A Unified Framework of Proactive Self-Learning Dynamic Pricing for High-Occupancy/Toll Lanes

Transcription

1 A Unfed Frmewor of Prove Self-Lernng Dynm Prng for Hgh-Oupny/oll Lnes Yngyn Lou Ph.D. Cndde Grdue eserh Asssn Deprmen of Cvl nd Cosl Engneerng Unversy of Flord Gnesvlle, FL Eml: Submed for Suden Pper Compeon nd Inernonl Symposum on Freewy nd ollwy Operons June -4, 009 Honolulu, Hw Aprl 009 Word Couns: Absr e: 3 Mnusrp e: 3970 Number of bles nd Fgures: words ol: 4970

2 Absr hs pper presens unfed frmewor o deermne dynm prng sreges for hgh-oupny/oll HO lnes. he frmewor onsss of wo rl seps, sysem nferene nd oll opmzon. he frs sep s o mne rff d rel me o lern moorss wllngness o py, esme rff se nd pred shor-erm rff demnd. he ned nowledge s hen used n he seond sep o eplly opmze oll res for he ne rollng horzon o mmze he freewy hroughpu whle ensurng free-flow rvel speed on HO lne. he pper dsusses how o merlze eh sep of he frmewor nd vlde n smulon epermen bsed on mul-lne hybrd ell rnsmsson model. I s demonsred h he frmewor s effen, effeve nd fleble, nd hs he poenl o be redly mplemened n pre.

3 . Inroduon he lerure on he use of prng s n effen wy o redue ongeson n be red b o he semnl wor by Pgou 90 nd Kngh 94 see reen revew by Lndsey 006. Congeson prng my e vrous forms suh s he re prng sheme n Sngpore nd ordon prng n London nd Soholm. A more prevlen form of ongeson prng n he U.S. s hgh-oupny/oll HO lnes. HO lnes refer o hgh-oupny vehle HOV fles h llow lower-oupny vehles LOV o py oll o gn ess. Sne he frs HO lne ws mplemened n 995 on Se oue 9 n Ornge Couny, Clforn, he onep hs been beomng populr mong governors nd rnsporon offls, n se legslures nd he med Ors, 006. Among oher fors, he resons of he populry nd wde epne of HO lnes re low ulzon of some HOV lnes nd he ddonl opon provded o LOV users. Operng poles of HO lnes re ofen o provde superor free-flow rff serve on he oll lnes whle mmzng he hroughpu of he freewy,.e., he ombned hroughpu of boh generl-purpose GP nd oll lnes Federl Hghwy Admnsron, 003. Beween hese wo objeves, he former usully hs hgher prory, beuse he HOV lnes re desgned frs nd foremos o provde less ongesed ondons for rpoolers nd rns users Munnh, 006. o heve hese objeves effenly, olls my be djused rel me, s ofen s every few mnues, n response o hnges n rff ondons. Currenly, here re hree uhores prng her oll lnes dynmlly, Clforn Deprmen of rnsporon DO on I-5, Mnneso DO on I-394, nd Flord DO on I-95 HO lnes. Dynm prng sreges doped n pre re mosly smple nd heurs. For emple, he oll res on I-394 re djused v loo-up ble, reed bsed on s rff ssgnmen nd ssumpons regrdng rvel demnd nd vlue of rvel me Hlvorson e l., 006. Alhough hese heurs dynm prng shemes ofen ouperforms fed olls Supern e l., 003, nd Mnneso DO, 006, here re mny opporunes o furher mprove hem. On he oher hnd, he reserh lerure does no offer prl nd sensble pproh 3

4 eher. Prevous sudes see, e.g., Arno e l., 998; Chu, 995; Lu nd MDonld, 999; Yng nd Hung, 997 nd Kuwhr, 00 hve emned me-vryng olls for bolenes. However, mos, f no ll, of hese sudes onsder hypohel nd delzed suons n whh nlyl soluons n be derved. eenly, Yn nd Lou 009 nd Lou e l. 007 delver proof of onep of reve self-lernng dynm prng pproh h, n sequenl fshon, lerns moorss wllngness o py WP nd eplly opmzes he oll re for he ne rollng horzon. hs pper furher dvnes he self-lernng pproh nd develops unfed frmewor of dynm prng sreges for HO lnes. he enhned frmewor ddresses severl ddonl rl ssues, nludng negred rff se esmon nd WP lernng, nd prove robus oll opmzon n oupled wh demnd predon. he overll frmewor s fleble for ddng new omponens nd s redly mplemenble n pre. I hs he poenl o me HO more rve o boh rnsporon uhores nd rvelers. For he remnder, Seon desrbes he proposed frmewor of prove self-lernng dynm prng pproh. Seon 3 presens resuls from smulon sudes o verfy nd vlde he frmewor. Fnlly Seon 4 onludes he pper nd dsusses fuure reserh dreons.. Frmewor of Prove Self-Lernng Prng Sreges for HO Lnes he proposed frmewor onsss of wo rl seps: sysem nferene nd oll opmzon. From mnng rel-me rff d suh s speed, flow or oupny olleed regulr nervls from loop deeors ofen lmed loons, he frs sep lerns rvelers WP n order o pred moorss reon o olls, omposes full pure of urren rff ondon for he enre freewy, nd foress shor-erm rff demnd. he ned nowledge wll hen be used n he seond sep o mmze he hroughpu re for he enre freewy whle mnnng superor serve on he oll lnes. Gven h fuure demnd s lely unern, sohs progrmmng pproh 4

5 s doped o deermne robus olls whose performne s no very sensve o dfferen relzons of unern rff demnd. Fgure sehes he frmewor for prove self-lernng dynm prng. Sysem Inferene oll opmzon oll re rff se esme WP lernng Demnd lernng rff mesuremens sensor loons HOV HO GP GP oll g reder Mnmum sensors requred rff sensors Downsrem Bolene Fgure : Sysem seh of he self-lernng dynm prng pproh. Sysem Inferene he frs sep of he frmewor, sysem nferene, s o nlyze rff d o esme he freewy sysem se nd lern hrerss of rvel demnd nd supply. he sep nludes hree omponens, WP lernng, rff se esmon nd demnd fores... Clbron of Wllngness o Py Undersndng moorss WP s rl for oll deermnon. Whle sed-preferene surveys ofen overesme moorss WP, he d olleed durng he ollng operon refle ul lne hoe behvors when rvelers re fng he olls beng hrged. Suh reveled-preferene nformon n be used o esme more urely moorss WP. Assumng moorss deson on wheher o py o gn ess o he 5

6 6 HO lnes follows log model, he relonshp beween he pprohng flow res nd he flows on HO nd GP lnes n be sed s follows: γ β α α μ μ ep where μ nd μ represen he pprohng flow res on HOV nd GP lnes durng me nervl respevely; nd re he flow res fer he lne hoe, nd nd re he verge rvel mes on HO nd GP lnes me nervl. In Equon, here re hree prmeers o be esmed: α, α nd γ, where α nd α nde respevely he mrgnl effe of rvel me nd oll on moorss uly, nd γ enpsules oher fors ffeng moorss WP. Noe h α α represens moorss rde-off beween me svngs nd olls,.e., he vlue of rvel me. Oher vrbles n Equon n eher be obned drely from loop deeors or esmed usng rff flow models, nd he oll re β s se by he operor. I should be poned ou h here we use volume spls μ μ o pprome he probbles of lne hoes. In rel-me operon, reursve les-squres ehnque or dsree Klmn flerng KF Klmn, 960 n be used o esme he onsn prmeers, α, α nd γ. o do so, Equon n be reformuled s follows: γ β α α μ μ ln Applyng KF o esme he prmeers, we hve: [ ] ˆ ˆ ˆ ln ˆ ˆ ˆ ˆ ˆ ˆ P G I P P P G G WP WP WP WP WP WP WP β σ β β β γ α α β μ μ γ α α γ α α 3

7 where he vrbles wh ^ re esmes. σ WP s he vrne of he lef-hnd sde of Equon ompued from he nown vrne of rndom deeor mesuremen error he vlue ould vry wh dfferen ypes of sensors, P WP s he epeed ovrne mr of he esmon errors nd G WP s he Klmn gn. Wh n nlzon of α 0, α 0, γ 0 nd P 0, Equon 3 n upde esmes of α, α nd γ rel me wh newly-obned nformon. As me evolves, he mp of he nlzon wll be dmnshng, nd ure esmes ofα, α nd γ re epeed... rff Se Esmon rff se esmon s o delver omplee mge of freewy rff ondons bsed on vlble rff d lmed loons. I s he bss for fly performne monorng nd provdng nl ondons for he oll opmzon n our frmewor. D obned from he deeors serves s dre mesures wh rndom mesuremen errors, whle he preded ses from rff flow models re vewed s n ndre mesure whh mgh no be very ure eher. he bs de s o reover he full mge by esblshng se h blnes boh he dre nd ndre mesuremens. Vrous flerng ehnques n be ppled bsed on he dfferen rff flow models doped e.g., Wng nd Ppgeorgou 005; Anonou e l., 007; Mhylov e l o more relslly pure rff dynms long he ollng fly, we dop he mul-lne hybrd ell rnsmsson model CM proposed by Lvl nd Dgnzo 006 n hs pper. he mul-lne hybrd CM s ble o pure boh he mndory nd he dsreonry lne-hngng behvors, nd o re lne hngng vehles s movng bolenes. he model llows us o eplly onsder he mps of he lne-hngng behvors on he rff ondons of he freewy. However, he model s non-lner non-dfferenble rnsformon from one rff se o noher, prulrly wh he lne-hoe probbly nvolved. We hus pply unsened Klmn Fler UKF Juler e l., 995 o pred he new se wh unsened smplng. 7

8 he se-spe model of hs flerng problem n be wren s: d f d vg z h d d f d vg e e e In he bove model, d s he veor of he denses long he freewy for boh HO nd GP lnes me pon, f represens he rnsformon from d o d, followng he mul-lne hybrd CM. Noe h he mesuremens from he rff deeors re usully ggreged,.e., drely mesured densy s n verge vlue ross ern me perod. We hen le d vg represens he verge denses for boh lnes beween wo d-pullng me pons nd, ompued from funon f bsed on he sme mul-lne hybrd CM. z s he veor of lmed dre mesuremens, nd h denoe he mr ndng loons of he deeors. e nd e re he orrespondng proess errors, nd e z s he mesuremen error. Whle e z n be sfely ssumed o be Gussn whe noses wh nown vrne mr σ z, furher nvesgon emprl sudy of he rff flow model s requred o me resonble ssumpons on e nd e. In he bove, he prmeers of he mul-lne hybrd CM, suh s he free flow speed, jm densy nd py, s well s he WP re no reed s unnown ses for beer effeny. Insed, he prmeers for he rff flow model n be lbred offlne, nd he WP n be esmed ndependenly wh he mnmum requremens of deeors see Fgure. he frs wo se equons n be ombned by ugmenng he se vrbles o ] z [ d, d vg, groupng wo CM rnsformons nd proess errors o f ] [ f, f nd e ] [ e, e, nd epndng h o H ordngly. A more onse form of he se-spe model beomes: f e z H ez Gven he esmes of he men nd he ovrne of se vrble me s ˆ nd P, he followng n pons wh he sme wegh n n be genered, where 8

9 n s he dmenson of he se vrble, nd he mr squre roo of n P : ˆ ˆ n P n P n P s he h row or olumn of,,..., n 4 Equon 4 s lled unsened smplng beuse hese pons hve he sme smple men nd ovrne s ˆ nd. Applyng UKF, he new esmes of he rff P ses re: ~ f,,...,n n ~ ~ men n ˆ ~ [ ~ men G z H men ] G Pz Pz n ~ ~ ~ ~ Pz men H H men n n P ~ ~ ~ ~ z H H men H H men n n ~ P ~ ~ ~ ~ men men σ n ~ P P G Pz G where σ s he ovrne mr of he proess error e. wo mos me-onsumng σ z 5 seps n model 5 re pplyng mul-lne CM o eh smple pon nd he mr nverson. Effeny of he former depends on he dmenson of he rff se vrble, nd he ler he number of deeors nslled...3 Demnd Lernng he hrd omponen of sysem nferene s o fores shor-erm rff demnd suh h oll deermnon n be more prove. he fores s nsrumened by Byesn lernng. I s ssumed h fuure rff rrvl follows Posson proess, wh unnown verge rrvl re, denoed s Λ. As me evolves, he deeed rrvl res re used o djus he esme of he verge rrvl re Λ, nd hus he dsrbuon of he rrvl 9

10 0 re. he operor n hen me use of he shor-erm demnd fores o deermne olls o beer heve he HO operng objeves. Ln 006 pples he bove Byesn lernng onep o dynmlly pre produ o mmze he epeed prof of frm. Bsed on he ssumpon, he number of vehles rrved durng me ] 0,, denoed s N, wll follow he Posson dsrbuon:!, n e n N P n Λ. Sne Λ s unnown, we my esme s pror dsrbuon from hsorl d s he followng gmm dsrbuon: e pdf Γ Λ where d e Γ 0 s he gmm funon, nd prmeers nd re seleed o f he hsorl smple men nd vrne of Λ : Λ Λ Vr E nd Λ Λ Vr E. o upde he pror dsrbuon of Λ durng operons, ssume h durng ] 0,, number of vehles hve been observed from loop deeors. By Byes heorem, poseror probbly densy funon of Λ s:!! 0 0 e d e e e e d N P pdf N P pdf pdf N Γ Γ Γ Λ Λ Λ Λ Λ, whh s noher gmm dsrbuon wh prmeers,. Wh he upded dsrbuon of Λ, he number of vehles h my rrve durng me nervl ], Δ n be esmed s below: n n n n d e n e n N N N P Δ Δ Δ Γ Γ Γ Δ Δ Δ!! 0

11 Noe h when s neger, he bove redues o negve bnoml dsrbuon: P N Δ N n N n n Δ Δ Δ n. oll Deermnon he frs sep n he frmewor, sysem nferene, hs reveled he urren sysem ondon nd preded fuure rff demnd. Wh hs nformon, he seond sep s o deermne oll res o mmze he ol hroughpu of he enre freewy whle ensurng he HO lne opere n free flow ondon. A rollng horzon sheme s used o opmze oll res. o beer represen he rff dynms, he mul-lne hybrd CM proposed by Lvl nd Dgnzo 006 s doped o ompue he objeve vlues... Deermns oll Opmzon Assume for now h rff rrvls re nown nd deermns for he ne rollng horzon M me nervls. Whn he horzon, oll res ould be fl or vry from one nervl o noher. Le q nd q represen he verge hroughpu of HO nd GP lnes he downsrem end of he freewy segmen respevely durng he rollng horzon, nd v he verge speed of HOV/HO lne. he onrol objeves n now be more speflly sed s mmzng he sum of he verge hroughpus he downsrem end whle empng o eep he verge speed of HOV/HO lne s free flow speed v f. hese mesuremens of performne n be ompued from he hybrd CM wh mndory lne-hngng demnd he HO enrne luled by Equon. Wh β beng he fl oll re or he oll veor, μ nd μ he sngleon nflow vlue or he veor of fuure nflows, ompung he objeve mesures s essenlly mppng ψ : q q, v ψ α, α, γ, β, μ, μ, problem n be wren s:. Consequenly, he oll opmzon m q q θ mn{ v v,0} β f s.. 0 β β 6 m

12 - Δβ β β Δ β 7 q q, v ψ α, α, γ, β, μ, μ, where θ s penly prmeer, β s he oll re of he prevous me nervl, Δ β nd β m re he mmum mrgn of oll vron for wo onseuve nervls due o sfey onsderon nd he bsolue mmum oll re spefed by he ollng uhory. he seond erm n he objeve funon s penly for speed reduon. If θ s seleed pproprely, hs erm should be equl o zero wh he opml soluon, ensurng h he verge speed of HO lne s free flow speed. Noe h he objeve funon s onnuous nd bounded bove, nd he fesbly se s omp, herefore opml soluon ess. A vrey of numerl lgorhms, suh s he golden-seon mehod, n be used o serh for lol opmum.... obus oll Opmzon wh Unern Demnd Gven he fuure rff demnd s lely unern, robus oll opmzon ddresses hs ssue by onsderng fuure nflow o be sohs nd sees for oll re h performs well under mos of he possble senros. Wh he fuure rrvl dsrbuon upded n rel me usng Byesn lernng dsussed n seon..3, he probbly densy funon of he rrvl dsrbuon n be ppromed by dsree se of possble senros, s,,,..., S, where eh senro spefes he rff rrvl res for eh nervl n he rollng horzon, denoed s μ s s nd μ, wh probbly of ourrene s p s. Sne mos deson mers re rs verse, senro-bsed sohs progrm s formuled o mnmze he epeed loss penly mnus hroughpu nurred by hgh-onsequene senros, whh s lled ondonl vlue--rs CV n fnnl engneerng ofellr nd Urysev, 000. he deson mers ude owrd rs n be onrolled by prmeer δ. More speflly, only senros wh loss greer hn he δ -perenle sy, 90%

13 vlue re onsdered, nd he epeed loss of hose senros s lled δ -CV. he senro-bsed sohs oll opmzon problem s wren s follows: S mn ξ β p s s m Ls ξ,0 δ, ξ,l s s s s s.. Ls q q θ m{ v f v,0} s s s s s q q, v ψ α, α, γ, β, μ, μ, Consrns 6 7 I n be proved h he opml vlue of he objeve funon s he mnmum δ -CV, nd he soluon ξ s he δ -perenle loss ofellr nd Urysev, 00. Noe h he bove sohs formulon s fleble nd n be eended o norpore oher soures of unerny. For emple, f moorss heerogeney n WP s onsdered, we my pprome he heerogeney by number of dsree senros. Afer negrng hem wh he senros of rff demnd, he bove sohs formulon wll be ble o onsder boh ypes of unerny. 3. Numerl Emple Gven es envronmen on rel fles s no redly vlble, we ondued smulon epermens o demonsre nd vlde he proposed frmewor. he developed smulon plform onsss of hree mjor modules: smulor, onroller nd monor. he smulor emps o reple he moorss lne-hoe behvors nd rff dynms. he onroller mplemens he proposed frmewor of sysem nferene nd oll opmzon. he monor serves s survellne sysem, olleng mesuremens of ul rff ondons eh nervl. In our smulon epermens, he monor lls he smulor wh ul prmeers o represen rel rff nd o produe perurbed rff mesures. Sne he onroller nd he smulor dop he sme behvor ssumpon,.e. he smulor represen he ul rff dynms urely, he performne of he proposed frmewor my be overesmed. In he followng, o fle he ey oneps of he proposed frmewor, we mplemened he sysem 3

14 nferene omponens of WP lernng nd rff se esmon, nd he deermns prove verson of oll opmzon. Fgure Smulon Sengs 3. Smulon Sengs he smulon se s freewy segmen shown n Fgure. he se s mles long, wh brrer-sepred HO slp rmp, one of he hree ypl HO ess desgns FHWA, 003. Fgure b presens he deled ell-represenon sheme of he se n he mul-lne hybrd CM. I s ssumed h eh lne obeys he sme rngulr fundmenl dgrm. he relevn prmeers re repored n Fgure, nd he smll rngle s for he downsrem bolene. Addonlly, ll lne-hngng vehles re 4

15 ssumed o hve n eleron re of. f/s,.e., wll e vehle 7. seonds o elere from zero speed o free-flow speed. he rue vlues of α, α nd γ used n he smulor re 0.5, nd 0.. α α 0. 5 ndes vlue of rvel me of 5 dollrs per hour sne he rvel me dfferene s represened n he un of one ollng nervl mnues. he enre smulon duron s 0 mnues. he dsree smulon nervl n hybrd CM model s 0.6 seonds, nd ll lnes re proned no smll ells of 0.0 mle, ledng o veor wh ol number of 40 elemens n he rff se esmon. o be onssen wh he pre, he monor repors ggreged rff mesuremens every sngle mnue, nd he rff se esmon s ondued one new mesuremens re reeved. Moreover, he oll re vres every wo mnues, nd he rollng horzon for oll opmzon s 4 mnues me nervls. he weghng for θ n Equon 5 s se s one nd he oll opmzon problem s solved usng he golden-seon mehod. In he smulon, rndom rrvls re genered from vrul soure wh n verge re of 400 vph for he GP lne, 600 vph nd 00 vph durng he frs nd ls 0 mnues for he HOV lne. α 0, α 0, γ 0 nd P WP 0 re nlzed s,, 0 nd n deny mr respevely. Inl vlues of rff ondons re genered rndomly, nd P 0 s se o deny mr. 3. Numerl esuls In hs seon, we nvesge he ury of he wo sysem nferene omponens: WP lernng nd rff se esmon, nd he overll performne of he resulng oll res n erms of boh he freewy hroughpu nd he HOV/HO lne rff ondon. Fgure 3 shows h rvelers WP n be grdully lerned durng he ollng operon. he lbred vlues of prmeers α, α nd γ re ble o onverge from he nl vlues o he rue vlues whn 6 mnues. he lbron me for eh nervl s less hn seond. Sne he sze of he rff se veor s very lrge, we do no 5

16 presen he deled omprson beween he esmed nd he ul vlues drely. Insed, n seon 3.3, we wll ompre he onrol resuls o he bse se where ul rff ondon s nown presely α Clbred Vlue Aul Vlue me Inervl mn α me Inervl mn γ Clbred Vlue Aul Vlue 0 Clbred Vlue Aul Vlue me Inervl mn Fgure 3 Clbron of Wllngness o Py he resulng oll res nd he performnes of he freewy segmen re presened n Fgure 4. I s demonsred h he onroller s ble o djus he oll re n response o he esmed rff ondon n rel me. he prove oll opmzon ensures h he oll re hnges smoohly from nervl o nervl. Fgure 4 ompres he resulng verge speed nd hroughpus from wo ses where HO lne s opered respevely under opml oll re nd whou ny onrol. he ler se mens h LOV n use he HO lne whou pyng ny oll. I n be observed h he onroller s ble o mnn hgh nd sble hroughpu whle prevenng he HOV/HO lne from beng ongesed. he mnmum verge speed long he HOV/HO lne s 55.4 mph under opml oll res; whle n he no-onrol se, he verge speed n be s low s 34 mph. On he oher hnd, he verge hroughpu s 39 vph under opml oll res, 6

17 whh s 9% of he downsrem bolene py, nd only slghly lower hn he hroughpu of 3350 vph whou ny onrol. On he oher hnd, 50% of he py would be wsed f LOV s no llowed o ess he HO lne. One of he resons h he HO lne hroughpu does no reh he py of he downsrem bolene s h lne-hngng vehles s movng bolenes whle elerng o he speed prevlng on he HO lne. hose vehles ree gps n he flow n fron of hem h propge forwrd, hereby redung he hroughpu. In he smulon, ddonl loss of hroughpu s due o nl nure esmes of moorss WP. oll re $ Averge speed long HOV/HO mph me Inervl mn me Inervl mn Op. oll No onroll Free flow speed Freewy hroughpu vph me Inervl mn Op. oll No onroll Fgure 4 Opml oll re nd s performne 3.3. Comprson beween Dfferen rff Se Esmors In he bove numerl emple, he ompuon me of he oll res for eh me nervl s less hn 9 seonds. However, updng he esmes of rff ondons one every mnue es more hn 90 seonds due o he lrge number of unsened smples requred n he UKF mehod. In order o opere he onroller n rel me, orser dsrezon of he se or smplfon of rff se esmon s neessry. 7

18 In our epermen, we esed wo smplfed rff se esmon proedures: one s o genere less smple pons n UKF, nd he oher s lner nerpolon. Noe h n he former smplfed proedure, lhough he vrne of he smple pons fer rnsformon my no be preserved, he smple men remns unsened. he mps of dfferen rff se esmors on he onrol objeves re nvesged by omprng hem o he bse se where ul rff ondon s nown presely. We ondued mulple runs nd our ssl ess nde h dfferen versons of rff se esmon do no ffe he resulng verge speed of HO lne nd he freewy hroughpu sgnfnly. One plusble reson s h he smplfed rff se esmors re sll ble o pure he prevlng rff ondons long he freewy beuse he mullne hybrd CM s mnly frs-order rff flow model. On he oher hnd, he wo smplfed versons re fr less demndng n ompuon wh n verge of less hn 8 nd 0.00 seonds respevely. her ompuonl effeny s desrble rbue n rel-me operon. 4. Conlusons We hve developed unfed frmewor of prove self-lernng dynm prng for HO lnes. hs frmewor norpores wo mjor seps, sysem nferene nd oll opmzon, nd s ble o genere robus nd prove dynm prng sreges he sysem nferene pples vrey of ehnques suh s regulr KF, UKF nd Byesn lernng o mne rff d o gn beer undersndng of moorss WP, rff se nd shor-erm rff demnd. he ned nowledge onrbues n he seond sep n deermnng opml oll res h led o effen ulzon of freewy py nd superor rvel serves for HO lnes. wo oll opmzon formulons hve been proposed. Smulon epermens onfrm h he negred frmewor s fesble, effen nd effeve. he proposed frmewor n be eended n mulple wys o e no oun more onsderons. For emple, noher more dvned dsree hoe model, suh s med log model, n be norpored s n lernve o desrbe he rvelers 8

19 heerogeneous lne-hoe behvors. Our fuure reserh wll lso negre rmp meerng wh dynm prng, nd develop oordned prng sreges for freewys wh mulple HO segmens. We lso pln o enhne he pbly of COSIM n smulng HO lne operons nd hen ondu smulon epermens o evlue he proposed frmewor n mrosop smulon plform. eferene Anonou, C., Ben-Av, M., nd Kousopoulos, H. 007 Nonlner Klmn flerng lgorhms for on-lne lbron of dynm rff ssgnmen models. IEEE rnsons on Inellgen rnsporon Sysems, Vol. 8, No. 4, Arno,., de Plm, A., nd Lndsey, een developmens n he bolene model. od Prng, rff Congeson nd he Envronmen: Issues of Effeny nd Sol Fesbly Kenneh J. Buon nd Er. Verhoef, eds, Chu, X Endogenous rp shedulng: he Henderson pproh reformuled nd ompred wh he Vrey pproh. Journl of Urbn Eonoms, 37, Federl Hghwy Admnsron 003. A Gude for HO Lne Developmen. U.S. Deprmen of rnsporon. Hlvorson,., Nool, M. nd Bueye, K Hgh oupny oll lne nnovons: I-394 MnPASS. he 85 h Annul Meeng of he rnsporon eserh Bord, Compendum of Ppers CD-OM, 06-65, Jnury 9 3, 006. Juler, S., Uhlmnn, J. nd Durrn-Whye H. 995 A new pproh for flerng nonlner sysems. Proeedngs of he Amern Conrol Conferene, Sele, June 995, Klmn,. 960 A new pproh o lner flerng nd predon problems. rnson of he ASME Journl of Bs Engneerng, 8D, Kngh, F.H. 94. Some flles n he nerpreon of sol os. Qurerly Journl of Eonoms, 38, Kuwhr, M. 00. A heorel nlyss on dynm mrgnl os prng. Proeedngs of he Sh Conferene of Hong Kong Soey for rnsporon Sudes,

20 Lvl, J.A. nd Dgnzo, C.F Lne-hngng n rff srems. rnsporon eserh, Pr B, 40, Ln, K. 006 Dynm prng wh rel-me demnd lernng. Europen Journl of Operonl eserh, 74, Lndsey, Do eonomss reh onluson on rod prng? he nelleul hsory of n de. Eon Journl Wh, 3, Lu, L.N., nd MDonld, J.F Eonom effeny of seond-bes ongeson prng shemes n urbn hghwy sysems. rnsporon eserh, Pr B, 33, Lou, Y., Yn, Y. nd Lvl, J. 007 Opml dynm prng sreges for hghoupny/oll lnes. rnsporon eserh, Pr C under re-revew. Mhylov, L., Boel,. nd Hegy, A Freewy rff esmon whn prle flerng frmewor. Auom, 43, Mnneso Deprmen of rnsporon 006. I-394 MnPASS ehnl Evluon Fnl epor. hp:// Munnh, L.W Esng he ommue: Anlyss shows MnPASS sysem urbng ongeson on Inerse 394. Downown Journl/Sywy News, <hp:// July 4, 006 Ors, C. K Hghwy ollng hs rehed he ppng pon. Innovons Brefs, 7. Supern, J., Seffey, D. nd Kshde, C. 003 Dynm vlue prng s n nsrumen of beer ulzon of HO lnes: he Sn Dego I-5 se. rnsporon eserh eord, 839, Pgou, A.C. 90. Welh nd Welfre, Mmlln, London. ofellr,.. nd Urysev, S Opmzon of ondonl vlue--rs. Journl of s,, -4. ofellr,.. nd Urysev, S. 00. Condonl vlue--rs for generl loss dsrbuon. Journl of Bnng nd Fnne, 6,

21 Wng, Y. nd Ppgeorgou, M. 005 el-me freewy rff se esmon bsed on eended Klmn fler: generl pproh. rnsporon eserh, Pr B, 39, Yng, H., nd Hung, H.J Anlyss of he me-vryng prng of bolene wh els demnd usng opml onrol heory. rnsporon eserh, Pr B, 3, Yn, Y. nd Lou, Y Dynm ollng sreges for mnged lnes. ASCE Journl of rnsporon Engneerng, Vol. 35, No., 45-5.