Session 3 Fractional Factorial Designs 4

Session 3 Fractional Factorial Designs 3 a Modification of a Bearing Example 3. Fractional Factorial Designs Two-level fractional factorial designs Confounding Blocking Two-Level Eight Run Orthogonal Array a b c ab ac bc abc Failure Rate run A B C D y 1 + + + 16 2 + + + 7 3 + + + 14 4 + + + 5 5 + + + 11 6 + + + 7 7 + + + 13 8 + + + + + + + 4 Last column, abc estimates two : D + ABC. Factors D and ABC are confounded: l D D + ABC, and D and ABC are aliases. a BHH2e Chapter 6. E. Barrios Design and Analysis of Engineering Experiments 3 1 E. Barrios Design and Analysis of Engineering Experiments 3 3 Session 3 Fractional Factorial Designs 2 Session 3 Fractional Factorial Designs 4 2 4 Full Factorial Design Modification of a Bearing Example I 1 2 3 4 12 13 14 23 24 34 123 124 134 234 1234 conversion + + + + + + + + 70 + + + + + + + + 60 + + + + + + + + 89 + + + + + + + + 81 + + + + + + + + 69 + + + + + + + + 62 + + + + + + + + 88 + + + + + + + + 81 + + + + + + + + 60 + + + + + + + + 49 + + + + + + + + 88 + + + + + + + + 82 + + + + + + + + 60 + + + + + + + + 52 + + + + + + + + 86 + + + + + + + + + + + + + + + + 79 We can accommodate 16 estimates: 1 mean; 4 main ; 6 two-factors interaction ; 4 three-factors interaction ; 1 four-factors interaction effect E. Barrios Design and Analysis of Engineering Experiments 3 2 E. Barrios Design and Analysis of Engineering Experiments 3 4

Session 3 Fractional Factorial Designs 5 Session 3 Fractional Factorial Designs 7 Modification of a Bearing Example Two-Level Eight Run Orthogonal Array a b c ab ac bc abc run A B C D 1 + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + + + + + Confounding pattern: l A A + (BCD) l AB AB + CD l B B + (ACD) l AC AC + BD l C C + (ABD) l BC BC + AD l D D + (ABC) Sometimes 3rd and higher order interactions are small enough to be ignored. The Anatomy of the Half Replicate If instead, we use the column ab to accommodate factor D, then AB = D and therefore I = ABD. Then, for this design, Note: A = BD, B = AD, C = ABCD, D = AB AC = BCD, BC = ACD, CD = ABC l A A + BD; l B B + AD; l D D + AB The defining relation contains three letters I = ABD, thus the design is of resolution III, 2 4 1 III. E. Barrios Design and Analysis of Engineering Experiments 3 5 E. Barrios Design and Analysis of Engineering Experiments 3 7 Session 3 Fractional Factorial Designs 6 Session 3 Fractional Factorial Designs 8 The Anatomy of the Half Replicate Generation: A = a; B = b; C = c and D = abc. Thus D = ABC is the generating relation of the design. Note that Thus, for this design: I = D D = D 2 = ABC D I = ABCD, B = ACD, C = ABD, D = ABC AB = CD, AC = BD, AD = BC This design is resolution 4, since the length of the defining relation has four letters (factors) I = ABCD. It is denoted as 2 4 1 IV where, the 2 means that factors of the design have 2 levels each; 4-1 because there are 4 factors and we are running only one have of the full factorial: 8 = 2 4 1 = 16/2; and IV because the design is resolution 4. E. Barrios Design and Analysis of Engineering Experiments 3 6 Justification for the use of fractional factorials: Redundancy: When high order interactions are considered negligible lower order are arranged to be confounded with them and thus are estimable. Parsimony: Effect sparsity; vital few, trivial many; Pareto effect. Projectivity: A 3D design project onto a 2 2 factorial design in all 3 subspaces of dimension 2. Then 3D designs are of projectivity 2. Similarly, 2 4 1 designs are of projectivity 3 since after dropping any factor a full 2 3 design is left for the remaining three factors. In general, for fractional factorials designs of resolution R, the projectivity P = R 1. Every subset of P = R 1 factors is a complete factorial (possibly replicated) in P factors. E. Barrios Design and Analysis of Engineering Experiments 3 8

Session 3 Fractional Factorial Designs 9 Session 3 Fractional Factorial Designs 11 3D Projectivity: Sequential Experimentation In sequential experimentation, unless the total number of runs is necessary to achieve a desired level of precision, it is usually best to start with a fractional factorial. The design could be later augmented if necessary To cover more interesting regions. To resolve ambiguities. A 2 3 1 III design showing projections into three 2 2 factorials. E. Barrios Design and Analysis of Engineering Experiments 3 9 E. Barrios Design and Analysis of Engineering Experiments 3 11 Session 3 Fractional Factorial Designs 10 Session 3 Fractional Factorial Designs 12 Note Eight-run Designs a It is recommended to dedicate just a modest amount of the budget to the first stages of the experimentation. Find or determine which factors to consider and appropriate responses Determine proper experimental region and factor ranges. Then you can dedicate to study deeper your experiment Estimate better factor Confirmatory experimentation Optimize product or process. a BHH2e Chapter 6, BHH Chapter 10. E. Barrios Design and Analysis of Engineering Experiments 3 10 E. Barrios Design and Analysis of Engineering Experiments 3 12

Session 3 Fractional Factorial Designs 13 Eight-run nodal designs Session 3 Fractional Factorial Designs 15 Sign switching, Foldover and Sequential Assembly After running a fractional factorial further runs may be necessary to resolve ambiguities. Folding over (changing signs) one column (main effect, say D) provide unaliased estimates of the main effect and all two-factor interactions involving factor D. E. Barrios Design and Analysis of Engineering Experiments 3 13 E. Barrios Design and Analysis of Engineering Experiments 3 15 Session 3 Fractional Factorial Designs 14 A Bicycle Example Session 3 Fractional Factorial Designs 16 A Bicycle Example. Second fraction E. Barrios Design and Analysis of Engineering Experiments 3 14 E. Barrios Design and Analysis of Engineering Experiments 3 16

Session 3 Fractional Factorial Designs 17 A Bicycle Example. Resulting 16-run design Main effect D and two-factor interactions involving D are free of aliasing. Session 3 Fractional Factorial Designs 19 Filtration Example For any given fraction one-column foldover will dealias a particular main effect and all its interactions E. Barrios Design and Analysis of Engineering Experiments 3 17 E. Barrios Design and Analysis of Engineering Experiments 3 19 Session 3 Fractional Factorial Designs 18 An Investigation Using Multiple-Column Foldover Filtration Example Session 3 Fractional Factorial Designs 20 Filtration Example E. Barrios Design and Analysis of Engineering Experiments 3 18 E. Barrios Design and Analysis of Engineering Experiments 3 20

Session 3 Fractional Factorial Designs 21 Session 3 Fractional Factorial Designs 23 Second Fraction: a 2III 7 3. (foldover all columns [mirror image]) 2D Projection over the [AE] subspace. A 2 2 design replicated 4 times E. Barrios Design and Analysis of Engineering Experiments 3 21 E. Barrios Design and Analysis of Engineering Experiments 3 23 Session 3 Fractional Factorial Designs 22 Session 3 Fractional Factorial Designs 24 Analysis of the resulting sixteen-run design: a 2 7 3 IV fractional factorial Increasing Design Resolution from III to IV by Foldover In general, any design of resolution III plus its mirror image becomes a design of resolution IV. Consider for example, the 2 8 5 III (= 1 8 : 2 7 4 III ) design and its mirror image. Their generation relations are respectively: and then, combining (1) and (2) I 8 = 8 = 124 = 135 = 236 = 1237 (1) I 8 = 8 = 124 = 135 = 236 = 1237 (2) I 16 = 1237 Also, from (1), I 8 = (8)(124) = 1248, and from (2), I 8 = ( 8)( 124) = 1248. Thus, I 16 = 1248. The four generators for this 2 8 4 III design are: I 16 = 1237 = 1248 = 1358 = 2368 E. Barrios Design and Analysis of Engineering Experiments 3 22 E. Barrios Design and Analysis of Engineering Experiments 3 24

Session 3 Fractional Factorial Designs 25 Sixteen-Run Designs Nodal Designs: Session 3 Fractional Factorial Designs 27 The 2 5 1 V Nodal Design. Reactor Example E. Barrios Design and Analysis of Engineering Experiments 3 25 E. Barrios Design and Analysis of Engineering Experiments 3 27 Session 3 Fractional Factorial Designs 26 Design Matrix and Alias Patterns Session 3 Fractional Factorial Designs 28 Alias Pattern E. Barrios Design and Analysis of Engineering Experiments 3 26 E. Barrios Design and Analysis of Engineering Experiments 3 28

Session 3 Fractional Factorial Designs 29 Normal plots for full and half fraction factorial designs. Session 3 Fractional Factorial Designs 31 The 2 8 4 IV Nodal Design. Paint Trial Example normal score 2 1 0 1 2 D:E E Full Factorial C:D B:E A:B:C A:B:C:E A:B:D A:B B:C A:C:D:E B:C:D A:C C:E A:D:E C:D:E A:B:D:E A:E B:C:E B:D:E A:B:C:D C A:B:C:D:E B:C:D:E A:C:D A:D A A:B:E A:C:E B:D D 10 0 10 20 B normal score 1 0 1 Fractional Factorial B D B:D C:E A:B B:C A:E B:E A:C C:D C A:D A E D:E 10 5 0 5 10 15 20 25 E. Barrios Design and Analysis of Engineering Experiments 3 29 E. Barrios Design and Analysis of Engineering Experiments 3 31 Session 3 Fractional Factorial Designs 30 3D Projectivity of 2 5 1 V design. Session 3 Fractional Factorial Designs 32 Normal plot of for glossiness and abrasion. E. Barrios Design and Analysis of Engineering Experiments 3 30 E. Barrios Design and Analysis of Engineering Experiments 3 32

Session 3 Fractional Factorial Designs 33 Contour plots for glossiness and abrasion. Session 3 Fractional Factorial Designs 35 Normal plot of for location and dispersion responses. Location Effects Dispersion Effects J H K O normal score 1 0 1 H N D A B G M E L P F normal score 1 0 1 B J E K G F N C D L P C A O 10 0 10 20 30 Location Effects M 1.0 0.5 0.0 0.5 1.0 1.5 Dispersion Effects 20 ME 1.5 ME 1.0 10 0.5 0 0.0 10 0.5 1.0 20 ME 1.5 ME A C E G J L N P factors A C E G J L N P factors E. Barrios Design and Analysis of Engineering Experiments 3 33 E. Barrios Design and Analysis of Engineering Experiments 3 35 Session 3 Fractional Factorial Designs 34 The 2 15 11 III Nodal Design. Shrinkage of Speedometer Example a. Session 3 Fractional Factorial Designs 36 Elimination of Block Effects Boys Shoes Example Two sample Comparison Paired Comparison 114 1.0 wear 112 110 wear difference 0.5 0.0 108 material A material B 0.5 1.0 2 4 6 8 10 boys 2 4 6 8 10 boys a Quinlan (1985) 10 + 10 observations 2 sample means 18 degrees of freedom 10 differences 1 sample mean 9 degrees of freedom E. Barrios Design and Analysis of Engineering Experiments 3 34 E. Barrios Design and Analysis of Engineering Experiments 3 36

Session 3 Fractional Factorial Designs 37 Session 3 Fractional Factorial Designs 39 Elimination of Block Effects Design Matrix 2 5 1 V Elimination of Block Effects Block what you can, randomize what you cannot Identify important extraneous factors within blocks and eliminate them. Representative variation between blocks should be encourage E. Barrios Design and Analysis of Engineering Experiments 3 37 E. Barrios Design and Analysis of Engineering Experiments 3 39 Session 3 Fractional Factorial Designs 38 Session 3 Fractional Factorial Designs 40 Two-Level Factorial Designs Blocking Arrangements for 2 k Factorial Designs a Number of Number of Block Variables Runs Size Block Interactions Confounded with Blocks 3 8 4 B 1 = 123 123 2 B 1 = 12,B 2 = 13 12, 13, 23 4 16 8 B 1 = 1234 1234 4 B 1 = 124,B 2 = 134 124, 134, 23 2 B 1 = 12,B 2 = 23, 12, 23, 34, 13, 1234,24, 14 B 3 = 34 5 32 16 B 1 = 12345 12345 8 B, = 123,B 2 = 345 123, 345, 1245 4 B 1 = 125,B 2 = 235, 125, 235, 345, 13,1234,24, 145 B 3 = 345 2 B 1 = 12,B 2 = 13, 12, 13, 34, 45, 23,1234, 1245,14, B 3 = 34,B 4 = 45 1345, 35, 24, 2345,1235, 15,25, i.e., all 2fi and 4fi 6 64 32 B 1 = 123456 123456 16 B 1 = 1236,B 2 = 3456 1236, 3456, 1245 8 B 1 = 135,B 2 = 1256, 135, 1256, 1234,236,245,3456, 146B 3 = 1234 4 B 1 = 126,B 2 = 136, 126, 136, 346, 456,23,1234,1245, B 3 = 346,B 4 = 456 14, 1345, 35, 246,23456, 12356, 156, 25 2 B 1 = 12,B 2 = 23, All 2fi, 4fi, and 6fi B 3 = 34,B 4 = 45, B 5 = 56 a BHH2e Table 5A.1 Minimal Aberration Two-Level Fractional Factorial Design for k Variables and N Runs a (Number in Parentheses Represent Replication). a BHH2e Table 6.22 E. Barrios Design and Analysis of Engineering Experiments 3 38 E. Barrios Design and Analysis of Engineering Experiments 3 40