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# Basic Statistical Meth ISYE 2028

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This 0 page Class Notes was uploaded by Maryse Thiel on Monday November 2, 2015. The Class Notes belongs to ISYE 2028 at Georgia Institute of Technology - Main Campus taught by Jye-Chyi Lu in Fall. Since its upload, it has received 10 views. For similar materials see /class/234204/isye-2028-georgia-institute-of-technology-main-campus in Industrial Engineering at Georgia Institute of Technology - Main Campus.

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Date Created: 11/02/15

Fractional Factorial Design Confounding Pattern or Alias Structure Example 1 2quot3 1 design with 3 12 X3 X1 X2 or C AB textbook notation Design layout or design matrix is given in page 335 of the DOE handouts See page 334 for a graphic presentation ofthe fractional factorial design also given in lecture notes on 082802 MA A A 0501 l 000 Multiply X3 to both sides of X3 X1 X2 Left hand side LHS becomes X3 X3 which is the same column of 1 and 1 multiplied by itself Thus it becomes a column ofall 1 s ie X3 X3 1 1 1 Right hand side RHS becomes X1 X2 X3 We thus obtain a generating relationship I X1 X2 X3 or 123 in short X3 X1 X2 or 3 12 is called the generator ofthe design Because there are three letters in the RHS ofthe generating relationship the resolution of the design is 3 From statement 4 ifone multiplies X1 to both sides of the generating relationship the resulted relationship becomes X1 X1X1X2 X3 X2 X3 that is 1 23 in short Similarly one can obtain 2 13 and 3 12 relationships You will find that the column of X2X3 ie 23 of the design layout will be the same as the column X1 verify it yourself Similarly X2 column is the same as X1X3 and X3 column and the same as X1 X2 10 In summary the confounding pattern ofthe design in this example is 123123213and312 11Then the Estimable Effects are Average for the column ofone s X1 which is equal to X2X3 X2 X1 X3 and X3 X1X2 ie only 4 estimable effects from the experiment of4 runs Example 2A 2quot6 2 resolution IV design with generators 5 123 and 6 234 1 N w h Design layout matrix 5 123 6234 12 35 1346 2456 1 1 1 1 71 71 71 71 71 71 71 71 1 1 1 1 1 1 1 1 71 71 71 71 71 71 71 71 1 1 1 1 1 1 1 1 71 71 71 71 71 71 71 71 1 1 1 1 1 1 1 1 71 71 71 71 71 71 71 71 1 1 1 1 Note 12 35 1346 2456 confounded Generating Relationships I 1235 2346 1456 where the last relationship I 1456 is from the multiplication of the first two relationships 1235 and 2346 Note that 2323 l Resolution in this design is IV 4 the minimum number of letters in the generating relationships is 4 Note that resolution IV design in this example is stronger than the resolution lll design in the first example This will be explained after the discussion of confounding patterns Confounding Pattern alias structure including the estimable effects 1 Background Based on the generating relationship we knowthat 1 235 moved 1 to the other side of the equality or multiply 1 in both side of the first equality Moreover we also knowthat 12 35 2 List of all 64 2A6 Effects Confounded to each other Since there are 16 runs with 16 data points there will be only 16 estimable effects marked in yellow below Average 1 estimable effect 23 done quot r 39 25 done 34 done 35 done 36 done 45 done 46 done 56 done ThreeFactor Interactions 2 estimable effects 123 done 125 done 134 done 135 done 136 done 145 done 146 done 156 done 234 done 235 done 236 done 245 done 246 done 256 done 345 done 346 done 356 done 456 done FourFactor Interactions all done 123445 1235 123614 124534 124613 125624 1345 24 1346 12 1356 34 1456 2345 56 2346 2356 45 2456 35 3456 25 FiveFactor Interactions all done 12345 4 12346 1 12356 6 12456 2 13456 3 23456 5 SixFactor Interactions done 123456 15 Note that every one of these 16 estimable effects have four items confounded to each other That is all 2A6 64 effects are confounded to each other in 16 sets of four effects Example 2B 2quot6 2 resolution lll design with generators 5 123 and 6 1234 1 Design layout matrix 5 123 61234 Run X5 X6 1 1 2 71 3 71 4 1 5 71 6 1 7 1 8 71 9 71 1O 1 1 11 1 12 lt 71 13 14 15 16 2 Generating Relationships 1235 2346 456 3 Resolution of this design is lll since I 456 has only 3 letters 4 Please work out all the confounding patterns for effects lower than fourfactor interactions ie include average main effects two and threefactor interactions and disregard the four five and sixfactor interactions Answer Review Lecture notes on 012104 and 012304 provide the Example 2 which is a resolution IV 2quot62 design with generators 5 123 and 6 234 Example 3 2quot6 2 design with generators 5 12 and 6 234 5 Generating Relationships I 125 2346 13456 where the last relationship I 13456 is from the multiplication of the first two relationships 125 and 2346 Note that 22 an Resolution in this design is III 3 the minimum number of letters in the generating relationships is 3 Note that resolution III design in this example is weaker than the resolution IV design in Example 2 presented in the class with another set of generators 5 123 and 6 234 This will be explained more clearly after the discussion of confounding patterns N Confounding Pattern alias structure including the estimable effects 3 Background Based on the generating relationship we knowthat 1 25 moved 1 to the other side of the equality or multiply 1 in both side of the first equality Moreover we also knowthat 2 15 4 Since there are 16 runs with 16 data points there will be only 16 estimable effects marked in yellow below Only effects lower than fourthorder interactions are presented below Average counted as 1 estimable effect TwoFactor Interactions 9 estimable effects 12 done I39 15 done 25 done 34 done a E 413113 g e In 41 46 done Remark We have all 16 estimable effects now There should not be any more Let us check it out below with threefactor interactions Note 1 Only the effects involve 5 has a minus sign Note 2 Compare with the 2quot62 5 123 and 6 234 resolution IV design presented in the previous lecture this resolution lll design is weaker in the sense that some main effects are confounded with twofactor interactions which will not happen in resolution IV design where main effects will only confounded with threefactor interactions which is less important as the twofactor interactions ThreeFactor Interactions 0 estimable effects 123 done 124 done 125 done 126 done 134 done 135 done 136 done 145 done 146 done 156 done 234 done 235 done 236 done 245 done 246 done 256 done 345 done 346 done 356 done 456 done Example 4 2quot6 2 design with generators 5 123 and 6 1234 5 Generating Relationships l 1235 2346 456 6 Resolution of this design is Iquot since I 456 has only 3 letters 7 Below give the confounding patterns for effects lower than fourfactor interactions ie include average main effects two and threefactor interactions and disregard the four five and sixfactor interactions Average m 1235 12346 T Main effects Note 3 Compare with the 2quot62 5 123 and 6 234 resolution IV design presented in the previous lecture this resolution lll design is weaker in the sense that some main effects are confounded with twofactor interactions which will not happen in resolution IV design where main effects will only confounded with threefactor interactions which is less important as the twofactor interactions Response Surface Method Section 96 on the DOE Handouts Step 1 Conduct an initial firstorder experiment with limited resources eg use the fractional factorial design and add a center point to validate the experimental results See class notes on 091002 for details Step 2Fit a linear model to the data collected in Step 1 Class notes on 091002 and 091202 explained how to construct this linear model from the estimates obtained via Yates algorithm Step 3Apply an optimization procedure to the above linear model for locating the direction points of the path of steepest ascent See lecture notes on 091202 for details Step 4Collect data on the path of steepest ascent and locate the best results on this path Step 5Conduct a followup secondorder experiment using the central composite design ideas See lecture notes on 091202 for details Step 6Construct a secondorder quadratic model based on procedures to be taught in Modeling Component of this class starting in November 2002 At this moment the model will be provided as forthe exam problems Step 7Use the above secondorder model to locate the optimal combination of process variables called process recipe and understand the characteristics of this optimum by conducting a characteristics analysis See lecture notes on 091202 and 091702 for details Details Step 1 Resolution fractional factorial design is recommended Step 2 If one applied Yates Algorithm to data collected from the fractional factorial design to obtain the estimates of effects the linear model will have the following form yhat estimate of overall average 12 estimate of X1 effect X1 12 estimate of X2 effect X2 12 estimate of Xp effect Xp where yhat is the prediction of process output X1 X2 Xp are p input variables coded into 1 1 format Because the estimate of effect is obtained forthe input variable being set at 1 and 1the change in the input is two units 1 1 2 which is in contrast to the single unit

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