Optimal Sparse Designs for Process Flexibility via Probabilistic Expanders

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 an optimal design, i.e., the 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 (2013) 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 much 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. Such a result indicates k-chain only achieves 1-Ω(1/k) of the full flexibility in expectation; while our design with an average degree k achieves at least 1-exp(-Ω(k)) of full feasibility w.h.p. Finally, we conduct numerical experiments to show the superior performance of our constructions under different demand distributions