In Jr. of Mhemcl Scences & Applcons Vol. 2, No. 2, My 2012 Copyrgh Mnd Reder Publcons ISSN No: 2230-9888 www.journlshub.com Orgn Desnon rnsporon Models: Mehods Jyo Gup nd 1 N H. Shh Deprmen of Mhemcs, Gujr Unversy, Ahmedbd, Gujr, Ind 1 Correspondng Auhor E-ml : nhshh@gml.com Absrc he orgn desnon (OD) mrx s key elemen of he clsscl four sges model, wdely used n rnsporon modelng. he four sge model cn be summrzed s () generon, () dsrbuon, () modl spl, nd (v) ssgnmen. OD mrx esmon, correspondng o he dsrbuon phse of he four sges process, s he mos populr represenon of rnsporon demnd. I consss of defnng wo enres ble, clled he demnd mrx or orgn desnon mrx, whose rows nd columns represen he zones of he sudy re. hs pper conns bref revew of he lerure on OD mrx esmon mehods. 1. Inroducon Choosng n deque represenon of rnsporon demnd comprses of rde-off beween model complexy nd d ccurcy. On one hnd, he ol number of rps n he re of sudy durng, sy, week, could be used s n ndcor of rnsporon demnd. Obvously, he prccl use of such nformon s lmed, f ll useful. On he oher hnd, deled descrpon of ech rp, ncludng he orgn, he desnon, ll nermede sops, he exc me, he purpose of he rp, ec, would provde suffcenly complee nformon. However, he fesbly of collecng such d s rher doubful, especlly for lrge survey res. Moreover, even f hs lrge pcure of rely ws vlble, he huge moun of nformon would be unmngeble, nd he mplude of mesuremen errors would probbly be unccepble. Consequenly, resonble represenon of he demnd should be somewhere n beween hese wo exremes. he mos populr compromse consss n defnng wo dmensonl ble, clled he demnd mrx or orgn-desnon (OD) mrx, whose rows nd columns represen ech zone of he sudy re. In hs pper, we dscuss exsng OD mrx esmon mehods. he de of reproducng he d s well s possble" hs severl mhemcl formulons. Bell (1984), Vn Zuylen nd Wllumsen (1980) dscussed log-lner models. Mher (1983) suggesed Byesn esmon echnques. Spess (1987) proposed mxmum lkelhood mehods. Generlzed les-squres lgorhms re consdered here, followng he pproches of Bell (1991) nd Csce (1984). he orgn desnon (OD) mrx s key elemen of he clsscl four-sge model, wdely used n rnsporon modelng. he four sge model cn be summrzed s follows: l Frs, he generon phse provdes, from soco-economcl d on populon, employmen, shoppng cenres or schools. l Secondly, he dsrbuon phse complees he demnd esmon by compung he OD mrx. l Nex, he choce of rnsporon mode s nlyzed durng he modl spl sge. l Fnlly, rffc flows for ech mode re ssgned on he nework. I s he ssgnmen sge. 819
OD mrx esmon, correspondng o he dsrbuon phse of he four - sge process, s he mos populr represenon of rnsporon demnd. I consss n defnng wo - enres ble, clled he demnd mrx or orgn-desnon (OD) mrx, whose rows nd columns represen he zones of he sudy re. A cell of he mrx refers herefore o prculr orgn-desnon pr, nd conns he ol number of people ccomplshng hs journey. hs number hs o be esmed from he d. We consder he problem of deermnng n OD mrx. he se of poenl rp orgns n he nework s denoed by O, whle D denoes he se of poenl desnons. he cell of he mrx correspondng o he -h row ( ) nd he j-h column ( j D) s denoed by. O Pror OD nformon s ssumed o be vlble from ps esmons or from he resuls of home- nd rodsde surveys, nd conned n pror mrx. Smlrly, represens he cell of row nd column j of. he ol number of rps levng orgn s denoed by O, whle he ol number of rps rechng desnon j s denoed by D j. hs nformon, provded by he generon phse, leds o he followng consrns on he OD mrx : O (1) Dj We wll dscuss O followng mehods for OD mrx esmon: v Smple Mehods: he growh fcor Mehod r proporonl Mehod v heorecl Models Grvy models Enropy models v Counng bsed mehods 1.1 Smple Mehods 1.1.1he Growh Fcor mehod (or Furness Dsrbuon Model or b proporonl mehod) hs smple mehod updes n old OD mrx usng he cul number O of rps orgnng n ech zone, nd/or he cul number D j of rps ermnng n ech zone. We denoe smlrly rp ends of he pror mrx s: nd o O (2) he mehod, proposed by Furness, bsed on he growh res dj nd O j, nd D o D j on dj (unknown) O blncng fcors A nd B j, s defned by I s deduced from (1) h A o O j D O A B o (4) j D BjDj dj dj (5) Bj hs echnque s somemes clled he b proporonl mehod. O AO o he Furness model generlly converges que quckly. Que ofen Furness s used for he exernl o exernl movemens or for forecsng goods vehcles nd fregh. Weknesses of Furness Model here re wo prmry weknesses of Furness. j D d j j (3) 820
he frs wekness s h f cell n he mrx s zero, no mer how much s fcored lwys remns zero. One mehod for geng round hs s o 'seed' ll he zero cells wh vlue (e.g. one rp, or o ssume dsrbuon of rps from he zone n queson o every oher zone nd from every oher zone o he zone n queson ). he resulng mrx for hs zone s herefore dependen on he npu ssumpons nd pus he onus on he modeler o ge he seedng rgh. he second wekness of he Furness s h s no sensve o chnges n he rnspor sysem. I s known h f he rnspor sysem s mproved people wll djus her choce of desnon o mke he mos of ddonl desnon opporunes whch hve suddenly become much more ccessble. A vrn of Furness ws developed clled me Funcon Ieron whch ook he dsnce or generlzed cos beween he orgn nd desnon zone s srng pon, from whch o pply Furness's row nd column blncng. 1.1.2 he r Proporonl mehod he growh - fcor mehod does no use nformon bou rvel coss. he r-proporonl mehod compenses from hs drwbck. In ddon o he consrns (1), d on ggrege cos dsrbuon s consdered, when vlble. hs d provdes he ol number of rps correspondng o ech cos nervl m, s llusred by Fgure 1. Addonl consrns mus herefore be verfed: Fgure 1 rp cos versus number of rps I δ RI where R I s he number of O, rps ssoced o nervl I nd les n nervl I, nd 0 oherwse. I s 1 f he I δ cos from o j 821
he phlosophy of hs mehod s very smlr o he Furness mehod. I consss n djusng ech consrn successvely, nd n erng unl ll of hem re verfed. 1.2 heorecl Models 1.2.1 Grvy Model When he growh-fcor mehod does no use cos nformon, nd he r-proporonl mehod uses only ggrege nformon, he grvy models nroduce ll rvel coss no he esmon of he demnd mrx. I llusres he mcroscopc relonshp beween plces. Grvy models, genered from n nlogy wh Newon's grvonl lw, were nroduced by Csey. I ges s nme from he de of grvy where he 'pull' beween wo srs s proporonl o he sze of he srs nd nversely proporonl o (some funcon of) he dsnce beween hem. hs s smlr o rvel beween res where he moun of rvel beween wo res cn be consdered s beng proporonl o her populon, numbers of jobs, schools, fcores, offces ec bu nversely proporonl o he dsnce (or some mesure of he sepron or deerrence) beween hem. So, n our noons, he number of rps beween orgn nd desnon j s proporonl o he number O of people levng, o he number D j of people rechng j, nd nversely proporonl o he squre of he (generlzed) cos c of rvelng beween nd j, h s j α 2 (6) c In prccl sudes, more flexble formulon s used, bsed on deerrence funcon f(:): ( ) (7) αodf j c Among populr deerrence funcons, we cn ce he exponenl funcon : he polynoml funcon : OD βc f ( c ) e η f ( c ) c ( ) he combned funcon : η βc fc ce where he vlue of prmeers β nd η mus be clbred, dependng on he prculr conex. he deerrence funcon hs clbron consn (or wo) whch mus be fed whle he rp ends hve row nd column fcors whch mus lso be fed. hs fng procedure s clled clbron nd s underken s pr of he process of buldng he rnspor model. Clbron cn gve you que dfferen mrx from h whch wen no he process (he bse yer observed rp mrx) nd he ccurcy of he model s dependen on geng 'good' clbron. When he grvy dsrbuon model hs been clbred, cn be used. We cn use he sme clbron consn for he deerrence funcon nd ssume h holds for he fuure yer. We lso need he fuure yer generlzed coss nd fuure yer rp ends nd wh hese nd he deerrence funcon, we cn forecs fuure yer rp mrx. hs fuure yer rp mrx wll be sensve o he rnspor neworks nd o he fuure levels of rp mkng from nd o ech zone. I wll lso overcome our oher problem (encounered wh Furness h of hvng zero cells n he bse yer) becuse we use he generlzed cos mrx s our srng pon nd hs comes from he rnspor neworks nd cn be clculed for every cell. he grvy model conns sensvy o he levels of rp mkng from nd o ech zone nd s sensve o he rnspor neworks. However cn be que volle model nd even when clbred he resulng rp mrx my be que dfferen from he observed rp mrx from whch ws derved. I s 822
herefore necessry o check he mrx produced gns he observed mrx secor o secor level o check h he model s workng correcly. 1.2.2 Enropy Models Enropy model hve been lso derved from lws of physcs (he second hermodynmcs prncple, n hs cse). he bsc ssumpon of enropy model s h, for gven OD mrx, ech correspondng mcro descrpon s equ-probble. herefore, he mos lkely OD s he one correspondng o he hghes number of deled descrpons. hs number, clled by nlogy he enropy of he sysem, s gven by ( E( ) O, C O, j D he mos lkely mrx s hus obned by )!! (8) subjec o mx ( O, C O O, j D O Dj 0 )!! (9) (10) 1.3 Counng Bsed Mehods Mrx esmon mehods bsed on observed flows hve been wdely developed, mnly becuse of he low cos of collecng such d. All of hem re bsed on he ssgnmen equons (11) V p where s he O, elemen n row nd column j of he demnd mrx, s he flow proporon beween nd j whch uses lnk nd V s he flow on lnk. Here, by lnk we men- n elemen of rnsporon p nework h connecs wo nodes. he mn problem rsng wh OD esmon from rffc coun d s under deermnon. Esmng he mrx from he couns mens solvng sysem of m equons (eq. (11)) wh n(n 1) unknowns, where m s he number of vlble couns, nd n s he number of cenrods. In mos cses n(n 1) s lrger hn m. he nformon provded by he couns s oo ggrege nd, herefore, nsuffcen o fully deermne n orgn-desnon mrx. herefore, ddonl echnques re requred o complee hs nformon. hree of hem re: Generlzed les-squres, Mnmzon of nformon, nd Enropy mxmzon. he generlzed les-squres mehod (see, for exmple, Csce, 1984, Csce nd Nguyen, 1988, Bell, 1991, or Bell, Inud, Lnge nd Mher, 1991) s bsed on he mnmzon of he dsnces from n pror esme o he new OD-mrx, nd from he observed rffc couns v o he ssgned flows V. More formlly, he objecve funcon h we wsh o mnmze ncludes erms of he form O, w ( ) + γ w ( V v ) (12) 2 A v 2 823
where A s he se of rcs for whch rffc couns re vlble, V s he flow on rc resulng from he ssgnmen of mrx on he nework, v s he observed flow (rffc coun) on rc, 824 w s he relve v confdence one hs n he vlue of, w s he relve confdence one hs n he vlue of v, nd s he globl relve wegh of he rffc couns s compred wh he pror OD-mrx. On he oher hnd, he nformon mnmzon mehod suggess n OD-mrx h dds s lle nformon s possble o he nformon conned n he couns. he nformon conned n se of N observons, where se k observed n k mes, s defned by Brlloun (1956) s I q n! nk k log en! (13) k k where q k s he pror probbly o observe se k. In our conex, se s he se for whch he observed vehcle rvels from o j. he number he pror probbly n of observons of hs se on lnk s n p (14) q of hs se s gven by p q (15) p he nformon I conned n V observons on he lnk s herefore where I p S p log ev! (16) ( ) p! Usng he Srlng pproxmon, we hve h he ol nformon s hen gven by S p (17) S I p (18) V S I I p (19) V he problem hen reduces o mnmzng hs nformon under he flow consrns (11). I cn be shown (Vn Zuylen nd Wllumsen, 1980) h, n hs conex, p g X ( 1+ λ) where g p nd X e S (20) wh λ beng he Lgrnge mulpler correspondng o he consrn on lnk. A hrd echnque s bsed on he enropy model. he funcon (9) s mxmzed subjec o he flow consrns (11). As shown by Vn Zuylen nd Wllumsen (1980), solvng hs progrm gves V
Wh X e λ X p where λ s he Lgrnge mulpler ssoced o he consrn on lnk. Inroducng pror mrx gves Wh X X 1 m p e λ where m s he number of used rffc couns. he grvy model hs been lso used n hs conex. he proporonly coeffcen α of (7) s deermned from equons (11). Hogberg (1976) uses non-lner regresson, Low (1972) lner regresson, nd Holm, Jensen, Nelsen, Chrsensen, Johnsen nd Ronby (1976) lkelhood mxmum mehod. References: Bell, M. G. H. (1983). he esmon of n orgn-desnon mrx from rffc couns, rnsporon Scence 17(2): 198-217. Bell, M. G. H. (1991). he esmon of orgn-desnon mrces by consrned generlzed les squres, rnsporon Reserch B 25(1): 13-22. Csce, E. (1984). Esmon of rp mrces from rffc couns nd survey d : generlzed les squres pproch esmor, rnsporon Reserch B 18(4/5): 289-299. Mher, M. J. (1983). Inferences on rp mrces from observons on lnk volumes: Byesn sscl pproch, rnsporon Reserch B 17(6): 435-447. Spess, H. (1987). A mxmum lkelhood model for esmng orgn-desnon mrces, rnsporon Reserch B 21(5): 395-412. Vn Zuylen, H. J. nd Wllumsen, L. G. (1980). he mos lkely rp mrx esmed from rffc couns, rnsporon Reserch B 14: 281-293. 825