Minimum Cost Linear Trend Free Fractional Factorial Designs

Abstract

Many fractional factorial experiments, especially those of exploratory type, are conducted sequentially in a specific run order. These experiments must be sequenced such that main effects and/or two-factor interactions are orthogonal (i.e., resistant) to the time trend or to any uncontrollable factor aliased with this trend. They must also be sequenced such that factors with expensive or difficult-to-vary levels are minimally varied during experimentation. This article employs the main effects–interactions assignment technique of Cheng and Jacroux (1988) and constructs a catalogue of minimum cost linear trend free regular 2 ^n−(n−k) fractional factorial designs of resolution at least 3. The construction is based on tabulating the complete 2 ^k factorial experiment such that main effects and interactions are in increasing number of level changes from 1 up to (2 ^k  − 1), then selecting subtables involving the first n nonmain effects and assigning them to new two-level factors, where (2 ^k−1 − (k − 1) ≤ n ≤ 2 ^k  − 1 − k) and k > 2. The paper provides for each linear trend free 2 ^n−(n−k) fraction: (1) the defining relation (i.e., the alias structure), (2) the k independent generators needed for the generalized fold-over run sequencing scheme, (3) the two-factor interactions (if any) that are linear trend free and (4) the total number of factor level changes (i.e., the experimental cost).

Keywords:

• 62K15

Keywords:

• Experimental cost
• Generalized fold-over scheme
• Main effects–interactions assignment
• Minimum factor level changes
• Sequential fractional factorial experimentation
• Time trend free run orders

Acknowledgment

This research was accomplished while the author was on sabbatical leave from the University of Jordan (2010–2011).