Solution method to user equilibrium traffic assignment

Following the formulation of equivalent user equilibrium programme in (2.10), a convex combination method based on the Frank-Wolfe algorithm has been widely used to solve the equivalent user equilibrium programme. In relation to the convex combination method, it can be carried out in the following two ways. First, a linear programme is formulated on

the basis of current travel times in deciding a feasible direction along which the value of objective function of (2.10) subject to constraint set can be reduced. Second, a sub- programme is formulated, in which the optimal move size along the feasible direction can be decided and hence the value of the objective function is minimised. In the linear programme, the first derivatives of the objective function with respect to path flows are used to determine the minimum path travel time for specified current traffic flows. This step is no more than a shortest path problem, in which the all-or-nothing traffic assignment is carried out in search for auxiliary flows on the basis of current travel times. In the sub- programme, a one-dimensional optimisation problem is carried out for which an optimal move size can be decided such that the value of objective function in terms of a linear combination of current flows and auxiliary flows is minimised. The linear programme and the corresponding sub-programme for the convex combination method can be carried out alternately in a series of iterative processes until a required criterion is satisfied.

<Linear programme>

Let be current link flows at iteration n and c , C be respectively current link

and path travel times at iteration n .

Find the shortest path at iteration n between each origin-destination pair w in W based on current traffic flows to

Minimise ^ ^ C ^

/ w e W p &

subject to

^

P G Pvv

/ “ > 0 p e P „ , V w e W

The linear programme (2.18) iS to find the shortest paths with the smallest travel times

at iteration n . i.e. = \ ^ for v/hich the travel demands of all origin-destination

flows and the corresponding current travel times c and C . In other words,

the resulting traffic assignment is no more than a problem of finding the shortest path for given specified travel time and thus it can be performed by the technique of all-or-nothing traffic assignment. The all-or-nothing traffic assignment for programme (2.18) can be described as follows.

For each origin-destination pair w in W and given trip rate over a period of analysis,

Ù , there exists a path h connecting origin-destination pair vu at iteration n , such

that

f T = . if , V p e

and = 0 , \f p ^ h

The corresponding auxiliary link flow can be derived in terms of the link-path

incidence matrix.

( « '" T = 5 y c y (2.19)

<Sub-programme>

Following the auxiliary flows q^f* obtained by expression (2.19), a one-dimensional line

search to find the optimal move length along the descent direction,

d = q^!^^ - q^Q^ which minimises (2 . 10) is to

Minimise Z(ç^^ -I- {q^f^ - ^[j^^))

(

2

20

subject to 0 < < 1

This represents a one-dimensional optimisation problem in searching for the optimal move size along the feasible direction; it is also a one-dimensional step size problem for a linear combination of auxiliary and current flows such that the value of objective function for mixed traffic flows can be minimised.

One of solution methods to a one-dimensional optimisation problem is a Bolzano search- bisection method. When this is applied to (2.20), the optimal move size can be found if the following condition is met.

= 0

0 < < 1 (2.21)

where - g^^)

The (2.21) corresponds to the ’cost equilibration’ as mentioned by Van Vliet (1987). When the sub-programme is solved by using a one-dimensional search process, new traffic flows can be obtained by the linear combination of auxiliary and current flows by means of the optimal move size and then the corresponding travel times can be updated accordingly. In relation to the stopping criterion for the convex combination method when carried out in a sequence of iterative processes by solving alternately the linear programme and sub-programme, it is suggested (Sheffi, 1985, p p ll9 ) to stop when the difference in the total origin-destination travel times between successive iterations is not greater than a predetermined threshold value, i.e..

E

e W

- 4 " - "I

. (n) <

8 (2.22)

where is the minimum travel time between a specified origin-destination pair w in

W at iteration n , which can be found by (2.18), and e , e > 0 , is a predetermined

value of threshold.

In summary, as referred to Sheffi (1985, pp 119-120) the procedure for carrying out the convex combination method to solve an equivalent user equilibrium programme can be stated in the following steps.

Step 0: Initializ.ation.

0.1 Set indicator n = 0 .

0.2 Perform all-or-nothing traffic assignment based on current travel times,

c = 0) by solving the linear programme (2.18) and yield the corresponding link

flows .

0.3 Set indicator n = \ .

Step 1: Update.

Set c = c

Step 2: Direction finding.

Perform all-or-nothing traffic assignment based on by solving the linear programme

(2.18) and yield the auxiliary link flows by means of (2.19).

Step 3: Line search.

Find along the descent direction d - q^^^ by solving (2.20), which also

can be carried out by means of (2.21).

Step 4: Move.

Set ^ + «<")

Step 5: Convergence test.

If condition (2.22) is met then the convex combination procedure is complete and

2.3 Equilibrium Network Design Problem