Global Optimization Method for Robust Pricing of Transportation Networks under Uncertain Demand

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1 Interntonl Journl of Trnportton Vol. 2, No. 2 (214), pp Globl Optmzton Method for Robut Prcng of Trnportton Network under Uncertn Demnd hun Wng 1, Luren M. Grdner 2 nd. Tr Wller 3 1 chool of Mthemtc nd Appled tttc, Unerty of Wollongong, Wollongong, NW 2522, Autrl 2 chool of Cl nd Enronmentl Engneerng, The Unerty of New outh Wle nd NICTA, ydney, NW 252, Autrl 3 chool of Cl nd Enronmentl Engneerng, The Unerty of New outh Wle nd NICTA, ydney, NW 252, Autrl 1 wnghun@gml.com, 2 grdner@unw.edu.u,.wller@unw.edu.u Abtrct We extend the extng toll prcng tude wth fxed demnd to tochtc demnd. A new nd prctcl econd-bet prcng problem wth uncertn demnd propoed nd formulted tochtc mthemtcl progrm wth equlbrum contrnt. In ew of the problem tructure, we deelop tlored globl mzton lgorthm. Th lgorthm ncorporte mple erge pproxmton cheme, relxton-trengthenng method, nd lnerzton pproch. The propoed globl mzton lgorthm ppled to three network: two-lnk network, een-eleen network nd the oux-fll network. The reult demontrte tht ung ngle fxed etmton of future demnd my oeretmte the future ytem performnce, whch content wth preou tude. Moreoer, the ml toll obtned by ung the men demnd lue my not be ml conderng demnd uncertnty. The propoed globl mzton lgorthm explctly cpture demnd uncertnty nd yeld oluton tht outperform thoe wthout conderng demnd uncertnty. Keyword: Toll Prcng; Trnportton Network Equlbrum; Congeton; Globl Optmzton 1. Introducton Trffc congeton h become gret concern n hely populted metropoltn re [9, 15]. When t not feble to ncree the cpcty of the trnportton network, mpong pproprte toll on rod cn reduce trffc congeton becue toll cn encourge treler to eek le drect route or to trel durng le congeted perod. Toll re ppled n mny cte uch London, ngpore, nd tockholm. In the lterture, the problem of determnng toll to reduce congeton/totl trel tme referred to the toll or congeton prcng problem. The toll prcng problem cn be clfed the frt nd econd bet. The frt-bet toll prcng problem ume tht eery rod or rc n trnportton network cn be tolled. The ytem mum (O) cn be cheed by ettng the toll on lnk t the dfference between the mrgnl ocl cot nd the uer cot [17]. The econd-bet toll prcng problem ume tht only ubet of rc n trnportton network cn be tolled becue of poltcl reon nd the hgh cot of ettng IN: IJT Copyrght 214 ERC

2 Interntonl Journl of Trnportton up the toll gntre. Toll under uch tuton generlly could not chee n O trffc flow nd hence re referred to econd-bet. There re number of reerch effort tht re deoted to the toll prcng problem. Howeer, lmot ll of them clculte the toll bed on ngle lue of trel demnd or determntc eltc demnd reltonhp [3, 5, 6, 7, 8, 12, 13, 14, 18, 21, 22, 23, 24, 25, 26]. Wller et l., [2] howed tht ung ngle fxed etmton of future demnd my oeretmte the future ytem performnce. Nge nd Akmtu [16] nd Chen nd ubprom [1] tuded the ml toll on ngle prte toll rod n rod frnchng conderng demnd uncertnty. L et l., [4] exmned the toll degn for mprong the relblty of trel tme under uncertn demnd. Grdner et l., [2] netgted dfferent technque for frt-bet toll prcng wth uncertn demnd. We relx the determntc demnd umpton by conderng tochtc demnd n the econd-bet prcng context. The rtonle behnd ung tochtc demnd threefold. Frt, the forected determntc demnd my not mtch the rel demnd. econd, the rel demnd ctully fluctute dy by dy, nd hence the rel demnd telf uncertn. Thrd, the lgorthm deeloped for model wth tochtc demnd re lo pplcble for fxed demnd becue model wth tochtc demnd net the ce of fxed demnd pecl ce. Accordng to the boe lterture reew, mxmzng the expected effcency of trnport network by econd-bet prcng new nd prctcl reerch topc. The contrbuton of th pper follow. () We propoe nd formulte the econd-bet prcng problem wth uncertn demnd. () We degn n exmple to demontrte tht the ml toll obtned by ung the men demnd lue my not be ml conderng demnd uncertnty. () We deelop tlored globl mzton lgorthm to ddre the econdbet prcng problem wth uncertn demnd. Th lgorthm ncorporte mple erge pproxmton cheme, relxton-trengthenng method, nd lnerzton pproch. The propoed globl mzton lgorthm ppled to three network. 2. Notton, Problem Decrpton nd Formulton The econd-bet prcng problem wth uncertn demnd m to mxmze the erge network performnce n ew of the tochtc nture of the demnd. It cn be formulted tochtc mthemtcl progrm wth equlbrum contrnt (MPEC). Before preentng the mthemtcl model, we lt the notton ued throughout the pper n Tble 1. In the trnportton network G ( N, A ) there re et of toll lnk repreented by A A. The trnport uthorty need to determne the ml toll leel from et of gen toll leel repreented by et I, A. The toll t leel I. For exmple my be equl to.5 UD, 1 UD, 1.5 UD, etc. If no toll lo condered, then we dd to et I prtculr 1 2 leel tfyng. For ntnce, f I {,1, 2} nd,.5, 1, then the trnport uthorty wll chooe ether toll leel (no toll) or leel 1 (.5 UD) or leel 2 (1 UD). We defne bnry decon rble z whch equl 1 f nd only f toll leel I leed on lnk A, nd otherwe. Defne toll ector z : ( z, A, I ) tht repreent the toll ettng. Defne et Z tht contn ll feble toll ettng: z I (1) Z : z {,1}, A, I ; z 1, A 34 Copyrght c 214 ERC

3 Interntonl Journl of Trnportton et A et of ll lnk n the trnport network A et of toll lnk, A A G et of the trnport network, G ( N, A ) Tble 1. Notton I et of poble toll leel on lnk A. N et of node n the network W et of OD pr ˆ et tht contn the lope nd ntercept lne for pproxmtng t ( ) et tht contn the lope nd ntercept lne for pproxmtng et tht contn the feble lnk flow for cenro z et tht contn ll the generted oluton et of demnd cenro. The probblty of ech cenro known. N Z et of N cnddte oluton Z et tht contn ll feble toll ettng t ( x ) d x V et of lnk flow tht tfy the flow conerton equton n cenro A mple of the demnd wth the ze A mple of the demnd wth the ze W et of OD pr Prmeter o b o A prmeter defned to be m f m o nd b : q otherwe m o m q ( o, d ) W od q Gen trel demnd for OD pr ( o, d ) W n cenro od M A lrge number t ( ) Trel tme functon on lnk A Toll t leel I UE T ( ) Totl ytem trel tme n demnd cenro O T Totl ytem ml trel tme n demnd cenro Decon Vrble z A bnry decon rble whch equl 1 f nd only f toll leel I lnk A, nd otherwe z Toll ector defned z : ( z, A, I ) Flow on lnk A A ector defned :, A o Flow on lnk A tht orgnte from node o N n demnd cenro Flow on lnk A n demnd cenro A ector defned : (, A ) ( z ) UE lnk flow n demnd cenro o Flow on lnk A tht re from node o N T ˆ An uxlry decon rble, A, T An uxlry decon rble, A, mpoed on Copyrght c 214 ERC 35

4 Interntonl Journl of Trnportton T An uxlry decon rble, A Other A pre-pecfed tolernce c The ml lue of [P] c The ml lue of [AA] c Men lue of c N Number of [AA] model to ole c The ml lue of [P], UE T ( z ) Totl ytem trel tme n demnd cenro when toll ector z leed ng n totl ytem trel tme by pplyng toll ector z rto of the mxmum poble ng n demnd cenro We ume tht the et of uncertn demnd h ery lrge number of cenro nd the probblty of ech cenro pror known. In demnd cenro, the trel demnd for OD pr ( o, d ) W repreented by q. Let repreent the ector of lnk od flow n cenro, : (, A ). Let V repreent the et of lnk flow tht tfy the flow conerton equton. V cn be formulted : o o o b, m N, o N ( m, n ) ( n, m ) m A A o, A, o N o, A, o N V (2) Note tht n Eq. (2) we ue orgn-bed lnk flow formulton rther thn the commonly ued pth flow formulton n mot tude on trffc gnment. A wll be hown lter, th lnk flow formulton could tke dntge of the tte-of-the-rt mxed-nteger lner progrmmng oler. UE Let T ( ) be the totl ytem trel tme n demnd cenro when no toll leed, leed, nd UE T ( z ) be the totl ytem trel tme n demnd cenro when toll ector z O T be the totl ytem ml trel tme n demnd cenro. Then the relte effcency for cenro wth toll ector z, repreented by ( z ) defned to be the ng n totl ytem trel tme rto of the mxmum poble ng (Grdner et l., 21). Mthemtclly, U E T ( z ) : T U E U E ( ) T ( z) ( ) T O (3) Note tht n prctce t rre tht T U E O T nd nce we re focung on the econd- bet prcng, ( z ) my be trctly le thn 1. Defne contnt lue ( ) UE T ( ) :, U E O T ( ) T (4) In fce of demnd uncertnty, the trnport uthorty m to mxmze the expected lue of the relte effcency ( z ) : 36 Copyrght c 214 ERC

5 Interntonl Journl of Trnportton U E U E U E T ( ) T ( z ) T ( z ) m x E [ ( z )] m x E m x U E O E U E O (5) z Z z Z T ( ) T Z z T ( ) T Therefore we could rewrte the objecte : [P] c E E (6) ( z) t ( ( z)) UE T ( z ) A : m n m n U E O U E O zz T ( ) T Z T ( ) T z Eq. (6) the negte of the expected relte effcency. The lnk flow ector ( z ) determned by the lower-leel uer equlbrum problem: [UE] ( z) rg m n t ( x ) d x [ t ( x ) z ] d x V I A \ A A rg m n t ( x ) d x z [ t ( x ) ] d x V A \ A A I (7) The = n Eq. (7) hold becue of the feble et of toll ector (1). Regrdng model [P], we he: Theorem 1: The ml toll obtned by ung the men demnd lue my not be ml conderng demnd uncertnty. We wll proe th theorem n ecton Globl Optmzton Algorthm The model [P] mxed-nteger non-conex tochtc mzton problem. There re three dffculte for ddreng [P]. Frt, the crdnlty of the uncertn demnd cenro my be ery lrge. For exmple, f there re 5 cenro for ech orgn-detnton (OD) pr, nd there re 1 OD pr, then the crdnlty of nd t would be mpoble to elute or mze the weghted um of o mny cenro; f the demnd dtrbuton contnuou, then there re n nfnte number of cenro, nd the expectton nole mult-dmenonl ntegrton, whch ntrctble. econd, een f there only one demnd cenro, [P] tll non-conex mzton problem where the reltonhp between the toll ector nd lnk flow mplctly defned by the UE problem. Thrd, the problem h both dcrete decon rble nd contnuou decon rble, complctng grdent-decent bed lgorthm. To oercome the frt dffculty, we pply the mple erge pproxmton pproch tht obtn good cnddte oluton long wth the tttcl etmte of t mlty gp. The econd dffculty ddreed by propong tlored relxton-trengthenng globl mzton lgorthm. The thrd dffculty oercome by degnng lner pproxmton cheme tht tke dntge of the conexty of the formulton nd tte-of-the-rt mxednteger lner progrmmng oler mple Aerge Approxmton The mple erge pproxmton (AA) method n pproch for olng tochtc mzton problem by ung Monte Crlo multon. In th technque the objecte functon of the tochtc progrm pproxmted by mple erge etmte dered from rndom mple. The reultng mple erge pproxmtng problem then oled Copyrght c 214 ERC 37

6 Interntonl Journl of Trnportton by determntc mzton pproche. Th proce repeted wth dfferent mple to obtn good cnddte oluton long wth the tttcl etmte of t mlty gp [11, 19]. To pply the AA method, we frt generte ndependent nd dentclly dtrbuted oberton of the uncertn demnd cenro from the upport ccordng to the jont probblty m functon or probblty denty functon. Thee cenro re denoted by 1, 2. Let : {1, 2 } nd the probblty of ech cenro 1/. Therefore, we ue new dtrbuton functon wth cenro of equl occurrence probblty to pproxmte the orgnl uncertn demnd whoe upport h n exponentl or n nfnte crdnlty. The AA model could be formulted : z t 1 A [AA] c : m n U E O where ( ) ( ( z)) T ( ) T z Z ( z ) rg m n t ( x ) d x z [ t ( x ) ] d x (9) V A \ A A I The ml lue c to model [AA] ctully rndom rble dependng on the et. The expected lue of c no greter thn c, nmely, E\ [ c ] c (Mk et l., 1999). Conequently, we cn generte N ndependent mple of the uncertn demnd, ech of ze, nd obtn N ml objecte lue of model [AA], denoted by tttcl lower bound for c cn be etmted by n 1 n n rndom rble. Let V r( c ) be the mple rnce of c (e.g., 2), cn be condered normlly dtrbuted n pekng, ( c [ c ]) / V r( c ) / N c N n : c / N c, n 1, 2,, N n c, n 1, 2,, N c. Note tht (8). A lo. When N lrge N o rm l( E\ [ c ],V r( c ) / N ) n (trctly E\ h t-dtrbuton wth N 1 degree of freedom). Therefore, n prctce we cn conder L B n : c 3 V r( c ) / N tochtc lower bound for the ml lue c of model [P] ccordng to the 3σ rule (the probblty tht LB lower bound 99.86%). A totl of N toll ector z re obtned fter olng the N [AA] model wth dfferent mple. It poble tht ome of the toll ector re dentcl. We cn chooe the one wth n the lowet objecte lue c, denoted by z, for derng n upper bound follow. Frt, new mple, whoe ze denoted by much lrger thn, generted. Then we compute the cot, denoted by c, wth fxed toll ector z for ech cenro ery lrge, U B : c : c / cn be condered n upper bound for c.. nce The forementoned pproch the tndrd AA procedure n the lterture. When pplyng the tndrd AA procedure for contnuou mzton problem, the obtned oluton lkely to be good one, howeer, the poblty tht t ml generlly. Howeer, n our problem the toll ector dcrete n tht t mut be choen from fnte et of cnddte toll ector (lthough the crdnlty of the et exponentl). Therefore, we tlor the AA pproch to our problem ettng, nd degn n mproed pproch. Frt, to obtn the upper bound, we elute ll the totl of N toll ector z, denoted by the et N Z, 38 Copyrght c 214 ERC

7 Interntonl Journl of Trnportton N tht re obtned fter olng the N [AA] model wth dfferent mple. For ech z Z, we generte new lrge mple nd compute the reultng expected cot. The toll ector wth the lowet cot mplemented nd t expected cot the bet upper bound UB. Although the computtonl effort wll be lrger, t worthwhle conderng tht () ettng the toll long-term decon; () the computtonl effort ncree t mot lnerly wth N ; nd () the mot computtonl effort le n olng the N [AA] model. The lower bound cn lo be trengthened. In fct, we only need lower bound for the N N cnddte oluton n et Z \ Z we he lredy eluted ll the oluton n et Z. Therefore, we gn ole the model [AA] wth nother N ndependent mple where N N z Z \ Z, nd dere tttcl lower bound LB. To exclude Z from Z, the followng contrn re dded to [AA]: N z (1 z ) 1, z Z (1) A I, z A I, z 1 If L B U B, then we re ure tht the obtned oluton ml wth probblty of t let 99.86% (ngle-ded 3σ rule). Otherwe, the mlty gp doe not exceed U B L B wth probblty of t let 99.86% A Relxton-trengthenng Globl Optmzton Method [AA] mxed-nteger progrmmng (MIP) model tht non-conex. To ttck the non-conexty of the UE contrnt, we frt relx [AA] [MIP-relxed] c R elx : m n 1 A U E O z Z, V T ( ) T t ( ) (11) Note tht n Eq. (11) we ue rther thn ( z ) becue the contrnt (9) remoed. We would mpoe the contrnt (9) dynmclly. The lgorthm : Algorthm 1: tep : Defne et : tht wll contn ll the generted toll ector oluton. Defne z et :, tht wll contn the feble lnk flow for cenro., tep 1: ole [MIP-relxed] wth the contrnt: t ( x ) d x z [ t ( x ) ] d x t ( x ) d x z [ t ( x ) ] d x A \ A A I A \ A A I, (12) Note tht the contrnt (12) ld due to the defnton of ( z ) n Eq. (9). If the ml oluton z concde wth one of the oluton n z, output z nd top. Otherwe et : z. ole [UE] by ettng z t z z z cenro. et : ( z ). Repet tep 1., nd obtn the lnk flow Theorem 2: When Algorthm 1 top, z the ml oluton. ( z ) for ech Copyrght c 214 ERC 39

8 Interntonl Journl of Trnportton Proof: When z generted econd tme, t UE flow lredy ext n. The correpondng contrnt (12) would enure tht the lnk flow for [MIP-relxed] the UE lnk flow. Tht, [MIP-relxed] wth contrnt (12) tght [AA] for oluton z. For nother oluton z Z nd z z, [MIP-relxed] wth contrnt (12) relxton of [AA] f the UE lnk flow of z Z doe not ext n, nd tght [AA] f the UE lnk flow of z Z ext n. Therefore, the objecte functon lue of (11) for z Z nd z z not greter thn the correpondng lue t UE. A the objecte lue of z whoe lnk flow t UE not greter thn the objecte lue of other oluton whoe lnk flow relxed or t UE, z ml. Theorem 3: Algorthm 1 termnte n fnte number of terton. Proof: Th theorem hold trlly becue the crdnlty of Z fnte. Note tht the dntge of dynmclly mpong contrnt (12) rther thn ung the rtonl nequlty (VI) formulton [1] tht contrnt (12) conex A Lnerzton Approch [MIP-relxed] mxed-nteger nonlner progrmmng model. Howeer, t lner progrmmng relxton conex. To tke dntge of tte-of-the-rt mxed-nteger lner progrmmng oler, we lnerze the nonlner term nd ue lner contrnt to pproxmte them. The nonlner term t ( ) n the objecte functon (11) cn be lnerzed hown n Fgure 1. The frt pproxmton lne generted from, nd the lope determned uch tht the mxmum gp between the pproxmton lne nd t ( ) when re from the frt nterecton pont (, ) to the econd nterecton pont equl pre-pecfed tolernce leel ε. Note tht ε not decon rble nd the lue of the nonnegte ε doe not ffect the precon of the fnl oluton. Howeer, one hould et n pproprte lue of ε: too mll ε led to too mny pproxmton lne, where too lrge ε reult n low pproxmton qulty. The econd pproxmton lne generted from the econd nterecton pont. Th proce repeted untl the end pont m x m x m x t reched. It (, ( )) m x hould be mentoned tht cn be et the totl mxmum demnd (upper bound of the upport) of ll OD pr. Defne et ˆ, A tht contn the lope nd ntercept lne for pproxmtng t ( ). The objecte functon (11) cn be formulted mxed-nteger lner progrmmng (MILP) model by ntroducng uxlry decon rble ˆ T : [MILP-relxed] ubject to c L P -R elx : m n 1 A Tˆ z Z, V, Tˆ U E O T ( ) T ˆ T k b, A,, ( k, b ) ˆ (13) (14) Contrnt (12) cn be lnerzed mlrly. Note tht the rght-hnd de of contrnt (12) lredy lner functon of z. To lnerze the left-hnd de, defne et, A 4 Copyrght c 214 ERC

9 Interntonl Journl of Trnportton tht contn the lope nd ntercept of lne for pproxmtng uxlry decon rble T nd T t ( x ) d x, contrnt (12) re lnerzed :. After ntroducng T lo p e n tercep t, A,, (lo p e, n tercep t ) (15) T M (1 z ), A,, I A A A \ A A I (16) T T t ( x ) d x z [ t ( x ) ] d x,, (17) A reult, [MILP-relxed] wth contrnt (14)-(17) MILP model nd cn be oled by tte-of-the-rt oler uch CPLEX. ˆ T Approxmton lne t ( ) Fgure 1. Lnerzton mx 4. Computtonl Experment The propoed model nd lgorthm re ppled to three network: one mple two-lnk network; the econd network h een node nd 11 lnk nd hence nmed een-eleen network; the thrd network the oux-fll network. The purpoe of the frt mple twolnk network to how how the lgorthm work nd the m of the een-eleen network nd the oux-fll network to demontrte the pplcblty of the propoed model to more complex ce. A peronl computer wth Intel Core (TM) Duo 2.7 GHz CPU, 4 GB RAM, nd Wndow 7 Profeonl opertng ytem ued for ll tet. The lgorthm re coded wth C++, cllng CPLEX 12.1 to ole the MILP problem A Two-Lnk Exmple We frt conder mple two-lnk exmple hown n Fgure 2 to exemplfy the mportnce of ncorportng demnd uncertnty n modelng nd the proce of the relxton-trengthenng globl mzton lgorthm. The uncertn demnd et h two 1 cenro: {, }, q wth probblty of 2/3, nd q 78 wth 1 2 1, 2 1, 2 probblty of 1/3. The lnk trel tme functon re defned follow: 4 t ( ) 6 [1.1 5 ( / 2 ) ] (18) 4 t ( ) 4 [1.1 5 ( / 8 ) ] (19) Copyrght c 214 ERC 41

10 Interntonl Journl of Trnportton The trnport uthorty conder mpong toll on lnk 2 t lue from {,.2 5,.5,.7 5,1,1.2 5,1.5,1.7 5}. Lnk Toll Lnk 2 Fgure 2. A Two-Lnk Exmple We frt ddre determntc model where the uncertn demnd replced wth t erge lue. In other word, we ume tht the demnd from node 1 to node 2 q ( 2 / 3) q (1 / 3) 1 3. The reult how tht the ml toll 1.5 nd the relte 1 2 1, 2 1, 2 effcency %. We then conder the tochtc model. nce there re only two demnd cenro, we could enumerte them n the model. Tht, the et {, } 1 2 nd the objecte functon : U E U E U E U E 2 T ( ) T ( z ) 1 T ( ) T ( z ) E z U E O U E O (2) m x [ ( )] z Z 3 T ( ) T 3 T ( ) T In the relxton-trengthenng globl mzton lgorthm, the frt terton yeld oluton z { (1,,,,,,, )}, tht, no toll mpoed. Note tht n the frt terton nce no contrnt (12) ext, the model ctully m to fnd the O flow. Therefore ny z Z ml n th terton. We then compute the correpondng lnk flow t UE when the toll fxed t z for ech of the two demnd cenro, nd updte the et. In the econd terton, oluton (,,,,,,1, ) obtned; the thrd terton yeld oluton (,,,,,1,, ) ; the fourth terton yeld oluton (,,,,,1,, ) whch dentcl to the one obtned n the thrd terton. Therefore th oluton ml. The ml toll conderng demnd uncertnty hence 1.25, nd the reultnt relte effcency 85.1%. Howeer, f we et the toll t 1.5, whch the one obtned by only conderng the men demnd, the reultnt relte effcency 78.%, whch mller thn 85.1%. Th exmple clerly demontrte the mportnce of ncorporton of demnd uncertnty n congeton prcng. Moreoer, th exmple h ctully proed Theorem 1. We lo fnd tht ung the men demnd lue, we et the toll t 1.5 nd the relte effcency %. Howeer, f demnd uncertnty condered, the erge relte effcency wth the toll of 1.5 only 78.%. Hence, th exmple upport the oberton n Wller et l. (21), tht, ung ngle fxed etmton of future demnd my oeretmte the future ytem performnce. Tble 2. Importnce of Conderng Demnd Uncertnty n the Two-Lnk Exmple trtegy Ue the men demnd Conder demnd uncertnty Optml toll Relte effcency 78.% 85.1% 42 Copyrght c 214 ERC

11 Interntonl Journl of Trnportton 4.2. The een-eleen Network The een-eleen network, hown n Fgure 3, h 7 node nd 11 lnk. There re 4 OD pr tbulted n Tble 3 wth ther repecte erge demnd. The demnd of dfferent OD pr re ndependent nd the demnd of ech OD pr h three relzton of equl probblty: erge lue hown n Tble 3, erge lue multpled by 11%, nd erge lue multpled by 9%. The trel tme functon on ech lnk follow the Bureu of Publc Rod (BRP) type functon: 4 t t , A C (21) where t denote the free flow trel tme nd C the cpcty of ech lnk. The pecfc lue of t nd C on ech lnk re proded n Tble 4. The et of toll lnk A {1, 2, 3, 4,1 1}, 1 2 nd poble toll leel I {,1, 2}, nd, 1, 2 for ll A. orgn 3 detnton toll lnk Fgure 3. een-eleen Network Tble 3. OD Demnd OD Pr Aerge trel Demnd (ehcle/hour) We generte N 1 ndependent mple of the uncertn demnd, ech of ze 1. Therefore N [AA] model re oled, nd we obtn N toll degn oluton. For ech toll degn oluton, we generte nother mple of ze 1 nd clculte the erge ng. The bet oluton ( z, z, z, z, z ) ( 2,,, 2, ) choen (tht, toll of mpoed on lnk 1 nd 4, nd no toll mpoed on lnk 2, 3, nd 11) whoe erge relte effcency 94.5% lower bound for the problem. Copyrght c 214 ERC 43

12 Interntonl Journl of Trnportton We then generte nother N 1 ndependent mple of the uncertn demnd, ech of. Therefore nother N [AA] model where the 1 eluted toll degn oluton ze 1 re excluded, re oled. The men of the relte effcency dered from the [AA] model 84.8%, the tndrd deton 1.6%, nd therefore n upper bound 89.8%. Therefore, f we exclude the oluton ( z, z, z, z, z ), then wth probblty of t let 99.86% the erge relte effcency cnnot exceed 89.8%. Howeer, the erge relte effcency of the oluton ( z, z, z, z, z ) 94.5%. Therefore, ( z, z, z, z, z ) the ml oluton wth probblty of t let 99.86%. Tble 4. Prmeter n Lnk Trel Tme Functon Lnk No. Free-flow trel tme (econd) t Cpcty (Vehcle/hour) C 44 Copyrght c 214 ERC

13 Interntonl Journl of Trnportton Fgure 4. oux-fll Network wth the top 5 Congeted Lnk 4.3. The oux-fll Network The oux-fll network, hown n Fgure 4, wdely ued n trnportton tude. It h 24 node nd 76 lnk, nd other prmeter cn be obtned from We frt gn the trffc n the network followng UE prncple. Lnk 19, 16, 48, 29, 49 re the mot congeted lnk n term of the rto of flow nd cpcty (2.557, 2.55, 2.28, 2.276, 2.237, repectely). Thee fe lnk re ndcted by thck lne n Fgure 4. Hence, 1 we et A {1 6,1 9, 2 9, 4 8, 4 9}, nd poble toll leel I {,1}, nd,.8 for ll A. The reult re hown n Tble 5. The oluton wth nd wthout the conderton of demnd uncertnty re dfferent. Conderng demnd uncertnty, the relte effcency mproed by (2.1%-1.88%)/1.88%=12%. It hould be noted tht nce the oux-fll network h 76 lnk, nd only 5 lnk re tolled (econd-bet prcng), we cnnot expect the relte effcency to be ner 1. Tble 5. Importnce of Conderng Demnd Uncertnty n the oux-fll Network trtegy Ue the men demnd Conder demnd uncertnty No toll on lnk 48 nd toll of.8 No toll on lnk 16 nd toll of.8 Optml toll on lnk 16, 19, 29, 49 on lnk 19, 29, 48, 49 Relte effcency 1.88% 2.1% Copyrght c 214 ERC 45

14 Interntonl Journl of Trnportton 5. Concluon We he exmned new nd prctcl econd-bet prcng problem wth uncertn demnd. Th problem cn be formulted tochtc mthemtcl progrm wth equlbrum contrnt. In ew of the problem tructure, we deelop tlored globl mzton lgorthm. Th lgorthm ncorporte mple erge pproxmton cheme, relxton-trengthenng method, nd lnerzton pproch. The propoed globl mzton lgorthm ppled to three network: two-lnk network, een-eleen network nd the oux-fll network. The reult demontrte tht ung ngle fxed etmton of future demnd my oeretmte the future ytem performnce. Moreoer, the ml toll obtned by ung the men demnd lue my not be ml conderng demnd uncertnty. The propoed globl mzton lgorthm explctly cpture demnd uncertnty nd yeld oluton tht outperform thoe wthout conderng demnd uncertnty. In th tudy the toll nformton n contnt, nd hence cn ely be known by uer. The demnd nformton my chnge from dy to dy: for ntnce, one pttern on Mondy, one pttern on Tuedy, etc. nce the purpoe of congeton prcng to llete congeton durng pek hour, whch re the tme for commutng to work nd bck home, t reonble to ume tht uer he enough experence bout the trffc condton (they trel to nd from work eery dy). Tht the rtonle behnd umng tht uer he full nformton. Although the core element of the pper requre hgh leel of mthemtcl experte, the fundmentl de of our model mple: nce there re mny demnd cenro, we try to fnd toll tht the bet for the erge outcome of ll thee demnd cenro. Howeer, t my not be ey to undertnd th de correctly. In prctce, trnport uthorte collect OD trel dt for mny dy. Edently, the collected dt on dfferent dy would be dfferent, nd nturl (where wrong) pproch to ue the erge trel demnd to replce the underlyng tochtc demnd. A demontrted by our pper, the ml toll obtned by ung the men demnd my not be ml conderng demnd uncertnty. In other word, ung the men demnd my led to ubml oluton. Tht rule on whch pecl ttenton hould be pd by prcttoner when ettng toll. To mplement the model n relty, the trnport uthorty need to do the followng: () Determne et of cnddte rod for toll prcng; () Collect the orgn-detnton trel nformton on dfferent dy; () Apply the propoed model nd lgorthm to clculte the ml toll chrge on ech cnddte rod; () Publh the toll nformton n dnce nd et up toll gntre to collect toll. There re few reerch drecton tht we wll explore n future. Frt, n th tudy we ume homogeneou treler n the network wth the me lue of tme (VOT). In relty treler wth hgher ncome generlly he hgher VOT. Moreoer, dfferent OD pr my he dfferent compoton of VOT (for exmple, f the detnton centrl bune dtrct wth mny bnk, then the mot treler he ery hgh VOT). econd, bede effcency conderton, ncorportng treler of dfferent VOT would led to nother mportnt ue: the equty between dfferent treler group. Equty ply centrl role n effectene, cceptblty nd ee of mplementton of toll prcng. Therefore, how to ncorporte equty n prcng n nteretng reerch topc. 46 Copyrght c 214 ERC

15 Interntonl Journl of Trnportton Reference [1] C. nd K. ubprom, Anly of regulton nd polcy of prte toll rod n buld-opertetrnfer cheme under demnd uncertnty, Trnportton Reerch Prt A, ol. 41, no. 6, (27), pp [2] L. M. Grdner, A. Unnkrhnn nd. T. Wller, oluton method for robut prcng of trnportton network under uncertn demnd, Trnportton Reerch Prt C, ol. 18, no. 5, (21), pp [3]. Lwphongpnch nd D. W. Hern, An MPEC pproch to econd-bet toll prcng, Mthemtcl Progrmmng, ol. 11, no. 1, (24), pp [4] H. L, M. C. J. Blemer nd P. H. L. Boy, Network relblty-bed ml toll degn, Journl of Adnced Trnportton, ol. 42, no. 3, (28), pp [5] L. N. Lu nd J. F. McDonld, Economc effcency of econd-bet congeton prcng cheme n urbn hghwy ytem, Trnportton Reerch Prt B, ol. 33, no. 3, (1999), pp [6] W. Lu, H. Yng nd Y. Yn, Trffc rtonng nd prcng n lner monocentrc cty, Journl of Adnced Trnportton, do: 1.12/tr.1219, (213). [7] Z. Lu, Probt-bed tochtc Uer Equlbrum nd Ther Applcton n Congeton Prcng, PhD The, Ntonl Unerty of ngpore, (211). [8] Z. Lu, Q. Meng nd. Wng, peed-bed toll degn for cordon-bed congeton prcng cheme, Trnportton Reerch Prt C, ol. 31, (213), pp [9] Z. Lu, Y. Yn, X. Qu nd Y. Zhng, Bu top-kppng cheme wth rndom trel tme, Trnportton Reerch Prt C, ol. 35, (213), pp [1] P. Luthep, A. umlee, W. H. K. Lm, Z. C. L nd H. K. Lo, Globl mzton method for mxed trnportton network degn problem: A mxed-nteger lner progrmmng pproch, Trnportton Reerch Prt B, ol. 45, no. 5, (211), pp [11] W. K. Mk, D. P. Morton nd R. K. Wood, Monte Crlo boundng technque for determnng oluton qulty n tochtc progrm, Operton Reerch Letter, ol. 24, (1999), pp [12] Q. Meng nd Z. Lu, Trl-nd-error method for congeton prcng cheme under de-contrned probt-bed tochtc uer equlbrum condton, Trnportton, ol. 38, no. 5, (211), pp [13] Q. Meng nd Z. Lu, Impct nly of cordon-bed congeton prcng cheme on mode-plt of bmodl trnportton network, Trnportton Reerch Prt C, ol. 21, no. 1, (212), pp [14] Q. Meng, Z. Lu nd. Wng, Optml dtnce toll under congeton prcng nd contnuouly dtrbuted lue of tme, Trnportton Reerch Prt E, ol. 48, no. 5, (212), pp [15] Q. Meng nd X. Qu, Bu dwell tme etmton t bu by: A probbltc pproch, Trnportton Reerch Prt C, ol. 36, (213), pp [16] T. Nge nd T. Akmtu, Dynmc reenue mngement of toll rod project under trnportton demnd uncertnty, Network ptl Economc,ol. 6, no. 3-4, (26), pp [17] A.C. Pgou, The Economc of Welfre, Mcmlln nd Co., London, (192). [18] E.T. Verhoef, econd-bet congeton prcng n generl network. Heurtc lgorthm for fndng econd-bet ml toll leel nd toll pont, Trnportton Reerch Prt B, ol. 36, no. 8, (22), pp [19] B. Verwej,. Ahmed, A. J. Kleywegt, G. Nemhuer nd A. hpro, The mple erge pproxmton method ppled to tochtc routng problem: computtonl tudy, Computtonl Optmzton nd Applcton, ol. 24, (23), pp [2]. T. Wller, J. L. chofer nd A. K. Zlkopoulo, Eluton wth trffc gnment under demnd uncertnty, Trnportton Reerch Record, ol. 1771, (21), pp [21]. Wng, Q. Meng nd Z. Lu, Fundmentl properte of olume-cpcty rto of prte toll rod n generl network, Trnportton Reerch Prt B, ol. 47, (213), pp [22] H. Yng, W. Xu nd B. Heydecker, Boundng the effcency of rod prcng, Trnportton Reerch Prt E, ol. 46, no. 1, (21), pp [23] H. Yng nd X. Zhng, Optml toll degn n econd-bet lnk-bed congeton prcng, Trnportton Reerch Record, ol. 1857, (23), pp [24] H. Yng, X. Zhng nd Q. Meng, Modelng prte hghwy n network wth entry-ext bed toll chrge, Trnportton Reerch Prt B, ol. 38, no. 3, (24), pp [25] X. Zhng, nd B. n Wee, Enhncng trnportton network cpcty by congeton prcng wth multneou toll locton nd toll leel mzton, Engneerng Optmzton, ol. 44, no. 4, (212), pp Copyrght c 214 ERC 47

16 Interntonl Journl of Trnportton [26] X. Zhng nd H. Yng, The ml cordon-bed network congeton prcng problem, Trnportton Reerch Prt B, ol. 38, no. 6, (24), pp Author hun Wng, obtned h PhD n Trnportton Engneerng t the Ntonl Unerty of ngpore n 212. H reerch nteret focued on mrtme contner trnportton, ntermodl freght ytem, nd trnportton nd trffc engneerng. Luren M. Grdner, lecturer of Trnport Engneerng t the Unerty of New outh Wle.. Tr Wller, the En & Peck Profeor of Trnport Innoton t the Unerty of New outh Wle. 48 Copyrght c 214 ERC