Optimal Design of Process Flexibility for General Production Systems

Process flexibility has been widely adopted as an effective strategy to improve responsiveness to demand uncertainty. Many algorithms for constructing sparse flexibility designs with good theoretical guarantees have been developed for balanced and symmetrical production systems, which assume that the number of plants equals the number of products, the supplies have the same capacity, and demands are independently and identically distributed.In this paper, we relax these restrictions and consider a very general class of production systems. We provide a simple construction of flexibility design to fulfill (1-ε) fraction of the expected total demand with high probability (w.h.p.) with an average degree of O (ln(1/ε)). To motivate our construction, we first consider a natural weighted probabilistic construction from Chou et. al. (2011), in which the degree of each node is proportional to its expected capacity. However, we show that this strategy is sub-optimal. To obtain an optimal construction, we develop a simple, yet effective thresholding scheme. Our theoretical analysis extends the classical analysis of expander graphs by overcoming several technical difficulties. The developed techniques might be useful for other applications that require certain expansion properties of graphs with non-uniform degree sequence