DIFFERENT POLICY OBJECTIVES OF THE ROAD AUTHORITY IN THE OPTIMAL TOLL DESIGN PROBLEM

DIFFERENT POLICY OBJECTIVES OF THE ROAD AUTHORITY IN THE OPTIMAL TOLL DESIGN PROBLEM Dusic Joksimovic*, Michiel Bliemer Delft University of Technology, The Netherlnds ARS Trnsport &Technology, The Netherlnds 1. INTRODUCTION In this pper we re deling with the optiml toll design problem for plnning purposes. Uniform, (qusi)uniform nd vrible (time-vrying) tolls during the pek re considered (see Section.1.), in which trvellers responses (route nd deprture time choice) to these tolls re tken into ccount. Chrging my in generl led to lower trvel demnd s well, however for simplicity the totl number of trvellers is ssumed constnt in this pper. We will consider tolling schemes in which the rod uthority hs lredy decided on which links to toll nd which periods to toll. Then for different tolling ptterns (e.g. uniform or time vrying), the im is to determine optiml toll levels given certin policy objective, such s minimizing totl trvel time or mximizing totl revenues. The focus of this pper is to describe the dynmic frmework of the optiml toll design problem nd to mthemticlly formulte the problem. Furthermore, it will be illustrted tht different objectives of the rod uthority nd different tolling schemes cn led to different optiml toll levels. The min contributions of this pper re the following. First, different objective functions of the rod uthority re explored, nmely mximizing revenues nd minimizing totl network trvel time. Secondly, different tolling schemes nd their impct on the objective of the rod uthority re nlysed. Finlly, dynmic insted of sttic trffic ssignment model with rod pricing is proposed. Dynmic pricing models in which network conditions nd link tolls re timevrying, hve been ddressed in Wie nd Tobin (1998), compring the effectiveness of vrious pricing policies (time-vrying, uniform, nd step-tolls). A limittion of these models is tht they re restricted to bottleneck or single destintion network. Mhmssni nd Hermn (1984) nd Ben-Akiv et l. (1986) developed dynmic mrginl (first-best) cost pricing models for generl trnsporttion networks. As indicted by these uthors, the ppliction of their models might not be esy to implement in prctice. Moreover, since tolls re bsed on mrginl cost pricing, it is implicitly ssumed tht ll links cn be priced dynmiclly, which is prcticlly infesible. In the work of Viti er l. (003) dynmic congestion-pricing model is formulted s bi-level progrmming problem, in which the prices re llowed to ffect the (sequentilly) modelled route nd deprture time choice of trvellers. Abou-Zeid (003) developed some models for pricing in dynmic Assocition for Europen Trnsport nd contributors 006

trffic networks. In the work of Joksimovic et l. (005), the time-vrying pricing model including route nd deprture choice is solved using simple lgorithm for the rod uthority s objective of optimising totl trvel time in the network. In this pper these models re extended to include different nd more generl tolling schemes nd different objectives of the rod uthority. The reminder of this pper is s follows. In Section, the problem formultion of the time-vrying optiml toll design is described. The rod uthority objectives re mthemticlly formulted in Section 3. In Section 4, solution procedure to the optiml toll design problem is proposed. Cse studies for both minimizing trvel time nd mximizing revenue objectives re provided in Section 5. This pper finishes with conclusion prt (Section 6).. OPTIMAL TOLL DESIGN PROBLEM In the optiml toll design problem described in this pper, the links nd time periods to toll re given, nd the im is to determine optiml toll levels for different tolling ptterns given specific objective of the rod uthority (such s minimizing congestion, mximizing totl toll revenues, mximizing ccessibility, mximizing socil welfre, etc.). The resulting rod-pricing scheme describes for ech link nd ech time period how much trveller hs to py for entering the link t tht time. Bi-level optiml toll design problem Tolling strtegies Policy objectives Upper level Rod pricing design tolls flows Lower level Trvel behvior Figure 1: Frmework of the bi-level optiml toll design problem The model frmework (see Figure 1) consists of two min prts, nmely rod pricing prt nd trvel behviour prt. This frmework essentilly describes bi-level problem. The upper level is the rod pricing problem given trvel responses, while the lower level is the trvel behviour prt for given toll levels. Both prts will be explined in more detil in the following subsections. Assocition for Europen Trnsport nd contributors 006

In the rod pricing model, the rod uthority ims to introduce the best tolling scheme depending on its gols. A tolling scheme is defined s pckge in which set of links nd time intervls is chosen to toll, together with the toll levels corresponding to certin time-vrying tolling ptterns. Rod uthorities my hve different gols, leding to different objective functions in the model. Depending on its gol, the rod uthority hs to select the best tolling scheme. We ssume tht the links to be tolled, A, re given, s well s the tolled time period, T, for ech tolled link. First, some possible link tolling ptterns will be described. Secondly, short description of the trvel behviour prt will be given..1 Link tolling ptterns Different link tolling ptterns over time cn be considered by the rod uthority. As illustrted in Figure, we distinguish (i) uniform tolling scheme (toll levels re constnt over the entire study time period T), (ii) qusiuniform tolling scheme (tolls levels re constnt over specified time period nd zero otherwise), nd (iii) vrible tolling scheme (tolls levels re timevrying). T T T T T uniform qusi-uniform vrible Figure Different tolling schemes with respect to time of dy The three different tolling ptterns cn be formulted s follows: Vrible: φ() t θ, if A, t T; θ ( t) = where 0 φ ( t) 1. (1) 0, otherwise, Qusi-uniform: Uniform: θ, if A, t T; θ () t = 0, otherwise. () θ, if A, t T; θ () t = 0, otherwise. (3) Where Assocition for Europen Trnsport nd contributors 006

θ (): t vrible toll vlue on link t time t, where A, t T φ (): t predefined function over time for ech tolled link θ : mximum toll to be pid t link A T : tolled time intervls t link A In ll cses, there is only single toll level θ to be determined for ech tolled link A, which indictes the mximum toll to be pid for tht link. Clerly, the qusi-uniform tolling pttern is specil cse of the vrible tolling pttern (ssuming tht φ () t = 1, t T ), while the uniform pttern is specil cse of the qusi-uniform pttern (by furthermore ssuming tht T = T ). In cse of uniform tolls, the toll levels for ll time intervls re set to toll level θ for tolled link. In cse of qusi-uniform tolls, only the tolls in the time intervls t T will be set to the toll level θ, nd will be zero outside tht time period. For vrible tolls we ssume there is given predefined function φ () t over time for ech tolled link. In other words, the proportions of the time-vrying tolls re fixed (hence, the shpe of the toll levels over time is given). All three toll ptterns will be used in the cse study in this pper (Section 5). A tolling scheme indictes combintion of tolling pttern nd corresponding mximum toll levels, hence describing the following vribles: () tolled links A (ssumed given); (b) tolled time intervls T, A (ssumed given); (c) link tolling pttern φ (), t A, t T (ssumed given); (d) mximum toll levels θ, A, t T (to be optimised). This mens tht for ech tolling scheme, toll levels θ () t re known for ech link nd ech time intervl t. In this pper the tolled links, the tolled time intervls, nd the link tolling ptterns re ssumed to be input. The rod uthority ims to find the set of optiml mximum toll levels θ * θ tht optimise some given objective. The next section (Section 3) will further discuss these rod uthority s policy objectives.. Modelling of trvel behviour For modelling trvel behviour frmework described in Joksimovic et. l. (006) will be used. Here we will stte only the generl pth cost function. Assocition for Europen Trnsport nd contributors 006

The generl pth cost function is extended to cpture tolls, heterogeneous users nd deprture time choice. ( k) k k k ( k) c rs ( k) = α τ rs + θ rs ( ) + β ς rs + χ + τ rs ξ rs (4) mp m p p p where rs c ( k): experienced generlized trvel cost for trveller m on pth p between mp origin r nd destintion s deprting in period k ; α : vlue of time for trveller clss m; τ m rs p ( k ) : trvel time on pth p t time t between origin r nd destintion s if deprting in period k; β, χ : deprture time prmeters rs rs ς, ξ : preferred deprture nd preferred rrivl time respectively for trip rs (equl for trveller clsses); For modelling of route nd deprture time choice MNL model is used. In this frmework we suppose tht tolls re dditive, s well s trvel time. It should be noted tht stochstic user-equilibrium (SUE) model is used. The dynmic link trvel time function is used: τ 0 () t = τ + bx() t where τ (): t Trvel time on link for trvellers entering t time t τ 0 : free-flow trvel time x (): t inflow in link entering t time t b : dely prmeter Eqn. (5) reltes for ech link the number of vehicles x () t to the trvel time τ () t on tht link s n incresing function, where ech link hs free-flow trvel time nd dely component (with b nonnegtive prmeter). For more informtion bout the modelling of the lower level of the optiml toll design problem, see Joksimovic et l. (006). 3. ROAD AUTHORITY OBJECTIVES For our experiments, two different objectives re chosen, nmely (i) mximiztion of totl toll revenues, nd (ii) minimiztion of the totl trvel time. The rod uthority seeks to optimise its objective by selecting the optiml mximum toll levels i.e. Assocition for Europen Trnsport nd contributors 006

Where * θ = rg min Z( θ ). (6) θ Θ θ * : vector of optiml mximum toll levels Z( θ ): the objective function to be optimised The set of fesible mximum toll levels, which typiclly includes upper nd lower bounds nd for ech tolled link, cn be expressed s follows: Where min mx { θ : θ θ θ, }. Θ : set of fesible mximum toll levels Θ= A (7) min θ : lower bound for ech tolled link A mx θ : upper bound for ech tolled link A For ech evlution of the objective function Z( θ )(Eqn, (5)) trvel behviour prt (modelled s dynmic trffic ssignment (DTA) problem) hs to be solved (see Figure 1). The totl revenues on the network re product of the inflows into tolled links nd the corresponding toll levels vlid t the link entrnce times. Given some mximum toll levels θ, the objective function (Eqn. (7)) describes the totl toll revenues to be mximized. Zrevenue( θ ) u( t) θ( t), = t (8) where Zrevenue( θ ): objective function of mximising revenues Insted of mximizing totl toll revenues, the rod uthority my be more interested in minimizing totl trvel time (e.g. s proxy for minimizing congestion or pollution). Objective function Z ( θ ) time describes the totl trvel time on the network, Z ( θ ) u () t τ (), t time = t (9) Assocition for Europen Trnsport nd contributors 006

4. SOLUTION ALGORITHM TO THE OPTIMAL TOLL DESIGN PROBLEM Ech component of the optiml toll design problem cn be solved using vrious types of lgorithms. The outline of the complete lgorithm for the cse of vrible tolls is s follows. The lgorithm strts with specifying the grid of considered toll levels for ll links to be tolled, stisfying the constrints (lower nd upper bounds). In ech itertion, the lgorithm solves DTA problem (i.e. finds dynmic stochstic multiclss user equilibrium solution) bsed on the current toll levels nd sets new tolls tht cn potentilly optimize the objective functions described in Eqn. (7) or Eqn. (8). Becuse the lgorithm is grid-serch method it stops fter ll fesible toll levels in the grid hve been considered stisfying equilibrium level. At this stge of the reserch, the focus is minly to investigte the frmework of the model nd the properties of the solutions for different objectives nd tolling schemes, nd not on the development of lgorithms. More efficient lgorithms will be developed in the future reserch. The two-stge itertive grid-serch procedure for the optiml time-vrying toll problem with DTA (including joint route nd deprture time choice) cn be outlined. Firstly, the input nd output is described. Secondly, the steps of the lgorithm re outlined. Input: Network G = ( N, A), set of tolled links A, set of tolled time intervls link tolling ptterns φ (), t trvel demnd, logit scle pr grid dimensions I,meter, freeflow link trvel times, number of DTA itertions J, grid dimensions I, rod uthority objective (Eqns. (8),(9)) Output: Optiml mximum toll levels θ *, optiml vlue of objective function Z * rev or Z *. time Grid Serch lgorithm Outer loop: PRICING Step 1: [Initiliztion] The mximum toll level grid for ech link is given by ( ) ( i) min i mx min θ = θ + I θ, θ i = 0, K, I. Assocition for Europen Trnsport nd contributors 006

All combintions of ll mximum toll levels for ll links determine the () i () i set of grid vectors θ [ θ ], which contins I ( I + 1) elements. Set i : = 1 nd set Step : [Set toll vlues] * Z =+. Select grid point i for the toll levels, yielding tolls () i θ () t Inner loop: DTA Step 3: [Dynmic trffic ssignment] In this step the DTA model is solved. Initilistion ssumes n empty network nd free flow conditions. The trvel costs ccording to Eqn. (4) re computed. The new intermedite dynmic route flow pttern is determined nd updtes using Method of Successive Averges (MSA) method. Finlly, dynmic network-loding (DNL) model is performed nd the convergence criteri for the lower level of the optiml toll design problem is checked. For more informtion see Joksimovic (006). Step 4: [Compute objective function] Compute the objective function () i Z ( ) θ using Eqn. (8) or Eqn. (9). If Z( ) () i * θ < Z, then set * i Z Z ( θ ( ) ) = nd set θ = θ * ( i). Step 5: [Convergence of rod pricing level] While i < N, set i: = i+1 nd return to Step. * Otherwise, the lgorithm is terminted nd θ is the set of optiml toll levels. Performing this simple itertive procedure, we explore ll toll level combintions nd find the optiml vlue of the objective function. Regrding the convergence of this lgorithm, the inner DTA loop using the widely used heuristic MSA procedure typiclly converges to n equilibrium solution, lthough convergence cnnot be proven. In the outer rod pricing loop the whole solution spce is investigted with certin grid ccurcy (yielding finite number of solutions tht re evluted). 5. CASE STUDIES Assocition for Europen Trnsport nd contributors 006

5.1 Network description, trvel demnd nd input prmeters The solution procedure proposed in the previous section hs been pplied for illustrtive resons smll network, see Figure. link 1 link 3 route 1 link 1 3 route Figure Network description The network consists of just single OD-pir connected by two nonoverlpping pths where only link is tolled. Since there is only one OD pir, we will ignore the OD subindices (,) rs in the vribles. Two user clsses with different vlue of time (VOT) re distinguished. The totl trvel demnd for deprture period K = 1, K, 0 from node 1 to node 3 is ssumed 86, of which { } 50% re high VOT trvellers nd 50% low VOT trvellers. The following prmeter vlues re used on route level: preferred deprture time ς = 10 [period] preferred rrivl time ξ = 15 [period], vlue of time for clss 1 α 1 = 0.5 [eur/periods], vlue of time for clss α = 0.75[eur/periods], penlty for deviting from preferred deprture time β = 0.5 [eur/periods], penlty for deviting from preferred rrivl time γ = 1, nd scle prmeter in MNL model is 0.8 On the link level, we ssume tht route 1 with free-flow trvel time of 7.0 time units is longer thn route (3.0 time units) by setting 0 0 0 the free-flow link trvel times in Eqn. (5) to τ1 = τ3 = 3.5 nd τ = 3.0. Furthermore, it is ssumed tht the first route will never show congestion, hence b = b = while congestion is possible on link for which we set b = 0.005 1 3 0, in Eqn. (5). 5. Optiml tolls schemes In Section.1 three different tolling schemes hve been mentioned, nmely the uniform, the qusi-uniform, nd the vrible tolling scheme, see Eqns. (1) (3). All three tolling regimes will be considered in the cse studies below. Note tht in this cse study with only single link (link ) tolled, determining the optiml toll for ech tolling regime (even for the vrible tolling scheme) only requires to find single optimum toll level, θ *. For the qusi-uniform tolling scheme we ssume tht toll will be levied in the pek period, i.e. the tolling period is T = {8,9,10,11,1} in Eqn. (1). In the vrible tolling scheme only periods 9, 10, nd 11 will be tolled with fixed proportions 0.6, 1.0, nd 0.6, tht is Assocition for Europen Trnsport nd contributors 006

1.0, if t = 10; φ( t) = 0.6, if t = 9,11; 0, otherwise. (11) 6.3 Optiml tolls for mximizing totl revenues Assume tht the rod uthority ims to mximize totl revenues, s formulted in Eqn. (8), by selecting the best tolling scheme nd the best toll level. The three different tolling regimes s mentioned bove will be considered. For ech tolling scheme nd for ech toll level, the dynmic trffic ssignment (DTA) problem cn be solved. In Figure 4 the totl revenues re plotted for ech tolling scheme for ll 0 θ 15 (lthough not shown here, in ll cses the DTA model converged). Z revenue 70 Z = mx revenue 64.1 60 50 Tolling schemes uniform qusi-uniform vrible 40 30 0 10 0 0 * θ 5 10 15 = 3.11 θ Figure 4 Totl revenues for different tolling schemes nd toll levels Assocition for Europen Trnsport nd contributors 006

In cse the toll level is zero, there re clerly no revenues. For very high toll levels, ll trvellers will choose to trvel on the untolled route, resulting in zero revenues s well. As cn be observed from Figure 4, uniform tolling with θ = 3.11 yields the highest revenues. The vrible tolling scheme is not ble to provide high revenues due to the smll number of tolled time periods. 5.3 Optiml tolls for minimizing totl trvel time In this cse study, the rod uthority ims t minimizing totl trvel time on the network (see Eqn (9)) by selecting the best tolling scheme nd the best toll level. Figure 5 depicts the totl trvel times for different tolling schemes nd toll levels. Z time 60 600 580 560 Tolling schemes uniform qusi-uniform vrible 540 50 500 min Z = time 496 480 Figure 5 0 * θ 5 10 15 = 3.4 Totl trvel time for different tolling schemes nd toll levels θ As cn be observed from Figure 5, it seems possible to decrese the totl trvel time on the network by imposing toll on congested route. High toll levels will push ll trvellers during the tolled period wy from route to the * longer route 1, yielding higher totl trvel times. Vrible tolling with θ = 3.4 (yielding θ * (10) = * * 3.4 nd θ(9) = θ(11) = 1.99 in the periods 10,9,11 respectively, zero otherwise) ccording to Eqns. (1) nd (11)) results in the lowest totl trvel time. The objective function looks somewht irregulr which Assocition for Europen Trnsport nd contributors 006

cn be explined by the rounding off of the link trvel times in flow propgtion 5.4 Discussion Results of both objectives (mximizing totl toll revenues nd minimizing totl trvel time) where different tolling schemes (uniform, qusi-uniform nd vrible) re pplied re given in Tble 1 Comprison of toll revenues nd totl trvel time for different objectives Objective: mximize totl toll revenue 5olling scheme Optiml toll Totl revenue Totl trvel time Uniform 3.11 64.15 534.48 Qusi-uniform.8 49.50 503.4 Vrible 4.04 37.13 498.6 Objective: minimize totl trvel time Tolling scheme Optiml toll Totl revenue Totl trvel time Uniform.41 60.81 53.56 Qusi-uniform.7 45.56 497.91 Vrible 3.4 35.5 496.0 These results show tht in the cse of mximizing totl toll revenues the best tolling scheme is uniform with toll level θ = 3.11. However, this toll will yield high totl trvel time (534.48). On the other hnd, in the cse of minimizing totl trvel time, the vrible tolling scheme with θ = 3.4 performs best. However, this toll will yield low totl toll revenue (35.5). In other words, mximizing totl toll revenues nd minimizing totl trvel time re opposite objectives. This cn be explined s follows. In mximizing toll revenues, the rod uthority would like to hve s mny s possible trvellers on the tolled route, hence trying to push s few s possible trvellers wy from the tolled lterntive by imposing toll. In contrst, when minimizing totl trvel time, the rod uthority would like to spred the trffic s much s possible in time nd spce, hence trying to influence s mny trvellers s possible to choose other deprture times nd routes. Using uniform tolling scheme, trvellers re not chnging their deprture times, mking it suitble for mximizing revenues, while in the vrible tolling scheme other deprture times re good lterntives, mking it suitble for minimizing trvel time. In ny cse, depending on the objectives of the rod uthority, there re different optiml tolling schemes with different toll levels. Assocition for Europen Trnsport nd contributors 006

6. CONCLUSIONS A mthemticl bi-level optimistion problem hs been formulted for the optiml toll network design problem. The rod uthority hs some policy objectives, which they my optimise by imposing tolls. Second-best scenrios re considered in this pper, ssuming tht only subset of links cn be tolled. Different tolling schemes cn be selected by the rod uthority, such s (qusi-)uniform nd vrible tolling schemes, ech hving different impct on the policy objective. Due to tolls, the trvellers my chnge their route nd deprture times. Heterogeneous trvellers with high nd low vlue of time re considered. The im of the reserch is to investigte the fesibility of the dynmic model frmework proposed in this pper nd to investigte properties of the objective function for different objectives nd tolling schemes. The complex optimistion problem hs been solved using simple grid serch method, but for more prcticl cse studies more sophisticted lgorithms will be developed in the future. In the cse studies the pper shows tht policy objectives cn indeed be optimised by imposing tolls, nd tht different policy objectives led to different optiml tolling schemes nd toll levels. Keeping the totl trvel demnd fixed, introducing uniform (fixed) toll, trvellers cn only void the toll by route chnges, not by chnging deprture time, leding to higher toll revenues. On the other hnd, hving vrible toll enbles trvellers to void tolls by chnging deprture time, yielding lower totl trvel times. References Abou-Zeid, M. (003). Models nd Algorithms for the Optimiztion of Trffic Flows nd Emissions using Dynmic Routing nd Pricing, M.Sc. Thesis, Msschusetts Institute of Technology, Cmbridge, MA, USA. Ben-Akiv, M., A. De Plm, nd P. Knroglou (1986), Dynmic Model of Pek Period trffic congestion with Elstic Arrivl Rtes, Trnsporttion Science, 0(), 164-181. Bliemer, M.C.J. nd P.H.L. Bovy (003), Qusi-Vritionl Inequlity Formultion of the Multiclss Dynmic Trffic Assignment Problem, Trnsporttion Reserch B, 37, 501-519. Joksimovic D., M.C.J. Bliemer nd P.H.L. Bovy (005), Optiml Toll Design Problem in Dynmic Trffic Networks- with Joint Route nd Deprture Time Choice, Trnsporttion Reserch Record, 193, 61-7. Joksimovic D., M. C.J. Bliemer nd P.H.L. Bovy (006). Dynmic optiml toll design problem trvel behviour nlysis including deprture time choice nd heterogonous user, presented t the 11 th Interntionl Conference on Tvle Behviour Reserch, Kyoto, Jpn. Assocition for Europen Trnsport nd contributors 006

Mhmssni H.S. nd R.Hermn (1984), Dynmic User Equilibrium Deprture time nd route choice in Idelized Trffic Arterils, Trnsporttion Science, 18 (4), 36-384. Rn, B. nd D. E. Boyce (1996), Modeling dynmic trnsporttion networks: n intelligent trnsporttion system oriented pproch, Springer-Verlg, nd edition, Berlin, Germny. Verhoef E.T. (00), Second-best Congestion Pricing in Generl Sttic Trnsporttion Networks with Elstic Demnd, Regionl Science nd Urbn Economics Journl, 3, 81-310. Viti, F., S.F. Ctlno, M. Li, C.D.R. Lindveld, nd H.J. vn Zuylen (003), An optimiztion Problem with dynmic route-deprture time choice nd Pricing, proceedings of 8nd Annul Meeting of the Trnsporttion Reserch Bord, Wshington, D.C., USA. Wie, B., nd R.L. Tobin (1998), Dynmic Congestion Pricing Models for Generl Trffic Networks, Trnsporttion Record B, 3(5), 313-37. Assocition for Europen Trnsport nd contributors 006