Benders decomposition for the large-scale probabilistic set covering problem

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.