Supplement 1 for LINGO
Contents
1. STDY J 5 Budget constraint BUDGET SUM ASSET i X i WEALTH Return requirement RETURN SUM ASSET i ERET i X i gt TARGET WEALTH ENDSUBMODEL CALC Do some pre processing of the data FOR ASSET I Compute yearly std dev from monthly STDY I STDM I 12 5 i Find min and max return over possible investments RET_MIN MIN ASSET I ERET I RET_MAX MAX ASSET I ERET I Print a heading for risk vs return trade off report WRITE NEWLINE 1 Efficient Frontier Portfolio Calculation NEWLINE 1 WRITE The possible investments NEWLINE 1 t CD risk fr rate NEWLINE 1 1 VGO40 SP500 stock index NEWLINE 1 1 VG058 Insured long term tax exempt NEWLINE 1 l VG072 Pacific stock index Q NEWLINE 1 VG079 European Stock index NEWLINE 1 1 VG102 Tax managed cap appreciation NEWLINE 1 k VG533 Emerging markets NEWLINE 2 WRITE Target Risk 1 sd Portfolio composition NEWLINE 1 WRITE Return Ye 13 Turn eff all but error output SET TERSEO 2 Write out the asset names in one row FOR ASSET I WRITE ASSET i i WRITE NEWLINE 1 N1 NPOINTS 1 K 0 Loop over different target returns WHILE K LT NPOINTS TARGET RET_MAX K N1 RET_MIN N1 K N1 Solve the model for this ne2. All 4 iji orm Windows views Clipboard ciji g Records Sort amp Filter Lo Window Find ES Machine I Machine ProdHoursA Roll 28 34 2 E Produa 0 Product gt Profit Produce P01 __ po2 10 __ P03 _ P04 E Marr Machine vi Product ProdHoursUsed POL Roll P02 Rolt P03 Roll P04 cut p01 Cut P02 Cut P03 Cut P04 Weld P01 Weld P02 Weld P03 Weld P04 Recordi 4 s 1of3 gt WH Wk Nori Datasheet View j Esek Figure 3S1 5 Database tables for product mix problem after solution A complication in using ODBC with Windows is that the user may need to register the database with the Windows ODBC administrator How one does this depends upon the version of Windows See Windows documentation for how to do this Doing Extra Processing Outside a Model in LINGO The previous LINGO examples illustrated only how to formulate a single model solve it and produce a standard solution report In practice you may want to a do some pre processing of data to prepare it for your model b do some post processing of a solution to produce a customized solution report or c solve several models automatically in one run as part of some comprehensive study LINGO includes a programming capability so that you can do a b and c We describe two additional section types CALC and SUBMODEL that you can use in LINGO CALC Executable statements pe
3. the objective function 4 Deir j l given earlier for the model The second key set looping function is the FOR function This function is used to generate constraints over members of a set The general form is FOR set constraints For example FOR Machine i SUM Product j ProdHoursUsed i j Produce j lt ProdHoursAvail i says to generate the constraint following the colon for each member of the set preceding the colon The less than or equal to symbol lt is not on the standard keyboard so LINGO treats the standard keyboard symbols lt as equivalent to lt This set is the set of machines Machine i for i 1 2 3 so this function loops over the index i For each i the constraint following the colon was expressed algebraically earlier as 4 D Soi Jal Therefore after the third section of the LINGO formulation the mathematical model itself is added we obtain the complete formulation shown below LINGO3h Product mix example Notice the SETS section says nothing about the number or names of the machines or products That information is determined completely by supplied data SETS The simple primitive sets Machine ProdHoursAvail Product Profit Produce A derived set MaPr Machine Product ProdHoursUsed ENDSETS DATA Get the names of the machines amp hours avail Machine ProdHoursAvail Roll 28 Cut 34 Weld 21 Get the names of
4. in hours on machine 9 Roll 28 28 17 2 1 1 4 2 4 10 Cut 34 33 5 14 2 5 17 2 6 11 Weld 21 21 1 6 1 3 1 6 0 8 12 13 14 15 16 Color codes 17 Data Decision Formula Constraint 18 19 This sheet is setup to be compatible with the What sBest solver see 20 http www lindo com to download the software 21 22 23 24 25 26 27 28 29 M4 M WB Status Sheeti I m Count 4 Sum 123 j0 Average 30 75 Ready Calculate C Figure 3S1 3 Screen shot showing data for the product mix example entered in a spreadsheet The LINGO spreadsheet connection can be pushed even further so that the LINGO model is stored on a tab in a spreadsheet Thus the user may think of the model as residing completely in the single spreadsheet One thus can combine the best features of a spreadsheet and a modeling language See LINGO manuals for details Importing and Exporting from a Database with LINGO Another common repository for data in a large firm is in a database In a manner similar to OLE LINGO has a connection function ODBC for transferring data from and to a databases This function is based around the Open DataBase Connectivity ODBC standard for communicating with SQL Structured Query Language databases Most popular databases such as Oracle Paradox DB 2 MS Access and SQL Server support the ODBC convention Let us illustrate the ODBC connection for our little product mix example Suppose that all the da
5. item For example the set of cells A9 A11 was given the range name of Machine Similarly the set of cells E4 H4 was given the range name Product The above example assumes that the appropriate spreadsheet has been opened in Excel and that this is the only spreadsheet opened If you want to make sure that the correct spreadsheet is referenced or you wish to retrieve data from several spreadsheets then you may put the file name in the OLE argument For example to make sure that the ProdHoursUsed data are retrieved from the file C Hillier9 wbest031 xls you would use the statement ProdHoursUsed OLE C Hillier9 wbhest03i xls in the DATA section s ago whestO3i xls Compatibility Mode Microsoft Excel z e x Home Insert Page Layout Formulas Data Review View Developer Add Ins ax amp Cut gt ea s lenis 73 a 7 Tam Fe Hh E Autosum S Arial z lAa oe Ei wrap Tet General a 55 7 z S Bs m amo Ad Fa Pate P Format Painter POCZ rea E E E EE E Merge amp Center paarina E matting as Tabie sies sss eare Fere Select Clipboard g Font g Alignment m Number g Styles Cells Editing Profit i fe 26 R A B c a a al Gla J K L M N o Q R z 1 WBest03i 2 Product Mix Example 3 Product 4 Total Profit contribution 75 P01 P02 P03 P04 5 Profit unit 26 35 25 37 6 Units to Produce 0 10 5 0 7 Hours Hours 8 Machine available Used Production time unit
6. months the supplement also introduces actual data into the formulation and then shows the complete solution The supplement goes on to discuss and illustrate the debugging and verification of this large model The supplement also describes further how to retrieve data from external files include spreadsheets and how to insert results in existing files In addition to this supplement the additional material on this website includes both a LINGO tutorial and LINGO LINDO files with numerous examples of LINGO formulations These LINGO LINDO files for most other chapters include LINGO formulations for problems that are outside the realm of linear programming In addition Sec 12 10 discusses a global optimization feature of LINGO for dealing with complicated nonlinear programming problems Also see Selected References 3 and 5 for Chap 3 for further details on LINGO 16
7. no such errors a solver will begin solving the problem during which time a solver status window will appear on the screen When the solver finishes a Solution Report will appear on the screen Note you can control whether the solution report appears automatically or not by clicking on LINGO Options Interface Output Level Terse Verbose The bottom half of Fig 3S1 1 shows the solution report for our example The Value column gives the optimal values of the decision variables The first entry in the Slack or Surplus column shows the corresponding value of the objective function The next two entries indicate the difference between the two sides of the respective constraints The Reduced Cost and Dual Price columns provide some sensitivity analysis information for the problem After discussing postoptimality analysis including sensitivity analysis in Sec 4 7 we will explain what reduced costs and dual prices are while describing LINDO in Appendix 4 1 These quantities provide only a portion of the useful sensitivity analysis information To generate a full sensitivity analysis report such as shown in Appendix 4 1 for LINDO the Range command in the LINGO menu would need to be chosen next Just as was illustrated with MPL in Sec 3 6 LINGO is designed mainly for efficiently formulating very large models by simultaneously dealing with all constraints or variables of the same type We soon will use the following example to illustrate how LINGO does
8. this Example Consider a production mix problem where we are concerned with what mix of four products we should produce during the upcoming week For each product each unit produced requires a known amount of production time on each of three machines Each machine has a certain number of hours of production time available per week Each product provides a certain profit per unit produced Figure 3S1 2 shows three types of data machine related data product related data and data related to combinations of a machine and product The objective is to determine how much to produce of each product so that total profit is maximized while not exceeding the limited production capacity of each machine x Ags wbest03a xls Compatibility Mode Microsoft Excel zex Home Inset Page Layout Formulas Data Review View Developer Add Ins a 5 E A cut ee or Seu Sawranite PESZE a s hha ee PA Autosum sie t u ore E oe s MEEI i en z a Si EREN ay i 3 ormat Painter Formatting as Table Styles 5 X z lt 2 Clear Filter Select Clipboard n Font Alignment w I Number E Styles Jt Cells _ Editing c D Renee SS a ey I dJ K L M E 1 2 Product Mix Example 3 Product 4 P01 P02 P03 P04 5 Profit unit 26 35 25 37 6 7 Hours d s Machine available Production time unit in hours on machine Roll 28 Cut 34 Weld 21 12 13 15 16 17 M4 M WB Status Sheeti I Ready Calcu
9. 31 0 3874 0 2924 0 3202 0 069 0 1035 0 2499 0 3413 0 4088 0 073 0 1244 0 1124 0 3902 0 4975 0 076 0 1457 0 3636 0 6364 0 080 0 1733 1 0000 Input data used Expected return yr 0 0400 0 0600 0 0600 0 0500 0 0650 0 0700 0 0800 Stdev in return yr 0 0000 0 0811 0 0911 0 0370 0 1010 0 1252 0 1733 Notice how as the desired or target return is increased the allocation of funds shifts from the less risky lower return investments on the left to the more risky higher return investments on the right More about LINGO Only some of the capabilities of LINGO have been illustrated in this supplement A summary of some of the additional capabilities beyond solution of simple linear programming problems is 1 Scalability of models The SETS subscripted variables capability allows you to formulate one model that can be applied to a small problem today and by only changing the input data applied to a huge problem tomorrow 2 Interface to outside data sources The examples in this section illustrate that you can retrieve input and store output for a model from a variety of data sources spreadsheets standard text files and databases 3 Ability to formulate and solve relatively arbitrary nonlinear problems 4 Solve problems with integer variables with either linear or nonlinear constraints 5 Solve problems with quadratic constraints with extra efficiency 6 Find and prove global optimality for complex nonlinear models 7 Programming capability
10. A ay Replace Ba copy E eBsave speling 44 YAdvanced n GoToy View Paste gt korat painter P EE ESE a X eiae Eek More E SV toogeriter param maa N Seree Views Clipboard ile Font m Rich Text Records Sort amp Filter Window Find All Ta 7 amy P Machine ProdHoursA Roll 28 Machine 2 Cut 34 E Machi Weld 21 MaPr a 0 E marr m e Product 2 Fa produc Product Profit Produce P01 26 0 P02 35 0 P03 25 0 P04 37 0 0 0 Machine Product ProdHoursUsed i POL 17 Roll P02 2 1 Roll P03 14 u Roll P04 24 Cut P01 11 Cut P02 2 5 Cut P03 1 7 Cut P04 2 6 Weld P01 1 6 Weld P02 13 Weld p03 1 6 Record 4 4 1of3 Weld p04 0 8 te g J D g Datasheet View CELE Figure 3S1 4 Initial data base tables for product mix example Notice that the Produce column has been initialized to zero in the Product table Once we solve the model the Produce amounts get inserted into the database and the Product table looks as in Fig 3S1 5 10 Ta bebe AaOtxy Table Tools Microsoft Access o x E ee i Home Create External Data Database Tools Datasheet e Se x A eae ahs 4 amp Cut Ihcatbri iu la New E Totals SLIS 7 MG Selection PE g a a Replace a copy save SF spelling X YJ Advanced GoTo Vii Pasti Ri h Filte Size ti Switch Find hew Paste omat Painter PATA eE E SE paeem Mores E Tener eaa mas Select
11. INGO program MODEL Computing an Efficient Frontier for a Portfolio Model in LINGO Generic Markowitz portfolio model in LINGO SETS Each asset has an expected return yr a standard deviation in return month a standard deviation yr to be computed and an amount to invest to be determined ASSET ERET STDM STDY X Each pair of assets has a correlation Only store the lower triangular part TMAT ASSET ASSET amp 1 GE amp 2 CORR ENDSETS DATA The initial wealth WEALTH 1 The investments available Vanguard funds ASSET CD VG040 VG102 VG058 VG079 VG072 VG533 Estimated future return per year ERET 04 06 06 05 065 s07 08 A Monthly std dev in return for each asset 12 computed from recent monthly data STDM 0 02341 02630 01067 02916 03615 05002 Correlation matrix computed from recent monthly data CORR VCD 1 VG040 0 1 VG102 0 98209 1 VG058 0 24279 31701 1 VG07 9 0 75201 76143 34311 1 VGO72 0 49386 49534 24055 68488 1 VG533 0 77011 76554 18415 84397 69629 1 Number different returns to evaluate NPOINTS 12 ENDDATA SUBMODEL HARRY The standard Markowitz portfolio model Minimize the variance in yearly portfolio return OBJ MIN RISK RISK SUM ASSET I STDY I STDY I X I 2 2 SUM TMAT I J I NE J X I X J CORR I J STDY I
12. NGO The Rows section of this report is slightly ambiguous in that you need to remember that Row 1 in the model concerns the objective function and the subsequent rows involve the constraints on machine capacities This association can be made more clear in the report by giving names to each constraint in the model This is done by enclosing the name in placed just in front of the constraint See the following modified fragment of the model TotProf MAX SUM Product i Profit i Produce i For each machine i FOR Machine i Hours used lt hours available Capc SUM Product j ProdHoursUsed i j Produce j lt ProdhoursAvail i i The solution report now contains these row names Row Slack or Surplus Dual Price TOTPROF 475 000000 1 000000 CAPC ROLL 0 000000 15 259740 CAPC CUT 0 500000 0 000000 CAPC WELD 0 000000 2 272727 An important feature of a LINGO model like this one is that it is completely scalable in products and machines In other words if you want to solve another version of this product mix problem with a different number of machines and products you would only have to enter the new data in the DATA section You would not need to change the SETS section or any of the equations This conversion could be done by clerical personnel without any understanding of the model equations Scalar vs Set Based Models The above is a set based formulation For simple models y
13. SUPPLEMENT 1 TO CHAPTER 3 The LINGO Modeling Language LINGO is a mathematical modeling language designed particularly for formulating and solving a wide variety of optimization problems including linear programming integer programming Chap 11 and nonlinear programming Chap 12 problems Simple problems are entered into LINGO in a fairly natural fashion To illustrate consider the following linear programming problem Maximize Z 20x 31y subject to 2x 5y lt 16 4x 3y 6 and x20 y 0 The screen shot in the top half of Fig 381 1 shows how this problem would be formulated with LINGO File Edit LINGO Window Help D slalS e 2 gt olele almal Sele 2e LINGO Model Text Only LINGO1 A simple linear program LINGO 11 0 LINGO Model Text Only LINGO1 DE Max 20 x 31 y 2 x 5 y lt 16 A x 3 y 6 Solution Report LINGO1 Global optimal solution found Objective value Infeasibilities Total solver iterations Variable x z Slack or Surplus 122 0000 1 000000 0 000000 7 076923 0 000000 1 461538 122 0000 0 000000 0 Value Reduced Cost 3 000000 0 000000 2 000000 0 000000 Dual Price Read FIGURE 381 1 Screen shots showing the LINGO formulation and the LINGO solution report for a simple linear programming problem The first line of this formulation is just a comment describing the model Note that the comment is preceded by an exclamation point and e
14. cribe our little product mix example We want to avoid retyping these data into our LINGO model The only part of the LINGO model that needs to be changed is the DATA section as shown below DATA Get the names of the machines amp hours avail Machine ProdHoursAvail OLE Get the names of the products amp profits Product Profit OLE Hours needed per unit of product ProdHoursUsed OLE Send the solution values back OLE Produce ENDDATA The OLE function acts as your plumbing contractor It lets the data flow from the spreadsheet to LINGO and back to the spreadsheet So called Object Linking and Embedding OLE is a feature of the Windows operation system LINGO exploits this feature to make a link between the LINGO model and a spreadsheet The first three uses of OLE above illustrate that this function can be used on the right of an assignment statement to retrieve data from a spreadsheet The last use above illustrates that this function can be placed on the left of an assignment statement to place solution results into the spreadsheet instead Notice from Fig 381 2 that the optimal solution has been placed back into the spreadsheet in cells E6 H6 One simple but hidden step that had to be done beforehand in the spreadsheet was to define range names for the various collections of cells containing the data Range names can be defined in Excel by using the mouse and the Insert Name Define menu
15. e members of these sets are 1 Attribute for each machine Number of hours of production time available per week 2 Attributes for each product Profit per unit produced Number of units produced per week Thus the first two types of attributes are input data that will become parameters of the model whereas the last type number of units produced per week of the respective products provides the decision variables for the model Let us abbreviate these attributes as follows Machine ProdHoursAvail Product Profit Produce One other key type of information is the number of hours of production time that each unit of each product would use on each of the machines This number can be viewed as an attribute for the members of the set of all combinations of a product and a machine Because this set is derived from the two simple sets it is referred to as a derived set Let us abbreviate the attribute for members of this set as follows MaPr Machine Product ProdHoursUsed A LINGO formulation typically has three sections 1 A SETS section that specifies the sets and their attributes You can think of it as describing the structure of the data 2 A DATA section that either provides the data to be used or indicates where it is to be obtained 3 A section that provides the mathematical model itself We begin by showing the first two sections for the example below l LINGO3h Product mix example Notice the SETS section says nothi
16. late Figure 3S1 2 Data needed for the product mix example In standard algebraic form the structure of the linear programming model for this problem is to choose the nonnegative production levels number of units produce during the upcoming week for the four products so as to 4 Maximize gt C Xj j l subject to 4 gt a x lt bi for i 1 2 3 j l where x production level for product P0 cj unit profit for product PO aij production time on machine i per unit of product PO b production time available per week on machine i This model is small enough with just 4 decision variables and 3 functional constraints that it could be written out completely term by term but it would tedious In some similar application there might instead be hundred of decision variables and functional constraints So writing out a term by term version of this model each week would not be practical LINGO provides a much more efficient and compact formulation comparable to the above summary of the model as we will see next Formulation of the Model in LINGO This model has a repetitive nature All the decision variables are of the same type and all the functional constraints are of the same type LINGO uses sets to describe this repetitive nature The simple sets of interest in this case are 1 The set of machines Roll Cut Weld 2 The set of products P01 P02 P03 P04 The attributes of interest for th
17. nded by a semicolon This is a requirement for all comments in a LINGO formulation The second line gives the objective function without bothering to include the Z variable and indicates that it is to be maximized Note that each multiplication needs to be indicated by an asterisk The objective function is ended by a semicolon as is each of the functional constraints on the next two lines The nonegativity constraints are not shown in this formulation because these constraints are automatically assumed by LINGO If some variable x did not have a nonnegative constraint you would need to add FREE x at the end of the formulation Variables can be shown as either lowercase or uppercase because LINGO is case insensitive For example a variable x can be typed in as either x1 or X1 Similarly words can be either lowercase or uppercase or a combination for clarity we will use uppercase for all reserved words that have a predefined meaning in LINGO Notice the menu bar at the top of LINGO window in Fig 3S1 1 The File and Edit menu items behave in a standard Windows fashion To solve a model once it has been entered click on the red bullseye icon If you are using a platform other than a Windows based PC instead type the GO command at the colon prompt and press the enter key Before attempting to solve the model LINGO will first check whether your model has any syntax errors and if so will indicate where they occur Assuming
18. ng about the number or names of the machines or products That information is determined completely by supplied data SETS The simple primitive sets Machine ProdHoursAvail Product Profit Produce A derived set MaPr Machine Product ProdHoursUsed ENDSETS DATA Get the names of the machines amp hours avail 4 Machine ProdHoursAvail Roll 28 Cut 34 Weld 21 Get the names of the products amp profits Product Profit P01 26 P02 35 P03 25 P04 2373 Hours needed per unit of product ProdHoursUsed 1 7 2 1 1 4 2 4 i Rolls Lied Aad Led 26 l Cut 1 6 1 3 1 6 0 8 Weld ENDDATA Before presenting the mathematical model itself we need to introduce two key set looping functions that enable applying an operation to all members of a set by using a single statement One is the SUM function which computes the sum of an expression over all members of a set The general form of SUM is SUM set expression For every member of the set the expression is computed and then they are all added up For example SUM Product j Profit j Product j sums the expression following the colon the unit profit of a product times the production rate of the product over all members of the set preceding the colon In particular because this set is the set of products Product j for j 1 2 3 4 the sum is over the index j Therefore this specific SUM function provides
19. ou may be satisfied with a simpler scalar formulation that does not include the SETS machinery In fact if for the above model you click on LINGO Generate Display model you will get the following display of the equivalent scalar model MODEL _1 MAX 26 PRODUCE_PO1 35 PRODUCE_P02 25 PRODUCE_P03 37 PRODUCE_P04 2 1 7 PRODUCE_PO1 2 1 PRODUCE_P02 1 4 PRODUCE_PO03 2 4 PRODUCE_P04 lt 28 _3 1 1 PRODUCE_PO1 2 5 PRODUCE_P02 1 7 PRODUCE_PO03 2 6 PRODUCE_P04 lt 34 4 1 6 PRODUCE_PO1 1 3 PRODUCE_P02 1 6 PRODUCE_P03 0 8 PRODUCE_P04 lt 21 END You could in fact enter this simple model in the above form 7 Importing and Exporting Spreadsheet Data with LINGO The previous SETS based example was completely self contained in the sense that all the data were directly incorporated into the LINGO formulation In some other applications a large body of data will be stored in some source and will need to be entered into the model from that source One popular place for storing data is spreadsheets LINGO has a simple function OLE for retrieving and placing data from and into spreadsheets To illustrate let us suppose the data for our product mix problem were originally entered into a spreadsheet as shown in Fig 3S1 3 For the moment we are interested only in the shaded cells in columns A B and E H The data in these cells completely des
20. rhaps including an SOLVE of some submodel ENDCALC 11 SUBMODEL Name Constraints describing a specific model ENDSUBMODEL Statements in a CALC section are treated as executable statements rather than constraints For example the statement xX x 1 is perfectly valid in a CALC section It means that the value of x should be increased by 1 If the same statement appeared as part of a model it would be interpreted as a constraint and the model would be declared infeasible You may also use various input and output statements in a CALC section The use of aSUBMODEL section allows you to refer to that particular sub model in a CALC section Thus ina CALC section you may solve a specific submodel several times with different data or solve several different submodels perhaps using the results from one submodel as input to another submodel The programming capabilities of LINGO are illustrated with the following example We want to compute a trade off curve or efficient frontier of risk vs expected return for a financial portfolio Notice that the CALC section a does some pre processing of input data to convert monthly standard deviations to yearly standard deviations b does some post processing of solution data and uses the WRITE statement to generate a customized solution report and c uses the SOLVE statement to repeatedly solve the portfolio submodel Notice in particular the solution report displayed after the L
21. ta describing our problem are stored in a database called acces03j The modification required in the LINGO model is almost trivial Only the Data section needs to be changed as illustrated by the following fragment form the LINGO model DATA Get the names of the machines amp hours avail Machine ProdHoursAvail ODBC acces03j Get the names of the products amp profits Product Profit ODBC acces03j 9 Hours needed per unit of product ProdHoursUsed OLE acces03j Send the solution values back OLE acces03j Produce ENDDATA Notice that similar to the spreadsheet based model the size of the model in terms of the number of variables and constraints is determined completely by what is found in the database The LINGO model automatically adjusts to what is found in the database Now let us show what is in the database considered above It contains three related tables We give these tables names to match those in the LINGO model namely Machine to hold machine related data Product to hold product related data and MaPr to hold data related to combinations of machines and products Fig 3S1 4 shows what the tables look like initially on the screen s CES s Table Tools Microsoft Access ox y Home Create External Data Database Tools Datasheet BZ B cut Sinn 1 E E EEEE New E Totals ah Y Wz Seletion 1G L
22. the products amp profits Product Profit P01 26 P02 35 P03 25 P04 37 Hours needed per unit of product ProdHoursUsed 1 7 2 1 1 4 2 4 t Rodis Velo ae LaF 2 6 l Cute 1 6 1 3 1 6 0 8 Weld ENDDATA Maximize total profit contribution MAX SUM Product i Profit i Produce i For each machine i FOR Machine i Hours used lt hours available SUM Product j ProdHoursUsed i j Produce j lt ProdhoursAvail i i The model is solved by pressing the bullseye button on the LINGO command bar Pressing the x button on the command bar produces a report that looks in part as follows Global optimal solution found Objective value 475 0000 Total solver iterations 3 Variable Value Reduced Cost PRODUCE P01 0 000000 3 577922 PRODUCE P02 10 00000 0 000000 PRODUCE P03 5 000000 0 000000 PRODUCE P04 0 000000 1 441558 Row Slack or Surplus Dual Price 1 475 0000 1 000000 2 0 000000 15 25974 3 0 5000000 0 000000 4 0 000000 2 272727 6 Thus we should produce 10 units of product P02 and 5 units of product P03 where Row 1 gives the resulting total profit of 475 Notice that this solution exactly used the available capacity on the first and third machines because Rows 2 and 4 give a Slack or Surplus of 0 and leaves the second machine with 0 5 hour of idleness We will discuss reduced costs and dual prices in Appendix 4 1 in conjunction with LINDO and LI
23. to allow the solution of several related models in a single run 8 Support for most mathematical functions one might need to use such as trigonometric functions and most probability functions such as the Normal and Binomial distributions LINGO contains extensive built in documentation To get to the User Manual when in LINGO click on Help Help topics Contents LINGO Users Manual Alternatively you may look up terms in the Index by clicking on Help Help topics Index 15 LINGO is available in a variety of sizes The version that is included with this text is limited to a maximum of 150 functional constraints and 300 decision variables When dealing with integer programming problems in Chap 11 or nonlinear programming problems in Chap 12 a maximum of 30 integer variables or 30 nonlinear variables are allowed In the case of global optimization problems discussed in Sec 12 10 the total number of nonlinear decision variables cannot exceed 5 The largest version called the Extended version is limited only by the computer memory available On a 64 bit computer linear programs with millions of variables and constraints have been solved LINGO is available for most major operating systems including Linux If you would like to see how LINGO can formulate a possibly huge model like the production planning example introduced in Sec 3 6 see the second supplement for this chapter By reducing the number of products plants machined and
24. w target return SOLVE HARRY Write another line in the report WRITE FORMAT TARGET 6 3f WRITE FORMAT RISK 7 4f FOR ASSET I If X is close to 0 print a blank rather than 0 0000 IFC X i GT 00001 WRITE FORMAT X i 7 4f ELSE WRITE se i WRITE NEWLINE 1 K K 4 1 While end WRITE NEWLINE 1 Input data used NEWLINE 1 WRITE Expected return yr FOR ASSET I QWRITE FORMAT ERET i 7 4f i WRITE NEWLINE 1 WRITE Stdev in return yr FOR ASSET I WRITE FORMAT STDY i 7 4f i ENDCALC END When we run the above LINGO program we get the following solution report Efficient Frontier Portfolio Calculation The possible investments CD risk fr rate VGO40 SP500 stock index VGO58 Insured long term tax exempt VGO72 Pacific stock index 14 VGO79 European Stock index VG102 Tax managed cap appreciation VG533 Emerging markets Target Risk l1 sd Portfolio composition Return 1 Yr CD VG040 VG102 VG058 VG0O79 VG072 VG533 0 040 0 0000 1 0000 0 044 0 0077 0 7443 0 0062 0 0193 0 1815 0 0301 0 0187 0 047 0 0154 0 4885 0 0124 0 0386 0 3631 0 0601 0 0373 0 051 0 0231 0 2328 0 0186 0 0579 0 5446 0 0902 0 0560 0 055 0 0309 0 0865 0 0174 0 6898 0 1240 0 0823 0 058 0 0452 0 0626 0 5705 0 1165 0 1703 0 0801 0 062 0 0635 0 4966 0 0655 0 2301 0 2078 0 065 0 08
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