Equilibria for Traffic Network with Capacity Constraints

Equilibria for Traffic Network with Capacity Constraints

Zhi Lin (Chongqing Jiaotong University, China) and Jiangtao Luo (University of Nebraska Medical Center, USA)
Copyright: © 2014 |Pages: 6
DOI: 10.4018/978-1-4666-5202-6.ch078
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Introduction

There have been several decades in the study of equilibria for optimizations related to road traffic due to its importance in practical applications and theoretical challenges. Wardrop (1952) studied the theoretical properties from mathematical and statistical viewpoints; since then many publications have been developed in this area. The classic book by Beckmann, McGuire and Winsten (1956) provides us equilibrium of traffic optimization with travel cost function. Quandt (1967) and Schneider (1968) first studied the equilibrium problem for traffic network with multicriteria. Dafermos (1972) studied traffic assignment problem in a network of multiclass-user model with travel units were divided into different classes, and an algorithm was given in this paper for optimizing the traffic patterns. Toll policies of the traffic network for multiclass-user were also studied (Dafermos, 1973). The conditions for the existence, uniqueness and stability of equilibria were studied by Smith (1979). Fisk introduced a network optimization problem for the traffic assignment with congestion effects, and the solution was then shown to be an equilibrium (Fisk, 1980). Wardrop’s equilibrium principle has been generalized to vector space (Chen & Yen, 1993), but the equivalence between the traffic equilibrium and variational inequality does not hold any more. Therefore, it is significant to study the properties of equilibria of traffic network in different settings. Daniel et al (1999) studied the existence, characterization and computation of a special traffic network in a Banach space setting. Khanh and Luu (2004 & 2005) introduced the weak and strong Wardrop equilibria for multivalued cost functions and generalized the vector equilibrium principle to capacity constraint of paths. Recently, Lin (2010a & b) has developed some methods for equilibria with capacity constraints of arcs and (weak) vector equilibrium principle. Lin and Yu (2005) have also considered the related quasi-equilibrium problems. There are also more publications related to vector equilibrium flows of multiclass multicriteria traffic equilibria (Nagurney, 2000; Nagurney & Dong, 2002; Yang & Huang, 2002; Li & Chen, 2006). More references are available for this area and weighted equilibrium methods (Li et al, 2008; Yang & Goh, 1997; Browder, 1968; Fan, 1961). There are three important elements in the real traffic network: (1) multiple classes of vehicles, (2) multiple criteria, (3) constraints for capacity. All the models studied in the above papers or books lack at least one of the three elements. Therefore it is important to study traffic model with these three elements. We call it multiclass multicriteria traffic equilibrium problem with capacity constraint of arc. The multiclass multicriteria traffic equilibrium problem with capacity constrain of arc

(MMTEPCCA) can be treated as a combination of the multiclass multicriteria traffic equilibrium problem and the multicriteria traffic problem with capacity constraint of arc.

*Supported by The National Science Foundation of China (11271389) and The Science Foundation of CQ CSTC (2011AC6104).

Key Terms in this Chapter

Feasible Flow: PCU flow does not exceed the capacity.

MMTEPCCA: Multiclass multicriteria traffic equilibrium problem with capacity constraints of arcs.

Vector Equilibrium Flow: The flow that gives us the optimal solution.

PCU: Passenger Car Unit.

Capacity Constraint of an Arc: The total traffic that an arc can hold for normal transportation.

Saturate Path (Arc): The traffic flow reaches the capacity.

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