Worst-Case Analysis of Process Flexibility Designs

Theoretical studies of process flexibility designs have mostly focused on expected sales. In this paper, we take a different approach by studying process flexibility designs from the worst-case point of view. To study the worst-case performances, we introduce the plant cover indices (PCIs), defined by bottlenecks in flexibility designs containing a fixed number of products. We prove that given a flexibility design, a general class of worst-case performance measures can be expressed as functions of the design's PCIs and the given uncertainty set. This result has several major implications. First, it suggests a method to compare the worst-case performances of different flexibility designs without the need to know the specifics of the uncertainty sets. Second, we prove that under symmetric uncertainty sets and a large class of worst-case performance measures, the long chain, a celebrated sparse design, is superior to a large class of sparse flexibility designs, including any design that has a degree of two on each of its product nodes. Third, we show that under stochastic demand, the classical Jordan and Graves (JG) index can be expressed as a function of the PCIs. Furthermore, the PCIs motivate a modified JG index that is shown to be more effective in our numerical study. Finally, the PCIs lead to a heuristic for finding sparse flexibility designs that perform well under expected sales and have lower risk measures in our computational study.