MIP reformulations of the probabilistic set covering problem

• 成果类型: Article
• 署名作者: Saxena, Anureet; Goyal, Vineet; Lejeune, Miguel A.
• 署名单位: Drexel University; Carnegie Mellon University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-008-0224-y
• 发表日期: 2010
• 页码: 1-31
• 关键词: programs

摘要

In this paper, we address the following probabilistic version (PSC) of the set covering problem: min{cx vertical bar P(Ax >= xi) >= p, x is an element of {0, 1}(N)} where A is a 0-1 matrix, xi is a random 0-1 vector and p is an element of (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.