Chance-Constrained Set Multicover Problem

TL;DR

This paper introduces a chance-constrained set multicover problem (CC-SMCP) addressing uncertainty in coverage, proposing exact reformulations, an outer-approximation algorithm, and sampling methods, with numerical validation showing superior performance in sparse scenarios.

Contribution

The paper develops a novel exact reformulation and an outer-approximation algorithm for CC-SMCP, advancing solution techniques for uncertain set covering problems.

Findings

The OA algorithm outperforms sampling methods in sparse probability scenarios.

Reformulations using log-transformation and binomial properties effectively handle special cases.

Numerical experiments confirm the practical efficiency of the proposed methods.

Abstract

We consider a variant of the set covering problem with uncertain parameters, which we refer to as the chance-constrained set multicover problem (CC-SMCP). In this problem, we assume that there is uncertainty regarding whether a selected set can cover an item, and the objective is to determine a minimum-cost combination of sets that covers each item $i$ at least $k_i$ times with a prescribed probability. To tackle CC-SMCP, we employ techniques of enumerative combinatorics, discrete probability distributions, and combinatorial optimization to derive exact equivalent deterministic reformulations that feature a hierarchy of bounds, and develop the corresponding outer-approximation (OA) algorithm. Additionally, we consider reducing the number of chance constraints via vector dominance relations and reformulate two special cases of CC-SMCP using the ``log-transformation" method and binomial distribution properties. Theoretical results on sampling-based methods, i.e., the sample average approximation (SAA) method and the importance sampling (IS) method, are also studied to approximate the true optimal value of CC-SMCP under a finite discrete probability space. Our numerical experiments demonstrate the effectiveness of the proposed OA method, particularly in scenarios with sparse probability matrices, outperforming sampling-based approaches in most cases and validating the practical applicability of our solution approaches.