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The Computer Journal
, Volume Advance Article (7) – Sep 18, 2017

23 pages

/lp/ou_press/reference-inspired-many-objective-evolutionary-algorithm-based-on-U4jm4jI0Ul

- Publisher
- Oxford University Press
- Copyright
- © The British Computer Society 2017. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
- ISSN
- 0010-4620
- eISSN
- 1460-2067
- D.O.I.
- 10.1093/comjnl/bxx077
- Publisher site
- See Article on Publisher Site

Abstract Keeping balance between convergence and diversity for many-objective optimisation problems (having four or more objectives) is a very difficult task as revealed in existing research in multiobjective evolutionary optimisation. In this paper, we propose a reference-inspired multiobjective evolutionary algorithm for many-objective optimisation. The main idea is (1) to summarise information inspired by a set of randomly generated reference points in the objective space to strengthen the selection pressure towards the Pareto front; and (2) to decompose the objective space into subregions for diversity management and recombination. We showed that the mutual relationship between a population of solution and the reference points provides not only a new dominance relation to producing fine selection pressure but also a balanced convergence-diversity information that is able to adapt search dynamics. The partition of the objective space into several subregions is able to preserve the Pareto front’s diversity. Moreover, a restricted stable match strategy is proposed to choose appropriate parent solutions from solution sets constructed at the subregions for high-quality offspring generation. Controlled experiments conducted on commonly used benchmark test suites have shown the effectiveness and competitiveness of the proposed algorithm compared with several state-of-the-art many-objective evolutionary algorithms. 1. INTRODUCTION In real world, a decision maker usually needs to consider optimising multiple objectives, e.g. in optimal design [1], economics [2] and engineering [3]. This is often referred to as optimisation for multiobjective problems (MOPs). Mathematically, a box-constrained1 MOP can be stated as minF(x)=(f1(x),…,fm(x))s.t.x∈Ω⊂Rn, (1) where Ω=∏i=1n[ai,bi] is the decision (variable) space, x=(x1,…,xn)∈Ω denotes candidate solution; and F:Ω→Rm constitutes m objectives. A solution F1=(f11,…,fm1) is said to dominate another F2=(f12,…,fm2) (denoted as F1⪯F2) iff fj1≤fj2 for all j∈{1,…,m} and there exists at least one index i such that fi1 is strictly less than fi2. A solution F* is Pareto optimal to (1) if there is no other F such that F⪯F*. The set of optimal solutions in Ω is called the Pareto set (PS). The image of PS is called the Pareto Front (PF). Without any preference information from a decision maker, ideally we wish to find an approximation set of objective vectors that are closest to the PF (i.e. convergence) and are most widely spread over the PF (i.e. diversity). These two goals are usually conflicting with each other [4]. Over the last two decades, evolutionary algorithms (EAs) have been found to be very suitable for solving MOPs in the sense that EAs can find an approximated PF in a single run. EAs have been proven to be very successful for problems with two and three objectives [5]. Existing multiobjective EAs (MOEAs) can be classified into three categories which are based on Pareto dominance relation, performance metric and decomposition [6]. Dominance-based MOEAs, such as NSGA-II [7], SPEA2 [8], PESA-II [9] and others, use a Pareto dominance relationship for fitness assignment. Performance metric-based MOEAs, such as HyPE [10], R2-IBEA [11] and others, employ some performance metrics (e.g. hypervolume) to guide the selection of solutions. Decomposition-based MOEAs, such as MOGLS [12], MSOPS [13], MOEA/D [14, 15], MOEA/D-M2M [16] and others, decompose a MOP into a set of single objective or multiobjective optimisation subproblems. These MOEAs attempt to optimise subproblems in a collaborative manner, and select solutions in terms of the objective values of the subproblems. In recent years, the development of MOEAs for many-objective optimisation problems (i.e. problems involving more than four objectives) have drawn much attention [17] as there often exists in many real-world applications, such as engineering design [18], air traffic control [19] and water supply portfolio planning [20]. The performance of MOEAs on two- and three-objective optimisation problems has been found deteriorated significantly on many-objective optimisation problems (MaOPs) [21–23]. It is thus of absolute necessity to develop effective optimisers for MaOPs. Due to the increasing number of objectives, it becomes very difficult for MOEAs to find an approximate PF that can well balance convergence and diversity. As discussed in [24, 25], there are generally six difficulties brought by many objectives. Except for the challenges of representation and visualisation of the PF, the primary four challenges on designing effective MOEAs are as follows: The proportion of non-dominated objective vectors in a population increases significantly along with the increased number of objectives [26]. The comparability between any two high-dimensional objective vectors becomes difficult, which lead to severe loss of selection pressure. As a result, Pareto dominance-based MOEAs cannot drive the search to the PF effectively. The conflicting between convergence and diversity becomes aggravated. When the individual solutions become almost incomparable to each other, diversity management schemes take control of the evolutionary process. As revealed in [4], present popular diversity management schemes, such as crowding distance [7], k th nearest distance [8], tend to select dominance resistance solutions. As a result, the obtained approximation set might be well distributed but be with a deteriorated convergence to the true PF. The efficiency of reproduction operators will be seriously weakened. The reason is that the population size cannot be very large in consideration of computational efficiency. In a high-dimensional space, it is highly likely that solutions are far away from each other. As a result, offspring generated will also be very far away from its parents. This brings great challenges to all categories of MOEAs. It becomes computationally expensive to evaluate the performance metrics due to the increased size of the objective space. As shown in [27], the computational cost required for the calculation of hypervolume increases exponentially. This will surely impede the performance of indicator-based MOEAs. Remarkable efforts have been devoted to addressing these challenges. First, one may consider to reduce the dimension and/or size of the objective space. It is possible since in practice although many objectives are to be considered, it often degenerates to a low-dimensional PF due to the existence of redundant objectives. Several efforts have been carried out on objective reduction, e.g. [28–31]. However, the redundant objectives assumption might not hold for some MaOPs. The applicability of these approaches is thus limited. On the other hand, even if there are redundant objectives, the number of active objectives might still be large. Alternatively, Jaimes et al. [32] proposed to partition the objective space using conflicting information gathered from the population. He et al. [33] proposed to first search for target boundary points to construct a constrained new objective space, and then employ a diversity improvement scheme to distribute solutions over the boundary. As discussed previously, a fundamental challenge is on the loss of selection pressure due to the failure of Pareto-dominance. A straightforward way is to improve the Pareto-dominance relation to accommodate a large number of objectives. A number of modified Pareto dominance has been proposed to increase the selection pressure toward the PF, such as α-dominance [34], ϵ-dominance [35], cone ϵ-dominance [36], grid-dominance [37], adaptive ε-ranking [38], preference order ranking [39], k-optimality [40], fuzzy-based Pareto optimality [41], and others. These new dominance relationships have gained reasonable success in solving MaOPs. The other way is to improve the diversity management scheme, or use a hierarchical fitness assignment (ranking) scheme. Adra et al. [42] proposed to activate/deactivate the diversity management scheme according to the degree of spread and crowding in the population, and to adaptively control the mutation magnitude of an individual solution. Li et al. [43] developed a shift-based density estimation strategy to cover both the diversity and convergence information of individuals. Wang et al. [44] proposed a two-archived evolutionary algorithm for many-objective optimisation problems in which the dominance-based and indicator-based approaches are combined. Li et al. [45] proposed to convert a multiobjective problem into a bi-goal optimisation problem considering the conflict between convergence and diversity. Third, efforts have been made to reduce the computational cost on calculating the performance indicator. Hypervolume-based MOEAs are probably the most successful of this type. Current improvement includes a fast calculation of the hypervolume contribution of a solution [46]; a quick way to determine which solutions contribute the least hypervolume to a front [47]; a fast search algorithm that uses Monte Carlo simulation to approximate the exact hypervolume [48, 10]. Other indicator-based MOEAs, such as distance-based (e.g. MyO-DEMR [49] based on IGD indicator, AGE using α-indicator [50], DDE incorporating δp indicator [51], ε-indicator based EA [52], etc.) and R2 based (e.g. R2MODE [53] and MOMBI [54], etc.), have also been developed for MaOPs and have achieved certain success. It is also worth mentioning MOEAs that are based on a set of predefined reference points in solving MaOPs. A set of reference points provides an external inspiration to measure diversity; and can be used to address the first two challenges. Lohn et al. [55] proposed to co-evolve a family of target vectors with a set of reference points to improve diversity across the PF, and provided a fitness assignment scheme where candidate solutions gain fitness by sharing among solutions that meet a set of target vectors. Wang et al. [56] also proposed to co-evolve a population of candidate objective vectors and a randomly generated reference set, where the candidate objective vectors and the reference set are evaluated against each other to provide comparability among the objective vectors. The proposed algorithm demonstrated somewhat promising performance for MaOPs. Figueira et al. [57] developed a two-stage parallel multi-reference point approach which connects the generation of reference points and the solving for each reference point in every processor. The two-archive algorithm proposed by Kata et al. [58] constructs the reference set using historical or current solutions in the convergence archive and diversity archive. TC-SEA developed by Moen et al. [59] uses a similar way to construct the reference set, while the principle of selection criteria is based on the Manhattan distance. The decomposition-based framework provides an efficient way to balance convergence and diversity. It can keep the selection pressure toward the PF while maintain the population diversity through the construction of uniformly distributed weight vectors and through selection based on scalarising aggregation functions. The convergence to the PF can be controlled by minimising the aggregation function at each subproblem; while the diversity can be measured by minimising the distance to the weight vectors. A number of MOEA/D-based algorithms have been developed and shown great success for bi- or three-objective problems. Their applicability and effectiveness on MaOPs has not yet been fully validated [60]. Some MOEAs based on generalised decomposition have been proposed, such as MAEA-gD [61], ranking-based [62], MDFA [63] and UMOEA/D [64]. Note that the difference among reference points, reference vectors and weight vectors is rather subtle.2 Recently, algorithms that incorporate these two concepts have achieved great success for MaOPs. Asafuddoula et al. [65] adopted the normal boundary intersection method to generate a set of uniformly distributed reference points in a hyperplane with unit intercepts; to construct reference directions based on these points through corner-sort [30]; and to use a simple preemptive distance comparison to balance diversity and convergence. Li et al. [25] used a set of uniformly distributed weight vectors to define not only subproblems but also subregions. Alternatively, Pareto-dominance and indicator/decomposition-based search can either be combined together to preserve diversity and promote convergence simultaneously; or the two criteria can be used to lead the population to co-evolve. For examples, Yuan et al. [23] proposed a similar approach in terms of reference-points generation and adaptive normalisation as in [24, 25], while the diversity preservation is realised by a proposed niche-preservation operator and θ-dominance. Deb et al. [24] used a set of reference points to partition the search space. In each subspace, Pareto-dominance is applied to rank individual solutions. Cheng et al. [66] proposed to use reference vectors to partition the search space and to guide evolutionary algorithm to preferred subset of the PF. Motivated by the success of the approaches that combine weight and reference points for MaOPs, we propose a reference-inspired multiobjective evolutionary algorithm (RIEA) for MaOPs. The major components of the developed algorithm can be summarised as follows: A new fitness assignment scheme is developed based on a set of randomly generated reference points. Under this scheme, at each generation, convergence and diversity information is collected through the comparison between the current population of solutions and the reference points. These information is then aggregated for the ranking of the population of solutions. A set of uniformly distributed reference vectors are generated and used to define subregions. Subregions are then employed to account for diversity management and recombination. First, each solution is assigned to only one subregion. Second, offspring are to be generated by controlling the selection of parent solutions from the subregions. This is important for the generation of offspring that promotes diversity among the PF. A restricted mating selection mechanism is proposed to select parent solutions for recombination. At each subregion, a convergence elite set and a diversity elite set are constructed based on the convergence and diversity information collected. Parent solutions are selected from the two sets. This accounts for the creation of high-quality offspring concerning both convergence and diversity, and thus is important for search speed/efficiency. The remainder of this paper is organised as follows. In Section 2, the proposed fitness assignment scheme is described. Section 3 presents the proposed algorithm. The algorithm settings, test problems and performance metrics used for the performance study are presented in Section 4. Experimental studies are presented afterwards including the comparison with some well-known MOEAs for MaOPs, the sensitivities to the algorithmic parameters and the investigation to some algorithmic components. Section 5 concludes the paper. 2. FITNESS ASSIGNMENT Most existing fitness assignment methods based on Pareto dominance assign fitness to individual objective vectors (solutions) by competing among these solutions. MOEAs based on reference points, such as NSGA-III [24] and MOEA/DD [15], rank individuals by associating these individuals to reference points and/or directions. These reference points are usually generated to be uniformly distributed on a (m−1)-dimensional unit simplex to ensure a uniform spread over the PF. In this section, we describe a new fitness assignment method based on a set of randomly generated reference points in the original m-dimensional objective space. In the sequel, we use Nr and Np to represent the set of reference points and the population of solutions, respectively. Our idea is that the ranking of the solutions in Np can be inspired by comparing them with the reference points in Nr. The relationship between the reference points and a solution in Np can be used to measure the closeness of the solution to the PF and its diversity within Np. In the following, we define a function L:Nr→Z+ to represent how many solutions in the population set Np dominate a reference point r∈Nr. L(r)=1 indicates that there is only one solution that dominates r. By this way, we can cluster the reference points according to their L values, which can result in several layers. The layers of the reference points provide us some useful information on the dominance relationship and convergence-diversity. We show an example in Fig. 1 to demonstrate our idea. In the figure, there are seven solutions ( Np={S1,…,S7}) in a bi-objective space filled with a set of 35 reference points (i.e. Nr) which are randomly generated inside the (hyper-)rectangle defined as ∏j=12[miniSij,maxiSij], (2) where Si=(Si1,Si2). In the figure, the reference points are clustered into five layers based on their L values. Further, we can see that there are four solutions ( S1, S2, S4, S5) that dominate some reference points in the first layer; while the other three solutions S3,S6 and S7 cannot dominate these reference points (they lie outside the first layer). It can also be observed that the four solutions are incomparable, while the other three solutions are dominated. From this demo, we see that reference points can indeed help with the comparability (or essentially ranking) among solutions. Figure 1. View largeDownload slide Illustration of the solutions’ L values inspired by a set of randomly generated reference points. In (a), Si,1≤i≤7 are the objective vectors, the filled pink dots are the randomly generated reference points. The reference points are clustered into 5 layers with different L value. The layer with L=k includes the reference points that are dominated by k solutions. In (b), S1 is moved to the lower right corner. The effect of this movement is shown in this plot. Figure 1. View largeDownload slide Illustration of the solutions’ L values inspired by a set of randomly generated reference points. In (a), Si,1≤i≤7 are the objective vectors, the filled pink dots are the randomly generated reference points. The reference points are clustered into 5 layers with different L value. The layer with L=k includes the reference points that are dominated by k solutions. In (b), S1 is moved to the lower right corner. The effect of this movement is shown in this plot. Further, we define a function D:Np→{0,1} to characterise the non-dominance of a solution in Np. If a solution s is non-dominated, we assign its D value as one, otherwise zero. The reason that we set D=1 for non-dominated solution will be described later. Before proceeding, we need to pay attention to a special case. Consider a solution which has the best performance in one dimension but the worst performance in other dimensions (i.e. the so-called dominance resistant solutions (DRS) [34]), normally it should be non-dominated. However, in our procedure, for such a solution, there will be no reference points that can be dominated by it since the reference points are all inside the hyper-rectangle which is partly determined by this solution (cf. Eq. (2)). For such solutions, we always set their D values to one. Moreover, information about a solution’s convergence can also be inspired according to the number of reference points dominated by this solution. It is intuitive that more reference points dominated by this solution, the closer it approximates to the PF. Further, information regarding a solution’s diversity can also be implied by the reference points dominated by it. Basically, if a reference point’s L value is large, it gives us hardly any information. For example, we cannot gain any useful information from the reference points in layer L=5 as shown in Fig. 1 since these points are dominated by all solutions. On the other hand, the less the L value, the more information we can gain. We thus propose to define a function V:Np→(0,1) to indicate the convergence-diversity information. That is, for a solution s∈Np, V(s)=1∣Nr∣∑r∈NrMrTr, (3) where Mr is the number of reference points dominated by s, and Tr is the number of solutions in Np that dominate r; ∣Nr∣ is the total number of reference points in Nr. Note that if we define Li={r:L(r)=i} as the i th layer, then we can rewrite Eq. (3) as follows: V(s)=1∣Nr∣∑r∈LiMri. (4) In Fig. 1(a), the dominated reference points are clustered into layers. From the figure, we can count that S1 dominates a total of 17 reference points in the five layers, while there are 14 reference points dominated by S2. We would expect that V(S1)>V(S2) since S1 dominates more reference points than S2 does. According to Eq. (4), we can compute the value of V(S1) and V(S2) as follows: V(S1)=1354·1+4·12+4·13+3·14+2·15≈0.24V(S2)=1351·1+3·12+5·13+3·14+2·15≈0.15. In a similar way, the values of V(S3) to V(S5) can be computed as 0.07, 0.12 and 0.07, respectively. From the figure, we see that the higher the V value, the closer the corresponding solution to the PF. This actually gives some indication information on convergence. Moreover, the V value also indicates some sort of diversity information. To illustrate clearly, we move S1 to a new position as shown in Fig. 1(b). The movement of S1 will decrease the diversity of S3; therefore, V(S3) should be increased. The re-calculated V(S3) value indeed rises up from 0.07 to 0.12. In general, the basic idea of the developed fitness assignment is to use a set of reference points to differentiate a solution’s quality within a population. We propose to aggregate the non-domination and convergence-diversity information for ranking solutions in a population. That is, for each s∈Np, we define R(s)=D(s)+V(s) (5) The value R will be used as the basis for mating selection and environmental selection.3 The immediate advantage of the proposed fitness assignment scheme is that the ranking of solutions can be automatically adapted to different evolution stages. At early stage, the D value dominates Eq. (5) since the non-dominated solutions have higher chances to survive than the dominated solutions. Note that the V value is always less than 1.0 since the total number of reference points dominated by the solutions is always smaller than the number of reference points as some reference points cannot be dominated. At later stage, solutions are almost non-dominated to each other (this means that all solutions will have D=1), which is especially the case in MaOPs as stated in the first challenge. Therefore, the V value will dominate the ranking based on Eq. (5). In this case, solutions will be differentiated based on their respect convergence and diversity information. We have discussed the usefulness of reference points on ranking solutions. The next question is about its scalability. Figure 2 demonstrates the effectiveness of the proposed fitness assignment scheme on scalability. To create the figure, we first randomly generate a set of 100 solutions in the m-dimensional ( m=2,…,10) objective space. We sort the 100 solutions into equivalent classes (or layers)4 by using the non-dominated sorting developed by Deb et al. [7] and by using the described fitness assignment, respectively. We run the experiment 500 times with different number of reference points. The average number of equivalent classes is shown in the figure. From the figure, it is evident that the proposed scheme can provide more comparability among solutions than the non-dominated sorting in the sense that more equivalent classes are differentiated. Further, we see that more reference points lead to improved comparability (or more equivalent classes). However, more reference points do not always improve the comparability as seen in the case when the number of reference points are set as 500, m×100 and m×200. In this paper, we choose the number of reference points to be m×100. Figure 2. View largeDownload slide The demonstration of the proposed fitness assignment in terms of the number of equivalent classes (layers). The x-axis shows the number of objectives, while in the y-axis, the average number of equivalent classes (layers) over 500 experiments obtained by using non-dominated sorting approach and the developed approach on 100 randomly generated solutions in the corresponding objective space is shown. Figure 2. View largeDownload slide The demonstration of the proposed fitness assignment in terms of the number of equivalent classes (layers). The x-axis shows the number of objectives, while in the y-axis, the average number of equivalent classes (layers) over 500 experiments obtained by using non-dominated sorting approach and the developed approach on 100 randomly generated solutions in the corresponding objective space is shown. 3. THE ALGORITHM The framework of the proposed reference-inspired multiobjective evolutionary algorithm (in short, RIEA) is summarised in Algorithm 1. RIEA maintains a set of N individuals S={x1,…,xN}, and their corresponding objective vectors ={F1,…,FN}. An initialisation procedure first generates N initial solutions, K reference directions (grouped in set V), M reference points (grouped in set R) (line 2). The neighbourhood index set B for each solutions is identified in the initialisation procedure and remains fixed during the evolution procedure. Within the main while-loop, a parameter δ is used to decide where to select parent solutions (lines 6–10) for offspring generation. An offspring is then generated by the variation of these parent solutions (line 11) taking two elite sets C¯ and D¯ into consideration (in the first generation, no elite sets are required). New population and elite sets are updated (line 12), by employing an elite-preserving mechanism. We will discuss the implementation details in the following subsections. Algorithm 1 The framework of RIEA. Require: a population size N Ensure: population 1: C←∅,D←∅,t←0; 2: [,V,R,B]←Initialisation(); 3: while termination not satisfied do 4: ′←∅; 5: for i←1 to Ndo 6: if rand()>δ or t=0then 7: B¯←{1,…,K},C¯←∅;D¯←∅; 8: else 9: B¯←Bi;C¯←Ci;D¯←Di; 10: end if 11: p←Reproduction (xi,B¯,C¯,D¯); 12: ′←′⋃F(p); 13: end for 14: ¯←⋃′; 15: [,{Ci,Di}i=1K]←Update_Population (¯,R,V) 16: t←t+1 17: end while 18: return Require: a population size N Ensure: population 1: C←∅,D←∅,t←0; 2: [,V,R,B]←Initialisation(); 3: while termination not satisfied do 4: ′←∅; 5: for i←1 to Ndo 6: if rand()>δ or t=0then 7: B¯←{1,…,K},C¯←∅;D¯←∅; 8: else 9: B¯←Bi;C¯←Ci;D¯←Di; 10: end if 11: p←Reproduction (xi,B¯,C¯,D¯); 12: ′←′⋃F(p); 13: end for 14: ¯←⋃′; 15: [,{Ci,Di}i=1K]←Update_Population (¯,R,V) 16: t←t+1 17: end while 18: return Algorithm 1 The framework of RIEA. Require: a population size N Ensure: population 1: C←∅,D←∅,t←0; 2: [,V,R,B]←Initialisation(); 3: while termination not satisfied do 4: ′←∅; 5: for i←1 to Ndo 6: if rand()>δ or t=0then 7: B¯←{1,…,K},C¯←∅;D¯←∅; 8: else 9: B¯←Bi;C¯←Ci;D¯←Di; 10: end if 11: p←Reproduction (xi,B¯,C¯,D¯); 12: ′←′⋃F(p); 13: end for 14: ¯←⋃′; 15: [,{Ci,Di}i=1K]←Update_Population (¯,R,V) 16: t←t+1 17: end while 18: return Require: a population size N Ensure: population 1: C←∅,D←∅,t←0; 2: [,V,R,B]←Initialisation(); 3: while termination not satisfied do 4: ′←∅; 5: for i←1 to Ndo 6: if rand()>δ or t=0then 7: B¯←{1,…,K},C¯←∅;D¯←∅; 8: else 9: B¯←Bi;C¯←Ci;D¯←Di; 10: end if 11: p←Reproduction (xi,B¯,C¯,D¯); 12: ′←′⋃F(p); 13: end for 14: ¯←⋃′; 15: [,{Ci,Di}i=1K]←Update_Population (¯,R,V) 16: t←t+1 17: end while 18: return 3.1. Initialisation and decomposition to subregions The initialisation procedure is presented in Algorithm 2. Individuals are to be generated within the search space by randomly sampling from Ω (line 1). The individuals are evaluated and their objective values are stored in (line 2). Next, we generate a set of randomly sampled points within [0,1]m (line 3). The generation of the reference vectors follows the approach developed in [16]. In this approach, reference vectors vi=(v1i,…,vmi),1≤i≤K are generated on a unit hypersphere: vki∈0H,1H,…,HH,s.t.∑i=1mvki=1, (6) where H is a positive integer. Note that for different H and m, the number of reference vectors K generated by the method is K=H+m−1m−1. Algorithm 2 The initialisation procedure (Initialisation). Ensure: ,V,R,B 1: Generate an initial population of individuals, where xji∈[aj,bj],1≤j≤N,1≤i≤n randomly. 2: Evaluate these individuals: Fi=F(xi),1≤i≤N 3: Randomly generate a set of M(=m×100) reference points, denoted by R, in [0,1]m. 4: Randomly generate a set of K reference vectors, denoted by V={v1,…,vK}, in a (m−1)-dimensional unit simplex. 5: For each i∈{1,…,N}, set Bi={i1,…,iT}, where vi1,…,viT are T closest reference vectors to vi 6: return ,V,R,B. Ensure: ,V,R,B 1: Generate an initial population of individuals, where xji∈[aj,bj],1≤j≤N,1≤i≤n randomly. 2: Evaluate these individuals: Fi=F(xi),1≤i≤N 3: Randomly generate a set of M(=m×100) reference points, denoted by R, in [0,1]m. 4: Randomly generate a set of K reference vectors, denoted by V={v1,…,vK}, in a (m−1)-dimensional unit simplex. 5: For each i∈{1,…,N}, set Bi={i1,…,iT}, where vi1,…,viT are T closest reference vectors to vi 6: return ,V,R,B. Algorithm 2 The initialisation procedure (Initialisation). Ensure: ,V,R,B 1: Generate an initial population of individuals, where xji∈[aj,bj],1≤j≤N,1≤i≤n randomly. 2: Evaluate these individuals: Fi=F(xi),1≤i≤N 3: Randomly generate a set of M(=m×100) reference points, denoted by R, in [0,1]m. 4: Randomly generate a set of K reference vectors, denoted by V={v1,…,vK}, in a (m−1)-dimensional unit simplex. 5: For each i∈{1,…,N}, set Bi={i1,…,iT}, where vi1,…,viT are T closest reference vectors to vi 6: return ,V,R,B. Ensure: ,V,R,B 1: Generate an initial population of individuals, where xji∈[aj,bj],1≤j≤N,1≤i≤n randomly. 2: Evaluate these individuals: Fi=F(xi),1≤i≤N 3: Randomly generate a set of M(=m×100) reference points, denoted by R, in [0,1]m. 4: Randomly generate a set of K reference vectors, denoted by V={v1,…,vK}, in a (m−1)-dimensional unit simplex. 5: For each i∈{1,…,N}, set Bi={i1,…,iT}, where vi1,…,viT are T closest reference vectors to vi 6: return ,V,R,B. The idea of using reference vectors comes from MOEA/D [15] in which an MOP is decomposed into a set of single-objective problems using these reference vectors. After generation of the reference vectors, we then identify the neighbourhoods of each reference vector (line 5). In this paper, the reference vectors will be used to divide solutions into subregions in the objective space. That is, for each reference vector vi, the subregion, denoted as Λi, can be defined as Λi={F∣⟨F,vi⟩≤⟨F,vj⟩for1≤j≤K,j≠i}, (7) where ⟨F,vi⟩ is the acute angle between F and the reference vector vi. That is, an objective vector F belongs to subregion Λi if it has the least acute angle value. We use Ωi to denote the corresponding solutions in the search space, that is Ωi={x∣F(x)∈Λi}. (8) The neighbourhood index set Bi of vi∈Λi is also considered as the index set of Ωi. The definition of subregion is the same as that in MOEA/DD [25] and MOEA/D-M2M [16]. However, in MOEA/DD, the subregions are used to facilitate local density estimation; while in MOEA/D-M2M, they are used to specify sub-populations for multiobjective subproblems. In this paper, the subregions are used to choosing parent solutions for offspring generation. 3.2. Reproduction procedure In this paper, we use differential evolution (DE) and polynomial mutation [67] for offspring generation. The reproduction procedure is illustrated in Algorithm 3. To generate an offspring, the mutation operator of DE variates an individual xi using two parent individuals xr1 and xr2. A mating control parameter δ∈[0,1] is used to decide where to choose the parent individuals. Specifically, with δ, parent individuals are selected from the neighbourhood of xi (the neighbourhood indices are predefined in Bi); otherwise the whole population is considered as the neighborhood. This means to balance exploration and exploitation. Algorithm 3 The reproduction procedure (Reproduction). Require: x=(x1,…,xn),B¯,C¯,D¯ Ensure: y 1: Randomly select two indexes i1 and i2 from B¯ 2: if C¯≠∅ and D¯≠∅then 3: Randomly select two individuals xr1 from C¯i1 of Λi1 and another individual xr2 from D¯i2 of Λi2; 4: Create a trial individual y=(y1,…,yn) by DE as follows: If rand()>CRg, set yi←xi; else yi←xi+Fg·(xir1−xir2)for1≤i≤n 5: else 6: Randomly select two individuals xr1 and xr2 from Λi1 and Λi2, respectively; 7: Create a trial individual y=(y1,…,yn) by DE as follows: If rand()>CRl, set yi←xi; else yi←xi+Fl·(xir1−xir2)for1≤i≤n 8: end if 9: Mutate y and repair it if necessary; 10: return y Require: x=(x1,…,xn),B¯,C¯,D¯ Ensure: y 1: Randomly select two indexes i1 and i2 from B¯ 2: if C¯≠∅ and D¯≠∅then 3: Randomly select two individuals xr1 from C¯i1 of Λi1 and another individual xr2 from D¯i2 of Λi2; 4: Create a trial individual y=(y1,…,yn) by DE as follows: If rand()>CRg, set yi←xi; else yi←xi+Fg·(xir1−xir2)for1≤i≤n 5: else 6: Randomly select two individuals xr1 and xr2 from Λi1 and Λi2, respectively; 7: Create a trial individual y=(y1,…,yn) by DE as follows: If rand()>CRl, set yi←xi; else yi←xi+Fl·(xir1−xir2)for1≤i≤n 8: end if 9: Mutate y and repair it if necessary; 10: return y Algorithm 3 The reproduction procedure (Reproduction). Require: x=(x1,…,xn),B¯,C¯,D¯ Ensure: y 1: Randomly select two indexes i1 and i2 from B¯ 2: if C¯≠∅ and D¯≠∅then 3: Randomly select two individuals xr1 from C¯i1 of Λi1 and another individual xr2 from D¯i2 of Λi2; 4: Create a trial individual y=(y1,…,yn) by DE as follows: If rand()>CRg, set yi←xi; else yi←xi+Fg·(xir1−xir2)for1≤i≤n 5: else 6: Randomly select two individuals xr1 and xr2 from Λi1 and Λi2, respectively; 7: Create a trial individual y=(y1,…,yn) by DE as follows: If rand()>CRl, set yi←xi; else yi←xi+Fl·(xir1−xir2)for1≤i≤n 8: end if 9: Mutate y and repair it if necessary; 10: return y Require: x=(x1,…,xn),B¯,C¯,D¯ Ensure: y 1: Randomly select two indexes i1 and i2 from B¯ 2: if C¯≠∅ and D¯≠∅then 3: Randomly select two individuals xr1 from C¯i1 of Λi1 and another individual xr2 from D¯i2 of Λi2; 4: Create a trial individual y=(y1,…,yn) by DE as follows: If rand()>CRg, set yi←xi; else yi←xi+Fg·(xir1−xir2)for1≤i≤n 5: else 6: Randomly select two individuals xr1 and xr2 from Λi1 and Λi2, respectively; 7: Create a trial individual y=(y1,…,yn) by DE as follows: If rand()>CRl, set yi←xi; else yi←xi+Fl·(xir1−xir2)for1≤i≤n 8: end if 9: Mutate y and repair it if necessary; 10: return y Moreover, to address the second challenge (i.e. the inefficiency of reproduction operators), we proposed a selection mechanism similar to the restricted mating selection mechanism proposed in [68] for choosing parent solutions from xi’s neighbourhood. That is, we first randomly select two subregions from the neighbourhood index set. Then the parent individuals are chosen from the convergence elite set and diversity elite set that are defined in the two selected subregions, respectively. How to construct the convergence elite set Ci and diversity elite set Di for each subregion Λi will be presented later. To be more specific, let us assume that xi locates in Ω3 (i.e. F(xi)∈Λ3), and its neighbourhood index set Bi={1,2,4,5}. Suppose Λ1 and Λ2 are two selected subregions for parent selection. One parent individual xr1 will be selected from the convergence set C1 of Λ1. xr2 will be selected from the diversity set D2 of Λ2. As a result, the generated solution will be moving along the direction pointing to xr1. Note xr1 locates in the convergence elite set. This reflects the search efforts in favour of convergence. On the other hand, a population’s diversity can be maintained by choosing xr2 from the diversity elite set D2. The random selection from neighbourhood index would help the search forward along various directions. In our implementation, two combinations of DE parameters are applied. This is to address different search purpose (exploration if the whole population is considered as neighbourhood and exploitation otherwise). 3.3. Environmental selection In our environmental selection procedure, we need to consider two main issues. The first is on how to rank solutions according to their R values (cf. Eq. (3)). The second is on how to construct the convergence elite set and diversity elite set. The pseudo-code of the environmental selection is given in Algorithm 4. In the algorithm, ¯ is a population of solutions combining the old population and the newly generated solutions. L is a parameter that defines the minimum number of solutions kept in a subregion for retaining population diversity. We first normalise the solutions in ¯ and the set of reference points in R to scale up the fitness assignment (line 1). Each Fi∈¯ is normalised as F¯i=(f¯1i,…,f¯mi) with each element j computed as follows: f¯ji=fji−fjminfjmax−fjmin,∀j∈{1,…,m}, (9) where fjmin and fjmax are the minimum and maximum values of ¯ at the j th objective. They will be used to normalise R as well. Algorithm 4 The environmental selection procedure (Update_Population). Require: ¯,R,V Ensure: ,C,D 1: ′←Normalise (¯,R); 2: Λ←Partition (′,V); 3: ←Population_Selection( Λ); 4: [{Ci,Di}i=1K]←Elite_Set_Construction (Λ); 5: return ,{Ci,Di}i=1K Require: ¯,R,V Ensure: ,C,D 1: ′←Normalise (¯,R); 2: Λ←Partition (′,V); 3: ←Population_Selection( Λ); 4: [{Ci,Di}i=1K]←Elite_Set_Construction (Λ); 5: return ,{Ci,Di}i=1K Algorithm 4 The environmental selection procedure (Update_Population). Require: ¯,R,V Ensure: ,C,D 1: ′←Normalise (¯,R); 2: Λ←Partition (′,V); 3: ←Population_Selection( Λ); 4: [{Ci,Di}i=1K]←Elite_Set_Construction (Λ); 5: return ,{Ci,Di}i=1K Require: ¯,R,V Ensure: ,C,D 1: ′←Normalise (¯,R); 2: Λ←Partition (′,V); 3: ←Population_Selection( Λ); 4: [{Ci,Di}i=1K]←Elite_Set_Construction (Λ); 5: return ,{Ci,Di}i=1K Second, we assign the combined population of solutions to subregions (line 2) (see Algorithm 5). During the assignment, we make sure that there are at least L solutions in each subregion. New population is then selected from the subregions (line 3) by selecting N solutions with the largest R (Eq. (5)) values from these subregions (Algorithm 6). Finally, for each subregion Λi, we construct two elite sets Ci and Di which will be used to generate offspring concerning convergence and diversity, respectively (Algorithm 7). In each subregion, we choose a solution from the combined population with the smallest acute angle to the reference vector in this subregion as the diversity elite set. Meanwhile, we select all non-dominated solutions in each subregion as the convergence elite set. If there is no non-dominated solution, solutions that are closest to the reference vector in that subregion will be selected as the convergence elite. Algorithm 5 The partition of solutions into subregions (Partition). Require: ,R,V Ensure: 1: // assign solutions to subregions 2: Set Λi←∅,∀i∈{1,…,K}; 3: for j←1 to Ndo 4: k←argmini⟨F¯j,vi⟩; 5: Λk←Λk⋃Fk; 6: end for 7: for j←1 to Kdo 8: // make sure there are at least Lsolutions in each subregion 9: if the number of solutions in Λj is less than Lthen 10: Set U←∅, S←∅; 11: for i←1 to Tdo 12: k←Bj(i); 13: Compute the R values for s∈Λk, store these values in Uk; 14: U←U⋃Uk and S←S⋃Λk; 15: end for 16: Select an index set I proportionally from U subject to ∣I∣=L−∣Λi∣; 17: Λi←Λi⋃SI; 18: end if 19: end for 20: return Λ Require: ,R,V Ensure: 1: // assign solutions to subregions 2: Set Λi←∅,∀i∈{1,…,K}; 3: for j←1 to Ndo 4: k←argmini⟨F¯j,vi⟩; 5: Λk←Λk⋃Fk; 6: end for 7: for j←1 to Kdo 8: // make sure there are at least Lsolutions in each subregion 9: if the number of solutions in Λj is less than Lthen 10: Set U←∅, S←∅; 11: for i←1 to Tdo 12: k←Bj(i); 13: Compute the R values for s∈Λk, store these values in Uk; 14: U←U⋃Uk and S←S⋃Λk; 15: end for 16: Select an index set I proportionally from U subject to ∣I∣=L−∣Λi∣; 17: Λi←Λi⋃SI; 18: end if 19: end for 20: return Λ Algorithm 5 The partition of solutions into subregions (Partition). Require: ,R,V Ensure: 1: // assign solutions to subregions 2: Set Λi←∅,∀i∈{1,…,K}; 3: for j←1 to Ndo 4: k←argmini⟨F¯j,vi⟩; 5: Λk←Λk⋃Fk; 6: end for 7: for j←1 to Kdo 8: // make sure there are at least Lsolutions in each subregion 9: if the number of solutions in Λj is less than Lthen 10: Set U←∅, S←∅; 11: for i←1 to Tdo 12: k←Bj(i); 13: Compute the R values for s∈Λk, store these values in Uk; 14: U←U⋃Uk and S←S⋃Λk; 15: end for 16: Select an index set I proportionally from U subject to ∣I∣=L−∣Λi∣; 17: Λi←Λi⋃SI; 18: end if 19: end for 20: return Λ Require: ,R,V Ensure: 1: // assign solutions to subregions 2: Set Λi←∅,∀i∈{1,…,K}; 3: for j←1 to Ndo 4: k←argmini⟨F¯j,vi⟩; 5: Λk←Λk⋃Fk; 6: end for 7: for j←1 to Kdo 8: // make sure there are at least Lsolutions in each subregion 9: if the number of solutions in Λj is less than Lthen 10: Set U←∅, S←∅; 11: for i←1 to Tdo 12: k←Bj(i); 13: Compute the R values for s∈Λk, store these values in Uk; 14: U←U⋃Uk and S←S⋃Λk; 15: end for 16: Select an index set I proportionally from U subject to ∣I∣=L−∣Λi∣; 17: Λi←Λi⋃SI; 18: end if 19: end for 20: return Λ Algorithm 6 The selection of solutions to form the new population (Population_Selection). Require: Λ Ensure: 1: Set ←∅; 2: for i←1 to Ndo 3: Randomly generate an integer j∈{1,…,K}; 4: s¯←argmaxs∈ΛjRs; 5: ←⋃p where p∈Ωj and F(p)=s¯; 6: Λj←Λj⧹s¯; 7: end for 8: return Require: Λ Ensure: 1: Set ←∅; 2: for i←1 to Ndo 3: Randomly generate an integer j∈{1,…,K}; 4: s¯←argmaxs∈ΛjRs; 5: ←⋃p where p∈Ωj and F(p)=s¯; 6: Λj←Λj⧹s¯; 7: end for 8: return Algorithm 6 The selection of solutions to form the new population (Population_Selection). Require: Λ Ensure: 1: Set ←∅; 2: for i←1 to Ndo 3: Randomly generate an integer j∈{1,…,K}; 4: s¯←argmaxs∈ΛjRs; 5: ←⋃p where p∈Ωj and F(p)=s¯; 6: Λj←Λj⧹s¯; 7: end for 8: return Require: Λ Ensure: 1: Set ←∅; 2: for i←1 to Ndo 3: Randomly generate an integer j∈{1,…,K}; 4: s¯←argmaxs∈ΛjRs; 5: ←⋃p where p∈Ωj and F(p)=s¯; 6: Λj←Λj⧹s¯; 7: end for 8: return Algorithm 7 The construction of convergence and diversity elite sets (Elite_Set_Construction). Require: ,V Ensure: C,D 1: for i←1 to Kdo 2: // construct the convergence elite set 3: Find the non-dominated solutions in Λi→U; 4: if ∣U∣=0then 5: Find out the nearest non-dominated solutions in the neighbourhood of Λi, i.e. ⋃j∈BiΛj; 6: Add the individuals associated with these solutions to Ci; 7: else 8: Ci←U; 9: end if 10: // construct the diversity elite set 11: s¯←argmins∈Λi⟨s,vi⟩; 12: Di←{p} where p∈Ωi and F(p)=s¯; 13: end for 14: return {Ci,Di}i=1K Require: ,V Ensure: C,D 1: for i←1 to Kdo 2: // construct the convergence elite set 3: Find the non-dominated solutions in Λi→U; 4: if ∣U∣=0then 5: Find out the nearest non-dominated solutions in the neighbourhood of Λi, i.e. ⋃j∈BiΛj; 6: Add the individuals associated with these solutions to Ci; 7: else 8: Ci←U; 9: end if 10: // construct the diversity elite set 11: s¯←argmins∈Λi⟨s,vi⟩; 12: Di←{p} where p∈Ωi and F(p)=s¯; 13: end for 14: return {Ci,Di}i=1K Algorithm 7 The construction of convergence and diversity elite sets (Elite_Set_Construction). Require: ,V Ensure: C,D 1: for i←1 to Kdo 2: // construct the convergence elite set 3: Find the non-dominated solutions in Λi→U; 4: if ∣U∣=0then 5: Find out the nearest non-dominated solutions in the neighbourhood of Λi, i.e. ⋃j∈BiΛj; 6: Add the individuals associated with these solutions to Ci; 7: else 8: Ci←U; 9: end if 10: // construct the diversity elite set 11: s¯←argmins∈Λi⟨s,vi⟩; 12: Di←{p} where p∈Ωi and F(p)=s¯; 13: end for 14: return {Ci,Di}i=1K Require: ,V Ensure: C,D 1: for i←1 to Kdo 2: // construct the convergence elite set 3: Find the non-dominated solutions in Λi→U; 4: if ∣U∣=0then 5: Find out the nearest non-dominated solutions in the neighbourhood of Λi, i.e. ⋃j∈BiΛj; 6: Add the individuals associated with these solutions to Ci; 7: else 8: Ci←U; 9: end if 10: // construct the diversity elite set 11: s¯←argmins∈Λi⟨s,vi⟩; 12: Di←{p} where p∈Ωi and F(p)=s¯; 13: end for 14: return {Ci,Di}i=1K It is worth noting that a solution’s D value is computed through the comparison within the entire population and its V value is computed with respect to the reference points within its subregion. Figure 3 demonstrates why we compute the fitness assignment in this way. In the figure, S1 and S2 are non-dominated to each other in Λ3. If we assign D(S1)=D(S2)=1, it may deteriorate the searching process since S1 is dominated by solutions in Λ2. To address this problem, we sort these solutions based on their R values in each subregion and select one by one from the subregions in turn. Figure 3. View largeDownload slide Illustration of the calculation of the D and V values in various space. S1 and S2 are non-dominated to each other in Λ3 but S1 is dominated by S5. This shows that we need to compute the solutions’ D values in the whole search space. Figure 3. View largeDownload slide Illustration of the calculation of the D and V values in various space. S1 and S2 are non-dominated to each other in Λ3 but S1 is dominated by S5. This shows that we need to compute the solutions’ D values in the whole search space. Algorithm 7 presents the construction of the convergence and diversity set at the subregions. To construct the convergence elite set for a subregion Λi, we first find all the non-dominated solution in that region (line 3). If there is no non-dominated solution, we search non-dominated solutions in its neighbourhood region and put these solutions in its convergence elite set (lines 4–8). To construct the diversity elite set, we locate the solution that is closest to the reference vector vi (line 11); and take this solution as the diversity elite set (line 12). 3.4. Time complexity In one generation of RIEA, the DE reproduction requires a total of O(N) computations. The normalisation of the current population (line 1 in Algorithm 4) requires O(N) computation. To partition the current population into subregions (line 2 in Algorithm 4), the time complexity is O(NK). To perform the fitness assignment (line 3 in Algorithm 4), we need to compare each solution with each reference point. Therefore, the time complexity of the fitness assignment method is O(MN). The construction of the convergence and diversity elite set requires a total of O(NK) computations. In summary, the total time complexity is O((K+M)N). 4. EXPERIMENTAL STUDY We carried out controlled experiments to test the performance of IREA on the DTLZ test suite [69] with 3, 5, 8, 10 and 15 objectives. The DTLZ test suite includes a variety of problems with various challenging characteristics, such as multi-modal, degenerate, non-uniform, disconnected PF and others. 4.1. Performance metrics To assess the performance, we choose two widely used performance metrics: the inverse generational distance (IGD) [70] and Hypervolume (HV) [71]. They are the metric representatives to measure the convergence and diversity of the obtained solutions [72]. The IGD metric is computed as follows: IGD(P,S)=∑i=1∣P∣diq1/q∣P∣, (10) where di is the shortest distance from pi to the PF. di=min∥F(pi)−F(s)∥,s∈S,pi∈P and ∥·∥ is the Euclidean distance. ∣P∣ is the number of elements in P. HV measures the quality of an approximation set to indicate the performance of convergence-diversity HV(S)=VOL(⋃x∈S[f1(x),z1r]×⋯[fm(x),zmr]), (11) where S is the approximation set. zr=(z1r,…,zmr) is the reference point. VOL(·) is the Lebesgue measure. In our experimental studies, to compute HV, we use (1.0,1.0,…,1.0) as the reference point for DTLZ1, for DTLZ2–DTLZ4, the reference point is (2.0,2.0,…,2.0). 4.2. General parameter settings We choose four state-of-the-art multiobjective evolutionary algorithms, including PICEA-g [56], GrEA [37], HypE [10] and MOEA/D [15] for comparison. The parameters of the compared algorithms are set as reported by their authors. All the codes were written in Matlab. We obtained the Matlab codes of the compared algorithms from the authors’ websites. For a fair comparison, each algorithm was executed independently 20 times on each test instance in our machine. To compute the performance metrics, the same Pareto optimal points for IGD and the reference points for HV were used in the compared algorithms. All the compared algorithms terminate at 30,000 function evaluations. The specific parameter settings of our proposed RIEA are summarised as follows: The number of reference points: 100×m. Population size: N=300 for all test instances except m=15 with N=900. K is obtained by choosing H to make it is close to 10m+1. Table 1 lists the H value and the corresponding number of reference vectors. Settings for reproduction operators: the mutation probability pm=1/n and its distribution index is set to be 20. The DE parameters: Fg=0.5,CRg=0.5;Fl=0.2,CRl=0.8. The neighbourhood size: T=5. The probability used to select parent solutions: δ=0.9. The minimum number of individuals in every subregion: L=5. Table 1. The number of reference vectors generated by the NBI approach for different number of objectives. d H K 3 7 36 5 4 70 8 3 120 10 2 55 15 2 120 d H K 3 7 36 5 4 70 8 3 120 10 2 55 15 2 120 View Large Table 1. The number of reference vectors generated by the NBI approach for different number of objectives. d H K 3 7 36 5 4 70 8 3 120 10 2 55 15 2 120 d H K 3 7 36 5 4 70 8 3 120 10 2 55 15 2 120 View Large The parameter settings for the compared algorithms are described as follows. For MOEA/D with DE recombination.5 The mutation probability is pm=1/n and its distribution index ηm=20; the DE recombination parameter: F=0.5,CR=0.5; the neighborhood size: 30; for 3, 5, 8, 10 and 15 dimensions, the population sizes are 105, 126, 120, 220 and 120, respectively. For PICEA-g, the mutation probability pm=1/n and its distribution index ηm=20; the number of preferences used to evaluate candidate solutions is set to m×100; the parameters used in the SBX operator include nc=15.0, the crossover probability pc=0.7, the probability of internal crossover is 0.5; and the population size is set the same as RIEA. For GrEA, the grid division in GrEA is summarised in Table 2. The population size is set to 100. For HYPE, the number of points in Monte Carlo sampling is set to 10,000. Table 2. The settings of the grid division in GrEA. Test instance m=3 m=5 m=8 m=10 m=15 DTLZ1 10 10 10 12 12 DTLZ2 10 9 8 8 10 DTLZ3 10 11 10 10 12 DTLZ4 10 9 8 9 10 Test instance m=3 m=5 m=8 m=10 m=15 DTLZ1 10 10 10 12 12 DTLZ2 10 9 8 8 10 DTLZ3 10 11 10 10 12 DTLZ4 10 9 8 9 10 View Large Table 2. The settings of the grid division in GrEA. Test instance m=3 m=5 m=8 m=10 m=15 DTLZ1 10 10 10 12 12 DTLZ2 10 9 8 8 10 DTLZ3 10 11 10 10 12 DTLZ4 10 9 8 9 10 Test instance m=3 m=5 m=8 m=10 m=15 DTLZ1 10 10 10 12 12 DTLZ2 10 9 8 8 10 DTLZ3 10 11 10 10 12 DTLZ4 10 9 8 9 10 View Large 4.3. Experimental results Table 3 shows the obtained results in terms of IGD. The best, mean and worst metric values are summarised in the tables. Moreover, the ranks of these algorithms on each problem are also presented. In the tables, best results are shaded in grey color. Table 3. Statistical results (best/mean/worst) obtained by RIEA, MOEA/D, PICEA-G, GrEA and HypE in 30 independent runs on DLTZ1 and DTLZ2 in terms of the IGD metric. m RIEA MOEA/D PICEA-G GrEA HypE RIEA MOEA/D PICEA-G GrEA HypE DTLZ1 3 1.658E−03 2.869E−03 3.529E−03 2.76E−02 1.82E+01 DTLZ3 9.467E−03 3.277E−03 1.093E−01 6.77E−02 1.65E+02 2.567E−03 2.904E−03 2.067E−02 3.34E−02 1.97E+01 2.450E−02 4.635E−02 2.335E−01 7.69E−02 1.70E+02 8.238E−03 2.942E−03 6.174E−02 1.35E−01 2.16E+01 6.114E−02 1.406E−01 5.502E−01 4.47E−01 1.76E+02 5 6.635E−03 8.596E−03 1.243E−02 7.37E−02 1.80E+01 1.155E−02 1.155E−02 5.789E−02 5.33E−01 1.83E+02 7.420E−03 8.953E−03 2.781E−02 3.36E−01 2.14E+01 3.112E−02 1.019E−01 1.894E−01 8.30E−01 2.17E+02 8.741E−02 9.504E−03 8.106E−02 4.94E−01 2.36E+01 9.316E−02 4.355E−01 3.935E−01 1.12E+00 2.28E+02 8 1.932E−02 2.073E−02 4.009E−02 1.02E−01 1.03E+01 4.910E−02 4.331E−02 1.173E−01 7.52E−01 2.20E+02 2.411E−02 5.742E−02 5.628E−02 1.20E−01 2.27E+01 7.554E−02 6.094E−01 2.988E−01 1.02E+00 2.70E+02 3.214E−02 3.747E−01 1.363E−01 3.85E−01 2.43E+01 1.270E−01 4.728E+00 8.273E−01 1.23E+00 2.95E+02 10 2.362E−02 2.498E−02 4.431E−02 1.18E−01 1.43E+01 4.898E−02 4.862E−02 1.224E−01 8.66E−01 1.72E+02 2.880E−02 3.779E−02 6.329E−02 1.59E−01 1.69E+01 6.757E−02 4.128E−01 3.737E−01 1.15E+00 2.89E+02 3.272E−02 1.051E−01 1.346E−01 5.11E−01 2.03E+01 1.492E−01 1.695E+00 8.444E−01 1.27E+00 3.39E+02 15 4.359E−02 5.702E−02 8.240E−02 8.06E−01 1.80E+01 1.024E−01 9.576E−02 1.732E−01 9.39E+01 2.36E+02 6.457E−02 6.862E−02 1.592E−01 2.06E+00 2.52E+01 3.384E−01 2.902E−01 4.897E−01 1.98E+02 2.64E+02 7.977E−02 1.077E−01 4.672E−01 6.31E+01 2.59E+01 8.964E−01 9.217E−01 9.605E−01 3.24E+02 3.45E+02 DTLZ2 3 2.531E−03 3.346E−03 2.539E−03 6.88E−02 6.73E−02 DTLZ4 4.233E−03 3.266E−03 2.638E−03 6.87E−02 6.66E−02 2.666E−03 3.376E−03 2.617E−03 7.18E−02 6.91E−02 4.825E−03 4.692E−03 1.071E−02 7.23E−02 7.07E−02 2.828E−03 3.418E−03 2.733E−03 7.44E−02 7.10E−02 5.268E−03 2.935E−02 2.938E−02 9.40E−01 5.27E−01 5 8.731E−03 1.201E−02 7.951E−03 1.41E−01 2.76E−01 8.777E−03 1.200E−02 8.444E−03 1.42E−01 2.60E−01 9.255E−03 1.202E−02 8.236E−03 1.47E−0 2.87E−01 9.338E−03 1.204E−02 1.847E−02 1.46E−01 2.68E−01 9.715E−03 1.205E−02 1.003E−02 1.56E−01 2.92E−01 9.854E−03 1.212E−02 2.778E−02 1.61E−01 5.30E−01 8 3.110E−02 3.445E−02 2.094E−02 3.45E−01 5.48E−01 2.274E−02 3.459E−02 2.704E−02 3.23E−01 4.79E−01 2.574E−02 3.769E−02 3.600E−02 3.73E−01 6.03E−01 2.946E−02 3.715E−02 2.991E−02 3.31E−01 4.96E−01 3.155E−02 4.550E−02 5.133E−02 4.13E−01 6.47E−01 3.369E−02 4.039E−02 3.707E−02 3.40E−01 5.39E−01 10 2.528E−02 4.477E−02 3.643E−02 4.11E−01 6.78E−01 2.494E−02 4.440E−02 3.734E−02 4.19E−01 6.76E−01 3.028E−02 4.890E−02 5.002E−02 4.51E−01 6.90E−01 2.764E−02 4.755E−02 4.511E−02 4.29E−01 6.83E−01 3.345E−02 5.646E−02 5.875E−02 5.16E−01 6.92E−01 3.364E−02 5.066E−02 5.829E−02 4.41E−01 6.88E−01 15 5.620E−02 9.018E−02 7.696E−02 5.09E−01 6.24E−01 5.449E−02 8.654E−02 8.341E−02 4.98E−01 5.99E−01 5.915E−02 9.541E−02 9.159E−02 5.29E−01 8.64E−01 5.756E−02 9.560E−02 8.993E−02 5.03E−01 6.10E−01 6.547E−02 1.019E−01 9.893E−02 5.38E−01 3.20E+00 6.046E−02 1.014E−01 9.763E−02 5.14E−01 6.13E−01 m RIEA MOEA/D PICEA-G GrEA HypE RIEA MOEA/D PICEA-G GrEA HypE DTLZ1 3 1.658E−03 2.869E−03 3.529E−03 2.76E−02 1.82E+01 DTLZ3 9.467E−03 3.277E−03 1.093E−01 6.77E−02 1.65E+02 2.567E−03 2.904E−03 2.067E−02 3.34E−02 1.97E+01 2.450E−02 4.635E−02 2.335E−01 7.69E−02 1.70E+02 8.238E−03 2.942E−03 6.174E−02 1.35E−01 2.16E+01 6.114E−02 1.406E−01 5.502E−01 4.47E−01 1.76E+02 5 6.635E−03 8.596E−03 1.243E−02 7.37E−02 1.80E+01 1.155E−02 1.155E−02 5.789E−02 5.33E−01 1.83E+02 7.420E−03 8.953E−03 2.781E−02 3.36E−01 2.14E+01 3.112E−02 1.019E−01 1.894E−01 8.30E−01 2.17E+02 8.741E−02 9.504E−03 8.106E−02 4.94E−01 2.36E+01 9.316E−02 4.355E−01 3.935E−01 1.12E+00 2.28E+02 8 1.932E−02 2.073E−02 4.009E−02 1.02E−01 1.03E+01 4.910E−02 4.331E−02 1.173E−01 7.52E−01 2.20E+02 2.411E−02 5.742E−02 5.628E−02 1.20E−01 2.27E+01 7.554E−02 6.094E−01 2.988E−01 1.02E+00 2.70E+02 3.214E−02 3.747E−01 1.363E−01 3.85E−01 2.43E+01 1.270E−01 4.728E+00 8.273E−01 1.23E+00 2.95E+02 10 2.362E−02 2.498E−02 4.431E−02 1.18E−01 1.43E+01 4.898E−02 4.862E−02 1.224E−01 8.66E−01 1.72E+02 2.880E−02 3.779E−02 6.329E−02 1.59E−01 1.69E+01 6.757E−02 4.128E−01 3.737E−01 1.15E+00 2.89E+02 3.272E−02 1.051E−01 1.346E−01 5.11E−01 2.03E+01 1.492E−01 1.695E+00 8.444E−01 1.27E+00 3.39E+02 15 4.359E−02 5.702E−02 8.240E−02 8.06E−01 1.80E+01 1.024E−01 9.576E−02 1.732E−01 9.39E+01 2.36E+02 6.457E−02 6.862E−02 1.592E−01 2.06E+00 2.52E+01 3.384E−01 2.902E−01 4.897E−01 1.98E+02 2.64E+02 7.977E−02 1.077E−01 4.672E−01 6.31E+01 2.59E+01 8.964E−01 9.217E−01 9.605E−01 3.24E+02 3.45E+02 DTLZ2 3 2.531E−03 3.346E−03 2.539E−03 6.88E−02 6.73E−02 DTLZ4 4.233E−03 3.266E−03 2.638E−03 6.87E−02 6.66E−02 2.666E−03 3.376E−03 2.617E−03 7.18E−02 6.91E−02 4.825E−03 4.692E−03 1.071E−02 7.23E−02 7.07E−02 2.828E−03 3.418E−03 2.733E−03 7.44E−02 7.10E−02 5.268E−03 2.935E−02 2.938E−02 9.40E−01 5.27E−01 5 8.731E−03 1.201E−02 7.951E−03 1.41E−01 2.76E−01 8.777E−03 1.200E−02 8.444E−03 1.42E−01 2.60E−01 9.255E−03 1.202E−02 8.236E−03 1.47E−0 2.87E−01 9.338E−03 1.204E−02 1.847E−02 1.46E−01 2.68E−01 9.715E−03 1.205E−02 1.003E−02 1.56E−01 2.92E−01 9.854E−03 1.212E−02 2.778E−02 1.61E−01 5.30E−01 8 3.110E−02 3.445E−02 2.094E−02 3.45E−01 5.48E−01 2.274E−02 3.459E−02 2.704E−02 3.23E−01 4.79E−01 2.574E−02 3.769E−02 3.600E−02 3.73E−01 6.03E−01 2.946E−02 3.715E−02 2.991E−02 3.31E−01 4.96E−01 3.155E−02 4.550E−02 5.133E−02 4.13E−01 6.47E−01 3.369E−02 4.039E−02 3.707E−02 3.40E−01 5.39E−01 10 2.528E−02 4.477E−02 3.643E−02 4.11E−01 6.78E−01 2.494E−02 4.440E−02 3.734E−02 4.19E−01 6.76E−01 3.028E−02 4.890E−02 5.002E−02 4.51E−01 6.90E−01 2.764E−02 4.755E−02 4.511E−02 4.29E−01 6.83E−01 3.345E−02 5.646E−02 5.875E−02 5.16E−01 6.92E−01 3.364E−02 5.066E−02 5.829E−02 4.41E−01 6.88E−01 15 5.620E−02 9.018E−02 7.696E−02 5.09E−01 6.24E−01 5.449E−02 8.654E−02 8.341E−02 4.98E−01 5.99E−01 5.915E−02 9.541E−02 9.159E−02 5.29E−01 8.64E−01 5.756E−02 9.560E−02 8.993E−02 5.03E−01 6.10E−01 6.547E−02 1.019E−01 9.893E−02 5.38E−01 3.20E+00 6.046E−02 1.014E−01 9.763E−02 5.14E−01 6.13E−01 View Large Table 3. Statistical results (best/mean/worst) obtained by RIEA, MOEA/D, PICEA-G, GrEA and HypE in 30 independent runs on DLTZ1 and DTLZ2 in terms of the IGD metric. m RIEA MOEA/D PICEA-G GrEA HypE RIEA MOEA/D PICEA-G GrEA HypE DTLZ1 3 1.658E−03 2.869E−03 3.529E−03 2.76E−02 1.82E+01 DTLZ3 9.467E−03 3.277E−03 1.093E−01 6.77E−02 1.65E+02 2.567E−03 2.904E−03 2.067E−02 3.34E−02 1.97E+01 2.450E−02 4.635E−02 2.335E−01 7.69E−02 1.70E+02 8.238E−03 2.942E−03 6.174E−02 1.35E−01 2.16E+01 6.114E−02 1.406E−01 5.502E−01 4.47E−01 1.76E+02 5 6.635E−03 8.596E−03 1.243E−02 7.37E−02 1.80E+01 1.155E−02 1.155E−02 5.789E−02 5.33E−01 1.83E+02 7.420E−03 8.953E−03 2.781E−02 3.36E−01 2.14E+01 3.112E−02 1.019E−01 1.894E−01 8.30E−01 2.17E+02 8.741E−02 9.504E−03 8.106E−02 4.94E−01 2.36E+01 9.316E−02 4.355E−01 3.935E−01 1.12E+00 2.28E+02 8 1.932E−02 2.073E−02 4.009E−02 1.02E−01 1.03E+01 4.910E−02 4.331E−02 1.173E−01 7.52E−01 2.20E+02 2.411E−02 5.742E−02 5.628E−02 1.20E−01 2.27E+01 7.554E−02 6.094E−01 2.988E−01 1.02E+00 2.70E+02 3.214E−02 3.747E−01 1.363E−01 3.85E−01 2.43E+01 1.270E−01 4.728E+00 8.273E−01 1.23E+00 2.95E+02 10 2.362E−02 2.498E−02 4.431E−02 1.18E−01 1.43E+01 4.898E−02 4.862E−02 1.224E−01 8.66E−01 1.72E+02 2.880E−02 3.779E−02 6.329E−02 1.59E−01 1.69E+01 6.757E−02 4.128E−01 3.737E−01 1.15E+00 2.89E+02 3.272E−02 1.051E−01 1.346E−01 5.11E−01 2.03E+01 1.492E−01 1.695E+00 8.444E−01 1.27E+00 3.39E+02 15 4.359E−02 5.702E−02 8.240E−02 8.06E−01 1.80E+01 1.024E−01 9.576E−02 1.732E−01 9.39E+01 2.36E+02 6.457E−02 6.862E−02 1.592E−01 2.06E+00 2.52E+01 3.384E−01 2.902E−01 4.897E−01 1.98E+02 2.64E+02 7.977E−02 1.077E−01 4.672E−01 6.31E+01 2.59E+01 8.964E−01 9.217E−01 9.605E−01 3.24E+02 3.45E+02 DTLZ2 3 2.531E−03 3.346E−03 2.539E−03 6.88E−02 6.73E−02 DTLZ4 4.233E−03 3.266E−03 2.638E−03 6.87E−02 6.66E−02 2.666E−03 3.376E−03 2.617E−03 7.18E−02 6.91E−02 4.825E−03 4.692E−03 1.071E−02 7.23E−02 7.07E−02 2.828E−03 3.418E−03 2.733E−03 7.44E−02 7.10E−02 5.268E−03 2.935E−02 2.938E−02 9.40E−01 5.27E−01 5 8.731E−03 1.201E−02 7.951E−03 1.41E−01 2.76E−01 8.777E−03 1.200E−02 8.444E−03 1.42E−01 2.60E−01 9.255E−03 1.202E−02 8.236E−03 1.47E−0 2.87E−01 9.338E−03 1.204E−02 1.847E−02 1.46E−01 2.68E−01 9.715E−03 1.205E−02 1.003E−02 1.56E−01 2.92E−01 9.854E−03 1.212E−02 2.778E−02 1.61E−01 5.30E−01 8 3.110E−02 3.445E−02 2.094E−02 3.45E−01 5.48E−01 2.274E−02 3.459E−02 2.704E−02 3.23E−01 4.79E−01 2.574E−02 3.769E−02 3.600E−02 3.73E−01 6.03E−01 2.946E−02 3.715E−02 2.991E−02 3.31E−01 4.96E−01 3.155E−02 4.550E−02 5.133E−02 4.13E−01 6.47E−01 3.369E−02 4.039E−02 3.707E−02 3.40E−01 5.39E−01 10 2.528E−02 4.477E−02 3.643E−02 4.11E−01 6.78E−01 2.494E−02 4.440E−02 3.734E−02 4.19E−01 6.76E−01 3.028E−02 4.890E−02 5.002E−02 4.51E−01 6.90E−01 2.764E−02 4.755E−02 4.511E−02 4.29E−01 6.83E−01 3.345E−02 5.646E−02 5.875E−02 5.16E−01 6.92E−01 3.364E−02 5.066E−02 5.829E−02 4.41E−01 6.88E−01 15 5.620E−02 9.018E−02 7.696E−02 5.09E−01 6.24E−01 5.449E−02 8.654E−02 8.341E−02 4.98E−01 5.99E−01 5.915E−02 9.541E−02 9.159E−02 5.29E−01 8.64E−01 5.756E−02 9.560E−02 8.993E−02 5.03E−01 6.10E−01 6.547E−02 1.019E−01 9.893E−02 5.38E−01 3.20E+00 6.046E−02 1.014E−01 9.763E−02 5.14E−01 6.13E−01 m RIEA MOEA/D PICEA-G GrEA HypE RIEA MOEA/D PICEA-G GrEA HypE DTLZ1 3 1.658E−03 2.869E−03 3.529E−03 2.76E−02 1.82E+01 DTLZ3 9.467E−03 3.277E−03 1.093E−01 6.77E−02 1.65E+02 2.567E−03 2.904E−03 2.067E−02 3.34E−02 1.97E+01 2.450E−02 4.635E−02 2.335E−01 7.69E−02 1.70E+02 8.238E−03 2.942E−03 6.174E−02 1.35E−01 2.16E+01 6.114E−02 1.406E−01 5.502E−01 4.47E−01 1.76E+02 5 6.635E−03 8.596E−03 1.243E−02 7.37E−02 1.80E+01 1.155E−02 1.155E−02 5.789E−02 5.33E−01 1.83E+02 7.420E−03 8.953E−03 2.781E−02 3.36E−01 2.14E+01 3.112E−02 1.019E−01 1.894E−01 8.30E−01 2.17E+02 8.741E−02 9.504E−03 8.106E−02 4.94E−01 2.36E+01 9.316E−02 4.355E−01 3.935E−01 1.12E+00 2.28E+02 8 1.932E−02 2.073E−02 4.009E−02 1.02E−01 1.03E+01 4.910E−02 4.331E−02 1.173E−01 7.52E−01 2.20E+02 2.411E−02 5.742E−02 5.628E−02 1.20E−01 2.27E+01 7.554E−02 6.094E−01 2.988E−01 1.02E+00 2.70E+02 3.214E−02 3.747E−01 1.363E−01 3.85E−01 2.43E+01 1.270E−01 4.728E+00 8.273E−01 1.23E+00 2.95E+02 10 2.362E−02 2.498E−02 4.431E−02 1.18E−01 1.43E+01 4.898E−02 4.862E−02 1.224E−01 8.66E−01 1.72E+02 2.880E−02 3.779E−02 6.329E−02 1.59E−01 1.69E+01 6.757E−02 4.128E−01 3.737E−01 1.15E+00 2.89E+02 3.272E−02 1.051E−01 1.346E−01 5.11E−01 2.03E+01 1.492E−01 1.695E+00 8.444E−01 1.27E+00 3.39E+02 15 4.359E−02 5.702E−02 8.240E−02 8.06E−01 1.80E+01 1.024E−01 9.576E−02 1.732E−01 9.39E+01 2.36E+02 6.457E−02 6.862E−02 1.592E−01 2.06E+00 2.52E+01 3.384E−01 2.902E−01 4.897E−01 1.98E+02 2.64E+02 7.977E−02 1.077E−01 4.672E−01 6.31E+01 2.59E+01 8.964E−01 9.217E−01 9.605E−01 3.24E+02 3.45E+02 DTLZ2 3 2.531E−03 3.346E−03 2.539E−03 6.88E−02 6.73E−02 DTLZ4 4.233E−03 3.266E−03 2.638E−03 6.87E−02 6.66E−02 2.666E−03 3.376E−03 2.617E−03 7.18E−02 6.91E−02 4.825E−03 4.692E−03 1.071E−02 7.23E−02 7.07E−02 2.828E−03 3.418E−03 2.733E−03 7.44E−02 7.10E−02 5.268E−03 2.935E−02 2.938E−02 9.40E−01 5.27E−01 5 8.731E−03 1.201E−02 7.951E−03 1.41E−01 2.76E−01 8.777E−03 1.200E−02 8.444E−03 1.42E−01 2.60E−01 9.255E−03 1.202E−02 8.236E−03 1.47E−0 2.87E−01 9.338E−03 1.204E−02 1.847E−02 1.46E−01 2.68E−01 9.715E−03 1.205E−02 1.003E−02 1.56E−01 2.92E−01 9.854E−03 1.212E−02 2.778E−02 1.61E−01 5.30E−01 8 3.110E−02 3.445E−02 2.094E−02 3.45E−01 5.48E−01 2.274E−02 3.459E−02 2.704E−02 3.23E−01 4.79E−01 2.574E−02 3.769E−02 3.600E−02 3.73E−01 6.03E−01 2.946E−02 3.715E−02 2.991E−02 3.31E−01 4.96E−01 3.155E−02 4.550E−02 5.133E−02 4.13E−01 6.47E−01 3.369E−02 4.039E−02 3.707E−02 3.40E−01 5.39E−01 10 2.528E−02 4.477E−02 3.643E−02 4.11E−01 6.78E−01 2.494E−02 4.440E−02 3.734E−02 4.19E−01 6.76E−01 3.028E−02 4.890E−02 5.002E−02 4.51E−01 6.90E−01 2.764E−02 4.755E−02 4.511E−02 4.29E−01 6.83E−01 3.345E−02 5.646E−02 5.875E−02 5.16E−01 6.92E−01 3.364E−02 5.066E−02 5.829E−02 4.41E−01 6.88E−01 15 5.620E−02 9.018E−02 7.696E−02 5.09E−01 6.24E−01 5.449E−02 8.654E−02 8.341E−02 4.98E−01 5.99E−01 5.915E−02 9.541E−02 9.159E−02 5.29E−01 8.64E−01 5.756E−02 9.560E−02 8.993E−02 5.03E−01 6.10E−01 6.547E−02 1.019E−01 9.893E−02 5.38E−01 3.20E+00 6.046E−02 1.014E−01 9.763E−02 5.14E−01 6.13E−01 View Large From Table 3, we see clearly that RIEA outperforms GrEA and HypE on all test instances with all numbers of objectives in terms of median IGD value. Especially, the performance of HypE is the worst on these problems in terms of IGD; while GrEA is the second worst. For DTLZ1 and DTLZ3, RIEA performs better than PICEA-g in terms of the best and median values of the IGD metric for all considered numbers of objectives. For DTLZ2, RIEA obtains better median IGD values than the other three algorithm on 8–10 objectives. PICEA-g performs the best on DTLZ2 with 3- and 5-objectives in terms of IGD, but RIEA obtains the best IGD values on the problem with 3-objectives. On DTLZ4, we see that RIEA performs the best over all the compared algorithm. Moreover, it is observed that along with the increase of the number of objectives, the performance of RIEA becomes better. This clearly shows that the proposed fitness assignment scheme can indeed increase the comparability of the solutions in high-dimensional objective space. Table 4 shows the comparison results on DTLZ1–DTLZ4 in terms of HV. From the table, it is observed that RIEA performs better along with the increase of the number of objectives. For DTLZ1, MOEA/D performs the best on 3- and 5-objectives, while RIEA performs the best on instances with ≥8 objectives. For DTLZ2 to DTLZ4, the best performance is obtained by GrEA and HypE for instances with small numbers ( ≤8) of objective; but RIEA achieves better performance on DTLZ2 with 10 and 15 objectives. Table 4. Statistical results (best/mean/worst) obtained by RIEA, MOEA/D, PICEA-G, GrEA and HypE in 30 independent runs on the DLTZ test suite in terms of the HV metric. m RIEA MOEA/D PICEA-G GrEA HypE RIEA MOEA/D PICEA-G GrEA HypE DTLZ1 3 0.9743 0.9729 0.9707 0.9674 0.0000 DTLZ3 0.8718 0.9310 0.8315 0.9247 0.0000 0.9690 0.9694 0.8559 0.9641 0.0000 0.6287 0.5660 0.0416 0.9227 0.0000 0.9428 0.9660 0.3951 0.8280 0.0000 0.0463 0.0000 0.0000 0.6212 0.0000 5 0.9977 0.9978 0.9805 0.9915 0.0000 0.9800 0.9842 0.3977 0.9630 0.0000 0.9957 0.9966 0.7916 0.8445 0.0000 0.7458 0.4481 0.0199 0.8081 0.0000 0.9938 0.9946 0.1648 0.5002 0.0000 0.0018 0.0000 0.0000 0.5000 0.0000 8 0.9995 0.9965 0.9595 0.9991 0.0000 0.9294 0.9521 0.0000 0.9535 0.0000 0.9959 0.8372 0.6577 0.9980 0.0000 0.5557 0.2421 0.0000 0.7912 0.0000 0.9825 0.0000 0.0000 0.9027 0.0000 0.0312 0.0000 0.0000 0.4986 0.0000 10 0.9996 0.9987 0.9184 0.9995 0.0000 0.9859 0.9546 0.0000 0.9622 0.0000 0.9971 0.9370 0.7467 0.9986 0.0000 0.7648 0.2262 0.0000 0.7360 0.0000 0.9901 0.2340 0.0680 0.5323 0.0000 0.4871 0.0000 0.0000 0.5000 0.0000 15 0.9987 0.9812 0.9298 0.1725 0.0000 0.9296 0.8990 0.0000 0.0000 0.0000 0.9810 0.9163 0.5061 0.0000 0.0000 0.3099 0.4690 0.0000+ 0.0000 0.0000+ 0.9097 0.6327 0.0000 0.0000 0.0000 0.2230 0.0000 0.0000 0.0000 0.0000 DTLZ2 3 0.9314 0.9277 0.9292 0.9242 0.9257 DTLZ4 0.9294 0.9297 0.9279 0.9246 0.9264 0.9254 0.9228 0.9256 0.9240 0.9257 0.9248 0.9175 0.8939 0.9241 0.9262 0.9204 0.9162 0.9178 0.9237 0.9255 0.9202 0.7999 0.7965 0.5000 0.8005 5 0.9882 0.9881 0.9886 0.9904 0.9879 0.9882 0.9881 0.9882 0.9905 0.9882 0.9845 0.9857 0.9856 0.9902 0.9877 0.9863 0.9853 0.9682 0.9904 0.9880 0.9793 0.9827 0.9824 0.9901 0.9875 0.9831 0.9828 0.9224 0.9902 0.9877 8 0.9984 0.9624 0.9988 1.0000 0.9974 0.9980 0.9639 0.9975 0.9991 0.9980 0.9936 0.9366 0.9886 0.9997 0.9966 0.9957 0.9428 0.9953 0.9990 0.9977 0.9843 0.9184 0.9664 0.9893 0.9958 0.9927 0.9267 0.9911 0.9990 0.9976 10 0.9998 0.9605 0.9938 0.9976 0.9990 0.9999 0.9671 0.9992 0.9997 0.9990 0.9993 0.9393 0.9854 0.9964 0.9989 0.9997 0.9262 0.9873 0.9996 0.9989 0.9983 0.9084 0.9690 0.9947 0.9994 0.9944 0.9035 0.9688 0.9995 0.9989 15 1.0000 0.9069 0.9952 0.9995 0.9999 0.9954 0.9050 0.9974 0.9996 0.9991 0.9997 0.8756 0.9767 0.9994 0.9970 0.9998 0.8616 0.9915 0.9995 0.9994 0.9893 0.8170 0.9270 0.9984 0.0000 0.9997 0.8086 0.9843 0.9995 0.9991 m RIEA MOEA/D PICEA-G GrEA HypE RIEA MOEA/D PICEA-G GrEA HypE DTLZ1 3 0.9743 0.9729 0.9707 0.9674 0.0000 DTLZ3 0.8718 0.9310 0.8315 0.9247 0.0000 0.9690 0.9694 0.8559 0.9641 0.0000 0.6287 0.5660 0.0416 0.9227 0.0000 0.9428 0.9660 0.3951 0.8280 0.0000 0.0463 0.0000 0.0000 0.6212 0.0000 5 0.9977 0.9978 0.9805 0.9915 0.0000 0.9800 0.9842 0.3977 0.9630 0.0000 0.9957 0.9966 0.7916 0.8445 0.0000 0.7458 0.4481 0.0199 0.8081 0.0000 0.9938 0.9946 0.1648 0.5002 0.0000 0.0018 0.0000 0.0000 0.5000 0.0000 8 0.9995 0.9965 0.9595 0.9991 0.0000 0.9294 0.9521 0.0000 0.9535 0.0000 0.9959 0.8372 0.6577 0.9980 0.0000 0.5557 0.2421 0.0000 0.7912 0.0000 0.9825 0.0000 0.0000 0.9027 0.0000 0.0312 0.0000 0.0000 0.4986 0.0000 10 0.9996 0.9987 0.9184 0.9995 0.0000 0.9859 0.9546 0.0000 0.9622 0.0000 0.9971 0.9370 0.7467 0.9986 0.0000 0.7648 0.2262 0.0000 0.7360 0.0000 0.9901 0.2340 0.0680 0.5323 0.0000 0.4871 0.0000 0.0000 0.5000 0.0000 15 0.9987 0.9812 0.9298 0.1725 0.0000 0.9296 0.8990 0.0000 0.0000 0.0000 0.9810 0.9163 0.5061 0.0000 0.0000 0.3099 0.4690 0.0000+ 0.0000 0.0000+ 0.9097 0.6327 0.0000 0.0000 0.0000 0.2230 0.0000 0.0000 0.0000 0.0000 DTLZ2 3 0.9314 0.9277 0.9292 0.9242 0.9257 DTLZ4 0.9294 0.9297 0.9279 0.9246 0.9264 0.9254 0.9228 0.9256 0.9240 0.9257 0.9248 0.9175 0.8939 0.9241 0.9262 0.9204 0.9162 0.9178 0.9237 0.9255 0.9202 0.7999 0.7965 0.5000 0.8005 5 0.9882 0.9881 0.9886 0.9904 0.9879 0.9882 0.9881 0.9882 0.9905 0.9882 0.9845 0.9857 0.9856 0.9902 0.9877 0.9863 0.9853 0.9682 0.9904 0.9880 0.9793 0.9827 0.9824 0.9901 0.9875 0.9831 0.9828 0.9224 0.9902 0.9877 8 0.9984 0.9624 0.9988 1.0000 0.9974 0.9980 0.9639 0.9975 0.9991 0.9980 0.9936 0.9366 0.9886 0.9997 0.9966 0.9957 0.9428 0.9953 0.9990 0.9977 0.9843 0.9184 0.9664 0.9893 0.9958 0.9927 0.9267 0.9911 0.9990 0.9976 10 0.9998 0.9605 0.9938 0.9976 0.9990 0.9999 0.9671 0.9992 0.9997 0.9990 0.9993 0.9393 0.9854 0.9964 0.9989 0.9997 0.9262 0.9873 0.9996 0.9989 0.9983 0.9084 0.9690 0.9947 0.9994 0.9944 0.9035 0.9688 0.9995 0.9989 15 1.0000 0.9069 0.9952 0.9995 0.9999 0.9954 0.9050 0.9974 0.9996 0.9991 0.9997 0.8756 0.9767 0.9994 0.9970 0.9998 0.8616 0.9915 0.9995 0.9994 0.9893 0.8170 0.9270 0.9984 0.0000 0.9997 0.8086 0.9843 0.9995 0.9991 View Large Table 4. Statistical results (best/mean/worst) obtained by RIEA, MOEA/D, PICEA-G, GrEA and HypE in 30 independent runs on the DLTZ test suite in terms of the HV metric. m RIEA MOEA/D PICEA-G GrEA HypE RIEA MOEA/D PICEA-G GrEA HypE DTLZ1 3 0.9743 0.9729 0.9707 0.9674 0.0000 DTLZ3 0.8718 0.9310 0.8315 0.9247 0.0000 0.9690 0.9694 0.8559 0.9641 0.0000 0.6287 0.5660 0.0416 0.9227 0.0000 0.9428 0.9660 0.3951 0.8280 0.0000 0.0463 0.0000 0.0000 0.6212 0.0000 5 0.9977 0.9978 0.9805 0.9915 0.0000 0.9800 0.9842 0.3977 0.9630 0.0000 0.9957 0.9966 0.7916 0.8445 0.0000 0.7458 0.4481 0.0199 0.8081 0.0000 0.9938 0.9946 0.1648 0.5002 0.0000 0.0018 0.0000 0.0000 0.5000 0.0000 8 0.9995 0.9965 0.9595 0.9991 0.0000 0.9294 0.9521 0.0000 0.9535 0.0000 0.9959 0.8372 0.6577 0.9980 0.0000 0.5557 0.2421 0.0000 0.7912 0.0000 0.9825 0.0000 0.0000 0.9027 0.0000 0.0312 0.0000 0.0000 0.4986 0.0000 10 0.9996 0.9987 0.9184 0.9995 0.0000 0.9859 0.9546 0.0000 0.9622 0.0000 0.9971 0.9370 0.7467 0.9986 0.0000 0.7648 0.2262 0.0000 0.7360 0.0000 0.9901 0.2340 0.0680 0.5323 0.0000 0.4871 0.0000 0.0000 0.5000 0.0000 15 0.9987 0.9812 0.9298 0.1725 0.0000 0.9296 0.8990 0.0000 0.0000 0.0000 0.9810 0.9163 0.5061 0.0000 0.0000 0.3099 0.4690 0.0000+ 0.0000 0.0000+ 0.9097 0.6327 0.0000 0.0000 0.0000 0.2230 0.0000 0.0000 0.0000 0.0000 DTLZ2 3 0.9314 0.9277 0.9292 0.9242 0.9257 DTLZ4 0.9294 0.9297 0.9279 0.9246 0.9264 0.9254 0.9228 0.9256 0.9240 0.9257 0.9248 0.9175 0.8939 0.9241 0.9262 0.9204 0.9162 0.9178 0.9237 0.9255 0.9202 0.7999 0.7965 0.5000 0.8005 5 0.9882 0.9881 0.9886 0.9904 0.9879 0.9882 0.9881 0.9882 0.9905 0.9882 0.9845 0.9857 0.9856 0.9902 0.9877 0.9863 0.9853 0.9682 0.9904 0.9880 0.9793 0.9827 0.9824 0.9901 0.9875 0.9831 0.9828 0.9224 0.9902 0.9877 8 0.9984 0.9624 0.9988 1.0000 0.9974 0.9980 0.9639 0.9975 0.9991 0.9980 0.9936 0.9366 0.9886 0.9997 0.9966 0.9957 0.9428 0.9953 0.9990 0.9977 0.9843 0.9184 0.9664 0.9893 0.9958 0.9927 0.9267 0.9911 0.9990 0.9976 10 0.9998 0.9605 0.9938 0.9976 0.9990 0.9999 0.9671 0.9992 0.9997 0.9990 0.9993 0.9393 0.9854 0.9964 0.9989 0.9997 0.9262 0.9873 0.9996 0.9989 0.9983 0.9084 0.9690 0.9947 0.9994 0.9944 0.9035 0.9688 0.9995 0.9989 15 1.0000 0.9069 0.9952 0.9995 0.9999 0.9954 0.9050 0.9974 0.9996 0.9991 0.9997 0.8756 0.9767 0.9994 0.9970 0.9998 0.8616 0.9915 0.9995 0.9994 0.9893 0.8170 0.9270 0.9984 0.0000 0.9997 0.8086 0.9843 0.9995 0.9991 m RIEA MOEA/D PICEA-G GrEA HypE RIEA MOEA/D PICEA-G GrEA HypE DTLZ1 3 0.9743 0.9729 0.9707 0.9674 0.0000 DTLZ3 0.8718 0.9310 0.8315 0.9247 0.0000 0.9690 0.9694 0.8559 0.9641 0.0000 0.6287 0.5660 0.0416 0.9227 0.0000 0.9428 0.9660 0.3951 0.8280 0.0000 0.0463 0.0000 0.0000 0.6212 0.0000 5 0.9977 0.9978 0.9805 0.9915 0.0000 0.9800 0.9842 0.3977 0.9630 0.0000 0.9957 0.9966 0.7916 0.8445 0.0000 0.7458 0.4481 0.0199 0.8081 0.0000 0.9938 0.9946 0.1648 0.5002 0.0000 0.0018 0.0000 0.0000 0.5000 0.0000 8 0.9995 0.9965 0.9595 0.9991 0.0000 0.9294 0.9521 0.0000 0.9535 0.0000 0.9959 0.8372 0.6577 0.9980 0.0000 0.5557 0.2421 0.0000 0.7912 0.0000 0.9825 0.0000 0.0000 0.9027 0.0000 0.0312 0.0000 0.0000 0.4986 0.0000 10 0.9996 0.9987 0.9184 0.9995 0.0000 0.9859 0.9546 0.0000 0.9622 0.0000 0.9971 0.9370 0.7467 0.9986 0.0000 0.7648 0.2262 0.0000 0.7360 0.0000 0.9901 0.2340 0.0680 0.5323 0.0000 0.4871 0.0000 0.0000 0.5000 0.0000 15 0.9987 0.9812 0.9298 0.1725 0.0000 0.9296 0.8990 0.0000 0.0000 0.0000 0.9810 0.9163 0.5061 0.0000 0.0000 0.3099 0.4690 0.0000+ 0.0000 0.0000+ 0.9097 0.6327 0.0000 0.0000 0.0000 0.2230 0.0000 0.0000 0.0000 0.0000 DTLZ2 3 0.9314 0.9277 0.9292 0.9242 0.9257 DTLZ4 0.9294 0.9297 0.9279 0.9246 0.9264 0.9254 0.9228 0.9256 0.9240 0.9257 0.9248 0.9175 0.8939 0.9241 0.9262 0.9204 0.9162 0.9178 0.9237 0.9255 0.9202 0.7999 0.7965 0.5000 0.8005 5 0.9882 0.9881 0.9886 0.9904 0.9879 0.9882 0.9881 0.9882 0.9905 0.9882 0.9845 0.9857 0.9856 0.9902 0.9877 0.9863 0.9853 0.9682 0.9904 0.9880 0.9793 0.9827 0.9824 0.9901 0.9875 0.9831 0.9828 0.9224 0.9902 0.9877 8 0.9984 0.9624 0.9988 1.0000 0.9974 0.9980 0.9639 0.9975 0.9991 0.9980 0.9936 0.9366 0.9886 0.9997 0.9966 0.9957 0.9428 0.9953 0.9990 0.9977 0.9843 0.9184 0.9664 0.9893 0.9958 0.9927 0.9267 0.9911 0.9990 0.9976 10 0.9998 0.9605 0.9938 0.9976 0.9990 0.9999 0.9671 0.9992 0.9997 0.9990 0.9993 0.9393 0.9854 0.9964 0.9989 0.9997 0.9262 0.9873 0.9996 0.9989 0.9983 0.9084 0.9690 0.9947 0.9994 0.9944 0.9035 0.9688 0.9995 0.9989 15 1.0000 0.9069 0.9952 0.9995 0.9999 0.9954 0.9050 0.9974 0.9996 0.9991 0.9997 0.8756 0.9767 0.9994 0.9970 0.9998 0.8616 0.9915 0.9995 0.9994 0.9893 0.8170 0.9270 0.9984 0.0000 0.9997 0.8086 0.9843 0.9995 0.9991 View Large Figures 4 and 5 present the final solutions’ distribution of the run with median IGD value. These figures visualise the parallel coordinates of the non-dominated fronts obtained by RIEA, PICEA-g and MOEA/D, on the 10-objectives DTLZ1 and DTLZ4. This particular run is connected with the results that is the closest to the median hypervolume value. From these plots on DTLZ1 (Fig. 4), it is observed that the solutions obtained by PICEA-g converge to the non-dominated front worse than RIEA, and the solutions achieved by MOEA/D are worse in terms of distribution than the other two algorithms. From the plots of DTLZ4, we see a similar phenomenon as shown in Fig. 5. Figure 4. View largeDownload slide Parallel coordinates of non-dominated fronts obtained on DTLZ1 with 10 objectives. Note that the ranges of the coordinates of the Pareto optimal solutions should be in [0,0.5]. (a) RIEA, (b) PICEA_G, (c) MOEA/D, (d) GrEA and (e) HypE. Figure 4. View largeDownload slide Parallel coordinates of non-dominated fronts obtained on DTLZ1 with 10 objectives. Note that the ranges of the coordinates of the Pareto optimal solutions should be in [0,0.5]. (a) RIEA, (b) PICEA_G, (c) MOEA/D, (d) GrEA and (e) HypE. Figure 5. View largeDownload slide Parallel coordinates of non-dominated fronts obtained on DTLZ4 with 10 objectives. Note that the ranges of the coordinates of the Pareto optimal solutions should be in [0,1]. (a) RIEA, (b) PICEA_G, (c) MOEA/D, (d) GrEA and (e) HypE. Figure 5. View largeDownload slide Parallel coordinates of non-dominated fronts obtained on DTLZ4 with 10 objectives. Note that the ranges of the coordinates of the Pareto optimal solutions should be in [0,1]. (a) RIEA, (b) PICEA_G, (c) MOEA/D, (d) GrEA and (e) HypE. To further study the performance of the developed algorithm, we carried out experiments on DTLZ5–DTLZ7. These problems are obtained by modifying DTLZ2 and DTLZ3, but much more difficult. Table 5 lists the results obtained by the compared algorithms. From the table, we see that RIEA performs clearly better than PICEA-G and HypE. It performs similar to MOEA/D on DTLZ5 and DTLZ6. GrEA performs the best on DTLZ7. Figure 6 shows the parallel coordinates of the median non-dominated fronts obtained by the compared algorithm on DTLZ7 with 10 objectives. It can be seen that RIEA got better front than PICEA_g and HypE, while it is not as good as the front by MOEA/D. Figure 6. View largeDownload slide Parallel coordinates of non-dominated fronts obtained on DTLZ7 with 10 objectives. (a) RIEA, (b) PICEA_G, (c) MOEA/D, (d) GrEA and (e) HypE. Figure 6. View largeDownload slide Parallel coordinates of non-dominated fronts obtained on DTLZ7 with 10 objectives. (a) RIEA, (b) PICEA_G, (c) MOEA/D, (d) GrEA and (e) HypE. Table 5. Statistical results (best/mean/worst) obtained by RIEA, MOEA/D, PICEA-G, GrEA and HypE on DLTZ5, DTLZ6 and DTLZ7 in terms of the IGD metric. 6 m RIEA MOEA/D PICEA-G GrEA HypE DTLZ5 4 1.034E−02 2.686E−02(3) 3.708E−02 1.249E−02 1.001E−01 1.734E−02 3.887E−02 3.752E−02 1.846E−02 1.216E−01 6.610E−02 8.625E−02 6.820E−02 8.060E−02 8.010E−01 5 8.272E−01 6.439E−02 8.472E−01 2.214E−02 1.274E−01 8.325E−01 8.499E−01 8.542E−01 4.333E−02 1.459E−01 8.788E−01 9.865E−01 8.722E−01 3.778E−01 9.190E−01 6 1.049E−01 9.151E−02 1.048E+00 2.097E−02 7.315E−01 1.120E−01 1.048E−01 1.209E+00 9.575E−02 1.734E−01 4.097E−01 4.806E−01 2.067E+00 8.060E−01 1.209E+00 8 1.245E−01 1.004E−01 1.245E+00 1.645E−01 1.504E−01 1.357E−01 1.245E−01 1.576E+00 2.327E−01 1.754E−01 2.334E−01 2.749E−01 1.952E+00 5.588E−01 1.565E+00 10 1.333E−01 1.012E−01 1.333E+00 2.244E−01 1.168E−01 1.373E−01 1.332E−01 1.689E+00 3.462E−01 1.560E−01 1.407E−01 3.373E−01 1.814E+00 1.157E+00 1.836E−01 DTLZ6 4 3.807E−02 3.744E−02 3.886E−02 3.440E−02 1.937E−01 8.094E−02 8.015E−02 8.939E−02 7.045E−02 3.919E+00 8.896E−02 9.284E−02 1.086E−01 9.398E−02 4.550E+00 5 8.314E−02 8.420E−02 8.499E−01 9.913E−02 4.806E+00 1.049E−01 1.194E−01 9.078E−01 1.429E−01 5.431E+00 3.364E−01 2.094E−01 2.391E+00 4.155E−01 7.076E+00 6 1.038E−01 1.048E−01 1.081E+00 4.172E−01 4.927E+00 4.340E−01 1.569E−01 1.210E+00 4.544E−01 5.679E+00 5.735E−01 1.991E−01 4.048E+00 8.060E−01 8.779E+00 8 1.152E−01 1.245E−01 1.543E+00 4.387E−01 4.395E+00 1.645E−01 1.830E−01 1.577E+00 5.971E−01 6.165E+00 6.234E−01 3.494E−01 3.245E+00 6.487E−01 9.274E+00 10 1.391E−01 1.419E−01 1.332E+00 5.406E−01 5.225E+00 2.514E−01 2.692E−01 1.689E+00 9.432E−01 6.428E+00 3.232E−01 2.756E−01 5.332E+00 1.077E+00 9.432E+00 DTLZ7 4 3.257E−01 3.234E−01 6.679E−01 1.427E−02 3.062E−01 3.432E−01 3.356E−01 7.685E−01 1.897E−02 4.846E−01 4.500E−01 4.397E−01 8.005E−01 4.970E−02 6.968E−01 5 3.792E−01 3.626E−01 9.438E−01 1.806E−02 8.478E−01 5.302E−01 5.326E−01 9.881E−01 3.328E−02 8.972E−01 7.341E−01 8.478E−01 1.048E+00 5.801E−02 9.312E−01 6 5.595E−01 5.101E−01 1.146E+00 4.387E−02 9.190E−01 8.370E−01 6.186E−01 1.201E+00 4.888E−01 9.894E−01 1.029E+00 7.632E−01 1.262E+00 7.084E−01 1.057E+00 8 8.876E−01 4.793E−01 1.599E+00 9.779E−02 1.010E+00 1.233E+00 6.096E−01 1.674E+00 7.643E−01 1.065E+00 1.613E+00 1.076E+00 1.745E+00 7.927E−01 2.627E+00 10 1.090E+00 4.020E−01 2.024E+00 1.014E+00 1.006E+00 1.094E+00 6.441E−01 2.156E+00 1.057E+00 1.224E+00 1.148E+00 1.038E+00 2.318E+00 1.259E+00 6.960E+00 6 m RIEA MOEA/D PICEA-G GrEA HypE DTLZ5 4 1.034E−02 2.686E−02(3) 3.708E−02 1.249E−02 1.001E−01 1.734E−02 3.887E−02 3.752E−02 1.846E−02 1.216E−01 6.610E−02 8.625E−02 6.820E−02 8.060E−02 8.010E−01 5 8.272E−01 6.439E−02 8.472E−01 2.214E−02 1.274E−01 8.325E−01 8.499E−01 8.542E−01 4.333E−02 1.459E−01 8.788E−01 9.865E−01 8.722E−01 3.778E−01 9.190E−01 6 1.049E−01 9.151E−02 1.048E+00 2.097E−02 7.315E−01 1.120E−01 1.048E−01 1.209E+00 9.575E−02 1.734E−01 4.097E−01 4.806E−01 2.067E+00 8.060E−01 1.209E+00 8 1.245E−01 1.004E−01 1.245E+00 1.645E−01 1.504E−01 1.357E−01 1.245E−01 1.576E+00 2.327E−01 1.754E−01 2.334E−01 2.749E−01 1.952E+00 5.588E−01 1.565E+00 10 1.333E−01 1.012E−01 1.333E+00 2.244E−01 1.168E−01 1.373E−01 1.332E−01 1.689E+00 3.462E−01 1.560E−01 1.407E−01 3.373E−01 1.814E+00 1.157E+00 1.836E−01 DTLZ6 4 3.807E−02 3.744E−02 3.886E−02 3.440E−02 1.937E−01 8.094E−02 8.015E−02 8.939E−02 7.045E−02 3.919E+00 8.896E−02 9.284E−02 1.086E−01 9.398E−02 4.550E+00 5 8.314E−02 8.420E−02 8.499E−01 9.913E−02 4.806E+00 1.049E−01 1.194E−01 9.078E−01 1.429E−01 5.431E+00 3.364E−01 2.094E−01 2.391E+00 4.155E−01 7.076E+00 6 1.038E−01 1.048E−01 1.081E+00 4.172E−01 4.927E+00 4.340E−01 1.569E−01 1.210E+00 4.544E−01 5.679E+00 5.735E−01 1.991E−01 4.048E+00 8.060E−01 8.779E+00 8 1.152E−01 1.245E−01 1.543E+00 4.387E−01 4.395E+00 1.645E−01 1.830E−01 1.577E+00 5.971E−01 6.165E+00 6.234E−01 3.494E−01 3.245E+00 6.487E−01 9.274E+00 10 1.391E−01 1.419E−01 1.332E+00 5.406E−01 5.225E+00 2.514E−01 2.692E−01 1.689E+00 9.432E−01 6.428E+00 3.232E−01 2.756E−01 5.332E+00 1.077E+00 9.432E+00 DTLZ7 4 3.257E−01 3.234E−01 6.679E−01 1.427E−02 3.062E−01 3.432E−01 3.356E−01 7.685E−01 1.897E−02 4.846E−01 4.500E−01 4.397E−01 8.005E−01 4.970E−02 6.968E−01 5 3.792E−01 3.626E−01 9.438E−01 1.806E−02 8.478E−01 5.302E−01 5.326E−01 9.881E−01 3.328E−02 8.972E−01 7.341E−01 8.478E−01 1.048E+00 5.801E−02 9.312E−01 6 5.595E−01 5.101E−01 1.146E+00 4.387E−02 9.190E−01 8.370E−01 6.186E−01 1.201E+00 4.888E−01 9.894E−01 1.029E+00 7.632E−01 1.262E+00 7.084E−01 1.057E+00 8 8.876E−01 4.793E−01 1.599E+00 9.779E−02 1.010E+00 1.233E+00 6.096E−01 1.674E+00 7.643E−01 1.065E+00 1.613E+00 1.076E+00 1.745E+00 7.927E−01 2.627E+00 10 1.090E+00 4.020E−01 2.024E+00 1.014E+00 1.006E+00 1.094E+00 6.441E−01 2.156E+00 1.057E+00 1.224E+00 1.148E+00 1.038E+00 2.318E+00 1.259E+00 6.960E+00 View Large Table 5. Statistical results (best/mean/worst) obtained by RIEA, MOEA/D, PICEA-G, GrEA and HypE on DLTZ5, DTLZ6 and DTLZ7 in terms of the IGD metric. 6 m RIEA MOEA/D PICEA-G GrEA HypE DTLZ5 4 1.034E−02 2.686E−02(3) 3.708E−02 1.249E−02 1.001E−01 1.734E−02 3.887E−02 3.752E−02 1.846E−02 1.216E−01 6.610E−02 8.625E−02 6.820E−02 8.060E−02 8.010E−01 5 8.272E−01 6.439E−02 8.472E−01 2.214E−02 1.274E−01 8.325E−01 8.499E−01 8.542E−01 4.333E−02 1.459E−01 8.788E−01 9.865E−01 8.722E−01 3.778E−01 9.190E−01 6 1.049E−01 9.151E−02 1.048E+00 2.097E−02 7.315E−01 1.120E−01 1.048E−01 1.209E+00 9.575E−02 1.734E−01 4.097E−01 4.806E−01 2.067E+00 8.060E−01 1.209E+00 8 1.245E−01 1.004E−01 1.245E+00 1.645E−01 1.504E−01 1.357E−01 1.245E−01 1.576E+00 2.327E−01 1.754E−01 2.334E−01 2.749E−01 1.952E+00 5.588E−01 1.565E+00 10 1.333E−01 1.012E−01 1.333E+00 2.244E−01 1.168E−01 1.373E−01 1.332E−01 1.689E+00 3.462E−01 1.560E−01 1.407E−01 3.373E−01 1.814E+00 1.157E+00 1.836E−01 DTLZ6 4 3.807E−02 3.744E−02 3.886E−02 3.440E−02 1.937E−01 8.094E−02 8.015E−02 8.939E−02 7.045E−02 3.919E+00 8.896E−02 9.284E−02 1.086E−01 9.398E−02 4.550E+00 5 8.314E−02 8.420E−02 8.499E−01 9.913E−02 4.806E+00 1.049E−01 1.194E−01 9.078E−01 1.429E−01 5.431E+00 3.364E−01 2.094E−01 2.391E+00 4.155E−01 7.076E+00 6 1.038E−01 1.048E−01 1.081E+00 4.172E−01 4.927E+00 4.340E−01 1.569E−01 1.210E+00 4.544E−01 5.679E+00 5.735E−01 1.991E−01 4.048E+00 8.060E−01 8.779E+00 8 1.152E−01 1.245E−01 1.543E+00 4.387E−01 4.395E+00 1.645E−01 1.830E−01 1.577E+00 5.971E−01 6.165E+00 6.234E−01 3.494E−01 3.245E+00 6.487E−01 9.274E+00 10 1.391E−01 1.419E−01 1.332E+00 5.406E−01 5.225E+00 2.514E−01 2.692E−01 1.689E+00 9.432E−01 6.428E+00 3.232E−01 2.756E−01 5.332E+00 1.077E+00 9.432E+00 DTLZ7 4 3.257E−01 3.234E−01 6.679E−01 1.427E−02 3.062E−01 3.432E−01 3.356E−01 7.685E−01 1.897E−02 4.846E−01 4.500E−01 4.397E−01 8.005E−01 4.970E−02 6.968E−01 5 3.792E−01 3.626E−01 9.438E−01 1.806E−02 8.478E−01 5.302E−01 5.326E−01 9.881E−01 3.328E−02 8.972E−01 7.341E−01 8.478E−01 1.048E+00 5.801E−02 9.312E−01 6 5.595E−01 5.101E−01 1.146E+00 4.387E−02 9.190E−01 8.370E−01 6.186E−01 1.201E+00 4.888E−01 9.894E−01 1.029E+00 7.632E−01 1.262E+00 7.084E−01 1.057E+00 8 8.876E−01 4.793E−01 1.599E+00 9.779E−02 1.010E+00 1.233E+00 6.096E−01 1.674E+00 7.643E−01 1.065E+00 1.613E+00 1.076E+00 1.745E+00 7.927E−01 2.627E+00 10 1.090E+00 4.020E−01 2.024E+00 1.014E+00 1.006E+00 1.094E+00 6.441E−01 2.156E+00 1.057E+00 1.224E+00 1.148E+00 1.038E+00 2.318E+00 1.259E+00 6.960E+00 6 m RIEA MOEA/D PICEA-G GrEA HypE DTLZ5 4 1.034E−02 2.686E−02(3) 3.708E−02 1.249E−02 1.001E−01 1.734E−02 3.887E−02 3.752E−02 1.846E−02 1.216E−01 6.610E−02 8.625E−02 6.820E−02 8.060E−02 8.010E−01 5 8.272E−01 6.439E−02 8.472E−01 2.214E−02 1.274E−01 8.325E−01 8.499E−01 8.542E−01 4.333E−02 1.459E−01 8.788E−01 9.865E−01 8.722E−01 3.778E−01 9.190E−01 6 1.049E−01 9.151E−02 1.048E+00 2.097E−02 7.315E−01 1.120E−01 1.048E−01 1.209E+00 9.575E−02 1.734E−01 4.097E−01 4.806E−01 2.067E+00 8.060E−01 1.209E+00 8 1.245E−01 1.004E−01 1.245E+00 1.645E−01 1.504E−01 1.357E−01 1.245E−01 1.576E+00 2.327E−01 1.754E−01 2.334E−01 2.749E−01 1.952E+00 5.588E−01 1.565E+00 10 1.333E−01 1.012E−01 1.333E+00 2.244E−01 1.168E−01 1.373E−01 1.332E−01 1.689E+00 3.462E−01 1.560E−01 1.407E−01 3.373E−01 1.814E+00 1.157E+00 1.836E−01 DTLZ6 4 3.807E−02 3.744E−02 3.886E−02 3.440E−02 1.937E−01 8.094E−02 8.015E−02 8.939E−02 7.045E−02 3.919E+00 8.896E−02 9.284E−02 1.086E−01 9.398E−02 4.550E+00 5 8.314E−02 8.420E−02 8.499E−01 9.913E−02 4.806E+00 1.049E−01 1.194E−01 9.078E−01 1.429E−01 5.431E+00 3.364E−01 2.094E−01 2.391E+00 4.155E−01 7.076E+00 6 1.038E−01 1.048E−01 1.081E+00 4.172E−01 4.927E+00 4.340E−01 1.569E−01 1.210E+00 4.544E−01 5.679E+00 5.735E−01 1.991E−01 4.048E+00 8.060E−01 8.779E+00 8 1.152E−01 1.245E−01 1.543E+00 4.387E−01 4.395E+00 1.645E−01 1.830E−01 1.577E+00 5.971E−01 6.165E+00 6.234E−01 3.494E−01 3.245E+00 6.487E−01 9.274E+00 10 1.391E−01 1.419E−01 1.332E+00 5.406E−01 5.225E+00 2.514E−01 2.692E−01 1.689E+00 9.432E−01 6.428E+00 3.232E−01 2.756E−01 5.332E+00 1.077E+00 9.432E+00 DTLZ7 4 3.257E−01 3.234E−01 6.679E−01 1.427E−02 3.062E−01 3.432E−01 3.356E−01 7.685E−01 1.897E−02 4.846E−01 4.500E−01 4.397E−01 8.005E−01 4.970E−02 6.968E−01 5 3.792E−01 3.626E−01 9.438E−01 1.806E−02 8.478E−01 5.302E−01 5.326E−01 9.881E−01 3.328E−02 8.972E−01 7.341E−01 8.478E−01 1.048E+00 5.801E−02 9.312E−01 6 5.595E−01 5.101E−01 1.146E+00 4.387E−02 9.190E−01 8.370E−01 6.186E−01 1.201E+00 4.888E−01 9.894E−01 1.029E+00 7.632E−01 1.262E+00 7.084E−01 1.057E+00 8 8.876E−01 4.793E−01 1.599E+00 9.779E−02 1.010E+00 1.233E+00 6.096E−01 1.674E+00 7.643E−01 1.065E+00 1.613E+00 1.076E+00 1.745E+00 7.927E−01 2.627E+00 10 1.090E+00 4.020E−01 2.024E+00 1.014E+00 1.006E+00 1.094E+00 6.441E−01 2.156E+00 1.057E+00 1.224E+00 1.148E+00 1.038E+00 2.318E+00 1.259E+00 6.960E+00 View Large 4.4. Parameter sensitivity study There are three main parameters in RIEA, i.e. the neighbourhood size ( T), the mating control probability ( δ) and the minimum number of solutions kept in a subregion ( L). To study how these parameters affect the performance of RIEA, we varied these parameters and tried a variety of combinations on DTLZ1 and DTLZ4 with 5 and 10 objectives. For T, we choose it from {5,15,25,35,45}; δ is chosen from {0,0.1,…,1.0} and L is chosen from {3,5,8,11,15}. Twenty independent runs were carried out for each combination of the two parameters on each instance. Figure 7 shows the median IGD values obtained by these parameter combinations on the selected test instances. From these figures, we can observe that different parameter combinations lead to different performances. Specifically, along with the increase of δ, the performance of RIEA becomes better, while the neighbourhood size does not affect the algorithmic performance much. We find that with δ=0.0, RIEA’s performance is always the worst. With δ=1.0, the performance is also not satisfactory. A proper chosen δ∈[0.6,0.9] is suggested. On the other hand, the neighbourhood size T does not affect the algorithmic performance much. We choose T=5 in the experiment. Figure 8 shows the boxplots of the IGD values obtained by RIEA with different L values on DTLZ1 and DTLZ4 with five and ten objectives. From the figure, we can see that L=5 is the best choice overall, but the optimal choice of L varies for different instances. Figure 7. View largeDownload slide Median IGD values found by RIEA with different combinations of parameters δ and T on DTLZ1 and DTLZ4 with 5 and 10 objectives, respectively. Figure 7. View largeDownload slide Median IGD values found by RIEA with different combinations of parameters δ and T on DTLZ1 and DTLZ4 with 5 and 10 objectives, respectively. Figure 8. View largeDownload slide The boxplots of the IGD values obtained by RIEA with different L values on DTLZ2 and DTLZ4 with 5 and 10 objectives, respectively. Figure 8. View largeDownload slide The boxplots of the IGD values obtained by RIEA with different L values on DTLZ2 and DTLZ4 with 5 and 10 objectives, respectively. 4.5. Component analysis In this section, we investigate the important components in the developed algorithm. First, we study the effect of the subregion decomposition to the performance of RIEA. We compare the performance of RIEA with and without subregion on the test instances with 3–15 objectives. RIEA without subregion was run 20 times on these test instances. The obtained IGD values were compared with the results obtained by RIEA with subregion. The boxplots of these results are shown in Fig. 9. From the plots, we can observe that RIEA with subregion clearly outperforms RIEA without subregion. Figure 9. View largeDownload slide The boxplots of the IGD values obtained by RIEA with and without subregion on DTLZ1–DTLZ4 with three to 15 objectives, respectively. In the plots, the results obtained by RIEA without subregion is shown in green. Figure 9. View largeDownload slide The boxplots of the IGD values obtained by RIEA with and without subregion on DTLZ1–DTLZ4 with three to 15 objectives, respectively. In the plots, the results obtained by RIEA without subregion is shown in green. Second, we study how the generation of the reference points affect the performance of RIEA. To do the study, under the same algorithmic framework, we replace the randomly generated reference points with a set of reference points that are uniformly distributed in the objective space. The RIEA with uniformly distributed reference points is called RIEA_U, while RIEA_R stands for the algorithm with randomly generated reference points. To generate the uniformly distributed points, we adopt the two-layer weight generation method developed in NSGA-III [24]. Note that a set of m×100 reference points is generated in RIEA. However, by using the two-layer method, we cannot get the exact number of reference points. Instead we choose the appropriate H values for the boundary and inside layers to approximate the number. RIEA_U was compared with RIEA_R on the DTLZ test suits in terms of the IGD and HV metrics. Table 6 summarises the obtained results. From the table, we see that the uniformly distributed reference points do not bring advantages over the randomly generated reference points significantly. Table 6. Comparison results of the mean values obtained by RIEA with reference points generated randomly and uniformly in 30 independent runs. Test instance m IGD HV Test instance m IGD HV RIEA_R RIEA_U RIEA_R RIEA_U RIEA_R RIEA_U RIEA_R RIEA_U DTLZ1 3 2.567E−03 2.150E−03 0.9690 0.9736 DTLZ3 3 2.450E−02 5.726E−02 0.6287 0.6889 5 7.420E−03 6.635E−03 0.9957 0.9953 5 3.112E−02 3.101E−02 0.7458 0.7584 8 2.411E−02 2.159E−02 0.9959 0.9983 8 7.554E−02 2.990E−02 0.5557 0.6233 10 2.880E−02 2.964E−02 0.9971 0.9987 10 6.757E−02 5.295E−02 0.7648 0.9750 15 6.457E−02 6.764E−02 0.9810 0.9821 15 3.384E−01 3.043E−01 0.3099 0.3310 DTLZ2 3 2.666E−03 2.948E−03 0.9254 0.9252 DTLZ4 3 4.825E−03 2.943E−03 0.9248 0.9269 5 9.255E−03 1.085E−02 0.9845 0.9804 5 9.338E−03 1.293E−02 0.9863 0.9761 8 2.574E−02 2.652E−02 0.9936 0.9949 8 2.946E−02 2.711E−02 0.9957 0.9970 10 3.028E−02 2.951E−02 0.9993 0.9991 10 2.764E−02 2.991E−02 0.9997 0.9996 15 5.915E−02 6.323E−02 0.9997 0.9974 15 5.576E−02 5.953E−02 0.9998 0.9994 Test instance m IGD HV Test instance m IGD HV RIEA_R RIEA_U RIEA_R RIEA_U RIEA_R RIEA_U RIEA_R RIEA_U DTLZ1 3 2.567E−03 2.150E−03 0.9690 0.9736 DTLZ3 3 2.450E−02 5.726E−02 0.6287 0.6889 5 7.420E−03 6.635E−03 0.9957 0.9953 5 3.112E−02 3.101E−02 0.7458 0.7584 8 2.411E−02 2.159E−02 0.9959 0.9983 8 7.554E−02 2.990E−02 0.5557 0.6233 10 2.880E−02 2.964E−02 0.9971 0.9987 10 6.757E−02 5.295E−02 0.7648 0.9750 15 6.457E−02 6.764E−02 0.9810 0.9821 15 3.384E−01 3.043E−01 0.3099 0.3310 DTLZ2 3 2.666E−03 2.948E−03 0.9254 0.9252 DTLZ4 3 4.825E−03 2.943E−03 0.9248 0.9269 5 9.255E−03 1.085E−02 0.9845 0.9804 5 9.338E−03 1.293E−02 0.9863 0.9761 8 2.574E−02 2.652E−02 0.9936 0.9949 8 2.946E−02 2.711E−02 0.9957 0.9970 10 3.028E−02 2.951E−02 0.9993 0.9991 10 2.764E−02 2.991E−02 0.9997 0.9996 15 5.915E−02 6.323E−02 0.9997 0.9974 15 5.576E−02 5.953E−02 0.9998 0.9994 View Large Table 6. Comparison results of the mean values obtained by RIEA with reference points generated randomly and uniformly in 30 independent runs. Test instance m IGD HV Test instance m IGD HV RIEA_R RIEA_U RIEA_R RIEA_U RIEA_R RIEA_U RIEA_R RIEA_U DTLZ1 3 2.567E−03 2.150E−03 0.9690 0.9736 DTLZ3 3 2.450E−02 5.726E−02 0.6287 0.6889 5 7.420E−03 6.635E−03 0.9957 0.9953 5 3.112E−02 3.101E−02 0.7458 0.7584 8 2.411E−02 2.159E−02 0.9959 0.9983 8 7.554E−02 2.990E−02 0.5557 0.6233 10 2.880E−02 2.964E−02 0.9971 0.9987 10 6.757E−02 5.295E−02 0.7648 0.9750 15 6.457E−02 6.764E−02 0.9810 0.9821 15 3.384E−01 3.043E−01 0.3099 0.3310 DTLZ2 3 2.666E−03 2.948E−03 0.9254 0.9252 DTLZ4 3 4.825E−03 2.943E−03 0.9248 0.9269 5 9.255E−03 1.085E−02 0.9845 0.9804 5 9.338E−03 1.293E−02 0.9863 0.9761 8 2.574E−02 2.652E−02 0.9936 0.9949 8 2.946E−02 2.711E−02 0.9957 0.9970 10 3.028E−02 2.951E−02 0.9993 0.9991 10 2.764E−02 2.991E−02 0.9997 0.9996 15 5.915E−02 6.323E−02 0.9997 0.9974 15 5.576E−02 5.953E−02 0.9998 0.9994 Test instance m IGD HV Test instance m IGD HV RIEA_R RIEA_U RIEA_R RIEA_U RIEA_R RIEA_U RIEA_R RIEA_U DTLZ1 3 2.567E−03 2.150E−03 0.9690 0.9736 DTLZ3 3 2.450E−02 5.726E−02 0.6287 0.6889 5 7.420E−03 6.635E−03 0.9957 0.9953 5 3.112E−02 3.101E−02 0.7458 0.7584 8 2.411E−02 2.159E−02 0.9959 0.9983 8 7.554E−02 2.990E−02 0.5557 0.6233 10 2.880E−02 2.964E−02 0.9971 0.9987 10 6.757E−02 5.295E−02 0.7648 0.9750 15 6.457E−02 6.764E−02 0.9810 0.9821 15 3.384E−01 3.043E−01 0.3099 0.3310 DTLZ2 3 2.666E−03 2.948E−03 0.9254 0.9252 DTLZ4 3 4.825E−03 2.943E−03 0.9248 0.9269 5 9.255E−03 1.085E−02 0.9845 0.9804 5 9.338E−03 1.293E−02 0.9863 0.9761 8 2.574E−02 2.652E−02 0.9936 0.9949 8 2.946E−02 2.711E−02 0.9957 0.9970 10 3.028E−02 2.951E−02 0.9993 0.9991 10 2.764E−02 2.991E−02 0.9997 0.9996 15 5.915E−02 6.323E−02 0.9997 0.9974 15 5.576E−02 5.953E−02 0.9998 0.9994 View Large We further study the effect of the recombination operators, namely DE and SBX, on the performance of the developed algorithm. Table 7 lists the obtained statistics in terms of the IGD metric. In the table, RIEA/DE (RIEA/SBX) denotes that DE (SBX) is used. From the table, it is observed that roughly speaking, DE works better on the test problems with fewer objectives, while SBX is good for problems with higher objective number under the proposed algorithmic framework. Table 7. Comparison results (best/mean/worst/std) obtained by RIEA with DE and SBX crossover operator on DLTZ1–DTLZ4 in terms of the IGD metric. m RIEA/DE RIEA/SBX RIEA/DE RIEA/SBX DTLZ1 3 2.227E−03 3.830E−03 DTLZ3 1.218E−02 1.209E−02 3.350E−03 6.960E−03 2.450E−02 2.526E−02 4.736E−03 2.774E−02 6.114E−02 1.036E−01 6.769E−04 5.198E−03 1.373E−02 2.072E−02 5 6.219E−03 9.115E−03 2.059E−02 2.842E−02 9.581E−03 1.216E−02 3.652E−02 3.626E−02 1.479E−02 1.731E−02 9.316E−02 6.009E−02 2.230E−03 1.858E−03 1.732E−02 1.160E−02 8 1.919E−02 2.506E−02 4.910E−02 5.043E−02 2.519E−02 2.880E−02 7.554E−02 6.657E−02 3.150E−02 3.688E−02 1.270E−01 9.671E−02 3.426E−03 3.054E−03 2.451E−02 1.751E−02 10 2.362E−02 2.697E−02 5.478E−02 4.872E−02 2.851E−02 3.335E−02 1.421E−01 7.097E−02 3.988E−02 4.216E−02 8.350E−01 1.597E−01 3.677E−03 4.108E−03 1.984E−01 2.627E−02 15 4.359E−02 4.873E−02 1.024E−01 8.878E−02 6.318E−02 6.203E−02 3.384E−01 1.403E−01 1.020E−01 9.609E−02 2.698E+00 2.432E−01 1.507E−02 1.031E−02 5.747E−01 4.189E−02 DTLZ2 3 2.531E−03 3.839E−03 DTLZ4 5.205E−03 9.866E−03 2.666E−03 6.550E−03 7.356E−03 1.378E−02 2.828E−03 1.210E−02 9.211E−03 1.812E−02 9.284E−05 2.180E−03 1.033E−03 1.670E−03 5 8.201E−03 8.898E−03 1.320E−02 1.772E−02 8.754E−03 1.443E−02 1.628E−02 2.206E−02 9.992E−03 1.996E−02 2.029E−02 2.318E−02 4.755E−04 2.761E−03 1.893E−03 1.741E−03 8 2.250E−02 2.704E−02 2.564E−02 2.785E−02 2.574E−02 3.581E−02 2.827E−02 3.615E−02 3.155E−02 4.310E−02 3.212E−02 4.318E−02 2.451E−03 5.369E−03 1.950E−03 4.068E−03 10 3.019E−02 2.574E−02 3.059E−02 2.636E−02 3.320E−02 3.040E−02 3.291E−02 3.090E−02 3.621E−02 4.084E−02 3.727E−02 3.625E−02 1.853E−03 3.574E−03 2.010E−03 2.512E−03 15 6.861E−02 5.535E−02 6.832E−02 5.238E−02 7.544E−02 6.720E−02 7.410E−02 6.340E−02 8.369E−02 7.683E−02 7.983E−02 7.763E−02 3.597E−03 6.766E−03 3.383E−03 6.057E−03 m RIEA/DE RIEA/SBX RIEA/DE RIEA/SBX DTLZ1 3 2.227E−03 3.830E−03 DTLZ3 1.218E−02 1.209E−02 3.350E−03 6.960E−03 2.450E−02 2.526E−02 4.736E−03 2.774E−02 6.114E−02 1.036E−01 6.769E−04 5.198E−03 1.373E−02 2.072E−02 5 6.219E−03 9.115E−03 2.059E−02 2.842E−02 9.581E−03 1.216E−02 3.652E−02 3.626E−02 1.479E−02 1.731E−02 9.316E−02 6.009E−02 2.230E−03 1.858E−03 1.732E−02 1.160E−02 8 1.919E−02 2.506E−02 4.910E−02 5.043E−02 2.519E−02 2.880E−02 7.554E−02 6.657E−02 3.150E−02 3.688E−02 1.270E−01 9.671E−02 3.426E−03 3.054E−03 2.451E−02 1.751E−02 10 2.362E−02 2.697E−02 5.478E−02 4.872E−02 2.851E−02 3.335E−02 1.421E−01 7.097E−02 3.988E−02 4.216E−02 8.350E−01 1.597E−01 3.677E−03 4.108E−03 1.984E−01 2.627E−02 15 4.359E−02 4.873E−02 1.024E−01 8.878E−02 6.318E−02 6.203E−02 3.384E−01 1.403E−01 1.020E−01 9.609E−02 2.698E+00 2.432E−01 1.507E−02 1.031E−02 5.747E−01 4.189E−02 DTLZ2 3 2.531E−03 3.839E−03 DTLZ4 5.205E−03 9.866E−03 2.666E−03 6.550E−03 7.356E−03 1.378E−02 2.828E−03 1.210E−02 9.211E−03 1.812E−02 9.284E−05 2.180E−03 1.033E−03 1.670E−03 5 8.201E−03 8.898E−03 1.320E−02 1.772E−02 8.754E−03 1.443E−02 1.628E−02 2.206E−02 9.992E−03 1.996E−02 2.029E−02 2.318E−02 4.755E−04 2.761E−03 1.893E−03 1.741E−03 8 2.250E−02 2.704E−02 2.564E−02 2.785E−02 2.574E−02 3.581E−02 2.827E−02 3.615E−02 3.155E−02 4.310E−02 3.212E−02 4.318E−02 2.451E−03 5.369E−03 1.950E−03 4.068E−03 10 3.019E−02 2.574E−02 3.059E−02 2.636E−02 3.320E−02 3.040E−02 3.291E−02 3.090E−02 3.621E−02 4.084E−02 3.727E−02 3.625E−02 1.853E−03 3.574E−03 2.010E−03 2.512E−03 15 6.861E−02 5.535E−02 6.832E−02 5.238E−02 7.544E−02 6.720E−02 7.410E−02 6.340E−02 8.369E−02 7.683E−02 7.983E−02 7.763E−02 3.597E−03 6.766E−03 3.383E−03 6.057E−03 View Large Table 7. Comparison results (best/mean/worst/std) obtained by RIEA with DE and SBX crossover operator on DLTZ1–DTLZ4 in terms of the IGD metric. m RIEA/DE RIEA/SBX RIEA/DE RIEA/SBX DTLZ1 3 2.227E−03 3.830E−03 DTLZ3 1.218E−02 1.209E−02 3.350E−03 6.960E−03 2.450E−02 2.526E−02 4.736E−03 2.774E−02 6.114E−02 1.036E−01 6.769E−04 5.198E−03 1.373E−02 2.072E−02 5 6.219E−03 9.115E−03 2.059E−02 2.842E−02 9.581E−03 1.216E−02 3.652E−02 3.626E−02 1.479E−02 1.731E−02 9.316E−02 6.009E−02 2.230E−03 1.858E−03 1.732E−02 1.160E−02 8 1.919E−02 2.506E−02 4.910E−02 5.043E−02 2.519E−02 2.880E−02 7.554E−02 6.657E−02 3.150E−02 3.688E−02 1.270E−01 9.671E−02 3.426E−03 3.054E−03 2.451E−02 1.751E−02 10 2.362E−02 2.697E−02 5.478E−02 4.872E−02 2.851E−02 3.335E−02 1.421E−01 7.097E−02 3.988E−02 4.216E−02 8.350E−01 1.597E−01 3.677E−03 4.108E−03 1.984E−01 2.627E−02 15 4.359E−02 4.873E−02 1.024E−01 8.878E−02 6.318E−02 6.203E−02 3.384E−01 1.403E−01 1.020E−01 9.609E−02 2.698E+00 2.432E−01 1.507E−02 1.031E−02 5.747E−01 4.189E−02 DTLZ2 3 2.531E−03 3.839E−03 DTLZ4 5.205E−03 9.866E−03 2.666E−03 6.550E−03 7.356E−03 1.378E−02 2.828E−03 1.210E−02 9.211E−03 1.812E−02 9.284E−05 2.180E−03 1.033E−03 1.670E−03 5 8.201E−03 8.898E−03 1.320E−02 1.772E−02 8.754E−03 1.443E−02 1.628E−02 2.206E−02 9.992E−03 1.996E−02 2.029E−02 2.318E−02 4.755E−04 2.761E−03 1.893E−03 1.741E−03 8 2.250E−02 2.704E−02 2.564E−02 2.785E−02 2.574E−02 3.581E−02 2.827E−02 3.615E−02 3.155E−02 4.310E−02 3.212E−02 4.318E−02 2.451E−03 5.369E−03 1.950E−03 4.068E−03 10 3.019E−02 2.574E−02 3.059E−02 2.636E−02 3.320E−02 3.040E−02 3.291E−02 3.090E−02 3.621E−02 4.084E−02 3.727E−02 3.625E−02 1.853E−03 3.574E−03 2.010E−03 2.512E−03 15 6.861E−02 5.535E−02 6.832E−02 5.238E−02 7.544E−02 6.720E−02 7.410E−02 6.340E−02 8.369E−02 7.683E−02 7.983E−02 7.763E−02 3.597E−03 6.766E−03 3.383E−03 6.057E−03 m RIEA/DE RIEA/SBX RIEA/DE RIEA/SBX DTLZ1 3 2.227E−03 3.830E−03 DTLZ3 1.218E−02 1.209E−02 3.350E−03 6.960E−03 2.450E−02 2.526E−02 4.736E−03 2.774E−02 6.114E−02 1.036E−01 6.769E−04 5.198E−03 1.373E−02 2.072E−02 5 6.219E−03 9.115E−03 2.059E−02 2.842E−02 9.581E−03 1.216E−02 3.652E−02 3.626E−02 1.479E−02 1.731E−02 9.316E−02 6.009E−02 2.230E−03 1.858E−03 1.732E−02 1.160E−02 8 1.919E−02 2.506E−02 4.910E−02 5.043E−02 2.519E−02 2.880E−02 7.554E−02 6.657E−02 3.150E−02 3.688E−02 1.270E−01 9.671E−02 3.426E−03 3.054E−03 2.451E−02 1.751E−02 10 2.362E−02 2.697E−02 5.478E−02 4.872E−02 2.851E−02 3.335E−02 1.421E−01 7.097E−02 3.988E−02 4.216E−02 8.350E−01 1.597E−01 3.677E−03 4.108E−03 1.984E−01 2.627E−02 15 4.359E−02 4.873E−02 1.024E−01 8.878E−02 6.318E−02 6.203E−02 3.384E−01 1.403E−01 1.020E−01 9.609E−02 2.698E+00 2.432E−01 1.507E−02 1.031E−02 5.747E−01 4.189E−02 DTLZ2 3 2.531E−03 3.839E−03 DTLZ4 5.205E−03 9.866E−03 2.666E−03 6.550E−03 7.356E−03 1.378E−02 2.828E−03 1.210E−02 9.211E−03 1.812E−02 9.284E−05 2.180E−03 1.033E−03 1.670E−03 5 8.201E−03 8.898E−03 1.320E−02 1.772E−02 8.754E−03 1.443E−02 1.628E−02 2.206E−02 9.992E−03 1.996E−02 2.029E−02 2.318E−02 4.755E−04 2.761E−03 1.893E−03 1.741E−03 8 2.250E−02 2.704E−02 2.564E−02 2.785E−02 2.574E−02 3.581E−02 2.827E−02 3.615E−02 3.155E−02 4.310E−02 3.212E−02 4.318E−02 2.451E−03 5.369E−03 1.950E−03 4.068E−03 10 3.019E−02 2.574E−02 3.059E−02 2.636E−02 3.320E−02 3.040E−02 3.291E−02 3.090E−02 3.621E−02 4.084E−02 3.727E−02 3.625E−02 1.853E−03 3.574E−03 2.010E−03 2.512E−03 15 6.861E−02 5.535E−02 6.832E−02 5.238E−02 7.544E−02 6.720E−02 7.410E−02 6.340E−02 8.369E−02 7.683E−02 7.983E−02 7.763E−02 3.597E−03 6.766E−03 3.383E−03 6.057E−03 View Large 5. CONCLUSION This paper presented a multiobjective evolutionary algorithm, called RIEA, inspired by a set of randomly generated reference points, for many objective optimisation problems. A new fitness assignment scheme induced from these reference points is developed to integrate dominance and convergence-diversity information for effective ranking of solutions. Empirical study has shown that the developed fitness scheme is capable to improve comparability among candidate solutions in a high-dimensional objective space. A set of reference vectors is employed to divide the search space into subregions to facilitate diversity management. A restricted mating selection strategy which involves the construction of convergence elite set and diversity elite set at each subregion is proposed for selecting mating solutions to improve search efficiency. Empirical studies have shown that the subregion decomposition and the mating selection scheme based on the decomposition can indeed improve the search efficiency in terms of diversity and convergence. Moreover, we have shown that RIEA compared favourably against some well-known MOEAs, including MOEA/D, PICEA-g, GrEA and HypE on the widely used DTLZ test suite with many objectives. Further research avenues may include (1) to design new fitness assignment scheme considering preference knowledge; (2) to apply to new test suite [73] which may better reflect real-world complexities than the DTLZ test suite; and (3) to apply to real-world many-objective optimisation problems induced in image processing [74], ordinal regression [75], classification [76], networking [77] and others will be our next goals. ACKNOWLEDGEMENTS The authors would like to thank all the reviewers for their helpful and constructive comments. FUNDING J.S. was supported by the National Science Foundation of China (NSFC) under Grant nos. 61573279, 6157332, 11301494 and 11626252; the State Key program of NSFC under Grant no. 91330204; the Major program of NSFC under Grant no. 11690011 and the 973 project under Grant no. 2013CB329404. Footnotes 1 That is to say that the decision variables are defined within its lower and upper bounds, i.e. x∈Ω=∏i=1n[ai,bi]. 2 Weight vectors are mainly used to denote the importance of objectives in the scalarising aggregation function, and reference points simply refer to points in the objective space. A reference vector refers to a point with direction. 3 Wang et al. [56] proposed to compute the fitness of a solution as the sum of the reciprocal of the number of solutions that dominate the reference points. It differs from our methods in the following two aspects. First, no layer is used in their method. 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( 2014 ) A variable threshold-value authentication architecture for wireless mesh networks . J. Internet Technol. , 15 , 929 – 936 . Author notes Handling editor: Fionn Murtagh © The British Computer Society 2017. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

The Computer Journal – Oxford University Press

**Published: ** Sep 18, 2017

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