MIP Reformulations of the Probabilistic Set Covering Problem

Abstract

In this paper we address the following probabilistic version (PSC) of the set cover-ing problem: $ min{cx | P(Ax ≥ ξ) ≥ p, x_j \in {0, 1}N }$ where A is a 0-1 matrix, ξ is arandom 0-1 vector and $p \in (0, 1]$ is the threshold probability level. We formulate (PSC)as a mixed integer non-linear program (MINLP) and linearize the resulting (MINLP)to obtain a MIP reformulation. We introduce the concepts of p-inefficiency and polaritycuts. While the former is aimed at reducing the number of constraints in our model,the later is used as a strengthening device to obtain stronger formulations. A hierarchy of relaxations for (PSC) is introduced, and fundamental relationships between therelaxations are established culminating with a MIP reformulation of (PSC) with noadditional integer constrained variables. Simplifications of the MIP model which resultwhen one of the following conditions hold are briefly discussed: A is a balanced matrix,A has the circular ones property, the components of ξ are pairwise independent, thedistribution function of ξ is a stationary distribution or has the so-called disjunctiveshattering 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 consistsof probabilistic variants of the set-covering model and capacitated versions of facilitylocation, warehouse location and k-median models. Our computational results showthat our procedure is orders of magnitude faster than any of the existing approachesto solve (PSC), and in many cases can reduce hours of computing time to fraction ofseconds.