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 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.