_{1}

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In the traffic equilibrium problem, we introduce capacity constraints of arcs, extend Beckmann’s formula to include these constraints, and give an algorithm for traffic equilibrium flows with capacity constraints on arcs. Using an example, we illustrate the application of the algorithm and show that Beckmann’s formula is a sufficient condition only, not a necessary condition, for traffic equilibrium with capacity constraints of arcs.

In [

For a traffic network, let V denote the set of nodes, E the set of directed arcs, and W the set of origin- destination O-D pairs. For each

by

demand on O-D pair ω. For each

let

is said to be a path flow (flow). Clearly, for

path k, otherwise

In this paper, we assume that for each

For the following definitions, see [

Definition 2.1. Assume that a flow

1) for

2) for

a saturated path of flow f, otherwise a nonsaturated path of flow f.

Definition 2.2. (Equilibrium principle with capacity constraints of arcs). A flow

f is said to be an equilibrium flow or solution of the TEPCCA. A TEPCCA is usually denoted by

For the TEPCCA

The above formula is a generalization of Beckmann’s formula. The next theorem shows that each solution of the generalization of Beckmann’s formula is an equilibrium flow for

Theorem 3.1. Consider the TEPCCA. Assume that for each

Proof. Set

where

When path k is a nonsaturated path of flow f, for each

Hence, when k is a nonsaturated path, we have

and when k is a saturated path, we have

In other words, if paths k is a nonsaturated path, then

From the generalization of Beckmann’s formula, it is easy to construct an algorithm to calculate the equilibrium flow for the TEPCCA

For the TEPCCA

Step 1. Find a feasible flow

Step 2. Solve the restricted problem

We obtain solution

Step 3. After deleting all saturated arcs of the flow

Step 4. If

Step 5. The equilibrium flow is

The following example shows the calculation process of the algorithm.

Example 4.1. Consider the TEPCCA (see

and

For O-D pair

Let

Next, we compute the equilibrium flow with capacity constraints of arcs.

1) It is easy to verify that

2) Solve the restricted problem

We obtain solution

3) There is no saturated arc of flow

thus

4) Since

We obtain solution

5) After deleting saturated arc

thus

6) Since

We obtain solution

7) After deleting saturated arc

8) Because

Note that

Thus the generalization of Beckmann’s formula Q is:

It is easy to verify that

Note that

solution of the generalization of Beckmann’s formula Q, i.e., Theorem 3.1 is a sufficient condition only, not a necessary condition.

This work was supported by National Natural Science Foundation of China (Grant No. 11271389).

ZhiLin, (2015) An Algorithm for Traffic Equilibrium Flow with Capacity Constraints of Arcs. Journal of Transportation Technologies,05,240-246. doi: 10.4236/jtts.2015.54022