Sparse Process Flexibility Designs: Is the Long Chain Really Optimal?

Sparse process flexibility and long chain has became an important concept in design flexible manufacturing systems since the seminal paper of Jordan and Graves (1995). In this paper, we study the performance of long chain in comparison to all designs with at most 2n edges over n supply and n demand nodes. We show that, surprisingly, long chain is not optimal in this class of networks even for i.i.d. demand distributions. In particular, we present a family of instances where a disconnected network with 2n edges has a strictly better performance than long chain even for i.i.d. demand distributions. This is quite surprising and contrary to the intuition that a connected design performs better than a disconnected one for symmetric distributions. Moreover, our family of instances show that the optimal design depends on the particular demand distribution. We also study the performance of long chain in comparison to connected designs with at most 2n arcs. We show that long chain is optimal in this class of designs for exchangeable demand distributions. Our proof is based on a coupling argument and a combinatorial analysis of the structure of maximum flow in directed networks. The analysis provides useful insights towards not just understanding the optimality of long chain but also towards designing more general sparse flexibility networks.