Models and Software for Urban and Regional Transportation Planning : The Contributions of the Center for Research on Transportation

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1 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on Mchel Florn Aprl 2008 CIRRELT

2 Models nd Softwre for Urbn Regonl Plnnng: The Contrbutons of the Center for Reserch on Mchel Florn 1,* 1 Interunversty Reserch Centre on Enterprse Networks, Logstcs nd (CIRRELT) nd Déprtement d nformtque et de recherche opértonnelle, Unversté de Montrél, C.P. 6128, succ. Centre-Vlle, Montrél, Cnd H3C 3J7 Abstrct. The m of ths rtcle s to gve sem-techncl nd somewht journlstc ccount of the contrbutons to the methods used for quntttve trnsportton plnnng by professors, reserchers nd grdute students who hve been ctve t the Centre for Reserch on (CRT) of the Unversty of Montrel snce ts ncepton. Keywords. plnnng, network optmzton models, trnst ssgnment, network equlbrum models. Acknowledgements. I would lke to thnk Dve Boyce nd Henz Spess for comments mde on prelmnry verson of ths pper. Also I would lke to thnk Glbert Lporte for skng me to wrte t. Results nd vews expressed n ths publcton re the sole responsblty of the uthors nd do not necessrly reflect those of CIRRELT. Les résultts et opnons contenus dns cette publcton ne reflètent ps nécessrement l poston du CIRRELT et n'enggent ps s responsblté. * Correspondng uthor: Dépôt légl Bblothèque ntonle du Québec, Bblothèque ntonle du Cnd, 2008 Copyrght Florn nd CIRRELT, 2008

3 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on 1. Introducton The m of ths rtcle s to gve sem-techncl nd somewht journlstc ccount of the contrbutons to the methods used for quntttve trnsportton plnnng by professors, reserchers nd grdute students who hve been ctve t the Centre for Reserch on (CRT) of the Unversty of Montrel snce ts ncepton. Ths pper s orgnzed s follows. The frst secton gves hstorcl bckground to the developments of models, lgorthms nd softwre t the CRT. The next sectons present the frmework for the models used for trnsportton plnnng nd then revew the CRT contrbutons to the stte-of-the-rt of network equlbrum models, trnst route choce models, combned mode nd clss equlbrum models nd lrge-scle trnsportton plnnng models. The presentton s techncl overvew of the mn models nd lgorthms developed durng ths perod. Next, the softwre development ctvtes n ths re re descrbed n chronologcl order. The model formultons re presented but the soluton lgorthms re only referred to. The gret vrety of lgorthms tht were developed for the models ddressed n ths rtcle mke t dffcult to descrbe them ll n detl. A smple of typcl pplctons s presented to llustrte the pplctons of these models. The fnl secton ddresses softwre developments tht mde t possble to trnsfer these reserch fndngs nto prctce. 2. Bckground In 1970, Trnsport Cnd ntted progrm tht med t estblshng trnsportton reserch centers t mjor unverstes cross the country. Even though the Centre for Reserch on (CRT) t the Unversty of Montrel ws estblshed n 1970, t dd not become ctve untl two yers lter. In 1972 the Ford Motor Compny of Cnd mde n unrestrcted grnt to the Unversty of Montrel, whch ws used to ntte bsc reserch progrm n trnsportton. Perre Robllrd, Mrc Gudry nd the uthor were the frst ctve professors to prtcpte n ths project. Most of the students who were ntegrted nto ths project were from the Deprtment of Computer Scence nd Opertons Reserch (Informtque et recherche opertonnelle). The uthor ws pponted s Drector of the CRT n the fll of The m of ths pper s to present the modelng nd softwre developments tht were crred out t the CRT, from ts ncepton, on urbn trnsportton plnnng models. In ddton, the model nnovtons tht resulted n the ntroducton of the EMME/2-Emme 3 softwre pckge re documented. The reserch drecton dopted n the Ford sponsored project ws to provde crtcl evluton of the methodology used n the trnsportton plnnng models of tht tme nd to explore new venues mde possble by the mthemtcl progrmmng nd computer scence competence of the tem. We strted by studyng the work of Dfermos (1968, 1971, 1972) nd Dl (1971). Ths ws the strtng pont of the reserch n ths CIRRELT

4 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on re t the CRT. Ths project produced severl doctorl theses: Mchel Trhn (1974), Renee Donne (1974), Sng Nguyen (1974), Clude Chrqu (1974) nd Robert Chpleu (1974), whch ddressed topcs relted to network equlbrum models nd lgorthms, trnst route choce models, stochstc route choce models nd network desgn models nd lgorthms. 3. Demnd nd Network Models for Plnnng In order to plce the modelng contrbutons mde by the CRT n the context of trnsportton plnnng nd trvel demnd forecstng, n overvew of the models used s presented next. 3.1 The Four-Step Plnnng Prdgm When consderng the choces tht trveller mkes n hs trp form n orgn to destnton, t s customry to explctly dentfy the followng: - destnton choce - mode choce - route choce on the rod or trnst network It hs become common to refer to four-step trvel demnd forecstng sequence of models: - The generton nd ttrcton of trps; these re econometrc models tht determne the number of trps deprtng from orgns, nd the number of trps rrvng t destntons. There my be more thn one model used for the generton nd ttrcton of trps, for nstnce when the trveller populton s subdvded by trp purpose (work, study, servce, etc.). - Destnton choce; one or more trp dstrbuton models. -Mode choce; one or more mode choce models. - Trffc ssgnment models re desgned to descrbe the trffc ptterns formed by users of trnsportton network such s n urbn street system. They lso dpt to serve s models for trvel on rl or rlne networks. It s ssumed tht the performnce chrcterstcs of the network re known nd tht the trvel demnd s defned by n orgn-destnton demnd mtrx, s descrbed n the precedng secton, or defned by demnd functons. The dgrm below shows schemtc dgrm of ths four-stge process. CIRRELT

5 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on Producton Attrcton Models Trp Dstrbuton Models Mode Choce Models Trffc Assgnment on Rod Network Trffc Assgnment on Trnst Network Fgure 1. The four-step trnsportton plnnng prdgm One of the mn contrbutons of the reserch done t the CRT ws the development of rgorous models nd of lgorthmc solutons for the trffc ssgnment models on the rod nd trnst networks. New ntegrted models tht would render ths process smultneous rther thn sequentl were lso sgnfcnt contrbutons. In the followng, the clsscl destnton choce model nd very generl formultons of the mode choce models used n prctce re presented. They re necessry component for the presentton of mult-modl network equlbrum models tht re ntroduced n lter sectons of ths rtcle. 3.2 Destnton choce models Destnton choce modellng uses trp dstrbuton or sptl ntercton models. These models ssume tht the totl trps from n orgn node nd the totl trps to destnton node re known. The trvel tmes (costs) re lso known, nd the result of the model s n orgn-destnton mtrx tht contns the trps from orgns to destntons n ts cells. The sptl ntercton models were developed pror to the 70s (see Wlson, 1970). Mny vrnts hve been mplemented n prctce. The model descrbed below s known s the entropy sptl ntercton model nd s perhps one of the most common n prctce. Consder trnsportton network tht permts the flow of one type of trffc (vehcles or pssengers) on ts lnks. The nodes n N represent orgns, destntons nd ntersectons on lnks. The lnks A N N represent the trnsportton nfrstructure. CIRRELT

6 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on If the number of trps tht strt from orgns p P N s O p nd the number of trps destned to destntons q Q N s D q, then the ssue of nterest s to determne (or g where ( p, q) = ) gven the tme or cost of trvel u. The clsscl model tht s used to determne the orgn-destnton mtrx g s known s the entropy model. Conservton of flow t orgns nd destntons mples tht g, = Op p P (1) nd q Q nd, evdently the demnd for trvel s nonnegtve g g, = Dq q Q (2) p P g 0, p P, q Q. (3) In the bsence of other nformton, t s postulted tht the orgn-destnton mtrx s the most lkely to occur, whch leds to the objectve functon mx g ln g (4) p q subject to (1)-(3). The objectve hs the nterpretton of entropy mxmzton; the formlsm orgntes from nformton theory (see Jynes (1957,1957b)). The model ws ntroduced to trnsportton nd regonl nlyss by Wlson (1967, 1970). When some pror nformton s known bout the mtrx, sy g 0, ( p, q), Kullbck (1959) nd Snckrs nd Webull (1977) hve suggested the use of the objectve g mx g ln 0 (5) p q g In order to chrcterze the dsperson of trps, constrnt s dded to the totl trvel tme, where C s n observed totl trvel tme gu = C (6) p q whch leds to the objectve functon mn g ( ln g +θu ). (7) p q θ hs the nterpretton of the dul vrble ssocted wth the constrnt (6). It s trvl to verfy tht, by pplyng the Krush-Kuhn-Tucker condtons, ny soluton of (7) subject to (1)-(3) hs the generl form gpg = exp( αp + βq 1) exp ( θu ), ( p, q) (8) = AB p qexp ( θ u), where α p nd β q re the dul vrbles ssocted wth the conservton of flow constrnts. Wth the conventon g ln g = 0 when g = 0, t s possble to obtn solutons to ths clss of problems by usng ny prml convex progrmmng lgorthm. CIRRELT

7 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on However, one of the propertes of ths problem s tht the prml vrbles my be expressed n terms of the dul vrbles, s wll be shown next. The Lgrngen dul problem of (7) subject to (1)-(2) s D( αβ,, g) = mx{ mn g ( lng + θu ) + αp Op g αβ, g p q p q (9) + βq Dq g q p By usng (8) to replce the prml vrbles g one obtns, fter some smplfctons D O D q u αβ, r s p q ( αβ, ) = mx αp p + βq q exp( αp + β 1 θ ) (10) Ths property led to the development of n effcent soluton procedure known s the blncng method, whch s dul scent method for one vrble t tme. The blncng method lgorthm my be referenced n stndrd texts. Ths method dtes bck to t lest 1937 when Kruthof used t for the predcton of telephone trffc dstrbuton. Demng nd Stephn (1940) ndependently redscovered ths method nd ppled t to cross-clssfcton problem n sttstcs s smplfcton of lest squres fttng. Evns nd Krby (1974) nd Andersson (1981) lso mde mportnt contrbutons to the nlyss of the mthemtcl structure of the model. The CRT mde severl contrbutons to the study of entropy trp dstrbuton models. Vrnts of the blncng method were studes by Robllrd nd Stewrt (1974). Erlnder, Nguyen nd Stewrt (1979) consdered the clbrton of the entropy model by usng survey dt. An lgorthm for ths model ws lso studed by Jefferson nd Scott (1979). Lmond nd Stewrt (1981) showed tht the blncng method nd ts vrnts my be vewed s specl cse of Bregmn s (1967) non-orthogonl projecton method to solve certn problems of convex progrmmng. An ccomplshment of the collborton between the CRT nd the Unversty of Lnkopng s the book by Erlnder nd Stewrt (1990). Ths text s n excellent synthess of the theory nd pplcton of trp dstrbuton models bsed on the entropy nd grvty prncples. 3.3 Mode choce models The formulton nd clbrton of mode choce models represents very lrge body of work. The text by Ben Akv nd Lermn (1980) s very good reference for the gret vrety of econometrc models tht re used n representng choces mong competng lterntves. One of the erly contrbutons to choce theory, by Domencch nd McFdden (1975), ws recognzed by wrdng Dn McFdden the 2000 Nobel Prze n Economcs. The smplest model cn be stted generclly s the probblty of usng prtculr mode CIRRELT

8 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on ' m mong set of modes m M s functon of the soco-economc chrcterstcs of the trveller, the trvel tmes nd the trvel costs: p (usng mode ' m ) = f (utlty of mode ' m / m utlty of mode m ) A common functonl form s tht of the logstc or logt functon. An exmple of smple mode choce functon mong two modes of trffc s 1 1 p(usng m ) =, 1+ exp( γx+ α Δ cost + β Δtme) where the γ x re the soco-economc vrbles tht chrcterze the trveller. Δ cost nd Δ tme re the dfferences n trvel cost nd trvel tmes between the two modes. The prmeters γ, α nd β re clbrted by usng survey dt. Mode choce functons ply centrl role n the formulton of mthemtcl models tht re used n trnsportton plnnng. 4. Trffc ssgnment models nd methods 4.1 The Network Equlbrum Model Trffc ssgnment models re desgned to descrbe the trffc ptterns formed by users of trnsportton network such s n urbn street system. They lso dpt to serve s models for trvel on rl or rlne networks. It s ssumed tht the performnce chrcterstcs of the network re known nd tht the trvel demnd s defned by n orgn-destnton demnd mtrx, s descrbed n the precedng secton, or t s defned by demnd functons. In order to hghlght the contrbutons mde by reserch stff t the CRT n the re of network equlbrum problems, t s necessry to ntroduce the bsc network equlbrum model s frst formulted by Beckmnn et l (1976). For nottonl smplcty, ssume trnsportton network model wth one type of vehculr flow on the drected lnks of the network. The nodes, N, represent orgns, destntons, nd ntersectons nd the rcs, A model the trnsportton lnks (streets, hghwys, ). Orgn to destnton (O-D) demnds gve rse to lnk flows v, A nd the cost of trvelng on lnk s gven by user cost (trvel tme) functon s () v, where v s the vector ( v ) of lnk flows over the entre network. Cost functons A model tme dely on lnk or more generl costs such s tolls or fuel consumpton, nd re ssumed to be nonnegtve. Let I be the set of O-D prs, K, I, be set of drected pths connectng O-D pr, nd K be the set of ll pths. The demnd between CIRRELT

9 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on O-D pr g, I uses drected pths nd the pth flows h k obey conservton of flow nd non negtvty hk = g, I (11) Lnk flows re gven by k K h 0 k K. (12) k (13) v = δ h A k k I k Kp where δ = 1 f lnk belongs to pth k nd s zero otherwse. k Defne Δ s the A K rc-pth ncdence mtrx ( δ ) so tht v= Δ h, where h s the vector ( ) hk k K then defned by nd u( u( h) ) of pth flows for ll O-D prs. The cost s ( ( ) ) k sk h k = for ech pth k s δ ( ) δ ( ),, (14) s = s v = s Δh k K I k k k A A = s by defnton the cost of the lest cost pth for O-D pr : u = mn s I. (15) k K k For ech I, the trvel demnd t p my be obtned from fxed O-D demnd mtrx, n whch cse we wrte g = g, or t s gven by specfc demnd functon g ( u ), where u s the vector of lest cost trvel vlues, ( ) I g g ( u) I. u for ll the O-D prs of the network: = (16) System optml trffc ssgnment models ssume tht trvel on the network follows pths such tht network utlzton s for the common good. If the demnds g re fxed, then the objectve s to stsfy normtve prncple tht sttes the verge trvel cost (or tme) s to be mnmzed. Snce totl demnd s constnt, t s equvlent to mnmze totl system cost, nd the fxed demnd system optmzton model s s ( v) v (17) mn A subject to (11), (12), nd (13) wth g = g. If, however, trvel demnd s elstc, tht s, dctted by demnd functons (6), the system-optmzton model ms to mxmze the net economc beneft to the network users. From stndrd economc prncples, the beneft to trvellers between ny O-D pr I g u. It s ssumed tht ths s mesured by the re under ther demnd curve ( ) CIRRELT

10 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on functon hs n nverse ( ) t p 0 w ( ) w g = u. Hence the economc beneft cn be expressed s y dy. Therefore, n ths cse, the system optmzton model becomes t p ( ) ( ) mx w 0 y dy s v v (18) A subject to (11), (12), (13) nd (16). These system problems hve counterprts n user-optmzton models tht m to more ccurtely descrbe the stuton where trvellers on the network dstrbute themselves so tht no sngle user cn unlterlly mprove trvel costs. The descrptve models of trffc flow, therefore, ssume the users re n Wrdrop equlbrum (Wrdrop, 1952), specl cse of Nsh equlbrum. The mthemtcl sttement s: Determne * h nd * u such tht the followng condtons re stsfed: ( ( ) ) * * ( ) s h u h = 0 k K, I (19) * * * k k sk h u 0 k K, I (20) * hk g = 0 I (21) k K * * h 0, u 0, (22) * where g = g for fxed demnd nd g = g( u ) when demnd s elstc. The equlbrum * * lnk flows, v re clculted from the pth flows h usng (13). Another equvlent wy to stte the equlbrum condtons (19),(20) s ( ) * * sk h = u f hk 0, k K, I * * sk ( h ) u f hk = 0, subject to (21), (22), whch s drect sttement of Wrdrop s user optml prncple. The frst two condtons ensure tht, for ll I, only mnmum cost pths re used, nd the thrd equtes the totl pth flows to the totl demnd, gven the mnmum pth costs. Ths generl verson of the problem s known s the network equlbrum model (NEM) whch hs pplctons n mny res, ncludng electrcl networks, wter ppe networks nd sptl prce equlbrum problems. Florn nd Hern (1995) provde numercl exmples of these pplctons. The bsc NEM reformultons used n trnsportton plnnng, however, re optmzton problems. The prmry ssumpton s tht the cost nd demnd functons re s v s v g u = g u, respectvely. In seprble, tht s, they hve the form ( ) = ( ) nd ( ) ( ) CIRRELT

11 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on other words, the cost on lnk depends only on the flow on tht lnk, nd the demnd for O-D pr p depends only on the mnmum trvel tme for tht O-D pr. It s further ssumed tht the cost functons re convex nd the demnd functons re strctly monotone. Under these condtons, the elstc demnd user-optmzton problem my be stted s the convex progrm: mn g ( ) ( ) v s 0 x dx w 0 y dy (23) A p P subject to (11), (12), (13) nd (16). In the fxed demnd cse, the user problem becomes mn v s ( x) dx (24) A 0 subject to (11), (12) nd (13). wth g = g. Tht the solutons of these problems re equvlent to (23) (26), n the elstc nd fxed demnd cses, respectvely, follows drectly from the Krush-Kuhn-Tucker condtons for the two problems. The connecton between the NEM condtons nd the system optml models s lso reveled by ther Krush-Kuhn-Tucker condtons. It s strght- * s h clculted from forwrd to verfy tht they hve the sme form, wth the terms k ( ) the mrgnl lnk costs, s ( * ) ( * ) * v s v v +. Thus, the nterestng nd mportnt connecton s tht the soluton to system optml model s n equlbrum wth respect to mrgnl costs, whle the soluton of user optml model s n equlbrum wth respect to verge costs. It s mportnt to menton tht the totl lnk flows of the NEM presented bove re unque under the ssumptons mde on the lnk cost functons. Any decomposton of the totl lnk flow by orgns or by pths s not necessrly unque. 4.2 Contrbutons to the study nd soluton of sngle clss network equlbrum models The frst contrbuton mde for the soluton of the NEM ws the doctorl thess of Nguyen (1974). He explored lgorthms for the network equlbrum model wth fxed nd vrble demnd n the spce of lnk flows, orgn flows nd orgn-destnton pth flows (see lso Nguyen, 1974, 1976). These lgorthms were the dptton of the clsscl methods of nonlner progrmmng such s the lner pproxmton method, the convex smplex method nd the reduced grdent method. At the tme, the computer memory (RAM) vlble ws qute lmted nd the most useful lgorthm ws the CIRRELT

12 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on dptton of the lner pproxmton method of Frnk nd Wolfe (1956) snce t requred reltvely lttle RAM nd ws qute smple to mplement. The heurstc methods used t the tme, such s the ncrementl ssgnment method or cpcty restrned method were to be replced eventully by rgorous methods; however the vldton of the method hd to be demonstrted. A clbrton nd vldton of the method ws crred out by usng the network of the Cty of Wnnpeg, Cnd (Florn nd Nguyen, 1976). Ths ws one of the frst studes tht showed tht rgorous soluton of the NEM could be clbrted nd vldted successfully. The CRT took the nttve to orgnze conferences on the topc of network equlbrum models nd ther lgorthmc soluton. The frst such meetng, whch took plce n 1974, brought together ll the reserchers who were nterested n ths emergng re of reserch. It s worth mentonng the prtcpton of Mrtn Beckmnn, Hrold Kuhn, Stell Dfermos, Mrvn Mnhem, Drck Vn Vlet, Suznne Evns nd Bob Dl who lso provded some support from UMTA. The proceedngs were publshed by Sprnger Verlg (see Florn, ed., 1976). Two other conferences on trffc network equlbr nd network modelng topcs took plce n 1977 nd 1981 tht helped to estblsh the CRT nterntonlly s ledng cdemc reserch centre n the feld. Severl ppers tht were presented t the 1977 conference were publshed n specl ssue of Reserch B (see Florn nd Gudry, 1980). In 1982, t the nvtton of the Ntonl Reserch Councl of Itly, the uthor orgnzed week long course n Amlf, Itly tht brought together some of the best reserchers n ths feld t tht tme nd ws ttended by mny young Itln reserchers who wll go on nd mke sgnfcnt contrbutons to ths feld of endevor. The lectures were publshed n book edted by the uthor (see Florn, ed., 1984). The nterest n soluton lgorthms for the NEM contnued wth developments tht ddressed vrnts of the lner pproxmton method such s the wy step (see Florn, 1977, Guelt nd Mrcotte, 1986), nd the PARTAN method (see Florn, Guelt nd Spess, 1987). Durng the sme perod, other results tht could be computed fter crryng out trffc ssgnment on the rod network, such s fuel consumpton, were studed (see Le-Vn Nguyen, 1982). A collborton wth Itln reserchers resulted n severl contrbutons. A dul shortest pth lgorthm (see Florn, Nguyen nd Pllottno, 1981) ws one of the frst such common works. A survey pper (Florn, 1986) summrzed some of the developments crred out t the C.R.T. nd elsewhere. However, one other mn re of reserch ws the development of mult-modl network equlbrum models tht could consder, n n ntegrted wy, the choces mde by trvelers s to mode nd routes on both the rod nd the trnst networks. Ths s descrbed n more detl below. The texts by Sheff (1985) nd the monogrph by Ptrckson (1994) provde more detl on network equlbrum models nd soluton lgorthms. CIRRELT

13 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on 4.3 Contrbutons to the study nd soluton of mult-clss network equlbrum models Trvellers re not homogeneous. They my be dstngushed by the vehcle they drve (cr, truck, etc..) or by ther soco-economc chrcterstcs. The extenson of the sngle clss NEM to multple clsses s reltvely strghtforwrd. Some more complex models rse when tolls re ntroduced nto the model snce the wllngness to py tolls depends on the soco-economc chrcterstcs of the populton. The modellng of the response to tolls hs been consdered n the context of dscrete multple clsses of trffc by Florn (2006). There s vlue of tme ssocted wth ech clss nd the resultng model s rther complex mult-clss network equlbrum model wth vrble demnd ssocted wth the choce between tolled lnks nd free lnks. The current trend of developng new hghwy fcltes by usng tolls s mens of fnncng the project hs led to very common use of such models. If one ssumes tht the vlue of tme s gven by contnuous dstrbuton n the populton then the model s dfferent nd ws solved by Mrcotte et l (1996). Mult-clss network equlbrum models nvolvng crs nd trucks hve been studed by Wu et l (2006), who consder the contrbuton of trucks to congeston to depend on the mx of trffc nd Noreg nd Florn (2007), who consder dfferent volume/dely functons for ech clss of trffc. 4.4 Contrbutons to the study nd soluton of network desgn models The network desgn problem hs ttrcted the ttenton of mny reserchers. Donne (1974) studed the optml desgn of network when the network s not subject to congeston. Mrcotte (1982) hs consdered versons of ths problem when congeston prevls n hs doctorl thess. He consdered both sngle level ( see Mrcotte, 1983) nd b-level (see Mrcotte,1986) versons of ths problem. 5. Trnst route choce models 5.1 Contrbutons to the study of un-congested trnst route choce models Trnst route choce models or trnst ssgnment models m to descrbe the trffc flows on network of trnst lnes tht operte t known frequences. The mn dfference wth the trffc ssgnment models for rod networks s the wtng phenomenon: trnst trvellers experence wtng tme for the frst vehcle (bus) of the lne whch they hve chosen. In ddton, the ccess to the trnst lne stop mples n ccess tme (whch s usully wlk tme), trnsfers between lnes f more then one lne s tken nd the nvehcle tme. The contrbutons of the CRT reserchers to the formulton nd soluton of ths problem re numerous nd sgnfcnt. CIRRELT

14 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on One of the frst contrbutons to the study of trnst route choce models ws the work of Chrqu nd Robllrd (1975) who consdered smple network of one orgn nd one destnton pr connected by severl non-overlppng trnst lnes, or common lnes. Ther semnl pper ntroduced the noton tht, on smple network of one orgn nd one destnton, pssengers cn select subset of ttrctve lnes nd bord the frst one of these tht rrves t stop n order to mnmze the expected sum of wtng plus trvel tmes. x O l l 1 2 D l n Fgure 2: The common-lne problem The des of Chrqu nd Robllrd were extended to generl trnst networks n two wys. Spess (1984) nd Spess nd Florn (1989) ntroduced the noton of strtegy, whch s the choce of n ttrctve set of lnes t ech decson pont; tht s, t ech node where bordng occurs. The resultng model nd lgorthm cheve the mnmzton of the expected vlue of the totl trvel tme, whch ncludes ccess, wt nd n-vehcle tme. Nguyen nd Pllottno (1988) provded grph theoretc nterpretton of strtegy s n cyclc drected grph, nd denoted t s hyperpth. These models consdered congeston bord the vehcles by ssoctng dscomfort functons wth ech segment of trnst lne, so tht the resultng equlbrum models could be solved by stndrd lgorthms for convex mnmzton. However, the wtng tmes re underestmted snce they do not consder the fct tht, n perod of hevy congeston, pssengers my not be ble to bord the frst vehcle to rrve t stop. The results of Chrqu nd Robllrd were used n dfferent wy by Chpleu (1974) nd DeCe nd Fernndez (1989) n trnst ssgnment model bsed on restrcted noton of strtegy, whch llows choces mong multple lnes t gven stop only f they ll shre the next stop to be served (for comprson wth the strtegy pproch, see DeCe et l. (1988)). A more forml study of congeston t bus stops bsed on results from queung theory ws ntted by Gendreu (1984) n hs doctorl thess. He ws the frst to formulte generl trnst ssgnment model wth congeston. For more recent results on the wtng processes t bus stops, see Bouzïene-Ayr et l. (2001) nd Comnett nd Corre (2001), s well s the recent thess by Ceped (2002). Consder trnst network tht conssts of set of nodes, set of trnst lnes, ech defned by n ordered lst of nodes t whch bordng nd lghtng re permtted, nd CIRRELT

15 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on set of wlk lnks, ech defned by two nodes. The tmes ssocted wth ech wlk lnk nd ech trnst lne segment re constnt. At ech node tht s on the tnerry of trnst lne, the dstrbuton of the nterrrvl tmes of the vehcles s known for ech lne tht serves the node. As consequence, one cn compute the combned expected tme for the rrvl of the frst vehcle, for ny subset of lnes ncdent t node, s well s the probblty tht ech lne rrves frst. In order to stte the mthemtcl model tht corresponds to the trnst route choce selecton, t s noted tht ech wlk lnk my be replced (conceptully) by trnst lne of one lnk wth zero wtng tme (nfnte frequency). Also, t s ssumed tht the underlyng network s strongly connected. The objectve s tken to be the mnmzton of expected wtng nd trvel tme, or the expected totl generlzed cost f wtng tmes nd trvel tmes hve dfferent weghts (e.g. wtng s more onerous tht n-vehcle tme). The network s composed of four types of rcs: wt rcs (no trvel tme), n-vehcle (no wtng), lghtng (no trvel tme, no wtng) nd wlk rcs (trvel tme, no wtng). Thus, the segment of trnst lne s n rc tht s served by vehcle t gven ntervls nd the trnst trveller wts for the lnk to be served by vehcle. b,b: n-vehcle rcs c: lght rc d: bordng rc c d Fgure 3. The lnk representton of trnst network The rcs tht wll be ncluded n soluton of the model re denoted by A A, where A s the set of rcs nd N s the set of nodes. Thus the soluton for sngle destnton s s b = N, A. The demnd for trvel from nodes, Nto destnton q s sub-grph s ( ) denoted g. Among the lnks ncluded n soluton A, t ech node, N, trveller + + bords the frst vehcle tht serves ny of the lnes n the A ( A= A ). The set A + corresponds to the lnes tht wll be chosen by the trveller to yeld one or more routes from to s n soluton of the model. At ech stop, t s convenent to refer to the set A + s the set of ttrctve lnes. Let W( A + ) be the expected wtng tme for the rrvl of the frst vehcle servng ny of the lnks A +, whch s denoted s the combned wtng tme of lnks A +. Let CIRRELT

16 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on ( ) P A + be the probblty tht lnk s the frst lne to be served mong the lnks A +. If n exponentl dstrbuton of nterrrvl tmes s ssumed then + 1 W( A ) = (25) f nd + A f P A = A ( ), f A where f s the frequency of lnk (lne) Snce A s not known pror, the sngle destnton model s formulted by usng bnry vrbles x 0 f A x =, A 1 f A The optmzton model for sngle destnton my be stted now s follows: V mn sv + f x subject to A I + AI (26) (27) x f + v =, A, N f x + A (28) V = v + g, N (29) A V 0, N (30) x = 0or1, A, (31) where s s the trvel cost on lnk nd V s the totl volume t node. At frst sght, the problem (15)-(19) s mxed nteger nonlner optmzton problem. Fortuntely, the problem my be reduced to much smpler lner progrmmng problem by mkng the followng observtons. (16) my be replced by the non-negtvty constrnts of the lnk volumes v 0, A snce v = V, N. Then, by ntroducng new vrbles w, + A V whch denote the totl wtng tme of ll trps t node, w =, N, one fx obtns the equvlent problem mn sv + w A N + A (32) CIRRELT

17 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on subject to + v = x fw, A, I (33) v v = t, N (34) + A A v 0, A. (35) The objectve functon (20) s now lner nd the 0-1 vrbles re only used n constrnts (21), whch re the only nonlner constrnts. These constrnts my be relxed by replcng (21) wth + v fw, A, N (36) whch yelds the lner progrmmng problem (32), (34), (35), (36). It my be shown, by usng the extreme pont propertes of the solutons of lner progrmmng model, tht ths problem s equvlent to (32)-(35). 5.2 Contrbutons to the study of congested trnst route choce models It s cler tht congeston t but stops does not only ncrese the wtng tmes but t lso ffects the flow shre of ech ttrctve lne. In the cse of ndependent rrvls wth exponentl dstrbuton, the flow splt s proportonl to the so-clled effectve frequency, tht s to sy, the nverse of the wtng tme of ech lne. The stop models re clled sem-congested f they consder only the ncrese n wtng tmes, nd full-congested f they lso nclude the effects on the flow splt. Wu, Florn nd Mrcotte (1994) consdered sem-congested trnst network model n whch the tme requred to bord vehcle ncreses wth flow, but the dstrbuton of flows mong ttrctve lnes s done n proporton to the nomnl frequences. It s worth mentonng the work of Wu nd Florn (1993) n solvng the sem-congested problem by usng smplcl decomposton pproch. Bouzïene-Ayr (1996) nd Bouzïene-Ayr et l. (1995) extended the ltter to fullcongested model tht combnes fxed-pont problem n the spce of rc flows wth vrtonl nequlty n the spce of hyperpth flows. An lgorthm remnscent of the method of successve verges ws lso proposed n Bouzïene-Ayr (1996) nd Bouzïene-Ayr et l. (1995), but the combntorl chrcter of hyperpths seems to lmt ts pplcblty to smll networks. Bouzïene-Ayr et l (2001) provded survey of the models used to represent the behvour of trnst pssengers t stops. More recently, Comnett nd Corre (2001) nlyzed full-congested verson of the common lnes problem of Chrqu nd Robllrd nd used t to develop trnst network model tht cn del wth generl rc trvel tmes s well s more relstc wtng tme functons wth symptotes t bus cpcty. The ltter re ntroduced by consderng effectve frequency functons tht vnsh when the flows exceed the cpcty of the lne. Although Comnett nd Corre (2001) estblshed the exstence of network equlbrum, they fl to propose n lgorthm to compute t. Nevertheless, snce the model s stted s fxed pont n the spce of rc flows only, t opens the wy to delng CIRRELT

18 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on wth lrge-scle networks. Buldng on ther work, Ceped (2002) devsed nondfferentble formulton of the strtegy trnst ssgnment wth flow-dependent trvel tmes nd perceved frequences. Ths model s descrbed n detl n Ceped et l (2005). In the followng, only the bsc models of Chrqu nd Robllrd (1975), Spess (1984) nd the extenson to the consderton of cpctes by Ceped et l (2005) re descrbed n detl. The strtegy model nd ts extensons for congested networks re descrbed n detl n the followng. It s possble to extend the strtegy lgorthm to nonlner verson of ths problem, where the lnk trvel tmes re no longer constnts, but re contnuous functons s( v), A of the rc flows v. The resultng model my be solved by n dptton of the lner pproxmton lgorthm. Further detls my be found n Spess (1984) nd Spess nd Florn (1989). v o 0 + v V A d D d d mn s ( x) dx w (37) d d + s.t. v w f d D, d, A (38) d v = v A. d D v 0, A, d D (39) d A stochstc verson of the trnst strtegy model ws developed by Nguyen, Pllottno, nd Gendreu (1998). Ths model consders logt bsed choce of strteges for route choce on trnst networks. Ths model consders congeston bord the vehcles but does not tke nto ccount the fct tht wtng tmes ncrese s the pssenger lod on the trnst vehcles ncreses. Pssengers my not be ble to bord the frst vehcle tht rrves t stop t whch they re wtng. The modellng of wtng tmes tht ncrese n congested condtons ws the topc of the doctorl thess of Ceped (2002). The results were lso reported n Ceped, Comnett nd Florn (2005). In order to extend the strtegy model to consder vehcle cpctes, some ddtonl notton s requred. A trnst lne s composed of severl trnst segments. Ech lne segment A s chrcterzed by n n-vehcle trvel tme functon s( v) nd sturton flow v, v > 0. The effectve frequency, whch s perceved by wtng trveller, s ssumed to be decresng n order to reflect the ncrement n wtng tme nduced by n ugmentton of flow. For ech d D, the effectve frequency f ( v) 0 d v wth f ( ) ( ) v s long s ( ) 0 when v strctly decresng wth d f v >. Note tht f v my tke the vlue 0, whch mkes t possble to model wtng tmes tht explode to nfnty beyond the lne cpcty. CIRRELT

19 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on The model tht consders trnst cpctes n ths wy my be restted n the form where Mnmze Gv ( ) v V0 d d v d d Gv () s() vv + mx gu () v + A d D A d f () v d (40) (41) d d d v v = g, I, d D + A A d d + α v f (), v A, N, d D (42) (43) v 0, A, d D d (44) Gv () s gp functon tht hs vlue of zero when n equlbrum flow s reched. Now, snce the optml vlue of s known, smple lgorthm s to use the Method of Successve Averges (MSA), whch hs been extensvely used n trnsportton pplctons s heurstc method, nd to evlute the devton from optmlty of the. At ech terton, the method computes trnst network equlbrum for the lner network obtned by fxng the trvel tmes nd the frequences t the vlues determned by the current flows, nd then updtes these flows by vergng the prevous terte nd the newly computed soluton. As mentoned erler, soluton for the lner cost network cn be found by solvng d d mn v V0 d tv w D + A d d d s.t. v w f d D, d, A +. Ths method hs been used successfully n prctce. 5.3 Contrbutons to the study nd soluton of network desgn models Trnst desgn problems, where the behvour of trnst trvellers s governed by strtegy ssgnment were consdered by Constntn (1986), Constntn nd Florn (1993), the un-congested cse, Noreg (2002) nd Noreg nd Florn (2003), the semcongested cse. These models optmze the frequency of set of trnst lnes gven normtve objectve of mnmzng the cost of the opertor. The resultng mn/mn optmzton problems re solved by tertve lgorthms tht use the projecton of the grdents of the upper level objectve functon. CIRRELT

20 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on 6. Combned nd Mult-Modl models 6.1 Smple combned models In ths secton, some of the contrbutons mde to the formulton nd soluton of multmode ntegrted demnd-trffc ssgnment models re explored n detl. The trnsportton choces offered n n urbn re nclude both rod nd trnst fcltes. The chllenge n the md 70 s ws to stte n ntegrted model tht would smultneously consder the choces mde by trvelers regrdng destnton, mode nd route. The ntegrton of demnd models wth network models n sngle model tht would descrbe ll choces mde n cty regrdng destnton, mode nd route choce ws n nnovtve de t the tme. An erly contrbuton ws the formulton of combned mode choce-trnst nd rod ssgnment model (Florn, 1977) nd the sttement of soluton lgorthm (tht turned out lter to be the dptton of the Jcob method) nd model tht combned trp dstrbuton, modl splt nd rod trffc ssgnment (Florn nd Nguyen, 1978). The motvton to formulte nd solve combned models comes from the smple fct tht the sequentl use of trvel demnd models nd network ssgnment models s nconsstent: the orgn to destnton trvel tmes tht re obtned fter the network ssgnment s crred out re not necessrly consstent wth the trvel tmes tht were the nput to the trvel demnd model. Hence, some equlbrton method ws necessry n order to ensure tht the models were consstent. The dgrm below dentfes the need for n equlbrton mechnsm n order to render the trvel demnd forecstng process consstent. CIRRELT

21 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on Producton Attrcton Models Trp Dstrbuton Models Mode Choce Models Equlbrton Method: Feedbck Trffc Assgnment on Rod nd Trnst Fgure 4. The four-step trnsportton plnnng prdgm wth equlbrton Ths ws frst pproched by sttng models tht combned the entropy type trp dstrbuton model nd the network equlbrum model for one trp purpose nd one clss of trffc on the rod mode. It s perhps one of the smplest combned models nd ws frst proposed by Evns. Its mthemtcl formulton s: mn A q Q 0 v s ( ) subject to g = O, p P p P x dx + ρ g ln g (45) g = D q Q p q, g 0, p P, q Q. h = g I k K k h 0, k K k p q Florn, Nguyen nd Ferlnd (1975) developed the dptton of the lner pproxmton method for ths model. Evns (1973, 1976) proposed soluton method, whch cn be nterpreted s prtl lner pproxmton lgorthm. It hs better emprcl convergence thn the former. A more elborte combned model ws formulted nd nlyzed by Florn nd Nguyen (1978). It s combned dstrbuton-ssgnment modl choce model bsed on entropy CIRRELT

22 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on type trp dstrbuton models for two modes: uto, u, nd trnst,tr. The trvel costs on the trnst network re not flow dependent whle the trvel tmes on the uto network re flow dependent. The convex cost optmzton problem: mn A u v 0 s u u x dx + ρ g ln g + + ρ g tr (ln tr tr g + u ) (46) ( ) p q u tr p subject to ( g + g ) = O, p P (47) q Q p u tr ( g + g ) = Dq, q Q (48) p P u tr g 0, g 0 p P, q Q. (49) u u hk = g, I (50) k K u tr v = δkhk + v, A (51) I k Kp h 0, k K, (52) k q yelds unque solutons nd hs the property tht the resultng mode choce s gven by logt functon: p( u ) = u exp(1/ ρ) u u exp(1/ ρ) u + exp(1/ ρ) u tr (53) Ths s reltvely smple mode choce functon, snce t does not nclude explntory vrbles other thn the trvel tmes from orgns to destntons u, u. Ths motvted the formulton of models tht would consder mode choce functons clbrted wth dt orgntng from surveys. Thus, demnd functon s ntegrted wth network ssgnment models. One of the frst such combned formultons ws contrbuted by Florn (1977). It consdered the dependency between modes of trffc shrng the sme fclty, e.g. buses slow down the speed of crs nd crs slow down the buses. Ths model ws reformulted s vrtonl nequlty by Florn nd Spess (1982). The fundmentl result of Smth (1979), tht the network equlbrum model cn be formulted s the vrtonl nequlty u tr subject to * s()( v v v) 0 (54) CIRRELT

23 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on nd hk = g, I (11) k K h 0 k K. (12) k v = δ h A (13) k k I k Kp nfluenced the reserch crred out t the CRT n ths re n sgnfcnt wy. The formulton of combned models tht would no longer stsfy the property tht there exsts n equvlent convex cost mnmzton problem ws gven new theoretcl frmework. The unqueness property of the equlbrum solutons would be stsfed only f the cost (trvel tme) functons stsfed monotoncty condtons nd the convergence of lgorthms would be ensured f certn suffcent condtons were stsfed. A combned mode choce-ssgnment wth dependency between modes my be formulted by postultng tht demnd functon hs been clbrted. It s monotone decresng wth trvel tme nd ts nverse depends on the trvel tmes of the two modes referred to bove s uto, u, nd trnst,tr. u tr u u u = w( g ), I (55) The trvel tmes by uto depend on the volume of trnst vehcles nd the trvel tme by trnst depends on the volume of crs. Let ( v ) be the vector of crs nd buses on lnk u tr u tr ( v ) =( v, v ). The totl demnd s fxed, hence g + g2 = g, I, nd t s ssumed tht the orgn to destnton trvel tmes u ( v) = mn s ( v), I, m= { u, tr} stsfy m k Km Wrdrop s user optml prncple. The vrtonl nequlty k (56) u * u u* tr tr tr* u* u u* s ( v)( v v ) + s ( v v ) w( g ) ( g g ) 0 Subject to: m hk = g, I, m= u, tr (57) m k K v = δkhk, A, m= u, tr (58) I k Km g 0, h 0, k K, I, m= u, tr (59) m k m δ ( ), (60) s () v = s v k K, I, m= u, tr k k A CIRRELT

24 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on yelds model tht hs the desred propertes. An equlbrum flow s estblshed on both the rod nd trnst networks nd the mode choce s consstent wth the orgn to destnton trvel tmes on both networks. Due to the dependence between modes, ths model s no longer equvlent to convex cost mnmzton model nd must be solved by lgorthms desgned for vrtonl nequltes. Florn nd Spess (1983) suggested the Jcob method nd provded suffcent condton for ts locl convergence (see Florn nd Spess, 1982). Snce 1979, the lterture on the soluton of vrtonl nequltes hs prctclly exploded nd vrety of lgorthms re vlble for solvng such models. For some other contrbutons from the CRT, see Nguyen nd Dupus (1984), Mrcotte (1986), Mrcotte nd Dussult (1987), Dussult nd Mrcotte (1989), Mrcotte (1991), Wu, Florn nd Mrcotte (1991), Mrcotte nd Zhu (1993), Zhu nd Mrcotte (1994), Mrcotte nd Wu (1995), Mrcotte nd Zhu (1995), Zhu nd Mrcotte (1996), Goffn, Mrcotte nd Zhu (1997), Crouzex, Mrcotte nd Zhu (2000) nd Mrcotte nd Zhu (2001). It s somewht regrettble tht these contrbutons nd those of other reserchers n ths re cnnot be ppled drectly to the soluton of more complex combned models becuse the complex mult-modl models formulted n prctce re rther d-hoc nd not menble to net mthemtcl model formulton. Ths wll be further dscussed below. Other mult-modl models consder trps by combned modes. These re trps tht strt on one mode, sy uto, nd end on nother mode, sy bus. These re qute common n ctes tht hve developed trnst system nd dequte prkng lots for prkng the cr before tkng trnst servce. Fernndez, DeCe nd Florn (1994) formulted nd solved b-modl model tht consdered pure modes nd combned modes. Such models re qute common n prctce now nd hve been enhnced to consder prkng restrctons (see Florn nd Los, 1980). They requre the computton of ntermedte orgn-destnton mtrces (see Florn nd Los, 1980b): from n orgn to trnsfer pont on frst mode, nd from the trnsfer pont to the destnton on nother mode. Florn, Wu nd He (2002) re-formulted nd developed n lgorthm for mult-clss mult-modl plnnng model for the Cty of Sntgo, Chle. 6.2 Lrge-Scle Combned Models The development of trnsportton plnnng models ws gretly nfluenced by the ncresng speed of vrous computng pltforms. In prtculr, greter speed, RAM vlblty nd dsk storge cpcty led to more lrge-scle models. The need to better model the demnd led to the prolferton of trp dstrbuton models, mode choce models nd mult-clss network equlbrum models to cheve trvel demnd forecstng n n urbn re. For nstnce, the Southern Clforn Assocton of Governments SCAG) model of the yer 2000 hs the followng chrcterstcs: 13 ctegores for trp generton nd ttrcton (by ncome, by trp purpose) CIRRELT

25 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on 7 trp dstrbuton models ( purpose) 7 mode choce models (by purpose) for 11 modes ( nd combned modes) 6 clsses of vehculr trffc (sngle occupncy, multple occupncy, lght trucks, hevy trucks, etc 6 trnst clsses (workers, school chldren, etc.) 2 combned modes: uto to locl bus, locl to express bus There re 3339 zones, 30,678 nodes, 109,770, lnks nd 1093 trnst lnes wth 65,417 trnst lne segments n ths network. The complexty of the models used s llustrted n the block dgrm of Fgure 5. Fgures 6 nd 7 re network plots of the SCAG rod network used n the plnnng model. The lgorthm mpled by the block dgrm n Fgure 5 computes ntl trvel tmes from orgns to destntons for ll the modes consdered, then verge trp trvel tmes re determned by usng the mode choce models (log sums); then the trp dstrbuton models re executed to obtn totl trvel demnd mtrces; then the mode choce models re ppled to determne the demnd by mode; then the resultng demnd mtrces for ech mode re used for the uto nd trnst ssgnment. The procedure s repeted by usng feed bck scheme n the serch of n pproxmte equlbrum soluton. The only wy to equlbrte such complex model s by usng heurstc method such s the Method of Successve Averges (MSA) to dmpen the osclltons n the lnk flows nd orgn to destnton trvel tmes. Such heurstc method s referred to s feedbck snce the lnk flows nd orgn to destnton trvel tmes re fed bck from one terton to the next n n vergng scheme. There re mny vrnts of the MSA tht re used n prctce. The bsc method strts from fesble soluton; the trvel tmes re updted nd new uxlry soluton s computed. An updted soluton s computed by combnng the current soluton nd the uxlry soluton by usng heurstc step sze λ. New soluton = Current soluton *(1- λ step sze)+ Auxlry Soluton * λ The most common choces for the step sze used re: 1/k, where k s the terton number nd 1/ const. where const s predetermned number (2,3,..5). The ltter s referred to s exponentl smoothng snce the erly solutons hve n exponentlly decresng weght. The convergence of these feedbck methods re montored by usng mesure of gp whch s nspred from the mesures of gp used n optmzton nd/or vrtonl nequlty models. If v s the vector of lnk flows for ll clsses, s s the vector of lnk trvel tmes for ll clsses, g s the demnd vector for ll clsses nd u s the vector of orgn to destnton trvel tmes for ll clsses, then mesure of reltve. When n equlbrum soluton s reched, ths gp s ( sv gu/ sv) mesure of gp s zero. In prctce, one ccepts solutons tht re of the order of 0.01 to CIRRELT

26 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on The convergence of vergng schemes hs been proven under some suffcent condtons, but s nerly mpossble to verfy on lrge-scle models. The mportnt pont to note s tht ll such complex models re bsed on the trp dstrbuton, mode choce, rod nd trnst ssgnment models descrbed erler. START uto skms for PK trnst skms for trnst skms for OP uto skms for OP Trp generton HBW Logsums for PK ( ) trp dstrbuton for PK (grvty) mode choce model for PK HBW Logsums for OP ( ) trp dstrbuton for OP (grvty) mode choce model for OP demnd computtons (tme of-dy model) uto-truck ssgnments for AM successve verge lnk volume for outer loop of AM uto-truck ssgnments for MD successve verge lnk volume for outer loop of Is convergence crteron stsfed?? uto-truck ssgnments for uto-truck ssgnments for trnst ssgnments for AM trnst ssgnments for MD END Fgure 5. The SCAG mult-modl mult-perod model CIRRELT

27 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on Fgure 6. AM pek uto volumes on the SCAG network Fgure 7. AM Pek truck flows on the SCAG network CIRRELT

28 Models nd Softwre for Urbn nd Regonl Plnnng : The Contrbutons of the Center for Reserch on 7. O-D mtrx djustment plnnng models re very dt hungry. The clbrton of demnd models s usully done wth dt orgntng from home surveys on the trvel ptterns of smple of households. Ths dt s qute expensve to collect n the developed world nd s not, n generl, done on regulr bss (the Montrel re s n excepton s home trvel surveys re done on regulr bss). Often, the orgn-destnton mtrces tht re used for the network ssgnments re out of dte. However, trffc counts on the rod network nd on trnst vehcles re crred out on regulr bss nd re much less costly thn home survey. Hence the nterest to use count dt to djust exstng orgndestnton mtrces. The djustment of n orgn-destnton (O-D) mtrx by usng observed flows (counts) on the lnks nd turns of trnsportton plnnng network hs ttrcted the ttenton of mny reserchers. The methods proposed my be subdvded nto two ctegores dependng on whether the network consdered s ssgned constnt trvel tmes or flow-dependent trvel tmes. Some of the contrbutons mde t the CRT for orgn-destnton mtrx djustment on uncongested networks nclude those of Spess (1987). However, the more mportnt contrbutons were mde for the stuton tht the underlyng route choce s tht of network equlbrum model. Nguyen (1977) proposed method tht ws precursor to future work s t only requred the djusted mtrx to replcte the orgn to destnton trvel tmes. Jornsten nd Nguyen (1979) explored ths pproch s well. A survey pper by Cscett nd Nguyen (1988), whch resulted from collborton between the CRT nd severl Itln Unverstes, provded frmework for ths clss of problems. However, some sgnfcnt contrbutons were stll to come. Ths problem ws formulted by Spess (1990) s b-level optmzton problem (or Mthemtcl Progrm wth Equlbrum Constrnts, MPEC). The mult-clss O-D djustment problem s gven by: 1 Mn Z( g) = 2 ( v v ) Subject to v = ssgn( g) (62) ˆ 2 (61) m M Aˆ where vˆ re the observed flows nd ssgn( g ) s the notton used to ndcte tht the vector of flows v s the result of the equlbrum ssgnment of demnd g. Ths ssgnment problem s: v Mn F() v s () v dv Subject to = (63) A 0 CIRRELT

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