The minimal covering location and sizing problem in the presence of gradual cooperative coverage

This paper introduces the capacitated gradual and cooperative minimal covering location problem with distance constraints (cGC-MCLPD). The cGC-MCLPD extends the location literature by implementing the concepts of gradual and cooperative coverage in the context of undesirable facility location problem with distance constraints. It also allows for variable coverage radii and capacity of facilities to assess the effect of facility size on the network performance. For the defined problem, we first develop a nonlinear mathematical model which seeks to determine the number, location and size of facilities such that the total population covered is minimized while the overall service requirement is met. Next, we propose three integer linear programming formulations that can be solved with off-the-shelf solvers. The first two are linear approximations that are based on a separable programming approach and a tangent line approximation method. The third is an exact reformulation which uses a special network mapping technique. Upon investigating the impact of linearization approximation error on the performance of the first two formulations, we carry out numerical experiments to compare formulations with respect to their solution time and quality. Solving them for a set of reasonably large problem instances, we found that approximations outperform the exact reformulation since they prove to achieve higher quality solutions at the expense of an acceptable level of objective function value error. Overall, the formulations developed for the cGC-MCLPD constitute a powerful portfolio of facility location selection techniques, enabling decision-makers to select the most appropriate balance of solution quality and computational speed.