Mathematical formulations and a Relax-and-Fix heuristic algorithm for capacitated reliable fixed-charge facility location problems.

The reliable fixed-charge facility location problem extends the fixed-cost facility location problem by incorporating facility unreliability. This paper addresses a novel capacitated version of the reliable fixed-charge facility location problem, where the failure probability of each facility is site-dependent, differing from existing literature models. Additionally, facilities are assumed to have limited capacity for demand allocation, with the option to increase capacity to a predetermined value in case of supply chain disruptions. To solve this problem, we develop a non-linear mixed-integer programming formulation and present its linear version. Given the NP-hard nature of the problem, we propose a novel Relax-and-Fix heuristic for its solution. We evaluate the efficiency of the proposed algorithm by solving a variety of experimental instances with different network sizes. Results demonstrate that the Relax-and-Fix heuristic improves upper bounds for problem instances and achieves this within a shorter computational time. Furthermore, sensitivity analysis is conducted on capacity and failure probabilities, and relevant results are presented.