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Optimal distance tolls under congestion pricing and continuously distributed value of time
Transportation Research Part E 48 (2012) 937–957
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Optimal distance tolls under congestion pricing and continuously distributed value of time Qiang Meng, Zhiyuan Liu ⇑, Shuaian Wang Department of Civil and Environmental Engineering, National University of Singapore, Singapore 117576, Singapore
a r t i c l e
i n f o
Article history: Received 12 August 2011 Received in revised form 13 February 2012 Accepted 27 March 2012
Keywords: Distance-based toll Cordon-based congestion pricing Stochastic user equilibrium Continuously distributed value-of-time Mathematical programming with equilibrium constraints Genetic algorithm
a b s t r a c t 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 nondecreasing 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. Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved.
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; 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; 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 tollcharge 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
⇑ Corresponding author. Tel.: +65 9864 6604; fax: +65 6779 1635. E-mail addresses: [email protected] (Q. Meng), [email protected] (Z. Liu). 1366-5545/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tre.2012.04.004
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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. 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; 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. 1.1. Relevant studies There are numerous achievements on the congestion pricing studies (Small, 1992; Yang and Huang, 2005; Lawphongpanich et al., 2006; Han and Yang, 2008; Verhoef et al., 2008; etc.). Among these studies, the cordon-based congestion pricing scheme is usually regarded as a particular second-best pricing solution (McDonald, 1995; Verhoef et al., 1996; Verhoef, 2002; Yang and Zhang, 2002; Meng et al., 2005; Meng and Liu, 2011a, 2012). Based on given cordon locations, optimal toll design problem for cordon-based congestion pricing aims to identify a proper toll fare solution that optimizes a system-wide objective function, such as the TSB. When assuming the flat toll-charge method, the optimal toll design problem for cordon-based congestion pricing schemes can be formulated as a mathematical program with equilibrium constraints (MPEC) or bi-level programming model
2.5
Bus Fares
2 1.5 1 0.5 0 0
5
10
15
20
25
30
35
40
45
50
Travel Distance Fig. 1. Bus fare structure for public transport system in Singapore.
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(Verhoef, 2002; Chen and Bernstein, 2004; Ho et al., 2005; Maruyama and Sumalee, 2007). In reality, these MPEC or bi-level programming models are similar to those built for the continuous network design or optimal signal setting problem with deterministic user equilibrium (DUE) constraints (Yang and Meng, 1998), because the flat toll-charge method with a fixed VOT leads to an additive link-based toll charge solution. There are several particular heuristic methods developed for the continuous network design or optimal signal setting problem, including iterative optimization-assignment algorithm (Allsop, 1974), equilibrium decomposed optimization (Suwansirikul et al., 1987), augmented Lagrangian algorithm (Meng et al., 2001), and gradient-based descent methods (Chiou, 2005). However, these methods are not available for the optimal distance-based toll design with continuously distributed VOT proposed in this study because of the non-additive distancebased toll charges and random VOT. The continuously distributed VOT is proven to be more rational than a fixed value (Lam and Small, 2001; Verhoef and Small, 2004). Traffic assignment problem with continuously distributed VOT has been comprehensively examined on a hypothetical two-route example, e.g., Mayet and Hansen (2000), Verhoef and Small (2004), Xiao and Yang (2008), and Nie and Liu (2010); to name a few. Some findings in these studies, however, may not be available when extended to realistic network with more than two links. Leurent (1993) and Dial (1996) investigated the traffic assignment problem with continuously distributed VOT on realistic networks by assuming DUE principle for the drivers’ route choice behavior. Cantarella and Binetti (1998) proposed a solution framework for the probit-based SUE problem with continuously distributed VOT. This framework assumed fixed OD demand and relies on a path-based simulation technique to cope with the probit-based stochastic network loading. This paper intends to extend the work of Cantarella and Binetti (1998) to the case of elastic demand and uses a link-based Monte Carlo simulation method to solve the probit-based stochastic network loading, which avoids path generation/enumeration. In addition to the second generation ERP system in Singapore, the importance and significance of the optimal distancedbased toll design addressed in this study can be further demonstrated by the experiments of a kilometer (KM)-based tollcharge method conducted in several European regions including Leeds, UK (Mitchell et al., 2005; Namdeo and Mitchell, 2008), Scotland (O’Mahony et al., 2000), Netherlands (Ubbels et al., 2002) and Germany (Hensher and Puckett, 2007; Link, 2008). As a special distance-based toll-charge method, the KM-based toll-charge method assumed that toll charge for each vehicle was linearly proportional to its travel distance in the pricing area (May and Milne, 2000), making the toll charge additive. Yet, previous studies and trials for KM charge all arbitrarily set a charging rate per kilometer, which is unlikely to achieve the maximal TSB. Therefore, this paper also takes the toll design problem for KM-based toll-charge method into analysis, which can be quickly solved by a brute-force method. In spite of its simplicity in modeling and calculation, KM-based toll-charge function is unlikely to be the optimal for all the transportation networks. Yet, no practical data is available for the analysis of a proper functional form of the toll-charge function or to calibrate such a function. It is therefore more reasonable to assume that the toll-charge function is generic to any positive and non-decreasing function, which includes the KM-based toll-charge function, linear function, and nonlinear functions. 1.2. Objectives and contributions This paper aims to propose a methodology that is able to estimate a positive and non-decreasing toll-charge function with maximal TSB by assuming a continuously distributed VOT. Assuming that drivers’ perception error on path travel time is normally distributed, it will first define a probit-based SUE problem with elastic demand, asymmetric link travel time functions and continuously distributed VOT, which is termed as the generalized probit-based SUE problem. The proposed generalized probit-based SUE problem will be subsequently formulated by a fixed-point model with unique solution. A network transformation technique is then proposed to circumvent the non-additive toll charges, which avoids path enumeration/generation. The Cost Averaging (CA) method proposed by Cantarella (1997) is then adopted to solve this SUE problem. The optimal distance-based toll design with the aim of maximizing TSB can be formulated as a MPEC model by taking the fixed-point model as a constraint. Since the distance-based toll-charge function is allowed to be any functional form, this study proposes a novel piecewise-linear function method that can approximate any form of the toll-charge function, which gives another mixed-integer MPEC model for the convenience of calculation. A Hybrid genetic algorithm (GA)–CA algorithm is designed for solving the mixed-integer MPEC model. Specially, in the case of KM-based toll-charge function, a more concise brute-force algorithm is adopted to solve the optimal distance-based toll design problem. The remaining sections are organized as follows. Section 2 defines the optimal distance-based toll design problem and Section 3 puts forward a fixed-point model and link-based solution method for the generalized probit-based SUE problem. Section 4 first develops a MPEC model for the toll design problem, and then introduces a mixed-integer MPEC model based on the piecewise-linear approximation function. Solution algorithms and numerical experiments are presented in Sections 5 and 6. Conclusions are finally provided in Section 7. 2. Notation, assumptions and problem description 2.1. Notation and Assumptions Consider a strongly connected network denoted by G = (N, A) where N and A are the sets of nodes and directed links, respectively. Let W be the set of the OD pairs and Rw be the set of paths between OD pair w e W. Travel demand between
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OD pair w e W is denoted by qw and q = (qw, w e W)T is a column vector for all these travel demands. Let fwk be traffic flow on path k e Rw between OD pair w e W, fw = (fwk, k e Rw)T be a column vector of all these path flows between OD pair w and T f ¼ ðf w ; w 2 WÞT be a column vector of all the path flows over the entire network. Let va denote traffic flow on link a e A and v = (va, a e A)T is a column vector of all these link flows. The following flow conservation equations should be fulfilled:
q ¼ Kf;
ð1Þ
v ¼ Df;
ð2Þ
f P 0:
ð3Þ
½dw ak jAjK
½dw k jWjK
where D ¼ and K ¼ are the incidence link/path and OD pair/path matrices, where |A|, |W| and K are the w w number of links, OD pairs and paths, respectively; dw ak ¼ 1 if link a is on path k e Rw, and dak ¼ 0, otherwise. dk ¼ 1 if path k connects OD pair w e W, and dw ¼ 0, otherwise. k The travel time on link a e A is assumed to be a function of link flow vector v, denoted by ta(v), which can flexibly represent a symmetric or asymmetric link travel time function. These link travel time functions are grouped into a column vector t(v) = (ta(v), a e A)T. It is assumed that vector function t(v) is non-negative, monotonically increasing and continuously differentiable. Travel time on path k e Rw is the sum of travel times of all the links on this path, denoted by cwk(v), namely,
cwk ðv Þ ¼
X ta ðv Þdw ak :
ð4Þ
a2A
These path travel times are grouped into vector cw(v) = (cwk(v), k e Rw)T. It is assumed that the travel time on path k e Rw perceived by drivers, denoted by Cwk(v), is a random variable with the expression:
C wk ðv Þ ¼ cwk ðv Þ þ fwk ;
ð5Þ
where fwk is a random perception error following normal distribution with zero mean and constant variance. 2.2. Toll-charge function, generalized path travel time and optimal distance-based toll design Let I be the number of congestion pricing cordons in network G. Network G = (N, A) is then divided into one external net^ and several cordon networks, denoted by G i Þ (i = 1, 2, . . . , I). The links (nodes) in any ^ ¼ ðN; i ¼ ðN ^ AÞ, i; A work, denoted by G ^ are termed ^ ¼ ðN; ^ AÞ cordon network are termed as internal links (nodes), while the other links (nodes) in external network G i # N and the exteras external links (nodes). Note that the boundary nodes of any cordon i exist both in its cordon node set N ^ # N. Let dw;i denote the length of the portion of path k e Rw in the congestion pricing cordon i and it can be nal node set N k expressed as follows: w;i
dk ¼
X w la dak ; k 2 Rw ; w 2 W;
ð6Þ
a2A i
i is the set of links in cordon i. where la is the length of link a and A A distance-based toll-charge method for the cordon-based congestion pricing scheme can be expressed by a distancebased toll-charge function /(d) where d is the distance traveled in a pricing cordon. It is assumed that the generic function /(d) is positive and non-decreasing but may not be continuous. As shown by Fig. 1, the toll-charge function for distancebased bus fare in Singapore is not continuous. /(d) is also allowed to follow any function form, including, linear function, quadratic function, exponential function, power function, etc. The kilometer (KM) based toll-charge function is a special case of /(d), where only the slope is a decision variable, shown as follows:
/ðdÞ ¼ qd;
ð7Þ
where q is the slope of KM based toll-charge function, which is the charging rate per kilometer. For any given toll-charge function /(d), the toll charge imposed on path k e Rw between OD pair w e W can be calculated as follows:
swk ð/Þ ¼
I X w;i /ðdk Þ:
ð8Þ
i¼1
Toll charge swk(/) shown in Eq. (8) can be converted into time-units by virtue of the drivers’ value of time (VOT), denoted by a. As mentioned in Section 1, a is assumed to be a continuously distributed variable across the whole population of drivers. The cumulative distribution function (CDF) of a can be obtained by the empirical curve plotting method if the VOT values of all the drivers are known. It is assumed that a is positive with constant mean and variance, and its probability density function (PDF) is continuously differentiable and strictly positive. Here, constant mean and variance of a imply that they are the same for all OD pairs. In the literature, a is usually assumed to follow uniform distribution, exponential distribution, or lognormal distribution. The distribution of a is also assumed to be independent to the link/path flows and other network attributes.
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wk ðv ; sw ð/ÞÞ, consists of the perceived path travel time shown by The overall travel impedance on path k e Rw, denoted by C wk ðv ; sw ð/ÞÞ is termed as the generalized path travel time. It can be Eq. (5) and a time-based value of the toll charges, and C expressed by
wk ðv ; sw ð/ÞÞ ¼ C wk ðv Þ þ swk ð/Þ=a ¼ cwk ðv Þ þ swk ð/Þ=a þ fwk ; C
ð9Þ
T
where vector sw = (swk(/), k e Rw) . Eq. (9) can be rewritten as follows:
wk ðv ; sw ð/ÞÞ ¼ cwk ðv Þ þ Eðswk ð/Þ=aÞ þ kwk ; k 2 Rw ; w 2 W; C
ð10Þ
where E(swk(/)/a) is the expectation of random term swk(/)/a. kwk , termed as generalized perception error, has the expression:
kwk ¼ swk ð/Þ=a Eðswk ð/Þ=aÞ þ fwk :
ð11Þ
It can be seen that kwk is also a flow-independent random variable with zero mean and fixed variance. All these generalized perception errors associated with OD pair w e W are grouped into a multivariate random variable kw ¼ ðkwk ; k 2 Rw ÞT , and evidently the PDF of kw is also continuously differentiable. Drivers are assumed to make their trip plans based on the generalized path travel time function defined by Eq. (10), and such a route choice principle is named as generalized probit-based SUE. Let Sw(cw, sw) be the expected value of minimum generalized path travel time between OD pair w e W, namely:
wk ðv ; sw Þg; Sw ðcw ; sw Þ ¼ E½minfC k2Rw
ð12Þ
where Sw(cw, sw) is usually referred to as satisfaction function (Sheffi, 1985). It is assumed that travel demand between OD pair w e W is a continuously differentiable, non-increasing and bounded function, denoted by Dw(), with respect to its satisfaction function:
w ; w 2 W; qw ¼ Dw ðSw ðcw ; sw ÞÞ 6 q
ð13Þ
w is a given upper bound of travel demand between OD pair w e W. This assumption is previously made by where parameter q Cantarella (1997) and Maher and Zhang (2000) for the conventional SUE problem with elastic demand. It shows that the OD demand function is defined at an aggregate level for the entire population, and there is no explicit connection between the OD demand and the PDF of VOT, which is different from the endogenous definitions in some previous studies, e.g., Small and Yan (2001), Verhoef and Small (2004), van den Berg and Verhoef (2011). All the toll charges on the network are determined by the toll-charge function shown in Eq. (8). Different toll-charge functions will result in different network equilibrium flows and consequently different total social benefit (TSB) values on the entire network. Let U be the set of all these positive and non-decreasing toll-charge functions. Assuming the generalized probit-based SUE principle for drivers’ route choice behavior, the optimal distance-based toll design problem aims to identify or estimate a proper toll-charge function with the maximal TSB from the set U. Model development and algorithm design for the proposed optimal distance-based toll design rely on solving a corresponding generalized probit-based SUE problem for any given toll-charge function. Thus, we first present the formulation and solution algorithm for such an SUE problem in the following section. 3. Generalized probit-based SUE problem According to the generalized path travel time defined in Eq. (10) with a particular toll-charge function / e U, the probability that generalized travel time on path k e Rw is perceived as the minimal one among all the paths between OD pair w e W, denoted by pwk(cw(v), sw(/)), can be expressed as follows:
wl ðv ; sw ð/ÞÞ; 8l 2 Rw and l–kjv ; sw ð/ÞÞ; k 2 Rw ; w 2 W: pwk ðcw ðv Þ; sw ð/ÞÞ ¼ PrðC wk ðv ; sw ð/ÞÞ 6 C
ð14Þ
The corresponding generalized probit-based SUE link flow pattern is a solution of the following fixed-point model:
v a ð/Þ ¼
XX
½Dw ðSw ðcw ðv Þ; sw ð/ÞÞÞ pwk ðcw ðv Þ; sw ð/ÞÞdw ak ; a 2 A:
ð15Þ
w2W k2Rw
The fixed-point model above is a variation of the fixed-point formulation proposed by Cantarella (1997) for the conventional SUE problem with elastic demand and asymmetric link travel time functions. Following the similar proof of Lemma 4.1 in Daganzo (1979), it can be shown that: Proposition 1. Satisfaction function Sw(cw(v), sw(/)) is concave with respect to path travel times cw(v) and
@ðSw ðcw ðv Þ; sw ð/ÞÞÞ ¼ pwk ðcw ðv Þ; sw ð/ÞÞ: @cwk ðv Þ
ð16Þ
In view of Proposition 1 as well as the monotonicity of demand functions and link travel time functions, the fixed-point model shown by Eq. (15) has a unique solution according to Theorems 1 and 2 of Cantarella (1997).
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The Cost Averaging (CA) method proposed by Cantarella (1997) is recognized as the only convergent algorithm for solving the fixed-point model shown by Eq. (15), which is provided in the Appendix for completeness. It can be seen from the Appendix that the stochastic network loading step is quite crucial to the CA method. The existing analytical approximation methods for the probit-based stochastic network loading problem (Daganzo and Sheffi, 1977; Maher and Hughes, 1997; Rosa and Maher, 2002) are inapplicable for the generalized probit-based stochastic network loading due to the random VOT. Hence, a Monte Carlo simulation based method is developed for solving the generalized probit-based stochastic network loading. As shown in Eqs. (5) and (8), both the drivers’ perception error in travel time, fwk, and the toll charges swk(/) are defined on paths. Thus, directly using Monte Carlo simulation to solve the generalized probit-based stochastic network loading problem necessitates path enumeration/generation, which would largely increase the computational time. To avoid path enumeration or generation, two interesting techniques are elaborated in Sections 3.1 and 3.2, which enable an efficient link-based Monte Carlo simulation method. 3.1. Link-based formulation for the generalized path travel time The path travel time perception error fwk in Eq. (9) can be formulated as the sum of perception errors on the link travel times by assuming that drivers’ perceived travel time on link a e A equals to (Sheffi, 1985):
T a ðv Þ ¼ t a ðv Þ þ na ; a 2 A;
ð17Þ
where na is a normally distributed random error term with zero mean and flow-independent variance, namely,
na Nð0; bt 0a Þ; a 2 A;
ð18Þ
where b is a proportionality constant parameter and t0a is the free-flow link travel time. With the perceived link travel time functions shown in Eq. (17), the generalized travel time on path k e Rw can be expressed as:
wk ðv ; sw ð/ÞÞ ¼ C
X X X w T a ðv Þdw t a ðv Þdw na dak : ak þ swk ð/Þ=a ¼ ak þ swk ð/Þ=a þ a2A
a2A
ð19Þ
a2A
P Let fwk ¼ a2A na dw ak . Since fwk is a summation of some normally distributed variables with zero mean and constant variance, it can inherit these properties, thus fwk is also normally distributed, which shows the validity of the link-based formulation. In addition, covariance of the generalized path travel times on two different paths takes into account their overlapped portions. The characteristics of the path travel time perception error fwk defined from the link-based formulation are summarized in Proposition 2 below: ðv ; sw ð/ÞÞ expressed by Eq. (19) has the form: Proposition 2. The generalized path travel time C wk
wk ðv ; sw Þ ¼ cwk ðv Þ þ swk ð/Þ=a þ fwk ; k 2 Rw ; w 2 W; C
ð20Þ
P
w a2A na dak
where fwk ¼ is a normally distributed random term which has zero mean and constant variance. Moreover, the covariance between the generalized path travel times on two different paths is given by
wk ; C wb Þ ¼ swk ð/Þswb ð/Þ var covðC
1
a
þ
X w bt 0a dw ak dab ; 8k; b 2 Rw ; w 2 W:
ð21Þ
a2A
3.2. Network transformation for non-additive path toll charges Let Ei denote the set of all the entry-exit pairs of cordon i e I. A pricing cordon usually has limited size, and in such cases it is effortless to enumerate all the simple paths (acyclic paths) between any entry-exit pair e e Ei, which are termed as internal paths. Thus, a network transformation can be efficiently conducted to convert the non-additive path-based toll charges into link-based: firstly, each internal path between entry-exit pair e e Ei is represented by a dummy link connecting its entry and i Þði ¼ 1; 2; . . . ; IÞ by corresponding dummy links. Hereby, i ¼ ðN i; A exit node; secondly, replace all the cordon networks G ^ and all the dummy links constitute a new composite network, denoted by G0 = (N0 , A0 ). We ^ ¼ ðN; ^ AÞ external network G can see that this network transformation converts the ‘‘path-based’’ toll charges to be ‘‘link-based’’ on each dummy link. 3.2.1. An illustrative example A small example shown in the upper part of Fig. 2 is used to illustrate the network transformation procedure. Links 1–6 ^ of this example. The area surrounded by a dashed ellipse is a pric^ ¼ ðN; ^ AÞ, and Nodes 1–5 constitute the external network, G Entry-exit pairs of the pric ¼ ðN; AÞ. ing cordon, and therefore Links 7–12 and Nodes 2, 4, 6 compose the cordon network, G ing cordon comprise 2 ? 4 and 4 ? 2, and any trip traversing this cordon is imposed by a distance-based toll charge. A new network, G0 = (N0 , A0 ), modified by the network transformation procedure is shown in the lower part of Fig. 2. It can be seen that the pricing cordon is replaced by four dummy links (dashed arcs), and as shown in Table 1 each of these dummy links represents a simple path.
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Note that a similar type of network transformation was used in Meng et al. (2004) and Yang et al. (2004) for solving optimal toll design problem of highways with non-additive toll charges. However, in these two papers, the non-additive toll charges are entry-exit based, thus only one dummy link exists for one entry-exit pair. e . Evidently, based on a given toll-charge function, All the internal paths between entry-exit pair, e e Ei are grouped in set R /, toll charge on each dummy link of composite network G0 = (N0 , A0 ) is a flow-independent constant, and generalized travel time on any path is additive to the links. Remark 1. In the case of a realistic network with large pricing cordons, there could be quite a number of simple paths between one entry-exit pair, thus considerably increasing the computational burden. Previous studies for path-based traffic assignment problems are confronted by this hurdle, see, e.g., Lo and Chen (2000) and Bekhor and Toledo (2005). Instead of enumerating all the simple paths, these studies only examine partial paths (a) explicitly in a pre-generated path choice set or (b) implicitly based on the column generation during iterative computations. These techniques of using partial paths are also recommended to cope with the internal paths on a realistic network. Remark 2. For the sake of presentation, the network transformation omits those OD pairs with origin or destination in the cordon area. However, this methodology presented in Section 3.2 can be simply modified for those OD pairs, by enumerating the paths from their origin or destination to each entry node of the cordon and then replacing those paths by dummy links. 3.3. Monte Carlo simulation based stochastic network loading method The stochastic network loading is invoked in each iteration of the CA method, where the real link travel times ta, a e A are fixed and the corresponding equilibrium link flows are taken as output. Based on the two techniques in Sections 3.1 and 3.2, we then manage to use a two-stage Monte Carlo simulation method for solving the generalized probit-based stochastic network loading problem. Main steps of this method are provided as follows: Stage 1: Monte-Carlo simulation for travel demand estimation. ð0Þ Step 1.0: (Initialization) Let the iteration counter be n = 1 and the initial estimated satisfaction be Sw ¼ 0; w 2 W. ðnÞ Step 1.1: (Sampling of link travel time) For each link a e A, sample a value for the perception error na from the normally ðnÞ ðnÞ distributed population Nð0; bt0a Þ; a 2 A. And then calculate the link travel time T~ a ¼ t a þ na . (n) Step 1.2: (Generalized travel time of dummy links) Sample a value for VOT a from its distribution function and calculate e ; e 2 E by the generalized travel time on all the internal paths l 2 R ðnÞ T~ el ¼
X /ðdel Þ e T~ ðnÞ ; a dal þ ðnÞ a2A
ð22Þ
a
Fig. 2. Illustrative example for network transformation.
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Q. Meng et al. / Transportation Research Part E 48 (2012) 937–957 Table 1 List of simple paths and corresponding dummy links. Entry-exit pairs
Simple paths
Represented by dummy link
2?4
2?4 2?6?4
[2–4–1] [2–4–2]
4?2
4?2 4?6?2
[4–2–1] [4–2–2]
and 0, otherwise. del is the length of internal path e contains internal link a 2 A, where deal equals to 1 if internal path l 2 R e . T~ ðnÞ is recorded as the travel time on the corresponding dummy link of composite network G0 = (N0 , A 0 ). l2R el ðnÞ Step 1.3: (Shortest path time calculation). With link travel time pattern fT~ a ; a 2 A0 g on the composite network G0 = (N0 , A0 ), ~ ðnÞ find the shortest path between each OD pair w e W, and record the total travel time on this path as C w . Step 1.4: (Satisfaction estimation). Estimate the satisfaction for each OD pair w e W by the following average scheme:
ðn1Þ ~ ðnÞ SðnÞ ¼ ðn 1ÞSw þ C w ; w 2 W: w n
ð23Þ
Step 1.5: (Accuracy checking). If the number of iterations n P n0 , where n0 is a predetermined sample size, go to Step 1.6; otherwise, set n = n + 1 and go to Step 1.1. Step 1.6: (OD demand calculation). Calculate OD travel demand pattern by the formulae:
qw ¼ Dw ðSðnÞ w Þ; w 2 W:
ð24Þ
Stage 2: Monte-Carlo simulation for calculating the generalized probit-based SUE link flow. ð0Þ Step 2.0: (Initialization). Let the initial link travel flow vector be v a ¼ 0; a 2 A and m = 1. ðmÞ 0 Step 2.1: (Sampling). Sample the perception error na from Nð0; bt a Þ; a 2 A based on normally distributed random number ðmÞ ðmÞ series, and then calculate the link travel time T~ a ¼ t a þ na ; a 2 A. (m) Step 2.2: (Sampling of VOT). Sample a value for VOT a from its distribution function and calculate the travel time on all e ; e 2 E by the internal paths l 2 R ðmÞ T~ el ¼
X /ðdel Þ e T~ ðmÞ : a dal þ ðmÞ
ð25Þ
a
a2A
ðmÞ Then, record T~ el as the travel time on corresponding dummy link of the composite network G0 = (N0 , A 0 ). ðmÞ Step 2.3: (All-or-nothing assignment). Based on link travel time pattern fT~ a ; a 2 A0 g on the composite network G0 = (N0 , A0 ), find the shortest path for each OD pair w, then assign qw to the shortest path. Flow on link a e A0 is therefore a summation of the travel demands of all the OD pairs whose shortest path uses link a e A0 . Then assign flows on all the dummy links to e ; e 2 E. This would generate an auxiliary flow pattern on all the links the links on their corresponding internal paths l 2 R fY ðmÞ ; a 2 Ag. Calculate the simulated link flow by a
ðm1Þ
v ðmÞ ¼ a
ðm 1Þv a m
þ Y ðmÞ a
; a 2 A:
ð26Þ
Step 2.4: (Convergence test). If the number of iterations m P m0 , where m0 is also a predetermined sample size, stop; otherwise, set m = m + 1 and go to step 2.1. This two-stage procedure here is an extension of our previous work (Meng and Liu, 2011b). Note that two-stage procedure of the Monte Carlo simulation is used here, rather than an integrated one-stage procedure that simulates the value of travel demand and SUE link flows based on the same set of samples at each iteration. This is because the two-stage procedure is more stable and thus efficient than the one-stage procedure. Note that these Monte Carlo simulation procedures can be easily modified for solving the stochastic network loading of DUE or logit-based SUE problem with continuously distributed VOT. Remark 3. The stop criterion at each stage of the Monte Carlo simulation is based on a predetermined sample size. Accuracy of the simulation is thereby determined by the sample size. With a given accuracy level, a lower bound for the sample size of each stage can be theoretically determined by the study of Meng and Liu (2011b). Alternatively, for any given network example, a sensitivity test can be performed in the first place to determine a proper sample size.
4. Two MPEC models for the optimal distance-based toll design 4.1. Total social benefit and the exact MPEC model We now proceed to investigate how to obtain a desirable toll-charge function / from the viewpoint of system optimum. A toll-charge function is regarded as optimum if it can give rise to the maximal total social benefit (TSB) on the network. In the context of generalized probit-based SUE, the following expression is adopted for the TSB:
Q. Meng et al. / Transportation Research Part E 48 (2012) 937–957
Zð/Þ ¼
XZ w2W
qw ðsÞ
0
D1 w ðxÞdx
X
qw ðsÞSw ðcw ðv ðsÞÞ; sw ð/ÞÞ þ
w2W
! XX f el ðsÞsel ð/Þ ; E e e2E l2R
a
945
ð27Þ
where E() is the expectation operator, and s = (sw(/), w e W)T. q(s) = (qw(s), w e W)T, v(s) = (va(s), a e A)T, and fðsÞ ¼ ðf ðsÞ; e 2 E; l 2 R e ÞT denote the column vectors for the corresponding equilibrium OD demands, link flows, and interel nal path flows, associated with a given toll-charge function /. Value of these vectors can be obtained by solving the generalized SUE problem discussed in Section 3. Note that in Eq. (27) the first term is the gross consumer benefit, the second term represents the overall cost borne by the commuters, and the last term is the total toll revenue that can be invested back to the transportation system or any other place with the most beneficial use. It is an aim of the authorities that implementing a congestion pricing scheme can enhance the TSB shown in Eq. (27) on the entire network. A MPEC model is hence built to find a toll-charge function maximizing the TSB:
maxZð/Þ
ð28Þ
/2U
s.t.
v a ð/Þ ¼
XX
½Dw ðSw ðcw ðv Þ; sÞÞ pwk ðcw ðv Þ; sÞdw ak ; a 2 A:
ð29Þ
w2W k2Rw
Constraint (29) is the fixed-point model proposed for the generalized probit-based SUE problem. It should be noted that the solution to this MPEC model is the system-optimal charge for the distance-based congestion pricing scheme. As mentioned in Section 2, the toll-charge function is merely required to be positive and non-decreasing, and it is generic to any linear or nonlinear functional forms. This type of toll-charge function has perfect rationality to the distance-based toll method, thus model (28) and (29) is termed as exact MPEC model. Yet, it is quite challenging to solve this model, since the toll-charge function has no specific functional form. In view of this, an approximation method is proposed in the following section, where a piecewise-linear function is used to approximate any positive and non-decreasing function. This approximation method then gives a mixed-integer MPEC model that can be simply solved. 4.2. A Mixed-integer MPEC Model with a piecewise-linear approximation function A piecewise-linear approximation function is indicated by Fig. 3. Suppose the nonlinear function in the left-hand-side of Fig. 3 is the optimal toll-charge function for one transportation network, it then can be approximated by a piecewise-linear function as shown in the right-hand-side of Fig. 3. The shape of a piecewise-linear function in each interval is uniquely determined by its two boundary values. Note that if the optimal toll-charge function /(d) is not continuous, it should also be approximated by a discontinuous piecewise-linear function. For conciseness, we only take into account the continuous piecewise-linear approximation function in this paper. Nevertheless, the methodology provided can be easily extended to the discontinuous cases. We proceed to talk about the theoretical definition of the piecewise-linear approximation function. Let dmin and dmax be the minimum and maximum lengths of all the internal paths. The range [dmin, dmax] can be uniformly divided into n intervals, where n is a positive integer number,
½dmin ; d2 ; ½d2 ; d3 ; . . . ; ½di ; diþ1 ; . . . ; ½dn ; dmax :
Fig. 3. An illustrative example for the piecewise-linear approximation function.
ð30Þ
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Note that the length of each sub-interval here is the same. The piecewise-linear approximation function contains n + 1 boundary values over these n intervals, and these boundary values are defined as y1 = /(dmin), yi = /(di), i = 2, 3, . . . , n and yn+1 = /(dmax). The piecewise-linear approximation function is defined accordingly as follows:
n ðd; yÞ ¼ y þ yiþ1 yi ðd di Þ; di 6 d < diþ1 ; i ¼ 1; 2; . . . n; / i diþ1 di
ð31Þ
where d1 = dmin and dn+1 = dmax. The column vector y is defined as y = (y1, y2, . . . , yn+1)T.The number of intervals, n, can influ n ðd; yÞ to the original toll-charge function /(d). It is reasonable to assume that: ence the fitness of /
n 2 ½nmin ; nmax ; and n is an integer:
ð32Þ
Herein, the lower bound nmin is imposed, because it will deteriorate the equity and efficiency of distance-based toll-charge method if the value of n is fairly small. And on the other hand, if the number of intervals is too large, computational cost of would geometrically increase, thus an upper bound nmax is also imposed for solving the optimal piecewise-linear function / n n. Based on the piecewise-linear function (31), the MPEC model (28) and (29) can be approximated by the following mixedinteger MPEC model with the decision variables y and n:
/ n ðd; yÞÞ maxZð
ð33Þ
y;n
s.t.
v a ð/ n ðd; yÞÞ ¼
XX
½Dw ðSw ðcw ðv Þ; sÞÞ pwk ðcw ðv Þ; sÞdw ak ; a 2 A;
ð34Þ
w2W k2Rw
ymin 6 y1 ; yi1 6 yi 6 yiþ1 ;
ð35Þ
8i ¼ 2; 3; . . . ; n 1;
ð36Þ
yn 6 ymax ;
ð37Þ
nmin 6 n 6 nmax and n is an integer;
ð38Þ
where ymin (ymax) is a predetermined lower (upper) bound of the toll charge. The inequalities (35)–(37) ensure that the piecewise-linear function is positive and non-decreasing. This mixed-integer MPEC model is then used to solve optimal distancebased toll design problem, solution algorithm of which is provided in the following section. Note that in practice, the value of n can be taken as a fixed input value for the model, which should be decided by the transport authorities for each specific cordon. 5. Solution algorithm Most of the solution algorithms for solving the MPEC model need to calculate the Jacobian matrixes of link flow vector v and OD demand vector q with respect to the vector of decision variables, which is, in this study, the piecewise-linear tollcharge function. Such Jacobian matrixes for the traffic assignment problem can be solved by the sensitivity analysis method (e.g., Yang, 1997). However, it is quite expensive to calculate the Jacobian matrixes in this paper, due to: (a) the complexity of generalized probit-based SUE problem with elastic demand, asymmetric link travel time functions and continuously distributed VOT; (b) the intricate connection between prototype network G = (N, A) and composite network G0 = (N0 , A0 ), for example, Eq. (15) is defined on G = (N, A), while the Monte Carlo simulation method for stochastic network loading is described on n . These hurdles prevent the G0 = (N0 , A0 ); (c) transformation between toll charges and piecewise-linear toll-charge function / usage of any gradient-based algorithm for solving the optimal distance-based toll design problem. However, for any given function type of toll-charge function with limited number of unknown variables, it would be an accurate and inexpensive way to enumerate and assess all the feasible variables with a small gap. For instance, considering the KM-based toll-charge function, shown in Eq. (7), only the slope q is a variable, and a reasonable range for q is usually not quite large, thus it is pleasant to test all the values of q in this range with a small increment. n ðd; yÞ discussed The brute-force enumeration method is unacceptable for the piecewise-linear approximation function / in Section 4.2. While, considering the discrete property of /n ðd; yÞ, when the value of n is given, it would be more straightforward to adopt Genetic Algorithm (GA) as a heuristic for solving the mixed-integer MPEC model (33)–(38). Thereby, we can first enumerate all the feasible n, and for any particular n, the optimal piecewise-linear function that yields maximal TSB can be solved by GA. Due to the space limit, rationale and detailed mechanism of GA are not covered here, and the readers are recommended to some masterpieces in this field for reference (e.g., Goldberg, 1989; Gen and Cheng, 1997). A feasible chromosome for the n ðd; yÞ is given as ðy ; y . . . y ; y . . . y ; y Þ piecewise-linear toll-charge function / 1 2 i iþ1 n1 n and yi here is defined as a gene of one chromosome. The yi is assumed to be integer-valued and measured in cents. Each chromosome will give rise to different toll charges on the network, which therefore leads to different values of total social
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benefit (TSB). Calculation of the TSB is, in itself, a generalized probit-based SUE problem, which can be solved by the Cost Averaging (CA) method. The solution algorithm for optimal piecewise-linear toll-charge function is then termed as Hybrid GA–CA method, which is summarized as follows: Step 1: (Initialization). Set initial value for the number of intervals to n(1) = nmin. And initialize the optimal TSB value to TSB⁄ = 0 and the optimal number of intervals to n⁄ = n(1). Set the number of iteration to p = 1. ðpÞ with maximal TSB Step 2: With the number of intervals equal to n(p), calculate the optimal piecewise-linear function / n value, denoted by TSB(p), by the following GA: Step 2.0: (Initial population). Randomly generate initial population of the piecewise-linear function, and each individual carries a feasible chromosome. Then, set the number of generation k = 1. The population size is defined as k. Step 2.1: (Evaluation). Based on the toll-charge function shown by one chromosome, calculate the value of toll charge on each internal path of the composite network G0 = (N0 , A0 ). Then, utilize the CA method to calculate the corresponding traffic assignment problem, results of which are used to evaluate the TSB for this toll-charge function. Conduct such evaluations for all the new generated chromosomes. chromosomes with higher TSB values as survivors for current generation and discard Step 2.2: (Selection). Select the first k the rest. 1 and Y 2 , two Step 2.3: (Crossover). Conduct pairing among survivors, where for each two chosen survivors, denoted by Y new chromosomes are generated by the following function:
Y^ 1 ¼ vY 1 þ ð1 vÞY 2 Y^ 2 ¼ vY 2 þ ð1 vÞY 1
v 2 ð0; 1Þ; v–0:5: v 2 ð0; 1Þ; v–0:5:
ð39Þ
Step 2.4: (Mutation). To guarantee the monotonicity of the toll-charge map, a mutation is conducted in this way: first, randomly choose some genes from existing chromosomes; second, for each chosen gene yi, update its value by sampling a uniformly distributed random variable in [ymin, ymax], then proportionally change the value of other genes of this chromosome in the interval [ymin, yi] and [yi, ymax]. Step 2.5: (Stop test). If a stop criterion is achieved, let the value of TSB(p) equal to the maximal TSB among the survivors in current generation and record the corresponding chromosome, and then go to Step 3; otherwise, set k = k + 1 and go to step 2.1. Step 3: If TSB(p) > TSB⁄, then let
TSB ¼ TSBðpÞ ; n ¼ nðpÞ ;
ð40Þ
and record the corresponding chromosome as the optimal toll-charge function. Step 4: If nðpÞ P nmax , then stop and output the optimal toll-charge function, TSB⁄ and n⁄; otherwise, let n(p+1) = n(p) + 1, p = p + 1 and go to Step 2. 6. Numerical examples To numerically validate the proposed methodology, two examples are adopted. The first example, named as Network C, was used by Meng et al. (2004) as well as Yang et al. (2004) for the non-additive entry-exit highway charge schemes. This example contains 14 nodes, 46 links and 8 OD pairs, and between each entry-exit pair of the pricing cordon there is only one simple path (dummy link). Using this concise example, properties of the proposed models and algorithms are comprehensively tested. The second example is designed based on the Orchard Road cordon of the Electronic Road Pricing (ERP) system in Singapore, and it has 33 nodes, 104 links and 12 OD pairs. For these two numerical examples, the travel demand between each OD pair w e W is assumed to be decided by the following function:
w expðc Sw ðcw ðv Þ; sÞÞ; w 2 W; qw ¼ q
ð41Þ
w is the upper bound of OD demand and c is a constant parameter. The link travel time function on link a e A is dewhere q fined as follows:
t a ðv Þ ¼
t0a
1 þ 0:15
v a þ 0:5v^ a^ 1:5ha
4 !
; a 2 A;
ð42Þ
where t0a is the free-flow travel time, and ha is the capacity of link flow. It is required that there is at least one pair of two-way ^ denote the opposite link of link a, and v ^ a^ in Eq. links that connect the same two nodes and go to different directions. Let a ^. Here, v ^ a^ is taken into consideration to calculate ta, which reflects the interference from (42) denotes the flow on link a opposing traffic due to turning and passing movements. By assuming such type of link travel time functions, the Jacobian matrix of link travel time vector t with respect to the link flows v is asymmetric, which is consistent with the assumption of asymmetric link travel time functions. This functional form follows that in Bar-Gera (2011).
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The value-of-time (VOT) is assumed to be continuously distributed across the whole population. Without loss of generality, we assume that the VOT is uniformly distributed in the range from 18.0 to 72.0 Singapore-Dollars (S$) per hour. For each given toll-charge function, the corresponding equilibrium network flow is calculated by the CA method provided in the Appendix, and the value of tolerance in the stop criterion of CA method is taken as e1 = 0.1. The CA method invokes the two-stage Monte Carlo simulation based stochastic network loading method in each iteration. Based on a trial-and-error trial, sample sizes for the two stages of Monte Carlo simulation (see, Section 3.3) are taken as 180 and 1000, respectively, to achieve an acceptable accuracy level. Note that each trial of the Monte Carlo simulation requires the calculation of a shortestpath-tree for each origin. The large sample size (totally 1180 for the two stages) has largely increased the computational time, especially for the second example. Note that the variance parameter b in Eq. (18) is taken as 0.1. As described in Section 5, optimal toll design problem for the KM charge is solved by a brute-force enumeration method and for the general nonlinear distance-based charge is solved by the Hybrid GA–CA method. Both of these two methods possess a good parallelism, thus it is quite convenient to use distributed computing to accelerate the computational speed in each case. For the enumeration method for KM charge, the range of slope q can be partitioned into many intervals and then evaluated by different processors in the distributed computing system. For the Hybrid GA–CA method, evaluation of each newly generated chromosome is independent and it thus can be conducted concurrently by different processors, which is known as parallel GA (see, Wong et al., 2001). With sufficient processors, computational speed of these two methods can be accelerated for tens of times. Thanks to the booming development of computer hardware and software, it is cost efficient to build a distributed computing system with several hundreds of processors. The programs for numerical tests in this section are all coded in FORTRAN 90 and performed at a Distributed Computing PC Cluster system in Civil and Environmental Engineering Department at National University of Singapore. Due to the space limit, profile of the computing platform is not described here, and any interested reader can refer to Liu and Meng (2011) for more information. For the numerical tests in this section, the number of processors involving the computation is fixed to be 50. 6.1. Network C As shown in Fig. 4, the Network C has one pricing cordon, circled by the dashed rectangular. Links in the cordon are grouped in the set:
¼ f33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46g: A Nodes 4, 5, 6 and 7 are entries as well as exits to this pricing cordon. w and values of the t 0a Some parameters for the network attributes, including the OD pairs, upper bound of OD demand q and ha on each link, are identical to those in Yang et al. (2004), which are not included here. The parameter c in Eq. (41) is properly taken as 0.01 for this example. Length of each link is given in Table 2, which is used to measure drivers’ travel distance in the pricing cordon. It should be pointed out that the length shown in Table 2 does not always jibe with that in Fig. 4. This is because the length of many links shown in Fig. 4 has high similarity, and when enumerating the inner paths for this example, most of the inner paths have the same length. This phenomenon has undermined the generality of this example. Thus, the length of each link has been reasonably adjusted in Table 2. Based on the link lengths, it is detected that the maximum and minimum path length, dmin and dmax, for this example is 405 and 2453 meters, respectively. Lower and upper bound of the toll charges, shown in Eqs. (35) and (37), are set to be ymin = 0.5 and ymax = 20.0 S$. These two values are taken to make the numerical example more consistent with practical conditions, in view that 0.5 S$ was adopted by Singapore’s ERP system as the initial toll rate and 20.0 S$ is a fairly large toll value compared with the current toll charges of the ERP system. 6.1.1. KM charge We first cope with the optimal KM-based toll-charge function, since it can be efficiently solved by the brute-force enumeration method and in addition it can be taken as a benchmark to evaluate the performance of optimal nonlinear distancebased charge. As shown by Eq. (7), only the slope q is an unknown parameter for the KM toll-charge function. Since the lower and upper bound of toll charges are 0.5 and 20.0 S$, q is reasonably assumed to range from 0.5 to 10.0 S$ per km. Thus, 951 successive values of q in this range with an increment of 0.01 are tested, and the corresponding values of total social benefit (TSB) are shown in Fig. 5. Fig. 5 indicates that the value of TSB achieves its maximum at around q = 0.92, and the optimal TSB value is 5.204 106. While, the smallest TSB value obtained during this enumeration is only 4.094 106. As shown in Fig. 5, when the value of q exceeds 2.10, the value of TSB stays at this smallest level. This is because when the toll charge of traveling in a pricing cordon is getting considerably high, no flow (in the Monte Carlo simulation) would be loaded to any link there. Hence, the pricing cordon is blocked from the transportation network. In such a case, the value for TSB is lower, due to the fact that: (a) there is no toll revenue; and (b) the drivers lose some options for route choice, such that their travel disutility increases. Another test is conducted for the case when there is no toll charge on the network, which can be directly solved by the CA algorithm. It shows that the value of TSB in this case is 5.145 106. However, Fig. 4 indicates that the values of TSB obtained
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Fig. 4. Structure of network C.
Table 2 Length of the links in the pricing cordon of network C. Link no. Length (m) Link no. Length (m)
33 790 40 38
34 1050 41 35
35 250 42 78
36 400 43 82
37 850 44 66
38 892 45 85
39 85 46 26
by most of the KM-based toll-charge functions are even lower than 5.145 106. This comparison confirms that when an arbitrary toll pattern is initially introduced to the transportation network, it may cause a reduction in total benefit/utility of the network. This phenomenon is consistent with the previous study of Wilson (1988) on the cordon-based congestion pricing scheme in Singapore. 6.1.2. Nonlinear distance-based charge The Hybrid GA–CA method can be used to calculate the optimal piecewise-linear approximation function for general nonlinear distance-based charge. Then, the network transformation is first performed, where links in the pricing cordon are replaced by some dummy links that connect each entry-exit pair. The resultant composite network is shown in Fig. 6. Since dmin and dmax for this example are found to be 405 and 2453 meters, the gap for the length-variety of all the internal paths is 2048 meters. As aforementioned, a range should be decided for the number of intervals, n, which is quite essential for the piecewise-linear approximation function. If n is too small, it would deteriorate the equity and efficiency of nonlinear distance-based charge, while when n is too large, it is computationally inhibitive to solve the optimal toll design problem. Thus, the upper and lower bound for n is properly decided to be 10 and 6, respectively. Namely, there are 5 scenarios with
TSB Value (Millions)
5.5
5
4.5
4
3.5
0.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
Slope Fig. 5. Sensitivity test for the slope of KM-based toll-charge function.
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the value of integer n changing from 6 to 10. Computational burden is in general increasing as the value of n increases, since larger n implies more genes in each chromosome and it subsequently increases the total number of mutations. of this numerical example. The stop criterion of GA, The Hybrid GA–CA method is then adopted for solving the optimal / n see Step 2.5 in Section 5, is taken as fixed number of generations, k = 50. Moreover, the total number of chromosomes (population) of one generation is chosen to be 200. Eventually, computational results of each scenario with different n are tabulated in Table 3. In Table 3, the first row of data is the maximal TSB value obtained by each scenario, and the second to fourth row is the value of each term in the expression R q ðsÞ P P for TSB, as in the right-hand-side of Eq. (27), namely, T1 ¼ w2W 0 w D1 weWqw(s)Sw(cw(v(s)), sw(/)), and w ðxÞdx, T2 = P P T3 ¼ e2E l2Re Eðf el ðsÞasel ð/ÞÞ. The fifth row entitled ‘‘Increase’’ is the improvement of nonlinear toll charge over KM charge in terms of the TSB obtained, recalling that the maximal TSB obtained by KM charge is 5.204 106. The sixth row shows the execution time for each scenario, yet this execution time may not be strictly proportional to the computational burden of each scenario. This is because the computation is performed on the distributed computing system, and each scenario with different n has different concurrency level affecting their parallelism. Most of the computational efforts are devoted onto evaluation of newly generated chromosome, thus the total number of chromosomes evaluated throughout the entire computation process could better reflect the total computational burden of each scenario, which is provided in the row six of Table 3. The last row of Table 3 is the efficiency of each scenario, denoted by xi, which is calculated by the following equations:
xi
TSBi TSB0 TSB TSB0
ð43Þ
;
where TSBi is the TSB value obtained for scenario i, TSB0 is the TSB value of un-tolled case, and TSB⁄ is the maximum TSB value obtained among all the scenarios; for this example, TSB⁄ is obtained by the scenario n = 10, herein, the TSB⁄ is an approximation of the system optimum for the nonlinear distance-based pricing. As per the data in Table 3, the optimum TSB value among all the scenarios is 5.877 106 when n = 10. Compared with the maximal TSB obtained by optimal KM charge, 5.204 106, the optimal nonlinear distance-based charge has a significant improvement, which is 12.94% as shown in Table 3. The optimal chromosome output from this scenario is depicted by Fig. 7, which has 10 intervals and 11 boundary values. Clearly, the optimal distance-based toll-charge function is a highly nonlinear one. 6.2. Orchard Road Network Example The proposed methodology is further tested in this sub-section via a realistic network example. As shown in Fig. 8, this example built from the Orchard Road Cordon of ERP system in Singapore has 33 nodes and 104 links, with a pricing cordon The cordon has 11 highlighted by the dashed ellipse. Totally there are 38 links in the cordon that are grouped into set A. entries and 11 exits connecting to the external network, which gives rise to 110 entry-exit pairs. As mentioned in Section 1, other than the cordon-based pricing schemes, the ERP system also has link-based tolls on some expressways and arterial
Fig. 6. Composite network for network C.
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Q. Meng et al. / Transportation Research Part E 48 (2012) 937–957 Table 3 Computational profile of different scenarios to network C.
a
Scenarios
n=6
n=7
n=8
n=9
n = 10
Max TSB T1a T2a T3a Increase CPU time (s) No. of total chromosomes Efficiency xi
55,41,560 69,16,080 27,00,789 13,26,270 6.48% 826 3369 54.12%
57,89,184 69,49,056 26,81,664 15,21,791 11.24% 868 3521 87.91%
57,28,330 69,39,036 26,87,571 14,76,864 10.07% 890 3628 79.60%
58,02,743 70,19,860 26,37,017 14,19,900 11.50% 906 3777 89.76%
58,77,805 69,64,026 26,72,666 15,86,445 12.94% 930 3961 100%
T1, T2, T3 are terms 1, 2, 3 in the right-hand-side of Eq. (27).
4.5
4.1
Toll Charge (S$)
4
3.5
3.5 3 2.5 2
1.6
1.5 1 0.5
1.1 0.5 0.5
0.5
0.5
0.6
0.6
0.8
0 0.41 0.61 0.81 1.02 1.22 1.43 1.63 1.84 2.04 2.25 2.45
Travel Distance Fig. 7. Optimal toll-charge function for network C.
Fig. 8. Network structure of Orchard Road example.
roads. For the future generation of ERP system, the distance-based toll charge method should only be adopted for the cordonbased tolls rather than link-based tolls. Data for the links are provided in Table 4. Travel time for this example is counted in seconds, and it would result in a comparatively larger magnitude to the satisfaction Sw(cw(v), s), thus the parameter c in Eq. (41) is decided to be 0.001 for
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Table 4 Link data for Orchard Road example. Link no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68
Tail node
Head node
Free-flow travel time t0a (s)
Link length
Capacity Ha (vehicles/h)
2 1 3 2 1 4 5 4 6 5 2 6 3 8 7 6 8 7 9 8 4 10 12 5 6 14 7 15 8 17 9 18 11 10 12 11 13 12 14 13 14 15 16 15 17 16 18 17 10 27 11 19 12 20 13 21 14 22 23 15 16 24 17 25 18 26 20 19
1 2 2 3 4 1 4 5 5 6 6 2 8 3 6 7 7 8 8 9 10 4 5 13 14 6 15 7 17 8 18 9 10 11 11 12 12 13 13 14 15 14 15 16 16 17 17 18 27 10 19 11 20 12 21 13 22 14 15 23 24 16 25 17 26 18 19 20
60 60 100 100 40 40 90 90 42 42 80 80 72 72 160 160 120 120 60 60 80 80 55 55 80 80 120 120 60 60 90 90 16 16 40 40 24 24 48 48 40 40 12 12 60 60 60 60 20 20 24 24 20 20 30 30 12 12 12 12 12 12 14 14 20 20 80 80
600 600 1000 1000 400 400 900 900 420 420 800 800 720 720 1600 1600 1200 1200 600 600 800 800 550 550 800 800 1200 1200 600 600 900 900 160 160 400 400 240 240 480 480 400 400 120 120 600 600 600 600 200 200 240 240 200 200 300 300 120 120 120 120 120 120 140 140 200 200 800 800
5400 5400 5400 5400 5400 5400 3600 3600 3600 3600 7200 7200 3600 3600 1800 1800 1800 1800 3600 3600 5400 5400 3600 3600 7200 7200 1800 1800 3600 3600 3600 3600 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 1800 1800 5400 5400 3600 3600 7200 7200 3600 3600 3600 3600 3600 3600 5400 5400 1800 1800
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Q. Meng et al. / Transportation Research Part E 48 (2012) 937–957 Table 4 (continued) Link no.
Tail node
Head node
69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104
20 21 22 21 23 22 24 23 25 24 26 25 19 28 20 29 22 30 23 31 24 32 26 33 28 27 29 28 30 29 31 30 32 31 33 32
21 20 21 22 22 23 23 24 24 25 25 26 28 19 29 20 30 22 31 23 32 24 33 26 27 28 28 29 29 30 30 31 31 32 32 33
Free-flow travel time t0a (s) 20 20 48 48 20 20 28 28 60 60 60 60 24 24 12 12 18 18 16 16 18 18 60 60 20 20 40 40 90 90 30 30 60 60 96 96
Link length 200 200 480 480 200 200 280 280 600 600 600 600 240 240 120 120 180 180 160 160 180 180 600 600 200 200 400 400 900 900 300 300 600 600 960 960
Capacity Ha (vehicles/h) 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 1800 1800 5400 5400 7200 7200 3600 3600 5400 5400 5400 5400 5400 5400 5400 5400 3600 3600 3600 3600 3600 3600 5400 5400
this example. It should be noted that the length of each link in Table 4 may not be exactly proportional with the link length in Fig. 8, since the locations of some nodes in Fig. 8 have been justified for a clearer view. There are 12 hypothetical OD pairs, and the upper bound values for the OD demands are tabulated in Table 5. Other than the data for network attributes listed in Tables 4 and 5, other settings for this example are similar to those for the Network C above, including the upper and lower bound for the toll charge, range of the intervals n, numbers of generation and population for the Hybrid GA–CA method, as well as the stop criteria for each approach involved, etc. These settings are thus not repeated here. Similarly, the computation is handled in the first place for this example regarding the optimal KM charge as well as the un-tolled case, which would act as benchmarks to evaluate the performance of nonlinear distance-based toll charge. It turns out that for this example, the TSB value with respect to the un-tolled case is as large as 2.608 107. While, the TSB obtained by the optimal KM charge is 2.489 107, where the slope q = 2.01, which is inferior to that of the un-tolled case. It further indicates that the implementation of congestion pricing schemes may cause a loss on total social benefit. Albeit the value of TSB decreases when implementing the optimal KM charge, the traffic congestions in cordon area (usually in the CBD) would be improved in view that toll charges limit the toll traffic volume in the cordon area. In addition, we can speculate that if the lower bound of the slope q is set to be 0.0 rather than 0.5, the optimal TSB of KM charge will increase, since when q = 0.0 it becomes the un-tolled case, thus would have a TSB value at 2.608 107. We then proceed to calculate the optimal nonlinear distance-based charge for this example using the Hybrid GA–CA method. A prerequisite for this method is the network transformation for the cordon area shown in Fig. 8. The network transformation requires enumeration of all the simple (cycle-free) paths between each entry-exit pair. However, as addressed in Section 3.2, it might be a computationally impossible task to enumerate all the simple paths on a realistic network with large cordons, and the pre-generated path set or column generation should be used to only test partial paths. For illustration, we adopt the technique of pre-generated path set on this realistic network example: only the first k ‘‘shortest’’ paths between each entry-exit pair are taken into the path choice set, and the first k ‘‘shortest’’ paths problem is solved by the well-known algorithm proposed by Yen (1971). We note that other than using the k ‘‘shortest’’ paths, some other alternatives can also be found in the literature for the path set generation problem, see, Cascetta et al. (2002), Bovy and Catalano (2007). For this example, only the first six shortest simple paths are generated between each entry-exit pair based on the free flow travel
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Q. Meng et al. / Transportation Research Part E 48 (2012) 937–957 Table 5 Upper bound of OD demand for Orchard Road example. OD pair w
w Total travel demand q (vehicles/h)
1 ? 33 9?1 3 ? 27 27 ? 9 2 ? 29 18 ? 28 4 ? 24 32 ? 14 33 ? 3 25 ? 4 28 ? 6 7 ? 23
5000 4000 5000 5000 2000 2000 3000 1000 2000 1000 3000 2000
Table 6 Computational profile of different scenarios to Orchard Road example.
a
Scenarios
n=6
n=7
n=8
n=9
n = 10
Max TSB T1a T2a T3a Increase CPU time (s) No. of total chromosomes Efficiency xi
261,45,078 334,11,508 81,38,525 872,095 5.06% 8632 3458 71.83%
261,55,314 334,19,832 81,24,970 860,451 5.10% 8744 3550 83.13%
261,70,594 334,22,782 81,19,750 867,562 5.16% 8818 3682 100%
261,62,038 334,20,758 81,23,377 864,656 5.12% 8982 3783 90.56%
261,60,578 334,20,770 81,23,398 863,206 5.12% 9582 3939 88.94%
T1, T2, T3 are terms 1, 2, 3 in the right-hand-side of Eq. (27).
4.0
3.6
Toll Charge (S$)
3.5 3.0 2.4
2.5 2.0 1.5 1.0 0.5
0.5 0.5
0.0 0.12
0.58
0.5
1.04
0.6
1.5
0.7
0.8
1.96
2.42
1.0
2.88
3.34
3.8
Travel Distance Fig. 9. Optimal Toll-charge function for Orchard Road example.
time of each link. These paths in the path choice set are then taken for the network transformation, each of which is replaced by a dummy link. Subsequently, the Hybrid GA–CA method is used to compute the optimal nonlinear distance-based toll charge function for this Orchard Road network example. The resultant data are tabulated in Table 6. It shows that the third scenario with n = 8 gives the maximal TSB value of 2.617 107. Compared with the optimal KM charge, it has increased the TSB value for 5.16%, which is also a non-negligible improvement. As the number of intervals, n, increases, the total execution time and computational burden (No. of total chromosomes) are also enlarged. The computational results for this example show that a larger value of n may not give larger maximal TSB value. The impacts of n on maximal TSB value are, in most cases, unpredicted due to the considerable diversity of path lengths in the cordon. The optimal nonlinear toll-charge function in terms of n = 8 is indicated by Fig. 9. Likewise to Fig. 7, the optimal nonlinear toll-charge function is a continuous non-decreasing function. Longer journeys in the cordon are charged up to nine times higher than the shorter journeys, which is (a) more equal than the commonly adopted pay-per-entry and licensing basis charge and (b) also helpful to encourage the drivers to wisely select a shorter itinerary in the cordon to avoid potential congestions. When implemented in practice, the equity property of distance-based
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charge makes it prone to be accepted by the public users. However, the nonlinear distance-based toll-charge function is comparatively more complicated in view that it has many different intervals. This complexity may cause some difficulties to the drivers in precisely perceiving the toll charge on each path and then in wisely making their route plan. To cope with this problem, some empirical policies executed in the initialization stage of many congestion pricing schemes can be followed. For instance, as used for the ERP system in Singapore (Foo, 2000), the authorities can (a) establish the nonlinear distancebased charge on partial pricing cordons and/or selected time of day as a trial and (b) frequently launch some publicity campaign for propagation. Such policies would help the drivers/users to be aware of and familiar with the updated charging rules, which is quite positive for the user acceptance of nonlinear distance-based toll charge. 7. Conclusions 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 probitbased 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; 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, congestionbased 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. Acknowledgements The authors are grateful to the editor and two anonymous reviewers for their constructive comments and suggestions, which have largely improved the work presented in this paper. Appendix A A.1. Cost Averaging (CA) method The Cost Averaging (CA) method can be regarded as a variation of the well-known Method of Successive Average (MSA) (see, Chapter 12 of Sheffi, 1985). Proposed by Cantarella (1997), CA method is the only convergent algorithm in the literature for solving SUE problem with elastic demand and asymmetric link travel time functions. Procedures of CA method are summarized as follows:
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Step 0: (Initialization) Let an initial link flow pattern v(1) = 0 and calculate the link travel times t(1) = (ta(v(1)), a e A)T. Let the number of iterations k = 1. Step 1: (Stochastic network loading) Based on the link travel time pattern t(k) and toll charge pattern sw, w e W, perform a stochastic network loading that yields a new link flow pattern v(k+1). Step 2: (Direction searching) Based on link flow pattern v(k+1), calculate an auxiliary link travel time pattern ðkþ1Þ ðkþ1Þ yðkþ1Þ ¼ ðya ; a 2 AÞT using link travel time functions, i.e., ya ¼ ta ðv ðkþ1Þ Þ. Step 3: (Convergence test) If the following condition is fulfilled, then stop and output v(k+1). Otherwise go to Step 4.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ðkþ1Þ ðt aðkÞ ya Þ2 a2A
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ðtaðkÞ Þ2
6 e1 ;
a2A
where e1 is a predetermined positive tolerance. Step 4: (Update) Find new generalized link travel time pattern tðkþ1Þ ¼ ðtðkþ1Þ ; a 2 AÞT according to the formula: a
taðkþ1Þ ¼ taðkÞ þ ð1=kÞðyðkþ1Þ taðkÞ Þ; a 2 A: a Let k = k + 1 and go to Step 1. References Akiyama, T., Okushima, M., 2006. Implementation of cordon pricing on urban network with practical approach. Journal of Advanced Transportation 40, 221– 248. Allsop, R.E., 1974. Some possibilities of using traffic control to influence trip distribution and route choice. In: Buckley, D.J. (Ed.), Proceedings of the 6th International Symposium on Transportation and Traffic Theory, pp. 345–375. Bar-Gera, H., 2011, Transportation Network Test Problems Website.
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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre
Optimal distance tolls under congestion pricing and continuously distributed value of time Qiang Meng, Zhiyuan Liu ⇑, Shuaian Wang Department of Civil and Environmental Engineering, National University of Singapore, Singapore 117576, Singapore
a r t i c l e
i n f o
Article history: Received 12 August 2011 Received in revised form 13 February 2012 Accepted 27 March 2012
Keywords: Distance-based toll Cordon-based congestion pricing Stochastic user equilibrium Continuously distributed value-of-time Mathematical programming with equilibrium constraints Genetic algorithm
a b s t r a c t 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 nondecreasing 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. Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved.
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; 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; 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 tollcharge 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
⇑ Corresponding author. Tel.: +65 9864 6604; fax: +65 6779 1635. E-mail addresses: [email protected] (Q. Meng), [email protected] (Z. Liu). 1366-5545/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tre.2012.04.004
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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. 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; 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. 1.1. Relevant studies There are numerous achievements on the congestion pricing studies (Small, 1992; Yang and Huang, 2005; Lawphongpanich et al., 2006; Han and Yang, 2008; Verhoef et al., 2008; etc.). Among these studies, the cordon-based congestion pricing scheme is usually regarded as a particular second-best pricing solution (McDonald, 1995; Verhoef et al., 1996; Verhoef, 2002; Yang and Zhang, 2002; Meng et al., 2005; Meng and Liu, 2011a, 2012). Based on given cordon locations, optimal toll design problem for cordon-based congestion pricing aims to identify a proper toll fare solution that optimizes a system-wide objective function, such as the TSB. When assuming the flat toll-charge method, the optimal toll design problem for cordon-based congestion pricing schemes can be formulated as a mathematical program with equilibrium constraints (MPEC) or bi-level programming model
2.5
Bus Fares
2 1.5 1 0.5 0 0
5
10
15
20
25
30
35
40
45
50
Travel Distance Fig. 1. Bus fare structure for public transport system in Singapore.
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(Verhoef, 2002; Chen and Bernstein, 2004; Ho et al., 2005; Maruyama and Sumalee, 2007). In reality, these MPEC or bi-level programming models are similar to those built for the continuous network design or optimal signal setting problem with deterministic user equilibrium (DUE) constraints (Yang and Meng, 1998), because the flat toll-charge method with a fixed VOT leads to an additive link-based toll charge solution. There are several particular heuristic methods developed for the continuous network design or optimal signal setting problem, including iterative optimization-assignment algorithm (Allsop, 1974), equilibrium decomposed optimization (Suwansirikul et al., 1987), augmented Lagrangian algorithm (Meng et al., 2001), and gradient-based descent methods (Chiou, 2005). However, these methods are not available for the optimal distance-based toll design with continuously distributed VOT proposed in this study because of the non-additive distancebased toll charges and random VOT. The continuously distributed VOT is proven to be more rational than a fixed value (Lam and Small, 2001; Verhoef and Small, 2004). Traffic assignment problem with continuously distributed VOT has been comprehensively examined on a hypothetical two-route example, e.g., Mayet and Hansen (2000), Verhoef and Small (2004), Xiao and Yang (2008), and Nie and Liu (2010); to name a few. Some findings in these studies, however, may not be available when extended to realistic network with more than two links. Leurent (1993) and Dial (1996) investigated the traffic assignment problem with continuously distributed VOT on realistic networks by assuming DUE principle for the drivers’ route choice behavior. Cantarella and Binetti (1998) proposed a solution framework for the probit-based SUE problem with continuously distributed VOT. This framework assumed fixed OD demand and relies on a path-based simulation technique to cope with the probit-based stochastic network loading. This paper intends to extend the work of Cantarella and Binetti (1998) to the case of elastic demand and uses a link-based Monte Carlo simulation method to solve the probit-based stochastic network loading, which avoids path generation/enumeration. In addition to the second generation ERP system in Singapore, the importance and significance of the optimal distancedbased toll design addressed in this study can be further demonstrated by the experiments of a kilometer (KM)-based tollcharge method conducted in several European regions including Leeds, UK (Mitchell et al., 2005; Namdeo and Mitchell, 2008), Scotland (O’Mahony et al., 2000), Netherlands (Ubbels et al., 2002) and Germany (Hensher and Puckett, 2007; Link, 2008). As a special distance-based toll-charge method, the KM-based toll-charge method assumed that toll charge for each vehicle was linearly proportional to its travel distance in the pricing area (May and Milne, 2000), making the toll charge additive. Yet, previous studies and trials for KM charge all arbitrarily set a charging rate per kilometer, which is unlikely to achieve the maximal TSB. Therefore, this paper also takes the toll design problem for KM-based toll-charge method into analysis, which can be quickly solved by a brute-force method. In spite of its simplicity in modeling and calculation, KM-based toll-charge function is unlikely to be the optimal for all the transportation networks. Yet, no practical data is available for the analysis of a proper functional form of the toll-charge function or to calibrate such a function. It is therefore more reasonable to assume that the toll-charge function is generic to any positive and non-decreasing function, which includes the KM-based toll-charge function, linear function, and nonlinear functions. 1.2. Objectives and contributions This paper aims to propose a methodology that is able to estimate a positive and non-decreasing toll-charge function with maximal TSB by assuming a continuously distributed VOT. Assuming that drivers’ perception error on path travel time is normally distributed, it will first define a probit-based SUE problem with elastic demand, asymmetric link travel time functions and continuously distributed VOT, which is termed as the generalized probit-based SUE problem. The proposed generalized probit-based SUE problem will be subsequently formulated by a fixed-point model with unique solution. A network transformation technique is then proposed to circumvent the non-additive toll charges, which avoids path enumeration/generation. The Cost Averaging (CA) method proposed by Cantarella (1997) is then adopted to solve this SUE problem. The optimal distance-based toll design with the aim of maximizing TSB can be formulated as a MPEC model by taking the fixed-point model as a constraint. Since the distance-based toll-charge function is allowed to be any functional form, this study proposes a novel piecewise-linear function method that can approximate any form of the toll-charge function, which gives another mixed-integer MPEC model for the convenience of calculation. A Hybrid genetic algorithm (GA)–CA algorithm is designed for solving the mixed-integer MPEC model. Specially, in the case of KM-based toll-charge function, a more concise brute-force algorithm is adopted to solve the optimal distance-based toll design problem. The remaining sections are organized as follows. Section 2 defines the optimal distance-based toll design problem and Section 3 puts forward a fixed-point model and link-based solution method for the generalized probit-based SUE problem. Section 4 first develops a MPEC model for the toll design problem, and then introduces a mixed-integer MPEC model based on the piecewise-linear approximation function. Solution algorithms and numerical experiments are presented in Sections 5 and 6. Conclusions are finally provided in Section 7. 2. Notation, assumptions and problem description 2.1. Notation and Assumptions Consider a strongly connected network denoted by G = (N, A) where N and A are the sets of nodes and directed links, respectively. Let W be the set of the OD pairs and Rw be the set of paths between OD pair w e W. Travel demand between
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OD pair w e W is denoted by qw and q = (qw, w e W)T is a column vector for all these travel demands. Let fwk be traffic flow on path k e Rw between OD pair w e W, fw = (fwk, k e Rw)T be a column vector of all these path flows between OD pair w and T f ¼ ðf w ; w 2 WÞT be a column vector of all the path flows over the entire network. Let va denote traffic flow on link a e A and v = (va, a e A)T is a column vector of all these link flows. The following flow conservation equations should be fulfilled:
q ¼ Kf;
ð1Þ
v ¼ Df;
ð2Þ
f P 0:
ð3Þ
½dw ak jAjK
½dw k jWjK
where D ¼ and K ¼ are the incidence link/path and OD pair/path matrices, where |A|, |W| and K are the w w number of links, OD pairs and paths, respectively; dw ak ¼ 1 if link a is on path k e Rw, and dak ¼ 0, otherwise. dk ¼ 1 if path k connects OD pair w e W, and dw ¼ 0, otherwise. k The travel time on link a e A is assumed to be a function of link flow vector v, denoted by ta(v), which can flexibly represent a symmetric or asymmetric link travel time function. These link travel time functions are grouped into a column vector t(v) = (ta(v), a e A)T. It is assumed that vector function t(v) is non-negative, monotonically increasing and continuously differentiable. Travel time on path k e Rw is the sum of travel times of all the links on this path, denoted by cwk(v), namely,
cwk ðv Þ ¼
X ta ðv Þdw ak :
ð4Þ
a2A
These path travel times are grouped into vector cw(v) = (cwk(v), k e Rw)T. It is assumed that the travel time on path k e Rw perceived by drivers, denoted by Cwk(v), is a random variable with the expression:
C wk ðv Þ ¼ cwk ðv Þ þ fwk ;
ð5Þ
where fwk is a random perception error following normal distribution with zero mean and constant variance. 2.2. Toll-charge function, generalized path travel time and optimal distance-based toll design Let I be the number of congestion pricing cordons in network G. Network G = (N, A) is then divided into one external net^ and several cordon networks, denoted by G i Þ (i = 1, 2, . . . , I). The links (nodes) in any ^ ¼ ðN; i ¼ ðN ^ AÞ, i; A work, denoted by G ^ are termed ^ ¼ ðN; ^ AÞ cordon network are termed as internal links (nodes), while the other links (nodes) in external network G i # N and the exteras external links (nodes). Note that the boundary nodes of any cordon i exist both in its cordon node set N ^ # N. Let dw;i denote the length of the portion of path k e Rw in the congestion pricing cordon i and it can be nal node set N k expressed as follows: w;i
dk ¼
X w la dak ; k 2 Rw ; w 2 W;
ð6Þ
a2A i
i is the set of links in cordon i. where la is the length of link a and A A distance-based toll-charge method for the cordon-based congestion pricing scheme can be expressed by a distancebased toll-charge function /(d) where d is the distance traveled in a pricing cordon. It is assumed that the generic function /(d) is positive and non-decreasing but may not be continuous. As shown by Fig. 1, the toll-charge function for distancebased bus fare in Singapore is not continuous. /(d) is also allowed to follow any function form, including, linear function, quadratic function, exponential function, power function, etc. The kilometer (KM) based toll-charge function is a special case of /(d), where only the slope is a decision variable, shown as follows:
/ðdÞ ¼ qd;
ð7Þ
where q is the slope of KM based toll-charge function, which is the charging rate per kilometer. For any given toll-charge function /(d), the toll charge imposed on path k e Rw between OD pair w e W can be calculated as follows:
swk ð/Þ ¼
I X w;i /ðdk Þ:
ð8Þ
i¼1
Toll charge swk(/) shown in Eq. (8) can be converted into time-units by virtue of the drivers’ value of time (VOT), denoted by a. As mentioned in Section 1, a is assumed to be a continuously distributed variable across the whole population of drivers. The cumulative distribution function (CDF) of a can be obtained by the empirical curve plotting method if the VOT values of all the drivers are known. It is assumed that a is positive with constant mean and variance, and its probability density function (PDF) is continuously differentiable and strictly positive. Here, constant mean and variance of a imply that they are the same for all OD pairs. In the literature, a is usually assumed to follow uniform distribution, exponential distribution, or lognormal distribution. The distribution of a is also assumed to be independent to the link/path flows and other network attributes.
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wk ðv ; sw ð/ÞÞ, consists of the perceived path travel time shown by The overall travel impedance on path k e Rw, denoted by C wk ðv ; sw ð/ÞÞ is termed as the generalized path travel time. It can be Eq. (5) and a time-based value of the toll charges, and C expressed by
wk ðv ; sw ð/ÞÞ ¼ C wk ðv Þ þ swk ð/Þ=a ¼ cwk ðv Þ þ swk ð/Þ=a þ fwk ; C
ð9Þ
T
where vector sw = (swk(/), k e Rw) . Eq. (9) can be rewritten as follows:
wk ðv ; sw ð/ÞÞ ¼ cwk ðv Þ þ Eðswk ð/Þ=aÞ þ kwk ; k 2 Rw ; w 2 W; C
ð10Þ
where E(swk(/)/a) is the expectation of random term swk(/)/a. kwk , termed as generalized perception error, has the expression:
kwk ¼ swk ð/Þ=a Eðswk ð/Þ=aÞ þ fwk :
ð11Þ
It can be seen that kwk is also a flow-independent random variable with zero mean and fixed variance. All these generalized perception errors associated with OD pair w e W are grouped into a multivariate random variable kw ¼ ðkwk ; k 2 Rw ÞT , and evidently the PDF of kw is also continuously differentiable. Drivers are assumed to make their trip plans based on the generalized path travel time function defined by Eq. (10), and such a route choice principle is named as generalized probit-based SUE. Let Sw(cw, sw) be the expected value of minimum generalized path travel time between OD pair w e W, namely:
wk ðv ; sw Þg; Sw ðcw ; sw Þ ¼ E½minfC k2Rw
ð12Þ
where Sw(cw, sw) is usually referred to as satisfaction function (Sheffi, 1985). It is assumed that travel demand between OD pair w e W is a continuously differentiable, non-increasing and bounded function, denoted by Dw(), with respect to its satisfaction function:
w ; w 2 W; qw ¼ Dw ðSw ðcw ; sw ÞÞ 6 q
ð13Þ
w is a given upper bound of travel demand between OD pair w e W. This assumption is previously made by where parameter q Cantarella (1997) and Maher and Zhang (2000) for the conventional SUE problem with elastic demand. It shows that the OD demand function is defined at an aggregate level for the entire population, and there is no explicit connection between the OD demand and the PDF of VOT, which is different from the endogenous definitions in some previous studies, e.g., Small and Yan (2001), Verhoef and Small (2004), van den Berg and Verhoef (2011). All the toll charges on the network are determined by the toll-charge function shown in Eq. (8). Different toll-charge functions will result in different network equilibrium flows and consequently different total social benefit (TSB) values on the entire network. Let U be the set of all these positive and non-decreasing toll-charge functions. Assuming the generalized probit-based SUE principle for drivers’ route choice behavior, the optimal distance-based toll design problem aims to identify or estimate a proper toll-charge function with the maximal TSB from the set U. Model development and algorithm design for the proposed optimal distance-based toll design rely on solving a corresponding generalized probit-based SUE problem for any given toll-charge function. Thus, we first present the formulation and solution algorithm for such an SUE problem in the following section. 3. Generalized probit-based SUE problem According to the generalized path travel time defined in Eq. (10) with a particular toll-charge function / e U, the probability that generalized travel time on path k e Rw is perceived as the minimal one among all the paths between OD pair w e W, denoted by pwk(cw(v), sw(/)), can be expressed as follows:
wl ðv ; sw ð/ÞÞ; 8l 2 Rw and l–kjv ; sw ð/ÞÞ; k 2 Rw ; w 2 W: pwk ðcw ðv Þ; sw ð/ÞÞ ¼ PrðC wk ðv ; sw ð/ÞÞ 6 C
ð14Þ
The corresponding generalized probit-based SUE link flow pattern is a solution of the following fixed-point model:
v a ð/Þ ¼
XX
½Dw ðSw ðcw ðv Þ; sw ð/ÞÞÞ pwk ðcw ðv Þ; sw ð/ÞÞdw ak ; a 2 A:
ð15Þ
w2W k2Rw
The fixed-point model above is a variation of the fixed-point formulation proposed by Cantarella (1997) for the conventional SUE problem with elastic demand and asymmetric link travel time functions. Following the similar proof of Lemma 4.1 in Daganzo (1979), it can be shown that: Proposition 1. Satisfaction function Sw(cw(v), sw(/)) is concave with respect to path travel times cw(v) and
@ðSw ðcw ðv Þ; sw ð/ÞÞÞ ¼ pwk ðcw ðv Þ; sw ð/ÞÞ: @cwk ðv Þ
ð16Þ
In view of Proposition 1 as well as the monotonicity of demand functions and link travel time functions, the fixed-point model shown by Eq. (15) has a unique solution according to Theorems 1 and 2 of Cantarella (1997).
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The Cost Averaging (CA) method proposed by Cantarella (1997) is recognized as the only convergent algorithm for solving the fixed-point model shown by Eq. (15), which is provided in the Appendix for completeness. It can be seen from the Appendix that the stochastic network loading step is quite crucial to the CA method. The existing analytical approximation methods for the probit-based stochastic network loading problem (Daganzo and Sheffi, 1977; Maher and Hughes, 1997; Rosa and Maher, 2002) are inapplicable for the generalized probit-based stochastic network loading due to the random VOT. Hence, a Monte Carlo simulation based method is developed for solving the generalized probit-based stochastic network loading. As shown in Eqs. (5) and (8), both the drivers’ perception error in travel time, fwk, and the toll charges swk(/) are defined on paths. Thus, directly using Monte Carlo simulation to solve the generalized probit-based stochastic network loading problem necessitates path enumeration/generation, which would largely increase the computational time. To avoid path enumeration or generation, two interesting techniques are elaborated in Sections 3.1 and 3.2, which enable an efficient link-based Monte Carlo simulation method. 3.1. Link-based formulation for the generalized path travel time The path travel time perception error fwk in Eq. (9) can be formulated as the sum of perception errors on the link travel times by assuming that drivers’ perceived travel time on link a e A equals to (Sheffi, 1985):
T a ðv Þ ¼ t a ðv Þ þ na ; a 2 A;
ð17Þ
where na is a normally distributed random error term with zero mean and flow-independent variance, namely,
na Nð0; bt 0a Þ; a 2 A;
ð18Þ
where b is a proportionality constant parameter and t0a is the free-flow link travel time. With the perceived link travel time functions shown in Eq. (17), the generalized travel time on path k e Rw can be expressed as:
wk ðv ; sw ð/ÞÞ ¼ C
X X X w T a ðv Þdw t a ðv Þdw na dak : ak þ swk ð/Þ=a ¼ ak þ swk ð/Þ=a þ a2A
a2A
ð19Þ
a2A
P Let fwk ¼ a2A na dw ak . Since fwk is a summation of some normally distributed variables with zero mean and constant variance, it can inherit these properties, thus fwk is also normally distributed, which shows the validity of the link-based formulation. In addition, covariance of the generalized path travel times on two different paths takes into account their overlapped portions. The characteristics of the path travel time perception error fwk defined from the link-based formulation are summarized in Proposition 2 below: ðv ; sw ð/ÞÞ expressed by Eq. (19) has the form: Proposition 2. The generalized path travel time C wk
wk ðv ; sw Þ ¼ cwk ðv Þ þ swk ð/Þ=a þ fwk ; k 2 Rw ; w 2 W; C
ð20Þ
P
w a2A na dak
where fwk ¼ is a normally distributed random term which has zero mean and constant variance. Moreover, the covariance between the generalized path travel times on two different paths is given by
wk ; C wb Þ ¼ swk ð/Þswb ð/Þ var covðC
1
a
þ
X w bt 0a dw ak dab ; 8k; b 2 Rw ; w 2 W:
ð21Þ
a2A
3.2. Network transformation for non-additive path toll charges Let Ei denote the set of all the entry-exit pairs of cordon i e I. A pricing cordon usually has limited size, and in such cases it is effortless to enumerate all the simple paths (acyclic paths) between any entry-exit pair e e Ei, which are termed as internal paths. Thus, a network transformation can be efficiently conducted to convert the non-additive path-based toll charges into link-based: firstly, each internal path between entry-exit pair e e Ei is represented by a dummy link connecting its entry and i Þði ¼ 1; 2; . . . ; IÞ by corresponding dummy links. Hereby, i ¼ ðN i; A exit node; secondly, replace all the cordon networks G ^ and all the dummy links constitute a new composite network, denoted by G0 = (N0 , A0 ). We ^ ¼ ðN; ^ AÞ external network G can see that this network transformation converts the ‘‘path-based’’ toll charges to be ‘‘link-based’’ on each dummy link. 3.2.1. An illustrative example A small example shown in the upper part of Fig. 2 is used to illustrate the network transformation procedure. Links 1–6 ^ of this example. The area surrounded by a dashed ellipse is a pric^ ¼ ðN; ^ AÞ, and Nodes 1–5 constitute the external network, G Entry-exit pairs of the pric ¼ ðN; AÞ. ing cordon, and therefore Links 7–12 and Nodes 2, 4, 6 compose the cordon network, G ing cordon comprise 2 ? 4 and 4 ? 2, and any trip traversing this cordon is imposed by a distance-based toll charge. A new network, G0 = (N0 , A0 ), modified by the network transformation procedure is shown in the lower part of Fig. 2. It can be seen that the pricing cordon is replaced by four dummy links (dashed arcs), and as shown in Table 1 each of these dummy links represents a simple path.
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943
Note that a similar type of network transformation was used in Meng et al. (2004) and Yang et al. (2004) for solving optimal toll design problem of highways with non-additive toll charges. However, in these two papers, the non-additive toll charges are entry-exit based, thus only one dummy link exists for one entry-exit pair. e . Evidently, based on a given toll-charge function, All the internal paths between entry-exit pair, e e Ei are grouped in set R /, toll charge on each dummy link of composite network G0 = (N0 , A0 ) is a flow-independent constant, and generalized travel time on any path is additive to the links. Remark 1. In the case of a realistic network with large pricing cordons, there could be quite a number of simple paths between one entry-exit pair, thus considerably increasing the computational burden. Previous studies for path-based traffic assignment problems are confronted by this hurdle, see, e.g., Lo and Chen (2000) and Bekhor and Toledo (2005). Instead of enumerating all the simple paths, these studies only examine partial paths (a) explicitly in a pre-generated path choice set or (b) implicitly based on the column generation during iterative computations. These techniques of using partial paths are also recommended to cope with the internal paths on a realistic network. Remark 2. For the sake of presentation, the network transformation omits those OD pairs with origin or destination in the cordon area. However, this methodology presented in Section 3.2 can be simply modified for those OD pairs, by enumerating the paths from their origin or destination to each entry node of the cordon and then replacing those paths by dummy links. 3.3. Monte Carlo simulation based stochastic network loading method The stochastic network loading is invoked in each iteration of the CA method, where the real link travel times ta, a e A are fixed and the corresponding equilibrium link flows are taken as output. Based on the two techniques in Sections 3.1 and 3.2, we then manage to use a two-stage Monte Carlo simulation method for solving the generalized probit-based stochastic network loading problem. Main steps of this method are provided as follows: Stage 1: Monte-Carlo simulation for travel demand estimation. ð0Þ Step 1.0: (Initialization) Let the iteration counter be n = 1 and the initial estimated satisfaction be Sw ¼ 0; w 2 W. ðnÞ Step 1.1: (Sampling of link travel time) For each link a e A, sample a value for the perception error na from the normally ðnÞ ðnÞ distributed population Nð0; bt0a Þ; a 2 A. And then calculate the link travel time T~ a ¼ t a þ na . (n) Step 1.2: (Generalized travel time of dummy links) Sample a value for VOT a from its distribution function and calculate e ; e 2 E by the generalized travel time on all the internal paths l 2 R ðnÞ T~ el ¼
X /ðdel Þ e T~ ðnÞ ; a dal þ ðnÞ a2A
ð22Þ
a
Fig. 2. Illustrative example for network transformation.
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Q. Meng et al. / Transportation Research Part E 48 (2012) 937–957 Table 1 List of simple paths and corresponding dummy links. Entry-exit pairs
Simple paths
Represented by dummy link
2?4
2?4 2?6?4
[2–4–1] [2–4–2]
4?2
4?2 4?6?2
[4–2–1] [4–2–2]
and 0, otherwise. del is the length of internal path e contains internal link a 2 A, where deal equals to 1 if internal path l 2 R e . T~ ðnÞ is recorded as the travel time on the corresponding dummy link of composite network G0 = (N0 , A 0 ). l2R el ðnÞ Step 1.3: (Shortest path time calculation). With link travel time pattern fT~ a ; a 2 A0 g on the composite network G0 = (N0 , A0 ), ~ ðnÞ find the shortest path between each OD pair w e W, and record the total travel time on this path as C w . Step 1.4: (Satisfaction estimation). Estimate the satisfaction for each OD pair w e W by the following average scheme:
ðn1Þ ~ ðnÞ SðnÞ ¼ ðn 1ÞSw þ C w ; w 2 W: w n
ð23Þ
Step 1.5: (Accuracy checking). If the number of iterations n P n0 , where n0 is a predetermined sample size, go to Step 1.6; otherwise, set n = n + 1 and go to Step 1.1. Step 1.6: (OD demand calculation). Calculate OD travel demand pattern by the formulae:
qw ¼ Dw ðSðnÞ w Þ; w 2 W:
ð24Þ
Stage 2: Monte-Carlo simulation for calculating the generalized probit-based SUE link flow. ð0Þ Step 2.0: (Initialization). Let the initial link travel flow vector be v a ¼ 0; a 2 A and m = 1. ðmÞ 0 Step 2.1: (Sampling). Sample the perception error na from Nð0; bt a Þ; a 2 A based on normally distributed random number ðmÞ ðmÞ series, and then calculate the link travel time T~ a ¼ t a þ na ; a 2 A. (m) Step 2.2: (Sampling of VOT). Sample a value for VOT a from its distribution function and calculate the travel time on all e ; e 2 E by the internal paths l 2 R ðmÞ T~ el ¼
X /ðdel Þ e T~ ðmÞ : a dal þ ðmÞ
ð25Þ
a
a2A
ðmÞ Then, record T~ el as the travel time on corresponding dummy link of the composite network G0 = (N0 , A 0 ). ðmÞ Step 2.3: (All-or-nothing assignment). Based on link travel time pattern fT~ a ; a 2 A0 g on the composite network G0 = (N0 , A0 ), find the shortest path for each OD pair w, then assign qw to the shortest path. Flow on link a e A0 is therefore a summation of the travel demands of all the OD pairs whose shortest path uses link a e A0 . Then assign flows on all the dummy links to e ; e 2 E. This would generate an auxiliary flow pattern on all the links the links on their corresponding internal paths l 2 R fY ðmÞ ; a 2 Ag. Calculate the simulated link flow by a
ðm1Þ
v ðmÞ ¼ a
ðm 1Þv a m
þ Y ðmÞ a
; a 2 A:
ð26Þ
Step 2.4: (Convergence test). If the number of iterations m P m0 , where m0 is also a predetermined sample size, stop; otherwise, set m = m + 1 and go to step 2.1. This two-stage procedure here is an extension of our previous work (Meng and Liu, 2011b). Note that two-stage procedure of the Monte Carlo simulation is used here, rather than an integrated one-stage procedure that simulates the value of travel demand and SUE link flows based on the same set of samples at each iteration. This is because the two-stage procedure is more stable and thus efficient than the one-stage procedure. Note that these Monte Carlo simulation procedures can be easily modified for solving the stochastic network loading of DUE or logit-based SUE problem with continuously distributed VOT. Remark 3. The stop criterion at each stage of the Monte Carlo simulation is based on a predetermined sample size. Accuracy of the simulation is thereby determined by the sample size. With a given accuracy level, a lower bound for the sample size of each stage can be theoretically determined by the study of Meng and Liu (2011b). Alternatively, for any given network example, a sensitivity test can be performed in the first place to determine a proper sample size.
4. Two MPEC models for the optimal distance-based toll design 4.1. Total social benefit and the exact MPEC model We now proceed to investigate how to obtain a desirable toll-charge function / from the viewpoint of system optimum. A toll-charge function is regarded as optimum if it can give rise to the maximal total social benefit (TSB) on the network. In the context of generalized probit-based SUE, the following expression is adopted for the TSB:
Q. Meng et al. / Transportation Research Part E 48 (2012) 937–957
Zð/Þ ¼
XZ w2W
qw ðsÞ
0
D1 w ðxÞdx
X
qw ðsÞSw ðcw ðv ðsÞÞ; sw ð/ÞÞ þ
w2W
! XX f el ðsÞsel ð/Þ ; E e e2E l2R
a
945
ð27Þ
where E() is the expectation operator, and s = (sw(/), w e W)T. q(s) = (qw(s), w e W)T, v(s) = (va(s), a e A)T, and fðsÞ ¼ ðf ðsÞ; e 2 E; l 2 R e ÞT denote the column vectors for the corresponding equilibrium OD demands, link flows, and interel nal path flows, associated with a given toll-charge function /. Value of these vectors can be obtained by solving the generalized SUE problem discussed in Section 3. Note that in Eq. (27) the first term is the gross consumer benefit, the second term represents the overall cost borne by the commuters, and the last term is the total toll revenue that can be invested back to the transportation system or any other place with the most beneficial use. It is an aim of the authorities that implementing a congestion pricing scheme can enhance the TSB shown in Eq. (27) on the entire network. A MPEC model is hence built to find a toll-charge function maximizing the TSB:
maxZð/Þ
ð28Þ
/2U
s.t.
v a ð/Þ ¼
XX
½Dw ðSw ðcw ðv Þ; sÞÞ pwk ðcw ðv Þ; sÞdw ak ; a 2 A:
ð29Þ
w2W k2Rw
Constraint (29) is the fixed-point model proposed for the generalized probit-based SUE problem. It should be noted that the solution to this MPEC model is the system-optimal charge for the distance-based congestion pricing scheme. As mentioned in Section 2, the toll-charge function is merely required to be positive and non-decreasing, and it is generic to any linear or nonlinear functional forms. This type of toll-charge function has perfect rationality to the distance-based toll method, thus model (28) and (29) is termed as exact MPEC model. Yet, it is quite challenging to solve this model, since the toll-charge function has no specific functional form. In view of this, an approximation method is proposed in the following section, where a piecewise-linear function is used to approximate any positive and non-decreasing function. This approximation method then gives a mixed-integer MPEC model that can be simply solved. 4.2. A Mixed-integer MPEC Model with a piecewise-linear approximation function A piecewise-linear approximation function is indicated by Fig. 3. Suppose the nonlinear function in the left-hand-side of Fig. 3 is the optimal toll-charge function for one transportation network, it then can be approximated by a piecewise-linear function as shown in the right-hand-side of Fig. 3. The shape of a piecewise-linear function in each interval is uniquely determined by its two boundary values. Note that if the optimal toll-charge function /(d) is not continuous, it should also be approximated by a discontinuous piecewise-linear function. For conciseness, we only take into account the continuous piecewise-linear approximation function in this paper. Nevertheless, the methodology provided can be easily extended to the discontinuous cases. We proceed to talk about the theoretical definition of the piecewise-linear approximation function. Let dmin and dmax be the minimum and maximum lengths of all the internal paths. The range [dmin, dmax] can be uniformly divided into n intervals, where n is a positive integer number,
½dmin ; d2 ; ½d2 ; d3 ; . . . ; ½di ; diþ1 ; . . . ; ½dn ; dmax :
Fig. 3. An illustrative example for the piecewise-linear approximation function.
ð30Þ
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Note that the length of each sub-interval here is the same. The piecewise-linear approximation function contains n + 1 boundary values over these n intervals, and these boundary values are defined as y1 = /(dmin), yi = /(di), i = 2, 3, . . . , n and yn+1 = /(dmax). The piecewise-linear approximation function is defined accordingly as follows:
n ðd; yÞ ¼ y þ yiþ1 yi ðd di Þ; di 6 d < diþ1 ; i ¼ 1; 2; . . . n; / i diþ1 di
ð31Þ
where d1 = dmin and dn+1 = dmax. The column vector y is defined as y = (y1, y2, . . . , yn+1)T.The number of intervals, n, can influ n ðd; yÞ to the original toll-charge function /(d). It is reasonable to assume that: ence the fitness of /
n 2 ½nmin ; nmax ; and n is an integer:
ð32Þ
Herein, the lower bound nmin is imposed, because it will deteriorate the equity and efficiency of distance-based toll-charge method if the value of n is fairly small. And on the other hand, if the number of intervals is too large, computational cost of would geometrically increase, thus an upper bound nmax is also imposed for solving the optimal piecewise-linear function / n n. Based on the piecewise-linear function (31), the MPEC model (28) and (29) can be approximated by the following mixedinteger MPEC model with the decision variables y and n:
/ n ðd; yÞÞ maxZð
ð33Þ
y;n
s.t.
v a ð/ n ðd; yÞÞ ¼
XX
½Dw ðSw ðcw ðv Þ; sÞÞ pwk ðcw ðv Þ; sÞdw ak ; a 2 A;
ð34Þ
w2W k2Rw
ymin 6 y1 ; yi1 6 yi 6 yiþ1 ;
ð35Þ
8i ¼ 2; 3; . . . ; n 1;
ð36Þ
yn 6 ymax ;
ð37Þ
nmin 6 n 6 nmax and n is an integer;
ð38Þ
where ymin (ymax) is a predetermined lower (upper) bound of the toll charge. The inequalities (35)–(37) ensure that the piecewise-linear function is positive and non-decreasing. This mixed-integer MPEC model is then used to solve optimal distancebased toll design problem, solution algorithm of which is provided in the following section. Note that in practice, the value of n can be taken as a fixed input value for the model, which should be decided by the transport authorities for each specific cordon. 5. Solution algorithm Most of the solution algorithms for solving the MPEC model need to calculate the Jacobian matrixes of link flow vector v and OD demand vector q with respect to the vector of decision variables, which is, in this study, the piecewise-linear tollcharge function. Such Jacobian matrixes for the traffic assignment problem can be solved by the sensitivity analysis method (e.g., Yang, 1997). However, it is quite expensive to calculate the Jacobian matrixes in this paper, due to: (a) the complexity of generalized probit-based SUE problem with elastic demand, asymmetric link travel time functions and continuously distributed VOT; (b) the intricate connection between prototype network G = (N, A) and composite network G0 = (N0 , A0 ), for example, Eq. (15) is defined on G = (N, A), while the Monte Carlo simulation method for stochastic network loading is described on n . These hurdles prevent the G0 = (N0 , A0 ); (c) transformation between toll charges and piecewise-linear toll-charge function / usage of any gradient-based algorithm for solving the optimal distance-based toll design problem. However, for any given function type of toll-charge function with limited number of unknown variables, it would be an accurate and inexpensive way to enumerate and assess all the feasible variables with a small gap. For instance, considering the KM-based toll-charge function, shown in Eq. (7), only the slope q is a variable, and a reasonable range for q is usually not quite large, thus it is pleasant to test all the values of q in this range with a small increment. n ðd; yÞ discussed The brute-force enumeration method is unacceptable for the piecewise-linear approximation function / in Section 4.2. While, considering the discrete property of /n ðd; yÞ, when the value of n is given, it would be more straightforward to adopt Genetic Algorithm (GA) as a heuristic for solving the mixed-integer MPEC model (33)–(38). Thereby, we can first enumerate all the feasible n, and for any particular n, the optimal piecewise-linear function that yields maximal TSB can be solved by GA. Due to the space limit, rationale and detailed mechanism of GA are not covered here, and the readers are recommended to some masterpieces in this field for reference (e.g., Goldberg, 1989; Gen and Cheng, 1997). A feasible chromosome for the n ðd; yÞ is given as ðy ; y . . . y ; y . . . y ; y Þ piecewise-linear toll-charge function / 1 2 i iþ1 n1 n and yi here is defined as a gene of one chromosome. The yi is assumed to be integer-valued and measured in cents. Each chromosome will give rise to different toll charges on the network, which therefore leads to different values of total social
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benefit (TSB). Calculation of the TSB is, in itself, a generalized probit-based SUE problem, which can be solved by the Cost Averaging (CA) method. The solution algorithm for optimal piecewise-linear toll-charge function is then termed as Hybrid GA–CA method, which is summarized as follows: Step 1: (Initialization). Set initial value for the number of intervals to n(1) = nmin. And initialize the optimal TSB value to TSB⁄ = 0 and the optimal number of intervals to n⁄ = n(1). Set the number of iteration to p = 1. ðpÞ with maximal TSB Step 2: With the number of intervals equal to n(p), calculate the optimal piecewise-linear function / n value, denoted by TSB(p), by the following GA: Step 2.0: (Initial population). Randomly generate initial population of the piecewise-linear function, and each individual carries a feasible chromosome. Then, set the number of generation k = 1. The population size is defined as k. Step 2.1: (Evaluation). Based on the toll-charge function shown by one chromosome, calculate the value of toll charge on each internal path of the composite network G0 = (N0 , A0 ). Then, utilize the CA method to calculate the corresponding traffic assignment problem, results of which are used to evaluate the TSB for this toll-charge function. Conduct such evaluations for all the new generated chromosomes. chromosomes with higher TSB values as survivors for current generation and discard Step 2.2: (Selection). Select the first k the rest. 1 and Y 2 , two Step 2.3: (Crossover). Conduct pairing among survivors, where for each two chosen survivors, denoted by Y new chromosomes are generated by the following function:
Y^ 1 ¼ vY 1 þ ð1 vÞY 2 Y^ 2 ¼ vY 2 þ ð1 vÞY 1
v 2 ð0; 1Þ; v–0:5: v 2 ð0; 1Þ; v–0:5:
ð39Þ
Step 2.4: (Mutation). To guarantee the monotonicity of the toll-charge map, a mutation is conducted in this way: first, randomly choose some genes from existing chromosomes; second, for each chosen gene yi, update its value by sampling a uniformly distributed random variable in [ymin, ymax], then proportionally change the value of other genes of this chromosome in the interval [ymin, yi] and [yi, ymax]. Step 2.5: (Stop test). If a stop criterion is achieved, let the value of TSB(p) equal to the maximal TSB among the survivors in current generation and record the corresponding chromosome, and then go to Step 3; otherwise, set k = k + 1 and go to step 2.1. Step 3: If TSB(p) > TSB⁄, then let
TSB ¼ TSBðpÞ ; n ¼ nðpÞ ;
ð40Þ
and record the corresponding chromosome as the optimal toll-charge function. Step 4: If nðpÞ P nmax , then stop and output the optimal toll-charge function, TSB⁄ and n⁄; otherwise, let n(p+1) = n(p) + 1, p = p + 1 and go to Step 2. 6. Numerical examples To numerically validate the proposed methodology, two examples are adopted. The first example, named as Network C, was used by Meng et al. (2004) as well as Yang et al. (2004) for the non-additive entry-exit highway charge schemes. This example contains 14 nodes, 46 links and 8 OD pairs, and between each entry-exit pair of the pricing cordon there is only one simple path (dummy link). Using this concise example, properties of the proposed models and algorithms are comprehensively tested. The second example is designed based on the Orchard Road cordon of the Electronic Road Pricing (ERP) system in Singapore, and it has 33 nodes, 104 links and 12 OD pairs. For these two numerical examples, the travel demand between each OD pair w e W is assumed to be decided by the following function:
w expðc Sw ðcw ðv Þ; sÞÞ; w 2 W; qw ¼ q
ð41Þ
w is the upper bound of OD demand and c is a constant parameter. The link travel time function on link a e A is dewhere q fined as follows:
t a ðv Þ ¼
t0a
1 þ 0:15
v a þ 0:5v^ a^ 1:5ha
4 !
; a 2 A;
ð42Þ
where t0a is the free-flow travel time, and ha is the capacity of link flow. It is required that there is at least one pair of two-way ^ denote the opposite link of link a, and v ^ a^ in Eq. links that connect the same two nodes and go to different directions. Let a ^. Here, v ^ a^ is taken into consideration to calculate ta, which reflects the interference from (42) denotes the flow on link a opposing traffic due to turning and passing movements. By assuming such type of link travel time functions, the Jacobian matrix of link travel time vector t with respect to the link flows v is asymmetric, which is consistent with the assumption of asymmetric link travel time functions. This functional form follows that in Bar-Gera (2011).
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The value-of-time (VOT) is assumed to be continuously distributed across the whole population. Without loss of generality, we assume that the VOT is uniformly distributed in the range from 18.0 to 72.0 Singapore-Dollars (S$) per hour. For each given toll-charge function, the corresponding equilibrium network flow is calculated by the CA method provided in the Appendix, and the value of tolerance in the stop criterion of CA method is taken as e1 = 0.1. The CA method invokes the two-stage Monte Carlo simulation based stochastic network loading method in each iteration. Based on a trial-and-error trial, sample sizes for the two stages of Monte Carlo simulation (see, Section 3.3) are taken as 180 and 1000, respectively, to achieve an acceptable accuracy level. Note that each trial of the Monte Carlo simulation requires the calculation of a shortestpath-tree for each origin. The large sample size (totally 1180 for the two stages) has largely increased the computational time, especially for the second example. Note that the variance parameter b in Eq. (18) is taken as 0.1. As described in Section 5, optimal toll design problem for the KM charge is solved by a brute-force enumeration method and for the general nonlinear distance-based charge is solved by the Hybrid GA–CA method. Both of these two methods possess a good parallelism, thus it is quite convenient to use distributed computing to accelerate the computational speed in each case. For the enumeration method for KM charge, the range of slope q can be partitioned into many intervals and then evaluated by different processors in the distributed computing system. For the Hybrid GA–CA method, evaluation of each newly generated chromosome is independent and it thus can be conducted concurrently by different processors, which is known as parallel GA (see, Wong et al., 2001). With sufficient processors, computational speed of these two methods can be accelerated for tens of times. Thanks to the booming development of computer hardware and software, it is cost efficient to build a distributed computing system with several hundreds of processors. The programs for numerical tests in this section are all coded in FORTRAN 90 and performed at a Distributed Computing PC Cluster system in Civil and Environmental Engineering Department at National University of Singapore. Due to the space limit, profile of the computing platform is not described here, and any interested reader can refer to Liu and Meng (2011) for more information. For the numerical tests in this section, the number of processors involving the computation is fixed to be 50. 6.1. Network C As shown in Fig. 4, the Network C has one pricing cordon, circled by the dashed rectangular. Links in the cordon are grouped in the set:
¼ f33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46g: A Nodes 4, 5, 6 and 7 are entries as well as exits to this pricing cordon. w and values of the t 0a Some parameters for the network attributes, including the OD pairs, upper bound of OD demand q and ha on each link, are identical to those in Yang et al. (2004), which are not included here. The parameter c in Eq. (41) is properly taken as 0.01 for this example. Length of each link is given in Table 2, which is used to measure drivers’ travel distance in the pricing cordon. It should be pointed out that the length shown in Table 2 does not always jibe with that in Fig. 4. This is because the length of many links shown in Fig. 4 has high similarity, and when enumerating the inner paths for this example, most of the inner paths have the same length. This phenomenon has undermined the generality of this example. Thus, the length of each link has been reasonably adjusted in Table 2. Based on the link lengths, it is detected that the maximum and minimum path length, dmin and dmax, for this example is 405 and 2453 meters, respectively. Lower and upper bound of the toll charges, shown in Eqs. (35) and (37), are set to be ymin = 0.5 and ymax = 20.0 S$. These two values are taken to make the numerical example more consistent with practical conditions, in view that 0.5 S$ was adopted by Singapore’s ERP system as the initial toll rate and 20.0 S$ is a fairly large toll value compared with the current toll charges of the ERP system. 6.1.1. KM charge We first cope with the optimal KM-based toll-charge function, since it can be efficiently solved by the brute-force enumeration method and in addition it can be taken as a benchmark to evaluate the performance of optimal nonlinear distancebased charge. As shown by Eq. (7), only the slope q is an unknown parameter for the KM toll-charge function. Since the lower and upper bound of toll charges are 0.5 and 20.0 S$, q is reasonably assumed to range from 0.5 to 10.0 S$ per km. Thus, 951 successive values of q in this range with an increment of 0.01 are tested, and the corresponding values of total social benefit (TSB) are shown in Fig. 5. Fig. 5 indicates that the value of TSB achieves its maximum at around q = 0.92, and the optimal TSB value is 5.204 106. While, the smallest TSB value obtained during this enumeration is only 4.094 106. As shown in Fig. 5, when the value of q exceeds 2.10, the value of TSB stays at this smallest level. This is because when the toll charge of traveling in a pricing cordon is getting considerably high, no flow (in the Monte Carlo simulation) would be loaded to any link there. Hence, the pricing cordon is blocked from the transportation network. In such a case, the value for TSB is lower, due to the fact that: (a) there is no toll revenue; and (b) the drivers lose some options for route choice, such that their travel disutility increases. Another test is conducted for the case when there is no toll charge on the network, which can be directly solved by the CA algorithm. It shows that the value of TSB in this case is 5.145 106. However, Fig. 4 indicates that the values of TSB obtained
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Fig. 4. Structure of network C.
Table 2 Length of the links in the pricing cordon of network C. Link no. Length (m) Link no. Length (m)
33 790 40 38
34 1050 41 35
35 250 42 78
36 400 43 82
37 850 44 66
38 892 45 85
39 85 46 26
by most of the KM-based toll-charge functions are even lower than 5.145 106. This comparison confirms that when an arbitrary toll pattern is initially introduced to the transportation network, it may cause a reduction in total benefit/utility of the network. This phenomenon is consistent with the previous study of Wilson (1988) on the cordon-based congestion pricing scheme in Singapore. 6.1.2. Nonlinear distance-based charge The Hybrid GA–CA method can be used to calculate the optimal piecewise-linear approximation function for general nonlinear distance-based charge. Then, the network transformation is first performed, where links in the pricing cordon are replaced by some dummy links that connect each entry-exit pair. The resultant composite network is shown in Fig. 6. Since dmin and dmax for this example are found to be 405 and 2453 meters, the gap for the length-variety of all the internal paths is 2048 meters. As aforementioned, a range should be decided for the number of intervals, n, which is quite essential for the piecewise-linear approximation function. If n is too small, it would deteriorate the equity and efficiency of nonlinear distance-based charge, while when n is too large, it is computationally inhibitive to solve the optimal toll design problem. Thus, the upper and lower bound for n is properly decided to be 10 and 6, respectively. Namely, there are 5 scenarios with
TSB Value (Millions)
5.5
5
4.5
4
3.5
0.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
Slope Fig. 5. Sensitivity test for the slope of KM-based toll-charge function.
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the value of integer n changing from 6 to 10. Computational burden is in general increasing as the value of n increases, since larger n implies more genes in each chromosome and it subsequently increases the total number of mutations. of this numerical example. The stop criterion of GA, The Hybrid GA–CA method is then adopted for solving the optimal / n see Step 2.5 in Section 5, is taken as fixed number of generations, k = 50. Moreover, the total number of chromosomes (population) of one generation is chosen to be 200. Eventually, computational results of each scenario with different n are tabulated in Table 3. In Table 3, the first row of data is the maximal TSB value obtained by each scenario, and the second to fourth row is the value of each term in the expression R q ðsÞ P P for TSB, as in the right-hand-side of Eq. (27), namely, T1 ¼ w2W 0 w D1 weWqw(s)Sw(cw(v(s)), sw(/)), and w ðxÞdx, T2 = P P T3 ¼ e2E l2Re Eðf el ðsÞasel ð/ÞÞ. The fifth row entitled ‘‘Increase’’ is the improvement of nonlinear toll charge over KM charge in terms of the TSB obtained, recalling that the maximal TSB obtained by KM charge is 5.204 106. The sixth row shows the execution time for each scenario, yet this execution time may not be strictly proportional to the computational burden of each scenario. This is because the computation is performed on the distributed computing system, and each scenario with different n has different concurrency level affecting their parallelism. Most of the computational efforts are devoted onto evaluation of newly generated chromosome, thus the total number of chromosomes evaluated throughout the entire computation process could better reflect the total computational burden of each scenario, which is provided in the row six of Table 3. The last row of Table 3 is the efficiency of each scenario, denoted by xi, which is calculated by the following equations:
xi
TSBi TSB0 TSB TSB0
ð43Þ
;
where TSBi is the TSB value obtained for scenario i, TSB0 is the TSB value of un-tolled case, and TSB⁄ is the maximum TSB value obtained among all the scenarios; for this example, TSB⁄ is obtained by the scenario n = 10, herein, the TSB⁄ is an approximation of the system optimum for the nonlinear distance-based pricing. As per the data in Table 3, the optimum TSB value among all the scenarios is 5.877 106 when n = 10. Compared with the maximal TSB obtained by optimal KM charge, 5.204 106, the optimal nonlinear distance-based charge has a significant improvement, which is 12.94% as shown in Table 3. The optimal chromosome output from this scenario is depicted by Fig. 7, which has 10 intervals and 11 boundary values. Clearly, the optimal distance-based toll-charge function is a highly nonlinear one. 6.2. Orchard Road Network Example The proposed methodology is further tested in this sub-section via a realistic network example. As shown in Fig. 8, this example built from the Orchard Road Cordon of ERP system in Singapore has 33 nodes and 104 links, with a pricing cordon The cordon has 11 highlighted by the dashed ellipse. Totally there are 38 links in the cordon that are grouped into set A. entries and 11 exits connecting to the external network, which gives rise to 110 entry-exit pairs. As mentioned in Section 1, other than the cordon-based pricing schemes, the ERP system also has link-based tolls on some expressways and arterial
Fig. 6. Composite network for network C.
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Q. Meng et al. / Transportation Research Part E 48 (2012) 937–957 Table 3 Computational profile of different scenarios to network C.
a
Scenarios
n=6
n=7
n=8
n=9
n = 10
Max TSB T1a T2a T3a Increase CPU time (s) No. of total chromosomes Efficiency xi
55,41,560 69,16,080 27,00,789 13,26,270 6.48% 826 3369 54.12%
57,89,184 69,49,056 26,81,664 15,21,791 11.24% 868 3521 87.91%
57,28,330 69,39,036 26,87,571 14,76,864 10.07% 890 3628 79.60%
58,02,743 70,19,860 26,37,017 14,19,900 11.50% 906 3777 89.76%
58,77,805 69,64,026 26,72,666 15,86,445 12.94% 930 3961 100%
T1, T2, T3 are terms 1, 2, 3 in the right-hand-side of Eq. (27).
4.5
4.1
Toll Charge (S$)
4
3.5
3.5 3 2.5 2
1.6
1.5 1 0.5
1.1 0.5 0.5
0.5
0.5
0.6
0.6
0.8
0 0.41 0.61 0.81 1.02 1.22 1.43 1.63 1.84 2.04 2.25 2.45
Travel Distance Fig. 7. Optimal toll-charge function for network C.
Fig. 8. Network structure of Orchard Road example.
roads. For the future generation of ERP system, the distance-based toll charge method should only be adopted for the cordonbased tolls rather than link-based tolls. Data for the links are provided in Table 4. Travel time for this example is counted in seconds, and it would result in a comparatively larger magnitude to the satisfaction Sw(cw(v), s), thus the parameter c in Eq. (41) is decided to be 0.001 for
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Table 4 Link data for Orchard Road example. Link no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68
Tail node
Head node
Free-flow travel time t0a (s)
Link length
Capacity Ha (vehicles/h)
2 1 3 2 1 4 5 4 6 5 2 6 3 8 7 6 8 7 9 8 4 10 12 5 6 14 7 15 8 17 9 18 11 10 12 11 13 12 14 13 14 15 16 15 17 16 18 17 10 27 11 19 12 20 13 21 14 22 23 15 16 24 17 25 18 26 20 19
1 2 2 3 4 1 4 5 5 6 6 2 8 3 6 7 7 8 8 9 10 4 5 13 14 6 15 7 17 8 18 9 10 11 11 12 12 13 13 14 15 14 15 16 16 17 17 18 27 10 19 11 20 12 21 13 22 14 15 23 24 16 25 17 26 18 19 20
60 60 100 100 40 40 90 90 42 42 80 80 72 72 160 160 120 120 60 60 80 80 55 55 80 80 120 120 60 60 90 90 16 16 40 40 24 24 48 48 40 40 12 12 60 60 60 60 20 20 24 24 20 20 30 30 12 12 12 12 12 12 14 14 20 20 80 80
600 600 1000 1000 400 400 900 900 420 420 800 800 720 720 1600 1600 1200 1200 600 600 800 800 550 550 800 800 1200 1200 600 600 900 900 160 160 400 400 240 240 480 480 400 400 120 120 600 600 600 600 200 200 240 240 200 200 300 300 120 120 120 120 120 120 140 140 200 200 800 800
5400 5400 5400 5400 5400 5400 3600 3600 3600 3600 7200 7200 3600 3600 1800 1800 1800 1800 3600 3600 5400 5400 3600 3600 7200 7200 1800 1800 3600 3600 3600 3600 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 1800 1800 5400 5400 3600 3600 7200 7200 3600 3600 3600 3600 3600 3600 5400 5400 1800 1800
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Q. Meng et al. / Transportation Research Part E 48 (2012) 937–957 Table 4 (continued) Link no.
Tail node
Head node
69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104
20 21 22 21 23 22 24 23 25 24 26 25 19 28 20 29 22 30 23 31 24 32 26 33 28 27 29 28 30 29 31 30 32 31 33 32
21 20 21 22 22 23 23 24 24 25 25 26 28 19 29 20 30 22 31 23 32 24 33 26 27 28 28 29 29 30 30 31 31 32 32 33
Free-flow travel time t0a (s) 20 20 48 48 20 20 28 28 60 60 60 60 24 24 12 12 18 18 16 16 18 18 60 60 20 20 40 40 90 90 30 30 60 60 96 96
Link length 200 200 480 480 200 200 280 280 600 600 600 600 240 240 120 120 180 180 160 160 180 180 600 600 200 200 400 400 900 900 300 300 600 600 960 960
Capacity Ha (vehicles/h) 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 5400 1800 1800 5400 5400 7200 7200 3600 3600 5400 5400 5400 5400 5400 5400 5400 5400 3600 3600 3600 3600 3600 3600 5400 5400
this example. It should be noted that the length of each link in Table 4 may not be exactly proportional with the link length in Fig. 8, since the locations of some nodes in Fig. 8 have been justified for a clearer view. There are 12 hypothetical OD pairs, and the upper bound values for the OD demands are tabulated in Table 5. Other than the data for network attributes listed in Tables 4 and 5, other settings for this example are similar to those for the Network C above, including the upper and lower bound for the toll charge, range of the intervals n, numbers of generation and population for the Hybrid GA–CA method, as well as the stop criteria for each approach involved, etc. These settings are thus not repeated here. Similarly, the computation is handled in the first place for this example regarding the optimal KM charge as well as the un-tolled case, which would act as benchmarks to evaluate the performance of nonlinear distance-based toll charge. It turns out that for this example, the TSB value with respect to the un-tolled case is as large as 2.608 107. While, the TSB obtained by the optimal KM charge is 2.489 107, where the slope q = 2.01, which is inferior to that of the un-tolled case. It further indicates that the implementation of congestion pricing schemes may cause a loss on total social benefit. Albeit the value of TSB decreases when implementing the optimal KM charge, the traffic congestions in cordon area (usually in the CBD) would be improved in view that toll charges limit the toll traffic volume in the cordon area. In addition, we can speculate that if the lower bound of the slope q is set to be 0.0 rather than 0.5, the optimal TSB of KM charge will increase, since when q = 0.0 it becomes the un-tolled case, thus would have a TSB value at 2.608 107. We then proceed to calculate the optimal nonlinear distance-based charge for this example using the Hybrid GA–CA method. A prerequisite for this method is the network transformation for the cordon area shown in Fig. 8. The network transformation requires enumeration of all the simple (cycle-free) paths between each entry-exit pair. However, as addressed in Section 3.2, it might be a computationally impossible task to enumerate all the simple paths on a realistic network with large cordons, and the pre-generated path set or column generation should be used to only test partial paths. For illustration, we adopt the technique of pre-generated path set on this realistic network example: only the first k ‘‘shortest’’ paths between each entry-exit pair are taken into the path choice set, and the first k ‘‘shortest’’ paths problem is solved by the well-known algorithm proposed by Yen (1971). We note that other than using the k ‘‘shortest’’ paths, some other alternatives can also be found in the literature for the path set generation problem, see, Cascetta et al. (2002), Bovy and Catalano (2007). For this example, only the first six shortest simple paths are generated between each entry-exit pair based on the free flow travel
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Q. Meng et al. / Transportation Research Part E 48 (2012) 937–957 Table 5 Upper bound of OD demand for Orchard Road example. OD pair w
w Total travel demand q (vehicles/h)
1 ? 33 9?1 3 ? 27 27 ? 9 2 ? 29 18 ? 28 4 ? 24 32 ? 14 33 ? 3 25 ? 4 28 ? 6 7 ? 23
5000 4000 5000 5000 2000 2000 3000 1000 2000 1000 3000 2000
Table 6 Computational profile of different scenarios to Orchard Road example.
a
Scenarios
n=6
n=7
n=8
n=9
n = 10
Max TSB T1a T2a T3a Increase CPU time (s) No. of total chromosomes Efficiency xi
261,45,078 334,11,508 81,38,525 872,095 5.06% 8632 3458 71.83%
261,55,314 334,19,832 81,24,970 860,451 5.10% 8744 3550 83.13%
261,70,594 334,22,782 81,19,750 867,562 5.16% 8818 3682 100%
261,62,038 334,20,758 81,23,377 864,656 5.12% 8982 3783 90.56%
261,60,578 334,20,770 81,23,398 863,206 5.12% 9582 3939 88.94%
T1, T2, T3 are terms 1, 2, 3 in the right-hand-side of Eq. (27).
4.0
3.6
Toll Charge (S$)
3.5 3.0 2.4
2.5 2.0 1.5 1.0 0.5
0.5 0.5
0.0 0.12
0.58
0.5
1.04
0.6
1.5
0.7
0.8
1.96
2.42
1.0
2.88
3.34
3.8
Travel Distance Fig. 9. Optimal Toll-charge function for Orchard Road example.
time of each link. These paths in the path choice set are then taken for the network transformation, each of which is replaced by a dummy link. Subsequently, the Hybrid GA–CA method is used to compute the optimal nonlinear distance-based toll charge function for this Orchard Road network example. The resultant data are tabulated in Table 6. It shows that the third scenario with n = 8 gives the maximal TSB value of 2.617 107. Compared with the optimal KM charge, it has increased the TSB value for 5.16%, which is also a non-negligible improvement. As the number of intervals, n, increases, the total execution time and computational burden (No. of total chromosomes) are also enlarged. The computational results for this example show that a larger value of n may not give larger maximal TSB value. The impacts of n on maximal TSB value are, in most cases, unpredicted due to the considerable diversity of path lengths in the cordon. The optimal nonlinear toll-charge function in terms of n = 8 is indicated by Fig. 9. Likewise to Fig. 7, the optimal nonlinear toll-charge function is a continuous non-decreasing function. Longer journeys in the cordon are charged up to nine times higher than the shorter journeys, which is (a) more equal than the commonly adopted pay-per-entry and licensing basis charge and (b) also helpful to encourage the drivers to wisely select a shorter itinerary in the cordon to avoid potential congestions. When implemented in practice, the equity property of distance-based
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charge makes it prone to be accepted by the public users. However, the nonlinear distance-based toll-charge function is comparatively more complicated in view that it has many different intervals. This complexity may cause some difficulties to the drivers in precisely perceiving the toll charge on each path and then in wisely making their route plan. To cope with this problem, some empirical policies executed in the initialization stage of many congestion pricing schemes can be followed. For instance, as used for the ERP system in Singapore (Foo, 2000), the authorities can (a) establish the nonlinear distancebased charge on partial pricing cordons and/or selected time of day as a trial and (b) frequently launch some publicity campaign for propagation. Such policies would help the drivers/users to be aware of and familiar with the updated charging rules, which is quite positive for the user acceptance of nonlinear distance-based toll charge. 7. Conclusions 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 probitbased 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; 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, congestionbased 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. Acknowledgements The authors are grateful to the editor and two anonymous reviewers for their constructive comments and suggestions, which have largely improved the work presented in this paper. Appendix A A.1. Cost Averaging (CA) method The Cost Averaging (CA) method can be regarded as a variation of the well-known Method of Successive Average (MSA) (see, Chapter 12 of Sheffi, 1985). Proposed by Cantarella (1997), CA method is the only convergent algorithm in the literature for solving SUE problem with elastic demand and asymmetric link travel time functions. Procedures of CA method are summarized as follows:
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Step 0: (Initialization) Let an initial link flow pattern v(1) = 0 and calculate the link travel times t(1) = (ta(v(1)), a e A)T. Let the number of iterations k = 1. Step 1: (Stochastic network loading) Based on the link travel time pattern t(k) and toll charge pattern sw, w e W, perform a stochastic network loading that yields a new link flow pattern v(k+1). Step 2: (Direction searching) Based on link flow pattern v(k+1), calculate an auxiliary link travel time pattern ðkþ1Þ ðkþ1Þ yðkþ1Þ ¼ ðya ; a 2 AÞT using link travel time functions, i.e., ya ¼ ta ðv ðkþ1Þ Þ. Step 3: (Convergence test) If the following condition is fulfilled, then stop and output v(k+1). Otherwise go to Step 4.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ðkþ1Þ ðt aðkÞ ya Þ2 a2A
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ðtaðkÞ Þ2
6 e1 ;
a2A
where e1 is a predetermined positive tolerance. Step 4: (Update) Find new generalized link travel time pattern tðkþ1Þ ¼ ðtðkþ1Þ ; a 2 AÞT according to the formula: a
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