A Pareto-Improving Hybrid Policy with Multiclass Network Equilibria

1 A Pareto-Improving Hybrid Policy with Multiclass Network Equilibria 2

3 4

Submitted by 5 6

Zhaoming Chu, Ph.D. Candidate (Corresponding Author) 7 School of Transportation, Southeast University 8

2 Sipailou, Nanjing, 210096, P.R. China 9 Tel: +86-13912984298 10

Email: [email protected] 11 12

Hui Chen, Lecturer 13 Department of Civil Engineering, Chengxian College, Southeast University 14

6 Dongda Road, Nanjing, 210088, P.R. China, 15 Tel: +86-13921439462 16

Email: [email protected] 17 18

Lin Cheng, Ph.D., Professor 19 School of Transportation, Southeast University 20

2 Sipailou, Nanjing, 210096, P.R. China 21 Tel: +86-13951716936 22 Email: [email protected] 23

24 Senlai Zhu, Ph.D., Candidate 25

School of Transportation, Southeast University 26 2 Sipailou, Nanjing, 210096, P.R. China 27

Tel: +86-15850602236 28 Email: [email protected] 29

30 AND 31

32 Chao Sun, Ph.D., Candidate 33

School of Transportation, Southeast University 34 2 Sipailou, Nanjing, 210096, P.R. China 35

Tel: +86-13276613357 36 Email: [email protected] 37

38 39 40

Total number of words: 5988 (text) + 1500 (5 tables, 1 figures) = 7,488 words 41 42

Submission Date: November 15, 2015 43 44

Submitted for Presentation at the 95rd Annual Meeting of Transportation Research Board and 45 Publication in Transportation Research Record: Journal of Transportation Research Board46

TRB 2016 Annual Meeting Original paper submittal - not revised by author.TRB 2016 Annual Meeting Original paper submittal - not revised by author.

mailto:[email protected]





Chu, Chen, Cheng, Zhu and Sun 2

ABSTRACT 1 This paper extends the recent work of Song et al. (2014) on Pareto-improving hybrid rationing 2 and pricing policy in general road networks by considering heterogeneous users with different 3 value of time (VOT). Mathematical programming models are proposed for finding a multiclass 4 Pareto-improving pure road space rationing schemes (MPI-PR) and multiclass hybrid rationing 5 and pricing schemes (MHPI and MHPI-S). A numerical example with a 9-nine node multimodal 6 network is provided for comparing both the efficiency and equity of the three proposed policies. 7 We discover that the MHPI-S scheme can bring the most total system delay reduction, the MPI-8 PR scheme can induce less inequitable than the two hybrid policies, and the MHPI policy is a 9 progressive policy which is appealing to policy makers. The class-specific link flow patterns 10 reveal that different classes of users react differently to the same hybrid policies. In general, low 11 VOT users tend to use transit mode in rationing days, while high VOT users prefer to use car 12 mode in restricted days. In addition, the numerical results also show that multiclass Pareto-13 improving hybrid schemes yield less delay reduction when compared to its single-class 14 counterparts. 15 16 Keywords: Pareto-improving, hybrid policy, rationing, congestion pricing, heterogeneous uses 17 18



1. INTRODUCTION 1 Congestion pricing has been advocated as an effective demand management strategy to reduce 2 traffic congestion and improve system performance, including environment effects, since it was 3 first proposed nearly a century ago (1). Although it has been gaining more support among 4 politicians, transportation officials, and those in legislatures and the media, getting the public to 5 accept congestion pricing is still a major obstacle (2). Hau point out that the marginal cost 6 pricing scheme, which is ubiquitous in the transportation literature, is “most likely doomed to be 7 political failure”, because it makes users worse off compared to the situations without pricing 8 (3). 9

To make congestion pricing more appealing to the public, the concept of Pareto 10 improvement was introduced to congestion pricing schemes in recent years. Pareto improving 11 congestion pricing schemes increase the social welfare without making any user worse off when 12 compared to the situation without any pricing intervention. Usually, Pareto improving congestion 13 pricing schemes can be achieved through a revenue refunding scheme (4-9). 14

Besides toll revenue redistribution, Lawphongpanich and Yin proposed a Pareto-15 improving congestion pricing scheme that leads a general transportation network to Pareto 16 improvement over the status quo (10). The existence of a nonnegative Pareto-improving toll 17 scheme relies on a fact that the original Wardropian user equilibrium (UE) flow distribution may 18 not be strongly Pareto optimal, which means the UE flow distribution may be dominated by 19 another distribution. It is possible to design a charging scheme that evolves flow distributions to 20 the one that is dominating in order to achieve a Pareto improvement. Song et al. (11) proposed 21 anonymous nonnegative Pareto-improving charging schemes with multiple user classes. Wu et al 22 (12) extended the Pareto-improving congestion pricing model to multimodal transportation 23 networks. 24

Computational experiments in Lawphongpanich and Yin (10) and Song et al. (11) suggest 25 that Pareto-improving tolls are relatively prevalent; however, they may not lead to significant 26 level of improvement. In order to improve the effectiveness of Pareto-improving congestion 27 pricing schemes, researchers turn to combining the congestion pricing policy with other demand 28 management instruments, especially the road space rationing strategy. Road space rationing has 29 been used to mitigate congestion and reduce air pollution in many cities worldwide, such as 30 Athens, Mexico City, Sao Paulo, Beijing and Guangzhou (13). The first hybrid rationing and 31 pricing scheme was proposed by Daganzo (14) to control traffic flow through a bottleneck. He 32 showed that the hybrid scheme can benefit everyone even without revenue redistribution. Later, 33 Daganzo and Garcia (15) extended the hybrid policy to cope with time-dependent bottleneck 34 congestion. They also showed that certain hybrid policy has the potential to achieve Pareto 35 improvement even if not returning the revenue to travelers. Liu et al. (16) presented a simple 36 spatial equilibrium model for a linear monocentric city to investigate the effects of rationing and 37 pricing on morning commuters’ travel cost and modal choice behavior in each location. Song et 38 al. (17) proposed a hybrid policy that integrates congestion pricing and road space rationing, 39 which extends the hybrid strategy proposed by Daganzo (14) from bottleneck level to general 40 congested transportation networks. 41

In the study of Song et al. (17), all road users are assumed homogeneous and have a 42 single (average) value of time (VOT). However, the value of time is not constant across 43 travelers: it varies by trip purpose (e.g., work versus non-work trips), by sociodemographics, and 44 so on, even for the same traveler at different times and locations (18). This is known as user 45 heterogeneity. Heterogeneity has profound implications for the procedures used to predict users’ 46



path, mode, and departure time choices (19). Because the VOT establishes a connection between 1 time and money, it plays a critical role in enabling the assignment models and more generally the 2 demand analysis tools, to provide policy makers with useful information for assessing the 3 economic and welfare impacts of proposed pricing-based schemes. Ignoring heterogeneity by 4 using a constant VOT is fundamentally incorrect and would lead to highly biased results (20). 5

In order to account for the effect of user heterogeneity on hybrid rationing and pricing 6 policies, this paper extended the work of Song et al. (2014) to the case with a discrete set of user 7 classes. The primary objective of this paper is to establish mathematical frameworks of designing 8 an optimal multiclass pure road space rationing policy and optimal multiclass hybrid policies for 9 a general network and compare both the efficiency and equity of these multiclass policies, as 10 well as compare the multiclass Pareto improving policies with their single class counterparts. In 11 particular, optimizations models are presented that determine the rationing ratio, anonymous 12 Pareto-improving toll vector, and the class-specific link flows. 13

The remainder of this paper is structured as follows. Section 2 formulates a mathematical 14 program to design an optimal multiclass Pareto-improving pure road space rationing scheme. 15 Section 3 formulates a mathematical program to design an optimal multiclass Pareto-improving 16 hybrid policy. Section 4 compares the efficiency and equity of different multiclass Pareto 17 improving policies and compares the multiclass hybrid policies with their single class 18 counterparts on a 9-node multimodal transportation network. The last section concludes the 19 paper and discusses extensions to this research. 20 21 2. A MULTICLASS PARETO-IMPROVING PURE ROAD SPACE RATIONING 22 POLICY 23 In this section, a multiclass Pareto-improving pure rationing is mathematically defined and the 24 problem of finding such a scheme as a mathematically programs with complementarity 25 constraints (MPCC) is formulated. To highlight key ideas, it is assumed that the travel demand 26 for every origin-destination (O-D) pair is fixed. However, travelers have the flexibility to choose 27 different transportation modes. 28 29 2.1 Preliminaries 30 Similar to many previous studies (21-23), this study represents user heterogeneity by a discrete 31 set of VOTs, and the users are classified accordingly into multiple classes. For each user class m, 32 let ,w md and mβ denote the travel demand between O-D pair w and the corresponding VOT, 33 respectively. The problem setting of the multiclass Pareto-improving pure road space rationing is 34 the same with the single-class Pareto-improving pure road space rationing policy, proposed by 35 Song et al. (17). Below is the list of sets, subscripts, parameters and variables used in this paper. 36 37

N Set of nodes L Set of directed links W Set of OD pairs K Set of user groups M Set of user classes A Node-arc incidence matrix of the network A Node-arc incidence matrix of the transit network



nL Set of links in the road network

tL Set of links in the transit network

gL Set of regular links in the road network

rL Set of restricted links in the road network

ijv Aggregate traffic flow on link ( ),i j mijv Class-specific traffic flow on link ( ),i j

( )ij ijt v Travel time function for link ( ),i j ,w md Number of passengers of user class m between OD pair w W∈ , ,w k m

ijx Link flow of user group k , user class m on link ( ),i j α Rationing ratio

wtl Direct transit link connecting the origin and destination of OD pairs w W∈

wA Set of road and transit links for users between OD pair w W∈ wE An input-output vector wρ A vector of Lagrange multipliers

,1,w mr

Φ Set of links for regular users on path r ( )d w Destination node of OD pair w ( )o w Origin node of OD pair w

,UEw mC Equilibrium travel cost for user class m before the policy implementation

, ,UEw k mC Equilibrium travel cost for user group k class m between OD pair w

ijτ Toll imposed on link ( , )i j

ijs Transit subsidy for transit users on link ( , )i j 0ijt Free-flow travel time (in minute) traversing link ( , )i j

ijb Capacity (in 100-vehicle) of link ( , )i j VOT value of time UE User Equilibrium OD Origin-Destination KKT Karush-Kuhn-Tucker MPCC Mathematical program with complementarity constraints MFCQ Magasarian-Fromovitz constraint qualification

1 In the presence of multiple user classes with different VOT, the system travel disutility 2

can be measured either in time unit (time-based disutility or total system travel time) or in cost or 3 monetary unit (cost-based disutility or total system cost). Absolutely, both time-based and cost-4 based system disutilities can be regarded as weighted sums of the travel times of all users in the 5 network. The former has a uniform weighting factor equal to unity, while the latter has non-6 uniform weighting factors equal to the VOT of respective user classes (7). From an economic 7 viewpoint, cost-based disutility is a more appropriate system disutility measure when users have 8



different VOT. However, in transportation context, time-based disutility has long been accepted 1 as a standard index of system performance. Therefore, in this study, we adopt time-based 2 disutility in all objective functions. 3

In order to avoid computationally troublesome path enumeration, all the models in this 4 paper are formulated in link-based forms, which can be effectively solved using commercial 5 optimization software. Moreover, in this study, we assumed all link travel costs are positive and 6 separable to avoid a cycle flow in equilibrium solutions, as Newell (25) and Patriksson (26) have 7 proved that a cycle flow cannot be present in an equilibrium solution when travel costs are 8 positive and separable. 9 10 2.2. Multiclass User Equilibrium under a pure road space rationing Policy 11 Based on the model P1 proposed by Song et al (17), the time-based multiclass UE problem under 12 a pure road space rationing policy (MUE-PR) can be formulated as: 13 MUE-PR:

0( , ) ( , ) ( , )min ( )ij

n t

v

ij ij ijx v i j L i j Lt d t vω ω

∈ ∈

+∑ ∑∫ 14

s.t. ( ),1, ,1 ,w w m w w mA x E d w mα = − ∀ (1) 15

, 2, , ,w m w w mAx E d w mα = ∀ (2) 16 ( ), , , , , ,w k m

ij ijw k mv x i j L w k m = ∀ ∈ ∑ ∑ ∑ (3) 17

( ), , 0 , , , ,w k mijx i j L w k m ≥ ∀ ∈ (4) 18

where constraints (1) and (2) describe flow balance constraints for each class of regular and 19 restricted users, respectively. wE is an input-output vector, that is, a vector, with exactly two 20 non-zero components, that specifies the origin and destination of the OD pair w. The component 21 of wE corresponding to the origin has a value of 1, and the one corresponding to the destination 22 has a value of -1. Constraint (2) states that each class of restricted users can only access the 23 transit network, A , which implies that , 2, 0w m

ijx = for all ( ), ni j L ∈ . Constraint (3) defines the 24 aggregate link flow ijv . Constraints (1)-(4) describe the feasible flow region of a multiclass 25 multimodal network. Observe that objective function of MUE-PR is strictly convex in aggregate 26 link flow ijv for monotonically increasing link travel time function ( )ij ijt v , but linear in class-27

specific link flow mijv and , ,w k m

ijx . Thus, under a given pure rationing policy, the equilibrium link 28 flow by user class m

ijv and , ,w k mijx are generally not unique, while the aggregate UE link flow ijv 29

is unique. 30 In order to demonstrate that Model MUE-PR is equivalent to the UE under a pure road 31

space rationing policy, the Karush-Kuhn-Tucker (KKT) of the program, which is both necessary 32 and sufficient, can be stated as follows, 33 ( ) ( ),1, ,1, 0 , , ,m w m w m

ij ij i j nt v i j L w mβ ρ ρ + − ≥ ∀ ∈ (5) 34

( ) ( ),1, ,1, ,1, 0 , , ,w m m w m w mij ij ij i j nx t v i j L w mβ ρ ρ + − = ∀ ∈ (6) 35

( ),1 ,1, 0 , , ,m w m w m wij i j tt i j l w mβ ρ ρ + − ≥ = ∀ ， (7) 36

( ),1, ,1, ,1, 0 , , ,w m m w m w m wij ij i j tx t i j l w mβ ρ ρ + − = = ∀ (8) 37



( ), 2, , 2, 0 , , ,m w m w m wij i j tt i j l w mβ ρ ρ + − ≥ = ∀ (9) 1

( ), 2, , 2, , 2, 0 , , ,w m m w m w m wij ij i j tx t i j l w mβ ρ ρ + − = = ∀ (10) 2

and (1)-(4), where wρ is a vector of Lagrange multipliers associated with the flow balance 3 constraints (1) and (2), which are also known as node potentials for OD pair w (24). 4 For regular user (user group k=1), there are two pairs of complementary constraints, that is, (5)-5 (8), involved because they are allowed to use all links in set wA . For class m users, when a link 6 ( ),i j is utilized, that is, ,1, 0w m

ijx > , constraint (6) forces the equation 7

( ) ,1, ,1, 0m w m w mij ij i jt vβ ρ ρ + − = to hold. Similarly, if ,1, 0w m

ijx > for regular class m users on the 8

transit link, then constraint (8) ensures ,1, ,1, 0m w m w mij i jtβ ρ ρ + − = . Summing up these two sets of 9

equations for each link along a utilized path together yields 10 11 ( )( ) ( ) ( ) ( ),1, ,1,

,1, ,1, ,1, ,1,, ,w m w m

r r

m w m w m w m w mij j i d w o wi j i j

tβ ρ ρ ρ ρ

∈Φ ∈Φ= − = −∑ ∑ 12

where, for OD pair w and class m users, ,1,w mr

Φ is a set containing links that are available to 13 regular users on path r and ( )d w and ( )o w denote the destination and origin nodes of OD pair 14

w. ( )ijt represents both road link and transit link travel times. Thus, constraints (6) and (8) 15

imply that the generalized travel cost of every utilized path equals ( ) ( ),1, ,1,w m w m

d w o wρ ρ − for regular 16

class m users between OD pair w. When a link ( ),i j is not utilized by class m users, that is, 17 ,1, 0w m

ijx = , constraints (5) and (6) imply that the inequality ( ) ,1, ,1, 0m w m w mij ij i jt vβ ρ ρ + − ≥ holds. 18

Similarly, if regular class m users are not using the transit link, that is, ,1, 0w mijx = , then constraints 19

(7) and (8) ensures ,1, ,1, 0m w m w mij i jtβ ρ ρ + − ≥ . Adding these two inequalities up along a non-20

utilized path yields ( )( ) ( ) ( ),1,,1, ,1,

, w mr

m w m w mij d w o wi j

tβ ρ ρ

∈Φ≥ −∑ . Therefore, we have the UE conditions 21

for each class of regular users. 22 For restricted users (user group k=2), because they are only allowed to access transit 23

links, one pair of complementarity constraints, that is, (9) and (10), applies. When a link ( ),i j is 24

utilized by class m users, that is, , 2, 0w mijx > , constraint (10) forces that , , , , 0m w m w m

ij i jtβ ρ ρ 2 2 + − = . 25 According to the preliminary, there is only one direct link connecting the origin and destination 26 of the OD pair w, and no transit path consists of multiple transit links. There, for restricted class 27 m users between OD pairs w, ( ) ( )

, 2, , 2,m w m w mij d w o wtβ ρ ρ = − . When a link ( ),i j is not utilized, that is 28

, 2, 0w mijx = , constraints (9) and (10) imply that the inequality , 2, , 2, 0m w m w m

ij i jtβ ρ ρ + − ≥ holds and 29

that ( ) ( ), 2, , 2,m w m w m

ij d w o wtβ ρ ρ ≥ − . Let ( ) ( ), , , ,

, ,UE w k m w k mw k m d w o wC ρ ρ = − ; then , ,

UEw k mC is the equilibrium travel cost 30

for users of group k and class m between OD pairs w. Thus, the UE conditions for each class of 31 restricted users are established. 32

It is noted that, in the bi-mode transportation network, the equilibrium conditions hold for 33 both road and transit users. Specifically, if regular class m users use both road and transit modes 34 for a given OD pair, w, then in the multiclass UE, travel cost for that OD pair will be the same 35 for both modes. Based on the above analysis, we prove that the KKT conditions of the 36



optimization problem MUE-PR are equivalent to the multiclass UE conditions under pure road 1 space rationing schemes. 2 3 2.3. Multiclass Pareto-improving pure road space rationing problem 4 On top of model P2 of Song et al. (17) and model MUE-PR, the multiclass Pareto-improving 5 pure road space rationing problem (MPI-PR) can be formulated as a mathematical program with 6 complementary constraints (MPCC). 7 MPI-PR:

( )( )

( ), , , ,min ij ij ijx v i j L

t v vρ α

∈∑ 8

s.t. (1)-(4) and (5)-(10) 9 ( ) ( ) ( ) ( ) ( )

,1, ,1, , 2, , 2,,1 ,w m w m w m w m UE

w md w o w d w o w C w mα ρ ρ α ρ ρ

− − + − ≤ ∀ (11) 10

where ,UEw mC is the equilibrium travel cost of class m users between OD pair w before the policy 11

implementation, that is, status quo. The objective function minimizes the total system travel cost. 12 Constraints (1)-(10) are the multiclass UE conditions under a pure road space rationing policy. 13 Each class of users may encounter different travel times depending on whether they are restricted 14 on a particular day. Constraint (11) guarantees that when averaged across rationing and regular 15 days, no users are made worse off compared with the status quo. Clearly, the MPI-PR problem 16 always has a feasible solution that is do-nothing solution, which consists of zero rationing ratio 17 value and the multiclass UE under the status quo. If the optimal objective function value is 18 strictly less than the total system travel cost under the status quo, we can conclude that a 19 multiclass Pareto-improving pure road space rationing (MPI-PR) scheme exists. 20

As formulated, the problem of MPI-PR is an MPCC, a class of problems difficult to solve 21 for mainly two reasons. One is because MPCC violates the Magasarian-Fromovitz constraint 22 qualification (MFCQ), and the other is due to the fact that the feasible region if non-convex (27, 23 28). An optimization with non-convex feasible region generally contains many local optimal 24 solutions and typically requires a time consuming branch-and-bound scheme to search for a 25 globally optimal solution. When an optimization problem violates MFCQ, one of the weaker 26 constraint qualifications, the KKT conditions may not hold and consequently, cannot be used to 27 verify whether a solution is optimal to the problem (10). 28 29 3. MULTICLASS PARETO IMPROVING HYBRID POLICIES 30 This section describes the mathematical formulation of the proposed Multi-class Pareto-31 improving Rationing and Pricing Hybrid Policy Problem (MHPI). The formulation presented 32 here also extends the model of P3, P4, and P5 in the paper of Song et al (17) by a discrete set of 33 user classes instead of one single class of users. Note that toll differentiation across user classes 34 is unrealistic and difficult to implement in reality, because users differ from one another in VOT 35 only, which is observationally indistinguishable (21). Therefore, in this section, we only consider 36 congestion pricing schemes with anonymous link flows, which means the same amount of toll is 37 levied on each link for all user classes. In addition, we assume all the link-based tolls and 38 subsidies are nonnegative. 39 40 3.1. Multiclass User Equilibrium under Hybrid Rationing and Pricing Policy 41 The time-based multiclass UE flow distribution in the presence of a hybrid rationing and pricing 42 policy (MUE-HRP) can be formulated by the following mathematical program. 43



MUE-HRP: ( )

, 2,

0( , ) ( , ) ( , ) ,min ( )ij

n t r

v w mij ij ij ij ijx v i j L i j L i j L w W m M

t d t v xω ω τ

∈ ∈ ∈ ∈ ∈

+ +∑ ∑ ∑ ∑ ∑∫ 1

s.t. ( ),1, ,1w w m w w mA x E d wα = − ∀ (12) 2 , 2, ,w m w w mAx E d wα = ∀ (13) 3 , ,w k m

ij ijw k mv x = ∑ ∑ ∑ (14) 4

( ), , 0 , , , ,w k mijx i j L w k m ≥ ∀ ∈ (15) 5

where constraints (12) and (13) describe flow balance constraints for regular and restricted users, 6 respectively. For restricted users, they have to pay nonnegative anonymous toll ijτ if they choose 7 to use the restricted links ( ), ri j L ∈ . 8 Constraints (12)-(15) describe the feasible flow region of a multiclass multimodal network, 9 which is convex. Also note that the objective function of MUE-HRP is strictly convex in 10 aggregate link flow ijv for monotonically increasing link travel time function ( )ij ijt v , but linear 11

in class-specific link flow mijv and , ,w k m

ijx . Thus, under a hybrid rationing and pricing policy, the 12 equilibrium link flow by user class m

ijv and , ,w k mijx are generally not unique, while the aggregate 13

UE link flow ijv is unique. In order to illustrate that the MUE-HPR problem is equivalent to the 14 multiclass UE under a hybrid rationing and pricing policy, the KKT conditions of the MUE-HRP 15 can be expressed as: 16 ( ) ( ), , , , 0 , , , ,m w k m w k m

ij ij i j gt v i j L w k mβ ρ ρ + − ≥ ∀ ∈ (16) 17

( ) ( ), , , , , , 0 , , , ,w k m m w k m w k mij ij ij i j gx t v i j L w k mβ ρ ρ + − ≥ ∀ ∈ (17) 18

( ), , , , 0 , , , ,m w k m w k m wij i j tt i j L w k mβ ρ ρ + − ≥ = ∀ (18) 19

( ), , , , , , 0 , , , ,w k m m w k m w k m wij ij i j tx t i j L w k mβ ρ ρ + − = = ∀ (19) 20

( ) ( ),1, , , 0 , , ,m w m w mij ij i j rt v i j L w mβ ρ ρ 1 + − ≥ ∀ ∈ (20) 21

( ) ( ), , ,1, , , 0 , , ,w k m m w m w mij ij ij i j rx t v i j L w mβ ρ ρ 1 + − ≥ ∀ ∈ (21) 22

( ) ( ), , , , 0 , , ,m w m w mij ij ij i j rt v i j L w mβ τ ρ ρ 2 2 + + − ≥ ∀ ∈ (22) 23

( ) ( ), , , 2, , , 0 , , ,w k m m w m w mij ij ij ij i j rx t v i j L w mβ τ ρ ρ 2 + + − ≥ ∀ ∈ (23) 24

and (12)-(15), where wρ is a vector of Lagrange multipliers (node potentials) associated with the 25 flow balance constraints (12) and (13). Definitely, the KKT conditions of MUE-HRP contain 26 four pairs of complementary constraints (16)-(23). Carrying out the same procedure in the MUE-27 PR problem, it can be easily proved that the above KKT conditions are equivalent to the 28 multiclass UE conditions under a hybrid rationing and pricing policy. For each class of regular 29 users (user group k=1), as they are free to access all links in set wA , UE conditions can be 30 established using complementarity constraints (16)-(21). While for each class of restricted users 31 (user group k=2), as they have to pay congestion tolls to access the restricted links; tolled UE 32 conditions can be obtained from complementary constrains (16)-(19), (22) and (23). 33 34 3.2. Multiclass Pareto-Improving Hybrid Rationing and Pricing Policy 35



Based on the formulation of MPI-PR and MUE-HRP, the multiclass Pareto-Improving hybrid 1 rationing and pricing problem (MHPI) can be formulated without difficult. In the MHPI 2 problem, the restriction ratio α , the set of restricted links, and their corresponding nonnegative 3 anonymous toll rates τ for restricted users have to be specified to minimize the time-based total 4 system delay. Following the same logic of Model P4 in Song et al. (17), we formulate the MHPI 5 problem as an MPCC in the following: 6 7 MHPI:

( )( )

( ) ( ), , , , , ,min

n t

ij ij ij ij ijx v i j L i j Lt v v t v

ρ α τ∈ ∈

+∑ ∑ 8

s.t. ( ) ( ),1, ,1, 0 , , ,m w m w mij ij i j nt v i j L w mβ ρ ρ + − ≥ ∀ ∈ (24) 9

( ) ( ),1, ,1, ,1, 0 , , ,w m m w m w mij ij ij i j nx t v i j L w mβ ρ ρ + − ≥ ∀ ∈ (25) 10

( ) ( ), 2, , 2, 0 , , ,m w m w mij ij ij i j nt v i j L w mβ τ ρ ρ + + − ≥ ∀ ∈ (26) 11

( ) ( ), 2, , 2, , 2, 0 , , ,w m m w m w mij ij ij ij i j nx t v i j L w mβ τ ρ ρ + + − = ∀ ∈ (27) 12

( ), , , , 0 , , , ,m w k m w k m wij i j tt i j l w k mβ ρ ρ + − ≥ = ∀ (28) 13

( ), , , , , , 0 , , , ,w k m m w k m w k m wij ij i j tx t i j l w k mβ ρ ρ + − = = ∀ (29) 14

( ) ( ) ( ) ( ) ( ),1, ,1, , 2, , 2, UE1 , ,w m w m w m w m

m wd w o w d w o w C w mα ρ ρ α ρ ρ β − − + − ≤ ∀ (30) 15

( )0 ,ij ni j Lτ ≥ ∀ ∈ (31) 16 and (12)-(15). 17 18

Constraints (24)-(29) are 3 pair of complementary conditions, which treat all road links as 19 potential restricted links. If the toll rate for restricted users is strictly positive on link ( ),i j , the 20 link belongs to rL ; otherwise, ( ), gi j L ∈ . Constraint (30) is the Pareto-improving conditions, 21 which means when averaged across restricted days and free days; no user is made worse off 22 compared to the status quo. When the optimal objective value of MHPI is strictly less than that 23 under the status quo, the solution vector is a multiclass Pareto-improving hybrid scheme. 24 Although, the existence of a strictly Pareto-improving solution cannot be guaranteed, we can 25 conclude that the MHPI problem has at least one feasible solution. Because if setting a 26 sufficiently large toll on all road links for restricted users so that they cannot afford using the 27 road network, the MHPI problem degraded to MPI-PR problem, and we have proved that the 28 MPI-PR problem always has a feasible solution in the last section. 29 30 3.3 The Multiclass Hybrid Pareto-Improving Policy with Transit Subsidy 31 The above multiclass hybrid Pareto improvement is attained without a toll revenue redistribution 32 procedure. In the literature, toll revenue redistribution has been proved an effective way to 33 achieve Pareto improvement in theory. This enlightens us to incorporate a certain revenue 34 redistribution plan in the MHPI scheme, so as to examine whether more substantial improvement 35 can be made. Similar to Song et al. (17), we adopt a transit subsidy plan that directly uses the toll 36 revenue to subsidize each class of transit users by reducing their transit fares. We formulated the 37 multiclass hybrid policy with transit subsidy (MHPI-S) problem as the following MPCC, 38 39



MHPI-S: ( )

( )( ) ( ), , , , , , ,

minn t

ij ij ij ij ijx v s i j L i j Lt v v t v

ρ α τ∈ ∈

+∑ ∑ 1

s.t. (12)-(15), (24)-(27), (30) and (31) 2 ( ) ( ), , , , 0, , , , ,m w k m w k m w

ij ij i j tt s i j l w k mβ ρ ρ − + − ≥ = ∀ (32) 3

( ) ( ), , , , , , 0, , , , ,w k m m w k m w k m wij ij ij i j tx t s i j l w k mβ ρ ρ − + − = = ∀ (33) 4

( ) ( )

, 2,

, ,t n

w mij ij ij ij

i j L i j L w W m Ms v xτ

∈ ∈ ∈ ∈

≤∑ ∑ ∑ ∑ (34) 5

( )0, ,ij ts i j L≥ ∀ ∈ (35) 6 where ijs denotes an anonymous nonnegative subsidy, that is, a reduction of transit fare, to each 7

class transit users using link ( ),i j . Constraint (31) and (35) require both tolls and subsidies to 8 be nonnegative. Nonnegative anonymous tolls, ijτ , are levied on each class of restricted users 9 who choose to use restricted links, whereas subsidies, ijs , are provided to both regular and 10 restricted users using the transit links. To guarantee the whole system can be self-sustained, we 11 use constraint (34) to force the total subsidy to transit users must less than or equal to the total 12 toll revenue collected. 13 14 4. NUMERICAL EXAMPLES 15 To explore the existence and properties of MPI-PR and MHPI, we use the same example as Song 16 et al. (17), as shown in Figure 1. To be self-contained, here we restate the characteristics of this 17 example. The 9-node multi-mode network contains four OD pairs [(1, 3), (1, 4), (2, 3) and (2, 18 4)], 18 auto-links and 4 transit links. The aggregate demand of each OD pair (in 100 19 passengers/hour) is also shown in Figure 1. The link performance functions associated with auto-20 links are assumed to follow Bureau of Public Roads (BPR) function, 21

( ) ( )40 1 0.15 /ij ij ij ij ijt v t v b = + , where 0

ijt is the free-flow travel time (in minutes) and ijb is the 22

capacity (in 100 vehicles) of link ( ),i j . The parenthesis near auto-link ( ),i j in the figure 23

denotes ( )0,ij ijt b . Four transit links (dotted lines) directly connect the origin and destination 24

nodes of each OD pair. Because the travel cost on transit link ( ),i j is represented by its 25 generalized travel cost, it is reasonable to assume ijt to be higher than the equilibrium travel cost 26 (time) on the road network for the same OD pair. The original network-related settings were 27 kept, two classes of users, m1 and m2, were introduced with VOTs of 0.6 $/minute and 1.4 28 $/minute, respectively. A uniform 50% and 50% split was assumed between the low- and high-29 VOT users for each O-D pair. 30

For MPI-PR, MHPI and MHPI-S are MPCC, we revise the manifold suboptimization 31 algorithm proposed by Lawphongpanich and Yin (10) to find strongly stationary solutions. The 32 algorithm is implemented on GAMS (29). Nonlinear subproblems involved are solved by the 33 solver of CONOPT (30). Due to the limitation of software license, some computations are 34 conducted on the NEOS Server (31). According to the above analysis, the optimal solution of 35 MPI-PR is a feasible solution of MHPI. Therefore, in order to solve MHPI more efficiently, we 36 use the resulting flow distribution of MPI-PR as an initial solution of MHPI problem. Having 37



solved all the models proposed in this paper, it is found that all the problems tested are solved 1 very efficiently. In fact, the longest elapsed time reported by GAMS is within 1 second. 2 3

1

2

5

6

7

8

3

4

9

(5, 12)

(1, 3)

(2, 4)

(2, 3)

(2, 11) (3, 25)

(9, 35) (6, 33) (6, 43)

(6, 18)

(3, 35)

(9, 20)

(8, 26)

(7, 32)

(4, 11)

(1, 4)

(4, 26)

(8, 30)

(2, 19) (4, 36)

(6, 24)

(8, 39)

OD pair: (1,3) (1,4) (2,3) (2,4)Demand: 30 30 40 50

4 FIGURE 1 The network layout for demonstrating example. 5

6 We assume that the travel cost of any given transit link between OD pair w W∈ is 7

exactly 150% of its corresponding equilibrium travel time using only road network. Having 8 solved problem MUE, MPI-PR, MHPI and MHPI-S, we summarized the results in TABLE 1-5. 9 The first scenario compares flow distributions and the system performance of the multiclass user 10 equilibrium (MUE) and three multiclass Pareto improving policies. TABLE 1 and TABLE 2 11 provide the link flow distribution under different policies. More specifically, the first four rows 12 of each table show the number of users using the direct transit links, which also coincide with the 13 number of transit users for the corresponding OD pairs. Note that, in the last column of TABLE 14 2, the first four rows represent subsidies to transit users, which are shown as negative values in 15 the “Toll” column of MHPI-S scheme. As we have proved, the aggregate link flow of MUE is 16 unique and the class-specific link flow of MUE is not unique. In TABLE 1, we only display the 17 aggregate link flow of MUE. However, in order to gain more insights form the multiclass 18 policies; we list both the unique aggregate link flow distribution and one of the class-specific 19 link flow distributions under different policies. 20

TABLE 3 provides (average) class-specific equilibrium travel costs, the optimal 21 rationing/restriction ratio (𝛼𝛼), total system delay and delay reduction under different policy 22 schemes. The “delay reduction (% of max)” refers to the ratio between the reduction in travel 23 delay and the maximum amount possible, that is, the difference of system delay under MUE and 24 multiclass system optimum (MSO). 25

When transit travel costs are 150% of the corresponding equilibrium travel costs using 26 the road network, the total system delay is 490 409.6 passenger minutes and 588 380.0 passenger 27



minutes under MSO and MUE conditions, respectively. As shown in TABLE 3, all the three 1 policies generate Pareto improvement. The hybrid policies provide substantially better system 2 performance than the pure rationing policy and the MHPI-S policy provides the best system 3 performance. 4

5 TABLE 1 Flow Distribution of MUE and Multiclass Pareto-Improving Pure Rationing 6

MUE Multiclass pure rationing policy Aggregate flow Regular user flow Restricted user flow

Link UEv 1kv 1, 1k mv 1, 2k mv 2kv 2, 1k mv 2, 2k mv (1, 3) 0.000 0.000 0.000 0.000 8.988 4.494 4.494 (1, 4) 0.000 0.000 0.000 0.000 8.988 4.494 4.494 (2, 3) 0.000 0.000 0.000 0.000 11.984 5.992 5.992 (2, 4) 0.000 0.000 0.000 0.000 14.980 7.490 7.490 (1, 5) 22.937 14.508 4.002 10.506 0.000 0.000 0.000 (1, 6) 37.063 27.516 17.010 10.506 0.000 0.000 0.000 (2, 5) 48.105 43.781 29.740 14.041 0.000 0.000 0.000 (2, 6) 41.895 19.254 1.778 17.476 0.000 0.000 0.000 (5, 6) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 (5, 7) 31.654 28.319 10.084 18.235 0.000 0.000 0.000 (5, 9) 39.388 29.970 23.658 6.312 0.000 0.000 0.000 (6, 5) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 (6, 8) 59.123 46.770 18.788 27.982 0.000 0.000 0.000 (6, 9) 19.834 0.000 0.000 0.000 0.000 0.000 0.000 (7, 3) 45.750 42.524 18.010 24.514 0.000 0.000 0.000 (7, 4) 28.321 15.765 15.732 0.033 0.000 0.000 0.000 (7, 8) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 (8, 3) 24.250 6.504 6.504 0.000 0.000 0.000 0.000 (8, 4) 51.679 40.266 12.284 27.982 0.000 0.000 0.000 (8, 7) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 (9, 7) 42.418 29.970 23.658 6.312 0.000 0.000 0.000 (9, 8) 16.805 0.000 0.000 0.000 0.000 0.000 0.000

7 We can further observe that most of class-specific average equilibrium travel cost under 8

MHPI is lower than that under the pure rationing scheme, except for OD pair (1, 4). All of the 9 class-specific average equilibrium travel cost under MHPI-S is lower than that under the pure 10 rationing scheme. 11

More importantly, from TABLE 1 and TABLE 2, we also find that class m2 users (high 12 VOT users) who are originally forced to use transit links under the pure rationing scheme, for 13 example, link (1, 3) and link (2, 4), choose to pay tolls to access road networks under hybrid 14 policies, which indicates that certain class of users do benefit from the flexibility brought by the 15 hybrid policy. However, in the same case, all of class m1 users (low VOT users) remain using the 16 transit lines in restricted days under MHPI and MHPI-S schemes, which indicate that different 17 classes of users react differently to the same hybrid policies. Another interesting scenario can be 18 seen under the MHPI-S scheme. A fraction of regular class m1 users use the transit links, for 19



example, link (1, 3), which indicate that the subsidy scheme is substantially attractive to the low 1 VOT travelers, so that they choose to use transit even in unrestricted days. This scenario shows 2 the potential that transit mode share can be improved by a hybrid policy with transit subsidies. 3



TABLE 2 Flow distributions of Multiclass Pareto-improving hybrid policies Multiclass Hybrid Policy Multiclass Hybrid Policy with Subsidy Regular users Restricted Users Regular Users Restricted Users

Link 1kv 1, 1k mv 1, 2k mv 2kv 2, 1k mv 2, 2k mv Toll ($) 1kv 1, 1k mv 1, 2k mv 2kv 2, 1k mv 2, 2k mv Toll ($) (1, 3) 0.000 0.000 0.000 4.690 4.690 0.000 0.000 0.263 0.263 0.000 5.507 5.507 0.000 -23.599 (1, 4) 0.000 0.000 0.000 9.380 4.690 4.690 0.000 0.000 0.000 0.000 5.507 5.507 0.000 -7.034 (2, 3) 0.000 0.000 0.000 12.508 6.254 6.254 0.000 0.000 0.000 0.000 7.343 7.343 0.000 -17.721 (2, 4) 0.000 0.000 0.000 7.817 7.817 0.000 0.000 0.000 0.000 0.000 9.179 9.179 0.000 -1.163 (1, 5) 14.854 6.194 8.660 1.524 0.000 1.524 16.890 8.651 8.651 0.000 10.778 0.000 10.778 7.371 (1, 6) 26.385 14.426 11.959 3.166 0.000 3.166 22.579 29.057 10.072 18.985 0.237 0.000 0.237 16.979 (2, 5) 43.297 29.551 13.746 0.000 0.000 0.000 0.000 42.753 28.478 14.275 0.000 0.000 0.000 0.000 (2, 6) 18.561 1.378 17.183 7.817 0.000 7.817 5.066 14.202 0.000 14.202 16.522 0.000 16.522 4.511 (5, 6) 0.000 0.000 0.000 1.524 0.000 1.524 0.000 0.000 0.000 0.000 7.158 0.000 7.158 0.000 (5, 7) 28.402 14.656 13.746 0.000 0.000 0.000 35.334 26.932 12.657 14.275 0.974 0.000 0.974 24.094 (5, 9) 29.749 21.089 8.660 0.000 0.000 0.000 21.380 24.472 24.472 0.000 2.646 0.000 2.646 10.052 (6, 5) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 (6, 8) 44.946 15.804 29.142 0.000 0.000 0.000 22.657 43.259 10.072 33.187 0.000 0.000 0.000 10.127 (6, 9) 0.000 0.000 0.000 12.507 0.000 12.507 13.025 0.000 0.000 0.000 23.917 0.000 23.917 0.351 (7, 3) 42.347 19.940 22.407 0.000 0.000 0.000 2.102 33.964 21.307 12.657 0.000 0.000 0.000 0.000 (7, 4) 15.805 15.805 0.000 1.686 0.000 1.686 2.099 17.440 15.821 1.619 6.096 0.000 6.096 12.709 (7, 8) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 (8, 3) 5.766 4.116 1.650 4.690 0.000 4.690 9.088 10.072 0.579 9.493 12.851 0.000 12.851 4.359 (8, 4) 39.181 11.688 27.493 6.131 0.000 6.131 9.710 33.188 9.493 23.695 8.590 0.000 8.590 22.165 (8, 7) 0.000 0.000 0.000 0.026 0.000 0.026 3.209 0.000 0.000 0.000 0.000 0.000 0.000 3.813 (9, 7) 29.749 21.089 8.660 1.660 0.000 1.660 13.954 24.472 24.472 0.000 5.122 0.000 5.122 14.042 (9, 8) 0.000 0.000 0.000 10.847 0.000 10.847 1.305 0.000 0.000 0.000 21.441 0.000 21.441 0.000



TABLE 3 Summary of MUE, MPI-PR, MHPI and MHPI-S

MUE MPI-PR MHPI MHPI-S

OD pair , 1UEw mC , 2

UEw mC -

, 1MPI PRw mC

-, 1

, 1

MPI PRw m

UEw m

CC

-, 2

MPI PRw mC

-, 2

, 2

MPI PRw m

UEw m

cc

, 1MHPIw mC , 1

, 1

MHPIw mUEw m

CC

, 2MHPIw mC , 2

, 2

MHPIw mUEw m

CC

-, 1

MHPI Sw mC

-, 1

, 1

MHPI Sw m

UEw m

cc

-, 2

MHPI Sw mC

-PR, 2

, 2

MPIw m

UEw m

CC

(1, 3) 27.378 63.882 25.065 0.916 58.485 0.916 17.468 0.638 52.312 0.819 24.300 0.888 56.701 0.888

(1, 4) 27.197 63.459 24.610 0.905 57.423 0.905 22.941 0.844 56.972 0.898 23.669 0.870 55.227 0.870

(2, 3) 21.134 49.314 20.672 0.978 48.234 0.978 13.850 0.655 40.993 0.831 20.443 0.967 47.699 0.967

(2, 4) 20.953 48.891 20.402 0.974 47.604 0.974 20.704 0.988 48.878 1.000 20.110 0.960 46.923 0.960

Optimal Rationing Ratio (𝛼𝛼)

0.0000 0.2996 0.3127 0.3672

Total system delay ($)

588380.0 534709.6 524 567.8 505 277.3

Delay reduction (% of max)

0 45.69 65.13 84.82

TABLE 4 Mode Split of Multiclass Pure Rationing and Multiclass Hybrid Policies

Multiclass pure rationing policy Multiclass hybrid policy Multiclass hybrid policy with subsidy

Demand of class m1

(100 passengers/hour) Demand of class m2





(100 passengers/hour)

OD pairs Car Transit Car Transit Car Transit Car Transit Car Transit Car Transit

(1, 3) 10.506 4.494 10.506 4.494 10.310 4.690 15 0 9.230 5.770 15 0

(1, 4) 10.506 4.494 10.506 4.494 10.310 4.690 10.310 4.690 9.493 5.507 15 0

(2, 3) 14.008 5.992 14.008 5.992 13.746 6.254 13.746 6.254 12.657 7.343 20 0

(2, 4) 17.510 7.490 17.510 7.490 17.183 7.817 25 0 15.821 9.179 25 0

Optimal Rationing Ratio (𝛼𝛼)

0.2996 0.3127 0.3672



TABLE 5 Summary of SHPI, MHPI, SHPI-S and MHPI-S SHPI (single-class ) MHPI (multiclass) SHPI-S (single-class) MHPI-S (multiclass)

Average Equilibrium cost ($) Average equilibrium cost ($) Average Equilibrium cost ($) Average equilibrium cost ($)

OD pair User class m1 User class m2 User class m1 User class m2

(1, 3) 43.131 25.065 58.485 35.529 17.468 52.312

(1, 4) 42.380 24.610 57.423 35.634 22.941 56.972

(2, 3) 33.760 20.672 48.234 30.655 13.850 40.993

(2, 4) 34.922 20.402 47.604 33.090 20.704 48.878

Optimal Rationing Ratio (𝛼𝛼) 0.3449 0.3127 0.4430 0.3672

Total system delay ($) 505 144.5 524 567.8 499 107.3 505 277.3

Total toll ($) 61 041.1 442 83.6. 38 043.7 61 227.3

Total subsidy ($) 0.000 0.000 35 590.6 31 571.5

Delay reduction (% of max) 84.96 65.13 91.12 84.82



Then, the equity issues of the proposed multiclass pure rationing policy and the 1 multiclass hybrid policies are explored. From TABLE 3, we can easily obtain the frequencies 2 and cumulative distributions of three ratios, -

, ,MPI PR UEw m w mC C , , ,

MHPI UEw m w mC C , and -

, ,MHPI S UEw m w mC C , 3

where -,

MPI PRw mC , ,

MHPIw mC , and -

,MHPI Sw mC are average equilibrium cost under MPI-PR, MHPI and 4

MHPI-S schemes, respectively. Then, we can calculate the Gini coefficients associated with the 5 ratio of -

, ,MPI PR UEw m w mC C , , ,

MHPI UEw m w mC C , and -

, ,MHPI S UEw m w mC C are 0.1003, 0.1624, and 0.1060, 6

respectively, suggesting that MPI-PR scheme induce less inequality than both the hybrid policies, 7 and incorporating the toll revenue redistribution procedure can improve the equity level of 8 MHPI scheme substantially. These scenarios are all accordance with the consensus that rationing 9 policy are more equitable than pricing-based policies and combining revenue refunding can 10 improve the equity of pricing schemes. Moreover, it is worth noting that, under MPI-PR scheme 11 and MHPI-S policy, the ratios -

, ,MPI PR UEw m w mC C and -

, ,MHPI S UEw m w mC C is the same for each class of 12

users and all OD pairs, while under the MHPI policy, the ratio , ,MHPI UEw m w mC C of low VOT users 13

are less than that of high VOT users for all OD pairs. This scenario can partially explain the 14 larger Gini coefficient value of MHPI scheme. More importantly, it indicates that low-VOT 15 users tend to be better off than the high-VOT users under the MHPI scheme, which makes MHPI 16 scheme a progressive policy and most attractive scheme to policy makers among the proposed 17 three schemes (32). 18

To further investigate the impact of different policies on the change of mode split, 19 TABLE 4 shows the class-specific mode split under pure rationing and hybrid policies. From 20 TABLE 4, we discover that the multiclass pure rationing policy imposes a uniform diversion rate 21 to transit for all OD pairs, which ignores the difference in network traffic conditions faced by 22 users from different OD pairs, and may inevitably cause inefficient allocation of road capacity 23 resources. Intuitively, the multiclass hybrid policies will induce more car demands than their 24 multiclass pure rationing counterpart. But it is worth noting that the optimal rationing ratios are 25 not the same under the three policies and the mode split change also depends on the optimal 26 rationing ratios under different policies. Therefore, the general tendency for the change of mode 27 split is difficult to depict. More specifically, in this case, on the aggregate level (sum all class of 28 users), even with higher optimal restriction ratios, MHPI policy yields more car demand splits in 29 half OD pairs, e.g. OD pair (1, 3) and (2, 4), and MHPI-S policy yields more car demands splits 30 in all the OD pairs, which indicate hybrid policies are more flexible and effective in managing 31 multimode network mobility. Furthermore, in the class-specific level, different classes of users 32 behave differently under the hybrid policies. Generally speaking, higher proportion of class m2 33 users (high VOT) choose car mode than class m1 users r (low VOT) under hybrid policies. In 34 particular, under MHPI-S policy, all the class m2 users, unrestricted and restricted, choose car 35 mode, meanwhile, a fraction of unrestricted class m1 users choose the transit mode. 36

In TABLE 5, we compared flow distributions and the system performance of the 37 multiclass hybrid policies with its single class counterpart. From TABLE 5, we observe that, 38 both MHPI and MHPI-S schemes yields less delay reduction compared to their single-class 39 counterpart. The existence and effectiveness multiclass Pareto-improving hybrid schemes not 40 only depend on the network configurations but also on demographic features (e.g., VOT values) 41 of the population. It is also worth noting that we solve all the models by manifold 42 suboptimization algorithm, which can only guarantee converging to strongly stationary 43 solutions, not necessarily global optimal solutions. Thus, the truly global optimal solutions of all 44



these models may yield less travel delay. 1 2

5. CONCLUSIONS 3 This study extended hybrid rationing and pricing Pareto-improving policy for homogeneous 4 users to heterogeneous users. The heterogeneous users refer to a discrete set of classes, each one 5 with a different VOT. The numerical results shows that, multiclass Pareto improving pure 6 rationing policies and multiclass hybrid Pareto improving policies are all exist. Like the single-7 class hybrid policies, the multiclass hybrid policies (MHPI and MHPI-S) also provide greater 8 flexibility than multiclass pure rationing (MPR-PR) policy. Under the MPR-PR policy, all class-9 specific travel demands are forced to use transit mode at the same ratio regardless of the traffic 10 conditions, which may cause inefficient allocation of road resources. On the contrast, under 11 multiclass hybrid policies, although the restriction ratio is still the same for all OD pairs and all 12 classes of users, link-based nonnegative anonymous tolls are introduced to adjust roadway 13 demands for different OD pairs and different user classes. 14

Comparing the equity level of the three policies, we discover that pure rationing policy 15 induce less inequalities than two hybrid policies and hybrid policies with transit subsidies can 16 improve the original MHPI policy substantially. MHPI policy is a progressive policy since it 17 favors the disadvantaged travelers (e.g. travelers with low VOT). 18

Different classes of users react differently to the same hybrid policies. As the numerical 19 example shows, low VOT users tend to use transit mode in rationing days while restricted high 20 VOT users may be better off paying tolls to access the road network on restricted days instead of 21 taking transit. Furthermore, under the hybrid Pareto-improving policy with subsidies, low VOT 22 users may use transit mode even in the unrestricted days. Although tolls are charged on restricted 23 users only, their route choice may influence both the mode choice and route choice of non-24 restricted users, hence achieve better system performance. 25

The numerical results also illustrate that, compared to the single-class hybrid polices, 26 multiclass hybrid policies yields less delay reduction. The existence and effectiveness multiclass 27 Pareto-improving hybrid schemes not only depend on the network configurations but also on 28 demographic features (e.g., VOT values) of the population. 29

In this paper, we assume that travel demand is deterministic to facilitate the presentation 30 of key ideas. The theory of elastic demand can be similarly established. Other extensions to the 31 model include adopting continuous VOT distributions instead of a discrete set of classes, 32 relaxing the strict Pareto-improving conditions to approximate Pareto-improving conditions, 33 considering more realistic network configurations, such as variable transit travel times, 34 intermodal scenarios, and applying the models to large scale real networks. 35

All in all, the multiclass hybrid Pareto-improving rationing and pricing schemes offer an 36 effective and equitable tool to manage network mobility, traffic congestion and environmental 37 quality. Its various potential applications remain largely unexplored. 38 39 ACKNOWLEDGEMENT 40 This research is supported by the National Natural Science Foundation of China (51178110, 41 51378119). Graduate Innovation Project of Jiangsu Province (CXZZ12_0113), the Ph.D. 42 Programs Foundation of Ministry of Education of China (20120092110062) and the Six Talent 43 Peaks Program of Jiangsu Province of China (2012-JZ-003). The authors are also grateful to 44 Prof. Siriphong Lawphongpanich and Prof. Yafeng Yin for providing us the source codes of their 45 original Manifold Suboptimization Algorihm and introducing the service of NEOS Server. 46



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