Benders decomposition for the large-scale probabilistic set covering problem

TL;DR

This paper introduces a Benders decomposition algorithm tailored for large-scale probabilistic set covering problems, effectively handling scenario complexity and providing high-quality solutions efficiently.

Contribution

The paper develops a novel Benders decomposition approach that reduces scenario dependence and enhances solution quality for large-scale probabilistic set covering problems.

Findings

The algorithm outperforms standard MIP solvers in efficiency.

It can solve instances with up to 500 rows, 5000 columns, and 2000 scenarios.

Enhanced cuts and heuristics improve solution quality.

Abstract

In this paper, we consider a probabilistic set covering problem (PSCP) in which each 0-1 row of the constraint matrix is random with a finite discrete distribution, and the objective is to minimize the total cost of the selected columns such that each row is covered with a prespecified probability. We develop an effective decomposition algorithm for the PSCP based on the Benders reformulation of a standard mixed integer programming (MIP) formulation. The proposed Benders decomposition (BD) algorithm enjoys two key advantages: (i) the number of variables in the underlying Benders reformulation is equal to the number of columns but independent of the number of scenarios of the random data; and (ii) the Benders feasibility cuts can be separated by an efficient polynomial-time algorithm, which makes it particularly suitable for solving large-scale PSCPs. We enhance the BD algorithm by using initial cuts to strengthen the relaxed master problem, implementing an effective heuristic procedure to find high-quality feasible solutions, and adding mixed integer rounding enhanced Benders feasibility cuts to tighten the problem formulation. Numerical results demonstrate the efficiency of the proposed BD algorithm over a state-of-the-art MIP solver. Moreover, the proposed BD algorithm can efficiently identify optimal solutions for instances with up to 500 rows, 5000 columns, and 2000 scenarios of the random rows.