A Modified Sand Cat Swarm Optimization Algorithm Based on Multi-Strategy Fusion and Its Application in Engineering Problems

Abstract

Addressing the issues of the sand cat swarm optimization algorithm (SCSO), such as its weak global search ability and tendency to fall into local optima, this paper proposes an improved strategy called the multi-strategy integrated sand cat swarm optimization algorithm (MSCSO). The MSCSO algorithm improves upon the SCSO in several ways. Firstly, it employs the good point set strategy instead of a random strategy for population initialization, effectively enhancing the uniformity and diversity of the population distribution. Secondly, a nonlinear adjustment strategy is introduced to dynamically adjust the search range of the sand cats during the exploration and exploitation phases, significantly increasing the likelihood of finding more high-quality solutions. Lastly, the algorithm integrates the early warning mechanism of the sparrow search algorithm, enabling the sand cats to escape from their original positions and rapidly move towards the optimal solution, thus avoiding local optima. Using 29 benchmark functions of 30, 50, and 100 dimensions from CEC 2017 as experimental subjects, this paper further evaluates the MSCSO algorithm through Wilcoxon rank-sum tests and Friedman’s test, verifying its global solid search ability and convergence performance. In practical engineering problems such as reducer and welded beam design, MSCSO also demonstrates superior performance compared to five other intelligent algorithms, showing a remarkable ability to approach the optimal solutions for these engineering problems.

1. Introduction

Optimization problems refer to finding the optimal solution from all possible scenarios under certain conditions, aiming to maximize or minimize performance indicators. These problems are primarily applied in practical engineering applications [1], image processing [2], path planning [3], and other fields.

Engineering optimization problems aim to solve real-life problems with constraints, enabling engineering systems’ parameters to reach their optimal states. During the development of the engineering field, numerous nonlinear and high-dimensional problems have emerged, making optimization problems increasingly complex, with corresponding increases in computational complexity. Traditional optimization algorithms were widely developed and applied to various optimization problems in the early stages. However, traditional optimization algorithms have certain limitations regarding problem form and scale. They face challenges such as high computational costs and slow convergence speeds for complex or large-scale problems, making it challenging to find the optimal solution within a reasonable time frame.

Metaheuristic algorithms have demonstrated remarkable robustness in addressing the shortcomings of traditional optimization algorithms when dealing with complex problems. These complex issues encompass multi-objective, single-objective, discrete, combinatorial, high-dimensional, or dynamic optimization, among others [4]. Metaheuristic algorithms are general-purpose optimization algorithms characterized by their global perspective when searching the solution space [5]. They are primarily divided into four categories: evolutionary-based algorithms, swarm-based algorithms, algorithms inspired by physics, mathematics, and chemistry, and algorithms based on human and social behaviors [6]. This article primarily focuses on swarm-based algorithms, i.e., swarm intelligence optimization algorithms, which mimic the behavior of natural swarms to achieve information exchange and cooperation, ultimately identifying the optimal solution to optimization problems. From early algorithms such as particle swarm optimization (PSO) [7], ant colony optimization (ACO) [8], and genetic algorithm (GA) [9] to later developments including the sparrow search algorithm (SSA) [10], dung beetle optimizer (DBO) [11], pelican optimization algorithm (POA) [12], Harris hawks optimization (HHO) [13], and subtraction-average-based optimization (SABO) [14], swarm intelligence optimization algorithms have gradually evolved into an optimization method with few design parameters and simple operation.

The sand cat swarm optimization (SCSO) algorithm is a novel swarm intelligence optimization algorithm, proposed by Seyyedabbasi in 2022 [15], inspired by the behavior of sand cats in nature. The SCSO algorithm is divided into an exploration phase and an exploitation phase, with the balance between the two phases adjusted by the parameter R. Due to its robust and powerful optimization capabilities, SCSO has been widely applied in various fields. For instance, it has been used to predict the scale of short-term rock burst damage [16] and design feedback controllers for nonlinear systems [17]. Nevertheless, the SCSO algorithm also has some drawbacks, such as slow convergence speed, poor global search ability during the exploration phase, and a tendency to fall into local optima.

Various experts and scholars have proposed improvements to address these shortcomings of the SCSO algorithm. Ref. [18] combined dynamic pinhole imaging and the golden sine algorithm to enhance the optimization performance of the sand cat population, improving global search ability through dynamic pinhole imaging and enhancing exploitation ability and convergence through the golden sine algorithm. Ref. [19] utilized a target encoding mechanism, elitism strategy, and adaptive T-distribution to solve dynamic path-planning problems, improving path-planning efficiency and introducing the ability to escape local optima. Ref. [20] proposed combining SCSO with reinforcement learning techniques to improve the algorithm’s ability to find optimal solutions in a given search space. Ref. [21] used chaotic mapping with chaos concepts to enhance global search ability and reduce issues like slow transitions between phases and early or late convergence. Ref. [22] defined a new coefficient affecting the exploration and exploitation phases and introduced a mathematical model to address the issues of slow convergence and late exploitation issues in SCSO. Although the experts and scholars above have significantly improved the SCSO algorithm’s performance, some areas still need further improvement, as shown in Table 1.

Table 1. Improved SCSO.

Improved SCSO	Research Gap
DGS-SCSO [18]	Time consumption is long
ISCSO [19]	Unable to apply multi-objective coordinated search
RLSCSO [20]	There exists premature convergence on specific problems
CSCSO [21]	Not suitable for multi-objective and multi-dimensional problems

In summary, while specific improvements have been made in the literature to address the shortcomings of the SCSO algorithm, according to the no free lunch theorem [23], despite the extensive research and enhancements made to the SCSO algorithm’s global search ability, convergence performance, and ability to escape local optima, existing optimization strategies are still insufficient to tackle the ever-changing and increasing complexity of real-world problems. Therefore, it is necessary to continue improving the SCSO algorithm.

To further optimize the SCSO algorithm’s performance and expand its applications in engineering problems, this paper proposes a multi-strategy fusion sand cat swarm optimization (MSCSO) algorithm.

• During the initial population phase, the good point set strategy is adopted to ensure the population is evenly distributed within the search space, enhancing population diversity.
• A nonlinear adjustment strategy is introduced in the exploration phase to improve the sand cat’s sensitivity range, dynamically adjusting the balance conversion parameters between the exploration and exploitation stages.it expands the search range of the sand cats, enhances the algorithm’s global search ability, and reduces the risk of falling into local optima.
• In the exploitation phase, the sparrow search algorithm’s warning mechanism is fused to update the sand cats’ positions, enabling them to respond quickly when approaching optimality, avoid falling into local optima, and accelerate the algorithm’s convergence speed.

2. SCSO: Sand Cat Swarm Optimization Algorithm

2.1. Population Initialization

Sand cats are randomly generated in the search area, with the position of the individual as indicated in Equation (1).

$$x=ub+a\times (ub-lb)$$

(1)

where the $ub$ and $lb$ represent the upper and lower limits of the search area. $a$ is a random number between 0 and 1, and d is the dimensionality of the algorithm. The upper and lower bounds of the search area are as shown in Equation (2).

$$ub=[{ub}_1,{ub}_2\cdots {ub}_d], lb=[{lb}_1,l{b}_2\cdots {lb}_d]$$

(2)

2.2. Search for Prey (Exploration Phase)

In the exploration stage, the sand cat’s search behavior and attack behavior were dynamically adjusted by changing the size of the R-value, as shown in Equation (4). When the |R| < 1, the sand cat performs a search behavior, and during the search process, the sand cat relies on the range of low-frequency noise to locate its prey, by effectively exploring the potential best prey locations with sand cat location updates, as Equation (6) indicates.

$$r_g=S_M-\frac{S_M\times t}{T_{max}}$$

(3)

$$\mathrm{R}=2\times r_g\times rand\left(0,1\right)-r_g$$

(4)

$$\mathrm{r}=r_g-rand(0,1)$$

(5)

where the $r_g$ value drops linearly from 2 to 0, $S_M$ is the auditory feature with a value of 2, $t$ is the current number of updates, $T_{max}$ is the maximum number of updates, $rand(0,1)$ is a random number from 0 to 1, and $\mathrm{r}$ is the sensitivity range [15].

$$\mathrm{x}=\mathrm{r}\times (x_{bc}-rand(0,1\left)\right)\times x_c$$

(6)

where $x_{bc}$ represents the current best location, and $x_c$ is the current location.

2.3. Attacking Prey (Development Phase)

When the |R| $\le$ 1, the sand cat starts to attack, gradually narrowing down the search area to improve the search accuracy when approaching the prey. At this stage, the sand cat selects a random angle $\theta$ between 0 and 360 in the search area and uses $x_{bc}$ and $x_c$ to generate a random position, making the sand cat’s attack behavior more flexible and changeable, increasing the probability of finding the global optimal solution, as Equation (8) indicates.

$$x_{rnd}=\left|rand(0,1)\times (x_{bc}-x_c)\right|$$

(7)

$$\mathrm{x}=x_b-r\times x_{rnd}\times \cos \left(\theta \right)$$

(8)

where $x_b$ is the optimal position and $x_{rnd}$ is the random position.

The pseudocode of SCSO is presented in Algorithm 1.

Algorithm 1 SCSO

1: Initialize the population

2: Calculate the fitness function based on the objective function

3: Use Equations (3)–(5) to initialize the parameters ${\mathit{r}}_{\mathit{g}}$, R, and r

4: while (t < T) do

5:  for Every individual do

6:   Take a random angle θ between 0 and 360°

7:     if |R| ≤ 1

8:    Use Equation (8) to update the position

9:    else

10:     Use Equation (6) to update the location

11:     end

12: end

13: end

3. MSCSO: Fusion of Multi-Strategy Sand Cat Swarm Optimization Algorithm

3.1. The Good Point Set Strategy

Based on Equation (1), it can be seen that the SCSO algorithm initializes the population using a random strategy, resulting in a random distribution of the population within the search space; it can lead to an uneven distribution of the population and an increased likelihood of falling into local optima. Therefore, this paper introduces the good point set strategy, which was proposed by Hua Luogeng in 1978. This strategy utilizes the distribution patterns of point sets to optimize search and solution efficiency [24]. Meanwhile, Ref. [25] employed the good point set strategy to distribute the population uniformly within the search space, effectively enhancing population diversity and reducing the probability of falling into local optima. Given the advantages of the good point set strategy in initializing populations, this paper adopts the good point set strategy for population initialization. The steps are as follows:

Step 1: Generate the smallest prime number within the z-dimensional Euclidean space, as Equation (9) depicts.

$$\mathrm{z}\le \frac{p-3}{2}$$

(9)

Step 2: Generate a good point within the z-dimensional space, as Equation (10) depicts.

$$\mathrm{r}=2\times \cos \left(\frac{2\times w\times \pi }{p}\right), 1 \le w \le z$$

(10)

Step 3: Let Gz be the unit cube in the z-dimensional Euclidean space, generating a good point set with a sample size of n, as Equation (11) depicts.

$$F_n=\left\{\left(\left\{r_1^n\times k\right\},\left\{r_1^n\times k\right\},\cdots ,r_z^n\times k\right)\right\},1 \le k \le n$$

(11)

Step 4: Replace Equation (1) with Equation (12) for population initialization.

$$x=ub+F_n\times (ub-lb)$$

(12)

In a two-dimensional plane, 300 points were randomly generated using the good point set and random strategy, as shown in Figure 1 and Figure 2. Based on the distribution of these 300 points, it can be observed that the good point set strategy results in a more uniform distribution across the entire search space, enhancing the diversity of the SCSO algorithm; it validates the effectiveness of the good point set strategy in initializing the population for the SCSO algorithm.

Figure 1. The good point set strategy.

Figure 2. The random strategy.

3.2. Nonlinear Adjustment Strategy

The SCSO algorithm prioritizes breadth during the exploration phase and emphasizes depth and speed during the exploitation phase. As indicated in Equation (3), the sensitivity range of the sand cat’s $r_g$ value gradually decreases linearly from 2 to 0. This linear decrease does not fully utilize the global search capability of Equation (3), resulting in a tendency for the algorithm to get trapped in local optima. Consequently, a nonlinear adjustment strategy is adopted to improve Equation (3). Studies in [26,27] have validated that transitioning from a linear method to a nonlinear adjustment strategy enables the algorithm to maintain both global search capability and local exploitation ability throughout the entire iteration process, thus fulfilling the algorithm’s requirements across iterations. In this paper, the improved nonlinear substitution formula for Equation (3) is utilized to modify the auditory range of the sand cat. The steps for enhancing the nonlinear adjustment strategy formula are outlined below. Additionally, the modified formula is presented as Equation (14).

Step 1: Design parameter f, as shown in Equation (13)

$$f=\sin (\frac{\pi }{4}\times \frac{t}{T_{max}})$$

(13)

Step 2: Introduce f into the original Equation (3); the improved Equation is shown as Equation (14).

$$\mathrm{R}=S_M-\frac{S_M\times t}{T_{max}}\times f$$

(14)

where the $S_M$ is the auditory feature with a value of 2, $t$ is the current number of updates, $T_{max}$ is the maximum number of updates.

As shown in Figure 3 and Figure 4, a comparative analysis was conducted on the search range capabilities of the linear and nonlinear adjustment strategies. The red line represents the process where $r_g$ changes linearly from 2 to 0, while the black line depicts the improved range. From the figures, it is evident that the exploitation phase of SCSO overly emphasizes local attacks, neglecting the global search capability. However, the improved formula enables extensive exploration during the exploration phase and facilitates both local attacks and global exploration during the exploitation phase. It can be concluded that Equation (14) significantly enhances the overall performance of the algorithm and reduces the risk of falling into local optima; it verifies the global search capability of the nonlinear adjustment strategy during the exploitation phase.

Figure 3. The global search scope of Equation (3).

Figure 4. The global search scope of Equation (14).

3.3. Integrate the Warning Mechanism of the Sparrow Algorithm

In the later iterations of the SCSO algorithm, the convergence speed may be relatively slow when dealing with some complex issues, requiring a longer computational time to find the optimal solution. The warning mechanism in the SSA can rapidly guide sparrows toward the optimal position based on their current locations [10], thereby enhancing the algorithm’s convergence during the iteration process and reducing computational time. Therefore, inspired by the sparrow search algorithm, this paper introduces the warning mechanism of the sparrow algorithm into the SCSO algorithm. The detailed position update Equation is shown in Equation (15).

$$x_b\left(t+1\right)=\left\{\begin{array}{c}{x}_b\left(t\right)+\beta \times \left|x_c\left(t\right)-x_b\left(t\right)\right|, if f_i{>f}_g\\ x_c\left(t\right)+K\times \left(\frac{x_c\left(t\right)-x_{worst}^t}{f_i-f_w+\epsilon }\right),if f_i\le f_g\end{array}\right.$$

(15)

where $x_b$ is the current global optimal position, $x_{worst}^t$ indicates the worst position of the T_th iteration, $\beta$ is the step-size control parameter, $K$∈ [−1.1] is a random number, $f_i$ is the current fitness value, $f_g$ is the current global optimal fitness value, $f_w$ is the minimum fitness value of the current population, $\epsilon$ is constant [10]. The steps for updating the position are as follows.

Step 1: In the sand cat population, 30% of the individuals are randomly selected to serve as warning sand cats.

Step 2: While $f_i{>f}_g$, this indicates that the sand cat is safe around $x_b$, and it can randomly move within this range to avoid falling into a local optimum. The specific movement position of the sand cat is represented as $x_b\left(t\right)+\beta \times \left|x_c\left(t\right)-x_b\left(t\right)\right|$.

Step 3: While $f_i{\le f}_g$, this indicates that the sand cat realizes it is in an unfavorable position and needs to move closer to other sand cats to find an optimal position. The sand cat updates its position according to the Equation $x_c\left(t\right)+K\times \left(\frac{x_c\left(t\right)-x_{worst}^t}{f_i-f_w+\epsilon }\right)$. The sand cats’ position updates are illustrated in Figure 5.

Figure 5. Movement diagram of the warning mechanism of the sand cat fusion with sparrow algorithm.

3.4. MSCSO Algorithm Flowchart and Pseudocode

Figure 6 shows the flow chart of the MSCSO algorithm after SCSO is integrated with the good point set strategy, the nonlinear adjustment strategy, and the early warning mechanism of the sparrow search algorithm. The MSCSO pseudocode is presented in Algorithm 2.

Algorithm 2 MSCSO

Parameters: the current fitness value ${\mathit{f}}_{\mathit{i}}$, $\mathbf{a}\mathbf{n}\mathbf{d}$ the current global optimal fitness value ${\mathit{f}}_{\mathit{g}}$

1: Initialize the population using Equation (12)

2: Calculate the location and fitness values for each individual

3: Use Equations (4) and (5) are used to initialize the parameters R and r and Equation (13) is used to calculate R

4: while (t < T) do

5: for Every individual do

6:   Take a random angle θ between 0 and 360°

7:    if |R| ≤ 1

8:    Use Equation (8) to update the position and fitness values of the sand cat

9:    else

10:      Use Equation (6) to update the location and fitness values of the sand cat

11:  Update the current best fitness value and posion

12:    end

13:  Randomly select 30% of the population size

14:  for i = 1 to number of warning sand cats do

15:    if ${\mathit{f}}_{\mathit{i}}$ > ${\mathit{f}}_{\mathit{g}}$

16:    Use ${\mathit{x}}_{\mathit{b}}\left(\mathit{t}\right)+\mathit{\beta }\times \left|{\mathit{x}}_{\mathit{c}}\left(\mathit{t}\right)-{\mathit{x}}_{\mathit{b}}\left(\mathit{t}\right)\right|$ to update the position

17:    else

18:    Use ${\mathit{x}}_{\mathit{c}}\left(\mathit{t}\right)+\mathit{K}\times \left(\frac{{\mathit{x}}_{\mathit{c}}\left(\mathit{t}\right)-{\mathit{x}}_{\mathit{w}\mathit{o}\mathit{r}\mathit{s}\mathit{t}}^{\mathit{t}}}{{\mathit{f}}_{\mathit{i}}-{\mathit{f}}_{\mathit{w}}+\mathit{\epsilon }}\right)$ to update the position

19:    end

20:  end

21:  Compare the updated position using the integrated early warning mechanism with the position in Step 11, and if the position is superior to the original one, proceed with the update.

22:  end for

23:  t = t + 1

24:  end while

Figure 6. MSCSO flow chart.

4. Experimental Results and Discussion

4.1. Experimental Design

In this experiment, the population size is set to 30, and the maximum number of iterations is 500. Each test function was independently calculated 30 times to obtain more accurate experimental results. The evaluation criteria are the data results’ mean value and standard deviation. The mean value represents the convergence accuracy of the algorithm, where a smaller value indicates higher accuracy. While the standard deviation represents the algorithm’s stability, with a more minor standard deviation indicating much better stability [28].

The CEC 2017 benchmark was adopted to evaluate the optimization algorithm’s global search ability and convergence performance. The CEC 2017 test functions incorporate complex and diverse optimization problems from real-life scenarios, mathematical models, and artificially constructed benchmark problems. Furthermore, the CEC 2017 test functions cover many optimization types, from simple local searches to complex global optimizations. As the dimensionality of the functions increases, the difficulty in solving them also escalates, posing more significant challenges to the performance of optimization algorithms. Therefore, this paper employs the 30-dimensional, 50-dimensional, and 100-dimensional functions from the CEC 2017 benchmark to verify the comprehensive performance of the optimization algorithm. Among the 29 essential test functions in CEC 2017, F1 and F2 are unimodal functions used to test the optimization capability and convergence speed. F3–F10 are simple multimodal functions with multiple local optima designed to evaluate the global search ability of the algorithm. F11–F20 are hybrid functions that combine unimodal and multimodal characteristics, requiring higher performance from the algorithm. F21–F30 are composition functions that reflect the comprehensive performance of the algorithm. However, F2 is not analyzed independently. Additionally, five intelligent optimization algorithms, namely DBO, POA, HHO, SABO, and SCSO, are utilized for comparative analysis with MSCSO. The specific parameters of these six intelligent algorithms are detailed in Table 2.

Table 2. Algorithm parameter settings.

Algorithm	Basic Parameters
DBO	P-Percent = 0.2
POA	I = round (1 + rand (1, 1))
HHO	E0 ∈ [−1, 1], E1 ∈ [0, 2]
SABO	I = round (1 + rand + rand)
SCSO	SM = 2, P = [1, 360]
MSCSO	SM = 2, P = [1, 360]

4.2. Results and Analysis

4.2.1. Analysis and Comparison of Experimental Results

As seen in Table 3, the MSCSO algorithm leads in terms of mean and standard deviation for the unimodal function F1. Among the simple multimodal functions F3–F10, MSCSO achieves the best mean and standard deviation for functions F3 and F4. The standard deviation of MSCSO for functions F7 and F10 is slightly higher than that of DBO and POA but lower than the original algorithm. It maintains a leading position for the remaining functions. In the simple multimodal functions, the average performance of MSCSO is average, ranking fifth and fourth in F5 and F6 and third in F7, F8, F9, and F10. For the mixed functions F11–F20, the MSCSO algorithm outperforms the other five in standard deviation for F11, F12, F13, F15, F16, and F19. The standard deviation of MSCSO for F14, F17, F18, and F120 is slightly higher than POA but better than the original algorithm. The average values rank fourth in F17 and second in F18. Among the composite functions F21–F30, MSCSO achieves the optimal mean and standard deviation for F23, F25, F27, F28, and F30. Although the mean value of MSCSO ranks fifth in F21, sixth in F22, second in F24, last in F26, and third in F29, its standard deviation remains leading among all composite functions.

Table 3. CEC2017 30D test results.

		DBO	POA	HHO	SABO	SCSO	MSCSO
F1	Mean	181061557.9	5660538295	255329045.4	4614031681	4187157339	6480.351978
	Std	226987212.7	16150983243	421644778.7	14425674652	9140251764	5760.1867
F3	Mean	17515.68987	8641.937849	7230.229332	9120.615487	13161.3602	5891.522648
	Std	92767.21213	42864.87112	58714.43472	61490.62122	57139.9279	23032.27802
F4	Mean	139.3806064	1180.492654	100.0683708	1538.350635	971.762774	17.53120497
	Std	702.7152913	2508.661121	730.5732048	2822.391812	1302.43027	499.858241
F5	Mean	55.88722149	36.71033452	34.90282077	42.86099805	46.5756173	49.52457744
	Std	754.5220169	771.7684659	769.3682924	821.7540847	765.931483	696.6663608
F6	Mean	14.22965423	6.420005693	6.813163342	14.04655116	8.55495747	9.141826164
	Std	649.3417836	661.4178179	667.1667533	665.3981712	662.780493	641.1724601
F7	Mean	102.4918613	79.59279925	67.41982703	46.32864246	102.703899	68.75953635
	Std	1043.000588	1265.842642	1289.164691	1137.536545	1151.54709	1049.194281
F8	Mean	65.30345964	28.52448851	25.83934647	39.91665502	35.8904039	28.88077642
	Std	1028.000156	1000.597586	982.3129753	1089.748856	1015.94441	964.2559725
F9	Mean	2480.28238	751.4759372	1110.636505	2021.913539	1090.50732	1064.581016
	Std	6502.388162	5694.540617	8781.793242	7867.602149	6263.03794	4356.192112
F10	Mean	845.120596	674.125884	955.1853536	310.1298527	695.423901	707.9633178
	Std	5844.366499	5179.793125	6074.498407	8875.343756	6308.35686	5578.120938
F11	Mean	383.3717531	984.7520981	131.0697791	1666.239461	1444.16239	65.31309442
	Std	1747.322418	2276.102084	1572.286193	4958.004962	3077.69663	1279.354045
F12	Mean	98614979.78	1532556837	70205784.97	701884840	391983407	716338.4703
	Std	51956094.49	1364675261	72935842.39	862414803	278116157	799837.9558
F13	Mean	16430067.87	67106598.52	1020004.509	422127043.3	568691225	32716.10554
	Std	6687687.648	24812467.87	1341056.187	210921857.1	166223617	36573.76204
F14	Mean	397818.3576	51345.75678	1006339.533	1078547.96	573708.149	49306.0385
	Std	396369.8551	31504.89097	1214146.75	1266375.926	507603.593	62704.4259
F15	Mean	9618753.619	27415.02495	69501.59766	3290687.005	13835517.6	26416.15645
	Std	2151740.612	40267.45615	126213.0482	2787500.82	4827029.27	16700.49309
F16	Mean	425.8795674	411.7597504	538.3952084	369.0523296	394.224764	286.8438364
	Std	3385.843813	3156.722445	3671.930327	4133.541783	3355.58979	2908.555732
F17	Mean	276.5561996	206.9603863	327.376406	272.2066037	253.446717	282.808772
	Std	2721.086218	2274.490088	2642.962091	2952.61335	2411.41041	2349.256087
F18	Mean	5019785.4	215328.4591	4829644.624	7611276.17	3100455.93	217570.0692
	Std	4500202.134	230891.6609	4833387.029	5627992.852	2951381.6	278289.2344
F19	Mean	2881341.697	1213028.798	1170113.043	3997783.174	12579710.7	19503.72515
	Std	1687089.263	1425064.215	1552859.14	3910735.162	6626031.24	12942.19377
F20	Mean	213.6571874	131.0908482	223.428119	150.6634982	208.000522	249.5415292
	Std	2793.25076	2528.405069	2820.833438	3084.784816	2663.15846	2561.539277
F21	Mean	37.6641326	30.9823029	74.42153325	30.75027983	45.0600895	48.3281029
	Std	2547.439668	2548.402841	2607.481715	2600.065147	2531.5595	2476.341655
F22	Mean	2294.055198	1506.850956	1337.270747	574.8287705	2056.71469	2509.271215
	Std	4708.058549	5269.136491	7169.615883	4278.399146	4749.34069	3880.26186
F23	Mean	77.54199525	91.00184094	133.3306507	77.40925511	63.5805378	59.34920959
	Std	3035.559958	3042.558105	3285.193148	3164.505177	2941.82475	2861.767827
F24	Mean	78.4541423	86.76109528	144.8806048	70.43495313	55.7774386	59.05235058
	Std	3177.184287	3205.654588	3484.85549	3274.996469	3085.01751	3017.376545
F25	Mean	241.7992819	204.2553378	38.69880114	187.0182296	145.279167	15.66963627
	Std	3023.565177	3278.42261	3009.436677	3424.132939	3160.50729	2892.210674
F26	Mean	926.4768522	932.3701957	1027.042403	470.4345926	1124.23286	2061.212617
	Std	6706.126862	7536.127707	8216.399113	8348.541515	6674.18073	4821.825556
F27	Mean	79.04577458	82.70004022	144.3232787	124.4876467	77.0816211	22.1711531
	Std	3327.025325	3376.418061	3556.386194	3506.911146	3401.59401	3251.414278
F28	Mean	805.8031815	405.9385201	66.17406326	465.5476469	248.150829	17.60517628
	Std	3810.441244	4016.377661	3461.71224	4425.980204	3709.25412	3222.277593
F29	Mean	279.9945813	313.0157891	416.3759905	632.9971989	240.354284	371.348704
	Std	4458.823837	4701.727384	4959.42317	5791.308845	4507.63297	4035.466042
F30	Mean	3286368.504	8636188.062	8439425.619	47899885.51	11321242.9	121760.9925
	Std	2209335.567	11889940.31	10203011.34	41653700.04	16812631.4	49818.54126

As seen in Table A1, in the 50-dimensional experiments, the MSCSO algorithm outperforms the other five algorithms regarding mean and standard deviation for the unimodal function F1. Among the multimodal functions F3–F10, the evaluation metrics reach the optimal values for F4. The standard deviation is slightly higher than the DBO algorithm for F7 but performs excellently for the remaining functions. The average values for F5–F10 remain average. In the mixed functions F11–F20, the MSCSO algorithm demonstrates strong optimization performance. MSCSO ranks first in average values and standard deviations for F11–16 and F18–19. The standard deviation of MSCSO is only higher than POA for F20 but lower than the original algorithm. The average values rank average only for F17 and F20. In the composite functions F21-F30, the MSCSO algorithm’s standard deviation also performs outstandingly, consistently ranking first. The average values rank average for F21, F22, F24, and F26, but MSCSO can always find the solution fastest for the remaining functions.

As seen in Table A2, in the 100-dimensional experiments, the MSCSO algorithm exhibits excellent performance in handling complex problems. For the unimodal function F1, the evaluation metrics maintain the optimal state. Among the multimodal functions F3–F10, the optimization performance of the MSCSO algorithm is similar to that in the 50-dimensional experiments. In the mixed functions F11–F20, the average value of the MSCSO algorithm ranks only fourth in F17 and F20, and the standard deviation is higher than POA only in F20. The average and standard deviation exhibits the best performance for the remaining functions. In the composite functions F21–F30, the average value of the MSCSO algorithm ranks poorly only in F21 and F26, but the evaluation metrics maintain a leading position for the remaining functions.

In summary, the MSCSO algorithm performs averagely in handling simple multimodal problems. Still, it exhibits excellent performance in dealing with unimodal, mixed, and composite function problems, especially when handling complex composite and high-dimensional functions.

4.2.2. Convergence Plot Analysis Discussion

In this paper, convergence curves of six intelligent optimization algorithms on the CEC2017 benchmark in 30-dimensional, 50-dimensional, and 100-dimensional spaces are introduced to demonstrate the performance of optimization algorithms.

As seen in Figure A1, it can be observed that DBO and POA surpass the MSCSO algorithm in the later iterations of F6 and F14. However, its performance is significantly improved compared to the SCSO algorithm. In F3 and F22, the convergence performance of the algorithm is not evident in the early to mid-iterations. However, it accelerates in the later iterations, surpassing the other five algorithms and ranking first; it fully demonstrates that the algorithm’s integration of an early warning mechanism accelerates its convergence. In functions such as F1, F4, F5, F8, F9, F10, F12, F13, F19–F23, F26, F28, and F30, MSCSO exhibits faster convergence speeds. This acceleration in convergence speed indicates that MSCSO leverages the good point set strategy and nonlinear adjustment strategy to enhance its global search capabilities, enabling the algorithm to locate optimal solutions more rapidly and improve its convergence performance.

As seen in Figure A2, the MSCSO algorithm is only surpassed by DBO in F7 and needs to find the optimal solution. Among the other algorithms, it exhibits strong convergence and optimization performance. Compared to the original algorithm, it accelerates the convergence of the algorithm.

As seen in Figure A3, the MSCSO algorithm fails to find the optimal solution only in F20, but it can quickly find the optimal solution in most functions. Moreover, it can conduct a broader global search in the early iterations and escape from local optima in the later iterations, demonstrating that the MSCSO algorithm has certain advantages in handling complex and high-dimensional functions.

In summary, while the MSCSO algorithm exhibits some improvement in addressing the premature convergence and stagnation issues in the 30-dimensional experimental convergence curve, this improvement is not particularly evident in the 30-dimensional experiments. However, as the dimensionality increased, the algorithm’s convergence performance saw significant improvement, demonstrating that the MSCSO algorithm has certain advantages in handling complex and high-dimensional problems and can escape from local optima.

4.2.3. Boxplots Analysis

This paper utilizes boxplots to compare MSCSO with five other algorithms, making the optimization performance of MSCSO more visually intuitive. The median represents the average of the sample results after 30 independent runs of the algorithm. The upper and lower quartiles indicate the degree of fluctuation in the sample data to some extent. Outliers suggest instability in the data. Figure A4, Figure A5 and Figure A6 show the example data results of the six intelligent algorithms in 30-dimensional, 50-dimensional, and 100-dimensional test functions.

In Figure A4, MSCSO exhibits the lowest average value on most functions, verifying its powerful optimization capability. For test functions F6-F9, F16-F17, F20, F24, F26, and F29, the upper and lower quartiles of MSCSO’s boxplots are evenly distributed, indicating low data dispersion. Although there are outliers in some functions, the overall performance remains stable.

In Figure A5 and Figure A6, the box plots of MSCSO in 50-dimensional and 100-dimensional spaces show significant improvements in their upper and lower quartile distributions compared to the 30-dimensional boxplots. The distribution range becomes shorter and more uniform, indicating that as the dimension increases, the dispersion of the algorithm decreases. Additionally, as the dimension increases, the number of functions with the lowest average value for MSCSO also rises, demonstrating that MSCSO is more reliable than the other five algorithms.

4.3. Non-Parametric Test

4.3.1. Wilcoxon Rank-Sum Test

This paper introduces the Wilcoxon rank-sum test to verify whether there are significant differences in performance between MSCSO and the other five algorithms. Each algorithm was independently tested 30 times in the experiment. The results are used to calculate the p-values. The null hypothesis is rejected when the p-value is less than 0.05, indicating a significant performance difference between the two algorithms. When the p-value is more significant than 0.05, it suggests that the search results of the two algorithms are statistically significant [29].

From Table 4, it can be observed that 5.5% of the algorithms are comparable to the MSCSO algorithm. In functions F6 and F7, MSCSO’s performance is statistically significant compared to DBO. In functions F17, F18, and F22, MSCSO’s performance is comparable to POA. In functions F10 and F20, the p-values for SCSO are more significant than 0.05, indicating statistical significance. Additionally, 5.5% of the algorithms are comparable to the MSCSO algorithm.

Table 4. Thirty-dimensional Wilcoxon rank-sum test of CEC2017 test function.

	DBO	POA	HHO	SABO	SCSO
F1	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05
F3	3.0 × 10−11 < 0.05	4.2 × 10−10 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05
F4	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05
F5	1.1 × 10−04 < 0.05	5.0 × 10−08 < 0.05	5.0 × 10−06 < 0.05	8.9 × 10−11 < 0.05	3.3 × 10−06 < 0.05
F6	5.0 × 10−01 > 0.05	4.1 × 10−07 < 0.05	3.1 × 10−10 < 0.05	7.1 × 10−05 < 0.05	9.8 × 10−08 < 0.05
F7	5.6 × 10−01 > 0.05	7.3 × 10−11 < 0.05	3.0 × 10−11 < 0.05	1.2 × 10−07 < 0.05	2.3 × 10−06 < 0.05
F8	4.0 × 10−08 < 0.05	1.1 × 10−06 < 0.05	3.1 × 10−03 < 0.05	4.0 × 10−11 < 0.05	2.1 × 10−07 < 0.05
F9	1.4 × 10−07 < 0.05	1.4 × 10−07 < 0.05	3.0 × 10−11 < 0.05	1.4 × 10−10 < 0.05	1.1 × 10−08 < 0.05
F10	7.9 × 10−03 < 0.05	4.8 × 10−03 < 0.05	2.0 × 10−01 > 0.05	3.0 × 10−11< 0.05	9.3 × 10−02> 0.05
F11	7.3 × 10−11 < 0.05	4.9 × 10−11 < 0.05	8.1 × 10−10 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05
F12	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05
F13	4.1 × 10−10 < 0.05	2.1 × 10−10 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	2.9 × 10−09 < 0.05
F14	2.0 × 10−06 < 0.05	7.2 × 10−03 < 0.05	1.1 × 10−09 < 0.05	3.0 × 10−11 < 0.05	1.4 × 10−07 < 0.05
F15	1.5 × 10−08 < 0.05	7.0 × 10−08< 0.05	8.1 × 10−11 < 0.05	3.0 × 10−11 < 0.05	1.2 × 10−10 < 0.05
F16	2.1 × 10−05< 0.05	3.6 × 10−03 < 0.05	4.8 × 10−07 < 0.05	3.0 × 10−11 < 0.05	1.3 × 10−05 < 0.05
F17	8.6 × 10−05 < 0.05	4.7 × 10−01 > 0.05	2.2 × 10−04 < 0.05	1.0 × 10−10 < 0.05	1.3 × 10−05 < 0.05
F18	4.3 × 10−08 < 0.05	1.8 × 10−01 > 0.05	2.5 × 10−07 < 0.05	2.0 × 10−08 < 0.05	3.8 × 10−06 < 0.05
F19	6.1 × 10−10 < 0.05	5.4 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05
F20	4.2 × 10−02 < 0.05	4.8 × 10−03 < 0.05	3.5 × 10−03 < 0.05	6.7 × 10−10 < 0.05	3.0 × 10−1 > 0.05
F21	5.9 × 10−05 < 0.05	5.9 × 10−05 < 0.05	8.3 × 10−08 < 0.05	2.2 × 10−09 < 0.05	9.5 × 10−04 < 0.05
F22	1.2 × 10−03 < 0.05	5.7 × 10−02 > 0.05	1.4 × 10−05 < 0.05	1.4 × 10−01 < 0.05	3.1×10−02 < 0.05
F23	9.8 × 10−08 < 0.05	4.1 × 10−09 < 0.05	3.3 × 10−11 < 0.05	4.5 × 10−11 < 0.05	4.0 × 10−05 < 0.05
F24	9.2 × 10−09 < 0.05	2.8 × 10−10 < 0.05	3.0 × 10−11 < 0.05	3.6 × 10−11 < 0.05	1.1×10−03 < 0.05
F25	2.6 × 10−09 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05
F26	1.2 × 10−09 < 0.05	7.7 × 10−09 < 0.05	8.9 × 10−11< 0.05	4.0 × 10−11 < 0.05	1.7 × 10−07 < 0.05
F27	1.6 × 10−08 < 0.05	4.5 × 10−09 < 0.05	1.2 × 10−11< 0.05	4.9 × 10−11 < 0.05	1.4 × 10−09 < 0.05
F28	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05
F29	7.6 × 10−05 < 0.05	1.3 × 10−07 < 0.05	5.4 × 10−09 < 0.05	3.0 × 10−11 < 0.05	1.7 × 10−07 < 0.05
F30	1.6 × 10−09 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05

Table A3 indicates that only 2% of the algorithms have comparable performance to MSCSO, while 98% show significant differences from MSCSO. Specifically, in function F7, MSCSO’s performance is comparable to DBO. In functions F3 and F20, the p-values for POA are greater than 0.05, suggesting statistical significance.

In Table A4, 98.6% of the data results are less than 0.05, suggesting that MSCSO exhibits significant differences from the other five algorithms.

In summary, as the dimensionality of the test functions continues to increase, the significance of the differences between MSCSO and DBO, POA, SCSO, HHO, SABO, and SCSO becomes increasingly pronounced.

4.3.2. Friedman Test

In this paper, the Friedman Test is used to provide an overall ranking of the algorithm experimental results. The Friedman Test aids in comprehensively evaluating the overall performance of different algorithms. In this ranking, a smaller serial number for an algorithm indicates better performance. Figure 7 displays the average Friedman ranking of six intelligent optimization algorithms: DBO, POA, HHO, SABO, SCSO, and MSCSO. As evident from the figure, MSCSO consistently ranks first in the average ranking across the 30-dimensional, 50-dimensional, and 100-dimensional test functions of CEC2017, validating the effectiveness of the improved algorithm and demonstrating the excellent comprehensive performance of MSCSO.

Figure 7. Friedman average rank.

5. Engineering Applications

In this section, two specific engineering examples, the reducer design problem, and the welded beam design problem, are used to verify the performance of the MSCSO algorithm [30].

5.1. Reducer Design Problems

The reducer design problem is a classic constraint engineering optimization problem, and its core goal is to minimize the weight of the reducer under the premise of satisfying a series of complex constraints. As can be seen from Figure 1, this problem involves seven structural parameters: face width (W₁), tooth die (W₂), number of pinion teeth (W₃), length of the first shaft between bearings (W₄), length of the second shaft between bearings (W₅), diameter of the first shaft (W₆) and diameter of the second shaft (W₇). By optimizing the combination of the seven parameters, the weight of the reducer is minimized, as shown in Figure 8.

Figure 8. Reducer design.

W₁, W₂, W₃, W₄, W₅, W₆, W₇ are the seven variables of the reducer design problem, and their objective functions are set as Equation (16) indicates.

$$\begin{gathered} f\left(w\right)=0.7854\times w_1\times w_2^2\times \left(3.3333w_3^2+14.9334w_3-43.0934\right)- \\ 1.508\times w_1\times \left(w_6^2+w_7^2\right)+7.4777\times \left(w_6^3+w_7^3\right)+0.7854\times (w_4{w}_6^2+w_5{w}_7^2) \end{gathered}$$

(16)

Restrictions:

$$\left\{\begin{array}{c}{f}_1\left(\overline{w}\right)=\frac{27}{w_1{w}_2^2{w}_3}-1\le 0\\ f_2\left(\overline{w}\right)=\frac{397.5}{w_1{w}_2^2{w}_3^2}-1\le 0\\ f_3\left(\overline{w}\right)=\frac{1.93w_4^3}{w_2{w}_7^4{w}_3}-1\le 0\\ f_4\left(\overline{w}\right)=\frac{1.93w_4^3}{w_2{w}_7^4{w}_3}-1\le 0\\ f_5\left(\overline{w}\right)=\frac{{\left[{\left(745\left(\frac{w_4}{w_2{w}_3}\right)\right)}^2+16.9\times {10}^6\right]}^{\frac{1}{2}}}{110w_6^3}-1\le 0\\ f_6\left(\overline{w}\right)=\frac{{\left[{\left(745\left(\frac{w_5}{w_2{w}_3}\right)\right)}^2+157.5\times {10}^6\right]}^{\frac{1}{2}}}{85w_7^3}-1\le 0\\ f_7\left(\overline{w}\right)=\frac{w_2{w}_3}{40}-1\le 0\\ f_8\left(\overline{w}\right)=\frac{{5w}_2}{w_1}-1\le 0\\ f_9\left(\overline{w}\right)=\frac{w_1}{{12w}_2}-1\le 0\\ f_{10}\left(\overline{w}\right)=\frac{{1.5w}_6+1.9}{w_4}-1\le 0\\ f_{11}\left(\overline{w}\right)=\frac{{1.5w}_7+1.9}{w_5}-1\le 0\end{array}\right.$$

The range of the variables is as follows:

$$\left\{\begin{array}{c}2.6\le w_1\le 3.6\\ 0.7\le w_2\le 0.8\\ 17\le w_3\le 28\\ 7.3\le w_4\le 8.3\\ 7.3\le w_5\le 8.3\\ 2.9\le w_6\le 3.9\\ 5.0\le w_7\le 5.5\end{array}\right.$$

MSCSO is compared with DBO, POA, HHO, SABO, and SCSO algorithms to verify the performance of MSCSO in the reducer design problem. From the data in Table 5, compared with the other five algorithms, the MSCSO algorithm is the closest to the optimal value of the reducer design problem, with a value of 1.6702, and the values of its seven variables are 3.5, 0.7, 17, 7.3, 7.71532, 3.35054 and 5.28665.

Table 5. Application of six intelligent algorithms in reducer design.

Algorithm	Optimal Values for Variables (w1, w2, w3, w4, w5, w6, w7)	Min f(w)
DBO	(3.5, 0.7, 17, 8.3, 8.3, 3.35253, 5.5)	1.7189
POA	(3.5, 0.7, 17, 8.3, 7.71541, 3.38461, 5.28665)	1.6718
HHO	(3.52428, 0.7, 17, 8.0053, 8.0053, 3.527, 5.28675)	2.6522
SABO	(3.54507, 0.7, 26.252, 8.26088, 8.1548, 3.9, 5.29138)	1.9847
SCSO	(3.50177, 0.700005, 17.0048, 7.35336, 8.16168, 3.35139, 5.28758)	1.6724
MSCSO	(3.5, 0.7, 17, 7.3, 7.71532, 3.35054, 5.28665)	1.6702

As seen in Figure 9, MSCSO can quickly find the global optimal solution compared to other algorithms, demonstrating its applicability in the reducer design problem.

Figure 9. Convergence curves of six algorithms in reducer design.

5.2. Welded Beam Design Problems

The welded beam design problem is a typical strongly constrained nonlinear programming problem, with its core objective being to reduce manufacturing costs by optimizing the design scheme. Figure 10 illustrates the structural parameters of the welded beam, where the optimal combination of joint thickness (h), tightening bar length (l), bar height (t), and bar thickness (b) is sought to minimize manufacturing costs, under the premise of satisfying various physical and engineering constraints.

Figure 10. Welded beam design.

The parameters h, l, t, and b are represented by the variables w₁, w₂, w₃, and w₄, and their objective functions are set as Equation (17) indicates.

$$f\left(w\right)=1.10471\times w_1^2{w}_2+0.04811\times w_3{w}_4(14.0+w_2)$$

(17)

Restrictions:

$$\left\{\begin{array}{c}{f}_1\left(\overline{w}\right)=\tau \left(\overline{w}\right)-{\tau }_{max}\le 0\\ f_2\left(\overline{w}\right)=\sigma \left(\overline{w}\right)-{\sigma }_{max}\le 0\\ f_3\left(\overline{w}\right)=\delta \left(\overline{w}\right)-{\delta }_{max}\le 0\\ f_4\left(\overline{w}\right)=w_1-w_4\le 0\\ f_5\left(\overline{w}\right)=p-p_c\left(\overline{w}\right)\le 0\\ f_6\left(\overline{w}\right)=0.125-w_1\le 0\\ f_7\left(\overline{w}\right)=1.10471w_1^2{w}_2+0.04811w_3{w}_4\left(14.0+w_2\right)-5.0\le 0\end{array}\right.$$

The range of variables is as follows:

$$\left\{\begin{array}{c}0.1\le w_1\le 2\\ 0.1\le w_2\le 10\\ 0.1\le w_3\le 10\\ 0.1\le w_4\le 2\\ \tau \left(\overrightarrow{w}\right)=\sqrt{({{\tau }^{\prime })}^2+2{\tau }^{\prime }{\tau }^{″}\frac{w_2}{2R}+({{\tau }^{″})}^2}\\ {\tau }^{\prime }=\frac{P}{\sqrt{2}{w}_1{w}_2}\\ {\tau }^{″}=\frac{MR}{J},M=P\left(L+\frac{w_2}{2}\right)\\ R=\sqrt{\frac{w_2^2}{4}+{\left(\frac{w_1+w_3}{2}\right)}^2}\\ J=2\left\{\sqrt{2}{w}_1{w}_2\left[\frac{w_2^2}{4}+{\left(\frac{w_1+w_3}{2}\right)}^2\right]\right\}\\ \sigma \left(\overrightarrow{w}\right)=\frac{6PL}{w_4{w}_3^2}\\ \delta \left(\overrightarrow{w}\right)=\frac{6P{L}^3}{E{w}_3^2{w}^4}\\ p_c\left(\overrightarrow{w}\right)=\frac{4.013E\sqrt{\frac{w_3^2{w}_4^6}{36}}}{L^2}\left(1-\frac{w_3}{2L}\sqrt{\frac{E}{4G}}\right)\end{array}\right.$$

As seen from the data in Table 6, MSCSO shows superior performance compared with the five algorithms of DBO, POA, HHO, SABO, and SCSO in solving the welded beam design problem. Specifically, the optimal solutions of the four variables of the MSCSO algorithm are 0.19883, 3.3374, 9.1920, and 0.19883, respectively. The combination of the four variables accurately approximates the optimal solution, and its value is 2994.4245.

Table 6. Application of six intelligent algorithms in welded beam design.

Algorithm	Optimal Values for Variables (w1, w2, w3, w4)	Min f(w)
DBO	(0.17595, 3.8169, 9.2549, 0.20014)	3158.6698
POA	(0.19893, 3.3362, 9.1899, 0.19893)	3012.1955
HHO	(0.20080, 3.5380, 9.0906, 0.20330)	3064.6047
SABO	(0.59404, 1.5484, 5.0911, 0.70551)	5208.4211
SCSO	(0.18774, 3.5610, 9.1921, 0.19883)	3007.0386
MSCSO	(0.19883, 3.3374, 9.1920, 0.19883)	2994.4245

As seen in Figure 11, MSCSO can quickly locate the global optimal solution in the early stages of iteration for the welded beam design and converge effectively. The presence of multiple inflection points on the MSCSO curve indicates that MSCSO can escape from local optima.

Figure 11. Convergence curves of six algorithms in welded beam design.

6. Conclusions

This paper proposed a multi-strategy fusion algorithm named MSCSO to improve the optimization performance of SCSO. The effectiveness of MSCSO was verified through simulation experiments using 29 benchmark test functions from CEC 2017. Through detailed analysis and comparison of the experimental data, the following enhancements in MSCSO’s performance are observed.

• Precision and convergence speed in unimodal functions: MSCSO demonstrated significantly higher precision and faster convergence speed in unimodal functions than other algorithms; it indicated MSCSO’s superiority in efficiently navigating and exploiting simple search landscapes.
• Robust optimization performance in hybrid and composition functions: MSCSO exhibited formidable optimization capabilities in hybrid and composition functions. It maintained a leading position in precision and convergence in most functions, underscoring its adaptability to diverse and complex optimization challenges. As the problem dimensionality increases, MSCSO’s performance became even more pronounced, demonstrating its prowess in tackling intricate and high-dimensional problems.
• Superiority confirmed by Wilcoxon rank-sum test: The Wilcoxon rank-sum test statistically validated that MSCSO outperforms other algorithms, providing robust evidence of its enhanced performance.
• Applicability in engineering problems: In practical engineering applications such as gearbox reducer and welded beam design problems, MSCSO produced solutions closer to the optimal solutions than five other algorithms. It highlighted MSCSO’s effectiveness in real-world scenarios and underscored its broad applicability in solving engineering optimization tasks.