Elsevier

Neurocomputing

Volume 150, Part A, 20 February 2015, Pages 58-66
Neurocomputing

An improved immigration memetic algorithm for solving the heterogeneous fixed fleet vehicle routing problem

https://doi.org/10.1016/j.neucom.2014.02.074Get rights and content

Abstract

This paper deals with the heterogeneous fixed fleet vehicle routing problem (HFFVRP) which is a generalization of the classical vehicle routing problem (VRP) in the sense that the fixed fleet of vehicles is assumed to be heterogeneous. The objective of HFFVRP is to find the best fleet composition and the collection of routes such that the total costs are minimized. To address this combinatorial optimization problem, we design and implement a hybrid heuristic model integrating a genetic algorithm, a local search mechanism and an immigration strategy. Several strategies for generating the initial population of the genetic algorithm in relation with six local search heuristics are considered. An important feature of the proposed approach refers to the immigration strategy used to ensure diversification by which the level of evolution for the new immigrant individuals increases along with the evolution of the population. The proposed algorithm is tested on a set of HFFVRP benchmark instances and the preliminary results point out that our approach is an attractive and appropriate method to explore the solution space of this complex problem leading to good solutions within reasonable computational times.

Introduction

Problems associated with determining optimal routes for vehicles from one or several depots to a set of locations/customers are known as Vehicle Routing Problems (VRPs) and have many practical applications in the field of distribution and logistics. A wide body of literature exists on the problem (for an extensive bibliography, see Laporte [9], [10] and the book edited by Ball et al. [1]).

Given a set of vehicles, a set of locations containing the depot location and the distance between each pair of locations, the VRP consists in finding the minimum cost tour for each vehicle such that all locations are visited and each vehicle returns to the depot. Because of the VRP simplicity, most attractive to many researchers have been the variations of the VRP, built on the basic VRP with extra features such as:

  • The Capacitated VRP [25] in which each vehicle has finite capacity and each location has a finite demand.

  • The VRP with Time Windows [19] in which there is a specified temporal window of opportunity in which to visit each location.

  • The VRP with Multiple Depots [3] generalizes the idea of a depot in such a way that there are several depots from which each customer can be served.

  • The Heterogeneous Fixed Fleet VRP [22] in which we have a fleet of heterogeneous (different types) vehicles using the depot as a starting base.

  • The Multi-Commodity VRP [18] in which each location has a demand for different commodities and each vehicle has a set of compartments in which only one commodity can be loaded. The problem then becomes that of deciding which commodities to place in which compartments in order to minimize distance traveled.

  • The Generalized Vehicle Routing Problem (GVRP) [5], [15] is the problem of designing optimal delivery or collection routes, subject to capacity restrictions, from a given depot to a number of predefined, mutually exclusive and exhaustive node-sets (clusters) with the property that exactly one node is visited from each cluster.

We are concerned in this paper with the heterogeneous fixed fleet vehicle routing problem (HFFVRP) introduced by Taillard [22]. The HFFVRP is an important variant of VRP, since usually the fleets are heterogeneous in most practical situations. Various heuristic and metaheuristic algorithms have been developed for solving the HFFVRP including an algorithm based on Tabu Search, adaptive memory and column generation described by Taillard [22], an algorithm that extends a number of VRP classical heuristics followed by a local search procedure based on the Steepest Descent Local Search and Tabu Search introduced by Prins [16], a threshold accepting procedure implemented by Tarantilis et al. [23] where a worse solution is only accepted if it is within a given threshold (the same authors [24] later presented another threshold accepting procedure to solve the same problem). Also, a record-to-record travel algorithm was proposed by Li et al. [11] and a multi-start adaptive memory procedure combined with Path Relinking and a modified Tabu Search was developed by Li et al. [12]. More recently, Brandao [2] proposed a Tabu Search algorithm for the HFFVRP which includes additional features such as strategic oscillation, shaking and frequency-based memory while Subramanian et al. [21] described a hybrid algorithm composed by an Iterated Local Search based heuristic and Set Partitioning formulation.

In this paper, we present an efficient memetic algorithm for solving the HFFVRP, obtained by combining an immigration-based genetic algorithm with a powerful local search procedure. The genetic algorithm is endorsed by an immigration strategy designed to ensure diversification of the genetic material by inserting new individuals (called immigrants) into the population every generation. The immigrants are not random but generated based on some heuristics allowing the evolution of immigrants along with the evolution of the main population. The initialization of population takes into account several strategies for inserting the route splitters leading to four different approaches tested. The strength of the local search mechanism integrated in the evolutionary process is ensured by six local search heuristics and their diversity. Computational experiments are performed for a set of HFFVRP benchmark instances and the results are presented, analyzed and compared with existing heuristic methods. The experimental results reveal that the solutions provided by the proposed memetic algorithm are of high quality and competent to those existing in the literature.

The rest of paper is structured as follows: the definition of the HFFVRP is presented in Section 2; the proposed memetic algorithm is described in Section 3 focusing on the general framework of the genetic algorithm (solution representation, fitness function, genetic search operators and strategies for population initialization), the immigration techniques integrated in the algorithm and the local search procedure; computational experiments and results are discussed in Section 4 and the conclusions of the study are depicted in Section 5.

Section snippets

Definition of the problem

Formally, the HFFVRP is defined on a directed graph G=(V,A) with V={0,1,2,,n} as the set of nodes, the set of arcs A={(i,j)|i,jV,ij} and a nonnegative distance cij associated with each arc (i,j)A. The set of nodes consists of vertex v=0 which represents the depot and the vertices v=1,,n which represent the customers. Each customer has a certain non-negative amount of demand. There exists a fleet of heterogeneous (different types) vehicles that are using the depot as a starting base. We

An improved immigration memetic algorithm for solving the HFFVRP

Memetic algorithms have been introduced by Moscato [14] to denote a family of metaheuristic algorithms that use a population-based approach with separate individual learning or local improvement procedures for problem search. Therefore, a memetic algorithm is a genetic algorithm (GA) hybridized with a local search procedure to intensify the search.

Genetic algorithms are not well suited for fine-tuning structures which are close to optimal solutions. Therefore, incorporating local improvement

Computational experiments and results

In order to assess the performance of the proposed memetic algorithm for solving the HFFVRP, we conducted experiments on two well-known sets of instances: one reported by Taillard [22] and the second one introduced by Li et al. [11].

For the computational experiments, we performed 30 independent runs of the proposed memetic algorithm for each problem instance. Results are analyzed based on the best and average solutions obtained over the 30 runs. The machine used for our experiments is equipped

Conclusions and future work

In this paper, we developed an improved immigration memetic algorithm for solving the heterogeneous fixed fleet vehicle routing problem (HFFVRP). The proposed hybrid heuristic integrates a number of original features: we combine a genetic algorithm with a powerful local search procedure and an immigration approach, the initial population of the GA is constructed using four different approaches and the search is further guided by generalized mutation operator. We used an efficient way of

Acknowledgments

This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS - UEFISCDI, project number PN-II-RU-TE-2011-3-0113. The authors are grateful to the anonymous referees for reading the manuscript very carefully and providing constructive comments which helped to improve substantially the paper.

Oliviu Matei obtained his Ph.D. in the field of practical applications of evolutionary computing in solving combinatorial optimization problems from Technical University Cluj-Napoca. Before that, he studied Artificial Intelligence at Vrije Universiteit, Amsterdam. Currently, he is a member of the Department of Electrical Engineering at North University Center of Baia Mare, Technical University of Cluj-Napoca, Romania. His research interests focus on evolutionary and natural computing methods

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    Oliviu Matei obtained his Ph.D. in the field of practical applications of evolutionary computing in solving combinatorial optimization problems from Technical University Cluj-Napoca. Before that, he studied Artificial Intelligence at Vrije Universiteit, Amsterdam. Currently, he is a member of the Department of Electrical Engineering at North University Center of Baia Mare, Technical University of Cluj-Napoca, Romania. His research interests focus on evolutionary and natural computing methods and their application to real-world optimization problems.

    Petrică C. Pop is a professor within the Department of Mathematics and Computer Science at North University Center of Baia Mare, Technical University of Cluj-Napoca, Romania. He published more than 80 scientific papers, majority of them being published in Theoretical Computer Science, Neurocomputing, Applied Mathematical Modelling, European Journal of Operational Research, Annals of Operations Research, Advances in Intelligent Systems and Computing, Lecture Notes in Computer Science. His scientific contributions have a very good visibility with more than 155 citations in ISI journals and international journals. He is a member of the Editorial Board of 5 international journals. His research interests include metaheuristics, nature-inspired computing and combinatorial optimization problems.

    Jozsef Laszlo Sas received B.S. degree in computer engineering from the North University Center at Baia Mare, Technical University of Cluj Napoca, Romania. Currently, he is a Master student in Informatics and Software Engineering at the same university and a junior software developer in Baia Mare.

    Camelia Chira received the M.Sc. (2002) and Ph.D. (2005) degrees from Galway-Mayo Institute of Technology, Ireland, in the area of agent-based systems for distributed collaborative design environments. From 2006 to 2013, Camelia was a researcher at Babes-Bolyai University, Romania, working on several research projects both as a member and principal investigator. She is now with Technical University of Cluj-Napoca, Department of Computer Science, Romania. Her main research interests include nature-inspired computing, complex systems and networks, multi-agent systems and bioinformatics. She published more than 70 papers in international conferences and ISI journals.

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