Two-Stage Archive Evolutionary Algorithm for Constrained Multi-Objective Optimization

Abstract

The core issue in handling constrained multi-objective optimization problems (CMOP) is how to maintain a balance between objectives and constraints. However, existing constrained multi-objective evolutionary algorithms (CMOEAs) often fail to achieve the desired performance when confronted with complex feasible regions. Building upon this theoretical foundation, a two-stage archive-based constrained multi-objective evolutionary algorithm (CMOEA-TA) based on genetic algorithms (GA) is proposed. In CMOEA-TA, First stage: The archive appropriately relaxes constraints based on the proportion of feasible solutions and constraint violations, compelling the population to explore more search space. Second stage: Sharing valuable information between the archive and the population, while embedding constraint dominance principles to enhance the feasibility of solutions. In addition an angle-based selection strategy was used to select more valuable solutions to increase the diversity of the population. To verify its effectiveness, CMOEA-TA was tested on 54 CMOPs in 4 benchmark suites and 7 state-of-the-art algorithms were compared. The experimental results show that it is far superior to seven competitors in inverse generation distance (IGD) and hypervolume (HV) metrics.

1. Introduction

With the advent of artificial intelligence, in real life, many situations involve constraints on multiple objectives. For example, issues such as allocation of water resources [1], vehicle scheduling [2], and route optimization [3,4]. In order to tackle the significant challenges posed by CMOPs [5], an increasing number of constrained multi-objective evolutionary algorithms (CMOEAs [6]) have been proposed over the past 20 years. These algorithms can generally be classified into those based on indicator-based selection mechanisms [7], penalty-based relaxation mechanisms [8], bi-archival memory schemes [9], and decomposition-based approaches [10]. Generally speaking, constrained multi-objective evolutionary algorithms face significant challenges in maintaining a balance between constraints and objectives [11]. Currently, it is difficult to simultaneously address both constraints and the demands of multiple objectives. Constrained Multi-Objective Optimization Problems (CMOPs) are typically defined as:   

$$\begin{array}{cc}\mathrm{Minimize}\hfill & F\left(x\right)=\left(f_1\left(x\right),\cdots ,f_m\left(x\right)\right)\hfill \\ \mathrm{Subject}\mathrm{to}\hfill & g_i\left(x\right)\le 0,i=1,\cdots ,p\hfill \\ \hfill & h_j\left(x\right)=0,j=p+1,\cdots ,q\hfill \\ \hfill & x\in \Omega \hfill \end{array}$$

(1)

where $x=\left(x_1,x_2,\dots ,x_n\right)$ is a solution vector. $g_i\left(x\right)$ represents the inequality constraints, where p is the number of inequality constraints. $h_j\left(x\right)$ represents the equality constraints, where $q-p$ is the number of equality constraints. $\Omega$ denotes the decision space and m represents the number of objective functions.

When addressing constraint problems, we need to know the constraint violation of solution x for each constraint. Therefore, the constraint violation of solution x on the j-th constraint is calculated as follows:

$$C{V}_j\left(x\right)=\left\{\begin{array}{c}max\left(0,g_j\left(x\right)\right),j=1,\cdots ,p\hfill \\ max\left(0,\left|h_j\left(x\right)\right|-\delta \right),j=p+1,\cdots ,q\hfill \end{array}\right.$$

(2)

In the above formula, $\delta$ serves as a small positive tolerance parameter, which in this paper is set to ${10}^{-6}$, aimed at converting equality constraints into inequality constraints. According to Formula (2), the constraint violation of the solution in each objective function is obtained, and then the total constraint violation is calculated as follows:

$$CV\left(x\right)=\sum _{j=1}^{q}C{V}_j\left(x\right)$$

(3)

The total constraint violation of each solution can be obtained using Formula (3). When $CV\left(x\right)=0$, x is a feasible solution; otherwise, it is an infeasible solution.

The primary challenge in solving CMOPs is to maintain a balance between objectives and constraints [12]. In some literature, researchers are making extensive efforts to achieve this balance, such as the dual population cooperative evolution framework [13], the multiobjective sorting framework [14], the multistage framework [15], and the archive cooperative evolution framework [16]. In the aforementioned frameworks, the multistage framework and archive cooperative evolution are receiving increasing attention. In a multistage framework, tasks are divided, and appropriate criteria and coping strategies are established at different stages to maintain a balance between constraints and objectives. The cooperative evolution framework of the archive aims to record useful information obtained by the algorithm, helping to obtain a complete constrained Pareto front (CPF) [17]. However, they appear to be somewhat inadequate in dealing with CMOPs with diverse characteristics [18]. These algorithms often tend to focus on searching for feasible solutions, neglecting information about infeasible solutions, which can easily lead the algorithm to local optima [19].

Building on the aforementioned theory, this paper integrates the concept of cooperative archive evolution into a dual-stage framework [15]. Different tasks and response strategies [20] are established at different stages, with a temporary archive mechanism implemented to record useful information obtained by the algorithm at each stage, further enhancing the overall performance of the algorithm. The proposed CMOEA-TA is an evolutionary algorithm based on genetic algorithms. After elite selection generates high-quality parents, high-quality offspring are produced through genetic operators, which promotes the population’s rapid convergence to the optimal solution. The main contributions of this paper are summarized as follows:

(1)

A novel two-stage archiving framework is proposed in this paper. In Stage 1, Constraints are appropriately relaxed based on the proportion of feasible solutions and the degree of constraint violation. Simultaneously, under the condition of relaxed constraints, the optimal solution (minimum $CV$ value) is stored in the archive. In Stage 2, an exchange between the archive and the population is conducted, giving equal consideration to objectives and constraints to enhance the convergence of the population.

(2)

A novel constraint learning relaxation mechanism is designed to enhance the algorithm’s exploration capability, prompting the population to attain the complete Pareto front (PF) [21]. At the same time, the archive is continually updated throughout the algorithm’s evolution process, encouraging the algorithm to obtain well-distributed solutions to enhance the convergence and diversity of the population.

(3)

Designing a strict constraint dominance principle for parent selection to generate superior offspring and compel the population to achieve the complete CPF. Simultaneously, a balancing mechanism was embedded to select candidate solutions, thereby enhancing the convergence of the population.

(4)

To validate the proposed CMOEA-TA, this paper conducted comparative experiments on 54 CMOPs against seven state-of-the-art algorithms. The experimental results indicate that CMOEA-TA significantly outperforms its competitors.

The rest of the work in this paper is arranged as follows. In Section 2, the related work is discussed to elucidate the motivation for this study. In Section 3, the proposed CMOEA-TA is presented in detail. Section 4 describes the experimental study of CMOEA-TA. Finally, Section 5 gives an overall summary and hopes for the future.

2. Related Work and Motivation

This section is divided into two parts. Section 2.1 elaborates on the current state of algorithms balancing constraints and objectives. Section 2.2 introduces the strengths and weaknesses of some algorithms, leading to the motivation for this work. To improve the readability of the paper,

Table 1. List of Abbreviations.

Abbreviation	Full Form
CMOP [5]	Constrained Multi-objective Optimization Problems
MOEA	Multi-objective Evolutionary Algorithms
CMOEA [6]	Constrained Multi-objective Evolutionary Algorithms
CMOEA-TA	Two-stage Archiving Constrained Multi-objective Evolutionary Algorithms
ToP [22]	Two-phase Optimization Procedure
PPS [23]	Push and Pull Search
CMOEA-MS [12]	Constrained Multi-Objective Evolutionary Algorithm with Multiple Strategies
CMOEA/D [24]	Constrained Multi-Objective Evolutionary Algorithm based on Decomposition
MSCMO [25]	Multi-Stage Constrained Multi-Objective
CCMO [26]	Coevolutionary Constrained Multiobjective Optimization
CMOQLMT [27]	Constrained Multi-Objective Quadratic Learning-based Multi-task
CCMODE [28]	Cooperative Constrained Multi-Objective Differential Evolution
C-TAEA [29]	Constrained Two-Archive Evolutionary Algorithm
C-TSEA [30]	Constrained Two-Stage Evolutionary Algorithm
CMOEA-TWC [31]	Constrained Multiobjective Evolutionary Algorithm with Two-Archive Weak Cooperation
IMTCMO [32]	Infeasibility-based Multi-Task Constrained Multi-Objective Optimization
DSPCMDE [33]	Dynamic Selection Preference-Assisted Constrained Multiobjective Differential Evolution
TSTI [34]	Two-Stage Three-Indicator
SPEA	Strength Pareto Evolutionary Algorithm
GA	Genetic Algorithm
SO	Swarm Optimization
PF [21]	Pareto Front
CPF [17]	Constrained Pareto Front
CV	Constraint Violation
CDP	Constraint Dominance Principle
FE	Function Evaluations
$fr$	Feasible Ratio
IGD [35]	Inverse Generation Distance
HV [36]	HyperVolume

outlines all the abbreviations.

2.1. Balancing Constraints and Objectives

To address the balance between objectives and constraints, current CMOEAs can be roughly classified into four categories.

2.1.1. Multi-Stage CMOEAs

This type of algorithm balances constraints and objectives by setting different tasks and responsing strategies at different stages. Wang et al. [22] proposed a two-stage framework called ToP. In the first stage, the weighted vector constrained multi-objective problem is decomposed into constrained single-objective problems, encouraging the population to discover more promising feasible regions. In the second stage, a CMOEA is executed to compel the population to obtain the complete constrained pareto front (CPF) [17]. Fan et al. [23] proposed a novel Push-Pull (PPS) framework mechanism, comprising two stages: the push stage and the pull stage. In the push stage, no constraints are considered, exploring the entire search space, while in the pull stage, constraints are relaxed appropriately to pull infeasible solutions into feasible regions. Tian et al. [12] proposed a two-stage algorithm named CMOEA-MS, which dynamically switches stages during the evolution process to promote a balance between objectives and constraints. Zhang et al. [37] introduced an escape strategy detection mechanism, where the task is to determine whether the population is trapped in a local optimum. If a local optimum is detected, the escape strategy is triggered to facilitate escaping. Tian et al. [25] proposed a multi-stage algorithm named MSCMO, which handles complex constraints one by one and sets corresponding strategies at different stages.

2.1.2. Multi-Population CMOEAs

This second category involves employing a multi-population strategy to balance constraints and objectives. Tian et al. [26] proposed a weak cooperative evolution framework called CCMO, in which two populations play a strong-weak collaboration role, enhancing the population’s ability for external exploration. Ming et al. [27] proposed a multi-task framework called CMOQLMT, which adopts different strategies for different tasks and employs a reinforcement learning mechanism to select corresponding auxiliary tasks. A differential evolution framework named CCMODE [28] is proposed, where constrained single-objective optimization and constrained multi-objective optimization are configured as separate archive populations.

2.1.3. Multi-Ranking CMOEAs

The third category achieves a balance between objectives and constraints through multi-ranking strategies. Wang et al. [38] used pareto dominance and constraint dominance relationships as evaluation metrics to assess the feasibility of algorithms. In [39], a three-objective framework for convergence, diversity, and feasibility was proposed, defining three metrics for evolution within the three-objective framework. Wang et al. [40] proposed a novel metric framework that integrates the metrics of MOEAs with constraint handling techniques, optimizing objectives through metrics and optimizing constraints through constraint handling techniques.

2.1.4. Archive-Based CMOEAs

The fourth category achieves a balance between objectives and constraints through archive-based strategies. Yao et al. [29] proposed a dual-archive framework called C-TAEA, where the CA archive simultaneously optimizes objectives and constraints, enhancing population convergence, while the DA archive only optimizes objectives, enhancing population diversity. Ming et al. [30] proposed a two-stage framework with a single archive called C-TSEA, where the archive assists population evolution, enhancing the algorithm’s exploration capability of feasible regions and facilitating the population’s attainment of a complete CPF. Li et al. [31] proposed a weakly cooperative dual-archive framework called CMOEA-TWC. The driving archive only considers objectives, while the general archive considers both objectives and constraints. The two archives weakly cooperate, sharing useful information to enhance the algorithm’s exploration capability and overall performance.

2.2. Motivation

To strike a balance between objectives and constraints, researchers have conducted a significant amount of research. In the initial research stage, Jain et al. [24] proposed the CMOEA/D algorithm, which decomposes constrained multi-objective problems into constrained single-objective subproblems and optimizes these subproblems to obtain a series of evenly distributed solutions. As shown in Figure 1a, although CMOEA/D obtained a relatively evenly distributed set of solutions on LIRCMOP14 [41], the density of the solutions is rather sparse, and it can not guarantee the algorithm to achieve a complete Pareto front. In ToP [22], although constrained multi-objective problems are transformed into constrained single-objective optimization problems by reference vectors, the handling process of subproblems is not sufficiently refined, resulting in slow convergence speed and affecting the overall performance of the algorithm. The performance of ToP on MW2 is illustrated in Figure 1b, where it’s evident that ToP fails to achieve complete convergence and overlooks many feasible regions. In the proposed algorithm MSCMO [25], the process of satisfying the constraints one by one to make the individual a feasible solution is very slow. Therefore, within a limited computational budget, it can not be guaranteed that the solutions will meet all the constraints. As shown in Figure 1c, the distribution of MSCMO [25] on MW12 reveals that the solutions are scattered and there are many infeasible individuals outside the feasible region. It hasn’t fully converged to the pareto front (PF) [21] and hasn’t obtained a complete constrained pareto front (CPF) [17]. Based on the above theory, it can be concluded that achieving a balance between constraints and objectives is extremely important.

Qiao et al. [32] proposed a multi-population cooperative algorithm called IMTCMO, which assists auxiliary populations to evolve along with the main population, significantly enhancing the algorithm’s performance. However, it neglects the information carried by individuals themselves, resulting in poor migration capability between populations, which easily traps the algorithm into local optima and greatly wastes the algorithm’s exploratory ability and resources. Faced with these challenges, this paper adds an archiving mechanism on top of the existing two-stage approach. As populations evolve continuously, the algorithm records information about solutions and preserves promising individuals, thereby enhancing the convergence of the populations. A balancing mechanism has been embedded in the process of parent selection, thereby enhancing the diversity and convergence of the population. To discover more exploration space and promising solutions, constraints are appropriately relaxed based on the proportion of feasible solutions, preventing the population from getting trapped in local optima and facilitating the population to reach a complete CPF.

3. The Proposed CMOEA-TA

3.1. Framework of CMOEA-TA

The algorithm in this paper is based on genetic algorithms. Under the algorithm framework designed in this paper, other algorithms can also be used as underlying algorithms, such as swarm optimization, differential evolution, etc. The framework of CMOEA-TA is presented in Algorithm 1. As shown in Algorithm 1, generate an initial population ($P_M$) arbitrarily in the decision space, apply constraint relaxation by factor $\epsilon$ to obtain the Archive. Compute the minimum constraint violation value in $P_M$ for subsequent $Archive$ $updates$. The algorithm then proceeds to iterate within the dual-stage framework we have established. Lines 7 to 13 constitute the first stage of the algorithm. In line 7, the Pareto front of $P_M$ is obtained through non-dominated sorting. In line 8, $P_F$ is combined with $P_M$ to form $P_1$. In line 9, through an elitist selection mechanism, parents are selected from the merged set $P_1$ and the $Archive$. In line 10, genetic operators are applied to generate offspring. At this point, our paper introduces the balancing mechanism of DSPCMDE [33], ensuring the convergence of the population with respect to the Pareto front. The specific formula for the balancing parameter is as follows:

$$a=0.5\times \left(1-cos\left((1-{\displaystyle \frac{t}{\mathrm{T}}})\pi \right)\right)$$

(4)

At the same time, CDP is used as a prerequisite for selecting parents. The specific formula is as follows:

$$x_1\prec x_2\iff \left\{\begin{array}{c}{f}_i\left(x_1\right)\le f_i\left(x_2\right),\forall i\in \{1,2,\dots ,m\},\hfill \\ \exists i\in \{1,2,\dots ,m\}:f_i\left(x_1\right)<f_i\left(x_2\right),\hfill \\ g_j\left(x_1\right)\ge 0,\forall j,\hfill \\ h_k\left(x_1\right)=0,\forall k,\hfill \end{array}\right.$$

(5)

When the condition $x_1\prec x_2$ is satisfied, solution $x_1$ dominates solution $x_2$ under the constraints. After obtaining the parents ($M_1$), genetic operators are applied to generate offspring ($Q_1$). Next, the minimum constraint violation value in the offspring is calculated for the next step of $Archive$ updating. After obtaining the new $Archive$, the population ($P_M$) and offspring ($Q_1$) are selected using angle-based environmental selection. At this point, the first stage concludes officially, and the algorithm proceeds to the second stage once the termination conditions are met.

When the algorithm enters the second stage, In line 15, there is an exchange between $P_M$ and $Archive$, ensuring that the population possesses high-quality candidate solutions. Lines 16 to 17 carry out the same parent selection and offspring generation operations, followed by the execution of elitist environmental selection for the final population $P_M$. To gain a more intuitive understanding of CMOEA-TA, we present its framework flowchart as shown in Figure 2.

Algorithm 1 Framework of CMOEA-TA.

Input: N (population size)  Output: P (final population)   1: $P_M\leftarrow$ Initialize the population (N);   2: Calculate the CV values of solutions in $P_M$ by the Formula (3);   3: $Archive\leftarrow P_M$($\left|CV\right|$<=$\epsilon$)   4: $Zmin\leftarrow$ Find the minimum constraint violation value of $P_M$;   5: while the terminal criterion is not fulfilled do   6:    if $FE\le \alpha MaxFE$ then   7:        $P_F\leftarrow$ Obtain the PF of $P_M$ by non-dominated sorting;   8:        $P_1\leftarrow$ $P_M\cup P_F$   9:        $M_1\leftarrow$ Perform the elite mating selection ($P_1$, $Archive$); 10:        $Q_1\leftarrow$ Genetic operator ($M_1$); 11:        $Zmin\leftarrow$ Find the minimum constraint violation value of $Q_1$; 12:        $Archive\leftarrow$ UpdateArchive ($P_M\cup Q_1$, $Archive$); 13:        $P_M\leftarrow$ Angle-based environmental selection ($P_M\cup Q_1$); 14:    else 15:        Exchange $P_M$ with Archive; 16:        $M_2\leftarrow$ Perform the elite mating selection ($P_M$, $Archive$); 17:        $Q_2\leftarrow$ Genetic operator ($M_2$); 18:        $P_M\leftarrow$ Perform the elite environment selection ($P_M\cup Q_2$, $Archive$); 19:    end if 20: end while

3.2. Relaxed the Constraint Learning Mechanism

The relaxed constraint learning mechanism is a commonly used approach aimed at relaxing the constraints of solutions to obtain more useful information from infeasible solutions. The relaxation method typically focuses on constraint violation and the search space, enhancing the exploration of the feasible region and improving the population’s search ability to obtain more solutions. However, during the relaxation process, there is a lack of evaluative indicators to control the boundary search criteria, leading to the unnecessary waste of algorithm resources. Therefore, it is important not only to select high-quality solutions from within the feasible boundary but also to use reference indicators as criteria for scaling the search. The mathematical model of constraint relaxation is as follows:

$$\begin{array}{cc}\mathrm{Minimize}\hfill & F^{\prime }\left(x\right)=\left(f_1\left(x\right),\cdots ,f_m\left(x\right),CV\left(x\right)\right)\hfill \\ \hfill & g\left(x\right)\le {\omega }^t\hfill \\ \hfill \mathrm{Subject}\mathrm{to}& {\omega }^t=\left({\omega }_1^t,\cdots ,{\omega }_q^t\right),t=0,\cdots ,T\hfill \\ \hfill & x\in \Omega \hfill \end{array}$$

(6)

${\omega }^t$ represents the constraint boundary value for lending, where T and t respectively represent the maximum and current generations. In this process, based on the obtained ${\omega }^t$, the inequality constraints are ignored and forcibly converted into equality constraints, with the aim of introducing more infeasible solutions, increasing the population’s diversity, and promoting its rapid convergence to the unconstrained Pareto front. The motivation for adopting relaxed constraints is as follows: (1) First, in order to increase the utilization of infeasible solutions in the archive and reduce the constraints on feasible solutions, we aim to provide the population with more valuable information. This further enhances the population’s exploratory capabilities, allowing it to find more feasible regions. (2) In the first stage, the purpose of relaxing constraints is to incorporate more infeasible solutions, obtaining more valuable information, and prompting the population to converge from feasible and infeasible regions to CPF. This is aimed at increasing the balance between constraints and objectives, and enhancing the algorithm’s convergence and diversity [37]. This lays the foundation for the migration and interaction in the second stage.

$${\gamma }^t=1-{\left({\displaystyle \frac{1}{(CV+1.0\times {10}^{-10})}}\right)}^{\left({\displaystyle \frac{3}{\left(\frac{t}{\mathrm{T}}\right)}}\right)}$$

(7)

$$\gamma =max({\gamma }_i^t,fr)$$

(8)

$${\omega }^t={\displaystyle \frac{\gamma (T-t)(-ln(\gamma )-4)}{Tln\left(t\right)}}$$

(9)

In the selection of relaxation values, we adopt an adaptive relaxation approach. The selection of ${\omega }^t$ is subject to constraints such as the proportion of feasible solutions and the degree of constraint violation. When $fr$ is relatively large, we choose $fr$ as the relaxation value to introduce more infeasible solutions and reduce their proportion of feasibility. Conversely, when ${\omega }_i^t$ is relatively large, we use ${\omega }_i^t$ as the relaxation value to obtain more information from infeasible solutions and employ constraint-handling techniques to remove redundant solutions.

3.3. The Mechanism of Elite Mating Pool Selection

Algorithm 2 describes the entire process of elite parent selection. First, calculate the number of feasible solutions, denoted as $N_f$, in set $Archive$. When the quantity of $N_f$ is greater than N or $Archive$ is less than N, supplement and eliminate using boundary binary tournaments. Otherwise, introduce the balancing mechanism from DSPCMDE to select high-quality solutions from the population and $Archive$. When $rand$ is less than a, select two solutions ($x_1$ and $x_2$) from the population, and use Formula (5) from lines 8 to 11 to identify dominated solutions. In this way, the algorithm’s exploration capability is enhanced while maintaining the convergence of the population. If two solutions are mutually non-dominated, select solutions with lower density for supplementation. To provide the population with more exploration space and enhance its diversity. Alternatively, we select two solutions from $Archive$ to perform the above operation.

Algorithm 2 Elite mating pool selection.

Input: $P_M$ (main population), $Archive$, N (population size), a  Output: $M_2$ (mating pool)   1: $U\leftarrow$ $P_M\cup Archive$;   2: $N_f\leftarrow$ Calculate the number of feasible solutions in $Archive$;   3: if $N_f>N$ or $Archive<N$ then   4:     $M_2\leftarrow$ Binary tournament mating selection (U);   5: else   6:      if $rand<a$ then   7:          Randomly select two solutions ($x_1$ and $x_2$) in $P_M$;   8:          $M_2\leftarrow$ Select solutions with stronger dominance based on Equation (5);   9:          $M_2\leftarrow$ Select solutions with better diversity based on Equation (11); 10:      else 11:          Randomly select two solutions ($x_1$ and $x_2$) in $Archive$; 12:          $M_2\leftarrow$ Select solutions with stronger dominance based on Equation (5); 13:          $M_2\leftarrow$ Select solutions with better diversity based on Equation (11); 14:      end if 15: end if

3.4. Updated Archive

Algorithm 3 elaborates on the entire process of updating the archive. Firstly, the number of feasible solutions in the combined population P (Algorithm 1 $P_M\cup Q_1$, $Archive$) is determined based on the relaxed constraint conditions. When the number of feasible solutions is less than N, the solutions in P are sorted in ascending order according to Formula (3), and the N-th solution is selected to supplement the update of the archive. When the number of feasible solutions exceeds N, the fitness indicator of SPEA2-CDP [42] is used as the evaluation criterion. By using Formula (10), the fitness of all solutions continues to be calculated, thereby removing redundant inferior solutions. The fitness evaluation formula is as follows:

$$fitness\left(x\right)=\sum _{y\in \mathcal{C},y\ne x}\left|R_x\right|+{\displaystyle \frac{1}{D_x}}$$

(10)

$R_x$ represents the number of solutions that can dominate x, and is used as an indicator of the convergence strength of x.

$$D\left(x\right)={\displaystyle \frac{1}{{\sum }_{i=1}^{k}d\left(x,{\Omega }_i\right)+2}}$$

(11)

Here, $D\left(x\right)$ represents the crowding degree of solution x. d(x, ${\Omega }_i$) represents the distance from K nearest neighboring solutions to solution x.

Algorithm 3 UpdatedArchive.

Input: N (population size) P (combined population) $Archive$  Output: Updated$Archive$   1: if  $Archive\le N$ then   2:      Sort P in ascending order according to CV;   3:      Updated$Archive\leftarrow$ Select the first N solutions;   4: else   5:     $F\leftarrow$ Calculate the fitness of $Archive$ according to CDP;   6:     $D\leftarrow$ Calculate the minimum Euclidean distance between individuals in $Archive$;   7:     $R\leftarrow$ F + ${\displaystyle \frac{1}{D}}$   8:     Updated$Archive\leftarrow$ Delete redundant solutions with maximum R;   9: end if 10: return Updated$Archive$

4. Experimental Study

The experimental section introduces the setup of the entire experimental process, elucidates the advantages of CMOEA-TA through IGD [35] and HV [36] metrics, and finally conducts ablation experiments to validate the importance of each innovation.

4.1. Experimental Setup

All experimental setups involved in this paper are as follows:

(1) Test problems

In this study, we employed four commonly used benchmark test suites: MW [43], LIRCMOP [41], ZXH_CF [44], and C_DTLZ [45]. A total of 54 test problems were considered, with the number of objective functions set to their default values without modification.

(2) Performance indicator

The evaluation metrics used in this paper are the most commonly used inverted generational distance (IGD) and hypervolume (HV). A smaller IGD value indicates better performance of the algorithm, while HV behaves inversely; a larger HV value indicates better performance of the algorithm. The black cells in the table represent the best values achieved by algorithms on the test problems.

(3) Compared algorithms

In the experimental comparison process of this paper, seven advanced algorithms were selected, namely ToP [22], C-TAEA [29], CMOEA_MS [12], CMOEA/D [24], TSTI [34], IMTCMO [32] and MSCMO [25]. Among them, C-TAEA, ToP, TSTI, and CMOEA_MS are two-stage algorithms, IMTCMO is an archive-based multi-population algorithm, and MSCMO is a two-stage archive algorithm.

(4) Algorithm parameters

The population size is set to 100, with 100,000 evaluations, and the algorithm runs for 20 iterations. To ensure fairness, the genetic and other parameters of the algorithm are kept unchanged as stated in their paper. Wilcoxon [46] tests are conducted for comparative experimental analysis at a significance level of 0.05. The transition parameter $\alpha$ for the two-stage conversion is set to 0.4. The symbols “+”, “−”, and “≈” in the text respectively represent “superior to”, “inferior to”, and “comparable to” our proposed CMOEA-TA, serving to facilitate readers’ quick comparison of performance differences between algorithms. The gray text in all the tables throughout the paper indicates the best values achieved.

(5) Platform and device requirements

The experiments in this paper were conducted using MATLAB software version 2021 or above, and all experiments were performed on the PlatEMO 4.3 [47] platform. The experiments were conducted on computers running Windows 10 or above, with X64-based processors. The analysis plots involved in this paper were created using the 2021 versions of Origin and draw software.

4.2. Comprehensive Analysis of IGD Values

From

Table 2. The IGD results (mean and standard deviation) obtained by eight algorithms on the MW, LIRCMOP, ZXH_CF and C_DTLZ problems.

Problem	M	D	CMOEA/D	ToP	TSTI	MSCMO	CMOEA_MS	CTAEA	IMTCMO	CMOEA-TA
MW1	2	15	2.3584e-2 (6.07e-2) −	NaN (NaN)	5.1794e-3 (9.85e-3) ≈	1.8263e-2 (3.83e-2) −	4.2037e-2 (6.54e-2) −	3.3830e-2 (3.48e-2) −	1.7055e-3 (2.97e-5) +	1.8848e-3 (1.21e-3)
MW2	2	15	2.1518e-2 (7.13e-3) ≈	1.7940e-1 (1.48e-1) −	1.7211e-2 (6.63e-3) ≈	2.2471e-2 (7.64e-3) ≈	2.4023e-2 (8.46e-3) ≈	1.8133e-2 (7.89e-3) ≈	5.9994e-2 (3.53e-2) −	2.0670e-2 (9.10e-3)
MW3	2	15	5.6961e-3 (6.97e-4) −	2.2097e-1 (3.23e-1) −	1.8927e-2 (3.28e-2) −	5.0488e-2 (1.37e-3) −	1.3648e-2 (5.31e-3) −	1.2954e-2 (2.52e-3) −	4.8148e-3 (1.13e-4) ≈	4.8462e-3 (1.91e-4)
MW4	3	15	4.1786e-2 (1.00e-3) −	7.2571e-1 (0.00e+0) ≈	4.1814e-2 (1.13e-3) −	6.1061e-2 (2.08e-2) −	4.9339e-2 (7.63e-3) −	6.4495e-2 (2.06e-2) −	4.8393e-2 (1.35e-2) −	4.0495e-2 (4.23e-4)
MW5	2	15	1.9579e-3 (1.83e-3) −	7.5416e-1 (0.00e+0) ≈	5.8355e-2 (1.62e-1) −	3.2066e-2 (1.18e-1) −	6.9010e-2 (1.22e-1) −	1.4127e-2 (4.15e-3) −	1.8159e-3 (5.34e-4) −	6.9501e-4 (6.65e-4)
MW6	2	15	1.8220e-2 (1.07e-2) ≈	7.3029e-1 (3.51e-1) −	6.3375e-2 (1.30e-1) ≈	2.3689e-2 (1.13e-2) ≈	1.7844e-1 (2.16e-1) −	1.0959e-2 (6.47e-3) +	1.5146e-1 (1.88e-1) −	2.4049e-2 (1.20e-2)
MW7	2	15	2.7204e-2 (1.00e-1) −	5.7367e-2 (9.94e-2) −	4.5487e-3 (4.55e-4) −	4.4436e-3 (3.77e-4) −	3.1495e-2 (2.92e-2) −	6.7262e-3 (5.04e-4) −	4.3318e-3 (2.04e-4) −	4.1605e-3 (2.18e-4)
MW8	3	15	5.7801e-2 (1.92e-2) −	6.9878e-1 (4.11e-1) −	5.0807e-2 (9.62e-3) −	5.1137e-2 (6.93e-3) −	5.2374e-2 (1.45e-2) −	5.4905e-2 (3.09e-3) −	7.4128e-2 (3.75e-2) −	4.4684e-2 (2.16e-3)
MW9	2	15	1.1867e-2 (1.21e-2) −	8.5441e-1 (8.52e-2) −	6.5016e-2 (1.80e-1) −	7.5498e-2 (2.17e-1) −	2.0160e-1 (2.60e-1) −	8.7774e-3 (1.69e-3) −	5.5130e-3 (9.16e-4) −	4.3101e-3 (1.66e-4)
MW10	2	15	1.2514e-1 (1.73e-1) ≈	NaN (NaN)	9.0594e-2 (1.29e-1) ≈	1.1679e-1 (1.85e-1) ≈	3.6383e-2 (1.90e-2) ≈	2.6026e-2 (1.65e-2) ≈	1.7727e-1 (5.62e-2) −	1.2531e-1 (2.14e-1)
MW11	2	15	8.1605e-2 (2.11e-1) −	5.2819e-1 (2.69e-1) −	6.0490e-3 (1.34e-4) ≈	6.0964e-3 (1.20e-4) ≈	1.4053e-2 (1.01e-2) −	2.0545e-2 (7.72e-3) −	5.9456e-3 (1.35e-4) +	6.0524e-3 (1.07e-4)
MW12	2	15	5.2969e-3 (1.87e-3) +	8.3690e-1 (0.00e+0) ≈	4.8703e-3 (3.82e-4) ≈	9.0572e-2 (2.39e-1) ≈	1.5524e-1 (2.82e-1) −	6.1014e-2 (1.82e-1) −	4.7654e-3 (1.32e-4) ≈	4.3137e-2 (1.72e-1)
MW13	2	15	1.4945e-1 (3.06e-1) −	1.0845e+0 (7.97e-1) −	1.1491e-1 (5.46e-2) −	1.2221e-1 (6.58e-2) −	5.0814e-1 (4.96e-1) −	6.3400e-2 (3.47e-2) ≈	1.6385e-1 (9.68e-2) −	5.6480e-2 (3.29e-2)
MW14	3	15	2.1253e-1 (1.58e-3) −	8.5539e-1 (8.65e-1) −	9.9443e-2 (3.47e-3) −	2.6988e-1 (2.13e-1) −	3.2031e-1 (1.88e-1) −	4.9614e-1 (3.99e-1) −	9.9032e-2 (1.80e-3) −	9.6494e-2 (1.31e-3)
$+/-/\approx$			1/10/3	0/9/3	0/8/6	0/9/5	0/12/2	1/10/3	2/10/2
LIRCMOP1	2	30	2.6605e-1 (3.37e-2) −	3.2853e-1 (2.16e-2) −	2.1023e-1 (1.61e-2) −	1.7366e-1 (6.64e-2) ≈	3.2481e-1 (4.71e-2) −	3.0813e-1 (9.45e-2) −	2.6521e-2 (8.07e-3) +	1.5082e-1 (3.57e-2)
LIRCMOP2	2	30	2.4059e-1 (3.20e-2) −	2.8906e-1 (2.18e-2) −	1.9674e-1 (2.60e-2) −	1.7570e-1 (4.37e-2) −	2.7883e-1 (4.45e-2) −	2.4246e-1 (9.34e-2) −	1.9316e-1 (4.29e-2) −	1.3974e-1 (3.25e-2)
LIRCMOP3	2	30	2.9268e-1 (4.09e-2) −	3.4399e-1 (1.64e-2) −	2.2665e-1 (2.50e-2) −	1.7748e-1 (7.59e-2) ≈	3.2129e-1 (4.27e-2) −	3.4241e-1 (1.64e-1) −	1.5362e-2 (7.41e-3) +	1.4615e-1 (2.23e-2)
LIRCMOP4	2	30	2.7739e-1 (2.83e-2) −	3.2278e-1 (1.21e-2) −	2.2184e-1 (2.40e-2) −	1.7877e-1 (6.06e-2) ≈	2.7799e-1 (4.96e-2) −	2.9330e-1 (5.13e-2) −	2.1581e-1 (4.24e-2) −	1.7325e-1 (2.87e-2)
LIRCMOP5	2	30	1.3606e+0 (4.15e-1) −	1.1161e+0 (2.81e-1) −	9.5232e-1 (4.14e-1) −	7.1017e-1 (1.20e+0) −	3.7233e-1 (2.04e-1) −	1.2229e+0 (2.19e-1) −	1.6409e+0 (5.60e-1) −	2.6910e-1 (6.15e-2)
LIRCMOP6	2	30	1.3462e+0 (4.96e-4) −	1.3467e+0 (4.70e-4) −	1.0774e+0 (4.22e-1) −	5.6711e-1 (4.75e-1) −	5.0159e-1 (3.69e-1) −	1.4359e+0 (3.20e-1) −	3.9458e-1 (4.69e-1) ≈	2.7307e-1 (1.03e-1)
LIRCMOP7	2	30	1.2294e+0 (7.10e-1) −	1.3804e+0 (6.23e-1) −	1.3641e-1 (3.25e-2) −	1.2684e-1 (5.15e-2) ≈	1.3986e-1 (3.98e-2) −	8.2863e-1 (7.43e-1) −	3.0093e-2 (4.74e-2) +	1.0432e-1 (2.42e-2)
LIRCMOP8	2	30	1.5407e+0 (4.37e-1) −	1.5448e+0 (4.26e-1) −	3.8337e-1 (4.46e-1) −	2.0599e-1 (5.03e-2) −	1.9733e-1 (3.83e-2) −	1.3794e+0 (7.07e-1) −	3.6623e-2 (5.69e-2) +	1.4401e-1 (5.24e-2)
LIRCMOP9	2	30	8.6590e-1 (1.40e-1) −	5.9781e-1 (1.38e-1) −	5.1760e-1 (5.28e-2) −	5.4604e-1 (2.37e-1) ≈	6.9448e-1 (1.45e-1) −	8.1715e-1 (2.97e-1) −	3.7098e-1 (7.51e-2) +	4.4033e-1 (9.00e-2)
LIRCMOP10	2	30	8.0151e-1 (1.49e-1) −	4.2407e-1 (5.83e-2) −	6.0129e-1 (2.28e-1) −	2.6753e-1 (1.36e-1) −	4.7126e-1 (1.46e-1) −	7.2617e-1 (4.02e-1) −	2.2463e-2 (4.56e-2) +	1.7029e-1 (6.94e-2)
LIRCMOP11	2	30	8.8053e-1 (7.32e-2) −	4.7327e-1 (1.14e-1) −	5.9713e-1 (1.50e-1) −	1.6791e-1 (1.34e-1) −	3.5132e-1 (1.61e-1) −	6.2374e-1 (4.03e-1) −	1.1186e-1 (9.19e-2) ≈	7.9458e-2 (4.69e-2)
LIRCMOP12	2	30	6.8445e-1 (1.79e-1) −	3.1726e-1 (7.74e-2) −	3.4999e-1 (8.22e-2) −	4.1603e-1 (1.96e-1) −	3.9776e-1 (1.40e-1) −	6.5436e-1 (4.89e-1) −	1.7933e-1 (4.18e-2) ≈	2.2480e-1 (9.26e-2)
LIRCMOP13	3	30	1.3034e+0 (2.63e-4) −	1.3129e+0 (9.95e-2) −	1.1956e+0 (3.75e-1) −	9.6046e-2 (2.33e-3) ≈	9.4959e-2 (2.80e-3) ≈	7.2146e-1 (6.26e-1) −	1.3102e+0 (1.54e-3) −	9.4648e-2 (1.17e-3)
LIRCMOP14	3	30	1.2595e+0 (2.97e-4) −	1.3028e+0 (6.03e-3) −	1.2719e+0 (1.85e-3) −	9.7539e-2 (2.42e-3) −	9.7374e-2 (2.81e-3) ≈	1.1114e-1 (1.37e-3) −	1.2842e+0 (4.32e-3) −	9.5547e-2 (9.27e-4)
$+/-/\approx$			0/14/0	0/14/0	0/14/0	0/8/6	0/12/2	0/14/0	6/5/3
ZXH_CF1	3	13	4.4992e-2 (8.92e-4) +	9.7660e-2 (1.56e-2) −	4.6274e-2 (8.80e-4) +	4.9205e-2 (8.82e-4) −	4.9308e-2 (1.25e-3) −	4.8756e-2 (1.50e-3) −	5.4677e-2 (1.77e-3) −	4.7450e-2 (9.62e-4)
ZXH_CF2	3	13	1.3602e-1 (2.14e-1) −	3.1536e-1 (1.70e-1) −	1.1518e-1 (1.49e-1) −	1.7290e-1 (2.02e-1) ≈	1.4038e-1 (2.63e-1) ≈	9.6657e-2 (8.65e-2) +	6.0289e-2 (1.06e-3) +	9.9641e-2 (1.40e-1)
ZXH_CF3	3	13	7.4126e-2 (2.04e-3) −	2.2845e-1 (5.99e-2) −	6.1754e-2 (1.43e-3) ≈	7.8090e-2 (1.78e-2) −	8.2462e-2 (3.86e-3) −	7.5642e-2 (2.67e-3) −	8.0494e-2 (2.65e-3) −	6.2459e-2 (1.16e-3)
ZXH_CF4	3	13	1.3128e-1 (9.08e-2) ≈	1.2052e+0 (2.55e-1) −	2.1409e-1 (2.62e-1) ≈	1.4814e-1 (1.13e-1) ≈	2.5642e-1 (3.04e-1) ≈	1.1208e-1 (6.64e-2) ≈	3.0602e-1 (4.02e-1) ≈	1.8278e-1 (1.43e-1)
ZXH_CF5	3	13	8.3063e-2 (1.22e-1) −	1.1952e-1 (5.06e-2) −	1.6079e-1 (2.42e-1) ≈	1.8633e-1 (2.35e-1) −	1.5666e-1 (3.91e-1) −	5.7842e-2 (3.25e-2) +	3.6755e-2 (6.76e-4) +	7.6004e-2 (1.41e-1)
ZXH_CF6	3	13	5.1424e-2 (1.64e-3) −	5.6768e-2 (7.17e-3) −	3.1093e-2 (7.51e-4) ≈	3.7145e-2 (1.53e-2) −	3.3786e-2 (9.61e-4) −	3.6313e-2 (1.60e-3) −	3.4249e-2 (8.42e-4) −	3.1484e-2 (6.36e-4)
ZXH_CF7	3	13	1.1754e-1 (8.18e-2) −	5.5079e-1 (2.46e-1) −	1.6633e-1 (9.83e-2) −	3.5776e-1 (2.13e-1) −	2.6985e-1 (2.08e-1) −	9.3481e-2 (6.01e-2) −	1.9554e-1 (2.42e-1) −	7.1806e-2 (7.68e-2)
ZXH_CF8	3	13	6.3914e-2 (4.64e-3) −	2.4528e-1 (3.62e-1) −	3.1273e-2 (4.59e-4) +	2.3968e-1 (1.34e-1) −	1.5169e-1 (2.76e-2) −	5.6277e-2 (5.62e-3) −	4.2737e-2 (1.05e-3) −	3.1794e-2 (6.94e-4)
ZXH_CF9	3	13	9.8523e-2 (6.10e-2) −	5.1171e-1 (1.21e-1) −	2.6399e-2 (5.85e-4) +	7.4746e-2 (3.12e-2) −	1.4279e-1 (1.06e-1) −	6.5038e-2 (1.40e-2) −	2.9977e-2 (7.86e-4) −	2.6735e-2 (4.45e-4)
ZXH_CF10	3	13	2.2796e-1 (2.55e-1) −	1.1566e+0 (4.25e-1) −	1.3016e-1 (1.00e-1) ≈	1.9627e-1 (1.66e-1) −	3.6530e-1 (3.20e-1) −	1.2220e-1 (8.04e-2) −	1.9639e-1 (1.98e-1) −	6.9362e-2 (7.11e-2)
ZXH_CF11	3	13	1.1475e-1 (1.00e-1) −	5.1396e-1 (2.09e-1) −	2.8028e-2 (4.51e-4) ≈	9.0487e-2 (2.33e-2) −	2.9348e-2 (3.45e-4) −	3.5532e-2 (9.70e-4) −	2.8493e-2 (3.77e-4) −	2.8123e-2 (4.54e-4)
ZXH_CF12	3	13	2.2195e-1 (2.05e-1) −	4.8357e-1 (1.01e-1) −	1.7894e-1 (3.07e-1) ≈	2.0889e-1 (3.48e-1) −	1.0819e-1 (2.19e-1) −	7.5879e-2 (1.50e-1) −	2.5987e-2 (8.12e-4) +	7.1211e-2 (1.17e-1)
ZXH_CF13	2	12	8.9725e-2 (1.49e-1) ≈	9.5274e-1 (3.91e-1) −	3.3227e-1 (3.33e-1) ≈	2.0733e-1 (3.29e-1) ≈	3.2273e-1 (2.31e-1) ≈	7.4464e-2 (1.04e-1) ≈	1.7974e-1 (2.36e-1) ≈	3.2395e-1 (4.67e-1)
ZXH_CF14	2	12	7.9689e-3 (1.83e-2) −	1.4504e-1 (2.09e-1) −	2.3831e-3 (5.13e-5) −	2.3952e-3 (5.04e-5) −	2.5264e-3 (5.04e-5) −	2.8541e-3 (1.22e-4) −	3.1063e-3 (7.60e-4) −	2.3435e-3 (3.74e-5)
ZXH_CF15	2	12	9.8882e-2 (1.54e-1) −	1.8225e-1 (2.06e-1) −	3.8100e-2 (1.25e-1) ≈	1.5372e-1 (3.10e-1) ≈	2.6988e-2 (5.14e-2) −	5.6692e-3 (7.02e-4) +	2.9271e-3 (7.37e-5) +	2.6042e-2 (1.04e-1)
ZXH_CF16	2	12	7.4565e-2 (4.78e-2) −	2.7397e-2 (4.87e-2) −	2.6260e-3 (3.94e-5) ≈	2.6375e-3 (3.60e-5) ≈	8.3846e-2 (5.56e-2) −	1.4255e-2 (5.13e-3) −	2.7845e-3 (5.03e-5) −	2.6406e-3 (5.53e-5)
$+/-/\approx$			1/13/2	0/16/0	3/3/10	0/11/5	0/13/3	3/11/2	4/10/2
C1_DTLZ1	3	7	2.0506e-2 (4.33e-5) −	NaN (NaN)	2.0642e-2 (5.89e-4) −	2.0558e-2 (5.80e-4) −	2.1095e-2 (5.92e-4) −	2.3158e-2 (3.07e-4) −	2.1200e-2 (2.82e-4) −	2.0057e-2 (1.11e-4)
C1_DTLZ3	3	12	2.4951e+0 (3.71e+0) −	2.6077e-1 (4.37e-1) −	6.0234e+0 (3.54e+0) −	4.5386e-1 (1.78e+0) ≈	5.4781e-2 (1.54e-3) ≈	9.4902e-2 (1.13e-1) −	6.0441e+0 (3.50e+0) −	5.4517e-2 (9.98e-4)
C2_DTLZ2	3	12	4.9310e-2 (2.24e-5) −	1.1085e-1 (2.15e-1) −	4.3699e-2 (1.29e-3) −	4.3685e-2 (1.09e-3) −	4.3628e-2 (1.44e-3) −	5.6604e-2 (1.15e-3) −	4.5503e-2 (5.78e-4) −	4.2557e-2 (5.74e-4)
C3_DTLZ4	3	12	9.1366e-2 (5.18e-5) +	1.4350e-1 (6.45e-3) −	9.8652e-2 (2.79e-3) −	1.7608e-1 (2.34e-1) ≈	6.5367e-1 (5.54e-2) −	1.1172e-1 (2.04e-3) −	9.7809e-2 (1.77e-3) −	9.6124e-2 (1.29e-3)
DC1_DTLZ1	3	7	1.9407e-2 (4.59e-3) −	2.0815e-2 (6.46e-3) −	4.5110e-2 (1.03e-1) ≈	1.7235e-2 (9.70e-3) −	1.5678e-2 (5.65e-3) −	3.0898e-2 (7.01e-2) −	1.2287e-2 (2.82e-4) −	1.1587e-2 (1.00e-4)
DC1_DTLZ3	3	12	4.5090e-2 (2.41e-5) −	8.1687e-1 (1.80e+0) −	3.4721e-2 (1.10e-3) ≈	2.0176e-1 (2.66e-1) −	5.0509e-2 (7.16e-2) ≈	4.3182e-2 (1.39e-3) −	3.4730e-2 (5.00e-4) −	3.4227e-2 (4.74e-4)
DC2_DTLZ1	3	7	7.1076e-2 (7.01e-2) −	NaN (NaN)	5.6862e-2 (6.41e-2) −	2.0659e-2 (5.52e-4) −	2.1261e-2 (5.66e-4) −	2.3208e-2 (1.88e-4) −	3.0650e-2 (3.26e-2) −	2.0124e-2 (1.35e-4)
DC2_DTLZ3	3	12	5.6089e-1 (3.52e-4) −	NaN (NaN)	5.6291e-1 (0.00e+0) ≈	2.4699e-1 (2.31e-1) −	4.6031e-1 (2.09e-1) −	1.6441e-1 (1.99e-1) −	5.6882e-1 (2.66e-3) −	5.3494e-2 (5.50e-4)
DC3_DTLZ1	3	7	1.2063e-1 (9.61e-2) −	1.4582e+0 (2.19e+0) −	2.7795e-1 (2.04e-1) −	3.9023e-2 (3.54e-2) −	3.5516e-2 (9.15e-3) −	9.3686e-3 (3.19e-4) −	4.5143e-2 (6.85e-2) −	6.9870e-3 (6.58e-5)
DC3_DTLZ3	3	12	1.1580e+0 (5.25e-1) −	6.8777e+0 (4.29e+0) −	1.8928e+0 (4.20e-1) −	8.6087e-1 (8.62e-1) −	2.6288e-1 (4.10e-1) −	2.6743e-2 (1.03e-3) +	1.0104e+0 (3.75e-1) −	4.7247e-2 (1.20e-1)
$+/-/\approx$			1/9/0	0/7/0	0/7/3	0/8/2	0/8/2	1/9/0	0/10/0
$Comprehensive$			3/46/5	0/46/3	3/32/19	0/36/18	0/45/9	5/44/5	12/35/7

, it can be observed that the IGD values of 8 algorithms are listed across 54 test problems in 4 benchmark test suites. It is evident from the table that CMOEA-TA outperforms its competitors significantly in the MW test suite. In the MW test problems, CMOEA-TA achieved 7 best values and outperformed CMOEA/D, ToP, TSTI, MSCMO, CMOEA_MS, CTAEA, and IMTCMO in 10, 9, 8, 9, 12, 10, and 10 test problems, respectively. This is primarily attributed to the application of the constraint learning mechanism, which enhanced its diversity and convergence.

In the LIRCMOP test suites, many infeasible regions are inconsistent with the algorithm’s Pareto front. As a result, many algorithms perform poorly when handling these test suites. In the LIRCMOP test cases, CMOEAD, ToP, TSTI, and CTAEA failed to find the optimal solution, while CMOEA_MS and MSCMO only achieved the best results in one test problem. To accelerate the population’s convergence toward the optimal PF, a strict dominance principle and archiving mechanism were embedded into the original algorithm to enhance its convergence. Compared to the competing algorithms, CMOEA-TA outperformed CMOEAD, ToP, TSTI, MSCMO, CMOEA_MS, CTAEA, and IMTCMO in 14, 14, 14, 8, 12, 4, and 5 test problems, respectively, fully demonstrating CMOEA-TA’s excellent convergence and robustness.

From the ZXH_CFtest problems, it can be seen that although CMOEA-TA did not perform satisfactorily in some test problems, overall, CMOEA-TA not only outperformed seven competing algorithms in 12, 16, 4, 11, 10, 11, and 10 test problems but also achieved the best results in 5 test problems, further validating the effectiveness of CMOEA-TA.

The purpose of the C_DTLZ test suite is to evaluate the performance of algorithms in many multi-objective scenarios. To handle ultra-high-dimensional objective scenarios, an archive update mechanism was embedded to filter higher-quality solutions. As seen from Table 2, CMOEA-TA significantly outperforms its competitors in the C_DTLZ test suite, demonstrating its strong capability in handling numerous multi-objective test problems. This further proves that CMOEA-TA possesses strong diversity and convergence. ToP did not find feasible solutions in five test problems, while CMOEA-TA achieved the best performance in 8 test problems.

In terms of comprehensive IGD values, CMOEA-TA demonstrates superior performance across 46, 46, 32, 36, 45, 44, and 35 test problems when compared to the other seven competing algorithms. However, in 3, 0, 3, 0, 0, 5, and 12 test problems, the seven competing algorithms outperform CMOEA-TA. This analysis reveals that while CMOEA-TA may not always achieve optimal performance in certain test problems, it consistently delivers significantly better results than its competitors across the majority of test cases.

4.3. Comprehensive Analysis of HV Values

From

Table 3. The HV results (mean and standard deviation) obtained by eight algorithms on the MW, LIRCMOP, ZXH_CF and C_DTLZ problems.

Problem	M	D	CMOEA/D	ToP	TSTI	MSCMO	CMOEA_MS	CTAEA	IMTCMO	CMOEA-TA
MW1	2	15	4.6948e-1 (5.16e-2) −	NaN (NaN)	4.8487e-1 (1.16e-2) ≈	4.6877e-1 (3.66e-2) −	4.4701e-1 (5.90e-2) −	4.4028e-1 (5.14e-2) −	4.8951e-1 (6.92e-5) −	4.8956e-1 (2.46e-3)
MW2	2	15	5.5149e-1 (1.16e-2) ≈	3.7181e-1 (1.54e-1) −	5.5898e-1 (1.12e-2) ≈	5.5054e-1 (1.29e-2) ≈	5.4790e-1 (1.25e-2) ≈	5.5749e-1 (1.36e-2) ≈	4.9835e-1 (4.47e-2) −	5.5353e-1 (1.41e-2)
MW3	2	15	5.4375e-1 (8.97e-4) −	3.8019e-1 (1.79e-1) −	5.3354e-1 (2.24e-2) ≈	5.2723e-1 (2.02e-3) −	5.3372e-1 (6.42e-3) −	5.3036e-1 (5.69e-3) −	5.4446e-1 (1.83e-4) ≈	5.4434e-1 (4.67e-4)
MW4	3	15	8.4106e-1 (8.60e-4) −	1.1720e-1 (0.00e+0) ≈	8.4078e-1 (9.66e-4) −	8.1638e-1 (2.74e-2) −	8.2816e-1 (1.25e-2) −	8.0781e-1 (3.59e-2) −	8.2920e-1 (1.99e-2) −	8.4186e-1 (3.92e-4)
MW5	2	15	3.2393e-1 (5.91e-4) −	5.6763e-2 (0.00e+0) ≈	3.0222e-1 (5.04e-2) −	3.1332e-1 (3.41e-2) −	2.9085e-1 (3.74e-2) −	3.1575e-1 (3.06e-3) −	3.2350e-1 (3.57e-4) −	3.2439e-1 (1.99e-4)
MW6	2	15	3.0532e-1 (1.52e-2) ≈	7.1210e-2 (5.91e-2) −	2.8970e-1 (4.04e-2) ≈	2.9767e-1 (1.54e-2) ≈	2.5440e-1 (6.20e-2) −	3.1273e-1 (9.65e-3) +	2.3016e-1 (6.84e-2) −	2.9732e-1 (1.65e-2)
MW7	2	15	4.0276e-1 (3.74e-2) −	3.7013e-1 (3.75e-2) −	4.1225e-1 (3.60e-4) −	4.1204e-1 (7.67e-4) −	3.9918e-1 (2.22e-2) −	4.0939e-1 (6.45e-4) −	4.1218e-1 (2.20e-4) −	4.1257e-1 (3.42e-4)
MW8	3	15	5.1728e-1 (4.40e-2) −	1.2738e-1 (1.54e-1) −	5.2085e-1 (2.46e-2) −	5.1916e-1 (1.92e-2) −	5.2514e-1 (1.37e-2) −	5.1979e-1 (1.35e-2) −	4.6974e-1 (6.71e-2) −	5.3614e-1 (9.65e-3)
MW9	2	15	3.8407e-1 (1.20e-2) −	0.0000e+0 (0.00e+0) −	3.5869e-1 (1.04e-1) −	3.5645e-1 (1.22e-1) −	2.5257e-1 (1.48e-1) −	3.9136e-1 (4.57e-3) −	3.9469e-1 (2.81e-3) −	3.9923e-1 (1.58e-3)
MW10	2	15	3.6891e-1 (8.88e-2) ≈	NaN (NaN)	3.8724e-1 (6.69e-2) ≈	3.7603e-1 (9.68e-2) ≈	4.1888e-1 (1.66e-2) ≈	4.2774e-1 (1.64e-2) ≈	3.3005e-1 (3.08e-2) −	3.7473e-1 (1.09e-1)
MW11	2	15	4.2815e-1 (5.34e-2) −	3.0696e-1 (5.99e-2) −	4.4764e-1 (2.47e-4) −	4.4753e-1 (1.55e-4) −	4.4094e-1 (6.73e-3) −	4.3958e-1 (3.89e-3) −	4.4783e-1 (1.22e-4) ≈	4.4785e-1 (1.47e-4)
MW12	2	15	6.0417e-1 (3.64e-3) ≈	0.0000e+0 (0.00e+0) ≈	6.0479e-1 (4.04e-4) +	5.3878e-1 (1.82e-1) ≈	4.8489e-1 (2.14e-1) −	5.5480e-1 (1.32e-1) −	6.0465e-1 (3.98e-4) +	5.7479e-1 (1.35e-1)
MW13	2	15	4.2945e-1 (5.41e-2) −	1.6784e-1 (1.39e-1) −	4.2529e-1 (2.80e-2) −	4.3416e-1 (2.05e-2) −	3.3448e-1 (1.18e-1) −	4.5178e-1 (1.60e-2) ≈	3.8739e-1 (6.02e-2) −	4.5126e-1 (1.59e-2)
MW14	3	15	4.4391e-1 (1.84e-3) −	2.4175e-1 (1.88e-1) −	4.7288e-1 (1.86e-3) +	4.0244e-1 (8.61e-2) ≈	3.9108e-1 (8.24e-2) −	2.8965e-1 (1.83e-1) −	4.6953e-1 (1.42e-3) −	4.7109e-1 (1.76e-3)
$+/-/\approx$			0/10/4	0/9/3	2/7/5	0/9/5	0/12/2	1/10/3	1/11/2
LIRCMOP1	2	30	1.2116e-1 (1.10e-2) −	1.0174e-1 (9.76e-3) −	1.4021e-1 (7.42e-3) −	1.4956e-1 (2.49e-2) ≈	1.0830e-1 (1.23e-2) −	1.1153e-1 (2.82e-2) −	2.2698e-1 (6.49e-3) +	1.6438e-1 (1.28e-2)
LIRCMOP2	2	30	2.3474e-1 (1.73e-2) −	2.1439e-1 (1.74e-2) −	2.5862e-1 (8.97e-3) −	2.8703e-1 (1.63e-2) ≈	2.2503e-1 (2.03e-2) −	2.4258e-1 (5.15e-2) −	2.5735e-1 (2.07e-2) −	2.8635e-1 (1.65e-2)
LIRCMOP3	2	30	1.0487e-1 (1.10e-2) −	9.3085e-2 (5.54e-3) −	1.2362e-1 (9.82e-3) −	1.3639e-1 (2.11e-2) −	9.8505e-2 (1.27e-2) −	9.9568e-2 (2.44e-2) −	2.0118e-1 (3.33e-3) +	1.4912e-1 (9.28e-3)
LIRCMOP4	2	30	1.9952e-1 (1.26e-2) −	1.7739e-1 (1.21e-2) −	2.2253e-1 (1.24e-2) −	2.4669e-1 (2.45e-2) ≈	1.9938e-1 (2.30e-2) −	1.9572e-1 (2.13e-2) −	2.2162e-1 (1.89e-2) −	2.4212e-1 (1.34e-2)
LIRCMOP5	2	30	0.0000e+0 (0.00e+0) −	2.0299e-2 (6.39e-2) −	4.3254e-2 (6.83e-2) −	1.2039e-1 (6.33e-2) −	1.3863e-1 (4.05e-2) −	7.1968e-3 (3.22e-2) −	0.0000e+0 (0.00e+0) −	1.6556e-1 (2.59e-2)
LIRCMOP6	2	30	0.0000e+0 (0.00e+0) −	0.0000e+0 (0.00e+0) −	2.7718e-2 (4.35e-2) −	8.6392e-2 (5.33e-2) −	8.7316e-2 (3.86e-2) −	0.0000e+0 (0.00e+0) −	1.2314e-1 (7.09e-2) ≈	1.2263e-1 (1.57e-2)
LIRCMOP7	2	30	7.0285e-2 (1.10e-1) −	4.8767e-2 (1.01e-1) −	2.4395e-1 (9.12e-3) −	2.4612e-1 (1.65e-2) ≈	2.4346e-1 (1.15e-2) −	1.3077e-1 (1.14e-1) −	2.8471e-1 (1.80e-2) +	2.5325e-1 (7.71e-3)
LIRCMOP8	2	30	2.1332e-2 (6.57e-2) −	1.9875e-2 (6.15e-2) −	2.0098e-1 (6.89e-2) −	2.2868e-1 (8.63e-3) −	2.3102e-1 (7.74e-3) −	5.8184e-2 (9.46e-2) −	2.8334e-1 (2.14e-2) +	2.4490e-1 (1.15e-2)
LIRCMOP9	2	30	1.6350e-1 (7.54e-2) −	2.8476e-1 (8.05e-2) −	3.5465e-1 (4.14e-2) −	3.4950e-1 (8.95e-2) ≈	2.6528e-1 (5.93e-2) −	1.9810e-1 (1.38e-1) −	4.1546e-1 (2.35e-2) +	3.9734e-1 (4.80e-2)
LIRCMOP10	2	30	1.5345e-1 (8.90e-2) −	4.7631e-1 (6.28e-2) −	3.3427e-1 (1.78e-1) −	5.3428e-1 (1.14e-1) −	4.2822e-1 (1.28e-1) −	2.8484e-1 (2.46e-1) −	6.9672e-1 (2.12e-2) +	6.2056e-1 (3.11e-2)
LIRCMOP11	2	30	1.9776e-1 (1.67e-2) −	3.7436e-1 (8.42e-2) −	3.3182e-1 (8.74e-2) −	6.0356e-1 (9.57e-2) −	4.6917e-1 (1.14e-1) −	3.7493e-1 (2.26e-1) −	6.2523e-1 (5.90e-2) ≈	6.5598e-1 (2.44e-2)
LIRCMOP12	2	30	3.4942e-1 (7.34e-2) −	4.6118e-1 (4.01e-2) −	4.4345e-1 (5.44e-2) −	4.6846e-1 (5.54e-2) −	4.3360e-1 (5.43e-2) −	3.3161e-1 (1.95e-1) −	5.2757e-1 (2.02e-2) ≈	5.1365e-1 (4.17e-2)
LIRCMOP13	3	30	4.3975e-4 (1.28e-5) −	5.3778e-3 (1.68e-2) −	5.4642e-2 (1.68e-1) −	5.5355e-1 (1.68e-3) +	5.5430e-1 (2.08e-3) +	2.7248e-1 (2.80e-1) −	1.6796e-4 (1.43e-4) −	5.5178e-1 (1.87e-3)
LIRCMOP14	3	30	9.8092e-4 (3.03e-5) −	2.5125e-4 (2.75e-4) −	5.1083e-4 (3.49e-4) −	5.5389e-1 (1.73e-3) −	5.5415e-1 (1.90e-3) ≈	5.4616e-1 (9.14e-4) −	5.2185e-4 (2.74e-4) −	5.5513e-1 (9.87e-4)
$+/-/\approx$			0/14/0	0/14/0	0/14/0	1/8/5	1/12/1	0/14/0	6/5/3
ZXH_CF1	3	13	8.3449e-1 (8.76e-4) +	7.6055e-1 (2.29e-2) −	8.3227e-1 (1.09e-3) +	8.2885e-1 (1.13e-3) −	8.2610e-1 (2.35e-3) −	8.3464e-1 (1.25e-3) +	8.1946e-1 (2.36e-3) −	8.3027e-1 (1.23e-3)
ZXH_CF2	3	13	4.7412e-1 (1.58e-1) −	2.3984e-1 (1.04e-1) −	4.7515e-1 (1.76e-1) ≈	4.1234e-1 (2.20e-1) ≈	4.8976e-1 (1.68e-1) ≈	4.9353e-1 (1.25e-1) −	5.3594e-1 (2.71e-3) +	5.0088e-1 (1.45e-1)
ZXH_CF3	3	13	5.0966e-1 (2.87e-3) −	2.5602e-1 (5.77e-2) −	5.1622e-1 (2.87e-3) +	4.8755e-1 (2.93e-2) −	5.0770e-1 (2.85e-3) −	5.1163e-1 (2.57e-3) −	4.6812e-1 (5.09e-3) −	5.1380e-1 (2.10e-3)
ZXH_CF4	3	13	3.2362e-1 (1.17e-1) ≈	0.0000e+0 (0.00e+0) −	2.7653e-1 (1.53e-1) ≈	2.9902e-1 (1.22e-1) ≈	2.5523e-1 (1.73e-1) ≈	3.4425e-1 (9.58e-2) ≈	2.3009e-1 (1.38e-1) ≈	2.6469e-1 (1.43e-1)
ZXH_CF5	3	13	2.7373e-1 (6.83e-2) −	1.9029e-1 (4.81e-2) −	2.2374e-1 (1.25e-1) −	1.9140e-1 (1.15e-1) −	2.7456e-1 (9.39e-2) −	2.7618e-1 (4.36e-2) −	2.9592e-1 (9.99e-4) +	2.7619e-1 (8.40e-2)
ZXH_CF6	3	13	2.1543e-1 (2.00e-3) −	1.9209e-1 (8.92e-3) −	2.2495e-1 (9.94e-4) ≈	2.1862e-1 (1.60e-2) −	2.2296e-1 (1.36e-3) −	2.2440e-1 (1.01e-3) ≈	2.1801e-1 (1.06e-3) −	2.2470e-1 (1.13e-3)
ZXH_CF7	3	13	2.4956e-1 (1.12e-1) −	2.8320e-2 (5.66e-2) −	1.8770e-1 (1.01e-1) −	6.1354e-2 (5.78e-2) −	1.1377e-1 (1.12e-1) −	2.6190e-1 (9.00e-2) −	2.1831e-1 (1.45e-1) −	3.0375e-1 (9.93e-2)
ZXH_CF8	3	13	2.2692e-1 (1.94e-3) −	1.0386e-1 (3.34e-2) −	2.3733e-1 (1.16e-3) +	8.6344e-2 (6.50e-2) −	1.2283e-1 (1.49e-2) −	2.1152e-1 (5.81e-3) −	2.2035e-1 (1.68e-3) −	2.3646e-1 (1.50e-3)
ZXH_CF9	3	13	2.5625e-1 (2.74e-2) −	8.0970e-2 (3.67e-2) −	2.7375e-1 (1.10e-3) +	1.9252e-1 (5.01e-2) −	1.6162e-1 (6.18e-2) −	2.3990e-1 (9.76e-3) −	2.5758e-1 (2.02e-3) −	2.7263e-1 (1.51e-3)
ZXH_CF10	3	13	2.4240e-1 (1.07e-1) −	1.2733e-2 (2.89e-2) −	1.7609e-1 (1.14e-1) −	1.3438e-1 (9.50e-2) −	1.2441e-1 (1.05e-1) −	1.8661e-1 (1.01e-1) −	1.6922e-1 (1.44e-1) −	2.7461e-1 (1.12e-1)
ZXH_CF11	3	13	3.9492e-1 (4.84e-2) ≈	1.9864e-1 (6.11e-2) −	4.2529e-1 (3.16e-3) ≈	3.2902e-1 (3.57e-2) −	4.2558e-1 (2.18e-3) ≈	3.8991e-1 (8.71e-3) −	4.2190e-1 (2.50e-3) −	4.2622e-1 (2.09e-3)
ZXH_CF12	3	13	6.1528e-1 (1.42e-1) −	3.0386e-1 (1.08e-1) −	5.7581e-1 (3.00e-1) ≈	5.3642e-1 (2.56e-1) −	6.5299e-1 (2.46e-1) ≈	6.7025e-1 (2.02e-1) −	7.4732e-1 (3.17e-3) +	6.7664e-1 (1.85e-1)
ZXH_CF13	2	12	2.4997e-1 (1.19e-1) ≈	4.6817e-3 (1.40e-2) −	1.2387e-1 (1.26e-1) ≈	1.8976e-1 (1.27e-1) ≈	1.3589e-1 (1.25e-1) ≈	2.4196e-1 (1.10e-1) ≈	1.9168e-1 (1.37e-1) ≈	1.5593e-1 (1.42e-1)
ZXH_CF14	2	12	5.3856e-1 (1.37e-2) −	3.8169e-1 (1.07e-1) −	5.4327e-1 (1.53e-4) −	5.4319e-1 (1.89e-4) −	5.4317e-1 (1.18e-4) −	5.4203e-1 (5.07e-4) −	5.4138e-1 (1.40e-3) −	5.4343e-1 (7.47e-5)
ZXH_CF15	2	12	6.1734e-1 (9.73e-2) −	4.4797e-1 (1.43e-1) −	6.2580e-1 (1.19e-1) ≈	5.3860e-1 (2.38e-1) ≈	6.4570e-1 (5.56e-2) +	6.6107e-1 (2.27e-4) +	6.6133e-1 (1.55e-4) +	6.3967e-1 (9.96e-2)
ZXH_CF16	2	12	7.6180e-1 (1.78e-2) −	7.6813e-1 (1.59e-2) −	7.7824e-1 (7.78e-5) ≈	7.7822e-1 (9.42e-5) ≈	7.5334e-1 (1.94e-2) −	7.6380e-1 (7.24e-3) −	7.7762e-1 (1.32e-4) −	7.7826e-1 (7.57e-5)
$+/-/\approx$			1/12/3	0/16/0	4/4/8	0/11/5	1/10/5	2/11/3	4/10/2
C1_DTLZ1	3	7	8.4085e-1 (7.92e-4) −	NaN (NaN)	8.4157e-1 (1.02e-3) −	8.3959e-1 (1.90e-3) −	8.3693e-1 (3.00e-3) −	8.3610e-1 (4.65e-3) −	8.3440e-1 (1.24e-3) −	8.4255e-1 (2.02e-4)
C1_DTLZ3	3	12	3.4236e-1 (2.62e-1) −	3.8957e-1 (1.78e-1) −	1.3949e-1 (2.48e-1) −	5.2777e-1 (1.24e-1) −	5.5818e-1 (2.49e-3) −	5.1571e-1 (1.20e-1) −	1.0815e-1 (2.16e-1) −	5.5969e-1 (1.64e-3)
C2_DTLZ2	3	12	5.1527e-1 (6.00e-5) −	4.3501e-1 (1.04e-1) −	5.1582e-1 (1.76e-3) ≈	5.1683e-1 (1.64e-3) ≈	5.1607e-1 (1.22e-3) ≈	5.0709e-1 (1.82e-3) −	5.0482e-1 (2.57e-3) −	5.1651e-1 (1.82e-3)
C3_DTLZ4	3	12	7.9583e-1 (6.88e-5) +	7.5308e-1 (5.29e-3) −	7.8874e-1 (1.66e-3) ≈	7.6208e-1 (7.91e-2) ≈	5.1934e-1 (5.39e-2) −	7.8570e-1 (1.11e-3) −	7.8859e-1 (1.43e-3) ≈	7.8908e-1 (1.02e-3)
DC1_DTLZ1	3	7	6.2348e-1 (1.54e-2) −	5.8569e-1 (2.55e-2) −	5.8770e-1 (1.36e-1) −	6.0932e-1 (4.00e-2) −	6.1898e-1 (2.01e-2) −	5.9650e-1 (1.34e-1) −	6.2804e-1 (1.73e-3) −	6.3290e-1 (4.59e-4)
DC1_DTLZ3	3	12	4.7268e-1 (2.25e-4) −	1.8706e-1 (1.70e-1) −	4.7299e-1 (1.27e-3) −	3.0680e-1 (2.05e-1) −	4.6759e-1 (2.65e-2) −	4.6132e-1 (2.49e-3) −	4.6921e-1 (1.72e-3) −	4.7419e-1 (8.05e-4)
DC2_DTLZ1	3	7	7.1317e-1 (1.77e-1) −	NaN (NaN)	7.5016e-1 (1.62e-1) −	8.4128e-1 (1.04e-3) −	8.3898e-1 (1.23e-3) −	8.3817e-1 (3.78e-4) −	8.1233e-1 (8.27e-2) −	8.4247e-1 (2.98e-4)
DC2_DTLZ3	3	12	8.0131e-3 (5.82e-5) −	NaN (NaN)	1.2137e-2 (0.00e+0) ≈	3.4596e-1 (2.45e-1) −	1.2253e-1 (2.24e-1) −	4.3140e-1 (2.13e-1) −	1.0616e-2 (1.34e-3) −	5.6110e-1 (9.14e-4)
DC3_DTLZ1	3	7	2.6120e-1 (2.14e-1) −	3.4257e-3 (1.53e-2) −	1.0511e-1 (1.53e-1) −	4.0398e-1 (1.03e-1) −	4.1284e-1 (3.87e-2) −	5.2341e-1 (2.79e-3) −	4.3237e-1 (1.82e-1) −	5.3563e-1 (1.02e-3)
DC3_DTLZ3	3	12	0.0000e+0 (0.00e+0) −	4.9278e-3 (2.20e-2) −	0.0000e+0 (0.00e+0) −	1.1254e-1 (1.40e-1) −	2.3868e-1 (1.80e-1) −	3.6003e-1 (4.54e-3) +	0.0000e+0 (0.00e+0) −	3.4979e-1 (8.23e-2)
$+/-/\approx$			1/9/0	0/7/0	0/7/3	0/8/2	0/9/1	1/9/0	0/9/1
$Comprehensive$			2/45/7	0/46/3	6/32/16	1/36/17	2/43/9	4/44/6	11/35/8

, we can see that the HV values of eight algorithms on 54 test problems. In the MW test set, CMOEA-TA achieved the best performance on 7 test problems, while ToP failed to find feasible solutions for the MW1 and MW10 test problems, demonstrating CMOEA-TA’s strong search capability, which enhances the algorithm’s diversity. CMOEA-TA outperformed the seven competing algorithms in 10, 9, 7, 9, 12, 10, and 11 test problems, indicating that embedding the dominance principle is the most effective choice, enabling the algorithm to exhibit superior diversity and robustness compared to its competitors.

On the LIRCMOP and ZXH_CF test suites, many algorithms struggled to achieve satisfactory results due to the presence of significant infeasible regions and discrete feasible regions on these two test suites. For example, in the LIRCMOP6 test problem, there are some large and discontinuous infeasible regions. Algorithms with weaker search capabilities cannot traverse these large infeasible regions, resulting in the failure to find feasible solutions. In contrast, CMOEA-TA was able to navigate through these infeasible regions and successfully find feasible solutions, demonstrating its robustness.

In the ZXH_CF test suite, CMOEA-TA demonstrated its superior performance by outperforming seven comparison algorithms in 12, 16, 4, 11, 10, 10, 11, and 10 test problems, while the seven competing algorithms surpassed CMOEA-TA in only 1, 0, 4, 0, 1, 2, and 4 test problems. This highlights the impressive superiority of CMOEA-TA. In the C_DTLZ test suite, among the 10 test problems, ToP failed to find feasible solutions for 2 problems, whereas CMOEA-TA attained the best performance in seven cases, significantly outperforming its competitors in terms of solution quality and robustness.

Looking at the 54 test problems in total, CMOEA-TA outperforms seven competitors in 45, 46, 32, 36, 43, 44, and 35 of the test problems. The seven competitors perform similarly to CMOEA-TA in 7, 3, 16, 17, 9, 6, and 8 of the test problems.

Therefore, based on the aforementioned quantitative analysis, it is evident that although CMOEA-TA may perform poorly on some test problems, overall, CMOEA-TA significantly outperforms its competitors, further demonstrating its advantages in diversity and convergence compared to the competing algorithms.

4.4. Performance Analysis

In this section, we conducted performance testing and analysis on CMOEA-TA. From

Table 4. Wilcoxon test results with respect to ${\mathrm{IGD}}^{+}$ and HV on the 54 test problems.

CMOEA-TA vs.	${\mathbf{IGD}}^{+}$			HV
	${\mathit{R}}^{+}$	${\mathit{R}}^{-}$	$\mathit{p}$-Value	${\mathit{R}}^{+}$	${\mathit{R}}^{-}$	$\mathit{p}$-Value
CMOEA/D	1322.0	109.0	0	1341.5	143.5	0
ToP	1225.0	−5.0	0	1225.0	−5.0	0
TSTI	1316.0	115.0	0	1275.5	155.5	0
MSCMO	1346.0	85.0	0	1366.0	119.0	0
CMOEA_MS	1447.5	37.5	0	1428.0	57.0	0
CTAEA	1254.0	231.0	0.00001	1227.5	203.5	0.000004
IMTCMO	1010.5	474.5	0.020649	1011.5	419.5	0.008449

, we can observe the Wilcoxon [46] test results of CMOEA-TA compared to seven competitors. In the table, $R^{+}$, $R^{-}$, and p-value respectively indicate superior to, inferior to, and approximate to competitors. ToP failed to find feasible solutions in some test problems and did not outperform or approximate CMOEA-TA in any test problem. While other competitors have all found feasible solutions, their Wilcoxon values are significantly lower compared to CMOEA-TA. Even though IMTCMO has the highest Wilcoxon value among the competitors, CMOEA-TA is noticeably superior to IMTCMO.

From Figure 3, we can observe the Pareto Front (PF) of eight algorithms on the ZXH_CF10 test problem. It is clear that ToP, IMTCMO, and TSTI did not find solutions on the PF, and most of the solutions found by CMOEA/D, CTAEA, MSCMO, and CMOEA_MS were not on the PF. In contrast, CMOEA-TA not only found all solutions on the PF but also distributed them evenly across it. Therefore, considering Table 4 and Figure 3 collectively, CMOEA-TA significantly outperforms the competitors.

4.5. Real-World Problems Analysis

To better highlight the applicability of CMOEA-TA, we compare CMOEA-TA with seven competitors on 15 real-world engineering problems in this section. Since CMOEA does not have a PF on real-world problems, HV is used as its evaluation metric in this paper.

As shown in

Table 5. The HV results (mean and standard deviation) obtained by eight algorithms on the real-world problems.

Problem	M	D	CMOEA/D	ToP	TSTI	MSCMO	CMOEA_MS	CTAEA	IMTCMO	CMOEA-TA
RWMOP1	2	4	1.0810e-1 (6.73e-5) −	6.0781e-1 (3.75e-4) −	6.0706e-1 (3.84e-4) −	6.0740e-1 (4.81e-4) −	7.0686e-1 (1.46e-1) +	6.0664e-1 (1.02e-3) −	6.0778e-1 (4.53e-4) −	6.0884e-1 (2.02e-4)
RWMOP2	2	5	3.1084e-1 (9.63e-2) −	3.9279e-1 (7.10e-4) −	2.5419e-1 (1.52e-1) −	3.9266e-1 (5.80e-4) −	3.1613e-1 (1.17e-1) −	2.5408e-1 (1.82e-1) −	3.9292e-1 (1.34e-5) −	3.9296e-1 (1.05e-5)
RWMOP3	2	3	1.0257e-1 (1.55e-2) −	9.0175e-1 (2.26e-4) −	8.9916e-1 (4.66e-4) −	8.9923e-1 (5.66e-4) −	8.6370e-1 (6.49e-3) −	8.8768e-1 (1.21e-2) −	8.9919e-1 (7.12e-4) −	9.0264e-1 (6.51e-5)
RWMOP4	2	4	1.9462e-2 (3.54e-2) −	8.6267e-1 (2.21e-4) −	8.5780e-1 (4.47e-3) −	7.8347e-1 (2.27e-1) −	8.0111e-1 (2.57e-2) −	8.5106e-1 (1.25e-2) −	8.5994e-1 (6.68e-4) −	8.6317e-1 (5.03e-4)
RWMOP5	2	4	4.2455e-1 (7.03e-3) −	4.3452e-1 (1.02e-4) −	4.3447e-1 (5.22e-4) −	4.3483e-1 (1.50e-4) −	4.2090e-1 (2.52e-3) −	4.3173e-1 (1.64e-3) −	4.3492e-1 (9.15e-5) −	4.3507e-1 (1.40e-4)
RWMOP6	2	7	2.7649e-1 (8.02e-6) −	2.7702e-1 (3.56e-4) −	2.7696e-1 (3.29e-5) −	2.1127e-1 (3.71e-2) −	2.7729e-1 (8.98e-6) +	2.4217e-1 (3.89e-2) −	2.7691e-1 (5.17e-5) −	2.7724e-1 (1.28e-5)
RWMOP7	2	4	4.8186e-1 (9.64e-4) −	4.8443e-1 (5.76e-5) −	4.8456e-1 (5.99e-5) ≈	4.8454e-1 (4.47e-5) ≈	4.8414e-1 (4.41e-4) −	4.8391e-1 (2.44e-4) −	4.8458e-1 (3.87e-5) +	4.8455e-1 (3.89e-5)
RWMOP8	3	7	1.0474e-2 (2.70e-4) −	2.5770e-2 (1.09e-4) −	2.5348e-2 (3.05e-4) −	2.4862e-2 (5.39e-4) −	2.6099e-2 (3.12e-4) −	2.6099e-2 (2.95e-5) ≈	2.6017e-2 (4.15e-5) −	2.6107e-2 (3.25e-5)
RWMOP9	2	4	5.3073e-2 (4.29e-5) −	4.0929e-1 (1.30e-4) −	4.0942e-1 (1.45e-4) −	4.0942e-1 (1.08e-4) −	4.0859e-1 (2.23e-4) −	4.0882e-1 (4.77e-4) −	4.0952e-1 (1.18e-4) −	4.0986e-1 (7.99e-5)
RWMOP10	2	2	7.9374e-2 (4.89e-4) −	8.4736e-1 (1.20e-4) −	8.4153e-1 (2.32e-3) −	8.4042e-1 (3.22e-3) −	8.3874e-1 (7.53e-4) −	8.4173e-1 (4.34e-3) −	8.4195e-1 (1.49e-3) −	8.4751e-1 (3.77e-5)
RWMOP11	5	3	5.8796e-2 (9.96e-5) −	8.6757e-2 (2.75e-3) −	9.3779e-2 (1.15e-3) −	6.2749e-2 (1.28e-2) −	8.9004e-2 (4.14e-3) −	9.4576e-2 (9.78e-4) −	9.0012e-2 (2.11e-3) −	9.7119e-2 (5.52e-4)
RWMOP12	2	4	7.0147e-2 (8.75e-3) −	5.6025e-1 (3.26e-4) −	5.5466e-1 (3.64e-3) −	5.5422e-1 (4.37e-3) −	4.6522e-1 (5.49e-2) −	5.4770e-1 (7.78e-3) −	5.5563e-1 (1.65e-3) −	5.6105e-1 (1.46e-4)
RWMOP13	3	7	8.8158e-2 (3.14e-5) +	8.7701e-2 (1.20e-4) −	8.7550e-2 (1.35e-4) −	5.7493e-2 (3.27e-2) −	8.2105e-2 (2.30e-3) −	8.6644e-2 (6.45e-4) −	8.7421e-2 (2.53e-4) −	8.8108e-2 (8.98e-5)
RWMOP14	2	5	1.0646e-1 (1.91e-2) −	6.1780e-1 (1.58e-4) ≈	6.1731e-1 (1.01e-3) −	6.0260e-1 (6.02e-2) −	3.8999e-1 (1.17e-1) −	6.1759e-1 (8.33e-4) −	6.1718e-1 (4.20e-4) −	6.1788e-1 (1.92e-3)
RWMOP15	2	3	6.6014e-2 (1.86e-6) −	5.4360e-1 (1.87e-5) +	5.4320e-1 (3.33e-4) −	3.6936e-1 (1.01e-1) −	5.4335e-1 (2.33e-4) ≈	5.3629e-1 (1.34e-2) −	5.4349e-1 (3.23e-5) +	5.4338e-1 (1.69e-4)
$+/-/\approx$			1/14/0	1/13/1	0/14/1	0/14/1	2/12/1	0/14/1	2/13/0

, the HV values of 8 CMOEAs on RWMOP1-15 are presented. It can be observed that CMOEA/D and ToP outperform CMOEA-TA only on RWMOP13 and RWMOP15, respectively. CMOEA_MS outperforms CMOEA-TA only on RWMOP1 and RWMOP6, while IMTCMO outperforms CMOEA-TA only on RWMOP7 and RWMOP10. In contrast, CMOEA-TA outperforms CMOEA/D, ToP, TSTI, MSCMO, CMOEA_MS, CTAEA, and IMTCMO on 14, 13, 14, 14, 12, 14, and 13 test problems, respectively.

To better highlight the superiority of CMOEA-TA, Figure 4 shows the HV plots of eight CMOEAs on RWMOP10. It is clear that CMOEA-TA reaches the optimal value the earliest, with a higher HV indicating stronger algorithm performance. In contrast, the other competitors fail to consistently maintain the optimal value throughout the evaluation process, and their HV values fluctuate significantly. Only CMOEA/D maintains the optimal value, but its HV value is too low, indicating poor algorithm performance. Therefore, based on the analysis of Table 5 and Figure 4, it can be concluded that CMOEA-TA outperforms its competitors.

4.6. Effectiveness of Core Components of CMOEA-TA

In this section, in order to further validate the effectiveness of the innovation, four variants (namely, CMOEA-TA1-4) were designed and compared with CMOEA-TA, with detailed information shown in

Table 6. List of Variant Abbreviations.

Variant Abbreviation	Description
CMOEA-TA1	Delete the newly designed two-stage framework
CMOEA-TA2	Delete the designed archiving mechanism
CMOEA-TA3	Delete the designed balancing mechanism, replaced by NSGAII binary tournament
CMOEA-TA4	Delete the relaxed constraint mechanism designed in the first stage, replaced by SPEA2-CDP

.

From

Table 7. The IGD results (mean and standard deviation) of CMOEA-TA and four variants of it on MW test suite.

Problem	M	D	CMOEA-TA1	CMOEA-TA2	CMOEA-TA3	CMOEA-TA4	CMOEA-TA
MW1	2	15	3.8071e-3 (1.97e-4) −	4.0026e-3 (2.33e-4) −	8.8589e-3 (2.90e-2) −	1.6173e-3 (1.46e-5) ≈	1.8848e-3 (1.21e-3)
MW2	2	15	2.1836e-2 (7.09e-3) ≈	2.3011e-2 (5.93e-3) ≈	1.6181e-1 (1.16e-1) −	9.3478e-2 (7.70e-2) −	2.0670e-2 (9.10e-3)
MW3	2	15	5.2364e-2 (1.10e-3) −	5.2738e-2 (1.27e-3) −	1.9094e-1 (1.96e-1) −	5.1435e-3 (3.92e-4) −	4.8462e-3 (1.91e-4)
MW4	3	15	4.7700e-2 (1.84e-3) −	4.5352e-2 (1.69e-3) −	5.6898e-2 (1.95e-2) −	4.9133e-2 (3.69e-2) ≈	4.0495e-2 (4.23e-4)
MW5	2	15	4.7190e-1 (1.39e-1) −	5.2974e-1 (9.85e-4) −	4.8963e-1 (2.29e-1) −	4.9571e-3 (5.81e-3) −	6.9501e-4 (6.65e-4)
MW6	2	15	4.9723e-2 (1.07e-1) ≈	1.9746e-2 (8.82e-3) ≈	7.3555e-1 (3.01e-1) −	3.2284e-1 (3.06e-1) −	2.4049e-2 (1.20e-2)
MW7	2	15	2.0531e-1 (2.28e-3) −	2.0691e-1 (2.07e-3) −	3.4686e-1 (2.55e-1) −	4.4718e-3 (3.35e-4) −	4.1605e-3 (2.18e-4)
MW8	3	15	6.0439e-2 (4.56e-3) −	6.6137e-2 (5.47e-3) −	6.5957e-1 (1.62e-1) −	9.8215e-2 (5.48e-2) −	4.4684e-2 (2.16e-3)
MW9	2	15	NaN (NaN)	NaN (NaN)	1.9261e-1 (3.38e-2) −	4.5752e-3 (3.84e-4) −	4.3101e-3 (1.66e-4)
MW10	2	15	4.0419e-2 (2.15e-2) ≈	4.5204e-2 (2.38e-2) ≈	3.2416e-1 (2.13e-1) −	2.8406e-1 (2.83e-1) ≈	1.2531e-1 (2.14e-1)
MW11	2	15	NaN (NaN)	NaN (NaN)	6.4271e-1 (3.62e-1) −	6.0197e-3 (1.14e-4) ≈	6.0524e-3 (1.07e-4)
MW12	2	15	NaN (NaN)	NaN (NaN)	4.1397e-1 (8.20e-2) −	4.7057e-3 (7.86e-5) ≈	4.3137e-2 (1.72e-1)
MW13	2	15	1.6612e-1 (3.19e-2) −	1.6253e-1 (2.69e-2) −	2.9258e-1 (2.40e-1) −	2.3967e-1 (2.27e-1) −	5.6480e-2 (3.29e-2)
MW14	3	15	9.9146e-2 (4.70e-3) −	9.2284e-2 (2.27e-3) +	3.4368e-1 (3.28e-2) −	1.8751e-1 (1.90e-1) −	9.6494e-2 (1.31e-3)
$+/-/\approx$			0/8/3	1/7/3	0/14/0	0/9/5

, we can observe the IGD values of CMOEA-TA and its four variants on the MW benchmark problem. CMOEA-TA1 and CMOEA-TA2 failed to find feasible solutions on three test problems, validating the importance of the two-stage framework and archiving mechanism. CMOEA-TA3 performs worse than CMOEA-TA on 14 test problems, further validating the effectiveness of the balancing mechanism. CMOEA-TA4 performs worse than CMOEA-TA on 8 test problems, indicating the irreplaceable role of the relaxed constraint mechanism in its design.

Table 8. The HV results (mean and standard deviation) of CMOEA-TA and four variants of it on MW test suite.

Problem	M	D	CMOEA-TA1	CMOEA-TA2	CMOEA-TA3	CMOEA-TA4	CMOEA-TA
MW1	2	15	4.8492e-1 (7.16e-4) −	4.8486e-1 (1.07e-3) −	4.8257e-1 (2.68e-2) −	4.9012e-1 (1.16e-5) ≈	4.8956e-1 (2.46e-3)
MW2	2	15	5.5135e-1 (1.26e-2) ≈	5.4924e-1 (1.03e-2) ≈	3.9373e-1 (1.12e-1) −	4.5897e-1 (8.69e-2) −	5.5353e-1 (1.41e-2)
MW3	2	15	5.3268e-1 (1.62e-3) −	5.3350e-1 (1.64e-3) −	4.0679e-1 (1.34e-1) −	5.4408e-1 (5.12e-4) ≈	5.4434e-1 (4.67e-4)
MW4	3	15	8.3386e-1 (1.78e-3) −	8.3447e-1 (2.02e-3) −	8.1867e-1 (2.32e-2) −	8.2897e-1 (5.60e-2) ≈	8.4186e-1 (3.92e-4)
MW5	2	15	1.8484e-1 (2.77e-2) −	1.7355e-1 (4.76e-6) −	1.4865e-1 (5.47e-2) −	3.2230e-1 (2.97e-3) −	3.2439e-1 (1.99e-4)
MW6	2	15	2.8527e-1 (4.56e-2) ≈	2.9956e-1 (1.25e-2) ≈	6.0458e-2 (7.22e-2) −	1.7525e-1 (9.53e-2) −	2.9732e-1 (1.65e-2)
MW7	2	15	3.4027e-1 (1.80e-3) −	3.4008e-1 (1.51e-3) −	2.4843e-1 (1.01e-1) −	4.1183e-1 (5.70e-4) −	4.1257e-1 (3.42e-4)
MW8	3	15	5.1542e-1 (1.38e-2) −	5.0686e-1 (1.75e-2) −	1.1196e-1 (8.53e-2) −	4.2565e-1 (9.57e-2) −	5.3614e-1 (9.65e-3)
MW9	2	15	NaN (NaN)	NaN (NaN)	2.5626e-1 (1.81e-2) −	3.9857e-1 (1.36e-3) ≈	3.9923e-1 (1.58e-3)
MW10	2	15	4.1517e-1 (1.85e-2) ≈	4.1123e-1 (2.12e-2) ≈	2.6251e-1 (1.04e-1) −	2.9240e-1 (1.45e-1) ≈	3.7473e-1 (1.09e-1)
MW11	2	15	NaN (NaN)	NaN (NaN)	2.7495e-1 (8.25e-2) −	4.4766e-1 (1.70e-4) −	4.4785e-1 (1.47e-4)
MW12	2	15	NaN (NaN)	NaN (NaN)	3.0925e-1 (2.24e-2) −	6.0503e-1 (1.20e-4) ≈	5.7479e-1 (1.35e-1)
MW13	2	15	4.2663e-1 (1.29e-2) −	4.2590e-1 (9.81e-3) −	3.3545e-1 (8.91e-2) −	3.5218e-1 (7.96e-2) −	4.5126e-1 (1.59e-2)
MW14	3	15	4.6987e-1 (2.39e-3) ≈	4.7530e-1 (1.73e-3) +	3.6245e-1 (1.73e-2) −	4.4506e-1 (7.44e-2) −	4.7109e-1 (1.76e-3)
$+/-/\approx$			0/7/4	1/7/3	0/14/0	0/8/6

presents the HV values of the four variants and CMOEA-TA across 14 test cases, where CMOEA-TA outperforms the four variants on 7, 7, 14, and 8 test problems respectively. In summary, both the framework and innovative mechanisms we proposed have achieved remarkable results, further validating the effectiveness of CMOEA-TA.

To validate the effectiveness of the proposed genetic algorithm-based framework on other evolutionary algorithms, we embedded swarm optimization and differential evolution algorithms as underlying algorithms into the CMOEA-TA framework, forming two independent variants (CMOEA-TA(SO) and CMOEA-TA(DE)).

As shown in

Table 9. The IGD results (mean and standard deviation) of CMOEA-TA(SO), CMOEA-TA(DE) variants, and their seven CMOEAs on the C_DTLZ test suite.

Problem	M	D	CMOEAD	ToP	TSTI	MSCMO	CMOEA_MS	CTAEA	IMTCMO	CMOEA-TA(SO)
C1_DTLZ1	3	7	2.0506e-2 (4.33e-5) −	NaN (NaN)	2.0642e-2 (5.89e-4) −	2.0558e-2 (5.80e-4) −	2.1095e-2 (5.92e-4) −	2.3158e-2 (3.07e-4) −	2.1200e-2 (2.82e-4) −	2.0012e-2 (9.49e-5)
C1_DTLZ3	3	12	2.4951e+0 (3.71e+0) −	2.6077e-1 (4.37e-1) −	6.0234e+0 (3.54e+0) −	4.5386e-1 (1.78e+0) −	5.4781e-2 (1.54e-3) ≈	9.4902e-2 (1.13e-1) −	6.0441e+0 (3.50e+0) −	5.4346e-2 (8.19e-4)
C2_DTLZ2	3	12	4.9310e-2 (2.24e-5) −	1.1085e-1 (2.15e-1) −	4.3699e-2 (1.29e-3) −	4.3685e-2 (1.09e-3) −	4.3628e-2 (1.44e-3) −	5.6604e-2 (1.15e-3) −	4.5503e-2 (5.78e-4) −	4.2386e-2 (4.36e-4)
C3_DTLZ4	3	12	9.1366e-2 (5.18e-5) +	1.4350e-1 (6.45e-3) −	9.8652e-2 (2.79e-3) −	1.7608e-1 (2.34e-1) −	6.5367e-1 (5.54e-2) −	1.1172e-1 (2.04e-3) −	9.7809e-2 (1.77e-3) −	9.5665e-2 (1.64e-3)
DC1_DTLZ1	3	7	1.9407e-2 (4.59e-3) −	2.0815e-2 (6.46e-3) −	4.5110e-2 (1.03e-1) ≈	1.7235e-2 (9.70e-3) −	1.5678e-2 (5.65e-3) −	3.0898e-2 (7.01e-2) −	1.2287e-2 (2.82e-4) −	1.1638e-2 (1.29e-4)
DC1_DTLZ3	3	12	4.5090e-2 (2.41e-5) −	8.1687e-1 (1.80e+0) −	3.4721e-2 (1.10e-3) ≈	2.0176e-1 (2.66e-1) −	5.0509e-2 (7.16e-2) ≈	4.3182e-2 (1.39e-3) −	3.4730e-2 (5.00e-4) −	3.4276e-2 (4.23e-4)
DC2_DTLZ1	3	7	7.1076e-2 (7.01e-2) −	NaN (NaN)	5.6862e-2 (6.41e-2) −	2.0659e-2 (5.52e-4) −	2.1261e-2 (5.66e-4) −	2.3208e-2 (1.88e-4) −	3.0650e-2 (3.26e-2) −	2.0151e-2 (1.71e-4)
DC2_DTLZ3	3	12	5.6089e-1 (3.52e-4) −	NaN (NaN)	5.6291e-1 (0.00e+0) ≈	2.4699e-1 (2.31e-1) −	4.6031e-1 (2.09e-1) −	1.6441e-1 (1.99e-1) −	5.6882e-1 (2.66e-3) −	5.3655e-2 (6.15e-4)
DC3_DTLZ1	3	7	1.2063e-1 (9.61e-2) −	1.4582e+0 (2.19e+0) −	2.7795e-1 (2.04e-1) −	3.9023e-2 (3.54e-2) −	3.5516e-2 (9.15e-3) −	9.3686e-3 (3.19e-4) −	4.5143e-2 (6.85e-2) −	6.9698e-3 (9.35e-5)
DC3_DTLZ3	3	12	1.1580e+0 (5.25e-1) −	6.8777e+0 (4.29e+0) −	1.8928e+0 (4.20e-1) −	8.6087e-1 (8.62e-1) −	2.6288e-1 (4.10e-1) −	2.6743e-2 (1.03e-3) −	1.0104e+0 (3.75e-1) −	2.0588e-2 (4.11e-4)
$+/-/\approx$			1/9/0	0/7/0	0/7/3	0/10/0	0/8/2	0/10/0	0/10/0
Problem	M	D	CMOEAD	ToP	TSTI	MSCMO	CMOEA_MS	CTAEA	IMTCMO	CMOEA-TA(DE)
C1_DTLZ1	3	7	2.0506e-2 (4.33e-5) +	NaN (NaN)	2.0642e-2 (5.89e-4) ≈	2.0558e-2 (5.80e-4) ≈	2.1095e-2 (5.92e-4) ≈	2.3158e-2 (3.07e-4) +	2.1200e-2 (2.82e-4) +	5.7965e-2 (1.34e-1)
C1_DTLZ3	3	12	2.4951e+0 (3.71e+0) +	2.6077e-1 (4.37e-1) +	6.0234e+0 (3.54e+0) ≈	4.5386e-1 (1.78e+0) +	5.4781e-2 (1.54e-3) +	9.4902e-2 (1.13e-1) +	6.0441e+0 (3.50e+0) ≈	7.8318e+0 (1.07e+1)
C2_DTLZ2	3	12	4.9310e-2 (2.24e-5) −	1.1085e-1 (2.15e-1) −	4.3699e-2 (1.29e-3) −	4.3685e-2 (1.09e-3) −	4.3628e-2 (1.44e-3) ≈	5.6604e-2 (1.15e-3) −	4.5503e-2 (5.78e-4) −	4.2815e-2 (5.85e-4)
C3_DTLZ4	3	12	9.1366e-2 (5.18e-5) +	1.4350e-1 (6.45e-3) −	9.8652e-2 (2.79e-3) ≈	1.7608e-1 (2.34e-1) ≈	6.5367e-1 (5.54e-2) −	1.1172e-1 (2.04e-3) −	9.7809e-2 (1.77e-3) +	9.9299e-2 (1.26e-3)
DC1_DTLZ1	3	7	1.9407e-2 (4.59e-3) −	2.0815e-2 (6.46e-3) −	4.5110e-2 (1.03e-1) ≈	1.7235e-2 (9.70e-3) −	1.5678e-2 (5.65e-3) −	3.0898e-2 (7.01e-2) −	1.2287e-2 (2.82e-4) −	1.1511e-2 (1.21e-4)
DC1_DTLZ3	3	12	4.5090e-2 (2.41e-5) −	8.1687e-1 (1.80e+0) −	3.4721e-2 (1.10e-3) ≈	2.0176e-1 (2.66e-1) −	5.0509e-2 (7.16e-2) ≈	4.3182e-2 (1.39e-3) −	3.4730e-2 (5.00e-4) −	3.3883e-2 (3.16e-4)
DC2_DTLZ1	3	7	7.1076e-2 (7.01e-2) ≈	NaN (NaN)	5.6862e-2 (6.41e-2) −	2.0659e-2 (5.52e-4) +	2.1261e-2 (5.66e-4) +	2.3208e-2 (1.88e-4) ≈	3.0650e-2 (3.26e-2) ≈	4.6899e-2 (7.41e-2)
DC2_DTLZ3	3	12	5.6089e-1 (3.52e-4) −	NaN (NaN)	5.6291e-1 (0.00e+0) ≈	2.4699e-1 (2.31e-1) −	4.6031e-1 (2.09e-1) −	1.6441e-1 (1.99e-1) −	5.6882e-1 (2.66e-3) −	5.3494e-2 (5.50e-4)
DC3_DTLZ1	3	7	1.2063e-1 (9.61e-2) −	1.4582e+0 (2.19e+0) −	2.7795e-1 (2.04e-1) −	3.9023e-2 (3.54e-2) −	3.5516e-2 (9.15e-3) −	9.3686e-3 (3.19e-4) −	4.5143e-2 (6.85e-2) −	6.8464e-3 (5.58e-5)
DC3_DTLZ3	3	12	1.1580e+0 (5.25e-1) −	6.8777e+0 (4.29e+0) −	1.8928e+0 (4.20e-1) −	8.6087e-1 (8.62e-1) ≈	2.6288e-1 (4.10e-1) +	2.6743e-2 (1.03e-3) +	1.0104e+0 (3.75e-1) −	4.7695e-1 (1.97e-1)
$+/-/\approx$			3/6/1	1/6/0	0/4/6	2/5/3	3/4/3	3/6/1	2/6/2

, the IGD values of variants CMOEA-TA(SO) and, CMOEA-TA(DE) and the seven CMOEAs on test instances C_DTLZ clearly indicate that the CMOEA-TA(SO) and CMOEA-TA(DE) variants achieved optimal values on 9 and 5 test problems, respectively, demonstrating that CMOEA-TA has strong applicability when faced with other underlying algorithms. Looking at the HV values in

Table 10. The HV results (mean and standard deviation) of CMOEA-TA(SO), CMOEA-TA(DE) variants, and their seven CMOEAs on the C_DTLZ test suite.

Problem	M	D	CMOEAD	ToP	TSTI	MSCMO	CMOEA_MS	CTAEA	IMTCMO	CMOEA-TA(SO)
C1_DTLZ1	3	7	8.4085e-1 (7.92e-4) −	NaN (NaN)	8.4157e-1 (1.02e-3) −	8.3959e-1 (1.90e-3) −	8.3693e-1 (3.00e-3) −	8.3610e-1 (4.65e-3) −	8.3440e-1 (1.24e-3) −	8.4254e-1 (2.29e-4)
C1_DTLZ3	3	12	3.4236e-1 (2.62e-1) −	3.8957e-1 (1.78e-1) −	1.3949e-1 (2.48e-1) −	5.2777e-1 (1.24e-1) −	5.5818e-1 (2.49e-3) −	5.1571e-1 (1.20e-1) −	1.0815e-1 (2.16e-1) −	5.6042e-1 (1.10e-3)
C2_DTLZ2	3	12	5.1527e-1 (6.00e-5) −	4.3501e-1 (1.04e-1) −	5.1582e-1 (1.76e-3) ≈	5.1683e-1 (1.64e-3) ≈	5.1607e-1 (1.22e-3) ≈	5.0709e-1 (1.82e-3) −	5.0482e-1 (2.57e-3) −	5.1631e-1 (1.19e-3)
C3_DTLZ4	3	12	7.9583e-1 (6.88e-5) +	7.5308e-1 (5.29e-3) −	7.8874e-1 (1.66e-3) ≈	7.6208e-1 (7.91e-2) ≈	5.1934e-1 (5.39e-2) −	7.8570e-1 (1.11e-3) −	7.8859e-1 (1.43e-3) ≈	7.8928e-1 (1.30e-3)
DC1_DTLZ1	3	7	6.2348e-1 (1.54e-2) −	5.8569e-1 (2.55e-2) −	5.8770e-1 (1.36e-1) −	6.0932e-1 (4.00e-2) −	6.1898e-1 (2.01e-2) −	5.9650e-1 (1.34e-1) −	6.2804e-1 (1.73e-3) −	6.3303e-1 (4.99e-4)
DC1_DTLZ3	3	12	4.7268e-1 (2.25e-4) −	1.8706e-1 (1.70e-1) −	4.7299e-1 (1.27e-3) −	3.0680e-1 (2.05e-1) −	4.6759e-1 (2.65e-2) ≈	4.6132e-1 (2.49e-3) −	4.6921e-1 (1.72e-3) −	4.7385e-1 (6.05e-4)
DC2_DTLZ1	3	7	7.1317e-1 (1.77e-1) −	NaN (NaN)	7.5016e-1 (1.62e-1) −	8.4128e-1 (1.04e-3) −	8.3898e-1 (1.23e-3) −	8.3817e-1 (3.78e-4) −	8.1233e-1 (8.27e-2) −	8.4248e-1 (2.89e-4)
DC2_DTLZ3	3	12	8.0131e-3 (5.82e-5) −	NaN (NaN)	1.2137e-2 (0.00e+0) ≈	3.4596e-1 (2.45e-1) −	1.2253e-1 (2.24e-1) −	4.3140e-1 (2.13e-1) −	1.0616e-2 (1.34e-3) −	5.6111e-1 (9.84e-4)
DC3_DTLZ1	3	7	2.6120e-1 (2.14e-1) −	3.4257e-3 (1.53e-2) −	1.0511e-1 (1.53e-1) −	4.0398e-1 (1.03e-1) −	4.1284e-1 (3.87e-2) −	5.2341e-1 (2.79e-3) −	4.3237e-1 (1.82e-1) −	5.3564e-1 (1.24e-3)
DC3_DTLZ3	3	12	0.0000e+0 (0.00e+0) −	4.9278e-3 (2.20e-2) −	0.0000e+0 (0.00e+0) −	1.1254e-1 (1.40e-1) −	2.3868e-1 (1.80e-1) −	3.6003e-1 (4.54e-3) −	0.0000e+0 (0.00e+0) −	3.6797e-1 (9.77e-4)
$+/-/\approx$			1/9/0	0/7/0	0/7/3	0/8/2	0/8/2	0/10/0	0/9/1
Problem	M	D	CMOEAD	ToP	TSTI	MSCMO	CMOEA_MS	CTAEA	IMTCMO	CMOEA-TA(DE)
C1_DTLZ1	3	7	8.4085e-1 (7.92e-4) +	NaN (NaN)	8.4157e-1 (1.02e-3) +	8.3959e-1 (1.90e-3) +	8.3693e-1 (3.00e-3) +	8.3610e-1 (4.65e-3) +	8.3440e-1 (1.24e-3) +	7.7134e-1 (2.32e-1)
C1_DTLZ3	3	12	3.4236e-1 (2.62e-1) +	3.8957e-1 (1.78e-1) +	1.3949e-1 (2.48e-1) ≈	5.2777e-1 (1.24e-1) +	5.5818e-1 (2.49e-3) +	5.1571e-1 (1.20e-1) +	1.0815e-1 (2.16e-1) ≈	1.2083e-1 (2.22e-1)
C2_DTLZ2	3	12	5.1527e-1 (6.00e-5) ≈	4.3501e-1 (1.04e-1) −	5.1582e-1 (1.76e-3) ≈	5.1683e-1 (1.64e-3) +	5.1607e-1 (1.22e-3) ≈	5.0709e-1 (1.82e-3) −	5.0482e-1 (2.57e-3) −	5.1490e-1 (1.75e-3)
C3_DTLZ4	3	12	7.9583e-1 (6.88e-5) +	7.5308e-1 (5.29e-3) −	7.8874e-1 (1.66e-3) +	7.6208e-1 (7.91e-2) ≈	5.1934e-1 (5.39e-2) −	7.8570e-1 (1.11e-3) −	7.8859e-1 (1.43e-3) ≈	7.8784e-1 (1.32e-3)
DC1_DTLZ1	3	7	6.2348e-1 (1.54e-2) −	5.8569e-1 (2.55e-2) −	5.8770e-1 (1.36e-1) ≈	6.0932e-1 (4.00e-2) −	6.1898e-1 (2.01e-2) ≈	5.9650e-1 (1.34e-1) −	6.2804e-1 (1.73e-3) −	6.3185e-1 (9.78e-4)
DC1_DTLZ3	3	12	4.7268e-1 (2.25e-4) −	1.8706e-1 (1.70e-1) −	4.7299e-1 (1.27e-3) ≈	3.0680e-1 (2.05e-1) −	4.6759e-1 (2.65e-2) ≈	4.6132e-1 (2.49e-3) −	4.6921e-1 (1.72e-3) −	4.7290e-1 (7.44e-4)
DC2_DTLZ1	3	7	7.1317e-1 (1.77e-1) ≈	NaN (NaN)	7.5016e-1 (1.62e-1) −	8.4128e-1 (1.04e-3) +	8.3898e-1 (1.23e-3) +	8.3817e-1 (3.78e-4) +	8.1233e-1 (8.27e-2) ≈	7.6135e-1 (2.18e-1)
DC2_DTLZ3	3	12	8.0131e-3 (5.82e-5) −	NaN (NaN)	1.2137e-2 (0.00e+0) ≈	3.4596e-1 (2.45e-1) −	1.2253e-1 (2.24e-1) −	4.3140e-1 (2.13e-1) −	1.0616e-2 (1.34e-3) −	5.6110e-1 (9.14e-4)
DC3_DTLZ1	3	7	2.6120e-1 (2.14e-1) −	3.4257e-3 (1.53e-2) −	1.0511e-1 (1.53e-1) −	4.0398e-1 (1.03e-1) −	4.1284e-1 (3.87e-2) −	5.2341e-1 (2.79e-3) −	4.3237e-1 (1.82e-1) −	5.3561e-1 (1.37e-3)
DC3_DTLZ3	3	12	0.0000e+0 (0.00e+0) ≈	4.9278e-3 (2.20e-2) ≈	0.0000e+0 (0.00e+0) ≈	1.1254e-1 (1.40e-1) ≈	2.3868e-1 (1.80e-1) +	3.6003e-1 (4.54e-3) +	0.0000e+0 (0.00e+0) ≈	5.4825e-2 (1.34e-1)
$+/-/\approx$			3/4/3	1/5/1	2/2/6	4/4/2	4/3/3	4/6/0	1/5/4

, the CMOEA-TA(SO) outperforms the seven competitors on 9, 7, 7, 8, 8, 10, and 9 test problems, while the CMOEA-TA(DE) variant outperforms them on 4, 5, 2, 4, 3, 6, and 5 test problems. Therefore, both the IGD and HV metrics show that CMOEA-TA(SO) and CMOEA-TA(DE) outperform the competitors, further validating the effectiveness and applicability of the proposed genetic algorithm-based CMOEA-TA.

5. Conclusions

In this research work, a two-stage archive-based framework based on genetic algorithms was designed to handle CMOP. In the first stage, a learning relaxation mechanism is introduced, where the relaxation of constraint values is adaptively adjusted based on the feasible ratio and the degree of constraint violation. Feasible solutions obtained are stored in an archive, which is continuously updated throughout the algorithm iterations. In the second stage, the relaxation of constraints is eliminated, and the balance between constraints and objectives is considered to ensure the feasibility of the algorithm. When selecting parents, a strict dominance constraint strategy is implemented to ensure the production of high-quality offspring.

The experimental results show that CMOEA-TA outperforms seven other competing algorithms across 54 test problems. Quantitatively, CMOEA-TA demonstrates superior performance in terms of HV and IGD on the majority of test problems, showing stronger convergence and diversity, which proves its robustness and effectiveness in solving CMOPs. Furthermore, the algorithm exhibits strong practicality when applied to 15 real-world engineering problems.

A limitation of CMOEA-TA is its preference for feasible solutions, which often leads to suboptimal results when dealing with large infeasible regions and sparse areas, as observed in the case of the LIR-CMOP problem, for example. Therefore, in future work, increasing the utilization of infeasible solution information and enhancing interaction with the archive will be key research directions.