Linear and piecewise linear formulations for a hierarchical facility location and sizing problem

This paper presents a hierarchical location and sizing problem in the presence of joint partial coverage and unreliable facilities. We consider a 3-level hierarchical production-distribution system of a supply chain network in which the lowest level facilities act as the first points of contact for customers and the upper level facilities supply the lower level facilities. For the problem, we first develop an integer nonlinear program which determines the number, location, and size of two types of facilities as well as their primary and backup assignments within the network such that the weighted total demand coverage is maximized under budget constraints. Adopting a special network mapping technique, we then develop an equivalent mixed integer linear programming formulation. Next, we propose two competing piecewise linear approximations, one based on a separable programming approach and the other on a tangent line approximation method. We finally assess the performance of the three proposed formulations via numerical experiments carried out for a variety of problem instances with different sizes under three optimality gap settings and two linearization approximation error level alternatives. Our results show that both approximations are promising and outperform the exact formulation both in terms of computation time and solution quality.