Self-Adaptive Artificial Bee Colony for Function Optimization

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In this paper, we used the advantage of ABC, presented by Karaboga [8], which mimics the hunting behaviors of honey bee swarm.Simulation results show that ABC is superior to many other population-based optimization methods, namely, genetic algorithm (GA), evolution strategies (ES), and particle swarm optimization (PSO) [9,10].ABC has caused wide attention and applications since its invention in 2005, owing to its simplicity and fewer control parameters.However, ABC faces some challenging problems, namely, low precision and slow convergence.As a result, many improved versions of ABC have been proposed to overcome these shortcomings.Zhu and Kwong [11] presented an improved Gbest-guided ABC (denoted as GABC) through combining the global best (Gbest) individual with the position-updated equation to enhance the ability of local search.Luo et al. [12] presented an improved position-updated equation by using global best individual information to generate offspring individuals.Inspired by DE, Gao et al. [13] developed an improved ABC variant applying the bees search only near the global best one to enhance the exploitation.Xiang and An [14] developed a modified position-updated equation to accelerate the convergence speed.Moreover, chaotic optimization mechanism is introduced to avoid being trapped in local minima.Li et al. [15] presented an improved ABC variant based on inertia weight and accelerating factors and to coordinate the ability of global and local search.Gao and Liu [16] developed two modified position-updated equations, namely, "ABC/best/1" and "ABC/rand/1."Kang et al. [17] developed a hybrid method which combines ABC algorithm and pattern search method to speed up convergence.Gao et al. [18] presented an improved version of ABC (denoted as CABC) based on a modified position-updated equation.In addition, the orthogonal experimental design is introduced.
As we know, for an optimization algorithm, both global and local search are necessary and they should be well coordinated to obtain better global performance [11].Different from the previous work, the position-updated equation is modified by self-adaptive adopting of the previous and global best solution to generate new candidate offspring to coordinate the ability of global and local search.Moreover, the goodpoint-set approach is used to generate initial population.The proposed SABC is tested on 12 benchmark global optimization problems.The experimental results show that our SABC algorithm is superior to basic ABC and other ABC variants.
The remainder of this study is organized as follows.The standard ABC is described in Section 2. The improved ABC called SABC algorithm is proposed and analyzed in Section 3. In Section 4, 12 benchmark global optimization problems are used to test the proposed SABC algorithm.Finally, Section 5 summarized the conclusions.

Artificial Bee Colony
ABC is metaheuristic optimization method inspired by hunting behavior of honey bee swarm.At the initialization step, generate randomly  solutions to construct an initial population: where  = 1, 2, . . ., ,  = 1, 2, . . ., ; rand(0, 1) is a uniformly distributed random number;  , and  , are the lower and upper bounds for the dimension , respectively.In onlooker bee stage, food source is chosen by probability where fit  denotes the fitness value of  →   and is defined as where (  →   ) denotes the objective function values of the decision vector  →   .

Self-Adaptive Artificial Bee Colony (SABC)
3.1.Population Initialization.Note that, for ABC algorithm, in order to find the existing area of optimal solutions faster, initial population should cover the whole search space.Good-point-set method is widely applied to generate many uniformly distributed candidate individuals [19].Therefore, we apply good-point-set technique for producing initial population of ABC to maintain diversity of population.The pseudocode of good-point-set technique is given in Algorithm 2.
The uniformity properties of random method and goodpoint-set method are compared as given in Figure 1 and it displays the 80 points on a unit square generated using random number generator and good-point-set approach.
From Figure 1, the candidate individuals produced through good-point-set technique are more uniform than the candidate individuals produced by random method.Thus, good-point-set method is preferred technique for generating initial population.

Modified Search Equation.
On the basis of the positionupdated equation depicted by (4), the offspring individual is produced through moving the previous individual towards (or far away from) another individual chosen randomly in population.Nevertheless, the randomly chosen individual is a good individual or a bad one; the probability is the same.Therefore, the offspring individual could not be guaranteed to be better than the previous one.In addition, the coefficient   in (4) is a random number over the range [−1, 1] and   is a randomly selected solution from population.Thus, the position-updated equation depicted by (4) is skilled in global search and bad at local search [11].
For the sake of enhancing the performance of ABC, one active research trend is to investigate its position-updated equation.As mentioned above, the characteristics of the search equations of ABC have been extensively investigated.ABC researchers have suggested many empirical guidelines for modifying search equation during the last decade.It has been clear that some search equations can speed up the convergence [11][12][13], and some others are suitable for the global search [20].Indubitably, these experiences are extremely helpful for improving the performance of ABC."ABC/rand/1" and "ABC/best/1" are invariably employed in lots of ABC variants and their characteristics have been commendably investigated.The equations of "ABC/rand/1" and "ABC/best/1" are stated as [16] "ABC/rand/1": "ABC/best/1": where  1 and  2 are randomly selected from {1, 2, . . ., }, and  1 ̸ =  2 ̸ = . best, is global best solution position vector and  ∈ {1, 2, . . ., } denotes  dimension.  is a uniformly distributed random number in [−1, 1].The "ABC/rand/1" search equation is one of many often used in the paper.In "ABC/rand/1" equation, all positions are randomly chosen in population and, consequently, it does not have any bias Generate a offspring individual by Eq. ( 4) and evaluate its quality.(07) Compareandselectthebetterone. (08) Calculate probabilities through Eq. ( 2). ( 09) Produce randomly a number in [0, 1].(10) Generate a offspring individual by Eq. ( 4) and evaluate its quality.(11) Compareandselectthebetterone.(12) Memorize the best solution achieved so far.(13) cycle = cycle + 1. ( 14) Until cycle=Maximum cycle number Algorithm 1: Pseudo-code of basic ABC algorithm.
(01) Set the population size ; the decision variables dimension ,  = 1,  = 1.(02) For  = 1 to  do (03) End if (13) End for (14) End while (15)  in particular positions and randomly selects new positions.Therefore, they often show better global search ability with slow convergence."ABC/best/1" has faster convergence based on global optima position.However, they are more likely to fall into local optima.
In the ABC algorithm, appropriate coordinate of global search and local search is very important for obtaining the optima effectively.In order to bring about a balance between global and local search abilities of ABC, we first introduces a parameter , which coordinates the effect of previous position on the current one, by modifying (4) to where Note that the parameter  is crucial in balancing the abilities of global and local search.When  is equal to 1, (7) becomes (5).When  decreases from 1 to 0, the global search ability of (7) will also decrease correspondingly.When  takes 0, ( 7) is (6).The search performance of algorithm will adaptively adjust through changing the parameter .From (7), with a large value of  in the early stage, individuals are admitted to move in all directions of solution space, instead of moving towards the best individual.A small value of  allows the population to converge to the global optimal solution in later stage.Therefore, a well-tuned  is very important.The parameter  in this paper is calculated according to the following function: where t denotes number of iterations and T denotes a predefined maximum number of iterations.Thus, the improved position-updated equation developed by ( 7) is able to balance the abilities of global and local search of SABC.

Rank Selection.
As we know, the roulette wheel selection mechanism is employed in classical ABC.This selection strategy makes more chance to select individuals with higher fitness [14].To keep population diversity better, a rank-based selection mechanism is introduced in this paper.Selection probability p i is defined [21]: where  is the population size, () is parameters,  is current iteration,  max is maximum iteration.

3.4.
The Proposed SABC Algorithm.The flow chart of proposed SABC algorithm is presented in Figure 2.

Benchmark Test Functions.
To evaluate the performance of SABC, 12 benchmark test problems from [14,15,22] are used and are shown in Table 1.

Comparison between SABC and Basic ABC Algorithm.
In SABC, population size N is 40, limit is 100, and the maximum iteration is 1000.Each experiment is repeated 30 runs independently.The experimental results of SABC and basic ABC are given in Table 2 regarding the best, the mean, the worst, the standard deviation (St.dev), and the convergence iteration (CI).
As seen from Table 2, SABC is able to obtain global optima for three functions (f 6 , f 8 , and f 10 ).Moreover, in seven functions (f 1 -f 4 , f 9 , f 11 , and f 12 ), the results obtained by SABC are pretty near to global optimal solution.Compared with ABC, SABC can obtain much better results than ABC on 12 functions, that is, f 1 -f 12 .The convergence performance of SABC and basic ABC on 12 functions are drawn in Figure 3 so as to show the performance of SABC more clearly.

Comparison between SABC and ABC Variants.
SABC is compared against four high-performance ABC algorithms under three performance evaluation criteria: Mean, St.dev, and CI.These selected ABC variants such as Gbestguided ABC (denoted as GABC) algorithm [11], an efficient and robust ABC (abbreviated as ERABC) algorithm [14], improved ABC (abbreviated as IABC) algorithm [15] and prediction and selection ABC (denoted as PSABC) [15].The parameters settings are the same as those of ERABC [14] and PSABC [15] together with SABC.The comparative results have been shown in Tables 3 and 4. The results provided by other algorithms were directly taken from the original references for each approach.
From Tables 3 and 4, compared to GABC, SABC could achieve much better "mean" and "standard deviation" results than GABC on 11 problems except for function f 7 .For the function f 7 , SABC algorithm obtained similar results.In addition, the convergence iteration obtained by SABC is smaller than those of ABC, except for function f 7 .
As can be seen from Tables 3 and 4, with respect to ERABC, SABC obtained better "mean" and "standard deviation" values on two functions (f 3 and f 4 ) and similar results on four test functions (f 6 , f 8 , f 9 , and f 10 ).However, ERABC algorithm provided better results on six problems (f 1 -f 2 , f 5 , f 7 , and f 11 -f 12 ) than SABC.
From Tables 3 and 4, compared to IABC, SABC obtained better results on five functions (f 3 , f 5 , f 6 , f 11 , and f 12 ) and similar solutions on four functions (f 7 , f 8 , f 9 , and f 10 ).However, IABC provided better solutions on three functions (f 1 , f 2 , and f 4 ) than SABC algorithm.
In comparison with PSABC, in Tables 3 and 4, SABC provided better solutions on four test functions (f 3 , f 5 , f 6 , and f 11 ) and similar results on five functions (f 7 , f 8 , f 9 , f 10 , and f 12 ).However, the PSABC algorithm found better solutions on three functions (f 1 , f 2 , and f 4 ).Compare V i and X i using greedy Compare V i and X i using greedy rand(0, 1) < p i cycle = cycle + 1 Generate an offspring (V i ) using ( 7) Generate an offspring (V i ) using ( 7)

Effects of Limit on the Performance of SABC.
To investigate the impact of limit, four different test functions with  = 30 are used.They are Sphere (f 1 ), Rastrigin (f 8 ), Ackley (f 9 ), and Griewank (f 10 ).Five different values of limit (i.e., 50, 100, 150, 200, and 250) are used to optimize the four high dimensional test functions.The mean values (mean) and the standard deviation values (St.dev) are shown in Table 5.The box plots of different limit values for four functions are presented in Figure 4. From Table 5, parameter limit is able to influence the performance of SABC.When limit is equal to 100, SABC achieves better performance for three functions (f 1 , f 8 , and f 10 ).For the Ackley function (f 9 ), the effect of limit on the performance of SABC is very little.From Figure 3 and Table 5, parameter limit = 100 is a suitable choice in SABC algorithm.

Conclusion
An improved version of ABC, called SABC, is developed by using good-point-set initialization employed to enhance the population distribution, rank-based selection strategy used to enhance the global search ability, self-adaptive positionupdated equation applying for balancing the exploration and exploitation.The proposed SABC is tested on 12 well-known global optimization problems.The simulation results show that our algorithm is superior to the conventional ABC and other ABC variants.The further work includes the studies on how to develop SABC to deal with those constrained and engineering design problems.

Figure 1 :
Figure 1: 80 points in the unit squares generated through random and good-point-set approaches.

Figure 2 :
Figure 2: Flow chart of proposed SABC algorithm.

Figure 4 :
Figure 4: Box plots of different limit values for function f 1 , f 8 , f 9 , and f 10 over 30 independent runs.

Table 2 :
Experimental results of SABC and ABC in  1 - 12 .

Table 5 :
The experimental results of SABC with different limit values.