Static Trip Assignment Models

Static assignment models assume that link flows and link trip times remain constant over the planning horizon of interest. Hence, a static origin-destination (O-D) matrix is given and assigned to the network links, results a link flow pattern that is intended to replicate the actual flow based on some behavior assumption. The static equilibrium assignment models are adequate for long-term planning analysis. Studies have shown that these formulations fail to capture the essential features of traffic congestion, including queue, departure time shift, traffic propagations, and etc.

(Herman and Lam, 1974; Lisco, 1983; Hendrickson and Planck, 1984). Early attempts ignored congestion and the equilibrium issue, assigning all trips between any given O-D pairs to the shortest travel cost path (all-or-nothing assignment). Refined approaches resulted from recognition of the need to incorporate congestion effects (Sheffi, 1985; Matsoukis, 1986)

User Equilibrium Concept

The first mathematical programming formulation for the static user equilibrium (UE) problem with fixed demand as an equivalent optimization problem is introduced by Beckmann et al. (1956). This formulation allows the derivation of existence and uniqueness properties of the solution, satisfying the Wardropian UE condition. While Wardrop’s UE condition indicates no user can improve his/her travel time/cost by unilaterally switching routes (Wardrop, 1952). The static UE flow pattern is obtained by solving the Beckmann equivalent optimization problem, stated as the following mathematical program:

8 

∑∫ ( )

=

a x

a

at d

Z( ) 0

min x ω ω (2.2a)

subject to

s r q f

k

rs rs

k = ∀ ,

∑

^(2.2b)

s r k

f_k^rs ≥0 ∀ , , (2.2c)

∑∑∑

^∀

=

r s k

rs k a rs k

a f a

x δ _, (2.2d)

where x is the vector of link flows, x_a represents the flow on link a, and t_a

( )

• is the link performance function for link a that specifies the link travel time as a function of the flow on the link. The link performance function, often refer to the BPR function, is a positive, increasing, and convex curve, as illustrated in Figure 2.1. The typical link performance function does not consider queued vehicles in the traffic stream nor the propagation of the traffic flow.

Figure 2.1 Typical Link Performance Function ta(xa)

The O-D demand between origin r and destination s is denoted by q_rs and the flow for O-D pair r-s assigned to path k is represented by f_k^rs. The static link-path incidence matrix relating path flows to link flows (equation 2.2d) are defined using

Link Flow 

Link Travel Time 

( )

a

a x

t

x

a

link-path incidence variable δ_a,^rs_k as follows:

⎩⎨

⎧ −

= 0,otherwise

pair D -O between path

on is link if , 1

,

s r k

rs a

k

δa (2.2e)

The objective function Z(x), which is the sum over all arcs of the integrals of the link performance functions, does not have an intuitive economic or behavioral interpretation and is viewed strictly as a mathematical construct to solve equilibrium problems. Equation 2.2b indicates the set of flow conservation constraint which imply that all O-D demand have to be assigned to the network. The non-negativity conditions (equation 2.2c) ensure the solution of the program is physically meaningful.

The network structure enters the formulation through the link-path incidence relationship (equation 2.2d) that relates the link-based objective function to the path-based constraint set.

Sheffi gives a comprehensive treatment of the static UE problem, addressing the conceptual, mathematical, algorithmic and computational aspects of the problem (Sheffi, 1985). A more difficult problem with asymmetric link interactions is addressed by Dafermos, and Fisk and Boyce by using variational inequality (VI) techniques (Dafermos, 1980, 1982; Fisk and Boyce, 1983). Nagurney, Mahmassani and Mouskos, and Patricksson address computational issues related to the VI problem (Nagurney, 1984,1986; Mahmassani and Mouskos, 1988, 1989).

System Optimal Concept

The other major class of assignment is system optimal (SO) formulation. The SO seeks a flow pattern that achieves some system-wide objectives. The static SO

10 

assignment problem can be formulated as follows:

∑

subject to

s

The objective function is the only difference from UE formulation with the interpretation of total system travel cost. The SO flow pattern does not always represent an equilibrium solution, as individual travelers may reduce their travel time by switching their routes. Hence, the SO flow pattern is not expected to hold without some control strategy such as road pricing or restriction. Consequently, the SO flow pattern is not an appropriate descriptive model of actual user behavior. It can be treated as a performance index of the network. The solution procedures for SO are identical to those of UE except that they differ in the specification of link cost functions (average cost function in UE; marginal cost function in SO).

Ben-Akiva enumerates the shortcomings of using static models in modeling congestions (Ben-Akiva, 1985). The static assignment has a major shortcoming of inadequately link congestion model. As discussed earlier in the thesis, the congestion is presented by a link performance function which gives the average trip time as a function of the average link flow. The average link flow can even exceed the actual capacity of the road section which is unrealistic for the control purpose. Another major problem lies on the description of traffic propagation. The static volume-delay

curve cannot describe the propagation of traffic flow, especially in high flow levels.

Thereby, static assignment models are inappropriate for real-time traffic control application, especially for congested networks.

2.2 Dynamic Trip Assignment Models

Dynamic network assignment is under intensive research, for both user equilibrium and system optimal problems. One common feature of these researches is that they differ from the standard static assignment assumptions to deal with time-varying flows. Another feature shared by these researches is that none presently provides a universal solution for general networks.

The first attempt to formulate the DTA problem as a mathematical program is introduced by Merchant and Nemhauser (1978a, 1978b). The model (referred to as the M-N model) is limited to the fixed-demand, single-destination, deterministic, system optimal scenario. A link exit function is utilized to propagate traffic and a static link performance function is introduced to present the travel cost as a function of link flow.

It results a flow-based, discrete time, non-convex non-linear programming formulation. The global solution can be derived by solving a piecewise linear version of the model.

System Optimal Concept

Carey (1987) reformulates the M-N model as a convex nonlinear program by the manipulation of exit function, which gives mathematical advantage over the original M-N model. The formulation differs from M-N model mainly in the consideration of multiple destinations, and the exit function g_a(•).

12 

subject to

( )

^{x b a t}

where x represents the number of vehicles on link a at the beginning of interval t. ^t_a

( )

a^t t a x

h represents the travel cost incurred by the volume x and assumed to be _a^t

continuous, convex, nondecreasing and nonnegative. The variable b and _a^t d _a^t denote the number of vehicle exiting and entering link a in interval t, respectively.

t

F is referred to the exogenous demand at node k in period t. k E_a is the initial volume on arc a. ^B

( )

^{k and C(k)} respectively represent the set of links incident from and to node k. g_a(•) the exit function define the maximum number of vehicles that can exit from link a and is a function of traffic conditions on the link; it is assumed to be a continuous, non-negative, non-decreasing, and concave function.

Figure 2.2 illustrates a possible shape of such link exit function.

Figure 2.2 A possible shape of link exit function

Although after the manipulation of exit function, the formulation is convex, but it remains problematic by the non-convexity issues arising from first-in first-out (FIFO) requirement. The FIFO violation implies some traffic physically jumps over another to reduce system cost which is inconsistent with traffic realism. The FIFO requirement is easily satisfied in single destination formulation. While facing general networks, the FIFO requirement would introduce additional constraints that yield a non-convex constraint set and increasing the computational burden severely (Carey, 1992). As SO flow patterns, it may often be advantageous to favor certain traffic movement over others to minimize system-wide travel cost. For example, traffic at minor approach of an intersection may be holding back in favor of the major approach.

That means vehicles may be artificially delayed for a time that might be considered as unfair or unreasonable; and the flow pattern may not be acceptable for real-world operation. Ziliaskopoulos introduces a linear programming formulation for the single destination system optimal DTA problem based on the cell transmission model (Ziliaskopoulos, 2000; Daganzo, 1994). This model circumvents the need for link performance function as the flow propagates according to the cell transmission model, hence is more sensitive to traffic realities.

t

x

a

Number of vehicles on link  Number of vehicle  exiting 

( )

a^t

a x

g

14 

User Equilibrium Concept

The user equilibrium formulation is generalized from the Wardrop condition for the static problem; it becomes the equilibration of the experienced path travel times of users. Janson represents one of the earliest attempts at modeling the UE dynamic traffic assignment problem as a mathematical program (Janson, 1991). The link-based UE model formulated as a mathematical program as follows:

( )

⁼

∑∑∫ ( )

t a

x t

a

t

a d

Z 0

min x λ ω ω (2.5)

subject to

Equation (2.4b) – (2.4f)

The λ^t_a represents the cost of traveling link a at the beginning of interval t when there exist x vehicles on the link. _a^t

Birge and Ho extend the M-N model to the stochastic case by relaxing the assumption that O-D demands are known for the entire planning horizon (Birge and Ho, 1993). It assumes a finite number of scenarios, defined as a possible combination of past O-D demands in every time interval, while assignment decisions are independent of future O-D demands.

Another approach is modeling dynamic traffic assignment problem in a continuous manner. The O-D demands are assumed to be known continuous functions of time; link flows are treated as continuous functions of time. Constraints of optimal control formulations are analogous to those of the mathematical programming formulations, but they are defined in a continuous-time manner. Friesz et al. discuss link-based optimal control formulation for both SO and UE objectives for the single

destination case (Friesz, Luque, Tobin, and Wie, 1989). The model assumes that changes from one system state to another may occur concurrently as the network conditions change; that implies the routing decision are made based on current network conditions, and can be continuously modified as conditions change. The SO model is represented as follows:

( )

⁼

∑∫ ( )

subject to

( )

x u

( )

t g

[

x

( )

t

]

a A t

[ ]

T

Equation 2.5b describe the rate of change of traffic volume with respect to time for link a will be considered as the difference of the flow entering link a, u_a

( )

t , and the flow exiting link a, g_a

[

x_a

( )

t

]

. In equation 2.6c, S_k

( )

t represents the traffic flow generated at node k, which is assumed to be a nonnegative and continuous function of time. ^A

( )

^{k and B(k)} respectively represent the set of links incident to and from node k. The nonnegative constraints of both traffic volume on the link and traffic volume entering the link are indicated in equation (2.6d) and (2.6e). As for the UE case, the objective function is illustrate as follows,

16 

( )

⁼

∑∫ ∫ ( ) ( )

^′

a

T x

a a a a a

ac g d dt

Z 0 0

min x ω ω ω (2.7)

subject to

equation (2.6b)-(2.6e).

The model proves that at the optimal solution, the instantaneous flow marginal costs on the used paths for an O-D pair are identical and less than or equal to the ones on the unused paths. As for the UE case, the model is in the form of equilibration of instantaneous user path costs.

Ran et al. use the optimal control approach to obtain a convex model for the instantaneous UE dynamic traffic assignment problem by defining link inflows and outflows to be control variables (Ran, Boyce, and LeBlanc, 1993) They recognize the inability of the usual cost functions to account for dynamic queuing and congestion costs, and propose splitting the link travel cost into moving and queuing parts. Boyce et al. proposed a methodology to solve the above model using Frank-Wolfe algorithm, but no implementations are illustrated (Boyce, Ran, and LeBlanc, 1995).

Because of the limitations of obtaining analytic mathematical properties, researchers focused on analytical DTA models have gradually migrated toward the variational inequality (VI) formulations. Variational inequality provides a general formulation platform for several different problems. Variational inequality approach is first introduced to the static traffic equilibrium by Dafermos (1980). Friesz et al.

introduce the VI formulation into network design problem and suggest a sensitivity analysis based heuristic algorithm (Friesz, Tobin, Cho, Mehta, 1990; Friesz, Cho, Mehta, Tobin, Anandalingam, 1992). Friesz et al. is the first to show there is a variational inequality formulation of dynamic user equilibrium with simultaneous route choice and departure time decisions (Friesz, Bernstein, Smith, Tobin, Wie,

1993). Wie et al. formulate the dynamic network user equilibrium problem as a variational inequality problem in discrete time in terms of unit path cost functions (Wie, Tobin, Friesz, Bernstein, 1995). They also demonstrate that, assuming certain regularity conditions hold, discrete time dynamic network user equilibrium is guaranteed to exist. They define the time varying flow pattern h* and associated minimum cost μ* is a discrete time dynamic network user equilibrium if the following conditions are satisfied:

( ) ( )

In equation 2.8, i and j are the origin and the destination node respectively. P _ij

denotes the set of all possible paths between origin i and destination j. )h_p(t is the number of vehicles entering the first link on path p in period t, and

[

^{h t p P t T}

]

h= _p( ): ∈ , =0,1,..., . ^c_p

( )

^t,h is the nonnegative unit travel cost incurred by travelers departing their origin in period t and choosing path p to their destination.

( )

^h =

[

μij

( )

^h :ⁱ∈^I,^j∈^J

]

μ is the vector of minimum unit travel costs. Q is the total _ij

fixed O-D demand between i and j during time interval 0≤t≤T. Since a complete path enumeration is required, an efficient method to identify a possible path set should be introduced to relief the computation burden.

Ran and Boyce propose a link-based discretized VI formulation SO DTA model with fixed departure time (Ran and Boyce, 1996). They equilibrate the experience

18 

travel time, the same as Friesz (1993). A queuing delay component is introduced in the model, but the capacity and oversaturation constraints increase the computation complexity significantly. Chen and Hsueh propose a link-based VI formulation UE DTA model and a nested diagonalization solution algorithm (Chen and Hsueh, 1998).

The constraint set of the model is nonlinear and nonconvex, and multiple local solutions might exist. The variational inequality approach gives greater analytical flexibility and convenience than other analytical approaches. Although it brings mathematical advantages, the variational inequality approach is much more computationally intensive, especially facing the complete path enumeration for path-based formulation.

Simulation based models are mostly based on the mathematical programming models, but the critical constraints that describe the traffic flow propagation (i.e. flow conservation, vehicular movement) are addressed through simulation instead of analytical representation. With the traffic simulation, these models can address the traffic flow more realistic. However, the theoretical insights cannot derive analytically, which is the key issue of simulation-based models. A deterministic DTA model, with both SO and UE solutions, is proposed by Mahmassani and Peeta (1993). A meso-scopic traffic simulator is used as part of an iterative algorithm, with complete priori information of O-D demands for the entire planning horizon. Ghali and Smith propose a deterministic SO DTA model with congestion arises exclusively at specified bottlenecks modeled as deterministic queues (Ghali and Smith, 1995). Simulation based models can describe the traffic flow more realistic, which is troublesome in analytical formulations. However, the limitation lies on the inability to derive the associated mathematical properties.

More recently, Ashok and Ben-Akiva introduced stochasticity to map the

assignment matrix between time-dependent O-D flows and link volumes both in off-line and real-time application (Ashok and Ben-Akiva, 2002). Ben-Akiva et al.

propose DynaMIT, a meso-scopic simulator, as a dynamic traffic assignment system to estimate and predict current and future traffic conditions (Ben-Akive, Koutsopoulos, Mishalani, Yang, 1997). The model considers both historical information and drivers’ response to information, supply and demand simulators work together to generate UE route guidance. Nie and Zhang proposed a relaxation approach for estimating static O-D matrix that minimizes a distance metric between measured and estimated traffic condition while the condition satisfies user equilibrium (Nie and Zhang 2008).

2.3 Summary and Discussion

To design and manage a transportation system, there is a need for efficient analyzing tool to describe the usage of the system. The traditional static method, four-stage planning process, includes trip generation, trip distribution, modal split, and trip assignment. Traditional sequential four-stage method is not suitable in the operation perspective. Since the travel costs used in the trip distribution stage are functions of the trip assignment outcomes; that is, the stages have to be repeated.

From the literature review, estimate the network flows directly from time-series of link flows seems to be a reasonable candidate for real-time traffic operation aspect.

This chapter has reviewed several topics relevant to the trip assignment problems.

Most existing research works assumed the existence of user-equilibrium or system-optimal conditions. However, the existence of equilibrium states in real traffic networks is questionable; an alternative approach to relax the assumption of user-equilibrium or system optimal might worth be established.

20 

Chapter 3

A Path-base Assignment Model with Time-invariant Coefficient Dynamic System

In this chapter, some essential concepts of the path-base assignment model with time-invariant coefficient dynamic systems are discussed. Begin with the brief introduction to dynamic system models, the model assumption and notation is addressed in section 3.1. A state space approach that modeled the path estimation is illustrated in section 3.2. Link dynamics that describe the propagation of traffic flows is introduced, in section 3.3, to modify the proposed model.

3.1 Assumption and Notation

Existing research works on traffic assignment usually assume the existence of user equilibrium status or pursuit some system optimal situation. However, it is questionable whether such equilibrium state really exists or not. In this study, we relax such assumption by some statistical approaches.

The basic assumption in this research is the existence of unknown relationship between consecutive path-flows; the assumption is similar to that of Okutani. While Okutani assumed a time invariant relation between time-series O-D flow exists; and this relationship can be estimated by some prior information. In this study, we do not estimate the relationship by prior information; the relationship is estimated simultaneously with path flows.

For convenience, we use the following notations in this chapter:

Table 3.1 Notations of Time-invariant State Space Model

3.2 Estimation of Path Flow by State Space Model without Prior Information

Isaac Newton introduced the differential equations in the 17^th century and provided mathematical models for many dynamic systems. Given a finite number of initial conditions, one can uniquely determine the system status for all time. The finite dimensional representation of a problem is the basic idea for the state-space approach to the representation of dynamic systems. The dependent variables of the equations are the state variable of the dynamic systems. The principal dynamic system models are listed in Table 3.2 below.

Notation Descriptions

F a p× path flow transition matrix p ψt a p×1 network path flows on time t

y t a 1q× link traffic observation vector on time t

H a q× zero-one matrix which denotes the path-link incidence matrix p p number of elements in path set P

q number of observations in the target network

t t γ

σ , independently and identically distributed Gaussian noise terms

T The transport of a matrix is mark by superscript T .

22 

Table 3.2 Principal dynamic system models

Reference: [Grewal & Andrews, 1993]

We focused on the discrete dynamic systems in this dissertation. In the dynamic model, F is a n×n dynamic coefficient matrix. The matrix

[

_n

]

^T

t ψ₁(t) ψ₂(t) ψ₃(t) ψ (t)

ψ = _L is called the state vector, where ψ_n(t)

denotes the n^th state variable in time t. The n-dimensional domain of the state vector is called the state space of the dynamic system. The state variables are related to the system outputs by a system of linear equations that can be represented in vector form, as follows.

t

The y_t is a l-vector called the measurement vector (also called observation vector)

Model Continuous Discrete

Time-invariant

Linear ψ_&(t)=Φψ(t)+Cu(t) ψ_t+1 =Fψ_t +Γu_t General ψ&(t)= f

(

ψ(t),u(t)

)

ψt₊₁ = f

(

ψt,ut

)

Time-varying

Linear ψ_&(t)=Φ(t)ψ(t)+C(t)u(t) ψ_t+1 =F_tψ_t +Γ_tu_t General ψ&(t)= f

(

t,ψ(t),u(t)

)

ψt₊₁ = f

(

t,ψt,ut

)

of the system. The matrix His a measurement sensitivity matrix with measurement sensitivities h measures the scale of _ln l output to ^th n state variable. ^th

Real world problems tend to have some kind of unpredictability in behaviors,

Real world problems tend to have some kind of unpredictability in behaviors,

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