- MIP Reformulations of the Probabilistic Set Covering Problem Anureet Saxena (anureetsandrew.cmu.edu) Vineet Goyal (vgoyalandrew.cmu.edu) Miguel A. Lejeune (mlejeuneandrew.cmu.edu) Abstract: In this paper we address the following probabilistic version (PSC) of the set covering problem: $min \{ cx \ |\ {\mathbb P} (Ax\ge \xi) \ge p,\ x_{j}\in \{0,1\}^N\}$ 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 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 polarity cuts. 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 the relaxations are established culminating with a MIP reformulation of (PSC) with no additional integer constrained variables. 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 so-called 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 fraction of seconds. Keywords: Probabilistic Programming, Set Covering, Mixed Integer Programming, Cutting Planes Category 1: Stochastic Programming Category 2: Integer Programming (0-1 Programming ) Category 3: Integer Programming (Cutting Plane Approaches ) Citation: To appear in Mathematical Programming. Download: Entry Submitted: 02/06/2007Entry Accepted: 02/06/2007Entry Last Modified: 03/20/2008Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society and by the Optimization Technology Center.