A two-stage diversity enhancement differential evolution algorithm for multi-objective optimization problem

Abstract

In order to solve the premature convergence of multi-objective evolutionary algorithm, a two-stage diversity enhancement differential evolution algorithm for multi-objective optimization problem(TSDE) is proposed. The offspring with better performance needs the generation of high-quality parent generation. In this paper, an improved cell density method is used to screen for the high quality parents by estimating the global distribution of the objective space. Moreover, Principal Component Analysis operator is introduced to the external archive to perturb the non-dominated solution, which not only ensures the convergence but also improves the diversity. In order to verify the effectiveness of the algorithm, TSDE and other advanced methods are run on 19 test functions. The results show that TSDE performs better than other algorithms.

Keywords

Multi-objective optimization differential evolution evolutionary algorithms principal component analysis

1 Introduction

Multi-objective optimization [1] is a mathematical problem involving simultaneous optimization of multiple objective functions, which has been applied to all aspects of real life. Yang et al. [2] proposed a multi-objective evolutionary algorithm (FACE) for solving steady-state constrained multi-objective problem. Ahmad et al. [3] proposed a method for systematic analysis of solid waste data. This approach makes lowest-cost recommendations for different locations based on the amount and frequency of waste in the residential grid. In order to obtain higher control accuracy in the rolling schedule [4], Hu et al. [5] proposed a multi-objective rolling optimization method. It not only predicts rolling force and rolling power in real time, but also provides solutions for multi-objective rolling plans.

Nowadays, metaheuristic algorithm [6] has become an effective solution to multi-objective optimization. Most metaheuristic algorithms come from the simulation of natural phenomena and the learning of biological intelligence. For example, ant colony algorithm (ACO) [7] simulates the process of actual ant colony foraging. The particle swarm optimization(PSO) [8] algorithm is derived from the research on the movement behavior of birds and fish groups. The simulated annealing(SA) [9] algorithm is derived from the cooling process of solids. New algorithms that have emerged recently, for example, Ant Lion Optimizer (ALO) [10] simulates the foraging behavior of an Ant lions. Moth-flame Optimisation Algorithm (MFO) [11] is inspired by the navigation method of moths in nature. The crow search algorithm(CSA) [12] simulates the behavior of crows storing excess food and extracting food when needed. The Salvia Swarm Algorithm (SSA) [13] simulates the foraging and navigation behavior of Salvia in the ocean. As a new type of metaheuristic algorithm, the multi-objective differential evolution algorithm (MODE) [14] has received extensive attention from academia and engineering because of its simple structure and fast convergence.

In recent years, researchers have proposed a variety of improved versions of MODE to optimize their performance. Guangzhi et al. [15] proposed a multi-stage perturbation differential evolution algorithm (MPDE). This algorithm introduced a normal random distribution with adjustable variance to perturb the current population. Moreover, the parameters are adjusted adaptively according to the current search status. To solve the problem that different subproblems has different requirements for exploration and exploitation, Li et al. [16] proposed the MOEA/D-MRS algorithm. According to the solving degree of each subproblem, a mating restriction probability is assigned to each subproblem to control the contribution of exploration and exploitation. Moreover, the mating restriction probability is updated according to the survival time of each generation. Meng et al. [17] proposed a Di-DE algorithm. The populations are grouped for novelty and the parameters are updated for each group. Moreover, the problem of premature convergence can be solved by finely detecting the external environment to improve the external archive(EXA). For the problem of slow or premature convergence in the evolution process, Yong-Jie et al. [18] proposed the MDADE algorithm. For diversity optimization, the algorithm divides the population into three subpopulations, and each subpopulation adopts a different mutation strategy. At the same time, a parameter adaptive mechanism is adopted to improve the convergence of the algorithm. At present, MODE algorithm has been applied in many fields and achieved good results.

Although many improvements have been proposed, MODE still has a lot of room for research. Table 1 shows The pros and cons of MODE.To overcome the shortcomings in Table 1, a two-stage diversity enhancement differential evolution algorithm for multi-objective optimization problem(TSDE) is proposed. The innovations of TSDE are as follows:

Table 1
The pros and cons of MODE

Pros Simple implementation, few control parameters, high convergence accuracy, fast convergence speed and strong robustness

Cons 1. The poor quality of the initial population leads to premature convergence of the population

2. The EXA cannot produce better solutions to update the population, which leads to the stagnation of the search

1. In order to screen for the high quality parent generation and produce the offspring population of better performance, the optimized cell density method is adopted. This method can estimate the global distribution of objective space and identify dense areas to increase the diversity of initial population of each generation.

2. Principal Component Analysis(PCA) operator is introduced into the EXA to perturb the non-dominated solutions and generate new solutions that around the original non-dominated solution set, so as to improve the diversity while ensuring convergence.

The remainder of this paper is organized as follows. Section 2 provides the background and related work involved in this paper. Section 3 describes the TSDE algorithm in detail. The analysis of the results compared with the other algorithms, the efficiency analysis of strategies and the discussion of parameter selection are presented in Section 4. Finally, the conclusion of this paper is drawn in the Section 5.

2 Background and related work

2.1 Multi-objective optimization problem

In the general case, a continuous or no constrained multi-objective optimization problem [19] can be described as:

$\begin{matrix} minF (x) = (f_{1} (x), f_{2} (x), . . ., f_{m} (x))^{T} \\ s . t . x \in Ω^{D} \end{matrix}$ (1)

Where x = (x¹, x², …, x^D) is one of the solutions to the D-dimensional decision space Ω^D. x = (x₁, x₂, …, x_N) ^T and N is the number of solutions in solution set. m is dimensions of the objective space. F : Ω^D → R^m defines m objective functions in the objective space R^m.

2.2 Differential evolutionary algorithm

Differential evolution algorithm has the characteristics of fast speed and good robustness. It is mainly divided into four steps: initial population, mutation, crossover and selection.

Generate initial population Randomly generate N_p individual to form initial population. $\begin{matrix} x_{i, 0} = x_{l} + random [0, 1] * (x_{h} - x_{l}) \end{matrix}$ (2)

Where x_h and x_l are the upper and lower of individual, i ∈ [1, 2, …, N_p].

Mutation For the i_th individual of g_th generation, randomly select three individuals except the i_th individual to generate new individual. $\begin{matrix} v_{i, g + 1} = x_{r 1, g} + F (x_{r 2, g} - x_{r 3, g}) \end{matrix}$ (3)

Where x_r1,g, x_r2,g and x_r3,g are the selected individuals, F is the scaling factor.

Crossover Cross the objective individual x_i,g and the mutated individual v_i,g+1 to obtain test individual. $u_{i, j, g + 1} = {\begin{matrix} x_{i, j, g} & rand (j) \leq C_{r} \\ x_{i, j, g} & otherwise \end{matrix}$ (4)

Where rand (j) is a random number between [0,1], C_r is the crossover probability.

Selection By selection to generate the (g + 1) _th generation of population. $x_{i, g + 1} = {\begin{matrix} u_{i, g + 1} & f (u_{i, g + 1}) \leq f (x_{i, g}) \\ x_{i, g} & otherwise \end{matrix}$ (5) Where f (u_i,g+1), f (x_i,g) are the fitness of u_i,g+1 and x_i,g.

2.3 Principal component analysis

Munteanu et al. [20] have proposed a novel PCA-driven method that performs variation in a real-coded genetic algorithm. This article shows that the PCA mutation operator has higher diversity in the search space than other mutation operators. Therefore, this paper introduces the PCA mutation operator into the archive update process to optimize the diversity of EXA.

PCA is a matrix compression algorithm. While reducing the dimension of the matrix, the information about the original matrix is kept as far as possible. For a n-dimensional sample set (m ∗ n) with m individuals, the PCA algorithm implementation is as follows:

(1) Standardized sample set, each sample minus the sample mean;

(2) Calculate the covariance of the standardized sample set;

(3) Calculate the eigenvalues and eigenvectors of covariance;

(4) Rearrange the eigenvectors in descending order of eigenvalues;

(5) Select the first k eigenvalues and get the transformation matrix(n ∗ k);

(6) Multiply the original sample set (m ∗ n) with the transformation matrix(n ∗ k), the reduced dimension matrix(m ∗ k) is obtained.

By analyzing the principal components of the non-dominated solutions, this algorithm generates a set of solutions that around the original non-dominated solution set. Then a better non-dominated solution set is generated based on the new solution set and the original solution set.

3 A two-stage diversity enhancement differential evolution algorithm

In this section, the TSDE algorithm is described in detail. Firstly, the framework of the algorithm is given. Then, population diversity enhancement strategy is introduced. This strategy is to make the population present a good diversity before evolution. Then there is the EXA diversity enhancement strategy, which optimizes population diversity by improving non-dominated solutions. Finally, the flow chart of the algorithm is given.

Algorithm 1
Framework of TSDE

1: Initialize the population P with N random solutions and update the EXA with P, F_mean = 1.

2: while the termination criterion is not reached do

3: Population diversity enhancement, which is described in detail in Algorithm 2.

4: Set P_new =∅,

5: for i = 1 to N do

6: C_r = Normal (C_r,mean, 0.1) where C_r,mean is the median value of successful C_r stored in a list L_{c
_r}.

7: v_i = mutation(x_i, F_mean).

8: u_i = crossover(v_i, x_i, C_r).

9: x_new = gaussmutation(u_i)(if x_new dominates x_i, add the value C_r into list L_{c
_r}, if the size of L_{c
_r} exceed L, then remove the first item in it).

10: Add x_new to P_new.

11: end for

12: Update population P with P_new.

13: EXA diversity enhancement, the detail is described in Algorithm 3.

14: Update EXA.

15: end while

The overall framework of the TSDE is presented in Algorithm 1. As shown in Algorithm 1, initialize the population first, and then optimize the initial population distribution pattern. The step is described in detail in Algorithm 2. In the optimization process of each individual, C_r is firstly initialized according to the gaussian distribution, and then new solutions are generated through the traditional evolutionary process. For the new solution, if it dominates the original solution, add its corresponding C_r to the L_{c
_r} list. Then remove the first C_r value of L_{c
_r} when the length of L_{c
_r} is greater than its maximum value L. The value of L is taken as 50. All the new solutions produced by evolution are put into the set P_new.

A new population P is formed by adding the new population P_new into the initial population P. The method of combining non-dominated sorting and crowding distance selection is applied to P. Finally, a new population P is selected for the next generation.

In the process of evolution, the optimization process of EXA is also emphasized. First, EXA is initialized at the same time as the population is initialized. By analyzing the non-dominated solution components in the EXA, a set of solutions near the original solution set are generated based on PCA mutation operator. The detailed process is in Algorithm 3.

Finally, merge the optimized EXA and P_new to obtain the final EXA through the non-dominated relationship.
Algorithm 2
Population diversity enhancement

1: Set the new solution set S_new to be empty.

2: for i = 1 to n_size do

3: Select solutions from population P randomly.

4: Apply the crossover operator on the selected solutions to generate a new solution x_i, and add x_i to S_new.

5: end for

6: P = P ⋃ S_new.

7: Calculate the fitness of each solution in P with optimal sequence method.

8: P_new =∅.

9: while P_new ≤ N do

10: Applying the cell density method to population P and get the population PE.

11: Remove PE from P.

12: P_new = P_new ∪ PE.

13: end while

14: P = P_new.
3.1 Population diversity enhancement

The simplicity and efficiency of the differential evolutionary algorithm have made a lot of achievements in the application [21] of the multi-objective problem. However, most of the improvements on the difference algorithm focus on the improvement on parameter selection and mutation operator. The diversity of the population involved in evolution is ignored. In order to produce offspring with better performance, high quality parents should be screened out first. An improved cell density method is introduced in this section. Screen individuals by locating individual positions to obtain a well-distributed parent population. The procedure of population diversity enhancement is presented in Algorithm 2.

As shown in Algorithm 2, first generate an empty set S_new. Then randomly select individuals from the population P for SBX crossover [22] until n_size(n_size=N/4) new solutions are generated. Put all these new solutions into the empty set S_new. After that, add S_new to the initial population P. For the new population P, the optimal sequence method in Section 3.1.1 is used. The location number of each individual is calculated and the location number is the corresponding fitness value. Then, the location of each individual in the objective space is located. The number of individuals in each cell is obtained by cell density method. Next, the individuals are screened based on the location number and dominance relationship. This process is described in detail in Section 3.1.2. Finally, the selected population P_new participated in the evolutionary process as P.

3.1.1 Optimal sequence method

There are three typical methods to calculate the fitness of multi-objective problems: weight sum approach [23], altering objective functions [24] and Pareto-ranking approach [25]. A new rank strategy based on optimal sequence method [26] is adopted, and the total optimal number K_i is used to sort the priorities of solutions.

Now we consider a population P with N solutions. xⁱ and x^j are random solutions in the population. f_l (xⁱ) is the value of xⁱ mapped to the objective space in the l-dimension(l∈(1, m) and m is the dimension of the objective space), and so is f_l (x^j). The value of solution xⁱ and x^j mapped to the objective space are compared in each dimension, as shown in Equation 6.

$a_{ijl} = {\begin{matrix} 1 & f_{l} (x^{i}) < f_{l} (x^{j}) \\ 0.5 & f_{l} (x^{i}) = f_{l} (x^{j}) \\ 0 & f_{l} (x^{i}) > f_{l} (x^{j}) ori = j . \end{matrix}$ (6)

Eq 6 obtains the comparison value a_ijl of f_l (xⁱ) and f_l (x^j). Then add the values a_ij1, a_ij2, …, a_ijm to obtain optimal number a_ij, as shown in Eq 7.

$\begin{matrix} a_{ij} = \sum_{l = 1}^{l = m} a_{ijl} i, j \in N \end{matrix}$ (7)

As shown in Eq 8, finally, compare xⁱ with all other solutions to obtain a_i1, a_i2, …, a_iN, and add these values to get total optimal number K_i.

$\begin{matrix} K_{i} = \sum_{j = 1}^{j = N} a_{ij} \end{matrix}$ (8)

The steps of this new rank strategy are as follows:

Step 1: Input the population P and its objective function value.

Step 2: Calculate the total optimal number K_i of each solution.

Step 3: Rank the solution according to its related K_i in descending order.

Step 4: Mark their location number. (The location number of the solution with the largest K_i is 1, the location number of the solution with the second largest K_i is 2, and so on. The better solution with the bigger K_i and the smaller location number.)

3.1.2 Individual selection process

The uniformly distributed solution set is very efficient for the differential process. In this algorithm, the improved cell density method is used to make the individual distribution uniform. Cell density method divides each dimension of the objective space into t equal parts, and finally forms t^m small spaces. Each small space is called a cell, and the number of individuals in each cell is called cell density.

The advantage of the cell density method is to find out the distribution of solutions over the whole space without adding too much computation. The improved cell density method is to combine the cell density method with dominance relationship. Then use dominance relationship to screen until the population number is N. The procedure of individual selection process is present as follows:

Step 1: Divides each dimension of the objective space into t equal parts. The width of each part is $w = \frac{h_{k}^{\max} - h_{k}^{\min}}{t}$ . $h_{k}^{\max}$ and $h_{k}^{\min}$ are the maximum and minimum values of the solution set in each dimension.

Step 2: Find the cell where each solution is located and calculate the number of solutions in each cell. Set up an empty set P_new.

Step 3: Each cell is examined in turn, if the cell density is not zero, the individual with the smallest location number will be select and placed in collection E.

Step 4: Apply the non-dominated sorting to E to get the non-dominated solution set PE.

Step 5: Remove PE from P and add PE to P_new. If the solutions in P_new are less than N, go back to Step 3.

Figure 1 illustrates the process of using cell density method to screen solutions and successively select solution set PE to enter P_new. As shown in Fig. 1, each solution has its corresponding location number. The larger the location number, the higher the priority of the corresponding solution to be selected. After dividing the cell regions, the number of solutions contained in each cell is determined. If a cell contains more than one individual, then the solution with the smallest location number is chosen. For example, solution A, B and C, D are in the same cell respectively. Since location number of A and D is the smallest in the cell, A and D are selected for solution set E. After selecting the optimal solution E in all cells, the non-dominated individuals in E are put into the solution set PE. The red dots represent the solution set PE₁ that is first selected and put into P_new. The green dots represent a solution set of PE₂ that is put into P_new after the second selection. Repeat the above steps until the number of solutions in P_new reaches N.

Fig. 1

Individual selection process.

3.2 EXA diversity enhancement

The EXA diversity enhancement strategy is to optimize the non-dominated solution in the EXA. The PCA operator [27] is used to generate a new solution set to perturb the non-dominated solutions. So as to increase population diversity. The detailed steps are written in Algorithm 3.

Algorithm 3
EXA diversity enhancement

1: Calculate the non-dominated solutions of EXA and put them into solution set X.

2: while individual number in X > Number do

3: Randomly select Number individuals in X and denoted these individuals as X.

4: end while

5: Applying the PCA operator on X and generate new solution set S_new.

6: EXA= EXA ⋃ S_new.

7: Applying non-dominated sorting and crowding distance on EXA to select N solutions, and set these solutions as new EXA.

As shown in Algorithm 3, the non-dominated solution set X of the EXA is firstly calculated. If the number of individuals in X exceeds Number(Number is taken as 20), randomly select Number solutions from X and denoted these solutions as X. For each solution in X, PCA operator is applied to generate the new solution. Then add the generated new solution set S_new into the original EXA. Apply non-dominated sorting and crowding distance on EXA to generate new EXA.

Figure 2 illustrates the process of generating new solutions through PCA mutation. As shown in Fig. 2, the black points represent the solutions in the non-dominated solution set, while the blue points represent the new solution set obtained by PCA mutation. From Fig. 2, it can be seen that the new solutions generated by PCA mutation formed around the old non-dominated solutions. The HV indicator [28] can evaluate the convergence, distribution and diversity of the set simultaneously. As shown in the Fig. 2, the area of the red line box represents the HV value of the new non-dominated solution set. The area formed by the dotted green line and the red line represents the HV value of the original solution set. It is obvious that the area of the former is larger than the latter. Therefore, through the original solution set and the new solution set generated by PCA, a new non-dominated solution set with better convergence and diversity is generated. The details of the PCA mutation are shown below.

Fig. 2
Non-dominated solutions expansion.

Step1: Get the n × 1 vector mean by calculate the average of x₁, …, x_k and the n × n covariance matrix ∑ by ∑ = E [(x - mean) (x - mean) ^T], where k is the number of solution in non-dominated set.

Step2: For covariance matrix ∑, compute the eigenvectors v₁, …, v_n and eigenvalues, and then arrange the corresponding eigenvectors in descending order according to the eigenvalues. Form a n × n matrix W = [v₁, …, v_n].

Step3: Form a n × h matrix V = [v₁, …, v_h] by selecting the first h eigenvectors from matrix W.

Step4: S_new = ∅.

Step5: While |S_new| < Number

Step5.1: y_i = V^T (x_i - mean).

Step5.2: $x_{i}^{'}$ = Vy_i + mean.

Step5.3: S_new = $S_{new} ⋃ x_{i}^{'}$

Step6: Output new solution set S_new.

The PCA algorithm is used as mutation operator to generates new solutions. The principal component of X is first computed. This procedure is mainly to analyze the distribution of the solution set in the decision space. The mapping of the solution set on the objective space is a combination of these principal components. The more the principal component column is used, the better the individual diversity and the worse the convergence of the solutions produced. Due to the contradictory relationship between diversity and convergence, it is very important to choose appropriate h. Therefore, a discussion on the value of h is set in Section 666. After obtaining the orthogonal basis V, an empty solution set S_new is initialized. Then the orthogonal basis V is used to calculate the projection solution of non-dominated solution x_i. Add the projected solution to S_new until |S_new| = Number.
3.3 Flow chart of TSDE

Figure 3 gives the flow chart of TSDE. First initialize the population P and EXA, and then operate on the P and EXA respectively. For the P, first apply the population diversity enhancement strategy proposed in Section 3.1, and the detailed steps are given on the left side of the flow chart. After optimizing the initial population, new individuals are produced by selection, crossover and mutation. After that, the new individuals are put into P, and the non-dominated sorting is used to update P. For the EXA, EXA diversity enhancement strategy which proposed in Section 3.2 is used to optimize the EXA. The detailed steps are given on the right side of the block diagram. Thereafter, put the P into EXA, and update the EXA with non-dominated sorting and crowding distance. Repeat the above steps until the termination condition is met, and output EXA.

Fig. 3

The flowchart of TSDE method.

4 Experimental results

Five multi-objective swarm algorithms, dMOPSO [29], MPSOD [30], MMOPSO [31], NMPSO [32] and CMO-PSO [33], are selected to verify the effectiveness of the algorithm. 19 test functions with different frontier characteristics are used to evaluate the performance of the algorithm. They respectively belong to ZDT series, DTLZ series and UF series. Different test functions correspond to decision variables of different dimensions. ZDT1-ZDT3 and UF1 series correspond to 30 dimension decision variables. ZDT4 and ZDT6 correspond to 10 dimensional decision variables. In DTLZ series, DTLZ1 has 7 dimensional decision variables. DTLZ2 and DTLZ4 have 12 dimensional decision variables. DTLZ7 has 22 dimensional decision variables. The number of individuals N of population P is set to 100. E_max (the maximum evaluations of functions) is set to 30000. All experimental results are obtained on a PC equipped with 2.50-GHz CPU and 8-GB memory. TSDE and comparison algorithms are all run on the MATLAB program. Moreover, all source codes of the comparison algorithms are provided by PlatEMO [34].

Two indicators are used to evaluate the performance of an algorithm. They are Inverted Generational Distance (IGD) [35] and Hypervolume(HV) [28]. The IGD is a commonly used indicator to evaluate the performance of algorithms. $\begin{matrix} IGD (P, Q) = \frac{\sum_{v \in P} d (v, Q)}{| P |} \end{matrix}$ (9)

As shown in Equation \ref eq:781, P is the set of uniform sampling points of the real Pareto front. v is random point in P. |P| is the number of points contained in P. Q is the set of Pareto points obtained by this algorithm. By calculating the average Euclidean distance between real Pareto front and the Pareto front obtained by this algorithm to to evaluate the overall performance. The smaller the IGD value, the better the overall performance.

HV represents the volume of the hypercube enclosed by the individual in the solution set and the reference point in the objective space. HV can simultaneously evaluate the convergence and distribution of the solution set. The larger the HV value, the better the overall performance of the algorithm. $\begin{matrix} HV = δ (⋃_{i = 1}^{| S |} v_{i}) \end{matrix}$ (10)

As shown in Equation 9, δ represents the Lebesgue measure, which is used to measure volume. |S| represents the number of non-dominated solution sets, v_i represents the super volume formed by the reference point and the i_th solution in the solution set.

4.1 Discussion of results

Tables 2 3 respectively show the IGD and HV values of TSDE and comparison algorithm after 30 runs of each test function. The value outside the brackets is the mean, and the value inside the brackets is the standard deviation(std). For each test function, the best result is shown in bold. Moreover, the Wilcoxon rank-sum test [36] is adopted at a significance level of 0.05. In the last row of the table, "+", "-", "=" respectively indicate the sum of TSDE results are significantly better, worse, or similar to those of a competitor.

Table 2
The mean IGD values and standard deviations obtained by the compared algorithms

dMOPSO MPSOD MMOPSO NMPSO CMOPSO TSDE

ZDT1 1.6616e-2 (5.61e-3)- 4.2032e-3 (4.70e-4)+ 4.8671e-3 (2.25e-4)- 2.9850e-2(1.30e-2)- 4.1856e-3 (1.54e-4)+ 4.5820e-3(1.50e-4)

ZDT2 5.7774e-2 (1.48e-1)- 3.8610e-3 (7.18e-5)+ 5.2334e-3 (4.74e-4) - 1.8430e-2(3.27e-3)- 4.0939e-3 (7.48e-5)+ 4.6550e-3(1.91e-4)

ZDT3 1.5704e-2 (1.81e-3)- 1.0393e-2 (3.20e-4)- 5.5300e-3 (2.33e-4) - 1.0125e-1(7.85e-4)- 4.6286e-3 (4.95e-5)+ 5.1540e-3(2.76e-4)

ZDT4 2.9168e-2 (4.90e-2)- 1.9314e+2 (2.06e+1)- 2.0438e+1 (1.57e+1)- 5.7215e+1(1.60e+1)- 1.9555e+1 (6.05e+0) - 4.8040e-3(2.69e-4)

ZDT6 1.9511e-2 (8.45e-3)- 3.1005e-3 (1.40e-5)= 4.2828e-3 (2.17e-4)- 4.4184e-3(5.05e-4)- 3.2116e-3 (8.71e-5)- 3.1490e-3(4.33e-4)

DTLZ1 3.1053e+0 (2.54e+0)- 1.4145e+0 (5.46e-1)- 4.7894e-1 (4.01e-1)- 6.4127e-2(1.04e-1)- 7.1838e+0 (4.65e+0) = 3.1820e-2(5.05e-2)

DTLZ2 1.3728e-1 (8.13e-3) - 5.4946e-2 (8.40e-5)+ 7.1469e-2 (2.49e-3)- 7.6885e-2(2.67e-3)- 5.8245e-2 (1.03e-3) + 6.3010e-2(1.80e-3)

DTLZ4 2.6230e-1 (2.73e-2)- 5.4963e-2 (1.78e-4)+ 7.0954e-2 (2.42e-3)- 1.0831e-1(1.18e-1)- 6.1298e-2 (1.60e-3)+ 6.5420e-2(2.13e-3)

DTLZ7 1.8944e-1 (1.91e-1)- 1.3081e-1 (3.53e-3)- 2.5684e-1 (2.87e-1)- 6.9539e-2(3.87e-3)+ 1.8362e-1 (2.58e-1)- 7.1760e-2(3.80e-3)

UF1 4.7411e-1 (1.31e-1) - 7.5428e-2 (1.14e-2)- 9.4911e-2 (4.29e-2)- 1.0502e-1(8.80e-3)- 7.9996e-2 (2.62e-2)= 6.6270e-2(1.74e-2)

UF2 9.7714e-2 (1.07e-2)- 4.3369e-2 (5.46e-3)- 3.8326e-2 (4.11e-3)- 6.1337e-2(1.11e-2)- 4.0682e-2 (8.58e-3)- 3.4490e-2(5.98e-3)

UF3 3.2650e-1 (8.25e-3) - 2.8961e-1 (3.02e-2)= 2.3344e-1 (2.64e-2)+ 2.4321e-1(4.46e-2)+ 1.7315e-1 (5.30e-2)+ 2.7900e-1(3.47e-2)

UF4 1.1941e-1 (7.26e-3) - 5.1149e-2 (1.84e-3)- 4.6947e-2 (2.36e-3) - 6.8012e-2(1.37e-2)- 9.4593e-2 (7.67e-3)- 4.3970e-2(5.17e-4)

UF5 2.4370e+0 (3.22e-1)- 1.0395e+0 (1.80e-1)- 9.0533e-1 (2.21e-1) - 7.9064e-1(2.21e-1)- 5.8777e-1 (1.13e-1) - 4.0000e-1(1.58e-1)

UF6 1.3143e+0 (2.76e-1)- 4.7317e-1 (3.36e-2)- 3.4299e-1 (1.30e-1)- 4.2027e-1(2.30e-1)- 2.5686e-1 (9.78e-2)= 2.7150e-1(1.14e-1)

UF7 3.4015e-1 (8.55e-2) - 5.4613e-2 (6.13e-2)- 1.7504e-1 (1.78e-1)- 2.5352e-1(2.39e-1)- 1.2422e-1 (1.59e-1) - 3.1360e-2(4.13e-3)

UF8 3.3756e-1 (3.56e-2) - 3.1319e-1 (1.62e-2)- 2.7748e-1 (4.40e-2) - 5.4052e-1(9.92e-2)- 6.0569e-1 (1.00e-1) - 1.9810e-1(2.25e-2)

UF9 6.3087e-1 (4.04e-2)- 4.6052e-1 (5.45e-2)- 3.5913e-1 (7.32e-2)- 3.8935e-1(4.53e-2)- 8.9530e-1 (1.59e-1)- 2.3100e-1(5.43e-2)

UF10 9.4525e-1 (3.01e-3) + 3.2108e+0 (2.67e-1)- 5.3285e-1 (6.11e-2) + 5.3279e-1(8.06e-2)+ 3.5005e+0 (5.41e-1)- 1.2640e+0(4.31e-1)

Nos.+/-/= 1/18/0 4/13/2 2/17/0 3/16/0 6/10/3

	dMOPSO	MPSOD	MMOPSO	NMPSO	CMOPSO	TSDE
ZDT1	1.6616e-2 (5.61e-3)-	4.2032e-3 (4.70e-4)+	4.8671e-3 (2.25e-4)-	2.9850e-2(1.30e-2)-	4.1856e-3 (1.54e-4)+	4.5820e-3(1.50e-4)
ZDT2	5.7774e-2 (1.48e-1)-	3.8610e-3 (7.18e-5)+	5.2334e-3 (4.74e-4) -	1.8430e-2(3.27e-3)-	4.0939e-3 (7.48e-5)+	4.6550e-3(1.91e-4)
ZDT3	1.5704e-2 (1.81e-3)-	1.0393e-2 (3.20e-4)-	5.5300e-3 (2.33e-4) -	1.0125e-1(7.85e-4)-	4.6286e-3 (4.95e-5)+	5.1540e-3(2.76e-4)
ZDT4	2.9168e-2 (4.90e-2)-	1.9314e+2 (2.06e+1)-	2.0438e+1 (1.57e+1)-	5.7215e+1(1.60e+1)-	1.9555e+1 (6.05e+0) -	4.8040e-3(2.69e-4)
ZDT6	1.9511e-2 (8.45e-3)-	3.1005e-3 (1.40e-5)=	4.2828e-3 (2.17e-4)-	4.4184e-3(5.05e-4)-	3.2116e-3 (8.71e-5)-	3.1490e-3(4.33e-4)
DTLZ1	3.1053e+0 (2.54e+0)-	1.4145e+0 (5.46e-1)-	4.7894e-1 (4.01e-1)-	6.4127e-2(1.04e-1)-	7.1838e+0 (4.65e+0) =	3.1820e-2(5.05e-2)
DTLZ2	1.3728e-1 (8.13e-3) -	5.4946e-2 (8.40e-5)+	7.1469e-2 (2.49e-3)-	7.6885e-2(2.67e-3)-	5.8245e-2 (1.03e-3) +	6.3010e-2(1.80e-3)
DTLZ4	2.6230e-1 (2.73e-2)-	5.4963e-2 (1.78e-4)+	7.0954e-2 (2.42e-3)-	1.0831e-1(1.18e-1)-	6.1298e-2 (1.60e-3)+	6.5420e-2(2.13e-3)
DTLZ7	1.8944e-1 (1.91e-1)-	1.3081e-1 (3.53e-3)-	2.5684e-1 (2.87e-1)-	6.9539e-2(3.87e-3)+	1.8362e-1 (2.58e-1)-	7.1760e-2(3.80e-3)
UF1	4.7411e-1 (1.31e-1) -	7.5428e-2 (1.14e-2)-	9.4911e-2 (4.29e-2)-	1.0502e-1(8.80e-3)-	7.9996e-2 (2.62e-2)=	6.6270e-2(1.74e-2)
UF2	9.7714e-2 (1.07e-2)-	4.3369e-2 (5.46e-3)-	3.8326e-2 (4.11e-3)-	6.1337e-2(1.11e-2)-	4.0682e-2 (8.58e-3)-	3.4490e-2(5.98e-3)
UF3	3.2650e-1 (8.25e-3) -	2.8961e-1 (3.02e-2)=	2.3344e-1 (2.64e-2)+	2.4321e-1(4.46e-2)+	1.7315e-1 (5.30e-2)+	2.7900e-1(3.47e-2)
UF4	1.1941e-1 (7.26e-3) -	5.1149e-2 (1.84e-3)-	4.6947e-2 (2.36e-3) -	6.8012e-2(1.37e-2)-	9.4593e-2 (7.67e-3)-	4.3970e-2(5.17e-4)
UF5	2.4370e+0 (3.22e-1)-	1.0395e+0 (1.80e-1)-	9.0533e-1 (2.21e-1) -	7.9064e-1(2.21e-1)-	5.8777e-1 (1.13e-1) -	4.0000e-1(1.58e-1)
UF6	1.3143e+0 (2.76e-1)-	4.7317e-1 (3.36e-2)-	3.4299e-1 (1.30e-1)-	4.2027e-1(2.30e-1)-	2.5686e-1 (9.78e-2)=	2.7150e-1(1.14e-1)
UF7	3.4015e-1 (8.55e-2) -	5.4613e-2 (6.13e-2)-	1.7504e-1 (1.78e-1)-	2.5352e-1(2.39e-1)-	1.2422e-1 (1.59e-1) -	3.1360e-2(4.13e-3)
UF8	3.3756e-1 (3.56e-2) -	3.1319e-1 (1.62e-2)-	2.7748e-1 (4.40e-2) -	5.4052e-1(9.92e-2)-	6.0569e-1 (1.00e-1) -	1.9810e-1(2.25e-2)
UF9	6.3087e-1 (4.04e-2)-	4.6052e-1 (5.45e-2)-	3.5913e-1 (7.32e-2)-	3.8935e-1(4.53e-2)-	8.9530e-1 (1.59e-1)-	2.3100e-1(5.43e-2)
UF10	9.4525e-1 (3.01e-3) +	3.2108e+0 (2.67e-1)-	5.3285e-1 (6.11e-2) +	5.3279e-1(8.06e-2)+	3.5005e+0 (5.41e-1)-	1.2640e+0(4.31e-1)
Nos.+/-/=	1/18/0	4/13/2	2/17/0	3/16/0	6/10/3

Table 3

The mean HV values and standard deviations obtained by the compared algorithms

	dMOPSO	MPSOD	MMOPSO	NMPSO	CMOPSO	TSDE
ZDT1	7.0197e-1 (6.77e-3) -	7.1866e-1 (1.06e-3) -	7.1927e-1 (1.98e-4) -	6.8835e-1 (1.64e-2)-	7.1934e-1 (3.69e-4) =	7.1939e-1(1.79e-4)
ZDT2	3.9477e-1 (8.35e-2) -	4.4467e-1 (1.49e-4) +	4.4400e-1 (1.99e-4) =	4.3606e-1 (1.92e-3)-	4.4404e-1 (2.50e-4) =	4.4400e-1(2.43e-4)
ZDT3	5.9940e-1 (3.20e-3) =	5.9679e-1 (1.18e-3) -	5.9943e-1 (9.32e-5) -	5.6673e-1 (3.20e-4)-	5.8972e-1 (1.05e-4) -	5.9933e-1(1.05e-4)
ZDT4	6.9264e-1 (3.94e-2) -	0.0000e+0 (0.00e+0) -	0.0000e+0 (0.00e+0) -	0.0000e+0 (0.00e+0)-	0.0000e+0 (0.00e+0) -	7.1720e-1(4.49e-4)
ZDT6	3.6778e-1 (1.08e-2) -	3.8822e-1 (1.86e-5) -	3.8771e-1 (2.20e-4) -	3.8778e-1 (4.20e-4)-	3.8804e-1(1.58e-4)-	3.8874e-1 (9.36e-5)
DTLZ1	3.2691e-2 (9.12e-2) -	8.7674e-4 (4.80e-3) -	2.4382e-1 (3.53e-1) -	7.5936e-1 (1.95e-1)-	0.0000e+0 (0.00e+0) -	7.9648e-1(1.25e-1)
DTLZ2	3.7925e-1 (1.58e-2) -	5.5694e-1 (3.64e-4) +	5.3143e-1 (3.18e-3) -	5.6046e-1 (1.07e-3)+	5.4148e-1 (2.95e-3) +	5.3639e-1(3.73e-3)
DTLZ4	4.1529e-1 (2.81e-2) -	5.5656e-1 (7.01e-4) +	5.3218e-1 (4.44e-3) =	5.4668e-1 (5.54e-2)	5.3282e-1 (4.60e-3) =	5.3259e-1(3.64e-3)
DTLZ7	2.4052e-1 (1.59e-2) -	2.3127e-1 (3.52e-3) -	2.5045e-1 (2.65e-2) -	2.7502e-1 (1.29e-3)-	2.5759e-1 (2.42e-2) -	2.7037e-1(1.96e-3)
UF1	1.6660e-1 (1.02e-1) -	6.0995e-1 (1.42e-2) -	5.9049e-1 (5.31e-2) -	5.9235e-1 (1.11e-2)-	6.0536e-1 (2.81e-2) -	6.2737e-1(1.90e-2)
UF2	6.1494e-1 (8.89e-3) -	6.6779e-1 (5.54e-3) -	6.7526e-1 (4.41e-3) -	6.3921e-1 (1.44e-2)-	6.7513e-1 (4.14e-3) -	6.8097e-1(5.37e-3)
UF3	3.0708e-1 (1.09e-2) -	3.7328e-1 (3.44e-2) -	4.1372e-1 (3.59e-2) =	4.2113e-1 (4.38e-2)+	4.7485e-1 (5.35e-2) +	3.9650e-1(3.82e-2)
UF4	2.8212e-1 (7.66e-3) -	3.7380e-1 (2.77e-3) -	3.8276e-1 (3.16e-3) -	3.5077e-1 (1.94e-2)-	3.1364e-1 (9.62e-3) -	3.8631e-1(6.40e-4)
UF5	0.0000e+0 (0.00e+0) -	0.0000e+0 (0.00e+0) -	6.5165e-3 (1.70e-2) -	2.4151e-2 (4.19e-2)-	6.5449e-2 (6.25e-2) -	1.2114e-1(1.07e-1)
UF6	0.0000e+0 (0.00e+0) -	7.4246e-2 (3.56e-2) -	2.1000e-1 (9.06e-2) =	1.8306e-1 (1.01e-1)=	2.6646e-1 (7.32e-2) +	2.0984e-1(7.54e-2)
UF7	1.8942e-1 (6.65e-2) -	5.1279e-1 (4.88e-2) -	4.2076e-1 (1.36e-1) -	3.8568e-1 (1.60e-1)-	4.6785e-1 (1.13e-1) -	5.4090e-1(5.40e-3)
UF8	2.9288e-1 (1.69e-2) =	2.3266e-1 (2.20e-2) -	2.6658e-1 (3.51e-2) -	3.2345e-1 (4.64e-2)+	1.3768e-2 (1.39e-2) -	2.9271e-1(2.85e-2)
UF9	2.0708e-1 (1.94e-2) -	2.8894e-1 (4.87e-2) -	4.1770e-1 (7.44e-2) -	5.0881e-1 (6.81e-2)=	2.2068e-2 (3.11e-2) -	4.8136e-1(7.81e-2)
UF10	9.0913e-2 (2.22e-5) +	0.0000e+0 (0.00e+0) =	9.4352e-2 (2.47e-2)+	1.6307e-1 (1.08e-1)+	0.0000e+0 (0.00e+0) =	0.0000e+0(0.00e+0)
Nos.+/-/=	1/16/2	3/15/1	1/14/4	4/13/2	3/12/4

It can be seen from Table 2 that the IGD results of TSDE are optimal in 10 of the 19 functions. This shows the advantages of TSDE in dealing with complex multi-objective optimization functions. Compared with dMOPSO, TSDE is inferior to dMOPSO on UF10, and is better than dMOPSO on the remaining 18 test functions. Compared with MPSOD, TSDE is inferior to MPSOD on UF1, UF2, DTLZ2 and DTLZ4, and there is no significant difference from MPSOD on ZDT6 and UF3, and is better than MPSOD on the remaining 13 test functions. Compared with MMOPSO, TSDE is inferior to MMOPSO on UF3 and UF10, and is better than MMOPSO on the remaining 17 test functions. Compared with NMPSO, TSDE is inferior to NMPSO on DTLZ7, UF3 and UF10, and is better than MMOPSO on the remaining 16 test functions. Compared with CMOPSO, TSDE is inferior to CMOPSO on ZDT1, ZDT2, ZDT3, DTLZ2, DTLZ4, and UF3, and there is no significant difference from CMOPSO on ZDT6 and DTLZ1, UF1 and UF6, and is better than CMOPSO on the remaining 10 test functions.

Table 3 shows the HV mean and std of TSDE and comparison algorithm after running 30 times independently on each test function. Among the 19 test functions, 11 of TSDE achieved the best results. Compared with dMOPSO, TSDE is inferior to dMOPSO on UF10, and there is no significant difference from dMOPSO on ZDT3 and UF8, and is better than dMOPSO on the remaining 16 test functions. Compared with MPSOD, TSDE is inferior to MPSOD on ZDT2, DTLZ2 and DTLZ4, and there is no significant difference from MPSOD on UF10, and is better than MPSOD on the remaining 15 test functions. Compared with MMOPSO, TSDE is inferior to MMOPSO on UF10, and there is no significant difference from MMOPSO on ZDT2, DTLZ4, UF3 and UF6, and is better than MMOPSO on the remaining 14 test functions. Compared with NMPSO, TSDE is inferior to NMPSO on DTLZ2, UF3, UF8 and UF10, and there is no significant difference from NMPSO on UF6 and UF9, and it is better than MMOPSO on the remaining 13 test functions. Compared with CMOPSO, TSDE is inferior to CMOPSO on DTLZ2, UF3 and UF6, and there is no significant difference from CMOPSO on ZDT1, ZDT2, DTLZ4 and UF10, and is better than CMOPSO on the remaining 12 test functions. It is worth noting that due to the different calculation methods of the indicators, the optimal IGD value and HV value are sometimes inconsistent. This does not affect the quality of the evaluation algorithm, but shows the necessity of using two evaluation indicators.

In order to show more vividly the superiority of the TSDE algorithm over the comparison algorithm, Figs. 4-12 shows the Pareto front of each algorithm in some tests. It can be seen that the Pareto front of TSDE on ZDT3 is slightly inferior to CMOPSO. On ZDT4, TSDE has the best distribution and convergence among all algorithms. In the DTLZ series, the Pareto front of TSDE is the best on DTLZ1 and DTLZ7. On DTLZ4, the Pareto front of TSDE is inferior to MPSOD and CMOPSO. In the UF series, the Pareto front of TSDE is the best on UF2,UF4 and UF7. On UF9, the Pareto front of TSDE is slightly inferior to NMPSO.

Fig. 4

Pareto front of TSDE, dMOPSO, MPSOD, MMOPSO, NMPSO, CMOPSO on ZDT3.

Fig. 5

Pareto front of TSDE, dMOPSO, MPSOD, MMOPSO, NMPSO, CMOPSO on ZDT4.

Fig. 6

Pareto front of TSDE, dMOPSO, MPSOD, MMOPSO, NMPSO, CMOPSO on DTLZ1.

Fig. 7

Pareto front of TSDE, dMOPSO, MPSOD, MMOPSO, NMPSO, CMOPSO on DTLZ4.

Fig. 8

Pareto front of TSDE, dMOPSO, MPSOD, MMOPSO, NMPSO, CMOPSO on DTLZ7.

Fig. 9

Pareto front of TSDE, dMOPSO, MPSOD, MMOPSO, NMPSO, CMOPSO on UF2.

Fig. 10

Pareto front of TSDE, dMOPSO, MPSOD, MMOPSO, NMPSO, CMOPSO on UF4.

Fig. 11

Pareto front of TSDE, dMOPSO, MPSOD, MMOPSO, NMPSO, CMOPSO on UF7.

Fig. 12

Pareto front of TSDE, dMOPSO, MPSOD, MMOPSO, NMPSO, CMOPSO on UF9.

In summary, it is not difficult to find that the solution obtained by TSDE has a good Pareto distribution and a good convergence rate in the process of evolution. Due to the emphasis on diversity enhancement, TSDE performs particularly well when it comes to dealing with discontinuous functions. The results show that TSDE algorithm is improved in both convergence and diversity and is potential in dealing with multi-objective problem.

4.2 Efficiency analysis of strategies

Two comparison algorithms TSDE₁ and TSDE₂ are provided to verify the effectiveness of the strategy in the algorithm. TSDE₁ is the algorithm without the population diversity enhancement strategy. TSDE₂ is the algorithm without the EXA diversity enhancement strategy. Table 4 shows the IGD mean and std of the two comparison algorithms and TSDE after 30 runs on the test function. It can be seen from Table 4 that TSDE gets better results in 11 of the 19 test functions. By comparing TSDE₁ and TSDE₂ respectively, it can be concluded that the applied strategy is effective.

Table 4
Efficiency analysis of strategies

TSDE₁ TSDE₂ TSDE

ZDT1 4.6132e-3(1.54e-4)- 4.4911e-3(1.87e-4)= 4.5820e-3(1.50e-4)

ZDT2 4.6590e-3(1.99e-4)= 4.5464e-3(1.42e-4)= 4.6550e-3(1.91e-4)

ZDT3 5.1296e-3(2.49e-4)= 5.1673e-3(2.19e-4)= 5.1540e-3(2.76e-4)

ZDT4 6.4284e-3(2.42e-3)- 5.0392e-3(1.42e-4)- 4.8040e-3(2.69e-4)

ZDT6 3.4137e-3(1.18e-3)- 2.8391e-3(3.96e-4)+ 3.1490e-3(4.33e-4)

DTLZ1 2.4032e-2(7.08e-4)+ 2.4192e-2(7.80e-4)+ 3.1820e-2(5.05e-2)

DTLZ2 6.4795e-2(2.25e-3)- 6.3613e-2(2.08e-3)- 6.3010e-2(1.80e-3)

DTLZ4 6.6106e-2(2.32e-2)= 6.5594e-2(6.35e-2)= 6.5420e-2(2.13e-3)

DTLZ7 7.3122e-2(3.05e-3)= 7.2264e-2(4.99e-2)= 7.1760e-2(3.80e-3)

UF1 6.7016e-2(1.48e-2)- 6.7355e-2(1.58e-2)- 6.6270e-2(1.74e-2)

UF2 3.6384e-2(5.59e-3)= 3.4833e-2(4.34e-3)= 3.4490e-2(5.98e-3)

UF3 3.9038e-1(2.29e-2)- 3.0901e-1(1.78e-2)- 2.7900e-1(3.47e-2)

UF4 4.2195e-2(2.86e-4)+ 4.3251e-2(3.70e-4)+ 4.3970e-2(5.17e-4)

UF5 4.3855e-1(1.46e-1)- 4.3572e-1(1.12e-1)- 4.0000e-1(1.58e-1)

UF6 2.5144e-1(1.14e-1)= 2.6431e-1(1.34e-1)= 2.7150e-1(1.14e-1)

UF7 4.1895e-2(8.29e-3)- 4.3141e-2(4.13e-2)- 3.1360e-2(4.13e-3)

UF8 2.3100e-1(3.47e-2)- 2.1046e-1(3.24e-2)- 1.9810e-1(2.25e-2)

UF9 2.5723e-1(5.76e-2)= 2.5491e-1(3.68e-2)- 2.3100e-1(5.43e-2)

UF10 1.2404e+0(4.99e+0)= 1.4154e+0(2.35e-1)= 1.2640e+0(4.31e-1)

Nos.+/-/= 2/9/8 3/8/8

	TSDE₁	TSDE₂	TSDE
ZDT1	4.6132e-3(1.54e-4)-	4.4911e-3(1.87e-4)=	4.5820e-3(1.50e-4)
ZDT2	4.6590e-3(1.99e-4)=	4.5464e-3(1.42e-4)=	4.6550e-3(1.91e-4)
ZDT3	5.1296e-3(2.49e-4)=	5.1673e-3(2.19e-4)=	5.1540e-3(2.76e-4)
ZDT4	6.4284e-3(2.42e-3)-	5.0392e-3(1.42e-4)-	4.8040e-3(2.69e-4)
ZDT6	3.4137e-3(1.18e-3)-	2.8391e-3(3.96e-4)+	3.1490e-3(4.33e-4)
DTLZ1	2.4032e-2(7.08e-4)+	2.4192e-2(7.80e-4)+	3.1820e-2(5.05e-2)
DTLZ2	6.4795e-2(2.25e-3)-	6.3613e-2(2.08e-3)-	6.3010e-2(1.80e-3)
DTLZ4	6.6106e-2(2.32e-2)=	6.5594e-2(6.35e-2)=	6.5420e-2(2.13e-3)
DTLZ7	7.3122e-2(3.05e-3)=	7.2264e-2(4.99e-2)=	7.1760e-2(3.80e-3)
UF1	6.7016e-2(1.48e-2)-	6.7355e-2(1.58e-2)-	6.6270e-2(1.74e-2)
UF2	3.6384e-2(5.59e-3)=	3.4833e-2(4.34e-3)=	3.4490e-2(5.98e-3)
UF3	3.9038e-1(2.29e-2)-	3.0901e-1(1.78e-2)-	2.7900e-1(3.47e-2)
UF4	4.2195e-2(2.86e-4)+	4.3251e-2(3.70e-4)+	4.3970e-2(5.17e-4)
UF5	4.3855e-1(1.46e-1)-	4.3572e-1(1.12e-1)-	4.0000e-1(1.58e-1)
UF6	2.5144e-1(1.14e-1)=	2.6431e-1(1.34e-1)=	2.7150e-1(1.14e-1)
UF7	4.1895e-2(8.29e-3)-	4.3141e-2(4.13e-2)-	3.1360e-2(4.13e-3)
UF8	2.3100e-1(3.47e-2)-	2.1046e-1(3.24e-2)-	1.9810e-1(2.25e-2)
UF9	2.5723e-1(5.76e-2)=	2.5491e-1(3.68e-2)-	2.3100e-1(5.43e-2)
UF10	1.2404e+0(4.99e+0)=	1.4154e+0(2.35e-1)=	1.2640e+0(4.31e-1)
Nos.+/-/=	2/9/8	3/8/8

First, compare TSDE₁ with TSDE. As can be seen from Table 4, TSDE is inferior to TSDE₁ in DTLZ1 and UF4, and there is no obvious difference from TSDE₁ in 8 test functions such as ZDT2. In the remaining 9 test functions, TSDE is better than TSDE₁. This result shows that the strategy of population diversity enhancement strategy is effective for the algorithm. The advantages of population diversity enhancement strategy are as follows:

1. The process of randomly generating new individuals can effectively prevent the algorithm from trapping in local optimal Pareto front.

2. Unlike the traditional sorting method, the optimal sequence method associates each individual with a value of K, which makes the selection process of individuals clear and fast.

3. The cell density method makes the individuals in the same cell not be selected at the same time, which increases the population diversity.

Then compare TSDE₂ with TSDE. TSDE is inferior to TSDE₂ in ZDT6, DTLZ1 and UF4, there is no obvious difference from it in 8 test functions such as ZDT1, and it is better than TSDE₂ in the other 8 test functions. The advantage of TSDE is mainly reflected in the test function with complex Pareto front, which effectively alleviates the search stagnation of the complex test function.

Figure 13 shows the IGD variation trends for some test function. The horizontal axis is the 100 uniform sampling points during operation and the vertical axis is the corresponding log(IGD) values. The green line represents the IGD variation trend of TSDE₁ and the red line represents the IGD variation trend of TSDE. The blue line represents the IGD variation trend of TSDE₂. It also can be seen that TSDE can get better results on DTLZ and UF series test functions and shows obvious advantages in dealing with irregular frontier. This should be the result of PCA operator producing more non-dominated solutions and increasing the coverage of the frontier. Anyway, the addition of strategies did not slow down the convergence speed of the algorithm, but improved the Pareto front performance of the algorithm.

Fig. 13

Evolution process of TSDE, TSDE₁ and TSDE₂.

4.3 Parameter selection

The parameter h in EXA diversity enhancement strategy represents the number of components of the original solution set when a new solution set is generated. The more eigenvectors are adopted, the larger the parameter h is. The influence of reference vector on the result is from left to right and from large to small. When the h value is too small, the generated solution may be quite different from the original solution, which can not retain the advantages of the original solution. When the h value is close to the objective space dimension, the new solution is almost the same as the original solution, which can not achieve better screening effect.

Table 5 shows the IGD results of the algorithm running on ZDT, DTLZ and UF series functions with different h values. In order to make the result clearer, only EXA diversity enhancement strategy is used. An h suitable for all test functions should be selected. Since the dimensions of different test functions are different in the decision space. The suitable h should not be greater than the test function DTLZ1 with the smallest dimension. The dimension of DTLZ1 in the decision space is 7, so h should be selected among 1, 3, and 7. h = 1 obtains the best value on the 2 test functions of DTLZ4 and UF6. h = 7 obtained the best value in the 4 test functions of ZDT4, ZDT6, UF1 and UF3. However, h = 3 obtains the best results on 8 test functions, so the final h takes 3.

Table 5
The effect of parameter h

h=1 h=3 h=7 h=10 h=15 h=20 h=25

ZDT1 4.5858e-3(1.57e-4) 4.6133e-3(1.54e-4) 4.5780e-3(1.30e-4) 5.2066e-3(3.01e-4) 4.5704e-3(1.63e-4) 5.1075e-3(2.13e-4) 5.0738e-3(2.44e-4)

ZDT2 4.6935e-3(1.82e-4) 4.6590e-3(1.99e-4) 4.7069e-3(1.73e-4) 5.3534e-3(3.73e-4) 4.7468e-3(3.20e-4) 5.2549e-3(2.21e-4) 5.2803e-3(2.79e-4)

ZDT3 5.1522e-3(2.61e-4) 5.1296e-3(2.49e-4) 5.1914e-3(2.57e-4) 5.7793e-3(3.26e-4) 4.7468e-3(3.20e-4) 5.4658e-3(3.25e-4) 5.5605e-3(2.78e-4)

ZDT4 1.6426e-2(2.51e-2) 6.4285e-3(2.42e-3) 5.9950e-3(5.33e-4) 3.5956e-2(4.04e-2) - - -

ZDT6 3.3958e-3(3.54e-4) 3.4137e-3(1.18e-3) 3.2302e-3(4.43e-4) 3.3333e-3(4.09e-4) - - -

DTLZ1 2.4803e-2(1.17e-3) 2.4032e-2(7.08e-4) 2.5298e-2(9.77e-4) - - - -

DTLZ2 6.4970e-2(2.18e-3) 6.4795e-2(2.25e-3) 6.5492e-2(2.50e-3) 6.5307e-2(2.32e-3) - - -

DTLZ4 6.5380e-2 (1.95e-3) 6.6106e-2(2.32e-3) 6.5975e-2(2.07e-3) 6.6000e-2(1.72e-3) - - -

DTLZ7 7.3928e-2 (4.37e-3) 7.3123e-2(3.05e-3) 7.2493e-2(3.57e-3) 7.1828e-2(4.26e-3) 7.2323e-2(4.50e-3) 7.2037e-2(3.39e-3) -

UF1 7.0070e-2 (1.53e-2) 6.7017e-2(1.48e-2) 6.6146e-2(1.40e-2) 7.0863e-2(1.12e-2) 6.8860e-2(2.00e-2) 6.6731e-2(1.98e-2) 7.3184e-2(2.46e-2)

UF2 3.9034e-2 (3.42e-3) 3.6385e-2(5.59e-3) 3.7568e-2(4.40e-3) 4.2360e-2(4.11e-3) 4.1990e-2(3.66e-3) 4.4902e-2(3.81e-3) 4.5019e-2(4.67e-3)

UF3 4.0216e-1(2.04e-2) 3.9039e-1(2.29e-2) 3.8759e-1(1.80e-2) 4.1708e-1(2.65e-2) 4.2554e-1(3.30e-2) 4.4130e-1(2.92e-2) 4.4144e-1(3.26e-2)

UF4 4.2240e-2(3.81e-4) 4.2195e-2(2.86e-4) 4.2564e-2(4.58e-4) 4.2199e-2(4.33e-4) 4.3288e-2(5.29e-4) 4.3132e-2(5.35e-4) 4.3304e-2(5.61e-4)

UF5 4.2128e-1 (1.31e-1) 4.3855e-1(1.46e-1) 3.4272e-1(7.27e-2) 4.4911e-1(1.42e-1) 2.8029e-1(9.85e-2) 3.5153e-1(7.65e-2) 3.7219e-1(8.08e-2)

UF6 2.4472e-1(1.26e-1) 2.5145e-1(1.14e-1) 2.8488e-1(9.69e-2) 2.9580e-1(1.29e-1) 2.8029e-1(9.85e-2) 3.5332e-1(1.47e-1) 2.7634e-1(8.86e-2)

UF7 4.4712e-2(6.27e-3) 4.1896e-2(8.29e-3) 4.8393e-2(1.86e-2) 4.9391e-2(6.44e-3) 5.0014e-2(1.54e-2) 5.4518e-2(1.63e-2) 5.9604e-2(2.05e-2)

UF8 2.4064e-1(3.58e-2) 2.3098e-1(3.47e-2) 2.4620e-1(3.76e-2) 2.6964e-1(4.80e-2) 3.0842e-1(5.21e-2) 3.1203e-1(5.24e-2) 3.1339e-1(5.14e-2)

UF9 3.3087e-1(5.21e-2) 2.5724e-1(5.76e-2) 2.6457e-1(5.78e-2) 3.5305e-1(6.34e-2) 4.4496e-1(9.46e-2) 4.5963e-1(8.36e-2) 4.6151e-1(8.17e-2)

UF10 1.2682e+0(5.23e-1) 1.2404e+0(4.99e-1) 1.2532e+0(4.28e-1) 1.5085e+0(5.28e-1) 1.1909e+0(3.71e-1) 1.3528e+0(8.36e-2) 1.4390e+0(6.29e-1)

	h=1	h=3	h=7	h=10	h=15	h=20	h=25
ZDT1	4.5858e-3(1.57e-4)	4.6133e-3(1.54e-4)	4.5780e-3(1.30e-4)	5.2066e-3(3.01e-4)	4.5704e-3(1.63e-4)	5.1075e-3(2.13e-4)	5.0738e-3(2.44e-4)
ZDT2	4.6935e-3(1.82e-4)	4.6590e-3(1.99e-4)	4.7069e-3(1.73e-4)	5.3534e-3(3.73e-4)	4.7468e-3(3.20e-4)	5.2549e-3(2.21e-4)	5.2803e-3(2.79e-4)
ZDT3	5.1522e-3(2.61e-4)	5.1296e-3(2.49e-4)	5.1914e-3(2.57e-4)	5.7793e-3(3.26e-4)	4.7468e-3(3.20e-4)	5.4658e-3(3.25e-4)	5.5605e-3(2.78e-4)
ZDT4	1.6426e-2(2.51e-2)	6.4285e-3(2.42e-3)	5.9950e-3(5.33e-4)	3.5956e-2(4.04e-2)	-	-	-
ZDT6	3.3958e-3(3.54e-4)	3.4137e-3(1.18e-3)	3.2302e-3(4.43e-4)	3.3333e-3(4.09e-4)	-	-	-
DTLZ1	2.4803e-2(1.17e-3)	2.4032e-2(7.08e-4)	2.5298e-2(9.77e-4)	-	-	-	-
DTLZ2	6.4970e-2(2.18e-3)	6.4795e-2(2.25e-3)	6.5492e-2(2.50e-3)	6.5307e-2(2.32e-3)	-	-	-
DTLZ4	6.5380e-2 (1.95e-3)	6.6106e-2(2.32e-3)	6.5975e-2(2.07e-3)	6.6000e-2(1.72e-3)	-	-	-
DTLZ7	7.3928e-2 (4.37e-3)	7.3123e-2(3.05e-3)	7.2493e-2(3.57e-3)	7.1828e-2(4.26e-3)	7.2323e-2(4.50e-3)	7.2037e-2(3.39e-3)	-
UF1	7.0070e-2 (1.53e-2)	6.7017e-2(1.48e-2)	6.6146e-2(1.40e-2)	7.0863e-2(1.12e-2)	6.8860e-2(2.00e-2)	6.6731e-2(1.98e-2)	7.3184e-2(2.46e-2)
UF2	3.9034e-2 (3.42e-3)	3.6385e-2(5.59e-3)	3.7568e-2(4.40e-3)	4.2360e-2(4.11e-3)	4.1990e-2(3.66e-3)	4.4902e-2(3.81e-3)	4.5019e-2(4.67e-3)
UF3	4.0216e-1(2.04e-2)	3.9039e-1(2.29e-2)	3.8759e-1(1.80e-2)	4.1708e-1(2.65e-2)	4.2554e-1(3.30e-2)	4.4130e-1(2.92e-2)	4.4144e-1(3.26e-2)
UF4	4.2240e-2(3.81e-4)	4.2195e-2(2.86e-4)	4.2564e-2(4.58e-4)	4.2199e-2(4.33e-4)	4.3288e-2(5.29e-4)	4.3132e-2(5.35e-4)	4.3304e-2(5.61e-4)
UF5	4.2128e-1 (1.31e-1)	4.3855e-1(1.46e-1)	3.4272e-1(7.27e-2)	4.4911e-1(1.42e-1)	2.8029e-1(9.85e-2)	3.5153e-1(7.65e-2)	3.7219e-1(8.08e-2)
UF6	2.4472e-1(1.26e-1)	2.5145e-1(1.14e-1)	2.8488e-1(9.69e-2)	2.9580e-1(1.29e-1)	2.8029e-1(9.85e-2)	3.5332e-1(1.47e-1)	2.7634e-1(8.86e-2)
UF7	4.4712e-2(6.27e-3)	4.1896e-2(8.29e-3)	4.8393e-2(1.86e-2)	4.9391e-2(6.44e-3)	5.0014e-2(1.54e-2)	5.4518e-2(1.63e-2)	5.9604e-2(2.05e-2)
UF8	2.4064e-1(3.58e-2)	2.3098e-1(3.47e-2)	2.4620e-1(3.76e-2)	2.6964e-1(4.80e-2)	3.0842e-1(5.21e-2)	3.1203e-1(5.24e-2)	3.1339e-1(5.14e-2)
UF9	3.3087e-1(5.21e-2)	2.5724e-1(5.76e-2)	2.6457e-1(5.78e-2)	3.5305e-1(6.34e-2)	4.4496e-1(9.46e-2)	4.5963e-1(8.36e-2)	4.6151e-1(8.17e-2)
UF10	1.2682e+0(5.23e-1)	1.2404e+0(4.99e-1)	1.2532e+0(4.28e-1)	1.5085e+0(5.28e-1)	1.1909e+0(3.71e-1)	1.3528e+0(8.36e-2)	1.4390e+0(6.29e-1)

5 Conclusions

In order to solve the problem of premature convergence, a two-stage diversity enhancement differential evolution algorithm for multi-objective optimization problem is proposed. In order to produce high quality offspring, an improved cell density method is used to screen the parent population. In addition, perturb non-dominated solutions in the EXA to increase diversity. After compared with the other advanced algorithms, it can be concluded that the TSDE can obtain better results in several different metrics. Especially on the diversity of the discontinuous problem, TSDE shows better performance.

Although TSDE has achieved certain results in the improvement of MODE, there is still a lot of research space. Here are several possible directions for future research.

(1) The TSDE algorithm shows good convergence and diversity when dealing with some complex multi-objective problems, but it still needs to continue to study in solving high-dimensional and dynamic problems.

(2) According to the evolutionary state of the population, the corresponding evolutionary operator is selected to balance exploration and exploitation. Moreover, change the parameters in the algorithm such as n_size and h with the environment.

(3) Take prevention of slippage and equal relative agreement as the objective function. Apply TSDE to optimize the rolling schedule to guide the production of products.

Footnotes

Acknowledgments

The author would like to thank her mentor and partner. This work was supported by National Key Research and Development Program of China (No. 2018YFB1702300), National Natural Science Foundation of China (No. 62003296), Natural Science Foundation of Hebei (F2020203031), Science and Technology Research Projects of Hebei University (QN2020225), Post-Doctoral Research Projects of Hebei (B2019003021) and Doctoral Foundation of Yanshan University (NO. BL18048).

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Pros	Simple implementation, few control parameters, high convergence accuracy, fast convergence speed and strong robustness
Cons	1. The poor quality of the initial population leads to premature convergence of the population
	2. The EXA cannot produce better solutions to update the population, which leads to the stagnation of the search