Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 165476, 11 pages

http://dx.doi.org/10.1155/2015/165476

## A Problem-Reduction Evolutionary Algorithm for Solving the Capacitated Vehicle Routing Problem

^{1}College of Information Engineering, Shenzhen University, Shenzhen 518060, China^{2}Shenzhen Key Lab of Communication and Information Processing, Shenzhen 518060, China

Received 30 April 2014; Accepted 13 October 2014

Academic Editor: Pandian Vasant

Copyright © 2015 Wanfeng Liu and Xia Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Assessment of the components of a solution helps provide useful information for an optimization problem. This paper presents a new population-based problem-reduction evolutionary algorithm (PREA) based on the solution components assessment. An individual solution is regarded as being constructed by basic elements, and the concept of acceptability is introduced to evaluate them. The PREA consists of a searching phase and an evaluation phase. The acceptability of basic elements is calculated in the evaluation phase and passed to the searching phase. In the searching phase, for each individual solution, the original optimization problem is reduced to a new smaller-size problem. With the evolution of the algorithm, the number of common basic elements in the population increases until all individual solutions are exactly the same which is supposed to be the near-optimal solution of the optimization problem. The new algorithm is applied to a large variety of capacitated vehicle routing problems (CVRP) with customers up to nearly 500. Experimental results show that the proposed algorithm has the advantages of fast convergence and robustness in solution quality over the comparative algorithms.

#### 1. Introduction

The large-scale NP-hard problems with typically exponential complexity are difficult to solve in polynomial time. In practice, evolutionary-based algorithms are proposed for searching near-optimal solutions. Many different types of evolutionary algorithms have been proposed so far, for example, the genetic algorithms (GAs) [1], particle swarm optimization (PSO) [2], shuffled frog-leaping algorithm (SFLA) [3], memetic algorithms (MAs) [4], differential evolution (DE) [5], ant-colony optimization (ACO) [6], extremal optimization (EO) [7], and so on. Among these algorithms, ACO allocates and modifies the pheromones of each edge, and the ants search the new tours under the guidance of the pheromones; in the extremal optimization, the fitness is defined on the components of the feasible solution and the undesirable components are more liable to be eliminated. Both algorithms attempt to evaluate quantitatively the components of a solution, which can be used to guide the optimization process. They have found wide applications in many fields, and the idea of exploring the intrinsic properties of each component in the feasible solutions forms the foundation of our work.

In this paper, a novel problem-reduction evolutionary algorithm (PREA) is proposed. The feasible solution of the problem is supposed to be composed of a series of basic elements with their respective acceptability defined. The PREA consists of the searching phase and the evaluation phase. In the evaluation phase, the acceptability of basic elements is calculated and passed to the searching phase. The searching phase attempts to search for better solutions with a group of delicately designed encapsulation processes and optimizers. Similar to the idea of “if the backbones (i.e., the easy part of problem) are held constant, the optimization process is able to concentrate on other parts, which are more difficult to solve” in searching for backbones [8], for each individual solution, the basic elements with higher acceptability (or better components) are encapsulated as a whole. The new solution is representative of a reduced-size optimization problem. Specifically, encapsulation probability is introduced into the encapsulation which helps to adjust the searching area size of the successive optimization. The acceptability for individual solutions is also defined in the optimization phase to guide the search direction, which provides a new way for an individual to learn from others. With the evolution of the algorithm, the original optimization problem is reduced to a series of smaller-size optimization problems, thus significantly improving the convergence speed.

In this paper, the problem-reduction evolutionary algorithm is applied to the capacitated vehicle routing problem (CVRP). The CVRP proposed by Dantzig and Ramser [9] is an extension of the well-known NP-hard traveling sales man problem (TSP) and has been studied extensively in the literature. It is reported that the best exact algorithm can solve instances involving approximately 100 customers [10]. Researchers mainly focus on heuristics which can find near-optimal solutions in acceptable time. The variable neighborhood method and the simple iterated local search (ILS) are commonly used for CVRP. Li et al. [11] proposed a VRTR algorithm which combines the record-to-record (RTR) principle with a variable length neighbor list. Chen et al. [12] presented an ILS algorithm together with variable neighborhood descent based on multioperator optimization. Subramanian et al. [13] designed an ILS-based heuristic incorporating a set partitioning (SP) approach. Sequences of SP models which represent routes found by a metaheuristic approach are solved by a mixed integer programming (MIP). To overcome the limitations of a single method, hybrid algorithms have been extensively proposed; for example, Luo et al. [14] presented an improved shuffled frog leaping algorithm (SFLA) combined with the power-law extremal optimization (-EO), Nagata and Bräysy [15] put forward a memetic algorithm (MA) with edge assembly (EAX) crossover in the local search, and an efficient modification algorithm is applied to address the constraint violation of the infeasible solutions. Some techniques have been put forward to reduce the computational complexity of an algorithm. Zachariadis and Kiranoudis [16] presented a penalized static move descriptors algorithm (PSMDA), in which a static move descriptor (SMD) data structure is constructed to reduce the computational cost for evaluating the solution neighborhoods. Liu and Li [17] provided a fast feasibility evaluation of solution neighborhoods by introducing the concepts of “preload” and “postload.” In this paper, we propose a new evolutionary algorithm in which the evolution of both individual solutions and reduced problems is considered. Experimental results show that it is computationally efficient and claim to find new best known solutions to 7 well-studied CVRP instances.

The rest of the paper is organized as follows. Section 2 briefly described the capacitated vehicle routing problems with the novel coding mechanism for the feasible solution. In Section 3, the problem-reduction evolutionary algorithm is presented in detail. Experimental results and analysis are reported in Section 4, followed by the conclusion in Section 5.

#### 2. The Capacitated Vehicle Routing Problem

The capacitated vehicle routing problem (CVRP) aims to find minimum total cost routes for a fleet of vehicles to serve the given customers with known locations and demands, subject to the constraint that vehicles assigned to the routes must carry no more than a fixed quantity of goods. For the capacity and distance VRP (CDVRP), the duration of each route must not exceed an upper bound .

In this paper, the variable length codes (VLC) are used. A feasible solution for the -customers CVRP is expressed as an integer sequence: where is the coding length defined later. Equation (1) implies that the first and the last elements of represent the central depot (denoted by 0), and the other elements in between can be either the depot or any customer (denoted by its index ) which appears only once. For example, is a solution of the CVRP, and there are two routes in the solution: Route 1: 0, 1, 3, 6, 8, 0, Route 2: 0, 4, 5, 7, 2, 0.

The coding length is equal to with being the number of vehicle routes. It should be noted that the number of vehicle routes is not fixed in the proposed algorithm which means that the coding length is variable. This implies that the CVRP with the proposed coding scheme is basically a multiobjective optimization problem which needs to be solved with both the number of vehicles and the traveling costs being optimized simultaneously.

#### 3. Problem-Reduction Evolutionary Algorithm

The PREA is a population-based evolutionary algorithm with two phases: the searching phase and the evaluation phase. During the searching phase, we start from randomly chosen initial feasible solutions. They are fed into individual optimizers to find “good” solutions. The optimizers may be the same or not. In the evaluation phase, analysis of the components of solutions is performed based on two factors. One is whether the solution is good (with big fitness) or not; another is the frequency of the component appearing in the individual solutions. Of course, components which are found available in most of the “good” solutions should be “reliable” components. Next, the evaluation values are fed back to the searching phase to reformulate the problem through encapsulation which will be described in detail later. As a consequence, the problem is reduced to a smaller-size one and the individual solutions can be optimized by respective optimizer with the help of the attained evaluation values of components. Again, the results are sent to the evaluation phase. The two phases are carried out alternately until all individuals converge to the same solution which is supposed to be the near-optimal solution. The novel point of PREA lies in the evaluation of the components extracted from the individual solutions and the delicate design of the encapsulation process which reduces the optimization scale. The framework of PREA can be described in Figure 1.