An improved artificial bee colony algorithm and its ...

Downloaded from http://iranpaper.ir

http://www.itrans24.com/landing1.html

Accepted Manuscript Title: An improved artificial bee colony algorithm and its application to reliability optimization problems Authors: Soheila Ghambari, Amin Rahati PII: DOI: Reference:

S1568-4946(17)30654-3 https://doi.org/10.1016/j.asoc.2017.10.040 ASOC 4535

To appear in:

Applied Soft Computing

Received date: Revised date: Accepted date:

8-1-2017 15-10-2017 24-10-2017

Please cite this article as: Soheila Ghambari, Amin Rahati, An improved artificial bee colony algorithm and its application to reliability optimization problems, Applied Soft Computing Journal https://doi.org/10.1016/j.asoc.2017.10.040 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.


1 2

An improved artificial bee colony algorithm and its application to reliability optimization problems

3 4 5 6 7


a Department

Soheila Ghambaria, Amin Rahatia,* of Computer Sciences, University of Sistan and Baluchestan, Zahedan, Iran1.

19

Abstract. Artificial bee colony (ABC) algorithm is a well-established swarm optimization technique that has been successfully applied for solving different kinds of optimization problems. In spite of its efficiency and wide use, ABC still suffers from slow convergence speed. To overcome this insufficiency, an improved version of ABC algorithm called IABC has been proposed in this paper. First, the proposed IABC incorporates a probabilistic population size reduction mechanism in order to accelerate the convergence speed. This mechanism transfers high quality solutions to the next cycle of the algorithm and discards the rest. Second, in addition to the original search operator of ABC, the IABC utilizes a new search operator which enhances the exploitation capability. This new search operator generates a new solution based on a randomly selected pair of solutions and the current best solution. Third, to better balance the trade-off between exploration and exploitation, the IABC unifies the employed and onlooker bee phases into an improved bee phase by using a self-adaptive probabilistic selection scheme. This helps the IABC to decide either to apply the original or the new search operator to produce a new solution. The performance of IABC is evaluated against CEC2014 test suite and eight well-known reliability optimization problems. Numerical experiments indicate that the IABC provides competitive results compared to several state-of-the-art algorithms in terms of convergence speed, robustness, and solution accuracy. Moreover, the IABC considerably improves the best-known solution for one reliability optimization problem.

20 21 22

Keywords: Artificial bee colony algorithm, Population size reduction mechanism, CEC2014 test suite, Reliability optimization problems.

23

Graphical abstract

8 9 10 11 12 13 14 15 16 17 18

24

* Corresponding author. Tel.: +98 (54)31136275; fax: +98 (54)33431070. Email addresses: [email protected], [email protected]



25 26 27 28 29 30 31 32 33

1. Introduction A large number of optimization problems exist in many engineering application domains and scientific fields which have a difficult and non-linear nature. It is well-known that solving these problems through classical mathematical approaches is often inefficient and requires strong math assumptions which are rarely met in some practical problems. Due to limitations of classical methods, the application of nature-inspired algorithms has gained increasing interest among researchers in the last two decades since they are easily applicable and have a flexible structure. Some examples of such algorithms are genetic algorithm (GA) [1], particle swarm optimization (PSO) [2], differential evolution (DE) [3], ant colony optimization (ACO) [4], and artificial bee colony algorithm (ABC) [5]. Among them, ABC is a relatively new swarm intelligence based meta-heuristic algorithm proposed by Karaboga in


34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

2005. This algorithm mimics the collective behavior of honey bees to locate food sources in the nature. ABC has been used in a wide range of real-world applications because of its robustness, effectiveness, and ease of implementation [6-19]. Also, recent studies have shown that the performance of ABC algorithm is competitive with other well-known algorithms such as GA, PSO, ACO, and DE in various numerical optimization problem [20, 21]. However, ABC still suffers from slow convergence and easily gets trapped in a local optimum when dealing with some highly complex problem [8, 22, 23]. In fact, according to the “no free lunch” theorem, there is not a specific optimization algorithm that achieves the best performance for all problems and ABC is not an exception [24]. The convergence speed of ABC, like other meta-heuristic algorithms, depends on a good trade-off between exploration and exploitation search behaviors. Exploration means the capability of an algorithm for seeking the unknown areas of search space to discover the optimal solution, whereas exploitation is the process of finding the optimum solution by taking advantage of previous good solutions in its neighborhood. These two capabilities are affected by several factors. An example of such factors is population size [25-27]. A large population gives the algorithm a higher chance to explore the search space but it causes the problem of slow convergence speed. On the contrary, a small population helps ABC to converge quickly toward good solutions but it may get trapped in a local optimum. The search operator of ABC is another influential factor. More precisely, this search operator produces a new candidate solution by a random change of only one parameter of a parent solution. Thus, the location of the new solution in the search space will be in the vicinity of the parent solution with a small perturbation. In addition, this search operator does not take into account the information of the best solution or other solutions that may have a positive effect on guiding ABC toward more promising areas of search space. Hence, the search operator of ABC favors exploration over exploitation which consequently results in a slow convergence speed. On the other hand, obtaining a good performance in solving optimization problems requires a proper balance between exploration and exploitation search behaviors. Thus, further studies are necessary for investigating schemes for ABC which will compensate the imbalance between these two capabilities. For this reason, various approaches are continuously being developed in the literature. Based on the above considerations, this study proposes an improved variant of ABC algorithm (IABC) which employs new search mechanisms. In brief, the main contributions of this research can be summarized as follows: 

59 60 61



62 63 64 65



66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83


First, a probabilistic population size reduction mechanism is introduced which transfers high quality solutions (food sources) to the next iteration and discards the rest. This mechanism efficiently guides the search process of IABC to focus on more promising search directions and consequently accelerates the convergence speed. Second, the proposed IABC employs a new search operator in addition to the original one used in ABC. The new search operator utilizes two random candidate solutions and the current best solution to generate a new candidate solution. The principle behind designing this search rule is improving the exploitation capability of IABC algorithm by bringing more local information into the new produced candidate solution. Third, an improved bee phase is constructed by unifying the employed and onlooker bee phases. This new phase uses a self-adaptive probabilistic selection scheme to decide either to apply the original or the new search operator for producing a new candidate solution. In this way, the balance between the exploration and exploitation capabilities can be properly managed. For this purpose, more chance is given to the original search operator to be selected in the early stages of a search process to enhance the exploration capability of IABC. Then, the probability of selecting the new search operator increases as the search process continues and thus the exploitation capability of IABC is improved as it reaches the final stages.

To validate the general properties of IABC, it is evaluated against CEC2014 test suite in the first step. Then, its performance is compared with four groups of algorithms that contain the standard ABC, several new variants of ABC, a few well-known state-ofthe-art algorithms, and some algorithms introduced in a special session at CEC2014. The experimental results demonstrate that the IABC algorithm can compete very well with the considered algorithms in terms of convergence speed, robustness, and solution accuracy. To further verify the application and efficiency of IABC algorithm, reliability optimization problems are used in this research because of their critical importance in almost all engineering (such as medical engineering) and industrial applications including military, nuclear, aerospace, and automotive industries. According to the obtained results, the proposed IABC is as effective in solving most of these problems as the algorithms in the literature and demonstrates a very favorable performance. Moreover, the efficiency of IABC is compared with ABC and its two recent variants. The results illustrate that the IABC is considerably more efficient than the compared algorithms in terms of computational time and accuracy.

86

The remainder of this paper is organized as follows. Section 2 starts by describing the standard ABC algorithm in Subsection 2.1. Next, in Subsection 2.2, a literature review about ABC improvements is presented. In Section 3, the proposed IABC algorithm is elaborated. The experimental setup and results are discussed in Section 4. Finally, the conclusions are summarized in Section 5.

87

2.

84 85

Review of ABC algorithm



89

This section first presents the optimization principle of ABC algorithm and then provides a brief overview of previous works on ABC improvements.

90

2.1. The standard ABC algorithm

91

Karaboga introduced ABC algorithm as a swarm intelligence based meta-heuristic and stochastic optimization algorithm [5]. This algorithm is inspired from the behavior of honey bees of a colony that seek food sources in the nature. The artificial bees of the colony are categorized into three main groups: employed bees, onlooker bees, and scout bees. The number of employed and onlooker bees are equal to the number of food sources and each bee is associated with each food source. The employed bees search for the food sources in their memory. Then they share their information with the onlooker bees. Onlooker bees wait in the hive and decide which food source to select. As such, more beneficial food sources have a higher probability to be selected. The scout bees are transformed from a few employed bees which abandon their food sources and search for new ones. Similar to other optimization algorithms, ABC is an iterative procedure. The main parts of the ABC algorithm are as follows:

88

92 93 94 95 96 97 98 99

-

100 101

Initialization: ABC algorithm includes a population of food sources or candidate solutions. Each food source consists of a multi-dimensional parameter vector. At first, the ABC starts the search process with a randomly distributed initial population. These initial solutions are generated by the following equation:

102

=

103

+

×(

−

)

(1)

104

where i and j are selected in [1, SN] and [1, D], respectively. SN is the number of food sources, D is the number of dimensions, rand is a uniform random number in [0, 1], and are the lower bound and upper bound, respectively.

105 106 107 108 109 110

-

111

Employed bee phase: at this stage, each employed bee is associated with only one specific food source ( ) and then produces a new food source ( ) through changing the food source ( ) in its memory based on a neighboring search. Thus, the new food source ( ) is defined as follows:

112 113

=

114

+

×(

−

)

(2)

115

where (or ) denotes the jth element of (or ), j is a random index, represents the jth element of another food source ( ) selected randomly from the population, and is a uniform random number in [-1,1]. In this phase, both food sources ( ) and ( ) are compared against each other and then the employed bee exploits the better one. Thereafter, employed bees will return to their hive and share the information on new food sources with onlooker bees through their waggle dances.

116 117 118 119 120 121 122 123

-

Onlooker bee phase: onlooker bees evaluate the nectar information received from all employed bees and choose a food source with a probability proportionate to its nectar amount. The value of probabilistic selection is calculated for the ith food source as follows:

124 125

= ∑

126

=

= 1,2, … , ≥ 0 1+

( )

limit then 25: Replace with a new randomly generated food source using Eq. (1) 26: End 27: Store the best food source found so far 28: iteration = iteration + 1 29: Until (iteration = maximum iteration) 145

Fig. 1. The pseudo code of ABC algorithm.

146 147

2.2. ABC improvements

148

So far, many studies have been done to improve the performance of ABC in various domains by different techniques. A good literature survey on the early works can be found in [20, 21]. Recent studies can be roughly classified into three categories as will be explained briefly below.

149 150


151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206


In the first category of studies on ABC, some methods for managing population size and diversity are employed. For example, with respect to the effect of population size on the performance of ABC, Ma et al. [28] introduced a new life-cycle in standard ABC such that each individual reproduces or dies dynamically throughout the searching process. Cui et al. [29] proposed an adaptive tuning method for the population size in ABC which is inspired by a natural principle that states the size of population is related to the availability of food sources. In their proposed approach, if the exploration behavior of the algorithm is good then the population size is shrunk by removing some unpromising solutions to enhance the exploitation capability; otherwise, it is enlarged by randomly adding some of the removed solutions in the previous iterations to increase the diversity. To mention a study about initial swarm distribution, Gao et al. [30, 31] used both chaotic system and opposition-based learning approaches to initialize the population. These techniques help the algorithm to enhance the convergence performance. Furthermore, researchers in [32, 33] suggested multi-population strategies to improve population diversity by dividing candidate solutions into several sub-populations and distributing them throughout the search landscape. Moreover, Nseef et al. [34] presented an adaptive multi-population strategy and a clearing scheme in a standard ABC framework to keep diversity in the population so that the number of sub-populations is modified over time. The second category of studies on the improvement of ABC introduces new search operators or procedures. For example, Sharma et al. [35] proposed a mean mutation operator as a new search operator in standard ABC which employs a linear combination of Gaussian and Cauchy distributions in order to enhance the convergence speed. Luo et al. [36] presented COABC algorithm in which a roulette wheel selection method is used to place all onlooker bees around the current global best food source. Moreover, the procedure of ABC is modified such that the employed bee phase is repeated several times in each iteration to improve the solution quality and speed up the convergence rate. In another study, Sharma et al. [37] introduced two versions of ABC called Intermediate ABC (I-ABC) and I-ABC greedy with the aim of producing potential solutions in the population. The I-ABC variant generates better solution locations in the initial step by utilizing opposition based learning approach and uniformly distributed random numbers. In I-ABC greedy, which is a variation of I-ABC, a greedy approach is applied to the search mechanism where the search is forced to move towards the direction of the best solution found so far in order to speed up the convergence. In order to make a better balance between the exploration and exploitation capabilities, Gao et al. [38] developed an enhanced ABC (EABC) which utilizes two new search rules to generate candidate solutions in employed and onlooker bee phases. More precisely, these two new search rules are designed based on some similarity with the DE/target-to-best/1 scheme but with different structures. To improve the exploitation ability of ABC, Karaboga et al. [39] introduced a quick ABC (qABC) which employs a new search equation in onlooker bee phase based on the information of best food source among the neighbors. Similarly, Kiran et al. [40] proposed a directed ABC algorithm (dABC) in which some directional information is utilized in the search operator to narrow the search space for obtaining better solutions. In another research, Kiran et al. [41] introduced ABCVSS algorithm in which an integration of multiple search operators is used in employed and onlooker bee phases. Yurtkuran et al. [42] presented an adaptive ABC (AABC) for global optimization which employs six unique search operators with different information sharing approaches to regulate the exploration-exploitation behavior. Cui et al. [22] presented an algorithm called DFSABC-elite which introduces a depth-first search framework with two novel search equations that are used in employed and onlooker bee phases. These new equations utilize the valuable information of the elite solutions in both phases. In another study, Loubiere et al. [43] attempted to improve the dimension selection process that is involved in neighborhood search of ABC algorithm using a sensitivity analysis approach. For this purpose, they applied Morris method [44] to find the dimensions which have a high impact on the objective function. Recently, Li et al. [45] recommended a novel gene recombination operator into ABC to speed up the convergence. The gene recombination operator generates new candidate solutions by using a part of potential solutions in the current population. Song et al. [23] improved ABC algorithm by utilizing two new solution search equations in employed and onlooker bee phases, respectively. The new search equation of employed bee phase uses the mid-point of two randomly selected solutions, the information of their objective function values, and the current best solution to improve the exploitation of algorithm on the basis of enhancing its exploration. The new solution search equation of onlooker bee phase is based on a modified trigonometric mutation operation that helps the algorithm to enhance the local exploitation capability. Liang et al. [46] proposed ABCADE algorithm that applies adaptive differential operators in employed bee phase to improve its convergence performance. Moreover, ABCADE uses a stair-step probability calculation approach in onlooker bee phase to exploit the promising areas of search space. Xue et al. [47] presented an SABC-GB algorithm that incorporates several candidate solution generating strategies with diverse characteristics. These strategies are adaptively applied during the different stages of search process to tune the balance between the convergence speed and the robustness of algorithm. Cui et al. [48] introduced a ranking-based search strategy into ABC to exploit the useful information of the food sources. To this end, they designed a new search equation that probabilistically selects parent food sources based on their ranking in the population where rankings for each food source is assigned according to its fitness. Furthermore, food sources are attracted by bees for exploitation based on their rankings. In another study, Zhong et al. [49] employed a modifiedneighborhood-based update operator in employed bee phase. This operator utilizes global-best and subset-best guided terms to balance between search capabilities of the algorithm. In order to increase the solution diversity, a subset partition approach is applied to generate perturbation term. In addition, an independent-inheriting-search strategy is employed in onlooker bee phase to enhance the exploitation capability of the algorithm.


207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222


Hybrid ABC algorithms belong to the third category. These algorithms are combinations of standard ABC with either some traditional methods or other evolutionary optimization algorithms. For example, Kang et al. [50] introduced a hybrid simplex ABC algorithm which combines Nelder-Mead Simplex method with ABC to improve the efficiency of the search process. Duan et al. [51] suggested a hybrid version of ABC with a quantum evolutionary algorithm. A novel hybrid swarm based on PSO and ABC was introduced by Shi et al. [52]. In another study, Kang et al. [53] put forward a hybrid Hooke Jeeves ABC algorithm with an intensified search process based on Hooke Jeeves pattern search. Also, a hybrid approach involving GA and ABC was recommended by Jatoth and Rajasekhar [54]. Ozturk and Karaboga [55] presented a hybrid ABC algorithm in which ABC is integrated with Levenberq-Marquardt for training an artificial neural network. Zhong et al. [56] described a hybrid ABC algorithm where the chemotaxis behavior of bacterial foraging optimization algorithm is incorporated into the employed and onlooker bee phases. Fister et al. [57] hybridized ABC with memetic search. Yildiz [58] presented a new hybrid ABC based on Taguchi method to solve manufacturing optimization problems. A different hybrid ABC was developed by Chen and Xiao [59] by employing an artificial immune network algorithm in ABC framework. In recent years, Tuba et al. [60] integrated ABC with firefly algorithm, Xiang et al. [61] combined a modified ABC with a modified version of DE algorithm, Li et al. [62] hybridized ABC algorithm with tabu search, Sharma et al. [63] developed a combination of ABC algorithm with shuffled frog-leaping algorithm, references [64, 65] mixed ABC with differential evolution algorithm, Shokouhifar et al. [66] introduced a hybrid of ABC algorithm with simulated annealing, and Goudarzi et al. [67] established an algorithm by integrating ABC with PSO algorithm.

226

Altogether, the aforementioned studies in the literature aimed to improve the performance of standard ABC algorithm in terms of convergence speed and global-local search capabilities. In this regard, it is worthwhile to point out that this study presents an effective algorithm which tries to improve the optimization potential of ABC algorithm for solving both numerical and practical real-world optimization problems.

227

3.

228

233

The proposed IABC extends the standard ABC algorithm by adding three features: (1) a probabilistic population size reduction mechanism in which good solutions survive in each iteration of the algorithm and are transferred to the next generation, (2) a new search operator that utilizes the information of a randomly selected pair of candidate solutions and the current best solution to increase the exploitation search behavior of ABC, and (3) an improved bee phase that unifies the employed and onlooker bee phases of ABC and also incorporates a self-adaptive probabilistic selection scheme to achieve a good balance between exploration and exploitation search behaviors. The following subsections elaborate these modifications.

234

3.1. Population size reduction startegy

223 224 225

229 230 231 232

The proposed approach (IABC algorithm)

235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253

The standard ABC algorithm utilizes a static population size during the whole search process. Thus, to accelerate the convergence speed of ABC, the only choice is to use a small population size. As a result, the diversity of population becomes low and the danger of falling into a local optimum becomes high. A solution for this issue is to incorporate a population size reduction strategy into the standard ABC algorithm. This strategy gives the algorithm a highly explorative search behavior at early stages of the search process by employing a large enough population size. Then, the population size is reduced progressively as the search process proceeds so that the algorithm exploits the promising search directions found in the previous stages. Bearing the above considerations in mind, this study introduces a new population size reduction strategy based on two concepts: a survival strategy and a survival rate (SR). The survival strategy determines which candidate solutions from the current population (or generation) survive and pass to the next population. To this end, an elitism-based survival strategy is employed that transfers solutions with higher fitness values to the next population and discards the rest. Then, it is assumed that the SR is a constant value within the range of [0, 1] which determines the rate of population size reduction. Obviously, a large value for the parameter leads to a mechanism that highly favors smaller population sizes and consequently a premature convergence problem occurs. On the other hand, a small value for this parameter neutralizes the positive effect of population size reduction on searching for the optimal solution. In previous studies [68-71], the size of population linearly decreases along the generations. However, finding optimal solutions for various optimization problems requires different exploration-exploitation behaviors during the search process. Thus, the proposed population size reduction strategy is in fact a probabilistic linear reduction population method that creates a more sophisticated search behavior in the ABC algorithm. The proposed strategy is expressed by Eq. (5) and can be thought as a mutation strategy that reduces the size of population with a probability equal to SR:

254 255 256 257

=

((

– ), ) ℎ

>

(5)



260

shows the current population size, is the smallest possible value for reducing the size of population, is a population threshold that determines the smallest possible population size after reduction, and finally ∈ [0,1] is a randomly produced number.

261

3.2. New search operator

258 259

where

262 263 264 265 266 267 268

According to Eq. (2), the standard ABC algorithm produces a new food source by moving the current one towards another food source which is randomly selected from the population. As stated before, this solution search equation gives ABC a good capability to explore the search space but is not helpful enough for ABC to exploit the information it has gathered so far during search. Therefore, to improve the exploitation behavior of ABC, a new search operator is introduced which makes use of more information from the food sources of the current population by selecting a random pair of them and also the current best food source. The following formula indicates how our new search operator computes a new food source:

269 270

=

+

.∗ (

,

−

+

−

,

,

)

(6)

271 272 273 274 275 276 277

th where (or ) represents the jth element of (or ) and j is an index that takes its value from [1, D], , refers to the j th element of a food source with the best fitness in the current population, , and , are the j elements of two randomly selected food sources where r1 and r2 are mutually exclusive integers randomly chosen from [1, SN] and r1 ≠ r2 ≠ i. Finally, sf is a random number with uniform distribution in [0, 1]. It is important to note that the IABC does not replace the ABC search operator (Eq. (2)) by the new search operator (Eq. (6)). In fact, one of these two search operators is selected in each iteration of IABC algorithm via a self-adaptive probabilistic selection scheme that is elaborated in the next subsection.

278 279

3.3. Improved bee phase and a self-adaptive probabilistic selection scheme

280 281 282 283 284 285 286 287

As explained before, a well-designed optimization algorithm properly balances the exploration and exploitation capabilities. Previous studies have shown that self-adaptive strategies have a great potential in providing such a balance [72, 73]. Therefore, this study introduces an improved bee phase by unifying the employed and onlooker bee phases of ABC. The improved bee phase employs a self-adaptive probabilistic selection scheme that utilizes the parameter to decide probabilistically whether to apply the new search equation (Eq. (6)) or the original search equation of ABC (Eq. (2)) to generate a new solution. In this way, a proper balance between the exploration and exploitation behaviors of IABC algorithm can be made. Eq. (7) shows a formula which calculates the parameter:

288

=

289

. exp(

(

)

.

)

(7)

290 291 292 293 294 295 296 297 298

Here, and are the maximum and minimum values of the parameter, respectively. Also, MaxIter is the maximum number of iterations and iter denotes the current number of iterations. At the early iterations, the calculated is a large value so it is more likely that the original ABC search equation is chosen which enhances the diversity of the solutions in the population. Then, as the optimization process proceeds, this value becomes smaller and the chance of selecting the new search equation increases. However, at the same time, the size of the population has been reduced through the probabilistic population size reduction mechanism and thus only the solutions with good information remain at the end. Therefore, at the final stages of the search process, the two random solutions in Eq. (6) are in fact two good solutions that can improve the exploitation capability of IABC. As a result, the search balance changes in favor of exploitation behavior.

299 300

3.4. The procedure of IABC algorithm

301 302 303 304 305 306

Fig. 2 shows the overall procedure of IABC. The key steps in the proposed algorithm are as follows: after setting the parameters, the algorithm is started by initializing a population of food sources using a random generation method (line 2). Next, the fitness of these generated food sources is evaluated (line 3). Thereafter, trial is set to zero for each food source (line 4). Then, the first iteration of IABC is started by running the probabilistic population size reduction strategy (line 7-10) and after that the value of SSP parameter is updated (line 12). Next, a selection scheme uses this value for choosing a search operator to generate a new food


307 308 309 310 311


source (line 14-18). The next steps are similar to those of the original ABC. In other words, a greedy selection process is applied to select the better food source by comparing the current food source with the newly generated food source (line 19). Also, the food sources that cannot be improved for a number of iterations (determined by the limit parameter) are abandoned and bees associated with these food sources become scout bees (line 22-25). Finally, the best food source is memorized at the end of each iteration (line 26). The execution of the main loop continues until a stopping criterion is met (line 28).

312 313 314 315 316 317 318 319 320 321 322 323

Algorithm 2. IABC ALGORITHM 1: Set Parameters: maximum iteration, SN, limit, SR, , , NPmin, NPThreshold 2: Initialize the population of food sources (solutions) using Eq. (1) where i=1,…, SN 3: Evaluate the population using Eq. (4) 4: ℎ , =0 end, where i=1,…, SN 5: Set iteration = 1 /* iteration counts IABC iteration*/ 6: repeat 7: /* Apply population size reduction strategy according to Eq. (5)*/ 8: if SR >random then 9: SN = max((SN - NPmin) , NPThreshold ) 10: end 11: /* Unified employed-onlooker bee phases equipped with a self-adaptive exploration-exploitation scheme*/ 12: Compute SSP parameter using Eq. (7) 13: For i = 1 to SN do 14: if random < SSP 15: Generate a new food source vi using Eq. (2) and evaluate its quality 16: else 17: Generate a new food source vi using Eq. (6) and evaluate its quality 18: end 19: Apply greedy selection process between vi & xi to select the better one 20: If solution did not improve then = + 1 otherwise =0 21: end 22: /* Scout bee phase*/ 23: if max (triali) > limit then 24: Replace food source with a new randomly produced one using Eq. (1) 25: End 26: Memorize the best food source achieved so far 27: iteration = iteration +1 28: Until (iteration = maximum iteration) 324 325

Fig. 2. The pseudo code of IABC algorithm.

326

4.

327

This section aims to investigate the efficiency and effectiveness of the recommended IABC algorithm through a series of experiments on numerical and reliability optimization problems. For this purpose, Subsection 4.1 starts with a description of the CEC2014 test suite. Next, the variations of parameter values introduced in the population size reduction mechanism are analyzed. Then, the effects of the selected random solutions in the newly introduced search equation and also the improved bee phase on the proposed IAB are studied. Thereafter, the compared algorithms, experimental setup, and statistical results obtained via experiments on CEC2014 are presented. In Subsection 4.2, the application of IABC on eight well-known classical problems of reliability

328 329 330 331 332

Experiments and discussions



334

optimization is investigated. The IABC algorithm is implemented via MATLAB2013a environment using windows 7 operating system. All experiments were performed on a computer with a 2.53GHz Intel(R) Core (TM) i3 processor, and 2GB of RAM.

335

4.1. Experiments on numerical optimization problems

333

336 337 338 339 340 341

4.1.1. Benchmark problems of CEC2014 This test suite contains 30 numerical minimization test functions that are divided into the following four groups: (1) 3 rotated unimodal functions (f1-f3), (2) 13 shifted and rotated multimodal functions (f4-f16), (3) 6 hybrid functions (f17-f22), and (4) 8 composition functions (f23-f30). These functions and their characteristics are summarized in Table 1. More details about these functions can be found in [74].

342 343 344 345 346 347 348 349 350

Table 1. Benchmark functions of CEC2014 test suite (U: Unimodal, M: Multimodal, NS: Non-separable, R: Rotated, A: Asymmetrical). Functions

No.

Unimodal

f1 f2 f3

Multimodal

Name Rotated High Conditioned Elliptic Function

Properties U, NS, Quadratic ill-conditioned

Optimal Value 100

Rotated Bent Cigar Function

U, NS, Smooth but narrow ridge

200

Rotated Discus Function

U, NS, With one sensitive direction

300

f4

Shifted and Rotated Rosenbrock’s Function

400

f5 f6 f7 f8 f9 f10

Shifted and Rotated Ackley’s Function

U, NS, Having a very narrow valley from local optimum to global optimum M, NS

Shifted and Rotated Weierstrass Function

M, NS, Continuous but differentiable only on a set of points

600

Shifted and Rotated Griewank’s Function

M, NS, R

700

Shifted Rastrigin’s Function

M, S, Local optima’s number is huge

800

Shifted and Rotated Rastrigin’s Function

M, NS, Local optima’s number is huge

900

Shifted Schwefel’s Function

1000

f11

Shifted and Rotated Schwefel’s Function

f12 f13 f14 f15

Shifted and Rotated Katsuura Function

M, NS, Local optima’s number is huge and the second best local optimum is far from the global optimum. M, NS, Local optima’s number is huge and the second best local optimum is far from the global optimum. M, NS, Continuous everywhere yet differentiable nowhere

Shifted and Rotated HappyCat Function

M, NS

1300

Shifted and Rotated HGBat Function

M, NS

1400

Shifted and Rotated Expanded Griewank’s plus Rosenbrock’s Function Shifted and Rotated Expanded Scaffer’s F6 Function

M, NS

1500

M, NS

1600

f17 f18 f19 f20 f21 f22

Hybrid Function 1 (N=3)

M or U, NS, Different properties for different variables subcomponents

1700



1800



1900



2000



2100



2200

f23 f24 f25 f26 f27 f28 f29

Composition Function 1 (N=5)

M, NS, A, Different properties around different local optima

2300


M, NS, Different properties around different local optima

2400



2500



2600



2700



2800


M, NS, A, Different properties around different local optima, Different properties for different variables subcomponents

2900

f16

Hybrid

Composition

500

1100 1200


f30



M, NS, A, Different properties around different local optima, Different properties for different variables subcomponents

3000

351 352

4.1.2.

Analysis of the variations of SR,

, and

353 354 355 356 357 358 359 360 361 362 363 364 365

Subsection 3.1 introduced the population size reduction mechanism as a contribution of this study for improving the convergence speed of IABC. This mechanism uses the following three parameters: SR, , and . Needless to say that the performance of IABC depends on how well these parameters are tuned. Thus, the effect of each configuration of these parameters is manually investigated by 51 independent runs of IABC on 10-dimensional problems of CEC2014 test suite. The sets of values for SR, , and are {0.995, 0.7475, 0.5, 0.2525, 0.005}, {5, 10, 15}, and {15, 20, 25}, respectively. The population size, limit, and the maximum number of function evaluations are set to 100, 200, and 1×105, respectively. Fig. 3 exhibits the total number of times that the IABC has obtained the smallest mean function error value for the benchmark problems using each configuration for the three parameters compared to other configurations. As Fig. 3 shows, regardless of the values for and , adopting the value 0.005 for the SR parameter has the biggest impact on the performance of IABC. Thereafter, the best values for and are set to 5 and 20, respectively. However, as can be seen, selecting = 10 and = 15 or = 10 and = 20 produces two other configurations that have a very close competition with the best one in improving the performance of the IABC. 18

Number of best performance

16 NP-threshold=15, NP-min=5

14

NP-threshold=15, NP-min=10


10


8


6

NP-threshold=20, NP-min=15 NP-threshold=25, NP-min=5

4



0 SR=0.995

SR=0.7475

SR=0.5

SR=0.2525

Fig. 3. Analysis of the variations of SR,

SR=0.005

, and

.

366

367

4.1.3.

368 369 370 371

As was mentioned before, the search equation proposed in this paper utilizes two random solutions and the current best solution. Thus, to investigate whether these two random solutions have a positive effect on the performance of IABC, a modified version of IABC (MIABC) is considered by eliminating them from Eq. (7) as follows:

372 373 374 375 376 377 378 379 380 381 382 383

The effect of random solutions in the new search equation

=

+

.∗ (

,

−

)

An experimental comparison among ABC, IABC, and MIABC is performed on CEC2014 test suite for 10-dimensional and 30dimensional benchmarks. Parameter configurations for this experiment are similar to the settings explained in the previous subsection. Table 2 reports the mean and standard deviation of function errors obtained for the benchmarks. The last row of this table summarizes the number of best performances achieved by the algorithms. According to the obtained results, the IABC outperforms ABC and MIABC in unimodal functions (f1- f3), multimodal functions (f6, f8, f9, f11, f13, f14), hybrid functions (f17f22), and composition functions (f24, f25, f27, f30). Also, Fig. 4 displays the convergence behavior of the algorithms for several challenging 10-dimensional and 30-dimensional functions (f1, f6, f9, f11, f17, f21, f24, f27 and f30). It is obvious that the IABC performs well and converges faster than the other two algorithms. This performance is achieved by all the three modifications. More precisely, as the search process proceeds, the population size reduction mechanism reduces the size of the population by transferring the elite solutions to the next population and discarding the rest. Thus, at the final stages of the search process, the



384 385 386 387

surviving solutions are in fact the good solutions found during search. On the other hand, the new search equation (Eq. (6)) has more chance to be selected in the final stages. Therefore, these two random solutions are more often two good solutions that increase the exploitation capability and decrease the chance of getting trapped into the current (global) best solution.

388

Table 2. The mean and standard deviation results for considering the impact of the new search rule. D=10 No. f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 f20 f21 f22 f23 f24 f25 f26 f27 f28 f29 f30

D=30

Algorithms ABC 9.24e+04±6.14e+04 7.00e+01±7.54e+01 1.79e+02±1.54e+02 1.20e-01±1.46e-01 1.80e+01±4.89e+00 2.37e+00±6.81e-01 1.37e-02±9.12e-03 1.29e-13±6.82e-14 7.56e+00±1.63e+00 2.39e-01±4.73e-01 8.38e+02±8.93e+01 2.15e-01±3.69e-02 1.19e-01±2.19e-02 1.59e-01±3.00e-02 9.03e-01±2.31e-01 2.28e+00±2.34e-01 1.53e+05±1.38e+05 5.39e+02±4.07e+02 4.69e-01±2.41e-01 1.74e+02±2.10e+02 8.63e+03±8.83e+03 2.74e+00±2.59e+00 8.21e+01±9.12e+01 1.19e+02±1.12e+01 1.37e+02±7.51e+00 9.95e+01±4.86e+00 3.71e+01±5.08e+01 3.73e+02±1.32e+01 2.94e+02±4.01e+01 5.93e+02±9.58e+01

MIABC 3.33e+06±4.43e+06 1.59e+04±4.44e+04 1.99e+02±6.15e+02 3.94e+01±2.53e+01 1.73e+01±6.28e+00 2.01e+00±7.94e-01 4.28e-01±3.01e-01 1.13e-10±3.63e-10 7.07e+00±2.70e+00 1.08e-01±8.16e-02 9.62e+02±1.26e+02 1.12e-01±4.14e-02 1.37e-01±5.02e-02 1.69e-01±8.65e-02 1.07e+00±8.32e-01 1.53e+00±2.64e-01 8.60e+04±1.09e+05 7.16e+03±7.95e+03 9.81e-01±5.37e-01 2.30e+03±4.33e+03 4.36e+03±7.20e+03 2.63e+01±4.97e+01 3.21e+02±4.74e+01 1.18e+02±7.39e+00 1.65e+02±2.77e+01 1.00e+02±1.25e-02 2.86e+01±1.82e+02 4.09e+02±5.68e+01 2.92e+02±7.81e+01 9.22e+02±2.24e+02

IABC 1.45e+04±1.11e+04 6.98e+00±9.09e+00 4.96e+00±8.47e+00 6.37e+00±1.24e+01 1.92e+01±3.65e+00 2.98e-05±5.13e-02 9.08e-03±7.57e-03 0.00e+00±0.00e+00 4.66e+00±1.31e+00 1.40e-01±6.78e-02 7.01e+02±9.57e+01 2.34e-01±5.35e-02 7.94e-02±1.63e-02 5.62e-02±1.72e-02 9.12e-01±1.40e-01 1.70e+00±3.73e-01 5.14e+02±2.92e+02 5.36e+01±4.75e+01 3.25e-01±2.25e-01 1.32e+01±1.58e+01 1.03e+02±3.04e+01 7.49e-01±2.43e+00 3.22e+02±4.61e+01 1.10e+02±2.23e+00 1.15e+02±5.13e+00 1.00e+02±1.79e-02 1.21e+01±1.72e+02 3.79e+02±3.09e+01 2.71e+02±2.60e+01 4.80e+02±1.52e+01

ABC 8.74e+06±3.16e+06 1.21e+02±1.25e+02 5.40e+02±5.63e+02 3.92e+01±2.66e+01 2.02e+01±3.44e-02 1.45e+02±1.48e+00 7.05e-04±4.94e-04 9.24e-11±3.24e-10 8.32e+01±9.88e+00 3.03e+00±1.49e+00 2.17e+03±2.29e+02 3.11e-01±4.02e-02 2.41e-01±2.53e-02 1.95e-01±1.23e-02 5.41e+00±1.38e+00 1.03e+01±2.83e-01 2.53e+06±1.16e+06 3.79e+03±2.16e+03 7.89e+00±7.72e-01 5.20e+03±2.37e+03 3.26e+05±1.53e+05 2.81e+02±8.11e+01 3.15e+02±8.45e-02 2.26e+02±4.32e+00 2.08e+02±1.05e+00 1.00e+02±4.98e-02 5.18e+02±3.14e+00 9.32e+02±5.21e+01 1.12e+03±9.19e+01 4.18e+03±1.16e+03

MIABC 6.70e+07±2.67e+07 9.54e+07±1.31e+08 2.29e+03±2.76e+03 2.43e+02±4.26e+01 2.01e+01±2.95e-02 1.73e+02±2.81e+00 1.83e+00±8.94e-01 5.53e+00±8.42e+00 9.25e+01±1.76e+01 1.23e+00±9.61e-01 2.84e+03±2.54e+02 1.81e-01±4.52e-02 3.26e-01±7.08e-02 1.88e-01±3.58e-02 3.76e+02±3.77e+02 9.03e+00±2.76e-01 4.06e+06±2.32e+06 1.30e+03±1.30e+03 7.28e+01±2.91e+01 8.56e+03±4.55e+03 5.08e+05±3.65e+05 5.40e+02±1.90e+02 3.22e+02±4.19e+00 2.29e+02±3.40e+00 2.19e+02±3.53e+00 1.00e+02±1.62e-02 5.93e+02±1.82e+02 1.34e+03±3.10e+02 4.49e+04±2.06e+05 4.40e+04±2.55e+04

IABC 3.06e+06±1.63e+06 3.95e-09±1.87e-08 1.49e-04±6.58e-04 5.17e+01±2.98e+01 2.03e+01±3.95e-02 9.88e+01±1.14e+00 3.10e-03±6.28e-03 4.79e-13±2.43e-13 3.60e+01±5.49e+00 1.57e+00±9.98e-01 2.00e+03±2.77e+02 3.65e-01±5.09e-02 1.30e-01±2.25e-02 7.91e-02±1.88e-02 7.13e+00±6.56e-01 1.01e+01±3.42e-01 1.07e+05±9.74e+04 2.22e+02±2.61e+02 5.40e+00±9.64e-01 1.08e+02±7.06e+01 2.41e+04±4.08e+04 1.20e+02±7.09e+01 3.15e+02±2.42e-12 2.25e+02±2.21e+00 2.04e+02±6.25e-01 1.00e+02±1.93e-02 5.15e+02±4.35e+01 8.48e+02±7.79e+01 5.36e+05±2.16e+06 3.36e+03±9.44e+02

6

3

21

4

5

21

Total 389 390

30D-F1 23

log(F(x)-F(X*))

30D-F6

6.47 ABC MIABC IABC

22

6.46

21

6.45

20

6.44

19

6.43

18

6.42

17

6.41

16

6.4

15

6.39

30D-F9

6.5 ABC MIABC IABC

ABC MIABC IABC

6

5.5

5

4.5

4

14

0

0.5

1

1.5 2 Function Evaluations

2.5

3 5

x 10

6.38

3.5 0

0.5

1


2.5

3 10 5

0

0.5

1


2.5

3 10 5



10D-F17

18

ABC MIABC IABC

16

log(F(x)-F(X*))

log(F(x)-F(X*))

14

12

10

8

6 0

1

2

3

4 5 6 Function Evaluations

7

8

9

10 10 4

10D-F27

7

ABC MIABC IABC

6

log(F(x)-F(X*))

5

4

3

2

1

0 0

1

2

3


7

8

9

10 10 4

30D-F27

8.5

ABC MIABC IABC

8.45 8.4

log(F(x)-F(X*))

8.35 8.3 8.25 8.2 8.15 8.1 8.05 8 0

0.5

1

1.5 Function Evaluations

2

2.5

3 10 5

Fig. 4. Convergence curve diagrams for the logarithmic scale of mean error values. 391

392

4.1.4.

The effect of the improved bee phase on the exploration-exploitation behavior

393 394 395 396 397 398 399 400 401 402

As illustrated in Subsection 3.3, the improved bee phase uses Eq. (7) to implement the self-adaptive selective probability scheme. This equation calculates the value of SSP parameter. Then, based on the SSP value, the improved bee phase probabilistically decides to apply either Eq. (2) or Eq. (6) to produce a new solution. Fig. 5 shows the curve of SSP variations during the search process. This figure shows that the SSP has a large value at the early iterations. Therefore, the original ABC search equation (Eq. (2)) has more chance to be chosen. As a result, the new solution generated by Eq. (2) enhances the diversity of the population. However, as the search process proceeds, the SSP value decreases and the selection chance of the new search equation (Eq. (6)) increases. Consequently, the improved bee phase favors exploitation over exploration. Fig. 6 displays the mean rate of selection for equations (2) and (6) during the search process. It clearly confirms that the improved bee phase chooses Eq. (2) and Eq. (6) with a higher frequency at the earlier and final stages, respectively. Thus, the proposed IABC algorithm can achieve a good trade-off between exploration and exploitation capabilities.



403

0.9 0.8

SSP curve

0.7 0.6 0.5 0.4 0.3 0.2 0.1

0

2

4

6

8

10

4 Function Evaluations x 10 Fig. 5. The variation curve for SSP parameter during the search process.

Mean rate of selecting search equations

2500

Search Eq. (2) Search Eq. (6)

2000

1500

1000

500

0

1e+04 2e+04 3e+04 4e+04 5e+04 6e+04 7e+04 8e+04 9e+04 1e+05

Function Evaluations Fig. 6. Mean rate of selecting search equations during the search process.

404

4.1.5.

Compared algorithms

405 406 407 408 409 410 411 412 413 414 415 416

The performance of IABC is compared with those of four groups of algorithms. The first group contains the standard ABC and two variants of ABC which are called COABC [36] and AABC [42]. Numerical results for these algorithms are obtained via an experiment in which 51 independent runs are conducted for each algorithm on 10-dimensional (10D) and 30-dimensional (30D) test functions. The second group of algorithms includes some other improved versions of ABC whose results for 25 independent runs on 10D test functions are available from the literature. These algorithms are known as qABC [39], EABC [38], ABCVSS [41], dABC [40], and DFSABC_elite [22]. The third group includes four well-known standard optimization algorithms, i.e. GA [75], PSO [2], DE [3], and CS [76]. GA and PSO were selected because they are extensively used in many optimization problems and have had a successful performance in many applications for decades. Also, DE and CS algorithms are studied because they are very popular swarm intelligence algorithms and are relatively new. The results of the third group of algorithms are obtained through an experiment in which 51 independent runs of each algorithm are performed on 30D test functions. Finally, the fourth group includes four algorithms that were used for comparison in the special session at CEC2014. For this group of algorithms, the results



420

of 51 independent runs on 30D test functions are available from the literature. These algorithms are NRGA [77], OPTBees [78], FWA-DM [79], and b3e3pbest [80]. NRGA is an improved variant of GA and OPTBees is an algorithm which has been inspired from the collective decision-making of bee colonies. Also, FWA-DM and b3e3pbest are the-state-of-the-art DE versions that have recently been evaluated on CEC2014 test suite with a great success.

421

4.1.6.

422

The previous subsection demonstrates the number of runs and the dimensionality of test functions for all experiments. Other settings for each run of the algorithms are as follows. The same search ranges [-100,100]D are considered for all test functions where D is the dimension size. The populations of all algorithms are initialized by a uniform random generator in the search space. The maximum number of function evaluations per run is 10,000 × D. A run of an algorithm terminates when the maximum number of function evaluations is reached or when the error value becomes smaller than 10-8. The population size is set to 100 for all runs. For GA, PSO, and CS, the parameters are adjusted as follows. For GA, no elitism has been utilized and its crossover and mutation probabilities are = 0.95, and = 0.05, respectively [75]. For PSO, the inertia weight is 1 and acceleration coefficients = = 2 [81]. The parameters of CS are set to the values determined in the original paper in which = 0.25, = 0.01, and = 1.5 [76]. The values for the parameters of other algorithms are the same as the ones considered in their corresponding references. The limit parameter of the first group of algorithms including the IABC is 200 [42]. Experimental setup for the proposed IABC algorithm is as follows: SR=0.005, NP =15, NP =10, and is a random number in the range of [0.1,0.9].

417 418 419

423 424 425 426 427 428 429 430 431 432

Experimental setup

433 434

4.1.7.

435 436

Table 3 shows the statistical results obtained from the experiment conducted on the first group of algorithms. In this experiment, the results for solving each test function with a specific dimension are presented by the best, worst, median, mean, and standard deviation of error function values obtained in all runs. The mean values in this table clearly indicate that the proposed IABC performs well in unimodal functions (f1-f3); most multimodal functions (f4, f6, f7, f9, f13- f15) except 30D case of f8 and 10D and 30D cases of f5, f10- f12, and f16; hybrid functions (f17-f22) except 10D case of f22; and composite functions (f23-f30) except 10D case of f23, f27 and 10D and 30D cases of (f28, f29). In addition, the IABC has a better performance than the considered algorithms in most cases in terms of all other values. Thus, the proposed IABC is superior to the algorithms of the first group in terms of convergence speed, robustness, and solution accuracy. As part of the first experiment, a non-parametric Wilcoxon signed-rank test with significance level of 0.05 is conducted which uses the mean results to perform a pair-wise comparison among IABC and the other algorithms. These results are also shown in Table 3. The last row of this table lists the number of better, similar, and worse performance of the mean value of IABC algorithm versus other ABC variants with symbols ‘+’, ‘≈’, and ‘-’, respectively. The results reflect that the proposed IABC has higher ‘+’ counts than the other competitors. Moreover, in Figs. 7 to 10, the logarithmic scale of best values of function errors are plotted versus the increasing number of function evaluations. According to these diagrams, in most cases the convergence behavior of the IABC is similar to or better than the compared algorithms.

437 438 439 440 441 442 443 444 445 446 447 448

Experimental results on CEC2014 test suite

449 450 451

Table 3. Results obtained for 51 independent runs of the first group of algorithms and IABC on 10D and 30D benchmarks of CEC2014.

No. Algorithms Unimodal

Best

10-dimensional Worst Median

Mean

S.D

Sign

Best

Worst

30-dimensional Median

Mean

S.D

Sign

f1

ABC COABC AABC IABC

1.778899e+04 2.185921e+04 7.678355e+03 1.096885e+03

5.135238e+05 1.858937e+07 3.157218e+05 3.696421e+04

8.679869e+04 2.834832e+05 6.946011e+04 1.035540e+04

1.152135e+05 1.787125e+06 9.025925e+04 1.211308e+04

9.117535e+04 3.669648e+06 6.590326e+04 7.766953e+03

+ + +

2.960252e+06 1.404094e+06 2.319418e+06 5.513189e+04

1.636920e+07 2.770196e+08 2.970186e+07 1.460876e+06

1.017800e+07 3.850485e+07 9.054943e+06 2.452644e+05

1.000122e+07 5.721845e+07 1.044267e+07 4.145073e+05

3.525243e+06 5.951451e+07 5.694583e+06 3.842276e+05

+ + +

f2

ABC COABC AABC IABC

3.810284e-01 6.392078e+00 1.369341e-01 2.195604e-02

3.084006e+02 1.179453e+04 5.835197e+02 3.620144e+01

3.147770e+01 1.142234e+03 6.250110e+01 2.499604e+00

5.679109e+01 2.830162e+03 1.005485e+02 4.589376e+00

6.337418e+01 3.285643e+03 1.168128e+02 7.087357e+00

+ + +

1.695516e+00 2.408136e+01 3.970982e+00 5.968559e-13

7.181331e+02 3.447533e+04 6.839115e+02 2.913652e-09

6.219380e+01 5.382113e+03 8.382419e+01 8.753887e-12

1.343845e+02 1.086112e+04 1.586465e+02 2.664318e-10

1.666064e+02 1.142890e+04 1.750950e+02 6.433265e-10

+ + +

f3

ABC COABC AABC IABC

2.374284e+00 1.418300e+01 3.260147e+00 5.856468e-02

8.064944e+02 1.815139e+04 5.367273e+02 2.181443e+01

1.014213e+02 1.951628e+03 6.834067e+01 2.693255e+00

1.760264e+02 2.912643e+03 1.029957e+02 3.584153e+00

1.849102e+02 3.233108e+03 1.146965e+02 3.992779e+00

+ + +

1.765915e+01 9.256848e+01 2.066341e+01 9.492851e-12

2.279674e+03 5.974671e+04 2.379690e+03 3.960206e-05

3.916203e+02 1.082936e+04 2.824382e+02 2.063189e-09

5.119510e+02 1.411434e+04 4.035647e+02 9.179552e-07

4.380987e+02 1.355721e+04 4.316757e+02 5.546256e-06

+ + +

Multimodal f4

ABC COABC AABC IABC

9.042024e-02 6.759793e-01 1.393282e-01 1.148467e-03

4.934918e+01 5.630155e+02 5.902747e+01 3.899725e+00

4.623013e+00 3.232364e+02 3.953312e+00 3.914832e-01

2.846629e+00 4.007547e+02 3.849119e+00 3.793384e-01

5.492955e-01 6.004102e+01 5.496305e+01 4.372920e-01

+ + +

8.731377e-01 7.409693e+01 2.591512e+00 4.232185e-03

8.178433e+01 3.419952e+02 8.843743e+01 1.382501e+01

2.417570e+01 1.595178e+02 5.480913e+01 6.858794e-01

3.320509e+01 1.669727e+02 4.491426e+01 4.515304e-01

2.487213e+01 5.833276e+01 2.907898e+01 4.110642e-01

+ + +

f5

ABC COABC

2.669206e+00 2.000132e+01

2.011080e+01 2.005434e+01

2.005396e+01 2.005434e+01

1.877357e+01 2.005917e+01

4.199053e+00 3.934023e-02

+ +

2.023996e+01 2.007684e+01

2.037452e+01 2.041254e+01

2.031876e+01 2.024179e+01

2.031667e+01 2.025387e+01

3.540090e-02 8.128881e-02

≈ ≈



AABC IABC

2.961087e-01 2.579928e+00

2.006186e+01 2.010498e+01

2.006186e+01 2.010498e+01

1.648909e+01 1.976247e+01

6.999335e+00 2.454283e+00

-

2.019847e+01 2.029489e+01

2.042485e+01 2.048672e+01

2.034850e+01 2.042094e+01

2.034540e+01 2.041549e+01

4.744315e-02 4.554709e-02

≈

f6

ABC COABC AABC IABC

8.086025e-01 2.983760e+00 4.430629e-01 0.000000e+00

3.538622e+00 7.972778e+00 2.522691e+00 1.861851e+00

2.358603e+00 5.298372e+00 1.292937e+00 6.435884e-05

2.277526e+00 5.371958e+00 1.355696e+00 1.891958e-03

5.936565e-01 1.210137e+00 5.074847e-01 4.132261e-02

+ + +

9.832504e+00 1.336432e+01 7.008517e+00 3.089938e+00

1.728759e+01 3.329604e+01 1.639085e+01 1.240439e+01

1.473931e+01 2.252044e+01 1.316984e+01 5.456591e+00

1.436006e+01 2.283199e+01 1.315668e+01 5.557274e+00

1.647342e+00 3.698236e+00 1.576418e+00 1.458170e+00

+ + +

f7

ABC COABC AABC IABC

3.089022e-04 5.422881e-02 3.982233e-06 1.136868e-13

3.594948e-02 8.684115e-01 2.659360e-02 2.140140e-02

1.323504e-02 2.582865e-01 9.001320e-03 8.294364e-03

1.473685e-02 3.019252e-01 8.083535e-03 6.872767e-03

1.052846e-02 1.984033e-01 7.524816e-03 6.757645e-03

+ + +

1.067043e-04 1.653127e-01 1.526879e-06 2.273737e-13

2.755686e-03 2.962032e+00 1.726514e-04 9.857287e-03

6.151577e-04 9.864512e-01 1.969837e-05 2.955858e-12

7.404299e-04 9.095497e-01 3.618565e-05 1.075910e-06

5.235766e-04 4.468334e-01 4.007756e-05 2.964378e-05

+ + +

f8

ABC COABC AABC IABC

0.000000e+00 4.547474e-13 0.000000e+00 0.000000e+00

3.410605e-13 3.979831e+00 0.000000e+00 0.000000e+00

1.136868e-13 9.949591e-01 0.000000e+00 0.000000e+00

9.808276e-14 9.232868e-01 0.000000e+00 0.000000e+00

6.024162e-14 1.007657e+00 0.000000e+00 0.000000e+00

+ + ≈

3.410605e-13 1.991167e+00 1.136868e-13 2.273737e-13

4.362505e-09 3.924737e+01 2.273737e-13 1.705303e-12

3.069545e-12 7.931030e+00 1.136868e-13 4.547474e-13

1.200020e-10 9.906358e+00 1.292909e-13 5.929549e-13

6.256554e-10 6.449380e+00 3.951077e-14 2.878893e-13

+ + -

f9

ABC COABC AABC IABC

4.117237e+00 7.016188e+00 3.005707e+00 1.371108e+00

1.280483e+01 6.234890e+01 7.020958e+00 6.809274e+00

8.193646e+00 2.489657e+01 4.669165e+00 3.190647e+00

8.281409e+00 2.544362e+01 4.800295e+00 3.368035e+00

2.034468e+00 9.931515e+00 1.095171e+00 1.038799e+00

+ + +

6.048955e+01 8.457545e+01 3.048064e+01 2.831343e+01

1.005161e+02 2.792644e+02 6.221869e+01 4.726951e+01

8.225028e+01 1.724004e+02 4.760625e+01 3.360235e+01

8.155141e+01 1.759970e+02 4.681604e+01 3.312576e+01

8.480750e+00 4.430987e+01 7.596356e+00 5.389911e+00

+ + +

f 10

ABC COABC AABC IABC

3.420338e-04 1.910589e-01 0.000000e+00 9.510358e-03

3.758289e-01 2.520587e+02 1.249089e-01 2.564792e-01

1.600839e-01 2.473019e+01 0.000000e+00 1.873637e-01

1.703391e-01 6.417881e+01 1.959355e-02 1.712494e-01

7.968339e-02 6.533279e+01 3.418534e-02 6.498753e-02

≈ + -

3.183938e-01 3.997483e+00 2.081925e-02 3.093229e-01

5.892930e+00 5.983209e+02 2.385882e-01 1.057621e+01

3.103980e+00 2.617369e+02 4.163849e-02 2.734639e+00

3.190226e+00 2.666416e+02 5.133799e-02 2.103381e+00

1.308580e+00 1.481047e+02 3.974105e-02 1.823938e+00

+ + -

f 11

ABC COABC AABC IABC

2.186634e+02 1.717649e+02 9.527263e+01 1.102057e+02

4.259253e+02 1.324592e+03 3.121600e+02 3.393658e+02

2.421492e+02 7.488425e+02 1.576428e+02 2.143542e+02

2.357141e+02 7.774876e+02 1.436893e+02 2.209779e+02

9.029836e+01 2.981849e+02 8.011874e+01 8.810690e+01

≈ + -

1.216108e+03 2.265168e+03 1.050734e+03 1.641429e+03

2.566949e+03 4.601915e+03 2.496702e+03 3.015540e+03

2.126456e+03 3.531515e+03 1.879966e+03 2.516819e+03

2.060585e+03 3.521096e+03 1.912703e+03 2.409268e+03

3.080328e+02 5.779536e+02 2.971767e+02 2.766855e+02

+ -

f 12

ABC COABC AABC IABC

1.765113e-01 2.162605e-01 2.075262e-01 2.582424e-01

3.115912e-01 6.595647e-01 4.449293e-01 4.541096e-01

2.364309e-01 3.249354e-01 2.621911e-01 3.262646e-01

2.300021e-01 3.441742e-01 2.714329e-01 3.175258e-01

4.402952e-02 1.348525e-01 6.712757e-02 7.973754e-02

+ -

2.808147e-01 6.619386e-01 3.839342e-01 3.884053e-01

4.216372e-01 7.988148e-01 6.336016e-01 6.537778e-01

3.391919e-01 6.940843e-01 4.204893e-01 5.253096e-01

3.363080e-01 6.815028e-01 4.231416e-01 5.255709e-01

5.323634e-02 1.100333e-01 7.260510e-02 6.331892e-02

+ -

f 13

ABC COABC AABC IABC

7.334797e-02 8.895561e-02 7.396900e-02 7.207098e-02

1.639379e-01 1.023602e+00 1.367312e-01 1.144166e-01

1.234765e-01 3.226774e-01 9.258913e-02 7.988779e-02

1.236380e-01 3.721802e-01 9.397754e-02 8.109499e-02

1.960307e-02 2.038747e-01 1.906755e-02 1.713098e-02

+ + +

1.646249e-01 2.643612e-01 1.188902e-01 8.391830e-02

2.863981e-01 2.121306e+00 2.742695e-01 1.782169e-01

2.494858e-01 5.036889e-01 1.902990e-01 1.211644e-01

2.415464e-01 5.351490e-01 1.873404e-01 1.231998e-01

2.979977e-02 2.593566e-01 2.874084e-02 2.008656e-02

+ + +

f 14

ABC COABC AABC IABC

3.564152e-02 1.966495e-01 3.274981e-02 2.432352e-02

3.890743e-01 1.074620e+00 1.717879e-01 7.685487e-02

1.672101e-01 3.729725e-01 9.219988e-02 4.724851e-02

1.616072e-01 4.255294e-01 9.126964e-02 4.793335e-02

3.300576e-02 1.912222e-01 2.961954e-02 1.207906e-02

+ + +

1.557780e-01 1.861703e-01 1.270495e-01 5.507531e-02

2.326313e-01 9.203604e-01 2.114455e-01 1.312572e-01

1.973191e-01 3.770252e-01 1.689290e-01 6.468989e-02

1.312572e-01 3.963479e-01 1.731363e-01 6.503503e-02

1.636300e-02 1.685275e-01 1.650143e-02 1.915848e-03

+ + +

f 15

ABC COABC AABC IABC

3.399915e-01 9.055116e-01 4.198059e-01 3.276677e-01

1.498815e+00 1.309209e+01 9.480715e-01 8.943980e-01

9.737966e-01 3.161341e+00 6.597973e-01 6.473324e-01

9.283723e-01 3.887552e+00 6.280895e-01 6.232262e-01

2.517144e-01 2.481845e+00 1.911597e-01 1.448536e-01

+ + ≈

8.038870e+00 1.019823e+01 3.704046e+00 2.828671e+00

1.348686e+01 4.817365e+01 7.788258e+00 5.285327e+00

1.001763e+01 1.931856e+01 5.757242e+00 4.238717e+00

1.009871e+01 2.230355e+01 5.802452e+00 4.166922e+00

1.224968e+00 9.670681e+00 1.049056e+00 5.631185e-01

+ + +

f 16

ABC COABC AABC IABC

1.561795e+00 2.044518e+00 1.231089e+00 1.127157e+00

2.662566e+00 3.767429e+00 2.360504e+00 2.327523e+00

2.267786e+00 3.052433e+00 1.753517e+00 1.924340e+00

2.252502e+00 3.047136e+00 1.766745e+00 1.629186e+00

2.409635e-01 3.855756e-01 3.035372e-01 2.884463e-01

+ + ≈

9.919239e+00 1.026096e+01 9.678074e+00 9.981289e+00

1.092717e+01 1.290534e+01 1.046803e+01 1.130073e+01

1.046041e+01 1.166243e+01 9.825350e+00 1.003741e+01

1.040626e+01 1.168304e+01 9.770976e+00 1.019262e+01

3.473417e-01 6.065626e-01 4.069720e-01 2.935927e-01

≈ + -

f 17

ABC COABC AABC IABC

4.783057e+03 4.778294e+03 3.556555e+03 4.720724e+02

4.653783e+05 2.118699e+06 3.540990e+05 3.257031e+03

7.476976e+04 1.802758e+05 3.705035e+04 2.023620e+03

1.087755e+05 4.181260e+05 6.474916e+04 2.053340e+03

1.089740e+05 5.661192e+05 6.284387e+04 3.916558e+02

+ + +

5.168922e+05 5.979466e+05 5.526378e+05 5.069413e+03

4.873078e+06 4.311230e+07 4.714793e+06 1.639473e+05

2.747149e+06 8.731310e+06 2.150908e+06 2.733514e+04

2.621783e+06 1.038358e+07 2.141677e+06 2.382731e+04

1.118112e+06 7.732630e+06 1.014088e+06 3.406620e+04

+ + +

f 18

ABC COABC AABC IABC

4.007632e+02 1.797398e+03 6.914920e+01 1.410891e+01

4.502944e+03 3.893596e+04 9.994820e+03 1.932176e+03

2.210696e+03 4.050172e+03 2.295358e+03 1.809912e+03

2.278284e+03 1.000331e+04 2.980439e+03 1.783378e+03

5.372395e+02 1.150039e+04 1.790245e+03 2.542784e+02

+ + +

2.780081e+03 2.314580e+02 1.927145e+03 6.585916e+01

2.197142e+04 2.082576e+05 3.885836e+04 2.742643e+02

8.302101e+03 5.174440e+03 5.374693e+03 1.339737e+02

8.967599e+03 1.101644e+04 6.927239e+03 1.071956e+02

4.026485e+03 2.879973e+04 5.254158e+03 1.164344e+02

+ + +

f 19

ABC COABC AABC IABC

1.641277e-01 4.343530e-01 2.758651e-01 7.401541e-02

8.543270e-01 5.899643e+00 7.518076e-01 1.009666e+00

4.670846e-01 1.471294e+00 3.192450e-01 2.378062e-01

4.772310e-01 1.904075e+00 3.392068e-01 2.249579e-01

1.635771e-01 1.222114e+00 1.125746e-01 1.062255e-01

+ + +

7.546205e+00 8.466635e+01 7.618043e+00 4.512291e+00

9.598664e+01 1.428347e+02 8.814128e+01 5.734203e+01

8.960457e+00 6.324122e+01 3.373208e+01 1.516626e+01

7.686074e+00 4.906223e+01 7.246172e+00 4.353376e+00

8.285698e-01 4.259634e+01 7.672341e-01 6.880272e-01

+ + +

f 20

ABC COABC AABC IABC

6.573805e+02 9.728541e+03 1.183709e+02 1.083962e+01

1.031544e+03 3.237714e+04 3.384543e+03 4.798809e+02

6.650544e+02 4.283651e+03 2.046334e+02 1.445187e+01

7.178897e+02 8.381662e+03 1.771250e+02 1.167164e+01

2.244669e+01 9.615242e+03 7.887938e+01 1.005949e+01

+ + +

2.134543e+03 4.175365e+03 2.340705e+03 1.384185e+02

1.100561e+04 8.692876e+04 9.684908e+03 2.289343e+03

7.062555e+03 2.653563e+04 4.872547e+03 2.105668e+03

7.370444e+03 3.185715e+04 4.824544e+03 2.070123e+03

2.025475e+03 1.819739e+04 1.219214e+03 2.795475e+02

+ + +

f 21

ABC COABC AABC IABC

1.619404e+03 2.529552e+03 1.981956e+03 6.905419e-01

4.230285e+04 5.488651e+05 1.435823e+04 2.571449e+03

6.105507e+03 2.769446e+04 3.714634e+03 4.105300e+02

9.215377e+03 8.312706e+04 4.705279e+03 2.524923e+03

8.151936e+03 1.220496e+05 2.704837e+03 3.133717e+02

+ + +

9.470746e+05 8.668941e+04 5.959572e+04 4.583839e+03

9.157722e+07 2.498430e+07 1.075880e+06 3.616400e+04

3.957964e+06 6.712642e+05 3.487404e+05 1.106637e+04

4.352818e+05 1.611000e+06 3.819996e+05 1.396256e+04

1.986625e+05 3.553731e+06 2.295112e+05 8.370606e+03

+ + +

f 22

ABC

5.283837e-01

1.935865e+01

2.959291e+00

3.657355e+00

3.557107e+00

+

5.363467e+02

4.788200e+03

8.958977e+02

6.787702e+02

9.968264e+01

+

Hybrid


COABC AABC IABC


9.158406e-01 5.942476e-02 8.478097e-02

2.994612e+02 5.861093e-01 2.033256e+01

2.876769e+01 1.567116e-01 2.294302e-01

7.872205e+01 2.147320e-01 6.909883e-01

8.352884e+01 1.419062e-01 2.814640e+00

+ -

1.486541e+03 4.299322e+02 2.107130e+01

1.297129e+04 4.636740e+03 4.096195e+02

4.986612e+03 6.793016e+02 8.779294e+01

3.588386e+03 4.599008e+02 1.213997e+02

2.880284e+02 1.019423e+02 9.470459e+01

+ +

Composition f 23

ABC COABC AABC IABC

8.546985e+00 2.209110e+02 6.221969e+02 0.000000e+00

3.294787e+02 3.334550e+02 3.295043e+02 3.294535e+02

1.952109e+01 3.294575e+02 3.294646e+02 1.172375e+02

8.321783e+01 3.166459e+02 2.423822e+02 1.100776e+02

9.946916e+01 6.460184e+01 1.437527e+02 7.829076e+01

+ +

3.152495e+02 3.153628e+02 3.152496e+02 3.152441e+02

3.156651e+02 3.968086e+02 3.165723e+02 3.152441e+02

3.153349e+02 3.220011e+02 3.153635e+02 3.152441e+02

3.153655e+02 3.264197e+02 3.154581e+02 3.152441e+02

9.335669e-02 1.394932e+01 2.703243e-01 3.669911e-12

≈ + ≈

f 24

ABC COABC AABC IABC

1.110325e+02 2.100555e+02 1.078827e+02 1.001889e+02

1.276925e+02 1.697905e+02 1.159550e+02 1.117987e+02

1.202444e+02 1.440813e+02 1.120954e+02 1.100452e+02

1.183363e+02 1.435460e+02 1.121807e+02 1.103886e+02

1.456678e+01 1.362735e+01 1.768526e+00 1.403151e+00

≈ + ≈

2.234014e+02 2.284975e+02 2.251087e+02 2.216464e+02

2.286685e+02 2.904668e+02 2.380860e+02 2.251464e+02

2.276284e+02 2.483569e+02 2.254747e+02 2.237054e+02

2.269924e+02 2.504957e+02 2.229628e+02 2.224652e+02

2.536469e+00 1.192977e+01 8.980855e+00 1.298304e-01

≈ + ≈

f 25

ABC COABC AABC IABC

1.217927e+02 1.260383e+02 1.144731e+02 1.000000e+02

1.540726e+02 2.043797e+02 1.414383e+02 1.211352e+02

1.373908e+02 2.005672e+02 1.280376e+02 1.141396e+02

1.369505e+02 1.890108e+02 1.277389e+02 1.133425e+02

8.067466e+00 2.156660e+01 6.412669e+00 5.702831e+00

+ + +

2.063421e+02 2.111868e+02 2.057961e+02 2.025347e+02

2.104339e+02 2.425096e+02 2.102455e+02 2.052809e+02

2.084684e+02 2.190253e+02 2.079757e+02 2.032900e+02

2.085651e+02 2.205741e+02 2.079872e+02 2.033559e+02

8.198829e-01 6.658576e+00 1.035167e+00 5.380786e-01

+ + +

f 26

ABC COABC AABC IABC

1.003547e+02 1.000881e+02 1.000509e+02 1.000501e+02

1.002630e+02 1.011632e+02 1.002016e+02 1.001218e+02

1.001995e+02 1.003668e+02 1.001549e+02 1.000906e+02

1.003793e+02 1.004191e+02 1.001492e+02 1.000913e+02

9.750841e+00 2.174479e-01 2.878642e-02 1.616283e-02

≈ ≈ ≈

1.003008e+02 1.003378e+02 1.002221e+02 1.000740e+02

1.005049e+02 2.020897e+02 1.003684e+02 1.001634e+02

1.004038e+02 1.005481e+02 1.002958e+02 1.001233e+02

1.004056e+02 1.104913e+02 1.002989e+02 1.001242e+02

4.730522e-02 3.024408e+01 3.494291e-02 1.702879e-02

+ + ≈

f 27

ABC COABC AABC IABC

3.545844e+00 2.567608e+00 4.784137e+00 7.617692e-01

1.820670e+01 4.917221e+02 3.545187e+02 4.002053e+02

8.278117e+00 4.013804e+02 5.123970e+00 1.412070e+00

8.818669e+00 2.777135e+02 3.212613e+01 4.253991e+01

2.708532e+00 1.930545e+02 9.225381e+01 1.498588e+02

+ -

4.045340e+02 4.064895e+02 4.034781e+02 4.009640e+02

4.206283e+02 1.197942e+03 4.169868e+02 4.039262e+02

4.111894e+02 4.498577e+02 4.104725e+02 4.020180e+02

4.115437e+02 6.521182e+02 4.107531e+02 4.008587e+02

3.162685e+00 2.872290e+02 3.220563e+00 2.638768e+00

+ + +

f 28

ABC COABC AABC IABC

3.588256e+02 3.793221e+02 3.571205e+02 3.508269e+02

4.042017e+02 8.775331e+02 3.763915e+02 4.248245e+02

3.741707e+02 5.297997e+02 3.594815e+02 3.638507e+02

3.746302e+02 5.420403e+02 3.618573e+02 3.665488e+02

7.669539e+00 1.044573e+02 5.367802e+00 2.084105e+01

≈ + ≈

8.331683e+02 1.111823e+03 7.414878e+02 6.651500e+02

1.117575e+03 3.810676e+03 9.878373e+02 1.133884e+03

9.512873e+02 2.186157e+03 8.499590e+02 8.405913e+02

9.610808e+02 2.253373e+03 8.234143e+02 7.414814e+02

5.012892e+01 7.435148e+02 4.930019e+01 6.924131e+01

+ + ≈

f 29

ABC COABC AABC IABC

2.363194e+02 2.307656e+02 2.416614e+02 2.309072e+02

4.618312e+02 1.457293e+03 4.385841e+02 3.922191e+02

3.090561e+02 4.037328e+02 2.619169e+02 2.659810e+02

3.127291e+02 4.593136e+02 3.021194e+02 2.707853e+02

4.506638e+01 2.140190e+02 4.464040e+01 2.939530e+01

+ + +

9.020645e+02 7.220528e+02 9.448372e+02 8.386699e+02

1.233876e+03 7.519447e+03 2.285267e+03 9.343914e+06

1.127913e+03 1.825555e+03 1.282827e+03 1.305856e+03

1.101461e+03 1.934110e+03 1.447273e+03 7.164552e+05

7.206474e+01 1.013374e+03 3.792110e+02 2.475878e+06

-

f 30

ABC COABC AABC IABC

4.874558e+02 5.047027e+02 4.650876e+02 4.596285e+02

8.095904e+02 2.022862e+03 9.114009e+02 5.107988e+02

5.825341e+02 1.017542e+03 5.274711e+02 4.661030e+02

5.843177e+02 1.079608e+03 5.466722e+02 4.620007e+02

6.470146e+01 3.367019e+02 7.976747e+01 1.294433e+01

+ + +

1.532854e+03 2.238362e+03 1.788627e+03 1.251640e+03

6.928183e+03 3.539746e+04 5.997521e+03 3.670521e+03

4.024838e+03 7.355395e+03 2.594459e+03 2.234437e+03

4.116756e+03 8.265810e+03 3.059493e+03 1.591829e+03

1.031882e+03 5.558008e+03 9.487774e+02 8.846875e+02

+ + +

ABC 22/5/3

COABC 29/1/0

AABC 18/6/6

ABC 23/4/3

COABC 28/1/1

AABC 19/5/6

The results of Wilcoxon signed-rank test based on the mean error IABC VS.

+/≈/452 453

10D-F1

10D-F2

22

18

20

log (F(x)-F(x*))

16 14 12

14 ABC COABC AABC IABC

15 10

10

5

10

1

2

3


7

8

9

-5 0

10 4 x 10

2

3

4 5 6 7 Function Evaluations

8

9

4

-2 0

10 4 x 10

1

2

3


30D-F2

18 16

20 10 0 -10

14

8

9

10 4 x 10


ABC COABC AABC IABC

15 10 log(F(x)-F(X*))

20

7

30D-F3


22

log(F(x)-F(X*))

6

0 1

30D-F1 24

5 0 -5 -10

-20

12 10 0

8

2 0

8 6 0

ABC COABC AABC IABC

12

log(F(x)-F(X*))

20

log(F(x)-F(X*))

10D-F3


log(F(x)-F(X*))

Sign

0.5

1


2.5

3 x 105

-30 0

-15 0.5

1


2.5

3 x 105

-20 0

0.5

1


2.5

3 x 105



10D-F4

5

3.06 ABC COABC AABC IABC

3.04

0 -5

log(F(x)-F(X*))

0

ABC COABC AABC IABC

3.05

log(F(x)-F(X*))

5 log(F(x)-F(X*))

10D-F6

10D-F5

10

3.03 3.02

-10 -15

3.01

-5

-10 0

1

2

3


8

9

2.99 0

10 x 104

1

2

3


30D-F4

9

-25 0

10 x 104

3.04 3.03


2.5

0.5

1


-10 -15

1

2

2.5

3 x 105

ABC COABC AABC IABC

3

2

1

-25 3


7

8

9

-30 0

10 x 104

1

2

3


8

9

0 0

10 4 x 10

0

0

-5

-5

-10 -15 ABC COABC AABC IABC

2

3


2.5

9

10 x 104

ABC COABC AABC IABC

5.5

-10 -15

5 4.5 4 3.5

-30 0

3 x 105

8

30D-F9

-25 1.5 2 Function Evaluations

7

6

-20

1

1

6.5 ABC COABC AABC IABC log(F(x)-F(X*))

5

log(F(x)-F(X*))

10

0.5

7

30D-F8

30D-F7

-30 0


4

ABC COABC AABC IABC

-15

5

-25

1

10D-F9

-10

10

-20

0.5

-20

ABC COABC AABC IABC

-30 0

1 0

3 5 x 10

5

-5

-5

log(F(x)-F(X*))

log(F(x)-F(X*))

0

log(F(x)-F(X*))

2.5

log(F(x)-F(X*))

0

10 x 104

ABC COABC AABC IABC

10D-F8 5

9

2

10D-F7 5

8

3

1.5

3 0

3 5 x 10

10

-25


2.5

3.01

-20

3

3.5

log(F(x)-F(X*))

3.05

3.02

0

1

2

30D-F6 ABC COABC AABC IABC

3.06

log(F(x)-F(x*))

5

0.5

1

4


10 log(F(x)-F(X*))

8

30D-F5

15

-5 0

ABC COABC AABC IABC

-20

3

0.5

1


2.5

3 0

3 x 105

0.5

1


2.5

3 x 105

Fig. 7. The convergence rate obtained by the logarithmic scale of the best value on f1-f9 with 10D and 30D benchmarks for the first group of algorithms and the IABC . 10D-F10

10D-F11

5

-10 -15

-25 1

2

6 5.5

ABC COABC AABC IABC

-20

6.5


7

8

9

10 4 x 10

4.5 0

1 0.5 0 -0.5 -1

5

3

ABC COABC AABC IABC

1.5

log(F(x)-F(X*))

7 log(F(x)-F(X*))

log(F(x)-F(X*))

0

2

ABC COABC AABC IABC

7.5

-5

-30 0

10D-F12

8

10

-1.5 1

2

3


7

8

9

10 x 104

-2 0

1

2

3


7

8

9

10 x 104



30D-F11

30D-F10 10

9 log(F(x)-F(X*))

6 4 2

ABC COABC AABC IABC

8.5

8

ABC COABC AABC IABC

2 1.5 1

log(F(x)-F(X*))

8

log(F(x)-F(X*))

30D-F12 2.5


0.5 0

0

-0.5

7.5 -2

-1

-4 0

0.5

1


2.5

7 0

3 x 105

0.5

1

10D-F13

2.5

-1.5 0

3 5 x 10

0.5

1


-1

-2

3 x 105

2

0

ABC COABC AABC IABC

12 10 8

log(F(x)-F(X*))

0

2.5


4 log(F(x)-F(X*))

1


10D-F15

10D-F14

2

log(F(x)-F(X*))


6 4 2

-2

0 -3 0

1

2

3


7

8

9

-4 0

10 x 104

1

2

3


30D-F13

8

9

-2 0

10 x 104

1

2

3


6

log(F(x)-F(X*))

1 0

ABC COABC AABC IABC

9

10 x 104

ABC COABC AABC IABC

15

4

log(F(x)-F(X*))

2

8

20


7

30D-F15

30D-F14

3

log(F(x)-F(X*))

7

2

-1

0

-2

-2

10

5

-3 0

0.5

1


2.5

-4 0

3 x 105

0.5

1

2.5

0 0

3 x 105

0.5

1

10D-F17


3 x 105

ABC COABC AABC IABC

15 log(F(x)-F(X*))

14 12 10

0.5

2.5


16

log(F(x)-F(X*))

1


10D-F18

18

1.5

log(F(x)-F(X*))


10

5 8

0 0

1

2

3


8

9

6 0

10 x 104

1

2

3


9

10 x 104

0 0

18

log(F(x)-F(X*))

2.55 2.5 2.45

14 12 10

2.35 1


3


2.5

3 5 x 10

8 0

8

9

10 x 104

ABC COABC AABC IABC

20

16

2.4

0.5

2

25

log(F(x)-F(X*))

2.6

1

30D-F18


2.65

log(F(x)-F(X*))

8

30D-F17

30D-F16 2.7

2.3 0

7

ABC COABC AABC IABC 0.5

15

10

5

1


2.5

3 x 105

0 0

0.5

1


2.5

3 x 105

Fig. 8. The convergence rate obtained by the logarithmic scale of the best value on f10-f18 with 10D and 30D benchmarks for the first group of algorithms and the IABC.



10D-F20


16 14 log(F(x)-F(X*))

log(F(x)-F(X*))

3 2 1

12 10 8

0

6

-1

4

1

2

3


8

9


2 0

10 x 104

1

2

3


5

8

9

-5 0

10 x 104

12 10

6

10

1 0

4 0

2.5

3 x 105

0.5

1

10D-F22

2.5

8 0

3 x 105

2

-5 -10 -15 ABC COABC AABC IABC

-25 7

8

9

-30 0

10 x 104

1

2

3


7

8

9

ABC COABC AABC IABC

5

1

2

3


30D-F23

8

9

10 x 104

30D-F24

7

6.5

ABC COABC AABC IABC

6.2 log(F(x)-F(X*))

7

7


7.5 log(F(x)-F(X*))

8

3 x 105

5.2

4.6 0

10 x 104


9

2.5

4.8

30D-F22 11 10


5.4

-20


1

10D-F24

0

0

10 x 104

5.6

log(F(x)-F(X*))

4

3

0.5

5

log(F(x)-F(X*))

6 log(F(x)-F(X*))



9

ABC COABC AABC IABC

10D-F23

8

8

14

2 1.5 2 Function Evaluations

7

16

12

3

2


18

8

1

3

30D-F21

log(F(x)-F(X*))

4

-2 0

2


14

log(F(x)-F(X*))

5

1

1

30D-F20

6 log(F(x)-F(X*))

7


7

log(F(x)-F(X*))

10

0

30D-F19 8

0.5

ABC COABC AABC IABC

15 log(F(x)-F(X*))

4

-2 0

10D-F21

18

5

6

5.8

6 6

5.6

5 4 0

0.5

1


2.5

5.5 0

3 x 105

0.5

1

2.5

3 x 105

5.4 0

0.5

1

10D-F26


4.72 4.7 log(F(x)-F(X*))

5.3 5.2 5.1

2.5

3 x 105


4.68 4.66

ABC COABC AABC IABC

6 5 log(F(x)-F(X*))

5.4

1.5 2 Function Evaluations 10D-F27

4.74

5.5

log(F(x)-F(x*))


4 3

5 4.64

2

4.9 4.62

1

4.8 4.7 0

1

2

3


7

8

9

10 x 104

0

1

2

3


7

8

9

10 x 104

0 0

1

2

3


7

8

9

10 x 104



8

6.5

ABC COABC AABC IABC

6 log(F(x)-F(X*))

6.2

ABC COABC AABC IABC

6 5.8

5.5

7

6.5

5

5.6

ABC COABC AABC IABC

7.5 log(F(x)-F(X*))

6.4

log(F(x)-F(X*))

30D-F27

30D-F26

30D-F25 6.6

6

5.4 4.5 0

5.2 0

0.5

1


2.5

0.5

1

3 x 10


2.5

3 x 105

5

5.5 0

0.5

1


2.5

3 x 105


10D-F29

10D-F28

6.5

15

ABC COABC AABC IABC

14 log(F(x)-F(X*))

log(F(x)-F(X*))

7

16

ABC COABC AABC IABC

ABC COABC AABC IABC

7.5 log(F(x)-F(X*))

10D-F30

20

8

12

10

10 8

6

5.5 0

5 0

1

2

3


7

8

9

1

2

3


10 x 104

30D-F28

8

9

6 0

10 x 104

2

3


7.5

9

10 x 104

18 16 14 12

ABC COABC AABC IABC

16

log(F(x)-F(X*))

8

8


20

log(F(x)-F(X*))

8.5

7

30D-F30


9

1

30D-F29

9.5

log(F(x)-F(X*))

7

14 12 10

10 7 6.5 0

8

8 0.5

1


2.5

3 x 105

6 0

0.5

1


2.5

3 x 105

6 0

0.5

1


2.5

3 x 105


454 455 456 457 458 459 460 461 462 463 464 465 466 467 468

Table 4 reports the results for the second group of algorithms in terms of mean and standard deviation values. The minimum mean value for each test function is marked in bold. In order to provide an overall comparison among these algorithms, a rank-based analysis is also conducted. In this way, the algorithms are ranked from the smallest mean value to the highest one. If the algorithms have the same mean value, the algorithm with the minimum standard deviation gets the first rank. The last two lines of this table present the number of best results and the overall mean ranks obtained by each algorithm, respectively. According to the mean results, the proposed IABC outperforms other ABC variants for all unimodal functions (f1- f3); multimodal functions (f6 and f14); most hybrid functions (f17, f18, f20, and f21); and compositions functions (f25, f28, and f30). For function f8, all ABC variants have the same performance. More precisely, by examining the best result achieved by each algorithm, the IABC wins with the maximum number of first ranks in comparison with the other algorithms as shown in Fig. 11. But according to the overall mean ranks, the DFSABC_elite is slightly better than the IABC. However, the proposed IABC algorithm is still superior to other ABC variants since it obtains better results for all unimodal and most hybrid functions. This achievement surely confirms the merits of employing improved bee phase for providing a delicate balance between exploration and exploitation capabilities.


469 470

NO.

Table 4. Mean and standard deviation of function errors obtained for 25 independent runs of the second group of algorithms and IABC on 10D benchmarks of CEC2014.

Algorithms Mean

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 f20 f21 f22 f23 f24 f25 f26 f27 f28 f29 f30


4.17e+04 3.39e+01 1.74e-01 8.95e+00 1.58e+01 2.24e-01 9.92e-03 0.00e+00 3.76e+00 6.06e-01 2.09e+02 3.23e-01 7.10e-02 6.55e-02 7.04e-01 1.54e+00 3.62e+02 1.12e+02 1.61e-01 1.74e+01 6.58e+01 2.60e-01 2.33e+02 1.10e+02 1.11e+02 1.00e+02 7.39e+01 3.59e+02 2.87e+02 4.96e+02

Total Mean Ranks

IABC S.D 2.21e+04 6.42e+01 3.04e-01 1.48e+01 5.20e+00 3.87e-01 8.78e-03 0.00e+00 1.26e+00 1.14e+00 1.17e+02 7.73e-02 1.56e-02 1.73e-02 1.92e-01 3.87e-01 3.62e+02 1.01e+01 9.00e-02 1.04e+01 6.06e+01 1.88e-01 6.58e+01 2.86e+00 7.11e+00 1.26e-02 1.50e+02 2.08e+01 4.71e+01 1.45e+01

R 1 1 1 6 4 1 4 1 3 5 5 6 2 1 4 3 1 1 2 1 1 2 3 2 1 2 5 1 5 1

DFSABC_elite Mean S.D 8.60e+04 1.98e+02 2.52e+02 3.65e-01 1.52e+01 6.88e-01 4.15e-03 0.00e+00 3.67e+00 3.49e-02 4.56e+01 1.22e-01 8.96e-02 1.30e-01 5.65e-01 1.28e+00 1.04e+03 2.56e+03 1.07e-01 1.18e+03 4.29e+03 2.51e-01 3.29e+02 1.12e+02 1.26e+02 1.00e+02 4.27e+01 3.63e+02 2.72e+02 5.86e+02

13 2.53

6.29e+04 3.98e+02 3.21e+02 8.91e-01 8.00e+00 3.26e-01 7.37e-03 0.00e+00 1.05e+00 4.06e-02 5.21e+01 3.62e-02 2.31e-02 3.36e-02 1.49e-01 3.25e-01 9.99e+02 2.51e+03 6.53e-02 1.51e+03 4.07e+03 1.57e-01 8.68e-04 2.99e+00 5.98e+00 2.95e-02 1.02e+02 5.23e+0 3.00e+01 8.20e+01

8 2.46

ABCVSS S.D

R

Mean

2 6 2 4 3 3 1 1 2 1 1 1 3 2 2 1 2 2 1 6 2 1 5 3 2 4 3 2 4 2

1.49e+05 7.25e+01 3.66e+02 2.74e-02 1.60e+01 2.13e+00 5.40e-03 0.00e+00 5.15e+00 4.75e-02 2.00e+02 1.45e-01 9.44e-02 1.58e-01 6.33e-01 1.79e+00 1.63e+05 7.46e+02 2.20e-01 4.78e+02 1.26e+04 2.76e-01 2.72e+02 1.19e+02 1.36e+02 1.00e+02 8.65e+01 3.78e+02 2.68e+02 6.24e+02

1.17e+05 1.41e+02 4.63e+02 3.03e-02 6.87e+00 6.42e-01 5.36e-03 0.00e+00 1.90e+00 5.19e-02 9.47e+01 4.02e-02 2.42e-02 3.08e-02 1.89e-01 2.71e-01 1.75e+05 6.10e+02 1.19e-01 6.87e+02 1.77e+04 1.49e-01 1.14e+02 4.23e+00 1.05e+01 4.36e-02 1.43e+02 1.58e+01 2.36e+01 8.73e+01

dABC S.D

R

Mean

5 4 6 2 5 4 2 1 4 3 4 4 4 4 3 4 6 6 4 4 5 3 4 4 5 6 6 4 2 3

2.15e+05 6.96e+01 2.85e+02 2.94e-02 1.68e+01 2.18e+00 1.97e-02 0.00e+00 7.80e+00 1.03e-01 2.09e+02 1.31e-01 1.15e-01 1.70e-01 8.66e-01 1.92e+00 2.36e+05 3.67e+02 3.58e-01 2.41e+02 2.50e+04 1.37e+00 1.79e+02 1.21e+02 1.40e+02 9.76e+01 2.12e+01 3.92e+02 2.71e+02 7.47e+02

1 4.03

2.02e+05 9.15e+01 2.30e+02 2.30e+02 6.69e+00 7.93e-01 1.03e-02 0.00e+00 2.22e+00 3.99e-02 1.32e+02 2.88e-02 2.31e-02 3.10e-02 2.23e-01 2.61e-01 1.49e+02 2.86e+02 1.57e-01 2.88e+02 2.77e+04 4.02e+00 1.23e+02 4.59e+00 1.39e+01 1.27e+01 7.05e+01 8.48e+01 1.93e+01 1.10e+02

3 4.20

qABC S.D

R

Mean

6 3 4 3 6 6 6 1 6 4 6 2 5 6 6 6 4 4 6 3 6 4 2 5 6 1 1 5 3 6

1.46e+05 6.56e+01 3.26e+02 2.34e-02 1.51e+01 2.15e+00 1.16e-02 0.00e+00 6.34e+00 4.34e-02 1.69e+02 1.39e-01 1.20e-01 1.64e-01 7.82e-01 1.90e+00 1.59e+05 4.38e+02 3.21e-01 1.67e+02 7.19e+03 1.59e+00 1.16e+02 1.22e+02 1.32e+02 1.00e+02 3.56e+01 4.01e+02 2.66e+02 6.35e+02

1.99e+05 1.12e+02 2.89e+02 3.50e-02 8.04e+00 5.92e-01 9.55e-03 0.00e+00 1.84e+00 5.37e-02 8.17e+01 2.52e-02 2.50e-02 2.81e-02 2.18e-01 3.00e-01 1.37e+05 3.76e+02 1.20e-01 2.12e+02 7.43e+03 3.33e+00 1.21e+02 4.65e+00 7.78e+00 4.01e-02 9.63e+01 2.14e+01 2.53e+01 6.60e+01

Mean

4 2 5 1 2 5 5 1 5 2 3 3 6 5 5 5 5 5 5 2 5 5 1 6 4 5 2 6 1 4

1.06e+05 1.94e+02 2.66e+02 2.05e+00 1.45e+01 3.08e-01 6.30e-03 0.00e+00 2.39e+00 6.31e-01 9.63e+01 2.46e-01 5.72e-02 1.57e-01 5.33e-01 1.10e+00 8.55e+04 3.01e+03 2.02e-01 1.15e+03 5.29e+03 7.80e+00 3.29e+02 1.10e+02 1.27e+02 1.00e+02 6.88e+01 3.73e+02 2.91e+02 6.68e+02

4 3.93

471 472

Numbers of the best solution

14 12 10 8 6 4 2 0 IABC

DFSABC_elite

ABCVSS

dABC

qABC

EABC

ABC variants

Fig. 11. Number of the first ranks achieved for the mean value by the second group of algorithms and the IABC. 473 474 475 476

EABC S.D

R

Also, Table 5 shows the experimental results for the third and fourth groups of algorithms in terms of the mean and standard deviation of function error values. Similar to the second group, each algorithm receives a rank in its group. The last row of this table indicates the mean ranks obtained by the algorithms of both groups. The best results are highlighted in bold.

7.46e+04 2.86e+02 3.20e+02 6.96e+00 8.97e+00 2.05e-01 9.05e-03 0.00e+00 1.38e+00 1.08e+00 1.03e+02 7.72e-02 1.66e-02 3.97e-02 1.43e-01 3.03e-01 6.86e+04 3.50e+03 3.50e+03 1.06e+03 5.84e+03 2.40e+01 1.07e-02 2.15e+00 8.16e+00 2.82e-02 1.23e+02 1.91e+01 4.02e+01 1.39e+02

6 3.26

R 3 5 3 5 1 2 3 1 1 6 2 5 1 3 1 2 3 3 3 5 3 6 6 1 3 3 4 3 6 5


477 478 479 480 481 482


As can be seen, compared to the third group of algorithms, the IABC has the first rank for all unimodal functions, all multimodal functions except f5 and f12, and all hybrid functions except f20 and f21. For composite functions, the IABC and DE have a close competition. However, the IABC has the first rank for functions f23, f27, f28, and f30 and DE has the first rank for functions f24, f25, and f26. Generally speaking, according to the overall average ranks, the IABC is superior to all compared algorithms of the third group. This is because the IABC is well-designed for managing both exploration and exploitation behaviors with respect to the improved bee phase.

487

As already stated, the performance of IABC is also compared with those of the fourth group of algorithms which includes b3e3pbest, OPTBees, FWA-DM, and NRGA. Table 5 exhibits the results for these algorithms. As can be seen, the IABC has achieved more accurate solutions than the other algorithms for multimodal functions that have many local optima. According to the overall average ranks in the last line of Table 5, b3e3pbest is slightly better than the IABC since it gives better results for unimodal and hybrid functions.

488 489 490

Table 5. Mean and standard deviation of function errors obtained for 51 runs by the third and fourth groups of algorithms on 30D functions (*: rank in the third group, **: rank in the fourth group).

483 484 485 486

Algorithm

IABC Group 3

Group 4

Group 3

Group 4

Group 3

Group 4

Group 3

Group 4

Group 3

GA PSO CS DE b3e3pbest FWA-DM NRGA OPTBees

Functions f1 4.14e+05 ± 3.84e+05 1.26E+08 ± 3.70E+07 1.38E+07 ± 9.84E+06 3.50E+07 ± 2.49E+07 9.68E+05 ± 8.23E-10 1.20E+04 ± 9.09 E+03 2.76E+05 ± 1.82E+05 1.31E+06 ± 7.06E+05 8.56E+04 ± 3.04E+05

IABC

f6 5.55e+00 ± 1.45e+00


2.70E+01 ± 1.57E+00 1.17E+01 ± 3.20E+00 3.23E+01 ± 3.27E+00 4.19E+01 ± 0.00E+00 1.42E+01 ± 1.19E+00 1.29E+01 ± 8.25E+00 1.78E+01 ± 2.18E+00 1.63E+01 ± 3.44E+00

IABC

f11 2.40e+03 ± 2.76e+02


6.02E+03 ± 4.37E+02 3.61E+03 ± 1.13E+03 4.13E+03 ± 5.35E+02 5.25E+03 ± 0.00E+00 2.42E+03 ± 2.40E+02 2.63E+03 ± 2.48E+02 3.42E+03 ± 6.47E+02 2.71E+03 ± 5.68E+02

IABC

f16 1.01e+01 ± 2.93e-01


1.20E+01 ± 2.97E-01 1.19E+01 ± 4.61E-01 1.27E+01 ± 5.01E-01 1.25E+01 ± 0.00E+00 9.94E+00 ± 3.79E-01 1.10E+01 ± 2.71E-01 1.15E+01 ± 6.56E−01 1.09E+01 ± 6.90E-01

IABC

f21 1.39e+04 ± 8.37e+03

GA PSO CS

1.13E+06 ± 9.18E+05 3.32E+05 ± 3.27E+05 3.54E+05 ± 3.48E+05

Rank

1* 4** 5 3 4 2 1 3 5 2 Rank

1* 1** 3 2 4 5 3 2 5 4 Rank

1* 1** 5 2 3 4 2 3 5 4 Rank

1* 2** 3 2 5 4 1 4 5 3 Rank

2* 3** 5 3 4

f2 2.66e-10 ± 6.43e-10 5.66E+09 ± 8.83E+08 8.44E+03 ± 1.00E+04 1.95E+07 ± 5.49E+07 2.50E+04 ± 0.00E+00 0.00E+00 ± 6.27E-10 1.08E-16 ± 1.87E-16 9.29E+03 ± 3.95E+03 3.317E-12 ± 1.14E-11 f7 1.07e-06 ± 2.96e-05 5.54E+01 ± 6.31E+00 1.15E-02 ± 1.23E-02 1.79E+00 ± 2.19E+00 3.16E-06 ± 4.71E-06 1.35E-03 ± 3.63E-03 8.55E-03 ± 9.81E-03 1.64E–02 ± 1.61E−02 3.74E-02 ± 3.81E-02 f12 5.25e-01 ± 6.33e-02 1.28E+00 ± 2.85E-01 2.14E+00 ± 4.37E-01 5.11E-01 ± 2.56E-01 7.06E-01 ± 0.00E+00 4.51E-01 ± 4.93E-02 3.71E-01 ± 6.66E-02 1.61E−01 ± 8.43E−02 1.81E-01 ± 6.11E-02 f17 2.38e+04 ± 3.40e+04 4.10E+06 ± 2.96E+06 9.56E+05 ± 6.49E+05 1.48E+06 ± 1.21E+06 6.53E+04 ± 2.29E+04 1.06E+03 ± 3.20E+02 6.29E+03 ± 5.95E+03 3.35E+05 ± 1.75E+05 2.74E+04 ± 4.04E+04 f22 1.21e+02 ± 9.97e+01 4.39E+02 ± 1.50E+02 2.61E+02 ± 1.67E+02 9.47E+02 ± 3.31E+02

Rank

1* 4** 5 2 4 3 1 2 5 3 Rank

1* 1** 5 3 4 2 2 3 4 5 Rank

2* 5** 4 5 1 3 4 3 1 2 Rank

1* 3** 5 3 4 2 1 2 5 4 Rank

1* 1** 3 2 4

f3 9.17e-07 ± 5.54e-06 2.63E+04 ± 1.57E+04 3.38E+03 ± 3.66E+03 3.10E+04 ± 1.36E+04 1.35E+02 ± 0.00E+00 0.00E+00 ± 5.30E-10 4.42E-16 ± 4.74E-16 4.91E+03 ± 3.77E+03 8.41E-03 ± 3.77E-02 f8 5.92e-13 ± 2.87e-13 8.54E+01 ± 8.62E+00 2.85E+01 ± 8.45E+00 1.71E+02 ± 3.46E+01 1.87E+02 ± 2.87E-14 0.00E+00 ± 8.91E-10 1.13E-13 ± 4.51E-13 3.01E+01 ± 8.80E+00 3.63E-13 ± 1.04E-13 f13 1.23e-01 ± 2.00e-02 1.12E+00 ± 1.77E-01 4.35E-01 ± 1.09E-01 4.81E-01 ± 1.17E-01 3.38E-01 ± 4.56E-02 2.80E-01 ± 5.47E-02 3.89E-01 ± 5.51E-02 2.81E-01± 5.64E−02 5.60E-01 ± 1.47E-01 f18 1.07e+02 ± 1.16e+02 3.83E+04 ± 3.06E+04 2.74E+04 ± 1.59E+05 7.67E+03 ± 6.70E+03 1.52E+02 ± 4.01E-13 7.41E+01 ± 3.18E+01 7.67E+01 ± 3.66E+01 5.50E+02 ± 7.16E+02 1.95E+02 ± 4.76E+02 f23 3.15e+02 ± 3.66e-12 3.35E+02±4.02E+00 3.16E+02 ± 3.40E-01 3.29E+02 ± 7.51E+00

Rank

1* 3** 4 3 5 2 1 2 5 4 Rank

1* 4** 3 2 4 5 1 2 5 3 Rank

1* 1** 5 3 4 2 2 4 3 5 Rank

1* 3** 5 4 3 2 1 2 5 4 Rank

1* 4** 5 3 4

f4 4.51e-01 ± 4.11e-01 5.81E+02 ± 8.93E+01 1.98E+02 ± 5.14E+01 2.03E+02 ± 6.69E+01 8.15E+01 ± 1.43E-14 3.72E+00 ± 1.51E+01 2.04E+01 ± 1.91E+01 9.36E+01 ± 3.02E+01 1.25+01 ± 1.37E+01 f9 3.31e+01 ± 5.38e+00 2.17E+02 ± 1.82E+01 6.56E+01 ± 1.38E+01 2.80E+02 ± 5.16E+01 1.69E+02 ± 2.29E-13 4.51E+01 ± 6.10E+00 5.66E+01 ± 1.08E+01 4.56E+01 ± 1.34E+01 1.37E+02 ± 3.24E+01 f14 6.50e-02 ± 1.91e-03 1.84E+01 ± 2.78E+00 3.37E-01 ± 1.68E-01 3.08E-01 ± 5.64E-02 2.55E-01 ± 2.39E-02 2.10E-01 ± 3.21E-02 2.69E-01 ± 7.76E-02 1.86E−01 ± 2.66E-02 3.99E-01 ± 2.31E-01 f19 4.35e+00 ± 6.88e-01 4.02E+01 ± 2.50E+01 1.09E+01 ± 1.42E+01 5.33E+01 ± 3.63E+01 8.66E+00 ± 4.99E-01 4.79E+00 ± 9.46E-01 9.95E+00 ± 1.93E+00 1.40E+01 ± 1.27E+00 7.89E+00 ± 1.87E+00 f24 2.22e+02 ± 1.29e-01 2.74E+02 ± 3.88E+00 2.33E+02 ± 6.81E+00 2.78E+02 ± 3.11E+01

Rank

1* 1** 5 3 4 2 2 4 5 3 Rank

1* 1** 4 2 5 3 2 4 3 5 Rank

1* 1** 5 4 3 2 3 4 2 5 Rank

1* 1** 4 3 5 2 2 4 5 3 Rank

2* 1** 4 3 5

f5 2.04e+01 ± 4.55e-02 2.08E+01 ± 6.40E-02 2.09E+01 ± 7.71E-02 2.00E+01 ± 2.28E-03 2.00E+01 ± 0.00E+00 2.03E+01 ± 2.90E-02 2.05E+01 ± 5.36E-02 2.00E+01 ± 1.52E−04 2.00E+01 ± 1.02E-05 f10 2.10e+00 ± 1.82e+00 1.27E+03 ± 1.77E+02 7.02E+02 ± 3.03E+02 2.66E+03 ± 5.34E+02 3.48E+03 ± 2.75E-12 2.44E-03 ± 7.95E-03 8.53E+00 ± 2.42E+00 1.27E+03 ± 5.35E+02 1.04E+03 ± 2.52E+02 f15 4.16e+00 ± 5.63e-01 6.13E+02 ± 4.03E+02 9.77E+00 ± 3.83E+00 9.80E+01 ± 3.02E+01 1.03E+02 ± 1.43E-13 5.66E+00 ± 6.17E-01 7.37E+00 ± 8.46E-01 1.40E+01 ± 4.72E+00 1.27E+01 ± 6.92E+00 f20 2.07e+03 ± 2.79e+02 2.00E+04 ± 1.67E+04 3.25E+03 ± 3.93E+03 3.93E+04 ± 2.20E+04 2.74E+02 ± 4.59E-13 2.82E+01 ± 2.89E+01 4.28E+01 ± 2.61E+01 1.20E+04 ± 5.70E+03 8.52E+02 ± 7.78E+02 f25 2.03e+02 ± 5.38e-01 2.26E+02 ± 3.66E+00 2.10E+02 ± 2.43E+00 2.23E+02 ± 9.39E+00

Rank

3* 4** 4 5 2 1 3 5 2 1 Rank

1* 2** 3 2 4 5 1 3 5 4 Rank

1* 1** 5 2 3 4 2 3 5 4 Rank

2* 4** 4 3 5 1 1 2 5 3 Rank

2* 3** 5 3 4


Group 4

Group 3

Group 4

Mean Ranks


DE b3e3pbest FWA-DM NRGA OPTBees

5.54E+03 ± 2.75E-12 3.19E+02 ± 1.63E+02 7.29E+02 ± 9.49E+02 2.11E+05 ± 1.09E+05 1.74E+04 ± 1.82E+04

1 1 2 5 4

2.42E+03 ± 1.14E-10 1.42E+02 ± 9.41E+01 1.46E+02 ± 8.83E+01 4.20E+02 ± 1.38E+02 2.32E+02 ± 9.24E+01

5 2 3 5 4

3.15E+02 ± 1.21E-10 3.15E+02 ± 2.30E-13 3.14E+02 ± 9.28E-14 3.15E+02 ± 2.96E−03 3.14E+02 ± 6.65E-02

2 3 1 5 2

2.00E+02 ± 0.00E+00 2.25E+02 ± 4.40.E+00 2.26E+02 ± 3.59E+00 2.28E+02 ± 4.53E+00 2.36E+02 ± 5.47E+00

1 2 3 4 5

2.00E+02 ± 0.00E+00 2.03E+02 ± 6.37E-01 2.01E+02 ± 1.98E-01 2.10E+02 ± 1.70E+00 2.00E+02 ± 1.68E-01

1 4 2 5 1

f26 1.00e+02 ± 1.70e-02

Rank

f28 7.41e+02 ± 6.92e+01

Rank

f29 7.16e+05 ± 2.47e+06

Rank

f30 1.59e+03 ± 8.84e+02

Rank


1.00E+02 ± 1.49E-01 1.04E+02 ± 1.96E+01 1.00E+02 ± 1.63E-01 1.00E+02 ± 0.00E+00 1.00E+02 ± 4.34E-02 1.00E+02 ± 5.35E-02 1.00E+02 ± 9.32E−02 1.00E+02 ± 1.72E-01

2* 1** 3 5 4 1 2 3 4 5

f27 4.00e+02 ± 2.63e+00

Rank

IABC

PSO 3.00

Second group CS 3.80

Algorithm GA 4.33

9.10E+02 ± 1.51E+02 5.92E+02 ± 1.56E+02 4.27E+02 ± 1.96E+01 4.18E+03 ± 6.88E-13 3.20E+02 ± 3.06E+01 4.01E+02 ± 3.06E+01 5.89E+02 ± 1.71E +02 4.02E+02 ± 9.76E-01

DE 2.63

1* 2** 4 3 2 5 1 3 5 4

1* 4** 3 2 4 5 3 1 5 2

1.38E+03 ± 5.90E+01 1.28E+03 ± 3.29E+02 3.49E+03 ± 5.48E+02 9.39E+03 ± 1.60E-12 7.46E+02 ± 4.76E+01 3.93E+02 ± 1.46E+01 1.60E+03 ± 5.88E+02 4.31E+02 ± 1.52E+01

IABC 1.36

8.46E+04 ± 5.75E+04 2.48E+06 ± 8.72E+06 5.44E+05 ± 2.61E+06 2.39E+05 ± 7.52E-09 7.56E+02 ± 5.22E+01 2.11E+02 ± 2.90E+00 1.33E+03 ± 2.05E+02 2.15E+02 ± 1.17E+00

4* 5** 1 5 3 2 3 1 4 2

OPTBees 3.400

Third group NRGA 4.400

b3e3pbets 2.00

4.28E+04 ± 2.78E+04 5.40E+03 ± 2.91E+03 2.49E+04 ± 2.26E+04 2.30E+03 ± 1.25E+03 2.84E+03 ± 4.58E+02 4.51E+02 ± 1.96E+02 3.22E+03 ± 5.99E+ 02 5.92E+02 ± 9.86E+01

FWA-DM 2.70

491 492 493 494 495 496 497 498 499 500 501

Moreover, Fig.12 displays the abundance of ranks achieved by the algorithms of the third and fourth groups. As mentioned before, a higher rank reflects the poor quality of the solution. For the third group of algorithms, the IABC algorithm has the maximum number of the first rank abundance and the minimum number of the last rank abundance which is equal to zero. For the fourth group of algorithms, Fig. 12 shows that the IABC wins with the maximum number of the first rank abundance after a close competition with b3e3pbest. However, it should be noted that b3e3pbest outperforms other algorithms since it achieves the maximum number of the second rank abundance and the minimum number of the last rank abundance. Furthermore, GA and NRGA have the poorest performance by obtaining the maximum number of the last rank abundance. Generally, these observations prove that the proposed IABC algorithm is superior to the algorithms of the third group and has a close competition with b3e3pbest as the best algorithm of the fourth group. The reason for this achievement is its well-balanced search abilities which are employed in the improved bee phase of IABC.

Group 3:

GA

PSO

DE

CS

IABC

Group 4:

FWA-DM

NRGA

OPTBees

b3e3pbest

IABC2

Number of rank abundance

25

20

15

10

5

0

Rank 1

Rank 2

Rank 3

Rank 4

Fig. 12. The abundance of ranks obtained by each algorithm in its group for 51 independent runs. 502 503 504 505 506

Rank 5

1* 3** 5 3 4 2 4 1 5 2

IABC 2.46



507

4.2. Experiments on reliability optimization problems

508

In this section, reliability optimization problems have been used in order to further investigate the performance of IABC algorithm on real-world applications. In fact, system reliability optimization which has been developed in industrial engineering is considered as a fundamental dimension of product quality. The aim of reliability optimization is to achieve a high-quality system that works safely and efficiently by considering several resource constraints such as cost, weight, and volume [82-84]. This is a very challenging goal to reach since it requires finding a delicate balance between resource constraints and reliability. On the other hand, demands for highly reliable systems or products are increasing due to competition between markets. Hence, reliability optimization problems have been an important research topic. The general formulation of a reliability optimization problem is described as follows:

509 510 511 512 513 514 515 516

517 518 519 520

= ( , )

(8)

( , ) ≤ , = 1,2, … , = ( , , … , , … , ) = ( , , … , … , 1 ≤ ≤ ; 0 ≤ ≤ 1 ; ∈ ; ∈

)

521 522 523 524 525 526

Here, is the reliability of the system, ( , ) is the objective function of overall system reliability, and are the vectors of the redundancy allocation and component reliabilities of the system, respectively, ( , ) is the constraint function which is usually associated with the system’s weight, volume, and cost, is the number of constraints, is the resource limitation, and are the reliability and the number of components in the ith subsystem, respectively, i stands for subsystems, and m is the number of subsystems in the system.

535

In this subsection, the IABC is validated against eight well-known reliability optimization problems. These problems have been studied previously as benchmark problems in the specialized literature [85-88]. They include complex (bridge) system (P1), series system (P2), series-parallel system (P3), over-speed protection system (P4), large-scale system reliability problem (P5), convex quadratic reliability problem (P6), incomplete fault detecting the switching (P7), and mixed series-parallel system (P8). The first four problems are mixed-integer programming problems which involve several real variables for the reliability allocation and several integer variables for redundancy allocation. The last four problems are integer programming problems that only allocate the number of system redundant components. These problems have a single objective function which aims to maximize the system reliability. Furthermore, to handle the constraints of reliability optimization problems, the IABC employs a constraint handling method that is described in the following subsection.

536

4.2.1.

527 528 529 530 531 532 533 534

Constraint handling method

537 538 539 540 541 542 543 544 545 546 547 548 549 550

As Eq. (8) demonstrates, reliability optimization problems belong to the category of constrained optimization problems where constraints are the conditions on the components and subsystems such as cost, weight, and volume. Such constraints make it more difficult to reach optimal solutions by meta-heuristic algorithms whose search operators stochastically alter some candidate solutions to generate better ones. In other words, some generated solutions by these algorithms may be infeasible solutions because they violate some constraints. Therefore, meta-heuristic algorithms must be equipped with some techniques to be able to find a global feasible solution in a reasonable time. One of the most popular methods for this purpose is to convert an optimization problem with constraints to a problem without constraints by defining a penalty function [89, 90]. In penalty approach, the cost or objective function of any candidate solution of the algorithm is modified based on some measure of violated constraints. There are two types of penalty function methods. First, exterior method which allows infeasible solutions in the population but penalizes their objective value or selection probability. Second, interior method which penalizes feasible solutions as they get closer to the constraint boundary and in this way prevents any infeasible solution from entering the population. In this study, an exterior method is employed which penalizes infeasible solutions proportional to their violation magnitude. Considering such a penalty function, the maximization constrained problem ( , ) is transformed into a minimization unconstrained problem ( , ) as follows:

551 552

553

= ( , ) = − ( , ) + ∑

max (0,

554 555 556

where

is a penalty coefficient with a constant value that is set to 105 in this study.

( , ) −

)

(9)



557

4.2.2.

558

The performance of IABC algorithm on reliability optimization problems is compared with those of two groups of algorithms. The first group includes original ABC and its two state-of-the-art variants, i.e. COABC and AABC. The results for these algorithms are obtained via an experiment in which 50 independent runs of each algorithm are conducted on the benchmark problems of reliability optimization. The second group consists of several meta-heuristic algorithms that have been mentioned in the literature as successful approaches for solving such problems. The following subsections present more details.

559 560 561 562

Compared algorithms

563 564

4.2.3.

565

568

The parameter configurations for the first group of algorithms are as follows: population size is 80, limit is 200, and maximum number of iterations is 2500. For the IABC algorithm, the parameter SSP will be in the range of [0.1,0.9]. NP , NP , and SR are set to 20, 5, and 0.005, respectively. Furthermore, all algorithms employ the constraint handling method described in Subsection 4.2.1 to handle the constraints of reliability optimization problems.

569

4.2.4.

570

Statistical results achieved for the first group of algorithms are reported in Table 6 in terms of best, worst, median, mean, and standard deviation (SD) of objective function value. The best results have been bolded. Also, the average running time of each algorithm for each problem is summarized in the last column of this table.

566 567

571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589

Experimental setup

Experimental results for the first group of algorithms

As shown in Table 6, the IABC algorithm has outperformed ABC and its variants according to all criteria in solving P1, P2, P3, P4, and P8 problems. Thus, the proposed mechanism for balancing the exploration and exploitation of improved bee phase has succeeded to enhance the search performance of IABC for finding optimal solutions for these reliability optimization problems. For low dimension (36D and 38D) cases of P5, the IABC and AABC have obtained the smallest best value but with respect to all other criteria (except the worst value of 36D case) the IABC performs better than the other algorithms. However, for high dimension cases (42D and 50D) of P5, the performance of IABC is superior to those of all the considered algorithms. This is the result of the population reduction mechanism that helps the IABC to inhibit the stagnant issue which is likely to occur in solving high dimensional optimization problems. For P6, all algorithms have obtained the smallest best value, the IABC and AABC have achieved the smallest median value, and the IABC has outperformed the other algorithms in the remaining criteria. For solving P7, the performance of all algorithms is similar. Remarkably, as shown in Fig. 13, the average computational time of the IABC for solving all problems is much less than that of the other algorithms. For example, the average computational time of ABC is at least twice as much as that of the IABC algorithm. Obviously, the population size reduction mechanism of the IABC is the main reason for this achievement. In addition, the diagrams of Fig.14 show the convergence curves obtained by the compared algorithms. These diagrams display that the IABC has a better convergence rate in most problems (P2, four cases of P5 with different dimensions, P6, and P8). Therefore, according to these simulation results, it can be concluded that the IABC algorithm is more powerful than the other ABC algorithms in terms of convergence speed, stability, and accuracy.

590 591

Table 6. Results obtained by the first group of algorithms and the IABC for reliability optimization problems (P1–P8). Problems

Algorithms

Best

Worst

Median

Mean

SD

AT

Complex (bridge) system (P1)

ABC COABC AABC IABC

0.99986461091612 0.99988588746145 0.99988895222483 0.99988963740661

0.99963074125896 0.99899794248103 0.99979399401957 0.99986524732603

0.99981504016275 0.99975374164781 0.99987498307057 0.99988935012970

0.99980222674114 0.99967240710063 0.99986911564391 0.99988894825699

0.00005061248414 0.00021404520756 0.00001954290626 0.00000342267890

32.191041 82.993869 39.525127 11.286299

Series system (P2)

ABC COABC AABC IABC

0.92783679073076 0.92563651230062 0.93113376004847 0.93168238886088

0.88933961669294 0.78338232759812 0.91660711316467 0.92336885891040

0.91043881791006 0.90328278811399 0.92644111521712 0.93168228089320

0.91020851069332 0.88851649606478 0.92548005979023 0.93060459371065

0.00884194641389 0.03657986890712 0.00354505172406 0.00153277871986

36.737946 94.503005 44.562147 10.256814

Series-parallel system (P3)

ABC COABC AABC IABC

0.99997888994380 0.99997627118103 0.99997960490296 0.99997982961494

0.99993399812213 0.99975877910955 0.99996803941443 0.99997664880861

0.99996649717054 0.99994931747144 0.99997562826690 0.99997982954541

0.99996484963599 0.99994175000175 0.99997545223718 0.99997963870455

0.00000885287266 0.00004310994447 0.00000292687136 0.00000076303103

25.126653 63.537496 31.086994 10.997443

Over-speed protection (P4)

ABC COABC AABC

0.99995309068234 0.99994562123045 0.99995433002549

0.99987700633090 0.99941790056396 0.99991556314412

0.99993331944366 0.99988939834849 0.99994819328206

0.99992541340052 0.99986174786704 0.99994527252412

0.00002020763069 0.00010022262737 0.00000886755847

36.042023 92.257332 43.702785



IABC

0.99995467566652

0.99994615116126

0.99995467463064

0.99995433367430

0.00000168720094

11.665228

ABC COABC AABC IABC

0.51795811162451 0.51695480601190 0.51997596538026 0.51997596538026

0.49909017139609 0.47897492893830 0.50776562723383 0.50287761739262

0.50507420922180 0.49751724994056 0.51921871882873 0.51947113434591

0.50620981919969 0.49933622921768 0.51735576556771 0.51761296302814

0.00443773761129 0.00708965298729 0.00461926318431 0.00364049866208

52.969201 132.386154 59.725085 20.035140

ABC COABC AABC IABC

0.50607051568387 0.50754166741586 0.51098859649712 0.51098859649712

0.48714549654464 0.47250655266680 0.50508312565297 0.50626874501369

0.49670515477736 0.48617231930301 0.50852026245916 0.50879042255786

0.49659539963943 0.48775596688026 0.50846628888751 0.50853940323779

0.00493835110504 0.00831808511711 0.00208288537147 0.00201301953199

53.720675 135.299305 60.547327 20.176788

ABC COABC AABC IABC

0.49941165361867 0.49651045824890 0.50329249306314 0.50599242124160

0.48084290761743 0.46615292905398 0.49651186501064 0.49700041523154

0.49195859746382 0.48189706593461 0.50159216155762 0.50256377545939

0.49115604127568 0.48149480708507 0.50114690095962 0.50205404519171

0.00457585204627 0.00729269982550 0.00184435560826 0.00014607788547

61.039208 152.954632 68.431416 22.833100

ABC COABC AABC IABC

0.46791641395795 0.47663108630828 0.47826647318126 0.47966355148656

0.45449845197715 0.43877996831973 0.46479677895660 0.46749388437386

0.46186229907359 0.45449235858095 0.47341590414215 0.47524953146626

0.46186416784146 0.45390575325417 0.47311546527464 0.47446834372235

0.00312514210054 0.00887616742542 0.00282146554944 0.00204335549495

55.955893 141.124506 62.720362 20.816273

ABC COABC AABC IABC

0.39204661470268 0.38824788080389 0.40617213985796 0.40695474513707

0.37518887098753 0.35864668363508 0.38899161444862 0.39434926306330

0.38467589084624 0.37526379264674 0.39586924053123 0.40203993459623

0.38451215931816 0.37482772517442 0.39569816668184 0.40226492636674

0.00424602081372 0.00731005604681 0.00366325677250 0.00337898875936

60.184348 149.293471 66.288018 21.169550

Convex quadratic reliability (P6)

ABC COABC AABC IABC

0.80884418963273 0.80884418963273 0.80884418963273 0.80884418963273

0.74681273794995 0.72521707461614 0.77888699742412 0.80884418968055

0.77553630479418 0.77194587851753 0.80884418963273 0.80884418963273

0.77566640415957 0.77152914496081 0.80259830957363 0.80884418966156

0.01551539293725 0.02031204285854 0.00877662844782 0.00000003409082

34.740335 88.659316 41.954146 13.225051

Incomplete fault detecting (P7)

ABC COABC AABC IABC

0.97456521646249 0.97456521646249 0.97456521646249 0.97456521646249

0.97456521646249 0.97456521646249 0.97456521646249 0.97456521646249

0.97456521646249 0.97456521646249 0.97456521646249 0.97456521646249

0.97456521646249 0.97456521646249 0.97456521646249 0.97456521646249

0.00000000000000 0.00000000000000 0.00000000000000 0.00000000000000

17.746305 45.497441 22.460371 8.221326

Mixed series-parallel system (P8)

ABC COABC AABC IABC

0.92769637069674 0.93075191423314 0.93410268160126 0.94561335745814

0.87409264326589 0.88241083409080 0.89121197360882 0.94474971679464

0.90958791672250 0.90726862017367 0.91693773959906 0.94561335745814

0.90851101639011 0.90727724189760 0.91382308292282 0.94544960752467

0.01197367254507 0.01250794087490 0.01114626941136 0.00003106719289

23.076201 59.036287 28.492323 10.425636

Large-scale system reliability (P5) Dim=36

Dim=38

Dim=40

Dim=42

Dim=50

592 593 594

Average computational time

180

ABC

COABC

AABC

160 140 120 100 80 60 40 20 0

Reliability optimization problems

Fig. 13. Comparison of ABC algorithms in terms of average computational times. 595 596

IABC



P1

1

0.996

COABC AABC

0.995 20

30

40

50

60

0.6 0.5 0.4 0.3 ABC COABC AABC IABC

0.2 0.1

IABC 10

0.7

0 0

70

500

P5 (n=36)

2000

0.48 ABC COABC AABC IABC 500

1000

1500

0.994 0.992 0.99

ABC COABC AABC IABC

0.988

20

2000

0.46

ABC COABC AABC IABC

0.42

500

1000

1500

P5 (n=50)

2000

ABC COABC AABC IABC

0.34

500

1000

1500

Iterations

2000

2500

150

P5 (n=42)

0.5 0.48 0.46

ABC COABC AABC IABC

0.44 500

1000

1500

2000

0.48

0.46


0.42

0.4 0

2500

Iterations

500

1000

1500

2000

2500

Iterations

P7

P8 0.96

0.85 0.8 0.75 0.7 0.65 ABC COABC AABC IABC

0.6 0.55 0.5 0

500

1000

1500

2000

2500

Iterations

0.9748 0.9746 0.9744 0.9742 0.974 0.9738 ABC COABC AABC IABC

0.9736 0.9734 0.9732 0

20

40

Objective function Value

0.36

100

Iterations

0.52

0.42 0

2500


0.38


0.4

50

0.5

P6

0.42

ABC COABC AABC IABC

0.99

0

0.9

0.44

0.992

0.988

0.54

Iterations

Iterations

0.994

P5 (n=40)

0.5

0.44

0.996

60

0.56

0.48

0.4 0

2500

40

0.998

Iterations



0.49

0.46

0.996

P5 (n=38)

0.5

0.47

0.998

0.986 0

2500

0.52

0.51

0.32 0

1500

Iterations

0.52

0.45 0

1000


ABC

0.8


0.997


0.998



0.999

Iterations


P4

1

0.9

0


P3

P2 1

1

60

Iterations

0.94 0.92 0.9 0.88 0.86 ABC COABC AABC IABC

0.84 0.82 0.8 0

500

1000

1500

2000

2500

Iterations

Fig. 14. Convergence curves of ABC algorithms for P1-P8.

597

4.2.5.

598 599 600

For further evaluation, this subsection compares the performance of the proposed IABC with those of the second group of algorithms that are some well-known meta-heuristic algorithms from the literature. The information of these algorithms has been presented in Table 7. The configuration setup of these algorithms can be found in their corresponding references. The best results obtained by these algorithms are reported in Tables 8-13 in bold face and the unused resources are shown with slacks.

601

Experimental results for the second group of algorithms

602 603 Table 7. Second group of algorithms (well-known meta-heuristic algorithms from the literature). Algorithms, authors, published year and references Methods

1. GAs- Hsieh and Chen (1998) [91] 2. GA- Yokota et al. (1996) [92] 3. NIP/NN-GA- Gen et al. (1998) [93] 4. SCA- Genand and Yun (2006) [94] 5. SAA- Kim et al. (2006) [95] 6. IAs- Chen. (2006) [96] 7. PSO1- Coelho. (2009) [97] 8. NGHS- Zou et al. (2010) [98] 9. MDE-HS- Liao. (2010) [99] 10.IA- Hsieh and You (2011) [100] 11. EGHS- Zou et al. (2011) [101] 12. ABC1- Yeh and Hsieh. (2011) [102] 13. IPSO- Wu et al. (2011) [103] 14. NMDE- Zou et al. (2011) [104] 15. CDEHS-Wang and Li. (2012) [105] 16. CS1- Valian and Valian. (2013) [106] 17. ICS- Valian et al. (2013) [107] 18. CS-GA- Kanagaraj et al. (2013) [108] 19. MICA- Afonso et al. (2013) [109] 20. GA-PSO- Sheikhalishahi et al. (2013) [51] 21. ABC2- Grag et al. (2013) [85] 22. LXPM-IPSO-GS Zhang et al. (2013) [110] 23. GA-SRS- Ardakan and Hamadani (2014) [111]

Genetic algorithms Genetic algorithm Hybrid neural network and genetic algorithm Soft computing approach Simulated annealing algorithm IAs-based approach Particle swarm optimization Novel global harmony search algorithm Two hybrid differential evolution algorithm An effective immune based two-phase approach Effective global harmony search algorithm Artificial bee colony algorithm Improved particle swarm optimization A novel modified differential evolution algorithm Co-evolutionary differential evolution with harmony search Cuckoo search algorithm Improved cuckoo search algorithm Hybrid cuckoo search and genetic algorithm Modified imperialist competitive algorithm Hybrid GA-PSO approach Artificial bee colony IPSO-based hybrid approaches for reliability–redundancy allocation problems Reliability–redundancy allocation problem with cold-standby redundancy strategy

Type of reliability optimization problems Mixed-integer Integer                         

    -


24. TS-DE- Lui and Qin. (2014) [112] 25. MPSO- Lui and Qin. (2014) [113] 26. INGHS- Ouyang et al. (2015) [87] 27. CS2- Grag. (2015) [114] 28. DE- Liu and Qin. (2015) [115] 29. EBBO- Grag. (2015) [116] 30. PSO/SSO/PSSO- Huang. (2015) [88] 31. NAFSA- He et al. (2015) [86] 32. PSFSA- Mellal and Zio. (2016) [117] 33. IMHS- Krishna and Ravi. (2016) [118] 34. MCS-AHGA -Yun et al (2017) [119]


A hybrid TS–DE algorithm for reliability redundancy optimization problem A modified particle swarm optimization algorithm for reliability redundancy optimization problem Improved novel global harmony search Cuckoo search algorithm A DE algorithm combined with levy flight An efficient biogeography based optimization algorithm A particle-based simplified swarm optimization algorithm A novel artificial fish swarm algorithm A penalty guided stochastic fractal search approach Modified harmony search Adaptive hybrid GA with modified CS for reliability optimization problems

          -

     

604 605 606 607 608 609 610 611 Algorithm SCA SAA GA GA-PSO IPSO IAs IA CS1 ICS CS-GA MICA CDEHS NGHS EGHS INGHS ABC1 ABC2 NMDE LXMP-IPSO-GS MPSO GA-SRS TS-DE DE CS2 EBBO PSO SSO PSSO NAFSA PSFS IABC

For complex (bridge) system (P1), as can be seen in Table 8, the best result achieved by the NMDE is 0.999889637550196, which is either better than or as good as the best result obtained by the other considered algorithms. However, rounding this value to a number with 8 significant digits gives a value of (0.99988964) that is equal to the best value reported by previous approaches such as GA-PSO, CS1, ICS, CS-GA, and CDEHS. Also, MPSO, TS-DE, DE, PSFS, and IABC algorithms have a close competition with each other for solving this problem. Table 8. Best results obtained by the second group of algorithms for complex (bridge) system (P1). (x1,x2,x3,x4,x5) (3, 3, 2, 3, 2) (3, 3, 3, 3, 1) (3, 3, 3, 3, 1) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1) (3, 3, 3, 3, 1) (3, 3, 3, 3, 1) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1) (3, 3, 3, 3, 1) (3, 3, 2, 4, 1) (3, 3, 3, 3, 1) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1) (3, 3, 2, 2, 3) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1) (3, 3, 2, 4, 1)

r1 0.814483 0.807263 0.814090 0.828134 0.82868361 0.812485 0.816624176 0.82809403 0.828094038 0.82808567 0.82764257 0.82812427 0.82983999 0.82983999 0.8279847911 0.828087 0.827970276262 0.8280864668 0.827974 0.8280816704 0.80457234 0.828086 0.82808641 0.827855652338 0.8280606892 0.77061588 0.82008362 0.82783292 0.82832179189 0.82812141729 0.8281009551

r2 0.821383 0.868116 0.864614 0.857831 0.85802567 0.867661 0.868767396 0.85800448 0.858004485 0.85780605 0.85747845 0.85784692 0.85798911 0.85798911 0.8576796813 0.857805 0.857874758586 0.8578047868 0.857818 0.8578118137 0.85717305 0.857805 0.85780478 0.857626105413 0.8580404545 0.90109253 0.85119629 0.85771241 0.85797450730 0.85781341076 0.8578100118

r3 0.896151 0.872862 0.890291 0.914192 0.91364616 0.861221 0.858748781 0.9141629 0.914162924 0.91424006 0.91419677 0.91420816 0.91333926 0.91333926 0.9141564522 0.704163 0.914186404228 0.9142407172 0.914166 0.9142411461 0.86734683 0.914241 0.91424067 0.914752916604 0.9141487486 0.89278651 0.91854858 0.91437458 0.91422098825 0.91423927822 0.9141983193

r4 0.713091 0.7126673 0.701190 0.648069 0.64803407 0.713852 0.710279379 0.64790779 0.647907792 0.64814375 0.64927379 0.64808425 0.64674479 0.64674479 0.6484814055 0.648146 0.648355386813 0.6481460609 0.648348 0.6481547109 0.72759162 0.648146 0.64814622 0.648217208595 0.6479689012 0.60083008 0.66072083 0.64861002 0.64775717018 0.64807680660 0.6482027361

r5 0.814091 0.751034 0.734731 0.704476 0.70227595 0.756699 0.753429200 0.70456598 0.704565982 0.70418228 0.70409200 0.70411685 0.70310972 0.70310972 0.7048654988 0.914240 0.703575311047 0.7041626466 0.704427 0.7040665038 0.76416666 0.704162 0.70416210 0.702670374782 0.7042048796 0.73451002 0.70275879 0.70287554 0.70300666185 0.70424641245 0.7039474153

f(r,x) 0.9997894 0.99988764 0.99987916 0.99988964 0.99988963 0.99988921 0.9998893505 0.99988964 0.99988964 0.99988964 0.99988963 0.99988964 0.99988960 0.99988960 0.9998896364 0.99988962 0.999889635809 0.999889637550196 0.999889636851887 0.999889637544786 0.999886726841704 0.999889637462077 0.999889637503023 0.999889631978 0.9998896364 0.99967140 0.99988862 0.999889635738075 0.99988963601 0.999889637512067 0.99988963740661

Slack(g1) 18 40 18 5 5 18 18 5 5 5 5 5 5 5 5 5 5 5 5 5 18 5 5 5 5 37 5 5 5 5 5

Slack(g2) 1.854075 0.007300 0.376347 0.00000 0.00000359 0.001494 4.0420871 ˟ 10-8 0.00007929 0.00007929 0.00000002 0.00004428 5.93e+07 0.00000594 0.00000594 0.00000189 - 25.43392577 3.74636763 ˟ 10-4 1.44763419 ˟ 10-8 3.35125258 ˟ 10-5 - 9.141729151 ˟ 10-9 6.77854912 ˟ 10-5 3.71181366 ˟ 10-5 1.987824296˟10-5 1.06723518 ˟ 10-10 1.4541 ˟ 10-4 16.54571312 0.00437599 2.50290205 ˟ 10-5 1.5485 ˟ 10-5 2.854643724 ˟ 10-6 0.00003291

Slack(g2) 4.264770 1.609289 4.264770 1.560466 1.56046629 4.264770 4.264770 1.56046628 1.560466288 1.56046629 1.56046629 1.560466 1.56046629 1.56046629 1.56046628 1.560466288 1.560466288 1.560466288 1.560466288 1.560466288 4.264769804 1.560466288 1.560466288 1.560466288 1.560466 1.411803224 1.560466288 1.560466288 1.56046629 1.560466288 1.560466288

612 613 614 615 616 617

Table 9 shows that the IABC has slightly improved the best-known solution found by the previous methods for solving series system. It is important to note that achieving even a very small improvement in the reliability of real-world and complex systems is very valuable. However, the effect of such a small change on the safety and efficiency of reliability systems is essential [87, 101, 107].

618 619

Table 9. Best results obtained by the second group of algorithms for series system (P2).

Algorithm SCA SAA GA IPSO IAs IA CS1 ICS CS-GA MICA INGHS ABC1 ABC2 GA-SRS LXPM-ISPO-GS MPSO

(x1,x2,x3,x4,x5) (3, 2, 2, 3, 3) (3, 2, 2, 3, 3) (3, 2, 2, 3, 3) (3, 2, 2, 3, 3) (3, 2, 2, 3,3) (3, 2, 2, 3,3) (3, 2, 2, 3,3) (3, 2, 2, 3, 3) (3, 2, 2, 3, 3) (3, 2, 2, 3, 3) (3, 2, 2, 3, 3) (3, 2, 2, 3, 3) (3, 2, 2, 3, 3) (3, 2, 2, 3, 3) (3, 2, 2, 3, 3) (3, 2, 2, 3, 3)

r1 0.777143 0.782391 0.779427 0.78037307 0.779266 0.779462304 0.77941693 0.779416938 0.779398871 0.779874 0.7793988710 0.779399 0.779403565208 0.76459335 0.779509 0.7793996871

r2 0.867514 0.866712 0.869482 0.87178343 0.872513 0.871883456 0.87183327 0.871833278 0.871837021 0.872057 0.8718370210 0.871837 0.87183320141 0.88752892 0.871859 0.8718379458

r3 0.896696 0.901747 0.902674 0.90240890 0.902634 0.902800879 0.90288508 0.902885082 0.902885355 0.903426 0.9028853550 0.902885 0.902886411643 0.91539527 0.902891 0.9028848599

r4 0.717739 0.717266 0.714038 0.71147356 0.710648 0.711350168 0.71139386 0.711393868 0.711402515 0.710960 0.7114025151 0.711403 0.711398061305 0.69350544 0.711345 0.7114027590

r5 0.793889 0.783795 0.786896 0.78738760 0.788406 0.787861587 0.78780371 0.787803712 0.787799488 0.786902 0.787799488032 0.787800 0.787808548579 0.77603145 0.787739 0.7877970932

f(r,x) 0.931363 0.931460 0.931680 0.931680 0.931678 0.931682340 0.93168238 0.931682387 0.931682388 0.93167939 0.931682387723 0.931682 0.931682387672 0.929082263568 0.93168236 0.931682387881029

Slack(g1) 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27

Slack(g2) 0.00000 0.053194 0.121454 0.000101 0.001559 5.284 ˟ 10-7 0.00000026 0.000000265 0.000000 0.000099 0.00000031 -2.183652 ˟ 10-4 2.258957 ˟ 10-10 2.26949936 ˟ 10-5 2.20 ˟ 10-7 2.279378463 ˟ 10-8

Slack(g3) 7.518918 7.518918 7.518918 7.518918 7.518918 7.518918 7.518918241 7.518918241 7.51891824 7.518918 7.518918241 7.518918241 7.518918241 7.5189182411 7.518918 7.51891824115


CS2 EBBO PSO SSO PSSO NAFSA PSFS IABC

(3, 2, 2, 3, 3) (3, 2, 2, 3, 3) (2, 3, 2, 4, 2) (3, 2, 2, 3, 3) (3, 2, 2, 3, 3) (3, 2, 2, 3, 3) (3, 2, 2, 3, 3) (3, 2, 2, 3, 3)

0.779439734086 0.7794057271 0.80059281 0.78271484 0.77946645 0.77938841387 0.77939888253 0.7793615527


0.871995212292 0.8718339252 0.74049316 0.8735199 0.87173278 0.87172098236 0.8718370134 0.8718512627

0.902873050171 0.9028816332 0.82914384 0.90264893 0.90284951 0.90303339184 0.90288535705 0.9028570058

0.711127088245 0.7114011594 0.63686144 0.71313477 0.71148780 0.71141836221 0.71140251717 0.7114247990

0.787986374473 0.7878058784 0.88704276 0.77729797 0.78781644 0.78778928811 0.7877994847351 0.787811636096

0.931682106582 0.9316823874 0.8885037 0.93150199 0.931682297215271 0.93168226855 0.931682387907 0.93168238886088

4.42986674 ˟ 10-7 4.7045 ˟ 10-7 16.775727 0.00182140 4.90819664 ˟ 10-5 6.7347 ˟ 10-9 4.13251655 ˟ 10-11 0.000018583

27 27 4 27 27 27 27 27

7.518918241 7.518918 4.8146147 7.51891824 7.51891824 7.518918 7.5189182411 7.518918241

620 621 622 623 624 625 626 627 Algorithm SCA SAA GA GA-PSO IPSO CS1 ICS CS-GA IAs IA MICA CDEHS INGHS ABC1 ABC2 GA-SRS TS-DE LXMP-IPSO-GS MPSO CS2 DE EBBO PSO SSO PSSO NAFSA PSFS IABC

As can be seen in Table 10, the best optimal solution for series-parallel system is 0.99997982961494 that is obtained by the IABC algorithm. In fact, the proposed IABC has succeeded to improve considerably the best-known solution found so far by the twentyseven competitive algorithms. This is due to the fact that by employing the new search operator, the IABC not only is able to provide more accurate solutions but also makes the search balance in favor of exploitation behavior. Table 10. Best results obtained by the second group of algorithms for series-parallel system (P3). (x1,x2,x3 ,x4,x5)

(3, 3, 1, 2, 3) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (3 ,3 ,2 ,1 ,3) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (4, 3, 2, 1, 2) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (2, 2, 2, 2, 4) (3, 3, 2, 2, 3)

r1 0.838193 0.812161 0.785452 0.819640 0.81918526 0.819927087 0.819927087 0.819660256 0.812485 0.819591561 0.82201264 0.819596 0.8198118626 0.8197457 0.819737753469 0.824846726 0.819659 0.819509 0.8196547522 0.819483232488 0.81965932 0.8196583448 0.84025282 0.81385803 0.81958939 0.819787575273 0.81965939118 0.827743712027

r2 0.855065 0.853346 0.842998 0.845091 0.84366421 0.845267657 0.845267657 0.844981615 0.843155 0.844951068 0.84365640 0.845000 0.8449506842 0.8450080 0.844991099776 0.842816570 0.844981 0.845312 0.8449752789 0.844783084455 0.84498074 0.8449101406 0.88865099 0.83912659 0.84458412 0.845671943728 0.84498085296 0.847138611794

r3 0.878859 0.897597 0.885333 0.895482 0.89472992 0.895491554 0.895491554 0.895519305 0.897385 0.895428548 0.89129092 0.895514 0.8956701585 0.8954581 0.895529543820 0.908173083 0.895507 0.895498 0.8955087772 0.895810553887 0.89550642 0.8954871713 0.62375055 0.89366150 0.89534134 0.894868363315 0.89550643076 0.856891035304

r4 0.911402 0.900710 0.917958 0.895517 0.89537628 0.895440692 0.895440692 0.895492245 0.894516 0.895522339 0.89869886 0.895519 0.8952327069 0.9009032 0.895433687206 0.898699000 0.895506 0.895568 0.8955091117 0.895220216915 0.89550643 0.8955148963 0.93984950 0.89845276 0.89581626 0.895908268569 0.89550645172 0.856634348041

r5 0.850355 0.866316 0.870318 0.868430 0.86912724 0.868318775 0.868318775 0.868447587 0.870590 0.868490229 0.86824939 0.868456 0.868438057445 0.8684069 0.868434824469 0.865463014 0.868448 0.868396 0.8684491638 0.868542486973 0.86844775 0.8684681613 0.75158691 0.87106323 0.86852902 0.868295830551 0.86844769346 0.875976531093

f(r,x) 0.99996875 0.99997631 0.99997418 0.99997665 0.99997664 0.999976649 0.999976649 0.99997665 0.99997658 0.999976649036520 0.99997661 0.99997665 0.9999766489 0.99997731 0.999976649054 0.999969638668474 0.9999766491 0.999976648938797 0.999976649066114 0.999976648818 0.999976649066076 0.999976649048875 0.99985845 0.99997657 0.999976648738107 0.999976648004 0.999976649066172 0.99997982961494

Slack(g1) 53 40 40 40 40 40 40 40 40 40 40 40 40 40 40 62 40 40 40 40 40 40 68 40 40 40 40 38

Slack(g2) 0.00000 0.007300 1.194440 0.000001 0.000561 0.0000161 0.0000161 0.000000017 0.002627 5.984537665 ˟ 10-4 0.000396 0.000007 0.00005305414 - 1.469522338 1.39152 ˟ 10-10 6.38655843 ˟ 10-5 - 2.66935542 ˟ 10-4 -1.71878053 ˟ 10-4 4.783657914 ˟ 10-9 2.7216628 ˟ 10-10 1.961642794 ˟ 10-7 1.748541669 ˟ 10-5 0.916915088 0.0024 8.11794613 ˟ 10-5 3.1248 ˟ 10-8 1.85082171 ˟ 10-10 0.0000183067

Slack(g2) 7.110849 1.609289 1.609289 1.609289 1.609289 1.6092890 1.6092890 1.60928897 1.609289 1.6092889667 1.609289 1.609289 1.609288966 1.6092889667 1.609288966 6.104140277 1.609288966 1.6092889667 1.6092889667 1.609288966 1.609288966 1.6092889667 4.017703641 1.609288966 1.609288966 1.609289 1.6092889667 0.705901612

628

631 632 633

For P4 problem, the results in Table 11 indicate that the best value which the proposed IABC has obtained is 0.99995467566652. Rounding this value to a number with 8 significant digits yields the value of 0.99995468 which is equal to the best value reported by previous approaches such as CS and ICS. Also, rounding it to a number with 10 significant digits produces the value of 0.9999546757 which is better than the values obtained by the INGHS, ABC2, and PSFS. Therefore, the IABC has equal or better performance than the other algorithms for solving over-speed protection system.

634 635

Table 11. Best results obtained by the second group of algorithms for over-speed protection system (P4).

629 630

Algorithm SAA PSO1 IPSO GA-PSO IAs IA CS1 ICS CS-GA MICA NGHS EGHS INGHS ABC2 DE TS-DE LXPM-IPSO-GS NMDE MPSO CS2 EBBO PSO SSO PSSO NAFSA PSFS

(x1,x2,x3,x4) (5, 5, 5, 5) (5, 5, 4, 5) (5, 5, 4, 5) (5, 5, 4, 6) (5, 5, 5, 5) (5, 5, 4, 6) (5, 5, 4, 6) (5, 5, 4, 5) (5, 5, 4, 6) (5, 5, 4, 5) (5, 5, 4, 5) (5, 6, 4, 5) (5, 5, 4, 6) (5, 6, 4, 5) (5, 6, 4, 5) (5, 6, 4, 5) (5, 5, 4, 6) (5, 6, 4, 5) (5, 6, 4, 5) (5, 5, 4, 6) (5, 5, 4, 6) (4, 6, 5, 5) (5, 6, 4, 5) (5, 5, 4, 6) (5, 6, 4, 5) (5, 6, 4, 5)

r1 0.895644 0.902231 0.90163164 0.901628 0.903800 0.901588628 0.90161459 0.901614595 0.901613407 0.90148988 0.900925066 0.900925066 0.9015565830 0.901614 0.90161482 0.901615 0.90163317 0.90161480 0.9016123483 0.901598077027 0.9015629232 0.92952331 0.90208435 0.90166461 0.90160779120 0.9016147331806

r2 0.885878 0.856325 0.84997020 0.888230 0.874992 0.888192380 0.888223369 0.888223369 0.888223375 0.85003526 0.851636929 0.851636929 0.8882438856 0.849920 0.84992114 0.849921 0.888251065 0.84992111 0.8499199719 0.888226184172 0.8882249447 0.81370356 0.85472107 0.88817296 0.84993077684 0.8499211645063

r3 0.912184 0.948145 0.94821828 0.948121 0.919898 0.948166022 0.94814102 0.948141029 0.948142110 0.94812952 0.948079849 0.948079849 0.9481110971 0.948143 0.94814139 0.948141 0.948141377 0.94814139 0.9481399512 0.948101861662 0.9481559581 0.88663747 0.94606018 0.94821033 0.94814603278 0.9481413898152

r4 0.887785 0.883156 0.88812885 0.849921 0.890609 0.849969792 0.84992089 0.849920899 0.849920787 0.88823833 0.887654500 0.887654500 0.8499817375 0.888223 0.88822284 0.888223 0.849854043 0.88822286 0.8882260306 0.849980778637 0.8499528951 0.89987183 0.88633728 0.84987084 0.88821809379 0.8882228732368

f(r,x) 0.999945 0.999953 0.99995467 0.99995467 0.999942 0.999954674554580 0.99995468 0.99995468 0.999954675 0.999954673 0.999955 0.999955 0.9999546743 0.999954674798458 0.99995467 0.999954674608111 0.999954674599810 0.99995467 0.999954674676319 0.999954674585 0.9999546746 0.99990474 0.99995416 0.99995467 0.99995467467 0.999954674676782

Slack(g1) 50 55 55 55 50 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 37 55 55 55 55

Slack(g2) 0.9380 0.975465 0.000009 0.000006 0.002152 1.24953721 ˟ 10-4 0.00000000 0.0000000096 0.0000001 0.00213782 0.00000584 0.00000584 0.0000505411 - 3.36378261 ˟ 10-4 1.00507377 ˟ 10-5 1.89670533 ˟ 10-4 5.30549334 ˟ 10-6 1.057 ˟ 10-5 1.901702262 ˟ 10-7 8.82494077 ˟ 10-10 2.7021 ˟ 10-5 11.5265677 0.109233104 3.2872253 ˟ 10-5 4.5195 ˟ 10-7 5.3387338 ˟ 10-10

Slack(g3) 28.8037 24.801882 24.081883 15.363463 28.803701 15.363463087 15.36346309 15.36346309 15.3634631 24.8018827 24.80188272 24.80188272 24.8018827221 24.80188272 24.80188272 24.80188272 15.36346308 24.80188272 24.8018827 15.3634630874 15.3634631 11.6447077 24.8018827 15.36346308 24.802 24.801882722


IABC

(5, 5, 4, 6)

55

0.99995467566652

0.0000220165

15.3634630874

Table 12. Best results obtained by the second group of algorithms for large-scale system reliability problem (P5).

40

42

50

642 643 644 645 646 647 648 649 650

P8

0.8499329515

640 641

38

P7

0.9481385252

Table 12 reports that the IABC and INGHS algorithms have been able to find the best solutions found so far for P5 problem with different dimensions. Here, VTV demonstrates the variables which adopt the value of 2 in optimum, and the other variables which have the value of 1 in optimum.

36

P6

0.8882184782

636 637 638 639

Dim

Problem

0.9016139540


Algorithm SCA CS1 ICS IPSO NGHS INGHS IABC

VTV {5, 10, 15, 21, 33} {5, 10, 15, 21, 33} {5, 10, 15, 21, 33} {5, 10, 15, 21, 33} {5, 10, 15, 21, 33} {5, 10, 15, 21, 33} {5, 10, 15, 21, 33}

f(x) 0.519976 0.51997597 0.519976 0.519976 0.519976 0.51997596538026 0.5199759653802567

Slack(g1) 1 1

Slack(g2) 49.125763519460 49.1257635194601790

Slack(g3) 109 109

Slack(g4) 301.353247018274 301.3532470182742600

SCA CS1 ICS IPSO NGHS INGHS IABC

{10, 13, 15, 21, 33} {10, 13, 15, 21, 33} {10, 13, 15, 21, 33} {10, 13, 15, 21, 33} {10, 13, 15, 21, 33} {10, 13, 15, 21, 33} {10, 13, 15, 21, 33}

0.510989 0.51098860 0.510989 0.510989 0.510989 0.51098859649712 0.5109885964971198

1 1

53.638550812459 53.6385508124589020

115 115

317.039538519290 317.0395385192896400


{5, 10, 13, 15, 33} {4, 10, 11, 21, 22, 33} {5, 10, 13, 15, 33} {5, 10, 13, 15, 33} {5, 10, 13, 15, 33} {4, 10, 11, 21, 22, 33} {4, 10, 11, 21, 22, 33}

0.503292 0.50599242 0.503292 0.503292 0.503292 0.505992421241597 0.5059924212415972

0 0

51.04714167016368 51.0471416701636830

119 119

333.24054864606615 333.2405486460661500


{4, 10, 11, 15, 21, 33} {4, 10, 11, 15, 21, 33} {4, 10, 11, 15, 21, 33} {4, 10, 11, 15, 21, 33} {4, 10, 11, 15, 21, 33} {4, 10, 11, 15, 21, 33} {4, 10, 11, 15, 21, 33}

0.479664 0.47966355 0.479664 0.479664 0.479664 0.47966355148656 0.4796635514865568

2 2

52.718250389045 52.7182503890448400

129 129

354.583694396574 354.5836943965737200

CS1 ICS IPSO NGHS INGHS IABC

{4, 10, 15, 21, 33, 42, 45} {4, 10, 15, 21, 33, 45, 47} {4, 10, 15, 21, 33, 45, 47} {4, 10, 15, 21, 33, 45, 47} {4, 10, 15, 21, 33, 42, 45} {4, 10, 15, 21, 33, 42, 45}

0.40695474 0.405390 0.405390 0.405390 0.40695474513707 0.4069547451370713

0 0

61.955982588824 61.9559825888243270

154.0 154.0

433.914646838262 423.9146468382616600

Table 13 exhibits the best results for P6, P7, and P8 problems found by the IABC and the other algorithms. As can be seen, the best results obtained by the IABC are the same as those found by INGHS algorithm. However, this table demonstrates that the proposed IABC algorithm outperforms the competitive algorithms in terms of worst and mean results for solving P6 and P8 problems, and in terms of standard deviation results for P7 and P8 problems. Thus, it can be concluded that the proposed IABC is more robust than these algorithms. Table 13. Best results obtained by the second group of algorithms for P6-P8 problems.

Algorithms SGA GAs [92] MDE-HS INGHS IABC

Best 0.808844 0.808844 0.808844 0.80884418963273 0.80884418963273

Worst 0.778631 0.803666 0.8088441896 0.80884418968055

Mean 0.805823 0.808276 0.808844 0.8088441896 0.80884418966156

SD 1.1215E-16 3.4090E-07

g(1) -0.9649307648E+13 -0.96493076480E+13

g(2) -0.0202998323E+13 -0.020299832320E+13

g(3) -4.3632406548E+13 -4.363240654848E+13

g(4) -0.0871498795E+13 -0.087149879500E+13

n (2, 2, 2, 1, 1, 2, 3, 2, 1, 2) (2, 2, 2, 1, 1, 2, 3, 2, 1, 2) (2, 2, 2, 1, 1, 2, 3, 2, 1, 2) (2, 2, 2, 1, 1, 2, 3, 2, 1, 2)

SGA GAs [92] MDE-HS INGHS IABC

0.974565 0.974565 0.974565 0.97456521646249 0.97456521646249

0.97456521646249 0.97456521646249

0.974565 0.97456521646249 0.97456521646249

6.7290E-16 0.0000E+00

-25 -25

-24.29939329763320 -24.29939329763320

-0.24945862219943 -0.24945862219943

-

(3, 3, 2, 3) (3, 3, 2, 3) (3, 3, 2, 3) (3, 3, 2, 3)

GA NIP/NN-GA IMHS INGHS MCS-AHGA IABC

0.9202 0.94471 0.945613 0.94561335745814 0.9456 0.94561335745814

0.847438 0.93296 0.9447484845 0.94474971679464

0.893046 0.944315 0.9454403828 0.94544960752467

3.6466E-04 3.1067E-04

-8 -8

0 0

-

-

(3, 4, 5, 3, 3, 2, 4, 5, 4, 3, 3, 4, 5, 5, 5) (3, 4, 5, 3, 3, 2, 4, 5, 4, 3, 3, 4, 5, 5, 5) (3, 4, 6, 4, 3, 2, 4, 5, 4, 2, 3, 4, 5, 4, 5) (3, 4, 6, 4, 3, 2, 4, 5, 4, 2, 3, 4, 5, 4, 5) (3, 4, 5, 3, 3, 2, 4, 5, 4, 3, 3, 4, 5, 5, 5) (3, 4, 6, 4, 3, 2, 4, 5, 4, 2, 3, 4, 5, 4, 5)

651 652 653 654 655

Generally speaking, the comparison results above confirm that the IABC algorithm is a promising meta-heuristic algorithm and can be a proper choice for solving real-world and complex reliability optimization problems.


656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694


5. Discussion and Conclusions This paper introduced an improved artificial bee colony (IABC) algorithm as a new ABC extension with the aim of solving both numerical and practical real-world optimization problems effectively and efficiently. In order to accelerate the convergence speed, the IABC algorithm utilized a probabilistic linear population size reduction mechanism. Thus, the proposed IABC started exploring the search space by a large enough population size. Then, as the search process proceeded, this mechanism progressively narrowed the search space and increasingly focused on more promising search areas by transferring the good solutions to the next populations. Furthermore, in addition to the original search operator of standard ABC, the IABC employed a new search operator which helped the algorithm to enhance the exploitation capability. This operator produced a new solution by utilizing the information of a randomly selected pair of solutions and the current best solution. Finally, the IABC presented an improved bee phase which was a fusion of the employed and onlooker bee phases. This new phase used a self-adaptive probabilistic selection scheme to decide how to select between two search operators during different stages of the search process. The original search operator was more likely to be selected by the self-adaptive probabilistic selection scheme at the early stages of the search process. However, as the search process proceeded, the selection probability of the new search operator increased which kept the good solutions in the population. Thus, the search balance changed in favor of exploitation behavior. To verify the general properties of IABC algorithm, some experiments were performed against CEC2014 test suite and the results were compared with 16 state-of-the-art meta-heuristic algorithms including ABC, COABC, AABC, EABC, qABC, dABC, ABCVSS, DFSABC_elite, GA, PSO, DE, CS, NRGA, OPTBees, FWA-DM, and b3e3pbest. The statistical results indicated that the IABC has a close competition with b3e3pbest and is superior to the other algorithms in terms of convergence speed, robustness, and solution accuracy. Also, the application of IABC was investigated through eight well-known reliability optimization problems (P1-P8) including four mixed integer programming problems (P1-P4) and four integer programming problems (P5-P8). First, the performance of IABC was compared with those of ABC, COABC, and AABC. According to the reported results, the IABC algorithm outperformed the other three algorithms for solving P1, P2, P3, P4, and P8 problems. Although for low dimension (36D and 38D) cases of P5, the IABC and AABC achieved the best solution, for high dimension (42D and 50D) cases of P5, the performance of IABC was superior to all the other considered algorithms. This means that the population size reduction mechanism played its role in inhibiting a stagnant situation for solving problems with high dimensionality. For P6 and P7 problems, the IABC had either equal or better performance than the compared algorithms. Furthermore, as a result of population size reduction mechanism, the average computational time spent by the IABC for solving each problem was considerably smaller than the other algorithms. For further validation, the efficiency of IABC was compared with thirty-four meta-heuristic algorithms from the literature. For P4, P5, P6, P7, and P8 problems, the best results achieved by the IABC algorithm were either better or as good as the best results obtained by other considered algorithms. Notably, because of the self-adaptive probabilistic selection scheme incorporated in the improved bee phase, the proposed IABC obtained an optimal solution for series system (P2) which was slightly better than the best-known solution found so far for this problem, and more importantly, it considerably improved the best-known solution for series-parallel system (P3). All the above evaluations confirm that the proposed IABC is a competitive algorithm which performs as well as or even better than the state-of-the-art algorithms in the literature for solving both numerical and reliability optimization problems.

695 696

References

697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

J.H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, MI, USA, 1975. J. Kennedy, R. Eberhart, Particle swarm optimization, in: Proceedings- IEEE International Conference Neural Networks. 4 (1995) 1942– 1948. R. Storn, K. Price, Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces, J. Glob. Optim. 11 (1997) 341–359. M. Dorigo, T. Stutzle, Ant Colony Optimization, Massachusetts Insititute of Technology Press, Cambridge, MA, 2004. D. Karaboga, An Idea Based on Honey Bee Swarm for Numerical Optimization, Technical Report-TR06, Erciyes University, Kayseri Turkey. (2005) 1-10. M.A. Awadallah, A.L.a. Bolaji, M.A. Al-Betar, A hybrid artificial bee colony for a nurse rostering problem, Appl. Soft Comput. 35 (2015) 726-739. A.L.a. Bolaji, A.T. Khader, M.A. Al-Betar, M.A. Awadallah, A hybrid nature-inspired artificial bee colony algorithm for uncapacitated examination timetabling problems, J. Intell. Syst. 24(1) (2015) 37-54. Y. Shi, C.M. Pun, H. Hu, H. Gao, An improved artificial bee colony and its application, Knowledge-Based. Syst. 107 (2016) 14-31. J.A. Delgado-Osuna, M. Lozano, C. García-Martínez, An alternative artificial bee colony algorithm with destructive–constructive neighbourhood operator for the problem of composing medical crews, Inf. Sci. 326 (2016) 215-226. P.N. Hong, C.W. Ahn, Fast artificial bee colony and its application to stereo correspondence, Expert Syst. Appl. 45 (2016) 460-470. A. Rubio-Largo, M.A. Vega-Rodríguez, D.L. González-Álvarez, Hybrid multiobjective artificial bee colony for multiple sequence alignment, Appl. Soft Comput. 41 (2016) 157-168. D. Karaboga, E. Kaya, An adaptive and hybrid artificial bee colony algorithm (aABC) for ANFIS training, Appl. Soft Comput. 49 (2016) 423-436.


717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786

[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

[27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]


M. Lozano, C. García-Martínez, F.J. Rodríguez, H.M. Trujillo, Optimizing network attacks by artificial bee colony, Inf. Sci. 377 (2017) 30-50. J. Zhou, X. Yao, Multi-population parallel self-adaptive differential artificial bee colony algorithm with application in large-scale service composition for cloud manufacturing, Appl. Soft Comput. 56 (2017) 379-397. K. Taheri, M. Hasanipanah, S.B. Golzar, M.Z.A Majid, A hybrid artificial bee colony algorithm-artificial neural network for forecasting the blast-produced ground vibration, Eng. Comput. 33(3) (2017) 689-700. C.B. Kalayci, O. Ertenlice, H. Akyer, H. Aygoren, An artificial bee colony algorithm with feasibility enforcement and infeasibility toleration procedures for cardinality constrained portfolio optimization, Expert Syst. Appl. 85 (2017) 61-75. M. Li, Y. Hei, Z. Qiu, Optimization of multiband cooperative spectrum sensing with modified artificial bee colony algorithm, Appl. Soft Comput. 57 (2017) 751-759. Y. Kumar, G. Sahoo, A two-step artificial bee colony algorithm for clustering, Neural Comput. Appl. 28(3) (2017) 537-551. S. Sundar, P.N. Suganthan, C.T. Jin, C.T. Xiang, C.C. Soon, A hybrid artificial bee colony algorithm for the job-shop scheduling problem with no-wait constraint, Soft Comput. 21(5) (2017) 1193-1202. D. Karaboga, B. Akay, A comparative study of artificial bee colony algorithm, Appl. Math. Comput. 214(1) (2009) 108-132. D. Karaboga, B. Gorkemli, C. Ozturk, N. Karaboga, A comprehensive survey: artificial bee colony (ABC) algorithm and applications, Artif. Intell. Rev. 42(1) (2014) 21-57. L. Cui, G. Li, Q. Lin, Z. Du, W. Gao, J. Chen, N. Lu, A novel artificial bee colony algorithm with depth-first search framework and elite-guided search equation, Inf. Sci. 367 (2016) 1012-1044. X. Song, Q. Yan, M. Zhao, An adaptive artificial bee colony algorithm based on objective function value information, Appl. Soft Comput. 55 (2017) 384-401. D.H. Wolpert, W.G. Macready, No free lunch theorems for optimization, IEEE Trans. Evol. Comput. 1 (1997) 67–82. Á.E. Eiben, R. Hinterding, Z. Michalewicz, Parameter control in evolutionary algorithms, IEEE Trans. on Evol. Comput. 3(2) (1999) 124–141. C.M. Fernandes, A.C. Rosa, Self-regulated population size in evolutionary algorithms, in: T.P. Runarsson, H.-G. Beyer, E.K. Burke, J.J.M. Guervós, L.D. Whitley, X. Yao (Eds.), Parallel Problem Solving from Nature - PPSN IX, 9th International Conference, Reykjavik, Iceland, September 9-13, 2006, Procedings, Lecture Notes in Computer Science, Springer. 4193 (2006) 920–929. S.K. Smit, A.E. Eiben, Comparing parameter tuning methods for evolutionary algorithms, in: IEEE Congress on Evolutionary Computation, IEEE Press, Trondheim. (2009) 399–406. L. Ma, Y. Zhu, D. Zhang, B. Niu, A hybrid approach to artificial bee colony algorithm, Neural Comput. Appl. 27(2) (2016) 387-409. L. Cui, G. Li, Z. Zhu, Q. Lin, Z. Wen, N. Lu, KC. Wong, J. Chen, A novel artificial bee colony algorithm with an adaptive population size for numerical function optimization, Inf. Sci. 414 (2017) 53-67. W. Gao, S. Liu, Improved artificial bee colony algorithm for global optimization, Inf. Processing Letters, 111(17) (2011) 871-882. W. Gao, S. Liu, L. Huang, A global best artificial bee colony algorithm for global optimization, J. Comput. Appl. Math. 236(11) (2012) 2741-2753. D. Bose, S. Biswas, S. Kundu, S. Das, A strategy pool adaptive artificial bee colony algorithm for dynamic environment through multipopulation approach, Swarm Evol. Memet. Comput. Springer. 7677 (2012) 611-619. S. Biswas, S. Kundu, D. Bose, S. Das, P.N. Suganthan, B.K. Panigrahi, Migrating forager population in a multi-population Artificial Bee Colony algorithm with modified perturbation schemes, in: IEEE Symposium on Swarm Intelligence (SIS). (2013) 248-255. S.K. Nseef, S. Abdullah, A.Turky, G. Kendall, An adaptive multi-population artificial bee colony algorithm for dynamic optimisation problems, Knowledge-Based. Syst. 104 (2016) 14-23. T.K. Sharma, M. Pant, J.C. Bansal, Artificial bee colony with mean mutation operator for better exploitation, in: IEEE Congress on Evolutionary Computation (CEC), Brisbane, QLD. (2012) 1-7. J. Luo, Q. Wang, X. Xiao, A modified artificial bee colony algorithm based on converge-onlookers approach for global optimization, Appl. Math. Comput. 219(20) (2013) 10253-10262. T.K. Sharma, M. Pant, Enhancing the food locations in an artificial bee colony algorithm, Soft Comput. 17(10) (2013) 1939-1965. W.-f. Gao, S.-y. Liu, L.-l. Huang, Enhancing artificial bee colony algorithm using more information-based search equations, Inf. Sci. 270 (2014) 112-133. D. Karaboga, B. Gorkemli, A quick artificial bee colony (qABC) algorithm and its performance on optimization problems, Appl. Soft Comput. 23 (2014) 227-238. M.S. Kıran, O. Fındık, A directed artificial bee colony algorithm, Appl. Soft Comput. 26 (2015) 454-462. MS. Kiran, H. Hakli, M. Gunduz, H. Uguz, Artificial bee colony algorithm with variable search strategy for continuous optimization, Inf. Sci. 300 (2015) 140-157. A. Yurtkuran, E. Emel, An adaptive artificial bee colony algorithm for global optimization, Appl. Math. Comput. 271 (2015) 10041023. P. Loubière, A. Jourdan, P. Siarry, R. Chelouah, A sensitivity analysis method for driving the Artificial Bee Colony algorithm's search process, Appl. Soft Comput. 41 (2016) 515-531. R.A. Cropp, R.D. Braddock, The New Morris Method: an efficient second-orderscreening method, Reliab. Eng. Syst. Safety. 78 (1) (2002) 77–83. G. Li, L. Cui, X.Fu, Z. Wen, N. Lu, J. Lu, Artificial bee colony algorithm with gene recombination for numerical function optimization, Appl. Soft Comput. 52 (2017) 146-159. Z. Liang, K. Hu, Q. Zhu, Z. Zhu, An enhanced artificial bee colony algorithm with adaptive differential operators, Appl. Soft Comput. 58 (2017) 480-494. Y. Xue, J. Jiang, B. Zhao, T. Ma, A self-adaptive artificial bee colony algorithm based on global best for global optimization, Soft Comput. (2017) 1-18. (doi: 10.1007/s00500-017-2547-1) L. Cui, G. Li, X. Wang, Q. Lin, J. Chen, N. Lu, J. Lu, A ranking-based adaptive artificial bee colony algorithm for global numerical optimization, Inf. Sci. 417 (2017) 169-185. F. Zhong, H. Li, S. Zhong, An improved artificial bee colony algorithm with modified-neighborhood-based update operator and independent-inheriting-search strategy for global optimization, Eng. Appl. Artif. Intell. 58 (2017) 134-156. F. Kang, J. Li, Q. Xu, Structural inverse analysis by hybrid simplex artificial bee colony algorithms, Computers & Structures. 87(13) (2009) 861-870.


787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855

[51] [52] [53] [54] [55] [56] [57] [58] [59]

[60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74]

[75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86]


H.-b. Duan, C.-f. Xu, Z.-H. Xing, A hybrid artificial bee colony optimization and quantum evolutionary algorithm for continuous optimization problems, I. J. Neural Syst. 20(01) (2010) 39-50. X. Shi, Y. Li, H. Li, R. Guan, L. Wang, Y. Liang, An integrated algorithm based on artificial bee colony and particle swarm optimization, in: Proceedings – IEEE International Conference on Neural Networks. 5 (2010) 2586–2590. F. Kang, J. Li, H. Li, Z. Ma, Q. Xu, An improved artificial bee colony algorithm, In: IEEE 2nd International Workshop on Intelligent Systems and Applications, Wuhan, China. (2010) 791–794. (doi: 10.1109/IWISA.2010.5473452). R.K. Jatoth, A. Rajasekhar, Speed control of PMSM by hybrid genetic artificial bee colony algorithm, In: IEEE International Conference on Communication Control and Computing Technologies (ICCCCT). (2010) 241–246. C. Ozturk, D. Karaboga, Hybrid artificial bee colony algorithm for neural network training, in: IEEE Congress on Evolutionary Computation (CEC). (2011) 84-88. Y. Zhong, J. Lin, J. Ning, X. Lin, Hybrid artificial bee colony algorithm with chemotaxis behavior of bacterial foraging optimization algorithm, in: Seventh International Conference on Natural Computation. 2 (2011) 1171–1174. I. Fister, I.Jr. Fister, J. Brest, V. Zumer, Memetic artificial bee colony algorithm for large-scale global optimization, in: IEEE Congress on Evolutionary Computation. (2012) 3038–3045. A.R. Yildiz, A new hybrid artificial bee colony algorithm for robust optimal design and manufacturing, Appl. Soft Comput. 13(5) (2013) 2906-2912. T. Chen, R. Xiao, Enhancing artificial bee colony algorithm with self-adaptive searching strategy and artificial immune network operators for global optimization, The Scientific World Journal, vol. 2014, Article ID 438260, 12 pages, 2014. (doi:10.1155/2014/438260). M. Tuba, N. Bacanin, Artificial bee colony algorithm hybridized with firefly algorithm for cardinality constrained mean-variance portfolio selection problem, Appl. Math. Inf. Sci. 8(6) (2014) 2831-2844. W. Xiang, S. Ma, M. An, Habcde: a hybrid evolutionary algorithm based on artificial bee colony algorithm and differential evolution, Appl. Math. Comput. 238 (2014) 370-386. J.-q. Li, Q.-k. Pan, Solving the large-scale hybrid flow shop scheduling problem with limited buffers by a hybrid artificial bee colony algorithm, Inf. Sci. 316 (2015) 487-502. T.K. Sharma, M. Pant, Shuffled artificial bee colony algorithm, Soft Comput. 21(20) (2017) 6085-6104. E. Zorarpacı, S.A. Özel, A hybrid approach of differential evolution and artificial bee colony for feature selection, Expert Syst. Appl. 62 (2016) 91-103. S.S. Jadon, R. Tiwari, H. Sharma, J.C. Bansal, Hybrid Artificial Bee Colony algorithm with Differential Evolution, Appl. Soft Comput. 58 (2017) 11-24. M. Shokouhifar, A. Jalali, Simplified symbolic transfer function factorization using combined artificial bee colony and simulated annealing, Appl. Soft Comput. 55 (2017) 436-451. S. Goudarzi, W.H. Hassan, M.H. Anisi, A. Soleymani, M. Sookhak, M.K. Khan, A.H. Hashim, M. Zareei, ABC-PSO for vertical handover in heterogeneous wireless networks, Neurocomputing. 256 (2017) 63-81. J. Arabas, Z. Michalewicz, J. Mulawka, GAVaPS-a genetic algorithm with varying population size, in: Proceedings IEEE International Conference on Evolutionary Computation, Orlando. (1994) 73–78. J. Brest, M.S. Maučec, Population size reduction for the differential evolution algorithm, Appl. Intell. 29(3) (2008) 228-247. J. Brest, M.S. Maučec, Self-adaptive differential evolution algorithm using population size reduction and three strategies, Soft Comput. 15(11) (2011) 2157-2174. U. Mlakar, I.Jr. Fister, I. Fister, Hybrid self-adaptive cuckoo search for global optimization, Swarm Evol. Comput. 29 (2016) 47-72. P.J. Angeline, Adaptive and self-adaptive evolutionary computations, in: M. Palaniswami, Y. Attikiouzel, R.J. Marks, D.B. Fogel, T. Fukuda (Eds.), Computational Intelligence: A Dynamic System Perspective, IEEE Press. (1995) 152–161. S. Aine, R. Kumar, P. Chakrabarti, Adaptive parameter control of evolutionary algorithms to improve quality-time trade-off, Appl. Soft Comput. 9(2) (2009) 527-540. J. Liang, B. Qu, P. Suganthan, Problem definitions and evaluation criteria for the CEC 2014 special session and competition on single objective real-parameter numerical optimization, Technical Report 201311, Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Nanyang Technological University, Singapore (2014). D.E. Goldberg, J.H. Holland, Genetic algorithms and machine learning, Machine learning. 3(2) (1988) 95-99. X.-S. Yang, S. Deb, Cuckoo search via Lévy flights, in: Proceeding of World Congress on Nature and Biologically Inspired Computing. (2009) 210–214. D. Yashesh, K. Deb, S. Bandaru, Non-uniform mapping in real-coded genetic algorithms, in: Evolutionary Computation (CEC), 2014 IEEE Congress on, IEEE. (2014) 2237–2244. R.D. Maia, L.N. de Castro, W.M. Caminhas, Real-parameter optimization with OptBees, in: Evolutionary Computation (CEC), 2014 IEEE Congress on, IEEE. (2014) 2649–2655. C. Yu, L. Kelley, S. Zheng, Y.Tan, Fireworks algorithm with differential mutation for solving the cec 2014 competition problems, in: Evolutionary Computation (CEC), 2014 IEEE Congress on, IEEE. (2014) 3238–3245. P. Bujok, J. Tvrdík, R. Polakova, Differential evolution with rotation-invariant mutation and competing-strategies adaptation, in: Evolutionary Computation (CEC), 2014 IEEE Congress on, IEEE. (2014) 2253–2258. R.C. Eberhart, Y. Shi, Particle swarm optimization: developments, applications and resources, in: Proceedings of Congress on Evolutionary Computation. 1 (2001) 81–86. W. Kuo, V.R. Prasad, An annotated overview of system-reliability optimization, IEEE Trans. Reliab. 49 (2000) 176–187. W. Kuo, V. R. Prasad, F. A. Tillman, C. L. Hwang, Optimal reliability design-fundamentals and applications, Cambridge University Press, Cambridge. 2001. W. Kuo, R. Wan, Recent advances in optimal reliability allocation, IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans. 37(2) (2007) 143-156. H. Garg, M. Rani, S. Sharma, An efficient two phase approach for solving reliability–redundancy allocation problem using artificial bee colony technique, Comput. Operat. Res. 40(12) (2013) 2961-2969. Q. He, X. Hu, H. Ren, and H. Zhang,A novel artificial fish swarm algorithm for solving large-scale reliability–redundancy application problem, ISA transactions. 59 (2015) 105-113.


856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923

[87] [88] [89] [90] [91] [92] [93]

[94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119]


H.B. Ouyang, L.Q. Gao, S. Li, and X.Y. Kong, Improved novel global harmony search with a new relaxation method for reliability optimization problems, Inf. Sci. 305 (2015) 14-55. C.-L. Huang, A particle-based simplified swarm optimization algorithm for reliability redundancy allocation problems, Reliab. Eng. Syst. Safety. 142 (2015) 221-230. N. Nahas, D. Thien-My, Harmony search algorithm: application to the redundancy optimization problem, Engineering Optimization. 42(9) (2010) 845-861. T.-J. Hsieh, W.-C. Yeh, Penalty guided bees search for redundancy allocation problems with a mix of components in series–parallel systems, Comput. Operat. Res. 39(11) (2012) 2688-2704. Y.-C. Hsieh, T.-C. Chen, D.L. Bricker, Genetic algorithms for reliability design problems, Microelectronics Reliability. 38(10) (1998) 1599-1605. T. Yokota, M. Gen, Y.-X. Li, Genetic algorithm for non-linear mixed integer programming problems and its applications, Comput. Indust. Eng. 30(4) (1996) 905-917. M. Gen, K. Ida, R. Kobuchi, C.Y. Lee, Hybridized neural network and genetic algorithms for solving nonlinear integer programming, in: L.C. Lain, R.K.Lain (Eds.), Second International Conference on Knowledge-Based Intelligent Electronic Systems, 21–23 April, (1998) 272–277. M. Gen, Y. Yun, Soft computing approach for reliability optimization: State-of-the-art survey, Reliab. Eng. Syst. Safety. 91(9) (2006) 1008-1026. H.G. Kim, C.O. Bae, D.J. Park, Reliability-redundancy optimization using simulated annealing algorithms, Journal of Quality in Maintenance Engineering. 12(4) (2006) 354-363. T.-C. Chen, IAs based approach for reliability redundancy allocation problems, Appl. Math. Comput. 182(2) (2006) 1556-1567. L. dos Santos Coelho, An efficient particle swarm approach for mixed-integer programming in reliability–redundancy optimization applications, Reliab. Eng. Syst. Safety. 94(4) (2009) 830-837. D. Zou, L. Gao, J. Wu, S. Li, and Y. ,Li, A novel global harmony search algorithm for reliability problems, Comput. Indust. Eng. 58(2) (2010) 307-316. T.W. Liao, Two hybrid differential evolution algorithms for engineering design optimization, Appl. Soft Comput. 10(4) (2010) 11881199. Y.-C. Hsieh, P.-S. You, An effective immune based two-phase approach for the optimal reliability–redundancy allocation problem, Appl. Math. Comput. 218(4) (2011) 1297-1307. D. Zou, L. Gao, S. Li, J. Wu, An effective global harmony search algorithm for reliability problems, Expert Syst. Appl. 38(4) (2011) 4642-4648. W.-C. Yeh, T.-J. Hsieh, Solving reliability redundancy allocation problems using an artificial bee colony algorithm, Comput. Operat. Res. 38(11) (2011) 1465-1473. P. Wu, L. Gao, D. Zou, S. Li, An improved particle swarm optimization algorithm for reliability problems, ISA transactions. 50(1) (2011) 71-81. D. Zou, H. Liu, L. Gao, S. Li, A novel modified differential evolution algorithm for constrained optimization problems, Comput. Math. Appl. 61(6) (2011) 1608-1623. L. Wang, L.-p. Li, A coevolutionary differential evolution with harmony search for reliability–redundancy optimization, Expert Syst. Appl. 39(5) (2012) 5271-5278. E. Valian, E. Valian, A cuckoo search algorithm by Lévy flights for solving reliability redundancy allocation problems, Engineering Optimization. 45(11) (2013) 1273-1286. E. Valian, S. Tavakoli, S. Mohanna, A. Haghi, Improved cuckoo search for reliability optimization problems, Comput. Indust. Eng. 64(1) (2013) 459-468. G. Kanagaraj, S. Ponnambalam, N. Jawahar, A hybrid cuckoo search and genetic algorithm for reliability–redundancy allocation problems, Comput. Indust. Eng. 66(4) (2013) 1115-1124. L.D. Afonso, V.C. Mariani, L. dos Santos Coelho, Modified imperialist competitive algorithm based on attraction and repulsion concepts for reliability-redundancy optimization, Expert Syst. Appl. 40(9) (2013) 3794-3802. H. Zhang, X. Hu, X. Shao, Z. Li, Z. Y. Wang, IPSO-based hybrid approaches for reliability-redundancy allocation problems, Science China Technological Sciences. 56(11) (2013) 2854-2864. M.A. Ardakan, A.Z. Hamadani, Reliability–redundancy allocation problem with cold-standby redundancy strategy, Simulation Modelling Practice and Theory. 42 (2014) 107-118. Y. Liu, G. Qin, A hybrid TS-DE algorithm for reliability redundancy optimization problem, J. Comput. 9(9) (2014) 2050-2058. Y. Liu, G. Qin, A modified particle swarm optimization algorithm for reliability redundancy optimization problem, J. Comput. 9(9) (2014) 2124-2131. H. Garg, An approach for solving constrained reliability-redundancy allocation problems using cuckoo search algorithm, Beni-Suef University, J. Basic. Appl. Sci. 4(1) (2015) 14-25. Y. Liu, G. Qin, A DE Algorithm Combined with Lévy Flight for Reliability Redundancy Allocation Problems, International Journal of Hybrid Information Technology. 8(5) (2015) 113-118. H. Garg, An efficient biogeography based optimization algorithm for solving reliability optimization problems, Swarm Evol. Comput. 24 (2015) 1-10. M.A. Mellal, E. Zio, A penalty guided stochastic fractal search approach for system reliability optimization, Reliab Eng. Syst. Safety. 152 (2016) 213-227. G.J. Krishna, V. Ravi, Modified Harmony Search Applied to Reliability Optimization of Complex Systems, Harmony Search Algorithm. (2016) 169-180. Y. Yun, J. Jo, M. Gen, Adaptive Hybrid Genetic Algorithm with Modified Cuckoo Search for Reliability Optimization Problem, in Proceedings of the Tenth International Conference on Management Science and Engineering Management, J. Xu, A. Hajiyev, S. Nickel, M. Gen (Eds), Advances in Intelligent Systems and Computing, Springer, Singapore. 502 (2017) 353-365.