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

Abstract

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.

Achieving High Individual Service-Levels without Safety Stock? Optimal Rationing Policy of Pooled Resources

“…It is clear that the performance of 3-chain is almost as good as the full flexibility design for all three types of service constraints. This phenomena is well-known when total capacity is fixed and the objective is to optimize demand fulfillment (Chou et al, 2011;Chen et al, 2015;Wei, 2015, 2012;Désir et al, 2016), and is also observed when the service constraints are of Type-II (Lyu et al, 2017a). Table 3 shows that even under Type-I constraints, the long chain design can achieve most of the pooling benefit of full flexibility, and the improvement from 3-chain to full flexibility is negligible.…”

On the Design of Sparse but Efficient Structures in Operations

“…They also proposed a heuristic to find such structures. Désir et al (2016) found that for some instances of demand distribution, a disconnected network performs strictly better than the long chain. They further proved that the long chain is indeed optimal among connected networks.…”

Online Demand Fulfillment Under Limited Flexibility

“…Wang and Zhang (2015) obtain a bound on the asymptotic performance of the long chain that only depends on the mean and variance of the demand distribution. Recently, Désir et al (2016) prove the optimality of the long chain among all connected structures with the same number of arcs. Research has also been conducted on the long chain using a graph expander.…”