Sand cat arithmetic optimization algorithm for global optimization engineering design problems

Abstract

Sand cat swarm optimization (SCSO) is a recently introduced popular swarm intelligence metaheuristic algorithm, which has two significant limitations – low convergence accuracy and the tendency to get stuck in local optima. To alleviate these issues, this paper proposes an improved SCSO based on the arithmetic optimization algorithm (AOA), the refracted opposition-based learning and crisscross strategy, called the sand cat arithmetic optimization algorithm (SC-AOA), which introduced AOA to balance the exploration and exploitation and reduce the possibility of falling into the local optimum, used crisscross strategy to enhance convergence accuracy. The effectiveness of SC-AOA is benchmarked on 10 benchmark functions, CEC 2014, CEC 2017, CEC 2022, and eight engineering problems. The results show that the SC-AOA has a competitive performance.

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1. Introduction

In recent years, using metaheuristics to solve complex non-linear optimization has become a hot research topic due to their superiority and efficiency in solving difficult optimization problems. Various metaheuristics have been developed and applied in several fields of science and engineering applications to test their ability to find the optimal solution. In the field of different applications such as data mining (P. Hu et al., 2020; Naik et al., 2020; Shetty et al., 2021), engineering optimization (Gupta et al., 2021; Hu, Zheng, et al., 2023; Singh & Bansal, 2022; Yin et al., 2022; Zare et al., 2023), path planning (Panwar & Deep, 2021; Qu et al., 2020a, b), image segmentation (Khairuzzaman & Chaudhury, 2017; Medjahed et al., 2016; Rajput et al., 2019), and so on. Compared with the traditional methods, metaheuristics are simpler, more efficient, and have also been recognized by many researchers. Some of the metaheuristics which are most common and widely used are the fruit fly optimization algorithm (Pan, 2012), krill herd algorithm (Gandomi & Alavi, 2012), dolphin echolocation (DE, Kaveh & Farhoudi, 2013), swallow swarm optimization (Neshat et al., 2013), animal migration optimization (X. Li et al., 2014), and so on. Some of the other algorithms that have recently been developed, such as sand cat swarm optimization (SCSO, Seyyedabbasi & Kiani, 2023), ant lion optimizer (Mirjalili, 2015a), crow search algorithm (CSA, Askarzadeh, 2016), whale optimization algorithm (WOA, Mirjalili & Lewis, 2016), sine cosine algorithm (SCA, Mirjalili, 2016), Harris hawks optimization (HHO, Heidari et al., 2019), arithmetic optimization algorithm (AOA, Abualigah, et al., 2021a), teaching–learning-based optimization (Rao et al., 2011), gaining–sharing knowledge-based algorithm (GSK, Mohamed et al., 2020), dwarf mongoose optimization algorithm (DMOA, Agushaka et al., 2022), golden jackal optimization (Chopra & Ansari, 2022), prairie dog optimization (Ezugwu et al., 2022), gazelle optimization algorithm (Agushaka et al., 2023), enhanced African vultures optimization algorithm (Zheng et al., 2023), modified beluga whale optimization (Jia et al., 2023), and running city game optimizer (Ma et al., 2023).

SCSO (Seyyedabbasi & Kiani, 2023) is a recently developed metaheuristic. The search and hunting behavior of the sand cats in nature inspires it. Sand cats have the feature to detect low-frequency noises. This critical feature allows it to find and catch its prey quickly. However, similar to other algorithms, there have been some drawbacks associated with SCSO, such as low convergence accuracy and the tendency to get stuck in local optima. Thus, the AOA is applied in this study to improve the SCSO. Here, we briefly review the major works of SCSO and AOA to understand the shortcomings of this study.

1.1. Literature review

Aiming at the limitations of SCSO, scholars have been inspired to conduct in-depth research and propose improved methods. D. Wu et al. (2022) proposed a modified SCSO based on the wandering strategy to enhance global exploration ability, and the proposed method was used to solve engineering issues. Kiani et al. (2023a) presented a novel version of the SCSO based on a political system to balance between exploration and exploitation, and the performance of the proposed method is analyzed on different benchmark functions. Talpur et al. (2023) proposed a new improved SCSO to solve local minima problems, and experimental results on the feature selection demonstrate the effectiveness of the proposed method. Alex Stanley Raja et al. (2023) proposed an improved SCSO based on Brownian random walk and chaotic tent drift strategy to enhance exploration and exploitation. Qtaish et al. (2023) proposed an enhanced SCSO called binary memory-based SCSO to overcome quickly falling into local optimal and low convergence accuracy when solving feature selection. Seyyedabbasi (2023) introduced reinforcement learning to improve the performance of the SCSO, and experimental results on the benchmark functions analyze the demonstration of the proposed method. Kiani et al. (2023b) proposed a chaotic SCSO to solve low search consistency, local optimum trap, inefficiency search, and low population diversity issues. Y. Hu et al. (2023) adapted the immune algorithm to improve the performance of the SCSO, and the proposed method was applied to solve double-layer spraying path parameters. X. Wang et al. (2023) proposed an adaptive SCSO based on Cauchy mutation and optimal neighborhood disturbance strategy to improve convergence precision.

The aforementioned literature used different strategies to improve SCSO. However, the AOA has been used by most researchers to improve the performance of methods because of its strong exploration and exploitation. For example, Mahajan et al. (2022a) presented a new hybrid method based on aquila optimizer and AOA and demonstrated the effectiveness of the proposed method on high-dimensional issues. Chauhan et al. (2022) proposed a hybrid algorithm that integrated the AOA and the proposed method was applied to solve the engineering issues. ÇetınbaŞ et al. (2022) proposed a hybrid Harris hawks optimizer-arithmetic optimization algorithm, which improves the performance of the HHO. Mahajan et al. (2022b) introduced AOA in the hunger games search to achieve high-quality performance. Thota and Sinha (2023) introduced AOA to improve the poor exploration of the grey wolf optimizer, which was applied to solve the maximum power point tracking. Erdemir (2023) adapted the exploration phase of the AOA to improve the salp swarm algorithm and demonstrated its better performance on different benchmark functions.

Although scholars have proposed that SCSO variants are superior to the basic SCSO, they still show limitations when faced with complex optimization issues. By inspiring the advantages of hybridizing two or more algorithms, in this study, the SCSO and AOA are hybridized to maintain a balance between exploration and exploitation, the proposed hybrid algorithm is named sand cat arithmetic optimization algorithm (SC-AOA). The SC-AOA is mainly based on the following factors. Firstly, SCSO has been found to have some adverse effects on its performance and high-dimensional issues. Secondly, although the existing literature has improved SCSO, the application of this method is relatively limited. Then, based on the “No Free Lunch” theory (Wolpert & Macready, 1997), no perfect method can fully resolve all issues, so researchers must constantly develop novel approaches compared with existing ones. Finally, according to the current literature, it is found that using a single operator is not very effective in improving the performance of the method, so using many different operators and combining other methods can effectively enhance the performance of the method, and it can be applied to solve most problems. The aforementioned factors motivate this study to propose a new method to solve more complex problems.

Thus, we present a hybrid algorithm named SC-AOA that combines a refracted opposition-based learning strategy, a multiplication and division operator of AOA, and a crisscross strategy. It aims to deal with low convergence accuracy, the need for more population diversity, and the poor balance between exploitation and exploration. The refracted opposition-based learning strategy is introduced to enhance the diversity of the population and minimize the probability that the method will be trapped in the optimal local state. Then, an AOA is added to update the position of the individual, which can be balanced exploration and exploitation. Lastly, the crisscross strategy can enhance convergence accuracy. All of these strategies are concerned with different aspects. Refracted opposition-based learning strategy focuses primarily on improving population diversity. The multiplication and division operator of the AOA mainly balances between exploitation and exploration. Moreover, the crisscross strategy is more concerned with enhancing convergence accuracy. Integrating those strategies removes the deficiencies of SCSO and dramatically enhances the performance of the primary method.

1.2. Contributions

To verify the efficiency of the SC-AOA, this study differs from other research in the following areas. Firstly, it is compared with 11 state-of-the-art algorithms, including SCSO, WOA, HHO, GSK, AOA, artificial hummingbird algorithm (AHA, W. Zhao et al., 2022), DMOA, honey badger algorithm (HBA, Hashim et al., 2022), Young’s double-slit experiment (YDSE, Abdel-Basset et al., 2023), multi-strategy improved slime mould algorithm (MSMA, Deng & Liu, 2023), and elite archives-driven particle swarm optimization (EAPSO, Zhang, 2023) on 10 classical benchmark functions, CEC 2014, CEC 2017, and CEC 2022. Then, three non-parametric tests (Wilcoxon signed-rank test, Friedman test, and Kruskal Wallis test) are examined. Moreover, SC-AOA is applied to eight challenging real-world engineering problems and compared with other algorithms. The experimental simulation results show that SC-AOA performs well, with high solution accuracy and robustness. The main contributions of this study can be summarized as follows.

• The refracted opposition-based learning strategy can enhance the initial population’s diversity and traversal.

• Combining with the multiplication and division operator of the AOA, a new update formula of the individual is proposed to balance exploration and exploitation.

• Considering the low convergence accuracy of SCSO, the crisscross strategy helps to improve the convergence accuracy.

• The performance of SC-AOA is compared with 11 state-of-the-art algorithms on 10 classical benchmark functions, CEC 2014, CEC 2017, CEC 2022, and eight challenging real-world engineering problems, respectively.

The rest of the study is organized as follows. Section 2 presents the basic SCSO and AOA. Section 3 discusses the proposed SC-AOA in detail. Section 4 analyzes the performance of the SC-AOA on benchmark functions. Section 5 applies the SC-AOA to solve real-world engineering problems. Finally, Section 6 summarizes the conclusions and future works.

2. Related Works

2.1. Sand cat swarm optimization

SCSO is inspired by the behavior of natural sand cats. The two main actions of the sand cats are foraging and attacking the prey. There are three main steps of sand cat hunting: initial population, searching, and attacking the prey.

• Initial population: In a D-dimensional optimization problem, the sand cat population size is N, and the solution for each sand cat is represented as |${X}_i = ( {{x}_{i1},\ {x}_{i2}, \cdots ,{x}_{iD}} )$|⁠. The initial population is calculated by Equation (1):

  $$\begin{eqnarray} {X}_{i,j}\left( t \right) = l{b}_j + U\left( {0,\ 1} \right) \times \left( {u{b}_j - l{b}_j} \right), \end{eqnarray}$$

  (1)

  where |${X}_{i,j}( t )$| indicates the jth dimension of the current position of |${X}_i$|⁠, t is the current iteration, |$U( {0,\ 1} )$| is random in [0, 1], |$l{b}_j$| indicates the jth dimension of the lower boundaries, and |$u{b}_j$| indicates the jth dimension of upper boundaries.

• Searching the prey: It is assumed that the sand cat sensitivity range starts from 2 kHz to 0. Each sand cat updates its position based on the best candidate position (⁠|$Po{s}_{\mathrm{ bc}}$|⁠), its current position (⁠|$Po{s}_\mathrm{ c}$|⁠), and its sensitivity range (r). Equations (2–4) describe the prey searching behavior:

  $$\begin{eqnarray} Pos\left( {t + 1} \right) = r \cdot \left( {Po{s}_{\mathrm{ bc}}\left( t \right) - rand\left( {0,\ 1} \right) \cdot Po{s}_\mathrm{ c}\left( t \right)} \right), \end{eqnarray}$$

  (2)

  $$\begin{eqnarray} r = {r}_G \cdot rand\left( {0,\ 1} \right), \end{eqnarray}$$

  (3)

  $$\begin{eqnarray} {r}_G = {s}_M - \left( {\frac{{{s}_M \cdot t}}{{{t}_{\mathrm{max}}}}} \right), \end{eqnarray}$$

  (4)

  where |${r}_G$| indicates the broad sensitivity range that is decreased linearly from 2 to 0, the |${s}_M$| value is inspired by the hearing characteristics of the sand cats, its value is assumed to be 2, rand (0, 1) is random in [0, 1], and |${t}_{\mathrm{max}}$| is maximum iterations.

• Attacking the prey: The sand cat sensitivity range is supposed as a circle. In this way, the direction of movement is determined by a random angle (⁠|$\theta $|⁠) on the circle. SCSO benefits the roulette wheel selection algorithm to select a random angle for each sand cat. Equations (5–7) describe the behavior of attacking the prey:

  $$\begin{eqnarray} X\left( {t + 1} \right) = \left\{ {\begin{array}{@{}*{1}{c}@{}} {Po{s}_\mathrm{ b}\left( t \right) - Po{s}_{\mathrm{ rnd}}\left( t \right) \cdot \cos \left( \theta \right) \cdot r\ \left| R \right| \le 1}\\ {r \cdot \left( {Po{s}_{\mathrm{ bc}}\left( t \right) - rand\left( {0,1} \right) \cdot Po{s}_\mathrm{ c}\left( t \right)} \right)\ \left| R \right| > 1\ } \end{array}} \right., \end{eqnarray}$$

  (5)

  $$\begin{eqnarray} Po{s}_{\mathrm{ rnd}} = \left| {rand\left( {0,1} \right) \cdot Po{s}_\mathrm{ b}\left( t \right) - Po{s}_\mathrm{ c}\left( t \right)} \right|, \end{eqnarray}$$

  (6)

  $$\begin{eqnarray} R = 2 \cdot {r}_G \cdot rand\left( {0,1} \right) - {r}_G, \end{eqnarray}$$

  (7)

where |$Po{s}_\mathrm{ b}$| indicates the best solution position, |$Po{s}_{\mathrm{ rnd}}$| indicates the random position, and |$\theta $| is between 0 and 360, and its value will be between −1 and 1.

2.2. Arithmetic optimization algorithm

AOA is proposed through the arithmetic operators in mathematical operations, namely addition (+), subtraction (−), multiplication (⁠|$\times $|⁠), and division (⁠|$\div $|⁠). AOA can solve optimization problems without calculating its derivatives. The optimization process of the AOA consists of two main phases: exploration and exploitation. The former refers to extensive coverage of search space using search agents of an algorithm to avoid local solutions. The latter is the improved accuracy of obtained solutions during the exploration phase.

• Exploration stage: According to the arithmetic operators, the division and multiplication operators are adopted to explore the searching mechanism. Equation (8) describes the exploration behavior:

  $$\begin{eqnarray} &&{X}_{i,j}\left( {t + 1} \right)\\ &&\quad= \left\{ {\begin{array}{@{}*{1}{c}@{}} {best\left( {{X}_j} \right) \div \left( {MOP + \epsilon } \right) \times \left( {\left( {u{b}_j - l{b}_j} \right) \times \mu + l{b}_j} \right)\ {r}_2 < 0.5\ }\\ {best\left( {{X}_j} \right) \times MOP \times \left( {\left( {u{b}_j - l{b}_j} \right) \times \mu + l{b}_j} \right)\ otherwise} \end{array}} \right.,\\ \end{eqnarray} $$

  (8)
  where |${X}_{i,j}( {t + 1} )$| denotes the jth dimension of the ith solution at the next iteration, and |$best( {{X}_j} )$| is the jth dimension in the best-obtained solution so far. |$\epsilon $| is a small integer number, and |$u{b}_j$| and |$l{b}_j$| denote the upper and lower bound values of the jth dimension, respectively. |$\mu $| is a control parameter to adjust the search process. |${r}_2$| is a random number between (0, 1), and MOP is a coefficient calculated by Equation (9):

  $$\begin{eqnarray} MOP = 1 - \frac{{{t}^{\frac{1}{\alpha }}}}{{{t}_{\mathrm{max}}^{\frac{1}{\alpha }}}}, \end{eqnarray}$$

  (9)

  where t is the current iteration, |${t}_{\mathrm{max}}$| denotes the maximum number of iterations, and |$\alpha $| is a sensitive parameter. It defines the exploitation accuracy over the iterations, which is fixed equal to 5.

• Exploitation stage: According to the arithmetic operators, the mathematical calculations using either subtraction or addition got high-dense results which refer to the exploitation search mechanism. Equation (10) describes the exploitation behavior:

  $$\begin{eqnarray} &&{X}_{i,j}\left( {t + 1} \right)\\ &&\quad=\left\{ {\begin{array}{@{}*{1}{c}@{}} {best\left( {{X}_j} \right) - MOP \times \left( {\left( {u{b}_j - l{b}_j} \right) \times \mu + l{b}_j} \right)\ {r}_3 < 0.5\ }\\ {best\left( {{X}_j} \right) + MOP \times \left( {\left( {u{b}_j - l{b}_j} \right) \times \mu + l{b}_j} \right)\ otherwise} \end{array}} \right.,\\ \end{eqnarray} $$

  (10)

where |${r}_3$| is a random number between (0, 1).

3. Proposed SC-AOA

3.1. Refracted opposition-based learning strategy

Tizhoosh proposed opposition-based learning in 2005 to improve the initialization of the population. It broadens the solution range by obtaining the opposition-based solution of the current solution to find a better alternative solution for a specific problem (Tizhoosh, 2005). The literature (Sharma & Pant, 2017; Z. Wang et al., 2022; Yu et al., 2021) combined the metaheuristic with opposition-based learning, proving that it can effectively improve the solution accuracy of the algorithm. Although introducing opposition-based learning in the early stage can enhance the convergence effect, it is easy to fall into premature convergence in the later stage. Thus, the principle of refraction (F. Zhao et al., 2020) is introduced into the opposition-based learning strategy to reduce the possibility of falling into premature convergence late in the search. The principle of refracted opposition-based learning is shown in Fig. 1.

where the search interval of the solution on the x-axis is distributed within the range [LB, UB], the origin O is the midpoint on [LB, UB], α and β are denoted as the angle of incidence, and the angle of refraction, respectively. m and m* are the lengths corresponding to the incident and refracted rays, respectively. The refracted index formula can be obtained as follows.

Let |$\sigma $| = |$\frac{m}{{{m}^*}}$| and n = 1. Substituted into Equation (11) and expanded to the high-dimensional space of SCSO yields the refracted direction solution |$x_{i,j}^*$|⁠, as follows:

where |${x}_{i,j}$| denotes the position of the ith sand cat in the population in jth dimensions (⁠|$i = 1,2, \cdots ,N,\ j = 1,2, \cdots ,D$|⁠), |$x_{i,j}^*$| denotes the refracted inverse solution of |${x}_{i,j}$|⁠, and |$L{B}_j$| and |$U{B}_j$| are the lower and upper bounds of the dynamic boundary, respectively.

3.2. The multiplication and division operator for the position component of SCSO

The multiplication and division operators in AOA have strong global exploration capability, facilitating the solution dispersion, thus avoiding premature convergence. When the sand cat agents perform position updates, their sensitivity takes different values of guides R parameter, making the sand cats choose different search strategies. In the SCSO, this balance is achieved by the guides R parameter, which allows the high exploration in the earlier iterations of the algorithm and exploits the discovered promising search areas in the later iterations.

where |$\epsilon $| is a small integer number, and |$ub$| and |$lb$| denote the upper and lower bound values, respectively. |$\mu $| is a control parameter to adjust the search process, fixed equal to 0.499. |$\rho $| is a random number between (0, 1), and MOP is a coefficient calculated by Equation (9).

3.3. Crisscross strategy

SCSO is improved by introducing the crisscross strategy (Meng et al., 2014), which enhances the solution accuracy of the algorithm.

3.3.1. Horizontal crossover

The horizontal crossover is an arithmetic crossover operated on all the dimensions between two agents. Suppose the ith parent sand cat |${X}_i$| and the kth parent sand cat |${X}_k$| are used to carry out the horizontal crossover operation at the jth dimension. Equations (16–17) can calculate their offspring:

where |$X_{ij}^{\prime}$| and |$X_{kj}^{\prime}$| are the moderation solutions that are the offspring of |${X}_{ij}$| and |${X}_{kj}$|⁠, respectively. |${r}_1$| and |${r}_2$| are random in [0, 1], |${c}_1$| and |${c}_2$| are random in [−1, 1]. The new solutions generated by horizontal crossover operation must be compared with the pre-crossover to retain better sand cats.

3.3.2. Vertical crossover

The vertical crossover is an arithmetic crossover operated on all the agents between two dimensions. Suppose the j₁th and the j₂th dimensions of the sand cat |${X}_i$| are used to carry out the vertical crossover operation. Equation (18) can calculate their offspring:

where |$X_{ij}^{\prime}$| is the offspring of |${X}_{i{j}_1}$| and |${X}_{i{j}_2}$|⁠, and r is random in [0, 1]. The new solutions generated by vertical crossover operation must be compared with the pre-crossover to retain better sand cats.

3.4. The proposed SC-AOA

The proposed SC-AOA benefits from three new search strategies. Firstly, the refracted opposition-based learning strategy enhances the diversity of the initial population. Then, the multiplication and division operator of the AOA mainly balances between exploitation and exploration. Moreover, the crisscross strategy is more concerned with enhancing convergence accuracy. Thus, cooperation among the three strategies can improve diversification, exploitation, exploration, and convergence accuracy. The pseudo-code and the detailed flowchart of the proposed SC-AOA are shown in Algorithm 1 and Fig. 2.

3.5. Computational complexity of SC-AOA

Assume that the population size is represented by N, the dimension is defined by D, and the number of maximum iterations is |${t}_{\mathrm{max}}$|⁠. The computational complexity of SCSO is O (⁠|$N \times D \times {t}_{\mathrm{max}}$|⁠). First, the computational complexity of introducing the refracted opposition-based learning to initialize the population is O (⁠|$N \times D$|⁠). Then the computational complexity of updating the population position is O (⁠|$N \times D \times {t}_{\mathrm{max}}$|⁠), and the computational complexity of introducing the crisscross strategy is O ((⁠|$\frac{N}{2} \times D + N$|⁠)|$\times {t}_{\mathrm{max}}$|⁠). Thus, the total computational complexity of the SC-AOA is O (⁠|$N \times D$| + |$N \times D \times {t}_{\mathrm{max}} + ( {\frac{N}{2} \times D + N} ) \times {t}_{\mathrm{max}}$|⁠) |$\cong $| O (⁠|$N \times D \times {t}_{\mathrm{max}}$|⁠).

4. Results and Analysis

This section analyzed the proposed SC-AOA on 10 classical benchmark functions. Among them, the first six functions are unimodal, while the remaining are multimodal. The corresponding formula of functions, dimension, range, and theoretical optimal value of these functions are given in Table 1. Dim indicates the function’s dimension, range shows the function’s domain, and |${f}_{\mathrm{min}}$| shows the function’s optimum value. All the functions used in this study are minimization problems. Several related works have used these benchmark functions (Abd Elaziz et al., 2017; Abualigah et al., 2023).

The parameters involved in the proposed SC-AOA and comparison algorithms are shown in Table 2. The comparison algorithms included SCSO, WOA, HHO, GSK, AOA, AHA, DMOA, HBA, YDSE, MSMA, and EAPSO, which are the native ones recently proposed, and some are ones with good performance. The values used are the same as those in the corresponding references.

4.1. Comparison with other algorithms on benchmark functions in different dimensions

In Tables 3–4, results are presented for 30 and 500-dimensional functions, respectively, to observe the robustness of the proposed SC-AOA on the scalability of benchmark functions. The performance of the proposed SC-AOA on 10 classical benchmark functions shows better exploitation than the other algorithms. In all the unimodal benchmark functions, the proposed SC-AOA generally gives the best results. In multimodal benchmark functions (F7–F10), the proposed SC-AOA also provides competitive results with the other algorithms. From Table 4, it can be seen that the proposed SC-AOA also outperforms the comparison algorithms in all high-dimensional benchmark functions. Overall, the proposed SC-AOA shows better efficacy regarding exploration and exploitation than the other algorithms.

Table 5 gives the average runtime obtained by each algorithm on 10 classical benchmark functions. To conclude more intuitively, the ranking of the runtime of each algorithm in most cases is as follows:

SCSO > AOA > SC-AOA > MSMA > HBA > YDSE > EAPSO > AHA > WOA > DMOA > HHO > GSK. From the results, SC-AOA consumes rank three. An explanation is that introduced strategies are added to the native algorithm. The high time consumption of SCSO itself is also the main reason. To improve the accuracy of solutions, we sacrifice some runtime.

4.2. Convergence behavior analysis

To observe the convergence behavior of algorithms, the convergence curves plotted in Fig. 3 correspond to the benchmark functions. From Fig. 3, the proposed SC-AOA shows a better convergence rate than the other algorithms. Moreover, for the F1, F6, F7, F9, and F10 functions, the convergence curve of the proposed SC-AOA shows noticeable decreases with the lapse of iteration, which indicates that the multiplication and division operator position update strategy can significantly escape from the local optimum and find better solutions. Overall, the proposed SC-AOA can be considered a better optimizer in terms of convergence rate.

4.3. Non-parametric statistical test analysis

Based on the statistical test analysis in the literature (M. W. Li et al., 2022), two methods, Wilcoxon signed-rank test and Friedman test, are used for statistical analysis to determine whether there is a significant difference between the proposed SC-AOA and the compared algorithms. Wilcoxon signed-rank test performs a signed rank test of the hypothesis that two independent samples and returns the P-value from the trial. In this study, the level of significance is set as 5%. Friedman test is used to detect significant differences between the behaviors of two or more algorithms. Moreover, this study uses the Kruskal Wallis test method (Azizi et al., 2023) for statistical test.

Table 6 shows the results of the Wilcoxon and Kruskal Wallis P-value test for the proposed SC-AOA and the compared algorithms. From them, it can be seen that the proposed SC-AOA is significantly different from the compared algorithms for all functions, which indicates that SC-AOA substantially outperforms the compared algorithms.

Table 7 and Fig. 4 present the results of the Friedman P-value test for the proposed SC-AOA and the compared algorithms. From them, it can be observed that the proposed SC-AOA gets the 1st rank compared with other algorithms. The Chi-square is 76.17, and the P-value is less than 0.05. It also proves significant differences between the proposed SC-AOA and the compared algorithms.

4.4. Impact of introduced strategies

In this subsection, to verify the impact of each strategy in the proposed SC-AOA, three different combinations are added to the SCSO named SCSO1, SCSO2, and SCSO3. In SCSO1, the improved refracted opposition-based learning is added to SCSO. In SCSO2, the AOA position update is added to SCSO. In SCSO3, the crisscross strategy is added to SCSO. The same benchmark functions (F1–F10) have been used to verify the impact of each strategy. The specific experimental results are shown in Table 8.

In all benchmark functions, the experimental results of Table 8 show that the proposed SC-AOA has performed better than other SCSO1, SCSO2, and SCSO3. Besides, SCSO2 is the second-best optimizer in these benchmark functions, which indicates the use of AOA position update to balance exploration and exploitation. On the one hand, global exploration in the early iteration makes the algorithm search the whole solution space, which improves the convergence speed. On the other hand, the sand cat population gathers near the optimal solution in the later iteration, local exploitation of the algorithm at this time can make the sand cat population continue to search for better solutions, which improves the convergence accuracy. Overall, combining the above three improvement strategies, the proposed SC-AOA can obtain high-level solutions quickly.

4.5. Comparison with other algorithms on CEC 2014 benchmark functions

In this section, 30 benchmark functions from CEC 2014 (Liang et al., 2013) are used to evaluate the proposed SC-AOA’s performance further. The name, the class, and the optimum of functions are given in Table 9. SCSO, WOA, HHO, GSK, AOA, AHA, DMOA, HBA, YDSE, MSMA, and EAPSO were selected to compare the optimization of CEC 2014 functions. The parameter settings of each algorithm are consistent with SCSO, D = 10. Table 10 is dedicated to conducting the experimental results in which the bold values represent the best-obtained solutions.

From Table 10, it is clear that the optimal values obtained by SC-AOA are better than the comparison algorithms. Specifically, SC-AOA obtains better optimal solutions than the other algorithms for the three unimodal functions from F01 to F03. Moreover, the multimodal function is used to verify the global exploration of the algorithm, and the optimal solutions obtained by SC-AOA on F04, F07, F08, F10, F11, and F15 are smaller than the other algorithms in terms of the mean. However, both hybrid and composition functions are more challenging functions than unimodal and multimodal functions, and from the obtained results, SC-AOA obtains good optimal solutions except for F24 to F27. Meanwhile, SC-AOA obtained the smallest standard deviation (Std) values in most of the benchmark functions, indicating that SC-AOA has good stability and robustness. Therefore, SC-AOA is in the first position compared with the other algorithms.

4.6. Comparison with other algorithms on CEC 2017 benchmark functions

In this section, 28 benchmark functions from CEC 2017 (G. Wu et al., 2017) are used to evaluate the proposed SC-AOA’s performance further. The name, the class, and the optimum of functions are given in Table 11. SCSO, WOA, HHO, GSK, AOA, AHA, DMOA, HBA, YDSE, MSMA, and EAPSO were selected to compare the optimization of CEC 2017 functions. The parameter settings of each algorithm are consistent with SCSO, D = 30. Table 12 is dedicated to conducting the experimental results in which the bold values represent the best-obtained solutions.

Table 12 gives the mean and the standard deviation (Std) of fitness obtained by each algorithm on 28 CEC 2017. In terms of mean, the proposed SC-AOA gets the best results on 71.43% of functions, proving that the proposed SC-AOA has better convergence accuracy. In terms of standard deviation (Std), the proposed SC-AOA obtains lower values on 53.57% of functions, which further proves that the proposed SC-AOA has better robustness. Furthermore, for unimodal functions, the proposed SC-AOA shows apparent advantages. The proposed SC-AOA provides very competitive results for the multimodal functions with the other algorithms except for F05, F07, and F08. For the hybrid functions, the proposed SC-AOA is better than the other algorithms on the F11, F12, F14, F16, and F18 to F20 functions regardless of the mean or standard deviation (Std). For composition functions, in terms of mean, the proposed SC-AOA also provides very competitive results with the other algorithms.

4.7. Comparison with other algorithms on CEC 2022 benchmark functions

In this section, 12 benchmark functions from CEC 2022 (Luo et al., 2022) are used to evaluate the proposed SC-AOA’s performance further. The name, the class, and the optimum of functions are given in Table 13. SCSO, WOA, HHO, GSK, AOA, AHA, DMOA, HBA, YDSE, MSMA, and EAPSO were selected to compare the optimization of CEC 2022 functions. The parameter settings of each algorithm are consistent with SCSO, D = 10.

Fig. 5 shows the convergence curves of 12 algorithms for solving CEC 2022. From Fig. 5, it can be seen that the results obtained by SC-AOA are significantly better than the comparison algorithms, and it is worth mentioning that the F01 to F05 benchmark functions are calculated to the global optimum.

5. Applications SC-AOA on Engineering Problems

Eight engineering design problems are used in this section of the experiments to assess further the efficiency of SC-AOA. They are welded beam design problem, pressure vessel design problem, three-bar truss design problem, tension/compression spring design problem, speed reducer, tubular column, piston lever, and heat exchanger. Then, it is compared with the literature (Abualigah et al., 2021a, b; Agushaka et al., 2022; Chen et al., 2019; Das et al., 2020; dos Santos Coelho, 2010; Gupta & Deep, 2019, 2020; Hu, Yang, et al., 2023; Kennedy & Eberhart, 1995; S. Li et al., 2020; Meng et al., 2014; Mirjalili, 2015b, 2016; Mirjalili & Lewis, 2016; Mirjalili et al., 2017; Nadimi-Shahraki et al., 2020; Rashedi et al., 2009; Seyyedabbasi & Kiani, 2023; Yang et al., 2021). The descriptions and mathematical models of all engineering problems are detailed below. Besides, the maximum iteration is 1000, the population size is 30.

5.1. Welded beam design problem

The objective of the welded beam design problem is to minimize the production cost of the welded beam (Coello, 2000). The problem has seven constraints such as |${g}_1( X )$|⁠, |${g}_2( X )$|⁠, |${g}_3( X )$|⁠, |${g}_4( X )$|⁠, |${g}_5( X )$|⁠, |${g}_6( X ),{\rm{\ }}\mathrm{and}\ {g}_7( X )$|⁠, and four decision variables including the width of the weld (⁠|${x}_1$|⁠), the length of the bar (⁠|${x}_2$|⁠), the height of the bar (⁠|${x}_3$|⁠), and the width of the bar (⁠|${x}_4$|⁠). The welded beam design problem is demonstrated in Fig. 6. Mathematically, this problem is defined as Equations (19–26).

Minimize:

Subject to:

Where:

Variable range:

Table 14 and Fig. 7 present the results of all the compared algorithms and SC-AOA to solve the welded beam design problem. From them, it can be seen that SC-AOA is a better algorithm compared with other compared algorithms by giving a more reliable solution where the optimal variables at |$[ {{x}_1,{x}_2,{x}_3,{x}_4} ]$|= [0.1668, 3.3980, 9.9995, 0.1680] with the best objective value |${f}_{\mathrm{min}}( x )$| = 1.5108. Although SC-AOA is not best for the optimization results of the properties of a single welded beam, its overall production cost is better than other compared algorithms, which further verifies the applicability and effectiveness of SC-AOA in practical applications.

5.2. Pressure vessel design problem

The pressure vessel design problem aims to reduce the overall weight of a particular cylindrical pressure vessel (Sandgren, 1990). The problem has four constraints such as |${g}_1( X )$|⁠, |${g}_2( X )$|⁠, |${g}_3( X )$|⁠, and |${\rm{\ }}{g}_4( X )$|⁠, and four decision variables including the width of the shell (⁠|${x}_1$|⁠), the width of the head (⁠|${x}_2$|⁠), internal radius (⁠|${x}_3$|⁠), and the height of the cylindrical part without studying the head (⁠|${x}_4$|⁠). The pressure vessel design problem is demonstrated in Fig. 8. Mathematically, this problem is defined as Equations (27–31).

Minimize:

Subject to:

Variable range:

Table 15 and Fig. 9 present the results of all the compared algorithms and SC-AOA to solve the pressure vessel design problem. From them, it can be seen that SC-AOA is a better algorithm compared with other compared algorithms by giving a more reliable solution where the optimal variables at |$[ {{x}_1,{x}_2,{x}_3,{x}_4} ]$| = [0.7813, 0.3979, 40.4768, 197.8236] with the best objective value |${f}_{\mathrm{min}}( x )$| = 5926.15.

5.3. Three-bar truss design problem

The three-bar truss design problem aims to reduce the overall weight of a particular three-bar truss (Save, 1983). The problem has three constraints such as |${g}_1( X )$|⁠, |${g}_2( X )$|⁠, and |${g}_3( X )$|⁠, and two decision variables including the cross-sectional areas of member 1 (⁠|${x}_1$|⁠) and the cross-sectional areas of member 2 (⁠|${x}_2$|⁠). The three-bar truss design problem is demonstrated in Fig. 10. Mathematically, this problem is defined as Equations (32–35).

Minimize:

Subject to:

Where:

Variable range:

Table 16 and Fig. 11 show the results of all the compared algorithms and SC-AOA to solve the three-bar truss design problem. From them, it can be seen that SC-AOA is a better algorithm compared with other compared algorithms by giving a more reliable solution where the optimal variables at |$[ {{x}_1,{x}_2} ]$| = [0.78 878, 0.40 794] with the best objective value |${f}_{\mathrm{min}}( x )$| = 263.8958. Although SC-AOA is not best for the optimization results of the properties of a single three-bar truss, its overall weight of a particular three-bar truss is better than other compared algorithms, which further verifies the applicability and effectiveness of SC-AOA in practical applications.

5.4. Tension/compression spring design problem

The goal of the tension/compression spring design problem is to reduce the overall weight of a particular spring (Coello, 2000). The problem has four constraints such as |${g}_1( X )$|⁠, |${g}_2( X )$|⁠, |${g}_3( X )$|⁠, and |${g}_4( X )$|⁠, and three decision variables including wire diameter (⁠|${x}_1$|⁠), mean coil diameter (⁠|${x}_2$|⁠), and the number of active coils (⁠|${x}_3$|⁠). The tension/compression spring design problem is demonstrated in Fig. 12. Mathematically, this problem is defined as Equations (36–40).

Minimize:

Subject to:

Variable range:

Table 17 and Fig. 13 show the results of all the compared algorithms and SC-AOA to solve the tension/compression spring design problem. From them, it can be seen that SC-AOA is a better algorithm compared with other compared algorithms by giving a more reliable solution where the optimal variables at |$[ {{x}_1,{x}_2,{x}_3} ]$| = [0.0520, 0.3650, 10.8197] with the best objective value |${f}_{\mathrm{min}}( x )$| = 1.2667714E−02. Although SC-AOA is not best for the optimization results of the properties of a single tension/compression spring, its overall weight of a particular spring is better than other compared algorithms, which further verifies the applicability and effectiveness of SC-AOA in practical applications.

5.5. Speed reducer

This problem aims to design a speed reducer with minimum weight (Agushaka et al., 2022). In this model, the constraints include |${g}_1( X )$| to |${\rm{\ }}{g}_{11}( X )$|⁠, and the speed reducer problem has seven variables: face width (⁠|${x}_1$|⁠), the module of teeth (⁠|${x}_2$|⁠), number of teeth in the pinion (⁠|${x}_3$|⁠), length of the first shaft between bearings (⁠|${x}_4$|⁠), size of the other shaft between bearings (⁠|${x}_5$|⁠), the diameter of the first shaft (⁠|${x}_6$|⁠), and diameter of the other shaft (⁠|${x}_7$|⁠); |${f}_{\mathrm{min}}( x )$| is the minimum weight of speed reducer. Equations (41–52) compute the mathematical formulation.

Minimize:

Subject to:

Variable range:

Table 18 and Fig. 14 show the results of all the compared methods and SC-AOA to solve the speed reducer problem. From them, it can be seen that SC-AOA is a better method compared with other methods by giving a more reliable solution where the optimal variables at |$[ {{x}_1,{x}_2,{x}_3,{x}_4,{x}_5,{x}_6,{x}_7} ]$| = [3.5003, 0.7000, 17.0000, 7.3023, 7.8025, 3.3502, 5.2870] with the best objective value |${f}_{\mathrm{min}}( x )$| = 2996.7149.

5.6. Tubular column

This problem aims to design a tubular column with minimum cost (Bayzidi et al., 2021). In this model, the constraints include |${g}_1( X )$| to |${\rm{\ }}{g}_6( X )$|⁠, and the tubular column problem has two variables, |${f}_{\mathrm{min}}( x )$| is the minimum cost of the tubular column. Equations (53–59) compute the mathematical formulation.

Minimize:

Subject to:

Variable range:

P = 2500 kgf, L = 250 cm, E = 0.85|$\times {10}^6$| kgf/|${\mathrm{cm}}^2$|⁠, and |${\sigma }_y = 500$| kgf/|${\mathrm{cm}}^2$|

Table 19 and Fig. 15 show the results of all the compared methods and SC-AOA to solve the tubular column problem. From them, it can be seen that SC-AOA is a better method than other methods by giving a more reliable solution where the optimal variables at |$[ {{x}_1,{x}_2} ]$| = [5.45 115, 0.29 197] with the best objective value |${f}_{\mathrm{min}}( x )$| = 26.49 954.

5.7. Piston lever

This problem aims to design a piston lever with minimum oil volume (Bayzidi et al., 2021). In this model, the constraints include |${g}_1( X )$| to |${\rm{\ }}{g}_4( X )$|⁠. The piston lever problem has four variables, |${f}_{\mathrm{min}}( x )$| is the minimum oil volume of the piston lever. Equations (60–64) compute the mathematical formulation.

Minimize:

Subject to:

Variable range:

Table 20 and Fig. 16 show the results of all the compared methods and SC-AOA to solve the piston lever problem. From them, it can be seen that SC-AOA is a better method compared with other compared methods by giving a more reliable solution where the optimal variables at |$[ {{x}_1,{x}_2,{x}_3,{x}_4} ]$| = [0.0500, 1.0081, 2.0163, 500.0000] with the best objective value |${f}_{\mathrm{min}}( x )$| = 1.0574.

5.8. Heat exchanger

Heat exchanger design is a benchmark minimization problem (Jaberipour & Khorram, 2010). This model’s constraints include |${g}_1( X )$| to |${\rm{\ }}{g}_6( X )$|⁠, and the heat exchanger problem has eight variables. |${f}_{\mathrm{min}}( x )$| is the minimum heat exchanger. Equations (65–71) compute the mathematical formulation.

Minimize:

Subject to:

Variable range:

Table 21 and Fig. 17 show the results of all the compared methods and SC-AOA to solve the heat exchanger problem. From them, it can be seen that SC-AOA is a better method compared with other methods by giving a more reliable solution where the optimal variables at |$[ {{x}_1,{x}_2,{x}_3,{x}_4,{x}_5,{x}_6,{x}_7,{x}_8} ]$| = [591.648, 1029.181, 5698.745, 168.644, 272.094, 212.269, 295.258, 372.091] with the best objective value |${f}_{\mathrm{min}}( x )$| = 7319.574.

6. Conclusions and Future Works

This study proposes a hybrid version of SCSO and AOA called SC-AOA to maintain an appropriate collaboration between the operators’ exploration and exploitation. In the proposed SC-AOA, first, the refracted opposition-based learning strategy is introduced to initialize the population to enhance diversity and traversal. Then, an AOA is added to update the agent’s position, which can be balanced exploration and exploitation. Furthermore, the crisscross strategy is used to enhance convergence accuracy. To verify the efficiency of SC-AOA, it is compared with 11 state-of-the-art algorithms on 10 classical benchmark functions, CEC 2014 functions, CEC 2017 functions, and CEC 2022 functions. The robustness of the proposed SC-AOA regarding scalability is also examined by increasing the dimension of the issues from 30 to 500. The analysis of results through non-parametric statistical tests and convergence curves shows better performance in SC-AOA than the other algorithms. Furthermore, SC-AOA is applied to eight challenging engineering problems. The results show that SC-AOA can find feasible solutions for all engineering instances and far outperforms most compared algorithms regarding solution accuracy. Therefore, SC-AOA is a promising algorithm for solving complex constrained optimization problems.

From the results gained by experimental performance evaluation, statistical analysis, and solutions found for engineering problems, the conclusions can be taken as follows:

• The results obtained by 10 classical benchmark functions, 30 CEC 2014 functions, 28 CEC 2017 functions, 12 CEC 2022 functions, and statistical tests verify the SC-AOA performance compared to other well-known methods.

• The comprehensive evaluation of SC-AOA in solving different dimensional problems shows that SC-AOA is highly portable and has great potential to handle different dimensional problems.

• The high efficiency of SC-AOA for eight engineering problems shows that SC-AOA is an excellent method for dealing with complicated contemporary problems.

However, like other optimization methods, the proposed SC-AOA also has some limitations that need to be improved. If the improved strategy is more complex, it will incur a higher computational cost, so high consumption is still the main limitation.

In future work, three main directions can be followed. First, the focus will be on reducing operating costs while maintaining the accuracy of the solution. In practical applications, we use SC-AOA to solve feature selection, reduce dimensionality, and improve model performance. Next, the binary and multi-objective versions of SC-AOA can be utilized to solve more complex problems. Finally, this work will motivate other researchers to work on new metaheuristics and optimization concepts.

Funding

This research was supported by the Fundamental Research Funds for the Central Universities and Graduate Student Innovation Fund of Donghua University (grant number: CUSF-DH-D-2023053).

Data availability

All data generated or analyzed during this study are included in this published article.

Conflict of interest statement

None declared.

References