DOE Fold Design

Objective

When we create 2^K fractional factorial , confounding will be an issue to be considered . If you aim of 2^K fractional factorial is to optimize the process , you must choose 2^K fractional factorial design with resolution V and above . Resolution V and above satisfied The Sparsity Of Effect Principle , thus can be used for optimization purpose . As for resolution III and IV , you will have difficulties to differentiate the effect of a main factor with 2-way interaction ( Resolution III ) and 2-way interaction with 2-way interaction ( Resolution IV ).

However, as for resolution III , if you would like to de-alias ( avoid confounding ) between a factor ( main effect ) with a 2-way interaction , you can changed from resolution III to resolution IV by fold your DOE design at the desired main effect .

Example:

Let’s consider a case where you have 5 factors : A , B , C, D and E . You decided to run 8 runs (1/4 fractional factorial ) . Without  “FOLDING” your design , you will create a DOE design with below minitab session window output :

Fractional Factorial Design

Factors:  5   Base Design:         5, 8   Resolution:  III

Runs:     8   Replicates:             1   Fraction:    1/4

Blocks:   1   Center pts (total):     0

* NOTE * Some main effects are confounded with two-way interactions.

Design Generators: D = AB, E = AC

Alias Structure

I + ABD + ACE + BCDE

A + BD + CE + ABCDE

B + AD + CDE + ABCE

C + AE + BDE + ABCD

D + AB + BCE + ACDE

E + AC + BCD + ABDE

BC + DE + ABE + ACD

BE + CD + ABC + ADE

Explanation:

In this case all the main effects ( factors ) are confounding with the 2-way interaction ; example factor A confounded with BC and CE . Let’s say if you want to ensure you can distinguish the effect of only factor A with 2-way interaction, then you should redesign your DOE with “FOLDED” design at factor A only . The result is as below :

Fractional Factorial Design

Factors:   5   Base Design:         5, 8   Resolution:   IV

Runs:     16   Replicates:             1   Fraction:    1/2

Blocks:    1   Center pts (total):     0

Design Generators (before folding): D = AB, E = AC

Folded on Factors: A

Alias Structure

I + BCDE

A + ABCDE

B + CDE

C + BDE

D + BCE

E + BCD

AB + ACDE

AC + ABDE

AD + ABCE

AE + ABCD

BC + DE

BD + CE

BE + CD

ABC + ADE

ABD + ACE

ABE + ACD

Obviously , factor A is free from 2-way confounding , in this FOLD design factor A is confounded with 5-way interaction, if A and ABCDE become significant to your DOE response ( Y ) , we can easily conclude that the effect is from A not from  ABCDE based on The Sparsity Of Effect Principle. How does minitab design a DOE with FOLD at factor A ?

When you set a fold DOE design at a factor minitab will add a replicate of original data with the sign of that factor switch, as below example:

¼ fractional factorial for 5 factor ( without fold )

DOE design ( 5 factor with FOLD at factor A only )

The total runs increased from 8 runs to 16 runs , thus the DOE design resolution changed from resolution III to resolution IV .

For above example , if you need to be able to distinguish the effect of main factor (A) with 2-way you have to modified your design from “WITHOUT FOLD” to “WITH FOLD” at factor A . This is the same if you initially create ½ fractional factorial at the design stage of your DOE, where all the main effect will be free from confounded with 2-way interaction ! . Thus before designing your DOE, you need to know the objective of your goal from the DOE itself.