to that point is returned If this limit is encountered and the model does not contain integer variables the returned solution is meaningless Please note that returning to any previously established best solution might take some time The default value is None signifying that no runtime limit imposed Constraint Display Select Slack or Indicator to prompt What sBest to return the slack value or indicator display for the constraint cells ADDITIONAL COMMANDS 51 Indicator mode places the sign of the equation gt lt in the constraint cell If an indicator is tightly satisfied i e the constraint contains no slack then What sBest will return an equal sign before the in
455. various option settings This chapter discusses these additional commands and options Options and Solvers Some options in What sBest pertain to the overall operating parameters of the What sBest solvers or to the user interface and data display These general options such as teration limit or Runtime Limit are set in the General Options dialog box Other options modulate the performance of a particular solver For example Model Reduction is a linear solver option and is set in the Linear Solver Options dialog box For most models the default option settings should offer the best performance in a standard running environment However as you build more complex models you may benefit from trying new option settings or be required to change the settings to get a valid of good solution In order to effectively use the options the user should understand the four solver architecture of What sBest and the types of options that are available The four classes of solvers available in What sBest are linear nonlinear global and integer The downloadable and solver suite versions of What sBest are supplied with a linear solver a nonlinear solver global solver and an integer solver Larger versions of What sBest are equipped with linear and integer solvers only A nonlinear solver and a global nonlinear solver are provided optionally for additional fees The linear solver generally runs in primal simplex mode but it can be set to dual simplex
456. ver Status 64 bit Lindo Systems Inc zi wy Copyright 2011 64 bit What sBest 11 0 0 2 Mar 31 2011 Library 7 0 1 129 Extended License Solver Status Model Type State Tries Infeasibility Objective Extended Solver Solver Type Best Obj Obj Bound Steps Active Classification Statistic Category Current Maximum Numerics Adjustables Integers Bin Formulas Constraints Nonlinears Coefficients Obj Direction Activity Extracting Data Reading File Elapsed Runtime 00 00 00 Hold Interrupt Help The solver status window contains useful information about the properties of your model and the progress of What sBest toward a solution The information about model properties will also appear on the WB Status worksheet tab added to your workbook after the model is solved so don t worry if the window doesn t remain long enough for you to read the information it contains See the Solve section for detailed information about the solver status window GETTING STARTED 17 The worksheet soon reappears indicating the highest possible profit The XYZ Worksheet After Optimization iMate bieles XYZ COMPUTER CORPORATION PRODUCTION PLAN Product Standard Deluxe PROFIT Quantity to Produce 60 30 Profit per Unit 300 500 Components Quantity Required Total Number Standard Deluxe Usage In Stock Standard Tower 1 0 60 lt Deluxe To
457. vested each period to equal the amount available This is enforced by the constraints in column E E9 WB D9 F9 E10 WB D10 F10 E11 WB D11 F11 The amount over goal at the end is computed with D15 D12 D3 D14 The constraint in E15 constrains this to be nonnegative E15 WB D15 gt F15 The amount under goal is in D15 The net utility to be maximized is given by D16 D5 D15 D4 D14 Step 2a We specify cells G9 H9 as a stage 0 half stage decision variable by inserting the expression WBSP_VAR 0 G9 H9 into cell L6 We specify cells B10 C10 as stage 1 random variables that are observed at the beginning of stage 1 by inserting in cell J7 the expression WBSP_RAND 1 B10 C10 Similar expressions are in cells L7 L8 L9 J8 and J9 for the remaining stages Step 2b We tell What sBest that the growth factors have a joint discrete distribution as listed in the scenario table 13 T14 by inserting the expressions WBSP_DIST_ DISCRETE SV M15 N16 B10 C10 WBSP_DIST_ DISCRETE SV M15 N16 B11 C11 WBSP_DIST_ DISCRETE SV M15 N16 B12 C12 into cells J12 313 and J14 Step 3 We tell What sBest how many scenarios to use in each stage i e the sample size by inserting the expression WBSP_STSC L20 M22 into cell J19 Step 4 We specify that we want a scenario by scenario report of the wealth and the allocation to Stocks and Bonds each period by placing the expression WBSP_REP F9 G9 H9 Wealth1 G10 H10 Wea
458. w is selected it is assumed the cells just above the row are the left hand side and the cells just below are the right hand side of the equation 26 CHAPTER2 If these are not the ranges of cells you want you can specify the cells or ranges by using the button on the right edge of each text box to bring up a cursor for cell selection or type the ranges in However you should use the following table when selecting cells or ranges If the then and the Left hand side Stored In Right hand side includes a should have a should have a single cell single cell single cell range same shape and size range single cell single cell range same shape and size range range same shape and size range same shape and size range You can relate a left hand side range to a single right hand side cell or a right hand side range to a single left hand side cell What sBest inserts a worksheet function called WB in the host cell s specified in the Stored In text box also known as the constraint cells By clicking on one of the constraint cells and looking at the Excel formula box you can see the WB function that was created For example select the B1 cell choose the WB Constraints command and press OK on the Constraints dialog box to insert the formula WB A1 lt C1 into the cell B1 This formula indicates that the value in Al should be less than or equal to that in C1 This could also be accomplished by
459. w minutes working with the spreadsheet you ll quickly understand how complex this problem really is You must make six decisions on the quantities of steel to ship from each mill to each plant and eighteen decisions on the amount of each product you will produce at each plant six products times three plants As your spreadsheet recalculates your What If projections be sure to keep in mind the following guidelines e Each plant must receive its minimum steel requirement Each steel mill cannot ship more than its capacity The use of any raw material cannot exceed the starting inventory SAMPLE MODELS 319 After you ve experimented on your own you re ready to solve The Shipping Worksheet After Optimization 2 put A 3 Plant B 5 Plant C 800 Output 1 200 6 400 PROFIT From ALL PLANTS less SHIPPING COSTS H B Wini 320 CHAPTER7 The Plant A Worksheet After Optimization a f l Profit at PLANTA e al SS EE EE ER KEN 2 E 2 EE GE HN EE EES ES EE a ar Sr AP SE 6 S30 45 ze as Se ap 120 D 220 160 20 50 800 Minimum Steel Requi Product Resource Requirements Steel Wood Plastic Rubber Glass Paint 8 9 18 14 12 13 KH 15 16 18 19 20 21 E EI 25 26 27 28 E Lan The best possible answer to this problem is a total profit of 35 860 Note how What sBest successfully met the minimum steel needs of each plant while
460. w necessary to use the relatively more expensive Grain 4 rich in Nutrient D to reach a 95 confidence in fulfilling all your nutritional requirements D Dual Values Dual values are included in two locations in this worksheet C20 F20 and J7 J9 J11 238 CHAPTER7 The dual values in cells J7 J9 J11 show the shadow prices or reward how much money you would save if each respective requirement 17 19 I11 is relaxed by one unit This means for example that you would save 97 per bushel if the requirement for Nutrient C is reduced to 4 A reduction to 4 9 saves you 0 97 If slightly less sprightly hogs are tolerable in return for such a saving relaxing this nutritional requirement would be a good decision Note that the dual values are zero for Nutrients A and B Lowering the requirement by one unit offers no savings because there is already an excess of Nutrients A and B in your optimum blend since constraints H7 and H8 are not tightly satisfied The other dual values in the Swine amp Roses model C20 F20 show the amount by which the price of an unused grain would have to be reduced before it would be cost effective to use it in the minimum cost blend For example the dual value for Grain 2 is 11 62 D20 This tells you how much the price of Grain 2 would have to be reduced in order for it to appear in the optimal solution of HOGCHANC This is a good number to have handy when negotiating to get a lower price Note In nonlinear mod
461. w to formulate an SP with more than two stages model JMVVESTMENTCOLLEGE XLSX This example was originally given by Birge and Louveaux in their book Introduction to Stochastic Programming You want to set aside 55 000 now hoping that it will grow to 80 000 in three periods so as to support your child in college Only two investments are available Stocks and Bonds At the beginning of each period or stage you re allocate you money between these two investments Each year the return on each investment is a random variable given by the table Stocks Bonds Scenario 1 25 1 14 Scenario 2 1 06 1 12 Stocks have the higher expected return so if we just wished to maximize expected return we would put all our money in Stocks at the beginning of each stage Stocks however also have the greater variability so if we are concerned about achieving our goal of 80 000 then Bonds might seem more attractive with their lower variability in return 216 CHAPTER 6 The INVESTMENTCOLLEGE Worksheet Before Optimization part 1 wo mes iment options each stage Stocks and Bonds Ref Birge amp Louveaux 80 Goal for wealth at beginning of period 4 4 Penalty unit for wealt under goal 1 Utility of wealth unit over goal Step 1 Core model Growth factor Beginning Total Invest in invested Stocks Bonds 0 0000 0 0000 0 0000 1 25 1 14 0 0000 0 0000 0 0000 1 25 1 14 0 0000 0 0000 0 0000 1 25 1 14 Under qoal 0 WESI
462. w to locate plants or warehouse facilities to minimize shipping expenses while meeting demand PLANTLOC XLSX SAMPLE MODELS 229 Staff Scheduling Models Staff Scheduling Meeting personnel needs at minimum cost STAFF XLSX Staff Sched Preferred Assignment Covering staff needs while meeting employee preferences for job or shift assignments ASSIGN XLSX Staff Sched Two Stage Fixed Shift Meeting two objectives in this case minimizing cost and maximizing employee satisfaction in scheduling FIXED1 XLSX FIXED2 XLSX Transportation Models Pipeline Optimization Moving resources along routes with limited capacities at minimum expense PIPELINE XLSX Shipping Cost Reduction Minimizing shipping costs on routes with fixed costs while meeting demand SHIPPING XLSX Traffic Congestion Cost Minimization Minimizing shipping costs on a network whose routes have costs that vary with the amount of traffic TRAFFIC XLSX Truck Loading Packing a container with objects of varying sizes to maximize efficiency a knapsack problem TRUCK XLSX Stochastic Models Newsvendor Stock A stochastic model to maximize net profit by deciding how much stock to hold over 2 stages and 1 random parameter for the demand NEWSVENDOR XLSX Investment Planning College A multi period stochastic model planning investments for going to college after 3 Periods INVESTMENTCOLLEGE XLSX Crop Allocation Optimization
463. wer 0 1 30 lt Hard Drive 2 120 e The What sBest solution yields the best possible profit attainable given your resources and constraints It also tells you the appropriate quantities to produce and the total usage of each component Note that profit is now 33 000 considerably higher than the 31 000 attained by simply building as many Deluxe models as possible What s Best If It s often useful to explore your optimization model from various angles For example what if you learn that the 20 Deluxe computer towers remaining from the What sBest answer found above G16 E16 will be obsolete after the current production run In that case it might be better to maximize the utilization of this inventory than to maximize profit Try this by making Total Usage of Deluxe computer towers cell E16 your best cell to be maximized This is done by selecting cell E16 choosing Best from the WB menu clicking OK and then re solving 18 CHAPTER 1 What sBest will maximize the value of cell E16 without regard to the former objective of maximizing Total Profit Note that the resulting production of 50 Deluxe units completely exhausts the obsolete inventory but that profit has dropped from 33 000 to 25 000 Now suppose for financial reasons profit must be at least 32 000 This constraint can be manually entered by simply typing WB G6 gt 32000 in a convenient cell in this case let s use H6 Optimizing agai
464. when the model is solved The next time you solve What sBest will generate a worksheet called WB Stochastic containing a listing of the cells and their locations and values of the variables solution or the random outcomes for the scenario path Click on the WB _ Stochastic tab to see the stochastic report Select button This button will open a new window To create the list of reporting cells specify the cell or cell range in the Select any cell to report field Once this is done pressing the Add button causes the selected cells to appear in the List of selected cells box You can decide to remove any selected cells by simply going to the list box select the cell reference to remove from the list of names and click on the Remove button The WBSP_REP function will be placed in the current selection of the field Place the function WBSP_REP in cell Output Report Select any cells to report multiple selection holding the Ctrl key SAS Cancel List of selected cells select and change by Add or Remove Place the function WBSP_REP in cell SAS1 Here s an example of the stochastic report for the NEWSVENDOR sample model ADDITIONAL COMMANDS 113 la 1 GSTS NEWSVENDOR xlsx Microsoft Excel bala File Home Insert Page La Formul Data Review View Develop Add Ins VY e o El ZS What sBest 11 0 0 2 Mar 31 2011 Library 7 0 1 129 B Cc D ch 2 Mar 31 2011 Library 7 0 1
465. won t consider any branch that doesn t yield an answer of at least 95 If in fact the best integer answer is only 93 you will eventually get the message Solution Status No Feasible Solution Found The default value for Hurdle is None 76 CHAPTER 3 Note When entering a hurdle value be sure that a solution exists that is at least as good or better than your hurdle If such a solution does not exist What sBest will not be able to find a feasible solution to the model Node Selection The branch and bound solver has a great deal of freedom in deciding how to search the branch and bound solution tree Use the Node Selection drop down box to control the order in which the solver selects branch nodes in the tree If you examine the pull down list for the node selection option you will see the following options Solver decides This is the default option What sBest makes an educated guess regarding the best node to branch on Depth first What sBest spans the branch and bound tree using a depth first strategy Worst bound What sBest picks the node with the worst bound Best bound What sBest picks the node with the best bound In general Solver decides will offer the best results Experimentation with the other options may be beneficial with some classes of models Strong Branch Use the Strong Branch text box to specify a more intensive branching strategy during the first n levels of the bran
466. x We specify cell B12 as a stage 1 random variable by inserting the expression WBSP_RAND 1 B12 into cell I8 MATHEMATICAL MODELING 211 Step 2b We tell What sBest that demand has a Normal distribution with mean 80 and standard deviation 20 by inserting the expression WBSP_DIST_NORMAL I15 I16 B12 into cell 113 Step 3 We tell What sBest how many scenarios to use in each stage De the sample size by inserting the expression WBSP_STSC 121 J21 into cell 119 Step 4 We specify that we want a scenario by scenario report on the amount ordered the demand lost sales and profit by placing the expression WBSP_REP B11 B12 B13 B22 in cell 124 We request a nine bin histogram of Profit by placing the expression WBSP_HIST 9 B22 in cell 125 There is also an Options Stochastic Solver dialog box for setting various SP related options Stochastic Solver Options Vi Specifications Optimization Method Solver Decides Seed for Random Generator stat Common Size per Stage a Iw Sampling on Continuous Distribution Only Report Information Calculation for Expected Value of Wait and See Calculation for Expected Value using Expected Value Policy Iw Calculation for Expected Value of Perfect Information Calculation for Expected Value of Modeling Uncertainty Print Scenarios Horizontally in Report e et x 212 CHAPTER 6 The NEWSVENDOR Weorksheet After Optimization
467. x applications The Problem in Words You are a manufacturer producing six products from six materials with each product requiring a different combination and amount of raw materials Background You know the profit per unit of each product and the quantity of each raw material required for a single unit of each product For instance each unit of product 2 contributes 45 to profit and uses 4 units of steel 5 of wood 3 of plastic etc in its manufacture Objective of Optimization The objective of optimization is to maximize the profits that can be realized from manufacturing products using only the current inventory of raw materials SAMPLE MODELS 311 The Worksheet Let s look at the PRODMIX sample file The PRODMIX Worksheet Before Optimization E EN TC AN wm D Cc D E F G H l J K b Er PRODUCT MIX Product 4 27 3 Af 5 6 Profit Unit 30 45 24 26 24 30 Quantity 0 0 0 0 0 0 Produced Product Resource Requirements Steel Wood Plastic Rubber Glass Paint PRODMIX 2 A Determine Adjustable Cells The adjustable cells in this model B8 G8 contain the Quantity Produced for each product B Define Best The best solution is the one that results in the maximum Total Profit A3 Take a moment to examine the formula in this cell SUMPRODUCT B6 G6 B8 G8 That is interpreted as Profit Unit of Product 1 Quantity Produced of Product 1 Profit Unit of Product 2 Quantity Produced of P
468. xe Usage Standard Tower 1 Deluxe Tower Hard Drive 1 GETTING STARTED 15 At this point let s briefly summarize what we have done so far The XYZ Worksheet Before Optimization Home Insert Page Layout Formulas Data Review View Developer Add Ins 8 7 X Product Quantity to Produce Profit per Unit 300 500 Product Component Requi Components Quantity Required Total Number Standard Deluxe Usage In Stock Standard Tower Deluxe Tower Hard Drive A Set Adjustable cells This lets What sBest know that the spreadsheet cells denoting Quantity to Produce C5 D5 are the variables and can therefore be adjusted as What sBest seeks an optimal answer B Define Best As XYZ s goal is to maximize profit the total profit G6 was specified as the objective of optimization This calls for maximizing the value of the formula in cell G6 through manipulation of the values in the adjustable cells C5 D5 C Specify Constraints XYZ s limitations are simply that Total Usage of inventory parts E15 E17 be less than or equal to the total Number in Stock G15 G17 These constraints are specified by means of the formulas in F15 F17 16 CHAPTER 1 Now you are ready to solve the XYZ model with What sBest When you choose Solve from the WB menu or press on the toolbar the What sBest solver is started and the solver status window appears like the one shown below g What sBest Sol
469. y be incorrectly formulated or with a nonlinear model the returned solution may be a local optimum rather than a global optimum Suggestions The following actions may help you in checking your formulation and or improving the solution returned by What sBest Use the Warnings feature to enable all warning messages set via Options General and re solve the model Check the returned status report for any indication of a problem A warning you 387 388 CHAPTER 8 consider trivial may be a sign of a more serious underlying problem Look at all of the constraints in the model to ensure that each was formulated properly all cell references are correct and their signs lt gt and are correct If you are using the Omit feature carefully check the contents of each omitted range For information on the Omit feature see the section entitled Advanced Omit If you know of a better solution than the one returned by What sBest input the new adjustable cell values and check that all the constraints are satisfied and the best cell has improved If the model contains nonlinear relationships check the model statistics in the status report consider re solving the model with different starting values for the adjustable cells Also check the scaling and consider tightening the bounds on your adjustable cells The section entitled Guidelines for Modeling with What sBest in Overview of Mathematical Modeling may be
470. y level of 9 has rated shift 8 as his highest preference After solving Tom has indeed been assigned to shift 8 356 CHAPTER7 Pipeline Optimization File name PIPELINE XLSX TYPE LINEAR OPTIMIZATION Application Profile This model is an example of a network problem requiring the movement of resources at minimum cost along different routes with which varying costs are associated With the addition of limits to capacity along the routes it becomes capacitated Oil and gas pipelines truck and air routes may be utilized at their highest cost efficiency by applying the principles demonstrated in this model The Problem in Words As the operator of an oil supply network you must choose among several available pipelines from two wells to three pumping stations and from the pumping stations to four refineries Wells have a monthly supply capacity that must not be exceeded capacities of the pipelines may be limited and costs vary among the pipelines In addition monthly refinery demand must be fully met Background You must decide how many barrels per month to pump along each pipeline At present no pipeline is operating between Well 1 and Pump C between Pump A and Refineries 3 and 4 and between Pump C and Refinery 1 The decision on how much material to send along a given pipeline is governed by the monthly cost per unit on that pipeline and by the necessity of satisfying refinery demand without exceeding monthly well s
471. y seeking a local optimum from the starting points supplied by the user This provides the user with the opportunity to explore the available local optima MATHEMATICAL MODELING 197 What sBest takes the initial values of the adjustable cells as starting points in its search for a locally optimum solution Therefore if you solve this model with different starting values for B3 and do not use the Global Solver then What Best may return a different local minimum Attempting to solve the model with a starting value between 5 and 6 for B3 is likely to lead to a local minimum at 5 518 which happens to be the global optimum You may try other starting values to find other local optima For some problems you may find that observing the results of solving the model several times with different initial values can help you find the best solution Convexity If a function is convex then it has a single global optimum If a function is not convex it may have multiple local optima The following example shows a convex function of a single variable Graph of 4 A1 3 2 5 40000 35000 30000 25000 20000 15000 10000 5000 0 Go CH L i 300 250 200 150 100 100 150 200 250 300 A geometric definition of convexity states that a function is convex if for any two points on or above the function a straight line connecting the two points lies entirely on or above the function As illustrated in the prec
472. y to speed overall solution times 0 Solver Decides 1 None 2 10 increasing levels of probing MaxCutsPasses 200 MaxCutsPasses sets the maximum number of iterative passes through a model determining appropriate constraint cuts to append to the formulation RelativeCutsLimit 0 5 RelativeCutsLimit is a fractional value imposing a relative limit on the number of constraint cuts that are generated The default limit is set to 0 5 times the number of true 170 CHAPTER 4 constraints in the original formulation CoefficientReduction No 1 True CoefficientReduction is used to enable or disable coefficient reduction cuts False Disable True Enable Disaggregation No 1 True Disaggregation used to enable or disable disaggregation cuts False Disable True Enable FlowCover No 1 True FlowCover is used to enable or disable flow cover cuts 0 Disable 1 Enable GCD No 1 True GCD is used to enable or disable greatest common denominator cuts 0 Disable 1 Enable Gomory No 1 True Gomory is used to enable or disable Gomory cuts 0 Disable 1 Enable GUB No 1 True GUB is used to enable or disable generalized upper bound cuts 0 Disable 1 Enable KnapsackCover No 1 True KnapsackCover is used to enable or disable knapsack cover cuts 0 Disable 1 Enable Lattice No 1 True Lattice is used to enable or dis
473. ype of any extended solver in use over and above the standard linear and nonlinear solvers Possible extended solvers are Global Branch and Bound or Multistart Tries This shows the current number of tries or iterations needed to solve the model Infeasibility This shows the total amount by which all constraints are violated When this value is reported as zero all constraints are satisfied However on integer models all integer restrictions may not be satisfied Best Objective Bound Assuming a best cell is present this shows the theoretical bound on the objective for integer programming models or any model solved with the global solver Steps This shows the number of solver steps required by any extended solver Active This shows the number of pending subproblems to be solved by any extended solver Solution Time This shows the length of time the solver has been running in hours minutes and seconds Additional detailed information about your model can be found by generating a solution report You can enable the solution report option via the Solution Report drop down box on the General Options dialog box 3 Additional Commands What sBest has a number of additional commands to allow you to do such things as perform sensitivity analysis identify integers variables change the type or appearance of a report or locate a particular type of cell Others will wish to tailor the performance of the solvers by manipulating

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