A new formulation and Benders decomposition for the multi-period maximal covering facility location problem with server uncertainty

Facility location problems reported in the literature generally assume the problem parameter values (like cost, budget, etc.) to be known with complete certainty, even if they change over time (as in multi-period versions). However, in reality, there may be some uncertainty about the exact values of these parameters. Specifically, in the context of locating primary health centers (PHCs) in developing countries, there is generally a high level of uncertainty in the availability of servers (doctors) joining the facilities in different time periods. For transparency and efficient assignment of the doctors to PHCs, it is desirable to decide the facility opening sequence (assigning doctors to unmanned PHCs) at the start of the planning horizon. We present a new formulation for a multi-period maximal covering location problem with server uncertainty. We further demonstrate the superiority of our proposed formulation over the only other formulation reported in the literature. For instances of practical size, we provide a Benders decomposition based solution method, along with several refinements. For instances that the CPLEX MIP solver could solve within a time limit of 20 hours, our proposed solution method turns out to be of the order of 150-250 times faster for the problems with complete coverage, and around 1000 times faster for gradual coverage.