Open AccessArticle

A Chaotic Decomposition-Based Approach for Enhanced Multi-Objective Optimization

Javad Alikhani Koupaei

^1,* and

Mohammad Javad Ebadi

^2,3,*

Department of Mathematics, Payame Noor University, Tehran P.O. Box 19395-3697, Iran

Department of Mathematics, Chabahar Maritime University, Chabahar 9971778631, Iran

Department of Law, Economics and Human Sciences, Mediterranea University of Reggio Calabria, 89125 Reggio Calabria, Italy

Authors to whom correspondence should be addressed.

Mathematics 2025, 13(5), 817; https://doi.org/10.3390/math13050817

Submission received: 9 January 2025 / Revised: 23 February 2025 / Accepted: 25 February 2025 / Published: 28 February 2025

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Abstract

Multi-objective optimization problems often face challenges in balancing solution accuracy, computational efficiency, and convergence speed. Many existing methods struggle with achieving an optimal trade-off between exploration and exploitation, leading to premature convergence or excessive computational costs. To address these issues, this paper proposes a chaotic decomposition-based approach that leverages the ergodic properties of chaotic maps to enhance optimization performance. The proposed method consists of three key stages: (1) chaotic sequence initialization, which generates a diverse population to enhance the global search while reducing computational costs; (2) chaos-based correction, which integrates a three-point operator (TPO) and a local improvement operator (LIO) to refine the Pareto front and balance the exploration–exploitation trade-offs; and (3) Tchebycheff decomposition-based updating, ensuring efficient convergence toward optimal solutions. To validate the effectiveness of the proposed method, we conducted extensive experiments on a suite of benchmark problems and compared its performance with several state-of-the-art methods. The evaluation metrics, including inverted generational distance (IGD), generational distance (GD), and spacing (SP), demonstrated that the proposed method achieves competitive optimization accuracy and efficiency. While maintaining computational feasibility, our approach provides a well-balanced trade-off between exploration and exploitation, leading to improved solution diversity and convergence stability. The results establish the proposed algorithm as a promising alternative for solving multi-objective optimization problems.

Keywords:

multi-objective optimization; decomposition-based evolutionary algorithm; chaotic maps; chaotic optimization

MSC:

90C29; 49M27; 58E17; 68W50

1. Introduction

Multi-objective optimization (MOO) is a fundamental area in computational optimization that deals with problems involving multiple conflicting objective functions. These problems arise in various domains, including engineering, economics, machine learning, and operations research. The primary goal of MOO is to identify a set of Pareto optimal solutions where no single solution is strictly superior across all objectives. A Pareto optimal solution ensures that improving one objective results in the degradation of another, forming the Pareto front (PF), while the corresponding decision variables constitute the Pareto set (PS).

Despite significant advancements, solving MOO problems efficiently remains a challenge due to the complexity of balancing multiple objectives. The difficulty is further compounded by the need for algorithms that can simultaneously ensure accuracy, diversity, and convergence of the obtained solutions. Evolutionary algorithms (EAs), particularly multi-objective evolutionary algorithms (MOEAs), have been widely employed for MOO problems [1,2]. However, these approaches often suffer from high computational costs and difficulties in maintaining diversity within the solution set, which can hinder the convergence process.

Chaotic maps, due to their nonlinear characteristics and complex behaviors, have been widely utilized in various fields such as optimization, cryptography, and signal processing. Their ergodic properties enable them to generate pseudo-random sequences, which play a crucial role in enhancing both global and local search capabilities in optimization algorithms. Numerous studies have demonstrated that incorporating chaotic systems can improve the efficiency of evolutionary algorithms and enhance security in cryptographic systems. For instance, Tang et al. proposed a two-dimensional cosine–sine-based chaotic system for secure communications, highlighting the significant potential of chaotic systems in data security enhancement. Furthermore, a study by Tang et al. introduced a simple chaotic model with complex behaviors and examined its hardware implementation, which can be beneficial for cryptographic system design and signal processing applications [3]. In the field of information security, Feng et al. presented a secure and efficient image transmission scheme based on two chaotic maps, demonstrating high performance in ensuring image data security [4]. Additionally, research conducted by Qian et al. introduced a multi-channel memristor-based chaotic map used for image encryption, showing a substantial increase in security and resistance against decryption attacks [5]. These studies emphasize that integrating chaotic maps with optimization and cryptographic algorithms can lead to improved performance, higher accuracy, and reduced computational costs.

To address these challenges, this paper proposes a novel chaotic decomposition-based method, referred to as the chaotic multi-objective optimization algorithm (CMOA). The CMOA integrates chaos theory with decomposition strategies to enhance both the convergence and diversity of Pareto optimal solutions. By leveraging key chaotic properties such as pseudo-randomness, ergodicity, and irregularity, the proposed algorithm improves the exploration and exploitation capabilities of the search process. The primary innovation of the CMOA lies in its chaotic sequence initialization and chaotic correction step, which reduce computational costs by generating non-iterative points, thereby improving efficiency and solution quality.

The main objective of this work was to develop the CMOA as an effective solution for unconstrained multi-objective optimization problems. By incorporating chaotic maps with decomposition strategies, this approach aims to balance local and global searches while mitigating computational complexity. The proposed algorithm is rigorously evaluated against well-established benchmark problems and compared with state-of-the-art MOEAs, demonstrating superior performance in both solution accuracy and computational efficiency.

The main contributions of this work can be summarized as follows:

We propose a novel chaotic decomposition-based multi-objective optimization algorithm that enhances solution diversity and convergence by incorporating chaotic maps.
TPO and LIO are introduced to refine the Pareto front and improve the accuracy of the solutions.
The effectiveness of the CMOA is validated through benchmark experiments on the ZDT [6] and CEC 2009 test suites [7], demonstrating superior performance compared to state-of-the-art MOEAs.
We analyze the impact of chaotic maps on optimization, highlighting their potential in balancing exploration and exploitation in multi-objective search spaces.

The remainder of this paper is structured as follows. Section 2 presents a review of related works, highlighting previous studies on evolutionary optimization methods and their applications in multi-objective problems. Section 3 provides an overview of essential concepts related to multi-objective optimization and introduces the fundamental principles of chaotic systems. Section 4 presents the proposed CMOA algorithm, detailing its chaotic sequence initialization, decomposition strategy, and correction mechanisms. Section 5 discusses experimental results, comparing the CMOA with existing MOEAs based on standard performance metrics. Finally, Section 6 concludes the paper with a summary of findings and potential directions for future research.

2. Related Work

Multi-objective optimization has been extensively studied, with numerous methods developed to identify Pareto optimal solutions. Among these, multi-objective evolutionary algorithms have gained significant attention due to their ability to tackle complex, high-dimensional optimization problems. Notable examples include the non-dominated sorting genetic algorithm (NSGA-II) [1] and the multi-objective evolutionary algorithm based on decomposition (MOEA/D) [2], both of which efficiently explore multiple solutions in parallel. The multi-objective genetic algorithm (MOGA) is a direct method that needs to synthetically consider the fitness, diversity and elitism of solutions [8]. The non-dominated moth-flame optimization (NS-MFO) algorithm, based on the search technique MFO, is capable of obtaining different non-domination levels and preserving the optimal set of solutions [9]. Zhan et al. presented an adaptive PSO for each swarm and designed co-evolutionary multi-swarm PSO (CMPSO) to MOO problems [10]. A multi-objective artificial bee colony (RMOABC) method with regulation operators was proposed by Huo and Liu [11]. Despite their effectiveness, MOEAs often suffer from high computational costs and difficulty in maintaining solution diversity, which can hinder convergence to the true Pareto front.

Decomposition-based methods address some of these challenges by breaking the original MOO problem into smaller, more manageable subproblems. The MOEA/D algorithm, for instance, decomposes MOO problems into scalar optimization tasks that are solved in parallel, leading to improved convergence [2]. However, maintaining diversity and avoiding premature convergence remain critical challenges.

In response to these limitations, researchers have explored chaos theory as a means of enhancing optimization algorithms. Chaos-based optimization techniques integrate the deterministic, yet unpredictable behavior of chaotic systems with traditional stochastic search methods. These approaches have shown promise in single-objective optimization, where chaotic maps such as the logistic map and the Chebyshev map have been used to generate pseudo-random sequences that enhance global search capabilities while reducing the risk of local optima entrapment [12].

In the multi-objective optimization domain, chaos theory has been leveraged to improve evolutionary algorithms. For example, the chaotic genetic algorithm (CGA) [13] employs chaotic sequences to initialize the population and adjust the search process dynamically, while chaotic particle swarm optimization (CPSO) [14] integrates chaotic dynamics to enhance the global search ability of traditional particle swarm optimization (PSO). These methods have demonstrated significant improvements in convergence speed and solution quality, particularly in complex, high-dimensional optimization problems.

More recently, researchers have introduced chaos-based correction strategies within MOEAs to balance exploration and exploitation. For instance, the chaotic multi-objective evolutionary algorithm (CMOEA) [15] integrates chaotic maps to refine the search process, thereby enhancing diversity and convergence. However, these approaches often rely on random perturbations, which may not fully utilize the structured benefits of chaos for systematic optimization.

To address these challenges, this paper proposes the CMOA, which introduces a structured approach by integrating chaotic decomposition with a correction mechanism based on two key operators: the three-point operator and the local improvement operator. By leveraging chaotic maps for population initialization and guided perturbations, the CMOA improves convergence, enhances solution diversity, and reduces computational costs, addressing the primary limitations of existing MOEAs.

3. Preliminaries

In this section, we first focus on some definitions in the multi-objective optimization and chaotic maps. Then, we introduce a brief of the CGA method that is applied in the literature.

3.1. Basic Concepts

A quadruple

< S, O, C, f >

is a multi-objective optimization problem whenever

S

is the search space,

O

is the objective space,

C

is the set of constraints, and

f

is the vector of

n

objective functions. In this paper, we concentrate on the solutions of certain categories of MOO problems, which can be formulated as follows:

\{\begin{matrix} \begin{matrix} m i n & f (\vec{x}) = (f_{1} (\vec{x}), \dots, f_{k} (\vec{x})) \end{matrix} \\ \begin{matrix} S . t . & \vec{x} = (x_{1}, x_{2}, \dots \end{matrix} x_{n}) \in \prod_{i = 1}^{n} [a_{i}, b_{i}], \end{matrix}

(1)

where

f : \prod_{i = 1}^{n} [a_{i}, b_{i}] \to R^{k}

and

- \infty < a_{i} < b_{i} < + \infty, i = 1, 2, \dots, n .

Because the objective functions in problem (1) often conflict with each other, there is not one single optimal solution for this kind of problem. Therefore, let us define a set of solutions based on the relationship of partial order that was described by Koopmans in 1951.

Definition 1

(domination). Let

\vec{x}, \vec{y} \in \prod_{i = 1}^{n} [a_{i}, b_{i}],

and

f (\vec{x}), f (\vec{y}) \in R^{k} .

f (\vec{x}) \neq f (\vec{y})

and

f_{i} (\vec{x}) \leq f_{i} (\vec{y}), i = 1, 2, \dots, k

then

f (\vec{x})

dominates

f (\vec{y}) .

Definition 2

(Pareto optimal solution). Let

{\vec{x}}^{*} \in \prod_{i = 1}^{n} [a_{i}, b_{i}]

in search space. If there is no

\vec{x} \in \prod_{i = 1}^{n} [a_{i}, b_{i}]

such that

f (\vec{x})

dominates

f ({\vec{x}}^{*})

, then

{\vec{x}}^{*}

is Pareto optimal.

Definition 3

(Pareto set). The set of all the Pareto optimal is the Pareto set (PS). In other words,

P S = \{\frac{\vec{x}}{\vec{x}} \in \prod_{i = 1}^{n} [a_{i}, b_{i}] a n d \vec{x} i s P a r e t o o p t i m a l\}

Definition 4

(non-dominated). The non-dominated vector is a point of

R^{k}

which is obtained by the element of Pareto set.

Definition 5

(Pareto front).

P F = \{(f_{1} (\vec{x}), \dots, f_{k} (\vec{x})) / \vec{x} \in P S\}

To make the paper more understandable, we present in Figure 1 a clear example of the above definitions for a two-objective optimization problem with two decision variables

X_{1}, X_{2} .

3.2. Chaotic Maps

Chaos refers to a form of pseudo-random behavior within a deterministic and nonlinear system that is sensitive to initial conditions. Chaos theory has been utilized in various domains, including optimization [16], synchronization, and machine vision [17]. Typically, a one-dimensional chaotic map can be represented as follows:

(m + 1) = φ (μ_{1}, μ_{2}, \dots, μ_{p}, x (m)), m = 0, 1, 2, 3, \dots .

(2)

where

μ_{i}, i = 1, \dots, p

are control parameters and

x

is a variable. Table 1 presents several significant examples of one-dimensional chaotic maps.

Chaotic maps produce chaotic sequences, whereas evolutionary algorithms typically rely on random processes to generate subsequent solutions. The deterministic nature of chaos can reduce complexity costs by generating non-repetitive solutions. A chaotic variable exhibits three fundamental characteristics: pseudo-randomness, ergodicity, and irregularity [22]. Consequently, chaos theory leverages insights from nonlinear dynamics and can be effectively applied to optimization problems.

3.3. CGA Algorithm

While the golden section search (GSS) algorithm offers several significant advantages, including rapid convergence, ease of implementation, and guaranteed convergence, it is primarily effective for optimizing unimodal objective functions [19]. The GSS method identifies the optimal solution by employing the concept of dividing regions.

The chaotic golden section search method (CGA) has been presented to handle some main weaknesses of GSS for solving nonlinear optimization problems [13]. This method has extended the GSS method to deal with multimodal optimization problems. The CGA method introduces the multimodal n-dimension GSS using a bipartite procedure. In the first step, CGA finds a search sub-space using chaotic concept to satisfy the unimodal condition of the optimization problem for the GSS algorithm. In the second step, the golden section search method is applied over the obtained search space as a local search. The general procedure of this method is given in algorithm CGA (Algorithm 1).

Algorithm 1: Chaotic Golden Section Search Method (CGA)
	Step (1) Initialization ${N_{1}, N_{2}, γ}_{1} (i) i = 1, \dots, n$ . $N_{1}$ is the number of iterations to achieve the proper feasible subspace. $N_{2}$ is the number of scanning steps in the golden section search algorithm. $γ_{1} (i)$ is a random value in the range $(0, 1) .$
Calculating the subarea under the global optimum	Step (2) Generating points $x_{l} (i), x_{u} (i)$ in the domain $[l (i), u (i)]$ by: $\begin{matrix} x_{l} (i) = l (i) + γ_{1} (i) (u (i) - l (i)), \\ x_{u} (i) = u (i) - γ_{1} (i) (u (i) - l (i)); \end{matrix} i = 1, \dots, n .$ Step (3) If $x_{u} (i)$ $<$ $x_{l} (i)$ then $l l \leftarrow x_{u} (i), x_{u} (i) \leftarrow x_{l} (i), x_{l} (i) \leftarrow l l$ ; $i = 1, \dots, n .$ Step (4) If $f (X_{l}) < f (l)$ then $l \leftarrow X_{l}$ else $l$ fix in place. Step (5) If $f (X_{u}) < f (u)$ then $u \leftarrow X_{u}$ else $u$ fix in place. Step (6) Generating $γ_{1} (i)$ by the $L o g i s t i c$ map, Step (7) Repeating steps 2thru-6 until the stop criteria $N_{1}$ is reached.
Golden section search algorithm	Suppose $[p, q]$ is the novel feasible space. Step (8) $c \leftarrow \frac{- 1 + \sqrt{5}}{2}$ , and calculate $x_{1} (j) = q (j) - c (q (j) - p (j))$ , $x_{2} (j) = p (j) + c (q (j) - p (j)); j = 1, \dots, n .$ Step (9) If $f (X_{1}) < f (X_{2})$ , the global optimum must be in $[X_{1}, q]$ thus $p \leftarrow X_{1}$ , $q$ fix in place, Else the global optimum must be in $[{p, X}_{2}]$ thus $q \leftarrow X_{2},$ $p$ fix in place. Step (10) If the stop criteria $N_{2}$ satisfied, go to stop and put out $p$ as the best solution, otherwise go to step 8.

4. The Proposed Chaotic Multi-Objective Algorithm

In this section, we introduce a novel multi-objective optimization method based on chaos theory and decomposition techniques, referred to as CMOA, designed to solve problem (1). The proposed method consists of three key stages: chaotic sequence initialization, chaos-based correction, and Tchebycheff-based updating. In the chaotic sequence initialization phase, a chaotic population is generated using a chaotic map such as a logistic map, leveraging its ergodicity and stochastic properties to ensure a diverse and well-distributed initial population while reducing computational complexity. In the chaos-based correction phase, the algorithm enhances the Pareto front by employing two specialized operators: the three-point operator and the local improvement operator. These operators work in synergy to refine the search process, balancing global exploration and local exploitation to improve solution quality. Finally, the Tchebycheff-based updating phase applies decomposition techniques to efficiently guide the search towards the Pareto-optimal front. To enhance clarity and reproducibility, a structured overview of the chaotic decomposition-based approach is illustrated in Figure 2, highlighting its key contributions and motivations.

4.1. Chaotic Sequence Initialization

The chaotic sequence initialization process is designed to generate an initial population that exhibits ergodicity and stochasticity, which helps to maintain diversity in the search space. We employ a logistic map to initialize the chaotic population:

Step 1: Parameter Initialization

Define the stopping criterion $k$ , the number of generated chaotic points $N$ , and the number of weight vectors $λ^{i}$ , where $i = 1, \dots, N$ .
Ensure the sum of weight vectors satisfies $\sum_{j = 1}^{s} λ_{j}^{i} = 1$ .
Compute the Euclidean distances between any two weight vectors and select the closest $T$ weight vectors.

Step 2: Chaotic Population Generation

Generate $N$ chaotic points $A = \{x_{1}, x_{2}, \dots, x_{N}\}$ using the logistic map.
Initialize the cost function $F (x_{0})$ with $x_{0}$ as the starting point.

The following pseudocode describes the initialization process (Algorithm 2):

Algorithm 2: Chaotic Initialization

1 . Input : N

, k

, weight vectors λ, Logistic map parameters

2 . Generate chaotic population A using :

x_{i} = 4 * x_{i} * (1 - x_{i}), x_{i} \in [0, 1] .

3 . Compute Euclidean distances and select closest T

weight vectors

4 . Compute initial \cos t function F (x_{0})

5. Output: Initialized chaotic population

4.2. Chaos-Based Correction

This phase refines the generated population using two operators:

Three-Point Operator (TPO): Enhances the search mechanism by utilizing three chaotic points to generate a new solution. TPO uses the information of three chaotic points to obtain a new point in search space based on a control parameter

B (x)

. The

B (x)

controller is a distribution function.

B (x) = \{\begin{matrix} {(2 x)}^{(\frac{1}{1 + μ})} i f 0 \leq x \leq \frac{1}{3} \\ \frac{1}{{(2 (τ - x))}^{{(\frac{1}{1 + μ})}^{2}}} i f \frac{1}{3} < x \leq τ \\ \frac{1}{{(2 (1 - x))}^{(\frac{1}{1 + μ})}} i f τ < x \leq 1 \end{matrix}, τ = \frac{2}{3}, μ = 2

(3)

Here, the interval [0, 1] can be divided into three partitions to create the new point

\bar{y}

from the three chaotic points

x_{i}, x_{j}, a n d x_{k}

\begin{matrix} \bar{y} = y_{A} + ε, \\ y_{A} = \frac{(x_{i} + x_{j} + x_{k})}{3}, ε = (\frac{1}{6}) x B (x) |2 x_{1} - 3 x_{2} + 2 x_{3}| . \end{matrix}

(4)

Here,

y_{A}

is the average of the three points. The error

ε

can be calculated using the control parameter

B (x)

. Figure 3 shows that the process of using the control parameter

B (x)

for the TPO operator.

Local Improvement Operator (LIO): Balances local and global search to improve convergence. In LIO, we used rich information of the three points to increase the accuracy of the search mechanism. To balance the local and global search of the proposed algorithm, we present a local improvement operator using the control parameter

ρ (x)

ρ (x) = \{\begin{matrix} {(2 x)}^{(\frac{1}{1 + μ})} - 1 i f 0 \leq x \leq \frac{1}{3} \\ 1 - {(2 (τ - x))}^{(\frac{1}{1 + μ})} i f \frac{1}{3} < x \leq τ \\ 1 - {(2 (1 - x))}^{(\frac{1}{1 + μ})} i f τ < x \leq 1 \end{matrix} τ = \frac{2}{3}, μ = 5

(5)

where the

ρ (x)

controller is the ratio.

ρ (x) = \frac{{\tilde{y}}_{j} - {\bar{y}}_{j}}{b_{j} - a_{j}} j = 1, 2, \dots, n i f 0 \leq r_{1} \leq \frac{1}{n}

(6)

Here,

{\tilde{y}}_{j}

is a mutation of

{\bar{y}}_{j}

, and

r_{1}

is used to generate

\bar{y} = ({\bar{y}}_{1}, {\bar{y}}_{2}, \dots, {\bar{y}}_{n})

of the logistic map.

\{\begin{matrix} r_{1} = 4 * r_{1} * (1 - r_{1}), \\ r_{1} \in [0, 1] \end{matrix}

(7)

This operator can reach the approximation Pareto front to fit the true

P F

. Figure 4 shows the process of the controller

ρ (x)

based on three points for the LIO operator.

Step 3: Three-Point Operator (TPO) (Algorithm 3)

Select three chaotic points $x_{i}, x_{j}, x_{k}$ from $A$ .
Apply the control parameter $B (x)$ for generating a new search point $\bar{y}$ .

Algorithm 3: Three Point Operator (TPO)

1 . Select chaotic points x_{i}, x_{j}, x_{k}

from A

2 . Compute \bar{y}

using : \bar{y} = \frac{(x_{i} + x_{j} + x_{k})}{3} + ε,

ε = (\frac{1}{6}) x B (x) |2 x_{1} - 3 x_{2} + 2 x_{3}| .

3. Output: New search point

\bar{y}

Step 4: Local Improvement Operator (LIO) (Algorithm 4)

Improve $\bar{y}$ by applying mutation using control parameter $ρ (x)$ .

Algorithm 4: Local Improvement Operator (LIO)

1. Apply mutation to

\bar{y}

using:

\tilde{y} = \bar{y}

ρ (x)

b_{j} - a_{j}

)

2. Ensure

\tilde{y}

remains within search space bounds

3. Output: Improved solution

\tilde{y}

4.3. Updating Based on Tchebycheff Decomposition

After generating improved solutions, we apply the Tchebycheff decomposition approach to update and optimize the Pareto front approximation.

Step 5: Update Process (Algorithm 5)

Compute the ideal vector solution by the CGA method: $z_{i}^{*} = m i n \{f_{i} (x)| x \in Ω\}, i = 1, 2, \dots, k$
Update the best solution using: $g (x / λ^{i}, z) = {m a x}_{1 \leq i \leq k} \{λ^{i} |f_{i} (x) - z_{i}|\}$

Algorithm 5: Tchebycheff-Based Updating

1 . Compute ideal vector solution z_{i}^{*}

2. Update solutions using:
If

g (\tilde{y} / λ^{i}, z) < g (x_{0} / λ^{i}, z)

then

x_{0} \leftarrow \tilde{y}

, i = 1, 2, \dots, T

3. Check stopping criteria, else repeat

Figure 5 presents a flowchart illustrating the overall process of CMOA. In fact, this approach is a straightforward extension of the decomposition-based multi-objective evolutionary algorithm (MOEA/D) based on the chaotic characteristics to solve problem (1), which aids in decreasing the complexity cost and increasing the convergence rate. Based on the LIO operator, the pdf of chaotic sequences of the logistic map with

γ = 4

can be derived as

ρ (z) = \frac{1}{π \sqrt{z (1 - z)}}

, which is called the Chebyshev distribution [23]. The Chebyshev distribution is bounded, with chaotic points with small differences T > 0 and

|B| \leq N

. Therefore, the evaluation function is bounded [24] and the proposed method can converge to an optimal solution.

5. Experimental Studies

In this section, we first introduce the experimental materials, including the test problems and the parameters of the applied algorithms. Then, two subsections are dedicated to evaluating the robustness and performance of CMOA. To ensure reproducibility and reliability, the experiments were conducted on a system equipped with an Intel Core i7 processor, 16 GB of RAM, and an integrated GPU. The software environment consisted of MATLAB R2023a running on Windows 10 Pro (64-bit). The benchmark datasets used in this study include standard multi-objective test problems, which are described in detail below.

5.1. Multi-Objective Test Problems and Parameter Settings

To investigate the performance of the proposed algorithm, the CMOA method was tested on benchmark problems with various features, including separability, multi-modality, different Pareto front shapes, and the presence of multiple local Pareto-optimal fronts. The test problems are categorized into two groups. The first group consists of five high-dimensional ZDT problems introduced by Zitzler et al. [6]. These problems are used to analyze the efficiency of chaotic points and chaos-based correction in the search mechanism of the proposed algorithm. To further evaluate the performance, ten unconstrained multi-objective problems (UF1-UF10) from CEC 2009 [7] are used as the second set of benchmark functions. Table 2 provides detailed information on the test problems. In this table, M and N represent the “number of objectives” and “number of variables”, respectively.

In this study, we compare the results of the proposed algorithm with those reported for several decomposition-based multi-objective evolutionary algorithms, including MOEA-DVA [21], MOEA/D-MS [25], MOEA/D [2], RVEA [26], MOEA/D-PaS [27], MMOPSO [28], and other state-of-the-art methods. The parameters of these algorithms were optimized by their authors in the original literature.

For CMOA, the computational parameters were set as follows: K = 10 as the stopping criterion, N = 200 for generating chaotic points, and T = 0.1 N. The proposed algorithm was executed 20 times with a total of 52,000 function evaluations for each test instance. These values were selected based on a combination of theoretical considerations and empirical evaluations to optimize the algorithm’s performance while maintaining computational efficiency.

To assess the performance of the proposed method, two key aspects were considered: accuracy and diversity. Generational distance (GD) and inverted generational distance (IGD) were used as accuracy metrics to measure the convergence of the obtained solutions [29]. These metrics help evaluate how close the obtained solutions are to the true Pareto-optimal front. Spacing (SP) and IGD were also used as diversity metrics to measure the distribution and spread of solutions [30].

5.2. Analysis and Discussion

In this section, the results are analyzed using GD as an accuracy metric, IGD as both an accuracy and diversity metric, and SP as a diversity criterion. To evaluate the effectiveness of the proposed method, we compare it with decomposition-based multi-objective evolutionary algorithms and other state-of-the-art approaches on five ZDT problems. Statistical analysis techniques, including mean and standard deviation, are used for comparisons, where the best mean value among all algorithms is highlighted in bold.

Table 3 presents the convergence and diversity evaluation of the proposed method alongside other approaches, including NSGA-II, MOEA-DVA, MOEA-IGD-NS, MOEA/D-MS, MOEA/D, RVEA, MOEA/D-PaS, MMOPSO, MOCMVO, and MOGNDO, using the IGD metric. The results in Table 3 indicate that the solutions obtained by CMOA are very close to the true Pareto front. Specifically, CMOA achieves the best IGD values on ZDT1, ZDT2, ZDT4, and ZDT6, demonstrating its strong convergence performance. It is worth noting that for the IGD metric, lower values indicate better performance in approximating the Pareto front. For ZDT3, the proposed method performs comparably to MOGNDO. These findings confirm that CMOA exhibits competitive performance across the ZDT benchmark problems.

The findings reveal that the chaotic sequence initialization step effectively generates a chaotic population using a deterministic model within the proposed method. The use of non-iterative generated points enhances the diversity of the algorithm. Additionally, the chaos-based correction strategy improves the Pareto front of the generated population through the implementation of two operators: TPO and LIO. These operations facilitate a balance between local and global searches in CMOA. The TPO operator averages three points to provide an interpolation or approximation, representing a subset of solutions for global exploration. Conversely, the LIO operator employs the logistic map within the interval [0, 1] to direct solutions toward maximizing the fitness evaluation function.

In Table 4, CMOA is compared with other leading multi-objective algorithms based on the mean value of spacing. The SP results indicate that CMOA surpasses the performance of state-of-the-art algorithms across most test benchmarks. Specifically, CMOA demonstrates strong results on the ZDT1 and ZDT3 problems and achieves the best outcomes on other benchmarking functions characterized by convex and concave Pareto fronts. Furthermore, CMOA exhibits high computational efficiency in solving ZDT problems, requiring only approximately 0.7 s per run. This efficiency stems from the deterministic chaotic sequence initialization, which quickly generates a diverse set of solutions, and the correction strategy, which efficiently refines the Pareto front without excessive computational overhead. Thus, the chaos-based correction step in CMOA effectively maintains a balance between exploration and exploitation, aiding in the convergence toward optimal solutions while preserving diversity within all populations.

Table 5 displays the mean GD values for CMOA and various state-of-the-art algorithms across four ZDT benchmarks. The results indicate that the proposed method consistently achieves better GD than all other algorithms in every instance. Thus, it can be concluded that CMOA effectively identifies true Pareto solutions for ZDTs with convex, concave, non-uniform concave, and disconnected convex Pareto fronts. Furthermore, the solutions suggest that the chaos-based strategy utilizing TPO and LIO operators enhances the efficiency of the algorithm.

To illustrate the performance of CMOA, the Pareto front solutions generated by the proposed algorithm for five ZDT problems are displayed in Figure 6. It is clear that the approximated Pareto front is uniformly distributed across the true Pareto front for these five test cases. Additionally, all optimal Pareto fronts for the ZDT problems converge toward the true Pareto front, yielding a diverse set of solutions, as demonstrated in Figure 6. These results suggest that the proposed algorithm is highly effective compared to existing state-of-the-art methods for solving multi-objective problems, due to the efficient local and global search capabilities provided by the chaotic population and chaotic correction mechanisms. As depicted in Figure 6, the chaos-based correction effectively searches the current population and explores different areas using two operators: TPO and LIO.

As shown in Table 6, the characteristics of the chaos-based method are presented for comparison with stochastic-based methods.

5.3. Comparison Experiments on CEC09 UF Problems

To see how the proposed algorithm performs in comparison with decomposition-based multi-objective evolutionary algorithms and state-of-the art methods, we use CEC09 UF problems in this section. IGD is applied to the analysis approach as a diversity and accuracy metric.

5.3.1. Comparison with Decomposition-Based MOEAs

To demonstrate the robustness and performance of CMOA, we conducted a comparison with various decomposition-based algorithms in multi-objective optimization. Table 7 presents a performance comparison of CMOA against MOEA/D-MS, MOEA/D-MRS, MOEA/DVA, MOEA/D-PaS, RVEA, SMOEA/D, and MOGWO/D, using the IGD metric as the evaluation criterion.

According to Table 7, CMOA outperforms other algorithms in six problems: UF2, UF4, UF6, UF8, UF9, and UF10. In problems UF1, UF3, UF5, and UF7, its performance is comparable to other methods. The Friedman test indicates a significant difference between the algorithms (

χ^{2} = 57.49, p = 6.43 \times 10^{- 8}

), and the Wilcoxon signed-rank test results show that CMOA performs significantly better than MOEA/D-PaS, RVEA, MOEA/D-MS, and MOGWO/D

(p < 0.05)

. However, no significant difference is observed between CMOA and algorithms such as MOEA/D-MRS, MOEA/DVA, and SMOEA/D. Additionally, the updated strategy of CMOA, incorporating TPO and LIO operators, enhances diversity maintenance and improves solution convergence compared to other methods. Moreover, its chaotic correction section leverages a logistic map to incorporate both the non-domination rank and crowding distance of three neighboring solutions, utilizing their rich information to preserve diversity and enhance search efficiency.

5.3.2. Comparison with State-of-the Art Methods

In Table 8, the performance of CMOA was compared against eight other algorithms for the UF1-UF10 problems using the IGD metric. These algorithms are based on evaluation strategies such as genetic algorithms and differential evolution. The IGD results indicate that CMOA outperforms all others in every problem except UF5 and UF1. Overall, the proposed method proves to be highly competitive with evolutionary-based algorithms, offering the advantages of low complexity and a high convergence rate. Despite its strong performance, CMOA maintains a reasonable computational cost, requiring approximately 5.5 s per run for UF problems. This efficiency is achieved through its chaotic strategy, which incorporates three fundamental characteristics-pseudo-randomness, ergodicity, and irregularity-enhancing the selection and updating processes in the chaos-based approach, helping to maintain solution diversity and improve accuracy.

6. Conclusions

This study introduces a novel chaotic decomposition-based optimization approach for solving unconstrained multi-objective optimization problems. The motivation behind this research stems from the limitations of traditional evolutionary algorithms, which often struggle with maintaining a balance between exploration and exploitation, leading to issues such as premature convergence or loss of solution diversity. To address these challenges, the proposed method leverages chaotic theory—specifically pseudo-randomness, ergodicity, and irregularity—to enhance solution diversity, convergence speed, and computational efficiency.

The effectiveness of CMOA was demonstrated through comprehensive benchmarking on both ZDT and UF problem sets. For ZDT problems, the method achieved superior performance compared to traditional evolutionary techniques, obtaining high-quality Pareto-optimal solutions with only 52,000 iterations, thus maintaining a low computational complexity. The chaotic point generation strategy facilitated the construction of an initial population with enhanced diversity, while the chaos-based correction mechanism—through TPO and LIO operators—enabled an effective balance between global exploration and local refinement. Similarly, for UF problems, CMOA outperformed existing stochastic-based approaches due to its robust local and global search capabilities, ultimately leading to better diversity preservation and improved convergence properties. Furthermore, despite its strong performance, CMOA maintains a reasonable computational cost, with an average execution time of 5.5 s per run for UF problems, making it a competitive alternative to traditional evolutionary algorithms.

Beyond its strengths, several limitations of this study should be acknowledged. First, the effectiveness of the chaotic maps used in the algorithm is highly dependent on the specific map selection and the phase of application. Additionally, the scale of sequences generated by chaotic maps may not always align perfectly with the requirements of certain problem types, potentially affecting convergence behavior. Future research should explore the impact of different chaotic models on optimization performance, investigate adaptive chaotic strategies, and extend the application of CMOA to real-world multi-objective problems, such as engineering design, financial modeling, and machine learning hyperparameter tuning. In summary, this study underscores the potential of chaos-based optimization as a powerful and efficient alternative to conventional approaches, paving the way for further advancements in multi-objective optimization research.

Author Contributions

J.A.K. and M.J.E. contributed to the conceptualization, methodology, and validation. J.A.K. handled software development, formal analysis, and writing the original draft. M.J.E. supervised the research and reviewed the manuscript. Both authors approved the final version. All authors have read and agreed to the published version of the manuscript.

Funding

The authors did not receive any specific funding for this research.

Data Availability Statement

Data are contained within the article, as no new datasets were generated or analyzed during the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Figure 1. Illustration of the search space and the objective space of a two-objective optimization problem.

Figure 2. The contributions of the proposed chaotic decomposition-based approach.

Figure 3. The distribution function

B (x)

used in the TPO operator.

Figure 3. The distribution function

B (x)

used in the TPO operator.

Figure 4. The distribution function

ρ (x)

used in the LIO operator.

Figure 4. The distribution function

ρ (x)

used in the LIO operator.

Figure 5. Flowchart of the overall CMOA process.

Figure 6. Comparison of CMOA’s Pareto front with the true Pareto front on five ZDT problems.

Table 1. Examples of 1-D chaotic maps.

Name	Equation	$C^{*}$	Source
Chebyshev	$x (n + 1) = \cos (k {c o s}^{- 1} (x (n))) x \in (- 1, 1)$	$k = 2$	[18]
Logistic	$x (n + 1) = a x (n) (1 - x (n)) x (0) \in (0, 1), x (0) \notin \{0, 0.25, 0.5, 0.75, 1\}$	$a = 4$	[18]
Circle	$x (n + 1) = x (n) + b - (a - 2 π) \sin (2 π x (n)) m o d (1)$	$a = 0.5, b = 0.2$	[19]
Gaussian	$x (n + 1) = \{\begin{matrix} 0 x (n) = 0 \\ \frac{μ}{x (n)} m o d 1 x (n) \neq 0 \end{matrix}$	$μ = 1$	[18]
ICMIC	$x (n + 1) = \sin (\frac{a}{x (n)}) x \neq 0, x \in [- 1, 1], a \in (0, \infty)$	$a = 2$	[20]
Sine	$x (n + 1) = (\frac{a}{4}) \sin (π x (n)) 0 < a \leq 4$	$a = 4$	[21]
Kent	$x (n + 1) = \{\begin{matrix} \frac{x (n)}{β} 0 < x (n) \leq β \\ \frac{(1 - x (n))}{(1 - β)} β < x (n) \leq 1 \end{matrix}$	$β = 0.4$	[22]

C^{*}

= value of control parameter when a chaotic sequence is generated by this map.

Table 2. Features of multi-objective test problems used in the experimental results.

Problem	M	N	Range	Characteristics
ZDT1	2	30	${[0, 1]}^{n}$	separable, unimodal, convex Pareto front
ZDT2	2	30	${[0, 1]}^{n}$	separable, unimodal, concave Pareto front
ZDT3	2	30	${[0, 1]}^{n}$	$separable, f_{1}$ is $unimodal and f_{2}$ is multimodal, discontinuous
ZDT4	2	30	$[0, 1] \times {[- 5, 5]}^{n - 1}$	$separable, f_{1}$ is $unimodal and f_{2}$ is $highly multimodal, convex Pareto front, with 2^{21}$ local Pareto-optimal fronts
ZDT6	2	30	${[0, 1]}^{n}$	separable, multimodal, concave Pareto front, with a non-uniform search space
UF1	2	30	$[0, 1] \times {[- 1, 1]}^{n - 1}$	complicated Pareto set
UF2	2	30	$[0, 1] \times {[- 1, 1]}^{n - 1}$	complicated Pareto set
UF3	2	30	${[0, 1]}^{n}$	complicated Pareto set
UF4	2	30	$[0, 1] \times {[- 2, 2]}^{n - 1}$	complicated Pareto set
UF5	2	30	$[0, 1] \times {[- 1,1]}^{n - 1}$	complicated Pareto set, discontinuous
UF6	2	30	$[0, 1] \times {[- 1, 1]}^{n - 1}$	complicated Pareto set, discontinuous
UF7	2	30	$[0, 1] \times {[- 1, 1]}^{n - 1}$	complicated Pareto set
UF8	3	30	${[0, 1]}^{2} \times {[- 2, 2]}^{n - 2}$	complicated Pareto set
UF9	3	30	${[0, 1]}^{2} \times {[- 2, 2]}^{n - 2}$	complicated Pareto set, discontinuous
UF10	3	30	${[0, 1]}^{2} \times {[- 2, 2]}^{n - 2}$	complicated Pareto set

M: number of objectives, N: number of variables.

Table 3. IGD mean and standard deviation (SD) for CMOA and decomposition-based MOEAs on ZDT problems.

IGD		ZDT1	ZDT2	ZDT3	ZDT4	ZDT6
CMOA	Mean SD	8.304 × 10⁻⁵ 2.949 × 10⁻⁷	2.943 × 10⁻⁴ 5.902 × 10⁻⁴	2.122 × 10⁻³ 1.303 × 10⁻³	9.125 × 10⁻⁵ 8.095 × 10⁻⁷	5.708 × 10⁻⁵ 1.678 × 10⁻⁸
NSGA-II [1]	Mean SD	4.760 × 10⁻³ 2.180 × 10⁻⁴	4.708 × 10⁻³ 1.870 × 10⁻⁴	1.894 × 10⁻² 3.380 × 10⁻²	9.921 × 10⁻³ 1.840 × 10⁻³	9.476 × 10⁻² 5.090 × 10⁻²
MOEA-DVA [21]	Mean SD	5.761 × 10⁻¹ 6.410 × 10⁻²	7.789 × 10⁻¹ 1.510 × 10⁻¹	7.428 × 10⁻¹ 1.760 × 10⁻¹	5.153 1.550	6.552 × 10⁻¹ 1.170 × 10⁻⁶
MOEA-IGD-NS [31]	Mean SD	3.823 × 10⁻³ 3.690 × 10⁻⁵	3.819 × 10⁻³ 1.860 × 10⁻⁵	1.469 × 10⁻² 1.450 × 10⁻²	1.350 × 10⁻² 2.390 × 10⁻³	3.105 × 10⁻¹ 1.010 × 10⁻¹
MOEA/D-MS [25]	Mean SD	3.826 × 10⁻³ 8.750 × 10⁻⁶	3.782 × 10⁻³ 1.110 × 10⁻⁵	5.125 × 10⁻³ 2.470 × 10⁻⁴	9.579 × 10⁻³ 3.220 × 10⁻³	3.082 × 10⁻³ 4.700 × 10⁻⁷
MOEA/D [2]	Mean SD	5.080 × 10⁻⁴ 1.900 × 10⁻⁴	4.510 × 10⁻⁴ 1.300 × 10⁻⁴	1.620 × 10⁻³ 6.800 × 10⁻⁴	1.080 × 10⁻² 7.100 × 10⁻³	1.440 × 10⁻⁴ 3.000 × 10⁻⁵
RVEA [26]	Mean SD	4.668 × 10⁻³ 3.760 × 10⁻⁴	4.947 × 10⁻³ 6.450 × 10⁻⁴	1.106 × 10⁻² 7.180 × 10⁻³	5.033 × 10⁻² 1.210 × 10⁻²	3.463 × 10⁻³ 2.800 × 10⁻⁴
MOEA/D-PaS [27]	Mean SD	3.953 × 10¹ 2.800 × 10¹	6.347 × 10¹ 2.840 × 10¹	4.080 × 10¹ 3.850 × 10¹	1.163 × 10² 1.350 × 10²	3.144 × 10⁻³ 2.960 × 10⁻⁵
MMOPSO [28]	Mean SD	1.870 × 10⁻³ 1.380 × 10⁻⁵	1.910 × 10⁻³ 2.100 × 10⁻⁵	2.100 × 10⁻³ 4.490 × 10⁻⁵	1.840 × 10⁻³ 1.870 × 10⁻⁵	1.560 × 10⁻³ 4.720 × 10⁻⁵
MOCMVO [32]	Mean SD	4.96 × 10⁻⁴ 1.81 × 10⁻⁴	4.05 × 10⁻⁴ 7.54 × 10⁻⁵	3.18 × 10⁻² 1.40 × 10⁻⁴	7.75 × 10⁻³ 4.75 × 10⁻³	3.52 × 10⁻³ 8.92 × 10⁻⁴
MOGNDO [33]	Mean SD	5.00 × 10⁻⁴ 6.00 × 10⁻⁵	5.01 × 10⁻⁴ 9.00 × 10⁻⁵	3.50 × 10⁻⁴ 3.00 × 10⁻⁵	5.72 × 10⁻² 7.82 × 10⁻²	6.33 × 10⁻³ 8.26 × 10⁻³

Table 4. Mean SP values of CMOA and state-of-the-art algorithms on five ZDT problems.

SP	ZDT1	ZDT2	ZDT3	ZDT4	ZDT6
CMOA	5.041 × 10⁻³	1.591 × 10⁻³	1.098 × 10⁻²	5.001 × 10⁻³	9.367 × 10⁻⁴
MOGWO [34]	7.709 × 10⁻³	7.138 × 10⁻³	1.078 × 10⁻²	4.734 × 10⁻²	9.773 × 10⁻³
RMOABC [11]	3.603 × 10⁻³	3.305 × 10⁻³	9.279 × 10⁻³	N/A	1.646 × 10⁻¹
MOEA/D [2]	2.430 × 10⁻²	2.245 × 10⁻²	4.087 × 10⁻²	N/A	4.250 × 10⁻²
NSGA-II [1]	6.247 × 10⁻³	7.629 × 10⁻³	7.031 × 10⁻³	7.600 × 10⁻³	1.770 × 10⁻¹
W-MOEA/D [35]	2.063 × 10⁻²	1.420 × 10⁻²	2.258 × 10⁻³	1.880 × 10⁻²	1.880 × 10⁻²
T-MOEA/D [35]	1.884 × 10⁻²	1.138 × 10⁻²	9.341 × 10⁻³	1.265 × 10⁻²	1.265 × 10⁻²
MOCMVO [32]	6.660 × 10⁻³	6.790 × 10⁻³	1.230 × 10⁻²	7.120 × 10⁻²	2.090 × 10⁻³
MOGNDO [33]	9.090 × 10⁻³	1.043 × 10⁻²	1.028 × 10⁻²	1.073 × 10⁻²	9.873 × 10⁻²
Xtornado-TM[15]	1.140 × 10⁻²	1.610 × 10⁻²	4.690 × 10⁻²	1.450 × 10⁻²	6.460 × 10⁻²

N/A = not available.

Table 5. Mean GD values for CMOA and state-of-the-art algorithms on four ZDT problems.

GD	ZDT1	ZDT2	ZDT3	ZDT6
CMOA	3.438 × 10⁻⁵	3.332 × 10⁻⁵	9.076 × 10⁻⁵	2.549 × 10⁻⁵
MBSO/D [36]	9.000 × 10⁻⁴	8.000 × 10⁻⁴	1.200 × 10⁻³	9.000 × 10⁻⁴
MBSO-DE [37]	1.100 × 10⁻³	8.000 × 10⁻⁴	1.200 × 10⁻³	4.000 × 10⁻³
MBSO-C [38]	9.120 × 10⁻²	9.050 × 10⁻²	5.890 × 10⁻²	8.130 × 10⁻²
NSGA-II [1]	3.330 × 10⁻²	7.240 × 10⁻²	1.140 × 10⁻¹	4.490 × 10⁻²
MODE [39]	5.800 × 10⁻³	5.500 × 10⁻³	2.150 × 10⁻²	N/A
MO-SCA [40]	2.850 × 10⁻⁴	2.480 × 10⁻⁴	1.210 × 10⁻³	N/A
MOEA/D [2]	1.384 × 10⁻²	2.575 × 10⁻²	7.918 × 10⁻³	6.100 × 10⁻³
RMOABC [11]	2.947 × 10⁻⁴	2.947 × 10⁻⁴	7.796 × 10⁻⁴	6.100 × 10⁻³
MOCMVO [32]	1.520 × 10⁻⁴	9.650 × 10⁻⁵	7.010 × 10⁻²	2.52 × 10⁻²
MOGNDO [33]	1.610 × 10⁻³	1.580 × 10⁻³	1.700 × 10⁻⁴	1.054 × 10⁻¹
Xtornado-TM [15]	2.380 × 10⁻³	6.610 × 10⁻⁴	2.450 × 10⁻³	3.880 × 10⁻³

N/A = not available.

Table 6. General comparison between CMOA and state-of-the art methods.

Categories	Algorithm	Reference	Characteristics
Evolutionary algorithm (stochastic-based methods)	NSGA-II	[1]	High complexity cost Need more iterations Low convergence rate Non-deterministic process in the global search step Weak local search Random population (iterative)
	MOEA-DVA	[21]
	MOEA-IGD-NS	[31]
	MOEA/D-MS	[25]
	MOEA/D	[2]
	RVEA	[26]
	MOEA/D-PaS	[27]
	MMOPSO	[28]
Chaotic-based method	CMOA		Low complexity cost Fast convergence rate Deterministic process in the global search step Balance local and global search with two operators Chaotic population (non-iterative)

Table 7. Statistical comparison of CMOA and decomposition-based MOEAs.

Algorithm		UF1	UF2	UF3	UF4	UF5	UF6	UF7	UF8	UF9	UF10	p-Value (Wilcoxon vs. CMOA)
CMOA	Mean Std.	4.03 × 10⁻³ 9.17 × 10⁻⁴	1.55 × 10⁻³ 1.88 × 10⁻³	1.15 × 10⁻² 5.41 × 10⁻⁴	1.78 × 10⁻³ 8.17 × 10⁻⁵	4.35 × 10⁻¹ 1.29 × 10⁻¹	2.33 × 10⁻² 8.77 × 10⁻³	6.96 × 10⁻³ 1.58 × 10⁻³	2.87 × 10⁻³ 7.36 × 10⁻⁴	1.52 × 10⁻³ 2.57 × 10⁻⁴	6.66 × 10⁻³ 1.72 × 10⁻⁵	-
MOEA/D-MS [25]	Mean Std.	5.46 × 10⁻² 1.47 × 10⁻²	9.43 × 10⁻³ 4.88 × 10⁻⁴	1.89 × 10⁻¹ 1.06 × 10⁻¹	7.51 × 10⁻² 6.36 × 10⁻³	2.12 × 10⁻¹ 2.45 × 10⁻²	1.22 × 10⁻¹ 1.18 × 10⁻²	2.70 × 10⁻² 2.42 × 10⁻³	1.68 × 10⁻¹ 6.77 × 10⁻²	2.41 × 10⁻¹ 5.26 × 10⁻²	3.62 × 10⁻¹ 6.48 × 10⁻²	0.0488 (Significant)
MOEA/D-MRS [41]	Mean Std.	1.08 × 10⁻³ 9.05 × 10⁻⁵	3.03 × 10⁻³ 4.68 × 10⁻⁴	1.11 × 10⁻² 8.10 × 10⁻³	4.68 × 10⁻² 1.74 × 10⁻³	2.54 × 10⁻¹ 2.91 × 10⁻²	6.77 × 10⁻² 8.06 × 10⁻³	2.44 × 10⁻³ 5.52 × 10⁻⁴	5.72 × 10⁻² 1.19 × 10⁻²	2.58 × 10⁻² 8.46 × 10⁻³	4.44 × 10⁻¹ 5.50 × 10⁻²	0.3223 (Not significant)
MOEA/DVA [21]	Mean Std.	4.13 × 10⁻³ 9.90 × 10⁻⁵	4.10 × 10⁻³ 4.90 × 10⁻⁵	2.27 × 10⁻² 7.26 × 10⁻³	3.50 × 10⁻² 1.01 × 10⁻³	3.25 × 10⁻² 4.68 × 10⁻³	5.61 × 10⁻² 1.37 × 10⁻²	3.76 × 10⁻³ 4.64 × 10⁻⁵	5.77 × 10⁻² 1.20 × 10⁻²	1.23 × 10⁻¹ 1.62 × 10⁻¹	1.03 × 10⁻¹ 3.30 × 10⁻³	0.1602 (Not significant)
MOEA/D-PaS [27]	Mean Std.	1.44 × 10⁻² 5.45 × 10⁻³	3.18 × 10⁻² 2.74 × 10⁻²	2.50 × 10⁻¹ 3.33 × 10⁻¹	7.79 × 10⁻² 6.17 × 10⁻³	6.36 × 10⁻¹ 1.60 × 10⁻¹	1.01 × 10⁻¹ 6.99 × 10⁻³	1.70 × 10⁻² 1.06 × 10⁻²	3.12 × 10⁻¹ 1.46 × 10⁻¹	3.09 × 10⁻¹ 2.25 × 10⁻²	3.09 × 10⁻¹ 1.31 × 10⁻¹	0.00195 (Highly significant)
RVEA [26]	Mean Std.	9.75 × 10⁻² 3.02 × 10⁻²	7.13 × 10⁻² 9.69 × 10⁻³	3.14 × 10⁻¹ 2.04 × 10⁻²	9.12 × 10⁻² 7.05 × 10⁻³	3.29 × 10⁻¹ 9.24 × 10⁻²	1.79 × 10⁻¹ 9.24 × 10⁻²	1.75 × 10⁻¹ 1.38 × 10⁻¹	3.42 × 10⁻¹ 1.03 × 10⁻²	3.30 × 10⁻¹ 1.03 × 10⁻²	5.60 × 10⁻¹ 1.04 × 10⁻¹	0.0137 (Significant)
SMOEA/D [42]	Mean Std.	1.71 × 10⁻³ 1.97 × 10⁻⁴	5.76 × 10⁻³ 2.32 × 10⁻³	7.92 × 10⁻³ 6.47 × 10⁻³	5.20 × 10⁻² 3.39 × 10⁻³	2.84 × 10⁻¹ 4.23 × 10⁻²	6.55 × 10⁻² 7.19 × 10⁻³	2.38 × 10⁻³ 5.23 × 10⁻⁴	4.16 × 10⁻² 8.78 × 10⁻³	3.13 × 10⁻² 6.75 × 10⁻³	7.40 × 10⁻¹ 1.20 × 10⁻¹	0.2754 (Not significant)
MOGWO/D [43]	Mean Std.	7.66 × 10⁻² 2.00 × 10⁻³	3.86 × 10⁻² 1.00 × 10⁻³	2.00 × 10⁻¹ 6.20 × 10⁻²	1.01 × 10⁻¹ 4.00 × 10⁻³	3.54 × 10⁻¹ 5.50 × 10⁻²	3.29 × 10⁻¹ 8.60 × 10⁻²	3.62 × 10⁻² 2.00 × 10⁻³	8.44 × 10⁻² 3.80 × 10⁻²	7.71 × 10⁻² 5.30 × 10⁻²	4.06 × 10⁻¹ 1.14 × 10⁻¹	0.0195 (Significant)

Table 8. Comparison of CMOA with state-of-the art methods.

Algorithm	UF1	UF2	UF3	UF4	UF5	UF6	UF7	UF8	UF9	UF10
NS-MFO [9]	4.21 × 10⁻³	7.62 × 10⁻³	6.72 × 10⁻²	3.29 × 10⁻²	6.29 × 10⁻²	4.54 × 10⁻²	2.02 × 10⁻²	6.29 × 10⁻²	2.00 × 10⁻¹	4.33 × 10⁻¹
MOEA-IGD-NS [31]	1.06 × 10⁻¹	5.23 × 10⁻²	2.45 × 10⁻¹	4.62 × 10⁻²	3.13 × 10⁻¹	2.51 × 10⁻¹	1.35 × 10⁻¹	2.51 × 10⁻¹	2.22 × 10⁻¹	3.71 × 10⁻¹
AMGA [44]	3.59 × 10⁻²	1.62 × 10⁻²	7.00 × 10⁻²	4.06 × 10⁻²	9.41 × 10⁻²	1.29 × 10⁻¹	5.71 × 10⁻²	1.71 × 10⁻¹	1.89 × 10⁻¹	3.24 × 10⁻¹
OW-MOSaDE [45]	1.22 × 10⁻²	8.10 × 10⁻³	1.03 × 10⁻¹	5.13 × 10⁻²	4.30 × 10⁻¹	1.92 × 10⁻¹	5.85 × 10⁻²	9.45 × 10⁻²	9.83 × 10⁻²	7.43 × 10⁻¹
MO-BBO_ACO [9]	5.79 × 10⁻³	7.84 × 10⁻³	6.93 × 10⁻²	3.59 × 10⁻²	3.79 × 10⁻²	5.53 × 10⁻²	2.14 × 10⁻²	9.93 × 10⁻²	1.17 × 10⁻¹	1.79 × 10⁻¹
MOGWO [34]	9.62 × 10⁻²	4.98 × 10⁻²	2.73 × 10⁻¹	5.64 × 10⁻²	8.57 × 10⁻¹	3.29 × 10⁻¹	8.48 × 10⁻²	1.10 × 10⁻⁰	2.57 × 10⁻¹	2.04 × 10⁻⁰
MOBMA [46]	1.20 × 10⁻²	2.67 × 10⁻³	1.38 × 10⁻²	5.29 × 10⁻³	1.08 × 10⁻⁴	3.49 × 10⁻²	1.71 × 10⁻²	5.91 × 10⁻²	3.59 × 10⁻²	2.12 × 10⁻²
MOGNDO [33]	3.89 × 10⁻³	2.43 × 10⁻³	1.47 × 10⁻²	2.18 × 10⁻³	6.14 × 10⁻¹	4.92 × 10⁻²	7.63 × 10⁻³	4.91 × 10⁻³	3.87 × 10⁻³	1.29 × 10⁻²
CMOA	4.03 × 10⁻³	1.55 × 10⁻³	1.15 × 10⁻²	1.78 × 10⁻³	4.35 × 10⁻¹	2.33 × 10⁻²	6.96 × 10⁻³	2.87 × 10⁻³	1.52 × 10⁻³	6.66 × 10⁻³

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Alikhani Koupaei, J.; Ebadi, M.J. A Chaotic Decomposition-Based Approach for Enhanced Multi-Objective Optimization. Mathematics 2025, 13, 817. https://doi.org/10.3390/math13050817

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Alikhani Koupaei J, Ebadi MJ. A Chaotic Decomposition-Based Approach for Enhanced Multi-Objective Optimization. Mathematics. 2025; 13(5):817. https://doi.org/10.3390/math13050817

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Alikhani Koupaei, Javad, and Mohammad Javad Ebadi. 2025. "A Chaotic Decomposition-Based Approach for Enhanced Multi-Objective Optimization" Mathematics 13, no. 5: 817. https://doi.org/10.3390/math13050817

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Alikhani Koupaei, J., & Ebadi, M. J. (2025). A Chaotic Decomposition-Based Approach for Enhanced Multi-Objective Optimization. Mathematics, 13(5), 817. https://doi.org/10.3390/math13050817

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