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# SPEPS Preprint

 Author(s): Anureet Saxena, Carnegie Mellon UniversityVineet Goyal, Carnegie Mellon UniversityMiguel Lejeune, Carnegie Mellon University Title: MIP Reformulations of the Probabilistic Set Covering Problem Date of Acceptance: 29.05.2007 Submission Date: 08.02.2007 Series Title: Stochastic Programming E-Print Series (SPEPS) Editors: Julie L. Higle; Werner Römisch; Surrajeet Sen Keywords (eng): Probabilistic Programming, Set Covering, Mixed Integer Programming, Cutting Planes in: Mathematical Programnming Metadata export: To export the complete metadata set as Endote or Bibtex format please click to the appropriate link. Endnote   Bibtex

Abstract (eng):
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 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 ξ are pairwise independent, the distribution function of ξ 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.
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