Particle swarm optimization and two solution representations for solving the capacitated vehicle routing problem

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Abstract

This paper presents two solution representations and the corresponding decoding methods for solving the capacitated vehicle routing problem (CVRP) using particle swarm optimization (PSO). The first solution representation (SR-1) is a (n + 2m)-dimensional particle for CVRP with n customers and m vehicles. The decoding method for this representation starts with the transformation of particle into a priority list of customer to enter route and a priority matrix of vehicle to serve each customer. The vehicle routes are then constructed based on the customer priority list and vehicle priority matrix. The second representation (SR-2) is a 3m-dimensional particle. The decoding method for this representation starts with the transformation of particle into the vehicle orientation points and the vehicle coverage radius. The vehicle routes are constructed based on these points and radius. The proposed representations are applied using GLNPSO, a PSO algorithm with multiple social learning structures, and tested using some benchmark problems. The computational result shows that representation SR-2 is better than representation SR-1 and also competitive with other methods for solving CVRP.

Introduction

The capacitated vehicle routing problem (CVRP) introduced by Dantzig and Ramser (1959), is a problem to design a set of vehicle routes in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum cost. The CVRP can be formally defined as follows (Cordeau et al., 2002, Lysgaard et al., 2004, Prins, 2004). A set of n customers require a delivery service from a depot. Each customer i has a non-negative demand qi and a service time si. A fleet of m identical vehicles of capacity Q and service time limit D is stationed at the depot. The depot and customers locations are known; therefore, the travel distance or travel cost between two locations (dij) and travel time between two locations (tij) are also known. The CVRP consists of designing a set of at most m delivery routes such that (1) each route starts and ends at the depot, (2) each customer is visited exactly once by exactly one vehicle, (3) the total demand of each route does not exceed Q, (4) the total duration of each route (including travel and service times) does not exceed a preset limit D, and (5) the total routing cost is minimized.

The CVRP is the key operational problem of the vehicle routing problems that must be solved in the daily operation of physical distribution and logistic. Hence, studying this basic problem and methods for finding solution of the problem is essential as the foundation to learn other advanced problem in this field and develop its solution methodology.

It is known that the CVRP is an NP-hard problem (Haimovich, Rinnooy Kan, & Stougie, 1988), in which finding the optimal solution of CVRP instance is very hard and usually requires very long computational time. As a consequence, evolutionary computing methods have been applied for CVRP to find a near optimal solution in a reasonable amount of time, for example: genetic algorithm (Baker and Ayechew, 2003, Berger and Barkaoui, 2003), ant colony optimization (Bullnheimer et al., 1999, Doerner et al., 2002) and particle swarm optimization (Chen et al., 2006, Ai and Kachitvichyanukul, 2007).

Particle swarm optimization (PSO), which first proposed by Kennedy and Eberhart (1995), is a population based search method that mimics the behavior of group organism as a searching method. In the PSO, a solution of a specific problem is being represented by multi-dimensional position of a particle and a swarm of particles is working together to search the best position which correspond to the best problem solution. In each PSO iteration, every particle moves from its original position to a new position based on its velocity, where particles’ velocity is influenced by the cognitive and social information of the particles. The cognitive information of a particle is the best position that has been visited by the particle, i.e. position that provides the best objective function, and the most common social information of the particles is called the global best position, the best position that has been visited by all particles in the swarm. A comprehensive survey on PSO mechanism, technique, and application is provided by Kennedy and Eberhart (2001) and also Clerc (2006).

Two previous researches on the application of PSO to CVRP had different features and characteristics, including the benchmark problems that had been used for testing the algorithms. Chen et al. (2006) applied the discrete version of PSO and combined the method with Simulated Annealing algorithm, while Ai and Kachitvichyanukul (2007) used the classical version of PSO without any hybridization. In term of computational result, Chen’s PSO could provide high quality solution for some benchmark problems with number of customers less than 134. However, their method required significantly larger computational time and even almost reached half an hour for the slowest case. On the other hand, Ai and Kachitvichyanukul’s PSO could provide solution within relatively fast computational time for some benchmark problems with number of customers less than 199. However, there were some variations on the solution quality.

In order to make PSO applicable to CVRP, the relationship between particle position and vehicle routes must be clearly defined. The definition of particle as an encoded solution is usually called a solution representation and the method to convert it to problem specific solution is usually called a decoding method. This paper proposes two specific solution representations, namely SR-1 and SR-2, and its corresponding decoding method to convert position in PSO into CVRP solution. The solution representation SR-1 is a direct extension of the work of Ai and Kachitvichyanukul (2007), in which a local improvement procedure is added to its decoding method in order to enhance solution quality. The solution representation SR-2 is a new proposed representation which expands the basic idea of SR-1. The decoding method for SR-2 is also incorporated some simple local improvement procedures for increasing solution quality. Both representations are designed for the classic variant of PSO, which is using real value of position. Hence, these representations are different with Chen’s work which was based on a discrete-valued representation.

The remainder of this paper is organized as follow: Section 2 reviews PSO framework for solving CVRP. Section 3 explains the proposed solution representations and decoding methods. Section 4 discusses the computational experiment of PSO that applied the solution representations on benchmark data set. Finally, Section 5 summarizes the result of this study and suggests further direction in this research.

Section snippets

PSO framework for solving CVRP

The PSO framework for solving CVRP is based on GLNPSO, a PSO Algorithm with multiple social structures (Pongchairerks & Kachitvichyanukul, 2005). In this framework, the particles are initialized in step 1, evaluated its corresponding fitness value within steps 2–3, updated its cognitive and social information within steps 4–7, and moved by step 8. Step 9 is controlling step for repeating or stopping the iteration. Note that the adjustment of this framework from the original GLNPSO algorithm are

Solution representation SR-1

The solution representation SR-1 of CVRP with n customers and m vehicles consists of (n + 2m) dimensional particle. Each particle dimension is encoded as a real number. The first n dimensions are related to customers, each customer is represented by one dimension. The last 2m dimensions are related to vehicles, each vehicle is represented by two dimensions as the reference point in Cartesian map. This solution representation is first proposed by Ai and Kachitvichyanukul (2007).

The decoding method

Computational result

Two set of computational experiments are conducted to test the performance of the PSO with the two solution representations for solving the CVRP. The first set of experiment is performed in order to compare the result of these proposed methods with PSO of Chen et al. (2006). In this experiment, the proposed methods are applied to the same sixteen benchmark problems that had been used by Chen. The second set of experiment is conducted in order to evaluate performance of these methods for the

Conclusion

This paper presents two solution representations, SR-1 and SR-2, and the corresponding decoding methods for solving the capacitated vehicle routing problem (CVRP) using particle swarm optimization (PSO). The representation SR-1 is a (n + 2m)-dimensional particle for CVRP with n customers and m vehicles. The representation SR-2 is a 3m-dimensional particle. The proposed representations are applied using a framework based on GLNPSO, a PSO algorithm with multiple social learning structures, and

Acknowledgements

The authors wish to thank the High Performance Computing Group at Asian Institute of Technology (AITHPC) and the Thai GRID Center for the access of computing facility and the technical support.

The first author expresses gratitude to Universitas Atma Jaya Yogyakarta, Indonesia for providing the financial support for his study in Asian Institute of Technology, Thailand.

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