Split-plot experiments with factor-dependent whole-plot sizes

In industrial split-plot experiments, the number of runs within each whole plot is usually determined independently from the factor settings. As a matter of fact, it is often equal to the number of runs that can be done within a given period of time or to the number of samples that can be processed in one oven run or with one batch. In such cases, the size of every whole plot in the experiment is fixed no matter what factor levels are actually used in the experiment. In this article, we discuss the design of a real-life experiment on the production of coffee cream where the number of runs within a whole plot is not fixed, but depends on the level of one of the whole-plot factors. We provide a detailed discussion of various ways to set up the experiment and discuss how existing algorithms to construct optimal split-plot designs can be modified for that purpose. We conclude the paper with a few general recommendations.