Fractional Factorial Experiments (Continued)

1. Fractional Factorial Experiments (Continued) • The concept of design resolution is a useful way to categorize fractional factorial designs. The higher the resolution of the design the better. However, as expected, to achieve higher resolution more experimentation is sometimes necessary. • Resolution III designs: in these designs no main effects are aliased with any other main effects but main effects are aliased with two factor interactions and two factor interactions may be aliased with other two factor interactions. Our 23-1 design (with I = ABC) is an example of a resolution III design. The notation for these designs typically uses the resolution as a subscript, e.g., 23-1III • Resolution IV designs have no main effect aliased with any other main effect or two factor effect, but two factor interactions are aliased with each other. The 24-1 design for the etch rate experiment is an example of a resolution IV design.

2. Resolution V designs have have no main effect or two factor interaction aliased with any other main effect or two factor interaction, but two factor interactions are aliased with three factor interactions. A 25-1 design with I = ABCDE is an example of a resolution V design.

3. 2k-p Designs • 2k-1 designs are useful for reducing the size of experiments, however we can often find even smaller fractions which will yield nearly as much information. A 2k can be run in a 1/2p fraction called a 2k-p fractional factorial design. So a 1/4 fraction is a 2k-2, a 1/8 fraction is 2k-3, etc. • Consider a 1/4 fraction of a design to study the effects of 6 factors e.g., a 2 k-2 design. To construct this design we first write down the full factorial design in 4 factors. Note that since we have 6 factors total we need two design generators to generate the columns for the remaining two factors (say E and F). • Suppose we choose I = ABCE and I = BCDF as our generators. Then column E would be equal to ABC column F would be equal to BCD. However, since the product of any two columns in a table of + and - signs for a full factorial design is another column in the table so the product of ABCE and BCDF is also an identity column. So the full defining relationship is: • I = ABCE = BCDF = ADEF • to find the alias of any effect multiply each word above by the effect.

4. The selection of the design generator will effect the aliases and therefore the resolution of the design. Tables have been developed to aid in selecting fractional factorial experiments with the maximum resolution.

5. Example • Parts manufactured in an injection molding process are experiencing excessive shrinkage. A quality improvement teams identifies 7 factors that they feel may impact the shrinkage problem. They decide to use a 27-3 design to investigate the problem with generators I = ABCE, I=BCDF, I = ACDG. Write out the design. What is the resolution of the design?

6. Taguchi Orthogonal Arrays • Taguchi has proposed a number of designs consisting of a collection of orthogonal arrays. • We will examine two of the simpler ones: the L 4(23) and the L8(27): L 4(23) 3 1 2

7. The L8(27)