Toll Pricing. Computational Tests for Capturing Heterogeneity of User Preferences. Lan Jiang and Hani S. Mahmassani

Transcription

1 Toll Pricing Coputational Tests for Capturing Heterogeneity of User Preferences Lan Jiang and Hani S. Mahassani Because of the increasing interest in ipleentation and exploration of a wider range of pricing alternatives, a study was done to copare approaches for capturing the heterogeneity of user preferences in forecasting the deand for tolled facilities resulting fro user responses to pricing schees. Existing tools used in practice typically deal with users value of tie (VOT) heterogeneity in one of two ways: (a) ignore VOT and use a constant value of tie and (b) recognize VOT by defining discrete user classes, each corresponding to a given VOT range, represented by its idpoint or average VOT. The first approach is fundaentally incorrect and would lead to highly biased estiates of network perforance; the second is only a coarse approxiation. A third approach, which recognizes that VOT is a continuous variable that is distributed probabilistically across the user population, is gaining greater attention. The following questions are exained: (a) What are the ipacts of different VOT assuptions on prediction biases in forecasts of toll road usage under different pricing schees? (b) Do the discrete treatents provide a good approxiation of odels with continuously distributed VOT, and if so how should VOT be assigned to user classes? (c) What are the relative coputational iplications of different VOT assuptions, and what is the role of efficient ipleentation techniques for largescale network applications? The results suggest that recognizing the continuous nature of the VOT distribution is justified both with regard to accuracy and on coputational grounds, especially in light of recent algorithic and ipleentation advances. To forecast the ipact of current sophisticated pricing schees, often in conjunction with operational easures, it is essential to capture two key eleents: (a) the responses of road users to these schees and (b) the resulting dynaics of traffic flow on the network. In capturing user responses in route and other traveler decisions, a generalized cost that cobines into a single easure the travel tie and out-of-pocket cost, that is, the toll incurred, is typically used. This is done by ultiplying the travel tie by corresponding values of tie (VOT) and thus converting the to cost units (oney). Existing practice in forecasting deand for toll facilities with conventional tools typically relies on a single (average) VOT for all road users; in soe cases different VOTs are assued for users of trucks and passenger cars. However, the value of tie is not a constant across travelers: it varies by trip Departent of Civil and Environental Engineering, Northwestern University, Transportation Center, 600 Foster Street, Evanston, IL Corresponding author: H. S. Mahassani, asah@northwestern.edu. Transportation Research Record: Journal of the Transportation Research Board, No. 2343, Transportation Research Board of the National Acadeies, Washington, D.C., 2013, pp DOI: / purpose (e.g., work versus nonwork trips), by socio deographics, and so on, even for the sae traveler at different ties and locations (1, 2). This is known as user heterogeneity people are different and have different preferences that result in different choices for pricing options. This heterogeneity is at the root of the econoic appeal and operational effectiveness of pricing at different prices, different responses are observed, and soe people are willing to pay the toll while others are unwilling in a direct reflection of their value of tie. Heterogeneity has profound iplications for the procedures used to predict users path, ode, and departure tie choices (3, 4). In particular, to the extent that finding least-generalized cost paths is required for the route choice assignent process, these paths ay be different for users with different values of tie, all else being equal. Solving for each and every user s least-generalized cost path would be coputationally prohibitive. Existing tools typically deal with this issue in one of two ways: ignore it and use a constant value of tie, or recognize it by defining discrete user classes that correspond each to a given VOT range, represented by its idpoint or average VOT. Yang et al. proposed an elastic deand ulticlass network equilibriu odel, which divided the whole road users population into a nuber of classes with an average VOT for each class (5). The odel was developed to investigate how VOT distribution affects traffic flow and profit forecasts of private toll roads. An advantage of their odel is its ability to assess the distribution of benefits generated by the toll road across classes of users between a given origin destination (O-D) pair. Later, Yang and Han suggested a ulticlass, ulticriterion traffic equilibriu assignent odel that used the sae logic of discrete VOT to analyze the effects of second-best tolls and derive bound for the toll levies (6). Yang and Zhang developed a unified fraework of ulticlass, ulticriteria user-equilibriu and Cournot Nash principle ixed equilibriu and found that unifor link tolls supporting such equilibriu not only exist but also lead to a syste optiu (7). Another approach for capturing heterogeneity is to recognize VOT as a continuous variable that is distributed probabilistically across the user population. Leurent was aong the first to propose a theoretical fraework for a cost-versus-tie (deterinistic static) equilibriu odel with variable deand, continuous lognoral distribution of VOT, and elastic travel tie functions (8). Cantarella and Binetti extended existing fixed-point odels for stochastic equilibriu assignent to deal with the heterogeneous VOT across users (9). Nielsen also cobined stochastic user equilibriu with continuous distributed VOT functions estiated on stated preference data to analyze the traffic condition for the Copenhagen region (10). All these forulations considered static networks with static prices. To reflect road users path choices in response to tie-varying toll charges with a siulation based algorith, Lu et al. presented a 105

2 106 Transportation Research Record 2343 bicriterion dynaic user equilibriu (BDUE) odel (4). At the core of the solution procedure, and its principal enabler as a practical ethodology for realistic networks, is a bicriterion path-finding algorith that does not require predefined user classes and instead paraetrically identifies the natural breakpoints in the VOT distribution for each tie-varying shortest path tree (11). Because the VOT establishes a connection between tie and oney, it plays a critical role in enabling the assignent odels, and ore generally the deand analysis tools, to provide policy akers with useful inforation for assessing the econoic and welfare ipacts of proposed road pricing facilities or schees. Ignoring heterogeneity by using a constant VOT is fundaentally incorrect and would lead to highly biased results; the discrete treatent, which is only a coarse approxiation, is difficult to calibrate a priori and ay becoe unwieldy coputationally. The third approach, which recognizes that VOT is a continuous variable that is distributed probabilistically across the user population, can capture greater realis in predictions of user responses to pricing but at the cost of higher coputational effort. The a priori discrete segentation, such as that proposed by Yang et al. (5) for the static equilibriu case, can be viewed as a discrete approxiation of the odel with continuously distributed VOT. Fro that perspective, several critical questions about solution quality, coputational perforance, and sensitivity of both solution quality and coputational perforance to the nuber of predefined user classes arise in connection with this approxiation. For instance, how any user classes ust be predefined so that the equilibriu odel can provide a good approxiation? How are VOT assigned to these user classes? How do these affect the resulting coputational effort? These questions were exained through coputational experients conducted on two actual networks of different sizes. The focus was on forecasts of toll road usage obtained under various user heterogeneity assuptions. The following questions were addressed: (a) What are the ipacts of different VOT assuptions on prediction biases in forecasts of toll road usage under different pricing schees? (b) Do the discrete treatents provide a good approxiation of odels with continuously distributed VOT, and if so how should VOT be assigned to user classes? (c) What are the relative coputational iplications of different VOT assuptions, and what is the role of efficient ipleentation techniques for large-scale network applications? Although the results cannot be generalized to all networks, they provide useful insight and guideposts for conteplating choice of ethodology to evaluate conteplated pricing schees. Following is a review of proble forulations for both continuous and discrete VOT cases, which is followed by an overview of the respective solution algoriths. In both cases the sae siulationbased network loading procedure is used, which controls for this eleent in the experients. The experiental setup and associated design are then described, followed by the results. Assuptions and Proble Stateent Consider a tie-dependent network G = (N, A) with a finite set of nodes N and a finite set of directed links (i, j), i N and j N of A. The period of interest (i.e., planning horizon) is discretized into a set of sall tie intervals, S = { 0, 0 + δ, 0 + 2δ,..., 0 + Mδ}, where 0 is the earliest possible departure tie fro any origin node, δ is a sall tie interval during which no perceptible change in traffic conditions or travel cost occurs, and M is a large nuber such that 0 to 0 + Mδ covers the planning horizon S. Without loss of generality, associated with each arc (i, j) and tie interval are two tie-dependent arc travel ipedance easures: tie (d ij (t)) and cost (c ij (t)) incurred when traveling fro node i to node j departing at tie t fro node i. Other notation is as follows: o = subscript for origin node, o O N; d = subscript for destination node, d D N; = superscript for departure tie interval, = 1,..., T; α = continuously distributed VOT, α = [α in, α ax ] and αax α in ϕ (α) = 1; β = discretized (ulticlass) VOT, β = [β 1,..., β M ]; = superscript for user class with β as corresponding VOT to user class ; P (o, d, ) = set of all feasible paths for given vehicle class and triplet (o, d, ); p = subscript for path p P (o, d, ); r = nuber of trips departing fro o to d at tie ; r = path flow vector; Ω = feasible set of path flow vector; GC = experienced generalized path travel cost departing fro o to d at tie that are assigned to path p P (o, d, ) evaluated by certain VOT; TT = experienced path travel tie for all trips departing fro o to d at tie that are assigned to path p P (o, d, ); and TC = experienced path travel cost (toll charges) for all trips departing fro o to d at tie that are assigned to path p P (o, d, ). The key behavioral assuption ade for the path choice decision is that each trip aker chooses a path that iniizes the generalized path cost functions. Regardless of the assuptions ade on heterogeneity about VOT, a path will be selected if and only if the following condition is satisfied: For ulticlass VOT: GC, *, ( β ) = in GC ( β ) (1) p P ( od,, ) For continuously distributed VOT: GC, ( α ) = in GC ( α) (2) * p Pod (,, ) Given this assuption, a bicriterion and dynaic extension of Wardrop s first principle (1952) is characterized in the usual anner as follows: For each origin destination (O-D) pair and for each departure tie interval, no trip aker can decrease the experienced path generalized cost with respect to that trip s particular VOT by unilaterally changing paths (12). This iplies that each trip aker is assigned to a path having the least tie-dependent generalized cost with respect to his or her own VOT. This study obtains tie-dependent path flows under a given pricing schee and assuptions on heterogeneous users in ters of their VOT. Model Forulation: Multiclass VOT Road users are divided into classes corresponding to groups with different socioeconoic characteristics (for exaple, incoe level). Let β denote the corresponding average VOT for users of class. Let r od be the deand of class between O-D pair (o, d),

3 Jiang and Mahassani 107 and r od = Σ M =1 r od. It is assued that r od is given and fixed and that the fixed-deand ulticlass traffic network equilibriu proble is being used. The generalized cost perceived by users of class for o to d at tie interval along path p P (o, d, ) is given by, GC ( β ) = TC +β TT where TT = d () t ij (,,) i j t p TC = c () t ( i,, j t) p ij Let Ω {r (β )} be the feasible set of path flow vectors r(β) satisfying the path flow conservation and nonnegativity constraints:, r = r od,, (6) od M = 1 od r od, 0 o, d,, (7) The ulticlass dynaic user equilibriu can be forulated as a variational inequality proble to find r* Ω such that T M, GC, r r, ( β, *) [ opd ( β ) r ( β )*] 0 (8) o O d D = 1 = 1 where r* is a ulticlass tie-varying DUE path flow vector and r, (β )* is the flow vector obtained by tie-varying iniu O-D generalized cost, evaluated at r* for the trips of vehicle class with the sae (o, d, ). Model Forulation: Continuously Distributed VOT The heterogeneous nature of the population is reflected by the treatent of the VOT as a continuous rando variable distributed across the user population with the density function ϕ(α) > 0, α [α in, α ax ], and αax α in ϕ(α) = 1, where the feasible range of VOT is given by the closed interval [α in, α ax ]. The distribution of VOT is assued known and could be estiated fro survey data. A paraetric for for the VOT distribution is not necessary, as discrete ass functions could be used instead. Alternatively, the distribution could be iplicit with individuals generated fro an activity-based deand icrosiulation process. The tie-dependent O-D deand for the entire feasible range of VOT for the planning horizon is also known a priori. (In practice, the O-D pattern and the VOT distribution will be considered independent of each other.) The experienced generalized cost perceived by trip akers with VOT α departing fro o to d in tie interval along path p is given by GC ( α ) = TC +α TT where TT = d () t ( i,, j t) p TC = c () t ( i,, j t) p ij ij (3) (4) (5) (9) (10) (11) Let Ω(α) {r(α)} be the feasible flow vectors r(α) satisfying the path flow conservation and nonnegativity constraints: r ( α ) = r ( α) od,, (12) od p P( od,, ) r (α) 0 o, d,, p Pod (,, ) (13) Lea 1. Solving for the flow pattern r* is equivalent to finding the solution of a syste of variational inequalities: r* (α) Ω(α) such that GC( α, rp) [ r( α) rp( α) ] 0 [ ] in ax r ( α) Ω( α), α α, α (14) The proofs of necessity and sufficiency that the solution of the preceding variational inequality satisfies the DUE conditions adapt the proofs of Lu et al. (4) and are not repeated here. Solution Algoriths Solution Algorith: Multiclass VOT A solution algorith is presented next that solves the ulticlass dynaic traffic assignent odel. Given the tie-dependent generalized costs, trip akers of all user classes choose the best path corresponding to their assigned VOT, which results in equilibriu path flows per user class. These flows propagate along the paths, yielding dynaic link flows. The propagation of these flows depends heavily on the link travel ties. The link travel ties, in turn, depend on the link flow rates; hence a fixed-point proble has to be solved. This proble can be solved with an algorith based on dynaic network loading, as depicted in Figure 1. Solution Algorith: Continuously Distributed VOT Lu et al. reforulated the variational inequality proble defined in Equation 14 via a gap function as a nonlinear iniization proble and provided a colun generation based approach for solving the proble (13). The colun generation based approach augents, in the outer loop, the subset of the feasible (extree efficient or nondoinated) paths and solves, in the inner loop, the restricted ulticlass DUE proble defined by the current subset of feasible paths. In each outer iteration k, the extree nondoinated pathfinding algorith the paraetric analysis ethod (PAM) proposed by Mahassani et al. (11) is activated. Essentially, PAM plays two iportant roles in solving the proble: (a) it obtains the breakpoints that partition the feasible ranges of VOT into subintervals and deterine the ultiple user classes, and (b) it finds the bicriterion tiedependent least-cost path (BTDLGCP) trees for each user class. New paths, if any, are added into the current path set. The algorith terinates if no new path is found for all user classes or if a preset convergence criterion is satisfied; otherwise the restricted ulticlass DUE proble is solved by adoption of the ulticlass path flow updating schee to equilibrate tie-varying and heterogeneous O-D deands on the current path set, before a return to the path-generation step (i.e., outer loop). A thorough presentation of the solution ethod and its ipleentation details is available elsewhere (11, 13).

4 108 Transportation Research Record 2343 FIGURE 1 Flowchart of unified algorith. Unifying Solution Algorith The steps of the unified solution algorith are as follows (Figure 1): Step 0. Initialization. Input: (i) tie-dependent O-D deands over the planning horizon, (ii) tie-dependent link tolls, (iii) VOT specification, and (iv) initial paths and path assignent. Set: (i) the iteration counter k = 0 and (ii) axiu nuber of iterations K ax. Step 1. Traffic siulation. Perfor dynaic network loading with the traffic siulation odel DYNASMART-P (14, 15) to evaluate the path assignent and obtain link travel ties and costs. Step 2. Generating path set. Search direction finding: Multiclass VOT: find the auxiliary least-generalized cost path evaluated by the assigned average VOT for each predefined user class. Continuously distributed VOT: evoke PAM to obtain the set of BTDLGCP trees, their corresponding least-generalized costs, and segents of VOT. Step 3. Update path assignent. Multiclass VOT: for each user class, update path flow by using the ethod of successive averages. Continuously distributed VOT: evoke restricted ulticlass DUE and deterine path flow by using the criterion-based selection technique (16). Step 4. Convergence checking. Multiclass VOT: the first convergence criterion is based on the coparison of path assignent over the ethod of successive averages proposed by Peeta and Mahassani (17), and the second is to check whether the iteration counter reaches the axiu nuber of iterations K ax. Continuously distributed VOT: the two convergence criteria are (a) if there is no new path found for all (o, d, ) triplets or (b) k = K ax. If either of the criteria is satisfied, the progra terinates; otherwise set k = k + 1 and go back to Step 1. Experiental Design and Model Results Test Network The test network, shown in Figure 2, is extracted fro the Fort Worth, Texas, network; it corresponds to one sector of I-35W between I-20 and I-30, and there is a surrounding network of signalized arterials and stop-controlled or yield-controlled local streets on both sides of the freeway. It consists of 180 nodes, 445 links, and 13 traffic analysis zones. The links shown in light blue in the figure are freeway links, and the green nubers represent the traffic analysis zones. The thickness of the red lines denotes the volue between a given O-D pair. A tie-varying O-D deand table for a 2-h (7:00 to 9:00 a..) weekday orning peak is extracted and loaded onto the network, having 37,760 vehicles in total.

5 Jiang and Mahassani 109 (a) (b) FIGURE 2 Forth Worth test: (a) network and (b) O-D deand volue. Experiental Design for Test Network A set of experients investigates the ipacts of tolled link coverage on PAM when finding the BTDLGCP trees and the cor - responding segents over the entire feasible range of VOT ([α in, α ax ]). The ulticlass VOT assuption can be regarded as a discrete approxiation of the odel using continuously distributed VOT. The question left unanswered pertains to the nuber of user classes that ust be predefined for the equilibriu odel to provide a good approxiation of the desired forecast quantities. Pricing Schees The siulation horizon is set to 120 in, and the axiu nuber of iterations is set to 13 for all experients in this section. Therefore, the hypothetical dynaic pricing schee divides 120 in into four periods with different toll levels, proportional to the lengths of the tolled links: Period 1. 7:00 to 7:30, $0.25/i; Period 2. 7:30 to 8:00, $0.50/i; Period 3. 8:00 to 8:30, $0.50/i; and Period 4. 8:30 to 9:00, $0.25/i. VOT Assuptions The continuous VOT distribution is assued to be truncated lognorally distributed with the feasible range of [0.3679, ], that is, α ~ Log N(3.1732, ). Multiclass segentation of VOT is created by dividing the entire feasible range into n segents and taking the average value of the end points to be the corresponding ean VOT for that class. Nuerical Results for Test Network User Responses to Pricing Schees Of 445 links, 29 are coded as freeway links on the tested network. Three sets of links are created to represent various tolled link coverage, and seven VOT specifications are ade, giving 21 experients in total.

6 110 Transportation Research Record 2343 TABLE 1 Results for Fort Worth Test Network allhighway a randselection6.5% a randselection15% b VOT Assuption Total Revenue Tolled Vehicles Total Revenue Tolled Vehicles Total Revenue Tolled Vehicles Continuous VOT 21, ,690 5, ,056 2, ,214 Mean 21, ,533 5, ,645 2, ,402 ulticlass 3 24, ,843 5, ,993 3, ,993 ulticlass 5 23, ,666 5, ,842 2, ,810 ulticlass 15 21, ,235 5, ,455 2, ,560 ulticlass 20 20, ,573 5, ,341 2, ,504 ulticlass 35 19, ,936 5, ,190 2, ,512 a 29 tolled links. b 67 tolled links. The ain results at Iteration 13, that is, the total revenue in dollars and the total nuber of vehicles passing through the tolled links, are presented in Table 1. As shown in Figure 2b, ost of the O-D deand is between Zones 1 and 2, connected directly by freeway links. This is why the allhighway case generates the ost total toll revenue. It is unexpected that randselection6.5% generates ore toll revenue and affects ore vehicles than predicted for randselection15%. In both cases, the tolled links are randoly selected. It turned out that randselection6.5% had five freeway links, but only two freeway links were selected in the randselection15% case. This observation suggested that a bad design for the pricing schee ay not satisfy the expected goals of road pricing, for exaple, revenue generation. However, the coverage of tolled links (both geographically and nuerically) has significant ipacts on PAM for solving the continuously distributed VOT cases, as discussed in the next subsection. Nuber of Segents Found by PAM PAM is evoked to find the set of BTDLGCP trees, to which all the trips with different VOT are assigned, and the corresponding set of breakpoints (i.e., VOT α = {α 0, α 0,..., α I α in = α 0 < α 1... < α I = α ax }) that partitions the entire feasible range of VOT ([α in, α ax ]) and hence defines the ultiple classes of trips, where each class includes the trips with VOT α {[α i 1, α i ), i = 1,..., I}. Starting fro the lowest possible VOT, the bicriterion tie-dependent shortest path algorith continuously solves for the BTDLGCP tree rooted at each destination for a given VOT interval and deterines the upper bound of that VOT interval, for which the BTDLGCP tree reains optial, until reaching the highest possible VOT. The ove fro the current VOT segent to the next and obtaining a different BTDLGCP tree require that a sall value Δ be added to the current breakpoint (α i ). This iplies that travelers cannot distinguish differences in VOT below Δ per inute. The value of Δ also iplicitly sets an upper bound for the nuber of VOT segents generated in PAM, with a value of ax in ( α α ) (15) For all three continuously distributed VOT experients, Δ is set to be Figure 3 deonstrates the nuber of segents per (o, d, ) triplet found by PAM in three pricing schees. The allhighway case gives the fewest segents for the entire feasible VOT range (one segent, 65.97%; two segents, 17.76%; three segents, 8.16%). Although the nuber of tolled links in the randselection6.5% case is exactly the sae as in the allhighway case, the coverage is greater, that is, the tolled links are scattered on the network; therefore ore segents are found by PAM in this case. As the coverage increases, PAM produces the ost segents in randselection15%. Coputational Tie and Selection of Optial Nuber of VOT Classes The ost intensive coputational operations in solving the proble are calculating the BTDLGCP tree at each user equilibriu iteration. Figure 4 depicts the coputational tie per iteration for all experients. In the ulticlass VOT assuption, the required coputational tie is proportional to the nuber of discrete classes specified, that is, for each class a BTDLGCP tree is coputed with its average VOT. Once the discrete set of VOT is given for a certain network, the coputational tie is essentially fixed, that is, no other factor is influencing the coputational tie. In the continuously distributed case, PAM is evoked to calculate the BTDLGCP tree and partitions the entire feasible range for each destination. Unlike the ulticlass VOT, PAM finds segents of VOT on the run given the experienced tie-dependent generalized cost for each destination. The nuber of segents found by PAM is significantly affected by the designs of the pricing schee. The question left unanswered is how any user classes ust be specified in the equilibriu odel to provide a good approxiation. For the answer to this question, the continuous cases are used as the bencharks, and the relative percentage change (shown in Figure 5) in toll revenue and nuber of vehicles passing through toll links is copared as follows: value %change = value benchark value ulticlass i benchark 100% (16) In the allhighway case, the 15-class VOT specification overpredicts the revenue by 2.14% and the nuber of tolled vehicles by 2.3%. The 20-class specification also produces good approxiation and underestiates the two indicators by 1.88% and 0.49%, respectively. As the geographical coverage of tolled links is changed, for exaple, in randselection6.5%, the pattern of the approxiation quality changes. Still, 15-class and 20-class VOT specifications give acceptable approxiation. In randselection15%, ore discrete

7 Jiang and Mahassani 111 Percent allhighway randselection6.5% (a) randselection15% Percent allhighway randselection6.5% randselection15% (b) FIGURE 3 Nuber of VOT segents found by PAM: (a) one to six segents and (b) seven to 25 segents. FIGURE 4 Coputational tie per iteration (iter.).

8 112 Transportation Research Record 2343 (a) (b) Percent Percent Percent (c) FIGURE 5 Relative percentage changes: (a) allhighway case, (b) randselection6.5% case, and (c) randselection15% case. classes of VOT are required to obtain a good approxiation. For 35-class, the gaps between the approxiation and the benchark are 1.92% and 1.4%, respectively. Large Network Application Although the heterogeneous user assuptions and pricing schees significantly affect predictions of user responses on the Fort Worth test network, they ust be tested against a large-scale real-life network. For this purpose, the Chicago network was selected (Figure 6). The network consists of 1,578 nodes and 4,805 links, covering the Chicago downtown area, and is bounded on the east by Lake Michigan and on the west by Cicero Avenue. The network is bounded by Harle Avenue and Roosevelt Road on the south and by Lake Avenue (in Evanston, Illinois) on the north. The network coprises the Kennedy Expressway and the Edens Expressway (light blue in the figure), along with signalized arterial streets and local streets. It is apped into 218 travel analysis zones. A 6-h (5:00 a.. to 11:00 a..) weekday tie-varying O-D deand table for every 5 in is loaded onto the network, containing approxiately 0.8 illion passenger vehicles. The algorith is coded and copiled in Intel Visual FORTRAN 10.0 and evaluated on a 64-bit Windows Vista platfor and a achine with Intel Xeon 3.2 GHz CPU and 32 GB RAM. Experiental Design The planning horizon is 6 h. To create the hypothetical dynaic pricing schee, all 136 freeway links are converted to the toll road with 12 periods having different toll levels proportional to the link length (starting at $1/i and increasing by $0.25/i every 30 in till 8:00 a.., then decreasing at the sae rate every 30 in until 11:00 a..). The VOT specified in the following was estiated in the Chicago activity-based odel with census data (P. Vovsha, personal counication, July 2012). The continuous VOT distribution (in dollars per hour) is assued to be truncated lognorally distributed with the feasible range of (0, 30], i.e., α ~ Log N(1.89,0.7818). The constant VOT is assued to be the ean VOT ($8.99/h) for all road users. The discrete VOT case segents the entire population into four groups according to incoe: $0 to $30,000 (35%), $6.01/h; $30,000 to $60,000 (25%), $8.81/h; $60,000 to $100,000 (21%), $10.44/; and $100,000+ (19%), $12.86/h. Nuerical Results Table 2 provides the toll road usage for the 6-h planning horizon predicted under various VOT assuptions. The benchark case is the continuously distributed VOT with the step size set to $1.50/h, yielding a axiu 20 segents for the feasible VOT range. Given the size of the network, this provides a fine resolution for the possible partitions. As reported in Table 2, the toll road usage predicted with discrete VOT is uch higher than the continuous assuption. The constant VOT assues hoogeneous users and fails to capture the fact that there are soe users with lower VOT who are not willing to pay a

9 Jiang and Mahassani 113 (a) (b) FIGURE 6 Chicago test: (a) network and (b) O-D deand volue. toll charge to avoid congestion. The four-user-class VOT specification acknowledges the heterogeneity to soe extent and thus gives relatively better predictions for toll revenue and the nuber of users (vehicles) using the toll roads. This is because underlying generalized costs are different for the continuously distributed VOT and the discrete case when path choices are ade. Thus the predicted toll road usage, in a given pricing schee, obtained fro constant or discrete VOT assuptions could be biased. The concerns in this real-life application are alleviating the coputational effort and eory requireents for solving the dynaic equilibriu proble with continuously distributed VOT on largescale networks. Two ipleentation techniques (adjusting the step size in PAM and the gap sorting technique) proposed by Jiang et al. are tested on the Chicago network (18), and coparison of the coputational tie is give in Figure 7. The CPU tie is defined as the average tie required by one operation to calculate the BTDLGCP tree per root node (destination). As discussed in the Fort Worth test, the required coputational tie is proportional to the nuber of discrete classes specified and reains essentially the sae for each iteration. The full run of PAM with $1.50/h as the step size requires a large nuber of coputations and thus takes the ost tie. When the step size is doubled to $3/h, the total coputational tie is reduced by 45.37%; however, the predictions of the toll revenue and nuber of vehicles traversing the toll roads copared with the benchark case (step size of $1.50/h) are quite accurate at within approxiately ±1%, as shown in Table 2. Although the gap sorting technique provides further iproveent in coputational tie, the predictions of toll road use deviate fro the benchark case. The gap sorting technique was shown to be efficient and effective for the New York etropolitan region s best practice network (18), but experients on the Chicago downtown network were ixed, suggesting that the ethod requires further refineent for general applicability. Concluding Rearks TABLE 2 Toll Road Revenue and Use for Chicago Core Network VOT Assuption Total Revenue ($) Tolled Vehicles Continuous VOT 17, ,095 (step size = 1.5) Continuous VOT 17, (1.01%) a 16,887 ( 1.22%) (step size = 3) Constant VOT 216, (22.65%) 21,593 (26.31%) ulticlass4 16, ( 8.02%) 19,260 (12.66%) a Values in parentheses indicate relative percentage change between benchark case and test case. Increasing interest in ipleenting and exploring a wider range of pricing alternatives and technologies otivated an exaination of heterogeneous users responses to pricing schees. A unified odeling fraework was presented, along with solution algoriths incorporating both discrete VOT and continuously distributed VOT assuptions for solving the dynaic traffic assignent odel. The nuerical results fro a Fort Worth test network showed that different VOT assuptions produce diverse predictions of toll road usage and required coputational efforts. The failure to realistically capture user heterogeneity leads to biased predictions of pricing ipacts. This observation was confired by the Chicago network application. To circuvent the intensive coputational efforts of

10 114 Transportation Research Record Minutes No. of Iterations contvot stepsize = 1.5 contvot stepsize = 1.5 & gapsorting contvot stepsize = 3 contvot stepsize = 3 & gapsorting constantvot ulticlass4 FIGURE 7 CPU tie (average tie required for calculating BTDLGCP tree per root node) for ipleentation techniques, Chicago core network (no. = nuber). the continuous assuption, two ipleentation techniques were tested on the Chicago network, and on the basis of final perforance easures, one of the two (adjust step size in PAM) was selected. The results confir that assuptions of a single, constant VOT value in forecasting the ipact of pricing schees could produce biased forecasts of toll road usage and associated revenues relative to recognizing the heterogeneity of user preferences confired in studies of user behavior. The assuption of a sall nuber of different VOT classes ay yield iproveent in the predictions of toll road usage and toll revenues. However, the nuerical results on the Fort Worth network indicate that unless the nuber of predefined classes is substantially large, in excess of 20 or even 30 classes, the quality of the results will reain a hit-or-iss proposition, while the associated heavy coputational cost will be a sure thing. With greater realis in the path choice behavior of trip akers, the iproved algorithic and ipleentation steps for the BDUE ethodology explicitly odel heterogeneous users with a continuously distributed VOT. Results fro the Chicago core network confired that a predefined nuber of discrete user classes could produce biased forecasts. The results also indicated that controlling the step size in the PAM path coputation procedure provides an effective eans to control the coputational effort with little corresponding decrease in prediction accuracy. The gap sorting technique, although proising, needs further refineent to iprove its predictive consistency with general toll road designs on a given network topology. Because of iproved algorithic and ipleentation steps for the continuously distributed BDUE ethodology, a case can be ade for the use of the conceptually and ethodologically preferable approach with continuously distributed VOT and eergent classes. A particularly notable advantage of the BDUE odel is that it offers sealess integration with activity-based deand icrosiulation procedures, which generate heterogeneous populations of individuals with associated preference attributes. The integration of transport supply and deand odels would be highly desirable in the context of toll pricing analysis. Acknowledgents This paper was based in part on work funded through a U.S. Departent of Transportation project conducted by Northwestern University as subcontractors to SAIC, Inc., and built on results of SHRP 2 Project C04, Iproving Our Understanding of How Highway Congestion and Pricing Affect Travel Deand, conducted by Northwestern University and PB Aericas, Inc. Peter Vovsha of PB Aericas provided recent VOT estiation results for the Chicago network. The authors thank Kerit Wies of the Chicago Metropolitan Agency for Planning for facilitating access to network and deand inforation for the Chicago network application. The authors also thank Darren Tiothy and John Halkias of the U.S. Departent of Transportation and Forrest Swisher of SAIC for coents and suggestions in connection with the ethodological tests. Several students contributed to the developent and calibration of the Chicago network; the authors thank Ali Zockaie, Oer Verbas, Jiwon Ki, and Meead Saberi. The authors acknowledge Jason C.-C. Lu and Kuilin Zhang for contributions to the algorithic developent and ipleentations related to the BDUE procedures. References 1. Sall, K. A., and C. Winston. The Deand for Transportation: Models and Applications. In Essays in Transportation Econoics and Policy: A Handbook in Honor of John R. Meyer (J. Goez-Ibanez, J. Tye, and C. Winston, eds.), Brookings Institution Press, Washington, D.C., 1999, pp Wardan, M. A Review of British Evidence on Tie and Service Quality Valuations. Transportation Research Part E, Vol. 37, No. 2 3, 2001, pp

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