A Hybrid Deployable Dynamic Traffic Assignment Framework for Robust Online Route Guidance

Transcription

1 A Hybrid Deployable Dynamic Traffic Aignment Framework for Robut Online Route Guidance Sriniva Peeta School of Civil Engineering, Purdue Univerity Chao Zhou Sabre, Inc. Sriniva Peeta School of Civil Engineering Purdue Univerity Wet Lafayette, IN Phone: (765) Fax: (765)

2 A Hybrid Deployable Dynamic Traffic Aignment Framework for Robut Online Route Guidance Sriniva Peeta 1 and Chao Zhou 2 Abtract Randomne in time-dependent origin-detination (O-D) demand and/or network upply condition, and the computational tractability of potential olution methodologie are two major concern for the online deployment of dynamic traffic aignment (DTA) under realtime traffic management ytem. Mot exiting DTA model ignore thee concern and/or make unrealitic aumption, precluding their online applicability. In thi paper, a hybrid approach coniting of offline and online trategie i propoed to addre the online tochatic dynamic traffic aignment problem. The baic idea i to addre the computationally intenive component offline, while efficiently and effectively reacting to the unfolding condition online. The offline component eek a robut initial olution vi-à-vi randomne in O-D demand uing hitorical O-D demand data. Termed the offline a priori olution, it i updated dynamically online baed on unfolding O-D demand and incident. The framework circumvent the need for accurate O-D demand and incident likelihood prediction model online, while exploiting hitorical O-D demand and incident data offline. Reult of imulation experiment highlight the robutne of the hybrid approach with repect to online variation in O-D demand, it ability to addre incident ituation effectively, and it online efficiency. Keyword: Hybrid deployment trategie, dynamic traffic aignment, real-time operation, tochatic optimization 1 School of Civil Engineering, Purdue Univerity, Wet Lafayette, IN USA 2 Sabre Inc., Southlake, TX USA

3 1. INTRODUCTION 1.1 Background and Motivation The traffic aignment problem aim to determine path to be aigned to network uer that atify certain objective, and/or to etimate the network tate. Traditional tatic traffic aignment model (Sheffi, 1985), motivated by long-term planning need, aume the origin-detination (O-D) demand, traffic flow, and network upply condition to be timeinvariant over the planning horizon of interet, implying contant trip time on network path. Thee aumption may uffice for non-peak network traffic condition. Randomne in uer behavior can be introduced, for example, by viewing uer perceived travel time a random variable, leading to the tochatic traffic aignment problem (Sheffi and Powell, 1982). If O-D demand, traffic flow, and network upply condition are viewed a time-dependent, it reult in the dynamic traffic aignment (DTA) problem. The DTA problem aim to determine the optimal time-dependent path (or path aignment proportion) and/or network tate that atify ome ytem-wide and/or uer objective. A key characteritic of DTA i the dependence of the current olution on future traffic condition. The ue of advanced technologie in recent year under the aegi of Advanced Traffic Management Sytem (ATMS) and Advanced Traveler Information Sytem (ATIS) ha generated ubtantial interet in DTA. It functional need range from the etimation of the dynamic network tate to peronalized route guidance. Several DTA model have been propoed in the literature, but ignificant difference exit among them (ee Peeta, 1994; Peeta and Ziliakopoulo, 2001) in term of: (i) aumption on the availability of information on O-D demand and/or network upply condition for the planning horizon, (ii) information availability to network uer, (iii) information upply trategie, and (iv) uer repone to the upplied routing information. If the time-dependent O-D demand and network upply condition are aumed known a priori for the entire planning horizon, the correponding model are termed determinitic dynamic traffic aignment (DDTA) model. Mot model in the literature fall into thi category (Janon, 1991; Mahmaani et al., 1993; Ran, et al., 1993; Boyce et al., 1995; Peeta and Mahmaani, 1995a; Wie et al., 1995). However, the tate-ofthe-art in O-D demand and incident likelihood prediction model cannot guarantee high level of accuracy, precluding the deployment of DDTA model online 3. In addition, they are computationally intenive online in a centralized architecture. Neverthele, they erve a ueful benchmark for online DTA trategie. If the ource of randomne inherent to traffic ytem are conidered, the aociated problem i labeled the tochatic DTA problem. Thee ource include randomne in O-D demand, network upply condition (primarily, incident), and uer repone to upplied information. Due to their computational burden, DDTA model can primarily be addreed offline. However, real-time operation entail the deployment of DTA online. Hence, online DTA model additionally eek computational tractability. When the focu i on incorporating the randomne in the unfolding condition online, it reult in the online tochatic dynamic traffic aignment (OSDTA) problem. Given the ignificant impact that randomne ha on the online ytem performance, it i imperative to addre OSDTA model to develop deployable trategie under ATIS/ATMS. To addre online the iue of randomne in O-D demand and network upply condition, Peeta and Mahmaani (1995b) propoe a tage-baed rolling horizon approach 3 Online i ued in the context of real-time operation. 1

4 for implementation. It divide the time horizon of interet into everal tage and olve a DDTA problem for each tage equentially over time. Each tage conit of a roll-period for which the optimal path aignment proportion are to be determined. The ret of the tage repreent the future duration conidered in obtaining thoe aignment proportion. The next tage i obtained by rolling the current tage by a time length equal to the roll-period. The rolling horizon approach i more realitic than a DDTA model a O-D demand forecat are required only for a tage rather than the entire horizon of interet. If highly accurate O-D demand forecat are available for the near-term future, a near-optimal olution can be expected for the unfolding traffic condition. Since it i tage-baed, the rolling horizon approach enure that unpredicted variation in online traffic condition (uch a incident) can be accounted for in ubequent tage. However, if the actual O-D demand in a tage deviate ignificantly from the forecat, thi approach can be ub-optimal. Another drawback i it computational inefficiency in a centralized ATIS/ATMS architecture. To addre the online computational burden, Hawa and Mahmaani (1995) propoe a reactive local heuritic rule baed approach within a decentralized architecture. An advantage of thi approach i it flexibility in deciding the territory ize of each controller baed on the controller computing capabilitie. However, due to it pure reactive nature, it doe not exploit hitorical data on O-D demand and incident. Hence, it performance can degrade ubtantially from that of the centralized rolling horizon approach in the abence of incident (Hawa and Mahmaani, 1997). Pavli and Papageorgiou (1999) ue a decentralized feedback control DTA trategy to enable computational tractability online. It react to real-time meaurement to etablih equal intantaneou travel time on alternative route for an O-D pair. Akin to Hawa and Mahmaani (1995), it circumvent the need for O-D demand and/or network upply prediction, and ue a decentralized logic. However, due to it pure reactive logic, uer repone behavior and other underlying procee are not conidered, retricting it robutne to pecific network topologie. Alo, hitorical data i not exploited. In thi paper, we preent a hybrid olution approach to addre two key online DTA concern: (i) the randomne in O-D demand and/or network upply condition, and (ii) the computational burden. Formulated a a multiple uer clae OSDTA model, the deployment problem i addreed through a combination of offline and online trategie. The baic idea of the approach i to addre the computationally intenive component offline by exploiting hitorical data to generate a robut initial olution that can be efficiently updated online baed on unfolding O-D demand and/or incident condition. A robut olution i viewed here a a et of time-dependent path aignment proportion that minimize the expected ytem travel time vi-à-vi the randomne in O-D demand. The hybrid approach circumvent the need for online O-D demand and incident likelihood prediction model. Thi i ignificant becaue exiting O-D demand and incident likelihood forecat model are unable to guarantee high level of accuracy. While other online iue uch a randomne in uer repone to upplied information alo exit, they are more effectively addreed online through conitency checking model (Peeta and Buluu, 1999) a they are dependent on the unfolding O-D demand and/or network upply condition. 1.2 The Online Stochatic Dynamic Traffic Aignment Problem 2

5 The deployment of DTA for real-time operation require capabilitie to addre the online iue dicued in Section 1.1. The idealized online DTA problem eek, in ub-real time, the optimal path aignment proportion for the next aignment interval that minimize the aociated ytem travel time while atifying uer cla objective. However, thi olution can be obtained only if a priori knowledge i available on the future O-D demand and network upply condition. Hence, to enable real-time deployment, we formulate an OSDTA problem (Zhou, 2002) where the focu i on minimizing the expected ytem travel time for the next aignment interval by conidering the randomne in O-D demand and network upply condition. The OSDTA problem i a follow: Conider a traffic network repreented by a directed graph G(N, A), node n N and directed arc a A, with multiple origin i I and detination j J. A node can repreent an origin, a detination, both, or jut a junction of phyical link. The time period of interet, labeled the planning horizon, i dicretized into mall equal interval called aignment interval, τ = 1,, T. For generality, aume multiple uer clae u U for vehicle in term of information acceibility, information upply trategy, and driver repone to the upplied information. Alo, aume that the hitorical time-dependent O-D demand ditribution u Rij τ u (µ, σ), with mean µ ij τ u and tandard deviation σ ij τ, are available i I, j J, u U and τ = 1, T. In addition, hitorical incident data i available in term location, tart time, duration, t and everity. For the current day, given the cumulative number of O-D deire ν ij for all O-D pair up to the current time t, t = 1, T, a well a the network tate and upply condition up to time t, the OSDTA problem eek to determine the path aignment proportion f τ u ijk( u ) to aign vehicle in the next aignment interval τ = t+1, uch that the aociated expected ytem travel time i minimized and uer cla objective are atified. Here, f τ u ijk( u ) i the proportion of vehicle of O-D pair (i, j) of cla u aigned to path k(u) Kij u in interval τ. Thi paper i organized a follow. Section 2 dicue the hybrid OSDTA olution framework and the aociated offline and online trategie to olve the problem. Section 3 analyze the effectivene of the framework through imulation experiment. Concluding comment are preented in Section THE HYBRID SOLUTION FRAMEWORK 2.1 The Hybrid Solution Logic Figure 1 illutrate the hybrid olution framework coniting of offline and online component to addre the OSDTA problem. It i baed on the a priori optimization concept (Jaillet, 1988; Bertima et al., 1990; Laporte et al., 1994) whereby a robut initial olution i determined offline which i then updated efficiently online baed on the unfolding condition on a given day. The randomne in O-D demand i addreed offline. The randomne in network condition, primarily incident, i addreed online. The offline component ue hitorical O-D demand data to determine a robut initial olution for online implementation. The online component ue an efficient and reactive online dynamic update heuritic (ODUH) to addre the demand and incident condition unfolding online. The hybrid approach ha everal advantage. The offline component i not contrained computationally and can eek 3

6 robutne uing intenive method (uch a Monte Carlo imulation) and by olving DDTA problem. For the ame reaon, it can alo exploit huge amount of hitorical data. Due to the robutne of the offline olution, it dynamic update online i enabled in ub-real time. It hould be noted that no DTA problem i olved online. Hitorical O-D demand and incident data are repreented a time-dependent probability ditribution (Peeta and Zhou, 1999). In the offline component, everal realization of O-D demand and incident are obtained from thee ditribution. Le likely O-D demand cenario are filtered out and the number of O-D demand realization for the offline trategie, L, i determined by the tolerable error in ytem travel time baed on the central limit theorem (Peeta and Zhou, 1999). Stochatic quai-gradient (SQG) method (Ermoliev, 1983; Ermoliev and Wet, 1988) are applied to obtain the robut initial olution, termed the offline a priori (OFAP) olution. It incorporate only the randomne in O-D demand and not incident. Incident are viewed in our approach a online event that can be more robutly addreed online. Thi i becaue the marginal effect of the conideration or excluion of an incident on the robutne of the OFAP olution and on the aociated network flow pattern i ubtantially larger than the conideration or excluion of an O-D deire. Incident have a potentially larger patial and temporal reach by affecting everal network path unlike an O-D deire. Therefore, the incluion of an incident that doe not occur online, or vice vera, may caue the offline olution to degrade ufficiently to overhadow the robutne gained by incorporating O-D demand randomne (Zhou, 2002). Hence, incident are addreed online when they occur. Two other computationally intenive component are executed offline for online uage. Firt, an offline heuritic (OFH) i ued to generate a robut et of path aignment proportion vi-à-vi O-D demand for ue in the ODUH. The DDTA olution uing the multiple uer clae time-dependent traffic aignment (MUCTDTA) algorithm (Mahmaani et al. 1993; Peeta, 1994) for each O-D demand realization i computed to determine it optimal path aignment proportion. The optimal path aignment proportion for all L O-D realization are combined (Peeta and Zhou, 1999) to generate the OFH olution for ue in the online component. The econd component involve the determination of the optimal path aignment proportion uing the MUCTDTA algorithm for likely ingle incident cenario under the mean O-D demand matrix. The mean O-D demand matrix i obtained uing the mean value of the time-dependent O-D demand ditribution. The aociated path aignment proportion are ued in the ODUH to addre incident ituation. The two component have two common characteritic. Both exploit hitorical data and involve computationally intenive DDTA olution computation. The OFAP olution i the default olution online. The ODUH ha two component, one for non-incident cenario (ODUH-NI) and another for incident cenario (ODUH-I). The ODUH-NI component ue the OFAP and OFH olution to update the path aignment proportion in the abence of incident. The ODUH-I component, ued under incident, combine the OFAP and OFH olution with the optimal path aignment proportion for the offline ingle incident cenario. In non-incident ituation, the focu i on enuring robutne with repect to the actual O-D demand pattern unfolding online. Thi i done by excluding thoe offline O-D demand realization which deviate (beyond predefined threhold level) from the actual O-D demand pattern known up to the current time on a pecific day, and re-computing the path 4

7 aignment proportion a dicued in Section A ynergitic feature in term of computational efficiency i the reduction of the online computational effort with the progre of the planning horizon, a more O-D demand realization are eliminated with increaing knowledge on the unfolding O-D demand pattern. If all offline realization are eliminated at ome point in the planning horizon, which can occur either toward the end of the planning horizon or if a very unlikely O-D pattern occur online, the OFAP olution i ued online for the relevant tage. The ODUH-I component i an extenion of ODUH-NI component to include incident cenario, and i dicued in Section Several related feature and advantage entail further dicuion. Only likely ingle incident cenario, baed on pat incident data, are olved offline. Of greater ignificance, they are olved only for the mean O-D demand pattern to enure offline computational tractability while addreing incident cenario. The augmentation of the ODUH-NI uing the offline ingle incident cenario olution ha two purpoe. Firt, robutne with repect to randomne in O-D demand i accounted for by the ODUH-NI component a the incident cenario olution i baed only on the mean O-D demand matrix. Second, recognizing that incident are ignificant online event in term of ytem performance, the offline incident cenario olution enure that ignificant flow pattern change due to an incident are incorporated in the ODUH-I. Therefore, if the incident effect i not ignificant, the effectivene of the online update olution i primarily baed on accounting for the randomne in O-D demand, and vice vera. Another favorable feature of ODUH-I i that multiple online incident cenario are alo addreed uing ingle incident offline cenario only. Given the potentially large number of incident combination for general network, olving them even offline may be: (i) prohibitively expenive, and (ii) unneceary given that incident are bet addreed online. 2.2 Offline Component The offline component primarily addree the offline tochatic DTA problem. A SQG algorithm i propoed to obtain the OFAP olution. In addition, the OFH and ingle incident DTA olution are computed The offline tochatic dynamic traffic aignment problem u Given a et of time-dependent O-D demand ditribution Rij τ (µ, σ) for the planning horizon, the offline tochatic DTA problem eek a vector of time-dependent path aignment proportion f τ u u ijk( u ), i, j, τ, u, and k(u)=1,... K ij, that minimize the expected ytem travel time while atifying uer cla objective. The number of uer from origin i to τ u,β detination j aigned to path k(u) in interval τ under O-D pattern β, r, ijk(u i obtained a: ) β β τu ijk( u) = rij f ijk ( u ), i, j, τ, u, and k(u) (1) r τ u,β where r ij i the number of vehicle of cla u who wih to depart from i to j in time interval τ under O-D pattern β. 5

8 2.2.2 The tochatic quai-gradient method The offline tochatic DTA problem i olved uing a SQG algorithm. SQG method are tochatic olution procedure for olving optimization problem with non-convex, nondifferentiable objective function and general contraint (Ermoliev, 1983). They generalize the tochatic approximation method for uncontrained optimization of the expectation of a random function to problem involving general contraint. The baic concept i to ue aymptotically conitent etimate, rather than precie value, for the value of function and their derivative while earching for the optimal olution. Under SQG, the tep direction are etimated by ampling. SQG method generally addre the following problem type: Minimize F(x) = E ω [g(x,ω)] (2a) ubject to x X R n (2b) Here, x repreent the vector of deciion variable to be optimized, and X i a et of contraint. ω i a random variable belonging to the appropriate probability pace. Hence, the objective function F(x) i the expected value of the function g(.) obtained by conidering the randomne in ω. The approach conider a limited number of obervation of the random function g(x,ω) at each iteration to determine the random tep direction ψ. Hence, the tep direction may be a tatitical etimate of the gradient (or ubgradient in the non-differentiable cae) of function F(x); then ψ ξ uch that: where the vector E ( x, x, x,..., x ) = Fx ( x ) + a a may depend on ( x, x, x,..., x ) ξ (3). For exact convergence to an optimal olution, at ome point we hould have a 0a. The vector tochatic quai-gradient when differentiable function) when a a ξ i called a 0, or tochatic ubgradient (tochatic gradient for 0. Generally: 1 ξ = L L l= 1 g x ( x l, ω ) for realization l = 1, L, and 1 ω,, L ω are random ample, when it i poible to obtain the gradient or ubgradient in a computationally inexpenive manner. If the gradient or ubgradient cannot be calculated, tatitical etimate of the gradient direction can be obtained uing finite-difference, random earch, or other heuritic method. The approximation of the optimal olution at tep +1 i computed a: +1 x = π X [ (4) x - ρ ξ ] (5) where π X i the projection operator which enure that all deciion variable value are within the feaible domain. A imple choice for ρ, the tep ize, i one that atifie the following propertie (Ermoliev and Wet, 1988): 6

9 ρ > 0, ρ =, 2 ρ < (6) =0 =1 A comprehenive expoition of the SQG approach for the offline tochatic DTA problem i provided in Zhou (2002). The following ub-ection dicu variou apect of the SQG method to determine the OFAP olution. Choice of tep direction Stochatic quai-gradient are uually calculated uing gradient or ubgradient obtained directly or from finite difference approximation. In our problem, the exact form of the objective function i unknown. Hence, convexity and differentiability are not guaranteed. Thee are inherent characteritic of DTA formulation that adequately incorporate the traffic flow apect (Peeta and Ziliakopoulo, 2001). Hence, the gradient or ubgradient cannot be obtained directly. We ue imulation to circumvent thee iue. Firt, imulation i ued to etimate the objective function (total ytem travel time) value for each realization l. Second, it i ued to determine the move direction for the deciion variable. Their computation typically entail projecting the change in ytem performance due to a mall change to each deciion variable while keeping the other deciion variable unchanged. However, thi implie one imulation for determining the decent direction for a ingle deciion variable in each earch iteration. Thi i prohibitively expenive for general network, even offline, a it implie multiple imulation per realization. The propoed SQG olution algorithm circumvent thi iue by conducting only one imulation per realization per iteration to etimate thee direction, and then compute the quai-gradient. The direction are obtained uing auxiliary olution within the SQG algorithm. Since the objective function can be non-convex and/or non-differentiable, there i no guarantee of a decent direction in each iteration, implying a random direction vector. However, thi i not a practical barrier a highlighted by everal previou imulation tudie on variou network (for example, Peeta and Mahmaani 1995a; Peeta and Zhou, 1999) which how mooth convergence of the objective function, or a decent direction on average. Thi i further corroborated by the experiment conducted in thi tudy even when tochatic quai-gradient are ued. For realization l in time interval τ, the leat marginal travel time path and hortet travel time path are obtained from the previou iteration olution. They are ued to determine an auxiliary olution in term of path aignment proportion, l, y ijk, ( u) uing an all-or-nothing aignment. The earch direction for realization l at tep i obtained a: l, d ijk = ( u) l, y ijk - ( u) f ijk ( u ) i, j, τ, u, k(u) (7) where l, d ijk i the direction for a given i, j, τ, k, l, and u, and ( u) Thee direction are averaged over all realization: f ijk ( u ) i the current olution. d τu, 1 l, ijk( u ) = d L ijk ( u) l i, j, τ, u, k(u) (8) 7

10 The vector of all d τu, ijk( u ) repreent the tochatic quai-gradient ξ. Choice of tep ize 1 ρ = atifie condition (6), and i ued a a imple tep ize in our algorithm in equation (5). It implie an increaing relative weighting of the current olution over the auxiliary olution a the iteration number increae. Thi provide an incentive for convergence a. However, there i no guarantee of convergence ince the objective function form i unknown precluding aumption of convexity and/or differentiability. The ue of a predetermined move ize circumvent the inability to analytically optimize the move ize. Solution update and projection operator The olution update i performed uing the method of ucceive average (MSA) (Wilde, 1964; Powell and Sheffi, 1982). Since it i a convex combination method, it enure that the new olution i in the feaible region, precluding the need for an explicit projection operator. Thi update procedure i ued for only ome uer clae u U *, a dicued in the next ection. The MSA i employed a follow:, 1 ijk u ( u + ) f τ = 1 f ijk + ( u) d τu, ijk( u ), i, j, τ, u U *, k(u) (9) Stopping criteria Several topping criteria are poible. One criterion could be to top when the improvement in the objective function i le than a pre-et value for ucceive iteration. Another criterion could be to top when the tep ize i maller than a pre-et value. Here, the former criterion i ued The SQG olution algorithm The SQG olution algorithm i a generalization of the determinitic MUCTDTA algorithm (Peeta and Mahmaani, 1995b) to the problem incorporating randomne in O-D demand. Four uer clae (Mahmaani et al., 1993) are conidered for algorithmic completene, a being repreentative of poible uer clae in term of information availability, information upply trategy, and uer repone behavior under ATIS/ATMS. The uer clae are: (1) PS; unequipped driver who follow pre-pecified path, which may be projected from hitorical databae or olved for exogenouly, (2) SO; equipped driver who follow precribed ytem optimum path, (3) UE; equipped driver who follow uer equilibrium route, and (4) BR; equipped driver who follow a boundedly-rational witching rule (Mahmaani and Stephan, 1988) in repone to decriptive information on prevailing traffic condition. The boundedly rational path witching rule tate that uer witch from the current path at a deciion point (typically a node) if the intantaneou travel time aving on an alternative route exceed a certain threhold. Figure 2 how the framework for the SQG algorithm. The algorithm firt generate L likely time-dependent O-D demand realization from hitorical data. At each iteration, a 8

11 traffic imulator i ued to evaluate the ytem performance under the current olution for each realization l. The imulation reult are ued to determine the earch direction for the SO and UE clae for that realization. The path of the BR uer cla are obtained from the traffic imulator while thoe of PS cla uer remain unchanged. The earch direction for all realization are ued to determine the tochatic quai-gradient for that iteration. The olution for the next tep i obtained from the tochatic quai-gradient and the current olution uing the MSA. The algorithm i a follow: Step 0: Generate L likely time-dependent O-D demand realization vector from hitorical ditribution. Set the iteration counter = 0. Step 1: For each generated O-D realization, aign the O-D deire of the equipped uer u,l,0 clae rij τ, i, j, τ, l, and u = 2,...4, to a time-dependent initial et of feaible path. The path of all unequipped vehicle are aumed to be known a priori and are part of the initial condition. Hence, τ1, l,0 r ijk ( 1), i, j, τ, and l, are known, and due to the lack of information acceibility are aumed to remain unchanged throughout the iterative earch proce. After computing the initial path for all realization, obtain the vector of the path aignment proportion averaged acro realization u,0 f τ ijk ( u, i, ) j, τ, u = 1, 4, and k(u). Thi repreent the initial olution. Set the realization counter l = 1. Step 2: For realization l, obtain the et of path aignment l, r ijk(u) uing u, f τ : ijk(u) and l, r ijk(u) = l, u, r ij f τ, i, j, τ, u=2-4, k(u) (10) ijk(u) τ1, l, r ijk(1) = τ1, l,0 r ijk(1), i, j, τ, k(1) (11) The et of path aignment l, r ijk obtained from ( u) f ijk ( u for the entire horizon of ) interet are imulated uing a traffic imulator, DYNASMART (Jayakrihnan et al., 1995). The imulation reult provide everal link level and aggregate performance meaure, including the ytem travel time for realization l. Step 3: Compute the link marginal travel time (Peeta, 1994; Peeta and Mahmaani, 1995a) for SO uer uing the time-dependent experienced link travel time and the number of vehicle on link obtained a pot-imulation data from Step 2. Step 4: Compute the time-dependent leat marginal travel time path and hortet travel time path. Step 5: Perform an all-or-nothing aignment of all O-D deire l, r ij for given i, j, τ, for the SO and UE clae. Aign the SO uer to the leat marginal travel time path and the UE uer to the hortet average travel time path. The reult i a et of auxiliary path aignment. The correponding proportion of vehicle are the auxiliary olution, 9

12 u y τ,, ijk ( l u, i, j, τ, k(u), u = 2, 3. Obtain the earch direction l, ) d ijk for realization ( u) l uing equation (7). Step 6: If l = L, go to Step 7; otherwie et l = l + 1, and go to Step 2. Step 7: Compute the average ytem travel time over all realization. Thi repreent the objective function value in iteration. Check for convergence uing the topping criterion (difference in objective function value in two ucceive iteration). If the criterion i atified, top the computation and pecify f ijk ( u, i, j, τ, u, and k(u), a ) the SQG olution. Otherwie, go to Step 8. Step 8: Calculate the tochatic quai-gradient ξ by computing d τu, ijk( u ), i, j, τ, k(u), u = 2, 3, uing equation (8). Update the proportion of vehicle to be aigned to path for the SO and UE cla uer, 1 fijk τu ( u + ) uing the MSA a hown in equation (9). Update the path aignment proportion for the BR uer cla by averaging acro realization, and normalize them. Set = +1, l = 1, and go to Step 2. The path aignment proportion for the SO and UE clae for a pecific realization under the a priori olution do not necearily atify the SO and UE principle, repectively (Zhou, 2002). Thee principle are atified only in the expected ene over all realization a the SQG algorithm determine an average quai-gradient uing all realization in an iteration. Thi i more robut and general compared to olving for path aignment proportion uing a ingle O-D demand pattern The offline heuritic The SQG olution provide a ingle path aignment proportion vector for all realization. However, a dicued in Section 2.1, the online reactive update for the unfolding O-D demand involve eliminating unlikely O-D demand pattern in each tage of the planning horizon for a given day and re-computing the path aignment proportion after excluding their contribution. The OFH enable thi by eparating the contribution of each realization while generating another robut offline olution, called the OFH olution. It i obtained by generating the DDTA olution for each realization, and then averaging them to obtain a vector of path aignment proportion (Peeta and Zhou, 1999). The reaoning for thi approach i that the optimal path aignment proportion for a realization provide a favorable path et for that realization, which can then be combined with uch path et for other realization to generate a robut vector of path aignment proportion relative to O-D demand. The OFH i imilar to the SQG algorithm. It i decribed comprehenively in Peeta and Zhou (1999). Here, it i decribed by elaborating the tep that are different from thoe in the SQG algorithm: Step 0: Set realization counter l = 1. Step 1: Set the iteration counter = 0. Generate the initial path et for realization l for u = 1, 4, akin to Step 1 of the SQG algorithm. Step 2: Similar to Step 2 in the SQG algorithm. Step 3: Similar to Step 3 in the SQG algorithm. Step 4: Similar to Step 4 in the SQG algorithm. 10

13 Step 5: Obtain the auxiliary path aignment u y τ,, ijk ( l u for the SO and UE clae for ) realization l, akin to Step 5 in the SQG algorithm. Obtain the earch direction l, d ijk ( u) uing equation (7). Step 6: The path aignment for the SO and UE cla uer for realization l are updated through a convex combination of the current path aignment, u r τ,, ijk ( l u and the ) auxiliary path aignment, u y τ,, ijk ( l u, i, j, and τ, uing the method of ucceive ) average (MSA). The BR cla path aignment proportion are obtained directly from the imulation output in Step 2. Obtain the updated et of path aignment, l, + 1 r ijk ( u, ) u = 1,, 4. Step 7: Check for convergence. The convergence criterion i baed on the difference in the number of vehicle aigned to variou path over ucceive iteration for the SO and UE clae. If the convergence criterion i atified, go to Step 8. Otherwie, et = +1 and go to Step 2. Step 8: If l= L, go to Step 9, otherwie update the realization counter l = l +1 and go to Step 1. Step 9: Compute the OFH olution a the proportion of the expected number of vehicle of uer cla u who wih to depart from node i to node j at time τ aigned to path k(u): τu ijk ( u ) f L l plr ijk ( u = l= 1 L τ p r l l= 1 k( u) ), i, j, u, k, and τ (12) u, l ijk ( u ) In the experiment of Section 3, we aume the probability of realization l, 1 p l =, for l = L 1,, L. Thi i reaonable given the large number of potential O-D demand cenario from which L are elected Offline ingle incident DDTA olution Likely ingle incident cenario obtained from hitorical incident data are olved offline uing the MUCTDTA algorithm for the mean O-D demand pattern. The aociated path aignment proportion for each incident are ued to compute correction to the OFH olution to account for the preence of that incident. The corrected path aignment proportion, tored offline, are ued in the ODUH-I when that incident occur, to capture it impact on the network flow pattern. The ue of the OFH path aignment proportion F H to compute the correction i baed on the ODUH logic, and i illutrated in Section The correction are computed (Zhou, 2002) a follow: C a τz = Oa τz τ, a, z (13) F H 11

14 where: C a τz = path aignment proportion correction in interval τ when an incident tart on link a in interval z, O a τz = the offline mean O-D demand baed path aignment proportion in interval τ for the ingle incident cenario in which an incident tart on link a in interval z. 2.3 Online Component Under the hybrid olution approach, the offline component addree the randomne in the O-D demand pattern, while the online olution react to unfolding O-D demand and incident. Given the lack of accurate O-D demand and incident prediction model in the literature for general network, a deirable characteritic for the online component i the circumvention of the need for uch prediction. Here, a reactive ODUH that avoid uch prediction i propoed to adjut the OFAP olution in repone to unfolding traffic condition. The ODUH-I i triggered only in the preence of incident. Hence, the propoed online component can be ued eamlely in the abence or preence of incident Non-incident online dynamic update heuritic (ODUH-NI) In the abence of incident, the primary factor that neceitate the online update of the OFAP olution i the actual O-D demand being realized online for a pecific day. Thi i done by updating the OFH olution determined in Section by excluding the unlikely O- D demand realization baed on the unfolding demand pattern. The actual demand from the tart of the planning horizon up to time interval τ are compared with the correponding total demand up to τ for each of the L realization, and thoe realization that do not atify the following two rule are excluded: Rule NI-1(enuring conitency in total O-D demand): For a given day β, if τ τβ Δ l N > Γ (14) then realization l i excluded a an unlikely cenario for day β tarting from interval τ. Here, τl Δ i the um of all the O-D deire from the beginning of the planning horizon up to time interval τ for realization l, and τβ N i the correponding um of online O-D deire up to τ for day β. Γ i a threhold parameter proportional to the total O-D demand in the mean O-D demand pattern. Rule NI-2 (enuring conitency in demand by O-D pair): τl τβ τβ If δ ij - ν ij > κ 1 ν ij, i, j; then l ϕ ij = 1, ele l ϕ ij= 0. If ϕ ij l > κ 2 Π (15) i j 12

15 then realization l i an unlikely cenario for day β. Here, τl δ ij i the um of O-D deire for O- D pair (i, j) from the beginning of the planning horizon up to time interval τ for realization l. τβ ν ij i the correponding um for day β. κ 1 and κ 2 are threhold parameter. Π i the total number of O-D pair in the network and l ϕ ij i an indicator variable. 1, if the O-D pair (i, j) demand for realization l i diqualified for day β l ϕ ij= 0, otherwie Rule NI-2 tate that for a given day β if the number of diqualified O-D pair for realization l exceed a threhold κ 2 Π, in time interval τ, then realization l i an unlikely realization for day β tarting from time interval τ. Any realization l and it aociated offline olution are dicarded for day β tarting in interval τ if at leat one rule i violated. Thi logic i alo attractive from an implementation perpective becaue the computational efficiency increae progreively over the planning horizon. Thi i becaue the number of likely O-D demand realization reduce with time online. Figure 3 illutrate the framework for the ODUH. The online implementation within a tage-baed framework tart with the aignment of the online O-D demand for the firt tage to path baed on the OFAP olution. Toward the end of the current tage γ, the actual timedependent O-D demand are compared with thoe of the O-D demand realization retained in tage γ-1, and rule NI-1 and NI-2 are applied. The realization for which the correponding O-D demand do not atify the rule are dicarded a unlikely realization for that day from that time interval τ. The OFH i then updated to generate the updated offline heuritic olution. The ODUH-NI olution i obtained uing the path aignment proportion of the OFH, the updated offline heuritic, and the OFAP olution. The tage i updated and path are aigned to O-D deire in the next tage uing the ODUH-NI olution. Thi procedure i repeated till the end of the planning horizon. Let: F = OFAP olution path aignment proportion γ F HO = Updated OFH path aignment proportion obtained by excluding unlikely O-D demand realization in tage γ online F γ = ODUH-NI path aignment proportion for tage γ Step 1: Set tage γ = 1. Step 2: Toward the end of tage γ, apply rule NI-1 and NI-2 to determine the unlikely O-D demand realization and dicard them. γ Step 3: Obtain the updated offline heuritic olution for tage γ, F HO, uing the optimal path aignment proportion of realization not dicarded in Step 2, according to (12). Step 4: Calculate the ODUH-NI olution for tage γ, F γ, uing the F and F H path aignment proportion for tage γ: 13

16 F γ F = F HO γ H Normalize F γ to enure that the um of all path aignment proportion for each O-D pair i equal to 1. Step 5: If the end of planning horizon i reached, top. Otherwie, et γ=γ+1, and go to Step 2. Ideally, the online olution for a tage hould be obtained uing the SQG algorithm for the O-D realization retained in that tage. However, thi i a computationally intenive proce. By contrat, the online update of the offline heuritic olution i very efficient a it γ involve manipulating proportion determined offline. Thereby, the proportion F HO / F H are ued to capture the change to the network flow pattern due to the excluion of ome timedependent O-D demand realization online Incident online dynamic update heuritic (ODUH-I) A capability to addre incident online i incorporated into the ODUH by extending the approach for the non-incident cenario. The ODUH-I i triggered by the detection of incident on a given day. A hown in Figure 3, the ODUH addree incident in a reactive manner uing offline and online component. In the offline component, likely ingle incident cenario are olved uing the MUCTDTA algorithm under the mean O-D demand matrix. The online component combine the ODUH-NI olution with the offline mean O-D demand ingle incident olution() correponding to the incident() detected online. Thi approach capture potential change to the traffic flow pattern due to the incident while enuring the robut incorporation of randomne in O-D demand. Of ignificance to deployment, unfolding multiple incident cenario are olved by combining ingle incident offline olution in a imple manner. The ODUH-I i a follow: Step 1: In tage γ, if M incident are preent in the network on link a 1, a M, tarting at time z 1, z M, repectively, compute the average path aignment proportion correction uing the ingle incident proportion correction dicued in Section C γ = M F 1 γz C a i i M i= 1 Step 2: Modify the ODUH-NI olution uing the correction in Step 1. (16) (17) γ γ γ F = C F (18) Step 3: Normalize γ F to obtain the ODUH-I path aignment proportion for tage γ under the incident(). The ODUH-I i attractive for online application under incident for everal reaon. Firt, it exploit hitorical incident data offline to conider only likely incident cenario. Second, it enure that the computationally intenive component are executed offline. Third, it ue combination of ingle incident cenario to capture the effect of multiple incident cenario, ignificantly reducing the number of incident cenario that need to be olved offline. 14

17 3. SIMULATION EXPERIMENTS 3.1 Experimental Setup Figure 4 illutrate the network tructure. It conit of 50 node, 168 link, and 320 O-D pair. All regular link are 0.4 kilometer long and have two lane. Freeway ramp have one lane. The freeway link have a mean free peed of 88 km/h while all other link have a mean free peed of 48 km/h. The 35-minute planning horizon i divided into even 5-minute aignment interval. In the experiment, all O-D ditribution are aumed to be truncated normal ditribution with upper and lower bound. For a given O-D pair, the mean and variance of the ditribution are aumed different in different aignment interval, but contant within that interval. A detailed explanation of the generation of the time-dependent O-D demand pattern i given in Peeta and Zhou (1999). Forty-one likely O-D demand realization are generated for the imulation experiment. An average of vehicle are generated acro realization over the 35 minute, repreenting a medium congetion level for thi network. 3.2 Performance Analyi Section 3.3 and 3.4 analyze the performance of the offline and online trategie in the preence or abence of incident. Thi ection dicue the characteritic of the aociated path aignment proportion in term of their online applicability, and their ignificance to the analyi and to the general DTA context. The term olution ha a different connotation in Section 3 unlike in the previou ection where it repreented the vector of path aignment proportion correponding to a particular trategy. For example, the OFAP olution implied the vector of SQG path aignment proportion. In Section 3, olution implie the average ytem (or vehicular) travel time over L realization when the correponding O-D demand pattern are imulated uing the vector of path aignment proportion for the pecified trategy. Hence, the offline heuritic olution (S3) repreent the average, over all O-D realization, of the ytem travel time obtained for each realization uing the vector of OFH path aignment proportion. Similar definition can be made for the other olution, S2, S4, S5, S6, and S7, by uing the correponding vector of path aignment proportion. MUCTDTA Solution (S1): Thi olution addree the DDTA problem in which the O-D demand pattern, and if relevant, the incident characteritic, are known a priori while olving the problem. Since randomne i excluded, it repreent the bet poible determinitic ytem performance and i a benchmark for the correponding cenario in which randomne in O-D demand and/or incident i encountered online. It i determined offline a it involve olving a computationally intenive DDTA problem for each O-D realization. The MUCTDTA olution, obtained by averaging the ytem travel time from the olution to the L (forty-one) realization, i ued a the benchmark for the correponding cenario in Section 3.3 and 3.4. OFAP Solution (S2): Thi olution incorporate the randomne in O-D demand realization and i baed on the SQG algorithm in Section It i obtained by computing the ytem travel time for each realization uing the SQG path aignment proportion, and then averaging them over all O-D realization. OFH Solution (S3): Thi olution alo incorporate the randomne in O-D demand realization offline and i baed on the path aignment proportion obtained according to the 15

18 procedure in Section The OFH olution i obtained by computing the ytem travel time for each realization uing the OFH path aignment proportion, and then averaging them over all O-D realization. Mean O-D Solution (S4): The mean O-D demand baed path aignment proportion are obtained by olving the MUCTDTA algorithm for the mean O-D demand pattern in the abence of incident. Of key ignificance, thee proportion repreent the olution ought by the vat majority of DTA model in the literature. Thi olution lack the robutne in term of O-D demand that i achieved by S2 and S3. However, it i computationally attractive a it involve olving only one DTA problem unlike S2 and S3. The mean O-D olution i obtained by computing the ytem travel time for each realization uing the mean O-D demand path aignment proportion, and then averaging them over all realization. The effectivene of the hybrid framework can be evaluated by comparing it with the S4 olution. Incident Mean O-D Solution (S5): Thi olution i imilar to S4 except that it incorporate ingle or multiple incident cenario. The path aignment proportion are obtained for everal likely incident cenario offline, and can be ued directly online for ingle or multiple incident cenario. However, due to the potentially large number of combination of multiple incident cenario, S5 may not repreent a realitic online olution under multiple incident. Even for ingle incident cenario, which can be addreed offline, S5 i beneficial only when the incident effect dominate the ytem performance. Further dicuion on thi trategy i provided in Section 3.4. Combined Single Incident Mean O-D Solution (S6): Thi i a relevant online olution for multiple incident cenario. Here, only ingle incident cenario are olved offline for the mean O-D demand pattern. The correponding path aignment proportion are combined efficiently online baed on the actual incident that occur on a given day. S6 i obtained by computing the ytem travel time for each realization uing the combined ingle incident mean O-D path aignment proportion, and then averaging them over all O-D demand realization. ODUH Solution (S7): The ODUH path aignment proportion are obtained uing the procedure in Section in the abence of incident, and the procedure in Section under incident. S7 i obtained by computing the ytem travel time for each realization uing the ODUH path aignment proportion, and then averaging them over all O-D demand realization. Rolling Horizon Solution (S8): Thi olution repreent the benchmark online olution baed on the tate-of-the-art in dynamic traffic aignment. It involve olving truncated DDTA problem for each tage uing predicted O-D demand and incident data. S8 i obtained by computing the ytem travel time for each realization uing the rolling horizon approach, and then averaging them over all O-D demand realization. 3.3 Analyi of the Offline Component Five cenario are conidered to analyze the OFAP olution. They are: Scenario I: Thi cenario explore the effectivene of the OFAP path aignment proportion by comparing it with the MUCTDTA path aignment proportion over the L O-D demand realization. The OFH path aignment proportion are alo determined. Forty-one timedependent O-D realization are generated from the O-D demand ditribution. The average uer cla fraction are aumed to be 0.25 for each cla. 16

19 Scenario II: Thi cenario examine the effectivene of the OFAP path aignment proportion when uer cla fraction are random variable. The fraction of uer clae SO, UE, and BR, are aumed to follow a normal ditribution with a mean of The fraction of PS cla uer i obtained by uing the property that the um of the four uer cla fraction hould equal 1. Thereby, each of the forty-one realization ha a different vector of uer cla fraction. Thi i a more realitic cenario. Scenario III: Here, a new et of forty-one time-dependent O-D demand realization are generated from the O-D demand ditribution. The OFAP and OFH path aignment proportion obtained under Scenario I are applied to the newly generated O-D demand realization to analyze their robutne. Scenario IV: Thi cenario examine the robutne of the OFAP and OFH path aignment proportion under incident. The location of the two incident are hown in Figure 4. Both incident tart at time 4 minute and lat for 45 minute. They are aumed to block 95 and 85 percent of the aociated link capacitie, repectively. Scenario V: Thi cenario tet the robutne of the OFAP path aignment proportion under different congetion level. Figure 5 illutrate the reult of Scenario I in which the performance of the SQG approach (OFAP olution) i compared with that of the benchmark MUCTDTA approach. The reult indicate that SQG converge fater than MUCTDTA, implying that for the ame number of iteration the SQG algorithm perform better. The SQG average travel time obtained in the fourth iteration i reached only after nine iteration of the MUCTDTA algorithm. Thi i becaue the SQG algorithm incorporate a key robutne-enhancing feature uggeted by previou reearch by the author (Peeta and Zhou, 1997, 1999): a large vector of good feaible path. Unlike the MUCTDTA and other determinitic DTA olution algorithm whoe path aignment vector are haped by a ingle O-D demand pattern (typically, the mean O-D demand pattern), the SQG algorithm ynergitically incorporate good feaible path generated for everal O-D demand pattern into the earch proce to determine one robut vector of path aignment proportion for all thoe O-D demand pattern. Of key ignificance to the generalization of the DDTA olution to the tochatic cae, the SQG algorithm provide a unique vector of path aignment proportion for all realization through the earch proce while the MUCTDA algorithm ha a different vector for each realization. Thu, the SQG path aignment proportion applied to an O-D demand pattern generated from the ame ditribution ued to compute them, enure robut ytem performance. However, ince MUCTDTA path aignment proportion are realization pecific, they cannot guarantee robut performance when applied to a different O-D demand pattern. The reult of Scenario II are highlighted in Figure 6. They mirror the concluion from Scenario I, and further indicate that robutne i conerved even when uer cla fraction vary acro realization while following known hitorical ditribution. In reality, uer cla fraction are random variable and can vary over time acro the planning horizon for a given day, a well a acro day. If their randomne can be captured and repreented through ditribution, it can be incorporated into the SQG algorithm. If it i difficult to obtain thee fraction online, then their mean value can be ued to obtain the olution, a in Scenario I. The conervation of ytem performance under random uer cla fraction, a indicated by Figure 5 and 6, i hence an important property for the realitic online implementation of the 17