APPLICATION OF QUEUING THEORY TO DYNAMIC VEHICLE ROUTING PROBLEM Wei-Ning Chen, Kainan University Taiwan

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1 GLOBAL JOURNAL OF BUSINESS RESEARCH VOLUME 3 NUMBER 009 APPLICATION OF QUEUING THEORY TO DYNAMIC VEHICLE ROUTING PROBLEM We-Nng Chen Knn Unersty Twn ABSTRACT In ths pper we eelope n nlyze ynmc moel of the ehcle routng problem. In the stte moel ehcle wth equte olume trels t constnt elocty n boune plne to proe serces to nepenent n unformly strbute emns. The ynmc emns rre ccorng to Posson process n on-ste serce tmes re generlly strbute nepenent of ther locton. A men repostonng polcy for the ynmc moel s propose to reuce system tme of the Frst Come Frst Sere polcy. The mproement of performnce s erfe by smulton results. JEL: C6 KEYWORDS: Queung Theory; Dynmc Vehcle Routng Problem INTRODUCTION T he routng n scheulng of ehcles represents n mportnt component of mny strbuton n trnsportton systems costs. The ehcle routng problem (VRP) noles the esgn of set of mnmum cost routes orgntng n termntng t centrl epot for ehcle tht serces set of customers wth known emns. Ech customer s serce exctly once n furthermore ll the customers must be ssgne to ehcles such tht the ehcle cpcty s not exceee. The generc VRP n mny of ts prctcl occurrences he been stue ntensely n the lterture (Bon et l. (983) Mgnnt (98) n Lporte n Nobert (987)). The st mjorty of the ehcle routng lterture s ecte to etermnstc n sttc moels where ll problem prmeters re ssume to be known n nce n ecsons re me only once t the begnnng of the plnnng horzon. Howeer strbuton opertons n prctce often nole uncertntes wth respect to loctons n szes of customer orers ehcle trel tmes etc. Whle n the pst reserch mnly concentrte on sttc rnts of VRP where ncresng nterest n ynmc VRPs cn be obsere n recent yers. Dynmc ehcle routng wth preously unknown customer requests rrng oer tme hs become ncresngly mportnt n the trnsportton nustry s new technologes such s globl postonng systems n wreless communctons llow the ssgnment of new requests to ehcles n rel tme. In ths reserch the objecte of the system s to mnmze the expecte customer wtng tme n eelop ynmc VRP rel tme optml polcy. By computng the expecte system tme n through smulte exmples we nten to show tht ths polcy outperforme the conentonl frst come frst sere (FCFS) polcy. The remner of ths pper s orgnze s follows. Frst secton proes bref lterture reew n generl moel ecte to ynmc ehcle routng problems. In secton 3 the rel tme serce problem s ntrouce wth exstng n propose moels. Secton 4 contns mthemtcl smulton of the moel. Conclusons n suggestons for future work re outlne n secton 5. LITERATURE REVIEW AND BACKGROUND Vehcle routng problems re wely present n toy's nustres rngng from strbuton problems to fleet mngement. They ccount for sgnfcnt porton of the opertonl costs of mny compnes. Opertons reserch technques he been use wth success n mny stutons for reucng such costs. 85

2 W. N. Chen Globl Journl of Busness Reserch Vol. 3 No. 009 Een f most rel nstnces of ehcle routng problems re sole wth heurstc methos the esre to prouce optml solutons hs gen rse to prolfc reserch re. Most results reltng to ynmc ehcle routng moels were publshe n the lst 5 yers. Sureys of exstng results cn be foun n Powell (988) Powell Jollet n Oon (995) Psrfts (995) Bertsms n Smch-Le (996) n Genreu n Potn (998). Compre wth sttc VRP moels chrcterstcs of ynmc VRP moels nclue: () nformton s prtlly known when plnnng ehcle routes; () new nformton wll rre urng the plnnng n executon process n orgnl nformton my lter; (3) t s not possble to obtn n scertne route through sngle ttempt. Below re the generl bckgrouns n moel efnton for the ynmc VRP. Geometrcl Probblty Gen two nepenently n unformly strbute rnom ponts z ( x ) z y z re A n length of. To fn the men n rnce of z z we let then * ( ) = ( ( ) + ( ) ) A xz (( x z x) ( yz y) ) yz E z z x x y y x y = + z z z z x y = Now replcng ( ) * z by z E z z x x y y = z + z + z z 6 z ( x ) z y z n squre of * z x y z be the fx pont we he E( ) = () 3 Smlrly we cn obtn E 0.5 () 0.06 Vr = E E. (3) Let z be unformly strbute rnom pont n re A z 0 be the men of A. By the boe metho we he E( z0 z ) = (4) 6 E z0 z 0.38 (5) = (6) V rz0 z E z0 z E z0 z 0.0 Pollczek-Khnchn (P-K) Formul for M/G/ Queung Moel For queung moel efne = Probblty of customer rre n one unt tme known s probblty ntensty s = Aerge customer serce tme 86

3 GLOBAL JOURNAL OF BUSINESS RESEARCH VOLUME 3 NUMBER 009 ρ = Serce ntensty ρ = s w = Aerge wtng tme T = Aerge system tme T = w+ s N = Aerge number of customer n the system nclung customers beng sere n those wtng for serce ccorng to Lttle s formul N = T For M/G/ moel wth sngle serer f the nput s the Posson process serce tme s s ny generl strbuton (where s n Vr( s) both exst) then we he + Vr() s N = ρ + (7) ( ρ) whch s the P-K formul for M/G/ queung moel. System Tme of Queung Moel for Dynmc VRP Dynmc VRP cn be escrbe s sngle serce ehcle trelng n squre re A of length t constnt elocty. For smplfcton of nlyss ehcle cpcty s ssume to be equte.e. the ehcle oes not nee to relo. Further ll emns re ynmc n emn loctons re nepenently n unformly strbute n the serce re A. Inter-rrl tmeu s negte exponentlly strbute wth u = n Vr ( u ) =. Let the on-ste serce tme s ' be generl strbute where s' n Vr( s ') both exst. Defne T s urton between rrl n completon of serce for emn n W s urton between strtng of serce n rrl of serce for emn. Thus we he W = T s. The stey stte system tme T= lm ET [ ] n the stey stte wtng * tmew = T s. The objecte of ths stuy s to fn serce polcy tht mnmze T enote s T. Bse on the boe efnton ths ynmc VRP cn now be trete s M/G/ queung moel. Notng tht for certn emn the serce tme s nclues the on-ste serce urton s ' n trelng tme of urton between receng emn request n trelng to emn poston where s the trelng stnce n s the trelng elocty. For stey stte system we he n n n lm n s = lm n s' + lm n n n n = = = tht s: s = s' +. (8) Assumng s ' n re not correlte rnom rbles s ' n re nepenently strbute then Vr( s) = Vr( s ') + Vr. (9) Combnng equton (7) (8) (9) n Lttle s formul we he 87

4 W. N. Chen Globl Journl of Busness Reserch Vol. 3 No. 009 Vr( ) s ' + + Vr( s ') + T = s' + +. (0) s ' + MODELS OF REAL TIME POLICY FOR DYNAMIC VRP Frst Come Frst Sere Polcy In the Frst Come Frst Sere (FCFS) polcy the ehcle serces emns n the orer of rrl. If the ehcle rops off emn n there s no wtng emn. Hence the ehcle wts t the lst emn s elery locton. Note tht n ths system successe serce tmes re not nepenent n so ny use of M/G/ results s n pproxmton. As emn loctons re nepenently n unformly strbute n serce re A then ccorng to Equton () n (3) we he = E( ) 0.5. () Vr ( ) = Vr ( ) 0.06 Substtute Equton () nto Equton (0) we wll he the system tme for FCFS polcy. 0.5 ' ( ') 0.06 s Vr s TFCFS = s' + + () 0.5 s ' + Men Reposton Polcy Dfferng from the sttc VRP t s usully more mportnt for ynmc VRP to mnmze the wtng tme for ynmc emn thn to mnmze the totl stnce trele or mke spn. Bertsms n Vn Ryzn (99) propose stochstc queue men polcy to mnmze the erge system tme. Bse on ths we propose mofcton polcy clle men reposton polcy. Thus when conserng there s no new emn wtng n lne the ehcle s wtng poston cn be repostone n orer to mnmze emn wtng tme. From Fgure we efne = n let =. When the ehcle completes the serce n z f there s n mmete emn n z the ehcle wll trel from z to z rectly. If there s no mmete emn n queue the ehcle wll then return to the men poston z 0. When trelng to poston z f new emn n z rre the ehcle wll rerect to z mmetely. Otherwse t wll return to z 0 wtng next orer. In ths cse we wll he ' Vr( ') s ' + + Vr( s ') + ' = s' + +. (3) ' s ' + 88

5 GLOBAL JOURNAL OF BUSINESS RESEARCH VOLUME 3 NUMBER 009 E z z = -T -T then As ( ) x = x x = Cx z z T z z z y = y y = Cy T z z z ( T) C = 0.38 Fgure : Men Reposton Polcy for Dynmc VRP y. (4) z z z 0 x z Ths fgure shows the grphcl representton of the notton use. In whch z0 represent the men poston z s the pror serce poston n z s the mmete followng emn poston. Follow the eucton of Equton () n () we he: E( ( ) ) = ( + C ) 6 C = ( x -Cx ) + ( y -Cy ) x y x y (5) z z z z z z z z ( ) Vr ( ) = E - THE SIMULATION EXAMPLE Tble stte the system prmeters for the exmple. 89

6 W. N. Chen Globl Journl of Busness Reserch Vol. 3 No. 009 Tble : Prmeters for the Exmple = = = 0.9 S follows [00.5] unform Dstrbuton SS = 0.5 VVVVVV (SS ) 0.0 Ths tble shows the prmeters use for the smulton exmple. We smulte the moels usng length()= ehcle elocty()= the probblty ntensty()=0.9 n n unformly strbute s of (0 0.5). From Equton () we cn etermne the system tme for FCFS polcy s TFCFS.76. From Equton (3) (4) n (5) we he.65. Wth the system prmeter n Tble we use Mntb softwre to generte smulte t n from the generte t we he: 50 T = T.74 T FCFS 50 FCFS = 50 men 50 = =.60 Mrgn of error: TFCFS -TFCFS 00%.5% T FCFS - 00% 3.% The smulte result of T FCFS n T men s consstent wth the theoretcl result n the system tme s -TFCFS reuce by 00% 8.05% for men reposton polcy. It hence shows tht men reposton polcy s more effecte n reucng the erge customer wtng tme thn the FCFS polcy. CONCLUSION Dynmc ehcle routng wth preously unknown customer requests rrng oer tme hs become ncresngly mportnt n the trnsportton nustry s new technologes such s globl postonng systems n wreless communctons llow the ssgnment of new requests to ehcles n rel tme. In ths pper we he stue ynmc VRP where new customer rres t unformly chosen rnom locton fter the ehcles he left the epot. We exmne the problem of choosng n pproprte serce polcy tht mnmzes the wtng tme n hence shortenng the totl mkespn. System tme for two rel tme strteges FCFS n men repostonng were clculte. Wth the smulte prmeters the obtne results clerly emonstrte the ntge of men reposton polcy: Compre wth the reference polcy of FCFS the men reposton polcy ws ble to reuce the men system tme by pproxmtely 8%. There remn seerl enues for future reserch. Frst our pper focuses on the cse of sngle tonl customer. A nturl next step woul be to explore further the problem rnts of nonunform strbuton of new customers n the rrl of more thn one new customer. Further our moel s bse on sngle serce ehcle. Some smplfctons he been tken through the process of formul eucton yet the result cn be ugmente nto more generl stuton such s multple ehcles wth lo lmttons. 90

7 GLOBAL JOURNAL OF BUSINESS RESEARCH VOLUME 3 NUMBER 009 REFERENCES Bertsms D. n G. Vn Ryzn (99) A stochstc n ynmc ehcle routng problem n the Euclen plne Opertons Reserch ol. 39 p Bertsms D. n D. Smch-Le (996) A new generton of ehcle routng reserch: Robust lgorthm ressng uncertnty Opertons Reserch ol. 44 p Bon L. B. Golen A. Ass n M. Bll (983) Routng n scheulng of ehcles n crews: The stte of the rt Computers n Opertons Reserch ol. 0 p. 6- Genreu M. n J. Potn (998) Dynmc ehcle routng n sptchng T. Crnc n G. Lporte (Es) Fleet Mngement n Logstcs Kluwer Boston MA U.S.A. p. 5-6 Lporte G. n Y. Nobert (987) Exct lgorthms for the ehcle routng problem Annls of Dscrete Mthemtcs ol. 3 p Mgnnt T. (98) Combntorl optmzton n ehcle fleet plnnng: Perspectes n prospects Networks ol. p Powell W. (988) A comprte reew of lternte lgorthms for the ynmc ehcle llocton problem B. Golen n A. Ass (Es) Vehcle Routng: Methos n Stues Elseer Scence Publshers Amsterm Netherlns p Powell W. P. Jollet n A. Oon (995) Stochstc n ynmc networks n routng M. Bll T. Mgnnt C. Monm n G. Nemhuser (Es) Network Routng Vol. 8 Hnbooks n Opertons Reserch n Mngement Scence North-Holln Amsterm Netherlns p Psrfts H. (995) Dynmc ehcle routng: Sttus n prospects Annls of Opertonl Reserch ol. 6 p BIOGRAPHY We-Nng Chen s n ssstnt professor n the Deprtment of Interntonl Busness t Knn Unersty Twn. He recee hs Ph.D. egree n Proucton n Opertons Mngement from The Unersty of Msssspp. Hs reserch nterests nclue scheulng moels forecstng methos stochstc tme seres fnncl nlyss. He cn be contcte e-ml t wnchen@ml.knu.eu.tw. 9