+ Design of experiment ERT 427 2 k-p fractional factorial Miss Hanna Ilyani Zulhaimi
+ OUTLINE n Limitation of full factorial design n The concept of fractional factorial, 2 k-p n One-half fraction factorial design, 2 k-1 n One-quarter fraction factorial design, 2 k-2 n General 2 k-p fractional factorial design
+ Fundamental Principles Regarding Factorial Effects n Suppose there are k factors (A,B,...,J,K) in an experiment. All possible factorial effects include n effects of order 1: A, B,..., K (main effects) n effects of order 2: AB, AC,...,JK (2-factor interactions) n Hierarchical Ordering principle n Lower order effects are more likely to be important than higher order effects. n Effects of the same order are equally likely to be important n Effect Heredity Principle n In order for an interaction to be significant, at least one of its parent factors should be significant.
+ Fractional Factorial Designs n Purpose: full factorial design can be very expensive n Large number of factors too many experiments n May not have sources (time, money etc) for full factorial design n Costly (Degrees of freedom wasted on estimating higher order terms) n Often only lower order effects are important n Number of runs required for full factorial grows quickly Consider 2 k design n If k=7 128 runs required n Can estimate 127 effects n Only 7 df for main effects, 21 for 2-factor interactions the remaining 99 df are for interactions of order 3 n A fraction of the full factorial design ( i.e. a subset of all possible level combinations) is sufficient.
+ Example 1 n Suppose mechanical engineer wants to design a new car and consider the following nine factors each with 2 levels 1. Engine Size 6. Shape 2. Number of cylinders 7. Tires 3. Drag 8. Suspension 4. Weight 9. Gas Tank Size 5. Automatic vs Manual n Only have resources for conducting is 64 runs n If you drop three factors for a 2 6 full factorial design, those factor and their interactions with other factors cannot be investigated. n Want investigate all nine factors in the experiment n A fraction of 2 9 factorial design will be used. n Confounding (aliasing) will happen because using a subset How to construct the fraction?
+ Example 2 Filtration rate experiment: Recall that there are four factors in the experiment (A, B, C and D), each of 2 levels. Suppose the available resource is enough for conducting 8 runs. 2 4 full factorial design consists of all the 16 level combinations of the four factors. We need to choose half of them. The chosen half is called 2 4 1 fractional factorial design Which half we should select (construct)?
+ Full Factorial of 2 Level for k=4
+ Continue For 2 4 1 Fractional Factorial Design n the number of factors: k = 4 n the fraction index: p = 1 n Thus, the number of runs (level combinations): n Construct 2 4 1 designs via confounding n Generate D with a high order interaction of A, B and C, where: D = ABC
+ Fractional Factorial for 2 4-1 The chosen fraction includes the following 8 level combinations: (,,, ), (+,,,+), (,+,,+), (+,+,, ), (,,+,+), (+,,+, ), (,+,+, ), (+,+,+,+) Note: 1 corresponds to + and 1 corresponds to
+ Aliasing in 2 3 1 Design (One-half Fraction) n Consider a situation with 3 factors, each at 2 level but the experiment cannot afford to run at 8 treatment combination. This will suggest one-half fraction of 2 3 design, which contains 2 3-1 =4 treatment combinations. n Note that I=ABC
+ DEFINING RELATION 11 I = ±ABC It is called the defining relation, or ABCD is called a defining word. In previous case, we select 4 treatment combination as our one-half fraction. Each half fraction with have Ø Principal fraction (positive) Ø Alternate fraction (negative) The experiment will be run either principal or alternate fraction. The two groups of runs can be combined to form a full factorial.
+ ASSIGNMENT: Construct a half fraction of 2 level factorial design matrix with 4 factors. Hint: Generator, Defining relation, D= ABC I = +ABCD
+ Aliasing in Fractional Design n For the principal fraction, notice that the contrast for estimating the main effect A is exactly the same as the contrast used for estimating the BC interaction. n This phenomena is called aliasing or confounding and it occurs in all fractional designs n Aliases can be found directly from the columns in the table of + and - signs
+ Design Resolution n Design of resolution III, IV and IV are particularly important. The definition is as follows: Resolution Definition Resolution III No main effects are aliased with other main effects, but main effect are aliased with two-factor interaction. Resolution IV No main effects are aliased with other main effects or 2-factor interaction, but two-factor interaction are aliased each other. Resolution V No main effect or two-factor interaction is aliased with other main effect or two-factor interaction, but two factor interactions are aliased with three-factor interactions.
+ The 2III3-1 Design
+ The 2 IV 4-1 Design 16
+ The 2v 5-1 Design
+ (continued)
+ Guide to choice of fractional factorial designs 19 Factors 2 3 4 5 6 7 8 4 runs Full 1/2 (III) - - - - - 8 2 rep Full 1/2 (IV) 1/4 (III) 1/8 (III) 1/16 (III) - 16 4 rep 2 rep Full 1/2 (V) 1/4 (IV) 1/8 (IV) 1/16 (IV) 32 8 rep 4 rep 2 rep Full 1/2 (VI) 1/4 (IV) 1/8 (IV) 64 16 rep 8 rep 4 rep 2 rep Full 1/2 (VII) 1/4 (V) 128 32 rep 16 rep 8 rep 4 rep 2 rep Full 1/2 (VIII
+ (continued) 20 Factors 9 10 11 12 13 14 15 4 runs - - - - - - - 8 - - - - - - - 16 1/32 (III) 1/64 (III) 1/128 (III) 1/256 (III) 1/512 (III) 1/1024 (III) 1/2048 (III 32 1/16 (IV) 1/32 (IV) 1/64 (IV) 1/128 (IV) 1/256 (IV) 1/512 (IV) 1/1024 (IV 64 1/8 (IV) 1/16 (IV) 1/32 (IV) 1/64 (IV) 1/128 (IV) 1/256 (IV) 1/512 (IV 128 1/4 (VI) 1/8 (V) 1/16 (V) 1/128 (IV) 1/64 (IV) 1/128 (IV) 1/128 (IV
+ Guide (continued) 21 n Resolution V and higher à safe to use (main and two-factor interactions OK) n Resolution IV à think carefully before proceeding (main OK, two factor interactions are aliased with other two factor interactions) n Resolution III à Stop and reconsider (main effects aliased with two-factor interactions). n See design generators for selected designs in the attached table.
+ Guide (continued) What is the maximum resolution criterion? n Fractional factorial design with maximum resolution is optimal! Why? n The higher the resolution, the less restrictive the assumptions that are required n Interactions are negligible to obtain a unique interpretation of result
+ One Quarter Fraction: 2 k 2 Design n Parts manufactured in an injection molding process are showing excessive shrinkage. A quality improvement team has decided to use a designed experiment to study the injection molding process so that shrinkage can be reduced. The team decides to investigate six factors n A: mold temperature B: screw speed C: holding time D: cycle time E: gate size F : holding pressure each at two levels, with the objective of learning about main effects and interactions. They decide to use 16-run fractional factorial design. a full factorial has 2 6 =64 runs. 16-run is one quarter of the full factorial How to construct the fraction?
+ One Quarter Fraction: 2 k 2 Design
+ General 2 k p Fractional Factorial Designs n k factors, 2 k level combinations, but want to run a 2 p fraction only. n Select the first k p factors to form a full factorial design (basic design). n Alias the remaining p factors with some high order interactions of the basic design. n Defining contrasts subgroup: G = { defining words} n Define alias structure that meet with the concern. n Use maximum resolution to choose the optimal design. n Choose important effect to form models, pool unimportant effects into error component
+ Thank you J