Optimization approaches for a fair multi-source capacitated facility location problem

We incorporate a fairness measure, known as the conditional β -mean in location theory, into the Multi-Source Capacitated Facility Location Problem (MSCFLP). The conditional β -mean measures the average distance traveled by the (100 · β )% of demand units that face the longest travel distances to a facility. Minimizing the conditional β -mean results in a less conservative and more flexible approach than a standard maximum distance minimization. We term the resulting problem the Fair MSCFLP, and we develop five mathematical formulations to model it: two bilevel models, one bilinear model, and two Mixed-Integer Linear Programming (MILP) formulations. We also devise two matheuristics: a Variable MILP Neighborhood Descent ( VMND ) algorithm, and a ternary search-based approach ( Ternary ). Extensive computational experiments conducted on benchmark data sets for the MSCFLP, consisting of up to 1000 facilities and 1000 customers, show that an off-the-shelf solver can solve, in reasonable computing times, only small instances. Both matheuristics found the optimal solution to each small instance within a few seconds. The experimental analysis indicates that no matheuristic outperforms the other across all instances: depending on the size, one matheuristic generally outperforms the other when run with a time limit. Ternary usually outperforms VMND solving instances that comprise up to 500 facilities and 500 customers, whereas VMND produces solutions of considerably better quality on larger instances.