An adaptive memory programming metaheuristic for the heterogeneous fixed fleet vehicle routing problem Xiangyong Li a, * , Peng Tian a , Y.P. Aneja b a Antai College of Economics & Management, Shanghai Jiao Tong University, Shanghai 200052, China b Odette School of Business, University of Windsor, Windsor, Ontario, Canada N9B 3P4 article info Article history: Received 16 January 2009 Received in revised form 6 December 2009 Accepted 3 February 2010 Keywords: Vehicle routing Heterogeneous fixed fleet Adaptive memory programming Path relinking Metaheuristic abstract This paper studies the heterogeneous fixed fleet vehicle routing problem (HFFVRP), in which the fleet is composed of a fixed number of vehicles with different capacities, fixed costs, and variable costs. Given the fleet composition, the HFFVRP is to determine a vehicle scheduling strategy with the objective of minimizing the total transportation cost. We pro- pose a multistart adaptive memory programming (MAMP) and path relinking algorithm to solve this problem. Through the search memory, MAMP at each iteration constructs multi- ple provisional solutions, which are further improved by a modified tabu search. As an intensification strategy, path relinking is integrated to enhance the performance of MAMP. We conduct a series of experiments to evaluate and demonstrate the effectiveness of the proposed algorithm. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction In this paper, we consider the heterogeneous fixed fleet vehicle routing problem (HFFVRP). Typically, the classical vehicle routing problem (VRP) has a fleet of homogeneous vehicles, which have the same capacity, fixed cost and unit variable cost. Moreover, the number of available vehicles is also assumed to be unlimited. Nevertheless, this assumption cannot cover the common practice in the distribution and logistics management. In general, vehicles may have different capacities, fixed costs, or variable costs. Additionally, because of resource constraints, there is a limited number of vehicles available for the delivery or collection task. Hence we must decide how to make the best use of a fixed fleet of heterogeneous vehicles. The HFFVRP is a more general case in practical distribution and transportation (Baldacci et al., 2008). From a graph theory point of view, we can define the HFFVRP by a digraph G ¼ðV ; AÞ where V ¼f0; 1; ... ; ng denotes node set, and A ¼fði; jÞ : i; j 2 V ; i – jg is arc set. Node 0 denotes the depot and other n nodes constitute a customer set C ¼f1; 2; ... ; ng. Each customer i has a demand q i and service time d i . With each arc ði; jÞ2 A, we associate a distance d ij . In the HFFVRP, there are several different vehicle types. For vehicle type k, the capacity is Q k , the fixed cost is f k , the variable cost per unit distance is v k , and the route-length restriction is L k . The number of available vehicles of type k is fixed and equals n k . We also assume that distance d ij and service time d i have the same dimension. Given a fleet composition, the HFFVRP is to determine a vehicle scheduling strategy such that the total transportation cost (fixed plus variable cost) is min- imized. Each vehicle route must satisfy some general assumptions: (1) Each route starts from the depot and returns to the depot after finishing the service for the last customer. (2) The demand of each customer must be fulfilled by one vehicle, exactly once. 1366-5545/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.tre.2010.02.004 * Corresponding author. Fax: +86 21 6293 2577. E-mail address: lixiangyong@163.com (X. Li). Transportation Research Part E 46 (2010) 1111–1127 Contents lists available at ScienceDirect Transportation Research Part E journal homepage: www.elsevier.com/locate/tre