Exploring K-best solutions to enrich network design decision-making

In this paper, we explore alternative solutions to the Capacitated Fixed Charge Facility Location problem (CFCFL) that usually arises in Supply Chain Network Design problems. More specifically, we aim to investigate in which cases these solutions can be considered as good as the optimal one from the point of view of decision-making in real-world problems. A method, as well as four enhancement variations, based on a mixed-integer programming (MIP) model is proposed, which allows K-best alternative solutions to be obtained. The method and its variations were applied to two benchmark instance sets available in the literature and the computational times were evaluated. The results have shown that the gap between the optimal solutions and the 20-best alternative ones were, on average, less than 1%; more surprisingly, 63.8% of all these alternative solutions had a gap smaller than 0.5%. This suggests that our approach may be used to identify whether near-optimal alternative solutions can yield to a better overall solution from the point of view of the decision-maker, by allowing other qualitative attributes to be considered. We were also able to rate the robustness of some selected facilities since many candidates have appeared in all 20 best solutions. In addition, the results may also suggest a way to measure the difficulty of benchmark instances for combinatorial problems and thus enhance the comparison of different heuristics proposed to solve them; not to mention that the uncertainty in input data of such strategic problems may reduce the relevance of the effort to find the best solution in the contexts in which several high-quality solutions arise.