Problem Formulation as a Variational Inequality

3.6 Dynamic Route Choice on a Road Network

3.5.1 Problem Formulation as a Variational Inequality

This section summarizes a dynamic route choice model, which can be extended to incorporate motorists’ departure/arrival time choice jointly with route choice (Ran and Boyce, 1996); see also Chen (1999) for a related approach. In the next subsection, some basic definitions and dynamic network flow constraints are stated; Table 1 provides a list of definitions and symbols for convenient reference. Based on actual link travel times, the ideal dynamic user-optimal (DUO) route choice model is then formulated. These formulations are presented to illustrate the concepts and formulation of the dynamic traffic assignment problem using VI methods.

Table 1 Definitions and Notation

( )

t

x_a number of vehicles on link a at time t

( )

t

x_ap^rs number of vehicles on link a and route p between O-D pair rs at time t

( )

t

u_a inflow rate of link a at time t

( )

t

v_a exit flow rate of link a at time t

( )

^t

v_ap^rs exit flow rate of link a on route p between O-D pair rs at time t

( )

t

E^rs_p cumulative number of vehicles arriving at destination s from origin r on route p by time t

( )

t

e^rs_p arrival flow rate at destination s from origin r on route p at time t

( )

^t

f^rs departure flow rate from origin r toward destination s at time t

( )

j

A set of links whose tail node is j

( )

j

B set of links whose head node is j

( )

t

τa actual travel time over link a for flows entering link a at time t

( )

t

τa estimated actual travel time over link a for flows entering link a at time t

( )

t

rs

ηp travel time actually experienced over route p by motorists departing origin r toward destination s at time t

( )

t

πrs minimal travel time actually experienced by motorists departing origin r for destination s at time t

( )

t

πrs estimated minimal travel time actually experienced by motorists departing origin r for destination s at time t

( )

t

rs

Ωa perceived actual travel time for flows departing origin r toward destination s over route p at time t

( )

^t

rj a

Ω ∗ difference of the minimal travel time from r to j and the travel time from r to j, via the minimal travel time route from r to i and link a for motorists departing from origin r at

time t

r

ξa departure time interval for flow from zone r on link a 3.5.1.1 Network Flow Constraints

The constraint conditions necessary for the formulation of dynamic route choice models are defined first. In addition to flow conservation conditions, flow propagation constraints are emphasized. Other important constraints include link capacity and spillback constraints. The constraint set for a typical VI model is summarized as follows.

Relationship between link status and link flow variables:

; , , , ) ( )

(t v t a p r s dt u

dx _rs

ap rs

ap rs

ap = − ∀ (1)

where x_ap^rs(t) is the number of vehicles on link a and route p between OD pair rs at time t, )

(t

u_ap^rs is the inflow rate into link a on route p between OD pair rs at time t, and )

(t

v_ap^rs is the exit flow rate from link a on route p between OD pair rs at time t.

; , , ) ) (

( e t p r s r

dt t

dE _rs

p rs

p = ∀ ≠ (2) where E^rs_p (t) is the cumulative number of vehicles arriving at destination s from origin r on

route p by time t, and )

(t

e^rs_p is the arrival flow rate at destination s from origin r on route p at time t.

Flow conservation constraints:

; , ) ( )

(

) (

s r t u t

f

p rs ap r A a

rs = ∑ ∑ ∀

∈ (3) where f^rs(t) is the departure flow rate from origin r toward destination s at time t (given);

; ,

; , , , ) ( )

(

) ( )

(

s r j s r p j t u t

v _ap^rs

j A a j

B a

rs

ap = ∑ ∀ ≠

∑ ∈

∈ (4)

;

; , ) ( ) (

) (

r s s r t e t

v ^rs

s B a

rs p ap

≠

∀

∑ ∑ =

∈ (5) where A(j) is the set of links whose tail node is j (after j), and B(j) is the set of links whose head node is j (before j); note that nodes r and s are specific cases of node j.

Flow propagation constraints:

)}

( )]

( [ { )}

( )]

( [ { )

(t ~ x t t x t E t t E t

x _{a bp}^rs _p^rs _a ^rs_p

p b

rs bp rs

ap = ∑ + − + + −

∈ τ τ ∀a∈B(j);j≠r;p,r,s; (6)

where p~ denotes the subroute from node j to destination s, and τ_a(t) is the actual travel time over link a for flows entering link a at time t.

Definitional constraints:

; ), ( ) ( ),

( )

(t u t v t v t a

u _a

rsp rs ap rsp a

rs

ap = ∑ = ∀

∑ (7)

; ),

( )

(t x t a

x _a

rsp rs

ap = ∀

∑ (8)

where )u_a(t is the inflow rate into link a at time t, )v_a(t is the exit flow rate from link a at time t, and )x_a(t is the number of vehicles on link a at time t.

Nonnegativity conditions:

; , , , 0 ) ( , 0 ) ( , 0 )

(t u t v t a p r s

x_ap^rs ≥ _ap^rs ≥ _ap^rs ≥ ∀ (9)

; , , 0 ) ( , 0 )

(t E t p r s

e^rs_p ≥ ^rs_p ≥ ∀ (10) Boundary conditions:

; , , , 0 ) 0

( p r s

E^rs_p = ∀ (11)

; , , , , 0 ) 0

( a p r s

x_ap^rs = ∀ (12) 3.5.1.2 Model Formulation

Dynamic route choice models can be formulated based on either actual or instantaneous travel times. The instantaneous link travel time at time t is defined as the travel time that would be experienced by motorists traversing a link if prevailing traffic conditions remain unchanged. The actual link travel time is the travel time over a link actually experienced by motorists. This time is also called the future or forecast time, since it is not observable at time t.

Let )τ_a(t be the actual travel time over link experienced by motorists entering link a at time t, which is assumed to depend on the number of vehicles x_a(t), the inflow u_a(t) and the exit flow

) (t

v_a on link a at time t. It follows that:

a t

v t u t x

t _{a a a a}

a()=τ [ ( ), (), ( )] ∀

τ (13) Similarly, the actual route travel time is the time over a route actually experienced by motorists.

Define η^rs_p (t) as the travel time actually experienced over route p by motorists departing origin r toward destination s at time t. Using a recursive formula, the route travel time η^rs_p (t) can be computed for all allowable routes. Assume route p consists of nodes (r,1,2,L,i,L,s). Let

)

rs(t

ηp denote the travel time actually experienced over route p from origin r to node i by motorists departing origin r at time t. Then, a recursive formula for route travel time η^rs_p (t) is:

).

, 1 (

; , , 2 , 1

; , , )]

( [

) ( )

(t ^r_p⁽ⁱ¹⁾ t _a t ^r_p⁽ⁱ¹⁾ t p r i i s a i i

ri

p =η ⁻ +τ +η ⁻ ∀ = L = −

η

The formulation of the ideal DUO route choice problem is based on the underlying choice criterion that each motorist uses the route that minimizes his/her future (actual) travel time when departing from the origin to destination. Thus, for any OD pair under ideal DUO route choice conditions, motorists departing the origin at the same time must arrive at the destination at the same time (Wardrop, 1952).

Route-Time-Based Model

In this problem, the time-dependent origin-destination trip pattern is assumed to be known a priori. That is, the departure times of motorists are given. The ideal dynamic user-optimal (DUO) route choice problem is to determine the dynamic status of links and the inflow and exit flow variables at each instant of time resulting from motorists using minimal-time routes, given the network, the link travel time functions and the time-dependent OD departure rate

requirements. Consider the following dynamic generalization of the conventional static user- optimal state.

Travel-Time-Based Ideal DUO State: If, for each OD pair at each instant of time, the actual travel times experienced by motorists departing at the same time are equal and minimal, the dynamic traffic flow over the network is in a travel-time-based ideal dynamic user-optimal state.

If the actual link travel time τ_a(t) is determined, the minimal travel time π^rs(t) actually experienced by motorists departing origin r for destination s at time t can be computed as

(

⁽⁾

)

min )

(t ^rs_p t

p

rs η

π = . Thus, π^rs(t) is a functional of all link flow variables at time w:

]

| ) ( ), ( ), ( [ )

(t ^rs u v x t

rs =π ω ω ω ω≥

π

This functional is not available in closed form. Nevertheless, it can be evaluated when )

( ), (ω v ω

u and x(ω) are temporarily fixed. The route-time-based ideal DUO route choice conditions can then be defined as follows:

; , , 0 ) ( )

(t ^rs t p r s

rs

p^∗ −π ≥ ∀

η (14)

; , , 0 )]

( ) ( )[

(t t t p r s

f_p^rs∗ η^rs_p^∗ −π^rs∗ = ∀ (15) .

, , 0 p r s

f_p^rs∗≥ ∀ (16) The asterisk in the above equations denotes that the flow variables are the optimal solutions under the route-time-based ideal DUO state. The equivalent variational inequality formulation of route-time-based ideal DUO route choice conditions (14)-(16) may be stated as follows:

The dynamic traffic flow pattern satisfying network constraint set (1)-(12) is in a route-time- based ideal DUO route choice state if and only if it satisfies the variational inequality problem:

∫ ∑ ∑ ^∗ − ^∗ ≥

T rs a

rs p rs

p rs

p t f t f t dt

0

0 )]

( ) ( )[

η ( (17) Solving variational inequality (17) is equivalent to solving ideal DUO route choice conditions (14)-(16). The properties of this formulation are described in Ran and Boyce (1996).

Link-Time-Based Model

Route-time-based VI models have an intuitive interpretation. With new solution methods, such as disaggregate simplicial decomposition (DSD) or gradient projection (GP), route enumeration is no longer required to obtain a solution to the route-time-based model. However, route-time- based models still require considerable computational effort for networks of realistic size.

Therefore, a link-time-based VI model is proposed. The link-time-based ideal DUO route choice conditions given below imply the route-time-based ideal DUO route choice conditions.

The set of dynamic network constraints defined above apply to both the link-time-based VI model and the route-time-based VI model. The basic difference between the two models is that the link-time-based model is formulated using link-based flow variables instead of the route- based variables in the route-time-based model. Let Ω^rj_a^∗(t) denote the difference of the minimal travel time from r to j and the travel time from r to j via minimal travel time route from r to i and link a for motorists departing from origin at time t:

).

, (

; , ) ( )]

( [

) ( )

(t ^ri t _a t ^ri t ^rj t a r a i j

rj

a = + + − ∀ =

Ω ^∗ π ^∗ τ π ^∗ π ^∗ (18) The link-time-based ideal DUO route choice conditions can then be stated as follows:

; ), , ( 0

)

(t a i j r

rj

a ≥ ∀ =

Ω ^∗ (19)

; , ), , ( 0

) ( ] ) (

[t t t a i j r s

u_a^rs∗ +π^ri∗ Ω^rj_a^∗ = ∀ = (20) .

, ), , ( 0

)]

(

[t t a i j r s

u_a^rs +π^ri∗ ≥ ∀ = (21) Link-time-based ideal DUO route choice conditions (19)-(21) imply route-time-based ideal DUO route choice conditions (14)-(16). Then, the equivalent variational inequality formulation of link-time-based ideal DUO route choice conditions (19)-(21) may be stated as follows:

The dynamic traffic flow pattern satisfying constraints (1)-(12) is in a link-time-based ideal DUO route choice state if and only if it satisfies the variational inequality problem:

∫ ∑ ∑Ω ^∗ ⋅ + ^∗ − ^∗ + ^∗ ≥

T rs a

ri rs

a ri

rs a rj

a t u t t u t t dt

0

0 )]}

( [

)]

( [

{ )

( π π (22) Solving variational inequality (22) is equivalent to solving link-time-based ideal DUO route choice conditions (19)-(21). Further properties are given in Ran and Boyce (1996). The above models are limited in the sense that the motorists are assumed to be homogenous. One way to

relax this assumption is to stratify them into classes, in which each class has distinctly different characteristics. Each class of motorists is associated with a disutility or generalized cost

function. Thus, the ideal DUO route choice conditions are defined for each class on the basis of travel disutilities instead of travel time only. Then, a multi-class DUO model can be formulated based on travel costs or disutilities instead of travel times.