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This paper addresses the optimal distance-based toll design problem for cordon-based congestion pricing schemes. The optimal distance tolls are determined by a positive and non-decreasing toll-charge function with respect to the travel distance. Each feasible toll-charge function is evaluated by a probit-based SUE (Stochastic User Equilibrium) problem with elastic demand, asymmetric link travel time functions, and continuously distributed VOT, solved by a convergent Cost Averaging (CA) method. The toll design problem is formulated as a mixed-integer mathematical programming with equilibrium constraints (MPEC) model, which is solved by a Hybrid GA (Genetic Algorithm)–CA method. Finally, the proposed models and algorithms are assessed by two numerical examples.

Acting as an economic lever for traffic demand management in urban metropolises, congestion pricing has received a lot of attention both academically and practically. Following two pioneer practices, Singapore in 1975 (Phang and Toh, 1997 and Li, 1999) and Norwegian cities in the mid 1980s (Langmyhr, 2001), implementations of congestion pricing have been accelerated when it comes to the new century, for instance, London in 2003 (Santos, 2008) and Stockholm in 2006 (Eliasson, 2009). Nearly all these congestion pricing practices adopt the cordon-based congestion pricing scheme: certain district in urban area is encircled by a pricing cordon and any vehicle passing through the cordon is charged. By affecting drivers’ route choice plans and subsequently restricting the total number of vehicles entering the encircled district, the cordon-based congestion pricing scheme is taken as an effective tool to mitigate traffic congestion, and it is also convenient for practical operations (May et al., 2002 and Akiyama and Okushima, 2006). All the implemented cordon-based congestion pricing schemes currently use a flat toll-charge method including the daily licensing basis charge (Santos, 2008) and the pay-per-entry basis charge, regardless of the travel distance or time in the pricing cordons. This flat toll-charge method, however, is inequitable because it undercharges long journeys and over-restrains short ones (May et al., 2008). In addition, the flat toll-charge is also not fully efficient for congestion mitigation, since some drivers may intentionally use more road segments in the cordon area, in order to maximize the utility of their investments. To cope with these drawbacks of flat toll-charge method, May and Milne (2000) examined three possible alternative toll-charge methods: (a) time-based method according to the time consumed in traversing a cordon; (b) congestion-based method in accordance with the travel time spent in congestion; and (c) distance-based method relied on the distance travelled. May and Milne (2000) concluded that these three methods outperformed the flat toll-charge method in terms of traffic congestion mitigation. The first two toll-charge methods, to some extent, encourage aggressive driving behaviors and may cause more traffic safety issues. They are hence not adopted in practical trials (Richards et al., 1996). The distance-based toll-charge method can be efficiently implemented with the aid of the global positioning system (GPS) and an in-vehicle unit integrating a GPS receiver, a digital map and a general packet radio service (GPRS) communication device. It is more preferable for the next generation of congestion pricing schemes. Note that toll charges for the distance-based toll-charge method should be a function of the travel distance in each pricing cordon, which is termed distance-based toll-charge function. It thus makes the toll charges, in most cases, non-additive (Bekhor and Toledo, 2005), i.e., the overall toll charge on an itinerary/path cannot be proportionally divided to be toll charges on its component links. Land Transport Authority (LTA) of Singapore has updated the bus fares for public transport system to be distance-based. The bus fares are determined by a universal fare structure table, and as shown by Fig. 1 the bus fare in such case is a nonlinear function of travel distance. The distance-based bus fare has eliminated the transfer fare penalty, thus it can encourage the use of public transport system. Meanwhile, LTA intends to convert its Electronic Road Pricing (ERP) system from the current pay-per-entry charge to the distance-based charge termed as the second generation ERP system (Ohno et al., 2007). Similar to the distance-based bus fare indicated in Fig. 1, a distance-based toll charge function is needed for second generation ERP system. It is essential to determine a proper toll-charge function that is beneficial for the whole road network. Therefore, it is a new research issue with practical importance to estimate a distance-based toll-charge function that maximizes the total social benefit (TSB). As a side note, we point out that apart from the cordon-based pricing schemes Singapore’s ERP system also has link-based tolls on partial expressways and arterial roads. Yet, the link-based tolls are not taken into consideration in this paper.Transportation Research Part E: Logistics and Transportation Review Volume 48, Issue 5, September 2012, Pages 937–957 Selected papers from the 14th ATRS and the 12th WCTR Conferences, 2010 Edited By Sveinn Vidar Gudmundsson, Cristina Barbot and Tae Hoon Oum Cover image Optimal distance tolls under congestion pricing and continuously distributed value of time * Qiang Meng E-mail the corresponding author, * Zhiyuan LiuCorresponding author contact information, E-mail the corresponding author, * Shuaian Wang * Department of Civil and Environmental Engineering, National University of Singapore, Singapore 117576, Singapore Received 12 August 2011 Revised 13 February 2012 Accepted 27 March 2012 Available online 10 May 2012 * http://dx.doi.org/10.1016/j.tre.2012.04.004, How to Cite or Link Using DOI * Permissions & Reprints Abstract This paper addresses the optimal distance-based toll design problem for cordon-based congestion pricing schemes. The optimal distance tolls are determined by a positive and non-decreasing toll-charge function with respect to the travel distance. Each feasible toll-charge function is evaluated by a probit-based SUE (Stochastic User Equilibrium) problem with elastic demand, asymmetric link travel time functions, and continuously distributed VOT, solved by a convergent Cost Averaging (CA) method. The toll design problem is formulated as a mixed-integer mathematical programming with equilibrium constraints (MPEC) model, which is solved by a Hybrid GA (Genetic Algorithm)–CA method. Finally, the proposed models and algorithms are assessed by two numerical examples. Highlights ► The nonlinear distance-based toll design problem is proposed. ► The continuously distributed value of time is assumed. ► A MPEC model is built. ► A GA-based efficient heuristic method is developed. Keywords * Distance-based toll; * Cordon-based congestion pricing; * Stochastic user equilibrium; * Continuously distributed value-of-time; * Mathematical programming with equilibrium constraints; * Genetic algorithm 1. Introduction Acting as an economic lever for traffic demand management in urban metropolises, congestion pricing has received a lot of attention both academically and practically. Following two pioneer practices, Singapore in 1975 (Phang and Toh, 1997 and Li, 1999) and Norwegian cities in the mid 1980s (Langmyhr, 2001), implementations of congestion pricing have been accelerated when it comes to the new century, for instance, London in 2003 (Santos, 2008) and Stockholm in 2006 (Eliasson, 2009). Nearly all these congestion pricing practices adopt the cordon-based congestion pricing scheme: certain district in urban area is encircled by a pricing cordon and any vehicle passing through the cordon is charged. By affecting drivers’ route choice plans and subsequently restricting the total number of vehicles entering the encircled district, the cordon-based congestion pricing scheme is taken as an effective tool to mitigate traffic congestion, and it is also convenient for practical operations (May et al., 2002 and Akiyama and Okushima, 2006). All the implemented cordon-based congestion pricing schemes currently use a flat toll-charge method including the daily licensing basis charge (Santos, 2008) and the pay-per-entry basis charge, regardless of the travel distance or time in the pricing cordons. This flat toll-charge method, however, is inequitable because it undercharges long journeys and over-restrains short ones (May et al., 2008). In addition, the flat toll-charge is also not fully efficient for congestion mitigation, since some drivers may intentionally use more road segments in the cordon area, in order to maximize the utility of their investments. To cope with these drawbacks of flat toll-charge method, May and Milne (2000) examined three possible alternative toll-charge methods: (a) time-based method according to the time consumed in traversing a cordon; (b) congestion-based method in accordance with the travel time spent in congestion; and (c) distance-based method relied on the distance travelled. May and Milne (2000) concluded that these three methods outperformed the flat toll-charge method in terms of traffic congestion mitigation. The first two toll-charge methods, to some extent, encourage aggressive driving behaviors and may cause more traffic safety issues. They are hence not adopted in practical trials (Richards et al., 1996). The distance-based toll-charge method can be efficiently implemented with the aid of the global positioning system (GPS) and an in-vehicle unit integrating a GPS receiver, a digital map and a general packet radio service (GPRS) communication device. It is more preferable for the next generation of congestion pricing schemes. Note that toll charges for the distance-based toll-charge method should be a function of the travel distance in each pricing cordon, which is termed distance-based toll-charge function. It thus makes the toll charges, in most cases, non-additive (Bekhor and Toledo, 2005), i.e., the overall toll charge on an itinerary/path cannot be proportionally divided to be toll charges on its component links. Land Transport Authority (LTA) of Singapore has updated the bus fares for public transport system to be distance-based. The bus fares are determined by a universal fare structure table, and as shown by Fig. 1 the bus fare in such case is a nonlinear function of travel distance. The distance-based bus fare has eliminated the transfer fare penalty, thus it can encourage the use of public transport system. Meanwhile, LTA intends to convert its Electronic Road Pricing (ERP) system from the current pay-per-entry charge to the distance-based charge termed as the second generation ERP system (Ohno et al., 2007). Similar to the distance-based bus fare indicated in Fig. 1, a distance-based toll charge function is needed for second generation ERP system. It is essential to determine a proper toll-charge function that is beneficial for the whole road network. Therefore, it is a new research issue with practical importance to estimate a distance-based toll-charge function that maximizes the total social benefit (TSB). As a side note, we point out that apart from the cordon-based pricing schemes Singapore’s ERP system also has link-based tolls on partial expressways and arterial roads. Yet, the link-based tolls are not taken into consideration in this paper. Full-size image (9 K) Fig. 1. Bus fare structure for public transport system in Singapore. Figure options * View in workspace * Download full-size image * Download as PowerPoint slide Analysis of the optimal distance-based toll design problem has to take into account the behavior of drivers in their route (or path) choice, which is assumed to obey the probit-based stochastic user equilibrium (SUE) principle, namely, drivers are assumed to have a normally distributed perception error on the actual path travel time. Other than the actual travel time and perception error, overall travel impedance on each path (termed as generalized path travel time) also includes a toll charge. The toll charge should be converted into time-units using the drivers’ value of time (VOT). It is well-known that the VOT varies among different drivers due to their different levels of income and trip emergency. To reflect this variation, it is more rational to formulate the VOT as a continuously distributed random variable rather than a fixed (mean) value (Verhoef and Small, 2004, Small et al., 2005 and van den Berg and Verhoef, 2011). The continuously distributed VOT results in another random term in the generalized path travel time, besides the perception error. A computational model is necessitated to describe the probit-based SUE principle with the continuously distributed VOT due to the inapplicability of the existing models developed for the SUE problems. Assuming the non-additive distance-based toll charges as well as a continuously distributed VOT, this paper aims to investigate the proposed optimal distance-based toll design problem in model development and algorithm design.

This paper solved the optimal toll design problem for the distance-based toll-charge method of cordon-based congestion pricing scheme. A mathematical programming with equilibrium constraints (MPEC) model was developed for the toll-charge function with maximal total social benefit (TSB) value. A fixed-point model was taken as a constraint of the MPEC model, which was proposed for the probit-based SUE problem with elastic demand, asymmetric link travel time functions and continuously distributed VOT. The toll-charge function is assumed to be generic to any positive and non-decreasing functional form. To solve the MPEC model, a piecewise-linear approximation function was first utilized to approximate any positive and non-decreasing toll-charge function, which gives a mixed-integer MPEC model. Then a Hybrid GA–CA method was adopted to solve the mixed-integer MPEC model. The proposed methodology was numerically validated by two numerical examples. The output optimal piecewise-linear toll-charge function is highly nonlinear and it can considerably improve the TSB compared with the un-tolled case and KM charge. In general, the contributions of this paper to the literature are: first, an integrated methodology is proposed to solve the distance-based toll design problem, which is a timely topic with practical significance. The distance-based charging method would make the cordon-based pricing scheme play a better role in urban transportation management, by improving its equity level. Second, the framework proposed for assessment of any toll charge pattern (generalized SUE), is more representative to the practical conditions, which makes the results more reasonable and suggestive for real-world implementations. This study is taken on a static transportation network, while the congestion pricing schemes in some cities (e.g., Singapore and Stockholm) are temporally dynamic and varying constantly at different time of day. It is thus necessary to extend the methodology proposed in this paper for the time-differentiable toll charges using dynamic or multi-period traffic assignment approaches (e.g., Sumalee et al., 2011). A link-based Monte Carlo simulation method was proposed for the stochastic network loading of the generalized probit-based SUE problem, which has considerably prolonged the execution time. Thus, the distributed computing was employed to accelerate the computational speed, but performance of the distributed computing approach is not fully discussed, since it is not a focus of the current study. While, it is a worthwhile future research to test and evaluate the sensitivity of computational speed to the number of processors used for the distributed computation as well as to the sample size of Monte Carlo simulation. Alternatively, to handle the issue of prolonged execution time, a (mixed) generalized-nested logit-based SUE can be used in future as a framework for the traffic assignment problem (e.g., Koppelman and Wen, 2000, Wen and Koppelman, 2001 and Daly and Bierlaire, 2006), where it only needs to simulate the value of random VOT. However, due to the existence of random VOT, the computational superiority of logit-based SUE to probit-based SUE would be weakened. It is interesting to further test the performance of these two SUE frameworks with continuously distributed VOT. Distance-based toll charge method is known to have better equity property and it is more convenient for practical implementations (see Singapore LTA masterplan, 2008), compared with other pricing schemes including time-based, congestion-based and link-based (each link in the cordon is levied by a toll) charge. However, the distance-based toll charge may not always be the most efficient scheme to improve the TSB of the transport system. Thus, another necessary future research topic is to fully investigate the efficiency of these different pricing schemes based on comprehensive numerical tests on multiple networks.