The capacity and distance constrained plant location problem

Abstract

This article introduces a new problem called the Capacity and Distance Constrained Plant Location Problem. It is an extension of the discrete capacitated plant location problem, where the customers assigned to each plant have to be packed in groups that will be served by one vehicle each. The constraints include two types of capacity. On the one hand plants are capacitated, and the demands of the customers are indivisible. On the other hand, the total distance traveled by each vehicle to serve its assigned customers in round trips plant–customer–plant is also limited. The paper addresses different modeling aspects of the problem. It describes a tabu search algorithm for its solution. Extensive computational tests indicate that the proposed heuristic consistently yields optimal or near-optimal solutions.

Introduction

One of the most determining strategic decisions in logistics concerns the location of facilities. Optimization problems for this type of decisions have been extensively studied, and a wide variety of problem specifications are now covered by the existing literature. In an important group of variants of such problems, the locations of facilities have to be chosen from a given finite set. These are known as discrete location problems [1], [2], as opposed to network or continuous location problems. As our knowledge of the basic discrete location problems improves, more attention is being paid to sophisticated versions that are closer to the needs of the real world. Thus, many authors have worked on problems that combine locational decisions with vehicle routing which are central to the design of logistic systems. Most of the advances made in this area have been summarized in the survey papers Laporte [3], Berman et al. [4], Min et al. [5], and more recently Nagy and Salhi [6].

The introduction of fleet management and routing decisions into location problems gives rise to an important increase in the difficulty of these problems. In this paper, we present a new problem that is halfway between pure discrete location and combined location-routing: the Capacity and Distance Constrained Plant Location Problem (CDCPLP). This problem captures some of the intricacies of routing decisions in location problems, but avoids some of the sources of complexity of classical combined location-routing problems.

As in the Single Source Capacitated Plant Location Problem (SSCPLP), customers are served from capacitated plants selected from a given candidate set. In addition, in the CDCPLP, each open plant houses a number of identical vehicles that will actually provide the service. It is assumed that customers are served by full return trips from the plant, but the same vehicle can be used for several services as long as its workload does not exceed a prespecified total driving distance. This situation occurs, for example, when planning health centers where patients are driven by ambulance to receive periodical treatments. Each patient must be transported from his home to a health center, and the number of hours a driver can work in a day is limited. In the CDCPLP, the goal is to select the set of plants to open, determine the number of vehicles needed at each open plant, and assign each customer to a plant and a vehicle, while ensuring that assignments are feasible both with respect to plant capacities and vehicle distance constraints and the total cost, which includes fixed costs for opening plants, fixed vehicle utilization costs and assignment costs, is minimized. We assume that all costs relate to the same planning horizon (one day, say).

We present several integer programming formulations for the CDCPLP and show how this problem relates to several well-known combinatorial optimization problems. These include location–allocation, vehicle routing, assignment, and bin packing. The problem is clearly $\mathcal{NP}$-hard, since it contains the SSCPLP as a particular case. In fact, we will show that general purpose methods fail to solve even small size instances within a reasonable CPU time. The success of tabu search (TS) [7], [8] on problems related to the CDCPLP [9], [10], [11], [12], [13] have lead us to design and implement a TS based algorithm for this new problem. Since in the case of the CDCPLP the different types of decisions to take are strongly hierarchized, we have designed a TS heuristic that respects the underlying hierarchy, as was done in Albareda-Sambola et al. [13] for a combined location-routing problem.

The remainder of this paper is organized as follows. In Section 1 we propose a variety of models and a relaxation of the CDCPLP. The different models allow us to relate the CDCPLP to other problems. In Section 2 we describe two proposed heuristics for this problem: a constructive method, and a TS improvement heuristic. Computational results are presented in Section 3. We present our conclusions in the last section.