An integer programming approach to the construction of trend-free experimental plans on split-plot designs

In many experimental designs, the standard procedure involves randomization of the factor-level combination run order. There are cases, however, where it is known that a time or position trend that can seriously compromise the results of the experiment may be present. These trends include wear of tooling and equipment, learning curves, change in temperatures, and so on, and the trends may show up as linear, quadratic, or even higher order trends. All previously published work has dealt with various methods of constructing trend-resistant run order plans on full and fractional factorial designs. These previous efforts have not addressed any additional dimensions in the trends that emerge when using hierarchical designs such as split-plot plans. These designs are common in many manufacturing experiments where complete randomization is not possible or is too expensive to be practical. The objective of this work is to establish the foundations of a method for constructing linear and quadratic trend-resistant plans in two-level split-plot designs that addresses the two-dimensional trends that may occur. The methodology involves development of a hybrid approach using the foldover method in each of the dimensions of interest and embedding these in a nonlinear integer programming model in the search for a feasible solution. Feasibility of this approach is shown for the particular case of a split-plot design (25 whole-plot factors and 31 × 21 split-plot factors) performed on abrasive machining. In this case study, an experimental plan that is robust against all linear trends and most quadratic trends was achieved.