Colored Traveling Salesman Problem and Solution

Transcription

1 Preprnts of the 19th World Congress The Internatonal Federaton of Autoatc Control Cape Town, South Afrca. August 24-29, 2014 Colored Travelng Salesan Proble and Soluton Jun L 1, 2, Qru Sun 1, 2, MengChu Zhou 3, Xaolong Yu 1, 2, and Xanzhong Da 1, 2 1. Key Laboratory of Measureent and Control of CSE, Mnstry of Educaton, Nanng , Chna 2. School of Autoaton, Southeast Unversty, Nanng Chna (e-al: {.l, xzda}@seu.edu.cn; {sunqr, xly }@163.co) 3. New Jersey Insttute of Technology, Newark, NJ USA (e-al: zhou@nt.edu) Abstract: Multple Travelng Salesan Proble (MTSP) s an portant cobnatoral optzaton proble. However, t s applcable to only the cases n whch ultple executng ndvduals (travelng salesan) share the coon workspace (cty set). It cannot be used to handle any ult-achne engneerng systes where ultple achnes workspaces are not the sae and partally overlap wth each other. Ths paper proposes and forulates a new MTSP called colored travelng salesan proble (CTSP). Each of ts salesen s assgned a prvate cty set and all salesen share a publc cty set. Every set of ctes s colored dfferently. To solve CTSP, we present two proved genetc algorths (GA) by cobnng the classc one wth a greedy algorth and hll-clbng one to acheve better perforance. Fnally, the algorths are appled and copared through a case study. The result shows that the hllclbng GA enoys the best perforance aong the nvestgated ones. Keywords: TSP, MTSP, Modellng, Genetc Algorth, Greedy Algorth, Hll-clbng Algorth 1. INTRODUCTION A ultple travelng salesan proble (MTSP) generalzed fro a travelng salesan proble (TSP) s a well-known cobnatoral optzaton proble. It as to deterne a faly of tours wth nal total cost for ultple salesen to vst each cty exactly once wthn a gven set and eventually return to the hoe cty. MTSP and TSP arse n a varety of applcatons that requre addressng schedulng, plannng, routng, and/or sequencng ssues. Applcaton exaples of TSP n achne schedulng and sequencng, and vehcle routng can be founded n (Gutn and Punnen, 2002). Other work reports addtonal applcatons n crcut wrng (Hrogak, et al., 2005) and n statstcal data analyss ncludng orderng and clusterng obects, e.g., gene orderng n (Ray, et al., 2007) and proten clusterng n (Johnson and Lu, 2007). The ost coprehensve survey on the applcatons of MTSP s gven n (Bektas, 2006). Carter and Ragsdale (2006) stress ts use n pre-prnt nsert advertseent schedulng. A slar applcaton n hot rollng schedulng s reported n (Tang, et al., 2000). Autonoous robot or vehcle oton plannng (Basu, et al., 2000, and Ryan, et al., 1998) represents other types of ts applcatons. Saleh and Chelouah (2004) apply t to satellte surveyng syste desgn. Toth and Vgo (2002) nvestgate a vehcle routng proble as the generalzaton of TSP. Cheong and Whte (2012) have nvestgated how to dynacally deterne a tour for TSP based on real-te traffc congeston data. In essence, MTSP s an abstracton of the practcal probles n whch ultple executng ndvduals (travelng salesen) are nvolved and they share the coon workspace (cty set). In other words, all the ctes of MTSP are dentcal for each salesan,.e., each cty can be vsted by any salesan. However, not all executng ndvduals have the sae workspace n soe applcaton probles. Take the schedulng of a ult-achne engneerng syste (MES) as an exaple. The workspaces of ndvdual achnes are not the sae but overlap partally wth each other. Thus, each achne has to perfor not only the operatons ndependently n ts prvate workspace, but also coplete all the operatons wth other achne(s) together n the overlapped workspace. A typcal MES,.e., a dual-brdge wateret cuttng achne tool, s llustrated n Fg. 1. Fg. 1. Dual-brdge Wateret Cuttng Machne Tool It conssts of two ndependent brdge systes. Ther cuttng areas have an overlapped secton,.e., the arked area on the workbench wth a red box, so as to prevent the presence of cuttng dead zone. Thus, the overlapped area allows both brdges to enter and the two areas out of t are ther exclusve cuttng areas. Due to the partally overlapped workspaces, a schedulng ethod for MTSP cannot be sply used to schedule MES. Copyrght 2014 IFAC 9575

2 Cape Town, South Afrca. August 24-29, 2014 On the other hand, the basc eleents of such a proble,.e., ndvduals, operatons, and workspaces, are stll slar to the salesen, cty vsts and cty set of TSP, respectvely. The dfference les n that each ndvdual (salesan) of the forer not only has a prvate workspace (cty set) but also shares a coon workspace wth others. To dstngush the dfferent ctes, we defne a new ultple travelng salesan proble by colorng the ctes, called Colored TSP (CTSP). CTSP frequently arses n real-lfe applcatons where soe closely dependent relatons between the salesen and the ctes ust be obeyed when one deternes a soluton. It s a sgnfcant proble n theory and practce. We call the sae proble as MTSP* by L, et al. (2013) and present a genetc algorth (GA) soluton. However, CTSP has not been forulated n a atheatcally rgorous way. Ths paper forally defnes CTSP and proves the pror ethod n (L, et al., 2013) by cobnng GA wth Greedy Algorth and Hll-clbng Algorth. Next, CTSP s forulated n Secton 2. Secton 3 presents two proved GAs. Secton 4 gves a case study wth the coparson results. The paper s concluded n Secton DEFINITION AND FORMULATION OF CTSP 2.1 CTSP Defnton Let n, 1,2,3,... Z and n. CTSP as to deterne a faly of tours wth the nal total cost for salesen to vst n ctes exactly once gven publc and prvate ones, and eventually return to the hoe cty (depot). Let V, Z 1, 2,..., be the prvate cty set assgned to the -th salesan and U, Z r, r 1 be the -th shared cty set, and W be the accessble cty set of the -th salesan,.e., the unon of sets of -th salesan s prvate ctes and shared ctes. The cty sets eet the followng constrants, gven U W : U,, Z r (1) W and W W,, Z (2) There are varous cases of the ntersectons aong the accessble cty sets n CTSP. A coon one s that there s only one coon cty set shared by salesen, as shown n Fg. 2. Fg. 2. Exaple of CTSP The nodes n the areas V 1, V 2, and V 3 represent the prvate ctes of salesen 1, 2, and 3. They have the only shared cty set V Integer Prograng Model CTSP s forulated over a coplete dgraph G ( V, E), where the vertex set V 0,1, 2,..., n 1 corresponds to the ctes and each edge n (, ) E,, s assocated wth a weght representng a vst cost (dstance) between two ctes and. The vertex 0 represents the hoe cty (depot). Let V V \ 0. V s dvded nto +1 sets,.e., V 0, the publc one, and V, the prvate ones of the salesen for all Z. The obectve of CTSP s to deterne Haltonan cycles or crcuts on G wth the least total cost such that any vertex of each prvate set s vsted exactly once by the specfed salesan and any vertex of the publc set s vsted by any salesan exactly once and eventually return to cty 0. Of course, t allows each salesan to have an exclusve hoe cty n ther prvate set, lke the wateret cuttng exaple shown n Fg. 1. However, ths work wll not dscuss such cases. Bnary varable x =1,,, V, and k Z k, f the k-th salesan passes through edge (, ); and otherwse, xk 0. u k s the nuber of nodes vsted on the k-th salesan s tour fro the depot up to node. The nteger prograng odel of CTSP s presented as follows. Mnze xk (3) k1 0 0 Subect to x0k 1, (4) x0k 1 1 =1, k Z, (5) xk 0, (6) V U0 1 V 2 x k 0, Vk, V \( V0 Vk), kz, (7) xl 0, (8) x l 0, Vk, V,, k l, lz, (9) V 3 xk 1, (10) 0 k1 9576

3 Cape Town, South Afrca. August 24-29, 2014 x k 1, V,, (11) 0 k1 x hk xk, 0 k 0 h V,, hv V, h, (12) uk uk nxk n 1,, V,, kz, (13) Equatons (4) and (5) requre that every salesan start fro and return to cty 0, and (6) ensures that salesan k cannot start fro hs own exclusve cty to vst a prvate cty of other salesen and (7) that another salesan s forbdden to vst a prvate cty of salesan k fro ts own prvate cty as well. Equatons (8) and (9) ensure that salesan l( k) can nether start fro a prvate cty of salesan k nor return to t. Each cty except cty 0 can be vsted exactly once as descrbed by (10) and (11). Equaton (12) represents that a publc cty can be vsted by any salesan whle (13) prohbts the foraton of any sub-tour aong nodes n V \{0}. Lke MTSP, CTSP s also NP-hard. Moreover, the restrcton on cty colours akes ts soluton ore dffcult and teconsung than that of MTSP. It s proven that the heurstcs are faster and ore effcent than the exact ethods n the soluton of MTSP wth respect to the proble sze. In any cases, however, the forer cannot be guaranteed to obtan the optal soluton and are thus applcable to solve those cases n whch good-qualty solutons suffce. Wth ths n nd, ths paper presents GA for CTSP. Fg. 3. Exaple of CC In Step 1, gven two parents, a secton of a cty ndvdual s selected at rando, and then ts genes are swapped wth those of another ndvdual, thereby resultng n two new ndvduals as shown n Step 2. The appng relatonshp of the selected sectons n two cty ndvduals s 8 3, 9 8, 5 2, 4 7, 7 1, and Step 3 exchanges the redundant genes accordng to the selected secton, and then fnds that prvate ctes 5, 3, 7, 1, and 6 n the left chroosoe and ctes 2, 5, and 4 n the rght one are assgned to the wrong salesen. Next, Step 4 reassgns the prvate ctes to the correct salesen and obtans two reasonable generatons. A cty utaton (CM) process n a dual-chroosoe s llustrated n Fg GENETIC ALGORITHMS FOR CTSP 3.1 GA and Its Ltaton Our pror work developed a basc GA to solve CTSP (L, et al., 2013). It represents a soluton va dual chroosoes that are decally coded,.e., cty and salesan chroosoes. Constrants (6)-(12) as a cty assgnent relaton are taken account nto the dual-chroosoe codng where each cty gene corresponds to a rght salesan gene at the sae poston. Suppose that prvate ctes of salesen 1-3 are ctes 1-2, 3-4, and 5-6, respectvely, and the shared ctes are ctes L, et al., (2013) adopt the cobnaton of Roulette Wheel ethod and Eltst strategy as the selecton operaton. Three pars of copostons of the crossover and utaton operators are copared and the result shows that the perforance of cty crossover and cty utaton (CCM) operator s the best. A cty crossover operator s a odfed partally atched crossover (PMX). Fgure 3 shows a crossover process of a dual-chroosoe wth a sngle crossover of cty chroosoes. Fg. 4. Exaple of cty utaton (CM) Frst, the gene ponts of ctes 8 and 7 are selected as swappng ones. After swappng, the cty assgnent relaton s satsfed and the utaton s over. The ftness functon takes the value of the nverse of the total tour cost f(x) equal to that of Eq. (3). 1 F( x)= 1+ f ( x ) (14) In (L, et al., 2013), the case study ndcates that the evoluton of GA s slow and t s easy to trap n a local optu. 3.2 Greedy GA The decson ade by usng Greedy Algorth at each step ay not reach the best n the global vew but the local optu. However, t can obtan the satsfactory soluton rapdly because t avods the great effort needed to exhaust all possbltes to fnd the optal soluton. We use t to 9577

4 Cape Town, South Afrca. August 24-29, 2014 optze the ndvduals of the ntal populaton generated randoly at the frst step of GA. Intal populaton of hgh qualty wll accelerate the populaton evoluton of GA and reach satsfactory soluton rapdly. We nae ths proved algorth as a greedy GA. Wth regard to CTSP, the crteron s defned as the shortest dstance between two ctes. Naely, a cty wll be selected as the next one that the correspondng salesan wll vst once t s nearest to the current cty n the vsted sequence. It can optze a soluton by reorderng ts sequence. For exaple as shown n Fg. 5, the randoly generated vst sequence s The su of dstances s =240. It can be optzed to be sequence by usng the greedy algorth. The proved dstance s =159. Obvously, t s a better soluton. algorth s very strong and t s a coon ethod used for the local optu search. GA n (L, et al., 2013) adopts the cobnatoral selecton strategy of elte reservaton and roulette n (Sun, 2013). After a certan perod of evoluton, t ay be trapped nto a local optu. To escape fro t, the best ndvdual of each generaton can be optzed by usng Hll-Clbng Algorth. Specfcally, f a better ndvdual s obtaned through hll-clbng, t replaces the orgnal one; and otherwse, the orgnal one reans n t. Note that the hllclbng GA adopts Greedy Algorth to optze the ntal populaton too. The neghbourhood pont selecton pacts greatly on the hll-clbng algorth and the paper adopts two pont swappng. Gven CTSP wth ( 2) salesen, t should select 2 genes by ths selecton strategy. The ftness ust be recalculated after every te of gene swappng. A hll-clbng GA ncludes the followng steps: Step 1: Deterne f the -th salesan perforng the current swappng s the -th salesan,.e., =. If so, end ths hllclbng; otherwse, go to the next step. Fg. 5. Dstances between ctes The generaton process of the ntal populaton n greedy GA s as follows. Step 1: Deterne f the nuber of ndvduals n the current ntal populaton s equal to the set nuber N or not. If t s true, ternate the process; otherwse, go to the next step. Step 2: Generate a cty sequence randoly and assgn the prvate ctes to the specfed salesan and the publc ctes to all the salesen randoly. It results n ndvdual a. Step 2: Select two cty genes assgned to the -th salesan, fro the cty chroosoe of a. Swap the and obtan ndvdual a, and go to the next step. Step 3: Deterne f the value of ftness of a s greater than that of a. If so, let a a ; and otherwse, gve up a and keep a. Step 4: Let 1, and return to Step 1. The an procedure of hll-clbng GA s suarzed n Fg. 6. Step 3: Reorder the cty sequence of a by the shortest dstance crteron to nze the vst cost and obtan ndvdual a. Step 4: Detect f a has already exsted n the populaton or not. If so, go back to Step 2; otherwse, nsert t nto the populaton and go back to Step Hll-Clbng GA Hll-clbng Algorth utlzes neghbourhood search technques to search, lke hll-clbng, n a sngle drecton that the qualty of a soluton s possble to be proved (L, et al., 2006). Startng fro an exstng node, t generates a new soluton wth a ethod of neghbourhood pont selecton and copares t wth the value of the exstng node. If the forer s larger, replaces the latter by the forer; otherwse, return the latter and set t as the axu. Repeat the process of clbng upward (to better soluton) untl the hghest pont s reached. The local search power of the Fg. 6. Flowchart of Hll-clbng GA 9578

5 Cape Town, South Afrca. August 24-29, CASE STUDY A CTSP wth n 51 and 4 s shown n Fg. 7, where V 0 s the publc cty set (vst area) and V 1 -V 4 are the prvate cty sets (vst areas) of Salesen 1-4, respectvely. All the algorths and processes are pleented n C++ on the platfor Mcrosoft Vsual Studo The coputer used s Dell Inspron620s havng Wndows 7 (32 bts) wth CPU Intel Core3 and 2GB RAM at 3.30GHz. Fg. 9. Evoluton perforance of GA wth epochs Fg. 7 CTSP and ts cty dstrbuton Next, the three algorths GA, greedy GA, and Hll-clbng GA are appled to solve the proble and ther perforances are copared. We set the sae paraeters,.e., the ndvdual nuber of a populaton to be 30, crossover probablty 0.7, and utaton probablty 0.1. Each algorth s run for fve tes and the axu of epochs s Convergence Rate Experents show that all the algorths are convergent. To copare ther convergence rates, we plot the best ndvdual of each generaton obtaned by the as shown n Fg. 8. The total path length of the best ndvdual of the ntal populaton of GA s about 1200k. Wth Greedy Algorth, the qualty of the ntal populaton can be greatly proved. For exaple, Greedy GA and Hll-clbng GA converge rapdly far before ther preset ternaton condton. However, t sees that the basc GA cannot coplete ts evoluton wthn 2000 epochs. Thus, we odfy the generaton count of GA to be The convergence s extreely slow and the evoluton ends at about the 14000th epoch and the evoluton plot s shown n Fg. 9. Opposte to t, Greedy GA and Hllclbng GA can accoplsh ther evoluton at about the 1300th generaton. Ths s agntude-fold savng n coputatonal te. In addton, Hll-clbng GA outperfors Greedy GA. Wthout a hll-clbng operaton, the latter needs about 1200 epochs to evolve to the result wth total tour length of 550(k); whle the forer spends about 200 epochs only to acheve the sae or better result. 4.2 Soluton Qualty The results obtaned n the tests are lsted n Table 1. Table 1. Results of three GAs (k) Tes GA Greedy GA Hll-clbng GA Mean tour length In TABLE 1, wth the sae set of paraeters, spendng 2000 epochs, Hll-clbng GA can reach the ean tour length of (k), copared to (k) of GA and (k) of Greedy GA. Fg. 8. Evoluton perforance of GAs wth 2000 epochs Fro Fg.9, we fnd that wthout the Hll-clbng operaton, GA traps n the local optu and t s hard to reach ts best 9579

6 Cape Town, South Afrca. August 24-29, 2014 result at about the 14000th epoch wth the total tour length of about 560(k). Wth the help of the Hll-clbng operaton, t s clear that after reachng the sae result of Greedy GA, Hll-clbng GA can contnue to obtan a better soluton as shown n Fg. 8. The total tour length of the best soluton wth Hll-clbng GA s (k). The soluton of vst tours s: Salesan 1: ; Salesan 2: ; Salesan 3: ; and Salesan 4: The vst routes are shown n Fg. 10. Fg. 10. Vst tours of four salesen In suary, (1) Hll-clbng GA overcoes the defcency of local search of GA to soe extent by keepng ts global search capablty. It possesses better optzaton ablty and yelds better results than Greedy GA; and (2). The convergence rate of Hll-clbng GA s the hghest aong the three consdered GAs. Ths attrbutes to the ntroducton of the hll-clbng operaton n GA. 5. CONCLUSION In ths paper, we forulate a new ultple travelng salesan proble, CTSP, where dfferent salesen have dfferent prvate cty sets and share a set of publc ctes. It s sgnfcant for odellng the applcatons where ultple ndvduals workspaces are not the sae but partally overlap wth each other. To overcoe the shortcongs of the classc GA, we present two proved genetc algorths (GA), called greedy GA and hll-clbng GA, by cobnng the classc one wth the greedy algorth and the hllclbng algorth to solve CTSP. Fnally, the presented GAs are used to solve an exaple CTSP and the result shows that the hll-clbng GA enoys the best perforance n ters of convergence rate and soluton qualty. In the future, we ntend to research other heurstcs of CTSP and explore the related applcatons. REFERENCES Basu, A., Elnaga, A., and Al-Ha, A. (2000). Effcent coordnated oton. Matheatcal and Coputer Modellng, vol. 31, pp Bektas, T. (Jun 2006). The ultple travelng salesan proble: an overvew of forulatons and soluton procedures. Oega, vol. 34, pp Carter, A.E., and Ragsdale, C.T. (2006). A new approach to solvng the ultple travelng salesperson proble usng genetc algorths. European ournal of operatonal research, vol. 175, pp Cheong, T., and Whte, C.C. (2012). Dynac travelng salesan proble: Value of real-te traffc nforaton. IEEE Transactons on Intellgent Transportaton Systes, vol. 13, pp Gutn, G., and Punnen, A. (2002). The Travelng Salesan Proble and Its Varatons. pp Kluwer, Dordrecht. Hrogak, T., Aoyaa, E., Ogawa, K., Hashoto, N., and Matsuura, M. (2005). CAM systes based on travelng salesan proble fro te perspectve for hgh densty through-hole drllng. In Pts A-C (ed.), Advances n Electronc Packagng, pp Johnson, O., and Lu, J. (2006). A travelng salesan approach for predctng proten functons. Source Code for Bology and Medcne, vol. 1, pp L, J., Sun, Q.R., Zhou, M.C., and Da, X.Z. (Oct 13-16, 2013). A New Multple Travelng Salesan Proble and ts Genetc Algorth-based Soluton. In Proc IEEE Int. Conf. Systes, Man and Cybernetcs, pp Manchester, UK. L, A., Rodrgues, B., and Zhang, X. (2006). A sulated annealng and hll-clbng algorth for the travelng tournaent proble. European Journal of Operatonal Research, vol. 174, pp Ray, S.S., Bandyopadhyay, S., and Pal, S.K. (2007). Gene orderng n parttve clusterng usng croarray expressons. Journal of Boscences, vol. 32, pp Ryan, J.L., Baley, T.G., Moore, J.T., and Carlton, W.B. (Dec 13-16, 1998). Reactve tabu search n unanned aeral reconnassance sulatons. In Proc. 30th Sulaton Conf. Wnter. Washngton, DC, vol.1, pp Saleh, H.A., and Chelouah, R. (2004). The desgn of the global navgaton satellte syste surveyng networks usng genetc algorths. Engneerng Applcatons of Artfcal Intellgence, vol. 17, pp Sun, Q.R. (2011). The research and applcaton of the colored travelng salesan proble. Master Dssertaton, School of Autoaton, Southeast Unversty. Tang, L., Lu, J., Rong, A., and Yang, Z. (2000). A ultple travelng salesan proble odel for hot rollng schedulng n Shangha Baoshan Iron & Steel Coplex. European Journal of Operatonal Research, vol. 124, pp ,. Toth, P., Vgo, D.(Eds.) (2002). The Vehcle Routng Proble. SIAM Monographs on Dscrete Matheatcs and Applcatons. SIAM, Phladelpha. 9580