An Evolutionary Multiple Heuristic with Genetic Local Search for Solving TSP

Transcription

1 An Evolutionary Multiple Heuriti with Geneti Loal Searh for Solving TSP Peng Gang Ihiro Iimura 2 and Shigeru Nakayama 3 Department of Information and Computer Siene Faulty of Engineering Kagohima Univerity Korimoto Kagohima Japan gang90@hotmail.om 2 Department of Adminitration Faulty of Adminitration Prefetural Univerity of Kumamoto Tukide Kumamoto Japan iiimura@pu-kumamoto.a.p 3 Department of Information and Computer Siene Faulty of Engineering Kagohima Univerity Korimoto Kagohima Japan hignaka@i.kagohima-u.a.p Abtrat Traveling aleman problem (TSP) i well tudied a one of the ombinatorial optimization problem. In thi paper we propoe an evolutionary multiple heuriti ombined with Geneti Loal Searh (GLS) for olving TSP. Two main operation of Complete 2-Opt () and Smallet Square (SS) are ombined in the new algorithm and applied to olve TSP. Another two operation of Deletion and Bet Part Colletor (BPC) are diued. A reaonable reult i preented by ombining thee operation with geneti operator in the propoed algorithm for the TSP in our experiment. Keyword: Heuriti Geneti Loal Searh (GLS) Traveling Saleman Problem (TSP) Fratal TSP. I. Introdution The Geneti Algorithm (GA) i an optimizing algorithm that model the proee of natural evolution []. The traditional GA uually onit of ome operator uh a roover eletion and mutation whih are imulated from biologial and geneti proee. However the traditional GA i ineffiient for olving large optimization problem. It i well known that mot of the ueful algorithm have often inorporated the GLS [2] [3] [4] [5] [6]. The GA i applied to the TSP whih i a well known and important ombinatorial optimization problem [7] [8]. In the TSP eah ditane between two itie i given for a et of n itie. The goal i to find the hortet tour that viit eah ity exatly one and then return to the tarting ity. Many book and paper introdue the proedure of how to find the hortet tour uually alled a the optimum olution in TSP earh algorithm [9] [0] []. There are urrently three general lae of heuriti for olving TSP: laial tour ontrution heuriti uh a the Nearet Neighbor heuriti the Greedy algorithm and loal earh algorithm baed on re-arranging egment of the tour [2]. The nearet-neighbor heuriti an be olved eaily in quadrati time and linear pae. However Bentley [3] ha hown that thi imple tehnique i outperformed by other eemingly harder to ompute method uh a the multiple fragment heuriti and heapet inertion [4].

2 P. Gang I. Iimura and S. Nakayama An Evolutionary Multiple Heuriti with Geneti Loal Searh for Solving TSP Multiple fragment heuriti onider all edge one at a time in orted order and inlude an edge if it onnet the endpoint of two fragment of tour onneted omponent of previouly added edge. Cheapet inertion maintain a tour of a ubet of the ite and at eah tep add a ite by replaing an edge of the tour by two edge through the new ite. Eah ueive inertion i hoen a the one auing the leat additional length in the augmented tour. Thee method onentrate on reontrution of the edge in the TSP tour. In thi paper we propoe an evolutionary multiple heuriti ombined with Geneti Loal Searh (GLS) to olve the TSP. Two main operation of Complete 2-Opt () and Smallet Square (SS) are ombined in our algorithm and applied to the TSP. The i baed on the 2-Opt heuriti earh and an remove all roed edge in the tour if the repetition i uffiiently great. The SS elet horter edge than the. The problem with the SS i that the ity order of the original tour are hanged whilt it i applied. Therefore the roed edge annot be removed ompletely. However it preent a good reult ombining the and the SS with geneti operator for the TSP in our experiment. Another two operation of Deletion and Bet Part Colletor (BPC) are employed in the propoed algorithm. The Deletion i effetive for removing dupliate from the population and the BPC i effetive for olleting the bet part among the individual to ultivate the elite individual. II. Outline of Evolutionary Multiple Heuriti The heuriti 2-Opt ha been ued to optimize TSP tour in onnetion with the GA [8]. Uually the 2-Opt i randomly applied by the GA to 20 30% of the individual in a population. K. Mathia et al. made a diuion on the topi of 2-Opt [5]. A deribed in their paper the 2-Opt remove two edge in a tour and then one of the reultant egment i revered and the two egment are reonneted. If the 2-Opt reult in an improved tour the hange i kept. Otherwie the tour i returned to it original form. The 2-Opt i typially applied to all n ( n ) 2 pair of edge. If one or more improvement are found the proe an be re-applied to the et of all edge. When no more further improvement are found the tour ha onverged on a loal optimum with repet to the 2- Opt. K. Mathia et al. alo tated that it i omputationally expenive to run the 2-Opt until onvergene [5]. In our previou work we preented a Multiple Heuriti Searh Algorithm and applied it to the TSP [6] [7] [8]. The reult how that the algorithm i effiient for loal improvement in the TSP intane but it ut emphaize on the performane of the multiple earh proe whih ue ome loal earh tehnique. Sine no geneti operator are employed to the earh proe it i not an evolutionary algorithm and eaily reahe the loal onvergene. Thi paper diue an evolutionary multiple heuriti algorithm ombined with the geneti operator baed on our previou work. The geneti operator uh a roover and mutation explore the diverity of earh pae and additionally improve the onvergene peed to avoid onverging prematurely to ome ub-optimal olution. peration of and SS are ombined in the algorithm for improving tour in the earh pae. The mean the 2-Opt i ompletely applied to an initial tour aording to the operation deribed in etion 3 until no roed edge are found in the tour. The i baed on the 2-Opt heuriti earh method. The tour reated by the uually i a loal optimum in whih probably ome of the ity order are the ame a they are in the optimal tour. Thi i a deirable attribute for loal earh operation. On the other hand ome horter edge might have been mied while the 2-Opt i being applied to remove the two roed edge. In a TSP tour two edge determine a quare exept for the neighboring one. The 2-Opt only hek roed edge and a pair of ide edge of a quare. Hene another pair of ide edge will be mied. The SS hek the roed edge and the two pair of the ide edge of a quare whih i the firt quare in our earh algorithm and then it hooe the hortet pair to improve the tour. If the tour i eparated a two ub tour another two edge whih form the eond quare between the two ub tour mut be found to reonnet the tour. The two edge hould not make the tour longer than the previou one. If uh edge do not exit the 2-Opt i 2

3 applied to the firt quare. In the algorithm the SS reate a tour horter than but it annot ompletely remove all the roed edge in the tour beaue the original ity order are hanged whilt the tour i reonneted in the eond quare. A better tour an be reated if the i et after the SS (whih we refer to a SS+). Another two operator ued in our algorithm are Deletion and Bet Part Colletor (BPC). Deletion keep an effiient pae for loal earh method by removing the individual whih have the ame ity order a other individual among the population. The individual removed i then re-generated by a preet mutation operator. The BPC i ued to ollet the bet part from different individual and ued to groom the elite individual. All the part olleted ontain the ame itie with the ame tart and end itie. If one part i found horter than that in the elite individual the part eleted in the elite individual i reombined with the horter one. The BPC an remarkably improve the tour by thi proe. In the traditional GA the eletion i an important operation. It reprodue the elite individual a new offpring. With the inreae of the elite individual from the eletion the population loe it diverity reulting in the earh pae of other geneti operator beoming narrow. The eleted elite uually form a loal optimum at the expene of finding an optimal olution. That i why we reeted employing the eletion operator in our algorithm. The roover and the mutation are implemented in our algorithm. The roover operator i an Edge Exhange Croover (EXX) [9]. The mutation operator i a Two Point Change method whih randomly hange the order of 2 itie. The EXX and the mutation interalate a new diverity into the population. The individual reombined by the roover and the mutation an ontain new roed edge and in their turn beome the new andidate for the and SS operation. III. Multiple Heuriti Algorithm a Loal Searh Our new algorithm of multiple heuriti i hown in Fig.. The terminative ondition of the algorithm i et to the total number of generation required. The order of the operation in the algorithm i ignifiant and related to the reult. The i hoen empirially to be et after the SS. A tour whih onit of n itie i expreed a = { 0 L i L L n }. Eah ditane d ( i ) i given for eah pair of itie i and. All itie are oded uing a path repreentation method. Start Initial Population Terminative Ye Condition No Bet Part Colletor Deletion Croover Mutation Smallet Square Complete 2-Opt End Fig.. Propoed algorithm of Evolutionary Multiple Heuriti with GLS. 3

4 P. Gang I. Iimura and S. Nakayama An Evolutionary Multiple Heuriti with Geneti Loal Searh for Solving TSP + i+ + i+ i i Fig. 2. Effet of (Left figure: Before appliation of Right figure: After appliation of ). A. Complete 2-Opt () N p tour are randomly initialized a one population. Here follow the tep to : L L L (the left figure of Fig. 2). The i (i) Chooe one tour { } = 0 i i+ + n and the are initialized to zero. for i < n. i i + + (ii) Chooe the edge No. ( ) (iii) Chooe the edge No. 2 ( ) for < n. (iv) If ( i +) 2 and d ( i ) + d( i+ + ) < d( i i+ ) + d( + ) the 2-Opt i applied namely the edge ( i i +) and ( +) are removed and the edge ( i ) and ( ) are onneted. The egment between i+ and i revered. i+ + + i+ i 2 k+ 2 q+ k 2 q Fig. 3. Effet of SS (Left figure: Before appliation of SS Right figure: After appliation of SS). (v) The tarting ity of the edge No. 2 i hanged to the next ity namely + and tep (iii) and (iv) are repeated until = n. If = n then + = 0. (vi) The tarting ity i of the edge No. i hanged to the next ity namely i + and tep (ii) (iii) (iv) and (v) are repeated until i = n. If i = n then i + = 0. (vii) The tep of (ii) (iii) (iv) (v) and (vi) are repeated for N time. If N i uffiient a tour without roed edge will be reated a hown in the right figure of Fig. 2. B. Smallet quare (SS) In the TSP a et of every two edge form a quare exept for the neighboring edge. There are only three way for the traveling aleman to pa through the 4 itie at the four orner of every quare (the left figure of Fig. 3): ( i i + ) + ( + ) ( i ) + ( i + + ) and ( i + ) + ( i + ). Obviouly the path of ( i + ) + ( i + ) i the hortet route in the quare. However if the aleman take ( i + ) + ( i + ) the tour will be eparated into two ub tour. Another two edge have to be found for reonneting the tour in the eond quare. N p tour are randomly initialized a one population. Here follow the tep to SS: (i) An initial tour = { 0 L i i+ L + L n } (the left figure of Fig. 3) i hoen in the population. The i and the are initialized to zero. The firt quare i earhed. for i < n. i i + + (ii) Chooe the edge No. ( ) (iii) Chooe the edge No. 2 ( ) for < n. 4

5 (iv) If ( i +) 2 d d the edge ( ) and ( i + ) + d( i + ) < d( i ) + d( i + + ) < d( i i + )+ ( +) i i + and ( +) ( ) and ( ) 5 are temporarily removed and the edge i + i + are onneted. The tour i eparated into two ub tour and 2 (the right figure of Fig. 3). The and 2 onit of t and u itie repetively where t + L L and = u = n. The ity order of and 2 are = { 0 k k+ t } { L L } repetively. 0 q q+ u (v) The eond quare i earhed. The k and the q are initialized to zero. from ub tour where k < t. (a) Chooe the edge No. 3 ( k k +) 2 (b) Chooe the edge No. 4 ( ) 2 q from ub tour q+ 2 where q < u. () The three pair of the edge in the eond quare are heked to find the hortet pair for reonneting the tour d d < d + d < d d () If ( k q ) + ( q+ k + ) ( k k + ) ( q q+ ) ( k q+ ) + ( q k + ) the edge ( 2 k k +) and ( 2 q q+) are removed and the edge ( 2 k q ) 2 ( q+ k + ) are onneted. A better tour i formed. Then go to the tep (vi) (2) If d( k q+ ) + d( q k + ) < d( k k + ) + d( q q+ ) < d( k q ) + d( q+ k + ) the edge ( 2 k k +) and ( 2 q q+) are removed and the edge ( 2 k q+) ( 2 q k +) are onneted. A better tour i formed. Then go to the tep (vi) (3) If d( ) d( ) < d( ) + d( ) < d( ) d( ) 2 and and k k + + q q+ k q q+ k + k q+ + q k + the tart ity of edge No. 4 i hanged to the next one namely q +. If q = u then q + = 0 and go to tep (b). When q = u the tarting ity of the edge No. 3 i hanged to the next one namely k + and go to tep (a). If no pair of edge in the eond quare i found to make the tour horter between and 2 until k = t and q = u the 2-Opt i applied to the i i + firt quare (Refer to the tep (iv)) namely the edge ( ) are removed and the edge ( i ) and ( i+ ) + + and ( ) are onneted. The egment between i+ and i revered. (vi) The tarting ity of the edge No. 2 i hanged to the next ity tep (iii) (iv) and (v) are repeated until = n-. If = n then + = 0. (vii) The tarting ity i of the edge No. i hanged to the next ity tep (ii) (iii) (iv) (v) and (vi) are repeated until i = n-. If i = n then i + = 0. (viii) The tep of (ii) (iii) (iv) (v) (vi) and (vii) are repeated N time. If N i uffiient a tour loe to the optimal olution will be obtained. In fat for ome TSP intane the optimal olution an be diretly obtained from or SS+ for example in our previou work [7] the optimal olution of C type of Conentri Cirle TSP wa obtained eaily by a ingle CPU even though the number of itie reahe to The repeat time (N) were experientially et to different number in our experiment. It i diffiult to deide how many repeat time are uffiient for running the or SS+ to remove all the roed edge in the tour beaue the amount of roed edge depend on the TSP intane itelf. But we found thi problem ould be olved while or SS+ i applied to the ame individual randomly for 2 N or more time. C. Deletion and bet part olletor (BPC) Deletion i effetive for removing the individual whih ha the ame ity order a the other individual among the population. Deletion i applied to the ix elite individual. The individual removed i re-generated by the mutation operator uing another individual eleted

6 P. Gang I. Iimura and S. Nakayama An Evolutionary Multiple Heuriti with Geneti Loal Searh for Solving TSP randomly from the population. Removing the dupliate from the population help to keep uffiient earh pae for the operation BPC a hown in Fig. 4. Part a Part b Individual I a Reombine Part a with Part b Individual I b Fig. 4. Effet of BPC. In our algorithm the and the SS+ diretly reate tour in whih roed edge are ompletely removed and partially ontain the ame ity order a they are in the optimal tour. The BPC i ued to ollet the partially optimal part for the elite individual from the other individual. Firtly we randomly and ontinuouly elet P itie in the elite individual 4 < P < n 2 (Fig. 4 Part a). The minimum range of P i et to 4 itie beaue it i the minimal improvable part whih ontain the ame tarting ity and ending ity. n 2 i et a the maximum range beaue it i large enough to over all improvable part in the tour whih roed edge are removed. Seondly the individual are examined for part whih ontain the ame itie with the ame tarting and ending itie a the part that i eleted from the elite individual. When one part i found to be horter than that in the elite individual the part eleted in the elite individual i reombined with the horter one (Fig. 4 Part b). The BPC i applied to the elite individual from No. 0 to No. 5. The EXX and mutation operator are randomly applied to eah perentage of the population; 60% and 5% repetively exept for the 6 hoen elite individual who remain unhanged to preerve their integrity. IV. Experiment and Conideration A. Experimental ondition The tour whih ontain n itie = { 0 i L L n } between two itie i and i d ( i ). Total ditane d total i given by L i given and the ditane n i= 0 ( ) d = d () total where n = 0. d total i the ame a the fitne of the individual. The horter the ditane i the higher the fitne i. The population ize i et to N p = 00. Our oure ode i written in Java and run on a PC (CPU: Pentium III.0GHz RAM: 256MB) with Mirooft Window 2000 Operating Sytem. The experiment are deigned with the following even ae and eah ae i applied to the TSP intane independently and number in the parenthee are the repeat time in eah generation; i i+ 6

7 Table. Seven ae in the experiment. ALGO No. Different loal earhe Fixed operation No. BPC (0) Deletion EXX Mutation No. 2 (0) Deletion EXX Mutation No. 3 SS (0) Deletion EXX Mutation No. 4 SS (5) (5) Deletion EXX Mutation No. 5 BPC (0) (0) Deletion EXX Mutation No. 6 BPC (0) SS (0) Deletion EXX Mutation No. 7 BPC (0) SS (5) (5) Deletion EXX Mutation The BPC i applied to the 5 hortet tour in every ae. The parameter of EXX Deletion and mutation are fixed in all ae. Two intane of the Eulidean Traveling Saleman Problem (ETSP) and three TSP intane from TSPLIB are ued in our experiment. One of the ETSP intane i the David tour a hown in Fig. 5 [20]. The intane from TSPLIB are att48 (Fig. 6) kroc00 (Fig. 7) and h30 (Fig. 8). Another one of the ETSP i the MNPeano tour a hown in Fig. 9 [2]. The oordinate of ETSP intane are generated with L-Sytem [22]. The David tour and the intane att48 are ued for examining the effiieny of the algorithm. The ae of BPC SS and i applied to the MNPeano tour and two other TSP intane (kroc00 and h30) to onfirm the robutne of the algorithm. Every optimal olution i onfirmed with the olution provided in TSPLIB. Fig. 5. The David tour. Fig. 6. The hortet tour of att48. Fig. 7. The hortet tour of kroc00. Fig. 8. The hortet tour of h30. 7

8 P. Gang I. Iimura and S. Nakayama An Evolutionary Multiple Heuriti with Geneti Loal Searh for Solving TSP Fig. 9. The MNPeano tour. B. Experiment of David tour and att48 Table 2 and 3 how the reult of different ae whih are applied to the David tour and the att48 repetively. The BPC i ued to ollet the optimal part from other individual and apply them to ultivate the elite individual. It i repeated 0 time and applied to the 5 hortet tour in eah generation in the algorithm. The reult indiate that it i impoible to obtain the optimal olution when only the BPC i et with the EXX Deletion and mutation. Six optimal olution are obtained in the att48 and none i found in the David tour in the ae of BPC and. The reult of ombination of BPC and SS are not alo atified in both David tour and att48. The ae of SS and preent a ignifiant reult: eleven optimal olution in the David tour and ixteen in the att48 are obtained. A mentioned in etion 3 the ould ompletely remove the roed edge it i benefiial to reate the andidate for the BPC. If ompared with the the SS took a muh longer time to run the ame number of generation. Thi happen beaue the SS hek three pair of the edge in the quare and one more quare i heked in the ame generation the tour i reonneted with the hortet pair of edge. The problem i that the ity order of the tour are hanged while the tour i reonneted in the eond quare. Some roed edge in the tour ould not be ompletely removed. The ae of SS and preent a better reult than ingle SS or beaue ould remove the roed edge whih remain in the tour after the appliation of SS. The bet ombination i BPC SS and whih form the new algorithm with the EXX Deletion and mutation. Optimal olution are obtained for all tour in the 20 run. The time and generation needed for produing the optimal olution beome horter and earlier than the other ae in both intane of the David tour and the att48. Table 2. Experimental reult of David tour with different loal earhe. TSP Citie ALGO Gen. MiniG MaxG MG ST LT Mt TT TS David tour 62 BPC David tour David tour 62 SS David tour 62 SS David tour 62 BPC David tour 62 BPC SS David tour 62 BPC SS Gen.: Total generation. MiniG: Minimum generation of the optimal olution. MaxG: Maximum generation of the optimal olution. MG: Mean generation of the optimal olution. ST: Shortet time for obtaining the optimal olution. LT: Longet time for obtaining the optimal olution. MT: Mean time for obtaining the optimal olution. TT: Mean time for running the total generation. The unit of time i eond. TS: Total optimal olution obtained in the 20 run. The mean no data wa obtained. 8

9 Table 3. Experimental reult of att48 with different loal earhe. TSP Citie ALGO Gen. MiniG MaxG MG ST LT Mt TT TS att48 48 BPC att att48 48 SS att48 48 SS att48 48 BPC att48 48 BPC SS att48 48 BPC SS Gen.: Total generation. MiniG: Minimum generation of the optimal olution. MaxG: Maximum generation of the optimal olution. MG: Mean generation of the optimal olution. ST: Shortet time for obtaining the optimal olution. LT: Longet time for obtaining the optimal olution. MT: Mean time for obtaining the optimal olution. TT: Mean time for running the total generation. The unit of time i eond. TS: Total optimal olution obtained in the 20 run. The mean no data wa obtained. C. Experiment of kroc00 h30 and MNPeano Furthermore the new algorithm i applied to the MNPeano tour whih onit of 364 itie and the other two TSP intane of kroc00 and h30. It i noted that all the olution are optimal in the 20 run (Table 4). For kroc00 tour the earliet generation for obtaining the optimal olution i at the 9 th and the latet one i found at the 28 th. The average value i the 87 th generation. The hortet time for obtaining the optimal olution i only 3 eond. The longet time i 42 eond. The average value i 60 eond. The average time for running 20 generation take only 84 eond. The optimal tour of the MNPeano tour i hown in Fig. 9. The earliet optimal olution i obtained at the 23 rd generation and take 494 eond of ingle CPU time. The imilar reult appear in the h30. The reult of the experiment indiate that the propoed evolutionary multiple heuriti ombining the GLS i an effetive algorithm for the TSP. Table 4. Experimental reult of different TSP intane with all loal earhe. TSP Citie ALGO Gen. MiniG MaxG MG ST LT Mt TT TS kroc00 00 BPC SS h30 30 BPC SS MNPeano 364 BPC SS Gen.: Total generation. MiniG: Minimum generation of the optimal olution. MaxG: Maximum generation of the optimal olution. MG: Mean generation of the optimal olution. ST: Shortet time for obtaining the optimal olution. LT: Longet time for obtaining the optimal olution. MT: Mean time for obtaining the optimal olution. TT: Mean time for running the total generation. The unit of time i eond. TS: Total optimal olution obtained in the 20 run. V. Summary The operation of the SS and the are looked at in depth revealing the pro and on of eah. The SS find orter edge but leave ome edge roed. The ueed in removing all the roed edge although olution are not a hort. By ombing firt the SS and then the after the BPC+Deletion and the geneti EXX+Mutation operation an optimal tour an be found in a reaonable number of generation whih i our prinipal reult. In the future we plan to look at how to ale up the algorithm for olving large intane of TSP. 9

10 P. Gang I. Iimura and S. Nakayama An Evolutionary Multiple Heuriti with Geneti Loal Searh for Solving TSP Aknowledgement The author would like to thank Mr. Chritopher J. Ahley for heking the idiomati uage very arefully in the paper. Referene [] J. H. Holland Adaptation in Natural and Artifiial Sytem (Univ. of Mihigan Pre 975). [2] A. Homaifar S. Guan and G. E. Liepin A New Approah to the Traveling Saleman Problem by Geneti Algorithm in Pro. 5th International Conferene on Geneti Algorithm (Morgan Kaufmann 993) [3] R. M. Brady Optimization Strategie Gleaned from Biologial Evolution Nature 37 (985) [4] T. G. Bui and B. R. Moon A New Geneti Approah for the Traveling Saleman Problem in Pro. t IEEE Conferene on Evolutionary Computation (994) 7 2. [5] G. Peng I. Iimura H. Turuawa and S. Nakayama Geneti Loal Searh Baed on Geneti Reombination: A Cae for Traveling Saleman Problem in Pro. 5th International Conferene on Parallel and Ditributed Computing Appliation and Tehnologie (PDCAT 2004) (Singapore 2004) [6] G. Peng T. Nakaturu and S. Nakayama Diuion on LGA with Parallel Sytem in Pro. Geneti and Evolutionary Computation Conferene (GECCO 2005) (Wahington DC 2005). [7] E. L. Lawler J. K. Lentra A. H. G. Rinnooy Kan and D. B. Shmoy The Traveling Saleman Problem: A Guided Tour of Combinatorial Optimization (John Wiley & Son 985). [8] S. Lin and B. Kernighan. An Effiient Heuriti Proedure for the Traveling Saleman Problem Operation Reearh 2(2) (973) [9] Mihael Hahler and Kurt Hornik. TSP - Infratruture for the traveling aleperon problem. Report 45 Reearh Report Serie Department of Statiti and Mathemati Wirthaftuniverität Wien Augae Wien Autria.(2006). [0] GUTIN TELOS. The Traveling Saleman Problem and it Variation. ISBN: (2007) 830 p. [] G.Reinelt. The Traveling Saleman: Computational Solution for TSP Appliation. Leture Note in Computer Siene 840 Springer (994) [2] P. C. Kanellaki and C. H. Papadimitriou Loal earh for aymmetri traveling aleman problem Operation Reearh 28(5) (980) [3] Bentley J. L. Experiment on traveling aleman heuriti. In Pro. t Symp. Direte Algorithm (990) pp ACM and SIAM. [4] Roenkrantz D. H. Stearn R. E. and Lewi P. M. II. An analyi of everal heuriti for the traveling aleman problem. SIAM J. Computing 6 3 (977) [5] K. Mathia and Darrell Whitley Geneti operator landape and the traveling aleman problem Parallel problem Solving from Nature 2 (992) [6] G. Peng and S. Nakayama Multiple Heuriti Searh in Geneti Algorithm for Traveling Saleman Problem in Pro. Geneti and Evolutionary Computation Conferene (GECCO 2003) Late-Breaking Paper (Chiago Illinoi USA 2003) [7] G. Peng I. Iimura and S. Nakayama A Multiple Heuriti Searh Algorithm for Solving Traveling Saleman Problem in Pro. 4th International Conferene on Parallel and Ditributed Computing Appliation and Tehnologie (PDCAT 2003) (Chengdu China 2003) [8] G. Peng I. Iimura and S. Nakayama Conideration on Multiple Heuriti Searh Method in Traveling Saleman Problem Tran. JSCES 6 (2004) (in Japanee) 0

11 [9] K. Maekawa H. Tamaki H. Kita and Y. Nihikawa A method for the traveling aleman problem baed on the geneti algorithm Tran. the Soiety of Intrument and Control Engineer 3(5) (995) (in Japanee) [20] P. Moato and M. G. Norman An analyi of the performane of traveling aleman heuriti on infinite-ize fratal intane in the Eulidean plane Tehnial report (CeTAD-Univeridad Naional de La Plata 994). [2] M. G. Norman and P. Moato The Eulidean Traveling Saleman Problem and a Spae- Filling Curve Chao Soliton and Fratal 6 (995) [22] A. Mariano P. Moato and M. G. Norman Uing L-Sytem to generate arbitrarily large intane of the Eulidean Traveling Saleman Problem with known optimal tour Anale del XXVII Simpoio Braileiro de Pequia Operaional (Vitoria Brazil Nov. 995) 6 8. Peng Gang reeived the Ph.D. degree in Agriulture and Ph.D. degree in Information Siene at Kagohima Univerity Japan in 200 and 2004 repetively. Hi reearh foue on the Evolutionary Computation. He i a member of International Soiety for Geneti and Evolutionary Computation (ISGEC) The Intitute of Eletroni Information and Communiation Engineer (IEICE) and Computer Aided Diagnoi of Medial Image (CADM). Ihiro Iimura wa born in Ibaraki Japan in 969. He reeived the B. Eng. and the M. Eng. from Sophia Univerity in 992 and 994 repetively and the Dr. Eng. from Kagohima Univerity in From 994 to 997 he wa Reearh Sientit for Hitahi Reearh Laboratory Hitahi Ltd. From 997 to 2002 he wa Leturer for Kumamoto Prefetural College of Tehnology. From 2002 to 2003 he wa Reearh Aoiate Prefetural Univerity of Kumamoto. From 2003 to 2006 he wa Senior Leturer and from 2006 he ha been Aoiate Profeor. He reeived Bet Paper Award for Young Reearher from IPSJ in 200 Certifiate of Merit for Bet Preentation from JSME in 2003 and Bet Paper Award for Young Reearher from IPSJ Kyuhu hapter in 2003 repetively. Hi reearh interet inlude evolutionary omputation warm intelligene and ditributed parallel proeing. He i a member of IPSJ IEICE and IEEJ et. Shigeru Nakayama wa born in Kyoto Japan in 948. He reeived the B. S. E. E. from Kyoto Intitute of Tehnology in 972 M. S. E. E. and Dotor of Engineering from Kyoto Univerity in 974 and 977repetively. From 977 to 98 he wa Reearh Aoiate for Sophia Univerity in Tokyo. From 98 to 987 he wa Reearh Aoiate for Kyoto Intitute of Tehnology. From 987 to 997 he wa Aoiate Profeor for Hyogo Univerity of Teaher Eduation. From 997 he beame Profeor of Information and Computer Siene for Kagohima Univerity. Hi reearh interet inlude ditributed parallel proeing parallel geneti algorithm parallel image proeing and ditributed obet. He i a member of Int. of Eletroni Information and Communiation Engineer Information Proeing Soiety and J.A.P.S.