A tabu search heuristic for the multi-depot vehicle routing problem

-This article describes a tabu search algorithm for the multi-depot vehicle routing problem with capacity and route length restrictions. The algorithm is tested on a set of 23 benchmark instances. It is shown to outperform existing heuristics. 1. I N T R O D U C T I O N The purpose of this article is to describe a new heuristic algorithm for the Mult i -Depot Vehicle Routing Problem (MDVRP) defined as follows. Let G = ( V, E) be a graph where V is the vertex set and E is the edge set. The vertex set V is partitioned into two subsets Vc = {Vl,.. . ,vn} and Vd = {Vn+l, . . . , Vn+p} representing, respectively, the set of cities or customers, and the set of depots. With each city v; ¢Vc is associated a non-negative demand qi and a service time ~i. A cost matrix C = (cij) corresponding to travel times is defined on E. We restrict our attention to problems for which C is symmetric and satisfies the triangle inequality, i.e. c U = cji for all i, j and cik <<, cij + cjk for all i, j , k. At each depot Vn+k C Vd are based mk identical vehicles of capacity Q, where mk belongs to some interval [m__k, Nk]. Here we assume that m__k = 0 and Nk = n ( k = 1 , . . . ,p), in other words not all depots are necessarily used. The MDVRP consists of constructing a set of vehicle routes in such a way that: (1) each route starts and ends at the same depot, (2) each customer is visited exactly once by a vehicle, (3) the total demand of each route does not exceed the vehicle capacity Q, (4) the total duration of each route (including travel and service time) does not exceed a preset limit L and (5) the total routing cost is minimized. The MDVRP is encountered in a large variety of contexts and has considerable economic importance. Documented case studies include the delivery of meals [1], of chemical products [2], of soft drinks [3], of machines [4], of industrial gases [5] of petroleum products [6], of packaged food [7], etc. These studies show that substantial savings can be achieved through the use of optimization techniques. In addition, the MDVRP arises naturally in a family of inventory-routing problems (see, t Jacques Renaud is Assistant Professor at Tflf-Universitf, Quebec City. This paper is part of his Ph.D. thesis on vehicle routing problems. {Gilbert Laporte is Professor at the Ecole des Hautes ]~tudes Commerciales, Montreal. His research interests lie in the fields of vehicle routing, location, and scheduling. He is currently the Editor of Transportation Science. §To whom correspondence should be addressed. ¶Fayez F. Boctor is Professor at the Faculty of Administration, Universit6 Lava1. His research interests include production and logistics problems.