An integer linear programing approach to find trend-robust run orders of experimental designs

Abstract

When a multi-factor experiment is carried out over a period of time, the responses may depend on a time trend. Unless the tests of the experiment are conducted in a proper order, the time trend has a negative impact on the precision of the estimates of the main effects, the interaction effects and the quadratic effects. A proper run order, called a trend-robust run order, minimizes the confounding between the effects' contrast vectors and the time trend's linear, quadratic and cubic components. Finding a trend-robust run order is essentially a permutation problem. We develop a multistage approach based on integer programming to find a trend-robust run order for any given design. The multistage nature of our algorithm allows us to prioritize the trend robustness of the main-effect estimates. In the literature, most of the methods used are tailored to specific designs, and are not applicable to an arbitrary design.Additionally, little attention has been paid to trend-robust run orders of response surface designs, such as central composite designs, Box-Behnken designs and definitive screening designs. Our algorithm succeeds in identifying trend-robust run orders for arbitrary factorial designs and response surface designs with two up to six factors.

Exact Optimal Designs of Experiments for Factorial Models via Mixed-Integer Semidefinite Programming

“…Recently, mixed-integer (linear) programming has been extensively used to find factorial designs for screening purposes, specifically in the construction of mixed-level orthogonal and two-level orthogonally blocked designs [30], orthogonal fractional factorial split-plot designs [31], trend robust run-order designs [32] and for breaking the symmetry of blocking two-level orthogonal experimental designs, which helps in finding optimal orthogonal blocking patterns [33].…”

Blocking OMARS designs and definitive screening designs

“…It is called mixed integer linear programming (MILP) if only some of the variables are required to be integer. In design of experiments, ILP was used earlier by Bulutoglu and Margot (2008) to classify orthogonal arrays, by Sartono et al (2015a) and Vo-Thanh et al (2018) to block given regular and nonregular orthogonal designs, by Capehart et al (2011) to construct regular two-level split-plot designs, by Sartono et al (2015b) to construct more general orthogonal fractional factorial split-plot designs, by Núñez Ares and Goos (2019) to identify trend-robust run orders for standard experimental designs, and by Vo-Thanh et al (2020) to find optimal row-column arrangements of twolevel orthogonal designs. In this section, we modify the MILP approach of Sartono et al (2015a) to find good blocking arrangements for OMARS designs, and the special case of DSDs.…”