MIP reformulations of the probabilistic set covering problem
 Anureet Saxena,
 Vineet Goyal,
 Miguel A. Lejeune
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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 01 matrix, ${\xi}$ is a random 01 vector and ${p \in (0,1]}$ is the threshold probability level. We introduce the concepts of pinefficiency 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 testbed consisting of almost 10,000 probabilistic instances. This testbed 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 kmedian 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.
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 Title
 MIP reformulations of the probabilistic set covering problem
 Journal

Mathematical Programming
Volume 121, Issue 1 , pp 131
 Cover Date
 20100101
 DOI
 10.1007/s101070080224y
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Probabilistic programming
 Set covering
 Mixed integer programming
 Cutting planes
 90C15
 9008
 90C10
 Industry Sectors
 Authors

 Anureet Saxena ^{(1)}
 Vineet Goyal ^{(1)}
 Miguel A. Lejeune ^{(2)}
 Author Affiliations

 1. Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, 15213, USA
 2. LeBow College of Business, Drexel University, Philadelphia, PA, 19104, USA