Reformulating Environmentally Constrained Traffic Equilibrium via a Smooth Gap Function

Abstract

Various government laws have recently been enacted to alleviate the environmental deterioration of transportation systems. Environmental constraint is a valid means to explicitly reflect various environmental protection requirements imposed by the government. In this paper, we examine the environmentally constrained traffic equilibrium problem (EC-TEP), which is a fundamental tool for modeling and evaluating environmental protection requirements. Specifically, we provide an equivalent reformulation for the EC-TEP. The proposed reformulation adapts the concept of gap function to simultaneously reformulate the nonlinear complementarity conditions associated with the generalized user equilibrium conditions, environmental constraints, and conservation constraints as an equivalent unconstrained optimization problem. This gap function reformulation has two desirable features: (1) it can handle a general environmental constraint structure (linear or nonlinear; link-based or area-based) and a general link and route cost structure, enhancing the modeling adaptability and flexibility; (2) it is smooth and unconstrained, permitting a number of existing efficient algorithms for its solution. A gradient-based solution algorithm with a self-regulated averaging stepsize scheme is customized to solve the reformulated unconstrained optimization problem. Numerical examples are also provided to demonstrate the modeling flexibility of the proposed EC-TEP reformulation.

Modelling Rail-Based Park and Ride with Environmental Constraints in a Multimodal Transport Network

“…Li et al [11] developed a toll design model by simultaneously considering congestion and environmental externalities. Recently, Xu, Chen, and Cheng [1] reformulated the environmental constrained traffic assignment model through a smooth gap function to enhance the modelling adaptability and flexibility.…”

“…This VI formulation is strictly monotone and Lipschitz continuous, permitting a number of existing efficient algorithms for its solution [10,26]. Third, even though the solution methods for the traffic assignment problems with side constraints have been well studied with numerous achievements [1,10,12,27], solutions for the multimodal traffic equilibrium with side constraints are relatively limited. Recently, Meng, Liu and Wang [28] proposed a CA incorporated self-adaptive PC algorithms to solve asymmetric stochastic user equilibrium (SUE) problem with elastic demand and link capacity constraints.…”

“…Equations (1) ensure the amount of pollution no larger than predetermined thresholds. Note that (1) are general formulation that could measure different types of vehicle pollutants [1]. Herein, we make assumption as follows.…”

“…(2) CO is one of the most critical pollutants among the various types of vehicular emissions; (3) the emission functions of other pollutants are similar to that of CO [11]. In addition, this consideration has also been used in a number of recent studies, e.g., Yin and Lawphongpanich [37]; Nagurney, Qiang and Nagurney [38]; Chen, Zhou and Ryu [10]; Li et al [11]; Xu, Chen and Cheng [1]; Yang, Ban and Ma [39]; Kitthamkesorn et al [2]. Therefore, we use CO as an illustration to model vehicle emission.…”

“…According to Hong Kong Air Pollutant Emission Inventory (2016), the amount of CO generated in the year of 2016 in Hong Kong is 58600 tones and among them, road transport contributes to 31500 tones and shares 54% of the CO emissions. Also, the US's environmental protection agency reported motorized vehicles in the United States contribute to 86% of CO emissions, 62% of NO x emissions, 12% of PM2.5 emissions, and 7% of SO 2 emissions [1]. Kitthamkesorn et al [2] pointed out that only about 20% of the world's town residents enjoy good enough air quality as measured by the levels of emissions and over 600 million people in urban areas worldwide were exposed to dangerous levels of traffic-generated emissions from congestion.…”