Adaptive Simplified Chicken Swarm Optimization Based on Inverted S‐Shaped Inertia Weight

Chinese Journal of ElectronicsVolume 31, Issue 2 p. 367-386 Computer Networks and Artificial IntelligenceOpen Access Adaptive Simplified Chicken Swarm Optimization Based on Inverted S-Shaped Inertia Weight Yanchun GU, Corresponding Author Yanchun GU [email protected] School of Science, Jiangnan University, Wuxi, 214122 ChinaSearch for more papers by this authorHaiyan LU, Corresponding Author Haiyan LU [email protected] School of Science, Jiangnan University, Wuxi, 214122 China Wuxi Engineering Technology Research Center for Biological Computing, Jiangnan University, Wuxi, 214122 ChinaSearch for more papers by this authorLei XIANG, Lei XIANG School of Science, Jiangnan University, Wuxi, 214122 ChinaSearch for more papers by this authorWanqiang SHEN, Corresponding Author Wanqiang SHEN [email protected] School of Science, Jiangnan University, Wuxi, 214122 China Wuxi Engineering Technology Research Center for Biological Computing, Jiangnan University, Wuxi, 214122 ChinaSearch for more papers by this author Yanchun GU, Corresponding Author Yanchun GU [email protected] School of Science, Jiangnan University, Wuxi, 214122 ChinaSearch for more papers by this authorHaiyan LU, Corresponding Author Haiyan LU [email protected] School of Science, Jiangnan University, Wuxi, 214122 China Wuxi Engineering Technology Research Center for Biological Computing, Jiangnan University, Wuxi, 214122 ChinaSearch for more papers by this authorLei XIANG, Lei XIANG School of Science, Jiangnan University, Wuxi, 214122 ChinaSearch for more papers by this authorWanqiang SHEN, Corresponding Author Wanqiang SHEN [email protected] School of Science, Jiangnan University, Wuxi, 214122 China Wuxi Engineering Technology Research Center for Biological Computing, Jiangnan University, Wuxi, 214122 ChinaSearch for more papers by this author First published: 01 March 2022 https://doi.org/10.1049/cje.2020.00.233 This work was supported by the National Natural Science Foundation of China (61772013) and the Natural Science Foundation of Jiangsu Province (BK20190578). AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Considering the issues of premature convergence and low solution accuracy in solving high-dimensional problems with the basic chicken swarm optimization algorithm, an adaptive simplified chicken swarm optimization algorithm based on inverted S-shaped inertia weight (ASCSO-S) is proposed. Firstly, a simplified chicken swarm optimization algorithm is presented by removing all the chicks from the chicken swarm. Secondly, an inverted S-shaped inertia weight is designed and introduced into the updating process of the roosters and hens to dynamically adjust their moving step size and thus to improve the convergence speed and solution accuracy of the algorithm. Thirdly, in order to enhance the exploration ability of the algorithm, an adaptive updating strategy is added to the updating process of the hens. Simulation experiments on 21 classical test functions show that ASCSO-S is superior to the other comparison algorithms in terms of convergence speed, solution accuracy, and solution stability. In addition, ASCSO-S is applied to the parameter estimation of Richards model, and the test results indicate that ASCSO-S has the best fitting results compared with other three algorithms. I. Introduction Swarm intelligence algorithms are random optimization algorithms constructed by simulating the living habits and behavior characteristics of various natural biological populations, such as genetic algorithm[1,2], particle swarm optimization algorithm (PSO)[3-5], grey wolf optimization algorithm (GWO)[6,7], artificial bee colony algorithm (ABC)[8,9], and so on. These algorithms provide a new approach to solving a large number of global optimization problems in the fields of computing science, engineering science, and management science. Chicken swarm optimization algorithm[10] (CSO) is an intelligent global optimization algorithm proposed by Meng et al. in 2014, which simulates the chicken swarm hierarchy and their food-searching behavior. CSO associates each member of the swarm with one of the three identities: rooster, hen, and chick, which act in different modes. The chickens with best several fitness values are regarded as roosters, the chickens with worst several fitness values are regarded as chicks, and the others are the hens. When solving optimization problems, each chicken in the swarm corresponds to a solution, and adopts different search strategy according to its identity in the swarm. In contrast with standard particle swarm optimization, differential evolution (DE), and bat algorithm (BA), CSO has advantage in either searching precision or convergence rate[10]. Therefore, CSO has been applied to many different fields. For example, Liang[11] used a hybrid cuckoo search-chicken swarm algorithm to optimize sidelobe-level suppression for linear and circular antenna arrays. Zouacheet al.[12] applied CSO to solve multi-objective optimization problems. Nieet al.[13] applied CSO to study parameter identification of static var compensator model. Tianet al.[14] applied CSO to solve the deadlock-free migration for virtual machine consolidation. Cao and Qiao[15] used CSO to adjust the space searching direction of wireless sensor network (WSN) in order to optimize the convergence of WSN location algorithm. Shadi and Safi-Esfahani[16] applied CSO for scheduling of tasks in cloud computing. While CSO has many advantages, it still has its inevitable disadvantages. In CSO, a random search strategy based on Gaussian distribution is adopted for particles of rooster swarm. This search strategy has strong local search ability, but its global search ability is weak, making it liable to fall into local optima. In addition, recall that the chicks follow the mother hens whereas the hens follow their mate roosters. Thus, if the roosters fall into local optima, then both hens and chicks will also fall into local optima, resulting in premature convergence of the algorithm. To improve the weak global search ability of chicken swarm algorithm, Huang et al.[17] introduced a global optimal guiding strategy and a dynamic inertia strategy into the evolution optimization of population. Han[18] introduced a dissipative structure into the roosters' position update equation and applied a differential mutation operator on randomly selected individuals to enhance the global search capability of the algorithm. In addition, due to the strong global search ability of the artificial bee colony algorithm (ABC), Shi and Gao[19] combined the local search phase of CSO with the global search phase of ABC to balance the global and local search abilities of CSO, making CSO easier to converge to the global optimal solution of high-dimensional problems. In order to prevent premature convergence, Kong[20] lets the chicks to learn not only from the hens but also from the roosters during foraging, which enhanced the ability of the chicks to escape from local optima. Furthermore, Han et al.[21] introduced an adaptive mutation strategy, a bias random walk strategy, and a target mutation strategy into the updating process of the roosters, hens, and chicks, respectively, which improved the overall performance of CSO. The optimization abilities of the above mentioned improved CSO algorithms are reinforced to some extent, but there are still some shortcomings. Ref.[20] only improved the updating mode of the chicks. Although the convergence speed is accelerated, the algorithm may still suffer from the premature convergence phenomenon. In Ref.[21], the inertial weight of bias random walk was added to the hens' update process, which enhanced the accuracy of algorithm. However, there is no much improvement on the convergence speed of the algorithm. In view of the above shortcomings, this paper proposes an adaptive simplified chicken swarm optimization algorithm based on inverted S-shaped inertia weights (ASCSO-S). Firstly, the simplified chicken swarm algorithm is presented, which improves solution accuracy and convergence speed of the algorithm. Secondly, the inverted S-shaped inertial weight and a global search guiding strategy are introduced into the update process of the roosters and the hens, which further enhances the accuracy and stability of the algorithm. Thirdly, an adaptive updating strategy is added to the update process of the hens, which greatly improves the ability of the algorithm to jump out of local optima. Simulation results show that ASCSO-S has great advantages in terms of solution accuracy, convergence speed, and stability. II. Chicken Swarm Optimization Algorithm CSO is a new intelligent bionic algorithm imitating various behaviors of roosters, hens, and chicks in the process of searching food. To solve an optimization problem, the chicken swarm is mapped as a population of specific particle individuals in the searching space. For simplicity, the chickens' behaviors can be described by the following rules. 1) There are some groups in the whole chicken population, and each group has one rooster, some hens, and some chicks. Therefore, the number of groups is determined by the number of roosters. 2) The identities of roosters, hens, and chicks are determined by their fitness values. The chickens with best several fitness values are regarded as roosters, the chickens with worst several fitness values are regarded as chicks, and the others are the hens. Each hen randomly chooses one rooster as her spouse, and each chick also randomly selects one hen as her mother. 3) The individual identity, spouse relationship and mother-child relationship in a group will remain unchanged within each update cycle, updating only every G iterations. 4) The hens follow their group-mate rooster to search for food and the chicks search for food around their mothers (the hens). 5) Let N be the total number of chickens, and N R , N H , N C , and N M be the number of the roosters, hens, chicks, and mother hens, respectively. The position update equation of individuals in the chicken group varies according to different chicken species. The roosters play a leading role in the whole foraging process and can search for food in a wider space. The position update equation of the roosters is as follows: x i , j ( t + 1 ) = x i , j ( t ) × ( 1 + randn ( 0 , σ 2 ) ) (1) σ 2 = 1 , f i < f k , exp f k − f i f i + ε , f i ≥ f k , k ∈ 1 , N R , k ≠ i (2) where randn ( 0 , σ 2 ) is a Gaussian distribution with mean 0 and variance σ 2 , ε is a small constant in the computer, k, a rooster's index, is randomly selected from the roosters group, and f i is the fitness value of individual i. The position update equation of the hens is as follows: x i , j ( t + 1 ) = x i , j ( t ) + S 1 × rand ( ) × ( x r 1 , j ( t ) − x i , j ( t ) ) + S 2 × rand ( ) × ( x r 2 , j ( t ) − x i , j ( t ) ) (3) S 1 = exp f i − f r 1 f i + ε (4) S 2 = exp ( f r 2 − f i ) (5) where rand ( ) is a uniform random number over [ 0 , 1 ] , r 1 is the index of the i th hen's group-mate rooster, and r 2 ( r 1 ≠ r 2 ) is the index of the chicken (rooster or hen) randomly chosen from the swarm. The position update equation of the chicks is as follows: x i , j ( t + 1 ) = x i , j ( t ) + F L × ( x m , j ( t ) − x i , j ( t ) ) (6) where x m , j ( t ) stands for the jth-dimensional position of the i th chick's mother ( m ∈ [ 1 , N ] ), and F L ( F L ∈ [ 0 , 2 ] ) is a parameter called following coefficient. III. Adaptive Simplified Chicken Swarm Optimization Algorithm Based on Inverted S-Shaped Inertia Weight 1. Simplified chicken swarm optimization algorithm (SCSO) The chicks are in the lowest position of the chicken group hierarchy, having neither the leading position of the roosters, nor the quantitative advantage of the hens. The existence of the position update equation of the chicks is merely to deal with the poor individuals in the group and to make them move towards the better ones. However, the existence of the chicks does not play a significant role in the algorithm, instead, it increases the running time of the algorithm. Therefore, in this paper, we first propose a simplified version of CSO, called SCSO, in which all the chicks are removed, and only the roosters and hens are remained searching for food. The updated equations of the roosters and hens in SCSO are consistent with that in CSO. 2. The inverted S-shaped inertia weight The inertial weight method[22] was first proposed by Shi and Eberhart in particle swarm optimization, which can greatly balance the exploration ability and exploitation ability of the algorithm. Because CSO has a weak global search ability, many scholars[13,17,20,21,23] have introduced different inertial weights into CSO to enhance its performance. In order to improve the optimization ability of SCSO, in this study an inertia weight is designed and introduced into the update equation of the roosters and hens, so that individuals can conduct regular self-learning to balance the exploration ability and exploitation ability. For this end, the inertia weight is expected to have the following characteristics: 1) The inertia weight should be a decreasing function. 2) In the early stage of iteration, the inertia weight maintains a larger value for a relatively long time, so as to enhance the global search ability of the algorithm. 3) In the later stage, it maintains a smaller value for a relatively long time so as to enhance the local search ability of the algorithm. According to the above analysis, we propose an inverted S-shaped inertia weight, which is expressed as follows: w ( t ) = w max − ( w max − w min ) × 1 1 + e a − b t (7) where w min and w max are the minimum and maximum values of the inertia weight, respectively; a and b are constants (i.e. a = 6 and b = 0.3 ). A large number of experiments (as discussed in Section IV) indicate that the algorithm performs well when w min = 0.1 and w max = 0.7. In order to intuitively demonstrate the characteristics of the above inverted S-shaped inertia weight, its image is given in Fig.1, where the maximum iteration number is 50. It is clear that, in the first 15 iterations, the inertia weight value is between 0.6 and 0.7, while in the last 15 iterations, the weight value is very small, being close to 0.1. These results satisfy the above mentioned requirements quite well. Fig. 1Open in figure viewerPowerPoint The inertia weight decline curve ( w min = 0.1 , w max = 0.7, a = 6 , and b = 0.3 ) 3. The adaptive updating strategy In order to solve the problem of premature convergence and poor solution effect of SCSO, an adaptive updating strategy is proposed in this paper for the population updating process. Firstly, a global search guiding strategy is added to the updating process of the roosters and hens, so that the individuals can learn from the global optimal individual. Then, if it is justified that the algorithm has fallen into local extrema, the hens will forage using a random learning strategy, which makes the hens more likely to jump out of the local optima. 1) The global search guiding strategy To search the global optimal solution faster, the roosters and hens not only need to learn from themselves, but also to learn from the current global optimal individual of the population, so as to make the search more directed and to avoid blind search. Consequently, the improved update equation of the roosters is as follows: x i , j ( t + 1 ) = w ( t ) × x i , j ( t ) + randn ( 0 , σ 2 ) × ( x best , j ( t ) − x i , j ( t ) ) (8) And the improved update equation of hens is as follows: x i , j ( t + 1 ) = w ( t ) × x i , j ( t ) + S 1 × rand ( ) × ( x r 1 , j ( t ) − x i , j ( t ) ) + S 3 × rand ( ) × ( x best , j ( t ) − x i , j ( t ) ) (9) S 3 = exp ( f best − f i ) (10) where x best , j ( t ) is the jth-dimensional position of the current optimal individual, S 3 is the learning coefficient, and f best is the fitness value of the current optimal individual. 2) The random learning strategy Although the global search guiding strategy can improve the convergence speed and solution accuracy of the algorithm, most individuals are prone to gathering around local extrema for high-dimensional multimodal functions. Therefore, when the algorithm is regarded as falling into local extrema, a random learning strategy is introduced, letting hens learn not only from optimal individuals, but also from random individuals (hens or roosters). Specifically, the random learning strategy works as follows. Firstly, we give a condition to judge whether the algorithm falls into local extrema. If the optimal solution does not change for K consecutive iterations, it can be considered that premature phenomenon takes place. And then, a random learning strategy is triggered to help the individuals to jump out of the potential local extrema. A large number of experiments show that the algorithm performs well for K = 10 . The position update equation of the hens with the random learning strategy is as follows: x i , j ( t + 1 ) = w ( t ) × x i , j ( t ) + rand ( ) × ( x h 1 , j ( t ) − x h 2 , j ( t ) ) (11) where x h 1 , j ( t ) and x h 2 , j ( t ) are the jth-dimensional positions of two randomly selected individuals h1 and h2, respectively. 4. The procedure of ASCSO-S The solution steps of ASCSO-S are described as follows: Step 1: Initialization. Set population parameters and random positions of individuals. Step 2: Evaluation. Evaluate the fitness value for each individual and find the optimal individual. Step 3: Establishment. If mod ( t , G ) = 1 , rank the chickens' fitness values and establish the hierarchal order in the swarm. Step 4: Update. Update the next position according to Eqs.(8) and (9). Step 5: Premature judgment. If the algorithm appears premature phenomenon, update the hens' position according to equation Eq.(11); otherwise, go to Step 6. Step 6: Termination. If the termination condition is reached, the algorithm stops; otherwise, go to Step 2. IV. Simulation Experiment and Result Analysis In this section, three main experiments are carried out. The first main experiment is to find the appropriate values of the parameters involved in our proposed algorithm, the second one is to test the effectiveness of the improved strategy introduced in the algorithm, and the third one is to compare the performance of ASCSO-S with that of several closely related algorithms. 1. Experimental setup To verify the optimization precision and convergence rate of the proposed algorithm, 21 standard test functions[24,25] are used for comparison experiments. The function names, expressions, search ranges, and optimal solutions are listed in Table 1. Among these functions, f 1 f 12 are unimodal functions, which are extremely hard for optimization algorithms to converge to a global optima, and are used to evaluate the searching precision, f 13 f 17 are multimodal functions with several local extreme points, and can be used to test the global search performance, and f 17 f 21 are fixed low-dimensional multimodal functions. To obtain unbiased results, in all comparisons, the population size and the maximum iteration number are set to 100 and 1000 respectively for all functions. All algorithms are tested 30 trials independently for each function. Table 1. The test functions Names Expressions Search ranges Optimal solutions Sphere f 1 = ∑ i = 1 n x i 2 [ − 100 , 100 ] 0 Sum of different powers f 2 = ∑ i = 1 n x i i + 1 [ − 1 , 1 ] 0 Sum squares f 3 = ∑ i = 1 n i x i 2 [ − 10 , 10 ] 0 Rosenbrock f 4 = ∑ i = 1 n − 1 [ 100 ( x i + 1 − x i 2 ) 2 + ( x i − 1 ) 2 ] [ − 5 , 10 ] 0 Dixon-price f 5 = ( x 1 − 1 ) 2 + ∑ i = 2 n i ( 2 x i 2 − x i − 1 ) 2 [ − 10 , 10 ] 0 Rotated hyper-ellipsoid f 6 = ∑ i = 1 n ∑ j = 1 i x j 2 [ − 65.536 , 65.536 ] 0 SchwefelsP2.21 f 7 ( x ) = max x i [ − 100 , 100 ] 0 SchwefelsP2.22 f 8 = ∑ i = 1 n x i + ∏ i = 1 n x i [ − 10 , 10 ] 0 Quartic f 9 = ∑ i = 1 n i x 4 + rand [ 0 , 1 ) [ − 1.28 , 1.28 ] 0 Step f 10 = ∑ i = 1 n ( [ x i + 0.5 ] ) 2 [ − 100 , 100 ] 0 Discus f 11 = 10 6 x 1 2 ∑ i = 1 n x i 2 [ − 100 , 100 ] 0 Zakharov f 12 = ∑ i = 1 n x i 2 + ∑ i = 1 n 0.5 i x i 2 + ∑ i = 1 n 0.5 i x i 4 [ − 5 , 10 ] 0 Griewank f 13 = 1 4000 ∑ i = 1 n x i 2 − ∏ i = 1 n cos ( x i i ) + 1 [ − 600 , 600 ] 0 Rastrigin f 14 = ∑ i = 1 n [ x i 2 − 10 cos ( 2 π x i ) + 10 ] [ − 5.12 , 5.12 ] 0 Ackley f 15 = − 20 exp ( − 0.2 ∑ i = 1 n x i 2 n ) − exp ( 1 n ∑ i = 1 n cos ( 2 π x i ) ) + 20 + e [ − 32 , 32 ] 0 Powell f 16 = ∑ i = 1 n / 4 [ ( x 4 i − 3 + 10 x 4 i − 2 ) 2 + 5 ( x 4 i − 1 − x 4 i ) 2 + ( x 4 i − 2 − 2 x 4 i − 1 ) 4 + 10 ( x 4 i − 3 + x 4 i ) 4 ] [ − 4 , 5 ] 0 Alpine f 17 = ∑ i = 1 n x i sin ( x i ) + 0.1 x i [ − 10 , 10 ] 0 Beale f 18 = ( 1.5 − x 1 + x 1 x 2 ) 2 + ( 2.25 − x 1 + x 1 x 2 2 ) 2 + ( 2.625 − x 1 + x 1 x 2 3 ) 2 [ − 4.5 , 4.5 ] 0 Eggholder f 19 = − ( x 2 + 47 ) sin ( x 2 + 0.5 x 1 + 47 ) − x 1 sin ( x 1 − ( x 2 + 47 ) ) + 959.6407 [ − 512 , 512 ] 0 Bukin f 20 = 100 x 2 − 0.01 x 1 2 + 0.01 x 1 + 10 x 1 ∈ [ − 15 , − 5 ] x 2 ∈ [ − 3 , 3 ] 0 Shekel f 21 = − ∑ i = 1 10 ( ∑ j = 1 4 ( x j − C ji ) 2 + β i ) − 1 , β = 1 10 1 , 2 , 2 , 4 , 4 , 6 , 3 , 7 , 5 , 5 , C = 4 1 8 6 3 2 5 8 6 7 4 1 8 6 7 9 3 1 2 3.6 4 1 8 6 3 2 5 8 6 7 4 1 8 6 7 9 3 1 2 3.6 [ − 10 , 10 ] −10.5364 All our experiments were conducted on a PC with Intel(R) Core(TM) i5-M420 CPU 8.00GB RAM and Windows 10 operating system, and all of the algorithms were implemented with MATLAB R2019a. 2. The selection of parameter G in SCSO The parameter G in the chicken swarm algorithm has a very important effect on the overall performance of the algorithm. A large value of G is disadvantageous for the algorithm to converge to the global optima, whereas a small value of G may lead the algorithm trap into local optima. In Ref.[10], Meng et al. demonstrated that the optimization ability of CSO is better with G ∈ [ 2 , 20 ] . In order to get the reasonable range of G in SCSO, we performed a series of extensive experiments on the test functions in Table 1 and obtained general consistent results. Here, for the sake of brevity, only the results of SCSO with different values of G on four functions are presented, as shown in Table 2. Of the four functions, f 1 and f 9 are unimodal, and f 13 and f 15 are multimodal. Table 2. The experimental results of SCSO (D=100) G Results f 1 f 9 f 13 f 15 10 Mean 7.993E−04 9.165E−02 3.906E−04 5.255E−04 20 Mean 2.347E−05 3.077E−03 9.318E−04 6.560E−05 30 Mean 1.584E−08 2.627E−03 7.505E−14 1.509E−14 40 Mean 1.050E−29 2.455E−03 0 5.410E−15 50 Mean 1.588E−30 1.295E−03 0 4.941E−15 60 Mean 8.417E−31 2.215E−03 0 8.440E−15 70 Mean 6.427E−31 1.908E−03 0 7.120E−15 80 Mean 3.792E−33 2.322E−03 0 4.440E−15 90 Mean 1.655E−29 1.476E−03 0 4.440E−15 100 Mean 1.715E−32 1.368E−03 0 4.796E−15 110 Mean 5.691E−09 1.368E−03 5.106E−12 7.231E−08 120 Mean 2.376E−05 5.169E−03 6.321E−04 6.781E−04 130 Mean 1.547E−02 8.348E−02 8.805E−03 2.567E−03 It is clear from Table 2 that when the dimension G is between 40 and 100, the optimization ability of SCSO is consistently among the best, and while G takes other values from the interval [ 10 , 130 ] , the performance of SCSO is basically much inferior, with worse results achieved when G getting closer to the two ends of the interval. Therefore, in terms of solution accuracy, the obtained reasonable range of G is [ 40 , 100 ] . It is worth noting that the reasonable value of G in SCSO is much larger than that of G (i.e. G ∈ [ 2 , 20 ] ) in CSO, which allows the population to take full advantage of the current hierarchy searching for the potential optima. On the other hand, in terms of efficiency, larger value of G can make SCSO converge quickly. Summing up the above analysis, to achieve best performance of SCSO in terms of both accuracy and efficiency, it is suggested that G = 100 in SCSO. 3. The selection of inertia weight range of inverted S-shaped in ASCSO-S In order to determine the appropriate values of parameters w min and w max of the inverted S-shaped inertia weight in ASCSO-S, we performed a series of extensive experiments on the test functions in Table 1 and obtained basically consistent results. Here, for the sake of brevity, only the results of ASCSO-S with different values of w on four functions are presented, as shown in Table 3. Of the four functions, f 1 and f 9 are unimodal, f 13 is multimodal, and f 21 is fixed low-dimensional multimodal. The values of w are uniformly selected from [ 0 , 1 ] with increment of 0.1, and the results are compared to obtain the favorable interval of w, then the left and right endpoints of the inverval will determine the values of w min and w max . It can be seen from Table 3 that when the inertia weight w is between 0.1 and 0.7, the solution effect of ASCSO-S is better. Consequently, the appropriate values of w min and w max are 0.1 and 0.7, respectively. Table 3. The test results of ASCSO-S with different values of w (D=100) w Results f 1 f 9 f 13 f 21 0.1 Mean 0 7.084E−06 0 −10.5364 0.2 Mean 0 3.806E−06 0 −10.5364 0.3 Mean 0 2.459E−05 0 −10.5364 0.4 Mean 0 1.433E−05 0 −10.5364 0.5 Mean 0 5.596E−06 0 −10.5364 0.6 Mean 0 8.611E−06 2.102E−306 −10.5364 0.7 Mean 1.579E−244 1.881E−05 3.514E−211 −10.5364 0.8 Mean 6.907E−159 4.887E−05 8.103E−178 −10.5362 0.9 Mean 9.529E−87 8.307E−05 1.149E−90 −10.5361 Furthermore, to intuitively show the influences of different values of w on the performance of ASCSO-S, Fig.2 gives the broken line graphs of the experimental results for f 9 and f 21 . It can be observed that when the w is between 0.1 and 0.7, the results of ASCSO-S are more stable and accurate. Therefore, we set w min = 0.1 and w max = 0.7 in ASCSO-S, so that the inverted S-shaped inertia weight can make the algorithm better balance the exploration ability and exploitation ability. Fig. 2Open in figure viewerPowerPoint The broken line graph of the solution results of ASCSO-S with different values of w To facilitate the follow-up comparison experiments, the parameter settings of CSO, PSO, SCSO, ICSO[13], ECSO[20], and ASCSO-S are summarized in Table 4. Table 4. The parameter settings of the six algorithms Algorithms Parameter settings PSO c 1 = c 1 = 2 , w = 0.8 CSO N R = 0.2 N , N H = 0.6 N , N = 100 , G = 10 , F L ∈ 0.4 , 1 SCSO N R = 0.4 N , N H = 0.6 N , N = 100 , G = 100 ICSO[13] N R = 0.2 N , N H = 0.7 N , N = 100 , G = 10 , F L ∈ 0.4 , 1 , w min = 0.4 , w max = 0.9 ECSO[20] N R = 0.2 N , N H = 0.6 N , N = 100 , G = 10 , F L ∈ 0.4 , 1 , w min = 0.4 , w max = 0.9 , k = 1000 ASCSO-S N R = 0.4 N , N H = 0.6 N , N = 100 , G = 100 , w min = 0.1 , w max = 0.7 , a = 6 , b = 0.3 4. The effectiveness testing of SCSO The 21 benchmark functions (shown in Table 1) are used to verify the effectiveness of SCSO compared with CSO. The mean values of 21 functions solved by SCSO and CSO are listed in Table 5. It is clearly seen that the accuracy of SCSO is significantly higher than that of CSO. Especially for f 10 and f 13 f 14 , the solution results of SCSO has reached the optimal value. For other functions, the results of SCSO are all up to 70 orders of magnitude higher than those of CSO. Table 5. The experimental results of CSO and SCSO Functions Dim Results CSO SCSO f 1 100 Mean 3.0815E+03 1.7152E−32 f 2 100 Mean 6.6712E−12 4.5430E−82 f 3 100 Mean 3.1720E+03 2.2053E−30 f 4 100 Mean 9.5488E+04 9.9205E+01 f 5 100 Mean 2.6444E+05 6.7067E−01 f 6 100 Mean 2.0763E+04 6.7397E−24 f 7 100 Mean 35.2580 26.8451 f 8 100 Mean 4.3475E−15 2.6767E−30 f 9 100 Mean 1.3302E+01 1.3685E−03