The Solution of the More General Traveling Salesman Problem

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1 AMSE JOURNALS 04-Series: Adces A; Vol. ; N ; pp -40 Submitted No. 0; Reised Dec., 0; Accepted July 0, 04 The Solutio of the More Geerl Trelig Slesm Problem C. Feg, J. Lig,.Deprtmet of Bsic Scieces d Applied Techique, Gugdog Uiersity of Sciece d Techology, 0 Gugdog P. R. Chi (040@qq.com).School of Applied Mthemtics, Gugdog Uiersity of Techology, 000 Gugzhou P. R. Chi (LigJP@.com) Abstrct I this pper, more geerl trelig slesm problem of which the trditiol trelig slesm problem is the specil cse hs bee discussed. We obti simple method to sole which ode is the best strtig ode of circuit to obti the miiml totl wor. Key Words: Trelig slesm problem, solutio, grph theory, miiml totl wor.. Itroductio The trditiol trelig slesm problem is the prologtio of the Hmiltoi problem. Let us te cities,,, d there is oe rod betwee ech pir of them. Let the legth of the rods be d( i, j ), such tht: d (, ) d (, ) + d (, ),,, i j i j The trelig slesm problem, or TSP for short, is to fid the shortest wy of isitig ll of the cities d returig to the strtig ode. Ech city is isited oly oce. Pper [] tell us if we gie -order side weighted udirected complete grph, where the odes re s cities d the sides re s the rods betwee cities, the trelig slesm problem is to fid Hmiltoi circuit i this grph to get the miiml sum of the weights of sides. It is esy to fid Hmiltoi Circuit of which the sum of the weights of sides is the miimum by the method of exhustio, but is ot possible. Becuse i -order side weighted udirected complete grph, there re ( )! Hmiltoi circuits. This is ery lrge umber d it is ofte cited s the reso the TSP seems to be so difficult to sole. It is true tht the rpidly growig

2 lue of ( )! =,there re rules out the possibility of checig ll tours oe by oe. For exmple, while 4! (. 0 ) Hmiltoi circuits eeded to be clculte. If it tes osecod ( 0 secod) per Hmiltoi circuit, it will tes us millios of yers to fid the shortest oe. Yet o effectie method is ow for the TSP. There is well-ow lgorithm clled Nerest Neighbor Algorithm. It s simple: While slesm is i city, The ext city he eed to go to is the oe, tht is the erest d he hs t bee to yet, d so o util ll the cities re isited. The method of the Nerest Neighbor Algorithm is: I -order side weighted udirected complete grph, we c freely choose y ode s the strtig ode x of the Hmiltoi Circuit, d the fid the erest ode to x s the secod ode x. The erest ode to x is s the third ode x d so o util ll the odes re chose. I order to express coeiet, we cll the circuit, obtied by Nerest Neighbor Algorithm, the Nerest Neighbor Circuit. Ad we c cll the Nerest Neighbor Circuit, with ode i s its strtig ode, the i - Nerest Neighbor Circuit. I the trditiol trelig slesm problem the grphs it discussed re oly sides weighted grphs. The grphs discussed here re complete weighted grphs, tht ll odes d sides re weighted. I this pper, we fid out tht, ot the sme totl wor (For exmple: cost) mybe obtied by differet strtig odes ee i sme circuit d the shortest Hmiltoi circuit my ot obti the miiml totl wor. I this pper, we obti simple method to sole which ode is the best oe s the strtig d ed ode of the circuit to obti the miiml totl wor. This is ery importt i simultio problem, for exmple i the urb plig dmiistrtio or i the trsporttio compy hedqurters buildig plig. Ad it is the mi result of this pper d the differece from the trditiol trelig slesm problem.

3 . Mi Results At first plce we gie the method: Method.. Cosider i -order complete weighted udirected grph, there re odes (,,, ). Let i mes the weight of ode i d mes the weight of side betwee i d j, where i, j,,,. All the i d re ow. Let mes the j th weight ode of the i - Nerest Neighbor Circuit. Let mes the j th side weight of the i - Nerest Neighbor Circuit. We c cll the wor of the j th ode i the i - Nerest Neighbor Circuit. Step By Nerest Neighbor Algorithm, we let ll the odes i,i,,,,s strtig ode seprtely, d fid out ll the d, j =,,,,i =,,,. Therefor we c get i (i,,, )-Nerest Neighbor Circuits d their totl wor: = j j, i i + i i + + i i =, () j= j= = j j. j= " $ & $ Step Compre (). If mi# j j, j j,, j j ' %$ j= j= j= ($ = j= i - Nerest Neighbor Circuit is the best Nerest Neighbor circuit whose totl wor is the, it shows tht the miimum. Ad the miiml totl wor is. j= Exmple. Cosider there is trsporttio compy hs goods distributig ceters:,,, 4,,. The dily mouts of trffic of goods distributig ceters re ow:

4 s qutity is, s qutity is, s qutity is, 4 s qutity is 4, s qutity is, s qutity is. Ad the erge uit trsporttio cost betwee ech pir of goods distributig ceters re ow. Now the trsporttio compy is goig to decide oe of the goods distributig ceters s the mi goods distributig ceter. Eery dy the fleet will strt from this mi goods distributig ceter, pss through ll the other goods distributig ceters oce d oly oce, d come bc to this mi goods distributig ceter. Suppose the goods of the former goods distributig ceter will be uloded i the lter. Which goods distributig ceter should be decided s the mi goods distributig ceter, such tht the totl trsporttio cost is the miimum? The solutio is: Accordig to the questio, costruct -order complete weighted udirected grph, s show i the fig.. Let the odes weight mes the goods dily trffic mout of goods distributig ceters d the sides weight mes the erge uit trsporttio cost betwee goods distributig ceters. Let mes the j th ode weight of the i - Nerest Neighbor Circuit, where j,i =,,,. Let mes the j th side weight of the i - Nerest Neighbor Circuit. () () 4 0 () 0 () 0 4 Fig.. -order Complete Weighted Udirected Grph Step. By Nerest Neighbor Algorithm, we let ll the odes s strtig ode seprtely, d 0

5 fid out ll the d, j =,,,, i =,,,. So we c get i - Nerest Neighbor Circuit, where i =,,,, d their totl wor (the totl trsporttio cost): ) Let the ode s the strtig ode. Becuse of mi{0,,,,} =, so the erest ode to is. The is s the secod ode d side is s the first side of the - Nerest Neighbor Circuit, s show i the Fig.. Therefore, we c get = d =. () Fig.. of the - Nerest Neighbor Circuit To the ode, Becuse of mi{,0,,} =, so the erest ode is, besides. The is s the third ode d side is s the secod side of the - Nerest Neighbor Circuit, s show i the Fig.. Therefore, we c get = d =. () () Fig.. Of the - Nerest Neighbor Circuit To the ode, Becuse of mi{4,,} =, so the erest ode is, besides d

6 . The is s the fourth ode d side is s the third side of the - Nerest Neighbor Circuit, s show i the Fig.4. Therefor, we c get = d =. () () () Fig.4. of the - Nerest Neighbor Circuit To the ode, Becuse of mi{0,} =, so the erest ode is, besides, d. The is s the fifth ode d side is s the fourth side of the - Nerest Neighbor Circuit, s show i the Fig.. Therefor, we c get 4 = d 4 =. () () () () Fig.. of the - Nerest Neighbor Circuit To the ode, the erest ode is 4, besides,, d. So 4 is s the sixth ode d side is s the fifth side of the 4 - Nerest Neighbor Circuit, s show i the

7 Fig.. Therefor, we c get = d =. () () () () 4 of the - Nerest Neighbor Circuit Fig.. 4 Li 4 to. So the side 4 is s the sixth side of the - Nerest Neighbor Circuit. Ad the - Nerest Neighbor Circuit is fiished lie tht: 4, s show i the Fig.. Therefore, we c get = 4, = d totl wor (the totl trsporttio cost) of the - Nerest Neighbor Circuit is: = j= j j Tht is = 4 ()

8 () () () Fig.. () 4 4 of the - Nerest Neighbor Circuit ) I the sme wy, let the ode s the strtig ode. We c get the - Nerest Neighbor Circuit lie tht: 4, s show i the Fig.. Therefor, we c get =, =, =, =, = 4, =, 4 =, 4 =, =, =, =, = d totl wor (the totl trsporttio cost) of the - Nerest Neighbor Circuit is: Tht is = j= j j = 4 () () () () 4 4

9 Fig.. 4 of the - Nerest Neighbor Circuit ) I the sme wy, let the ode s the strtig ode. We c get the - Nerest Neighbor Circuit lie tht: 4, s show i the Fig.. Therefor, we c get =, =, =, =, =, =, 4 =, 4 =, =, =, = 4, = d totl wor (the totl trsporttio cost) of the - Nerest Neighbor Circuit is: = j= j j Tht is = () () () () 4 of the - Nerest Neighbor Circuit Fig.. 4 4) I the sme wy, let the ode 4 s the strtig ode. We c get the 4 - Nerest Neighbor Circuit lie tht: 4 4 s show i the Fig.0. Therefor, we c get: 4 = 4, 4 =, 4 =, 4 =, 4 =, 4 =, 44 =, 44 =, 4 =, 4 =, 4 =, 4 =. Ad totl wor (the totl trsporttio cost) of the 4 - Nerest Neighbor Circuit is: = j= 4 j 4 j

10 Tht is = 4 () () () () () 4 Fig of the 4 - Nerest Neighbor Circuit ) I the sme wy, let the ode s the strtig ode. We c get the - Nerest Neighbor Circuit lie tht: 4, s show i the Fig.. Therefore, we c get 0 0 =, =, =, =, =, = 0, 4 =, 4 =, =, = 0, = 4, = d totl wor (the totl trsporttio cost) of the - Nerest Neighbor Circuit is: = j= j j Tht is = ()

11 () () 0 () () 4 of the -Nerest Neighbor Circuit Fig ) I the sme wy, let the ode s the strtig ode. We c get the -Nerest Neighbor Circuit lie tht: 4, s show i the Fig.. Therefor, we c get =, =, =, =, =, =, 4 =, 4 =, =, =, = 4, = d totl wor (the totl trsporttio cost) of the -Nerest Neighbor Circuit is: = j= j j Tht is = () () () () () 4

12 of the -Nerest Neighbor Circuit Fig.. 4 We c see tht, i fct, the -Nerest Neighbor Circuit, the -Nerest Neighbor Circuit d the 4 -Nerest Neighbor Circuit re the sme circuit lthough by differet strtig odes,, 4. The -Nerest Neighbor Circuit d the -Nerest Neighbor Circuit re the sme circuit lthough by differet strtig odes,, but their totl wor (totl trsporttio cost) jj =, jj j= j= = re differet. Step Compre ()-(). Becuse of mi {() ()} = jj =, so the -Nerest Neighbor Circuit is the best circuit whose totl wor (totl trsporttio cost) is the miimum d we should choose goods distributig ceter s the mi goods distributig ceter, such tht the totl wor (the totl trsporttio cost) is the miimum. I fct, i the trditiol trelig slesm problem, the shortest Hmiltoi Circuit obtied by Nerest Neighbor Algorithm is the circuit with or or s strtig ode, ot. The mi result of this exmple. is: ) We fid out the best Nerest Neighbor Circuit whose totl wor is the miimum; ) The strtig ode d the ed ode of the best Nerest Neighbor Circuit hs bee soled, tht is, which goods distributig ceter should be decided s the mi goods distributig ceter hs bee soled. This is ery importt i simultio problem. j=. Coclusios ) I method., whe the odes weight d sides weight re ow, we let ll the odes s strtig ode seprtely d fid out ll the d,j =,,,, i =,,, by Nerest Neighbor Algorithm. The we c get Nerest Neighbor Circuit d their totl wor. By compre totl wor, the best Nerest Neighbor Circuit whose totl wor is the miimum is soled. ) I exmple.,we fid out tht, ot the sme totl wor my be obti by differet strtig odes ee i sme circuit d the shortest Hmiltoi circuit my ot obti the miiml totl wor.

13 ) I pper [], the grphs it discussed re oly sides weighted grphs. The grphs discussed i this pper re complete weighted grphs, tht ll odes d sides re weighted; I pper [], the trditiol trelig slesm problem is to fid the shortest Hmiltoi Circuit such tht the sum of the weights of sides is the miimum. I this pper, more geerl trelig slesm problem hs bee discussed to fid the best Nerest Neighbor Circuit such tht the totl wor is the miimum. 4) The Nerest Neighbor Algorithm is the specil cse of method., whe = = = =. Refereces. Xueci Sho, Togyig She (00) Discrete Mthemtics [M], BeiJig, Tsighu Uiersity Press, pp. -.. Hou MegShu, Liu DiBo (0) A Noel Method for Solig the Multiple Trelig Slesme Problem with Multiple Depots, Chiese Sciece Bulleti, Vol. 0, pp. -.. Rogwei G, Qigshu Guo,Huiyou Chg, Yg Yi (00) Improed At Coloy Optimiztio Algorithm for The Trelig Slesm Problems,Jourl of Systems Egieerig d Electroics Vol., pp Heow Pueh Lee (00) Solig Trelig Slesm Problems Usig Geerlized Chromosome Geetic Algorithm, Progress i Nturl Sciece, pp. -.. J. Lig (00) The Limit Cycle of Clss Cubic System III, Jourl of Systems Sciece d Mthemticl Sciece, Vol., pp. -.. Lig, Ychu (00) Solig Trelig Slesm Problems by Geetic Algorithms, Progress i Nturl Sciece, Vol. 4, pp J. Lig (0) The Uiqueess of Limit Cycle i A Clss for Qutic Polyomil System, AMSE Jourls, Adces A-Mthemtics, Vol. 4, pp. -.. Zhou, Tie Ju (0) A Multi - Aget Approch for Solig Trelig Slesm Problem, Jourl of Wuh Uiersity,Vol., pp. -.. Jeogho Bg, Jughee Ryu (0) A Qutum Heuristic Algorithm for the trelig Slesm Problem, Jourl of the Kore Physicl Society, Vol., pp Gild Brch (0) Hugo Fort Iformtio i the trelig Slesm Problem, Applied Mthemtics, Vol. 4, pp J.Lig (00) The Limit cycle i clss of Qutic Polyomil System II, AMSE Jourls, Adces A-Mthemtics, Vol. 0, pp. -.. Bes A, Alslibi, Mrizeh Bebei Jelodr, Ibrhim Vet (0) A Comprtie Study betwee the Nerest Neighbor d Geetic Algorithms: A reisit to the Trelig Slesm Problem,

14 Itertiol Scietific Acdemy of Egieerig d Techology, Vol.., Beig.. Murizio Mrchese (00) A At Coloy Optimiztio Method for Geerlized TSP Problem, Vol., Progress i Nturl Sciece, pp Xiojig Wg, Jiyig Li, Liwei Xiu (0) Electro-Hydrulic Sero System Idetifictio of Cotiuous Rotry Motor Bsed o the Itegrtio Algorithm of Geetic Algorithm d At Coloy Optimiztio, Jourl of Doghu Uiersity (Eglish Editio), Vol., pp Ho Wu, Guolig Li, Lizhu Zhou (0) Giix: Geerlized Ierted Idex for Keyword Serch, Tsighu Sciece d Techology, pp