A many-objective evolutionary algorithm based on three states for solving many-objective optimization problem

Abstract

In recent years, researchers have taken the many-objective optimization algorithm, which can optimize 5, 8, 10, 15, 20 objective functions simultaneously, as a new research topic. However, the current research on many-objective optimization technology also encounters some challenges. For example: Pareto resistance phenomenon, difficult diversity maintenance. Based on the above problems, this paper proposes a many-objective evolutionary algorithm based on three states (MOEA/TS). Firstly, a feature extraction operator is proposed. It can extract the features of the high-quality solution set, and then assist the evolution of the current individual. Secondly, based on Pareto front layer, the concept of “individual importance degree” is proposed. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer. Then, a repulsion field method is proposed. The diversity of the population in the objective space is maintained by the repulsion field, so that the population can be evenly distributed on the real Pareto front. Finally, a new concurrent algorithm framework is designed. In the algorithm framework, the algorithm is divided into three states, and each state focuses on a specific task. The population can switch freely among these three states according to its own evolution. The MOEA/TS algorithm is compared with 7 advanced many-objective optimization algorithms. The experimental results show that the MOEA/TS algorithm is more competitive in many-objective optimization problems.

Introduction

In reality, many optimization problems involve multiple conflicting objectives, such as the design of urban public transport routes¹, production scheduling², securities portfolio management³ and so on. These types of optimization problems are called multi-objective optimization problems (MOPs). This means that there is no one solution to make all the objectives reach the optimum simultaneously, that is, the optimization of one objective may lead to the deterioration of other objectives^4,5. Consequently, the solutions of MOPs are usually a set of compromise solutions that weigh all objectives. The definition of MOPs is as follows:

$$minf\left(x\right)={\left({f}_{1}\left(x\right),{f}_{2}\left(x\right),{f}_{3}\left(x\right),...,{f}_{m}\left(x\right)\right)}^{T}\in {R}^{m},$$

$$x={\left({x}_{1},{x}_{2},{x}_{3},...,{x}_{n}\right)}^{T}\in \Omega .$$

(1)

Among them, $f\left(x\right)$ is the m-dimensional objective vector, which contains m conflicting objective functions; ${f}_{i}\left(x\right)$ represents the i-th objective function; x represents the n-dimensional decision variable; $\Omega$ represents decision space; R^m represents the objective space.

In the field of multi-objective optimization, problems with 2 or 3 optimization objectives are called general multi-objective optimization problems (GMOPs). Problems with more than 3 optimization objectives are called many-objective optimization problems (MaOPs)^6,7,8. GMOPs aren’t the focus of our attention, as there have been many reports about GMOPs^9,10. On the contrary, MaOPs are the focus of our attention, as there are still some challenges to be solved. The fundamental difference between GMOPs and MaOPs is the number of optimization objectives. Assuming that the number of optimization objectives is m, the probability that one individual dominates another is $1/{2}^{m-1}$ in theory^11,12. This means that with the increase of the number of optimization objectives, traditional Pareto dominance will fail, Pareto resistance will occur, and most multi-objective optimization algorithms will lose selection pressure in terms of convergence.

In recent years, with the research and exploration of MaOPs, many-objective optimization technology has been developed to a certain extent, and basically 4 mainstream many-objective optimization algorithms have been formed¹³. The first is many-objective optimization algorithm based on dominance. The algorithm modifies the definition of traditional Pareto domination by domination relaxation technique to enhance the selection pressure of the algorithm in terms of convergence. $\alpha$-Dominance, $\upepsilon$-Dominance and Cone $\upepsilon$-Dominance are all common domination relaxation techniques. Compared with traditional Pareto dominance, the effectiveness of dominance relaxation technology has been reported in many works. Therefore, dominance relaxation technology has been widely used to solve MaOPs. However, the current domination relaxation technique also faces two problems: (1) With the increase of the number of optimization objectives, the effect of the domination relaxation technique is getting worse and worse; (2) The domination relaxation technique tends to make the population converge to a certain sub-region of the real Pareto front (PF).

The second is many-objective optimization algorithm based on index. The algorithm guides the selection and evolution of the population by integrating convergence and diversity into one index (such as IGD, HV). Its representative work includes: HypE, MaOEA/IGD, SMS-EMOA. However, the algorithm faces some problems when it is used to solve MaOPs, such as complex index calculation, difficult selection of reference point or reference PF.

The third is many-objective optimization algorithm based on decomposition. The algorithm transforms MaOPs into several single-objective optimization sub-problems through an aggregation function, and then drives the individuals in the neighborhood to update by neighborhood strategy, finally realizes the evolution of the whole population. Its representative work includes: MOEA/D, MOEA/D-D, MOEA/D-DU. However, many-objective optimization algorithm based on decomposition is only suitable for MaOPs with regular PF (such as the DTLZ1 problem). When dealing with MaOPs with irregular PF, many-objective optimization algorithm based on decomposition often performs poorly.

The fourth is many-objective optimization algorithm based on hybrid strategy. The algorithm adopts different search strategies in different environments (different stages or different sub-populations), and uses the advantages of their respective search strategies to deal with complex MaOPs. Its representative work includes: AHM, eMOFEOA, CPSO. In many reports, many-objective optimization algorithm based on hybrid strategy is more suitable for solving MaOPs.

According to the above analysis, this paper considers using the many-objective optimization algorithm based on hybrid strategy, and further proposes the many-objective evolutionary algorithm based on three states (MOEA/TS). The innovations and contributions of this paper are as follows: (1) A feature extraction operator is proposed. The feature extraction operator is a feature extractor, which can extract the features of the high-quality solution set, and then assist the evolution of the current individual. (2) Based on the Pareto front layer, the concept of “individual importance degree” is proposed. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer, and effectively solve the phenomenon of Pareto resistance. (3) A repulsion field method is proposed. The repulsion field is used to maintain the diversity of the population in the objective space, so that the population can be evenly distributed on the real PF. (4) Design a new concurrent algorithm framework. In the framework, the algorithm is divided into three states, and each state focuses on a specific task. The population can freely switch among these three states according to its own evolution.

The remainder of this paper is organized as follows: Sect. "Preparatory work" introduces the basic definition, related work and research motivation. Sect. "Basic definition" introduces each part of the MOEA/TS algorithm in detail. Sect. "Related work" introduces the test results of MOEA/TS algorithm and 7 advanced many-objective optimization algorithms on various test problems, and then analyzes and summarizes them according to the test results. Sect. " Many-objective optimization algorithm based on dominance"summarizes this article and looks forward to future work.

Preparatory work

Basic definition

In this section, we will introduce some basic definitions related to many-objective optimization technology.

Definition of dominance: if solution x isn’t worse than solution y in all objectives and solution x is better than solution y in at least one objective, it is said that x dominates y. That is, if $\forall i\in \left\{\text{1,2},3,...,m\right\}$ satisfies ${f}_{i}\left(x\right)\le {f}_{i}\left(y\right)$ and $\exists j\in \left\{\text{1,2},3,...,m\right\}$ satisfies ${f}_{j}\left(x\right)<{f}_{j}\left(y\right)$, it is said that x dominates y.

Definition of non-dominated solution: if there are no solutions that can dominate x in the decision space, then x is called a Pareto optimal solution or a non-dominated solution. That is, if $\nexists {x}^{*}\in \Omega$ makes x* dominate x, then x is called a Pareto optimal solution or a non-dominated solution.

Definition of Pareto optimal solution set: the set composed of Pareto optimal solutions is called the Pareto optimal solution set (PS). The mathematical description of PS is as follows:

$$PS=\left\{x\in \Omega |x\, is\, pareto \,optimal\, solution\right\}.$$

(2)

Definition of Pareto front: the mapping of PS in the objective space is called Pareto front (PF). The mathematical description of PF is as follows:

$$PF=\left\{f\left(x\right)\in {R}^{m}|x\in PS\right\}.$$

(3)

The goal of the many-objective optimization technology is to find a set of non-dominated solutions that are close to the real PF (convergence) and make them well distributed on the real PF (diversity).

Related work

In recent years, many scholars have conducted in-depth research and exploration in the many-objective optimization technology.

Many-objective optimization algorithm based on dominance

Considering the limitations of Pareto dominance relationship in high-dimensional objective space, Zhou et al¹⁴ proposed a many-objective optimization algorithm based on dominance relation selection. Firstly, they introduced an angle domination relationship with higher selection pressure based on the traditional Pareto domination relationship, and designed a new dominance selection strategy. Additionally, they proposed an angle-based individual distribution method to ensure even population distribution in the objective space. The algorithm shows strong competitiveness in solving MaOPs. Wang et al¹⁵ believed that as the number of objectives increased, the traditional dominance relationship would become invalid. Therefore, they proposed a modified dominance relation. That is, they used penalty-based adaptive matrix regions to assist the traditional dominance relationship. Further, for MaOPs with irregular Pareto fronts, they introduced a population-based adaptive adjustment method to replace the predefined weight vector. On this basis, for MaOPs, they developed a many-objective optimization algorithm based on modified dominance relation and adaptive adjustment method. Zhang et al¹⁶ believed that the current many-objective optimization algorithms focused too much on convergence, which would cause the population to converge to a certain sub-region of the real Pareto front. In order to solve this problem, they proposed a many-objective optimization algorithm based on double distance domination. In this algorithm, double distance can not only measure the convergence of the algorithm to adapt to different Pareto fronts, but also combine angle-based niche technology to emphasize the diversity of the algorithm. In addition, they also designed a special mutation operator. This operator can generate high-quality individuals in sparse areas to improve the diversity of the algorithm.

Many-objective optimization algorithm based on index

Aiming at the high complexity problem of hypervolume computation, Shang et al¹⁷ proposed a new multi-objective evolutionary algorithm (MOEA) based on R2 index, namely the R2HCA-EMOA algorithm. The core idea of this algorithm is to use R2 index variables to approximate the contribution of hypervolume. The basic framework of the proposed algorithm is similar to that of SMS-EMOA. In order to improve the calculation efficiency of the algorithm, the utility tensor structure is introduced to calculate R2 index variables. In addition, the normalization mechanism is incorporated into the R2HCA-EMOA algorithm to improve its performance. Zhang et al¹⁸ believed that the loss of selection pressure was the core reason for the poor performance of the algorithm. In order to solve this problem, they proposed a many-objective optimization algorithm based on fitness evaluation and hierarchical grouping. The fitness evaluation method combined the convergence measure based on the cos function and the diversity measure based on angle to create the selection pressure of convergence and diversity. In order to further strengthen the selection pressure, they proposed a hierarchical grouping strategy. Firstly, individuals are divided into different layers by front index, and then individuals in the same layer are divided into different groups by R2 index. Although some indexes can approximate the contribution of HV, However, Nan et al¹⁹ believed that the key of performance evaluation was to find the worst solution rather than accurately approaching the HV value of each solution. In order to improve the ability to identify the worst solution, they proposed a two-stage R2 index evaluation method. In the first stage, the R2 indexes of all individuals are roughly evaluated to select some candidate solutions. In the second stage, these candidate solutions are accurately evaluated. Finally, they proposed a many-objective optimization algorithm based on the two-stage R2 index.

Many-objective optimization algorithm based on decomposition

In order to balance the convergence and diversity of the decomposition-based algorithm and reduce its dependence on the real PF direction, Wu et al²⁰ developed a many-objective optimization algorithm based on antagonistic decomposition method. This method utilizes the complementary characteristics of different sub-problems in a single example. Specifically, two populations are co-evolved by two sub-problems with different contours and opposite search directions. In order to avoid allocating redundant computing resources to the same area of PF, two populations are matched into one-to-one pairing according to their working areas on PF. In mating selection, each solution pair can only contribute one parent at most. In order to improve the performance of decomposition-based algorithms, Fan et al²¹ proposed a differential multi-objective optimization algorithm based on decomposition. Firstly, they designed a neighborhood intimacy factor to improve the diversity of the algorithm based on the characteristics of neighborhood search. Then, they introduced a Gaussian mutation operator with dynamic step size to enhance the algorithm’s ability to escape from local optimal regions and improve convergence. Finally, they combined a difference strategy with the decomposition-based multi-objective optimization algorithm to further strengthen its evolutionary ability. Peng et al²² believed that data dimensionality reduction could be applied to the objective space. Based on this consideration, they proposed a many-objective optimization algorithm based on projection. Firstly, they used the idea of data dimensionality reduction and spatial decomposition to divide the objective space into projection plane and free dimension. Then, a double elite strategy was used to maintain the balance between convergence and diversity of the algorithm. Finally, the algorithm based on decomposition was used as the algorithm of free dimension to solve MaOPs.

Many-objective optimization algorithm based on hybrid strategy

Aiming at convergence problem and diversity problem of the algorithm, Sun et al²³ proposed a many-objective optimization algorithm based on two independent stages. The algorithm deals with convergence and diversity problems in two independent and successive stages. Firstly, they introduced a non-dominated dynamic weight aggregation method, which is capable of identifying the Pareto optimal solutions of MaOPs. Then, they used these solutions to learn the Pareto optimal subspace in order to solve the convergence problem. Finally, the diversity problem was solved by using reference lines in the Pareto optimal subspace. Considering the advantages of the multi-objective and multi-population (MPMO) framework in solving MaOPs, Yang et al²⁴ proposed an algorithm based on the MPMO framework. The algorithm adopts the deviation sorting (BS) method to solve MaOPs, so as to obtain good convergence and diversity. In terms of convergence, the BS method is applied to each population in the MPMO framework, and the effect of non-dominant sorting is enhanced by the optimization objectives of the corresponding population. In terms of diversity, the maintenance method based on reference vector is used to save the diversity solutions. Aiming at the five-objective job shop scheduling problem (JSSP), Liu et al²⁵ proposed a new genetic algorithm based on the MPMO framework. Firstly, five populations are used to optimize five objectives, respectively. Secondly, in order to prevent each population from focusing only on its corresponding single objective, an archive sharing technology (AST) is proposed to store the elite solutions collected from five populations, so that the population can obtain the optimization information of other objectives from the archive. Thirdly, the archive updating strategy (AUS) is proposed to further improve the quality of the solutions in the archive.

Research motivation

Based on the related work, we believe that there are still the following problems in the current many-objective optimization technology:

• (1) The diversity and convergence of the algorithm are difficult to balance. Most algorithms can’t coordinate the balance between them well, and they either emphasize convergence or diversity too much, which leads to poor quality of the non-dominated solution set.

• (2) It is difficult to maintain the convergence of the algorithm. When the number of optimization objectives is large, the algorithm will produce Pareto resistance, and the traditional Pareto dominance may fail.

• (3) It is difficult to maintain the diversity of the algorithm. Especially when the real PF is complex or the latitude of the objective space is high, individuals may have the clustering effect, and the population may not be evenly distributed on the real PF.

• (4) The evolution efficiency of the algorithm is low. The traditional evolution operators have strong randomness and low evolution efficiency, and aren’t suitable for dealing with MaOPs.

Therefore, solving these problems and providing a good many-objective optimization algorithm constitute the research motivation of this paper.

For problem 1, some work attempts to separate the convergence optimization and diversity optimization of the algorithm, thus designing a concurrent algorithm architecture. Concurrent algorithm architecture means that only one of convergence or diversity is considered in one iteration instead of considering both convergence and diversity simultaneously. In order to solve GMOPs, Professor Ye Tian²⁶ tried to design a concurrent algorithm architecture and proposed the MSEA algorithm, and the experimental results were satisfactory. Therefore, it seems to be a feasible path to solve MaOPs by using concurrent algorithm architecture. However, recent research²³ shows that in MaOPs, the concurrent algorithm architecture seems to be unstable, and the experimental results fluctuate greatly (such as MaOEA/IT algorithm). Because when the algorithm only considers the convergence of the population, it often affects the diversity of the population; Similarly, when the algorithm only considers the diversity of the population, it often affects the convergence of the population. If a coordination intermediary can be added to the concurrent algorithm architecture to alleviate the contradiction between diversity and convergence, the concurrent algorithm architecture will become stable and its superiority will be truly reflected. Based on this motivation, this paper proposes a new concurrent algorithm framework. In the new algorithm framework, the algorithm is divided into three states, namely, convergence maintenance state, diversity maintenance state and coordination state. Each state focuses on a specific task. That is, the convergence maintenance state is responsible for improving the population convergence; Diversity maintenance state is responsible for improving population diversity; the coordination state is responsible for coordinating the contradiction between diversity and convergence. The population can freely switch among these three states according to its own evolution.

For problem 2, some scholars try to modify the definition of traditional Pareto dominance by using dominance relaxation technology to enhance the selection pressure of the algorithm in terms of convergence. However, with the increase of the number of optimization objectives, the effect of dominance relaxation technology is getting worse and worse. They only focus on the modification of Pareto domination definition, but ignore the difference between objective values. If we can distinguish the importance of different individuals by using the difference between the objective values, we can further create the selection pressure of the algorithm in terms of convergence, and finally Pareto resistance will be eliminated. Therefore, based on Pareto front layer, this paper proposes the concept of “individual importance degree”. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer, and effectively solve the phenomenon of Pareto resistance. Obviously, compared with domination relaxation technique, individual importance degree has greater advantages.

For problem 3, the traditional diversity maintenance technology isn’t suitable for high-dimensional objective space. For instance: the niche method, the density evaluation method, and the weight vector method. In the field of microphysics, when the distance between particles is too close, repulsion will push the particles away from their neighbors. On the contrary, when the distance between particles is too great, the repulsion will decrease and the particles tend to be close to the neighboring particles. This way makes the distribution of particles present a state of mutual coordination. Based on the characteristics of particle distribution, a repulsion field method is proposed in this paper. The repulsion field is used to maintain the diversity of the population in the objective space, so that the population can be evenly distributed on the real PF.

For problem 4, traditional evolution operators aren’t suitable for dealing with MaOPs. Because traditional evolution operators have strong randomness and low evolution efficiency. For instance: the binary crossover operator, the polynomial mutation operator, and the differential evolution operator. In principal component analysis, the decomposition of the covariance matrix and correlation matrix is a very important step. By decomposing the covariance matrix or the correlation matrix, we can obtain a set of orthogonal bases. These orthogonal bases are the most important features of the original data²⁷. Therefore, this paper designs a feature extraction operator based on Cholesky decomposition²⁸. The feature extraction operator can be understood as a feature extractor. It can extract the features of the high-quality solution set, and then assist the evolution of the current individual. Obviously, compared with traditional evolution operators, the feature extraction operator has higher evolution efficiency.

MOEA/TS algorithm

Feature extraction operator

The feature extraction operator is a feature extractor, which can extract the features of the high-quality solution set, and then assist the evolution of the current individual. The workflow of the feature extraction operator is shown in Fig. 1.

Figure 1

The workflow of feature extraction operator.

In the first step, W high-quality solutions ${x}^{1},{x}^{2},{x}^{3},...,{x}^{W}$ are selected from the population. These W solutions will form the high-quality solution set S.

In the second step, calculate the mean $\overline{x}$ and covariance matrix A of the high-quality solution set S:

$$\overline{x}=\frac{\sum_{i=1}^{W}{x}^{i}}{W}={\left(\frac{{\sum }_{i=1}^{W}{x}_{1}^{i}}{W},\frac{{\sum }_{i=1}^{W}{x}_{2}^{i}}{W},\frac{{\sum }_{i=1}^{W}{x}_{3}^{i}}{W},...,\frac{{\sum }_{i=1}^{W}{x}_{n}^{i}}{W}\right)}^{T},$$

(4)

$$A=\left[\begin{array}{ccc}cov({x}_{1},{x}_{1})& \cdots & cov({x}_{1},{x}_{n})\\ \vdots & \ddots & \vdots \\ cov({x}_{n},{x}_{1})& \cdots & cov({x}_{n},{x}_{n})\end{array}\right]=\left[\begin{array}{ccc}cov({\left({x}_{1}^{1},...,{x}_{1}^{W}\right)}^{T},{\left({x}_{1}^{1},...,{x}_{1}^{W}\right)}^{T})& \cdots & cov({\left({x}_{1}^{1},...,{x}_{1}^{W}\right)}^{T},{\left({x}_{n}^{1},...,{x}_{n}^{W}\right)}^{T})\\ \vdots & \ddots & \vdots \\ cov({\left({x}_{n}^{1},...,{x}_{n}^{W}\right)}^{T},{\left({x}_{1}^{1},...,{x}_{1}^{W}\right)}^{T})& \cdots & cov({\left({x}_{n}^{1},...,{x}_{n}^{W}\right)}^{T},{\left({x}_{n}^{1},...,{x}_{n}^{W}\right)}^{T}\end{array}\right].$$

(5)

Among them, ${x}^{i}={\left({x}_{1}^{i},{x}_{2}^{i},{x}_{3}^{i},...,{x}_{n}^{i}\right)}^{T}, i\in (1,...,W); {x}_{j}={\left({x}_{j}^{1},{x}_{j}^{2},{x}_{j}^{3},...,{x}_{j}^{W}\right)}^{T}, j\in (1,...,n)$

In the third step, Cholesky decomposition is performed on the covariance matrix A. That is, the covariance matrix A is decomposed into the product of the lower triangular matrix and the transposition of the lower triangular matrix. Assuming that the lower triangular matrix is L, there is

$$A=L*{L}^{T}.$$

(6)

Now let

$$A=\left[\begin{array}{ccc}cov({x}_{1},{x}_{1})& \cdots & cov({x}_{1},{x}_{n})\\ \vdots & \ddots & \vdots \\ cov({x}_{n},{x}_{1})& \cdots & cov({x}_{n},{x}_{n})\end{array}\right]=\left[\begin{array}{ccc}{a}_{11}& \cdots & {a}_{1n}\\ \vdots & \ddots & \vdots \\ {a}_{n1}& \cdots & {a}_{nn}\end{array}\right],$$

(7)

$$L=\left[\begin{array}{ccc}{l}_{11}& \cdots & 0\\ \vdots & \ddots & \vdots \\ {l}_{n1}& \cdots & {l}_{nn}\end{array}\right].$$

(8)

Then

$${L}^{T}=\left[\begin{array}{ccc}{l}_{11}& \cdots & {l}_{n1}\\ \vdots & \ddots & \vdots \\ 0& \cdots & {l}_{nn}\end{array}\right].$$

(9)

Through formula $A=L*{L}^{T}$, we can calculate ${a}_{11}={l}_{11}^{2}$, that is, ${l}_{11}=\sqrt{{a}_{11}}$. Then, according to ${a}_{i1}={l}_{i1}*{l}_{11}$, we can get ${l}_{i1}={a}_{i1}/{l}_{11}$, so we can get the first column element of matrix L.

Assuming that we have calculated the first k-1 column elements of the matrix L. Through

$${a}_{kk}=\sum_{i=1}^{k}{l}_{ki}^{2}.$$

(10)

We can get

$${l}_{kk}=\sqrt{{a}_{kk}-\sum_{i=1}^{k-1}{l}_{ki}^{2}}.$$

(11)

Further

$${a}_{ik}={l}_{ik}*{l}_{kk}+\sum_{j=1}^{k-1}{l}_{ij}*{l}_{kj}, (i=k+1,...,n).$$

(12)

Then

$${l}_{ik}=\frac{{a}_{ik}-\sum_{j=1}^{k-1}{l}_{ij}*{l}_{kj}}{{l}_{kk}}, \left(i=k+1,...,n\right).$$

(13)

In this way, we can solve the k-th column element of matrix L through the first k-1 column elements of matrix L. Then, we can solve matrix L by recursion.

In the fourth step, the sampling vector $s={\left({s}_{1},...,{s}_{n}\right)}^{T}$ is generated by Gaussian distribution $N\left(\text{0,0.7}\right)$. Then, a feature solution is generated.

$${x}^{feature}=\overline{x}+L*s.$$

(14)

Among them, ${x}^{feature}={\left({x}_{1}^{feature},...,{x}_{n}^{feature}\right)}^{T}$

It should be noted that the standard deviation std is an important parameter of the Gaussian distribution. In this paper, the standard deviation std is set to 0.7. The parameter analysis verifies that 0.7 is a reasonable standard deviation. For more details on parameter analysis, please browse the experiment chapter (Parameter sensitivity analysis section).

In the fifth step, assuming that the selected individual is ${x}^{i}({x}_{1}^{i},...,{x}_{n}^{i})$. Based on binary crossover operator²⁹ and feature solution, the formula of generating offspring individual is as follows:

$${c}_{k}=\frac{{x}_{k}^{i}+{x}_{k}^{feature}}{2}+\frac{{\beta }_{k}*\left({x}_{k}^{i}-{x}_{k}^{feature}\right)}{2}.$$

(15)

Among them, ${c({c}_{1},...,{c}_{n})}^{T}$ is the offspring individual. ${\beta }_{k}$ is dynamically determined by the feature factor $\mu$:

$${\beta }_{k}^{\prime}=\left\{\begin{array}{c}{(rand*2)}^{\frac{1}{1+\mu }}, if \,rand\le 0.5\\ ({\frac{1}{2-rand*2})}^{\frac{1}{1+\mu }}, if\, rand>0.5\end{array}\right.,$$

(16)

$${\beta }_{k}^{*}=\left\{\begin{array}{c}1*{\beta }_{k}^{\prime}, if\, randi(\text{0,1})==0\\ -1*{\beta }_{k}^{\prime},if\, randi(\text{0,1})==1\end{array}\right.,$$

(17)

$${\beta }_{k}=\left\{\begin{array}{c}1,if \,rand\le 0.5\\ {\beta }_{k}^{*},if\, rand>0.5\end{array}\right..$$

(18)

Among them,$rand$ is used to generate a random number between 0 and 1;$r and i(\text{0,1})$ is used to generate 0 or 1 randomly.

For the design principle of formula (15), please browse the Supplementary Information Document.

In the sixth step, the individual $c$ is detected and repaired. When some components in individual $c$ exceed the upper bound or lower bound, these components need to be repaired. The repair formula is as follows:

$${c}_{i}^{\prime}=\left\{\begin{array}{c}{c}_{i}^{u}, {c}_{i}\ge {c}_{i}^{u} \\ {c}_{i}, {{ c}_{i}^{l}<c}_{i}<{c}_{i}^{u}\\ {c}_{i}^{l}, {c}_{i}\le {c}_{i}^{l}\end{array}\right..$$

(19)

Among them, ${c}_{i}^{u}$, ${c}_{i}^{l}$ represent the upper bound and lower bound of the i-th component of individual $c$, respectively. ${{c}{\prime}({c}_{1}{\prime},...,{c}_{n}{\prime})}^{T}$ represents the repaired individual.

Individual importance degree based on the Pareto front layer

When the number of optimization objectives is large, the algorithm will produce Pareto resistance, and the traditional Pareto dominance may fail. Some scholars try to modify the definition of traditional Pareto dominance by using dominance relaxation technology to enhance the selection pressure of the algorithm in terms of convergence. However, with the increase of the number of optimization objectives, the effect of dominance relaxation technology is getting worse and worse. They only focus on the modification of Pareto domination definition, but ignore the difference between objective values. Figure 2 shows 4 non-dominant individuals. Among them, individual B is the closest to Origin O, individual C is second, individual A is third, and individual D is the farthest from Origin O. This means that in the population, individual B is the most important, individual C is the second most important, individual A is the third most important, and individual D is the least important. In addition, we can also find from Fig. 2 that there is a significant difference between the objective values of individual B and the objective values of other individuals, that is, $\sum_{X\in \left\{A,C,D\right\}}\sum_{i=1}^{2}{f}_{i}(B)-{f}_{i}(X)$ is the smallest. This shows that there is a special relationship between the importance of individuals and the difference of the objective values. Based on this discovery, if we can distinguish the importance of different individuals by using the difference between the objective values, we can further create the selection pressure of the algorithm in terms of convergence, and finally Pareto resistance will be eliminated. Therefore, based on Pareto front layer, we propose the concept of “individual importance degree”.

Figure 2

Schematic diagram of individual importance.

Assuming that there are n solutions in a certain Pareto front layer, the objective function values of these solutions are normalized to [0,1] based on the maximum and minimum values of each objective function. ${f}_{k}{\prime}({x}^{i})$ represents the k-th normalized objective function value of individual ${x}^{i}$.

Define the Pareto dominance function

$$PDF\left(x\right)=\left\{\begin{array}{c}1, x<-1\\ {e}^{{-2\left(x+1\right)}^{2}}, -1\le x\le 1.\\ 0, x>1\end{array}\right.$$

(20)

The trend of the Pareto dominance function is shown in Fig. 3.

Figure 3

The trend of the Pareto dominance function.

Pareto dominance function can be used to reflect the dominance degree among different individuals. For example, $PDF({f}_{k}{\prime}({x}^{i})-{f}_{k}{\prime}({x}^{j}))$ represents the dominance degree of individual ${x}^{i}$ to individual ${x}^{j}$ on the k-th objective function; $PDF({f}_{k}{\prime}({x}^{j})-{f}_{k}{\prime}({x}^{i}))$ represents the dominance degree of individual ${x}^{j}$ to individual ${x}^{i}$ on the k-th objective function; Obviously, the greater the dominance degree, the better one individual is than another on one objective function. Therefore, on one objective function, the dominance degree of one individual to another can be expressed as:

$${P}_{k}\left({x}^{i},{x}^{j}\right)=PDF\left({f}_{k}{\prime}\left({x}^{i}\right)-{f}_{k}^{\prime}\left({x}^{j}\right)\right).$$

(21)

On this basis, the dominance degree of one individual to another can be expressed as:

$$P\left({x}^{i},{x}^{j}\right)=\prod_{k=1}^{m}{P}_{k}\left({x}^{i},{x}^{j}\right).$$

(22)

Further, the importance degree of one individual to another can be expressed as:

$$Imp\left({x}^{i},{x}^{j}\right)=\frac{P\left({x}^{i},{x}^{j}\right)}{P\left({x}^{i},{x}^{j}\right)+P\left({x}^{j},{x}^{i}\right)}.$$

(23)

Importance degree can indicate the importance of one individual to another. The greater the importance degree, the more important one individual is than another.

Since a certain Pareto front layer has n solutions, each solution needs to be compared with other n-1 solutions, so as to construct n-1 competing pairs. Assuming that an individual is ${x}^{i}$, then the n-1 competing pairs are $\left({x}^{i},{x}^{1}\right),\left({x}^{i},{x}^{2}\right),...,\left({x}^{i},{x}^{j}\right),...,\left({x}^{i},{x}^{n}\right)$, respectively (note: $i\ne j$). Thus, the importance degree of individual ${x}^{i}$ to the other n-1 individuals is $Imp\left({x}^{i},{x}^{1}\right),Imp\left({x}^{i},{x}^{2}\right),...,Imp\left({x}^{i},{x}^{j}\right),...,Imp\left({x}^{i},{x}^{n}\right)$, respectively (note: $i\ne j$).

Finally, the importance degree of the individual ${x}^{i}$ can be expressed as:

$$Imp\left({x}^{i}\right)=\frac{\sum_{j=1,j\ne i}^{n}Imp\left({x}^{i},{x}^{j}\right)}{n-1}.$$

(24)

The importance degree of one individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer. The greater the importance degree of one individual, the more important it is in the same Pareto front layer.

Figure 4 shows the use of individual importance degree. Firstly, based on a certain Pareto front layer, the competition pools and competition pairs are constructed. Then, the individual importance degree of different individuals is calculated by Formula (24). Finally, the importance of different individuals in the same Pareto front layer is obtained.

Figure 4

The use of individual importance degree.

Taking the two-objective optimization problem as an example, it is assumed that there are 4 non-dominant individuals. They are $A\left(\text{17,5}\right)$, $B\left(\text{9,7}\right)$, $C\left(\text{7,15}\right)$ and $D\left(\text{5,25}\right)$, respectively. It means that these 4 individuals belong to the first non-dominant layer, and their advantages and disadvantages can’t be compared by the non-dominant rank. The distribution of 4 individuals in the objective space is shown in Fig. 5(a).

Figure 5

The distribution of 4 individuals.

In order to better compare the advantages and disadvantages of these 4 individuals, we use the individual importance degree to deal with these 4 individuals. Firstly, the objective function values of these 4 individuals are normalized to [0,1]. After normalization, the coordinates of these 4 individuals are $A\left(\text{1,0}\right)$, $B\left(\text{0.333,0.1}\right)$, $C\left(\text{0.167,0.5}\right)$ and $D\left(\text{0,1}\right)$, respectively. The distribution of 4 individuals in the normalized objective space is shown in Fig. 5(b). Next, according to Fig. 4, the competition pools and competition pairs are constructed. Then, according to formula (22) and formula (23), the $P\left({x}^{i},{x}^{j}\right)$ and $Imp\left({x}^{i},{x}^{j}\right)$ of each competition pair are calculated. The calculation results are shown in Table 1. Finally, according to the formula (24), the importance degree of these 4 individuals is 0.1918, 0.9488, 0.6673 and 0.1921, respectively. The results show that individual B is the most important, individual C is the second most important, individual D is the third most important, and individual A is the least important. This result is consistent with the intuitive perception that we get from Fig. 5(b). Based on the above example, we believe that the concept of individual importance degree and related process are effective and can achieve the desired goals.

Table 1 Competition pools and competition pairs.

Repulsion field method

In the field of microphysics, when the distance between particles is too close, repulsion will push the particles away from their neighbors. On the contrary, when the distance between particles is too great, the repulsion will decrease and the particles tend to be close to the neighboring particles. This way makes the distribution of particles present a state of mutual coordination (As shown in Fig. 6). Based on the characteristics of particle distribution, a repulsion field method is proposed in this paper. The repulsion field is used to maintain the diversity of the population in the objective space, so that the population can be evenly distributed on the real PF.

Figure 6

The uniform distribution of microscopic particles.

Firstly, according to the maximum and minimum values of each objective function, the objective function values of all solutions in the population are normalized to [0,1]. ${f}_{k}{\prime}({x}^{i})$ represents the k-th normalized objective function value of individual ${x}^{i}$.

Then, a repulsion potential field with repulsion radius r is constructed around each individual. Assuming that a repulsion potential field has been constructed for individual ${x}^{i}$, then all individuals within the repulsion potential field will be subject to the repulsion potential from individual ${x}^{i}$. The magnitude of the repulsion potential depends on the distance between other individuals and individual ${x}^{i}$. When other individuals are outside the repulsion potential field of individual ${x}^{i}$, the repulsion potential is 0. When other individuals are within the repulsion potential field of individual ${x}^{i}$, the closer the other individuals are to individual ${x}^{i}$, the greater the repulsion potential that they obtain. Assuming that there is individual ${x}^{j}$, then the repulsion potential that individual ${x}^{j}$ obtains is

$$Repfield\left({x}^{j},{x}^{i}\right)=\left\{\begin{array}{c}\frac{1}{2}\rho {\left(\frac{1}{dis\left({x}^{j},{x}^{i}\right)}-\frac{1}{r}\right)}^{2}, dis\left({x}^{j},{x}^{i}\right)\le r\\ 0, dis\left({x}^{j},{x}^{i}\right)>r\end{array}\right..$$

(25)

Among them, $\rho$ is the gain coefficient of the repulsion potential field, usually set to 1; $r$ is the radius of the repulsion potential field; $dis\left({x}^{j},{x}^{i}\right)$ represents the euclidean distance between individual ${x}^{j}$ and individual ${x}^{i}$ in the objective space. The formula is as follows:

$$dis\left({x}^{j},{x}^{i}\right)=\sqrt{\sum_{k=1}^{m}{\left({f}_{k}^{\prime}\left({x}^{j}\right)-{f}_{k}^{\prime}\left({x}^{i}\right)\right)}^{2}}.$$

(26)

Further, the repulsion $Rep\left({x}^{j},{x}^{i}\right)$ that individual ${x}^{j}$ obtains is the negative gradient of the repulsion potential $Repfield\left({x}^{j},{x}^{i}\right)$. The formula is as follows:

$$Rep\left({x}^{j},{x}^{i}\right)=-\nabla Repfield\left({x}^{j},{x}^{i}\right)=-\frac{d\left(Repfield\left({x}^{j},{x}^{i}\right)\right)}{d\left(dis\left({x}^{j},{x}^{i}\right)\right)}=\left\{\begin{array}{c}\rho \left(\frac{1}{dis\left({x}^{j},{x}^{i}\right)}-\frac{1}{r}\right)\frac{1}{{\left(dis\left({x}^{j},{x}^{i}\right)\right)}^{2}}, dis\left({x}^{j},{x}^{i}\right)\le r\\ 0, dis\left({x}^{j},{x}^{i}\right)>r\end{array}\right..$$

(27)

It means that when $dis\left({x}^{j},{x}^{i}\right)\le r$, the smaller $dis\left({x}^{j},{x}^{i}\right)$ is, the larger $Rep\left({x}^{j},{x}^{i}\right)$ is. when $dis\left({x}^{j},{x}^{i}\right)>r$, $Rep\left({x}^{j},{x}^{i}\right)=0$.

Based on the repulsion potential field, the total repulsion potential that individual ${x}^{j}$ obtains is

$$Repfield\left({x}^{j}\right)=\sum_{i=1,i\ne j}^{N}Repfield\left({x}^{j},{x}^{i}\right).$$

(28)

Finally, the total repulsion that individual ${x}^{j}$ obtains is

$$Rep\left({x}^{j}\right)=-\nabla Repfield\left({x}^{j}\right)=\sum_{i=1,i\ne j}^{N}Rep\left({x}^{j},{x}^{i}\right).$$

(29)

It should be noted that the repulsion potential and repulsion proposed in this paper are both vectors. It means that repulsion potential and repulsion have both magnitude and direction. The addition of different repulsion is the vector synthesis of repulsion, rather than the pure numerical addition. This is also an obvious feature that the repulsion field method is different from other scalar function methods (such as niche method). Figure 7 shows the vector synthesis process of repulsion in a two-dimensional space environment. Among them, F_SUM is the total repulsion that individual A obtains; F_BA is the repulsion generated by individual B to individual A; F_CA is the repulsion generated by individual C to individual A.

Figure 7

The vector synthesis process of repulsion.

In the repulsion field method, the individual with large repulsion usually means that the individual is located in the multiple repulsion potential field that other individuals construct. It indicates that the individual is located in a dense area in the objective space and is close to other individuals. Therefore, individuals with large repulsion aren’t conducive to maintaining population diversity. Naturally, we hope that individuals with large repulsion can move away from dense areas in the objective space along the direction of repulsion. Based on this idea, firstly, we need to find some individuals closest to the direction of the repulsion to construct a high-quality solution set. Then, the feature extraction operator is used to extract the location features of the high-quality solution set. Finally, based on these features, individuals with large repulsion can evolve along the direction of repulsion. As shown in Fig. 8, individual D and individual E are the individuals closest to the direction of repulsion. The feature extraction operator is used to extract the position features of these two individuals. Based on these features, individual A evolves into individual A*, which is far away from the previous dense area.

Figure 8

Individual A is far away from the dense area.

It should be noted that the feature extraction operator has the randomness caused by Gaussian sampling. Therefore, the evolution of individuals also has a certain degree of randomness.

Framework of MOEA/TS algorithm

The framework of the MOEA/TS algorithm is shown in Fig. 9. Firstly, the relevant parameters of the algorithm are initialized; secondly, judge which state the algorithm is in. If the algorithm is in the convergence maintenance state, the following steps are adopted to improve the convergence of the algorithm: (1) Randomly select the parent individual. (2) Use feature extraction operator to generate offspring individuals. (3) If the offspring individual is superior to the individual with the worst convergence in the population, the worst individual is replaced by the offspring individual. If the algorithm is in the diversity maintenance state, the following steps are adopted to improve the diversity of the algorithm: (1) Select the individual with the worst diversity in the population. (2) Use feature extraction operator to generate offspring individuals. (3) If the offspring individual is superior to the individual with the worst diversity in the population, the worst individual is replaced by the offspring individual. If the algorithm is in the coordination state, the following steps are adopted to coordinate the convergence and diversity of the algorithm: (1) Randomly select the parent individual. (2) Use the Gaussian mutation operator to generate offspring individuals. (3) If the offspring individual is superior to the parent individual in convergence and diversity, the parent individual is replaced by the offspring individual. Then, it is judged whether the algorithm has completed the i-th iteration. If the algorithm doesn’t complete the i-th iteration, the corresponding maintenance step or coordination step is re-executed. If the algorithm completes the i-th iteration, the current state of the algorithm is updated. Finally, it is judged whether the algorithm ends. If the algorithm doesn’t end, the corresponding maintenance step or coordination step is performed according to the current state of the algorithm. If the algorithm is finished, the population is output.

Figure 9

The framework of MOEA/TS algorithm.

Description of MOEA/TS algorithm

Main framework

This section describes the main framework of the MOEA/TS algorithm. The pseudo-code of the main framework of the MOEA/TS algorithm is shown in Algorithm 1. The main steps include: in line (1), initializing population P, repulsion field radius r, and state value (state=1 means that the algorithm is in convergence maintenance state, state=2 means that the algorithm is in diversity maintenance state, and state=3 means that the algorithm is in coordination state.); In line (2), the Front value, Imp value and Rep value of each solution in population P are calculated (The Front value is calculated by the fast non-dominated sorting method.); In line (3), it is judged whether the algorithm meets the termination condition (The termination condition is usually the maximum iterations.); In line (4), the count value is initialized. The count value is used to count the number of updated solutions in the i-th iteration; in lines (5)-(11), according to the current state of the algorithm, the update way of the population is selected. When state=1, the convergence of the population is updated. When state=2, the diversity of the population is updated. When state=3, the convergence and diversity of the population are coordinated; in line (12), the state value of the algorithm is updated according to the current state of the algorithm and the count value.

Algorithm 1

Main framework.

Convergence maintenance

This section mainly describes the convergence maintenance of the population. The pseudo-code of convergence maintenance is shown in Algorithm 2. The main steps include: in line (1), the algorithm enters the i-th iteration; In line (2), one parent individual is randomly selected from population P; In lines (3)–(4), based on Front, Imp, the high-quality solution set S is constructed; In line (5), the feature extraction operator is used to extract the features of the high-quality solution set S, and then assist the evolution of the parent individual; In line (6), the individual with the worst convergence in the population is found; In lines (7)–(13), if the offspring individual is superior to the individual with the worst convergence in the population, the worst individual is replaced by the offspring individual and flag is marked as 1. If the offspring individual is inferior to the individual with the worst convergence in the population, the flag is marked as 0. Among them, the flag value is used to indicate whether the population P has changed (flag=0 means that the population P hasn’t changed, flag=1 means that the population P has changed.); In lines (14)–(16), it is judged whether flag equals 1. If flag equals 1, the count value is updated, and the Front value, Imp value and Rep value of each solution in population P are updated.

Algorithm 2

Convergence maintenance.

Diversity maintenance

This section mainly describes the diversity maintenance of the population. The pseudo-code of diversity maintenance is shown in Algorithm 3. The main steps include: in line (1), the algorithm enters the i-th iteration; In line (2), the individual with the worst diversity in the population is found; In line (3), according to the direction of total repulsion that the worst individual obtains, the distance ${dis}_{j}$ can be calculated; In line (4), based on ${dis}_{j}$, the high-quality solution set S is constructed; In line (5), the feature extraction operator is used to extract the features of the high-quality solution set S, and then assist the evolution of the worst individual; In lines (6)–(12), if the offspring individual is superior to the individual with the worst diversity in the population, the worst individual is replaced by the offspring individual and flag is marked as 1. If the offspring individual is inferior to the individual with the worst diversity in the population, the flag is marked as 0; In lines (13)–(15), it is judged whether flag equals 1. If flag equals 1, the count value is updated, and the Front value, Imp value and Rep value of each solution in population P are updated.

Algorithm 3

Diversity maintenance.

Coordination of convergence and diversity

This section mainly describes the coordination of convergence and diversity of the population. The pseudo-code of coordination of convergence and diversity is shown in Algorithm 4. The main steps include: in line (1), the algorithm enters the i-th iteration; In line (2), one parent individual is randomly selected from population P; In line (3), based on the parent individual, the Gaussian mutation operator is used to generate the offspring solution; In lines (4)–(10), if the offspring individual is superior to the parent individual in convergence and diversity, the parent individual is replaced by the offspring individual and flag is marked as 1. If the offspring individual is inferior to the parent individual in convergence and diversity, the flag is marked as 0; In lines (11)–(13), it is judged whether flag equals 1. If flag equals 1, the count value is updated, and the Front value, Imp value and Rep value of each solution in population P are updated.

Algorithm 4

Coordination.

Feature extraction operator

This section mainly describes the feature extraction operator. The pseudo-code of the feature extraction operator is shown in Algorithm 5. The main steps include: in line (1), the features $\overline{x},L$ of the high-quality solution set S are extracted by formula (4) and formula (13); In line (2), the sampling vector $s={\left({s}_{1},...,{s}_{n}\right)}^{T}$ is generated by the Gaussian distribution $N\left(\text{0,0.7}\right)$; In line (3), based on $\text{s},\overline{x},L$, the feature solution ${x}^{feature}$ is generated by formula (14); In line (4), based on parent, ${x}^{feature}$, the offspring solution O is generated by formula (15).

Algorithm 5

Feature extraction.

Update of algorithm state

In this paper, the algorithm state is further updated according to the current state of the algorithm and the stability of the population. The pseudo-code of the update of the algorithm state is shown in Algorithm 6. When the algorithm is in the convergence maintenance state and the number of updated solutions in the i-th iteration is less than or equal to 5%*N, it is considered that the population tends to be stable in terms of convergence, then the algorithm turns to the diversity maintenance state; When the algorithm is in the diversity maintenance state and the number of updated solutions in the i-th iteration is less than or equal to 5%*N, it is considered that the population tends to be stable in terms of diversity, then the algorithm turns to the coordination state; When the algorithm is in the coordination state and the number of updated solutions in the i-th iteration is less than or equal to 5%*N, it is considered that the population tends to be stable in terms of coordination, then the algorithm turns to the convergence maintenance state. It should be noted that the threshold value T is a key parameter in measuring whether the population tends to be stable or not. In this paper, the threshold value T is set to 5%. The parameter analysis verifies that 5% is a reasonable threshold. For more details on parameter analysis, please browse the experiment chapter (Parameter sensitivity analysis section).

Algorithm 6

Determination state.

Computational complexity of one iteration of MOEA/TS algorithm

Assuming that the size of the population is N, the number of the objective function is m, the dimension of the decision variable is n, and the size of the high-quality solution set is W, then the computational complexity of Rep is O(mN2), the computational complexity of Front is O(mN2), and the computational complexity of Imp is O(mN2). The core steps of the feature extraction operator (Algorithm 5) include the construction of the covariance matrix and Cholesky decomposition. The computational complexity of covariance matrix construction is O(Wn2) and the computational complexity of Cholesky decomposition is O(n3/6). Therefore, the computational complexity of the feature extraction operator (Algorithm 5) is O(Wn2+n3/6). The core steps of convergence maintenance (Algorithm 2) include population ranking, feature extraction operator, selection of the worst individual, and updating of Front, Imp and Rep. Their computational complexity is O(N2), O(Wn2+n3/6), O(N), O(mN2), O(mN2), O(mN2), respectively. Therefore, the computational complexity of convergence maintenance (Algorithm 2) is O(N(N2+Wn2+n3/6+N+3mN2)). The core steps of diversity maintenance (Algorithm 3) include selection of the worst individual, distance calculation, population ranking, feature extraction operator, and updating of Front, Imp and Rep. Their computational complexity is O(N), O(nN), O(N2), O(Wn2+n3/6), O(mN2), O(mN2), O(mN2), respectively. Therefore, the computational complexity of diversity maintenance (Algorithm 3) is O(N(N+nN+N2+Wn2+n3/6+3mN2)). The core steps of coordination of convergence and diversity (Algorithm 4) include the Gaussian mutation operator, and updating of Front, Imp and Rep. Their computational complexity is O(n), O(mN2), O(mN2), O(mN2), respectively. Therefore, the computational complexity of coordination of convergence and diversity (Algorithm 4) is O(N(n+3mN2)). The computational complexity of Determination State (Algorithm 6) is O(1). Based on the above computational complexity analysis, the computational complexity of one iteration of the MOEA/TS algorithm is max{O(N(N2+Wn2+n3/6+N+3mN2)), O(N(N+nN+N2+Wn2+n3/6+3mN2)), O(N(n+3mN2))}+O(1)≈max{O(NWn2+Nn3+mN3), O(NWn2+Nn3+mN3), O(mN3)}= O(NWn2+Nn3+mN3). In this paper, N>>max{W, n, m}. Therefore, the computational complexity of the MOEA/TS algorithm is O(mN3). As a reference algorithm, the computational complexity of the NSGA-III algorithm is O(mN2). The computational complexity of the MOEA/TS algorithm is an order of magnitude higher than that of the NSGA-III algorithm. This shows that the MOEA/TS algorithm is an expensive many-objective optimization algorithm.

It should be noted that although MOEA/TS algorithm has a higher computational complexity. But compared with the NSGA-III algorithm, the MOEA/TS algorithm also has greater advantages. In terms of convergence optimization, the NSGA-III algorithm adopts the traditional definition of Pareto domination. Obviously, the traditional definition can’t solve the problem of Pareto resistance. MOEA/TS algorithm uses the concept of “individual importance degree”. Individual importance degree can solve the problem of Pareto resistance. In terms of diversity optimization, the NSGA-III algorithm uses predefined reference points. The predefined reference points can’t solve the problem that the population can't be evenly distributed on the real PF in the high-dimensional objective space. MOEA/TS algorithm uses the repulsion field method. The repulsion field method can solve the problem that the population can’t be evenly distributed on the real PF in the high-dimensional objective space. In terms of algorithm architecture, the NSGA-III algorithm adopts the serial algorithm architecture. The serial algorithm architecture is difficult to coordinate the convergence optimization and diversity optimization of the algorithm. MOEA/TS algorithm adopts the concurrent algorithm architecture. The concurrent algorithm architecture can coordinate the convergence optimization and diversity optimization of the algorithm. In terms of operators, the NSGA-III algorithm uses the traditional binary crossover operator and polynomial mutation operator. The evolutionary ability of these two operators is weak. MOEA/TS algorithm uses feature extraction operator. Feature extraction operator has strong evolutionary ability. Therefore, the MOEA/TS algorithm has better performance than the NSGA-III algorithm. The comparison results support our conclusion. For the comparison results of these two algorithms, please browse Supplementary Information Document.

Experimental results and analysis

Experimental settings

Configuration of experimental software and hardware

The hardware and software configurations of the experiment are shown in Table 2. Among them, PlatEMO³⁰ is a professional many-objective optimization experiment platform. The platform includes multiple test function sets and many-objective optimization algorithms.

Table 2 Software and hardware configurations.

Test function

The test functions used in the experiment include: DTLZ test function set (DTLZ1-7), MAF test function set (MAF1-6) and WFG test function set (WFG1-9). Literature³¹ describes the characteristics of related test functions. The parameter settings of the related test functions are shown in Table 3.

Table 3 Parameter settings of the related test functions.

Comparison algorithm

In order to verify the performance of MOEA/TS algorithm in the many-objective optimization field, this paper compares MOEA/TS algorithm with 7 advanced many-objective optimization algorithms. These 7 many-objective optimization algorithms include: VMEF³², BiGE-BEW³³, MOEA/DG³⁴, MOEA/D³⁵, LSMaODE³⁶, MaOEA/IT²³ and MaOEA/IGD³⁷.

For all test cases, Wilcoxon rank sum test at 5% significance level³⁸ is used to compare the significance of the difference between the MOEA/TS algorithm and the comparison algorithms. The symbol “+” indicates that the comparison algorithms are significantly better than the MOEA/TS algorithm; the symbol “-“indicates that the comparison algorithms are significantly inferior to the MOEA/TS algorithm. The symbol “=” indicates that there is no significant difference between the MOEA/TS algorithm and the comparison algorithms.

Performance evaluation

In the aspect of performance evaluation, this paper uses inverted generational distance plus (IGD+) and hypervolume (HV)³⁹ to measure the performance of many-objective optimization algorithm. The smaller the IGD+ value that the algorithm obtains, the better the performance of the algorithm. The larger the HV value that the algorithm obtains, the better the performance of the algorithm.

In order to facilitate observation, we provide the normalized HV value of each algorithm relative to the best HV result. This normalization makes all the results lie in the range [0,1], and 1 represents the best value.

Considering the length of the paper, we only show the IGD+ values of different algorithms in the experiment chapter. For the HV values of different algorithms, please browse the Supplementary Information Document.

Parameter setting

In terms of algorithm parameters, according to some existing parameter research results^13,40, the feature factor $\mu$ is set to 20 in this paper. According to the parameter sensitivity analysis, the number of high-quality solutions W is set to 9 in this paper. The parameter sensitivity analysis of W is detailed in the subsequent chapters.

The algorithm parameters of the 7 comparison algorithms are determined according to the best parameters provided by the corresponding literature.

Performance comparison under benchmark test functions

Performance comparison under DTLZ test function set

In this paper, each algorithm is executed 30 times to get the average data as shown in Table 4. As can be seen from Table 4, MOEA/TS algorithm wins the first place in 15 test cases; BiGE-BEW algorithm wins the first place in 5 test cases; MOEA/D algorithm wins the first place in 15 test cases. In the 35 test cases, the number of MOEA/TS algorithm is significantly superior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 21, 27, 25, 16, 32, 35 and 31, respectively. The number of MOEA/TS algorithm is significantly inferior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 6, 5, 5, 15, 1, 0 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 8, 3, 5, 4, 2, 0 and 4, respectively. Therefore, in the DTLZ test function set, MOEA/TS algorithm and MOEA/D algorithm have the best performance. The performance of VMEF algorithm, MOEA/DG algorithm, BiGE-BEW algorithm and LSMaODE algorithm decreases in turn. The performance of MaOEA/IGD algorithm and MaOEA/IT algorithm is similar and the worst.

Table 4 The IGD + values of each algorithm under DTLZ test function set.

Based on Table 4, we further analyze the performance of these algorithms. In the DTLZ test function set, MOEA/TS algorithm performs poorly on DTLZ1, DTLZ5 and DTLZ6 test functions. The possible reasons are that the DTLZ1 test function has multiple local optima, and the DTLZ5 and DTLZ6 test functions have a narrow convergence curve. In the DTLZ1 test function, although the repulsion field method of the MOEA/TS algorithm makes the population widely distributed. However, its population distribution isn’t uniform and regular. The population distribution of some algorithms using predefined weight vectors is uniform and regular. In the DTLZ5 and DTLZ6 test functions, the coordination mechanism of MOEA/TS algorithm fails. The narrow convergence curve makes the population more concentrated, but the repulsion field method will disperse the population. The coordination mechanism is difficult to play a role.

The real Pareto front of DTLZ test function set is regular and the function complexity isn’t high. Therefore, algorithms with better diversity may be more popular. MOEA/D algorithm uses predefined weight vectors to maintain diversity and aggregation functions to maintain convergence. Therefore, it has good performance. VMEF algorithm uses different convergence ranking methods to deal with different test problems. Therefore, VMEF algorithm is good in convergence and poor in diversity. Based on the convergence measure and diversity measure, BiGE-BEW algorithm transforms the many-objective optimization problem into a two-objective optimization problem. In theory, the algorithm should perform well. However, there are defects in its convergence and diversity measurement formula. Finally, the experimental results of the algorithm aren’t as good as the expected results. MOEA/DG algorithm still uses the traditional dominance relationship to maintain the convergence of external archives. Therefore, MOEA/DG algorithm is poor in convergence and good in diversity. LSMaODE algorithm divides the population into two subpopulations and uses different strategies to optimize them. Because the real Pareto front of DTLZ test function set isn’t complex, the advantage of this multi-population algorithm architecture isn’t obvious. Therefore, compared with other algorithms, its performance is mediocre. MaOEA/IT algorithm optimizes convergence and diversity through two independent phases. However, the algorithm's performance is always poor because it doesn’t alleviate the contradiction between convergence and diversity. The reference Pareto front of MaOEA/IGD algorithm is poor. Therefore, the algorithm’s performance is always poor.

Performance comparison under MAF test function set

In this paper, each algorithm is executed 30 times to get the average data as shown in Table 5. As can be seen from Table 5, MOEA/TS algorithm wins the first place in 10 test cases; BiGE-BEW algorithm wins the first place in 8 test cases; MOEA/DG algorithm wins the first place in 2 test cases; MOEA/D algorithm wins the first place in 5 test cases; LSMaODE algorithm wins the first place in 5 test cases. In the 30 test cases, the number of MOEA/TS algorithm is significantly superior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 22, 18, 25, 21, 20, 27 and 30, respectively. The number of MOEA/TS algorithm is significantly inferior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 6, 11, 2, 5, 9, 1 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 2, 1, 3, 4, 1, 2 and 0, respectively. Therefore, in the MAF test function set, MOEA/TS algorithm has the best performance. The performance of BiGE-BEW algorithm, LSMaODE algorithm, VMEF algorithm, MOEA/D algorithm, MOEA/DG algorithm and MaOEA/IT algorithm decreases in turn. The performance of MaOEA/IGD algorithm is the worst.

Table 5 The IGD + values of each algorithm under MAF test function set.

Based on Table 5, we further analyze the performance of these algorithms. In the MAF test function set, MOEA/TS algorithm performs poorly on MAF2 and MAF3 test functions. The possible reasons are that the MAF2 test function greatly increases the difficulty of convergence on the basis of the DTLZ2 test function, and the MAF3 test function has a convex Pareto front and many local fronts. In the MAF2 test function, although the MOEA/TS algorithm can recognize the advantage and disadvantage of different individuals in the same front layer, the evolutionary efficiency of the MOEA/TS algorithm isn’t ideal. In other words, after the algorithm is finished, the population still has the large evolution potential in convergence. In the MAF3 test function, MOEA/TS algorithm can effectively deal with the convex Pareto front. However, MOEA/TS algorithm is difficult to deal with multiple local fronts because feature extraction operator of MOEA/TS algorithm is difficult to extract features of multiple local fronts.

MAF test function set is the variety of DTLZ test function set. It adds a lot of characteristics to the DTLZ test function set. For example, degenerate, convex, concave, partial, multimodal, deceptive, et al. Therefore, the MAF test function set is more difficult in terms of convergence and diversity. Based on the convergence measure and diversity measure, BiGE-BEW algorithm transforms the many-objective optimization problem into a two-objective optimization problem. Although there are some defects in its diversity and convergence measurement formula, BiGE-BEW algorithm shows good performance in convergence when dealing with more complex MaOPs. VMEF algorithm uses different convergence ranking methods to deal with different test problems. However, the complex Pareto fronts and diversified characteristics still pose a great challenge to VMEF algorithm. Therefore, the performance of VMEF algorithm is mediocre. MOEA/DG algorithm still uses the traditional dominance relationship to maintain the convergence of external archives. Therefore, MOEA/DG algorithm is poor in convergence. MOEA/D algorithm uses predefined weight vectors to maintain diversity and aggregation functions to maintain convergence. MOEA/D algorithm can easily deal with the DTLZ test function set. However, its performance isn’t ideal when dealing with more complex MAF test function set. Surprisingly, LSMaODE algorithm shows good performance. We speculate that the possible reason is that the real Pareto front of the MAF test function set is complex, and then the advantages of multi-population algorithm architecture can be reflected. MaOEA/IT algorithm optimizes convergence and diversity through two independent phases. However, the algorithm’s performance is always poor because it doesn’t alleviate the contradiction between convergence and diversity. The reference Pareto front of MaOEA/IGD algorithm is poor. Therefore, the algorithm’s performance is always poor.

Performance comparison under WFG test function set

In this paper, each algorithm is executed 30 times to get the average data as shown in Table 6. As can be seen from Table 6, MOEA/TS algorithm wins the first place in 27 test cases; VMEF algorithm wins the first place in 8 test cases; BiGE-BEW algorithm wins the first place in 6 test cases; MOEA/DG algorithm wins the first place in 1 test case; LSMaODE algorithm wins the first place in 3 test cases. In the 45 test cases, the number of MOEA/TS algorithm is significantly superior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 26, 29, 42, 45, 39, 45 and 43, respectively. The number of MOEA/TS algorithm is significantly inferior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 10, 9, 3, 0, 3, 0 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 9, 7, 0, 0, 3, 0 and 2, respectively. Therefore, in the WFG test function set, MOEA/TS algorithm has the best performance. The performance of VMEF algorithm, BiGE-BEW algorithm, LSMaODE algorithm and MOEA/DG algorithm decreases in turn. The performance of MaOEA/IGD algorithm, MOEA/D algorithm and MaOEA/IT algorithm is similar and the worst.

Table 6 The IGD + values of each algorithm under WFG test function set.

Based on Table 6, we further analyze the performance of these algorithms. MOEA/TS algorithm performs well in all WFG test functions. The possible reason is that the problem characteristics of the WFG test function set are bias, fraud and degradation. The WFG test function set is more difficult than the DTLZ test function set. However, the problem characteristics of the WFG test function set don’t include multiple local fronts (From the previous analysis, we know that MOEA/TS algorithm isn’t good at dealing with multiple local fronts.). MOEA/TS algorithm can deal with these problem characteristics. Therefore, MOEA/TS algorithm performs well in all WFG test functions. It should be noted that the WFG3 test function has a narrow convergence curve, but the performance of MOEA/TS algorithm is still the best. This is an interesting phenomenon. Because from the previous analysis, we know that MOEA/TS algorithm isn’t good at dealing with test functions with narrow convergence curves (such as DTLZ5 and DTLZ6 test functions). Based on the convergence difficulty of the WFG test function set, we speculate that the performance of the other 7 algorithms is worse, thus highlighting the performance of MOEA/TS algorithm.

Compared with the DTLZ test function set, the MAF test function set is more difficult in terms of convergence and diversity. VMEF algorithm uses different convergence ranking methods to deal with different test problems. This approach helps VMEF algorithm to deal with different problem characteristics. Therefore, the performance of VMEF algorithm is good. Based on the convergence measure and diversity measure, BiGE-BEW algorithm transforms the many-objective optimization problem into a two-objective optimization problem. Although there are some defects in its diversity and convergence measurement formula, BiGE-BEW algorithm shows good performance in convergence when dealing with more complex MaOPs. MOEA/DG algorithm still uses the traditional dominance relationship to maintain the convergence of external archives. Therefore, MOEA/DG algorithm is poor in convergence. MOEA/D algorithm uses predefined weight vectors to maintain diversity and aggregation functions to maintain convergence. This approach isn’t suitable for dealing with test functions with bias characteristic. Therefore, the performance of MOEA/D algorithm is the worst. LSMaODE algorithm divides the population into two subpopulations and uses different strategies to optimize them. Because most WFG test functions have bias characteristic, LSMaODE algorithm doesn’t consider the bias problem. Therefore, the performance of LSMaODE algorithm is mediocre. MaOEA/IT algorithm optimizes convergence and diversity through two independent phases. However, the algorithm’s performance is always poor because it doesn’t alleviate the contradiction between convergence and diversity. The reference Pareto front of MaOEA/IGD algorithm is poor. Therefore, the algorithm’s performance is always poor.

Comparison and analysis

By synthesizing Tables 4, 5, 6, we can obtain the data shown in Table 7. As can be seen from Tables 4, 5, 6, MOEA/TS algorithm wins the first place in 52 test cases; VMEF algorithm wins the first place in 8 test cases; BiGE-BEW algorithm wins the first place in 19 test cases; MOEA/DG algorithm wins the first place in 3 test cases; MOEA/D algorithm wins the first place in 20 test cases; LSMaODE algorithm wins first place in 8 test cases. As can be seen from Table 7, in the 110 test cases, the number of MOEA/TS algorithm is significantly superior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 69, 74, 92, 82, 91, 107 and 104, respectively. The number of MOEA/TS algorithm is significantly inferior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 22, 25, 10, 20, 13, 1 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 19, 11, 8, 8, 6, 2 and 6, respectively. Based on the above data, we can get the following conclusions: MOEA/TS algorithm has the best performance; the performance of BiGE-BEW algorithm, VMEF algorithm, MOEA/D algorithm, LSMaODE algorithm, MOEA/DG algorithm and MaOEA/IT algorithm decreases in turn. MaOEA/IGD algorithm has the worst performance.

Table 7 Comprehensive data.

In addition to the above conclusions, we can also observe 3 interesting phenomena:

• (1) In the MAF test function set and WFG test function set, MOEA/TS algorithm has no competitors. However, in the DTLZ test function set, MOEA/TS algorithm and MOEA/D algorithm are competitors, and they have similar performance. This is because most DTLZ test functions have regular PF, while most MAF test functions and WFG test functions have more complex PF. It can be seen from Sect. "Introduction" that MOEA/D algorithm is suitable for MaOPs with regular PF. Therefore, in the DTLZ test function set, MOEA/D algorithm can compete with MOEA/TS algorithm. In the MAF test functions and WFG test functions, only MOEA/TS algorithm shows excellent performance.

• (2) The performance of MOEA/TS algorithm is better on the test cases with 10 objectives, 15 objectives and 20 objectives. The performance of MOEA/TS algorithm is relatively ordinary on the test cases with 5 objectives and 8 objectives. This is because when the number of optimization objectives is small, most many-objective optimization algorithms perform well. Compared with other many-objective optimization algorithms, the advantages of MOEA/TS algorithm aren’t obvious. However, with the increase of the number of optimization objectives, the performance of other many-objective optimization algorithms becomes worse and worse. In contrast, the performance of MOEA/TS algorithm isn’t significantly affected. Therefore, compared with other many-objective optimization algorithms, MOEA/TS algorithm has obvious advantages. This shows that MOEA/TS algorithm is more suitable for solving MaOPs with more than 10 objectives.

• (3) Without considering MOEA/TS algorithm, MOEA/D algorithm has the best performance in the DTLZ test function set. BiGE-BEW algorithm has the best performance in the MAF test function set. VMEF algorithm has the best performance in the WFG test function set. This shows that different many-objective optimization algorithms are suitable for different test function sets. However, MOEA/TS algorithm can show excellent performance on three test function sets. This indicates that MOEA/TS algorithm has strong universality and applicability.

Distribution diagram of solutions in the objective space

In order to describe the distribution of solutions in the high-dimensional objective space more intuitively, this paper draws the distribution diagram of solutions in the objective space. Considering the length of the paper, it is unrealistic to show the distribution diagrams of all test functions. Therefore, this section only shows the distribution diagrams of 3 representative test cases. These 3 test cases are DTLZ2 test case with 20 objectives, MAF1 test case with 15 objectives and WFG3 test case with 10 objectives, respectively.

Figure 10 shows the distribution diagrams of each algorithm on DTLZ2 test case with 20 objectives. It can be seen from Fig. 10 that distribution diagrams of MOEA/TS algorithm, BiGE-BEW algorithm, MOEA/DG algorithm and MOEA/D algorithm are similar, which indicates that these 4 algorithms are excellent in convergence and diversity; VMEF algorithm and LSMaODE algorithm are good in diversity, but poor in convergence; MaOEA/IT algorithm and MaOEA/IGD algorithm are very poor in convergence and diversity.

Figure 10

Distribution diagrams of each algorithm on DTLZ2 test case with 20 objectives.

Figure 11 shows the distribution diagrams of each algorithm on MAF1 test case with 15 objectives. It can be seen from Fig. 11 that MOEA/TS algorithm and VMEF algorithm are good in convergence, but poor in diversity; BiGE-BEW algorithm and LSMaODE algorithm are good in diversity, but poor in convergence. MOEA/DG algorithm, MOEA/D algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm are very bad in convergence and diversity.

Figure 11

Distribution diagrams of each algorithm on MAF1 test case with 15 objectives.

Figure 12 shows the distribution diagrams of each algorithm on WFG3 test case with 10 objectives. It can be seen from Fig. 12 that MOEA/TS algorithm has the best convergence and diversity; LSMaODE algorithm is also excellent, only slightly worse than MOEA/TS algorithm in terms of diversity; BiGE-BEW algorithm and MOEA/DG algorithm are good in diversity, but poor in convergence. VMEF algorithm is good in convergence, but poor in diversity. MOEA/D algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm are very bad in convergence and diversity.

Figure 12

Distribution diagrams of each algorithm on WFG3 test case with 10 objectives.

Evolution curve analysis of the algorithm

This section takes DTLZ2 test case with 20 objectives, MAF1 test case with 15 objectives and WFG3 test case with 10 objectives as examples to display the evolution curves of 8 algorithms (as shown in Figs. 13, 14, 15).

Figure 13

Evolution curve of each algorithm on DTLZ2 test case with 20 objectives.

In Figure 13, in terms of the final IGD+ value of the algorithm, MOEA/TS algorithm has the smallest IGD+ value, while the IGD+ values of MOEA/DG algorithm, BiGE-BEW algorithm, MOEA/D algorithm, LSMaODE algorithm, VMEF algorithm and MaOEA/IGD algorithm successively increase, and MaOEA/IT algorithm has the largest IGD+ value. This shows that MOEA/TS algorithm has the best convergence and diversity within the specified number of iterations. In terms of the evolution of the algorithm, the final IGD+ values of all algorithms are smaller than the initial IGD+ values. This shows that all algorithms have strong evolution ability, especially MOEA/TS algorithm has the strongest evolution ability. In terms of algorithm fluctuation, MaOEA/IT algorithm fluctuates greatly. This shows that MaOEA/IT algorithm isn’t stable. Based on the above analysis, we believe that MOEA/TS algorithm has the best comprehensive performance on DTLZ2 test case with 20 objectives, and is suitable for solving DTLZ2 test problem with 20 objectives.

In Figure 14, in terms of the final IGD+ value of the algorithm, MOEA/TS algorithm has the smallest IGD+ value, while the IGD+ values of BiGE-BEW algorithm, VMEF algorithm, LSMaODE algorithm, MaOEA/IGD algorithm, MOEA/D algorithm and MOEA/DG algorithm successively increase, and MaOEA/IT algorithm has the largest IGD+ value. This shows that MOEA/TS algorithm has the best convergence and diversity within the specified number of iterations. In terms of the evolution of the algorithm, the final IGD+ values of all algorithms are smaller than the initial IGD+ values. This shows that all algorithms have strong evolution ability, especially MOEA/TS algorithm has the strongest evolution ability. In terms of algorithm fluctuation, MaOEA/IT algorithm fluctuates greatly. This shows that MaOEA/IT algorithm isn’t stable. Based on the above analysis, we believe that MOEA/TS algorithm has the best comprehensive performance on MAF1 test case with 15 objectives, and is suitable for solving MAF1 test problem with 15 objectives.

Figure 14

Evolution curve of each algorithm on MAF1 test case with 15 objectives.

In Fig. 15, in terms of the final IGD+ value of the algorithm, MOEA/TS algorithm has the smallest IGD+ value, while the IGD+ values of LSMaODE algorithm, MOEA/DG algorithm, VMEF algorithm, BiGE-BEW algorithm, MaOEA/IGD algorithm and MOEA/D algorithm successively increase, and MaOEA/IT algorithm has the largest IGD+ value. This shows that MOEA/TS algorithm has the best convergence and diversity within the specified number of iterations. In terms of the evolution of the algorithm, the final IGD+ values of the MaOEA/IT algorithm, VMEF algorithm, MaOEA/IGD algorithm, BiGE-BEW algorithm and VMEF algorithm are all greater than the initial IGD+ values. This shows that the performance of these 5 algorithms deteriorates during evolution, and they aren’t suitable for dealing with WFG3 test problem with 10 objectives. The initial IGD+ value of MOEA/DG algorithm is close to the final IGD+ value, and the IGD+ value of MOEA/DG algorithm fluctuates little during the evolution. This shows that MOEA/DG algorithm is insensitive to evolution. Only the final IGD+ values of LSMaODE algorithm and MOEA/TS algorithm are less than the initial IGD+ values. This shows that LSMaODE algorithm and MOEA/TS algorithm have strong evolution ability, especially MOEA/TS algorithm has the strongest evolution ability. In terms of algorithm fluctuation, MOEA/D algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm have greater fluctuation. This shows that these 3 algorithms aren’t stable. Based on the above analysis, we believe that MOEA/TS algorithm has the best comprehensive performance on WFG3 test case with 10 objectives, and is suitable for solving WFG3 test problem with 10 objectives.

Figure 15

Evolution curve of each algorithm on WFG3 test case with 10 objectives.

In addition, we can also observe an interesting phenomenon from Fig. 13 to Fig. 15: the IGD+ values of some algorithms sometimes increase significantly with the increase of iterations. That is, the performance of some algorithms sometimes deteriorates seriously with the increase of iterations. The reasons for this phenomenon may include three aspects: (1) The algorithm doesn’t adopt the elite preservation strategy. Some high-quality solutions may gradually disappear; (2) Due to the complexity of the optimization problems, the evolutionary direction of the population may be misled by some pseudo-elite individuals; (3) The convergence optimization and diversity optimization of the algorithm aren’t coordinated. The optimization of convergence may affect the optimization of diversity or the optimization of diversity may affect the optimization of convergence. It can be seen from the pseudo-code of the algorithm in Section 3.5 that the MOEA/TS algorithm proposed in this paper considers the above three aspects. Therefore, MOEA/TS algorithm can effectively alleviate this phenomenon.

Effectiveness verification of innovation part

In order to verify the effectiveness of the innovative parts, 4 variants are designed in this section. As follows:

MOEA/TS-1 algorithm: The feature extraction operator in MOEA/TS algorithm is changed to the binary crossover operator and polynomial mutation operator;

MOEA/TS-2 algorithm: The repulsion field method in MOEA/TS algorithm is removed;

MOEA/TS-3 algorithm: The concurrent architecture in MOEA/TS algorithm is changed to serial architecture;

MOEA/TS-4 algorithm: The individual importance degree in MOEA/TS algorithm is removed.

This paper takes WFG test function set (45 test cases) as samples, and then verifies the performance of 5 algorithms. In this paper, 5 algorithms are executed 30 times to get the average data as shown in Table 8. As can be seen from Table 8, MOEA/TS algorithm wins the first place in 24 test cases; MOEA/TS-1 algorithm wins the first place in 13 test cases; MOEA/TS-2 algorithm wins the first place in 7 test cases; MOEA/TS-3 algorithm wins the first place in 1 test case. In the 45 test cases, the number of MOEA/TS algorithm is significantly superior to MOEA/TS-1 algorithm, MOEA/TS-2 algorithm, MOEA/TS-3 algorithm and MOEA/TS-4 algorithm is 21, 30, 40 and 45, respectively. The number of MOEA/TS algorithm is significantly inferior to MOEA/TS-1 algorithm, MOEA/TS-2 algorithm, MOEA/TS-3 algorithm and MOEA/TS-4 algorithm is 11, 6, 0 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to MOEA/TS-1 algorithm, MOEA/TS-2 algorithm, MOEA/TS-3 algorithm and MOEA/TS-4 algorithm is 13, 9, 5 and 0, respectively. The average ranking of MOEA/TS algorithm is about 1.64; the average ranking of MOEA/TS-1 algorithm is about 2.02; the average ranking of MOEA/TS-2 algorithm is about 2.62; the average ranking of MOEA/TS-3 algorithm is about 3.71; the average ranking of MOEA/TS-4 algorithm is 5.

Table 8 The IGD + values and rankings of 5 algorithms under WFG test function set.

Therefore, we think that the 4 innovative parts of MOEA/TS algorithm are necessary and indispensable. The lack of any innovative parts will seriously affect the performance of MOEA/TS algorithm. This shows that our innovations are effective. In addition, based on the above data, we can also find that “individual importance degree” has the greatest influence on the algorithm; the algorithm architecture ranks second; the repulsion field method ranks third; the feature extraction operator ranks fourth.

Ablation experiment of selection approach

In the feature extraction operator, we select W high-quality solutions. To prove the effectiveness of this selection approach over random selection, the ablation experiment will be performed in this sect. "Introduction" variant is designed in this section. As follows:

MOEA/TS-5 algorithm: W solutions are randomly selected in the feature extraction operator.

This paper takes WFG test function set (45 test cases) as samples, and then verifies the performance of 2 algorithms. In this paper, 2 algorithms are executed 30 times to get the average data as shown in Table 9. As can be seen from Table 9, MOEA/TS algorithm wins the first place in 45 test cases. In the 45 test cases, the number of MOEA/TS algorithm is significantly superior to MOEA/TS-5 algorithm is 42. The number of MOEA/TS algorithm is significantly inferior to MOEA/TS-5 algorithm is 0. Statistically, the number of MOEA/TS algorithm is similar to MOEA/TS-5 algorithm is 3. Therefore, we believe that the performance of MOEA/TS algorithm is better than MOEA/TS-5 algorithm in the WFG test function set. It proves that the selection approach that we use is better than random selection in the feature extraction operator.

Table 9 The IGD + values of 2 algorithms under WFG test function set.

In addition, the performance of MOEA/TS-5 algorithm isn’t as good as that of MOEA/TS-1 algorithm. It means that the performance of the feature extraction operator based on random selection is even worse than that of some classical operators. The possible reason is that the randomly selected solution set will cause the feature extraction operator to extract many bad features. These bad features hinder individual evolution, which makes the convergence maintenance state and diversity maintenance state of MOEA/TS algorithm fail for a long time, and only the coordination state can play some role. The architecture of the MOEA/TS algorithm is undermined by some bad features.

Parameter sensitivity analysis.

The algorithm parameters analyzed in this paper are mainly the number of high-quality solutions W, threshold value T, standard deviation std. Due to the high complexity of the WFG3 test case with 10 objectives, it is difficult for the population of each algorithm to cover the real Pareto front, so this paper considers the WFG3 test case with 10 objectives as the main function of parameter analysis.

The initial value and value range of each parameter are shown in Table 10.

Table 10 The initial value and value range of each parameter.

As shown in Fig. 16, when $W<9$, the IGD + value of the algorithm decreases significantly with the increase of W. It means that when $W<9$, the performance of the feature extraction operator is greatly improved with the increase of W. This is because the features extracted by the feature extraction operator are closer to the ideal situation. When $W=9$, the IGD + value of the algorithm is minimum. This shows that when $W=9$, the feature extraction operator performs best. When $W>9$, the IGD + value of the algorithm increases slowly. It means that when $W>9$, the performance of the feature extraction operator deteriorates gradually with the increase of W. This is because some features are over-extracted by feature extraction operators. Therefore, for WFG3 test case with 10 objectives, $W=9$ is the best parameter selection.

Figure 16

The corresponding relationship between IGD + value and W.

As shown in Fig. 17, when $T<5\%$, the IGD + value of the algorithm decreases significantly with the increase of T. This is because if the threshold value T is too small, the algorithm will remain in the same state for a long time, and it is difficult to be adjusted to other states. Convergence and diversity of algorithm will also be difficult to balance. This situation will be improved with the increase of T. When $T=5\%$, the IGD + value of the algorithm is minimum. This shows that when $T=5\%$, the algorithm has the best performance. When $T>5\%$, the IGD + value of the algorithm increases gradually with the increase of T. This is because if the threshold value T is too large, the algorithm’s state will be adjusted frequently. Even if the population isn’t stable in one state (convergence, diversity, coordination), the algorithm will also be adjusted to other states. This isn’t conducive to improving the convergence and the diversity of the algorithm. The efficiency of the algorithm will also be affected. Therefore, for WFG3 test case with 10 objectives, $T=5\%$ is the best parameter selection.

Figure 17

The corresponding relationship between IGD + value and T.

As shown in Fig. 18, when $std<0.7$, the IGD + value of the algorithm decreases significantly with the increase of std. This is because if std is too small, the results of Gaussian sampling are too concentrated in the middle region, and the randomness of the sampling vector is weak, which isn’t conducive to the use of features and generation of diversified feature solutions. When $std=0.7$, the IGD + value of the algorithm is minimum. This shows that when $std=0.7$, the feature extraction operator performs best. When $std>0.7$, the IGD + value of the algorithm increases significantly with the increase of std. This is because if the std is too large, the result of Gaussian sampling is too scattered, the randomness of the sampling vector is strong, some components are easy to exceed the upper bound or lower bound, and some features are easy to be eliminated by the repair operator. Therefore, for WFG3 test case with 10 objectives, $std=0.7$ is the best parameter selection.

Figure 18

The corresponding relationship between IGD + value and std.

Based on the above analysis of algorithm parameters, we think $W=9, T=5\%, std=0.7$ are the best parameter combinations in WFG3 test case with 10 objectives. Further, we test the performance of the above parameter combinations in more test cases. The experimental results show that the above parameter combinations perform well in most test cases. Therefore, this paper sets the number of high-quality solutions $W$, the threshold value $T$ and the standard deviation $std$ to 9, 5% and 0.7, respectively.

Practical problem testing

This section mainly explores the performance of MOEA/TS algorithm in practical problems. The practical problem selected in this section is the industrial internet optimization problem based on the blockchain provided in reference⁴⁰.

The industrial internet can support effective control of the physical world through a large amount of industrial data, but data security has always been a challenge due to various interconnections and accesses. Blockchain technology supports the security and privacy protection of industrial internet data with its trusted and reliable security mechanism. Fragmentation technology can help improve the overall throughput and scalability of the blockchain network. However, due to the uneven distribution of malicious nodes, the effectiveness of fragmentation is still challenging. In addition, the conflict between multiple industrial network indicators is also a problem we have to consider. Therefore, the industrial internet optimization problem based on blockchain is an important research problem.

In this section, the industrial internet optimization problem based on blockchain has the following 4 optimization objectives:

• (1) Minimizing the shard invalidation probability (SIP);

• (2) Minimizing the transmission delay (TD);

• (3) Maximizing the throughput (TP);

• (4) Minimizing the load of Malicious Nodes (LMN).

The research background of the industrial internet based on blockchain and the calculation formulas of these 4 objectives are detailed in reference⁴⁰.

In this section, we set the population size to 220, the number of iterations to 300, and the number of function evaluations to 66000. We still use inverted generational distance plus (IGD+) to measure the performance of many-objective optimization algorithms. However, the real PF of the practical problem is unknown. Therefore, we run these algorithms many times to obtain the different non-dominated solution sets. The non-dominated union set of the different non-dominated solution sets is considered as the real PF. The relevant parameters of these algorithms are shown in Section 4.1.

In this section, each algorithm is executed 30 times to get the data as shown in Table 11. As can be seen from Table 11, MOEA/TS algorithm has absolute advantages. The performance of BiGE-BEW algorithm and MOEA/DG algorithm is good and similar. The performance of VMEF algorithm and MOEA/D algorithm in practical problems is obviously not as good as that in benchmark test functions. This is because the real PF of the practical problem is more complex. The performance of LSMaODE algorithm is close to that of MOEA/D algorithm. The performance of MaOEA/IT algorithm and MaOEA/IGD algorithm is the worst. Based on the above observations and analysis, we believe that MOEA/TS algorithm still has excellent performance and strong applicability in practical problems.

Table 11 The IGD + values of each algorithm in practical problem.

Considering that the solutions obtained by the many-objective optimization algorithms are the population, it is unrealistic to compare different network indicators of different algorithms intuitively. However, in practical applications, we only need to make choices according to the specific needs or preferences of users or enterprises. In this section, we first select the individuals with the largest throughput in each algorithm, and then compare the MOEA/TS algorithm with other algorithms on the basis of ensuring the maximum throughput. The network indicators obtained by these 8 algorithms are shown in Table 12. As can be seen from Table 12, in terms of SIP and TP, MOEA/TS algorithm has the best performance; In terms of TD, MOEA/TS algorithm ranks second; In terms of LMN, MOEA/TS algorithm ranks third. Therefore, we believe that the MOEA/TS algorithm has the best comprehensive performance in the industrial internet optimization problem based on blockchain, and various network indicators are at the forefront.

Table 12 The network indicators obtained by each algorithm.

Based on the experimental analysis from Section 4.2 to Section 4.8, we can obtain the following conclusions:

• (1) In the benchmark test cases, MOEA/TS algorithm is superior to the other 7 advanced many-objective optimization algorithms.

• (2) MOEA/TS algorithm is more suitable for dealing with the MaOPs with more than 10 objectives.

• (3) MOEA/TS algorithm can show excellent performance in different test function sets, and has strong universality and applicability.

• (4) MOEA/TS algorithm has the best convergence and diversity, the strongest evolution ability and the fastest convergence speed.

• (5) The 4 innovative parts of MOEA/TS algorithm are necessary and indispensable. The lack of any innovative parts will seriously affect the performance of MOEA/TS algorithm.

• (6) MOEA/TS algorithm still has excellent performance and strong applicability in practical problems.

Summary and future work

Aiming at some difficulties in the many-objective optimization field, this paper proposes a many-objective evolutionary algorithm based on three states (MOEA/TS). Firstly, a feature extraction operator is proposed. The feature extraction operator is a feature extractor, which can extract the features of the high-quality solution set, and then assist the evolution of the current individual. Secondly, in terms of convergence maintenance, this paper doesn’t consider using domination relaxation technique, because the current domination relaxation technique still faces some problems. Based on Pareto front layer, this paper proposes the concept of “individual importance degree”. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer, and effectively solve the phenomenon of Pareto resistance. Then, in terms of diversity maintenance, this paper considers maintaining the diversity of the population in the objective space by repulsion field, so that the population can be evenly distributed on the real PF. Finally, a new concurrent algorithm framework is designed. In the framework, the algorithm is divided into three states, namely, convergence maintenance state, diversity maintenance state and coordination state. Each state focuses on a specific task. That is, the convergence maintenance state is responsible for improving the convergence of population; Diversity maintenance state is responsible for improving the diversity of population; the coordination state is responsible for coordinating the contradiction between diversity and convergence. The population can freely switch among these three states according to its own evolution. The experimental results show that MOEA/TS algorithm is superior to the other 7 advanced many-objective optimization algorithms. In addition, the effectiveness of the innovation parts is further verified.

However, MOEA/TS algorithm also has obvious defects: MOEA/TS algorithm isn’t good at dealing with test problems with narrow convergence curves or multiple local fronts. Therefore, in the future, we will further improve MOEA/TS algorithm, so that MOEA/TS algorithm can deal with test problems with narrow convergence curve or multiple local fronts. In addition, constrained MOPs and high-dimensional MOPs are also the focus of our future research.