A Localization and Reformulation Discrete Programming Approach for the Rectilinear Distance Location-Allocation Problem

This paper is concerned with the rectilinear distance location-allocation problem (RDLAP), which seeks the location of capacitated facilities, along with the allocation of their products to customers, so as to minimize total costs that are proportional to rectilinear distances and the amount shipped. Using a localization result, we present a new formulation of this problem as a mixed-integer bilinear programming problem. The problem is then reformulated using a reformulation linearization technique (RLT) as a linear mixed-integer problem that is shown to possess a tight linear programming relaxation. A branch-and-bound algorithm is then designed to implicitly enumerate over the location decision variable space. The special structure of the underlying linear programming relaxation is exploited to derive quick lower bounds via a suitable Lagrangian dual formulation. This methodology enables us to solve larger problem than heretofore solvable, and affords provably good quality heuristic solutions upon premature termination. An illustrative example and computational experience are provided to demonstrate the efficiency of the proposed algorithm.