On-Line Marginal-Cost Pricing across Networks: Incorporating Heterogeneous Users and Stochastic Equilibria

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1 On-Line Mrginl-Cost Pricing cross Networks: Incorporting Heterogeneous Use nd Stochstic Equilibri by Yong Zho Wilbur Sith Assocites 99 Mountin Ridge Drive Austin, TX nd Kr Mri Kockeln Clre Boothe Luce Associte Professor of Civil Engineering The Univeity of Texs t Austin 6.9 E. Cockrell Jr. Hll Austin, TX kkockel@il.utexs.edu Phone: FAX: (Corresponding Author) Originl subitted 13 Februry 24 for publiction in Trnsporttion Reserch, Prt B Revised Pper subitted on 7 October 24 Second Revision subitted 2 Mrch 25 Abstrct This pper discusses n on-line, tril-nd-error ipleenttion of rginl-cost pricing for networks with use whose vlues of trvel tie vry, whose dend functions re unknown, nd whose route choices confor to rndo-utility xiiztion. It is n extension of Yng et l s (24) clcultions of optil congestion tolls with hoogenous trvele nd shortest-pth choices. The nuericl exple on n ctul, lrge-scle network suggests the heuristic itertive procedure does converge in serching for optil tolls. Key Words: Mrginl cost pricing, congestion pricing, toll rods, vlue of trvel tie 1

2 1. INTRODUCTION Rod tolls bsed on rginl socil dely costs hve long been considered n econoiclly efficient solution to highwy congestion (Vickrey 1969). In ost rkets, goods should not be llocted beyond the point where rginl gins equl the rginl cost (MC) to furnish the good. And in certin, iperfect rkets, the presence of unpriced externlities styies this reltionship. Such is the cse of rod use, where trvele re generlly oblivious to the delys iposed on those following the during dense trffic conditions, nd consider only the verge trvel tie, or verge cost (AC), they experience directly. A rginl-cost-pricing (MCP) strtegy chrges the user ny difference between verge cost nd rginl costs, theoreticlly driving the rket to level of flow where rginl costs nd benefits equte. Knowing dend for trvel cross network, one cn itertively solve for the set of prices tht equte MC nd rginl benefits (MB) on ll links. In prctice, however, dend functions re unknown. Li (22) initited nd Yng et l (24) expnded tril-nd-error ipleenttion of MCP on network without knowledge of dend functions but with known link perfornce functions, observed flows, nd observed responses to pricing decisions. The procedure they propose ssues single vlue of tie for ll use nd coputes the optil prices t ny deonstrted flow levels. It then relies on diinishing frction of the difference in optil nd current toll vlues, in order to djust current tolls. Yng et l s clcultions re bsed on soe iportnt ssuptions, including user equilibriu (UE) or shortest-pth route choices nd single, known vlue of trvel tie (VOTT) for ll vehicles. The UE ssuption requires full infortion of rodwy conditions (nd prices) on the prt of ll drive nd focus on trvel tie (rther thn other eleents of trvel, such s nuber of stops, relibility, nd route esthetics). A ore relistic network ssignent ssuption is believed to be stochstic user equilibriu (SUE), where ech user y perceive different pth costs or benefits, nd rndo vrition in route nd/or trveler chrcteristics results in distribution of route choices, for the se origin-destintion (O-D) pir t the se tie of dy. In ddition, trvele vlue their tie differently, depending on the purpose of their trip nd their willingness nd/or bility to py. To relx this restriction, one cn segent the entire popultion of use into nuber of groups or clsses, ccording to their incoes nd other deogrphic chrcteristics relted to VOTT. For ech trveler group, the use shre the se the VOTT probbility distributions. The distributions cn be odeled s either continuous or discrete (see, for exple, Cntrell nd Binetti, 1998). This pper extends Yng et l s odel, lgorith nd ppliction by (1) llowing SUE network ssignent, (2) recognizing group-bsed vritions in VOTT, nd (3) pplying the odel to relistic, regionl network. As in virtully ll MCP network pplictions, this work ssues tht MC nd MB re bsed on trvele own VOTTs, s evidenced through their willingness to py (to sve tie). However, one cn rgue tht soe other, socil VOTT pplies for definition of MCs nd MBs, nd for selection of optil tolls. In such situtions, the optiizing-toll equtions used here cn be ltered, to reflect the MC nd MB definitions; the solution lgoriths would rein the se. The following section presents soe nottion nd ssuptions coonly dopted in SUE nlysis nd introduces the odel for deterining MCP tolls. Section 3 describes the trilnd-error ipleenttion lgorith used to clculte the optil tolls, nd Section 4 reports the results of the heuristic lgorith s ppliction to rel, lrge-scle network. Conclusions nd future reserch directions re provided in Section 5. 2

3 2. MCP MODEL: SUE WITH MULTIPLE USER CLASSES Network flow equilibri represent n interction between congestion nd trvel decisions. The sitution is chrcterized by two sets of functions: (1) perfornce functions for ll network links, describing how trvel ties rise with dend; nd (2) dend functions, illustrting how trvel dend responds to trvel-tie increses (Sheffi, 1985). These dend functions re cobintion of trip-king, ode choice, destintion choice, route choice, nd deprture-tie decisions. In the stble, equilibriu condition, individul trvele cnnot iprove their trvel ties by unilterlly chnging routes. This is known s Wrdrop s fit principle (Wrdrop, 1952), nd the result is UE. Since it is unlikely tht ll trvele hve full infortion bout iniu trvel ties on every possible route nd lwys ke shortest-tie route choices, the SUE equilibriu condition is ore populr principle. It is described s network condition where no trveler cn iprove his/her perceived trvel tie by unilterlly chnging routes. (Dgnzo nd Sheffi, 1977). SUE ssignent hs been forulted s optiiztion (Dgnzo nd Sheffi, 1977), vritionl inequlity (Bell nd Iid, 1993), nd fixed-point probles (Dgnzo, 1983; nd Cntrell, 1997). Here, the optiiztion forultion is followed. Dgnzo (1983) proposed n SUE frework with ultiple user clsses, nd Mher nd Zhng (2) provided forultion nd lgoriths for SUE with elstic dend. Ros nd Mher (22) extended these forultions to develop SUE with both ultiple user clsses nd elstic dend. This pper s forultion is built on the ltter three works. Let G( N, A) be trnsporttion network with the node set N nd the link set A, with positive onotoniclly incresing link perfornce (trvel tie) functions t ( x ) of flows 1 x for ech link A. The link perfornce function is ssued to be differentible (with respect to flow x ). Network use re segented into M clsses nd ech clss shres the se socioeconoic chrcteristics. Clss s dend for trvel between O-D pir r nd s t prticulr tie of dy is denoted by q, nd it is ssued to be continuously decresing function of trvel cost between tht O-D pir 2. The ulti-user clss SUE proble hs n equivlent iniiztion forultion, which cn be written s follows: v x v in v Z( x) = q S ( x) + t ( x ) x t ( w) dw (1) s.t. where: x x k K, = q, r, s = f δ, (2) f ; nd (3) f. (4) K is the set of pths between O-D pir r nd s; δ, = 1 if the pth flow C f is the flow of (dend by) clss use on pth k between O-D pir r nd s; S f uses link, nd otherwise; is the trvel cost of clss use on pth k between O-D pir r nd s; nd v v v ( x) = E[in{ C } C ( x)] (5) k K 3

4 Note tht eqution (5) is the expected vlue of the perceived iniu trvel cost. It is concve with respect to C v (x) (peritting the iniiztion proble to hve unique solution), nd its fit derivtive is the following: S ( C v ) = P C, (6) where P is the probbility of clss use choosing pth k between r nd s. The pth choice probbility, P, is ssued to depend on the utility function: U = θ C + ε, (7) where θ is behviorl preter nd ε is rndo ter. If ε follows Gubel distribution nd is iid for ll routes of interest 3 (in ech individul trveler s pth choice decision), route choice follows logit odel. If ε follows norl distribution, choice confirs to probit odel. Both re vlid, nd, in fct, the result should hold for ny concve behviorl odel of route choice. In ny specifiction, the following should hold: P = Pr[ ε ε j θc + θc j j K ]. (8) The VOTT dispeion ong use cn be odeled using continuous rndo vribles (with known en vlue) within ech clss, or fixed vlue per clss. Without loss of generlity, single fixed-vlue pproch (per clss) is used here 4. Assuing trvel cost dditivity, pth cost is: C VOTT t( x δ τ, (9) δ, =, ) + where τ is the toll chrged on link ; nd VOTT is the VOTT of clss use. The fit derivtive of the objective function with respect to the pth flow yields: v Z( f ) ' = ( q P + f ) δ, t ( x ), (1) f which ust be zero t sttionry equilibriu point, nd thus f = q P, which is the SUE condition. This SUE odel cn be extended to cses of vrible dend between O-D pi. If dend functions D re nonnegtive nd strictly decresing in own-pth cost, then q = D ( S ) nd S 1 ( = D q ). Ros nd Mher (22) proposed n SUE odel with elstic dend tht cn be forulted s the following unconstrined optiiztion proble: v v x 1 v in Z( x, q) = t ( x ) x t ( w) dw + D ( q ) D S ( x) v v x, q S v ( x) D q v 1 1 ( S ( x) ) + D ( q) dq q D ( q ). ( ) (11) where ll vribles re s defined erlier. Additionlly, x ' t( x) x = ( t ( ) ( )) w + wt w dw. (12) 4

5 Yng (1999) deonstrted, bsed on econoic benefit xiiztion, tht the MCP principle is still pplicble in network under SUE. As in Yng et l. (24) nd other works, * the optil tolls, τ, for hoogenous use, re set equl to totl rginl costs s follows (ssuing single, unitry VOTT): * * ' * τ = xt( x), (13) * ' where x is the optil flow level (such tht rginl cost equls rginl benefits) nd t is the derivtive of trvel tie with respect to flow on tht link. Of coue, eqution (13) is not pplicble when considering heterogeneous use with different VOTTs. Insted, the optil toll is the dend (flow)-weighted verge, representing the true rginl cost of n dditionl rod user: * ' * fk δ, VOTTxt ( x) * τ =. (14) f δ k, Yng nd Hung (24) suggested tht when link flow-weighted VOTTs re used in deterining n MCP link toll pttern to support syste optiu s UE flow pttern, the results will differ when the optiu is esured in units of cost veus tie. Here the cost (onetry) unit is used. The objective function in eqution (11) is esured in tie units nd needs to be rewritten. After substituting (14) into (9) to get C nd then S v (x) for the heterogeneous use conditions, the resulting ulti-user SUE link flows with different VOTTs nd dend ptterns cn be identified by solving the following unconstrined optiiztion proble: v v x 1 v in Z( x, q) = VOTT δ t ( x ) x VOTT δ t ( w) dw + VOTT D ( q ) D S ( x) vv xq,,, v v + q ( ) 1 1 S ( x) D S ( x) VOTT ( ) ( ) D q dq qvott D q ( ) (15) Essentilly, the flow solution to optiiztion proble (15) results in the theoreticlly optil tolling sitution provided by eqution (14) s tolls. This is the MCP odel objective under SUE nd with heterogeneous use. However, the dend functions D(q) re not known in prctice, coplicting the solution for tolls nd flows. A tril-nd-error procedure for coputing these given observed flows nd flow responses to toll choices, bsed on Li (22) nd Yng et l. (23), provides the reedy. 3. TRIAL-AND-ERROR PROCEDURE One lgorith for solving SUE probles is the ethod of successive verges (MSA), which cretes weighted cobintion of the flow vlues of the current itertion nd the previous 1 itertion. The weighting fctor, or step size, t ech itertion n is α = (Sheffi, 1985); nd the n MSA solution ethod hs been proven to converge to the unique solution (Sheffi nd Powell, 1982). Ros nd Mher (22) showed tht the MSA pplies to eqution (11) s proble nd is gurnteed to converge in tht ppliction. In the bsence of dend functions D, Li (22) proposed n itertive, tril-nd-error 1 procedure. Fit, n initil known/observed or desired/trgeted dend q v is loded onto the 5

6 1 network with resulting flow pttern x v v v 1 v v 1 v1 nd tolls re set to τ = MC( q ) AC( q ), where verge trvel cost is the su of tie nd toll costs: v1 1 AC ( q ) = VOTTδ, t( x ) + δ, τ, nd rginl cost is the totl dditionl trvel cost k k v v v T v v AC( q ) q iposed by the rginl user: MC( q) v. Once v 1 τ is iposed, the reveled trffic q 1' 1 dend becoes q v v < q 5, suggesting v tht the initil tolls re too high. By djusting the dend 1' v 2 v1 to new level, for exple, q < q < q, the trget toll is reclculted nd the resulting dend 2' level q v 2 is copred with q v. This genertes n itertive procedure for optil toll clcultions. Of coue, network flows increse when dends rise, nd these flows cn be used s indicto of the unknown dends. Through electronic identifiction of rod use (e.g., vi siple trnsponde), rod nge cn scertin who is using prticulr link t ny tie nd estite their VOTTs, bsed on prior responses to pricing odifictions. This is lwys done with soe uncertinty, of coue 6. But, ssuing tht nge know the VOTT of those on ech link t ny oent in tie, optil tolls cn be clculted using eqution (14). Therefore, the itertive clcultion cn be ipleented without knowing the dend function. To solve the optiiztion proble (15), one needs to know the dend function nd its invee. Here, siilr pproch s Li s (22) nd Yng et l s (24) tril-nd-error procedure is developed to find the optil link flows nd tolls defined by eqution (14). One ust solve for the SUE ssignent with ultiple user groups fter iposing tril set of link tolls. Step. (Initiliztion) Set n initil set of fesible link flows{ x, A}. Let n=1. Step 1. (Clcultion of Toll) For ech link, clculte link toll using: ( n) *( n) ' *( n) f k δ, VOTT x t ( x ) *( n) τ =. ( n) f δ k, ( Step 2. (Observing flows) Observe the reveled link flows { x n )', A} ( n)' x x Step 3 (Convergence check). If ( n) x stop. Otherwise, proceed to the next step. ( n) fter iposing the tolls. φ (where φ is pre-defined sll vlue), then Step 4 (Link flow updtes) Updte link flows s follows: ( n+ 1) ( n)' ( n) ( n)' ( n) x = x + α ( x x ), A, (16) nd set n=n+1. Go to Step 1. Yng et l (24) proved tht this tril-nd-error procedure does converge nd converges to the optil solution, if the dend functions is onotoniclly decresing function of trvel cost nd re differentible. A proof tht the MCP odel with SUE nd heterogeneous use converges will be siilr. While fully, forl proof is not presented here, brief discussion of three of Yng et l s (24) propositions is helpful to undetnding the siilrity nd sequence. Fit, it is esy to see tht t the convergent solution to the MCP odel with SUE nd heterogeneous use, the corresponding flow nd toll ptterns re the optil solution to the 6

7 optiiztion proble (15). A ulti-clss SUE with elstic dend (15) nd without constrints is strictly convex proble, so its iniu is unique (Ros nd Mher, 22). Second, by ssuing tht (1) dend functions re nonnegtive nd strictly decresing in own-pth costs nd (2) link cost functions re positive nd onotoniclly incresing with link flows, one cn prove tht tril-nd-error s updting procedure genertes fesible descent direction for the objective function. Finlly, given these two prior prepositions nd the fct tht (1) the flow pttern is bounded nd (2) the initil dend nd flow pttern re fesible nd not t the boundry, the MSA ethod does converge for link flows (Sheffi nd Powell, 1982). Therefore, the tril-nd-error itertions will converge to the optil flow nd tolling solution. A prcticl explntion for the success of this rther siple tril-nd-error procedure seeking optil tolls cn be surized s follows. Assuing tht there is trnsporttion fcility opertor who seeks to chrge rginl-cost tolls on ll rod sections (i.e., links) in the network, he/she will fit obtin (vi observtion nd/or estition) flow volues nd link use infortion, including VOTTs. Then he/she chrges eqution (14) s tolls, which result in higher tolls for higher flows. Since different use hve different VOTTs, the overll rginl cost of dditionl use is function of ll current use VOTTs, nd their representtion on the link. Therefore, the MCP toll is flow-weighted verge of VOTTs. After the tolls re pplied, soe use will shift route choices nd forego trips. Overll, flows re likely to fll. The opertor observes the new flow levels nd use nd coputes new set of optil tolls, using eqution (14). After nuber of itertions pplying nd observing the results of these tolls, the opertor finds tht flows hve stbilized, nd optil tolls re in effect. This procedure is bsed on severl ssuptions, of coue. It will be hrd to know the link use different VOTTs with significnt certinty, especilly in the short run. Moreover, use cnnot ke optil decisions without sufficient forewrning of link tolls, nd trvel dend is constntly shifting nd lwys soewht unknown, even for the se tie of dy on the se dy of the week one week prt. Thus, this prcticl pproch to optil tolling y never chieve true optiu. However, this is the sitution operto re likely to be in, nd recognition of heterogenous use nd SUE route choice behvior certinly dds relis to Yng et l s (24) pricing proposl. 4. NUMERICAL EXAMPLE To illustrte the results of this MCP ethod, the pproch is pplied to Austin, Texs s rod network (see Figure 1) with 14,491 nodes, 31,34 links, 1,74 trffic nlysis zones, nd two user clsses. All links re subject to MCP tolls. The link perfornce function is bsed on the stndrd Bureu of Public Rods forul (BPR, 1964): 4 x t ( x ) t = , A, G (17) where t is the free-flow trvel tie on link nd G is the cpcity (which y be levelof-service flow rte, rther thn true cpcity). Then the link tolls re clculted by: τ f δ VOTT.6t x k, * G = fk δ, 4, A. (18) 7

8 The initil O-D dend trices, for network ssignent, re borrowed fro the region s plnning odel for orning pek hour pssenger trips. 7% of the dend eleents re ssigned to the fit user clss nd 3% to the second user clss. The VOTT of the two user clsses ( = 1, 2) is ssued to be $.15 nd $.25 per inute (or $9 nd $12 per hour), respectively. In order to predict the resulting link flows, Yng et l (24) solved n elstic dend ssignent exple (s in eqution (15)); nd, of coue, this required knowledge of the dend functions. Siilrly, n exple ppliction is puued here. The dend functions re specified s the following: ( n) D = D exp(.3c ), for ll r, s,, (19) where D is the initil dend eleent for user clss between r nd s. Yng et l (1994) pointed out tht O-D trix estition fro UE (or SUE) solution is not unique. However, s long s the estited O-D flows yield the current link flows (which re criticl to toll clcultions), the uniqueness of the O-D trix is not concern in this pper. TrnsCAD (Cliper Co., 1996) ws used here for the ulti-clss SUE trip ssignent with different VOTTs. In the ulti-clss SUE ssignent, logit odel of route choice is used (see Dil, 1971). Averge perceived link costs re ssued to equl trvel ties ultiplied by VOTTs, for ech of the two user groups; nd their iid Gubel error ters re ssued to hve loction preter of nd scle preter of 1. By siultion, the link choice probbilities were clculted, then the flow ssigned to the network, one user clss t tie. Once the flow solution ws obtined, the tolls were clculted nd the O-D trix for ech use clss ws recoputed nd re-ssigned to the network, with new/updted tolls on ll links. The convergence threshold ws set t φ =. 1, nd two different initil-dend ssuptions were tested, one with n ctul dend trix nd nother with hlf of the ctul dend trix. Both cses converge using the MSA lgorith fter bout 5 itertions. The resulting tolls on the four selected links re shown in Tble 1, nd they re virtully identicl (differing due to rounding erro). This suggests tht the clculted MCP tolls re independent of the initil dend vlues. Using the originl ctul dend trix, two step sizes (α ) were tested. The choice of ( n) 1 1 α = is equivlent to stndrd MSA lgorith, while vlue α = serves s check on n 2 Li s (22) bisection lgorith. The convergence ptterns re shown in Figure 2. As one cn see, with φ =. 1, the MSA converged t the 51st itertion. The bisection lgorith did not converge, even fter 1 itertions. This confirs the nuericl results of Yng et l (24). 5. CONCLUSIONS This pper expnds upon Yng et l s (24) odel nd lgoriths to clculte optil tolls with SUE nd heterogeneous use without knowing network dend functions, but knowing VOTTs of ll use. Essentilly, the se tril-nd-error procedure ws shown to be pplicble in ipleenting this ore relistic MCP odel, nd nuericl exple with n ctul, lrge-scle network suggests tht the tril-nd-error procedure does converge when using the MSA lgorith but does not when using the bisection lgorith. This is consistent with Yng et l s (24) results for cse of UE nd hoogeneous use. This pper lso discusses proof of convergence of the heuristic, tril-nd-error procedure, for n SUE ppliction with heterogeneous use. Mking stndrd ssuptions of 8

9 nonnegtive, onotonic cost nd dend functions, nd uniqueness in SUE solution with elstic dend, the proof should be very siilr to Yng et l s (24). To extend both these works, it would be useful to hve link costs interct with other links flows (recognizing flow interctions t inteections, for exple) nd to be ble to ipose tolls only on selected links. In ddition, the ssuption of known VOTTs y not be relistic. While infortion linked to trnsponde or vehicle identifiction tgs will ke such estites possible (bsed on vehicle type, route selection, tie of dy, nd pst trvel choices, for exple), it would be interesting to estite such VOTTs within the context of the toll-setting lgorith, bsed on route choice decisions, in the presence of tolls nd in rel tie, nd it would be helpful to evlute the suitbility of solution serch lgoriths other thn the MSA. The MCP odel with SUE nd heterogeneous use provides reltively relistic frework for studying optil tolls, even when O-D dend functions re unknown. The trilnd-error ethod proposed here is strightforwrd nd likely to be of iedite use in plces like Singpore nd elsewhere, if the objective of rodwy operto is true rginl cost pricing. This work lso deonstrtes the pplicbility of coercilly vilble trnsporttion softwre (TrnsCAD) for ipleenting the odel lgoriths to lrge-scle network. The results point to n ttrctive future for prcticl pplictions of MCP. ACKNOWLEDGEMENTS The utho re grteful to nonyous reviewe of this work for their coents nd suggestions. ENDNOTES 1 The coon ter for link dend in the network equilibriu literture is flow. However, flows cnnot exceed cpcities, while dend lods routinely do, resulting in queuing nd significnt delys. Here the coon ter flow will be used. This will help void confusion with the dend vrible q, which signifies totl dend between zones, rther thn link-specific dends. 2 Dend for trvel is ssued to depend on the origin nd destintion only, nd the cost between the, regrdless of other trvel pttern costs. This ssuption is stndrd in network odels but is unfortuntely unrelistic, since trvele often cn (nd do) substitute destintions (nd even origins) in ny of their trvel decisions. 3 In relity, route vritions re unlikely to be independent, prticulrly when soe routes overlp ore thn othe. There hs been soe work on such specifictions where routes re correlted, bsed on logit frework (e.g., Koppeln nd Wen 2). 4 Rede y refer to Cntrell nd Binetti (1998) nd Yng nd Zhng (22) for ore detiled discussion of VOTT distribution ssuptions, nd their role in the forultion. 5 In relity, fter iposition of tolls, soe links y exhibit higher dend, due to substitution. The ssuption of dend D depending only on O-D pir s trvel ties results in this reduced-dend result, nd is stndrd odel shortcoing. 6 Rod use cn trde toll tgs/identifiction units ong theselves, soe use re new/unknown, nd user s VOTT cn chnge (fro dy to dy nd fro one trip to the next). 9

10 REFERENCES Bureu of Public Rods (BPR) (1964). Trffic Assignent Mnul. U.S. Dept. of Coerce, Urbn Plnning Division, Wshington D.C. Cliper Co, (1996) TrnsCAD User s Guide. Cntrell, G.E. (1997) A Generl Fixed-Point Approch to Multi-ode Multi-User Equilibriu Assignent with Elstic Dend. Trnsporttion Science 31, pp Cntrell, G.E. nd M. Binetti (1998) Stochstic Equilibriu Trffic Assignent with Vlue of Tie Distributed Aong Use. Interntionl Trnsctions in Opertionl Reserch, 5(6), pp Dgnzo, C.F. (1983) Stochstic Network Equilibriu with Multiple Vehicle Types nd Asyetric, Indefinite Link Cost Jcobins. Trnsporttion Science 17, pp Dgnzo, C.F. nd Y. Sheffi (1977) On Stochstic Models of Trffic Assignent. Trnsporttion Science, 11(3), pp Dferos, S.C. nd F.T. Sprrow (1968) The Trffic Assignent Proble for Generl Network. Ntionl Bureu of Stndrds. Journl of Reserches. 73B, pp Devrjn, S. (1981) A Note on Network Equilibriu nd Noncoopertive Ges. Trnsporttion Reserch, Prt B. 15(6), pp Dil, R.B. (1971) A Probbilistic Multi-Pth Trffic Assignent Algorith Which Obvites Pth Enuertion. Trnsporttion Reserch, 5 (2), Frnk, M. nd P. Wolfe (1956) An Algorith for Qudrtic Progring. Nvl Reserch Logistics Qurterly 3 (1-2), pp Koppeln, F.S., nd C-H. Wen (2) The pired cobintoril logit odel: properties, estition nd ppliction. Trnsporttion Reserch 34B (2), pp Li, M. Z. F. (22) The Role of Speed-Flow Reltionship in Congestion Pricing Ipleenttion with n Appliction to Singpore. Trnsporttion Reserch 35E, pp Mher M.J. nd X. Zhng (2) Forultion nd Algorith for the Proble of Stochstic User Equilibriu with Elstic Dend. 8 th EURO Working Group Meeting on Trnsporttion, Roe, Septeber 2. Ros, A. nd M. Mher (22) Stochstic User Equilibriu Trffic Assignent with Multiple User Clsses nd Elstic Dend. The Proceedings of the 13 th Mini-Euro Conference nd 9 th Meeting of the Euro Working Group on Trnsporttion, Bri, Itly. Sheffi, Y. (1985) Urbn Trnsporttion Network, Prentice-Hll. Englewood Cliffs, NJ. Sheffi, Y. nd W. B. Powell (1982) An Algorith for the Equilibriu Assignent Proble with Rndo Link Ties. Network, 12(2), pp Sith, M. J. (1979) The Existence, Uniqueness nd Stbility of Trffic Equilibriu. Trnsporttion Reserch, 13B, pp Vickrey, W. S. (1969) Congestion Theory nd Trnsport Investent. Aericn Econoic Review 59 (2), pp

11 Wrdrop J G. (1952) Soe Theoreticl Aspects of Rod Reserch. Proceedings of the Institute of Civil Enginee. (2), pp Yng, H. (1999). Syste Optiu, Stochstic User Equilibriu, nd Optil Link Tolls. Trnsporttion Science 33(4), pp Yng, H. nd H. Hung (24) "The Multi-Clss, Multi-Criteri Trffic Network Equilibriu nd Syste Optiu Proble, Trnsporttion Reserch, 38B, pp Yng, H. nd Zhng, X. (22) The Multi-Clss Network Toll Design Proble with Socil nd Sptil Equity Constrints. Journl of Trnsporttion Engineering, 128(5), pp Yng, H., Y. Iid, nd T. Sski (1994) The Equilibriu-Bsed Origin-Destintion Mtrix Estition Proble. Trnsporttion Reserch, 28B, pp Yng, H. nd H.J. Hung (1998) Principle of Mrginl-Cost Pricing: How Does It Work in Generl Network? Trnsporttion Reserch, 32A, pp Yng, H., Q. Meng, nd D-H. Lee (24) Tril-nd-Error Ipleenttion of Mrginl-Cost Pricing on Networks in the Absence of Dend Functions. Trnsporttion Reserch, 38B, pp

12 LIST OF TABLES AND FIGURES Tble 1. Clculted Tolls on Selected Links nd their fter 51 Itertions Figure 1. Network Used for Nuericl Exple: Austin, Texs Rodwys Figure 2. Convergence of the Itertive Procedure, using Two Step-Size Rules Tble 1. Clculted Tolls on Selected Links nd Their Flows fter 51 Itertions Initil Dend =.5 Actul Initil Dend = Actul Dend Selected Links Dend Link Toll ($) Flow (vph) Link Toll ($) Flow (vph) 1 (US 183 North) (IH 35) (US 29 E) (Loop 1 S)

13 Figure 1. Network Used for Nuericl Exple Austin, Texs Rod Network Figure 1. Network Used for Nuericl Exple: Austin, Texs Rodwys 13

14 Convergence Distnce Step Size=1/n Step Size=1/ Itertion Figure 2. Convergence of the Itertive Procedure, using Two Step-Size Rules 14