Transit Assignment Model Incorporating Dwell Time

Transcription

1 Sun et l Trnst Assgnment Moel Incorportng Dwell Tme Lele Sun PhD. Cnte School of Economcs n Mngement Chng n Unversty Nn Er Hun Zhong Dun, X n, Shnnx 00, Chn Phone: (+) 00 Fx: (+) Eml: jerry_sl@hotml.com Qng Meng Assocte Professor Deprtment of Cvl n Envronmentl Engneerng, Ntonl Unversty of Sngpore, Sngpore Phone: (+) Fx: (+) Eml: ceemq@nus.eu.sg Zhyun Lu Lecturer Insttute of Trnsport Stues Deprtment of Cvl Engneerng Monsh Unversty, Clyton, Vctor 00, Austrl Phone: (+) 0 Fx: (+) 0 Eml: zhyun.lu@monsh.eu To the commttee of Bus Trnst Systems (AP00) for revew Pper submtte to the st Annul Meetng of Trnsportton Reserch Bor to be consere for presentton n publcton n Trnsportton Reserch Recor Submsson Dte: July, 0 Totl Number of Wors Number of wors n text: = 0 wors Number of tbles: ( 0) = 0 wors equvlent Number of fgures: ( 0) = 0 wors equvlent Totl number of wors: = 0 wors equvlent Corresponng uthor: Qng Meng TRB 0 Annul Meetng Pper revse from orgnl submttl.

2 Sun et l. 0 ABSTRACT The trnst ssgnment problem wth conserton of bus well tme s nlyze n ths pper. The lnks n trnst network cn be clssfe nto four types: on-bor lnks, lghtng lnks, borng lnks n ummy lnks reflectng the bus wellng behvor t the bus sttons. Accorngly, the trvel tme functons on these lnks cn be nlyze. Ths pper frst proposes new equton for the bus well tme t ech bus stton, whch s the mxmum vlue between the pssengers borng tme n lghtng tme. Then, ynmc progrmmng bse pproch s use to escrbe the equlbrum pssenger flow n the trnst network. A gp functon s subsequently propose to formulte the equlbrum pssenger flow, whch s convenent to be solve by the Revse Optml Strtegy Algorthm n the Metho of Successve Averge. Fnlly, two numercl exmples re opte to vlte the propose methoology, where the bus well tme functon s clbrte usng rel survey t. KEY WORDS Trnst Assgnment; User Equlbrum; Dwell Tme; Gp Functon; Metho of Successve Averge TRB 0 Annul Meetng Pper revse from orgnl submttl.

3 Sun et l INTRODUCTION Trnst system plys sgnfcnt role n crryng the trvel emns of urbn ctzens n mny ense ctes; for nstnce, n Sngpore, the moe shre of publc trnsport systems s more thn 0% (). Wth the fst expnson of populton growth, the pssenger emn of trnst system s lso lrgely ncrese, whch brngs much pressure on the current nfrstructure systems of publc trnsport. Thus, n orer to mprove the trnst system to cope wth the new/precte emn, the stues on trnst ssgnment s crre out to evlute ny new pln of the trnst system; see the monogrphs by Ceer () n Lm (). Trnst ssgnment s, per se, result of the pssengers route choce ecsons, thus the nlyss of pssengers trvel behvors re of conserble mportnce. The pssengers m to reuce the totl trvel tme by properly etermne ther route plns. Here, the totl trvel tme for pssenger conssts of two components: on-bor tme n tme spent t the bus sttons. The tme spent t bus sttons cn be further clssfe nto wtng tme for the bus, n well tme of the bus (borng n lghtng tme of pssengers). However, n the prevous stues, the well tme of the bus s rrely tken nto ccount, n few stues only took well tme s lner functon of the totl number of pssengers. Nevertheless, n prctce, for the buses wth two oors (one for borng n one for lghtng), the well tme s ffecte by both the borng n lghtng tme. In fct, the well tme shoul be the mxmum vlue between borng n lghtng tme. The trnst ssgnment wth ths new type of well tme s stll n open queston, whch s resse n ths stuy. Lterture Revew Erler stues by Dl (), Fernse n Drper () n Le Clerq () hve recognze tht the tme of pssengers spent t bus stops ply n mportnt role n the trnst route choces of the pssengers n ssume tht the pssengers chose the shortest pths from ther orgns to estntons. It s worth mentonng the stuy by Chqu n Robllr (), whch ntrouce the concept of common bus lnes - some routes shre common sectons n ech pssenger must select the buses he/she wll probbly use - whch s mlestone work for the trnst network ssgnment stuy. Chqu n Robllr lso propose network moel tht, on smple network of one orgn-estnton (OD) pr, pssengers cn select subset of the lnes connectng ths OD pr to mnmze ther totl trvel tme (wtng tme plus on-bor tme). De Ce n Fernnez () hve extene the stuy by Chqu n Robllr () v proposng mthemtcl formulton of the trnst ssgnment problem, where pssengers trvel by followng sequence of ntermete trnsfer noes. Ths moel ws extene by De Ce n Fernnez () to new moel whch conucte successfully the congeston. They ssume tht the congeston t the bus stop n the bor vehcles ffect the tme spent t bus stton for the pssengers. An they moele the congeston by efnng sgnfcnt concept effectve frequency whch ecrese s the pssenger flow ncresng, n h profoun nfluence for further reserch. Fnlly they efne the user equlbrum (UE) n got the equvlent vrtonl nequlty formulton whch ws solve by gonlzton metho. Ther work ws extene by Lm (0) usng logt-bse stochstc user equlbrum (SUE) n conserng cpcty constrnt t ech trnsfer noe. Spess n Florn () extene the common lne problem n nother wy. They ntrouce the concept of strtegy, whch efne s set of coherent ecson rules tht, when pple, llowe the pssenger to trvel from hs/her orgn to estnton. The moel n lgorthm they propose were to mnmze the totl expecte trvel tme, nclung tme spent t TRB 0 Annul Meetng Pper revse from orgnl submttl.

4 Sun et l bus stton n on-bor tme. The concept strtegy s expresse n grth-therotetc lnguge n enote t s hyperpth, nmely, n cyclc recte grph by Nguyen n Pllotno (). Both of ther moels consere the congeston bor the vehcle by usng scomfort functons wth ech segment of trnst lne. However, they not tke the effect of congeston on tme spent t bus stton nto ccount, when ssumng tht ll the pssengers cn bor the frst rrvng vehcle t bus stop. More recently, Comnett n Corre () evelope frequencybse trnst equlbrum moel whch cn el wth generl lnk trvel tmes s well s more relstc wtng tme functon wth symptotes t bus cpcty. However t not propose n lgorthm to compute the network equlbrum. Cepe et l. () extene the work of Comnett n Corre () to obtn new chrcterzton of the equlbrum through whch the uthors formulte n equvlent optmzton problem. Ths moel formulton cn el wth the problem whch the lnk flow ws epenent trvel tme. Menwhle, the lgorthm of the moel cn pply successfully on lrge network. However, ll the bove stues not tke nto ccount the tme tht pssengers bore n lghte from the vehcles, whch my hve remrkble nfluence uner the stuton of huge pssenger flow n hevy congeston. Lrrn n Munoz () consere the reltonshp between lnk flows n trvel tmes nclung the tme t trnst stop becuse of pssengers borng the vehcles. They nlyze the exstence of equlbrum n whch the pssengers who h the sme estnton t bus stop n opt fferent trvel strteges, n propose metho to etermnng the equlbrum. However, n ther moel, they thought the bus well tme only etermne by the tme pssengers borng the vehcles. They n t tke nto ccount the tme pssengers lghtng from the vehcles. The bus well tme my be etermne by the tme pssenger lghtng from the vehcles s the tme pssengers lghtng tme s bgger enough thn the tme pssengers borng. Szeto et l. () ssume tht the well tme on the bus stops s constnt, whch my be resonble when the number of borng n lghtng pssengers s smll. Actully, the well tmes on the bus stops often fluctute wth the number of borng n lghtng pssengers. We then brefly revew the stues for the well tme. Bse on the number of borng n lghtng pssengers, Levson et l. () reporte the representtve well tme relton for fferent ctes, ln use, tme-to-y. Guenthner n Snh () propose two moels whch epene on the totl number of pssengers borng n lghtng exceeng or not exceeng. Guenthner n Hmt () nlyze the reltonshp between the well tme n fre structure. Ln n Wlson (0) propose formulton of well tme for lght rl trnst, whch bse on the number of borng pssengers, lghtng pssengers n stnees. Rjbhnr et l. () stue the bus well tme n the mpct of borng n lghtng pssengers on well tme usng the t erve from the tbse of Automtc Pssenger Counter. Trchn () propose multple regresson moels to nlyze the nfluence of fferent pyment methos, the exstence of steps t oors, the ge of pssengers n the possble frcton between users borng, lghtng n strnng, on the bus well tme. Objectves n Contrbutons Accorng to the lterture revew, we foun tht there were few stues on the trnst ssgnment wth well tme, or they merely consere the well tme n n unresonble wy. To fully nlyss the mpct of well tme on trnst ssgnment, ths pper buls moel for trnst ssgnment wth well tme, whch extens the chrcterzton of trnst equlbrum propose by Cepe et l. () to the trnst network wth well tme, uses the chrcterzton to obtn n TRB 0 Annul Meetng Pper revse from orgnl submttl.

5 Sun et l equvlent optmzton moel. To solve ths moel, we frstly propose Revse Optml Strtegy Algorthm bse on the lgorthm ntrouce by Spess n Florn () n exten t to compute the optml strtegy for the trnst network wth well tme. An then the metho of successve verge (MSA) s use to solve the optmzton moel, n to mprove the convergence spee of MSA, the metho of successve weghte verge (MSWA) propose by Lu et l. () s employe n ths pper. The contrbutons of ths pper re threefol: ) bul trnst ssgnment wth well tme whch s equl to the lrger one between borng tme n lghtng tme; b) propose the Revse Optml Strtegy Algorthm for computng the optml strtegy for trnst network wth well tme n MSA for solvng the optmzton moel; c) nlyze the nfluence of well tme on trnst ssgnment n pssengers routng strteges. The rest of ths pper s orgnze s follows. In secton, we el wth trnst network trnsformton to represent the bus well phenomenon t bus stops on trnst system properly n ntrouce the generl ssumptons relevnt to our stuy. Secton presents the lnk trvel tme formulton to escrbe the bus well tme t the bus stop cuse by the pssengers borng n lghtng from bus, n the chrcterstcs of the trnst equlbrum. In secton, we scuss bout the soluton lgorthm for our moel. The survey n numercl stues re shown n the secton, n secton conclues ths pper. PROBLEM DESCRIPTION In ths pper, the strtegy-bse trnst ssgnment frmework s opte. The strtegy of one pssenger s efne s set of coherent ecson rules tht, when pple, llows the pssenger to trvel from hs/her orgn to estnton; see Spess n Florn, () for etle explnton of the strtegy. Fgure shows smple exmple of trnst network, use n Spess n Florn (). There re four bus lnes n four bus stops n ths smple network. For pssenger wtng t the stop A towr estnton B, possble strtegy my be: opton, tke the bus lne, n ext t B; n opton, tke lne to noe Y; trnsfer to lne n pproch noe B. Thus, when wtng t stop A, the pssenger woul tke bus lne or lne whchever rrves erler. Actully, there re fferent strteges for fferent pssengers trvelng from noe A to noe B on ths smple network. LINE LINE LINE LINE A X Y B FIGURE An exmple of trnst network. Let G( N, A) enote the trnst network, where N n A represent the set of noes n lnks, respectvely. It shoul be ponte out tht ths trnst network G( N, A ) s fferent from the physcl network: lthough one ro lnk my be use by fferent bus lnes, they re TRB 0 Annul Meetng Pper revse from orgnl submttl.

6 Sun et l. regre s fferent lnks n the trnst network G( N, A ), s shown n Fgure. The set of noes N contns two types of noes: stop-noes Ns N n lne-noes Nc N. Bus trnsfer s only llowe t the stop-noes rther thn lne-noes. The tnerry of ech bus lne l s represente by the sequence of noes serve by ths lne, whch re nclue n set N N. The frequency of bus lne l s enote by f l. As forementone, ths pper consers the pssengers borng n lghtng behvors t the bus stops, thus for the ske of presentton, network representton s crre out on the trnst network, by ng three new types of lnks: borng lnks, lghtng lnks n ummy lnks. Fgure shows smple exmple of network representton of the trnst network. Hence, 0 the lnks on the trnst network nclue: on-bor lnks Ao A, ummy lnks Ap A, lghtng lnks Am A n borng lnks An A. As shown n Fgure, the otte ummy lnk between two lne-noes s opte to reflect the well tme of the bus t the corresponng bus stop. Ech lnk A s chrcterze wth trvel tme t () v n frequency f. Here, f s use to reflect the wtng tme on lnk, where the wtng tme s equl to / f. The expressons for trvel tme t () v n frequency f on fferent types of lnks re fferent, whch re elborte s follows. Frst, on the on-bor lnks oao A, t s ssume tht the trvel tme t () v re constnt, whch s nepenent wth ts lnk flow (), n ts frequency, o o f, snce the wtng tme s zero. l Lne-noe Lne-noe Lne-noe Lne-noe On-bor On-bor Alghtng Borng Alghtng Stop-noe Stop-noe 0 FIGURE Representton of trnst network. Secon, regrng the ummy lnks pa A: n fct, the well tme s nfluence both by the borng n lghtng pssengers, whch shoul be the mxmum vlue between the tme pssengers lghtng from the vehcles n the tme pssengers borng the vehcles. Hence, the trvel tme of the ummy lnks t () v shoul be: p p m n m n p t ( v, v )=mx v +, v + () where v m n v n re the number of lghtng pssengers n borng pssengers, respectvely. Stll, the wtng tme on ummy lnk s zero, so let f p. As forementone, the well tme my evently nfluence the route choce of pssengers, n vew tht the pssengers my obvte trnsferrng t the crowe bus stops wth long well tme. TRB 0 Annul Meetng Pper revse from orgnl submttl.

7 Sun et l. Thr, on the Alghtng lnks mam A : the trvel tme of the lght lnks tm() v s ssume to be lner functon of the flow on the lght lnks, t ( v )= v + () m m m n f m for these lnks. Then, on the borng lnks nan A, t s obvous tht the trvel tme shoul be equl to the well tme,.e., t ( v, v )=mx{ v +, v + } () 0 n m n m n regrng the frequency, f n shoul be the frequency of the trnst lne ssoctng wth the borng lnk. In ths stuy, we nten to use ynmc progrmmng metho to express the trnst equlbrum, thus estnton-bse term s efne here for the pssenger flows. The set of estntons s enote by D N, n let g 0 enote the pssenger flow from the noe to estnton D, let g g. Then, we use v to enote the flow on lnk A, whch s ssocte wth estnton D. The V 0 enotes the set of fesble flow such tht v 0 for ll A (.e. no flow wth estnton levng noe ) n stsfyng the flow conservton constrnts. v v g Here, A { : } n + () - + A A A { : j } re the set of lnks levng n enterng noe N wth n j enotng respectvely the tl n he noes of lnk A. Let s A 0 enote strtegy from noe to the estnton. T s n re the expecte trvel tme from noe to the estnton usng strtegy s n the mnmum expecte trvel tme from to the respectvely. Let ys enote the pssenger flow from noe to the estnton usng strtegy s, n () x v g v A the totl flow from to. We hve the followng flow conservton equtons: ys x () v () 0 s where stns for the set of non-empty subsets s A. When pssenger s trvelng t bus stop, he/she wll conser set of ttrctve lnes n bor the frst bus lne from the ttrctve lnes. In generl, fferent pssengers cn select mny fferent strteges between ther orgn n estnton. Ech pssenger s ssume to select strtegy tht mnmzes hs/her totl trvel tme, whch nclues the on-bor (n-vehcle) trvel tme, wtng n lghtng/borng tme t bus sttons. We lso mke the ssumpton tht the pssengers re fully wre of the frequency of ech bus lne, n hve perfect nformton on the trvel tmes. Bse on long-term y-by-y justment, the pssengers route choce on the entre network woul come to the equlbrum,.e., no pssenger cn further reuce hs/her totl trvel tme by unlterlly chngng the routng strtegy. The objectve of the trnst ssgnment resse n ths pper s to solve such kn of equlbrum pssenger flows, when conserng the well tme shown n eqn. (). TRB 0 Annul Meetng Pper revse from orgnl submttl.

8 Sun et l. 0 MATHEMATICAL MODELS Network Equlbrum A ynmc progrmmng bse pproch s use to escrbe the network equlbrum n ths secton. As shown n Fgure, conser pssenger heng towrs estnton n rechng n ntermete noe n hs trp. He shoul ece trvel to whch noe for the next move to + mnmze the excepte trvel tme from to, usng the lnks A from the noe. The clcultons for ynmc progrmmng re conucte recursvely, where the optmzton of one subproblem s tken s n nput to the next subproblem. Here we ssume () v s the mnmum expecte trvel tme from j to, n let t () v enote the trvel tme on lnk A. If t () v n () v re fxe, the ecson problem for the pssenger t the noe s common-lne j + problem (-) bse on trvel tme t( v) j ( v) n frequency of lnk, A. The soluton of ths common-lne problem () v cn etermne the mnmum expecte trvel tme from noe to the estnton, whch cn be use recursvely to solve the common lne problem of the upstrem noes. j t () v j j x () v 0 FIGURE The -to- common-lne problem. Gven the frequency vector f ( f) A n the trvel tme vector t ( t) A wth t [0, ), the tme-to-estnton ( ) N s the unque soluton of the generlze Bellmn equtons (): 0, () [ ( ) j ] t v f s mn for ll s f s where s the set of non-empty subset s A. In prtculr, f strtegy s contns one or more lnks wth nfnte frequency, the mnmum tme s the verge vlue of the lnk trvel tmes. The quntty t ( v) t ( v) ( v) represents the mnml tme to estnton when j usng lnk. The set of V () v enote the set of locl equlbrum flows corresponng to common-lne problem efne by the lnks n t A, wth totl flow x () v, constnt trvel tme () v, n frequency. A flow v s globl equlbrum f t s locl equlbrum wth respect to tself. Hence, the network equlbrum s efne s: fesble flow pttern v V0 s clle s equlbrum flows ff for ll D n the flow ( ) v belong to V () A v. The set of equlbrum flows s enote * V 0. TRB 0 Annul Meetng Pper revse from orgnl submttl.

9 Sun et l The bove efnton of network equlbrum s expresse by usng lnk flows. It cn be rewrtten s follows, usng the efnton of common-lne: the vector v ( v ) D s n equlbrum ff for ll D n there exst R n strtegy flow ys 0 for s stsfyng the flow conservton equtons n: f v ys A () f s bs b n the Wrrop s contons over the trnst network f ys 0 Ts f ys 0 k where T s s gven by [ ( ) ] t v f s Ts f j Gp Functon Due to the hgh complexty of the trnst ssgnment problem resse n ths pper, t s qute chllengng to rectly solve eqn. (). Inspre by the gp functon metho use by Cepe et l. (), we nten to propose smlr gp functon for the equlbrum pssenger flow, whch s ntrouce s follows. Cepe et l. () put forwr the followng property of trnst network equlbrum: * Proposton. v V o f n only f v V0 n there exst 0 such tht for ll D n f t ( v) ( v) v f t ( v) ( v) (0) f 0 f t ( v) ( v) where t t( v) j ( v). Then, we cn erve gp functon ssurng tht ts mnmum equls to network equlbrum bse on the followng proposton: Proposton. For ll v V0, D n the followng nequlty hols: n moreover v A * V 0 ff v V0 s v [ t ( v) ( v)] v mx ( v) v f j A () n ll these nequltes boun. Hence, the trnst equlbrum flow cn be obtne by solvng the followng moel v mn Gv t( v) v mx g ( v) A D A I f () The mnmum of G v s known to be 0, whch cn be use to montor the progress of mnmzton lgorthm nt to erve stoppng rule. () () TRB 0 Annul Meetng Pper revse from orgnl submttl.

10 Sun et l SOLUTION ALGORITHM The metho of successve verge (MSA) s use to solve the trnst ssgnment problem n ths pper. At ech terton, trnst network equlbrum bse on fxe trvel tmes s solve frstly, n then the MSA uptes the lnk flows by tkng the verge vlue of lnk flows from the prevous tertons. Here, the optml strtegy s clculte seprtely for ech estnton, n then t etermnes the equlbrum lnk flows. However, the exstng soluton lgorthms of the optml strtegy (-) for trnst network cnnot be opte rectly to solve the propose moel ue to the network representton. We then evelop new lgorthm bse on the Optml Strtegy Algorthm of to Spess n Florn () to hnle ths problem, whch s terme s Revse Optml Strtegy Algorthm. For the estnton D, the Revse Optml Strtegy Algorthm s summrze s follows: The Revse Optml Strtegy Algorthm: Step 0: (Intlzton) Let for N {}, n f 0 for N. A enotes the set of the lnk whch wll be use n the optml strtegy n S enotes the set of lnks whch re not selecte n Step. An let A : n S: A. Step : (Termnton crteron) If S then stop. Step : (Fnng the next lnk) Fn (, j) S whch stsfes t t, (, j) S j j then let S : S { }. Step : (Uptng) If s ummy lnk n t j, then mx{ j t, j tb} where b (, j) s the lght lnk connecte the sme noe wth the ummy lnk n set f, { } f fc A A c, where c enote the lnk whch hs the bgger vlue between j t n t j b. If s not ummy lnk n t, then j f f ( j t ), f f f, n A A {}. f f Go to step. Note tht the lgorthm s sequentlly pple for ech estnton. Then, the metho of successve verge s summrze s followng (). 0 Metho of Successve Averge: Step 0: (Intlzton) Compute the optml strtegy usng the bove lgorthm, n ssgn the emn usng the metho propose by Spess n Florn (). Get the ntl soluton 0 v V 0 n set 0. Step : (Termnton crteron) If Gv ( ), then stop. Step : (Drecton fnng) Compute the trvel tme t t ( v ), then compute optml strtegy for ech D to etermne the nuce flows v ˆ. Step : (Uptng) Upte the lnk flow Go to step. v v ( vˆ v ) n set. TRB 0 Annul Meetng Pper revse from orgnl submttl.

11 Sun et l. 0 The step sze n Step s usully tken s. The MSA cn fn the equlbrum soluton untl the gp functon stsfyng Gv ( ). However, t s well known tht the MSA suffers slow convergent spee ue to the preetermne suboptml step szes. Thus, the step k sze of MSA ws extene by Lu et l. () usng. Nme s metho of k k k... successve weghte verge (MSWA), ths metho s lso opte n the followng numercl secton to clculte the trnst ssgnment problem. NUMERICAL STUDIES In ths secton, two numercl exmples re opte to further vlte the propose moel n lgorthm. Snce the nnovton n focus of ths stuy s the well tme functon t bus stops, before we ntrouce the etls of the numercl exmple, the prmeters n the well tme functon s frst clbrte bse on prctcl survey n Sngpore. Ths survey ws conucte t Clement bus stton n Sngpore urng :0pm to :0pm n th Februry 0, wth smple sze of 0 bus vehcles. All these vehcles hve two oors, the front oor for pssengers borng n the rer oor for pssengers lghtng. The t collecte by ths survey nclue the pssenger borng tme, pssengers lghtng tme, the number of borng pssengers, n the number of lghtng pssengers. Due to the spce lmt, t s mpossble to show the full etls of these t, thus only the pssenger numbers n well tme of the frst 0 bus vehcles re prove n Fgure, for the reers nterests. 0 Pssenger number Tme (s) Borng Alghtng Bus vehcle Bus vehcle FIGURE The number n tme of pssengers borng n lghtng. TRB 0 Annul Meetng Pper revse from orgnl submttl.

12 Sun et l. 0 0 Bse on the rel survey t, the prmeters of the well tme functon cn be obtne s follows, usng the lner regresson nlyss: Borng tme: BT.. A () Alghtng tme: AT.0 0.B () where A n B enote the number of borng pssengers n lghtng pssengers, n the R vlues for the lner regresson re 0. n 0., respectvely. Accorng the eqn.(), the formulton of well tme shoul be DW mx{.. A,.0+0. B} () where DW enotes the well tme of the bus t the bus stop. A Smll Exmple Conser smll network wth three stops A, B, C, n three OD prs: 00 pssengers from A to B, 00 pssengers from B to C, n 00 pssengers from A to C. There re two trnst lnes connectng A n C: the express lne A-C rectly from A to C n locl lne A-B-C connectng A n C by two consecutve segments, s shown n Fgure. Bse on the network representton, the express bus lne A-C contns only one lnk (A, C) wth trvel tme of mnutes. An the locl lne A-B-C contns two consecutve lnks (A, B) n (B, C), whose trvel tmes re both 0 mnutes. In Fgure, the ummy lnk (B, B) enote the bus well tme whch s cuse by the pssengers borng n lght from the bus t the bus stop B. The frequency of the express lne A-C s buses per hour, n the locl lne s 0. A express C A locl- AB B ` B locl- BC C Alghtng Borng 0 A B C FIGURE A smll network. The MSA cn then be use to solve the trnst ssgnment problem n ths network. The propose soluton lgorthm s coe n C++, n teste on computer wth n Intel Core.GHz, n GB RAM. Snce ths pper focus on the trnst ssgnment wth well tme, therefore to fully test the effects of well tme on the trnst ssgnment results, two scenros re exmne: the trnst ssgnment wthout well tme n trnst ssgnment wth well tme. Tble shows the optml strtegy n the corresponng trnst ssgnment of the trnst network. We cn see tht the bus well tme oes not only ffect the trnst ssgnment, but lso nfluence the serch of the optml strtegy. It s cler tht the trnst ssgnment s nfluence by the well tme. The optml strtegy from A to C contns (A, A), (A, B), (B, B), (B, C), (C, C), (A, A), (A, C), (C, C) for the trnst network wthout well tme. However, the optml strtegy only contns lnks of (A, A), (A, C), (C, C) for the trnst network wth well tme. The totl trvel tme for the OD whch s from A to C s 0 mnutes for the trnst network wthout well tme, whle the totl trvel tme ncrese to. mnutes s the well tme s consere n the trnst ssgnment. TRB 0 Annul Meetng Pper revse from orgnl submttl.

13 Sun et l. TABLE The Optml Strtegy for the Network Trnst network wthout well tme Trnst network wth well tme Lnks Optml Strtegy Lnks Optml Strtegy Lnks OD OD OD OD OD OD Tme Flow Tme Flow A-B B-C A-C A-B B-C A-C (A, A) 0. yes no yes yes no no (A, A) 0. no no yes. 00 no no yes (A, B) 0. yes no yes 0 00 yes no no (B, B) 0. no no yes. 0 no no no (B, B) 0 00 yes no no yes no no (B, B) 0 00 no yes no. 00 no yes no (B, C) 0 0. no yes yes 0 00 no yes no (A, C). no no yes 00 no no yes (C, C) 0 0. no yes yes. 00 no yes no (C, C) 0. no no yes. 00 no no yes A Rel-sze Exmple 0 FIGURE Smplfe Sngpore bus network. We lso test the MSA metho n the evelope optml strtegy on the lrger exmple s shown n Fgure. The trnst network use for the numercl test presente below bse on the Sngpore bus network but nclues only mjor stops n mjor servce offere by Sngpore Bus servce (SBS) Trnst Lmte, whch ws use n the pper of Szeto et l. (). Ths trnst network conssts of noes, bus lnes, OD prs n lne segments. The represente network of ths trnst network use the exmple n ths pper hs 0 noes n lnks. The shng noes re the orgn or estnton noes, n fferent bus lnes re ncte by fferent types of lnes. Due to the spce lmt, the etls of OD emn re not prove here. TRB 0 Annul Meetng Pper revse from orgnl submttl.

14 Sun et l. 0 0 FIGURE Convergence Tren of the Conventonl MSA n MSWA. Fgure shows the logrthmc vlue of Gp functons n the frst 0 tertons of MSA n MSWA, where the gp Gv ( ) s expresse s percentge of the totl trvel tme k k g ( v ). It clerly shows tht the convergence spee of MSA s much slower thn tht of kk j j jk MSWA. Lkewse to the frst exmple, the computton results lso show tht the optml strtegy of the trnst ssgnment wth well tme s much fferent from the cse wthout well tme, the etls of whch re not nclue here ue to the spce lmt. CONCLUSIONS In ths pper, new formulton of the bus well tme, whch s the mxmum vlue between the pssengers borng tme n lghtng tme t the bus stops, were gven. We extene the chrcterzton bout the trnst equlbrum propose by Cepe et l. () to our network wth bus well tme. Bse the chrcterzton, the problem of fnng the equlbrum flow cn be retrete s mnmzton problem wth the optmzton vlue 0. A revse optml strtegy ws propose to fn the optml strtegy for the network wth bus well tme. MSA ws presente to fn the equlbrum flow for the trnst network. In the future, one cn exten the propose moel to tke nto ccount the cpcty constrnts n the congeston n the bus. Menwhle, the esgn of trnst network must tke nto the nfluence of well tme ccount. Therefore, t s necessry to nlyze the nfluence of well tme on trnst network esgn n stuy the esgn of trnst frequency bse on the well tme n the future. REFERENCES. Lm, R., M. L. C. Sn. Ln Trnsport Msterpln. Sngpore Ln Trnsport Authorty, 00.. Ceer, A. Publc Trnst Plnnng n Operton. Butterworth-Henemnn, Oxfor, 00.. Lm, W. H. K., M. G. H. Bell. Avnce Moelng For Trnst Opertons An Servce Plnnng. Emerl Group Publshng Lmte, Brfor, 00. TRB 0 Annul Meetng Pper revse from orgnl submttl.

15 Sun et l Dl, R. B. Trnst Pthfner Algorthms. Hghwy Reserch Recor, Vol. 0,, pp. -.. Fernse, K., D. P. Drper. Publc Trnsport Assgnment--A New Approch. Trffc Engneerng Control, Vol.,, pp. -.. Le Clercq, F. A Publc Trnsport Assgnment Moel. Trffc Engneerng Control, Vol.,, pp. -.. Chrqu, C., P. Robllr. Common Bus Lnes. Trnsportton Scence, Vol.,, pp. -.. De Ce, J., E. Fernánez. Trnst Assgnment for Congeste Publc Trnsport Systems: An Equlbrum Moel. Trnsportton Scence, Vol.,, pp De Ce, J., E. Fernánez. Trnst Assgnment to Mnml Routes: An Effcent New Algorthm. Trffc Engneerng n Control, Vol.,, pp Lm, W. H. K., Z. Y. Go, K. S. Chn, H. Yng. A Stochstc User Equlbrum Assgnment Moel for Congeste Trnst Networks. Trnsportton Reserch Prt B, Vol.,, pp. -.. Spess, H., M. Florn. Optml Strteges: A New Assgnment Moel for Trnst Networks. Trnsportton Reserch Prt B, Vol. B, No.,, pp Nguyen, S., S. Pllottno. Equlbrum Trffc Assgnment for Lrge Scle Trnst Networks. Europen Journl of Opertonl Reserch, Vol.,, pp. -.. Comnet, R., J. Corre. Common-Lnes n Pssenger Assgnment n Congeste Trnst Networks. Trnsportton Scence, Vol., No., 00, pp Cepe, M., R. Comnett, M. Florn. A Frequency-Bse Assgnment Moel for Congeste Trnst Networks wth Strct Cpcty Constrnts: Chrcterzton n Computton of Equlbr. Trnsportton Reserch Prt B, Vol. 0, 00, pp. -.. Lrrn, H., J. C. Muñoz. Publc Trnst Corror Assgnment Assumng Congeston Due to Pssenger Borng n Alght. Network n Sptl Economcs, Vol., 00, pp. -.. Szeto, W. Y., Y. Jng, K. I. Wong, M. Solyppn. Relblty-Bse Stochstc Trnst Assgnment wth Cpcty Constrnts: Formulton n Soluton Metho. Trnsportton Reserch Prt C, 0.. Levson, H. S. Anlyzng Trnst Trvel Tme Performnce. In Trnsportton Reserch Recor: Journl of the Trnsportton Reserch Bor, No., TRB, Ntonl Reserch Councl, Wshngton, D.C.,, pp. -.. Guenthner, R. P., K. C. Snh. Moelng Bus Delys ue to Pssenger Borngs n Alghtngs. In Trnsportton Reserch Recor: Journl of the Trnsportton Reserch Bor, No., TRB, Ntonl Reserch Councl, Wshngton, D.C.,, pp. -.. Guenthner, R. P., K. Hmt. Trnst Dwell Tme uner Complex Fre Structure. Journl of Trnsportton Engneerng, Vol., No.,, pp Ln, T. M., N. H. M. Wlson. Dwell Tme Reltonshps for Lght Rl Systems. In Trnsportton Reserch Recor: Journl of the Trnsportton Reserch Bor, No., TRB, Ntonl Reserch Councl, Wshngton, D.C.,, pp. -.. Rjbhnr, R., S. I. Chen, J. R. Dnel. Estmton of Bus Dwell Tmes wth Automtc Pssenger Counter Informton. In Trnsportton Reserch Recor: Journl of the Trnsportton Reserch Bor, No. 0, TRB, Ntonl Reserch Councl, Wshngton, D.C., 00, pp Trchn, A. Bus Dwell Tme: the Effect of Dfferent Fre Collecton Systems, Bus Floor Level n Age of Pssengers. Trnsportmetrc, 0, pp. -. TRB 0 Annul Meetng Pper revse from orgnl submttl.

16 Sun et l.. Lu, H. X., X. He, B. He. Metho of Successve Weghte Averges (MSWA) n Self- Regulte Avergng Schemes for Solvng Stochstc User Equlbrum Problem. Network n Sptl Economcs, Vol., 00, pp Wu, J. H., M. Florn, P. Mrcotte. Trnst Equlbrum Assgnment: A Moel n Soluton Algorthms. Trnsportton Scence, Vol.,, pp Szeto, W. Y., M. Solppn, Y. Jng. Relblty-Bse Trnst Assgnment for Congeste Stochstc Trnst Network. Computer-Ae Cvl n Infrstructure Engneerng, Vol., 0, pp. -. TRB 0 Annul Meetng Pper revse from orgnl submttl.