A Hybrid Mutation-based Artificial Bee Colony for Traveling Salesman Problem

Transcription

1 A Hybrd Mutaton-based Artfcal Bee Colony for Travelng Salesman Problem Shma Sabet and Fardad Farokh Islamc Azad Unversty of Central Tehran Branch, Tehran, Iran Emal: Mohammad Shokouhfar Shahd Behesht Unversty, Tehran, Iran Emal: Abstract Travellng Salesman Problem (TSP) belongs to the class of P-Complete problems. It has been proved that evolutonary algorthms are effectve and effcent, wth respect to the tradtonal methods for solvng P-Complete problems lke TSP, wth avodance trappng n local mnma areas. Artfcal Bee Colony (ABC) s a new swarm-based optmzaton algorthm, whch nspred by the foragng behavor of honey bees. In ABC, the neghborhood search strategy s employed n order to fnd better solutons around the prevous ones. In ths paper, a hybrd mutaton based mechansm s proposed to perform the neghborhood searchng for the bees for the TSP. The ntroduced approach leads to effcently reduce the complexty and runnng tme of the ABC algorthm. Expermental results were carred out on the benchmarks from TSPLIB for valdaton. The fndngs mply that the proposed approach s able to acheve optmal tours very quckly. Index Terms travelng salesman problem (TSP), artfcal bee colony (ABC), mutaton I. ITRODUCTIO Travelng salesman problem (TSP) s one of the most popular combnatoral optmzaton problems whch s classfed as a P-Complete problem. The TSP represents the salesman who wants to vst a set of ctes exactly once and fnally turns back to the startng cty. The objectve s to determne the tour wth mnmum total dstance. TSP s a sub-problem ssue of many applcaton domans such as network communcaton [1], transportaton [2], schedulng problems [3], semconductor ndustres and ntegrated crcuts desgns [4], and physcal mappng problems [5]. In recent years many approaches have been developed to solve TSP that can be dvded to exact [6] and approxmate approaches [7]. Exact methods can only be used for very small sze nstances, and for real-world problems, researchers have to apply approxmate methods Manuscrpt receved December 30, 2012; revsed Aprl M. Shokouhfar was wth the Electrcal Engneerng Department, Islamc Azad Unversty of Central Tehran Branch, Tehran, Iran. He s now wth the Department of Electrcal and Computer Engneerng, Shahd Behesht Unversty, Tehran, Iran (Tel: , e-mal: m_shokouhfar@eee.org). that fnd near-optmal solutons n a reasonable tme rather than an exact method that guarantees to fnd the optmal soluton n an exponental tme. The smplest exact method s to make all possble tours, and then select the optmal tour wth the mnmum total cost. All possble permutatons of ctes are equal to!; however, every tour can be represented n 2 dfferent manner depends on the ntal cty and the length of tour. So the sze of search space s equal to 1!/2 [7]. It s obvously that ths evaluaton s not possble n term of computatonal tme even for 20 ctes. Among proposed approaches for TSP, populatonbased evolutonary algorthms effcently have been developed for solve TSP. Moon et. al [8] descrbed a genetc algorthm (GA) whch s formulated by network model and used encodng scheme for TSP based on topologcal sort to form orderng of vertces n a drected graph. Albayrak et al. [9] has ntroduced a GA based approach for TSP wth a new mutaton operator as greedy sub tour mutaton (GSTM). Zhao et. al [10] ntroduced a new method called balancng exploraton and explotaton GA, ams to balance the explotaton of excellent ndvduals and the exploraton of dverse tours usng the GA. Dstnct methods s publshed to mprove GA on TSP so far, varety mutaton operators were employed, such as swap [11], adjacent-change, GSTM [9], shftchange [12], exchange [13], stochastc hll clmbng [14], and route mutaton [15]. Ant colony optmzaton (ACO), s a well-known algorthm desgned for TSP and has good performance. Shang et al. [16] proposed a hybrd approach whch ntegrates ACO and assocaton rule (AR) to solve TSP. L et al. [17] presented an mproved ACO, whch apples a new probablty selecton mechansm by usng Held- Karp lower bound to determne the trade-off between the nfluence of the heurstc nformaton and the pheromone trals. Sh et al. [18] proposed a modfed partcle swarm optmzaton (PSO) algorthm whch appled a crossover elmnated technque and two local search technques. Recently, ABC algorthm was developed for solve optmzaton problems. Zhang suggested a novel algorthm to solve TSP so that t mxtures the route constructon mechansm of the ABC algorthm wth Path do: /lnt

2 Relnkng [19]. An approach called Combnatoral ABC s llustrated by Karaboga et al. [20], whch used a mechansm for neghborhood searchng based on GSTM mutaton. In ths paper, a new Hybrd Mutaton-based ABC algorthm (named HMABC) s proposed to quckly fnd an optmal tour. The rest of paper s organzed as follows: In secton 2, the ABC algorthm has revewed. Secton 3 presents the methodology of HMABC algorthm. In secton 4, the expermental results are shown, and consequently, a concluson s descrbed n secton 5. II. ARTIFICIAL BEE COLOY ALGORITHM Artfcal bee colony (ABC) s a novel populatonbased algorthm whch nspred of the ntellgent foragng behavor of real honey bees swarm, proposed by Karaboga et al. n 2006 [21]. To solve TSP usng ABC algorthm, the poston of a food source represents a possble tour and the nectar amount of a food source corresponds to the qualty of the tour. In ABC model, the colony conssts of three groups of bees: employed bees, onlookers and scouts. Each employed bee goes to the food source area n her memory and determnes a neghbor source, and then come back to the hve and begn dance n dance area. Ths dance s essental for colony communcaton, and contans three peces of nformaton regardng a flower patch: the drecton n whch t wll be found, ts dstance from the hve and ts qualty ratng (ftness). Provded that the nectar amount of the new selected source s hgher than the prevous one, the bee memorzes the new source poston and forgets the old one. Otherwse she keeps the prevous poston n her memory. Each onlooker watches the dances of employed bees, and makes decson to choose the food source through the shared nformaton from the dances. Then, some employed bees whch found hgh qualty solutons, are chosen as scouts. For each scout, some employed bees are determned to search for fndng new food sources n the neghborhood of the soluton of the scout. III. A. Problem Representaton METHODOLOGY TSP can be represented as a graph G = (V, E), where V = {1, 2,..., } s a set of nodes, and E =, j, j ε V} s the set of all connecton edges between them. Each node represents a cty, and each edge means the possble path between two related ctes. The dstance d j s assocated wth edge (, j) and represents the Eucldean dstance from cty to cty j Eq. (1). The heurstc nformaton s calculated before performng the man algorthm. So, the dstances of all edges were calculated and saved. d j = (x x j ) 2 + (y y j ) 2 (1) Iteratvely, after constructon the solutons, the qualtes of the gathered solutons are evaluated by the cost functon, whch s calculated by sum of the Eucldean dstances of the consecutve edges wthn the tour as Eq. (2). ote that the Cost k s correspondng to the bee k. Cost k = l=1 d l (2) B. Used Mutaton Operators for eghborhood Search As noted above, there are varety mutaton operatons for TSP, such as swap, adjacent-change, route and shftchange. In ths study, the eghbor Swap (called S), eghbor Shft-Change (called SC), and the proposed eghbor-change (called C) operators are used n the presented hybrd mutaton strategy. In S operator, two neghbor agents of the bee are selected randomly, and then are replaced wth each other. The SC operator frstly selects two neghbor agents randomly, and then t replaces one of them wth the other and shfts all remaned agents between them wth the same sequence. The neghborhood search by the bee usng the S and SC operatons can be seen respectvely n Fg. 1, and Fg. 2. Fgure 1. The S mutaton operaton. Fgure 2. The SC mutaton operaton. And fnally the proposed C operator works as follow: One cty s chosen randomly (for example the cty ). A cty must be selected n the neghborhood of the cty, and the one whch has less dstance, has more probablty to be selected. The probablty of the selected cty j s computed by Eq. (3). The postons of the cty and the newly selected cty are determned for the bee. So the correspondng agents n the bee are determned. The shft-change operator s appled to the two selected neghbor ctes. P j 1 d j 1 d h C. Constructon of Intal Solutons To solve TSP usng ABC algorthm, at the frst step, a dstrbuted ntal populaton (food source postons) s generated. The way whch s used for achevng ntal solutons, was set up n a way that, t led to acheve better qualty solutons than random selecton. In HMABC algorthm, the ntal solutons are not determned completely random. In ths manner a random cty s selected as ntal of the tour (e. g. cty ). Then the next ctes consecutvely are added to the rout, wth a probablty whch corresponds to the nverse of ther dstance from cty. jh (3) 100

3 D. Selecton of Scout Bees by Onlookers After all employed bees complete ther search, they come back to the hve and share ther nformaton about the nectar amount of ther food sources wth the onlookers watng there; so the qualty of the gathered solutons are evaluated. In ths step, a selecton strategy s appled to choose some food source by onlookers. The probablty of selectng bee k as a scout, s computed by Eq. (4). Where k s the amount of nectar n food =1 k s the total amount of acheved source k, and nectar. In HMABC, the k was consdered as 1/Cost k. The ncrescent n the value of μ (μ>1) leads to select the better bees. So, the onlookers play the role as an objectve functon to evaluate generated solutons. E. Equatons P k (4) k 1 After selectng scout bees by onlookers, the recrutment process s performed such that some employed bees (solders) are assgned for each scout. The number of solders determned for each scout bee s determned as: S ( P H ) (5) where S s the percentage of solders that are selected for the th selected scout. P s the preference number of the th scout bee, and H s the number of all determned scout bees. The ß and γ are two parameters that may be constant, or may change teratvely. The ß and γ determne the recrutment model. The value of γ s revealed postve. If ß=0, the number of solders for all scouts s determned unformly. If ß>0, for the scout whch has better qualty, less number of solders are assgned; and f ß<0, more solders are consdered for the better scout bee. F. eghborhood Search Usng Hybrd Mutaton Scheme In ths step, each scout bee wth her assgned solders goes to the neghborhood area of the food source whch was found by that scout. In ths way, for selecton of a new soluton surroundng the prevous one, some part of the tour wll be changed a few, to fnd better solutons. In order to do that, the proposed hybrd mutaton scheme s appled for each bee. As noted above three mutaton operators are used n HMABC, whch each of them has the own advantage and prevents the search path can avod from local mnma. Each solder uses the C operator wth %50 probablty, and the chances of the S and the SC operators are %20 and %30 respectvely. As noted above, the man steps of proposed ABC-based algorthm can be summarzed as follows: Step 1: Intalzaton. Step 2: Generaton of ntal solutons by employed bees, based on nearest neghborhood search strategy. Step 3: Evaluaton of acheved solutons. Step 4: The global best soluton found so far, s regstered. Step 5: Selecton of hgh-qualty solutons as scout bees by onlookers, and performng the recrutment process for them. Step 6: Check the stop crteron: Stop, f the number of teratons s more than the maxmum, otherwse contnue. Step 7: Search the neghborhood usng the proposed hybrd mutaton-based scheme. Step 8: go to Step 3 and contnue. Intalzaton and Settng up the Parameters Selecton of Intal Solutons based on earest eghborhood by Employed Bees Evaluaton of Gathered Solutons: Sharng the nformaton obtaned by the employed bees wth the onlookers Check the stop crtera Stop Fnal Selected Tour Fgure 3. The overall flowchart of HMABC algorthm. IV. Searchng the neghborhood of hgh-qualty tours by applyng the proposed hybrd mutaton scheme Recrutment Process Selectng the scout bees by onlookers The global best found tour so far, s regstered. EXPERIMETAL RESULT All of experments tested on an Intel PC wth 2.53 GHZ processor and 4 MB memory runnng MATLAB R2009a n wndows 7. In order to evaluate the performance of HMABC algorthm, we tested t on the benchmark problems from TSPLIB [22]. For tunng the parameters, dfferent values were tested, and the best values were consdered for experments, whch can be summarzed n Table I. We compare the performance of HMABC wth two ACO-based methods gven n [7]. The expermental results are summarzed n Table II. The frst column s the problem names, the 2 nd column s the optmum tour lengths. The 3 th and 4 th columns are the soluton of the 101

4 two compared algorthms, and the last column s the acheved tour lengths by proposed method. An optmum soluton acheved by HMABC for El76, Berln52, El51, and St70 problems can be seen respectvely n Fg. 4 to Fg. 7. And n Table III a comparson of HMABC algorthm employng the dfferent mutaton operators, s shown. TABLE I. PARAMETER SETTIG Parameter Value Max Iteraton 1000 umber of Employed Bees 100 umber of Scout Bees 5 Max Dstance of two eghbor agents for Mutaton /5 Probablty of S mutaton %20 Probablty of SC mutaton %30 Probablty of C mutaton %50 µ 10 Β -20 Γ 32 H 5 Fgure 6. ote how the capton s centered n the column. TABLE II. COMPARISO BETWEE BEST FOUD TOUR LEGTH Problem Optmal Soluton BWAS RBAS HMABC El Berln El St , Rat Fgure 7. ote how the capton s centered n the column. TABLE III. COMPARISO THE RESULT WITH DIFFERET MUTATIO SCHEMES, AVERAGE O 10 RUS Problem El51 Berln52 El76 eghbor-swap (S) eghbor-shft-change (SC) eghbor-change (C) Proposed Hybrd Mutaton Fgure 4. ote how the capton s centered n the column. V. COCLUSIO In ths paper, a novel ABC algorthm based on a hybrd mutaton scheme (named HMABC algorthm) was ntroduced for TSP, ams to effcently reduce the complexty of evolutonary algorthms. Proposed algorthm has a strong search capablty n the graph space and can effectvely fnd an optmal tour. Expermental results show that ths approach consders both runnng tme and soluton qualty as well. As a future work, the algorthm wll be hybrdzed wth ACO algorthm to fnd better results. Moreover the HMABC quckly acheves good performance n compare wth the other evolutonary algorthms; and t s qute smple and easy to apply. REFERECES Fgure 5. ote how the capton s centered n the column. [1] J. R. L. Fourner and S. Perre, Assgnng cells to swtches n moble networks usng an ant colony optmzaton heurstc, Computer Communcaton Conf., 2005, vol. 28, pp

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