A Hybrid Deployable Dynamic Traffic Assignment Framework for Robust

Online Route Guidance

Srinivas Peeta

School of Civil Engineering, Purdue University

Chao Zhou Sabre, Inc.

Srinivas Peeta School of Civil Engineering

Purdue University West Lafayette, IN 47907-1284

Phone: (765)-494-2209 Fax: (765)-496-1105

E-mail:peeta@purdue.edu

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

Srinivas Peeta1 and Chao Zhou2

Abstract

Randomness in time-dependent origin-destination (O-D) demands and/or network supply conditions, and the computational tractability of potential solution methodologies are two major concerns for the online deployment of dynamic traffic assignment (DTA) under real-time traffic management systems. Most existing DTA models ignore these concerns and/or make unrealistic assumptions, precluding their online applicability. In this paper, a hybrid approach consisting of offline and online strategies is proposed to address the online stochastic dynamic traffic assignment problem. The basic idea is to address the computationally intensive components offline, while efficiently and effectively reacting to the unfolding conditions online. The offline component seeks a robust initial solution vis-à-vis randomness in O-D demands using historical O-D demand data. Termed the offline a priori solution, it is updated dynamically online based on unfolding O-D demands and incidents. The framework circumvents the need for accurate O-D demand and incident likelihood prediction models online, while exploiting historical O-D demand and incident data offline. Results of simulation experiments highlight the robustness of the hybrid approach with respect to online variations in O-D demand, its ability to address incident situations effectively, and its online efficiency. Keywords: Hybrid deployment strategies, dynamic traffic assignment, real-time operations,

stochastic optimization

1 School of Civil Engineering, Purdue University, West Lafayette, IN 47907-1284 USA 2Sabre Inc., Southlake, TX 76092 USA

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1. INTRODUCTION 1.1 Background and Motivation

The traffic assignment problem aims to determine paths to be assigned to network users that satisfy certain objectives, and/or to estimate the network state. Traditional static traffic assignment models (Sheffi, 1985), motivated by long-term planning needs, assume the origin-destination (O-D) demands, traffic flows, and network supply conditions to be time-invariant over the planning horizon of interest, implying constant trip times on network paths. These assumptions may suffice for non-peak network traffic conditions. Randomness in user behavior can be introduced, for example, by viewing users’ perceived travel times as random variables, leading to the stochastic traffic assignment problem (Sheffi and Powell, 1982). If O-D demands, traffic flows, and network supply conditions are viewed as time-dependent, it results in the dynamic traffic assignment (DTA) problem. The DTA problem aims to determine the optimal time-dependent paths (or path assignment proportions) and/or network states that satisfy some system-wide and/or user objectives. A key characteristic of DTA is the dependence of the current solution on future traffic conditions.

The use of advanced technologies in recent years under the aegis of Advanced Traffic Management Systems (ATMS) and Advanced Traveler Information Systems (ATIS) has generated substantial interest in DTA. Its functional needs range from the estimation of the dynamic network states to personalized route guidance. Several DTA models have been proposed in the literature, but significant differences exist among them (see Peeta, 1994; Peeta and Ziliaskopoulos, 2001) in terms of: (i) assumptions on the availability of information on O-D demands and/or network supply conditions for the planning horizon, (ii) information availability to network users, (iii) information supply strategies, and (iv) user response to the supplied routing information. If the time-dependent O-D demands and network supply conditions are assumed known a priori for the entire planning horizon, the corresponding models are termed deterministic dynamic traffic assignment (DDTA) models. Most models in the literature fall into this category (Janson, 1991; Mahmassani et al., 1993; Ran, et al., 1993; Boyce et al., 1995; Peeta and Mahmassani, 1995a; Wie et al., 1995). However, the state-of-the-art in O-D demand and incident likelihood prediction models cannot guarantee high levels of accuracy, precluding the deployment of DDTA models online3. In addition, they are computationally intensive online in a centralized architecture. Nevertheless, they serve as useful benchmarks for online DTA strategies. If the sources of randomness inherent to traffic systems are considered, the associated problem is labeled the stochastic DTA problem. These sources include randomness in O-D demands, network supply conditions (primarily, incidents), and user response to supplied information. Due to their computational burden, DDTA models can primarily be addressed offline. However, real-time operations entail the deployment of DTA online. Hence, online DTA models additionally seek computational tractability. When the focus is on incorporating the randomness in the unfolding conditions online, it results in the online stochastic dynamic traffic assignment (OSDTA) problem. Given the significant impact that randomness has on the online system performance, it is imperative to address OSDTA models to develop deployable strategies under ATIS/ATMS.

To address online the issue of randomness in O-D demand and network supply conditions, Peeta and Mahmassani (1995b) propose a stage-based rolling horizon approach 3 “Online” is used in the context of real-time operations.

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for implementation. It divides the time horizon of interest into several stages and solves a DDTA problem for each stage sequentially over time. Each stage consists of a roll-period for which the optimal path assignment proportions are to be determined. The rest of the stage represents the future duration considered in obtaining those assignment proportions. The next stage is obtained by rolling the current stage by a time length equal to the roll-period. The rolling horizon approach is more realistic than a DDTA model as O-D demand forecasts are required only for a stage rather than the entire horizon of interest. If highly accurate O-D demand forecasts are available for the near-term future, a near-optimal solution can be expected for the unfolding traffic conditions. Since it is stage-based, the rolling horizon approach ensures that unpredicted variations in online traffic conditions (such as incidents) can be accounted for in subsequent stages. However, if the actual O-D demands in a stage deviate significantly from the forecasts, this approach can be sub-optimal. Another drawback is its computational inefficiency in a centralized ATIS/ATMS architecture. To address the online computational burden, Hawas and Mahmassani (1995) propose a reactive local heuristic rules based approach within a decentralized architecture. An advantage of this approach is its flexibility in deciding the territory size of each controller based on the controller’s computing capabilities. However, due to its pure reactive nature, it does not exploit historical data on O-D demand and incidents. Hence, its performance can degrade substantially from that of the centralized rolling horizon approach in the absence of incidents (Hawas and Mahmassani, 1997).

Pavlis and Papageorgiou (1999) use a decentralized feedback control DTA strategy to enable computational tractability online. It reacts to real-time measurements to establish equal instantaneous travel times on alternative routes for an O-D pair. Akin to Hawas and Mahmassani (1995), it circumvents the need for O-D demand and/or network supply predictions, and uses a decentralized logic. However, due to its pure reactive logic, user response behavior and other underlying processes are not considered, restricting its robustness to specific network topologies. Also, historical data is not exploited.

In this paper, we present a hybrid solution approach to address two key online DTA concerns: (i) the randomness in O-D demand and/or network supply conditions, and (ii) the computational burden. Formulated as a multiple user classes OSDTA model, the deployment problem is addressed through a combination of offline and online strategies. The basic idea of the approach is to address the computationally intensive components offline by exploiting historical data to generate a robust initial solution that can be efficiently updated online based on unfolding O-D demand and/or incident conditions. A robust solution is viewed here as a set of time-dependent path assignment proportions that minimizes the expected system travel time vis-à-vis the randomness in O-D demand. The hybrid approach circumvents the need for online O-D demand and incident likelihood prediction models. This is significant because existing O-D demand and incident likelihood forecast models are unable to guarantee high levels of accuracy. While other online issues such as randomness in user response to supplied information also exist, they are more effectively addressed online through consistency checking models (Peeta and Bulusu, 1999) as they are dependent on the unfolding O-D demands and/or network supply conditions.

1.2 The Online Stochastic Dynamic Traffic Assignment Problem

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The deployment of DTA for real-time operations requires capabilities to address the online issues discussed in Section 1.1. The idealized online DTA problem seeks, in sub-real time, the optimal path assignment proportions for the next assignment interval that minimize the associated system travel time while satisfying user class objectives. However, this solution can be obtained only if a priori knowledge is available on the future O-D demands and network supply conditions. Hence, to enable real-time deployment, we formulate an OSDTA problem (Zhou, 2002) where the focus is on minimizing the expected system travel time for the next assignment interval by considering the randomness in O-D demands and network supply conditions. The OSDTA problem is as follows:

Consider a traffic network represented by a directed graph G(N, A), nodes n∈N and directed arcs a∈A, with multiple origins i∈I and destinations j∈J. A node can represent an origin, a destination, both, or just a junction of physical links. The time period of interest, labeled the planning horizon, is discretized into small equal intervals called assignment intervals, τ = 1,…, T. For generality, assume multiple user classes u∈U for vehicles in terms of information accessibility, information supply strategy, and driver response to the supplied information. Also, assume that the historical time-dependent O-D demand distributions uijRτ (µ, σ), with means u

ijτµ and standard deviations u

ijτσ , are available ∀ i∈I, j∈J, u∈U and τ

= 1,…T. In addition, historical incident data is available in terms location, start time, duration, and severity. For the current day, given the cumulative number of O-D desires t

ijν for all O-D pairs up to the current time t, t = 1,…T, as well as the network state and supply conditions up to time t, the OSDTA problem seeks to determine the path assignment proportions u

uijkfτ

)( to

assign vehicles in the next assignment interval τ = t+1, such that the associated expected system travel time is minimized and user class objectives are satisfied. Here, u

uijkfτ

)( is the

proportion of vehicles of O-D pair (i, j) of class u assigned to path k(u)∈ uijK in interval τ.

This paper is organized as follows. Section 2 discusses the hybrid OSDTA solution framework and the associated offline and online strategies to solve the problem. Section 3 analyzes the effectiveness of the framework through simulation experiments. Concluding comments are presented in Section 4. 2. THE HYBRID SOLUTION FRAMEWORK 2.1 The Hybrid Solution Logic Figure 1 illustrates the hybrid solution framework consisting of offline and online components to address the OSDTA problem. It is based on the a priori optimization concept (Jaillet, 1988; Bertsimas et al., 1990; Laporte et al., 1994) whereby a robust initial solution is determined offline which is then updated efficiently online based on the unfolding conditions on a given day. The randomness in O-D demands is addressed offline. The randomness in network conditions, primarily incidents, is addressed online. The offline component uses historical O-D demand data to determine a robust initial solution for online implementation. The online component uses an efficient and reactive online dynamic update heuristic (ODUH) to address the demand and incidents conditions unfolding online. The hybrid approach has several advantages. The offline component is not constrained computationally and can seek

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robustness using intensive methods (such as Monte Carlo simulation) and by solving DDTA problems. For the same reason, it can also exploit huge amounts of historical data. Due to the robustness of the offline solution, its dynamic update online is enabled in sub-real time. It should be noted that no DTA problem is solved online. Historical O-D demand and incident data are represented as time-dependent probability distributions (Peeta and Zhou, 1999). In the offline component, several realizations of O-D demands and incidents are obtained from these distributions. Less likely O-D demand scenarios are filtered out and the number of O-D demand realizations for the offline strategies, L, is determined by the tolerable error in system travel time based on the central limit theorem (Peeta and Zhou, 1999). Stochastic quasi-gradient (SQG) methods (Ermoliev, 1983; Ermoliev and Wets, 1988) are applied to obtain the robust initial solution, termed the offline a priori (OFAP) solution. It incorporates only the randomness in O-D demand and not incidents. Incidents are viewed in our approach as online events that can be more robustly addressed online. This is because the marginal effect of the consideration or exclusion of an incident on the robustness of the OFAP solution and on the associated network flow pattern is substantially larger than the consideration or exclusion of an O-D desire. Incidents have a potentially larger spatial and temporal reach by affecting several network paths unlike an O-D desire. Therefore, the inclusion of an incident that does not occur online, or vice versa, may cause the offline solution to degrade sufficiently to overshadow the robustness gained by incorporating O-D demand randomness (Zhou, 2002). Hence, incidents are addressed online when they occur.

Two other computationally intensive components are executed offline for online usage. First, an offline heuristic (OFH) is used to generate a robust set of path assignment proportions vis-à-vis O-D demands for use in the ODUH. The DDTA solution using the multiple user classes time-dependent traffic assignment (MUCTDTA) algorithm (Mahmassani et al. 1993; Peeta, 1994) for each O-D demand realization is computed to determine its optimal path assignment proportions. The optimal path assignment proportions for all L O-D realizations are combined (Peeta and Zhou, 1999) to generate the OFH solution for use in the online component. The second component involves the determination of the optimal path assignment proportions using the MUCTDTA algorithm for likely single incident scenarios under the mean O-D demand matrix. The mean O-D demand matrix is obtained using the mean values of the time-dependent O-D demand distributions. The associated path assignment proportions are used in the ODUH to address incident situations. The two components have two common characteristics. Both exploit historical data and involve computationally intensive DDTA solution computations.

The OFAP solution is the default solution online. The ODUH has two components, one for non-incident scenarios (ODUH-NI) and another for incident scenarios (ODUH-I). The ODUH-NI component uses the OFAP and OFH solutions to update the path assignment proportions in the absence of incidents. The ODUH-I component, used under incidents, combines the OFAP and OFH solutions with the optimal path assignment proportions for the offline single incident scenarios.

In non-incident situations, the focus is on ensuring robustness with respect to the actual O-D demand pattern unfolding online. This is done by excluding those offline O-D demand realizations which deviate (beyond predefined threshold levels) from the actual O-D demand pattern known up to the current time on a specific day, and re-computing the path

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assignment proportions as discussed in Section 2.3.1. A synergistic feature in terms of computational efficiency is the reduction of the online computational effort with the progress of the planning horizon, as more O-D demand realizations are eliminated with increasing knowledge on the unfolding O-D demand pattern. If all offline realizations are eliminated at some point in the planning horizon, which can occur either towards the end of the planning horizon or if a very unlikely O-D pattern occurs online, the OFAP solution is used online for the relevant stages.

The ODUH-I component is an extension of ODUH-NI component to include incident scenarios, and is discussed in Section 2.3.2. Several related features and advantages entail further discussion. Only likely single incident scenarios, based on past incident data, are solved offline. Of greater significance, they are solved only for the mean O-D demand pattern to ensure offline computational tractability while addressing incident scenarios. The augmentation of the ODUH-NI using the offline single incident scenario solutions has two purposes. First, robustness with respect to randomness in O-D demands is accounted for by the ODUH-NI component as the incident scenario solution is based only on the mean O-D demand matrix. Second, recognizing that incidents are significant online events in terms of system performance, the offline incident scenario solutions ensure that significant flow pattern changes due to an incident are incorporated in the ODUH-I. Therefore, if the incident effect is not significant, the effectiveness of the online update solution is primarily based on accounting for the randomness in O-D demands, and vice versa. Another favorable feature of ODUH-I is that multiple online incident scenarios are also addressed using single incident offline scenarios only. Given the potentially large number of incident combinations for general networks, solving them even offline may be: (i) prohibitively expensive, and (ii) unnecessary given that incidents are best addressed online. 2.2 Offline Component The offline component primarily addresses the offline stochastic DTA problem. A SQG algorithm is proposed to obtain the OFAP solution. In addition, the OFH and single incident DTA solutions are computed. 2.2.1 The offline stochastic dynamic traffic assignment problem

Given a set of time-dependent O-D demand distributions uijRτ (µ, σ) for the planning

horizon, the offline stochastic DTA problem seeks a vector of time-dependent path assignment proportions u

uijkfτ

)( , ∀ i, j, τ, u, and k(u)=1,... uijK , that minimize the expected

system travel time while satisfying user class objectives. The number of users from origin i to destination j assigned to path k(u) in interval τ under O-D pattern β, βτ ,

)iuu(jkr , is obtained as:

uuijk

uij

uuijk frr τβτβτ

)(,,

)( = , ∀ i, j, τ, u, and k(u) (1)

where βτ ,uijr is the number of vehicles of class u who wish to depart from i to j in time

interval τ under O-D pattern β.

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2.2.2 The stochastic quasi-gradient method The offline stochastic DTA problem is solved using a SQG algorithm. SQG methods

are stochastic solution procedures for solving optimization problems with non-convex, non-differentiable objective functions and general constraints (Ermoliev, 1983). They generalize the stochastic approximation method for unconstrained optimization of the expectation of a random function to problems involving general constraints. The basic concept is to use asymptotically consistent estimates, rather than precise values, for the values of functions and their derivatives while searching for the optimal solution. Under SQG, the step directions are estimated by sampling. SQG methods generally address the following problem type:

Minimize F(x) = Eω [g(x,ω)] (2a)

subject to x ∈ X ⊆ Rn (2b) Here, x represents the vector of decision variables to be optimized, and X is a set of constraints. ω is a random variable belonging to the appropriate probability space. Hence, the objective function F(x) is the expected value of the function g(.) obtained by considering the randomness in ω. The approach considers a limited number of observations of the random function g(x,ω) at each iteration s to determine the random step direction sψ . Hence, the step direction may be a statistical estimate of the gradient (or subgradient in the non-differentiable case) of function F(x); then sψ ≡ sξ such that:

( ) ssx

ss axFxxxxE += )(....,,,, 210ξ (3)

where the vector sa may depend on ( )sxxxx ....,,,, 210 . For exact convergence to an

optimal solution, at some point we should have 0→sa as s → ∞. The vector sξ is called a

stochastic quasi-gradient when sa ≢ 0, or stochastic subgradient (stochastic gradient for

differentiable function) when sa ≡ 0. Generally:

),(1

1

lsL

xs xg

Lωξ ∑

==

ℓ (4)

for realizations ℓ = 1,…L, and 1ω , …, Lω are random samples, when it is possible to obtain the gradients or subgradients in a computationally inexpensive manner. If the gradients or subgradients cannot be calculated, statistical estimates of the gradient directions can be obtained using finite-difference, random search, or other heuristic methods.

The approximation of the optimal solution at step s+1 is computed as:

1+sx = πX[ sx - sρsξ ] (5)

where πX is the projection operator which ensures that all decision variable values are within the feasible domain. A simple choice for sρ , the step size, is one that satisfies the following properties (Ermoliev and Wets, 1988):

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sρ > 0, ∑∞

=0ssρ = ∞, ∑

∞

=1

2

ssρ < ∞ (6)

A comprehensive exposition of the SQG approach for the offline stochastic DTA problem is provided in Zhou (2002). The following sub-sections discuss various aspects of the SQG method to determine the OFAP solution. Choice of step direction

Stochastic quasi-gradients are usually calculated using gradients or subgradients obtained directly or from finite difference approximations. In our problem, the exact form of the objective function is unknown. Hence, convexity and differentiability are not guaranteed. These are inherent characteristics of DTA formulations that adequately incorporate the traffic flow aspects (Peeta and Ziliaskopoulos, 2001). Hence, the gradients or subgradients cannot be obtained directly. We use simulation to circumvent these issues. First, simulation is used to estimate the objective function (total system travel time) value for each realization ℓ. Second, it is used to determine the move directions for the decision variables. Their computation typically entails projecting the change in system performance due to a small change to each decision variable while keeping the other decision variables unchanged. However, this implies one simulation for determining the descent direction for a single decision variable in each search iteration. This is prohibitively expensive for general networks, even offline, as it implies multiple simulations per realization. The proposed SQG solution algorithm circumvents this issue by conducting only one simulation per realization per iteration to estimate these directions, and then computes the quasi-gradients. The directions are obtained using auxiliary solutions within the SQG algorithm.

Since the objective function can be non-convex and/or non-differentiable, there is no guarantee of a descent direction in each iteration, implying a random direction vector. However, this is not a practical barrier as highlighted by several previous simulation studies on various networks (for example, Peeta and Mahmassani 1995a; Peeta and Zhou, 1999) which show smooth convergence of the objective function, or a descent direction on average. This is further corroborated by the experiments conducted in this study even when stochastic quasi-gradients are used. For realization ℓ in time interval τ, the least marginal travel time paths and shortest travel time paths are obtained from the previous iteration solution. They are used to determine an auxiliary solution in terms of path assignment proportions, su

uijky,,)(ℓτ ,

using an all-or-nothing assignment. The search direction for realization ℓ at step s is obtained as:

suuijkd,,)(ℓτ = su

uijky,,)(ℓτ - su

uijkf,)(

τ ∀ i, j, τ, u, k(u) (7)

where suuijkd,,)(ℓτ is the direction for a given i, j, τ, k, ℓ, and u, and su

uijkf,)(

τ is the current solution.

These directions are averaged over all realizations:

suuijkd,)(

τ = ∑ℓ

ℓ suuijkdL,,)(

1 τ ∀ i, j, τ, u, k(u) (8)

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The vector of all suuijkd,)(

τ represents the stochastic quasi-gradient sξ .

Choice of step size

sρ = s1 satisfies conditions (6), and is used as a simple step size in our algorithm in

equation (5). It implies an increasing relative weighting of the current solution over the auxiliary solution as the iteration number increases. This provides an incentive for convergence as s → ∞. However, there is no guarantee of convergence since the objective function form is unknown precluding assumptions of convexity and/or differentiability. The use of a predetermined move size circumvents the inability to analytically optimize the move size. Solution update and projection operator

The solution update is performed using the method of successive averages (MSA) (Wilde, 1964; Powell and Sheffi, 1982). Since it is a convex combinations method, it ensures that the new solution is in the feasible region, precluding the need for an explicit projection operator. This update procedure is used for only some user classes u∈U*, as discussed in the next section. The MSA is employed as follows:

1,)(+suuijkf

τ = suuijkf,)(

τ +s1 su

uijkd,)(

τ , ∀ i, j, τ, u∈U*, k(u) (9)

Stopping criteria Several stopping criteria are possible. One criterion could be to stop when the improvement in the objective function is less than a pre-set value for successive iterations. Another criterion could be to stop when the step size is smaller than a pre-set value. Here, the former criterion is used. 2.2.3 The SQG solution algorithm

The SQG solution algorithm is a generalization of the deterministic MUCTDTA algorithm (Peeta and Mahmassani, 1995b) to the problem incorporating randomness in O-D demands. Four user classes (Mahmassani et al., 1993) are considered for algorithmic completeness, as being representative of possible user classes in terms of information availability, information supply strategy, and user response behavior under ATIS/ATMS. The user classes are: (1) PS; unequipped drivers who follow pre-specified paths, which may be projected from historical databases or solved for exogenously, (2) SO; equipped drivers who follow prescribed system optimum paths, (3) UE; equipped drivers who follow user equilibrium routes, and (4) BR; equipped drivers who follow a boundedly-rational switching rule (Mahmassani and Stephan, 1988) in response to descriptive information on prevailing traffic conditions. The boundedly rational path switching rule states that users switch from the current path at a decision point (typically a node) if the instantaneous travel time savings on an alternative route exceed a certain threshold.

Figure 2 shows the framework for the SQG algorithm. The algorithm first generates L likely time-dependent O-D demand realizations from historical data. At each iteration, a

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traffic simulator is used to evaluate the system performance under the current solution for each realization ℓ. The simulation results are used to determine the search directions for the SO and UE classes for that realization. The paths of the BR user class are obtained from the traffic simulator while those of PS class users remain unchanged. The search directions for all realizations are used to determine the stochastic quasi-gradients for that iteration. The solution for the next step is obtained from the stochastic quasi-gradients and the current solutions using the MSA. The algorithm is as follows: Step 0: Generate L likely time-dependent O-D demand realization vectors from historical

distributions. Set the iteration counter s = 0. Step 1: For each generated O-D realization, assign the O-D desires of the equipped user

classes 0,,ℓuijrτ , ∀ i, j, τ, ℓ, and u = 2,...4, to a time-dependent initial set of feasible

paths. The paths of all unequipped vehicles are assumed to be known a priori and are part of the initial conditions. Hence, 0,,1

)1(ℓτ

ijkr , ∀ i, j, τ, and ℓ, are known, and due to the

lack of information accessibility are assumed to remain unchanged throughout the iterative search process. After computing the initial paths for all realizations, obtain the vector of the path assignment proportions averaged across realizations 0,

)(uuijkf

τ , ∀ i,

j, τ, u = 1,… 4, and k(u). This represents the initial solution. Set the realization counter ℓ = 1.

Step 2: For realization ℓ, obtain the set of path assignments s,,u)u(ijkrℓτ using s,u

)u(ijkfτ :

s,,u)u(ijkrℓτ = s,,u

ijrℓτ s,u

)u(ijkfτ , ∀ i, j, τ, u=2-4, k(u) (10)

and

s,,1)1(ijkrℓτ = 0,,1

)1(ijkrℓτ , ∀ i, j, τ, k(1) (11)

The set of path assignments suuijkr,,)(ℓτ obtained from su

uijkf,)(

τ for the entire horizon of

interest are simulated using a traffic simulator, DYNASMART (Jayakrishnan et al., 1995). The simulation results provide several link level and aggregate performance measures, including the system travel time for realization ℓ.

Step 3: Compute the link marginal travel times (Peeta, 1994; Peeta and Mahmassani, 1995a) for SO users using the time-dependent experienced link travel times and the number of vehicles on links obtained as post-simulation data from Step 2.

Step 4: Compute the time-dependent least marginal travel time paths and shortest travel time paths.

Step 5: Perform an all-or-nothing assignment of all O-D desires suijr

,,ℓτ for given i, j, τ, for the

SO and UE classes. Assign the SO users to the least marginal travel time paths and the UE users to the shortest average travel time paths. The result is a set of auxiliary path assignments. The corresponding proportions of vehicles are the auxiliary solutions,

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suuijky,,)(ℓτ , ∀ i, j, τ, k(u), u = 2, 3. Obtain the search directions su

uijkd,,)(ℓτ for realization

ℓ using equations (7). Step 6: If ℓ = L, go to Step 7; otherwise set ℓ = ℓ + 1, and go to Step 2. Step 7: Compute the average system travel time over all realizations. This represents the

objective function value in iteration s. Check for convergence using the stopping criterion (difference in objective function values in two successive iterations). If the criterion is satisfied, stop the computation and specify su

uijkf,)(

τ , ∀ i, j, τ, u, and k(u), as

the SQG solution. Otherwise, go to Step 8. Step 8: Calculate the stochastic quasi-gradients sξ by computing su

uijkd,)(

τ , ∀ i, j, τ, k(u), u = 2,

3, using equations (8). Update the proportions of vehicles to be assigned to paths for the SO and UE class users 1,

)(+suuijkf

τ using the MSA as shown in equations (9). Update

the path assignment proportions for the BR user class by averaging across realizations, and normalize them. Set s = s+1, ℓ = 1, and go to Step 2. The path assignment proportions for the SO and UE classes for a specific realization

under the a priori solution do not necessarily satisfy the SO and UE principles, respectively (Zhou, 2002). These principles are satisfied only in the expected sense over all realizations as the SQG algorithm determines an average quasi-gradient using all realizations in an iteration. This is more robust and general compared to solving for path assignment proportions using a single O-D demand pattern. 2.2.4 The offline heuristic

The SQG solution provides a single path assignment proportion vector for all realizations. However, as discussed in Section 2.1, the online reactive update for the unfolding O-D demands involves eliminating unlikely O-D demand patterns in each stage of the planning horizon for a given day and re-computing the path assignment proportions after excluding their contributions. The OFH enables this by separating the contribution of each realization while generating another robust offline solution, called the OFH solution. It is obtained by generating the DDTA solution for each realization, and then averaging them to obtain a vector of path assignment proportions (Peeta and Zhou, 1999). The reasoning for this approach is that the optimal path assignment proportions for a realization provide a “favorable” path set for that realization, which can then be combined with such path sets for other realizations to generate a robust vector of path assignment proportions relative to O-D demands. The OFH is similar to the SQG algorithm. It is described comprehensively in Peeta and Zhou (1999). Here, it is described by elaborating the steps that are different from those in the SQG algorithm: Step 0: Set realization counter ℓ = 1. Step 1: Set the iteration counter s = 0. Generate the initial path set 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.

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Step 5: Obtain the auxiliary path assignments suuijky,,)(ℓτ for the SO and UE classes for

realization ℓ, akin to Step 5 in the SQG algorithm. Obtain the search directions suuijkd,,)(ℓτ

using equations (7). Step 6: The path assignments for the SO and UE class users for realization ℓ are updated

through a convex combination of the current paths assignments, suuijkr,,)(ℓτ and the

auxiliary path assignments, suuijky,,)(ℓτ , ∀ i, j, and τ, using the method of successive

averages (MSA). The BR class path assignment proportions are obtained directly from the simulation output in Step 2. Obtain the updated set of path assignments, 1,,

)(+su

uijkrℓτ ,

u = 1,…, 4. Step 7: Check for convergence. The convergence criterion is based on the difference in the

number of vehicles assigned to various paths over successive iterations for the SO and UE classes. If the convergence criterion is satisfied, go to Step 8. Otherwise, set s = s+1 and go to Step 2.

Step 8: If ℓ= L, go to Step 9, otherwise update the realization counter ℓ = ℓ +1 and go to Step 1.

Step 9: Compute the OFH solution as the proportions of the expected number of vehicles of user class u who wish to depart from node i to node j at time τ assigned to path k(u):

∑ ∑

∑=

=

=L

l uk

uuijk

L

l

uuijk

uuijk

rp

rpf

1 )(

,)(

1

,)(

)(ℓ

ℓ

ℓℓ

τ

ττ , ∀ i, j, u, k, and τ (12)

In the experiments of Section 3, we assume the probability of realization ℓ, ℓp = L1 , for ℓ =

1,…, L. This is reasonable given the large number of potential O-D demand scenarios from which L are selected. 2.2.5 Offline single incident DDTA solutions

Likely single incident scenarios obtained from historical incident data are solved offline using the MUCTDTA algorithm for the mean O-D demand pattern. The associated path assignment proportions for each incident are used to compute corrections to the OFH solution to account for the presence of that incident. The corrected path assignment proportions, stored offline, are used in the ODUH-I when that incident occurs, to capture its impact on the network flow pattern. The use of the OFH path assignment proportions HF to compute the corrections is based on the ODUH logic, and is illustrated in Section 2.3.2. The corrections are computed (Zhou, 2002) as follows:

zaτC =

HFO zaτ

∀ τ, a, z (13)

12

where: zaτC = path assignment proportion corrections in interval τ when an incident starts on link a in

interval z, zaτO = the offline mean O-D demand based path assignment proportions in interval τ for the

single incident scenario in which an incident starts on link a in interval z. 2.3 Online Component

Under the hybrid solution approach, the offline component addresses the randomness in the O-D demand pattern, while the online solution reacts to unfolding O-D demands and incidents. Given the lack of accurate O-D demand and incident prediction models in the literature for general networks, a desirable characteristic for the online component is the circumvention of the need for such predictions. Here, a reactive ODUH that avoids such predictions is proposed to adjust the OFAP solution in response to unfolding traffic conditions. The ODUH-I is triggered only in the presence of incidents. Hence, the proposed online component can be used seamlessly in the absence or presence of incidents. 2.3.1 Non-incident online dynamic update heuristic (ODUH-NI)

In the absence of incidents, the primary factor that necessitates the online update of the OFAP solution is the actual O-D demands being realized online for a specific day. This is done by updating the OFH solution determined in Section 2.2.4 by excluding the unlikely O-D demand realizations based on the unfolding demand pattern. The actual demands from the start of the planning horizon up to time interval τ are compared with the corresponding total demands up to τ for each of the L realizations, and those realizations that do not satisfy the following two rules are excluded: Rule NI-1(ensuring consistency in total O-D demand):

For a given day β, if

|| τβτ N−Δ ℓ > Γ (14)

then realization ℓ is excluded as an unlikely scenario for day β starting from interval τ. Here, ℓτΔ is the sum of all the O-D desires from the beginning of the planning horizon up to time

interval τ for realization ℓ, and τβN is the corresponding sum of online O-D desires up to τ

for day β. Γ is a threshold parameter proportional to the total O-D demand in the mean O-D demand pattern. Rule NI-2 (ensuring consistency in demand by O-D pair):

If | ℓτδ ij - τβν ij | > κ1τβν ij , ∀ i, j; then ℓ

ijϕ = 1, else ℓijϕ = 0. If

∑∑i j

ijℓϕ > κ2Π (15)

13

then realization ℓ is an unlikely scenario for day β. Here, ℓτδ ij is the sum of O-D desires for O-D pair (i, j) from the beginning of the planning horizon up to time interval τ for realization ℓ. τβν ij is the corresponding sum for day β. κ1 and κ2 are threshold parameters. Π is the total

number of O-D pairs in the network and ℓijϕ is an indicator variable.

1, if the O-D pair (i, j) demand for realization ℓ is disqualified for day β ℓijϕ =

0, otherwise Rule NI-2 states that for a given day β if the number of disqualified O-D pairs for realization ℓ exceeds a threshold κ2Π, in time interval τ, then realization ℓ is an unlikely realization for day β starting from time interval τ.

Any realization ℓ and its associated offline solution are discarded for day β starting in interval τ if at least one rule is violated. This logic is also attractive from an implementation perspective because the computational efficiency increases progressively over the planning horizon. This is because the number of likely O-D demand realizations reduces with time online.

Figure 3 illustrates the framework for the ODUH. The online implementation within a stage-based framework starts with the assignment of the online O-D demand for the first stage to paths based on the OFAP solution. Towards the end of the current stage γ, the actual time-dependent O-D demands are compared with those of the O-D demand realizations retained in stage γ-1, and rules NI-1 and NI-2 are applied. The realizations for which the corresponding O-D demands do not satisfy the rules are discarded as unlikely realizations for that day from that time interval τ. The OFH is then updated to generate the updated offline heuristic solution. The ODUH-NI solution is obtained using the path assignment proportions of the OFH, the updated offline heuristic, and the OFAP solutions. The stage is updated and paths are assigned to O-D desires in the next stage using the ODUH-NI solution. This procedure is repeated till the end of the planning horizon. Let: F = OFAP solution path assignment proportions γHOF = Updated OFH path assignment proportions obtained by excluding unlikely O-D

demand realizations in stage γ online Fγ = ODUH-NI path assignment proportions for stage γ Step 1: Set stage γ = 1. Step 2: Towards the end of stage γ, apply rules NI-1 and NI-2 to determine the unlikely O-D

demand realizations and discard them. Step 3: Obtain the updated offline heuristic solution for stage γ, γ

HOF , using the optimal path assignment proportions of realizations not discarded in Step 2, according to (12).

Step 4: Calculate the ODUH-NI solution for stage γ, Fγ, using the F and HF path assignment proportions for stage γ:

14

Fγ FFF

H

HO ⋅=γ

(16)

Normalize Fγ to ensure that the sum of all path assignment proportions for each O-D pair is equal to 1.

Step 5: If the end of planning horizon is reached, stop. Otherwise, set γ=γ+1, and go to Step 2. Ideally, the online solution for a stage should be obtained using the SQG algorithm for

the O-D realizations retained in that stage. However, this is a computationally intensive process. By contrast, the online update of the offline heuristic solution is very efficient as it involves manipulating proportions determined offline. Thereby, the proportions γ

HOF / HF are used to capture the changes to the network flow pattern due to the exclusion of some time-dependent O-D demand realizations online. 2.3.2 Incident online dynamic update heuristic (ODUH-I)

A capability to address incidents online is incorporated into the ODUH by extending the approach for the non-incident scenario. The ODUH-I is triggered by the detection of incidents on a given day.

As shown in Figure 3, the ODUH addresses incidents in a reactive manner using offline and online components. In the offline component, likely single incident scenarios are solved using the MUCTDTA algorithm under the mean O-D demand matrix. The online component combines the ODUH-NI solution with the offline mean O-D demand single incident solution(s) corresponding to the incident(s) detected online. This approach captures potential changes to the traffic flow pattern due to the incident while ensuring the robust incorporation of randomness in O-D demands. Of significance to deployment, unfolding multiple incident scenarios are solved by combining single incident offline solutions in a simple manner. The ODUH-I is as follows: Step 1: In stage γ, if M incidents are present in the network on links a1,… aM, starting at times

z1, …zM, respectively, compute the average path assignment proportion corrections using the single incident proportion corrections discussed in Section 2.2.5.

∑==

M

i

zM 1

1 iia

CC γγ (17)

Step 2: Modify the ODUH-NI solution using the corrections in Step 1. γγγ ⋅= FCF (18)

Step 3: Normalize γF to obtain the ODUH-I path assignment proportions for stage γ under the incident(s). The ODUH-I is attractive for online application under incidents for several reasons.

First, it exploits historical incident data offline to consider only likely incident scenarios. Second, it ensures that the computationally intensive components are executed offline. Third, it uses combinations of single incident scenarios to capture the effect of multiple incident scenarios, significantly reducing the number of incident scenarios that need to be solved offline.

15

3. SIMULATION EXPERIMENTS 3.1 Experimental Setup

Figure 4 illustrates the network structure. It consists of 50 nodes, 168 links, and 320 O-D pairs. All regular links are 0.4 kilometers long and have two lanes. Freeway ramps have one lane. The freeway links have a mean free speed of 88 km/h while all other links have a mean free speed of 48 km/h. The 35-minute planning horizon is divided into seven 5-minute assignment intervals. In the experiments, all O-D distributions are assumed to be truncated normal distributions with upper and lower bounds. For a given O-D pair, the mean and variance of the distributions are assumed different in different assignment intervals, but constant within that interval. A detailed explanation of the generation of the time-dependent O-D demand patterns is given in Peeta and Zhou (1999). Forty-one likely O-D demand realizations are generated for the simulation experiments. An average of 21856 vehicles are generated across realizations over the 35 minutes, representing a medium congestion level for this network. 3.2 Performance Analysis

Sections 3.3 and 3.4 analyze the performance of the offline and online strategies in the presence or absence of incidents. This section discusses the characteristics of the associated path assignment proportions in terms of their online applicability, and their significance to the analysis and to the general DTA context. The term “solution” has a different connotation in Section 3 unlike in the previous sections where it represented the vector of path assignment proportions corresponding to a particular strategy. For example, the OFAP solution implied the vector of SQG path assignment proportions. In Section 3, “solution” implies the average system (or vehicular) travel time over L realizations when the corresponding O-D demand patterns are simulated using the vector of path assignment proportions for the specified strategy. Hence, the offline heuristic solution (S3) represents the average, over all O-D realizations, of the system travel time obtained for each realization using the vector of OFH path assignment proportions. Similar definitions can be made for the other solutions, S2, S4, S5, S6, and S7, by using the corresponding vector of path assignment proportions. MUCTDTA Solution (S1): This solution addresses the DDTA problem in which the O-D demand pattern, and if relevant, the incident characteristics, are known a priori while solving the problem. Since randomness is excluded, it represents the best possible deterministic system performance and is a benchmark for the corresponding scenarios in which randomness in O-D demand and/or incidents is encountered online. It is determined offline as it involves solving a computationally intensive DDTA problem for each O-D realization. The MUCTDTA solution, obtained by averaging the system travel times from the solutions to the L (forty-one) realizations, is used as the benchmark for the corresponding scenarios in Sections 3.3 and 3.4. OFAP Solution (S2): This solution incorporates the randomness in O-D demand realizations and is based on the SQG algorithm in Section 2.2.3. It is obtained by computing the system travel time for each realization using the SQG path assignment proportions, and then averaging them over all O-D realizations. OFH Solution (S3): This solution also incorporates the randomness in O-D demand realizations offline and is based on the path assignment proportions obtained according to the

16

procedure in Section 2.2.4. The OFH solution is obtained by computing the system travel time for each realization using the OFH path assignment proportions, and then averaging them over all O-D realizations. Mean O-D Solution (S4): The mean O-D demand based path assignment proportions are obtained by solving the MUCTDTA algorithm for the mean O-D demand pattern in the absence of incidents. Of key significance, these proportions represent the solution sought by the vast majority of DTA models in the literature. This solution lacks the robustness in terms of O-D demands that is achieved by S2 and S3. However, it is computationally attractive as it involves solving only one DTA problem unlike S2 and S3. The mean O-D solution is obtained by computing the system travel time for each realization using the mean O-D demand path assignment proportions, and then averaging them over all realizations. The effectiveness of the hybrid framework can be evaluated by comparing it with the S4 solution. Incidents Mean O-D Solution (S5): This solution is similar to S4 except that it incorporates single or multiple incident scenarios. The path assignment proportions are obtained for several likely incident scenarios offline, and can be used directly online for single or multiple incident scenarios. However, due to the potentially large number of combinations of multiple incident scenarios, S5 may not represent a realistic online solution under multiple incidents. Even for single incident scenarios, which can be addressed offline, S5 is beneficial only when the incident effect dominates the system performance. Further discussion on this strategy is provided in Section 3.4. Combined Single Incidents Mean O-D Solution (S6): This is a relevant online solution for multiple incident scenarios. Here, only single incident scenarios are solved offline for the mean O-D demand pattern. The corresponding path assignment proportions are combined efficiently online based on the actual incidents that occur on a given day. S6 is obtained by computing the system travel time for each realization using the combined single incidents mean O-D path assignment proportions, and then averaging them over all O-D demand realizations. ODUH Solution (S7): The ODUH path assignment proportions are obtained using the procedure in Section 2.3.1 in the absence of incidents, and the procedure in Section 2.3.2 under incidents. S7 is obtained by computing the system travel time for each realization using the ODUH path assignment proportions, and then averaging them over all O-D demand realizations. Rolling Horizon Solution (S8): This solution represents the benchmark online solution based on the state-of-the-art in dynamic traffic assignment. It involves solving truncated DDTA problems for each stage using predicted O-D demand and incident data. S8 is obtained by computing the system travel time for each realization using the rolling horizon approach, and then averaging them over all O-D demand realizations.

3.3 Analysis of the Offline Component

Five scenarios are considered to analyze the OFAP solution. They are: Scenario I: This scenario explores the effectiveness of the OFAP path assignment proportions by comparing it with the MUCTDTA path assignment proportions over the L O-D demand realizations. The OFH path assignment proportions are also determined. Forty-one time-dependent O-D realizations are generated from the O-D demand distributions. The average user class fractions are assumed to be 0.25 for each class.

17

Scenario II: This scenario examines the effectiveness of the OFAP path assignment proportions when user class fractions are random variables. The fractions of user classes SO, UE, and BR, are assumed to follow a normal distribution with a mean of 0.25. The fraction of PS class users is obtained by using the property that the sum of the four user class fractions should equal 1. Thereby, each of the forty-one realizations has a different vector of user class fractions. This is a more realistic scenario. Scenario III: Here, a new set of forty-one time-dependent O-D demand realizations are generated from the O-D demand distributions. The OFAP and OFH path assignment proportions obtained under Scenario I are applied to the newly generated O-D demand realizations to analyze their robustness. Scenario IV: This scenario examines the robustness of the OFAP and OFH path assignment proportions under incidents. The locations of the two incidents are shown in Figure 4. Both incidents start at time 4 minutes and last for 45 minutes. They are assumed to block 95 and 85 percent of the associated link capacities, respectively. Scenario V: This scenario tests the robustness of the OFAP path assignment proportions under different congestion levels. Figure 5 illustrates the results of Scenario I in which the performance of the SQG approach (OFAP solution) is compared with that of the benchmark MUCTDTA approach. The results indicate that SQG converges faster than MUCTDTA, implying that for the same number of iterations the SQG algorithm performs better. The SQG average travel time obtained in the fourth iteration is reached only after nine iterations of the MUCTDTA algorithm. This is because the SQG algorithm incorporates a key robustness-enhancing feature suggested by previous research by the authors (Peeta and Zhou, 1997, 1999): a large vector of “good” feasible paths. Unlike the MUCTDTA and other deterministic DTA solution algorithms whose path assignment vectors are shaped by a single O-D demand pattern (typically, the mean O-D demand pattern), the SQG algorithm synergistically incorporates “good” feasible paths generated for several O-D demand patterns into the search process to determine one robust vector of path assignment proportions for all those O-D demand patterns. Of key significance to the generalization of the DDTA solution to the stochastic case, the SQG algorithm provides a unique vector of path assignment proportions for all realizations through the search process while the MUCTDA algorithm has a different vector for each realization. Thus, the SQG path assignment proportions applied to an O-D demand pattern generated from the same distributions used to compute them, ensure robust system performance. However, since MUCTDTA path assignment proportions are realization specific, they cannot guarantee robust performance when applied to a different O-D demand pattern. The results of Scenario II are highlighted in Figure 6. They mirror the conclusion from Scenario I, and further indicate that robustness is conserved even when user class fractions vary across realizations while following known historical distributions. In reality, user class fractions are random variables and can vary over time across the planning horizon for a given day, as well as across days. If their randomness can be captured and represented through distributions, it can be incorporated into the SQG algorithm. If it is difficult to obtain these fractions online, then their mean values can be used to obtain the solution, as in Scenario I. The conservation of system performance under random user class fractions, as indicated by Figures 5 and 6, is hence an important property for the realistic online implementation of the

18

SQG algorithm under the hybrid approach. The randomness in the user class fractions can also be addressed through online consistency models (see Peeta and Bulusu, 1999). Table 1 shows the results of Scenarios I through V in terms of average system travel times. The MUCTDTA solution for each scenario is benchmarked at 100%, and other solutions are specified using it as the basis. Solutions S1, S2, and S3 are based on forty-one O-D demand realizations generated from the historical distributions. The results for Scenario III indicate that robustness is conserved even when a different set of O-D demand realizations from the same distributions are used to obtain the path assignment proportions than the O-D demand patterns actually encountered. The results of Scenarios I through III indicate that in the absence of incidents, the OFAP and OFH solutions are very robust while the mean O-D solution performs slightly worse. Scenario IV, which incorporates incidents, emphasizes the superiority of the offline solutions S2 and S3 compared to the mean O-D solution S4. S2 and S3 significantly outperform S4 though neither of them incorporates the incidents in determining the path assignment proportions. Only the corresponding benchmark MUCTDTA solution incorporates the presence of the incidents. The gap between it and solutions S2 or S3 suggests the need for an online procedure to enhance system performance when incidents occur on a given day, further reinforcing the logic of the hybrid approach.

Scenario V illustrates the robustness of the various approaches under different congestion levels in the absence of incidents. While none of the solutions S2 through S4 is significantly different from S1 under low congestion, S4 is perceptibly different under high congestion levels. Thereby, the robustness associated with the offline approaches S2 and S3 is especially valuable under high congestion levels.

In summary, the offline analysis yields the following insights. The SQG and offline heuristic approaches are significantly robust in the absence of incidents as well as under high congestion levels. They need to be augmented online to respond to incident situations, especially when the incidents are severe. The SQG algorithm is especially close in performance to the MUCTDTA algorithm, and represents a generalization of it to the DTA problem that incorporates randomness in O-D demands. 3.4 Analysis of the Online Component

The analysis of the online component addresses the application of the hybrid approach to a specific day. The online component reacts to randomness in unfolding O-D demands and incidents. Three scenarios are addressed to analyze the ODUH. The forty-one O-D demand realizations of Scenario I are assumed to represent the actual O-D demand patterns encountered online. Scenario VI: This scenario examines the performance of the ODUH-NI, by comparing it with the offline solutions S2 and S4, and the rolling horizon solution (S8). Scenario VII: This scenario analyzes the effectiveness of the ODUH-I under a single incident. The location of the incident is indicated in Figure 4. It starts at time 4 minutes and lasts for 35 minutes, and is assumed to block 95 percent of the associated link capacity. Scenario VIII: In this scenario, two incidents are assumed to occur simultaneously. It tests the effectiveness of the ODUH-I under multiple incident situations. The locations of the two incidents are shown in Figure 4. Both incidents start at time 4 minutes and last for 27 and 28

19

minutes, respectively. They are assumed to block 95 and 85 percent of the associated link capacities, respectively.

Table 2 illustrates the results of Scenarios VI through VIII. As before, the results are compared with the corresponding MUCTDTA benchmarks. Scenario VI results suggest that the ODUH may be unnecessary and the offline SQG approach may suffice in the absence of incidents. Here, the ODUH is likely to be valuable only when the unfolding O-D demand pattern substantially deviates from historical O-D demand data. O-D demand patterns associated with special events can be addressed using the historical O-D demand database for such events only, suggesting that under the hybrid approach historical O-D databases can be further refined before applying the offline and online components.

Of significance for online computational tractability, the rolling horizon solution (S8) requires substantially larger computational time compared to the online heuristic. This is because the rolling horizon procedure involves solving truncated DDTA problems online. The rolling horizon solution is also less robust. The CPU times on a SUN Ultra 2 200MHz machine indicate that the ODUH is substantially more efficient, by a factor of 60, compared to the rolling horizon procedure. Even in terms of robustness, the value of the ODUH is further highlighted when it is noted that the effectiveness of the rolling horizon approach is highly dependent on the accuracy of O-D demand and incident likelihood predictions. In Scenario VI, the results of the rolling horizon approach are rather optimistic as it is assumed that the predicted O-D demands are also realized online. The hybrid approach being reactive, obviates these concerns.

The single incident situation addressed in Scenario VII highlights the advantages of the ODUH. First, the non-consideration of an incident in the solution procedure could lead to a substantial deterioration of system performance online, especially when incident effects dominate the flow pattern, as indicated by the offline non-incident solutions S2 and S4. S7 is slightly more robust than S5. However, when the notion that S5 is a restrictive offline strategy is factored in, S7 stands out as the only robust and computationally tractable online approach for general networks among the strategies considered. S5 is restrictive because it is unrealistic computationally even as an offline strategy under multiple incident scenarios. It is also restrictive because it is valuable primarily when the incident effects dominate significantly to reduce the gap between S5 and S7, as is the case here. In the context of unfolding online conditions, it is difficult to generalize when incident effects dominate the effects of randomness in O-D demand with regard to system performance. Also, the robustness in S7 with regard to randomness in O-D demands is beneficial even when incident effects dominate. Hence, it is worthwhile to perform the online update as compared to the procedurally simpler approach S5, for general networks.

Scenario VIII analyzes the ODUH under multiple incident situations. S7 is substantially more robust than S2 and S4. While the deterioration in system performance under S2 and S4 indicates the dominance of the incident effects, S5 as a strategy is computationally even more untenable here compared to Scenario VII. It implies that all incidents are considered offline in obtaining the optimal path assignment proportions under the mean O-D demand pattern, requiring the offline determination of DDTA solutions for a very large number of combinations of incidents, an intractable proposition for general networks. A more realistic approach computationally is to combine the offline single incident mean O-D demand path assignment proportions online when the incidents occur, represented

20

here by S6. However, its performance is poor, and worse than even the OFAP solution where incidents are not considered. This implies that incorporation of randomness in O-D demands is beneficial as indicated by the superiority of S2 over S6. Hence, the ODUH solution is substantially better than the next best computationally tractable strategy under multiple incident situations. 4. CONCLUDING COMMENTS A hybrid deployment framework that combines offline and online strategies is proposed to solve the OSDTA problem. It involves the determination of an OFAP solution that serves as a robust initial solution online, thereby enabling computationally efficient updates online in response to unfolding traffic conditions. The framework addresses the computationally intensive components offline, thereby ensuring that the online approach is efficient as well as responsive. It circumvents the need for online O-D demand and/or incident likelihood predictions while exploiting available historical data offline. The approach ensures that computational efficiency increases progressively over the planning horizon. The offline component is designed to incorporate the randomness in time-dependent O-D demands. A SQG algorithm is proposed to solve the offline problem. A comprehensive theoretical discussion of the SQG approach is provided in Zhou (2002). It represents a generalization of the DDTA problem to the stochastic case where randomness in O-D demands are incorporated. The SQG algorithm can outperform even the deterministic MUCTDTA algorithm. This is because the MUCTDTA algorithm enumerates only a limited number of “good” feasible paths based on the single O-D pattern assumed (typically, the mean O-D demand pattern in the current DTA literature). The SQG approach uses multiple O-D demand patterns, thereby ensuring a much larger feasible path set, implying a more efficient search for the optimal solution.

The online component is a reactive dynamic update heuristic which excludes unlikely O-D realizations online, considers incidents only when they occur, and ensures a real-world deployment capability by requiring offline solutions to only single incident scenarios, even for multiple incident situations online.

Simulation experiments highlight the two key advantages of the hybrid framework: (i) its robustness vis-à-vis randomness in online O-D demands and incidents, and (ii) its highly efficient online computational tractability. While the offline a priori solution is very robust in the absence of incidents online, the online update heuristic is especially effective under incident scenarios, both single and multiple. ACKNOWLEDGEMENTS This material is based upon work supported by the National Science Foundation under Grant No. 9702612. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Additional support was provided through a research grant from the Purdue Research Foundation.

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Figure 1 Hybrid Solution Framework

Generate likely O-D demand realizations

Use SQG algorithm to obtain OFAP

solution

Online O-D demand and

network supply conditions for the

current stage

Use ODUH to determine

path assignment proportions

MUCTDTA solution for each O-D demand

realization

Historical time-dependent O-D

demand distributions and

incident data

Online Component

Offline Component

MUCTDTA path assignment proportions under likely single

incidents for the mean O-D demand matrix

Assign paths to vehicles in the next

stage

Generate L time-dependent O-D realizations from historical data

Set iteration counter s = 0

24

Figure 2 Framework for the SQG Solution Algorithm

Convergence criteria satisfied?

Update iteration counter s = s + 1

ℓ = L? Update realization counter ℓ = ℓ +1

All-or-nothing assignment of O-D desires to auxiliary paths: obtain s

ijky,,3)3(ℓτ , ∀i, j, τ, k(3)

All-or-nothing assignment of O-D desires to auxiliary paths: obtain s

ijky,,2)2(ℓτ , ∀ i, j, τ, k(2)

Yes

No

Set realization counter ℓ = 1

Traffic simulator (DYNASMART)

Obtain the shortest travel time paths

Obtain the least marginal travel time paths

Normalize to ensure 1)4(

1,4)4( =∑ +

k

sijkfτ , ∀ i, j, τ

Stop; specify suuijkf,)(

τ , ∀ i, j,

τ, u, and k(u)

Compute stochastic quasi-gradients sξ : suuijkd,)(

τ = ( )∑ −ℓ

ℓ suuijk

suuijk fy

L,)(

,,)(

1 ττ ∀ i, j, τ, k(u), u=2,3

Use MSA to obtain the new solution, 1,)(+suuijkf

τ , ∀ i, j, τ, k(u), u=2,3

Paths of BR class users

1,4)4(+s

ijkfτ = ∑

ℓ

ℓ sijkfL

,,4)4(

1 τ , ∀ i, j, τ, k(4)

No Yes

25

Figure 3 Framework for the Online Dynamic Update Heuristic

MUCTDTA solutions for single incident

scenarios under mean O-D matrix

Did incident occur? Yes

Likely O-D demand realizations and single incident scenarios from historical data OFAP solution F

γ = 1

Online traffic assignment

Is the end of planning horizon

reached?

Stop

Yes

Online O-D demand towards the end of stage γ

Exclude unlikely O-D realizations

Calculate updated OFH solution, γ

HOF γ = γ + 1

Calculate ODUH solution Fγ or γF

MUCTDTA solutions for O-D demand realizations

OFH solution

FH

No

Offline component

Online component

44

1 2 3 4 5 6

43 45

26

Figure 4 Network Structure

Two online incidents

Single online incident

Two offline incidents

27

Figure 5 Comparison of SQG and MUCTDTA Approaches (Scenario I)

5.5

6

6.5

7

7.5

8

8.5

1 2 3 4 5 6 7 8 9

Iteration Number

Aver

age T

rave

l Tim

e (Mi

nutes

)

MUCTDTA

S QG

28

Figure 6 Comparison of SQG and MUCTDTA Approaches Under Random User Class Fractions (Scenario II)

Table 1 Comparison of the Offline Performance of Solution Approaches for Various Scenarios, as a Percentage of the Corresponding MUCTDTA Solutions

5

5.5

6

6.5

7

7.5

8

8.5

9

1 2 3 4 5 6 7 8 9

Iteration Number

Aver

age

Trav

el T

ime

(Min

utes

)

MUCTDTA

SQG

29

Scenario MUCTDTA solution

(S1)

OFAP solution

(S2)

OFH solution

(S3)

Mean O-D solution

(S4)

I 100 99.2 101.0 109.4

II 100 99.8 100.5 108.0

III 100 100.2 100.2 109.2

IV 100 139.7 140.0 193.9

V

Low congestion

level

100

100.7

101.9

102.7

High congestion

level

100

100.1

104.0

114.0

Table 2 Comparison of the Online Performance of Solution Approaches for Various Scenarios, as a Percentage of the Corresponding MUCTDTA Solutions

30

Scenario MUCTDTA solution

(S1)

OFAP solution

(S2)

Mean O-D

solution

(S4)

Incidents mean O-D

solution

(S5)

Combined single

incidents mean O-D solution

(S6)

ODUH solution

(S7)

Rolling horizon solution

(S8)

VI 100 99.2 109.4 - - 99.0 110.7

VII 100 128.3 138.1 119.9 - 115.5 -

VIII 100 156.0 178.3 134.9 161.2 121.4 -

Average CPU time (sec.)

18

1103