A Novel Many-Objective Sine–Cosine Algorithm (MaOSCA) for Engineering Applications

Abstract

In recent times, numerous innovative and specialized algorithms have emerged to tackle two and three multi-objective types of problems. However, their effectiveness on many-objective challenges remains uncertain. This paper introduces a new Many-objective Sine–Cosine Algorithm (MaOSCA), which employs a reference point mechanism and information feedback principle to achieve efficient, effective, productive, and robust performance. The MaOSCA algorithm’s capabilities are enhanced by incorporating multiple features that balance exploration and exploitation, direct the search towards promising areas, and prevent search stagnation. The MaOSCA’s performance is evaluated against popular algorithms such as the Non-dominated sorting genetic algorithm-III (NSGA-III), the Multi-objective Evolutionary Algorithm based on Decomposition (MOEA/D) integrated with Differential Evolution (MOEADDE), the Many-objective Particle Swarm Optimizer (MaOPSO), and the Many-objective JAYA Algorithm (MaOJAYA) across various test suites, including DTLZ1-DTLZ7 with 5, 9, and 15 objectives and car cab design, water resources management, car side impact, marine design, and 10-bar truss engineering design problems. The performance evaluation is carried out using various performance metrics. The MaOSCA demonstrates its ability to achieve well-converged and diversified solutions for most problems. The success of the MaOSCA can be attributed to the multiple features of the SCA optimizer integrated into the algorithm.

1. Introduction

Numerous real-world situations involve multi-objective problems (MOPs) or many-objective problems (MaOPs). MOPs generally consist of two or three objectives, while MaOPs possess a higher objective dimension. In various domains such as industrial production scheduling, aerospace energy concerns, and logistical cost issues, MOPs (or MaOPs) frequently arise. These problems can be mathematically represented as MOPs (or MaOPs) and addressed using multi-objective evolutionary algorithms (MOEAs) or many-objective evolutionary algorithms (MaOEAs). In general, many-objective optimization problems are formulated as follows in Equation (1):

$$\genfrac{}{}{0pt}{}{\frac{MinF}{Max}F\left(\overrightarrow{x}\right)=\left\{f_1\left(\overrightarrow{x}\right),f_2\left(\overrightarrow{x}\right),\dots ,f_o\left(\overrightarrow{x}\right)\right\}}{\begin{array}{c}\mathrm{Subject} \mathrm{to}:g_i\left(\overrightarrow{x}\right)\ge 0,i=\mathrm{1,2},\dots ,m\\ h_i\left(\overrightarrow{x}\right)=0,i=\mathrm{1,2},\dots ,p\\ {Lb}_i\le x_i\le {Ub}_i,i=\mathrm{1,2},...,n\end{array}}$$

(1)

where [${Lb}_i$, ${Ub}_i$] denotes the lower and upper boundaries of an $i$th variable, $h_i$ shows the $i$th equality constraints, $g_i$ denotes ith inequality constraints, $p$ signifies the number of constraint limits, $m$ signifies the number of unconstraint limits, $o$ represents the total number of objective functions, and $n$ represents the number of design parameters. The essential definitions in Equations (2) to (5) are as follows:

Definition 1.

Pareto Optimality [1]:

$$\nexists \overrightarrow{y}\in X|F\left(\overrightarrow{y}\right)\prec F\left(\overrightarrow{x}\right)$$

(2)

Definition 2.

Pareto Dominance [1]:

$$\forall i\in \left\{\mathrm{1,2},\dots ,k\right\}:f_i\left(\overrightarrow{x}\right)\le f_i\left(\overrightarrow{y}\right)\wedge \exists i\in \left\{\mathrm{1,2},\dots ,k\right\}:f_i\left(\overrightarrow{x}\right)<f_i\left(\overrightarrow{y}\right)$$

(3)

Definition 3.

Pareto-optimal set [1]:

$$P_s≔\{x,y\in X|\exists F\left(\overrightarrow{y}\right)\succ F\left(\overrightarrow{x}\right)\}$$

(4)

Definition 4.

Pareto-optimal front [1]:

$$P_f≔\left\{F\left(\overrightarrow{x}\right)|\overrightarrow{x}\in P_s\right\}$$

(5)

Early multi-objective algorithms rely on Pareto dominance, employing the Pareto dominance relationship as the evaluative benchmark for two solutions throughout the iterative process to achieve the ultimate outcomes. The Non-dominated sorting genetic algorithm (NSGA-II) [1] represents a more traditional method, adding crowding distance as a supplementary indicator following Pareto dominance to improve diversity. The NSGA-III [2,3] uses a uniform distribution reference point to facilitate the optimal direction. The Non-dominated sorting genetic algorithm-III (NSGA-III) is a widely used many-objective optimization algorithm proposed by Kalyanmoy Deb and his colleagues in 2012. As research advances, the shortcomings of Pareto-based algorithms become more evident, especially when dealing with a large number of objectives. With the increase in objectives, the performance of Pareto-based algorithms progressively deteriorates in aspects such as search capacity and ultimate outcomes. This is mainly attributed to the expanding number of Pareto-optimal solutions as the objectives multiply. In certain cases, the entire population may consist of Pareto-optimal solutions. In these situations, relying on the Pareto dominance relationship as a yardstick for assessing solution quality loses its significance, leading to a decline in the performance of Pareto-based algorithms. In response to this challenge, researchers have developed various approaches, which can be generally grouped into three main types. These techniques aim to balance convergence and diversity, eventually producing Pareto-optimal solution sets that faithfully depict the Pareto frontier parametric and objective spaces, as illustrated in Figure 1.

The first strategy entails redefining Pareto domination or adding supplementary metrics to devise more suitable methods for tackling MaOPs. Examples include generalized Pareto dominance [4] and fuzzy-based techniques [5]. Importantly, both the crowding distance in the NSGA-II [1] and the reference vector in the NSGA-III [2] are part of this approach. Recently, associated algorithms have emerged, such as the grid-based evolutionary algorithm (GrEA) [6], which presents grid dominance as a secondary criterion for selecting solutions while retaining Pareto dominance as the primary indicator. Experimental outcomes demonstrate that the GrEA performs exceptionally well when managing MaOPs. Despite their enhanced performance, these methods also possess certain limitations, such as the propensity to converge to local optima.

The second strategy focuses on utilizing innovative metrics for selecting solutions. Rather than employing Pareto dominance, these methods depend on new indicators. One prevalent algorithm is the fast hypervolume-based algorithm (HypE) [7], which was developed using Monte Carlo simulation. The HypE’s optimization process strives to maximize the hypervolume value of the complete optimal solution set. Monte Carlo simulation is used to approximate exact hypervolume values, enabling a balance between the precision of the estimates and available computational resources [7]. Experimentation validates the efficacy of this technique. Additionally, the HPEA [8] and hpaEA [9] adopt a hyperplane strategy.

The third strategy is based on decomposition, as illustrated by the Evolutionary Algorithm based on Decomposition (MOEA/D) [10]. Various enhanced versions have been introduced in recent years, such as the MaOEA-RD [11]. Moreover, to overcome the constraints of generating reference vectors, the MOEA/D-URAW [12] can be updated throughout the iterative process.

Although evolutionary algorithms have demonstrated efficacy in tackling a variety of problems, they are not without limitations, prompting the proposal of diverse improvement strategies. These limitations may include issues related to scalability, the handling of many-objective problems, and difficulties in maintaining diversity, coverage, and convergence. One such strategy is the information feedback model (IFM) [13], which has shown encouraging results in single-objective evolutionary algorithms. Unlike many optimization algorithms, the MaOSCA recognizes the importance of this information and retains data from previous populations. First, the MaOSCA algorithm is compared to the NSGA-III [2], MOADDE [14], MaOPSO [15], and MaOJAYA [16] on DTLZ1-DTLZ7 [17] problems. The experimental results indicate that the MaOSCA outperforms several cutting-edge multi-objective algorithms. This paper includes the development of the novel Many-objective Sine–Cosine Algorithm (MaOSCA), its application to engineering problems, and the insights gained from its performance compared to other existing algorithms.

The paper is organized as follows: Section 2 provides an overview of the related work. Section 3 presents the Single- and Many-objective Sine–Cosine algorithms. Section 4 outlines the experimental setup, including benchmark problems, performance metrics, and parameter settings. The results discuss the performance of the MaOSCA vis-à-vis other popular algorithms. Finally, Section 5 concludes the paper and suggests future research directions.

2. Related Work

Optimization problems involving multiple conflicting objectives, known as many-objective optimization problems, frequently arise in various domains such as engineering design, machine learning, and scheduling [18,19,20]. These problems require simultaneously optimizing multiple objectives, often leading to a set of trade-off solutions rather than a single optimal solution. Metaheuristic optimization algorithms have been widely employed to address such complex problems, as they can effectively explore the search space and identify a set of Pareto-optimal solutions. Real-life problems predominantly involve multiple objectives, necessitating a holistic approach that considers aspects such as production rate, process time, and technology [21,22]. Solving multi-objective problems (MOPs) is inherently difficult, as achieving optimal solutions for each objective concurrently is not possible [23,24,25]. Addressing multi-objective problems requires weighing the benefits and drawbacks according to the specific context, as highlighted by the well-known no-free-lunch theorem [26]. Multi-objective evolutionary algorithms (MOEAs) have been utilized to tackle a variety of practical problems. For example, Mohammadi et al. [27] developed a multi-objective genetic algorithm to optimize model predictive control (MPC) weightings in driving simulators. This method identifies optimal tuning for MPC cost function weights while minimizing user burden and incorporating user satisfaction feedback. However, MOEAs exhibit limited performance when handling many-objective problems, mainly due to the exponential increase in non-dominated solutions as objectives grow. Consequently, there is a growing interest in MaOEAs. Numerous MaOEAs are designed based on original MOEAs, incorporating new strategies such as the NSGA-III [2] and the UMOEA/D [28]. The effectiveness of these new strategies has been proven, motivating researchers to explore diverse strategies for algorithm development. For instance, Sieni et al. [29] described the NSGA (M-NSGA) and Sieni et al. [30] described the self-adaptive migration NSGA (SA-M-NSGA).

Despite these advancements, few researchers have exploited historical information from previous iterations’ individuals to enhance algorithm performance. Real-life problems often involve more than three objectives, with decision variables numbering in the thousands. To address real-world requirements, it is crucial to improve algorithms capable of solving many-objective problems.

In this paper, we integrate the information feedback model (IFM) [13] into the original Sine–Cosine Algorithm framework [31,32] to tackle many-objective optimization problems. The model’s underlying concept involves using information from prior iterations of individuals to influence offspring generation. The primary process consists of two steps. First, in the current iteration, an individual is generated using the original SCA algorithm. Subsequently, the historical information from individuals is randomly or fixedly chosen to produce a newly updated result for the information feedback model (IFM), which has been effectively applied to a variety of single-objective optimization algorithms, such as the PSO [33], the ACO [34], and the bat algorithm (BA) [35]. The experimental results indicate that incorporating information feedback models improves the performance of these algorithms. To date, there has been no reported application of IFM to many-objective optimizers, which has led us to investigate their potential integration with many-objective optimizers. IFMs depend on the fitness function and use information from previous iterations to update current individuals while maintaining a balance between current individuals and past information.

In this paper, we present the Many-objective Sine–Cosine Algorithm (MaOSCA), which is an extension of the SCA specifically designed for solving many-objective optimization problems. In the MaOSCA, the original SCA is modified to handle multiple conflicting objectives by incorporating Pareto-based ranking and reference points with an information feedback mechanism to maintain diversity among the solutions. The algorithm maintains a set of non-dominated solutions (i.e., Pareto-optimal solutions) and iteratively updates them using the sine and cosine components. The MaOSCA has demonstrated promising performance in various many-objective optimization problems, outperforming other popular algorithms in some cases.

3. Algorithm Design

3.1. Sine–Cosine Algorithm (SCA)

The Sine–Cosine Algorithm (SCA) is a nature-inspired metaheuristic optimization algorithm that draws its inspiration from the mathematical concepts of sine and cosine functions. Proposed by Seyedali Mirjalili in 2016 [32], the SCA is designed to effectively solve complex optimization problems and constrained and unconstrained challenges. The SCA is based on the principle that sine and cosine functions can be utilized to model the oscillatory behavior of particles searching for an optimal solution in a search space. The algorithm simulates the exploration and exploitation processes by adjusting the amplitude and frequency of the sine and cosine functions, leading to efficient convergence towards an optimal solution as presented below:

$$Z_i^{t+1}=\left\{\begin{array}{c}{Z}_i^t+r_1\ast \sin \left(r_2\right)\ast \left|r_3{P}_i^t-Z_i^t\right|,r_4<0.5\\ Z_i^t+r_1\ast \cos \left(r_2\right)\ast \left|r_3{P}_i^t-Z_{ii}^t\right|,r_4\ge 0.5\end{array}\right.$$

(6)

The current solution and the destination point locations are $Z_i^t$ and $P_i^t$, respectively, in the $t$th dimension; $r_2\in \left[\mathrm{0,2}\pi \right]$; $r_3=[-\mathrm{2,2}]$ and $r_1$ represents the next search region, which is updated as follows:

$$r_1=a-t\frac{a}{{Iter}_{max}}$$

(7)

In the above equation, $t$ and ${Iter}_{max}$ are the current and maximum iteration numbers, respectively, while $a$ is a constant controlled by the designer.

Here is a step-by-step explanation of the Sine–Cosine Algorithm:

Population Initialization: A set of candidate solutions, also known as particles, is generated randomly within the problem’s search space. Each solution is represented as a vector of decision variables.

Amplitude and Frequency: The SCA uses amplitude “r1” and a variable “r2” to define the sine and cosine functions, respectively. The amplitude linearly decreases over iterations, while “r2” is a random number in the range [0, 1]. These parameters guide the particles’ movement within the search space.

Exploration and Exploitation: The algorithm balances exploration and exploitation by adjusting the sine and cosine functions. Exploration refers to the global search, where particles move randomly in the search space to find new promising areas. Exploitation is the local search, in which particles update their positions based on the best solution found so far to fine-tune the search.

Position Update: Particles’ positions are updated using the sine and cosine functions, and the best solution is found. The updated positions are then evaluated to determine the new best solution using Equation (6).

Stopping Criteria: The algorithm iterates through the above steps until a predefined stopping criterion is met. This could be reaching a maximum number of iterations, achieving a desired level of solution accuracy, or any other user-defined criteria.

3.2. Many-Objective Sine–Cosine Algorithm (MaOSCA)

The Many-objective Sine–Cosine Algorithm (MaOSCA) predominantly utilizes reference points used in the NSGA-III [2] and reduces the computational complexity by introducing a new information feedback mechanism (IFM) for addressing many-objective optimization problems (MaOPs)

Initially, establish reference points and a population (size N) using Das and Dennis’s method. Generate the offspring population $Q_t$ using SCA operators on the parent population $P_t$. Initially, employ the original SCA algorithm and the parent population $P_t$ to produce $u_i^{t+1}$ and assess its fitness value using test problems. Compute according to $x_i^{t+1}$ and Equation (8), where $x_i^{t+1}$ represents the ith individual at generation (t + 1) and ${\partial }_1$ and ${\partial }_2$ are weight coefficients satisfying ${\partial }_1+{\partial }_2=1$. ${\partial }_1$ and ${\partial }_2$ can be calculated using Equation (9).

$$x_i^{t+1}={\partial }_1{u}_i^{t+1}+{\partial }_2{x}_k^t$$

(8)

$$\begin{array}{c}{\partial }_1=\frac{f_k^t}{f_i^{t+1}+f_k^t}\\ {\partial }_2=\frac{f_i^{t+1}}{f_i^{t+1}+f_k^t}\end{array}$$

(9)

The current generation is t, and $x_i^t \mathrm{and} x_k^t$ represent the $i$th/$k$th individual at the $t$th generation. $u_i^{t+1}$ represents the $i$th individual at the (t + 1)-th generation generated through the MaOSCA, and the fitness value of $u_i^{t+1}$ is $f_i^{t+1}$. This information feedback model utilizes the best global solution found so far to direct the search process towards promising areas in the search space. By incorporating information about the best solution, the algorithm can focus on areas with higher potential for improvement. Next, arrange solutions in a non-dominant sequence, assuming it is the t-th iteration of the population. Merge $P_t$ and $Q_t$ $(R_t=P_t\cup Q_t)$ to generate $R_t$, which would be a new 2N-sized population. Rank $R_t$ solutions and categorize them into various non-dominant levels ($F_1,F_2,\dots ,F_l,\dots ,F_w$). Subsequently, build a new population $S_t$ by iteratively transferring non-dominant levels into it until $\left|S_t\right|\ge N$ is first reached. If $F_l$ attains $\left|S_t\right|\ge N$, continue to the following step. Lastly, select the solutions from the $S_t$ population to create the forthcoming parent population $P_{t+1}$. If $\left|S_t\right|=N$ is achieved, treat it as the immediate next parent population $P_{t+1}$. Otherwise, incorporate the solutions into $S_t$\$F_l$ and choose the remaining solutions from $P_{t+1}$ according to the selection mechanism.

The selection mechanism in the MaOSCA is based on reference points. Start by identifying the population’s ideal point from $S_t$, normalizing the population and reference points, and computing each solution’s vertical distance to each reference line in $S_t$. Link the solution to the reference point with a minimal vertical distance. Utilize a novel niche-preserving operation to pick individuals in $F_l$. The niche count ${\rho }_j$ equals the number of individuals of the $S_t$\$F_l$ populations connected to the jth reference point. To boost the MaOSCA’s diversity, locate reference point $i$ with the lowest niche count value ${\rho }_i$ and inspect if any individuals in $F_l$ are linked to reference point i. If so, select one individual as a member of $P_{t+1}$ based on the value of ${\rho }_i$. If not, redo the niche-preserving operation with another reference point exhibiting the smallest niche count value until $\left|P_{t+1}\right|=N$ is fulfilled. Finally, ascertain if the termination condition has been satisfied. If so, output the final population; otherwise, revert to the second step, as shown in Figure 2.

Step 1: Initialization. Generate a reference point set $\Lambda$, randomly initialize the population $P_0$, and establish the ideal point $Z_{min}$.

Step 2: Updating. Assume it is the t-th generation of the population.

Step 2.1: Offspring population generation $Q_t$. Initially, employ the original SCA algorithm and the parent population $P_t$ to produce $u_i^{t+1}$ with the IFM. Use the historical information of s individuals from previous generations at a time to influence the generation of a new individual and reduce the computational complexity.

Step 2.2: Acquire $R_t$ by combining the parent population $P_t$ with the offspring population $Q_t$. To obtain the subsequent parent population $P_{t+1}$, first execute non-dominated sorting on $R_t$. Through this sorting, $R_t$ is partitioned into w distinct non-dominant levels ($F_1,F_2,\dots ,F_l,\dots ,F_w$). Starting from $F_1$, transfer one non-dominant level into the population $S_t$ each time until $F_l$ is reached, which is the first level to achieve $\left|S_t\right|\ge N$. Then, verify if $\left|S_t\right|=N$ is met. If so, it will directly be considered the next generation $P_{t+1}$. Otherwise, include the solutions in $S_t$\$F_l$ in $P_{t+1}$ and choose the remaining solutions $P_{t+1}$ from $F_l$.

Step 2.3: To select individuals from $F_l$, employ the selection mechanism, reference point information, normalization operation, and niche technology.

Step 2.4: Assess if the termination condition $t=t+1$ has been met. If not, return to Step 2.2. If it has, proceed to Step 3.

Step 3: Output. Generate the final population $P_{t+1}$.

The MaOSCA incorporates an information feedback model to improve its performance in many-objective optimization problems. By utilizing these feedback models, the MaOSCA can effectively guide the search process, ensuring a balance between exploration and exploitation. This leads to improved convergence and diversity preservation, which are crucial aspects of many-objective optimization.

4. Evaluation and Interpretation of Results

To validate the effectiveness of the proposed MaOSCA algorithm in addressing many-objective optimization challenges, a series of tests were performed on unconstrained DTLZ1–DTLZ7 benchmarks [17] with 5, 9, and 15 objectives. The efficacy of the MaOSCA optimizer was measured through various quality indicators. The outcomes of the MaOSCA algorithm were contrasted with four cutting-edge optimization techniques, including the NSGAIII [2], MOEADDE [14], MaOPSO [15], and MaOJAYA [16]. The upcoming subsection provides a concise overview of the selected test problems and performance measures. Subsequently, the specifics of the experimental configurations for the NSGA-III, MOEADDE, MaOPSO, and MaOJAYA comparison algorithms are outlined. Finally, the experimental outcomes, encompassing the empirical observations and comparative analyses, are presented and deliberated.

4.1. Evaluation Benchmarks

In this study, we employ a diverse set of DTLZ1–DTLZ7 benchmark problems representing various complexities and characteristics to evaluate the performance of the MaOSCA. These DTLZ problems are chosen as they represent optimal benchmarks for better comparisons, thereby evaluating the effectiveness of the MaOSCA optimizer in addressing many-objective optimization problems (MOOPs). The number of objectives, M, includes {5, 9, 15}. The test suites display a diverse range of attributes, including linear, combined, partially separable, concave, multimodal, disjoint, biased, and degenerate Pareto-optimal fronts, all of which can be found in DTLZ.

4.2. Evaluation Metrics

To evaluate the performance of the MaOSCA, we used five widely accepted performance metrics: inverted generational distance (IGD), hypervolume (HV), generational distance (GD), runtime (RT), and spread (Δ/SD) metrics [2]. The efficiency of many-objective optimization algorithms is typically evaluated through the various parameters used. IGD and HV offer better convergence to the true Pareto front with diversity preservation, while the GD and SD (Δ) metrics serve as convergence and diversity measures. The runtime (RT) metric is assessed based on the average simulation time, thereby demonstrating an algorithm’s computational efficiency in solving the optimization problem. These metrics enabled us to assess the convergence and diversity preservation capabilities of the MaOSCA and compare its performance with the NSGAIII, MOEADDE, MaOPSO, and MaOJAYA. The equations for calculating all the quality indicators are displayed in Figure 3.

4.3. Experimental Configuration

The parameter settings for the MaOSCA, NSGAIII, MOEADDE, MaOPSO, and MaOJAYA were chosen based on a preliminary parameter tuning study. The selected parameters were kept consistent across all algorithms for a fair comparison. The main parameters considered were a 30-time run, a population size of 100, and a maximum function evaluation of 30,000. These parameter settings were used in the experiments conducted on the DTLZ1-DTLZ7 benchmark problems to assess the performance of the MaOSCA in comparison to the NSGAIII, MOEADDE, MaOPSO, and MaOJAYA.

Scripts for all the algorithms were developed in-house in MATLAB v2019. The computational experiments were run on a Windows platform with an Intel (R) Core (TM) i7 CPU @3.40 GHz and 24 GB of RAM.

4.4. Evaluation of DTLZ Test Problems

The performance of the MaOSCA was compared with that of the NSGAIII, MOEADDE, MaOPSO, and MaOJAYA using the benchmark problems in the DTLZ1–DTLZ7 test suite with 5, 9, and 10 objectives in terms of convergence, divergence, and computational cost metrics.

Table 1. GD results by MaOSCA, NSGAIII, MOEADDE, MaOPSO, and MaOJAYA algorithms on DTLZ1-DTLZ7 test suite with 5, 9, and 15 objectives.

Problem	M	D	MaOSCA	NSGAIII	MOEADDE	MaOPSO	MaOJAYA
DTLZ1	5	9	0.4312 (±0.561)	0.5402 (±0.448)	3.8244 (±0.78)	0.2789 (±0.475)	5.3629 (±1.47)
	9	13	0.1438 (±0.133)	13.95 (±4.22)	1.2855 (±0.699)	0.7581 (±0.668)	1.2667 (±1.43)
	15	19	0.1183 (±0.119)	33.224 (±2.82)	1.5799 (±1.25)	1.7768 (±3.03)	3.5872 (±1.04)
DTLZ2	5	14	0.0063 (±0.0002)	0.0066 (±0.0023)	0.0282 (±0.0052)	0.0208 (±0.0034)	0.0424 (±0.004)
	9	18	0.0197 (±0.0008)	0.0886 (±0.006)	0.0306 (±0.0025)	0.0596 (±0.0238)	0.0353 (±0.0014)
	15	24	0.0557 (±0.0007)	0.0762 (±0.075)	0.0707 (±0.0181)	0.0581 (±0.0194)	0.0859 (±0.0214)
DTLZ3	5	14	3.281 (±1.07)	14.914 (±2.89)	18.768 (±1.67)	18.415 (±15.9)	25.826 (±4.31)
	9	18	2.3056 (±0.87)	119.4 (±8.88)	5.9621 (±4.7)	16.932 (±14.7)	21.912 (±1.02)
	15	24	5.1607 (±1.29)	188.15 (±36.6)	2.3762 (±3.99)	0.0388 (±0.0182)	29.469 (±4.51)
DTLZ4	5	14	0.006 (±0.0013)	0.0099 (±0.0017)	0.0082 (±0.0006)	0.0065 (±0.0006)	0.009 (±0.0004)
	9	18	0.0196 (±0.0015)	0.1444 (±0.0121)	0.0174 (±0.0019)	0.0889 (±0.0749)	0.0143 (±0.0006)
	15	24	0.0515 (±0.0038)	0.3105 (±0.0263)	0.0558 (±0.0073)	0.0384 (±0.0112)	0.0456 (±0.0168)
DTLZ5	5	14	0.047 (±0.0165)	0.1292 (±0.0096)	0.2291 (±0.0294)	0.2424 (±0.0417)	0.243 (±0.0254)
	9	18	0.0347 (±0.0014)	0.1616 (±0.0118)	0.132 (±0.0112)	0.1228 (±0.213)	0.1098 (±0.0315)
	15	24	0.0324 (±0.0284)	0.3182 (±0.0239)	0.0135 (±0.0115)	0.2005 (±0.347)	0.0497 (±0.0101)
DTLZ6	5	14	0.1956 (±0.0248)	0.527 (±0.032)	0.5092 (±0.0042)	0.6445 (±0.0762)	0.5933 (±0.139)
	9	18	0.2433 (±0.0327)	0.9672 (±0.0184)	0.2334 (±0.0095)	1.0195 (±0.134)	0.3319 (±0.128)
	15	24	0.3069 (±0.0855)	1.69 (±0.0412)	0 (0)	0.5587 (±0.968)	0.2877 (±0.049)
DTLZ7	5	24	0.0476 (±0.0102)	0.0279 (±0.0058)	0.7771 (±0.226)	0.0416 (±0.0428)	0.6878 (±0.097)
	9	28	0.263 (±0.0982)	1.6127 (±0.142)	1.0144 (±0.375)	0.1737 (±0.0683)	1.2591 (±0.0976)
	15	34	0.3597 (±0.0673)	8.4952 (±0.917)	2.0948 (±0.207)	0.4783 (±0.0712)	2.6417 (±0.547)

and

Table 2. SD results by MaOSCA, NSGAIII, MOEADDE, MaOPSO, and MaOJAYA algorithms on DTLZ1-DTLZ7 test suite with 5, 9, and 15 objectives.

Problem	M	D	MaOSCA	NSGAIII	MOEADDE	MaOPSO	MaOJAYA
DTLZ1	5	9	0.824 (±0.783)	0.9161 (±0.186)	1.5301 (±0.118)	0.8067 (±0.924)	1.058 (±0.381)
	9	13	0.5455 (±0.29)	0.678 (±0.147)	1.5669 (±0.394)	1.6479 (±0.287)	1.4848 (±0.628)
	15	19	0.8676 (±0.104)	0.9724 (±0.115)	1.4307 (±0.222)	1.8118 (±5.5)	1.4523 (±0.56)
DTLZ2	5	14	0.1185 (±0.0042)	0.1672 (±0.0065)	1.1912 (±0.0758)	0.1932 (±0.0072)	1.0228 (±0.0783)
	9	18	0.1863 (±0.0135)	0.2432 (±0.0181)	0.9476 (±0.118)	0.5282 (±0.136)	0.8709 (±0.237)
	15	24	0.6986 (±0.0869)	1.1216 (±0.356)	1.0689 (±0.0246)	1.2823 (±0.111)	1.0157 (±0.0194)
DTLZ3	5	14	0.5898 (±0.257)	0.7484 (±0.0868)	1.6056 (±0.0645)	1.5573 (±0.135)	0.8975 (±0.132)
	9	18	0.4515 (±0.0838)	0.3262 (±0.0396)	1.8334 (±0.137)	1.5845 (±0.675)	0.9657 (±0.0788)
	15	24	0.7809 (±0.0481)	0.8711 (±0.205)	1.0872 (±0.0486)	1.2572 (±0.0492)	1.4375 (±0.312)
DTLZ4	5	14	0.235 (±0.195)	0.209 (±0.0293)	1.8473 (±0.0452)	0.3003 (±0.122)	1.6363 (±0.0257)
	9	18	0.1303 (±0.0254)	0.1761 (±0.0119)	1.2036 (±0.109)	0.5143 (±0.203)	1.1875 (±0.0677)
	15	24	0.5972 (±0.113)	0.5354 (±0.0117)	1.0556 (±0.0679)	0.7882 (±0.113)	1.0925 (±0.0714)
DTLZ5	5	14	0.3971 (±0.0635)	0.3364 (±0.0138)	1.3799 (±0.0152)	0.6794 (±0.122)	1.2332 (±0.101)
	9	18	0.415 (±0.0504)	0.4118 (±0.0528)	1.6244 (±0.126)	1.2018 (±0.343)	1.5054 (±0.111)
	15	24	1.3133 (±0.348)	0.8704 (±0.148)	2.3754 (±1.12)	1.7192 (±1.25)	1.1872 (±0.109)
DTLZ6	5	14	0.4312 (±0.0544)	0.5891 (±0.0242)	1.6819 (±0.0877)	0.4279 (±0.0393)	1.4817 (±0.199)
	9	18	0.5227 (±0.0073)	0.2024 (±0.0055)	1.8463 (±0.076)	0.7049 (±0.184)	1.4945 (±0.0947)
	15	24	0.8707 (±0.0692)	0.6142 (±0.067)	2.6485 (±0.0314)	1.5048 (±0.829)	1.0965 (±0.0445)
DTLZ7	5	24	0.3326 (±0.0471)	0.4256 (±0.0214)	1.1555 (±0.179)	1.1516 (±0.194)	1.0674 (±0.124)
	9	28	0.3986 (±0.0406)	0.5774 (±0.141)	1.0963 (±0.0743)	4.049 (±2.62)	0.989 (±0.12)
	15	34	0.8912 (±0.0234)	0.9103 (±0.0235)	1.3099 (±0.181)	3.3837 (±4.61)	1.2514 (±0.183)

display the comparative outcomes for GD and SD, respectively. Convergence and diversity were assessed using the IGD and HV indicators depending on the DTLZ benchmarks, with the results presented in

Table 3. IGD results by MaOSCA, NSGAIII, MOEADDE, MaOPSO, and MaOJAYA algorithms on DTLZ1-DTLZ7 test suite with 5, 9, and 15 objectives.

Problem	M	D	MaOSCA	NSGAIII	MOEADDE	MaOPSO	MaOJAYA
DTLZ1	5	9	0.5076 (±0.315)	0.9788 (±0.52)	2.2224 (±0.35)	0.0745 (±0.0052)	0.2735 (±0.0428)
	9	13	0.475 (±0.167)	19.261 (±3.44)	1.7302 (±2)	0.1362 (±0.0043)	0.2927 (±0.173)
	15	19	0.5023 (±0.275)	56.1 (±25.3)	2.8889 (±3.37)	0.1944 (±0.0268)	0.996 (±0.914)
DTLZ2	5	14	0.2166 (±0.0007)	0.2199 (±0.0073)	0.4284 (±0.0049)	0.2623 (±0.0069)	0.458 (±0.0356)
	9	18	0.4338 (±0.0346)	0.7918 (±0.0412)	0.6934 (±0.0544)	0.5921 (±0.102)	0.7252 (±0.0923)
	15	24	0.7873 (±0.0531)	0.8367 (±0.18)	1.0484 (±0.0709)	1.1103 (±0.14)	1.1078 (±0.0058)
DTLZ3	5	14	10.637 (±2.95)	32.52 (±5.72)	9.7917 (±13.4)	1.4866 (±0.908)	26.765 (±22.5)
	9	18	10.716 (±3.45)	429.48 (±162)	1.6408 (±0.715)	1.1477 (±0.369)	4.4102 (±6.06)
	15	24	23.651 (±8.35)	374.01 (±232)	9.9361 (±15.4)	1.178 (±0.0134)	35.773 (±26.5)
DTLZ4	5	14	0.3621 (±0.242)	0.2358 (±0.0018)	0.474 (±0.0013)	0.4471 (±0.106)	0.4756 (±0.0327)
	9	18	0.4648 (±0.0371)	1.1116 (±0.0927)	0.8153 (±0.0319)	0.6391 (±0.0371)	0.8067 (±0.103)
	15	24	0.815 (±0.0384)	1.7389 (±0.182)	1.0916 (±0.0767)	0.7835 (±0.0163)	1.1219 (±0.0239)
DTLZ5	5	14	0.0428 (±0.0029)	0.1464 (±0.017)	0.094 (±0.013)	0.1095 (±0.0266)	0.0544 (±0.0174)
	9	18	0.0811 (±0.0263)	0.2765 (±0.0669)	0.0966 (±0.0213)	0.584 (±0.265)	0.0604 (±0.0048)
	15	24	0.1047 (±0.0354)	0.6118 (±0.246)	0.259 (±0.26)	0.646 (±0.166)	0.1708 (±0.0229)
DTLZ6	5	14	0.0329 (±0.0001)	1.2422 (±0.546)	0.4056 (±0.429)	2.2123 (±1.16)	0.2128 (±0.261)
	9	18	0.0359 (±0.0087)	7.4301 (±0.975)	0.3286 (±0.145)	2.7745 (±0.602)	0.4732 (±0.252)
	15	24	0.0703 (±0.0016)	6.1177 (±1.3)	0.9623 (±0.714)	0.9101 (±0.406)	0.541 (±0.0312)
DTLZ7	5	24	1.02 (±0.0661)	0.4164 (±0.0025)	0.4781 (±0.0684)	1.1508 (±0.0993)	1.6128 (±0.979)
	9	28	1.4141 (±0.138)	5.7524 (±2.59)	2.2147 (±0.636)	2.1271 (±0.674)	3.7978 (±2.45)
	15	34	2.3098 (±0.111)	19.311 (±6.38)	5.3051 (±1.77)	2.9026 (±0.207)	10.36 (±4.7)

and

Table 4. HV results by MaOSCA, NSGAIII, MOEADDE, MaOPSO, and MaOJAYA algorithms on DTLZ1-DTLZ7 test suite with 5, 9, and 15 objectives.

Problem	M	D	MaOSCA	NSGAIII	MOEADDE	MaOPSO	MaOJAYA
DTLZ1	5	9	0.2102 (±0.351)	0.0008 (±0.0014)	0 (0)	0.9401 (±0.0166)	0.3233 (±0.0316)
	9	13	0.152 (±0.179)	0 (0)	0.2127 (±0.368)	0.7652 (±0.064)	0.5917 (±0.473)
	15	19	0.2814 (±0.409)	0 (0)	0.0925 (±0.16)	0.7057 (±0.0595)	0.1189 (±0.192)
DTLZ2	5	14	0.7401 (±0.0007)	0.7499 (±0.0147)	0.4236 (±0.0124)	0.5831 (±0.0119)	0.3572 (±0.0658)
	9	18	0.7491 (±0.0616)	0.1315 (±0.0656)	0.3885 (±0.0319)	0.5123 (±0.0531)	0.3569 (±0.0244)
	15	24	0.5437 (±0.0409)	0.4905 (±0.27)	0.1889 (±0.0236)	0.2391 (±0.1)	0.1412 (±0.0138)
DTLZ3	5	14	0.1047 (±0.181)	0 (0)	0 (0)	0 (0)	0.0296 (±0.0513)
	9	18	0.2079 (±0.222)	0 (0)	0.0974 (±0.169)	0 (0)	0.0594 (±0.0605)
	15	24	0.2289 (±0.0131)	0 (0)	0.1341 (±0.117)	0 (0)	0 (0)
DTLZ4	5	14	0.67 (±0.11)	0.7264 (±0.0078)	0.5208 (±0.0263)	0.611 (±0.0797)	0.5231 (±0.0198)
	9	18	0.8802 (±0.0387)	0.0569 (±0.0267)	0.5197 (±0.0277)	0.732 (±0.0505)	0.5642 (±0.0866)
	15	24	0.7389 (±0.0247)	0 (0)	0.2928 (±0.0321)	0.7686 (±0.0263)	0.3182 (±0.0597)
DTLZ5	5	14	0.0909 (±0.0113)	0.0379 (±0.016)	0.1196 (±0.0006)	0.1084 (±0.0018)	0.1188 (±0.0009)
	9	18	0.0881 (±0.0052)	0.0046 (±0.0074)	0.0971 (±0.001)	0.0916 (±0.0012)	0.0968 (±0.0017)
	15	24	0.092 (±0.0008)	0.0272 (±0.0314)	0.0873 (±0.0041)	0.0909 (0)	0.0466 (±0.0417)
DTLZ6	5	14	0.0392 (±0.0486)	0 (0)	0.1218 (±0.0005)	0 (0)	0.1172 (±0.0116)
	9	18	0.0079 (±0.0133)	0 (0)	0.0999 (±0.0001)	0 (0)	0.0972 (±0.0054)
	15	24	0.0307 (±0.0531)	0 (0)	0.0939 (±0.0003)	0.0611 (±0.0529)	0.0941 (±0.0001)
DTLZ7	5	24	0.0318 (±0.0309)	0.1777 (±0.0191)	0.0014 (±0.0015)	0.1393 (±0.0176)	0.0003 (±0.0001)
	9	28	0.0469 (±0.0126)	0.0001 (±0.0002)	0.0001 (±0.0001)	0.0041 (±0.0037)	0 (0)
	15	34	0.0309 (±0.0204)	0 (0)	0.1335 (0)	0.0001 (±0.0001)	0 (0)

, respectively.

Table 5. RT results by MaOSCA, NSGAIII, MOEADDE, MaOPSO, and MaOJAYA algorithms on DTLZ1-DTLZ7 test suite with 5, 9, and 15 objectives.

Problem	M	D	MaOSCA	NSGAIII	MOEADDE	MaOPSO	MaOJAYA
DTLZ1	5	9	1.5839 (±1.27)	7.125 (±0.843)	3.442 (±0.0276)	0.975 (±0.0586)	3.8261 (±0.0885)
	9	13	0.9498 (±0.0385)	8.9573 (±0.235)	3.5711 (±0.0964)	0.9269 (±0.049)	3.9969 (±0.0173)
	15	19	1.2184 (±0.0099)	6.7288 (±0.319)	3.4148 (±0.058)	1.6715 (±0.174)	3.6621 (±0.0344)
DTLZ2	5	14	1.1849 (±0.0307)	3.3841 (±4.42)	3.5069 (±0.0906)	0.8423 (±0.0097)	3.7923 (±0.0148)
	9	18	1.1583 (±0.052)	9.428 (±0.154)	3.5708 (±0.128)	1.0919 (±0.0822)	3.8016 (±0.102)
	15	24	1.3945 (±0.125)	3.4075 (±2.71)	3.4185 (±0.0588)	3.2486 (±1.35)	3.6233 (±0.0491)
DTLZ3	5	14	0.7513 (±0.0143)	6.9638 (±0.195)	3.6298 (±0.142)	1.0067 (±0.107)	3.9314 (±0.0987)
	9	18	0.9598 (±0.0188)	9.406 (±0.191)	3.6974 (±0.0486)	2.105 (±1.67)	3.9199 (±0.101)
	15	24	1.2522 (±0.0151)	6.5482 (±0.0989)	3.5396 (±0.0574)	4.5001 (±0.574)	3.8462 (±0.129)
DTLZ4	5	14	1.1093 (±0.0801)	9.0846 (±0.0856)	3.6915 (±0.025)	2.8377 (±0.523)	3.9762 (±0.0444)
	9	18	1.2388 (±0.0364)	10.404 (±0.0677)	4.3269 (±0.323)	1.6983 (±0.126)	4.1482 (±0.0711)
	15	24	1.3758 (±0.0393)	6.8722 (±0.248)	3.6542 (±0.0202)	1.3693 (±0.0726)	3.8139 (±0.0801)
DTLZ5	5	14	0.8137 (±0.0267)	8.8386 (±0.0987)	3.8432 (±0.296)	2.3402 (±0.15)	3.9709 (±0.0414)
	9	18	0.8398 (±0.0272)	9.7619 (±0.0656)	3.7695 (±0.0771)	5.8826 (±2.73)	4.0757 (±0.0882)
	15	24	1.116 (±0.0271)	6.656 (±0.0416)	3.6226 (±0.0879)	5.4733 (±1.12)	3.8067 (±0.0683)
DTLZ6	5	14	0.911 (±0.0359)	8.6916 (±0.0337)	3.817 (±0.109)	2.7058 (±0.341)	4.1625 (±0.0561)
	9	18	0.9546 (±0.0174)	10.435 (±0.0445)	3.9033 (±0.13)	2.3165 (±0.149)	4.0934 (±0.075)
	15	24	1.3514 (±0.119)	6.8517 (±0.0533)	3.8025 (±0.179)	5.0249 (±2.11)	3.9948 (±0.129)
DTLZ7	5	24	0.8936 (±0.0215)	8.5428 (±0.0614)	3.6669 (±0.0906)	3.4362 (±0.798)	3.95 (±0.0607)
	9	28	0.9593 (±0.0066)	10.341 (±0.289)	3.7188 (±0.0616)	5.0424 (±0.0757)	3.9639 (±0.0387)
	15	34	1.2661 (±0.0114)	6.8473 (±0.215)	3.7129 (±0.177)	5.9195 (±0.448)	3.7478 (±0.0584)

shows the computational efficiency in terms of run time. The best result in each instance is highlighted in blue within each table cell. To further emphasize the advantages of the proposed MaOSCA algorithm, the Pareto front and decision variable’s front obtained by each algorithm on DTLZ1–DTLZ7 with 5, 9, and 10 objectives are visualized in Figure 4 and Figure 5. The results indicate that the MaOSCA Pareto front is well distributed in most cases.

The MaOSCA algorithm surpasses the convergence performance in comparison to the NSGAIII, MOEADDE, MaOPSO, and MaOJAYA for the GD indicator at 12/21, 1/21, 3/21, 4/21, and 1/21, respectively. Regarding the diversity result, the MaOSCA outperforms the NSGAIII, MOEADDE, MaOPSO, and MaOJAYA for the spread indicator at 10/21, 8/21, 0/21, 3/21, and 0/21, respectively. The MaOSCA optimizer also excels in balancing convergence and diversity results compared to the NSGAIII, MOEADDE, MaOPSO, and MaOJAYA for the IGD metrics at 11/21, 2/21, 0/21, 7/21, and 1/21, respectively, and the HV indicator at 9/21, 3/21, 4/21, 4/21, and 1/21, respectively. Moreover, the MaOSCA algorithm exhibits lower computational complexity than the NSGAIII, MOEADDE, MaOPSO, and MaOJAYA for the RT indicator at 16/21, 0/21, 0/21, 5/21, and 0/21, respectively, as shown in Table 5. The quality indicator performance indicates that the MaOSCA optimizer achieves a superior balance between convergence and diversity maintenance while requiring a minimum computational cost.

DTLZ1-DTLZ7 is a scalable benchmark that poses greater challenges than classical and ZDT test suites. Benchmark functions from DTLZ1 to DTLZ4 are multimodal, making it difficult to converge to the true Pareto front. However, the MaOSCA attains good convergence and diversity of solutions compared to competing optimizers. DTLZ5 and DTLZ6 feature degenerate PFs, making it easier to converge than to distribute non-dominated solutions (NDS) along the complete PFs. Competing optimizers struggle to spread NDS along the full PF. The MaOPSO fails to explore the end part of the true PF on both problems. In contrast, the MaOSCA covers the whole PF, including the last results, achieving both convergence and diversity on DTLZ5 and DTLZ6. DTLZ7, a benchmark function with a disconnected Pareto front, combines convex and concave types of PFs and features a disconnected search space. On DTLZ benchmarks, the performance of the NSGAIII, MOEADDE, MaOPSO, and MaOJAYA does not meet expectations, whereas the MaOSCA exhibits significantly better results. On the DTLZ test suite, the MaOSCA again exceeds in terms of convergence and diversity and is on par with other algorithms for most problems. The DTLZ test suite results reveal that the MaOSCA successfully handles mixed and non-separable data. To assess the scalability of the MaOSCA, we conducted additional experiments with an increasing number of objectives and decision variables. The results indicate that the MaOSCA maintains its performance advantage over the NSGA-III, MOEADDE, MaOPSO, and MaOJAYA as the number of objectives and decision variables increases, showcasing its robustness and scalability. A deeper analysis of the results reveals that the integration of the IFM in the MaOSCA significantly contributes to its improved performance.

4.5. Discussion of the Many-Objective Engineering Problems

In this section, the MaOSCA optimizer is employed to solve the car cab design, water resources management, car side impact, marine design, and 10-bar truss design. They are compared with the NSGA-III, MOEADDE, MaOPSO, and MaOJAYA algorithms.

Table 6. GD/SD/IGD/HV/RT results taken by MaOSCA, NSGAIII, MOEADDE, MaOPSO, and MaOJAYA algorithms on RWMOP1-RWMOP5 engineering design problems with 9, 5, and 4 objectives.

GD Metric
Problem	MaOSCA	NSGAIII	MOEADDE	MaOPSO	MaOJAYA
RWMOP1	70,741 (±16,700)	3730.3 (±48,900)	470.79 (±9050)	1348.7 (±31,400)	12.29 (±208)
RWMOP2	1370 (±57,200)	4569.8 (±1100)	16.9 (±983)	2282.1 (±1750)	55.29 (±28.5)
RWMOP3	1006.1 (±20,700)	3628.7 (±1550)	482.98 (±159)	55,517 (±19,300)	10.44 (±309)
RWMOP4	9499.8 (±148)	31.92 (±22.4)	6.06 (±1.34)	142.84 (±8750)	17.97 (±2.51)
RWMOP5	3081.5 (±1540)	12.21 (±417)	240.16 (±280)	3582.6 (±1400)	23.13 (±17.7)
SD Metric
Problem	MaOSCA	NSGAIII	MOEADDE	MaOPSO	MaOJAYA
RWMOP1	48.25 (±334)	68.07 (±775)	1.02 (±701)	64.02 (±10.6)	76.48 (±605)
RWMOP2	32.84 (0)	76.82 (±11.2)	1.01 (±167)	74.9 (±26.4)	99.31 (±139)
RWMOP3	50.17 (±547)	66.98 (±555)	1.06 (±423)	66.2 (±893)	93.36 (±379)
RWMOP4	68.7 (±778)	93.08 (±14)	1.67 (±12.5)	95.83 (±975)	1.27 (±12.5)
RWMOP5	41.94 (±12.3)	1.38 (±19.9)	93.75 (±88.6)	51.91 (±749)	1.33 (±29.9)
IGD Metric
Problem	MaOSCA	NSGAIII	MOEADDE	MaOPSO	MaOJAYA
RWMOP1	155.28 (±4390)	506.67 (±172)	41.61 (±10.5)	584.33 (±593)	68.29 (±10.9)
RWMOP2	46.25 (±29.6)	349.07 (±5220)	83.29 (±689)	11.63 (±11.5)	1.27 (±34.5)
RWMOP3	147.83 (±3060)	419.09 (±136)	40.74 (±14)	114.82 (±1710)	66.96 (±11.2)
RWMOP4	16.28 (±15.8)	55.71 (±12.7)	5.63 (±2.48)	35.8 (±20.2)	19.06 (±1.29)
RWMOP5	276.03 (±134)	19.08 (±367)	7402.9 (±8410)	304.39 (±113)	32.07 (±21.9)
HV Metric
Problem	MaOSCA	NSGAIII	MOEADDE	MaOPSO	MaOJAYA
RWMOP1	70.24 (±5300)	66.45 (±9000)	26.17 (±829)	67.86 (±348)	755.83 (±523)
RWMOP2	17.17 (±16.6)	39.18 (±8450)	0 (0)	32.86 (±10.3)	0 (0)
RWMOP3	58.83 (±2940)	57.36 (±118)	31.5 (±929)	61.08 (±2020)	13.84 (±823)
RWMOP4	58.73 (±12.9)	13.22 (±662)	0 (0)	44.75 (±14)	0 (0)
RWMOP5	35.42 (±173)	17.58 (±289)	38.33 (±105)	34.94 (±143)	16.36 (±15.1)
RT Metric
Problem	MaOSCA	NSGAIII	MOEADDE	MaOPSO	MaOJAYA
RWMOP1	2.47 (±10.4)	23.41 (±84.1)	15.69 (±12.8)	2.58 (±47.6)	17.06 (±1.03)
RWMOP2	2.25 (±643)	23.61 (±2.36)	15.38 (±19.3)	2.59 (±42)	16.14 (±15.1)
RWMOP3	2.17 (±14.6)	22.95 (±37.4)	15.6 (±46.7)	2.44 (±23.1)	16.19 (±36.7)
RWMOP4	2.26 (±53)	22.29 (±98.7)	15.26 (±26.1)	2.17 (±299)	15.9 (±26.2)
RWMOP5	1.66 (±604)	21.62 (±23.3)	15.32 (±28.6)	2.46 (±50.7)	17.95 (±3.8)
FRNT Test	17/6/2	2/18/5	2/20/3	4/16/5	0/25/0

shows the statistical results of the tested engineering design problems for the GD/SD/IGD/HV/RT performance metrics. The MaOSCA had better results in terms of convergence and divergence for maximum engineering design problems. Based on the results obtained, we can conclude that the MaOSCA algorithm achieved better results and a balanced convergence and divergence capacity to solve complex engineering problems with many objectives.

4.5.1. The Car Cab Design (RWMOP1)

The car cab design (RWMOP1) is a nine-objective constrained optimization problem [36]. The detailed objective function and constraints are presented in Appendix A.1.

Figure 6 shows the obtained Pareto front and dimension curves for car cab design problems with nine objective functions using the MaOSCA, NSGA-III, MOEADDE, MaOPSO, and MaOJAYA algorithms. The PF results of the proposed MaOSCA optimizer and the NSGA-III, MOEADDE, MaOPSO and MaOJAYA algorithms can be seen in this figure, with the MaOSCA obtaining better convergence and coverage than the NSGA-III, MOEADDE, MaOPSO, and MaOJAYA algorithms.

4.5.2. Water Resources Management (RWMOP2)

The water resources management (RWMOP2) problem encompasses five objectives aimed at optimizing water resource planning [37]. These objectives include minimizing the expenses related to the drainage system $\left(f_1\right)$, storage facilities $\left(f_2\right)$, treatment plants $\left(f_3\right)$, projected flood damage $\left(f_4\right)$, and anticipated financial losses resulting from flooding events $\left(f_5\right)$. The detailed objective functions and constraints are presented in Appendix A.2.

Figure 7 shows the obtained PF and dimension curves for water resources management problems with five objective functions using the MaOSCA, NSGA-III, MOEADDE, MaOPSO, and MaOJAYA algorithms. The PF results of the proposed MaOSCA optimizer and the NSGA-III, MOEADDE, MaOPSO, and MaOJAYA algorithms can be seen in this figure, with the MaOSCA obtaining better coverage and spread than the NSGA-III, MOEADDE, MaOPSO, and MaOJAYA algorithms.

4.5.3. Car Side-Impact Design (RWMOP3)

The car side-impact design (RWMOP3)—initially employed to minimize the automobile’s weight $\left(f_1\right)$, the passenger’s pubic-force exposure $\left(f_2\right)$, and the average velocity of the V-pillar responsible for bearing impact loads $\left(f_3\right)$, as well as the reduction in constraint violations $\left(f_4\right)$—has been previously described [36]. The detailed objective functions and constraints are presented in Appendix A.3.

Figure 8 shows the obtained PF and dimension curves for car side-impact design problems with four objective functions using the MaOSCA, NSGA-III, MOEADDE, MaOPSO, and MaOJAYA algorithms. The PF results of the proposed MaOSCA optimizer and the NSGA-III, MOEADDE, MaOPSO, and MaOJAYA algorithms can be seen in this figure, with the MaOSCA obtaining better well-distributed solutions than the NSGA-III, MOEADDE, MaOPSO, and MaOJAYA algorithms.

4.5.4. Marine Design (RWMOP4)

The first, second, and third objectives of the RWMOP4 problem are to minimize the transportation cost $\left(f_1\right)$, the lightship weight $\left(f_2\right)$, the annual cargo transport capacity $\left(f_3\right)$, and the constraint violations $\left(f_4\right)$ [38]. The details of the problem are presented in Appendix A.4.

Figure 9 shows the obtained PF and dimension curves for marine design problems with four objective functions using the MaOSCA, NSGA-III, MOEADDE, MaOPSO, and MaOJAYA algorithms. The PF results of the proposed MaOSCA optimizer and the NSGA-III, MOEADDE, MaOPSO, and MaOJAYA algorithms can be seen in this figure, with the MaOSCA obtaining better coverage and spread than the NSGA-III, MOEADDE, MaOPSO, and MaOJAYA algorithms.

4.5.5. 10-Bar Truss Design (RWMOP5)

The objectives of the RWMOP5 problem are to reduce truss mass, minimize compliance, optimize the first natural frequency, and decrease the maximum buckling factor [39]. The details of the problem are presented in Appendix A.5.

Figure 10 shows the obtained PF and dimension curves for 10-bar truss bar design problems with four objective functions using the MaOSCA, NSGA-III, MOEADDE, MaOPSO, and MaOJAYA algorithms. The PF results of the proposed MaOSCA optimizer and the NSGA-III, MOEADDE, MaOPSO, and MaOJAYA algorithms can be seen in this figure, with the MaOSCA obtaining better convergence and spread than the NSGA-III, MOEADDE, MaOPSO, and MaOJAYA algorithms.

In Table 6, the MaOSCA optimization method demonstrates superior convergence towards PF performance in comparison to the NSGAIII, MOEADDE, MaOPSO, and MaOJAYA, as evidenced by the GD metric results of 4/5, 0/5, 0/5, 1/25, and 0/5, respectively. Additionally, the MaOSCA technique exhibits improved coverage performance when compared to the aforementioned algorithms, with spread metric results of 5/5, 0/5, 0/5, 0/25, and 0/5, respectively. The MaOSCA method also displays a better balance between convergence and coverage performance relative to the NSGAIII, MOEADDE, MaOPSO, and MaOJAYA. This is indicated by the IGD metric results of 2/5, 1/5, 1/5, 1/25, and 0/5, respectively, and the HV metric results of 2/5, 1/5, 1/5, 1/25, and 0/5, respectively. Furthermore, the computational complexity of the MaOSCA algorithm is lower than its counterparts, as demonstrated by the RT metric results of 4/5, 0/5, 0/5, 1/25, and 0/5, respectively, as seen in Table 6. The statistical analyses reveal that the proposed MaOSCA algorithm maintains a more favorable balance between convergence and diversity while requiring fewer computational resources. In conclusion, the MaOSCA algorithm is comparatively the most effective solution for many-objective engineering design problems when evaluated against the NSGA-III, MOEADDE, MaOPSO, and MaOJAYA algorithms.

4.6. Constraints and Potential Solutions

The prospective advantages of the proposed MaOSCA method include: (1) obtaining a uniformly distributed PF without a single sensitive parameter adjustment; (2) employing a reference point approach in the MaOSCA to guarantee the best solution for each population while eliminating superfluous NDS; (3) maintaining a better balance between coverage and convergence; and (4) precisely assessing the distribution of NDS in the search space. Despite the promising performance of the MaOSCA across a broad spectrum of DTLZ1-DTLZ7 with 5-, 9-, and 15-objective problems, it possesses inherent limitations, as do most other algorithms.

It should be noted that the decision vector’s dimension for some degenerated and biased MOOPs studied herein is relatively small. Consequently, the MaOSCA’s performance may decline with increasing dimensionality. This may be attributed to the exponential increase in local PFs as the dimension grows, making it considerably more challenging for the algorithm to escape local PFs. A potential approach to improving the MaOSCA’s performance involves incorporating a deep learning strategy. This is due to the fact that these methods can discern the structure of Pareto solutions, subsequently accelerating convergence speed.

5. Conclusions

In prior many-objective optimization algorithms, the utilization of population information from preceding iterations was minimal or even nonexistent. As a result, valuable information was likely to be discarded during the optimization process. This study presented the Many-objective Sine–Cosine Algorithm (MaOSCA), which incorporates an information feedback model to improve its performance in many-objective optimization problems. These feedback models compensate for the deficiencies of neglecting useful information by preserving historical data and selecting individuals randomly or in a predetermined manner. The performance of the MaOSCA algorithm was evaluated on DTLZ1-DTLZ7 many-objective test problems with 5, 9, and 15 objectives, as well as on solved car cab design, water resources management, car side impact, marine design, and 10-bar truss engineering design problems, and compared with the NSGA-III, MOEADDE, MaOPSO, and MaOJAYA. The experimental results demonstrated the MaOSCA’s superior performance compared to the NSGA-III, MOEADDE, MaOPSO, and MaOJAYA on a diverse set of DTLZ1-DTLZ7 benchmark problems. The integration of the information feedback model contributed to better convergence and diversity preservation, which are crucial aspects of many-objective optimization.

Future work may explore the following directions:

• Hybridization with other optimization algorithms: combining MaOSCA with other popular many-objective optimization algorithms, such as the MOEA/D-DRA or the SPEA2, may lead to more robust and efficient hybrid algorithms that can tackle a wider range of problems;
• The MaOSCA’s unique capacity to handle multiple conflicting objectives makes it a promising tool for application in the machine learning and biomedical fields. In the context of machine learning, the MaOSCA could be applied to hyperparameter tuning, feature selection, and model selection problems. These are inherently many-objective in nature, as they involve trade-offs between various performance metrics such as accuracy, precision, recall, and computational cost. In biomedical research, the MaOSCA can be applied in areas such as drug discovery and personalized medicine. These fields often involve many-objective optimization problems, where the goal is to maximize therapeutic effects while minimizing side effects and costs. For example, in drug discovery, researchers might need to optimize the chemical properties of potential drug molecules to ensure they are effective, safe, stable, and easy to manufacture and administer. It is important to note that while the potential of the MaOSCA in these fields is promising, rigorous experimental studies are needed to validate its effectiveness and advantages over other existing algorithms in these specific applications;
• Application to real-world problems: evaluating the performance of the MaOSCA on real-world MOOPs, such as scheduling or environmental management, can provide valuable insights into its practical applicability and effectiveness;
• Adaptation to dynamic environments: extending the MaOSCA to handle dynamic MOOPs where problem constraints vary with time could broaden its applicability to a wider range of scenarios;
• Parallelization and distributed computing: investigating parallel and distributed computing techniques to further enhance the scalability and efficiency of the MaOSCA, especially for large-scale and computationally expensive problems, would be beneficial.

The proposed MaOSCA demonstrates the potential of incorporating the information feedback model into MOOPs, paving the way for further research and improvements in this field. The MaOSCA provides a more efficient and effective tool for tackling real-world engineering problems and contributes to the ongoing research in many-objective optimization, offering a new perspective and inspiring further developments in this field.