A many-objective evolutionary algorithm based on vector angle distance scaling

Abstract

In the past two decades, multi-objective evolutionary algorithms (MOEAs) have achieved great success in solving two or three multi-objective optimization problems. As pointed out in some recent studies, however, MOEAs face many difficulties when dealing with many-objective optimization problems(MaOPs) on account of the loss of the selection pressure of the non-dominant candidate solutions toward the Pareto front and the ineffective design of the diversity maintenance mechanism. This paper proposes a many-objective evolutionary algorithm based on vector guidance. In this algorithm, the value of vector angle distance scaling(VADS) is applied to balance convergence and diversity in environmental selection. In addition, tournament selection based on the aggregate fitness value of VADS is applied to generate a high quality offspring population. Besides, we adopt an adaptive strategy to adjust the reference vector dynamically according to the scales of the objective functions. Finally, the performance of the proposed algorithm is compared with five state-of-the-art many-objective evolutionary algorithms on 52 instances of 13 MaOPs with diverse characteristics. Experimental results show that the proposed algorithm performs competitively when dealing many-objective with different types of Pareto front.

Keywords

Vector angle distance scaling evolutionary algorithm many-objective optimization problem

1 Introduction

Multi-objective evolutionary algorithm (MOEAs) is a population-based heuristic algorithm that allows the search of Pareto approximation sets in a single run, becoming a universal and effective method to solve multi-objective optimization problems (MOPs) with two or three objectives [1, 2]. Because MOPs exists widely in the real world especially in optimal back-propagation artificial neural network [3, 4], MOEAs may generate contribution in missing data management task in smart systems [5, 6]. Therefore, multi-objective optimization has become an interesting research field. In the past decades, many classical Pareto dominance-based algorithms have been proposed to solve multi-objective optimization problems, such as NSGA-II [7], SPEA2 [8] and PESA-II [9]. However, Pareto-dominance-based algorithms are not well-suited to tackle MOPs with more than three objectives called as the many-objective problems(MaOPs) [10]. The selection pressure towards Pareto front based on Pareto dominance decreases sharply with the increase of the number of objectives. When the number of objectives is large, almost all of the solutions in the population are non-dominant, which makes the Pareto-based selection indistinguishable from the individuals [11].

Faced with such challenges, various strategies are put forward to improve the performance of multi-objective algorithm in solving many-objective problems. In the existing research, the most common methods can be divided into the following five categories [12].

Dominance-based methods

Indicator-based methods

Objective reduction-based methods

Preference-based methods

Decomposition-based methods

In recent years, several different forms of dominance-based methods have been proposed to improve the selective pressure. Several new dominance concepts have been well studied and applied, such as, ɛ-dominance [13, 14], α-dominance [15], fuzzy Pareto dominance [16, 17] and grid-based dominance [18]. These algorithms modify the dominance criterion to accelerate the convergence in solving many-objective problems. However, the main disadvantage of these Pareto advantages in heuristic selection is that it often involves one or more parameters and the parameters associated with these problems. Another typical method is to combine Pareto-dominance-based method with convergence-based metrics. For example, ε-indicator preference [19], shift-based density estimation strategy [20], a mating selection based on favorable convergence [21] and knee point-based secondary selection [22].

The second category is known as the indicator-based methods which convert a MOP into the problem of optimizing an indicator by evaluating the solutions using performance indicators. For example, methods based on HV indictors include S-metric selection-based multi-objective evolutionary algorithm(SMS-EMOA)[23], HV contribution based on Monte Carlo sampling algorithm(HypE) [24], HV contribution-based multi-objective particle swarm optimizer and HV-based dynamic neighborhood method [25]. Unfortunately, when the number of objective increases, the computational cost for the calculation of hypervolume is prohibitively more expensive. Apart from the hypervolume indicator, R2 indicator, which achieves a comprehensive balance between convergence and diversity, can be regarded as the mutual preference based on the contribution to each reference vector, such as R2-EMOA [26], R2MOPSO [27], R2-IBEA [28], MOMOBI-II [29] and TS-R2EA [30]. Besides, some researchers are exploring other indicators to replace the dominance-based approach, such as GD [31], IGD [32], IGD+ [33] and ▵_p [34].

The third category is known as reduce the number of objectives, which finds the relevant objectives and eliminate these redundant ones that do not need to describe the Pareto front. some techniques, such as feature selection [35], principal component analysis [37] and Pareto corner search [38], are proposed in recent years. However, objective reduction is inevitably accompanied by the loss of information of the original problem. Therefore, whether this technique is suitable for solving MaOPs depends on the characteristics of the problem itself.

The forth category is based on preference-based approaches [38]. In this case, preference ordering-based approaches include k-optimality relation [16] and maximum ranking [39]. The main characteristic of the preference based ranking method is that it emphasizes certain target areas to perform better, but the main shortcomings is that it performs worse overall. Another method is preference incorporation-based approaches. In this situation, decision makers usually preferred only the Pareto frontier in which they were most interested and can obtain the effective work, such as PBEA [40], r-NSGA-II [41] and TKR-NSGA-II [42]. However, premature convergence could contribute to the inferior results.

The fifth category is the most popular decomposition-based approaches. MOEA/D [43] is the classical decomposition-based multi-objective evolutionary algorithm which decomposes a MOP into a number of subproblems and use a set of weight vectors to optimize these subproblems based on a aggregated function. This method becomes a turning point on the path of multi-objective optimization algorithm exploration, and some decomposition-based algorithms are proposed gradually, such as MOEA/D-DE [44], MOEA/D-DD [45] and MOEA/D-M2M [46]. These algorithms are variations on the MOEA/D framework to solve many-objective problems and the researchers are not satisfied with this single research direction. Another category based on decomposition method is NSGA-III [47] which need a set of predefined evenly distributed reference points to maintain the diversity of the population. Then, reference vectors is referred as vectors that is use to decompose the original objective space, such as [48, 49]. In recent years, Cheng [50] proposed a reference vector-guided EA, which designed angle-penalized distance to balance the convergence and diversity of the candidate solutions. Maha [51] proposed a new decomposition-based NSGA-II for MaOPs, in this case, reference point-based dominance enables a set of well-distributed reference points to create a strict partial order over a set of non-dominant solutions. Jiang [52] introduced an efficient reference direction-based density estimator to enhance the diversity of candidate solutions. Liu [53] proposes a fuzzy decomposition-based MOEA to address the problem of the similarity metric affection.

Most of the existing MOEAs focus on enhancing convergence and maintaining diversity, it is noted that the use of reference vector will become particulary important for MaOEAs, not only can users choose a solution with better distribution based on the distribution of the reference vector, but they can also choose a solution with better convergence based on the fitness value generated by the reference vector.

As shown in the existing research [47, 50], reference points or reference vectors are evenly distributed in the whole objective space that can guide the candidate solutions to toward the Pareto front and evenly distributed. In addition, the value of vector angle distance scaling is applied in [54] to assign scores for each function ranking the individuals. Motivated by the methods of reference vectors and the aim to use the fitness value to achieve the convergence and diversity, we propose a many-objective evolutionary algorithm based on vector angle distance scaling. Compared with existing decomposition-based methods, the main contribution of this algorithm can be summarized as follows.

A scalarization method, termed as the new fitness value of vector angle distance function. In the proposed method, the distance between the candidate solutions and the ideal point is used as the convergence criterion, and the cosine of the angle between the candidate solutions and reference vector is regarded as the diversity criterion.

Scaling the fitness of vector angle distance, relative to the best performance solutions on the reference vector. In this process, scale each column by the minimum found, and then use the second minimum to scale the result. This method can deal with many-objective problems by using a compact analytic method to obtain the well solutions.

A final aggregate fitness will be found in each population and will be designed into tournament selection. This method improve the tournament selection in both convergence and diversity.

An adaptation strategy is applied to adjust the distribution of reference vector, which can improve the ability of algorithm to solve the PF with asymmetric objective function.

The rest of this paper is organized as follows. Section 2 describes some background knowledge. Section 3 introduces the proposed MaOEA-VADS 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 decision can be mathematically defined as $\min F (x) = (f_{1} (x), f_{2} (x), . . ., f_{m} (x))^{T}$ (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 a n-dimensional decision various from Ω, m is a the number of objective functions.

Definition 1. (Pareto dominance) x_a and x_b belong to Ω and 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 solution) A solution x^* ∈ Ω is called Pareto optimal solution (Non-dominated solution) 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 solution, which is called Pareto front(PF).

2.2 Motivated MaOEAs

The aggregation-based methods uses an aggregation function to assign scores for each solution, and then these solutions will be selected according to the scores. MSOPS-II [54] proposed that the combination of weighted min-max function with vector angle distance scaling is used to assign scores for each function to rank the individuals. In MSOPS-II, the design of weighted min-max focuses on the convergence and the vector angles distance scaling contributes to the diversity. In this case, current population is used to generate target vectors to improve the distribution of solutions. MOEA/D [43] is proposed that decomposes a multi-objective problem into a set of scalar subproblems using uniformly distributed weight vectors, where weighted sum method, weighted Tchebycheff method and penalty-based boundary intersection were discussed to solve MOPs and MaOPs. In MOEA/D, neighborhood information defined according to the Euclidean distance between weight vectors is used to optimize subproblems, where the distributed weight vectors contributes to diversity of solutions.

The reference vectors-based methods uses the association between reference vectors and solutions to select these solutions with good convergence and distribution. NSGA-III [47] based on the framework of NSGA-II is proposed that each solution is associated with a reference vector according to the perpendicular distance defined by a solution to reference vector. In this case, these solutions with small niche counts will have more changes to be selected, which improve the diversity performance in solving MaOPs. RVEA [50] uses a set reference vectors to partition the objective space into a number of subspaces and the angle penalized distance is calculated in each subspace for one solution. Then, a reference vector adaptation strategy is adopted to deal with scaled problems with no-well normalized objective functions for MaOPs.

Existing references describe that uniformly distributed reference vector paly the key role which contribute algorithm to achieving the good convergence and distribution. Motivated by MSOPS-II, MOEA/D and RVEA, the selection pressure is improved by calculating the vector angles distance scaling between these reference vectors and solutions. In addition, a aggregation function based on vector angles distance scaling is applied into the mating selection for improve the selection pressure and the quality of offspring population. Moreover, a well distributed reference vectors in the whole objective space is necessary to adjust adaptively for various problems.

3 The framework of MaOEA-VADS algorithm

3.1 Framework of MaOEA-VADS aglorithm

The main framework of the proposed MaOEA-VADS is given in the Algorithm 1, where the population size N, the number of objectives M and the maximal number of iterations t_max are given. Firstly, we initialize a population P₀ with N individuals randomly and a set of unit reference vectors. In the main loop, a tournament selection is applied to select some promising solutions for generating offspring O_t. Then, environmental selection is performed to select a new population P_t+1 from the union of current population P_t and their offspring Q_t+1. The main loop of MaOEA-VADS is similar to most MaOEAs, but the proposed algorithm is improved in mating selection and environmental selection.

Algorithm 1 Main Loop of MaOEA-VADS algorithm

Input: the population size (N), the number of objectives M, the maximal number of iterations (t_max), parameter f_r = 0.1

Output: final population P_{t
_max}

1: Initialization

2: Initializes the population P₀ with N individuals Initializes a set of unit reference vector (W₀ = w₁, w₂, . . . , w_N)

3: [P₀, r]= Environmental selection(P₀,W,N)

4: Main Loop

5: while t < t_max do

6: Q_t+1= Create offspring(P_t, N, r)

7: R_t = P_t ∪ Q_t+1

8: [P_t+1, r]= Environmental selection(R_t,W,N)

9: t = t + 1

10: if mod (t, f_r * t_max) =0

11: W = W₀ * (z_max - z_min)

12: $W = \frac{W}{∥ W ∥}$

13: end

14: end while

3.2 Initialization procedure

The initialization procedure of MaOEA-VADS includes the initial population P₀, a set of reference vectors W. In this paper, the initial population is randomly sampled from the decision space using a uniformly distribution. Then, a set of uniform distributed reference vectors ia generated. In most existing algorithms, a systematic approach from Das and Dennis’s method is used to generate reference vectors. The number of reference vectors is N = $(\begin{matrix} H + m - 1 \\ m - 1 \end{matrix})$ , where H > 0 is the number of divisions considered along each objective coordinate [47]. However, there are a large number of reference vectors in high-dimensional objective space, which increases the computational burden of the algorithm. Therefore, we adopt two layers of reference points with H₁ and H₂, where H₁ is applied in the boundary layer and H₂ is used in the inside layer. According to the two layers method, we obtain the number of reference vectors is N = $(\begin{matrix} H_{1} + m - 1 \\ m - 1 \end{matrix})$ + $(\begin{matrix} H_{2} + m - 1 \\ m - 1 \end{matrix})$ . Therefore, the reference vector W is obtained by a coordinate transformation. In addition, motivated by RVEA, adaptation strategy including line 10-12 is designed to adjust the distribution of reference vectors, which can improve the ability of algorithm to solve the PF with asymmetric objective function [50].

3.3 Environmental selection

Environmental selection is applied to survive N better candidate solutions to regenerate new population for the next generation from the union of the current parent population and offspring population. In this situation, the selected candidate solutions should have the good convergence and diversity performance. However, it is not easy to maintain convergence and diversity in solving high dimensional problems. In this paper, we propose vector angle distance scaling method based on reference vectors. The procedure of environmental selection is listed in Algorithm 2.

Algorithm 2 Environmental selection

Input: the parent population (P_t), the offspring population (Q_t), the reference vectors (W), the population size (N)

Output: the selected population P_t+1, fitness value r′

1: if Q_t =∅ then

2: R_t = P_t

3: else

4: R_t = P_t ∪ Q_t

5: end if

6: R_t = P_t ∪ Q_t

7: for each reference vector w_i ∈ W do

8: $f (i) = \frac{| F_{t} |}{(W \cdot \frac{F_{t}}{| F_{t} |})^{b}}$

9: end for

10: Scale each column by the minimum value found S₁ (f)

11: Scale the rows which have the minimum value by the second minimum value (S₁ (f))

12: Calculate the aggregate fitness r of each solutions

13: [P_t+1, r′]=VADS-based selection(S₁, W, N)

14: return P_t+1, r′

In the proposed MaOEAs-VADS, environmental selection operation is based on a set of reference vector to calculate the fitness to select quality solutions with good convergence and diversity. Firstly, the offspring population and the parent population are combined to form set R_t. Then, the metric of the vector angle distance scaling between each solution and vector is calculated. After obtaining the vector angle distance scaling metric S, the improved associated metric S₁ is generated, and the aggregate fitness r of each solution are calculated. In Algorithm 2, lines 10 through lines 12 describe the calculation process. First of all, we find the minimum value and index for each reference vector, and all the value of each reference vector are scaled by the minimum value found. Then, the rows which have the minimum value is scaled by the second minimum value, and the scaled metric S₁ is obtained. Secondly, the final aggregate fitness r can be found by the minimum scaled value. Finally, the next generation population will be selected based on the scaled metric S₁, and the detail selection process is shown in Algorithm 3.

Algorithm 3 VADS-based selection

Input: the population (R_t), the reference vectors (W), the population size (N), the scaled metric S₁

Output: the selected population P_t+1, fitness r

1: Set P_t+1 =∅, r′ =∅

2: if P_t+1 < N then

3: [temp, index] = min S₁ (f)

4: Find a solution closest to the reference vector x_i based on index and temp

5: P_t+1 = P_t+1 ∪ {x_i}

6: Delete the reference vector corresponding to the solution x_i and reference vector w_j, and get the new S₁

7: Aggregate values corresponding to selected individuals are also selected to be retained

8: r′ = r′ ∪ {r_i}

9: end if

10: return P_t+1, r′

In Algorithm 3, the selection operation of the population is carried out based on the scaled vector angle distance scale value(VADS). Vector angle distance scaling(VADS) has been introduced in [54]. The VADS value is calculated by the following formula: $f = \frac{| F |}{(W \cdot \frac{F}{| F |})^{b}}$ (2)

In this formula, the numerator is the magnitude of the vector of objectives |F| and the denominator is the cosine value of the angle between the objective vector and reference vector, where the constant factor b is contributed to scaling the ability of the angle cosine. The distance along the objective vector is assigned as the direction of algorithm convergence. The magnitude of angle cosine can not only affect the VADS value, but also represent the distance between a solution and the reference vector, so as to show the diversity between solution sets. In Fig. 1, an example showing how to associate a solution with a reference vector is listed.

Fig. 1

Example showing how to associate a solution with a reference vector.

In this example, solution A and solution B are two objective vector, and w_i is a reference vector. θ₁ and θ₂ are the angles between w_i and A and B, respectively. Since θ₂ < θ₁, the solution B is associated with reference vector w_i. Low θ for a solution represents the a high angle cosine value, then the low VADS value will be obtained. Therefore, low VADS value may lead to a solution with good convergence and diversity.

Firstly, the solution will be found in S₁ based on the minimum VADS value, the solution x_i will be stored into next generation population P_t+1 and the corresponding reference vector w_j will be deleted. Then the aggregate fitness corresponding to the selected individual is selected to generate the offspring population. This process ensures the selection of the solution for the corresponding VADS value of each vector and indicates the solution with good convergence and distribution. In this situation, this selection process is carried out until the population size |P_t+1| is satisfied to the population size N.

3.4 Create offspring

After the initialization is completed, the main loop begins to renew the population in each generation. In this paper, a tournament selection based on the new aggregate fitness r is designed to select better parent individuals. A low r value represents the solution with good convergence and diversity so as to improve the quality of parent generation matching and effect the quality of offspring population. Then, the simulated binary crossover(SBX) and the polynomial mutation(PM) are used to generate the offspring population [7]. The simulated binary crossover(SBX) is calculated as follows: $\begin{matrix} β = {\begin{matrix} (2 μ)^{\frac{1}{η + 1}}, & if μ \leq 0.5 \\ (\frac{1}{2 (1 - μ)})^{\frac{1}{η + 1}}, & otherwise \end{matrix} \end{matrix}$ (3) Then, the offspring solutions are calculated as follows: $x_{i}^{1, t + 1} = 0.5 [(1 + β) x_{i}^{t} + (1 - β) x_{i}^{2, t}]$ (4) $x_{i}^{2, t + 1} = 0.5 [(1 - β) x_{i}^{t} + (1 + β) x_{i}^{2, t}]$ (5)

Besides, polynomial mutation(PM) is used to enhance the diversity and the formula is as follows. δ is calculated as follows: $\begin{matrix} δ = {\begin{matrix} (2 u)^{\frac{1}{η_{m} + 1}} - 1, & μ \leq 0.5 \\ 1 - (2 (1 - u))^{\frac{1}{η_{m} + 1}}, & otherwise \end{matrix} \end{matrix}$ (6)

The offspring individual is generated when a random ind ≤ p_m is true, where p_m = 1/n is mutation probability. $x (ind) = x (ind) + δ (Max (ind) - Min (ind))$ (7)

4 Experimental design

4.1 Test problems

The thirteen test problems incorporate with two test suits: Deb-Thiele-Laumanns-Zitzler(DTLZ) [55] and Walking Fish Group(WFG) [56] are applied to conduct the experiments. The DTLZ test suite consists of four problems (DTLZ1-4) while WFG test suite consists of nine problems (WFG1-9). In addition, the test problems needs to set a certain number of objectives and dimensions during the experiment, so the number of objectives m, the decision variables n and the parameters k and l and reference point z^ref are listed in Table 1. Therefore, using such a set of test problems can reflect the universality of the algorithm.

Table 1
The characteristics and parameter settings of 13 test problems

problems (m) (n) k and l z ^ref characteristic

DTLZ1 3,5,8,10 k + m - 1 k = 5 (1, . . . . , 1) linear, multi-modal

DTLZ2 3,5,8,10 k + m - 1 k = 10 (2, . . . . , 2) concave

DTLZ3 3,5,8,10 k + m - 1 k = 10 (2, . . . . , 2) concave, multi-modal

DTLZ4 3,5,8,10 k + m - 1 k = 10 (2, . . . . , 2) concave, biased

WFG1 3,5,8,10 k + l k = 2 * (m - 1) , l = 20 (3, 5, 7, . . . , 2m + 1) mixed, biased

WFG2 3,5,8,10 k + l k = 2 * (m - 1) , l = 20 (3, 5, 7, . . . , 2m + 1) convex,discontinuous, non-separable

WFG3 3,5,8,10 k + l k = 2 * (m - 1) , l = 20 (3, 5, 7, . . . , 2m + 1) liner,non-separable

WFG4 3,5,8,10 k + l k = 2 * (m - 1) , l = 20 (3, 5, 7, . . . , 2m + 1) concave,multi-modal

WFG5 3,5,8,10 k + l k = 2 * (m - 1) , l = 20 (3, 5, 7, . . . , 2m + 1) concave,deceptive

WFG6 3,5,8,10 k + l k = 2 * (m - 1) , l = 20 (3, 5, 7, . . . , 2m + 1) concave,non-separable

WFG7 3,5,8,10 k + l k = 2 * (m - 1) , l = 20 (3, 5, 7, . . . , 2m + 1) concave,biased

WFG8 3,5,8,10 k + l k = 2 * (m - 1) , l = 20 (3, 5, 7, . . . , 2m + 1) concave,biased,non-separable

WFG9 3,5,8,10 k + l k = 2 * (m - 1) , l = 20 (3, 5, 7, . . . , 2m + 1) concave,biased,deceptive,multi-modal

problems	(m)	(n)	k and l	z ^ref	characteristic
DTLZ1	3,5,8,10	k + m - 1	k = 5	(1, . . . . , 1)	linear, multi-modal
DTLZ2	3,5,8,10	k + m - 1	k = 10	(2, . . . . , 2)	concave
DTLZ3	3,5,8,10	k + m - 1	k = 10	(2, . . . . , 2)	concave, multi-modal
DTLZ4	3,5,8,10	k + m - 1	k = 10	(2, . . . . , 2)	concave, biased
WFG1	3,5,8,10	k + l	k = 2 * (m - 1) , l = 20	(3, 5, 7, . . . , 2m + 1)	mixed, biased
WFG2	3,5,8,10	k + l	k = 2 * (m - 1) , l = 20	(3, 5, 7, . . . , 2m + 1)	convex,discontinuous, non-separable
WFG3	3,5,8,10	k + l	k = 2 * (m - 1) , l = 20	(3, 5, 7, . . . , 2m + 1)	liner,non-separable
WFG4	3,5,8,10	k + l	k = 2 * (m - 1) , l = 20	(3, 5, 7, . . . , 2m + 1)	concave,multi-modal
WFG5	3,5,8,10	k + l	k = 2 * (m - 1) , l = 20	(3, 5, 7, . . . , 2m + 1)	concave,deceptive
WFG6	3,5,8,10	k + l	k = 2 * (m - 1) , l = 20	(3, 5, 7, . . . , 2m + 1)	concave,non-separable
WFG7	3,5,8,10	k + l	k = 2 * (m - 1) , l = 20	(3, 5, 7, . . . , 2m + 1)	concave,biased
WFG8	3,5,8,10	k + l	k = 2 * (m - 1) , l = 20	(3, 5, 7, . . . , 2m + 1)	concave,biased,non-separable
WFG9	3,5,8,10	k + l	k = 2 * (m - 1) , l = 20	(3, 5, 7, . . . , 2m + 1)	concave,biased,deceptive,multi-modal

4.2 Performance metrics

Some metrics are needed to evaluate the quality of the optimization algorithm. In the existing research, two metrics, the inverted generation distance (IGD) [57] and Hypervolume(HV) [24], are widely used performance metrics, which can measure both convergence and diversity.

Inverted Generation Distance (IGD) measures te average Euclidean distance from the a set of uniformly distributed points across PF to the closet solution in its approximation set. $IGD = \frac{\sum_{v \in P} d (v, P^{*})}{| P |}$ (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) 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 set the same parameters including the number of objectives m, population size N, the maximal number of generation t_max, the parameters of generating uniformly distributed reference vectors H₁ and H₂ in Table 2.

Table 2
Parameters setting of algorithm running

m N t _max H₁ and H₂

3 210 300 H₁ = 19, H₂ = 0

5 210 500 H₁ = 6, H₂ = 0

8 156 800 H₁ = 3, H₂ = 2

10 276 1000 H₁ = 3, H₂ = 2

m	N	t _max	H₁ and H₂
3	210	300	H₁ = 19, H₂ = 0
5	210	500	H₁ = 6, H₂ = 0
8	156	800	H₁ = 3, H₂ = 2
10	276	1000	H₁ = 3, H₂ = 2

Besides, we employ the standard toolkits to implement the peer algorithms and use their original study to set the parameters of compared algorithms. Parameters include neighborhood size T, crossover probability p_c, mutation probability p_m, distribution index η_c of simulated binary crossover (SBX), distribution index η_m of mutation operation. Then, some other special parameters are also indicated in Table 3.

Table 3

Parameter settings of compared algorithms

Algorithm	Parameter settings
MOEA/D	F = 0.5, CR = 1.0, T = 20, p_m = 1/n, η_m = 20, δ = 0.9, n_r = 0.2
MOMOBI-II	The threshold of variances: α = 0.5, Tolerance threshold: ε = 0.001, recordsize = 5
NSGA-III	p_c = 0.9, p_m = 1/n, η_c = 20, η_m = 20
RVEA	The rate of change of penalty: α = 2, the frequency: f_r = 0.1
RPD-NSGA-II	p_c = 0.9, p_m = 1/n, η_c = 20, η_m = 20

5 Experimental results

In this section, aiming to demonstrate the superiority of MaOEA-VADS implementation required for convergence and diversity in dealing with many-objective problems, the proposed algorithm is to be validated to compare with MOEA/D [43], MOMOBI-II [29], NSGA-III [47], RVEA [50] and RPD-NSGA-II [51] on DTLZ and WFG benchmark test problems. In addition, we analyze the experimental results from performance comparisons on many-objective test problems and parameter b.

5.1 Performance comparisons

5.1.1 The analysis of data

Firstly, we run all the algorithms 20 times for each problem with different number of objectives independently and compare the performance metrics of the algorithms. Table 4 and 6 list the mean and standard deviation of IGD and HV respectively of the six algorithms on DTLZ and WFG problems suits. In addition, Wilconxon’s rank sum test [58] is carried out between MaOEA-VADS and each compared algorithm on each test problem at 0.05 significance level, which is aimed to investigate whether MaOEA-VADS is superior to compared algorithms in a statistical sense. In this case, +, ≈ and - is applied to indicate that MaOEA-VADS is superior to similar to and inferior to the compared algorithm in the corresponding column, respectively. In the statistical, the minimum IGD and maximum HV are recorded, where the best result based on the mean values for each test problem is highlighted in boldface and Bold italic font represents that the indicator value of the compared algorithm is superior to MaOEA-VADS when there is no significant difference.

Table 4
Comparison of the MOEAs on IGD value with means and standard deviations for DTLZ and WFG benchmark with 4 different dimensions

M MOEA/D MOMOBI-II NSGA-III RVEA RPD-NSGA-II MaOEA-VADS

DTLZ1 3 1.847E-02(4.917E-05)+ 1.322E-02(9.689E-05)+ 1.317E-02(1.849E-04)+ 1.356E-02(2.070E-04)+ 1.370E-02(3.281E-04)+ 1.246E-02(2.876E-05)

5 1.102E-01(3.496E-04)+ 5.337E-02(2.891E-04)+ 5.362E-02(4.417E-04)+ 5.104E-02(1.053E-04)+ 5.512E-02(2.400E-03)+ 5.064E-02(5.107E-05)

8 1.589E-01(7.623E-03)+ 1.991E-01(4.879E-02)+ 1.799E-01(1.281E-01)+ 9.657E-02(4.607E-04)+ 1.067E-01(7.720E-03)+ 9.291E-02(2.237E-04)

10 1.736E-01(1.382E-02)+ 2.254E-01(3.760E-02)+ 1.633E-01(5.351E-02)+ 1.050E-01(8.923E-04)- 1.383E-01(1.859E-03)+ 1.070E-01(6.646E-04)

DTLZ2 3 4.568E-02(1.200E-04)+ 3.476E-02(7.669E-05)+ 3.463E-02(3.985E-05)+ 3.483E-02(3.782E-04)+ 3.550E-02(6.501E-04)+ 3.448E-02(4.439E-06)

5 3.184E-01(6.671E-04)+ 1.663E-01(4.436E-04)+ 1.658E-01(1.386E-04)+ 1.631E-01(4.029E-05)- 1.701E-01(2.966E-03)+ 1.644E-01(3.030E-05)

8 5.817E-01(3.196E-02)+ 3.362E-01(3.125E-03)+ 3.358E-01(4.958E-02)+ 3.160E-01(1.585E-04)+ 3.395E-01(6.946E-03)+ 3.156E-01(2.021E-04)

10 6.726E-01(3.208E-02)+ 4.379E-01(2.390E-03)+ 4.771E-01(5.295E-02)+ 4.083E-01(6.520E-04)- 4.627E-01(1.902E-03)+ 4.203E-01(7.570E-04)

DTLZ3 3 4.668E-02(7.188E-04)+ 3.795E-02(8.165E-04)+ 4.779E-02(3.684E-02)+ 8.233E-02(2.607E-02)+ 3.801E-02(1.324E-03)+ 3.643E-02(2.619E-03)

5 3.199E-01(1.621E-03)+ 1.752E-01(2.671E-03)+ 3.913E-01(5.075E-01)+ 1.688E-01(1.998E-03)+ 1.768E-01(5.445E-03)+ 1.669E-01(4.142E-03)

8 6.355E-01(6.390E-02)+ 4.575E-01(7.515E-02)+ 5.221E+00(3.920E+00)+ 3.217E-01(2.325E-03)- 8.475E-01(7.731E-01)+ 3.532E-01(4.556E-02)

10 7.534E-01(5.838E-02)+ 5.759E-01(1.514E-01)+ 3.994E+00(2.399E+00)+ 4.249E-01(2.158E-03)- 6.013E-01(2.423E-01)+ 4.300E-01(3.028E-03)

DTLZ4 3 1.949E-01(2.324E-01)+ 3.499E-02(1.661E-04)+ 3.465E-02(5.346E-05)+ 3.498E-02(8.365E-04)+ 3.554E-02(6.737E-04)+ 3.448E-02(8.501E-06)

5 3.971E-01(1.045E-01)+ 2.153E-01(9.852E-02)+ 3.913E-01(9.915E-02)+ 1.642E-01(4.722E-05)- 1.714E-01(3.283E-03)+ 1.648E-01(4.305E-05)

8 6.221E-01(4.110E-02)+ 3.977E-01(5.019E-02)+ 4.495E-01(4.967E-02)+ 3.389E-01(4.603E-02)≈ 3.474E-01(2.112E-02)+ 3.165E-01(3.767E-04)

10 6.882E-01(2.813E-02)+ 4.736E-01(2.488E-02)+ 5.197E-01(3.344E-02)+ 4.211E-01(8.285E-04)- 4.642E-01(2.010E-03)+ 4.223E-01(9.012E-04)

WFG1 3 1.164E+00(2.045E-02)+ 1.245E+00(1.792E-02)+ 1.377E+00(1.019E-02)+ 1.424E+00(1.513E-02)+ 1.377E+00(1.174E-02)+ 1.150E+00(9.338E-03)

5 1.615E+00(8.794E-02)- 1.552E+00(8.775E-02)- 1.901E+00(1.149E-02)- 1.861E+00(1.713E-02)- 1.929E+00(2.414E-02)- 1.939E+00(1.250E-02)

8 2.294E+00(7.587E-02)- 2.246E+00(3.614E-01)- 2.532E+00(2.286E-02)- 2.366E+00(4.520E-02)- 2.565E+00(2.919E-02)- 2.813E+00(3.610E-02)

10 2.491E+00(1.327E-01)- 2.542E+00(1.532E-01)- 2.882E+00(2.835E-02)- 2.653E+00(4.239E-02)- 2.772E+00(6.328E-02)- 3.150E+00(5.609E-02)

WFG2 3 4.398E-01(1.363E-01)+ 2.261E-01(1.617E-01)+ 1.130E-01(2.631E-03)- 1.716E-01(1.033E-01)+ 1.996E-01(1.459E-01)+ 1.697E-01(8.042E-02)

5 1.415E+00(1.697E-01)+ 7.126E-01(2.210E-01)+ 5.248E-01(3.230E-01)+ 6.048E-01(3.968E-01)+ 5.248E-01(2.640E-01)+ 4.540E-01(2.303E-01)

8 2.586E+00(4.937E-01)+ 2.387E+00(6.109E-01)+ 1.426E+00(6.279E-01)+ 1.593E+00(8.015E-01)≈ 1.255E+00(5.080E-01)+ 1.178E+00(3.610E-02)

10 3.223E+00(8.835E-01)+ 3.845E+00(1.092E+00)+ 1.340E+00(6.525E-01)+ 1.712E+00(1.061E+00)≈ 2.772E+00(7.212E-01)+ 1.111E+00(2.509E-02)

WFG3 3 8.096E-02(1.293E-02)- 4.658E-02(3.608E-03)- 1.013E-01(9.651E-03)- 1.955E-01(2.324E-02)+ 1.810E-01(3.000E-02)+ 1.199E-01(1.050E-02)

5 1.267E+00(1.144E-01)+ 7.697E-01(1.044E-01)+ 4.677E-01(6.458E-02)- 5.413E-01(7.431E-02)≈ 5.929E-01(2.578E-02)+ 5.117E-01(3.184E-02)

8 1.751E+00(1.038E-01)- 1.397E+00(9.851E-02)- 7.682E-01(1.906E-01)- 2.042E+00(5.625E-01)+ 1.596E+00(6.187E-02)- 1.904E+00(1.339E-01)

10 1.917E+00(1.674E-01)- 1.315E+00(6.391E-02)- 1.317E+00(4.789E-01)- 2.607E+00(5.707E-01)+ 1.936E+00(5.705E-02)- 2.423E+00(1.482E-01)

WFG4 3 2.148E-01(2.835E-03)+ 1.522E-01(2.272E-03)+ 1.623E-01(3.506E-03)+ 1.766E-01(2.512E-03)+ 1.650E-01(1.935E-03)+ 1.441E-01(1.153E-03)

5 2.441E+00(2.379E-01)+ 1.479E+00(1.271E-01)+ 9.752E-01(7.033E-03)+ 9.557E-01(2.618E-03)+ 1.054E+00(4.814E-02)+ 9.512E-01(1.602E-03)

8 5.010E+00(2.578E-01)+ 5.379E+00(8.527E-01)+ 3.019E+00(1.295E-01)≈ 2.980E+00(2.701E-02)+ 3.227E+00(8.242E-02)+ 2.956E+00(7.010E-03)

10 7.343E+00(5.748E-01)+ 7.585E+00(5.143E-01)+ 4.265E+00(3.543E-01)+ 4.337E+00(4.054E-02)≈ 4.515E+00(5.716E-02)+ 4.221E+00(4.703E-02)

WFG5 3 2.225E-01(2.792E-03)+ 1.811E-01(1.707E-03)+ 1.767E-01(1.183E-03)+ 1.863E-01(2.127E-03)+ 1.822E-01(1.498E-03)+ 1.630E-01(8.914E-04)

5 2.170E+00(1.234E-01)+ 1.559E+00(9.241E-02)+ 9.4743E-01(5.144E-03)+ 9.414E-01(1.613E-03)+ 1.042E+00(2.696E-02)+ 9.404E-01(1.273E-03)

8 4.929E+00(1.472E-01)+ 3.828E+00(2.262E-01)+ 2.9915E+00(2.188E-01)≈ 2.960E+00(1.937E-02)≈ 3.177E+00(6.590E-02)+ 2.952E+00(6.537E-03)

10 6.762E+00(2.115E-01)+ 5.887E+00(7.047E-01)+ 4.393E+00(3.875E-02)+ 4.453E+00(3.033E-02)+ 4.461E+00(7.406E-02)+ 4.173E+00(2.422E-02)

WFG6 3 2.283E-01(7.110E-03)+ 1.746E-01(4.890E-03)+ 1.878E-01(4.838E-03)+ 2.204E-01(6.603E-03)+ 1.893E-01(7.095E-03)+ 1.699E-01(5.898E-03)

5 2.333E+00(2.579E-01)+ 1.686E+00(2.195E-01)+ 9.6623E-01(3.384E-03)+ 9.619E-01(1.475E-03)+ 1.051E+00(1.417E-02)+ 9.502E-01(1.175E-03)

8 5.056E+00(2.299E-01)+ 3.677E+00(2.991E-02)+ 2.966E+00(9.348E-03)- 3.024E+00(3.564E-02)+ 3.230E+00(4.145E-02)+ 2.974E+00(9.867E-03)

10 6.802E+00(2.362E-01)+ 5.330E+00(3.713E-02)+ 4.299E+00(3.257E-02)+ 4.372E+00(8.762E-02)+ 4.474E+00(4.900E-02)+ 4.234E+00(2.455E-02)

WFG7 3 1.030E+00(7.155E-01)+ 2.024E-01(1.602E-01)+ 1.462E-01(6.810E-04)+ 1.887E-01(1.291E-01)+ 1.825E-01(1.190E-01)+ 1.397E-01(1.682E-04)

5 3.890E+00(1.130E+00)+ 2.598E+00(8.829E-01)+ 1.505E+00(2.940E-01)+ 2.400E+00(8.532E-01)+ 1.342E+00(2.437E-01)+ 9.522E-01(1.250E-03)

8 1.014E+01(1.419E+00)+ 8.469E+00(1.136E+00)+ 4.844E+00(1.053E+00)+ 9.102E+00(1.407E+00)+ 5.318E+00(2.108E+00)+ 3.081E+00(9.245E-02)

10 1.244E+01(1.876E+00)+ 1.144E+01(7.732E-01)+ 5.875E+00(1.385E+00)+ 1.252E+01(1.440E+00)+ 7.101E+00(1.431E+00)+ 4.594E+00(2.579E-02)

WFG8 3 7.273E-01(5.689E-02)+ 6.239E-01(2.376E-02)≈ 6.735E-01(1.684E-02)+ 7.936E-01(1.951E-02)+ 7.035E-01(2.814E-02)+ 6.322E-01(2.029E-02)

5 2.909E+00(6.321E-02)+ 2.404E+00(2.745E-02)+ 1.510E+00(2.857E-02)+ 1.502E+00(4.589E-02)+ 1.448E+00(1.953E-02)+ 1.403E+00(1.983E-02)

8 5.658E+00(2.540E-01)+ 4.876E+00(1.270E+00)+ 3.763E+00(2.273E-01)+ 3.678E+00(4.334E-01)+ 3.653E+00(6.761E-02)+ 3.429E+00(4.279E-02)

10 8.284E+00(4.430E-01)+ 7.711E+00(3.269E+00)+ 4.918E+00(3.679E-01)+ 4.563E+00(1.084E-01)- 4.961E+00(1.506E-01)≈ 4.608E+00(3.074E-02)

WFG9 3 4.316E-01(4.275E-01)+ 2.190E-01(1.447E-03)+ 6.735E-01(5.184E-04)+ 2.011E-01(1.477E-03)- 2.157E-01(1.609E-02)+ 2.022E-01(1.181E-04)

5 2.597E+00(3.080E-01)+ 1.777E+00(1.120E-01)+ 1.157E+00(5.402E-01)≈ 1.025E+00(4.088E-01)+ 1.038E+00(2.479E-02)+ 9.245E-01(1.262E-03)

8 7.114E+00(1.587E+00)+ 5.478E+00(1.123E+00)+ 3.425E+00(4.409E-01)+ 5.046E+00(2.442E+00)+ 3.342E+00(5.172E-01)+ 2.933E+00(2.913E-02)

10 8.825E+00(1.439E+00)+ 8.120E+00(1.463E+00)+ 4.614E+00(4.089E-01)+ 6.607E+00(2.450E+00)+ 4.528E+00(4.040E-01)≈ 4.480E+00(1.771E-02)

	M	MOEA/D	MOMOBI-II	NSGA-III	RVEA	RPD-NSGA-II	MaOEA-VADS
DTLZ1	3	1.847E-02(4.917E-05)+	1.322E-02(9.689E-05)+	1.317E-02(1.849E-04)+	1.356E-02(2.070E-04)+	1.370E-02(3.281E-04)+	1.246E-02(2.876E-05)
	5	1.102E-01(3.496E-04)+	5.337E-02(2.891E-04)+	5.362E-02(4.417E-04)+	5.104E-02(1.053E-04)+	5.512E-02(2.400E-03)+	5.064E-02(5.107E-05)
	8	1.589E-01(7.623E-03)+	1.991E-01(4.879E-02)+	1.799E-01(1.281E-01)+	9.657E-02(4.607E-04)+	1.067E-01(7.720E-03)+	9.291E-02(2.237E-04)
	10	1.736E-01(1.382E-02)+	2.254E-01(3.760E-02)+	1.633E-01(5.351E-02)+	1.050E-01(8.923E-04)-	1.383E-01(1.859E-03)+	1.070E-01(6.646E-04)
DTLZ2	3	4.568E-02(1.200E-04)+	3.476E-02(7.669E-05)+	3.463E-02(3.985E-05)+	3.483E-02(3.782E-04)+	3.550E-02(6.501E-04)+	3.448E-02(4.439E-06)
	5	3.184E-01(6.671E-04)+	1.663E-01(4.436E-04)+	1.658E-01(1.386E-04)+	1.631E-01(4.029E-05)-	1.701E-01(2.966E-03)+	1.644E-01(3.030E-05)
	8	5.817E-01(3.196E-02)+	3.362E-01(3.125E-03)+	3.358E-01(4.958E-02)+	3.160E-01(1.585E-04)+	3.395E-01(6.946E-03)+	3.156E-01(2.021E-04)
	10	6.726E-01(3.208E-02)+	4.379E-01(2.390E-03)+	4.771E-01(5.295E-02)+	4.083E-01(6.520E-04)-	4.627E-01(1.902E-03)+	4.203E-01(7.570E-04)
DTLZ3	3	4.668E-02(7.188E-04)+	3.795E-02(8.165E-04)+	4.779E-02(3.684E-02)+	8.233E-02(2.607E-02)+	3.801E-02(1.324E-03)+	3.643E-02(2.619E-03)
	5	3.199E-01(1.621E-03)+	1.752E-01(2.671E-03)+	3.913E-01(5.075E-01)+	1.688E-01(1.998E-03)+	1.768E-01(5.445E-03)+	1.669E-01(4.142E-03)
	8	6.355E-01(6.390E-02)+	4.575E-01(7.515E-02)+	5.221E+00(3.920E+00)+	3.217E-01(2.325E-03)-	8.475E-01(7.731E-01)+	3.532E-01(4.556E-02)
	10	7.534E-01(5.838E-02)+	5.759E-01(1.514E-01)+	3.994E+00(2.399E+00)+	4.249E-01(2.158E-03)-	6.013E-01(2.423E-01)+	4.300E-01(3.028E-03)
DTLZ4	3	1.949E-01(2.324E-01)+	3.499E-02(1.661E-04)+	3.465E-02(5.346E-05)+	3.498E-02(8.365E-04)+	3.554E-02(6.737E-04)+	3.448E-02(8.501E-06)
	5	3.971E-01(1.045E-01)+	2.153E-01(9.852E-02)+	3.913E-01(9.915E-02)+	1.642E-01(4.722E-05)-	1.714E-01(3.283E-03)+	1.648E-01(4.305E-05)
	8	6.221E-01(4.110E-02)+	3.977E-01(5.019E-02)+	4.495E-01(4.967E-02)+	3.389E-01(4.603E-02)≈	3.474E-01(2.112E-02)+	3.165E-01(3.767E-04)
	10	6.882E-01(2.813E-02)+	4.736E-01(2.488E-02)+	5.197E-01(3.344E-02)+	4.211E-01(8.285E-04)-	4.642E-01(2.010E-03)+	4.223E-01(9.012E-04)
WFG1	3	1.164E+00(2.045E-02)+	1.245E+00(1.792E-02)+	1.377E+00(1.019E-02)+	1.424E+00(1.513E-02)+	1.377E+00(1.174E-02)+	1.150E+00(9.338E-03)
	5	1.615E+00(8.794E-02)-	1.552E+00(8.775E-02)-	1.901E+00(1.149E-02)-	1.861E+00(1.713E-02)-	1.929E+00(2.414E-02)-	1.939E+00(1.250E-02)
	8	2.294E+00(7.587E-02)-	2.246E+00(3.614E-01)-	2.532E+00(2.286E-02)-	2.366E+00(4.520E-02)-	2.565E+00(2.919E-02)-	2.813E+00(3.610E-02)
	10	2.491E+00(1.327E-01)-	2.542E+00(1.532E-01)-	2.882E+00(2.835E-02)-	2.653E+00(4.239E-02)-	2.772E+00(6.328E-02)-	3.150E+00(5.609E-02)
WFG2	3	4.398E-01(1.363E-01)+	2.261E-01(1.617E-01)+	1.130E-01(2.631E-03)-	1.716E-01(1.033E-01)+	1.996E-01(1.459E-01)+	1.697E-01(8.042E-02)
	5	1.415E+00(1.697E-01)+	7.126E-01(2.210E-01)+	5.248E-01(3.230E-01)+	6.048E-01(3.968E-01)+	5.248E-01(2.640E-01)+	4.540E-01(2.303E-01)
	8	2.586E+00(4.937E-01)+	2.387E+00(6.109E-01)+	1.426E+00(6.279E-01)+	1.593E+00(8.015E-01)≈	1.255E+00(5.080E-01)+	1.178E+00(3.610E-02)
	10	3.223E+00(8.835E-01)+	3.845E+00(1.092E+00)+	1.340E+00(6.525E-01)+	1.712E+00(1.061E+00)≈	2.772E+00(7.212E-01)+	1.111E+00(2.509E-02)
WFG3	3	8.096E-02(1.293E-02)-	4.658E-02(3.608E-03)-	1.013E-01(9.651E-03)-	1.955E-01(2.324E-02)+	1.810E-01(3.000E-02)+	1.199E-01(1.050E-02)
	5	1.267E+00(1.144E-01)+	7.697E-01(1.044E-01)+	4.677E-01(6.458E-02)-	5.413E-01(7.431E-02)≈	5.929E-01(2.578E-02)+	5.117E-01(3.184E-02)
	8	1.751E+00(1.038E-01)-	1.397E+00(9.851E-02)-	7.682E-01(1.906E-01)-	2.042E+00(5.625E-01)+	1.596E+00(6.187E-02)-	1.904E+00(1.339E-01)
	10	1.917E+00(1.674E-01)-	1.315E+00(6.391E-02)-	1.317E+00(4.789E-01)-	2.607E+00(5.707E-01)+	1.936E+00(5.705E-02)-	2.423E+00(1.482E-01)
WFG4	3	2.148E-01(2.835E-03)+	1.522E-01(2.272E-03)+	1.623E-01(3.506E-03)+	1.766E-01(2.512E-03)+	1.650E-01(1.935E-03)+	1.441E-01(1.153E-03)
	5	2.441E+00(2.379E-01)+	1.479E+00(1.271E-01)+	9.752E-01(7.033E-03)+	9.557E-01(2.618E-03)+	1.054E+00(4.814E-02)+	9.512E-01(1.602E-03)
	8	5.010E+00(2.578E-01)+	5.379E+00(8.527E-01)+	3.019E+00(1.295E-01)≈	2.980E+00(2.701E-02)+	3.227E+00(8.242E-02)+	2.956E+00(7.010E-03)
	10	7.343E+00(5.748E-01)+	7.585E+00(5.143E-01)+	4.265E+00(3.543E-01)+	4.337E+00(4.054E-02)≈	4.515E+00(5.716E-02)+	4.221E+00(4.703E-02)
WFG5	3	2.225E-01(2.792E-03)+	1.811E-01(1.707E-03)+	1.767E-01(1.183E-03)+	1.863E-01(2.127E-03)+	1.822E-01(1.498E-03)+	1.630E-01(8.914E-04)
	5	2.170E+00(1.234E-01)+	1.559E+00(9.241E-02)+	9.4743E-01(5.144E-03)+	9.414E-01(1.613E-03)+	1.042E+00(2.696E-02)+	9.404E-01(1.273E-03)
	8	4.929E+00(1.472E-01)+	3.828E+00(2.262E-01)+	2.9915E+00(2.188E-01)≈	2.960E+00(1.937E-02)≈	3.177E+00(6.590E-02)+	2.952E+00(6.537E-03)
	10	6.762E+00(2.115E-01)+	5.887E+00(7.047E-01)+	4.393E+00(3.875E-02)+	4.453E+00(3.033E-02)+	4.461E+00(7.406E-02)+	4.173E+00(2.422E-02)
WFG6	3	2.283E-01(7.110E-03)+	1.746E-01(4.890E-03)+	1.878E-01(4.838E-03)+	2.204E-01(6.603E-03)+	1.893E-01(7.095E-03)+	1.699E-01(5.898E-03)
	5	2.333E+00(2.579E-01)+	1.686E+00(2.195E-01)+	9.6623E-01(3.384E-03)+	9.619E-01(1.475E-03)+	1.051E+00(1.417E-02)+	9.502E-01(1.175E-03)
	8	5.056E+00(2.299E-01)+	3.677E+00(2.991E-02)+	2.966E+00(9.348E-03)-	3.024E+00(3.564E-02)+	3.230E+00(4.145E-02)+	2.974E+00(9.867E-03)
	10	6.802E+00(2.362E-01)+	5.330E+00(3.713E-02)+	4.299E+00(3.257E-02)+	4.372E+00(8.762E-02)+	4.474E+00(4.900E-02)+	4.234E+00(2.455E-02)
WFG7	3	1.030E+00(7.155E-01)+	2.024E-01(1.602E-01)+	1.462E-01(6.810E-04)+	1.887E-01(1.291E-01)+	1.825E-01(1.190E-01)+	1.397E-01(1.682E-04)
	5	3.890E+00(1.130E+00)+	2.598E+00(8.829E-01)+	1.505E+00(2.940E-01)+	2.400E+00(8.532E-01)+	1.342E+00(2.437E-01)+	9.522E-01(1.250E-03)
	8	1.014E+01(1.419E+00)+	8.469E+00(1.136E+00)+	4.844E+00(1.053E+00)+	9.102E+00(1.407E+00)+	5.318E+00(2.108E+00)+	3.081E+00(9.245E-02)
	10	1.244E+01(1.876E+00)+	1.144E+01(7.732E-01)+	5.875E+00(1.385E+00)+	1.252E+01(1.440E+00)+	7.101E+00(1.431E+00)+	4.594E+00(2.579E-02)
WFG8	3	7.273E-01(5.689E-02)+	6.239E-01(2.376E-02)≈	6.735E-01(1.684E-02)+	7.936E-01(1.951E-02)+	7.035E-01(2.814E-02)+	6.322E-01(2.029E-02)
	5	2.909E+00(6.321E-02)+	2.404E+00(2.745E-02)+	1.510E+00(2.857E-02)+	1.502E+00(4.589E-02)+	1.448E+00(1.953E-02)+	1.403E+00(1.983E-02)
	8	5.658E+00(2.540E-01)+	4.876E+00(1.270E+00)+	3.763E+00(2.273E-01)+	3.678E+00(4.334E-01)+	3.653E+00(6.761E-02)+	3.429E+00(4.279E-02)
	10	8.284E+00(4.430E-01)+	7.711E+00(3.269E+00)+	4.918E+00(3.679E-01)+	4.563E+00(1.084E-01)-	4.961E+00(1.506E-01)≈	4.608E+00(3.074E-02)
WFG9	3	4.316E-01(4.275E-01)+	2.190E-01(1.447E-03)+	6.735E-01(5.184E-04)+	2.011E-01(1.477E-03)-	2.157E-01(1.609E-02)+	2.022E-01(1.181E-04)
	5	2.597E+00(3.080E-01)+	1.777E+00(1.120E-01)+	1.157E+00(5.402E-01)≈	1.025E+00(4.088E-01)+	1.038E+00(2.479E-02)+	9.245E-01(1.262E-03)
	8	7.114E+00(1.587E+00)+	5.478E+00(1.123E+00)+	3.425E+00(4.409E-01)+	5.046E+00(2.442E+00)+	3.342E+00(5.172E-01)+	2.933E+00(2.913E-02)
	10	8.825E+00(1.439E+00)+	8.120E+00(1.463E+00)+	4.614E+00(4.089E-01)+	6.607E+00(2.450E+00)+	4.528E+00(4.040E-01)≈	4.480E+00(1.771E-02)

Table 5

Comparison of the MOEAs on HV value with means and standard deviations for DTLZ and WFG benchmark with 4 different dimensions

	M	MOEA/D	MOMOBI-II	NSGA-III	RVEA	RPD-NSGA-II	MaOEA-VADS
DTLZ1	3	9.718E-01(5.561E-04)+	9.755E-01(6.823E-04)-	9.755E-01(6.174E-04)-	9.752E-01(4.899E-04)-	9.754E-01(4.814E-04)-	9.702E-01(6.095E-04)
	5	9.970E-01(1.498E-04)+	9.990E-01(1.062E-04)-	9.989E-01(1.096E-04)-	9.990E-01(1.268E-04)-	9.989E-01(8.765E-05)-	9.984E-01(1.242E-03)
	8	9.960E-01(8.064E-04)≈	9.928E-01(1.059E-02)≈	9.711E-01(1.178E-01)+	1.000E-01(9.987E-06)-	1.000E-01(1.576E-05)-	9.996E-01(1.479E-03)
	10	9.965E-01(1.110E-03)+	9.943E-01(8.169E-03)≈	9.998E-01(5.714E-04)+	1.000E+00(3.078E-06)-	1.000E+00(0.000E+00)-	1.000E+00(7.543E-04)
DTLZ2	3	9.269E-01(6.201E-04)+	9.298E-01(6.511E-04)≈	9.295E-01(7.686E-04)+	9.293E-01(1.441E-03)+	9.294E-01(6.624E-04)+	9.297E-01(8.047E-04)
	5	9.857E-01(4.593E-04)+	9.906E-01(2.763E-04)≈	9.903E-01(3.120E-04)≈	9.907E-01(2.692E-04)≈	9.903E-01(2.6050E-04)≈	9.906E-01(2.671E-04)
	8	9.551E-01(1.277E-02)+	9.994E-01(7.475E-05)≈	9.990E-01(8.828E-04)+	9.994E-01(1.082E-04)≈	9.993E-01(9.240E-05)+	9.9940E-01(5.986E-05)
	10	9.572E-01(1.574E-02)+	9.999E-01(2.700E-05)≈	9.997E-01(6.733E-04)+	9.999E-01(3.385E-05)≈	9.999E-01(3.760E-05)+	9.999E-01(2.140E-05)
DTLZ3	3	9.257E-01(1.138E-03)+	9.283E-01(1.117E-03)≈	9.232E-01(1.291E-02)+	9.132E-01(8.765E-03)+	9.271E-01(1.267E-03)≈	9.277E-01(1.779E-03)
	5	9.854E-01(4.085E-04)+	9.900E-01(3.449E-04)≈	8.604E-01(3.133E-01)+	9.895E-01(4.388E-04)≈	9.889E-01(7.173E-04)+	9.897E-01(7.537E-04)
	8	9.557E-01(6.253E-03)+	9.976E-01(2.636E-03)+	2.909E-02(1.301E-01)+	9.993E-01(1.115E-04)-	8.032E-01(3.285E-01)+	9.989E-01(1.168E-03)
	10	9.578E-01(6.842E-03)+	9.945E-01(1.150E-02)+	2.713E-02(6.422E-02)+	9.999E-01(2.849E-05)≈	9.665E-01(9.842E-02)+	9.999E-01(3.093E-05)
DTLZ4	3	8.893E-01(5.861E-02)+	9.298E-01(8.401E-04)≈	9.296E-01(8.745E-04)≈	9.290E-01(8.109E-04)+	9.293E-01(7.863E-04)+	9.297E-01(7.314E-04)
	5	9.739E-01(1.835E-02)+	9.873E-01(6.888E-03)+	9.868E-01(6.926E-03)+	9.906E-01(2.950E-04)≈	9.902E-01(3.055E-04)+	9.906E-01(3.430E-04)
	8	9.594E-01(7.910E-03)+	9.988E-01(4.757E-04)+	9.975E-01(1.313E-03)+	9.992E-01(3.453E-04)≈	9.992E-01(2.256E-04)+	9.994E-01(8.197E-05)
	10	9.611E-01(1.618E-02)+	9.999E-01(7.666E-05)+	9.995E-01(4.659E-04)+	9.999E-01(3.059E-05)≈	9.999E-01(3.591E-05)+	9.999E-01(2.452E-05)
WFG1	3	5.247E-01(8.229E-03)+	5.031E-01(5.780E-03)+	4.789E-01(2.582E-03)+	4.636E-01(1.001E-02)+	4.776E-01(2.739E-03)+	5.462E-01(2.810E-03)
	5	5.160E-01(2.612E-02)-	5.589E-01(2.056E-02)-	4.028E-01(2.877E-03)+	4.062E-01(4.947E-03)+	4.195E-01(4.503E-03)+	4.512E-01(2.649E-03)
	8	4.457E-01(2.235E-02)-	4.178E-01(9.014E-02)≈	3.271E-01(3.861E-03)+	3.391E-01(6.813E-03)+	3.772E-01(2.256E-04)-	3.657E-01(3.684E-03)
	10	4.661E-01(3.967E-02)-	3.812E-01(3.062E-02)-	2.999E-01(2.964E-03)+	3.183E-01(6.647E-03)+	3.2895E-01(5.572E-03)+	3.468E-01(3.063E-03)
WFG2	3	8.322E-01(5.904E-02)+	9.108E-01(6.309E-02)≈	9.481E-01(3.037E-03)-	9.209E-01(4.263E-02)+	9.195E-01(5.875E-02)+	9.362E-01(3.312E-02)
	5	8.455E-01(7.704E-02)+	9.714E-01(5.313E-02)≈	9.427E-01(6.258E-02)≈	9.300E-01(7.660E-02)+	9.613E-01(5.288E-02)-	9.524E-01(4.546E-02)
	8	8.194E-01(7.583E-02)+	8.353E-01(8.354E-02)≈	9.367E-01(8.517E-02)+	8.903E-01(8.254E-02)+	9.400E-01(6.104E-02)≈	9.375E-01(6.461E-02)
	10	8.334E-01(8.687E-02)+	7.789E-01(5.599E-02)+	9.706E-01(6.080E-02)-	9.201E-01(7.706E-02)+	9.626E-01(5.461E-02)+	9.696E-01(6.247E-03)
WFG3	3	7.130E-01(3.706E-03)-	7.189E-01(1.712E-03)-	7.062E-01(2.921E-03)-	6.747E-01(4.845E-03)+	6.822E-01(9.740E-03)+	6.971E-01(3.022E-03)
	5	5.874E-01(9.896E-03)+	5.971E-01(1.501E-02)+	6.535E-01(7.986E-03)≈	6.157E-01(1.122E-02)+	6.440E-01(5.382E-03)+	6.523E-01(6.533E-03)
	8	5.474E-01(1.584E-02)-	4.179E-01(5.251E-03)+	6.007E-01(1.642E-02)-	3.863E-01(7.794E-02)+	5.372E-01(8.840E-03)-	5.040E-01(1.943E-02)
	10	5.498E-01(1.734E-02)-	4.445E-01(2.312E-03)+	5.700E-01(2.404E-02)-	3.617E-01(5.585E-02)+	4.982E-01(9.343E-03)-	4.657E-01(1.952E-02)
WFG4	3	7.180E-01(2.482E-03)+	7.266E-01(3.681E-03)≈	7.113E-01(3.203E-03)+	7.024E-01(2.747E-03)+	7.149E-01(2.182E-03)+	7.250E-01(2.427E-03)
	5	7.299E-01(2.696E-02)+	7.859E-01(2.591E-02)+	7.905E-01(4.967E-03)+	8.085E-01(6.236E-03)+	8.2150E-01(5.223E-03)≈	8.259E-01(6.288E-03)
	8	6.170E-01(5.892E-02)+	6.152E-01(6.509E-02)+	8.026E-01(1.961E-02)+	8.117E-01(2.735E-02)+	8.393E-01(1.375E-02)≈	8.474E-01(1.240E-02)
	10	5.639E-01(4.745E-02)+	6.770E-01(3.533E-02)+	8.379E-01(2.942E-02)+	8.674E-01(2.327E-02)≈	8.922E-01(6.266E-03)-	8.647E-01(1.422E-02)
WFG5	3	6.802E-01(3.417E-03)+	6.819E-01(2.406E-03)+	6.863E-01(2.160E-03)+	6.804E-01(2.159E-03)+	6.866E-01(1.757E-03)+	6.966E-01(2.083E-03)
	5	7.367E-01(1.820E-02)+	6.963E-01(1.707E-02)+	7.897E-01(3.709E-03)+	8.033E-01(2.833E-03)+	8.012E-01(2.375E-03)+	8.119E-01(2.945E-03)
	8	5.935E-01(2.543E-02)+	7.065E-01(5.772E-02)+	7.829E-01(2.635E-02)+	8.571E-01(6.317E-03)-	8.128E-01(7.370E-03)+	8.416E-01(7.127E-03)
	10	6.108E-01(2.262E-02)+	6.645E-01(6.388E-02)+	8.285E-01(9.957E-03)+	8.889E-01(3.502E-03)-	8.558E-01(2.773E-03)+	8.668E-01(5.895E-03)
WFG6	3	6.910E-01(8.037E-03)+	7.000E-01(4.184E-03)≈	6.895E-01(3.586E-03)+	6.770E-01(4.538E-03)+	6.908E-01(5.303E-03)+	6.998E-01(5.144E-03)
	5	7.112E-01(3.041E-02)+	7.551E-01(3.711E-02)+	7.950E-01(5.521E-03)+	8.168E-01(5.256E-03)+	8.1573E-01(6.664E-03)+	8.242E-01(4.244E-03)
	8	6.137E-01(2.469E-02)+	8.208E-01(1.336E-02)+	8.175E-01(1.649E-02)+	8.618E-01(2.126E-02)+	8.6457E-01(8.658E-03)+	8.782E-01(1.070E-02)
	10	6.214E-01(4.092E-02)+	8.579E-01(8.768E-03)+	8.571E-01(1.047E-02)+	8.977E-01(1.485E-02)+	8.9475E-01(6.660E-03)+	9.072E-01(6.929E-03)
WFG7	3	5.194E-01(1.495E-01)+	7.151E-01(5.592E-02)+	7.281E-01(1.639E-03)+	7.077E-01(4.319E-02)+	7.212E-01(4.086E-02)+	7.368E-01(1.502E-03)
	5	4.466E-01(1.342E-01)+	6.085E-01(1.274E-01)+	7.312E-01(5.572E-02)+	5.604E-01(1.304E-01)+	7.753E-01(6.000E-02)+	8.620E-01(4.936E-03)
	8	3.025E-01(6.600E-02)+	4.129E-01(8.054E-02)+	6.620E-01(1.124E-01)+	3.286E-01(8.279E-02)+	6.456E-01(1.819E-01)+	8.054E-01(2.939E-02)
	10	3.471E-01(9.146E-02)+	4.575E-01(5.363E-02)+	7.647E-01(1.205E-01)+	3.301E-01(8.647E-02)+	7.029E-01(1.087E-01)+	8.642E-01(1.968E-02)
WFG8	3	4.533E-01(2.878E-02)≈	4.980E-01(7.638E-03)-	4.634E-01(6.830E-03)+	4.139E-01(8.130E-03)+	4.444E-01(1.175E-02)+	4.824E-01(9.245E-03)
	5	2.279E-01(3.102E-02)+	3.025E-01(3.166E-02)+	4.449E-01(1.352E-02)+	4.350E-01(1.792E-02)+	4.733E-01(1.484E-02)+	4.967E-01(1.294E-02)
	8	2.155E-01(4.972E-02) +	3.655E-01(5.228E-02)+	3.517E-01(3.160E-02)+	2.090E-01(3.339E-02)+	3.359E-01(3.447E-02)+	4.461E-01(2.607E-02)
	10	2.332E-01(4.813E-02) +	2.874E-01(7.019E-02)+	3.260E-01(3.294E-02)+	2.206E-01(2.157E-02)+	3.500E-01(4.804E-02)+	4.516E-01(2.586E-02)
WFG9	3	6.019E-01(8.751E-02) +	6.467E-01(1.882E-03)+	6.539E-01(1.502E-03)≈	6.539E-01(1.280E-03)≈	6.481E-01(1.340E-02)+	6.545E-01(1.773E-03)
	5	4.638E-01(4.550E-02) +	6.094E-01(3.114E-02)+	6.988E-01(9.577E-02)+	7.390E-01(5.748E-02)≈	7.360E-01(3.267E-03)+	7.514E-01(2.082E-03)
	8	4.024E-01(7.873E-02)+	4.806E-01(8.593E-02)+	6.855E-01(4.754E-02)+	5.920E-01(1.677E-01)+	7.166E-01(5.930E-02)+	7.557E-01(1.121E-02)
	10	4.418E-01(4.818E-02) +	4.944E-01(8.668E-02)+	7.107E-01(3.228E-02)+	6.351E-01(1.363E-01)+	7.360E-01(2.752E-02)+	7.693E-01(4.622E-03)

Table 6

The comparison statistical summary of MaOEA-VADS and five algorithms on the DTLZ and WFG test problems

	Indicator	MOEA/D	MOMOBI-II	NSGA-III	RVEA	RPD-NSGA-II	MaOEA-VADS
+/≈/-	IGD	46/0/6	45/1/6	40/3/9	34/6/12	44/2/5	-
+/≈/-	HV	44/2/6	31/14/7	40/5/7	35/10/7	38/5/9	-
Best/All	IGD	1/52	5/52	4/52	9/52	0/52	33/52
Best/All	HV	2/52	10/52	6/52	8/52	3/52	23/52

Table 7

Comparison of the MOEAs on HV value with means and standard deviations for WFG benchmark with 4 different dimensions

Problems	-	IGD			HV
	M	b=100	b=50	b=10	b=100	b=50	b=10
WFG1	3	1.178E+00(8.778E-03)	1.150E+00(9.338E-03)	1.088E+00(1.426E-02)	5.379E-01(4.024E-03)	5.462E-01(2.810E-03)	5.557E-01(4.366E-03)
	5	1.963E+00(1.434E-02)	1.939E+00(1.250E-02)	1.740E+00(1.917E-02)	4.484E-01(3.969E-03)	4.512E-01(2.649E-03)	4.543E-01(3.805E-03)
	8	2.817E+00(4.271E-02)	2.813E+00(3.610E-02)	2.590E+00(3.470E-02)	3.645E-01(3.509E-03)	3.657E-01(3.684E-03)	3.603E-01(5.117E-03)
	10	3.151E+00(6.468E-02)	3.150E+00(5.609E-02)	2.927E+00(2.761E-02)	3.506E-01(4.647E-03)	3.468E-01(3.063E-03)	3.344E-01(3.057E-03)
WFG2	3	1.304E-01(2.325E-03)	1.697E-01(8.042E-02)	3.328E-01(8.885E-02)	9.456E-01(3.233E-03)	9.362E-01(3.312E-02)	8.917E-01(3.678E-02)
	5	3.809E-01(4.848E-03)	4.540E-01(2.303E-01)	5.591E-01(8.353E-03)	9.703E-01(4.427E-03)	9.524E-01(4.546E-02)	9.250E-01(3.469E-03)
	8	9.313E-01(2.241E-02)	1.178E+00(3.610E-02)	1.205E+00(3.186E-01)	9.555E-01(1.376E-02)	9.375E-01(6.461E-02)	9.284E-01(3.787E-02)
	10	1.091E+00(2.275E-02)	1.111E+00(2.509E-02)	1.371E+00(4.654E-01)	9.697E-01(8.880E-03)	9.696E-01(6.247E-03)	9.441E-01(3.930E-02)
WFG3	3	1.256E-01(9.403E-03)	1.199E-01(1.050E-02)	3.455E-01(1.670E-02)	6.982E-01(2.855E-03)	6.971E-01(3.022E-03)	6.598E-01(3.419E-03)
	5	5.688E-01(3.957E-02)	5.117E-01(3.184E-02)	6.686E-01(3.264E-02)	6.439E-01(6.664E-03)	6.523E-01(6.533E-03)	6.327E-01(6.901E-03)
	8	2.007E+00(1.193E-01)	1.904E+00(1.339E-01)	1.720E+00(1.094E-01)	4.912E-01(1.670E-02)	5.040E-01(1.943E-02)	5.337E-01(1.910E-02)
	10	2.481E+00(1.538E-01)	2.423E+00(1.482E-01)	2.448E+00(6.575E-01)	4.505E-01(2.094E-02)	4.657E-01(1.952E-02)	4.938E-01(2.902E-02)
WFG4	3	1.487E-01(1.807E-03)	1.441E-01(1.153E-03)	1.406E-01(7.042E-04)	7.205E-01(2.912E-03)	7.250E-01(2.427E-03)	7.346E-01(1.996E-03)
	5	9.619E-01(3.688E-03)	9.512E-01(1.602E-03)	1.150E+00(2.114E-02)	8.164E-01(8.456E-03)	8.259E-01(6.288E-03)	8.000E-01(4.536E-03)
	8	2.964E+00(4.760E-03)	2.956E+00(7.010E-03)	3.241E+00(3.362E-02)	8.417E-01(1.242E-02)	8.474E-01(1.240E-02)	7.611E-01(1.108E-02)
	10	4.215E+00(9.362E-03)	4.221E+00(4.703E-02)	4.280E+00(3.775E-02)	8.687E-01(1.451E-02)	8.647E-01(1.422E-02)	7.659E-01(8.375E-03)
WFG5	3	1.666E-01(1.470E-03)	1.630E-01(8.914E-04)	1.582E-01(7.058E-04)	6.945E-01(2.623E-03)	6.966E-01(2.083E-03)	6.978E-01(1.767E-03)
	5	9.476E-01(2.031E-03)	9.404E-01(1.273E-03)	1.135E+00(7.503E-03)	8.082E-01(3.768E-03)	8.119E-01(2.945E-03)	7.727E-01(1.994E-03)
	8	2.955E+00(4.128E-03)	2.952E+00(6.537E-03)	3.320E+00(1.598E-02)	8.347E-01(1.053E-02)	8.416E-01(7.127E-03)	7.323E-01(5.599E-03)
	10	4.160E+00(1.344E-02)	4.173E+00(2.422E-02)	4.281E+00(2.800E-02)	8.654E-01(7.104E-03)	8.668E-01(5.895E-03)	7.315E-01(4.961E-03)
WFG6	3	1.774E-01(6.112E-03)	1.699E-01(5.898E-03)	1.577E-01(4.905E-03)	6.952E-01(4.533E-03)	6.998E-01(5.144E-03)	7.092E-01(4.963E-03)
	5	9.574E-01(1.383E-03)	9.502E-01(1.175E-03)	1.115E+00(1.665E-02)	8.239E-01(6.390E-03)	8.242E-01(4.244E-03)	7.916E-01(4.877E-03)
	8	2.988E+00(1.017E-02)	2.974E+00(9.867E-03)	3.379E+00(4.489E-02)	8.783E-01(7.892E-03)	8.782E-01(1.070E-02)	7.634E-01(9.464E-03)
	10	4.217E+00(1.315E-02)	4.234E+00(2.455E-02)	4.368E+00(3.806E-02)	9.068E-01(1.327E-02)	9.072E-01(6.929E-03)	7.633E-01(7.156E-03)
WFG7	3	1.418E-01(4.886E-04)	1.397E-01(1.682E-04)	1.412E-01(3.928E-04)	7.331E-01(1.837E-03)	7.368E-01(1.502E-03)	7.406E-01(1.261E-03)
	5	9.581E-01(9.701E-04)	9.522E-01(1.250E-03)	1.069E+00(1.260E-02)	8.525E-01(6.703E-03)	8.620E-01(4.936E-03)	8.231E-01(2.138E-03)
	8	3.089E+00(8.866E-02)	3.081E+00(9.245E-02)	3.164E+00(4.728E-01)	7.953E-01(2.806E-02)	8.054E-01(2.939E-02)	8.081E-01(3.068E-02)
	10	4.613E+00(3.207E-02)	4.594E+00(2.579E-02)	4.425E+00(6.703E-01)	8.517E-01(2.138E-02)	8.642E-01(1.968E-02)	8.606E-01(4.384E-02)
WFG8	3	6.609E-01(1.363E-02)	6.322E-01(2.029E-02)	6.021E-01(1.641E-02)	4.730E-01(7.147E-03)	4.824E-01(9.245E-03)	4.954E-01(8.697E-03)
	5	1.416E+00(1.778E-02)	1.403E+00(1.983E-02)	1.357E+00(1.766E-02)	4.944E-01(1.050E-02)	4.967E-01(1.294E-02)	5.073E-01(1.762E-02)
	8	3.441E+00(4.734E-02)	3.429E+00(4.279E-02)	3.426E+00(4.695E-02)	4.425E-01(2.806E-02)	4.461E-01(2.607E-02)	3.951E-01(3.580E-02)
	10	4.611E+00(3.143E-02)	4.608E+00(3.074E-02)	4.433E+00(5.323E-02)	4.582E-01(2.749E-02)	4.516E-01(2.586E-02)	3.145E-01(2.847E-02)
WFG9	3	2.028E-01(1.463E-04)	2.022E-01(1.181E-04)	2.013E-01(1.494E-04)	6.543E-01(1.790E-03)	6.545E-01(1.773E-03)	6.544E-01(1.275E-03)
	5	9.303E-01(2.651E-03)	9.245E-01(1.262E-03)	1.060E+00(1.246E-02)	7.500E-01(3.689E-03)	7.514E-01(2.082E-03)	7.167E-01(2.852E-03)
	8	2.942E+00(2.018E-02)	2.933E+00(2.913E-02)	3.127E+00(1.770E-01)	7.564E-01(1.028E-02)	7.557E-01(1.121E-02)	7.021E-01(1.882E-02)
	10	4.532E+00(1.525E-02)	4.480E+00(1.771E-02)	4.333E+00(2.610E-01)	7.680E-01(6.460E-03)	7.693E-01(4.622E-03)	7.311E-01(2.535E-02)
+/-		29/7	-	21/15	27/9	-	25/11
Best/All		7/36	14/36	15/36	9/36	16/36	11/52

Besides, the comparison statistical summary of MaOEA-VADS and five algorithms on the DTLZ and WFG test problems in Table 8. In this table, the number of test problems where MaOEA-VADS is significantly better than, similar to and worse than the compared algorithms is recorded and the statistics of the best results for each algorithm in all test problems are described. As an be seen from Table 8, the optimal value of MaOEA-VADS occupies half or more of the quantity, and has obvious advantages compared with other algorithms.

From Table 4, it can be seen that MaOEA-VADS obtained 33 best IGD values out of 52 test problems, lager than the IGD values obtained by the compared algorithms(i.e., MOEA/D, MOMOBI-II, NSGA-III, RVEA, and RPD-NSGA-II got the number of best result are 1, 5, 4, 9, 0, respectively). The penultimate row of Table 4 records that MaOEA-VADS shows significantly better performance than any the compared algorithm over more than half of the test problems. From Table 6, it can be seen that MaOEA-VADS has achieved 23 best HV values out of 52 test problems, lager than the HV values the other algorithms(i.e., MOEA/D, MOMOBI-II, NSGA-III, RVEA, and RPD-NSGA-II got the number of best result are 2, 10, 6, 8, 3, respectively). In addition, MaOEA-VADS performs significantly preponderance than any the compared algorithm. Therefore, the results of IGD and HV value statistics prove that the algorithm has certain competitiveness when dealing with high-dimensional problems.

DTLZ1 is characterized by linear and multi-modal, and its biggest challenge is to jump out of multiple fronts and converge to the true front. As evidenced by statistical results of IGD and HV in Table 4 and Table 6, MaOEA-VADS algorithm does not show significant competitiveness over other algorithms especially compared to RVEA. In HV statistics, MaOEA-VADS performs better than MOEA/D for the four objectives, better than NSGA-III for 8- and 10-objective and similar to MOMOBI-II for 8- and 10-objective but worse than MOMOBI-II and NSGA-III for 3- and 5-objective. However, MaOEA-VADS shows the poor performance in dealing with DTLZ1 when compared with RVEA and RPD-NSGA-II. DTLZ2 is a simple concave problem. For the statical results in Table 4, MaOEA-VADS shows the competitive performance in IGD statistics than the compared algorithms for 3-, 5-, 8- and 10-objectives expect RVEA for 5- and 10-objective. However, for the statical results in Table 6, MaOEA-VADS obtains the HV mean values that are all superior to and similar to other algorithms for 3-, 5-, 8- and 10-objectives. What’s more, the best results come from MaOEA-VADS expect 3-objective. The statistical results indicate that MaOEA-VADS performs well in convergence and diversity for dealing with concave problems. Concerning DTLZ3, which is featured by concave and multi-modal. Jumping out of local optimality is a great challenge for all algorithms to solve this problem. As shown in Table 4 and Table 6, MaOEA-VADS behaves the superior performance for the four objectives than other algorithms expect RVEA for 8- and 10-objective. On the remaining problems, MaOEA-VADS presents the advantage over the compared algorithms, which indicates that MaOEA-VADS has certain advantages in handling concave and multi-modal problem. DTLZ4 is characterized by concave and biased, where the challenge lies in how to get the biased solutions. In Table 4, MaOEA-VADS shows the superior performances than the other algorithms but worse than RVEA for 5-objective and 10-objective. In Table 6, MaOEA-VADS shows the superior and similar performance to compared algorithms. These results indicate that MaOEA-VADS obtains the competitiveness for solving concave and biased PF.

WFG1 is introduced to test the ability of algorithm to solve mixed and biased PF. As shown in In Table 4, MaOEA-VADS shows inferior performance than other five algorithms for 5-, 8- and 10-objective. In Table 6. MaOEA-VADS behaves the superior performance to NSGA-III and RVEA but inferior to MOEA/D and MOMOBI-II for 3-, 5- and 10-objective. In general, MaOEA-VADS still need to be improved in handling the WFG1 problem. Regarding WFG2, which is a disconnected test problem. MaOEA-VADS displays the competitive performance than other algorithms for four objective expect NSGA-III 3-objective in Table 4. As indicated by the statistical results in Table 6, MaOEA-VADS performs superior and similar performance than compared algorithms but inferior to NSGA-III for 3- and 10-objective and RPD-NSGA-II for 5-objective. Overall, MaOEA-VADS algorithm can be quite competitive in dealing with discontinuous problems. However, as for WFG3, MaOEA-VADS behaves worse performance than MOEA/D, MOMOBI-II, NSGA-III and RPD-NSGA-II and superior to RVEA. Form this results, MaOEA-VADS have yet to be improved in dealing with liner and non-separable problems. WFG4 is recommended to test the ability of algorithm to deal with concave and multi-modal problem. As shown in Table 4, MaOEA-VADS displays the superior and similar performance to compared algorithms. And in Table 6, MaOEA-VADS shows that almost all of these problems are superior to other algorithms expect for RDP-NSGA-II for 10-objective. Considering WFG5, which is a concave and deceptive test problem. As shown in Table 4 and Table 6, MaOEA-VADS behaves the better results than other algorithms and only slightly worse than RVEA in HV data statistics for 8- and 10-objective. WFG6 is introduced to concave and non-separable PF. As recorded in Table 4, MaOEA-VADS performs better than MOEA/D, MOMOBI-II, RVEA and RPD-NSGA-II, but it shows the worse performance than NSGA-III for 8-objective. In Table 6, MaOEA-VADS shows similar performance than MOMOBI-II for 3-objective and all of the four objectives are superior to other algorithms. Regarding WFG7, which is adopted to test ability of algorithm to solve concave and biased PF. As shown by the statistical results in Table 4 and Table 6, MaOEA-VADS performs the best results for 3-, 5-, 8- and 10-objective than other algorithms and obtains the best results for four objectives. WFG8 is featured by concave, biased and non-separable, MaOEA-VADS only shows the worse HV value than MOMOBI-II for 3-objective in Table 6. WFG9 is characterized by concave, biased, deceptive and multi-modal. As shown in Table 4 and Table 6, MaOEA-VADS achieves competitive performance for 3-, 5-, 8- and 10-objective than other algorithms, which indicate the proposed algorithm has good convergence and distribution when solving the problem of high dimensional correlation.

5.1.2 The analysis of figures

The statistical results can only be compared from the accuracy of the algorithm, but it is difficult to see the convergence process and distribution of the algorithm. Therefore, we analyze the figures about Pareto front and convergence of six algorithms in some distinct problems. In the process of drawing the convergence graph, the x-coordinate is the maximum iterative algebra, and the y-coordinate is log(IGD) and log(HV), which is used to make the graph clearer.

For 3-objective test problems, Fig. 2 lists the optimal solutions obtained by six algorithms on DTLZ3 and WFG5 for the 3-objective. As can be seen in Fig. 2 (a)-(f), the obtained PF of MaOEA-VADS on DTLZ3 is significantly more uniform, and the distribution of solutions is better. However, in Fig. 2 (g)-(l), the Pareto fronts of MaOEA-VADS, NSGA-III and RVEA show similar distribution, and the distribution of solutions is relatively uniform. Fig. 3 and Fig. 4 list the representation of IGD and HV convergence trend by six algorithms on partial test problems for 3-objective. MaOEA-VADS obtains the best IGD accuracy on DTLZ1, DTLZ2, DTLZ3, DTLZ4, WFG1, WFG4, WFG5 and WFG7. However, MaOEA-VADS obtains the best HV accuracy on DTLZ2, DTLZ3, DTLZ4, WFG1, WFG5 and WFG7, the second best values on WFG4 and WFG6. Although MaOEA-VADS algorithm may not have the fastest convergence speed in the process of convergence, but it has more advantages in convergence accuracy, which indicates the possibility of this algorithm in dealing with the 3-objective problems.

Fig. 2

The representation of the optimal solutions obtained by each algorithm on the 3-objective DTLZ3(a)-(f) and WFG5(g)-(l).

Fig. 3

The representation of IGD convergence trend by each algorithm on the 3-objective of partial functions.

Fig. 4

The representation of HV convergence trend by each algorithm on the 3-objective of partial functions.

For 5-objective test problems, Fig. 5 lists the optimal solutions obtained by six algorithms on WFG2 (a)-(f) and WFG8 (g)-(l) for the 5-objective. MOEA/D and MOMOBI-II are obviously very poorly distributed, MaOEA-VADS performs the similar convergence and distribution to RVEA. Fig. 6 and Fig. 7 list the representation of IGD and HV convergence trend by six algorithms on partial test problems for 5-objective. MaOEA-VADS obtains the best IGD accuracy on DTLZ1, DTLZ2, DTLZ4, WFG2, WFG7, WFG8 and WFG9 and achieves the best HV accuracy on DTLZ2, DTLZ4, WFG4, WFG5, WFG7, WFG8 and WFG9. Although the convergence accuracy of the two indicators is different on the same problem, which is caused by the different calculation methods, it also indicates that the algorithm has good convergence. In addition, as can be seen from Fig. 6 and Fig. 7, MaOEA-VADS also has a certain advantage in convergence rate for WFG problems.

Fig. 5

The representation of the optimal solutions obtained by each algorithm on the 3-objective WFG2(a)-(f) and WFG8(g)-(l).

Fig. 6

The representation of IGD convergence trend by each algorithm on the 5-objective of partial functions.

Fig. 7

The representation of HV convergence trend by each algorithm on the 5-objective of partial functions.

For 8-objective test problems, Fig. 8 shows the representation of optimal PF by six algorithms on DTLZ4 (a)-(f) and WFG7 (g)-(l) for 8-objective. It is obvious that the distribution of RPD-NSGA-II is more uniform than MOEA/D, MOMOBI-II, NSGA-III and RVEA, it is still slightly worse than MaOEA-VADS. As shown in Fig. 9, MaOEA-VADS achieves the best IGD accuracy on DTLZ4, WFG2, WFG7, WFG8 and WFG9. RVEA converges the fastest on DTLZ1 and DTLZ2 but with the same accuracy as MaOEA-VADS. As shown in Fig. 10, MaOEA-VADS displays significant superiority in HV accuracy on WFG4, WFG6, WFG7 and WFG9. For 10-objective test problems, Fig. 11 lists the optimal PF on DTLZ2 (a)-(f) and WFG10 (g)-(l) for 10-objective. Fig. 11(a) and (g) shows that the distribution of DTLZ2 and WFG9 in MOEA/D was poor. MOMOBI-II, NSGA-III, RVEA and RPD-NSGA-II can obtain the relatively uniform solutions when solving DTLZ2, but some solutions have not converged to the true Pareto front when solving WFG9. However, MaOEA-VADS performs the good convergence and distribution on DTLZ2 and WFG9. As can be seen from Fig. 12, the convergence accuracy and convergence rate of MaOEA-VADS are the highest on WFG2, WFG7, WFG9. For the problems of DTLZ1, DTZL2, DTLZ3 and DTLZ4, MaOEA-VADS can maintain the accuracy of RVEA, but it is inferior to the convergence speed. As shown in Fig. 13, MaOEA-VADS obtains the best HV accuracy on WFG7 and WFG9. In the course of convergence diagram of DTLZ2, DTLZ4, WFG2, WFG5 and WFG6, MaOEA-VADS show slightly worse convergence speed, but it is superior to other algorithms. For WFG5, RVEA is superior to MaOEA-VADS and RPD-NSGA-II in convergence accuracy, but inferior to them in convergence speed.

Fig. 8

The representation of the optimal solutions obtained by each algorithm on the 3-objective DTLZ4(a)-(f) and WFG7(g)-(l).

Fig. 9

The representation of IGD convergence trend by each algorithm on the 8-objective of partial functions.

Fig. 10

The representation of HV convergence trend by each algorithm on the 8-objective of partial functions.

Fig. 11

The representation of the optimal solutions obtained by each algorithm on the 3-objective DTLZ2(a)-(f) and WFG9(g)-(l).

Fig. 12

The representation of IGD convergence trend by each algorithm on the 10-objective of partial functions.

Fig. 13

The representation of HV convergence trend by each algorithm on the 10-objective of partial functions.

To sum up, we can conclude that MaOEA-VADS has greater potential when dealing with higher dimensional problems and can achieve good convergence and distribution in most problems. However, according to the law of no free lunch [59], MaOEA-VADS algorithm is not guaranteed to be superior to other algorithms on all problems. At least for most problems, the proposed algorithm obtains more best results.

5.2 Sensitivity of performance to the parameter b

In this experiment, the parameter b ia a key factor for the calculation of vector angle distance. Therefore, in order to study the sensitivity of performance to the parameter b, WFG1-WFG9 characterized by different types have been taken to verify the impact of parameter b. The mean IGD and HV of 20 independent runs with varying the parameter value b=10, 50, 100. In order to observe the impact of b on the performance of the algorithm, Table 9 shows the mean IGD and HV obtained using different values of b for WDFG1-WFG9.

In this experiment, b = 10 has the significant advantage in solving WFG1. For WFG2, b = 100 obviously obtains the best results in both IGD and HV, but b = 100 performs the worst performance for WFG3. Due to the different calculation methods of indicators, b = 50 has a certain advantage in IGD statistics, while b = 10 occupies the best value in 8- and 10-objective for WFG3. Considering WFG4, b = 50 produced the best results in IGD statistics for 5- and 8-objectives, but it performs slightly worse than b = 10 for 3-objective and b = 100 for 10-objective. Taking into WFG5 and WFG6, b = 10 shows the superior performance for 3-objective in both IGD and HV, b = 50 only performs the best values for 10-objective in IGD statistics. In view of WFG7, b = 50 shows the best IGD value for 3-, 5- and 8-objective and obtains the better HV value for 5- and 10-objective. However, b = 50 performs worse than b = 10 for WFG8 but better than b = 100. Then, in consideration of WFG9, b = 50 obtains the best IGD value for 5- and 8-objective and the best HV values for 3-, 5- and 10-objectives.

In the last row of Table 9, we list the number of problems that have the best results out of 36 functions. As can be seen from the statistical results, the number of the best IGD and best HV for b = 50 are 14 and 16, respectively. However, the number of the best IGD and best HV for b = 10 are 15 and 11, respectively. In order to filter b = 100 and b = 50 clearly, we counts the number of superior solutions and inferior solutions between b = 50 and b = 100 and b = 10, respectively. In the Table 9, +/- represents that b = 50 is superior to and inferior to other parameters, respectively. As can be seen from the penultimate row in the Table 9, there are 29 and 21 IGD values better than b = 100 for b = 50, and there are 27 and 25 HV values better than b = 100 for b = 50. Therefore, through the overall data analysis, we finally choose b = 50 as the VADS calculation, even though b = 100 is the common value in this article [54].

6 Conclusions and future work

In order to deal with MaOPs with various types of Pareto fronts, a many-objective evolutionary algorithm based on vector angle distance scaling is presented, termed MaOEA-VADS. It contains three main improvements, i.e., calculate the new fitness value of vector angle distance function and scaling the fitness in environmental selection, the final aggregate fitness update the population. In this experiment, the calculation of vector angle distance reflects the convergence and diversity criterion and the reference vectors are adopted to guide solutions towards to accurate Pareto front. Then, scaling fitness of vector angle distance process is a compact analytic method to obtain better solutions. Besides, a final aggregate fitness will be applied into mating selection, which improve matching process to select better parent generation to produce quality offspring. What’s more, vector angle distance scaling strategy guarantees the effectiveness of the proposed algorithm on MaOPs. MaOEA-VADS is compared with five algorithms, i.e. one classical MOEAs and four MaOEAs. Experimental results have demonstrated that MaOEAs-VADS outperforms other compared algorithms on overall 52 test problems. Therefore, MaOEA-VADS is a promising and versatile algorithm with good convergence and diversity.

It is worth mentioning that MaOEA-VADS is a promising algorithm to solve MaOPs. However, it is a hindrance when handling PFs with complicated mixed geometries, non-separable and deceptive(e.g. WFG1, WFG2 and WFG3). In the future, we would like to focus on dealing with more complex PF shapes.

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 work was supported by National Key Research and Development Project under Grant 2018YFC1602704, and National Natural Science Foundation of China under Grant 61873006.

References

Zhang

and Xing

, A Survey of Multiobjective Evolutionary Algorithms, 2017 IEEE International Conference on Computational Science and Engineering (CSE) and IEEE International Conference on Embedded and Ubiquitous Computing(EUC) 1 (2017), 93–100. doi: 10.1109/CSE-EUC.2017.27in.

Wang

, Su

, Lin

, Ma

, Gong

, Li

and Ming

, A survey of decomposition approaches in multiobjective evolutionary algorithms, Neurocomputing 408 (2020), 308–330. ISSN 0925-2312, doi: https://doi.org/10.1016/j.neucom.2020.01.114.

, Wang

, Jia

and Cui

, An intelligent braking system composed single-pedal and multi-objective optimization neural network braking control strategies for electric vehicle, Applied Energy 259 (2020), 114172. ISSN 0306-2619, doi: https://doi.org/10.1016/j.apenergy.2019.114172.

, Vaddi

and Wong

, Multi-objective Precision Optimization of Deep Neural Networks for Edge Devices, in: 2019 Design, Automation Test in Europe Conference Exhibition(DATE), 1100–1105, doi:10.23919/DATE.2019.8714785,2019

Tkachenko

, Izonin

, Kryvinska

, Dronyuk

and Zub

, An Approach towards Increasing Prediction Accuracy for the Recovery of Missing IoT Data Based on the GRNN-SGTM Ensemble, Sensors 20(9) (2020), 2625.

Izonin

, Tkachenko

, Verhun

and Zub

, An approach towardsmissing datamanagement using improved GRNNSGTM ensemble method, Engineering Science and Technology, an International Journal ISSN 2215–0986. doi: https://doi.org/10.1016/j.jestch.2020.10.005.

Deb

, Pratap

, Agarwal

and Meyarivan

, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation 6(2) (2002), 182–197. ISSN 1089-778X, doi: https://doi.org/10.1109/4235.996017.

Zitzler

, Laumanns

and Thiele

, SPEA2: Improving the strength Pareto evolutionary algorithm, TIK-report 103.

Corne

D.W.

, Jerram

N.R.

, Knowles

J.D.

and Oates

M.J.

, PESAII: Region-based selection in evolutionary multiobjective optimization, in: Proceedings of the 3rd Annual Conference on Genetic and Evolutionary Computation, Morgan Kaufmann Publishers Inc., (2001), 283–290.

10.

Ishibuchi

, Tsukamoto

and Nojima

, Evolutionary Many-Objective Optimization: A Short Review, (2008), 2419–2426, doi:10.1109/CEC.2008.4631121.

11.

, Li

, Tang

and Yao

, Many-Objective Evolutionary Algorithms: A Survey, ACM Comput Surv 48(1) ISSN 0360-0300, doi:10.1145/2792984, URL https://doi.org/10.1145/2792984.

12.

, Wang

, Zhang

and Ishibuchi

, Evolutionary Many-Objective Optimization: A Comparative Study of the Stateofthe-Art, IEEE Access 6 (2018), 26194–26214. ISSN 2169-3536, doi: https://doi.org/10.1109/ACCESS.2018.2832181.

13.

Laumanns

, Thiele

, Deb

and Zitzler

, Combining Convergence and Diversity in Evolutionary Multiobjective Optimization, Evolutionary Computation 10(3) (2002), 263–282.

14.

Deb

, Mohan

and Mishra

, Evaluating the epsilondomination based multi-objective evolutionary algorithm for a quick computation of Pareto-optimal solutions, Evolutionary Computation 13(4) (2014), 501–525.

15.

Ikeda

, Kita

and Kobayashi

, Failure of Pareto-based MOEAs: Does non-dominated really mean near to optimal?, 2 (2001), 957–962. doi: 10.1109/CEC.2001.934293

16.

Farina

and Amato

, A fuzzy definition of “optimality” for many-criteria optimization problems, IEEE Transactions on Systems Man and Cybernetics Part A Systems and Humans 34(3) (2004), 315–326.

17.

K?ppen

, Vicente-Garcia

and Nickolay

, Fuzzy-Pareto-Dominance and its Application in Evolutionary Multiobjective Optimization, in: C. A. Coello Coello, A. Hernĺćndez Aguirre, E. Zitzler (Eds.), Evolutionary Multi-Criterion Optimization, Springer Berlin Heidelberg, Berlin, Heidelberg, (2005), 399–412.

18.

Yang

, Li

, Liu

and Zheng

, A Grid-Based Evolutionary Algorithm for Many-Objective Optimization, IEEE Transactions on Evolutionary Computation 17(5) (2013), 721–736. doi: 10.1109/TEVC.2012.2227145

19.

Ląŕopez

, Coello

C.A.C.

, Oyama

and Fujii

, An Alternative Preference Relation to Deal with Many-Objective Optimization Problems, in: R. C. Purshouse, P. J. Fleming, C. M. Fonseca, S. Greco, J. Shaw (Eds.), Evolutionary Multi-Criterion Optimization, Springer Berlin Heidelberg, Berlin, Heidelberg, (2013), 291–306.

20.

, Yang

and Liu

, Shift-Based Density Estimation for Pareto-Based Algorithms in Many-Objective Optimization, IEEE Transactions on Evolutionary Computation 18(3) (2014), 348–365.

21.

Cheng

, Yen

G.G.

and Zhang

, A Many-Objective Evolutionary AlgorithmWith Enhanced Mating and Environmental Selections, IEEE Transactions on Evolutionary Computation 19(4) (2015), 592–605.

22.

Zhang

, Tian

and Jin

, A Knee Point-Driven Evolutionary Algorithm for Many-Objective Optimization, IEEE Transactions on Evolutionary Computation 19(6) (2015), 761–776. ISSN 1089-778X, doi: https://doi.org/10.1109/TEVC.2014.2378512.

23.

Beume

, Naujoks

and Emmerich

, SMS-EMOA: Multiobjective selection based on dominated hypervolume, European Journal of Operational Research 181(3) (2007), 1653–1669. ISSN 0377-2217, doi: https://doi.org/10.1016/j.ejor.2006.08.008.

24.

Bader

and Zitzler

, HypE: An algorithm for fast hypervolumebased many-objective optimization, Evolutionary Computation 19(1) (2011), 45–76.

25.

Coello

C.A.C.

, MOPSOhv: A new hypervolume-based multi-objective particle swarm optimizer, in: 2014 IEEE Congress on Evolutionary Computation (CEC) 1 (2014), 266–273. doi: 10.1109/CEC.2014.6900540.

26.

Trautmann

, Wagner

and Brockhoff

, R2-EMOA: Focused Multiobjective Search Using R2-Indicator-Based Selection, in: G. Nicosia, P. Pardalos (Eds.), Learning and Intelligent Optimization, Springer Berlin Heidelberg, Berlin, Heidelberg, (2013), 70–74.

27.

, Liu

, Tan

and Yu

, R2-MOPSO: A multi-objective particle swarm optimizer based on R2-indicator and decomposition, in: 2015 IEEE Congress on Evolutionary Computation (CEC), (2015), 3148–3155, doi:10.1109/CEC.2015.7257282

28.

Phan

D.H.

and Suzuki

, R2-IBEA: R2 indicator based evolutionary algorithm for multiobjective optimization, in: 2013 IEEE Congress on Evolutionary Computation (2013), 1836–1845. doi:10.1109/CEC.2013.6557783

29.

Coello Coello

C.A.

, Improved Metaheuristic Based on the R2 Indicator for Many-Objective Optimization, in: The, (2015), 679–686.

30.

, Cheng

, Liu

and Jin

, A two-stage R2 indicator based evolutionary algorithm for many-objective optimization, Applied Soft Computing 67 (2018), 245–260. ISSN 1568-4946, doi: https://doi.org/10.1016/j.asoc.2018.02.048.

31.

Villalobos

and Coello

, A New Multi-Objective Evolutionary Algorithm Based on a Performance Assessment Indicator 505–512. doi:10.1145/2330163.2330235, 2012

32.

Sun

, Yen

G.G.

and Yi

, IGD Indicator-Based Evolutionary Algorithm for Many-Objective Optimization Problems, IEEE Transactions on Evolutionary Computation 23(2) (2019), 173–187. doi: 10.1109/TEVC.2018.2791283

33.

Lopez

E.M.

and Coello

C.A.C.

, Improving the integration of the IGD+ indicator into the selection mechanism of a Multi-objective Evolutionary Algorithm, in: 2017 IEEE Congress on Evolutionary Computation (CEC), (2017), 2683–2690. doi:10.1109/CEC.2017.7969633

34.

Schutze

, Esquivel

, Lara

and Coello

C.A.C.

, Using the Averaged Hausdorff Distance as a Performance Measure in Evolutionary Multiobjective Optimization, IEEE Transactions on Evolutionary Computation 16(4) (2012), 504–522. doi: 10.1109/TEVC.2011.2161872

35.

Zhang

, Gong

, Hu

and Zhang

, Feature selection algorithm based on bare bones particle swarm optimization, Neurocomputing 148 (2015), 150–157.

36.

, Li

and Huang

, A Multi-Objective Evolutionary Algorithm Based on Principal Component Analysis and Grid Division, in: 2018 14th International Conference on Computational Intelligence and Security (CIS), (2018), 201–204. doi:10.1109/CIS2018.2018.00051

37.

Singh

H.K.

, Isaacs

and Ray

, A Pareto Corner Search Evolutionary Algorithm and Dimensionality Reduction in Many-Objective Optimization Problems, IEEE Transactions on Evolutionary Computation 15(4) (2011), 539–556.

38.

Bechikh

, Kessentini

, Said

L.B.

and Ghąäl?edira

, Preference Incorporation in Evolutionary Multiobjective Optimization: A Survey of the State-of-the-Art, Advances in Computers 98 (2015), 141–207.

39.

Bentley

P.J.

and Wakefield

J.P.

, Finding Acceptable Solutions in the Pareto-Optimal Range using Multiobjective Genetic Algorithms, in: P.K. Chawdhry, R. Roy, R.K. Pant (Eds.), Soft Computing in Engineering Design and Manufacturing, Springer London, (1998), 231–240.

40.

Thiele

, Miettinen

, Korhonen

P.J.

and Molina

, A Preference-Based Evolutionary Algorithm for Multi-Objective Optimization, Evolutionary Computation 17(3) (2009), 411–436.

41.

Ben Said

S.K.

and Bechikh

, The r-Dominance: A New Dominance Relation for Interactive Evolutionary Multicriteria Decision Making, IEEE Transactions on Evolutionary Computation 14(5) (2010), 801–818.

42.

Bechikh

, Said

L.B.

and Ghąäl?edira

, Searching for knee regions of the Pareto front using mobile reference points, Soft Computing 15(9) (2011), 1807–1823.

43.

Zhang

and Li

, MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition, IEEE Transactions on Evolutionary Computation 11(6) (2007), 712–731. ISSN 1089-778X, doi: https://doi.org/10.1109/TEVC.2007.892759.

44.

and Zhang

, Multiobjective optimization problems with complicated Pareto sets, MOEA/D and NSGA-II, IEEE Transactions on Evolutionary Computation 13(2) (2009), 284–302.

45.

, Deb

, Zhang

and Kwong

, An Evolutionary Many-Objective Optimization Algorithm Based on Dominance and Decomposition, IEEE Transactions on Evolutionary Computation 19(5) (2015), 694–716.

46.

Liu

, Gu

and Zhang

, Decomposition of a Multiobjective Optimization Problem Into a Number of Simple Multiobjective Subproblems, IEEE Transactions on Evolutionary Computation 18(3) (2014), 450–455.

47.

Deb

and Jain

, An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints, IEEE Transactions on Evolutionary Computation 18(4) (2014), 577–601. ISSN 1089-778X, doi: https://doi.org/10.1109/TEVC.2013.2281535.

48.

Yuan

, Xu

and Wang

, An improved NSGA-III procedure for evolutionary many-objective optimization, in: Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation (2014), 661–668.

49.

Zhao

, Zhang

, Bin

and Peibo

, Decomposition-based sub-problem optimal solution updating direction-guided evolutionary many-objective algorithm, Information ences: An International Journal (2018), 448–449.

50.

Cheng

, Jin

, Olhofer

and Sendhoff

, A Reference Vector Guided Evolutionary Algorithm for Many-Objective Optimization, IEEE Transactions on Evolutionary Computation 20(5) (2015).

51.

Elarbi

, Bechikh

, Gupta

, Ben Said

and Ong

, A New Decomposition-Based NSGA-II for Many-Objective Optimization, IEEE Transactions on Systems, Man, and Cybernetics: Systems 48(7) (2018), 1191–1210. doi: 10.1109/TSMC.2017.2654301

52.

Jiang

and Yang

, A Strength Pareto Evolutionary Algorithm Based on Reference Direction for Multiobjective and Many-Objective Optimization, IEEE Transactions on Evolutionary Computation 21(3) (2017), 329–346. doi: 10.1109/TEVC.2016.2592479

53.

Liu

, Lin

, Tan

K.C.

, Gong

and Coello

C.A.C.

, A Fuzzy Decomposition-Based Multi/Many-Objective Evolutionary Algorithm, IEEE Transactions on Cybernetics (2020). 1ĺC15doi:10.1109/TCYB.2020.3008697

54.

Hughes

E.J.

, MSOPS-II: A general-purpose Many-Objective optimiser, in: 2007 IEEE Congress on Evolutionary Computation (2007), 3944–3951. doi:10.1109/CEC.2007.4424985.

55.

Deb

, Thiele

, Laumanns

and Zitzler

, Scalable Test Problems for Evolutionary Multiobjective Optimization (2006), 105–145.

56.

Huband

, Hingston

, Barone

and While

, A review of multiobjective test problems and a scalable test problem toolkit, IEEE Transactions on Evolutionary Computation 10 (2006), 477–506. doi: 10.1109/TEVC.2005.861417

57.

Neri

and Cotta

, Memetic algorithms and memetic computing optimization: A literature review, Swarm and Evolutionary Computation 2 (2012), 1–14.

58.

Wilcoxon

, Some Rapid Approximate Statistical Procedures, Annals of the New York Academy of ences 52(The Place of Statistical Methods in Biological and Chemical Experimentation) (2010), 808–814.

59.

Wolpert

D.H.

and Macready

W.G.

, No free lunch theorems for optimization, IEEE Transactions on Evolutionary Computation 1(1) (1997), 67–82.

A many-objective evolutionary algorithm based on vector angle distance scaling

Abstract

Keywords

1 Introduction

2 Background

2.1 Basic definition

3 The framework of MaOEA-VADS algorithm

3.1 Framework of MaOEA-VADS aglorithm

3.2 Initialization procedure

3.3 Environmental selection

4.1 Test problems

Table 2 Parameters setting of algorithm running m N t max H1 and H2 3 210 300 H1 = 19, H2 = 0 5 210 500 H1 = 6, H2 = 0 8 156 800 H1 = 3, H2 = 2 10 276 1000 H1 = 3, H2 = 2

5.1 Performance comparisons

5.1.1 The analysis of data

6 Conclusions and future work

Footnotes

Acknowledgment

References

Table 2
Parameters setting of algorithm running

m N t _max H₁ and H₂

3 210 300 H₁ = 19, H₂ = 0

5 210 500 H₁ = 6, H₂ = 0

8 156 800 H₁ = 3, H₂ = 2

10 276 1000 H₁ = 3, H₂ = 2