Discrete-time dynamic road congestion pricing under stochastic user optimal principle

Transportation Research Part E 131 (2019) 24–36

Contents lists available at ScienceDirect

Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

Discrete-time dynamic road congestion pricing under stochastic user optimal principle

T

Linghui Hana, Chengjuan Zhub, David Z.W. Wangc, , Huijun Sund, Zhijia Tana, Meng Menge ⁎

a

School of Maritime Economics and Management, Dalian Maritime University, Dalian, China School of Traffic & Transportation Engineering, Dalian Jiaotong University, Dalian, China c School of Civil & Environmental Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798, Singapore d Key Laboratory of Urban Transportation Complex Systems Theory and Technology, Ministry of Education, Beijing Jiaotong University, Beijing 100044, China e Faculty of Business, University of Greenwich, London SE10 9LS, UK b

ARTICLE INFO

ABSTRACT

Keywords: Road congestion pricing Traffic dynamic control Day-to-day traffic dynamic model Stochastic user equilibrium

Road pricing is believed to be an effective instrument for alleviating roadway congestion. Most existing road pricing schemes are developed based on traditional static traffic equilibrium models, in which a fixed toll can be obtained to support the corresponding traffic flow pattern as an equilibrium. However, static models cannot consider the evolution process of traffic flow caused by the day-to-day fluctuations of road users’ route choices. Under practical traffic conditions, the multiple traffic equilibria may exist (e.g., due to the asymmetric travel cost function). Indeed, the fixed road pricing scheme derived from the equilibrium model cannot guarantee that the dynamic traffic system can converge to the desired equilibrium state from any initial traffic state. This study, assuming that travelers follow the stochastic user optimal principle, develops a day-to-day dynamic road pricing scheme that can drive the traffic dynamic system to converge to a given stochastic user equilibrium (SUE) even when the traffic system has multiple SUE states. The characteristic of this dynamic road pricing scheme is verified by rigorous proof and numerical tests in this study.

1. Introduction 1.1. Motivation Road congestion is one of the most challenging problems faced by major cities. Many researchers believe that the enormous economic losses and environmental pollution caused by traffic congestion can be avoided in principle, as they mostly result from socially inefficient choices by individual drivers (Yang and Huang, 2005). Road congestion pricing, as an effective traffic management measure, significantly influences the road users’ routing choices and thus can drive the traffic assignment on the network towards the requirement of social optimum. Indeed, the suitable tolls charged on certain links can ensure that the limited road capacities are efficiently utilized (Pigou, 1924; Vickrey, 1969).

Corresponding author. E-mail addresses: [email protected] (L. Han), [email protected] (C. Zhu), [email protected] (D.Z.W. Wang), [email protected] (M. Meng). ⁎

https://doi.org/10.1016/j.tre.2019.09.009 Received 27 December 2018; Received in revised form 24 August 2019; Accepted 16 September 2019 1366-5545/ © 2019 Elsevier Ltd. All rights reserved.

Transportation Research Part E 131 (2019) 24–36

L. Han, et al.

Existing road congestion pricing schemes (the first-best and second-best pricing schemes) are mostly developed based on the static traffic equilibrium models with assumption that, by implementing the road pricing scheme, a corresponding static traffic equilibrium state can be achieved. In this study, we call this type of road congestion pricing scheme as equilibrium congestion pricing scheme. However, before an equilibrium traffic state is reached, the traffic network indeed involves a dynamic process, in which, road users make day-to-day route choices resulting in the daily fluctuations of traffic state. The fluctuations of users’ day-to-day route adjustment behavior are not explicitly considered in the equilibrium congestion pricing schemes. Indeed, an equilibrium congestion pricing scheme supports the desired traffic state as a traffic equilibrium. However, it may not guarantee that the desired traffic state will be achieved through the road users’ day-to-day routing adjustment process. To be more specific, when the traffic system has multiple equilibria (e.g., the asymmetric travel cost function may result in multiple equilibrium states), each equilibrium state has its own attraction domain (Bie and Lo, 2010; Han et al., 2011). If the initial traffic state does not fall into the attraction domain of the target equilibrium state, the traffic system cannot converge to the target equilibrium from the initial traffic state via the dynamic traffic state adjustment process. Therefore, to ensure the traffic system can be directed to converge to a desired target traffic equilibrium, it is imperative to develop dynamic congestion control approaches with consideration of traffic dynamics. Specifically, this study aims to develop a day-to-day dynamic road congestion pricing scheme based on the dayto-day traffic dynamic (DTD) model to drive the traffic dynamic system to converge to a given stochastic user equilibrium (SUE) even when the traffic system has multiple SUE states. 1.2. Literature review The congestion pricing schemes based on static traffic equilibrium can be classified into two groups: the first-best and the secondbest pricing schemes. The first-best pricing or the marginal cost pricing scheme on the road network is the theoretical basis of road congestion pricing. This pricing scheme entails that the road users should pay for road usage to internalize the congestion externality or additional cost they impose upon other users. By doing so, each road user will also bear the marginal social cost of road use other than the marginal private cost (Yang and Huang, 2005; Chen et al., 2018), so that the system optimum can be achieved. Besides, with more practical considerations, the second-best pricing scheme is applied to optimize the system performance under certain practical and institutional constraints (Yang and Huang, 2005; Lawphongpanich et al., 2006). Based on the static traffic equilibrium, researchers also develop other traffic management measures to improve congestion. These measures include the vehicle plate number-based traffic rationing currently implemented in Beijing, China (Han et al., 2010), tradable travel credit scheme (Yang and Wang, 2011; Nie, 2012), the congestion call option (Friesz et al., 2008), auction-based congestion pricing (Teodorović et al., 2008), the reservation-based management schemes(Yang et al., 2013; Liu et al., 2015), parking permits distribution and trading (Zhang et al., 2011), route guidance system (Jahn et al., 2005), traffic congestion management considering air pollution and travel distance (Tzeng and Chen, 1993), and multiple model traffic system (Poudel et al., 2019). Some researchers also believe that the increase in logistics demand deteriorates traffic congestion (Niu et al., 2019; Stefanello et al., 2017). They presented some measures to improve traffic congestion by logistic management. More details can be referred to Wu et al. (2019) and Hu et al. (2018). As compared to static traffic assignment model which only pays attention to the final traffic equilibrium, DTD model describes the day-to-day evolution of traffic state incurred by the road users’ travel choices adjustment on the day-to-day basis. It allows transportation planners and operational managers to evaluate the transportation network performance under disequilibrium and help them develop reliable traffic control measures. There is a large body of research on DTD models in the literature, and one can refer to Watling and Hazelton (2003) & Xiao et al. (2016) for more details. In contrast to the ample research on DTD models in the literature, the studies on DTD based dynamic road congestion pricing scheme are much less, and most of them are based on DTD models under deterministic user optimal principle. Some of them are based on continuous-time DTD models (Friesz et al., 2004; Tan et al., 2015). Guo et al. (2015) is the first study in the literature work on the discrete-time dynamic road pricing scheme. The convergence of their dynamic pricing scheme is verified based on the assumption of the uniqueness of the deterministic user equilibrium state. To address this issue, Han et al. (2017) presented a discrete-time dynamic road pricing scheme which can drive the traffic system to converge to a given target equilibrium from any initial traffic state when multiple equilibria exist. In addition, Rambha and Boyles (2016) proposed a discrete-time dynamic pricing scheme to reduce the expected total system travel time of the logit route choice model using an average cost Markov decision process model. Dynamic management measures are also studied in other areas, such as dynamic power tariff (Huang et al., 2018), dynamic congestion control in the communication channel (Zemouri et al., 2018), dynamic product development and launch (Sunar et al., 2019) and dynamic revenue management (Xia et al., 2019). While most of the existing research works developed the dynamic road pricing schemes by assuming the deterministic user optimal behavior principle, nevertheless, very few previous studies considered the stochastic user optimal principle as the road users’ behavior principle. Stochastic user optimal behavior principle captures and incorporates road users’ perception error. Under this behavior principle, the travel cost of each route is perceived differently by each road user because of variations in perception and exogenous factors. The perceived travel cost of each route is modeled as a random variable distributed across the population of road users. As the theoretical basis is the random utility theory, one can find all existing DTD models under stochastic user optimal behavior principle are path-based models. However, for practical application, the road congestion pricing scheme should be linkbased. Therefore, this study would focus on how to develop the link-based dynamic road pricing scheme based on the DTD model under stochastic user optimal behavior principle so as to drive the traffic system to converge to a target traffic state with given link flow pattern. Besides, based on the results of Smith et al. (1995), one can find that not all feasible traffic link flow patterns can be 25

Transportation Research Part E 131 (2019) 24–36

L. Han, et al.

supported as a SUE state by positive road pricing. Thus, this study would first discuss the characteristics of the target traffic link flow pattern that can be supported as a SUE by a road pricing scheme. Then, we would examine how to develop a specific link-based DTD dynamic road pricing scheme based on the DTD model of Watling (1999). Through implementing the link-based dynamic pricing scheme, the traffic system can converge to the given target traffic flow pattern. Besides, we also discuss some practical problems on how to implement the link-based dynamic pricing scheme in the real application. The organization of this paper is as follows: Section 2 shows the notations, the formulation of the applied DTD model following stochastic user optimal behavior principle, as well as the discussions on the target link flow pattern. Section 3 gives the formulation of the link-based dynamic road pricing scheme. Section 4 discusses some problems related to the real application of the dynamic road pricing scheme. In Section 5, numerical tests are conducted to verify the dynamic road pricing scheme. Finally, Section 6 presents the conclusions of this study and some future works. 2. Model description 2.1. Notation Let G (N , A) denote a general traffic network with a set N of nodes and a set A of directed links. W is the set of OD pairs. Rw is the set of feasible acyclic routes between OD pair w W , and R = w W Rw . M = w W R is the cardinality of route set R and assumed to be finite. d = (d w > 0, w W ) denotes the traffic demand vector, whose element d w ( > 0) is the travel demand between OD pair (t ) w W and fixed. Let frw ( 0) be the traffic flow on route r Rw and va(t ) ( 0) be the flow on link a A at time t. (t ) ( t ) f = (f rw , r Rw , w W ) and v (t ) = (va(t ), a A) denote the vectors of route flows and link flows, respectively. = ( aw, r , a A, r Rw , w W ) denotes the link-route incidence matrix, where aw, r = 1 if route r uses link a and 0 otherwise, i.e., (t ) (t ) w = ( rw , r Rw , w W ) be the OD-route incidence matrix, where rw = 1 if route r connects OD pair va = w W r Rw f rw a, r . Let w and 0 otherwise. Therefore, we have v (t ) = f (t ) and d = f (t ) . ca (v (t ) ) is the travel time cost on link a A at time t, which is a continuous and positive function of the link flow vector v (t ) , i.e., the link cost may be not separable. Cr (f (t ) ) is the travel time cost of path r Rw caused by traffic flow at time t. c (v (t ) ) = (ca (v (t ) ), a A) and C (f (t ) ) = T c (v (t ) ) are respectively the link travel time cost vector and the path travel time cost vector at time t. Let the set v be the feasible link flow and f be the feasible path flow: f

= {f d =

f, f

v

= {v v = f , f

(2.1)

0}

(2.2)

f}

This study assumes that the real travel cost of each road user at time t consists of two parts: the travel time cost depending on traffic flow pattern v (t ) at time t and the link road pricing (t ) = { a(t ), a A} charged by traffic management authorities at time t. Before departing from the trips origins on each day, users are assumed to have received the information on intraday link tolls.

c (v (t ) )

2.2. The day-to-day traffic dynamics under stochastic user optimal behavior principle In stochastic user optimal behavior principle, as was defined in Sheffi (1985), the road users’ perceived travel cost is comprised of two parts: a systematic or deterministic component (the mean perceived travel cost) and an additive random “error term”. Let (t ) CC (t ) = {CCrw , r Rw , w W } be the vector of the mean perceived path travel time cost depending on network traffic flow. According to the assumption on the travel cost in this study, the mean perceived path travel cost vector Ut can be expressed as:

U (t ) = CC (t ) +

T (t ).

In the day-to-day traffic dynamic process wherein a dynamic road pricing is implemented, after completing their trips at time t, road users experienced their real path travel time costs as C (f (t ) ) = T c (f (t ) ) . Then, through certain learning mechanism, road users update CC (t ) to acquire CC (t + 1) at time t + 1. In general, CC (t + 1) may be influenced by road users’ all real experienced travel costs at time t , t 1, …, 0. However, road users have limited memory and therefore, they only use a part of their experienced travel information to predict their travel time costs on the next time. Therefore, this study applies the learning mechanism as defined in Watling (1999) to update CC (t ) , i.e.,

CC (t + 1) = (1

(2.3)

) CC (t ) + C (f (t ) ),

(0, 1] implies the weight of the actual travel time cost in updating the mean perceived travel time cost incurred by the where network traffic flow pattern. Eq. (2.3) shows that CC (t + 1) on day t + 1 is a weighted average of the mean perceived travel time costs CC (t ) caused by traffic flow and their experienced real travel time costs C (f (t ) ) at time t. Combining the road tolls (t + 1) levied at time t + 1, the mean perceived path travel cost U (t + 1) on day t + 1 can be described as: U (t + 1) = CC (t + 1) +

(2.4)

T (t + 1).

Under stochastic user optimal behavior principle, f (t + 1) can be described by the probabilistic assignment model, i.e., (2.5)

f (t + 1) = dp (U (t + 1) ),

where p (U (t + 1) ) = {prw (U (t + 1) ), r Rw , w W } is the route choice probability vector, in which prw (U (t + 1) ) (r Rw , w W ) is the (t + 1) + rw is the minimum perceived travel cost between OD pair w W . The expression of prw (U (t + 1) ) is dependent probability that Urw 26

Transportation Research Part E 131 (2019) 24–36

L. Han, et al.

on the specific distribution of the random term rw . By the relation between path flow pattern f (t + 1) and link flow pattern v (t + 1) , v (t + 1) = (va(t + 1), a A) at time t + 1 can be calculated by the following equation:

va(t + 1) =

(t + 1) f rw

w ar

w Rw r Rw

d w pr (U (t + 1) )

=

w ar ,

a

A.

(2.6)

w Rw r Rw

From Eqs. (2.3)–(2.5), one can observe that

CC (t + 1)

) CC (t )

= (1

+

can also be described as:

CC (t + 1)

(2.7)

C (dp (U (t ) )).

Assuming that the road pricing vector is when the dynamic system in Eqs. (2.3)–(2.5) becomes stable. Then, f point of the dynamic system in Eqs. (2.3)–(2.5) under if and only if f satisfies the following condition:

CC = C ( dp (CC +

T

is a steady (2.8)

)).

where CC is the mean perceived travel time costs depending on flow f . This conclusion can be directly inferred from the conclusion of Watling (1999), i.e., under the road pricing , f is the fixed point solution of the dynamic system in Eqs. (2.3)–(2.5) if and only if

U = C ( dp (U )) +

T

.

Combining Eq. (2.4) and the above equation, one can find that

U = CC +

T

,

and

CC +

T

= C ( dp (CC +

T

)) +

T

,

Therefore, Eq. (2.8) is proved. From Eq. (2.8), one can find that, under a certain fixed road pricing, the dynamic traffic system in Eqs. (2.3)–(2.5) becomes stable if and only if road users’ mean perceived travel time costs incurred by traffic flow are equal to their real experienced travel time costs without the road pricing. With the assumption of users’ route choice behavior – stochastic user optimal behavior principle, one may ask whether any feasible traffic state desired by traffic management authorities can be achieved by the dynamic traffic system under a certain dynamic traffic control measure. In the next subsection, some characteristics of the target flow of this study desired by traffic management authorities are discussed. 2.3. The target traffic flow pattern Compared to path flow, the link flow pattern is much easier to be observed and to be measured in practice. Therefore, the target flow pattern desired by traffic management authorities is assumed to be a link flow pattern in this study. In addition, from the standpoint of traffic management authorities, an ideal target link flow pattern should fully exert the capacity of urban traffic network so that the traffic congestion is alleviated. Traffic management authorities also desire to implement a certain traffic management measure to support the target flow pattern as a traffic equilibrium flow pattern so that the traffic system, once reached, would stay at a stable state. That is, the target link flow pattern may be obtained by solving a certain traditional static traffic equilibrium model. For real-world applications, if the traffic management authorities charge tolls to influence road users’ route choices, the tolls should be positive rather than negative tolls or subsidy on some links. Indeed, as stated in Smith et al. (1995), for a given feasible link flow distribution, it is not true there must exist nonnegative link tolls under which the given link flow distribution can be supported as a SUE flow pattern. Therefore, this paper presents some assumptions upon the target link flow pattern v obj v. Definition 2.1. For any travel demand vector d = {d w > 0, w r f (r Rw ) is positive. where f f and the element f w

W } , a link flow pattern v is designated as fully positive if v = f ,

Definition 2.2. The link cost function ca (v) > 0 for any link a A is designated as regular if and only if ca (v) is continuously differentiable and nondecreasing in all its arguments (i.e., the gradient of ca satisfies ca (v) 0 ) Assumption 1. The target link flow pattern v obj

v

desired by traffic management authorities is fully positive in Definition 2.1.

Assumption 1 is very natural since traffic management authorities desire that paths in the urban traffic network should be fully used by road users so that no path flow is zero. Assumption 2. The link travel cost ca (v) (a

A ) is regular as in Definition 2.2.

Based on Assumption 1, 2 as well as Theorem 4.6 in Smith et al. (1995), one can observe that the target link flow pattern in this study is supportable by a nonnegative link toll obj as a SUE, i.e.,

vaobj =

d w pr c (v obj) +

obj

w ar ,

a

A.

(2.9)

w W r Rw

27

Transportation Research Part E 131 (2019) 24–36

L. Han, et al.

In principle, a candidate target traffic flow pattern should satisfy some realistic traffic network conditions, as well as traffic system indexes desired by traffic management authorities. It should be supported as a SUE flow pattern under a nonnegative road pricing so that traffic system following stochastic user optimal behavior principle can stay stable under the proposed dynamic road pricing scheme. In this study, the target traffic flow pattern and the toll scheme to support it as a SUE state are both given by traffic management authorities in prior. 3. The dynamic road pricing scheme 3.1. The formulation In general urban traffic network, there may exist multiple equilibrium states. From the studies of Bie and Lo (2010) and Watling (1999), each stable traffic equilibrium state x has itself attraction domain H (x ) . If the initial traffic state x (0) does not belong to the attraction domain H (x ) , the traffic system cannot converge to the traffic equilibrium state x without external forces according to the dynamic adjustment process. It should be noted that the target traffic state discussed in this study is a SUE state under the toll obj . Therefore, the attraction domain of the target traffic state x obj should be noted as H (x obj , obj ) in this study. For the convenience of illustration, the definition of attraction domain H (x obj , obj ) of the target traffic state x obj under the toll obj is presented as follows: Definition 3.1. The attraction domain H (x obj , under the toll obj , i.e.,

H x obj , where

obj

= x (0)

obj )

of the target state x obj is the collection of all states that will evolve toward x obj

, lim x (t ) = x obj , t

(3.1)

+

is the feasible domain of traffic state x .

One can observe that the target traffic state x obj in Definition 3.1 is not the desired traffic flow (link flow or path flow) pattern in this study. It is a representation which can describe the transient traffic states along the evolution process of the traffic dynamic system with the dynamic road pricing scheme. That is, before the traffic system becomes stable at time t, the traffic state x (t ) is not equal to the traffic state x obj corresponding to the target link flow pattern v obj . The specific formulation of the traffic state x (t ) (t = 0, 1…) will be given in Section 4. This study aims to develop a dynamic road pricing scheme to drive the traffic system to converge to the desired target link flow pattern under stochastic user optimal behavior principle from any initial traffic state even when multiple SUE exist. By Definition 3.1, one can observe that the dynamic road pricing scheme should be able to drive the traffic state to enter the attraction domain H (x obj , obj ) of x obj corresponding to the target link flow pattern v obj . In addition, within the attraction domain H (x obj , obj ) , the pricing can be fixed and equal to obj . Therefore, the framework of the dynamic road pricing scheme { (t + 1), t = 0, 1, …} can be described as: (t + 1) = (t + 1) , if x (t ) /H (x obj, obj ); (t + 1) = obj , otherwise. Theorem 3.1 presents the sufficient and necessary conditions for the convergence of traffic states under the dynamic road pricing scheme following above framework. Theorem 3.1. Under the dynamic road pricing scheme { (t + 1) } (t = 0, 1, …) wherein (t + 1) = obj for x (t ) H (x obj, obj ) , the sequence {x (t ) } (t = 0, 1, …) of traffic state of the dynamic system in Eqs. (2.3)–(2.5) can converge to x obj corresponding to the target link flow pattern v obj if and only if there is a positive integer N , for any t > N , x (t ) H (x obj, obj ) . Proof. For necessity: Because limt

x (t )

x obj

x (t ) = x obj . Given

> 0 , there is a positive integer N , for any t > N ,

<

Therefore, there is also a positive integer N, for any t > N , x (t ) H (x obj, obj ) . For sufficiency: Because there is a positive integer N, for any t > N , x (t ) H (x obj, obj ) . The traffic state x (t ) (t > N ) enters into the attraction domain H (x obj , obj ) . Then, (t + 1) = obj for t > N . By Definition 3.1 on the attraction domain H (x obj , obj ) , one can find that

lim x (t ) = x obj.

t

The proof is completed.

□

Next, we discuss how to develop the dynamic road pricing scheme based on Theorem 3.1. From Eq. (2.3), one can observe that, if the road toll (t + 1) at time t + 1 is able to drive v (t + 1) to reach v obj , then

CC (t + 2) = (1 where C (v obj) =

CC(t + 2)

) CC (t + 1) + Cobj

T c (v obj) .

The distance between CC (t + 2) and C (v obj) can be expressed as

C (v obj) = (1

) CC (t + 1)

C (v obj) .

Thus, CC(t + 2) C (v obj) is less than CC(t + 1) C (v obj) . Based on the above discussions, we can propose the following dynamic link-based road pricing scheme { (t + 1), t = 0,1 …} , in which the (t + 1) at time t + 1 (t = 0, 1, …) is expressed as follows: (t + 1)

=

T (t + 1), ifx (t ) obj, otherwise,

x /H (C (v

obj),

obj ),

(3.2) 28

Transportation Research Part E 131 (2019) 24–36

L. Han, et al.

0, v obj = dp (CC(t + 1) + T )} . where T (t + 1) = { Next, a specific road pricing (t + 1) (t = 0, 1, …) at time t + 1 which follows Eq. (3.2) is developed to verify that T (t + 1) is nonempty when the traffic state x (t ) at time t does not belong to the attraction domain H (x obj , obj ) of x obj corresponding to the target link flow pattern v obj . Before presenting the specific formulation of (t + 1) , we make one assumption on the traffic network G (N , A) as follows: Assumption 3. The traffic network has a full positive link flow pattern v (Definition 2.1), which is a deterministic user equilibrium flow distribution under a certain positive road toll , i.e.,

(c (v ) + )T (v

v)

0,

y

(3.3)

v.

Taking advantage of the full positive deterministic user equilibrium link flow pattern v , for x (t ) the following linear programming problem in Eq. (3.4) to obtain the toll (t + 1) charged at time t + 1.

x / H (C (v

obj),

obj ) ,

we design

min(CC(t + 1) )T f

(3.4a)

s. t .

(3.4b)

f

f

v

f rw = d w , w

W

(3.4c)

r Rw

frw

0, r

Rw , w

(3.4d)

W

wherein constraints (3.4c) and (3.4d) are respectively demand conservation and nonnegative constraints of path traffic flows. Constraint (3.4b) is introduced to obtain the partial toll µ(t + 1) of the toll (t + 1) at time t + 1 in Eq. (3.7). The partial toll µ(t + 1) can make each path having the same mean perceived travel cost caused by traffic flow, i.e., for any (t + 1) (t + 1) r l Rw , CCrw + a µa(t + 1) arw = CClw + a µa(t + 1) alw . It can be explained as follows: Let f be the optimal solution of the linear programming (3.5). Then f follows Karush–Kuhn–Tucker conditions as follows: (t + 1) CCrw + a A

(t + 1) CCrw +

µa(t + 1)

µa(t + 1)

w ar

w ar

(t + 1) w

(t + 1) , w

r

w

f r = 0, r

Rw , w

Rw , w

W,

(3.5a)

W,

(3.5b)

a A

w

f r = dw, w

W,

(3.5c)

r

µ(t + 1)

(3.5d)

T

(3.5e)

v ) µ(t + 1) = 0,

( f

f

0,

(3.5f)

0,

where (w W ) is the Lagrange multiplier of constraint (3.4c), fact, the constraint (3.4b) implies f = v . Because, if link a A follows (t + 1) w

µ(t + 1)

is the Lagrange multiplier vector of constraint (3.4b). In f w < va in constraint (3.4b), then it can be r Rw rw ar

w W

v ) < 0 , which contradicts with the characteristic of v in Eq. (3.3). Thus, the path flow vector following derived that (c (v ) + )T ( f constraint (3.4b) is the one which can produce link flow pattern v . Moreover, from Assumption 3, the path flow pattern f > 0 . Therefore, Eqs. (3.5a) and (3.5b) can be rewritten as: (t + 1) CCrw +

µa(t + 1)

w ar

=

(t + 1) . w

(3.6)

a A

By the linear programming in Eq. (3.4), we present a formulation of time t + 1 can be defined as follows: (t + 1)

= µ(t + 1) + c (v obj) +

(t + 1)

when

x (t )

/H (C (v obj),

obj ) .

The road pricing

(t + 1)

at

(3.7)

obj,

where µ(t + 1) is the Lagrange multiplier vector of constraint (3.4b), and c (v obj) + obj is the link travel cost vector corresponding to the target link flow pattern v obj given by traffic management authorities. Then, the following conclusion can be obtained: Proposition 3.1. Based on Assumption 3, when the traffic state x (t ) does not belong to the attraction domain H (C (v obj), obj ) of the traffic state x obj corresponding to the target link flow pattern v obj at time t , under the link-based road toll (t + 1) in Eq. (3.7), the link flow pattern v (t + 1) at time t + 1 (t = 0, 1, …) is equal to the target link flow pattern v obj . Proof. Under the road pricing can be expressed as follows:

(t + 1)

(t + 1) defined in Eq. (3.7), the mean perceived travel cost Urw of path r

29

Rw (w

W ) at time t + 1

Transportation Research Part E 131 (2019) 24–36

L. Han, et al.

(t + 1) (t + 1) Urw = CCrw + a A

µa(t + 1)

w ar

ca (v obj) +

+

obj a

w ar .

a A

(t + 1) From Eq. (3.6), Uwr can be rewritten as: (t + 1) Urw =

obj a

ca (v obj) +

w ar

+

(t + 1) , w

a A obj = Urw +

(t + 1) . w

(3.8)

By the probability assignment model, the choice probability pr (U (t + 1) ) of route r follows: (t + 1) pr (U (t + 1) ) = p {Urw +

=

obj p {Urw

+

rw

rw

>

(t + 1) > Ulw + obj Ulw

+

lw ,

lw ,

l

l

r, l r, l

Rw , w

Rw , w

Rw (w

W ) at time t + 1 is calculated as

W}

W}

= pr (Uobj). Therefore,

v (t + 1) = dp (U (t + 1) ) = dp (Uobj) = v obj. The proof is completed. □ The convergence on the dynamic system in Eqs. (2.3)–(2.5) under the dynamic link-based road pricing scheme in Eq. (3.2) is presented in the following theorem. Theorem 3.2. The dynamic link-based road pricing scheme in Eq. (3.2) can drive the traffic dynamic system under stochastic user optimal behavior principle in Eqs. (2.3)–(2.5) to converge to the target link flow distribution v obj . Proof. When the traffic state x (t ) at time t (t = 0, 1, …) does not belong to the attraction domain H (x obj , obj ) of the traffic state x obj corresponding to the target link flow patter v obj , the link-based dynamic road pricing scheme in Eq. (3.2) can ensure the link flow pattern v (t + 1) = v obj . Therefore, for any time t > 0 , the link flow pattern v t is the target link pattern v obj if the traffic state x (t 1) /H (C (v obj), obj ) at time t 1, and

CC (t + 1) = (1

) CC (t ) +

Supposing the traffic state

CC(t + 1)

As

= (1

) CC(t )

(3.9)

T c (v obj).

x (t ) +

/H (C (v obj),

obj )

(t = 0, 1, …), one can obtain the following equation on

CC (n)

from Eq. (3.9),

T c obj

= (1

)2CC(t

1)

+ [ (1

)+ ]

= (1

)3CC (t

2)

+ [(1

)3 + (1

= (1

)t CC(1) + [1

(1

)t ]

T c (v obj)

)2 + 1]

T c (v obj)

(3.10)

T c (v obj).

(0, 1], n

lim CC(t + 1) = +

T c (v obj).

(3.11)

In this study, we use to represent the traffic state at time t (t = 0, 1, 2, …). Thus, Eq. (3.11) implies that the traffic state represented by CC (t + 1) finally enters the attraction domain H (x obj , obj ) , i.e, there is a time t > 0 , for any t > t , CC(t + 1) H (x obj, obj ) . Then, by the dynamic pricing scheme in Eq. (3.2), the road pricing (t + 1) = obj (t > t ). Therefore, under the dynamic pricing scheme in Eq. (3.2), the mean perceived travel cost U (t + 1) = CC (t + 1) + T (t + 1) converges to T c (v obj) + obj . Eventually, the target link flow pattern v obj will become the fixed point of the dynamic system in Eqs. (2.3)–(2.5) under the road pricing obj . The proof is completed.

CC (t )

Although Theorem 3.2 has verified the convergence of the dynamic system in Eqs. (2.3)–(2.5) under the link-based dynamic road pricing scheme in Eq. (3.2), there are still some problems needed to be further addressed. For example, why the traffic state x (t ) (t = 0, 1, …) is represented by CC (t ) , and how to determine the attraction domain H (C (v obj), obj ) of the traffic state x obj corresponding to the target link flow pattern v obj . These problems are discussed in the following section. 4. Some discussions on the representation of traffic state and attraction domain In this section, we firstly discuss on the specific representation of the traffic state x (t ) at time t (t = 0, 1, …). Ideally, the traffic state x (t ) should possess the following characteristics: (i) The daily fluctuations of the traffic state x (t ) can reflect the day-to-day variations of the road users’ route choices under stochastic user optimal behavior principle. 30

Transportation Research Part E 131 (2019) 24–36

L. Han, et al.

(ii) Traffic state x (t ) is not equal to the traffic state x obj corresponding to the target link flow pattern v obj when traffic dynamic system has not stabilized under the dynamic pricing scheme. From Eq. (2.8), one can find that, under a fixed road pricing, the traffic state x is a stable state if and only if the mean perceived path travel time cost vector CC depending on network traffic flow is equal to the real path travel time cost vector C (x ) , which indicates that road users’ mean perceived travel time costs determine whether the traffic dynamic system is stable or not (condition ii). Therefore, it is intuitive to choose the mean perceived travel time cost vector CC (t ) incurred by traffic flow to represent the travel state at time t. From Eqs. (2.5) and (2.6), the link flow pattern v (t ) is determined by CC (t ) and the road pricing (t ) ; thus, CC (t ) also satisfies the condition i. Indeed, the link flow vector v (t ) is not proper for describing traffic state at time t in this study. The reasons are as follows: from the dynamic pricing scheme in Eq. (3.2), one can find that the toll (t + 1) can make v (t + 1) = v obj at time t + 1 (t = 0, 1, …) when CC (t + 1) does not belong to the attraction domain H (C (v obj), obj ) . That is, the link flow v (t + 1) (t = 0, 1, …) can be equal to the objective link flow pattern v obj whether the traffic dynamic system becomes stable or not. Therefore, one cannot judge whether the traffic dynamic system becomes stable from the variance of the link flow under the dynamic road pricing scheme. Meanwhile, one can observe that, before CC (t ) entering the attraction domain H (C (v obj), obj ) , the link flow v (t + 1) is invariant so that the path travel cost vector C(t + 1) is also invariant. Therefore, despite that Bie and Lo (2010) used the path cost vector C(t ) at time t to represent the traffic state X(t ) , it is not appropriate to apply it for traffic state representation in this study.. As mentioned above, this study chooses the mean path perceived travel time cost vector CC (t ) incurred by traffic flow to represent the traffic state at time t (t = 0, 1, …). However, it should be noted that the feasible domain of CC (t ) may vary with time t. So, there are some difficulties in determining the feasible domain of CC (t ) though users’ perceived travel costs can be estimated by the results from survey interview. In real application, such defect on CC (t ) can be omitted. By Definition (3.1), in the process of implementing dynamic road pricing scheme, one can find that if CC(t ) is sufficiently close to C (v obj) at time t , then fixing the road pricing, i.e., (t ) = obj for any time t t , can direct CC (t ) to converge to C (v obj) . So, it is not even necessary to know the feasible domain of CC (t ) (t = 0, 1, …). In Bie and Lo (2010), some topological properties on the attraction domain H (C (v obj), obj ) are presented. For example, the attraction domain of a stable equilibrium is always open, and its boundary is formed by trajectories towards unstable equilibria. However, these results may not be sufficient in determining the exact boundary of the attraction domain H (C (v obj), obj ) . Fortunately, we do not need to know the exact H (C (v obj), obj ) in this study. From the definition on attraction domain in Eq. (3.1), one can observe that, from any mean perceived travel time cost vector CC (t ) caused by traffic flow belonging to a neighborhood H (C (v obj), ) H (C (v obj), obj ) , the fixed road pricing obj can eventually drive CC (t ) to converge to C (v obj) . Therefore, we only need to find a proper to obtain a neighborhood H (C (v obj), ) for implementing the link-based dynamic road pricing scheme, where H (C (v obj), ) = {CC CC C (v obj) 2 } . A very simple method to obtain is presented as follows: step 1: Given initial value of n > 0 , parameter 0 < < 1, let iterative number n = 0 ; step 2: Randomly produce N (500 or larger positive integer) vector , in which its each element belongs to set ( 1, 1) ; , judge whether it can converge to C (v obj) under the fixed road pricing obj step 3: For each random vector , let CC = C (v obj) + by the traffic dynamic model in Eqs. (2.3)–(2.5). step 4: If, for N random vector , all CC = C (v obj) + can converge to C (v obj) , then stop; otherwise, let n+ 1 = n , n = n + 1. Go to step 2. 5. Numerical tests This section presents some numerical results through a simple example traffic network shown in Fig. 1. The traffic network consists of 8 nodes and 10 links. There are two OD pairs: OD (1, 8) and OD (2, 8) . Between each OD pair, there are 4 paths, which are shown in Table 1 and denoted as the sequence of nodes. The traffic demands of these two OD pairs are both 4 units. To validate the dynamic road pricing scheme of this study when multiple SUE exist, the link cost functions of the example network in Fig. 1 are asymmetrical. Their specific formulations are given as follows:

Fig. 1. The example traffic network.

31

Transportation Research Part E 131 (2019) 24–36

L. Han, et al.

Table 1 The paths between OD (1,8) and (2,8). Paths between OD (1,8)

Paths between OD (2,8)

1 1 1 1

2 2 2 2

3 3 4 4

5 6 7 8

9 9 10 10

3 3 4 4

5 6 7 8

9 9 10 10

c1 = v1 + 1; c2 = v2 + 1; c3 = v3 + 0.5; c4 = v4 + 0.5; c5 = v5 + 3v6 + 0.5: c6 = 2v5 + v6 + 0.5; c7 = v7 + 3v8 + 0.5: c8 = 2v7 + v8 + 0.5; c9 = v9 + 0.5; c10 = v10 + 0.5;

(5.1)

For simplicity, the probability assignment model in our numerical tests is assumed to be a logit model, i.e.,

pwr (U (t + 1) ) =

exp( exp

(t + 1) ) Uwr

,

(t + 1) Uwl

(5.2)

l Rw

where the parameter with relation to road users’ perceived error is assumed to be = 0.7 in the following numerical tests. The parameter used to determine the neighborhood H (C (v obj), ) is = 0.5. The parameter in Eq. (2.3) to update the mean path travel cost vector CC (t ) caused by traffic flow is set to be = 0.2 . For = 0.7 , link flow vectors

v I = (4.00, 4.00, 4.88, 3.12, 0.01, 4.87, 0.04, 3.08, 4.88, 3.12) and

v II = (4.00, 4.00, 4.79, 3.21, 4.50, 0.29, 0.04, 3.17, 4.79, 3.21) are both SUE link flow patterns under the link-based toll

= (0, 0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5). The corresponding path travel cost vectors are respectively:

C (v I ) = (25.50, 15.76, 22.53, 16.41, 25.50, 15.76, 22.53, 16.41) and

C (v II ) = (16.19, 20.12, 22.91, 16.63, 16.19, 20.12, 22.91, 16.63) First, we let v I = (4, 4, 4.88, 3.12, 0.01, 4.87, 0.04, 3.08, 4.88, 3.12) be the objective SUE link flow pattern. The specific dynamic pricing scheme applied is the one in Eq. (3.7). Fig. 2 illustrates the evolution process of the link flows with the initial link flow vector v (0) I = (4.00, 4.00, 4.69, 3.31, 2.31, 2.38, 1.57, 1.74, 4.69, 3.31) . The fluctuations of the mean perceived path travel costs caused by traffic flow of paths between OD (1,8) are depicted in Fig. 3. The evolutions of the link-based dynamic tolls are shown in Fig. 4 corresponding to the same initial link flow. Fig. 5 displays the trajectories of the perceived path travel costs of path 3 and path 4 between OD (1,8) for 300 random feasible initial link flow vectors. Fig. 6 depicts the evolution process of link flows with the initial link flow vector v (0) when the objective SUE link flow pattern is II = (4.00, 4.00, 1.71, 6.29, 1.00, 0.71, 1.48, 4.80, 1.71, 6.29) v II = (4.00, 4.00, 4.79, 3.21, 4.50, 0.29, 0.04, 3.17, 4.79, 3.21) . Fig. 7 shows the corresponding evolutions of the mean perceived path travel costs. In Fig. 8, the evolutions of the tolls are shown corresponding to the same initial link flow pattern. Fig. 9 illustrates the trajectories of the mean perceived path costs for 300 random feasible initial link flow vectors. From Proposition 3.1, one can find that, when the traffic state CC (t ) does not belong to the attraction domain of the target traffic state, the link flow v (t + 1) should be equal to the target link flow pattern v obj (t = 0, 1, …) under the dynamic road pricing scheme in Eq. (3.7). After CC (t ) enters the attraction domain, the road toll is fixed and set to be obj . This dynamic variances of the road toll may cause the fluctuations of the link flows, which can be observed in Fig. 2 and 6. Fig. 3 and 7 show that the mean perceived path travel cost finally converges to the real path travel cost of the target link flow pattern. The evolutions of the tolls in Figs. 4 and 8 as well as the trajectories of CC (t ) for 300 random initial link flow can further illustrate the characteristic process of evolution of the mean perceived path travel cost under the specific dynamic pricing scheme in Eq. (3.7). That is, the traffic dynamic system under stochastic user optimal behavior principle can converge to the target link flow pattern under the dynamic road pricing scheme of this study. 32

Transportation Research Part E 131 (2019) 24–36

L. Han, et al. 6

5

Flow

4

3 link 3 link 4 link 5 link 6 link 7 link 8

2

1

0

0

5

10

15

20

25

30

35

40

Day I Fig. 2. The evolution process of link flow with initial link flow v (0) I when the objective equilibrium is v .

26

Perceived Cost

24

22 path 1 path 2 path 3 path 4

20

18

16

14

0

5

10

15

20

25

30

35

40

Day

Fig. 3. The fluctuations of the mean perceived path travel costs caused by traffic flow evolution process with initial link flow v (0) I when the objective equilibrium is v I .

60 50

Toll

40

link 1 link 2 link 3 link 4 link 5 link 6 link 7 link 8 link 9 link 10

30 20 10 0 0

5

10

15

20

Day

25

30

35

40

I Fig. 4. The corresponding toll evolution process with initial link flow v (0) I when the objective equilibrium is v .

33

Transportation Research Part E 131 (2019) 24–36

L. Han, et al. 18.5

18

path 4

17.5

17

16.5

16 20

20.5

21

21.5

22

22.5

23

path 3

Fig. 5. The trajectories of the mean perceived path travel costs caused by traffic flow for random 300 initial link flow when the objective equilibrium is v I .

7 link 3

link 4

link 5

link 6

link 7

link 8

link 9

link 10

6 5

Flow

4 3 2 1 0 0

5

10

15

20

25

30

35

40

Day II Fig. 6. The evolution process of link flow with initial link flow v (0) II when the objective equilibrium is v .

40

path 1 path 2 path 3 path 4

Perceived Cost

35

30

25

20

15

10

0

5

10

15

20

25

30

35

40

Day

Fig. 7. The fluctuations of the mean perceived path travel costs caused by traffic flow evolution process with initial link flow v (0) II when the objective equilibrium is v II .

34

Transportation Research Part E 131 (2019) 24–36

L. Han, et al. 70 60

link 1 link 2 link 3 link 4 link 5 link 6 link 7 link 8 link 9 link 10

50

Toll

40 30 20 10 0 0

5

10

15

20

25

30

35

40

Day II Fig. 8. The corresponding toll evolution process with initial link flow v (0) II when the objective equilibrium is v .

28

26

path 4

24

22

20

18

16 22

24

26

28

30

32

34

36

path 3

Fig. 9. The trajectories of the mean perceived path travel costs caused by traffic flow for random 300 initial link flow when the objective equilibrium is v II .

6. Conclusions This study discusses how to develop the dynamic road pricing scheme to drive traffic system, wherein multiple SUE states exist, to converge to a given desired SUE state when road users follow stochastic user optimal behavior principle for routing choices. Under stochastic user optimal behavior principle, road users make daily route choices according to their perceived travel costs which are different from their real experienced travel costs as road users have perception errors. This study applies the users’ mean perceived travel time costs, instead of traffic flow pattern or the real travel time costs, to represent the traffic state in the process of implementing the dynamic pricing scheme. Through rigorous proof and numerical tests, one can find the dynamic road pricing scheme proposed in this study is able to drive the traffic system with multiple SUE states to converge to the target SUE state when road users make route choice following stochastic user optimal behavior principle. Dynamic traffic management is the key instrument for relieving traffic congestion and driving the traffic system to converge to a traffic state desired by traffic management authorities. This study, as an early attempt in this research topic, only applies the road pricing as the traffic management measure. In the future works, we would consider other dynamic traffic control measures such as the dynamic signal control scheme, the dynamic speed limit control scheme as well as the responsive transit service in the multimodal transportation system (Zhang et al., 2018; Wang et al., 2017). Acknowledgments Dr. Wang was supported by the Singapore Ministry of Education Academic Research Fund Tier 2 MOE2015-T2-2-076. Dr. Sun was supported by the National Natural Science Foundation of China (71771018). Dr. Tan was supported by the National Natural Science 35

Transportation Research Part E 131 (2019) 24–36

L. Han, et al.

Foundation of China (71871037, 71831002). Dr. Han was supported by the program for innovative research team of Ministry of Education of China (IRT-17R13) and the Fundamental Research Funds for the Central Universities (3132019501, 3132019502). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.tre.2019.09.009. References Bie, J., Lo, H.K., 2010. Stability and attraction domains of traffic equilibria in a day-to-day dynamical system formulation. Transport. Res. Part B: Methodol. 44, 90–107. Chen, D., Ignatius, J., Sun, D., Goh, M., Zhan, S., 2018. Impact of congestion pricing schemes on emissions and temporal shift of freight transport. Transport. Res. Part E: Logist. Transport. Rev. 118, 77–105. Friesz, T.L., Bernstein, D., Kydes, N., 2004. Dynamic congestion pricing in disequilibrium. Networks Spatial Econ. 4, 181–202. Friesz, T.L., Mookherjee, R., Yao, T., 2008. Securitizing congestion: the congestion call option. Transport. Res. Part B: Methodol. 42, 407–437. Guo, R.Y., Yang, H., Huang, H.J., Tan, Z., 2015. Day-to-day flow dynamics and congestion control. Transport. Sci. Han, D., Yang, H., Wang, X., 2010. Efficiency of the plate-number-based traffic rationing in general networks. Transport. Res. Part E: Logist. Transport. Rev. 46, 1095–1110. Han, L., Sun, H., Wu, J., Zhu, C., 2011. Day-to-day evolution of the traffic network with advanced traveler information system. Chaos, Solitons Fractals 44, 914–919. Han, L., Wang, D.Z., Lo, H.K., Zhu, C., Cai, X., 2017. Discrete-time day-to-day dynamic congestion pricing scheme considering multiple equilibria. Transport. Res. Part B: Methodol. 104, 1–16. Hu, L., Zhu, J.X., Wang, Y., Lee, L.H., 2018. Joint design of fleet size, hub locations, and hub capacities for third-party logistics networks with road congestion constraints. Transport. Res. Part E: Logist. Transport. Rev. 118, 568–588. Huang, S., Wu, Q., Shahidehpour, M., Liu, Z., 2018. Dynamic power tariff for congestion management in distribution networks. IEEE Trans. Smart Grid PP, 1. Jahn, O., Möhring, R.H., Schulz, A.S., Stier-Moses, N.E., 2005. System-optimal routing of traffic flows with user constraints in networks with congestion. Oper. Res. 53, 600–616. Lawphongpanich, S., Hearn, D., Smith, M.J., 2006. Mathematical and Computational Models for Congestion Charging, vol. 101 Springer Science & Business Media. Liu, W., Yang, H., Yin, Y., 2015. Efficiency of a highway use reservation system for morning commute. Transport. Res. Part C: Emerg. Technol. 56, 293–308. Nie, Y.M., 2012. Transaction costs and tradable mobility credits. Transport. Res. Part B: Methodol. 46, 189–203. Niu, B., Mu, Z., Li, B., 2019. O2O results in traffic congestion reduction and sustainability improvement: Analysis of “online-to-store” channel and uniform pricing strategy. Transport. Res. Part E: Logist. Transport. Rev. 122, 481–505. Pigou, A.C., 1924. The Economics of Welfare. Transaction Publishers. Poudel, S.R., Quddus, M.A., Marufuzzaman, M., Bian, L., et al., 2019. Managing congestion in a multi-modal transportation network under biomass supply uncertainty. Ann. Oper. Res. 273, 739–781. Rambha, T., Boyles, S.D., 2016. Dynamic pricing in discrete time stochastic day-to-day route choice models. Transport. Res. Part B: Methodol. Sheffi, Y., 1985. Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Prentice-Hall, Englewood Cliffs, NJ. Smith, T.E., Eriksson, E.A., Lindberg, P.O., 1995. Existence of optimal tolls under conditions of stochastic user-equilibria. In: Road Pricing: Theory, Empirical Assessment and Policy. Springer, pp. 65–87. Stefanello, F., Buriol, L.S., Hirsch, M.J., Pardalos, P.M., Querido, T., Resende, M.G., Ritt, M., 2017. On the minimization of traffic congestion in road networks with tolls. Ann. Oper. Res. 249, 119–139. Sunar, N., Birge, J.R., Vitavasiri, S., 2019. Optimal dynamic product development and launch for a network of customers. Oper. Res. Tan, Z., Yang, H., Guo, R.Y., 2015. Dynamic congestion pricing with day-to-day flow evolution and user heterogeneity. Transport. Res. Part C: Emerg. Technol. 61, 87–105. Teodorović, D., Triantis, K., Edara, P., Zhao, Y., Mladenović, S., 2008. Auction-based congestion pricing. Transport. Plann. Technol. 31, 399–416. Tzeng, G.H., Chen, C.H., 1993. Multiobjective decision making for traffic assignment. IEEE Trans. Eng. Manage. 40, 180–187. Vickrey, W.S., 1969. Congestion theory and transport investment. Am. Econ. Rev. 251–260. Wang, W.W., Wang, D.Z., Zhang, F., Sun, H., Zhang, W., Wu, J., 2017. Overcoming the downs-thomson paradox by transit subsidy policies. Transport. Res. Part A: Policy Practice 95, 126–147. Watling, D., 1999. Stability of the stochastic equilibrium assignment problem: a dynamical systems approach. Transport. Res. Part B: Methodol. 33, 281–312. Watling, D., Hazelton, M.L., 2003. The dynamics and equilibria of day-to-day assignment models. Networks Spatial Econ. 3, 349–370. Wu, X., Nie, L., Xu, M., Zhao, L., 2019. Distribution planning problem for a high-speed rail catering service considering time-varying demands and pedestrian congestion: A lot-sizing-based model and decomposition algorithm. Transport. Res. Part E: Logist. Transport. Review 123, 61–89. Xia, Y., Yang, J., Zhou, T., 2019. Revenue management under randomly evolving economic conditions. Naval Res. Logist. (NRL) 66, 73–89. Xiao, F., Yang, H., Ye, H., 2016. Physics of day-to-day network flow dynamics. Transport. Res. Part B: Methodol. 86, 86–103. Yang, H., Huang, H.J., 2005. Mathematical and economic theory of road pricing. Yang, H., Liu, W., Wang, X., Zhang, X., 2013. On the morning commute problem with bottleneck congestion and parking space constraints. Transport. Res. Part B: Methodol. 58, 106–118. Yang, H., Wang, X., 2011. Managing network mobility with tradable credits. Transport. Res. Part B: Methodol. 45, 580–594. Zemouri, S., Djahel, S., Murphy, J., 2018. An altruistic prediction-based congestion control for strict beaconing requirements in urban vanets. IEEE Trans. Syst., Man, Cybernet.: Syst. Zhang, F., Zheng, N., Yang, H., Geroliminis, N., 2018. A systematic analysis of multimodal transport systems with road space distribution and responsive bus service. Transport. Res. Part C: Emerg. Technol. 96, 208–230. Zhang, X., Yang, H., Huang, H.J., 2011. Improving travel efficiency by parking permits distribution and trading. Transport. Res. Part B: Methodol. 45, 1018–1034.

36