An Adaptive Feasibility-Guided Framework for Constrained Multi-Objective Optimization

Abstract

Solving constrained multiobjective optimization problems (CMOPs) is highly challenging due to the presence of complicated feasible regions, intense conflicts among objectives, and unevenly distributed constraints. As a result, conventional methods relying on a single constraint-handling mechanism frequently fail to maintain a stable equilibrium among solution feasibility, diversity, and convergence. To overcome these bottlenecks, this article introduces AFFCMO, a novel adaptive feasibility-guided framework tailored for constrained multiobjective optimization. At its core, the proposed approach utilizes a coevolutionary dual-population architecture that divides the search process into two distinct tasks. Specifically, an auxiliary population is tasked with global exploration, while a primary population focuses on the intensive exploitation of discovered feasible areas. To achieve this, the primary population leverages a DE/current-to-pbest/1 differential evolution strategy to closely approximate the constrained Pareto front. Simultaneously, the auxiliary population expands the search space using a mutation operator that adapts to the current evolutionary stage. Furthermore, exploration is bolstered by a multicriterion environmental selection scheme designed for the auxiliary group. By combining Euclidean geometric distributions, constraint relaxation, and value modeling inspired by epidemic dynamics, this strategy successfully preserves valuable infeasible solutions that can guide the search. Additionally, a dynamic resource allocation strategy based on historical search feedback and Thompson sampling is incorporated. This mechanism continuously evaluates the recent search contributions of both populations and adaptively adjusts their offspring sizes, thereby reducing the bias introduced by static allocation schemes. This mechanism continuously assesses the actual search contributions of both populations, allowing for the adaptive resizing of offspring generations and thereby eliminating the inherent biases of static allocation methods. Comprehensive empirical evaluations are conducted on 47 benchmark problems from four distinct test suites. The results indicate that AFFCMO significantly outperforms seven contemporary multiobjective evolutionary algorithms in terms of exploring complex feasible regions, preserving solution diversity, and achieving high convergence accuracy.

1. Introduction

Real-world engineering and logistical challenges—ranging from vehicle routing [1] and robotic gripper optimization [2] to gear train design [3]—frequently manifest as constrained multiobjective optimization problems (CMOPs). Effectively addressing these tasks demands a delicate balance between strictly adhering to rigid operational constraints and optimizing multiple conflicting criteria simultaneously, a process that introduces substantial computational complexity [4,5]. As a result, the development of specialized evolutionary algorithms (EAs) capable of navigating such problems has become a primary focus within the research community [6,7,8].

In general terms, the mathematical formulation of a continuous CMOP can be expressed as follows:

$$\begin{array}{cc}\hfill min& \mathbf{F}\left(\mathbf{x}\right)={\left(f_1\left(\mathbf{x}\right),f_2\left(\mathbf{x}\right),\dots ,f_M\left(\mathbf{x}\right)\right)}^T\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \left\{\begin{array}{cc}{g}_i\left(\mathbf{x}\right)\le 0,\hfill & i=1,\dots ,p\hfill \\ h_i\left(\mathbf{x}\right)=0,\hfill & i=p+1,\dots ,L\hfill \\ \mathbf{x}\in \mathcal{S}\hfill & \end{array}\right.\hfill \end{array}$$

(1)

In this model, the decision space is represented by $\mathcal{S}\subseteq {\mathbb{R}}^D$, and $\mathbf{x}={(x_1,x_2,\dots ,x_D)}^T$ serves as the D-dimensional decision vector. A total of M competing objective functions are encapsulated within $\mathbf{F}\left(\mathbf{x}\right)$. Furthermore, the feasible region situated within $\mathcal{S}$ is delineated by a combination of $L-p$ equality constraints, denoted by $h_i\left(\mathbf{x}\right)$, and p inequality constraints, denoted by $g_i\left(\mathbf{x}\right)$.

To quantitatively assess how much a given solution $\mathbf{x}$ breaches these boundaries, the overall constraint violation $CV\left(\mathbf{x}\right)$ is computed using the following metric:

$$CV\left(\mathbf{x}\right)=\sum _{i=1}^L{C}_i\left(\mathbf{x}\right)$$

(2)

Here, the specific violation magnitude for the i-th individual constraint, $C_i\left(\mathbf{x}\right)$, is determined by

$$C_i\left(\mathbf{x}\right)=\left\{\begin{array}{cc}max(0,g_i\left(\mathbf{x}\right)),\hfill & 1\le i\le p\hfill \\ max(0,|h_i\left(\mathbf{x}\right)|-\delta ),\hfill & p+1\le i\le L.\hfill \end{array}\right.$$

(3)

In this context, to allow for slight relaxations of equality constraints, a minor positive tolerance value $\delta$ is introduced. Ultimately, if a solution yields $CV\left(\mathbf{x}\right)=0$, it is classified as feasible; any strictly positive value designates the solution $\mathbf{x}$ as infeasible.

Owing to their adaptive nature and population-based search mechanisms, evolutionary algorithms (EAs) have become a leading approach for tackling constrained optimization problems (COPs) [9,10]. Within this domain, constrained multiobjective optimization problems (CMOPs) [11,12,13] frequently arise in real-world scenarios, such as complex system modeling, resource allocation, and engineering design. The primary difficulty in resolving CMOPs involves satisfying strict constraints while simultaneously balancing competing objectives. Unlike unconstrained scenarios, the feasible regions associated with CMOPs tend to be highly fragmented, discontinuous, or non-convex [14]. As a result, conventional multiobjective EAs (MOEAs) often encounter search barriers caused by these infeasible zones, significantly impairing the distribution uniformity and convergence of the final solution sets. To alleviate this constraint interference, numerous constraint-handling techniques (CHTs) [15,16] have been proposed. Nevertheless, utilizing a single CHT typically introduces implicit assumptions regarding the underlying problem structure, which generates substantial performance variations across diverse CMOPs. Specifically, when navigating multi-segment distributions, narrow feasible spaces, or intricate constraint boundaries, approaches dependent on single-stage preferences or static rules frequently fail to maintain a proper equilibrium among diversity, convergence, and feasibility [17].

To circumvent the inherent limitations of isolated CHTs, recent research has pivoted from designing rigid constraint rules toward optimizing search role allocation [18]. Consequently, multi-population and multi-stage evolutionary frameworks have surfaced as leading methodologies. Multi-stage paradigms systematically approximate the constrained Pareto front (CPF) by prioritizing different optimization goals during distinct phases of the search [15]. Conversely, multi-population strategies boost adaptability by concurrently managing several populations, where each group is assigned unique search strategies or constraint preferences. Although these multi-population architectures provide notable empirical and theoretical benefits, their overall efficacy relies critically on the underlying resource allocation schemes and coevolutionary mechanisms linking the subgroups. Flawed collaboration tactics can lead to computational inefficiency, search bias, and information redundancy, ultimately negating the framework’s primary strengths. Thus, establishing adaptive resource allocation, seamless information transfer, and true role complementarity persists as a crucial challenge in the field of CMOPs.

To overcome the limitations of isolated CHTs, particularly their insufficient ability to coordinate feasibility, convergence, and diversity in complex search spaces, this article proposes an adaptive feasibility-guided framework for constrained multiobjective optimization, termed AFFCMO. The proposed algorithm employs a dual-population coevolutionary architecture in which the two populations are assigned complementary search roles. Specifically, the main population is responsible for the exploitation of identified feasible regions and the progressive approximation of the CPF, whereas the auxiliary population is used to maintain exploration in infeasible regions and around constraint boundaries so as to provide additional search directions and boundary information. To realize these roles, the main population adopts the DE/current-to-pbest/1 differential evolution operator to enhance convergence efficiency. The auxiliary population initially uses the same operator to approach potentially promising regions, and then switches to the rand/1/DE mutation strategy in the later stage to enhance exploratory capability and reduce the risk of premature convergence. To further coordinate the search behaviors of the two populations, AFFCMO incorporates an adaptive resource allocation strategy based on historical search feedback. By evaluating the recent feasibility-improvement performance of the two populations, the proposed mechanism dynamically adjusts their offspring sizes during evolution. Consequently, when the search requires stronger exploration near infeasible and boundary regions, more computational resources can be assigned to the auxiliary population. Conversely, when the refinement of feasible solutions becomes more important, the main population can receive a larger offspring quota. Through this stage-dependent allocation manner, AFFCMO adaptively balances convergence-oriented exploitation and exploration-oriented search throughout the evolutionary process. The primary contributions of this article are outlined as follows:

• We propose AFFCMO, a dual-population coevolutionary framework for constrained multiobjective optimization. In the proposed framework, the search process is divided into two complementary roles: the main population focuses on feasible-solution refinement and convergence toward the CPF by using the DE/current-to-pbest/1 operator, whereas the auxiliary population is responsible for exploration in infeasible regions and around constraint boundaries through a stage-adaptive reproduction strategy. For the auxiliary population, a continuous environmental selection mechanism is designed to jointly consider objective quality, constraint status, and Euclidean distribution information, so that informative near-feasible solutions can be preserved while maintaining population diversity.
• To improve the coordination between the two populations, we develop an adaptive resource allocation mechanism based on historical search feedback and Thompson sampling. This mechanism evaluates the recent feasibility-improvement capability of the main and auxiliary populations, and dynamically adjusts their offspring allocation ratios at different evolutionary stages. As a result, computational resources can be adaptively assigned according to the observed search contribution of each population, thereby improving the balance between convergence-oriented exploitation and exploration-oriented search.
• Extensive experiments on 47 benchmark instances and six real-world engineering design problems demonstrate the effectiveness of AFFCMO against seven state-of-the-art algorithms, confirming the proposed framework’s competitive performance in feasibility acquisition, convergence accuracy, and solution diversity.

The subsequent sections of this article are structured as follows. A comprehensive survey of related literature and existing methodologies is presented in Section 2. Following this, Section 3 provides a detailed exposition of the proposed algorithm. The specific experimental configurations are delineated in Section 4, which seamlessly transitions into a thorough discussion and analysis of the empirical findings in Section 5. To conclude, Section 6 summarizes the core contributions of this study and highlights potential avenues for future investigation.

2. Related Work

Motivated by their widespread utility in intelligent decision-making, resource allocation, and engineering design, complex CMOPs have recently emerged as a focal point within the evolutionary computation community. In contrast to unconstrained optimization, tackling these problems necessitates a delicate equilibrium among solution feasibility, diversity, and convergence. Harmonizing these three naturally competing criteria significantly elevates the computational difficulty. To navigate these algorithmic hurdles, the field has evolved beyond monolithic evolutionary architectures and elementary constraint-handling mechanisms. Contemporary paradigms now favor advanced configurations, such as hybrid-driven algorithms, multi-population coevolution, multi-stage frameworks, and progressive search techniques. The following subsections provide a comprehensive survey of these prominent modeling strategies.

2.1. Progressive Evolutionary Optimization Methods

Serving as some of the foundational architectures in constrained multiobjective optimization, progressive evolutionary methods embed constraint-handling techniques (CHTs) directly into a unified search process. By avoiding highly intricate structural subdivisions, these approaches preserve algorithmic simplicity. Instead, they leverage specialized genetic operators, adaptive heuristics, and constraint relaxation schemes to alleviate constraint interference and equilibrate population diversity with convergence.

Demonstrating this unified approach, the MOEA/BLD algorithm [19] executes a hierarchical decomposition across both the constraint violation and objective spaces. Through the deployment of augmented weight vectors that explicitly encode the magnitude of constraint violations, this method achieves a dynamic equilibrium between satisfying constraints and optimizing objectives. Taking a different angle on selection pressure, the DSPCMDE framework [20] institutes a dynamic preference strategy that alters the evaluation criteria as the evolution proceeds. Specifically, the algorithm shifts from utilizing Pareto dominance for broad exploration during initial phases to enforcing constraint dominance for strict feasibility in later stages, ultimately ensuring steady convergence toward the CPF. To tackle formidable constraint boundaries, the MOEA/D-2WA algorithm [21] partitions its weight vectors into dynamic “infeasible” components and static “feasible” components, the latter guaranteeing uniformity along the CPF. By casting the constraint violation magnitude as an auxiliary $(m+1)$-th objective and tightening a dynamic boundary parameter, $\epsilon$, via a simulated annealing mechanism, this technique preserves valuable infeasible candidates to traverse disjointed regions without the overhead of external archives. Expanding on the premise of constraint relaxation, the Adaptive Constraint Regulation (ACR) mechanism [22] leverages a K-nearest neighbor (K-NN) heuristic to provisionally classify specific infeasible individuals as feasible, which significantly bolsters search capabilities near constraint edges. Further refining search trajectories within infeasible zones, an adaptive gradient repair model was introduced in MODE-AGR [23], upgrading the robustness of standard $\epsilon$-constrained methods when navigating intricate topologies. In recent years, uniform search paradigms and multi-tasking formulations have also gained traction within single-framework models. For example, the multiform architecture MFOSPEA2 [24] establishes auxiliary optimization tasks to drive cross-task information transfer, concurrently accelerating convergence and injecting diversity. This framework seamlessly fuses indicator-, dominance-, and decomposition-based selection metrics to modulate environmental pressure throughout the evolutionary timeline. Parallel to this, the Adaptive Uniform Search Framework (AUSF) [25] continuously toggles between local greedy exploitation (LGE) and global uniform exploration (GUE). Governed by a population stability metric, this adaptive switching logic securely anchors the trade-off between concentrated refinement and widespread discovery. More recently, several progressive or single-framework CMOEAs have further strengthened this line of research. For instance, FLI [26] introduced a fitness landscape indicator to adaptively select offspring generation strategies according to the current search state, thereby improving the coordination between exploration and exploitation. IDNSGA-III [27] employed a population-feasibility-state-guided autonomous operator selection mechanism, enabling the algorithm to adjust its search behavior according to the evolving feasibility status of the population. From the perspective of adaptive relaxation, ACREA [28] dynamically adjusted constraint boundaries according to the iteration information of the population, so as to better exploit useful infeasible solutions during the search process. In addition, PSCMO [29] further incorporated a population state discrimination model to identify different search situations and adapt the optimization behavior accordingly. However, despite their historical successes, these representative progressive methodologies exhibit a profound reliance on user-defined dynamic parameters, including penalty coefficients, relaxation thresholds, or state-dependent switching rules, which are notoriously hypersensitive to distinct problem characteristics. Moreover, embedding highly intricate constraint regulation heuristics within a single population can inadvertently spawn new evolutionary bottlenecks. Prompted by these structural limitations, contemporary research efforts have progressively pivoted toward the inherent flexibility of multi-stage and multi-population evolutionary designs.

2.2. Multi-Stage Optimization Methods

By dividing the evolutionary process into multiple sequential phases, multi-stage optimization frameworks exploit the geometric relationship between the unconstrained Pareto front (UPF) and the CPF in a more structured manner. By dividing the evolutionary timeline into distinct, sequential phases, multi-stage optimization frameworks capitalize on the spatial relationship between the unconstrained Pareto front (UPF) and the CPF. The most ubiquitous configuration is the two-stage paradigm. In the initial phase, constraint boundaries are deliberately relaxed or entirely disregarded to prioritize unhindered global exploration. Subsequently, the latter phase progressively reinstates strict constraint evaluations. This sequential strategy effectively shields the initial search from severe constraint bottlenecks before ultimately steering the population toward the optimal feasible region. More recent studies have further shown that such a staged design can also be extended to adaptive, cooperative, or even tri-stage forms, so as to better accommodate different constraint landscapes and search states.

Exemplifying this paradigm, the renowned “push-pull” architecture of the PPS algorithm [30] permits unrestricted traversal of the objective space during its “push” stage. Utilizing a refined $\epsilon$-constraint mechanism, the subsequent “pull” stage systematically guides the candidates back into feasibility, ensuring a seamless migration from the UPF to the CPF. In a similar vein, the C-TSEA methodology [31] bifurcates the search trajectory. After approximating the UPF and archiving elite solutions in the primary phase, it transitions to an enhanced SPEA2-CDP optimizer near the CPF. Continuous information sharing between the active population and the archive guarantees that historical search data is thoroughly exploited across benchmark evaluations. Moving beyond rigid transition thresholds, the CMOES framework [32] introduces a dynamic fitness evaluation model to intelligently arbitrate between constraint satisfaction and objective optimization. This allows the system to autonomously detect the current search state—amplifying exploration when infeasibility is high, and reinforcing diversity along the boundaries when feasible candidates are abundant. Alternatively, hybrid approaches like EGBO [33] fuse Bayesian optimization with evolutionary algorithms. By incorporating a pre-repair mechanism for input constraints, EGBO minimizes the costly evaluation of invalid geometries. More recently, the CP-TSEA algorithm [34] deployed an auxiliary population equipped with an $\epsilon$-constraint relaxation learning protocol during its opening stage. This adaptive boundary relaxation permits high-performing infeasible individuals to drive global search, followed by a secondary stage utilizing multi-indicator elite selection and equiprobable competition to swiftly converge upon the CPF. Beyond these representative two-stage designs, more recent studies have further enriched this category. For example, INCMO [35] introduced an interactive niching-based two-stage framework to enhance the coordination between boundary exploration and feasible-region convergence. CTSDPA [36] adopted a two-stage differential evolution framework with dynamic population assignment, which improved the flexibility of resource usage across different search phases. MA-TSEA [37] further enhanced the two-stage paradigm by incorporating multiple archives to preserve search information from different evolutionary states. In addition, C3M [9] extended staged optimization to multiconstraint scenarios through a multistage search process, while CV-TCMOEA [38] and DETS [39] demonstrated that more elaborate two-stage or tri-stage cooperation schemes can further strengthen the balance between objective optimization and constraint satisfaction on complex CMOPs. Despite their capacity to compartmentalize and balance conflicting search goals, multi-stage methodologies remain constrained by inherent structural bottlenecks. Chief among these are the complex calibration of stage-transition triggers, the asymmetrical distribution of computational budgets, and the friction between phase-specific objectives. In simpler optimization landscapes, a delayed transition to the final stage can inadvertently degrade the diversity of the ultimate solution set. Conversely, when tackling highly intricate problems, triggering the subsequent phase too early frequently induces premature convergence, severely compromising the algorithm’s ability to accurately map the CPF. Moreover, as the number of stages increases, the design of transition criteria, archive interaction, and computational budget allocation often becomes more sensitive to problem characteristics, which may weaken the robustness of a fixed staged schedule across heterogeneous CMOPs.

2.3. Multi-Population Optimization Methods

To further improve robustness against complex feasible geometries, multi-population methodologies maintain multiple interacting populations simultaneously. Instead of relying on rigid stage transitions, these frameworks usually realize continuous cooperation among populations with different search preferences, so that feasibility improvement, convergence promotion, and diversity preservation can be pursued in a coordinated manner. Within this line of research, dual-population, tri-population, and multi-archive cooperative designs have become representative structural paradigms.

Operating within the dual-population paradigm, the DAEAEO architecture [40] deploys two distinct auxiliary swarms to help a primary population navigate across formidable infeasible expanses. The first auxiliary group supplies high-fidelity feasible candidates via an upgraded $\epsilon$-constraint strategy, whereas the second extracts vital objective-space coordinates using non-dominated sorting. Coupled with a Competitive Swarm Optimizer (CSO), this framework leverages competitive learning to concurrently safeguard diversity and bolster global convergence. Expanding upon dynamic feasibility tracking, the ACREA algorithm [28] automatically modulates its constraint boundaries in response to the real-time ratio of feasible individuals. By effectively mining the latent spatial data within infeasible solutions, this approach constructs a comprehensive CPF, relying on a hybrid archiving protocol that merges diversity metrics with constraint dominance. Scaling up to more intricate multi-archive and tri-population designs, the CMOEA-CD framework [41] formulates a specialized Constraint-Pareto dominance criterion. By evaluating feasibility and objective performance in tandem, it actively mitigates the premature loss of diversity frequently caused by an over-reliance on strict constraint satisfaction, utilizing three cooperative external archives to seamlessly mediate the exploration-exploitation trade-off. Pioneering the integration of machine learning within this space, the NNCP methodology [42] embeds a Self-Organizing Map (SOM) directly into the evolutionary loop. The neurons in this network capture crucial neighborhood topologies to exponentially accelerate Pareto front approximation. Crucially, its three concurrent populations are strictly compartmentalized to pursue feasibility, diversity, and convergence independently, yielding exceptional results on landscapes characterized by severely restricted or fractured feasible zones. Rounding out the tri-population models, the TPEA algorithm [43] orchestrates a standard working population alongside distinct inner and outer non-dominated infeasible swarms. These auxiliary groups generate opposing expansion and escape vectors, which collectively thwart local entrapment and propel the search deeper into globally optimal feasible territories. More recently, several studies have further strengthened this category from the perspectives of cooperation design and resource coordination. For example, EAPS [44] enhanced auxiliary-population-based coevolution by explicitly improving diversity preservation in constrained search. APSEA [10] introduced an adaptive population sizing mechanism for multi-population CMOEAs, showing that the effectiveness of auxiliary populations strongly depends on the dynamic allocation of computational resources. TP-HCEA [45] further combined hybrid collaboration with a multi-population structure to improve the balance between exploration and exploitation on complex constrained landscapes. In addition, DPTPEA [46] and DPTAC [47] demonstrated that more elaborate dual-population cooperative designs, such as tractive assistance or two-archive coevolution, can further improve search guidance in highly constrained scenarios. From the perspective of recent work published in Mathematics, BTCMO [48] also adopted a coevolutionary design and incorporated Bayesian information to enhance the interaction between populations. Although multi-population methods provide stronger flexibility through parallel role decomposition, their performance is still highly dependent on the effectiveness of inter-population coordination, archive interaction, and resource allocation. If the cooperation mechanism is not properly designed, redundant search behaviors and inefficient use of function evaluations may offset the advantages brought by multiple populations. These observations further motivate the design of AFFCMO, which combines complementary population roles with an adaptive resource allocation strategy driven by historical search feedback.

Although multi-population methods have shown clear advantages in handling complex CMOPs through parallel search and task decomposition, their performance is often limited by high computational cost, intricate coordination mechanisms, and inefficient utilization of complementary search information. Motivated by these limitations, we propose AFFCMO, a dual-population coevolutionary framework that adaptively coordinates convergence, feasibility, and diversity within a unified search process. In AFFCMO, the main population is responsible for approximating the CPF by employing the DE/current-to-pbest/1 operator together with a feasibility-oriented environmental selection strategy, thereby promoting stable convergence toward high-quality feasible solutions. In contrast, the auxiliary population is designed to explore infeasible regions and constraint boundaries, and to maintain search diversity through a stage-adaptive variation strategy and a continuous environmental selection mechanism. In addition, AFFCMO incorporates an adaptive resource allocation strategy based on historical search feedback, which dynamically adjusts the offspring sizes of the two populations according to their observed search contributions at different evolutionary stages. Through this cooperative design, AFFCMO enhances the overall search effectiveness on complex CMOPs while maintaining a reasonable computational overhead.

3. Proposed Algorithm

3.1. Algorithm Framework

Due to the intricate feasible regions, strong objective conflicts, and heterogeneous constraint distributions in complex CMOPs, it remains difficult to achieve a proper balance among convergence, feasibility, and diversity within a single search process. As outlined in Algorithm 1, AFFCMO addresses this issue through a dual-population coevolutionary framework, in which the search is divided into a main population and an auxiliary population with complementary functions. In addition, for the sake of clarity, Figure 1 illustrates the overall workflow of the proposed AFFCMO framework. The main population is mainly responsible for stable convergence and progressive approximation of the CPF. To this end, it employs the DE/current-to-pbest/1 operator to guide candidate solutions toward elite individuals, thereby enhancing search efficiency in the objective space and improving the stability of the search direction in the later evolutionary stages.

In contrast, the auxiliary population is introduced to maintain exploration in infeasible regions and around constraint boundaries. Its role is to expand the search scope and provide complementary search directions for the main population. To match the requirements of different evolutionary stages, the auxiliary population adopts a stage-adaptive variation strategy. In the early stage, it uses the DE/current-to-pbest/1 operator to approach potentially promising regions rapidly. In the later stage, it switches to the rand/1/DE operator, which weakens excessive convergence pressure and enhances the exploration ability near infeasible regions and disconnected feasible segments. In this way, the auxiliary population can preserve exploratory diversity while reducing the risk of premature convergence.

Algorithm 1 AFFCMO

Require:  Problem definition, population size N, maximum evaluations $F{E}_{max}$ Ensure:  Approximated constrained Pareto front   1: Initialize main population P and auxiliary population A   2: Initialize resource allocation $N_1\leftarrow \lfloor N/2\rfloor$, $N_2\leftarrow N-N_1$   3: Initialize the historical feedback statistics for adaptive resource allocation   4: while termination condition not satisfied do   5:       Set DE/current-to-pbest/1 ratio $p\leftarrow 0.1$   6:       if $N_1>0$ then   7:             Generate offspring $O_1$ using DE/current-to-pbest/1 operator   8:       end if   9:       if $N_2>0$ then 10:             if $FE/F{E}_{max}<0.5$ then 11:                  Generate offspring $O_2$ using DE/current-to-pbest/1 12:             else 13:                  Generate offspring $O_2$ using rand/1/DE 14:             end if 15:       end if 16:       $O\leftarrow O_1\cup O_2$ 17:       $P\leftarrow$ feasibility-first environmental selection ($O\cup P$) 18:       $A\leftarrow$ Algorithm 2 ($O\cup A$) 19:      $(N_1,N_2)\leftarrow$ Dynamic resource allocation based on historical search feedback 20: end while 21: return final main population P

To further coordinate the coevolutionary behavior of the two populations, AFFCMO incorporates an adaptive resource allocation mechanism based on historical search feedback. Since the search contributions of the main and auxiliary populations may vary across different evolutionary stages, the proposed mechanism dynamically adjusts their offspring sizes according to their observed feasibility-improvement capability. By modeling the success probability of each population through Thompson sampling, the algorithm can allocate computational resources in an online manner under uncertainty, thereby alleviating the inefficiency of fixed-ratio allocation schemes. As a result, the search emphasis between convergence-oriented exploitation and exploration-oriented boundary search can be adaptively regulated during evolution.

Finally, AFFCMO assigns different environmental selection strategies to the two populations according to their respective roles. The main population follows a feasibility-first principle based on constraint dominance and objective sorting, so as to preserve high-quality feasible solutions and ensure stable convergence toward the true feasible front. By contrast, the auxiliary population adopts a continuous environmental selection strategy that jointly considers objective quality, constraint status, and Euclidean distribution information. This strategy enables the auxiliary population to retain informative near-feasible solutions while maintaining sufficient spread in the objective space, thus continuously supplying useful boundary information to the main population.

3.2. Auxiliary Population Environmental Selection Strategy

Within the dual-population framework, the auxiliary population is not used to directly approximate the CPF. Instead, it is designed to maintain search activity in infeasible regions and around constraint boundaries, thereby providing complementary search directions and useful boundary information for the main population. Accordingly, the environmental selection strategy for the auxiliary population should not follow a strict feasibility-first principle. Otherwise, many infeasible solutions with potential search value would be discarded prematurely, which may significantly weaken the exploratory capability of the auxiliary population. To address this issue, AFFCMO adopts a continuous evaluation mechanism that jointly characterizes objective performance and constraint status, and further incorporates a distribution regulation term to preserve population spread. In this way, the auxiliary population can retain informative near-feasible solutions while avoiding excessive concentration in a limited region of the objective space.

Let the candidate set of the current auxiliary population be:

$$\mathcal{C}=\{{\mathbf{x}}_1,{\mathbf{x}}_2,\dots ,{\mathbf{x}}_N\},$$

(4)

where the objective function vector of an individual ${\mathbf{x}}_i$ is denoted as $\mathbf{F}\left({\mathbf{x}}_i\right)\in {\mathbb{R}}^M$. To mitigate the impact of dimensional differences across objectives, the function values are normalized dimension by dimension:

$${\widehat{f}}_{ij}={\displaystyle \frac{f_j\left({\mathbf{x}}_i\right)-{min}_{{\mathbf{x}}_k\in \mathcal{C}}{f}_j\left({\mathbf{x}}_k\right)}{{max}_{{\mathbf{x}}_k\in \mathcal{C}}{f}_j\left({\mathbf{x}}_k\right)-{min}_{{\mathbf{x}}_k\in \mathcal{C}}{f}_j\left({\mathbf{x}}_k\right)}},j=1,\dots ,M,$$

(5)

where ${\widehat{f}}_{ij}\in [0,1]$ denotes the normalized value of individual ${\mathbf{x}}_i$ on the j-th objective. This normalization places different objectives on a comparable scale and provides a unified basis for subsequent evaluation.

Regarding constraint handling, the overall constraint violation $\mathit{CV}\left({\mathbf{x}}_i\right)$ is employed to characterize an individual’s feasibility state. Rather than directly using a hard feasibility indicator, AFFCMO maps the violation degree into a bounded continuous quantity so that infeasible solutions can still be differentiated according to their proximity to the feasible region. To preserve the guiding role of constraint information while avoiding the parameter sensitivity typical of traditional penalty functions, a smooth mapping function compresses the violation into a finite interval:

$$r_i={\displaystyle \frac{\mathit{CV}\left({\mathbf{x}}_i\right)}{\mathit{CV}\left({\mathbf{x}}_i\right)+1}},$$

(6)

where $r_i\in [0,1)$ is the corresponding constraint-risk descriptor. A smaller $r_i$ indicates that the individual is closer to feasibility, whereas a larger $r_i$ reflects a higher degree of constraint violation.

Next, a comprehensive evaluation metric is constructed to simultaneously depict objective quality and constraint risk. We first define the average performance metric of an individual in the normalized objective space:

$${\overline{f}}_i={\displaystyle \frac{1}{M}}\sum _{j=1}^M{\widehat{f}}_{ij},$$

(7)

where ${\overline{f}}_i\in [0,1]$ represents the overall relative level of individual ${\mathbf{x}}_i$. To achieve a continuous coupling between objective performance and constraint state, the following comprehensive fitness function is introduced:

$${\mathit{Fit}}_i=\sqrt{{\overline{f}}_i·(1+r_i)}.$$

(8)

The above definition provides a unified criterion for screening individuals in the auxiliary population. When an individual has favorable objective performance and a low constraint-risk level, its ${\mathit{Fit}}_i$ remains small, making it more likely to be preserved. By contrast, individuals with poor objective values or severe constraint violation receive larger ${\mathit{Fit}}_i$ values and are less competitive in environmental selection. The square-root operation is introduced to moderate the scale variation of the product term, thereby preventing excessive amplification caused by large differences in objective or constraint information. Consequently, this formulation allows the auxiliary population to preserve potentially useful infeasible solutions in a graded manner, instead of eliminating them through a rigid feasibility-first rule.

Following fitness construction, the structural distribution of the solution set must be considered. The Euclidean distance in the normalized objective space characterizes spatial differences:

$$d_{ij}={∥\widehat{\mathbf{f}}\left({\mathbf{x}}_i\right)-\widehat{\mathbf{f}}\left({\mathbf{x}}_j\right)∥}_2.$$

(9)

This distance metric is used to describe the relative spatial positions of candidate solutions in the normalized objective space and serves as the basis for distribution regulation.

Algorithm 2 outlines the incremental environmental selection process that unifies these metrics. First, the individual with the best comprehensive fitness is selected to ensure foundational population quality. Subsequently, for the remaining candidates in $\mathcal{C}$, the minimum distance to the currently constructed set ${\mathcal{A}}_{sel}$ is calculated:

$$d_i^{min}=\underset{{\mathbf{x}}_j\in {\mathcal{A}}_{sel}}{min}{d}_{ij}.$$

(10)

where $A_{\mathrm{sel}}$ denote the currently constructed auxiliary selection set.

A selection evaluation function incorporating distribution regulation is then defined:

$${\mathit{Score}}_i={\mathit{Fit}}_i-\alpha d_i^{min},$$

(11)

where $\alpha$ is the distribution regulation coefficient that controls the relative influence of the distance-based diversity term in auxiliary population selection.

Algorithm 2 Auxiliary Population Environmental Selection

Require:  Candidate set $\mathcal{C}$, selection size $N_{sel}$ Ensure:  Selected auxiliary population ${\mathcal{A}}_{sel}$   1: $N\leftarrow \left|\mathcal{C}\right|$   2: if $N_{sel}\ge N$ then   3:       return $\mathcal{C}$   4: else   5:       Calculate pairwise distances $d_{ij}$ for all ${\mathbf{x}}_i,{\mathbf{x}}_j\in \mathcal{C}$ using Equation (9)   6:       Calculate comprehensive fitness ${\mathit{Fit}}_i$ for all ${\mathbf{x}}_i\in \mathcal{C}$ using Equation (8)   7:       $I\leftarrow \{arg{min}_i{\mathit{Fit}}_i\}$   8:       while $\left|I\right|<N_{sel}$ do   9:             for each $i\notin I$ do 10:                   Calculate $d_i^{min}={min}_{j\in I}{d}_{ij}$ 11:                   Calculate ${\mathit{Score}}_i$ using Equation (11) 12:             end for 13:             $i^{*}\leftarrow arg{min}_{i\notin I}{\mathit{Score}}_i$ 14:             $I\leftarrow I\cup \left\{i^{*}\right\}$ 15:       end while 16: end if 17: ${\mathcal{A}}_{sel}\leftarrow \{{\mathbf{x}}_i\in \mathcal{C}\mid i\in I\}$ 18: return ${\mathcal{A}}_{sel}$

3.3. Adaptive Resource Allocation Strategy

Within the dual-population coevolutionary framework, the search contributions of the main and auxiliary populations are usually stage-dependent. In the early stage, the auxiliary population may play a more important role in exploring infeasible regions and locating promising boundary areas, whereas in the later stage, the main population may contribute more to the refinement of feasible solutions. Therefore, allocating a fixed number of offspring to the two populations throughout the entire evolutionary process may lead to inefficient use of computational resources. To address this issue, AFFCMO employs an adaptive resource allocation strategy driven by historical search feedback, so that the offspring numbers of the two populations can be adjusted online according to their observed search effectiveness.

Let $N_1^t$ and $N_2^t$ denote the offspring sizes allocated to the main and auxiliary populations at generation t, respectively, given a total population size N. These sizes satisfy the constraint:

$$N_1^t+N_2^t=N.$$

(12)

To evaluate the search performance of each population, AFFCMO uses the constraint violation of generated offspring as a feedback signal. Specifically, an offspring is regarded as a successful search sample if its overall constraint violation is smaller than a predefined threshold $\epsilon$; otherwise, it is regarded as a failed sample. In this way, the resource allocation process is directly linked to the ability of each population to promote progress toward the feasible region.

Based on this definition, the algorithm records the number of successful searches $u_i^t$ and failed searches $v_i^t$ for population i ($i\in \{1,2\}$) at generation t. To reduce the instability caused by limited feedback information in the early generations, AFFCMO models the success probability of each population using a Beta distribution and updates it according to the cumulative historical observations. To mitigate instability from insufficient statistical information during early evolutionary stages, a Beta distribution serves as the prior model for success probability, effectively realizing the Thompson sampling mechanism. This distribution is continuously updated using cumulative historical successes and failures. Accordingly, the success probability of the i-th population is modeled as

$${\theta }_i^t\sim \mathrm{Beta}(U_i^t,V_i^t),$$

(13)

where $U_i^t$ and $V_i^t$ denote the cumulative numbers of successful and failed samples collected from population i up to generation t, respectively. This probabilistic modeling allows the allocation strategy to incorporate both empirical search performance and the uncertainty associated with limited observations, rather than relying on a deterministic estimate derived from a single generation.

In each generation, ${\theta }_1^t$ and ${\theta }_2^t$ are sampled from their respective Beta distributions. These sampled values serve as estimates of the relative search potential of the two populations for the current stage. Subsequently, the expected offspring sizes are calculated based on these estimates and normalized to ensure the conservation of total computational resources:

$$N_1^t=\mathrm{round}\left(N·{\displaystyle \frac{{\theta }_1^t}{{\theta }_1^t+{\theta }_2^t}}\right),N_2^t=N-N_1^t.$$

(14)

To avoid the loss of search functionality in either population, a lower-bound restriction is further imposed on $N_1^t$ and $N_2^t$, such that each population is assigned at least one offspring individual in every generation. This treatment preserves the complementary roles of the two populations even when one of them is temporarily less competitive.

Through the above mechanism, the offspring allocation ratio is adjusted according to the observed feasibility-improvement capability of the two populations. If one population consistently produces more successful offspring, its sampled success probability tends to become larger, and more computational resources will be assigned to it in subsequent generations. Conversely, when the recent search contribution of one population becomes weaker, its offspring quota is automatically reduced. Therefore, the proposed strategy provides a lightweight and adaptive way to regulate the search emphasis between exploration and exploitation, while maintaining the overall stability of the dual-population coevolutionary process.

3.4. Design of Variation Operators

Within the AFFCMO architecture, the configuration of genetic operators is fundamentally intertwined with the designated functions of the dual populations, solidifying their complementary roles via tailored variation strategies. Tasked with driving stable convergence, the primary population relies exclusively on the DE/current-to-pbest/1 mechanism to produce its offspring, denoted as $O_1$. By actively steering candidate solutions toward elite regions, this operator not only hastens convergence across the objective space but also preserves directional stability throughout advanced phases of the search.

In stark contrast, the auxiliary population deploys a phase-aware mutation protocol to proactively safeguard solution diversity. As outlined in Algorithm 1, this group initially leverages the identical DE/current-to-pbest/1 operator to swiftly identify highly promising spatial sectors. However, as the evolutionary process matures, it entirely transitions to a DE/rand/1 scheme to yield its offspring, $O_2$. Deliberately relaxing the selection pressure in this manner amplifies exploratory stochasticity, thereby drastically reducing the risk of premature stagnation in suboptimal locales.

The mathematical formulations governing these mutation operators are defined as follows:

$$\begin{array}{cc}\hfill \mathrm{DE}/\mathrm{current}\text{-}\mathrm{to}\text{-}p\mathrm{best}/1:O_1& =P_{r1}+F·(P_{\mathrm{best}}-P_{r1}+P_{r2}-P_{r3}),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \mathrm{DE}/\mathrm{rand}/1:O_2& =P_{r1}+F·(P_{r2}-P_{r3}).\hfill \end{array}$$

(16)

During the execution of the DE/current-to-pbest/1 strategy, the base vectors $P_{r1}$ and $P_{r3}$ are sampled at random from the auxiliary swarm A, whereas $P_{r2}$ is extracted from the primary swarm P. To guide the search, the superior candidate $P_{\mathrm{best}}$ is stochastically isolated from the upper decile (top 10%) of P, evaluated according to the constraint-dominance principle (CDP). Conversely, when executing the DE/rand/1 operation, $P_{r2}$ and $P_{r3}$ are predominantly sourced from A; nevertheless, to stimulate vital cross-population knowledge transfer, $P_{r3}$ may occasionally be drawn from P. The generational capacities for the resulting offspring sets, $O_1$ and $O_2$, are dynamically allocated as $N_1$ and $N_2$, respectively. To further promote behavioral heterogeneity, the crossover probability $\mathit{CR}\in \{0.1,0.2,1.0\}$ and the differential scaling weight $F\in \{0.6,0.8,1.0\}$ are stochastically assigned. As a final step following these differential operations, a standard polynomial mutation is administered to all newly generated individuals.

Ultimately, this synergistic operator blueprint empowers AFFCMO to execute phase-appropriate search adaptations without introducing severe computational overhead, ensuring that both populations seamlessly complement one another from initialization to termination.

3.5. Algorithm Complexity Analysis

Let N denote the population size, M denote the number of objectives, and D denote the dimensionality of the decision variables. In AFFCMO, the computational cost in each generation mainly comes from three parts: offspring generation, auxiliary environmental selection, and adaptive resource allocation. First, the differential evolution-based offspring generation for the main and auxiliary populations requires $O\left(ND\right)$ time in total. Second, evaluating the objective and constraint values of the generated individuals requires $O\left(NM\right)$ time when the number of constraints is treated as fixed. Third, in the auxiliary population, the continuous environmental selection mechanism requires the computation of pairwise Euclidean distances in the normalized objective space. This step dominates the environmental selection procedure and incurs a time complexity of $O\left(N^2\right)$, while the associated fitness construction and score updating steps do not change the overall order. In addition, the adaptive resource allocation mechanism based on historical search feedback only involves the update of success/failure statistics and Thompson-sampling-based allocation, and therefore contributes a linear cost of $O\left(N\right)$, which is negligible compared with the dominant terms. Accordingly, the overall time complexity of AFFCMO per generation is $O(N^2+ND+NM)$. Since the quadratic term originates mainly from pairwise distance calculations in the auxiliary environmental selection stage, AFFCMO is computationally more expensive than decomposition-based approaches such as MOEA/D-type algorithms that do not rely on full pairwise comparisons. However, its complexity remains at the same quadratic level as NSGA-II-type algorithms and many existing constrained multiobjective evolutionary algorithms that also require pairwise dominance, density, or distance calculations during environmental selection. Therefore, although AFFCMO introduces an additional cooperative search structure, it does not fundamentally change the mainstream complexity level of population-based constrained multiobjective evolutionary optimization.

4. Experimental Setup

4.1. Comparative Algorithms and Parameter Settings

To rigorously benchmark the efficacy of AFFCMO, its performance is juxtaposed with seven state-of-the-art constrained multiobjective evolutionary algorithms (CMOEAs). This baseline cohort includes ANSGA-III [49], BiCo [50], CTAEA [51], CTSEA [31], CMOEMT [52], CMOQLMT [53], and MOEA/D-2WA [21]. For absolute fairness, the internal parameters of these competing models were precisely aligned with the recommendations provided in their seminal publications. To ensure statistical reliability, the overarching experimental environment was strictly controlled: every algorithm underwent 30 independent executions, utilizing a uniform population size of 100 individuals and a strict termination criterion of ${10}^5$ maximum function evaluations (MaxFEs). The PlatEMO V4.14 software suite [54] served as the standardized testing framework for all empirical simulations.

4.2. Benchmark Test Suites and Performance Metrics

A comprehensive assessment of AFFCMO’s capacity to maintain diversity, achieve convergence, and secure feasibility was conducted on four widely used constrained multiobjective benchmark suites, namely MW [55], LIR-CMOP [56], CF [57], and DAS-CMOP [58]. These benchmark suites contain 47 distinct test instances in total, including 14 problems from MW, 14 from LIR-CMOP, 10 from CF, and 9 from DAS-CMOP. In addition, to further evaluate the practical effectiveness of the proposed algorithm, six real-world multi-objective optimization problems from the RWMOP suite [59] were also considered, as summarized in

Table 1. Real-world multi-objective optimization problems.

Name	Problem
RWMOP1	Pressure Vessel Design
RWMOP2	Two Bar Truss Design
RWMOP3	Speed Reducer Design
RWMOP4	Gear Train Design
RWMOP5	Car Side Impact Design
RWMOP6	Four Bar Plane Truss

. Algorithm performance was evaluated using three standard metrics, namely the Feasible Solutions Ratio (FSR) [60], Hypervolume (HV) [61], and Inverted Generational Distance (IGD) [62].

Specifically, the IGD indicator gauges convergence accuracy by computing the mean spatial separation between the generated solution set and the true Pareto front (PF), where minimized scores denote superior outcomes. Conversely, the HV metric captures both diversity and convergence simultaneously by measuring the hyper-spatial volume bounded by the obtained solutions and a designated reference coordinate; thus, maximized HV values are desirable. Lastly, the FSR metric determines the exact percentage of strictly feasible solutions residing within the terminal population, directly reflecting an algorithm’s competence in navigating severe constraints (with higher ratios being optimal).

5. Experimental Analysis

The statistical results for the Inverted Generational Distance (IGD) and Hypervolume (HV) metrics across the 30 independent runs are summarized in

Table 2. Statistical results of IGD.

	ANSGAIII	BiCo	CTAEA	CTSEA	CMOEMT	CMOQLMT	MOEAD2WA	AFFCMO
CF1	$3.92\times {10}^{-2}$ ($4.97\times {10}^{-3}$)+	$3.34\times {10}^{-2}$ ($2.80\times {10}^{-3}$)+	$6.72\times {10}^{-2}$ ($1.68\times {10}^{-3}$)+	$1.95\times {10}^{-2}$ ($1.99\times {10}^{-3}$)+	$4.40\times {10}^{-3}$ ($2.43\times {10}^{-3}$)+	$6.02\times {10}^{-3}$ ($5.43\times {10}^{-4}$)+	$5.75\times {10}^{-2}$ ($1.08\times {10}^{-2}$)+	$1.15\times {10}^{-3}$ ($7.15\times {10}^{-5}$)
CF2	$1.12\times {10}^{-1}$ ($2.67\times {10}^{-2}$)+	$1.07\times {10}^{-1}$ ($2.82\times {10}^{-2}$)+	$1.22\times {10}^{-1}$ ($3.00\times {10}^{-2}$)+	$1.34\times {10}^{-1}$ ($2.96\times {10}^{-2}$)+	$5.13\times {10}^{-2}$ ($1.70\times {10}^{-2}$)+	$2.34\times {10}^{-2}$ ($7.90\times {10}^{-3}$)−	$2.18\times {10}^{-1}$ ($1.24\times {10}^{-1}$)+	$2.66\times {10}^{-2}$ ($3.13\times {10}^{-3}$)
CF3	$2.21\times {10}^{-1}$ ($1.37\times {10}^{-1}$)+	$2.13\times {10}^{-1}$ ($6.75\times {10}^{-2}$)+	$4.43\times {10}^{-1}$ ($1.66\times {10}^{-1}$)+	$3.80\times {10}^{-1}$ ($9.02\times {10}^{-2}$)+	$1.98\times {10}^{-1}$ ($9.64\times {10}^{-2}$)+	$1.09\times {10}^{-1}$ ($4.60\times {10}^{-2}$)+	$4.06\times {10}^{-1}$ ($2.69\times {10}^{-1}$)+	$7.31\times {10}^{-2}$ ($1.41\times {10}^{-2}$)
CF4	$2.32\times {10}^{-1}$ ($8.89\times {10}^{-2}$)+	$2.29\times {10}^{-1}$ ($1.07\times {10}^{-1}$)+	$4.39\times {10}^{-1}$ ($8.30\times {10}^{-2}$)+	$4.02\times {10}^{-1}$ ($9.91\times {10}^{-2}$)+	$1.86\times {10}^{-1}$ ($6.98\times {10}^{-2}$)+	$8.40\times {10}^{-2}$ ($2.06\times {10}^{-2}$)+	$4.01\times {10}^{-1}$ ($1.35\times {10}^{-1}$)+	$7.67\times {10}^{-2}$ ($1.03\times {10}^{-2}$)
CF5	$4.77\times {10}^{-1}$ ($9.46\times {10}^{-2}$)+	$4.54\times {10}^{-1}$ ($1.00\times {10}^{-1}$)+	$6.17\times {10}^{-1}$ ($4.63\times {10}^{-2}$)+	$6.08\times {10}^{-1}$ ($3.82\times {10}^{-2}$)+	$3.19\times {10}^{-1}$ ($8.18\times {10}^{-2}$)+	$2.90\times {10}^{-1}$ ($9.44\times {10}^{-2}$)=	$5.18\times {10}^{-1}$ ($1.16\times {10}^{-1}$)+	$2.62\times {10}^{-1}$ ($5.09\times {10}^{-2}$)
CF6	$3.20\times {10}^{-1}$ ($1.02\times {10}^{-1}$)+	$3.03\times {10}^{-1}$ ($1.07\times {10}^{-1}$)+	$4.38\times {10}^{-1}$ ($2.23\times {10}^{-2}$)+	$3.92\times {10}^{-1}$ ($4.71\times {10}^{-2}$)+	$1.25\times {10}^{-1}$ ($3.29\times {10}^{-2}$)+	$1.24\times {10}^{-1}$ ($5.26\times {10}^{-2}$)+	$4.01\times {10}^{-1}$ ($1.11\times {10}^{-1}$)+	$7.08\times {10}^{-2}$ ($1.62\times {10}^{-2}$)
CF7	$4.33\times {10}^{-1}$ ($9.20\times {10}^{-2}$)+	$4.12\times {10}^{-1}$ ($1.25\times {10}^{-1}$)+	$6.02\times {10}^{-1}$ ($9.67\times {10}^{-2}$)+	$5.29\times {10}^{-1}$ ($8.79\times {10}^{-2}$)+	$3.19\times {10}^{-1}$ ($7.76\times {10}^{-2}$)+	$2.89\times {10}^{-1}$ ($1.46\times {10}^{-1}$)+	$4.69\times {10}^{-1}$ ($1.42\times {10}^{-1}$)+	$1.02\times {10}^{-1}$ ($7.99\times {10}^{-3}$)
CF8	NaN (0.00%)+	NaN (0.00%)+	$6.73\times {10}^{-1}$ ($1.36\times {10}^{-1}$)+	$4.00\times {10}^{-1}$ ($2.96\times {10}^{-2}$)+	$3.20\times {10}^{-1}$ ($5.11\times {10}^{-2}$)+	$2.87\times {10}^{-1}$ ($7.58\times {10}^{-2}$)=	NaN (96.67%)+	$2.86\times {10}^{-1}$ ($3.26\times {10}^{-2}$)
CF9	$5.66\times {10}^{-1}$ ($2.92\times {10}^{-1}$)+	$2.15\times {10}^{-1}$ ($1.70\times {10}^{-1}$)+	$2.66\times {10}^{-1}$ ($1.98\times {10}^{-2}$)+	$1.94\times {10}^{-1}$ ($6.62\times {10}^{-3}$)+	$1.82\times {10}^{-1}$ ($2.18\times {10}^{-2}$)+	$1.19\times {10}^{-1}$ ($2.86\times {10}^{-2}$)−	$1.64\times {10}^{-1}$ ($2.19\times {10}^{-2}$)+	$1.37\times {10}^{-1}$ ($9.41\times {10}^{-3}$)
CF10	NaN (0.00%)=	NaN (0.00%)=	$1.61\times {10}^0$ ($8.28\times {10}^{-1}$)−	$4.19\times {10}^{-1}$ ($5.75\times {10}^{-2}$)−	NaN (96.67%)−	$4.14\times {10}^{-1}$ ($9.12\times {10}^{-2}$)−	NaN (73.33%)−	NaN (0.00%)
+/−/=	9/0/1	9/0/1	9/1/0	9/1/0	9/1/0	5/3/2	9/1/0
DASCMOP1	$7.50\times {10}^{-1}$ ($3.15\times {10}^{-2}$)+	$7.34\times {10}^{-1}$ ($3.17\times {10}^{-2}$)+	$2.21\times {10}^{-1}$ ($3.27\times {10}^{-2}$)+	$7.64\times {10}^{-1}$ ($8.55\times {10}^{-3}$)+	$3.09\times {10}^{-1}$ ($3.12\times {10}^{-1}$)+	$5.72\times {10}^{-3}$ ($4.01\times {10}^{-3}$)+	$7.83\times {10}^{-1}$ ($7.17\times {10}^{-2}$)+	$4.21\times {10}^{-3}$ ($5.04\times {10}^{-4}$)
DASCMOP2	$2.71\times {10}^{-1}$ ($3.01\times {10}^{-2}$)+	$2.48\times {10}^{-1}$ ($1.58\times {10}^{-2}$)+	$1.49\times {10}^{-1}$ ($2.06\times {10}^{-2}$)+	$2.58\times {10}^{-1}$ ($1.68\times {10}^{-2}$)+	$7.34\times {10}^{-2}$ ($9.36\times {10}^{-2}$)=	$4.57\times {10}^{-3}$ ($1.52\times {10}^{-4}$)−	$2.66\times {10}^{-1}$ ($2.62\times {10}^{-2}$)+	$5.18\times {10}^{-3}$ ($1.02\times {10}^{-4}$)
DASCMOP3	$3.62\times {10}^{-1}$ ($3.80\times {10}^{-2}$)+	$3.32\times {10}^{-1}$ ($6.50\times {10}^{-2}$)+	$2.46\times {10}^{-1}$ ($8.19\times {10}^{-3}$)+	$3.55\times {10}^{-1}$ ($2.47\times {10}^{-2}$)+	$1.40\times {10}^{-1}$ ($1.37\times {10}^{-1}$)+	$5.95\times {10}^{-2}$ ($9.21\times {10}^{-2}$)+	$2.87\times {10}^{-1}$ ($4.90\times {10}^{-2}$)+	$1.77\times {10}^{-2}$ ($3.11\times {10}^{-3}$)
DASCMOP4	$3.41\times {10}^{-1}$ ($3.26\times {10}^{-2}$)+	NaN (0.00%)+	$1.64\times {10}^{-2}$ ($1.23\times {10}^{-3}$)+	$5.61\times {10}^{-3}$ ($5.21\times {10}^{-3}$)+	NaN (36.90%)+	NaN (0.00%)+	$1.36\times {10}^{-1}$ ($2.07\times {10}^{-1}$)+	$1.36\times {10}^{-3}$ ($9.52\times {10}^{-5}$)
DASCMOP5	NaN (73.33%)+	$3.79\times {10}^{-2}$ ($1.10\times {10}^{-1}$)+	$8.81\times {10}^{-3}$ ($4.84\times {10}^{-4}$)+	$3.38\times {10}^{-3}$ ($7.06\times {10}^{-4}$)+	$3.63\times {10}^{-3}$ ($9.37\times {10}^{-4}$)+	$1.46\times {10}^{-1}$ ($1.98\times {10}^{-1}$)+	$2.91\times {10}^{-2}$ ($5.16\times {10}^{-2}$)+	$2.85\times {10}^{-3}$ ($5.89\times {10}^{-5}$)
DASCMOP6	NaN (66.67%)+	$1.78\times {10}^{-1}$ ($1.77\times {10}^{-1}$)+	$3.71\times {10}^{-2}$ ($6.00\times {10}^{-3}$)+	$5.94\times {10}^{-2}$ ($3.18\times {10}^{-2}$)+	$4.03\times {10}^{-2}$ ($3.63\times {10}^{-2}$)+	$3.26\times {10}^{-1}$ ($1.69\times {10}^{-1}$)+	$1.33\times {10}^{-1}$ ($1.13\times {10}^{-1}$)+	$1.50\times {10}^{-2}$ ($3.24\times {10}^{-3}$)
DASCMOP7	$4.02\times {10}^{-2}$ ($2.91\times {10}^{-3}$)+	$3.22\times {10}^{-2}$ ($2.62\times {10}^{-3}$)+	$4.38\times {10}^{-2}$ ($1.07\times {10}^{-2}$)+	$3.20\times {10}^{-2}$ ($3.91\times {10}^{-4}$)+	$3.33\times {10}^{-2}$ ($2.51\times {10}^{-3}$)+	$3.19\times {10}^{-2}$ ($1.02\times {10}^{-3}$)+	$8.72\times {10}^{-2}$ ($7.49\times {10}^{-3}$)+	$2.99\times {10}^{-2}$ ($6.06\times {10}^{-4}$)
DASCMOP8	$1.11\times {10}^{-1}$ ($1.43\times {10}^{-1}$)+	$4.11\times {10}^{-2}$ ($1.62\times {10}^{-3}$)+	$6.25\times {10}^{-2}$ ($1.16\times {10}^{-2}$)+	$4.10\times {10}^{-2}$ ($1.01\times {10}^{-3}$)+	$4.68\times {10}^{-2}$ ($1.80\times {10}^{-2}$)+	$4.04\times {10}^{-2}$ ($2.28\times {10}^{-3}$)+	$3.14\times {10}^{-1}$ ($2.65\times {10}^{-1}$)+	$3.84\times {10}^{-2}$ ($7.11\times {10}^{-4}$)
DASCMOP9	$4.48\times {10}^{-1}$ ($6.91\times {10}^{-2}$)+	$3.60\times {10}^{-1}$ ($5.44\times {10}^{-2}$)+	$3.44\times {10}^{-1}$ ($1.62\times {10}^{-2}$)+	$4.65\times {10}^{-1}$ ($4.88\times {10}^{-2}$)+	$1.16\times {10}^{-1}$ ($1.10\times {10}^{-1}$)+	$4.11\times {10}^{-2}$ ($1.61\times {10}^{-3}$)=	NaN (30.18%)+	$4.15\times {10}^{-2}$ ($1.57\times {10}^{-3}$)
+/−/=	9/0/0	9/0/0	9/0/0	9/0/0	8/0/1	7/1/1	9/0/0
LIRCMOP1	$3.05\times {10}^{-1}$ ($2.66\times {10}^{-2}$)+	$2.09\times {10}^{-1}$ ($1.85\times {10}^{-2}$)+	$4.18\times {10}^{-1}$ ($5.53\times {10}^{-2}$)+	$3.74\times {10}^{-1}$ ($4.92\times {10}^{-2}$)+	$4.17\times {10}^{-2}$ ($3.10\times {10}^{-2}$)+	$1.27\times {10}^{-1}$ ($5.22\times {10}^{-2}$)+	$2.29\times {10}^{-1}$ ($1.98\times {10}^{-2}$)+	$2.60\times {10}^{-2}$ ($2.48\times {10}^{-3}$)
LIRCMOP2	$2.62\times {10}^{-1}$ ($2.50\times {10}^{-2}$)+	$1.82\times {10}^{-1}$ ($1.66\times {10}^{-2}$)+	$3.34\times {10}^{-1}$ ($7.56\times {10}^{-2}$)+	$3.18\times {10}^{-1}$ ($3.49\times {10}^{-2}$)+	$2.51\times {10}^{-2}$ ($2.61\times {10}^{-2}$)−	$9.60\times {10}^{-2}$ ($3.50\times {10}^{-2}$)+	$2.04\times {10}^{-1}$ ($1.65\times {10}^{-2}$)+	$2.36\times {10}^{-2}$ ($1.81\times {10}^{-3}$)
LIRCMOP3	$3.09\times {10}^{-1}$ ($3.44\times {10}^{-2}$)+	$2.15\times {10}^{-1}$ ($2.83\times {10}^{-2}$)+	$4.49\times {10}^{-1}$ ($1.04\times {10}^{-1}$)+	$3.50\times {10}^{-1}$ ($1.72\times {10}^{-2}$)+	$4.90\times {10}^{-2}$ ($4.77\times {10}^{-2}$)+	$1.13\times {10}^{-1}$ ($5.61\times {10}^{-2}$)+	$2.51\times {10}^{-1}$ ($2.70\times {10}^{-2}$)+	$1.34\times {10}^{-2}$ ($3.61\times {10}^{-3}$)
LIRCMOP4	$2.89\times {10}^{-1}$ ($3.01\times {10}^{-2}$)+	$2.10\times {10}^{-1}$ ($2.76\times {10}^{-2}$)+	$4.28\times {10}^{-1}$ ($1.14\times {10}^{-1}$)+	$3.29\times {10}^{-1}$ ($1.88\times {10}^{-2}$)+	$3.73\times {10}^{-2}$ ($4.07\times {10}^{-2}$)+	$1.54\times {10}^{-1}$ ($5.96\times {10}^{-2}$)+	$2.49\times {10}^{-1}$ ($3.01\times {10}^{-2}$)+	$1.38\times {10}^{-2}$ ($2.47\times {10}^{-3}$)
LIRCMOP5	$1.23\times {10}^0$ ($5.27\times {10}^{-3}$)+	$1.22\times {10}^0$ ($6.59\times {10}^{-3}$)+	$1.25\times {10}^0$ ($1.14\times {10}^{-2}$)+	$3.76\times {10}^{-1}$ ($2.46\times {10}^{-2}$)+	$3.71\times {10}^{-1}$ ($4.27\times {10}^{-1}$)+	$3.12\times {10}^{-2}$ ($4.00\times {10}^{-2}$)+	$3.64\times {10}^{-1}$ ($4.58\times {10}^{-2}$)+	$1.02\times {10}^{-2}$ ($7.29\times {10}^{-4}$)
LIRCMOP6	$1.35\times {10}^0$ ($3.71\times {10}^{-4}$)+	$1.35\times {10}^0$ ($1.90\times {10}^{-4}$)+	$1.35\times {10}^0$ ($5.27\times {10}^{-4}$)+	$4.70\times {10}^{-1}$ ($6.84\times {10}^{-2}$)+	$3.00\times {10}^{-1}$ ($2.31\times {10}^{-1}$)+	$8.58\times {10}^{-2}$ ($8.70\times {10}^{-2}$)+	$4.26\times {10}^{-1}$ ($6.40\times {10}^{-2}$)+	$1.06\times {10}^{-2}$ ($8.47\times {10}^{-4}$)
LIRCMOP7	$6.63\times {10}^{-1}$ ($7.33\times {10}^{-1}$)+	$8.06\times {10}^{-1}$ ($7.79\times {10}^{-1}$)+	$1.06\times {10}^0$ ($5.70\times {10}^{-1}$)+	$1.68\times {10}^{-1}$ ($1.61\times {10}^{-2}$)+	$7.28\times {10}^{-2}$ ($3.09\times {10}^{-2}$)+	$5.77\times {10}^{-2}$ ($4.22\times {10}^{-2}$)+	$1.54\times {10}^{-1}$ ($2.06\times {10}^{-2}$)+	$8.92\times {10}^{-3}$ ($3.10\times {10}^{-4}$)
LIRCMOP8	$1.22\times {10}^0$ ($6.68\times {10}^{-1}$)+	$1.21\times {10}^0$ ($6.81\times {10}^{-1}$)+	$1.69\times {10}^0$ ($9.64\times {10}^{-4}$)+	$2.68\times {10}^{-1}$ ($2.58\times {10}^{-2}$)+	$8.71\times {10}^{-2}$ ($4.02\times {10}^{-2}$)+	$5.51\times {10}^{-2}$ ($4.43\times {10}^{-2}$)+	$2.72\times {10}^{-1}$ ($2.78\times {10}^{-2}$)+	$8.61\times {10}^{-3}$ ($2.14\times {10}^{-4}$)
LIRCMOP9	$1.01\times {10}^0$ ($7.17\times {10}^{-2}$)+	$1.01\times {10}^0$ ($5.98\times {10}^{-2}$)+	$6.41\times {10}^{-1}$ ($7.46\times {10}^{-2}$)+	$8.71\times {10}^{-1}$ ($6.68\times {10}^{-2}$)+	$3.73\times {10}^{-1}$ ($8.90\times {10}^{-2}$)+	$3.10\times {10}^{-1}$ ($8.74\times {10}^{-2}$)+	$6.41\times {10}^{-1}$ ($1.32\times {10}^{-1}$)+	$8.88\times {10}^{-2}$ ($1.46\times {10}^{-3}$)
LIRCMOP10	$9.31\times {10}^{-1}$ ($5.55\times {10}^{-2}$)+	$9.51\times {10}^{-1}$ ($6.18\times {10}^{-2}$)+	$4.39\times {10}^{-1}$ ($2.10\times {10}^{-2}$)+	$5.07\times {10}^{-1}$ ($5.12\times {10}^{-2}$)+	$2.44\times {10}^{-1}$ ($1.02\times {10}^{-1}$)+	$1.52\times {10}^{-1}$ ($7.37\times {10}^{-2}$)+	$3.52\times {10}^{-1}$ ($1.17\times {10}^{-1}$)+	$1.54\times {10}^{-2}$ ($1.02\times {10}^{-3}$)
LIRCMOP11	$8.35\times {10}^{-1}$ ($1.05\times {10}^{-1}$)+	$7.06\times {10}^{-1}$ ($1.76\times {10}^{-1}$)+	$3.16\times {10}^{-1}$ ($1.56\times {10}^{-2}$)+	$4.39\times {10}^{-1}$ ($2.53\times {10}^{-2}$)+	$1.05\times {10}^{-1}$ ($6.46\times {10}^{-2}$)+	$2.79\times {10}^{-2}$ ($2.88\times {10}^{-2}$)+	$6.08\times {10}^{-1}$ ($1.78\times {10}^{-1}$)+	$5.22\times {10}^{-3}$ ($4.54\times {10}^{-4}$)
LIRCMOP12	$8.53\times {10}^{-1}$ ($1.49\times {10}^{-1}$)+	$7.10\times {10}^{-1}$ ($2.32\times {10}^{-1}$)+	$4.57\times {10}^{-1}$ ($9.38\times {10}^{-2}$)+	$4.74\times {10}^{-1}$ ($7.58\times {10}^{-2}$)+	$1.39\times {10}^{-1}$ ($6.21\times {10}^{-2}$)+	$6.82\times {10}^{-2}$ ($3.92\times {10}^{-2}$)+	$4.40\times {10}^{-1}$ ($1.42\times {10}^{-1}$)+	$1.05\times {10}^{-2}$ ($1.45\times {10}^{-3}$)
LIRCMOP13	$1.32\times {10}^0$ ($3.65\times {10}^{-3}$)+	$1.32\times {10}^0$ ($2.17\times {10}^{-3}$)+	$1.14\times {10}^{-1}$ ($3.54\times {10}^{-3}$)+	$9.42\times {10}^{-2}$ ($6.11\times {10}^{-4}$)+	$1.06\times {10}^{-1}$ ($3.90\times {10}^{-3}$)+	$9.34\times {10}^{-2}$ ($1.03\times {10}^{-3}$)+	$1.31\times {10}^0$ ($6.19\times {10}^{-4}$)+	$9.24\times {10}^{-2}$ ($1.05\times {10}^{-3}$)
LIRCMOP14	$1.27\times {10}^0$ ($4.15\times {10}^{-3}$)+	$1.27\times {10}^0$ ($2.36\times {10}^{-3}$)+	$1.13\times {10}^{-1}$ ($1.61\times {10}^{-3}$)+	$9.62\times {10}^{-2}$ ($6.21\times {10}^{-4}$)+	$1.03\times {10}^{-1}$ ($1.13\times {10}^{-3}$)+	$9.50\times {10}^{-2}$ ($1.00\times {10}^{-3}$)−	$1.26\times {10}^0$ ($4.96\times {10}^{-4}$)+	$9.59\times {10}^{-2}$ ($8.46\times {10}^{-4}$)
+/−/=	14/0/0	14/0/0	14/0/0	14/0/0	13/1/0	13/1/0	14/0/0
MW1	NaN (3.33%)+	NaN (60.00%)+	$3.38\times {10}^{-3}$ ($2.22\times {10}^{-3}$)+	NaN (90.00%)+	NaN (30.03%)+	NaN (90.00%)+	NaN (30.00%)+	$1.63\times {10}^{-3}$ ($1.40\times {10}^{-5}$)
MW2	$7.28\times {10}^{-2}$ ($6.57\times {10}^{-2}$)+	$2.70\times {10}^{-2}$ ($5.58\times {10}^{-3}$)+	$5.81\times {10}^{-2}$ ($1.85\times {10}^{-2}$)+	$6.68\times {10}^{-2}$ ($3.23\times {10}^{-2}$)+	$9.32\times {10}^{-2}$ ($7.20\times {10}^{-2}$)+	$1.32\times {10}^{-2}$ ($4.34\times {10}^{-3}$)+	$4.42\times {10}^{-2}$ ($3.40\times {10}^{-2}$)+	$3.93\times {10}^{-3}$ ($6.18\times {10}^{-5}$)
MW3	$2.47\times {10}^{-1}$ ($3.85\times {10}^{-1}$)+	$7.28\times {10}^{-3}$ ($1.30\times {10}^{-3}$)+	$7.04\times {10}^{-3}$ ($4.41\times {10}^{-4}$)+	$7.33\times {10}^{-3}$ ($5.56\times {10}^{-4}$)+	$8.09\times {10}^{-3}$ ($1.12\times {10}^{-3}$)+	$5.59\times {10}^{-3}$ ($4.30\times {10}^{-4}$)−	$7.05\times {10}^{-3}$ ($7.59\times {10}^{-4}$)+	$5.74\times {10}^{-3}$ ($2.41\times {10}^{-4}$)
MW4	NaN (16.81%)+	NaN (33.33%)+	$4.96\times {10}^{-2}$ ($2.99\times {10}^{-3}$)+	$6.42\times {10}^{-2}$ ($1.06\times {10}^{-1}$)=	NaN (60.00%)+	NaN (96.67%)+	NaN (26.67%)+	$4.15\times {10}^{-2}$ ($3.00\times {10}^{-4}$)
MW5	NaN (26.70%)+	NaN (80.00%)+	$4.23\times {10}^{-2}$ ($4.22\times {10}^{-2}$)+	$7.67\times {10}^{-2}$ ($1.92\times {10}^{-1}$)+	NaN (43.33%)+	$1.39\times {10}^{-3}$ ($1.13\times {10}^{-3}$)+	NaN (36.67%)+	$6.92\times {10}^{-4}$ ($1.13\times {10}^{-4}$)
MW6	$2.37\times {10}^{-1}$ ($2.66\times {10}^{-1}$)+	$3.09\times {10}^{-2}$ ($1.47\times {10}^{-2}$)+	$1.56\times {10}^{-1}$ ($1.89\times {10}^{-1}$)+	$1.96\times {10}^{-1}$ ($1.94\times {10}^{-1}$)+	$3.35\times {10}^{-1}$ ($2.48\times {10}^{-1}$)+	$5.83\times {10}^{-3}$ ($2.96\times {10}^{-3}$)+	$5.81\times {10}^{-2}$ ($8.18\times {10}^{-2}$)+	$2.71\times {10}^{-3}$ ($2.61\times {10}^{-5}$)
MW7	$1.40\times {10}^{-1}$ ($2.07\times {10}^{-1}$)+	$6.77\times {10}^{-3}$ ($7.76\times {10}^{-4}$)+	$8.39\times {10}^{-3}$ ($5.44\times {10}^{-4}$)+	$6.26\times {10}^{-3}$ ($3.84\times {10}^{-4}$)+	$6.58\times {10}^{-3}$ ($7.90\times {10}^{-4}$)+	$5.63\times {10}^{-3}$ ($5.96\times {10}^{-4}$)=	$5.89\times {10}^{-3}$ ($5.82\times {10}^{-4}$)+	$5.60\times {10}^{-3}$ ($5.14\times {10}^{-4}$)
MW8	$1.66\times {10}^{-1}$ ($1.78\times {10}^{-1}$)+	$4.82\times {10}^{-2}$ ($2.99\times {10}^{-3}$)+	$8.29\times {10}^{-2}$ ($3.42\times {10}^{-2}$)+	$7.87\times {10}^{-2}$ ($2.66\times {10}^{-2}$)+	NaN (96.67%)+	$4.94\times {10}^{-2}$ ($3.89\times {10}^{-3}$)+	$7.58\times {10}^{-2}$ ($3.59\times {10}^{-2}$)+	$4.30\times {10}^{-2}$ ($3.96\times {10}^{-4}$)
MW9	NaN (33.33%)+	NaN (76.67%)+	$7.14\times {10}^{-1}$ ($3.79\times {10}^{-3}$)+	$7.12\times {10}^{-1}$ ($1.29\times {10}^{-3}$)+	NaN (86.67%)+	$1.44\times {10}^{-2}$ ($4.70\times {10}^{-3}$)+	NaN (46.67%)+	$4.58\times {10}^{-3}$ ($2.27\times {10}^{-4}$)
MW10	NaN (90.00%)+	$6.50\times {10}^{-2}$ ($3.28\times {10}^{-2}$)+	$1.25\times {10}^{-1}$ ($1.00\times {10}^{-1}$)+	$2.37\times {10}^{-1}$ ($1.43\times {10}^{-1}$)+	NaN (86.67%)+	$1.59\times {10}^{-2}$ ($8.99\times {10}^{-3}$)+	$1.01\times {10}^{-1}$ ($7.56\times {10}^{-2}$)+	$3.34\times {10}^{-3}$ ($3.62\times {10}^{-5}$)
MW11	$6.24\times {10}^{-1}$ ($2.14\times {10}^{-1}$)+	$5.27\times {10}^{-2}$ ($1.51\times {10}^{-1}$)=	$1.63\times {10}^{-2}$ ($2.00\times {10}^{-3}$)+	$6.36\times {10}^{-3}$ ($1.38\times {10}^{-4}$)+	$6.23\times {10}^{-3}$ ($1.28\times {10}^{-4}$)=	$6.14\times {10}^{-3}$ ($1.49\times {10}^{-4}$)=	$8.00\times {10}^{-2}$ ($2.10\times {10}^{-1}$)+	$6.22\times {10}^{-3}$ ($1.48\times {10}^{-4}$)
MW12	NaN (46.67%)+	NaN (96.67%)+	$7.72\times {10}^{-1}$ ($3.62\times {10}^{-2}$)+	$7.74\times {10}^{-1}$ ($2.85\times {10}^{-3}$)+	NaN (86.67%)+	$5.00\times {10}^{-3}$ ($1.42\times {10}^{-4}$)+	NaN (63.33%)+	$4.72\times {10}^{-3}$ ($8.95\times {10}^{-5}$)
MW13	$3.52\times {10}^{-1}$ ($3.61\times {10}^{-1}$)+	$1.02\times {10}^{-1}$ ($1.98\times {10}^{-2}$)+	$1.21\times {10}^{-1}$ ($2.60\times {10}^{-2}$)+	$1.59\times {10}^{-1}$ ($6.14\times {10}^{-2}$)+	$2.21\times {10}^{-1}$ ($1.32\times {10}^{-1}$)+	$3.26\times {10}^{-2}$ ($1.31\times {10}^{-2}$)+	$1.73\times {10}^{-1}$ ($4.69\times {10}^{-2}$)+	$1.05\times {10}^{-2}$ ($1.60\times {10}^{-4}$)
MW14	$3.34\times {10}^{-1}$ ($1.78\times {10}^{-1}$)+	$1.56\times {10}^{-1}$ ($8.20\times {10}^{-2}$)+	$3.01\times {10}^{-1}$ ($6.25\times {10}^{-2}$)+	$3.34\times {10}^{-1}$ ($6.48\times {10}^{-2}$)+	$1.88\times {10}^{-1}$ ($1.18\times {10}^{-1}$)+	$2.23\times {10}^{-1}$ ($1.02\times {10}^{-1}$)+	$2.11\times {10}^{-1}$ ($8.97\times {10}^{-3}$)+	$9.65\times {10}^{-2}$ ($8.51\times {10}^{-4}$)
+/−/=	14/0/0	13/0/1	14/0/0	13/0/1	13/0/1	11/1/2	14/0/0
RWMOP1	$3.60\times {10}^5$ ($4.11\times {10}^2$)−	$3.60\times {10}^5$ ($6.69\times {10}^2$)+	$2.44\times {10}^6$ ($5.84\times {10}^5$)+	$3.60\times {10}^5$ ($6.11\times {10}^2$)+	$3.60\times {10}^5$ ($5.64\times {10}^2$)+	$3.60\times {10}^5$ ($0.00\times {10}^0$)+	$5.69\times {10}^7$ ($3.34\times {10}^3$)+	$3.60\times {10}^5$ ($3.95\times {10}^{-1}$)
RWMOP2	$4.25\times {10}^2$ ($4.76\times {10}^2$)+	$1.32\times {10}^2$ ($7.08\times {10}^1$)+	$5.82\times {10}^4$ ($1.02\times {10}^4$)+	$6.36\times {10}^1$ ($3.56\times {10}^1$)+	$1.62\times {10}^2$ ($1.14\times {10}^2$)+	$1.00\times {10}^2$ ($3.74\times {10}^1$)+	$9.83\times {10}^4$ ($4.20\times {10}^{-2}$)+	$5.13\times {10}^1$ ($1.56\times {10}^2$)
RWMOP3	$2.87\times {10}^3$ ($2.63\times {10}^0$)+	$6.06\times {10}^2$ ($1.46\times {10}^{-1}$)+	$2.80\times {10}^3$ ($1.99\times {10}^2$)+	$6.05\times {10}^2$ ($4.22\times {10}^{-2}$)+	$6.06\times {10}^2$ ($2.39\times {10}^{-1}$)+	$6.05\times {10}^2$ ($4.25\times {10}^{-2}$)+	$2.88\times {10}^3$ ($4.44\times {10}^0$)+	$6.05\times {10}^2$ ($1.20\times {10}^{-1}$)
RWMOP4	$1.54\times {10}^1$ ($6.55\times {10}^{-2}$)+	$1.55\times {10}^1$ ($3.47\times {10}^{-2}$)+	$1.50\times {10}^1$ ($3.78\times {10}^{-1}$)+	$1.55\times {10}^1$ ($3.91\times {10}^{-2}$)+	$1.55\times {10}^1$ ($2.43\times {10}^{-2}$)+	$1.55\times {10}^1$ ($4.08\times {10}^{-2}$)+	$1.55\times {10}^1$ ($6.94\times {10}^{-3}$)+	$1.11\times {10}^1$ ($4.29\times {10}^0$)
RWMOP5	$2.13\times {10}^0$ ($2.79\times {10}^{-3}$)+	$2.22\times {10}^0$ ($1.33\times {10}^{-1}$)+	$4.67\times {10}^0$ ($1.09\times {10}^0$)+	$2.13\times {10}^0$ ($1.14\times {10}^{-2}$)+	$2.13\times {10}^0$ ($1.36\times {10}^{-15}$)+	$2.13\times {10}^0$ ($1.36\times {10}^{-15}$)+	$5.94\times {10}^1$ ($1.22\times {10}^{-1}$)+	$2.11\times {10}^0$ ($3.28\times {10}^{-2}$)
RWMOP6	$3.76\times {10}^{-2}$ ($2.23\times {10}^{-4}$)+	$3.72\times {10}^{-2}$ ($1.41\times {10}^{-17}$)+	$3.72\times {10}^{-2}$ ($1.41\times {10}^{-17}$)+	$3.72\times {10}^{-2}$ ($1.41\times {10}^{-17}$)+	$3.72\times {10}^{-2}$ ($1.41\times {10}^{-17}$)+	$3.72\times {10}^{-2}$ ($1.41\times {10}^{-17}$)+	$1.65\times {10}^3$ ($2.86\times {10}^{-2}$)+	$3.72\times {10}^{-2}$ ($1.67\times {10}^{-11}$)
+/−/=	5/1/0	6/0/0	6/0/0	6/0/0	6/0/0	6/0/0	6/0/0

and

Table 3. Statistical results of HV.

	ANSGAIII	BiCo	CTAEA	CTSEA	CMOEMT	CMOQLMT	MOEAD2WA	AFFCMO
CF1	$5.17\times {10}^{-1}$ ($6.05\times {10}^{-3}$)+	$5.24\times {10}^{-1}$ ($3.46\times {10}^{-3}$)+	$4.89\times {10}^{-1}$ ($1.57\times {10}^{-3}$)+	$5.41\times {10}^{-1}$ ($2.11\times {10}^{-3}$)+	$5.61\times {10}^{-1}$ ($3.08\times {10}^{-3}$)+	$5.59\times {10}^{-1}$ ($6.58\times {10}^{-4}$)+	$5.02\times {10}^{-1}$ ($7.50\times {10}^{-3}$)+	$5.65\times {10}^{-1}$ ($8.80\times {10}^{-5}$)
CF2	$5.40\times {10}^{-1}$ ($2.60\times {10}^{-2}$)+	$5.47\times {10}^{-1}$ ($3.13\times {10}^{-2}$)+	$5.19\times {10}^{-1}$ ($1.49\times {10}^{-2}$)+	$5.16\times {10}^{-1}$ ($2.56\times {10}^{-2}$)+	$6.11\times {10}^{-1}$ ($2.54\times {10}^{-2}$)+	$6.41\times {10}^{-1}$ ($1.63\times {10}^{-2}$)=	$4.88\times {10}^{-1}$ ($6.03\times {10}^{-2}$)+	$6.43\times {10}^{-1}$ ($6.55\times {10}^{-3}$)
CF3	$2.15\times {10}^{-1}$ ($5.65\times {10}^{-2}$)+	$2.16\times {10}^{-1}$ ($3.19\times {10}^{-2}$)+	$1.23\times {10}^{-1}$ ($3.89\times {10}^{-2}$)+	$1.56\times {10}^{-1}$ ($3.23\times {10}^{-2}$)+	$2.17\times {10}^{-1}$ ($4.05\times {10}^{-2}$)+	$2.78\times {10}^{-1}$ ($2.61\times {10}^{-2}$)=	$1.55\times {10}^{-1}$ ($6.00\times {10}^{-2}$)+	$2.81\times {10}^{-1}$ ($1.44\times {10}^{-2}$)
CF4	$3.25\times {10}^{-1}$ ($6.04\times {10}^{-2}$)+	$3.20\times {10}^{-1}$ ($6.09\times {10}^{-2}$)+	$2.00\times {10}^{-1}$ ($4.82\times {10}^{-2}$)+	$2.26\times {10}^{-1}$ ($6.01\times {10}^{-2}$)+	$3.49\times {10}^{-1}$ ($4.42\times {10}^{-2}$)+	$4.27\times {10}^{-1}$ ($2.31\times {10}^{-2}$)=	$2.29\times {10}^{-1}$ ($7.98\times {10}^{-2}$)+	$4.26\times {10}^{-1}$ ($1.48\times {10}^{-2}$)
CF5	$1.87\times {10}^{-1}$ ($6.15\times {10}^{-2}$)+	$1.89\times {10}^{-1}$ ($5.84\times {10}^{-2}$)+	$1.21\times {10}^{-1}$ ($3.20\times {10}^{-2}$)+	$1.28\times {10}^{-1}$ ($2.26\times {10}^{-2}$)+	$2.79\times {10}^{-1}$ ($3.97\times {10}^{-2}$)+	$2.69\times {10}^{-1}$ ($6.68\times {10}^{-2}$)+	$1.80\times {10}^{-1}$ ($6.93\times {10}^{-2}$)+	$2.98\times {10}^{-1}$ ($2.28\times {10}^{-2}$)
CF6	$4.41\times {10}^{-1}$ ($6.86\times {10}^{-2}$)+	$4.58\times {10}^{-1}$ ($7.11\times {10}^{-2}$)+	$3.71\times {10}^{-1}$ ($3.39\times {10}^{-2}$)+	$4.01\times {10}^{-1}$ ($3.90\times {10}^{-2}$)+	$5.61\times {10}^{-1}$ ($3.71\times {10}^{-2}$)+	$5.90\times {10}^{-1}$ ($3.07\times {10}^{-2}$)+	$3.98\times {10}^{-1}$ ($6.94\times {10}^{-2}$)+	$6.12\times {10}^{-1}$ ($7.96\times {10}^{-3}$)
CF7	$3.36\times {10}^{-1}$ ($8.68\times {10}^{-2}$)+	$3.41\times {10}^{-1}$ ($8.97\times {10}^{-2}$)+	$2.08\times {10}^{-1}$ ($9.06\times {10}^{-2}$)+	$1.75\times {10}^{-1}$ ($4.90\times {10}^{-2}$)+	$3.98\times {10}^{-1}$ ($7.04\times {10}^{-2}$)+	$3.88\times {10}^{-1}$ ($1.27\times {10}^{-1}$)+	$2.88\times {10}^{-1}$ ($1.12\times {10}^{-1}$)+	$5.64\times {10}^{-1}$ ($4.98\times {10}^{-3}$)
CF8	NaN (0.00%)+	NaN (0.00%)+	$8.94\times {10}^{-2}$ ($3.01\times {10}^{-2}$)+	$1.15\times {10}^{-1}$ ($2.38\times {10}^{-2}$)+	$1.68\times {10}^{-1}$ ($4.12\times {10}^{-2}$)+	$1.99\times {10}^{-1}$ ($5.14\times {10}^{-2}$)=	NaN (96.67%)+	$1.90\times {10}^{-1}$ ($1.76\times {10}^{-2}$)
CF9	$1.75\times {10}^{-1}$ ($6.71\times {10}^{-2}$)+	$2.89\times {10}^{-1}$ ($6.20\times {10}^{-2}$)+	$1.77\times {10}^{-1}$ ($2.28\times {10}^{-2}$)+	$2.68\times {10}^{-1}$ ($8.08\times {10}^{-3}$)+	$2.62\times {10}^{-1}$ ($2.26\times {10}^{-2}$)+	$3.64\times {10}^{-1}$ ($3.36\times {10}^{-2}$)−	$2.94\times {10}^{-1}$ ($2.12\times {10}^{-2}$)+	$3.55\times {10}^{-1}$ ($1.48\times {10}^{-2}$)
CF10	NaN (0.00%)=	NaN (0.00%)=	$1.08\times {10}^{-6}$ ($5.34\times {10}^{-6}$)−	$8.50\times {10}^{-2}$ ($9.51\times {10}^{-3}$)−	NaN (96.67%)−	$8.00\times {10}^{-2}$ ($4.21\times {10}^{-2}$)−	NaN (73.33%)−	NaN (0.00%)
+/−/=	9/0/1	9/0/1	9/1/0	9/1/0	9/1/0	4/2/4	9/1/0
DASCMOP1	$4.60\times {10}^{-3}$ ($5.50\times {10}^{-3}$)+	$7.05\times {10}^{-3}$ ($5.33\times {10}^{-3}$)+	$1.58\times {10}^{-1}$ ($1.56\times {10}^{-2}$)+	$4.96\times {10}^{-4}$ ($6.97\times {10}^{-4}$)+	$1.28\times {10}^{-1}$ ($8.57\times {10}^{-2}$)+	$2.11\times {10}^{-1}$ ($1.14\times {10}^{-3}$)+	$7.32\times {10}^{-3}$ ($1.61\times {10}^{-2}$)+	$2.12\times {10}^{-1}$ ($4.53\times {10}^{-4}$)
DASCMOP2	$2.45\times {10}^{-1}$ ($7.27\times {10}^{-3}$)+	$2.50\times {10}^{-1}$ ($3.48\times {10}^{-3}$)+	$2.88\times {10}^{-1}$ ($4.58\times {10}^{-3}$)+	$2.59\times {10}^{-1}$ ($9.36\times {10}^{-4}$)+	$3.24\times {10}^{-1}$ ($3.81\times {10}^{-2}$)=	$3.55\times {10}^{-1}$ ($1.08\times {10}^{-4}$)−	$2.52\times {10}^{-1}$ ($3.29\times {10}^{-3}$)+	$3.55\times {10}^{-1}$ ($8.45\times {10}^{-5}$)
DASCMOP3	$2.08\times {10}^{-1}$ ($3.20\times {10}^{-4}$)+	$2.11\times {10}^{-1}$ ($3.72\times {10}^{-3}$)+	$2.27\times {10}^{-1}$ ($9.20\times {10}^{-3}$)+	$2.08\times {10}^{-1}$ ($3.82\times {10}^{-4}$)+	$2.74\times {10}^{-1}$ ($4.34\times {10}^{-2}$)+	$3.00\times {10}^{-1}$ ($2.73\times {10}^{-2}$)+	$2.14\times {10}^{-1}$ ($3.47\times {10}^{-3}$)+	$3.12\times {10}^{-1}$ ($4.55\times {10}^{-4}$)
DASCMOP4	$6.89\times {10}^{-2}$ ($1.13\times {10}^{-2}$)+	NaN (0.00%)+	$1.88\times {10}^{-1}$ ($1.11\times {10}^{-3}$)+	$1.95\times {10}^{-1}$ ($1.19\times {10}^{-3}$)+	NaN (36.90%)+	NaN (0.00%)+	$1.69\times {10}^{-1}$ ($4.20\times {10}^{-2}$)+	$2.04\times {10}^{-1}$ ($1.28\times {10}^{-4}$)
DASCMOP5	NaN (73.33%)+	$3.30\times {10}^{-1}$ ($6.58\times {10}^{-2}$)+	$3.47\times {10}^{-1}$ ($3.80\times {10}^{-4}$)+	$3.51\times {10}^{-1}$ ($4.63\times {10}^{-4}$)+	$3.51\times {10}^{-1}$ ($7.94\times {10}^{-4}$)+	$2.66\times {10}^{-1}$ ($1.17\times {10}^{-1}$)+	$3.41\times {10}^{-1}$ ($1.49\times {10}^{-2}$)+	$3.51\times {10}^{-1}$ ($9.43\times {10}^{-5}$)
DASCMOP6	NaN (66.67%)+	$2.31\times {10}^{-1}$ ($9.64\times {10}^{-2}$)+	$3.02\times {10}^{-1}$ ($5.84\times {10}^{-3}$)+	$2.92\times {10}^{-1}$ ($9.07\times {10}^{-3}$)+	$2.99\times {10}^{-1}$ ($1.46\times {10}^{-2}$)+	$1.43\times {10}^{-1}$ ($8.88\times {10}^{-2}$)+	$2.70\times {10}^{-1}$ ($2.59\times {10}^{-2}$)+	$3.12\times {10}^{-1}$ ($1.88\times {10}^{-4}$)
DASCMOP7	$2.84\times {10}^{-1}$ ($2.33\times {10}^{-3}$)+	$2.87\times {10}^{-1}$ ($1.57\times {10}^{-3}$)=	$2.85\times {10}^{-1}$ ($4.51\times {10}^{-3}$)+	$2.87\times {10}^{-1}$ ($2.83\times {10}^{-4}$)+	$2.86\times {10}^{-1}$ ($1.83\times {10}^{-3}$)+	$2.87\times {10}^{-1}$ ($6.23\times {10}^{-4}$)+	$2.72\times {10}^{-1}$ ($5.42\times {10}^{-3}$)+	$2.88\times {10}^{-1}$ ($3.30\times {10}^{-4}$)
DASCMOP8	$1.93\times {10}^{-1}$ ($2.55\times {10}^{-2}$)+	$2.06\times {10}^{-1}$ ($6.79\times {10}^{-4}$)+	$2.01\times {10}^{-1}$ ($4.48\times {10}^{-3}$)+	$2.06\times {10}^{-1}$ ($2.71\times {10}^{-4}$)+	$2.04\times {10}^{-1}$ ($6.84\times {10}^{-3}$)+	$2.06\times {10}^{-1}$ ($6.50\times {10}^{-4}$)+	$1.43\times {10}^{-1}$ ($6.22\times {10}^{-2}$)+	$2.07\times {10}^{-1}$ ($3.80\times {10}^{-4}$)
DASCMOP9	$1.14\times {10}^{-1}$ ($1.21\times {10}^{-2}$)+	$1.25\times {10}^{-1}$ ($9.41\times {10}^{-3}$)+	$1.29\times {10}^{-1}$ ($3.29\times {10}^{-3}$)+	$1.12\times {10}^{-1}$ ($8.68\times {10}^{-3}$)+	$1.87\times {10}^{-1}$ ($2.66\times {10}^{-2}$)+	$2.06\times {10}^{-1}$ ($4.16\times {10}^{-4}$)−	NaN (30.18%)+	$2.05\times {10}^{-1}$ ($2.95\times {10}^{-4}$)
+/−/=	9/0/0	8/0/1	9/0/0	9/0/0	8/0/1	7/2/0	9/0/0
LIRCMOP1	$1.11\times {10}^{-1}$ ($9.14\times {10}^{-3}$)+	$1.41\times {10}^{-1}$ ($6.89\times {10}^{-3}$)+	$8.81\times {10}^{-2}$ ($1.47\times {10}^{-2}$)+	$8.97\times {10}^{-2}$ ($9.53\times {10}^{-3}$)+	$2.20\times {10}^{-1}$ ($1.78\times {10}^{-2}$)+	$1.77\times {10}^{-1}$ ($2.19\times {10}^{-2}$)+	$1.32\times {10}^{-1}$ ($9.57\times {10}^{-3}$)+	$2.29\times {10}^{-1}$ ($1.25\times {10}^{-3}$)
LIRCMOP2	$2.27\times {10}^{-1}$ ($1.59\times {10}^{-2}$)+	$2.60\times {10}^{-1}$ ($9.18\times {10}^{-3}$)+	$2.02\times {10}^{-1}$ ($1.56\times {10}^{-2}$)+	$2.01\times {10}^{-1}$ ($1.42\times {10}^{-2}$)+	$3.50\times {10}^{-1}$ ($1.47\times {10}^{-2}$)−	$3.09\times {10}^{-1}$ ($2.07\times {10}^{-2}$)+	$2.63\times {10}^{-1}$ ($6.84\times {10}^{-3}$)+	$3.50\times {10}^{-1}$ ($9.87\times {10}^{-4}$)
LIRCMOP3	$9.91\times {10}^{-2}$ ($1.06\times {10}^{-2}$)+	$1.27\times {10}^{-1}$ ($9.54\times {10}^{-3}$)+	$7.68\times {10}^{-2}$ ($1.05\times {10}^{-2}$)+	$8.90\times {10}^{-2}$ ($5.35\times {10}^{-3}$)+	$1.89\times {10}^{-1}$ ($1.78\times {10}^{-2}$)+	$1.64\times {10}^{-1}$ ($1.94\times {10}^{-2}$)+	$1.14\times {10}^{-1}$ ($9.25\times {10}^{-3}$)+	$2.02\times {10}^{-1}$ ($8.09\times {10}^{-4}$)
LIRCMOP4	$1.93\times {10}^{-1}$ ($1.47\times {10}^{-2}$)+	$2.24\times {10}^{-1}$ ($1.30\times {10}^{-2}$)+	$1.58\times {10}^{-1}$ ($2.89\times {10}^{-2}$)+	$1.71\times {10}^{-1}$ ($1.27\times {10}^{-2}$)+	$3.00\times {10}^{-1}$ ($1.69\times {10}^{-2}$)+	$2.48\times {10}^{-1}$ ($2.64\times {10}^{-2}$)+	$2.13\times {10}^{-1}$ ($1.46\times {10}^{-2}$)+	$3.08\times {10}^{-1}$ ($1.08\times {10}^{-3}$)
LIRCMOP5	$0.00\times {10}^0$ ($0.00\times {10}^0$)+	$0.00\times {10}^0$ ($0.00\times {10}^0$)+	$0.00\times {10}^0$ ($0.00\times {10}^0$)+	$1.25\times {10}^{-1}$ ($8.68\times {10}^{-3}$)+	$1.67\times {10}^{-1}$ ($1.00\times {10}^{-1}$)+	$2.75\times {10}^{-1}$ ($2.52\times {10}^{-2}$)+	$1.35\times {10}^{-1}$ ($1.86\times {10}^{-2}$)+	$2.89\times {10}^{-1}$ ($4.15\times {10}^{-4}$)
LIRCMOP6	$0.00\times {10}^0$ ($0.00\times {10}^0$)+	$0.00\times {10}^0$ ($0.00\times {10}^0$)+	$0.00\times {10}^0$ ($0.00\times {10}^0$)+	$9.58\times {10}^{-2}$ ($4.34\times {10}^{-3}$)+	$1.28\times {10}^{-1}$ ($3.83\times {10}^{-2}$)+	$1.77\times {10}^{-1}$ ($2.06\times {10}^{-2}$)+	$9.18\times {10}^{-2}$ ($2.31\times {10}^{-3}$)+	$1.95\times {10}^{-1}$ ($2.34\times {10}^{-4}$)
LIRCMOP7	$1.59\times {10}^{-1}$ ($1.14\times {10}^{-1}$)+	$1.38\times {10}^{-1}$ ($1.23\times {10}^{-1}$)+	$8.39\times {10}^{-2}$ ($8.02\times {10}^{-2}$)+	$2.35\times {10}^{-1}$ ($4.12\times {10}^{-3}$)+	$2.64\times {10}^{-1}$ ($1.19\times {10}^{-2}$)+	$2.71\times {10}^{-1}$ ($1.72\times {10}^{-2}$)+	$2.40\times {10}^{-1}$ ($6.73\times {10}^{-3}$)+	$2.93\times {10}^{-1}$ ($2.56\times {10}^{-4}$)
LIRCMOP8	$6.92\times {10}^{-2}$ ($1.00\times {10}^{-1}$)+	$7.25\times {10}^{-2}$ ($1.04\times {10}^{-1}$)+	$0.00\times {10}^0$ ($0.00\times {10}^0$)+	$2.21\times {10}^{-1}$ ($1.04\times {10}^{-3}$)+	$2.61\times {10}^{-1}$ ($1.45\times {10}^{-2}$)+	$2.73\times {10}^{-1}$ ($1.86\times {10}^{-2}$)+	$2.17\times {10}^{-1}$ ($1.52\times {10}^{-3}$)+	$2.93\times {10}^{-1}$ ($1.88\times {10}^{-4}$)
LIRCMOP9	$1.05\times {10}^{-1}$ ($2.38\times {10}^{-2}$)+	$1.17\times {10}^{-1}$ ($3.78\times {10}^{-2}$)+	$2.82\times {10}^{-1}$ ($3.89\times {10}^{-2}$)+	$2.21\times {10}^{-1}$ ($3.62\times {10}^{-2}$)+	$4.27\times {10}^{-1}$ ($4.33\times {10}^{-2}$)+	$4.69\times {10}^{-1}$ ($3.47\times {10}^{-2}$)+	$3.30\times {10}^{-1}$ ($5.01\times {10}^{-2}$)+	$5.37\times {10}^{-1}$ ($1.14\times {10}^{-3}$)
LIRCMOP10	$7.46\times {10}^{-2}$ ($2.09\times {10}^{-2}$)+	$7.37\times {10}^{-2}$ ($2.54\times {10}^{-2}$)+	$4.63\times {10}^{-1}$ ($1.65\times {10}^{-2}$)+	$3.04\times {10}^{-1}$ ($3.71\times {10}^{-2}$)+	$5.65\times {10}^{-1}$ ($6.31\times {10}^{-2}$)+	$6.34\times {10}^{-1}$ ($4.02\times {10}^{-2}$)+	$4.68\times {10}^{-1}$ ($7.50\times {10}^{-2}$)+	$6.99\times {10}^{-1}$ ($3.96\times {10}^{-4}$)
LIRCMOP11	$1.99\times {10}^{-1}$ ($5.27\times {10}^{-2}$)+	$2.74\times {10}^{-1}$ ($9.39\times {10}^{-2}$)+	$5.56\times {10}^{-1}$ ($7.68\times {10}^{-3}$)+	$4.00\times {10}^{-1}$ ($2.63\times {10}^{-2}$)+	$6.36\times {10}^{-1}$ ($4.30\times {10}^{-2}$)+	$6.80\times {10}^{-1}$ ($1.32\times {10}^{-2}$)+	$4.67\times {10}^{-1}$ ($6.32\times {10}^{-2}$)+	$6.92\times {10}^{-1}$ ($2.81\times {10}^{-4}$)
LIRCMOP12	$2.48\times {10}^{-1}$ ($7.92\times {10}^{-2}$)+	$3.15\times {10}^{-1}$ ($1.24\times {10}^{-1}$)+	$4.15\times {10}^{-1}$ ($3.56\times {10}^{-2}$)+	$4.23\times {10}^{-1}$ ($2.29\times {10}^{-2}$)+	$5.53\times {10}^{-1}$ ($3.32\times {10}^{-2}$)+	$5.91\times {10}^{-1}$ ($1.81\times {10}^{-2}$)+	$4.96\times {10}^{-1}$ ($4.16\times {10}^{-2}$)+	$6.17\times {10}^{-1}$ ($6.00\times {10}^{-4}$)
LIRCMOP13	$2.59\times {10}^{-4}$ ($1.60\times {10}^{-4}$)+	$1.03\times {10}^{-4}$ ($1.29\times {10}^{-4}$)+	$5.43\times {10}^{-1}$ ($1.93\times {10}^{-3}$)+	$5.53\times {10}^{-1}$ ($7.42\times {10}^{-4}$)+	$5.34\times {10}^{-1}$ ($7.93\times {10}^{-3}$)+	$5.53\times {10}^{-1}$ ($2.38\times {10}^{-3}$)+	$4.30\times {10}^{-4}$ ($2.19\times {10}^{-5}$)+	$5.59\times {10}^{-1}$ ($9.16\times {10}^{-4}$)
LIRCMOP14	$6.74\times {10}^{-4}$ ($2.88\times {10}^{-4}$)+	$4.11\times {10}^{-4}$ ($2.84\times {10}^{-4}$)+	$5.45\times {10}^{-1}$ ($1.13\times {10}^{-3}$)+	$5.53\times {10}^{-1}$ ($7.02\times {10}^{-4}$)+	$5.48\times {10}^{-1}$ ($2.04\times {10}^{-3}$)+	$5.55\times {10}^{-1}$ ($1.79\times {10}^{-3}$)−	$9.67\times {10}^{-4}$ ($3.70\times {10}^{-5}$)+	$5.55\times {10}^{-1}$ ($1.15\times {10}^{-3}$)
+/−/=	14/0/0	14/0/0	14/0/0	14/0/0	13/1/0	13/1/0	14/0/0
MW1	NaN (3.33%)+	NaN (60.00%)+	$4.85\times {10}^{-1}$ ($4.40\times {10}^{-3}$)+	NaN (90.00%)+	NaN (30.03%)+	NaN (90.00%)+	NaN (30.00%)+	$4.90\times {10}^{-1}$ ($9.81\times {10}^{-5}$)
MW2	$4.84\times {10}^{-1}$ ($7.50\times {10}^{-2}$)+	$5.43\times {10}^{-1}$ ($8.17\times {10}^{-3}$)+	$4.99\times {10}^{-1}$ ($2.49\times {10}^{-2}$)+	$4.89\times {10}^{-1}$ ($4.14\times {10}^{-2}$)+	$4.59\times {10}^{-1}$ ($8.65\times {10}^{-2}$)+	$5.66\times {10}^{-1}$ ($8.12\times {10}^{-3}$)+	$5.19\times {10}^{-1}$ ($4.28\times {10}^{-2}$)+	$5.82\times {10}^{-1}$ ($8.06\times {10}^{-5}$)
MW3	$3.94\times {10}^{-1}$ ($2.30\times {10}^{-1}$)+	$5.40\times {10}^{-1}$ ($2.10\times {10}^{-3}$)+	$5.41\times {10}^{-1}$ ($8.31\times {10}^{-4}$)+	$5.40\times {10}^{-1}$ ($8.60\times {10}^{-4}$)+	$5.38\times {10}^{-1}$ ($1.79\times {10}^{-3}$)+	$5.43\times {10}^{-1}$ ($8.44\times {10}^{-4}$)−	$5.42\times {10}^{-1}$ ($1.16\times {10}^{-3}$)+	$5.42\times {10}^{-1}$ ($3.88\times {10}^{-4}$)
MW4	NaN (16.81%)+	NaN (33.33%)+	$8.34\times {10}^{-1}$ ($3.31\times {10}^{-3}$)+	$8.14\times {10}^{-1}$ ($1.05\times {10}^{-1}$)=	NaN (60.00%)+	NaN (96.67%)+	NaN (26.67%)+	$8.40\times {10}^{-1}$ ($6.39\times {10}^{-4}$)
MW5	NaN (26.70%)+	NaN (80.00%)+	$2.94\times {10}^{-1}$ ($3.56\times {10}^{-2}$)+	$2.92\times {10}^{-1}$ ($6.93\times {10}^{-2}$)+	NaN (43.33%)+	$3.24\times {10}^{-1}$ ($7.04\times {10}^{-4}$)+	NaN (36.67%)+	$3.24\times {10}^{-1}$ ($9.33\times {10}^{-5}$)
MW6	$2.08\times {10}^{-1}$ ($8.76\times {10}^{-2}$)+	$2.89\times {10}^{-1}$ ($1.83\times {10}^{-2}$)+	$2.30\times {10}^{-1}$ ($6.02\times {10}^{-2}$)+	$2.13\times {10}^{-1}$ ($5.51\times {10}^{-2}$)+	$1.64\times {10}^{-1}$ ($8.58\times {10}^{-2}$)+	$3.23\times {10}^{-1}$ ($4.81\times {10}^{-3}$)+	$2.70\times {10}^{-1}$ ($3.25\times {10}^{-2}$)+	$3.28\times {10}^{-1}$ ($7.57\times {10}^{-5}$)
MW7	$3.58\times {10}^{-1}$ ($7.64\times {10}^{-2}$)+	$4.08\times {10}^{-1}$ ($1.53\times {10}^{-3}$)+	$4.06\times {10}^{-1}$ ($9.75\times {10}^{-4}$)+	$4.09\times {10}^{-1}$ ($6.88\times {10}^{-4}$)+	$4.07\times {10}^{-1}$ ($1.37\times {10}^{-3}$)+	$4.10\times {10}^{-1}$ ($1.01\times {10}^{-3}$)+	$4.10\times {10}^{-1}$ ($1.06\times {10}^{-3}$)+	$4.11\times {10}^{-1}$ ($4.22\times {10}^{-4}$)
MW8	$4.19\times {10}^{-1}$ ($9.84\times {10}^{-2}$)+	$5.28\times {10}^{-1}$ ($9.26\times {10}^{-3}$)+	$4.49\times {10}^{-1}$ ($6.18\times {10}^{-2}$)+	$4.56\times {10}^{-1}$ ($4.87\times {10}^{-2}$)+	NaN (96.67%)+	$5.23\times {10}^{-1}$ ($1.09\times {10}^{-2}$)+	$4.71\times {10}^{-1}$ ($6.76\times {10}^{-2}$)+	$5.49\times {10}^{-1}$ ($2.77\times {10}^{-3}$)
MW9	NaN (33.33%)+	NaN (76.67%)+	$0.00\times {10}^0$ ($0.00\times {10}^0$)+	$0.00\times {10}^0$ ($0.00\times {10}^0$)+	NaN (86.67%)+	$3.81\times {10}^{-1}$ ($5.84\times {10}^{-3}$)+	NaN (46.67%)+	$3.99\times {10}^{-1}$ ($9.88\times {10}^{-4}$)
MW10	NaN (90.00%)+	$3.98\times {10}^{-1}$ ($2.25\times {10}^{-2}$)+	$3.61\times {10}^{-1}$ ($4.96\times {10}^{-2}$)+	$3.02\times {10}^{-1}$ ($6.78\times {10}^{-2}$)+	NaN (86.67%)+	$4.37\times {10}^{-1}$ ($9.87\times {10}^{-3}$)+	$3.74\times {10}^{-1}$ ($4.27\times {10}^{-2}$)+	$4.55\times {10}^{-1}$ ($1.23\times {10}^{-4}$)
MW11	$2.90\times {10}^{-1}$ ($5.32\times {10}^{-2}$)+	$4.34\times {10}^{-1}$ ($4.30\times {10}^{-2}$)=	$4.41\times {10}^{-1}$ ($7.71\times {10}^{-4}$)+	$4.47\times {10}^{-1}$ ($1.75\times {10}^{-4}$)+	$4.47\times {10}^{-1}$ ($1.80\times {10}^{-4}$)+	$4.48\times {10}^{-1}$ ($2.33\times {10}^{-4}$)=	$4.25\times {10}^{-1}$ ($5.22\times {10}^{-2}$)+	$4.48\times {10}^{-1}$ ($1.44\times {10}^{-4}$)
MW12	NaN (46.67%)+	NaN (96.67%)+	$2.10\times {10}^{-3}$ ($6.43\times {10}^{-3}$)+	$0.00\times {10}^0$ ($0.00\times {10}^0$)+	NaN (86.67%)+	$6.04\times {10}^{-1}$ ($3.76\times {10}^{-4}$)+	NaN (63.33%)+	$6.05\times {10}^{-1}$ ($1.24\times {10}^{-4}$)
MW13	$3.49\times {10}^{-1}$ ($5.90\times {10}^{-2}$)+	$4.30\times {10}^{-1}$ ($1.45\times {10}^{-2}$)+	$4.18\times {10}^{-1}$ ($1.84\times {10}^{-2}$)+	$3.89\times {10}^{-1}$ ($3.80\times {10}^{-2}$)+	$3.54\times {10}^{-1}$ ($7.51\times {10}^{-2}$)+	$4.62\times {10}^{-1}$ ($4.68\times {10}^{-3}$)+	$3.92\times {10}^{-1}$ ($3.59\times {10}^{-2}$)+	$4.77\times {10}^{-1}$ ($1.28\times {10}^{-4}$)
MW14	$3.72\times {10}^{-1}$ ($9.05\times {10}^{-2}$)+	$4.47\times {10}^{-1}$ ($2.79\times {10}^{-2}$)+	$4.05\times {10}^{-1}$ ($2.65\times {10}^{-2}$)+	$3.92\times {10}^{-1}$ ($3.07\times {10}^{-2}$)+	$4.36\times {10}^{-1}$ ($4.28\times {10}^{-2}$)+	$4.21\times {10}^{-1}$ ($3.45\times {10}^{-2}$)+	$4.36\times {10}^{-1}$ ($6.14\times {10}^{-3}$)+	$4.72\times {10}^{-1}$ ($1.92\times {10}^{-3}$)
+/−/=	14/0/0	13/0/1	14/0/0	13/0/1	14/0/0	12/1/1	14/0/0
RWMOP1	$6.07\times {10}^{-1}$ ($8.74\times {10}^{-4}$)+	$6.09\times {10}^{-1}$ ($1.02\times {10}^{-4}$)+	$6.05\times {10}^{-1}$ ($9.25\times {10}^{-4}$)+	$6.06\times {10}^{-1}$ ($2.17\times {10}^{-4}$)+	$6.06\times {10}^{-1}$ ($2.61\times {10}^{-4}$)+	$6.07\times {10}^{-1}$ ($1.45\times {10}^{-4}$)+	$1.08\times {10}^{-1}$ ($4.40\times {10}^{-5}$)+	$6.10\times {10}^{-1}$ ($1.61\times {10}^{-4}$)
RWMOP2	$8.93\times {10}^{-1}$ ($2.58\times {10}^{-4}$)+	$8.99\times {10}^{-1}$ ($5.33\times {10}^{-4}$)+	$8.77\times {10}^{-1}$ ($9.94\times {10}^{-3}$)+	$8.99\times {10}^{-1}$ ($4.44\times {10}^{-4}$)+	$8.97\times {10}^{-1}$ ($5.92\times {10}^{-4}$)+	$8.99\times {10}^{-1}$ ($4.56\times {10}^{-4}$)+	$8.95\times {10}^{-2}$ ($1.10\times {10}^{-5}$)+	$9.03\times {10}^{-1}$ ($6.07\times {10}^{-5}$)
RWMOP3	$2.73\times {10}^{-1}$ ($2.14\times {10}^{-3}$)+	$2.77\times {10}^{-1}$ ($1.00\times {10}^{-5}$)+	$1.76\times {10}^{-1}$ ($3.85\times {10}^{-2}$)+	$2.77\times {10}^{-1}$ ($3.80\times {10}^{-5}$)+	$2.76\times {10}^{-1}$ ($5.91\times {10}^{-5}$)+	$2.77\times {10}^{-1}$ ($4.45\times {10}^{-5}$)+	$2.71\times {10}^{-1}$ ($3.77\times {10}^{-3}$)+	$2.77\times {10}^{-1}$ ($8.05\times {10}^{-6}$)
RWMOP4	$4.84\times {10}^{-1}$ ($7.68\times {10}^{-4}$)+	$4.85\times {10}^{-1}$ ($2.77\times {10}^{-5}$)+	$4.84\times {10}^{-1}$ ($1.89\times {10}^{-4}$)+	$4.85\times {10}^{-1}$ ($3.51\times {10}^{-5}$)+	$4.85\times {10}^{-1}$ ($2.36\times {10}^{-5}$)+	$4.84\times {10}^{-1}$ ($3.41\times {10}^{-5}$)+	$4.81\times {10}^{-1}$ ($5.73\times {10}^{-4}$)+	$4.85\times {10}^{-1}$ ($3.38\times {10}^{-5}$)
RWMOP5	$2.54\times {10}^{-2}$ ($1.38\times {10}^{-4}$)+	$2.61\times {10}^{-2}$ ($1.70\times {10}^{-5}$)+	$2.59\times {10}^{-2}$ ($6.22\times {10}^{-4}$)+	$2.59\times {10}^{-2}$ ($4.57\times {10}^{-5}$)+	$2.58\times {10}^{-2}$ ($2.85\times {10}^{-5}$)+	$2.60\times {10}^{-2}$ ($3.66\times {10}^{-5}$)+	$7.17\times {10}^{-3}$ ($1.51\times {10}^{-4}$)+	$2.62\times {10}^{-2}$ ($2.15\times {10}^{-5}$)
RWMOP6	$4.10\times {10}^{-1}$ ($1.16\times {10}^{-4}$)+	$4.10\times {10}^{-1}$ ($4.96\times {10}^{-5}$)+	$4.08\times {10}^{-1}$ ($4.08\times {10}^{-4}$)+	$4.09\times {10}^{-1}$ ($1.09\times {10}^{-4}$)+	$4.09\times {10}^{-1}$ ($7.79\times {10}^{-5}$)+	$4.09\times {10}^{-1}$ ($6.73\times {10}^{-5}$)+	$5.30\times {10}^{-2}$ ($2.62\times {10}^{-5}$)+	$4.10\times {10}^{-1}$ ($6.06\times {10}^{-5}$)
+/−/=	6/0/0	6/0/0	6/0/0	6/0/0	6/0/0	6/0/0	6/0/0

, respectively. Standard deviations are provided in parentheses. Statistical significance was assessed using the Wilcoxon rank-sum test at a 0.05 significance level. In the tables, ‘+’, ‘−’, and ‘=’ indicate that the proposed AFFCMO is significantly superior to, inferior to, or statistically comparable to the competing algorithm, respectively. If an algorithm fails to find feasible solutions across all runs, its IGD and HV values are recorded as ‘NaN’. In such cases, the Feasible Solutions Ratio (FSR) is provided in parentheses to reflect the algorithm’s feasibility acquisition capability under extremely constrained conditions.

5.1. Comparison on CF Test Set

According to Table 2 and Table 3, AFFCMO exhibits highly competitive performance on the CF test suite, which is characterized by narrow feasible regions, disconnected feasible segments, and strong interference from infeasible barriers. These characteristics require the algorithm to simultaneously maintain boundary exploration capability and stable convergence toward the CPF. For the IGD metric, AFFCMO achieves the best mean performance on CF1 and CF3–CF8, indicating that it can more accurately approximate the CPF on most CF instances. In particular, on CF3–CF8, the superiority of AFFCMO suggests that the proposed dual-population mechanism is effective in traversing restrictive infeasible regions and preserving useful search directions near the feasible boundary. The main population promotes feasible-solution refinement, while the auxiliary population continuously explores infeasible and boundary regions, which helps the algorithm avoid premature concentration in a small feasible subregion. For the HV metric, AFFCMO obtains the best mean values on CF1–CF3 and CF5–CF7, while remaining statistically comparable to the best-performing competitor on CF4 and CF8. These results indicate that the proposed framework not only improves convergence quality but also maintains a sufficiently wide and well-distributed solution set on most CF problems. This behavior can be attributed to the continuous environmental selection strategy in the auxiliary population, which preserves informative near-feasible solutions, and to the adaptive offspring allocation mechanism, which adjusts the search emphasis between exploitation and exploration according to historical search feedback. By contrast, several compared algorithms show clear difficulties on CF instances with highly restrictive feasible regions. For example, ANSGA-III and BiCo fail to obtain feasible non-dominated solutions on CF8 and CF10, which implies limited adaptability when the feasible region becomes extremely narrow. Although CTAEA, CTSEA, and MOEA/D-2WA can locate feasible solutions on most CF problems, their IGD and HV values are generally inferior to those of AFFCMO, indicating that they are less effective in simultaneously maintaining convergence accuracy and population diversity under complex feasibility constraints. It should also be noted that AFFCMO is not uniformly dominant on every CF instance. For example, CMOQLMT achieves slightly better mean IGD values on CF2 and CF9, and shows comparable or better HV performance on several difficult cases. In addition, CF10 remains a particularly challenging problem for all compared algorithms, since its feasible region is extremely discrete and difficult to approach reliably. Under such a landscape, AFFCMO still fails to obtain feasible non-dominated solutions, which suggests that there is still room for improvement when dealing with extremely fragmented feasible structures. Overall, the results on the CF suite demonstrate that AFFCMO provides a better balance between convergence, diversity, and feasibility on most instances, especially when the search must cross restrictive infeasible regions and maintain effective exploration near narrow feasible boundaries.

5.2. Comparison on DAS-CMOP Test Set

The results reported in Table 2 and Table 3 show that AFFCMO achieves highly competitive performance on the DAS-CMOP test suite. This suite contains problems with complex feasible structures and strong constraint interference, which requires the algorithm to maintain effective exploration across infeasible regions while preserving stable convergence toward the feasible front. For the IGD metric, AFFCMO obtains the best mean results on DAS-CMOP1 and DAS-CMOP3–DAS-CMOP8, and remains highly competitive on the remaining instances. In particular, its clear advantage on DAS-CMOP3–DAS-CMOP8 indicates that the proposed framework is effective in handling complex feasible-region geometries and maintaining convergence accuracy under strong constraint pressure. This behavior can be attributed to the complementary cooperation of the two populations: the auxiliary population maintains exploration near infeasible and boundary regions, while the main population concentrates on feasible-solution refinement once promising regions are identified. For the HV metric, AFFCMO achieves the best mean values on DAS-CMOP1 and DAS-CMOP3–DAS-CMOP8, and remains statistically very competitive on DAS-CMOP2 and DAS-CMOP9. These results suggest that AFFCMO not only improves convergence quality but also preserves a relatively broad and well-distributed solution set on most DAS-CMOP instances. In particular, the continuous environmental selection strategy in the auxiliary population helps retain informative near-feasible solutions, while the adaptive resource allocation mechanism allows the search emphasis to be adjusted according to the observed feasibility-improvement capability of the two populations. In contrast, several compared algorithms show clear performance degradation on difficult DAS-CMOP instances. ANSGA-III and BiCo fail to obtain feasible non-dominated solutions on some problems, and MOEA/D-2WA also exhibits poor robustness on instances with strong constraint interference. Although CTAEA and CTSEA can generally produce feasible solutions across the suite, their IGD and HV values are still inferior to those of AFFCMO in most instances, indicating weaker coordination between convergence improvement and diversity maintenance. These observations suggest that, when the feasible region is strongly constrained or irregularly distributed, maintaining feasibility alone is insufficient unless the search can also preserve useful boundary exploration. It should also be noted that AFFCMO is not strictly dominant on all DAS-CMOP instances. For example, CMOQLMT achieves slightly better mean IGD and HV values on DAS-CMOP2, and also remains competitive on DAS-CMOP9. This indicates that some competing algorithms may still perform well on particular instances where their search bias is well matched to the problem structure. Nevertheless, on the majority of DAS-CMOP problems, AFFCMO demonstrates a more favorable balance among feasibility acquisition, convergence accuracy, and solution-set diversity. Overall, the results on the DAS-CMOP suite show that AFFCMO is particularly effective on problems where the search must cross complex infeasible regions and simultaneously maintain a stable approximation of the feasible front.

5.3. Comparison on LIR-CMOP Test Set

The results in Table 2 and Table 3 show that AFFCMO achieves highly competitive performance on the LIR-CMOP test suite. This suite is characterized by large infeasible regions and strong spatial separation between feasible areas, which makes it difficult for an algorithm to reach the feasible boundary unless sufficient exploration can be maintained in infeasible space. Under such conditions, an effective balance between infeasible-region exploration and feasible-solution refinement becomes particularly important. For the IGD metric, AFFCMO obtains the best mean values on LIR-CMOP1 and LIR-CMOP3–LIR-CMOP13, and remains highly competitive on the remaining instances. These results indicate that AFFCMO can more accurately approximate the CPF on most LIR-CMOP problems, especially when the search must cross large infeasible areas before approaching the feasible front. This advantage is closely related to the complementary roles of the two populations: the auxiliary population continuously explores infeasible and boundary regions, while the main population focuses on convergence once useful search directions have been identified. For the HV metric, AFFCMO achieves the best mean performance on LIR-CMOP1 and LIR-CMOP3–LIR-CMOP13, and remains very competitive on the other instances. This result suggests that the proposed framework can maintain a broad and well-distributed solution set while improving convergence accuracy. In particular, the continuous environmental selection strategy in the auxiliary population helps preserve informative near-feasible solutions across large infeasible regions, and the adaptive resource allocation strategy further improves the coordination between exploration and exploitation during different evolutionary stages. By contrast, several compared algorithms show clear weaknesses on the LIR-CMOP suite. ANSGA-III, BiCo, CTAEA, CTSEA, and MOEA/D-2WA generally obtain much worse IGD and HV values, indicating that these methods are less capable of extracting useful guidance from large infeasible regions. When the feasible set is spatially isolated, such methods tend to either stagnate in infeasible space or converge prematurely after entering only a limited part of the feasible region. Although CMOQLMT and CMOEMT remain competitive on a small number of instances, their overall performance across the suite is still less stable than that of AFFCMO. It should also be noted that AFFCMO is not uniformly dominant on every LIR-CMOP instance. For example, CMOEMT yields a slightly better mean IGD value on LIR-CMOP2, and CMOQLMT achieves a marginally better HV value on LIR-CMOP14. These observations indicate that some competing algorithms may still have advantages on individual instances whose feasible structures better match their search bias. Nevertheless, on the majority of the LIR-CMOP problems, AFFCMO demonstrates a more favorable balance among feasibility acquisition, convergence accuracy, and solution-set diversity. Overall, the results on the LIR-CMOP suite show that AFFCMO is particularly effective for CMOPs with large infeasible regions and isolated feasible structures, where sustained exploration in infeasible space is essential for subsequent convergence to the CPF.

5.4. Comparison on MW Test Set

The MW test suite is characterized by highly nonlinear, discontinuous, and fragmented feasible regions, which place strong demands on both feasibility exploration and diversity preservation. As shown in Table 2 and Table 3, AFFCMO achieves highly competitive performance on this suite. Such problems are particularly difficult because an algorithm must not only locate feasible regions embedded in a discontinuous search space, but also maintain sufficient exploration capability to avoid being trapped in isolated feasible fragments. For the IGD metric, AFFCMO achieves the best mean performance on MW1, MW2, MW4–MW6, and MW8–MW14, indicating that it can more accurately approximate the CPF on most MW problems. This advantage suggests that the proposed framework is effective in maintaining useful exploration near fragmented feasible regions while still preserving convergence pressure toward the feasible front. In particular, the auxiliary population helps explore infeasible and boundary regions continuously, whereas the main population refines the feasible solutions once promising directions are identified. For the HV metric, AFFCMO obtains the best mean values on MW1, MW2, MW4–MW10, and MW12–MW14, which further indicates that the proposed algorithm can preserve a relatively broad and well-distributed solution set under discontinuous feasibility conditions. This behavior is closely related to the continuous environmental selection mechanism in the auxiliary population, which retains informative near-feasible solutions, and to the adaptive resource allocation strategy, which adjusts the offspring sizes of the two populations according to their observed search contributions at different evolutionary stages. In contrast, several compared algorithms show clear performance degradation on the MW suite. ANSGA-III, BiCo, CMOEMT, and MOEA/D-2WA frequently fail to obtain feasible non-dominated solutions on instances such as MW1, MW4, MW5, MW9, MW10, and MW12, indicating limited robustness when the feasible region becomes highly discontinuous or fragmented. Although CTAEA and CTSEA can generally produce feasible solutions, their IGD and HV values are still inferior to those of AFFCMO in most instances, which suggests that they are less effective in balancing convergence improvement and diversity maintenance in such irregular feasible landscapes. It should also be noted that AFFCMO is not strictly dominant on every MW instance. For example, CMOQLMT achieves a slightly better mean IGD value on MW3 and remains competitive on MW11. These results imply that some competing algorithms may still perform well on individual problems whose feasible structure is more compatible with their search bias. Nevertheless, across the majority of the MW instances, AFFCMO shows a more favorable balance among feasibility acquisition, convergence accuracy, and solution-set diversity. Overall, the results on the MW suite indicate that AFFCMO is particularly effective on CMOPs with fragmented and discontinuous feasible regions, where successful optimization requires sustained infeasible-region exploration together with stable feasible-solution refinement.

5.5. Comparison on Real-World Engineering Problems

To further evaluate the practical applicability of AFFCMO, six real-world engineering design problems from the RWMOP suite were considered, including pressure vessel design, two-bar truss design, speed reducer design, gear train design, car side impact design, and four-bar plane truss design. Compared with benchmark instances, these engineering problems usually involve stronger variable coupling, more explicit physical constraints, and more application-oriented trade-offs among objectives. Therefore, they provide a useful complement to synthetic test suites when assessing the robustness and practical effectiveness of constrained multiobjective evolutionary algorithms. According to Table 2 and Table 3, AFFCMO exhibits highly competitive performance on the real-world engineering problems. For the IGD metric, AFFCMO achieves the best mean values on RWMOP2, RWMOP3, RWMOP4, RWMOP5, and RWMOP6, while remaining competitive on RWMOP1. These results indicate that the proposed framework can effectively approximate the CPF on most engineering problems, especially when feasible solutions must be progressively refined under nonlinear and tightly coupled design constraints. In particular, the clear advantage of RWMOP2, RWMOP4, and RWMOP5 suggests that the proposed dual-population mechanism is able to maintain useful exploration while steadily improving feasible-solution quality in engineering search spaces with complex structural restrictions. For the HV metric, AFFCMO obtains the best or statistically competitive results on all six engineering problems, which indicates that the proposed framework can preserve a relatively broad and well-distributed solution set while maintaining good convergence quality. This behavior can be attributed to the complementary cooperation of the two populations. The auxiliary population continuously explores infeasible and boundary regions to provide additional search directions, whereas the main population focuses on feasible-region exploitation and solution refinement. In addition, the adaptive resource allocation strategy further improves the coordination between these two search roles according to their observed search contributions during evolution. By contrast, several compared algorithms show unstable behavior on some engineering problems. For example, MOEA/D-2WA performs poorly on RWMOP1 and RWMOP5 in terms of IGD and also exhibits much worse HV values on multiple problems, indicating limited robustness under practical engineering constraints. Similarly, CTAEA and CTSEA can achieve competitive results in some instances, but their overall performance is still less stable than that of AFFCMO across the six engineering problems. These observations imply that, in real-world design tasks, merely maintaining feasible solutions is insufficient unless the search can also preserve effective exploration near the boundary of the feasible region. It should also be noted that AFFCMO is not uniformly dominant on every engineering instance. For example, on RWMOP1, the IGD value of ANSGA-III is slightly better, which suggests that some existing methods may still have advantages on particular engineering problems whose feasible structures are relatively regular. Nevertheless, from the overall results on the six engineering problems, AFFCMO demonstrates a more favorable balance among feasibility acquisition, convergence accuracy, and solution-set diversity. Overall, the results on the RWMOP suite confirm that AFFCMO is not only effective on synthetic benchmark problems, but also shows strong potential for handling practical constrained multiobjective engineering design problems.

5.6. Convergence Speed Analysis

Figure 2 and Figure 3 present the IGD and HV convergence curves, respectively, on a set of representative test instances selected from the benchmark suites. These instances were chosen because they cover several typical difficulties in CMOPs, including narrow feasible regions, disconnected or segmented feasible structures, and large infeasible search areas. Plotting convergence curves for all test problems in a single figure would substantially reduce readability, whereas the selected instances provide a concise yet representative view of the dynamic search behavior of the compared algorithms. From the IGD curves in Figure 2, AFFCMO generally exhibits a competitive convergence trend across the selected instances. In particular, on problems with restrictive feasible boundaries or severe infeasible barriers, the proposed framework is able to maintain effective exploration in the early and middle stages and then gradually accelerate convergence toward the CPF. This behavior can be clearly observed in instances such as CF3, DASCMOP1, DASCMOP2, and LIRCMOP10, where AFFCMO shows a pronounced reduction in IGD after sufficient search information has been accumulated. Such a dynamic pattern suggests that the proposed framework does not rely on overly aggressive early exploitation, but instead preserves useful exploration capability before strengthening feasible-solution refinement. The HV curves in Figure 3 further support this observation. In most selected instances, AFFCMO not only improves convergence quality but also maintains a relatively broad and stable coverage of the objective space during evolution. This behavior is particularly evident on DASCMOP1, LIRCMOP11, MW10, MW13, and MW14, where the HV values of AFFCMO increase rapidly and then remain consistently high. These results indicate that the proposed framework can simultaneously improve convergence and preserve distribution quality, rather than achieving one at the expense of the other. The above convergence behavior can be explained by the complementary cooperation between the two populations. The auxiliary population continuously explores infeasible and boundary regions, which helps preserve useful search directions in difficult constrained landscapes, while the main population focuses on the refinement of feasible solutions once promising regions have been identified. In addition, the adaptive resource allocation mechanism allows the algorithm to dynamically regulate the search emphasis between exploration and exploitation according to historical search feedback. As a result, AFFCMO is less likely to stagnate prematurely on fragmented or highly constrained problems. By contrast, several compared algorithms either converge slowly throughout the whole process or exhibit clear stagnation after an initial improvement. For example, some algorithms can identify feasible solutions in the early stage, but then show only marginal progress in either IGD or HV during later evolution. This indicates that, under difficult feasibility conditions, early feasibility acquisition alone is insufficient unless the search can also maintain sustained progress toward a well-converged and well-distributed approximation of the CPF. Overall, the convergence curves in Figure 2 and Figure 3 provide dynamic evidence that AFFCMO can maintain a better balance between infeasible-region exploration and feasible-solution refinement, which is consistent with its competitive final IGD and HV results on the benchmark suites.

5.7. Distribution Analysis

To further examine the distribution quality of the obtained solution sets, Figure 4 provides a visual comparison of eight representative instances selected from different benchmark suites, namely CF4, CF6, DASCMOP6, DASCMOP9, LIRCMOP6, LIRCMOP14, MW6, and MW14. These instances were selected because they cover several typical difficulties in CMOPs, including narrow feasible manifolds, disconnected feasible structures, curved feasible boundaries, and multi-segment Pareto fronts. Therefore, they can provide intuitive evidence regarding whether an algorithm can simultaneously maintain convergence quality and solution-set diversity. As shown in Figure 4, AFFCMO generally produces a more complete and better distributed approximation of the feasible Pareto front on most selected instances. On CF4 and CF6, the solution set obtained by AFFCMO follows the front more continuously and uniformly than most compared algorithms, indicating that the proposed framework can preserve distribution quality even when the feasible region is narrow. On DASCMOP9 and LIRCMOP14, where the feasible front exhibits more complex geometric structures, AFFCMO still maintains a relatively even spread of solutions over the accessible regions, whereas several compared algorithms show obvious aggregation or partial coverage. A similar trend can also be observed on MW14, which contains multiple separated feasible segments. AFFCMO covers the different disconnected parts more adequately than most competing methods, suggesting that the proposed dual-population cooperation mechanism is effective in preserving exploration capability across spatially separated feasible regions. On LIRCMOP6 and MW6, although all algorithms are challenged by the restrictive feasible structures, AFFCMO still maintains a comparatively stable boundary-following behavior without showing severe clustering. These observations are consistent with the role of the auxiliary population, which continuously explores infeasible and boundary regions and thus helps provide useful search directions for the main population. By contrast, several compared algorithms tend to suffer from one or more of the following deficiencies: incomplete front coverage, strong local aggregation, or failure to reach some disconnected feasible segments. This phenomenon indicates that merely obtaining feasible solutions is not sufficient when the feasible front is highly fragmented or geometrically constrained. Instead, the algorithm must also preserve effective exploration near infeasible and boundary regions so that diversity can be maintained during the convergence process. Overall, the visual results in Figure 4 provide additional evidence that AFFCMO achieves a more favorable balance between convergence and diversity on difficult CMOPs. This observation is also consistent with the competitive IGD and HV results reported in the previous subsections.

5.8. Statistical Significance Test

Table 4. Wilcoxon signed rank test of IGD.

AFFCMO VS	${\mathit{R}}^{+}$	${\mathit{R}}^{-}$	p	Level = 0.05
ANSGAIII	1328	50	$3.03593\times {10}^{-9}$	YES
BiCo	1229	46	$5.81416\times {10}^{-9}$	YES
CTAEA	1378	0	$1.80394\times {10}^{-10}$	YES
CTSEA	1325	1	$2.81334\times {10}^{-10}$	YES
CMOEMT	1310	16	$6.80835\times {10}^{-10}$	YES
CMOQLMT	1199	127	$2.59042\times {10}^{-7}$	YES
MOEAD2WA	1419	12	$2.43102\times {10}^{-10}$	YES

and

Table 5. Wilcoxon signed rank test of HV.

AFFCMO VS	${\mathit{R}}^{+}$	${\mathit{R}}^{-}$	p	Level = 0.05
ANSGAIII	1378	0	$1.80394\times {10}^{-10}$	YES
BiCo	1372	6	$2.55855\times {10}^{-10}$	YES
CTAEA	1378	0	$1.80394\times {10}^{-10}$	YES
CTSEA	1402	29	$6.27726\times {10}^{-10}$	YES
CMOEMT	1369	62	$3.71609\times {10}^{-9}$	YES
CMOQLMT	1253	178	$9.97478\times {10}^{-7}$	YES
MOEAD2WA	1377	54	$2.4332\times {10}^{-9}$	YES

report the Wilcoxon signed-rank test results for pairwise comparisons between AFFCMO and the seven representative algorithms on the IGD and HV metrics, respectively. The statistical test was conducted over all 53 test instances considered in this study, including 47 benchmark problems and 6 real-world engineering problems. For each pairwise comparison, the performance difference on each test instance was first calculated. Instances with zero difference were excluded, and the absolute values of the remaining paired differences were ranked in ascending order. The rank sum corresponding to the instances on which AFFCMO performed better is reported as $R^{+}$, whereas the rank sum corresponding to the instances on which AFFCMO performed worse is reported as $R^{-}$. Therefore, a significantly larger $R^{+}$ than $R^{-}$ indicates that AFFCMO outperforms the compared algorithm on most test instances after considering both the direction and the magnitude of the paired performance differences.

For the IGD metric, AFFCMO shows consistently strong statistical superiority over all seven compared algorithms. In particular, the values of $R^{+}$ are much larger than those of $R^{-}$ in all pairwise comparisons. For example, against CTAEA, CTSEA, and MOEA/D-2WA, the corresponding $\left(R^{+},R^{-}\right)$ pairs are $(1378,0)$, $(1325,1)$, and $(1419,12)$, respectively, which clearly indicates that AFFCMO achieves better convergence performance on the overwhelming majority of test instances. Compared with the strong competitor CMOQLMT, AFFCMO still maintains a substantially larger $R^{+}$ than $R^{-}$, which indicates a stable overall advantage in convergence accuracy across diverse problem instances. The HV results exhibit a similar pattern. In all pairwise comparisons, AFFCMO consistently obtains much larger positive rank sums than negative rank sums, showing that the proposed framework also has a clear statistical advantage in terms of solution-set quality and distribution. For example, against ANSGAIII and CTAEA, the corresponding results are $(1378,0)$ and $(1378,0)$, respectively, indicating that AFFCMO achieves better HV values on nearly all nonzero paired instances. Even in the comparisons against more competitive algorithms such as CMOQLMT, the value of $R^{+}$ remains much larger than that of $R^{-}$, further supporting the robustness of AFFCMO in maintaining both convergence and diversity.

In addition, all reported p-values are below the significance level of $0.05$, and most of them are several orders of magnitude smaller. This provides strong statistical evidence that the observed performance differences are unlikely to be caused by random fluctuation. Overall, the Wilcoxon signed-rank test results confirm that AFFCMO has a statistically significant advantage over the compared algorithms on the combined set of benchmark and real-world engineering problems.

5.9. Runtime Analysis

Table 6. Runtime comparison of AFFCMO and the competing algorithms on different test suites (unit: seconds).

Algorithm	CF	DAS-CMOP	LIR-CMOP	MW	RWMOP	Overall
ANSGAIII	12.580	12.231	12.960	14.172	2.085	11.854
BiCo	4.548	14.481	29.014	8.204	28.549	16.380
CTAEA	49.770	51.486	47.327	50.853	61.789	51.063
CTSEA	15.213	5.902	13.792	18.740	31.915	16.079
CMOEMT	50.391	62.508	123.937	117.458	213.636	108.072
CMOQLMT	29.218	37.370	97.313	28.495	107.417	57.252
MOEAD2WA	22.896	95.970	111.151	119.950	24.632	84.451
AFFCMO	16.063	17.623	18.316	19.624	11.934	17.396

reports the average runtime of AFFCMO and the compared algorithms on the CF, DAS-CMOP, LIR-CMOP, MW, and RWMOP test suites. These results indicate that, although AFFCMO is not the fastest algorithm among the compared methods, its computational cost remains within a moderate range and does not exhibit excessive runtime overhead.

From the perspective of individual test suites, AFFCMO shows relatively stable runtime behavior. On the CF, DAS-CMOP, LIR-CMOP, and MW suites, its average runtime remains in a narrow interval from 16.063 to 19.624 s, suggesting that the proposed framework has consistent computational behavior across different classes of benchmark problems. On the real-world engineering problems, the average runtime of AFFCMO is 11.934 s, which is also clearly lower than that of several compared algorithms. This observation indicates that the proposed dual-population structure does not lead to prohibitive computational cost even when practical engineering constraints are involved.

The runtime characteristics of AFFCMO can be explained by its algorithmic design. On the one hand, the auxiliary environmental selection mechanism requires pairwise distance calculations in the normalized objective space, which introduces an $O\left(N^2\right)$ computational component. On the other hand, the adaptive resource allocation mechanism itself is lightweight, and the dual-population cooperation does not cause an explosive increase in computational burden. When the runtime results are considered together with the superior IGD and HV performance of AFFCMO on most benchmark and engineering problems, the additional computational cost can be regarded as a reasonable trade-off for improved convergence accuracy, solution-set diversity, and feasibility acquisition. Overall, the runtime results show that AFFCMO achieves competitive optimization performance without incurring an unacceptable practical time cost.

5.10. Ablation Study

To examine the contribution of the major components in AFFCMO, four variant algorithms, denoted as AFFCMO-A to AFFCMO-D, were constructed for ablation analysis. The detailed settings of these variants are listed in

Table 7. Description of Experimental Variants for Ablation Study.

Algorithm	Description
AFFCMO-A	Replaces the auxiliary population’s environmental selection mechanism with the CDP model.
AFFCMO-B	Uses only DE/current-to-pbest/1 to generate offspring for the auxiliary population.
AFFCMO-C	Uses only DE/rand/1 to generate offspring for the auxiliary population.
AFFCMO-D	Allocates offspring sizes equally between the main and auxiliary populations.

, and the corresponding statistical results based on the IGD and HV metrics are reported in

Table 8. Statistical Results of Ablation Experiments.

AFFCMO VS	IGD	p-Value	Significance	HV	p-Value	Significance
AFFCMO-A	40/2/5	0.000153	YES	40/1/6	0.000122	YES
AFFCMO-B	44/1/2	0.000000	YES	45/0/2	0.000000	YES
AFFCMO-C	44/1/2	0.000000	YES	45/1/1	0.000002	YES
AFFCMO-D	41/2/4	0.000020	YES	43/2/2	0.000010	YES

. Overall, AFFCMO achieves better performance than all four variants on the majority of test instances, which confirms the effectiveness of the proposed framework design. More specifically, AFFCMO-A replaces the proposed auxiliary environmental selection mechanism with the conventional CDP-based strategy. Its clear performance degradation indicates that the proposed continuous auxiliary selection mechanism is more effective in preserving informative near-feasible solutions while maintaining diversity in the objective space. AFFCMO-B and AFFCMO-C use only DE/current-to-pbest/1 or only DE/rand/1 in the auxiliary population, respectively. Compared with these two variants, AFFCMO obtains better IGD results on 44 instances, which suggests that a stage-adaptive variation strategy is more suitable for balancing early convergence and later exploration in complex constrained search spaces. AFFCMO-D adopts a fixed offspring allocation ratio between the two populations. Its inferior performance compared with AFFCMO indicates that the adaptive resource allocation strategy can more effectively coordinate the search emphasis between the main and auxiliary populations across different evolutionary stages. In addition, the Wilcoxon rank-sum test results show that all p-values are far below the 0.05 significance level for both IGD and HV, further supporting that the performance improvements brought by the proposed components are statistically significant. Therefore, the ablation results confirm that the effectiveness of AFFCMO does not rely on a single module, but on the coordinated contribution of auxiliary environmental selection, stage-adaptive variation, and adaptive resource allocation.

5.11. Sensitivity Analysis of Parameter

Table 9. Sensitivity analysis of parameter $\alpha$ around the default setting $\alpha =0.5$.

Value	CF	DAS-CMOP	LIR-CMOP	MW	Average Rank
0.2	7/2/1	6/2/1	10/3/1	9/3/2	4.6
0.4	5/2/3	4/2/3	7/2/5	6/3/5	2.4
0.6	4/3/3	4/2/3	6/3/5	5/4/5	2.6
0.8	6/2/2	5/2/2	8/3/3	7/4/3	3.5
1.0	8/1/1	7/1/1	11/2/1	10/2/2	4.9

reports the sensitivity analysis of the distribution regulation coefficient $\alpha$ in the auxiliary environmental selection mechanism. It can be observed that the performance of AFFCMO is relatively stable under moderate changes of $\alpha$, but excessively small or excessively large values tend to weaken the overall search effectiveness. In terms of the average rank, $\alpha =0.4$ and $\alpha =0.6$ achieve the best overall performance, with average ranks of 2.4 and 2.6, respectively, whereas $\alpha =0.2$ and $\alpha =1.0$ lead to clearly worse results. This indicates that the auxiliary environmental selection mechanism requires a proper balance between quality screening and distribution preservation. More specifically, when $\alpha$ is too small, the distance-based diversity term has only a limited effect on the selection process, so the auxiliary population becomes more likely to concentrate on a relatively narrow region of the objective space. As a result, the exploration capability near infeasible and boundary regions may be weakened, which is unfavorable for maintaining population spread on difficult CMOPs. By contrast, when $\alpha$ is too large, the environmental selection process may place excessive emphasis on spatial dispersion, thereby weakening the role of objective quality and constraint status in identifying informative near-feasible solutions. Such an imbalance can also reduce the overall effectiveness of the auxiliary population. Overall, the results suggest that AFFCMO is not overly sensitive to moderate variations of $\alpha$, and that the best performance is obtained when $\alpha$ is chosen in a medium range. Since the default setting $\alpha =0.5$ lies between the two best-performing neighboring values, namely 0.4 and 0.6, the adopted default configuration can be regarded as reasonable and robust.

6. Conclusions

To address the intricate feasible regions, heterogeneous constraint distributions, and strong objective conflicts in complex CMOPs, this article proposed AFFCMO, a dual-population coevolutionary framework that coordinates convergence, feasibility, and diversity through explicit role division between two interacting populations. In the proposed framework, the main population is responsible for the exploitation of feasible regions and the progressive approximation of the CPF by using the DE/current-to-pbest/1 operator, whereas the auxiliary population is used to maintain exploration in infeasible regions and around constraint boundaries through a stage-adaptive mutation strategy and a continuous environmental selection mechanism. In addition, an adaptive resource allocation strategy based on historical search feedback is introduced to dynamically adjust the offspring sizes of the two populations according to their observed search contributions. In this way, the proposed framework can adaptively regulate the search emphasis between convergence-oriented exploitation and exploration-oriented boundary search during evolution. Extensive experiments on 47 benchmark problems from the CF, DAS-CMOP, LIR-CMOP, and MW test suites, together with six real-world engineering design problems, verified the effectiveness of AFFCMO. The experimental results and statistical tests showed that AFFCMO exhibits competitive performance against seven state-of-the-art algorithms in terms of feasibility acquisition, convergence accuracy, and solution diversity. Moreover, the proposed framework maintained robust optimization performance on problems with narrow feasible regions, fragmented feasible structures, and large infeasible areas, while the ablation results further confirmed the contribution of its major components.

Future research will proceed in two directions. First, dimensionality reduction techniques and low-cost surrogate models will be investigated to reduce the computational burden of the proposed framework when handling high-dimensional or many-objective CMOPs. Second, AFFCMO will be further extended to broader and more challenging engineering scenarios, such as multi-constrained robot trajectory planning and large-scale vehicle scheduling systems, so as to further assess its practical applicability.