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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 507376, 5 pages
Well-Posedness of MultiCriteria Network Equilibrium Problem
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.
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.
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 , 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  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.
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 ). 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 .
Lemma 3 (see ). 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)  published in German.
Definition 6. Let and . Gerstewitz’s function is defined by
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 ). 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,
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.
This research was partially supported by the National Natural Science Foundation of China (Grant no. 11226231).
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