Hybrid Artificial Bee Colony Algorithm with Simulated Annealing for Numerical Optimization

Hybrid Artificial Bee Colony Algorithm with Simulated Annealing for Numerical Optimization

• 1.
  Hybrid Artificial BeeColony Algorithm with Simulated Annealing for Numerical Optimization Saad T Alharbi Department of Computer Science, Taibah University Medina, KSA stharbi@taibahu.edu.sa Abstract—The artificial bee colony (ABC) is one of the most powerful swarm intelligence (SI) approaches and it simulates honeybees? foraging behavior. A well-known limitation of ABC is getting trapped at the local optima owing to its exploitation process. On the contrary, one of the effective metaheuristics approaches that is known to avoid such a problem is simulated annealing (SA) because of its effective search mechanism. There- fore, this paper presents a new hybrid algorithm called SA-ABC that exploits the advantages of SA and ABC to improve the performance of ABC. The proposed algorithm was tested on seven numerical benchmark functions, and its performance was compared with the original ABC. The results were promising in that SA-ABC found lower objective values when compared with well-known approaches. Furthermore, SA-ABC outperformed ABC in the majority of the benchmark functions. However, various issues were highlighted in the experiment, and these need to be further investigated in the future. Index Terms—Artificial Bee Colony , Hybrid Systems, Opti- mization , Swarm Intelligence, Simulated Annealing I. INTRODUCTION In recent years, swarm intelligence (SI), an artificial intelli- gence approach, has become one of the dominant approaches for solving optimization problems. SI is concerned with the adoption of the collective behavior of animals and social insects such as birds, fish, bees, and ants to develop systems that can solve complex problems [9, 10, 17]. Currently, various SI approaches such as particle swarm optimization (PSO) and ant colony optimization are widely used. Furthermore, one of the most recent SI approaches is ABC, which imitates the behavior of honeybees foraging. This approach is dominantly used by researchers because of its simplicity and robustness. In addition, different versions of bee-colony approaches can be found in the literature and are most likely based on the mating behavior of bees. In fact, ABC was adopted successfully to solve optimization problems in different domains (see Section II). However, one of the well-known limitations of ABC is getting trapped in the local optima, which is most likely owing to its exploration process. Therefore, the aim of this paper is to fill this gap by proposing a hybrid algorithm by taking the advantages of a powerful metaheuristics approach with simulated annealing (SA). Re- cently, it was adopted successfully to solve real-world prob- lems. These problems come from numerous domains. Owing to space limitations, we will highlight only a few of the well- known problems. For instance, various studies were performed to solve the traveling salesman problem (TSP) using SA, including [3] [41] [50]. In fact, it has been adopted to solve one of the most recent versions of the TSP, which is called the multiple maximum scatter traveling salesperson problem [15]. Furthermore, several approaches and techniques adopted SA to solve various transportation problems, including freight transportation systems [36], green transportation [49], and vehicle routing [54]. The majority of these studies presented encouraging results for the adoption of SA in optimization. The rest of the paper is structured as follows. Section II sheds light on ABC and related studies. In section III, the basic version of ABC is described. Then, SA is defined in section IV. The experimental design and setup are presented in section VI. Experimental results and discussion are presented in section VII. Finally, the paper concludes with highlights on future works and concluding remarks in section VIII. II. RELATED WORK In 2005, Dervis Karaboga proposed a new artificial al- gorithm based on a honeybee swarm called ABC[27]. The honeybee swarm was adopted because it satisfied the two fundamental concepts of SI: self-organization and division of labor. ABC basically comprises three types of bees: employed bees, onlookers, and scouts. The algorithm will be described in more detail in the following sections. Since then, researchers have been investigating ABC from different perspectives such as studying its performance and adopting it to solve real problems. For instance, various studies were performed to investigate the performance of ABC in numerical optimization and compare it to other evolutionary algorithms. Karaboga and Basturk [26] and Karaboga and Basturk [29], for example, compared the performance of ABC with differential evolution (DE) and PSO using 13 benchmark functions. The results demonstrated that ABC found the global minimum in the majority of test functions and was near the global minimum in the others. In addition, ABC outperformed DE and PSO in several test functions. During the same year, the researchers published a similar study that compared ABC with the genetic algorithm (GA), PSO, and particle swarm inspired evolution- ary algorithm (PS-EA) for multivariable numerical functions International Journal of Computer Science and Information Security (IJCSIS), Vol. 16, No. 6, June 2018 205 https://sites.google.com/site/ijcsis/ ISSN 1947-5500
• 2.
  [28]. Similarly, theresults demonstrated that ABC was able to avoid the global minimum and outperformed the other algorithms in several functions. In contrast, Krishnanand et al. [33] carried out a comparative study with five bio-inspired evolutionary algorithms, including ABC. The performance of these algorithms was tested using standard benchmark functions. The results indicated that ABC was not of the best algorithms, although it reached the optimum value in some test functions[33]. Numerous modified ABC versions were introduced in the literature. For instance,TSai et al. [52] claimed that the on- looker bees’ movements consider only the relation between an employed bee and the selected one randomly. Therefore, the researchers proposed the adoption of the universal gravitation law in the exploitation process between the onlooker bees and the selected employed one. The proposed algorithm was called the interactive ABC (IABC). It was compared with the original ABC and PSO using five benchmark functions, and the results indicated that (IABC) outperformed the other algorithms for only a specific n value in test functions. Furthermore,Zhang et al. [58] claimed that the convergence of ABC is slow and can be stuck in the local minima easily because of its random mechanism. Therefore, these researchers proposed a modified version of ABC wherein two modifications were applied: 1) combined greedy selection and 2) using sensibility to produce new solutions in onlooker bees instead of using random numbers to calculate probability. To examine the performance of the proposed modifications, they were inde- pendently compared to the original ABC, and then combined and again compared to ABC. The results showed that the searching ability was improved in the modified versions of ABC. Likewise,Guo et al. [19] proposed a modification to the search process of ABC to avoid being trapped in local optima by applying the concept of global search, which was inspired by PSO. Previous information of found solutions such as position and nectar amount (fitness value) was employed, and onlooker bees were used instead of neighbor information. The results demonstrated an improved performance for the proposed method over the original ABC and PSO. In fact, various studies were conducted to improve the exploitation process of ABC, including [4] [45]. Parameter selection and tuning are crucial factors in the success of ABC in solving optimization problems. These factors affect its performance. Three parameters are mainly used in ABC: scout limit, maximum cycle number, and the number of food sources. Reference Akay and Karaboga [2] performed several experiments to investigate the effect of these parameters in addition to others such as colony size, and found that the value of these parameters can significantly affect the performance of the algorithm. Specifically, it was observed that the scout limit for a small population is recommended to be relatively small to reduce the number of random solutions. In addition, it was found that initialization ranges do not considerably affect ABC in producing solutions. Population size was also found to affect the behavior and performance of ABC [1][14]. Furthermore, The selection mechanism adopted in the ABC was found to significantly affect its performance. For instance,Bao and Zeng [7] performed a comparative study among different selection strategies adopted in the ABC such as tournament, rank, and disruptive strategy. The results demonstrated differences in the performances of the algorithms among the three selection strategies and that originally used with ABC. Specifically, the experimental results revealed that the performance of ABC with the three proposed strategies outperformed the original algorithm [7]. ABC in its original or modified form was adopted to solve different optimization problems. Various researchers adopted it to solve multiobjective optimization problems such as [20][60][55], and others used it to solve constrained problems such as [40]. In addition, ABC was used frequently to solve combinatorial optimization and real-world problems, although it was originally designed for numerical optimization. In order to solve such problems, the ABC was mostly modified or hybridized with other evolutionary algorithms or local search methods. For instance, Banharnsakun et al. [5] proposed a hybrid ABC with a Greedy Subtour Crossover to solve the TSP. The purpose of this hybrid was to enhance the exploita- tion process of the ABC, and the results showed that the proposed method was able to solve the problem. In addition, this method demonstrated an improved performance in terms of precision and computational time when compared with other algorithms. Furthermore, Li et al. [35] proposed a hybrid ABC algorithm in order to solve the well-known job-shop scheduling problem. In this study, an ABC was hybridized with a Pareto archive set to keep track of good solutions during the process of optimization. The results indicated that the proposed algorithm was able to solve the job-shop problem; however, it did so with almost the same performance as other approaches used for comparison in the study. Recently, Sundar et al. [51] proposed a hybrid ABC algorithm to solve an extended version of the same problem, which consists of a no-wait between operations in machine jobs. In this study, an ABC was hybridized with a local search algorithm to find the best neighborhood solutions. The proposed approach was compared with two approaches that reported the best results in the literature, and showed an improved performance. Hybrid ABC algorithms were also used for the optimization of cloud computing in different aspects. For example, Prashanth et al. [44] proposed a hybrid ABC with a cuckoo search to optimize cloud vulnerabilities. By contrast,Zhou and Yao [61] proposed a Leavy-flight-based ABC for solving cloud service compositions and optimal selection problems, which are well- known cloud manufacturing problems. In artificial intelligence, a hybrid ABC was able to successfully optimize the parameters of an adaptive network fuzzy inference system (ANFIS)[7]. In addition, an ABC was successfully hybridized with other approaches to solve real-life problems such as engineering and manufacturing challenges [46][59][6][57]. Table I lists some of the approaches that were hybridized with an ABC, and the problems that these approaches were adopted to solve. As mentioned earlier, one of the reasons behind International Journal of Computer Science and Information Security (IJCSIS), Vol. 16, No. 6, June 2018 206 https://sites.google.com/site/ijcsis/ ISSN 1947-5500
• 3.
  TABLE I: Examplesof typical hybrid ABC approaches Algorithm Problem References bacterial foraging Numerical optimization Zhong et al. [60] Pareto archive set job shop scheduling Li et al. [35] Cuckoo search Cloud Security Prashanth et al. [44] artificial neural network pavement classification Banharnsakun [6] Taguchi method vehicle structural design Yildiz [57] Quantum evolutionary combinatorial optimization Duan et al. [16] Greedy Randomized Adaptive Search Procedure clustering Marinakis et al. [38] particle swarm Numerical optimization Shi et al. [48] genetic algorithms Permanent Magnet Synchronous Motors Jatoth and Rajasekhar [23] tabu search job shop scheduling Li et al. [34] graph node coloring University Course Scheduling Oner et al. [42] TABLE II: Examples of typical hybrid SA approaches Algorithm Problem References Whale Optimization feature selection techniques Mafarja and Mirjalili [37] Genetic Algorithm Traveling Salesman Problems, flex- ible flow shop scheduling, Cloud manufacturing Xu et al. [56], Dai et al. [13],Hu et al. [22] Ant colony Distribution network Chen et al. [12] particle swarm optimization flow shop problem, Wireless Mesh Networks Meziani et al. [39],Sakamoto et al. [47] linear programming Distribution network Popovi´c et al. [43] brain storm optimization continuous optimization Jia et al. [24] imperialist competitive algorithms facility layout problem Hosseini et al. [21] hybridization in an ABC is to solve combinatorial optimization problems. Moreover, one of the main drawbacks of the ABC is getting trapped in local optima. Therefore, efforts have been devoted to overcome this limitation by hybridization and modification. From studies described above and by taking a look at the literature, it can be noticed that no significant efforts have been carried out to improve the greedy selection mechanism used in the ABC. We believe this is one of the main factors that contribute to the local optima issue. SA [53] is one of the powerful evolutionary approaches. It takes into consideration the worst candidate solutions during the optimization process. In fact, it was successfully used for solving real-life problems in its original form in studies such as [25][32][8], and was hybridized with other approaches such as [11][56][22][13][37]. Table II lists some of the approaches that were hybridized with SA. To the best of our knowledge, ABC and SA have not yet been hybridized. Therefore, in this paper, a hybrid ABC-and-SA algorithm is proposed. III. ARTIFICIAL BEE COLONY ALGORITHM The previous section highlighted the applications of the ABC algorithm and shed light on some studies that adopted it. The ABC is briefly defined and discussed in this section, although it is well described in the literature [30]. As mentioned earlier, the ABC is one of the approaches that mimic the behavior of honeybees. Three types of bees construct the colony of ABC: employed, onlooker, and scout bees [30]. Candidate solutions in an ABC are representations of food sources, and the amount of nectar in each food source is its fitness value. The number of employed bees in the colony is the same as the food sources, which means that each employed bee is associated with a food source. Similar to other swarm approaches, several parameters need to be adjusted before running an ABC. Most important are the colony size, maximum cycle number, and scout limit [30]. In addition, random candidate solutions are generated during the initialization phase, and their fitness values are calculated. Such solutions are usually generated based on the lower and upper limits of the parameters. Employed bees, then, try to improve food sources (solutions) by generating new solutions from the neighborhood and computing their fitnesses values. New solutions are generated using equation 1, where vij is the new solution, xijandxik are randomly selected solutions where i is the position of the selected employed bee, j ∈ {1, 2, ..., D}, and k ∈ {1, 2, ..., N}. Here, D is the dimension of the problem, which is the number of parameters in each solution; whereas N is the number of candidate solutions, which is equal to the number of employed and onlooker bees. φij indicates a random number in the range [−1, 1]. Then, new generated solutions are compared with the one initially associated, and a greedy selection mechanism is applied. In greedy selection, if the fitness value of the initial solution is lower than that in the new solution, then it is replaced with the new one. vij = xij + φij(xij − xkj) (1) After completing this process, employed bees share the information of the food sources and their fitness values with the onlooker bees, which choose food sources depending on their fitness values in a probabilistic approach. The International Journal of Computer Science and Information Security (IJCSIS), Vol. 16, No. 6, June 2018 207 https://sites.google.com/site/ijcsis/ ISSN 1947-5500
• 4.
  probability of afood source is computed using equation 2, where Pi indicates the probability of the solution and fiti the fitness value of the selected solution. Onlookers generate a new solution based on the selected one using equation 1. In a similar manner to the employed bees, a greedy selection concept is applied to the selected and current solutions. The scout limit parameter is updated in the two phases (i.e., employed and onlooker bees) throughout the optimization process, where its value is increased if the food source has not improved and is decreased if an improvement occurs. Pi = fiti N n=1 fiti (2) Finally, scout bees will be abundant and replace the food sources that did not improve by onlooker and employed bees. Such food sources are determined using the scout limit pa- rameters, where food sources with parameter values exceeding the predefined limit will be abundant and replaced by a new food source using equation 3. xi is the solution selected for abundance according to its limit value, and j ∈ {1, 2, ..., D}. All three phases are repeated until the number of iterations reaches the value of the maximum cycle number parameter or if a termination criterion is met. vij = xmin j + rand[0, 1](xmax j − xmin j ) (3) Algorithm 1: Simulated annealing pseudocode Set initial temperature T; Define cooling function α(T) ∈ [0, T]; Generate a random solution x0 for initialization; while not(termination condition) do Generate a candidate solution x; if f(x) < f(x0) then x0 ← x; else r ← U[0, 1]; if r < exp[f((x0) − f(x))/T] then x0 ← x; end end T ← α(T); end IV. SIMULATED ANNEALING SA is a metaheuristic artificial algorithm inspired by the concept of annealing solids and the crystallizing behavior of chemical substances [53]. It was first introduced in 1983 by Kirkpatrick et al. [31]. One of the main advantages of SA is its ability to avoid getting trapped in local optima by accepting worse solutions within a certain probability [53] [31]. Several parameters need to be tuned before running SA. Most important is the initial temperature. At the beginning, a random solution is generated. SA usually starts with a high temperature value, which is gradually reduced throughout the iterations. This is called the cooling process. In each iteration, a new candidate solution from the neighborhood is generated using a predefined method depending on the problem under consideration. The fitness values of candidate solutions should also be computed using a predefined objective function. Better neighborhood solutions (i.e., a higher fitness value) are always accepted. However, if the fitness value of the new solution is less than the best value so far, it is accepted based on a probability factor that is computed based on the difference between the fittingness values of the two solutions. The current temperature value is (equation 4), where E(s) denotes the power of any two solutions, k is the Boltzmann constant, and T is the temperature. This probability is compared with a random number r between 0 and 1. When new solutions are accepted, the probability is greater than r. P(s) = exp[−E(s)/(kT)] w exp[−E(w)/(kT)] (4) When the algorithm reaches the end, the temperature value becomes low. Consequently, the probability of accepting worse solutions gets smaller. Algorithm 1 shows the pseudocode for SA. TABLE III: Parameters and notations of SA-ABC Category Parameter Meanings Colony size (CS) number of candidate solutions Employed bees 50% of colony size ABC Onlooker bees 50% of colony size Maximum cycles (C) Number of iterations Dimension (D) Number of solution parameters Scout limit limit to abundance a solution SA Temperature (T) value for accepting worse solutions Cooling rate (K) mechanism for temperature reduction V. PROPOSED SA-ABC ALGORITHM As seen in previous sections, ABC is a powerful optimization approach. However, it is likely to get trapped in local optima, most likely because of the greedy selection mechanism adopted for selecting candidate solutions. SA, on the other hand, is capable of avoiding such a problem by accepting worse solutions within some probability. In this section, we describe a hybrid algorithm called SA- ABC that takes the advantages of the two approaches and combines them. In fact, SA-ABC applies the complete exploration process (employed, onlooker, and scout) of ABC and combines it with the selection methods used in SA, in addition to various modifications. The algorithm starts with the initialization of all parameters needed by ABC such as dimensions, colony sizes, and limits for scout bees. In addition, the temperature and cooling rate are initialized at the beginning and are used to select new candidate solutions during the exploration process. The temperature International Journal of Computer Science and Information Security (IJCSIS), Vol. 16, No. 6, June 2018 208 https://sites.google.com/site/ijcsis/ ISSN 1947-5500
• 5.
  Algorithm 2: SA-ABCpseudocode SA(CurrentTemprature,CurrentSolutions,NewSolutions) while not(termination condition) do p ← exp[f((x0) − f(x))/T]; if f(NewtSolution) > f(CurrentSolution) then ScoutLimit ← 0; CurrentSolution ← NewSolution; else r ← U[0, 1]; if r < p then ScoutLimit ← 0; CurrentSolution ← NewSolution; else ScoutLimit + +; end end end /* Parameter initialization */ temperature ← initial temperature; MaxCycles ← Maximum Cycle Number; ScoutLimit ← set limit for abundance of solutions; ColonyZize ← set size of colony (number of solutions) Generate random solutions v in Colony while cycle <= MaxCycles do /* Employed bee phase */ for i <= ColonyZize do Generate new solutions from neighborhood; vi = xij + φij(xij − xkj) ; Evaluate fitness value of v[i]; end SA (temperature,X,V]); /* Onlooker bee phase */ while t <= ColonyZize do /* Select soultions based on the probability of fitness values */ for i = 1toColonyZize do r ← U[0, 1]; if r < f(xi) then x[t] = x[i]; break;; end end Generate new solutions from neighborhood; vt = xij + φij(xij − xkj) ; Evaluate fitness value of v[i]; SA (temperature,x[i],v[i]); end /* Scout bee phase */ find solution with maximum scout limit x ← max(x); if x > ScoutLimit then ScoutLimit ← 0; Generate new solutions; v = xmin j + rand[0, 1](xmax j − xmin j ); end temperature ← intialTemperature/cycle; cycle + +; end International Journal of Computer Science and Information Security (IJCSIS), Vol. 16, No. 6, June 2018 209 https://sites.google.com/site/ijcsis/ ISSN 1947-5500
• 6.
  TABLE IV: Sevenbenchmark functions used in experiment Function Formula Range F1 Griewangk f(x) = − n i=1 cos xi√ i + n i=1 x2 i 4000 + 1 −600 ≤ xi ≤ 600 F2 Sphere f(x) = n i=1 x2 i −100 ≤ xi ≤ 100 F3 Rosenbrock f(x) = n−1 i=1 100 (xi+1 − x2 i )2 + (xi − 1)2 −30 ≤ xi ≤ 30 F4 Rastrigin f(x) = n ∗ 10 + n i=1(x2 i − 10 cos(2πxi)) −5.12 ≤ xi ≤ 5.12 F5 Schwefel f(x) = n i=1 −xi sin( |xi|) −500 ≤ xi ≤ 500 F6 Ackley f(x) = −20 exp(−0.2 1 d d i=1 x2 i ) − exp( 1 d d i=1 cos(2πxi) + 20 + exp(1) −30 ≤ xi ≤ 30 F7 Levy f(x) = sin2(πw1) + d−1 i=1 (wi − 1)2[1 + 10 sin2(πwi + 1)] + (wd − 1)2[1 + sin2(2πwd)], where wi = 1 + xi−1 4 −10 ≤ xi ≤ 10 parameter is initialized with a large value, and it is reduced throughout the running of the algorithm so that the closer the cycle number to the maximum cycle number, the lower the probability of accepting a worse solution. Table III lists the main parameters used and their meanings. In addition, SA-ABC starts with randomly generated solutions. Then, employed bees generate new solutions in the same way as the original ABC. However, after computing the fitness values of these solutions, the SA method is called with the current temperature and current and new solutions. In contrast to the original ABC, a worse solution can be accepted by employed bees based on a probability p, which is computed based on the current temperature using equation 4. Furthermore, if the new solution is accepted, the abundance limit will be assigned zero, whereas it is increased by 1 if it is not accepted, which will increase its probability to be selected by scout bees for abundance. Algorithm 2 shows the pseudocode of SA-ABC. The onlooker bee phase works the same way as with the employed bees; however, candidate solutions are selected based on their fitness values. Similarly, greedy selection is replaced with SA, where a worse solution can be selected based on the current temperature in a probabilistic manner. Then, solutions that did not improve by the employed and onlooker bees are abandoned by scout bees irrespective of the current temperature. Here, solutions with scout limit values exceeding the value initialized at the beginning of the algorithm are selected for abundance, and a new solution is generated using equation 3. At the end of each cycle, the temperature is reduced. Various cooling mechanisms can be adopted with SA. Inverse linear cooling was adopted here using equation 5. α(T) = T0/k (5) VI. EXPERIMENT To evaluate SA-ABC and compare its performance with the original ABC, an experiment was performed using seven well- known benchmark functions. Table IV lists these functions with their ranges. All were used with 20 dimensions (d). To ensure a valid comparison between ABC and SA-ABC, the TABLE V: Parameter settings Parameter Value Colony size (CS) 10 Maximum cycles (C) 2000 Employed bees (no) 50% of colony Onlookers bees (ne) 50% of colony Scout bees (ns) 1 Scout limit ne ∗ d TABLE VI: Best values obtained in literature; values obtained from [28] [18] Functions GA PSO BA ABC DE F1 1.0139 0.030565 36.55 2.01E-08 5.16 F2 19.90 25.08 64.92 8.66 6.43 F3 7.17 12.59 26.15 7.17 7.06 F4 6.81 7.57 12.03 1.45E-08 7.44 F5 1 6.9 9.75 3.61 5.05 F6 9.924 1.1778E-6 7.32 1.6E-11 4.36 values of the parameters were set the same for all functions in the two algorithms. Table V lists the parameter values used in ABC and SA-ABC. In addition, temperature is one of the main parameters used in SA-ABC but is not required in ABC. Its value was set to 2000. VII. RESULTS AND DISCUSSION The two algorithms were run 10 times for each benchmark function, and the best objective value obtained during each run for each algorithm was recorded. The mean, maximum, and minimum were calculated to evaluate and compare the performances of both approaches. Table VII lists the results obtained from the experiment. The best values obtained by well-known approaches in the literature were also recorded for comparison purposes. Table VI lists the best values obtained for six benchmark functions (F1:F2). To perform a valid comparison, all values recorded in Table VI were obtained from functions with 20 variables (i.e., dimension = 20), which is the same number used in our experiment, as mentioned previously. Unfortunately, we were unable to find values for F7 that complied with our experiment. International Journal of Computer Science and Information Security (IJCSIS), Vol. 16, No. 6, June 2018 210 https://sites.google.com/site/ijcsis/ ISSN 1947-5500
• 7.
  TABLE VII: ExperimentalResults Functions ABC SA-ABC Mean Max Min Mean Max Min F1 3.12E-12 1.44E-11 2.22E-16 7.01E-06 3.50E-05 1.11E-16 F2 2.12E-16 3.15E-16 9.97E-17 2.85E-16 4.96E-16 1.00E-17 F3 2.51E+00 6.70E+00 1.02E-01 4.17E-01 1.49E+00 4.83E-02 F4 6.02E-08 2.92E-07 0 2.06E-03 1.93E-02 1.42E-14 F5 -7.32E+02 -3.93E+02 -1.02E+03 -7.62E+02 -2.22E+02 -1.20E+03 F6 2.11E-14 4.35E-14 7.99E-15 1.28E-03 1.06E-02 1.87E-14 F7 -7.32E+02 -3.93E+02 -1.02E+03 3.86E-16 2.15E-16 2.15E-16 Fig. 1: Convergence of Griewangk function Fig. 2: Convergence of Sphere function In fact, SA-ABC was able to find lower objective values when compared with well-known approaches in the literature such as those in Table VI. Furthermore, it can be noticed from Table VII that SA-ABC obtained smaller objective values than ABC in all functions except for Rastrigin and Ackley. Smaller values are bolded in Table II. Although the mean values of the objective values obtained from ABC were smaller than those for SA-ABC in four functions, it can be noticed that the difference between the maximum and minimum values for ABC in all functions is minor. This confirms that the algorithm stayed within a specific range. By contrast, SA-ABC was able to find better solutions for most of the benchmark functions. Furthermore, the convergence of five functions in the two approaches are presented in Figures 5-2. Owing to space limitations, the convergences of all functions are not presented. In these figures, the best fitness value of each iteration in one run was recorded and plotted. To simplify the understanding of the algorithms’ performances, the fitness value is presented on the Y axis in the power of base 10. SA-ABC demonstrated better convergence for the Fig. 3: Convergence of Rosenbrock function Fig. 4: Convergence of Rastrigin function Fig. 5: Convergence of Levy function majority of functions, whereas ABC outperformed it in only one function (Rastrigin). VIII. CONCLUSION In this paper, we proposed a new hybrid algorithm called SA-ABC that uses ABC and SA. The purpose of proposing such an algorithm was to improve the performance of ABC to International Journal of Computer Science and Information Security (IJCSIS), Vol. 16, No. 6, June 2018 211 https://sites.google.com/site/ijcsis/ ISSN 1947-5500
• 8.
  avoid getting trappedin local optima. The performance of SA- ABC was tested on seven well-known benchmark functions and was compared with the original ABC. The experimental results were promising and competitive with ABC. In fact, SA-ABC was able to find better solutions for the majority of the benchmark functions. However, further investigation is needed in the future. The effect of the algorithm parameters and function dimensions on the performance of the proposed approach should be investigated. In addition, different design options should be examined, for example, certain types of bees would adopt SA for neighborhood generation. Finally, the performance of SA-ABC would also be investigated on n-variable benchmark functions. REFERENCES [1] Andrej Aderhold, Konrad Diwold, Alexander Scheidler, and Martin Middendorf. Artificial Bee Colony Optimiza- tion: A New Selection Scheme and Its Performance, pages 283–294. Springer Berlin Heidelberg, Berlin, Heidelberg, 2010. ISBN 978-3-642-12538-6. [2] Bahriye Akay and Dervis Karaboga. Parameter tuning for the artificial bee colony algorithm. ICCCI, 2009: 608–619, 2009. [3] Zulaiha Ali Othman, Nasser Hamed Al-Dhwai, Ayman Srour, and Wudi Yi. Water flow-like algorithm with simulated annealing for travelling salesman problems. International Journal on Advanced Science, Engineering and Information Technology, 7(2):669–675, 2017. [4] B. Babayigit and R. Ozdemir. A modified artificial bee colony algorithm for numerical function optimization. In 2012 IEEE Symposium on Computers and Communica- tions (ISCC), pages 000245–000249, 2012. ISBN 1530- 1346. doi: 10.1109/ISCC.2012.6249302. [5] A. Banharnsakun, T. Achalakul, and B. Sirinaovakul. Abc-gsx: A hybrid method for solving the traveling salesman problem. In 2010 Second World Congress on Nature and Biologically Inspired Computing (NaBIC), pages 7–12, 2010. doi: 10.1109/NABIC.2010.5716308. [6] Anan Banharnsakun. Hybrid abc-ann for pavement surface distress detection and classification. Inter- national Journal of Machine Learning and Cybernet- ics, 8(2):699–710, 2017. ISSN 1868-808X. doi: doi.org/10.1007/s13042-015-0471-1. [7] L. Bao and J. c. Zeng. Comparison and analysis of the selection mechanism in the artificial bee colony algorithm. In 2009 Ninth International Conference on Hybrid Intelligent Systems, volume 1, pages 411–416, 2009. [8] Sule Birim. Vehicle routing problem with cross docking: A simulated annealing approach. Procedia - Social and Behavioral Sciences, 235(Supplement C):149–158, 2016. ISSN 1877-0428. doi: d. [9] Christian Blum and Xiaodong Li. Swarm intelligence in optimization, pages 43–85. Springer, 2008. [10] Eric Bonabeau, Marco Dorigo, and Guy Theraulaz. Swarm intelligence: from natural to artificial systems. Oxford university press, 1999. ISBN 0195131592. [11] K. Bouleimen and H. Lecocq. A new efficient simu- lated annealing algorithm for the resource-constrained project scheduling problem and its multiple mode ver- sion. European Journal of Operational Research, 149(2):268–281, 2003. ISSN 0377-2217. doi: https://doi.org/10.1016/S0377-2217(02)00761-0. [12] E. K. Chen, S. Zhang, and T. Wang. Research on distribution network reconstruction based on improved simulated annealing x2014; ant colony algorithm. In 2017 Chinese Automation Congress (CAC), pages 3575– 3579, Oct 2017. doi: 10.1109/CAC.2017.8243401. [13] Min Dai, Dunbing Tang, Adriana Giret, Miguel A. Salido, and W. D. Li. Energy-efficient scheduling for a flexible flow shop using an improved genetic-simulated annealing algorithm. Robotics and Computer-Integrated Manufacturing, 29(5):418–429, 2013. ISSN 0736-5845. doi: doi.org/10.1016/j.rcim.2013.04.001. [14] Konrad Diwold, Andrej Aderhold, Alexander Scheidler, and Martin Middendorf. Performance evaluation of artificial bee colony optimization and new selection schemes. Memetic Computing, 3(3):149, 2011. ISSN 1865-9292. doi: doi.org/10.1007/s12293-011-0065-8. URL https://. [15] Wenyong Dong, Xueshi Dong, and Yufeng Wang. The improved genetic algorithms for multiple maximum scat- ter traveling salesperson problems. In Jianzhong Li, Huadong Ma, Keqiu Li, Li Cui, Limin Sun, Zenghua Zhao, and Xiaofei Wang, editors, Wireless Sensor Net- works, pages 155–164, Singapore, 2018. Springer Singa- pore. ISBN 978-981-10-8123-1. [16] Haibin Duan, Zhihui Xing, and Chunfang Xu. An improved quantum evolutionary algorithm based on artificial bee colony optimization. In Wen Yu and Edgar N. Sanchez, editors, Advances in Computational Intelligence, pages 269–278, Berlin, Heidelberg, 2009. Springer Berlin Heidelberg. ISBN 978-3-642-03156-4. [17] Simon Garnier, Jacques Gautrais, and Guy Theraulaz. The biological principles of swarm intelligence. Swarm Intelligence, 1(1):3–31, 2007. ISSN 1935-3812. [18] Lihong Guo, Gai-Ge Wang, Amir H Gandomi, Amir H Alavi, and Hong Duan. A new improved krill herd algorithm for global numerical optimization. Neurocom- puting, 138:392–402, 2014. [19] P. Guo, W. Cheng, and J. Liang. Global artificial bee colony search algorithm for numerical function opti- mization. In 2011 Seventh International Conference on Natural Computation, volume 3, pages 1280–1283, 2011. ISBN 2157-9555. [20] R. Hedayatzadeh, B. Hasanizadeh, R. Akbari, and K. Ziarati. A multi-objective artificial bee colony for optimizing multi-objective problems. In 2010 3rd International Conference on Advanced Computer Theory and Engineering(ICACTE), volume 5, pages V5–277–V5–281, 2010. ISBN 2154-7491. doi: International Journal of Computer Science and Information Security (IJCSIS), Vol. 16, No. 6, June 2018 212 https://sites.google.com/site/ijcsis/ ISSN 1947-5500
• 9.
  10.1109/ICACTE.2010.5579761. [21] Seyedmohsen Hosseini,Abdullah Al Khaled, and Satish Vadlamani. Hybrid imperialist competitive algorithm, variable neighborhood search, and simulated annealing for dynamic facility layout problem. Neural Computing and Applications, 25(7):1871–1885, Dec 2014. [22] Yanjuan Hu, Xingfu Chang, Yao Wang, Zhanli Wang, Chao Shi, and Lizhe Wu. Cloud manufacturing re- sources fuzzy classification based on genetic simulated annealing algorithm. Materials and Manufacturing Pro- cesses, 32(10):1109–1115, 2017. ISSN 1042-6914. doi: doi.org/10.1080/10426914.2016.1269921. [23] Ravi Kumar Jatoth and A. Rajasekhar. Speed control of pmsm by hybrid genetic artificial bee colony algorithm. In 2010 International Conference on Communication Control and Computing Technologies, pages 241–246, Oct 2010. doi: 10.1109/ICCCCT.2010.5670559. [24] Zhengxuan Jia, Haibin Duan, and Yuhui Shi. Hybrid brain storm optimisation and simulated annealing algo- rithm for continuous optimisation problems. Interna- tional Journal of Bio-Inspired Computation, 8(2):109– 121, 2016. [25] Zheng Jingjing, Shen Zhihang, Gao Huaien, Chen Bianna, Zheng Weida, and Xiong Xiaoming. An im- proved soc test scheduling method based on simulated annealing algorithm. Journal of Physics: Conference Series, 806(1):012011, 2017. ISSN 1742-6596. [26] D. Karaboga and B. Basturk. On the performance of artificial bee colony (abc) algorithm. Applied Soft Computing, 8(1):687–697, 2008. ISSN 1568-4946. doi: doi.org/10.1016/j.asoc.2007.05.007. [27] Dervis Karaboga. An idea based on honey bee swarm for numerical optimization. Report, Technical report- tr06, Erciyes university, engineering faculty, computer engineering department, 2005. [28] Dervis Karaboga and Bahriye Basturk. A powerful and efficient algorithm for numerical function optimization: artificial bee colony (abc) algorithm. Journal of Global Optimization, 39(3):459–471, 2007. ISSN 1573-2916. doi: doi.org/10.1007/s10898-007-9149-x. [29] Dervis Karaboga and Bahriye Basturk. Artificial bee colony (abc) optimization algorithm for solving con- strained optimization. In Problems, LNCS: Advances in Soft Computing: Foundations of Fuzzy Logic and Soft Computing, Springer-Verlag, IFSA (2007). Citeseer, 2007. [30] Dervis Karaboga, Beyza Gorkemli, Celal Ozturk, and Nurhan Karaboga. A comprehensive survey: artificial bee colony (abc) algorithm and applications. Artificial Intelligence Review, 42(1):21–57, 2014. ISSN 0269- 2821. [31] Scott Kirkpatrick, C Daniel Gelatt, and Mario P Vec- chi. Optimization by simulated annealing. science, 220 (4598):671–680, 1983. [32] Lucas Kletzander and Nysret Musliu. A Multi-stage Simulated Annealing Algorithm for the Torpedo Schedul- ing Problem, pages 344–358. Springer International Publishing, 2017. [33] Krishnanand, Nayak, Panigrahi, and Rout. Compara- tive study of five bio-inspired evolutionary optimization techniques. In 2009 World Congress on Nature & Biologically Inspired Computing (NaBIC), pages 1231– 1236, 2009. [34] J. Li, Q. Pan, and S. Xie. Flexible job shop schedul- ing problems by a hybrid artificial bee colony al- gorithm. In 2011 IEEE Congress of Evolutionary Computation (CEC), pages 78–83, June 2011. doi: 10.1109/CEC.2011.5949601. [35] Jun-qing Li, Sheng-xian Xie, Quan-ke Pan, and Song Wang. A hybrid artificial bee colony algorithm for flexible job shop scheduling problems. International Journal of Computers Communications Control, 6(2): 286–296, 2011. ISSN 1841-9836. [36] Boliang Lin, Jiaxi Wang, Ruixi Lin, Chang Liu, Jie Xiao, Siqi Liu, and Jianping Wu. Simultaneous optimization of railcar itinerary and train formation plan. Technical report, 2018. [37] Majdi M. Mafarja and Seyedali Mirjalili. Hy- brid whale optimization algorithm with simulated an- nealing for feature selection. Neurocomputing, 260 (Supplement C):302–312, 2017. ISSN 0925-2312. doi: doi.org/10.1016/j.neucom.2017.04.053. [38] Yannis Marinakis, Magdalene Marinaki, and Nikolaos Matsatsinis. A hybrid discrete artificial bee colony-grasp algorithm for clustering. In Computers & Industrial Engineering, 2009. CIE 2009. International Conference on, pages 548–553. IEEE, 2009. [39] Nadjat Meziani, Mourad Boudhar, and Ammar Oula- mara. Pso and simulated annealing for the two-machine flowshop scheduling problem with coupled-operations. European Journal of Industrial Engineering, 12(1):43– 66, 2018. [40] E. Mezura-Montes and R. E. Velez-Koeppel. Elitist arti- ficial bee colony for constrained real-parameter optimiza- tion. In IEEE Congress on Evolutionary Computation, pages 1–8, 2010. ISBN 1089-778X. [41] Yassine Moumen, Otman Abdoun, and Ali Daanoun. Parallel approach for genetic algorithm to solve the asymmetric traveling salesman problems. In Proceedings of the 2nd International Conference on Computing and Wireless Communication Systems, page 24. ACM, 2017. [42] A. Oner, S. Ozcan, and D. Dengi. Optimization of university course scheduling problem with a hybrid arti- ficial bee colony algorithm. In 2011 IEEE Congress of Evolutionary Computation (CEC), pages 339–346, June 2011. doi: 10.1109/CEC.2011.5949638. [43] ˇZN Popovi´c, V Dj Kerleta, and DS Popovi´c. Hybrid simulated annealing and mixed integer linear program- ming algorithm for optimal planning of radial distribution networks with distributed generation. Electric Power Systems Research, 108:211–222, 2014. [44] S. K. Prashanth, N. S. Rao, and C. S. Kumar. Hy- International Journal of Computer Science and Information Security (IJCSIS), Vol. 16, No. 6, June 2018 213 https://sites.google.com/site/ijcsis/ ISSN 1947-5500
• 10.
  brid cuckoo search-abc algorithm based vulnerabilities mapping and security in clouds. In 2016 International Conference on Electrical, Electronics, and Optimization Techniques (ICEEOT), pages 2569–2572, 2016. [45] A. Rajasekhar, A. Abraham, and M. Pant. Levy mutated artificial bee colony algorithm for global optimization. In 2011 IEEE International Conference on Systems, Man, and Cybernetics, pages 655–662, 2011. ISBN 1062- 922X. [46] Anguluri Rajasekhar, Ajith Abraham, and Ravi Jatoth. Controller tuning using a cauchy mutated artificial bee colony algorithm. In Soft Computing Models in Indus- trial and Environmental Applications, 6th International Conference SOCO 2011, pages 11–18. Springer, 2011. [47] Shinji Sakamoto, Kosuke Ozera, Admir Barolli, Leonard Barolli, Vladi Kolici, and Makoto Takizawa. Perfor- mance evaluation of wmn-psosa considering four dif- ferent replacement methods. In Leonard Barolli, Fatos Xhafa, Nadeem Javaid, Evjola Spaho, and Vladi Kolici, editors, Advances in Internet, Data & Web Technologies, pages 51–64, Cham, 2018. Springer International Pub- lishing. ISBN 978-3-319-75928-9. [48] X. Shi, Y. Li, H. Li, R. Guan, L. Wang, and Y. Liang. An integrated algorithm based on artificial bee colony and particle swarm optimization. In 2010 Sixth International Conference on Natural Computa- tion, volume 5, pages 2586–2590, Aug 2010. doi: 10.1109/ICNC.2010.5583169. [49] Amir Abbas Shojaie and Hossein Raoofpanah. Solving a two-objective green transportation problem by using meta-heuristic methods under uncertain fuzzy approach. Journal of Intelligent & Fuzzy Systems, 34(1):1–10, 2018. [50] Łukasz Strak, Wojciech Wieczorek, and Arkadiusz Nowakowski. Simulated annealing for finding tsp lower bound. In Ngoc Thanh Nguyen, George A. Papadopou- los, Piotr Jedrzejowicz, Bogdan Trawi´nski, and Gottfried Vossen, editors, Computational Collective Intelligence, pages 45–55, Cham, 2017. Springer International Pub- lishing. ISBN 978-3-319-67077-5. [51] Shyam Sundar, P. N. Suganthan, Chua Tay Jin, Cai Tian Xiang, and Chong Chin Soon. A hybrid artifi- cial bee colony algorithm for the job-shop schedul- ing problem with no-wait constraint. Soft Comput- ing, 21(5):1193–1202, 2017. ISSN 1433-7479. doi: doi.org/10.1007/s00500-015-1852-9. URL https://. [52] Pei-Wei TSai, Jeng-Shyang Pan, Bin-Yih Liao, and Shu- Chuan Chu. Enhanced artificial bee colony optimization. International Journal of Innovative Computing, Informa- tion and Control, 5(12):5081–5092, 2009. [53] Peter J. M. van Laarhoven and Emile H. L. Aarts. Simulated annealing, pages 7–15. Springer Netherlands, 1987. [54] Chao Wang, Dong Mu, Fu Zhao, and John W. Sutherland. A parallel simulated annealing method for the vehicle routing problem with simultaneous pickupdelivery and time windows. Computers & Industrial Engineering, 83: 111 – 122, 2015. ISSN 0360-8352. [55] Z. Wang, M. Li, L. Dou, Y. Li, Q. Zhao, and J. Li. A novel multi-objective artificial bee colony algorithm for multi-robot path planning. In 2015 IEEE International Conference on Information and Automation, pages 481– 486, 2015. doi: 10.1109/ICInfA.2015.7279336. [56] Mingji Xu, Sheng Li, and Jian Guo. Optimization of multiple traveling salesman problem based on simulated annealing genetic algorithm. In MATEC Web of Con- ferences, volume 100, page 02025. EDP Sciences, 2017. ISBN 2261-236X. [57] Ali R. Yildiz. A new hybrid artificial bee colony algorithm for robust optimal design and manufacturing. Applied Soft Computing, 13 (5):2906–2912, 2013. ISSN 1568-4946. doi: https://doi.org/10.1016/j.asoc.2012.04.013. [58] D. Zhang, X. Guan, Y. Tang, and Y. Tang. Modified artificial bee colony algorithms for numerical optimiza- tion. In 2011 3rd International Workshop on Intelligent Systems and Applications, pages 1–4, 2011. [59] Shuzhu Zhang, C. K. M. Lee, K. L. Choy, William Ho, and W. H. Ip. Design and development of a hybrid arti- ficial bee colony algorithm for the environmental vehicle routing problem. Transportation Research Part D: Trans- port and Environment, 31(Supplement C):85–99, 2014. ISSN 1361-9209. doi: doi.org/10.1016/j.trd.2014.05.015. [60] Y. Zhong, J. Lin, J. Ning, and X. Lin. Hybrid artifi- cial bee colony algorithm with chemotaxis behavior of bacterial foraging optimization algorithm. In 2011 Sev- enth International Conference on Natural Computation, volume 2, pages 1171–1174, 2011. ISBN 2157-9555. [61] Jiajun Zhou and Xifan Yao. Multi-objective hybrid arti- ficial bee colony algorithm enhanced with lvy flight and self-adaption for cloud manufacturing service composi- tion. Applied Intelligence, 47(3):721–742, 2017. ISSN 1573-7497. doi: doi.org/10.1007/s10489-017-0927-y. International Journal of Computer Science and Information Security (IJCSIS), Vol. 16, No. 6, June 2018 214 https://sites.google.com/site/ijcsis/ ISSN 1947-5500