Optimal Design of Process Flexibility for General Production Systems

TL;DR

This paper develops a new, optimal process flexibility design for general production systems, relaxing previous assumptions and ensuring high demand fulfillment with minimal complexity.

Contribution

It introduces a simple thresholding scheme for flexible design that outperforms previous probabilistic methods in general, unbalanced production systems.

Findings

Achieves $(1-psilon)$-demand fulfillment with $O(\u2206\, ext{ln}(1/psilon))$ average degree.

Demonstrates the sub-optimality of previous weighted probabilistic constructions.

Extends classical expander graph analysis to non-uniform degree sequences.

Abstract

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