In ABC algorithm (given in Algorithm 1), there are three types of honey bees as employed, onlooker and scout. The employed bees are responsible for calculating the nectar amount of every possible food sources. The number of the employed bees in the population is equal to the number of food sources in the feeding area. The onlookers are charged with choosing the food source which has a great amount of nectar. The number of the onlookers in the population is equal to the number of employed bees. The scouts are responsible for discovering new food sources. If a food source is exhausted, the employed bee becomes onlooker and the new food source discovered by the scout bee replaces the old food source.

Position of the food source represents the possible solution of the optimization problem aimed to be solved. High nectar amount of the food source means that the possible solution of the optimization problem is good. Therefore, quality of the possible solution is represented by the nectar amount and this value is called fitness value in ABC algorithm. At initialization step, SN (number of food sources) is established randomly by using the Eq. (1): (1) x i , j = x j min + φ i , j ( x j max − x j min ) where φ i , j is a uniform random number in [ 0 , 1 ], x j min is the minimum value of the jth dimension while x j max is the maximum value of that dimension. Moreover, each solution (or each food source) has a fitness and limit value. limit value is used to control whether the food source has been exhausted or not.

In ABC algorithm, fitness value is calculated in accordance with the Eq. (2): (2) fitness i = 1 1 + f i , f i ⩾ 0 , 1 + a b s ( f i ) , f i < 0 where f i is the objective value of ith solution.

Following the initialization stage, employed bees try to find the new candidate solution based on the Eq. (3): (3) v i , j = x i , j + φ i , j ( x i , j − x k , j ) , i ≠ k where v i , j represents the new solution to be found, x i , j is the previous solution, φ i , j is the random number between the range of [ − 1 , 1 ], and x k , j is the neighbouring solution. k is rated between 1 and S N, while j is rated between 1 and D (maximum number of dimensions). If the candidate solution found is better than the previous one, then it supersedes. Later on, like employed bees, onlooker bees also try to find new good food sources. However, in selecting food sources, onlooker bees as different from the employed bees prefer the ones above a certain probability value p i . This probability value can be seen in the Eq. (4): (4) p i = f itness i ∑ n = 1 SN fitness n .

Observing the behaviour of employed and onlooker bees, it is evident that they search for good food sources and focus on good solutions. Algorithm should explore new solutions to avoid local optimums. If employed and onlooker bees cannot improve a food source for limit times, it means exhaustion of that food source. Exhausted food resource is a source that is not good enough to produce new solutions. In ABC algorithm, scout bees are responsible for the exploration of new solutions searched to avoid local optimums, and they explore new food sources by utilizing the Eq. (1) instead of exhausted food sources.

However, it is seen that in certain problems, convergence speed of ABC algorithm is slow in comparison with the other population based algorithms and sticks to local optimums. Thus, new ABC algorithm versions have been proposed to increase its convergence speed.

In IABC algorithm, Gao and Liu (2011) proposed some modifications on the steps of original ABC algorithm. The first is an efficient initialization strategy that leads algorithm to find good initial population. Instead of the initialization with normal distribution in original ABC, the initialization approach of IABC utilizes chaotic systems and opposition-based learning. In detail, a chaotic random generator is used to create initial solutions. Then, the initial solutions are duplicated by opposition-based initialization. Finally, the initial population is obtained by selecting the best solutions. The procedure of this initialization strategy is given in Algorithm 2.

In search equations of the employed and onlooker bees steps, IABC offered two modifications as well. Contrary to original ABC, IABC emphasizes in controlling the number of decision variables changed in search equation by adding a new parameter called m. Second modification in search mechanism is that IABC comes with a probabilistic selection of two search equations, called as “ABC/best/1” (Eq. (5)) and “ABC/rand/1” (Eq. (6)) (Gao and Liu, 2011). (5) v i , m = x best , m + φ i , j ( x i , m − x r 1 , m ) , (6) v i , m = x r 1 , m + φ i , j ( x i , m − x r 2 , m ) where m parameter is used to control how many dimensions will be updated. x best , m represents the best solution till then. x r 1 , m and x r 2 , m represent the randomly selected solutions. While “ABC/best/1” is used to produce new solutions based on the information of the best-so-far solutions “ABC/rand/1” is used to search the whole population. Moreover, “ABC/best/1” ensures the finding of the optimum solution quickly, while “ABC/rand/1” enables the avoidance from local optimums. Utilizing these two equations together leads to an increase in the convergence speed towards global optimum and helps to avoid local optimums. By this it means, it has been aimed to balance exploration and exploitation abilities of IABC. The probabilistic selection of these two search equations is controlled by p ( 0 ⩽ p ⩽ 1) parameter. The procedure of this new searching strategy is given in Algorithm 3.

The additional parameters of IABC, m and p, have direct effect on IABC performance. Therefore, authors of IABC tried to find out appropriate values of these parameters from one problem to another. Therefore, with the appropriate parameter values, IABC shows good performance on low-dimensional benchmark functions. However, Liao et al. (2013) have detected that its performance worsens in high dimensional problems. The reason of producing good results in low dimensional problems while putting forward bad results in high dimensional problems is that the value of m and p parameters used in the proposed search mechanisms do not change depending on the problem. The m parameter shows the number of parameters to be updated in a repeat cycle in employed and onlooker bee stages. The effect of m parameter on performance is seen directly in line with the increase in the dimension of the problem.

AIABC tries to increase performance by making online updates for m and p strategic parameters having a direct effect on the performance of IABC through adaptive parameter control. In adaptive parameter control, strategic parameters are updated according to the feedbacks to be obtained in the search process during the performance of the algorithm. With this approach, the algorithm will become adaptive to the problem and it will be ensured that it displays good performance in high dimensional optimization problems.

In this version, randomly generated candidate values of m and p parameters are kept in two separate arrays (mArray and pArray). In onlooker bee stage, the values in the array are used in turn for m and p parameters. If the best solution found so far has changed by using m and p values, these successful m and p values are kept in two separate arrays (sMarray and sParray). Figure 1 shows the relations of these four arrays. When all candidate values in sMarray and sParray are used, re-initialization of these arrays is made by using random values beside the ones in sMarray and sParray. By this it means, it is aimed to achieve good parameter values through the utilization of successful experiences in different stages of the problem during the performance of the algorithm. The pseudo-code of AIABC can be seen in Algorithm 4. The proposed modifications are presented in italic form in the Algorithm 4.

The method developed to update the online values of m and p parameters is realized through the determination of new values by utilizing previous successful experiences during the performance of the algorithm. There are arrays including the possible ranges for each parameter (mArray for m and pArray for p). mArray i = ( m i 1 , m i 2 , … , m i maxCount ) , m i j ∈ [ 1 , 2 ] , pArray i = ( p i 1 , p i 2 , … , p i maxCount ) , p i j ∈ [ 0 , 1 ] . Each array includes values as many as maxCount. The process begins with cnt = 1, which is used as a counter. Before each employed bee stage, values are determined as m = m i cnt , p = p i cnt . In employed bee stage, the algorithm continues its performance with these values. If the best-so-far solution is improved further, m and p values used in that cycle are transferred to sMarray i and sParray i . When cnt counter reaches to maxCount value representing the size of arrays, re-initialization process is started for mArray and pArray. In the re-initialization process, mArray and pArray are updated by taking values randomly selected within the adaptively adjusted ranges or by taking from a certain number of successful values in sMarray i and sParray i . The selection mechanism is controlled randomly by using re-initialization probability value (RP). RP is increased gradually in the running of algorithm and thus, successful m and p values can be selected more than random values in later iterations. Moreover, when all the elements of mArray and pArray are used but successful arrays are still empty, the stagnation is detected. This time, mArray and pArray are re-initialized with new random values by utilizing the ranges of parameters adaptively.

Determination of the possible range of m parameter properly will also have a positive effect on the performance of the algorithm. For instance, it is detected that the high values for m may have a negative effect on the performance of the algorithm specifically for high dimensional functions (Liao et al., 2013; Aydın, 2015). Therefore, the range of these parameters is adjusted adaptively while algorithm is running.

In the first random initialization, the range of m is kept [ 1 , 2 ]. In the following initializations, an adaptive approach is applied in the determination of the range of m parameter. If all elements of mArray and pArray are used and any of them is not transferred to sMarray and sParray in onlooker bee stage and the stagnation is detected, range of m parameter is changed. If the range of m is between 1 and D ∗ factor ( LARGE strategy), it is changed to between 1 and 2 ( SMALL strategy). On the contrary, when the range of m is between 1 and 2, it is changed to 1 and D ∗ factor after stagnation. The initial value of factor variable is 0.1 and is increased by 0.1 after each stagnation. When this variable reaches 1, it is then reduced to 0.1 again. In other words, the maximum value of m is 10% of the problem dimension in the first stage; it is increased by 10% after each stagnation. When range of m parameter reaches to problem dimension, it is then reduced to 10% of the problem dimension again. Moreover, in each stagnation of the algorithm, mArray and pArray are re-initialized according to the range determined above. So, values leading to stagnation are changed. The whole process of adaptive parameter control procedure mentioned here is presented in Algorithm 5.

In the self-adaptive parameter control (Eiben et al., 1999), parameters to be controlled are encoded to solution chromosome as gene in evolutionary algorithms while they are encoded as a new dimension to the solution vector in SaIABC (Fig. 2). m and p parameters are added to each candidate solution as two new dimensions. If in the employed and onlooker bee search equations, these dimension values can be modified just as the dimensions of the problem and m and p values can be changed. Therefore, the control of these parameters depends on the algorithm itself.

SaIABC algorithm uses values found in two additional dimensions of the best-so-far solution as m and p values for each iteration. Search equations in employed and onlooker bee stages are performed according to these parameter values. The range of m parameter is as well adjusted adaptively as in AIABC. The only difference is determination of algorithm stagnation. Here, if the algorithm cannot improve the best-so-far solution at some consecutive iterations (i.e. 25 iterations), this means algorithm stagnates and changes the strategy from LARGE to SMALL or vice versa.

In this section, the proposed adaptive IABC algorithms are applied to minimize a set of 19 scalable functions presented by Lozano et al. (2011) for Soft Computing (SOCO) special issue. Some of them ( f 1 , f 2 , f 7 , f 8 , f 9 , f 10 , f 11 ) are unimodal and the others are multimodal functions. These benchmark functions and their peculiarities are described in Table 1.

All experiments were conducted under the same conditions as SOCO special issue; all algorithms were run 25 times for each function and each run stops when the maximum number evaluations ( 5000 ∗ D, where D is a problem dimension) or error value is lower than a threshold ( 10 − 14 ), which is approximated to zero. An error value of a solution, x, found on a function f i was defined as: ( f i ( x ) − f i ∗ ), where f i ∗ is the known optimum value of function f i .

In order to extensively test the adaptive IABC algorithms, three experiments are conducted. The first experiment compares the average results of the proposed approaches with respect to the IABC and the other ABC variants by using default parameter values. At the second experiment, same comparisons were made by using tuned parameter values. These tuned parameter values were obtained by using Iterated F-race (Birattari et al., 2010) parameter tuning tool. Finally, third experiment compares the average results of proposed adaptive IABC algorithms with the state-of-the art algorithms on the same SOCO benchmark functions.

The Friedman Test (Friedman, 1940), a nonparametric test that detects significant differences between the behaviour of two or more algorithms, was applied to all experiments to see whether the results are significantly better or not. The test gives us a chance to compare between all algorithms statistically. We used a level of significance α = 0.05 and compared p-values.

In this section, experiments are done with default parameter values of the algorithms. Comparison of adaptive IABCs with other ABC variants is given in the following subsections.

In this section, experiments are done with tuned parameter values of the algorithms. In the following subsections, comparison of adaptive IABCs with other ABC variants is given.

In this section, we presented the comparison results of SaIABC and AIABC algorithms with the SOCO competitors. The results of compared algorithms are taken directly from their papers or competition website (http://sci2s.ugr.es/EAMHCO). The detailed comparison results on 50, 100 and 500 dimensional function are listed in supplementary document (Afşar et al., 2016). On the other hand, Figs. 9 and 10 show the boxplots representing the median error distributions of the 19 SOCO functions obtained with SaIABC and AIABC, and the competitor algorithms published in the special issue of Soft Computing journal and the algorithms provided as base-reference techniques, CHC (Eshelman and Schaffer, 1993), GCMAES (Auger and Hansen, 2005) and DE (Storn and Price, 1997) for dimensions 100 and 500. As seen in Figs. 9 and 10, it is noteworthy that the proposed algorithms perform competitively to state-of-the-art algorithms. If one considers the comparison results 50 and 100 SOCO functions listed in supplementary document (Afşar et al., 2016), median errors of SaIABC and AIABC are below the optimal threshold 13 or 14 times of the 19 SOCO functions. Only MOS-DE matches such a performance. For 500 dimensional functions, SaIABC and AIABC algorithms’ performances decrease. However, the average performance of the algorithms is very similar to SOCO competitors except MOS-DE. Finally, we conducted Friedman test on the results of compared algorithm for 50, 100 and 500 dimensional functions. Friedman test confirms significant improvements of proposed algorithms over CHC, GCMAES and EVOPROpt for all cases. Moreover, no other considered algorithms are better than the proposed algorithms according to statistical results for 50 and 100 dimensional functions. Only MOS-DE which is the winner algorithm of the SOCO competition is significantly better than SaIAB and AIABC for 500 dimensional functions.