Abstract and Applied Analysis

Volume 2014 (2014), Article ID 507376, 5 pages

http://dx.doi.org/10.1155/2014/507376

Research Article

Well-Posedness of MultiCriteria Network Equilibrium Problem

W. Y. Zhang

The School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 610074, China

Received 19 October 2013; Accepted 6 January 2014; Published 20 February 2014

Academic Editor: Bashir Ahmad

Copyright © 2014 W. Y. Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

New notions of ϵ-equilibrium flow and -ϵ-equilibrium flow of multicriteria network equilibrium problem are introduced; an equivalent relation between vector ϵ-equilibrium pattern flow and -ϵ-equilibrium flow is established. Then, the well-posedness of multicriteria network equilibrium problem is discussed.

1. Introduction

For a long time, real-valued functions have played a central role in network equilibrium problems. Recently, motivated by applications to real-world situations, much attention has been attracted to multicriteria network equilibrium problems, that is, equilibrium problems with vector-valued cost functions. Different concepts of vector equilibrium flow have been introduced and the existence of such flows has been investigated by various authors (refer, first, to [1], in which Chen and Yen generalized Wardrop’s scalar equilibrium principle to vector equilibrium principle, which asserts that users only choose Pareto optimal or efficient routes to travel on, and, among the others, to [2–6]).

Traffic equilibrium problem always depends on some parameters, because people’s intuitive judgment plays a central role in route choices and it is impossible to have a precise estimation of the trip cost available to many paths. Therefore, some of the factors involved in transportation networks may be regarded as perturbing parameters. We were motivated to study the behavior of perturbations of multicriteria network equilibrium problems and cope with well-posed issues in the framework of traffic equilibrium problems. The notion of well-posedness, which can be useful for numerical purposes, is based on the behavior of either minimizing or maximizing sequences (Tychonov well-posedness). These notions for scalar optimization have been extensively investigated in many papers (cf., e.g., Dontchev and Zolezzi [7] in which a good list of basic references can be found). In the last decades, some extensions of this concept for vector optimization problems appeared; see [8–10] and the references therein. In the paper, we introduce the concept of well-posedness of multicriteria network equilibrium problems. As such, we obtain a sufficient condition that multicriteria network equilibrium problems are well-posed.

We now outline the remainder of the paper. Section 2 is devoted to the detailed description of the traffic network model. In Section 3, we introduce new notions of parametric equilibrium flows and obtain an equivalent relation between -equilibrium pattern flow and --equilibrium flow. Finally, we discuss the well-posedness of multicriteria network equilibrium problem.

2. Preliminaries

Consider a traffic network , where denotes the set of origin and destination (O/D) pairs and is the set of directed arcs. We will suppose that has members. Let , which connects origin point with destination point . Then a sequence

of contiguous links in is called a path (or chain) from to . Denote by the set of routes from to which traverse no link twice. Let be the set of paths that connect an O/D pair . Then is a finite set and

is a finite set too. We will suppose that has members.

For a given path , let denote the traffic flow on this path and let be a flow of network. We will assume, for the rest of this paper, that the demand of network is fixed for each O/D pair; that is, , where is a given demand for each O/D pair . A flow satisfying the demand is called a feasible flow. is clearly a convex and compact set in . Let there be given a vector of demands .

We will consider that the network system maintains an expected amount of flow in every path. That is, for every O/D pair and every path , an expected flow is given. For every O/D pair and every path , we assume that . Let . Suppose that the set-valued mapping is defined as .

Let be a Hausdorff topological vector space ordered by a pointed, closed convex cone with , . We denote by “” the ordering induced by ; that is,

For every and  , we define the cost function of the path as a vector-valued function ; the mapping is called the cost function of the network.

Definition 1 (see [11]). A vector is said to be an equilibrium pattern flow with a vector-valued cost function if and only if and

for each and any .

Remark 2. Notice the fact that and and the relationship in (4) actually is equivalent to . Therefore, (4) is equivalent to

Lemma 3 (see [11]). A vector flow is an equilibrium pattern flow with a unilateral constraint if and only if is a solution to the quasivariational inequality: to find such that

We now introduce a concept of approximate equilibria for the family of network equilibrium problems. Let be a flow of the network , and let . For each , denote

Definition 4. A vector is said to be an -equilibrium pattern flow with a vector-valued cost function if and only if and

Remark 5. When , the -equilibrium pattern flow reduces to the equilibrium pattern flow as shown in Definition 1. Namely, a vector is a -equilibrium pattern flow if and only if is an equilibrium pattern flow.

In fact, assume that is a -equilibrium pattern flow but not an equilibrium pattern flow. Then, there exist and , satisfying and . However, by Definition 4 and being a -equilibrium pattern flow, . Hence, we have, for all , , which is a contradiction. Conversely, assume that is an equilibrium pattern flow but not a -equilibrium pattern flow. Then, there exists with . From the definition of , there exists satisfying . From (5), . This is a contradiction.

3. Well-Posedness of Multicriteria Network Equilibrium Problem

The following real-valued function is of fundamental importance to our current analysis. The original version is due to what Gerstewitz (Tammer) [12] published in German.

Definition 6. Let and . Gerstewitz’s function is defined by

By Theorem 2.1 of [13] and Lemmas 3 and 4 of [14], we have the following results.

Lemma 7. Let and . For each and , we have the following results:

(i);(ii);(iii);(iv);(v) is a continuous and strictly monotone function; namely,

(vi) is subadditive; namely,

(vii), for all ;(viii), for all , .

Definition 8 (see [11]). A vector is said to be a -equilibrium pattern flow if and only if and

for each and .

Now let be a flow of network , and let . For each , denote

Definition 9. Let . A vector is said to be a --equilibrium pattern flow if and only if and

Theorem 10. If is a --equilibrium pattern flow, , and , are continuous mappings, then is a -equilibrium pattern flow.

Proof. Let a sequence be a --equilibrium pattern flow, let , and let be arbitrarily chosen. From the continuity of , compactness of , and , we have . Moreover, it follows from the continuity of and the compactness of that, for every , there exists a sequence satisfying .

Let and ; define

Clearly, .

By Definition 9, we have

For and , we have

Since , then, for every , when . By , , . Thus, for each ,

Letting , we conclude that

Therefore, from Lemma 3 (take ), is a -equilibrium pattern flow.

Set in the following form:

where .

Theorem 11. Let and be defined as (20) for all , . is an -equilibrium pattern flow with a vector-valued cost function if and only if is a --equilibrium pattern flow.

Proof. Assume that is an -equilibrium pattern flow but not a --equilibrium pattern flow. Then, there exist an and a pair of , satisfying

From , we have

By Lemma 7,

Thus,

This implies that . It follows from being an -equilibrium pattern flow that

which contradicts (22).

Conversely, assume that is a --equilibrium pattern flow. Then, for any and every path , we want to deduce .

It follows from that there exists satisfying

Then,

Since , we have . Therefore,

Hence, . By Definition 9, we have .

Definition 12. Let be a sequence, let be an -equilibrium flow for each , and let ; the network is called well-posed if is a traffic equilibrium flow.

Theorem 13. Let be defined as (20). If and are continuous mappings, then the network is well-posed.

Proof. Let be an -equilibrium pattern flow. From Theorem 11, is a --equilibrium pattern flow. From Theorem 10 and , is a -equilibrium pattern flow. By Theorem 11 (take ), is an equilibrium pattern flow for the vector network equilibrium problem.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was partially supported by the National Natural Science Foundation of China (Grant no. 11226231).

References

G. Y. Chen and N. D. Yen, “On the variational inequality model for network equilibrium,” Internal Report 196, Department of Mathematics, University of Pisa, 1993.

P. Q. Khanh and L. M. Luu, “On the existence of solutions to vector quasivariational inequalities and quasicomplementarity problems with applications break to traffic network equilibria,” Journal of Optimization Theory and Applications, vol. 123, no. 3, pp. 533–548, 2004.

C. J. Goh and X. Q. Yang, “Vector equilibrium problem and vector optimization,” European Journal of Operational Research, vol. 116, no. 3, pp. 615–628, 1999.

S. J. Li and G. Y. Chen, “On relations between multiclass, multicriteria traffic network equilibrium models and vector variational inequalities,” Journal of Systems Science and Systems Engineering, vol. 15, no. 3, pp. 284–297, 2006.

S. J. Li, K. L. Teo, and X. Q. Yang, “A remark on a standard and linear vector network equilibrium problem with capacity constraints,” European Journal of Operational Research, vol. 184, no. 1, pp. 13–23, 2008.

X. Q. Yang and C. J. Goh, “On vector variational inequalities: application to vector equilibria,” Journal of Optimization Theory and Applications, vol. 95, no. 2, pp. 431–443, 1997.

A. Dontchev and T. Zolezzi, Well-Posed Optimization Problems, vol. 1543 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1993.

E. M. Bednarczuk, “An approach to well-posedness in vector optimization problems: consequences to stability,” Control and Cybernetics, vol. 23, pp. 107–122, 1994.

X. X. Huang, “Extended well-posedness properties of vector optimization problems,” Journal of Optimization Theory and Applications, vol. 106, no. 1, pp. 165–182, 2000.

E. Miglierina, E. Molho, and M. Rocca, “Well-posedness and scalarization in vector optimization,” Journal of Optimization Theory and Applications, vol. 126, no. 2, pp. 391–409, 2005.

Y. N. Wu, G. Y. Chen, and T. C. E. Cheng, “A vector network equilibrium problem with a unilateral constraint,” Journal of Industrial and Management Optimization, vol. 6, no. 3, pp. 453–464, 2010.

C. Gerstewitz (Tammer), “Nichtkonvexe dualität in der vektoroptimierung,” Wissenschaftliche Zeitschrift TH Leuna-Merseburg, vol. 25, pp. 357–364, 1983.

C. Gerth and P. Weidner, “Nonconvex separation theorems and some applications in vector optimization,” Journal of Optimization Theory and Applications, vol. 67, no. 2, pp. 297–320, 1990.

S. J. Li, X. Q. Yang, and G. Y. Chen, “Vector Ekeland variational principle,” in Vector Variational Inequalities and Vector Equilibria, F. Giannessi, Ed., pp. 321–333, Kluwer Academic, 2000.