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A Link-Node Nonlinear Complementarity Model for a Multiclass Simultaneous Transportation Dynamic User Equilibria Mohamad K. Hasan Associate Professor Department of Quantitative Methods and Information Systems College of Business Administration, Kuwait University P.O. Box 5486, Safat, 13055, Kuwait Fax: (965) 2483-9406 Tel.: Work: + (965)2498-8453, Mobile: + (965)9782-2073 Email: mkamal@cba.edu.kw And Xuegang (Jeff) Ban Assistant Professor Department of Civil and Environmental Engineering Rensselaer Polytechnic Institute Jonsson Engineering Center, room: 4034 110 8th Street, Troy, New York 12180 Fax: 518-276-4833, Tel.: 518-276-8043 Email: banx@rpi.eduAccepted for Publication in International Journal of Operations Research and Information Systems (IJORIS), IGI Global, USA, May 2011
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Presentation Outlines1. Introduction2. The Nonlinear Complementarity Formulation for DUE3. Multiclass Simultaneous Transportation Equilibrium Model (MSTEM)4. Discrete-Time Domain Finite-Dimensional Nonlinear Complementarity DUE Multiclass Simultaneous Transportation Equilibrium Model (DMSTEM)5. Solution Algorithm6. Conclusions and Future Research
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1. Introduction Transportation Network Equilibrium Models Traffic (Passengers ) Models Freight Models (Hasan UN-ESCWA) Dynamic Models Stochastic Models Deterministic Models Traffic Assignment LU-MUEDSTEM (Hasan and Departure Traffic MUE LU-MUE MSTEM Time (Jeff and et. Assignment (Hasan)Work in and Jeff ) Simultaneous al. 2008) (Jeff and et. al. progress Sequential Models 2008) Models 1- Trip Generation 2- Trip Distribution 3- Modal Split Multiclass Single Class 4- Traffic assignment Variational Inequality Variational Equivalent Generation, Distribution, Inequality Convex program Distribution , Modal Split Modal split, Assignment, and and Assignment (De Cea PGSTEM, GSTEM Departure Time (Hasan and et. al 2003) (Hasan 1991) Dashti 2007) ESTRAUS MSTEM Generation, Distribution, Distribution , Modal Distribution Modal split, and Assignment Split and Assignment and (Safwat and Magnanti 1988) ( Florian and Nguyen Assignment STEM 1978) (Evans 1976)
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Dynamic Traffic Assignment (DTA) models, which aim to predictfuture dynamic traffic states in a short-term fashion, have beenextensively studied for decades. DTA studies accelerated in the lastfifteen years with the advent of intelligent transportation systems(ITS).Within DTA studies, the dynamic user equilibrium (DUE)problem, which tries to determine the distribution of time-dependenttraffic flow of a traffic network assuming travelersfollow rational route choices (such as to minimize travel time or otherforms of cost), is one of the most challenging.Two distinct approaches have dominated the methodologiesapplied to DUE research: The simulation-based (microscopic/mesoscopic) approach. The analytical (macroscopic) approach. Variational Inequality (VI) and Nonlinear Complementarity (NC) have been shown to be the most effective approaches.
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The simulation-based (microscopic/mesoscopic) approach.
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Multiclass DUE models have also been studied with theseanalytical approaches (Ran and Boyce, 1996; Belimer etal., 2003) or simulation-based approaches(Ben-Akiva et al, 2001; Mahmassani et al., 2001).The early studies, however, focused on different vehicleclasses and a few social-economic factors of drivers, suchas value of times (VOD) preferences (Lu et al., 2008).With this goal, Ramadurai and Ukkusuri (2010)integrate activity-based models for demand analysis intoDUE. The model is solved using a super-networkrepresentation of activities and traffic networks.
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By investigating further how socio-economic factors ofdrivers may affect trip generation, trip distribution, modechoice, and departure-times in a dynamic (i.e. time-dependent) fashion, and integrating the findings into DUEmodeling, we expect to improve the practicality andaccuracy of DUE models.With this objective, we combine the NCP-based DUE modelin Ban et al. (2008) and the Multiclass SimultaneousTransportation Equilibrium Model (MSTEM) in Hasan andDashti (2007) to develop a multiclass combined TripGeneration (TG)-Trip Distribution (TD)-Nested Modal Split(NMS)-Modal Split (MS)-Dynamic Trip Assignment (DTA)-Departure Time (DT) Model (NCPDUE-TG-TD-NMS-MS-DTA-DT).
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2. The Nonlinear Complementarity Formulation for DUEBan et al. (2008) evenly divided the entire study period into K time intervals by introducing ththe length of each time interval such that K T . We use index k to denote the k- timeinterval [(k 1) , k ) for 1 k K . The notation for the discrete-time model is first listed asfollows (Ban et al., 2008): ku as u as ((k 1) ) : t he inflow rat e t o link a t owardsdest inat io s at t hebeginningof t imeint erval , n k k wh is also assumed t o be const antduring t heent iret imeint erval , u ich k (u as ) a , s ,k kv as vas ((k 1) ) : t heexit flow rat efrom link a t owardsdest inat io s at t hebeginningof t imeint erval , n k k whi is also assumed t o be const antduring t heent iret imeint erval , v ch k (vas ) a ,s ,k ku as : t heinflows t o link a t owardsdest inat io s during t imeint erval n k kvas : t heexit flows from link a t owardsdest inat io s during t imeint erval n k kxas xas ((k 1) ) : t heflows of link a t owardsdest inat io s at t hebeginningof t imeint erval , n k k which is assumed t o be const antduring t heent ireint erval , x k ( xas ) a ,s ,k k ku s u : t heaggregat edinflow rat e t o link a at t hebeginningof t imeint erval , as k s S which is assumed t o be const antduring t heent ire int erval , u A k k (u a ) a ,k k kv a v : t heaggregat edexit flow rat efrom link a at t hebeginningof t imeint erval , as k s S wh is assumed t o be const antduring t heent ire int erval , v A ich k k ( va ) a ,k k kxa xas : t heaggregat edflows of link a t owardsdest inat io s at t hebeginningof t imeint erval , n k s S wh is assumed t o be const antduring t heent ire int erval , x A ich k k ( xa ) a ,k kd is d is ((k 1) ) : t hedemand rat egenerat edfrom node i t o dest inat io s at t hebeginningof t imeint erval , n k k wh is assumed t o be const antduring t heent ire int erval , d ich k ( d is ) i , s , k ,i s kd is : t hedemands generat ed from node i t o dest inat io s during t imeint erval n k
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C k (u ) C (k ) : t he t ravelt imeof link a at t he end of t imeint ervalk , a funct ionof u, C (C k ) a a a a, k k (k ) : t he minimum t ravelt imefrom node i t o dest inat io s at t heend of t imeint ervalk , n ( k) is is is i, s, k , i s e k (u ) (k 1) C k 1(u ) : t heexit t ime for vehicl ent eringa at t he beginningof t imeint ervalk , es a a a funct ionof u, e (e k ) a a, k Ban et al. (2008) showed that the above discretization scheme leads to the following NCP formulation for DUEwith fixed demand: find (u, ) such that 0 u u (u, ) 0 0 (u, ) 0where the two functions u (u, ) and (u, ) are defined as: (u, ) C k (u ) 3, k , l (u ) l [1 3, k , l (u )] l 1 k ,u a a h s a h s l s l 1 ek 1(u ) / l a a a a a, s , k (1) (u, ) uk dk [ 1, k (u ) 2, k , k (u ) u k 1, k 1(u ) (1 2, k , k (u )) u k 1] as is a a as a a as a A(i ) a B(i ) k : ek (u ) (k 1) ek 1(u ) a a i, s, i s, k (2) ( −1) ℎ = − 1 ∆ + is the exit time of vehicles entering a at the ( −1) beginning of the k-th interval, which is a function of the inflow vector since is so.
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He also define three types of indicator functions on : 1, , 2, , and 3, , . 1,First, flow propagation constraints can be represented by , which is defined as: ∆ 1, = −1 , ∀, (3) − +∆The relation between link inflow and exit flow rates can be represented by both 1, and 2, , with the latter defined as: + +1− ∆ +1 2, = −1 ∀, , , ≤ − 1 Δ (4) − +∆ 3,Similarly, is used to discretize (and interpolate) the minimum travel timeℎ + in equation (1). It can be defined as: +1 3, = − − , ∀ − 1 ≤ (5) ∆ ∆The three indicator functions satisfy the following: 1, 0, ∀, +1 0 2, , ≤ 1, ∀, , , ≤ − 1 Δ +1 3, , 0 ≤ 1, ∀ − 1 ≤ ∆Φ is a vector function for any combination of link a, destination s, and time interval k.Each of its component represents, for the given a; s; k, the difference of the minimumtravel times via two sets of paths from the starting node (or tail node) of link a todestination s at k. The first set of paths must traverse link a, which is a subset of thesecond set that includes all paths from the starting node of a to destination s. This difference should be zero if link a is selected (i.e. the inflow to link a at time k, , isnonzero). Φ , on the other hand, simply represents the flow conservation constraint atany node i, destination ≠ and time k. Detailed discussion on how Φ and Φ arederived can be found in Ban et al. (2008), which also showed (Theorem 3) that NCPDUEhas a nonempty and compact solution set under certain conditions.
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3. Multiclass Simultaneous Transportation Equilibrium Model (MSTEM)A brief description of MSTEM model that developed by Hasan and Dashti (2007).The notation for MSTEM model is first listed as follows:( N , A) = A multimodal traffic network consisting of a set of N nodes and a set of A linksl = User class (e.g., income level, car availability, etc.)L = Set of all user classeso = Trip purpose (e.g., home-based-work, home-based-shopping, etc.)O = Set of all trip purpose I lo = Set of origin nodes for user class l and trip purpose o i = An origin node in the set I lo for user class l with trip purpose o Dilo = Set of destination nodes that are accessible from a given origin i for user class l with trip purpose o s = A destination node in the set Dilo for user class l with trip purpose oR lo = Set of origin-destination pairs is for user class l with trip purpose o , i.e., the set of all origins i I lo and destinations s Dilom = Any transportation mode in the urban arean = Nest of transportation modes m that has a specific characteristics (e.g., pure modes including private and public or combined modes) that are available for user class l with trip purpose o travel between origin-destination pairs is lo is = Set of all nests of modes n that are available for user class l with trip purpose o travel between origin-destination pairs is loM n = Set of all transportation modes m in the nest n for user class l with trip purpose o travel between origin-destination pairs isk = Departure time period for user class l with trip purpose o using mode m in the nest n to travel between origin-destination pairs is
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loKm = Time horizon of the departure time periods k for users of class l with trip purpose o using mode m between origin-destination pairs isa = A link in the set A in the multimodal network ( N , A) = the travel time of vehicles entering link a at the end of the th time interval (=Departure time period k ) for user class l with trip purpose o using mode m in the nest n to travel between origin-destination pairs is Awslo = the value of the w th socio-economic variable that influences trip attraction at destination s for users of class l with trip purpose o W lo lo lo loA s iw g w ( Aws ) =a composite measure of the effect that socio-economic variables, which, w 1 are exogenous to the transport system, have on trip attraction at destination s for users of class l with trip purpose o . lo gw ( Aws ) = a given function specifying how the w th socio-economic variable Awj lo lo influences trip attraction at destination s for users of class l with trip purpose o , and
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The quantities ilo and iw for w 1, 2, ...,W are coefficients to be estimated, where ilo 0 . loDuring the time period required to achieve the short-run equilibrium predicted in the model,socio-economic activities in the system will remain essentially unchanged; the composite effectAlo for users of class l with trip purpose o of these activities is assumed to be a fixed constant. jE loi = the value of the th socio-economic variable that influences the number of trips generated from origin i for users of class l with trip purpose o ,q Eloi = a given function specifying how the th socio-economic variable, E loi , influences the number of trips generated from origin i for users of class l with trip purpose o , andEilo lo q ( E loi ) = a composite measure of the effect the socio-economic variables, 1 which are exogenous to the transport system, have on the number of trips generated from origin i for users of class l with trip purpose oThe quantities lo and lo for 1, 2, ..., are coefficients to be estimate l L, o O .
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loSimilar to As , Eilo is assumed to be a fixed constant during the time period required toachieve short-run equilibrium. = the inflow rate to link a towards destination s at the beginning of the -th time interval for user class l with trip purpose o using mode m in the nest n ,which is assume to be constant during the entire th time interval = the minimum travel time from node i to s destination at the end of time interval k for user class l with trip purpose o using mode m in the nest nS ilo = the accessibility of origin i I lo as perceived from user of class l with trip purpose o traveling from that origin.Gilo = the number of trips generated from origin i for users of class l with trip purpose o .d is k = the number of trips of users of class l with trip purpose o traveling from the origin lonm node i I lo to the destination node s Dilo and whose already chose the mode of lo lo transport m M n from the nest of modes n is and start their trip at the time lo interval k Km . lonmd is = the number of trips of users of class l with trip purpose o traveling from the origin node i I lo to the destination node j Dilo and whose already chose the mode of lo lo transport m M n from the nest of modes n is lond is = the number of trips of users of class l with trip purpose o traveling from the origin node i I lo to the destination node s Dilo and whose already chose the nest of lo modes n is . lod is = the number of trips of users of class l with trip purpose o traveling from the origin node i I lo to the destination node s Dilo .
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MSTEM:S ilo max 0, ln exp ( θ ilo lonmk is lo As ) i I lo , l L, o O s Dilo n lo is lo m Mn k lo KmGilo lo Silo Eilo i I lo , l L, o O exp( θ ilo lonmk is lo As ) lo lo lo lo n m M n k Kmd is Gilo is is R lo , l L, o O exp( θ i lo lonmk is A lo s ) s Dilo n lo is lo lo m M n k Km exp( θ ilo lonmk is lo As ) lo lo lon lo m M n k Km lod is d is , n is , is R lo , l L, o O exp( θ i lo lonmk is A lo s ) lo lo lo n is m M n k Km exp( θ ilo lonmk is lo As ) lo lonm lon k Km lo lod is d is , m Mn , n is , is R lo , l L, o O exp( θ i lo lonmk is A lo s ) lo lo m M n k Km lo lonmk lo lonmk lonm exp( As ) lo lo lod is d is i is , k Km , m Mn , n is , is R lo , l L, o O exp ( θ i lo lonmk is lo As ) lo k Km lonmkd is can also be given by the following combined trip generation, trip distribution, modal split,and departure time demand model (function): lo lonmk lo lonmk lo lo lo exp( i is As )d is ( S i E ) i exp( θilo lonmk is lo As ) j Dilo n lo is lo lo m M n t Km lo lo lo k Km , m Mn , n is , is R lo , l L, o O
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4. Discrete-Time Domain Finite-Dimensional Nonlinear Complementarity DUE Multiclass Simultaneous Transportation Equilibrium Model (DMSTEM)Following the same thought of Ban et al. (2008), the dynamic user equilibrium traffic assignment (NCPDUE-DTA) can be represented by thefollowing two NCPs:0 ≤ ⊥ Φ ( , , , , , , , , , , , ) ≥ 0, ∀, , , (6) 0 ≤ ⊥ Φ ( , , , , , , , , , , , ) ≥ 0, ∀, , , (7) To extend the DTA model above to model simultaneous route and departure time, we assume the early/late arrival penalty function for each user class l with trip purpose o using mode m in the nest n to travel between origin-destination pairs is departing at time (denoted as )i 2 = 0.5 ∇ + − , ∀, , , ≠ ∀ , , , where , ∀, , , ≠ is the desired arrival time for user class l with trip purpose o using mode m in the nest n to travel between origin-destination pairs is departing at time . Therefore, ∇ + − represents early or late arrival for travelers of class lwith trip purpose o using mode m in the nest n departing to at time . In other words, the cost (disutility) for such a traveler at time is 2 Ψ = + = 0.5(Δ + − ) , ∀, , , ≠ , ∀, , , Further, we denote the minimum disutility between and for user class l with trip purpose o using mode m in the nest n(for all time intervals) as = min Ψ , ∀, , ≠ , ∀, , , Therefore, the departure time choice condition can then be expressed as: ≥ 0, Ψ = , ∀, , , ≠ , ∀, , , = 0, Ψ , ∀, , , ≠ , ∀, , , This condition simply states that travelers are trying to minimize their disutility when choosing departure times.
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Combining equations (6) and (7) with the following two NCPs we can combined the dynamic traffic assignment withdeparture time ((NCPDUE-DTA-DT) :0 ≤ ⊥ Φ ( , , , , , , , , , , , ) ≥ 0, ∀, , , (8) 0 ≤ ⊥ Φ ( , , , , , , , , , , , ) ≥ 0, ∀, , , (9) Similarly, combining equations (6)-(9) with the following two NCPs we can combined the Dynamic TrafficAssignment, Departure Time and Modal Split ((NCPDUE-MS-DTA-DT) :0 ≤ ⊥ Φ ( , , , , , , , , , , , ) ≥ 0, ∀, , (10)0 ≤ ⊥ Φ ( , , , , , , , , , , , ) ≥ 0, ∀, , (11) Combining equations (6)-(11) with the following two NCPs we can combined the Dynamic TrafficAssignment, Departure Time, Modal Split, and Nested Modal Split (NCPDUE-NMS-MS-DTA-DT):0 ≤ ⊥ Φ ( , , , , , , , , , , , ) ≥ 0, ∀, (12)0 ≤ ⊥ Φ ( , , , , , , , , , , , ) ≥ 0, ∀, (13) Combining equations (6)-(13) with the following two NCPs we can combined the Dynamic TrafficAssignment, Departure Time, Modal Split, Nested Modal Split, and Trip Distribution(NCPDUE-TD-NMS-MS-DTA-DT):0 ≤ ⊥ Φ ( , , , , , , , , , , , ) ≥ 0, ∀, (14) 0 ≤ ⊥ Φ ( , , , , , , , , , , , ) ≥ 0, ∀, (15)
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Similarly to the formulation of Ban et al. (2008), all Φs functions can be defined as follows:Φ ( , , , , , , , , , , , ) = 3, , ( + { ℎ +1 −1≤ ∆ +[1 − 3, , ]ℎ , +1 } − ℎ ), ∀, , ), ∀, , , (18)Φ ( , , , , , , , , , , , ) = ( – − { [1, 2, , +1 : ≤ −1 Δ 1, +1 +1 + 1 − 2, , ]}, ∀, , ), ∀, , , (19)whereΦ ( , , , , , , , , , , , ) = Ψ − , ∀, , , ≠ ∀, , , (20)Φ ( , , , , , , , , , , , ) = lo lonmk lo exp( As ) − ( lo S ilo E ilo ) i is 0 , ∀, , ≠ ∀, , , , exp ( θ ilo lonmk is lo As ) s Dilo n lo is m lo Mn k lo Km(21) Equations (21) ensure that if the demand and a period time ( departure time ) is positive it shouldbe given by the MSTEM combined trip generation, trip distribution, modal split, and the departure time demanmodel (function) mentioned before.Following the same concept we can define all others Φ ‘s and Φ ‘s functions where all Φ ‘s given byMSTEM model equations.
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Φ ( , , , , , , , , , , , ) = μ − , ∀, , , ≠ ∀, , (22)Φ ( , , , , , , , , , , , ) = exp( θ ilo lonmk is lo As ) lo k Km lonm d is lon d is 0 ∀, , , , ∀, , ≠ (23) exp( θ ilo lonmk is lo As ) lo lo m M n k KmΦ ( , , , , , , , , , , , ) = μ − , ∀, , , ≠ ∀, (24)Φ , , , , , , , , , , , = exp( θ ilo lonmk is lo As ) lo lo lon lo m M n k Km lo d is d is 0 , n is , is R lo , l L, o O exp( θ i lo lonmk is A lo s ) lo lo lo n is m M n k Km , ∀, , ≠ (25)Φ , , , , , , , , , , , = μ − , ∀, , ≠ ∀, (26)
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Φ ( , , , , , , , , , , , ) = exp( θ ilo lonmk is lo As ) lo lo lo lo lo n m Mn k Km d is ( S ilo E ilo ) is 0 is R lo , l L, o O exp( θ i lo lonmk is A lo s ) s Dilo n lo is lo m Mn k lo Km , ∀, ≠ (27)Φ ( , , , , , , , , , , , ) = − , ∀ ∀, (28)Φ , , , , , , , , , , , =([ S ilo max 0, ln exp ( θ ilo lonmk is lo As ) 0] i I lo , l L, o O (29) s Dilo n lo is lo m Mn k lo Km Theorem: The NCPDUE-TG-TD-NMS-MS-DTA-DT model (6) - (17) has a nonempty and compact solution set if the following four conditions are satisfied. (a) The link travel time is positive and finite for any finite ; (b) 1 is bounded from above; (c) ∀, , , ≠ ∀ , , , is positive and bounded from above; and (d) , , , , , , , , , , , are continuous with respect to , , , , , , , , , , , .
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5. Solution Algorithmwe presented a heuristic approach to solve the link-node based simultaneous TripGeneration (TG) - Trip Distribution (TD) - Nested Modal Split (NMS) - Modal Split(MS) - Dynamic Trip Assignment (DTA) - and Departure Time (DT) Model(NCPDUE-TG-TD-NMS-MS-DTA-DT). The solution convergence condition of thealgorithm cannot be established mainly due to the fact, as shown in the definitions offunctions , , , , , , , , , , , in Equations (18)-(19), that the model is not close-formed. However, for agiven demands vectors , , , , (called base demand) and its resulting inflow vector (called base inflow), all the indicator functions and theexit times (e) can be calculated and fixed via a so-called dynamic network loadingprocedure (see Ban et al. 2008). This leads to the so-called relaxed sub-problems thatare explained in details in the appendix.
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DSTEM Algorithm:Step 1: Initialization. Assign a base demands vectors () () () () () () , , , , and a base inflow vector Step 2: Major Iteration. Set = . Step 2.1: Construct and solve RNCPDUE-DTA (A1) - (A2). Repeat this () times. Each time using as the fixed demand. For the first () time, use as the base inflow. For other times, use inflow from the previous solve as the base inflow. Denote the final inflow (after () the solves) as . Step 2.2: Construct and solve RNCPDUE-DTA-DT (A5) - (A8). Repeat this () times. For the first time, use as the base inflow and use () as the initial demand. For other times, use inflow and demand obtained from the previous solve as the base inflow and initial demand. Denote the final inflow and demand (after the solves) as and respectively.
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Step 2.3: Construct and solve RNCPDUE-DTA-DT-MS (A10) - (A15). Repeat () () this times. For the first time, use and as the () base inflow and base demand respectively and use as the initial demand. For other times, use inflow and demands obtained from the previous solve as the base inflow and initial demand. Denote the final () () inflow and demands (after the solves) as , , and () respectively.Step 2.4: Construct and solve RNCPDUE-DTA-DT-MS-NMS (A17) - (A24). () () Repeat this times. For the first time, use , , and () () as the base inflow and base demands respectively and use as the initial demand. For other times, use inflow and demand obtained from the previous solve as the base inflow and initial demand. Denote () the final inflow and demands (after the solves) as , () () () , , and respectively.Step 2.5: Construct and solve RNCPDUE-DTA-DT-MS-NMS-TD (A26) - () (A35). Repeat this times. For the first time, use , () () () , , and as the base inflow and base demands () respectively and use as the initial demand. For other times, use inflow and demand obtained from the previous solve as the base inflow and initial demand. Denote the final inflow and demands (after the () () () () () solves) as , , , , and respectively .
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Step 2.6: Construct and solve RNCPDUE-DTA-DT-MS-NMS-TD-TG (A37) - (A48). () () () () Repeat this for times. For the first time, use , , , , () () and as the base inflow and base demands respectively and use as the initial demand. For other times, use inflow and demand obtained from the previous solve as the base inflow and initial demand. Denote the final inflow and () () () () () demands (after the solves) as , , , , , () and respectively . Call them the candidate solutionStep 2.7: Convergence Test. If certain convergence criterion is satisfied at the candidate solution, go to Step 3; otherwise, go to Step 2.8. Step 2.8: Update and Move. Set (+) () = + ( − ) (+) () = + ( − ) (+) () = + ( − ) (+) () = + ( − ) (+) () () () = + ( − ) (+) () () () = + ( − ) Set n = n + 1 and go to Step 2.1. () () () () () ()Step 3 Find an optimal solution , , , , , and
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In Step 2.1 and 2.2 of DSTEM Algorithm, the calculation of the indicator functionsand exit times are done via dynamic network loading, as detailed in Nie and Zhang(2005) and Ban et al. (2008). The relaxed sub-problems RNCPDUE-DTA andRNCPDUE-DTA-DT, RNCPDUE-DTA-DT-MS, RNCPDUE-DTA-DT-MS-NMS,RNCPDUE-DTA-DT-MS-NMS-TD, and RNCPDUE-DTA-DT-MS-NMS-TD-TG aresolved directly by the PATH solver in GAMS (Ferris and Munson 1998). For detaileddiscussions, one can refer to Ban et al. (2008) In Step 2.7, there are several commonly used gap functions that can be used inthe convergence test. The first one is based on the difference of the candidate solutionand the base solution. In particular, we can check whether the following conditionholds: = − + − + − − + − + − ∀, (30) Especially if = , = , − , = , = , and = we obtain a fix point solution to the original problem NCPDUE-DTA-DT-MS-NMS-TD-TG. We can also check directly the conditions of route choice, departure time choice,nest mode choice, mode choice from the nest, trip distribution choice, and tripgeneration choice by defining the second gap function: = + + + + + ∀, () Similarly, if = , both conditions are satisfied, the obtained candidatesolution is indeed an optimal solution. In (30) and (31), and are both smallscalars representing user-defined convergence accuracy.
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6. Conclusions and Future ResearchIn this paper we presented a link-node based discrete-time NCP model forDUE that combined multiclass trip generation, trip distribution, nestedmodal split, specific modal split, dynamic trip assignment and departuretime (DMSTEM) in a unified formulation.This model combined the dynamic link-node based discrete-time NCPmodel that developed by Ban et al. (2008) and the static MulticlassSimultaneous Transportation Equilibrium Model (MSTEM) that developed bHasan and Dashti (2007).We also developed an iterative solution algorithm for the proposed modelby solving several relaxed NCP in each iteration.
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Most of the recent DUE models deal only with the traffic assignment and fixeddemand flows between each O-D pair at each time period of the time horizon or anelastic dynamic demand (input to DTA models) that is usually modeled as departure-time choice for single user (traveler) class.To apply DTA with departure-time choice (DTA-DT) in practice, we require fixeddemand flows between each O-D pair during the whole time horizon and the DTA-DT tobe embedded in more comprehensive transportation planning framework that results inan inconsistence prediction process.The DMSTEM model will generate a better solution than this inconsistent predictionprocess within a comprehensive planning and operations framework.
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We would apply the DMSTEM and it solution algorithm to a prototypeexample to get more insight and implication of its predictions andvalidity.This prototype solution will encourage applying the model to realworld transportation network. A comparative study betweenDMSTEM as a dynamic model and MSTEM as static model in term oftheir prediction accuracy and solution complexity is very desirable inthe transportation planning in general and in particular, theIntelligence Transportation System (ITS).Transportation network analysis tools can be broadly categorized asstatic analysis tools and dynamic analysis tools. The former focuses onlong-term, steady traffic states, often referred as the four-step demandanalysis models (Sheffi, 1985), while the latter focuses on short-term,dynamic traffic states, often called dynamic network analysis ordynamic traffic assignment (DTA, see Peeta and Ziliaskopoulos (2001)).Traditionally, these two types of models have been applied in quitedistinct scenarios: static tools for transportation planning purposes anddynamic tools for traffic operation purposes.
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It would be ideal to combine the planning and operation tools to addresstransportation problems in a more comprehensive manner. In practice,such combination has been recently experimented and tested. Examplesinclude the corridor management plan demonstration project (CCIT,2006) and the state-wide corridor system management plan that iscurrently underway in California, which aims to integrate long-termplanning tools (such as the static four-step demand analysis models) withoperational tools such as microscopic traffic simulation and dynamictraffic assignment. The ICM program by FHWA also focuses oncombining different transportation analysis tools for integrated corridorplanning and operations.Combining planning and DTA models in theory is not trivial because thetwo types of models are based on quite distinct assumptions. In thispaper we present a combined model to simultaneously consider theorigin, destination, mode, departure, and route choices of multiclasstravelers.
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