A Capacitated and distance-limited continuous location-allocation problem for spatial planning of decentralized distribution systems

Abstract

We introduce a new continuous location-allocation problem with capacitated and distance-limited facilities for spatial planning of decentralized distribution systems to contribute the green eld development. Our aim is to determine the number, location and allocation con gurations of facilities to be opened as stand-alone systems such as solar panels, windmills, diesel generators, water pumps and etc. The problem minimizes the sum of xed facility opening costs and distribution costs such as cable, pipeline or transportation costs which are linearly dependent to their physical lengths. To be able to model some physical facts such as energy loss, voltage drop, pressure drop and etc., facilities to be opened are assumed to be distance limited. Due to the fact that green eld areas are undeveloped regions and wide open spaces, where generally no previous structures exist, facilities can be located at almost any site in the eld. Therefore, the domain of the problem is not restricted by the discrete space and it is subject to the continuous space. We provide a quadratically constrained mixed integer programming model and a heuristic approach for the problem. In the heuristic, rstly discrete counterpart of the problem is solved with augmented set of candidate location points. Alternative augmented sets of candidate locations are formed in accordance with the spatial distribution of demand points. After solving the discrete counterpart of the problem with augmented domain, Cooper's well known heuristic that alternates between allocation and location steps is followed to nd better allocation con guration of customers to the facilities. Since the facilities considered in the problem are capacitated, some of the demand points may not be assigned to their nearest facility. Therefore, allocation steps are performed with respect to the priority measure of each demand point based on a regret function. After each allocation step, to determine the new locations of the facilities, Weiszfeld's iterative convergence algorithm is applied with some modi cations and adjustments to conform the distance limitation constraints.