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Design and Analysis of Experiments. Dr. Tai- Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC. Two-Level Fractional Factorial Designs. Dr. Tai- Yue Wang Department of Industrial and Information Management

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Design and Analysis of Experiments Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC Two-Level Fractional Factorial Designs Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC Outline Introduction The One-Half Fraction of the 2k factorial Design The One-Quarter Fraction of the 2k factorial Design The General 2k-p Fractional Factorial Design Alias Structures in Fractional Factorials and Other Designs Resolution III Designs Resolution IV and V Designs Supersaturated Designs Introduction(1/5) As the number of factors in 2k factorial design increases, the number of runs required for a complete replicate of the design outgrows the resources of most experimenters. In 26factorial design, 64 runs for one replicate. Among them, 6 df for main effects, 15 df for two-factor interaction. That is, only 21 of them are majorly interested in. Introduction(2/5) The remaining 42 df are for three of higher interactions. If the experimenter can reasonably assume that certain high-order interactions are negligible, information on the main effects and low-order interactions may be obtained by running only a fraction of the complete factorial design. Introduction(3/5) The Fractional Factorial Designs are among the most widely used types of designs for product and process design and process improvement. A major use of fraction factorials is in screening experiments. Introduction(4/5) Three key ideas that fractional factorial can be used effectively: The sparsity of effects principle – When there are several variables, the system or process is likely to be driven primarily by some of the main effects an lower-order interactions. The projection property -- Fractional factorials can be projected into stronger designs in the subset of significant factors. Introduction Sequential experimentation – It is possible to combine the runs of two or more fractional factorials to assemble sequentially a larger design to estimate the factor effects and interactions interested. The One-Half Fraction of the 2kDesign – Definitions and Basic Principles Consider a 23 factorial design but an experimenter cannot afford to run all (8) the treatment combinations but only 4 runs. This suggests a one-half fraction of a 23 design. Because the design contains 23-1=4 treatment combinations, a one-half fraction of the 23 design is often called a 23-1 design. The One-Half Fraction of the 2kDesign – Definitions and Basic Principles Consider a 23 factorial The One-Half Fraction of the 2kDesign – Definitions and Basic Principles We can have tow options: One is the “+” sign in column ABC and the other is the “-” sign in column ABC. The One-Half Fraction of the 2kDesign – Definitions and Basic Principles For the “+” in column ABC, effects a, b, c, and abc are selected. For the “-” in column ABC, effects ab, ac, bc, and (1) are selected. Since we use ABC to determine which half to be used, ABC is called generator. The One-Half Fraction of the 2kDesign – Definitions and Basic Principles We look further to see if the “+” sign half is used, the sign in column I is identical to the one we used. We call I=ABC is the defining relation in our design. Note: C=AB is factor relation. C=AB I=ABC The One-Half Fraction of the 2kDesign – Definitions and Basic Principles In general, the defining relation for a fractional factorials will always be the set of all columns that are equal to the identity column I. If one examines the main effects: [A]=1/2(a-b-c+abc) [B]= 1/2(-a+b-c+abc) [C]= 1/2(-a-b+c+abc) The One-Half Fraction of the 2kDesign – Definitions and Basic Principles The two-factor interactions effects: [BC]=1/2(a-b-c+abc) [AC]= 1/2(-a+b-c+abc) [AB]= 1/2(-a-b+c+abc) Thus, A = BC, B = AC, C = AB The One-Half Fraction of the 2kDesign – Definitions and Basic Principles So [A]A+BC [B]B+AC [C]C+AB The alias structure can be found by using the defining relation I=ABC. AI = A(ABC) = A2BC = BC BI =B(ABC) = AC CI = C(ABC) = AB The One-Half Fraction of the 2kDesign – Definitions and Basic Principles The contrast for estimating the main effect A is exactly the same as the contrast used for estimating the BC interaction. In fact, when estimating A, we are estimating A+BC. This phenomena is called aliasing and it occurs in all fractional designs. Aliases can be found directly from the columns in the table of + and – signs. The One-Half Fraction of the 2kDesign – Definitions and Basic Principles This one-half fraction, with I=ABC, is usually called the principal fraction. That is, we could choose the other half of the factorial design from Table. This alternate, or complementary, one-half fraction (consisting the runs (1), ab, ac, and bc) must be chosen on purpose. The defining relation of this design is I=-ABC The One-Half Fraction of the 2kDesign – Definitions and Basic Principles So [A]’A-BC [B]’B-AC [C]’C-AB The alias structure can be found by using the defining relation I=-ABC. AI = A(-ABC) = A2BC = -BC BI =B(-ABC) = -AC CI = C(-ABC) = -AB The One-Half Fraction of the 2kDesign – Definitions and Basic Principles In practice, it does not matter which fraction is actually used. Both fractions belong to the same family. Two of them form a complete 23 design. The two groups of runs can be combined to form a full factorial – an example of sequential experimentation The One-Half Fraction of the 2kDesign – Design Resolution The 23-1 design is called a resolution III design. In this design, main effects are aliased with two-factor interactions. In general, a design is of resolution R if no p factor effect is aliased with another effect containing less than R-p factors. For a 23-1 design, no one (p) factor effect is aliased with one (less than 3(R) – 1(p)) factor effect. The One-Half Fraction of the 2kDesign – Design Resolution Resolution III designs – These are designs in which no main effects are aliased with any other main effect. But main effects are aliased with two-factor interactions and some two-factor interactions maybe aliased with each other. The 23-1 design is a resolution III design. Noted as The One-Half Fraction of the 2kDesign – Design Resolution Resolution IV designs – These are designs in which no main effects are aliased with any other main effect or with any two-factor interaction. But two-factor interaction are aliased with each other. The 24-1 design with I=ABCD is a resolution IV design. Noted as The One-Half Fraction of the 2kDesign – Design Resolution Resolution V designs – These are designs in which no main effects or two-factor interaction is aliased with any other main effect or with any two-factor interaction. But two-factor interaction are aliased with three-factor interaction. The 25-1 design with I=ABCDE is a resolution V design. Noted as The One-Half Fraction of the 2kDesign – Construction and analysis of the one-half fraction Example: C‧I=C‧ABC=AB The One-Half Fraction of the 2kDesign – Construction and analysis of the one-half fraction The one-half fraction of the 2k design of the highest resolution may be constructed by writing down a basic design consisting of the runs for a full 2k-1 factorial and then adding the kth factor by identifying its plus and minus levels with the plus and minus signs of the highest order interaction ABC..(K-1). The One-Half Fraction of the 2kDesign – Construction and analysis of the one-half fraction Note: Any interaction effect could be used to generate the column for the kth factor. However, use any effect other than ABC…(K-1) will not product a design of the highest possible resolution. The One-Half Fraction of the 2kDesign – Construction and analysis of the one-half fraction Any fractional factorial design of resolution R contains complete factorial designs (possibly replicated factorials) in any subset of R-1 factors. Important and useful !!! Example, if an experiment has several factors of potential interest but believes that only R-1 of them have important effects, the a fractional factorial design of resolution R is the appropriate choice of design. The One-Half Fraction of the 2kDesign – Construction and analysis of the one-half fraction Because the maximum possible resolution of a one-half fraction of the 2k design is R=k, every 2k-1 design will project into a full factorial in any (k-1) of the original k factors. the 2k-1 design may be projected into two replicates of a full factorial in any subset of k-2 factors., four replicates of a full factorial in any subset of k-3 factors, and so on. The One-Half Fraction of the 2kDesign – example (1--1/7) Y=filtration rate Fours factors: A, B, C, and D. Use 24-1 with I=ABCD The One-Half Fraction of the 2kDesign – example (1--2/7) STAT > DOE > Factorial > Create Factorial Design Number of factors 4 Design ½ fraction OK Factors Fill names for each factor Fractional Factorial Design Factors: 4 Base Design: 4, 8 Resolution: IV Runs: 8 Replicates: 1 Fraction: 1/2 Blocks: 1 Center pts (total): 0 Design Generators: D = ABC Alias Structure I + ABCD A + BCD B + ACD C + ABD D + ABC AB + CD AC + BD AD + BC The One-Half Fraction of the 2kDesign – example (1--3/7) The One-Half Fraction of the 2kDesign – example(1--4/7) The One-Half Fraction of the 2kDesign – example (1--5/7) After collecting data STAT > DOE > Factorial > Analyze Factorial Design Response Filtration OK Estimated Effects and Coefficients for Filtration (coded units) Term Effect Coef Constant 70.750 Temperature 19.000 9.500 Pressure 1.500 0.750 Conc. 14.000 7.000 Stir Rate 16.500 8.250 Temperature*Pressure -1.000 -0.500 Temperature*Conc. -18.500 -9.250 Temperature*Stir Rate 19.000 9.500 The One-Half Fraction of the 2kDesign – example (1--6/7) Obviously, no effect is significant B is less important Try A, C, and D projection 23 with A, C, D The One-Half Fraction of the 2kDesign – example (1--7/7) Factorial Fit: Filtration versus Temperature, Conc., Stir Rate Estimated Effects and Coefficients for Filtration (coded units) Term Effect Coef SE Coef T P Constant 70.750 0.7500 94.33 0.007 Temperature 19.000 9.500 0.7500 12.67 0.050 Conc. 14.000 7.000 0.7500 9.33 0.068 Stir Rate 16.500 8.250 0.7500 11.00 0.058 Temperature*Conc. -18.500 -9.250 0.7500 -12.33 0.052 Temperature*Stir Rate 19.000 9.500 0.7500 12.67 0.050 Conc.*Stir Rate -1.000 -0.500 0.7500 -0.67 0.626 S = 2.12132 PRESS = 288 R-Sq = 99.85% R-Sq(pred) = 90.62% R-Sq(adj) = 98.97% Prediction equation: Coded variable : The One-Half Fraction of the 2kDesign – example (2—1/8) 5 Factors 25-1 design Response: Yield The One-Half Fraction of the 2kDesign – example (2--2/8) The One-Half Fraction of the 2kDesign – example (2--3/8) The One-Half Fraction of the 2kDesign – example (2--4/8) Factorial Fit: Yield versus Aperture, Exposure, Develop, Mask, Etch Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef Constant 30.3125 Aperture 11.1250 5.5625 Exposure 33.8750 16.9375 Develop 10.8750 5.4375 Mask -0.8750 -0.4375 Etch 0.6250 0.3125 Aperture*Exposure 6.8750 3.4375 Aperture*Develop 0.3750 0.1875 Aperture*Mask 1.1250 0.5625 Aperture*Etch 1.1250 0.5625 Exposure*Develop 0.6250 0.3125 Exposure*Mask -0.1250 -0.0625 Exposure*Etch -0.1250 -0.0625 Develop*Mask 0.8750 0.4375 Develop*Etch 0.3750 0.1875 Mask*Etch -1.3750 -0.6875 S = * PRESS = * Analysis of Variance for Yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 5 5562.8 5562.8 1112.56 * * 2-Way Interactions 10 212.6 212.6 21.26 * * Residual Error 0 * * * Total 15 5775.4 The One-Half Fraction of the 2kDesign – example (2--5/8) Reduced to A, B, C, AB The One-Half Fraction of the 2kDesign – example (2--6/8) Factorial Fit: Yield versus Aperture, Exposure, Develop Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant 30.313 0.4002 75.74 0.000 Aperture 11.125 5.562 0.4002 13.90 0.000 Exposure 33.875 16.937 0.4002 42.32 0.000 Develop 10.875 5.437 0.4002 13.59 0.000 Aperture*Exposure 6.875 3.438 0.4002 8.59 0.000 S = 1.60078 PRESS = 59.6364 R-Sq = 99.51% R-Sq(pred) = 98.97% R-Sq(adj) = 99.33% Analysis of Variance for Yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 3 5558.19 5558.19 1852.73 723.02 0.000 2-Way Interactions 1 189.06 189.06 189.06 73.78 0.000 Residual Error 11 28.19 28.19 2.56 Lack of Fit 3 9.69 9.69 3.23 1.40 0.313 Pure Error 8 18.50 18.50 2.31 Total 15 5775.44 The One-Half Fraction of the 2kDesign – example (2--7/8) The One-Half Fraction of the 2kDesign – example (2--8/8) Collapse into two replicate of a 23 design The One-Half Fraction of the 2kDesign – Construction and analysis of the one-half fraction Using fractional factorial designs often leads to greater economy and efficiency in experimentation. Particularly if the runs can be made sequentially. For example, suppose that we are investigating k=4 factors (24=16 runs). It is almost always preferable to run 24-1IV fractional design (four runs), analyze the results, and then decide on the best set of runs to perform next. The One-Half Fraction of the 2kDesign – Construction and analysis of the one-half fraction If it is necessary to resolve ambiguities, we can always run the alternate fraction and complete the 24 design. When this method is used to complete the design, both one-half fractions represent blocks of the complete design with the highest order interaction (ABCD) confounded with blocks. Sequential experimentation has the result of losing only the highest order interaction. Possible Strategies for Follow-Up Experimentation Following a Fractional Factorial Design The One-Half Fraction of the 2kDesign – example (3—1/4) From Example 1, 24-1IV design Use I=-ABCD STAT>DOE>Factorial>Create Factorial Design Create base design first 2-level factorial(specify generators) Number of factors 3 Design Full factorial Generators D=-ABC OK The One-Half Fraction of the 2kDesign – example (3—2/4) The One-Half Fraction of the 2kDesign – example (3—3/4) The One-Half Fraction of the 2kDesign – example (3—4/4) Estimated Effects and Coefficients for Filtration (coded units) Term Effect Coef Constant 69.375 Temperature 24.250 12.125 Pressure 4.750 2.375 Conc. 5.750 2.875 Stir Rate 12.750 6.375 Temperature*Pressure 1.250 0.625 Temperature*Conc. -17.750 -8.875 Temperature*Stir Rate 14.250 7.125 S = * PRESS = * Analysis of Variance for Filtration (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 4 1612 1612 403.1 * * 2-Way Interactions 3 1039 1039 346.5 * * Residual Error 0 * * * Total 7 2652 The One-Half Fraction of the 2kDesign – Construction and analysis of the one-half fraction Adding the alternate fraction to the principal fraction may be thought of as a type of confirmation experiment that will allow us to strengthen our initial conclusions about the two-factor interaction effects. A simple confirmation experiment is to compare the results from regression and actual runs. The One-Quarter Fraction of the 2kDesign For a moderately large number of factors, smaller fractions of the 2kdesign are frequently useful. One-quarter fraction of the 2k design 2k-2 runs called 2k-2 fractional factorial The One-Quarter Fraction of the 2kDesign Constructed by writing down a basic design consisting of runs associated with a full factorial in k-2 factors and then associating the two additional columns with appropriately chosen interactions involving the first k-2 factors. Thus, two generators are needed. I=P and I=Q are called generating relations for the design. The One-Quarter Fraction of the 2kDesign The signs of P and Q determine which one of the one-quarter fractions is produced. All four fractions associated with the choice of generators ±P or ±Q are members of the same family. +P and +Q are principal fraction. I=P=Q=PQ P, Q, and PQ are defining relation words The One-Quarter Fraction of the 2kDesign Example: P=ABCE, Q=BCDF, PQ=ADEF Thus A=BCE=ABCDF=DEF When estimating A, one is really estimating A+BCE+DEF+ABCDF The One-Quarter Fraction of the 2kDesign Complete defining relation: I = ABCE = BCDF = ADEF The One-Quarter Fraction of the 2kDesign The One-Quarter Fraction of the 2kDesign Factor relations: E=ABC, F=BCD I=ABCE=BCDF=ADEF The One-Quarter Fraction of the 2kDesign STAT>DOE>Factorial>Create factorial Design Design Full factorialOKOK The One-Quarter Fraction of the 2kDesign The One-Quarter Fraction of the 2kDesign Alternate fractions of 26-2 design P=ABCE, -Q=-BCDF -P=-ABCE, Q=BCDF -P=-ABCE, -Q=-BCDF [A] A+BCE-DEF-ABCDF The One-Quarter Fraction of the 2kDesign A 26-2 design will project into a single replicate of a 24 design in any subset of fours factors that is not a word in the defining relation. It also collapses to a replicated one-half fraction of a 24 in any subset of four factors that is a word in the defining relation. The One-Quarter Fraction of the 2kDesign Projectionof the design into subsets of the original six variables Any subset of the original six variables that is not a word in the complete defining relation will result in a full factorial design Consider ABCD (full factorial) Consider ABCE (replicated half fraction) Consider ABCF (full factorial) The One-Quarter Fraction of the 2kDesign In general, any 2k-2 fractional factorial design can be collapsed into either a full factorial or a fractional factorial in some subset of r≦k-2 of the original factors. Those subset of variables that form full factorials are not words in the complete defining relation. The One-Quarter Fraction of the 2kDesign— example(4—1/10) Injection molding process Response: Shrinkage Factors: Mold temp, screw speed, holding time, cycle time, gate size, holding pressure. Each at two levels To run a 26-2 design, 16 runs The One-Quarter Fraction of the 2kDesign— example(4—2/10) The One-Quarter Fraction of the 2kDesign— example(4—3/10) Full model The One-Quarter Fraction of the 2kDesign— example(4—4/10) Factorial Fit: Shrinkage versus Temperature, Screw, ... Estimated Effects and Coefficients for Shrinkage (coded units) Term Effect Coef Constant 27.313 Temperature 13.875 6.937 Screw 35.625 17.812 Hold Time -0.875 -0.437 Cycle Time 1.375 0.688 Gate 0.375 0.187 Pressure 0.375 0.187 Temperature*Screw 11.875 5.938 Temperature*Hold Time -1.625 -0.813 Temperature*Cycle Time -5.375 -2.688 Temperature*Gate -1.875 -0.937 Temperature*Pressure 0.625 0.313 Screw*Cycle Time -0.125 -0.062 Screw*Pressure -0.125 -0.063 Temperature*Screw*Cycle Time 0.125 0.062 Temperature*Hold Time*Cycle Time -4.875 -2.437 The One-Quarter Fraction of the 2kDesign— example(4—5/10) Reduced model The One-Quarter Fraction of the 2kDesign— example(4—6/10) Reduced model Factori

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