A multi-objective evolutionary algorithm based on niche selection in solving irregular Pareto fronts

Abstract

The key characteristic of multi-objective evolutionary algorithm is that it can find a good approximate multi-objective optimal solution set when solving multi-objective optimization problems(MOPs). However, most multi-objective evolutionary algorithms perform well on regular multi-objective optimization problems, but their performance on irregular fronts deteriorates. In order to remedy this issue, this paper studies the existing algorithms and proposes a multi-objective evolutionary based on niche selection to deal with irregular Pareto fronts. In this paper, the crowding degree is calculated by the niche method in the process of selecting parents when the non-dominated solutions converge to the first front, which improves the the quality of offspring solutions and which is beneficial to local search. In addition, niche selection is adopted into the process of environmental selection through considering the number and the location of the individuals in its niche radius, which improve the diversity of population. Finally, experimental results on 23 benchmark problems including MaF and IMOP show that the proposed algorithm exhibits better performance than the compared MOEAs.

Keywords

Niche selection multi-objective optimization diversity irregular Pareto front

1 Introduction

Multi-objective optimization problems(MOPs) with two or three objectives to be optimized simultaneously in real-world applications [1]. Several objectives in a MOP are in conflict with each other, meaning that there is no single solution that optimizes all objectives. Similarly, the goal of solving a MOP is to find a set of solutions as a tradeoff between different objectives, which is known as Pareto optimal solutions. The obtained tradeoff solutions should not only be close to Pareto front (PF) for better convergence performance in the objective space, but also have good diversity.

In the last two decades, various evolutionary algorithms have demonstrated high effectiveness in dealing with MOPs, which known as multi-objective evolutionary algorithms(MOEAs). In the process of population evolution, MOEAs allow the search of Pareto approximation solutions in a single run and can not only guide the population to evolve in the PF direction, but also spread along the PF direction with good distribution and expansibility [2]. Population diversity is preserved in the process of environmental selection by choosing different solutions from existing populations and passing them on to the next generation.

Generally, MOEAs are mainly considered from two directions of convergence and diversity. NSGAII [3] is a classical dominance-based MOEA, which adopts a non-dominated sorting mechanism and a diversity maintenance scheme(crowding distance) to select candidates. MOEA/D [4] is the classical decomposition-based MOEA, which transforms a MOP into a set of subproblems and optimizes them simultaneously. RVEA [5], NSGAIII [6] et al update the solutions according to the predefined reference vectors, and each subproblem corresponds to a solution to form the solution set. In this situation, the presupposed uniformly distributed reference vector plays an important role in population distribution, and the scalarizing function can not only ensure the convergence of the algorithm, but also balance the diversity. In addition, a class of algorithms based on indicators are also favored by researchers, such as Hypervolume (HV) [7], R2 indicator [8] and Inverted Generation Distance (IGD) indicator [9] and so on. As for indicator-based algorithms, the solutions with the least contribution to the population indicator is iteratively deleted, but the computational complexity has always been a difficult problem to solve.

Besides, the quality of the offspring is also a key factor affecting the performance of the algorithm. Therefore, considering the convergence and diversity of the algorithm, the construction of matching pool needs to be analyzed. For instance, DAEA [10] uses a duplication analysis-based method to modified the reproduction process so as to improve the quality of offsprings. DGEA [11] uses the dominated solutions to generate the descendant solutions of the Pareto optimal set to enhance convergence, and uses the non-dominated solutions to disperse the solutions on Pareto optimal set to maintain diversity. RSEA [12] suggests a region search strategy to deal with different kinds of benchmark problems, where the region search strategy not only enhance the diversity of population, but also improve the quality of offsprings. Therefore, both convergence and diversity are necessary to produce high-quality progeny.

The most MOEAs have witnessed great success in solving the regular PFs, however, a large number of literature studies show that most of the existing MOEAs diversity preservation methods encounter difficulties in solving the MOPs of extremely concave, extremely convex or other highly irregular PFs. While some MOEAs are specifically tailored to mop with irregular PFs, most MOEAs are less versatile [13]. For example, decomposition-based MOEAs using a set of predefined fixed and uniformly distributed weight vectors are difficult to search the entire irregular Pareto front because the weight vectors are only distributed on part of the irregular front [14]. Most diversity maintenance in dominance-based MOEAs assume that PF is regular, so when PF is irregular, it does not work properly [3]. Indicator-based MOEAs are based on the selection of individuals with greater contributions. However, for the degradation front, the contribution value of the indicator may mislead the environmental selection, because many dominated individuals scattered in other regions of the objective space may contribute more to the indicator [7].

In the existing studies, MOEAs that solve irregular front problems can be divided into four types [15].

The first type is the fixed vector based MOEAs with auxiliary methods, such as non-dominated sorting, archive. For example, BCE-MOEA/D [16] uses non-dominated sorting and decomposition to exchange the information during the search process, which make it possible to solve irregular problems. MOEA/D-AED [17] uses archive to store the elite solutions with better fitness values by MOEA/D. However, the idea is still that each subproblem is associated with a different Pareto optimal solution, which is difficult to adapt to solving irregular front problems. Therefore, MOEA/D-SAS [18] and ASEA [19] allow multiple individuals to be associated with the same active reference vector. Besides, two sets of fixed reference vectors, one set starting from the ideal point and the other from the lowest point, are used to solve irregular problems, such as PAEA [20] and MOEA/AD [21]. However, although the two sets of reference vectors are used together, there is only one solution associated with each vector. Therefore, MOEA/D-MR [22] uses two sets of reference vectors, and each solution can be associated to several nearest reference vectors. However, the predefined uniformly distributed reference vectors may not match well the shape of the irregular fronts.

The second type is MOEAs based on reference vectors adjustment. In the existing studies, MOEA/D [4] and RVEA [5] adopt different scalarizing function to guide the search process, where the scalarizing function plays a key role to balance the convergence and diversity. However, the adjustment of adaptive reference vectors should take into account both the shape of PFs and the scalarizing function adopted in solving irregular problems. In this case, the main two questions are difficult to deal with: which and when the reference vectors should be adjusted. In the existing research, the current population or archive population is used to adjust reference vectors, such as MOEA/D-AWA [23], MOEA/D-M2M [24]. However, a rapid change in the distribution of the population may lead to lost the promising solutions. Some researchers propose to use the existing reference vectors adjustment, for example, in A-NSGAIII [6], the niche count is more than one around the reference vectors. However, this method performs not well when dealing with degenerate fronts. Besides, self-organising map (SOM) network or growing neural gas (GNG) is used to learn the topology of fronts, such as MOEA/D-SOM [25], DEA-GNG [26]. However, this method needs a huge number of reference vectors to be sampled.

The third type is MOEAs based on reference points. One method uses the initially generated points to adjust reference points, for example, AR-MOEA [27] uses the higher contribution of individuals as new reference points and delete the individuals which is not associated with reference points. However, the limited number of iterations may result in the waste of reference points. Therefore, the current population with good convergence and diversity is used as the reference points to guide the population update, such as CAMOEA [28] and RPEA [29]. However, reference points resulting from frequent changes in population distribution in each generation may reduce the rate of convergence.

The forth type is MOEAs based on grid or clustering, which divides the population into a number of subregions and selects only one solution from each subregion. RdEA [30] uses the grid method to deal with irregular front problems and obtains certain competitiveness. However, this kind of algorithms face the great difficulties, such as the number of grids, the selection of dividing points and the prediction of individual distribution, when the grid method is adopted. In addition, clustering divides the population based on the similarity of the solutions, which can effectively group the solutions into one subproblem. For example, F-DEA [31] regards the reference points generated by Das and Dennis’s as the cluster centers, but the distribution of reference points will effect the selection of population. Therefore, EMyO/C [32] uses a bottom-up hierarchial clustering and selects the solution nearest to the ideal point from each cluster. Clustering method can automatically recognize the distribution of the population and is an effective approach to deal with irregular fronts, however, the clustering is easily affected by the dominated solutions in the updating process [15].

Through a review of the existing literature, it is simple and effective to guide individual selection to solve irregular fronts according to population information. In evolutionary algorithm, niche technology is proposed to simulate the phenomenon of "birds of a feather flock together" in nature in order to maintain the diversity of evolutionary population. In this method, individuals with similar characteristics cluster together and the fitness values are calculated to define the level of similarity within the niche radius. Therefore, this paper proposes a multi-objective evolutionary algorithm based on niche selection of population. The contribution of this paper includes the following three aspects:

Construct a matching pool based on the convergence and distribution performance. In the process of generating offspring, when the non-dominated solutions do not converge to the first front, the dominated relations of the main objective functions of the matching pool are chosen as parent population mainly by the front layers F and convergence values Con. When the non-dominated solutions converges to the first front, the mating pool is designed by the diversity values Div calculated by the niche to select parents and produce better offsprings.

Environment selection is based on niche selection. In the environmental selection process, a normalization procedure is used to eliminate the effects of different ranges on the directional density estimation. Then, niching method is used to estimate the crowding degree of a solution by considering both the number and the location of the individuals in its niche.

The diversity performance of the MOEANS algorithm is verified by comparing with 15 representative MOEAs on 23 benchmark problems. The experimental results show that MOEANS has better diversity performance than MOEAs.

The rest of this paper is organized as follows. Section 2 describes some background knowledge. Section 3 introduces the proposed MOEANS algorithm. Then, section 4 presents and discusses the experimental results. Finally, section 5 concludes this paper.

2 Background

2.1 Basic definition

Generally, a minimization multi-objective optimization problem(MOP) with m objectives and n-dimensional decisions can be mathematically defined as [3]: $\min F (x) = (f_{1} (x), f_{2} (x), . . ., f_{m} (x))^{T}$ (1) s . t .x∈ Ω

where F is a mapping from the decision space Ω to the objective space R^m. x = (x₁, x₂, . . . , x_n) is the n-dimensional decision variables from Ω, m is the number of objective functions.

In addition, some basic definitions are shown below [33]:

Definition 1. (Pareto dominance) If x_a and x_b belonging to Ω are two feasible solutions of multi-objective optimization problems, then x_a is said to dominate x_b (x_a ≺ x_b) if and only if f_i (x_a) ≤ f_i (x_b) for every i ∈ (1, 2, . . . , m) and f_j (x_a) < f_j (x_b) for at least one j ∈ (1, 2, . . . , m).

Definition 2.(Non-dominated solutions) A solution x^* ∈ Ω is called Pareto optimal if x satisfies condition ¬ ∃ x ∈ Ω : x ≺ x^*.

Definition 3.(Pareto optimal set) The Pareto optimal solution set is the set of all Pareto optimal solutions, P^* = {x^*| ¬ ∃ x ∈ Ω : x ≺ x^*}.

Definition 4.(Pareto front) The Pareto optimal solution set is the front of the objective vector corresponding to the Pareto optimal solutions, which is called Pareto front(PF).

2.2 Motivation

The traditional MOEAs, such as NSGAII [3], make extensive use of crossover and mutation operators in the process of evolution to generate offspring. However, due to the high randomness of parental mating and progeny reproduction, the process of progeny generation is uncertain without proper guidance. MOEA/D [4] generates offspring according to the subproblem neighborhood, which is beneficial to local space search, but it is difficult to search all degenerate PFs or inverted PFs for the distribution of fixed vectors. RSEA [12] uses the region search method, which is applied to constrain the updating process and is associated with each solution in a region. However, this algorithm shows the algorithm’s reliability for solving both convex and concave problems. DAEA [10] proposes that a method of duplication analysis method in decision space is adopted to filter out the redundant solutions in the objective space. DGEA [11] proposes an adaptive offspring generation method which constructs the direction vectors in the decision space to reproduce promising offspring solutions. However, the distribution of solutions in the decision space is not consistent with the distribution of functions in the objective space, which may not be able to search all PFs in the objective space well.

As for dealing with irregular PFs, RVEA [5] uses a set reference vectors to divided the objective space into a number of subspaces and adopts the angle penalized distance to maintain the diversity. Then, a reference vector adaptation strategy is adopted to deal with scaled problems with no-well normalized objective functions. BCE-MOEA/D [16] proposes a bi-criterion evolution (BCE) framework of the Pareto criterion (PC) and Non-Pareto criterion (NPC), which attempts to make use of their strengths and compensates for each other’s weaknesses for solving convex, concave, linear, mixed, discontinuous and complex PFs. However, vector evolution-based individuals are affected by the distribution of weight vectors in solving degenerate and inverted problems. MOEA/D-PaS [34] proposes Pareto adaptive scalarizing (PaS) approximation to approximate the optimal p value, which avoids an estimation of the PF geometrical shape and is computationally efficient. A-NSGAIII [6] generates more reference vectors around the promising reference vectors, but this method does not shows well in solving degenerate PFs. MOEA/D-AWA [23] utilizes the historical non-dominated solutions in archive to detect promising regions, and a set of solutions are used to generate the reference vectors. However, the rapid change in distribution of the non-dominated solutions may lose the promising regions. Besides, DEA-GNG [26] and RVEA-iGNG [35] trains growing neural gas (GNG) network to adjust the reference vectors in order to adopt to the shape of PFs. However, a large number of reference vectors are needed to be sampled when solving irregular PFs. ARMOEA [27] adjusts the reference points generated with Das and Dennis’s approach by using the contribution of individuals, however, this method can not be well adapted to the population distribution in finite iterations. RPEA [29] adopts the current population as reference points to guide evolution, however, the frequent changes of reference points with population distribution may reduce the convergence rate. In order to avoid predefining the clustering centers, EMyO-C [32] uses a bottom-up hierarchical clustering method and selects the solution nearest to the ideal point from each cluster. Clustering method can automatically identify the population, however, evolution is very susceptible to the influence of dominated solutions.

The main purpose of the MOEAs is to converge the solution set to the real PFs and have uniform distribution on the PFs, no matter what methods are taken. According to the above existing literature, MOEAs based on population information have better performance in solving irregular front problems. Inspired by the above research methods, we propose a multi-objective evolutionary algorithm based on niche selection in solving irregular Pareto fronts. In this algorithm, NGSAII algorithm is used as the main framework, the Pareto criterion in BCE-MOEA/D is used to construct the mating pool and maintain the diversity in environmental selection.

3 Proposed multi-objective evolutionary algorithm

3.1 Framework of MOEA

In this section, the framework of MOEANS is given in Algorithm 1, where some parameters are given, such as the population size N, the number of objectives m and the number of evaluations FEs. In addition, the final expected population P is when the termination condition is satisfied. The framework of MOEANS algorithm mainly includes two parts: Initialization and Main loop.

Algorithm 1 Main Loop of MOEANS algorithm

Input: the population size (N), the number of evaluations (FEs)

Output: final population P

1: Initialization

2: Randomly initialized population (P)

3: Non-dominated sorting method is used to divide individuals into different layers (F = F₁, F₂ . . .), i = 1, 2, . . .

4: Calculate the values of convergence (Con) and diversity (Div)

5: Main Loop

6: while termination criterion is not satisfied do

7: If F_i > F₁ then

8: Construct the mating pool based on TournamentSelection according to P, F_i, Con.

9: else

10: Construct the mating pool based on TournamentSelection according to P and Div.

11: end if

12: Use genetic algorithm to reproduce offspring Q according to matingpool

13: Execute the environmental selection to obtain P, F_i, Div, Con according to P, Q.

14: end while

15: Return the final population P

(1) Initialization: firstly, initialized population P is randomly generated in the range. In this population, non-dominated sorting method is used to divide individuals into different layers (F = F₁, F₂ . . .) according to NSGAII. Then, the values of convergence (Con) and diversity (Div) are calculated. Firstly, the objective function is normalized into [0,1] and the formula is as follow: $f_{i}^{'} (x) = \frac{f_{i} (x) - f_{\min}}{f_{\max} - f_{\min}}$ (2) where f is the objective function values, f_min = min (f) and f_min = max (f) are the minimal values and the maximal of the objectives of all non-dominated solutions in the first front F₁.

Then, the convergence values (Con) and diversity values (Div) are calculated with the normalized objectives. In the existing research, the sum of of the normalized objective values is defined as the convergence degree of the solution x, and the formula of convergence values is as follows: $Con (x) = \sum_{i = 1}^{m} (f_{i}^{'} (x))$ (3) where m is the number of objectives and $f_{i}^{'} (x)$ is the i-th normalized objective value of solution x.

The diversity values is calculated by the population information, which is based on the number and location of the solutions in its niche[16]. The diversity values is calculated as follows: $Div (x) = 1 - \prod (R (x, y))$ (4)

$\begin{matrix} R (x, y) = {\begin{matrix} d (x, y) / r, & if d (x, y) < r \\ 1, & otherwise \end{matrix} \end{matrix}$ (5) where r the niche radius and the parameter k = 3, d (x, y) is the Euclidean distance between solution x and solution y, the d (x, y) is defined as: $d (x, y) = ∥ f^{'} (x) - f^{'} (y) ∥$ (6)

(2) Main Loop: the algorithm enters the main loop until the termination criterion is satisfied, where the selection process of mating pool depends on whether the number of front layers of non-dominated solutions is greater than 1, because when the number of front layers is equal to 1, it means that all solutions are non-dominated. In this process, the convergence or distribution performance of the population is considered to choose which method to construct the matching pool. Therefore, if the condition (F_i > F₁) is met, the mating pool by using a binary tournament selection is based on the different layers F and the convergence values Con, which is advantageous to produce more converge progeny promotion. Otherwise, a binary tournament selection is executed based on the diversity values Div, which is contributed to produce progeny with more uniform local distribution. In each selection, if two solutions are randomly selected from the population, one solution having the smaller front will be selected as the parent. Then, the offsprings population Q is reproduced by using crossover and mutation operators of genetic algorithm. Finally, the parent population P and offspring population Q are merged into environmental selection for the next generation and the final population P is obtained.

3.2 Environmental Selection

Environmental selection plays a key role in maintaining the population diversity and selecting the mating pool. In this paper, we adopt the niche selection strategy to select the uniformly distributed solutions, where N candidate solutions are survived to reproductive new population from the combined set of parent and offspring population. The procedure of environmental selection is shown in Algorithm 2.

Algorithm 2 Environmental Selection

Input: the population (P), the offspring population (Q), the population size (N)

Output: P, F, Con, Div

1: Execute non-dominated sorting to divide individuals into different layers (F = F₁, F₂ . . .), i = 1, 2, . . .

2: NP = |F₁ ∪ F₂ ∪ . . . ∪ F_k| > = N Normalized the objective values and obtain f′ according to Algorithm 3

3: while |NP| > N do

4: Calculate the distance d (x, y) according to Eq. 6

5: Determine the niche radius r = 3

6: Calculate R (x, y) according to Eq. 5

7: Delete the candidate solution with the worse Div value from F_k

8: Update the radius of niche of all the remaining candidate solutions in F_k

9: end while

10: Store the N non-dominated solutions in set P and k layers into F={F₁, F₂, . . . , F_k}

11: Calculate the Con and Div values by the stored solutions

12: Return P, F, Con, Div

In the proposed MOEANS, the niche selection between the candidate solutions is used in the environment selection to ensure the good distribution performance of the algorithm and to update the convergence values of the algorithm. In this situation, F = [F₁, F₂, . . .] is constructed via non-dominated sorting method in NSGAII, and the maximal front number k is found in the the set of F so that the number of candidate solutions from F₁ to F_k is greater than or equal to the population size N.

In the loop of Algorithm 2, when the non-dominated population size NP is great than population size N, the loop is carried out. Then, the Euclidean distance d (x, y) between any two solutions is calculated according to Eq. 6. In this situation, we determine the radius of the niche after sorting d (x, y) in ascending order and find these solutions in radius, where the radius is composed of the nearest three solutions. Next, these candidate solution with the worst radius value from F_k is found and deleted so that the overcrowded candidate solution is screened and deleted one by one in the form of radius niche to make the distribution of solution set better. In addition, it is worth noting that, since the radius value of each candidate solution is related to the radius of others solutions remaining in the current population, the radius of niche of all the remaining candidate solutions in F_k is updated in the process of loop. Finally, the next parent population P stores N non-dominated solutions from F₁ to F_k, and the new Con and Div is calculated by the P to generated next offspring population.

The detail projection is carried out in Algorithm 3.

Algorithm 3 Projection

Input: the non-dominated population (NP), the number of Front (F)

1: Determine ideal point z^min=min (f)

2: find the extreme solutions from F₁

3: Calculate the intercept a

4: Normalize the objectives by Eq 7

Firstly, extreme points is needed to be found according to the the candidate solutions in F₁, and the projection is executed by the formula [6]: $f^{'} = \frac{f - z^{m i n}}{a - z^{m i n}}$ (7) where the f′ is projected by z^min and the intercept a, where the intercept a and the hyperplane can be calculated according to analytic geometry. The ideal point $z^{\min} = (z_{1}^{zim}, z_{2}^{zim}, . . ., z_{m}^{zim})$ is determined by the minimum value. Then, f′ is transformed by Eq 7 and projected normalized point onto unit hyperplane toward the origin.

In order to prove the effectiveness of the insertion normalization method, Table 1 records the comparison between MOEANS of projection normalization and max/min normalization. In Table 1, MOEANS1 adopts the max/min normalization method and MOEANS adopts the projection normalization method. Experimental results show that MOEANS performs better than MOEANS1.

Table 1

IGD values of MOEANS algorithm compared with different normalization methods

	MOEANS1	MOEANS
MaF1	4.0701e-2 (3.67e-4) -	4.0232e-2 (2.56e-4)
MaF2	3.2206e-2 (1.01e-3) -	2.9059e-2 (2.22e-4)
MaF3	3.5930e-2 (2.82e-3) +	4.4919e-2 (6.21e-4)
MaF4	2.7128e-1 (2.88e-2) +	3.0451e-1 (8.98e-3)
MaF5	2.4929e-1 (2.36e-3) +	6.4028e-1 (7.06e-1)
MaF6	4.2307e-3 (7.35e-5) ≈	4.2976e-3 (2.14e-4)
MaF7	6.1957e-2 (1.05e-3) -	6.1200e-2 (1.13e-3)
MaF8	6.7851e-2 (1.07e-3) -	6.6784e-2 (1.69e-3)
MaF9	1.7989e-1 (1.56e-1) -	5.9661e-2 (3.66e-4)
MaF10	1.5392e-1 (2.68e-3) -	1.4220e-1 (2.88e-3)
MaF11	1.8196e-1 (3.79e-3) -	1.6119e-1 (1.46e-3)
MaF12	2.2757e-1 (2.84e-2) -	2.1824e-1 (2.57e-3)
MaF13	8.6420e-2 (7.22e-3) -	7.3875e-2 (8.00e-3)
MaF14	5.4421e-1 (9.40e-2) ≈	5.4424e-1 (9.04e-2)
MaF15	2.9904e-1 (2.69e-2) -	2.5732e-1 (8.80e-2)
IMOP1	5.2394e-3 (1.27e-4) +	1.0942e-1 (4.40e-3)
IMOP2	9.6671e-3 (7.91e-3) ≈	6.8761e-3 (1.92e-3)
IMOP3	3.5399e-3 (3.59e-5) +	3.9739e-3 (5.47e-5)
IMOP4	7.2011e-3 (1.29e-4) -	6.9803e-3 (6.59e-5)
IMOP5	3.4251e-2 (3.58e-4) +	3.4491e-2 (2.25e-4)
IMOP6	3.3278e-2 (5.90e-4) ≈	3.3026e-2 (3.78e-4)
IMOP7	3.7842e-2 (6.49e-4) +	5.0067e-1 (4.30e-1)
IMOP8	7.9473e-2 (2.79e-3) -	6.9807e-2 (9.96e-4)
+/-/≈	7/12/4	-
Best/All	9/23	14/23

3.3 Complexity analysis

In MOEANS, the computational cost is consist of individual exploration and population maintenance. In population maintenance, the Euclidean distance between any two individuals in the population is first calculated and the computations is O (mN²). Then, the niche selection needs to calculate the niche radius, which needs O (N²) computations. In individual selection, the convergence values or diversity values needs O (N²) computations, and O (N) is needed by exploring which individuals will be selected as parents. Therefore, the complexity of MOEANS is O (mN²).

4 Experimental design

4.1 Test problems

In this paper, 23 test problems with diverse properties test suite(MaF1-MaF15) [36] and IMOPs [37] are used, which is the good representation of various real-world scenarios. The characteristics and parameter settings of MaF and IMOPs test problems is shown in Table 2, where m is the number of objective, n is the dimension of decision variable.

Table 2
The characteristics and parameter settings of 23 test problems

problems (m) (n) characteristic

MaF1 3 12 Linear, inverted

MaF2 3 12 Concave

MaF3 3 12 Convex, multi-modal

MaF4 3 12 Concave, multi-modal

MaF5 3 12 Convex, biased

MaF6 3 12 Concave, degenerate

MaF7 3 22 Mixed, disconnected, Multimodal

MaF8 3 2 Linear, degenerate

MaF9 3 2 Linear, degenerate

MaF10 3 12 Mixed, biased

MaF11 3 12 Convex, disconnected, nonseparable

MaF12 3 12 Concave, nonseparable, biased deceptive

MaF13 3 5 Concave, unimodal, nonseparable,degenerate

MaF14 3 60 Linear, partially separable, large scale

MaF15 3 60 Convex, partially separable, large scale

IMOP1 2 10 Convex with sharp tails

IMOP2 2 10 Concave curves with sharp tails

IMOP3 2 10 Discontinuous curve

IMOP4 3 10 Wavy line

IMOP5 3 10 Consists of eight circles

IMOP6 3 10 Separated into several grids

IMOP7 3 10 A part of a unit sphere in the first octant

IMOP8 3 10 100 discontinuous subregions

problems	(m)	(n)	characteristic
MaF1	3	12	Linear, inverted
MaF2	3	12	Concave
MaF3	3	12	Convex, multi-modal
MaF4	3	12	Concave, multi-modal
MaF5	3	12	Convex, biased
MaF6	3	12	Concave, degenerate
MaF7	3	22	Mixed, disconnected, Multimodal
MaF8	3	2	Linear, degenerate
MaF9	3	2	Linear, degenerate
MaF10	3	12	Mixed, biased
MaF11	3	12	Convex, disconnected, nonseparable
MaF12	3	12	Concave, nonseparable, biased deceptive
MaF13	3	5	Concave, unimodal, nonseparable,degenerate
MaF14	3	60	Linear, partially separable, large scale
MaF15	3	60	Convex, partially separable, large scale
IMOP1	2	10	Convex with sharp tails
IMOP2	2	10	Concave curves with sharp tails
IMOP3	2	10	Discontinuous curve
IMOP4	3	10	Wavy line
IMOP5	3	10	Consists of eight circles
IMOP6	3	10	Separated into several grids
IMOP7	3	10	A part of a unit sphere in the first octant
IMOP8	3	10	100 discontinuous subregions

4.2 Performance metrics

The quality of the optimization algorithm is needed to be evaluated via some metrics. In the existing research, the inverted generation distance(IGD) [38] is widely used, which can measure both convergence and diversity. The IGD value is calculated from a set of uniformly distributed points across PF to the closet solution in its approximation set. The formula is shown as follows: $IGD = \frac{\sum_{v \in P} d (v, P^{*})}{| P |}$ (8) where P is the uniformly distributed points and P^* is the approximation set. d (v, P^*) is the Euclidean distance between v ∈ P and P^*, |P| the cardinality of P. In this case, the lower IGD value shows that the quality of p^* for approximating the whole PF.

Hypervolume (HV) [39] is defined as tge volume of the hypercube enclosed in the objective space by the reference point and every vector in the Pareto approximation set A. This is mathematically defined as: $HV = {\cup_{i} {vol}_{i} | {vec}_{i} \in A}$ (9) where vol_i is the volume from the hypercube formed by the reference point and the non-dominated vector vec_i. vec_i is a non-dominated vector from Pareto approximation A and reference point is z^ref in the objective space.

4.3 Parameter setting

In this section, we employ the standard toolkits to implement the peer algorithms and the original study to set parameters of compared algorithms. The proposed algorithm is to be validated to compare with classical algorithms, which includes promoting matching pools, based on fixed reference vectors, based on reference vector adjustment, based on reference point adjustment and based on clustering. The comparison of algorithms can be divided into three groups: the first group is NSGAII [3] and MOEA/D [4], DGEA [11], DAEA [10] and RSEA [12]. The second group is BCE-MOEA/D [16], MOEA/D-AWA [23], MOEA/D-PaS [34], A-NSGAIII [6] and RVEA [5]. The third group is EMyO-C [32], ARMOEA [27], RPEA [29], RVEA-iGNG [35] and DEA-GNG [25].

In addition, the number of evaluations FEs = 100000, the population size N = 100. All algorithms run on PlatEMO [40], running all algorithms 20 times per problem to avoid surprises.

5 Experimental results

In this section, experimental results is shown to demonstrate the superiority of MOEANS for convergence and diversity in dealing with irregular front problems.

5.1 The analysis of data

Tables 3-8 list the means and standard deviations of IGD and HV respectively of the 16 algorithms on MaF and IMOP problems suits. In addition, Wilconxon’s rank sum test [41] is carried out between MOEANS and each compared algorithm on each test problem at 0.05 significance level, which is aimed to investigate whether MOEANS is superior to compared algorithms in a statistical sense. In this case, +/-/≈ is used to indicate that the compared algorithm is superior to, inferior to and similar to MOEANS algorithm respectively, where the best results based on the mean values for each test problem are highlighted in boldface. In addition, the last and penultimate row of the tables show the proportion of the optimal value in the total between MOEANS and the compared algorithms on the MaF and IMOP test problems.

Table 3
Comparison of the MOEAs on IGD value with means and standard deviations for MaF and IMOP benchmark problems

NSGAII MOEA/D RSEA DGEA DAEA MOEANS

MaF1 5.6909e-2 (2.12e-3) - 7.0475e-2 (8.08e-7) - 4.1300e-2 (2.78e-4) - 8.2845e-2 (1.54e-4) - 1.1576e-1 (1.88e-2) - 4.0232e-2 (2.56e-4)

MaF2 4.6114e-2 (3.11e-3) - 3.8936e-2 (5.43e-4) - 2.8997e-2 (2.59e-4) ≈ 3.9649e-2 (2.04e-4) - 3.7216e-2 (2.03e-3) - 2.9059e-2 (2.22e-4)

MaF3 1.6445e-1 (4.79e-1) - 5.2470e-2 (1.86e-3) - 5.0478e-2 (2.85e-3) - 2.7033e+3 (4.24e+3) - 5.5325e+3 (2.77e+3) - 4.4919e-2 (6.21e-4)

MaF4 2.9587e-1 (1.00e-2) + 6.8595e-1 (4.18e-2) - 3.5164e-1 (5.22e-2) - 1.5679e+2 (1.61e+2)- 1.8293e+2 (5.41e+1) - 3.0451e-1 (8.98e-3)

MaF5 7.5992e-1 (1.41e+0) - 8.7760e-1 (1.19e+0) - 2.4420e-1 (1.50e-3) + 3.0074e-1 (2.26e-3) + 2.0074e+0 (4.48e-1) - 6.4028e-1 (7.06e-1)

MaF6 5.2817e-3 (2.19e-4) - 3.3929e-2 (5.56e-6) - 1.4030e-2 (3.87e-2) - 3.3742e-2 (1.48e-4) - 8.5672e+0 (3.02e+0) - 4.2976e-3 (2.14e-4)

MaF7 7.6959e-2 (4.10e-3) - 1.5424e-1 (1.25e-3) - 9.2521e-2 (8.92e-2) - 1.9152e-1 (1.41e-1) - 1.8963e-1 (3.79e-2) - 6.1200e-2 (1.13e-3)

MaF8 9.8035e-2 (6.48e-3) - 1.0879e-1 (3.73e-3) - 7.7470e-2 (6.85e-3) - 1.6587e-1 (1.65e-3) - 5.0565e-1 (1.14e-16) - 6.6784e-2 (1.69e-3)

MaF9 3.4891e-1 (1.95e-1) - 6.4974e-2 (4.84e-3) - 6.5975e-2 (1.10e-2) - 9.4949e-2 (9.52e-2) - 4.7825e-1 (2.28e-16) - 5.9661e-2 (3.66e-4)

MaF10 2.1919e-1 (1.44e-2) - 2.3109e-1 (5.81e-3) - 1.6091e-1 (1.08e-2) - 1.0277e+0 (1.11e-1) - 2.1430e+0 (8.64e-2) - 1.4220e-1 (2.88e-3)

MaF11 2.1666e-1 (1.29e-2) - 2.3416e-1 (1.61e-2) - 1.5609e-1 (2.45e-3) + 1.9852e-1 (1.17e-2) - 4.0217e-1 (9.55e-2) - 1.6119e-1 (1.46e-3)

MaF12 2.8143e-1 (2.55e-2) - 2.7008e-1 (3.08e-2) - 2.1432e-1 (2.08e-3) + 2.4348e-1 (5.20e-3) - 4.6239e-1 (3.93e-2) - 2.1824e-1 (2.57e-3)

MaF13 1.1860e-1 (1.53e-2) - 9.7549e-2 (1.07e-2) - 9.7069e-2 (3.20e-2) - 6.1829e-2 (3.19e-3) + 1.8845e-1 (3.38e-2) - 7.3875e-2 (8.00e-3)

MaF14 6.2159e-1 (1.34e-1) ≈ 5.8506e-1 (1.41e-1) ≈ 5.0512e-1 (1.04e-1) ≈ 6.7433e-1 (7.41e-2) - 8.6072e-1 (1.14e-16) - 5.4424e-1 (9.04e-2)

MaF15 6.4978e-1 (6.39e-2) - 3.2685e-1 (1.11e-1) - 7.1141e-1 (6.06e-2) - 1.4116e-1 (1.33e-2) + 3.5399e-1 (4.52e-2) - 2.5732e-1 (8.80e-2)

IMOP1 6.1240e-3 (2.45e-4) + 3.5255e-1 (2.73e-3) - 1.3312e-1 (3.11e-2) - 1.4378e-1 (9.86e-3) - 3.1150e-1 (6.22e-2) - 1.0942e-1 (4.40e-3)

IMOP2 5.8977e-3 (2.37e-4) + 7.8497e-1 (3.94e-6) - 6.0051e-3 (3.93e-4) ≈ 1.9520e-1 (3.03e-1) - 1.9252e-1 (1.94e-2) - 6.8761e-3 (1.92e-3)

IMOP3 4.4098e-3 (1.48e-4) - 1.4099e-1 (1.15e-1) - 1.2005e-2 (3.66e-3) - 4.3055e-2 (6.77e-3) - 2.7708e-1 (2.26e-2) - 3.9739e-3 (5.47e-5)

IMOP4 9.0658e-3 (4.21e-4) - 1.9682e-2 (6.21e-4) - 1.9777e-2 (1.89e-3) - 5.5879e-2 (2.17e-3) - 2.8510e-1 (4.06e-2) - 6.9803e-3 (6.59e-5)

IMOP5 4.7432e-2 (3.03e-3) - 5.7737e-1 (1.62e-4) - 3.7708e-2 (1.13e-3) - 6.6893e-2 (6.53e-3) - 4.0846e-1 (5.94e-2) - 3.4491e-2 (2.25e-4)

IMOP6 5.8187e-2 (6.35e-3) - 1.1928e-1 (1.64e-1) - 3.3710e-2 (5.46e-4) - 1.0122e-1 (1.12e-1) - 3.8397e-1 (5.19e-2) - 3.3026e-2 (3.78e-4)

IMOP7 4.8697e-2 (4.43e-3) ≈ 9.3867e-1 (6.39e-5) - 3.8373e-2 (5.55e-4) + 8.9326e-1 (1.94e-1) - 4.3273e-1 (3.56e-2) ≈ 5.0067e-1 (4.30e-1)

IMOP8 1.0970e-1 (5.49e-3) - 1.0612e+0 (2.57e-3) - 7.6285e-2 (2.54e-3) - 5.8227e-1 (3.77e-1) - 3.7072e-1 (5.86e-2) - 6.9807e-2 (9.96e-4)

+/-/≈ 3/18/2 0/22/1 4/16/3 3/20/0 0/22/1 -

Best/All 3/23 0/23 6/23 2/23 0/23 12/23

	NSGAII	MOEA/D	RSEA	DGEA	DAEA	MOEANS
MaF1	5.6909e-2 (2.12e-3) -	7.0475e-2 (8.08e-7) -	4.1300e-2 (2.78e-4) -	8.2845e-2 (1.54e-4) -	1.1576e-1 (1.88e-2) -	4.0232e-2 (2.56e-4)
MaF2	4.6114e-2 (3.11e-3) -	3.8936e-2 (5.43e-4) -	2.8997e-2 (2.59e-4) ≈	3.9649e-2 (2.04e-4) -	3.7216e-2 (2.03e-3) -	2.9059e-2 (2.22e-4)
MaF3	1.6445e-1 (4.79e-1) -	5.2470e-2 (1.86e-3) -	5.0478e-2 (2.85e-3) -	2.7033e+3 (4.24e+3) -	5.5325e+3 (2.77e+3) -	4.4919e-2 (6.21e-4)
MaF4	2.9587e-1 (1.00e-2) +	6.8595e-1 (4.18e-2) -	3.5164e-1 (5.22e-2) -	1.5679e+2 (1.61e+2)-	1.8293e+2 (5.41e+1) -	3.0451e-1 (8.98e-3)
MaF5	7.5992e-1 (1.41e+0) -	8.7760e-1 (1.19e+0) -	2.4420e-1 (1.50e-3) +	3.0074e-1 (2.26e-3) +	2.0074e+0 (4.48e-1) -	6.4028e-1 (7.06e-1)
MaF6	5.2817e-3 (2.19e-4) -	3.3929e-2 (5.56e-6) -	1.4030e-2 (3.87e-2) -	3.3742e-2 (1.48e-4) -	8.5672e+0 (3.02e+0) -	4.2976e-3 (2.14e-4)
MaF7	7.6959e-2 (4.10e-3) -	1.5424e-1 (1.25e-3) -	9.2521e-2 (8.92e-2) -	1.9152e-1 (1.41e-1) -	1.8963e-1 (3.79e-2) -	6.1200e-2 (1.13e-3)
MaF8	9.8035e-2 (6.48e-3) -	1.0879e-1 (3.73e-3) -	7.7470e-2 (6.85e-3) -	1.6587e-1 (1.65e-3) -	5.0565e-1 (1.14e-16) -	6.6784e-2 (1.69e-3)
MaF9	3.4891e-1 (1.95e-1) -	6.4974e-2 (4.84e-3) -	6.5975e-2 (1.10e-2) -	9.4949e-2 (9.52e-2) -	4.7825e-1 (2.28e-16) -	5.9661e-2 (3.66e-4)
MaF10	2.1919e-1 (1.44e-2) -	2.3109e-1 (5.81e-3) -	1.6091e-1 (1.08e-2) -	1.0277e+0 (1.11e-1) -	2.1430e+0 (8.64e-2) -	1.4220e-1 (2.88e-3)
MaF11	2.1666e-1 (1.29e-2) -	2.3416e-1 (1.61e-2) -	1.5609e-1 (2.45e-3) +	1.9852e-1 (1.17e-2) -	4.0217e-1 (9.55e-2) -	1.6119e-1 (1.46e-3)
MaF12	2.8143e-1 (2.55e-2) -	2.7008e-1 (3.08e-2) -	2.1432e-1 (2.08e-3) +	2.4348e-1 (5.20e-3) -	4.6239e-1 (3.93e-2) -	2.1824e-1 (2.57e-3)
MaF13	1.1860e-1 (1.53e-2) -	9.7549e-2 (1.07e-2) -	9.7069e-2 (3.20e-2) -	6.1829e-2 (3.19e-3) +	1.8845e-1 (3.38e-2) -	7.3875e-2 (8.00e-3)
MaF14	6.2159e-1 (1.34e-1) ≈	5.8506e-1 (1.41e-1) ≈	5.0512e-1 (1.04e-1) ≈	6.7433e-1 (7.41e-2) -	8.6072e-1 (1.14e-16) -	5.4424e-1 (9.04e-2)
MaF15	6.4978e-1 (6.39e-2) -	3.2685e-1 (1.11e-1) -	7.1141e-1 (6.06e-2) -	1.4116e-1 (1.33e-2) +	3.5399e-1 (4.52e-2) -	2.5732e-1 (8.80e-2)
IMOP1	6.1240e-3 (2.45e-4) +	3.5255e-1 (2.73e-3) -	1.3312e-1 (3.11e-2) -	1.4378e-1 (9.86e-3) -	3.1150e-1 (6.22e-2) -	1.0942e-1 (4.40e-3)
IMOP2	5.8977e-3 (2.37e-4) +	7.8497e-1 (3.94e-6) -	6.0051e-3 (3.93e-4) ≈	1.9520e-1 (3.03e-1) -	1.9252e-1 (1.94e-2) -	6.8761e-3 (1.92e-3)
IMOP3	4.4098e-3 (1.48e-4) -	1.4099e-1 (1.15e-1) -	1.2005e-2 (3.66e-3) -	4.3055e-2 (6.77e-3) -	2.7708e-1 (2.26e-2) -	3.9739e-3 (5.47e-5)
IMOP4	9.0658e-3 (4.21e-4) -	1.9682e-2 (6.21e-4) -	1.9777e-2 (1.89e-3) -	5.5879e-2 (2.17e-3) -	2.8510e-1 (4.06e-2) -	6.9803e-3 (6.59e-5)
IMOP5	4.7432e-2 (3.03e-3) -	5.7737e-1 (1.62e-4) -	3.7708e-2 (1.13e-3) -	6.6893e-2 (6.53e-3) -	4.0846e-1 (5.94e-2) -	3.4491e-2 (2.25e-4)
IMOP6	5.8187e-2 (6.35e-3) -	1.1928e-1 (1.64e-1) -	3.3710e-2 (5.46e-4) -	1.0122e-1 (1.12e-1) -	3.8397e-1 (5.19e-2) -	3.3026e-2 (3.78e-4)
IMOP7	4.8697e-2 (4.43e-3) ≈	9.3867e-1 (6.39e-5) -	3.8373e-2 (5.55e-4) +	8.9326e-1 (1.94e-1) -	4.3273e-1 (3.56e-2) ≈	5.0067e-1 (4.30e-1)
IMOP8	1.0970e-1 (5.49e-3) -	1.0612e+0 (2.57e-3) -	7.6285e-2 (2.54e-3) -	5.8227e-1 (3.77e-1) -	3.7072e-1 (5.86e-2) -	6.9807e-2 (9.96e-4)
+/-/≈	3/18/2	0/22/1	4/16/3	3/20/0	0/22/1	-
Best/All	3/23	0/23	6/23	2/23	0/23	12/23

Table 4

Comparison of the MOEAs on HV value with means and standard deviations for MaF and IMOP benchmark problems

	NSGAII	MOEA/D	RSEA	DGEA	DAEA	MOEANS
MaF1	2.0467e-1 (1.68e-3) -	1.9823e-1 (8.84e-7) -	2.2084e-1 (3.32e-4) -	1.8345e-1 (6.24e-4) -	1.4004e-1 (1.91e-2) -	2.2166e-1 (4.66e-4)
MaF2	2.3507e-1 (1.44e-3) -	2.3888e-1 (6.94e-4) -	2.4457e-1 (9.02e-4) ≈	2.3183e-1 (7.47e-4) -	2.4127e-1 (1.55e-3) -	2.4481e-1 (5.11e-4)
MaF3	8.9855e-1 (2.12e-1) -	9.5445e-1 (1.67e-3) -	9.5757e-1 (2.08e-3) -	5.0295e-2 (2.13e-1) -	0.0000e+0 (0.00e+0) -	9.5869e-1 (6.47e-4)
MaF4	5.2265e-1 (3.27e-3) -	4.9982e-1 (4.43e-3) -	5.2388e-1 (1.29e-2) -	4.1117e-2 (1.27e-1)-	0.0000e+0 (0.00e+0) -	5.3328e-1 (2.12e-3)
MaF5	4.9150e-1 (1.37e-1) -	4.7673e-1 (1.32e-1) -	5.5766e-1 (8.96e-4) +	5.3953e-1 (1.84e-3) +	1.6569e-1 (1.18e-1) -	5.0257e-1 (9.36e-2)
MaF6	1.9974e-1 (1.50e-4) ≈	1.8185e-1 (3.30e-6) -	1.9851e-1 (3.46e-3) -	1.8162e-1 (8.98e-5) -	0.0000e+0 (0.00e+0) -	1.9977e-1 (5.12e-5)
MaF7	2.6782e-1 (2.24e-3) -	2.5721e-1 (6.88e-4) -	2.7393e-1 (1.13e-2) ≈	2.4963e-1 (1.14e-2) -	2.4246e-1 (1.39e-2) -	2.7736e-1 (8.94e-4)
MaF8	2.6556e-1 (2.41e-3) -	2.6081e-1 (1.28e-3) -	2.7086e-1 (2.83e-3) -	2.2699e-1 (1.17e-3)-	1.0969e-1 (5.70e-17) -	2.7808e-1 (7.24e-4)
MaF9	6.0053e-1 (1.44e-1) -	8.3363e-1 (4.47e-3) -	8.3163e-1 (6.66e-3) -	8.0962e-1 (7.48e-2) -	3.8375e-1 (1.14e-16) -	8.3862e-1 (3.56e-4)
MaF10	9.3095e-1 (4.13e-3) -	9.0833e-1 (1.21e-2) -	9.4012e-1 (1.03e-3) -	4.6018e-1 (4.56e-2)-	1.3911e-2 (1.38e-2) -	9.4468e-1 (1.12e-3)
MaF11	9.2013e-1 (2.21e-3) -	9.1575e-1 (3.71e-3) -	9.3298e-1 (8.19e-4) +	8.9507e-1 (1.14e-2) -	7.7680e-1 (3.67e-2) -	9.3225e-1 (6.09e-4)
MaF12	4.9672e-1 (2.62e-2) -	4.9261e-1 (2.66e-2) -	5.4964e-1 (2.37e-3) +	5.1823e-1 (7.68e-3) -	3.5569e-1 (2.02e-2) -	5.2650e-1 (3.94e-3)
MaF13	4.5628e-1 (2.28e-2) -	4.7553e-1 (1.80e-2) -	4.8394e-1 (3.22e-2) -	5.4103e-1 (5.55e-3) +	4.1422e-1 (2.42e-2) -	5.1267e-1 (1.06e-2)
MaF14	2.0619e-1 (1.10e-1) ≈	2.2711e-1 (7.41e-2) ≈	2.4839e-1 (1.06e-1) ≈	1.0527e-1 (2.75e-2) -	9.0909e-2 (4.27e-17) -	2.6208e-1 (9.01e-2)
MaF15	2.1944e-2 (2.73e-2) -	3.5833e-1 (7.43e-2) +	5.8767e-2 (1.30e-2) -	3.7808e-1 (1.91e-2) +	1.8434e-1 (3.58e-2) -	2.4978e-1 (5.48e-2)
IMOP1	9.8713e-1 (9.71e-5) +	9.6967e-1 (7.11e-4) -	9.8375e-1 (2.38e-3) -	9.8649e-1 (1.49e-4) +	4.7634e-1 (1.29e-1) -	9.8495e-1 (2.94e-4)
IMOP2	2.3145e-1 (1.52e-4) +	9.0909e-2 (1.82e-9) -	2.2977e-1 (1.90e-3) -	2.0107e-1 (5.65e-2) -	1.2068e-1 (4.32e-2) -	2.3135e-1 (1.14e-4)
IMOP3	6.5842e-1 (7.92e-5) -	5.5351e-1 (1.13e-1) -	6.5581e-1 (1.51e-3) -	6.1078e-1 (1.54e-2)-	2.6008e-1 (4.97e-2) -	6.5870e-1 (2.66e-5)
IMOP4	4.3265e-1 (5.55e-4) -	4.2823e-1 (2.80e-4) -	4.3007e-1 (3.87e-4) -	4.0211e-1 (1.42e-3)-	1.2389e-1 (3.67e-2) -	4.3410e-1 (1.26e-4)
IMOP5	4.8818e-1 (6.39e-3) -	4.1943e-1 (1.33e-3) -	5.1370e-1 (1.07e-2) +	4.9916e-1 (1.14e-2) ≈	4.1798e-1 (7.73e-2) -	5.0160e-1 (1.08e-3)
IMOP6	5.0313e-1 (6.07e-3) -	4.5147e-1 (1.45e-1) -	5.2665e-1 (6.12e-4) -	4.7558e-1 (6.60e-2) -	1.5959e-1 (5.17e-2) -	5.3123e-1 (5.64e-4)
IMOP7	4.9487e-1 (6.51e-3) ≈	9.0909e-2 (2.95e-8) -	5.2714e-1 (8.75e-4) ≈	1.1093e-1 (8.95e-2) -	2.0039e-1 (3.68e-2) ≈	2.9066e-1 (2.21e-1)
IMOP8	4.7182e-1 (5.94e-3) -	7.0426e-2 (8.02e-4) -	5.2744e-1 (1.51e-2) -	2.4452e-1 (1.96e-1) -	3.2194e-1 (6.32e-2) -	5.2977e-1 (1.06e-3)
+/-/≈	2/18/3	1/21/1	4/15/4	4/18/1	0/22/1	-
Best/All	2/23	0/23	5/23	2/23	0/23	14/23

Table 5

Comparison of the MOEAs on IGD value with means and standard deviations for MaF and IMOP benchmark problems

	BCE-MOEA/D	MOEA/D-AWA	MOEA/D-PaS	A-NSGAIII	RVEA	MOEANS
MaF1	4.3772e-2 (5.10e-4) -	4.3415e-2 (2.52e-4) -	9.0883e-2 (1.02e-3) -	4.4799e-2 (2.02e-4) -	8.2069e-2 (3.09e-4) -	4.0232e-2 (2.56e-4)
MaF2	3.1282e-2 (4.30e-4) -	3.0503e-2 (1.76e-4) -	5.4788e-2 (1.97e-3) -	3.1068e-2 (6.49e-4) -	4.2800e-2 (8.55e-4) -	2.9059e-2 (2.22e-4)
MaF3	4.9246e-2 (1.23e-2) ≈	3.8228e-2 (1.70e-3) +	4.4508e+0 (1.87e+1) -	5.0425e-2 (5.17e-3) -	5.2271e-2 (4.91e-2) -	4.4919e-2 (6.21e-4)
MaF4	3.9862e-1 (5.70e-2) -	2.7078e-1 (7.18e-3) +	1.8405e+0 (5.77e+0) -	3.3876e-1 (1.84e-2) -	4.3012e-1 (1.27e-1) -	3.0451e-1 (8.98e-3)
MaF5	2.6116e-1 (3.65e-3) +	2.5608e-1 (2.05e-3) +	1.0026e+0 (8.68e-1) -	6.0317e-1 (1.08e+0) +	2.5975e-1 (5.12e-6) +	6.4028e-1 (7.06e-1)
MaF6	4.5907e-3 (6.27e-5) -	5.4304e-3 (1.13e-4) -	2.3843e-2 (1.45e-3) -	1.2110e-2 (1.52e-3)-	3.6106e-2 (4.04e-3) -	4.2976e-3 (2.14e-4)
MaF7	6.1782e-2 (1.42e-3) ≈	1.0619e-1 (1.01e-2) -	1.0387e+0 (1.93e+0) -	1.0509e-1 (9.13e-2) -	1.0536e-1 (1.30e-3) -	6.1200e-2 (1.13e-3)
MaF8	6.9820e-2 (7.88e-4) -	7.4275e-2 (1.61e-3) -	4.1045e+1 (2.17e+1) -	9.7733e-2 (5.64e-3)-	1.4197e-1 (1.01e-2) -	6.6784e-2 (1.69e-3)
MaF9	1.1818e-1 (2.41e-2) -	6.4777e-2 (2.29e-3) -	1.0581e-1 (2.49e-2) -	6.4167e-2 (2.20e-3) -	6.2714e-2 (2.01e-3) -	5.9661e-2 (3.66e-4)
MaF10	1.6042e-1 (3.35e-3) -	1.9304e-1 (7.32e-3) -	5.1354e-1 (1.02e-1) -	1.6397e-1 (5.75e-3)-	1.6375e-1 (4.61e-3) -	1.4220e-1 (2.88e-3)
MaF11	1.7929e-1 (5.62e-3) -	1.6866e-1 (5.74e-3) -	3.1131e-1 (9.81e-3) -	1.7203e-1 (6.16e-3)-	1.7939e-1 (4.07e-3) -	1.6119e-1 (1.46e-3)
MaF12	2.2418e-1 (2.96e-3) -	2.1923e-1 (3.99e-3) ≈	3.1300e-1 (3.74e-2) -	2.4656e-1 (7.00e-3)-	2.2470e-1 (2.54e-3) -	2.1824e-1 (2.57e-3)
MaF13	8.6459e-2 (6.15e-3) -	7.9702e-2 (9.37e-3) -	1.2397e-1 (4.67e-2) -	1.1553e-1 (7.36e-2) -	8.3904e-2 (8.71e-3) -	7.3875e-2 (8.00e-3)
MaF14	5.1360e-1 (3.79e-2) ≈	4.8819e-1 (7.95e-2) +	1.2854e+0 (1.17e+0) -	5.5696e-1 (9.11e-2) ≈	8.8446e-1 (2.29e-1) -	5.4424e-1 (9.04e-2)
MaF15	2.1831e-1 (2.12e-2) ≈	1.9016e-1 (2.09e-2) +	4.6637e-1 (8.69e-2) -	4.9724e-1 (1.46e-1) -	9.2886e-1 (1.88e-1) -	2.5732e-1 (8.80e-2)
IMOP1	5.7965e-3 (2.41e-4) +	1.0486e-2 (1.70e-3) +	7.9542e-1 (6.43e-3) -	1.5348e-1 (4.61e-2) -	1.3973e-1 (5.25e-3) -	1.0942e-1 (4.40e-3)
IMOP2	5.0528e-3 (3.37e-4) +	4.8527e-1 (1.44e-1) -	5.2886e-1 (2.43e-1) -	1.3360e-2 (4.85e-3) -	7.8080e-1 (1.87e-2) -	6.8761e-3 (1.92e-3)
IMOP3	3.6826e-3 (1.06e-4) +	4.2610e-3 (3.97e-4) -	7.5175e-1 (1.76e-1) -	6.3055e-3 (2.41e-3) -	7.5921e-3 (6.68e-4) -	3.9739e-3 (5.47e-5)
IMOP4	7.8181e-3 (8.96e-5) -	8.9877e-3 (4.82e-4) -	2.8521e-2 (7.82e-4) -	1.6263e-2 (1.49e-3) -	4.4303e-2 (6.01e-3) -	6.9803e-3 (6.59e-5)
IMOP5	3.7468e-2 (4.18e-3) -	3.6804e-2 (1.38e-3) -	6.8025e-1 (1.46e-1) -	4.7351e-2 (2.28e-3) -	6.8306e-2 (4.42e-3) -	3.4491e-2 (2.25e-4)
IMOP6	3.2852e-2 (5.26e-4) ≈	3.5731e-2 (1.76e-3) -	6.0271e-2 (1.99e-3) -	4.2523e-2 (9.62e-4) -	3.3207e-1 (1.99e-1) -	3.3026e-2 (3.78e-4)
IMOP7	3.8704e-2 (3.39e-4) ≈	4.0673e-2 (9.76e-4) ≈	1.6621e-1 (2.01e-1) ≈	6.7560e-1 (3.77e-1) -	4.3504e-1 (4.31e-1) ≈	5.0067e-1 (4.30e-1)
IMOP8	8.4977e-2 (3.28e-3) -	1.3048e-1 (1.26e-2) -	1.0861e-1 (4.07e-3) -	1.0850e-1 (1.01e-1) -	1.3561e-1 (3.15e-3) -	6.9807e-2 (9.96e-4)
+/-/≈	4/13/6	6/15/2	0/22/1	1/21/1	1/21/1	-
Best/All	5/23	5/23	0/23	0/23	0/23	13/23

Table 6

Comparison of the MOEAs on HV value with means and standard deviations for MaF and IMOP benchmark problems

	BCE-MOEA/D	MOEA/D-AWA	MOEA/D-PaS	A-NSGAIII	RVEA	MOEANS
MaF1	2.1933e-1 (5.99e-4) -	2.2074e-1 (2.18e-4) -	1.8351e-1 (8.89e-4) -	2.2070e-1 (1.74e-4) -	1.8608e-1 (1.98e-3) -	2.2166e-1 (4.66e-4)
MaF2	2.4311e-1 (7.82e-4) -	2.4504e-1 (2.65e-4) ≈	2.3148e-1 (1.21e-3) -	2.3869e-1 (1.09e-3) -	2.3029e-1 (1.08e-3) -	2.4481e-1 (5.11e-4)
MaF3	9.5579e-1 (7.40e-3) ≈	9.6093e-1 (7.22e-4) +	7.1985e-1 (3.79e-1) -	9.5525e-1 (2.82e-3) -	9.5195e-1 (3.24e-2) ≈	9.5869e-1 (6.47e-4)
MaF4	5.0551e-1 (1.37e-2) -	5.2504e-1 (2.46e-3) -	4.5872e-1 (1.57e-1) -	5.1545e-1 (5.07e-3)-	4.9224e-1 (3.82e-2) -	5.3328e-1 (2.12e-3)
MaF5	5.5667e-1 (9.93e-4) +	5.6019e-1 (6.48e-4) +	4.4506e-1 (1.31e-1) -	5.1457e-1 (1.10e-1) +	5.5961e-1 (2.78e-5) +	5.0257e-1 (9.36e-2)
MaF6	1.9991e-1 (4.59e-5) +	1.9927e-1 (9.40e-5) -	1.8952e-1 (1.03e-3) -	1.9529e-1 (9.45e-4) -	1.7881e-1 (2.84e-3) -	1.9977e-1 (5.12e-5)
MaF7	2.7728e-1 (7.67e-4) ≈	2.6500e-1 (3.06e-3) -	2.1213e-1 (9.40e-2) -	2.6596e-1 (1.08e-2) -	2.6492e-1 (1.19e-3) -	2.7736e-1 (8.94e-4)
MaF8	2.7674e-1 (4.17e-4) -	2.7568e-1 (5.53e-4) -	0.0000e+0 (0.00e+0) -	2.6251e-1 (2.55e-3)-	2.3731e-1 (4.75e-3) -	2.7808e-1 (7.24e-4)
MaF9	7.7410e-1 (2.56e-2) -	8.2846e-1 (3.38e-3) -	8.0351e-1 (1.63e-2) -	8.3405e-1 (2.63e-3) -	8.3647e-1 (1.48e-3) -	8.3862e-1 (3.56e-4)
MaF10	9.4481e-1 (6.55e-4) ≈	9.3339e-1 (2.22e-3) -	7.8134e-1 (4.17e-2) -	9.4021e-1 (1.25e-3) -	9.4057e-1 (1.24e-3) -	9.4468e-1 (1.12e-3)
MaF11	9.3385e-1 (1.12e-3) +	9.2666e-1 (3.90e-3) -	9.2158e-1 (3.65e-3) -	9.2720e-1 (2.03e-3) -	9.2796e-1 (9.46e-4) -	9.3225e-1 (6.09e-4)
MaF12	5.3129e-1 (5.62e-3) +	5.2976e-1 (6.54e-3) +	4.9469e-1 (4.62e-2) -	5.0820e-1 (4.38e-3) -	5.3960e-1 (2.06e-3) +	5.2650e-1 (3.94e-3)
MaF13	5.0160e-1 (1.19e-2) -	5.1317e-1 (1.29e-2) ≈	5.0018e-1 (3.15e-2) ≈	4.6707e-1 (3.42e-2) -	5.0418e-1 (1.30e-2) -	5.1267e-1 (1.06e-2)
MaF14	2.5466e-1 (5.12e-2) ≈	2.9033e-1 (5.23e-2) ≈	8.8695e-2 (6.28e-2) -	2.4828e-1 (8.31e-2) ≈	7.6561e-2 (7.22e-2) -	2.6208e-1 (9.01e-2)
MaF15	3.2609e-1 (2.35e-2) +	3.8177e-1 (1.59e-2) +	4.4743e-2 (8.74e-2) -	9.6745e-2 (8.50e-2) -	6.0386e-3 (9.96e-3) -	2.4978e-1 (5.48e-2)
IMOP1	9.8722e-1 (5.50e-5) +	9.8765e-1 (2.84e-5) +	1.0472e-1 (6.33e-3) -	9.8284e-1 (3.22e-3) ≈	9.8643e-1 (1.15e-4) +	9.8495e-1 (2.94e-4)
IMOP2	2.3170e-1 (9.28e-5) +	1.0587e-1 (3.98e-2) -	1.0343e-1 (6.39e-2) -	2.3112e-1 (9.22e-4) ≈	9.0910e-2 (3.27e-6) -	2.3135e-1 (1.14e-4)
IMOP3	6.5881e-1 (8.79e-5) +	6.5863e-1 (1.34e-4) -	1.1219e-1 (1.29e-1) -	6.5765e-1 (1.03e-3) -	6.5523e-1 (9.02e-4) -	6.5870e-1 (2.66e-5)
IMOP4	4.3320e-1 (3.08e-4) -	4.3281e-1 (4.03e-4) -	4.2328e-1 (2.95e-4) -	4.2596e-1 (2.37e-3)-	4.1181e-1 (1.91e-3) -	4.3410e-1 (1.26e-4)
IMOP5	5.0657e-1 (9.96e-3) +	5.0652e-1 (4.05e-3) +	2.5594e-1 (5.16e-2) -	4.8993e-1 (3.70e-3) -	4.6775e-1 (1.29e-2) -	5.0160e-1 (1.08e-3)
IMOP6	5.2819e-1 (3.97e-4) -	5.2723e-1 (1.61e-3) -	5.0505e-1 (1.67e-3) -	5.1404e-1 (1.22e-3) -	2.5819e-1 (1.74e-1) -	5.3123e-1 (5.64e-4)
IMOP7	5.2773e-1 (5.98e-4) ≈	5.2918e-1 (6.16e-4) +	4.6150e-1 (1.01e-1) ≈	2.0213e-1 (1.90e-1) -	3.2352e-1 (2.04e-1) ≈	2.9066e-1 (2.21e-1)
IMOP8	5.2002e-1 (2.47e-3) -	4.9088e-1 (7.05e-3) -	4.9791e-1 (2.95e-3) -	5.0019e-1 (2.73e-2) -	4.8486e-1 (9.31e-3) -	5.2977e-1 (1.06e-3)
+/-/≈	9/9/5	7/13/3	0/21/2	1/19/3	3/18/2	-
Best/All	6/23	8/23	0/23	0/23	1/23	8/23

Table 7

Comparison of the MOEAs on IGD value with means and standard deviations for MaF and IMOP benchmark problems

	EMyO/C	AR-MOEA	RPEA	RVEA-iGNG	DEA-GNG	MOEANS
MaF1	4.1909e-2 (6.72e-4) -	4.3608e-2 (3.83e-4) -	9.9663e-2 (3.11e-2) -	4.4003e-2 (4.02e-4) -	4.2198e-2 (4.89e-4) -	4.0232e-2 (2.56e-4)
MaF2	2.9522e-2 (3.70e-4) -	3.2849e-2 (6.35e-4) -	6.5930e-2 (5.34e-3) -	3.0452e-2 (3.83e-4) -	3.0105e-2 (4.89e-4) -	2.9059e-2 (2.22e-4)
MaF3	5.0040e-2 (4.53e-3) -	4.7122e-2 (5.73e-4) -	2.3878e-1 (4.98e-3) -	3.4376e-2 (6.30e-4) +	6.0342e-2 (2.35e-2) -	4.4919e-2 (6.21e-4)
MaF4	3.6575e-1 (3.12e-2) -	3.4217e-1 (2.03e-3) -	4.4683e-1 (2.37e-2) -	2.6558e-1 (6.38e-3) +	4.2424e-1 (1.90e-1) -	3.0451e-1 (8.98e-3)
MaF5	3.0631e-1 (1.06e-2) +	1.9607e+0 (1.66e+0) -	1.6141e+0 (1.55e+0) -	2.6237e-1 (3.75e-3) +	2.5771e-1 (5.05e-3) +	6.4028e-1 (7.06e-1)
MaF6	4.6248e-3 (1.09e-4) -	5.0368e-3 (7.47e-5) -	3.4488e-2 (1.20e-2) -	4.9887e-3 (1.53e-4)-	5.3524e-3 (2.98e-4) -	4.2976e-3 (2.14e-4)
MaF7	6.3830e-2 (2.85e-3) -	1.7942e-1 (1.41e-1) -	3.9959e-1 (2.39e-1) -	8.8775e-2 (8.95e-2) -	5.9493e-2 (2.83e-3) +	6.1200e-2 (1.13e-3)
MaF8	7.2091e-2 (2.74e-2) ≈	7.5765e-2 (2.93e-3) -	1.1399e-1 (9.30e-3) -	6.4683e-2 (1.24e-3) +	8.8674e-2 (1.65e-2) -	6.6784e-2 (1.69e-3)
MaF9	1.5747e-1 (2.10e-1) -	7.3199e-2 (4.96e-2) -	6.6358e-1 (1.98e-1) -	6.3475e-2 (6.70e-4) -	2.6249e-1 (1.44e-1) -	5.9661e-2 (3.66e-4)
MaF10	9.7292e-1 (5.62e-2) -	1.4932e-1 (3.47e-3) -	9.4003e-1 (2.16e-1) -	1.5193e-1 (2.59e-3) -	1.8289e-1 (1.99e-2) -	1.4220e-1 (2.88e-3)
MaF11	1.7837e-1 (5.94e-3) -	1.6450e-1 (9.64e-4) -	3.5908e-1 (2.95e-2) -	1.5906e-1 (2.87e-3)+	1.6522e-1 (1.14e-2) ≈	1.6119e-1 (1.46e-3)
MaF12	2.5437e-1 (2.97e-2) -	2.2090e-1 (9.40e-4) -	3.6263e-1 (2.37e-2) -	2.2701e-1 (3.38e-3) -	2.2019e-1 (2.14e-3) -	2.1824e-1 (2.57e-3)
MaF13	7.7195e-2 (8.27e-3) ≈	8.1611e-2 (9.24e-3) -	3.6290e-1 (1.23e-1) -	8.2062e-2 (9.19e-3) -	1.2808e-1 (1.56e-2) -	7.3875e-2 (8.00e-3)
MaF14	8.5721e-1 (5.66e-3) -	4.8502e-1 (6.99e-2) +	8.5067e-1 (7.48e-1) -	5.1461e-1 (1.24e-1) ≈	5.6360e-1 (1.05e-1) ≈	5.4424e-1 (9.04e-2)
MaF15	1.5345e-1 (4.89e-3)+	2.4860e-1 (3.22e-2) ≈	7.1364e-1 (4.12e-2) -	3.3578e-1 (7.89e-2) -	5.4932e-1 (1.21e-1) -	2.5732e-1 (8.80e-2)
IMOP1	1.5742e-1 (1.46e-2) -	1.0579e-1 (1.19e-3) +	1.0378e-1 (1.34e-2) ≈	6.0236e-2 (9.39e-3) +	2.5124e-2 (8.26e-3) +	1.0942e-1 (4.40e-3)
IMOP2	1.3022e-1 (8.18e-3) -	1.3260e-2 (5.29e-3) -	8.2667e-2 (1.09e-2) -	9.7062e-3 (1.79e-3) -	1.2170e-2 (4.39e-3) -	6.8761e-3 (1.92e-3)
IMOP3	1.5327e-1 (1.76e-2) -	4.8729e-3 (1.18e-4) -	6.1834e-2 (3.53e-2) -	5.8594e-3 (4.39e-3) ≈	1.3422e-2 (6.96e-3) -	3.9739e-3 (5.47e-5)
IMOP4	1.0187e-1 (1.82e-2) -	8.9372e-3 (1.26e-4) -	1.7185e-2 (2.66e-3) -	7.7056e-3 (1.23e-4) -	1.2215e-2 (2.31e-3) -	6.9803e-3 (6.59e-5)
IMOP5	1.8330e-1 (2.97e-2) -	3.6140e-2 (8.53e-4) -	5.8357e-2 (4.10e-3) -	3.5750e-2 (3.58e-4) -	4.1200e-2 (2.02e-3) -	3.4491e-2 (2.25e-4)
IMOP6	9.4882e-2 (1.87e-2) -	1.6451e-1 (2.28e-1) -	1.9819e-1 (2.09e-1) -	3.3873e-2 (4.24e-4) -	3.2680e-2 (7.65e-4) ≈	3.3026e-2 (3.78e-4)
IMOP7	2.1846e-1 (1.44e-1) ≈	8.4539e-1 (1.90e-1) -	8.8789e-1 (2.15e-6) -	3.9761e-2 (1.18e-3) ≈	4.0795e-2 (3.35e-3) ≈	5.0067e-1 (4.30e-1)
IMOP8	1.2728e-1 (1.08e-2) -	1.2245e-1 (9.66e-2) -	2.7880e-1 (2.14e-1) -	7.9215e-2 (1.97e-3) -	7.3462e-2 (2.56e-3) -	6.9807e-2 (9.96e-4)
+/-/≈	2/18/3	2/20/1	0/22/1	6/14/3	3/16/4	-
Best/All	1/23	1/23	0/23	5/23	4/23	12/23

Table 8

Comparison of the MOEAs on HV value with means and standard deviations for MaF and IMOP benchmark problems

	EMyO/C	AR-MOEA	RPEA	RVEA-iGNG	DEA-GNG	MOEANS
MaF1	2.1932e-1 (6.19e-4) -	2.1808e-1 (4.94e-4) -	1.7685e-1 (2.14e-2) -	2.1916e-1 (4.10e-4) -	2.1876e-1 (5.87e-4) -	2.2166e-1 (4.66e-4)
MaF2	2.4192e-1 (1.47e-3) -	2.3972e-1 (7.25e-4) -	2.2979e-1 (2.08e-3) -	2.4473e-1 (3.77e-4) ≈	2.4171e-1 (1.20e-3) -	2.4481e-1 (5.11e-4)
MaF3	9.5746e-1 (1.98e-3) ≈	9.5713e-1 (6.73e-4) -	8.4791e-1 (2.80e-3) -	9.6161e-1 (5.84e-4) +	9.4464e-1 (1.12e-2) -	9.5869e-1 (6.47e-4)
MaF4	5.2737e-1 (3.21e-3) -	5.2137e-1 (1.67e-3) -	5.0911e-1 (5.46e-3) -	5.3349e-1 (1.45e-3) ≈	5.0394e-1 (5.48e-2) -	5.3328e-1 (2.12e-3)
MaF5	5.3601e-1 (4.43e-3) +	3.5820e-1 (1.67e-1) ≈	4.0327e-1 (1.64e-1) -	5.5773e-1 (8.73e-4) +	5.5114e-1 (1.95e-3) +	5.0257e-1 (9.36e-2)
MaF6	1.9954e-1 (1.15e-4) -	1.9949e-1 (8.24e-5) -	1.8247e-1 (5.20e-3) -	1.9970e-1 (9.24e-5) -	1.9973e-1 (9.39e-4) -	1.9977e-1 (5.12e-5)
MaF7	2.7789e-1 (1.44e-3) ≈	2.6229e-1 (1.68e-2) -	2.3388e-1 (2.30e-2) -	2.7449e-1 (1.09e-2) ≈	2.7704e-1 (9.69e-4) ≈	2.7736e-1 (8.94e-4)
MaF8	2.7436e-1 (1.16e-2) -	2.7297e-1 (1.46e-3) -	2.5747e-1 (3.65e-3) -	2.7816e-1 (5.16e-4) ≈	2.6544e-1 (6.57e-3) -	2.7808e-1 (7.24e-4)
MaF9	7.4557e-1 (1.75e-1) -	8.2871e-1 (3.84e-2) -	4.3411e-1 (1.39e-1) -	8.3349e-1 (8.63e-4) -	6.9273e-1 (8.24e-2) -	8.3862e-1 (3.56e-4)
MaF10	6.9454e-1 (6.36e-2) -	9.4266e-1 (9.25e-4) -	8.5343e-1 (1.58e-2) -	9.4291e-1 (1.57e-3) -	9.2734e-1 (5.08e-3) -	9.4468e-1 (1.12e-3)
MaF11	9.2458e-1 (2.87e-3) -	9.3053e-1 (9.51e-4) -	8.8013e-1 (7.87e-3) -	9.3419e-1 (7.96e-4) +	9.2691e-1 (3.35e-3) -	9.3225e-1 (6.09e-4)
MaF12	4.9915e-1 (2.78e-2) -	5.3755e-1 (2.59e-3) +	5.1514e-1 (5.33e-3) -	5.4254e-1 (2.79e-3) +	5.2847e-1 (3.49e-3) ≈	5.2650e-1 (3.94e-3)
MaF13	5.3223e-1 (9.97e-3) +	5.0332e-1 (1.82e-2) ≈	3.9160e-1 (4.16e-2) -	5.2496e-1 (1.05e-2) +	4.2842e-1 (2.25e-2) -	5.1267e-1 (1.06e-2)
MaF14	9.0910e-2 (2.48e-6) -	2.7311e-1 (5.78e-2) ≈	2.4905e-1 (1.25e-1) ≈	2.4402e-1 (8.20e-2) ≈	2.6145e-1 (1.00e-1) ≈	2.6208e-1 (9.01e-2)
MaF15	3.4697e-1 (6.19e-3) +	3.2404e-1 (2.64e-2) +	3.3991e-2 (1.33e-2) -	1.8084e-1 (6.68e-2) -	4.8851e-2 (4.32e-2) -	2.4978e-1 (5.48e-2)
IMOP1	9.8423e-1 (7.89e-4) -	9.8495e-1 (3.73e-5) ≈	9.8303e-1 (1.39e-3) -	9.8760e-1 (4.96e-5) +	9.8753e-1 (8.37e-5) +	9.8495e-1 (2.94e-4)
IMOP2	2.0620e-1 (3.22e-3) -	2.3203e-1 (9.12e-5) +	2.2442e-1 (1.18e-3) -	2.3206e-1 (6.05e-5) +	2.3070e-1 (9.92e-4) -	2.3135e-1 (1.14e-4)
IMOP3	4.6660e-1 (1.72e-2) -	6.5835e-1 (4.43e-5) -	5.9674e-1 (4.96e-2) -	6.5779e-1 (1.83e-3) -	6.5422e-1 (3.15e-3) -	6.5870e-1 (2.66e-5)
IMOP4	3.1066e-1 (2.22e-2) -	4.3358e-1 (1.56e-4) -	4.2849e-1 (1.36e-3) -	4.3342e-1 (1.57e-4)-	4.3072e-1 (1.37e-3) -	4.3410e-1 (1.26e-4)
IMOP5	4.8277e-1 (1.34e-2) -	5.0105e-1 (4.05e-3) ≈	4.8099e-1 (3.27e-3) -	5.0056e-1 (2.36e-3) -	5.1219e-1 (8.68e-3) +	5.0160e-1 (1.08e-3)
IMOP6	4.9167e-1 (1.03e-2) -	4.7927e-1 (8.51e-2) -	4.5038e-1 (6.97e-2) -	5.2736e-1 (8.03e-4) -	5.2387e-1 (9.59e-4) -	5.3123e-1 (5.64e-4)
IMOP7	4.0444e-1 (7.66e-2) ≈	1.1648e-1 (9.53e-2) -	9.5185e-2 (5.39e-7) -	5.2654e-1 (1.03e-3) ≈	5.0305e-1 (6.75e-3) ≈	2.9066e-1 (2.21e-1)
IMOP8	4.8684e-1 (1.08e-2) -	4.9737e-1 (2.59e-2) -	4.5949e-1 (6.30e-2) -	5.2888e-1 (1.85e-3) ≈	5.4192e-1 (1.27e-2) ≈	5.2977e-1 (1.06e-3)
+/-/≈	3/17/3	3/15/5	0/22/1	7/9/7	3/15/5	-
Best/All	3/23	1/23	0/23	9/23	2/23	8/23

Tables 3 and 4 record the IGD and HV means and standard deviations of NSGAII, MOEA/D, DGEA, DAEA and RSEA, respectively. From Table 3, MOEANS obtains 12 best IGD values out of 23 test problems, lager than the compared algorithms (i.e., NSGAII, MOEA/D, DGEA, DAEA and RSEA obtained the number of IGD best values are 3, 0, 6, 2, 0, respectively). From Table 4, it can be seen that MOEANS obtains 14 best HV values out of 23 test problems, lager than the compared algorithms (i.e., NSGAII, MOEA/D, DGEA, DAEA and RSEA obtained the number of HV best values are 2, 0, 5, 2, 0, respectively). In this situation, NSGAII uses the non-dominated sorting and crowding distance to balance the convergence and diversity. MOEA/D uses the decomposition method to transform a MOP into a group of single objective and optimize them simultaneously. Compared with NSGAII and MOEA/D, MOEANS algorithm adopts the non-dominated sorting and niche selection to update the population. As can be seen from the statistical results in Tables 3 and 4, MOEANS algorithm performs well than NSGAII and MOEA/D in solving irregular problems. DGEA, DAEA and RSEA algorithms are mainly constructed from matching pools to improve the quality of offspring and have been successful in dealing with regular PFs while being ungenerous to irregular PFs. It can be seen from the data statistics that MOEANS algorithm using niche selection strategy is more competitive in dealing with irregular PFs.

Table 5 and Table 6 record the IGD and HV means and standard deviations of BCE-MOEA/D, MOEA/D-AWA, MOEA/D-PaS, A-NSGAIII and RVEA, respectively. From Table 5, MOEANS obtains 13 best IGD values out of 23 test problems, lager than the compared algorithms (i.e., NSGAII, MOEA/D, DGEA, DAEA and RSEA obtained the number of IGD best values are 5, 5, 0, 0, 0, respectively). From Table 6, it can be seen that MOEANS obtains 8 best HV values out of 23 test problems, lager than the compared algorithms (i.e., BCE-MOEA/D, MOEA/D-AWA, MOEA/D-PaS, A-NSGAIII and RVEA obtained the number of HV best values are 6, 8, 0, 0, 1, respectively). In this group, the algorithms mainly are based on the reference vectors. BCE-MOEA/D and MOEA/D-PaS adopt auxiliary selection methods working together with a set of fixed reference vectors to solve irregular PFs for improving the diversity of population. However, although the auxiliary method can adjust the distribution of the solution set, it is still affected by the evenly distributed weight vector, so it is difficult to search the real front for extremely irregular problems, such as MaF1-MaF15, IMOP4-IMOP5 and IMOP8. RVEA algorithm uses the angle penalized distance (APD) under the guidance of reference vectors to maintain population diversity. A-NSGAIII generates denser reference vectors around promising regions by using the information of the active reference vector for improving the distribution of population. However, the distribution of solution set is still affected by weight vector, and it is still difficult to search all PFs for degradation problems MaF6, MaF8, MaF9 and MaF13. MOEA/D-AWA adopts all historical non-dominated solutions in current archive population to make the adaptation of reference vectors. From the statistical data in Table 5 and Table 6, the number of MOEA/D-AWA superior to, inferior to and similar to MOEANS is 6/15/2 and 7/13/3 respectively, which show the good performance in solving irregular PFs. However, the generation of the reference vectors using the solutions in archive is only triggered at the late stage of the search process, and the promising regions may not be found if the population distribution changes dramatically. MOEANS constructs niches based on current population information and selects high-quality solutions directly within niches radius. According to the data statistics, compared with these algorithms for adjusting the reference vectors, MOEANS algorithm has a certain improvement in the search of the whole front in solving irregular front problems.

Table 7 and Table 8 record the IGD and HV means and standard deviations of EMyO-C, ARMOEA, RPEA, RVEA-iGNG and DEA-GNG, respectively. From Table 7, MOEANS obtains 12 best IGD values out of 23 test problems, lager than the compared algorithms (i.e., EMyO-C, ARMOEA, RPEA, RVEA-iGNG and DEA-GNG obtained the number of IGD best values are 1, 1, 0, 5, 4, respectively). From Table 8, it can be seen that MOEANS obtains 8 best HV values out of 23 test problems, lager than the compared algorithms (i.e., EMyO-C, ARMOEA, RPEA, DEA-GNG obtained the number of HV best values are 3, 1, 0, 2, respectively) but less than RVEA-iGNG. In this set, the compared algorithms mainly are based on learning the distribution of PFs, reference points and clustering. RVEA-iGNG and DEA-GNG adjust the reference vectors by training a growing neural gas (GNG) network. DEA-GNG trains the GNG network and adjusts the all reference vectors in each generation, and RVEA-iGNG is designed for timely yet steadily adjusting the reference vectors. In Table 7 and Table 8, the number of RVEA-iGNG superior to, inferior to and similar to MOEANS is 6/14/3 and 7/9/7 respectively, and the number of DEA-GNG superior to, inferior to and similar to MOEANS is 3/16/4 and 3/15/5 respectively. It can be seen from the data that RVEA-iGNG algorithm performs almost as well as MOEANS in solving irregular PFs in HV data statistics. However, the effect of using GNG to adjust reference vectors on inverted PF, IMOP4-IMOP7 and extremely irregular PFs is not as good as the niche selection method adopted by MOEANS. ARMOEA adjusts the reference points according to the uniformly distributed reference points are initially produced with Das and Dennis’s approach. In Table 8, ARMOEA obtains the similar performance than MOEANS on MaF5, MaF13, MaF14, IMOP1 and IMOP5, superior than MOEANS on MaF12, MaF15 and IMOP2. However, in the limited number of iterations, it can not well adapt to the distribution of population, resulting in the waste of reference points. RPEA adopts the current population to adjust reference points and adaptively adjusts the convergence and distribution of the solution. However, the experimental results show that the algorithm is obviously inferior to MOEANS in solving irregular front problems. EMyO-C uses the individuals to adaptively clustering, which performs the better convergence and diversity on MaF7, MaF13 and MaF15 than MOEANS. EMyO-C selects the solution closest to the ideal point from each cluster to ensure the convergence of the algorithm, but the distribution performance needs to be improved.

The IGD and HV values of 16 algorithms were calculated according to the data in Table 3-Table 8. Although the statistical results of the two indicators are different, this is due to the different calculation methods of indicators. According to the law of no free lunch [42], MOEANS algorithm is not guaranteed to solve all the problems well. However, at least for most problems, the proposed MOEANS algorithm obtains more best results.

5.2 The analysis of figures

The analysis of Pareto front figures can intuitively see the convergence and distribution of the algorithm on each problem and the indicator convergence figures can show the convergence speed and accuracy of the algorithm.

Fig. 1 lists the the representation of the optimal solutions obtained by each algorithm on MaF1, MaF6 and IMOP5, which is characterized by inverted, degenerate and eight circles. From Fig. 1, MOEA/D, DGEA and DAEA do not search the whole PFs. MOEA/D uses the uniformly distributed reference vectors and is obviously difficult to maintain the diversity. DGEA converges to real PFs on MaF1 and IMOP5, but it is constrained by direction vector and cannot cover all fronts. Besides, DGEA is difficult to converge on degradation problems. DAEA has the worst performance in both convergence and diversity, indicating that the influence of diversity of decision space on objective space is difficult to solve the MaF1, MaF6 and IMOP5 problems. In this group, RSEA converges to and covers the true PFs, but it’s still not as evenly distributed as MOEANS.

Fig. 1

The representation of the optimal solutions obtained by each algorithm on the MaF1, MaF6 and IMOP5.

Fig. 2 lists the representation of IGD convergence trend by each algorithm on MaF1, MaF3, MaF8 and MaF10. Fig. 3 lists the representation HV convergence trend by each algorithm on MaF7, IMOP3, IMOP4 and IMOP6. The results show that MOEANS has the highest convergence accuracy even though the convergence speed is not the fastest on some functions. In a word, MOEANS can obtain the better convergence and diversity than NSGAII, MOEA/D, RSEA, DGEA and DAEA.

Fig. 2

The representation of IGD convergence trend by each algorithm on the partial functions.

Fig. 3

The representation of HV convergence trend by each algorithm on the partial functions.

Fig. 4 lists the representation of the optimal solutions obtained by each algorithm on MaF2, MaF12 and IMOP8. In this group, these compared algorithms mainly use fixed reference vectors and adaptive reference vectors to solve irregular front problems. BCE-MOEA/D, MOEA/D-PaS and RVEA do not converge to the true the PF of MaF2, which indicates the influence of fixed reference vector on convergence and distribution when the number of extremum points exceeds the number of objectives. MOEA/D-AWA and A-NSGAIII use the adjusted reference vector, and the adjustment based on the archived elite solution performs better than the interpolation based on the original reference vector. According to the results, MOEANS applies the niche selection performs better than the compared algorithms. MaF12 is characterized by concave, nonseparable, biased deceptive. MOEANS and RVEA are superior to other algorithms based on population adjustment or auxiliary strategies. As for IMOP8, there are 100 discontinuous regions and there are identical individuals in each region. The results in Fig. 4 show that the compared algorithms are difficult to search all discontinuous regions, while the 100 individuals of MOEANS proposed in this paper are just evenly distributed in 100 regions.

Fig. 4

The representation of the optimal solutions obtained by each algorithm on the MaF2, MaF12 and IMOP8.

Fig. 5 lists the representation of IGD convergence trend by each algorithm on MaF2, MaF6, MaF9 and MaF13. Fig. 6 lists the representation of HV convergence trend by each algorithm on MaF4, IMOP4, IMOP6 and IMOP8. Experimental results show that MOEANS obtains the highest accuracy on these eight functions. In addition, IGD and HV can show the convergence and diversity of the algorithm at the same time, so MOEANS shows better convergence and diversity in this group of compared algorithms.

Fig. 5

The representation of IGD convergence trend by each algorithm on the partial functions.

Fig. 6

The representation of HV convergence trend by each algorithm on the partial functions.

Fig. 7 lists the the representation of the optimal solutions obtained by each algorithm on MaF9, IMOP4 and IMOP6. In this group, the compared algorithms include the neural network adjustment of reference vector algorithms, adaptive adjustment of reference point algorithms and clustering based algorithms. As for MaF9, it is characterized by a liner and degenerate. MOEANS and ARMOEA converge to the true front and uniformly cover the entire front, which indicates that niche selection and the use of individual contributions can handle linearity and degradation very well. EMyO-C, RVEA-iGNG and DEA-GNG can converge to the true PF, but they cannot cover the whole front, especially DEA-GNG. When RPEA algorithm is solving MaF9, the strategy with the non-dominated solutions as the reference points cannot converge to the real front. IMOP4 is a wavy line and IMOP6 is separated into several grids, which are difficult to maintain the diversity. As can be seen from the Fig. 7, EMyO-C and RPEA perform the worst distribution. In this comparison, ARMOEA, RVEA-iGNG and DEA-GNG can cover the entire PF of IMOP4 and IMOP6, but the distribution of waves in IMOP4 front is not uniform.

Fig. 7

The representation of the optimal solutions obtained by each algorithm on the MaF9, IMOP4 and IMOP6.

Fig. 8 lists the representation of IGD convergence trend by each algorithm on MaF1, MaF10, IMOP4 and IMOP5. Fig. 9 lists the representation of HV convergence trend by each algorithm on MaF6, MaF9, MaF10 and IMOP6. RPEA obtains the worst IGD convergence and accuracy on MaF1, MaF10 and HV accuracy on MaF9 and IMOP6. EMyO-C shows good convergence accuracy and speed only on MaF1 and MaF6, and the convergence accuracy is inferior to MOEANS on other problems. ARMOEA, RVEA-iGNG and DEA-GNG are slightly inferior to MOEANS in convergence speed and accuracy.

Fig. 8

The representation of IGD convergence trend by each algorithm on the partial functions.

Fig. 9

The representation of HV convergence trend by each algorithm on the partial functions.

Based on data and figures analysis, MOEANS algorithm has the ability to deal with irregular Pareto front problems. However, MOEANS algorithm is not guaranteed to solve all the problems well [42]. At least for most problems, the proposed algorithm obtains more best results.

6 Conclusions and future work

In this paper, a multi-objective evolutionary algorithm based on niche selection in solving irregular Pareto fronts, termed MOEANS. The main contribution of this paper includes three aspects. The first one is constructing a matching pool based on the convergence and diversity values. In this situation, the convergence values is designed to construct the matching pool when the non-dominated solutions do not converge to the first front, which improve the population convergence. The diversity values calculated by the niche is used to select parents when the non-dominated solutions are converge to the first front, which enhances the quality of offsprings in searching local regions. The second contribution is the adoption of niche selection strategy in environmental selection process. In this part, the normalization procedure is used to eliminate the effects of different ranges on the directional density estimation, and niche selection strategy is designed to estimate the crowding degree of a solution and choose the solution with good diversity. The third contribution is that the algorithm is compared with 15 representative algorithms. Experimental results have demonstrated that MOEANS outperforms other compared algorithms on overall 23 test problems. Therefore, MOEANS is a promising and versatile algorithm with good convergence and diversity.

In further studies, we need to deal with not only large-scale problems, but also high-dimensional space problems.

Footnotes

Acknowledgment

The authors would like to thank the editor and reviewers for their helpful comments and suggestions to improve the quality of this paper. This study is support by the National Natural Science Foundation of China(61873006) and Beijing Natural Science Foundation(4212040, 4204087).

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