Cordon tolling for mixed traffic flow

Abstract

This study develops a cordon tolling model for mixed traffic flow including motorcycle and automobile trips with multiple origins to one destination. A simultaneous differential equation system is employed to solve two trip functions with elastic demand. This model can thus analyze the interaction between different road users on the same road. No-toll equilibrium, optimal pricing by maximizing welfare, and cordon tolling solutions are analyzed. It is shown that the tolls of automobile trips and motorcycle trips are related to the size of the modes for any location of users in the first-best optimum regime. In the case of the Taipei metropolitan area, the cordon pricing scheme yields a similar relation for the tolls of both modes regardless of the differences in the parameters of demand and cost for the two types of trips. This study can provide a reference for decision-making for transportation policies in some Asian cities with mixed traffic of motorcycles and automobiles.

KEYWORDS:

• Mixed traffic flow
• cordon tolling
• motorcycle
• optimal congestion pricing

Acknowledgements

The authors wish to thank the editor and several anonymous referees for their helpful suggestions and critical comments on an earlier version of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Marchand (1968) originated second-best pricing by charging road users on one road only in a road system consisting of two parallel two roads. Wilson (1983) and D’Ouville and McDonald (1990) analyze the road capacity under the second-best pricing scheme. Verhoef, Nijkamp, and Rietveld (1995) explore the second-best pricing issue with heterogeneous road users. Liu and McDonald (1999) analyze the second-best congestion pricing scheme in peak and non-peak periods.

2 This approach is extended to a non-monocentric city in which the assumption of identical destination is released (Mun, Konishi, and Yoshikawa 2005). In addition, the applications of this approach to the congestion tolling on an elevated road, and the cordon tolling issue with a HOV lane are proposed (Chu and Tsai 2008; Chu, Tsai, and Hu 2012).

3 One exception is Tsai, Chu, and Hu (2015), who explore the congestion and accident externality for mixed flow with motorcycles and automobiles.

4 Clark et al. (2009) use the demand from an origin to a destination (O-D) is a function of the cost associated with the O-D with different demand functions for different user classes (with different values of time). Similar to Clark et al. (2009), our study consists of two types of users with different values of time.

5 Assume the area of the CBD is negligible and thus the CBD can be treated as a point.

6 These two modes can be treated as two types of modes with different size and different cost parameter for travelling (see eqs. (6) and (7)). In the case study in Section 3, accident costs, and fuel costs as well as time costs are considered in the cost function to reflect the features of motorcycle trips.

7 Note that the linear cost function is for every unit distance (a very small distance) of a trip. However, the cost function of any trip is nonlinear in distance of the trip. The reason is because the traffic flow at x is formulated by cumulating every trip from x to B (boundary).

8 For this case study, it is difficult to get $x_m$ from the optimal conditions (40)-(43) directly. Instead, we set a value for $x_m$ at first. Then the optimal cordon tolls for automobiles and motorcycles ($\tau$, and $\delta$) can be solved by a numerical approach. The associated social surplus in eq. (40) can be obtained after the values of $x_m$, $\tau$, and $\delta$ are inserted. Next, we compare the social surplus levels associated with different values of $x_m$. Finally, we find the maximum of social surplus associated with the value of $x_m$. The Maple software is used for the numerical calculation.

9 This fairness index is from Tsai and Lu (2018).

10 In this case, the commuting city boundary is the new town in Danshui, 26 km away from the CBD.

11 These four parameters of demand functions are explained as follows. The maximum willingness-to-pay, a, is estimated as the sum of the value of commuting time, fuel costs and accident costs of the farthest commuter for both modes. The value of commuting time is estimated for 60 min of travel by automobile and motorcycle users at NT$4.85 and NT$2.905 per min., respectively. The fuel costs are NT$ 27.2 per liter for both modes, with 0.0717 liters/km for automobiles and 0.0208 liters/km for motorcycles (Lin et al. 2011). The accident costs are NT$ 0.67/km for automobile users and twice that for motorcycle users (Tsai, Chu, and Hu 2015). A higher value of accident costs for motorcycle users from the literature above is not used due to the consideration of users’ expectation in reality. The slope of the inverse demand function, b, is estimated by parameter a divided by the maximum number of trips per person. The automobile trips per person and motorcycle trips per person per day for Taipei city are 0.624 and 0.716, respectively and 0.585 and 0.672 for New Taipei city, respectively. We used half a day as the time period. The distances from these two cities to the CBD are 3.0  and 10.0 km, respectively. The maximum number of automobile and motorcycle trips is thus 0.3205 and 0.3675 by linear extrapolation.

12 The parameters of driving cost in (6) and (7) include fixed and marginal cost for driving one unit of distance. The speed of free flow is assumed to be 60  and 50 km/h for automobiles and motorcycles, respectively. The time is thus 1.0  and 1.2 min/km for the two modes. This value is multiplied by the time value plus the fuel and accident costs to yield $f_1$  NT$7.470/km, and $f_2$ = NT$5.39176/km. The marginal cost is 0.4860 min/km (see Tsai and Lu 2018). It was estimated by fitting the travel time from 31 zones to the CBD by the cordon-pricing model to the actual travel time in the morning rush hour. This value of marginal cost per km is multiplied by the time value for each mode to yield $c_1$ = 2.357, $c_2$ = 1.412.

13 The value for the analysis is under normalization. For the Taipei metropolis, the number of residents was 4,666,431 in 2018 (and it is assumed that half of residents in New Taipei city commute to the CBD in Taipei). Nearly 60% of commuting trips are by private vehicles. In addition, about 60% of private trips are by motorcycles and 40% are by automobiles. The normalization multiplier is thus equal to 4,666,431 (60%)(60%)/26 = 64,612 for motorcycle trips and 43,075 for automobile trips. The net benefit is NT$19,413,322 for motorcycle trips and NT$19,466,454 for automobile trips. The social surplus is thus NT$ 38,879,776 with no toll. This multiplier can be also applied to the other two regimes.

14 These studies employing different settings for cordon pricing such as Zhang and Yang (2004), Akiyama, Mun, and Okushima (2004), and Sumalee, Shepherd, and May (2009) obtain a lower relative efficiency.

Additional information

Funding

The work presented in this paper was supported by a grant from the Ministry of Science and Technology of the Republic of China (MOST 104-2410-H-305-074).