A Learning Automata based Algorithm for Solving Traveling Salesman Problem improved by Frequency-based Pruning

Transcription

1 Internatonal Journal of Computer Applcatons ( ) Volume 46 No.7, May 202 A Learnng Automata based Algorthm for Solvng Travelng Salesman Problem mproved by Frequencybased Prunng Mr Mohammad Alpour Department of Computer Engneerng, Unversty of Bonab, Bonab , ran ABSTRACT Many real world ndustral applcatons nvolve fndng a Hamltonan path wth mnmum cost. Some nstances that belong to ths category are transportaton routng problem, scan chan optmzaton and drllng problem n ntegrated crcut testng and producton. Dstrbuted learnng automata, that s a general searchng tool and s a solvng tool for varety of NPcomplete problems, together wth 2opt local search s used to solve the Travelng Salesman Problem (TSP). Two mechansms named frequencybased prunng strategy (FBPS) and fxedradus near neghbour (FRNN) 2opt are used to reduce the hgh overhead ncurred by 2opt n the DLA algorthm proposed prevously. Usng FBPS only a subset of promsng solutons are proposed to perform 2opt. Invokng geometrc structure, FRNN 2opt mplements effcent 2opt n a permutaton of TSP sequence. Proposed algorthms are tested on a set of TSP benchmark problems and the results show that they are able to reduce computatonal tme, whle mantanng the average soluton qualty at 0.62% from known optmal. Keywords Travelng Salesman Problem, Dstrbuted Learnng Automata, Frequencybased prunng strategy, Fxedradus near neghbour.. INTRODUCTION Travelng Salesman Problem (TSP) s about fndng a Hamltonan path (tour) wth mnmum cost. TSP s one of the dscrete optmzaton problems whch s classfed as NPhard []. It s common n areas such as logstcs, transportaton and semconductor ndustres. For nstance, fndng an optmzed scan chans route n ntegrated chps testng, parcels collecton and sendng n logstcs companes, are some of the potental applcatons of TSP. Effcent soluton to such problems wll ensure the tasks are carred out effectvely and thus ncrease productvty. Due to ts mportance n many ndustres, TSP s stll beng studed by researchers from varous dscplnes and t remans as an mportant test bed for many newly developed algorthms. Varous technques have been used to solve TSP. In general, these technques are classfed nto two broad categores: exact and approxmaton algorthms [2]. Exact algorthms are methods whch utlze mathematcal models whereas approxmaton algorthms make use of heurstcs and teratve mprovements as the problem solvng process. Some nstances n the exact methods category are Branch and Bound, Lagrangan Relaxaton and Integer Lnear Programmng. Approxmaton algorthms can be further classfed nto two groups: constructve heurstcs and mprovement heurstcs. Instances n constructve heurstcs nclude Nearest Neghbourhood, Greedy Heurstcs, Inserton Heurstcs [3], Chrstofdes Heurstcs [4] etc. Instances n mprovement heurstc nclude kopt [5], LnKernghan Heurstcs [6], Smulated Annealng [7], Tabu Search [8], Evolutonary Algorthms [9], Ant Colony Optmzaton [2,3] and Bee System [4]. In ths paper an optmzed Dstrbuted Learnng Automata (DLA) algorthm s mentoned that sgnfcantly mproves the computatonal speed of algorthm proposed prevously by the author [5]. Each soluton generated by the prevous DLA algorthm was locally optmzed by an exhaustve 2opt and proposed algorthm was tested on a set of TSP benchmark nstances. The 2opt heurstc s mplemented accordng to the basc dea whch elmnates two arcs n order to obtan two dfferent paths. These two paths are then reconnected n the other possble way f the new path results n a shorter tour length. Although the ntegraton of 2opt gave promsng results, the results presented showed that the executon tme could be mproved further. To acheve ths, two mechansms, namely FBPS and FRNN 2opt, are presented n ths paper. FBPS s a strategy that allows only a subset of promsng solutons to perform 2opt and FRNN 2opt s an effcent mplementaton of 2opt. Ths paper organzed as follows: In the next secton the Learnng Automata and Dstrbuted Learnng Automata and 2opt Local Search Heurstc s ntroduced and then DLA wth 2opt local search algorthm s descrbed. Secton 3 descrbes FBPS and ts workng mechansm. In secton 4 the FRNN 2opt adapted from [6] s descrbed. Computatonal results and fndngs of ths paper are presented and analyzed n secton 5 whle n the last secton conclusons are gven. 2. LEARNING AUTOMATA The automaton approach to learnng nvolves determnaton of an optmal acton from a set of allowable actons. An automaton can be regarded as an abstract obect that has fnte number of actons. It selects an acton from ts fnte set of actons. Ths acton s appled to a random envronment. The random envronment evaluates the appled acton and gves a grade to the selected acton of automata. The response from envronment s used by automata to select ts next acton. By contnung ths process, the automaton learns to select an acton wth best response. The learnng algorthm used by automata to determne the selecton of next acton from the response of envronment. An automaton actng n an unknown random envronment and mproves ts performance n some specfed manner, s refered to as learnng automata (LA) [722]. 7

2 Internatonal Journal of Computer Applcatons ( ) Volume 46 No.7, May 202 The envronment can be descrbed by a trple E {,, c}, where {, 2,..., r} represents the fnte set of the nputs,,,..., } denotes the set of the values can { 2 m be taken by the renforcement sgnal, and c { c, c2,..., cr} denotes the set of the penalty probabltes, where the element c s assocated wth the gven acton. If the penalty probabltes are constant, the random envronment s sad to be a statonary random envronment, and f they vary wth tme, the envronment s called a non statonary envronment. The envronments dependng on the nature of the renforcement sgnal can be classfed nto Pmodel, Qmodel and Smodel. The envronments n whch the renforcement sgnal can only take two bnary values 0 and are referred to as Pmodel envronments. Another class of the envronment allows a fnte number of the values n the nterval [0, ] can be taken by the renforcement sgnal. Such an envronment s referred to as Qmodel envronment. In Smodel envronments, the renforcement sgnal les n the nterval [a, b]. The relatonshp between the learnng automaton and ts random envronment has been shown n Fg.. n Fg.. The relatonshp between the learnng automaton and ts random envronment. Learnng Automata can be classfed nto two man famles: fxed structure learnng automata and varable structure learnng automata. A varable structure learnng automaton s represented by trple < α, β, T >, where β s a set of nputs, α s a set of actons and T s learnng algorthm. The learnng algorthm s a recurrence relaton and s used to modfy the acton probablty vector. It s evdent that the crucal factor affectng the performance of the varable structure learnng automata, s learnng algorthm for updatng the acton probabltes. Let α(k) and p(k) denote the acton chosen at nstant k and the acton probablty vector on whch the chosen acton s based, respectvely. The recurrence equaton shown by Eq. () and Eq. (2) s a lnear learnng algorthm by whch the acton probablty vector p s updated. Let (k) be the acton chosen by the automaton at nstant k. p n + = p n + a p n = a p n () When the taken acton s rewarded by the envronment (.e., β n = 0) and p n + = Random Envronment Learnng Automata ( b)p n = b r + b p n (2) When the taken acton s penalzed by the envronment (.e., β n = ). r s the number of actons can be chosen by the automaton, a(k) and b(k) denote the reward and penalty parameters and determne the amount of ncreases and decreases of the acton probabltes, respectvely. n If a(k) = b(k), the recurrence equatons () and (2) are called lnear rewardpenalty ( ) algorthm, f a(k) >> b(k) the LR P LR P gven equatons are called lnear rewardε penalty ( ), and fnally f b(k) = 0 they are called lnear rewardinacton ). In the latter case, the acton probablty vectors ( LR I reman unchanged when the taken acton s penalzed by the envronment. Learnng automata have been found to be useful n systems where ncomplete nformaton about the envronment, wheren the system operates, exsts. Learnng automata are also proved to perform well n dynamc envronments. It has been shown that the learnng automata are capable of solvng the NPhard problems [23]. 2. Dstrbuted Learnng Automata (DLA) A DLA s a network of the learnng automata whch collectvely cooperate to solve a partcular problem [23]. In DLA the number of actons for any automaton n the network s equal to the number of outgong edges from that automaton. When an automaton selects one of ts actons, another automaton on the other end of edge correspondng to the selected acton wll be actvated. The envronment evaluates the appled actons and emts a renforcement sgnal to the DLA. The actvated learnng automata along the chosen path update ther acton probablty vectors on the bass of the renforcement sgnal by usng the learnng schemes. Formally, a DLA can be defned by a quadruple < A, E, T, A 0 > where A = {A,, A n } s the set of learnng automata, E A A s the set of the vertces n whch vertex e (, ) corresponds to the acton of the automaton A, T s the set of learnng schemes wth whch the learnng automata update ther acton probablty vectors and A 0 s the root automaton of DLA from whch the automaton actvaton s started. For example n Fg. 2, every automaton has two actons, f automaton A selects acton α then automaton A 3 wll be actvated. The actvated automaton A 3 chooses one of ts actons whch n turn actvates one of the automata connected to A 3. At any tme only one automaton n the network wll be actve. α A A 3 α 2 A 2 Fg. 2. Dstrbuted learnng automata. 2.2 Local Search: 2OPT Heurstc The method of 2opt heurstc s appled frequently to solve TSP. Applyng 2opt heurstc n TSP has some advantages such as smplcty n ts mplementaton and ts ablty to obtan near optmal results. The basc dea of 2opt heurstc s to elmnate two arcs n R n order to obtan two dfferent paths. These two paths are then reconnected n the other possble way. Let s consder a feasble soluton, R, wth the permutaton of A, B, C, D, E, F, A wth total tour length of 8 unts as shown n Fg. 3(a). Ths closed tour s then further transformed by frstly elmnatng two arcs (B, C) and (E, F) and thus producng two separate paths F, A, B and C, D, E. Next, these two paths are then reconnected to produce another closed tour, R = A, B, E, D, C, F, A, as shown n Fg. 3(b). Note that there s only one possble way to 8

3 Internatonal Journal of Computer Applcatons ( ) Volume 46 No.7, May 202 reconnect these two paths n order to preserve the length of the other four arcs. By performng a twoarc transformaton, the total length of the closed tour s reduced from 8 unts to 6 unts. The 2opt transformaton mechansm can be generalzed to cater for more arcs n ts elmnateandreconnect process [5]. Fg. 3. (a) Orgnal closed tour, R. (b) R after 2opt transformaton 2.3 DLA Algorthm wth 2opt for TSP In ths secton, we descrbe the DLA2opt algorthm proposed n [5]. At frst a network of learnng automata whch s somorphc to the graph of TSP nstance s created. In ths network each node s a learnng automaton and each outgong edge of ths node (connectng ths cty to other cty) s one of the actons of ths learnng automaton. At frst, one automaton selected (actvated) randomly as startng cty, then at the each stage, actve automaton chooses one of ts actons based on ts acton probablty vector and heurstc nformaton (t prefer to choose short edges) accordng to the probablty dstrbuton gven n Eq. (4). P { p p F B [ p r [ p W W (, )] (, )] :,2,..., r} where P s the s the probablty wth whch automaton chooses to move from cty to cty, W (, ) s the nverse of the dstance between cty to cty, s the one of ctes that can be vsted by automaton on cty (to make the soluton feasble), and s a parameter whch determnes the relatve mportance of (a) (4) P versus dstance. In Eq. () we multply the P by the correspondng heurstc value W (, ). In ths way we favor the choce of edges whch are shorter and whch have a greater C 2 2 A D E P. Ths acton actvates automaton on the other end of edge. The process of choosng an acton and actvatng an automaton s repeated untl a tour s created, ths means, every node of the graph s vsted and comng back at startng cty (or makng a feasble tour s mpossble). In order to exclude loops from the traversed paths, the algorthm meets every node along a path beng traversed once (except startng cty). To mplement ths, f an automaton chooses acton k from the lst of ts actons, then all nactvated automatons (unvsted nodes) wll dsable acton n ther lst of actons. However, at the next teraton of the algorthm, all the dsabled actons wll be enabled. The acton of a DLA s a sequence of actons that represents a partcular tour n the graph. After ths, the soluton s mproved by 2opt heurstc. The outlne of our DLA 2opt algorthm s shown n Fg. 4. A F B (b) C E D Procedure TSP Begn Repeat Produce a tour by DLA. Perform 2opt on ths tour. Compute the tour length. f CurrentTour < BestTour then Reward the selected actons of all LAs along the tour accordng to the LR I. Modfy the Probablty vector. End f Enable all the dsabled actons of each LAs. Untl (stop condton reached) End Procedure Fg. 4. DLA wth 2opt local search for TSP The envronment uses the length of ths tour to produce ts response. Ths response, dependng on whether t s favorable or unfavorable, causes the actons along the traversed path be rewarded f t s favorable (the acton probablty vectors reman unchanged f t s unfavorable) accordng to lnear RewardInacton learnng algorthm L ). ( R I 3. FREQUENCYBASED PRUNING STRATEGY FBPS s based on buldng blocks concept that has been found n many areas. One of them s the Genetc Algorthms (GA) n computer scence. Gero et al [24,25] studed the buldng blocks n GA chromosomes. A populaton of chromosomes s dvded nto two groups, one above a threshold (better ftness), and the other one below the threshold (lower ftness). They proposed a mechansm to dentfy prevalent buldng blocks n the better ftness group and absent buldng blocks n the lower ftness group. The dentfed buldng blocks are then combned as a sngle gene and they are prevented from further modfcatons by crossover and mutaton. The proposed approach was tested on space layout plannng and results showed that f these blocks are frequently reused n the evolutonary process, a sgnfcant reducton of computatonal tme can be acheved. The FBPS proposed n ths paper s able to dentfy some common buldng blocks n the TSP solutons generated by DLAs. The dentfed buldng blocks wll be used to decde whch solutons need to be further mproved by 2opt. If every soluton generated by DLAs locally be optmzed usng 2opt, computatonal tme for obtanng good results s hghly ncreased. Hence, a prunng strategy based on the accumulated frequency of buldng blocks s proposed to prohbt 2opt operatons from beng performed on some solutons. Consder the followng example that explans the workng mechansm of the prunng strategy. Frstly, an n n matrx s created wth each entry of the matrx assgned to zero. n s the number of ctes of a TSP nstance. If a TSP nstance wth 5cty s beng solved, a matrx F 5 5 s requred: A F = B C D E F s used to record the accumulated frequency of the smallest buldng blocks of every solutons generated by DLAs. In ths case, the smallest buldng block contans two elements. Consder the soluton A, B, C, D, E, A and how F wll be updated: 9

4 Internatonal Journal of Computer Applcatons ( ) Volume 46 No.7, May 202 A F = B C D E In ths paper only the smallest edges (edge between two nodes) s consdered n a soluton, but t can be extended to consder buldng blocks n dfferent szes e.g. threeelement block and so on. For every buldng block found n the permutaton, two entres of F wll be updated. For nstance, when the buldng block AB s encountered, the entres of AB and BA of F are both ncremented by. Consder another soluton A, B, E, D, C, A s generated, result s: A F= B C D E Assume the algorthm generates several solutons and then fnal value of F s as below: A F = B C D E Lookng at F, we can dentfy the buldng blocks wth hgh frequency. Consder the frst row of F as an example, where t records the frequency of the buldng blocks wth A as the frst element {AB, AC, AD and AE}. It has a total frequency of 69 (summaton of 25, 84, 32 and 98). The buldng block of AA s dsregarded as the system does not allow any recursve vst. Of these four buldng blocks, AB contrbutes 0.69% of the total occurrences ( 25 00%). AC, AD and 69 AE contrbute 69.63%, 2.73% and 6.93% respectvely. Rest of entres n F are calculated smlarly and the resultng matrx s as below: A B F% = C D E Based on F%, buldng blocks that are q% and above are dentfed as hot spots. q s a user defned threshold value. For nstance, f q = 5%, AB, AC and AE are the hot spots wth A as ts frst element. The dentfed hot spots wll then be employed n the decson makng process. The prunng crteron s set as follow: f any soluton contans k% or more dssmlar buldng blocks, t wll be prohbted from performng the FRNN 2opt. Otherwse, t wll be further optmzed by the FRNN 2opt. If a permutaton of A, D, C, B, E, A s consdered (based on F% and q = 5%), buldng blocks of AD and DC are not hot spots. And CB, BE and EA are hot spots. If k s confgured at 25%, ths partcular permutaton wll be pruned from performng 2 opt. If another permutaton of A, D, E, B, C, A s encountered, t wll be accepted for 2opt optmzaton as only one dssmlar buldng blocks s observed (20% < κ where k s set at 25%). k s a prunng threshold that defned by user and results of tunng t wll be mentoned n Secton FxedRadus Near Neghbour 2OPT In ths paper, Fxedradus Near Neghbour (FRNN) 2opt s used for effcent mplementng 2opt [6]. The 2opt swap could not decrease the tour length as both edges ncrease n length. Because of ths observaton, the exploraton of the second edge can be restrcted to a crcle centered at a node of the frst edge wth a radus equal to the edge length. The tour wth a permutaton of E, F, I, H, C, G, A, B, D, E n Fg. 5 wll be optmzed by 2opt. Whle the edge that lnks E and F, E EF, serves as the frst edge, a search of the second edge based on FRNN method centered at E s performed wth a radus of d EF. All the nodes wthn the vcnty wll be consdered to form the second edge together wth ts sngle approprate neghbour. In Fg. 5, node B s found wthn the vcnty. E BA and E BD wll then be pcked as potental canddates for the second edge n 2opt swap. C D H I Fg. 5. Fndng the second edge where E EF s the frst edge based on FRNN method. Usng FRNN 2opt, only two comparsons are needed. However, f the exhaustve 2opt local search s appled on the same permutaton of E, F, I, H, C, G, A, B, D, E and E EF s chosen as the frst edge, the potental canddates for second edge are E IH, E HC, E CG, E GA, E AB and E BD. In the worst case scenaro, the exhaustve 2opt has to perform sx comparsons before a successful swap. Consderng FRNN 2opt and exhaustve 2opt local search, FRNN 2opt can 67% reducton n terms of the number of comparsons. 5. EXPERIMENTAL RESULTS In ths secton proposed algorthm s tested on a set of benchmark problems taken from TSPLIB and some expermental results n ths study are presented. The dmenson of the problems ranges from 48 to 38 ctes. The numercal fgure denotes the dmenson of the problem. For example, ATT48 s a 48cty problem from the ATT seres. The results for each benchmark nstance are computed as an average tour length of fve replcatons. Frstly, results for the experments of tunng k are presented. Sx experments were carred out wth the settngs as follows: a = 2 (n 2) (n s the n (n ) number of ctes) and β = 0. FBPS and FRNN 2opt s ntegrated and each experment conssts of fve replcatons. LIN38 ( as ts known optmal) s used as the test dataset for ths set of experments. The results of the experments are summarzed n Table. Whenever a replcaton acheves the optmum tour length, the executon wll be halted and the results wll be collected. Otherwse, t wll be executed tll termnaton for teratons (cycles). G A E B F 0

5 For each replcaton, the table shows the best tour length, executon tme and at whch cycle the best tour length s acheved. #2opt denotes the total number of 2opt operatons f the FRNN 2opt s performed on every soluton. For cases where the optmum tour length s obtaned before teratons, #2opt s equal to. For nstance, #2opt for replcaton n experment I equals to 24. For cases where the executon s run for teratons tll termnaton, #2opt s equal to. denotes the total number of pruned solutons and ts percentage. In the last column we gve the error percentage (%Error), a measure of the qualty of the best result found by proposed algorthm (%(Avg Optmal_Result)/Optmal_Result). Consderng the results shown n Table, experment 2 produces the most satsfactory results. It shows the lowest values for the average tour length and the average computatonal tme, that are and 2403s respectvely. In terms of the total number of pruned solutons, Table shows that experments 3, 4, 5 and 6 fal to prune any soluton. Ths makes ther computatonal tmes much longer than experment 2 (nearly.5 tmes of experment 2), although ther average tour lengths are not very much dfferent from experment 2. In experment, by settng k to a low value, t s expected that the executon tme wll be reduced. However, the results show otherwse. The FBPS prunes all solutons towards the end of the algorthm executon and ths does not contrbute to fndng the optmal solutons. Hence, more tme s needed to search n the space. Ths makes the performance of experment not comparable to the rest. All these observatons show that k should not be confgured ether too hgh or too low. To acheve a reasonable trade off between soluton qualty and computatonal cost, k should be set wthn a bound of [0%, 30%]. The expermental results that show the performance of the DLA algorthm ntegrated wth FBPS and FRNN 2opt n terms of soluton qualty and computatonal tmng on a set of Exp k% Attr. Tme(s) #2opt Tme(s) #2opt Tme(s) #2opt Tme(s) #2opt Tme(s) #2opt Tme(s) #2opt Rep Table : Adustng k and ts Results n experments. Rep Rep Internatonal Journal of Computer Applcatons ( ) Volume 46 No.7, May selected problem nstances from TSPLIB are presented n Table 2. The table llustrates results of three dfferent sets of experment. Parameter settngs for these three experments are the same as what were used n prevous secton, except t wll be executed for 0000 teratons. Experment A uses the 2opt whch requres an exhaustve comparson and swap n ts operaton. Experment B uses the FRNN 2opt, but not the FBPS. Experment C uses both FRNN 2opt and FBPS and the k s set at 0%. The results for experment A n Table 2 has been prevously reported n [5] and they were comparable wth other seven exstng approaches for solvng TSP. As the soluton qualty s mantaned after the FRNN 2 opt and FBPS ntegraton, such comparson study as n [5] s not ncluded n ths paper. The focus of ths paper s to nvestgate the effect on computatonal tmng when such the ntegraton s made. As shown n Table 2, the three approaches are able to obtan the optmal or near to optmal tour length for all the problems. Consderng all 20 problems together, Table 2 shows that the soluton qualty s sustaned whle a maor reducton n computatonal tme s observed. 6. CONCLUSION A new DLA algorthm optmzed by frequencybased prunng strategy and effcent mplementaton of 2opt has been proposed n ths paper. Based on the expermental results, the addton of both FBPS and FRNN 2opt n the DLA algorthm s able to sgnfcantly mprove ts executon performance compared to the DLA algorthm proposed n [5]. The proposed algorthm wll ensure that only a set of promsng solutons are locally optmzed by FRNN 2opt, and hence avod performng an exhaustve 2opt on every soluton generated by DLA. Rep Rep Avg Error %

6 Internatonal Journal of Computer Applcatons ( ) Volume 46 No.7, May 202 Table 2: Performance of the DLA Algorthm terms of Soluton Qualty. Exp. A Exp. B Exp. C Problem DLA + 2opt DLA + FRNN 2opt DLA + FRNN 2opt + FBPS Optmal (ctes) Tour Avg. Tour Avg. Tour Avg. %Err %Err %Err Best Avg Tme(s) Best Avg Tme(s) Best Avg Tme(s) ATT(48) EIL(5) BERLIN (52) ST(70) EIL(76) PR(76) KROA (00) KROB (00) KROC (00) KROD (00) KROE (00) EIL(0) LIN (05) KROA (50) KROA (200) TSP (225) A(280) LIN (38) Average: REFERENCES [] G. Laporte, "The travelng salesman problem: An overvew of exact and approxmate algorthms," European Journal of Operatonal Research, vol. 59, no. 2, pp , 992. [2] G. Laporte, "The travelng salesman problem: An overvew of exact and approxmate algorthms," European Journal of Operatonal Research, vol. 59, no. 2, pp , 992. [3] D. J. Rosenkrantz, R. E. Stearns, and P. M. Lews, "An analyss of several heurstcs for the travelng salesman problem," SIAM Journal on Computng, vol. 6, no. 3, pp , 977. [4] A. M. Freze, "An extenson of Chrstofdes heurstc to the kperson travellng salesman problem," Dscrete Appled Mathematcs, vol. 6, no., pp. 7983, 983. [5] B. Chandra, H. Karloff, and C. Tovey, "New results on the old kopt algorthm for the travelng salesman problem," SIAM Journal on Computng, vol. 28, no. 6, pp , 999. [6] S. Ln and B. W. Kernnghan, "An effectve heurstc algorthm for the travelng salesman problem," Operatons Research, vol. 2, no. 2, pp , 973. [7] E. H. L. Aarts, J. H. M. Korst, and P. J. M. Vanlaarhoven, "A quanttatve analyss of the smulated annealng algorthm A case study for the travelng salesman problem," Journal of Statstcal Physcs, vol. 50, no. 2, pp , 988. [8] J. Knox, "Tabu search performance on the symmetrc travelng salesman problem," Computers & Operatons Research, vol. 2, no. 8, pp , 994. [9] B. Fresleben and P. Merz, "A genetc local search algorthm for solvng symmetrc and asymmetrc travelng salesman problems," n Proceedngs of Internatonal Conference on Evolutonary Computaton, 996. pp [0] P. Merz and B. Fresleben, "Genetc local search for the TSP: New results," n Proceedngs of the 997 IEEE Internatonal Conference on Evolutonary Computaton, 997. pp [] C. M. Whte and G. G. Yen, "A hybrd evolutonary algorthm for travelng salesman problem," n Proceedngs of Congress on Evolutonary Computaton, 2004, CEC2004., pp [2] L. M. Gambardella and M. Dorgo, "Solvng symmetrc and asymmetrc TSPs by ant colones," n Proceedngs of IEEE Internatonal Conference on Evolutonary Computaton, pp [3] T. Stützle and H. Hoos, "MAXMIN ant system and local search for the travelng salesman problem," n Proceedngs of ICEC' IEEE 4th Internatonal Conference on Evolutonary Computaton, 997. pp

7 Internatonal Journal of Computer Applcatons ( ) Volume 46 No.7, May 202 [4] P. Lucc and D. Teodorovc, "Computng wth Bees: Attackng Complex Transportaton Engneerng Problems," Internatonal Journal on Artfcal Intellgence Tools, vol. 2, no. 3, pp , [5] M. Alpour, "Solvng Travelng Salesman Problem Usng Dstrbuted Learnng Automata mproved by 2opt local search heurstc," Proceedngs of Frst CSUT Conference on Computer, Communcaton and Informaton Technology, Computer Scence Department, Tabrz Unversty, Tabrz, Iran, pp Nov. 20. [6] J. L. Bentley, Fast algorthms for geometrc travelng salesman problems, ORSA Journal on Computng, vol. 4, no. 4, pp , 992. [7] K. S. Narendra and K. S. Thathachar, "Learnng Automata: An Introducton," (PrentceHall, New York, 989). [8] M. A. L. Thathachar and P. S. Sastry, "A herarchcal system of learnng automata that can learn the globally optmal path," Informaton Scence 42 (997) [9] M. A. L. Thathachar and B. R. Harta, "Learnng automata wth changng number of actons," IEEE Trans. Systems, Man, and Cybernetcs SMG7 (987) [20] M. A. L. Thathachar and V. V. Phansalkar, "Convergence of teams and herarches of learnng automata n connectonst systems," IEEE Trans. Systems, Man and Cybernetcs 24 (995) [2] S. Lakshmvarahan andm. A. L. Thathachar, "Bounds on the convergence probabltes of learnng automata," IEEE Trans. Systems, Man, and Cybernetcs, SMC6 (976) [22] K. S. Narendra and M. A. L. Thathachar, "On the behavor of a learnng automaton n a changng envronment wth applcaton to telephone traffc routng," IEEE Trans. Systems, Man, and Cybernetcs SMCl0(5) (980) [23] M. Alpour, "A learnng automata based algorthm for solvng capactated vehcle routng problem," Internatonal Journal of Computer Scence Issues Vol. 9, Issue 2, No, pp March 202. [24] J. S. Gero and V. A. Kazakov, Evolvng desgn genes n space layout plannng problems, Artfcal Intellgence n Engneerng, vol. 2, no. 3, pp , 998. [25] J. S. Gero, V. A. Kazakov, and T. Schner, Genetc engneerng and desgn problems, n Evolutonary Algorthms n Engneerng Applcatons, D. Dasgupta and Z. Mchalewcz, Eds. Berln: Sprnger, 997, pp