Elsevier

Applied Soft Computing

Volume 19, June 2014, Pages 343-361
Applied Soft Computing

An Improved Decomposition-Based Memetic Algorithm for Multi-Objective Capacitated Arc Routing Problem

https://doi.org/10.1016/j.asoc.2014.03.005Get rights and content

Highlights

  • ID-MAENS is proposed in this paper.

  • The replacement accelerates the convergence speed.

  • The implemented elitism maintains the diversity of the solutions.

  • The experimental results show the effectiveness of the proposed algorithm.

Abstract

Capacitated Arc Routing Problem (CARP) has attracted the attention of many researchers during the last few years, because it has a wide application in the real world. Recently, a Decomposition-Based Memetic Algorithm for Multi-Objective CARP (D-MAENS) has been demonstrated to be a competitive approach. However, the replacement mechanism and the assignment mechanism of the offspring in D-MAENS remain to be improved. First, the replacement after all the offspring are generated decreases the convergence speed of D-MAENS. Second, the representatives of these sub-problems are reassigned at each generation by only considering one objective function. In response to these issues, this paper presents an improved D-MAENS for Multi-Objective CARP (ID-MAENS). The two improvements of the proposed algorithm are as follows: (1) the replacement of the solutions is immediately done once an offspring is generated, which references to the steady-state evolutionary algorithm. The new offspring will accelerate the convergence speed; (2) elitism is implemented by using an archive to maintain the current best solution in its decomposition direction during the search, and these elite solutions can provide helpful information for solving their neighbor sub-problems by cooperation. Compared with the Multi-Objective CARP algorithm, experimental results on large-scale benchmark instances egl show that the proposed algorithm has performed significantly better than D-MAENS on 23 out of the total 24 instances. Moreover, ID-MAENS find all the best nondominated solutions on 13 egl instances. In the last section of this paper, the ID-MAENS also proves to be competitive to some state-of-art single-objective CARP algorithms in terms of quality of solutions and computational efficiency.

Graphical abstract

The original Multi-Objective Capacitated Arc Routing Problem is firstly decomposed into N scalar sub-problems with a set of uniformly distributed weight vectors λ1, …, λN. Each region represents one sub-population, and individuals are classified into different sub-populations according to their own direction vectors. When solving each problem respectively, two improvements of the new algorithm are as follows: (1) the replacement of the solutions is immediately done once an offspring solution is generated so as to accelerate the convergence; (2) elitism is implemented by using an archive to maintain the optimal solution in its decomposition direction during the search and these elite solutions can provide helpful information for solving their neighbor sub-problems by cooperation.

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Introduction

Both Arc Routing Problem (ARP) and Vehicle Routing Problem (VRP) are classical combinatorial optimization problems. VRP is to design routes for a fleet of vehicles, which are to serve a set of geographically dispersed points (customers, stores, schools, cities, warehouses, etc.) at the least cost [1]. While ARP is to design routes for a fleet of vehicles, which are to serve a set of task arcs (salting route, mail delivering route, street sweeping route, school bus scheduling route, etc.) at the least cost [2]. Both ARP and VRP have many extensive models [3], [4]. In this paper, a special kind of ARP called Capacitated ARP (CARP) is discussed. Considering an important constraint called capacity constraint, this new model has a more practical application [5], [6]. CARP is defined on an undirected and connected graph representing the road network, where each edge has a travel cost and a non-negative demand. A number of vehicles are based at the depot with limited capacity [2]. The aim of CARP is to seek a set of minimum cost trips for the vehicles to serve all the positive-demand edges on the conditions: (1) the total demand served by any trip cannot exceed the vehicle's capacity; and (2) each vehicle must start and end at the depot.

For many applications in reality, there is a wide gap between the classical CARP model and the real world situations. Hence, researchers pay more attention to other extended versions of CARP [7], [8], [9], [10], [11]. These models are more complex and pose greater challenges for researchers. In 2006, Lacomme firstly proposed a Multi-Objective CARP (MO-CARP) model which not only minimizes the total-cost but also balances these trips [11]. That is minimizing the total-cost and makespan (i.e. the cost of the longest trip) simultaneously. These two objectives considered by Lacomme are conflicted with each other. Thus, no unique global optimal solution exists in this case. Closed to the actual application, this model has attracted the attention of many researchers [12]. Above all, Lacomme proposed a Multi-Objective Genetic Algorithm (LMOGA) to solve MO-CARP [11], which uses the fast nondominated sorting and the crowding distance approach of a commonly Multi-Objective Evolutionary Algorithm (MOEA), namely Nondominated Sorting Genetic Algorithm II (NSGA-II) [13]. Through maintaining a set of solutions which are good “tradeoffs” between the two objectives, this problem is solved as a whole in LMOGA. Furthermore, a comparison is made between LMOGA and LMA (an approach for SO-CARP) [14]. Recently, a new Memetic Algorithm (MA) called Decomposition-Based MA with Extended Neighborhood Search (D-MAENS) was proposed by Yi Mei [2] et al. D-MAENS adopts a decomposition-based framework which is similar to that of MOEA/D [15], and D-MAENS adopts the MAENS approach for SO-CARP [16]. Through following the advanced features of evolution strategy which is NSGA-II [13], D-MAENS shows a superior performance than LMOGA [2]. Experimental studies on three well-known benchmark sets (gdb, val, egl) also demonstrate that D-MAENS is a competitive approach.

However, the replacement mechanism and the assignment mechanism of the offspring in D-MAENS remain to be improved. First of all, the solutions’ replacement is done when all the offspring are generated, and it is a one-time replacement. In this way, changing the order of solving the sub-problems does not impact the algorithm, while it will decrease the convergence speed of D-MAENS. Second, in D-MAENS, the representatives are reassigned to sub-problems at each generation by only considering one objective function. In this mechanism, it can make better use of the information of the current population during the search process, while the representative solution of each sub-problem may not be a better solution in its decomposition direction. In response to these issues, this paper presents an Improved Decomposition-Based Memetic Algorithm for Multi-Objective Capacitated Arc Routing Problem (ID-MAENS). ID-MAENS makes two improvements over the existing D-MAENS. One is the use of a steady-state evolutionary algorithm (SSEA) [17], [18], [19], [20], and the other is the introduction of an elitism archive [13], [21] for the best-so-far solutions. SSEAs are overlapping systems, because parents and offspring compete for survival. Each new offspring will either replace an existing population member, or it will die, depending on the selection pressure [22]. Some representatives of SSEA may be summarized as follows: the reducing genetic drift in SSEA by Branke [22], the median-selection for parallel SSEA by Wakunda [23] and a simple evolutionary algorithm for multi-objective optimization (SEAMO) by Valenzuela [24]. The elitism archiving mechanism is an evolutionary strategy commonly used in some EAs. By retaining the best individual in the current population, the elitism archiving mechanism can accelerate the convergence of the algorithm [13], [21]. This shows that these high fitness individuals (elite individuals) play an important role in the evolution of the population. The followings are the specific improvements of the improved algorithm: (1) the replacement is immediately done once an offspring solution is generated, which can be regarded as an online mode, while that of D-MAENS is more similar to a batch mode; (2) elitism is implemented by using an archive to maintain the current best solution in its decomposition direction during the search, and these elite solutions can provide helpful information for solving their neighbor sub-problems by cooperation.

The rest of the paper is organized as follows. The related works are introduced in Section 2, which includes the literature review, the similarities and differences in the problem formulation between MO-CARP and classical CARP and the main frame of D-MAENS. In Section 3, the ID-MAENS is proposed. Afterwards, some experimental studies and a comparative analysis are shown in Section 4. Finally, the conclusions and future work are described in Section 5.

Section snippets

Literature review

Both CARP and ARP are NP-hard problems, which have been proven by Golden and Wong [25]. Because exact algorithms are only available for very small instances (the branch-and-bound approach [26] can deal with the instances of 20–30 edges), large-scale instances must be solved in practice with heuristics in early stage. To name a few: the Augment-Merge [25] proposed by Golden et al. in 1981, the Path-Scanning [27] proposed by Golden et al. in 1983, the Ulusoy's splitting technique [28] proposed by

The improved D-MAENS

ID-MAENS uses the algorithm framework of D-MAENS and improves the algorithm's two aspects which are the replacement mechanism of the offspring and the sub-populations partition mechanism. First, in ID-MAENS, the solution replacement is immediately done once an offspring is generated at each generation and this will speed up the algorithm's convergence. Second, in ID-MAENS, an elitist strategy is used to retain the best-so-far solutions in each decomposed sub-problem according to the direction

Experimental results and analysis

In order to test the validity of ID-MAENS, three different experiments are carried out. In the first experiment, a key comparison between two MO-CARP algorithms is made, which are ID-MAENS and D-MAENS. After that, three different versions of ID-MAENS are tested in order to evaluate the impact of different improvements on the algorithm's overall performance. In the last experiment, since all test instances in this paper are used for SO-CARP, it is also necessary to compare ID-MAENS with some

Conclusion and future work

In this paper, the advantages and disadvantages of D-MAENS are analyzed and an improved algorithm called ID-MAENS is proposed. ID-MAENS is more in line with the theory which is based on the decomposition framework to solve MOPs. By retaining the current best solution for each sub-problem and adding it to the original sub-population of D-MAENS, the improved algorithm not only keeps the original population characteristics of D-MAENS but also speeds up the convergence with the neighbors sharing

Acknowledgements

We would like to express our sincere appreciation to the anonymous reviewers for their valuable comments, which have greatly helped us in improving the quality of the paper. This work was partially supported by the National Basic Research Program (973 Program) of China under Grant 2013CB329402, the National Natural Science Foundation of China, under Grants 61001202, 61371201, and 61272279, the National Research Foundation for the Doctoral Program of Higher Education of China, under Grants

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