Heuristics for a continuous multi-facility location problem with demand regions

Abstract

We consider a continuous multi-facility location problem where the demanding entities are regions in the plane instead of points. Each region may consist of a finite or an infinite number of points. The service point of a station can be anywhere in the region that is assigned to it. We do not allow fractional assignments, that is, each region is assigned to exactly one facility. The problem we consider can be stated as follows: given m demand regions in the plane, find the locations of q facilities and allocate regions to the facilities so as to minimize the sum of squares of the maximum Euclidean distances of the demand regions to the facility locations they are assigned to. We assume that the regions are closed polygons as any region can be approximated within any desired accuracy with a polygon.We first propose mathematical programming formulations of single and multiple facility location problems. The single facility location problem is formulated as a second order cone program (SOCP) which can be solved in polynomial time. The multiple facility location problem is formulated as a mixed integer SOCP. This formulation is weak and does not even solve medium-size problems. We therefore propose heuristics to solve larger instances of the problem. We develop three heuristics that work when the regions are polygons. When the demand regions are rectangles with sides parallel to coordinate axes, a special heuristic is developed. We compare our heuristics in terms of both solution quality and computational time.

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“…Since then, this problem has become a contentious issue of debate among scientists and researchers. A large number of studies on location-allocation problems have been conducted in the literature; for instance, incapacitated single allocation planar hub location problem (Damgacioglu et al, 2015), continuous multi-facility location allocation problem (Dinler et al, 2014), multi-period location-allocation problem of engineering emergency blood supply systems (Sha and Huang, 2012), multi-source facility location-allocation and inventory problem (Yao et al, 2010), a dynamic multi-period location-allocation problem (Gebennini et al, 2009) and a multi-objective facility location-allocation problem (Arabzad et al, 2015).…”

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