Design of experiment ERT k-p fractional factorial. Miss Hanna Ilyani Zulhaimi

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1 + Design of experiment ERT k-p fractional factorial Miss Hanna Ilyani Zulhaimi

2 + 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

3 + 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.

4 + 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.

5 + 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?

6 + 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 fractional factorial design Which half we should select (construct)?

7 + Full Factorial of 2 Level for k=4

$factors: k = 4 n the fraction index: p = 1 n Thus, the number of$ runs (level combinations): n Construct 2 4 1 designs via

8 + Continue For 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 designs via confounding n Generate D with a high order interaction of A, B and C, where: D = ABC

9 + Fractional Factorial for The chosen fraction includes the following 8 level combinations: (,,, ), (+,,,+), (,+,,+), (+,+,, ), (,,+,+), (+,,+, ), (,+,+, ), (+,+,+,+) Note: 1 corresponds to + and 1 corresponds to

10 + Aliasing in 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 =4 treatment combinations. n Note that I=ABC

11 + 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.

12 + ASSIGNMENT: Construct a half fraction of 2 level factorial design matrix with 4 factors. Hint: Generator, Defining relation, D= ABC I = +ABCD

13 + 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

14 + 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.

15 + The 2III3-1 Design

16 + The 2 IV 4-1 Design 16

17 + The 2v 5-1 Design

18 + (continued)

19 + Guide to choice of fractional factorial designs 19 Factors runs Full 1/2 (III) rep Full 1/2 (IV) 1/4 (III) 1/8 (III) 1/16 (III) 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) rep 8 rep 4 rep 2 rep Full 1/2 (VII) 1/4 (V) rep 16 rep 8 rep 4 rep 2 rep Full 1/2 (VIII

20 + (continued) 20 Factors runs /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

21 + 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.

22 + 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

23 + 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?

24 + One Quarter Fraction: 2 k 2 Design

25 + 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

26 + Thank you J

Lecture 12: 2 k p Fractional Factorial Design

Lecture 12: 2 k p Fractional Factorial Design Montgomery: Chapter 8 Page 1 Fundamental Principles Regarding Factorial Effects Suppose there are k factors (A,B,...,J,K) in an experiment. All possible factorial