Chicken Swarm Optimization Based on Elite Opposition-Based Learning

Qu, Chiwen; Zhao, Shi’an; Fu, Yanming; He, Wei

doi:https://doi.org/10.1155/2017/2734362

Mathematical Problems in Engineering

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Abstract Introduction Results Conclusion Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2017 | Article ID 2734362 | https://doi.org/10.1155/2017/2734362

Chicken Swarm Optimization Based on Elite Opposition-Based Learning

Chiwen Qu,¹Shi’an Zhao,²Yanming Fu,³and Wei He¹

Academic Editor: George S. Dulikravich

Received14 Oct 2016

Revised13 Feb 2017

Accepted01 Mar 2017

Published26 Mar 2017

Abstract

Chicken swarm optimization is a new intelligent bionic algorithm, simulating the chicken swarm searching for food in nature. Basic algorithm is likely to fall into a local optimum and has a slow convergence rate. Aiming at these deficiencies, an improved chicken swarm optimization algorithm based on elite opposition-based learning is proposed. In cock swarm, random search based on adaptive distribution is adopted to replace that based on Gaussian distribution so as to balance the global exploitation ability and local development ability of the algorithm. In hen swarm, elite opposition-based learning is introduced to promote the population diversity. Dimension-by-dimension greedy search mode is used to do local search for individual of optimal chicken swarm in order to improve optimization precision. According to the test results of 18 standard test functions and 2 engineering structure optimization problems, this algorithm has better effect on optimization precision and speed compared with basic chicken algorithm and other intelligent optimization algorithms.

1. Introduction

Many problems in areas of scientific computing, engineering science, and business management can be concluded as global optimum problems. The consumption time of traditional accurate computation approaches for solving large-scale optimization problems increases exponentially. So this approach cannot meet the real requirements. To solve these problems, many scholars, by simulating life habits of creatures in the nature, presented intelligent swarm optimization algorithms including particle swarm optimization (PSO) [1], ant colony optimization (ACO) [2], artificial bee colony (ABC) [3], invasive weed colonization optimization (IWO) [4], firefly algorithm (FA) [5, 6], cuckoo search (CS) algorithm [7, 8], fish swarm algorithm (FSA) [9], bat algorithm (BA) [10, 11], monkey algorithm (MA) [12], krill herd (KH) algorithm, and flower pollination algorithm (FPA) [13]; all these have gained favorable results. As a new kind of burgeoning metaheuristic algorithm, intelligent swarm optimization algorithms have characteristics of high precision, fast convergence rate, and good stability and can obtain the exact solution or approximate solution of large-scale optimum problems within limited time.

Chicken swarm optimization (CSO) is a new intelligent bionic algorithm proposed by Meng et al. [14] in 2014, which simulates chickens swarm hierarchy and their food search behavior. The whole chicken swarm is divided into cock swarm, hen swarm, and chick swarm. Chickens with highest fitness values and lowest fitness values are taken as cock swarm and chick swarm, respectively, and the rest are taken as hen swarm. When solving optimization problems, each chicken in the swarm corresponds to a solution. Different search strategies are adopted for different subswarm according to different population. In contrast with standard particle swarm optimization, differential evolution, and bat algorithm, chicken swarm has advantage in either searching precision or convergence rate [14, 15]. In basic chicken swarm algorithm, a random search strategy based on Gaussian distribution is adopted for particles of cock swarm. This search strategy has strong ability of local development, but its global development ability is weak to make it liable to fall into a local optimum. For hen swarm particles, the searching is done by common guidance of cocks of their own population and other populations, which helps hen particles close to global optimum. However, the population diversity will lack and the hen particles will fall into a local optimum when most cock particles (with cock particles of own race and other races) are in a local optimum. Consequently, the global optimal solution cannot be obtained and its search performance will be affected for basic chicken swarm algorithm.

In this paper, a chicken swarm optimization algorithm on the basis of elite opposition-based learning (EOCSO) is presented to solve global optimum problems. A random search strategy based on dynamic adaptive distribution is adopted in this algorithm for cock swarm to replace the random search based on Gaussian distribution. The local exploitation ability and global development ability are balanced. In order to improve the optimization precision and convergence rate of the algorithm, an opposition-based learning method is used to improve population diversity for hen swarm and a greedy dimension-by-dimension search mode is applied to individual of optimal chicken swarm for local search. Through experiments of 18 basic test functions and 2 engineering structure optimization measurement problems, the comparison among improved chicken swarm algorithm, basic chicken swarm algorithm, and other typical intelligent algorithms is conducted to show that improved chicken swarm algorithm has more excellent optimization precision, convergence rate, and robustness.

2. Chicken Swarm Algorithm

Chicken swarm optimization is a new intelligent bionic algorithm proposed according to various behaviors of cocks, hens, and chicks in the process of searching food. In this algorithm, chicken swarm in searching space is mapped as specific particle individual. Cock particle swarm, hen particle swarm, and chicken particle swarm are sorted according to fitness value of particle, and each subswarm uses different searching mode [16, 17].

In this algorithm, several particles with best fitness are selected as cock particle swarm, which is given bywhere and are the position of th dimension of particle in and iterations, respectively, and is a random number of Gaussian distribution whose variance is . The parameter can be calculated bywhere and . represents the number of cock swarms. and are the fitness values of cock particle and , respectively; represents a number which is small enough.

Moreover, most particles with good fitness are selected as hen swarm. Its random search is done via cocks of population of hen and that of others, which can be expressed as where and are the position of cock individual in the population of hen and cock individual in the other population, respectively. is an uniform random number over . and denote the weight calculated bywhere and are, respectively, the fitness value of cock individual in the population of hen and cock individual in the other population.

All individuals, except for cock swarm and hen swarm, are defined as chick swarm. Its search mode follows that of hen swarm, which is given bywhere stands for a parameter, meaning that the chick would follow its mother to forage for food. represents the position of the th chick’s mother ().

3. Chicken Swarm Optimization Based on Elite Opposition-Based Learning

3.1. Random Search Strategy Based on Dynamic Adaptive Distribution

In chicken swarm optimization, a random search mode on the basis of Gaussian distribution is adopted for cock particle searching. This search mode has strong ability in local development, but it is weak in global exploitation. Therefore, enlargement of search space for cock particle will effectively reduce the risk to fall into a local optimum. Thus, the search mode with distribution is used for cock particle in order to balance the global development ability and local exploitation ability of the algorithm.

The distribution, also called student distribution [18], owns degree of freedom parameter ; its probability density function is as follows:where is the Gamma function and .

Apparently, the distribution is defined as Cauchy distribution density function () when degree of freedom parameter and is defined as Gaussian distribution density function when (). Thus, Cauchy distribution and Gaussian distribution are two special boundary cases of distribution. The characteristics of Cauchy distribution and Gaussian distribution are integrated into distribution, and different mutation range can be obtained via changing degree of freedom parameter .

In the algorithm, random search based on dynamic adaptive distribution is selected for the position of cock particles , which is expressed aswhere is the position after random search of dynamic adaptive distribution used by th dimension of cock particle , is the position of th dimension of th cock particle, is random function of distribution with iteration as its degree of freedom, and is mutation control factor. Each dimension value performs differently in searching process of multidimensional objective function. Thus, the mutation control factor has the same control strategy to each dimension, and it cannot meet the requirement of algorithm obviously. In this paper, dynamic adaptive mutation control factor is used to adjust mutation range, as given bywhere is proportionality coefficient, is the number of cock members, is mean value of th dimension of cock particle in the current iteration, and is a mutated value of th dimension of cock particle in optimal position of the current iteration.

In random search of dynamic adaptive distribution, iteration is taken as degree of freedom parameter, and dynamic adaptive mutation control factor is used to adjust searching range. In the early stage of evolution, the search is conducted by distribution (which is similar to that via Cauchy distribution) and maintains population diversity due to small . Thus, the algorithm has good ability of global exploitation. With increase of , random search of distribution transits gradually from that of Cauchy distribution to that of Gaussian distribution, and random search of distribution is similar to that of Gaussian distribution with good ability of local development in the later stage.

3.2. Elite Opposition-Based Learning

Elite opposition-based learning proposed by Tizhoosh [19] is a new technology applied to intelligent computing area, and it has been successfully applied in many intelligent algorithm optimizations [20–22]. Theoretically verified by Zhong and others [23], opposition-based learning can acquire a solution that closes to global optimal solution with a higher probability. In the paper, elite opposition-based learning is adopted to enlarge search range for hen swarm and enhance population diversity so as to improve searching performance of the algorithm. The basic thoughts are as follows.

Set the optimal cock position and its opposite position in th iteration as and , respectively; thenwhere , is a coefficient which denotes uniformly distributed random variable, and is the boundary of th dimension of search space in the current population. The result is obtained bywhere and . Equation (10) is applied to replace fixed boundary, which ensures that cocks generate opposite solution in decreased space range. Besides, threshold value of the opposite solution may go beyond the range of to form a nonfeasible solution. Traditional handling is to set the searching particle which goes beyond borders as boundary value. As a result, numerous searchers gather on the border. To avoid above problem, border buffering wall is used for handling, and its basic thoughts are as follows.

Suppose that is upper bound of th dimension of search range in the population and is the lower bound. If opposite solution of the optimal individual in the population is , the value after border buffering wall handling will bewhere sgn is sign function and is a proper constant, which relates to the thickness of buffering wall.

With border buffering wall to process off normal individual, search individuals are able to decide their values dynamically according to the practical searching conditions and can better overcome deficiencies brought by traditional handling. The search mode of hen swarm using elite opposition-based learning can be expressed as follows:where , , and are weights, and have the same value, is an uniform random number over , and is the opposite solution of the optimal particle of the population of chick swarm particle in th iteration.

3.3. Local Search via Greedy Dimension-by-Dimension Search

In basic chicken swarm optimization algorithm, the cock swarm, the hen swarm, and the chick swarm use different random search modes to acquire comparatively high-quality solution and convergence rate. But the three subswarms are all assessed by overall upgrading assessment method. For high-dimensional global optimization problem, overall upgrading assessment method will affect the quality of solution and convergence rate due to the interference among each dimension [24]. Suppose that the objective global optimization function is (sphere test function) and independent variable equals . Then, . In iterations, the independent variable is updated as after overall upgrading. Then, , and will be abandoned in chicken swarm optimization as . The overall upgrading assessment method will cause a slow convergence rate for the algorithm as the first-dimensional contribution is ignored due to deterioration of the second- and third-dimensional contributions.

The greedy dimension-by-dimension search mode in this paper makes full use of updated achievement of each dimension. The basic idea is as follows. (1) Suppose that is the value after updating of the value of th dimension of chicken swarm particle (fitness value is ). Then, and the values of other dimensions of chicken swarm particle before updating are integrated into a new chicken swarm particle . (2) Calculate its fitness value . If , then the updated result will be saved. Otherwise, it will be abandoned and the updating of th dimension will be conducted. The procedures are shown in Algorithm 1.

(1) Set the generation counter , the maximum generation , dynamic adaptive
change step , max step , the chicken individual .
(2) (*)
(3) for
(4) ;
(5) ;
(6) if
(7) ;
(8) end
(9) end

To ensure that each dimension of chicken particles in initial period of iteration updates in a relatively big step-length and in a small step-length in later period of iteration, dynamic adaptive step-length updating as (*) in Algorithm 1 is adopted to further improve the algorithm’s convergence rate and quality of solution.

3.4. Algorithm Flow

Based on Sections 3.1–3.3, the procedure of the chicken swarm optimization algorithm is shown in Algorithm 2.

() Set the initial parameters, including the total population size , the roosters accounts ,
the hens accounts , the mother hens accounts , updating frequency of the
chicken swarm and the maximum number of generations et al.
() Generate a population of chickens with random solutions.
() Calculate the fitness and find the best solution of the population.
() for
() if iter% or
() Sort all population individuals according to their fitness.
() Divide total population individuals into three subpopulations (called rooster population, hen
population, and chickens population) according to their sort criteria, and establish
the relationship between the chickens and its mother (hens).
() end
() Update the rooster population individuals according to Eq. (7) and calculate their fitness.
() Update the hen population individuals according to Eq. (12) and calculate their fitness.
() Update the chicken population individuals according to Eq. (5) and calculate their fitness.
() Update the personal best position and the global optimal position .
() Perform local search for the global optimal individual according to Section 3.3.
() end
() Output the best solution

4. Results and Discussions

4.1. Parameter Setting

To test the algorithm’s performance, seven typical intelligent algorithms, bat algorithm (BA) [10, 11], particle swarm optimization (PSO) [1], differential evolution (DE) [25], ant colony optimization (ACO) [2], cuckoo search (CS) algorithm [7], flower pollination algorithm (FPA) [13], and chick swarm algorithm (CSO) [14], are selected to do contrast experiments with EOCSO algorithm. The parameters of EOCSO algorithm are set as follows. Population size is set as 100, and each of the scale of cock swarm and that of chick swarm occupies 15%, while the scale of hen swarm takes up 70%. All of the common parameters of these algorithms (including the population size, dimensions, and maximum number of generations) are set to be the same for a fair comparison. Other parameters are described in Section 3. The parameters set of all other five algorithms is shown in Table 1.

4.2. Test Function

To verify the optimization precision and convergence rate of the proposed algorithm, 18 standard test functions in [26, 27] are chosen for contrast experiments. are high-dimensional unimodal functions and are extremely hard to converge to a global optimum, which are used to inspect the searching precision. are high-dimensional multimodal functions with several local extreme points. So they are used to test the global searching performance and avoid premature of the algorithms [28–30]. are low-dimensional multimodal functions. The standard test functions are seen in Table 2.

4.3. Influence of Dynamic Adaptive Distribution Search Strategy on Bird Swarm Algorithm

Influence of dynamic adaptive distribution search strategy on bird swarm algorithm contains two parts. The first part is the influence of different parameter values of variable control factor on Gaussian distribution algorithm. The second part is the empirical comparison and analysis of CSO under dynamic adaptive distribution (ATD-CSO) and dynamic adaptive Gaussian distribution (AGD-CSO) as well as the original algorithm (O-CSO). High-dimensional unimodal functions , high-dimensional multimodal functions , and low-dimensional multimodal functions are selected as test functions.

4.3.1. Impact Analysis of Control Factor

Other parameters remain invariant, 0.2, 0.5, and 0.8, and dynamic adaptive change is used as control factor for Gaussian distribution algorithm. Test function parameters are set the same as Section 4.1, and test result is assessed from optimal value, mean value, worst value, and standard variance. Table 3 shows the statistical result of different ψ on the test functions.

As can be seen from Table 3, search effect is better when is smaller. But it is not quite ideal for Gaussian distribution with fixed to improve algorithm performance. Through dynamic adaptive parameter setting, the algorithm can be ensured to maintain a large search step size in initial search stage. In later stage, dynamic adaptive step size of control factor is reduced, while the local search ability is enhanced. The test result proved that dynamic adaptive control factor has more advantages than single fixed control factor.

4.3.2. Comparison of ATD-CSO, AGD-CSO, and O-CSO

Table 4 indicates the statistical result of dynamic adaptive distribution, dynamic adaptive Gaussian distribution, and original CSO algorithm on three different kinds of test function. As the data shows, in terms of optimal values, 6 and 4 orders of magnitude are improved for and separately by ATD-CSO compared with AGD-CSO. For function , searching precision of ATD-CSO is higher than that of AGD-CSO. For , , and , optimal solution is searched by both ATD-CSO and AGD-CSO. Also, for mean value, worst value, and standard variance, and obtain the same through these two algorithms. But, for the standard variance for , ATD-CSO is better than AGD-CSO. Compared with O-CSO, ATD-CSO produces optimal mean value, worst value, and variance. It can be seen from Figures 1–6 that, in early evolution period, convergence rate of ATD-CSO is faster than that of AGD-CSO when calculating the above 6 functions. In later period, the convergence rate is similar, but the overall convergence rate of ATD-CSO is more optimal. However, the convergence rate of O-CSO is slower than that of ATD-CSO and AGD-CSO. Therefore, ATD-CSO has some advantages in terms of solution precision, robustness, and convergence rate generally.

4.4. The Influence of Dynamic Adaptive Distribution and Opposition-Based Learning on CSO (ATD-EO-CSO)

As demonstrated in Table 5, the best value, mean value, worst value, and variance of obtained by dynamic adaptive distribution and elite opposition-based learning are apparently better than those obtained by original CSO. Target value can be searched on function by both ATD-EO-CSO and O-CSO. Therefore, the O-CSO is suitable to solve this function, and the improvement of the solution after optimization is not obvious. For function , the best value can be searched by both. However, in terms of mean value, worst value, and variance, the results will be better based on cowork of both distribution and elite opposition-based learning. Since searching can be done more widely via elite opposition-based learning, boundary buffering wall is adopted to deal with individuals crossing the border, avoiding the deficiency that individuals gather on boundary value to improve the diversity of population. As a result, ATD-EO-CSO is superior in solution precision.

4.5. Influence of Local Search via Greedy Dimension-by-Dimension Search on Bird Swarm Algorithm

Influence of local search via greedy dimension-by-dimension search on bird swarm algorithm contains two parts. The first part is the influence of fixed step size , , , and the adaptive step on algorithm. The second part is the comparison between adaptive step strategy and original CSO. Table 6 shows the statistical result by means of various step values. According to Table 6, when step value is small, higher search precision can be obtained, but it is liable to get into premature convergence. In case of larger step value, the range of search step size is larger, and it is liable to skip optimal solution range, which causes lower searching effect in later stage of evolution and reduced local search ability. Through adaptive step, the algorithm’s global research can be ensured to conduct in long step size at initial stage and in short step size at later stage, which improves the search efficiency of dimension-by-dimension search. Hence, adaptive step dimension-by-dimension search is superior to the fixed step size strategy in terms of search precision.

4.6. Analysis of Experimental Results

In order to avoid being influenced by random factors, the experimental test for each case is conducted by 20 trials independently. The algorithm’s searching performance is assessed according to the best value, mean value, standard deviation, and the worst value produced by the test result. The number of iterations of and is 50 and 1000 in each trial, respectively. The algorithm is examined by Matlab 2012a on the platform with Win 8 OS, Intel Core i5-4210U 2.4 GHZ CPU, and 4 GB memory. The test statistical results are shown in Tables 7, 8, and 9.

The test statistical results of functions are shown in Table 7. As can be seen from Table 7, the best value, mean value, standard variance, and the worst value of EOCSO are all superior to those of other intelligent algorithms (including BA, PSO, DE, ACO, CS, FPA, and CSO). Particularly, global minimum of the test function lies in the bottom of parabola, which is quite reliable to fall into a local optimum in the searching process. The optimal result of EOCSO is , which is 8, 11, 10, 10, 7, 3, and 10 orders of magnitude higher than that of BA, PSO, DE, ACO, CS, FPA, and CSO, respectively. The mean value of EOCSO is , which is 5, 7, 4, 4, 4, 4, and 6 orders of magnitude higher than that of the compared algorithms, respectively. Besides, EOCSO algorithm is also superior to seven other intelligent algorithms in terms of the worst value and standard variance. It comes to a conclusion that EOCSO algorithm has high precision and good robustness in searching high-dimensional unimodal function.

The test statistical result of functions is demonstrated in Table 8, from which we can find that EOSCO algorithm can obtain global optimal extreme value for , , and , which is free from interference of local extreme value. For other algorithms, only CSO can obtain global optimum in searching of and . For , the result is almost equivalent to that of EOCSO, CSO, and CS. Therefore, the original CSO is suitable to solve this function; the improvement of the solution after optimization is not obvious. With regard to , EOSCO does better than five other algorithms in precision and standard variance of its solution, which reflects the advantage of EOCSO.

In Table 9, the test statistical result of low-dimension multimodal functions is described. EOCSO and seven other algorithms can be seen to be able to get the global optimum in searching process, but EOCSO has fixed advantages in its mean value, worst value, and standard variance. “Error” on -axis is the error between actual global minimum and converged minimum. The convergence graphs of the 6 algorithms are expressed in Figures 7–24. In EOCSO algorithm, its convergence rate of all 18 functions is more excellent than that in the other five algorithms, especially for , , and . The convergence curve is quite smooth and drops rapidly, reflecting its good convergence rate.

4.7. Structural Design Examples

In order to validate the performance of proposed method for constraint problems, EOCSO is examined by solving constrained engineering design problems, such as speed reducer design problem and pressure vessel design problem.

4.7.1. Speed Reducer Design Problem

Speed reducer design problem is proposed by the famous scholar Mezura-Montes, which is a classic constrained optimization and is used to verify the design engineering constrained optimization algorithm performance, as shown in Figure 25. There are 7 variables and 11 inequality constraints of the problem. The mathematical model is represented as follows:where , , , , and . This case study has been previously solved using other methods such as PSO-DE [31], MBA [32], HEAA [33], and HGA [34]. The best results of the various methods for solving the speed reducer design problem are shown in Table 7. Table 8 shows the statistical results of the different methods.

It can be seen from Table 10 that the optimal solution of the EOCSO algorithm is better than the other 4 kinds of algorithms. From Table 11, the optimal value, the average value, and the worst value of EOCSO are better than those of the other 4 algorithms. In terms of stability, the standard deviation of EOCSO is smaller than those of MBA and HEAA algorithms by 6 and 4 orders of magnitude, respectively.

4.7.2. Pressure Vessel Design Problem

Another benchmark structural optimization problem is the pressure vessel design problem proposed by Kannan and Kramer. Figure 26 shows a pressure vessel design problem which has four variables () and four nonlinear inequality constraints (). The objective function of the problem can be expressed as follows:where and . The approaches have previously been applied to solve this problem including many different numerical optimization techniques, such as the new modification approach on bat algorithm (EBA) [35], the cuckoo search (CS) algorithm [7], interior search algorithm (ISA) [36], the mine blast algorithm (MBA) [32], the improved ant colony optimization (IACO) [37], an effective hybrid cuckoo search algorithm for constrained global optimization (HCS-LSAL) [38], the hybrid Nelder-Mead simplex search and particle swarm optimization (NM-PSO) [39], an improved accelerated PSO algorithm [40], and crow search algorithm (CSA) [41].

The best solutions obtained by the various methods are reported in Table 12. Table 13 shows the statistical results. It is shown in Tables 12 and 13 that EOCSO performance for the pressure vessel design problem surpassed the other 10 methods in terms of the minimum obtained value, solution average, and the standard deviation.

5. Conclusion

In this paper, an improved chicken swarm algorithm based on elite opposition-based learning is proposed to overcome the disadvantages of the deficiencies of lack of population diversity, being easy to stick to “premature,” and low searching precision in later stage of evolution in chicken swarm algorithm. Search mode of dynamic adaptive distribution is applied for cock swarm to balance the global development ability and local exploitation ability of the algorithm. Search mode of elite opposition-based learning is used to enrich population diversity for hen swarm. Greedy dimension-by-dimension searching is used for individual of optimal chicken swarm to do local search, which improves the search precision and convergence rate of the algorithm. Numerical experiments of 18 standard test functions and 2 engineering structure optimization problems are conducted to verify the availability and feasibility of the algorithm. The search precision, convergence rate, and robustness are all better than those in other typical intelligent algorithms.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is financially supported by the Natural Science Foundation of Guangxi Province (Grant no. 2014GXNSFBA118283) and the Higher School Scientific Research Project of Guangxi Province (Grant no. 2013YB247).

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Copyright © 2017 Chiwen Qu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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