Capacitated Mobile Facility Location Problem with Mobile Demand: Efficient relief aid provision to en route refugees

As a humanity crisis, the tragedy of forced displacement entails relief aid distribution efforts among en route refugee to alleviate their migration hardships. This study aims to assist humanitarian organizations in cost-efficiently optimizing the logistics of capacitated mobile facilities utilized to deliver relief aid to transiting refugees in a multi-period setting. The problem is referred to as the Capacitated Mobile Facility Location Problem with Mobile Demands (CMFLP-MD). In CMFLP-MD, refugee groups follow specific paths, and meanwhile, they receive relief aid at least once every fixed number of consecutive periods, maintaining continuity of service. To this end, the overall costs associated with capacitated mobile facilities, including fixed, service provision, and relocation costs, are minimized. We formulate a mixed integer linear programming (MILP) model and propose two solution methods to solve this complex problem: an accelerated Benders decomposition approach as an exact solution method and a matheuristic algorithm that relies on an enhanced fix-and-optimize agenda. We evaluate our methodologies by designing realistic instances based on the Honduras migration crisis that commenced in 2018. Our numerical results reveal that the accelerated Benders decomposition excels MILP with a 46% run time improvement on average while acquiring solutions at least as good as the MILP across all instances. Moreover, our matheuristic acquires high-quality solutions with a 2.4% average gap compared to best-incumbents rapidly. An in-depth exploration of the solution properties underscores the robustness of our relief distribution plans under varying migration circumstances. Across several metrics, our sensitivity analyses also highlight the managerial advantages of implementing CMFLP-MD solutions.