# Three Categories of Minimum Cost Systematic Full 2^{n} Factorial Designs

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## Abstract

Full 2^{n} factorial experiments are often conducted sequentially run after run. A full 2^{n} factorial experiment has a total of 2^{n}! permutations among its 2^{n} runs but not all of these 2^{n}! permutations produce runs sequences with good statistical properties. In fact, the standard runs order is not economic (requiring a large number of factor level changes between runs), and does not produce time-trend-resistant main effects. Four main algorithms exist for sequencing runs of the full 2^{n} factorial experiment such that: (1) main effects and/or two-factor interactions are orthogonal to the linear/quadratic time trend and/or (2) the number of factor level changes between runs (i.e., cost) is minimal = (2^{n} − 1) or minimum. This article proposes, through using the generalized foldover scheme and the interactions-main effects assignment, three categories of systematic full 2^{n} factorial designs where main effects and/or two-factor interactions are linear/quadratic trend free and where the number of factor level changes is minimal (i.e., (2^{n} − 1)) or less than that of existing algorithms. These three categories are called (i) minimal-cost full 2^{n} factorial designs, (ii) minimum-cost linear-trend-free full 2^{n} factorial designs, and (iii) minimum-cost linear and quadratic-trend-free full 2^{n} factorial designs. A comparison with existing systematic full 2^{n} factorial designs reveals that the proposed systematic 2^{n} designs compete well regarding minimization of the number of factor level changes (i.e., cost) or regarding protection of main effects and/or two-factor interactions against the linear/quadratic time trend.

## Keywords

Sequential factorial experimentation Time-trend-free run orders Number of factor level changes and the experimental cost Runs sequencing algorithms Interactions-main effects assignment Generalized foldover scheme## AMS Subject Classification

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