Optimal Sparse Designs for Process Flexibility via Probabilistic Expanders

Abstract

We study the problem of how to design a sparse flexible process structure in a balanced and symmetrical production system to match supply with random demand more effectively. Our goal is to provide a sparsest design to achieve (1 − ϵ)-optimality relative to the fully flexible system. In a balanced system with n plants and n products, Chou et al. (2011) proved that there exists a graph expander with O(n/ϵ) arcs to achieve (1 − ϵ)-optimality for every demand realization. Wang and Zhang (2015) showed that the simple k-chain design with O(n/ϵ) arcs can achieve (1 − ϵ)-optimality in expectation. In this paper, we introduce a new concept called probabilistic graph expanders. We prove that a probabilistic expander with O(n ln(1/ϵ)) arcs guarantees (1 − ϵ)-optimality with high probability (w.h.p.), which is a stronger notion of optimality as compared to the expected performance. Easy-to-implement randomized and deterministic constructions of probabilistic expanders are provided. We show our bound is best possible in the sense that any structure should need at least Ω(n ln(1/ϵ)) arcs to achieve (1 − ϵ)-optimality in expectation (and hence w.h.p.). We also show that in order to achieve (1 − ϵ)-optimality in the worst case, any design would need at least Ω(n/ϵ) arcs; and in order to achieve (1 − ϵ)-optimality in expectation, k-chain needs at least Ω(n/ϵ) arcs.

Optimal Design of Process Flexibility for General Production Systems

“…to a few generalized expansion properties of graphs constructed using the TPC (see Section 4.1). These generalized expansion properties extend the notion of "probabilistic expanders" in Chen et al (2015). In fact, they can be viewed as a continuous generalization of the probabilistic expansion property, which is based on the cardinality of a set of nodes.…”

“…By choosing the linkage probability so that each node has an average degree of O(ln(1/ )), Chen et al (2015) showed that the uniform probabilistic construction achieves (1 − )optimality w.h.p. Moreover, such a construction is asymptotically optimal, which means it has the fewest possible edges for achieving (1 − )-optimality w.h.p.…”

“…Due to the importance of this problem in manufacturing, a number of papers tackle this question by constructing sparse designs and analyzing their performances (see, e.g., Chou et al (2010), Chou et al (2011), Simchi-Levi and Wei (2012), Simchi-Levi and Wei (2015), Deng and Shen (2013), Wang and Zhang (2015), Chen et al (2015), Shi et al (2015), Tsitsiklis and Xu (2015), Bidkhori et al (2016), Désir et al (2016), and references therein). Most papers study the special setting where (1) the production system 2 is balanced (the number of supply and demand nodes are equal, i.e., m = n) and (2) symmetrical (the random supplies s(u) share the same deterministic capacity or are identically independently distributed (i.i.d.)…”

Reliable Flexibility Design of Supply Chains Via Extended Probabilistic Expanders

“…The expansion property has been used as a surrogate metric for structure connectivity, and has been proved to be effective in balanced flexibility design problems (Chen et al 2015, Chou et al 2011. In this section, we show that the expansion property, with a proper extension, is also critical to reliable flexibility design problems.…”

“…They prove by a randomized algorithm that there exist graph expanders with O(n/) edges that achieve (1 À )-optimality in the worst case in balanced and symmetric systems with bounded demands. Chen et al (2015) develop a new variant of the graph expandersprobabilistic expanders, and show that the probabilistic expanders are sparser than the graph expanders, with only O(n ln (1/)) edges necessary, to achieve (1 À )-optimality with high probability, which is a stronger concept of optimality relative to the (1 À )-optimality in expectation.…”