Improved Slime Mould Algorithm Hybridizing Chaotic Maps and Differential Evolution Strategy for Global Optimization

Slime Mould Algorithm (SMA) is a new meta-heuristics algorithm that is inspired by the behaviors of slime mould from nature. Due to its effective performance, SMA has shown its competitive performance among other meta-heuristics algorithms and has been used in many mathematical optimization and real-world problems. However, SMA tends to sink into local optimality and lacks the diversity of the population. Therefore, to cope with the drawbacks of the classical SMA, this paper proposes an improved SMA algorithm named CHDESMA. First of all, the chaotic maps methods have the characteristics of ergodicity and randomness, and we used chaotic maps methods to replace the original random initialization to improve the diversity of the algorithm, which is more suitable for exploring the potential areas in the early stage. Then, based on the superior searching ability of the differential evolution algorithm (DE), the crossover and selection operations of DE are applied to CHDESMA, and the position is updated by the combination of the original SMA operator and the mutation strategy of DE, which effectively avoids the algorithm falling into local optimum. CHDESMA was evaluated using CEC2014 and CEC2017 test suits and four real-world engineering problems. CHDESMA was compared with advanced algorithms and DE variants. The experimental results and statistical analysis indicate that CHDESMA has competitive performance compared with the state-of-the-art algorithms.


I. INTRODUCTION
The swarm intelligence algorithms are mainly inspired by the hunting, foraging, and survival processes based on evolution and population in the natural environment [1]. In recent years, these meta-heuristics algorithms have been widely developed in various fields because of their outstanding advantages in many large-scale and real-life problems, such as engineering problems [2], feature selection [3], biological information processing [4], and cost-effective scheduling problems [5]. The most popular optimizers include Evolution Strategies (ES) [6], Grey Wolf Optimizer (GWO) [7], Bat Algorithm (BA) [8], Differential Evolution (DE) [9], Genetic Algorithms (GA) [10] and Salp Swarm Algorithm (SSA) [11], etc.
Li et al. [12] introduced a new optimizer which is a random method based on viscous oscillation mode called Slime Mould Algorithm. Many experiments have proved that the The associate editor coordinating the review of this manuscript and approving it for publication was Hongwei Du. algorithm is effective in solving global optimization problems. For example, Djekidel et al. [13] employed SMA to affect the magnetic coupling between high-voltage lines and metal pipes. Mostafa et al. [14] applied SMA to extract photovoltaic panel model parameters. Gush et al. [15] utilized SMA to enhance the photovoltaic capacity on the issue of distribution networks. Tiacht et al. [16] applied SMA for damage detection, location, and quantification. Deb et al. [17] adopted SMA to alleviate the transmission congestion of generators. Chen et al. [18] proposed a chaotic mud model algorithm (CSMA) to predict accuracy and computational complexity. Kumar et al. [19] applied SMA to estimate photovoltaic cell parameters. Kadry et al. [20] designed a multiscale matched filter using SMA. Agarwal et al. [21] improved SMA to generate the best collision-free path for a mobile robot.
There have been many improvement mechanisms to be proposed to boost the performance of SMA. For instance, Houssein et al. [22] proposed to combine VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ SMA with an adaptive guided differential evolution algorithm [23]. In [24], the author Yousri proposed a hybrid of SMA and Marine Predator Algorithm (MPA) [25] to simulate photovoltaic systems. Premkumar et al. [26] proposed a multi-objective slime mould algorithm (MOSMA). Abdel-Basset et al. [27] mixed SMA with Whale Optimization Algorithm (WOA) [28] to resolve the image segmentation issue of COVID-19 [29].
Although the performance of SMA has been improved in some areas, it still has certain limitations when dealing with complex problems. Table 1 summarizes the SMA variants in recent years and illustrates their features, merits, and limitations. For this reason, an enhanced SMA algorithm called CHDESMA has been proposed in this paper, which presents Chaotic Maps [30] and DE strategy into SMA simultaneously. First, Logistic Map [31] is used to initialize the population, which is beneficial to accelerate the convergency at the initial stage of the iteration. Then use the Differential Evolution strategy to improve the local search ability of the group agent, increase the diversity of slime mold populations, and prevent the premature maturity of the entire iterative process. The chaotic maps strategy and DE operator have been successfully applied to deal with various algorithm optimization problems. These successful literature are presented in Table 2 and Table 3, respectively. Chaotic initialization has been shown to improve the quality of the solution. However, the drawbacks are that it lacks robustness in dealing with various problems and the performance of the algorithm combined with the chaotic maps approach is not competitive when dealing with discrete problems. The DE algorithm shows good exploration capabilities, but individuals can easily fall into local optimization as the iterations proceed. The proposed CHDESMA was compared with seven advanced algorithms and seven well-known DE variants. The performance test was carried out on CEC2014 [32] and CEC2017 [33] benchmark functions. CHDESMA was then used to solve four engineering design problems, such as tension/compression spring design problem (TCSD), welded beam design problem (WBD), pressure vessel design problem (PVD), and three-bar truss design problem (TBTD). The experimental results and statistical analysis show that CHDESMA can achieve satisfactory results in the population iteration's convergence speed and quality of optimization problems.
The following shows the main contributions of this research: a) In the early stage, CHDESMA applies Logistic Map initialization to improve the diversity of the population and find a promising area. b) The DE algorithm is used to enhance the searching ability and prevent the algorithm from stagnation. c) The proposed CHDESMA was evaluated by CEC2014 and CEC2017 test suits and four real-world engineering problems. Experimental results and statistical analysis indicate that CHDESMA is more effective than other state-of-the-art algorithms.
d) This study is promising for the study, analysis, improvement, and expansion of SMA, and the study of the evolutionary optimizer is of great significance. The main components of the paper are distributed as follows: the basic versions of SMA, Differential Evolution, and Chaotic Maps are explained in Section 2. Section 3 describes in detail the proposed variant CHDESMA. In Section 4, the experimental simulation results are studied and discussed. In the end, Section 5 summarizes the article and analyzes possible future work.

A. THE BASIC VERSION OF SMA
SMA is a new optimizer based on the natural slime moulds oscillation mode. SMA uses weights as the positive and negative feedback generated by simulating the foraging process of slime moulds, composed of the following three different forms: approaching food, wrapping food, and oscillating.

1) APPROACH FOOD
Slime moulds rely on the smell of the air to get close to the desired food. Its approach to food is simulated as follows: where W is the weight of slime moulds. Where p is shown as follows: where Fit(i) is the fitness of each slime mould, BF is the best fitness of slime mould in the current iteration. The value of vb and vc are shown as follows: Among them, t indicates the current iteration number, and maxiter is the maximum iterations.
The W is expressed as follows: SmellIndex is the sequence of fitness values after sorting. r is a random number in the range of [0, 1], condition indicates that Fit(i) is ranked in the first half.

2) WRAP FOOD
This part uses a mathematical model to simulate the shrinkage of slime mold mathematically. The positive and negative feedback between food concentration is simulated by Eq. (5). When the food concentration of the area is high, the weight nearby is greater, and when food concentration near the area is low, the weight nearby decreases to explore other areas.
The formula of each slime mould is calculated as follows: where UB and LB represent the upper and lower bounds of the total space. z is usually taken as a tiny number, such as 0.03 [12]. The definitions of vb, vc, W , p, x b , x A , and x B are shown in Eq. (1).

3) OSCILLATION
The slime moulds for the most part change the flow of cytoplasm in the vein by the propagation wave generated by the biological vibrator so that it can be placed in a better food concentration position. In order to simulate the changes in the pulse width of slime bacteria, W , vb, and vc are used by us to achieve these changes. Mathematical methods are used to simulate the vibrational frequencies of different food concentrations. vb is generated in the range of [−a, a] randomly and approaches zero by degrees with the increase of the number of iterations. vc is in the range of [-1, 1] and finally reaches zero.
Algorithm 1 is the pseudo-code of SMA.

B. THE BASIC VERSION OF DE
DE based on genetic algorithms and other evolutionary thoughts was proposed by Storn et al. [43] in 1997. The primary process of DE consists of three steps, mutation, crossover, and selection.

1) INITIALIZATION
We initialize the main population randomly in the search space. For example, the j th individual in the i th dimension is generated at first-generation (G = 0) and is obtained as follows: where UB and LB represent the upper and lower bounds of the total space.

2) MUTATION OPERATION
The mutation vector of each generation is generated with the following formula: where indicators r 1 , r 2 , and r 3 are randomly chosen from the main population [1, 2, . . . , N ], which are not equal to i. Storn et al. [43] mentioned that the range of F is within [0, 2]. F is the control parameter of the proportional vector. The following formulas are the other most commonly used mutation strategies: where r 1 , r 2 , r 3 , r 4 , and r 5 are randomly chosen from the main population [1, 2, . . . , N ], which are not similar to index i. x G best is the best fitness value, and F is the control parameter of the proportional vector.

3) CROSSOVER OPERATION
In this part, the target vector and the mutation vector are mixed, and the following scheme is used to obtain the test vector: where CR is the control crossing number, which controls the probability of generating parameters from the mutation vector.

4) SELECTION OPERATION
In this part, the greedy strategy is used to discuss whether to choose u G i or x G i . If the fitness value of u G i is better, it will be retained; otherwise, it will be replaced by x G i . The options are as follows: Chaotic methods have various characteristics, such as ergodicity, randomness, and irregularity. Moreover, they are efficacious initial dependency criteria [38], [39]. Due to the different characteristics mentioned above, a chaotic mapping method composed of different equations is constructed to change random variables in optimization methods. Wang et al. [44] called this process the Chaos Optimization Algorithm (COA) [45]. The optimization type gives the intensity traversal of the chaos theory. The optimization method can effectively help the algorithm avoid premature maturity, and at the same time, can speed up the convergence.

III. THE PROPOSED CHDESMA
This section mainly introduces the composition and structure of CHDESMA. In the traditional SMA algorithm, the algorithm tends to fall into local optimization and cannot maintain the balance between exploration and exploitation. We hybridized the original SMA algorithm with a differential evolution algorithm and introduced chaotic mapping to solve these problems. In the early steps of the algorithm, we used Logistic Map to initialize the population. In the whole optimization process of the algorithm, we use the operators of DE instead of the operators of the original SMA to improve the local search capabilities of slime mould, increase the diversity of the population and prevent CHDESMA from stagnating prematurely.

A. CHAOTIC INITIALIZATION
Among the current chaotic search methods, logistic chaotic mapping is widely used [46]. Due to its simple operation and good optimization performance, CHDESMA uses Logistic Map to initialize the population to achieve acceleration in the early steps of the algorithm. The formulation of the Logistic Map is calculated by Eq. (17), and the pseudo-code of chaotic initialization is shown in Algorithm 2.
where a is a real number (a = 4), k indicates the current iteration, and x k represents the k th chaotic number.

Algorithm 2 Pseudo-Code of Chaotic Initialization
Generate variable x k according to the Logistic map shown in Eq. (17)

B. USE THE OPERATORS OF DE TO UPDATE THE POPULATION
The component mark CR controls the direction of exploration and exploitation. We mixed the DE strategy in the optimization process to enhance the local search ability of slime mould. Each position of the slime mould is calculated based on the following formula: where rand j,i is a random number in the range of [0,1], CR is the crossover probability, and parameter W is calculated as follows: (20) where maxiter is the maximum number of iterations and t is the current iteration. The parameters BF and WF are the best and worst solutions in the current iteration, respectively, and Fit(i) represents the fitness of a current individual. The formulation of parameter v is defined as follows: where r 1 , r 2 , and r 3  The main steps of the CHDESMA are shown in Algorithm 3, and Figure 1 shows the flowchart of CHDESMA.

IV. EXPERIMENTS AND DISCUSSION
The performance of CHDESMA under different functions is evaluated, and the data are analyzed. We set up three sets of experimental schemes. The first experimental scheme was carried out on 29 CEC2017 benchmark functions to verify the effectiveness of the chaotic maps and DE strategy used. The second experiment was to test the performance of CHDESMA on 30 CEC2014 benchmark functions. The experiment compared CHDESMA with published DE variants, which were composed of the latest DE variants and recognized DE variants with better performance. In the third experiment, in order to test the scalability of CHDESMA, CHDESMA was compared with the advanced algorithms in different dimensions. In addition, Wilcoxon rank-sum test was used in each experiment to study the statistical significance of the results. To further analyze the results, the box diagram was used to show the dispersion of data.  CHDESMA was used to solve four practical engineering problems. For the sake of fairness, referring to [47], algorithmic hyper parameters of all algorithms must be tuned in the same manner on the selected problems. Therefore, the parameters of the competition optimizer in all experiments were based on the most appropriate parameter settings in the published articles, as shown in Table 4. In this paper, the maximum number of fitness evaluations (MAXFES) was set to 10000 × D.

A. SYSTEM DETAILS
In this study, all the involved algorithms were tested using python 3.9 under MacOS10.15, with Intel (R) Core (TM) i5 CPU @ 3.8 GHz, and with 16 GB of RAM.

B. VALIDITY OF CHAOTIC MAPS AND DE STRATEGIES
CHDESMA algorithm is mainly composed of chaotic maps, DE strategy, and the original SMA algorithm. This section aims to verify the effectiveness of the use of chaotic maps and DE strategy. CHDESMA was compared with DESMA (SMA algorithm with DE operator), CHSMA (SMA algorithm with chaotic initialization), and SMA in Table 5. The experiment was carried out on the 30-dimensional CEC2017 test suite. The population number was set to 30. For the sake of fairness, each test function was run 30 times independently, and the average (Mean) and standard deviation (Std) were taken. In addition, Table 5 shows the statistical results (S.R.) obtained at the 5% significance level by Wilcoxon ranksum test. ''−'', ''≈'', and ''+'' respectively indicate that the performance of competitors is poor, similar, and better than CHDESMA.
As can be seen from the SMA variants in Table 5, CHDESMA ranks first overall. The overall performance of the CHSMA algorithm is better than the original SMA algorithm, which shows that the chaotic maps method can effectively increase the diversity of slime mold populations and improve the performance of the original SMA algorithm. The DESMA algorithm is better than the SMA algorithm, which means that using the DE strategy can prevent SMA from falling into the local optimal value and enable SMA to find the global optimal solution. Wilcoxon rank-sum tests also show that this improvement is statistically significant.

C. COMPARISON WITH DE VARIANTS ON CEC2014
In order to test the performance of the proposed CHDESMA, CHDESMA was compared with DE variants in Table 6, including adaptive guided DE (AGDE) [23], ensemble sinusoidal differential covariance matrix adaptation with Euclidean neighborhood (LSHADE_cnEpSi) (named as LS_cnEpSi) [48], DE with strategy adaptation (SADE) [49], composite differential evolution (CoDE) [50], hybrid SMA with AGDE (SMA-AGDE) [22], modified BA hybridizing by DE (MBADE) [40], and JSO algorithm for solving single-objective real-parameter optimization problems (JSO) [51]. The experiment was carried out on the 30-dimensional CEC2014 test suites. For the sake of fairness, each test function was run 30 times independently. In addition, Table 6 shows the S.R. obtained at the 5% significance level by the Wilcoxon rank-sum test.
It can be seen from Table 6 that among these DE variants, CHDESMA ranks first. The results show that the CHDESMA has reached the global optimum value on the F2, F3, F7, F8,     F15, F19, and F23 to F30, and the CHDESMA algorithm has achieved minimum optimization on F9, F11, F16, and F22. According to the Wilcoxon rank-sum test in Table 6, the p-value of most functions is ≤0.05 (5% significance level), so the experimental results are statistically significant for the CEC2014 test suite, verifying that the performance of CHDESMA is not random.
The convergence of CHDESMA and DE variants on CEC2014 test suites is shown in Figure 2. It can be observed that the CHDESMA converges faster than other advanced algorithms in most cases. On F8, F9, F16, F24, F25, F27, and F28, the convergence rate of CHDESMA is fast, and CHDESMA has reached the global optimum on functions F2, F3, F7, F8, F15, F19, and F23 to F30. On F11 F22 and F29, the convergence rate of CHDESMA is not the fastest, but the final optimization results of CHDESMA are much smaller than other competitors. On F2, F15, F18 to F20, and F22, the convergence speed of CHDESMA is relatively slow at the beginning of the iteration. Still, as the number of iterations increases, the final optimization result of CHDESMA is much smaller than the competition. The chaotic maps method can effectively increase the diversity of slime mold populations and improve the performance of the original SMA algorithm, and the DE strategy can prevent SMA from falling into the local optimal value and enable SMA to find the global optimal solution. The boxplots of the CHDESMA and DE variants are shown in Figure 3. It can be seen from the distribution of the data that in most functions, the line shape of CHDESMA is narrow, and the distribution of the data is relatively concentrated. Only on F11, F14, and F16 the data of this algorithm is relatively scattered.

D. SCALABILITY OF CHDESMA AND ADVANCED ALGORITHMS
In this part, different dimensions (dim = 10, 30, 50, and 100) are used to test the ductility of the proposed CHDESMA. The test results are used to compare with the advanced algorithms proposed on the CEC2017 test functions in Table 7, including the CMA Evolution Strategy (CMAES) [52], memory-based grey wolf optimizer (mGWO) [53], modified Sine Cosine Algorithm (mSCA) [54], timevarying hierarchical salp swarm algorithm (TVBSSA) [55], modified global flower pollination algorithm (MGFPA) [56], phasor particle swarm optimization (PPSO) [57] and improved opposition-based whale optimization algorithm (OBWOA) [58]. For the sake of fairness, each test function was run 30 times independently. In addition, Table 7 shows the S.R. obtained at the 5% significance level by Wilcoxon rank-sum test.  Table 7 shows the comparison and statistical results between CHDESMA and the advanced algorithms with different dimensions. As shown in Table 7, CHDESMA ranks first across different dimensions. For unimodal functions (F1 and F3), CHDESMA converges very quickly, and the quality of its solution is very high. For multi-modal functions, CHDESMA performs better at higher dimensions. It is worth noting that for composition functions (F21 to F30), the performance of CHDESMA did not deteriorate seriously as the dimension increased. These results indicate that the searchability of CHDESMA is effective. Moreover, CHDESMA can avoid falling into local optimum, and the optimization performance of solving high dimensional functions is strong. The results of p-values in Table 7 show that the p-values of CHDESMA on most functions are less than 0.05, which suggests that the proposed algorithm has obvious advantages over the other algorithms. It is not difficult to see from Figure 4 that of these advanced algorithms, CHDESMA has the best optimization effect. As shown in Figure 4, CHDESMA converges fast and outperforms the other algorithms on most benchmark functions. There is close competition among all algorithms in dealing with hybrid functions (F16, F17, F19, and F20), but VOLUME 10, 2022 CHDESMA can find a better solution during the overall steps. It is because the chaotic map method in CHDESMA can increase the diversity of the population, and the DE strategy can prevent SMA from falling into the local optimal value and accelerate the algorithm's convergence rate. All in all, the experimental results of CHDESMA are superior to other advanced algorithms.
The boxplots of the CHDESMA and DE variants are shown in Figure 5. It can be seen from the distribution of the data that in most functions, the line shape of CHDESMA is narrow, and the distribution of the data is relatively concentrated.

E. CHDESMA FOR ENGINEERING DESIGN PROBLEMS
The performance tested by the optimizer on engineering constraint problems shows the potential efficiency in other types of problems. Therefore, engineering constraint problems are often used to verify the algorithm's performance [59]. This article uses the proposed CHDESMA to solve four classic engineering design problems: TCSD, WBD, PVD, and TBTD. The CHDESMA's results were compared to results obtained with SMA [12], ILSHADE [60], JADE [61], JSO [51], LSHADE_cnEpSi [48], LSHADE [62], SADE [49], SHADE [63], MPEDE [64], and mGWO [53]. The constrained optimization problem in the real world can be represented by the following mathematical formulation [65].
In general, one equation constraint can be transformed into two-equation constraints, as follows: where ε is a small value (close to zero). The ranking of algorithms in the table is based on the best value. The penalty function method is as follows [66], [67]: where f (x) represents the objective function, and g c (x) represents its constraints. λ is the penalty coefficient, and N c is the number of constraints.

1) ENGINEERING PROBLEM 1 (TCSD)
The primary purpose of the TCSD problem is to obtain the value of the minimum spring weight [68]. Therefore, the following three parameters must be optimized in modeling: average coil diameter (D), wire diameter (d), and effective coil number (N ), as shown in Figure 6. The detailed description of the TCSD is shown in Appendix A. Table 8 shows the best design solution for the TCSD problem, and Table 9 summarizes the comparison results between CHDESMA and other algorithms. CHDESMA has the smallest value for the best solution to the TCSD problem and ranks first for the worst solution and average value. Therefore, the CHDESMA algorithm has good performance and stability in solving the TCSD problem.

2) ENGINEERING PROBLEM (WBD)
The primary purpose of the WBD problem is to minimize the economic cost [69]. There are the following four optimization constraints parameters: the buckling load on the rod (P c ), the end deflection of the beam (δ), the shear stress (τ ), and the bending stress in the beam (θ). Reducing costs requires controlling the following four variables: connecting steel bar length (l), weld thickness (h), steel bar thickness (b), and steel bar height (t), as shown in Figure 7. The detailed descriptions of the WBD are shown in Appendix B. Table 10 and Table 11 show the best design for the WBD problem and the statistical results of the comparison algorithms, respectively. It can be seen from these tables that CHDESMA ranks first in the best value among all algorithms. In addition, the worst solution and average value are also the smallest. The SMA algorithm ranks second on the WBD problem. Therefore, the performance of the proposed CHDESMA in solving the WBD problem is better than the original SMA and other competing optimizers.

3) ENGINEERING PROBLEM (PVD)
The PVD problem's primary purpose is to obtain the smallest value of the total cost [70]. Economic cost calculation needs to consider three factors welding, forming, and material. To minimize the financial cost, four design variables need to be constrained: head thickness (T h ), thickness (T s ), container cylindrical section length (L), and inner radius (R), as shown in Figure 8. The detailed description of the PVD is shown in Appendix C. VOLUME 10, 2022 Table 12 and Table 13 show the best design for the PVD problem and the statistical results of the comparison algorithms, respectively. CHDESMA can find the best optimization solution. But the statistical results show that CHDESMA's worst value and average ranking are both 9 th , indicating that CHDESMA does not find the best solution every time.

4) ENGINEERING PROBLEM (TBTD)
The primary purpose of TBTD is to ensure that the volume of the truss should be minimized under static pressure and to meet the stress (σ ) constraints on each truss member. This problem can be transformed into an optimal cross-sectional area (x 1 , x 2 ) problem, as shown in Figure 9. The detailed description of the TBTD is shown in Appendix D. Table 14 and Table 15 show the optimized design and statistical results on the TBTD problem. CHDESMA gets the best function value in this problem. However, ILSHADE, JSO, SADE, and MPEDE rank first in the worst solution and    average value. Therefore, CHDESMA has a close competitive relationship with these DE variants.

F. DISCUSSION OF EXPERIMENTAL RESULTS
In the proposed algorithm, we mainly use two methods (chaotic initialization and DE algorithm) to further improve the performance of the original SMA. Chaotic initialization has been shown to improve the quality of the solution. However, it lacks robustness in dealing with various problems, and the performance of the proposed algorithm is not competitive when dealing with discrete problems.
The DE algorithm shows good exploration capabilities, but individuals easily fall into local optima as the iterations proceed. From the effectiveness test of the components (Section IV.B), the effectiveness of the chaotic maps and DE components in CHDESMA can be confirmed. From the statistical results of the CEC2014 and CEC2017 test functions (Sections IV.C and IV.D), the proposed algorithm CHDESMA shows excellent performance in dealing with multimodal and mixed functions. However, the convergence of the algorithm is not fast enough when dealing with unimodal functions. In general, CHDESMA is applicable to multimodal and mixed functions, but not to unimodal functions. It can be seen from the different dimensions (10, 30, 50, and 100) that the higher the dimensionality, the better the performance of CHDESMA compared to other competing algorithms. From the statistical results (Section IV.E) of four real-world engineering problems (TCSD, WBD, PVD, and TBTD), the CHDESMA algorithm exhibits superior performance and can better solve the four engineerings constrained problems.

V. CONCLUSION
This paper proposes a hybrid SMA algorithm based on Chaotic Maps and Differential Evolution. The proposed CHDESMA introduces the Chaotic Maps strategy to accelerate the convergence in the initial iteration process. The operators of differential evolution effectively avoid prematurely, enhance the local searchability, and avoid stagnation in local optima. CHDESMA conducted three sets of test experiments on the CEC2014 and CEC2017 benchmark functions. Experiments compared with DE variants and advanced algorithms verify the effectiveness of the proposed improved strategies, including SMA-AGDE, MBADE, AGDE, LSHADE_cnEpSi, SADE, CoDE, JSO, CMAES, mGWO, mSCA, TVBSSA, MGFPA, PPSO, and OBWOA. In addition, CHDESMA was used to solve four practical engineering design problems. Experimental results and statistical analysis show that CHDESMA algorithm is more suitable for solving multi-model and hybrid function problems, and CHDESMA performs well in high dimensions, which can effectively solve global optimization problems and complex engineering practical problems.
In future work, based on the advantages of CHDESMA, we will apply CHDESMA to large-scale and highdimensional problems. CHDESMA can also be used for multi-peaked complex optimization problems, such as constraint engineering optimization, photovoltaic design, and other related problems. In addition, the binary version of the variant can be further enhanced and used for feature selection.

APPENDIX A TENSION/COMPRESSION SPRING DESIGN PROBLEM (TCSD)
The mathematical model of the Tension/compression spring design problem is as follows:

APPENDIX B WELDED BEAM DESIGN PROBLEM (WBD)
The mathematical model of the Welded beam design problem is as follows: ι (x) = (ι ) 2 + 2ι ι

APPENDIX D THREE-BAR TRUSS DESIGN PROBLEM (TBTD)
The mathematical model of the Three-bar truss design problem is as follows: JUNWEI DONG received the bachelor's degree in mathematics and applied mathematics from Zhejiang Normal University, China, where he is currently a graduate student (majoring computer science and technology). His current research interests include machine learning and intelligent optimization algorithm.