#### Abstract

Traffic network equilibrium problems with capacity constraints of arcs are studied. A (weak) vector equilibrium principle with vector-valued cost functions, which are different from the ones in the work of Lin (2010), and three kinds of parametric equilibrium flows are introduced. Some necessary and sufficient conditions for a (weak) vector equilibrium flow to be a parametric equilibrium flow are derived. Relationships between a parametric equilibrium flow and a solution of a scalar variational inequality problem are also discussed. Some examples are given to illustrate our results.

#### 1. Introduction

The earliest traffic network equilibrium model was proposed by Wardrop [1] for a transportation network. After getting Wardrop’s equilibrium principle, many scholars have studied variant kinds of network equilibrium models, see, for example, [2–5]. However, most of these equilibrium models are based on a single criterion. The assumption that the network users choose their paths based on a single criterion may not be reasonable. It is more reasonable to assume that no user will choose a path that incurs both a higher cost and a longer delay than some other paths. In other words, a vector equilibrium should be sought based on the principle that the flow of traffic along a path joining an *O*-*D* pair is positive only if the vector cost of this path is the minimum possible among all the paths joining the same *O*-*D* pair. Recently, equilibrium models based on multiple criteria or on a vector cost function have been proposed. In [6], Chen and Yen first introduced a vector equilibrium principle for vector traffic network without capacity constraints. In [7, 8], Khanh and Luu extended vector equilibrium principle to the case of capacity constraints of paths. For other results of vector equilibrium principle with capacity constraints of paths, we refer to [9–17].

Very recently, in [18, 19], Lin extended traffic network equilibrium principle to the case of capacity constraints of arcs and obtained a sufficient condition and stability results of vector traffic network equilibrium flows with capacity constraints of arcs. In [20], Xu et al. also considered that vector network equilibrium problems with capacity constraints of arcs. By virtue of a function, which was introduced by Zaffaroni [21], the authors introduced a -equilibrium flow and a weak -equilibrium flow, respectively, and obtained sufficient and necessary conditions for a weak vector equilibrium flow to be a (weak) -equilibrium flow.

In this paper, our aim is to further investigate traffic network equilibrium problems with capacity constraints for arcs. We introduce a (weak) vector equilibrium principle with vector-valued cost functions, which are more reasonable from practical point of view than the ones in [18, 19]. In order to obtain necessary and sufficient conditions for a (weak) vector equilibrium, we introduce three kinds of parametric equilibrium flows. Simultaneously, we also discuss relationships between a parametric equilibrium flow and a solution of a scalar variational inequality problem.

The outline of the paper is as follows. In Section 2, a (weak) equilibrium principle with capacity constraints of arcs is introduced. In Section 3, three kinds of parametric equilibrium flows are introduced. Some sufficient and necessary conditions for a (weak) vector equilibrium flow are obtained. Relationships between a parametric equilibrium flow and a solution of a scalar variational inequality problem are also discussed.

#### 2. Preliminaries

For a traffic network, let and denote the set of nodes and directed arcs, respectively, and let denote the capacity vector, where (>0) denotes the capacity of arc . Let denote the set of origin-destination (*O*-*D*) pairs and let denote the demand vector, where (>0) denotes the demand of traffic flow on *O*-*D* pair . A traffic network with capacity constraints of arcs is usually denoted by . For each arc , the arc flow needs to satisfy the capacity constraints: , for each . For each , let denote the set of available paths joining *O*-*D* pair . Let . For a given path , let denote the traffic flow on this path and is called a path flow. The path flow vector induces an arc flow on each arc given by
where if the arc is contained in path and , otherwise. Suppose that the demand of network flow is fixed for each *O*-*D* pair . We say that a path flow satisfies demand constraints
A path flow satisfying the demand constraints and capacity constraints is called a feasible path flow. Let : for all , and for all , : for all , and for all , and let . Clearly, is convex and compact. Let be a vector-valued cost function for the path on the arc . Let be a vector-valued cost function along the path . Then the vector-valued cost on the path is equal to the sum of the all costs of the flow through arcs, which belong to the path , that is,
Let .

*Remark 2.1. *In [18, 19], Lin defined the vector cost function along the path as follows:
where be a vector-valued cost function for arc . If the paths have common arcs, then the definition is unreasonable. The following example can illustrate the case.

*Example 2.2. * Consider the network problem depicted in Figure 1. , , , , . The cost functions of arcs from to are, respectively, as follows:

For *O*-*D* pair (1, 4): includes path and path , for *O*-*D* pair includes path and path . And by (2.4), we have
Then, for flow , we have that arc flows
It follows from (2.4) that
However, from the practical point of view, the cost values of the path 1 and path 3 with respect to are, respectively, as follows:
So, in this paper, we define the vector-valued cost function on a path as (2.3).

In this paper, the cost space is an -dimensional Euclidean space , with the ordering cone , a pointed, closed, and convex cone with nonempty interior . We define the ordering relation as follows: The orderings and are defined similarly. In the sequel, we let the set , for all be the dual cone of . Denote the interior of by

Lemma 2.3 (see [22]). *Consider
*

*Definition 2.4 (see [18, 19]). *Assume that a flow , (i)for , if , then arc is said to be a saturated arc of flow , otherwise a nonsaturated arc of flow . (ii)for , if there exists a saturated arc of flow such that belongs to path , then path is said to be a saturated path of flow , otherwise a nonsaturated path of flow .

We introduced the following vector equilibrium principle and weak vector equilibrium principle.

*Definition 2.5 (vector equilibrium principle). *A flow is said to be a vector equilibrium flow if for all , for all , we have

*Definition 2.6 (weak vector equilibrium principle). *A flow is said to be a weak vector equilibrium flow if for all , for all , we have
If for all , , then the capacity constraints of arcs are invalid, in this case, the traffic equilibrium problem with capacity constraints of arcs reduces to the traffic equilibrium problem without capacity constraints of arcs.

#### 3. Sufficient and Necessary Conditions for a (Weak) Vector Equilibrium Flow

In this section, we introduce an -parametric equilibrium flow, a -parametric equilibrium flow and a -parametric equilibrium flow, respectively. By using the three new concepts, we can obtain some sufficient and necessary conditions of a vector equilibrium flow and a weak vector equilibrium flow, respectively.

*Definition 3.1. * A flow is said to be in -parametric equilibrium if for all , for all and for all , we have

*Definition 3.2. * A flow is said to be in -parametric equilibrium if for all , for all and for all , we have

*Definition 3.3. *Let a be given. A flow is said to be in -parametric equilibrium flow if for all and for all , we have

The -equilibrium flow and -parametric equilibrium flow for some are defined in Definitions 3.1 and 3.2, respectively. They can be used to characterize vector equilibrium flow in the following theorems.

Theorem 3.4. * A flow is in vector equilibrium if and only if the flow is in -parametric equilibrium. *

*Proof. * It can get immediately the above conclusion by Lemma 2.3. Thus the proof is omitted here.

Theorem 3.5. * If there exists such that a flow is in -parametric equilibrium, then the flow is in vector equilibrium. *

* Proof. * Suppose that for any *O*-*D* pair , for all , we have
By and Lemma 2.3, we get immediately
Since is in -parametric equilibrium, we have
Thus, the flow is in vector equilibrium.

Now, we give the following example to illustrate Theorem 3.5.

*Example 3.6. *Consider the network problem depicted in Figure 2. , , , , . The cost functions of arcs from to are defined as follows:

Then, we have

Taking , then there exists such that the flow is in -parametric equilibrium. Thus, by Theorem 3.5, we have that the flow is in vector equilibrium.

For weak vector equilibrium flows, we have following similar results.

Theorem 3.7. * A path flow is in weak vector equilibrium if and only if the flow is in -parametric equilibrium. *

Theorem 3.8. *If there exists such that a path flow is in -parametric equilibrium, then the flow is in weak vector equilibrium. *

From Theorems 3.4–3.8, we can get immediately the following corollaries.

Corollary 3.9. *If there exists such that a flow is in -parametric equilibrium, then the flow is in int-parametric equilibrium. *

Corollary 3.10. * If there exists such that a flow is in -parametric equilibrium, then the flow is in -parametric equilibrium. *

*Remark 3.11. * When a flow is in -parametric equilibrium, then, the flow may not be in -parametric equilibrium for some . Of course, when a flow is in -parametric equilibrium, then, the flow may not be in -parametric equilibrium for some . The following example can explain these cases.

*Example 3.12. *Consider the network problem depicted in Figure 1. , , , , . Let the cost functions of arcs are defined as follows:

Then, we have

Taking we have Thus, by Definitions 3.1 and 3.2, we know that the flow is a -parametric equilibrium flow and is a -parametric equilibrium flow as well. On the other hand, for , there exists and path , we have But, and path 1 is nonsaturated path of . Thus, it follows from Definition 3.3 that the flow is not in -parametric equilibrium.

Theorem 3.13. * Let be given. A flow is in -parametric equilibrium if the flow solves the following scalar variational inequality:
*

*Proof. *Assume that solves above scalar variational inequality problem. For all , for all , if and path is nonsaturated path of flow , we need to prove that . Denote that belongs to path . If the conclusion is false, then
Construct a flow as follows:
It is easy to verify that
It follows readily that
which contradicts (3.14). Thus, is in -parametric equilibrium and the proof is complete.

From Theorems 3.4–3.13, we can get the following corollary.

Corollary 3.14. * If there exists () such that a flow is a solution of the following scalar variational inequality:
**
then the flow is in (weak) vector equilibrium. *

*Remark 3.15. * We can prove that the the converse of Theorem 3.13 is valid when the traffic network equilibrium problem without capacity constraints of arcs, such as traffic network equilibrium problems without capacity constraints or with capacity constraints of paths. The result will be showed on Theorem 3.18. But, if the traffic network equilibrium problem with capacity constraints of arcs, then the converse of Theorem 3.13 may not hold. The following example is given to illustrate the case.

*Example 3.16. *Consider the network problem depicted in Figure 1. , , , , . Let the cost functions of arcs are defined as follows:

Then, we have Taking we have Then for any (), we have and path 1 is a saturated arc path of , and path 3 is a saturated arc path of as well. Thus, the flow is a -parametric equilibrium flow by Definition 3.3. However, taking , we have Thus, for any (), we can always get Therefore, the converse of Theorem 3.13 is not valid.

The following theorem shows that the converse of Theorem 3.13 is valid when the traffic equilibrium problem with capacity constraints of paths. The proof is similar when the traffic network equilibrium problem without capacity constraints. Let be the feasible set of traffic network equilibrium problem with capacity constraints of paths, where and are lower and upper capacity constraints of paths, respectively. The -parametric equilibrium principle of traffic equilibrium problem with capacity constraints of paths is as follows.

*Definition 3.17. * Let a be given. A flow is said to be in -parametric equilibrium flow if for all and for all , we have

Theorem 3.18. *Let be given. A path is in -parametric equilibrium if and only if the flow solves the following scalar variational inequality:
*

*Proof. *From Theorem 3.13, we only prove necessity. So, we set
It follows from the definition of the -parametric equilibrium flow that
Thus, there exists a such that
Let be arbitrary. Then, for every , we consider three cases.*Case 1*. If , then . Hence, , and
*Case 2*. If , then . Hence, , and
*Case 3.* If , then we have
From (3.33), (3.34), and (3.35), we have
Thus, the proof is complete.

#### 4. Conclusions

In this paper, we have studied traffic network equilibrium problems with capacity constraints of arcs. We have introduced some new parametric equilibrium flows, such as: -parametric equilibrium flows, -parametric equilibrium flows, and -parametric equilibrium flows. By using these new concepts, we have characterized vector equilibrium problems on networks and derived some necessary and sufficient conditions for a (weak) vector equilibrium flow.