MIP reformulations of the probabilistic set covering problem

Abstract

In this paper, we address the following probabilistic version (PSC) of the set covering problem: ${\min\{cx\,|\,{\mathbb P}(Ax \ge \xi) \ge p, x \in \{0, 1\}^N\}}$ where A is a 0-1 matrix, ${\xi}$ is a random 0-1 vector and ${p \in (0,1]}$ is the threshold probability level. We introduce the concepts of p-inefficiency and polarity cuts. While the former is aimed at deriving an equivalent MIP reformulation of (PSC), the latter is used as a strengthening device to obtain a stronger formulation. Simplifications of the MIP model which result when one of the following conditions hold are briefly discussed: A is a balanced matrix, A has the circular ones property, the components of ${\xi}$ are pairwise independent, the distribution function of ${\xi}$ is a stationary distribution or has the disjunctive shattering property. We corroborate our theoretical findings by an extensive computational experiment on a test-bed consisting of almost 10,000 probabilistic instances. This test-bed was created using deterministic instances from the literature and consists of probabilistic variants of the set covering model and capacitated versions of facility location, warehouse location and k-median models. Our computational results show that our procedure is orders of magnitude faster than any of the existing approaches to solve (PSC), and in many cases can reduce hours of computing time to a fraction of a second.