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User Manual - Variation.com

1. User s Guide VaR TRAN Version 3 1 Dr Wayne A Taylor Copyright 2004 Taylor Enterprises Inc All Rights Reserved Taylor Enterprises Inc 5510 Fairmont Road Suite A Libertyville IL 60048 847 367 1032 Fax 847 367 1037 E mail info variation com Web www variation com VarTran is a registered trademark of Taylor Enterprises Inc This software package was created by Dr Wayne A Taylor Dr Taylor is the founder and chairman of Taylor Enterprises Inc where he is responsible for the development of the VarTran Change Point Analyzer and Sampling Plan Analyzer software packages as well as providing consulting and training on the Six Sigma approach and tools including Robust Design Tolerance Analysis Design of Experiments SPC Acceptance Sampling and Statistics Dr Taylor retired from his position as Director of Quality Technologies at Baxter Healthcare Corporation where he was responsible for implementing Baxter s Six Sigma program He had been with Baxter for 22 years Dr Taylor is author of the books Optimization and Variation Reduction in Quality and Guide to Acceptance Sampling The first book outlines the approach implemented at Baxter for achieving six sigma quality The second book provides an easy to use approach for selecting valid sampling plans and for reducing inspection costs that has saved Baxter over 30 million dollars annually His two courses Successful Acceptance Sampling and Robust
2. 0 0430545023685393 90 MT 0 00749437527699365 175 THB 2 83 3209058456264 0 75 DT 2 Keypad 1 03469576191631 175 THB 0 75 DT E2 Paste Target 26 Lower Spec Limit 20 Upper Spec Limit 2 Method of Estimating Average and Standard Deviation Taylor Series 2nd Order z V Include In Optimizations with Multiple Outputs X Cancel f Delete View Eq Help Try the same for temperature of the hot bar The dialog box shown below is displayed The standard deviation is 2 F This standard deviation represents the full variation expected during production A tolerance has been entered that is 3 standard deviations The standard deviation is then specified as 1 6 the width of the tolerance The minimum and maximum targets have been set to the response surfaces region of study This avoids extrapolation by restricting the range of alternative targets considered when the process is later optimized to the region of study The target is initially set to the production target It will change later when the process is optimized The Include in Optimization check box is checked signifying that the target for this input can be changed during optimization Click the OK button to close the dialog box Continuous Input Variable Dialog Box x Symbol THE Name TEMPERATURE_HOT_BAR Units de9 F 2X Tolerance Graphic Symbol Statistics fe Target 190 f TEMPERATURE_HOT_BAR THB
3. SS Average Y StandardDeviation and O2 Cpk Properties are used to specify characteristics of output variables and categories of category input variables Rounded Parentheses and are used to control the order that operations are performed Without parentheses the operations are performed in the order determined by the operator precedence and association rules If in doubt about the order the operations will be performed use parenthesis Special types of expressions are constant expressions and differentiable expressions Jointly refers to continuous input variables and category input variables Model fitting both products and process consisting of one or more output variables along with all the input variables affecting these outputs Our goal is optimize and reduce the variation of the outputs by setting targets and tolerances on the inputs Glossary Interval for Values Kurtosis Objective Function Optimization and Variation Reduction Output Output Variable Precedence and Association The interval for values represents a range containing almost all of the individual values Methods are given for calculating the interval for individual values for worst case statistical and process tolerances in Sections 5 3 to 5 5 Measure of how heavy the tails of a distribution are The normal distribution has a kurtosis of 3 Higher values mean the tails are heavier than the normal distrib
4. The Source radio box is used to specify the source of the equation When the equation is pasted VarTran will automatically determine the source and sets this radio box If the Create Inputs check box is checked VarTran will automatically create any input variables referenced in the equation which don t already exist When generating these variables VarTran uses the name field to determine if a variable already exists Name fields should not be edited until all equations have been converted Once the equation to be converted has been pasted into the Equation edit box and the appropriate options selected click the Convert button to complete the conversion process The converted equation will appear in the Equation edit box in the Output Variable dialog box or Objective Function edit box in the Objective Function dialog box To abort the conversion process click the Cancel button press the Esc key or click the X in the upper right corner of the dialog box For help click the Help button or press the F1 key 4 6 Objective Functions An objective function is a function used for optimization that can contain characteristics of several output variables This allows the simultaneous optimization of several outputs A commonly used objective function for this purpose is the minimum C of the outputs Maximizing this objective function helps to ensure that all outputs have acceptable values of Cpr Objective functions can also be written for Tagu
5. Introduction Design offers the first and best opportunity for reducing variation To illustrate the strategies and tools required consider the task of designing a new pump Suppose that the pump must be capable of delivering solution at a constant rate of 10 ml min Customer usage requires that the flow rate remain between 9 and 11 ml min The first step is to develop a design concept Taguchi calls this the system design Suppose we decide to use a piston to push the solution This concept requires a motor to drive the piston and a valve to control the direction of the flow This is the creative part of design Now the work begins We need to determine all the specifics such as How far should the piston travel and How fast should the motor turn To accomplish this task we need to make a list of all the factors affecting flow rate Three obvious factors are the piston radius R the stroke length L and the motor speed S Another possible factor is the amount of backflow through the valves B Factors affecting flow rate are referred to as the input variables Flow rate is referred to as the output variable It is the output for which we want to optimize the average and reduce the variation This is accomplished by establishing requirements and controls for the inputs Initial Design To accomplish our task we must be able to predict how the output will behave This requires knowledge of how the inputs behave and how the inputs affect
6. Tutorials 15 Optimizing Seal Strength to Achieve Robustness One way of reducing the seal strength variation is through what Taguchi calls robustness Robustness is obtained by setting up the heat sealer to be less sensitive to the variation of the inputs both process parameters and material properties The inputs continue to vary but less of this variation is transmitted to the output Robust design is accomplished through the setting of the inputs targets Since it is generally less costly to adjust targets than tighten tolerances robustness is the preferred method of reducing variation A robust process can be obtained by finding the set of targets for the inputs that minimizes the defect rate Start by clicking on the Optimize button on the toolbar The Find Optimal Targets dialog box shown below is displayed Specify that the seal strength percent defective is to be minimized and then click on the Simplex Optimize button The optimization will take just a couple of seconds Analysis Find Optimal Targets Find Optimal Targets Dialog Box x Optimization Settings Output ss x Characteristic Percent Defective Minimize ba fay Optimize Using Simplex GE Optimize Using Interval Optimization completed Better point found F Precision Y Options Help Once the optimization is completed exit the Find Optimal Targets dialog box by clicking on the Close button The capability study is
7. Type of Tolerance Statistical Tolerance Worst Case Tolerance Values LSL T USL Statistical Tolerance Variation C Process Tolerance Average Variation Variation Distribution Normal zl Standard Deviation USL LSL 6 c Min 150 00000 Max 200 00000 V Include In Optimizations X Cancel f Delete Options Help 14 Chapter 2 Predicting Seal Strength Variation If you know how the inputs behave and you know how the inputs affect the output you can predict how the output will behave While this can be accomplished using a simulation VarTran uses analytic methods instead To view the predicted results at the original operating conditions click on the Capability Study button on the toolbar The predicted performance is shown below zat Analysis Capability Study Capability Study SS el Ed Seal Strength SS Statistical Tolerance LSL il USL 12 329 37 286 Characteristic Value Average 24 808 Standard Deviation 2 7729 Cp 0 72 Cc 0 20 Cpk 0 58 Def Rate normal 4 62 Z Score short term 1 68 Sigma Level 1 68 Interval for Values 12 329 37 286 4 5SD i Simulated Worst Case Conditions Ontians amp Print Copy Help in Lock The predicted performance is far from that desired The process is only slightly off target The most significant issue is the excessive variation Leave the Capability Study window open
8. 165 Glossary S Sensitivity analysis See variation transmission analysis Sigma level 91 99 165 Simulation 43 136 166 Simulation dialog box 136 Six sigma quality 88 99 165 Skewness 166 Statgraphics 69 Statistical tolerance 44 52 90 166 Benderizing 93 Example 22 Pros and cons 24 93 Status line 50 58 Summary Table window 76 Symbol 51 61 62 63 71 166 T Taguchi 8 40 Inner outer arrays 12 146 Loss function 36 73 Signal to noise ratios 73 146 Target Input 52 Minimum and maximum 56 Output 65 Taylor series approach 66 Tolerance Benderizing 93 Individual values 104 Process See process tolerance Selecting type to use 47 58 Statistical See statistical tolerance Unified approach 26 Worst case See worst case tolerance Toolbar 50 157 Transmission of variation 37 U Uninstalling software 2 Updating windows 58 155 Usage conditions 41 47 Vv Variable Add 50 74 Editing 74 Input See input variable Off axis 133 Output See output variable Symbol See Symbol Variables menu 50 150 Variation inducing parameter 38 113 143 146 166 Variation reduction Benefits 35 Strategy 37 Variation transmission analysis 43 79 166 View Equations dialog box 67 VIP See variation inducing parameter W Wear and deterioration 41 Window menu 153 Worst case tolerance 44 94 166 Example 20 Pros and cons 24 96 Z Z score 87 91 99 167 173 Taylo
9. 4 3 Continuous Input Variables 4 4 Category Input Variables 4 5 Output Variables 4 6 Objective Functions 4 7 VO System Window and Editing Variables 4 8 Summary Table Chapter 5 Predicting Capability 5 1 Capability Study Window 5 2 Cp Cc Cpk and Six Sigma 5 3 Statistical Tolerances 5 4 Worst Case Tolerances 5 5 Process Tolerances 5 6 Objective Functions PO po DS ale me m Oro al 10h Chapter6 Optimization and Robustness 6 1 Find Optimal Targets Dialog Box 6 2 Simplex Method 6 3 Interval Method 6 4 Optimization Options Chapter 7 Tightening Tolerances 7 1 Pareto Chart Window 7 2 Working With More Than One Design Chapter 8 Understanding the Results 8 1 Effects Tables 8 2 Plots Dialog Box 8 3 Plot Window 8 4 Checking Results Using Simulations Chapter9 Using VarTran with Designed Experiments 9 1 Types of Equations 9 2 Dual Response Approach 9 3 Tolerance Analysis Approach Chapter 10 Menus and Toolbar 10 1 File Menu 10 2 Variables Menu 10 3 Analysis Menu 10 4 Window Menu 10 5 Options Menu 10 6 Help Menu 10 7 Toolbar Glossary References Index 103 103 106 107 109 111 111 116 121 121 125 127 136 139 139 143 147 147 150 152 153 154 155 157 159 169 171 Program Installation This section describes the system resources required by VarTran and how to install VarTran on your computer 1 1 System Requirements VarTran runs on any 32
10. 9 3092 10 666 3SD Calculated Worst Case Conditions E Ontions amp Print Fe Copy Help v Lock 20 Motor E Entering the tolerances for the 2 value and repeating the optimization improves the Cpk to 2 31 and the sigma level to 8 43 This meets our objective while only adding 1 to the cost of the design 1 valve to 2 valve This is the better option Capability Study F 2 value OO x Flow Rate F Process Tolerance 9 ml min 11 Characteristic On Target Worst Case Average Minimum 9 985 9 6981 Maximum 10 277 Standard Deviation 0 098918 0 10067 Cp 3 37 3 31 Ce 0 01 0 30 Cpk 3 32 2 31 Def Rate normal 1 22 10 17 dpm 0 00000204 dpm Z Score 9 95 6 93 Sigma Level 9 95 8 43 Interval for Values 9 3961 10 579 3SD Calculated Worst Case Conditions Bi print 5 Copy Help v Lock 2 value 3F Bestore 118 Chapter 7 Now we have four different designs displayed the original the optimized without tightening any tolerances the 20 motor and the 2 value We can switch between the designs by clicking the Restore buttons on each of the Capability Study windows Closing a Capability Study window erases the design associated with that window Saving the file saves all four versions of the design together Flow Rate F Process Tolerance LSL ii USL 8 5048 mimin 11 501 Characteristic On Target Worst Case Average Minimum 9
11. A short cut is to right mouse click on the plot Understanding the Results 127 E os S Brin Prints the plot Shortcuts are to click the Print button on the toolbar or to select the Print menu item on the File menu I Copy Copies the plot to the clipboard in graphic format Shortcuts are to click the Copy to Clipboard button on the toolbar or to select the Copy to Clipboard menu item on the File menu i ik SREE Displays help on using this window Short cut is to press F1 key You cannot change the inputs or output variables plotted in a Plot window A new plot must be generated using the Plots dialog box Further the Plot window will automatically close itself if a change is made to the I O system that makes it impossible to update the plot Examples include the deletion of one of the variables plotted and a change in an equation that adds or removes an input The Plot window draws a red X through itself when it is not current Clicking on the window will cause it to redraw itself All plots are drawn for the currently selected category of any category input variables If a new category is selected the plot is updated Plot Options Dialog Box Selecting the Plot Options menu item from the popup menu associated with the Options button displays the Plot Options dialog box shown on the next page There are up to nine tabs displayed for changing different features of the plot The tabs are summarized in the table below Most of these
12. Closer to Target the Better case An example of this case is fill volume The target represents the ideal value There are also upper and lower specification limits since we are guarding against both high and low fills In this case optimization of the average means to center the average on target A second case shown below is called the Larger the Better case Seal strength is an example In this case the ideal value is as large as possible or infinity There is only a lower specification limit guarding against low values In this case optimization of the average means to maximize the average Lower Specification Limit Larger gt Larger the Better Case Overview 33 A third case shown below is called the Smaller the Better case An example is harmful emissions In this case the ideal value is zero There is only an upper specification limit guarding against high values In this case optimization of the average means to minimize the average Upper Specification Limit 0 lt _ Smaller Smaller the Better Case Optimization of the average can mean to maximize the average minimize the average or to center it at target In all cases the variation should also be reduced Otherwise some product can be outside the specification limits VarTran is based on the input output system model The I O system could be a product like a pump The output might be flow rate The inputs could be piston radius st
13. My Documents Stack ST vta Stack WC vta FE Weight Original vta My Computer My Network Places File name Purp vta 7 Files of type VerTran files ta 7 Save Saves changes to a session back in the file which was originally opened to start the session If no file is shown the Save As menu item is invoked instead The file associated with a session is displayed on the caption bar of the main window Shortcut is Ctrl S Save File 7 x Save in S VarTran ef Ey Fe Bridge vta J FloForm vta My Recent HeatSeal vta Documents HS2 vta E Math vta Pump vta Desktop Pump PR vta Pump ST vta RAN Pump WC vta afl Stack PRvta My Documents Stack ST vta Stack WC vta FE Weight Original vta My Computer mn My Network Places File name pump2 vta zl Save as type VerTran vta x Cancel Z 148 Chapter 10 Save As Print Print Setup Copy to Clipboard Exit Saves the session in a new file The Save File dialog box on the previous page is displayed for selecting the file to save the session in An extension of vta will automatically be added to indicate it is a VarTran file If the file already exists prompts to prevent accidental overwriting Prints the contents of the displayed tab of the active child window Shortcut is Ctrl P Displays the Print Setup dialog box for selecting the printer to use and for changing printer options such as portrait versus landscape paper si
14. The Other tab is used to change whether automatic scaling is performed the plot s resolution and the number of lines Plot Options x Tite Axis Style Y Axis x Axis Key Lines Of Axis Other Auto Scale V Auto scale plots whenever axis scale changes Resolution Resolution for lines Low Resolution 50 pts v Number of Contours Number of Contours B X Cancel Help Understanding the Results 135 The Automatic Scaling check box is used to select whether automatic scaling is active Changing an off axis variable the x axis scale and so on can cause the curves to shift so that they are no longer visible on the plot Automatic scaling automatically changes the plot s scales whenever such a change occurs so that the curves remain visible For line and interaction plots the y axis changes whenever the x axis an off axis variable or a category input variable changes For contour plots the contour values change whenever the x axis y axis an off axis variable or a category input variable changes Uncheck this box if you want to override the default scaling The Resolution list box is used to select the plot s resolution For line and interactions plots the resolution is the number of points used to construct the curve For contour plots the resolution is the number of points along each axis used to construct the grid for locating the contours The default is low resolution which speeds up the drawi
15. it discards trailing zeros The Objective Function edit box is used to enter the equation representing the objective function It must be a valid expression This equation can include the symbols of other objective functions as well as properties of the input and output variables A complete list of the operators function and constructs allowed in the objective function are Numbers 2 2 0 2 02 10 3 Unary Operators Binary Operators multiplication and power Input Variables Symbol Property where the following properties and abbreviations are allowed Target T Average A StandardDeviation SD UpperSpecLimit USL and LowerSpecLimit LSL Category Inputs Symbol Category Output Variables Symbol Property where the following properties are allowed Average A StandardDeviation SD Cp Cc Cpk Cpm RootMeanSquareError RMSE PercentDefective PD ZScore Z CoefficientVariation CV LowerInterval Values LIV UpperInterval Values UIV DistanceInsideSpec DIS DistanceFromTarget DFT LowerSpecLimit LSL UpperSpecLimit USL and Target T Objective Functions Symbol Functions Log param Log10 param Exponential param SquareRoot param Sine param Cosine param Tangent param ArcSine param ArcCosine param ArcTangent param HyperbolicSine param HyperbolicCosine param HyperbolicTangent param Gamma param LogGamma param Entering Information 71 Beta param1
16. window The Show All Parenthesis check box is used to indicate how to display parentheses When checked parentheses are display around each operator so that the order of calculation is clear When not checked only the parentheses necessary to ensure the correct order of calculation are displayed The operator precedence and association rules apply to all other operators Entering Information 67 The Show Multiplications check box is used to indicate how multiplications are to be displayed When checked multiplication signs are displayed When not checked no multiplication signs are displayed Close the dialog box by clicking the OK button Alternatives are to press the Enter and Esc keys Pressing the Copy button causes a copy of the equation to be added to the clipboard for pasting into other programs These equations can become very lengthy Equations more than nine printed pages are truncated The Help button is used to display help specific to the View Equations dialog box Keypad Equation Dialog Box The Keypad Equation dialog box is used to enter an equation for either an output variable or objective function It is displayed by clicking on the Keypad button in either the Output Variable dialog box or Objective Function dialog box Keypad Equation Dialog Box x Equation 16 388 pif R 2 L B S Variables Functions Pll il eg S Kea AE aA Cosine 4 2 3 El D oana Gamma Boar E HyperbolicCosine x kom 2
17. 0 05 ml 0 005 ml Targets must be selected for piston radius stroke length and motor speed Values of R 0 1 and L 0 5 were selected based on judgment Plugging these values into the equation for flow rate setting the equation equal to 10 ml min and solving for motor speed results in a motor speed of 48 2 rpm This completes the preliminary design The average should be on target but what about the variation Using VarTran For a demonstration of VarTran s abilities start VarTran and open the file pump vta This is accomplished by selecting the Open menu item from the File menu Highlight pump vta and then click on the OK button The I O System window shown below will be displayed The input variables are shown on the left of the window and the output variables are shown on the right 0 System Window Ee x Medical Pump amp Name E Options amp Print Copy Help All of the information about the input and output variables has already been entered into VarTran To view the information entered for a variable double click on it in the 7 0 System window For example double clicking on the output variable F displays the dialog box shown on the next page This dialog box is used to enter the equation target and specification limits for flow rate Press the OK button to close the dialog box Tutorials 5 Output Variable Dialog Box x Symbol F Name Flow Rate Units ml min OOK Equation 16 388 pij R
18. 0 C Historically the quantity C xC has been referred to as k resulting in the following formula from which C derives its symbol 86 Chapter 4 _ USL LSL 2 eee Cx C k where k rc Defect Rate and Z Score Cpk measures the relative distance to the nearest specification limit An alternative measure is the Z Score For a lower specification limit the Z score is LSL Z Score wer e o For an upper specification limit the Z score is USL Z Score ne ZOnE o For the one sided case Z Score a ae For the two sided case a te Minimum Z SCOT e ower Z Score pk 3 Assuming the normal distribution Z scores can be used to determine the defect rate by looking up the defect rate that corresponds to the Z score Below is an abbreviated table Normal Curve Percentage Outside Specification LSL 0 3 million 3 4 million 31 million 15 8 233 million 0 135 0 6 40 30 26 Io 0 Io 26 30 40 2 3 Z Score 1 6 7 15 8 30 9 50 Predicting Capability 87 The function that relates the Z score and defect rate is called the normal distribution function It is represented by the symbol The defect rate as a fraction defective is calculated as follows Lower Specification Limit Defect Rate Z Score upper Upper Specification Limit Defect Rate z Score 2 Sided Specification Defect Rate Z Score pwe z Score ne Multiply the def
19. 119 8 Understanding the Results The optimizations performed by VarTran make it easy to determine the optimal targets Understanding why these targets are best is much more difficult Effects tables and plots can be used to help you develop an understanding of why 8 1 Effects Tables Effects tables are used to determine which variables have the largest effects on the average and the variation It also helps to identify when strong nonlinear effects and interactions are present which need to be understood Using the effects table you can determine which plots are of greatest interest The Effects Table window is a child window used to display the effects that changing the targets of individual inputs has on the average standard deviation and coefficient of variation CV of the selected output variable or the value of the selected objective function It is displayed by clicking on the E button on the toolbar This is a shortcut for the Effects Table menu item on the Analysis menu If there is only one output or objective function the results will be displayed immediately If there is more than one a menu will appear as shown below for specifying which variable to display the results for Analysis ale Find Optimal Targets Robust Design E LAl Capability Study Tolerance Stackup wW beh Pareto Chart of Variation Sensitivity Analysis i Simulation Lock All Unlocked Windows An example of the E
20. 2 L B S a an 123 Target fio Lower Spec Limit 3 Upper Spec Limit 11 Method of Estimating Average and Standard Deviation Taylor Series 2nd Order V Include In Optimizations with Multiple Outputs X Cancel f Delete View Eq Help Try the same for the input variable R The dialog box shown below is displayed This dialog box is used to enter the target and standard deviation for R A tolerance has been entered that is 3 standard deviations The standard deviation is then specified as 1 6 the width of the tolerance A maximum and minimum target has also been entered representing the range of alternative targets considered when optimizing the design later The Include in Optimization check box is checked indicating that the target for this input can be changed during design optimization Press the OK button to close the dialog box Continuous Input Variable Dialog Box x Symbol P Name Radius of Piston Units inches xx Tolerance Graphic Symbol Statistics 10 0015 Radius of Piston R WE Statistical Tolerance LSL T USL m Type of Tolerance C Worst Case Tolerance Values Statistical Tolerance Variation Process Tolerance Average Variation Variation Distribution Normal ag Standard Deviation VSL LSL 6 Cp Optimization Options for Target Min 0 01 Max fi V Include In Optimizations X Cancel f Delete Options H
21. 8 S linear 1183 1 1 8646 0 013766 quadratic 0 0 25365 0 0059823 R L 2099 5 9726 0 26007 R B 0 0 0066486 1 0813 R S 1734 6 1 8072 0 014071 L B 0 0 0062349 0 4411 L S 587 89 0 60144 0 012342 B S 2 25 0 0019336 0 0000049944 Maximum Effect 2845 5 9 9543 1 0813 Diagram Table Sprint P cory Help The general strategy is to start with those inputs affecting the variation Their targets are set to reduce the variation in order to achieve a robust design Then the targets of those inputs affecting the average but not the variation are set to optimize the average Finally if inputs exist affecting neither the average nor the variation their targets are set to their low cost values in order to minimize costs As a general rule if the average and standard deviations sections appear different then use those two columns and ignore the CV If the average the standard deviation sections appear very similar then ignore the standard deviation and use the average and CV columns Examining the effects tables for flow rate given previously you should notice that the average and standard deviation columns look similar Those variables with large effects on the average also have large effects on the standard deviation Looking in the CV column you should notice that motor speed S does not have much of an effect on the CV The linear and quadratic terms as well as all its interactions R S L S B S have minimal effects This suggests the s
22. Add menu item on the Variables menu and to double click the yellow region in the center of the graphic 74 Chapter 4 _O nane Displays the Name I O System dialog box for changing the name at the top of the graphic Short cut is to double click on the name in the graphic p j gt Clicking this button displays the popup menu to the a right associated with this window This popup menu has menu items that correspond to each of the buttons shown at the bottom of the window It also has items that can be used to change how the graphic is sized within the window A short cut is to right Print mouse click on the graphic Copy to Clipboard Help e Size To Fit Fixed Size Normal Fixed Size Custom gt Print Prints the I O system graphic Shortcuts are to click the Print button on the toolbar or to select the Print menu item on the File menu _B cow Copies the I O system graphic to the clipboard Shortcuts are to click the Copy to Clipboard button on the toolbar or to select the Copy to Clipboard menu item on the File menu 7 Helo Displays help on using this window Short cut is to press F7 key Name I O System Dialog Box Clicking the Name button displays the Name I O System dialog box shown below Double clicking on the name in the O System window has the same effect Name I O System x Name I O System Heat Sealing Process X Cancel Help Enter the new name in the Name I O System edit box All char
23. Average x Length Average Area SD SquareRoot Length Average 2 Width SD 2 Width Average 2 Length SD 2 Width SD 2 Length SD 2 Using VarTran with Designed Experiments 139 These equations represent the average and standard deviation of the area of a rectangle as a function of the averages and standard deviations of width and length VarTran derives these two property equations from the value equation given above Now let us consider what happens when a designed experiment is run Suppose a response surface study was run on a heat seal machine to investigate the effect that time and temperature has on seal strength The following equation was obtained SS b0 b1 TIME b2 TEMP b3 TIME 2 b4 TEMP 2 b5 TIME TEMP where SS seal strength and b0 b5 are constants estimated by the response surface study What type of equation is this a value equation or a property equation The response surface study was run by adjusting the two inputs to different target combinations For each set of targets a number of seals were produced tested and the average of these results calculated The above equation was then fit to these averages In which case this equation describes how the targets of the inputs affect the average of the output It is in fact a property equation It is more correctly written as SS Average b0 b1 TIME Target b2 TEMP Target b3 TIME Target 2 b4 TEMP Target 2 b5 TIME Targe
24. Box x Symbol D1 Name Dimension 1 Units inches 2x Graphic Symbol Statistics Tolerance Target fi i 0 03 Type of Tolerance Dimension 1 D1 Worst Case Tolerance Values Worst Case Tolerance El C Statistical Tolerance Variation Process Tolerance Average Variation Optimization Options for Target Min fo Max le V Include In Optimizations X Cancel Delete Options Help Tutorials 21 Double clicking on overall height displays the dialog box shown below This dialog box contains the equation for overall height along with its target and specification limits Again press the OK button to close the dialog box Output Variable Dialog Box x Symbol H Name Overall Height Units inches DX Equation D D2 D3 D4 ha Paste Target 4 Lower Spec Limit f s Upper Spec Limit a1 Method of Estimating Average and Standard Deviation Taylor Series 2nd Order a V Include In Optimizations with Multiple Outputs Statistical Tolerancing Approach A second commonly used approach is called statistical tolerancing The results of a statistical tolerance analysis can be displayed by opening the file stack st vta The results are shown below Overall Height H Statistical Tolerance LSL T USL 3 9 inches 4 1 Characteristic Value Average 4 Standard Deviation 0 02 Cp 1 67 Cc 0 00 Cpk 1 67 Def Rate normal 0 573
25. Dual Response Approach To use the equations for the average and standard deviation directly you should construct two objective functions one for the average and one for the standard deviation For the heat seal case study presented in Section 2 2 the objective functions for the seal strength average SSM and seal strength standard deviation SSS are shown below Notice that the equations are written in terms of the targets of the inputs Objective Function Dialog Box x Symbol SSM Name Seal Strength Mean Units xox Objective 27 9029646803808 0 0822867601868433 175 T_HB Target 9 00263649461868 Function 0 75 DT Target 0 0458833145964137 100 P Target 0 0430545023685393 90 MT Target 0 00749437527699365 175 T_HB Target 2 83 3209058456264 0 75 DT Target 2 1 03469576191631 175 T_HB Target 0 75 DT Target X Cancel Delete Help Objective Function Dialog Box x Symbol ss_D Name Seal Strength Standard Deviation Units OK Objective 11 40686679950494 0 0120234421 491 467 175 T_HB Target 1 26768598066296 Wi Function 0 75 DT Target 0 000138594984121135 100 P Target 0 00331996521199702 90 MT Target 0 00087221 4574975831 175 Keypad T_HB Target 2 26 82493421 40873 0 75 DT Target 2 0 286725843393249 175 T_HB Target 0 75 DT Target ie E Paste X Cancel f Dele
26. The Value edit box is only shown for interaction plots and contour plots It is used to change the values associated with the line For interaction plots this value represents the value of the second input selected For contour plots this value represents the value of the output associated with the contour Plot Options Dialog Box Off Axis Tab The Off Axis tab is used to change the values of continuous input variables that are not displayed along an axis Double clicking on an off axis variable in the key displays this tab The Size of Off Axis Variables combo box is used to select the size for displaying the off axis variables in the key The Display Off Axis Variables check box is used to specify whether to display the values of off axis variables in the key Generally this information is desired but can be removed The Variable tabs are used to select which off axis variable to change The values displayed in the other controls are for the off axis variable whose tab is selected Understanding the Results 133 Plot Options x Title Axis Style Y Axis x Axis Key Lines OfAxis Other Size of Off Axis Variables Normal x V Display Off Axis Variables I Im Variable Name B 48 2 Value X Cancel Help The Name edit box is used to change the name of the off axis variable This field is initially set to the variable s symbol The Value edit box is used to change the value of the off axis
27. To show that the window is no longer current a red X is drawn through it For edit boxes as long as the cursor remains in the edit box no update is performed The update occurs when the edit box is exited by using the tab key or clicking on another control If an error is made entering data for an output an error message will be displayed on the status line It should be handled as described in Section 4 3 One important use of objective functions is in the case of multiple output variables To simultaneously optimize several outputs combine them into a single objective function Assuming two outputs with symbols O1 and O2 the following objective functions might be used Minimum O1 Cpk 02 Cpk O1 PercentDefective O2 PercentDefective Another use of objective functions is to construct Taguchi loss functions like the one below The value 3 2 represents the k value for output O1 If several outputs exist the total loss can be obtained by summing the losses for each output 3 2 O1 Average O1 Target 2 O1 StandardDeviation 2 Taguchi signal to noise ratios can also be constructed for the outputs Several commonly used objective functions are Nominal The BestI 10 Log10 O1 Average 2 O1 StandardDeviation 2 Nominal The Best II 10 Log10 O01 StandardDeviation 2 Smaller The Better 10 Log10 O1 Average 2 O1 StandardDeviation 2 Entering Information 73 4 7 V O System Window and Editing Variables The O Syste
28. a Z score above 4 5 The value shown assumes the inputs are exactly on target so represents the best Z score obtainable Lower Z scores will result in practice as the inputs vary from their selected targets To maintain a Z score of 4 5 requires an on target Z score of 6 or more Sigma Level Predicted sigma level For statistical tolerances the sigma level is the same as the Z score An interval for individual values is displayed at the bottom of the Capability Study window This tolerance represents an interval containing almost all the individual units of product For a statistical tolerance it is calculated as follows Upper End Interval for Values Average 4 5 Standard Deviations Lower End Interval for Values Average 4 5 Standard Deviations The number 4 5 in these formulas is selected so that no more than 3 4 units per million will fall outside each end of the interval The value 4 5 can be changed using the Number Standard Deviations for Interval for Values menu item from the Options menu The graphic visually represents the relationship between the interval for individual values and the specification limits The width of the normal curve displayed always matches the interval for individual values To perform statistical tolerancing VarTran derives an equation for the output s average and standard deviation These equations can be viewed using the View Equation button at the bottom of the Output Variable dialog box as descri
29. and Nelson L S 1975 The Nelder Mead Simplex Procedure for Function Minimization Technometrics 17 1 45 51 Snyder John M 1992 Interval Analysis For Computer Graphics Computer Graphics 26 2 121 130 Taguchi Genichi 1986 Introduction to Quality Engineering Designing Quality into Products and Processes Tokyo Asian Productivity Organization Taylor Wayne A 1991 Optimization and Variation Reduction in Quality Libertyville IL Taylor Enterprises Inc References 169 A Add Category dialog box 62 Analysis menu 152 Association 65 72 163 Automatic scaling 136 C Capability study 79 90 Capability Study window 79 Caption bar 49 Category Input dialog box 61 Category input variable See input variable Cc 84 91 98 159 Child window 147 159 Clipboard 159 Coefficient of variation 123 160 Constant expression 160 Continuous input variable See input variable Continuous Input Variable dialog box 51 Contour plot 30 125 Costs Reducing 39 42 Cp 81 90 98 160 Cpk 85 91 98 104 160 Cpm 161 Curve fit approach 66 CV See coefficient of variation D Defect rate 88 91 98 104 161 Dialog box Add Category 62 Category Input 61 Continuous Input Variable 51 Enter Note 153 Find Optimal Targets 103 Keypad Equation 68 Name I O System 75 Number of Duplicate Copies 150 Number Standard Deviations for Interval for Values 154 Objective Function 70 Optimization Options 1
30. button or press the Enter key when done The symbol is checked and an error message displayed if the symbol is not valid Otherwise the category is added or changed and the dialog box closed Clicking the Cancel button or pressing the Esc key closes the dialog box without adding or changing the changes Clicking on the Help button displays help concerning the Add or Edit Category dialog box 4 5 Output Variables An output variable is a measurable characteristic of direct interest to the user of the product or process Examples include seal strength of a heat sealer and flow rate of a pump Before the analysis can be performed an equation must be entered representing the effect that the inputs have on the output When an output variable is added an Output Variable dialog box like the one below is displayed This dialog box is used to enter and change information concerning the output variable Output Variable Dialog Box x Symbol IF Name Flow Rate Units ml min vox Equation 16 388 pi R 2L B S E 3 Keypad a E2 Paste Target fio Lower Spec Limit 3 Upper Spec Limit E Method of Estimating Average and Standard Deviation Taylor Series 2nd Order x V Include In Optimizations with Multiple Outputs X Cancel f Delete D View Eq Help The Symbol edit box is used to enter the symbol The variable s symbol is used to represent the variable in equations Each variable must have a unique symbol It
31. calculate only that part of the equation and omit the minimum A negative value means that the interval for individual values is outside the specification limits Glossary 161 Expression Input Input Output System 162 Expressions consist of numbers operators functions variables properties and parentheses Numbers Examples are 1 1 0 1 1 1 10 2 and 1 1 10 2 Operators Unary operators and Binary operators and If the parentheses are omitted the operations are performed in the order determined by the operator precedence and association rules Multiplication operators may be omitted Functions Of form function paraml param2 The following functions are supported Log Logl0 Exponential SquareRoot Sine Cosine Tangent ArcSine ArcCosine ArcTangent HyperbolicSine HyperbolicCosine HyperbolicTangent Gamma LogGamma Beta LogBeta AbsoluteValue Minimum Maximum NormalDistribution InverseNormalDistribution pi and e Most functions require one parameter Exceptions are Beta LogBeta Minimum and Maximum which require 2 and pi and e which require zero Squared brackets must be used with functions Variables String consisting of characters the digits 0 9 and underscores The first character cannot be a digit Examples are SS Seal Strength Zone_1 and _1 Properties Of form Symbol Symbol Examples are
32. can be of any length but is best limited to 3 characters or less The symbol must consist of letters the digits 0 9 and underscores It cannot begin with a digit Entering Information 63 The Name edit box is used to enter a more detailed name or description of the variable It may contain any characters and be of any length While the symbol should be short the name should be descriptive The Units edit box can optionally be used to enter the units of measure associated with this variable It can be any combination of characters and any length The xx xx button displays a popup menu for specifying the number of digits to report for this variable The default is five significant digits You can choose between 2 12 significant digits and from 0 to 12 digits after the decimal point When VarTran displays numbers it discards trailing zeros The Equation edit box is used to enter the equation representing the effect of the inputs on this output It must be a valid expression This equation can include the symbols of continuous input variables and other output variables It can also include references to the categories of category input variables A complete list of the operators function and constructs allowed in the equation are Numbers 2 2 0 2 02 10 3 Unary Operators Binary Operators multiplication and power Input Variables Symbol Output Variables Symbol Category Inputs Symbol Category Functions
33. click the Add or Remove Programs icon e Select the program VarTran and click the Change Remove button This will remove the program user manual tutorial and example files It will not affect any files you created using the software 2 Chapter 1 Tutorials VarTran has numerous capabilities It can be used to perform tolerance analysis VarTran performs both worst case and statistical tolerancing It can be used to solve tough nonlinear problems for which the classical tolerancing formulas fail VarTran also implements an entirely new approach to tolerancing called process tolerancing This novel approach to tolerancing represents a unified approach that allows worst case tolerances to be used for some variables statistical tolerances for others and a combination of the two for the rest Process tolerancing solves the age old debate of worst case versus statistical tolerances VarTran however is much more than a tolerancing package It can be used to reduce variation through robust designs You can ask VarTran to find the set of targets that maximizes Cpk minimizes the Taguchi loss or that achieves some other objective You can then ask VarTran which tolerances contribute the most to the remaining variation so that tolerances can selectively be tightened based on cost and benefit The result is high quality at a low cost VarTran can be used in product design process development material development and manufacturing Of all the va
34. deleted all other windows are immediately notified of the change Some windows such as the 1 O System window will immediately display the change However analysis windows will generally wait to update their results until the analysis window is selected by clicking on it To show that the window is no longer current a red X is drawn through it For edit boxes as long as the cursor remains in the edit box no update is performed The update occurs when the edit box is exited by using the tab key or clicking on another control Add or Edit Category Dialog Box The Add or Edit Category dialog box shown below is used to add a new category and to edit an existing category A symbol and description of the category must be specified This dialog box is displayed as a result of clicking the Add Category or the Edit Category buttons on the Category Input dialog box Add or Edit Category x Old Symbol Description 1PL142 Xc Help The Symbol edit box is used to enter the symbol associated with the category This symbol is used in equations referencing this variable It must be a valid symbol consisting entirely of 62 Chapter 4 letters the digits 0 9 and underscores The symbol cannot begin with a digit The symbol can be of any length but is best limited to 1 3 characters The Description edit box is used to enter a more detailed description of the category It can be any combination of characters and any length Click the OK
35. deviation It has the same units as the average and standard deviation The root MSE is always greater than or equal to zero It can only be zero if the average is on target and the standard deviation is zero Minimizing the root MSE drives the average to target with minimum variation Minimizing the root MSE is equivalent to minimizing the Taguchi Loss Measure of quality that directly relates to the defect rate Six Sigma corresponds to 3 4 defects per million Three sigma to 0 13 defective or 1300 defects per million See Section 5 2 Six sigma quality requires that for each measurable characteristic such as fill volume seal strength and so on the defect rate to be no more than 3 4 defects per million In a table of the normal curve the value 3 4 defects per million is found at 4 5 standard deviations Six sigma quality requires that the process average be kept at least 4 50 from the nearest specification limit See Section 5 2 This is accomplished if Cpk gt 1 5 e C gt 2 and C lt 0 25 e The Interval for Values based on 4 5 SD is within the specification limits Glossary 165 Skewness Statistical Tolerance Symbol Variation Transmission Analysis Variation Inducing Parameter VIP Worst case Tolerance 166 Measure of whether the distribution is symmetric i e both tails of the distribution are of the same size A value of zero means the distribution is symmetrical A positive value means
36. dom ZScore short term 4 86 Sigma Level 4 86 Interval for Values 3 94 4 06 3SD 22 Chapter 2 The resulting statistical tolerance for overall height is from 3 94 to 4 06 This easily meets the specification limits The tolerance for the components does not require tightening Statistical tolerancing results in the opposite conclusion as worst case tolerancing Statistical tolerancing assumes that the components come from a process that is centered on the target and that traditionally has a maximum standard deviation equal to 1 6 the tolerance In this case the tolerance of 0 03 results in a maximum standard deviation of 0 01 For tolerance stack up problems statistical tolerances are obtained for the outputs by using the root sum of squares formula In this case the resulting tolerance for overall height is 10 03 0 03 0 03 0 03 0 06 The resulting statistical tolerance of 4 0 0 06 is exactly half that of the worst case tolerance To view how the information entered for input D1 has changed double click on D1 in the O System window The dialog box shown below will appear This dialog box contains the target and tolerance of D1 In the standard deviation field the standard deviation is defined to be 1 6 the width of the tolerance To close the dialog box press the OK button Continuous Input Variable Dialog Box Eg Symbol 1 Name Dimension 1 Units finches wox Tolerance Graphic Symbol Stat
37. figure below shows how the target of an input can be adjusted to reduce the amount of transmitted variation Reducing the variation by adjusting targets is called robust design Changing targets is generally less costly than tightening tolerances It may cost nothing As a result robustness is generally the preferred approach to reducing variation 4IcUvACO Shift Target INPUT Reducing Variation by Adjusting Targets of Input Variables There is a third way of reducing variation This is to change the design concept For example instead of designing a pump using a piston to move the solution you might decide to use roller bars to push the solution through a tube This results in a different set of inputs and a different relationship between the inputs and outputs The figure below summarizes the three approaches to reducing variation 3 Change Equation 4cUUvACO 1 Change Tolerance INPUT 2 Change Target Three Approaches to Reducing Variation Overview 39 3 4 Robustness The concept of robustness was introduced by Taguchi 1986 It is such an important concept it requires further discussion Robustness reduces the variation of the output by making the output less sensitive to the variation of the inputs As a result the inputs can continue to vary but less of this variation is transmitted to the output As shown below robustness is achieved by identifying targets for the inputs were
38. is shown on the next page It has 3 tabs containing various results The results displayed depend on the type of tolerances that were specified for the inputs Section 5 3 covers statistical tolerances Section 5 4 covers worst case tolerances and Section 5 5 covers process tolerances Prior to this the next section provides a summary of the different capability indices displayed in the Capability Study window Click on the X in the upper right corner to close this window The Capability Study window automatically updates itself when information affecting the displayed results is changed When the program is in immediate update mode the updates occur immediately following the change being made If the program is in delayed update mode the Capability Study window draws a red X through itself when it is not current Clicking on the window will cause it to update and redraw itself The update mode is selected using the Options menu Double clicking on the title in the window displays the variable s Output Variable dialog box This is a short cut to selecting the Edit menu item on the Variables menu Predicting Capability 79 Capability Study F O x Flow Rate F Statistical Tolerance LSL T USL 8 7544 ml min 11 242 Characteristic Value Average 9 9981 Standard Deviation 0 27638 Cp 1 21 Ce 0 00 Cpk 1 20 Def Rate normal 297 dpm Z Score short term 3 43 Sigma Level 3 43 Interval for Values 8 7544
39. of 33 4 samples before a point goes outside the control limits The process can run for days off target by 20 before signaling a problem If the process is lo off target it takes an average of 4 5 samples before a point goes outside the control limits The process can run much of a shift off target by this amount Such a control chart does not detect shifts of 1o or less in reasonable periods of time 82 Chapter 4 1000 100 Average Number Samples to Detect 10 1 0 0 560 oO 1 50 20 Amount That Average Shifted Ability of X R Control Chart to Detect Shifts n 5 Now suppose that the process is 1 50 off target The control chart only averages 1 6 samples before detecting the condition In fact 64 of the time the first point plotted following the shift falls outside the control limits Such a control chart does a good job of detecting shifts of 1 50 and above Therefore when using a control chart to keep the process centered a 1 50 operating window should be allowed Take a second example Suppose a capability study was performed on the process and that this capability study indicated that the process was stable i e all points were within the control limits Since a control chart was used to determine whether the process was stable and since this control chart is only assured of detecting shifts of 1 50 or greater a stable process is in essence a process capable of holding a 1 50 operating window Unstable processes require wid
40. of targets has been found If a better set have been found the current targets of the inputs are updated To close the Find Optimal Targets dialog box click the Close button or press the Esc key To view the results perform a capability study To view the new targets display a Summary window The optimization is performed for the current set of tolerances If these tolerances are later changed the optimization should be repeated The optimization is also performed only for the currently selected categories of the category input variables To determine which category is best separate optimizations should be performed for each category and the category with the best results selected Optimization and Robustness 105 In the event that worst case or process tolerances are specified VarTran actually performs a minimax or maximin optimization For example to maximize Cpk VarTran would determine for each set of input targets the minimum C across the worst case tolerances and operating windows for the average It would then find the set of input targets maximizing the minimum Cx This requires that one optimization routine be nested within another which takes longer The Precision button at the bottom of the dialog box is used to increase the precision of the results This button performs an additional simplex optimization beginning at the current set of targets The termination conditions of this optimization are set to improve the precision of
41. outputs that have a 1 sided spec limit Systematic approach to setting targets and tolerances to achieve a specified level of quality at the lowest possible cost The process involves e Determining the key outputs and their specifications the key inputs and their low cost tolerances and the relationship between the inputs and outputs e Find the optimal targets starting with the low cost tolerances to achieve a robust design e Determine if the capability meets specified level of quality e If not determine cause of variation and tighten its tolerance e Repeat previous 3 steps until the specified level of quality is achieved Glossary Robustness Root Mean Square Error Root MSE Sigma Level Six Sigma Quality Taguchi has emphasized the importance of robust designs A robust product or process is one that is less sensitive to variation of the inputs That way the inputs can continue to vary but less of this variation is transmitted to the output This minimizes the need to tighten tolerances Robustness is achieved by careful selection of targets for the inputs See Section 3 4 In VarTran robust designs are achieved by finding the set of targets minimizing the defect rate maximizing Cx or maximizing the distance inside specification This minimizes the transmitted variation while driving the average to its ideal value The root MSE is Root MSE 4 u Target 07 where u is the average and o is the standard
42. param2 LogBeta param1 param2 AbsoluteValue param Minimum param1 param2 Maximum param1 param2 NormalDistribution param InverseNormalDistribution param pi and e Parentheses and The different properties available for the inputs and outputs all have abbreviations associated with them For example to refer to the percent defective of output Ol you can type O1 PercentDefective or O1 PD For a category input variable a reference of the form Cav _2 refers to category _2 of the category input variable Cav This reference has the value 1 when the given category is the selected category and a value of 0 when it is not Categories are selected using the Category Input dialog box Multiplication signs may be omitted A reference like 2 A is interpreted as two times the variable A A number like 2 02 10 3 is interpreted as 2 02 times 10 to the negative three Squared brackets are used to enclose the parameters of a function For example Log 2 0 refers the log of 2 0 The reference Log 2 0 would be interpreted as meaning the variable Log times 2 0 The operator precedence and association rules are the same as for the equation of an output variable This order can always be overridden using parentheses While an equation can be directly typed into the Objective Function edit box it is generally simpler to click the Keypad button and use the calculator style interface provided by the Keyboard Equations dialog box to enter
43. preceding section An exact copy of the displayed plot will be copied to a new Plot window You can create as many Plot windows as you like Contour Plot F Std Dev by R and L ioj x F Std Dev V _ F Std Dev 17 F Sid Dev 13 F Sid Dev 9 F Std Dev 5 J 1 005 F Std Dev 1 S 48 2 B 0 05 0 01 0 01 0 505 1 R Eerint Py Copy 2 Help 1 Once in the Plot window you can modify the plot including rescaling changing the values of any off axis input variables and changing the style To change an item double click on it or select the Plot Options menu item from the popup menu associated with the Options button The buttons are Clicking this button displays the popup menu to Ba i Z i Number Lines the right associated with this window This popup menu has menu items that correspond to each of Size To Fit the buttons shown at the bottom of the window It Boe ele also has items that can be used to change how the Fixed Size Custom graphic is sized within the window and for Print changing the plot options The Plot Options menu Copy to Clipboard item display the Plot Options dialog box described Help in the next section The Number Lines menu item also displays the Plot Options dialog box with the tab displayed for changing the number of lines on either an interaction or contour plot It only appears on the popup menu for these two types of plots
44. prevents the algorithm from fruitlessly trying to meet the relative error criteria when the optimum is zero For the interval method the magnitude of the optimum is guaranteed to be below the specified value For the simplex method the optimization terminates when the magnitude of all corners of the simplex are below the specified value This does not provide a guarantee In both cases the default value is 1 0 10 40 The Stop when above edit box is used to terminate the optimization when it appears to be converging to either infinity or negative infinity The optimization terminates when the magnitude absolute value of the optimum is bounded above the specified value This prevents the algorithm from fruitlessly trying to meet the relative error criteria when the optimum is infinity For the interval method the magnitude of the optimum is guaranteed to be above the specified value For the simplex method the optimization terminates when the magnitude of all corners of the simplex are above the specified value This does not provide a guarantee In both cases the default value is 1 0 10 40 When the desired termination conditions have been entered click on the OK button or press the Enter key to save the values and close the Optimization Options dialog box If illegal values are specified for any of the options a message is displayed and the user is asked to correct the problem first To exit without saving the changes click the Cancel button
45. quality product Units barely within spec while they might function initial could fail early in the life of the product latent defect Lower Upper Spec Spec Limit Limit 20 L O S 10 S 0 Taguchi Loss Function The goal post model and the Taguchi loss function are models Neither one might perfectly fit your particular product However more products more closely fit the Taguchi loss function than the goal post model 36 Chapter 3 3 3 How to Reduce Variation So how do we reduce the variation of the outputs Variation of the outputs is caused by variation of the inputs As shown below variation of the inputs is transmitted to the output through the functional relationship between the inputs and the output This is called transmission of variation Transmitted Variation lt Relationship Between Input and Output 4IcUUvaACcCO Variation of Input INPUT Transmission of Variation One way of reducing variation is to reduce the variation of the inputs This amounts to tightening the tolerances of the inputs Transmitted Variation lt Relationship Between Input and Output 4cUUvACO j lt Variation of Input INPUT Reducing Variation by Tightening Tolerances Overview 37 While all the key inputs contribute to the variation they generally do not contribute equally Applying the Pareto principle 80 of the variation is due to 20 of the inputs As a result
46. simulation along with the predicted values using the analytic approach See Section 4 8 for further information on interpreting the summary table The simulation results are also shown on the second tab of the Capability Study window as described in Sections 5 3 to 5 5 If discrepancies exist between the simulated values and analytic values consider choosing a different method for estimating the average and standard deviation using the Output Variables dialog box Targets and Tolerances of Outputs Output Tyee Foerance value aa Tanger LSL ust Interval tor Values 35D Cp ce Cpk oererrate zscore sima Units F Statistical Note Second row of statistical and process tolerances show simulation results Tolerances japu Er optone Bein B Copy 2 Holp l Lock The Close button is used to close the Simulation dialog box without viewing the results A short cut is to press the Esc key Pressing the Reset button causes previous simulation results to be deleted This allows a new simulation to be run from scratch Otherwise new trials will be added to the previous trials Clicking the Help button displays help on using the Simulation dialog box Understanding the Results 137 Using VarTran with Designed Experiments Performing a variation transmission analysis requires an equation for how the inputs affect the outputs If you cannot obtain this equation any other means you can alway
47. target Set up errors and process shifts can result in performance worse than that predicted Take as an example the case of overall height of a part created by stacking four identical components of top of each other Suppose the specification limits are from 3 94 to 4 06 Each component has an average of 1 with a standard deviation of 0 01 Using 3 standard deviations to form a tolerance results in a tolerance of 1 0 03 for the components The resulting statistical tolerance for the overall height is 4 0 06 These are shown in light gray in the two figures below The tolerance for overall height falls with the specifications Actually Suppose Actually Suppose Behaves to Behave Behaves to Behave Lower J Upper J Upper Toler Toler Spec 0 97 1 0 1 03 3 94 4 0 4 06 INPUTS OUTPUT How Statistical Tolerances Can Result in Out of Spec Product Now suppose that a batch of components is received with an average of 0 0985 and a standard deviation of 0 004 The variation is much smaller than required However the process is not centered a violation of the assumptions of statistical tolerancing All the components are within the tolerance for the components and would pass an attribute sampling plan or a 100 inspection In this case the overall height will average 3 94 with a standard deviation of 0 008 The actual behavior of the inputs and output are shown in dark gray above Despite the fact that the i
48. temperature of the hot bar The heat sealer continuously monitors hot bar temperature and automatically adjusts it This ensures the temperature remains centered A record of hot bar temperature is printed out in the form of a strip chart Using this chart the standard deviation of hot bar temperature is estimated to be 5 F A statistical tolerance is appropriate since the process remains centered The target was initially set to 150 However this can be changed so the Include in Optimization check box is checked A designed experiment was used to fit an equation for seal strength The minimum and maximum targets for hot bar temperature were set to the limits of the study region The Input Variable dialog box appears as below Continuous Input Variable Dialog Box x Symbol HE Name Hot Bar Temperature Units aeg F AES Tolerance Graphic Symbol Statistics fis Target fi 50 M Type of Tolerance C Worst Case Tolerance Values ie Hot Bar Temperature HB Statistical Tolerance LSL T USL Statistical Tolerance Variation Process Tolerance Average Variation m Variation Distribution Normal x Standard Deviation 5 c Optimization Options for Target 175 Min 150 Max 200 V Include In Optimizations X Cancel f Delete Options Help These three examples demonstrate that all three types all tolerances have their applications 60 Chapter 4 4 4 Categ
49. the average from the response surface study It also requires additional information about how much the inputs vary during long term production Based on this information VarTran can perform a robust tolerance analysis to predict the behavior of the output to find the optimal targets and to tighten tolerances as needed The tolerance analysis approach requires that an output variable be constructed based on the equation for the average from the response surface study As shown on the next page the property equation for the average has been converted to a value equation Notice that the equations are written in terms of the inputs themselves and not their targets as before Transferring the equations from your response surface package is accomplished using the Paste button The input variables must also by entered This time tolerances must be entered which are representative of long term manufacturing not the variation during the study An example is shown on the next page In this case a temperature transducer was used to continuously record the temperature over many days of production Temperature was seen to fluctuate up and down 6 degrees from target This included startups following breaks etc and not just during steady production Using VarTran with Designed Experiments 143 Output Variable Dialog Box x Symbol ss Name Seal Strength Units 0x Equation 2 7 9029646803808 0 0822867601668433 175 THB 9 002636
50. the equation The Keyboard Equations dialog box was described in Section 4 5 If an equation was fit to a set of data using a statistics package you can paste the equation by clicking the Paste button and using the Paste Equation dialog box to automatically convert the equation into the format required by VarTran The Paste Equation dialog box was described in Section 4 5 When an equation is pasted into an objective function the equation can be evaluated and optimized directly but cannot be used for a tolerance analysis An example of such as equation is a model for the standard deviation or percent defective fit using a response surface study Chapter 9 covers this topic in greater depth To close the dialog box and save the changes click the OK button The symbol is checked to verify that it is valid and not already in use If an error is encountered a message is displayed To cancel the changes including the addition of a new variable click the Cancel button press the Esc key or click the X in the upper right corner of the dialog box To delete the variable click the Delete button For help click the Help button or press the F1 key 72 Chapter 4 When an edit box is exited all other windows are immediately notified of the change Some windows such as the I O System window will immediately display the change However analysis windows will generally wait to update their results until the analysis window is selected by clicking on it
51. the graphic The following buttons and controls are available i Options Print Copy B Help F1 Lock check box 112 Clicking this button displays the popup menu associated with this window This popup menu has menu items that correspond to each of the buttons shown at the bottom of the window It also has items that can be used to change how the graphic is sized within the window A short cut is to right mouse click on the graphic Prints the graphic on the selected tab Shortcuts are to click the Print button on the toolbar or to select the Print menu item on the File menu Copies the graphic on the selected tab to the clipboard in graphic format Shortcuts are to click the Copy to Clipboard button on the toolbar or to select the Copy to Clipboard menu item on the File menu Displays help on using this window Short cut is to press F key Checking this box locks the window so that it does not update as changes are made to the variables Locking the window causes a complete copy of the I O System to be saved It also causes the Restore button to be displayed which can be used to restore the I O System to the state it was in when the Lock check box was clicked Unlocking a window deletes the saved copy of the I O System Chapter 7 Note edit box Used to enter a note that appears on the caption bar It is primarily intended to be used when the window is locked to describe the version of the design being display
52. the output For a pump it is possible to use geometry to derive the following equation for flow rate F 16 388 n R L B S The expression 1 RL represents the volume of a cylinder of radius R and height L Multiplying by 16 388 converts the quantity from cubic inches to milliliters The volume is then reduced by the backflow and multiplied by the motor speed to obtain the final flow rate We also need information on how the inputs behave This requires us to call the plant to obtain process capability information on drilling the piston hole They tell us the standard deviation of their process is 0 0005 We must also call the motor supplier to determine the capability of their process in controlling motor rpm They have both 5 and 20 motors Our initial design will be based on low cost components and processes so we request information on the 5 motor Changes can be made later if necessary They inform us that the standard deviation of their process is 0 17 rpm The valve supplier must also be contacted to determine their process capability information for valve backflow There is a choice between a 1 and a 2 valve Again we start with the 1 valve The valve manufacturer provides us with both an average and standard deviation The table on the next page shows the information collected 4 Chapter 2 Data from Plants and Suppliers Piston Radius R po 0 0005 Stroke Length L po 0 0017 Motor Speed S a 0 17 rpm Backflow B
53. the previous result Since the interval analysis approach terminates after determining 2 significant digits this button can be used to obtain more exact results The Options button is used to change the termination parameters used by the two methods of optimization Clicking this button results in the Optimization Options dialog box being displayed This dialog box is covered in Section 6 4 Clicking the Help button displays further help concerning Find Optimal Targets dialog box including definitions of the different characteristics to optimize 6 2 Simplex Method This section describes the simplex method of optimization It is for information only and can be skipped For purpose of explanation it will be assumed that we are minimizing a function The simplex method starts at a randomly selected point in the region of optimization and then travels downhill until it can find no direction resulting in further improvement The weakness of this approach is that it may locate the bottom of one valley and terminate without recognizing that another valley exists with an even better point To reduce this error the simplex optimization should be performed multiple times The presence of local optimums can also be determined by careful examination of plots generated using the Plots dialog box The simplex method is based on a geometric object called a simplex In two dimensions a simplex is a triangle three corner points In three dimensions it is a
54. the slope is shallow Shallow slopes mean less sensitivity less transmitted variation and more robustness Steep slopes mean increased sensitivity more transmitted variation and less robustness Shallow Slope a Steep Slope gt JAC UAcCoO INPUT Robustness by Making Less Sensitive Opportunities for achieving robustness abound We have a product a process and materials Each can be designed to be robust to the others In addition all can be designed to be robust to the usage conditions and environment as well as to wear and deterioration of the components and materials A Material amp Components Wear amp Usage amp Deterioration Environment Opportunities for Robustness 40 Chapter 3 You can design the product to be robust to its components and materials For example in the pump tutorial the pump was designed to be robust to motor speed Besides improving quality this allows the use of less expensive components You can design the product to be robust to the process Taguchi gives an example where there is excessive warping of tiles due to a large temperature gradient across the drying oven One solution is to reduce the temperature gradient which requires a large capital investment However the solution actually implemented was to reformulate the tiles to make them less sensitive to the temperature gradient As a result quality was improved without a large expenditure This a
55. tolerances are specified for some of the inputs the capability study reports the minimum and maximum values of the objective function over the worst case tolerances and operating windows for the averages of the continuous input variables It also gives the value of the objective function when the inputs are at their target values In all cases it is assumed that the category input variables are set to their selected categories An example is shown below Capability Study T ioj x Taguchi Loss Characteristic Value Value At Target 0 0022547 Minimum 2 4158 10 12 Maximum 1 34 J Simulated Worst Case Conditions amp Print D coy Help C tok sis Rather than display separate capability studies for each output you can display a Summary Table window showing the capability study results of all the output variables and objective functions in a single window See Section 4 8 What if the capability study results are not acceptable There are two approaches to improving the design e Selecting new targets for the inputs to optimize the design and make it robust Chapter 6 e Reducing the variation by tightening the tolerances of the inputs Chapter 7 102 Chapter 4 6 Optimization and Robustness One way of improving the performance of your product or process is to change the targets of the inputs to optimize the average and to reduce the variation to obtain a more robust design In VarTran this can be acco
56. unreasonable to expect to be able to produce millions of ee ee ee oe units nearly free of defects Predicting Capability 5 3 Statistical Tolerances We return to the main topic of this chapter performing capability studies When a capability study is performed by clicking on the w button the results displayed will depend on the type of tolerances specified for the input variables When statistical tolerances are specified for the input variables a statistical tolerance is calculated for the output The results are shown on 3 tabs Tab 1 Calculated Results The first tab shows the results of a tolerance analysis performed using the analytic approach Capability Study F OO x Flow Rate F Statistical Tolerance LSL ah USL 8 7544 ml min 11 242 Characteristic Value Average 9 9981 Standard Deviation 0 27638 Cp 1 21 Ce 0 00 Cpk 1 20 Def Rate normal 297 dpm Z Score short term 3 43 Sigma Level 3 43 Interval for Values 8 7544 11 242 4 5SD Calculated Simulated WorstCase Conditions Options E Print Copy l The name of the output variable and the type of tolerance are displayed at the top Further down a table of the capability study results is displayed The following values are displayed Average The predicted average of the output variable Assumes the continuous input variables are at their selected targets and that the category input variables are s
57. updating the child windows Help menu Used to display the help index help on specific topics registration and support information and the About dialog box containing copyright information 10 1 File Menu File O New Ctri N Open Ctri O E Save Ctri S T Save As amp Print Ctri P SS Print Setup Copy To Clipboard Ctrl C 1 C Program Files Taylor Enterprises VarTran Pump vta 2 C Program Files Taylor Enterprises WarTran Hs2 vta 3 C Program Files Taylor Enterprises VarTran HeatSeal vta Fi Exit The File menu is the first menu on the menu bar It is used to start a new I O system open an existing I O system save an I O system print and to exit the program The first group of six buttons on the toolbar contains shortcuts for those menu items with icons except for Print Setup and Exit The available menu items are Menus and Toolbar 147 New Starts a new session Shortcut is Ctrl N First asks whether to save the current session Open Reads previously saved session from a file The Open File dialog box is displayed for selecting the file to open VarTran files have the extension vta Shortcut is Ctrl O You will first be asked whether to save the current session Open File 2 x Look in VarTran x e ck Ey k Bridge vta FloForm vta My Recent HeatSeal vta Documents HS2 vta Math vta a4 Pump vta Desktop Pump PR vta Pump ST vta amp Pump WC vta J Stack PR vta
58. we can explore the tolerances and decide which to tighten The Pareto chart is shown below Tightening Motor Speed S has the biggest effect This requires the use of the 20 motor Value Backflow B has a smaller effect but is considerably cheaper We will explore both options Pareto Chart F Optimized Ol x Flow Rate Interval for Values 9 0926 10 881 3SD Input Transmitted Effect of Setting Input Tolerance to 0 Variable Variation New Interval for Values Reduction 0 9105 9 3662 10 599 31 1 9 2907 10 679 22 4 0 73999 0 57973 9 2334 10 735 16 0 0 35938 Percent Reduction Transmitted Variation Table Plot for Input Options amp Print Copy Help r Lock Optimized ZF Bestore Entering the tolerances for the 20 motor and repeating the optimization improves the Cpx to 1 80 and the sigma level to 7 17 This meets our objective but requires adding 15 to the cost of the design 5 motor to 20 motor 9 1694 10 801 8 7 Tightening Tolerances 117 Capability Study F 20 Motor O x Flow Rate F Process Tolerance LSL T USL 9 ml min 11 Characteristic On Target Worst Case Average Minimum 9 9846 9 6563 Maximum 10 319 Standard Deviation 0 11361 0 11572 Cp 2 93 2 88 Ce 0 02 0 34 Cpk 2 89 1 89 Def Rate normal 2 42 10 12 dpm 0 00707 dpm Z Score 8 66 5 67 Sigma Level 8 66 7 17 Interval for Values
59. 09 Output Variable 63 Paste Equation 69 Plot Options 128 Plots 125 Print Setup 149 Registration 156 Simulation 136 View Equations 67 Differentiable expression 161 Dimensional stack up problems 20 Distance from target 104 161 Distance inside spec 104 161 Distribution 53 Dual response approach 12 141 143 E Effects table 121 Effects Table window 121 Enter Note dialog box 153 Environment See usage conditions Equation See expression Errors 58 EXCEL 69 Expression 162 Constant 160 Differentiable 161 Objective function 71 Output variable 64 Value versus property 139 F File Open 148 Save 50 148 Save as 149 File menu 147 Find Optimal Targets dialog box 103 Functions 64 71 162 H Help menu 156 I V O System window 74 Input variable 34 162 Category 61 134 159 171 Continuous 51 160 Input Variable dialog box 51 Input output system 34 162 Installing software 1 Interaction 41 123 125 Interval for values 91 95 99 104 154 163 J JMP 69 K Keypad Equation dialog box 68 Kurtosis 163 M Main window 49 Manufacturability 11 48 Math capabilities 28 Mathematica 69 Menu 49 147 Analysis 152 File 147 Help 156 Options 154 Plot 127 Variables 50 150 Window 153 Minitab 69 N Name I O System dialog box 75 Normal distribution 53 88 Notes window 151 Number of Duplicate Copies dialog box 150 Number Standard Deviatio
60. 1 Each component has a tolerance of 1 0 03 The resulting statistical tolerance for the overall height is 4 0 06 This meets the spec The resulting worst case tolerance for overall height is 4 0 12 This is twice as wide as the statistical tolerance and fails to meet the spec Therefore the inputs tolerances must be tightened increasing costs Which is correct In the last section we saw that statistical tolerancing makes the unlikely assumption that the process is kept perfectly centered As a result the statistical tolerance for overall height is probably narrower than what will be experienced in production However worst case tolerancing also has its flaw The worst case conditions it guards against are unlikely if not impossible to occur in production As a result the worst case tolerance for overall height is wider than will be experienced in production Returning to the overall height example if the components have a standard deviation of 0 005 the average must be in the range 0 985 to 1 015 in order to meet the tolerance of 0 97 to 1 03 below left At these two extremes the resulting distributions of overall height are shown below on the right Because of the variation it is unlikely that an assembly will be produced with an overall height close to either 3 88 or 4 12 Worst case tolerancing results in tolerances wider than will be experienced in production Worse Case Worse Case Wors
61. 1 or 2 of the key inputs are responsible for most of the variation Resulting Total Variation gt OmaAaA SsoO2zrPII4 zZzO oA gt DUPSP lt X3 X6 X9 X2 X8 X5 X7 X1 X10 X4 INPUTS IN ORDER OF IMPORTANCE Pareto Chart of the Contributions of the Inputs to the Total Variation Those key inputs that make significant contributions to the variation of the output will be referred to as Variation Inducing Parameters VIP s When tightening tolerances only tightening the tolerances of the VIP s does any good Other tolerances should be set to their low cost values i e based on the low cost materials and processes The VIP s are the limiting factors of the process They prevent the process from performing better It is important to identify the VIP s Otherwise one is forced to specify tight tolerances for all inputs or to resort to trial and error EFFECT OF EFFECT OF ELIMINATING VIP S ELIMINATING BOTTOM 80 Resulting Total o Variation Resulting Total o Variation omaa zZ0zZz gt J4 Z20 HA gt VP lt OmaAA SH2PRIH 20O AP VPp lt c X3 X6 X9 X2 X8 X5 X7 X1 X10 X4 INPUTS IN ORDER OF IMPORTANCE X3 X6 X X2 XB XS X7 X X0 X4 INPUTS IN ORDER OF IMPORTANCE Effects of Tightening the Tolerances of the VIPs and the Other Inputs 38 Chapter 3 Tightening tolerances is not the only way of reducing variation A second approach is to design the product or process to be less sensitive to the variation of the inputs The
62. 11 242 44 5SD Calculated Simulated Worst Case Conditions Optians Print Copy The buttons and controls at the bottom can be hidden by removing the check from the Show Buttons menu item on the Options menu When hidden the popup menu can still be used to perform the functions of these buttons and controls To display the popup menu right mouse click over the graphic The following buttons and controls are available Clicking this button displays the popup menu associated with this window This popup menu has menu items that correspond to each of the buttons shown at the bottom of the window It also has items that can be used to change how the graphic is sized within the window A short cut is to right mouse click on the graphic S Prints the graphic on the selected tab Shortcuts are to click the Print button on the toolbar or to select the Print menu item on the File menu _B cow Copies the table on the selected tab to the clipboard in graphic format Shortcuts are to click the Copy to Clipboard button on the toolbar or to select the Copy to Clipboard menu item on the File menu 7 Hele Displays help on using this window Short cut is to press F7 key 80 Chapter 4 Lock check box Checking this box locks the window so that it does not update as changes are made to the variables Locking the window causes a complete copy of the I O System to be saved It also causes the Restore button to be displayed which can be used to
63. 49461868 0 75 DT 0 04588331 4596413 100 P 0 0430545023685393 90 MT 0 00749437527699365 175 THB 2 83 3209058456264 0 75 DT 2 1 03469576191631 175 THB 0 75 DT Target 26 Lower Spec Limit 20 Upper Spec Limit 32 Taylor Series 2nd Order v Method of Estimating Average and Standard Deviation V Include In Optimizations with Multiple Outputs Continuous Input Variable Dialog Box x Symbol THE Name TEMPERATURE_HOT_BAR Units deg F KOX Graphic Symbol Statistics Tolerance fe Target 190 Ee fe TEMPERATURE_HOT_BAR THB Type of Tolerance Statistical Tolerance C Worst Case Tolerance Values LSL T USL fe Statistical Tolerance Variation Process Tolerance Average Variation gt Variation Distibuion Nomal tsi ation USL LSL 6 c Standard Deviation p T min 150 00000 wax 200 00000 V Include In Optimizations X Cancel f Delete EJ Options Help 144 Chapter 9 Once the data is entered VarTran can be used to find the optimal targets predict the capability and determine the effects of the different tolerances The optimal targets and the predicted capability are shown below Summary Tables Tolerances Input Tolerances Outputs Entries Continut Jo amp Print h Copy Help F Capability Study SS Seal Strength SS Statistical Tolerance L
64. 9 9 2876 10 714 52 3 R 1 1223 8 7793 11 221 18 4 L 0 37223 8 5761 11 427 4 8 S 0 31113 8 565 11 434 4 2 Percent Reduction Transmitted Variation Table Plot for Input Options amp Print gt Copy i 114 Chapter 7 Tab 4 Plot for Input Tab 4 displays a plot of the effect of tightening the tolerance of one of the inputs on the width of interval for values of the output variable The input plotted is initially that having the largest effect The input plotted can be changed using the Input to Plot menu item on the associated popup menu This popup menu is displayed by clicking the Options button or right mouse clicking over the plot For worst case tolerances the plots are linear For statistical and process tolerances the plots are curved Percent Reduction of Variation of F From Tightening Tolerance of B 100 B 0 T T T T T T T T T 0 50 100 Tolerance Tightened Optians amp Print Copy Help B Lock On Tab 3 we saw that tightening the tolerance for B to zero 100 reduction resulted in a 52 reduction in the width of the interval of values for F In the plot above we can see that tightening the tolerance for B by 50 shortens the interval of values for F by around 30 fa z ej 3 g In both the table and graphics the inputs are listed in order of importance The top or left most inputs have the largest effects If tolerances must be tightened t
65. 9981 9 3414 Maximum 10 664 Standard Deviation 0 27638 0 27886 Cp 1 21 1 20 Ce 0 00 0 66 Cpk 1 20 0 40 Def Rate normal 297 dpm 114 Z Score 3 43 1 20 Sigma Level 3 43 2 70 Interval for Values 8 5048 11 501 3SD v Original Flow Rate F Process Tolerance LSL T USL 9 mimin 11 Characteristic On Target Worst Case Average Minimum 9 9846 9 6563 Maximum 10 319 Standard Deviation 0 11361 0 11572 Cp 2 93 2 88 Ce 0 02 0 34 Cpk 2 89 1 89 Def Rate normal 242 10 12 dpm 0 00707 dpm ZScore 8 5 67 Sigma Level 8 66 ATi Interval for Values 9 3092 10 666 3SD v 20 Motor Flow Rate F Process Tolerance LSL T USL 9 mimin 1 Characteristic On Target Worst Case Average Minimum 9 9817 9 5554 Maximum 10 418 Standard Deviation 0 15072 0 15426 Cp 2 21 2 16 Cc 0 02 0 44 Cpk an 1 20 Def Rate normal 0 0000438 dpm 159 dpm Z Score 6 49 3 60 Sigma Level 649 5 10 Interval for Values 9 0926 10 881 3SD v Optimized Flow Rate F Process Tolerance LSL E USL 9 mimin 11 Characteristic On Target Worst Case Average Minimum 9 985 9 6981 Maximum 10 277 Standard Deviation 0 098918 0 10067 Cp 3 37 3 31 Ce 0 01 0 30 Cpk 3 32 231 Def Rate normal 1 22 10 17 dpm 0 00000204 dpm ZScore 9 95 6 93 Sigma Level 9 95 843 Interval for Values 9 3961 10 579 3SD iv s2value Tightening Tolerances
66. Chapter 2 single value This can be represented in VarTran by a worst case tolerance of width zero as shown below To display this dialog box double click on x in the I O System window Continuous Input Variable Dialog Box Input x Variable is Constant Value 0 The values of the inputs can be viewed in the Summary Tables window Initially both values are set to zero i Variables Summary Tables Summary Tables Tutorials 29 When constant values are entered for the inputs performing a capability study amounts to evaluating the function To display a capability study double click on the Capability Study button on the toolbar When x and y are zero the function is equal to 1 Leave the Capability Study window open mi Analysis Capability Study Capability Study f iof x Output f Output is Constant Options E Print Copy Help L Lock Eai VarTran can be used to obtain line interaction and contour plots of this function One such contour plot is shown below This contour plot shows that this function has numerous local minimums and maximums f Value 1 Z f Value 4 f Value 3 3333 f Value 2 6667 O f Value 2 f Value 1 3333 f Value 0 66667 f Value 6 6613 10 16 i f Value 0 66667 1o NAN Q f Value 1 3333 4 f Value 2 _ To generate your own plots click on the Plot button on the toolbar T
67. Clicking the View Equations button displays the View Equations dialog box for viewing the derived equations for the average the standard deviation as described later in this section For help click the Help button or press the F1 key When an edit box is exited all other windows are immediately notified of the change Some windows such as the I O System window will immediately display the change However analysis windows will generally wait to update their results until the analysis window is selected by clicking on it To show that the window is no longer current a red X is drawn through it For edit boxes as long as the cursor remains in the edit box no update is performed The update occurs when the edit box is exited by using the tab key or clicking on another control If an error is made entering data for an output an error message will be displayed on the status line It should be handled as described in Section 4 3 66 Chapter 4 View Equations Dialog Box The View Equations dialog box is used to display the derived equations for the average and standard deviation as well as the full equation for the output It is displayed by clicking the View Equations button on the Output Variable dialog box View Equations Dialog Box x Type of Equation Standard Deviation SquareRoot 102 96884061 4059 SfAverage LfAverage RfAverage R StandardDeviation 2 51 4844204070295 S Average RiAverage 2 L StandardDeviation 2 S Averag
68. Log param Log10 param Exponential param SquareRoot param Sine param Cosine param Tangent param ArcSine param ArcCosine param ArcTangent param HyperbolicSine param HyperbolicCosine param HyperbolicTangent param Gamma param LogGamma param Beta param1 param2 LogBeta param1 param2 AbsoluteValue param Minimum param1 param2 Maximum param1 param2 NormalDistribution param InverseNormalDistribution param pi and e Parentheses and A reference of the form Cav _2 refers to category _2 of the category input variable whose symbol is Cav It has the value 1 when the given category is the selected category and a value of 0 when it is not Categories are selected using the Category Input dialog box Multiplication signs may be omitted A reference like 2 A is interpreted as two times the variable A A number like 2 02 10 3 is interpreted as 2 02 times 10 to the negative three Squared brackets are used to enclose the parameters of a function For example Log 2 0 refers the log of 2 0 The reference Log 2 0 would be interpreted as meaning the variable Log times 2 0 64 Chapter 4 The operator precedence and association rules determine the default order that the operators in expressions are performed This order can always be overridden using parentheses The operations are first performed in the order of precedence The highest precedence operators are performed first The precedence of the differ
69. Number or Label 94x z zao E z X Cancel Help The Axis Style radio buttons are used to select one of four different axis styles The Tic Style radio buttons are used to select from three different tic styles Tic marks can appear next to the values on an axis as well as in between values The Tic Size Number or Label combo box is used to select the size for displaying tics marks appearing next to values The Tic Size Between combo box is used to select the size for displaying tics marks appearing between values The default is to make the tic marks appearing next to values 0 4 times normal size and the tic marks appearing between values smaller at 0 2 times normal size The Ratio X Axis to Y Axis edit box is used to change the plot s aspect ratio A value of 1 results in a square plotting region A value of 2 makes the x axis twice as long as the y axis A value of 0 5 makes the x axis half as long as the y axis The y axis size remains fixed Plot Options Dialog Box Y Axis Tab The Y Axis tab is used to change the y axis label scale tic marks and orientation Double clicking on the y axis displays this tab The Text edit box is used to change the axis label By default this is the symbol of the variable displayed on the y axis The Label Size combo box is used to select the size for displaying the axis label The default is 1 5 times normal size The Display Label Vertically check box allows the label to be d
70. SL T Jya 20 32 Characteristic Value Average 26 Standard Deviation 0 79353 Cp 2 52 Cc 0 00 Cpk 2 52 Def Rate normal 3 99 10 8 dpm Z Score short term 7 47 Sigma Level 7 47 Interval for Values 22 429 29 571 4 5SD Simulated Worst Case Conditions Ex options i erin P Cony Help CES In this case the tolerance analysis approach produced results similar to those of the dual response approach in the previous section This is because study conditions were representative of long term manufacturing Had this not been the case the two approaches could produce very different results Using VarTran with Designed Experiments 145 What the dual response approach did not find was the cause of the variation VIP The VIP can be identified by clicking on the Pareto Chart button The result is displayed below The VIP is dwell time If the variation needed to be reduced further only tightening the tolerance of dwell time would do any good Pareto Chart SS Pel EY Seal Strength Interval for Values 22 429 29 571 4 5SD 100 Percent Reduction Percent He Transmitted Variation Plot for Input E a E Print Copy Help eo r Lock The tolerance analysis approach using response surface studies has several important advantages over Taguchi s approach based on inner outer arrays and signal to noise ratios First the studies are less costly to run Second while both requi
71. Summary Tables Notes Window Edit Continuous Input Variable gt B i Sear Mee Variable Edt ORC T Displays additional menus for changing the tolerances of all continuous inputs variables at one time Variables badd Duplicate Variable Edit gt Make All Tolerance Case Make All Tolerances ATF Make All Tolerances Process Make All Tolerances Cp Make All Tolerances Cp 1 5 Make All Tolerances Cp 1 Make All Tolerances Cc 1 4 Make All Tolerances Cc 1 3 Make All Tolerances Cc 1 2 Make All Tolerances Average 1 5 SD Make All Tolerances Six Sigma Process Cp 2 Cc 1 4 Modify All Tolerances 2F 1 0 System Window a Summary Tables Notes Window If the I O System window is closed this menu item will open a new one Otherwise it will make the O System window the top child window See Section 4 7 Displays a new Summary Tables window for displaying the data entered for the variables See Section 4 8 Displays a Notes window for typing notes and comments An example of the Notes window is shown below This child window is a text editor The information typed into the Notes window is saved in the same file as the rest of the data You can use it as a to do list to keep track of what additional data is required You can also enter a message before emailing the file to someone else Menus and Toolbar 151 Notes Window Sources of Data Temperature Examinat
72. The maximum value will now appear in the Capability Study window The settings of the inputs that result in the maximum are displayed in the Summary Tables window Ifa red X appears in these windows click on them to update them The function f is maximized when x 0 9082 and y 0 73633 Its maximum value is 4 0619 Try resetting the inputs and repeating the optimization You get the same answer every time The Interval optimization routine is never fooled by local maximums Capability Study f Summary Tables _ ol x Output f Output is Constant Tolerance Category Constant 0 9082 0 Constant 0 73633 0 3 0619 5 0619 Value 4 0619 gt gt Ce 50 Simulated Worst Case Conditions i Tolerances Outputs Entries Continuous Inputs 4 gt Options amp Print Copy Help Lock EJ Options J amp Print ip Copy Help E Lock Try minimizing f as well The function f is minimized when x 0 48828 and y 0 71094 Its minimum value is 2 2202 Try repeating the optimization Again the optimization routine is never fooled by local minimums Most optimization routines have difficulties with functions containing local minimums and maximums Capability Study f ol x H Summary Tables Output f Output is Constant Type Tolerance Category Constant 0 48828 0 Constant 0 71094 0 Value 2 2202 Tolerances Outputs Entrie
73. Tolerance Analysis have been attended by thousands of engineers and scientists Dr Taylor is a leading expert on acceptance sampling and process validation in the pharmaceutical medical device and diagnostics industries His articles on selecting statistically valid sampling plans have become standards in the industry He is one of the authors of the Global Harmonization Task Force guideline on Process Validation The VarTran software and Dr Taylor s course Robust Tolerance Analysis have rapidly become a key component of many companies DFSS Design for Six Sigma programs Honeywell the Six Sigma Academy and numerous other companies have adopted VarTran as an essential tool for designing high quality products Dr Taylor received his Ph D in Statistics from Purdue University He is a fellow of the American Society for Quality ale e m Oro CLE Chapter 1 Program Installation 1 1 System Requirements 1 2 Installation VarTran 1 3 Registering the Software 1 4 Uninstalling VarTran Chapter 2 Tutorials 2 1 Design of a Pump 2 2 Optimization of a Heat Sealer 2 3 Dimensional Stack Up Problem 2 4 VarTran as a Mathematics Package Chapter3 Overview 3 1 Optimization amp Variation Reduction 3 2 The Importance of Reducing Variation 3 3 How to Reduce Variation 3 4 Robustness 3 5 Variation Transmission Analysis 3 6 Behavior of the Inputs 3 7 Strategy Chapter4 Entering Information 4 1 Main Window 4 2 Adding Variables
74. acters are allowed The title can be of any length It cannot be blank When done click on the OK button or press the Enter key To cancel without making any changes click on the Cancel button or press the Esc key For further help click on the Help button Entering Information 75 4 8 Summary Table The Summary Table window is a child window for displaying information about all the variables at once The table has 7 tabs containing various displays It is displayed by selecting the Summary Tables menu item on the Variables menu A short cut is to press the I button on the toolbar Click on the X in the upper right corner to close this window The Summary Table window automatically updates itself when information affecting the displayed results is changed When the program is in immediate update mode the updates occur immediately following the change being made If the program is in delayed update mode the Summary Table window draws a red X through itself when it is not current Clicking on the window will cause it to update and redraw itself The update mode is selected using the Options menu Double clicking on a variable in the window results in the appropriate variable dialog box being displayed so that the variable can be edited This is a short cut to selecting the Edit menu item on the Variables menu There are seven tabs displaying the following information Tab 1 Tolerances Inputs Displays sp
75. al Negative Lognormal LSL T USL LSL T USL LSL T USL 0 099775 0 100225 0 099775 0 100225 0 099775 0 100225 The Fit to Spec distributions determine their parameters from the upper and lower spec limits When one of these distributions is selected values must be entered in both the 4 and 4 edit boxes The width of the spec limits determines the standard deviation as follows Distribution Triangular Fit to Spec Standard Deviation 0 20412 USL LSL Uniform Fit to Spec 0 28868 USL LSL Uniform Minus Center 25 Fit to Spec Uniform Minus Center 1 3 Fit to Spec 0 33072 USL LSL Uniform Minus Center 50 Fit to Spec 0 38188 USL LSL Uniform Minus Center 2 3 Fit to Spec Uniform Minus Center75 Fit to Spec 0 41944 USL LSL 0 34694 USL LSL 0 43899 USL LSL Entering Information 53 For all the others distributions the parameters of the distribution must be entered directly All require that the standard deviation be entered The lognormal and gamma distributions require that a second parameter be entered namely a lower bound The negative lognormal and negative gamma distributions require that an upper bound be entered The beta distribution requires both a lower and upper bound In general the distribution does not have much of an effect compared to the standard deviation However for the Fit to Spec distributio
76. ances or statistical tolerances for all inputs With VarTran you can select worst case tolerances for some inputs statistical tolerances for others and process tolerances for the rest 3 6 4 Which Type of Tolerance Should I Use Worst case tolerances should be specified for environmental inputs For example suppose that one factor affecting the flow rate of a pump is usage temperature An advertising claim is made that the pump s flow rate is between 9 and 11 ml min over a temperature range of 40 F to 125 F A worst case tolerance should be specified for usage temperature such as 75 35 50 F This ensures all pumps operate in very hot and very cold environments Statistical tolerances are frequently used for machine parameters that are automatically controlled For example suppose that one input affecting seal strength of a heat sealer is die temperature A strip chart of die temperature indicates that the temperature averages 150 F with a standard deviation of 5 F A statistical tolerance based of this information could be specified for die temperature The automatic controller ensures continual centering of the process Process tolerances are frequently used for components and manually controlled processes Suppose a supplier is being considered for the pistons in a pump A capability study indicates that the radius of the supplier s pistons average 0 1 with a standard deviation of 0 0005 rpm Specifying a statistical tolerance based o
77. and flow rate of a pump The operator precedence and association rules determine the default order that the operators in expressions are performed This order can always be overridden using parentheses See Section 4 5 Glossary 163 Process Tolerance Relative Distance Inside Spec Robust Tolerance Analysis 164 A process tolerance is a tolerance that specifies requirements for a process producing large quantities of parts A process tolerance specifies a minimum and maximum average and a maximum standard deviation Equivalently a process tolerance can be defined by specifying values for Cp and Ce See Section 5 5 The relative distance inside spec is the relative distance that the interval for values is from the specification limits Its formula is USL UIV ini USL LSL seg 2 Minimum LIV _LSL If 2 sided spec USL LSL g EUREEN If upper spec and target USL Target EINES If lower spec and target Target LSL where USL is the upper specification limit LSL is the lower specification limit UIV is the upper end of the interval for values and LIV is the lower end of the interval for values It represents the ratio of the distance from the interval for values to the nearest spec and half the width of the spec The relative distance inside spec is independent of the units of measure and is defined for all three types of tolerances including worse case tolerances It requires that a target be specified for those
78. ard deviation are 0 0 01 1 3 Log 10 1 2 pif 0 03 Target 0 03 T USL LSL 12 The value or expression entered must be greater than or equal to zero for all values of the target between the minimum and maximum targets To reference this field in other edit boxes use the symbol StandardDeviation or SD Clicking the Cp button displays a popup menu shown to the right for automatically entering various equations into Bi i z the Standard Deviation edit box including equations to Cp 2 specify a Cp of 1 1 5 and 2 and a CV of 1 There are E E ALE also menu items for specifying a Cp value or CV value CV 1 CY X replace X with value Six Sigma Cp 2 Cc 0 25 For the beta distribution the standard deviation must also be below J Average Min Max Average This condition must be satisfied for all averages in the interval from the minimum target to the maximum target If process tolerances are specified this interval must be expanded by the width of the operating window for the average 54 Chapter 4 These two menu items insert equations for the standard deviation with an X at the place to insert the value Finally there is also a menu item to specify Six Sigma quality which sets the tolerance type to process Cp 2 and Ce 1 4 The Min edit box is used to enter a lower bound for the lognormal gamma and beta distributions This edit box is only displayed when one of these distributions is s
79. ariable or objective function that is to be plotted For output variables the following characteristics are available value average standard deviation percent defective C Co Cpk Z Score CV Cpm root MSE distance inside spec distance from target lower end of interval of values and upper end of the interval of values Plotting the output s value plots the equation for the output exactly as entered in the Output Variable dialog box While all these characteristics are available not all will be appropriate for your particular problem For example Cpk requires at least one specification limit and the use of statistical or process tolerancing Selecting a characteristic that is not defined results in a blank plot For an objective function the only characteristic available is the value The plots are initially scaled based on the minimum and maximum targets specified in the Continuous Input Variable dialog box All plots are drawn assuming the inputs not included in the plot are at their selected target or selected category The value of these inputs is displayed along the right edge of the plot The characteristics percent defective Cp Ce Cpr Z Score CV Cpm root MSE distance inside spec distance from target lower end of interval of values and upper end of the interval of values are calculated ignoring any worst case tolerances The values plotted represent their values when all inputs are on target The Plots dialog box is used to q
80. ariation be considered when determining the inputs targets The initial design should be based on low cost tolerances This ensures manufacturability of the product Tolerances are selectively tightened only if needed Instead designers frequently specify overly tight tolerances for the inputs to insure design performance which in turn drives up costs Tutorials 11 2 2 Optimization of a Heat Sealer Introduction A heat sealer is used to form the top seal of a pouch protecting the product The top seal must be strong enough to survive shipping and handling but weak enough to be easily opened by the customer The target seal strength is 26 pounds with spec limits of 20 to 32 pounds A response surface study was run to optimize the performance of the heat sealer Eight inputs variables were studied Of these hot bar temperature dwell time pressure and material temperature had significant effects on the average The seal strength standard deviation was analyzed separately Hot bar temperature and dwell time were found to also affect the standard deviation Contour plots were then used to identify the input targets minimizing the variation and achieving an average seal strength of 26 pounds Selecting targets for the inputs that minimize the seal strength variation is called a robust process This approach is called the dual response average and standard deviation approach to robustness Studies like the one above have proven highly success
81. ated WorstCase Conditions Ey Print Py Copy Help v Lock Original ZF Bestore Locking a window saves a copy of the design at the point the window was locked It also adds a Restore button Clicking the Restore button resets the current design back to the point that the window was locked It is a good idea to enter a note in the slot provided to keep track of the designs To lock several windows at once and give them all the same description use Lock All Unlocked Windows menu item on the Analysis menu The original design has a worst case Cpk of 0 4 and is 2 70 sigma It needs improvement Now suppose we go ahead and optimize the design We can open a new Capability Study window to view the design as shown on the next page Once again we might want to lock it and add a label The Cpk has improved to 1 20 and the sigma level to 5 10 It is still not Six Sigma 116 Chapter 7 Capability Study F Optimized O x Flow Rate F Process Tolerance LSL T USL 9 ml min 11 Characteristic On Target Worst Case Average Minimum 9 9817 9 5554 Maximum 10 418 Standard Deviation 0 15072 0 15426 Cp 2 21 2 16 Ce 0 02 0 44 Cpk 2 17 1 20 Def Rate normal 0 0000438 dom 159 dpm Z Score 6 49 3 60 Sigma Level 6 49 5 10 Interval for Values 9 0926 10 881 3SD Calculated Simulated Worst Case Conditions Options Print Copy Help v Lock Optimized 3F Bestore Next
82. automatically updated It will now appear as shown on the next page The variation has been cut by 70 while achieving an average seal strength of 26 pounds The predicted defect rate is well below 1 dpm defects per million The capability study predicts that nearly all units will fall in the range of 22 429 and 29 571 pounds This is well within the specification limits All this was accomplished by adjusting the targets at which the process is operated This simply requires adjusting a few knobs No tolerances were tightened 16 Chapter 2 Capability Study SS oO x Seal Strength SS Statistical Tolerance T io 20 32 Characteristic Value Average 26 Standard Deviation 0 79353 Cp 2 52 Ce 0 00 Cpk 2 52 Def Rate normal 3 99 10 8 dpm Z Score short term 7 47 Sigma Level 7 47 Interval for Values 22 429 29 571 4 5SD ET Simulated Worst Case Conditions Options amp Print Copy Help L Lock To view the targets selected for the inputs click on the Summary Tables button on the toolbar The result is shown below The new targets are a hot bar temperature of 191 F a dwell time of 0 596 seconds and a pressure of 55 28 pounds Material temperature remains at 75 F Material temperature did not change because its Include in Optimizations check box was not checked The optimization determined Without preheating of the material what is the best setup Since the process safel
83. bed in Section 4 6 Predicting Capability 91 Tab 2 Simulation Results The second tab shows the results of a tolerance analysis performed by a Monte Carlo simulation To perform a simulation use the Simulation menu item on the Analysis menu as described in Section 8 4 Capability Study F oO x Flow Rate F Statistical Tolerance LSL ul USL 8 7573 ml min 11 238 Characteristic Value Average 9 9979 Standard Deviation 0 27569 Cp 1 21 Cc 0 00 Cpk 1 21 Def Rate 400 dpm Z Score short term 3 35 Sigma Level 3 35 Interval for Values 8 7573 11 238 4 5SD Calculated Simulated Worst Case Conditions Ex Options print Gy copy 2 Help T to If no simulation was performed the message No Simulation Data will appear Otherwise the results will appear as above The results are almost identical to those for the analytic approach on the first tab with the following exceptions e At the top a histogram of the data is shown e The defect rate is determined by counting the number of units outside the specifications It does not assume the normal distribution Tab 3 Worst Case Conditions The third tab does not apply to statistical tolerances so displays the message Only Applicable to Worst Case and Process Tolerances 92 Chapter 4 Pros and Cons of Statistical Tolerancing Statistical tolerancing should be used with care It assumes that all the inputs are exactly at
84. bility functions and do not have the ability to handle the variation and more complex objective functions like Cpk VWarTran can be a useful addition The dual response approach has two major weaknesses First it optimizes the process for the conditions that existed while the study was run If the study s variation is not representative of actual manufacturing variation the wrong targets may result For example if only one roll of plastic sheeting was used roll to roll variation of sheeting thickness is not captured in the study Taguchi solves this problem by running noise arrays to mimic actual variation However this makes the study more expensive and complicated to run The second weakness of the dual response approach is that it does not identify the root cause of the variation VIP As a result it does not support the next step of tightening tolerances Since there is no guarantee that the quality objectives can be met by robustness alone the dual response approach frequently falls short of the final objective without guidance for how to proceed further VarTran can be used to overcome both these shortcomings The real benefit of VarTran is not as an aid in performing the dual response approach to robustness but as an alternative approach to achieving robustness which will be referred to as the tolerance analysis approach to robust design 9 3 Tolerance Analysis Approach The tolerance analysis approach only uses the equation for
85. bit Windows operating system including Windows 95 98 2000 XP and NT It requires 5 MB of hard disk space to install 1 2 Installing VarTran VarTran is a shareware program A full version of the software can be downloaded for a 30 day trial If you want to continue using the software after trial period you must register it To install VarTran download the file vta31 exe from www variation com vta Executing this file will run the installation program The software will be automatically installed to the directory C Program Files Taylor Enterprises VarTran Along with the software the user manual a tutorial and numerous example files are copied to this directory VarTran data files end with the extension vta A shortcut is added to the desktop for starting the program It appears as below Double clicking this shortcut will start the program z VarTran In addition three menu items are added to Programs menu Click on the amp Start button in the lower left corner of the screen and then click on the Programs menu item Find the VarTran folder and click on it This displays the following three menu items Vartran 4 Tutorials LA User Manual hh YarTran The third menu item starts the program The first two menu items display program documentation in pdf format You must have a copy of the free Adobe Acrobat Reader to view these files Program Installation 1 1 3 Registering the Software VarTran can be used f
86. ble is one of the form bo by x1 b2 x2 b3 X34 where the b s are constants and the x s are input variables This type of equation is generally limited to dimensional stack up problems In a dimensional stack up problem you might be interested in controlling the overall height of a product where the overall height is the sum of the heights of the components In all other cases robustness can be used to improve quality and reduce costs 42 Chapter 3 3 5 Variation Transmission Analysis So how does one optimize and reduce the variation of a product or process By setting targets and tolerances for the inputs This is 2 step process First starting with the low cost tolerances the optimal targets are identified The effects of the targets on both the average and variation are considered when finding the optimal targets This reduces the variation resulting in a robust design The second step is to tighten tolerances to achieve the desired performance The trick is to only tighten those tolerances necessary and only to the degree necessary to meet the customer requirements The result is high quality and low cost Before you can use this 2 step process you first have to gather some information First you must identify the output variables of concern Next you need to determine which inputs affect these outputs Then you must gather the following information about these variables 1 The target and specification limits for th
87. can be adjusted over a continuous scale Examples include dwell time and temperature on a heat seal machine and stroke length and motor speed on a pump Targets and tolerances are to be determined for these inputs Capability index comparing the process variation with the width of the specification limits Its formula is _ USL LSL C k 60 where USL is the upper specification limit LSL is the lower specification limit and o is the standard deviation Cp is independent of the units of measure See Section 5 2 Capability index measuring the ability of the process to meet the specifications Both variation and centering are considered One way of writing its formula is USL u u LSL o 30 C Minimum where USL is the upper specification limit LSL is the lower specification limit u is the average and o is the standard deviation If only one specification limit exists calculate only that part of the equation and omit the minimum C x is independent of the units of measure See Section 5 2 Glossary Defect Rate Differentiable Expression Distance From Target Distance Inside Spec Capability index measuring the ability of the process to meet the specifications Both variation and centering are considered The formula is USL LSL Co 2 2 6 0 u T where USL is the upper specification limit LSL is the lower specification limit T is the target is the average and o is the standard deviation Cpm is indep
88. ch sets the tolerance type to process Cp 2 and Ce 1 4 Optimizations Options for Target Group Box The group box is used to specify options for performing an optimization per the Find Optimal Targets dialog box covered in Section 6 1 The values entered do not affect the current design including the Capability Study and Pareto Chart windows The Minimum and Maximum Target edit boxes are used to specify limits for the target They must be constant expressions The minimum must be less than or equal to the maximum There may be further restrictions depending on the distribution selected Optimizations effects tables and plots are all performed over the region specified by the minimum and maximum targets The minimum and maximum may represent physical or practical limits In event that a response surface study was used to obtain equations for the outputs the minimum and maximum targets should be no wider than the region of study When the Include in Optimizations check box is checked the target for this input will change when an optimization is performed using the Find Optimal Targets dialog box When unchecked the target for the input is not optimized By default it is checked Certain targets may not be adjustable in which case the check should be removed Other inputs may have preferred values By setting the input to its preferred value and removing the check the product process can be optimized for the preferred setting and the result
89. chi s signal to noise ratios allowing them to be maximized When an objective function is added an Objective Function dialog box like the one below is displayed This dialog box is used to enter and change information concerning the objective function Objective Function Dialog Box x Symbol Cp Name Cp of Seal Strength Units sox E SS UpperSpecLimit SS LowerSpecLimit 6 SS StandardDeviation unction X Cancel f Delete Help 70 Chapter 4 The Symbol edit box is used to enter the symbol The variable s symbol is used to represent the variable in equations Each variable must have a unique symbol It can be of any length but is best limited to 3 characters or less The symbol must consist of letters the digits 0 9 and underscores It cannot begin with a digit The Name edit box is used to enter a more detailed name or description of the objective function It may contain any characters and be of any length While the symbol should be short the name should be descriptive The Units edit box can optionally be used to enter the units of measure associated with this variable if any It can be any combination of characters and any length The xx xx button displays a popup menu for specifying the number of digits to report for this variable The default is five significant digits You can choose between 2 12 significant digits and from 0 to 12 digits after the decimal point When VarTran displays numbers
90. click the OK button Print Copy to Clipboard The symbol is checked to verify that it is valid and not already in use If an error is encountered a message is displayed To cancel the changes including the addition of a new variable click the Cancel button press the Esc key or click the X in the upper right corner of the dialog box To delete the variable click the Delete button For help click the Help button or press the F1 key Entering Information a7 Updating Analysis and Reporting Errors When an edit box is exited or a new item selected from a combo box all other windows are immediately notified of the change Some windows such as the I O System window will immediately display the change However analysis windows will generally wait to update their results until the analysis window is selected by clicking on it To show that the window is no longer current a red X is drawn through it For edit boxes as long as the cursor remains in the edit box no update is performed The update occurs when the edit box is exited by using the tab key or clicking on another control If an error is made entering data for an input an error message will be displayed on the status line The status error message initially displays a list of variables with errors However when a variable dialog box like the Input Variable dialog box is opened the status error message changes to a list of controls in that dialog box with errors When a control is
91. clicked on for which an error is reported the status error message again changes to a description of what is wrong In this way errors can be quickly tracked down Determining Which Type of Tolerance to Use The purpose of a tolerance analysis is to accurately predict the performance of the output Key to this goal is to accurately describe the behaviors of the inputs However not all inputs behave the same Some are uncontrolled environmental variables while others may be carefully controlled including close loop feedback to keep them on target For this reason neither worst case nor statistical tolerances are adequate on their own VarTran s ability to do a process tolerance analysis allows it to combine these two approaches and to cover behaviors not adequately captured by either This added flexibility will allow more accurate predictions of the output s behavior to be made For each input separately we must decide which type of tolerance to use Our choice will be based on which best describes the behavior of the input Three examples follow Example 1 Manufactured Component Suppose you are designing a pump and must specify a tolerance for piston radius A capability study was run on the piston radius which resulted in an estimated standard deviation of 0 0005 Since the radius has a lower bound of zero the lognormal distribution was selected Because this is a manufactured component an operating window should be allowed Th
92. d by counting the number of units outside the specifications It does not assume the normal distribution Tab 3 Worst Case Conditions The third tab shows the settings of the inputs that result in the minimum and maximum average of the output reported on the first tab These are the settings used for the two worst case simulations shown on the second tab 100 Chapter 4 Capability Study F iojx Flow Rate F Process Tolerance Input Settings Producing Worst Case Conditions Input Minimum Maximum R Radius of Piston 0 09925 0 10075 L Stroke Length 0 4975 0 5025 B Black Flow 0 0575 0 0425 S Motor Speed 47 95 48 45 Inputs not listed are either constant or statistical Calculated Simulated Worst Case Conditions amp Print gt Copy Help L Lock Pros and Cons of Process Tolerancing Statistical tolerancing assumes that all processes are on target an unlikely assumption Worst case tolerances guards against all units simultaneously being at the worst case conditions which can only occur if there is no variation This is just as unlikely Process tolerancing provides middle ground by allowing both variation and off target processes Like worst case tolerancing process tolerancing is safe If the inputs all meet their stated process tolerances the output is guaranteed to meet its resulting process tolerance Process tolerancing is less expensive than worst case tolerancing While not as inexpensi
93. d of specifying requirements for individual units of products a process tolerance specifies requirements for the process producing the product A process tolerance consists of a minimum and maximum average and a maximum standard deviation These are specified using four parameters the target T delta upper A delta lower A_ and the maximum standard deviation o Process tolerances are interpreted as meaning that the average must be in the interval T A_ T A and that the standard deviation must be below o For piston radius suppose that T 0 1 A 0 001 A 0 001 and o 0 0067 The upper and lower ends of the interval for the average are then UIA T A 0 101 LIA T A 0 099 This process tolerance is shown below The two normal curves represent the maximum variation at the minimum and maximum averages Overview 45 Maximum Variation i Lower pper Toler rA X Toler Limits for Average Process Tolerance The upper and lower ends of the interval for individual values can be determined as follows UIV T A 30 0 103 LIV T A 3o 0 097 Process tolerances are represented by the two equivalent drawing symbols shown below The symbol is used to indicate the tolerance is not a worst case tolerance An informational box must be added providing clarification One should specify either an interval for the average or the capability index C in an information box One should also speci
94. d order to fitting a quadratic polynomial etc The default is second order Taylor series approach Further information on the methods for deriving such equations can be found in Cox 1986 Evans 1975 and Taylor 1991 When the Include in Optimization with Multiple Outputs check box is checked the output will be included in optimizations involving multiple outputs When using the Find Optimal Targets dialog box and there are multiple outputs one has an option to optimize all outputs This option will construct an objective function to optimize that includes all the output functions that have the Include in Optimization with Multiple Outputs check box checked The default is to check this box If you wish to exclude an output from the optimization remove the check mark Some outputs might represent intermediate calculations used by other outputs and not be appropriate for use in the optimization Other times you may wish to exclude outputs of lesser importance that appear to prevent other more important outputs from reaching acceptable values To close the dialog box and save the changes click the OK button The symbol is checked to verify that it is valid and not already in use If an error is encountered a message is displayed To cancel the changes including the addition of a new variable click the Cancel button press the Esc key or click the X in the upper right corner of the dialog box To delete the variable click the Delete button
95. d process makes good product if the process average is anywhere in the center half of the specification limits Operating windows are important if one wants to consistently make good product OPERATING WINDOW NEW OPERATING WINDOW Reducing Variation Results in Wider Operating Windows Overview 35 There is another reason for continuing to reduce variation We frequently view specification limits as goal posts Units of product inside the specification limits are perfectly good Units of product outside the specification limits are bad It does not matter where the product falls within the specification limits so long as it is within the limits That is all that counts 20 Bad Good Bad Product Product Product 0 Goal Post Model for Specification Limits Your customers may view it differently They may find it hard to distinguish between a good unit barely inside the specification limit and a bad unit barely outside the specification limit The customer is more likely to notice the difference between a unit that is right on target and one that is near the specification limits Customers are more likely to view your product in terms of the Taguchi loss function shown below According to the Taguchi loss function the further the product deviates from the ideal value the greater the loss associated with it In this case reducing the variation results in more units closer to the ideal value The customer views this as a higher
96. d restore it 4 1 Main Window When the program is first started the main window shown below is displayed The main window consists of from top to bottom the caption bar the menu bar the toolbar the interior containing the I O System window and the status bar YarTran untitled lofx Eile Variables Analysis Window Options Help lb a m o 7 a VO System Window Input Output System Continuous Input Variable Output Variable Category Input Varaible Objective Function ri Name Options amp Print Copy Help New VO System Begin by adding variables to I O system Caption Bar The caption bar contains the program name and the name of any associated file When the program is not maximized to cover the entire screen dragging the caption bar moves the window Double clicking on the caption bar maximizes restores the window On the left of the caption bar is the system menu button J Clicking this button displays the system menu containing items to move size and close the window The program can also be closed by double clicking on this button On the right of the caption bar are the minimize button El the maximize restore button ol and the close button I Menu Bar The menu bar provides a list of drop down menus containing menu items These menu items serve as the primary means of telling the program what to do Menu items exist for adding variables performing analysis printing the re
97. dually not taking into account interactions The inputs are listed in order of importance For exact values see the third tab Tightening Tolerances 113 Pareto Chart F O x Flow Rate Interval for Values 8 5061 11 5 3SD 2 9936 Transmitted Variation 0 B R L S Total Percent Reduction Transmitted Variation Plot for Input amp Print Copy Help r Lock sts Tab 3 Table Tab 3 displays in tabular form both the transmitted variation and percent reduction in the total variation of each input variable The interval for values of the output variable is shown at the top and is the same as shown on the Capability Study window Each row in the table shows the results of setting that s input tolerance width to zero while leaving all the other input tolerances at their original values The resulting interval for values for the output variable is shown along with the percent reduction in the width of the original interval for values Also shown is the transmitted variation for each input This the width of the resulting interval for values for the output due to a single input variable alone The values in this table were used to construct the graphics on first two tabs Pareto Chart F o x Flow Rate Interval for Values 8 5061 11 5 3SD Transmitted Effect of Setting Input Tolerance to 0 Variation New Interval for Values Input Variable Reduction B 2 16
98. e The Equation edit box displays the current equation It is initialized to the equation appearing in either the Output Variable dialog box or Objective Function dialog box While the equation can be typed directly into this edit box it is easier to enter it by double clicking on the variables and functions below as well as clicking on the character keys The Variables list box displays a list of available variables for entering into the equation Double click a variable to insert it into the equation For an output variable the inputs and other output variables are listed For an objective function all characteristics of the inputs and outputs are listed along with other objective functions 68 Chapter 4 The Functions list box displays a list of all available functions Double click a function to insert it into equation The character keys are used to enter numbers operators and delimiters To close the dialog box and save the changes click the OK button To cancel the changes click the Cancel button press the Esc key or click the X in the upper right corner of the dialog box For help click the Help button or press the F1 key Paste Equation Dialog Box The Paste Equation dialog box is used to paste and convert an equation from another software package Supported packages include EXCEL JMP Mathematica Minitab and Statgraphics The equation is converted into the format required for either an output variable or objective f
99. e B StandardDeviation 2 51 4844204070295 RiAverage 2 LfAverage BfAverage S StandardDeviation 2 0 5 102 96884081 4059 S Average LfAverage R StandardDeviation 4 102 96884081 4059 SfAveraqe RiAverage 2 R StandardDeviation 2 L StandardDeviation 2 102 96884081 4059 L Average RfAverage 2 R StandardDeviation 2 S StandardDeviatian 2 51 4844204070295 RiAverage 2 2 L StandardDeviation 2 S StandardDeviation 2 B StandardDeviation 2 S StandardDeviation 2 Equation Show All Parenthesis Show Multiplications Py Copy Help The Type of Equation list box is used to select which equation to display The options are Value Original Equation Value Oriqinal Equation Average e Value Original Equation This is the full equation used for Stande d Deviation the output in terms of the inputs It is constructed by taking the equation entered and substituting the equations for other outputs into it e Average Equation derived for the average of the output in terms of the averages and standard deviations of the inputs This equation is evaluated to calculate the value of the average displayed in the Capability Study window e Standard Deviation Equation derived for the standard deviation of the output in terms of the averages and standard deviations of the inputs This equation is evaluated to calculate the value of the standard deviation displayed in the Capability Study
100. e Edit Output Variable Modify All Tolerances Objective Function 3F VO System Window U Summary Tables 2 Notes Window Both cause a menu to appear for specifying the type of variable to add Selecting a variable type will display a dialog box for entering further information about the variable This process must be repeated for each variable Four types of variables are available continuous input variable categorical input variable output variable and objective function Continuous input variables are inputs that can be adjusted over a continuous scale Examples include temperature length and viscosity Category input variables are inputs that are restricted to specific levels One example is cavity where the only possible values are 1 2 3 and 4 Another example is material with levels A and B Output variables are outputs such as flow rate and seal strength Objective functions combine several outputs into a single desirability function which can be optimized An example is Minimum O1 Cpk O02 Cpk where O1 and O2 are output variables Once the variables have been added it is a good idea to save the data immediately Do this by selecting the Save menu item from the File menu 50 Chapter 4 4 3 Continuous Input Variables A continuous input is an input that can be adjusted over a continuous scale Examples include dwell time and temperature on a heat seal machine and stroke length and motor speed on a pump When a con
101. e Case Worse Case Without Variation With Variation Without Variation With Variation Lower Upper J Toler Toler 0 97 0 1 1 03 3 9 4 0 4 1 INPUTS OUTPUT Worst case Conditions When Variation is Present 96 Chapter 4 5 5 Process Tolerances Statistical tolerancing results in tolerances for the outputs that are narrower than what will be experienced in production Worst case tolerancing results in tolerances for the outputs that are wider than what will be experienced in production The truth lies somewhere in between Process tolerances offer a solution to this dilemma When process tolerances are specified for the input variables or a combination of statistical worst case and process tolerances are specified a process tolerance is calculated for the output The results are shown on 3 tabs Tab 1 Calculated Results The first tab shows the results of a tolerance analysis performed using the analytic approach Capability Study F Ol x Flow Rate F Process Tolerance LSL T USL 8 0865 ml min 11 919 Characteristic On Target Worst Case Average Minimum 9 9981 9 3414 Maximum 10 664 Standard Deviation 0 27638 0 27886 1 21 1 20 0 00 0 66 1 20 0 40 Def Rate normal 297 dpm 11 4 Z Score 3 43 1 20 Sigma Level 3 43 2 70 Interval for Values 8 0865 11 919 4 5SD Ey Si Worst Case Conditions E Options Print h Copy Help B Lock The name of the output variable and the type of tolerance a
102. e capability study indicates that the process is stable Therefore a 1 50 operating window is sufficient The target was initially set to 0 1 However this can be changed so the Include in Optimization check box is checked Zero was specified for the minimum target since a negative radius is impossible The maximum target was set to 1 since at this point the pump is getting larger than desired The Input Variable dialog box appears as on the next page 58 Chapter 4 Continuous Input Variable Dialog Box Fame FecusorPsen it freres l Radius of Piston R Process Tolerance lognoms 0 0005 Example 2 Usage Environment Input Suppose one factor effecting flow rate of a pump is the usage temperature An advertising claim is made that the pump s flow rate will be between 9 and 11 ml min over a temperature range of 40 F to 125 F A worst case tolerance should be used to ensure the pumps give the desired performance at each temperature in this range In addition the Include in Optimization check box should be unchecked to prevent the target from being changed during optimizations The Input Variable dialog box appears as below Continuous Input Variable Dialog Box Te neme UsegeTenporctre Po Usage Temperature Te 9 Worst Case Tolerance z k 40 125 Entering Information 59 Example 3 Machine Parameter A factor affecting seal strength of a heat seal machine is
103. e interval analysis It states that as the size of the interval shrinks the calculated bound converges to the true bound This allows optimization to be performed by splitting the intervals After each split those intervals that cannot contain the maximum or minimum are discarded The remaining intervals are split again This continues until the desired accuaracy is achieved Further details can be found in Snyder 1992 The key feature of this approach is that it bounds the true maximum or minimum each step of the way As the optimization proceeds the message box displays these bounds The optimization is set to terminate when two significant digits are assured Bounding the maximum or minimum to 2 digits accuracy generally results in 3 or 4 digits accuracy 108 Chapter 6 6 4 Optimization Options The Optimization Options dialog box shown below is used to change the termination conditions for the two methods of optimization It is displayed by clicking on the Options button of the Find Optimal Targets dialog box Optimization Options Dialog Box Ea Simplex Interval l Simplex Termination Options Stop when relative error below fo oo00001 Stop after number of trials fiooooo Stop when below foio Stop when above foia X Cancel Y Defaults Help There are two tabs Simplex tab and Interval tab Different termination conditions can be entered for each method of optimization The Stop when relative error below ed
104. e multiple output variables with specifications one can also select All Included Output Variables This will result is the simultaneous optimization of all output variables with specifications that have the Include in Optimizations with Multiple Outputs check box in the Output Variable dialog box checked The Characteristic list box is used to select which characteristic to optimize The items in the list change depending on the selection in the Output list box If you selected an output variable in the Output list box you have the following choices Percent Defective The percent defective or defect rate will be minimized Only available for statistical and process tolerances with at least one specification limit Cpk Cpk is maximized Only available for statistical and process tolerances with at least one specification limit Distance Inside Spec Distance inside specification is maximized Available for all three types of tolerances with at least one specification limit This is the counterpart to Cpk when worst case tolerances are used Distance From Target Distance from target is minimized Available for all three types of tolerancing when a target for the output is specified Lower End of Interval for Values The lower end of interval for values is maximized Available for all three types of tolerances even when no targets or specification limits are specified Upper End of Interval for Values The upper end of interval for value
105. e output variables These are the customer requirements that must be met 2 A standard deviation and initial target and for each of the input variables These can be obtained by performing capability studies on the inputs This information describes how the inputs behave 3 Equations describing the effects of the inputs on the outputs In some cases these equations can be derived or are available in books or articles However in other cases designed experiments will be necessary to empirically fit equations This information describes how the inputs affect the outputs As shown below if you know how the inputs behave and you know how the inputs affect the outputs you can predict how the outputs will behave Such an analysis will be referred to as a Variation Transmission Analysis from which the name VarTran is derived Behavior of Output Effect of Input 4cuvaACO Behavior of Input INPUT Variation Transmission Analysis One method of predicting the behavior of the output is to perform a simulation While VarTran can perform such simulations it is also capable of much more powerful analytic methods allowing alternate designs to be quickly evaluated and the optimal design selected Overview 43 3 6 Behavior of the Inputs If you know how the inputs behave and you know how the inputs affect the outputs you can predict how the outputs will behave The behaviors of the inputs are described using tolerances More than one
106. e popup menu right mouse click over the tables The following buttons and controls are available Options i Print if Copy Help F1 i Lock check box Note edit box 35 Restore i Clicking this button displays the popup menu associated with this window This popup menu has menu items that correspond to each of the buttons shown at the bottom of the window It also has items that can be used to change how the graphic is sized within the window A short cut is to right mouse click on the graphic Prints the table of selected tab Shortcuts are to click the Print button on the toolbar or to select the Print menu item on the File menu Copies the table on the selected tab to the clipboard in both text tab delimited and graphic formats Shortcuts are to click the Copy to Clipboard button on the toolbar or to select the Copy to Clipboard menu item on the File menu Displays help on using this window Short cut is to press F7 key Checking this box locks the window so that it does not update as changes are made to the variables Locking the window causes a complete copy of the I O System to be saved It also causes the Restore button to be displayed which can be used to restore the I O System to the state it was in when the Lock check box was clicked Unlocking a window deletes the saved copy of the I O System Used to enter a note that appears on the caption bar It is primarily intended to be used when the wi
107. ecified tolerances and settings for all continuous input variables and category input variables Shows the same information as displayed on the tabs in the Continuous Input Variable dialog boxes and in the Category Input Variable dialog boxes Tab 2 Tolerances Outputs Displays calculated tolerances and values for all output variables and objective functions Shows the same information as displayed in the Capability Study windows Also displays simulation results if available Tab 3 Continuous Inputs Displays all values and settings specified for continuous input variables in their Continuous Input Variable dialog boxes Tab 4 Category Inputs Displays all values and settings specified for category input variables in their Category Input Variable dialog boxes 76 Chapter 4 Tab 5 Outputs Displays all values and settings specified for output variables in their Output Variable dialog boxes Tab 6 Objective Functions Displays all values and settings specified for objective functions in their Objective Function dialog boxes Tab 7 Simulation Inputs Displays simulation results for the continuous input variables The corresponding simulation results for the outputs are on Tab 2 The buttons and controls at the bottom can be hidden by removing the check from the Show Buttons menu item on the Options menu When hidden the popup menu can still be used to perform the functions of these buttons and controls To display th
108. ect rate by 100 to get a percent defective Multiply the defect rate by 1 000 000 to get the defect per million dpm Previously formulas where given for the Z score for the case of a single specification limit In the event of a 2 sided specification limit it is calculated as follows Z Score 0 oz SCOT e ower olz SCOTE pper where is the inverse of the normal distribution function The Z score is determined by calculating the defect rate at each specification limit added them together to get the total defect rate and then looking up the Z score corresponding to the total defect rate Six Sigma Quality Six sigma quality requires that each measurable characteristic such as fill volume and seal strength have a defect rate of no more than 3 4 defects per million In the table of the normal curve shown previously the value 3 4 defects per million corresponds to a Z score of 4 5 and a Cp of 1 5 Six sigma quality requires that the process average be kept at least 4 50 from the nearest specification limit Six sigma quality can be translated into the process tolerance shown on the next page This process tolerance requires 1 Maximum o lt ao or equivalently Cp gt 2 2 Average be kept within 1 50 of the target or equivalently C lt 0 25 The second requirement is met if the process is stable or a control chart is used to keep it centered on the target If the process average comes off target by the maximum am
109. ed ZF Bestore This button is only displayed when the window is locked Clicking it restores the I O System to the exact state it was in when this window was locked Tab 1 Percent Reduction The first tab graphically displays the percent reduction in the width of interval for values of the output variable that results from tightening the tolerance of each of the individual input variables The original interval for values of the output variable is shown at the top and is the same as shown on the Capability Study window The inputs are listed in order of importance For example B has the largest effect making it the VIP Tightening its tolerance can shorten the interval for values of the output by about 50 For exact values see the third tab Pareto Chart F OO x Flow Rate Interval for Values 8 5061 11 5 3SD 100 Percent Reduction a B R L S Inputs Percent Reduction Transmitted Variation Table Plot for Input E Print Copy Help C Lock Tab 2 Transmitted Variation Tab 2 graphically displays the variation transmitted by each input variable to the output variable along with the total variation of the output variable The interval for values of the output variable is shown at the top and is the same as shown on the Capability Study window The figure at the right of the chart represents this interval The figures for the inputs represent the variation transmitted by each input indivi
110. ed below Overall Height H Worst Case Tolerance LSL T USL 3 88 inches 4 4 2 Characteristic Value At Target 4 Minimum 3 88 Maximum 4 12 Interval for Values 3 88 4 12 The resulting worst case tolerance for overall height is from 3 88 to 4 12 This fails to meet the specification limits for the overall height As a result tolerances for the inputs must be tightened if the specification limits for overall height are to be met Worst case tolerancing assumes that all four components can simultaneously be at the same extreme value For example if all four components have the minimum height of 0 97 the overall height will be 3 88 Likewise if all four components have the maximum height of 20 Chapter 2 1 03 the overall height will be 4 12 This yields a worst case tolerance for overall height of 3 88 to 4 12 If the tolerances for the individual components are all met the resulting worst case tolerance for overall height is guaranteed to be met To view the information that was entered into VarTran double click on the different variables in the O System window shown below I0 System Window Input Output System f E Name E Options amp Print Copy 2 Help For example double clicking on input D1 displays the dialog box shown below This dialog box contains the target and worst case tolerance for D1 To close the dialog box press the OK button Continuous Input Variable Dialog
111. elected Traditionally these distributions have a lower bound of zero However it can be modified to other values These modifications are referred to as the three parameter lognormal three parameter gamma and four parameter beta distributions The lower bound must be less than the minimum target specified in the optimization panel The Max edit box is used to enter an upper bound for the negative lognormal negative gamma and beta distributions This edit box is only displayed when one of these distributions is selected Traditionally the beta distribution has an upper bound of one However it can be modified to other values This modification is referred to as the four parameter beta distribution The upper bound must be less than the maximum target specified in the optimization panel If bounds exist not specifying them can lead to erroneous results To see why consider the case where Y SquareRoot X and X has a normal distribution with a target average of 3 and a standard deviation of 1 Performing a simulation will eventually result in the random number generator generating a negative value for X causing an error in calculating Y When this happens the program will not be able to print any simulation results Average Group Box This section is only displayed for process tolerances It is used to specify an operating window for the average The average must stay within the interval Target AverageDeltaLower Target AverageDeltaUp
112. elp 6 Chapter 2 For input variable B valve backflow the Include in Optimization check box is not checked This prevents the backflow target from changing during optimization The target for backflow was provided by the supplier and cannot be adjusted by the designer Evaluating the Initial Design If you know how the inputs behave and you know how the inputs affect the output you can predict how the output will behave While this can be accomplished using a simulation VarTran uses analytic methods instead To view the results click on the Capability Study button on the toolbar This performs a tolerance analysis and displays the results shown below wv Analysis Capability Study Capability Study F iojx Flow Rate F Statistical Tolerance LSL T USL 8 7544 ml min 11 242 Characteristic Value Average 9 9981 Standard Deviation 0 27637 Cp 1 21 Cc 0 00 Cpk 1 20 Def Rate normal 297 dpm Z Score short term 3 43 Sigma Level 3 43 Interval for Values 8 7544 11 242 4 5SD Calculated Worst Case Conditions Options Print I Copy Help a Lock In designing this pump our goal is six sigma quality This requires a defect rate of 3 4 defects per million dpm or below The capability study predicts that the defect rate will be 297 dpm and that some pumps will have flow rates as low as 8 7544 and as high as 11 242 ml min Both these values are outside the specifications of 9 to 11 ml m
113. endent of the units of measure The denominator of Cpm contains the Root MSE Maximizing Cpm is equivalent to minimizing Taguchi loss The defect rate is a measure of the relative number of units that are defective It is measured either in terms of percent defective or defects per million dpm units Expressions restricted to those function that are differentiable Differentiable functions are Log Logl0 Exponential SquareRoot Sine Cosine Tangent ArcSine ArcCosine ArcTangent HyperbolicSine HyperbolicCosine Hyperbolic Tangent NormalDistribution InverseNormalDistribution pi and e Entering a differentiable expression for the equation of an output variable allows VarTran to use the Taylor Series approach in the Output Variable dialog box Otherwise one is restricted to the curve fitting approach The distance from target is the maximum amount that the interval for values is from the target Its formula is Maximum UIV Target Target LIV where UIV is the upper end of the interval for values and LIV is the lower end of the interval for values The distance inside spec is the amount that the interval for values is from the specification limits Its formula is Minimum USL UIV LIV LSL where USL is the upper specification limit LSL is the lower specification limit UIV is the upper end of the interval for values and LIV is the lower end of the interval for values If only one specification limit exists
114. ent operators are given below starting with those with the highest precedence 1 negative 2 3 4 subtraction Operators that have the same precedence are performed according to their order Sometimes they are performed from left to right and sometimes from right to left The order they are performed in is called their association The associations of the different operators are given below Left to Right subtraction Right to Left negative If in doubt about the order that the operations will be performed use parentheses Some examples are 1 2 3 1 2 3 7 142 S22 1 8 2 3 8 2 3 12 24142 24 142 2 While an equation can be directly typed into the Equation edit box it is generally simpler to click the Keypad button and use the calculator style interface provided by the Keyboard Equations dialog box to enter the equation The Keyboard Equations dialog box is described later in this section If the equation was fit to a set of data using a statistics package you can paste the equation by clicking the Paste button and using the Paste Equation dialog box to automatically convert the equation into the format required by VarTran The Paste Equation dialog box is also described later in this section The Target edit box is used to enter the target if one exists Otherwise leave it blank It must be a valid constant expression and must be between the lower and upper specification lim
115. ent to 0 1 defective Minimum Z Score Predicted minimum Z Score Value reported is the Z Score associated with the maximum defect rate Minimum Sigma Level Predicted minimum sigma level Value reported is minimum Z Score plus 1 5 An interval for individual values is displayed at the bottom of the Capability Study window This tolerance represents an interval containing almost all the individual units of product For a process tolerance it is calculated as follows Upper End Interval for Values Max Average 4 5 Max Std Dev Lower End Interval for Values Min Average 4 5 Max Std Dev The number 4 5 in these formulas is selected so no more than 3 4 units per million will fall outside the interval This value can be changed using the Number Standard Deviations for Intervals for Values menu item from the Options menu The graphic visually represents the relationship between the process tolerance the tolerance for individual values and the specification limits The outside two normal curves represent the minimum and maximum averages and the maximum standard deviation i e the two worst case distributions The width of the normal curve equals the number of standard deviations used to calculate the interval for values Six sigma quality was introduced in Section 5 2 Meeting any of the following criteria assures Six sigma quality e Minimum C2 1 5 e Maximum defect rate lt 3 4 dpm e Sigma Level gt 6 e Tolerance for indi
116. er number of trials it is not necessary to change this value Just begin the simulation and halt it when sufficient trials have been completed Click on the Simulation button to begin the simulation The simulation will run until the specified number of trials has been performed The message window displays the progress The simulation can be stopped at any time by clicking on the Halt button The Halt button is only displayed while the simulation is being performed It does not initially appear on the dialog box 136 Chapter 8 Simulation Dialog Box x Simulation Number Trials f 00000 Exclude Out Of Spec Inputs Values Ep Simulation D View Results There are no previous trials Y Beset Help For statistical tolerances the simulation is performed at the current set of targets for the continuous input variables and the currently selected categories for the category input variables For process tolerances three simulations are performed one at the current targets one at the targets producing the minimum average and one at the targets producing the maximum average The targets producing the minimum and maximum averages are displayed on the third tab of the Capability Study window Simulations are not performed for worst case tolerances Once the simulations are completed the results can be viewed by clicking on the View Results button A Summary Tables window is displayed showing the results of
117. er operating windows If automatic controllers with real time feedback are used to keep the process centered a narrower operating window might be achieved Now consider the case where Cp is 1 5 Now if the process shifts off target by 1 5o it is still 30 from the nearest specification limit This worst case would result in 0 135 defective or 1350 defects per million dpm A C of 1 5 is generally considered acceptable 1 50 LSL USL 0 135 Defective 9o 4 50 0 4 50 90 Cp 1 5 Predicting Capability 83 Finally consider the case where Cp is 2 Now if the process shifts off target by 1 50 it is still 4 50 from the nearest specification limit This would result in no more than 3 4 defects per million dpm As seen later this represents a level of quality known as Six Sigma 1 50 LSL USL 3 4 Defects Million 120 60 0 60 120 C only evaluates the variation It does not consider the average In fact the average could be outside the spec limits and still yield a great Cp value A second capability index called C is used to evaluate the average Capability Index C C stands for capability of centering It is a measure of how well the process is centered relative to the specification limits Its formula is Cy Maximum Het tH j USL T T LSL where USL is the upper specification limit LSL is the lower specification limit T is the target and u is the process average C is independent of the unit
118. et to their selected categories Standard Deviation The predicted standard deviation of the output variable Cp Predicted value of C Six sigma quality requires a C gt 2 0 90 Chapter 4 Ce Predicted value of C e Six sigma quality requires a Ce lt 0 25 or below For a statistical tolerance C is calculated assuming all the inputs are exactly on target As such it will frequently be zero and represent the best case It tends to underestimate what will occur in practice Cpk Predicted value of Cpk Six sigma quality requires maintaining Cpk above 1 5 The value shown assumes the inputs are exactly on target so represents the best C obtainable Lower Cpk values will result in practice as the inputs vary from their selected targets To maintain a Cpk of 1 5 requires an on target Cpk of 2 or more Defect Rate Predicted defect rate as either a percent defective or in defects per million dpm 10 000 dpm is equivalent to 1 defective 1000 dpm is equivalent to 0 1 defective The defect rate is calculated assuming the normal distribution Six sigma quality requires 3 4 dpm or better The value shown assumes the inputs are exactly on target so represents the best defect rate obtainable Higher defect rates will result in practice as the inputs vary from their selected targets To average 3 4 dpm requires an on target defect rate that is considerable lower Z Score Predicted value of Z score Six sigma quality requires maintaining
119. ex of the help system A short cut is to press the F1 key Displays glossary of terms that is a part of help system Search for help based on a key word Displays help on using the help system Displays instructions for registering the software Registration and payment is required to continue using the software after the 30 day evaluation period Displays a registration form that can be printed out and used to register your copy of VarTran by Fax or mail Displays the Registration dialog box for entering user and registration codes so that VarTran can continue to be used after the 30 day free trial period See Section 1 3 Displays instructions for obtaining technical support If you have an internet connection and a web browser displays the home page for VarTran containing downloads of the most recent version and up to date information If you have an internet connection and a web browser displays the web page for registering your copy of VarTran over our secure web server using either a VISA or Master Card Displays copyright and registration information in the About dialog box Chapter 10 10 7 Toolbar tole 8 Sal 2 a6 OW big al o a The toolbar buttons provide short cuts for the more frequently used menu items Holding the mouse cursor over a toolbar button displays a popup description of the button The menu items associated with each button is given in the table below Menus and Toolbar 157 158 Cha
120. ffects Table window is shown on the next page It has 2 tabs containing the results Click on the X in the upper right corner to close this window Double clicking on a variable in the window displays the variable s dialog box for editing This is a shortcut to selecting the Edit menu item from the Variables menu The Effects Table window automatically updates itself when information affecting the displayed results is changed When the program is in immediate update mode the updates occur immediately following the change being made If the program is in delayed update mode the Effects Table window draws a red X through itself when it is not current Clicking on the window will cause it to update and redraw itself The update mode is selected using the Options menu Understanding the Results 121 The buttons and controls at the bottom can be hidden by removing the check from the Show Buttons menu item on the Options menu When hidden the popup menu can still be used to perform the functions of these buttons and controls To display the popup menu right mouse click over the graphic or table The following buttons and controls are available Options i Print Copy iif Help F1 Clicking this button displays the popup menu to the right associated with this window This popup menu has menu items that correspond to each of the buttons shown at the bottom of the window It also has items that can be used to change how the graphic
121. formed for the currently selected category The List of Categories list box is used to display all categories and used to select categories for the edit and delete operations To add a new category to this list press the Add Category button This displays the Add or Edit Category dialog box shown on the next page To Entering Information 61 change the symbol or name associated with a category select the category and press the Edit Category button This displays the Add or Edit Category dialog box below To delete a category select the category and press the Delete Category button A prompt will appear to confirm that you want to delete the selected category Suppose the symbol for the category input variable is M and there are two categories Old and New These are referenced as M Old and M New in an equation When New is the selected category the symbol M New has value 1 and M Old has value 0 To close the dialog box and save the changes click the OK button The symbol is checked to verify that it is valid and not already in use If an error is encountered a message is displayed To cancel the changes including the addition of a new variable click the Cancel button press the Esc key or click the X in the upper right corner of the dialog box To delete the variable click the Delete button For help click the Help button or press the F1 key When an edit box is changed new category is selected or a category is added edited or
122. ful However they also have their limitations First the dual response approach optimizes the process performance under the conditions that were present when the study was run Frequently the variation present during the study is not representative of the full range of variation that will be experienced over more extended periods of production For example if only one roll of plastic sheeting is used in the study roll to roll variation of sheeting thickness would not be captured as part of the variation This can result in making the process robust to the wrong sources of variation Taguchi suggest an alternative approach called the inner outer array approach that overcomes this limitation His outer array corresponds to the design array of the response surface study He suggests that for each trial in the outer array those inputs potentially contributing to the variation be purposely varied in a pattern that mimics their variation in production This pattern is called the inner or noise array This approach increases the costs of performing the study but ensures the process is optimized for production conditions Using VarTran the process can be optimized for actual production conditions without the added expense of running an inner outer array type study The second limitation of the dual response approach is that it does not identify the root cause of the variation If adjusting targets fails to obtain the desired performance further reduction i
123. fy either the maximum standard deviation or the capability index Cp 0 01 0 003 A lt 0 001 o lt 0 00067 0 01 0 003 C lt 1 C 21 5 Drawing Symbols for Process Tolerances A process tolerance is like a statistical tolerance but with an operating window for the average added A process tolerance is also like a worst case tolerance but with the worst case conditions defined in terms of process conditions rather than for individual units of product Process tolerances are much more than simply a third approach to specifying tolerances Process tolerances represent a unified approach to tolerancing that encompasses both worst case and statistical tolerances A process tolerance with o 0 is in effect a worst case tolerance Worst case tolerances guard against the event that all units are at the same extreme value This is unlikely if not impossible when variation is present A process tolerance with A 0 is in effect a statistical tolerance Statistical tolerances assume that the process is perfectly centered i e that no operating window is required 46 Chapter 3 Again this is unlikely if not impossible in most situations Process tolerancing allows more realistic tolerances to be developed Further comparisons will be made in Chapter 5 An important benefit of process tolerancing is the flexibility that it allows in specifying the behavior of the inputs Traditionally you had to select between using worst case toler
124. he input is adjusted from minimum target to the maximum target in the Input Variable dialog box The quadratic effects represent the distance the value at the midpoint deviates off of a straight line drawn between the values at the minimum and maximum targets The interaction effect is calculated using the four high low combinations of the two inputs involved Three of the points are used to predict what the forth point would be if there was no interaction effect i e if the effects where additive The interaction effect represents the distance between what one would expect at this fourth point and its actual value The effects table assumes all categorical variables are at their selected categories Since the effects are calculated over the region defined by the minimum and maximum targets in the Input Variable dialog box changing these values will change the effects table Wide regions should be selected for the input targets However care should also be taken to exclude impossible values such as a radius of zero and a motor speed of zero Understanding the Results 123 Effects Table F Oo x Flow Rate Effects of Adjusting Inputs Targets Term Change in Average Change in Std Dev Change in CV R linear 2845 5 9 9543 0 11581 quadratic 697 29 2 3572 0 058314 L linear 1437 1 4 1305 0 27242 quadratic 0 0 40349 0 13593 B linear 5 5 0 012607 0 00001304 quadratic 0 0 000005129 3 1893 10
125. hese are the inputs one should begin with Of course one should also consider the costs associated with tightening each tolerance to obtain the best value To tighten a tolerance associated with an input double click on that input in the graphic or table A Continuous Input Variable dialog box showing this input is displayed for editing Whenever a tolerance is changed the process should be re optimized Remember when performing an optimization you are finding the best targets for the specified set of tolerances The Pareto chart assumes all categorical input variables are at their selected categories Tightening Tolerances 115 7 2 Working With More Than One Design VarTran allows you to play What if games What effect does changing to a 2 dollar value have How about a 20 motor It is a good idea so save each design so that they can be compared and you can revert back to a previous design To save a copy of the design lock a Capability Study window a Pareto Chart window or a Summary Table window as shown below Capability Study F Original Flow Rate F Process Tolerance LSL T USL 8 5048 ml min 11 501 Characteristic On Target Worst Case Average Minimum 9 9981 9 3414 Maximum 10 664 Standard Deviation 0 27638 0 27886 Cp 1 21 1 20 Ce 0 00 0 66 Cpk 1 20 0 40 Def Rate normal 297 dpm 11 4 Z Score 3 43 1 20 Sigma Level 3 43 2 70 Interval for Values 8 5048 11 501 3SD Calculated Simul
126. his displays the Plots dialog box shown on the next page Try selecting the different types of plots To copy a plot to a separate window click on the Window button Close the dialog box by pressing the Close button 30 Chapter 2 bs Analysis Plots Plot Dialog Box a x Type of Plat _ Variables to Plat Characteristic C Line Plot Vaiue Interaction Plot Output H C Average Contour Plot Input 1 x z C Std Dev z C Defecti e window Input2 1 A cles vec f Value C Cpk f Value 4 Z Score f Value 2 5 f Value 1 Cy f Value 0 5 f Value 2 C Cpm C Root MSE C Dist Target C Dist Spec 1 o 1 Lower Value x Upper Value E Print Py Copy 2 Help To find the maximum and minimum values of f click on the Find Optimal Targets button on the toolbar This displays the Find Optimal Targets dialog box shown below Initially this dialog box is set to maximize the value of f Click on the Optimize Using Interval button to start the optimization analysis Find Optimal Targets Optimization Settings Output f x Characteristic Lower End of Interval for Values Maximize DZ amp Optimize Using Simplex Optimize Using Interval Optimization completed Better point found JA Close f Brecision Options Help Tutorials 31 Once the optimization is completed close the dialog box by clicking the Close button
127. ic Symbal Statistics DZ Dimension 1 D1 0 03 Process Tolerance LSL T USL r Type of Tolerance C Worst Case Tolerance Values C Statistical Tolerance Variation Process Tolerance Average Variation r Variation Distribution Normal x Standard Deviation USLLSL 3 Cp Optimization Options for Target Max Ave Target 1 5 SD ao amp n OQ Mex 2 Min Ave Target 1 5 SD Min x V Include In Optimizations X Cancel f Delete Options Help For the specified process tolerances for the individual components it can be determined that the average overall height will remain between 3 96 and 4 04 and that the standard deviation for overall height will be at or below 0 013333 Extending the interval for the average 3 standard deviations in either direction results in a tolerance for overall height of 3 92 to 4 08 So long as the process producing the components meets the process tolerance overall height is guaranteed to be within the specification limits This falls between the 4 0 12 from worst case tolerancing and 4 0 06 from statistical tolerancing Like worst case tolerancing process tolerancing is safe However process tolerancing is much less expensive A Unified Approach to Tolerancing Process tolerancing is much more than simply a third approach to tolerancing It represents a unified approach to tolerancing that combines both of the previous approache
128. icate Variable 150 Used to add a new variable Displays menu for selecting type of variable shown below The appropriate variable dialog box is then displayed so that the variable s information can be entered Variables Continuous Input Variable Category Input Variable Output Variable Objective Function Duplicate Variable Edit k Modify All Tolerances gt 3F 1 0 System Window a Summary Tables Notes Window Displays Number Duplicate Copies dialog box for making duplicate copies of a variable in order to save time when there are a large number of similar variables that must be entered Number of Duplicate Copies Variable to Duplicate R E fi v X Cancel Help Select which variable to duplicate and specify the number of copies Click the OK button or press the Enter key to close the Number of Copies Chapter 10 Edit Modify All Tolerances I O System Window Summary Tables Notes Window dialog box and make the duplicate copies Clicking the Cancel button or pressing the Esc key instead closes the dialog box without duplicating a variable Displays additional menus for selecting a variable to edit as shown below Shortcut is to double click on the variable in the I O System window or any of the other child windows that displays information about the variables Variables add Duplicate Variable Edit Modify All Tolerances 3F 1 0 System Window Mt
129. ification limit and a target If there are two outputs O1 and O2 uses the objective function Minimum O1 RDIS 02 RDIS If the desired characteristic to optimize does not appear it is either because it is not appropriate for the method of tolerancing specified or because a required value has not been entered For example Cpk is not appropriate if worst case tolerancing is used and cannot be calculated if at least one spec limit has not been entered Once the output and characteristic to optimize has been selected press either the Optimize Using Simplex button or Optimize Using Interval button to perform the optimization The interval method is safer but slower The simplex method is faster but does not always yield the global optimum Start with the interval method It is guaranteed to find the global optimum with at least 2 digits accuracy If it takes too long to converge press the Halt button to stop the optimization and use the simplex method instead The simplex method while faster may converge to a local optimum which is not the global optimum Examination of the plots as described in Chapter 8 can help to determine if local optimums exist If so the simplex optimization should be repeated several times These two methods of optimization are covered in detail in Sections 6 2 and 6 3 As the optimization proceeds status messages will be displayed in the message box When completed the message box will notify you whether a better set
130. in The initial design does not achieve our six sigma goal Since the flow rate average is on target the variation must be reduced Leave the Capability Study window open Tutorials 7 Robust Design One way of reducing the flow rate variation is through what Taguchi calls robustness Robustness is obtained by designing the pump to be less sensitive to the variation of the inputs The inputs continue to vary but less of this variation is transmitted to the output Robust design is accomplished through the setting of the inputs targets Since it is generally less costly to adjust targets than tighten tolerances building in robustness is the preferred method of reducing variation This is what Taguchi calls the second stage of design which he refers to as Parameter Design A robust design can be obtained by finding the set of targets for the inputs that minimizes the defect rate Start by clicking on the Optimize button on the toolbar The Find Optimal Targets dialog box shown below is displayed Specify that the flow rate percent defective is to be minimized and then click on the Simplex Optimize button The optimization will take just a couple of seconds analysis Find Optimal Targets Optimization Settings Output F Characteristic Percent Defective Minimize 7 amp Optimize Using Simplex 4 Optimize Using Interval Optimization completed Better point found J Close Ff Precision Y O
131. ing the analytic approach Flow Rate F Worst Case Tolerance LSL Ti USL 8 6038 ml min 11 34 Characteristic Value At Target 9 9977 Minimum 8 6938 Maximum 11 34 Interval for Values 8 6938 11 34 Calculated Simulated WorstCase Conditions s E Print Copy Help B Lock The name of the output variable and the type of tolerance are displayed at the top Further down a table of the capability study results is displayed The following values are displayed At Target The value of the output when all the inputs are at their targets Minimum The minimum value of the output variable over the worst case tolerances specified for the continuous input variables Assumes that the category input variables are set to their selected categories 94 Chapter 4 Maximum The maximum value of the output variable over the worst case tolerances specified for the continuous input variables Assumes that the category input variables are set to their selected categories An interval for individual values is displayed at the bottom of the Capability Study window For a worst case tolerance it is calculated as follows Upper End Interval for Values Maximum Value Lower End Interval for Values Minimum Value The graphic visually represents the relationship between the worst case tolerances and the specification limits Tab 2 Simulation Results The second tab does not apply to worst case tolerances so di
132. ion of strip chart determined range of 194 206 Standard deviation estimated as 1 6 this range Radius Per phone coversation with Bill Smith 123 456 7890 Options amp Print Py Copy Help 10 3 Analysis Menu Analysis Find Optimal Targets Robust Design LAI Capability Study Tolerance Stackup be Pareto Chart of Variation Sensitivity Analysis amp Simulation amp Lock All Unlocked Windows EEF Effects Table Ps Plots The Analysis menu is the third menu on the menu bar It is used to specify which analysis to perform The third and fourth groups of buttons on the toolbar are shortcuts for the menu items from the Analysis menu The available menu items are 152 Find Optimal Targets Capability Study Variation Pareto Chart Simulation Robust design option Displays Find Optimal Targets dialog box for determining optimal targets See Section 6 1 Tolerance analysis option Performs capability study on the selected output assuming inputs are at their current targets Shows results in a Capability Study window described in Section 5 1 Sensitivity analysis option Draws Pareto chart of the contributions of the individual inputs to the total variation for the selected output Assumes the inputs are at their current targets Shows results in a Pareto Chart window described in Section 7 1 Displays Simulation dialog box for performing simulation of all outputs assumi
133. ions When Symbols Change If checked and the symbol for a variable changes automatically changes all reference to the variable in any equations The update symbols mode is the default mode and is in effect each time the program is started Show Buttons If checked shows buttons at bottom of child windows The state of this menu item is stored in the system registry and remains in effect until changed Show Toolbar If checked shows toolbar The state of this menu item is stored in the system registry and remains in effect until changed 10 6 Help Menu Help g Contents g Glossary Search for Help On How to Use Help How to Register Registration Form ts Enter Registration Code Technical Support VarTran Home Page Register Using Web Site ssh About Menus and Toolbar 155 The Help menu is the last menu on the menu bar It is used to used to display the help index help on specific topics registration and support information and the About dialog box containing copyright information The last group of buttons on the toolbar provides short cuts for some of the more commonly used menu items on the Help menu The available menu items are Contents Glossary Search for Help On How to Use Help How to Register Registration Form Enter Registration Code Technical Support VarTran Home Page Register Using Web Site About VarTran 156 Displays table of contents or ind
134. is results in fewer process adjustments as the room conditions change You can design a material or component that is robust to itself In rubber compounds an important characteristic is elongation When mixing batches of the compound there will be variation in the amount of the ingredients added As a result the elongation will vary Targets should be selected that makes elongation insensitive to variations in the amounts of the ingredients You can design a material or component that is robust to the environment Temperature often has an effect on the strength of a plastic When formulating the plastic targets should be selected for the ingredients that make the strength insensitive to fluctuations in temperature This results in a more reliable product i e one less likely to encounter conditions under which it will fail The potential applications of robustness are endless Besides improving quality robustness allows the use of less expensive components and processes improves reliability increases the range of conditions under which the product or process can operate successfully reduces capital expenses and makes a process easier to operate Almost all products and processes can be made robust So long as nonlinear effects like squared square root log and trigonometric terms exist or so long non additive effects like multiplication division and power exist robustness is possible The only equation for which robustness is not possi
135. is sized within the window A short cut is to right mouse click on the graphic Prints the graphic or table on the selected tab Shortcuts are to click the Print button on the toolbar or to select the Print menu item on the File menu Copies the graphic or table on the selected tab to the clipboard in graphic format Shortcuts are to click the Copy to Clipboard button on the toolbar or to select the Copy to Clipboard menu item on the File menu Displays help on using this window Short cut is to press F key Tab 1 Diagram Effects Table F _ lolx Flow Rate Effects of Adjusting Inputs Targets Change in Average Change in Std Dev Change in CV R linear sl SSS em quadratic E L linear e E z quadratic E B linear quadratic S linear D 1 quadratic I RALE ee 5 R B lt lt R S I l L B i L oO B S Maximum Effect 2845 5 9 9543 1 0813 Diagram 122 Chapter 8 The first tab shows the results in diagram form The first column lists the terms Both linear and quadratic effects are calculated for each input as well as interaction effects for all possible interactions between two inputs These interaction effects are denoted with an separating the symbols of the two inputs An interaction between two variables is when their combined effect either exceeds or is below their individual effects The second column shows the effects of the different terms o
136. isplayed either vertically or horizontally The Minimum Value and Maximum Value edit boxes are used to change the scale of the y axis The minimum value must be less than the maximum value When changing the scale automatic scaling should be on to avoid shifting the lines or contours off the edge of the plot Automatic scaling is turned on and off using the Other tab The Value Size combo box is used to select the size for displaying axis values The default is normal size 130 Chapter 8 Plot Options x Title Axis Style YAxis xAxis Key Lines OftAxis Other IV Display Label Vertically Label Scale Text p Minimum Value 0 01 Size 15x x Maximum Value B Size Normal Layout Number of Values on Axis f Number of Tics Between Values 4 X Cancel Help The Number of Values on Axis edit box is used to change the number of values displayed along the y axis The minimum is two The default is 3 The Number of Tics Between Values edit box is used to change the number of tic marks displayed between each value on the y axis A value of zero can be entered The default is 4 Plot Options Dialog Box X Axis Tab The X Axis tab is used to change the x axis label scale and tic marks Double clicking on the x axis displays this tab It functions identical to the Y Axis tab except it affects the bottom or X axis Plot Options x Tile Axis Style Y Axis Axis Key Li
137. istics Dimension 1 D1 0 03 Statistical Tolerance LSL T USL M Type of Tolerance C Worst Case Tolerance Values Statistical Tolerance Variation Process Tolerance Average Variation Variation Distribution Nomal ssi Standard Deviation USL LSL 6 Cp Optimization Options for Target Min fo Max 2 V Include In Optimizations X Cancel f Delete Options Help Tutorials 23 Statistical Versus Worst Case Tolerances The dilemma faced when tolerancing is which approach to tolerancing should be used As the above example illustrates the approach selected can make a large difference So what are the pros and cons Worst case tolerancing is a safe approach If the 4 components all meet their tolerances the overall height is guaranteed to be within the calculated worst case tolerance However worst case tolerancing guards against the unlikely event where all four components are simultaneously at the same extreme either all at the lower tolerance value or all at the upper tolerance value This is unlikely in any case However it is especially unlikely if the components come from a process with variation When variation is present the average height of the components must be safely within the tolerance limits to ensure that the individual components all remain within the tolerances As a result only a small number of components are actually at the ex
138. it box is used to specify the relative error Optimization terminates when the relative error is less than the specified value For the interval method the relative error is guaranteed to be below the value specified Generally it is significantly better The default value is 0 01 ensuring two significant digits For the simplex method the relative error is estimated using the corners of the current simplex This underestimates the relative error because these corners do not generally bound the optimum The default value is 0 0000001 which frequently results in 4 or 5 significant digits However there is no guarantee and it is possible for the simplex method to terminate with only one or no significant figures The Stop after number trials edit box is used to specify the maximum number of trials to allow before forcing the optimization routine to end The optimization terminates when the number of trials exceeds the specified number even if none of the other termination conditions have been met For the interval method a trial is a splitting of an interval For the Optimization and Robustness 109 simplex method a trial is a contraction expansion or shifting of the simplex In both cases the default value is 100 000 The Stop when below edit box is used to terminate the optimization when it appears to be converging to zero The optimization terminates when the magnitude absolute value of the optimum is bounded below the specified value This
139. its if they exist The target is needed to calculate the distance from target The Lower Spec Limit edit box and Upper Spec Limit edit box are used to enter the specification limits if they exist If a specification limit does not exist leave the edit box blank The values entered must be valid constant expressions The lower spec limit must be less than the upper spec limit At least one specification limit is required to calculate Cpr the defect rate Z score sigma level and the distance inside spec Both specification limits are required to calculate Cp Entering Information 65 The Method of Estimating Average and Standard Deviation list box is used to specify the analytic approach to be used to derive equations for the average and standard deviation of the output There are two basic approaches used curve fit and Taylor series Both approaches locally approximate the equation by a polynomial for which the average and standard deviation can be derived The Taylor series approach uses Taylor series expansions to generate the polynomial The curve fit approach fits a curve through specific points The Taylor series approach is generally faster to execute but requires continuously differentiable functions so is not applicable when the following functions are used Minimum Maximum and AbsoluteValue The higher the order used the better the approximation but the longer the execution time A first order corresponds to fitting a straight line secon
140. l values are determined by adding and subtracting a specified number of the standard deviations from the target Traditionally 3 standard deviations is used For piston radius you might specify T 0 1 and o 0 001 The upper and lower ends of the interval for values are then determined as follows UIV T 30 0 103 LIV T 30 0 097 44 Chapter 3 0 097 lt Values gt 0 103 Statistical Tolerance Based on the normal curve 99 73 of the units will fall within 30 of the average One can use values other than 3 for calculating the upper and lower tolerances In VarTran the default value is 4 5 In this case only 6 8 units per million fall outside the interval for values Statistical tolerances are represented by the two equivalent drawing symbols shown below The symbol is used to indicate the tolerance is a statistical tolerance Optionally an informational box can be added providing clarification It is recommended that one specify either the standard deviation of the capability index C in an information box so that there is no confusion about the interpretation of the statistical tolerance 0 01 0 003 0 01 0 003 Drawing Symbols for Statistical Tolerances The above statistical tolerances are interpreted as meaning that the average radius is 0 1 and that the individual parts vary around this average with a maximum standard deviation of 0 001 3 6 3 Process Tolerances Instea
141. llows the use of less expensive processes You can design the product to be robust to wear and deterioration of its components For a car door closure force is an important characteristic Closure force is highly dependent upon the strength of the spring used As the spring deteriorates over time the closure force will change When the car is 5 years old we would still like the door to open and close with ease One solution is to use a more expensive spring that deteriorates less Another solution is to design the hinge to be less sensitive to the spring deterioration This results in smaller changes in the closure force as the spring deteriorates The effect is the same Robustness to wear and deterioration results in more reliable products and processes You can design the product to be robust to usage conditions and the environment Suppose one is designing a copy machine The user would like to use a variety of different weights of papers as well as different materials such as transparencies If excessive variation in copy darkness results from changes in the weight of the paper the customer will complain The designer cannot tighten the tolerance for paper weight because it is based on the customers requirements Robustness can be used to solve this problem The trick is to find an input whose target you can adjust that interacts with the weight of the paper An interaction is where adjusting one input changes the effect that the other input ha
142. m window is a child window that displays the symbols and names of the variables It is initially displayed If closed it can be reopened by clicking on the EF button on the toolbar or by selecting the O System Window menu item on the Variables menu Click on the X in the upper right corner to close this window 1 0 System Window Medical Pump Nome E Options Print fy Copy 2 Help The continuous input variables and category input variables are displayed on the left The output variables and objective functions are displayed to the right The title at the top can be changed The primary purpose of the O System window is to allow easy access to the variables for editing Double clicking on a variable in the I O System window results in the appropriate dialog box being displayed so that the variable can be edited This also works in many of the other windows as well This is a short cut to selecting the Edit Variables menu item on the Variables Menu The buttons at the bottom can be hidden by removing the check from the Show Buttons menu item on the Options menu When hidden the popup menu can still be used to perform the functions of these buttons The following buttons are available Used to add a new variable Displays a menu for selecting the type of variable The appropriate variable dialog box is then displayed so that the variable s information can be entered Shortcuts are the Add button on the toolbar the
143. mplished by simply asking the software to maximize Cpk or some other objective Within minutes the optimization will be completed and the results displayed Don t be fooled by the simplicity of accomplishing this task While this is a much simpler approach to robust design than using inner outer arrays and signal to noise ratios the strategy is the same and the results are superior to those of Taguchi Methods 6 1 Find Optimal Targets Dialog Box The Find Optimal Targets dialog box is used to find the set of input targets optimizing an objective function or a characteristic of an output variable It is displayed by clicking on the button on the toolbar This is a shortcut for the Find Optimal Targets menu item on the Analysis menu The optimal targets for the continuous input variables are found for the current tolerances and selected categories of the category input variables The optimization is performed over the region bounded by the Minimum Target and Maximum Target edit boxes on the Continuous Input Variable dialog box for those inputs whose Use for Optimization check box is checked Find Optimal Targets Dialog Box x Optimization Settings Output F Characteristic CPk Maximize amp Optimize Using Simplex Optimize Using Interval F Precision Y Options Help Optimization and Robustness 103 The Output list box is used to select which output variable or objective function to optimize If there ar
144. n copies graphics in a form called a Metafile to the clipboard These graphics can then be inserted into a word processing document by selecting the Paste menu item on the Edit menu of the word processor For tables it copies them to the clipboard in both text and Metafile formats The text is tab delimited to allow easy pasting into EXCEL In Word pasting will result in the text being pasted To paste the graphic instead in Word select the Paste Special menu item from the Edit menu and then check the picture option Glossary 159 Coefficient of Variation CV Constant Expression Continuous Input Variable Cp 160 The coefficient of variation or CV of a variable is its standard deviation divided by its average It is frequently multiplied by 100 and reported on a percentage basis For many products and processes the standard deviation increases as the average is increased An example is fill volume for a filling process This type of process is better characterized by its CV For example a CV of 3 means regardless of the average the standard deviation is 3 of the average Expressions containing only numbers operators functions and parenthesis No variables or properties are allowed Constant expressions represent a numerical constant such as a fraction Examples include 2 3 1 2 pi and e Within VarTran when a constant or number is required one can enter a constant expression Input of I O system that
145. n the flow rate average The effects are calculated by adjusting each input s target over the region bounded by the minimum and maximum targets specified in the Continuous Input Variable dialog box The longer the bar is the greater the effect The maximum effect shown at the bottom of the column is the size of the effect of the term with the longest bar The third column shows the effects that changing the input targets has on the flow rate standard deviation Achieving robust designs requires identifying inputs whose targets affect the output s standard deviation The targets of these inputs are generally set to minimize the variation The targets of the other inputs are then set to achieve the desired average The last column shows the effects of the inputs on the flow rate coefficient of variation or CV for short CV is the standard deviation divided the average On many processes the variation increases as the averages does Therefore all inputs that affect the average also affect the standard deviation One can achieve a robust design if one can identify an input that affects the CV Setting this input to minimize the CV reduces the variation for a given average Tab 2 Table The second tab shows the results in table form Everything appears the same except the bars are replaced with numbers The linear effects for the inputs represent the amount that the output characteristic average standard deviation or CV changes as the target for t
146. n the variation must be achieved by tightening the tolerances of the inputs However it is not known which tolerances to tighten VarTran can be used to identify the root cause of the variation allowing further reduction of the variation through the tightening of tolerances The following tutorial shows how VarTran can be used to overcome both these limitations Response Surface Tolerance Analysis Approach The first step using this approach is to run a response surface study The purpose of this response surface study is to obtain an equation for how the inputs affect the output The equation shown below was obtained 12 Chapter 2 SS 27 9029646803808 0 0822867601868433 THB 175 9 00263649461868 DT 0 75 0 0458833145964137 P 100 0 0430545023685393 MT 90 0 00749437527699365 THB 175 83 3209058456264 DT 0 75 1 03469576191631 THB 175 DT 0 75 where SS seal strength THB temperature of the hot bar D dwell time P pressure and MT material temperature This equation is for the average seal strength We also need information on how the inputs behave This requires collecting additional data from the process To obtain information on temperature of the hot bar a temperature transducer was attached to the hot bar and the results recorded over an extended production run From this data the standard deviation of the hot bar temperature was estimated Likewise standard deviations for the other input
147. n these values would only be appropriate if all future lots have the same average However in reality the average of each lot will be slightly different An allowance for an operating window needs to be made One approach is to specify the process tolerance T 0 1 A 1 50 A 1 50 and o 0 0005 This amounts to allowing a 1 50 operating window for the average This approach sets a clear requirement on the average that can be inspected for or enforced using a control chart It is appropriate to use a value of 1 50 for stable processes and processes controlled using a control chart A larger operating window should be used if the process is unstable A narrower operating window may be achievable using online controllers Process tolerances can also be used to account for product wear Suppose a car door hinge is being designed and that one input variable is the spring force of the spring A capability study of springs from a supplier indicates an average of 10 lbs with a standard deviation of 0 2 lbs Life testing of the springs predicts that the springs will weaken by 1 pound over a five year period Specifying a process tolerance of T 10 A 0 A 1 and o 0 2 describes both the manufacturing variation and the variation over time due to deterioration of the springs The purpose of a tolerance analysis is to accurately predict the behavior of the output This requires accurate descriptions of the behaviors of the inputs Not all inp
148. ndividual components are within their tolerances the final assemblies are outside of specifications Statistical tolerancing requires centered processes If control charts are used to keep processes centered it can perform well However if your plants or your suppliers let the processes wander throughout the specifications even though the process remains within the specification limits statistical tolerancing can have disastrous effects Since it is impossible to keep processes perfectly centered all the time some allowance needs to be made for the average deviating from the target In other words an operating window must be allowed One practice called Benderizing tolerances inflates the input standard deviations by 50 multiplies by 1 5 before performing statistical tolerancing See Bender Predicting Capability 93 1962 A better approach is to use process tolerances and specify the allowable operating window for the average This results in a clearer and enforceable set of requirements Despite its serious shortcoming statistical tolerancing is less expensive than worst case tolerancing as will be seen in the next section It deserves serious consideration 5 4 Worst Case Tolerances When worst case tolerances are specified for the input variables a worst case tolerance is calculated for the output The results are shown on 3 tabs Tab 1 Calculated Results The first tab shows the results of a tolerance analysis performed us
149. ndow is locked to describe the version of the design being displayed This button is only displayed when the window is locked Clicking it restores the I O System to the exact state it was in when this window was locked Entering Information 77 Predicting Capability Once the targets and tolerances for the inputs have been entered and once the equations for the outputs have been entered it is time to perform the different analysis often starting with a tolerance stack up to determine how the outputs are expected to behave The result of a tolerance analysis is displayed in the form of a capability study 5 1 Capability Study Window The Capability Study window is a child window used to display the results of a tolerance stack up for an output variable or objective function Click on the l button on the toolbar to display a Capability Study window This is a shortcut for the Capability Study menu item on the Analysis menu If there is only one output or objective function the results will be displayed immediately If there is more than one a menu will appear as shown below for specifying which variable to display the results for Analysis l lAl Find Optimal Targets Robust Design E ji Capability Study Tolerance Stackup wW be Pareto Chart of Variation Sensitivity Analysis W fae Simulation Lock All Unlocked Windows EEF Effects Table gt P lt Plots An example of the Capability Study window
150. nds above the target It must be greater than or equal to zero You can enter an expression that includes only the variables T or Target both referring to the value entered into the Target edit box Some examples are 0 1 1 12 1 0 10 5 Log 30 and 0 1 Target To reference this field in other edit boxes use the symbol ToleranceDeltaUpper or TDU The upper spec limit is equal to Target TDU To reference the upper spec limit in other edit boxes use the symbols UpperSpecLimit or USL For statistical and process tolerances this field may generally be left blank The 4 edit box is used to specify the distance the tolerance extends below the target It must be greater than or equal to zero You can enter an expression that includes only the variables T or Target Some examples are 0 1 1 12 1 0 10 5 Log 30 and 0 1 Target To reference this field in other edit boxes use the symbol ToleranceDeltaLower or TDL The lower spec limit is equal to Target TDL To reference the lower spec limit in other edit boxes use the symbols LowerSpecLimit or LSL For statistical and process tolerances this field may generally be left blank The Type of Tolerance radio group is used to specify whether the tolerance is a worst case tolerance statistical tolerance or process tolerance If either a statistical or process tolerance is selected a group box for specifying the variation is displayed If a pr
151. nes OftAxis Other Label Scale Text R Minimum Value 0 01 Size 1 5x bd Maximum Value fi Size Normal v Layout Number of Values on Axis E Number of Tics Between Values 4 X Cancel Help Understanding the Results 131 Plot Options Dialog Box Key Tab The Key tab is used to change the key title and style On an interaction plot the key describes how the lines correspond to the different values of the second input variable For a contour plot the key describes which value of the output each line represents Double clicking on the key title displays this tab Plot Options x Title Axis Style Y Axis x Axis Key Lines OftAxis Other gt KeyTitle Key Title Size Normal z Color Black z Display Key To Lines v Display Key to Lines Size of Line Labels in Key Normal z X Cancel Help The Key Title edit box is used to change the text for the key title It may be left blank if no title is desired The Key Title Size combo box is used to select the size for displaying the key title Normal size is the same size as the system font and is the default size You can select one of the values in the list or type your own value Entering a value of 1 3 results in the title being 1 3 times normal size The Key Title Color list box is used to change the color of the key title The Display Key To Line check box is used to specify whether t
152. ng inputs are at current targets See Section 8 4 Chapter 10 Lock All Unlocked Windows Locks all unlocked Capability Study Pareto Chart and Summary Table windows First displays Enter Note dialog box for entering a note to add to the windows being locked See Section 7 2 Note Original Design X Cancel Help First enter a note to display on the newly locked windows When done click the OK Button To exit without locking any windows click the Cancel button Effects Table Draws effects table for the selected output Shows result in Effects Table window See Section 8 1 Plot Displays Plots dialog box for displaying plots of the effects of the inputs on the outputs See Section 8 2 10 4 Window Menu Window Tile Shift F4 Cascade Shift F5 Arrange All Y 11 0 System Window The Window menu is the fourth menu on the menu bar It is used to rearrange the child windows or to select a particular child window A list of opened child windows is appended to the bottom of this menu These can be used to select a child window Selecting a child window brings it to the top Clicking on a child window has the same effect but the menu items can be used when the child window is buried under other windows Tile Displays all the child windows in a tile pattern The windows are reduced in size so that none of the windows overlap A shortcut is to press shift F4 Cascade Stacks the child windows slightly off
153. ng of the plots Higher resolutions can be used to produce smoother plots The Number of Contours Lines edit box is used to change the number of lines for interaction plots and the number of contours for contour plots The number of lines entered must be at least 2 For interaction plots the default is 3 Generally 2 3 is best For contour plots the default is 5 Generally 5 10 looks best 8 4 Checking Results Using Simulations VarTran analytically derives equations for the average and standard deviation of the output variables These equations can be viewed using the View Equations dialog box Some approximation is involved in deriving these equations A simulation can be performed to check the accuracy of the derived equations The Simulation dialog box is used to simulation to behavior of the outputs for comparison with the analytic results The results of the simulation can be viewed using the Summary Tables and Capability Study windows The Simulation dialog box is displayed by clicking on the gl button on the toolbar This is a shortcut for the Simulation menu item on the Analysis menu The Number Trials edit box specifies the number of trials to perform For each trial random values are generated for each of the input variables and the resulting values of the output variables calculated The default number of trials is 100 000 This is the recommended number of trials Minimally 10 000 trials should be performed To perform a small
154. ns for Interval for Values dialog box 154 Numbers Writing 64 71 162 O Objective function 70 163 Capability 102 Equation 71 139 Objective Function dialog box 70 Off axis variable 133 Operating window 35 82 Operator 64 71 162 Association 65 72 163 172 Precedence 65 72 163 Optimization 32 103 105 Excluding inputs 56 Interval method 32 107 109 Multiple responses 66 104 Simplex 106 109 Optimization and variation reduction 33 163 Closer to target the better 33 Larger the better 33 Smaller the better 34 Optimization Options dialog box 109 Options menu 154 Output 163 Output variable 34 63 163 Equation 64 139 Output Variable dialog box 63 P Pareto Chart window 111 Paste Equation dialog box 69 Percent defective See defect rate Plot menu 127 Plot Options dialog box 128 Plot window 127 Plots dialog box 125 Precedence 65 72 163 Print Setup dialog box 149 Process tolerance 45 97 164 Example 25 Pros and cons 26 101 Six sigma 88 Propagation of errors See variation transmission analysis Properties 162 R Registering software 2 Registration dialog box 156 Relative distance inside spec 164 Reliability 41 47 Response surface study 12 139 Robust tolerance analysis 48 164 Robustness 8 11 39 40 103 165 Dual response approach 12 141 143 Taguchi methods 12 146 Tolerance analysis approach 12 19 143 146 Root mean square error
155. ns the standard deviation depends on the distribution Therefore changing distributions changes the standard deviation Care should be taken when using these distributions to ensure the standard deviation fits the input It is better practice to obtain an estimate of the standard deviation and use a distribution that allows the standard deviation to be entered The Fit to Spec distributions are primarily intended for use as part of a preliminary analysis prior to obtaining good capability data on the inputs For the other distributions the standard deviation must be specified In this case changing distributions from normal to uniform to lognormal will generally have little effect The exception is when the input is limited by some bound For example piston radius is bounded below by zero Either the lognormal or gamma distributions should be selected in the event of a lower bound Otherwise negative values could be generated for which the equation may not be defined This can be especially troublesome when performing simulations The Standard Deviation edit box is used to enter a constant or expression for the standard deviation This edit box does not appear if you selected one of the Fit to Spec distributions You may enter any differentiable expression referencing the following variables Target T ToleranceDeltaUpper TDU UpperSpecLimit USL ToleranceDeltaLower TDL LowerSpecLimit and LSL Examples of valid entries for the stand
156. o display the key to the lines Generally the key is desired but can be removed for purposes of printing and copying to the clipboard The Size of Line Labels in Key combo box is used to select the size for displaying the line labels in the key The default is normal size Plot Options Dialog Box Lines Tab The Lines tab is used to change the label style and value of a line on either an interaction or contour plot A short cut is to double click on the key for the lines Double clicking on a line in either the plot or key displays this tab 132 Chapter 8 Plot Options x Title Axis Style Y Axis x Axis Key Lines Off Axis Other T E Line Info Value zlat F Std Dev 7 a Value i Color Black ha Style El Thickness X Cancel Help The Line tabs are used to select which line to change The values displayed in the other controls are for the line whose tab is selected The Label edit box is used to change the label shown next to the line in the key for lines The Color list box is used to change the color of the line The Style list box is used to select the line style Ifa line style other than a solid line is selected the line thickness is set to hairline The Thickness combo box is used to change the line thickness of solid lines Only solid lines can have a thickness other than hairline If a thickness other than hairline is selected the line style will be set to solid
157. ocess tolerance is selected a second group box for specifying an operating window for the average is displayed as well 52 Chapter 4 Variation Group Box This section is only displayed for statistical and process tolerances distribution and the maximum standard deviation o required depending on the distribution selected The Distribution list box is used to select the distribution of the individual units around the average The normal distribution is the default distribution and is sufficient for most purposes The full list of distributions is shown to the right All these distributions are available for statistical tolerances Those distributions with the Fit to Spec designation cannot be used with process tolerances To see what the different distributions look like select the distribution and view the results on the Graphics tab Below are a few of the distributions Triangular Fit to Spec Uniform Fit to Spec LSL T USL You must specify the Other information might also be Normal z Beta Gamma Lognormal Negative Lognormal Negative Gamma Triangular Fitto Spec Uniform Uniform Fitto Spec Uniform Minus Center 25 Fitto q Uniform Minus Center 1 3 Fitto Uniform Minus Center 50 Fit to 4 Uniform Minus Center 2 3 Fit to S Uniform Minus Center 75 Uniform Minus Center 50 LSL T USL Aa a U 0 099724432 0 10027557 0 099610289 0 10038971 0 09948446 0 10051554 Normal Lognorm
158. of 3 has been used historically in conjunction with statistical tolerancing but this practice results in high levels of defects Once the number of standard deviations has been entered click the OK button to save the change If the information entered is incorrect an error message will be displayed Otherwise the setting is updated and all effected child windows are notified of the change Clicking the Make Default button makes the value displayed in the Number Standard Deviations combo box the new default value An error message is displayed if it is 154 Chapter 10 not a valid number between 1 0 and 6 0 This new default value will effect all new sessions but does not effect previously saved sessions It also will not effect the current session unless the OK button is also selected Clicking the Cancel button closes the dialog box without making a change Immediate Update Mode If checked causes all child windows to be immediately updated following each change This may result in slower execution if a series of changes are made or if numerous windows must be updated Removing the check delays updating child windows When in delay update mode and a child window requires updating a red X is drawn through it Clicking on the window will cause the window to update itself The top child window is always automatically updated The delay update mode is the default mode and is in effect each time the program is started Update Equat
159. of less than ideal practices One approach used is called Benderizing tolerances It basically is to do statistical tolerancing and then inflate the resulting tolerance by 50 Another approach is to split the difference This approach involves doing both worst case and statistical stack ups and taking the midpoint between them As unscientific as either of these approaches sounds both are probably closer to the correct answer than either the worst case or the statistical stack up A Solution Process Tolerancing Worst case tolerancing results in tolerances for the outputs that are too wide It is safe but expensive Statistical tolerancing results in tolerances for the outputs that are too narrow It is less expensive but risky Ideally we want to be somewhere between these two extremes 24 Chapter 2 A new method of tolerancing called process tolerancing has been implemented in VarTran that allows just this The results of a process tolerance analysis can be displayed by opening the file stack pr vta The results are shown below Overall Height H Process Tolerance LSL T USL 3 9 inches 4 1 Characteristic On Target Worst Case Average Minimum 4 3 96 Maximum 4 04 Standard Deviation 0 013333 0 013333 Cp 2 50 2 50 Cc 0 00 0 40 Cpk 2 50 1 50 Def Rate normal 6 38 10 8 dom 3 37 dpm Z Score 7 41 4 50 Sigma Level 7 41 6 00 Interval for Values 3 92 4 08 3SD The resulting process tolerance calculated for o
160. on Pareto Chart Pareto Chart F O x Flow Rate Interval for Values 9 3211 10 676 4 5SD 100 Percent Reduction Ontions Print Copy Help B Lock Motor speed variation S is the largest contributor to the flow rate variation Tightening its tolerance requires using the 20 motor One could enter the tolerance standard deviation for the 20 motor re optimize the process and then view the new capability One should also consider tightening the tolerance for valve backflow This requires using the 2 valve which is significantly cheaper The Pareto chart indicates the effect is nearly the same as tightening the tolerance of motor speed The option representing the best value to the customer should be selected This is what Taguchi refers to as the third stage of design which he calls Tolerance Design 10 Chapter 2 Summary The above example illustrates the following strategies 1 2 3 The designer must take the time to identify e The key outputs Customer requirements for the key outputs The key inputs How the key inputs behave manufacturing and suppliers capabilities How the key inputs affect the outputs equations Without this knowledge the design process breaks down into one of trial and error Designing in robustness results in higher quality and lower costs Otherwise variation must be reduced by tightening tolerances which drives up costs This requires that v
161. or press the Esc key Clicking the Default button resets all options to their default values Clicking the Help button displays help on using the Optimize Options dialog box 110 Chapter 6 Tightening Tolerances Optimization of the input targets results in a robust design This reduces the variation or width of the tolerance of the outputs However there is no guarantee that robustness alone will allow you to achieve the desired results Following optimization further improvements must be achieved by tightening tolerances of the inputs Deciding which input tolerances to tighten and by how much requires knowledge of the relative costs of tightening the different tolerances as well as the effects of tightening the tolerances The effects of tightening the tolerances can be determined using the Pareto Chart window 7 1 Pareto Chart Window The Pareto Chart window is a child window used to display the contributions of individual input variables to the variation or tolerance of a selected output variable It is displayed by clicking on the bd button on the toolbar This is a shortcut for the Pareto Chart of Variation menu item on the Analysis menu If there is only one output the results will be displayed immediately If there is more than one a menu will appear as shown below for specifying which variable to display the results for Analysis bob Find Optimal Targets Robust Design E LAl Capability Study Tolerance Stackup
162. orials 2T 2 4 VarTran as a Mathematics Package For a demonstration of VarTran s mathematical capabilities start VarTran and open the file math vta This is accomplished by selecting the Open menu item from the File menu Highlight math vta and then click on the OK button The O System window shown below is displayed There is one output variable f shown on the right of the window The output is the function of two inputs variables x and y These are shown on the left of the window This represents a function f in two parameters x and y VO System Window Function Q Name E Options amp Print Py Copy Help F1 The equation for f is shown below This dialog box is displayed by double clicking on the box labeled f in the previous window Close this dialog box by pressing the OK button Output Variable Dialog Box x Symbol f Name Units 2K se 123 Keypad Cl Equation 2 y 2 Cosine 2 pil x Sine 2 pil y Sine 2 pil x 2 Cosine 2 pil y 2 E Paste Target Lower Spec Limit Upper Spec Limit Method of Estimating Average and Standard Deviation Taylor Series 2nd Order z V Include In Optimizations with Multiple Outputs X Cancel Delete DP View Eq Help In tolerance analysis inputs have behaviors described by the tolerances selected These tolerances describe a range of possible values In a mathematical analysis inputs have a 28
163. ory Input Variables A category input variable is an input variable that is restricted to specific choices or levels referred to as categories One example is material type A and B Types A and B represent the categories Other examples are mold cavity 1 2 3 and 4 and vendor A B and C When a category input variable is added a Category Input dialog box like the one shown below is displayed This dialog box is used to enter and change information concerning category input variables Category Input Dialog Box x Symbol M Name Material Type Selected Category Old List of Categories Old PL 142 Add Category New PL 457 Edit Category Delete Category X Cancel i Delete Help The Symbol edit box is used to enter the symbol The variable s symbol is used to represent the variable in equations Each variable must have a unique symbol It can be of any length but is best limited to 3 characters or less The symbol must consist of letters the digits 0 9 and underscores It cannot begin with a digit The Name edit box is used to enter a more detailed name or description of the variable It can be any combination of characters and any length While symbols should be short names should be descriptive The Selected Category combo box is used to select a category to use for the analysis Optimizations capability studies Pareto charts effects tables plots and simulations are all per
164. ount of 1 50 the process average will still be at least 4 50 from the nearest specification limit ensuring six sigma quality Per the notation introduced in 3 6 3 a six sigma tolerance is written as T A 6 lt 1 6 32 88 Chapter 4 1 56 Window Lower Upper Spec Spec Limit Limit At least 4 50 At least 4 56 60 4o 20 0 20 40 60 12c Wide Spec Limits Six Sigma Process Tolerance At first appearance the requirement for six sigma quality may appear to be too stringent First 100 Samples If 100 samples are selected from a six sigma process they generally fall in a range of 30 around the average A histogram of these units might appear as to the right The units all fit in the middle half of the specs However the histogram of these 100 units does not display all the variation in the lot Complete Lot Suppose the sample was selected from a lot of 10 000 units One unit has the highest value and another has the lowest The chance of including both these units in your sample is small There are probably dozens of units in the lot more extreme than the 100 samples The entire lot might appear as to the right Year s Production Not all lots will have exactly the same average If you can hold a 1 50 operating window the year s production might appear as to the right It fills the specifications The truth is that if your first 100 units do not fit within the middle half of the specs it is
165. per The Average Delta Upper edit box is used to enter the maximum amount that the average can be above the target You may enter any differentiable expression referencing the variables Target T ToleranceDeltaUpper TDU UpperSpecLimit USL ToleranceDeltaLower TDL LowerSpecLimit LSL StandardDeviation and SD Examples of valid entries are 0 0 01 1 3 1 2 pif 0 25 TDU USL LSL 8 0 01 Target 0 03 T 1 5 StandardDeviation The value or expression entered must be greater than or equal to zero for all possible values of the target between the minimum and maximum specified in the optimization panel The Average Delta Lower edit box is used to enter the maximum amount that the average can be below the target Its requirements are then same as the Average Delta Upper edit box Entering Information 55 Clicking the Ce button displays a popup menu shown to the Cc right for automatically entering various equations into the Ce 14 Average Delta Upper and Average Delta Lower edit boxes Cc 1 3 including equations to specify a Cc of 1 4 1 3 and 1 2 and Cc 1 2 a 1 5 standard deviation operating window There is Cc X replace X s with value 1 5 SD Operating Window Six Sigma Cp 2 Cc 0 25 also a menu item to specify a Cc value This menu item inserts equations for the Average Delta Upper and Average Delta Lower with an X at the place to insert the value of Cc Finally there is a menu item to specify Six Sigma quality whi
166. pter 10 Glossary Category Input Variable Child Window Clipboard Input of I O system that is restricted to specific values or levels which are referred to as categories One example is material type A and B Types A and B represent the categories Other examples are mold cavity 1 2 3 and 4 and vendor A B and C Capability index comparing the process average with the target and spec limits to assess the centering of the process Its formula is C Maximum pel be USL T T LSL where USL is the upper specification limit LSL is the lower specification limit T is the target and u is the process average Ce is independent of the units of measure See Section 5 2 Child windows are displayed in the interior of the main window They are used to display the data inputted and analysis results The available child windows are O System window Summary Table window Notes window Capability Study window Pareto Chart window Effects Table window Plots dialog box and Plot window The top child window is called the active window Its caption bar is highlighted To make a child window active click on any part of the child window One can also select a child window using the Window menu A third way of selecting a child window is to hold down the Ctrl key and repeatedly press the Tab key till the appropriate child window is made active Utility provided by Windows operating system for sharing data with other programs VarTra
167. ptions Help Once the optimization is completed exit the Find Optimal Targets dialog box by clicking on the Close button The Capability Study window will automatically update itself It will now appear as shown on the next page The variation has been cut in half while maintaining an average flow rate of 10 ml sec Since the defect rate is below 3 4 dpm our design is now six sigma The capability study predicts that nearly all units will fall in the range of 9 3224 to 10 678 ml min This is well within the specification limits All this was accomplished using the low cost components Your results may differ slightly due to the properties of the optimization routine but should be very close 8 Chapter 2 Capability Study F oO x Flow Rate F Statistical Tolerance LSL T USL 9 ml min 11 Characteristic Value Average 10 Standard Deviation 0 15059 Cp 2 21 Cc 0 00 Cpk 2 21 Def Rate normal 0 0000312 dpm Z Score short term 6 54 Sigma Level 6 54 Interval for Values 9 3224 10 678 4 5SD Calculated Simulated Worst Case Conditions Bassssssccosessssesecccsssssssssccedl Options amp Print Py Copy Help Lock To view the new design click on the Summary Table button on the toolbar The result is shown below The new targets are R 0 1737 L 0 40942 and S 17 064 rpm The target for valve backflow did not change because it was excluded from the optimization It was provided by the
168. pyramid four corner points In n dimensions a simplex is represented by n 1 points In VarTran each input variable being optimized represents a dimension An initial simplex is randomly generated and the function being optimized evaluated at each corner of the simplex The simplex then moves downhill through a series of shifts expansions and contractions as illustrated below The first move is to shift away from the highest point If this results in an improvement an expansion is tried If a worst point is found a contraction is tried The idea is to continuously replace the highest point with a better point 106 Chapter 6 Original Simplex Shift Expansion Contraction lt High Point A Low Point As the optimization proceeds the simplex is able to wind and twist itself downhill changing shapes as it goes until it can go no further Unfortunately the stopping point may be a local minimum not the global minimum Another difficulty with this approach is recognizing when the minimum has been found The corners of the simplex do not necessary bound the optimum so it cannot be estimated with certainty what the relative error is Unless well designed stopping criteria are selected the optimization can terminate before reaching the minimum VarTran uses a complex combination of stopping criteria which rarely terminates prematurely Further details on the simplex method can be found in Olsson 1974 and Olsson and Nelson 1975 6 3 Inter
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170. re displayed at the top Further down a table of the capability study results is displayed There are two columns of numbers The first column gives the values of the characteristics when all the input averages are equal to the target This amounts to doing a statistical stack up ignoring the operating windows for Predicting Capability 97 the averages and ignoring any worst case tolerances It is represented in the graphic by the center normal curve The second column represents the worst case values for the different characteristics as the input averages vary over their allowed operating windows The three values minimum average maximum average and maximum standard deviation define the resulting process tolerance for the output In the graphic the left hand normal curve represents the minimum average with the maximum standard deviation and the right hand normal curve represents the maximum average with the maximum standard deviation The other worst case values are calculated from these three The following worst case values are displayed Minimum Average Maximum Average Maximum Std Dev Minimum Cp Maximum Cc Minimum Cpk Maximum Defect Rate 98 The minimum average of the output variable Using the derived equation for the average the minimum average is found over the operating windows for the averages and over any worst case tolerances for the continuous input variables It assumes that the category input va
171. re knowledge of how the inputs behave Taguchi uses this information to design the study The tolerance analysis approach does not use this information until the analysis phase This allows alternate sets of tolerances to be evaluated and optimized without requiring additional data The best practice is to use both the dual response and tolerance analysis approaches Both approaches require exactly the same data to be collected Comparison of the results can yield significant insights For example if the predicted variation from the tolerance analysis approach is below that actually observed by the dual response approach it means an important source of variation was not included in the study 146 Chapter 9 1 O Menus and Toolbar This chapter covers the menus and toolbar Many of the menu items have already been covered elsewhere However a few have not yet been described This chapter explains those remaining menu items The menu bar contains the menus listed below File menu Used to start a new I O system open an existing I O system save an I O system print and to exit the program Variables menu Used to add and edit variables and display the information entered Analysis menu Used to perform the various analyses Window menu Used to rearrange the child windows or to select a particular child window Options menu Used to specify options for calculating intervals for values displaying buttons and the toolbar and for
172. ree of charge for 30 days After the 30 days trial period you must register the software to continue to use it To register the software go to www variation com vta and click the Register button Up to date pricing information will be provided along with further instructions The registration process can be completed online The Help menu of VarTran also contains further details about registering the software Once the software is registered you will receive a user name and registration code by email These must be entered into the software The user name and registration code can be manually entered into VarTran by selecting the Enter Registration Code menu item on the Help menu This displays the Registration dialog box shown below The user name and registration code must both be entered exactly as provided case sensitive Once done click the OK button Help Contents Glossary Search for Help On How to Use Help Registration User Name Registration Code J OK X Cancel Help How to Register Registration Form isk Enter Registration Code Technical Support VarTran Home Page Register Using Web Site ssh About 1 4 Uninstalling VarTran VarTran can be uninstalled using the Add or Remove Programs icon on the Control Panel Step by step instructions follow e Click the Start button e Select the Control Panel menu item from the Settings menu e Double
173. restore the I O System to the state it was in when the Lock check box was clicked Unlocking a window deletes the saved copy of the I O System Note edit box Used to enter a note that appears on the caption bar It is primarily intended to be used when the window is locked to describe the version of the design being displayed ZF Bestore This button is only displayed when the window is locked Clicking it restores the I O System to the exact state it was in when this window was locked 5 2 Cp Co Cpk and Six Sigma To interpret the results of a capability study you must first have an understanding of Cp Ce Cpk and Six Sigma These terms are defined in this section Capability Index C C stands for capability of the process It is a measure of how well the process variation fits within the specification limits Its formula is C USL LSL 60 where USL is the upper specification limit LSL is the lower specification limit and o is the process standard deviation C is independent of the units of measure The numerator is the distance between the specification limits The denominator is the width of the process For a normal distribution 99 73 of the product falls within 30 of the average Therefore 99 73 of the product falls within an interval that is 66 wide Cp compares the width of the specification limits to the width of the process When C is 1 the process variation is equal to the width of the specification limits a
174. riables are set to their selected categories The maximum average of the output variable Using the derived equation for the average the maximum average is found over the operating windows for the averages and over any worst case tolerances for the continuous input variables It assumes that the category input variables are set to their selected categories The maximum standard deviation of the output variable Using the derived equation for the standard deviation the maximum standard deviation is found over the operating windows for the averages and over any worst case tolerances for the continuous input variables It assumes that the category input variables are set to their selected categories Predicted minimum value of C It is calculated as follows USL LSL Min Cp 4 6 x Max Std Dev Predicted maximum value of C It is calculated as follows Max Ave T T Min ave Max Cc Max USL T T LSL Predicted minimum value of Cpx It is calculated as follows Min USL Max Ave Min Ave LSL Min Cpk 3 x Max Std Dev Predicted maximum defect rate as either a percent defective or in defects per million dpm The reported value assumes the normal distribution The defect rate is calculated assuming the Chapter 4 maximum standard deviation at both the minimum and maximum averages The larger of the two defect rates is reported 10 000 dpm is equivalent to 1 defective 1000 dpm is equival
175. riation reduction tools it is the one that can be used earliest in the design process Despite its state of the art capabilities VarTran is easy to use The typical analysis including optimization can be performed in less than 15 minutes This type of analysis has traditionally taken days to complete The following four tutorials can be used to quickly explore VarTran s capabilities Design of a Pump Demonstrates the use of VarTran in product development Optimization of a Heat Sealer Demonstrates the use of VarTran on a process following a response surface study Dimensional Stack up Problem Demonstrates the variety of methods available in VarTran for addressing dimensional tolerance stack up problems VarTran as a Mathematics Package Demonstrates VarTran s mathematical capabilities including its state of the art optimization routines based on interval analysis The tutorials can be explored in any order Start with the tutorial of greatest interest However it is a good idea to explore all four tutorials Each highlights a different aspect of VarTran s capabilities Once the tutorials are completed turn to Chapter 3 to understand the basic principles and strategies for using VarTran The remaining chapters provide complete documentation on the software They can be read in their entirety or referenced as needed Much of this same documentation is also available through the program s help system Tutorials 3 2 1 Design of a Pump
176. roke length motor speed and valve backflow The I O system could also be a process like a heat sealer The output might be seal strength The inputs could be time temperature pressure and material thickness SYSTEM OQACUVZ oOoxAcCVACO System Input Output Model It is the outputs for which optimization and variation reduction is desired The variation of the outputs is caused by variation of the inputs Optimization and variation reduction of the outputs is accomplished by identifying the key input variables and determining proper controls including targets nominal and tolerances process windows Further information can be found in the book Optimization and Variation Reduction in Quality Taylor 1991 Much of the material in this chapter is borrowed with permission from this book 34 Chapter 3 3 2 The Importance of Reducing Variation Why reduce variation The obvious answer is to reduce the number of defectives as shown below Lower Upper Spec Spec Limit Limit GOOD PRODUCT KR 7 Aa 7 DEFECTIVE PRODUCT NO DEFECTIVE PRODUCT Reducing Variation Results in Fewer Defects Now assume we have achieved the right hand picture above Are there additional benefits from reducing the variation further The answer is yes Further reduction of the variation results in a wider operating window as shown below The left hand process only makes good product if the process is close to being perfectly centered The right han
177. s Continuous Input 4 E al amp Print Py Copy Help Lock i E Options amp Print Py Copy Help F T Lock VarTran has extensive mathematical capabilities including a novel new method of optimization based on interval analysis However it differs from traditional math packages in the way that it handles the input variables In most math packages input variables represent values These values are used to calculate the value of the function output variable In VarTran the input variables represent behaviors For example an input variable might be assigned a statistical distribution statistical tolerance or an interval that its value must fall within worst case tolerance VarTran then determines how the function or output variable will behave When dealing with products and processes this represents the difference between how the product or process performs in theory value versus how it will perform in practice behavior 32 Chapter 2 Overview 3 1 Optimization amp Variation Reduction The goal is to optimize the product process performance and to reduce its variation There are two parts to this objective 1 optimization of the average i e drive the average to its ideal value and 2 reduction of the variation This goal is shown below Upper Specification Limit Lower Target Specification Limit Closer to Target the Better Case The above figure shows what is called the
178. s Worst case tolerances are special cases of process tolerances where the standard deviation is zero Statistical tolerances are special cases of process tolerances where the minimum average and maximum average are set equal to the target Neither of these two special cases is very realistic Using process tolerances you are no longer forced to make a choice between worst case tolerances with its increased costs or statistical tolerances with its increased risks With VarTran you can also use a combination of all three approaches You can specify worst case tolerances for environmental variables and usage conditions over which you have no control You can specify statistical tolerances for machine parameters that are controlled using automatic controllers to ensure they remain centered You can also specify process tolerances for components from a supplier where you can not guarantee that the process remains perfectly centered With VarTran you can choose the appropriate type of tolerance for each input No longer are you forced to treat all inputs the same 26 Chapter 2 VarTran is not restricted to performing tolerancing on just stack up problems It can perform all three methods of tolerancing for any equation including complex nonlinear equations This allows one to perform tolerancing on problems impossible to solve by hand and for which the tolerance stack up formulas do not apply The pump and heat seal tutorials serve as two examples Tut
179. s compared to the overall optimum The dial control that appears in this group box shows the position of the target in the interval determined by the minimum and maximum targets It can also be used to change the current target by double clicking on the desired value 56 Chapter 4 Graphics and Buttons The 3 tabs represent pictorially the type of tolerance specified The information on the first and third tabs is similar to that displayed for an output variable in a Capability Study window For further information on interpreting the values see Section 5 1 and the Glossary Graphic Statistics Radius of Piston R Radius of Piston R Statistical Tolerance Statistical Tolerance e Characteristic Value LSL T USL Average 0 1 Standard Deviation 0 0005 Cp 1 0000 Cec 0 0000 Cpk 1 0000 Def Rate normality 0 27 Z Score short term 2 78 Sigma Level 2 78 0 09775 0 10225 Interval for Values 0 09775 0 10225 4 5SD The middle tab displays how the tolerance would be represented on a drawing The two equivalent forms are given Symbol Radius of Piston R Statistical Tolerance 0 1 4 0 0015 5D lt 0 0005 0 1 4 0 0015 Cp gt 1 0000 To resize print or copy these tabs click the Options menu This Size To Fit displays the popup to the right An alternative is to right mouse click Bediko Nawa over the graphic Fixed Size Custom gt To close the dialog box and save the changes
180. s is minimized Available for all three types of tolerances even when no targets or specification limits are specified Only those characteristics applicable to your problem are displayed For example Cpk is only displayed when at least one specification limit has been entered for the output variable and if statistical or process tolerances are specified The value initially displayed is the characteristic the software is recommending be optimized based on the type of tolerance and the specification limits and target entered If an objective function has been selected you can choose between maximizing or minimizing the objective function s value If you selected All Included Output Variables in the Output list box the Characteristic list box will contain the following choices Total Percent Defective The total percent defective is minimized Only available for statistical and process tolerances with at least one specification limit If there are 104 Chapter 6 two outputs O1 and O2 uses the objective function O1 PD 02 PD Worst Cpk The worst Cpk is maximized Only available for statistical and process tolerances with at least one specification limit If there are two outputs O1 and O2 uses the objective function Minimum O1 Cpk 02 Cpk Worst Relative Distance Inside Spec The worst relative distance inside spec is maximized Available for all 3 types of tolerances with either two specification limits or 1 spec
181. s obtain it using a designed experiment such as a response surface study This section addresses some of the special issues involved with performing a robust tolerance analysis following a designed experiment 9 1 Types of Equations VarTran allows two types of equations the equations associated with output variables and the equations associated with objective functions What is the difference between these two types The equations associated with output variables are the direct result of the physical laws of nature They describe how the values of the inputs affect the value of the output For example the equation for the area of a rectangle is Area Width x Length For a particular rectangle if its length is 2 and its width is 3 then its area is 6 This equation applies to individual units of product Such equations will be referred to as value equations The equations associated with objective functions on the other hand do not deal with individual units of product Instead they deal with properties of the inputs and outputs For example one objective function is OF Minimum O1 Cpk O02 Cpk This equation is defined in terms of the Cpk of the output variables O1 and O2 It does not make reference to individual values of these two outputs Instead it references their behavior Such equations will be referred to as property equations Two other examples of property equations are given below Area Average Width
182. s of measure The numerator is the distance off target The denominator is the distance from target to the spec limit C measures the relative distance off target A value of 0 25 means the average is in the middle quarter of the spec A value of 0 50 means the average is in the middle half of the spec Smaller values of C are better Ideally a C of 0 25 should be obtained This in combination with a Cp of 2 results in Six Sigma Quality C can only be calculated in the event of two specification limits and target C only evaluates the average It does not consider the process variation In fact the average could be right on target C 0 but with a large variation resulting in a large number of units out of spec Variation is evaluated using Cp A third capability index C jointly evaluates the average and variation 84 Chapter 4 Capability Index Cx Cpk stands for capability of the process after adjusted for centering It is a measure of how well the process fits within the specification limits The formula for Cp is min USL p u LSL 30 Cx where USL is the upper specification limit LSL is the lower specification limit u is the process average and o is the process standard deviation The term USL u is the distance to the upper specification limit The term u LSL is the distance to the lower specification limit The numerator is the smaller of these two values which represents the distance to the nearest specifica
183. s on the output The figure below shows how adjusting arm gap from 0 10 to 0 11 can make the product robust to the weight of the paper Through interactions you can design your product or process to be robust to inputs that you cannot control 0 11 ARM GAP 0 10 OuNMZADPEO PAPER WEIGHT Using Interactions to Achieve Robustness Overview 41 You can design the process to be robust to materials and components On an injection molding machine the melt point of the material has a large impact on part quality By adjusting zone temperatures pressures and times the injection molder can be set up to be robust to the melt point This results in fewer process adjustments as the material changes Robustness can be used to make processes easier to control You can design a process to be robust to itself In the heat sealer tutorial it was determined that the primary cause of the seal strength variation was dwell time variation By adjusting the hot bar temperature and dwell time the process was made robust to the dwell time variation This resulted in less variation and higher quality By achieving maximum performance from your processes it can also result in fewer capital expenses You can design a process to be robust to the environment Suppose you are manufacturing gloves By identifying the proper temperatures and line speed the glove line can be setup so that is robust to daily fluctuations in room temperature and room humidity Th
184. s shown on the next page A C of 2 means the specification limits are twice as wide as the process variation A C of 2 means the specification limits are 2 the width of the process The larger C is the better C can only be calculated in the case where there are two specification limits Predicting Capability 81 USL aaa Interpreting Cp SL Each company has its own guidelines as to acceptable values of C However some general guidance will be given here First suppose that C is 1 At first appearances this might seem OK However a Cp of 1 only produces good product if the process is kept perfectly centered If the process shifts off target high levels of defects can result No process remains perfectly centered all the time Suppose the process shifts off target by 1 50 As shown below the average would now be only 1 50 from the specification limit resulting in 6 7 defective 1 50 LSL USL 6 7 Defective 60 30 0 30 6c When interpreting Cp it is important to keep in mind the need for an operating window An operating window of 1 50 is frequently allowed This operating window is based on the ability of a control chart to keep the process centered Suppose you were using an X R control chart Every hour a sample of 5 units is selected and plotted on the control chart The figure below shows the average number of samples required to detect different size shifts If the process is 20 off target it takes an average
185. s used to change the text color and size of the title and subtitle at the top of the plot Double clicking on the title displays this tab Plot Options x Tile Axis Style Y Axis x Axis Key Lines Of Axis Other MainTitle Main Tite F Std Dev Size v Color Bue sts Sub Title Sub Title 1 5x v Size Color Black z X Cancel Help The Main Title edit box is used to change the text for the main title It may be left blank if no title is desired The Main Title Size combo box is used to select the size for displaying the main title Normal size is the same size as the system font The default is twice the normal size You can select one of the values in the list or type your own value Entering a value of 1 3 results in the title being 1 3 times normal size The Main Title Color list box is used to change the color of the main title The Sub Title edit box Sub Title Size combo box and Sub Title Color list box are used to enter an optional subtitle By default it is blank They work the same as the main title controls Plot Options Dialog Box Axis Style Tab The Axis Style tab is used to change the axis style tic mark style tic mark size and the plot s aspect ratio Understanding the Results 129 Plot Options x Tite Axis Style y Axis XAxis Key Lines Other Axis Style Tic Style ae C Both Sides a C Inside Es fe 2 f Outside ces Tic Size oe
186. s were also obtained These standard deviations should be representative of long term production The results are shown below Also displayed are the targets for the inputs that are used in production Data on the Behavior of the Inputs Inputs Production Target Standard Deviation Temperature of the Hot Bar THB 190 F Dwl Time O We are now ready to perform a tolerance analysis using VarTran For a demonstration of VarTran s abilities start VarTran and open the file heatseal vta This is accomplished by selecting the Open menu item from the File menu Highlight heatseal vta and then click on the OK button The I O System window below is displayed showing the inputs and outputs The inputs are displayed on the left followed by the outputs on the right VO System Window Heat Sealing Process Name E Options amp Print h Copy Help To view the information entered for a variable double click on it in the O System window For example double clicking on the output SS displays the dialog box on the next page This dialog box is used to enter the equation from the response surface study along with the target and specification limits Press the OK button to close the dialog box Tutorials 13 Output Variable Dialog Box x Symbol ss Name 528 Strength Units zox Equation 27 9029646803808 0 0822867601868433 175 THB 9 00263649461868 0 75 DT 0 04588331 45964137 100 P
187. se tolerance of 3 4 The output Y log X X2 Then the worst case tolerance for Y is 0 602 0 778 since 0 778 max log X X 0 602 min log X X X 3 4 X 3 4 The first equation is read as the maximum of the function log X X2 for X between 1 and 2 and for X2 between 3 and 4 The minimum and maximum could be calculated using the simplex routine of the previous section Interval analysis can be used to determine the worst case tolerance for Y by simply performing interval operations on the worst case tolerances for the inputs This results in Y log Xi X2 log 1 2 3 4 log 4 6 log 4 log 6 0 602 0 778 The resulting interval gives the minimum and maximum Determining maximum and minumus cannot really be this easy can it The spoiler is that this approach only works if each input appears in the equation exactly once Take as an example Y X X In this case Y is always zero Therefore the maximum and minimum of Y when X is between 1 and 2 is also zero However Y X X 1 2 1 2 1 1 The resulting interval contains zero but does not equal zero In general interval calculations bound the minimum and maximum They may be wider than necessary when variables appear more than once in the equation This is because interval calculations treat each occurance of a variable as a new variable which can vary independent of all others The saving grace is the fundamental law of th
188. set from each other so that the caption of each window is visible A shortcut is to press shift F5 Arrange Icons Places the icons of all child windows that are minimized along the bottom of the parent window Menus and Toolbar 153 10 5 Options Menu options Number Standard Deviations for Intervals for Values Immediate Update Mode V Update Equations When Symbols Change vV Show Buttons iv Show Toolbar The Options menu is the fifth menu on the menu bar It is used to specify options for calculating intervals for values displaying buttons and the toolbar and for updating the child windows The available menu items are Number of Standard Deviations for Intervals for Values Display the Number of Standard Deviations for Intervals for Values dialog box for specifying number of standard deviations to use when calculating intervals for individual values The formulas appear in Sections 5 3 to 5 5 Number of Standard Deviations for Intervals for Values 45 v Number of standard deviations X Cancel Y Make Defauit Help The Number Standard Deviations combo box is used to enter a new value for the number of standard deviations Type a value or select one from the list Values are limited to the range 1 0 to 6 0 A value of 4 5 when combined with process tolerancing results in an interval for values that if it is inside the specification limits means that Six Sigma quality has been achieved A value
189. splays the message Simulation only applicable to statistical and process tolerances Tab 3 Worst Case Conditions The third tab shows the settings of the inputs that result in the minimum and maximum values of the outputs shown on the first tab Capability Study F ojx Flow Rate F Worst Case Tolerance Input Settings Producing Worst Case Conditions Input Minimum Maximum R Radius of Piston 0 0985 0 1015 L Stroke Length 0 495 0 505 B Black Flow 0 065 0 035 S Motor Speed 47 1 48 7 Inputs not listed are either constant or statistical Calculated Simulated Worst Case Conditions i amp Print Copy Help i Lock Predicting Capability 95 Pros and Cons of Worst Case Tolerancing Worst case tolerancing is completely safe It guards against the situation where all inputs are simultaneously at their worst case conditions If the inputs meet their tolerances the output is assured of being within its worst case tolerance Worst case tolerancing is more expensive than statistical tolerancing Given the same tolerances for the inputs worst case tolerancing results in a wider tolerance for the output This in turn requires more frequent tightening of the inputs tolerances in order to meet the specifications driving up costs Take as an example the case of overall height of a part created by stacking four identical components of top of each other Suppose the specification limits are from 3 9 to 4
190. sults and much more Chapter 10 gives a complete description of all the menus Entering Information 49 Toolbar The toolbar contains buttons serving as shortcuts for the most commonly used menu items To see what a particular button does hold the mouse cursor over the top of the button A description of the button will appear Interior The interior initially contains the O System child window Later we will encounter other child windows that can also be displayed in the interior If a child window extends outside the interior of the main window scroll bars are provided for shifting the child windows up down and left right Status Line The status line is at the bottom of main window It contains a description of the current status of the session If there are input errors the status line will initially display a list of the variables containing errors Handling errors is covered in Section 4 3 A sizing border surrounds the window except when the main window is maximized Dragging the border causes the window to be resized 4 2 Adding Variables The first step is to add the different variables Begin by clicking the Add Variable button on the toolbar This button serves as a short cut to selecting the Add Variable menu item on the Variables menu Variables Add Continuous Input Variable Add Category Input Variable Add Output Variable Add Objective Function Duplicate Variable Category Input Variabl
191. supplier and cannot be adjusted by the designer i Variables Summary Tables Summary Tables Targets and Tolerances of Inputs Input Type Tolerance Category LSL USL Average and Variation Requirements Interval for Values 4 55D R Statistical 0 1737 i 0 0015 0 1722 0 1752 Average 0 1737 Std Dev lt 0 0005 0 17145 0 17595 Statistical 0 40942 i 0 005 0 40442 0 41442 Average 0 40942 Std Dev lt 0 0016667 0 40192 0 41692 aa Statistical 17 064 i 0 5 16 564 17 564 Average 17 064 Std Dev lt 0 16667 16 314 17 814 za Statistical 0 05 0 015 0 035 0 065 Average 0 05 Std Dev lt 0 005 0 0275 0 0725 Ti Glerance S Input Tolerances Outputs ARA Entries Continuous Inputs Entries Category Inputs Entries Outputs Entries Objective Functions Simulation Ir EE amp Print B copy Help Dok Tutorials 9 Tightening Tolerances What if the desired performance had not been achieved The next step would be to tighten tolerances This generally requires using more expensive components and processes The question is which tolerances and by how much In deciding which tolerances to tighten you should consider both the cost and the effect of tightening each tolerance To determine the effect of tightening the tolerances on the different inputs click on the Variation Pareto Chart button on the toolbar The window shown below is displayed bl Analysis Variati
192. t TEMP Target Frequently equations are also fit to the standard deviations of the outputs as well This might result in a second property equation SS SD c0 cl TIME Target c2 TEMP Target c3 TIME Target 2 c4 TEMP Target 2 c5 TIME Target TEMP Target While the equation for the seal strength average is in fact a property equation it generally closely approximates the underlying value equation However there are differences so this introduces some error For the area equation on the previous page the value equation and the property equation for the average are identical so no distinction is needed For the pump example in Chapter 2 the value equation is F 16 388 pi R42 L B S The property equation for the average is F Average 16 388 pil R A 2 L A B A S A 0 5 32 776 pif S A L A R SD 42 where A denotes average and SD denotes standard deviation These equations are not identical although when plotted they are very similar Following a response surface study VarTran can be used to two ways First it can be used to plot and optimize the resulting property equations for the average and standard deviation This approach is referred to as the dual response approach and covered in Section 9 2 You can also treat the property equation for the average as a value equation and perform a robust tolerance analysis This approach is covered in Section 9 3 140 Chapter 9 9 2
193. tabs can be displayed automatically by double clicking on the appropriate part of the plot For example to display the Title tab double click on the title Title Changes title text and style Axis Style Changes the axis style tic mark style and the plot s aspect ratio Y Axis Changes y axis label scale style and orientation X Axis Changes x axis label scale and style Key Used to specify whether to display the key and for setting style options Lines Changes the labels values and styles associated with the lines Off Axis Changes the value and labels of the off axis variables The off axis variables are those continuous input variables not displayed on an axis This tab only appears if there are off axis variables Comments Changes the comments displayed on the plot The comments describe the selected levels of category input variables This tab only appears if there are category input variables Other Changes the resolution of the plot whether to automatically update axis scales and the number of lines 128 Chapter 8 When you are done click the OK button to close the dialog box and update the plot An alternative is to press the Enter key Clicking the Cancel button instead closes the dialog box without saving any changes An alternative is to press the Esc key The Help button provides help concerning the Plot Options dialog box Plot Options Dialog Box Title Tab The Title tab i
194. tailed name or description of the variable It can contain any characters and be of Significant Digits any length While the symbol should be short the name should be pe eens descriptive 0 Digits 1 Digits The Units edit box can optionally be used to enter the units of a measure associated with this variable It can be any combination of 4 Digits characters and any length 5 Digits 6 Digits The xx xx button displays the popup menu to the right for CONE specifying the number of digits to report for this variable The aa default is five significant digits You can choose between 2 12 10 Digits significant digits and from 0 to 12 digits after the decimal point 11 Digits 12 Digits When VarTran displays numbers it discards trailing zeros Tolerance Group Box The Target edit box is used to specify the current target or nominal Capability studies variation Pareto charts and simulations are all performed at the target The target must be between the minimum and maximum targets specified in the Optimizations Options for Target group box It must be a constant expression meaning it can include operators and functions but not variables Some examples are 1 5 2 3 1 4 10 5 and Log 30 Further information on expressions can be found in the glossary To reference this field in other edit boxes use the symbols Target or T The A edit box is used to specify the distance the tolerance exte
195. tarting with the low cost tolerances to achieve a robust design use Find Optimal Targets dialog box in Section 6 1 e Determine if the capability meets specified level of quality perform capability study per Section 5 1 e If not determine cause of variation and tighten its tolerance perform Pareto chart of variation per Section 7 1 e Repeat previous 3 steps until the specified level of quality is achieved Key to this process is gathering good information on the behavior of the inputs and the effects of the inputs on the outputs Capability studies can be used to study the behavior of the inputs Equations describing the effects of the inputs on the outputs can always be obtained using designed experiments While gathering all this information takes time without such knowledge one is reduced to trial and error With this information mistakes can be avoided and the time to market reduced Starting with the low cost tolerances helps to ensure the design s manufacturability The use of robust design methods and the selective tightening of tolerances only when necessary reduces costs The result is high quality and low cost Further design issues are found earlier in the design process resulting in fewer design iterations This speeds up time to market 48 Chapter 3 Entering Information This section describes how to enter information into VarTran It also describes how to display this information print it edit it save it an
196. te Help Transferring the equations from your response surface package is accomplished using the Paste button This automatically constructs the inputs as well For each input you will need to enter the current target plus the minimum and maximum targets The minimum and maximum targets should be set to the edges of the study region The current target can initially be set to the middle of the study region It will be optimized later The tolerances do not have to be entered Using VarTran with Designed Experiments 141 In order to find the optimal targets a third objective function should be constructed combining the average and standard deviation One choice is to use Cpk as shown below Objective Function Dialog Box Minimum SS_M 20 32 SS_M 3 SS_D You can then use the Find Optimal Targets dialog box to maximize the value of the objective function Cpk Once the optimization is completed use the summary table to view the new settings of the inputs and a capability study to view the expected performance of the output These are shown below Summary Tables et tom oer Oe ec Capability Study Cpk Objective Function Cpk Value 1 9457 142 Chapter 9 You can also obtain effects tables and plots of these objective functions to further understand their relationships Many design of experiment packages now have some ability to find optimal targets However most are based on desira
197. the right tail is longer A negative value means the left tail is longer A statistical tolerance is a tolerance that specifies requirements for the process variation while assuming a centered process A statistical tolerance is determined by its target standard deviation and distribution A statistical tolerance is a special case of a process tolerance with no operating window i e minimum average maximum average target Unique identifier for variable It is used in equations to represent the variable It can be any combination of letters the digits 0 9 and underscores so long as the first character is not a digit It can be of any length but is best limited to 1 4 characters Examples of valid symbols are A Al al Res_1 Examples if illegal symbols are AB No embedded blanks allowed A 1 1_ Digit not allowed as first character Illegal character Jointly refers to methods of predicting the behavior of an output based on the behaviors of inputs and the effect of the inputs on the outputs Several approaches include Monte Carlo simulation propagation of errors and statistical tolerancing Input variable primarily responsible for inducing or causing the variation of an output variable While all the input variables are potential sources of variation the Pareto principle implies that most of the variation is the result of just 1 or 2 inputs A worst case tolerance specifies a minimum and maximum value for a unit of prod
198. tinuous input variable is added a Continuous Input Variable dialog box like the one shown below is displayed This dialog box is used to enter and change information concerning the continuous input variable Continuous Input Yariable Dialog Box x Symbol R Name Radius of Piston Units finches vox Tolerance Graphic Symbol Statistics a Radius of Piston R z fo o015 Statistical Tolerance Type of Tolerance LSL T USL C WorstCase Tolerance Values Statistical Tolerance Variation Process Tolerance Average Variation Variation Distribution Nomal ss iatiqn USL LSL 6 C Standard Deviation p EE Min 0 01 Max fi V Include In Optimizations X Cancel f Delete EJ Options Help The Symbol edit box is used to enter the symbol The variable s symbol is used to represent the variable in equations Each variable must have a unique symbol It can be of any length but is best limited to 3 characters or less The symbol must consist of letters the digits 0 9 and underscores It cannot begin with a digit Some examples of valid symbols are A _1 Al and A_1 1A and 1_are not valid symbols because they begin with a digit A B is not a valid symbol because it contains an illegal character A B is not a valid symbol because it contains an embedded blank which is an illegal character Entering Information 51 The Name edit box is used to enter a more de
199. tion limit The denominator is the half the width of the process Cpk compares the distance to the nearest specification limit to the process variation LSL 30 u LSL Calculating Cpk When C is 1 the process average is exactly 30 from the nearest specification The figure below shows several processes with Cpk of 1 In each case one or both ends of the normal curve is touching the specification limits USL m LSL Processes with Cpk of 1 When C is 2 the distance from the process average to the nearest specification limit is twice the process variation The figure below shows several processes with Cpk of 2 Predicting Capability 85 USL Processes with Cpk of 2 Cpk is always less than or equal to Cp They are equal only when the process is perfectly centered The figure below shows a series of processes with a Cp of 2 As the process average shifts further off target the value of C decreases USL Processes with C of 2 While C requires two specification limits only one specification limit is required to calculate Cpk If only a lower specification exists Cp is u LSL C i 30 If only an upper specification limit exists Cp is USL wu ee 35 When two specification limits exist Cp is the lesser of these two quantities When there two specification limits and the target is in the middle of specification limits the following special formula can be used to calculate Cpx Cx 2 C
200. tive e Window Cp F Average C Cpk 1300 L 0 5 C ZScore S 48 2 B 0 05 C Cy C Cpm 600 C Root MSE Dist Target Dist Spec 100 a6 0 505 C Lower Value R C Upper Value 7 E Print Copy 2 Help F1 The Type of Plot radio buttons are used to specify whether a line interaction or contour plot is desired Line plots are easiest to interpret but can be misleading in the event of strong interactions Generally you should get interaction or contour plots for those pairs of input variables involved in strong interactions Then obtain line plots for the remaining input variables not involved in any interactions Next you need to specify the variables to plot The Output list box is used to select the output variable or objective function to plot You also need to specify the continuous input variables A line plot requires a single input variable Interaction and contour plots require two input variables The Input 1 list box is used to select the first input variable For all three types of plots this input variable appears along the bottom axis The Input 2 list box only appears if you have selected an interaction or contour plot For interaction plots several Understanding the Results 125 lines appear on the plot representing different values of this input For contour plots this input appears on the y axis The Characteristic radio buttons are used to specify the characteristic of the specified output v
201. trategy of setting the targets of the other three inputs to minimize the CV Then adjust the target of motor speed to achieve the desired target This important information indicates that if in the future we wanted to design the pump to deliver a different flow rate or wanted to add an adjustment for flow rate we should use motor speed to make the adjustment To gain further understanding it would be useful to plot the effects of piston radius R stroke length L and valve backflow B on the CV as well as plot of the effect of motor speed on the average Using the effects table and plots to develop a better understanding requires patience but can yield enormous benefits 124 Chapter 8 8 2 Plots Dialog Box The Plots dialog box is used to quickly view different plots of the effects of the inputs on the output variables and objective functions Plots of particular interest can then be transferred to their own window and modified as desired The effects table in the previous section is useful in determining which plots will be most meaningful To display the Plots dialog box click on the amp button on the toolbar This is a shortcut for the Plots menu item on the Analysis menu The dialog box shown below will then appear Plot Dialog Box me x Type of Plot Variables to Plot Characteristic Line Plot C Value Interaction Plot Output F El Average Contour Plot Input 1 IR C Std Dev 7 C Defec
202. trates that VarTran can be used to 1 Optimize the process for production rather than study conditions 2 Identify the root cause of the variation 3 Be used to tighten tolerances to achieve the desired performance even when the response surface fails to find targets that achieve the desired result Any time a response surface study is run one should use VarTran to complete the task of establishing operating windows 18 Chapter 2 The response surface tolerance analysis approach to robustness offers a third alternative to the dual response and Taguchi inner outer array approaches The response surface tolerance analysis approach has all the advantages of Taguchi s approach without the added cost of running a more complex study It also has another significant advantage over both the dual response and Taguchi approaches The estimates of the variation resulting from the response surface tolerance analysis approach are generally more accurate and precise than estimates resulting from the other two approaches The response surface tolerance analysis approach is the best demonstrated practice for achieving robustness The response surface study is only required if an equation is not otherwise available If the equation is already known VarTran can be used directly For the infamous Wheatstone bridge example used by Taguchi the equation was first derived This equation can be entered directly into VarTran and the problem solved within minutes
203. treme value The probability of selecting four components all at the same extreme is therefore extremely small Worst case tolerancing guards against a worst case scenario that can only occur if the variation is zero As a result worst case tolerances for the output are wider than will be experienced in practice This can result in the unnecessary tightening of tolerances for the components driving up costs Worst case tolerancing while safe is overly expensive Statistical tolerancing results in much narrower tolerances for the outputs As a result the tolerances for the inputs do not require as much tightening resulting in lower costs However there is a risk involved Statistical tolerancing assumes that the process producing the components remains centered at the selected target No process remains perfectly centered all the time With statistical tolerancing off center processes can result in components that meet the tolerance for the individual components but when assembled together into a finished unit result in a finished unit that exceeds specifications As a result statistical tolerances for the output can be narrower than will be experienced in practice As such statistical tolerancing might be better called best case tolerancing Statistical tolerancing while less expensive is more risky The real answer generally falls somewhere between the worst case and statistical approaches This long recognized dilemma has lead to a number
204. uct Use of worst case tolerances is safe However because they guard against a worst case scenario that is unlikely if not impossible worst case tolerancing results in the setting of tolerances tighter than necessary As a result worst case tolerancing unnecessarily drives up costs A worst case tolerance is a special case of a process tolerance with zero standard deviation allowing all units to be at either extreme of the operating window Glossary Z score Measure of how close the process average is to the specification relative to the variation For a single specification limit its formula is LSL H or Z ITER aol Oo Oo Z Score where USL is the upper specification limit LSL is the lower specification limit u is the average and o is the standard deviation In the 1 sided spec case the Z Score is 3 times Cpr The z score is independent of the units of measure See Section 5 2 Glossary 167 References Bender A 1962 Benderizing Tolerances A Simple Practical Probability Method of Handling Tolerances for Limit Stack Ups Graphic Science Dec Cox N D 1986 How to Perform Statistical Tolerance Analysis Milwaukee ASQC Quality Press Evans D H 1975 Statistical Tolerancing The State of the Art Journal of Quality Technology 7 1 1 12 Olsson D M 1974 A Sequential Simplex Program for Solving Minimization Problems Journal of Quality Technology 6 1 53 57 Olsson D M
205. uickly view a variety of plots You can save a copy of a plot by clicking on the Windows button Pressing this button displays a copy of the plot in a separate Plot window This allows multiple plots to be viewed at the same time The Plot window is described in the next section Once in the Plot window almost every aspect of the plot can be modified including changing the values of the inputs variables not included in the plot The Plots dialog box differs from most dialog boxes in that it may be left open and multiple copies can be opened at once It is in fact a child window which can be minimized shown in the background etc The Plot dialog box draws a red X through the plot when it is not current Clicking on the dialog box will cause it to redraw itself To close the Plots dialog box click the Close button or press the Esc key Other buttons are Fl ox Print Prints the plot Shortcuts are to click the Print button on the toolbar or to select the Print menu item on the File menu h Copy Copies the plot to the clipboard in graphic format Shortcuts are to click the Copy to Clipboard button on the toolbar or to select the Copy to Clipboard menu item on the File menu PE cel shah Displays help on using this window Short cut is to press F key 126 Chapter 8 8 3 Plot Window The Plot window is a child window used to display plots To create a Plot window click on the Windows button in the Plots dialog box as described in the
206. unction For each of these 5 packages specific instructions are available in the help system for generating the equation and copying it into the clipboard ready for pasting Start by clicking on the Help button and then click on the package you are using to locate these instructions Paste Equation Dialog Box Ea The equation you wantto paste and convert must first be copied to the clipboard from the other application Once in the clipboard click the Paste button to the left The equation will appear below Selectthe source and options and click the Convert button Equation 40 0394110958585 32 9872894925798 SILICA 1 78807650994367 SILANE 21 6130769261925 SULFUR SILANE 50 SILICA 1 2 1 02500000000005 SULFUR 2 3 SILICA 1 2 28 4999999999925 SULFUR 2 3 SILANE 50 1 57499999999915 SILICA 1 2 SILICA 1 2 16 0384036168402 SILANE 50 SILANE 50 0 0344710561510968 SULFUR 2 3 SULFUR 2 3 6 28848527241052 f r Source Excel C Mathematica Statgraphics fe JMP C Minitab V Create Inputs JA Convert X Cancel Help Once the equation is in the clipboard click the Paste button This copies the equation from the clipboard into the Equation edit box The Equation edit box displays the equation as copied from the clipboard before any conversion This control is read only Entering Information 69
207. ution i e more units fall outside the specifications than predicted using the normal distribution Smaller values indicate lighter tails than the normal distribution i e fewer units fall outside the specifications than predicted using the normal distribution For example the uniform distribution with no tails has a kurtosis of 1 8 Objective functions are used to simultaneously optimize several output variables They can reference any of the characteristics of the outputs and inputs One example is Minimum O1 Cpk 02 Cpk where O1 and O2 are output variables Maximizing this objective function ensures all the outputs have acceptable values of Cpk Objective functions can also be written for Taguchi s signal to noise ratios allowing them to be maximized Goal of VarTran Jointly refers to optimizing the average and minimizing the variation It is the outputs of the input output system for which optimization and variation reduction is desired Depending on the nature of the outputs optimization of the average can mean to maximize the average seal strength minimize the average harmful emissions or the center the average on target fill volume In all three cases variation must also be reduced to ensure all units are within specifications Jointly refers to output variable and objective function Part of input output system Output of I O system that is a measurable characteristic Examples include seal strength of a heat sealer
208. uts behave the same Overview 47 Some inputs are best described using statistical tolerances Other inputs are best described using worst case tolerances Some inputs don t adequately fit either of these two behaviors and are better represented with process tolerances VarTran allows you to mix all these different behaviors The type of tolerance specified for each input should be that which best describes its behavior When specifying the tolerances of the inputs be sure to include variation due to manufacturing and materials variation due to wear and deterioration and variation due to usage and environmental conditions This helps to ensure that all units of product perform as required over their entire useful life and under all conditions they might encounter 3 7 Robust Tolerance Analysis VarTran you can set targets and tolerances on the inputs to optimize average and reduce the variation of the outputs It is recommended you use the strategy below called a Robust Tolerance Analysis A Robust Tolerance Analysis is a systematic approach to setting targets and tolerances for the inputs to achieve a specified level of quality at the lowest possible cost The process involves e Determining the key outputs and their specifications the key inputs and their low cost tolerances and the relationship between the inputs and outputs This is the information that must be entered into VarTran to perform an analysis e Find the optimal targets s
209. val Method This section describes the interval method of optimization Itis for information only and can be skipped The interval method of optimization is based on a new branch of mathematics called interval analysis Interval analysis redefines the arithmetic operators for use with intervals For example let a b and c d be two intervals Then the addition operator is a b c d a c b d The interval a c b d is the set of all real values x y where x is from the interval a b and y is from the interval c d The interval log function is defined to be log a b log a log b The interval log a log b is the set of all real values log x such that x is from the interval a b Likewise interval operators and functions can be defined as extensions of the real operators and functions Real arithmetic is a special case of interval arithmetic Real numbers can be represented by intervals of length 0 For example the real number 2 is equivalent to 2 2 In this case the interval functions and operators reduce to the real operators and functions For example 2 3 2 2 3 3 5 5 5 Optimization and Robustness 107 log 6 log 6 6 log 6 log 6 1 792 1 792 1 792 Interval analysis has important applications in optimization It can frequently be used to directly calculate the minimum and maximum Take the case of two inputs X which has a worst case tolerance of 1 2 and X2 which has a worst ca
210. variable This field is initially set to the current target Changing the values of the off axis variables can dramatically alter the shape of the plot if the variable changed interacts with the plotted inputs When changing off axis variables automatic scaling should be on to avoid the lines or contours from shifting off the plot Automatic scaling is turned on and off using the Other tab Plot Options Dialog Box Comments Tab The Comments tab is used to change the comments displayed to the right of the plot VarTran uses these comments for displaying the selected levels of the category input variables Double clicking on a comment displays this tab The Size of Comments combo box is used to select the size for displaying the comments The default is normal size The Display Comments check box is used to specify whether to display comments on the key Generally this information is desired but can be removed The Comment tabs are used to select which comment to change The Comment edit box is used to change the text of the comment on the selected tab This does not change the selected category of the category input variables That must be done using the Category Input dialog box 134 Chapter 8 Plot Options x Title Axis Style Y Axis XAxis Key Lines Of Axis Comments Other Size of Comments Normal gt V Display Comments Comment M New X Cancel Help Plot Options Dialog Box Other Tab
211. ve as statistical tolerancing process tolerancing is as inexpensive as possible while maintaining guaranteed safety Process tolerancing has another distinct advantage When performing statistical tolerancing statistical tolerances must be specified for all inputs When performing worst case tolerancing worst case tolerances must be specified for all inputs However when performing process tolerancing statistical tolerances can be specified for some inputs worst case tolerances for others and process tolerances for still others No longer are you forced to select the same type of tolerance for all inputs Process tolerancing represents a unified approach to tolerancing encompassing all the previous approaches A process tolerance where the minimum and maximum averages are equal represents a statistical tolerance A process tolerance where the maximum variation is zero represents a worst case tolerance This is why VarTran is so easily able to handle all three approaches to tolerancing Predicting Capability 101 5 6 Objective Functions Capability studies can also be performed for objective functions In the event that statistical tolerances are specified for the inputs the capability study simply gives the value of the objective function as shown below Capability Study T loj x Taguchi Loss Value 0 27638 Cale Worst Case Conditions E amp Print I Copy Help C Lock it If worst case tolerances or process
212. verall height is from 3 92 to 4 08 This meets the specification limits for overall height The process tolerance falls between the worst case and statistical tolerances and is more representative of what will actually occur in production Process tolerances are obtained by specifying requirements for the processes that produce the components rather than for the individual components Process tolerances are specified by giving a minimum average a maximum average and a maximum standard deviation They do not assume that the standard deviation is zero nor do they assume that the process is perfectly centered To see how a process tolerance was specified for D1 click on D1 in the I O System window The dialog box shown below is displayed In the example above it was specified that the average height must be in the range 1 0 01 It was also specified that the standard deviation must be less than or equal to 0 006667 This was done by specifying a standard deviation equal to 1 9 the width of the tolerance and an operating window for the average of 1 5 standard deviations The three normal curves in the graphic represent the minimum average on target average and maximum average all with the maximum standard deviation This results in the same overall tolerance for the individual components as before namely 1 00 0 03 Tutorials 25 Continuous Input Variable Dialog Box x Symbol D1 Name Dimension 1 Units inches OK Tolerance Graph
213. vidual tolerances fits inside specification limits when calculated using 4 5 standard deviations In each case meeting the criteria above ensures six sigma quality throughout the entire operating window specified for the inputs Tab 2 Simulation Results The second tab shows the results of three simulations performed at the two worst case case conditions and when all input averages are at their targets To perform a simulation use the Simulation menu item on the Analysis menu as described in Section 8 4 The worst case conditions are determined analytically and displayed on the third tab Predicting Capability 99 Capability Study F oO x Flow Rate F Process Tolerance LSL USL willl iil 8 0826 ml min 11 924 Characteristic On Target Worst Case Average Minimum 9 9967 9 3407 Maximum 10 666 Standard Deviation 0 27668 0 27956 Cp 1 20 1 19 Ce 0 00 0 67 Cpk 1 20 0 40 Def Rate 330 dpm 10 7 Z Score 3 41 1 24 Sigma Level 3 41 2 74 Interval for Values 8 0826 11 924 4 5SD Calculated Simuleted amp Print Copy Help L Lock lz If no simulation was performed the message No Simulation Data will appear Otherwise the results will appear as above The results are almost identical to those for the analytic approach on the first tab with the following exceptions e At the top histograms of the three simulations are shown e The defect rate is determine
214. wW Pareto Chart of Yariation Sensitivity Analysis E Simulation amp Lock All Unlocked Windows EFF Effects Table P lt Plots An example of the Pareto Chart window is shown on the next page It has 4 tabs containing the results Click on the X in the upper right corner to close this window The Pareto Chart window automatically updates itself when information affecting the displayed results is changed When the program is in immediate update mode the updates occur immediately following the change being made If the program is in delayed update mode the Pareto Chart window draws a red X through itself when it is not current Clicking on the window will cause it to update and redraw itself The update mode is selected using the Options menu Double clicking on the title in the window displays the variable s Output Variable dialog box Double clicking on any of the inputs displays their Continuous Input Variable dialog box These are shortcuts to selecting the Edit menu item on the Variables menu Tightening Tolerances 111 Pareto Chart F o x Flow Rate Interval for Values 8 5061 11 5 3SD 100 Percent Reduction The buttons and controls at the bottom can be hidden by removing the check from the Show Buttons menu item on the Options menu When hidden the popup menu can still be used to perform the functions of these buttons and controls To display the popup menu right mouse click over
215. way exists to describe the behaviors of the inputs This section explores three different approaches called worst case statistical and process tolerances Tolerances are typically given in the form of an interval For example it might be specified that the tolerance for piston radius is 0 1 0 003 This means that the radius should be between 0 097 and 0 103 A unit exceeding this tolerance is called a defective unit These limits represent the limits for individual values and will be denoted by LIV lower end interval for values and UIV upper end interval for values respectively While a standard notation exists for expressing tolerances the interpretation of these tolerances can vary greatly 3 6 1 Worst Case Tolerances For a worst case tolerance you must specify three values the target T delta upper A and delta lower A In the above example T 0 1 and A A 0 003 The upper and lower ends of the interval for values are determined as follows UIV T A 0 103 LIV T A 0 097 Worst case tolerances are represented by the symbols shown below 0 1 0 003 S gt 0 097 0 103 Worst Case Tolerance Worst case tolerances are interpreted as meaning that the input can take any value over the specified range 3 6 2 Statistical Tolerances A statistical tolerance is specified using two values the target T and the standard deviation o The upper and lower intervals for individua
216. without needing a designed experiment Tutorials 19 2 3 Dimensional Stack up Problem Suppose that four identical components are to be stacked on top of each other and that we are interested in controlling the overall height The height will be denoted by the symbol H and will be referred to as the output variable We desire an overall height of 4 0 The customer will not tolerant a deviation from target of more than 0 1 This results in a lower specification limit of 3 9 and an upper specification limit of 4 1 The four components are identical Their manufacturing tolerance is 1 00 0 03 or stated another way from 0 97 to 1 03 Essentially all components fall within these tolerances The heights of the four components will be denoted D1 D2 D3 and D4 and will be referred to as the inputs The overall height is equal to H D1 D2 D3 D4 The question we want to answer is whether the overall height requirement will be met There are several different approaches to answering this question These are illustrated below Worst Case Tolerancing Approach One commonly used approach is called worst case tolerancing For a demonstration of VarTran s abilities in calculating worst case tolerances start VarTran and open the file stack wc vta This is accomplished by selecting the Open menu item from the File menu Highlight stack wc vta and then click on the OK button The results of a worst case tolerance analysis are display
217. y fits between the specification limits preheating of the material is not necessary SUMM Variables Summary Tables Targets and Tolerances Statistical 191 14 i 6 185 14 197 14 Average 191 14 Std Dev 2 Statistical 0 59579 0 24 0 35579 0 83579 Average 0 59579 Std Dev lt 0 08 P Statistical 55 28 43 52 28 58 28 Average 55 28 Std Dev lt 1 Statistical 76 41 12 Average 75 Std Dev lt 4 put Tolerances Outputs Entries Continuous Inputs Entries Category Inputs Entries Outputs Entries Objectivi coal E Print Copy 2 Help Tutorials 17 Cause of the Variation We have still not answered the question of what is causing the variation This is especially important when the desired objective has not been achieved The answer can be obtained by clicking on the Variation Pareto Chart button on the toolbar The results are shown below Dwell time variation turns out to be the largest source of variation Using VarTran one could enter a new tolerance for dwell time re optimize the process and then view the new predicted capability bl Analysis Variation Pareto Chart Pareto Char SS ioj x Seal Strength Interval for Values 22 429 29 571 4 5SD 100 Percent Reduction 0 DT MT THB P Inputs Percent Reduction Transmitted Variation Table Plotfor Input E Ontions E Print Help B Lock Summary The above example illus
218. ze and paper source Print Setup 21x Printer Name WOFFICE Lexmark Tray 2 v Properties Status Ready Type Lexmark Optra 5 1620 Where WOFFICE Lexmark Tray 2 Comment Paper Orientation Size Letter 814x11 in x Portrait passed Source Auto Select v C Landscape Network Cancel The Name list box is used to select which printer to print to Clicking on the Properties button displays a dialog box for modifying printer options such as landscape versus portrait mode and the number of copies When done click on the OK button To exit without changing the selected printer click on the Cancel button Copies the contents of the displayed tab of the active child window to the clipboard Shortcut is Ctrl C Exits the program First asks whether to save the current session Same as double clicking the system menu button icon in top left comer or clicking the close button x in top right corner A shortcut is to press Alt F4 Menus and Toolbar 149 10 2 Variables Menu Variables Add Duplicate Variable Edit k Modify All Tolerances 3F VO System Window W Summary Tables E Notes Window The Variables menu is the second menu on the menu bar It is used to add and edit variables and to display the information entered The second group of four buttons on the toolbar provides short cuts for those menu items with icons The available menu items are Add Dupl

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