**Suggested Citation:**"Appendix A - Mathematical and Procedural References." National Academies of Sciences, Engineering, and Medicine. 2012.

*Improving Our Understanding of How Highway Congestion and Pricing Affect Travel Demand*. Washington, DC: The National Academies Press. doi: 10.17226/22689.

**Suggested Citation:**"Appendix A - Mathematical and Procedural References." National Academies of Sciences, Engineering, and Medicine. 2012.

*Improving Our Understanding of How Highway Congestion and Pricing Affect Travel Demand*. Washington, DC: The National Academies Press. doi: 10.17226/22689.

**Suggested Citation:**"Appendix A - Mathematical and Procedural References." National Academies of Sciences, Engineering, and Medicine. 2012.

*Improving Our Understanding of How Highway Congestion and Pricing Affect Travel Demand*. Washington, DC: The National Academies Press. doi: 10.17226/22689.

**Suggested Citation:**"Appendix A - Mathematical and Procedural References." National Academies of Sciences, Engineering, and Medicine. 2012.

*Improving Our Understanding of How Highway Congestion and Pricing Affect Travel Demand*. Washington, DC: The National Academies Press. doi: 10.17226/22689.

**Suggested Citation:**"Appendix A - Mathematical and Procedural References." National Academies of Sciences, Engineering, and Medicine. 2012.

*Improving Our Understanding of How Highway Congestion and Pricing Affect Travel Demand*. Washington, DC: The National Academies Press. doi: 10.17226/22689.

**Suggested Citation:**"Appendix A - Mathematical and Procedural References." National Academies of Sciences, Engineering, and Medicine. 2012.

*Improving Our Understanding of How Highway Congestion and Pricing Affect Travel Demand*. Washington, DC: The National Academies Press. doi: 10.17226/22689.

**Suggested Citation:**"Appendix A - Mathematical and Procedural References." National Academies of Sciences, Engineering, and Medicine. 2012.

*Improving Our Understanding of How Highway Congestion and Pricing Affect Travel Demand*. Washington, DC: The National Academies Press. doi: 10.17226/22689.

**Suggested Citation:**"Appendix A - Mathematical and Procedural References." National Academies of Sciences, Engineering, and Medicine. 2012.

*Improving Our Understanding of How Highway Congestion and Pricing Affect Travel Demand*. Washington, DC: The National Academies Press. doi: 10.17226/22689.

**Suggested Citation:**"Appendix A - Mathematical and Procedural References." National Academies of Sciences, Engineering, and Medicine. 2012.

*Improving Our Understanding of How Highway Congestion and Pricing Affect Travel Demand*. Washington, DC: The National Academies Press. doi: 10.17226/22689.

**Suggested Citation:**"Appendix A - Mathematical and Procedural References." National Academies of Sciences, Engineering, and Medicine. 2012.

*Improving Our Understanding of How Highway Congestion and Pricing Affect Travel Demand*. Washington, DC: The National Academies Press. doi: 10.17226/22689.

**Suggested Citation:**"Appendix A - Mathematical and Procedural References." National Academies of Sciences, Engineering, and Medicine. 2012.

*Improving Our Understanding of How Highway Congestion and Pricing Affect Travel Demand*. Washington, DC: The National Academies Press. doi: 10.17226/22689.

**Suggested Citation:**"Appendix A - Mathematical and Procedural References." National Academies of Sciences, Engineering, and Medicine. 2012.

*Improving Our Understanding of How Highway Congestion and Pricing Affect Travel Demand*. Washington, DC: The National Academies Press. doi: 10.17226/22689.

**Suggested Citation:**"Appendix A - Mathematical and Procedural References." National Academies of Sciences, Engineering, and Medicine. 2012.

*Improving Our Understanding of How Highway Congestion and Pricing Affect Travel Demand*. Washington, DC: The National Academies Press. doi: 10.17226/22689.

**Suggested Citation:**"Appendix A - Mathematical and Procedural References." National Academies of Sciences, Engineering, and Medicine. 2012.

*Improving Our Understanding of How Highway Congestion and Pricing Affect Travel Demand*. Washington, DC: The National Academies Press. doi: 10.17226/22689.

**Suggested Citation:**"Appendix A - Mathematical and Procedural References." National Academies of Sciences, Engineering, and Medicine. 2012.

*Improving Our Understanding of How Highway Congestion and Pricing Affect Travel Demand*. Washington, DC: The National Academies Press. doi: 10.17226/22689.

**Suggested Citation:**"Appendix A - Mathematical and Procedural References." National Academies of Sciences, Engineering, and Medicine. 2012.

*Improving Our Understanding of How Highway Congestion and Pricing Affect Travel Demand*. Washington, DC: The National Academies Press. doi: 10.17226/22689.

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160 Mathematical and Procedural References Method for Synthesizing a Distribution of Consistent Path-Dependent Originâ Destination Travel Times from the Known Distribution of Link Traffic Counts Problem This method is suggested for the generation of originâ destination (O-D) travel time distribution for the base year, which is needed for calculating travel time reliability measures. These reliability measures are used in travel demand models to explain travel choices along with the average travel time and cost. The current memo explains generation of O-D time distribution for model estimation for the base year. The method is designed to produce a distribution of travel times for a full regional O-D matrix for a certain time of day period or hour (i.e., period-specific âreliability skimâ). The following inputs are assumed given: a â A = highway network links; a â A~ â A = links for which traffic counts are known; {wna}aâA~,nâNa = sets of traffic counts (discrete distribution) for each link; wâa = average traffic count for each link; ca(va) = link volume-delay function (VDF) cali- brated to reproduce observed travel time distribution; and Tij = seed trip table adjusted to reproduce average traffic counts. Without a loss of generality, we assume that a set of traffic counts for each link has the same size (say, 10 or 20 observa- tions). If a link has fewer observations than the established number, additional counts are created by interpolation to preserve the observed distribution. For the algorithm implementation, it is convenient to order counts for each link from the smallest to the largest: â¤ â¤ â¤ â¤ â¤. . . . . . . (A.1)1 2w w w wa a an aN In general, it is assumed that traffic counts for different links are taken on different days and maybe during different seasons and even in different years. For this reason, some of them can undergo certain adjustments to bring them to a common denominator. Thus, in general, it is impossible to establish a linkage or a priori correlation pattern between them (it might be possible only for some of them taken on the same day in parallel; this partial information can be used in the proposed algorithm as an additional constraint). The method generates the following output: s â S = demand fluctuation scenarios, each associ- ated by a trip table Tsij; xans = 0, 1 = counts assortment by scenarios; and Csij = a set of travel time skims (discrete distribu- tion) by scenarios. Approach Travel time skims by scenarios are generated by scenario- specific trip tables Tsij pivoted off the seed trip table. The scenario-specific trip tables are constructed by adjustment to scenario-specific subsets of traffic counts {wan(s,a)}aâA~ where n(s, a) represents a count value from the link distribution that is selected for scenario s. Scenarios can be meaningfully asso- ciated with certain demand fluctuations reflected in traffic counts (e.g., a rainy day with generally low traffic, special event with the corresponding counts taking very high values). These scenarios, however, cannot cover cases where the sup- ply side, that is, network capacity, was reduced (like a traffic accident or lane closure because of road work). A P P e n D i x A

161 When forming the network scenarios from link count dis- tributions, two main principles should be followed: â¢ The observed distribution of traffic counts for each link should be preserved across the network scenarios. This can be easily achieved by constructing scenarios as combinations of permutations of the numbered counts 1, 2, . . . , N for each link. In this case, the number of sce- narios is equal to the number of counts on each link (say, 10). Each count value is used once in one scenario and cannot be reused in a different scenario. Although this strategy guarantees a replication of the distribution of counts for each link, it alone is not enough to construct realistic network scenarios portraying demand fluctuations. For example, one could envision constructing simplified sce- narios fully ordered by traffic counts (from the smallest to the largest, i.e., setting s = n). In this case, the first scenario would correspond to all lowest count values while the last scenario will correspond to all highest count values. This, however, would create an unrealistic pattern of cor- relations where all links in the network are assumed to fluctuate simultaneously, even though it is true only for links in the same corridor that have a substantial mutual traffic flow. â¢ The correlation pattern between links across scenarios should follow the logic of traffic flows. This means that adjacent links in the same highway corridor with a substan- tial mutual traffic flow (or parallel links competing for the same O-D pairs) should have a highly correlated pattern (both should have either a high or a low value in the same scenario). Contrary to that, links that do not have a sub- stantial mutual flow and do not serve the same O-D pairs (i.e., correspond to different demand segments) should have a random correlation pattern where each link can take a value independently from the other in each scenario. In particular, one link can have a very high value and the second one can have a very low value in the same scenario. The correlation pattern is established in the assignment procedure with a trip table adjusted to traffic counts. In particular, several count values can be easily replicated by a small adjustment of the trip table if they are consistent with the flow pattern. On the other hand, if a particular count value persists not to be replicated, it is an indication of the contradiction between this count and the traffic flow pattern defined by the rest of the counts and O-D demand. Program Formulation We assume without essential loss of generality that the number of scenarios S is equal to the number of count values available for the links N. The problem can be stated as the following mathematical program: min , , ,T x v ij s ij s ij s a ns a s F T F T> ={ } { }( )+0 0 1 1 2 2Âµ { }+ { }( )( )[ ]Âµ3 3 2 F x vans as, , ( . )A where â{ })( = ln , (A.3)1 , F T T T T ij s ij s ij s ijs ij â{ })( = Ï , (A.4)2 2 , F T T Tijs ij kl ijkl ij kl ââ( ){ } = âï£«ï£ï£¬ ï£¶ï£¸ï£· â , , (A.5)3 2 , Ë F x v v w xans as as an ans ns a A Âµ2 > 0, Âµ3 > 0 represents weights for the second and third criterion relative to the first criterion, under constraints: â = ï£«ï£ ï£¶ï£¸1, each scenario uses only onecount value for each link (A.6)xansn â = ï£«ï£ ï£¶ï£¸1, each count value for each linkis used in only one scenario (A.7)xans s ââ )(= Î´ , flow conservation (A.8)v T pas ijs ijrs ars rij where psijr â¥ 0 = path probabilities, dsar = 0,1 = linkâpath incidence matrix. The meaning of the first criterion (Equation A.3) is that all else being equal, the scenario-specific trip tables Tsij should be structurally similar to the original/base trip table Tij. The meaning of the second criterion (Equation A.4) is that all else being equal, the scenario-specific variations of trip tables should not be systematically correlated but rather randomized across O-D pairs. In other words, we assume that demand fluctuations in different cells are independent unless the constraints and/or other two criteria indicate cor- relation. Randomization of the scenario-specific trip tables pre- serves the necessary independence of traffic flows across links except for links that have a significant mutual flow of vehicles (i.e., significant demand from the same O-D pairs). Traffic vol- umes on those links will always be fluctuating in parallel, even if the demand fluctuations for the corresponding O-D pairs are independent from each other. The pair-wise measure of demand correlation between O-D pairs can be written as follows: . (A.9), 2 2 T T T T T T T T ij kl ij s ij kl s kl s ij s ij s kl s kl s â â â ( ) ( ) ( ) ( ) Ï = â â â ï£® ï£°ï£¯ ï£¹ ï£»ï£º â ï£® ï£°ï£¯ ï£¹ ï£»ï£º

162 In the program formulation, this measure is squared (because independence is violated by both positive and nega- tive correlations), weighted by the demand, and summed over all pairs of O-D cells. The original correlation coefficient is -1 â¤ rij,kl â¤ 1. The squared coefficient is in the unit interval. It should be noted that, in general, there is no contradiction between the first and second criterion. The first criterion (Equation A.3) states that the deviations of the scenario- specific trip tables from the original average trip table should be minimal. The second criterion (Equation A.4) is not dependent on the magnitude of the deviations, but rather requests from them to be independent across O-D pairs. The third criterion (Equation A.5) expresses the desired rep- lication of distributions of traffic counts observed for each link. The mathematic program formulated in Equations A.2 through A.9 is a complicated, nonlinear program of a very large size (in real-world networks) with both continuous and discrete variables. It cannot be solved by direct formal methods. This formulation serves only as a useful conceptual frame- work, within which a practically effective heuristic algorithm is proposed. Heuristic Algorithm The following algorithm is suggested to incorporate both the principles and the resulting three formal criteria (Equations A.3â A.5) described in the previous section. Step 0. Define a seed trip table Tij and adjust it to replicate average traffic counts {wâa}. This is an auxiliary step to prepare a good starting condition. Set initially all scenario-specific trip tables to be equal to the seed trip table Tsij = Tij. Step 1. Randomly and independently vary each scenario- specific trip table Tsij. The variation should be implemented independently in each cell to address the second criterion (Equation A.4). The magnitude of variation is approximately equal to the average level of variation in counts. In each cell and for each scenario, an independent draw is implemented from a log normal distribution with the mean equal to Tsij and sched- uled time of departure (STD) equal to average STD observed in traffic counts. The lognormal distribution ensures positive val- ues and avoids a truncation problem associated with the normal distribution. Assign the scenario-specific trip tables (each one separately) to obtain scenario-specific link volumes vsa. Step 2. Optimize the assortment of traffic count values by scenarios in terms of xans and taking into account the third criterion (Equation A.5). Define xans in such a way that the order of traffic counts on each link would correspond to the order of link volumes. It means that for each link, scenarios are ordered as follows: â¤ â¤ â¤ â¤ â¤( ) ( ) ( ) ( ). . . . . . . (A.10)1 2v v v vas a as a as a as an N Then the assortment variables are calculated to ensure that the order of traffic counts corresponds to order of link volumes: ( ) ( )= = â ï£±ï£²ï£³ï£´ 1, if 0, if . (A.11)x s a n s a n a ns n n After the assortment, each link obtains a scenario-specific set of counts {wsa} by the following formula: âÏ = . (A.12)w xas an ans n Step 3. Adjust the scenario-specific trip tables Tsij to the cor- responding scenarios of traffic counts {w sa}. Matrix adjustment corresponds to the original program (Equations A.2âA.9) with fixed count assortment variables {xans}. If the assortment variables are fixed, the program is reduced to optimization by two criteria (Equations A.3, A.5) under constraint (Equation A.8). Existing effective heuristic algorithms can be employed for this sub-problem. Save the adjusted scenario-specific trip tables Tsij, traffic volumes {vsa}, and O-D travel times {C s ij}. Step 4. Possible feedback can be implemented to Step 1 if the trip table adjustment after multiple iterations [i.e., com- promising criterion (Equation A.3)] has not produce a good match to traffic counts. Persistent counts that cannot be matched indicate a possible inconsistency in the counts assort- ment by scenarios that can be re-sorted at the second iteration. The essence of this algorithm is to reorder the traffic count values by consistent demand fluctuation scenarios where each scenario has a consistent flow pattern that replicates the chosen subset of count values. It is essential to fully exploit the switching mechanism to create consistent scenarios rather than overadjust the trip table to replicate inconsistent scenarios. For this reason, the matrix adjustment subroutine will be used with a small number of internal iterations (Equations A.2âA.3) expecting a large number of feedback iterations between Steps 1 and 3. Implementation The entire algorithm can be implemented using the script language of any transportation software package (Geographic Information System Developerâs Kit [GISDK] for TransCAD or Macro-language for EMME). The only substantial subroutine required is a trip matrix adjustment to traffic counts available in all transportation software packages. Mathematical Formulation of the integrated Multidimensional network Choice Model This section provides additional detail on the mathematical formulation of the integrated multidimensional network travel choice model, with multimodal route and mode choice,

163 introduced in Chapter 4 and applied to the New York Regional Best Practice Network. The basic model assumptions were stated in Chapter 4 within the Integrated Model Framework to Evaluate Pric- ing and Reliability section. Given a time-varying network G = (N, A), where N is a finite set of nodes and A is a finite set of directed links, the time period of interest (planning horizon) is discretized into a set of small time intervals, H = {t0, t0 + Dt, t0 + 2Dt, . . . , t0 + TDt}, where t0 is the earliest possible departure time from any origin node, Dt is a small time interval during which no perceptible changes in traffic conditions and/or travel cost occur, and T is a large number such that the inter- vals from t0 to t0 + TDt cover the planning horizon H. The time-dependent zonal demand qwt over the study horizon represents the number of individual travelers of an O-D pair w(w â W) at departure time t(t â T). A set of available modes is denoted as M. The integrated model in this study is to find a dynamic network equilibrium modeâpath flow pattern by recognizing multiple dimensions of network choice behaviorâ that is, mode choice decision, highway user heterogeneity, and reliability of route choice. The integrated multidimensional network choice model includes two main submodels, namely, the time-dependent mode choice stochastic user equilibrium (TDMSUE) model, and the multiclass multicriterion dynamic user equilibrium (MDUE) route choice model. TDMSUE Model Based on the weak law of large numbers, a mode choice prob- ability pmwt(y) can be obtained through mode flow ymwt divided by total O-D demand, qwt, as follows: ( ) = â â â â, , , (A.13)p y y q m M w W t Tmwt m wt wt The TDMSUE conditions can be stated mathematically as follows: y q p y m M w W t Tmwt wt mwt= Ã ( ) â â â â, , , ( . )A 14 Therefore, the TDMSUE problem of interest can be for- mulated as the following fixed point problem. Let W(y) be a feasible set of mode flows. ( ) ( )â ââ¦ â = Ã âFind , satisfying . (A.15)y y y q p y Solving this system of nonlinear equations (Equations A.13â A.15) will give a set of mode flows y*, which is also the solu- tion of the TDMSUE problemâthat is, y* would satisfy the TDMSUE condition in Equation A.14. To solve this prob- lem by using advanced optimization-based procedures, we reformulate this problem as a gap function based nonlinear programming (NLP) in Equations A.16 through A.18, which was proposed in connection with a generalized dynamic stochastic use equilibrium problem (Zhang et al. 2008). âââ [ ]( ) ( )= Ã â ÃMin 1 2 (A.16) 2 g y y q p yM mwt wt mwt mtw subject to y q w W t Tmwt wt m = â â ââ , , ( . )A 17 0, , , (A.18)y w W t T m Mmwt â¥ â â â â where the objective function in Equation A.16 is a gap mea- sure defined by summation of the square difference between the assigned mode flow, ymwt, and the expected mode flow, qwt Ã pmwt(y), over all O-D pairs, departure times, and modes. Constraints in Equation A.17 are flow balance constraints for each O-D pair and departure time. Constraints in Equation A.18 are nonnegative mode flow constraints. MDUE Route Choice Model Based on the MDUE definition, the MDUE conditions can be mathematically stated as a nonlinear complementary problem (NCP) in the following: â âï£¯ï£° ï£ºï£»Î± Î± Î±min max, , , , 0, , , , , , (A.19) x GC x a x k K w t m w W t T m M k wtm k wtm wtm[ ]( ) ( ) ( ) ( )Î± Î± â pi = â â â â â ( ) ( ) ( ) Î± â pi â¥ â â â â â , , 0, , , , , , (A.20) GC x a x k K w t m w W t T m M k wtm wtm â ( ) ( )Î± = Î± â â â â ( )â , , , (A.21) , , x y w W t T m Mkwtm wtm k K w t m x k K w t m w W t T m Mkwtm Î±( ) â¥ â â ( ) â â â0 22, , , , , , ( . )A where x = {xk wtm(a)ï£¦"k â K(w, t, m), w â W, t â T, m â M, a â [amin, amax]} is a multiclass time-varying MDUE route flow vector, and pwtm(a, x) is the time-varying minimum O-D generalized travel cost for each mode, evaluated at x, for the trips with the same (w, t, m, a). Constraints in Equations A.19 and A.20 are complementary constraints. Constraints in Equation A.21 are flow balance constraints. Constraints in Equation A.22 are nonnegative path flow constraints. Given the assumptions and definition, the multiclass dynamic route choice model aims at solving the MDUE

164 problem, under a given road pricing scheme, to obtain a time- varying route flow vector satisfying the MDUE conditions. Based on the aforementioned NCP formulation (Equa- tions A.19âA.22), we can derive an equivalent gap function based on NLP formulation in Equation A.23, which is an exten- sion of an equivalent gap functionâbased reformulation for the dynamic user equilibrium problem (Lu et al. 2009). â«ââââ ( )( )( ) ( )= Î± Ã Î±â pi Î±ï£®ï£°ï£¯ ï£¹ï£»ï£º Î±Î± Î± Min , , (A.23) min max g x x GC x x dR kwtm k wtm wtm kmtw subject to Equations A.20 through A.22. Integrated Multidimensional Network Choice Model As shown in the TDMSUE and MDUE models in the previous sections, the integrated multidimensional network choice model essentially aims to seamlessly and correctly connect the mode choice model (TDMSUE) and multimodal dynamic route choice model (MDUE). The connection between these two models is defined by the flow balance conditions as follows: â« ââ«( ) ( )= Î± Î± = ï£®ï£° Î± ï£¹ï£» Î± â â â â ( )Î± Î± âÎ± Î± , , , (A.24) , ,min max min max y y d x d w W t T m M m wt wtm k K w t m k wtm Accordingly, the objective of the mode choice model is to obtain a TDMSUE mode flow pattern, and the objective of the multiclass dynamic route choice model is to obtain a MDUE route flow pattern with an input of the given TDMSUE mode flow pattern; in turn, the mode travel attributes (Level of Services) resulted in the MDUE route flow pattern leading to a new TDMSUE mode flow pattern. Therefore, the integrated multidimensional network choice model is mathematically formulated as follows: Min Ag y x y q p yI mwt wt mwt mtw , ( .( ) = Ã â Ã ( )[ ]âââ1 2 2 25) subject to y q w W t Tmwt wt m = â â ââ , , ( . )A 26 0, , , (A.27)y w W t T m Mmwt â¥ â â â â ââ«â« ( ) ( )= Î± Î± = ï£®ï£° Î± ï£¹ï£» Î± â â â â ( )âÎ± Î± Î± Î± , , , (A.28) , ,min max min max y y d x d w W t T m M m wt wtm k K w t m k wtm , , 0, , , , , , (A.29) x GC x x k K w t m w W t T m M k wtm k wtm wtm[ ]( ) ( ) ( ) ( )Î± Î± â pi Î±ï£¯ï£° ï£ºï£» = â â â â â p ( ) ( ) ( ) Î± â pi Î± â¥ â â â â â , , 0, , , , , , (A.30) GC x x k K w t m w W t T m M k wtm wtm â ( ) ( )Î± = Î± â â â â ( )â , , , (A.31) , , x y w W t T m Mkwtm wtm k K w t m x k K w t m w W t T m Mkwtm Î±( ) â¥ â â ( ) â â â0 32, , , , , , ( . )A Essentially, Equations A.28 through A.32 define dynamic as a multimodal route flow vector with respect to each value of time (VOT), x* = {xk wtm(a)*ï£¦"k â K(w, t, m), w â W, t â T, m â M, a â [amin, amax]}, which can be obtained by solving the multimodal bicriterion dynamic use equilibrium problem. The solution algorithm for this formulation, presented in Chapter 4, is described in greater detail in the next section of this appendix. Solution Algorithm for the integrated Multidimensional network Choice Model This section provides additional detail on the algorithmic procedures developed for the integrated multidimensional network travel choice model, with multimodal route and mode choice, introduced in Chapter 4. The algorithmic procedures were implemented for the large-scale New York Regional Best Practice Network, as described in Chapter 4. Projected Gradient-Based Descent Direction Method to Solve the TDMSUE Problem The TDMSUE Problem described in Equations A.16 through A.18 of this appendix can be decomposed into each O-D pair and departure time. In light of Zhang et al. (2008), we can derive a cross-set gradient of the decomposed problem as follows. â â [ ] [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) â = â + â = â Ã Ã â Ã â â ï£® ï£°ï£¯ ï£¹ ï£»ï£º + â Ã Ã â Ã â â ï£® ï£°ï£¯ ï£¹ ï£»ï£º = â Ã + â Ã Ã â Ã â â ï£® ï£°ï£¯ ï£¹ ï£»ï£º â â² â² â² â²â â² â² â²â â² 1 (A.33) 3 \ \ g g g y q p q p y y q p q p y y q p y q p q p y y M wt y m wt x M m wt m wt wt m wt wt m wt m wt m wt wt m wt wt m wt m wt m M m m wt wt m wt m wt wt m wt m M wt m wt m wt m wt m wt m wty y y y y y y y y y

165 Therefore, a projected gradient-based descent direction mode flow update scheme is as follows. ( ) ( )= â â â Ã( ) ( ) ( ) ( ) (A.34)1 1 1 1 g y g y n n n nd y y y y â( ) ( )( )= = â â² â² Ã( ) ( ) ( )â (A.35)1 1 1 1yn yn M yn D d d I E EE E d = + Î» Ã( ) ( ) ( ) ( )+ (A.36)1 1 1 1 1n n n yny y d Multimodal Parametric Analysis MethodâBased Path Generation In actual transportation networks, the number of available routes that can carry vehicles for each O-D pair, departure time, and mode is a finite number. For a given feasible route set of (w, t, m), the least experienced generalized cost route, will be different for different travelers due to heterogeneous VOT. Following the same approach of Lu et al. (2008), we adopt the parametric shortest path approach to obtain a set of break- points, or values of a corresponding to the changes in the least experienced generalized cost path, that partition the fea- sible range of VOT a, ï£°[amin, amax]ï£», into a set of subintervals, aI = {a0, a1, . . . , ai, . . . , aIï£¦amin = a0 < a1 < . . . < ai < . . . < aI = amax}, and within each subinterval of VOT, a â [ai-1, ai), "i = 1, . . . , I, travelers are assumed to have the same path as their least experienced generalized cost path. For each ai, O-D pair w, departure time t, and mode m, let K ~ (w, t, m, ai) be a restricted route set. Integrating time-varying and heterogeneous O-D demand ywtm(a) and route flow xk wtm(a) over each subinterval [ai-1, ai), "i = 1, . . . , I, we obtain the time-varying demand vector ywtm(aI) = {ywtm(ai)ï£¦"i = 1, . . . , I}, and route flow vector, xk wtm(aI) = {xk wtm(ai)ï£¦"i = 1, . . . , I} for each user class i, as in Equations A.37 and A.38, respectively. â« â« [ ] ( ) ( ) ( ) ( ) ( ) Î± = Î± Î± = Ã Ï Î± Î± = Ã Î¦ Î± â Î¦ Î± Î± Î± Î± Î± â â â (A.37)1 1 1 y y d y d y wtm i wtm wtm wtm i i i i i i â« â« [ ] ( ) ( ) ( ) ( ) ( ) Î± = Î± Î± = Ã Ï Î± Î± = Ã Î¦ Î± â Î¦ Î± Î± Î± Î± Î± â â â (A.38)1 1 1 x x d x d x k wtm i k wtm k wtm k wtm i i i i i i Equations A.37 and A.38 can be rewritten as Equations A.39 and A.40, respectively. y y w W t T m Mwtm wtm i i I = ( ) â â â â = â Î± , , , ( . ) 1 39A â ( )( )= Î± â â â â â = , , , , , , (A.40) 1 x x k K w t m w W t T m Mkwtm kwtm i i I Restricted MDUE Problem In light of the multimodal parametric analysis method and the integrated multidimensional network choice model, we can derive a restricted multiclass dynamic use equilibrium (RMDUE) problem as follows in Equation A.41: Min , , , (A.41) 4g x x GC x x M I k wtm i ikmtw k wtm i wtm i âââââ [ ] ( ) ( ) ( ) ( )Î± = Î± Ã Î± â pi Î± â subject to ( ) ( ) ( ) Î± â pi Î± â¥ â â Î± â â â â , , 0, , , , , , , , (A.42) GC x x k K w t m w W t T m M i I k wtm i wtm i i â ( ) ( )Î± = Î± â â â â â ( )â Î± , , , , (A.43) , , , x y w W t T m M i Ikwtm i wtm i k K w t m i ( )( )Î± â¥ â â Î± â â â â0, , , , , , , , (A.44) x k K w t m w W t T m M i Ikwtm i i As such, we can transform the continuous complementarity NLP problem into a discrete problem, which can be solved in a column generation solution framework. Projected Gradient-Based Descent Direction Method to Solve the RMDUE Problem The RMDUE problem can be decomposed into each O-D pair and departure time. Following Lu et al. (2008), we can derive a projected gradient-based descent direction method to solve the decomposed problem as follows in Equations A.45 and A.46: d GC x x GC x n i k wtm i wtm i k wtm 2 1( ) ( ) = â Ã ( )â ( )Î± Î± pi Î±, , x i, ( . ) Î±( ) A 45 â ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )Î± = Î± + Î» Ã Î± â â Î± Î± Î± + Î± Î± â â Î± ï£± ï£² ï£´ï£´ï£´ï£´ ï£³ ï£´ï£´ï£´ï£´ ( ) ( ) ( ) ( ) pi â²â Î± Î± pi â² pi ( ) ( ) ( ) ( )+ pi + , , , , \ , , , , , , , , , , (A.46) 2 2 , , , \ , , , 2 1 2 2 2 1 x x d k K w t m K w t m x K w t m x k K w t m k wtm i k wtm i n x n i i i k wtm i k K w t m K w t m i k wtm i i n n n i i n

166 Simulation-Based Iterative Solution Framework The TDMSUEâMDUE problem consists of finding both equilibrium travelersâ mode choice and equilibrium vehiclesâ route choice with a given time-dependent traveler demand. The TDMSUE problem is solved by a projected gradient-based descent direction method. However, it is difficult to enumer- ate the complete set of feasible routes for solving the MDUE problem in a practical size transportation network. To cap- ture the individual choice behavior and traffic dynamics, the simulation-based dynamic traffic assignment (DTA) algorith- mic framework disaggregates the O-D demands into individual vehicles, and only a portion of paths would have a nonzero probability to carry vehicles in an MDUE solution. This study uses the trajectories of vehicles as a proxy to store the feasible path set, namely, the vehicle-based implementation technique, to save computer memory and eliminate some unrealistic paths. To avoid explicit enumeration of all feasible routes, this study applies a column generation approach to generate a represen- tative subset of paths with competitive cost and augment the feasible path set. The parametric analysis method (PAM) is applied to obtain a set of breakpoints that partition the entire VOT interval into multiple subintervals. A projected descent direction method is used to solve the resulting MDUE problem in a restricted/reduced path set, called an RMDUE problem, in Equations A.41 through A.44. The simulation-based column generation iterative solu- tion framework for the TDMSUEâMDUE problem includes four main steps: (1) input and initialization, (2) nested logit mode choice, (3) ride-sharing choice and vehicle genera- tion, and (4) multidimensional simulation-based dynamic micro-assignment. This solution algorithm is shown in Figure A.1. Application to new York Regional network: Calibration of Time-Dependent O-D Demand with Multiple Vehicle Types This section describes the steps and procedures involved in developing the application of the integrated user choice processes and network modeling procedures presented in Chapter 4 to the New York region âbest practiceâ network. As noted, this is the largest actual network application of simulation-based network equilibrium procedures reported to date. Applications of this scale and magnitude pose additional challenges beyond the usual steps encountered in applying dynamic modeling procedures using data initially developed for static model application. These are documented in this sec- tion, with particular emphasis on the innovations developed to estimate unknown time-dependent demand patterns for the DTA model. Building a Large-Scale Network Model: Summary of Challenges In general, because of their ability to represent network operational characteristics, simulation-based dynamic traffic assignment models require more in-depth network informa- tion than comparable static assignment models. Traffic control signs and signals, left turns, and other movement capabilities at a node are mainly (only crudely) represented in the link performance function in a static network, whereas a dynamic model requires more accurate information on junction control and allowed movements at each phase at a signalized inter- section, as well as careful definition of each downstream movement at a node. Basically, there is no direct method of (correctly) converting a static network model into a dynamic network model in a single attempt by only using the existing data obtained from the provided static network model database. Smart conversion of existing database, use of external information sources, and more importantly, use of engineering judgment, are essential elements of building a large-scale dynamic network model. In summary, models developed for static assignment appli- cation generally lack several essential elements for dynamic network analysis, including â¢ Oversimplified representation of junctions, especially freeway interchanges for correct operational simulation; â¢ Absence or incorrect control information at junctions, and lack of reliable electronic database of control devices and control parameters at signalized junctions; â¢ Definition of origin and destination zones, including treatment/connection of centroids and external traffic generators; â¢ Insufficient specification of the operational attributes of links and junctions for the purpose of traffic simulation; and â¢ Absence of time-varying O-D information, which must be synthesized from available static matrices, coupled with traffic counts sometimes taken in mutually different time periods. Conversion of Existing Network for Dynamic Analysis The regional New York best practice model (NYBPM) includes 28 counties from three different states divided into 3,586 inter- nal traffic analysis zones (TAZs): â¢ 10 counties from the New York Metropolitan Transportation Council (NYMTC) area;

167 Step 1. Input and Initialization 1.1 Input: Time-dependent multimodal traveler O-D demand with individual characteristics (income, auto ownership, and purpose), network, and initial network level of services (time, cost, and reliability etc.) 1.2 VOT generation: Generate VOT for each traveler based on Monte Carlo simulation with given VOT distribution Step 2. Nested Logit Mode Choice 1.3 Initialization: Set mode choice loop ml = 0 2.1 Input of travelers with individual characteristics and mode attributes 2.2 Mode choice set construction systematic utility calculation 2.3 Nested logit choice probability calculation 2.4 Descent direction finding and mode choice update Step 3. Ride Sharing Choice and Vehicle Generation 3.1 Input of travelers with mode choice 3.2 Ride sharing choice and vehicle generation 3.3 Append external vehicles 2.5 Output travelers with mode choice 3.4 Output vehicles Step 4. Multidimensional Simulation-Based Dynamic Micro-Assignment 4.3 Solving the restricted MDSUE problem. 4.3.4 DTA inner loop stop checking: g(x)<= , or il=ilMax 4.3.1 Initialization. Set inner loop il=0. Read network performance and assignment from last outer loop 4.3.2 Multiclass path assignment 4.3.3 Multiclass dynamic network loading 4.1 Input and initialization 4.1.1 Input: Time-dependent vehicle demand, network, road pricing scheme 4.1.2 Initialization: Set DTA out loop ol = 0. Perform a dynamic network loading to obtain network performance 4.2 Parametric analysis of VOT and path generation 4.2.1 Bi-criterion dynamic shortest path calculation to define multiple user classes and shortest path trees 3.4 DTA out loop stop checking: no new path, or ol=olMax 3.5 Mode choice loop stop checking: ml=mlMax, or g(y)<= Stop 4.2.2 Relabeling shortest path by including reliability Y Y Y N N N Figure A.1. Simulation-based column-generation solution framework.

168 â¢ 2 other counties from New York State; â¢ 13 counties from the North Jersey Transportation Planning Authority (NJTPA) area; â¢ 1 other county from the state of New Jersey; and â¢ 2 counties from the state of Connecticut. The zones are mainly concentrated in New York City (NYC). See Figure A.2 for a general view of the NYBPM network modeled in TransCAD. â¢ 2,449 zones from New York State; â¢ 740 zones from the state of New Jersey; â¢ 397 zones from the state of Connecticut; and â¢ 111 external zones for travel entries to and exits from the network. Furthermore, the NYBPM network consists of â¢ 53,395 links; and â¢ 31,812 nodes. The DTA model converted from the static TransCAD model can be seen in Figures A.2 and A.3. The existing network was made available in TransCAD but is limited for static assignment application, and hence suffers from the limitations just noted. These had to be overcome through a systematic process of verifying and correcting the topology and connectivity of the network, assigning correct operational attributes for simulation purposes, determining appropriate junction control (often with visual verification through aerial photography, using sources such as Google Earth), defaulting signal control parameters, and estimating time-dependent O-D patterns using a state of the art meth- odology developed for this purpose. In the existing static model, most arterial-freeway inter- changes are designed properly in NYC, whereas the interchanges in New Jersey and Connecticut are designed as at-grade inter- sections. In a dynamic model, if an arterial and a freeway link, or two freeway links directly meet at a node, then the move- ments will be incorrect; for example, a left-turning vehicle would be able to block the opposing traffic, which is impos- sible on a full-access controlled road. This would result in an unrealistic dynamic traffic assignment. Therefore, conversion of these intersections into interchanges was necessary and has been done manually using the âCreate Interchangesâ toolbox of TransCAD. A beforeâafter example of such a case is shown in Figure A.4. Table A.1 lists the required and optional input data files for DYNASMART-P. Figure A.5 depicts the flowchart of the conversion methodology. Figure A.2. TransCAD model of the NYBPM network.

169 Figure A.3. DYNASMART-P model of the New York network. Figure A.4. Ramp correction for realistic turn movements.

170 Table A.1. Input Files for DYNASMART-P Input File Description Status Network Data xy.dat Contains the coordinates of the physical nodes to be used by the DYNASMART-P GUI Optional linkxy.dat Contains linksâ starting and ending nodes. For more accurate representation, users can specify as many feature points as needed to create curvatures Optional linkname.dat Contains names of links (i.e., street names) Optional network.dat Contains information regarding the network configuration, zoning, node, and link characteristics Required movement.dat Contains information regarding the allowed movements of vehicles (right turns, left turns, through, and others) Required TrafficFlowModel.dat Contains information regarding the type of speed-density models specified and their corresponding parameters Required GradeLengthPCE.dat Contains information regarding the effect of grade length and truck percentage on passenger car equivalent (PCE) factors Required Control Data control.dat Contains information regarding the type of traffic control at each node; phasing information, if the inter- section is signalized; major and minor approaches if the intersection has yield or two-way stop sign Required leftcap.dat Contains information regarding the left-turn capacity at signalized intersections Required StopCap2Way.dat Contains information regarding the capacity of approaches at two-way stop-controlled intersections Required StopCap4Way.dat Contains information regarding the capacity of approaches at all-way stop-controlled intersections Required YieldCap.dat Contains information regarding the capacity of approaches at yield-controlled intersections Required Traffic Management Data scenario.dat Contains information regarding the basic simulation variables Required vms.dat Contains information regarding the location, type, and time of activation for variable message signs May be empty if no signs WorkZone.dat Contains information regarding the number of work zone links, duration, capacity reduction, new speed limit, and discharge rate May be empty incident.dat Contains information about capacity reduction (fraction) and duration of incident on links May be empty pricing.dat Contains information regarding the pricing on the regular and high-occupancy toll (HOT)/high-occupancy vehicle (HOV) lanes May be empty ramp.dat Contains information regarding ramp locations, detector locations, metering rate, and its timing May be empty bus.dat Contains information regarding the buses, including the trajectories, location of stops, dwell time May be empty output_option.dat Contains information regarding the frequency of writing to output files Required system.dat Contains information regarding selection of the solution mode, length of the planning horizon, aggregation interval, and assignment interval Required Demand Data zone.dat Contains information that describes the zonal boundaries Optional demand.dat Contains information regarding the temporal and spatial distribution of vehicular demand Required demand_HOV.dat Contains information regarding the temporal and spatial distribution of HOV demand Required demand_truck.dat Contains information regarding the temporal and spatial distribution of truck demand Required destination.dat Contains information regarding destinations in the network Required origin.dat Contains information regarding the generation links in the network Required vehicle.dat Contains information regarding the number of vehicles to be loaded May be empty path.dat Contains the vehicle trajectory to be used in conjunction with vehicle.dat May be empty SuperZone.dat Contains the mapping of zones to super zones Optional

171 TransCAD data Google Earth and other sources Manual corrections on TransCAD and output of data Use the conversion tool to prepare input data for DYNABUILDER Create network using DYNABUILDER Run the network on DYNASMART- P Properly running? Y Manual correction possible? N STOP Correct DYNASMART-P files manually Y Modify the conversion tool and correct the errors N Figure A.5. Flowchart for the conversion from the static to the dynamic network model.

172 Preparation of DYNASMART-P data files for the simulation and graphical user interface (GUI) consists of several steps. The main tool for this conversion is a software package called DYNABUILDER, which is capable of converting many networks from different platforms into a DYNASMART-P network. However, DYNABUILDER requires input files in a certain format. The conversion of the TransCAD dataset into DYNABUILDER input was established using several different codes and macros based on the need. The package of these custom-made codes and macros are called âconversion toolâ in the flowchart. Signal information was provided as a link feature, which is actually a node feature. Using the conversion tool, the down- stream nodes of the links containing signals were assigned traffic signals, ensuring that the node has more than one down- stream movement and no freeway link is involved. Arterials with more than three lanes are also assigned signals at their downstream nodes using the same conditions. The phasing and movements at the signalized intersections are done by DYNABUILDER. Because no further information is available for the vast number of signalized intersections, all of them are assigned to be actuated signals with default minimum and maximum green times and amber times. Two-way stop and yield signs are assigned also by the conversion tool according to intersection configuration. Ramps entering highways are assigned yield signs, and arte- rial intersections with different lane numbers on the upstream approaches are assigned two-way stop signs. The major/minor approach assignment is also done by the conversion tool. Geometric configuration of the network and movements at the nodes are done by DYNABUILDER using the provided input by the conversion tool. Another important point is the removal of internal and external centroid nodes and internal and external centroid connector links. DYNASMART-P only requires information for physical nodes and links. It creates centroids and connectors implicitly for each zone. Furthermore, all bidirectional links were dualizedâthat is, converted into two one-directional links, which is another requirement for DYNASMART-P. As a result, the DYNASMART-P network has the following properties. Zone Information â¢ 3,697 zones 44 3,586 internal; and 44 111 external. Node and Control Information â¢ 28,406 nodes 44 3,816 uncontrolled; 44 2,625 yield signed; 44 12,944 all-way stop signed; 44 8,054 actuated controlled; and 44 967 two-way stop signed. Link and Type Information â¢ 68,490 links 44 6,026 freeways; 44 169 freeway HOV links; 44 56,102 arterials; 44 37 HOV arterial links; 44 150 highways; 44 2,688 on-ramps; and 44 3,318 off-ramps. Pricing Information There are 297 tolled links: â¢ 291 static tolling; and â¢ 6 dynamic tolling. As seen in Figure A.6, most of the pricing is nondistance based, except along the I-95 New Jersey Turnpike corridor. â¢ The George Washington Bridge, Lincoln Tunnel, Holland Tunnel, Goethals Bridge, Outerbridge Crossing, and Bay- onne Bridge are dynamically tolled bridges. â¢ The Verrazano-Narrows Bridge, BronxâWhitestone Bridge, BrooklynâBattery Tunnel, Queens Midtown Tunnel, Throgs Neck Bridge, Triborough Bridge, Marine Parkway-Gil Hodges Memorial Bridge, Cross Bay Veterans Memorial Bridge, and Henry Hudson Bridge are the bridges and tunnels tolled in New York Metropolitan Area. â¢ The Tappan Zee Bridge, Bear Mountain Bridge, Kings- ton Rhinecliff Bridge, Mid Hudson Bridge, and Newburgh Beacon Bridge are the tolled bridges in New York State. Methodology for Calibration of O-D Demand for Dynamic Analysis Given static O-D demand information and time-dependent link measurements, the dynamic O-D demand estimation pro- cedure aims to find a consistent time-dependent O-D demand table that minimizes (1) the deviation between estimated link flows and observed link counts, and (2) the deviation between the estimated demand and the target demand (based on the static demand matrix). The induced network flow pattern can be expressed in terms of path flows and link flows. In a dynamic context, especially in congested networks, elements of the mapping matrix between O-D demand and link flows are not constant and are, themselves, a function of the unknown O-D demand values. A bi-level dynamic O-D estimation formulation is adapted here to integrate the

173 dynamic traffic assignment constraint. Specifically, the upper- level problem seeks to estimate the dynamic O-D trip desires based on given link counts and flow proportions, subject to non negativity constraints for demand variables. The flow pro- portions are, in turn, generated from the dynamic traffic net- work loading problem at the lower level, which is solved by a DTA simulation program (DYNASMART-P), with a dynamic O-D trip table calculated from the upper level. The weights w1 = (1 - w) and w2 = w in the combined deviations could be interpreted as the decision makerâs relative preference or importance belief for the different objectives; they could also be considered as the dispersion scales for the first and second error terms in the ordinary least-squares estimation procedure. Upper Level: Constrained Ordinary Least-Squares Problem ââ ââÃ âï£®ï£° ï£¹ï£» + âï£®ï£° ï£¹ï£»Ï ÏÏ ÏÏ1 , , , , , , ,, , 2 , 2 , , , 2 , w p q c w q dl t i j i j l t i jl t i j i j i j Subject to nonnegativity constraints qi,j,t â¥ 0 "i, j, t where qi,j,t = estimated traffic demand from zone i to zone j at departure interval t; cl,t = measured traffic flows on link l at observation interval t; di,j = induced traffic demand from zone i to zone j (target demand); pl,t,i,j,t = time-dependent link-flow proportionsâthat is, fraction of vehicular demand flows from origin i to destination j, starting their trips during departure interval t, contributing to the flow on link l during observation interval t; and w1, w2 = weighting factors for the first and second objective functions in the weighted objective function, where w1 is weight for the deviations from observed link flows and the w2 is the deviations from target demand. Lower Level: Dynamic Traffic Assignment Problem [ ] ( )=P DTA Q where P = link proportion matrices contain link-flow propor- tions pl,t,i,j,t; Q = time-dependent O-D demand vector contains elements [qi,j,t] for the subarea network; and DTA = function of dynamic traffic assignment process. Figure A.6. Pricing information.

174 Three types of input are required for this task, namely the link proportions, link observations, and an initial estimate of the O-D demand matrix (target matrix). DYNASMART-P is first run with the target matrix. Then the vehicle trajectory file (DYNASMART-P output) is post-processed to determine the link proportions. The O-D estimation module is then executed and a more-consistent O-D demand matrix is obtained and fed back into the system until convergence. The general procedure for this task is depicted in Figure A.7. Calibration Results and Validation Tests Much effort was spent to come up with the best representative O-D demand matrix. Recognizing some of the data limitations described earlier, it was still possible to develop and cali- brate a reliable dynamic traffic assignment tool that repre- sents the dynamics of traffic in the study area to a reasonable degree, and allows meaningful comparative analysis of alternative scenarios. Figure A.7. Schematic of the procedure for O-D demand estimation. Run DTA and OUTPUT: -Simulated Link Proportions (linkprop.inc) -Simulated link flows RUN GAMS OPTIMIZATION PROGRAM Given: -demand.inc -linkcount.inc -linkprop.inc -sets.inc -index.inc Find: Consistent OD demand matrix OUTPUT New TD OD matrix in GAMS format TD OD ESTIMATION PROCEDURE NO YES STOP OUTPUT final TD OD matrix CONVERT -demand.dat to demand.inc -observed measurements to linkcount.inc INPUT: -Historic (static OD) matrix -Partial link counts GENERATE: -Initial TD OD matrix -LinkWithObs.dat Stopping Criteria RMSE between observed and simulated link flows UPDATE TD OD matrix with estimated new matrix CONVERT Demand matrix from GAMS to DYNASMART-P format (demand.dat)

175 To evaluate the performance of the procedure, the root mean squared error between observed link volumes and simulated link volumes is used as an overall measure of effectiveness. Validation against individual link counts was performed for selected links. Cumulative curves provide insight into the ability of the resulting assignment to capture the link flow volumes. The results are satisfactory in light of the available data, from the aggregate initial demand matrix to the link counts used, and pro- vide encouraging indications for the ability of the DTA tool to support the intended analysis of traffic patterns under various scenarios. For an example of the results of the simulated link volumes compared with observed link volumes, see Figure A.8. Figure A.8. Sample of simulated link volumes versus observed link volumes.