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. 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.

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.

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.

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

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. 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. 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

, 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.

As seen in

Table A1. CEC2017 50D test results.

		DBO	POA	HHO	SABO	SCSO	MSCSO
F1	Mean	3302398088	10390489898	1484653454	8799881201	8778375893	196284819
	Std	3899252845	49823118066	5512734336	42119582938	30042244589	61728185.7
F3	Mean	66786.35118	22023.41482	18437.34617	16561.57008	13980.77039	17413.1313
	Std	268106.3887	117684.1181	167032.023	182813.6126	133707.5058	99157.0174
F4	Mean	524.8249642	2726.350905	480.1093044	3027.911409	1656.421951	53.2610363
	Std	1409.957618	7812.684022	1805.635567	8077.440603	4571.652584	567.295001
F5	Mean	84.95671361	40.22758238	31.12382383	43.52121811	36.853378	43.369063
	Std	960.315895	926.2623231	922.5744492	1103.391257	948.8158352	838.558676
F6	Mean	11.32040522	6.237167532	4.065757496	12.3553266	7.190901281	7.57760074
	Std	668.8507287	672.129517	679.9782754	686.2532994	677.4816775	656.270736
F7	Mean	135.3215707	90.41802472	69.8349611	79.64051781	110.3563116	131.381464
	Std	1412.10116	1764.55676	1899.381916	1632.291613	1665.295017	1464.57607
F8	Mean	83.01299842	39.82495196	38.7444479	43.10136231	51.18661919	50.2141428
	Std	1321.315603	1255.066828	1244.088564	1426.266704	1290.516019	1161.60697
F9	Mean	7710.747487	1529.036855	2603.718659	3922.344162	3210.250677	2219.92498
	Std	25806.01922	18620.38525	32122.69238	31402.25016	21158.10829	12664.2894
F10	Mean	2272.307432	630.9933193	950.8373144	368.7032054	1119.804333	1573.49553
	Std	11510.69442	9181.627844	10355.83564	15130.59108	10190.86836	8705.85798
F11	Mean	1322.890937	2470.236447	526.0597284	2745.167283	2255.918509	82.9280624
	Std	4033.575553	6728.049967	2988.033843	11606.62784	7861.105679	1379.4339
F12	Mean	538853153.4	7237740245	567912624.7	7960953201	2208829317	6744695.91
	Std	830186029.6	14648371591	954810132.7	13876884946	3833950125	10351644.2
F13	Mean	114150217.4	4494947129	41254837.95	2425906523	1903746602	35670.1713
	Std	92502949.45	4707351617	33200051.09	2580833263	1474877114	56441.5596
F14	Mean	3974759.282	607675.3837	3910591.193	5374681.808	2703986.653	146418.047
	Std	3918959.364	649914.5906	4748048.359	6304012.675	2681790.879	310747.588
F15	Mean	23480267.91	175874736.5	955411.5079	296542849.3	424171520.7	14866.0908
	Std	9825049.542	88883732.08	1495111.31	238992978.7	189176839.5	28855.0623
F16	Mean	628.3633522	692.8244518	665.4755948	562.5161157	569.3277895	532.561722
	Std	4989.421503	4372.086747	4990.462575	6001.400479	4672.412437	3592.24632
F17	Mean	429.73778	408.269056	485.9001181	365.0381958	479.430001	426.899651
	Std	4413.380306	3774.015327	3828.298961	4618.758234	3919.009562	3492.97037
F18	Mean	14964943.25	4418438.238	6842181.004	28288404.34	24711295.34	1272072.92
	Std	15796585.34	4262892.242	10795307.23	40890071.68	23837109.2	1564630.47
F19	Mean	13933122.64	362744484.7	3677685.849	50424696.53	222217039.8	14766.5931
	Std	9116184.623	149962044.9	3069100.459	50048950.38	70718954.71	37005.8523
F20	Mean	410.2578286	213.5439526	308.4450941	168.3324002	294.2973357	345.601964
	Std	3710.28514	3063.836687	3668.520885	4212.271348	3565.674877	3335.5459
F21	Mean	83.14094219	61.95986229	101.6349972	63.59623969	73.23544841	66.0088371
	Std	2876.18541	2827.935196	2958.650285	2934.355452	2794.594213	2648.16974
F22	Mean	2015.922742	1074.838903	823.2283951	2343.895269	985.7160627	1602.4466
	Std	12502.87764	11437.60931	12417.40035	16309.13028	12327.33491	10992.8653
F23	Mean	132.4932719	147.379751	206.4727144	180.1562827	123.2249366	114.317005
	Std	3523.633457	3590.806985	3990.493109	3943.872193	3384.520633	3182.09634
F24	Mean	158.7447433	109.0159033	255.7003902	158.4743871	104.2052828	106.811743
	Std	3695.170402	3733.672181	4419.94957	3913.251521	3537.007824	3320.05097
F25	Mean	1569.240404	1094.525409	162.705011	1287.475791	663.4378573	27.3972524
	Std	3861.611951	6999.033404	3816.319194	7075.942069	5232.677992	3114.66829
F26	Mean	1118.232735	1865.600314	1342.11311	842.5372236	1614.13788	3234.80923
	Std	11100.88082	12768.05974	11536.63164	13553.13631	11748.54079	5729.92126
F27	Mean	308.0431984	291.1735528	683.8444814	492.0369643	207.7881354	169.730378
	Std	4010.032961	4339.722563	5155.284814	4642.827783	4274.549997	3611.67258
F28	Mean	2254.867558	755.4207857	441.8933227	825.6899025	570.2134973	87.1245666
	Std	6131.061848	6815.875429	4758.024815	7617.282993	5661.331436	3377.80703
F29	Mean	1002.694976	621.4094541	1096.522217	1507.201481	822.3748579	485.91776
	Std	6231.20715	7022.556572	7345.788008	9940.701573	6901.555629	4970.4202
F30	Mean	47489390.7	242380028.8	51564123.09	603825274.3	112349961.3	1336896.06
	Std	60009001.81	346537086.2	128720404.5	734334649	220873074	2431763.66

, 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. CEC2017 100D test results.

		DBO	POA	HHO	SABO	SCSO	MSCSO
F1	Mean	64439785350	14679470611	7123064242	12786225157	13574141598	6191835519
	Std	88946907692	1.44787 × 1011	48058863472	1.56287 × 1011	1.06645 × 1011	1.8053 × 1010
F3	Mean	327312.9072	31671.50162	122549.3883	14531.95724	17342.92473	30730.8664
	Std	730924.6607	305881.7838	372984.6991	348932.4274	320105.4844	298781.525
F4	Mean	15071.01887	6198.775365	1998.389853	6240.446945	4582.407811	1073.78027
	Std	16471.56686	25602.81592	9838.835433	31928.30398	15787.59256	1733.29016
F5	Mean	255.9942286	64.41293008	46.42333274	113.6177764	60.19254203	50.4321657
	Std	1737.151357	1606.074738	1673.752052	1930.352906	1632.976958	1371.77767
F6	Mean	13.17570887	3.748854711	3.847419004	5.791203439	4.652130833	3.18444647
	Std	682.8830424	681.3642251	690.6116046	705.539359	686.9988035	664.088596
F7	Mean	230.5354516	80.11767378	131.0653554	183.3490029	204.1486638	207.995493
	Std	2978.277946	3470.650798	3776.686954	3405.051846	3424.583953	2882.10396
F8	Mean	203.1690199	69.75688334	67.34965118	97.75941497	76.22128493	120.07328
	Std	2215.08076	2086.033586	2132.86936	2360.188236	2085.977332	1762.19817
F9	Mean	6031.929811	3543.233864	4754.054793	4463.180529	5964.901628	1649.51376
	Std	78882.20914	41704.00481	68787.89439	78575.29059	44974.51707	26104.6671
F10	Mean	5235.205338	1140.992805	1997.673166	731.0527176	1614.762398	2533.84918
	Std	27335.04082	21101.19419	24224.20078	32359.4073	22747.04663	17293.8057
F11	Mean	68644.9515	18627.26318	38471.07732	42124.32695	23270.27405	6659.60843
	Std	223448.5264	88617.44402	144597.4059	205655.4621	94315.73489	28493.7276
F12	Mean	2268580864	16365928649	2676065431	18164454902	13817550434	441613219
	Std	7277633037	68930637601	11619528549	54872552598	36736964033	255071073
F13	Mean	192201584.9	5459325140	196797367.5	4378193430	3222776818	30025.803
	Std	264566938.2	15150913501	340976541.6	8883083561	5668646842	65185.9529
F14	Mean	11153472.54	3516377.125	2237802.183	9886433.727	5827627.548	1513169.62
	Std	18865348.02	7659394.616	9054348.956	23213173.86	10840847.18	2950801.58
F15	Mean	109477038	2800000225	10706452.68	1617740396	1859453612	24423.7711
	Std	91000114.11	5414412363	18070615.52	2194201582	2129450783	44422.7137
F16	Mean	1437.527393	1401.778021	1127.143	1675.072261	1129.010461	687.019308
	Std	9830.457777	11418.37002	10048.44809	14272.76565	10546.5916	6353.67028
F17	Mean	1446.023266	60266.75664	2015.379609	87948.15107	55169.84024	647.352039
	Std	9522.423148	44338.51265	8111.732752	58510.76338	30618.69714	5891.41214
F18	Mean	13836695.43	3976344.651	4860168.729	14620039.89	5884800.631	3282615.36
	Std	24279018.43	9314116.26	10333720.11	29683396.12	11376427.43	4933958.16
F19	Mean	104306523.7	2451264977	36239372.08	1655661585	1570265191	91064.6918
	Std	139026655.4	4336804118	40238023.7	2111376240	1058246800	36491.2231
F20	Mean	873.1403828	485.1281919	391.7970499	288.1319208	574.8553324	492.113677
	Std	7283.694865	5452.846214	6232.521204	7911.433915	6246.751869	5591.37673
F21	Mean	194.6733628	151.272736	240.8060467	174.772987	158.0561619	150.778371
	Std	4034.475166	3900.882946	4336.474513	4609.40168	3732.258314	3344.92746
F22	Mean	4982.55286	1240.221994	1250.068268	1073.874386	1304.752149	2262.16708
	Std	29603.2738	24731.09808	27107.14681	34789.05817	26263.40432	21557.4056
F23	Mean	228.9527889	228.6376536	463.797077	382.3292313	187.8860342	170.356698
	Std	4774.38515	4964.923155	5861.948017	5563.140243	4534.221736	3861.13035
F24	Mean	434.7971439	293.7675921	601.9109496	544.0433336	292.4855238	254.89335
	Std	6174.067484	6366.875405	8242.826952	7426.1978	5655.794757	4576.10926
F25	Mean	4971.625162	2146.970151	469.6290082	1718.771909	1580.137642	280.365032
	Std	8528.152533	13375.00475	6799.205617	15072.28933	9867.156162	4069.11079
F26	Mean	2716.297116	2893.558169	2197.696998	2627.738246	2775.087158	7306.39958
	Std	26555.3795	36793.64805	31943.81382	39199.66094	33818.41939	19838.3825
F27	Mean	517.8607132	624.4369159	1066.926374	907.8012227	519.8501265	142.25572
	Std	4796.677845	6170.744599	6821.516679	7226.379513	5855.303843	3782.70941
F28	Mean	6867.524167	2361.314563	803.4499535	2055.769571	1887.503296	420.03276
	Std	18557.442	18477.30251	9319.445458	19527.98794	13378.17802	4602.70205
F29	Mean	2818.558281	10708.19759	1042.989965	8864.106404	3914.703263	617.239425
	Std	12278.1268	22161.34028	12642.23408	24025.07565	15093.16939	7535.46986
F30	Mean	109769797.4	4948234972	294385332.6	3570975523	2380766605	930694.966
	Std	241833483.5	13085971393	669153209.3	8978885602	4603212747	981372.554

, 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. 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

, 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 A3. Fifty-dimensional Wilcoxon rank-sum test of CEC2017 test function.

	DBO	POA	HHO	SABO	SCSO
F1	3.6 × 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	8.9 × 10−11 < 0.05	3.2 × 10−01 > 0.05	1.1 × 10−05 < 0.05	2.8 × 10−06 < 0.05	3.2 × 10−02 < 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	2.2 × 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
F6	7.5 × 10−07 < 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
F7	6.5 × 10−01 > 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	4.6 × 10−10 < 0.05	1.4 × 10−08 < 0.05
F8	9.7 × 10−10 < 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
F9	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
F10	7.3 × 10−10 < 0.05	5.1 × 10−07 < 0.05	4.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	1.5 × 10−09 < 0.05
F11	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
F12	2.3 × 10−10 < 0.05	3.0 × 10−11 < 0.05	6.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05
F13	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
F14	4.6 × 10−08 < 0.05	1.6 × 10−05 < 0.05	1.1 × 10−09 < 0.05	3.0 × 10−11 < 0.05	7.3 × 10−10 < 0.05
F15	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
F16	8.9 × 10−11 < 0.05	3.3 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05
F17	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.3 × 10−11 < 0.05	3.0 × 10−11 < 0.05	1.2 × 10−10 < 0.05
F18	1.8 × 10−09 < 0.05	1.8 × 10−06 < 0.05	7.5 × 10−07 < 0.05	3.0 × 10−11 < 0.05	1.4 × 10−07 < 0.05
F19	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
F20	7.7 × 10−09 < 0.05	3.4 × 10−01 > 0.05	5.3 × 10−03 < 0.05	3.0 × 10−11 < 0.05	5.5 × 10−03 < 0.05
F21	4.5 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	1.6 × 10−10 < 0.05
F22	2.0 × 10−08 < 0.05	5.1 × 10−07 < 0.05	1.7 × 10−10 < 0.05	3.0 × 10−11 < 0.05	4.6 × 10−08 < 0.05
F23	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	1.2 × 10−10 < 0.05
F24	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	4.5 × 10−11 < 0.05
F25	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
F26	6.9 × 10−03 < 0.05	3.0 × 10−11 < 0.05	3.6 × 10−11 < 0.05	3.0 × 10−11 < 0.05	1.4 × 10−10 < 0.05
F27	1.9 × 10−10 < 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
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	3.6 × 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
F30	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

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. One-hundred-dimensional Wilcoxon rank-sum test of CEC2017 test function.

	DBO	POA	HHO	SABO	SCSO
F1	1.8 × 10−09 < 0.05	1.8 × 10−06 < 0.05	7.5 × 10−07 < 0.05	3.0 × 10−11 < 0.05	1.4 × 10−07 < 0.05
F3	8.9 × 10−11 < 0.05	4.9 × 10−01 > 0.05	1.1 × 10−05 < 0.05	2.8 × 10−06 < 0.05	3.2 × 10−02 < 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	2.2 × 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
F6	7.5 × 10−07 < 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
F7	7.5 × 10−07 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	4.6 × 10−10 < 0.05	1.4 × 10−08 < 0.05
F8	9.7 × 10−10 < 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
F9	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
F10	7.3 × 10−10 < 0.05	5.1 × 10−07 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	1.5 × 10−09 < 0.05
F11	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
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	2.3 × 10−10 < 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
F14	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
F15	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
F16	8.9 × 10−11 < 0.05	3.3 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05
F17	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	1.2 × 10−10 < 0.05
F18	1.8 × 10−09 < 0.05	1.8 × 10−06 < 0.05	7.5 × 10−07 < 0.05	3.0 × 10−11 < 0.05	1.4 × 10−07 < 0.05
F19	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
F20	7.7 × 10−09 < 0.05	2.3 × 10−01 > 0.05	5.3 × 10−03 < 0.05	3.0 × 10−11 < 0.05	5.5 × 10−03 < 0.05
F21	4.5 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	1.6 × 10−10 < 0.05
F22	2.0 × 10−08 < 0.05	5.1 × 10−07 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	4.6 × 10−08 < 0.05
F23	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	1.2 × 10−10 < 0.05
F24	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
F25	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
F26	6.9 × 10−03 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	3.0 × 10−11 < 0.05	1.4 × 10−10 < 0.05
F27	1.9 × 10−10 < 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
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	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

, 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.

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.

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. 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

, 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.

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.

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.

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. 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

, 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.

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.

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.