A memetic algorithm for a vehicle routing problem with backhauls
Introduction
This paper considers an extension of a vehicle routing problem with backhauls (VRPB). In this problem, a set of costumers are divided in two subsets consisting of linehaul and backhaul costumers. Each linehaul costumer requires delivering its demands from the depot. In addition, a specified quantity of products should be picked up from the backhaul nodes to the depot. A good example of this costumer set can be the grocery industry. An instance of this costumer partitioning is represented by the grocery industries. In this case, supermarkets and shops are the linehaul nodes and grocery suppliers are the backhaul nodes. In recent years, it is discovered that a great amount of savings have been achieved with combining the pickup/delivery context and exactly by visiting backhaul costumer along the distribution route. For example, the Interstate Commerce Commission estimated to save $160 million each year in USA grocery industries due to the introduction of backhauling [1].
More precisely, the VRPB can be defined as a problem of determining a set of vehicle routes visiting all costumers subject to the constraints as follows: (i) each vehicle performs just one route; (ii) for each route, the total load assigns to linehaul and backhaul node, in which should not separately exceed the vehicle capacity; (iii) one each route, backhaul nodes should be visited after all linehaul routes; and (iv) the total transportation cost should be minimized. The third condition (i.e., precedence constraint) generally puts here by the fact that in many applications, linehaul costumers have higher service priority than backhauls.
The model and algorithm presented here consider the heterogeneous fleet and associated networks can be symmetric or asymmetric. In the heterogeneous fleet, different type of vehicle with dissimilar capacities may be introduced. More over in symmetric networks, the distance between two nodes is same in two directions, whereas in asymmetric network this assumption does not hold. The VRPB is NP-hard in strong sense, since it generalizes the capacitated VRP arising when there is no backhaul node available.
Section snippets
Relevant literature
All previous researches from the literature have been considered only the homogeneous fleet version of the VRP. Toth and Vigo [2] developed a branch-and-bound algorithm in which a lower bound on the optimal solution is derived from a Lagrangean relaxation of some constraints of their linear programming (LP) formulation. Iteratively, the Lagrangean relaxation bound is further strengthened by adding valid inequalities in a cutting plane fashion. Yano et al. [3] developed a set covering based on
Mathematical model
In contrast to the previous researches, in this paper a heterogeneous VRPB model is presented. The proposed model considers different types of vehicles in the fleet in terms of the capacity and transportation cost. A mixed-integer programming (MIP) formulation of the problem is presented below.
Memetic algorithm for the VRPB
Memetic Algorithms (MAs) belong to the class of evolutionary algorithms (EAs) that apply a separate local search process to refine individuals (i.e. improve their fitness by hillclimbing). These methods are inspired by models of adaptation in natural systems that combine evolutionary adaptation of populations of individuals with individual learning within a lifetime. Additionally, MAs are inspired by Dawkin’s concept of a meme [15], which represents a unit of cultural evolution that can exhibit
Experiments and computational results
This section presents experimental results. Three experiments are carried out to test the performance of the proposed memetic algorithm (MA). The first experiment evaluates the effectiveness of using the nearest neighbor algorithm in order to generate the initial population. The second experiment examines the performance of MA in comparison with the mathematical programming method. Following tests investigate in functioning MA related to different heuristic algorithms that propose to solve the
Conclusion
In this paper, we have presented a memetic algorithm (MA) to solve the vehicle routing problem with backhaul (VRPB) and heterogeneous VRPB. The proposed algorithm used a greedy heuristic method to generate initial solutions in order to improve its performance in terms of the solution quality and computational time. The proposed MA employs different types of evolutionary operators such as PMX, OX, PBX, OBX, and several mutations which have already been applied for the traveling salesman problem
Acknowledgements
The authors would like to acknowledge the Iran National Science Foundation (INSF) for the financial support of this work. We would also thank the scholars who recommended and helped through the preparation of this paper.
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