- Email: [email protected]
General framework for localised multi-objective evolutionary algorithms
Information Sciences 258 (2014) 29–53
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Information Sciences journal homepage: www.elsevier.com/locate/ins
General framework for localised multi-objective evolutionary algorithms Rui Wang a,b,⇑, Peter J. Fleming a, Robin C. Purshouse a a b
Department of Automatic Control & Systems Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK Department of Systems Engineering, College of Information Systems and Management, National University of Defense Technology, Chang Sha 410073, China
a r t i c l e
i n f o
Article history: Received 15 November 2012 Received in revised form 23 July 2013 Accepted 17 August 2013 Available online 30 August 2013 Keywords: Multi-objective optimisation Evolutionary algorithm Framework Clustering
a b s t r a c t Many real-world problems have multiple competing objectives and can often be formulated as multi-objective optimisation problems. Multi-objective evolutionary algorithms (MOEAs) have proven very effective in obtaining a set of trade-off solutions for such problems. This research seeks to improve both the accuracy and the diversity of these solutions through the local application of evolutionary operators to selected sub-populations. A local operation-based implementation framework is presented in which a population is partitioned, using hierarchical clustering, into a pre-defined number of sub-populations. Environment-selection and genetic-variation are then applied to each sub-population. The effectiveness of this approach is demonstrated on 2- and 4-objective benchmark problems. The performance of each of four best-in-class MOEAs is compared with their modified local operation-based versions derived from this framework. In each case the introduction of the local operation-based approach improves performance. Further, it is shown that the combined use of local environment-selection and local genetic-variation is better than the application of either local environment-selection or local genetic-variation alone. Preliminary results indicate that the selection of a suitable number of sub-populations is related to problem dimension as well as to population size. 2013 Elsevier Inc. All rights reserved.
1. Introduction Multi-objective problems (MOPs) regularly arise in real-world design scenarios, where two or more objectives are required to be optimised simultaneously. As such objectives are often in competition with one another, the optimal solution of MOPs is a set of trade-off solutions, rather than a single solution. Due to the population-based approach, multi-objective evolutionary algorithms (MOEAs) are well suited for solving MOPs since this leads naturally to the generation of an approximate trade-off surface (or Pareto front [22]) in a run [6,9]. A variety of MOEA approaches has been proposed [63]: (1) Pareto dominance (and modified Pareto dominance) based MOEAs, e.g. MOGA [17,19,21],NSGA-II [12], SPEA2 [65], -MOEA [11] and Pareto Cone -dominance based MOEA [2], (2) indicator (hypervolume [66]) based MOEAs, e.g. IBEA [64], SMS-EMOA [4], HypE [1], (3) scalarising function based MOEAs, e.g. MSOPS [26], MOEA/D [62], and, latterly, (4) preference-inspired co evolutionary algorithms (PICEAs), e.g. PICEA-g [48,58,59]. All of these approaches strive to converge quickly to a satisfactory approximation to the Pareto front (convergence) and that this approximation be well distributed with a good coverage of the front (diversity).
⇑ Corresponding author at: Department of Automatic Control & Systems Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK. E-mail addresses: [email protected], cop10rw@sheffield.ac.uk (R. Wang). 0020-0255/$ - see front matter 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2013.08.049
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Pareto dominance based MOEAs were one of the earliest approaches and it is accepted that they perform well on MOPs with 2 and 3 objectives. However, their search capability often degrades significantly as the number of objectives increase [46,27]. This is because the proportion of Pareto-optimal (or non-dominated) objective vectors in the population grows large when MOPs have more than 3 objectives i.e. many-objective problems. As a result, insufficient selection pressure can be generated toward the Pareto front [47,36]. Relatively recently, there has been considerable effort invested in other types of MOEAs that might perform more effectively on many-objective problems. In particular, PICEA-g, HypE and MSOPS are MOEAs that claim to perform better on many-objective problems [55,58]. In general, a MOEA can be described as
Pðt þ 1Þ ¼ Ss ðv ðSv ðPðtÞÞÞ; PðtÞÞ
ð1Þ
where P(t) are the candidate solutions (population) at iteration t, Sv is the selection-for-variation operator, v is the genetic-variation (recombination and mutation) operator, Ss is the environment-selection operator, and P(t + 1) are the newly generated solutions [47]. Thus, a set of candidate solutions is evolved by successively applying recombination, mutation, and selection to yield better solutions in an iterative process. In many cases, Ss and v are executed on the entire population (described as a global operation). Accordingly, an evolutionary operator that is executed on sub-populations is denoted a local operation. Global operations are held to be beneficial for speed of convergence. However, global operation increases the probability of recombining solutions distant from one another and so produce lower performance offspring known as lethals [19,47,31]. The issue of lethals were originally considered in single objective optimisation by Deb & Goldberg as a response to the problems of fitness sharing [22]. The superfluous production of lethals, known as dominance resistance [47], will consequently reduce the efficiency of the optimisation process and affect convergence towards the Pareto optimal front. It has been argued that the use of local operations may achieve a better overall performance, taking account of both convergence and diversity [7,37]. For example, Sato et al. [49,50] demonstrated that the execution of Pareto dominance and recombination on neighbouring solutions could produce more fruitful solutions and so improve the performance (convergence and diversity) of MOEAs. Although local operation has been investigated in some studies1 [39,49,50,31,57], no generalised MOEA framework is available for the implementation of local operations. In this study we first propose a local operation based framework and then present a specific implementation for using this framework. In this implementation, a clustering technique first partitions the global population into a number of sub-populations. Then, the operators, Sv and v, are executed on each sub-population independently. The proposed framework shares some similarities with parallel MOEAs, particularly, the island model based MOEAs (in which the population is divided into sub-populations, each sub-population is associated with an island and an optimiser evolves the sub-population with occasional migration of individuals between sub-populations). A specific instantiation of the framework is evaluated by comparing the performance of MOEAs and their modified local versions (LMOEAs) derived from the new framework on 2- and 4-objective real-parameter test problems. The MOEAs selected for the evaluation are four best-in-class algorithms: NSGA-II (Pareto dominance), HypE (indicator-based), MSOPS (scalarising function) and PICEA-g (co-evolutionary). Since MOEA/D [62] includes local operators – local selection and local replacements – it is not selected for this study [32]. Overall, the main contributions of this paper are as follows: A local operation based framework is proposed and a specific implementation for using this framework is presented. The effectiveness of this local framework is demonstrated to work well on four different types of MOEAs. The probability p of doing local operations is discussed. Experimental results show using fixed setting of p is worse than using different p (changed according to a relationship defined in Eq. (3)) during the search. The effect of the independent use of local environment-selection or local genetic-variation is discussed. Experimental results show applying both the local operations jointly is better than applying one of the operations on its own. Moreover, local genetic-variation is more effective. The effect of the number of sub-populations (k) to the performance of the framework is discussed. It is found k is impacted by both population size and problem dimension. Some suggestions are provided for the choice of a suitable k. The remainder of the paper is structured as follows. In Section 2, MOPs and MOEAs are introduced and related work is briefly reviewed. This is followed, in Section 3, by a description of the new MOEA framework for implementing local operations. Section 4 provides a systematic performance comparison of individual MOEAs and LMOEAs. Section 5 examines the effectiveness of this new framework over other existing local operation based algorithms. Section 6 provides a further discussion of this framework. Section 7 offers concluding remarks and future research. 2. Basic features of MOPs and MOEAs In this Section, the formulation of MOPs and some fundamental concepts in multi-objective optimisation are first introduced, then some features relating to the development of existing MOEAs are described and some related studies on the use of local operations are briefly reviewed. 1
In [57], local operators are implemented in PICEA-g as part of a preliminary study.
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2.1. MOP formulation and fundamental concepts A multi-objective problem has a number of objective functions that are to be minimised/maximised. Without loss of generality, a minimisation MOP is defined as follows:
minimise fm ðxÞ m ¼ 1; 2; . . . ; M subject to g j ðxÞ 6 0;
j ¼ 1; 2; . . . ; J ð2Þ
hk ðxÞ ¼ 0;
k ¼ 1; 2; . . . ; K
xli 6 xi 6 xui ;
i ¼ 1; 2; . . . ; n
A solution x is a vector of n decision variables: x ¼ ðx1 ; x2 ; ; xn Þ; x 2 Rn . Each xi is subject to a lower xli and an upper xui bound. fi the ith objective function. M is the number of objectives (generally, M > 2). J and K are the number of inequality and equality constraints, respectively. Definition 2.1 (Pareto dominance). For two feasible decision vectors x, y, x is said to Pareto dominate y (denoted as x y) if and only if "i 2 1, 2, . . ., n, fi(x) 6 fi(y) and $i 2 1, 2, . . ., n, fi(x) < fi(y)
Definition 2.2 (Pareto optimality). A solution x 2 Rn is said to be Pareto optimal in Rn if and only if 9 = y 2 Rn ; y x. Definition 2.3 (Pareto optimal set). The Pareto optimal set (PS) is defined as the set of all Pareto optimal solutions, i.e. PS ¼ fx 2 Rn j 9 = y 2 Rn ; y xg. Definition 2.4 (Pareto optimal front). The Pareto optimal front (PF) is defined as the set of all objective functions values corresponding to the solutions in PS, i.e., PF = {(f1(x), f2(x), . . ., fM(x)):x 2 PS}. 2.2. MOEA development MOEAs have gradually been refined over the last two decades [8]. In the first decade, MOEAs were mainly based on two pioneering concepts, i.e. Pareto dominance selection and a niching technique [22]. Pareto dominance selection serves to promote convergence by favouring solutions closer to the Pareto optimal front. The niching technique increases diversity by favouring solutions in a sparse region, thus maintaining well distributed solutions along the entire Pareto front. Representative MOEAs are Multi-Objective Genetic Algorithm (MOGA [17]), Non-dominated Sorting Genetic Algorithm (NSGA [51]), and the Niched-Pareto Genetic Algorithm (NPGA [24]). Second generation MOEAs may be said to arise with the introduction of elitism as a standard mechanism. SPEA [66] popularized the notion of using elitism and these second-generation algorithms feature a combination of Pareto dominance selection, diversity maintenance and the implementation of an elitism strategy. The incorporation of an elitism strategy has since become common practice in the design of MOEAs. Elitism can be implemented in a (l + k) framework (see Fig. 1 and Algorithm 1).
Fig. 1. (l + k) elitist framework.
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Algorithm 1. A general (l + k) elitist Framework
In Algorithm 1, a population of candidate solutions, P is evolved for a fixed number of generations, maxGen. At each generation gen, parents P are manipulated by (representation-appropriate) genetic-variation operators to produce new offspring, Pc. P and Pc, are then pooled and the combined population is sorted according to fitness of individuals. Environment-selection is applied to select the set of solutions that will be the new parent population, P. NSGA-II [12] is a typical MOEA representative of the (l + k) elitist framework. Elitism can also be implemented with an external archive which retains all the nondominated solutions found so far, see SPEA2 [65]. 2.3. MOEAs employing local operations The application of local operations in MOEAs is distinct from the practice of applying a standard local search. Local search is typically implemented on some selected good solutions after an evolutionary search, while environment-selection and genetic-variation are still operated on the entire population. However, for local operations, the environment-selection and genetic-variation are executed on sets of pre-selected sub-populations separately. A number of strategies have been proposed for the execution of local operations in MOEAs. A direct way is to execute evolutionary operators on neighbouring solutions in the objective space. The earliest studies of this type of approach are [17,22] in which solutions are prohibited from recombination with each other if the distance between them lies beyond a defined limit. Watanabe et al. [60] suggested recombination on solutions that are adjacent to each other in one of the objectives. Ishibuchi and Shibata implemented local recombination based on a modified tournament selection [33]. Specifically, in this case, when performing recombination, an individual is first selected, then among the individuals selected by multiple standard binary tournaments, the individual closest to it is selected as its mate, i.e. selecting the mate based on the similarities to first parent. Other studies that also investigated the issues of similarity based mating can be found in [34,35,31]. In another approach, local operations are executed on individuals with a similar search direction in the objective space. Representative approaches are multi-objective genetic local search (MOGLS), as proposed by Ishibuchi et al. [30,37] or Jaszkiewicz. [38]. A concern here is how to distribute the search effort uniformly toward all search directions. The coarse grained parallel MOEAs (i.e. islands) might implicitly encourage executions of local operations. Specifically, the island paradigm divides the overall population into a number of independent sub-populations. Each sub-population is evolved by an optimiser in isolation for the majority of algorithm execution. Individuals occasionally migrate between an island and its neighbour(s) based on some selection criteria [6, p. 455]. If the division is implemented based on the similarities of solutions then operations executed on each sub-population can be seen as local operations. This is because solutions in each sub-population are similar to each other and therefore recombination between these solutions are more likely to produce higher performance offspring, i.e. reducing the possibility of lethals [31,47]. However, often more attention is paid to defining suitable migration/replacement policies2 rather than the division strategies [61,42] and [6, pp. 493–500]. MOEA/D, a decomposition-based approach, performs evolutionary operations locally [62]. It treats a MOP as a collection of single-objective problems (SOPs), which are defined by a scalarising function with different weight vectors. Each scalarising fitness function (defined by a specific weight vector) identifies a single solution which is optimal with respect to that scalarising fitness function. For each SOP, a new solution is generated by performing genetic-variation operations (recombination and mutation) on several solutions amongst its neighbours. Neighbours are defined based on the distance between the weight vectors. The newly generated solution is only compared with its neighbours. If the new solution is better, then some (or all) of its corresponding neighbours are replaced by the new solution (environment-selection). Apart from these approaches, Purshouse and Fleming [44] proposed dividing the objectives themselves into sets of subobjectives according to the relations (conflict, harmony or independent) of the objectives [45]. Evolutionary operations are then performed on these sub-problems independently. The novelty of this approach is that the division is not based on individuals but objectives. 2 These policies include the migration frequency, the number of solutions to migrate/replacement, the selection of migration/replacement solutions, and so on [6, pp. 455–458].
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One specific question relating to the effectiveness and efficiency of the use of local operations is when and how often should local operations be executed. Experience [7] suggests that local operations should be (1) executed after a predefined number of generations, and (2) used in a probabilistic manner. When many generations are assigned for global operation before applying local operations, this may result in too much emphasis on exploration in the search, possibly resulting in unsatisfactory convergence rates. In contrast, frequent application of local operations may place too much emphasis on exploitation of the search space, possibly leading to premature convergence. In other words, the balance between global and local operations is very important. This was first pointed out in [37], and then was clearly demonstrated in [29,35]. If this balance is not appropriately specified, the performance of MOEAs would often be degraded by hybridisation with local operations. Furthermore, it is accepted that an increase in the number of objectives in a MOP generates increased numbers of Pareto optimal solutions and their diversity. Furthermore, solutions in the current population are often quite dissimilar from each other. As stated in a study [28] that a large population diversity has negative effects on the performance of MOEAs. This is because recombining dissimilar solutions often cannot produce good offspring. This issue was also implicitly demonstrated in [32], where the performance of MOEA/D on many-objective 0/1 knapsack problems was degraded by increasing the size of a neighbourhood structure for parent selection. These results emphasise the importance of applying local operations in manyobjective problems as recombining similar solutions often generates high-quality offspring [34,35,28]. 3. A framework for integrating local evolutionary operators into the design of MOEAs The global application of evolutionary operators has proved to be successful in identifying Pareto front efficiently. However, local operations which emphasise locality of selection and variation may achieve even better solutions. A generalised local operation based MOEA framework is described in this section. In this framework evolutionary operators are probabilistically executed either locally or globally during the search process. Both the benefits of global operation and local operation are expected to be realised. This local framework is based on the (l + k) elitist strategy. The flow chart of the framework is shown in Fig. 2 and the pseudo-code is described in Algorithm 2. As shown in Fig. 2, parents P compete with their offspring Pc, probabilistically, the better individuals are more likely to be propagated as new parents. However, on the operations of genetic-variation and environment-selection, condition A and condition B are checked respectively to determine whether the operation is to be executed locally (on each of a set of sub-populations) or globally (on the entire population). Algorithm 2. A New local operation based (l + k) elitist MOEA Framework
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Fig. 2. Local operations based elitist framework.
If condition A is satisfied, candidate solutions are then partitioned into a number of sub-populations, Pk, and genetic-variations are performed on each Pk independently to produce new offspring, Pck. The entire set of offspring, Pc is comprised of all the Pck. Fitness evaluation, FitJointP is conducted on all solutions, JointP, which is the combination of P and Pc. If condition B is satisfied, then in a similar way, JointP is partitioned into a number of sub-populations, JointPk. and environment-selection is executed on each member of JointPk to select individuals to be new parents, Pk according to their fitness FitJointPk. This process is repeated until the stopping criterion is satisfied. For the implementation of this framework, there are two important issues: (1) when and how often should local operations be executed? (2) how to choose an appropriate partition method so as to prepare a suitable sub-population for the local operation? For issue (1), the conditions for performing local operations can be set by a user-defined rule or, alternatively, the frequency can be subject to self-adaptation. For issue (2), solutions which are similar to one other should be members of the same sub-population since the aim is to enhance the effect of recombination [49]. Similarities can be defined by any distance measure, e.g. Euclidean distance, Manhattan distance, Chebyshev distance and Minkowski distance [41]. The proposed implementation is as follows: (1) As described in [7], the genetic-variation and environment-selection are executed locally when p > rand is satisfied, i.e. conditions A and B. rand is a random number generated in [0, 1]. p is related to the stage reached in the search process, see Eq. (3):
pðgenÞ ¼
gen maxGen
ð3Þ
(2) A hierarchical clustering method [40] is used to partition candidate solutions. Specifically, the Chebyshev function is used to measure the distance between two solutions in the objective space. The Ward-linkage clustering [5] is chosen as the linkage criteria. Before computing the Chebyshev distance, the objective vector of each individual, F(s) is normalised by Eq. (4), in order to avoid the influence of different ranges of objectives [54].
FðsÞi ¼
FðsÞi lFðsÞi dFðsÞi
!2 ;
i ¼ 1; 2; . . . ; M
ð4Þ
where lFðsÞi is the mean value of F(s)i and dFðsÞi represents the standard deviation of F(s)i. A fuller description of the hierarchical clustering method is provided in Appendix A. In order to use the hierarchical clustering method, the user has to specify the number of clusters, k to be created. In this study, k is set equal to M. This may not be the ideal setting as LMOEAs with different choices of k are likely to result in different levels of performance. However, as will be demonstrated, such a setting leads to good outcomes. A discussion on the ideal setting for k is provided in Section 6.3. The proposed local framework shares some commonality with parallel MOEAs, particularly, the island model based MOEAs, in that the population is divided into sub-populations and an optimiser evolves each sub-population. However, the differences between these two schemes are as follows:
R. Wang et al. / Information Sciences 258 (2014) 29–53
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(1) The motivation of parallelizing MOEAs is to take advantages of the distributed computation. The proposed framework aims at improving the convergence and diversity performance of MOEAs. (2) The main issue of island model based MOEAs is how to choose suitable migration/replacement policies. The main issue of the proposed framework is how to group solutions, i.e. combine similar solutions into the same group. (3) In the island model, individuals occasionally migrate between islands. In the proposed framework, the entire population can be reorganised via clustering. (4) In the island model, fitness calculation is based on each sub-population, independently [6, pp. 447–448]. In our framework the fitness calculation is based on the entire population. We are also aware of some studies that considered using clustering techniques to divide the population for island model based MOEAs [14,52]. For example, in [52] the authors also suggested a divide-and-conquer approach beyond an island model MOEA with migration parameters. In their approach, sub-populations are limited in specific regions by zone constraints based upon the dominance principle using k-means cluster centroids. However, the aim of these studies is to demonstrate the superiority of the ‘divide-and-conquer’ based island model over a general island model with migration parameters. The authors have not attempted to utilise this clustering plus divide-and-conquer strategy to encourage local operations. In this work, we for the first time generalise this idea as a framework for localised MOEAs in order to handle the issue of lethals and thereby improve algorithm performance. 4. Comparative studies of the local framework on MOEAs 4.1. Comparative studies: experiment description To demonstrate the effectiveness of this framework, we compared four best-in-class algorithms (NSGA-II, HypE, MSOPS and PICEA-g) with their corresponding modified versions, denoted as LNSGA-II, LHYPE, LMSOPS and LPICEA-g, respectively, on some leading real-parameter benchmark problems. First, the experimental setup is presented and then the selected performance metrics and statistical treatments are introduced. The section concludes by briefly introducing the four baseline MOEAs and describing the settings for the corresponding LMOEAs. 4.1.1. Experiment setup Many test suites have been proposed to test the performance of MOEAs. Widely used ones include the ZDT [66], the DTLZ [13] and the WFG [25] suites. The ZDT test suite contains only 2-objective test problems. The DTLZ and WFG test suites are scalable and can be used to construct many-objective test problems. In this study, problems 2–9 from the WFG test suite are invoked in 2 and 4-objective instances. In each case, the WFG position parameter and distance parameter are set to 18 and 14, providing a constant number of decision variables (18 + 14 = 32) for each problem instance. Problem attributes include separability/non-separability, unimodality/multimodality, unbiased/biased parameters and convex/concave geometries. Note that in this study WFGn-Y refers to problem WFGn with Y objectives. For each test problem, 20,000 function evaluations are undertaken. 50 runs of each algorithm test are performed in order to subject to statistical analysis. Other general parameter settings are listed in Table 1 and are fixed across all algorithm runs. 4.1.2. Performance assessment For qualitative performance assessment, median attainment surfaces [18] are plotted to visualise the performance of algorithms on 2-objective instances. For quantitative performance assessment, a broad range of performance metrics is chosen to measure the obtained Pareto front [67], comprising unary metrics (generational distance (GD [53]), spread metric (D [12]), inverse generational distance (IGD [38]) and hypervolume (HV [66])) and a binary metric (the coverage of two sets (C [66])): (1) the GD metric measures the average distance from objective vectors in the approximation set to the nearest neighbour in the reference set (Pareto optimal front), i.e. convergence, (2) D measures the degree of dispersion on the distribution of the obtained approximation set, i.e. diversity, (3) HV measures the volume of space dominated by the approximation set and one reference point, (4) IGD takes the average distance for all members in the reference set (Pareto optimal front) to their nearest solutions in the obtained approximation (exactly the inverse process of GD), and Table 1 General parameter settings. Objectives, M The number of clusters, k Population size, N Max generations, maxGen Decision variables, n Simulated binary crossover (SBX [12]) Polynomial mutation (PM [12])
2, 4 k=M 200 100 32 (pc = 1, gc = 15 ) (pm = 1/n, gm = 20)
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R. Wang et al. / Information Sciences 258 (2014) 29–53 Table 2 Performance assessment. Test problem
Shape of Pareto front
Performance metrics
WFG2 WFG3 WFG4-WFG9
Disconnected, convex Linear, degenerate Continuous, concave
C, HV GD, D, IGD GD, D, HV
(5) the C metric provides complementary information on convergence. For example, assuming that A and B are the sets of non-dominated solutions found by two algorithms, C(A, B) is the fraction of solutions in B that are dominated at least by one solution in A. A smaller GD value implies good proximity and also a smaller spread metric value implies good diversity. A favourable hypervolume (larger, for a minimisation problem) and IGD (smaller) implies good proximity with diversity. GD, D and IGD metrics are based on the assumption that the true Pareto front of a test problem is known. For the C metric, C(A, B) < C(B, A), for example, indicates a better convergence of the B set. This combination of performance metrics provides a clear indication as to the quality of an approximation set. With regard to the test functions used, according to [25], the Pareto optimal front of WFG2 is not a regular geometry (disconnected and convex), the Pareto optimal front of WFG3 is a linear line, and the Pareto optimal front of WFG4-WFG9 is the surface of an M-dimension hyper-ellipsoid with radius ri = 2i, i = 1. . .M in the first quadrant. Based on the properties of performance metrics and the shape of Pareto fronts of the selected test problems, the specific settings of performance assessment are shown in Table 2. Performance comparisons of the algorithms are made using a rigorous non-parametric statistical framework, drawing on recommendations in [67]. The approximation sets used in the comparisons are the members of the off-line archive of all nondominated points found during the search. For reasons of computational feasibility, prior to analysis, the set is pruned to a maximum size of 100 using the SPEA2 truncation procedure [65]. (The approximation sets used for performance assessment are available for download at [56].) For each problem instance, performance metric values for each algorithm are calculated for each approximation set, the averages and standard deviations of performance metrics are presented. The non-parametric statistical approach introduced in [23] is performed on the performance metric values. Note that prior to calculating the performance metrics for each approximate set, we normalise all objectives to be within the range [0, 1] by the nadir point [10] (which assumes equal relative importance of all the objectives). For the hypervolume metric, the reference point is set as ri = 1.2, i = 1, 2, . . ., M and the software developed by [20] is used to calculate the hyper volume. For unary metrics, the null hypotheses used are those of equality of median values, against the two-sided alternative hypotheses. For the binary C metric, the null hypotheses are defined as the difference between the median values of the mutual coverages of pairs of algorithms. The non-parametric Wilcoxon-ranksum two-sided comparison [23] procedure at the 95% confidence level is employed to test the working hypotheses. 4.1.3. Overview of selected MOEAs In this subsection, the four selected MOEAs, i.e. NSGA-II, HypE, MSOPS and PICEA-g are briefly introduced. and related parameter settings adopted for this study are explained. (1) NSGA-II [12] is a well-known Pareto dominance based MOEA in which parents and offspring are combined and evaluated using a fast non-dominated sorting approach (based on Pareto dominance). An efficient diversity promotion mechanism, the crowding distance, is used as a secondary criterion to discriminate among equally ranked solutions, such that solutions in less crowded regions are preferred. (2) HypE [1] is an efficient indicator based MOEA [64], where a hypervolume metric is used as the indicator. Often the high computational effort required for hypervolume calculation [3] inhibits the full exploitation of hypervolume based MOEAs [16,4]. HypE uses a Monte Carlo simulation to approximate the hypervolume values, thus the accuracy of these estimates can be traded off against the available computing resources. It has been shown that HypE is effective for both multi- and many-objective problems [1]. In common with standard MOEAs, it is based on fitness assignment schemes and consists of successive application of mating selection, variation and environmental selection. The hypervolume indicator is applied in environmental selection. In this study, we strictly follow the hypervolume calculation method described in [1]. For 2-objective problems, the exact hypervolume is used, whereas, for 4-objective problems, a Monte Carlo simulation method with 2000 sampling points is used to calculate the estimated hypervolume contribution. (3) MSOPS [26] can be classified as a scalarising function based MOEA. In MSOPS, all objectives are handled in parallel. A set of target vectors is used to guide the search process in multiple directions simultaneously. Two scalarising functions are used, weighted min–max (weighted Chebyshev) and Vector-Angle-Distance-Scaling (VADS) and T uniformed distributed target vectors are generated priori to the search process. For each solution, 2T scores are computed, based
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WFG2−2
5 4.5 4
WFG2−2
5
True PF NSGAII LNSGAII
4.5 4
3.5
WFG2−2
5
True PF HypE LHypE
4.5 4
3.5
4 3.5
3.5
3
3
3
3
f 2 2.5
f 2 2.5
f 2 2.5
2
2
2
2
1.5
1.5
1.5
1.5
1
1
1
1
0.5
0.5
0.5
0.5
0
1
2
3
0
0
1
WFG3−2
4
0
0
1
4
0
4
4
3
3
3
f 2 2.5
f 2 2.5
2
2
2
2
1.5
1.5
1.5
1.5
1
1
1
1
0.5
0.5
0.5
0.5
2
3
0
0
1
WFG4−2
4
0
0
1
4
0
4
4
3.5
3
3
3
f 2 2.5
f 2 2.5
2
2
2
2
1.5
1.5
1.5
1.5
1
1
1
1
0.5
0.5
0.5
0.5
2
3
0
0
1
f1
4
0
0
1
4
0
4
4
3.5
3
3
3
f 2 2.5
f 2 2.5
2
2
2
2
1.5
1.5
1.5
1.5
1
1
1
1
0.5
0.5
0.5
0.5
2
f1
3
0
0
1
2
f1
3
0
3
3.5
f 2 2.5
1
2
True PF PICEA−g LPICEA−g
4.5
3
0
1
WFG5−2
5
f 2 2.5
0
0
f1
True PF MSOPS LMSOPS
4.5
3.5
3.5
3
WFG5−2
5
True PF HypE LHypE
4.5
2
f1
WFG5−2
5
True PF NSGAII LNSGAII
4.5
3
f1
WFG5−2
5
2
3
3.5
f 2 2.5
1
2
True PF PICEA−g LPICEA−g
4.5
3
0
1
WFG4−2
5
f 2 2.5
0
0
f1
True PF MSOPS LMSOPS
4.5
3.5
3.5
3
WFG4−2
5
True PF HypE LHypE
4.5
2
f1
WFG4−2
5
True PF NSGAII LNSGAII
4.5
3
f1
f1 5
2
3
3.5
3.5
f 2 2.5
1
2
True PF PICEA−g LPICEA−g
4.5
3
0
1
WFG3−2
5
f 2 2.5
0
0
f1
True PF MSOPS LMSOPS
4.5
3.5
3.5
3
WFG3−2
5
True PF HypE LHypE
4.5
2
f1
WFG3−2
5
True PF NSGAII LNSGAII
4.5
3
f1
f1 5
2
True PF PICEA−g LPICEA−g
4.5
f 2 2.5
0
WFG2−2
5
True PF MSOPS LMSOPS
0
1
2
f1
Fig. 3. Attainment surfaces for WFG2-2 to WFG5-2.
3
0
0
1
2
f1
3
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R. Wang et al. / Information Sciences 258 (2014) 29–53
WFG6−2
5 4.5 4
WFG6−2
5
True PF NSGAII LNSGAII
4.5 4
3.5
WFG6−2
5
True PF HypE LHypE
4.5 4
3.5
4
3.5
3.5
3
3
3
3
f 2 2.5
f 2 2.5
f 2 2.5
2
2
2
2
1.5
1.5
1.5
1.5
1
1
1
1
0.5
0.5
0.5
0.5
0
1
2
3
0
0
1
f1
4
0
0
1
4
3.5
3
0
4.5 4
3.5
4
3.5
3
3
3
f 2 2.5
f 2 2.5
2
2
2
2
1.5
1.5
1.5
1.5
1
1
1
1
0.5
0.5
0.5
0.5
2
3
0
0
1
f1
4
0
0
1
4
3.5
3
0
4.5 4
3.5
4
3.5
3
3
3
f 2 2.5
f 2 2.5
2
2
2
2
1.5
1.5
1.5
1.5
1
1
1
1
0.5
0.5
0.5
0.5
2
3
0
0
1
f1
4
0
0
1
4
3.5
3
0
4.5 4
3.5
4
3.5
3
3
3
f 2 2.5
f 2 2.5
2
2
2
2
1.5
1.5
1.5
1.5
1
1
1
1
0.5
0.5
0.5
0.5
2
f1
3
0
0
1
2
f1
3
0
3
3.5
f 2 2.5
1
2
True PF PICEA−g LPICEA−g
4.5
3
0
1
WFG9−2
5
True PF MSOPS LMSOPS
f 2 2.5
0
0
f1
WFG9−2
5
True PF HypE LHypE
4.5
2
f1
WFG9−2
5
True PF NSGAII LNSGAII
4.5
3
f1
WFG9−2
5
2
3
3.5
f 2 2.5
1
2
True PF PICEA−g LPICEA−g
4.5
3
0
1
WFG8−2
5
True PF MSOPS LMSOPS
f 2 2.5
0
0
f1
WFG8−2
5
True PF HypE LHypE
4.5
2
f1
WFG8−2
5
True PF NSGAII LNSGAII
4.5
3
f1
WFG8−2
5
2
3
3.5
f 2 2.5
1
2
True PF PICEA−g LPICEA−g
4.5
3
0
1
WFG7−2
5
True PF MSOPS LMSOPS
f 2 2.5
0
0
f1
WFG7−2
5
True PF HypE LHypE
4.5
2
f1
WFG7−2
5
True PF NSGAII LNSGAII
4.5
3
f1
WFG7−2
5
2
True PF PICEA−g LPICEA−g
4.5
f 2 2.5
0
WFG6−2
5
True PF MSOPS LMSOPS
0
1
2
f1
Fig. 4. Attainment surfaces for WFG6-2 to WFG9-2.
3
0
0
1
2
f1
3
39
R. Wang et al. / Information Sciences 258 (2014) 29–53 Table 3 The mean/deviation values of HV metric for 2-objective problems. Problem
NSGA-II
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.3996/0.0076 0.3721/0.0062 0.3706/0.0098 0.3433/0.0066 0.2607/0.0094 0.3708/0.0187
Problem
HypE
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.4029/0.0062 0.3785/0.0052 0.3712/0.0080 0.3450/0.0079 0.2600/0.0063 0.3700/0.0166
LNSGA-II
MSOPS
< < = < < <
0.4072/0.0061 0.3815/0.0046 0.3735/0.0080 0.3557/0.0080 0.2908/0.0207 0.3866/0.0212
0.4026/0.0047 0.3778/0.0051 0.3716/0.0071 0.3511/0.0066 0.2611/0.0071 0.3662/0.0148
LHypE
PICEA-g
< < < < < <
0.4149/0.0051 0.3883/0.0035 0.3802/0.0069 0.3605/0.0095 0.2979/0.0174 0.3903/0.0231
0.4167/0.0033 0.3931/0.0028 0.3803/0.0059 0.3655/0.0075 0.2554/0.0082 0.3751/0.0190
LMSOPS < < < < < <
0.4206/0.0053 0.3895/0.0033 0.3786/0.0078 0.3663/0.0064 0.2991/0.0167 0.3867/0.0231
< < < < < <
0.4286/0.0032 0.3992/0.0024 0.3856/0.0062 0.3752/0.0070 0.2693/0.0172 0.3959/0.0219
< < < < < <
0.0002/0.0001 0.0018/0.0001 0.0022/0.0002 0.0025/0.0002 0.0092/0.0023 0.0026/0.0016
< < = < < <
0.0009/0.0001 0.0025/0.0001 0.0035/0.0004 0.0036/0.0003 0.0148/0.0043 0.0030/0.0016
< < < < < <
0.2080/0.0379 0.2856/0.0247 0.3349/0.0454 0.5085/0.0357 0.6738/0.0477 0.2434/0.0251
< < < < < <
0.1194/0.0196 0.2062/0.0183 0.2590/0.0293 0.4266/0.0452 0.7510/0.0956 0.1981/0.0208
LPICEA-g
Table 4 The mean/deviation values of the GD metric for 2-objective problems. Problem
NSGA-II
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.0003/0.0001 0.0019/0.0001 0.0021/0.0004 0.0028/0.0001 0.0093/0.0021 0.0036/0.0014
Problem
HypE
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.0003/0.0001 0.0019/0.0001 0.0022/0.0003 0.0027/0.0001 0.0090/0.0013 0.0038/0.0013
= < = < < <
= < = < < <
LNSGA-II
MSOPS
0.0002/0.0002 0.0016/0.0001 0.0020/0.0003 0.0022/0.0003 0.0081/0.0024 0.0024/0.0017
0.0004/0.0001 0.0019/0.0001 0.0024/0.0004 0.0030/0.0001 0.0120/0.0016 0.0040/0.0009
LHypE
PICEA-g
0.0003/0.0001 0.0018/0.0001 0.0021/0.0003 0.0023/0.0002 0.0075/0.0019 0.0023/0.0018
0.0012/0.0001 0.0026/0.0001 0.0036/0.0005 0.0040/0.0005 0.0168/0.0030 0.0044/0.0014
LMSOPS
LPICEA-g
Table 5 The mean/deviation values of the D metric for 2-objective problems. Problem
NSGA-II
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.3479/0.0511 0.4108/0.0429 0.3969/0.0505 0.6317/0.0363 0.8394/0.0420 0.2705/0.0222
Problem
HypE
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.3274/0.0430 0.3673/0.0380 0.3889/0.0423 0.6284/0.0434 0.8448/0.0395 0.2620/0.0201
LNSGA-II
MSOPS
< < < < < <
0.2959/0.0412 0.3448/0.0337 0.3721/0.0421 0.5729/0.0457 0.7168/0.0641 0.2564/0.0181
0.3240/0.0317 0.3710/0.0368 0.3835/0.0381 0.5854/0.0360 0.8357/0.0386 0.2727/0.0202
LHypE
PICEA-g
< < < < < <
0.2479/0.0361 0.2957/0.0257 0.3290/0.0388 0.5511/0.0535 0.6876/0.0484 0.2380/0.0148
0.1922/0.0206 0.2499/0.0206 0.2932/0.0255 0.4803/0.0463 0.8253/0.0738 0.2333/0.0133
LMSOPS
LPICEA-g
on different scalarising functions with different weight vectors. The scores are recorded in a [N 2T] matrix, S_M. Each row of S_M belongs to a candidate solution and each column represents a score for the candidate solutions as measured by the specific scalarising function. Each column of the matrix S_M is ranked, giving the best performing individual rank 1 and the rank values are stored in a matrix R. Each row of R is sorted in ascending order, providing a lexicographical order of the individuals. (4) PICEA-g [48,58] refers to a preference-inspired co-evolutionary algorithm using goal vectors. PICEA is a novel class of MOEA, where a family of preferences are co-evolved with the usual population of candidate solutions. Different forms of preferences used in multi-criteria decision-making (e.g. goals and weights) can be realised in the PICEA approach. PICEA-g is a realisation that takes goals as preferences [58]. Candidate solutions gain fitness by dominating a particular
40
R. Wang et al. / Information Sciences 258 (2014) 29–53
Table 6 The mean/deviation values of the HV metric for 4-objective problems. Problem
NSGA-II
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.5817/0.0145 0.5295/0.0177 0.5398/0.0183 0.5415/0.0170 0.4392/0.0166 0.5648/0.0107
Problem
HypE
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.7026/0.0056 0.6754/0.0055 0.6725/0.0133 0.7200/0.0097 0.5664/0.0086 0.6081/0.0053
LNSGA-II
MSOPS
< < < < < <
0.6086/0.0110 0.5591/0.0152 0.5555/0.0161 0.5662/0.0157 0.4594/0.0142 0.5728/0.0079
0.6624/0.0075 0.6301/0.0076 0.6223/0.0136 0.6158/0.0161 0.5213/0.0091 0.5888/0.0065
LHypE
PICEA-g
< < = < < <
0.7122/0.0051 0.6839/0.0046 0.6730/0.0138 0.7387/0.0085 0.5890/0.0125 0.6169/0.0128
0.7102/0.0058 0.6830/0.0054 0.6746/0.0154 0.7204/0.0103 0.5744/0.0076 0.6080/0.0050
LMSOPS < < = < < <
0.6789/0.0069 0.6525/0.0061 0.6281/0.0158 0.6442/0.0159 0.5491/0.0075 0.5931/0.0064
< < = < < <
0.7279/0.0061 0.6990/0.0040 0.6739/0.0130 0.7388/0.0121 0.5940/0.0133 0.6291/0.0128
= < = < < =
0.0052/0.0002 0.0054/0.0002 0.0069/0.0006 0.0074/0.0003 0.0175/0.0012 0.0062/0.0005
< = = < > <
0.0041/0.0001 0.0043/0.0001 0.0051/0.0001 0.0037/0.0001 0.0123/0.0029 0.0045/0.0005
= < = = < =
1.4535/0.0670 1.4651/0.0474 1.5668/0.0719 1.4745/0.0643 1.9485/0.0693 1.5215/0.0564
LPICEA-g
Table 7 The mean/deviation values of GD metric for 4-objective problems. Problem
NSGA-II
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.0075/0.0004 0.0091/0.0009 0.0095/0.0008 0.0111/0.0008 0.0250/0.0007 0.0076/0.0008
Problem
HypE
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.0043/0.0001 0.0045/0.0002 0.0051/0.0003 0.0038/0.0002 0.0119/0.0032 0.0055/0.0002
LNSGA-II
MSOPS
< < = < < <
0.0068/0.0003 0.0084/0.0003 0.0094/0.0007 0.0101/0.0004 0.0241/0.0010 0.0071/0.0006
0.0053/0.0003 0.0058/0.0003 0.0068/0.0005 0.0076/0.0007 0.0201/0.0009 0.0063/0.0003
LHypE
PICEA-g
< < = < < <
0.0041/0.0001 0.0042/0.0001 0.0050/0.0003 0.0035/0.0001 0.0112/0.0012 0.0047/0.0006
0.0043/0.0001 0.0043/0.0001 0.0050/0.0002 0.0039/0.0001 0.0095/0.0036 0.0056/0.0003
LMSOPS
LPICEA-g
Table 8 The mean/deviation values of D metric for 4-objective problems. Problem
NSGA-II
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
1.6673/0.0922 1.7658/0.0756 1.8149/0.0985 1.7555/0.1037 2.2171/0.0947 1.5957/0.0652
Problem
HypE
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
1.3947/0.0608 1.3922/0.0497 1.4773/0.0675 1.3328/0.0725 2.2025/0.0959 1.4836/0.0566
< < < < < =
= = = = < <
LNSGA-II
MSOPS
1.5628/0.0729 1.6447/0.0834 1.7229/0.0844 1.6110/0.1046 2.1134/0.0809 1.5822/0.0684
1.4584/0.0636 1.5033/0.0576 1.5574/0.0616 1.5029/0.0667 2.1679/0.0663 1.5285/0.0507
LHypE
PICEA-g
1.3741/0.0713 1.3941/0.0770 1.4679/0.0577 1.3196/0.0672 1.8935/0.0710 1.4575/0.0590
1.3796/0.0499 1.3930/0.0564 1.4546/0.0685 1.3345/0.0635 2.0912/0.1787 1.4767/0.0587
LMSOPS
LPICEA-g = < = = = <
1.3805/0.0684 1.2851/0.0301 1.4691/0.0734 1.3176/0.0687 2.1039/0.0759 1.4477/0.0659
set of goals in objective space, but the fitness contribution is shared between other solutions that also satisfy those goals. Goals only gain fitness by being dominated by a candidate solution, but the fitness is reduced according to how often each goal is dominated by other solutions in the population. In this way, the candidate solution population and the goal population co-evolve toward the Pareto optimal front. In this study, the number of goal vectors, Ngoal, is set to be N M. For each of the four MOEAs, two versions are built: one is within the general (l + k) elitist framework, the other is within the proposed local (l + k) elitist framework. Each pair of MOEA versions are compared.
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R. Wang et al. / Information Sciences 258 (2014) 29–53 Table 9 The mean/deviation values of performance metrics for WFG3 (a: WFG3-2 b: WFG3-4). NSGA-II a GD D IGD
0.0005/0.0002 0.3504/0.0572 0.0025/0.0006
LNSGA-II
MSOPS
< < <
0.0003/0.0001 0.2779/0.0367 0.0017/0.0004
0.0006/0.0002 0.3125/0.0458 0.0020/0.0004
LHypE
PICEA-g
< < <
0.0002/0.0001 0.2215/0.0393 0.0013/0.0004
0.0013/0.0002 0.1712/0.0283 0.0011/0.0003
LNSGA-II
MSOPS
< > <
0.0905/0.0038 2.2403/0.1008 0.0052/0.0007
0.0712/0.0021 2.1974/0.0952 0.0034/0.0004
LHypE
PICEA-g
< = =
0.0730/0.0032 2.1462/0.0947 0.0030/0.0004
0.0932/0.0045 2.2304/0.1077 0.0030/0.0003
HypE GD D IGD
0.0004/0.0001 0.2806/0.0436 0.0018/0.0005 NSGA-II
b GD D IGD
0.0924/0.0039 2.1798/0.1313 0.0058/0.0009 HypE
GD D IGD
0.0885/0.0051 2.1550/0.0949 0.0031/0.0004
LMSOPS < < <
0.0004/0.0001 0.2216/0.0317 0.0013/0.0002
< < <
0.0011/0.0003 0.1524/0.0284 0.0009/0.0002
LPICEA-g
LMSOPS < < <
0.0694/0.0032 2.1329/0.1007 0.0032/0.0002
< = =
0.0882/0.0054 2.2005/0.0925 0.0029/0.0003
LPICEA-g
Table 10 The mean/deviation values of performance metrics for WFG2 (a: WFG2-2 b: WFG2-4). NSGA-II a C HV
0.4624/0.3664 0.5609/0.0053
LNSGA-II
MSOPS
0.2257/0.2007 0.5751/0.0033
0.5357/0.3735 0.5630/0.0183
LHypE
PICEA-g
0.1748/0.2389 0.5774/0.0021
0.5724/0.2427 0.5653/0.0110
LNSGA-II
MSOPS
< <
0.0729/0.0610 0.7963/0.0061
0.2358/0.1380 0.7988/0.0378
LHypE
PICEA-g
< >
0.0694/0.0713 0.8065/0.0067
0.1652/0.0520 0.8071/0.0042
< <
HypE C HV
0.6470/0.2846 0.5727/0.0030
< <
NSGA-II b C HV
0.2996/0.1362 0.7802/0.0079 HypE
C HV
0.1478/0.0877 0.8137/0.0296
LMSOPS < <
0.2288/0.2420 0.5698/0.0039 LPICEA-g
< =
0.2721/0.2543 0.5639/0.0059 LMSOPS
< <
0.1283/0.2358 0.8264/0.0371 LPICEA-g
= =
0.1480/0.0788 0.8032/0.0063
4.2. Comparative studies: experiment results Median attainment surfaces (the 25th) across the 50 runs of each algorithm are shown in Figs. 3 and 4. Tables 3–8 provide comparison results of corresponding MOEAs and LMOEAs in terms of GD, D and HV metrics for problems, WFG4-WFG9. Table 9 shows the comparison results of MOEAs and LMOEAs on WFG3 in terms of GD, D and IGD metrics; Table 10 presents the results for WFG2. In each table, the mean/deviation values of each performance metric across the 50 independent runs are shown. The symbol ‘<’, ‘=’ or ‘>’ denotes whether a MOEA is statistically worse, equal or better than LMOEA at the 95% confidence level. Superior results in each comparison are in boldface.
4.2.1. Median attainment surfaces for 2-objective problems In Figs. 3 and 4, the plots of median attainment surfaces of NSGA-II and LNSGA-II, HypE and LHypE, MSOPS and LMSOPS, PICEA-g and LPICEA-g for each problem are shown in the rectangles from left to right.3 Visually, MOEAs and LMOEAs have comparable performance on convergence for WFG2 to WFG7. On WFG8 and WFG9, the Pareto approximation sets obtained by LMOEAs are closer to the true Pareto front than those obtained by MOEAs. Regarding diversity, LMOEAs perform better than MOEAs on all the problems except for WFG2 where they perform comparably. 3
Colour reproduction available at [56].
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R. Wang et al. / Information Sciences 258 (2014) 29–53
4.2.2. HV comparison results for 2-objective problems From Table 3, it is clear that the mean HV values of MOEAs are smaller than their variants (LMOEAs) for all the test problems. This superiority of LMOEA is also confirmed by the non-parametric statistical test. Each LMOEA performs significantly better than its corresponding MOEA on all the test benchmarks with only one exception, WFG6, for which NSGA-II and LNSGA-II perform comparably. Such results indicate that this new framework works effectively, that is, for 2-objective problems, the performance of MOEAs can be improved by implementing MOEAs within the local framework. 4.2.3. GD and D comparison results for 2-objective problems To further study the effect of the new framework, we have also separately calculated proximity (as measured by GD, shown in Table 4) and diversity (as measured by the spread metric, D shown in Table 5) measures for problems WFG4– WFG9. (1) Table 4 shows that the mean GD values of LMOEAs are lower than the associated MOEAs on all the 2-objective test problems. The non-parametric statistical results prove that the difference is significant on every test problem for MSOPS and LMSOPS; on 5 out of 6 test problems for PICEA-g and LPICEA-g; on 4 out of 6 test problems for NSGA-II and LNSGA-II, and HypE and LHypE. PICEA-g and LPICEA-g perform comparably on WFG6; NSGA-II and LNSGA-II show equivalent performance on WFG4 and WFG6, as do HypE and LHypE. (2) Table 5 demonstrates all LMOEAs have better diversity on all the test problems in terms of average D value. Moreover, the difference is significant for all the problems according to the non-parametric statistical test results. 4.2.4. HV comparison results for 4-objective problems The HV comparison results for 4-objective problems WFG4–WFG9 (see Table 6) demonstrate that the LMOEAs achieve larger mean HV values for all problems. According to the non-parametric statistical tests, the difference is significant for every pair of algorithms on all problems, except for WFG6 for which the difference is significant only for NSGA-II and LNSGA-II. 4.2.5. GD and D comparison results for 4-objective problems From the GD metric results (see Table 7), the following results are observed: (1) LNSGA-II has smaller mean values on all six 4-objective problems. LNSGA-II is statistically better than NSGA-II at the 95% confidence level for all problems, except for WFG6. (2) HypE and LHypE behave similarly to NSGA-II and LNSGA-II. (3) Compared with MSOPS, LMSOPS achieves smaller average GD values on five out of the six problems; for only WFG6 is the average value of MSOPS lower. Differences are significant only for WFG5, 7 and 8. (4) Compared with PICEA-g, LPICEA-g has a smaller average value on WFG4, 5, 7 and 9. The non-parametric statistical test results show that LPICEA-g gives better convergence on WFG4, 7 and 9, equivalent performance on WFG5, 6 and inferior performance on WFG8. With respect to the spread metric (see Table 8), the following results are observed: (1) The average values of LNSGA-II are smaller than NSGA-II for all six problems. Statistically, LNSGA-II performs better than NSGA-II on all problems, except for WFG9, where the difference is not significant. (2) Compared with MSOPS, LMSOPS achieves smaller mean D values on all problems, except for WFG6, where MSOPS has a smaller mean value. LMSOPS performs statistically better than MSOPS on WFG5 and WFG8, and performs comparably to MSOPS on the remaining four problems. (3) Compared with HypE, LHypE has smaller mean D values on five out of six problems (the exception being WFG5). The statistical test results show LHypE has significantly better performance on WFG8 and WFG9, and comparable performance on the other four problems. (4) Compared with LPICEA-g, PICEA-g has smaller mean values on WFG4, 6 and 8; while LPICEA-g has smaller values on WFG5, 7 and 9. PICEA-g is statistically inferior to LPICEA-g on WFG5 and WFG8 and equivalent to LPICEA-g on the other four problems. 4.2.6. Comparison results for 2- and 4-objective WFG3 problems Since the true Pareto front of the WFG3 problem is a degenerated straight line it may not be appropriate to use hypervolume as a performance metric on WFG3-4 because an algorithm that converges close to the line may achieve a smaller hypervolume value compared with an algorithm that does not converge closely to the line but is spread over the objective space. To avoid this problem, we choose the IGD metric instead of then HV metric to evaluate the overall performance of MOEAs. Results are shown in Table 9. Compared their global counterparts, the LMOEAs have smaller average values and perform statistically better in terms of GD, D and IGD on WFG3-2. For WFG3-4, the LMOEAs have better convergence in terms of the mean GD value. The difference is significant according to the statistical tests. The same is true for the D metric, except that
R. Wang et al. / Information Sciences 258 (2014) 29–53
43
LNSGA-II performs statistically worse than NSGA-II, otherwise every LMOEA achieves statistically better (LMSOPS) or comparable (LHypE and LPICEA-g) diversity than its corresponding MOEA. Regarding IGD, LMOEAs perform statistically better (LNSGA-II and LMSOPS) than or at least equivalently to (LHypE and LPICEA-g) to their corresponding MOEAs. 4.2.7. Comparison results for WFG2-2 and WFG2-4 Since the true Pareto front of WFG2 is disconnected and not a regular geometry, the C metric is chosen, instead of GD, to compare the convergence of MOEAs and LMOEAs; HV is chosen to compare the overall performance of the algorithms. In Table 10, the C metric is calculated by C(MOEA,LMOEA) for MOEAs and by C(LMOEA, MOEA) for LMOEAs. For WFG2-2, the mean value of C(LMOEA,MOEA) is smaller than that of C(MOEA, LMOEA) and the difference is significant at the 95% level. This implies that LMOEAs achieve statistically better convergence than MOEAs. Regarding HV, LNSGA-II, LHypE and LMSOPS have larger mean values and perform statistically better than NSGA-II, HypE and MSOPS, respectively. However, LPICEA-g has a slightly smaller mean HV value than PICEA-g but the difference is not significant. For WFG2-4, LNSGA-II and LMSOPS perform statistically better than NSGA-II and MSOPS in terms of C and HV metrics. LHypE has significantly better convergence than HypE, however, the overall performance (as measured by the HV) of LHypE is worse than HypE. The average value of C(LPICEA-g,PICEA-g) is slightly smaller than C(PICEA-g,LPICEA-g) but the average HV value of PICEA-g is larger than LPICEA-g. According to the statistical results, LPICEA-g performs comparably with PICEA-g in terms of both performance metrics. 4.3. Summary From these results, we claim that this local operation based (l + k) elitist framework works more effectively than the general (l + k) elitist framework. In most cases, the performance (convergence, diversity or both) of MOEAs are improved by implementing them within this framework. Key observations are: (1) For most problems convergence performance is improved for almost all of the MOEAs. (2) For all of the 2-objective problems, diversity is improved for all of the MOEAs modified by the local framework. However, for most of the 4-objective problems, the improvement of diversity performance for HypE, MSOPS and PICEA-g is not significant. (3) Compared to MSOPS, HypE and PICEA-g, the effect of the framework on NSGA-II is much more significant. The performance of NSGA-II, modified by the local framework, is improved for almost all of the test problems. 5. Comparison between LMOEAs and other local operations based algorithms As mentioned earlier, local operation has been investigated in some earlier studies [62,39,49,50,31,57]. Zhang and Li’s MOEA/D [62] is inherently a local operation based algorithm, that is, environment-selection and genetic-variations are executed on neighbouring individuals [28]. In [39], Jaszkiewicz improves the IMMOGLS algorithm [30] by applying local recombination. In [49] Sato et al. applied local dominance and local recombination to NSGA-II in an implementation (which we call SNSGA-II) in which local dominance implicitly realises a local selection. There have also been general studies of local dominance and local recombination by, respectively, Sato et al. [50] and Ishibuchi et al. [31]. We have also, in earlier work [57], generated proof-of-principle results (using the PICEA-g optimiser) for the local framework proposed in Section 3. In order to further validate the effectiveness of our local framework, we further compare some of our local instantiations with two representative local operations based algorithms mentioned above – specifically, MOEA/D [62] and SNSGA-II [49]. We omit comparison with MOGLS [39] since Zhang and Li have already demonstrated that this algorithm is outperformed by MOEA/D [62]. 5.1. Comparison between LMOEAs and MOEA/D In this section we compare MOEA/D with the four LMOEAs from Section 4 using the same test problems. For MOEA/D, 100 uniform weights are used for 2-objective problems and 455 uniform weights are used for 4-objective problems. The comparison results, in terms of HV (or IGD), are shown in Table 11 and we observe that: Compared to LNSGA-II, MOEA/D performs better for WFG2-2 and WFG6-2, performs comparably for WFG4-2, WFG7-2 and WFG9-2, and performs worse for the remaining three 2-objective problems. Regarding 4-objective problems, MOEA/D performs better only for WFG2-4, performs comparably only for WFG5-4, and performs worse for the remaining six problems. Compared to LMSOPS, MOEA/D performs better only for WFG2-2, performs comparably for WFG4-2, WFG7-2 and WFG92, and performs worse for the remaining four 2-objective problems. Its performance is worse than LMSOPS on all the 4objective problems. Compared to LHypE, MOEA/D performs better only for WFG6-2, and performs worse for the remaining seven problems. Its performance is worse than LHypE on all the 4-objective problems.
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Table 11 The HV (or IGD) comparison results between MOEA/D and other MOEAs for 2-objective problems.
WFG2-2 WFG3-2 WFG4-2 WFG5-2 WFG6-2 WFG7-2 WFG8-2 WFG9-2
WFG2-4 WFG3-4 WFG4-4 WFG5-4 WFG6-4 WFG7-4 WFG8-4 WFG9-4
LNSGA-II
LMSOPS
LHypE
LPICEA-g
> < = < > = < =
> < = < < = < =
< < < < > < < <
= < < < < < = <
LNSGA-II
LMSOPS
LHypE
LPICEA-g
> < < = < < < <
< < < < < < < <
< < < < < < < <
< < < < < < < <
Compared to LPICEA-g, MOEA/D performs better only for WFG6-2, and performs worse for the remaining seven problems. Its performance is worse than LPICEA-g on all the 4-objective problems. The reason for the poor performance of MOEA/D for 4-objective problems might be that the number of weights used in the search is insufficient and therefore performance of MOEA/D might be improved by using more weights [58]. However, since MOEA/D requires the population size to be equal to the number of weights, it is not easy to strike an effective balance between population size (i.e. diversity) and generations (i.e. convergence) under a fixed computational budget – with a larger the population size, the beneficial dynamics of evolution are curtailed. 5.2. Comparison between LNSGA-II and SNSGA-II The local strategies proposed in [50] only apply to Pareto-dominance based algorithms. Therefore, by necessity, we restrict our comparison to a local instantiation of NSGA-II. In SNSGA-II local environment-selection is realised by the use of local dominance. Specifically, the dominance level of each individual is only determined by its neighbours. The neighbourhood size (nLD) is set by the user. Note that, before applying the non-dominated sorting, the individual and its neighbours are rotated by p/4 radians in objective-space. The local geneticvariation in SNSGA-II is executed as follows: for each individual, an offspring is generated by recombining this individual with one of its neighbours. Again, the neighbourhood size (nLR) needs to be defined by the user. Based on the study of Sato et al. [49], nLD = nLR = 10 is used here. Other parameter settings remain as before in Table 1. From Table 12 we can observe that, for 2-objective problems, LNSGA-II and SNSGA-II perform comparably on six out of the eight problems. On WFG8-2 LNSGA-II outperforms SNSGA-II while on WFG5-2 the situation is reversed. For the 4-objective problems LNSGA-II performs better than SNSGA-II on five problems (from WFG3 to WFG7). For the other three problems the two algorithms have comparable performance. These results indicate that for bi-objective problems our local framework does not have significant superiority over the local operations proposed in [49]. However, our proposed framework appears to perform better on 4-objective problems.
Table 12 The mean/deviation values of the HV (or IGD) metric for 2- and 4-objective problems. M=2
M=4
SNSGA-II WFG2 WFG3 WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.5734/0.0021 0.0019/0.0003 0.4086/0.0038 0.3881/0.0049 0.3698/0.0076 0.3543/0.0066 0.2713/0.0083 0.3778/0.0187
= = = > = = < =
LNSGA-II
SNSGA-II
0.5751/0.0033 0.0017/0.0004 0.4072/0.0061 0.3815/0.0046 0.3735/0.0080 0.3557/0.0080 0.2908/0.0207 0.3866/0.0212
0.7892/0.0086 0.0067/0.0009 0.5814/0.0129 0.5351/0.0106 0.5279/0.0194 0.5438/0.0119 0.4392/0.0166 0.5742/0.0107
LNSGA-II = < < < < < = =
0.7963/0.0061 0.0052/0.0007 0.6086/0.0110 0.5591/0.0152 0.5555/0.0161 0.5662/0.0157 0.4594/0.0142 0.5728/0.0079
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The reason for the poor performance of SNSGA-II on 4-objective problems is likely to be that the local dominance has deleterious effect on convergence. This operation can potentially assign higher fitness to some dominated solutions. Although this is helpful from a diversity perspective [49], it slows down the convergence speed as some dominated solutions are stored during the search. This side effect becomes more significant on higher-dimension problems since convergence is inherently more difficult to achieve in many-objective fitness landscapes. Another reason might be that the choices of neighbourhood size for local dominance and local recombination are not the optimal settings for problems with more than two objectives [49].
5.3. Summary The results suggest that our proposed local operation based framework is generally more effective than existing similar algorithms. Key observations are: (1) Compared to MOEA/D, the four LMOEAs offer better performance on most of the 2-objective problems and all the 4objective problems. (2) Compared to the approach in [49], LNSGA-II offers better performance on the 4-objective problems.
WFG4−2: =
0.42
WFG4−2: = 0.425
0.415
0.42 0.41 0.415 0.41
HV
HV
0.405 0.4 0.395
0.405 0.4
0.39
0.395
0.385
0.39 0
0.25
0.5
0.75
1
0
Linear
0.25
WFG4−2: <
0.425
0.5
0.75
1
Linear
1
Linear
WFG4−2: < 0.435
0.42 0.43 0.415
HV
HV
0.425 0.41 0.405
0.42
0.415
0.4
0.41
0.395 0
0.25
0.5
0.75
1
Linear
0
0.25
0.5
0.75
Fig. 5. Box plots of HV results for WFG4-2 for algorithms with different p. The symbol ‘<’, ‘=’ or ‘>’ denotes LMOEA with p = 0.75 is statistically worse, equal or better than LMOEA with the relationship described in Eq. (3) at the 95% confidence level.
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WFG4−4: <
WFG4−4: = 0.63
0.72
0.62
0.71
0.61
0.7
HV
HV
0.6 0.59
0.69 0.68
0.58
0.67
0.57
0.66
0.56
0.65
0.55 0
0.25
0.5
0.75
1
Linear
0
0.25
WFG4−4: =
0.75
1
Linear
1
Linear
WFG4−4: < 0.75
0.725
0.74
0.72 0.715
0.73
0.71
HV
HV
0.5
0.72
0.705 0.71
0.7
0.7
0.695 0.69
0
0.25
0.5
0.75
1
Linear
0
0.25
0.5
0.75
Fig. 6. Box plots of HV results for WFG4-4 for algorithms with different p. The symbol ‘<’, ‘=’ or ‘>’ denotes LMOEA with p = 0.75 is statistically worse, equal or better than LMOEA with the relationship described in Eq. (3) at the 95% confidence level.
6. Discussion The empirical comparison results have identified the effectiveness of the local operations based framework. In this section, three further issues are studied. Section 6.1 studies the influence of the probability p to the performance of algorithms. Section 6.2 concerns the independent effect of local genetic-variation and local environment-selection. Section 6.3 studies the number of clusters into which a population is partitioned. LMOEAs with different values of k may perform differently. We study this configuration issue, as part of a wider discussion for the effect of this local framework. 6.1. The effect of the probability p In the above comparative study, the probability p for doing local operations is linearly increased from 0 to 1 (see Eq. (3)). Here, we provided a further discussion on the influence of different p values to the performance of the algorithms. p is set to 0, 0.25, 0.5, 0.75, 1. The 50 HV results of the four algorithms are box-plotted in Figs. 5 and 6 for WFG4-2 and WFG4-4 respectively. The upper and lower ends of the box are the upper and lower quartiles, while a thick line within the box encodes the median. Dashed
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x10-3 WFG3− 2
WFG2− 2
0.64
WFG4− 2
2
0.435
1.5
0.425
0.62
0.43
0.6 0.58
0.4 0.395
0.42 1
0.56
0.39
0.415 0.41
0.54 1
2
3
1
4
WFG6− 2
2
3
4
0.39
1
0.39
0.29
0.38
0.28
0.36
0.25
0.35
0.24
4
4
2
3
4
2
3
4
WFG9− 2 0.41 0.4 0.39 0.38
0.26
1
1
0.42
0.27
0.37 3
3
WFG8− 2
0.37
2
2
0.3
0.38
1
0.385
WFG7− 2 0.4
0.4
WFG5− 2
0.405
0.37 0.36 1
2
3
4
1
2
3
4
Fig. 7. Box plots of HV (or IGD) comparison results for 2-objective WFG problems. (1: PICEA-g 2: LPICEAsel-g 3: LPICEAvar-g 4: LPICEA-g).
WFG2− 4
4
x10−3 WFG3− 4
0.815 0.81
3.5
0.805 0.8
2
3
1
4
WFG6− 4
2
3
4
WFG7− 4
1
2
3
4
0.68
0.69
1
2
3
4
4
3
4
0.64 0.62 0.6
0.54 3
2
0.66
0.56 2
1
WFG9− 4
0.58
1
0.67
WFG8− 4
0.6
0.7
0.64
0.69
0.62
0.72
0.66
0.7
0.64
0.74
0.68
0.73
0.66
0.76
0.7
0.71
0.7
2.5 1
0.74
0.71
0.79 0.785
WFG5− 4
0.72
3
0.795
WFG4− 4
0.75
1
2
3
4
1
2
3
4
Fig. 8. Box plots of HV (or IGD) comparison results for 4-objective WFG problems. (1: PICEA-g 2: LPICEAsel-g 3: LPICEAvar-g 4: LPICEA-g).
appendages indicate the spread and shape of the distribution. Outlying values are marked as +. The average value of each algorithm is shown as } (and is connected by a line through algorithms with different p). Each graph in the Figure contains six box plots representing the HV (or IGD) results (Y-label) of algorithms with five different p values, and the relationship described in Eq. (3). From Figs. 5 and 6 we can clearly observe that among the five different p values, p = 0.75 is the best setting, that is, LMOEAs with p = 0.75 perform better than LMOEAs with other p values in terms of the HV metric on both WFG2-4 and WFG4-4. However, comparing the results between p = 0.75 and the relationship for p described in Eq. (3), it is found that LMOEAs with the relationship (Eq. (3)) perform better than or at least comparably to p = 0.75 at the 95% confidence level. Similar results are obtained for other test problems, and can be downloaded at [56]. These results indicate that neither p = 1 (i.e. apply local operations throughout the search process) nor p = 0 (i.e. never apply local operations) is a good choice for the test problems. The global operation and the local operation should be
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R. Wang et al. / Information Sciences 258 (2014) 29–53 Table 13 Parameter settings. Parameter
Value
N k
200, 400 1 2 3 4 5 6 7 8 9 10 11 12 13 14
2 − objective WFG tests: N=200 WFG2− 2
−3
x 10
WFG3− 2
4 3.5
0.62
2
0.41
1.5
0.4
0.54
1
0.39
0.52
0.5
0.56
0.39 0.385 0.38 0.375 0.37 0.365
1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
WFG6− 2
WFG7− 2
WFG8− 2
WFG9− 2
0.4 0.38
0.38
0.37
0.37
0.36 0.34
0.35
0.33
0.34 1 2 3 4 5 6 7 8 9 1011121314
0.41 0.4
0.3
0.39 0.28
0.36
0.35
0.42
0.32
0.39
0.39
0.32
0.4 0.395
0.42
2.5
0.58
WFG5− 2 0.405
0.43
3
0.6
WFG4− 2 0.44
0.38 0.37
0.26
1 2 3 4 5 6 7 8 9 1011121314
0.24
0.36 0.35 1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
Fig. 9. Boxplots of HV (or IGD) distribution of LPICEA-g with N = 200 and different k for 2-objective WFG problems.
appropriately balanced [37,35] in order to obtain a good approximation of the Pareto front. The relationship used in this study offers better results than all the considered constant settings and appears to be a good choice for this framework. 6.2. The effect of individual local operations PICEA-g is selected as the algorithm to be studied in this context. The three compared algorithms are LPICEA-g, LPICEAselg and LPICEAvar-g. LPICEAsel-g and LPICEAvar-g apply local environment-selection and local genetic-variation on PICEA-g, respectively. Parameter settings are the same as those described in Section 4.1. All algorithms are submitted to 50 runs for each experiment. WFG problems (WFG2–WFG9) with 2 and 4 objectives are chosen as test problems. HV is used to evaluate the performance. Note that the IGD metric is used instead of HV on WFG3 problems, for reasons already given in the previous section. Box plots are used in Figs. 7 and 8 in order to visualise the distribution of the 50 HV or IGD values for the associated problems. The average value of each algorithm is shown as } (and is connected by a line through all four algorithm versions). Each graph in the Figure contains four box plots representing the HV (or IGD) results (Y-label) of the four compared versions of PICEA-g. (1) Fig. 7 clearly shows that for most of the 2-objective problems, the performance of LPICEAvar-g is better than PICEA-g; LPICEAsel-g also performs comparably with PICEA-g. However, neither LPICEAsel-g nor LPICEAvar-g outperforms LPICEA-g. Comparing LPICEAvar-g and LPICEAsel-g, LPICEAvar-g, by and large, exhibits better performance. (2) For 4-objective test problems (see Fig. 8), LPICEAvar-g performs much better than LPICEAsel-g and PICEA-g on the majority of problems. Compared with LPICEA-g, its performance is worse for all the problems except for WFG3-4. As for LPICEAsel-g, it performs comparably to PICEA-g on most of problems.
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2− objective WFG tests: N=400 0.65
−4
x 10
WFG2− 2
WFG4− 2
WFG5− 2
14
0.445
12
0.44
10
0.435
8
0.43
6
0.425
0.398
4
0.42
0.396
0.6
0.55 1 2 3 4 5 6 7 8 9 1011121314
WFG6− 2
WFG3− 2
0.408 0.406 0.404 0.402 0.4
1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
WFG7− 2
0.43
0.41
0.394
WFG8− 2
WFG9− 2
0.41
0.42
0.36
0.42
0.4
0.41
0.34
0.41
0.4
0.32
0.39
0.39
0.38 0.37 0.36
0.4 0.39
0.3
0.38
0.38
0.28
0.37
0.37
0.26
0.36 1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
0.36 1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
Fig. 10. Boxplots of HV (or IGD) distribution of LPICEA-g with N = 400 and different k for 2-objective WFG problems.
4− objective WFG tests: N=200 WFG2− 4 0.84
−3
x 10
WFG3− 4
6
0.75
5
0.68
0.7
4.5
0.8
0.66
4
0.78
3.5
0.76
3
0.64
0.65
0.62
2.5
0.74
0.6 1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
WFG6− 4
0.68
0.6
1 2 3 4 5 6 7 8 9 1011121314
WFG8− 4
WFG9− 4
0.64
0.66
0.62
0.65 0.64 0.63
0.56
0.65
0.6
1 2 3 4 5 6 7 8 9 1011121314
0.58
0.66
0.62
1 2 3 4 5 6 7 8 9 1011121314
0.6
0.7
0.64
0.6
WFG7− 4 0.75
0.7
WFG5− 4 0.7
5.5
0.82
0.58
WFG4− 4
1 2 3 4 5 6 7 8 9 1011121314
0.54
0.62
0.52
0.61
0.5
0.6
0.48
0.59 1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
Fig. 11. Boxplots of HV (or IGD) distribution of LPICEA-g with N = 200 and different k for 4-objective WFG problems.
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4− objective WFG tests: N=400 WFG2− 4 0.95
−3
x 10
WFG3− 4
3.2 3
0.76
0.725
0.85
0.71 0.73
2.2 1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
WFG6− 4
0.72 0.71 0.7
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1 2 3 4 5 6 7 8 9 1011121314
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0.72
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1 2 3 4 5 6 7 8 9 1011121314
0.7
0.7
0.77
0.68
0.76
0.66
0.66
0.64
0.65
0.68 0.67
0.64 0.63
0.6 1 2 3 4 5 6 7 8 9 1011121314
WFG9− 4
0.69
0.62
0.74
0.68
0.705
0.78
0.75
0.69
0.715
0.74
2.4
0.8
0.72
0.75
2.6
WFG5− 4 0.73
2.8
0.9
0.73
WFG4− 4 0.77
0.62 1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
Fig. 12. Boxplots of HV (or IGD) distribution of LPICEA-g with N = 400 and different k for 4-objective WFG problems.
Overall, algorithms that apply both local genetic-variation and local environment-selection perform better than algorithms apply local genetic-variation or local environment-selection alone. Although only comparison results on PICEA-g are provided above, similar experiments were conducted on NSGA-II, MSOPS and HypE, with similar outcomes. Results can be downloaded at [56].
6.3. The influence of parameter k In this study, k = 2 and k = 4 are used. That is, the entire population is partitioned into 2 and 4 sub-populations for 2- and 4-objective problems, respectively. With such setting, LMOEAs perform better than MOEAs. However, it is not known whether these settings are the best choices for are k. Therefore, the influence on LMOEAs of the choice of number of sub-populations is studied. Again, results for PICEA-g are presented here. Similar results are obtained for the other three algorithms, which can be downloaded at [56]. N 100 function evaluations are taken as the stopping criterion. Similarly, WFG benchmarks with 2 and 4 objectives are used as test problems. LPICEA-g is subjected to 50 runs on each test problem for different values of k. Table 13 presents the settings of k. Figs. 9–12 use box plots to illustrate the distribution of HV (or IGD for WFG3) metrics (Y-label) for LPICEA-g implemented with different values of k (X-label). The implementation of PICEA-g is represented by k = 1. Moreover, in order to know whether the population size influences the settings of k, the analysis is also performed on LPICEA-g with N = 200 and N = 400. Figs. 9 and 10 show the results on 2-objective problems with N = 200 and N = 400, respectively. Figs. 11 and 12 show the results on 4-objective problems with N = 200 and N = 400, respectively. (1) From the results (N = 200) shown in Fig. 9, we find that k = 2 (used in the comparative study) is not the optimum setting for solving 2-objective problems. Better results can be obtained using k = 4. Again, from Fig. 11, it can be seen that k = 4 is not the best setting, however, this setting produces relatively good results for most of the problems. (2) Comparing the results obtained by using different N, we find that, given a larger population size, the optimum value of k increases. For example, k = 3 is the optimum setting for WFG7-2 when N = 200 (See Fig. 9), however, the optimum setting is k = 4 for WFG7-2 when N = 400 (see Fig. 10). Moreover, a larger range of k values is available for producing better results (compared with the results obtained by k = 1), when a larger population size is used. (3) The optimum setting for k is also influenced by the problem dimension, M. For example, when N = 400, k = 5 is the optimum setting for WFG4-2 (see Fig. 10), however, the optimum setting of k for WFG4-4 is 8 (see Fig. 12).
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Overall, as illustrated in Figs. 9–12, there is an optimum setting of k for most of the problems. We suggest that applying a value of k in the interval [2,2M] will produce good results. Although the optimum value of k may not lie within this interval, it provides a guide for selecting an appropriate value of k. In addition, it is observed that local operations have no significant positive effect on some problems, especially for WFG2. The reason for this may be that the partition method used in this study is not appropriate. Also for the WFG2 problem, another reason may be that a disconnected Pareto front weakens the effect of local operations. This issue deserves further investigation.
7. Conclusions This study proposes a local operation based framework for the design of MOEAs. In this framework, evolutionary operators are executed on sets of sub-populations, separately. The effect of this framework is systematically studied by comparing four best-in-class algorithms with their modified versions derived from this local framework. A local operation based elitist framework is proposed for designing MOEAs. A specific implementation of the new framework is provided. Specifically, a clustering technique is applied to partition the evolutionary population into a pre-defined number (k) of sub-populations. Then the evolutionary operations genetic-variation and environment-selection are executed separately on each sub-population. MOEAs (NSGA-II, MSOPS, HypE and PICEA-g) within the general elitist framework have been compared with their corresponding LMOEAs derived from this new framework. The experimental results demonstrate that this new framework works effectively on most of the test problems, and unlike other existing local operations based methods, this framework suits for different types of MOEAs. Therefore, this local operation based elitist framework could be considered for designing future multi-objective evolutionary algorithms. The main findings are as follows. (1) The performance (convergence, diversity or both) of MOEAs can be improved by applying MOEAs within the local operation based framework. (2) This framework works more effectively on MOPs with a continuous Pareto front than MOPs with disconnected Pareto fronts, such as the WFG2 problems. (3) Applying local genetic-variation and local environment-selection jointly to MOEAs is better than applying one of these operations on its own. Comparing the two local operations, the effect of local genetic-variation is more significant. (4) The performance of LMOEAs is varies with different settings of k. A good setting of k is impacted by both the problem dimension and population size. The setting of k would be more flexible if a larger population is used. The main limitation of this study is that its findings are based on real-parameter function optimisation problems. It is also important to assess the effect of this local framework on other problem types, e.g. multi-objective combinatorial problems. With respect to further research, first, it would be valuable to understand how to set the appropriate number, k, of subpopulations. k might be adjusted either subject to a pre-defined rule or self-adaptively. Second, approaches (other than the use of clustering methods) for partitioning the population should be investigated. Thirdly, it would be valuable to investigate the effect of this local framework on multi-objective combinatorial problems.
Acknowledgments The work of Rui Wang was supported by the China Scholarship Council and the National Science Foundation of China (No. 70971132). Also, the authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the manuscript.
Appendix A. Hierarchical clustering There are many clustering algorithms such as connectivity based clustering, centroid based clustering, distribution based clustering, and density based clustering [43]. Hierarchical clustering [40] belongs to the connectivity based clustering, which seeks to build a hierarchy of clusters. The core idea is that objects being more related to nearby objects than to objects farther away. Thus, the clusters are formed by connecting objects according to their distances. Note that instead of giving a single partitioning of the data set, the algorithms provide an extensive hierarchy of clusters that merge with each other at certain distances. At different distances, different clusters will be formed, which is often described by a dendrogram. As for the computation of distances, there are many choices such as the Euclidean distance, Squared Euclidean distance, Chebyshev distance, Mahalanobia distance, cosine similarity and so on. Apart from this, we also need to define the linkage criterion. Several choices are available such as single-linkage clustering (the nearest distance), complete linkage clustering (the furthest distances), average-linkage clustering (the un-weighted average distance) and ward-linkage clustering (the inner squared distance). More details can be found in [15].
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Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
General framework for localised multi-objective evolutionary algorithms Rui Wang a,b,⇑, Peter J. Fleming a, Robin C. Purshouse a a b
Department of Automatic Control & Systems Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK Department of Systems Engineering, College of Information Systems and Management, National University of Defense Technology, Chang Sha 410073, China
a r t i c l e
i n f o
Article history: Received 15 November 2012 Received in revised form 23 July 2013 Accepted 17 August 2013 Available online 30 August 2013 Keywords: Multi-objective optimisation Evolutionary algorithm Framework Clustering
a b s t r a c t Many real-world problems have multiple competing objectives and can often be formulated as multi-objective optimisation problems. Multi-objective evolutionary algorithms (MOEAs) have proven very effective in obtaining a set of trade-off solutions for such problems. This research seeks to improve both the accuracy and the diversity of these solutions through the local application of evolutionary operators to selected sub-populations. A local operation-based implementation framework is presented in which a population is partitioned, using hierarchical clustering, into a pre-defined number of sub-populations. Environment-selection and genetic-variation are then applied to each sub-population. The effectiveness of this approach is demonstrated on 2- and 4-objective benchmark problems. The performance of each of four best-in-class MOEAs is compared with their modified local operation-based versions derived from this framework. In each case the introduction of the local operation-based approach improves performance. Further, it is shown that the combined use of local environment-selection and local genetic-variation is better than the application of either local environment-selection or local genetic-variation alone. Preliminary results indicate that the selection of a suitable number of sub-populations is related to problem dimension as well as to population size. 2013 Elsevier Inc. All rights reserved.
1. Introduction Multi-objective problems (MOPs) regularly arise in real-world design scenarios, where two or more objectives are required to be optimised simultaneously. As such objectives are often in competition with one another, the optimal solution of MOPs is a set of trade-off solutions, rather than a single solution. Due to the population-based approach, multi-objective evolutionary algorithms (MOEAs) are well suited for solving MOPs since this leads naturally to the generation of an approximate trade-off surface (or Pareto front [22]) in a run [6,9]. A variety of MOEA approaches has been proposed [63]: (1) Pareto dominance (and modified Pareto dominance) based MOEAs, e.g. MOGA [17,19,21],NSGA-II [12], SPEA2 [65], -MOEA [11] and Pareto Cone -dominance based MOEA [2], (2) indicator (hypervolume [66]) based MOEAs, e.g. IBEA [64], SMS-EMOA [4], HypE [1], (3) scalarising function based MOEAs, e.g. MSOPS [26], MOEA/D [62], and, latterly, (4) preference-inspired co evolutionary algorithms (PICEAs), e.g. PICEA-g [48,58,59]. All of these approaches strive to converge quickly to a satisfactory approximation to the Pareto front (convergence) and that this approximation be well distributed with a good coverage of the front (diversity).
⇑ Corresponding author at: Department of Automatic Control & Systems Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK. E-mail addresses: [email protected], cop10rw@sheffield.ac.uk (R. Wang). 0020-0255/$ - see front matter 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2013.08.049
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Pareto dominance based MOEAs were one of the earliest approaches and it is accepted that they perform well on MOPs with 2 and 3 objectives. However, their search capability often degrades significantly as the number of objectives increase [46,27]. This is because the proportion of Pareto-optimal (or non-dominated) objective vectors in the population grows large when MOPs have more than 3 objectives i.e. many-objective problems. As a result, insufficient selection pressure can be generated toward the Pareto front [47,36]. Relatively recently, there has been considerable effort invested in other types of MOEAs that might perform more effectively on many-objective problems. In particular, PICEA-g, HypE and MSOPS are MOEAs that claim to perform better on many-objective problems [55,58]. In general, a MOEA can be described as
Pðt þ 1Þ ¼ Ss ðv ðSv ðPðtÞÞÞ; PðtÞÞ
ð1Þ
where P(t) are the candidate solutions (population) at iteration t, Sv is the selection-for-variation operator, v is the genetic-variation (recombination and mutation) operator, Ss is the environment-selection operator, and P(t + 1) are the newly generated solutions [47]. Thus, a set of candidate solutions is evolved by successively applying recombination, mutation, and selection to yield better solutions in an iterative process. In many cases, Ss and v are executed on the entire population (described as a global operation). Accordingly, an evolutionary operator that is executed on sub-populations is denoted a local operation. Global operations are held to be beneficial for speed of convergence. However, global operation increases the probability of recombining solutions distant from one another and so produce lower performance offspring known as lethals [19,47,31]. The issue of lethals were originally considered in single objective optimisation by Deb & Goldberg as a response to the problems of fitness sharing [22]. The superfluous production of lethals, known as dominance resistance [47], will consequently reduce the efficiency of the optimisation process and affect convergence towards the Pareto optimal front. It has been argued that the use of local operations may achieve a better overall performance, taking account of both convergence and diversity [7,37]. For example, Sato et al. [49,50] demonstrated that the execution of Pareto dominance and recombination on neighbouring solutions could produce more fruitful solutions and so improve the performance (convergence and diversity) of MOEAs. Although local operation has been investigated in some studies1 [39,49,50,31,57], no generalised MOEA framework is available for the implementation of local operations. In this study we first propose a local operation based framework and then present a specific implementation for using this framework. In this implementation, a clustering technique first partitions the global population into a number of sub-populations. Then, the operators, Sv and v, are executed on each sub-population independently. The proposed framework shares some similarities with parallel MOEAs, particularly, the island model based MOEAs (in which the population is divided into sub-populations, each sub-population is associated with an island and an optimiser evolves the sub-population with occasional migration of individuals between sub-populations). A specific instantiation of the framework is evaluated by comparing the performance of MOEAs and their modified local versions (LMOEAs) derived from the new framework on 2- and 4-objective real-parameter test problems. The MOEAs selected for the evaluation are four best-in-class algorithms: NSGA-II (Pareto dominance), HypE (indicator-based), MSOPS (scalarising function) and PICEA-g (co-evolutionary). Since MOEA/D [62] includes local operators – local selection and local replacements – it is not selected for this study [32]. Overall, the main contributions of this paper are as follows: A local operation based framework is proposed and a specific implementation for using this framework is presented. The effectiveness of this local framework is demonstrated to work well on four different types of MOEAs. The probability p of doing local operations is discussed. Experimental results show using fixed setting of p is worse than using different p (changed according to a relationship defined in Eq. (3)) during the search. The effect of the independent use of local environment-selection or local genetic-variation is discussed. Experimental results show applying both the local operations jointly is better than applying one of the operations on its own. Moreover, local genetic-variation is more effective. The effect of the number of sub-populations (k) to the performance of the framework is discussed. It is found k is impacted by both population size and problem dimension. Some suggestions are provided for the choice of a suitable k. The remainder of the paper is structured as follows. In Section 2, MOPs and MOEAs are introduced and related work is briefly reviewed. This is followed, in Section 3, by a description of the new MOEA framework for implementing local operations. Section 4 provides a systematic performance comparison of individual MOEAs and LMOEAs. Section 5 examines the effectiveness of this new framework over other existing local operation based algorithms. Section 6 provides a further discussion of this framework. Section 7 offers concluding remarks and future research. 2. Basic features of MOPs and MOEAs In this Section, the formulation of MOPs and some fundamental concepts in multi-objective optimisation are first introduced, then some features relating to the development of existing MOEAs are described and some related studies on the use of local operations are briefly reviewed. 1
In [57], local operators are implemented in PICEA-g as part of a preliminary study.
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2.1. MOP formulation and fundamental concepts A multi-objective problem has a number of objective functions that are to be minimised/maximised. Without loss of generality, a minimisation MOP is defined as follows:
minimise fm ðxÞ m ¼ 1; 2; . . . ; M subject to g j ðxÞ 6 0;
j ¼ 1; 2; . . . ; J ð2Þ
hk ðxÞ ¼ 0;
k ¼ 1; 2; . . . ; K
xli 6 xi 6 xui ;
i ¼ 1; 2; . . . ; n
A solution x is a vector of n decision variables: x ¼ ðx1 ; x2 ; ; xn Þ; x 2 Rn . Each xi is subject to a lower xli and an upper xui bound. fi the ith objective function. M is the number of objectives (generally, M > 2). J and K are the number of inequality and equality constraints, respectively. Definition 2.1 (Pareto dominance). For two feasible decision vectors x, y, x is said to Pareto dominate y (denoted as x y) if and only if "i 2 1, 2, . . ., n, fi(x) 6 fi(y) and $i 2 1, 2, . . ., n, fi(x) < fi(y)
Definition 2.2 (Pareto optimality). A solution x 2 Rn is said to be Pareto optimal in Rn if and only if 9 = y 2 Rn ; y x. Definition 2.3 (Pareto optimal set). The Pareto optimal set (PS) is defined as the set of all Pareto optimal solutions, i.e. PS ¼ fx 2 Rn j 9 = y 2 Rn ; y xg. Definition 2.4 (Pareto optimal front). The Pareto optimal front (PF) is defined as the set of all objective functions values corresponding to the solutions in PS, i.e., PF = {(f1(x), f2(x), . . ., fM(x)):x 2 PS}. 2.2. MOEA development MOEAs have gradually been refined over the last two decades [8]. In the first decade, MOEAs were mainly based on two pioneering concepts, i.e. Pareto dominance selection and a niching technique [22]. Pareto dominance selection serves to promote convergence by favouring solutions closer to the Pareto optimal front. The niching technique increases diversity by favouring solutions in a sparse region, thus maintaining well distributed solutions along the entire Pareto front. Representative MOEAs are Multi-Objective Genetic Algorithm (MOGA [17]), Non-dominated Sorting Genetic Algorithm (NSGA [51]), and the Niched-Pareto Genetic Algorithm (NPGA [24]). Second generation MOEAs may be said to arise with the introduction of elitism as a standard mechanism. SPEA [66] popularized the notion of using elitism and these second-generation algorithms feature a combination of Pareto dominance selection, diversity maintenance and the implementation of an elitism strategy. The incorporation of an elitism strategy has since become common practice in the design of MOEAs. Elitism can be implemented in a (l + k) framework (see Fig. 1 and Algorithm 1).
Fig. 1. (l + k) elitist framework.
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Algorithm 1. A general (l + k) elitist Framework
In Algorithm 1, a population of candidate solutions, P is evolved for a fixed number of generations, maxGen. At each generation gen, parents P are manipulated by (representation-appropriate) genetic-variation operators to produce new offspring, Pc. P and Pc, are then pooled and the combined population is sorted according to fitness of individuals. Environment-selection is applied to select the set of solutions that will be the new parent population, P. NSGA-II [12] is a typical MOEA representative of the (l + k) elitist framework. Elitism can also be implemented with an external archive which retains all the nondominated solutions found so far, see SPEA2 [65]. 2.3. MOEAs employing local operations The application of local operations in MOEAs is distinct from the practice of applying a standard local search. Local search is typically implemented on some selected good solutions after an evolutionary search, while environment-selection and genetic-variation are still operated on the entire population. However, for local operations, the environment-selection and genetic-variation are executed on sets of pre-selected sub-populations separately. A number of strategies have been proposed for the execution of local operations in MOEAs. A direct way is to execute evolutionary operators on neighbouring solutions in the objective space. The earliest studies of this type of approach are [17,22] in which solutions are prohibited from recombination with each other if the distance between them lies beyond a defined limit. Watanabe et al. [60] suggested recombination on solutions that are adjacent to each other in one of the objectives. Ishibuchi and Shibata implemented local recombination based on a modified tournament selection [33]. Specifically, in this case, when performing recombination, an individual is first selected, then among the individuals selected by multiple standard binary tournaments, the individual closest to it is selected as its mate, i.e. selecting the mate based on the similarities to first parent. Other studies that also investigated the issues of similarity based mating can be found in [34,35,31]. In another approach, local operations are executed on individuals with a similar search direction in the objective space. Representative approaches are multi-objective genetic local search (MOGLS), as proposed by Ishibuchi et al. [30,37] or Jaszkiewicz. [38]. A concern here is how to distribute the search effort uniformly toward all search directions. The coarse grained parallel MOEAs (i.e. islands) might implicitly encourage executions of local operations. Specifically, the island paradigm divides the overall population into a number of independent sub-populations. Each sub-population is evolved by an optimiser in isolation for the majority of algorithm execution. Individuals occasionally migrate between an island and its neighbour(s) based on some selection criteria [6, p. 455]. If the division is implemented based on the similarities of solutions then operations executed on each sub-population can be seen as local operations. This is because solutions in each sub-population are similar to each other and therefore recombination between these solutions are more likely to produce higher performance offspring, i.e. reducing the possibility of lethals [31,47]. However, often more attention is paid to defining suitable migration/replacement policies2 rather than the division strategies [61,42] and [6, pp. 493–500]. MOEA/D, a decomposition-based approach, performs evolutionary operations locally [62]. It treats a MOP as a collection of single-objective problems (SOPs), which are defined by a scalarising function with different weight vectors. Each scalarising fitness function (defined by a specific weight vector) identifies a single solution which is optimal with respect to that scalarising fitness function. For each SOP, a new solution is generated by performing genetic-variation operations (recombination and mutation) on several solutions amongst its neighbours. Neighbours are defined based on the distance between the weight vectors. The newly generated solution is only compared with its neighbours. If the new solution is better, then some (or all) of its corresponding neighbours are replaced by the new solution (environment-selection). Apart from these approaches, Purshouse and Fleming [44] proposed dividing the objectives themselves into sets of subobjectives according to the relations (conflict, harmony or independent) of the objectives [45]. Evolutionary operations are then performed on these sub-problems independently. The novelty of this approach is that the division is not based on individuals but objectives. 2 These policies include the migration frequency, the number of solutions to migrate/replacement, the selection of migration/replacement solutions, and so on [6, pp. 455–458].
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One specific question relating to the effectiveness and efficiency of the use of local operations is when and how often should local operations be executed. Experience [7] suggests that local operations should be (1) executed after a predefined number of generations, and (2) used in a probabilistic manner. When many generations are assigned for global operation before applying local operations, this may result in too much emphasis on exploration in the search, possibly resulting in unsatisfactory convergence rates. In contrast, frequent application of local operations may place too much emphasis on exploitation of the search space, possibly leading to premature convergence. In other words, the balance between global and local operations is very important. This was first pointed out in [37], and then was clearly demonstrated in [29,35]. If this balance is not appropriately specified, the performance of MOEAs would often be degraded by hybridisation with local operations. Furthermore, it is accepted that an increase in the number of objectives in a MOP generates increased numbers of Pareto optimal solutions and their diversity. Furthermore, solutions in the current population are often quite dissimilar from each other. As stated in a study [28] that a large population diversity has negative effects on the performance of MOEAs. This is because recombining dissimilar solutions often cannot produce good offspring. This issue was also implicitly demonstrated in [32], where the performance of MOEA/D on many-objective 0/1 knapsack problems was degraded by increasing the size of a neighbourhood structure for parent selection. These results emphasise the importance of applying local operations in manyobjective problems as recombining similar solutions often generates high-quality offspring [34,35,28]. 3. A framework for integrating local evolutionary operators into the design of MOEAs The global application of evolutionary operators has proved to be successful in identifying Pareto front efficiently. However, local operations which emphasise locality of selection and variation may achieve even better solutions. A generalised local operation based MOEA framework is described in this section. In this framework evolutionary operators are probabilistically executed either locally or globally during the search process. Both the benefits of global operation and local operation are expected to be realised. This local framework is based on the (l + k) elitist strategy. The flow chart of the framework is shown in Fig. 2 and the pseudo-code is described in Algorithm 2. As shown in Fig. 2, parents P compete with their offspring Pc, probabilistically, the better individuals are more likely to be propagated as new parents. However, on the operations of genetic-variation and environment-selection, condition A and condition B are checked respectively to determine whether the operation is to be executed locally (on each of a set of sub-populations) or globally (on the entire population). Algorithm 2. A New local operation based (l + k) elitist MOEA Framework
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R. Wang et al. / Information Sciences 258 (2014) 29–53
Fig. 2. Local operations based elitist framework.
If condition A is satisfied, candidate solutions are then partitioned into a number of sub-populations, Pk, and genetic-variations are performed on each Pk independently to produce new offspring, Pck. The entire set of offspring, Pc is comprised of all the Pck. Fitness evaluation, FitJointP is conducted on all solutions, JointP, which is the combination of P and Pc. If condition B is satisfied, then in a similar way, JointP is partitioned into a number of sub-populations, JointPk. and environment-selection is executed on each member of JointPk to select individuals to be new parents, Pk according to their fitness FitJointPk. This process is repeated until the stopping criterion is satisfied. For the implementation of this framework, there are two important issues: (1) when and how often should local operations be executed? (2) how to choose an appropriate partition method so as to prepare a suitable sub-population for the local operation? For issue (1), the conditions for performing local operations can be set by a user-defined rule or, alternatively, the frequency can be subject to self-adaptation. For issue (2), solutions which are similar to one other should be members of the same sub-population since the aim is to enhance the effect of recombination [49]. Similarities can be defined by any distance measure, e.g. Euclidean distance, Manhattan distance, Chebyshev distance and Minkowski distance [41]. The proposed implementation is as follows: (1) As described in [7], the genetic-variation and environment-selection are executed locally when p > rand is satisfied, i.e. conditions A and B. rand is a random number generated in [0, 1]. p is related to the stage reached in the search process, see Eq. (3):
pðgenÞ ¼
gen maxGen
ð3Þ
(2) A hierarchical clustering method [40] is used to partition candidate solutions. Specifically, the Chebyshev function is used to measure the distance between two solutions in the objective space. The Ward-linkage clustering [5] is chosen as the linkage criteria. Before computing the Chebyshev distance, the objective vector of each individual, F(s) is normalised by Eq. (4), in order to avoid the influence of different ranges of objectives [54].
FðsÞi ¼
FðsÞi lFðsÞi dFðsÞi
!2 ;
i ¼ 1; 2; . . . ; M
ð4Þ
where lFðsÞi is the mean value of F(s)i and dFðsÞi represents the standard deviation of F(s)i. A fuller description of the hierarchical clustering method is provided in Appendix A. In order to use the hierarchical clustering method, the user has to specify the number of clusters, k to be created. In this study, k is set equal to M. This may not be the ideal setting as LMOEAs with different choices of k are likely to result in different levels of performance. However, as will be demonstrated, such a setting leads to good outcomes. A discussion on the ideal setting for k is provided in Section 6.3. The proposed local framework shares some commonality with parallel MOEAs, particularly, the island model based MOEAs, in that the population is divided into sub-populations and an optimiser evolves each sub-population. However, the differences between these two schemes are as follows:
R. Wang et al. / Information Sciences 258 (2014) 29–53
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(1) The motivation of parallelizing MOEAs is to take advantages of the distributed computation. The proposed framework aims at improving the convergence and diversity performance of MOEAs. (2) The main issue of island model based MOEAs is how to choose suitable migration/replacement policies. The main issue of the proposed framework is how to group solutions, i.e. combine similar solutions into the same group. (3) In the island model, individuals occasionally migrate between islands. In the proposed framework, the entire population can be reorganised via clustering. (4) In the island model, fitness calculation is based on each sub-population, independently [6, pp. 447–448]. In our framework the fitness calculation is based on the entire population. We are also aware of some studies that considered using clustering techniques to divide the population for island model based MOEAs [14,52]. For example, in [52] the authors also suggested a divide-and-conquer approach beyond an island model MOEA with migration parameters. In their approach, sub-populations are limited in specific regions by zone constraints based upon the dominance principle using k-means cluster centroids. However, the aim of these studies is to demonstrate the superiority of the ‘divide-and-conquer’ based island model over a general island model with migration parameters. The authors have not attempted to utilise this clustering plus divide-and-conquer strategy to encourage local operations. In this work, we for the first time generalise this idea as a framework for localised MOEAs in order to handle the issue of lethals and thereby improve algorithm performance. 4. Comparative studies of the local framework on MOEAs 4.1. Comparative studies: experiment description To demonstrate the effectiveness of this framework, we compared four best-in-class algorithms (NSGA-II, HypE, MSOPS and PICEA-g) with their corresponding modified versions, denoted as LNSGA-II, LHYPE, LMSOPS and LPICEA-g, respectively, on some leading real-parameter benchmark problems. First, the experimental setup is presented and then the selected performance metrics and statistical treatments are introduced. The section concludes by briefly introducing the four baseline MOEAs and describing the settings for the corresponding LMOEAs. 4.1.1. Experiment setup Many test suites have been proposed to test the performance of MOEAs. Widely used ones include the ZDT [66], the DTLZ [13] and the WFG [25] suites. The ZDT test suite contains only 2-objective test problems. The DTLZ and WFG test suites are scalable and can be used to construct many-objective test problems. In this study, problems 2–9 from the WFG test suite are invoked in 2 and 4-objective instances. In each case, the WFG position parameter and distance parameter are set to 18 and 14, providing a constant number of decision variables (18 + 14 = 32) for each problem instance. Problem attributes include separability/non-separability, unimodality/multimodality, unbiased/biased parameters and convex/concave geometries. Note that in this study WFGn-Y refers to problem WFGn with Y objectives. For each test problem, 20,000 function evaluations are undertaken. 50 runs of each algorithm test are performed in order to subject to statistical analysis. Other general parameter settings are listed in Table 1 and are fixed across all algorithm runs. 4.1.2. Performance assessment For qualitative performance assessment, median attainment surfaces [18] are plotted to visualise the performance of algorithms on 2-objective instances. For quantitative performance assessment, a broad range of performance metrics is chosen to measure the obtained Pareto front [67], comprising unary metrics (generational distance (GD [53]), spread metric (D [12]), inverse generational distance (IGD [38]) and hypervolume (HV [66])) and a binary metric (the coverage of two sets (C [66])): (1) the GD metric measures the average distance from objective vectors in the approximation set to the nearest neighbour in the reference set (Pareto optimal front), i.e. convergence, (2) D measures the degree of dispersion on the distribution of the obtained approximation set, i.e. diversity, (3) HV measures the volume of space dominated by the approximation set and one reference point, (4) IGD takes the average distance for all members in the reference set (Pareto optimal front) to their nearest solutions in the obtained approximation (exactly the inverse process of GD), and Table 1 General parameter settings. Objectives, M The number of clusters, k Population size, N Max generations, maxGen Decision variables, n Simulated binary crossover (SBX [12]) Polynomial mutation (PM [12])
2, 4 k=M 200 100 32 (pc = 1, gc = 15 ) (pm = 1/n, gm = 20)
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R. Wang et al. / Information Sciences 258 (2014) 29–53 Table 2 Performance assessment. Test problem
Shape of Pareto front
Performance metrics
WFG2 WFG3 WFG4-WFG9
Disconnected, convex Linear, degenerate Continuous, concave
C, HV GD, D, IGD GD, D, HV
(5) the C metric provides complementary information on convergence. For example, assuming that A and B are the sets of non-dominated solutions found by two algorithms, C(A, B) is the fraction of solutions in B that are dominated at least by one solution in A. A smaller GD value implies good proximity and also a smaller spread metric value implies good diversity. A favourable hypervolume (larger, for a minimisation problem) and IGD (smaller) implies good proximity with diversity. GD, D and IGD metrics are based on the assumption that the true Pareto front of a test problem is known. For the C metric, C(A, B) < C(B, A), for example, indicates a better convergence of the B set. This combination of performance metrics provides a clear indication as to the quality of an approximation set. With regard to the test functions used, according to [25], the Pareto optimal front of WFG2 is not a regular geometry (disconnected and convex), the Pareto optimal front of WFG3 is a linear line, and the Pareto optimal front of WFG4-WFG9 is the surface of an M-dimension hyper-ellipsoid with radius ri = 2i, i = 1. . .M in the first quadrant. Based on the properties of performance metrics and the shape of Pareto fronts of the selected test problems, the specific settings of performance assessment are shown in Table 2. Performance comparisons of the algorithms are made using a rigorous non-parametric statistical framework, drawing on recommendations in [67]. The approximation sets used in the comparisons are the members of the off-line archive of all nondominated points found during the search. For reasons of computational feasibility, prior to analysis, the set is pruned to a maximum size of 100 using the SPEA2 truncation procedure [65]. (The approximation sets used for performance assessment are available for download at [56].) For each problem instance, performance metric values for each algorithm are calculated for each approximation set, the averages and standard deviations of performance metrics are presented. The non-parametric statistical approach introduced in [23] is performed on the performance metric values. Note that prior to calculating the performance metrics for each approximate set, we normalise all objectives to be within the range [0, 1] by the nadir point [10] (which assumes equal relative importance of all the objectives). For the hypervolume metric, the reference point is set as ri = 1.2, i = 1, 2, . . ., M and the software developed by [20] is used to calculate the hyper volume. For unary metrics, the null hypotheses used are those of equality of median values, against the two-sided alternative hypotheses. For the binary C metric, the null hypotheses are defined as the difference between the median values of the mutual coverages of pairs of algorithms. The non-parametric Wilcoxon-ranksum two-sided comparison [23] procedure at the 95% confidence level is employed to test the working hypotheses. 4.1.3. Overview of selected MOEAs In this subsection, the four selected MOEAs, i.e. NSGA-II, HypE, MSOPS and PICEA-g are briefly introduced. and related parameter settings adopted for this study are explained. (1) NSGA-II [12] is a well-known Pareto dominance based MOEA in which parents and offspring are combined and evaluated using a fast non-dominated sorting approach (based on Pareto dominance). An efficient diversity promotion mechanism, the crowding distance, is used as a secondary criterion to discriminate among equally ranked solutions, such that solutions in less crowded regions are preferred. (2) HypE [1] is an efficient indicator based MOEA [64], where a hypervolume metric is used as the indicator. Often the high computational effort required for hypervolume calculation [3] inhibits the full exploitation of hypervolume based MOEAs [16,4]. HypE uses a Monte Carlo simulation to approximate the hypervolume values, thus the accuracy of these estimates can be traded off against the available computing resources. It has been shown that HypE is effective for both multi- and many-objective problems [1]. In common with standard MOEAs, it is based on fitness assignment schemes and consists of successive application of mating selection, variation and environmental selection. The hypervolume indicator is applied in environmental selection. In this study, we strictly follow the hypervolume calculation method described in [1]. For 2-objective problems, the exact hypervolume is used, whereas, for 4-objective problems, a Monte Carlo simulation method with 2000 sampling points is used to calculate the estimated hypervolume contribution. (3) MSOPS [26] can be classified as a scalarising function based MOEA. In MSOPS, all objectives are handled in parallel. A set of target vectors is used to guide the search process in multiple directions simultaneously. Two scalarising functions are used, weighted min–max (weighted Chebyshev) and Vector-Angle-Distance-Scaling (VADS) and T uniformed distributed target vectors are generated priori to the search process. For each solution, 2T scores are computed, based
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WFG2−2
5 4.5 4
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Fig. 3. Attainment surfaces for WFG2-2 to WFG5-2.
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Fig. 4. Attainment surfaces for WFG6-2 to WFG9-2.
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R. Wang et al. / Information Sciences 258 (2014) 29–53 Table 3 The mean/deviation values of HV metric for 2-objective problems. Problem
NSGA-II
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.3996/0.0076 0.3721/0.0062 0.3706/0.0098 0.3433/0.0066 0.2607/0.0094 0.3708/0.0187
Problem
HypE
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.4029/0.0062 0.3785/0.0052 0.3712/0.0080 0.3450/0.0079 0.2600/0.0063 0.3700/0.0166
LNSGA-II
MSOPS
< < = < < <
0.4072/0.0061 0.3815/0.0046 0.3735/0.0080 0.3557/0.0080 0.2908/0.0207 0.3866/0.0212
0.4026/0.0047 0.3778/0.0051 0.3716/0.0071 0.3511/0.0066 0.2611/0.0071 0.3662/0.0148
LHypE
PICEA-g
< < < < < <
0.4149/0.0051 0.3883/0.0035 0.3802/0.0069 0.3605/0.0095 0.2979/0.0174 0.3903/0.0231
0.4167/0.0033 0.3931/0.0028 0.3803/0.0059 0.3655/0.0075 0.2554/0.0082 0.3751/0.0190
LMSOPS < < < < < <
0.4206/0.0053 0.3895/0.0033 0.3786/0.0078 0.3663/0.0064 0.2991/0.0167 0.3867/0.0231
< < < < < <
0.4286/0.0032 0.3992/0.0024 0.3856/0.0062 0.3752/0.0070 0.2693/0.0172 0.3959/0.0219
< < < < < <
0.0002/0.0001 0.0018/0.0001 0.0022/0.0002 0.0025/0.0002 0.0092/0.0023 0.0026/0.0016
< < = < < <
0.0009/0.0001 0.0025/0.0001 0.0035/0.0004 0.0036/0.0003 0.0148/0.0043 0.0030/0.0016
< < < < < <
0.2080/0.0379 0.2856/0.0247 0.3349/0.0454 0.5085/0.0357 0.6738/0.0477 0.2434/0.0251
< < < < < <
0.1194/0.0196 0.2062/0.0183 0.2590/0.0293 0.4266/0.0452 0.7510/0.0956 0.1981/0.0208
LPICEA-g
Table 4 The mean/deviation values of the GD metric for 2-objective problems. Problem
NSGA-II
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.0003/0.0001 0.0019/0.0001 0.0021/0.0004 0.0028/0.0001 0.0093/0.0021 0.0036/0.0014
Problem
HypE
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.0003/0.0001 0.0019/0.0001 0.0022/0.0003 0.0027/0.0001 0.0090/0.0013 0.0038/0.0013
= < = < < <
= < = < < <
LNSGA-II
MSOPS
0.0002/0.0002 0.0016/0.0001 0.0020/0.0003 0.0022/0.0003 0.0081/0.0024 0.0024/0.0017
0.0004/0.0001 0.0019/0.0001 0.0024/0.0004 0.0030/0.0001 0.0120/0.0016 0.0040/0.0009
LHypE
PICEA-g
0.0003/0.0001 0.0018/0.0001 0.0021/0.0003 0.0023/0.0002 0.0075/0.0019 0.0023/0.0018
0.0012/0.0001 0.0026/0.0001 0.0036/0.0005 0.0040/0.0005 0.0168/0.0030 0.0044/0.0014
LMSOPS
LPICEA-g
Table 5 The mean/deviation values of the D metric for 2-objective problems. Problem
NSGA-II
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.3479/0.0511 0.4108/0.0429 0.3969/0.0505 0.6317/0.0363 0.8394/0.0420 0.2705/0.0222
Problem
HypE
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.3274/0.0430 0.3673/0.0380 0.3889/0.0423 0.6284/0.0434 0.8448/0.0395 0.2620/0.0201
LNSGA-II
MSOPS
< < < < < <
0.2959/0.0412 0.3448/0.0337 0.3721/0.0421 0.5729/0.0457 0.7168/0.0641 0.2564/0.0181
0.3240/0.0317 0.3710/0.0368 0.3835/0.0381 0.5854/0.0360 0.8357/0.0386 0.2727/0.0202
LHypE
PICEA-g
< < < < < <
0.2479/0.0361 0.2957/0.0257 0.3290/0.0388 0.5511/0.0535 0.6876/0.0484 0.2380/0.0148
0.1922/0.0206 0.2499/0.0206 0.2932/0.0255 0.4803/0.0463 0.8253/0.0738 0.2333/0.0133
LMSOPS
LPICEA-g
on different scalarising functions with different weight vectors. The scores are recorded in a [N 2T] matrix, S_M. Each row of S_M belongs to a candidate solution and each column represents a score for the candidate solutions as measured by the specific scalarising function. Each column of the matrix S_M is ranked, giving the best performing individual rank 1 and the rank values are stored in a matrix R. Each row of R is sorted in ascending order, providing a lexicographical order of the individuals. (4) PICEA-g [48,58] refers to a preference-inspired co-evolutionary algorithm using goal vectors. PICEA is a novel class of MOEA, where a family of preferences are co-evolved with the usual population of candidate solutions. Different forms of preferences used in multi-criteria decision-making (e.g. goals and weights) can be realised in the PICEA approach. PICEA-g is a realisation that takes goals as preferences [58]. Candidate solutions gain fitness by dominating a particular
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R. Wang et al. / Information Sciences 258 (2014) 29–53
Table 6 The mean/deviation values of the HV metric for 4-objective problems. Problem
NSGA-II
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.5817/0.0145 0.5295/0.0177 0.5398/0.0183 0.5415/0.0170 0.4392/0.0166 0.5648/0.0107
Problem
HypE
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.7026/0.0056 0.6754/0.0055 0.6725/0.0133 0.7200/0.0097 0.5664/0.0086 0.6081/0.0053
LNSGA-II
MSOPS
< < < < < <
0.6086/0.0110 0.5591/0.0152 0.5555/0.0161 0.5662/0.0157 0.4594/0.0142 0.5728/0.0079
0.6624/0.0075 0.6301/0.0076 0.6223/0.0136 0.6158/0.0161 0.5213/0.0091 0.5888/0.0065
LHypE
PICEA-g
< < = < < <
0.7122/0.0051 0.6839/0.0046 0.6730/0.0138 0.7387/0.0085 0.5890/0.0125 0.6169/0.0128
0.7102/0.0058 0.6830/0.0054 0.6746/0.0154 0.7204/0.0103 0.5744/0.0076 0.6080/0.0050
LMSOPS < < = < < <
0.6789/0.0069 0.6525/0.0061 0.6281/0.0158 0.6442/0.0159 0.5491/0.0075 0.5931/0.0064
< < = < < <
0.7279/0.0061 0.6990/0.0040 0.6739/0.0130 0.7388/0.0121 0.5940/0.0133 0.6291/0.0128
= < = < < =
0.0052/0.0002 0.0054/0.0002 0.0069/0.0006 0.0074/0.0003 0.0175/0.0012 0.0062/0.0005
< = = < > <
0.0041/0.0001 0.0043/0.0001 0.0051/0.0001 0.0037/0.0001 0.0123/0.0029 0.0045/0.0005
= < = = < =
1.4535/0.0670 1.4651/0.0474 1.5668/0.0719 1.4745/0.0643 1.9485/0.0693 1.5215/0.0564
LPICEA-g
Table 7 The mean/deviation values of GD metric for 4-objective problems. Problem
NSGA-II
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.0075/0.0004 0.0091/0.0009 0.0095/0.0008 0.0111/0.0008 0.0250/0.0007 0.0076/0.0008
Problem
HypE
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.0043/0.0001 0.0045/0.0002 0.0051/0.0003 0.0038/0.0002 0.0119/0.0032 0.0055/0.0002
LNSGA-II
MSOPS
< < = < < <
0.0068/0.0003 0.0084/0.0003 0.0094/0.0007 0.0101/0.0004 0.0241/0.0010 0.0071/0.0006
0.0053/0.0003 0.0058/0.0003 0.0068/0.0005 0.0076/0.0007 0.0201/0.0009 0.0063/0.0003
LHypE
PICEA-g
< < = < < <
0.0041/0.0001 0.0042/0.0001 0.0050/0.0003 0.0035/0.0001 0.0112/0.0012 0.0047/0.0006
0.0043/0.0001 0.0043/0.0001 0.0050/0.0002 0.0039/0.0001 0.0095/0.0036 0.0056/0.0003
LMSOPS
LPICEA-g
Table 8 The mean/deviation values of D metric for 4-objective problems. Problem
NSGA-II
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
1.6673/0.0922 1.7658/0.0756 1.8149/0.0985 1.7555/0.1037 2.2171/0.0947 1.5957/0.0652
Problem
HypE
WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
1.3947/0.0608 1.3922/0.0497 1.4773/0.0675 1.3328/0.0725 2.2025/0.0959 1.4836/0.0566
< < < < < =
= = = = < <
LNSGA-II
MSOPS
1.5628/0.0729 1.6447/0.0834 1.7229/0.0844 1.6110/0.1046 2.1134/0.0809 1.5822/0.0684
1.4584/0.0636 1.5033/0.0576 1.5574/0.0616 1.5029/0.0667 2.1679/0.0663 1.5285/0.0507
LHypE
PICEA-g
1.3741/0.0713 1.3941/0.0770 1.4679/0.0577 1.3196/0.0672 1.8935/0.0710 1.4575/0.0590
1.3796/0.0499 1.3930/0.0564 1.4546/0.0685 1.3345/0.0635 2.0912/0.1787 1.4767/0.0587
LMSOPS
LPICEA-g = < = = = <
1.3805/0.0684 1.2851/0.0301 1.4691/0.0734 1.3176/0.0687 2.1039/0.0759 1.4477/0.0659
set of goals in objective space, but the fitness contribution is shared between other solutions that also satisfy those goals. Goals only gain fitness by being dominated by a candidate solution, but the fitness is reduced according to how often each goal is dominated by other solutions in the population. In this way, the candidate solution population and the goal population co-evolve toward the Pareto optimal front. In this study, the number of goal vectors, Ngoal, is set to be N M. For each of the four MOEAs, two versions are built: one is within the general (l + k) elitist framework, the other is within the proposed local (l + k) elitist framework. Each pair of MOEA versions are compared.
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R. Wang et al. / Information Sciences 258 (2014) 29–53 Table 9 The mean/deviation values of performance metrics for WFG3 (a: WFG3-2 b: WFG3-4). NSGA-II a GD D IGD
0.0005/0.0002 0.3504/0.0572 0.0025/0.0006
LNSGA-II
MSOPS
< < <
0.0003/0.0001 0.2779/0.0367 0.0017/0.0004
0.0006/0.0002 0.3125/0.0458 0.0020/0.0004
LHypE
PICEA-g
< < <
0.0002/0.0001 0.2215/0.0393 0.0013/0.0004
0.0013/0.0002 0.1712/0.0283 0.0011/0.0003
LNSGA-II
MSOPS
< > <
0.0905/0.0038 2.2403/0.1008 0.0052/0.0007
0.0712/0.0021 2.1974/0.0952 0.0034/0.0004
LHypE
PICEA-g
< = =
0.0730/0.0032 2.1462/0.0947 0.0030/0.0004
0.0932/0.0045 2.2304/0.1077 0.0030/0.0003
HypE GD D IGD
0.0004/0.0001 0.2806/0.0436 0.0018/0.0005 NSGA-II
b GD D IGD
0.0924/0.0039 2.1798/0.1313 0.0058/0.0009 HypE
GD D IGD
0.0885/0.0051 2.1550/0.0949 0.0031/0.0004
LMSOPS < < <
0.0004/0.0001 0.2216/0.0317 0.0013/0.0002
< < <
0.0011/0.0003 0.1524/0.0284 0.0009/0.0002
LPICEA-g
LMSOPS < < <
0.0694/0.0032 2.1329/0.1007 0.0032/0.0002
< = =
0.0882/0.0054 2.2005/0.0925 0.0029/0.0003
LPICEA-g
Table 10 The mean/deviation values of performance metrics for WFG2 (a: WFG2-2 b: WFG2-4). NSGA-II a C HV
0.4624/0.3664 0.5609/0.0053
LNSGA-II
MSOPS
0.2257/0.2007 0.5751/0.0033
0.5357/0.3735 0.5630/0.0183
LHypE
PICEA-g
0.1748/0.2389 0.5774/0.0021
0.5724/0.2427 0.5653/0.0110
LNSGA-II
MSOPS
< <
0.0729/0.0610 0.7963/0.0061
0.2358/0.1380 0.7988/0.0378
LHypE
PICEA-g
< >
0.0694/0.0713 0.8065/0.0067
0.1652/0.0520 0.8071/0.0042
< <
HypE C HV
0.6470/0.2846 0.5727/0.0030
< <
NSGA-II b C HV
0.2996/0.1362 0.7802/0.0079 HypE
C HV
0.1478/0.0877 0.8137/0.0296
LMSOPS < <
0.2288/0.2420 0.5698/0.0039 LPICEA-g
< =
0.2721/0.2543 0.5639/0.0059 LMSOPS
< <
0.1283/0.2358 0.8264/0.0371 LPICEA-g
= =
0.1480/0.0788 0.8032/0.0063
4.2. Comparative studies: experiment results Median attainment surfaces (the 25th) across the 50 runs of each algorithm are shown in Figs. 3 and 4. Tables 3–8 provide comparison results of corresponding MOEAs and LMOEAs in terms of GD, D and HV metrics for problems, WFG4-WFG9. Table 9 shows the comparison results of MOEAs and LMOEAs on WFG3 in terms of GD, D and IGD metrics; Table 10 presents the results for WFG2. In each table, the mean/deviation values of each performance metric across the 50 independent runs are shown. The symbol ‘<’, ‘=’ or ‘>’ denotes whether a MOEA is statistically worse, equal or better than LMOEA at the 95% confidence level. Superior results in each comparison are in boldface.
4.2.1. Median attainment surfaces for 2-objective problems In Figs. 3 and 4, the plots of median attainment surfaces of NSGA-II and LNSGA-II, HypE and LHypE, MSOPS and LMSOPS, PICEA-g and LPICEA-g for each problem are shown in the rectangles from left to right.3 Visually, MOEAs and LMOEAs have comparable performance on convergence for WFG2 to WFG7. On WFG8 and WFG9, the Pareto approximation sets obtained by LMOEAs are closer to the true Pareto front than those obtained by MOEAs. Regarding diversity, LMOEAs perform better than MOEAs on all the problems except for WFG2 where they perform comparably. 3
Colour reproduction available at [56].
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R. Wang et al. / Information Sciences 258 (2014) 29–53
4.2.2. HV comparison results for 2-objective problems From Table 3, it is clear that the mean HV values of MOEAs are smaller than their variants (LMOEAs) for all the test problems. This superiority of LMOEA is also confirmed by the non-parametric statistical test. Each LMOEA performs significantly better than its corresponding MOEA on all the test benchmarks with only one exception, WFG6, for which NSGA-II and LNSGA-II perform comparably. Such results indicate that this new framework works effectively, that is, for 2-objective problems, the performance of MOEAs can be improved by implementing MOEAs within the local framework. 4.2.3. GD and D comparison results for 2-objective problems To further study the effect of the new framework, we have also separately calculated proximity (as measured by GD, shown in Table 4) and diversity (as measured by the spread metric, D shown in Table 5) measures for problems WFG4– WFG9. (1) Table 4 shows that the mean GD values of LMOEAs are lower than the associated MOEAs on all the 2-objective test problems. The non-parametric statistical results prove that the difference is significant on every test problem for MSOPS and LMSOPS; on 5 out of 6 test problems for PICEA-g and LPICEA-g; on 4 out of 6 test problems for NSGA-II and LNSGA-II, and HypE and LHypE. PICEA-g and LPICEA-g perform comparably on WFG6; NSGA-II and LNSGA-II show equivalent performance on WFG4 and WFG6, as do HypE and LHypE. (2) Table 5 demonstrates all LMOEAs have better diversity on all the test problems in terms of average D value. Moreover, the difference is significant for all the problems according to the non-parametric statistical test results. 4.2.4. HV comparison results for 4-objective problems The HV comparison results for 4-objective problems WFG4–WFG9 (see Table 6) demonstrate that the LMOEAs achieve larger mean HV values for all problems. According to the non-parametric statistical tests, the difference is significant for every pair of algorithms on all problems, except for WFG6 for which the difference is significant only for NSGA-II and LNSGA-II. 4.2.5. GD and D comparison results for 4-objective problems From the GD metric results (see Table 7), the following results are observed: (1) LNSGA-II has smaller mean values on all six 4-objective problems. LNSGA-II is statistically better than NSGA-II at the 95% confidence level for all problems, except for WFG6. (2) HypE and LHypE behave similarly to NSGA-II and LNSGA-II. (3) Compared with MSOPS, LMSOPS achieves smaller average GD values on five out of the six problems; for only WFG6 is the average value of MSOPS lower. Differences are significant only for WFG5, 7 and 8. (4) Compared with PICEA-g, LPICEA-g has a smaller average value on WFG4, 5, 7 and 9. The non-parametric statistical test results show that LPICEA-g gives better convergence on WFG4, 7 and 9, equivalent performance on WFG5, 6 and inferior performance on WFG8. With respect to the spread metric (see Table 8), the following results are observed: (1) The average values of LNSGA-II are smaller than NSGA-II for all six problems. Statistically, LNSGA-II performs better than NSGA-II on all problems, except for WFG9, where the difference is not significant. (2) Compared with MSOPS, LMSOPS achieves smaller mean D values on all problems, except for WFG6, where MSOPS has a smaller mean value. LMSOPS performs statistically better than MSOPS on WFG5 and WFG8, and performs comparably to MSOPS on the remaining four problems. (3) Compared with HypE, LHypE has smaller mean D values on five out of six problems (the exception being WFG5). The statistical test results show LHypE has significantly better performance on WFG8 and WFG9, and comparable performance on the other four problems. (4) Compared with LPICEA-g, PICEA-g has smaller mean values on WFG4, 6 and 8; while LPICEA-g has smaller values on WFG5, 7 and 9. PICEA-g is statistically inferior to LPICEA-g on WFG5 and WFG8 and equivalent to LPICEA-g on the other four problems. 4.2.6. Comparison results for 2- and 4-objective WFG3 problems Since the true Pareto front of the WFG3 problem is a degenerated straight line it may not be appropriate to use hypervolume as a performance metric on WFG3-4 because an algorithm that converges close to the line may achieve a smaller hypervolume value compared with an algorithm that does not converge closely to the line but is spread over the objective space. To avoid this problem, we choose the IGD metric instead of then HV metric to evaluate the overall performance of MOEAs. Results are shown in Table 9. Compared their global counterparts, the LMOEAs have smaller average values and perform statistically better in terms of GD, D and IGD on WFG3-2. For WFG3-4, the LMOEAs have better convergence in terms of the mean GD value. The difference is significant according to the statistical tests. The same is true for the D metric, except that
R. Wang et al. / Information Sciences 258 (2014) 29–53
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LNSGA-II performs statistically worse than NSGA-II, otherwise every LMOEA achieves statistically better (LMSOPS) or comparable (LHypE and LPICEA-g) diversity than its corresponding MOEA. Regarding IGD, LMOEAs perform statistically better (LNSGA-II and LMSOPS) than or at least equivalently to (LHypE and LPICEA-g) to their corresponding MOEAs. 4.2.7. Comparison results for WFG2-2 and WFG2-4 Since the true Pareto front of WFG2 is disconnected and not a regular geometry, the C metric is chosen, instead of GD, to compare the convergence of MOEAs and LMOEAs; HV is chosen to compare the overall performance of the algorithms. In Table 10, the C metric is calculated by C(MOEA,LMOEA) for MOEAs and by C(LMOEA, MOEA) for LMOEAs. For WFG2-2, the mean value of C(LMOEA,MOEA) is smaller than that of C(MOEA, LMOEA) and the difference is significant at the 95% level. This implies that LMOEAs achieve statistically better convergence than MOEAs. Regarding HV, LNSGA-II, LHypE and LMSOPS have larger mean values and perform statistically better than NSGA-II, HypE and MSOPS, respectively. However, LPICEA-g has a slightly smaller mean HV value than PICEA-g but the difference is not significant. For WFG2-4, LNSGA-II and LMSOPS perform statistically better than NSGA-II and MSOPS in terms of C and HV metrics. LHypE has significantly better convergence than HypE, however, the overall performance (as measured by the HV) of LHypE is worse than HypE. The average value of C(LPICEA-g,PICEA-g) is slightly smaller than C(PICEA-g,LPICEA-g) but the average HV value of PICEA-g is larger than LPICEA-g. According to the statistical results, LPICEA-g performs comparably with PICEA-g in terms of both performance metrics. 4.3. Summary From these results, we claim that this local operation based (l + k) elitist framework works more effectively than the general (l + k) elitist framework. In most cases, the performance (convergence, diversity or both) of MOEAs are improved by implementing them within this framework. Key observations are: (1) For most problems convergence performance is improved for almost all of the MOEAs. (2) For all of the 2-objective problems, diversity is improved for all of the MOEAs modified by the local framework. However, for most of the 4-objective problems, the improvement of diversity performance for HypE, MSOPS and PICEA-g is not significant. (3) Compared to MSOPS, HypE and PICEA-g, the effect of the framework on NSGA-II is much more significant. The performance of NSGA-II, modified by the local framework, is improved for almost all of the test problems. 5. Comparison between LMOEAs and other local operations based algorithms As mentioned earlier, local operation has been investigated in some earlier studies [62,39,49,50,31,57]. Zhang and Li’s MOEA/D [62] is inherently a local operation based algorithm, that is, environment-selection and genetic-variations are executed on neighbouring individuals [28]. In [39], Jaszkiewicz improves the IMMOGLS algorithm [30] by applying local recombination. In [49] Sato et al. applied local dominance and local recombination to NSGA-II in an implementation (which we call SNSGA-II) in which local dominance implicitly realises a local selection. There have also been general studies of local dominance and local recombination by, respectively, Sato et al. [50] and Ishibuchi et al. [31]. We have also, in earlier work [57], generated proof-of-principle results (using the PICEA-g optimiser) for the local framework proposed in Section 3. In order to further validate the effectiveness of our local framework, we further compare some of our local instantiations with two representative local operations based algorithms mentioned above – specifically, MOEA/D [62] and SNSGA-II [49]. We omit comparison with MOGLS [39] since Zhang and Li have already demonstrated that this algorithm is outperformed by MOEA/D [62]. 5.1. Comparison between LMOEAs and MOEA/D In this section we compare MOEA/D with the four LMOEAs from Section 4 using the same test problems. For MOEA/D, 100 uniform weights are used for 2-objective problems and 455 uniform weights are used for 4-objective problems. The comparison results, in terms of HV (or IGD), are shown in Table 11 and we observe that: Compared to LNSGA-II, MOEA/D performs better for WFG2-2 and WFG6-2, performs comparably for WFG4-2, WFG7-2 and WFG9-2, and performs worse for the remaining three 2-objective problems. Regarding 4-objective problems, MOEA/D performs better only for WFG2-4, performs comparably only for WFG5-4, and performs worse for the remaining six problems. Compared to LMSOPS, MOEA/D performs better only for WFG2-2, performs comparably for WFG4-2, WFG7-2 and WFG92, and performs worse for the remaining four 2-objective problems. Its performance is worse than LMSOPS on all the 4objective problems. Compared to LHypE, MOEA/D performs better only for WFG6-2, and performs worse for the remaining seven problems. Its performance is worse than LHypE on all the 4-objective problems.
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Table 11 The HV (or IGD) comparison results between MOEA/D and other MOEAs for 2-objective problems.
WFG2-2 WFG3-2 WFG4-2 WFG5-2 WFG6-2 WFG7-2 WFG8-2 WFG9-2
WFG2-4 WFG3-4 WFG4-4 WFG5-4 WFG6-4 WFG7-4 WFG8-4 WFG9-4
LNSGA-II
LMSOPS
LHypE
LPICEA-g
> < = < > = < =
> < = < < = < =
< < < < > < < <
= < < < < < = <
LNSGA-II
LMSOPS
LHypE
LPICEA-g
> < < = < < < <
< < < < < < < <
< < < < < < < <
< < < < < < < <
Compared to LPICEA-g, MOEA/D performs better only for WFG6-2, and performs worse for the remaining seven problems. Its performance is worse than LPICEA-g on all the 4-objective problems. The reason for the poor performance of MOEA/D for 4-objective problems might be that the number of weights used in the search is insufficient and therefore performance of MOEA/D might be improved by using more weights [58]. However, since MOEA/D requires the population size to be equal to the number of weights, it is not easy to strike an effective balance between population size (i.e. diversity) and generations (i.e. convergence) under a fixed computational budget – with a larger the population size, the beneficial dynamics of evolution are curtailed. 5.2. Comparison between LNSGA-II and SNSGA-II The local strategies proposed in [50] only apply to Pareto-dominance based algorithms. Therefore, by necessity, we restrict our comparison to a local instantiation of NSGA-II. In SNSGA-II local environment-selection is realised by the use of local dominance. Specifically, the dominance level of each individual is only determined by its neighbours. The neighbourhood size (nLD) is set by the user. Note that, before applying the non-dominated sorting, the individual and its neighbours are rotated by p/4 radians in objective-space. The local geneticvariation in SNSGA-II is executed as follows: for each individual, an offspring is generated by recombining this individual with one of its neighbours. Again, the neighbourhood size (nLR) needs to be defined by the user. Based on the study of Sato et al. [49], nLD = nLR = 10 is used here. Other parameter settings remain as before in Table 1. From Table 12 we can observe that, for 2-objective problems, LNSGA-II and SNSGA-II perform comparably on six out of the eight problems. On WFG8-2 LNSGA-II outperforms SNSGA-II while on WFG5-2 the situation is reversed. For the 4-objective problems LNSGA-II performs better than SNSGA-II on five problems (from WFG3 to WFG7). For the other three problems the two algorithms have comparable performance. These results indicate that for bi-objective problems our local framework does not have significant superiority over the local operations proposed in [49]. However, our proposed framework appears to perform better on 4-objective problems.
Table 12 The mean/deviation values of the HV (or IGD) metric for 2- and 4-objective problems. M=2
M=4
SNSGA-II WFG2 WFG3 WFG4 WFG5 WFG6 WFG7 WFG8 WFG9
0.5734/0.0021 0.0019/0.0003 0.4086/0.0038 0.3881/0.0049 0.3698/0.0076 0.3543/0.0066 0.2713/0.0083 0.3778/0.0187
= = = > = = < =
LNSGA-II
SNSGA-II
0.5751/0.0033 0.0017/0.0004 0.4072/0.0061 0.3815/0.0046 0.3735/0.0080 0.3557/0.0080 0.2908/0.0207 0.3866/0.0212
0.7892/0.0086 0.0067/0.0009 0.5814/0.0129 0.5351/0.0106 0.5279/0.0194 0.5438/0.0119 0.4392/0.0166 0.5742/0.0107
LNSGA-II = < < < < < = =
0.7963/0.0061 0.0052/0.0007 0.6086/0.0110 0.5591/0.0152 0.5555/0.0161 0.5662/0.0157 0.4594/0.0142 0.5728/0.0079
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The reason for the poor performance of SNSGA-II on 4-objective problems is likely to be that the local dominance has deleterious effect on convergence. This operation can potentially assign higher fitness to some dominated solutions. Although this is helpful from a diversity perspective [49], it slows down the convergence speed as some dominated solutions are stored during the search. This side effect becomes more significant on higher-dimension problems since convergence is inherently more difficult to achieve in many-objective fitness landscapes. Another reason might be that the choices of neighbourhood size for local dominance and local recombination are not the optimal settings for problems with more than two objectives [49].
5.3. Summary The results suggest that our proposed local operation based framework is generally more effective than existing similar algorithms. Key observations are: (1) Compared to MOEA/D, the four LMOEAs offer better performance on most of the 2-objective problems and all the 4objective problems. (2) Compared to the approach in [49], LNSGA-II offers better performance on the 4-objective problems.
WFG4−2: =
0.42
WFG4−2: = 0.425
0.415
0.42 0.41 0.415 0.41
HV
HV
0.405 0.4 0.395
0.405 0.4
0.39
0.395
0.385
0.39 0
0.25
0.5
0.75
1
0
Linear
0.25
WFG4−2: <
0.425
0.5
0.75
1
Linear
1
Linear
WFG4−2: < 0.435
0.42 0.43 0.415
HV
HV
0.425 0.41 0.405
0.42
0.415
0.4
0.41
0.395 0
0.25
0.5
0.75
1
Linear
0
0.25
0.5
0.75
Fig. 5. Box plots of HV results for WFG4-2 for algorithms with different p. The symbol ‘<’, ‘=’ or ‘>’ denotes LMOEA with p = 0.75 is statistically worse, equal or better than LMOEA with the relationship described in Eq. (3) at the 95% confidence level.
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WFG4−4: <
WFG4−4: = 0.63
0.72
0.62
0.71
0.61
0.7
HV
HV
0.6 0.59
0.69 0.68
0.58
0.67
0.57
0.66
0.56
0.65
0.55 0
0.25
0.5
0.75
1
Linear
0
0.25
WFG4−4: =
0.75
1
Linear
1
Linear
WFG4−4: < 0.75
0.725
0.74
0.72 0.715
0.73
0.71
HV
HV
0.5
0.72
0.705 0.71
0.7
0.7
0.695 0.69
0
0.25
0.5
0.75
1
Linear
0
0.25
0.5
0.75
Fig. 6. Box plots of HV results for WFG4-4 for algorithms with different p. The symbol ‘<’, ‘=’ or ‘>’ denotes LMOEA with p = 0.75 is statistically worse, equal or better than LMOEA with the relationship described in Eq. (3) at the 95% confidence level.
6. Discussion The empirical comparison results have identified the effectiveness of the local operations based framework. In this section, three further issues are studied. Section 6.1 studies the influence of the probability p to the performance of algorithms. Section 6.2 concerns the independent effect of local genetic-variation and local environment-selection. Section 6.3 studies the number of clusters into which a population is partitioned. LMOEAs with different values of k may perform differently. We study this configuration issue, as part of a wider discussion for the effect of this local framework. 6.1. The effect of the probability p In the above comparative study, the probability p for doing local operations is linearly increased from 0 to 1 (see Eq. (3)). Here, we provided a further discussion on the influence of different p values to the performance of the algorithms. p is set to 0, 0.25, 0.5, 0.75, 1. The 50 HV results of the four algorithms are box-plotted in Figs. 5 and 6 for WFG4-2 and WFG4-4 respectively. The upper and lower ends of the box are the upper and lower quartiles, while a thick line within the box encodes the median. Dashed
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R. Wang et al. / Information Sciences 258 (2014) 29–53
x10-3 WFG3− 2
WFG2− 2
0.64
WFG4− 2
2
0.435
1.5
0.425
0.62
0.43
0.6 0.58
0.4 0.395
0.42 1
0.56
0.39
0.415 0.41
0.54 1
2
3
1
4
WFG6− 2
2
3
4
0.39
1
0.39
0.29
0.38
0.28
0.36
0.25
0.35
0.24
4
4
2
3
4
2
3
4
WFG9− 2 0.41 0.4 0.39 0.38
0.26
1
1
0.42
0.27
0.37 3
3
WFG8− 2
0.37
2
2
0.3
0.38
1
0.385
WFG7− 2 0.4
0.4
WFG5− 2
0.405
0.37 0.36 1
2
3
4
1
2
3
4
Fig. 7. Box plots of HV (or IGD) comparison results for 2-objective WFG problems. (1: PICEA-g 2: LPICEAsel-g 3: LPICEAvar-g 4: LPICEA-g).
WFG2− 4
4
x10−3 WFG3− 4
0.815 0.81
3.5
0.805 0.8
2
3
1
4
WFG6− 4
2
3
4
WFG7− 4
1
2
3
4
0.68
0.69
1
2
3
4
4
3
4
0.64 0.62 0.6
0.54 3
2
0.66
0.56 2
1
WFG9− 4
0.58
1
0.67
WFG8− 4
0.6
0.7
0.64
0.69
0.62
0.72
0.66
0.7
0.64
0.74
0.68
0.73
0.66
0.76
0.7
0.71
0.7
2.5 1
0.74
0.71
0.79 0.785
WFG5− 4
0.72
3
0.795
WFG4− 4
0.75
1
2
3
4
1
2
3
4
Fig. 8. Box plots of HV (or IGD) comparison results for 4-objective WFG problems. (1: PICEA-g 2: LPICEAsel-g 3: LPICEAvar-g 4: LPICEA-g).
appendages indicate the spread and shape of the distribution. Outlying values are marked as +. The average value of each algorithm is shown as } (and is connected by a line through algorithms with different p). Each graph in the Figure contains six box plots representing the HV (or IGD) results (Y-label) of algorithms with five different p values, and the relationship described in Eq. (3). From Figs. 5 and 6 we can clearly observe that among the five different p values, p = 0.75 is the best setting, that is, LMOEAs with p = 0.75 perform better than LMOEAs with other p values in terms of the HV metric on both WFG2-4 and WFG4-4. However, comparing the results between p = 0.75 and the relationship for p described in Eq. (3), it is found that LMOEAs with the relationship (Eq. (3)) perform better than or at least comparably to p = 0.75 at the 95% confidence level. Similar results are obtained for other test problems, and can be downloaded at [56]. These results indicate that neither p = 1 (i.e. apply local operations throughout the search process) nor p = 0 (i.e. never apply local operations) is a good choice for the test problems. The global operation and the local operation should be
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R. Wang et al. / Information Sciences 258 (2014) 29–53 Table 13 Parameter settings. Parameter
Value
N k
200, 400 1 2 3 4 5 6 7 8 9 10 11 12 13 14
2 − objective WFG tests: N=200 WFG2− 2
−3
x 10
WFG3− 2
4 3.5
0.62
2
0.41
1.5
0.4
0.54
1
0.39
0.52
0.5
0.56
0.39 0.385 0.38 0.375 0.37 0.365
1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
WFG6− 2
WFG7− 2
WFG8− 2
WFG9− 2
0.4 0.38
0.38
0.37
0.37
0.36 0.34
0.35
0.33
0.34 1 2 3 4 5 6 7 8 9 1011121314
0.41 0.4
0.3
0.39 0.28
0.36
0.35
0.42
0.32
0.39
0.39
0.32
0.4 0.395
0.42
2.5
0.58
WFG5− 2 0.405
0.43
3
0.6
WFG4− 2 0.44
0.38 0.37
0.26
1 2 3 4 5 6 7 8 9 1011121314
0.24
0.36 0.35 1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
Fig. 9. Boxplots of HV (or IGD) distribution of LPICEA-g with N = 200 and different k for 2-objective WFG problems.
appropriately balanced [37,35] in order to obtain a good approximation of the Pareto front. The relationship used in this study offers better results than all the considered constant settings and appears to be a good choice for this framework. 6.2. The effect of individual local operations PICEA-g is selected as the algorithm to be studied in this context. The three compared algorithms are LPICEA-g, LPICEAselg and LPICEAvar-g. LPICEAsel-g and LPICEAvar-g apply local environment-selection and local genetic-variation on PICEA-g, respectively. Parameter settings are the same as those described in Section 4.1. All algorithms are submitted to 50 runs for each experiment. WFG problems (WFG2–WFG9) with 2 and 4 objectives are chosen as test problems. HV is used to evaluate the performance. Note that the IGD metric is used instead of HV on WFG3 problems, for reasons already given in the previous section. Box plots are used in Figs. 7 and 8 in order to visualise the distribution of the 50 HV or IGD values for the associated problems. The average value of each algorithm is shown as } (and is connected by a line through all four algorithm versions). Each graph in the Figure contains four box plots representing the HV (or IGD) results (Y-label) of the four compared versions of PICEA-g. (1) Fig. 7 clearly shows that for most of the 2-objective problems, the performance of LPICEAvar-g is better than PICEA-g; LPICEAsel-g also performs comparably with PICEA-g. However, neither LPICEAsel-g nor LPICEAvar-g outperforms LPICEA-g. Comparing LPICEAvar-g and LPICEAsel-g, LPICEAvar-g, by and large, exhibits better performance. (2) For 4-objective test problems (see Fig. 8), LPICEAvar-g performs much better than LPICEAsel-g and PICEA-g on the majority of problems. Compared with LPICEA-g, its performance is worse for all the problems except for WFG3-4. As for LPICEAsel-g, it performs comparably to PICEA-g on most of problems.
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R. Wang et al. / Information Sciences 258 (2014) 29–53
2− objective WFG tests: N=400 0.65
−4
x 10
WFG2− 2
WFG4− 2
WFG5− 2
14
0.445
12
0.44
10
0.435
8
0.43
6
0.425
0.398
4
0.42
0.396
0.6
0.55 1 2 3 4 5 6 7 8 9 1011121314
WFG6− 2
WFG3− 2
0.408 0.406 0.404 0.402 0.4
1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
WFG7− 2
0.43
0.41
0.394
WFG8− 2
WFG9− 2
0.41
0.42
0.36
0.42
0.4
0.41
0.34
0.41
0.4
0.32
0.39
0.39
0.38 0.37 0.36
0.4 0.39
0.3
0.38
0.38
0.28
0.37
0.37
0.26
0.36 1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
0.36 1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
Fig. 10. Boxplots of HV (or IGD) distribution of LPICEA-g with N = 400 and different k for 2-objective WFG problems.
4− objective WFG tests: N=200 WFG2− 4 0.84
−3
x 10
WFG3− 4
6
0.75
5
0.68
0.7
4.5
0.8
0.66
4
0.78
3.5
0.76
3
0.64
0.65
0.62
2.5
0.74
0.6 1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
WFG6− 4
0.68
0.6
1 2 3 4 5 6 7 8 9 1011121314
WFG8− 4
WFG9− 4
0.64
0.66
0.62
0.65 0.64 0.63
0.56
0.65
0.6
1 2 3 4 5 6 7 8 9 1011121314
0.58
0.66
0.62
1 2 3 4 5 6 7 8 9 1011121314
0.6
0.7
0.64
0.6
WFG7− 4 0.75
0.7
WFG5− 4 0.7
5.5
0.82
0.58
WFG4− 4
1 2 3 4 5 6 7 8 9 1011121314
0.54
0.62
0.52
0.61
0.5
0.6
0.48
0.59 1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
Fig. 11. Boxplots of HV (or IGD) distribution of LPICEA-g with N = 200 and different k for 4-objective WFG problems.
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R. Wang et al. / Information Sciences 258 (2014) 29–53
4− objective WFG tests: N=400 WFG2− 4 0.95
−3
x 10
WFG3− 4
3.2 3
0.76
0.725
0.85
0.71 0.73
2.2 1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
WFG6− 4
0.72 0.71 0.7
WFG7− 4
1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
0.72
WFG8− 4
1 2 3 4 5 6 7 8 9 1011121314
0.7
0.7
0.77
0.68
0.76
0.66
0.66
0.64
0.65
0.68 0.67
0.64 0.63
0.6 1 2 3 4 5 6 7 8 9 1011121314
WFG9− 4
0.69
0.62
0.74
0.68
0.705
0.78
0.75
0.69
0.715
0.74
2.4
0.8
0.72
0.75
2.6
WFG5− 4 0.73
2.8
0.9
0.73
WFG4− 4 0.77
0.62 1 2 3 4 5 6 7 8 9 1011121314
1 2 3 4 5 6 7 8 9 1011121314
Fig. 12. Boxplots of HV (or IGD) distribution of LPICEA-g with N = 400 and different k for 4-objective WFG problems.
Overall, algorithms that apply both local genetic-variation and local environment-selection perform better than algorithms apply local genetic-variation or local environment-selection alone. Although only comparison results on PICEA-g are provided above, similar experiments were conducted on NSGA-II, MSOPS and HypE, with similar outcomes. Results can be downloaded at [56].
6.3. The influence of parameter k In this study, k = 2 and k = 4 are used. That is, the entire population is partitioned into 2 and 4 sub-populations for 2- and 4-objective problems, respectively. With such setting, LMOEAs perform better than MOEAs. However, it is not known whether these settings are the best choices for are k. Therefore, the influence on LMOEAs of the choice of number of sub-populations is studied. Again, results for PICEA-g are presented here. Similar results are obtained for the other three algorithms, which can be downloaded at [56]. N 100 function evaluations are taken as the stopping criterion. Similarly, WFG benchmarks with 2 and 4 objectives are used as test problems. LPICEA-g is subjected to 50 runs on each test problem for different values of k. Table 13 presents the settings of k. Figs. 9–12 use box plots to illustrate the distribution of HV (or IGD for WFG3) metrics (Y-label) for LPICEA-g implemented with different values of k (X-label). The implementation of PICEA-g is represented by k = 1. Moreover, in order to know whether the population size influences the settings of k, the analysis is also performed on LPICEA-g with N = 200 and N = 400. Figs. 9 and 10 show the results on 2-objective problems with N = 200 and N = 400, respectively. Figs. 11 and 12 show the results on 4-objective problems with N = 200 and N = 400, respectively. (1) From the results (N = 200) shown in Fig. 9, we find that k = 2 (used in the comparative study) is not the optimum setting for solving 2-objective problems. Better results can be obtained using k = 4. Again, from Fig. 11, it can be seen that k = 4 is not the best setting, however, this setting produces relatively good results for most of the problems. (2) Comparing the results obtained by using different N, we find that, given a larger population size, the optimum value of k increases. For example, k = 3 is the optimum setting for WFG7-2 when N = 200 (See Fig. 9), however, the optimum setting is k = 4 for WFG7-2 when N = 400 (see Fig. 10). Moreover, a larger range of k values is available for producing better results (compared with the results obtained by k = 1), when a larger population size is used. (3) The optimum setting for k is also influenced by the problem dimension, M. For example, when N = 400, k = 5 is the optimum setting for WFG4-2 (see Fig. 10), however, the optimum setting of k for WFG4-4 is 8 (see Fig. 12).
R. Wang et al. / Information Sciences 258 (2014) 29–53
51
Overall, as illustrated in Figs. 9–12, there is an optimum setting of k for most of the problems. We suggest that applying a value of k in the interval [2,2M] will produce good results. Although the optimum value of k may not lie within this interval, it provides a guide for selecting an appropriate value of k. In addition, it is observed that local operations have no significant positive effect on some problems, especially for WFG2. The reason for this may be that the partition method used in this study is not appropriate. Also for the WFG2 problem, another reason may be that a disconnected Pareto front weakens the effect of local operations. This issue deserves further investigation.
7. Conclusions This study proposes a local operation based framework for the design of MOEAs. In this framework, evolutionary operators are executed on sets of sub-populations, separately. The effect of this framework is systematically studied by comparing four best-in-class algorithms with their modified versions derived from this local framework. A local operation based elitist framework is proposed for designing MOEAs. A specific implementation of the new framework is provided. Specifically, a clustering technique is applied to partition the evolutionary population into a pre-defined number (k) of sub-populations. Then the evolutionary operations genetic-variation and environment-selection are executed separately on each sub-population. MOEAs (NSGA-II, MSOPS, HypE and PICEA-g) within the general elitist framework have been compared with their corresponding LMOEAs derived from this new framework. The experimental results demonstrate that this new framework works effectively on most of the test problems, and unlike other existing local operations based methods, this framework suits for different types of MOEAs. Therefore, this local operation based elitist framework could be considered for designing future multi-objective evolutionary algorithms. The main findings are as follows. (1) The performance (convergence, diversity or both) of MOEAs can be improved by applying MOEAs within the local operation based framework. (2) This framework works more effectively on MOPs with a continuous Pareto front than MOPs with disconnected Pareto fronts, such as the WFG2 problems. (3) Applying local genetic-variation and local environment-selection jointly to MOEAs is better than applying one of these operations on its own. Comparing the two local operations, the effect of local genetic-variation is more significant. (4) The performance of LMOEAs is varies with different settings of k. A good setting of k is impacted by both the problem dimension and population size. The setting of k would be more flexible if a larger population is used. The main limitation of this study is that its findings are based on real-parameter function optimisation problems. It is also important to assess the effect of this local framework on other problem types, e.g. multi-objective combinatorial problems. With respect to further research, first, it would be valuable to understand how to set the appropriate number, k, of subpopulations. k might be adjusted either subject to a pre-defined rule or self-adaptively. Second, approaches (other than the use of clustering methods) for partitioning the population should be investigated. Thirdly, it would be valuable to investigate the effect of this local framework on multi-objective combinatorial problems.
Acknowledgments The work of Rui Wang was supported by the China Scholarship Council and the National Science Foundation of China (No. 70971132). Also, the authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the manuscript.
Appendix A. Hierarchical clustering There are many clustering algorithms such as connectivity based clustering, centroid based clustering, distribution based clustering, and density based clustering [43]. Hierarchical clustering [40] belongs to the connectivity based clustering, which seeks to build a hierarchy of clusters. The core idea is that objects being more related to nearby objects than to objects farther away. Thus, the clusters are formed by connecting objects according to their distances. Note that instead of giving a single partitioning of the data set, the algorithms provide an extensive hierarchy of clusters that merge with each other at certain distances. At different distances, different clusters will be formed, which is often described by a dendrogram. As for the computation of distances, there are many choices such as the Euclidean distance, Squared Euclidean distance, Chebyshev distance, Mahalanobia distance, cosine similarity and so on. Apart from this, we also need to define the linkage criterion. Several choices are available such as single-linkage clustering (the nearest distance), complete linkage clustering (the furthest distances), average-linkage clustering (the un-weighted average distance) and ward-linkage clustering (the inner squared distance). More details can be found in [15].
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