Chaotic sparrow search algorithm with manta ray spiral foraging for engineering optimization

Targeting the problem of traditional sparrow search algorithms being prone to falling into local optima, a new algorithm called the Chaotic Sparrow Search Algorithm with Manta Ray Spiral Foraging (abbreviated as MSSA) is proposed. The Logistic-Sine-Cosine chaotic map and elite Reverse learning strategy are fused to initialize the population. It is experimentally demonstrated that this hybrid strategy outperforms the population after random initialization in reducing ineffective sparrow individuals. In the vigilante update stage, the spiral foraging behaviour of the manta ray population in the integrated manta ray optimization algorithm, the sparrows search around the best food source, which enhances the sparrow search algorithm's ability to explore the optimal solution. To enhance the stability of the algorithm to search for the optimum, a mixed Gaussian variational and logistic perturbation strategy is proposed to further improve the performance of the algorithm. Finally, using 12 commonly used benchmark test functions and the Wilcoxon rank sum test, MSSA was compared with other original algorithms and advanced improved algorithms, it is demonstrated that MSSA has higher accuracy and convergence performance, and the improved MSSA algorithm is applied to three types of engineering optimization problems with constraints, demonstrating its feasibility and effectiveness.


Introduction
With the continuous progress of the current society, UAV, medical industry, autonomous driving and other fields are constantly developing, as the times require, many complex optimization problems arise.For the traditional optimization algorithms, such as ant colony optimization algorithms (ACO) (Christian, 2005;Dorigo & Di Caro.,1999),Particle Swarm Optimization Algorithm (PSO)(Montes de Oca et al., 2009) and so on have been unable to efficiently solve the problems needed to be solved, Therefore, improving the structure and content of the algorithm in many ways can improve the convergence accuracy and low convergence speed of the original algorithm, Zhu and Yousefi (2021) proposed a modified bare bone particle swarm optimization to solve the DNA sequence design problem, Qian et al. (2023) successfully achieved the selection of piston sealing grooves for domestic pneumatic cylinders by introducing a search space contraction mechanism to accelerate the convergence speed of the algorithm.Du et al. (2023) used the rotation matrix to fine tune the yawing particles, alleviated the risk of poor local search ability and easy precocity CONTACT Hong Yang cherryyh@126.comthrough inertia weight and learning factor, and reduced the charging cost and peak valley difference of electric vehicles.Kumari and Sahana (2022) reduced the probability of packet loss by heuristic initialization of the ant population.At the same time, more and more intelligent biological optimization algorithms have been put forward to solve more optimization problems, such as the Mountain Gazelle Optimizer(MGO) (Abdollahzadeh et al., 2022)ïĳŇand the Grasshopper Optimisation Algorithm (GOA) (Saremi et al., 2017) in recent years, locust optimization algorithm compared with the traditional intelligent algorithm, the new optimization algorithm has fewer parameters, high precision, simple logic characteristics, in some multi-objective optimization problems than the traditional algorithm has greatly improved the effect.So more and more researchers are going into it and applying it in many ways, Such as image processing (Ma & Yue, 2022), path planning (Saraswathi et al., 2018;Zhang et al., 2011), photovoltaic power prediction (Ma et al., 2022), mechanical structure optimization design (Wu et al., 2023a).Through the above introduction, it can be found that the innovation and application of intelligent optimization algorithms are major research topics for many scholars.The chaotic spiral foraging sparrow search algorithm proposed in this article is in line with the background of this era and can handle optimization problems more accurately compared to other classic algorithms.
The Sparrow Search Algorithm is an intelligent algorithm created by Xue and Shen (2020) in 2020 to handle various optimization problems.It is superior to many swarm intelligent algorithms in performance, but SSA also has the disadvantage of easily falling into local optimization.Many scholars have proposed improvements to solve this problem.Yan et al. (2021b) proposed a game predation and suicide mechanism to improve the ability and speed of sparrow search algorithm in local optimization.Ouyang et al. (2021) obtained the initial population through the reverse learning of the lens, and combined the variable spiral strategy and simulated annealing strategy of the whale optimization algorithm to increase the activity of the follower and the accuracy of optimization, which was applied to the three-dimensional UAV path planning to verify its effectiveness.Yuan et al. (2021) proposed a mechanism based on barycentric reverse learning to initialize sparrow populations, and introduced a learning coefficient into the discoverer update part, so as to obtain the maximum power point more quickly in photovoltaic microgrids.Zhu and Yousefi (2021) used an adaptive sparrow search algorithm to identify optimal parameters and minimize errors.Yan et al. (2021a) used SSA algorithm BP neural network to optimize and process coal mine water source data so as to prevent the harm caused by coal mine emergencies.Zhou et al. (2021) applied cross mutation and mutation to a focusing optimization method of wavefront shaping based on phase modulation and achieved good results.Zhang et al. (2021) fused the SSA algorithm with the SSA to establish a semi-supervised AdaBoost monitor for data classification problems, which has good performance on both labelled and unlabelled lung CT images.Xiong et al. (2021) proposed an improved SSA algorithm based on fractional chaos to improve the accuracy of iris recognition.Aiming at the low accuracy of sparrow search algorithm, Wu et al. (2023b) introduced quantum computation into sparrow initialization, adopted enhanced search mechanism to accelerate convergence, and verified its superiority in practical applications of several scenic spots.Luan et al. (2022) initialized the hybrid search (HS) population, introduced quantum rotating gate (QRS) and sine cosine algorithm to get rid of local optimality, and applied it to energy-saving flexible job-shop scheduling problem to verify the superiority of the algorithm.Han et al. (2022) initialized sparrow population by sin chaotic mapping, introduced adaptive weights, OBL strategy and local and global search capabilities of Cauchy variation harmonic algorithm, and predicted power load data by constructing improved sparrow algorithm and optimizing hyperparameters of long and short-term memory networks.Sun et al. (2023) introduced CAT chaotic mapping to initialize sparrow population, which enhanced the randomness and ergo of the initial population.At the same time, Cauchy variation and Tent chaotic disturbance were introduced to expand the local search capability, which was applied to network security situation prediction and improved the accuracy of prediction.Liu and Mo (2022) introduced the Latin hypercube sampling technique to obtain the initial population in the initialization process of sparrow, then introduced the adaptive adjustment sine and cosine algorithm and Levy flight strategy to improve the convergence efficiency of the algorithm, and finally introduced a mutation interference mechanism to optimize the poor individuals in the population.The performance of the improved algorithm is verified by applying it to the coverage problem of wireless sensor networks.The current research attaches great importance to the process of population initialization of sparrow search algorithm.The quality of population distribution directly affects the search for the global optimal solution.The most commonly used form in the above literature is the low dimensional chaotic map, Although common low dimensional chaotic maps enrich the diversity of the population, they also have uncertainty and randomness.For traditional reverse learning, it can only be solved in a certain reverse space and lacks certain flexibility.Therefore, using these strategies alone for population initialization may not necessarily achieve satisfactory solutions.while the Logistic-Sine-Cosine chaotic map used in this paper can solve the problem of poor distribution of low dimensional maps, and the distribution of population after chaos is again processed by elite Reverse learning, The combination of the two forms a new population, further improving the disadvantage of poor initialization of a single chaos in the aforementioned literature.Moreover, the aforementioned literature does not have sufficient advantages in finding the optimal value for sparrow search algorithms.However, this article can improve the movement mode of sparrow population vigilantes by integrating the spiral feeding strategy of manta rays, so that the vigilantes are distributed in locations with more food, It can be demonstrated through experiments that the optimization results of certain test functions are multiple orders of magnitude higher than the previous algorithm, thereby improving the optimization performance of the algorithm.
Based on this, a multi-strategy fusion of chaotic spiral sparrow search algorithm is proposed.Under the disadvantage of uneven distribution of low-dimensional chaotic mapping sequences, a strategy of fusion of high-dimensional Logistic-Sine-Cosine mapping and elite backward learning is proposed to solve the problem of population diversity loss easily caused by uneven lowdimensional chaotic sequences and broaden the breadth of search.In the vigilant update stage, a spiral foraging vigilant position update strategy is proposed to fuse the manta ray optimization algorithm, so that the sparrows at the boundary that are vulnerable to predation move to the safety zone and search around the best food source, which reduces the number of sparrows at the population boundary and allows more sparrows to aggregate together to obtain the optimal solution at the best food source, improving the accuracy of the algorithm for finding the best.In order to control a degree of stability of the algorithm, Gaussian variation and logistic chaos perturbation are proposed to deal with the sparrow positions that are not around the mean fitness, which improves the robustness of the algorithm.Finally, the superior performance of MSSA is verified on the experimental results of 12 test functions with other algorithms and their improved variants, and the feasibility of the algorithm is demonstrated by applying it to three engineering problems.The main work of this paper is as follows: Firstly, the research background of intelligent optimization algorithms in recent years is introduced, as well as the improvement measures taken by many researchers for new and old algorithms.The focus is on various improvement methods and applications of the sparrow search algorithm.Secondly, introduce the basic sparrow search algorithm.Then, three measures to improve the algorithm are introduced in order, and the comparison between the chaotic elite Reverse learning and the random initialization is given, as well as the schematic diagram of the movement mode of the sparrow watchman.Then the pseudocode and flow chart of the improved algorithm are given.Subsequently, a time complexity analysis of the algorithm was conducted, followed by testing of 12 benchmark test functions, Wilcoxon rank sum test, Friedman test to verify the advantages and disadvantages of MSSA.Algorithm ablation experiments were conducted to verify the impact of each strategy on the overall algorithm.Then verify the practicality and effectiveness of MSSA through three specific engineering examples.Finally, some planning and outlook were made for the work of this article.

Sparrow search algorithm
By imitating the predatory and anti-predatory behaviour of sparrows in nature, sparrow search algorithms have been created to handle various optimization problems.
The sparrow population consists of three types of sparrows.The first type is the discoverer, which is the part of the sparrow population looking for food, accounting for 20% of the sparrow population, The discoverer's updated formula is as follows: where X t i,d represents the position of sparrow i in dimension d, t is the current number of iterations, α and Q are random numbers in the range of (0, 1] and obey normal distribution, respectively, λ and ST are warning values and safety values respectively, When λ < ST the population is in a safe period, and no predators appear, so the discoverer can forage extensively; when λ < ST there are predators near the population, and the follower and watchman will flee to a farther place to forage as the discoverer is far away from the range of the predator.
In the process of foraging, if the fitness value of a finder is less than that of a follower, then the follower will turn into a finder, but the ratio between the finder and the follower is constant.The update formula of the follower is as follows: X t+1 P represents the optimal position of the finder in the global search process, X t worst represents the sparrow's worst position in the t generation global search, n is the number of species, A stands for 1 × d dimensional matrix, Formula of the A + = A T (AA T ) −1 , another kind is 10% of the population 20% of alert, It's position update formula is as follows: where X t best is an optimal position of the global iteration up to this point; In the formula, γ is a random number and follows normal distribution, and f i represents the fitness value of the population iterated to this time.f g represents the global optimal fitness value, f w represents the global worst fitness value, R represents a random number, the range is between (0, 1), ε represents an extremely small constant that prevents the denominator from being zero, In this paper, ε = 1e − 50.When f i > f g , the sparrow is situated on the edge of the populations vulnerable to predators, when f i = f g danger of the sparrow sparrows will be close to other prevent predation.

Logistic Sine-Cosine Chaotic mapping
Chaotic mapping is often used nowadays to generate initial populations generated by random chaotic sequences in intelligent optimization algorithms.At present, this technology is widely used in image processing, image encryption and other fields, and many scholars also apply it in the initial population generating intelligent optimization algorithm (Hua et al., 2019).Through chaotic mapping, the diversity of biological population can be improved and local optimal can be prevented.The most commonly used chaotic mappings were Sine mapping and Logistic mapping, but these low-dimensional chaotic mappings were difficult to form effective population distribution, so the Sine, Logistic low-dimensional chaotic mappings were mixed together with Cosine mapping to form a new complex chaotic mapping to overcome the disadvantages of poor distribution of low-dimensional mapping.Logistics-sine-cosine chaotic mapping expression is as follows: where p ∈ [0, 1], each individual in the population can be represented by rows and columns of a matrix to represent their position in space.Each point in the image is considered an individual in initialization, and the position and dispersion of duplicate individuals can be observed through observation.It can be seen from Figure 1 that the mixed chaotic mapping model is more evenly distributed, and the initialization effect of sparrow population is better than the mapping effect of low-dimensional chaos alone.

Elite opposition-based learning
Opposition-based Learning (OBL) is a new strategy proposed by Tizhoosh (2005) in 2005, The main idea of this strategy is to improve the diversity of the population by evaluating and comparing the reverse solution of the current solution.OBL can be defined as Suppose there exists a viable solution X = (x 1 , x 2 , x 3 . . . . . .x D ) for a population in the D dimensional space, The inverse solution Elite Opposition-Based Learning (EOBL) is proposed under the factor that it may be difficult to find the optimal value in the search space.EOBL forms the reverse solution through the elite individuals in the algorithm, and then selects excellent individuals from the reverse population and elite population to form a new population, which is better than the original population in terms of diversity and quality.EOBL can be defined as assuming that the extreme value of the average individual in the current population is the elite individual, ). Can be expressed as: where δ ∈ [0, 1] is a random value, X E i,j ∈ [Lb j + Ub j ], Lb j = Min(X i,j ), Ub j = Max(X i,j ), where Lb j , Ub j are the lower and upper bounds of the dynamic boundary, respectively.If − X E i,j is out of bounds, It will be corrected using the random rule.It can be expressed as:  Through the above analysis, the population with good distribution randomness obtained by Logistic-Sine-Cosine mapping can be obtained by elite reverse learning, and the population distribution with the best initialization can be obtained, and the diversity of the population cannot be easily lost in the later iteration of the sparrow search algorithm.The population after chaotic mapping and elite reverse learning can be expressed as: The two populations are combined into a new population and initialized, the fitness is ranked, and the sparrows with the maximum population are selected for iteration.Where X i,j is the merged population, X i,j is the population after the chaotic mapping, − X E i,j is the population after reverse learning of X i,j elites.The population distribution initialized by random initialization and Logistic-Sine-Cosine elite reverse learning is shown in Figure 2 below.

Improved alert update formula
In the sparrow search algorithm, the alert will quickly move to the safe area when there is a threat from natural enemies.At this time, the movement mode of sparrows determines the probability of predation by natural enemies.The movement mode of the original sparrow population is that each individual flees the danger alone, so it cannot avoid the probability of some single sparrows being predation by natural enemies.Thus, it imitating a behaviour mode of manta ray group spiral foraging in manta ray optimization algorithm (Zhao et al., 2020), in which a single manta ray individual slowly converges into a spiral to converge to the food source.This idea is applied to the sparrow search algorithm, in which the alert flees from the predation of natural enemies, so that the alert converges into a spiral to move to the safe area.It reduces the risk of individual free movement being captured by natural enemies and helps the algorithm to jump out of local optimum and find the best solution.
The improved alert update formula is as follows: where ω = 2 • e r (Tmax−t+1) t • sin(2π r), r is the rotation factor of the manta ray and follows the uniform random number r ∈ [0, 1], and ω is the weight coefficient of the spiral motion.T max is the maximum number of iterations.t is the current number of iterations.Through the integration with the foraging behaviour of the manta ray, the movement mode of the vigilance of the sparrow search algorithm is changed from individual random motion to spiral motion, which improves the convergence accuracy of the algorithm.The schematic diagram is shown in Figure 3.
In the experiment, the number of sparrows was set to 500, and the number of iterations was set to 500.It can be seen from a in Figure 4 that in the alert update position of the original sparrow search algorithm, there are two places where sparrows gather, which represent the safest place (with the best solution).Moreover, sparrows are not uniformly distributed near the optimal solution, so it is easy to produce the local optimal solution.From b in Figure 5, it can be seen that, there is only one sparrow gather, and sparrows uniformly explore around the optimal value.The spiral position protects the whole population, reduces the loss of population diversity caused by natural enemies, and can find the optimal solution in the subsequent iteration process.

Logistic Chaotic perturbation
In the process of solving the optimal value of the intelligent optimization algorithm, there will always be the problem of falling into the local optimal value, so the random sequence generated by the chaotic mapping and the Gaussian mutation disturbance are not near the average value of the sparrow position to solve the local optimal problem, improve the global search ability of the algorithm.The random sequence generated by Chaotic perturbation can solve the sparrow position at the local optimal point.The Logistic Chaos formula is as follows: where X 0 / ∈ {0, 0.25, 0.5, 0.75, 1.0}, X 0 represents the initial chaotic value, μ ∈ [0, 4] is a random number between 0 and 4. The idea of Chaotic perturbation can be explained as follows: firstly, the chaotic sequence C s is generated through the above Equation ( 9), and then the generated chaotic sequence is carried to the solution space of the optimal value problem to be solved: n represents the new position in the d dimensional space, x max , x min is the maximum and minimum in this space.Then select the individuals who need to be disturbed for perturbation: X n2 = (X * + X n )/2, the X n2 for individual after disturbance, X n2 as the need for disturbance of the individual, X n is chaos mapping to produce a quantity.

Gaussian mutation
In the sparrow search algorithm, some inferior individuals will be generated to affect the results of optimization, and increasing the mutation operation of individuals in the population will reduce the influence of inferior individuals, accelerate the convergence speed and the ability to jump out of local optimum.Liu et al. (2021) enhanced the sparrow search algorithm's ability to break free from stagnation by introducing Gaussian Cauchy mutation, which has a significant effect on unmanned aerial vehicle path planning.Therefore, this paper proposes to use Gaussian mutation and chaotic perturbation to improve the sparrow algorithm's ability to jump out of local optima.Gaussian mutation (GM) is an optimization strategy, which uses random numbers subject to Normal distribution to act on the original position vector to generate new positions.Most mutation operators are distributed around the original position, which is equivalent to performing neighbourhood searches within a small range.This mutation not only improves the optimization accuracy of the optimization algorithm, but also facilitates the algorithm to jump out of the local optimal region.At the same time, a few operators are far from the current position, which enhances the diversity of the population and is conducive to better searching for potential regions, thereby improving search speed and accelerating the convergence trend of optimization algorithms.The Gaussian function used in Gaussian mutation is as follows: where α ∈ [0, 1] is a random number, σ = 1, and the individual after Gaussian mutation is represented as follows: If the fitness value after mutation is better than the previous fitness value, replace the sparrow at this time, otherwise, do not replace.

Time complexity analysis
The time complexity of intelligent algorithm is one of the important indicators to evaluate the performance and running time of an algorithm.Assuming that the dimension of data in the original algorithm is D, the maximum iteration number is T max , and the population number is M, then the time complexity of SSA algorithm is H(D × T max × M).The time complexity analysis of MSSA is carried out.Firstly, the logic-Sine-Cosine mapping and elite reverse learning are used for initialization, and the time complexity is H 1 = H(D × M), Then introduce the discoverer proportion update formula, and the time complexity is is the proportion of improved finders, then the manta ray spiral foraging strategy is introduced to improve the SSA alert update formula, The time complexity is Finally, Logistic chaotic map and Gaussian mutation are introduced, and the time complexity is Based on the above analysis of the time complexity of the increased improved strategy, the time complexity of MSSA algorithm is , which is consistent with the original algorithm and does not increase the time complexity, so as to prove that the improved algorithm does not increase the consumption.

MSSA algorithm pseudocode
Import M: maximum population size.T max : maximum number of iterations.λ: prewarning value.SD: Number of sparrows aware of the danger.D: The dimensions of the problem must be addressed.PD: Number of discoverers.Output: the global optimal position X best and the corresponding fitness f gbest ; Use Equation ( 8) to initialize the sparrow population M,sorted by best fitness, The first half of the population was selected and assessed for fitness.t = 1; for t = 1 : T max The fitness is ranked to obtain the best and worst fitness; R 2 = rand(1) for i = 1 : PD Update the follower position according to Equation (2); endfor for i = 1 : SD Update the alert location using Equation 9; endfor Fitness f i and average fitness f avg were calculated for each sparrow; if f i > f avg Find the sparrow position where f i > f avg and calculate its fitness f l , Logistic perturbation is carried out according to Formula 10 to obtain the fitness f L after perturbation; Find the sparrow position where f i < f avg and calculate its fitness f g , Gaussian mutation is carried out according to Formula 12 to obtain the fitness f G after mutation; if f g < f G Output f G ; end for end for Update the location and fitness of the optimal individual; t = t + 1; End while Returns the global optimal position X best and the corresponding fitness f gbest .

Algorithm performance testing
In order to verify the performance of MSSA, 12 benchmark test functions were selected to test the accuracy and stability of the algorithm.By comparing the original SSA and two variants proposed in the literature, XSSA and HSSA (Wang et al., 2022a;2022b), as well as the Gray Wolf optimizer (GWO) (Mirjalili et al., 2014b) proposed in 2014, the Manta ray foraging optimization (MRFO) proposed in 2019 and the Whale Optimization Algorithm (WOA) (Mirjalili & Lewis, 2016) proposed in 2016, The Sine cosine algorithm (SCA) (Mirjalili, 2016) proposed in 2016, Slime mould algorithm (SMA) (Li et al. 2020a) proposed in 2020, and the Harris Hawks Optimizer (HHO) (Heidari et al., 2019)  in 2019 were compared with MSSA to verify the effectiveness of the improved algorithm from multiple perspective.In this paper, the experimental environment for 12th -Gen -Intel(R) -Core(TM)i5 -12500H@3.10GHz.The software platform is MATLAB 2021a.The parameter settings and 12 test functions are recorded in Tables 1  and 2. Table 3 records the experimental results of MSSA with seven other algorithms and two SSA variants, (Mark the optimal value of the data in bold) for functions F1-F4, SSA has obvious advantages over the other three variants, which can find the optimal value quickly and accurately in different dimensions.With the increase of dimensions, SMA and MRFO still maintain high search accuracy, and the convergence speed is lower than that of the four SSA.For F5, the convergence accuracy of MSSA is much higher than that of other comparison algorithms in the 30th and 50th dimensions, and the optimal value is slightly worse than that of HSSA in the 100-dimensional test.However, MSSA can find the optimal value more stably, and the mean and standard deviation are several orders of magnitude higher than that of HSSA.For F6, MSSA has strong optimization performance in dimensions 30, 50 and 100 due to other comparison algorithms.For F7 and F12, the four SSAs still maintain strong optimization performance, and the speed of mining the optimal value is much higher than other algorithms.For F8, MSSA still maintains stability and optimization accuracy far exceeding other algorithms in dimensions 30, 50 and 100.Due to the specificity of dimension, MRFO has higher stability than MSSA when testing F9 function with 100 dimensions, but its performance is much better than MRFO and other algorithms in 30 and 50 dimensions.For function F10, the search characteristics better than SSA are inferior to HHO and MRFO in terms of stability, but MSSA is better than SSA and other variants in finding the optimal value due to the integration of the spiral foraging mechanism of MRFO.Similarly, for F11, MSSA still maintains strong performance and leads other algorithms in each dimension.Through the convergence curves of various algorithms under different dimensions shown in Figures 6-8, it is more intuitive to  30,50,100 [−10,10] 0 u(x i , 10, 100, 4) u(x i , 5, 100, 4) 30,50,100 [−50,50] 0 30,50,100 [−65.536, 65.536] 0 see that MSSA has stronger convergence speed and optimization accuracy than other algorithms.Therefore, it can be judged that the overall performance of MSSA is better.
Compared with other algorithms, MSSA is more suitable for dealing with complex optimization problems.
The study confirms that the optimal value, mean and standard deviation alone cannot prove the comprehensive performance of the algorithm, so the difference between the algorithm and the other nine algorithms is verified by Wilcoxon rank sum test.When P < 0.05 is set at the significance level of 5%, it can be concluded that there is a significant difference between the two algorithms; when P > 0.05, it can be concluded that there is no obvious difference between the two algorithms.The results of different dimensions obtained in the experiment are recorded in Table 4 below, and those with higher algorithm similarity are represented by NaN.
In addition, the average value and standard deviation of 10 algorithms recorded in 30 experiments in different dimensions are tested by Friedman test to verify the comprehensive ranking of the whole 12 test functions, and recorded in Table 5 below, So as to observe the performance under different test functions.
Compared with other SSA variants, the effect is significantly improved, and it will be applied to more complex optimization problems in the future.However, the superior performance of MSSA cannot be proved by these basic algorithms.Therefore, it is tested and compared with the improved variants of these basic algorithms by domestic and foreign researchers in recent years (Arulkumar & Chandrasekaran, 2022).In the case of 30,50,100 dimensions respectively, Improved Gray Wolf algorithm for hunting search strategy based on dimensional learning (IGWO) (Nadimi-Shahraki et al., 2021), equilibrium slime mould algorithm (ESMA) (Naik et al., 2021a), Leader Harris Eagle Algorithm (LHHO) (Naik et al., 2021b), Dynamic Control Cuckoo Bird Algorithm (DCCS) (Naik et al., 2022), Improved autonomous particle swarm optimization (IPSO) (Mirjalili et al., 2014a), and tent mapping is compared with the improved whale algorithm of tournament strategy (IWOA) (Li et al., 2020b).The parameter Settings of each improved algorithm are also set to 500 for the population size and iteration number of 30 experiments as in the original.The experiment in Table 6 below proves that in terms of the optimization effect of 12 common benchmark functions (Mark the optimal value of the data in bold), MSSA is slightly better than the improved variants of other algorithms.With the increase of the dimension, the optimization effect of each algorithm decreases significantly, but MSSA still leads the variants of other algorithms in the test results of F5-F9 and F11 in each dimension.Due to the defects of SSA itself, the optimization effect of MSSA on F10 is not as good as that of other algorithms, but it is far higher than the optimization accuracy and stability of its basic algorithm.And it can find the same optimal solution as other algorithm variants, so it can be considered that the optimization effect of MSSA is better than the improved algorithm variants proposed by some researchers recently.
Through the results of basic algorithms and recent advanced algorithms on these 12 test functions, it can be seen that MSSA has good application prospects.In various dimensions, the search for the optimal solution of the test function can obtain a solution closer to the theoretical optimal value, indicating that MSSA has good performance in optimizing problems.Similarly, the three indicators of optimal value, mean value and standard deviation alone cannot show the difference between MSSA and other algorithm variants, so Wilcoxon rank sum test and Friedman test need to be performed on them.The comparison results shown in Tables 7 and 8 below show the significance level of MSSA and other algorithm variants and the average ranking of the 12 benchmark functions.
As can be seen from Tables 7 and 8 above, there are only slight differences between MSSA and other improved algorithm variants, which shows the uniqueness of MSSA.In addition, MSSA ranks first in terms of 1.8750, 1.8750 and 2.0625 in the environment of 30, 50 and 100 dimensions.It is proved that MSSA has better performance than other variants of algorithms in recent years, and has a good research prospect.
In order to further demonstrate the performance of the improved algorithm, it will be demonstrated from the perspectives of diversity, development, and exploration (Morales-Castañeda et al., 2020).The unimodal test function F5, multimodal test function F8, and dimensionality test function F11 were selected to observe from different function types.The population size was set to 30, and the number of iterations was 100.The purpose of this is to  better see the transition process between development and exploration.The above Figures 9-11 respectively represent the diversity of SSA, the proportion of development and exploration, the diversity of MSSA, and the proportion of development and exploration.From the figures, it can be seen that as the number of iterations increases, the diversity of SSA does not decrease significantly, and the reduction is not as large as that of MSSA.MSSA maintains high population diversity in the early stages of iterations for development in different types of test functions.As iterations increase, diversity gradually decreases, The purpose is to improve the Rate of convergence and facilitate exploration.From (b) in the above figure, it can be seen that SSA does not transition well from the development stage to the exploration stage.In the later stages of iteration, there will still be situations where the development proportion is greater than the exploration proportion, which can easily lead to the algorithm falling into local optima.Because of its fast Rate of convergence, MSSA entered the development stage after a few iterations.When MSSA no longer converges, the proportion of development decreases, the proportion of exploration increases, and exploration gradually takes the lead to prevent the algorithm from falling into local optimization.Due to the combined effect of the three strategies, MSSA has better search ability and the ability to jump out of local optima, which can enable MSSA to find the optimal  were used for ablation experiments and the results were recorded in Table 9.The same experiment was conducted 30 times, the dimension was set to 30, and both the number of populations and the number of iterations were set to 500.It can be seen from Table 9 that MSSA performs best in 12 test functions, and the three indexes are ahead of other strategies and their combinations.It can be seen from the results of CSSA test that the introduction of spiral foraging strategy can increase the accuracy and robustness of optimization to the greatest extent.ESSA and GSSA, which are combined with them for testing, produce better results than the single strategy without integrating spiral foraging strategy, so it can be seen that the introduction of CSSA has greatly improved the optimization ability of the whole algorithm.Moreover, BSSA integrated with Logistic-sine-cosine chaotic map and elite reverse learning can be slightly better than original SSA in terms of optimization accuracy by increasing the diversity of population.The DSSA with Gaussian variation and Logistic chaotic disturbance can also be slightly better than the original SSA in the test results of each test function.Therefore, by improving the initialization population, mutation and disturbance operations, the stability of the overall algorithm and the ability to explore the optimal value can be further improved on the basis of the introduction of spiral foraging strategy.Prove the reliability of each strategy after integration.

Application in engineering design problems
Through the performance analysis of the above algorithms, the performance of MSSA is theoretically better than that of other latest intelligent optimization algorithms.Therefore, the practicability of MSSA is further verified by three optimization constraints in engineering.

Three bar truss design problem
The three-bar truss design problem is a structural optimization problem in the field of civil engineering (Xue & Shen, 2022) Its main purpose is to minimize the weight of the stress, disturbance and buckling constraint between two bars, and to minimize the volume of the three-bar truss by adjusting the cross-sectional area (x 1 , x 2 ).The constraints for adjustment are as follows: Reduce to a minimum: The following conditions must be met: Variable range of parameters: where l = 100cm, P = 2KN/cm 2 , σ = 2KN/cm 2 , (13)

Tension/compression spring design
The tension/compression spring design problem is to obtain the minimum tension/compression spring weight, and should meet the minimum disturbance, vibration frequency and shear stress constraints.The model for this case is expressed as follows: Variable: Need to reach the minimum: The following conditions need to be met: Domain of variables:

Design problem of welded beam
The purpose of the welding beam design problem is to estimate the best value of the thickness of the bar (b), the length of the attached part of the bar (l), the height of the bar (t) and the thickness of the weld (h), so as to minimize the consumables and reduce the construction cost.The model expression is as follows: Variable: x The following conditions need to be met: Domain of variables: where: Tables 10-12 record the data of three classic engineering problems under different algorithms.Through the comparison data of three classical engineering constrained optimization problems: three-bar truss design problem, tension/compression spring design problem, and welded beam design problem, it is not difficult to see that MSSA has a slight advantage over the new optimization algorithms and their variants that have emerged in recent years, and has better performance than the basic SSA algorithm, which proves that MSSA can deal with practical engineering problems.

Results
This paper initializes the sparrow population through the fusion of Logistic-sine-cosine chaotic map and elite reverse learning, selects the sparrow with high fitness as the new population, improves the diversity of the sparrow population, improves the distribution of the sparrow population, and is conducive to the development in the later iteration process.Considering the low convergence accuracy of the basic sparrow search algorithm, the spiral foraging mode of the manta ray group in the manta ray optimization algorithm is used to improve the movement update mode of the alert in the sparrow population, which makes it more difficult for the individual sparrow to Min 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 F1 Mean 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 Std 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 Min 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 F2 Mean 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 Std 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 Min 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 F3 Mean 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 Std 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 Min 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 F4 Mean 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 Std 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 be captured by natural enemies, maintains the diversity towards the optimal solution, strengthens the ability of local development, and after one iteration, logistic chaos disturbance and Gaussian mutation operation are carried out for sparrows not near the average fitness value, which strengthens the stability of the overall algorithm.Through 12 benchmark functions, it is not difficult to conclude that the effect of the multimodal test function is  multimodal testing function, MSSA can find the optimal value size and stability that are multiple orders of magnitude higher than some common basic algorithms and advanced improved algorithms, which can prove the strong ability of MSSA to handle complex optimization problems.On the other hand, through three engineering problems with constraints, MSSA can also outperform other comparative algorithms in solving results, and can save resource costs and reduce resource waste in practical engineering.Therefore, MSSA has good application prospects.However, MSSA only considers how to improve the simple constrained optimization problem, that is, introduce the highly convergent foraging behaviour of manta ray optimization algorithm, and does not consider whether this scheme is feasible in the face of complex multiobjective and multi constraint optimization problems.The effectiveness of the algorithm has not been verified for practical engineering problems, so future work will be to apply it to optimization experiments in reality based on this article to increase the universality and practicality of the improved algorithm.Dealing with complex multiobjective optimization problems will be the main task of the next step, and existing algorithms will be continuously improved through experiments to handle more complex problems and broaden the application fields of the algorithm.

Figure 2 .
Figure 2. Panel a shows the population distribution before the improvement, and Panel b shows the distribution after the improvement.

Figure 3 .
Figure 3. Schematic diagram of the movement mode of sparrow in the improved watchman formula.

Figure 4 .
Figure 4. Panel a presents the location map of the unimproved alert, while Panel b presents the location map of the improved alert.

Figure 9 .
Figure 9. Single peak test function F5 diversity and development Exploratory testing.(a)SSA diversity (b) SSA development and exploration (c) MSSA diversity.(d) MSSA development and exploration.

Figure 10 .
Figure 10.Multi peak test function F8 Diversity and development Exploratory testing. g

Figure 11 .
Figure 11.Diversity and development Exploratory testing of fixed dimension test function F11.

Table 1 .
algorithm proposed Parameter setting.

Table 3 .
Test function results comparison.

Table 4 .
Wilcoxon rank sum test results.

Table 6 .
Test function results with other algorithm variants.In order to verify the influence of MSSA strategies on the optimization process, BSSA fused with Logistic Sine-Cosine chaotic map and elite reverse learning, CSSA fused with MRFO's spiral foraging mechanism, DSSA with Gaussian variation and Logistic chaotic disturbance were used.ESSA fused with BSSA strategy and CSSA strategy, FSSA fused with BSSA strategy and DSSA strategy, GSSA fused with CSSA and DSSA, and MSSA fused with all strategies

Table 7 .
Results of Wilcoxon rank sum test with other algorithm variants.

Table 8 .
Friedman test results compared to other algorithm variants.

Table 9 .
Results of ablation experiment.

Table 10 .
Best results of the Three bar truss design problem.

Table 11 .
Best results of the Tension/compression spring design.farsuperiorto the basic SSA algorithm, WOA, GWO, SCA and other intelligent algorithms and SSA variants, and has obvious advantages compared with various intelligent algorithms that have emerged in recent years.Moreover, the practicability of the improved algorithm in this paper is proved in three engineering optimization problems.It is suitable to be applied to practical engineering problems and has high research value.On the one hand, benchmark function testing experiments have shown that for a certain high-dimensional

Table 12 .
Best results of the Design problem of welded beam.