An improved artificial bee colony algorithm with local search for ...

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AN IMPROVED ARTIFICIAL BEE COLONY ALGORITHM WITH LOCAL SEARCH FOR TRAVELING SALESMAN PROBLEM a

Hasan Erdinc Kocer & Melike Ruhan Akca a

b

Selcuk University, Technical Education Faculty , Konya , Turkey

b

Nazilli Vocational High School , Aydin , Turkey Published online: 28 Oct 2014.

To cite this article: Hasan Erdinc Kocer & Melike Ruhan Akca (2014) AN IMPROVED ARTIFICIAL BEE COLONY ALGORITHM WITH LOCAL SEARCH FOR TRAVELING SALESMAN PROBLEM, Cybernetics and Systems: An International Journal, 45:8, 635-649, DOI: 10.1080/01969722.2014.970396 To link to this article: http://dx.doi.org/10.1080/01969722.2014.970396

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Cybernetics and Systems: An International Journal, 45:635–649 Copyright # 2014 Taylor & Francis Group, LLC ISSN: 0196-9722 print=1087-6553 online DOI: 10.1080/01969722.2014.970396

An Improved Artificial Bee Colony Algorithm with Local Search for Traveling Salesman Problem HASAN ERDINC KOCER1 and MELIKE RUHAN AKCA2 1

Downloaded by [Selcuk Universitesi] at 01:20 29 October 2014

Selcuk University, Technical Education Faculty, Konya, Turkey 2 Nazilli Vocational High School, Aydin, Turkey

This study aims to solve the traveling salesman problem for small, medium, and large traveling salesman problems taken from the TSPLIB with known solutions, by using an improved artificial bee colony algorithm that is a swarm intelligence-based heuristic algorithm. The improvement process is achieved by using a loyalty function that is used in bee colony optimization as a fitness function used in ABC algorithms. Obtained solutions are compared to solutions from the TSPLIB and the results of Çunkas¸ and O¨zsag˘lam’s (2009) study, which includes the solutions for benchmark problems and cities and counties in Turkey according to the genetic algorithm and particle swarm optimization. KEYWORDS artificial bee colony algorithm, opt-2 local search, opt-2 local search, traveling salesman problem

INTRODUCTION As developments in computer technology keep increasing, optimization methods are used for solutions of many applications in various fields such as social, health, and applied sciences. The heuristic algorithm is an important alternative method that is used especially for problems that have more than one solution. One of these problems is the traveling salesman problem (TSP), which is a problem of finding the shortest route and is also studied rather frequently. The TSP is to discover the shortest route possible that a Address correspondence to Hasan Erdinc Kocer, Selcuk University, Technical Education Faculty, 42003, Konya, Turkey. E-mail: [email protected] Color versions of one or more of the figures in the article can be found online at www. tandfonline.com/ucbs. 635


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H. E. Kocer and M. R. Akca

salesman can take, starting from one point and visiting every point on the route and returning to the starting point in the end. Heuristic algorithms are various and are based on two main concepts: development and swarm intelligence. The artificial bee colony (ABC) algorithm is based on swarm intelligence and is a quite popular and efficient method that is influenced by the behavior of honey bees searching for food. In this study, an improved ABC algorithm, developed with Opt-2 local search, was applied to small-, medium-, and large-size symmetric and asymmetric traveling TSPs and a large-scale problem including cities (81 points) and counties (888 points) in Turkey. Although the ABC algorithm herein is a new method, it is an algorithm that has been studied heuristically in many fields and has a proven performance. Paths obtained for the TSP according to the loyalty value used in the bee colony optimization were assessed in terms of fitness. The Opt-2 search algorithm was included in order to make improvements in the local area. Experimental findings obtained with the ABC algorithm were compared with the known best results of the literature and results of a genetic algorithm (GA) and particle swarm optimization (PSO) sta¨ zsag˘lam (2009). Small-, medium-, and large-size ted in a study of Çunkas¸ and O symmetric and asymmetric traveling salesman benchmark problems and the known best results were taken from the TSP library (TSPLIB)1A software program was developed to perform the experiments in a MATLAB environment. Until now, many studies about solving the TSP using algorithms based on bee’s behaviors have been performed. Lucic and Tedorovic (2001, 2002, 2003a, 2003b) implemented the algorithm based on a bee system for TSP benchmark problems. Bee colony optimization has also been applied to solve TSP problems (Teodorovic et al. 2006; Fenglei et al. 2007; Yang et al. 2007; Wong et al. 2008, 2009, 2010). Karaboga (2005) first developed an ABC algorithm to solve problems such as the TSP. Karaboga and Gorkemli (2011) applied ABC optimization to solve TSP benchmark problems. Kiran et al. (2012) and Li et al. (2012) proposed discrete ABC optimization for solving TSP problems. As a result of experiments, the proposed ABC algorithm shows better performance than PSO and GA in TSP problems. This article continues with an explanation of bee colony optimization in the following section and goes on to discuss the ‘‘Artificial Bee Colony Algorithm’’ in the next section. Then ‘‘The Proposed Algorithm’’ is presented. ‘‘Experimental Results’’ follow the presentation of the ABC algorithm, and the paper concludes with ‘‘Discussion.’’

BEE COLONY OPTIMIZATION The concept of a ‘‘bee system’’ is a heuristic algorithm developed by Tomoya Sato and Masafumi Hagiwara who were inspired by the behaviors of real bees. 1

http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/index.html

Improved Artificial Bee Colony Algorithm

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The bee colony algorithm was created as a deriative of the bee system (Lucic and Teodorovic 2003b; Li et al. 2012). The working principle of the bee colony algorithm is as follows: At the beginning of the search process, all bees are in the hive and they directly communicate with one another during this process. While each bee creates one or several possible solutions for itself, it also prepares the components of the general search process. The search process is composed of repeats. The first circle ends once bees find one or several possible solutions. The most ideal solution is recorded at the first cycle, and the second cycle starts. At the second cycle, bees form the solutions step by step. One or more partial solution(s) exist(s) at the end of each cycle. The decision maker determines the total number of cycles. This working principle originated completely from the concept of a herd mentality of real bees.

Bees and Herd Mentality In the herd mentality, the ability to organize and share the work are important. Each bee found in the hive successfully accomplishes the work assigned to it without any authority. Each bee has its own work, and the work sharing, which is a factor of the herd mentality, is clearly seen in bees. The fact that no authority regulates this work sharing demonstrates that bees have the ability to organize. In the hive, there are many bees undertaking various tasks. Task distributions of the worker bees, onlookers, and scouts mentioned in the bee colony algorithm are as follows: .

.

.

The worker bee goes out and searches for a food source and it informs the onlookers about the distance, direction, and quality of the source through various dances that it performs in the dancing area of the hive. At the end of this, if it wants to go to the source, it continues as a worker bee, but if it wants to find new sources, it becomes a scout and looks for new sources or it becomes an onlooker, stays in the hive, and makes use of one of the sources. The onlooker is a kind of decision-making mechanism as it decides on what to do depending on the information that the worker bees give in the dancing area. As a result of this decision, it can go to any source as a worker bee or can wait a bit longer for source selection. The scout bee is charged with exploring new food sources. The worker bee that finishes its task or the onlooker that wants to find its own source can transfer itself to the position of the scout bee. After the scouts inform the onlookers in the dancing area about the food sources that they find, if they want to set to the nectar transporting work, they pass to the position of worker bee.

Owing to aforementioned characteristics, we can easily conclude that there is a certain level of work sharing among bees. During this work-sharing

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process, the worker bee–onlooker–scout balance in the hive is constantly maintained without a need for an authority. This attribute proves that bees can be organized.


Finding the Food Source Information sharing among the bees is of paramount importance in reaching the correct information. Bees transmit the information regarding the food source and its quality to each other in the dancing area. While the bees bringing the information are dancing, the onlookers touch them with their antennas and receive the information about the taste and smell of the source from them. Factors affecting the dance are closeness of the source to the hive, amount of nectar, sweetness of the nectar (the most important one), easy extraction of the nectar, time of the day, and the weather. In short, all factors that can impede or facilitate the work of the bees affect the type and pace of the dance. When the bees are orientated toward a source after they are given the information about the food source through the dance, they need directions. Thanks to their adjacent eyes, bees can calculate the angle between the sun and their own orbits (Akay 2009). Bees display various dances in order to transmit the information about the food source. These dances can be listed as round dance, wagtail dance, and waggle dance. Round and wagtail dances are used distinctively to indicate how far the source is. The round dance implies that a source is available in a distance of 50–100 m; it does not contain direction and angle information. However, the wagtail is a dance type performed to express a food source within an extensive area ranging from 100 m to 10 km. The round dance is an eight-shaped dance type giving almost all information about the food source, such as its place and direction. Direction information is obtained from the angle of the dance. Finally, the waggle dance is performed only when the detected nectar is considerably high quality and abundant in terms of quantity. It means ‘‘I have found a source, it has an abundant amount of nectar in it, I want to start to proceed toward it.’’ Its aim is to maintain the balance between the food capacity within the hive and the food transporting activity. After all the information is transmitted to the onlookers in the dancing area according to the aforementioned dance types, the onlookers make a decision considering the distance between the sources and the amount and quality of the nectar. After the decision is made, the worker bees either start to bring nectar from their own sources or start to look for new sources. As to the scouts, they go out to look for sources randomly. The onlooker can fly toward the source to transport nectar depending on the selected dance or continues to make observations. There is a numeric balance among the bees. In other words, all of the bees do not go out of the hive to work. There is


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always a constant balance between the working bees and those waiting in the hive. Bees maintain this balance in the following manner (Akay 2009): . . . .

Positive feedback: As the quality of the source increases, the number of onlookers choosing this source also increases. Negative feedback: The source is abandoned when it starts to be consumed away. Vibrations: The scout bees make random searches to find food sources. Multiple interactions: Information sharing is realized in the dancing area.


ARTIFICIAL BEE COLONY ALGORITHM The artificial bee colony algorithm is a kind of optimization algorithm developed in 2005 by Karabog˘a et al., who were inspired by the food search system of real bees (Fenglei et al. 2007). In this algorithm, some assumptions are made to facilitate the process. One is that each source is transported by only one worker bee. This means one worker bee for each source. Thus, we can conclude that the number of worker bees is equal to the number of sources. Another assumption is that the number of worker bees is equal to the number of onlookers. The worker bee that consumes the nectar in its source becomes a scout and starts to look for new sources. If we evalute these assumptions in terms of the optimization problems, sources indicate the possible solutions, and the quality of the nectar in the source gives the quality (fitness) of the solution. In other words, the bee finds the best solution when it finds the source containing the highest amount of nectar. Process steps of the ABC algorithm (Fenglei et al. 2007; Aslantas and Kurban 2010): 1. Food sources representing the possible solutions are produced by the worker bees. 2. The following steps are repeated until the condition is fulfilled (until the desired result is obtained): a. All workers bees go to the source and determine the surrounding sources, this nectar is evaluated. b. Each onlooker selects a worker bee. After the neighboring sources are selected around this source, nectar amount is evaluated. c. Some sources are abandoned and the scout bees repeat the search process. d. The best food source is found in the memory. The criterium for fulfilling the condition is to fulfill the iteration number or the fitness function. Food source is the representative of the possible solutions and the food amount in the source represents the conformity of

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the problem to its solution. The number of solutions is equal to the number of worker bees. At the first step, uniformly distributed random source points are created. At the second step, this population is evaluated by the worker, onlooker, and scout bees in the form of a repeated circle. The onlookers select a source according to the probability (pi) obtained from the Eq. (1). Fitness expression, included in this equation, is the fitness function and is determined according to the applied problem.


fitnessi : pi ¼ PN n fitnessi

ð1Þ

After the onlooker selects a source, the worker bee changes the place of the source according to Eq. (2). In this equation, x represents the existing source and v represents the new strong (stable) source. N is the number of the sources and D represents the dimension of the problem. vij ¼ xij þ r and ð0; 1Þðxij xkj Þ k 2 f1; 2; . . . ; N g

ð2Þ

J 2 f1; 2; . . . ; Dg If the amount of nectar is excessive in the new source, it is kept in the memory. After completing the source-searching process, worker bees communicate the results to the onlookeers. The onlookers select the new source by making an evaluation based on the nectar amounts in the sources. Some sources are abandoned when they are assessed in terms of the limit value, and the scout bees produce new random sources.

THE PROPOSED ALGORITHM In this study, an improved artificial bee colony algorithm was applied to the TSP. In this approach, loyalty value, which is used in the bee colony optimization, was employed instead of the fitness function used in the ABC algorithm. Moreover, a threshold value was determined in the work sharing between the scouts, onlookers, and worker bees in the ABC algorithm while the bees were aligned and a total distribution was made. Also, the Opt-2 local search algorithm was included in our algorithm. Nontheless, by including the local search algorithm Opt-2 technique in this modified structure, local development is realized. Aflow chart of the algorithm recommended in this study is given in Figure 1. According to the main assumptions in our algorithm, there are worker bees as many as the number of the points to be visited. Besides, there are onlooker bees as many as the worker bees. Each point to be visited is a food source for the bees. The total length of the path taken to visit the food sources represents the quality of the food source. As our aim is to complete



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FIGURE 1 Flow chart of the proposed algorithm.

the tour from the shortest route possible, the total path length is calculated according to the length equation (Eq. (3)) used in the mathematics. Here, the shortest route means the highest-quality source. ftp ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ðxi xiþ1 Þ2 þðyi yiþ1 Þ2 þ ðxn x1 Þ2 þðyn y1 Þ2

ð3Þ

i¼1

After the total path length is calculated by evaluating the possible solutions for all bees, the objective function is calculated in line with the formula specified in Eq. (4) to understand the quality of the found solution. As the second step following the calculation of the objective function, the threshold value that will be used in deciding on the tasks of the bees (onlooker, worker

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bee) and the loyalty value that will be included in the decision-making mechanism are calculated according to the Eq. (5). .

Objective Function: It is the calculation of the total length of the path taken by each bee until that moment according to Eq. (3). Because the aim is to find the shortest route for the TSP, the solution with a low objective function value is regarded to be close to the optimal solution. Thus, the solution whose objective function value is low is a high-quality solution. OFVx ¼

n X

Bx;i

ð4Þ


i¼1

.

OFVx is the objective function value of the xth bee, n is the number of the visited points and Bx represents the xth bee. Loyalty value: This is a mechanism that decides which bee will be a worker and which bee will be an onlooker in line with the threshold value produced or created depending on the objective function value ranging between 0 and 1 (Teodorovic et al. 2006). Sx ¼

.

.

Bx;OFV MinOFV : MaxOFV MinOFV

ð5Þ

Here, Bx represents loyalty of the xth bee, Bx,OFV represents objective function value of the xth bee, MinOFV represents minimum objective function value, and MaxOFV represents the maximum objective function value. Sx is the loyalty value of the xth bee. Threshold value: The threshold value is used to decide which bee will be a worker and which bee will be an onlooker; it is obtained after the loyalty value of the bees is calculated. Threshold value is randomly obtained between the minimum loyalty value possessed by all bees and the average loyalty value. Bees that have higher loyalty values than threshold values are determined as worker bees whereas those that have lower loyalty values than threshold values are determined as onlookers. Afterwards, the Opt-2 local search algorithm is applied to the bees that will remain in the position of worker. Local search algorithm is a method frequently applied in the heuristic algorithms to prevent obsession with the local optimum. In this study, the Opt-2 technique was used out of the local search algorithms in order to avoid of the local optimum. In this way, the probability of the onlookers being obsessed with the local optimum because of their preferences to become worker bees, and the time spent on assessing the same solution once more, were minimized. Opt-2 Local Search Algorithm: Opt-2 aims at removing the contingencies on the created path. As the contingencies are reduced, the desired minimum path comes closer. If the new path to be obtained from the selected path is shorter than the existing path, the shorter one is preferred and the

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FIGURE 2 Process steps of Opt-2 local search algorithm.

points to be visited are rearranged in line with the new path (Chandra et al. 1999). Process steps of the Opt-2 local search algorithm applied in the A-B-E-C-D arrangement are given in Figure 2. Depending on the quality of the solution that the worker bees find or in proportion to the objective function values, the onlookers select worker bees for themselves and adopt the solutions that they have produced until that moment. These steps are repeated until the preset stop condition is met.

EXPERIMENTAL RESULTS The recommended ABC algorithm was applied to the small-, medium-, and large-size TSP problems in the literature. In our first experiment, the algorithm was applied to the small-size benchmark problems. The number of points of the small-size TSPs, which are syntactic, does not exceed 41. The obtained results were compared with the optimal results available in the ¨ zsag˘lam’s (2009) study as shown literature and the results of Çunkas¸ and O in Table 1. The error rates were calculated according to Eq. (6) as follows: Error Rate ¼

OBV OPV 100; OPV

ð6Þ

TABLE 1 Experimental Results for Syntactic Data TSP problem name C20 C30 C40 F32 F41 S21

The best known solution

Optimal result (ABC)

Average lengths

Error rates (%)

ABC

PSO

GA

ABC

PSO

GA

62,575 62,716 62,768 84,180 68,168 60,000

62,575 62,716 62,768 84,180 68,168 60,000

62,575 62,716 62,768 84,180 68,168 60,000

63,276 63,625 64,212 85,535 69,995 60,786

63,188 63,356 63,753 85,392 69,702 60,648

0 0 0 0 0 0

1.12 1.45 2.3 1.61 2.68 1.31

0.98 1.02 1.57 1.44 2.25 1.08

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TABLE 2 Experimental Results for Medium Size TSP TSP problem name


Burma14 Dantzig42 Eil51 Eil76 Eil101 KroA100


Optimal results (ABC)

3,323 699 426 538 629 21,282

3,323 699 432 561 657 21,688

Average lengths ABC 3,334 704 432 572 663 21,742

Relative error (%)

PSO

GA

ABC

PSO

GA

3,360 713 436 555 652 22,071

3,353 713 434 551 645 21,852

0.33 0.71 1.89 6.31 5.4 2.16

1.14 2.08 2.45 3.16 3.66 3.71

0.92 2.05 2.11 2.56 2.7 2.68

where OBV is the obtained value (our result) and OPV is the optimal value (known best result). The proposed ABC algorithm was run 20 times, and 100% success was achieved in the small-size traveling salesman test problems by reaching the known optimum at every turn. As stated in previous parts of our study, small-size test problems were obtained from the symmetric problems of the literature. It was observed that the ABC algorithm ended with 0% margin of error in the problems displaying symmetric distribution. In the second part of our experiment, different performance rates were obtained in the medium-size asymmetric TSPs. The obtained results were given comparatively in Table 2. As can be clearly seen in the table, solutions of test problems Burma14, Dantzig42 Eil51 and KroA100 through the proposed ABC algorithm produced shorter paths than solutions through PSO and GA algorithms. This result can be accepted as an indicator of the high performance of the proposed ABC algorithm. It was also observed in the other medium-size test problems that the ABC algorithm displayed results close to those of the best known result. The next experiment of this study was to apply the proposed algorithm to the large-size benchmark problems. The number of points of the large size benchmark problems varies between 131 and 436. The obtained results are given in Table 3. As can be clearly seen in the table, our solutions of large-size benchmark problems have the worst results. It was detected that the performance of the proposed ABC algorithm was poor in the large-size asymmetric TSPs. TABLE 3 Experimental Results for Large-Size TSP TSP problem name XQF131 XQG237 BCL380 PBM436


Optimal results (ABC)

Average lengths

Relative error (%)

ABC

PSO

GA

ABC

PSO

GA

564 1,019 1,621 1,443

587 1,110 1,807 1,612

598 1,143 1,841 1,659

584 1,070 1,774 1,634

576 1,068 1,748 1,574

6.02 12.16 13.57 14.96

3.59 5.19 9.44 10.33

2.13 4.84 7.89 9.1

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Improved Artificial Bee Colony Algorithm TABLE 4 Optimal Results for TR81

ABC PSO GA

The best path found

Average length

Relative error (%)

3,781 3,869 3,869

3,919 4,239 4,176

3.64 9.56 7.93


The next experiments are applied to cities and counties of Turkey that are named as TR81 and TR888, respectively. The program was run 10 times again and the obtained solutions for TR81 and TR888 are recorded. The length of the 5 optimal path, the average lengths, and the relative error for TR81 are shown in Table 4.

FIGURE 3 The optimal paths for TR81 (a) ABC (b) PSO-GA.

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H. E. Kocer and M. R. Akca TABLE 5 Optimal Results for TR888 Average length

Relative error (%)

15,643 16,330 16,242

16,834 18,385 18,142

7.61 12.58 11.69


ABC PSO GA

The best path found

FIGURE 4 The optimal paths for TR888 (a) ABC, (b) PSO, and (c) GA.



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Table 4 shows us that the proposed ABC algorithm has found the shortest path and given the best results for the TR81 problem according to PSO and GA algorithms. The optimal path for TR81 achieved by ABC algorithm and PSO-GA algorithms is illustrated in Figure 3a and 3b, respectively. In addition, the proposed algorithm is implemented in the solution of the TR888 problem. As mentioned before, 888 indicates the number of counties in Turkey. After the program is run five times, the achieved results for the TR888 problem are shown in Table 5. In this table, the length of the optimal path, the average lengths, and the relative error are shown. As can be seen from Table 5, the proposed ABC algorithm has found the best result for the TR888 problem. The optimal paths for TR888 achieved by proposed ABC, PSO, and GA algorithms are illustrated in Figure 4a, 4b, and 4c, respectively.

DISCUSSION It is thought that this performance decline originates from the fact that points included in the large-size datasets are relatively close to each other. For, during the path determination in the ABC algorithm, the next point is randomly selected from the points at a certain distance from the current point. In this selection, one of the points available between the minimum and the average distance is randomly selected out of all the points. Because the selection range is narrow, the possibility of cross selection of the points increases. This, in turn, results in the elongation of the path. When the results are analyzed, it can be argued that closeness and distantness, in other words, arrangement of the points, plays a crucial role in the recommended approach regardless of the number of points. This situation may be attributed to the process step of going to the closest random point, which we developed from the food-searching behavior of the bees. Even though the number of points is slight, if the coordinates of the points are too close to each other, errors can occur in the selection. If the distances of the points are below and above a certain threshold value, considerable success can be achieved in the TSP problems with high numbers of points.

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