A multi-population cooperative coevolutionary algorithm...

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A multi-population cooperative coevolutionary algorithm for multi-objective capacitated arc routing problem Ronghua Shang a,, Yuying Wang a , Jia Wang a , Licheng Jiao a , Shuo Wang b , Liping Qi a a Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education of China, Xidian University, Xi’an 710071, China b Cercia, School of Computer Science, The University of Birmingham Edgbaston, Birmingham B15 2TT, UK article info Article history: Received 9 March 2013 Received in revised form 5 February 2014 Accepted 3 March 2014 Available online 11 March 2014 Keywords: Capacitated arc routing problem Coevolution Multi-objective optimization Evolutionary algorithm abstract Capacitated Arc Routing Problem (CARP) has drawn much attention during the last few years. In addition to the goal of minimizing the total cost of all the routes, many real-world applications of CARP also need to balance these routes. The Multi-objective CARP (MO- CARP) commonly exists in practical applications. In order to solve MO-CARP efficiently and accurately, this paper presents a Multi-population Cooperative Coevolutionary Algo- rithm (MPCCA) for MO-CARP. Firstly, MPCCA applies the divide-and-conquer method to decompose the whole population into multiple subpopulations according to their different direction vectors. These subpopulations evolve separately in each generation and the adja- cent subpopulations can share their individuals in the form of cooperative subpopulations. Secondly, multiple subpopulations are used to search different objective subregions simultaneously, so individuals in each subpopulation have a different fitness function, which can be modeled as a Single Objective CARP (SO-CARP). The advanced MAENS approach for single-objective CARP can be used to search each objective subregion. Thirdly, the internal elitism archive is used to construct evolutionary pool for each subregion, which greatly speeds up the convergence. Lastly, the fast nondominated ranking and crowding distance of NSGA-II are used for selecting the offspring and keeping the diversity. MPCCA is tested on 91 CARP benchmarks. The experimental results show that MPCCA obtains better generalization performance over the compared algorithms. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction Capacitated Arc Routing Problem (CARP) focuses on servicing task edges of an undirected and connected network graph under certain conditions, which is a classic combinatorial optimization problem [5]. Many applications in real-word can be modeled as CARP, such as school bus scheduling [4], sprinkler path planning [13] and garbage cleaning [31,42]. Since CARP is NP-hard, many classical heuristic and meta-heuristic algorithms were proposed to solve this problem in the past few years. The heuristics include Path-Scanning [21], Augment-Merge [14] and Ulusoy’s tour splitting technique [56]. The meta- heuristics include the tabu search algorithms [24], the tabu scatter search algorithm [22], the variable neighborhood search algorithm [25], the guided local search algorithm [6], memetic algorithms (MA) [33,54], and the global repair operator [36]. With the goal of minimizing the total cost, CARP has been formulated as a single-objective optimization problem (SOP) traditionally. However, more than one objective needs to be considered in practical applications. For example, in actual http://dx.doi.org/10.1016/j.ins.2014.03.008 0020-0255/Ó 2014 Elsevier Inc. All rights reserved. Corresponding author. Tel.: +86 02988202279. E-mail address: [email protected] (R. Shang). Information Sciences 277 (2014) 609–642 Contents lists available at ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins

Transcript of A multi-population cooperative coevolutionary algorithm...

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Information Sciences 277 (2014) 609–642

Contents lists available at ScienceDirect

Information Sciences

journal homepage: www.elsevier .com/locate / ins

A multi-population cooperative coevolutionary algorithmfor multi-objective capacitated arc routing problem

http://dx.doi.org/10.1016/j.ins.2014.03.0080020-0255/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author. Tel.: +86 02988202279.E-mail address: [email protected] (R. Shang).

Ronghua Shang a,⇑, Yuying Wang a, Jia Wang a, Licheng Jiao a, Shuo Wang b, Liping Qi a

a Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education of China, Xidian University, Xi’an 710071, Chinab Cercia, School of Computer Science, The University of Birmingham Edgbaston, Birmingham B15 2TT, UK

a r t i c l e i n f o a b s t r a c t

Article history:Received 9 March 2013Received in revised form 5 February 2014Accepted 3 March 2014Available online 11 March 2014

Keywords:Capacitated arc routing problemCoevolutionMulti-objective optimizationEvolutionary algorithm

Capacitated Arc Routing Problem (CARP) has drawn much attention during the last fewyears. In addition to the goal of minimizing the total cost of all the routes, many real-worldapplications of CARP also need to balance these routes. The Multi-objective CARP (MO-CARP) commonly exists in practical applications. In order to solve MO-CARP efficientlyand accurately, this paper presents a Multi-population Cooperative Coevolutionary Algo-rithm (MPCCA) for MO-CARP. Firstly, MPCCA applies the divide-and-conquer method todecompose the whole population into multiple subpopulations according to their differentdirection vectors. These subpopulations evolve separately in each generation and the adja-cent subpopulations can share their individuals in the form of cooperative subpopulations.Secondly, multiple subpopulations are used to search different objective subregionssimultaneously, so individuals in each subpopulation have a different fitness function,which can be modeled as a Single Objective CARP (SO-CARP). The advanced MAENSapproach for single-objective CARP can be used to search each objective subregion. Thirdly,the internal elitism archive is used to construct evolutionary pool for each subregion,which greatly speeds up the convergence. Lastly, the fast nondominated ranking andcrowding distance of NSGA-II are used for selecting the offspring and keeping the diversity.MPCCA is tested on 91 CARP benchmarks. The experimental results show that MPCCAobtains better generalization performance over the compared algorithms.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Capacitated Arc Routing Problem (CARP) focuses on servicing task edges of an undirected and connected network graphunder certain conditions, which is a classic combinatorial optimization problem [5]. Many applications in real-word can bemodeled as CARP, such as school bus scheduling [4], sprinkler path planning [13] and garbage cleaning [31,42]. Since CARP isNP-hard, many classical heuristic and meta-heuristic algorithms were proposed to solve this problem in the past few years.The heuristics include Path-Scanning [21], Augment-Merge [14] and Ulusoy’s tour splitting technique [56]. The meta-heuristics include the tabu search algorithms [24], the tabu scatter search algorithm [22], the variable neighborhood searchalgorithm [25], the guided local search algorithm [6], memetic algorithms (MA) [33,54], and the global repair operator [36].With the goal of minimizing the total cost, CARP has been formulated as a single-objective optimization problem (SOP)traditionally. However, more than one objective needs to be considered in practical applications. For example, in actual

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610 R. Shang et al. / Information Sciences 277 (2014) 609–642

applications, the relevant departments not only want to get the vehicle arrangement with the minimum route consumptionbut also need to consider makespan (the cost of the longest route) [32]. The two objectives are conflicting with each other. Nounique global optimal solution exists in this case. Thus, the algorithm should return a set of solutions which produce good‘‘tradeoffs’’ between the two goals. Among all existing solutions for multi-objective optimization problems (MOPs), themajority focused on numerical optimization [38–50]. Very few discussed MO-CARP. MO-CARP combines MOP with the com-binatorial optimization problem, which makes it a very challenging problem. Lacomme first proposed to use the geneticalgorithm (GA) for MO-CARP, named LMOGA [32]. Inspired by the nondominated sorted GA (NSGA-II) [12], the procedureof LMOGA was improved by using good constructive heuristics to seed the initial population and by including a local searchstrategy. Lacomme made a comparative experiment between NSGA-II and LMOGA for MO-CARP and SO-CARP in terms ofsolution quality and computational efficiency. After Lacomme’s work, decomposition-based MA with extended neighbor-hood search (D-MAENS) was proposed to solve MO-CARP by Mei et al. [35]. D-MAENS combines the advantages of multi-objective evolutionary algorithms based on decomposition (MOEA/D) [62] and the MAENS approaches [54] for SO-CARP.With the same evolution strategy as in NSGA-II, D-MAENS shows a superior performance over LMOGA on 81 CARP instances.However, D-MAENS is still inadequate because it uses the framework of MOEA/D to solve MO-CARP. MOEA/D makes theassumption that each Pareto optimal solution of original MOPs is the global optimal solution to a scalar optimization sub-problem [15]. However, for most multi-objective combinatorial optimization problems including MO-CARP, this assumptionis no longer valid. There exist solutions which are not optimal for any weighted sum of the objectives in MO-CARP. Further-more, due to the high discreteness of the Pareto front (PF) in MO-CARP, one Pareto optimal solution may corresponds to theoptimal solution of multiple decomposed subproblems. The search process can be greatly hindered, and many computingresources can be wasted. Therefore, the optimization strategy based on problem decomposition is not suitable for MO-CARP[35].

Coevolutionary Algorithm (CA) is a new class of evolutionary algorithm (EA). Based on the theory of coevolution over thepast decade, it shows great advantages over traditional EAs [58]. The main differences between CA and EA are: 1. in tradi-tional EAs, the fitness function is predefined and immutable during the evolutionary process. However, the real fitnessshould be local. Representing the struggle between individual and environment, it is susceptible to the changes of environ-ment. CAs pay more attention on the coordination between the population and the environment, and between the popula-tions. 2. EAs only consider the competition between populations, without considers the possibility of collaboration betweenpopulations. On the contrary, both competition mechanism and collaboration mechanism exist in CAs, which is the so-calledcoevolution. In a broad sense, most CAs fall into three categories depending on interaction ways: the cooperative CA based onsymbiotic mechanism, the competitive CA based on population competition and the CA based on predator–prey mechanism[19,60]. The representatives of cooperative CAs include the cooperative coevolutionary GA (CCGA) proposed by Potter and DeJong [43,44], the cooperative particle swarm optimizer (CPSO-SK) [57] proposed by Van den Bergh and Engelbrecht, and thedistributed cooperative CA (DCCEA) [53] proposed by Tan et al. Their main idea is to utilize ‘‘divide and conquer’’ to solvehigh-dimensional numerical optimization problems by dividing the n-dimensional decision vector into the n componentsand using n subpopulations to optimize each of these n components [61]. The fitness of a particular individual in a specificsubpopulation depends on its ability of cooperating with other individuals to generate good solutions. The representatives ofcompetitive CAs include Coevolutionary augmented Lagrangian methods for constrained optimization proposed by Tahk andSun [52] and the GA based on multi-population competitive coevolution (GAMCC) proposed by Li et al. [34]. The main idea ofcompetitive CAs is to reserve two populations: one population representing the solutions of the problem and the other pop-ulation putting the violation degree to a condition of the first population as the fitness [47]. The stringent requirements willpromote the evolution of the first population. There are also some works about the CAs based on predator–prey mechanism.For example, Hillis was the first to build the model of predator–prey coevolution and then propose the coevolutionary GA(CGA) [26]; Paredis proposed a life-time fitness evaluation CGA [41].

In our recent work, we proposed a Coevolutionary Multi-Objective Optimization Algorithm based on Direction Vectors(DVCMOA) [28]. Our simulation results showed that it outperforms other MOEAs in the field of numerical optimization.Although DVCMOA can get good performance on numerical optimization filed, it cannot be used to solve MO-CARP directly.To overcome this issue, a Multi-population Cooperative CA (MPCCA) for MO-CARP is proposed in this paper. The proposedalgorithm divides the whole objective space into a plurality of subareas by a set of uniformly distributed direction vectors.The individuals located in different areas constitute different subpopulations. These subpopulations evolve independently.At the beginning of each iteration, all the individuals are combined together and reassigned to different subpopulationsaccording to the size of each direction vector. Using this strategy, it can make better use of the information of the currentpopulation during the search process and can assign a more appropriate individual to each subpopulation. More importantly,during the evolution of each subpopulation, its adjacent subpopulations can provide useful information through neighbor-hood sharing. Incorporated with various features like multi-elite archiving mechanism, neighborhood sharing, the fast non-dominated sorting and the crowding distance approach of NSGA-II, the MPCCA is capable of maintaining archive diversityand a fast convergence in the evolution. Compared with the state-of-art algorithms, our experiments show that MPCCAare superior in convergence speed and the closeness to the PF.

The remainder of this paper is organized as follows: the related works are given in Section 2, including the detailedintroduction of MO-CARP model, the definitions of evolutionary multi-objective optimization and the description ofdirection vector. In Section 3, our new method MPCCA is proposed, followed by the experimental study in Section 4. Finally,the conclusions and future work are included in Section 5.

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2. Related works

2.1. The model of MO-CARP

CARP was proposed by Golden and Wong in 1981 [20]. Given an undirected and connected graph, including a series oftask edges and a special vertex called depot, several vehicles with the same capacity start from the depot to service thosetask edges and then come back to the same depot. The goal of CARP is to determine a reasonable scheme with the minimumtotal cost on the conditions that all task edges should be serviced and each task edge should only be serviced once by onevehicle. Fig. 1 shows a simple scheme for CARP including 3 vehicle routes in total, in which the straight line indicates the taskedge, the dotted line indicates the non-task edge, the arrow direction is the vehicle traveling direction, the red node repre-sents the depot and the black node represents the intersection point between different edges.

In many real-world applications, however, some other factors must be taken into account in addition to the total routeconsumptions. In the garbage cleaning example of Troyes city in [32], all the vehicles have to leave the depot at the sametime. In order to improve efficiency, the sanitation department wants to make the entire garbage clean-up work to endas soon as possible. With the above considerations, the authors ignored the parameter of vehicle speed in the modeling pro-cess and used a second objective-makespan (the cost of the longest route) to reduce the duration time, based on the assump-tion that the route consumption is proportional to the time [32].

For an easy understanding, we define the following functions and symbols in a MO-CARP model, given a graph G(V, E):

1. V = {v0, v1, . . . , vn} is a collection of vertices in the graph G where v0 is the depot and the remaining nodes represent thecross-points.

2. E = {(vi, vj)|vi 2 V, vj 2 V, i – j} is the set of edges. For each e 2 E, there are three non-negative attributes, such as thedemand of e d(e), the service cost of e s(e) and the traveling cost of e c(e).

3. ER = (e 2 E|d(e) > 0) is the set of task edges.4. For each edge task (vi,vj), it is assigned two positive integers t1 and t2, one for each direction.5. A CARP solution can be represented as a set of routes x = (T1, T2, ... , Tm), and Tk = (0, tk1, tk2, . . . , tk|Tk|, 0) is the task sequence

of the k-th route, where 0 denotes the depot and |Tk| denotes the total number of task edges serviced by the k-th vehicle.6. For each route Tk, its total cost and total demand can be calculated as follows:

costðTkÞ ¼XjTk�1j

i¼1

sðtkiÞ þ dist tki

; tkiþ1

� �� �

dðTkÞ ¼XjTk j

i¼1

dðtkiÞ

ð1Þ

where the function dist(t1, t2) is the shortest distance (with the minimum traveling cost) from t1 to t2.At this point, according to the above definitions, the MO-CARP model can be established.

min f1ðxÞ ¼Xm

k¼1

costðTkÞ

Min f2ðxÞ ¼ max16k6m

ðcostðTkÞÞ

whereXm

i¼1

jTij ¼ jERj

Ti \ Tj ¼ U; i–j; i; j ¼ 1;2; . . . ;mdðTkÞ 6 Q ; 1 6 k 6 m

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð2Þ

Fig. 1. A simple scheme for CARP and its coding.

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where the f1 is the total-cost of all vehicle routes and f2 is the makespan. |ER| is the total number of task edges and Q is thecapacity of each vehicle. In Fig. 1, each task edge is assigned two integers, one for each direction. The solution of scheme inFig. 1 can be denoted as x = (0, 1, 2, 0, 3, 4, 5, 0, 6, 7, 8, 0).

2.2. Related definitions of multi-objective optimization

In the fields of scientific research and engineering applications, many problems involve multiple targets. MOP thus hasbecome important. MOPs aim to optimize multiple objectives simultaneously. These objectives are dependent and conflict-ing. Optimizing one objective will deteriorate other objectives [8,11,46,55,59]. Nowadays, the solutions for MOPs includemathematical programming methods and a variety of search methods. Because EA can optimize multiple solutions in a sin-gle run, it becomes the most commonly used method for MOPs.

The minimization MOPs can be formulated as follows:

min y ¼ f1ðxÞ; f2ðxÞ; . . . ; fnðxÞ½ �s:t: x 2 X

�ð3Þ

where X is the feasible solution region and f1(x), f2(x), . . . , fn(x) are n objective functions which are conflicting with eachother [63]. In SOPs, the optimal solution is usually unique, while the optimal solution for MOPs is often a set due to the irrec-oncilable nature between different objective functions. In order to solve MOPs, it is necessary to understand followingconcepts.

Definition 1 (Pareto-Dominance). Based on the formula (3), given x, x� e X, x� is said dominate (Pareto-optimal) anothersolution x (denoted x� � x) iff it satisfies the following two conditions:

8i 2 f1;2; . . . ;ng; fiðx�Þ 6 fiðxÞ9k 2 f1;2; . . . ;ng; fkðx�Þ < fkðxÞ

�ð4Þ

Definition 2 (Pareto-Optimal and Pareto-Optimal Set). A solution x⁄ e X is said to be Pareto-Optimal (nondominated) iff

:9x 2 X : x � x� ð5Þ

The set of all the Pareto-Optimal solutions is called Pareto-Optimal Set (PS).

Definition 3 (Pareto-Optimal Front). The set composed by objective function vectors corresponding to Pareto-Optimal Set isPF:

PF ¼ fy ¼ ½f1ðxÞ; f2ðxÞ; . . . ; fnðxÞ�T jx 2 PSg ð6Þ

Based on the above definitions [1,2,48], a MOP can be seen as the method of looking for the Pareto-Optimal solutions orapproaching the PF. The solutions found by a good MOP algorithm should approach the Pareto-Optimal Set and have a gooddiversity.

2.3. The description of direction vector

The direction vector of x can be denoted by k = [k1, k2, . . . , kn]T, where ki is the cosine of the angle di formed by the vector ofx in objective space with the i-th axis. R = [f1min, f2min, . . . , fnmin]T is the reference point. fimin represents the minimum value ofall individuals in the i-th dimension of the objective space. Obviously, direction vector is a unit vector and the sum of squareof each component is equal to 1. The specific definition of the direction vector x can be expressed as follows:

ki ¼fiðxÞ � fimin

�� ��ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPnj¼1 fjðxÞ � fjmin

� �q ; 1 6 i 6 n ð7Þ

Direction vector is an important concept in analytic geometry. There already exist approaches using sort of direction vec-tor, hyper-spheres or spherical coordinates in the literature [3,27,37]. In 2003, Messac et al. proposed the normal constraint(NC) method for MOP [37]. In this NC method, the so-called ‘‘Utopia line’’ is drawn between two anchor points in objectivespace, and then the ‘‘Utopia line’’ is divided into multiple segments, resulting in multiple points. One of the generic pointsintersecting the segments is used to define a normal to the ‘‘Utopia line’’, which is called the normal line. This normal line isused to reduce the feasible space. By translating the normal line, a corresponding set of solutions will be generated. Hughesproposed a Multiple Single Objective Pareto Sampling method (MSOPS) and its improved version MSOPS-II in 2007 [27]. InMSOPS, it firstly generates a set of a-priori direction vectors, and then each individual in the population will be evaluatedunder each direction vector, which indicates how well the population member satisfy the range of target conditions. Thekey advantage of this algorithm is that it does not rely on Pareto ranking to provide selective pressure. In 2009, Oliver

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R. Shang et al. / Information Sciences 277 (2014) 609–642 613

and Koch proposed a novel evolutionary optimization technique with a geometric based selection scheme [39]. This schemeis designed to produce approximately equidistant solutions on the Pareto front, which called rake selection. The rakes lieequidistantly in the objective space and guide the evolutionary selection process. In 2010, Gilberto et al. used the multi-objective differential evolution algorithm with spherical pruning to design the continuous controllers [17]. The idea of spher-ical pruning can be understood as that if the decision maker encounters any desired solution, he will be searching for thenearest non-dominated solution in the PF in any possible direction using discrete arc increments. In 2011, Batista et al. pro-posed the concept of cone e-dominance [3], which is a variant of the e-dominance. Depending on the hyper grid, several via-ble solutions may be lost in the e-dominance. However, the cone e-dominance uses a mechanism to control the hypervolume dominated by a specific cone. Experimental validation of the proposed cone e-dominance shows a significantimprovement in the diversity of solutions over both the regular Pareto-dominance and the e-dominance. In 2007, Zhangand Li proposed a multi-objective evolutionary algorithm based on decomposition (MOEA/D) [62], in which it uses a setof weight vectors to decompose a MOP into a number of scalar optimization subproblems and optimizes them simulta-neously. Direction vector and weight vector both assign weights to all the objective functions, therefore they have samefunctionality in aggregation methods. But their physical meanings are different. Weight vector means the weight of theweight sum, and direction vector denotes the direction of solution vector. In addition to, weight vector is distributed on ahyper plane, but direction vector is distributed on a hyper sphere. Recently, DVCMOA was proposed for MOPs. The main ideaof DVCMOA is to solve MOPs by dividing the entire population into several subpopulations on the basis of the initial directionvectors in the objective space and individuals are classified according to their different direction vectors. Here is an examplefor the MOP based on direction vector.

In Fig. 2, the yellow points are the initial individuals, purple points are the final nondominated solutions andR = [min(f1(x)), min(f2(x))]T is the reference point. Supposing there are an infinite number of direction vectors, the objectivespace is divided into infinite subregions. For each direction vector through R, there is always one point which is the closest toR in this objective subregion. In Fig. 2, the three purple points are respectively the closest in direction ki-1, ki and ki+1. All thepurple points form a set W. Obviously, the nondominated set in W is the PF [8]. Similarly, for the maximization problems, thereference point R can be selected to [max(f1(x)), max(f2(x))]T. The goal of MOP based on direction vectors is to find the optimalsolution in each direction vector.

3. The MPCCA for MO-CARP

Based on the model of CA, MPCCA uses a set of direction vectors to divide the entire population into different subpopu-lations. The size of a population is 2 times larger than that of the subpopulations. Two individuals in the current populationcorrespond to a unique subpopulation. MPCCA assigns fitness to different subpopulations, and an individual’s fitness islinked with a reference point in an implicit way. These subpopulations evolve separately in each generation and mergetogether to reassign representatives to each subpopulation before starting each generation. In particular, the adjacent sub-populations can share their individuals in the form of cooperative subpopulations. Incorporated with various features likemulti-elite archiving, neighbors sharing, the fast non-dominated sorting and the crowding distance approach of NSGA-II,the MPCCA is capable of maintaining archive diversity and guaranteeing a fast convergence in the evolution.

3.1. Initial population and subpopulations partition

In the process of population initialization, 2N nonclone individual solutions are generated and inserted into population X.As described in the problem definition of MO-CARP, each individual is denoted as x = (T1, T2, . . . , Tm), where Tk = (0, tk1, tk2, . . . ,tk|Tk|, 0), and t is the integer number assigned to task edge. Most MOEAs start from the initial population with random

Fig. 2. The illustration of MOP based on direction vector.

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Table 1The production of uniformly distributed direction vectors.

The production of uniformly distributed direction vectorsInput: The number of direction vectors N.Output: A set of uniformly distributed direction vectors {k1, k2, . . . , kN}, where ki = [ki1, ki2]T.1: Set x1 = 0, d = 0.5 � p/(N � 1);2: For i = 1 to N do3: Set x2 = 0.5 � p �x1, ki1 = cosx1, ki2 = cosx2;4: Set x1 = x1 + d;5: End for6: End

Table 2The assignment of individuals to different subpopulations.

The assignment of individuals to different subpopulationsInput: An unsorted population X = {x1, x2, . . . , x2N}.Output: A sorted population Y = {y1, y2, . . . , y2N}.1: For i = 1 to 2N � 1 do2: For j = i + 1 to 2N do

3: Iff1ðxjÞ�f1minffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðf1ðxjÞ�f1minÞ2þðf2ðxjÞ�f2min

Þ2p >

f1ðxiÞ�f1minffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf1ðxi Þ�f1min

Þ2þðf2ðxiÞ�f2minÞ2

p !

then

4: Swap xi and xj;5: End if6: End for7: End for8: For i = 1 to 2N do9: Set yi = xi;10: End for11: End

614 R. Shang et al. / Information Sciences 277 (2014) 609–642

individuals. Including a few good individuals, however, can help to accelerate the convergence speed. Therefore, we usethree heuristic methods to produce three elite individuals during initialization of the MPCCA. The heuristic methods are:Path-Scanning [21], Augment-Merge [14] and Ulusoy’s heuristic [56]. The remaining individuals are all generated randomly.In MPCCA, the whole objective space is divided into a number of subregions. By assigning different fitnesses, the algorithmcan conduct local search in different subregions of the objective space. MPCCA maintains multiple populations, each for aseparate objective subregion. To construct the N subpopulations, we need N uniformly distributed direction vectors. BecauseMO-CARP is a two-objective optimization problem, the initialization procedure of direction vectors for two objectives is gi-ven in this paper. In the initialization program, the angle between the direction vector k1 and the axis f1 is 0; the angle be-tween the direction vector kN and the axis f1 is p/2. We define the progressive angle d = 0.5 � p/(N � 1). These directionvectors are uniformly increasing because of a certain angle d. The detailed initialization steps are given in Table 1.

Through the N evenly distributed direction vectors, the entire objective space has been divided into N subregions. Next,we need to assign these 2N individuals to N different subpopulation according to the above subregions. The ideal allocation isthe attribution of individual xi determined by the closeness between the solution vector of xi and the N evenly distributeddirection vectors. By using this strategy, it may happen that individuals are distributed non-uniformly. For example, this sit-uation occurred in DVCMOA, where the individuals in intensive subregions are retained and the individuals in sparse sub-regions are deemphasized. Therefore, the ideal subpopulations partition mechanism is not suitable. In order to ensure thatthe number of each subpopulation is same, we propose a fast and simple allocation scheme. According to the definition ofthese N direction vectors, as i increases, the angles between these direction vectors and the axis f1 increase, while the anglesdecrease with f2. In other words, subpopulation 1 focuses on the area with a low f2 and subpopulation N focuses on the areawith a low f1. Based on this idea, we develop the algorithm shown in Table 2 to sort the 2N individuals based on the anglesbetween the individuals and the axis f1. After the sorting is completed, the 2i-th and (2i � 1)-th individuals in the sorted pop-ulation will be assigned to subpopulation i.

In Fig. 3, there are 5 evenly distributed direction vectors k1–k5. The population X includes ten individuals A–J. The distri-bution of solutions in MPCCA is: B, F ? subpop 1, D, G ? subpop 2, A, C ? subpop 3, H, I ? subpop 4, E, J ? subpop 5. In thissubpopulation partition mechanism, the computing resources are evenly distributed. It increases the diversity of PF at thecost of slowing down the convergence speed. Hence, we use two elitism archive strategies to speed up the convergence.The elitism archiving mechanism will be given in Section 3.3.

3.2. The fitness evaluation in each subpopulation

As MPCCA adopts multiple subpopulations, the individuals in different subpopulations have different fitness functions.According to Wiegand [60], the fitness of an individual mainly depends on its ability to collaborate with other subpopulations

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Fig. 3. The division of subpopulations in MPCCA.

0 f1

f2

R

PFd2d1

A

Fig. 4. The fitness assignment in the direction of k.

R. Shang et al. / Information Sciences 277 (2014) 609–642 615

in CA. There are a lot of interactions among subpopulations and the changes of one subpopulation may cause the changes ofother subpopulations or even entire population. Based on the above ideas, the PBI ((penalty-based boundary intersection) [62]is used as the standard of evaluation.

In MPCCA, the reference point R is updated by every individual. If R is changed, the fitness of all individuals within thepopulation will be affected. As shown in Fig. 4, k is the direction vector of the subpopulation which individual A belongs to. d1

is the projection of the direction vector of individual A in the direction of k, and d2 is the offset distance between F(xA) � Rand k. The fitness of individual A in the direction of k can be formulated as follows:

fitnessðxAÞ ¼ jjðFðxAÞ � RÞTkjj þ rjjFðxAÞ � ðRþ kjjðFðxAÞ � RÞTkjjÞjj ð8Þ

where r is the penalty parameter and its value usually is 5. From the formula (8), we can see that the fitness of one individualdepends on the other individuals implicitly. Once R changes, all individuals’ direction vectors will change accordingly.Moreover, different individuals located in different subpopulations are not comparable and the fitness function reflectsthe interaction among these populations.

3.3. The elitism archiving mechanism

The elitism archiving mechanism is an evolutionary strategy commonly used in EAs. By retaining the best individual incurrent population, the elitism archiving mechanism can accelerate the convergence of the algorithm [26,35]. It suggeststhat these high fitness individuals (elite individuals) play an important role in the evolution of the population. In MPCCA,two elitism archiving mechanisms are used: the external elitism archive and the internal elitism archive. The differencebetween the external elitism archive and the internal elitism archive is whether the elitism archive participates in theevolution. More details about these two archiving mechanisms will be given next.

3.3.1. The external elitism archiveStored in the external, the external elitism archive does not participate in evolution, and the external elitism archive is

mainly used to save the current nondominated individuals during the evolutionary process. In the initialization phase,

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Table 3The external elitism archive.

The external elitism archiveInput: An individual x.Output: An external elitism archive X�.1: If (|X�| == 0) then2: Add x to X�;3: Else if (x dominates any archive member) then4: Delete dominated members in X�;5: Add x to X�;6: Else if (x is dominated by any archive member) then7: Exit;8: Else9: Add x to X�;10: End if11: End

616 R. Shang et al. / Information Sciences 277 (2014) 609–642

the external elitism archive X� is an empty set. Whenever a new individual x is generated, we firstly determine whether xdominates any individuals of the current external elite population. If this situation exists, the individuals dominated by x willbe removed out of X�, and x will be added to X�. If there is no individual in X� dominated by x, then we determine whetherthere are individuals in X� dominate x. If there is no individual dominate x, then x and all individuals in X� are mutuallynondominated. x will then become a new elite individual and will be added to X�. At the end of the algorithm, X� is justthe Pareto-Optimal Set. The detailed steps are shown in Table 3.

3.3.2. The internal elitism archiveStored in the internal, the internal elitism archive participates in evolution. It is mainly used to speed up the convergence

of the algorithm. The size of the internal elitism archive is fixed at N and the i-th individual in the internal elitism archive Z�

corresponds to the current best individual in the direction ki. In MPCCA, at the beginning of each iteration, these 2N individ-uals are resorted according to their direction vectors and evenly reassigned to N subpopulations. Although the computingresources are evenly distributed, it increases the diversity of PF at the cost of slowing down the convergence speed. Hence,we use the internal elitism archive to participate in the evolution process. In this way, when searching the i-th subregion ofki, the i-th individual in the internal elitism archive Z� can merge with the original individuals of subpopulation i and someother ‘‘adjacent’’ individuals to construct an evolutionary pool for the i-th objective subregion. The internal elite individualsplay a guiding role in searching the different objective subregions. In the initialization phase, the internal elitism archive Z� isalso an empty set. Whenever a new individual x is generated, we calculate the direction vector of x. Next, according to itsdirection vector, the objective subregion j which x belongs to is found. Making a comparison between x and the j-th individ-ual in Z�, the winner is kept as the current best individual in the j-th objective subregion. Table 4 gives the details of theinternal elitism archive.

3.4. The cooperative coevolutionary process

MPCCA uses multiple subpopulations to search different objective subregions simultaneously. The individuals in eachsubpopulation have a special fitness function, which can be modeled as a SO-CARP. For each SO-CARP, a special evolutionary

Table 4The internal elitism archive.

The internal elitism archiveInput: An individual x.Output: An internal elitism archive Z�.1: Calculate the direction vector of individual x in the objective space;

2: kx ¼jf1ðxÞ�f1min

jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP2

j¼1ðfjðxÞ�fjmin

Þ2q ;

jf2ðxÞ�f2minjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP2

j¼1ðfjðxÞ�fjmin

Þ2q ;

24

35

T

;

3: For i = 1 to N do4: Calculate the size of |ki � kx|;5: End for6: The j-th subregion of objective space which individual x belongs to has the minimum value of |ki � kx|;7: If (the j-th individual in Z� is empty) then8: Add x to the j-th individual in Z�;9: Else if (x is better than the j-th individual in Z�) then10: Add x to the j-th individual in Z�;11: End if12: End

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Fig. 5. The illustration of multi-subpopulations’neighbor coevolution in MPCCA.

R. Shang et al. / Information Sciences 277 (2014) 609–642 617

pool is firstly built, and then some evolutionary operators are carried on this evolutionary pool. The first cooperative coevo-lutionary mechanism is to construct the evolutionary pool for each SO-CARP (or each subregion). In the framework of MOPbased on direction vector, the optimal individual in the i-th objective subregion should be close to that in the j-th objectivesubregion if ki is close to kj. Hence, the individuals of those subregions whose direction vectors are close to that of the currentsubregion should be helpful for search the current subregion. The second cooperative coevolutionary mechanism is the fit-ness evaluation in local search among different subregions. As stated in Section 3.2, the fitness functions of different subpop-ulations have a weak coevolutionary relationship, because once R updated, all individuals’ fitness will change. In otherwords, the fitness of an individual depends on all the populations, which is a cooperative coevolution between populations.Next, we will briefly describe the cooperative coevolutionary process.

3.4.1. Construct evolutionary pool for each subregionThe coevolution between subpopulations is emphasized in MPCCA. The entire objective space is divided into N subregions

by a set of uniformly distributed direction vectors and the 2N individuals are evenly assigned to N subpopulations accordingto their different direction vectors. The i-th subpopulation is the representative of the i-th objective subregion. The first coop-erative coevolutionary mechanism is to build evolutionary pool for each subregion. In MPCCA, the evolutionary pool of thecurrent objective subregion is composed of the subpopulations and current best elite individuals of 5 closest objective sub-regions (including its own).

In Fig. 5, the (i � 2)-th subregion, the (i � 1)-th subregion, the (i + 1)-th subregion and the (i + 2)-th subregion are 4 closestobjective subregions of the i-th subregion. When we searching the i-th subregion, the (i � 2)-th subpopulation, the (i � 1)-thsubpopulation, the i-th subpopulation, the (i + 1)-th subpopulation, the (i + 2)-th subpopulation and the current best individ-uals in i � 2, i � 1, i, i + 1, and i + 2 subregions are merged together to construct the evolutionary pool. By neighbor coevo-lution, the evolution of one subregion or subpopulation will result in the response of the others, which is the emphasis incooperative CAs.

3.4.2. CrossoverWhen we search each subregion independently, an evolutionary pool which is associated with this subregion is built. For

the i-th evolutionary pool, we apply the sequence based crossover (SBX) operator in the evolution process. First, two routesT1 and T2 are randomly selected from two parents x1 and x2, one route for each parent. Second, both T1 and T2 are split intotwo parts randomly, for example T1 = (T11, T12) and T2 = (T21, T22). Finally, a new individual x3 is obtained through the follow-ing 3 changes: 1. Replace T12 with T22; 2. Remove the duplicated tasks; 3. Reinsert the missing tasks into the new individual.

3.4.3. Local searchIn 2009, a MA with extended neighborhood search (MAENS) was proposed by Tang et al. [54], which is novel in terms of

the utilization of a large-step local search operator, namely Merge-Split. In general, any local search methods for SO-CARPcan be embedded into MPCCA. We use Merge-Split operator considering its excellent performance. Single Insertion, DoubleInsertion and Swap are the three traditional move operators for local search, which are widely used in CARP. While thesethree local search operators have ‘‘small’’ search step and thus are only capable of searching within a ‘‘small’’ neighborhood.The Merge-Split operator has an extended step size, which can easily jump out of the local optimum solution. Because CARPhas a large solution space and the capacity constraints are tight, we use the local search strategy of MAENS to search eachsubregion in MPCCA. The detailed procedure of local search is shown in Fig. 6.

When searching each subregion of the objective space, it is inevitable to encounter infeasible solutions. The totalconstraint violation of an individual x can be calculated through the following equation:

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Fig. 6. The local search of MAENS.

618 R. Shang et al. / Information Sciences 277 (2014) 609–642

tcvðxÞ ¼Xm

k¼1

maxðdðTkÞ � Q ;0Þ ð9Þ

When comparing two individuals, we judge whether they are viable individuals. If tcv > 0, the individual is unviable; else, theindividual is viable. In MPCCA, the viable individuals are prior to unviable individuals. Given two feasible candidate solu-tions, the fitness evaluation in this direction vector is used as criteria. Given two infeasible candidate solutions, the individ-ual with a smaller tcv is considered to be better than the other individual.

3.4.4. The selection of offspring solutions and diversity preservation mechanismIn MPCCA, the correspondences between the individuals and the subpopulations are not static. Before the beginning of

each iteration, the 2N parent individuals and N offspring individuals found in the last iteration will be merged together. Thenthe algorithm picks 2N outstanding individuals reassigned to N subpopulations and goes to the next generation. Because thefitness evaluation is not universal in different subregions, we need other strategies to choose offspring. In this step, we useNSGA-II as the selection mechanism of offspring solutions. Proposed by Deb in 2002, NSGA-II is one of the best MOEAs so far.The main idea of fast nondominated sorting is: first, sort all the solutions in the population according to the relations of dom-ination. The nondominated solutions in the front row are assigned with level 1. Second, the solutions less dominated rankbefore the solutions which are dominated by more solutions and the levels are in a descending order. A lot of previous workin numerical optimization has proven that this selection mechanism based on the fast nondominated ranking is better in theuniformity and broadness of the PF. So it can be applied to MO-CARP.

Diversity is another important performance aspect of MOEA. There are three common existing strategies for diversitypreservation so far: the niching technique proposed by Srinivas and Deb [51], the cell-based method proposed by Knowlesand Corne [29], and the crowding distance method proposed by Deb et al. [12]. The popular niching technique uses theparameter r as the threshold to evaluate the distance between different solutions, and the solutions whose distance is lessthan the threshold r will be punished. In other words, the solutions in intensive areas will face a more severe punishmentthan the solutions in sparse subregions. By dividing the whole objective space into many cells with the same size, the cell-based method can control the distribution of solutions by limiting the number of solutions in each cell. In the crowding dis-tance method, we first calculate the distance between each solution and its two nearest solutions in objective space. Thesedistances are then normalized and summed. The smaller the value, the more chance it will get to retain. Among the methodsdescribed above, the performance of the niching technique largely depends on the sharing parameter rshare. The cell-basedmethod largely depends on the cell size. Different from these two methods, the crowding distance method has a wider appli-cability because it has no user-defined parameter. In summary, after the fast nondominated sorting procedure and thecrowding distance method of NSGA-II, the best 2N individuals are kept to form the population X and the MPCCA entersthe next generation.

3.5. The MPCCA for Multi-objective CARP

Generally speaking, this paper studies the MO-CARP model, in which both the total cost of the routes and the cost of thelongest trip (makespan) need to be optimized. Inspired by the divide and conquer strategy in CA, we propose a Multi-pop-ulation Cooperative Coevolutionary Algorithm for MO-CARP. In this algorithm, the whole objective space is divided into mul-tiple subregions by a set of uniformly distributed direction vectors, and different subregions correspond to differentsubpopulations. At the beginning of each iteration, all the individuals in current population are sorted according to their dif-ferent direction vectors and then assigned to N subpopulations evenly. These subpopulations evolve separately, while theadjacent subpopulations can share their individuals in the form of cooperative subpopulations. By referencing some otherevolutionary strategies, such as the elitism archiving, the NSGA-II and the MAENS for SO-CARP, MPCCA shows good diversityand fast convergence. The detailed steps of MPCCA are given in Table 5.

3.6. The differences between MPCCA and DVCMOA, MCCA

Although MPCCA and DVCMOA have a similar framework, there are some significant differences:

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Table 5A Multi-population Cooperative Coevolutionary Algorithm for MO-CARP.

A Multi-population Cooperative Coevolutionary Algorithm for MO-CARPInput: An instance of MO-CARP s, the number of subpopulations N, a set of uniformly distributed direction vectors k1, . . . , kN, the maximum of

generation Gmax.Output: A set of nondominated solutions X�.1: Initialize a population X = {x1, . . . , x2N}, set the external elitism archive X� = Ø and the internal elitism archive Z� = Ø;2: Using the Direction Vectors Generating Mechanism to generate N uniformly distributed direction vectors k1, . . . , kN;

3: Set it = 0;4: While (it < Gmax) do5: According to The Subpopulations Partition Mechanism, these 2N individuals are assigned to N subpopulations evenly.6: For i = 1 to N do7: Construct an evolutionary pool for the i-th SO-CARP(or i-th subregion);8: Randomly select two individuals from the pool and apply the crossover and local search operators of MAENS to find the offspring yi;9: Update the archive Z� and X�;10: End for11: P = X [ Y, where Y = {y1, . . . , yN} are the offspring solutions. Sort the individuals in P by the fast nondominated sorting procedure and crowding

distance approach of NSGA-II, Then, let X be the top of 2N solutions in the sorted P;16: End while17: Export X�;18: End

R. Shang et al. / Information Sciences 277 (2014) 609–642 619

1. In MPCCA, individuals of each subpopulation are reassigned at each generation, while DVCMOA carries out the individ-uals’ assignment at the initialization phase only and the subpopulations no longer change in the evolution. In this way,each subpopulation of MPCCA can be assigned some more appropriate individuals according to the information of thecurrent population during the search process, while DVCMOA does not have this feature.

2. In DVCMOA, the individual assignment operation is used to allocate individuals according to the closeness between thedirection vectors of individuals and the direction vector in each subpopulation. This assignment may lead to uneven dis-tributions among these subpopulations, and the allocation of computing resources will be uneven too. While MPCCAcombines all the solutions together regardless of which subpopulation they belong to. After that, we sort the individualsin accordance with the size of their direction vectors and evenly assign the sorted individuals to these subpopulations.MPCCA pays more attention on those undeveloped areas and increases the diversity.

3. The crowding distance approach of NSGA-II prevents the diversity loss. The elitist strategy in MPCCA of using both exter-nal and internal archives maintains elite solutions during the search. Moreover, the fast nondominated ranking is used toselect the offspring. This makes it more likely to capture the whole PF when the number of Pareto optimal solutions islarger than the population size.

4. Finally, the chromosome representation and evolutionary operators of DVCMOA is used for multi-objective numericaloptimization, while MPCCA applies the sequence based crossover and some classical local search operators, which areused for CARP.

Very recently, another algorithm called a Multi-population Cooperative Cultural Algorithm (MCCA) is proposed [23].MPCCA and MCCA are similar in the following aspects:

1. Both MPCCA and MCCA divide the whole population into multiple subpopulations and evolutionary operations areperformed on each subpopulation respectively.

2. Both MPCCA and MCCA use elite individuals to guide each subpopulation’s evolution.3. They both adopt some kinds of coevolution mechanisms for sub-populations’ cooperation.

The differences between MPCCA and MCCA are:

1. MCCA is a SOP numerical algorithm, while MPCCA belongs to MOEAs for MO-CARP.2. The mode of dividing the whole population into multiple subpopulations is different between MPCCA and MCCA. In

MPCCA, N uniformly distributed direction vectors are used to divide the objective space and individuals are assignedto these subpopulations according to their different direction vectors. However, it is not clear for the division of subpop-ulation in MCCA or it only just uses multiple populations.

3. For MCCA, in each subpopulation, GA is adopted to search the optimal solution. Moreover, all the individuals in these sub-populations have the same fitness function. In contrast, MPCCA uses MAENS approach to search each objective subregionand the fitness evaluation is different in different subpopulation.

4. Although both MPCCA and MCCA use elite individuals to guide each subpopulation’s evolution, their manners are differ-ent. In MCCA, each kind of the helpful information (which is called knowledge in MCCA) extracted from elite individuals isutilized to induce the mutation operator, while the elite individuals of MPCCA are mainly used to construct evolutionarypool for each objective subregion.

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620 R. Shang et al. / Information Sciences 277 (2014) 609–642

4. Experimental results and discussions

4.1. Test problems and experimental setup

The experiments discuss four benchmark problems, which are gdb, val, egl and Brandão&Eglese. gdb contains 23 small sizeinstances [10]. val contains 34 instances in 10 groups composed of 10 different graphs. In each graph, different instances aregenerated by changing the capacity of the vehicle. Based on the data from a winter gritting application in Lancashire, eglcontains 24 instances which are in two graphs [14]. Each graph corresponds to 12 instances. These three groups of instancesare commonly used in the assessment of the algorithm for solving CARP. Because the CARP instances in real life are usually

Table 6The statistical results of ID on the gdb benchmark test set after 51 independent runs.

gdb V T E MPCCA D-MAENS DVCMOA

Mean Std Mean Std p h Mean Std p h

1 12 22 22 0 0 0.0784 0.2169 0.0313 1 0.1307 0.2673 0.0020 12 12 26 26 0.4482 0.6164 2.4786 1.3102 7.0e�10 1 2.4152 1.0212 9.7e�10 13 12 22 22 0.1373 0.4752 0.1855 0.5061 0.6865 0 0.6025 0.7959 0.0049 14 11 19 19 0 0 0.1503 0.3848 0.0156 1 0.0131 0.0934 1 05 13 26 26 0.1343 0.1998 1.0119 0.5937 3.4e�9 1 1.2141 0.7281 1.1e�9 16 12 22 22 0.0482 0.3442 0.6869 0.8366 3.4e�5 1 1.0460 1.0193 2.3e�6 17 12 22 22 0 0 0.0392 0.0802 0.002 1 0.0235 0.0651 0.0313 18 27 46 46 3.3327 0.8426 3.3283 0.9364 0.5300 0 3.3228 0.8860 0.8734 09 27 51 51 0.6727 0.9508 3.6258 0.8797 7.5e�10 1 3.7786 0.8724 5.1e�10 110 12 25 25 0.2055 0.4283 1.7958 1.7703 2.6e�9 1 2.1253 1.6581 9.3e�10 111 22 45 45 2.2848 1.8376 7.8611 3.2722 8.8e�10 1 7.3531 3.3839 1.2e�9 112 13 23 23 0.7579 0.3959 0.9924 0.9953 0.2165 0 1.2714 1.2200 0.0047 113 10 28 28 12.1311 0 11.6975 1.7624 1 0 11.9719 1.1366 0.25 014 7 21 21 0 0 1.8189 2.2124 3.8e�6 1 1.5013 2.1047 1.3e�6 115 7 21 21 0.0098 0.0396 0.4620 0.2962 1.7e�9 1 0.5047 0.2517 5.8e�10 116 8 28 28 0.2052 0.2319 1.6257 1.0922 6.9e�10 1 2.0823 1.2834 1.6e�9 117 8 28 28 0.7457 0.2827 2.6243 0.9217 5.3e�10 1 2.6768 0.8509 4.7e�10 118 9 36 36 3.3915 1.6823 6.5813 2.5215 9.6e�8 1 6.8393 2.5748 1.8e�7 119 8 11 11 0 0 0 0 1 0 0 0 1 020 11 22 22 0.6920 0.6239 1.3733 0.4927 8.5e�7 1 1.2225 0.5789 1.4e�4 121 11 33 33 1.2453 0.8422 2.4881 1.0228 6.4e�7 1 2.4494 0.9857 3.7e�7 122 11 44 44 0.4422 0.3686 1.6175 0.4994 6.1e�10 1 1.8371 0.5148 5.5e�10 123 11 55 55 1.0077 0.1909 10.9106 46.3249 5.1e�10 1 1.6812 0.2407 6.5e�10 1

Table 7The statistical results of HD on the gdb benchmark test set after 51 independent runs.

gdb V T E MPCCA D-MAENS DVCMOA

Mean Std Mean Std p h Mean Std p h

1 12 22 22 0 0 0.2353 0.6508 0.0313 1 0.3922 0.8020 0.0020 12 12 26 26 1.4857 1.8353 6.2835 3.5732 1.6e�9 1 6.0232 2.0335 2.5e�9 13 12 22 22 0.5490 1.9007 0.7418 2.0245 0.6865 0 2.4101 3.1836 0.0049 14 11 19 19 0 0 0.5490 1.3902 0.0156 1 0.0784 0.5601 1 05 13 26 26 0.8426 1.3251 4.4301 2.5261 4.0e�9 1 5.9595 4.2136 2.3e�9 16 12 22 22 0.3264 1.1625 2.8572 2.9177 8.9e�6 1 3.7788 2.9420 4.7e�7 17 12 22 22 0 0 0.1961 0.4010 0.0020 1 0.1176 0.3254 0.0313 18 27 46 46 11.6078 4.9843 8.8326 3.2038 8.1e�4 1 8.4362 3.6009 0.0011 19 27 51 51 1.7069 2.2602 6.1553 1.3397 1.2e�9 1 7.7635 3.5076 1.1e�9 1

10 12 25 25 1.4665 2.7796 10.9778 12.2612 7.0e�9 1 12.7811 12.7207 2.6e�9 111 22 45 45 13.9776 13.2771 41.7110 20.2859 2.5e�9 1 39.1951 20.8129 9.1e�8 112 13 23 23 3.4633 1.1515 3.8517 2.4875 0.5693 0 4.7308 3.3304 0.0150 113 10 28 28 24.3516 0 23.9635 2.9371 1 0 24.2134 1.8882 0.5 014 7 21 21 0 0 8.5905 10.1532 2.5e�6 1 7.0737 9.4733 1.4e�6 115 7 21 21 0.0588 0.2376 2.0825 1.0472 9.1e�10 1 2.0986 0.7797 3.5e�10 116 8 28 28 1.1029 0.9377 6.8224 4.1910 1.9e�9 1 8.0297 4.6125 1.6e�9 117 8 28 28 2.1601 0.8004 7.8465 2.2406 6.7e�10 1 7.7994 2.3984 8.8e�10 118 9 36 36 18.8893 9.3744 31.9063 9.4206 8.7e�7 1 32.9721 10.0586 5.3e�7 119 8 11 11 0 0 0 0 1 0 0 0 1 020 11 22 22 4.3421 3.1574 7.1999 2.4545 1.2e�5 1 6.4817 2.8508 0.0022 121 11 33 33 5.6635 5.8825 12.7591 6.4568 2.5e�5 1 12.8653 6.5980 1.9e�6 122 11 44 44 1.7905 0.8660 4.0525 1.5711 8.1e�9 1 4.5645 1.6579 2.0e�9 123 11 55 55 5.9318 2.2650 13.0816 46.6782 3.4e�4 1 3.2103 1.3445 7.4e�8 1

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R. Shang et al. / Information Sciences 277 (2014) 609–642 621

ultra-large-scale, Brandão and Eglese generated 10 ultra-large-scale instances based on the traffic topology of Lancashire [7].Their graph includes 255 vertices and 375 edges, which is the largest test set in our experiment.

In order to verify the effectiveness of the algorithm, we make a comparison among MPCCA, DVCMOA and D-MAENS. D-MAENS is the ‘‘best-so-far algorithm’’ for MO-CARP, which was shown to outperform all other existing MO-CARP algorithms.Since MPCCA is different with DVCMOA, it is also necessary to compare it with DVCMOA. For this reason, we also hybridizethe framework of DVCMOA with SO-CARP MAENS algorithm and consider it as a reference. The comparison is carried out onthe four benchmark test sets of CARP instances. For a fair comparison, MPCCA and DVCMOA adopt the same parameters as D-MAENS. For all the 3 methods, the maximum number of iterations Gmax = 200, the size of the offspring population osize = 60,

Table 8The statistical results of e-indicator on the gdb benchmark test set after 51 independent runs.

gdb V T E MPCCA D-MAENS DVCMOA

Mean Std Mean Std p h Mean Std p h

1 12 22 22 1.0000 0 1.0036 0.0099 0.0313 1 1.0059 0.0122 0.0020 12 12 26 26 1.0066 0.0063 1.0166 0.0025 1.3e�8 1 1.0164 0.0027 1.7e�8 13 12 22 22 1.0013 0.0046 1.0032 0.0072 0.2075 0 1.0081 0.0099 1.8e�4 14 11 19 19 1.0000 0 1.0019 0.0048 0.0156 1 1.0003 0.0020 1 05 13 26 26 1.0029 0.0044 1.0134 0.0076 3.0e�7 1 1.0155 0.0078 3.8e�9 16 12 22 22 1.0003 0.0019 1.0063 0.0100 2.9e�5 1 1.0112 0.0143 1.9e�6 17 12 22 22 1.0000 0 1.0033 0.0068 0.0020 1 1.0020 0.0055 0.0313 18 27 46 46 1.0230 0.0035 1.0240 0.0027 0.1764 0 1.0240 0.0028 0.2135 09 27 51 51 1.0065 0.0092 1.0247 0.0042 8.8e�10 1 1.0269 0.0055 5.0e�10 1

10 12 25 25 1.0064 0.0118 1.0330 0.0117 1.1e�8 1 1.0340 0.0111 1.1e�9 111 22 45 45 1.0291 0.0059 1.0578 0.0299 1.1e�9 1 1.0557 0.0288 5.4e�10 112 13 23 23 1.0060 0.0014 1.0067 0.0034 0.4312 0 1.0081 0.0056 0.0128 113 10 28 28 1.0149 0 1.0147 0.0018 1 0 1.0146 0.0016 0.5 014 7 21 21 1.0000 0 1.0340 0.0370 2.1e�6 1 1.0341 0.0351 1.3e�6 115 7 21 21 1.0020 0.0079 1.1056 0.0671 1.1e�9 1 1.0974 0.0594 3.2e�10 116 8 28 28 1.0229 0.0126 1.0787 0.0519 1.2e�8 1 1.1046 0.0608 3.5e�9 117 8 28 28 1.0250 0.0178 1.2549 0.0976 4.8e�10 1 1.2671 0.0937 3.9e�10 118 9 36 36 1.0660 0.0313 1.1662 0.0598 2.6e�9 1 1.1613 0.0622 1.1e�8 119 8 11 11 1.0000 0 1.0000 0 1 0 1.0000 0 1 020 11 22 22 1.0165 0 1.0162 0.0016 0.5 0 1.0157 0.0034 0.25 021 11 33 33 1.0373 0.0264 1.0841 0.0557 2.7e�6 1 1.0830 0.0501 1.3e�6 122 11 44 44 1.0251 0.0400 1.1501 0.0650 2.1e�9 1 1.1664 0.0673 1.5e�9 123 11 55 55 1.0120 0.0112 1.0410 0.2161 1.5e�7 1 1.0767 0.0413 2.1e�9 1

Table 9The statistical results of HV on the gdb benchmark test set after 51 independent runs.

gdb V T E MPCCA D-MAENS DVCMOA

Mean Std Mean Std p h Mean Std p h

1 12 22 22 112.0 0.0 108.7 9.1 0.0313 1 106.5 11.2 0.0020 12 12 26 26 690.8 88.9 674.2 83.5 8.3e�9 1 674.7 86.2 3.8e�9 13 12 22 22 267.9 13.9 266.9 15.2 0.2075 0 264.3 15.5 1.9e�4 14 11 19 19 294.9 73.9 293.1 69.4 0.0156 1 294.7 73.6 1 05 13 26 26 878.8 291.6 859.6 283.1 1.2e�7 1 852.6 270.2 6.2e�9 16 12 22 22 425.6 167.9 417.8 160.9 0.0015 1 407.6 149.2 4.5e�5 17 12 22 22 420.0 0.0 414.9 10.4 0.0020 1 416.9 8.5 0.0313 18 27 46 46 301.1 111.5 274.9 104.0 3.4e�9 1 269.7 107.6 2.1e�9 19 27 51 51 169.9 88.2 129.6 74.5 1.1e�9 1 125.3 71.4 5.1e�10 110 12 25 25 3358.7 172.9 3312.8 178.2 4.9e�9 1 3303.4 173.4 1.5e�9 111 22 45 45 6835.4 1181.6 6548.5 1168.3 6.1e�10 1 6485.5 1119.6 5.5e�10 112 13 23 23 95.6 86.8 94.3 87.5 0.1446 0 90.7 72.2 0.0038 113 10 28 28 0.0 0.0 0.0 0.0 1 0 3.4118 24.3 1 014 7 21 21 158.8 22.9 157.0 23.2 2.4e�4 1 156.4 22.4 1.6e�4 115 7 21 21 40.0 6.4 38.2 6.0 6.6e�8 1 37.7 5.8 9.2e�9 116 8 28 28 240.3 34.8 230.6 33.9 3.4e�9 1 227.6 31.4 2.7e�9 117 8 28 28 25.5 4.9 18.2 4.7 3.2e�9 1 17.3 4.5 6.3e�10 118 9 36 36 481.8 88.7 439.4 84.5 5.1e�10 1 439.2 79.2 7.1e�10 119 8 11 11 0.0 0.0 0.0 0.0 1 0 0.0 0.0 1 020 11 22 22 116.7 43.5 114.9 41.7 4.2e�4 1 114.7 41.2 0.0023 121 11 33 33 412.0 83.2 399.2 79.1 1.7e�8 1 399.6 79.2 2.2e�9 122 11 44 44 98.1 26.4 86.9 23.8 1.1e�9 1 84.4 23.2 6.6e�10 123 11 55 55 64.8 36.5 50.3 36.2 1.5e�7 1 52.3 34.0 3.5e�9 1

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622 R. Shang et al. / Information Sciences 277 (2014) 609–642

and the local search probability pls = 0.1. The Because MPCCA and DVCMOA emphasize the coevolution between differentsubpopulations, the size of their parent population is set to 120. It is set to 60 in D-MAENS. All then algorithms are run51 times independently.

4.2. Metrics

In order to evaluate the performance of MOEAs, two aspects should be considered – convergence and diversity. In termsof convergence, the nondominated solutions should be close to the PF. In terms of diversity, these solutions should be dis-tributed as uniformly and diversely as possible. Convergence and diversity can hardly be measured by a single metric, so thefollowing four metrics are used. 1. The distance (ID) between the obtained nondominated set and the true PF is used to mea-sure the convergence. The algorithm with a smaller ID has a better convergence. 2. The hausdorff distance (HD) is a measuredescribing the degree of similarity between the obtained PF and the true PF. It indicates the maximum mismatch betweenthe two point sets. 3. e-indicator is also an important metric to evaluate the quality of approximation sets. 4. Hypervolume(HV) is used to evaluate the broadness and diversity of the obtained nondominated set. In the following statements, we sup-pose that set A = {x1, . . . , xP} is the approximation set obtained by an MO-CARP algorithm and R = {y1, . . . , yW} is reference PF.Because it is difficult to find the true PF in a MO-CARP instance, the PF itself may be not uniformly distributed. For each in-stance, the nondominated solutions obtained by the three algorithms in 51 runs are collected together, and those solutionsremained nondominated in this set are used as the reference set R.

1. Distance From Reference Set (ID)

This metric was proposed by Czyzzak and Jaszkiewicz [9] and used to evaluate the distance between the achieved PF andthe true PF. The formula is listed below.

Table 1The sta

Val

1A1B1C2A2B2C3A3B3C4A4B4C4D5A5B5C5D6A6B6C7A7B7C8A8B8C9A9B9C9D10A10B10C10D

IDðAÞ ¼PW

j¼1ðmin dðxi; yjÞÞjRj 1 6 i 6 P ð10Þ

where d (xj, yi) is the Euclidean distance between xj and yi in objective space.

0tistical results of ID on the val benchmark test set after 51 independent runs.

V T E MPCCA D-MAENS DVCMOA

Mean Std Mean Std p h Mean Std p h

24 39 39 0.7347 0.2399 1.5033 0.5141 4.4e�9 1 1.7059 0.5517 8.8e�10 124 39 39 0.3169 0.4355 2.2771 0.5148 6.1e�10 1 2.2742 0.6172 5.1e�10 124 39 39 0 0 0.5980 0.7452 1.8e�5 1 0.6739 0.7529 5.1e�6 124 34 34 2.0143 0.3104 2.6054 0.6675 6.2e�6 1 2.7963 0.6000 4.4e�9 124 34 34 1.4359 0.5600 2.0857 0.8298 6.5e�5 1 2.1876 0.7298 8.6e�7 124 34 34 0.3333 1.3515 3.5249 3.4506 1.5e�5 1 3.0997 2.8601 4.9e�5 124 35 35 0.1046 0.1245 0.2778 0.2463 1.7e�4 1 0.4297 0.3023 8.0e�8 124 35 35 0.0609 0.1117 0.0788 0.1674 0.5442 0 0.2473 0.2848 2.1e�5 124 35 35 0 0 0 0 1 0 0.0196 0.1400 1 041 69 69 2.2892 0.7232 4.8302 0.5724 5.5e�10 1 5.3878 0.7233 5.1e�10 141 69 69 1.8104 0.6475 3.9310 0.4954 5.1e�10 1 4.3662 0.6860 5.5e�10 141 69 69 2.8570 0.3942 4.6028 1.0646 1.2e�9 1 5.3038 1.1189 5.1e�10 141 69 69 3.3622 1.0986 6.9950 3.0449 1.0e�8 1 8.7749 3.5113 1.2e�9 134 65 65 7.3482 1.4432 10.2012 1.8477 3.7e�9 1 10.3154 1.5779 1.2e�8 134 65 65 4.9838 1.1593 8.2064 1.6260 1.2e�9 1 8.6129 1.7630 5.1e�10 134 65 65 4.8847 1.3528 7.1668 1.1533 9.3e�10 1 7.7889 1.5778 9.3e�10 134 65 65 5.1665 1.9545 6.8559 1.8516 3.7e�5 1 8.6582 2.4944 2.0e�7 131 50 50 0.5451 0.5290 2.5362 0.5720 7.8e�10 1 2.8089 0.6033 5.1e�10 131 50 50 0.5815 0.4350 2.6359 0.8980 6.1e�10 1 3.0997 0.6713 5.1e�10 131 50 50 0.7440 0.4243 3.7135 1.4936 1.5e�9 1 4.1556 1.5542 5.5e�10 140 66 66 1.7721 0.9027 5.1106 1.2917 6.9e�10 1 5.5599 1.1842 5.5e�10 140 66 66 2.2281 1.1430 4.4361 1.2064 5.7e�8 1 4.9894 1.2725 2.6e�9 140 66 66 0.7734 0.7086 1.8326 0.7575 1.4e�7 1 2.7528 1.4495 1.3e�8 130 63 63 4.8313 1.3814 6.0429 1.0333 3.2e�6 1 6.5455 1.2592 4.5e�7 130 63 63 4.2505 0.8620 6.3853 1.3347 1.2e�9 1 6.8588 1.4776 5.8e�10 130 63 63 5.6968 1.5593 9.4106 2.1106 3.7e�9 1 9.6163 2.0671 3.5e�9 150 92 92 4.8723 1.0923 8.7414 2.1818 9.9e�10 1 8.7541 1.7669 6.2e�10 150 92 92 3.8456 1.0268 6.6365 1.5565 5.2e�9 1 7.0523 1.4506 1.8e�9 150 92 92 3.0642 0.6392 5.8339 1.3928 5.5e�10 1 6.6473 1.0276 5.1e�10 150 92 92 2.2825 0.5508 4.0440 0.8158 5.5e�10 1 4.5067 0.8574 5.1e�10 150 97 97 4.2827 1.2032 12.4337 4.9076 5.1e�10 1 11.9129 4.4633 5.1e�10 150 97 97 7.0197 2.9660 22.8875 8.0258 5.1e�10 1 21.4672 6.3699 8.3e�10 150 97 97 4.2129 1.3248 8.4750 2.7262 3.7e�9 1 9.9041 3.9178 1.8e�9 150 97 97 9.1537 2.7449 14.9530 5.8712 1.3e�6 1 14.1472 4.9689 2.3e�7 1

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R. Shang et al. / Information Sciences 277 (2014) 609–642 623

1. Hausdorff Distance (HD)Proposed by Oliver et al., this metric describes the degree of similarity between the obtained PF and the true PF [40]. HD(A,

R) = max(hD(A, R), hD(R, A)), which indicates the maximum mismatch between A and R. hD(A, R) can be calculated by the fol-lowing equation:

Table 1The sta

Val

1A1B1C2A2B2C3A3B3C4A4B4C4D5A5B5C5D6A6B6C7A7B7C8A8B8C9A9B9C9D10A10B10C10D

hDðA;RÞ ¼maxðxi 2 AÞminðyj 2 RÞkxi � yjk ð11Þ

where ||xi � yj|| represents the Euclidean distance between xi and yj. In Eq. (11), for each point xi in set A, we firstly find thepoint yj in set R which has the closest distance between them and then sort ||xi � yj||. The maximum distance is hD(A, R).1. e indicator

This metric [65] was proposed by Zitzler et al. Ie(A, R) can be stated as follows:

IeðA;RÞ ¼maxðyj 2 RÞminðxi 2 AÞmaxx1

i

y1j

;x2

i

y2j

; . . . ;xn

i

ynj

!ð12Þ

In Eq. (12), for each point yj in set R, the corresponding dimension divisions are done between all the P points of set A andyj. The minimum value of the above operation is as the representative of yj. For all the representatives of y in set R, The max-imum distance is e-indicator.

1. Hypervolume (HV)

Hypervolume was propoesed by Zizler et al. [64] and it calculates the volume in the objective domain covered by theobtained nondominated solution set. It is used to evaluate the broadness of the nondominated individuals. It is defined asfollows:

HVðAÞ ¼ volume [nPFi¼1v i

� �ð13Þ

1tistical results of HD on the val benchmark test set after 51 independent runs.

V T E MPCCA D-MAENS DVCMOA

Mean Std Mean Std p h Mean Std p h

24 39 39 4.1809 0.4128 5.7480 2.6494 0.0013 1 5.7953 2.1219 6.0e�5 124 39 39 4.4902 4.1346 9.1292 2.1191 5.9e�7 1 8.5812 3.4279 3.2e�6 124 39 39 0 0 1.6972 2.5103 4.3e�5 1 1.6741 2.1957 8.9e�6 124 34 34 12.0416 0 12.4679 1.3578 1.7e�4 1 12.9687 1.5129 2.2e�8 124 34 34 7.2097 0.9901 9.2222 3.6042 0.0010 1 9.0679 3.1921 0.0026 124 34 34 0.3529 1.4258 3.7059 3.6016 8.4e�6 1 3.1569 2.9487 6.5e�5 124 35 35 0.5686 0.6084 1.8079 2.0943 3.2e�5 1 1.7408 0.9215 9.9e�8 124 35 35 0.2549 0.4835 0.3333 0.6831 0.4461 0 0.8885 0.8688 3.2e�5 124 35 35 0 0 0 0 1 0 0.0196 0.1400 1 041 69 69 41.2904 13.7049 38.9536 22.7958 0.4937 0 39.9596 23.1092 0.5238 041 69 69 37.9412 24.5385 46.3224 22.0734 0.0131 1 44.3922 19.8606 0.0664 041 69 69 19.5333 12.5100 28.9649 15.8529 0.0023 1 35.4520 13.5041 2.0e�7 141 69 69 10.3380 4.8608 12.3090 4.8145 0.0240 1 14.7585 3.5903 3.1e�5 134 65 65 29.0165 14.1217 52.5634 22.5045 1.0e�6 1 51.4594 24.1787 1.2e�5 134 65 65 34.7819 16.1798 44.3976 21.5868 0.0078 1 46.8965 21.0778 0.0012 134 65 65 79.7487 32.6401 45.5142 28.4803 1.4e�5 1 43.7966 23.1173 6.9e�8 134 65 65 19.2339 13.2189 18.5366 9.6104 0.7173 0 22.0516 8.9589 0.0345 131 50 50 6.5402 5.6081 11.5567 5.5563 1.8e�5 1 13.2916 6.2976 1.5e�6 131 50 50 3.1063 1.5484 10.8950 5.3782 6.3e�10 1 11.3013 2.8683 5.7e�10 131 50 50 2.7976 1.5927 7.4841 2.8579 2.0e�8 1 8.4251 2.5763 1.1e�9 140 66 66 12.4976 7.0617 20.4471 7.8406 4.0e�6 1 23.8881 9.3480 4.3e�7 140 66 66 15.1203 7.5196 21.1682 8.7031 0.0018 1 21.0545 9.2963 0.0015 140 66 66 5.5098 2.4688 6.3218 2.8453 0.0083 1 8.1451 4.8115 4.1e�5 130 63 63 47.2776 18.4951 44.3044 23.1129 0.4041 1 37.2549 21.8122 0.0081 130 63 63 23.6253 7.2224 36.7240 20.9192 3.0e�4 1 40.7226 24.0255 1.1e�4 130 63 63 17.2168 3.0520 20.6097 6.9011 0.0030 1 22.8681 8.3205 6.3e�6 150 92 92 21.6969 7.1630 27.1384 13.7061 0.1242 0 23.3300 10.1648 0.4202 050 92 92 12.0714 5.7469 21.1183 11.7574 2.9e�5 1 21.0584 10.5828 2.0e�6 150 92 92 31.5294 20.8292 17.7729 12.1831 6.9e�4 1 20.4775 11.5761 0.0062 150 92 92 5.3554 2.0363 10.2296 3.9398 6.9e�9 1 14.5057 5.9963 1.5e�9 150 97 97 103.0965 35.8829 53.2182 22.0331 4.2e�8 1 47.0568 23.0081 2.3e�8 150 97 97 49.6910 23.6042 119.0107 27.6532 5.1e�10 1 113.9498 23.4405 1.9e�9 150 97 97 19.9117 8.4330 31.1383 15.3902 8.3e�5 1 34.0938 19.1255 7.1e�5 150 97 97 55.5097 19.7287 64.7398 26.2366 0.0676 0 59.0643 22.9670 0.1832 0

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624 R. Shang et al. / Information Sciences 277 (2014) 609–642

where nPF is the number of nondominated solutions in the set. The reference point is found by constructing a vector of theworst objective function values among all the comparative algorithms. HV reflects the size of space between the obtainednondominated solution set and the reference point, and it can represent the diversity of the solutions. Moreover, HV isone of the most commonly used metrics for evaluating the performance of MOPs. If a solution set dominates another one,it always has a better HV [7].

4.3. Experimental results and analysis

The following tables show the experimental results of MPCCA, D-MAENS and DVCMOA. All the experiments areconducted for 51 independent runs. The mean and standard deviation values of each metric are included in the tables.For a statistical understanding, the Wilcoxon signed rank test [18] has been carried out at a significance level of 0.05 betweenMPCCA and other algorithms. Tables 6–21 show the statistical test results. In the following analysis, some notations are usedfor clarity: 1. V, T and E respectively denote the number of vertices, the number of tasks and the total number of edges in theinstance. The edges in gdb and val are all tasks, while there are task edges and non-task edges in egl and Brandão&Eglese. 2.For the Wilcoxon signed rank test, p is the probability of a hypothesis of equal median for two paired samples. h is the resultof the test. If the median of the difference between MPCCA and another compared algorithm is zero, h = 0; Otherwise, a sig-nificant difference, then h = 1[18]. 3. The best results of average values and standard deviations are indicated in bold in eachtest instance. The winner of these algorithms has a smallest mean for each metrics. 4. For the most ID, HD and e-indicatorobtained by the three algorithms, they are accurate to the fourth decimal place, while HV is accurate to the first decimalplace.

Table 6 shows the statistics results of performance indicator ID among the three algorithms on small-scale test set gdb.As seen in Table 6, gdb test set contains 23 small-scale instances. The interval of vertex varies from 7 to 27, and the num-

ber of tasks is up to 46. It shows that MPCCA obtains better results on 20 out of 23 gdb instances than the remaining twoalgorithms, while D-MAENS and DVCMOA only obtain the winner just on one gdb instance. Moreover, it is noteworthy thatMPCCA not only get the smallest means but also obtain the smallest standard deviations on most of the gdb instances. As tothe Wilcoxon signed rank test, on most test instances, the median of the differences between MPCCA and the compared

Table 12The statistical results of e-indicator on the val benchmark test set after 51 independent runs.

Val V T E MPCCA D-MAENS DVCMOA

Mean Std Mean Std p h Mean Std p h

1A 24 39 39 1.0216 0.0022 1.0270 0.0071 4.0e�7 1 1.0289 0.0066 1.3e�9 11B 24 39 39 1.0130 0.0138 1.0346 0.0015 8.3e�9 1 1.0350 0.0030 1.1e�8 11C 24 39 39 1.0000 0 1.0049 0.0059 1.0e�5 1 1.0056 0.0061 2.2e�6 12A 24 34 34 1.0107 0.0016 1.0174 0.0065 1.2e�8 1 1.0202 0.0059 1.6e�9 12B 24 34 34 1.0088 0.0048 1.0154 0.0070 2.1e�6 1 1.0151 0.0074 2.7e�6 12C 24 34 34 1.0008 0.0031 1.0081 0.0079 8.4e�6 1 1.0068 0.0063 6.5e�5 13A 24 35 35 1.0056 0.0061 1.0137 0.0107 5.3e�5 1 1.0204 0.0128 4.5e�8 13B 24 35 35 1.0023 0.0042 1.0035 0.0074 0.1831 0 1.0094 0.0100 1.3e�5 13C 24 35 35 1.0000 0 1.0000 0 1 0 1.0001 0.0010 1 04A 41 69 69 1.0157 0.0035 1.0317 0.0056 7.9e�10 1 1.0341 0.0053 5.1e�10 14B 41 69 69 1.0155 0.0054 1.0301 0.0051 1.0e�9 1 1.0334 0.0057 6.2e�10 14C 41 69 69 1.0162 0.0039 1.0295 0.0061 9.2e�10 1 1.0329 0.0057 7.2e�10 14D 41 69 69 1.0118 0.0051 1.0232 0.0067 2.3e�9 1 1.0268 0.0064 1.1e�9 15A 34 65 65 1.0212 0.0067 1.0471 0.0109 1.3e�9 1 1.0469 0.0086 5.1e�10 15B 34 65 65 1.0183 0.0056 1.0417 0.0095 1.5e�9 1 1.0435 0.0093 5.1e�10 15C 34 65 65 1.0260 0.0048 1.0416 0.0069 1.1e�9 1 1.0448 0.0081 7.5e�10 15D 34 65 65 1.0137 0.0027 1.0230 0.0050 1.5e�9 1 1.0259 0.0051 1.4e�9 16A 31 50 50 1.0191 0.0052 1.0292 0.0102 1.4e�8 1 1.0303 0.0087 1.1e�8 16B 31 50 50 1.0163 0.0053 1.0265 0.0054 2.4e�9 1 1.0297 0.0055 7.4e�10 16C 31 50 50 1.0061 0.0057 1.0198 0.0045 2.1e�9 1 1.0218 0.0044 5.6e�10 17A 40 66 66 1.0239 0.0052 1.0418 0.0128 1.2e�9 1 1.0499 0.0262 7.9e�10 17B 40 66 66 1.0240 0.0030 1.0358 0.0107 5.2e�9 1 1.0431 0.0115 7.5e�10 17C 40 66 66 1.0151 0.0022 1.0234 0.0024 3.5e�10 1 1.0255 0.0040 4.9e�10 18A 30 63 63 1.0290 0.0075 1.0490 0.0099 1.2e�9 1 1.0504 0.0102 2.6e�9 18B 30 63 63 1.0228 0.0094 1.0470 0.0095 5.1e�10 1 1.0504 0.0109 9.8e�10 18C 30 63 63 1.0194 0.0058 1.0356 0.0050 6.5e�10 1 1.0383 0.0058 5.8e�10 19A 50 92 92 1.0255 0.0116 1.1102 0.0376 5.4e�10 1 1.1012 0.0406 5.2e�10 19B 50 92 92 1.0264 0.0203 1.0882 0.0353 2.2e�9 1 1.0859 0.0341 6.1e�9 19C 50 92 92 1.0265 0.0086 1.0731 0.0240 5.1e�10 1 1.0774 0.0238 5.0e�10 19D 50 92 92 1.0126 0.0051 1.0273 0.0055 7.5e�10 1 1.0324 0.0071 5.4e�10 110A 50 97 97 1.0208 0.0184 1.1647 0.0640 5.1e�10 1 1.1563 0.0642 5.1e�10 110B 50 97 97 1.0654 0.0354 1.2373 0.0573 5.3e�10 1 1.2365 0.0475 5.7e�10 110C 50 97 97 1.0257 0.0114 1.0981 0.0418 7.5e�10 1 1.1102 0.0469 5.8e�10 110D 50 97 97 1.0586 0.0252 1.1031 0.0299 2.5e�8 1 1.0978 0.0287 1.1e�7 1

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Table 13The statistical results of HV on the val benchmark test set after 51 independent runs.

Val V T E MPCCA D-MAENS DVCMOA

Mean Std Mean Std p h Mean Std p h

1A 24 39 39 626.7 69.2 594.5 64.1 6.5e�10 1 583.9 66.7 5.1e�10 11B 24 39 39 873.8 256.4 829.5 239.7 1.0e�9 1 823.3 237.0 8.8e�10 11C 24 39 39 4.5 8.4 1.2 4.9 2.4e�4 1 1.4 4.1 9.8e�4 12A 24 34 34 4684.6 133.6 4621.9 133.2 7.5e�10 1 4604.2 141.6 5.4e�10 12B 24 34 34 3599.6 329.0 3560.8 329.6 3.7e�8 1 3555.3 335.5 2.5e�8 12C 24 34 34 1.7 5.2 0.1 0.4 0.0313 1 1.0 4.1 0.2500 13A 24 35 35 221.5 26.8 219.1 25.5 5.2e�6 1 216.5 26.5 1.8e�8 13B 24 35 35 36.8 8.2 36.7 8.4 0.3352 0 36.0 7.6 3.4e�4 13C 24 35 35 0 0 0 0 1 0 0 0 1 04A 41 69 69 6893.5 1335.8 6561.5 1295.4 5.1e�10 1 6472.3 1314.3 5.1e�10 14B 41 69 69 3624.6 1218.9 3367.0 1211.7 5.5e�10 1 3335.6 1199.9 6.1e�10 14C 41 69 69 1455.3 622.3 1335.4 599.2 1.4e�9 1 1278.6 570.9 6.1e�10 14D 41 69 69 87.9 73.7 43.0 32.9 3.8e�8 1 31.1 39.7 2.9e�9 15A 34 65 65 1.6e+4 0.2e+4 1.5e+4 0.2e+4 6.1e�10 1 1.5e+4 0.2e+4 5.1e�10 15B 34 65 65 9095.6 1787.9 8624.5 1727.7 5.8e�10 1 8547.7 1761.8 5.1e�10 15C 34 65 65 6077.5 1556.2 5780.5 1577.1 5.1e�10 1 5710.9 1513.7 5.8e�10 15D 34 65 65 1027.6 321.8 921.1 301.0 8.3e�10 1 897.8 298.3 5.5e�10 16A 31 50 50 2106.9 353.0 2047.7 343.9 1.8e�9 1 2030.8 346.4 5.1e�10 16B 31 50 50 1187.7 165.3 1143.8 159.2 7.3e�10 1 1130.4 156.5 5.1e�10 16C 31 50 50 78.1 30.9 48.3 29.5 3.7e�9 1 42.5 25.8 5.1e�10 17A 40 66 66 5646.9 863.6 5462.5 845.8 5.1e�10 1 5443.3 852.1 5.1e�10 17B 40 66 66 2073.5 419.5 1976.8 416.5 6.5e�10 1 1939.7 409.6 5.1e�10 17C 40 66 66 460.2 91.5 446.8 88.7 9.7e�10 1 438.2 83.6 2.2e�9 18A 30 63 63 1.4e+4 0.1e+4 1.3e+4 0.1e+4 5.5e�10 1 1.3e+4 0.1e+4 5.1e�10 18B 30 63 63 7077.3 1210.3 6559.0 1157.0 5.1e�10 1 6483.5 1147.8 5.8e�10 18C 30 63 63 986.3 372.8 813.4 346.6 7.8e�10 1 786.9 322.1 5.1e�10 19A 50 92 92 6938.4 1195.2 6236.8 1192.2 5.1e�10 1 6133.6 1165.6 5.1e�10 19B 50 92 92 4203.0 794.9 3737.3 752.8 7.3e�10 1 3664.0 727.8 5.8e�10 19C 50 92 92 2838.1 653.1 2467.1 629.4 5.1e�10 1 2400.2 620.6 5.5e�10 19D 50 92 92 363.1 176.5 280.4 158.7 5.1e�10 1 257.8 147.6 5.1e�10 110A 50 97 97 2.3e+4 0.4e+4 2.1e+4 0.3e+4 5.1e�10 1 2.1e+4 0.3e+4 5.5e�10 110B 50 97 97 1.0e+4 0.2e+4 0.9e+4 0.1e+4 5.5e�10 1 0.9e+4 0.1e+4 5.1e�10 110C 50 97 97 6226.8 1011.1 5591.0 989.8 5.1e�10 1 5474.7 1012.4 5.1e�10 110D 50 97 97 2498.3 1112.1 2095.1 923.1 5.5e�10 1 2073.4 921.0 5.1e�10 1

Table 14The statistical results of ID on the egl benchmark test set after 51 independent runs.

egl V T E MPCCA D-MAENS DVCMOA

Mean Std Mean Std p h Mean Std p h

E1-A 77 51 98 6.4871 1.3813 7.3649 2.2318 0.0127 1 8.0623 1.9873 5.1e�5 1E1-B 77 51 98 4.0821 7.3082 14.1238 6.4036 4.5e�7 1 17.5598 5.8098 2.7e�8 1E1-C 77 51 98 37.3669 4.6199 52.8392 12.4102 2.8e�8 1 54.5251 14.9810 4.0e�8 1E2-A 77 72 98 19.1655 18.7182 52.6231 16.2404 9.1e�9 1 62.6490 14.7111 6.2e�10 1E2-B 77 72 98 25.0122 5.0953 42.1731 6.2965 5.1e�10 1 45.5208 8.4915 6.5e�10 1E2-C 77 72 98 19.1610 5.8300 39.1281 11.6592 6.2e�10 1 48.1858 13.1415 6.2e�10 1E3-A 77 87 98 44.2949 14.1913 76.3673 18.0139 6.2e�9 1 82.8305 20.4652 4.4e�9 1E3-B 77 87 98 46.2759 6.6165 57.0486 11.3180 4.2e�7 1 61.3614 12.3690 1.3e�8 1E3-C 77 87 98 39.8912 15.5999 58.3136 22.3946 3.2e�7 1 77.9532 29.5696 9.1e�9 1E4-A 77 98 98 50.0045 7.7156 73.7701 31.1991 5.9e�6 1 86.6980 22.4186 2.4e�9 1E4-B 77 98 98 28.0476 5.0138 66.4717 20.3543 5.5e�10 1 87.8175 24.9333 5.1e�10 1E4-C 77 98 98 56.6145 40.9926 156.5018 30.4954 1.2e�9 1 171.7349 36.7921 7.3e�10 1S1-A 140 75 190 49.6998 10.4353 61.2755 14.3539 8.3e�5 1 70.4663 18.5674 2.9e�7 1S1-B 140 75 190 37.9986 4.6869 65.9309 16.1134 8.8e�10 1 75.1488 15.8693 6.9e�10 1S1-C 140 75 190 15.2154 7.5144 67.8097 23.2621 5.5e�10 1 80.0359 25.3047 5.1e�10 1S2-A 140 147 190 74.0856 14.3440 98.1120 21.5575 0.5178 0 114.9619 26.6838 8.9e�5 1S2-B 140 147 190 63.2305 15.3847 108.8641 34.1791 2.8e�9 1 127.9945 39.6104 1.7e�9 1S2-C 140 147 190 55.3809 34.8551 244.2146 76.7322 5.1e�10 1 259.8283 101.5085 5.5e�10 1S3-A 140 159 190 108.4623 40.8624 109.9562 25.9053 0.7500 0 121.8685 26.3229 0.0916 0S3-B 140 159 190 60.1846 19.5550 93.7351 28.3941 3.0e�8 1 109.7255 33.9281 3.0e�9 1S3-C 140 159 190 48.9827 42.1472 413.4969 101.2926 5.1e�10 1 441.7966 120.5941 5.1e�10 1S4-A 140 190 190 25.7847 17.0052 166.1725 57.0508 5.1e�10 1 213.7841 69.4853 5.1e�10 1S4-B 140 190 190 55.0863 69.0522 366.7897 68.9704 5.1e�10 1 406.5299 75.8416 5.1e�10 1S4-C 140 190 190 132.2291 113.4577 178.9059 105.0823 0.0236 1 231.7199 101.7923 2.8e�4 1

R. Shang et al. / Information Sciences 277 (2014) 609–642 625

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Table 15The statistical results of HD on the egl benchmark test set after 51 independent runs.

egl V T E MPCCA D-MAENS DVCMOA

Mean Std Mean Std p h Mean Std p h

E1-A 77 51 98 28.5116 4.0783 31.4928 9.1121 0.0416 1 38.0247 32.6647 4.8e�4 1E1-B 77 51 98 7.9095 12.3535 26.8993 9.1595 9.1e�8 1 29.0267 8.5206 2.4e�8 1E1-C 77 51 98 97.1961 11.0671 91.6067 20.0966 0.1441 0 89.0267 23.2006 0.0323 1E2-A 77 72 98 97.4745 72.9626 167.8926 49.2996 9.7e�6 1 169.1747 38.6051 2.8e�7 1E2-B 77 72 98 72.7137 16.5495 96.6061 26.7999 2.2e�7 1 108.2279 52.2063 4.2e�8 1E2-C 77 72 98 53.5129 9.1593 98.0197 41.5989 2.6e�9 1 117.5124 44.6675 5.1e�10 1E3-A 77 87 98 405.2380 154.0212 343.6737 152.1630 0.2707 0 383.5235 160.3466 0.5988 0E3-B 77 87 98 224.1765 62.5743 171.8445 50.5824 1.1e�4 1 192.2160 52.1967 0.0120 1E3-C 77 87 98 72.7657 22.2059 126.4162 47.5738 6.5e�9 1 158.5613 53.2070 3.1e�9 1E4-A 77 98 98 256.9723 78.0565 175.5811 58.0883 9.0e�7 1 202.4104 80.5474 1.5e�4 1E4-B 77 98 98 117.9671 59.0283 207.1934 59.7920 2.0e�7 1 245.6479 76.0016 1.8e�8 1E4-C 77 98 98 69.9631 42.2860 190.3197 61.6919 1.2e�9 1 201.7735 50.1291 5.5e�10 1S1-A 140 75 190 216.5171 41.2460 208.6146 37.1198 0.3296 0 222.9418 38.8751 0.4202 0S1-B 140 75 190 125.4321 65.6542 151.8245 31.7142 0.0013 1 189.6108 45.6355 1.7e�6 1S1-C 140 75 190 96.6921 26.1935 137.3454 31.3900 1.5e�7 1 157.0492 30.9862 2.8e�9 1S2-A 140 147 190 880.5437 215.1362 828.3104 139.7331 8.6e�7 1 876.1899 194.3676 0.0168 1S2-B 140 147 190 268.1963 80.3161 293.2413 102.6327 0.3296 0 375.3449 123.9633 2.1e�5 1S2-C 140 147 190 203.1035 100.9511 425.9716 147.3816 5.5e�9 1 453.4176 114.6301 5.1e�10 1S3-A 140 159 190 525.2300 143.0199 560.2045 132.4524 0.1254 0 575.1783 147.0931 0.7287 0S3-B 140 159 190 325.0705 68.5573 365.5464 101.0717 0.0236 1 401.6146 112.6542 2.3e�4 1S3-C 140 159 190 431.7255 122.3735 635.7187 113.9763 1.6e�8 1 681.6041 101.0549 8.8e�10 1S4-A 140 190 190 118.2133 99.5803 315.0452 103.9147 3.9e�9 1 412.6433 158.6847 5.8e�10 1S4-B 140 190 190 61.4510 73.4260 387.3848 67.1425 5.1e�10 1 434.1176 80.7390 5.1e�10 1S4-C 140 190 190 155.5637 108.5719 186.5294 108.7773 0.1265 0 258.2549 103.4675 1.1e�4 1

Table 16The statistical results of e-indicator on the egl benchmark test set after 51 independent runs.

egl V T E MPCCA D-MAENS DVCMOA

Mean Std Mean Std p h Mean Std p h

E1-A 77 51 98 1.0074 0.0011 1.0075 0.0015 1 0 1.0080 0.0015 0.0872 0E1-B 77 51 98 1.0013 0.0026 1.0045 0.0023 1.9e-6 1 1.0057 0.0021 3.7e-8 1E1-C 77 51 98 1.0172 0.0019 1.0164 0.0032 0.2153 0 1.0158 0.0040 0.0421 1E2-A 77 72 98 1.0083 0.0044 1.0163 0.0049 2.6e-8 1 1.0202 0.0048 1.4e-9 1E2-B 77 72 98 1.0087 0.0005 1.0134 0.0014 4.7e-10 1 1.0152 0.0023 4.6e-10 1E2-C 77 72 98 1.0062 0.0012 1.0118 0.0045 1.5e-9 1 1.0139 0.0049 5.1e-10 1E3-A 77 87 98 1.0159 0.0041 1.0267 0.0052 2.5e-9 1 1.0307 0.0048 5.1e-10 1E3-B 77 87 98 1.0137 0.0023 1.0193 0.0031 9.8e-10 1 1.0205 0.0027 6.8e-10 1E3-C 77 87 98 1.0065 0.0027 1.0134 0.0052 7.7e-9 1 1.0169 0.0069 3.2e-9 1E4-A 77 98 98 1.0156 0.0019 1.0230 0.0053 4.5e-9 1 1.0256 0.0051 1.3e-9 1E4-B 77 98 98 1.0058 0.0019 1.0168 0.0046 6.1e-10 1 1.0206 0.0038 5.1e-10 1E4-C 77 98 98 1.0052 0.0036 1.0151 0.0040 1.2e-9 1 1.0164 0.0038 6.5e-10 1S1-A 140 75 190 1.0253 0.0020 1.0381 0.0085 5.5e-10 1 1.0466 0.0098 5.1e-10 1S1-B 140 75 190 1.0097 0.0033 1.0233 0.0058 5.5e-10 1 1.0261 0.0060 5.1e-10 1S1-C 140 75 190 1.0088 0.0030 1.0163 0.0033 6.9e-10 1 1.0174 0.0040 5.5e-10 1S2-A 140 147 190 1.0132 0.0053 1.0237 0.0035 1.8e-9 1 1.0269 0.0042 5.1e-10 1S2-B 140 147 190 1.0085 0.0022 1.0238 0.0051 5.1e-10 1 1.0259 0.0054 5.1e-10 1S2-C 140 147 190 1.0061 0.0034 1.0228 0.0043 5.1e-10 1 1.0251 0.0053 5.1e-10 1S3-A 140 159 190 1.0120 0.0038 1.0270 0.0043 5.5e-10 1 1.0338 0.0056 5.5e-10 1S3-B 140 159 190 1.0082 0.0024 1.0196 0.0044 5.1e-10 1 1.0242 0.0040 5.1e-10 1S3-C 140 159 190 1.0138 0.0051 1.0317 0.0056 5.1e-10 1 1.0347 0.0058 5.1e-10 1S4-A 140 190 190 1.0117 0.0037 1.0231 0.0045 2.5e-9 1 1.0277 0.0057 1.7e-9 1S4-B 140 190 190 1.0036 0.0043 1.0228 0.0041 5.1e-10 1 1.0252 0.0047 5.1e-10 1S4-C 140 190 190 1.0024 0.0057 1.0086 0.0050 1.8e-6 1 1.0118 0.0045 6.7e-8 1

626 R. Shang et al. / Information Sciences 277 (2014) 609–642

algorithms are significant. Summing up the above, in the light of ID, MPCCA has a stronger capability of reaching the areawhich is closer to the true PF than D-MAENS and DVCMOA.

Table 7 shows the statistics results of performance indicator HD among the three algorithms on small-scale test set gdb.For the HD, MPCCA is superior to D-MAENS and DVCMOA on 19 test instances. Hausdorff distance indicates the maximummismatch between the obtained PF and the true PF. The smaller the HD, the closer it approaches to the true PF. Note that ongdb1, gdb4, gdb7, gdb14 and gdb19, MPCCA reaches the theoretical optimal value 0 of HD. It shows that MPCCA finds all thebest nondominated solutions in 51 independent runs on these instances. In addition, the median differences between MPCCA

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Table 17The statistical results of HV on the egl benchmark test set after 51 independent runs.

egl V T E MPCCA D-MAENS DVCMOA

Mean Std Mean Std p h Mean Std p h

E1-A 77 51 98 24697.1 3987.0 23605. 4 4044.0 0.0024 1 23473. 2 3864.4 7.5e�6 1E1-B 77 51 98 950.9 783.2 606.0 590.5 1.6e�5 1 417.8 389.9 4.1e�7 1E1-C 77 51 98 691.7 1209.2 715.9 914.9 0.9043 0 832.3 1135.3 0.1738 0E2-A 77 72 98 14.7e+4 2.6e+4 14.1e+4 2.5e+4 8.2e�9 1 13.9e+4 2.5e+4 5.1e�10 1E2-B 77 72 98 2.4e+4 1.2e+4 2.0e+4 1.1e+4 5.1e�10 1 1.9e+4 0.9e+4 5.1e�10 1E2-C 77 72 98 4764.0 2180.2 3291.1 1900.9 5.1e�10 1 2696.9 1600.5 5.1e�10 1E3-A 77 87 98 1.6e+5 0.4e+5 1.5e+5 0.4e+5 1.6e�9 1 1.4e+5 0.4e+5 5.1e�10 1E3-B 77 87 98 2.0e+4 1.1e+4 1.7e+4 1.2e+4 6.5e�9 1 1.6e+4 1.1e+4 6.5e�10 1E3-C 77 87 98 1.3e+4 0.3e+4 1.0e+4 0.3e+4 7.3e�9 1 0.7e+4 0.4e+4 2.8e�9 1E4-A 77 98 98 1.1e+5 0.2e+5 1.0e+5 0.2e+5 5.5e�10 1 1.0e+5 0.2e+5 5.5e�10 1E4-B 77 98 98 4.1e+4 1.3e+4 3.1e+4 1.1e+4 1.8e�9 1 2.8e+4 1.1e+4 5.8e�10 1E4-C 77 98 98 8093.8 5735.7 2550.6 2718.5 5.1e�10 1 1837.4 2197.4 5.1e�10 1S1-A 140 75 190 3.7e+5 1.4e+5 3.5e+5 1.3e+5 5.1e�10 1 3.4e+5 1.3e+5 5.1e�10 1S1-B 140 75 190 1.1e+5 0.4e+5 1.0e+5 0.4e+5 5.1e�10 1 1.0e+5 0.3e+5 5.1e�10 1S1-C 140 75 190 6.0e+4 0.5e+4 5.0e+4 0.5e+4 5.1e�10 1 5.0e+4 0.5e+4 5.1e�10 1S2-A 140 147 190 2.2e+5 0.7e+5 2.0e+5 0.6e+5 5.8e�10 1 1.8e+5 0.6e+5 5.1e�10 1S2-B 140 147 190 7.6e+4 2.8e+4 5.7e+4 2.3e+4 5.1e�10 1 5.3e+4 2.1e+4 5.1e�10 1S2-C 140 147 190 3.7e+4 1.3e+4 1.4e+4 0.8e+4 5.5e�10 1 1.2e+4 0.9e+4 5.1e�10 1S3-A 140 159 190 2.9e+5 0.8e+5 2.6e+5 0.8e+5 5.1e�10 1 2.5e+5 0.8e+5 5.5e�10 1S3-B 140 159 190 8.9e+4 3.1e+4 6.9e+4 2.4e+4 5.1e�10 1 6.4e+4 2.4e+4 5.1e�10 1S3-C 140 159 190 4.8e+4 1.7e+4 1.8e+4 1.1e+4 5.1e�10 1 1.4e+4 1.0e+4 5.1e�10 1S4-A 140 190 190 3.2e+4 1.7e+4 1.9e+4 1.3e+4 5.5e�10 1 1.4e+4 1.1e+4 7.3e�10 1S4-B 140 190 190 9081.3 8974.5 1774.3 2782.0 7.6e�9 1 688.3 1808.7 7.6e�9 1S4-C 140 190 190 5227.5 3772.2 1663.7 2192.8 3.5e�8 1 619.8 1187.6 2.8e�8 1

Table 18The statistical results of ID on the Brandão&Eglese benchmark test set after 51 independent runs.

EGL V T E MPCCA D-MAENS DVCMOA

Mean Std Mean Std p h Mean Std p h

1-A 255 347 375 0.9e+4 0.5e+4 2.3e+4 0.6e+4 5.8e�10 1 2.8e+4 0.6e+4 5.1e�10 11-B 255 347 375 1.9e+4 1.1e+4 4.0e+4 0.6e+4 1.6e�9 1 4.4e+4 0.5e+4 9.9e�10 11-C 255 347 375 1.4e+4 1.3e+4 3.1e+4 0.4e+4 5.2e�8 1 3.7e+4 0.6e+4 2.3e�9 11-D 255 347 375 1.9e+4 0.9e+4 3.4e+4 0.4e+4 2.8e�9 1 3.5e+4 0.6e+4 7.8e�10 11-E 255 347 375 1.4e+4 0.5e+4 2.7e+4 0.6e+4 5.1e�10 1 2.6e+4 0.5e+4 5.5e�10 12-A 255 375 375 1.1e+4 0.9e+4 3.0e+4 0.5e+4 3.3e�9 1 3.1e+4 0.6e+4 1.9e�9 12-B 255 375 375 1.4e+4 0.7e+4 2.8e+4 0.4e+4 1.4e�9 1 2.8e+4 0.5e+4 9.9e�10 12-C 255 375 375 0.1e+5 0.1e+5 0.2e+5 0.1e+5 7.3e�9 1 1.4e+5 3.9e+5 4.2e�9 12-D 255 375 375 1.6e+4 0.3e+4 3.3e+4 0.7e+4 5.1e�10 1 3.6e+4 0.7e+4 5.1e�10 12-E 255 375 375 1.3e+4 1.0e+4 4.3e+4 0.9e+4 6.2e�10 1 4.1e+4 1.1e+4 1.7e�9 1

Table 19The statistical results of HD on the Brandão&Eglese benchmark test set after 51 independent runs.

EGL V T E MPCCA D-MAENS DVCMOA

Mean Std Mean Std p h Mean Std p h

1-A 255 347 375 2.0e+4 1.3e+4 4.2e+4 0.9e+4 5.9e�9 1 5.3e+4 1.2e+4 1.8e�9 11-B 255 347 375 2.4e+4 1.4e+4 7.9e+4 1.9e+4 5.1e�10 1 8.7e+4 1.7e+4 5.1e�10 11-C 255 347 375 2.0e+4 2.2e+4 8.4e+4 2.2e+4 8.3e�10 1 8.5e+4 1.7e+4 6.1e�10 11-D 255 347 375 2.3e+4 1.3e+4 6.3e+4 0.9e+4 5.1e�10 1 6.4e+4 1.0e+4 5.1e�10 11-E 255 347 375 2.2e+4 0.9e+4 5.2e+4 0.8e+4 6.5e�10 1 5.3e+4 0.9e+4 5.8e�10 12-A 255 375 375 2.0e+4 0.9e+4 5.7e+4 1.7e+4 7.3e�10 1 6.6e+4 2.1e+4 5.5e�10 12-B 255 375 375 1.7e+4 1.1e+4 5.7e+4 1.9e+4 6.1e�10 1 6.7e+4 2.2e+4 5.1e�10 12-C 255 375 375 0.2e+5 0.1e+5 0.4e+5 0.1e+5 1.7e�9 1 1.7e+5 3.9e+5 1.2e�9 12-D 255 375 375 2.8e+4 1.2e+4 6.4e+4 1.0e+4 5.1e�10 1 6.1e+4 0.8e+4 5.1e�10 12-E 255 375 375 3.5e+4 2.0e+4 7.5e+4 1.6e+4 8.2e�9 1 7.2e+4 1.4e+4 3.0e�9 1

R. Shang et al. / Information Sciences 277 (2014) 609–642 627

and D-MAENS are significant on 19 gdb instances. The median differences between MPCCA and DVCMOA are significant on20 gdb instances.

Table 8 shows the statistics results of performance indicator e-indicator among the three algorithms on small-scale testset gdb. During all the experiments about e-indicator, we have been regarding reference PF as the comparing set. So, the value

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Table 20The statistical results of e-indicator on the Brandão&Eglese benchmark test set after 51 independent runs.

EGL V T E MPCCA D-MAENS DVCMOA

Mean Std Mean Std p h Mean Std p h

1-A 255 347 375 1.0256 0.0097 1.0400 0.0063 1.7e�8 1 1.0458 0.0069 5.8e�10 11-B 255 347 375 1.0200 0.0109 1.0447 0.0062 9.8e�10 1 1.0484 0.0059 1.0e�9 11-C 255 347 375 1.0145 0.0153 1.0452 0.0074 2.0e�9 1 1.0516 0.0082 7.6e�10 11-D 255 347 375 1.0146 0.0086 1.0438 0.0059 5.1e�10 1 1.0435 0.0074 5.5e�10 11-E 255 347 375 1.0034 0.0062 1.0296 0.0061 5.1e�10 1 1.0277 0.0067 5.5e�10 12-A 255 375 375 1.0133 0.0112 1.0350 0.0048 1.7e�9 1 1.0394 0.0067 9.9e�10 12-B 255 375 375 1.0054 0.0110 1.0336 0.0052 1.2e�9 1 1.0386 0.0056 5.0e�10 12-C 255 375 375 1.0073 0.0084 1.0298 0.0059 8.3e�10 1 0.9569 0.2820 1.7e�5 12-D 255 375 375 1.0089 0.0048 1.0331 0.0045 5.1e�10 1 1.0348 0.0046 5.1e�10 12-E 255 375 375 1.0093 0.0074 1.0298 0.0053 5.8e�10 1 1.0282 0.0065 1.3e�9 1

Table 21The statistical results of HV on the Brandão&Eglese benchmark test set after 51 independent runs.

EGL V T E MPCCA D-MAENS DVCMOA

Mean Std Mean Std p h Mean Std p h

1-A 255 347 375 1.4e+9 0.5e+9 1.2e+9 0.3e+9 3.7e�9 1 1.1e+9 0.4e+9 6.2e�10 11-B 255 347 375 6.9e+8 4.7e+8 5.2e+8 3.4e+8 3.9e�9 1 4.9e+8 3.5e+8 6.5e�10 11-C 255 347 375 5.2e+8 3.0e+8 3.2e+8 2.4e+8 2.7e�8 1 3.0e+8 2.2e+8 1.1e�9 11-D 255 347 375 1.8e+8 2.6e+8 1.0e+8 1.5e+8 1.3e�9 1 1.0e+8 1.5e+8 2.2e�9 11-E 255 347 375 3.6e+7 3.8e+7 1.1e+7 1.7e+7 1.3e�9 1 0.9e+7 1.6e+7 6.2e�10 12-A 255 375 375 9.1e+8 4.4e+8 7.4e+8 3.8e+8 1.4e�9 1 6.9e+8 3.7e+8 6.5e�10 12-B 255 375 375 4.9e+8 2.7e+8 3.1e+8 1.8e+8 7.8e�10 1 2.8e+8 1.8e+8 5.1e�10 12-C 255 375 375 1.8e+8 1.3e+8 1.0e+8 0.7e+8 6.5e�9 1 79.5e+8 272.4e+8 5.6e�5 12-D 255 375 375 4.3e+7 4.5e+7 1.4e+7 2.1e+7 5.1e�10 1 1.6e+7 2.0e+7 9.9e�10 12-E 255 375 375 3.6e+7 4.2e+7 0.9e+7 2.0e+7 4.4e�9 1 1.4e+7 3.0e+7 3.3e�9 1

628 R. Shang et al. / Information Sciences 277 (2014) 609–642

of e-indicator is always greater than or equal to 1. As seen from Table 8, the comparison results of e-indicator are similar to ID

and HD. MPCCA gets most of the winners once again. Moreover, the Wilcoxon signed rank test returns 1 on most gdb testinstances, which shows that the median of the differences between MPCCA and the compared algorithm are significant.

Table 9 shows the statistics results of performance indicator HV among the three algorithms on small-scale test set gdb.Because the reference point selected for calculating HV in each independent run is different, the results of HV change greatly.

As shown in the statistical results in Table 9, the standard deviations of HV change greatly, which do not reflect the sta-bility of the algorithms. However, to some extent, the average values of HV still can represent the diversity and convergenceof three algorithms. As seen in Table 9, on most of gdb instances, the means of MPCCA about HV are always the largest. As tothe Wilcoxon signed rank test, on most test instances, the median of the differences between MPCCA and compared algo-rithms are significant.

Table 10 shows the statistics results of performance indicator ID among the three algorithms on medium-scale test set val.val test set contains totally 34 medium-scale instances. The number of vertices varies from 24 to 50 and the number of tasksvaries from 34 to 97.

As can be seen from Table 10, MPCCA performs significantly better than D-MAENS and DVCMOA on 33 out of the total 34val instances. On the remaining val3C, MPCCA also has obvious advantages. Moreover, we can conclude that MPCCA has abetter stability because of the smallest standard deviations on most of the val instances. Only on few instances, the mediandifferences between MPCCA and D-MAENS, DVCMOA are not significant.

Table 11 shows the statistics results of performance indicator HD among the three algorithms on test set val. Taking acloser look at the HD, it can be found that MPCCA obtains a better result on 27 out of 34 val instances than D-MAENS andDVCMOA. Only on 6 test instances, MPCCA gets a worse result than the other two algorithms. D-MAENS gets the winner onlyon val4A, val5D and val9C. DVCMOA gets the winner only on val5C, val8A and val10A. Moreover, on total 24 val instances,MPCCA finds the smallest standard deviations of HD, while D-MAENS and DVCMOA fail to be the smallest on most of theval instances. As seen from Table 11, on most val instances, the median of the statistical results of HD between MPCCAand compared algorithms (D-MAENS and DVCMOA) are significantly different.

Table 12 shows the statistics results of performance indicator e-indicator among the three algorithms on test set val. InTable 12, the statistics results of e-indicator are similar to the results of ID. On 34 val instances, the total number of winnersof MPCCA is 33. It is also only on val3C that MPCCA does not win, which indirectly shows that there are some intrinsic linksbetween ID and e-indicator. On almost all the test instances, the results of the Wilcoxon signed rank test return 1.

Table 13 shows the statistics results of performance indicator HV among the three algorithms on test set val. As can beseen from Table 13, MPCCA can find the largest average value of HV on most of val instances. It shows that MPCCA has a

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stronger ability to keep diversity and converging to the true Pareto-Optimal Front. As described above, the standard devia-tions of HV have no rules to follow. As to the Wilcoxon signed rank test, the median results among the three algorithms aresignificantly different on most val instances

Table 14 shows the statistics results of performance indicator ID among the three algorithms on test set egl. egl set wasgenerated by Eglese which is a large-scale test set for CARP. Based on the data from a winter gritting application in Lanca-shire, it includes 24 instances based on two graphs and the number of tasks varies from 51 to 190. ID reflects the distributionof the nondominated solutions and the extent of the convergence to the true Pareto front. In Table 14, it can be observed thatMPCCA performs significantly better than the others on total 24 egl instances. Moreover, on most of egl instances, D-MAENScan get a better result than DVCMOA. In addition, on egl-s2-A and egl-s3-A, the Wilcoxon signed rank tests between MPCCAand D-MAENS output 0. Also on egl-e3-A, the Wilcoxon signed rank tests between MPCCA and DVCMOA return 0. On theother test instances, the results of the Wilcoxon signed rank test return 1.

Table 15 shows the statistics results of performance indicator HD among the three algorithms on test set egl.It can be seen from table 15, for HD, MPCCA performs significantly better than the others on 18 egl instances. D-MAENS is

significantly better on 5 egl instances and DVCMOA only obtain the winner on egl-e1-C. As to the Wilcoxon signed rank test,there are total 6 instances on which the test returns 0 for D-MAENS, while the number is 3 for DVCMOA.

Table 16 shows the statistics results of performance indicator e-indicator among the three algorithms on test set egl. InTable 16, MPCCA performs significantly better than the others on 23 egl instances. DVCMOA is significantly better on 1egl instances and D-MAENS fails to be the best on all the egl instances. On almost all the test instances, the results of theWilcoxon signed rank test return 1.

Table 17 shows the statistics results of performance indicator HV among the three algorithms on test set egl. Much like theresults of e-indicator in Table 16, the total number of winners of MPCCA is 23. It is also only on egl-e1-C that MPCCA does notwin. On almost all the test instances, the median differences between MPCCA and the compared algorithm are significant.

Fig. 7. Box plots of the three algorithms on metric ID, HD, e-indicator and HV for part of gdb set.

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Fig. 7 (continued)

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Table 18 shows the statistics results of performance indicator ID among the three algorithms on test set Brandão&Eglese.Brandão&Eglese includes 255 vertices and 375 edges, which is the largest among all these test sets. As can be seen from Ta-ble 18, MPCCA has more obvious advantages over the other two algorithms on all the 10 Brandão&Eglese instances. As for themean of ID, D-MAENS is not as good as MPCCA. While D-MAENS has a little better stability than the other two algorithmsbecause of it obtains 5 smallest standard deviations on total Brandão&Eglese10 instances. On all the Brandão&Eglese test in-stances, the results of the Wilcoxon signed rank test return 1, which indicates that the median differences between MPCCAand the compared algorithm are significant.

Table 19 shows the statistics results of performance indicator HD among the three algorithms on test set Brandão&Eglese.As can be seen from Table 19, we can conclude that the obtained PF of MPCCA is the most similar to the true PF among thethree algorithms because of it finds 10 smallest means on all the Brandão&Eglese instances. On all the Brandão&Eglese testinstances, the median differences between MPCCA and the compared algorithm are significant.

Table 20 shows the statistics results of performance indicator e-indicator among the three algorithms on test set Bran-dão&Eglese. First, we focus on the means of e-indicator. MPCCA gets the smallest means on all the Brandão&Eglese instances.Then, we focus on the standard deviations of e-indicator. It is shown from the table that D-MAENS finds the smallest standarddeviations on 9 out of the total 10 instances of Brandão&Eglese set. On all the Brandão&Eglese test instances, the e-indicatorresults of the Wilcoxon signed rank test are the same as that of ID and HD.

Table 21 shows the statistics results of performance indicator HV among the three algorithms on test set Brandão&Eglese.In Table 21, MPCCA performs significantly better than the others on 9 Brandão&Eglese instances. The results of the Wilcoxonsigned rank test also demonstrated the advantage of MPCCA. DVCMOA is significantly better on 1 Brandão&Eglese instancesand D-MAENS fails to be the best on all the Brandão&Eglese instances.

For a more visual comparison of the three algorithms, the box plots of all the experimental results are given in Figs. 7–10,followed by a detailed discussion. Because the number of test instances used in this experiment is huge and the space of

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Fig. 8. Box plots of the three algorithms on metric ID, HD, e-indicator and HV for part of val set.

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paper is limited, here we select parts of box plots to show according to certain laws. Box plots of the three algorithms forparts of gdb test set are given in Fig. 7. For all the 23 gdb instances, we select one of every three instances to show. InFig. 7, for e-indicator, from the first column of eight graphics, it is clear that the results of MPCCA have smaller e-indicatorthan the other algorithms on most of instances except gdb20. Especially on gdb14, gdb17 and gdb23, MPCCA can obtainthe optimal results stably during all 51 independent runs, while D-MAENS and DVCMOA do not have this feature. For HD,from the second column of eight graphics, on gdb 8 and gdb23, the result of MPCCA is frustrating, which can also be verifiedby the statistical numerical results in Table 7. However, most of the HD of MPCCA is encouraging. For the box plots of HV, theadvantage of MPCCA is no longer obvious. Only on gdb8, gdb11 and gdb17, the distribution of HV is better than the other twoalgorithms. At last, for ID, MPCCA has the smaller values than the other algorithms on most of instances except gdb8 andgdb23.

Box plots of the three algorithms for parts of val test set are given in Fig. 8. val contains 34 instances based on ten differentgraphs. We choose one instance from each graph and show the box plot. First, we focus on the e-indicator. From the firstcolumn of ten graphics, there is no doubt that MPCCA is the best. On most of val instances, the upper quartiles of MPCCA

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Fig. 8 (continued)

632 R. Shang et al. / Information Sciences 277 (2014) 609–642

are even smaller than the lower quartiles of the other two algorithms. Then for HD, from the second column of ten graphics,the advantage of MPCCA is not so obvious, especially on val10A, val9A, val8A and val4A. At last for the HV and ID, the perfor-mance of three algorithms is obvious. MPCCA gets the best performance, and then followed by D-MAENS, and DVCMOA isthe worst.

Box plots of the three algorithms for parts of egl test set are given in Fig. 9. egl consists of 24 instances based on twographs. Much like these box plots on gdb, we also select one of every three instances to show.

In Fig. 9, for e-indicator, the results of MPCCA are significantly better than the others on most of instances except egl-e1-A.For HD, the performance of MPCCA and D-MAENS is neck and neck. On egl-e1-A, egl-e2-A, egl-s3-A and egl-s4-A, MPCCA showsignificantly better performance than the other two. While on the remaining 4 instances, D-MAENS is the winner. From thethird column of 8 graphics, it is clear that MPCCA performs better than the other algorithms on the HV. At last for the boxplots of ID, MPCCA is still the most competitive.

Box plots of the three algorithms for parts of Brandão&Eglese test set are given in Fig. 10. Brandão&Eglese contains 10instances which are based on 2 graphs. We choose all the five instances belonging to the first graph to show.

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Fig. 9. Box plots of the three algorithms on metric ID, HD, e-indicator and HV for part of egl set.

R. Shang et al. / Information Sciences 277 (2014) 609–642 633

From Fig. 10, it is clear to see that MPCCA gets the best performance followed by D-MAENS, and DVCMOA is the worst,based on these Brandão&Eglese box plots. The relationship between the above box plots and the quantitative results inTables 6–21 deserves more discussion. D-MAENS adopts a decomposition-based framework. It decomposes the originalMO-CARP into a number of scalar subproblems using the weighted sum approach, while MPCCA uses the mechanism ofcooperative coevolution. Coevolutionary algorithms have been proved to be effective in solving large-scale optimizationproblems. Although both MPCCA and D-MAENS use neighbor relations to promote evolution, they adopt different methods.MPCCA uses the internal elite archive to construct evolutionary pool for each subregion, which is different from the situ-ation in D-MAENS. Thus, the convergence of MPCCA is greater than the convergence in D-MAENS. With the expansion ofdata, the advantage of MPCCA is more and more obvious. In most cases, DVCMOA does not perform well. In summary,there are two main reasons for the worst results of DVCMOA: 1. DVCMOA carries out the assignment of subpopulationsat the initialization phase only and it is no longer changing during the search process. This mechanism may lead to unevendistributions among these subpopulations, and the allocation of computing resources will be uneven too. While in D-MAENS and MPCCA, individuals of each subpopulation are reassigned at each generation, which pays more attention onthose undeveloped areas and increases the diversity. 2. In DVCMOA, a solution of a subpopulation can only be replacedby a solution generated for the same subpopulation. While D-MAENS and MPCCA combine all the solutions togetherand compare them regardless of which subpopulation they belong to. After that, the fast nondominated sorting procedureand the crowding distance method of NSGA-II are used to selected offspring, and then these offspring are reassigned toeach subpopulation.

In order to conduct a more comprehensive comparison of the three algorithms, we use the method of attainment surfaceto drawn the graphical results, which shows the advantages of one algorithm converges to the PF. Fonseca and Fleming [16]proposed the concept of attainment surface and used it as a performance assessment for stochastic MOEAs. After that, in

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Fig. 9 (continued)

634 R. Shang et al. / Information Sciences 277 (2014) 609–642

2005, Knowles reported an algorithm that can be conveniently used to plot summary attainment surface in any number ofdimensions [30]. The interested readers can refer to http://www.cs.man.ac.uk/~jknowles/plot_attainments/. Usually, whenwe want to display the PF of multiple runs of one or more algorithms, it cannot give the idea of how good the individualpoints found in each run tend to be. Moreover, the information of how these nondominated individuals found in each rundistributed along the trade-off surface is also unclear. However, when we use the attainment surface, the above troubles be-come very simple. The attainment surface conveys the information about the location, dispersion, and even skewness of thedistributions of the results, much like the box plot used in statistics. In the attainment surface plots, the 1-th attainment sur-face indicates the best trade-off approximation (the area located down or left of the 1-th attainment surface is composed ofall those objective vectors which are never attained in all 51 runs). Second, the upper line is the 51-th attainment surface (thearea located up or right of the 51-th attainment surface is composed of all those objective vectors which are all attained in all51 runs). Third, the 26-th attainment surface provides an estimate of the real 50%-attainment surface of each algorithm,which likes the median in box plot [16,30].

In order to improve the attainment surface interpretability to show in the same graph results comparing all algorithms,we only present the 50%-attainment surface for each instance. The attainment surface plots of the three algorithms (51 runs)for part of gdb and val set are shown in Fig. 11. Here we select the instances with a relatively large number of nondominatedsolutions to show.

In Fig. 11, on some small and medium-scale instances, such as gdb11, val5A, val5B, val5C, val6A, val6B, val7A, val7B andval7C, the 50%-attainment surface of the three algorithms are almost coincided with each other. The advantage of MPCCA isnot obvious. While on the remaining instances, it is clear that MPCCA not only have a good convergence but also is very sta-ble. Because the 50%-attainment surface in the attainment surface plots of MPCCA are more close to the real Pareto front,while the other two algorithms do not have this phenomenon. That is to say MPCCA can always get the nondominated

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Fig. 10. Box plots of the three algorithms on metric ID, HD, e-indicator and HV for part of Brandão&Eglese set.

R. Shang et al. / Information Sciences 277 (2014) 609–642 635

solutions which are the closest to the real Pareto front during the 51 independent runs. It also shows that MPCCA wouldappear to produce better results more often than D-MAENS and DVCMOA.

The 50%-attainment surface plots of the three algorithms (51 runs) for part of egl and Brandão&Eglese set are shown inFig. 12. Due to the limited space, here we also select the instances with a relatively large number of nondominated solutionsto show.

It can be seen from Fig. 12 that the difference of the three 50%-attainment surfaces are very obvious. Only on egl-e2-A, the50%-attainment surfaces of the three compared algorithms are almost coincided with each other in the attainment surfaceplots. On most of egl and Brandão&Eglese test instances, the 50%-attainment surface obtained by MPCCA are closer to the leftand low region in the objective space than the 50%-attainment surfaces found by the other two compared algorithms. Itshows MPCCA can find a better f1 on these instances.

In order to comprehensively evaluate the performance of MPCCA, D-MAENS and DVCMOA, the nondominated solutionsobtained by them on some instances are plotted in the objective space. For those small and medium scale data instances (gdb

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(11-1-a) (11-1-b) (11-1-c)

(11-2-a) (11-2-b) (11-2-c)

(11-3-a) (11-3-b) (11-3-c)

(11-4-a) (11-4-b) (11-4-c)

(11-5-a) (11-5-b) (11-5-c)

Fig. 11. Representation of the statistical performance of the three algorithms (51 runs) for part of gdb and val set.

636 R. Shang et al. / Information Sciences 277 (2014) 609–642

and val), these nondominated solutions are overlaps and it is different to distinguish. So it is more important to visualize thePF for the instances with larger scale than those with less scale. In practice, for each instance, all the solutions obtainedthroughout multiple runs are first combined together. Then, the nondominated solutions are identified and chosen for show.Due to the limited space, here we take egl set as representative and select a few instances to show in Fig. 13.

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(11-7-a) (11-7-b) (11-7-c)

(11-8-a) (11-8-b) (11-8-c)

(11-9-a) (11-9-b) (11-9-c)

(11-6-a) (11-6-b) (11-6-c)

Fig. 11 (continued)

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As can be seen form Fig. 13, on egl-e3-A, MPCCA finds all the nondominated solutions in the intermediate region (thatwith the value of makespan from 830 to 870 and the value of total-cost from 6300 to 6500). On egl-e2-B, MPCCA also hasthe advantage of a strong convergence in the intermediate region (6350 < total-cost < 6650 and 825 < makespan < 855). Onegl-e4-B and egl-s1-C, MPCCA pays more attentions on the low total-cost region, while D-MAENS and DVCMOA does not havethis advantage. On egl-e3-B, MPCCA converges better than the others, and the solutions found by the other algorithms aremostly dominated by MPCCA. Besides, on egl-s2-C, egl-s3-C and egl-s4-A, MPCCA finds almost all the nondominated solutions,and the PF of MPCCA is more close to the true PF. For the rest test instances of egl, MPCCA still has a stronger capability ofconverging to the true PF than other two algorithms.

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(12-1-a) (12-1-b) (12-1-c)

(12-2-a) (12-2-b) (12-2-c)

(12-3-a) (12-3-b) (12-3-c)

(12-4-a) (12-4-b) (12-4-c)

(12-5-a) (12-5-b) (12-5-c)

Fig. 12. Representation of the statistical performance of the three algorithms (51 runs) for part of egl and Brandão&Eglese set.

638 R. Shang et al. / Information Sciences 277 (2014) 609–642

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(12-6-a) (12-6-b) (12-6-c)

(12-7-a) (12-7-b) (12-7-c)

(12-8-a) (12-8-b) (12-8-c)

(12-9-a) (12-9-b) (12-9-c)

(12-10-a) (12-10-b) (12-10-c)

Fig. 12 (continued)

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Fig. 13. Nondominated solutions obtained by all 51 runs of three algorithms for part of egl set.

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5. Conclusion and future work

In this paper, we investigate MO-CARP within the framework of CA. First, a set of uniformly distributed direction vectorsis generated to divide the whole objective space into multiple subregions. The individuals in different subregions form

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different subpopulations. These subpopulations are not static. Before each iteration, all the individuals in current populationare sorted according to their different direction vectors and then assigned to N subpopulations evenly. These subpopulationsevolve separately, while the adjacent subpopulations can share their individuals in the form of cooperative subpopulations.By referencing some other evolutionary strategies, such as the elitism archiving mechanism, the NSGA-II and the MAENS forSO-CARP, a multi-population CA for MO-CARP (MPCCA) is proposed. Compared with the state-of-art algorithm D-MAENS andDVCMOA, MPCCA shows a better diversity and a faster convergence, especially on those large scale instances.

In future, we will consider other factors in MO-SARP, such as the time window constraints, multi-depot and multi-vehicles. Meanwhile, we will focus on more practical applications.

Acknowledgements

We would like to express our sincere appreciation to the anonymous reviewers for their insightful and valuable com-ments, which have greatly helped us in improving the quality of the paper. This work was partially supported by the NationalBasic Research Program (973 Program) of China, under Grant 2013CB329402, the National Natural Science Foundation ofChina, under Grants 61371201, 61001202, 61203303 and 61272279, the Fundamental Research Funds for the CentralUniversities, under Grant K5051302028, the Fund for Foreign Scholars in University Research and Teaching Programs(the 111 Project) under Grant B07048, and the Program for Cheung Kong Scholars and Innovative Research Team inUniversity under Grant IRT1170.

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