Parallel Artificial Bee Colony Algorithm for the Traveling Salesman Problem

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1 Parallel Artfcal Bee Colony Algorthm for the Travelng Salesman Problem Kun Xu, Mngyan Jang, Dongfeng Yuan The School of Informaton Scence and Engneerng Shandong Unversty, Jnan, , Chna E-mal: correspondng author: Abstract Artfcal Bee Colony Algorthm (ABCA) s a novel swarm ntellgence algorthm whch a colony of artfcal bees cooperate n fndng good solutons for numercal optmzaton problems and combnatoral optmzaton problems. Travelng Salesman Problem (TSP) s a famous combnatoral optmzaton problem whch has been used n many felds such as network communcaton, transportaton, manufacturng and logstcs. However, t requres a consderably large amount of computatonal tme and resources for solvng TSP. To dealng wth ths problem, we present a Parallel Artfcal Bee Colony Algorthm () n several computers whch operaton system s Lnux based on the Message Passng Interface (MPI). The entre artfcal bee colony s dvded nto several subgroups by equally. Each subgroup performs an ABCA for TSP on each processor node, respectvely. Each sub-colony on every processor node communcates the current best ftness functon and parameters of current best ftness functon accordng to rng topologcal structure durng calculaton process. Some well-known benchmark problems n TSP are used to evaluate the performance of ABCA and. Meanwhle, the performance of s compared wth Genetc Algorthm (GA) and Partcle Swarm Optmzaton (PSO). Expermental results show that the can obtan solutons wth equal precson and reduce the tme of computaton obvously n comparson wth seral ABCA. And have much better performance n contrast wth GA and PSO. Keyword: parallel algorthm; artfcal bee colony algorthm; travelng salesman problem; message passng nterface I. INTRODUCTION The Travelng Salesman Problem s a famous combnatoral optmzaton problem known to be NP-complete. It has been used n varous applcaton felds such as network communcaton, transportaton, manufacturng and logstcs [1-4]. In recent years, many researchers have already proposed many methods n order to solve TSP, such as Artfcal Fsh Swarm Algorthm (AFSA) [5], Genetc Algorthm [6] and Partcle Swarm Optmzaton [7]. Artfcal Bee Colony Algorthm s a novel heurstc algorthm nspred by collectve behavor of honeybees to fnd food sources around the hve. Ths algorthm has been appled to problems such as optmal tunng of PID controller [8], tranng neural networks [9] and mage edge enhancement [10], etc. A bee colony conssts of three knds of bees: employed bees, onlooker bees and scouts. Each knd of bee s used to execute a knd of functon n ABCA. Parallel mplementaton can roughly be categorzed nto two models [11]: one s shared memory archtecture; the other s dstrbuton memory archtecture. The shared memory archtecture manly has one dsadvantage. Parallel algorthm must be confned n only one computer and t cannot make use of dfferent computaton resources n dfferent computers, so the shared memory archtecture has no scalablty. A Parallel Artfcal Bee Colony Algorthm based on Message Passng Interface n Lnux operaton system s proposed to overcome the dsadvantage. In our parallel algorthm, the entre artfcal bee colony s dvded nto several subgroups by equally. Each subgroup performs an ABCA for TSP on each processor node, respectvely, so t s dstrbuton memory archtecture obvously. Each subgroup communcates parameters and data whch are defned and used by wth others usng MPI durng the calculaton process. MPI s a lbrary specfcaton for message-passng, proposed as a standard by a broadly based commttee of vendors, mplementers, and users. It s desgned for hgh performance on both massvely parallel machnes and on workstaton clusters. [12] In ths paper, t s manly used to mplement sendng and recevng of data and allocaton of tasks n. All of algorthms and programs are executed n Lnux operaton system. II. TRAVELING SALESMAN PROBLEM The TSP can be descrbed as follows. Gven a collecton of ctes and the cost of travelng between each par of them, the TSP s to fnd the cheapest way of vstng all of the ctes and returnng to ts startng pont. When a salesman takes hs tour he must vst every pont once and each pont must be vsted only one tme. In the standard verson we study, the travel costs are symmetrc n the sense that travelng from cty A to cty B costs ust as much as travelng from B to A. The cost of tour drectly depends on the tour length. So the cost between cty A and cty B s calculated as Eucldean dstance n (1). 2 2 d( A, B) = ( xa xb) + ( ya yb). (1) Where the parameter x A, x B, y A, yb represents the x coordnate value of pont A, x coordnate value of pont B, y coordnate value of pont A, y coordnate value of pont B, respectvely. d( A, B ) refers to the dstance between cty A and cty B. The total length of n cty can be shown n (2). n 1 total _ f = L 1 + L1 n. (2) The parameter total _ f descrbes the total length whch a salesman travels around all ctes ust one tme and return to hs start pont. Parameter n refers to the total number of 0663

2 ctes. L s the Eucldean dstance between the cty and the cty and L 1n s the dstance between the last cty and the frst cty. III. ARTIFICIAL BEE COLONY ALGORITHM FOR TRAVELING SALESMAN PROBLEM In ABCA, the colony of artfcal bees contans three groups of bees: employed bees, onlookers and scouts. Employed bees carry out randomly search n the food sources when the frst operaton s executed and conduct the neghbor search based on ther local postons to fnd out good sources. Onlookers are watng n the dancng area for makng decson on whch food sources to be chosen and then execute neghbor search followng the employed bee they choose before. Scouts are placed randomly n other food source when the current employed bee or onlooker s trappng nto a local soluton. There are three parameters should be defned by users before the executon of algorthm. They are named as populaton, teraton and lmt, respectvely. The populaton defnes the number of bees n an entre colony; the teraton s the teratve tmes whch are expected by users; the parameter lmt defnes a lmted value. If a varable exceeds ths value, an employed bee wll gve up the current neghbor searchng and a scout should be released. In general, the number of employed bees and the number of onlookers are equal to half of the populaton and ust allow one scout to be sent out n an teratve process. The man steps of ABCA are showed as follows: A. Intalze Artfcal Bee Colony Intalze populaton, teraton tmes and lmt tmes. Generate a set of ntal solutons and calculate ther ftness functon. B. Search for Food Sources Send employed bees to carry out the neghbor search. Onlookers make udgments dependng on ftness functon. Place scouts on the food source randomly. C. Judge the Termnal Condton If the termnal condton s satsfed, the algorthm s over. Otherwse, the algorthm go back to B In ABCA, every teratve process contans three stages. Frst of all, employed bees are ntalzed randomly and then to evaluate ther functon ftness. In the next stage, employed bees return to ther prevous food source and go on explorng the neghborhood of the source, meanwhle, every onlooker chooses a food source offered by employed bees accordng to ts ftness, and starts to explore ts neghborhood. Durng the neghborhood exploraton, the employed bees would be substtuted by the bees wth better ftness. In the last stage, the scouts are sent to the possble food source randomly [9]. The equaton whch calculates the probablty s showed n (3). And the onlookers make decson whether to follow the employed bees or not depend on (3). ft p = SN. (3) ft Where the ft represents the ftness value of th employed bee, p represents the probablty whch the th employed bee can be chosen by the onlookers, SN s the total number of employed bees. The equaton whch produces canddate food source around eb s descrbed n (4). The employed bees and onlookers conduct the neghbor search based on (4). V = eb + ζ ( eb ebk ). (4) The eb represents the th parameter of the th employed bee, the V represents a new soluton of the th employed bee; the parameter ndcates the th employed bee; represents the th parameter n ths employed bee. The parameter ζ s a random number from -1 to 1, and k {1,2,..., NP} ; k ; {1,2,..., D}. NP s the total number of artfcal bees and D s the amount of functon dmensons. ABCA for TSP can be descrbed concretely as follows: The entre parameters of a bee refer to a path whch traverses all ctes from the startng pont to the end pont. The value of ftness functon s showed n (5): ftness _ f = 1/ total _ f. (5) Where total _ f s the total length whch a salesman travels around all ctes ust one tme and return to hs start pont. From (5) we can know that, the bee whch has hgher value of ftness functon has the shorter path. IV. PARALLEL ARTIFICIAL BEE COLONY ALGORITHM ABCA s a novel heurstc ntellgent search method wdely used to fnd proper solutons to a varety of NP problems wthn a reasonable amount of tme (e.g. Travelng Salesman Problem, Job Shop Schedule Problem and so on). However, when they are appled to more complcated problems (e.g. the number of ctes ncreases n TSP), the tme requred to fnd an adequate soluton ncreases dramatcally. In order to reduce the tme, we proposed a by makng use of the parallelsm hdden n ABCA. s an algorthm that combnes several computers wth the nherent parallelsm of ABCA, and enhances the speed of algorthm to obtan a best soluton. A. Parallel Stratagy of ABCA has parallel attrbuton n nature. In ths paper, we dvde the entre bee colony nto several sub-colones equally accordng to the number of processor nodes whch engaged n calculaton. Every sub-colony s assgned to a processor node. Each processor node executes an ABCA to fnd the

3 global best soluton ndependently. Durng the calculaton process, each sub-colony or processor node communcates ther current best soluton and gets rd of the current worst soluton n fxed number of teraton to make sure that successful solutons can be spread to other nodes and the algorthm can obtan a best soluton as far as possble. In the concrete mplementaton of, we choose a computer to be the master node n charge of results dsplay, parameter settng and computatonal resource allocaton. And other nodes are ust used to engage n calculaton. However, the master node also takes part n calculaton as the same as the other nodes wth the excepton of above works we talk about. The equaton whch descrbes parameters relatonshp between the entre colony and the sub-colony s shown as follows. SN sub _ SN =. (6) num total _ teraton sub _ teraton =. comun _ num (7) sub _ LIMIT = LIMIT. (8) Equaton (6) shows the populaton relatonshp between the entre colony and every sub-colony. sub _ SN s the populaton n every sub-colony; SN s the total number of entre colony and num refers to the number of processor partcpated n calculaton. total _ teraton s the total number of teraton whch s expected n. sub _ teraton refers to the number of teraton n every sub-colony. comun _ num s the tmes of communcaton of every sub-colony. The relatonshp between total _ teraton and sub _ teraton s descrbed n (7). The parameter LIMIT defnes a lmted value. If a varable exceeds ths value, an employed bee wll gve up the current neghbor searchng and a scout should be released. The parameter LIMIT and sub_limit s equal n. B. Adusment of ABCA to Adapt to Parallel Algorthm The mathematcs mechansm descrpton of the employed bees s (4). Equaton (4) descrbes that the ABCA conduct the neghbour search by usng the correlaton between the current bee and any other bee n the same colony. However, because the entre bee colony has been dvded nto several sub-colony, the populaton of bees n every sub-colony s much less than the populaton of entre bees. If we contnue to use (4) to execute the functon of employed bees, the dversty of wll declnes obvously and the algorthm s very easy to trap nto the local best soluton. In order to mprove ths stuaton, we use a new strategy to conduct neghbour search nstead of usng (4). Because the entre parameters of a bee refer to a path whch traverses all ctes from the startng pont to the end pont, the new strategy s that the algorthm pck up two parameters randomly n entre parameters of a bee and exchange them to generate a new possble soluton for the neghbour search. C. Topologcal Structure and Communcaton Mechansm Topologcal structure and communcaton mechansm among every sub-colony s of vtal mportantce n and t affects the effcency and effectveness of drectly. The rng topologcal structure s ntroduced to mplement data communcaton among each processor node n. Every processor node wll be gven a seral number at the begnnng of algorthm executon. Every processor node uses ths seral number to determne the communcated obect. Each processor node ust acheves data communcaton wth the node whose seral number next to t. Each node receves a current best soluton whch s send by the former one and sends out the current best soluton tself to the latter one. In addton, each node has a ablty to fnd out ts current worst soluton and replace t wth a best soluton whch the node receves from the former one. Data communcaton happens after the algorthm excutng a fxed tmes of teraton. All operatons of communcatons are accomplshed by functons whch are offered by MPI. D. The Procedure of a) Intalze parameters of n a master node. It ncludes populaton of entre bees, teratons tmes, lmtaton value and tmes of communcaton durng the computaton. b) Calculate parameter for sub-colony. It ncludes populaton of sub-colony, tmes of sub-teraton and those ponts executed data communcatons. c) Every sub-colony receves the data whch s sendng from the master node. d) Every sub-colony generates a seres of random solutons and calcuates ther ftness functons accordng to parameters whch are send from the mater node.fnd out the mnmun and maxmun value of ftness functons n each sub-colony and replace the varable global best soluton n every sub-colony wth ther maxmun ftness functons. e) Each sub-colony exeuctes the model of employed bees. f) Every sub-colony calcuates the probablty whch depends on (3). 0665

4 TABLE I. PARAMETER OF AND CALCULATION TIME OF populat on teraton commun caton Tme of Seral ABCA Tme of n two processor nodes Tme of n three processor nodes Tme of n four processor nodes ch ch pr TABLE II. MINIMUM VALUE OF TSP OBTAINED BY DIFFERENT METHORDS Best Soluton Seral ABCA two nodes of three nodes of four nodes of ch ch pr g) Onlookers make decsons based on the probablty obtaned n f). And then, t conducts the neghbour search followng the employed bee whch t chooses. h) Every sub-colony fnds out the current best soluton and current worst soluton, updates the global best soluton n each sub-colony. ) Every sub-colony carrys out the model of scouts. Judge varables whch records tmes of neghbor search of every employed bee. If some varables exceed a value we set n the begnnng of the algorthm, the varable return to zero and correpongng employed bees of ths varable s replaced by a new randomly generatng soluton. Otherwse, the varable add one. ) Every sub-colony udges the teraton tmes, f the teraton tmes doesn t reach the lmt tmes, go to step e). If the teraton tmes reaches the lmt tmes, go to step k). k) Every sub-colony communcates the best ftness functon wth each other accordng to the rng topologcal structure we talk above. Replace the worst ftness functon by usng the best ftness functon recevng from other colones and the varable whch records tmes of neghbor search of every employed bee returns to zero. l) Every sub-colony udges the communcaton tmes, f the communcaton tmes doesn t reach the lmt tmes we set before, go to step e). If the teraton tmes reaches the lmt tmes, go to step m). m) Algorthm s over. V. EXPERIMENTS AND RESULTS Several benchmark problems are used to evaluate the performance of ABCA and. The benchmark problems are used n ths paper ncludng ch130, ch150 and pr107. All benchmark problems are obtaned from Internatonal standard database of tsp database. We mplemented the wth C language and MPI functon lbrary. And s executed on an expermental platform whch composes of four computers. GA PSO Each computer has one processor of 3.0 GHz domnant frequency and 1.93GB memory. Parameters should be set n the experments and calculaton tme of s showed n tableⅠ. From ths we can get a concluson that tme of calculaton declne as the number of computers ncreases. The multple of reducng tme s equal to the multple of ncreasng processor nodes bascally. However, the mprovement of tme consumng descends and the multple relatonshp between tme and processor nodes s not correspondng wth each other strctly, because tme n communcaton ncreases as the computer whch engaged n calculaton ncreases. Three curves whch descrbe the relatonshp between tme of and processor number s showed n Fg. 1. To evaluate the qualty of obtaned soluton n TSP, we compare the soluton of wth known best soluton, the soluton of Genetc Algorthm and the soluton of Partcle Swarm Optmzaton. Mnmum value of TSP obtaned by these dfferent methods s dsplayed n Table Ⅱ. It s obvously showed n Table Ⅱ that wth multple processor nodes has approxmately same qualty soluton n comparson wth seral ABCA and has much better performance n contrast wth the soluton whch obtaned by Genetc Algorthm and Partcle Swarm Optmzaton. However, s stll trappng nto a local best soluton wth lmted tme n comparson wth the known best soluton. VI. CONCLUSION AND DISCUSSION Artfcal bee colony algorthm s a novel heurstc algorthm whch excels at searchng for a global soluton. TSP s a famous combnatoral optmzaton problem and a NP-hard problem. In the paper, we solve TSP by usng. Three benchmark problems are used to evaluate the performance of ABCA and. And the performance of n TSP s compared wth GA and PSO. The expermental results demonstrate that 0666

5 tme x 108 tme of curve of ch130 curve of ch150 curve of pr processor number Fgure 1. Tme of n dfferent TSP problems. can mprove the speed of calculaton and ensure the qualty of soluton n comparson wth seral ABCA and has much better performance n soluton n contrast wth GA and PSO. In the future, we manly have two assgnments need to do. On the one hand, the also traps nto a local best soluton n comparson wth known best soluton. So we need to mprove the to make sure that t can fnd a global best soluton. On the other hand, allocates the calculaton resource equally n ths paper. In the next step, we wll propose a strategy to allocate the calculaton resource dynamcally accordng the current calculaton load of each processor node. ACKNOWLEDGMENT Ths work s supported by the Natonal Natural Scence Foundaton of Shandong Provnce of Chna under Grant No.ZR2010FM040 and Specal Fundng Proect for Independent Innovaton Achevements transform of Shandong Provnce under Grant No.2009ZHZX1A0108, No.2010ZHZX1A1001. REFERENCES [1] J.R.L. Fourner, S.Perre, "Assgnng cells to swtches n moble networks usng an ant colony optmzaton heurstc, " Computer Communcatons., vol. 28, pp , [2] J.E. Bella, P.R. McMullen, "Ant colony optmzaton technques for the vehcle routng problem," Advanced Engneerng Informatcs., vol. 18, pp , [3] T.P. Bagch, IN.D. Gupta, C. Srskandaraah, "A revew of TSP based approaches for flowshop schedulng," European Journal of Operatonal Research, pp , [4] C.A. Slvaa, J.M.C. Sousaa, T.A. Runkler, "Reschedulng and optmzaton of logstc processes usng GA and ACO," Engneerng Applcatons of Artfcal Intellgence, vol. 21, pp , [5] Jang Mngyan, Yuan Dongfeng, "Artfcal Fsh Swarm Algorthm and ts applcatons". Scence Press, Beng, Chna, [6] Jang-we Zhang,Wen-an S, "Improved Enhanced Self-Tentatve PSO algorthm for TSP," Proceedngs of the Sxth Internatonal Conference on Natural Computaton, pp , 2010 [7] We-xn Lng, Huan-png Luo, "An Adaptve Parameter Control Strategy for Ant Colony Optmzaton," Proceedngs of the 2007 Internatonal Conference on Computatonal Intellgence and Securty, pp , 2007 [8] Kun Xu, Mngyan Jang, "Optmal Tunng of Robust Controller Based on Artfcal Bee Colony Algorthm," Advanced Materals Research Vols , pp , 30 Aprl [9] Derv Karaboga and Bahrye Akay, "Artfcal Bee Colony (ABC) Algorthm on Tranng Artfcal Neural Networks," Sgnal Processng and Communcatons Applcatons, SIU IEEE 15th, Eskşehr, pp. 1-4, June [10] Trmula Rao Benala and Sree Durga Jampala, "A novel Approach to Image Edge Enhancement Usng Artfcal Bee Colony Optmzaton Algorthm for Hybrdzed Smoothenng Flters," 2009 World Congress on Nature & Bologcally Inspred Computng (NaBIC 2009), combatore, Inda, pp , 2-4 Spet [11] A Grama, A Gupta, G. Karyps, V. Kumar, "Introducton to parallel computng," New York: Addson-Wesley, [12]

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