A Novel Genetic Algorithm for Constrained Multimodal Multi-Objective Optimization Problems

Abstract

This paper proposes a multitasking-based genetic algorithm (MTGA-CMMO) to solve constrained multimodal multi-objective optimization problems (CMMOPs). In MTGA-CMMO, the main task is assisted by two auxiliary tasks to obtain all the feasible Pareto solution sets. The constraint boundaries of auxiliary task 1 are dynamically adjusted, facilitating the main task’s population in crossing infeasible regions early in the evolution and providing more evolutionary direction later in the evolution. Auxiliary task 2 can contribute to the exploitation ability of the main task. Meanwhile, a probability-based leader mating selection mechanism is devised to improve the global search capability of MTGA-CMMO. Additionally, three environmental selection strategies are designed to correspond to the different tasks in MTGA-CMMO. Extensive experimental verification demonstrates that MTGA-CMMO outperforms other comparative algorithms across multiple test instances and one practical application problem.

1. Introduction

Constrained multi-objective optimization problems (CMOPs) are widely present in real life and have attracted increasing attention [1,2] in areas such as multi-source compressed-air pressure optimization [3], gas pipeline design and operation [4], and aerodynamic shape optimization [5], etc. The CMOPs require the concurrent optimization of multiple competing objectives subject to various constraint conditions. Without sacrificing generality, a CMOP is represented as in Formula (1):

$$\begin{array}{c}min\mathbf{F}\left(\mathbf{x}\right)=\{f_1\left(\mathbf{x}\right),f_2\left(\mathbf{x}\right),\dots ,f_M\left(\mathbf{x}\right)\}\hfill \\ \mathrm{subject}\mathrm{to}:\hfill \\ \left\{\begin{array}{c}{g}_i\left(\mathbf{x}\right)\le 0,i=1,\dots ,l\hfill \\ h_i\left(\mathbf{x}\right)=0,i=l+1,\dots ,n\hfill \end{array}\right.\hfill \end{array}$$

(1)

where $\mathbf{x}=(x_1,\dots ,x_D)$ is a D-dimensional decision vector, $\mathbf{F}\left(\mathbf{x}\right)$ is an M-dimensional objective vector, $g_i\left(\mathbf{x}\right)\le 0(i=1,\dots ,l)$ is the ith inequality constraint, and $h_i\left(\mathbf{x}\right)=0$ $(i=l+1,\dots ,n)$ denotes the $(i-l)$th equality constraint. The constraint violation value of each constraint can be determined in Formula (2).

$$\begin{array}{c}C{V}_i\left(\mathbf{x}\right)=\left\{\begin{array}{c}max(0,g_i\left(\mathbf{x}\right)),i=1,\dots ,l\hfill \\ max(0,\left|h_i\left(\mathbf{x}\right)\right|-\xi ),i=l+1,\dots ,n\hfill \end{array}\right.\hfill \end{array}$$

(2)

where $\xi$ is a positive value for relaxing the equality constraint. When the sum of all the CV values of $\mathbf{x}$ is equal to 0, as shown in Formula (3), $\mathbf{x}$ is considered to be a feasible solution, in other cases, it is deemed infeasible.

$$\begin{array}{c}CV\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^n}C{V}_i\left(\mathbf{x}\right)=0\hfill \end{array}$$

(3)

In CMOPs, because of the conflicting relationship between various objective functions, multiple non-domination solutions consist of an optimal solution set, called the Pareto-optimal solution set (PS), and the set obtained by mapping PS to the objective space is called the Pareto front (PF). Due to the existence of constraints, the PS can be further categorized into the unconstrained Pareto-optimal solution set (UPS) and the constrained Pareto-optimal solution set (CPS), which correspond to the unconstrained Pareto front (UPF) and the constrained Pareto front (CPF) in the objective space, respectively.

However, in real-world scenarios, many CMOPs exhibit multimodality [6,7], i.e., one CPF in the objective space maps to multiple CPSs in the decision space. Reference [8] may be the first to define this problem type as constrained multimodal multi-objective optimization problems (CMMOPs). To facilitate an intuitive understanding of CMMOPs, we employ a real-world location-selection problem as an example. Figure 1 is an actual example of a CMMOP, where the problem aims to enable the tenant to not only find the location with the shortest distance to reach schools, hospitals, and convenience stores but also the selected location that avoids the constraints of roads, bridges, and rivers. In Figure 1, there are four locations satisfying the shortest distance, i.e., Location 1–4. Since Location 1 and Location 4 are both positioned in the river, it is obvious that they are infeasible siting schemes, while Location 2 and Location 3 are both located within the feasible region and from two different PSs; thus, these two siting schemes can satisfy the needs of the tenants.

From Figure 1, it can be seen that CMMOPs are challenging to solve because of the need to simultaneously consider the constraints, multimodality, and multiple objective functions and maintain the diversity in both the decision space and the objective space. Addressing CMMOPs is a valuable research area; unfortunately, there are few studies on CMMOPs so far. Liang [8] introduced a differential evolution approach with speciation for tackling CMMOPs, where an improved speciation strategy was first employed to partition the population into a number of subpopulations, which evolved independently to identify all the CPSs. Moreover, a novel environmental selection mechanism was designed which balanced the diversity, constraints, and convergence. Ming [9] devised a coevolutionary framework (CMMOCEA), Solving CMMOPs by exchanging useful information between two related populations. Consequently, there is plenty of room for research into CMMOPs.

Evolutionary multitasking (EMT) [10] is a novel optimization paradigm, migrating potentially superior solutions among multiple tasks to achieve the goal of solving multiple related optimization tasks simultaneously. Specifically, in EMT, the main task is to obtain better-performing solutions by mining valid information from other tasks. Due to the obvious correlation between CMOPs and multi-objective optimization problems (MOPs), In [11,12,13], some novel algorithms based on the EMT framework were proposed to address CMOPs, and the relevant experiments have confirmed the significant advantages of these EMT-based algorithms over the earlier proposed algorithms [14,15,16,17].

CMMOPs inherit all the characteristics of CMOPs and also feature multimodality. Considering the superiority of the algorithms that employ the EMT in addressing CMOPs, this study proposes a multitasking-based genetic algorithm (MTGA-CMMO) to solve CMMOPs. The main contributions of this study are as follows:

(1)

A multitasking-based genetic algorithm (MTGA-CMMO) is devised to solve CMMOPs. MTGA-CMMO consists of one main task and two auxiliary tasks. By transferring useful knowledge among the different tasks, the main task can obtain all the CPSs of CMMOPs.

(2)

A novel mating selection mechanism based on the decision space probabilistic information is proposed, which ensures the individuals with better decision space diversity have a higher probability as parents for reproduction, thus improving the exploration capability of MTGA-CMMO.

(3)

Three distinct environmental selection strategies are developed, each tailored to fulfill the specific functionalities required by different tasks of MTGA-CMMO.

The remainder of this article is organized as follows. Section 2 provides a brief overview of the relevant works. Section 3 explains MTGA-CMMO in detail. The experimental settings are described in Section 4. The experimental data and analysis are presented in Section 5. Finally, the conclusions and future works are discussed in Section 6.

2. Related Works

In the related works, we provide a concise introduction to constrained multi-objective optimization evolutionary algorithms (CMOEAs), multimodal multi-objective evolutionary algorithms (MMEAs), and EMT.

2.1. CMOEAs

In the last two decades, many CMOEAs have emerged to solve CMOPs, which can be organized into two types according to whether utilizing UPF information.

Algorithms not utilizing UPF information mainly include (1) penalty function-based methods, (2) methods that separate objectives and constraints, and (3) multi-objective methods. Penalty function-based methods construct the penalty terms based on CV values. By incorporating the penalty terms into the objective functions, a CMOP is turned into a multi-objective optimization problem without constraints (MOP). In [18], a strategy for dynamically adjusting the penalty term was applied to the MOEA/D structure for solving CMOPs. Vaz et al. [19] proposed a three-step penalty that employed three different penalty factors at various phases of evolution. Methods that separate objectives and constraints treat the objective functions and constraints as two independent factors and thoroughly assess the convergence and feasibility of algorithms by various approaches. Typical approaches are the constrained dominance principle (CDP) [14], $ϵ$ constrained method [15], stochastic ranking method (SR) [20], etc. Saha [21] integrated the CDP method with a clustering method for repairing infeasible solutions to address equality constrained optimization problems. Fan [22] developed an angle-based CDP method, known as ACDP, which was subsequently combined with the MOEA/D structure to effectively address CMOPs. Yang [23] proposed a multi-objective differential evolutionary algorithm based on an improved $ϵ$ constraint-handling method. The algorithm dynamically adjusts the $ϵ$ value to prevent the $ϵ$ value setting from being unreasonable. In [24], a multi-objective evolutionary algorithm incorporating decomposition and a dynamic constraint-handling strategy was proposed, in which the search patterns were classified into two categories based on whether considered the constraints. An enhanced $ϵ$ approach is employed in the constrained search mode to increase population diversity. Gu [25] developed a surrogate-assisted evolutionary algorithm that incorporates an improved SR strategy. This strategy, which leverages a fitness mechanism and an adaptive probability operator, aims to strike a balance between convergence and diversity, thereby enhancing the quality of candidate solutions. Multi-objective methods consider the constraints single or multiple additional objective functions, transforming a CMOP into an MOP. In [26], a tri-goal evolutionary framework was devised to address constrained many-objective problems, in which three new objectives are formed by combining the convergence, diversity, and feasibility indicators based on the transformation of constraints.

Algorithms utilizing UPF information include two-stage optimization algorithms and two-population optimization algorithms. Two-stage optimization algorithms coordinate objectives and constraints between different evolutionary phases. Fan [16] proposed an algorithm with a push–pull architecture for solving CMOPs. In the push stage, the constraints are not considered, facilitating the population to quickly pass through the infeasible area and approach the UPF; in the pull stage, an improved $ϵ$ method is used to handle the infeasible solutions in the push stage to reach the CPF. Ma [27] developed a multi-stage evolutionary algorithm, where the partial constraints are taken into account in the early stages, allowing the population to efficiently approach the potentially feasible areas in a well-distributed manner. During evolutionary development, more constraints are examined to find the best solutions through the solutions acquired in earlier stages. In [28], the fitness assessment strategy is adjusted during the evolution process to adaptively tune the weights between the objective functions and the constraints. Two-population optimization algorithms utilize two populations to identify feasible solutions and promising infeasible ones. The collaboration between different populations can facilitate the trade-off of constraints and objectives. Tian [17] developed a co-evolutionary framework for solving CMOPs, consisting of two populations: one population is employed to deal with the original CMOP, while an auxiliary population aims to solve a problem derived from the original problem and transfer valuable knowledge to the other population. Wang [29] developed a cooperative multi-objective evolutionary algorithm including propulsive and normal populations, with the propulsive population focusing on the convergence and the normal population prioritizing feasibility and maintaining diversity. In [30], a bidirectional coevolution was designed, called BiCo, maintaining two populations, and the CPF was approached from both feasible and infeasible sides by updating two populations to solve the CMOPs.

2.2. MMEAs

Over the past decade, plenty of MMEAs have been devised to address multimodal multi-objective optimization problems (MMOPs). Omni-optimizer [31] is an NSGAII-based algorithm and employs the crowding distance method to balance the diversity between the decision and objective space. Yue [32] proposed an index-based ring topology niching technique to foster stable niches, enabling the identification of all Pareto sets (PSs), and adopted a special crowding distance method (SCD) to balance the density between the decision space and the objective space. In addition, the same author developed a differential evolution method based on improved crowding distance [33] to solve MMOPs, where the differential vector is generated by diversity indicators, and the improved crowding distance method allows well-distributed individuals ranked relatively low in the non-dominated sorting to participate in the evolution process, enhancing the exploration ability of the algorithm. Liang [34] designed a novel MMEA based on the decision space niching technique, in which the crowding method is used in the decision space to search the large number of PSs. In [35], a weighted indicator-based evolutionary algorithm (MMEA-WI) was developed, which incorporates diversity information from the decision space into an objective space performance indicator to maintain the decision space diversity and uses a special convergence archive to guarantee the algorithm’s convergence. Li [36] presented a multimodal multi-objective coevolutionary algorithm (CoMMEA), introducing dual archives and coevolving them simultaneously using effective knowledge transfer to obtain both global and local PSs. Liu [37] proposed an MMEA with two-archive and recombination strategies (TriMOEA-TA&R), where the characteristics of the decision variables are evaluated to employ better and different recombination strategies to produce equivalent PSs. In a separate work [38], this author developed an evolutionary algorithm utilizing a convergence-penalized density method (CPDEA) to deal with imbalanced MMOPs. Zhang [39] proposed a two-stage double-niched evolution strategy. In the first stage, the niche technique is applied solely in the decision space, whereas in the second stage, it is utilized in both the decision and objective spaces. Additionally, a self-adaptive adjustment method is introduced to address imbalances in the decision space, thereby enhancing the uniformity of solutions on different Pareto sets (PSs). In [40], a ring hierarchy-based evolutionary algorithm for solving MMOPs was presented, including a ring-based niche method that is utilized according to the Pareto ranking hierarchy. Additionally, a dual-crowding distance is employed in distance-based dominance selection to identify diverse individuals. Reference [41] systematically summarized the test problems, existing MMEAs, and current performance indicators for the MMOPs.

2.3. EMT

EMT is an optimization framework that transfers valuable knowledge between various tasks to simultaneously deal with multiple relevant optimization tasks. The first work about EMT is the multifactorial evolutionary algorithm (MFEA) [10]. Although only a single population exists in MFEA, each individual of the population can be assigned to different tasks in terms of skill factor, which forms an implicit EMT framework. Subsequently, Li [42] presented an explicit EMT framework for solving multifactorial optimization problems. The explicit EMT framework offers an advantage over the implicit EMT framework by focusing the population more intently on the target task, thereby mitigating the adverse effects of negative transfer.

Inspired by reference [42], many studies based on the explicit EMT framework have been devised to deal with different types of optimization problems [43,44,45,46,47]. Recently, several novel EMT frameworks have been proposed for solving CMOPs [11,12,13]. Reference [11] is mainly dedicated to addressing the question of what to transfer between different optimization tasks. In [12], the constrained boundary of the auxiliary task is gradually reduced to continuously improve the similarity between the main task and auxiliary task, facilitating solving the CMOPs for CPF away from UPF. An evolutionary multitasking with global and local auxiliary tasks was presented in [13] to prevent the population from becoming trapped in local regions. In addition, a large number of the latest studies on EMT were introduced in [48].

3. MTGA-CMMO

This section elaborates upon the proposed MTGA-CMMO in detail. First, the procedure of MTGA-CMMO is introduced. Next, the probability-based leader mating selection mechanism is detailed. Moreover, various environmental selection strategies corresponding to different tasks are analysed. Finally, the computational complexity of MTGA-CMMO is discussed.

3.1. Procedure of MTGA-CMMO

There are three optimization tasks in MTGA-CMMO: (1) main task, (2) auxiliary task 1, and (3) auxiliary task 2. These optimization tasks are related since they have identical objective functions; only the constraints are distinct. Specifically, the constraints in the main task remain the same as those of the original problem, while the constraint boundary values of auxiliary task 1 are adjusted in terms of a gradually decreasing strategy. Meanwhile, the constraint boundary values of auxiliary task 2 are determined based on the average CV values of the infeasible offspring in the main task. Using knowledge transfer between different tasks, the main task, in collaboration with auxiliary task 1, can pass through infeasible obstacles to obtain all the CPSs early in the evolutionary process and obtain more promising evolutionary directions late in the evolutionary process due to the high relatedness. In addition, auxiliary task 2 performs further mining around the main task population, which enhances the local search capability of MTGA-CMMO. Figure 2 shows the framework of MTGA-CMMO.

The pseudocode of MTGA-CMMO is illustrated in Algorithm 1. $Population1$, $Population2$, and $Population3$, corresponding to the main task, auxiliary task 1, and auxiliary task 2, respectively, are randomly generated in the separate decision space, and the population size is $NP$. Meanwhile, the objective functions and CV values of the three populations are evaluated. Next, the initial values of the constraint boundaries for the auxiliary tasks 1 and 2, denoted as ${\alpha }_0$ and ${\beta }_0$, are the maximum of the individual CV values in the three populations and the mean of the CV values of the infeasible individuals in $Population1$, respectively. During the primary loop, the procedure below is executed sequentially until the current iteration number t equals the maximum iteration number $T_{max}$.

•

The probability value $Prob$ of each individual of three populations is calculated as in Equations (7) and (8) in Section 3.2, which reflects the decision space diversity of the individuals (line 5).

•

For each optimization task, based on the $Prob$ value, the well-distributed individuals in the decision space are likely to be selected as parents from the population using the roulette-wheel selection mechanism (lines 6–8). It is worth noting that the number of parents selected from auxiliary task 2 is the minimum value between the size of $Population3$ and $NP/2$, while the number of parents selected from the other two tasks is both set to $NP/2$. This difference is caused by the environmental selection mechanism of auxiliary task 2, which retains only the feasible individuals in $Population3$ and $Offspring$ (Explained further in Section 3.3).

•

$Offspring1$, $Offspring2$, and $Offspring3$, corresponding to the three optimization tasks, are reproduced via GA operators based on the selected parents and evaluated. Then, a temp-pop $Offspring$ is generated by combining $Offspring1$, $Offspring2$, and $Offspring3$ (lines 9–13).

•

$Population1$, $Population2$, and $Population3$ are combined with their respective $Offspring$ and the ideal individuals are selected for the next generation population using different environmental selection strategies based on the various roles of the tasks in MTGA-CMMO (lines 14–16).

Algorithm 1 MTGA-CMMO

Input: $Population1$ (population for main task), $Population2$ (population for auxiliary task 1), $Population3$ (population for auxiliary task 2), $NP$ (population size), $T_{max}$ (maximum iteration number), and t (current iteration number); Output: $Population1$;   1: Initialize $Population1$, $Population2$, and $Population3$ on the $NP$ size;   2: Evaluate these populations;   3: ${\alpha }_0$ and ${\beta }_0$ ⟵ Calculate the initial values of the constraint boundary of auxiliary task 1 and 2, respectively;   4: for $t=1:T_{max}$ do   5:     $Prob$ ⟵ Calculate the probability-based diversity indicator in the decision space of each individual in $Population1$, $Population2$, and $Population3$ as in Equations (7) and (8).   6:     $Parent1$ ⟵ Choose parents from $Population1$ using roulette-wheel selection based on $Prob$ with size $NP/2$.   7:     $Parent2$ ⟵ Choose parents from $Population2$ using roulette-wheel selection based on $Prob$ with size $NP/2$.   8:     $Parent3$ ⟵ Choose parents from $Population3$ using roulette-wheel selection based on $Prob$ with the minimum value between $\left|Population3\right|$ and $NP/2$.   9:     $Offspring1$ ⟵ Generate $NP/2$ offspring using GA operator on $Parent1$. 10:     $Offspring2$ ⟵ Generate $NP/2$ offspring using GA operator on $Parent2$. 11:     $Offspring3$ ⟵ Generate $\left|Parent3\right|$ offspring using GA operator on $Parent3$. 12:     Evaluate three offspring populations; 13:     $Offspring$ ⟵ $Offspring1$⋃$Offspring2$⋃$Offspring3$. 14:     $Population1$ ⟵ Select $NP$ individuals from $Population1$⋃$Offspring$ by Algorithm 2 and 3. 15:     $Population2$ ⟵ Select $NP$ individuals from $Population2$⋃$Offspring$ by Algorithm 2 and 4. 16:     $Population3$ ⟵ Select feasible individuals from $Population3$⋃$Offspring$ by Algorithm 2 and 5. 17:     $t=t+1$; 18: end for 19: return $Population1$.

Figure 2 and Algorithm 1 show that the population of the main task ($Population1$) in MTGA-CMMO is considered the output after specified evolutionary iterations. Through knowledge transfer, two auxiliary tasks assist the main task to obtain the individuals that are feasible, well-convergent, and well-distributed.

Auxiliary task 1 is depicted as Formula (4):

$$\begin{array}{c}min\mathbf{F}\left(\mathbf{x}\right)={\{f_1\left(\mathbf{x}\right),f_2\left(\mathbf{x}\right),\dots ,f_M\left(\mathbf{x}\right)\}}^{\mathrm{T}}\hfill \\ \mathrm{subject}\mathrm{to}:\hfill \\ \mathrm{CV}\left(x\right)\le {\alpha }_t\hfill \end{array}$$

(4)

where t is the current iteration number, ${\alpha }_t$ is the constraint boundary value of auxiliary task 1 at the tth iteration, which is dynamically adjusted by a gradually decreasing strategy as shown in Formula (5).

$${\alpha }_t={\alpha }_0·(1-{\displaystyle \frac{log\left(t\right)}{log\left(T_{max}\right)}})$$

(5)

where t and $T_{max}$ are the current iteration number and the maximum iteration number, respectively. ${\alpha }_0$ equals the maximum initial CV value of three populations. In the initial phases of evolution, ${\alpha }_t$ is greater than the value of the constraint boundary of the main task, which implies that the feasible domain of auxiliary task 1 is larger. Meanwhile, mating selection is performed based on the decision space diversity indicator $Prob$ of auxiliary task 1, facilitating the full exploration of the decision space to obtain all the CPSs. Through knowledge transfer, the main task can cross the barriers formed by infeasible regions with the help of auxiliary task 1, improving the global search capability of the main task. As the evolution progresses, ${\alpha }_t$ gradually decreases, which indicates that the degree of correlation continually increases between the main task and auxiliary task 1, providing additional promising search directions for the main task.

Inspired by [13], auxiliary task 2 is described as Formula (6):

$$\begin{array}{c}min\mathbf{F}\left(\mathbf{x}\right)={\{f_1\left(\mathbf{x}\right),f_2\left(\mathbf{x}\right),\dots ,f_M\left(\mathbf{x}\right)\}}^{\mathrm{T}}\hfill \\ \mathrm{subject}\mathrm{to}:\hfill \\ \mathrm{CV}\left(x\right)\le {\beta }_t\hfill \end{array}$$

(6)

where ${\beta }_t$ is the constraint boundary value of the auxiliary task 2. The method for determining ${\beta }_t$ is based on the mean CV value of all the infeasible individuals in $Offspring1$, with the aim of generating a number of promising infeasible solutions around the main population. Additionally, using the environmental selection mechanism of auxiliary task 2 (introduced in Section 3.3.3), the individuals with a uniform distribution of decision space are retained and assist the main task through knowledge transfer, further mining the region around the main population and improving the local search ability of the main task.

3.2. Probability-Based Leader Mating Selection Mechanism

In MTGA-CMMO, a probability-based leader mating selection mechanism is proposed to comprehensively search the decision space in all three optimization tasks and locate more CPSs.

Firstly, the sparsity of each individual in the decision space ($sp$) is characterized based on the sum of the distances of the closest k neighborhood individuals to the individual in the decision space. Assuming the population size is $NP$, the decision space $sp$ value of the ith individual can be calculated by Formula (7):

$$s{p}_i={\sum }_{j=1}^{k}dis(i,j),i\in \left\{1,2,\dots ,NP\right\}$$

(7)

where $dis(i,j)$ denotes the Euclidean distance between the ith individual and the jth individual.

Then, the selected probability of individual i can be calculated as follows:

$$Pro{b}_i={\displaystyle \frac{s{p}_i}{{\sum }_{s=1}^{NP}s{p}_s}},i\in \left\{1,2,\dots ,NP\right\}$$

(8)

Finally, the mating selection mechanism is carried out using the roulette-wheel [49,50] method based on the $Prob$ value, i.e., if an individual i satisfies the following:

$$\sum _{s=1}^{i-1}Pro{b}_s<rand\le \sum _{s=1}^{i}Pro{b}_s,i=1,2,\dots ,NP$$

(9)

the individual i is selected as a parent. $rand$ is a randomly generated number ranging from 0 to 1.

3.3. Environmental Selection Strategies in Various Optimization Tasks

In MTGA-CMMO, the role of the main task is to obtain all the CPSs. Auxiliary task 1 can facilitate the main task in crossing the barriers formed by infeasible regions in the initial phases of iteration—improving the global search ability of the main task—and provide the main task with more promising evolutionary directions late in the evolutionary stage. Auxiliary task 2 generates additional potential individuals around the main population, enhancing the local search capability of the main task. Therefore, according to the different roles of each task in MTGA-CMMO, we design corresponding environmental selection strategies.

3.3.1. Environmental Selection Strategy in the Main Task

To realize the effect of the main task in MTGA-CMMO, the environmental selection of the main task first adopts the CDP method [14] to guarantee the feasibility and convergence of $Population1$; then, for the individuals with the same domination level, we employ the SR [20] method to balance the diversity between the decision space and the objective space.

The SR method is a constraint handling technique originally used to reconcile objective functions and constraints in CMOPs. The $Pf$ value in SR is employed to control the criterion when comparing different individuals, and the comparison process is performed by the bubble sorting method [51], as shown in Algorithm 2. The SR method sorts all the individuals by scanning a union population $U_t$ (population size is $2NP$) $NP$ times, and at each scanning session, all individuals are compared by a randomly selected indicator ($I_1$ or $I_2$). The selection metric is controlled according to the parameter $Pf$, and the $Pf$ value is a random number between 0.4 and 0.6.

During the environmental selection phase of the main task, the diversity between the decision space and the objective space is accommodated through the SR method. The diversity indicators of the ith individual of the population in the decision space and objective space, named $s{p}_{dec}\left(i\right)$ and $s{p}_{obj}\left(i\right)$, respectively, are as follows:

$$s{p}_{dec}\left(i\right)={\sum }_{j=1}^{k}di{s}_{dec}(i,j),i\in \left\{1,2,\dots ,s\right\}$$

(10)

$$s{p}_{obj}\left(i\right)={\sum }_{j=1}^{k}di{s}_{obj}(i,j),i\in \left\{1,2,\dots ,s\right\}$$

(11)

where $di{s}_{dec}(i,j)$ and $di{s}_{obj}(i,j)$ denote the Euclidean distance between the ith individual and the jth individual in the decision space and objective space, respectively. The difference between these and Equation (7) is that the ith individual in Equations (10) and (11) takes a value from 1 to s, with s denoting the number of individuals at the same level of dominance.

Algorithm 2 SR Algorithm

Input: $U_t=(u_1,\dots ,u_{2NP})$ (union population), $Pf$ (control parameter); Output: $P_{t+1}$ (sorted population);   1: for $ScanningTime=1:\left|U_t\right|/2$ do   2:     for $j=1:\left|U_t\right|-1$ do   3:         $rand$∈$U(0,1)$;   4:         if $rand<Pf$ then   5:            if $I_1\left(u_j\right)$ inferiors to $I_1\left(u_{j+1}\right)$ then   6:                exchange($u_j$,$u_{j+1}$)   7:            end if   8:         else   9:            if $I_2\left(u_j\right)$ inferiors to $I_2\left(u_{j+1}\right)$ then 10:                exchange($u_j$,$u_{j+1}$) 11:            end if 12:         end if 13:     end for 14:     if no exchange then 15:         break; 16:     end if 17: end for 18: return $P_{t+1}$ ⟵ the top $|U_t|/2$ individuals of $U_t$.

Algorithm 3 demonstrates the environmental selection process in the main task. First, $Population1$ is mixed with $Offspring$ to obtain a temporary population (line 1), named temp-pop, where $Offspring$ consists of the offspring of the three optimization tasks. Then, the temp-pop is separated into multiple groups using the CDP method (line 2). Next, according to Equations (10) and (11), the decision space and objective space diversity indicators of individuals with the same dominance level are respectively calculated, and  the individuals with the same dominance level are sorted by Algorithm 2 (lines 3–6). Finally, the top $NP$ individuals of the sorted temporary population are selected as the next generation of $Population1$, where $NP$ is the size of $Population1$ (line 7).

From Algorithm 3, the CDP method ensures the population feasibility and convergence, and the SR method balances diversity indicators in the decision space and the objective space.

Algorithm 3 Environmental selection strategy of the main task

Input: $Population1$ (the main population), $Offspring$ (the combined offspring), $NP$ (size of $Population1$); Output: $Population1$;   1: temp-pop $⟵Population1\bigcup Offspring$;   2: the temp-pop is separated into multiple groups by the CDP method, defining ${\mathbf{F}}_1,{\mathbf{F}}_2,\dots ,{\mathbf{F}}_k$ (k is the number of groups).   3: for $i=1:k$ do   4:     Obtain the diversity indicators of all the individuals in ${\mathbf{F}}_i$ according to Equations (10) and (11);   5:     Sort all the individuals in ${\mathbf{F}}_i$ by Algorithm 2;   6: end for   7: return $Population1$ ⟵ the top $NP$ solutions of temp-pop.

3.3.2. Environmental Selection Strategy in Auxiliary Task 1

By transferring useful knowledge to the main task, auxiliary task 1 assists the main task in crossing the barriers formed by infeasible regions in the early iteration and obtaining more CPSs; meanwhile, in the late iterations, more promising evolutionary directions are provided for the main task due to increasing relevance to the main task. Thus, we design the constraint boundary of auxiliary task 1 as gradually decreasing with evolution time as Equation (5). Moreover, in the environmental selection of auxiliary task 1, more attention is paid to diversity information in the decision space while considering the diversity both the decision space and the objective space. The environmental selection strategy in auxiliary task 1 is represented as Algorithm 4.

Algorithm 4 Environmental selection strategy of auxiliary task 1

Input: $Population2$ (population of auxiliary task 1), $Offspring$ (combined offspring), $NP$ (size of $Population2$), ${\alpha }_0$ (initial value of the constraint boundary), t (current iteration number), and $T_{max}$ (maximum iteration number); Output: $Population2$;   1: ${\alpha }_t$ ⟵ Calculate the constraint boundary value of the tth iteration as Equation (5);   2: temp-pop $⟵Population2\bigcup Offspring$;   3: Categorize temporary population into the infeasible subpopulation ($ifsp$) and feasible subpopulation ($fsp$) based on the relationship between the CV value and ${\alpha }_t$;   4: if $\left|fsp\right|$ = 0 then   5:     Sort the temporary population using the CDP method and Algorithm 2;   6:     if $ND-num$ < $NP$ then   7:         // $ND-num$ denotes the number of non-dominated individuals.   8:         Retain the top $NP$ individuals of temporary population as $Population2$.   9:     else 10:         Use a truncation method similar to SPEA2 [52] based on the decision space diversity and retain $NP$ non-dominated individuals as $Population2$. 11:     end if 12: else if $\left|fsp\right|$ < $NP$ then 13:     Retain $fsp$ as part of $Population2$. 14:     s = $NP$ − $\left|fsp\right|$. 15:     Sort $ifsp$ by the CDP method and Algorithm 2; 16:     if $ND-num$ < s then 17:         Retain the top s individuals of $ifsp$ as the remainder of $Population2$. 18:     else 19:         Use a truncation method similar to SPEA2 based on the decision space diversity and retain s non-dominated individuals as the remainder of $Population2$. 20:     end if 21: else if $\left|fsp\right|$ > $NP$ then 22:     Sort $fsp$ by the multi-objective approach [27] and Algorithm 2; 23:     if $ND-num$ < $NP$ then 24:         Retain the top $NP$ individuals of $fsp$ as $Population2$. 25:     else 26:         Use a truncation method similar to SPEA2 based on the decision space diversity and retain $NP$ non-dominated individuals as $Population2$. 27:     end if 28: end if

From Algorithm 4, the tth generation constraint boundary value of auxiliary task 1 (${\alpha }_t$) is calculated according to Equation (5), and the combined population (temp-pop) is divided into an infeasible sub-population ($ifsp$) and a feasible sub-population ($fsp$) using the CV value and ${\alpha }_t$ (lines 1–3). According to the number of feasible solutions in temp-pop, three cases can be classified as follows:

•

Case 1, where there are no feasible individuals in the temp-pop (lines 4–11). The CDP method is first used to sort the temp-pop in terms of non-domination, then the individuals with the same domination level are ranked by the SR method based on the diversity metrics. When the number of non-dominated individuals ($ND-num$) is less than $NP$, the top-ranked $NP$ individuals from temp-pop are taken as the updated $Population2$, otherwise, a truncation method similar to that of SPEA2 [52] is used to trim the number of non-dominated individuals to $NP$ based on the decision space diversity indicator.

•

Case 2, where the number of feasible individuals in the temp-pop falls below $NP$ (lines 12–20). Firstly, the feasible subpopulation $fsp$ is stored as part of the updated $population2$, and s denotes the deviation between $NP$ and the number of feasible solutions. Then, for infeasible individuals, the method in Case 1 is used to retain s well-convergent and well-distributed individuals as the remaining individuals in $population2$.

•

Case 3, where the number of feasible individuals in the temp-pop exceeds $NP$ (lines 21–28). First, we perform a non-dominated ranking for the feasible subpopulation $fsp$ using a multi-objective approach [27], which treats the CV value as an auxiliary objective function, thus the individuals who perform well on feasibility and objective functions have more possibilities to be ranked at the top of the subpopulation $fsp$. Next, the individuals whose dominance relationships are at the same level are sorted based on diversity indicators using the SR method. When the number of non-dominant individuals $ND-num$ falls below $NP$, the top-ranked $NP$ individuals in $fsp$ are retained as $Population2$, otherwise, a truncation method similar to that of SPEA2 is used to trim the number of non-dominated individuals to $NP$ based on the decision space diversity indicator.

3.3.3. Environmental Selection Strategy in Auxiliary Task 2

The role of auxiliary task 2 is to generate additional individuals around the main task population to increase the local search capability of the main task. Therefore, the average CV value of all infeasible $Offspring1$ is considered to be the constraint boundary value for auxiliary task 2, and the environmental selection focuses on the decision space diversity indicator, accounting for the diversity in the decision space and the objective space. Algorithm 5 is the environmental selection procedure for auxiliary task 2.

In Algorithm 5, the input is $Population3$, the combined $Offspring$ consisting of offspring of the three optimization tasks, and since the population size of $Population3$ may change with the evolutionary process, $NP$ is determined according to the main population size. The output is the updated $Population3$. Firstly, the current constraint boundary value ${\beta }_t$ is calculated based on the mean CV value of all infeasible individuals in $Offspring1$ (line 1). Next, a temporary population, named temp-pop, is generated by combining $Population3$ and $Offspring$, and the feasible subpopulation $fsp$ in the temp-pop is found in terms of the relationship between the ${\beta }_t$ value and the CV values of the temp-pop (lines 2–3). When the population size of $fsp$ falls below $NP$, $fsp$ is treated as an updated $Population3$; otherwise, $fsp$ is first evaluated for convergence of each individual using a multi-objective approach, and then the individuals at the same domination level are sorted through the SR method based on diversity indicators. If the number of non-dominated individuals, denoted as $ND-num$, is less than $NP$, the top $NP$ individuals in the $fsp$ are retained as the updated $Population3$; otherwise, a truncation similar to SPEA2 was used to trim the size of $fsp$ to $NP$ based on the decision space diversity indicator.

Algorithm 5 Environmental selection strategy of auxiliary task 2

Input: $Population3$ (population of auxiliary task 2), $Offspring$ (combined offspring), $NP$ (size of $Population1$); Output: $Population3$;   1: ${\beta }_t$ ⟵ Calculate the constraint boundary value at the tth iteration based on the average CV value of all infeasible individuals of $Offspring$;   2: temp-pop $⟵Population3\bigcup Offspring$;   3: Obtain the feasible subpopulation $fsp$ from $Population3$ based on the relationship between the CV value and ${\beta }_t$;   4: if $\left|fsp\right|$ < $NP$ then   5:     Retain $fsp$ as $Population3$.   6: else   7:     Sort $fsp$ by the multi-objective approach and Algorithm 2;   8:     if $ND-num$ < $NP$ then   9:         Retain the top $NP$ individuals of $fsp$ as $Population3$. 10:     else 11:         Use a truncation method similar to SPEA2 based on the decision space diversity and retain $NP$ non-dominated individuals as $Population3$. 12:     end if 13: end if

3.4. Computational Complexity of MTGA-CMMO

The main operators of MTGA-CMMO are mating selection, offspring generation, and environmental selection. The assumption is that $NP$ represents the population size, while D and M denote the dimensions of the decision space and objective space, respectively. The computational complexities of the above three operators are $O\left(NP\right)$, $O(NP\ast D)$, and $O(M\ast N{P}^2)$, respectively. Thus, the worst complexity of MTGA-CMMO is $3\ast O\left(NP\right)$ + $3\ast O(0.5\ast NP\ast D)$ + $3\ast O(2\ast M\ast N{P}^2)$.

4. Experimental Settings

This section presents the experimental settings to validate the effectiveness of MTGA-CMMO. First, the benchmark test problems employed in the experiments are introduced. Next, the performance indicators utilized are explained. Finally, the parameters of comparison algorithms and the experimental setting are presented.

4.1. Benchmark Test Problems

To validate MTGA-CMMO’s performance, two representative benchmark test suits, i.e., CMMOP [9] and CMMF [8], are utilized. The CMMOP is generated using a similar construction method for the MW and MMF test sets, and the CMMF test set was proposed by Liang based on real-world problems with both multimodal characteristics and different CPF properties.

4.2. Performance Indicators

In this paper, we employ IGD [53], IGDX [40], and PSP [54] indicators to provide a comprehensive evaluation of the experimental results. The IGD requires a reference vector set in the objective space which is usually uniformly distributed over the true PFs. With the measurement carried out on the distance between the reference vector set and the obtained solution set, the IGD can comprehensively quantify the convergence and the objective space diversity of the obtained solution set. The IGDX can be regarded as the IGD in the decision space, and the smaller values of both IGDX and IGD indicate a superior performance of the obtained solution set. The PSP value can reflect the similarity between the obtained solution set and the reference vector set [32]. A larger PSP value signifies the superior performance of the obtained solution set in the decision space.

4.3. Comparison Algorithms

To validate the effectiveness of MTGA-CMMO in solving CMMOPs, MTGA-CMMO is compared with nine other algorithms. These comparison algorithms include two CMOEAs, namely, C-TAEA [55] and PPS [16]; three MMEAs, i.e., DN-NSGA-II [34], MRPS [32], and MMEA-WI [35]. These are combined with two CHTs, i.e., CDP [14] and the $ϵ$ constrained method [15], to form the six comparison algorithms, as well as a novel CMMEA, i.e., CMMODE [8]. C-TAEA consists of two archives, one of which drives the population to approximate the true PF, while the other explores unexplored areas to improve the global search capability of the algorithm. PPS is a two-stage CMOEA. In the previous phase, the constraints are not considered, enabling the population to quickly pass through the infeasible region. In the later phase, an improved $ϵ$ method is employed to handle the infeasible solutions in the previous phase to reach the feasible non-dominated region. DN-NSGA-II, MRPS, and MMEA-WI are three representative MMEAs. DN-NSGA-II combines the NSGA-II [56] with the crowding niching technique [53] to obtain more PSs. MRPS is based on a particle swarm optimization algorithm (PSO) [57] and employs an index-based niching method to generate multiple stable subpopulations. MMEA-WI incorporates the decision space diversity information into an objective space performance indicator to preserve the decision space diversity and introduces a convergence archive to guarantee algorithm convergence. To deal with CMMOPs efficiently, these MMEAs are combined with the CDP method [14] and the $ϵ$ constraint method [15]. CMMODE is a novel CMMEA that employs speciation techniques to obtain more CPSs and maintains the diversity of the population by improving the environmental selection mechanisms.

Table 1. The algorithm parameters.

Algorithms	Parameters Setting
C-TAEA [55]	${\eta }_c={\eta }_m=20, p_c=0.9, p_m=1/n.$
PPS [16]	$\delta =0.9, nr=2.$
DN-NSGA-II [34]	${\eta }_c={\eta }_m=20, p_c=0.9, p_m=1/n.$
MRPS [32]	$\omega =0.7298, c_1=c_2=2.05.$
MMEA-WI [35]	$p=0.4, {\eta }_c={\eta }_m=20, p_c=0.9, p_m=1/n.$
MTGA-CMMO	${\eta }_c={\eta }_m=20, p_c=0.9, p_m=1/n.$

presents the relevant parameters for all comparison algorithms.

To reflect the fairness of the experiments, the population size of the compared experiment is fixed at 100, the maximum evaluation number is 100,000, and every algorithm is executed independently 21 times. Meanwhile, to more significantly represent the difference between the experimental results of MTGA-CMMO and other comparative algorithms, we employ the Wilcoxon rank-sum test, setting a significance threshold of 0.05, to evaluate the experimental results. All experiments are implemented based on a PC configured with an Intel i9-9900X @ 3.50 GHz and 64 G RAM, and the experimental platform is PlatEMO V3.5.

5. Experiment Results and Analysis

To fully evaluate the performance of MTGA-CMMO, we designed different types of experiments, including comparison, ablation, and practical application. Here, the settings of the critical parameters are discussed.

5.1. Comparison Experiment

To examine the effectiveness of MTGA-CMMO in solving CMMOPs, MTGA-CMMO is compared with nine other algorithms.

The mean value and standard deviation values of the IGD indicator for MTGA-CMMO and other compared algorithms on the 31 CMMOP test instances are shown in

Table 2. Experimental data based on the IGD metric of all the comparison algorithms on 31 test problems.

Problem	DN-NSGA-II-CDP	DN-NSGA-II-Epsilon	MRPS-CDP	MRPS-Epsilon	MMEA-WI-CDP	MMEA-WI-Epsilon	C-TAEA	PPS	CMMODE	MTGA-CMMO
CMMF1	9.2967 $\times {10}^{-3}$ (7.96 $\times {10}^{-3}$) −	1.2214 $\times {10}^{-2}$ (7.76 $\times {10}^{-3}$) −	1.1876 $\times {10}^{-2}$ (4.59 $\times {10}^{-3}$) −	1.6004 $\times {10}^{-2}$ (5.92 $\times {10}^{-3}$) −	6.2626 $\times {10}^{-3}$ (4.77 $\times {10}^{-4}$) −	1.1279 $\times {10}^{-2}$ (5.26 $\times {10}^{-3}$) −	3.6883 $\times {\mathbf{10}}^{-\mathbf{3}}$ (1.40 $\times {\mathbf{10}}^{-\mathbf{4}}$) +	9.1591 $\times {10}^{-3}$ (1.01 $\times {10}^{-2}$) −	5.2961 $\times {10}^{-3}$ (3.52 $\times {10}^{-4}$) −	4.8796 $\times {10}^{-3}$ (2.14 $\times {10}^{-4}$)
CMMF2	7.2629 $\times {10}^{-3}$ (4.53 $\times {10}^{-4}$) −	1.0367 $\times {10}^{-2}$ (7.66 $\times {10}^{-4}$) −	8.7403 $\times {10}^{-3}$ (1.52 $\times {10}^{-3}$) −	1.1392 $\times {10}^{-2}$ (2.35 $\times {10}^{-3}$) −	6.8357 $\times {10}^{-3}$ (4.78 $\times {10}^{-4}$) −	1.0345 $\times {10}^{-2}$ (9.07 $\times {10}^{-4}$) −	4.4557 $\times {\mathbf{10}}^{-\mathbf{3}}$ (1.48 $\times {\mathbf{10}}^{-\mathbf{4}}$) +	6.5334 $\times {10}^{-3}$ (5.40 $\times {10}^{-4}$) −	5.6377 $\times {10}^{-3}$ (4.38 $\times {10}^{-4}$) =	5.4308 $\times {10}^{-3}$ (2.88 $\times {10}^{-4}$)
CMMF3	1.0416 $\times {10}^{-2}$ (5.84 $\times {10}^{-3}$) −	1.9681 $\times {10}^{-2}$ (2.87 $\times {10}^{-3}$) −	7.7246 $\times {10}^{-3}$ (9.96 $\times {10}^{-4}$) −	9.1336 $\times {10}^{-3}$ (1.93 $\times {10}^{-3}$) −	8.0003 $\times {10}^{-3}$ (9.30 $\times {10}^{-4}$) −	2.4947 $\times {10}^{-2}$ (3.64 $\times {10}^{-3}$) −	1.1201 $\times {10}^{-2}$ (1.04 $\times {10}^{-2}$) −	9.6998 $\times {10}^{-3}$ (8.51 $\times {10}^{-3}$) −	5.5099 $\times {10}^{-3}$ (3.14 $\times {10}^{-4}$) −	5.3280 $\times {\mathbf{10}}^{-\mathbf{3}}$ (3.15 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF4	6.3859 $\times {10}^{-3}$ (3.73 $\times {10}^{-4}$) −	1.5535 $\times {10}^{-2}$ (1.97 $\times {10}^{-3}$) −	7.4818 $\times {10}^{-3}$ (1.07 $\times {10}^{-3}$) −	8.4758 $\times {10}^{-3}$ (1.34 $\times {10}^{-3}$) −	6.8058 $\times {10}^{-3}$ (3.89 $\times {10}^{-4}$) −	1.6097 $\times {10}^{-2}$ (3.72 $\times {10}^{-3}$) −	7.0008 $\times {10}^{-3}$ (1.07 $\times {10}^{-4}$) −	6.4582 $\times {10}^{-3}$ (7.23 $\times {10}^{-4}$) −	5.6825 $\times {10}^{-3}$ (3.05 $\times {10}^{-4}$) −	5.3478 $\times {\mathbf{10}}^{-\mathbf{3}}$ (3.29 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF5	3.7001 $\times {10}^{-3}$ (3.26 $\times {10}^{-4}$) −	8.5807 $\times {10}^{-3}$ (1.81 $\times {10}^{-3}$) −	3.6760 $\times {10}^{-3}$ (2.38 $\times {10}^{-4}$) −	3.3264 $\times {10}^{-3}$ (2.54 $\times {10}^{-4}$) −	3.5987 $\times {10}^{-3}$ (2.36 $\times {10}^{-4}$) −	9.2461 $\times {10}^{-3}$ (1.58 $\times {10}^{-3}$) −	3.0067 $\times {10}^{-3}$ (2.60 $\times {10}^{-4}$) =	2.5482 $\times {\mathbf{10}}^{-\mathbf{3}}$ (1.38 $\times {\mathbf{10}}^{-\mathbf{4}}$) +	3.0361 $\times {10}^{-3}$ (2.17 $\times {10}^{-4}$) =	2.9636 $\times {10}^{-3}$ (1.80 $\times {10}^{-4}$)
CMMF6	2.9787 $\times {10}^{-3}$ (2.77 $\times {10}^{-4}$) −	4.0393 $\times {10}^{-3}$ (5.19 $\times {10}^{-4}$) −	2.4320 $\times {10}^{-3}$ (2.11 $\times {10}^{-4}$) −	2.3970 $\times {10}^{-3}$ (1.38 $\times {10}^{-4}$) −	2.5962 $\times {10}^{-3}$ (2.25 $\times {10}^{-4}$) −	2.9791 $\times {10}^{-3}$ (2.26 $\times {10}^{-4}$) −	1.3649 $\times {\mathbf{10}}^{-\mathbf{3}}$ (6.07 $\times {\mathbf{10}}^{-\mathbf{6}}$) +	1.7318 $\times {10}^{-3}$ (5.14 $\times {10}^{-5}$) +	1.9025 $\times {10}^{-3}$ (8.79 $\times {10}^{-5}$) −	1.8069 $\times {10}^{-3}$ (5.82 $\times {10}^{-5}$)
CMMF7	1.0447 $\times {10}^{-2}$ (1.13 $\times {10}^{-3}$) −	1.3203 $\times {10}^{-2}$ (2.94 $\times {10}^{-3}$) −	9.9226 $\times {10}^{-3}$ (1.49 $\times {10}^{-3}$) −	1.0035 $\times {10}^{-2}$ (1.99 $\times {10}^{-3}$) −	9.4371 $\times {10}^{-3}$ (1.58 $\times {10}^{-3}$) −	1.4170 $\times {10}^{-2}$ (2.62 $\times {10}^{-3}$) −	6.8650 $\times {10}^{-3}$ (6.49 $\times {10}^{-5}$) −	9.4925 $\times {10}^{-3}$ (1.60 $\times {10}^{-2}$) =	6.0534 $\times {10}^{-3}$ (3.83 $\times {10}^{-4}$) =	5.9114 $\times {\mathbf{10}}^{-\mathbf{3}}$ (2.43 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF8	2.3141 $\times {10}^{-3}$ (1.19 $\times {10}^{-4}$) −	4.5898 $\times {10}^{-2}$ (1.64 $\times {10}^{-2}$) −	2.3051 $\times {10}^{-3}$ (1.33 $\times {10}^{-4}$) −	2.3409 $\times {10}^{-3}$ (1.51 $\times {10}^{-4}$) −	2.5518 $\times {10}^{-3}$ (1.24 $\times {10}^{-4}$) −	3.1866 $\times {10}^{-2}$ (7.48 $\times {10}^{-3}$) −	1.3657 $\times {\mathbf{10}}^{-\mathbf{3}}$ (1.34 $\times {\mathbf{10}}^{-\mathbf{5}}$) +	1.9479 $\times {10}^{-3}$ (6.54 $\times {10}^{-5}$) −	1.9578 $\times {10}^{-3}$ (9.20 $\times {10}^{-5}$) =	1.9039 $\times {10}^{-3}$ (6.57 $\times {10}^{-5}$)
CMMF9	2.1666 $\times {10}^{-3}$ (1.11 $\times {10}^{-4}$) −	2.3600 $\times {10}^{-3}$ (1.18 $\times {10}^{-4}$) −	2.5008 $\times {10}^{-3}$ (2.21 $\times {10}^{-4}$) −	2.6113 $\times {10}^{-3}$ (1.67 $\times {10}^{-4}$) −	2.6185 $\times {10}^{-3}$ (2.11 $\times {10}^{-4}$) −	2.8657 $\times {10}^{-3}$ (2.48 $\times {10}^{-4}$) −	1.3882 $\times {\mathbf{10}}^{-\mathbf{3}}$ (3.58 $\times {\mathbf{10}}^{-\mathbf{5}}$) +	1.7154 $\times {10}^{-3}$ (3.92 $\times {10}^{-5}$) +	1.9555 $\times {10}^{-3}$ (6.70 $\times {10}^{-5}$) =	1.8717 $\times {10}^{-3}$ (7.91 $\times {10}^{-5}$)
CMMF10	2.4182 $\times {10}^{-2}$ (9.61 $\times {10}^{-2}$) −	6.4483 $\times {10}^{-3}$ (8.36 $\times {10}^{-4}$) −	3.5893 $\times {10}^{-3}$ (4.34 $\times {10}^{-4}$) −	3.7705 $\times {10}^{-3}$ (4.34 $\times {10}^{-4}$) −	3.4170 $\times {10}^{-3}$ (2.20 $\times {10}^{-4}$) −	7.5719 $\times {10}^{-3}$ (9.40 $\times {10}^{-4}$) −	4.5388 $\times {10}^{-2}$ (1.33 $\times {10}^{-1}$) −	2.8676 $\times {10}^{-3}$ (1.60 $\times {10}^{-3}$) −	2.9398 $\times {10}^{-3}$ (1.50 $\times {10}^{-4}$) −	2.7995 $\times {\mathbf{10}}^{-\mathbf{3}}$ (1.75 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF11	4.9391 $\times {10}^{-3}$ (6.97 $\times {10}^{-4}$) −	1.0081 $\times {10}^{-2}$ (1.85 $\times {10}^{-3}$) −	5.6235 $\times {10}^{-3}$ (8.11 $\times {10}^{-4}$) −	7.8024 $\times {10}^{-3}$ (4.42 $\times {10}^{-3}$) −	4.6781 $\times {10}^{-3}$ (2.54 $\times {10}^{-4}$) −	1.0138 $\times {10}^{-2}$ (1.91 $\times {10}^{-3}$) −	7.7085 $\times {10}^{-3}$ (4.94 $\times {10}^{-4}$) −	3.4431 $\times {\mathbf{10}}^{-\mathbf{3}}$ (1.03 $\times {\mathbf{10}}^{-\mathbf{4}}$) +	4.4078 $\times {10}^{-3}$ (7.03 $\times {10}^{-4}$) −	3.9739 $\times {10}^{-3}$ (3.77 $\times {10}^{-4}$)
CMMF12	5.1439 $\times {10}^{-3}$ (1.13 $\times {10}^{-2}$) −	3.5236 $\times {10}^{-3}$ (1.15 $\times {10}^{-3}$) −	2.2837 $\times {10}^{-3}$ (1.92 $\times {10}^{-4}$) −	2.5174 $\times {10}^{-3}$ (2.15 $\times {10}^{-4}$) −	2.4797 $\times {10}^{-3}$ (1.31 $\times {10}^{-4}$) −	2.8224 $\times {10}^{-3}$ (7.74 $\times {10}^{-4}$) −	1.3566 $\times {\mathbf{10}}^{-\mathbf{3}}$ (1.21 $\times {\mathbf{10}}^{-\mathbf{5}}$) +	1.9099 $\times {10}^{-3}$ (6.20 $\times {10}^{-5}$) =	2.0130 $\times {10}^{-3}$ (7.73 $\times {10}^{-5}$) −	1.9440 $\times {10}^{-3}$ (6.51 $\times {10}^{-5}$)
CMMF13	4.1862 $\times {10}^{-3}$ (4.40 $\times {10}^{-4}$) −	3.8572 $\times {10}^{-2}$ (4.00 $\times {10}^{-2}$) −	7.2704 $\times {10}^{-3}$ (1.35 $\times {10}^{-3}$) −	8.8674 $\times {10}^{-3}$ (2.46 $\times {10}^{-3}$) −	4.6420 $\times {10}^{-3}$ (4.52 $\times {10}^{-4}$) −	5.4349 $\times {10}^{-2}$ (4.97 $\times {10}^{-2}$) −	3.3323 $\times {10}^{-3}$ (3.95 $\times {10}^{-4}$) +	3.0362 $\times {\mathbf{10}}^{-\mathbf{3}}$ (1.05 $\times {\mathbf{10}}^{-\mathbf{4}}$) +	3.5518 $\times {10}^{-3}$ (1.96 $\times {10}^{-4}$) =	3.5724 $\times {10}^{-3}$ (1.87 $\times {10}^{-4}$)
CMMF14	3.8776 $\times {10}^{-3}$ (1.08 $\times {10}^{-3}$) −	4.3509 $\times {10}^{-2}$ (1.06 $\times {10}^{-3}$) −	3.6620 $\times {10}^{-3}$ (4.98 $\times {10}^{-4}$) −	4.6795 $\times {10}^{-3}$ (7.89 $\times {10}^{-4}$) −	5.1048 $\times {10}^{-3}$ (4.50 $\times {10}^{-4}$) −	4.1870 $\times {10}^{-2}$ (7.88 $\times {10}^{-4}$) −	2.5320 $\times {10}^{-3}$ (2.44 $\times {10}^{-4}$) +	2.4994 $\times {\mathbf{10}}^{-\mathbf{3}}$ (6.54 $\times {\mathbf{10}}^{-\mathbf{5}}$) +	3.9718 $\times {10}^{-3}$ (1.11 $\times {10}^{-3}$) −	3.2643 $\times {10}^{-3}$ (2.94 $\times {10}^{-4}$)
CMMF15	5.0715 $\times {10}^{-3}$ (3.54 $\times {10}^{-4}$) −	1.4685 $\times {10}^{-1}$ (1.36 $\times {10}^{-1}$) −	4.8295 $\times {10}^{-3}$ (5.27 $\times {10}^{-4}$) −	5.0213 $\times {10}^{-3}$ (5.30 $\times {10}^{-4}$) −	6.2789 $\times {10}^{-3}$ (5.04 $\times {10}^{-4}$) −	1.3756 $\times {10}^{-1}$ (1.66 $\times {10}^{-1}$) −	5.4931 $\times {10}^{-3}$ (1.29 $\times {10}^{-3}$) −	4.0184 $\times {10}^{-3}$ (2.58 $\times {10}^{-4}$) =	4.0782 $\times {10}^{-3}$ (1.95 $\times {10}^{-4}$) =	3.9931 $\times {\mathbf{10}}^{-\mathbf{3}}$ (3.20 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF16	2.3521 $\times {10}^{-2}$ (2.94 $\times {10}^{-3}$) −	4.3821 $\times {10}^{-1}$ (5.98 $\times {10}^{-4}$) −	2.6262 $\times {10}^{-2}$ (3.11 $\times {10}^{-3}$) −	3.1470 $\times {10}^{-2}$ (6.98 $\times {10}^{-3}$) −	2.2696 $\times {10}^{-2}$ (1.82 $\times {10}^{-3}$) −	4.3804 $\times {10}^{-1}$ (8.29 $\times {10}^{-4}$) −	1.8265 $\times {10}^{-2}$ (2.65 $\times {10}^{-3}$) =	3.7091 $\times {10}^{-2}$ (4.16 $\times {10}^{-3}$) −	1.8890 $\times {10}^{-2}$ (1.379 $\times {10}^{-3}$) −	1.7651 $\times {\mathbf{10}}^{-\mathbf{2}}$ (7.32 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF17	1.4709 $\times {10}^{-2}$ (8.75 $\times {10}^{-4}$) −	2.7018 $\times {10}^{-1}$ (1.26 $\times {10}^{-1}$) −	1.5562 $\times {10}^{-2}$ (1.65 $\times {10}^{-3}$) −	2.0102 $\times {10}^{-2}$ (2.67 $\times {10}^{-3}$) −	1.7915 $\times {10}^{-2}$ (1.30 $\times {10}^{-3}$) −	3.4565 $\times {10}^{-1}$ (7.64 $\times {10}^{-3}$) −	1.0852 $\times {\mathbf{10}}^{-\mathbf{2}}$ (1.41 $\times {\mathbf{10}}^{-\mathbf{3}}$) +	1.4598 $\times {10}^{-2}$ (1.34 $\times {10}^{-3}$) −	1.3465 $\times {10}^{-2}$ (6.23 $\times {10}^{-4}$) −	1.1690 $\times {10}^{-2}$ (3.88 $\times {10}^{-4}$)
CMMOP1	7.4703 $\times {10}^{-3}$ (4.38 $\times {10}^{-4}$) −	1.0488 $\times {10}^{-2}$ (7.72 $\times {10}^{-3}$) −	8.4974 $\times {10}^{-3}$ (6.20 $\times {10}^{-4}$) −	8.9947 $\times {10}^{-3}$ (7.00 $\times {10}^{-4}$) −	6.9937 $\times {10}^{-3}$ (4.93 $\times {10}^{-4}$) −	1.7400 $\times {10}^{-2}$ (8.21 $\times {10}^{-3}$) −	6.4392 $\times {10}^{-3}$ (5.82 $\times {10}^{-4}$) −	6.1653 $\times {10}^{-3}$ (3.10 $\times {10}^{-4}$) −	6.7131 $\times {10}^{-3}$ (2.76 $\times {10}^{-4}$) −	5.3383 $\times {\mathbf{10}}^{-\mathbf{3}}$ (2.51 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMOP2	1.1097 $\times {10}^{-2}$ (7.65 $\times {10}^{-3}$) −	1.1439 $\times {10}^{-2}$ (8.42 $\times {10}^{-3}$) =	1.1003 $\times {10}^{-2}$ (1.91 $\times {10}^{-3}$) −	3.2174 $\times {10}^{-2}$ (2.09 $\times {10}^{-2}$) −	1.0038 $\times {10}^{-2}$ (1.16 $\times {10}^{-3}$) −	1.3780 $\times {10}^{-2}$ (1.06 $\times {10}^{-2}$) −	1.2007 $\times {10}^{-2}$ (5.54 $\times {10}^{-3}$) −	6.1830 $\times {\mathbf{10}}^{-\mathbf{3}}$ (5.10 $\times {\mathbf{10}}^{-\mathbf{4}}$) +	6.7670 $\times {10}^{-3}$ (3.93 $\times {10}^{-4}$) −	6.7338 $\times {10}^{-3}$ (1.14 $\times {10}^{-3}$)
CMMOP3	7.1893 $\times {10}^{-3}$ (2.98 $\times {10}^{-4}$) −	1.5754 $\times {10}^{-2}$ (7.89 $\times {10}^{-3}$) −	8.0897 $\times {10}^{-3}$ (4.59 $\times {10}^{-4}$) −	8.4557 $\times {10}^{-3}$ (5.21 $\times {10}^{-4}$) −	7.1815 $\times {10}^{-3}$ (6.29 $\times {10}^{-4}$) −	1.1116 $\times {10}^{-2}$ (7.62 $\times {10}^{-3}$) −	5.7780 $\times {10}^{-3}$ (4.35 $\times {10}^{-4}$) −	6.8275 $\times {10}^{-3}$ (6.74 $\times {10}^{-4}$) −	1.2604 $\times {10}^{-2}$ (8.36 $\times {10}^{-4}$) −	5.3735 $\times {\mathbf{10}}^{-\mathbf{3}}$ (1.68 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMOP4	1.0291 $\times {10}^{-1}$ (4.55 $\times {10}^{-4}$) +	9.0852 $\times {10}^{-2}$ (7.58 $\times {10}^{-3}$) +	1.0351 $\times {10}^{-1}$ (2.03 $\times {10}^{-4}$) =	1.0357 $\times {10}^{-1}$ (3.29 $\times {10}^{-4}$) =	1.0268 $\times {10}^{-1}$ (3.96 $\times {10}^{-4}$) +	9.0817 $\times {\mathbf{10}}^{-\mathbf{2}}$ (8.58 $\times {\mathbf{10}}^{-\mathbf{3}}$) +	1.0377 $\times {10}^{-1}$ (8.87 $\times {10}^{-5}$) =	1.0374 $\times {10}^{-1}$ (1.93 $\times {10}^{-4}$) =	1.0601 $\times {10}^{-1}$ (3.59 $\times {10}^{-3}$) =	1.0531 $\times {10}^{-1}$ (2.88 $\times {10}^{-3}$)
CMMOP5	6.0286 $\times {10}^{-3}$ (4.48 $\times {10}^{-4}$) −	2.0866 $\times {10}^{-2}$ (8.49 $\times {10}^{-3}$) −	6.2854 $\times {10}^{-3}$ (7.04 $\times {10}^{-4}$) −	6.3228 $\times {10}^{-3}$ (9.48 $\times {10}^{-4}$) −	4.9302 $\times {10}^{-3}$ (3.25 $\times {10}^{-4}$) =	4.4158 $\times {10}^{-2}$ (3.79 $\times {10}^{-2}$) −	3.9023 $\times {\mathbf{10}}^{-\mathbf{3}}$ (4.79 $\times {\mathbf{10}}^{-\mathbf{5}}$) +	6.5248 $\times {10}^{-3}$ (9.76 $\times {10}^{-4}$) −	5.3330 $\times {10}^{-3}$ (2.75 $\times {10}^{-4}$) −	4.8651 $\times {10}^{-3}$ (3.45 $\times {10}^{-4}$)
CMMOP6	1.0470 $\times {10}^{-1}$ (1.43 $\times {10}^{-3}$) +	1.0309 $\times {\mathbf{10}}^{-\mathbf{1}}$ (3.72 $\times {\mathbf{10}}^{-\mathbf{3}}$) +	1.0861 $\times {10}^{-1}$ (5.34 $\times {10}^{-3}$) =	1.2400 $\times {10}^{-1}$ (1.78 $\times {10}^{-2}$) −	1.0477 $\times {10}^{-1}$ (3.40 $\times {10}^{-3}$) +	1.0390 $\times {10}^{-1}$ (2.15 $\times {10}^{-3}$) +	1.0589 $\times {10}^{-1}$ (1.72 $\times {10}^{-3}$) =	1.0503 $\times {10}^{-1}$ (2.52 $\times {10}^{-4}$) =	1.0786 $\times {10}^{-1}$ (3.73 $\times {10}^{-3}$) =	1.0777 $\times {10}^{-1}$ (4.55 $\times {10}^{-3}$)
CMMOP7	8.3220 $\times {10}^{-3}$ (3.55 $\times {10}^{-4}$) −	1.4203 $\times {10}^{-2}$ (8.52 $\times {10}^{-3}$) =	1.0021 $\times {10}^{-2}$ (7.37 $\times {10}^{-4}$) −	1.0475 $\times {10}^{-2}$ (8.44 $\times {10}^{-4}$) −	9.7509 $\times {10}^{-3}$ (7.62 $\times {10}^{-4}$) −	1.4146 $\times {10}^{-2}$ (7.36 $\times {10}^{-3}$) =	7.9168 $\times {10}^{-3}$ (5.21 $\times {10}^{-4}$) −	8.1703 $\times {10}^{-3}$ (5.39 $\times {10}^{-4}$) −	3.2404 $\times {10}^{-2}$ (6.32 $\times {10}^{-4}$) −	6.9729 $\times {\mathbf{10}}^{-\mathbf{3}}$ (2.19 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMOP8	6.3606 $\times {10}^{-3}$ (4.32 $\times {10}^{-4}$) −	2.2700 $\times {10}^{-2}$ (1.35 $\times {10}^{-2}$) −	6.0683 $\times {10}^{-3}$ (6.17 $\times {10}^{-4}$) −	6.1497 $\times {10}^{-3}$ (7.00 $\times {10}^{-4}$) −	6.4407 $\times {10}^{-3}$ (9.27 $\times {10}^{-4}$) −	1.8586 $\times {10}^{-2}$ (9.66 $\times {10}^{-3}$) −	4.0286 $\times {\mathbf{10}}^{-\mathbf{3}}$ (1.73 $\times {\mathbf{10}}^{-\mathbf{4}}$) +	6.2601 $\times {10}^{-3}$ (5.74 $\times {10}^{-4}$) −	5.3013 $\times {10}^{-3}$ (3.69 $\times {10}^{-4}$) −	4.6826 $\times {10}^{-3}$ (3.06 $\times {10}^{-4}$)
CMMOP9	6.3587 $\times {10}^{-3}$ (3.19 $\times {10}^{-4}$) −	9.8536 $\times {10}^{-3}$ (6.61 $\times {10}^{-3}$) −	7.4770 $\times {10}^{-3}$ (5.89 $\times {10}^{-4}$) −	7.2024 $\times {10}^{-3}$ (6.37 $\times {10}^{-4}$) −	6.2420 $\times {10}^{-3}$ (3.78 $\times {10}^{-4}$) −	8.1202 $\times {10}^{-3}$ (4.98 $\times {10}^{-3}$) −	3.9268 $\times {\mathbf{10}}^{-\mathbf{3}}$ (3.23 $\times {\mathbf{10}}^{-\mathbf{4}}$) +	5.7185 $\times {10}^{-3}$ (5.19 $\times {10}^{-4}$) −	5.7122 $\times {10}^{-3}$ (3.69 $\times {10}^{-4}$) −	4.9626 $\times {10}^{-3}$ (1.76 $\times {10}^{-4}$)
CMMOP10	5.3291 $\times {10}^{-3}$ (6.27 $\times {10}^{-4}$) −	1.0233 $\times {10}^{-2}$ (3.71 $\times {10}^{-3}$) −	7.7381 $\times {10}^{-3}$ (1.15 $\times {10}^{-3}$) −	7.6587 $\times {10}^{-3}$ (7.47 $\times {10}^{-4}$) −	5.9153 $\times {10}^{-3}$ (5.54 $\times {10}^{-4}$) −	1.0428 $\times {10}^{-2}$ (4.07 $\times {10}^{-3}$) −	3.5166 $\times {\mathbf{10}}^{-\mathbf{3}}$ (1.29 $\times {\mathbf{10}}^{-\mathbf{4}}$) +	5.1528 $\times {10}^{-3}$ (5.00 $\times {10}^{-4}$) −	5.2524 $\times {10}^{-3}$ (3.31 $\times {10}^{-4}$) −	4.6691 $\times {10}^{-3}$ (2.61 $\times {10}^{-4}$)
CMMOP11	9.5011 $\times {10}^{-3}$ (5.42 $\times {10}^{-4}$) −	3.7036 $\times {10}^{-2}$ (3.48 $\times {10}^{-3}$) −	1.4897 $\times {10}^{-2}$ (1.51 $\times {10}^{-3}$) −	2.1053 $\times {10}^{-2}$ (3.04 $\times {10}^{-3}$) −	9.1192 $\times {10}^{-3}$ (6.28 $\times {10}^{-4}$) −	3.3391 $\times {10}^{-2}$ (1.20 $\times {10}^{-2}$) −	9.7707 $\times {10}^{-3}$ (6.85 $\times {10}^{-4}$) −	6.8229 $\times {10}^{-3}$ (2.68 $\times {10}^{-4}$) −	1.0273 $\times {10}^{-2}$ (1.46 $\times {10}^{-3}$) −	6.2979 $\times {\mathbf{10}}^{-\mathbf{3}}$ (2.89 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMOP12	8.3553 $\times {10}^{-3}$ (4.34 $\times {10}^{-4}$) −	1.1834 $\times {10}^{-2}$ (1.20 $\times {10}^{-2}$) −	1.3817 $\times {10}^{-2}$ (1.30 $\times {10}^{-3}$) −	1.8620 $\times {10}^{-2}$ (2.05 $\times {10}^{-3}$) −	1.0931 $\times {10}^{-2}$ (1.05 $\times {10}^{-3}$) −	7.6453 $\times {10}^{-3}$ (1.08 $\times {10}^{-3}$) −	7.2634 $\times {10}^{-3}$ (3.35 $\times {10}^{-4}$) −	6.0760 $\times {\mathbf{10}}^{-\mathbf{3}}$ (2.75 $\times {\mathbf{10}}^{-\mathbf{4}}$) +	1.0599 $\times {10}^{-2}$ (1.21 $\times {10}^{-3}$) −	6.2736 $\times {10}^{-3}$ (2.61 $\times {10}^{-4}$)
CMMOP13	7.5756 $\times {10}^{-3}$ (4.69 $\times {10}^{-4}$) −	1.2705 $\times {10}^{-2}$ (8.39 $\times {10}^{-3}$) −	8.5514 $\times {10}^{-3}$ (5.43 $\times {10}^{-4}$) −	8.8509 $\times {10}^{-3}$ (6.34 $\times {10}^{-4}$) −	6.9558 $\times {10}^{-3}$ (4.65 $\times {10}^{-4}$) −	1.4466 $\times {10}^{-2}$ (7.66 $\times {10}^{-3}$) −	6.6750 $\times {10}^{-3}$ (5.68 $\times {10}^{-4}$) −	6.2855 $\times {10}^{-3}$ (2.76 $\times {10}^{-4}$) −	7.0711 $\times {10}^{-3}$ (3.75 $\times {10}^{-4}$) −	5.3416 $\times {\mathbf{10}}^{-\mathbf{3}}$ (2.10 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMOP14	8.2301 $\times {10}^{-3}$ (3.43 $\times {10}^{-4}$) −	6.4814 $\times {10}^{-3}$ (3.27 $\times {10}^{-4}$) −	8.7533 $\times {10}^{-3}$ (4.50 $\times {10}^{-4}$) −	9.2627 $\times {10}^{-3}$ (6.94 $\times {10}^{-4}$) −	7.3968 $\times {10}^{-3}$ (5.39 $\times {10}^{-4}$) −	7.0643 $\times {10}^{-3}$ (4.09 $\times {10}^{-3}$) −	7.7750 $\times {10}^{-3}$ (4.71 $\times {10}^{-4}$) −	6.4324 $\times {10}^{-3}$ (2.71 $\times {10}^{-4}$) −	7.6243 $\times {10}^{-3}$ (4.37 $\times {10}^{-4}$) −	5.5061 $\times {\mathbf{10}}^{-\mathbf{3}}$ (2.23 $\times {\mathbf{10}}^{-\mathbf{4}}$)
+/−/=	2/29/0	2/27/2	0/29/2	0/30/1	2/28/1	2/28/1	13/14/4	8/18/5	0/22/9

. The bold portions in the table highlight the best experimental results. The marks “−”, “+”, and “=” in the final row of the table denote the number of test instances for which MTGA-CMMO is significantly better, significantly worse, or approximately the same, respectively, compared to the other comparison algorithms. From Table 2, MTGA-CMMO obtains the best average IGD values in 12 of 31 test instances. C-TAEA and PPS, which are two representative CMOEAs, win 11 and 6 test problems, respectively. DN-NSGA-II-epsilon and MMEA-WI-epsilon win one test instance each, while DN-NSGA-II-CDP, MRPS-CDP, MRPS-epsilon, and MMEA-WI-CDP do not outperform the other algorithms in any test instances. Furthermore, according to the Wilcoxon rank-sum test analysis, MTGA-CMMO performs similarly to C-TAEA and is slightly better than PPS but superior to the other comparison algorithms. This is because C-TAEA and PPS have stronger convergent selection pressures that can push the population closer to the CPF. Based on the above experimental data and analysis, the objective space performance of MTGA-CMMO is close to that of C-TAEA but has a significant advantage over the other comparison algorithms.

Table 3. Experimental data based on the IGDX metric of all the comparison algorithms on 31 test problems.

Problem	DN-NSGA-II-CDP	DN-NSGA-II-Epsilon	MRPS-CDP	MRPS-Epsilon	MMEA-WI-CDP	MMEA-WI-Epsilon	C-TAEA	PPS	CMMODE	MTGA-CMMO
CMMF1	1.0830 $\times {10}^{-1}$ (7.06 $\times {10}^{-2}$) =	8.6562 $\times {10}^{-2}$ (4.17 $\times {10}^{-2}$) =	1.2771 $\times {10}^{-1}$ (4.74 $\times {10}^{-2}$) −	1.8086 $\times {10}^{-1}$ (5.56 $\times {10}^{-2}$) −	7.4451 $\times {10}^{-2}$ (2.79 $\times {10}^{-3}$) +	6.9631 $\times {10}^{-2}$ (3.02 $\times {10}^{-2}$) =	1.2623 $\times {10}^{-1}$ (3.61 $\times {10}^{-2}$) −	3.4092 $\times {10}^{-1}$ (8.60 $\times {10}^{-2}$) −	2.9499 $\times {\mathbf{10}}^{-\mathbf{2}}$ (2.69 $\times {\mathbf{10}}^{-\mathbf{2}}$) +	7.8217 $\times {10}^{-2}$ (3.18 $\times {10}^{-3}$)
CMMF2	3.0064 $\times {10}^{-2}$ (4.41 $\times {10}^{-3}$) −	3.9545 $\times {10}^{-2}$ (5.33 $\times {10}^{-3}$) −	2.9634 $\times {10}^{-2}$ (3.21 $\times {10}^{-3}$) −	3.6396 $\times {10}^{-2}$ (8.56 $\times {10}^{-3}$) −	2.6475 $\times {10}^{-2}$ (2.21 $\times {10}^{-3}$) =	4.0739 $\times {10}^{-2}$ (4.06 $\times {10}^{-3}$) −	2.3177 $\times {10}^{-1}$ (1.07 $\times {10}^{-1}$) −	6.6569 $\times {10}^{-2}$ (6.24 $\times {10}^{-2}$) −	3.1184 $\times {10}^{-2}$ (6.33 $\times {10}^{-3}$) −	2.5860 $\times {\mathbf{10}}^{-\mathbf{2}}$ (2.53 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMF3	3.4633 $\times {10}^{-2}$ (8.17 $\times {10}^{-2}$) =	3.4979 $\times {10}^{-2}$ (4.87 $\times {10}^{-3}$) −	1.7194 $\times {10}^{-2}$ (2.44 $\times {10}^{-3}$) =	4.0446 $\times {10}^{-2}$ (1.45 $\times {10}^{-2}$) −	1.9190 $\times {10}^{-2}$ (1.50 $\times {10}^{-3}$) −	4.2701 $\times {10}^{-2}$ (5.96 $\times {10}^{-3}$) −	1.8438 $\times {10}^{-1}$ (2.00 $\times {10}^{-1}$) −	6.3166 $\times {10}^{-2}$ (1.73 $\times {10}^{-1}$) −	1.6698 $\times {\mathbf{10}}^{-\mathbf{2}}$ (1.84 $\times {\mathbf{10}}^{-\mathbf{3}}$) =	1.7117 $\times {10}^{-2}$ (1.79 $\times {10}^{-3}$)
CMMF4	2.1777 $\times {10}^{-2}$ (1.38 $\times {10}^{-3}$) −	4.4922 $\times {10}^{-2}$ (1.21 $\times {10}^{-2}$) −	3.4416 $\times {10}^{-2}$ (1.08 $\times {10}^{-2}$) −	7.4282 $\times {10}^{-2}$ (5.28 $\times {10}^{-2}$) −	2.3807 $\times {10}^{-2}$ (1.28 $\times {10}^{-3}$) −	4.6453 $\times {10}^{-2}$ (4.98 $\times {10}^{-3}$) −	7.9266 $\times {10}^{-2}$ (7.95 $\times {10}^{-2}$) −	1.0163 $\times {10}^{-1}$ (6.53 $\times {10}^{-2}$) −	2.1897 $\times {10}^{-2}$ (1.56 $\times {10}^{-3}$) −	2.0717 $\times {\mathbf{10}}^{-\mathbf{2}}$ (1.44 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMF5	4.7442 $\times {10}^{-2}$ (6.87 $\times {10}^{-2}$) −	3.5223 $\times {10}^{-2}$ (6.09 $\times {10}^{-3}$) −	1.7808 $\times {10}^{-2}$ (1.77 $\times {10}^{-3}$) −	5.2721 $\times {10}^{-1}$ (1.75 $\times {10}^{-1}$) −	1.5938 $\times {10}^{-2}$ (1.37 $\times {10}^{-3}$) −	3.4315 $\times {10}^{-2}$ (3.75 $\times {10}^{-3}$) −	3.6940 $\times {10}^{-1}$ (2.08 $\times {10}^{-1}$) −	3.1269 $\times {10}^{-1}$ (2.14 $\times {10}^{-1}$) −	1.3738 $\times {\mathbf{10}}^{-\mathbf{2}}$ (1.42 $\times {\mathbf{10}}^{-\mathbf{3}}$) =	1.4004 $\times {10}^{-2}$ (1.17 $\times {10}^{-3}$)
CMMF6	1.8493 $\times {10}^{-2}$ (1.52 $\times {10}^{-3}$) −	2.3934 $\times {10}^{-2}$ (2.51 $\times {10}^{-3}$) −	2.0669 $\times {10}^{-2}$ (2.00 $\times {10}^{-3}$) −	2.0134 $\times {10}^{-2}$ (3.68 $\times {10}^{-3}$) −	1.5077 $\times {\mathbf{10}}^{-\mathbf{2}}$ (1.25 $\times {\mathbf{10}}^{-\mathbf{3}}$) +	1.8862 $\times {10}^{-2}$ (1.41 $\times {10}^{-3}$) −	5.1165 $\times {10}^{-2}$ (6.62 $\times {10}^{-2}$) =	2.8970 $\times {10}^{-2}$ (2.74 $\times {10}^{-2}$) −	3.5158 $\times {10}^{-2}$ (2.89 $\times {10}^{-2}$) −	1.6277 $\times {10}^{-2}$ (1.70 $\times {10}^{-3}$)
CMMF7	1.8236 $\times {10}^{-2}$ (2.05 $\times {10}^{-3}$) −	3.1646 $\times {10}^{-2}$ (4.68 $\times {10}^{-3}$) −	1.8447 $\times {10}^{-2}$ (1.69 $\times {10}^{-3}$) −	1.7817 $\times {10}^{-2}$ (1.74 $\times {10}^{-3}$) =	2.0451 $\times {10}^{-2}$ (1.29 $\times {10}^{-3}$) −	3.1696 $\times {10}^{-2}$ (9.13 $\times {10}^{-3}$) −	8.0975 $\times {10}^{-1}$ (1.28 $\times {10}^{-1}$) −	2.8232 $\times {10}^{-1}$ (3.61 $\times {10}^{-1}$) −	1.7862 $\times {10}^{-2}$ (1.49 $\times {10}^{-3}$) =	1.7017 $\times {\mathbf{10}}^{-\mathbf{2}}$ (1.52 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMF8	3.8639 $\times {10}^{-2}$ (1.48 $\times {10}^{-1}$) =	6.4425 $\times {10}^{-2}$ (1.19 $\times {10}^{-2}$) −	7.2328 $\times {10}^{-3}$ (6.97 $\times {10}^{-4}$) −	7.1567 $\times {10}^{-3}$ (4.54 $\times {10}^{-4}$) −	8.3809 $\times {10}^{-3}$ (5.06 $\times {10}^{-4}$) −	5.3496 $\times {10}^{-2}$ (8.70 $\times {10}^{-3}$) −	6.4995 $\times {10}^{-2}$ (1.44 $\times {10}^{-1}$) −	5.3025 $\times {10}^{-1}$ (2.80 $\times {10}^{-1}$) −	7.0849 $\times {10}^{-3}$ (8.56 $\times {10}^{-3}$) −	6.3266 $\times {\mathbf{10}}^{-\mathbf{3}}$ (2.63 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF9	1.0216 $\times {10}^{-1}$ (2.44 $\times {10}^{-1}$) =	5.2581 $\times {10}^{-3}$ (2.34 $\times {10}^{-4}$) =	5.7614 $\times {10}^{-3}$ (3.14 $\times {10}^{-4}$) −	6.3618 $\times {10}^{-3}$ (6.00 $\times {10}^{-4}$) −	6.8937 $\times {10}^{-3}$ (6.48 $\times {10}^{-4}$) −	7.4924 $\times {10}^{-3}$ (1.26 $\times {10}^{-3}$) −	4.4183 $\times {10}^{-1}$ (3.14 $\times {10}^{-1}$) −	4.6749 $\times {10}^{-1}$ (3.06 $\times {10}^{-1}$) −	6.2928 $\times {10}^{-3}$ (1.69 $\times {10}^{-3}$) −	5.2460 $\times {\mathbf{10}}^{-\mathbf{3}}$ (3.44 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF10	1.2660 $\times {10}^{-2}$ (1.18 $\times {10}^{-3}$) =	2.4519 $\times {10}^{-2}$ (3.90 $\times {10}^{-3}$) −	1.5188 $\times {10}^{-2}$ (1.80 $\times {10}^{-3}$) −	3.4581 $\times {10}^{-2}$ (5.14 $\times {10}^{-2}$) −	1.4441 $\times {10}^{-2}$ (1.01 $\times {10}^{-3}$) −	2.7352 $\times {10}^{-2}$ (5.11 $\times {10}^{-3}$) −	1.9690 $\times {10}^{-1}$ (1.59 $\times {10}^{-1}$) −	3.2129 $\times {10}^{-1}$ (1.61 $\times {10}^{-1}$) −	1.3011 $\times {10}^{-2}$ (9.93 $\times {10}^{-4}$) =	1.2497 $\times {\mathbf{10}}^{-\mathbf{2}}$ (8.43 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF11	3.8356 $\times {10}^{-2}$ (6.09 $\times {10}^{-2}$) −	1.9238 $\times {10}^{-2}$ (3.77 $\times {10}^{-3}$) −	2.2802 $\times {10}^{-2}$ (1.21 $\times {10}^{-2}$) −	1.6883 $\times {10}^{-1}$ (1.30 $\times {10}^{-1}$) −	1.0877 $\times {10}^{-2}$ (9.42 $\times {10}^{-4}$) −	2.0589 $\times {10}^{-2}$ (2.06 $\times {10}^{-3}$) −	4.0135 $\times {10}^{-1}$ (1.20 $\times {10}^{-1}$) −	1.4177 $\times {10}^{-1}$ (2.11 $\times {10}^{-1}$) −	1.0919 $\times {10}^{-2}$ (9.09 $\times {10}^{-4}$) −	9.5716 $\times {\mathbf{10}}^{-\mathbf{3}}$ (7.87 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF12	6.6881 $\times {10}^{-2}$ (1.53 $\times {10}^{-1}$) −	3.5177 $\times {10}^{-2}$ (8.18 $\times {10}^{-3}$) −	2.0743 $\times {10}^{-2}$ (4.99 $\times {10}^{-2}$) −	8.9484 $\times {10}^{-3}$ (1.32 $\times {10}^{-3}$) −	8.4790 $\times {10}^{-3}$ (6.12 $\times {10}^{-4}$) −	2.7395 $\times {10}^{-2}$ (3.62 $\times {10}^{-3}$) −	1.5972 $\times {10}^{-1}$ (2.15 $\times {10}^{-1}$) −	2.7746 $\times {10}^{-1}$ (1.35 $\times {10}^{-1}$) −	7.7346 $\times {10}^{-3}$ (7.26 $\times {10}^{-4}$) −	6.9204 $\times {\mathbf{10}}^{-\mathbf{3}}$ (5.04 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF13	3.7735 $\times {10}^{-2}$ (5.15 $\times {10}^{-2}$) =	1.3548 $\times {10}^{-1}$ (8.15 $\times {10}^{-2}$) −	1.3416 $\times {10}^{-2}$ (2.69 $\times {10}^{-3}$) −	1.4742 $\times {10}^{-2}$ (4.38 $\times {10}^{-3}$) −	7.8750 $\times {10}^{-3}$ (4.85 $\times {10}^{-4}$) =	5.1743 $\times {10}^{-2}$ (6.00 $\times {10}^{-2}$) −	7.3996 $\times {10}^{-2}$ (8.22 $\times {10}^{-2}$) −	3.0376 $\times {10}^{-1}$ (1.62 $\times {10}^{-1}$) −	1.4385 $\times {10}^{-2}$ (2.04 $\times {10}^{-2}$) −	7.4532 $\times {\mathbf{10}}^{-\mathbf{3}}$ (7.47 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF14	1.5146 $\times {10}^{-1}$ (3.10 $\times {10}^{-1}$) −	5.2693 $\times {10}^{-1}$ (1.59 $\times {10}^{-1}$) −	4.6779 $\times {10}^{-3}$ (4.59 $\times {10}^{-4}$) =	5.8133 $\times {10}^{-3}$ (6.98 $\times {10}^{-4}$) −	6.3431 $\times {10}^{-3}$ (5.00 $\times {10}^{-4}$) −	1.5795 $\times {10}^{-1}$ (3.62 $\times {10}^{-2}$) −	3.0043 $\times {10}^{-1}$ (3.46 $\times {10}^{-1}$) −	6.3234 $\times {10}^{-1}$ (2.62 $\times {10}^{-1}$) −	5.5312 $\times {10}^{-3}$ (1.15 $\times {10}^{-3}$) −	4.6091 $\times {\mathbf{10}}^{-\mathbf{3}}$ (3.93 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF15	4.3896 $\times {10}^{-2}$ (1.21 $\times {10}^{-1}$) −	3.0521 $\times {10}^{-1}$ (1.12 $\times {10}^{-1}$) −	5.9979 $\times {10}^{-3}$ (1.86 $\times {10}^{-3}$) −	2.8164 $\times {10}^{-2}$ (1.86 $\times {10}^{-2}$) −	1.0513 $\times {10}^{-2}$ (5.02 $\times {10}^{-3}$) −	2.2944 $\times {10}^{-1}$ (1.43 $\times {10}^{-1}$) −	8.1945 $\times {10}^{-3}$ (2.97 $\times {10}^{-3}$) −	9.0175 $\times {10}^{-3}$ (4.47 $\times {10}^{-3}$) −	7.8701 $\times {10}^{-3}$ (2.77 $\times {10}^{-3}$) −	4.7162 $\times {\mathbf{10}}^{-\mathbf{3}}$ (5.73 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF16	6.0094 $\times {10}^{-1}$ (5.39 $\times {10}^{-1}$) −	1.8116 $\times {10}^{-1}$ (8.28 $\times {10}^{-3}$) −	6.2757 $\times {10}^{-2}$ (1.38 $\times {10}^{-2}$) −	1.0760 $\times {10}^{-1}$ (3.71 $\times {10}^{-2}$) −	2.7310 $\times {10}^{-2}$ (1.78 $\times {10}^{-3}$) −	1.8164 $\times {10}^{-1}$ (3.37 $\times {10}^{-3}$) −	5.8692 $\times {10}^{-1}$ (3.45 $\times {10}^{-1}$) −	9.1101 $\times {10}^{-1}$ (3.43 $\times {10}^{-1}$) −	1.3505 $\times {10}^{-1}$ (4.23 $\times {10}^{-2}$) −	2.4193 $\times {\mathbf{10}}^{-\mathbf{2}}$ (2.31 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMF17	3.4862 $\times {10}^{-2}$ (9.55 $\times {10}^{-3}$) −	3.3044 $\times {10}^{-1}$ (9.25 $\times {10}^{-2}$) −	4.1291 $\times {10}^{-2}$ (8.85 $\times {10}^{-3}$) −	1.5252 $\times {10}^{-1}$ (5.51 $\times {10}^{-2}$) −	2.6121 $\times {\mathbf{10}}^{-\mathbf{2}}$ (2.71 $\times {\mathbf{10}}^{-\mathbf{3}}$) +	3.2463 $\times {10}^{-1}$ (9.88 $\times {10}^{-2}$) −	4.5470 $\times {10}^{-2}$ (3.03 $\times {10}^{-2}$) =	3.8877 $\times {10}^{-1}$ (8.58 $\times {10}^{-2}$) −	6.9368 $\times {10}^{-2}$ (3.31 $\times {10}^{-2}$) −	3.0234 $\times {10}^{-2}$ (5.78 $\times {10}^{-3}$)
CMMOP1	7.2170 $\times {10}^{-2}$ (2.87 $\times {10}^{-3}$) −	7.6454 $\times {10}^{-2}$ (8.07 $\times {10}^{-3}$) −	8.9924 $\times {10}^{-2}$ (7.51 $\times {10}^{-3}$) −	1.2490 $\times {10}^{-1}$ (1.56 $\times {10}^{-2}$) −	6.9218 $\times {10}^{-2}$ (2.16 $\times {10}^{-3}$) =	6.9859 $\times {10}^{-2}$ (7.20 $\times {10}^{-3}$) =	1.2005 $\times {10}^{-1}$ (3.13 $\times {10}^{-2}$) −	1.6188 $\times {10}^{-1}$ (5.09 $\times {10}^{-2}$) −	8.1839 $\times {10}^{-2}$ (3.16 $\times {10}^{-3}$) −	6.7959 $\times {\mathbf{10}}^{-\mathbf{2}}$ (2.56 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMOP2	5.8105 $\times {10}^{-2}$ (3.53 $\times {10}^{-2}$) −	5.0546 $\times {10}^{-2}$ (4.27 $\times {10}^{-2}$) −	2.0085 $\times {10}^{-2}$ (4.72 $\times {10}^{-3}$) =	8.9831 $\times {10}^{-2}$ (4.26 $\times {10}^{-2}$) −	2.2155 $\times {10}^{-2}$ (8.91 $\times {10}^{-3}$) =	1.7498 $\times {10}^{-2}$ (5.80 $\times {10}^{-3}$) +	8.1982 $\times {10}^{-2}$ (4.20 $\times {10}^{-2}$) −	1.5910 $\times {10}^{-1}$ (6.96 $\times {10}^{-2}$) −	1.7277 $\times {\mathbf{10}}^{-\mathbf{2}}$ (2.51 $\times {\mathbf{10}}^{-\mathbf{3}}$) =	2.0920 $\times {10}^{-2}$ (8.15 $\times {10}^{-3}$)
CMMOP3	1.3053 $\times {10}^{-1}$ (6.22 $\times {10}^{-3}$) −	1.3454 $\times {10}^{-1}$ (1.24 $\times {10}^{-2}$) −	1.6324 $\times {10}^{-1}$ (2.10 $\times {10}^{-2}$) −	2.2802 $\times {10}^{-1}$ (3.74 $\times {10}^{-2}$) −	1.0789 $\times {\mathbf{10}}^{-\mathbf{1}}$ (4.72 $\times {\mathbf{10}}^{-\mathbf{3}}$) +	1.1249 $\times {10}^{-1}$ (8.17 $\times {10}^{-3}$) +	2.0636 $\times {10}^{-1}$ (4.35 $\times {10}^{-2}$) −	2.7491 $\times {10}^{-1}$ (4.32 $\times {10}^{-2}$) −	6.0561 $\times {10}^{-1}$ (1.27 $\times {10}^{-2}$) −	1.2052 $\times {10}^{-1}$ (5.69 $\times {10}^{-3}$)
CMMOP4	4.3800 $\times {10}^{-2}$ (5.85 $\times {10}^{-3}$) −	7.5722 $\times {10}^{-2}$ (1.45 $\times {10}^{-2}$) −	9.4876 $\times {10}^{-2}$ (2.25 $\times {10}^{-2}$) −	2.0662 $\times {10}^{-1}$ (6.25 $\times {10}^{-2}$) −	3.6649 $\times {\mathbf{10}}^{-\mathbf{2}}$ (2.01 $\times {\mathbf{10}}^{-\mathbf{3}}$) =	7.0901 $\times {10}^{-2}$ (1.38 $\times {10}^{-2}$) −	1.3421 $\times {10}^{-1}$ (4.80 $\times {10}^{-2}$) −	2.0092 $\times {10}^{-1}$ (7.60 $\times {10}^{-2}$) −	1.5635 $\times {10}^{-1}$ (3.91 $\times {10}^{-2}$) −	3.8025 $\times {10}^{-2}$ (3.01 $\times {10}^{-3}$)
CMMOP5	1.0606 $\times {10}^{-1}$ (1.77 $\times {10}^{-2}$) −	1.6664 $\times {10}^{-1}$ (5.74 $\times {10}^{-2}$) −	3.1553 $\times {10}^{-1}$ (1.02 $\times {10}^{-1}$) −	5.8575 $\times {10}^{-1}$ (1.84 $\times {10}^{-1}$) −	8.3948 $\times {\mathbf{10}}^{-\mathbf{2}}$ (7.19 $\times {\mathbf{10}}^{-\mathbf{3}}$) +	2.5759 $\times {10}^{-1}$ (1.38 $\times {10}^{-1}$) −	9.9951 $\times {10}^{-1}$ (3.68 $\times {10}^{-1}$) −	1.1020 $\times {10}^0$ (4.41 $\times {10}^{-1}$) −	5.8120 $\times {10}^{-1}$ (2.83 $\times {10}^{-1}$) −	9.0748 $\times {10}^{-2}$ (1.12 $\times {10}^{-2}$)
CMMOP6	5.4838 $\times {10}^{-2}$ (5.09 $\times {10}^{-2}$) −	4.2890 $\times {10}^{-2}$ (4.74 $\times {10}^{-2}$) −	1.4571 $\times {\mathbf{10}}^{-\mathbf{2}}$ (2.86 $\times {\mathbf{10}}^{-\mathbf{3}}$) +	4.2791 $\times {10}^{-2}$ (3.70 $\times {10}^{-2}$) −	1.5835 $\times {10}^{-2}$ (3.71 $\times {10}^{-3}$) =	1.9733 $\times {10}^{-2}$ (1.47 $\times {10}^{-2}$) −	7.8047 $\times {10}^{-2}$ (3.44 $\times {10}^{-2}$) −	1.4674 $\times {10}^{-1}$ (3.72 $\times {10}^{-2}$) −	3.0621 $\times {10}^{-2}$ (1.94 $\times {10}^{-2}$) −	1.9053 $\times {10}^{-2}$ (6.28 $\times {10}^{-3}$)
CMMOP7	1.1924 $\times {10}^{-1}$ (6.47 $\times {10}^{-3}$) −	1.1803 $\times {10}^{-1}$ (7.98 $\times {10}^{-3}$) −	1.4295 $\times {10}^{-1}$ (1.76 $\times {10}^{-2}$) −	1.9317 $\times {10}^{-1}$ (2.73 $\times {10}^{-2}$) −	9.5075 $\times {\mathbf{10}}^{-\mathbf{2}}$ (3.87 $\times {\mathbf{10}}^{-\mathbf{3}}$) +	9.8018 $\times {10}^{-2}$ (6.31 $\times {10}^{-3}$) +	2.0965 $\times {10}^{-1}$ (3.43 $\times {10}^{-2}$) −	2.1759 $\times {10}^{-1}$ (2.34 $\times {10}^{-2}$) −	7.0134 $\times {10}^{-1}$ (4.46 $\times {10}^{-3}$) −	1.0783 $\times {10}^{-1}$ (5.54 $\times {10}^{-3}$)
CMMOP8	2.7548 $\times {10}^{-2}$ (1.69 $\times {10}^{-3}$) =	7.8138 $\times {10}^{-2}$ (3.44 $\times {10}^{-2}$) −	6.4008 $\times {10}^{-2}$ (2.52 $\times {10}^{-2}$) −	6.9071 $\times {10}^{-2}$ (1.86 $\times {10}^{-2}$) −	3.0718 $\times {10}^{-2}$ (1.69 $\times {10}^{-3}$) −	7.1433 $\times {10}^{-2}$ (2.72 $\times {10}^{-2}$) −	6.6548 $\times {10}^{-2}$ (1.84 $\times {10}^{-2}$) −	9.7853 $\times {10}^{-2}$ (5.05 $\times {10}^{-2}$) −	3.1092 $\times {10}^{-2}$ (2.28 $\times {10}^{-3}$) −	2.7043 $\times {\mathbf{10}}^{-\mathbf{2}}$ (1.53 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMOP9	6.9497 $\times {10}^{-2}$ (2.20 $\times {10}^{-3}$) −	7.1125 $\times {10}^{-2}$ (3.78 $\times {10}^{-3}$) −	9.4871 $\times {10}^{-2}$ (1.37 $\times {10}^{-2}$) −	1.1485 $\times {10}^{-1}$ (1.99 $\times {10}^{-2}$) −	6.6395 $\times {10}^{-2}$ (3.06 $\times {10}^{-3}$) −	6.8850 $\times {10}^{-2}$ (6.34 $\times {10}^{-3}$) −	1.0116 $\times {10}^{-1}$ (1.76 $\times {10}^{-2}$) −	1.5413 $\times {10}^{-1}$ (5.47 $\times {10}^{-2}$) −	7.6829 $\times {10}^{-2}$ (3.99 $\times {10}^{-3}$) −	6.4313 $\times {\mathbf{10}}^{-\mathbf{2}}$ (2.14 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMOP10	5.3780 $\times {10}^{-2}$ (5.48 $\times {10}^{-3}$) =	6.4282 $\times {10}^{-2}$ (8.57 $\times {10}^{-3}$) −	8.2822 $\times {10}^{-2}$ (2.30 $\times {10}^{-2}$) −	1.3834 $\times {10}^{-1}$ (4.11 $\times {10}^{-2}$) −	5.4036 $\times {10}^{-2}$ (2.89 $\times {10}^{-3}$) −	6.3319 $\times {10}^{-2}$ (6.14 $\times {10}^{-3}$) −	9.4199 $\times {10}^{-2}$ (4.99 $\times {10}^{-2}$) −	1.6430 $\times {10}^{-1}$ (7.41 $\times {10}^{-2}$) −	6.2487 $\times {10}^{-2}$ (3.17 $\times {10}^{-3}$) −	5.1718 $\times {\mathbf{10}}^{-\mathbf{2}}$ (2.43 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMOP11	1.1005 $\times {10}^{-1}$ (5.93 $\times {10}^{-3}$) −	1.8644 $\times {10}^{-1}$ (2.29 $\times {10}^{-2}$) −	1.7456 $\times {10}^{-1}$ (1.82 $\times {10}^{-2}$) −	2.7020 $\times {10}^{-1}$ (4.94 $\times {10}^{-2}$) −	1.0138 $\times {10}^{-1}$ (5.93 $\times {10}^{-3}$) −	1.8037 $\times {10}^{-1}$ (3.02 $\times {10}^{-2}$) −	1.9046 $\times {10}^{-1}$ (5.52 $\times {10}^{-2}$) −	2.1467 $\times {10}^{-1}$ (6.94 $\times {10}^{-2}$) −	1.2420 $\times {10}^{-1}$ (9.34 $\times {10}^{-2}$) −	9.4769 $\times {\mathbf{10}}^{-\mathbf{2}}$ (3.40 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMOP12	1.5576 $\times {10}^{-1}$ (4.62 $\times {10}^{-2}$) −	1.4389 $\times {10}^{-1}$ (2.38 $\times {10}^{-2}$) −	1.9414 $\times {10}^{-1}$ (2.59 $\times {10}^{-2}$) −	3.9282 $\times {10}^{-1}$ (8.64 $\times {10}^{-2}$) −	1.1186 $\times {\mathbf{10}}^{-\mathbf{1}}$ (1.02 $\times {\mathbf{10}}^{-\mathbf{2}}$) =	1.1545 $\times {10}^{-1}$ (1.86 $\times {10}^{-2}$) =	3.6138 $\times {10}^{-1}$ (8.92 $\times {10}^{-2}$) −	5.2785 $\times {10}^{-1}$ (1.10 $\times {10}^{-1}$) −	1.4344 $\times {10}^{-1}$ (2.09 $\times {10}^{-2}$) −	1.1658 $\times {10}^{-1}$ (1.70 $\times {10}^{-2}$)
CMMOP13	7.4174 $\times {10}^{-2}$ (3.06 $\times {10}^{-3}$) −	7.9406 $\times {10}^{-2}$ (7.47 $\times {10}^{-3}$) −	9.2540 $\times {10}^{-2}$ (1.35 $\times {10}^{-2}$) −	1.2291 $\times {10}^{-1}$ (1.64 $\times {10}^{-2}$) −	6.9285 $\times {10}^{-2}$ (2.01 $\times {10}^{-3}$) −	7.0371 $\times {10}^{-2}$ (7.00 $\times {10}^{-3}$) =	1.2231 $\times {10}^{-1}$ (3.20 $\times {10}^{-2}$) −	1.7390 $\times {10}^{-1}$ (3.91 $\times {10}^{-2}$) −	8.4145 $\times {10}^{-2}$ (5.49 $\times {10}^{-3}$) −	6.6456 $\times {\mathbf{10}}^{-\mathbf{2}}$ (2.64 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMOP14	7.6405 $\times {10}^{-2}$ (3.05 $\times {10}^{-3}$) −	6.9473 $\times {10}^{-2}$ (2.31 $\times {10}^{-3}$) −	9.1532 $\times {10}^{-2}$ (9.80 $\times {10}^{-3}$) −	1.2137 $\times {10}^{-1}$ (1.45 $\times {10}^{-2}$) −	7.0439 $\times {10}^{-2}$ (1.96 $\times {10}^{-3}$) −	6.5676 $\times {\mathbf{10}}^{-\mathbf{2}}$ (3.59 $\times {\mathbf{10}}^{-\mathbf{3}}$) +	1.3596 $\times {10}^{-1}$ (3.96 $\times {10}^{-2}$) −	1.7637 $\times {10}^{-1}$ (4.86 $\times {10}^{-2}$) −	8.6605 $\times {10}^{-2}$ (3.84 $\times {10}^{-3}$) −	6.7299 $\times {10}^{-2}$ (2.81 $\times {10}^{-3}$)
+/−/=	0/23/8	0/29/2	1/27/3	0/30/1	6/18/7	4/23/4	0/29/2	0/31/0	1/25/5

shows the experimental data in terms of the IGDX indicator obtained by the comparison algorithms on the 31 test problems. MTGA-CMMO obtains the smallest IGDX average values on 18 test problems, as well as the second and third smallest IGDX average values on 7 and 4 test instances, respectively. MMEA-WI-CDP has the best performance in terms of IGDX indicator on 7 test instances, but the statistical analysis showed that MMEA-WI-CDP does not perform substantially differently from MTGA-CMMO on CMMOP4 and CMMOP12, which are 2 of 7 test instances. Compared to MTGA-CMMO, the advantages of MMEA-WI-CDP are mainly in CMMOP3-5 and CMMOP12, which have smaller spacing between various CPSs and a high overlap of the CPSs. The reason why MMEA-WI-CDP can identify different CPSs and obtain more CPSs in these four instances is that the decision space diversity information is integrated with the objective space performance indicator in the environmental selection strategy of MMEA-WI-CDP. CMMODE has the smallest average IGDX values on CMMF1, CMMF3, CMMF5, and CMMOP2, and, likewise, based on the statistical analyses, CMMODE significantly outperforms MTGA-CMMO on only the CMMF1 test instance. The experimental results demonstrate how to reasonably decompose a whole population into multiple sub-populations by speciation strategy, which may be decisive for the performance of the CMMODE. In addition, MRPS-CDP and MMEA-WI-epsilon outperform the other comparison algorithms on CMMOP6 and CMMOP14, respectively. According to the statistical analysis, MTGA-CMMO bests DN-NSGA-II-CDP, DN-NSGA-II-epsilon, MRPS-CDP, MRPS-epsilon, MMEA-WI-CDP, MMEA-WI-epsilon, C-TAEA, PPS, and CMMODE in 23, 29, 27, 30, 18, 23, 29, 31, and 25 test instances, respectively.

Table 4. Experimental data based on the PSP metric of all the comparison algorithms on 31 test problems.

Problem	DN-NSGA-II-CDP	DN-NSGA-II-Epsilon	MRPS-CDP	MRPS-Epsilon	MMEA-WI-CDP	MMEA-WI-Epsilon	C-TAEA	PPS	CMMODE	MTGA-CMMO
CMMF1	1.1281 $\times {10}^1$ (4.01 $\times {10}^0$) =	1.5385 $\times {10}^1$ (9.14 $\times {10}^0$) =	8.3981 $\times {10}^0$ (2.36 $\times {10}^0$) −	5.8580 $\times {10}^0$ (1.85 $\times {10}^0$) −	1.3193 $\times {10}^1$ (4.95 $\times {10}^{-1}$) +	1.7883 $\times {10}^1$ (8.94 $\times {10}^0$) =	8.3315 $\times {10}^0$ (2.19 $\times {10}^0$) −	3.0695 $\times {10}^0$ (8.55 $\times {10}^{-1}$) −	5.4483 $\times {\mathbf{10}}^{\mathbf{1}}$ (2.48 $\times {\mathbf{10}}^{\mathbf{1}}$) +	1.2452 $\times {10}^1$ (5.14 $\times {10}^{-1}$)
CMMF2	3.3827 $\times {10}^1$ (4.19 $\times {10}^0$) −	2.5761 $\times {10}^1$ (3.78 $\times {10}^0$) −	3.3952 $\times {10}^1$ (3.57 $\times {10}^0$) −	2.8304 $\times {10}^1$ (6.60 $\times {10}^0$) −	3.7895 $\times {10}^1$ (3.29 $\times {10}^0$) =	2.4500 $\times {10}^1$ (2.47 $\times {10}^0$) −	5.1732 $\times {10}^0$ (6.56 $\times {10}^0$) −	1.9777 $\times {10}^1$ (7.97 $\times {10}^0$) −	3.2529 $\times {10}^1$ (5.44 $\times {10}^0$) −	3.8560 $\times {\mathbf{10}}^{\mathbf{1}}$ (4.29 $\times {\mathbf{10}}^{\mathbf{0}}$)
CMMF3	5.7129 $\times {10}^1$ (1.37 $\times {10}^1$) =	2.9091 $\times {10}^1$ (3.82 $\times {10}^0$) −	5.8932 $\times {10}^1$ (7.99 $\times {10}^0$) =	2.6628 $\times {10}^1$ (8.50 $\times {10}^0$) −	5.2369 $\times {10}^1$ (3.91 $\times {10}^0$) −	2.3778 $\times {10}^1$ (3.33 $\times {10}^0$) −	1.6036 $\times {10}^1$ (1.69 $\times {10}^1$) −	4.3239 $\times {10}^1$ (2.08 $\times {10}^1$) −	6.0107 $\times {\mathbf{10}}^{\mathbf{1}}$ (7.09 $\times {\mathbf{10}}^{\mathbf{0}}$) =	5.8784 $\times {10}^1$ (5.69 $\times {10}^0$)
CMMF4	4.6093 $\times {10}^1$ (2.88 $\times {10}^0$) −	2.4446 $\times {10}^1$ (8.83 $\times {10}^0$) −	3.0996 $\times {10}^1$ (7.51 $\times {10}^0$) −	1.7475 $\times {10}^1$ (7.83 $\times {10}^0$) −	4.1946 $\times {10}^1$ (2.14 $\times {10}^0$) −	2.1696 $\times {10}^1$ (2.38 $\times {10}^0$) −	2.1412 $\times {10}^1$ (1.28 $\times {10}^1$) −	1.2887 $\times {10}^1$ (5.91 $\times {10}^0$) −	4.5123 $\times {10}^1$ (3.19 $\times {10}^0$) −	4.8006 $\times {\mathbf{10}}^{\mathbf{1}}$ (3.27 $\times {\mathbf{10}}^{\mathbf{0}}$)
CMMF5	5.5468 $\times {10}^1$ (2.58 $\times {10}^1$) −	2.9455 $\times {10}^1$ (6.71 $\times {10}^0$) −	5.6653 $\times {10}^1$ (5.41 $\times {10}^0$) −	1.9032 $\times {10}^0$ (3.32 $\times {10}^0$) −	6.3170 $\times {10}^1$ (5.24 $\times {10}^0$) −	2.9475 $\times {10}^1$ (3.22 $\times {10}^0$) −	5.3289 $\times {10}^0$ (1.06 $\times {10}^1$) −	9.9218 $\times {10}^0$ (1.64 $\times {10}^1$) −	7.3582 $\times {\mathbf{10}}^{\mathbf{2}}$ (8.13 $\times {\mathbf{10}}^{\mathbf{0}}$) =	7.1876 $\times {10}^1$ (6.01 $\times {10}^0$)
CMMF6	5.4433 $\times {10}^1$ (4.57 $\times {10}^0$) −	4.2229 $\times {10}^1$ (4.50 $\times {10}^0$) −	4.8774 $\times {10}^1$ (4.68 $\times {10}^0$) −	5.0609 $\times {10}^1$ (6.18 $\times {10}^0$) −	6.6748 $\times {\mathbf{10}}^{\mathbf{1}}$ (5.36 $\times {\mathbf{10}}^{\mathbf{0}}$) +	5.3306 $\times {10}^1$ (4.07 $\times {10}^0$) −	6.4774 $\times {10}^1$ (4.79 $\times {10}^1$) =	4.6628 $\times {10}^1$ (1.61 $\times {10}^1$) −	3.9489 $\times {10}^1$ (1.58 $\times {10}^1$) −	6.2104 $\times {10}^1$ (6.79 $\times {10}^0$)
CMMF7	5.5474 $\times {10}^1$ (6.25 $\times {10}^0$) =	3.2266 $\times {10}^1$ (4.84 $\times {10}^0$) −	5.4622 $\times {10}^1$ (5.00 $\times {10}^0$) −	5.6597 $\times {10}^1$ (5.33 $\times {10}^0$) =	4.8894 $\times {10}^1$ (3.01 $\times {10}^0$) −	3.3143 $\times {10}^1$ (7.21 $\times {10}^0$) −	4.9086 $\times {10}^{-1}$ (4.72 $\times {10}^{-1}$) −	2.0342 $\times {10}^1$ (2.40 $\times {10}^1$) −	5.5789 $\times {10}^1$ (4.79 $\times {10}^0$) =	5.8894 $\times {\mathbf{10}}^{\mathbf{1}}$ (5.21 $\times {\mathbf{10}}^{\mathbf{0}}$)
CMMF8	1.4885 $\times {10}^2$ (3.56 $\times {10}^1$) =	1.6021 $\times {10}^1$ (2.87 $\times {10}^0$) −	1.3887 $\times {10}^2$ (1.28 $\times {10}^1$) −	1.3932 $\times {10}^2$ (9.09 $\times {10}^0$) −	1.1912 $\times {10}^2$ (6.75 $\times {10}^0$) −	1.9239 $\times {10}^1$ (3.58 $\times {10}^0$) −	4.1201 $\times {10}^1$ (3.02 $\times {10}^1$) −	6.2805 $\times {10}^0$ (1.19 $\times {10}^1$) −	1.3971 $\times {10}^2$ (1.66 $\times {10}^1$) −	1.5651 $\times {\mathbf{10}}^{\mathbf{2}}$ (6.77 $\times {\mathbf{10}}^{\mathbf{0}}$)
CMMF9	1.7022 $\times {10}^2$ (7.16 $\times {10}^1$) =	1.9054 $\times {\mathbf{10}}^{\mathbf{2}}$ (8.60 $\times {\mathbf{10}}^{\mathbf{0}}$) =	1.7366 $\times {10}^2$ (9.63 $\times {10}^0$) −	1.5700 $\times {10}^2$ (1.49 $\times {10}^1$) −	1.4511 $\times {10}^2$ (1.24 $\times {10}^1$) −	1.3528 $\times {10}^2$ (1.83 $\times {10}^1$) −	8.7429 $\times {10}^0$ (1.37 $\times {10}^1$) −	1.4956 $\times {10}^1$ (3.03 $\times {10}^1$) −	1.6267 $\times {10}^2$ (3.28 $\times {10}^1$) −	1.8830 $\times {10}^2$ (1.19 $\times {10}^1$)
CMMF10	7.9581 $\times {10}^1$ (6.78 $\times {10}^0$) =	4.2150 $\times {10}^1$ (9.31 $\times {10}^0$) −	6.6689 $\times {10}^1$ (7.68 $\times {10}^0$) −	4.3907 $\times {10}^1$ (1.52 $\times {10}^1$) −	6.9572 $\times {10}^1$ (5.00 $\times {10}^0$) −	3.7612 $\times {10}^1$ (6.03 $\times {10}^0$) −	2.4398 $\times {10}^1$ (3.15 $\times {10}^1$) −	1.0089 $\times {10}^1$ (2.16 $\times {10}^1$) −	7.7043 $\times {10}^1$ (5.80 $\times {10}^0$) =	8.0104 $\times {\mathbf{10}}^{\mathbf{1}}$ (5.53 $\times {\mathbf{10}}^{\mathbf{0}}$)
CMMF11	7.8285 $\times {10}^1$ (4.32 $\times {10}^1$) −	8.3876 $\times {10}^1$ (1.06 $\times {10}^1$) −	5.0220 $\times {10}^1$ (1.79 $\times {10}^1$) −	9.4640 $\times {10}^0$ (9.39 $\times {10}^0$) −	9.1864 $\times {10}^1$ (7.25 $\times {10}^0$) −	4.8767 $\times {10}^1$ (4.80 $\times {10}^0$) −	1.0865 $\times {10}^0$ (5.40 $\times {10}^{-1}$) −	2.9803 $\times {10}^1$ (2.62 $\times {10}^1$) −	8.8801 $\times {10}^1$ (8.06 $\times {10}^0$) −	1.0359 $\times {\mathbf{10}}^{\mathbf{2}}$ (9.34 $\times {\mathbf{10}}^{\mathbf{0}}$)
CMMF12	1.0747 $\times {10}^2$ (5.31 $\times {10}^1$) −	9.9805 $\times {10}^1$ (6.34 $\times {10}^0$) −	1.0005 $\times {10}^2$ (2.87 $\times {10}^1$) −	1.1380 $\times {10}^2$ (1.53 $\times {10}^1$) −	1.1849 $\times {10}^2$ (8.53 $\times {10}^0$) −	9.7183 $\times {10}^1$ (5.55 $\times {10}^0$) −	6.7052 $\times {10}^1$ (7.74 $\times {10}^1$) −	9.8200 $\times {10}^0$ (2.09 $\times {10}^1$) −	1.3013 $\times {10}^2$ (1.19 $\times {10}^1$) −	1.4509 $\times {\mathbf{10}}^{\mathbf{2}}$ (9.85 $\times {\mathbf{10}}^{\mathbf{0}}$)
CMMF13	1.0245 $\times {10}^2$ (6.16 $\times {10}^1$) =	2.0001 $\times {10}^1$ (3.04 $\times {10}^1$) −	7.6809 $\times {10}^1$ (1.57 $\times {10}^1$) −	7.2227 $\times {10}^1$ (1.83 $\times {10}^1$) −	1.2721 $\times {10}^2$ (7.76 $\times {10}^0$) =	6.3219 $\times {10}^1$ (4.08 $\times {10}^1$) −	2.8395 $\times {10}^1$ (2.48 $\times {10}^1$) −	7.4493 $\times {10}^0$ (1.21 $\times {10}^1$) −	9.9572 $\times {10}^1$ (2.96 $\times {10}^1$) −	1.3484 $\times {\mathbf{10}}^{\mathbf{2}}$ (1.36 $\times {\mathbf{10}}^{\mathbf{1}}$)
CMMF14	1.7128 $\times {10}^2$ (8.61 $\times {10}^1$) −	1.5076 $\times {10}^0$ (1.12 $\times {10}^0$) −	2.1480 $\times {10}^2$ (2.15 $\times {10}^1$) =	1.7320 $\times {10}^2$ (1.93 $\times {10}^1$) −	1.5754 $\times {10}^2$ (1.30 $\times {10}^1$) −	9.9397 $\times {10}^0$ (1.80 $\times {10}^1$) −	1.3826 $\times {10}^1$ (1.41 $\times {10}^1$) −	1.4226 $\times {10}^0$ (3.85 $\times {10}^0$) −	1.8449 $\times {10}^2$ (3.53 $\times {10}^1$) −	2.1607 $\times {\mathbf{10}}^{\mathbf{2}}$ (1.70 $\times {\mathbf{10}}^{\mathbf{1}}$)
CMMF15	1.6434 $\times {10}^2$ (5.76 $\times {10}^1$) −	1.3059 $\times {10}^1$ (3.46 $\times {10}^1$) −	1.7634 $\times {10}^2$ (4.34 $\times {10}^1$) −	4.6535 $\times {10}^1$ (2.43 $\times {10}^1$) −	1.0754 $\times {10}^2$ (3.49 $\times {10}^1$) −	3.3304 $\times {10}^1$ (4.93 $\times {10}^1$) −	1.3231 $\times {10}^2$ (3.86 $\times {10}^1$) −	1.2973 $\times {10}^2$ (4.63 $\times {10}^1$) −	1.3678 $\times {10}^2$ (4.06 $\times {10}^1$) −	2.1077 $\times {\mathbf{10}}^{\mathbf{2}}$ (2.65 $\times {\mathbf{10}}^{\mathbf{1}}$)
CMMF16	5.4190 $\times {10}^0$ (6.73 $\times {10}^0$) −	5.3866 $\times {10}^0$ (2.98 $\times {10}^{-1}$) −	1.6393 $\times {10}^1$ (3.92 $\times {10}^0$) −	9.5751 $\times {10}^0$ (3.60 $\times {10}^0$) −	3.6509 $\times {10}^1$ (2.47 $\times {10}^0$) −	5.5066 $\times {10}^0$ (1.01 $\times {10}^{-1}$) −	1.4419 $\times {10}^0$ (7.11 $\times {10}^{-1}$) −	7.8463 $\times {10}^{-1}$ (5.51 $\times {10}^{-1}$) −	7.5289 $\times {10}^0$ (2.48 $\times {10}^0$) −	4.1348 $\times {\mathbf{10}}^{\mathbf{1}}$ (4.03 $\times {\mathbf{10}}^{\mathbf{0}}$)
CMMF17	3.0191 $\times {10}^1$ (6.06 $\times {10}^0$) =	4.3921 $\times {10}^0$ (6.69 $\times {10}^0$) −	2.4568 $\times {10}^1$ (4.90 $\times {10}^0$) −	6.8391 $\times {10}^0$ (3.40 $\times {10}^0$) −	3.8602 $\times {\mathbf{10}}^{\mathbf{1}}$ (3.61 $\times {\mathbf{10}}^{\mathbf{0}}$) +	5.8945 $\times {10}^0$ (9.80 $\times {10}^0$) −	2.6904 $\times {10}^1$ (1.09 $\times {10}^1$) −	2.4522 $\times {10}^0$ (1.13 $\times {10}^0$) −	1.65414 $\times {10}^1$ (6.53 $\times {10}^0$) −	3.3852 $\times {10}^1$ (6.19 $\times {10}^0$)
CMMOP1	1.3874 $\times {10}^1$ (5.51 $\times {10}^{-1}$) −	1.3214 $\times {10}^1$ (1.35 $\times {10}^0$) −	1.1070 $\times {10}^1$ (8.50 $\times {10}^{-1}$) −	7.8770 $\times {10}^0$ (1.08 $\times {10}^0$) −	1.4437 $\times {10}^1$ (4.59 $\times {10}^{-1}$) =	1.4427 $\times {10}^1$ (1.39 $\times {10}^0$) =	8.5536 $\times {10}^0$ (1.92 $\times {10}^0$) −	6.3517 $\times {10}^0$ (1.69 $\times {10}^0$) −	1.2137 $\times {10}^1$ (4.59 $\times {10}^{-1}$) −	1.4634 $\times {\mathbf{10}}^{\mathbf{1}}$ (5.40 $\times {\mathbf{10}}^{-\mathbf{1}}$)
CMMOP2	2.1102 $\times {10}^1$ (1.17 $\times {10}^1$) −	3.5669 $\times {10}^1$ (2.25 $\times {10}^1$) −	4.9379 $\times {10}^1$ (8.92 $\times {10}^0$) =	1.2829 $\times {10}^1$ (7.33 $\times {10}^0$) −	4.9387 $\times {10}^1$ (1.35 $\times {10}^1$) =	6.1929 $\times {\mathbf{10}}^{\mathbf{1}}$ (1.60 $\times {\mathbf{10}}^{\mathbf{1}}$) +	1.4545 $\times {10}^1$ (7.75 $\times {10}^0$) −	7.9150 $\times {10}^0$ (4.38 $\times {10}^0$) −	5.8662 $\times {10}^2$ (7.69 $\times {10}^0$) =	5.2218 $\times {10}^1$ (1.53 $\times {10}^1$)
CMMOP3	7.6719 $\times {10}^0$ (3.60 $\times {10}^{-1}$) −	7.4863 $\times {10}^0$ (6.78 $\times {10}^{-1}$) −	6.0420 $\times {10}^0$ (7.76 $\times {10}^{-1}$) −	4.1956 $\times {10}^0$ (8.60 $\times {10}^{-1}$) −	9.2675 $\times {\mathbf{10}}^{\mathbf{0}}$ (3.97 $\times {\mathbf{10}}^{-\mathbf{1}}$) +	8.9249 $\times {10}^0$ (6.46 $\times {10}^{-1}$) +	4.7809 $\times {10}^0$ (9.73 $\times {10}^{-1}$) −	3.6121 $\times {10}^0$ (6.51 $\times {10}^{-1}$) −	1.15991 $\times {10}^0$ (2.99 $\times {10}^{-2}$) −	8.2449 $\times {10}^0$ (3.72 $\times {10}^{-1}$)
CMMOP4	2.3192 $\times {10}^1$ (2.87 $\times {10}^0$) −	1.3852 $\times {10}^1$ (3.60 $\times {10}^0$) −	1.0723 $\times {10}^1$ (2.34 $\times {10}^0$) −	4.9611 $\times {10}^0$ (2.15 $\times {10}^0$) −	2.7302 $\times {\mathbf{10}}^{\mathbf{1}}$ (1.42 $\times {\mathbf{10}}^{\mathbf{0}}$) +	1.4859 $\times {10}^1$ (4.42 $\times {10}^0$) −	7.9857 $\times {10}^0$ (2.46 $\times {10}^0$) −	5.2429 $\times {10}^0$ (1.42 $\times {10}^0$) −	6.7603 $\times {10}^0$ (1.84 $\times {10}^0$) −	2.6214 $\times {10}^1$ (2.06 $\times {10}^0$)
CMMOP5	9.6176 $\times {10}^0$ (1.45 $\times {10}^0$) −	6.8357 $\times {10}^0$ (2.62 $\times {10}^0$) −	3.1639 $\times {10}^0$ (1.11 $\times {10}^0$) −	1.6646 $\times {10}^0$ (6.63 $\times {10}^{-1}$) −	1.1920 $\times {\mathbf{10}}^{\mathbf{1}}$ (9.95 $\times {\mathbf{10}}^{-\mathbf{1}}$) +	6.0505 $\times {10}^0$ (4.30 $\times {10}^0$) −	8.9117 $\times {10}^{-1}$ (3.06 $\times {10}^{-1}$) −	8.7083 $\times {10}^{-1}$ (4.42 $\times {10}^{-1}$) −	2.0983 $\times {10}^0$ (1.03 $\times {10}^0$) −	1.0987 $\times {10}^1$ (1.38 $\times {10}^0$)
CMMOP6	2.6362 $\times {10}^1$ (1.61 $\times {10}^1$) −	3.6082 $\times {10}^1$ (1.70 $\times {10}^1$) −	6.9578 $\times {10}^1$ (9.65 $\times {10}^0$) +	2.9358 $\times {10}^1$ (1.09 $\times {10}^1$) −	6.5341 $\times {10}^1$ (1.06 $\times {10}^1$) =	6.9879 $\times {\mathbf{10}}^{\mathbf{1}}$ (2.82 $\times {\mathbf{10}}^{\mathbf{1}}$) +	1.3120 $\times {10}^1$ (6.12 $\times {10}^0$) −	7.1864 $\times {10}^0$ (2.87 $\times {10}^0$) −	4.0449 $\times {10}^1$ (1.65 $\times {10}^1$) −	5.5005 $\times {10}^1$ (1.75 $\times {10}^1$)
CMMOP7	8.3995 $\times {10}^0$ (4.73 $\times {10}^{-1}$) −	8.5059 $\times {10}^0$ (5.54 $\times {10}^{-1}$) −	6.9413 $\times {10}^0$ (7.67 $\times {10}^{-1}$) −	5.0039 $\times {10}^0$ (7.95 $\times {10}^{-1}$) −	1.0507 $\times {\mathbf{10}}^{\mathbf{1}}$ (4.16 $\times {\mathbf{10}}^{-\mathbf{1}}$) +	1.0219 $\times {10}^1$ (6.57 $\times {10}^{-1}$) +	4.5239 $\times {10}^0$ (9.63 $\times {10}^{-1}$) −	4.4835 $\times {10}^0$ (6.73 $\times {10}^{-1}$) −	8.1939 $\times {10}^{-1}$ (7.65 $\times {10}^{-3}$) −	9.1943 $\times {10}^0$ (4.55 $\times {10}^{-1}$)
CMMOP8	3.6424 $\times {10}^1$ (2.14 $\times {10}^0$) =	1.7183 $\times {10}^1$ (1.07 $\times {10}^1$) −	1.6429 $\times {10}^1$ (6.25 $\times {10}^0$) −	1.4478 $\times {10}^1$ (3.97 $\times {10}^0$) −	3.2560 $\times {10}^1$ (1.75 $\times {10}^0$) −	1.6842 $\times {10}^1$ (8.05 $\times {10}^0$) −	1.5468 $\times {10}^1$ (5.18 $\times {10}^0$) −	1.1769 $\times {10}^1$ (4.73 $\times {10}^0$) −	3.2175 $\times {10}^1$ (2.28 $\times {10}^0$) −	3.7025 $\times {\mathbf{10}}^{\mathbf{1}}$ (2.09 $\times {\mathbf{10}}^{\mathbf{0}}$)
CMMOP9	1.4398 $\times {10}^1$ (4.61 $\times {10}^{-1}$) −	1.4090 $\times {10}^1$ (7.01 $\times {10}^{-1}$) −	1.0457 $\times {10}^1$ (1.49 $\times {10}^0$) −	8.6446 $\times {10}^0$ (1.60 $\times {10}^0$) −	1.5071 $\times {10}^1$ (6.49 $\times {10}^{-1}$) −	1.4592 $\times {10}^1$ (1.19 $\times {10}^0$) −	9.8716 $\times {10}^0$ (1.74 $\times {10}^0$) −	6.5821 $\times {10}^0$ (2.01 $\times {10}^0$) −	1.2888 $\times {10}^1$ (6.35 $\times {10}^{-1}$) −	1.5449 $\times {\mathbf{10}}^{\mathbf{1}}$ (4.87 $\times {\mathbf{10}}^{-\mathbf{1}}$)
CMMOP10	1.8763 $\times {10}^1$ (1.81 $\times {10}^0$) =	1.5813 $\times {10}^1$ (2.11 $\times {10}^0$) −	1.2516 $\times {10}^1$ (2.71 $\times {10}^0$) −	7.3361 $\times {10}^0$ (2.15 $\times {10}^0$) −	1.8512 $\times {10}^1$ (9.80 $\times {10}^{-1}$) −	1.5875 $\times {10}^1$ (1.46 $\times {10}^0$) −	1.2147 $\times {10}^1$ (3.92 $\times {10}^0$) −	6.7149 $\times {10}^0$ (2.53 $\times {10}^0$) −	1.5867 $\times {10}^1$ (8.27 $\times {10}^{-1}$) −	1.9232 $\times {\mathbf{10}}^{\mathbf{1}}$ (9.15 $\times {\mathbf{10}}^{-\mathbf{1}}$)
CMMOP11	9.0907 $\times {10}^0$ (4.79 $\times {10}^{-1}$) −	5.4663 $\times {10}^0$ (9.04 $\times {10}^{-1}$) −	5.6851 $\times {10}^0$ (5.86 $\times {10}^{-1}$) −	3.5535 $\times {10}^0$ (7.59 $\times {10}^{-1}$) −	9.8805 $\times {10}^0$ (5.58 $\times {10}^{-1}$) −	5.7618 $\times {10}^0$ (1.53 $\times {10}^0$) −	5.3411 $\times {10}^0$ (1.35 $\times {10}^0$) −	4.7655 $\times {10}^0$ (1.57 $\times {10}^0$) −	8.0768 $\times {10}^0$ (7.22 $\times {10}^{-1}$) −	1.0523 $\times {\mathbf{10}}^{\mathbf{1}}$ (3.78 $\times {\mathbf{10}}^{-\mathbf{1}}$)
CMMOP12	6.7522 $\times {10}^0$ (1.66 $\times {10}^0$) −	7.1289 $\times {10}^0$ (1.17 $\times {10}^0$) −	5.0606 $\times {10}^0$ (7.50 $\times {10}^{-1}$) −	2.4436 $\times {10}^0$ (5.98 $\times {10}^{-1}$) −	8.9942 $\times {\mathbf{10}}^{\mathbf{0}}$ (7.88 $\times {\mathbf{10}}^{-\mathbf{1}}$) =	8.8270 $\times {10}^0$ (1.22 $\times {10}^0$) =	2.7194 $\times {10}^0$ (7.33 $\times {10}^{-1}$) −	1.7815 $\times {10}^0$ (4.54 $\times {10}^{-1}$) −	7.1164 $\times {10}^0$ (1.10 $\times {10}^0$) −	8.6462 $\times {10}^0$ (1.19 $\times {10}^0$)
CMMOP13	1.3502 $\times {10}^1$ (5.55 $\times {10}^{-1}$) −	1.2706 $\times {10}^1$ (1.27 $\times {10}^0$) −	1.0783 $\times {10}^1$ (1.46 $\times {10}^0$) −	7.9266 $\times {10}^0$ (1.12 $\times {10}^0$) −	1.4421 $\times {10}^1$ (4.08 $\times {10}^{-1}$) −	1.4315 $\times {10}^1$ (1.35 $\times {10}^0$) =	8.3687 $\times {10}^0$ (1.74 $\times {10}^0$) −	5.6008 $\times {10}^0$ (1.70 $\times {10}^0$) −	1.1835 $\times {10}^1$ (7.88 $\times {10}^{-1}$) −	1.4976 $\times {\mathbf{10}}^{\mathbf{1}}$ (6.16 $\times {\mathbf{10}}^{-\mathbf{1}}$)
CMMOP14	1.3092 $\times {10}^1$ (4.98 $\times {10}^{-1}$) −	1.4401 $\times {10}^1$ (4.82 $\times {10}^{-1}$) −	1.0907 $\times {10}^1$ (1.06 $\times {10}^0$) −	8.1026 $\times {10}^0$ (1.00 $\times {10}^0$) −	1.4194 $\times {10}^1$ (3.88 $\times {10}^{-1}$) −	1.5239 $\times {\mathbf{10}}^{\mathbf{1}}$ (7.57 $\times {\mathbf{10}}^{-\mathbf{1}}$) +	7.5031 $\times {10}^0$ (1.81 $\times {10}^0$) −	5.8310 $\times {10}^0$ (1.73 $\times {10}^0$) −	1.1459 $\times {10}^1$ (5.25 $\times {10}^{-1}$) −	1.4803 $\times {10}^1$ (6.21 $\times {10}^{-1}$)
+/−/=	0/21/10	0/29/2	1/27/3	0/30/1	7/18/6	5/22/4	0/30/1	0/31/0	1/25/5

presents the experimental data based on the PSP indicator on the 31 test instances. Similar to Table 3, MTGA-CMMO obtains the maximum average PSP value on 17 test problems. According to the Wilcoxon ranksum test, MTGA-CMMO outperforms DN-NSGA-II-CDP, DN-NSGA-II-epsilon, MRPS-CDP, MRPS-epsilon, MMEA-WI-CDP, MMEA-WI-epsilon, C-TAEA, PPS, and CMMODE on 21, 29, 27, 30, 18, 22, 30, 31, and 25 test instances, respectively. The experimental results confirm that MTGA-CMMO can obtain well-convergent and uniformly distributed multiple CPSs in most of the test instances.

To further observe and analyse the performance of MTGA-CMMO, we plot the CPSs and CPFs found by all the comparison algorithms on two typical test problems, namely CMMF11 and CMMF12, which are shown in Figure 3, Figure 4, Figure 5 and Figure 6. Figure 3 represents the distribution of CPSs obtained by all the comparison algorithms on the CMMF11, where the red line segments denote the true CPSs and the blue solid dots denote the obtained CPSs. From Figure 3, C-TAEA and MRPS-epsilon do not obtain all the CPSs of CMMF11. Furthermore, PPS, MRPS-CDP, and MMEA-WI-epsilon obtain all the CPSs, while their distributions are significantly worse than those of MTGA-CMMO and the other four algorithms. The CPFs obtained by MTGA-CMMO and other algorithms on CMMF11 are shown in Figure 4. MTGA-CMMO performs only slightly worse than PPS in the objective space and outperforms the remaining comparison algorithms. On CMMF12, PPS is the only algorithm that does not find all the CPSs as shown in Figure 5. In addition, the CPSs obtained by MTGA-CMMO are more well-distributed than those of MRPS-CDP, MRPS-epsilon, and C-TAEA. Figure 6 shows that the CPF obtained by MTGA-CMMO in the objective space on CMMF12 is slightly worse than C-TAEA and similar to PPS, but outperforms the remaining seven algorithms.

Based on the comparison experiment results and analysis, It can be concluded that MTGA-CMMO outperforms the other nine comparison algorithms for the majority of the test problems in both the objective space and the decision space.

5.2. Ablation Experiment

MTGA-CMMO mainly consists of three strategies:

(1)

A novel multitasking optimization structure, including three optimization tasks.

(2)

Mating selection strategy based on the probabilistic information of the decision space.

(3)

Environmental selection strategies in different optimization tasks.

A crucial question arises regarding the necessity of concurrently retaining the proposed strategies in MTGA-CMMO. Five MTGA-CMMO variants are devised to address this question. First, to validate the efficacy of the novel multitasking optimization structure, MTGA-CMMO-V1 and MTGA-CMMO-V2 are designed. In MTGA-CMMO-V1, auxiliary task 1 is used without constraints to verify the effect of dynamically adjusting the constraint boundaries in auxiliary task 1 on the performance of the algorithm. MTGA-CMMO-V2, which is a variant without auxiliary task 2, is used to test the improvement of auxiliary task 2 on the local exploitation capability of MTGA-CMMO. Next, to confirm the validity of the mating selection strategies in MTGA-CMMO, MTGA-CMMO-V3 performs mating selection according to a fitness value similar to that in SPEA2, where the convergence and the diversity in both the objective space and the decision space are simultaneously considered in the fitness value calculation. In MTGA-CMMO-V4, mating selection is based on a random selection mechanism. Finally, MTGA-CMMO-V5 utilizes the SCD method [57] instead of the SR method [58] employed in MTGA-CMMO and balances the diversity of the decision space and objective space in two auxiliary tasks. To ensure the fairness of the experimental validation, MTGA-CMMO and its five variants were executed on 31 test instances. The population size and the maximum evaluation number are 200 and 200,000, respectively. MTGA-CMMO and its five variants were run independently 21 times on every test instance. The experimental results are analysed based on the IGDX indicator, while the experimental data are compared using the Wilcox rank-sum test.

Table 5. The mean IGDX indicator generated by MTGA-CMMO and its five variants on 31 test problems.

Problem	MTGA-CMMO-V1	MTGA-CMMO-V2	MTGA-CMMO-V3	MTGA-CMMO-V4	MTGA-CMMO-V5	MTGA-CMMO
CMMF1	7.4737 $\times {10}^{-2}$ (2.26 $\times {10}^{-3}$) −	7.0340 $\times {\mathbf{10}}^{-\mathbf{2}}$ (2.51 $\times {\mathbf{10}}^{-\mathbf{3}}$) =	7.0478 $\times {10}^{-2}$ (2.10 $\times {10}^{-3}$) =	7.3367 $\times {10}^{-2}$ (1.90 $\times {10}^{-3}$) −	7.5422 $\times {10}^{-2}$ (1.99 $\times {10}^{-3}$) −	7.0460 $\times {10}^{-2}$ (1.78 $\times {10}^{-3}$)
CMMF2	1.6349 $\times {10}^{-2}$ (1.69 $\times {10}^{-3}$) =	1.5434 $\times {10}^{-2}$ (1.65 $\times {10}^{-3}$) =	1.6278 $\times {10}^{-2}$ (1.46 $\times {10}^{-3}$) =	1.5318 $\times {\mathbf{10}}^{-\mathbf{2}}$ (1.44 $\times {\mathbf{10}}^{-\mathbf{3}}$) =	1.5688 $\times {10}^{-2}$ (1.49 $\times {10}^{-3}$) =	1.5533 $\times {10}^{-2}$ (1.06 $\times {10}^{-3}$)
CMMF3	1.0467 $\times {10}^{-2}$ (7.88 $\times {10}^{-4}$) =	1.0310 $\times {10}^{-2}$ (8.70 $\times {10}^{-4}$) =	1.0448 $\times {10}^{-2}$ (7.94 $\times {10}^{-4}$) =	1.0247 $\times {10}^{-2}$ (7.94 $\times {10}^{-4}$) =	1.0582 $\times {10}^{-2}$ (9.35 $\times {10}^{-4}$) =	1.0136 $\times {\mathbf{10}}^{-\mathbf{2}}$ (7.48 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF4	1.2469 $\times {10}^{-2}$ (5.93 $\times {10}^{-4}$) =	1.2557 $\times {10}^{-2}$ (6.80 $\times {10}^{-4}$) =	1.2427 $\times {10}^{-2}$ (5.43 $\times {10}^{-4}$) =	1.2299 $\times {10}^{-2}$ (4.60 $\times {10}^{-4}$) =	1.2236 $\times {10}^{-2}$ (4.74 $\times {10}^{-4}$) =	1.2177 $\times {\mathbf{10}}^{-\mathbf{2}}$ (4.72 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF5	8.1636 $\times {10}^{-3}$ (5.48 $\times {10}^{-4}$) −	7.9052 $\times {10}^{-3}$ (5.45 $\times {10}^{-4}$) =	8.2119 $\times {10}^{-3}$ (5.19 $\times {10}^{-4}$) −	8.1911 $\times {10}^{-3}$ (5.44 $\times {10}^{-4}$) −	7.9942 $\times {10}^{-3}$ (4.79 $\times {10}^{-4}$) =	7.7269 $\times {\mathbf{10}}^{-\mathbf{3}}$ (5.84 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF6	1.0769 $\times {10}^{-2}$ (8.90 $\times {10}^{-4}$) −	1.0331 $\times {10}^{-2}$ (8.44 $\times {10}^{-4}$) =	1.0326 $\times {10}^{-2}$ (7.23 $\times {10}^{-4}$) −	1.0637 $\times {10}^{-2}$ (1.02 $\times {10}^{-3}$) −	1.0111 $\times {10}^{-2}$ (9.77 $\times {10}^{-4}$) =	9.8954 $\times {\mathbf{10}}^{-\mathbf{3}}$ (6.13 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF7	1.1031 $\times {10}^{-2}$ (8.57 $\times {10}^{-4}$) =	1.0354 $\times {10}^{-2}$ (7.83 $\times {10}^{-4}$) =	1.0476 $\times {10}^{-2}$ (7.98 $\times {10}^{-4}$) =	1.0539 $\times {10}^{-2}$ (4.74 $\times {10}^{-4}$) =	1.0142 $\times {\mathbf{10}}^{-\mathbf{2}}$ (7.27 $\times {\mathbf{10}}^{-\mathbf{4}}$) +	1.0600 $\times {10}^{-2}$ (7.86 $\times {10}^{-4}$)
CMMF8	3.6832 $\times {10}^{-3}$ (1.44 $\times {10}^{-4}$) =	3.6402 $\times {10}^{-3}$ (9.96 $\times {10}^{-5}$) =	3.6383 $\times {10}^{-3}$ (1.58 $\times {10}^{-4}$) =	3.6189 $\times {10}^{-3}$ (1.19 $\times {10}^{-4}$) =	3.4708 $\times {\mathbf{10}}^{-\mathbf{3}}$ (1.10 $\times {\mathbf{10}}^{-\mathbf{4}}$) +	3.6121 $\times {10}^{-3}$ (1.52 $\times {10}^{-4}$)
CMMF9	3.2177 $\times {10}^{-3}$ (1.46 $\times {10}^{-4}$) −	2.9785 $\times {\mathbf{10}}^{-\mathbf{3}}$ (1.35 $\times {\mathbf{10}}^{-\mathbf{4}}$) =	2.9851 $\times {10}^{-3}$ (1.21 $\times {10}^{-4}$) =	2.9980 $\times {10}^{-3}$ (1.56 $\times {10}^{-4}$) =	3.0448 $\times {10}^{-3}$ (1.42 $\times {10}^{-4}$) =	3.0276 $\times {10}^{-3}$ (1.78 $\times {10}^{-4}$)
CMMF10	7.9716 $\times {10}^{-3}$ (5.20 $\times {10}^{-4}$) =	7.6641 $\times {10}^{-3}$ (4.50 $\times {10}^{-4}$) =	7.8615 $\times {10}^{-3}$ (6.00 $\times {10}^{-4}$) =	7.6732 $\times {10}^{-3}$ (5.91 $\times {10}^{-4}$) =	7.5794 $\times {\mathbf{10}}^{-\mathbf{3}}$ (5.53 $\times {\mathbf{10}}^{-\mathbf{4}}$) =	7.8563 $\times {10}^{-3}$ (6.23 $\times {10}^{-4}$)
CMMF11	6.3617 $\times {10}^{-3}$ (3.28 $\times {10}^{-4}$) =	6.4003 $\times {10}^{-3}$ (5.27 $\times {10}^{-4}$) =	6.3819 $\times {10}^{-3}$ (4.11 $\times {10}^{-4}$) =	6.1927 $\times {\mathbf{10}}^{-\mathbf{3}}$ (5.38 $\times {\mathbf{10}}^{-\mathbf{4}}$) =	6.2740 $\times {10}^{-3}$ (4.64 $\times {10}^{-4}$) =	6.3504 $\times {10}^{-3}$ (4.10 $\times {10}^{-4}$)
CMMF12	4.2619 $\times {10}^{-3}$ (2.04 $\times {10}^{-4}$) =	4.1286 $\times {\mathbf{10}}^{-\mathbf{3}}$ (2.26 $\times {\mathbf{10}}^{-\mathbf{4}}$) +	4.1526 $\times {10}^{-3}$ (1.96 $\times {10}^{-4}$) =	4.2009 $\times {10}^{-3}$ (1.99 $\times {10}^{-4}$) =	4.1451 $\times {10}^{-3}$ (2.10 $\times {10}^{-4}$) =	4.2626 $\times {10}^{-3}$ (2.43 $\times {10}^{-4}$)
CMMF13	4.6510 $\times {10}^{-3}$ (5.34 $\times {10}^{-4}$) =	4.4461 $\times {10}^{-3}$ (4.86 $\times {10}^{-4}$) =	4.3816 $\times {\mathbf{10}}^{-\mathbf{3}}$ (5.58 $\times {\mathbf{10}}^{-\mathbf{4}}$) =	4.4353 $\times {10}^{-3}$ (3.94 $\times {10}^{-4}$) =	4.6890 $\times {10}^{-3}$ (5.76 $\times {10}^{-4}$) =	4.5413 $\times {10}^{-3}$ (5.11 $\times {10}^{-4}$)
CMMF14	2.9241 $\times {10}^{-3}$ (2.51 $\times {10}^{-4}$) −	2.4730 $\times {10}^{-3}$ (2.16 $\times {10}^{-4}$) =	2.5879 $\times {10}^{-3}$ (1.97 $\times {10}^{-4}$) −	2.5218 $\times {10}^{-3}$ (1.86 $\times {10}^{-4}$) =	2.7017 $\times {10}^{-3}$ (2.13 $\times {10}^{-4}$) −	2.4726 $\times {\mathbf{10}}^{-\mathbf{3}}$ (1.50 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF15	3.6034 $\times {10}^{-3}$ (4.07 $\times {10}^{-4}$) −	3.5779 $\times {10}^{-3}$ (5.04 $\times {10}^{-4}$) −	3.3340 $\times {10}^{-3}$ (4.20 $\times {10}^{-4}$) =	3.8691 $\times {10}^{-3}$ (4.86 $\times {10}^{-4}$) -	3.3916 $\times {10}^{-3}$ (3.68 $\times {10}^{-4}$) =	3.2063 $\times {\mathbf{10}}^{-\mathbf{3}}$ (4.78 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF16	1.4674 $\times {10}^{-2}$ (1.48 $\times {10}^{-3}$) −	1.3370 $\times {10}^{-2}$ (1.18 $\times {10}^{-3}$) −	1.3143 $\times {10}^{-2}$ (6.30 $\times {10}^{-4}$) −	1.3173 $\times {10}^{-2}$ (7.41 $\times {10}^{-4}$) −	1.5530 $\times {10}^{-2}$ (1.67 $\times {10}^{-3}$) −	1.2512 $\times {\mathbf{10}}^{-\mathbf{2}}$ (7.21 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMF17	2.1326 $\times {10}^{-2}$ (3.85 $\times {10}^{-3}$) −	1.8632 $\times {10}^{-2}$ (2.98 $\times {10}^{-3}$) =	1.9846 $\times {10}^{-2}$ (2.97 $\times {10}^{-3}$) −	2.2464 $\times {10}^{-2}$ (3.07 $\times {10}^{-3}$) −	2.1607 $\times {10}^{-2}$ (5.42 $\times {10}^{-3}$) −	1.8100 $\times {\mathbf{10}}^{-\mathbf{2}}$ (3.90 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMOP1	4.2209 $\times {10}^{-2}$ (8.16 $\times {10}^{-4}$) −	4.3156 $\times {10}^{-2}$ (1.10 $\times {10}^{-3}$) −	4.3708 $\times {10}^{-2}$ (9.56 $\times {10}^{-4}$) −	4.1852 $\times {10}^{-2}$ (7.84 $\times {10}^{-4}$) −	4.4888 $\times {10}^{-2}$ (1.60 $\times {10}^{-3}$) −	4.1093 $\times {\mathbf{10}}^{-\mathbf{2}}$ (9.83 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMOP2	1.2929 $\times {10}^{-2}$ (9.76 $\times {10}^{-3}$) −	1.5051 $\times {10}^{-2}$ (4.88 $\times {10}^{-3}$) −	1.5351 $\times {10}^{-2}$ (6.03 $\times {10}^{-3}$) −	1.3547 $\times {10}^{-2}$ (1.54 $\times {10}^{-2}$) −	1.4194 $\times {10}^{-2}$ (5.17 $\times {10}^{-3}$) −	1.0865 $\times {\mathbf{10}}^{-\mathbf{2}}$ (6.50 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMOP3	7.5887 $\times {10}^{-2}$ (2.33 $\times {10}^{-3}$) =	7.6169 $\times {10}^{-2}$ (3.71 $\times {10}^{-3}$) =	7.6195 $\times {10}^{-2}$ (3.42 $\times {10}^{-3}$) =	7.5367 $\times {10}^{-2}$ (3.41 $\times {10}^{-3}$) =	8.3770 $\times {10}^{-2}$ (4.82 $\times {10}^{-3}$) −	7.4812 $\times {\mathbf{10}}^{-\mathbf{2}}$ (3.48 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMOP4	1.9531 $\times {10}^{-2}$ (1.43 $\times {10}^{-3}$) −	1.9294 $\times {10}^{-2}$ (1.05 $\times {10}^{-3}$) −	1.9195 $\times {10}^{-2}$ (1.63 $\times {10}^{-3}$) =	1.8545 $\times {10}^{-2}$ (1.49 $\times {10}^{-3}$) =	2.1985 $\times {10}^{-2}$ (2.16 $\times {10}^{-3}$) −	1.8358 $\times {\mathbf{10}}^{-\mathbf{2}}$ (1.41 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMOP5	4.4711 $\times {10}^{-2}$ (4.76 $\times {10}^{-3}$) =	4.9070 $\times {10}^{-2}$ (8.37 $\times {10}^{-3}$) −	4.5038 $\times {10}^{-2}$ (6.67 $\times {10}^{-3}$) =	4.4512 $\times {10}^{-2}$ (5.34 $\times {10}^{-3}$) =	6.9768 $\times {10}^{-2}$ (1.89 $\times {10}^{-2}$) −	4.3344 $\times {\mathbf{10}}^{-\mathbf{2}}$ (5.92 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMOP6	1.3091 $\times {10}^{-2}$ (5.72 $\times {10}^{-3}$) =	1.6635 $\times {10}^{-2}$ (5.29 $\times {10}^{-3}$) −	2.2968 $\times {10}^{-2}$ (9.26 $\times {10}^{-3}$) −	1.2814 $\times {10}^{-2}$ (5.12 $\times {10}^{-3}$) =	1.3629 $\times {10}^{-2}$ (5.52 $\times {10}^{-3}$) =	1.2437 $\times {\mathbf{10}}^{-\mathbf{2}}$ (4.35 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMOP7	7.1475 $\times {10}^{-2}$ (2.82 $\times {10}^{-3}$) −	7.1602 $\times {10}^{-2}$ (2.60 $\times {10}^{-3}$) −	7.1252 $\times {10}^{-2}$ (2.23 $\times {10}^{-3}$) −	7.1135 $\times {10}^{-2}$ (2.69 $\times {10}^{-3}$) −	7.5489 $\times {10}^{-2}$ (3.86 $\times {10}^{-3}$) −	6.9087 $\times {\mathbf{10}}^{-\mathbf{2}}$ (2.97 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMOP8	1.4449 $\times {10}^{-2}$ (5.24 $\times {10}^{-4}$) −	1.4498 $\times {10}^{-2}$ (5.50 $\times {10}^{-4}$) −	1.4393 $\times {10}^{-2}$ (5.55 $\times {10}^{-4}$) −	1.4259 $\times {10}^{-2}$ (6.37 $\times {10}^{-4}$) =	1.4082 $\times {10}^{-2}$ (5.59 $\times {10}^{-4}$) =	1.4035 $\times {\mathbf{10}}^{-\mathbf{2}}$ (5.61 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMOP9	3.8051 $\times {10}^{-2}$ (1.06 $\times {10}^{-3}$) −	3.8519 $\times {10}^{-2}$ (1.08 $\times {10}^{-3}$) −	3.7925 $\times {10}^{-2}$ (8.61 $\times {10}^{-4}$) −	3.8177 $\times {10}^{-2}$ (1.09 $\times {10}^{-3}$) −	3.9216 $\times {10}^{-2}$ (1.00 $\times {10}^{-3}$) −	3.6851 $\times {\mathbf{10}}^{-\mathbf{2}}$ (8.59 $\times {\mathbf{10}}^{-\mathbf{4}}$)
CMMOP10	3.0856 $\times {10}^{-2}$ (1.91 $\times {10}^{-3}$) =	3.0901 $\times {10}^{-2}$ (1.17 $\times {10}^{-3}$) =	2.9747 $\times {\mathbf{10}}^{-\mathbf{2}}$ (1.06 $\times {\mathbf{10}}^{-\mathbf{3}}$) =	2.9752 $\times {10}^{-2}$ (1.24 $\times {10}^{-3}$) =	3.2940 $\times {10}^{-2}$ (1.75 $\times {10}^{-3}$) −	3.0142 $\times {10}^{-2}$ (1.50 $\times {10}^{-3}$)
CMMOP11	6.1527 $\times {10}^{-2}$ (1.70 $\times {10}^{-3}$) =	6.3946 $\times {10}^{-2}$ (1.71 $\times {10}^{-3}$) −	6.8483 $\times {10}^{-2}$ (1.97 $\times {10}^{-3}$) −	6.1282 $\times {10}^{-2}$ (1.72 $\times {10}^{-3}$) =	6.9445 $\times {10}^{-2}$ (2.11 $\times {10}^{-3}$) −	6.1192 $\times {\mathbf{10}}^{-\mathbf{2}}$ (1.62 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMOP12	6.7884 $\times {10}^{-2}$ (6.79 $\times {10}^{-3}$) −	6.5079 $\times {10}^{-2}$ (7.65 $\times {10}^{-3}$) −	6.8100 $\times {10}^{-2}$ (9.25 $\times {10}^{-3}$) −	6.6301 $\times {10}^{-2}$ (9.84 $\times {10}^{-3}$) =	9.0540 $\times {10}^{-2}$ (1.73 $\times {10}^{-2}$) −	6.0199 $\times {\mathbf{10}}^{-\mathbf{2}}$ (5.72 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMOP13	4.2168 $\times {10}^{-2}$ (8.31 $\times {10}^{-4}$) =	4.3815 $\times {10}^{-2}$ (1.17 $\times {10}^{-3}$) −	4.4049 $\times {10}^{-2}$ (6.82 $\times {10}^{-4}$) −	4.1934 $\times {10}^{-2}$ (1.06 $\times {10}^{-3}$) =	4.5647 $\times {10}^{-2}$ (1.58 $\times {10}^{-3}$) −	4.1632 $\times {\mathbf{10}}^{-\mathbf{2}}$ (1.06 $\times {\mathbf{10}}^{-\mathbf{3}}$)
CMMOP14	4.3114 $\times {10}^{-2}$ (9.26 $\times {10}^{-4}$) −	4.4529 $\times {10}^{-2}$ (8.15 $\times {10}^{-4}$) −	4.5706 $\times {10}^{-2}$ (1.25 $\times {10}^{-3}$) −	4.6162 $\times {10}^{-2}$ (8.68 $\times {10}^{-4}$) −	4.6794 $\times {10}^{-2}$ (1.37 $\times {10}^{-3}$) −	4.1993 $\times {\mathbf{10}}^{-\mathbf{2}}$ (8.78 $\times {\mathbf{10}}^{-\mathbf{4}}$)
+/−/=	0/16/15	1/14/16	0/15/16	0/11/20	2/16/13

presents the average IGDX values obtained by MTGA-CMMO and the different variants on 31 test problems. From Table 5, it can be seen that MTGA-CMMO obtained the smallest average IGDX values on 21 of 31 test problems. MTGA-CMMO-V1 does not derive the smallest average IGDX value for any of the test problems, suggesting that auxiliary task 1 in MTGA-CMMO does not consider the constraints. However, it may assist the main task population in traversing infeasible obstacle areas and increase the algorithm’s global searching ability. Some of the infeasible solutions in the later stages of iteration do not deliver valuable knowledge to the main population; thus, the strategy of dynamically adjusting the constraint boundaries of auxiliary task 1 in MTGA-CMMO is necessary. MTGA-CMMO-V2 obtains the smallest average IGDX value on CMMF1, CMMF9, and CMMF12; however, following statistical analysis, MTGA-CMMO-V2 significantly outperforms MTGA-CMMO only on the CMMF12 problem, while the experimental results on the other two test problems are approximately the same as those of MTGA-CMMO. This indicates that auxiliary task 2 helps develop the exploitation capability of MTGA-CMMO. MTGA-CMMO-V3 and MTGA-CMMO-V4 attain the smallest average IGDX values on two test problems each, even though following statistical analysis they do not have any significant advantage over MTGA-CMMO. This demonstrates that under the multitasking optimization framework designed in this paper, mating selection based on the decision space probabilistic information can contribute to obtaining more CPSs with better distribution and convergence. MTGA-CMMO-V5 obtains the smallest average IGDX value for only three test problems, significantly less than that of MTGA-CMMO. The experimental results illustrate that under the novel multitasking optimization architecture, designing the specific environmental selection strategies for different optimization tasks is more effective than adopting the SCD approach in each optimization task. Moreover, according to the Wilcoxon rank-sum test, MTGA-CMMO has a significant advantage over MTGA-CMMO-V1, MTGA-CMMO-V2, MTGA-CMMO-V3, and MTGA-CMMO-V5 on nearly half of 31 test instances, and the results are similar to those of these four variants on the remaining test problems. Compared to MTGA-CMMO-V4, MTGA-CMMO is superior on only 11 test questions. The reason why the advantage of MTGA-CMMO over MTGA-CMMO-V4 is not highly significant may be that MTGA-CMMO-V4 employs a random mating selection strategy, which increases the diversity of the algorithm and may contribute to searching for all the CPSs.

To visualize the advantages of MTGA-CMMO over the other variants, we plot the distribution of CPSs and CPFs obtained by MTGA-CMMO and the other five variants on CMMOP2, a representative CMMOP with two CPSs, as shown in Figure 7. MTGA-CMMO and its five variants all obtained two CPSs for CMMOP2. Nevertheless, the number of optimal solutions obtained for MTGA-CMMO-V3, MTGA-CMMO-V4, and MTGA-CMMO-V5 on the two CPSs differs considerably. Although the number of optimal solutions obtained by MTGA-CMMO-V1 and MTGA-CMMO-V2 on the two CPSs is relatively balanced, the distribution of optimal solutions on each CPS is not as uniform as that obtained by MTGA-CMMO. Additionally, as seen in Figure 8, the PF obtained by MTGA-CMMO and the five other variant algorithms on CMMOP2 showed no significant difference.

It can be seen in Table 5 and Figure 7 and Figure 8 that MTGA-CMMO outperforms its other variants, validating that the three strategies used in MTGA-CMMO are effective.

5.3. Parameter Analysis

In Section 5.3, we discuss the effects of population size and the auxiliary task 2 constraint boundary value on the performance of MTGA-CMMO.

First, the population size is selected sequentially as 100, 200, 300, and 400 and tested based on the average IGDX values on six representative test problems; the test results are demonstrated in Figure 9. As the population size increases, the average IGDX values obtained by MTGA-CMMO on all six test problems exhibit a decreasing trend, indicating that the performance of MTGA-CMMO improves with the growing population size. It is worth noting that the decrease in the mean IGDX value for increasing the population size from 100 to 200 displayed greater changes on all six test problems than those caused by the increase in population size in the other cases. Additionally, an increase in population size will bring about an increase in computational cost, therefore, we deem it more suitable to establish the population size at 200.

Secondly, the constraint boundary value of auxiliary task 2 may impact the algorithm’s performance. MTGA-CMMO employs the average CV value of infeasible solutions in $Offspring1$ as the constraint boundary of auxiliary task 2. To determine a more appropriate approach to the constraint boundary, we compare MTGA-CMMO to five other algorithms with different approaches to value-taking, which are in light of the maximum and minimum values of CV values of infeasible solutions in $Offspring1$ (i.e., MTGA-CMMO-A1 and MTGA-CMMO-A2) and based on the maximum, minimum, and average values of CV values of infeasible solutions in the union population formed by mixing $Population1$ and $Offspring1$ (i.e., MTGA-CMMO-A3, MTGA-CMMO-A4, and MTGA-CMMO-A5). Figure 10 illustrates the average IGDX metrics for these algorithms using different value-taking methods on 20 test problems. MTGA-CMMO obtains the minimum average IGDX value on 10 test problems, i.e., CMMF3-6, CMMOP1, CMMOP3, CMMOP4, CMMOP7-9, and CMMOP12-14. MTGA-CMMO-A2 outperforms four test problems, and MTGA-CMMO-A1 does not perform optimally on each test problem. The comparison results indicate that either large or small local search regions can not satisfy sufficient mining, and thus taking the mean is a more reasonable approach. In addition, MTGA-CMMO-A3, MTGA-CMMO-A4, and MTGA-CMMO-A5 only obtain the smallest IGDX value on one test problem, respectively, demonstrating that utilizing the union population reduces the efficiency of updating the infeasible solutions, actually narrows down the range of the local search area. Therefore, the average of the CV values of the infeasible solutions of $Offspring1$ is most reasonable as the constraint boundary of auxiliary task 2.

5.4. Application

The constrained multimodal multi-objective location-selection problem (CMMLP) [8] is a classical CMMOP in practice. The CMMLP aims to find the multiple closest siting schemes for schools, hospitals, and convenience stores simultaneously in feasible areas. The CMMLP is shown as follows

$$\begin{array}{c}\left\{\begin{array}{c}{f}_1\left(\mathbf{x}\right)=mindist(\mathbf{x},S_a),a=1,\dots ,5\hfill \\ f_2\left(\mathbf{x}\right)=mindist(\mathbf{x},H_b),b=1,\dots ,4\hfill \\ f_3\left(\mathbf{x}\right)=mindist(\mathbf{x},C_c),c=1,\dots ,10\hfill \end{array}\right.\hfill \\ \mathrm{subject}\mathrm{to}:\hfill \\ \mathbf{x}\notin I{F}_1\cup I{F}_2\cup I{F}_3\hfill \end{array}$$

(12)

where $S_a(a=1,\cdots ,5)$, $H_b(b=1,\cdots ,4)$ and $C_c(c=1,\cdots ,10)$ represent the locations of 5 schools, 4 hospitals, and 10 convenience stores, respectively. The lake, river, and bridges in the search space, which compose the infeasible regions, are denoted as $I{F}_1$, $I{F}_2$, and $I{F}_3$, respectively. The function $dist\left(\right)$ stands for the Euclidean distance between the candidate location and the objective site. Figure 11 shows the true CPS distribution on the CMMLP.

In this subsection, MTGA-CMMO is compared with nine comparison algorithms on the CMMLP to validate the ability of MTGA-CMMO to solve real-world problems. The population size and the maximum evaluation number are 200 and 200,000, respectively. Each comparison algorithm runs 21 times independently. Figure 12 displays the box plots of the comparison algorithms on the CMMLP based on the IGDX, PSP, and IGD metrics, respectively. In Figure 12a,b, MTGA-CMMO obtains both the minimum value of the median of IGDX and the maximum value of the median of PSP as well as the smallest size of the box plot, indicating that MTGA-CMMO can obtain better decision space experimental results than the other nine comparison algorithms on the CMMLP. In addition, as shown in Figure 12c, MTGA-CMMO significantly outperforms the other eight comparison algorithms except for C-TAEA in IGD metrics. The performance of C-TAEA in the objective space is similar to that of MTGA-CMMO, which may be attributed to the fact that C-TAEA constructs a diversity archive for retaining the additional promising infeasible solutions. To visualize the experimental results, the distribution of the obtained CPSs based on the median IGDX for all comparison algorithms is illustrated in Figure 13. Obviously, DN-NSGA-II-CDP, DN-NSGA-II-epsilon, MMEI-WI-CDP, MMEI-WI-epsilon, CMMODE, and MTGA-CMMO can obtain all the CPSs. Furthermore, MTGA-CMMO has a more even distribution in all the CPSs, and the reason for this may be that two auxiliary tasks migrate the individuals with better performance in the decision space to the main task to facilitate the evolutionary process of the main task. From Figure 12 and Figure 13, it is evident that the performance of MTGA-CMMO is superior to the other nine comparison algorithms on CMMLP.

6. Conclusions and Future Work

In this work, we propose MTGA-CMMO to address CMMOPs. In MTGA-CMMO, the main task can obtain all the well-distributed and well-convergent CPSs with the assistance of two auxiliary tasks. Auxiliary task 1 adopts the strategy of dynamically adjusted constraint boundaries, facilitating the main population in traversing the infeasible areas earlier in the evolution and providing more promising directions in the late stage of evolution. Auxiliary task 2 can generate additional individuals around the main task population to improve the local diversity of the main task. Meanwhile, a probability-based leader mating selection mechanism is designed to refine the exploratory ability of MTGA-CMMO. Furthermore, in the environmental selection phase, different strategies are devised to balance convergence, diversity, and feasibility depending on the various roles of these tasks in MTGA-CMMO. MTGA-CMMO is evaluated in comparison with nine other algorithms on two distinct types of CMMOP test sets including 31 test problems. Extensive experiments illustrate that MTGA-CMMO has better or similar performance in IGDX, IGD, and PSP metrics. In addition, different MTGA-CMMO variants validate the effectiveness of MTGA-CMMO, and important parameters of MTGA-CMMO are discussed. Finally, MTGA-CMMO outperforms the other comparison algorithms in one practical constrained multimodal multi-objective location-selection problem. Given that MTGA-CMMO employs distance-based diversity metrics in both its mating selection and environmental selection processes, we conclude that its performance degrades when solving many-objective problems (MaOPs) compared to its efficacy in low-dimensional optimization scenarios.

In future work, a CMMOPs test set with local CPSs, which has been not studied and is closer to the real case, will be devised. Moreover, we will study more efficient multitasking optimization frameworks for solving different types of CMMOPs.

In the future, we will further design more efficient multitasking structures for solving MMOPLs as well as further develop effective local convergence metrics.