
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39237632
71581
10.1038/s41598-024-71581-2
Article
An improved sand cat swarm optimization with lens opposition-based learning and sparrow search algorithm
Cai Yanguang 12
Guo Changle guochangle0212@163.com

1
Chen Xiang 1
1 https://ror.org/04azbjn80 grid.411851.8 0000 0001 0040 0205 School of Automation, Guangdong University of Technology, Guangzhou, 511400 China
2 School of Intelligent Manufacturing and Electrical Engineering, Guangzhou Institute of Science and Technology, Guangzhou, 510540 China
5 9 2024
5 9 2024
2024
14 2069025 5 2024
29 8 2024
© The Author(s) 2024
2024
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The sand cat swarm optimization (SCSO) is a recently proposed meta-heuristic algorithm. It inspires hunting behavior with sand cats based on hearing ability. However, in the later stage of SCSO, it is easy to fall into local optimality and cannot find a better position. In order to improve the search ability of SCSO and avoid falling into local optimal, an improved algorithm is proposed - Improved sand cat swarm optimization based on lens opposition-based learning and sparrow search algorithm (LSSCSO). A dynamic spiral search is introduced in the exploitation stage to make the algorithm search for better positions in the search space and improve the convergence accuracy of the algorithm. The lens opposition-based learning and the sparrow search algorithm are introduced in the later stages of the algorithm to make the algorithm jump out of the local optimum and improve the global search capability of the algorithm. To verify the effectiveness of LSSCSO in solving global optimization problems, CEC2005 and CEC2022 test functions are used to test the optimization performance of LSSCSO in different dimensions. The data results, convergence curve and Wilcoxon rank sum test are analyzed, and the results show that it has a strong optimization ability and can reach the optimal in most cases. Finally, LSSCSO is used to verify the effectiveness of the algorithm in solving engineering optimization problems.

Keywords

Sand cat swarm optimization
Lens opposition-based learning
Sparrow search algorithm
Engineering optimization problems
Subject terms

Computer science
Computational science
Science and Technology Program of Guangdong ProvinceProvince under grant No. 2016A050502060 Province under grant No. 2020B1010010005 Science and Technology Program of GuangzhouNo. 202206010011 No. 2023B03J1339 issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

As engineering applications have become more sophisticated, numerous complex optimization problems have been introduced. Optimization problems in engineering frequently encompass intricate model structures, a multitude of decision variables, demanding constraints, and challenging objective functions. Constantly locating their solutions is an arduous undertaking. Solving problems in a constrained amount of time using conventional optimization techniques, such as the Lagrange multiplier, and dynamic programming methods, is typically challenging.

In order to better solve the above problems, the researchers proposed a meta-heuristic algorithm. Meta-heuristic algorithms are divided into four main categories according to their inspiration: evolution-based algorithms, physics-based algorithms, human-based algorithms, and population-based algorithms.

Evolution-based algorithms are created by modeling the way organisms evolve in nature. Genetic algorithm (GA)1 is the most representative algorithm. The algorithm takes inspiration from Darwin’s theory of evolution and includes operators such as selection, reproduction and mutation, generating new populations with each iteration. Memetic algorithm (MA)2 is inspired by cultural evolution and combines global search with local search for individuals. Differential evolution (DE)3 is actually a further enhancement of the genetic algorithm, which uses individual differences in the current population to construct differential mutation patterns of mutants, and the algorithm includes four main concepts: random population, mutation, crossover, and selection.

Physics-based algorithms are inspired based on physical phenomena or laws of physics. For example, light, heat, temperature, gravitational motion, electromagnetic field, gas solubility, etc. Inspired by the law of universal gravity, a new type of group intelligence optimization algorithm is introduced – the Gravitational search algorithm (GSA)4. Simulated annealing (SA)5 is proposed based on the process of heat degradation of high-temperature objects. Kepler optimization algorithm (KOA)6 is a new type of optimization based on Kepler’s law of planetary motion. Thermal Exchange Optimization (TEO)7 is a new optimization algorithm based on Newton’s law of cooling. Furthermore, the Equilibrium Optimizer (EO), Electromagnetic Field optimization (EFO)8, Electromagnetic Field optimization (HGSO)9, and Flow Regime Algorithm (FRA)10 all belong to this category.

Human-based algorithms are a type of optimization algorithms designed to take inspiration from human social behavior. Human learning optimization (HLO)11, simulates human learning processes to solve complex problems. Social group optimization (SGO)12, proposals based on the learning capacity of social groups and improving their own problem-solving capacity, Social network search (SNS)13, is a new optimization algorithm based on the social behavior of human groups. Harmony Search (HS)14 was inspired by the observation that the purpose of music is to seek a state of perfect consonance. Trying to find consonance in music is similar to finding the optimal solution in the process of optimization. In other words, the improvisation process of a jazz musician can be likened to the search process in optimization.The Brain Storm Optimization Algorithm (BSOA)15 was inspired by brainstorming, a situation in which a group of people generate new ideas around a particular area of interest. Brainstorming is a method to fully exploit human creative thinking to solve problems. It has the characteristics of strong evolutionary ability, fast search speed and strong optimization ability.In addition, there are many algorithms that fall into this category, such as Imperialist Competitive Algorithm (ICA)16, Kidney inspired algorithm (KA)17, Interior Search Algorithm (ISA)18, Teaching Learning Based Optimization (TLBO)19, etc.

Population-based algorithms simulates the behavior and biological characteristics of biological groups in nature, and builds mathematical models and solves them according to the cooperation mode between individuals and the information sharing mode between groups. Ant colony optimization (ACO)20 is very famous algorithm which models the solution based on the fact that ants will release pheromones on their path to find food. Particle swarm optimization (PSO)21 is inspired by the search for optimal solutions based on the foraging behavior of birds through collaboration and information sharing among individuals in the group. Gray wolf optimization (GWO)22 imitates the hunting actions of gray wolves, structured based on the hierarchical structure within a wolf group, where the alpha wolf leads the group in searching for the optimal solution. Meta-heuristic algorithms are frequently employed for resolving intricate engineering issues because they are easy to construct, have rapid processing speed, offer great flexibility, and exhibit robust resilience. Engineering sectors utilize meta-heuristic algorithms to address optimization difficulties. The issues involve the scheduling of workshops, planning of paths, optimization of parameters, identification of systems, and processing of signals. The research of meta-heuristic algorithm has important academic and practical significance.

The main difference between meta-heuristic algorithms is how to balance the exploration and exploitation phases in the optimization process. During the exploration phase, a global search is performed across the entire solution space to find the region that may contain the best solution. Exploration refers to analyzing a problem on a global level and identifying certain areas of the search space with the best global solution. During the exploration phase, the local search is intensified to find more precise solutions in the most promising areas. Exploration and exploration are two opposite turning points. Raising the results of one stage lowers the results of another stage.

Although meta-heuristic algorithms have been successful in many areas. But the capabilities, applications, and advantages of algorithms, for a wide range of optimizations, there are always questions about the usefulness and applicability of meta-heuristics. According to the NFL theorem, no single algorithm can accurately solve all optimization problems. In other words, optimization algorithms can solve a range of problems, but not others. Given the emergence of new optimization problems and the rapid growth in the size and complexity of real-world and engineering design problems, the development of new and more powerful optimization algorithms seems inevitable.

In view of the above problems, the existing theoretical literature mainly focuses on three aspects: proposing new algorithms, improving existing algorithms and integrating multiple algorithms.

Over the past few decades, numerous meta-heuristic algorithms have been suggested, including Sparrow Search Algorithm (SSA)23, Elephant Herding Optimization (EHO)24, Snake Optimizer (SO)25, Dung Beetle Optimizer (DBO)26, Whale Optimization Algorithm (WOA)27, Harris Hawks Optimization (HHO)28, Artificial Bee Colony (ABC)29, Dragonfly algorithm (DA)30, Bat Algorithm (BA)31, Cuckoo Search (CS)32, Ant Lion Optimizer (ALO)33, Woodpecker Mating Algorithm (WMA)34. These algorithms possess a straightforward structure, are easily implemented, exhibit resilience, and are extensively employed in resolving diverse intricate optimization problems.

In the second method, in addition to using different mathematical or stochastic operators to improve the performance of the meta-heuristic algorithm, the use of various chaotic mapping and mutation strategies to increase the performance of the algorithm during the initial phase of the algorithm and the position update phase is another popular way to solve the optimization problem, which can obtain promising results. For example, Levy flight trajectory based whale optimization algorithm (LWOA)35 is proposed by combining Levy flight strategy with WOA. It helps to strike a better balance between exploration and exploration of WOA, making the algorithm faster, more robust, and avoiding premature convergence.In order to improve the search capability of GWO, Gupta and Deep proposed an improved algorithm based on random walk (RW-GWO)36. In the improved algorithm, the leader plays an exploratory role by taking a random walk in the entire search space. At the same time, the remaining wolves update their positions based on the leader’s position. Hu et al. proposed a Binary Grey Wolf Optimizer (BGWO)37 specially designed for feature selection problems.Liu et al. proposed a modified sparrow search algorithm named CASSA38 has been presented to deal with unmanned aerial vehicle (UAV) route planning problem. Chaotic strategy and Cauchy-Gaussian mutation are added to increase the effectiveness of the algorithm.Abdelazim G. Hussien and Mohamed Amin proposed an improved HHO, called IHHO39. The algorithm combines opposition-based learning (OBL), Chaotic Local Search (CLS) and adaptive strategy to enhance the ability of the algorithm to jump out of the local optimal and improve the performance of the algorithm.

The last enhancement method is the hybrid meta-heuristic algorithm. This method integrates the solving strategies and location updating methods of different meta-heuristic algorithms through a collaborative mechanism, which improves the competitiveness of the original algorithm. Hybrid meta-heuristic algorithm is the latest research direction for solving optimization problems. It can not only overcome the weakness exploration strategy of one algorithm, but also overcome the weakness exploitation mechanism of another algorithm, so it becomes an attractive research field for solving constrained unknown search space problems. A hybrid WOA-WMA algorithm (HWMWOA)40 embedded WMA modified position update equation based on advanced detection capability into WOA algorithm, and added modified Cauchy mutation operator into the proposed hybrid method. Finally, an arithmetic spiral motion guide algorithm using a new search-oriented mode is used to guide the movement in the search space. The WOA algorithm is easy to fall into local optimization and miss the diversity of the population, resulting in premature convergence. In order to accelerate the global search phase of the existing Harris Hawks optimizer and enable it to jump out of the local search space, Kamboj et al. proposed a hybrid variant of the Harris Hawks optimizer was developed using the sine and cosine algorithm, named Hybrid Harris Hawke-Sine and Cosine algorithm (hHHO-SCA)41. Combining WMA with SCA, a hybrid meta-heuristic algorithm HSCWMA42 for optimization problems is proposed. The sine and cosine relationships used in SCA provide good exploration capabilities for this algorithm, and SCA solves the problem of the lack of an efficient process for effective exploration. HSCWMA was proposed to leverage the exploration capabilities of WMA and Levy, exploit the development sensitivity and local search memory of SCA, to develop exploration and development quality, and provide a dynamic balance between the two phases.

Sand Cat Swarm Optimization (SCSO)43 is a novel meta-heuristic optimization algorithm proposed in 2022 based on the biological characteristics of sand cats. The key idea behind the algorithm is to imitate the cooperative hunting behavior of the sand cats. This is accomplished by facilitating the exchange of information and working together among individuals to arrive at the most practical solution to the issue at hand. During the process, each sand cats updates their location and status based on their own experiences and information with the other group members. This allows them to progressively get closer to the optimal solution to the issue, characterized by structural simplicity, a reduced number of parameter variables, and the benefits of optimization. However, the drawbacks of the SCSO are rather evident. It does not converge quickly and may not succeed in finding global optimal solutions, because it starts late in the development phase. In addition, the performance of the SCSO is also affected by the incorrect location, as it depends on the globally optimal location. The algorithm is weak in the balance between exploration and development, and the convergence rate is slow, so it may not be effective in finding the global optimal solution, especially for complex problems.

Thus, to resolve such drawbacks, various researches have been made in the literature to increase the capability of the basic SCSO. Wu et al.44 proposed a modified sand cat swarm optimization algorithm (MSCSO) that incorporates multiple strategies, adding levy flight strategy45,46 in the exploitation phase of the algorithm. In the exploitation phase of the algorithm, the levy flight strategy is added to increase the chances of sand cats to find better positions and avoid falling into local optimization. In the exploration phase of the algorithm, a triangle wandering strategy is added to increase the global search capability of the algorithm. Finally, the lens opposition-based learning strategy is introduced to increase the convergence speed of the algorithm. Wang et al.47 proposed an adaptive sand cat swarm optimization algorithm based on Cauchy mutation and optimal neighborhood disturbance strategy (COSCSO). The algorithm introduces a nonlinear adaptive parameter balancing algorithm in the exploitation phase and the exploration phase, and uses the Cauchy mutation strategy and optimal neighborhood disturbance strategy to prevent it from falling into a local optimum, to improve the exploitation of the algorithm, and to increase the convergence speed of the algorithm. Hu et al.48 proposed the improved sand cat swarm operation (ISCSO), which employs SPM chaotic mapping for population initialization. The objective of this strategy is to improve the quality of the starting population and get a more evenly distributed set of individuals over the search space. A non-linear adjustment method is included in the algorithm to achieve a balance between the exploitation phase and the exploration phase. In addition, the fusion immunization algorithm proposes a novel approach for updating locations in order to boost convergence accuracy, expedite the convergence process, avoid being trapped in local optimum solutions, and enhance the overall performance of the algorithm. Li et al49 proposed the intensified sand cat swarm optimization algorithm with multi-strategy strengthening (IMSCSO). IMSCSO uses a dynamic stochastic search technique to enhance the algorithm’s convergence speed. A approach that utilizes hybrid dyadic learning is suggested as a means to enhance population variety and prevent early convergence of the algorithm. Finally, a novel approach is introduced to effectively address the trade-off between exploration and exploitation in the algorithm by utilizing a joint pairwise selection mechanism. Chen et al.50 proposed an improved SCSO based on arithmetic optimization algorithm (AOA), based on refractive opposition-based learning and crisscross strategy called sand cat arithmetic optimization algorithm (SC-AOA), which introduces AOA to balance the exploration and exploitation, reduces the likelihood of falling into a local optimum, and uses a crossover strategy to improve the convergence accuracy. Kiani et al.51 proposed a SCSO with multidisciplinary principles and called it PSCSO. The algorithm draws inspiration from political systems. Randomly selecting the position between the candidate best solution and the current position during the exploitation phase increases the chance of finding a global solution. For this purpose, a new coefficient affecting the exploration and exploitation phases is defined. In addition, a new mathematical model is introduced for the exploration phase.

Inspired by the SCSO and SSA, a novel hybrid metaheuristic algorithm is proposed in this paper to solve real-world complex optimization problems. In the proposed hybrid algorithm that called an improved sand cat swarm optimization with lens opposition-based learning and sparrow search algorithm (LSSSCSO). In the process of the algorithm, the sand cat is always in the alert state, guarding around, and when it is aware of the danger, it immediately stays away and sends an alarm to remind other sand cats, so that the sand cat can search more fully in the search space. In the local development stage, the spiral search strategy is added to make the sand cat’s search route show the characteristics of spiral, so that the sand cat can better adjust its position in the search space through the spiral route, so as to improve the convergence accuracy of the algorithm. Finally, the lens opposition-based learning is embedded to expand the search range of the sand cat, increase the diversity of the sand cat population, and avoid falling into the local optimal.

Section "Sand cat swarm optimization" introduces the sand cat swarm optimization algorithm. Section "The improved sand cat swarm optimization" introduces the improvement strategy of LSSCSO algorithm. Section "Experimental results and discussion" introduces the experimental results of LSSCSO on CEC test function. Section "Constrained engineering design problems" is the application of LSSCSO to engineering design problems. Section "Conclusions and future works" summarizes and looks forward to the future work.

Sand cat swarm optimization

The sand cat is a species of mammal that is mainly found in the desert environment of the Sahara of Central Asia, the Sahara of Africa and the Arabian Peninsula. The living behavior of a sand cat is unlike the domestic cat although, there is no big difference in appearance between a sand cat and a domestic cat. The sand cats do not live in a group like many felines. In order to escape the harsh conditions during the day, they rest underground during the day and hunt at night. They can meet these needs thanks to their special hunting mechanisms. Their excellent hearing also allows them to pick up low frequency sounds. In this way, their ability to detect prey (insects and rodents) moving underground is high. Additionally, they are capable of traversing considerable distances without respite. In alignment with these distinctive attributes, the sand cat is able to discern sounds (prey movement) with heightened acuity, meticulously track the prey, and launch an effective attack contingent upon the prey’s location.

Initialize population

In the SCSO, a location matrix is first created based on the size of the optimization problem and the defined population of sand cats, where the location of each sand cat represents a solution to the optimization problem. Assuming that the size of the defined sand cat population is N and the dimension of the search space is d, then the location X of the sand cat population is an N×d matrix, assuming that the location of the i-th sand cat is Xi=xi,1,xi,2,⋯,xi,d. The location vector of the sand cat population is shown in Eq. (1).1 X=x1,1,x1,2,⋯,x1,dx2,1,x2,2,⋯,x2,d⋮⋮⋯⋮xN,1,xN,2,⋯,xN,d

Random initialization of the positions of the sand cat clusters according to Eq. (2) makes the sand cat clusters uniformly distributed in the search space.2 Xi,j=lbj+rand(0,1)·(ubj-lbj)

where Xi,j refers to the value of the jth dimensional variable of the ith sand cat individual, lbj refers to the lower bound of the jth dimensional variable, ubj refers to the upper bound of the jth dimensional variable, and rand(0, 1) is a random number obeying a uniform distribution on (0,1).

Prey-seeking strategy (exploration)

During the prey exploration phase, sand cats use their keen hearing ability to search or track prey, detect the movement of prey, and determine the general direction of prey. They have a unique ear canal structure that can sense low-frequency noise below 2khz. In this way, the sand cat is able to detect prey (insects and rodents) moving underground. According to this characteristic, the sensitivity range of the sand cat is defined, and in the optimization process, the sensitivity range decreases linearly from 2 to 0 with the progress of iteration. To improve the performance of the algorithm, it is assumed that each sand cat has a different sensitivity range. The sensitivity range of each sand cat is determined by the best candidate position, the current position of the sand cat, and the current position of the sand cat, and each sand cat updates its position towards its prey based on the best candidate position, the current position, and the sensitivity range. This approach above ensures that the location is updated according to the best candidate location and the current location, and that the updated location is between its current location and the prey location, while increasing the chances of finding a new optimal solution in the search space. This is shown in Eqs. (3)–(5).3 rg=SM-(SM·t)T

4 r=rg·rand(0,1)

5 Xt+1=r·Xbct-rand0,1·Xct

where, rg is the sensitivity range of sand cat; r is the sensitivity range of each sand cat; SM is the maximum sensitivity, which is set to 2 according to the hearing properties of sand cats; t is the current number of iterations, and T is the maximum number of iterations; Xbc and Xc is the best candidate and current position during the t iteration.

Prey-seeking strategy (exploitation)

When the sand cat determines the general location of the prey based on the sound, it will quickly move towards the prey and catch the prey. The sand cat also has a curious ability to rapidly dig if the prey is underground. Assuming that the sensitivity range of the sand cat is a circle, its moving direction can be determined by a random angle. The SCSO randomly selects an angle (θ) between (0, 360) by a roulette algorithm. By this way, it makes each sand cat able to move in different directions according to its respective circular area in the search space. To ensure that the sand cat can approach the prey, a random position is selected between the global optimal position and the current position and the position of the sand cat is updated according to Eqs. (6) and (7).6 Xrnd=rand(0,1)·Xb(t)-Xc(t)

7 X(t+1)=Xb-Xrnd·cos(θ)·r

where, Xrnd is a random position in the t iteration; X(t+1) is the position of the sand cat after the update; Xb is the global optimal position; Xc is the current position in the t iteration. t represents the number of iterations. rand(0, 1) is a random number obeying a uniform distribution on (0,1); cos(θ) is the cosine of a random angle.

Exploration and exploitation

During the whole hunting process of the sand cat, it is important to have a better balance between the exploration and exploration phases. To this end, during the optimization process, SCSO defines the parameter (R) to balance the conversion between the exploration phase and the exploration phase, which is defined as shown in Eq. (8).8 R=2·rg·rand(0,1)-rg

From Eq.(8), the value of the parameter R depends on the sensitivity range rg. As mentioned earlier, rg decreases linearly from 2 to 0 as the iteration progresses, therefore, R is a random value between [-2,2]. When R>1, the algorithm enters into the exploration phase, otherwise, the algorithm enters into the exploitation phase as shown in Eq. (9). The pseudo-code of the SCSO is shown in Algorithm 1.9 X(t+1)=Xb-Xrnd·cos(θ)·r|R|≤1r·Xbc(t)-rand(0,1)·Xc(t)|R|>1

Algorithm 1 SCSO Pseudo-Code.

The improved sand cat swarm optimization

In SCSO, the sand cat searches for prey by detecting low-frequency noise emitted by the prey, and after determining the general range of the prey, it quickly digs for the prey. In this process, the R parameter is used to balance the exploration stage and the development stage, so that the sand cat can carry out seamless conversion between the two stages, so that the algorithm has high efficiency. However, it should not be ignored that its convergence rate is not fast. At the same time, SCSO does not perform well in the exploitation phase for complex problems. However, it is worth noting that according to many studies in the literature, the algorithm performs well in many situations. On this basis, it is considered that the improvement of the algorithm will make a greater contribution to the literature. In addition, the performance of SCSO is also affected by the global optimal location, as it depends on the global optimal location. Therefore, this paper improves the SCSO. A dynamic spiral search strategy is introduced in the exploitation phase, which enables individual sand cats to approach the prey in a spiral manner in the search space, increasing the diversity of the population and improving the convergence accuracy of the algorithm. In the exploration phase of the algorithm, the lens opposition-based learning strategy and the sparrow alert mechanism are introduced to increase the search range of the sand cat, accelerate the convergence speed, improve the search efficiency, and enhance the global search capability of the algorithm.

Dynamic spiral search

In the exploration stage of SCSO, the search space will gradually decrease with the progress of iteration, but the reduced search space means the decrease of population diversity, and the ability of individuals to explore unknown areas will be reduced. In order to solve this problem, the inspiration of WOA27 spiral search is eliminated, and a dynamic spiral search52,53 method is added. The addition of this method makes the search process dynamically change with the change of the number of iterations, thus improving the performance of the algorithm. Spiral search was first proposed in WOA, where humpback whales swim around their prey in a reduced circle while following a spiral path to encircle the prey. In the spiral renewal model, the parameter controlling the shape of the helix is a constant. Although it can make the population converge quickly in the later stage, it will lead to a single movement path of the search individual, which is easy to fall into the local optimal solution, and reduce the optimization accuracy of the algorithm. In order to solve this problem, dynamic spiral search is added to SCSO. The updated formula is shown in Eqs.(10) and (11).10 z=ek·cos(π·tT)

11 X(t+1)=ezl·cos(2πl)·r·Xbc(t)-rand·Xc(t)

where, k is a constant for defining the shape of the logarithmic spiral, l is a random number in [ 1,1], t represents the number of iterations, T represents the maximum number of iterations, r is the sensitivity range of each sand cat.

On the basis of the original spiral search strategy, the shape parameters of the spiral are improved, and the iteration parameters are added to make the shape of the spiral dynamically adjust with the number of iterations. At the beginning of the iteration, the global search should be carried out with a larger spiral shape to speed up the convergence. In the later iteration period, with the increase of the number of iterations, the value of z should gradually decrease, and the target should be searched with a smaller spiral shape to improve the optimization accuracy of the algorithm.

Lens opposition-based learning

Opposition-based learning54 is a common enhancement strategy, the main idea is to expand the exploration ability of an algorithm by searching for reverse points in space. However, they are all in the early stage of algorithm execution, and the effect is not obvious in the later stage. Because in the late iteration, most of the search individuals are concentrated in a small range area, and the area where the reverse point is located is also concentrated in this area, that is, the adaptation value of the reverse point is not much better than the original value, so it is difficult to jump out of the current extreme value area by trying to reverse in the late iteration. Therefore, lens opposition-based learning (LOBL)55,56 is introduced in this paper.

LOBL is derived from the inverse learning strategy(ILS)55, which utilizes the lens refraction theorem to produce the inverse solution. The scaling factor k is used to regulate the relationship between the current solution and the inverse solution. This allows the ILS to exhibit superior dynamic performance. Integrating the lens opposition-based learning technique into the exploration phase of SCSO can increase the variety of the population, strengthen the algorithm’s search ability, and expedite the convergence speed of the algorithm.

For example, find the reversal point of point A in a two-dimensional plane, where the transverse coordinates are searched in the range (a,b) and the y-axis represents a convex lens. Suppose that the point A has transverse coordinates of x and height of h. The reverse point A∗ has transverse coordinates of x∗ and height of h∗. The schematic is shown in Fig. 1.Fig. 1 Two-dimensional planar lens opposition-based learning.

x centered at the origin and transformed according to the refraction theorem to obtain the corresponding reversal point x∗, as shown in Eq. (12).12 (a+b)/2-xx∗-(a+b)/2=hh∗

Let the scaling factor k=hh∗, then the inverse point x∗ can be obtained by the transformation as shown in Eq. (13).13 x∗=a+b2+a+b2k-xk

In the optimization process, assume that the current solution is X(x1,x2,...,xn) and the generated inverse solution is X∗x1∗,xx∗,...,xn∗. If the fitness value of f(X∗) is better than that of f(X), the inverse solution X∗ is used instead of the current solution X, so that the sand cat individual can obtain a better position in the search space and avoid falling into a local optimum.

Sparrow alert mechanism

The key to whether the SCSO can obtain the global optimal solution lies in the ability of the algorithm to jump out of the local optimum. If only the introduction of a single strategy could lead to the algorithm falling into local optimum again while accelerating the convergence rate. Therefore, inspired by the sparrow search algorithm, the sparrow alert mechanism is added to SCSO to expand the search space, avoid falling into local optimal value, and enhance the search capability. A subset of sand cat individuals was selected for the role of vigilantes, and these sand cat individuals were located at the edge of the group. Once detecting potential threats, the vigilant sand cat positioned at the edge will promptly notify its fellow sand cats and afterwards move to a safer location to evade the danger. The vigilantes will quickly move to a safe area to obtain a better position when they realize the danger, while the sand cat individuals in the central area will move randomly. This is shown in Eq. (14).14 X(t+1)=Xb+β·Xc(t)-Xbc(t),fc(t)>fgtX(t)+K·Xc(t)-Xw(t)fc(t)-fw(t)+ε,fc(t)=fgt

where β is the step control parameter, whose value is a random number obeying the standard normal distribution, and K is a random number between[-1,1]; fc is the fitness value of the current individual, fg is the fitness value of the optimal individual of the current iteration, fw is the fitness value of the worst individual of the current iteration, and Xw is the position of the worst individual of the current iteration, and εis the constant avoiding the denominator to be zero.

When fc=fg, it means that the vigilant senses the danger and warns the surrounding,and the population moves towards the safe position; when fc>fg, it means that the surrounding is safe and the population gradually approaches the optimal.

Implementation of the LSSCSO

The LSSCSO algorithm proposed in this paper adds the spiral search strategy in the exploitation stage, so that individuals can find better positions in the search space and improve the convergence accuracy. In the later stage of the algorithm, the lens opposition-based learning strategy and sparrow alert mechanism are introduced to improve the ability of the algorithm to explore the unknown region and jump out of the local optimum. The pseudo-code of the LSSCSO algorithm is shown in Algorithm 2.Algorithm 2 LSSCSO Pseudo-Code.

Computational complexity analysis

Computational complexity is an important index to evaluate the meta-heuristic algorithm, which is related to the structure and implementation of the algorithm. The computational complexity of LSSCSO mainly depends on the population size (N), the problem dimension (d), the maximum number of iterations (T) and the evaluation cost required by the fitness function (C). The complexity of the initialization phase is O(N×d). The time complexity of updating of positions is O(T×N×d). Time required for position update of lens opposition-based learning O(T×N×d). The cost time of the calculation function includes the calculation time cost of the algorithm itself O(T×N×C), the calculation time cost of Sparrow Alert Mechanism O(T×N×C),calculation time cost of lens opposition-based learning O(T×N×C). Therefore, the complexity of LSSCSO is O((3×T×N×C+2×T×N×D).

Experimental results and discussion

In order to ensure the fairness of the algorithm comparison, the experimental environment is as follows:the operating system is Windows 11, the CPU is Intel(R)Core(TM)i5-12450H 2.00 GHz, the RAM is 16GB, and the software is MATLAB R2023b. The algorithm was tested on CEC2005 benchmark functions and CEC2022benchmark functions.

The population size of each algorithm is set to 100, and the number of iterations is set to 500. In order to reduce the randomness effect, each algorithm is run independently for 30 times. The optimal value(Min), standard deviation(Std), mean(Mean) and worst value(Worse) of the results are recorded to measure the performance of the algorithms, and at the end, the ranking of the results corresponding to each algorithm is counted, and the optimal results are bolded in the table.

Experiments on the CEC2005 functions

To verify the optimization effect of the LSSCSO algorithm, this paper uses CEC2005 benchmark functions57 to verify the performance of the LSSCSO algorithm. In order to further confirm the efficacy of the proposed algorithms. This study employs PSO, GA, SCSO, HHO, Subtraction-Average-Based-Optimizer (SABO)58, WOA, WMA, MSCSO and the newly developed LSSCSO to compare the dataset discussed in the previous paper. The parameters settings of each algorithm are shown in Table 1.Table 1 Simulation parameter for each optimization algorithms.

Algorithm	Parameter	References	
LSSCSO	rg∈[0,2]		
MSCSO	rg∈[0,2],C=0.35	44	
WOA	b=1	27	
SCSO	rg∈[0,2]	43	
HHO	J∈[0,2]	28	
SABO	v∈1,2	58	
PSO	c1=2, c2=2	58,59	
GA	pc= 0.8, pm= 0.05	44,58	

In CEC2005 benchmark functions, F1-F7 are uni-modal benchmark functions (Table 2), which has only one global optimum in the range of the definition domain to test the convergence speed and exploitation ability; F8-F13 are multi-modal benchmark functions (Table 3), which has multiple local optimums in the range of the definition domain, and the higher the dimensions are, the more local optimums there are, so that the algorithm can be tested for its ability of global search and avoiding falling into the local optimums; F14-F23 are fixed-dimension multi-modal benchmark functions (Table 4), which can test the ability of the algorithm to balance the global search and local exploitation.Table 2 Uni-modal functions.

Function	Dim	Range	Fmin	
F1(x)=∑i=1nxi2	30	[-100,100]	0	
F2(x)=∑i=1n|xi|+∏i=1n|xi|	30	[-10,10]	0	
F3(x)=∑i=1n(∑j=1ixj)2	30	[-100,100]	0	
F4(x)=max{|xi|,1≤i≤n}	30	[-100,100]	0	
F5x=∑i=1n-1[100(xi+1-xi2)2+(xi-1)2]	30	[-30,30]	0	
F6(x)=∑i=1nxi+0.52	30	[-100,100]	0	
F7(x)=∑i=1ni×xi4+random[0,1)	30	[-1.28,1.28]	0	

Table 3 Multi-modal functions.

Function	Dim	Range	Fmin	
F8(x)=∑i=1n-xisin(|xi|)	30	[-500,500]	-418.98×D	
F9(x)=∑i=1n[xi2-10cos(2πxi)+10]	30	[5.12,5.12]	0	
F10(x)=-20exp(-0.21n∑i=1nxi2)-exp(1n∑i=1ncos2πxi)+20+e	30	[-32,32]	0	
F11(x)=14000∑i=1nxi2-∏i=1ncos(xii)+1	30	[–600,600]	0	
F12(x)=πn{10sin(πy1)+∑i=1n-1(yi-1)2[1+10sin2(πyi+1)]+(yn-1)2}	
+∑i=1nu(xi,10,100,4)	
yi=1+xi+14,u(xi,α,k,m)=k(xi-α)nxi>α0-α<xi<αk(-xi-α)nxi<-α	30	[-50,50]	0	
F13(x)=0.1{sin2(3πx1)+∑i=1n(xi-1)2[1+sin2(3πxi+1)]	
+(xn-1)2[1+sin2(2πxn)]}+∑i=1nu(xi,5,100,4)	30	[-50,50]	0	

Table 4 Fixed-dimension multi-modal function.

Function	Dim	Range	Fmin	
F14(x)=(1500+∑j=1251j+∑i=12(xi-αij)6)-1	2	[-65,65]	1	
F15(x)=∑i=111[ai-x1(bi2+b1x2)bi2+b1x3+x4]2	4	[-5,5]	0.0003	
F16(x)=4x12-2.1x14+13x16+x1x2-4x22+4x24	2	[-5,5]	-1.0316	
F17(x)=(x2-5.14π2x12+5πx1-6)2+10(1-18π)cosx1+10	2	[-5,5]	0.398	
F18(x)=[1+(x1+x2+1)2(19-14x1+3x12-14x2+6x1x2+3x22)]	
×[30+(2x1-3x2)2(18-32x1+12x12+48x2-36x1x2+27x22)]	5	[-2,2]	3	
F19(x)=-∑i=14ciexp(-∑j=13aij(xj-pij)2)	3	[1,3]	-3.86	
F20(x)=-∑i=14ciexp(-∑j=16aij(xj-pij)2)	6	[0,1]	-3.32	
F21(x)=-∑i=1s[(X-ai)(X-ai)T+ci]-1	4	[0,10]	-10.1532	
F22(x)=-∑i=1r[(X-ai)(X-ai)r+ci]-1	4	[0,10]	-10.4028	
F23(x)=-∑i=110[(X-ai)(X-ai)T+ci]-1	4	[0,10]	-10.5363	

Result Statistics and Convergence Curve Analysis

The experimental results are shown in Table 5. The experimental results show that LSSCSO achieves the optimal solution in 19 out of 23 benchmark functions, accounting for about 82.61% of all test functions, and has the best standard deviation, mean, and worst value on 16 test functions, accounting for 69.57% of all test functions, which is the first place in the overall ranking.Table 5 Comparison results between LSSCSO and other algorithms on 23 benchmark functions.

F	Metric	PSO	GA	SCSO	HHO	SABO	WOA	MSCSO	LSSCSO	
F1	Min	4.43E6	17.92696	3.80E-132	7.00E-124	1.80E-206	6.50E-107	0	0	
Std	0.046942	10.46286	1.10E-121	1.20E-101	0	6.65E-95	0	0	
Mean	0.022702	30.50201	2.10E-122	2.20E-102	1.10E-204	1.62E-95	0	0	
Worse	0.158149	56.92799	6.10E-121	6.60E-101	5.90E-204	3.62E-94	0	0	
F2	Min	0.089719	1.745356	3.97E-70	1.03E-62	1.00E-117	9.88E-63	0	0	
Std	0.534508	0.544788	5.09E-65	3.22E-55	2.40E-116	1.93E-55	0	0	
Mean	0.731055	2.788395	1.76E-65	7.31E-56	2.00E-116	3.66E-56	0	0	
Worse	1.908597	3.806556	2.29E-64	1.77E-54	8.80E-116	1.06E-54	0	0	
F3	Min	36.45082	1424.187	2.10E-116	2.60E-109	5.73E-60	992.8246	0	0	
Std	1118.449	639.6914	1.30E-105	2.83E-93	1.00E-17	8203.642	0	0	
Mean	643.4302	2168.983	2.40E-106	5.53E-94	1.83E-18	13273.82	0	0	
Worse	5210.771	3458.935	7.00E-105	1.55E-92	5.49E-17	33884.04	0	0	
F4	Min	3.435068	2.216469	3.47E-60	2.58E-61	3.38E-79	3.15E-08	0	0	
Std	2.439277	0.466936	1.39E-54	2.00E-53	4.05E-78	19.9513	0	0	
Mean	6.431779	2.829395	5.52E-55	5.14E-54	4.63E-78	18.96073	0	0	
Worse	14.35043	3.992738	7.12E-54	9.81E-53	1.66E-77	68.2743	0	0	
F5	Min	11.41045	228.808	25.14437	2.04E-06	27.74649	26.03261	23.20624	21.38153	
Std	44.9786	424.9867	0.95396	0.002722	0.334131	0.333655	1.269703	0.289346	
Mean	84.19977	595.3854	27.10865	0.001413	28.25606	26.80216	25.75	22.16023	
Worse	178.5254	2257.058	28.74677	0.013314	28.82224	27.70283	27.88987	22.87163	
F6	Min	5.23E5	15.61244	8.26E-06	5.75E-09	0.154356	0.001154	5.94E-07	5.62E-12	
Std	0.345263	13.54999	0.338369	1.87E-05	0.544483	0.002798	0.269803	1.97E-10	
Mean	0.082637	34.14746	0.522692	1.69E-05	1.48912	0.005065	0.268171	1.33E-10	
Worse	1.549095	62.76702	1.27459	7.54E-05	2.457135	0.013009	1.250101	7.13E-10	
F7	Min	0.074626	0.003032	3.02E-07	5.30E-07	2.81E-06	0.000108	3.25E-07	2.46E-06	
Std	0.060668	0.004819	5.69E-05	5.95E-05	3.65E-05	0.001076	1.86E-05	1.20E-05	
Mean	0.168645	0.010589	4.67E-05	5.58E-05	4.22E-05	0.001091	2.14E-05	1.76E-05	
Worse	0.293086	0.021939	0.000256	0.000262	0.000187	0.005636	6.15E-05	5.90E-05	
F8	Min	7862.41	9681.18	-8669.529	-12569.49	-4189.735	-12569.26	-10132.51	-9975.995	
Std	885.9265	641.2242	586.0692	0.113375	338.3872	1182.994	743.8707	656.5691	
Mean	6500.96	8421.5	-7328.178	-12569.41	-3363.099	-11592.83	-8756.989	-8888.443	
Worse	4751.67	7028.99	-6401.768	-12569.01	-2829.591	-8657.469	-7268.969	-7702.789	
F9	Min	29.84883	23.23239	0	0	0	0	0	0	
Std	21.62223	13.80758	0	0	0	0	0	0	
Mean	60.31323	54.68123	0	0	0	0	0	0	
Worse	113.4265	76.90086	0	0	0	0	0	0	
F10	Min	1.778035	2.881962	4.44E-16	4.44E-16	4.00E-15	4.44E-16	4.44E-16	4.44E-16	
Std	0.715238	0.396644	0	0	0	2.55E-15	0	0	
Mean	2.739329	3.5751	4.44E-16	4.44E-16	4.00E-15	4.12E-15	4.44E-16	4.44E-16	
Worse	4.38263	4.641967	4.44E-16	4.44E-16	4.00E-15	7.55E-15	4.44E-16	4.44E-16	
F11	Min	0.001467	1.288095	0	0	0	0	0	0	
Std	0.360388	0.123868	0	0	0	0.008494	0	0	
Mean	0.156717	1.473471	0	0	0	0.002097	0	0	
Worse	1.662839	1.725859	0	0	0	0.042542	0	0	
F12	Min	0.074149	0.060841	4.54E-06	1.49E-08	0.042076	8.59E-05	7.29E-08	6.62E-14	
Std	1.251155	0.138648	0.020569	1.51E-06	0.048723	0.002561	0.008024	1.63E-11	
Mean	1.583141	0.274894	0.033825	9.03E-07	0.113659	0.001105	0.009869	9.97E-12	
Worse	5.095104	0.650842	0.09144	8.07E-06	0.24632	0.014426	0.030831	7.14E-11	
F13	Min	0.04709	1.291959	0.518373	5.91E-09	0.76237	0.001816	2.53E-06	0.010987	
Std	4.549049	0.754476	0.5766	1.89E-05	0.632386	0.041307	0.546125	0.603717	
Mean	4.690251	2.707835	1.911005	1.17E-05	2.710504	0.026837	0.782037	0.657032	
Worse	14.57619	3.940231	2.598632	9.64E-05	2.977476	0.234159	1.8927	2.371097	
F14	Min	0.998004	0.998004	0.998004	0.998004	0.998045	0.998004	0.998004	0.998004	
Std	4.397024	0.222066	0.807208	0.252193	0.88212	0.181484	0.951304	0.605406	
Mean	3.9306	1.048667	1.394824	1.064272	1.898728	1.031138	1.659371	1.196414	
Worse	15.50382	1.992037	2.982105	1.992031	3.324152	1.992031	2.982105	2.982105	
F15	Min	0.000307	0.000782	0.000307	0.000308	0.000316	0.000308	0.000307	0.000307	
Std	0.013444	0.016221	0.000232	0.000169	0.000144	0.000341	2.21E-10	0.000167	
Mean	0.005546	0.015388	0.000369	0.000356	0.000463	0.000656	0.000307	0.000338	
Worse	0.056543	0.066917	0.001223	0.001246	0.000951	0.001318	0.000307	0.001223	
F16	Min	1.03163	1.03163	-1.031628	-1.031628	-1.031604	-1.031628	-1.031628	-1.031628	
Std	8.82E17	4.78E6	3.56E-11	6.64E-16	0.00206	1.23E-11	6.49E-13	6.12E-16	
Mean	1.03163	1.03163	-1.031628	-1.031628	-1.029536	-1.031628	-1.031628	-1.031628	
Worse	1.03163	1.03161	-1.031628	-1.031628	-1.02363	-1.031628	-1.031628	-1.031628	
F17	Min	0.397887	0.397887	0.397887	0.397887	0.397892	0.397887	0.397887	0.397887	
Std	0.254527	0.302731	5.96E-09	9.69E-08	0.090173	1.91E-07	9.20E-11	0	
Mean	0.52702	0.466023	0.397887	0.397887	0.425501	0.397887	0.397887	0.397887	
Worse	1.130918	1.75218	0.397887	0.397888	0.898592	0.397888	0.397887	0.397887	
F18	Min	3	3	3	3	3.001179	3	3	3	
Std	2.58E15	10.54375	8.49E-07	7.98E-11	0.800923	1.34E-06	1.18E-08	1.69E-15	
Mean	3	7.302903	3.000001	3	3.610022	3.000001	3	3	
Worse	3	34.94955	3.000003	3	5.548131	3.000007	3	3	
F19	Min	3.862782	3.862782	-3.862782	-3.862782	-3.858954	-3.862777	-3.862782	-3.862782	
Std	2.06E15	0.000295	0.002716	5.87E-05	0.100869	0.002038	9.49E-10	2.67E-15	
Mean	3.86278	3.86262	-3.861734	-3.862769	-3.750619	-3.861677	-3.862782	-3.862782	
Worse	3.86278	3.86183	-3.854901	-3.86246	-3.458089	-3.854901	-3.862782	-3.862782	
F20	Min	-3.321995	3.32163	-3.321995	-3.321691	-3.319842	-3.321988	-3.321995	-3.321995	
Std	0.057012	0.078203	0.075409	0.074531	0.050779	0.065528	0.059241	0.060867	
Mean	3.29494	3.2283	-3.250951	-3.245629	-3.26986	-3.262063	-3.250659	-3.262167	
Worse	3.13764	2.99723	-3.137427	-3.11769	-3.068654	-3.164102	-3.203102	-3.197383	
F21	Min	10.1532	9.73855	-10.1532	-10.15162	-5.793284	-10.1532	-10.1532	-10.1532	
Std	3.494447	2.711083	2.375135	1.754592	0.140571	0.930686	1.95E-07	6.45E-15	
Mean	7.15161	6.26023	-6.586115	-5.731773	-5.066039	-9.982856	-10.1532	-10.1532	
Worse	2.63047	2.38578	-5.055197	-5.054853	-4.904617	-5.055197	-10.1532	-10.1532	
F22	Min	10.40294	9.9828	-10.40294	-10.40135	-9.070798	-10.40293	-10.40294	-10.40294	
Std	3.465308	1.916626	2.679323	1.61852	0.948608	2.994183	2.87E-07	0.970431	
Mean	5.31476	7.37187	-6.542989	-5.618068	-5.363548	-8.660674	-10.40294	-10.22576	
Worse	2.75193	2.67682	-0.909805	-5.087499	-4.7001	-1.837293	-10.40294	-5.087672	
F23	Min	10.5364	10.1845	-10.53641	-10.53278	-7.293956	-10.53641	-10.53641	-10.53641	
Std	3.772667	2.608634	2.5929	1.646801	0.574618	2.41796	1.79E-07	1.23E-15	
Mean	5.56226	6.36016	-6.931121	-5.668131	-5.036805	-9.225872	-10.53641	-10.53641	
Worse	2.42734	2.38229	-5.128481	-5.128181	-3.932434	-3.835383	-10.53641	-10.53641	
Summary	Min	8	4	14	11	2	9	17	19	
Std	0	0	3	6	4	2	9	16	
Mean	1	0	5	9	3	4	15	16	
Worse	1	0	5	8	2	3	16	16	
Significant values are in bold.

In the uni-modal benchmark functions, LSSCSO and MSCSO find the theoretical optimal values of F1, F2, F3 and F4, the standard deviation of LSSCSO, MSCSO and SABO is minimized, and the mean and worst values of LSSCSO and MSCSO are better than those of SCSO, HHO, SABO, WOA, PSO and GA. HHO achieves the optimal results on F5 that has better performance. LSSCSO results in F6 and F7 for standard deviation, mean and worst values are closer to the theoretical values and the mean and standard deviation are much better than the other algorithms, which proves that it is more stable. It can be shown that the introduced spiral search strategy can effectively improve the performance of the algorithm in the exploitation stage, which can effectively avoid falling into local optima and improve the convergence accuracy of the algorithm.

For the multi-modal benchmark functions F9-F13, LSSCSO has the best results for all four metrics on F9, F10, F11, F12, and F13, and in F12, LSSCSO has a more obvious advantage over the other optimization algorithms, with a higher optimization-seeking accuracy and stronger robustness. HHO has the results on F8 and F13 that are closest to the theoretical optimums, with stronger optimization-seeking ability. It can be shown that the sparrow alert mechanism and lens opposition-based learning strategy introduced in this paper enhances the population diversity of the algorithm, expands the search range, and has better performance in the exploration phase.

In the fixed-dimension multi-modal benchmark functions, LSSCSO achieves the best results in the optimal values of F14-F23, and the results obtained are the closest to the theoretical optimal solution;WOA has the best performance and the best stability in F14;MSCSO has the strongest energy-seeking ability in F15, with a standard deviation that is much smaller than that of other algorithms, and a better stability;in F16, F17, F18, F19, F21 and F23, the four indexes of LSSCSO are better than other algorithms, with obvious advantages.

In addition, in order to show the performance of the proposed algorithm and the experimental comparison results more intuitively, among the 23 benchmark functions, a part of them is selected on the uni-modal benchmark function, multi-modal benchmark function, and fixed-dimension multi-modal benchmark function to show their function images and convergence curves, and the specific results are shown in Figs. 2, 3, 4.

Fig.2 Convergence curves of uni-modal benchmark functions.

Fig. 3 Convergence curves of multi-moda benchmark functions.

Fig. 4 Convergence curves of fixed-dimension multi-modal benchmark functions.

Analysis of the Wilcoxon rank sum test results

Wilcoxon rank sum test is a non-parametric statistical analysis that can be used to compare the variability between two data samples. In this paper, Wilcoxon rank sum test is used to analyze whether the LSSCSO is significantly different from the other seven algorithms or not, using the significance level p-value to determine the significant difference and setting a confidence level of 0.05. When the p-value is less than 0.05, it means that there is a significant difference between the two algorithms. Table 6 gives the results of Wilcoxon rank sum test for LSSCSO and seven other different algorithms in standard benchmark functions and the corresponding results are counted at the end, where “+” indicates that the proposed algorithm is significantly different from the other algorithms, and “–” means that the proposed algorithm is not significantly different from the other algorithms and “=” means that the proposed algorithm is similar to the other algorithms.Table 6 Wilcoxon rank sum test statistical results.

F	PSO	GA	SCSO	HHO	SABO	WOA	MSCSO	
F1	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1	
F2	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1	
F3	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1	
F4	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1	
F5	8.48E-09	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	
F6	6.70E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	
F7	3.02E-11	3.02E-11	0.12597	0.010763	0.000168	3.02E-11	0.899995	
F8	7.39E-11	3.02E-11	8.10E-10	3.02E-11	3.02E-11	7.77E-09	0.482517	
F9	1.21E-12	1.21E-12	1	1	1	1	1	
F10	1.21E-12	1.21E-12	1	1	1.69E-14	3.86E-09	1	
F11	1.21E-12	1.21E-12	1	1	1	0.160802	1	
F12	1.20E-08	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	
F13	4.98E-11	3.02E-11	2.60E-08	3.02E-11	5.07E-10	7.12E-09	0.233989	
F14	0.718896	4.47E-08	9.17E-09	1.93E-07	5.33E-09	4.47E-08	3.34E-09	
F15	5.57E-10	4.97E-11	4.61E-10	5.07E-10	5.57E-10	3.16E-10	5.57E-10	
F16	0.003011	1.14E-11	1.14E-11	1.56E-09	1.14E-11	1.14E-11	1.14E-11	
F17	1	1.21E-12	1.21E-12	8.87E-07	1.21E-12	1.21E-12	1.21E-12	
F18	0.002116	2.20E-11	2.20E-11	1.05E-08	2.20E-11	2.20E-11	2.20E-11	
F19	0.654333	3.15E-12	3.15E-12	3.15E-12	3.15E-12	3.15E-12	3.15E-12	
F20	0.900445	1.78E-11	0.002225	0.002582	0.659071	0.004183	0.000479	
F21	0.00222	1.45E-11	1.45E-11	1.45E-11	1.45E-11	1.45E-11	1.45E-11	
F22	0.386157	3.16E-12	8.44E-12	4.39E-12	6.79E-12	3.37E-11	7.75E-11	
F23	0.688753	1.45E-11	1.45E-11	1.45E-11	1.45E-11	1.45E-11	1.45E-11	
+/=/-	17/1/5	23/0/0	19/3/1	20/3/0	20/2/1	21/1/1	13/7/3	

According to the data in Table 6 the statistical analysis reveals that LSSCSO exhibits significant variations in over 20 test functions when compared to GA, HHO, SABO, and WOA. This accounts for 86.96% of the total test functions, indicating a strong convergence in those specific test functions. The test results of MSCSO, SCSO, HHO, SABO, and WOA in F9-F11 are all equal to 1. This is because the majority of algorithms are able to identify the optimal value within the range of F9-F11, leading to a resemblance in the final best value. This similarity is also observed in the test results of F1-F4.

In summary, LSSCSO showed superior performance in the Wilcoxon ranking sum test and has been shown to be highly effective in optimizing 23 benchmark functions. In comparison to SCSO, LSSCSO has shown significant progress. The LSSCSO offers significant advantages in comparison to other similar algorithms.

Experiments on the CEC2022 functions

The benchmark function of CEC2005 is a simple test function, which can not fully prove the optimization performance of LSSCSO. In order to thoroughly verify LSSCSO, the optimization effect of CEC2022 benchmark function was selected for testing in this section. Table 7 shows the CEC2022 benchmark function. Set the population size N = 30, the maximum number of iterations T = 500, and the dimensions dim= 10 and dim=20 to verify the performance of LSSCSO algorithm in different dimensions. In this experiment, MSCSO, SCSO, HHO, SABO, SAO and LSSCSO were independently run for 30 times, and the mean value and variance were recorded.Table 7 CEC2022 Functions.

	No.	Functions	Fi	
Unimodal function	1	Shifted and full Rotated Zakharov Function	300	
Basic functions	2	Shifted and full Rotated Rosenbrock’s Function	400	
	3	Shifted and full Rotated Expanded Schaffer’s f6 Function	600	
	4	Shifted and full Rotated Non-Continuous Rastrigin’s Function	800	
	5	Shifted and full Rotated Levy Function	900	
Hybrid functions	6	Hybrid Function 1(N=3)	1800	
	7	Hybrid Function 2(N=6)	2000	
	8	Hybrid Function3(N=5)	2200	
Composition functions	9	Composition Function 1 (N=5)	2300	
	10	Composition Function 2(N=4)	2400	
	11	Composition Function 3(N=5)	2600	
	12	Composition Function 4(N=6)	2700	
		Search range: [-100,100]D		

Result statistics and convergence curve analysis

Table 8 shows the statistical results of the benchmark functions of LSSCSO algorithm and six comparison algorithms in CEC2022. It can be seen from the data in the table that the LSSCSO algorithm has achieved good results in both dimensions of the CEC2022 benchmark function.Table 8 Comparison results between LSSCSO and other algorithms on CEC2022 functions.

F	Metric	LSSCSO	MSCSO	SCSO	HHO	SABO	WOA	SAO	
D=10	D=20	D=10	D=20	D=10	D=20	D=10	D=20	D=10	D=20	D=10	D=20	D=10	D=20	
F1	std	0.001006	3710.084	33.19597	3359.465	1320.048	4392.227	1.426765	2036.464	1151.832	3247.313	9397.578	4094.133	610.4817969	11755.79212	
avg	300.0004	11267.13	333.9848	3706.222	1125.541	7971.977	302.2484	3951.808	3546.622	26422.43	15227.39	14334.36	770.0634902	25779.49051	
F2	std	19.82193	14.03187	24.7397	29.80288	23.98453	56.2761	37.18442	39.48498	21.30382	38.24298	62.15825	64.81191	23.23001392	18.79780589	
avg	413.7303	451.263	416.5974	479.3797	425.447	518.5478	419.9517	498.0673	441.8594	642.3093	433.4233	547.1691	417.8835105	457.0246587	
F3	std	1.362945	7.449596	5.538323	8.995595	7.695667	11.58354	8.706493	8.666247	3.359544	20.88197	13.26425	9.908413	1.525192207	0.270880728	
avg	600.9413	610.0954	605.1739	626.692	612.0723	638.2213	628.8599	655.8683	609.9444	638.9074	626.6499	659.2116	600.0046037	600.0989357	
F4	std	6.35706	11.54978	7.68786	17.6841	6.357043	19.22017	8.577024	11.26087	5.612878	18.61652	14.45958	34.58608	7.04123646	42.32332482	
avg	827.7925	880.6835	826.5609	884.3725	826.7928	891.4087	824.4824	882.1657	838.6457	931.7824	833.9308	927.1963	831.1123028	865.9040682	
F5	std	28.07347	587.7759	176.9026	311.11	60.11538	347.6968	175.8294	344.5618	17.96533	455.6779	386.2443	1075.702	0.155358946	1.926719519	
avg	913.2646	1859.581	1079.72	2345.087	987.5394	2226.12	1309.117	2609.742	929.3786	1455.326	1424.268	3845.651	920.0725136	900.9898039	
F6	std	2252.342	4212.493	2288.501	502056.6	1992.964	3036583	1636.622	56877.88	7894.036	7276264	1766.104	225124.9	2844.045093	1596.216811	
avg	4429.272	4522.685	4261.38	125089.7	3978.835	1018433	3319.767	100134.7	16444.46	5200448	3714.773	165672.3	4335.237717	3390.425804	
F7	std	6.565561	26.61793	10.01525	35.74016	11.34236	40.55791	23.12101	54.65847	15.32121	38.11344	30.03223	73.60265	9.245566366	32.25751477	
avg	2021.948	2072.129	2027.288	2099.825	2035.352	2128.545	2052.709	2167.853	2070.547	2195.265	2066.354	2198.819	2022.800418	2057.223099	
F8	std	5.657833	9.04427	2.184735	29.94718	4.59488	46.80036	11.60414	32.01956	13.2214	62.08107	6.982015	65.26759	4.351668217	49.75186107	
avg	2220.207	2229.843	2224.523	2240.673	2225.654	2259.752	2232.164	2253.875	2251.289	2334.647	2232.494	2287.531	2220.767011	2251.808067	
F9	std	0	0.035806	10.10534	9.441386	38.92305	28.86664	36.39541	7.964608	33.84701	33.85985	41.71977	29.14782	0	0.067014469	
avg	2529.284	2480.799	2532.036	2487.693	2576.964	2528.656	2552.379	2493.078	2613.932	2699.246	2556.761	2528.963	2529.284383	2480.844783	
F10	std	51.13094	60.90794	48.77727	272.5671	53.43406	1117.457	91.5728	716.4588	34.29574	1349.944	62.76787	1206.242	58.34439531	762.6636645	
avg	2528.149	2523.927	2521.759	2644.594	2526.736	3234.34	2577.854	3666.459	2608.959	5878.29	2537.657	4374.102	2561.60316	3284.290715	
F11	std	80.22664	80.08806	95.19721	476.358	145.6835	607.9969	141.6093	668.8304	89.85985	675.1478	106.3232	78.53076	97.93044196	0.028343887	
avg	2645.401	2934.129	2639.394	3351.082	2778.789	3847.271	2811.329	3175.378	3244.986	4870.91	2862.834	3161.714	2856.752099	2900.030275	
F12	std	1.474382	20.65419	7.495774	23.21877	1.76279	49.41301	24.55478	111.383	1.864829	35.76339	27.69214	116.8659	4.157094913	19.72018652	
avg	2863.695	2965.601	2869.047	2977.722	2865.104	3013.923	2890.901	3093.542	2870.768	3053.166	2884.171	3074.969	2864.927678	2955.282999	
Rank	std	7	4	1	0	1	0	1	2	2	0	0	0	2	5	
avg	7	6	2	1	0	0	2	0	0	0	1	0	1	4	

In the case of dimension 10, in F1, F2, F7, F9 and F12, compared with other comparison algorithms, LSSCSO algorithm can obtain better lowest value and is more stable, with strong robustness. In F5 and F8, the LSSCSO algorithm finds the smallest value among the algorithms, but is less stable than SAO and MSCSO. However, the solutions found are generally superior to other algorithms. A total of 7 functions with minimum values and minimum standard deviations are found, accounting for 58.33% of the CEC2022 functions.

In the case of dimension 20, LSSCSO obtains the minimum standard deviation on four functions F2, F7, F8 and F9, and the minimum value on seven functions F1, F2, F5, F7, F8, F9, F10 and F12. It can be seen that with the increase of dimension, the solution becomes more difficult, but LSSCSO still performs well in the algorithm. The minimum values of 6 functions are obtained in the test functions.

In order to see the convergence curve of functions more directly, the convergence curve of functions is given below to show their convergence process, and the specific results are shown in Figs. 5 and 6.

Fig. 5 The convergence curve analysis of each algorithm in CEC2022 10 dimension.

Fig. 6 The convergence curve analysis of each algorithm in CEC2022 20 dimension.

Analysis of the Wilcoxon rank sum test results

According to statistical analysis of the data in Table. 9, for CEC2022 functions, compared with MSCSO, SCSO, HHO, SABO, WOA and SAO, the P-value of more than half of the functions in dimension 10 is less than 0.05. There are more than 8 functions in the 20 dimension with a P-value less than 0.05. This shows that there is a strong convergence in these specific test functions. It shows that LSSCSO is different from the comparison algorithm. As a result, LSSCSO showed superior performance in Wilcoxon rank sum tests and proved to be highly effective in optimizing CEC2022 functions. Compared with SCSO, LSSCSO has made significant progress. Compared with other similar algorithms, LSSCSO has significant advantagesTable 9 Wilcoxon rank sum test statistical results.

F	MSCSO	SCSO	HHO	SABO	WOA	SAO	
D=10	D=20	D=10	D=20	D=10	D=20	D=10	D=20	D=10	D=20	D=10	D=20	
F1	3.02E-11	5.00E-09	3.02E-11	0.000399	3.02E-11	2.15E-10	3.02E-11	5.49E-11	3.02E-11	0.00077	6.52774E-08	0.045146208	
F2	0.852865	1.64E-05	0.179351	1.07E-09	0.62966	9.76E-10	2.33E-07	3.02E-11	0.047619	3.47E-10	4.41E-03	2.16E-03	
F3	4.42E-06	1.20E-08	1.61E-10	1.78E-10	3.02E-11	3.02E-11	3.69E-11	6.12E-10	3.02E-11	3.02E-11	7.39E-11	3.02E-11	
F4	0.401139	0.233982	0.567248	0.015013	0.116945	0.673491	2.62E-08	5.49E-11	0.080803	4.44E-07	9.21E-09	1.11E-03	
F5	5.46E-09	0.001302	1.56E-08	0.036439	4.08E-11	6.05E-07	1.61E-06	0.032651	6.07E-11	3.16E-10	3.28E-11	3.02E-11	
F6	0.684323	0.004033	0.78446	5.19E-07	0.10547	4.08E-11	2.03E-09	3.02E-11	0.510598	8.15E-11	5.30E-01	4.64E-01	
F7	0.000812	0.001174	1.73E-06	2.38E-07	8.89E-10	1.07E-09	3.02E-11	4.98E-11	8.15E-11	1.96E-10	6.28E-06	6.74E-06	
F8	1.39E-06	0.004427	3.09E-06	6.77E-05	5.97E-09	1.03E-06	3.34E-11	3.02E-11	2.15E-10	5.46E-09	9.07E-03	3.71E-01	
F9	1.21E-12	3.95E-11	1.21E-12	2.92E-11	1.21E-12	2.92E-11	1.21E-12	2.92E-11	1.21E-12	2.92E-11	2.88E-05	9.12E-01	
F10	0.830255	0.684323	0.420386	8.15E-05	9.53E-07	5.57E-10	4.11E-07	3.02E-11	0.000337	2.03E-09	1.49E-04	4.73E-01	
F11	0.004966	6.36E-05	3.70E-06	3.02E-11	8.41E-09	0.00168	2.10E-11	3.02E-11	2.62E-09	3.02E-11	3.51E-02	4.50E-11	
F12	0.000148	0.011711	0.000853	1.86E-06	5.97E-11	1.69E-09	2.97E-11	1.09E-10	1.39E-09	1.29E-09	0.082306067	1.24E-03	
+/=/-	8/0/4	10/0/2	8/0/4	12/0/1	9/0/3	11/0/1	12/0/0	12/0/0	12/0/0	12/0/0	8/0/4	7/0/5	

Constrained engineering design problems

Tension/compression spring design problem

The objective of the tension/compression spring design problem60 is to minimize the mass of the spring,which consists of three variables, the wire diameter d, the average coil diameter D and the effective number of coils P, and four constraints, the minimum deviation, the shear stress, the surge frequency and the outer diameter limit. The exact structure of the spring is shown in Fig.7. The mathematical formulation of this problem is shown below:

Consider:15 x→=[x1,x2,x3]=[d,D,P]

Minimize:16 f(x→)=x3+2x2x12

Subject to:17 g1(x→)=1-x23x371785x14≤0

18 g2(x→)=4x22-x1x212566x2x13-x14+15108x12-1≤0

19 g3(x→)=1-140.45x1x22x3≤0

20 g4X=x1+x21.5-≤0

Boundaries:21 0.05≤x1≤2.00.25≤x2≤1.32.0≤x3≤15.0

Fig. 7 The Tension/Compression Spring engineering design problem.

The comparative results for the tension/compression spring design optimization problem are given in Table 10, with some data referenced to the corresponding literature. The comparison results show that the LSSCSO algorithm achieves the best results among the eight algorithms,with the optimal variable values of [d, D, P]=[0.052626, 0.368202, 10.39837] and the optimal spring mass of fmin(x→)=0.012643. The LSSCSO algorithm has the smallest mean and standard deviation, and can effectively solve the tension/compression spring design optimization problem. compression spring design optimization problem.Table 10 Comparison of tension/compression spring design problem.

Algorithms	d	D	P	Best	
PSO36	0.051728	0.357644	11.24454	0.012675	
GA44	0.05148	0.0351661	11.6322	0.012705	
SCSO43	0.05	0.31752	14.02001	0.012717	
HHO61	0.051796	0.359305	11.13886	0.012665	
SABO58	0.051689	0.356718	11.28897	0.012665	
WOA27	0.051207	0.345215	12.00403	0.012676	
MSCSO	0.051584	0.354202	11.43828	0.012666	
LSSCSO	0.052626	0.368202	10.39837	0.012643	

Pressure vessel design problem

The objective of the pressure vessel design problem is to minimize the total cost of fabricating a cylindrical pressure vessel. The schematic diagram of the pressure vessel is shown in Figure. 8. The pressure vessel is capped with caps at both ends and the cap at the head end is hemispherical. The variables discussed are shell thickness Ts, head thickness Th, inner radius R and vessel length L. By optimizing the design of the pressure vessel structure, the cost of the pressure vessel is minimized. The mathematical formulation of this problem is shown below:

Consider:22 x→=[x1,x2,x3,x4]=[Ts,Th,R,L]

Minimize:23 f(x→)=0.6624x1x2x3+1.7781x2x32+3.1661x4x12+19.84x3x12

Subject to:24 g1(x→)=-x1+0.0193x3≤0

25 g2(x→)=-x2+0.00954x3≤0

26 g3(x→)=-πx32x4-43πx33+1296000≤0

27 g4(x→)=x4-240≤0

Boundaries:28 0≤x1,x2≤9910≤x3,x4≤200

Fig. 8 The pressure vessel design problem.

The comparison results of pressure vessel design problem are given in Table 11 and some data are referred to the corresponding literature. The comparison results show that the LSSCSO algorithm achieves the best results among the eight algorithms with the optimal variable values [Ts,Th,R,L]=[0.778295,0.384712,40.32617,199.9089] and the optimal cost fmin(x→)=5885.549. The LSSCSO algorithm achieves the smallest total cost as compared to the other algorithms.Table 11 Comparison of pressure vessel design problem.

Algorithms	Ts	Th	R	L	Best	
PSO61	0.8125	0.4375	42.0913	176.7465	6061.078	
GA44	0.8125	0.4375	42.0974	176.6541	6059.946	
SCSO	0.778779	0.384973	40.34571	199.6671	5887.831	
HHO	0.952325	0.461767	48.2196	113.1299	6334.94	
SABO	1.20702	0.576749	59.6872	36.8875	7202.897	
WOA	0.888486	0.411355	42.6635	169.7582	6428.847	
MSCSO	0.781602	0.386949	40.49751	197.5671	5893.609	
LSSCSO	0.778295	0.384712	40.32617	99.9089	5885.549	

Three-bar truss design problem

The objective of the three-bar truss design problem is to minimize the volume of a three-bar truss, which contains two parameters A1 and A2. This three-bar truss is constrained by stress σ at each truss member. As shown in Fig. 9. The mathematical formulation of this problem is shown below:

Consider:29 x→=[x1,x2]=[A1,A2]

Minimize:30 fx→=L×22x1+x2

Subject to:31 g1x→=2x1+x22x12+2x1x2P≤σ

32 g2x→=x22x2+x1P≤σ

Boundaries:33 0.01≤x1≤10.01≤x2≤1

34 L=100cm,P=2km/cm2,σ=2km/cm2.

Fig. 9 The three-bar truss design problem.

The comparison results of three-bar truss design problem are given in Table 12 and some data refer to the corresponding literature. The comparison results show that LSSCSO algorithm achieves the best result among the eight algorithms, the optimal variable value is[A1,A2]=[0.788591,0.408487], and the optimal cost is fmin(x→)=263.8958.Compared with other algorithms, LSSCSO performs better than other algorithms.Table 12 Comparison of three-bar truss design problem.

Algorithms	A1	A2	Best	
PSO-DE26	0.788675	0.408248	263.8958	
CS62	0.78867	0.40902	263.9716	
SCSO	0.788924	0.407545	263.8959	
HHO	0.78764	0.411183	263.8966	
SABO	0.791564	0.400847	263.9729	
WOA	0.795858	0.388302	263.9328	
MSCSO	0.788586	0.408501	263.8959	
LSSCSO	0.788591	0.408487	263.8958	

Welded beam design problem

The objective of the welded beam design problem is to minimize the production cost of the welded beam. The four variables of this problem are the thickness of the weld h, the length of the part connected to the rod l, the height of the rod t and the thickness of the rod b. The optimized design constraints are shear stress τ, bending stress in the beam θ, deflection of the beam δ and buckling load pc. The design problem for the welded beam is shown in Fig. 10. The mathematical formulation of this problem is shown below:Fig. 10 The welded beam design problem.

Consider:35 x→=[x1,x2,x3,x4]=[h,l,t,b]

Minimize:36 fx→=1.10471x12x2+0.04811x3x414.0+x2

Subject to:37 g1x→=τx→-τmax≤0

38 g2x→=σx→-σmax≤0

39 g3x→=δx→-δmax≤0

40 g4x→=x1-x4≤0

41 g5x→=p-pcx→≤0

42 g6x→=0.125-x1≤0

43 g7x→=1.10471x12+0.04811x3x4(14.0+x2)-5≤0

Boundaries:44 0.1≤x1≤2,0.1≤x2≤10,0.1≤x3≤10,0.1≤x4≤2

45 τx→=τ′2+2τ′τ′′x22R+τ′′2

46 τ′=P2x1x2,τ′′=MRJ

47 M=PL+x22,R=x224+x1+x322

48 J=22x1x2x224+x1x322

49 σx→=6PLx4x32,δx→=6PL3Ex4x32

50 pcx→=4.013Ex32x4636L21-x32LE4G

51 P=6000lb,L=14in.δmax=0.25in.E=30×106psi,G=12×106psi,τmax=13600psi,σmax=30000psi.

The comparison results of welded beam design problem are given in Table 13 and some data are referred to the corresponding literature. The comparison results show that the LSSCSO algorithm achieves the best results among the eight algorithms, with the optimal variable values of [h, l, t, b]=[0.205638,3.236887,9.036462,0.205737] and the optimal cost of fmin(x→)= 0.205737. The experimental results prove that LSSCSO can solve the welded beam design problem effectively.Table 13 Comparison of welded beam design problem.

Algorithms	h	l	t	b	Best	
PSO58	0.407489	5.001097	5.120335	0.644527	3.934221	
GA62	0.2489	6.173	8.1789	0.2533	2.4331	
SCSO	0.20568	3.236597	9.036638	0.205744	1.693033	
HHO	0.184328	3.727919	9.06471	0.207555	1.744578	
SABO	0.185815	3.475918	9.616606	0.203908	1.781242	
WOA	0.209631	3.321579	8.702858	0.221812	1.769934	
MSCSO	0.205737	3.234806	9.036901	0.205751	1.69297	
LSSCSO	0.205638	3.236887	9.036462	0.205737	1.692928	

Conclusions and future works

Aiming at the problem that SCSO has insufficient exploratory ability in the late stage and is easy to fall into the local optimum, which leads to the difficulty of convergence of the algorithm, this paper proposes a sand cat swarm algorithm based on Lens Opposition-Based Learning and Sparrow Alerting Mechanism, which is named as LSSCSO. A dynamic spiral search strategy is introduced in the exploitation phase of the algorithm to improve the convergence accuracy of SCSO by finding better positions within the search range through the path of the spiral. Inspired by SSA, the lens opposition-based learning strategy and the sparrow alert mechanism are introduced in the later stages of the algorithm to improve the global search capability and the ability to jump out of the local optimum of SCSO. In order to validate performance algorithms, choose CEC2005 and CEC2022 benchmark functions and compare the results with those of competitive PSO, GA, MSCSO, SCSO, HHO, SABO and WOA. The above results are statistically analyzed by Wilcoxon rank sum test, and the results show that LSSCSO has better performance with higher accuracy of optimization search and better robustness. Finally, LSSCSO is applied to four engineering optimization problems, and the results show that LSSCSO achieves the best results, proving that the proposed algorithm is practical in solving engineering problems.

In the future, we will carry out further research and continue to conduct more in-depth analysis and research on SCSO algorithm. We will also try to apply the algorithm to more complex real-world engineering problems, such as the Traveling Provider Problem(TSP), Vehicle Route Planning Problem(VRP), etc. and extend the existing SCSO to multi-objective optimization problems.

Acknowledgements

This work is partially supported by the Science and Technology Program of Guangdong Province under grant No. 2016A050502060 and No. 2020B1010010005, the Science and Technology Program of Guangzhou under grant No. 202206010011 and No. 2023B03J1339.

Author contributions

C.G. conceived the experiments, C.G. and X.C. conducted the experiments, C.G. analysed the results. All authors reviewed the manuscript.

Data availability

Data is provided within the manuscript and in supplementary information files.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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