Integrating link-based discrete credit charging scheme into discrete network design problem

Accepted Manuscript

Integrating link-based discrete credit charging scheme into discrete network design problem Guangmin Wang , Ziyou Gao , Meng Xu PII: DOI: Reference:

S0377-2217(18)30488-0 10.1016/j.ejor.2018.05.069 EOR 15182

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

1 August 2017 9 May 2018 29 May 2018

Please cite this article as: Guangmin Wang , Ziyou Gao , Meng Xu , Integrating link-based discrete credit charging scheme into discrete network design problem, European Journal of Operational Research (2018), doi: 10.1016/j.ejor.2018.05.069

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Highlights The link-based discrete credit charging scheme is integrated into the discrete NDP.

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Discrete credit charging scheme with locations are used to manage the travel demand.

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The number of addition lanes with the locations is used to increase the road supply.

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Interactions of TCS and DNDP amplify individual effects of separate TCS and DNDP.

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The integrated model can outperformance than the sequential decision problems.

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ACCEPTED MANUSCRIPT Integrating link-based discrete credit charging scheme into discrete network design problem

Guangmin Wang*1, ZiyouGao2, Meng Xu3 1. School of Economics and Management, China University of Geosciences, Wuhan 430074, P.R. China 2. School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, P.R. China

100044, China

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3. State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing

Abstract: In this paper, we present an optimization model for integrating link-based discrete credit charging scheme into the discrete network design problem, to improve the transport

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performance from the perspectives of both transport network planning and travel demand management. The proposed model is a mixed-integer nonlinear bilevel programming problem, which includes an upper level problem for the transport authority and a lower level problem for the network users. The lower level sub-model is the traffic network user equilibrium (UE) formulation

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for a given network design strategy determined by the upper level problem. The network user at the lower level tries to minimize his/her own generalized travel cost (including both the travel time and

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the value of the credit charged for using the link) by choosing his/her route. While the transport authority at the upper level tries to find the optimal number of lanes and credit charging level with

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their locations to minimize the total system travel time (or maximize the transportation system performance). A genetic algorithm is used to solve the proposed mixed-integer nonlinear bilevel

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programming problem. Numerical experiments show the efficiency of the proposed model for traffic congestion mitigation, reveal that interaction effects across the tradable credit scheme and

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the discrete network design problem which amplify their individual effects. Moreover, the integrated model can achieve better performance than the sequential decision problems.

Keywords: Transportation, Discrete network design problem (DNDP), Bilevel programming problem, Tradable credit scheme (TCS)

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Corresponding author. Email: [email protected] 2

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1.

Introduction

Urban transport has been playing an overriding role in economic activity and growth, and a major role in the quality of individuals' lives. However, the spatial in-coordination between the traffic demand and urban road capacity leads to some negative effects, such as traffic congestion, vehicle emissions, noise pollution and traffic accidents. Much attention has been focused on dealing

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with these problems from the perspective of the transportation network planning and the mobility management.

Typically, the classical network design problem (NDP) involves the optimal allocation of budget to expand existing links and/or to add new candidate links so as to optimize the network

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performance (e.g. minimizing the total system travel time or maximizing the social total welfare), while accounting for the travelers’ route choice behavior represented by a user equilibrium (UE) or stochastic user equilibrium (SUE) manner (Yang and Bell, 2001). Depending on the nature of road capacity change, NDP can be categorized into the continuous network design problem (CNDP), the discrete network design problem (DNDP) and the mixed network design problem (MNDP), which

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is a combination of CNDP and DNDP (Farahani et al., 2013). Owning to the underlying complexity

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of NDP, the majority of existing solution methods address CNDP as opposed to DNDP, which more directly deals with the addition of lane, a road closure scheme, and the provision of a new public

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transport service. However, Boyce and Janson (1980) pointed out that the DNDP model is more appropriate for transportation network improvement problems because CNDP models give the

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solution in terms of fractions of highway lanes. Improving only the road space has been proven to be self-defeating in congested areas

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considering that increased capacity will soon be absorbed by the induced travel demand (Hansen and Huang, 1997). The focus is currently turning to the travel demand management. As a typical approach, the congestion pricing has gained much attention both theoretically and practically. However, its applicability is questioned from the viewpoints of social equity (Giuliano, 1994) and economic efficiency (Viegas, 2001). More importantly, people may be against paying for congestion because they may prefer to pay for things they wish to acquire rather than for things they wish to avoid (i.e., traffic congestion) (Kahneman and Tversky, 2003). Therefore, Yang and Wang (2011) introduced another quantity control method, namely, the tradable credits scheme 3

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(TCS), for the mobility management considering its advantages over tolls. Traditionally, these two subjects, that is, transportation network planning and mobility management, have been addressed separately in the scientific literature. However, the development of integrated transport strategies has gained increasing interest in recent years. Particularly, studies of simultaneous determination of the road toll pricing (the tradable credit) and road capacity

(credit) and the improvement capacity on the given links.

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development from the perspective of modeling focused on investigating the amount of both toll

In this paper, we present an optimization model for integrating the transportation network planning and the mobility management with respect to determining the number of lanes and the credit charging level with their locations.

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The model aims at answering the following two questions:

(1) How to determine the location of added lanes and design the TCS simultaneously? (2) How many lanes should be added to the selected links and what credit level (i.e. zero, low, median, or high) should be charged for the selected links to improve the transportation network

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performance?

As will be shown in the modelling approach, the proposed DNDP consider both adding new

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links and capacity improvement in existing links. In addition, the discrete credit charging scheme simply the complexity of determining credit charging compared with the continuous one.

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The proposed optimization problem is a nonlinear bilevel programming model, which consists of an upper level problem for the transport authority and a lower level problem for network users.

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The lower level problem represents the network user equilibrium problem for a given decision for the optimal network design strategy and credit charging level with their locations determined by the

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upper level problem. The network users at the lower level try to minimize their own generalized travel cost by choosing their routes, whilst the transport authority at the upper level tries to find the optimal network design strategy and credit charging level with their locations to minimize the total system travel time. The upper level decision variables (i.e. the number of added lanes and the charging credit level) are integers and the lower level decision variables (i.e. the traffic flows of links) are continuous. Therefore, the proposed model falls into the mixed-integer nonlinear bilevel programming problem (MIBLP). There are some existing algorithms (Dempe, 2002; Colson et al., 2007, and therein) are 4

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available to solve the nonlinear bilevel programming problem. However, methods to solve the mixed-integer bilevel programming problem have so far addressed a very restricted class of problems. One of the major difficulties is that conventional solution approaches to solve continuous bilevel programming problems cannot be applied to solve the bilevel programming problem with discrete variables. There exists another challenge in the presented model i.e. the non-differentiable constraint functions to determine the locations for both adding lanes and charging credits. A genetic

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algorithm (GA) is therefore used for solving the MIBLP given in this paper as it is easier to deal with those non-convex and non-differential problems owing to its virtues of simplicity and minimal problem restrictions (Goldberg, 1989).

The main contribution of this work is the development of a mixed-integer bilevel optimization

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model to identify the optimal network design strategy and credit charging scheme for minimizing the total system travel time. Specifically, the features of the proposed model are as follows: (1) discrete credit charging schemes with the locations are considered to manage the travel demand; (2) the number of addition lanes with the link locations is considered to increase the road supply; (3)

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two ways (increase road supply and manage travel demand) are integrated to alleviate the traffic congestion.

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The rest of this paper is organized as follows. Section 2 provides the literature review. In Section 3, the mixed-inter nonlinear bilevel programming model is proposed to describe DNDP to

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simultaneously optimize both the link-based credit charging level and the road lane improvement with their locations. Section 4 presents a GA based on the Frank-Wolfe (FW) method for solving

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the proposed model. Numerical experiments are illustrated by using the Sioux Falls network in

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Section 5. Finally, Section 6 concludes this paper and provides future research directions.

2.

Literature review

2.1 Discrete network design problem

DNDP addresses the selection of link additions to an existing road network for minimizing the total system travel time while accounting for the route choice behaviors of network users. Gao et al. (2005) proposed a new solution algorithm by using the concept of support function to express the relationship between improvement flows and the new additional links in the existing urban network. 5

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Farvaresh and Sepehri (2013) designed a branch and bound algorithm to obtain the exact solution to DNDP. Generally, DNDP is formulated as a bi-level programming. Farvaresh and Sepehri (2011) presented a single-level mixed integer linear programming (SL-MILP) formulation for bi-level DNDP to provide the optimal solution. Later, Wang et al. (2013) extended the conventional DNDP to the DNDP with multiple capacity levels, in which the transport authority needs to determine how many lanes should be added to each link from a set of candidate links. Then, two novel and

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efficient global optimization methods were developed based on the relationship between the UE and system optimization (SO) principles after relaxing the proposed bi-level programming model into a single-level problem using the SO-relaxation. Zhang et al. (2014) proposed an active-set algorithm to solve a single-level mathematical program with complementarity constraint for

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multimodal DNDP. Fontaine and Minner (2014) solved DNDP by using a Benders decomposition algorithm by reformulating the bilevel problem into a single-level problem. Fontaine and Minner (2017) also used the Benders decomposition algorithm to solve a dynamic model for network maintenance planning after extending the DNDP. Wang et al. (2015) addressed a novel yet general

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DNDP by simultaneously determining the optimal new link addition and their optimal capacities. An assumption was made that the capacity increase and construction cost of each road are based on

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the number of lanes. Haas and Bekhor (2017) addressed DNDP with emphasis on the environmental benefits by using an alternative approach for approximating the environmental

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impact of traffic. Time-dependent and multi-objective DNDPs were also researched (Szeto et al.,

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2010; Hosseininasab and Shetab-Boushehri, 2015).

2.2 Tradable credit scheme (TCS)

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Recently, Yang and Wang (2011) introduced a type of sophisticated quantity control

method-termed the tradable credit- into transportation and examined a simple but forceful tradable credit distribution and charging scheme. They defined an origin-destination (OD)-specific credit distribution scheme (the first best pricing scheme) for a Pareto-improving SO flow pattern. Then,This scheme has received extensive attention. Wang et al. (2012) extended it to a tradable credit scheme (TCS) with heterogeneous travelers. Bao et al. (2014) studied travelers’ loss aversion behavior for credit charging during the route choice process under a given TCS. The properties of 6

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the multiclass traffic network equilibria under a TCS were also researched (Zhu et al., 2015). Based on the researches on UE under TCS, Wang et al. (2014b) researched CNDP under a given tradable credit scheme. Only a subset of links are charged to credits, which is a second-best pricing scheme, because it is impractical to charge users on each network link in view of the operating cost and public acceptance. Further, Wang et al. (2014c) investigated joint link-based credit charging and road capacity improvement in CNDP under fixed OD demands. In reality, discrete toll rates, as

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opposed to the continuous toll rates, are generally considered. For example, the toll rates should always be rounded to the closest 25¢ for most real-world applications (Fan and Gurmu, 2014). Flat toll rates on existing toll roads are changed to a variable toll schedule so that the toll is higher during peak travel hours and lower during off-peak. Moreover, the variation of the flat toll rates is

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discrete in reality (FHWA, 2006). For example, the standard charge in London was raised 15% from £10 per day to £11.50 in June 2014. Therefore, discrete toll rates have also been considered for determining the second-best link-based optimal toll locations and toll levels simultaneously (Fan, 2016). Recent reviews on tradable credits scheme include the road capacity management (Fan

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and Jiang, 2013), the road traffic congestion management (Grant-Muller and Xu, 2014), and

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empirical research and relevant behavioural approaches (Dogterom et al., 2017).

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2.3 Integrated transport strategies

Verhoef et al. (2010) introduced capacity and tolls in congested networks via using ‘long-run

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cost functions’ to solve the second-best network problems. Furthermore, some researchers were concerned with how the private sector chooses the toll and capacity to meet the self-financialing

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under the Build-Operate-Transfer (BOT) framework from the perspectives of both policy and technology (Koh et al, 2009, Yang and Meng, 2002, and therein). Wang et al. (2014c) proposed the bilevel programming models to examine the joint optimal link-based tradable credit charging scheme and road capacity improvement under the case of fixed OD demand. Fan and Gurmu (2014) determined the optimal solution for combined congestion pricing and capacity expansion problems by using a bi-level GA-based optimization solution method. Xu et al. (2017) investigated a general DNDP with a road toll pricing scheme to minimize the total travel time under a budget constraint. The 7

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revenue from a road toll charge is assumed to be a part of the infrastructure financing. Bao et al., (2017) employed the tradable credits scheme for traffic mobility management and private provision of public transportation infrastructure through the build-equity-credit (BEC) scheme. Past research into the problem of simultaneous determination of the road toll pricing (the tradable credit) and road capacity development has focused on investigating the amount of both the toll (credit charged) and the improvement capacity on the given links. In fact, the locations (the set of links) for adding

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lanes and charging credit (or pricing toll) should be considered before determining the toll pricing (credit charging) level and the number of lanes. Little attention has been paid to this issue. Zhang and Wee (2012) determined the optimal selection of both toll levels and toll locations by using the bi-level modes with mixed variables. Zhang and Zhou (2009) integrated the social and spatial

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equity constraints to solve the toll locations problem for a congestion pricing scheme in a multi-class network. Ekstrom et al. (2012) minimized the total travel time and the toll-point cost by finding the toll locations and levels in a congestion pricing scheme considering the principle of user

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equilibrium under the fixed OD demand.

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2.4 Algorithms for the mixed-integer nonlinear bilevel programming problem

The majority of methods to solve the mixed-integer bilevel programming problem are

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restricted to the linear problems and problems where the upper level decision variables only appear in the lower level problem objective and are separable (Saharidis and Ierapetritou, 2009). Moreover,

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the mixed-integer nonlinear bilevel programming problem has received a little attention in the literature, such as an algorithm using parametric analysis by Jan and Chern (1994), a stochastic

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simulated annealing algorithm proposed by Sahin and Ciric (1998) and a parametric integer programming algorithm by Köppe et al. (2010). Then, Gümüs and Floudas (2005) introduced two deterministic global optimization methods that solve mixed-integer nonlinear bilevel programming problems. Later, Mitsos (2010) extended the deterministic algorithm for the global solution of continuous bilevel programs with a nonconvex lower-level program to the mixed-inter case. Recently, Li and Guo (2017) proposed the algorithm for the mixed integer bilevel programming problem via reformulating it as a mixed integer mathematical program with complementarity 8

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constraints. However, it notes that no rigorous approaches are available for solving bilevel mixed-integer programming problems. Lozano and Smith (2017) proposed an exact finite algorithm based on an optimal-value-function reformulation for solving bilevel mixed-integer programs, in which all upper-level variables and a subset of the lower-level variables are integer.

Problem formulation

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3.

In a transportation network, the total system travel time is minimized at the given total investment on adding lanes and constructing credit charging facilities. The transport authority should find the optimal network design strategy and credit charging level with their locations to

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maximize the transportation network performance. It is assumed that the travel demand between each OD pair is fixed and known because the network design and credit charging are both long-term decisions. Network users are considered homogeneous, and they have the same value of time (VOT). They follow the UE principle in selecting their paths. The proposed model minimizes

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the total system travel time under a budget constraint by determining joint optimal credit charging scheme and road capacity improvement in the discrete case. Moreover, we also consider selecting

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the locations for both lanes adding and credit charging, then optimizing the number of lane additions and the level of credit charging. That is to say, we will both optimize the set of links to

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adding lanes as well as the set of links to charge credit and determine the number of lane additions as well as the level of credit charging.

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The notations are introduced before presenting the problem formulation.

3.1 Notations

Sets:

𝐺 = (𝑁, 𝐴)

transportation network

𝑁

set of all nodes in the network

𝐴

set of all links in the network

𝐴̅

set of potential links that may be expanded, 𝐴̅ ⊆ 𝐴 9

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𝐴̂

set of potential links that may be charged, 𝐴̂ ⊆ 𝐴

𝐼𝑎

set of the number of lanes to be added to link 𝑎 ∈ 𝐴̅ For example, 𝐼𝑎 = {0,1,2,3} means that the transport authority determines to add either zero (do-nothing), one, two or three lanes to link 𝑎 ∈ 𝐴̅

𝐽𝑎

set of the lever of charging credit for using link 𝑎 ∈ 𝐴̂.For example, means that

medium or high for using link 𝑎 ∈ 𝐴̂

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𝐽𝑎 = {0,1,2,3}the transport authority determines to charge either zero, low,

𝑊

set of OD pairs

𝑅𝑊

set of all simple paths (or routes) connecting OD pair 𝑤 ∈ 𝑊

Parameters

quantity of credits issued for each traveler between OD pair 𝑤 ∈ 𝑊 , which is

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𝑞𝑤

given and fixed

travel demand between OD pair w W , which is given and fixed

δ𝑎𝑟

δ𝑎𝑟 = 1 if path 𝑟 uses link 𝑎, and 0 otherwise, 𝑎 ∈ 𝐴, 𝑟 ∈ 𝑅𝑤

𝐶𝑎

construction cost of adding each lane to link 𝑎 ∈ 𝐴̅.

𝑒𝑎

construction and operation cost of the tradable credit charging equipment on link

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𝑑𝑤

𝑡𝑎 (𝑣𝑎 ) 𝐾

link travel time function on link 𝑎 when adding 𝑧𝑎 lane(s) to link 𝑎 ∈ 𝐴̅ link travel time function on link 𝑎 ∈ 𝐴\𝐴̅

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𝑡𝑎 (𝑣𝑎 , 𝑧𝑎 )

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𝑎 ∈ 𝐴̂ if the tradable credit charging level is not zero.

total amount of credit for a given and fixed OD demand 𝑑𝑤 and a given credit

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distribution scheme 𝑞𝑤 , 𝑤 ∈ 𝑊

𝐵

a predetermined budget the cost of adding lanes and the cost of constructing and

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operating the tradable credit charging equipment

𝑛1

the number of the links to add lanes

𝑛2

the number of the links to charge credits

𝑔𝑎 (𝑢𝑎 )

the amount of credit for charging levels, which is a given piecewise function. For example,

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ua  0( zero)   0,   5, ua  1(low)  g a (ua )    10, ua  2(medium)    ua  3(high)   20, Decision variables 𝑧𝑎

the number of lanes is added to link 𝑎 ∈ 𝐴̅ , 𝑧𝑎 ∈ 𝐼𝑎 is an integer decision

𝑢𝑎

the credit level is charged for using link 𝑎 ∈ 𝐴̂, 𝑢𝑎 ∈ 𝐽𝑎 is an integer decision variable, 𝐮 = (𝑢𝑎 ), 𝑎 ∈ 𝐴

𝑝 𝑓𝑟𝑤

the unit credit price in the credit market

traffic flow on path 𝑟 ∈ 𝑅𝑤 ,between OD pair 𝑤 ∈ 𝑊 , 𝐟 = (𝑓𝑟𝑤 ), 𝑟 ∈ 𝑅𝑤 ,

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𝑤∈𝑊 𝑣𝑎

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variable, 𝐳 = (𝑧𝑎 ), 𝑎 ∈ 𝐴

aggregate traffic flow of link 𝑎 ∈ 𝐴 ,𝐯 = (𝑣𝑎 ), 𝑎 ∈ 𝐴

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3.2 Formulation of the model

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A separable link travel time function, ta (va ) , is considered to minimize complexity. This function is assumed to be nonnegative, differentiable, convex and monotonically increasing with

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respective to the traffic flow va on link a  A . This study continues to use the link travel time

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   va   function ta (va )  Ta 1      developed by the US Bureau of Public Roads (1964) to   ba   

determine the travel time on each link, where 𝑇𝑎 and 𝑏𝑎 are parameters representing the

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free-flow travel time and original capacity for link 𝑎 ∈ 𝐴 . 𝛼 and

are volume/delay

coefficients. The traditional Bureau of Public Roads (BPR) values for 𝛼 and

are 0.15

and 4.0, respectively. In the conventional DNDP that determines whether each link in a set of new candidate links should be added to a transport network, we assume that each of these new candidate links already exists by associating it with an infinite (or extremely large) free flow travel time. If one lane or more than one lane is added, then the actual free flow travel time is assumed in the corresponding link travel time function. Therefore, the link travel time function on the 11

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candidate links has the following form

 if ba  0 and za  0   4  ta (va , za )    ,a A    va   T 1  0.15 otherwise   a    za  CL  ba     

(1)

where 𝐶𝐿 denotes the capacity in one added lane.

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Note: New links are added when 𝑏𝑎 = 0 and 𝑧𝑎 ≠ 0. Capacity improvements for the existing links are described by lanes additions when 𝑏𝑎 ≠ 0 and 𝑧𝑎 ≠ 0. Neither happens when 𝑏𝑎 ≠ 0 and 𝑧𝑎 ≠ 0. Thus, DNDP in this paper includes both adding new links and improving lanes in existing links.

Further, the tradable credit scheme, used in this paper, includes the following aspects:

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(1) Credits are initially distributed freely to all travelers and the OD specific credit distribution is adopted. The quantity of distributed credit to each traveler for every OD pair is given and fixed.

(2) The total amount of credits is predetermined by the transport authority and the credits are

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monthly specific to avoid banking or stocking credits for the future. (3) The link-specific credits are used to pay for the credit charges for using the link.

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(4) A free market for credit trading is assumed for travelers to buy or sell credits according to their travel demands. The transport authority merely acts as a manager to monitor the

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system without earnings in trading credits, so that the transport authority will make up one's mind to alleviate the traffic congestion and improve environmental quality via the

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tradable credits scheme.

(5) The transaction cost and administrative cost are both zero and are ignored in this study.

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Hence, a uniform and constant credit price prevails based on the assumption that the market is

competitive (reasonable in view of the large number of agents involved) (Yang and Wang, 2011). Therefore, for the given and fixed OD demand 𝑑𝑤 (𝑤 ∈ 𝑊) and a given credit distribution scheme 𝑞𝑤 (𝑤 ∈ 𝑊), the total amount of credits is given by 𝐾 = ∑ 𝑑𝑤 𝑞𝑤 , which is determined by the transport authority to achieve a policy target in terms of fix-quantity travel credits. Thus, the tradable credits scheme can enhance the income distribution within a predefined group of travelers when credits flow from the higher income groups to the lower ones while the money flows 12

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inversely. Thus, the tradable credits scheme is more acceptable than the toll pricing scheme. Ω𝑓 represents the set of feasible path flow patterns defined as follows:

   f  f | f rw  0,  f rw  d w , w W  rRw  

(2)

Ω𝑣 represents the set of feasible link flow patterns defined as follows:

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  v   v | va    f rw a ,r , f  f , a  A wW rRw  

(3)

Any feasible point 𝒛 in the following set is termed as a scenario for the given credit distribution 𝐾.

(4)

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  S  z |  ca za   ea min{ua ,1}  B  aAˆ  aA 

The transport authority needs to select the set of links for capacity improvement from a set of candidate links 𝐴̅ and determine how many lanes should be added to each selected link. The decision variable representing the number of lanes is discrete. At the same time, the transport

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authority also needs to select the set of links for charging credit from a set of candidate links 𝐴̂ and to determine the credit charging level (e.g. zero, low, medium or high) for using the selected

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links. That is, the lever of credit charged is also discrete. Further, the amount of credit for each charging level can be given by the transport authority according to the real situation. The following

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bilevel programming model can describe this type of DNDP:

( BLP) min F (z, u, v) 

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z ,u , v



aA

A

vata (va )   vata (va , za )

(5)

aA

subject to

c z  e aA

a a

aAˆ

a

min{ua ,1}  B

(6)

 min{z ,1}  n

1

(7)

 min{u ,1}  n

2

(8)

za  I a , a  A

(9)

ua  J a , a  Aˆ

(10)

aA

aAˆ

a

a

where 𝐯 solves the following problem 13

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min G(z, u, v)  v

va

 t

aA

A 0

va

a

( )d    ta ( , za )d

(11)

aA 0

subject to

 v g (u )  K   d a

a

a

f

w r

va 

f

rRw

wW

q

(12)

w w

 d w , w W

wW rRw

(13)

 ,a A

w r ar

f rw  0, w W , r  Rw

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aAˆ

where (5) and (11) are the upper and lower level objective functions, respectively. The upper level

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decision-maker (the transport authority) determines the credit charging level and number of lanes with their locations considering the response of network users (the lower level decision-makers) to the resulting changes in form of user equilibrium.

Formulations (5)-(15) are based on the link-path representation in which a matrix, named the link-path adjacency matrix, play an important role in converting path flows to link flows. The

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budgetary constraint is presented in Constraint (6); Constraint (7) describes the number of links to

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add lanes; Constraint (8) describes the number of links to charge credit; Constraint (9) is the decision variable for lanes added to links; Constraint (10) is the credit level charged for using links;

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Constraints (11)-(15) assert UE conditions with the tradable credit scheme. Constraint (12) is the credit feasibility constraint; Constraint (13) is related to the travel demand for an OD pair to the

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path flows connecting the OD pair; Constraint (14) is path flow conservation and expresses the relationship between the link flows and the path flows; Finally, Constraint (15) shows the

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non-negativity of path flows.

4.

Solution algorithm

The bilevel programming problem is a non-convex programming problem with an implicitly determined feasible set (Dempe, 2002). Solving the mixed-integer bilevel programming problem will be more difficult because of the inapplicability of conventional solution approaches to solve continuous bilevel programming problems when discrete variables exist. In addition, there also exists the challenge of non-differentiable constraint functions for determining the locations of both 14

(14)

(15)

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adding lanes and charging credits in our proposed model. A GA is proposed for solving MIBLP in this paper considering it has no special requirement for the differentiability of the function (Goldberg, 1989). In this proposed GA, the binary encoding is used to represent only the upper level decision variables. The fitness value for each chromosome can be obtained after solving the lower level by using Frank-Wolfe method (Frank and Wolfe, 1956). The crossover, mutation and selection procedures are used after initializing the population. It can be easily checked that whether

charging credits are satisfied after decoding the chromosomes.

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the non-differentiable constraint functions for determining the locations of both adding lanes and

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4.1 Frank-Wolfe algorithm for solving the lower level problem

For the fixed upper level decision variable 𝑍𝑎∗ (𝑎 ∈ 𝐴̅) and 𝑢𝑎∗ (𝑎 ∈ 𝐴̂) , the lower level problem is

min H ( v)  G (U* ,Z* ,V)  v



aA

va

A 0

aA 0

*  ta ( )d    ta ( , za )d

M

subject to

va

(16)

 v g (u )  K

(17)

f

(18)

rRw

w r

* a

 d w , w W

f

wW rRw

 ,a A

w r ar

(19)

PT

va 

a

ED

aAˆ

a

frw  0, w W , r  Rw

CE

(20)

This problem describes the traffic assignment problem (TAP) with the tradable credit charging

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scheme (Yang and Wang, 2011). The link flow pattern at equilibrium, 𝐯 ∗ , is unique because it is a convex nonlinear programming problem with linear constraints. The classical model that is commonly used in practice is a static deterministic TAP. This classical model was developed in the 1950s and since then various algorithms have been proposed to solve it. Perederieieva et al. (2015) implemented 11 algorithms for traffic assignment and identified the advantages and disadvantages of different groups of algorithms by analyzing and comparing the most promising approaches for solving TAP from the viewpoint of a framework that can be shared by different algorithms. Historically, the first algorithms developed for solving TAP were link-based. The most well-known 15

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algorithm is the Frank-Wolfe, a general algorithm for convex optimization problems (Frank and Wolfe, 1956). Owing to its simplicity and low memory requirements, it is used even at present and is implemented in different commercial software packages. In the iteration of the FW algorithm, a sub-problem of minimizing the linearized objective function is solved by assigning all traffic flows to minimum generalized cost paths, where generalized link costs are determined by the link flows in the current solution for the main problem. A new solution is obtained by minimizing the original

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objective function over the line segment connecting the current solution and the sub-problem solution.

The steps of FW algorithm for solving the TAP with the tradable credit scheme at the given 𝑍𝑎∗ and 𝑢𝑎∗ are given as follows:

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Step 1 (Feasible path flow solution). Given 𝑍𝑎∗ and 𝑢𝑎∗ , generate the feasible path flow solution 𝐟 𝟎 by assigning all traffic flows to the shortest paths. The corresponding link flow 𝐯 𝟎 can be obtained. Set iteration counter 𝑘 = 0;

Step 2 (Descent direction). Find direction of descent 𝐬 𝒌 by solving the linear programming

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problem using CPLEX; min H ( v ) f k T

(21)

ED

Subject to (17)-(20)

The path flow 𝐬 𝒌 denotes the best solution, which indicates the direction along which an improved

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solution will be obtained. Let 𝑽𝒌𝒔 denote the corresponding link flow.

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Step 3 (Line search). Line search method is applied to find the largest step size:

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min H ( vk   ( vks  vk ))

s.t.

(22)

0   1

(23)

The above problem corresponds to a one-dimensional optimization (Antoniou and Lu, 2010)(golden section, Fibonacci, etc.) to determine the step size   along Step 4(Update). Update solution

s

k

fk  .

f k 1  f k    (sk  f k ) .

Step 5(Stop criteria) The iteration stops, if the convergence criteria are met, and the optimal path flow is f k 1 ; otherwise, let k  k  1 , go to Step 2. 16

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4.2 Genetic algorithm for solving the model

In this subsection, the steps of the proposed algorithm are described in detail. The main parts of this algorithm are the single point crossover operator, the one-point mutation operator, and the

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ranking selection scheme with elitism after adopting the binary encoding scheme. (1) The binary encoding scheme:

The upper level decision variables are (𝑍; 𝑈) = (𝑧1 , 𝑧2 , ⋯ , 𝑧|𝐴̅| ; 𝑢1 , 𝑢2 , ⋯ , 𝑢|𝐴̂| ), where |𝐴̅| and |𝐴̂| are the numbers of the links for adding lanes and charging credits in the network, respectively.

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The binary encoding of chromosome is used to represent the upper level decision variables. If the maximal lane added to each link is no more than 3, and there are 4 credit charging levels such as zero, low, medium, high, only two genes can represent the decision variable for lanes added to the links and credit charging level for using links, respectively. Thus, a chromosome to denote the

z

1

z

2

A

u

1

u

2

0 1 0 1 1 0

u

Aˆ

1 1

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1 1 1 0

z

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upper level decision variables is as follows:

Figure 1: Encoding of the chromosome

The(2𝑖 − 1)-th and (2𝑖)-th genes denote the number of lanes added to the 𝑖-th link, 𝑖 =

PT

1,2, ⋯ , |𝐴̅|. That is,

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00, if the added lane is zero 01, if the added lane is one (2i-1)-th and (2i)-th genes= { 10, if the added lanes are two 11, if the added lanes are three

(24)

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where 𝑖 = 1,2, ⋯ , |𝐴̅|. The (2|𝐴̅| + 2𝑗 − 1)-th and (2|𝐴̅| + 2𝑗)-th genes denote the credit charging level for using the 𝑗-th link, 𝑗 = 1,2, ⋯ , |𝐴̅|. That is,

̅ |+2j-1)-th and (2|A ̅|+2j)-th genes = { (2|A

00, if credit charging level is zero 01, if credit charging level is one 10, if credit charging level is average 11, if credit charging level is high

(25)

where 𝑗 = 1,2, ⋯ , |𝐴̅|. Therefore, the length of the chromosome is 𝑙 = 2(|𝐴̅| + |𝐴̂|). If more than four lanes are 17

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added to the links or more than four credit levels are charged for using the links, the decoding scheme of the chromosome to represent the upper level decision variables is still applied only by using more genes. (2) Crossover procedure The one-point crossover is performed for the selected two parent chromosomes at the crossover point as follows:

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The first 𝑚 components of the children are the same components as the respective parents (i.e. the first child from the first parent and the second child from the second parent.). The remaining components are selected according to the following rules: i.

The (m  i)-th component of the first child is replaced by the (l  i  1)-th

ii.

, l  m ).

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component of the second parent (for i  1, 2,

The (m  i)-th component of the second child is replaced by the (l  i  1)-th component of the first parent (for i  1, 2,

, l  m ).

For example, three links (Links 1, 2, 3) are chose as candidate links for adding lanes and

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charging credits. The number of links for adding lanes and charging credits are all two. Two feasible decision variables (added lanes scheme and charged credit scheme) can be coded into two

ED

parents. If the 9th gene is chose as the crossover point, the children can be obtained by use of the crossover procedure in the proposed GA. They are decoded into two feasible decision variables.

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The procedure is depicted in Figure 2. After decoding the chromosomes, we can easily check whether the corresponding feasible decision variables satisfy Constraints (7) and (8), which

AC

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describe the number of links for adding lanes and charging credit.

Figure 2: Crossover procedure From the figure, we can see that the proposed operator generates chromosomes with more 18

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variety, as this operator can generate different children with their parents even from the same parents. (3) Mutation procedure The gene is mutated as follows: The value of the chosen gene is changed to 0, if it was initially 1 and to 1 if it was initially 0. For example, three links (Links 1, 2, 3) are chosen as candidate links for adding lanes and

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charging credits. Further, two links are available for adding lanes and charging credits. Two feasible decision variables (added lanes scheme and charged credit scheme) can be coded into two parents. If the 3rd gene is chosen as the mutation point, child 1 can be obtained by using the mutation procedure in the proposed GA. After decoding it, it is easily checked that child 1 is not

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feasible because it does not satisfy Constraint (7). Then, if the 7th gene is chosen as the mutation point, child 2 can be obtained by using the same way. After decoding, it is also checked that child 2

CE

PT

ED

M

is feasible because it satisfies Constraints (7) and (8). The procedure is illustrated in Figure 3.

Figure 3: Mutation procedure

(4) Selection procedure

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The new population can be generated by ranking selection. The advantage of this selection is that it allows only comparing the fitness values of every two chromosomes without considering the sign of the fitness value of each chromosome (Back, 1992). The proposed algorithm has the following steps: Step 1:

Initialing the parameters of the algorithm. Set the parameters, such as the population

size 𝑀 , probability of crossover Pc , probability of mutation Pm and the maximal generation of terminating the algorithm T. Set the counter of generation 𝑡 = 0; Step 2:

Generating the initial population P(0) . The initial population P(0) consists of a 19

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set of feasible chromosomes. The chromosome is randomly selected. If the chromosome satisfies the upper level constraints, then the generated chromosome is feasible. The lower level problem is solved by using the FW algorithm in Subsection 4.1. The upper level objective function value is assigned to the fitness value of the chromosome. After generating sufficient such chromosomes, go to the next step; Step 3:

Keeping the current best chromosome. If the counter of current generation 𝑡 equals

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to 0, then keep the best chromosome of the initial population as the current best chromosome; Otherwise, compare the best chromosome of the current population with the current best chromosome, and if the former is better than the latter, then keep the former as the current best chromosome; otherwise, the current best chromosome is not

Step 4:

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changed.

Crossover. Crossover (also called recombination) is the main method to generate

new chromosomes. This involves exchanging some genes of the two chromosomes selected for recombination. The random matching is adopted as the matching strategy, in

M

which the 𝑀 chromosomes are randomly divided into ⌊𝑀/2⌋ matching group. For every two chromosomes in each matching group, the 𝑚𝑡ℎ gene is assigned as the

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crossover point by randomly generating an integer m in the interval [1, l ] . Then, the one-point crossover is performed for them at the crossover point according to the

PT

probability of crossover Pc . If the new chromosomes are not in the feasible or infeasible lists, they are evaluated as the way in Step 2 for feasibility and are assigned the fitness

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values. If the generated chromosomes are not feasible, they are added to the infeasible list and are eliminated. Mutation. Mutation is a genetic operator for maintaining genetic diversity from one

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Step 5:

generation of a population to the next. The aim of mutation is to allow the algorithm to avoid local convergence by preventing chromosomes in populations from being too similar to each other. The mutation point is chosen according to the probability of mutation Pc and the chromosome is mutated. If the new chromosome is not in the feasible or infeasible lists, it is evaluated as the way in Step 2 for feasibility and is assigned a fitness value. If the generated chromosome is not feasible, it is added to the infeasible list and is eliminated. 20

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Step 6:

Selecting. The chromosomes are arranged in the ascending order of their fitness

values. The selected probability is set as the order, and then, the next population corresponding to the size of the original population is selected by the ranking selection. Step 7:

Terminating criterion. The algorithm terminates at the maximal iteration number.

The best generated solution, which can be obtained by the current best chromosome kept

AC

CE

PT

ED

M

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The flowchart of the proposed algorithm is as follows:

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in all iterations in the earliest time, is reported as the solution by the proposed GA.

Figure 4: Flowchart of the proposed algorithm

Note: The advantage of this type of generating rule is that the generated chromosomes are all feasible. Further, the feasibility of the chromosome is checked in the crossover and mutation steps to guarantee the chromosomes obtained by them are all feasible. This rule will save time, although it is time consuming to generate feasible chromosomes and checking their feasibility of the chromosomes in the crossover and mutation steps. 21

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5.

Numerical experiments

In order to test the feasibility and performance of the proposed model, we use the Sioux Falls network as a test example. As shown in Figure 5, the network consists of 24 nodes, 76 links and 552 OD pairs. The OD demand and the link travel functions are the same as those presented in Suwansirikul et al. (1987). The parameters of GA were set as follows: the population number is 80,

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𝑃𝑐 =0.85, 𝑃𝑚 =0.1, and the number of generations is 500. The algorithm was coded by Matlab 7.0 and the test was carried out on a laptop with an Intel Core 2 Duo 2.10GHz CPU, 2 GB RAM, and Windows XP operating system. Cplex is used in the FW algorithm for solving the lower level problem. To avoid any bias in the comparisons, the results of different models except UE and SO

algorithm presented in Subsection 4.1. 3

1

1 2

5

2

4

8

3

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were obtained by using the GA in this paper. The results of UE and SO were obtained by the FW

4

6

11 9

5

15

12

M

10 31

ED

35

33

11

32

CE

34 40

21

25 26 10

23

22 47 48

16

41

18 54

55

18

51 49 52 30 17 56 60

15

57

19

45 46 67

72

22

59 61 63

73 76

39

7

50

29

70

74

17 20

28 43

42 71

13

8

24

44

AC

16 19

53 58

14

37 38

9

27

PT

36

6

12

13 23

7

14

69 65

68

62 21 20 64 75 Potential links that may be expanded and charged Other links 24

66

Figure 5: The Sioux Falls network 22

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5.1 Experiments on the result comparison under different models In this part of experiment, the solutions to UE and SO were obtained by using FW. Other models except UE and SO were solved 20 times by the GA. We recorded corresponding results,

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such as the lanes added schemes, the credit charging schemes and link flows for every run. Further, the ratios of flow and capacity were computed. We chose the best one (not the average) of 20 runs as our results in this part of experiments because the added lanes and toll level are all integers.

Firstly, we assigned the traffic in the network following the UE principle to show the initial congestion situation by solving the TAP (Huang, 1994). Considering the lower level programming

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is UE, it can be solved only assigning the upper level decision variables to zero in our proposed model. The ratio of flow and capacity of all links are illustrated in Figure 6. These 16 links, which are indicated by dotted lines in Figure 5, are the most congested links in terms of the ratio of flow and capacity (more than 2.0). So, the flow pattern and ratios of flow and capacity for those 16 most

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congested links are listed in Table 2. The total system travel time is 10062.74 without any scheme

AC

CE

PT

ED

to alleviate the traffic congestion.

Figure 6: Ratio of flow and capacity of all the links under UE Secondly, we charged the discrete credit level on these 16 most congested links indicated by dotted

lines

in

Figure

5.

Therefore, 23

the

credit

charging

links

set

is

ACCEPTED MANUSCRIPT 𝐴̂ = {16,19,29,34,39,40,48,49,52,53,58,66,70,72,74,75} . The number of links for charging credits is set to be 10. The costs for constructing the credit charging facilities are assumed to be zero. This TCS proposed in this paper is the second-best pricing scheme because only 16 links (not all links) in the transportation network are considered for toll. The results under TCS can be obtained to examine its performance for alleviating congestion only not considering the capacity improvement scheme in our proposed model (i.e. adding 0 lane to links). The link flow pattern with

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their ratios of flow and capacity and the credit charging level are listed only for the 16 most congested links in Table 2. The total system travel time is 10003.99, which is lower than that under UE. TCS can reduce the traffic congestion, while it is not obviously effective to alleviate the traffic congestion because the decrease in total system travel time is only 58.75, which is a reduction of

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0.58% compared with results for UE.

Thirdly, the total system travel time is 9826.453 when the SO is achieved by solving TAP. The SO is obtained to show the performance of the first-best toll pricing scheme (FBTP) because SO and FBTP can realize the same performance, as established by Yang and Wang (2011). The total

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system travel time under SO can be reduced by 2.4% compared with results for UE. Therefore, the SO (FBTP) is better than SBTP in alleviating the traffic congestion when only implementing the

ED

pricing scheme.

Fourthly, we choose the improvement links set identical to the credit charging links set without

PT

loss of generality, that is, 𝐴̅ = 𝐴̂ . Thus, the transport authority considers whether it needs to add 0, 1, 2 or 3 lanes to links in A , and the travel time function on these 16 most congested links are

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adopted the form defined by Equation (1). The number of links for adding lanes is set to be 10. We assume that the budget available is set as B=5000 and each lane has a capacity of 3. Construction

AC

costs are considered as linear functions of the lanes added to links. This scheme is called to be DNDP because the discrete capacity improvements on links are considered. The construction costs for adding one lane to link a  A are as follows: Table 1: Construction costs for adding one lane to links Link No.

𝑐𝑎

Link No.

𝑐𝑎

16

6-8

260

49

16-17

330

19

8-6

260

52

17-16

330

29

10-16

380

53

17-19

280

48

16-10

380

58

19-17

280

24

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34

11-14

320

66

21-24

210

40

14-12

320

75

24-21

210

39

13-24

240

70

22-23

290

74

24-13

240

72

23-22

290

The total system travel time is 6976.82, which is calculated by solving the DNDP. This DNDP model can be obtained by issuing sufficient amount of credits for everyone to travel in our proposed model. The total system travel time can be cut down by 30.67% by DNDP compared with

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the results for UE. Therefore, increasing road supply can greatly reduce the traffic congestion.

Fifthly, considering the combination of TCS and DNDP proposed in this paper, the total system travel time is only 6895.141. The reduction can reach up to 31.48% comparing with results for UE. Thus, the integration is the most effective scheme for alleviating the traffic congestion.

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Moreover, the reduction achieved by our proposed model is more than the sum of those achieved by separately using TCS and DNDP. That is, the interaction effect of TCS and DNDP schemes is positive in alleviating the traffic congestion. The combination of these two schemes can yield better effect than sum of these two separate schemes. In brief, the integration of TCS and DNDP has the

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‘synergy’ that the simultaneous use of two or more instruments gives a greater benefit than the sum of the benefits of using either one of them alone.

ED

To illustrate the effect of our integrated model further, we consider two sequential decision models. One model considers TCS after implementing DNDP (denoted as Scenario 1), and the

PT

other one considers DNDP after obtaining the optimal TCS (denoted as Scenario 2). The total system travel times under Scenarios 1 and 2 are 6940.688 and 7263.077, respectively. The

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reductions can reach up to 31.03% and 27.82% comparing with the results for UE. Moreover, the performance of our proposed integrated model is better than Scenarios 1 and 2 in minimizing the

AC

total system travel time. That is, the integration of two schemes (DNDP and TCS) can get better effect than these two sequential schemes (first DNDP then TCS and first TCS then DNDP). To sum up, TCS as the second-best toll pricing scheme is not effective in alleviating the traffic

congestion. The total system travel time under TCS is less than that under UE, but more than that under SO. DNDP can greatly alleviate the traffic congestion because of increasing traffic supply by adding lanes to links. Our proposed model can reduce more congestion than the above schemes as our model combines the discrete credit-charging level and discrete added lanes. Furthermore, our scheme can achieve less total system travel time than the sum of those by using separate TCS and 25

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DNDP. It is good to combine TCS and DNDP schemes and consider their interaction for alleviating the traffic congestion because interaction effects across TCS and DNDP amplify the individual effects of TCS and DNDP. Furthermore, the integrated model is better than the sequential decision

AC

CE

PT

ED

M

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problems in decreasing the total system travel time.

26

Table 2: Results under different schemes UE Link No.

SO

TCS

DNDP

Our model

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Scenario 1 (First DNDP, Then TCS)

ra

ua

va

ra

va

ra

za

va

ra

za

ua

va

14.017

2.862

1

13.957

2.849

13.786

2.814

3

19.04

1.37

2

1

17.737

19

8-6

14.017

2.862

14.09

2.876

13.786

2.814

2

18.473

1.695

2

1

29

10-16

11.926

2.323

2

11.805

2.3

11.769

2.293

1

14.965

1.84

2

2

34

11-14

10.624

2.179

1

10.489

2.151

10.326

2.117

1

8.614

1.767

39

13-24

11.728

2.304

3

11.17

2.194

11.64

2.286

19.332

1.372

40

14-11

10.624

2.179

1

10.481

2.149

10.326

2.117

1

8.619

1.767

48

16-10

11.926

2.323

1

11.93

2.324

11.769

2.293

1

15.068

2

17.247

1.549

49

16-17

12.441

2.379

2

12.101

2.314

11.591

2.216

1

14.608

2

11.106

52

17-16

12.441

2.379

11.913

2.278

11.591

2.216

1

14.618

53

17-19

10.169

2.108

9.424

1.954

8.822

1.829

58

19-17

10.169

2.108

9.16

1.899

8.822

1.829

66

21-24

11.089

2.27

10.932

2.238

10.739

2.198

70

22-23

10.373

2.075

10.309

2.062

10.179

2.036

72

23-22

10.373

2.075

10.436

2.087

10.179

2.036

74

24-13

11.728

2.304

3

11.202

2.2

75

24-21

11.089

2.27

1

10.734

2.197

10062.74

10003.99

p

-

1.795

19.075

CE AC

za

ua

va

ra

za

ua

va

ra

3

1

18.758

1.35

1

1

15.979

2.023

17.87

1.64

2

1

17.928

1.65

2

18.184

1.668

17.042

1.531

1

2

14.591

1.79

2

2

15.041

1.351

2

7.856

1.68

1

1

12.736

1.354

3

1.559

19.394

1.38

1

3

10.989

1.776

1.358

8.175

1.61

1

1

12.280

1.853

2

1.617

1

14.977

1.84

2

1

15.964

1.775

1.434

2.124

1

14.702

1.79

2

10.678

1.776

2.042

11.448

2.189

1

2

14.378

1.75

10.450

1.998

7.488

1.552

6.820

1.414

7.889

1.615

3

1

2.123

2

8.509

1.764

3

9.815

2.04

M

8.66

1.774

3

8.712

1.806

2

9.491

1.97

1

13.215

1.676

16.112

1.48

9.116

1.72

2

17.375

1.422

8.594

1.82

3

19.914

1.580

19.848

1.41

1

3

10.642

1.315

15.874

1.46

1

1

6.246

1.279

10.242

2

10.252

2.125

16.349

1.502

1

9.199

1.84

1

12.812

1.602

9.19

1.838

1

12.663

1.583

2

2

11.64

2.286

3

19.234

1.365

3

19.081

1.354

3

10.739

2.198

2

16.208

1.489

1

13.615

1.727

2

1

3

9826.453

6976.82

6895.141

6940.688

7263.077

-

-

1.3

1.37

4.605

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3

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3

8.649

ra

1.627

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va

6-8

16

Scenario 2 (First TCS, Then DNDP)

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5.2 Ratios of flow and capacity under different schemes To compare the efficiency of different schemes in alleviating the traffic congestion in detail, For UE and SO, the ratio of flow and capacity for each link in the set 𝐴 is obtained by running FW one time. For models except UE and SO, the ratio of flow and capacity for each link in the set 𝐴 is obtained by computing the average values of 20 runs of GA. Then, the ratios' descriptive statistics such as maximum, mean, and standard deviation (SD)

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were computed for links in the sets 𝐴̂ ∪ 𝐴̅ and 𝐴 to demonstrate the effect of different schemes on the local network (the 16 most congested links) and the global network (all links in the network). These values are listed in Table 3. ‘Maximum’ denotes the maximal ratios of flow and capacity among links in the sets 𝐴̂ ∪ 𝐴̅ and 𝐴 to demonstrate the worst case of the local and global networks respective to the ratios of flow and capacity. ‘Mean’ denotes the mean ratios of flow and capacity among links in the sets 𝐴̂ ∪ 𝐴̅ and 𝐴 to demonstrate the average case of the local and

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global networks respective to the ratios of flow and capacity. ‘SD’ denotes the SD of ratios of flow and capacity among links in the sets 𝐴̂ ∪ 𝐴̅ and 𝐴 to demonstrate the road load uniformity of the local and global networks respective to the ratios of flow and capacity.

The TCS scheme can decrease the average ratios of flow and capacity for both the local and

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global networks. However, it increases the maximum of ratios of flow and capacity for both the local and global networks. That is, the TCS scheme will increase the traffic congestion of the worst

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link in both the local and global networks respective to the ratio of flow and capacity. The TCS scheme will increase the SD of the local network while decreasing that of the global network. In

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other words, the TCS scheme will increase the road load non-uniformity of the local network and the road load uniformity of the global network. The SO, the same as the first-best toll pricing

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scheme, can lower down the maximum and mean of the ratios of flow and capacity for both the local and global networks compared with the TCS, which is the second-best toll pricing scheme.

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Further, with regard to the SD of the ratios of flow and capacity, SO can yield the same outcome as TCS for the local network and a better one than TCS in the global network. Thus, SO (FBTP) can be better than TCS (the second-best toll pricing scheme) in alleviating congestion when they are separately implemented. The DNDP scheme can greatly reduce the maximum, mean and SD of the ratios of flow and capacity in both the local and global networks than TCS and SO. However, DNDP does not reduce the SD of the ratios of flow and capacity in the local network. Our scheme can decrease greatly the average congestion and increase the road load uniformity of the local and global networks compared with other schemes respective to the ratios of flow and capacity because 28

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its mean and SD are less than those of other schemes. However, because the TCS scheme is integrated, our scheme increases the worst case of the local network compared with the DNDP, although it is better than UE, TCS and SO. The reason is that TCS will worsen some link(s) respective to the ratio of flow and capacity. Scenarios 1 and 2 have better performance than our model on the local network while they have the same effect on the global network as our model. Thus, sequential decision models exhibit better performance on the local network than our

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integrated model although they have higher total system travel times.

The descriptive

UE

TCS

SO

Scenario 1

Scenario 2

(First DNDP,

(First TCS,

Then TCS)

Then DNDP)

2.04

2.02

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Table 3: Descriptive statistics of the ratios in the local and global networks

DNDP

statistics Maximum

2.86

2.88

2.81

local

Mean

2.31

2.25

2.22

network

SD

0.24

0.27

0.27

The

Maximum

2.86

2.88

2.81

global

Mean

1.59

1.59

1.59

network

SD

0.59

0.58

0.54

2.13

model 2.19

1.72

1.69

1.67

1.64

0.24

0.22

0.21

0.20

2.19

2.19

2.19

2.02

1.42

1.41

1.41

1.42

0.45

0.44

0.44

0.43

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The

Proposed

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5.3 Sensitivity analysis of the budget under our scheme

Figure 7：Variations of the total system travel time as increasing the budget for the capacity improvement To analyze the sensitivity of the budget, we solved this model by increasing the budget from 0 29

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to 9000 in increments of 1000. For each value of the budget, the model was solved for 20 times. The total system travel time is the average of the results from these 20 runs. Figure 7 shows that the total system travel time varies as the budget increases. We can see that the total system travel time will decrease as the budget increases. The traffic congestion can be alleviated by investing more funds to increase the road supply. However, when the budget exceeds 7760, the total system travel

improvement scheme is as follows:

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time reaches the minimum because the maximal lanes added to links is 3. The capacity

Table 4：Capacity improvement scheme when 𝑩 ≥ 𝟕𝟕𝟔𝟎

za

Link

16

2

52

19

3

53

29

0

34

0

39

0

40

2

48

3

49

2

za 3 3

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Link

3

66

3

70

0

72

0

74

3

75

0

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Further, the number of lanes added to links 29, 34, 39, 70, 72 and 75 is only 2, and even the

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budget exceeds 7760.When the number of lanes to links 29, 34, 39, 70, 72 and 75 increases up to 3, the cost of adding lanes reaches 8670, and the total system travel time is 7289.267 (the red point in

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Figure 7), which is more than the case when the budget is 7760. This is the proof of the paradox

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time.

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that link capacity improvement does not necessarily lead to a decrease in the total system travel

5.4 Effects of combinations of 𝒏𝟏 , 𝒏𝟐 on the network under our scheme

We solved the proposed model by choosing different combination of number of links for adding lanes and charging credits and recorded the corresponding total system travel time. For each combination of 𝑛1 and 𝑛2 , we solved the model 20 times, and computed the average of the total system travel time from these 20 runs. The 𝑛1 and 𝑛2 axes denote the number of links to add 30

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lanes and charge credits. The circles with coordinates (𝑛1 , 𝑛2 ) denotes the total system travel time (the values are numbers above the circles) when implementing 𝑛1 links for adding lanes and 𝑛2 links for charging credits in our proposed model. The color and size of the circles denote the value of the total system travel time (see the legend in Figure 8). The levels of the bubble scale are set by increment of the value of the total system travel time. The increment is determined by one in eight

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of the gap between the maximum and the minimum for 16 (𝑛1 , 𝑛2 ) combinations.

Figure 8: Total system travel time when choosing different number of links to add lanes and

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charge credits

From Figure 8, we observe the followings: (1). When the number of links for adding lanes is small (i.e. the number is 6), the total system

travel time will decrease as the number of links for charging credits increases. In particular, the number of links for adding lanes is 0, that is only TCS scheme is implemented. The first-best toll pricing scheme (SO) is better than the second-best toll pricing scheme, which has been illustrated by the experiment described in Subsection 5.1. (2). When the number of links for adding lanes is higher, (i.e. the number is 8), the total 31

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system travel time will be less that that when charging credits for more links. However, it changes only slightly as the number of links for charging credits increases. (3). When the number of links for adding lanes is sufficiently high (i.e. the number is 10 and 12), the total system travel time will increase as the number of links for charging credits increases. That is, when DNDP is integrated, a greater number of links for charging credits may result in worse performance. Therefore, we choose the second-best toll pricing scheme in our proposed

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scheme.

To sum up, the location of links for adding lanes and charging credits is very important for alleviating the traffic congestion. We should coordinate the number of links for adding lanes and

6.

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charging credit to achieve better performance for alleviating the traffic congestion.

Conclusions and future research directions

In this paper, the discrete bilevel model is proposed as an approach to how to mitigate traffic

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congestion. The upper level decision-maker chooses an optimal link-based discrete credit charging level and the number of lanes to be added to links to minimize the total system travel time under a

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budget constraint while accounting for the route choice behavior of users. This is done by considering the generalized travel time in the fixed OD demand case as the lower level model. The

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proposed bilevel programming model provides a new development on how to combine the ways of increasing the road supply and decreasing the travel demand to enhance the road network

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performance. More importantly, the proposed model considers not only the locations of both charging credits and adding lanes but also determine the credit charging level and the number of

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lanes. Numerical experiments on the Sioux Falls network included two parts: the comparison of our proposed model with other separate schemes (i.e. UE, TCS, SO and DNDP) and two sequential schemes (first DNDP then Toll, and first Toll then DNDP) from the viewpoints of the total system travel time and the ratios of flow and capacity. A sensitivity analysis was conducted with respect to the budget for capacity improvements and the combination of the numbers of links for charging credit and adding lanes. The GA is used only to solve our proposed mix-integer bilevel programming problem. Our 32

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main focus is on how to use GA to obtain the solutions to compare them under different schemes. Thus, we pay little attention on the efficiency and calibration of the algorithm, which is a limitation. Thus, the calibration of GA including the sensitive analysis on the parameters, the proof of convergence, rate of convergence and comparison with other existing heuristic algorithms can be left to future work. As an extension of the fixed OD demand case discussed in this paper, it can continue to discuss the elastic demand case. Moreover, extending the model to heterogeneous

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travelers will be more complicated but more interesting. The influence of the transaction costs on the results will be also a future direction for better implementation of the tradable credits scheme.

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Acknowledgement

The authors would like to thank the anonymous reviewers for the constructive comments and suggestions. This research was funded by the National Natural Science Foundation of China (Nos. 71471167, 71422010) and the Science Fund for Creative Research Groups of the National Natural

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Science Foundation of China (71621001).

References

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[1] Back, T., 1992. The interaction of mutation rate, selection and self-adaptation within a genetic algorithm, in: R. Manner, B. Manderick (Eds.), Parallel ProblemSolving from Nature, Elsevier,

CE

Amsterdam, 85–94.

[2] Bao, Y., Gao, Z., Xu, M., Yang, H., 2014. Tradable credit scheme for mobility management

AC

considering travelers’ loss aversion. Transportation Research Part E, 68, 138–154. [3] Bao, Y., Gao, Z., Yang, H., Xu, M., Wang, G., Rose, J., 2017. Private financing and mobility management of road network with tradable credits. Transportation Research Part A, 97, 158-176.

[4] Boyce, D. E., Janson, B. N., 1980. A discrete transportation network design problem with combined trip distribution and assignment. Transportation Research Part B, 14(1-2), 147-154. [5] Colson, B., Marcotte, P.,Savard, G., 2007. An overview of bilevel optimization, Annals of Operations Research, 153, 235–256. [6] Dempe, S., 2002. Foundations of Bilevel Programming. Kluwer Academic Publishers. 33

ACCEPTED MANUSCRIPT

[7] Dogterom, N., Ettema, D., Dijst, M., 2017. Tradable credits for managing car travel: a review of empirical research and relevant behavioural approaches. Transport Reviews, 37(3), 322-343. [8] Ekström, J., Sumalee, A., Hong, K. L., 2012. Optimizing toll locations and levels using a mixed integer linear approximation approach. Transportation Research Part B, 46(7), 834-854. [9] Fan, W., 2016. Optimal congestion pricing toll design for revenue maximization: comprehensive numerical results and implications. Canadian Journal of Civil Engineering, 42,

CR IP T

544-551.

[10] Fan, W., Gurmu, Z. K., 2014. Combined decision making of congestion pricing and capacity expansion: genetic algorithm approach. Journal of Transportation Engineering, 140(8): 04014031.

[11] Fan, W.B., Jiang, X.G., 2013. Tradable mobility permits in roadway capacity allocation: review and appraisal. Transportation Policy 30 (6), 132–142.

AN US

[12] Farahani, R. Z., Miandoabchi, E., Szeto, W. Y., Rashidi, H., 2013. A review of urban transportation network design problems. European Journal of Operational Research, 229(2), 281–302.

[13] Farvaresh, H., Sepehri, M. M., 2013. A branch and bound algorithm for bi-level discrete network design problem. Networks and Spatial Economics, 13(1), 67-106.

M

[14] Farvaresh, H., Sepehri, M. M., 2011. A single-level mixed integer linear formulation for a bi-level discrete network design problem. Transportation Research Part E, 47(5), 623-640.

ED

[15] FHWA, 2006. Congestion pricing: a primer, publication number: FHWA-HOP-07-074. Federal Highway Administration, Office of Transportation Management, p. 6. (retrieved

PT

20.01.10).

[16] Fontaine, P., Minner, S., 2014. Benders decomposition for discrete-continuous linear bilevel problems with application to traffic network design. Transportation Research Part B, 70(12),

CE

163-172.

[17] Fontaine, P., Minner, S., 2017. A dynamic discrete network design problem for maintenance

AC

planning in traffic networks. Annals of Operations Research, 253(2), 757-772.

[18] Frank, M., Wolfe, P., 1956. An algorithm for quadratic programming. Naval Research Logistics, 3(1-2), 95-110.

[19] Gao, Z., Wu, J., Sun, H., 2005. Solution algorithm for the bi-level discrete network design problem. Transportation Research Part B, 39(6), 479-495. [20] Giuliano, G., 1994. Equity and fairness considerations of congestion pricing. In: Special Report 242: Curbing Gridlock: Peak-Period Feesto Relieve Traffic Congestion, vol. 2. Transportation Research Board, Washington DC. [21] Goldberg, D., 1989. Genetic Algorithms in Search, Optimization and Machine Learning. 34

ACCEPTED MANUSCRIPT

Reading, MA: Addison-Wesley Professional. [22] Grant-Muller, S., Xu, M., 2014. The role of tradable credit schemes in road traffic congestion management. Transport Reviews, 34(2), 128-149. [23] Gümüs, Z. H., Floudas, C.A., 2005. Global optimization of mixed-integer bilevel programming problems, Computational Management Science, 2(3), 181-212. [24] Haas, I., Bekhor, S. 2017. An alternative approach for solving the environmentally-oriented discrete network design problem. Networks and Spatial Economics, 17(4),1-26.

Research Part A, 31(3), 205-218.

CR IP T

[25] Hansen M, Huang Y., 1997. Road supply and traffic in California urban areas. Transportation

[26] Hosseininasab, S. M., Shetab-Boushehri, S. N. 2015. Integration of selecting and scheduling urban road construction projects as a time-dependent discrete network design problem. European Journal of Operational Research, 246(3), 762-771.

AN US

[27] Huang, H.J., 1994. Urban Transportation Network Equilibrium Analysis: Theory and Practice. China Communications Press, Beijing. (in Chinese)

[28] Jan, R.H., Chern, M.S., 1994. Nonlinear integer bilevel programing. European Journal of Operational Research, 72(3), 574-587.

[29] Kahneman, D., Tversky, A., 2003. The rational choice, values and frames. Psikhologicheskii

M

Zhurnal, 24 (4), 31-42.

[30] Koh, A., Shepherd, S.P., Sumalee, A., 2009. Second best toll and capacity optimization in

ED

networks: Solution algorithm and policy implications. Transportation, 36 (2), 147-165. [31] Köppe, M., Queyranne, M., Ryan, C.T., 2010. A parametric integer programming algorithm

137-150.

PT

for bilevel mixed integer programs. Journal of Optimization Theory and Applications, 146(1),

[32] Li, C.J., Guo, L., 2017. A single-level reformulation of mixed integer bilevel programming

CE

problems. Operations Research Letters, 45(1), 1-5. [33] Lozano, L., Smith, J.C., 2017. A value-function-based exact approach for the bilevel mixed-integer programming problem. Operations Research, 65(3), 768-786.

AC

[34] Mitsos, A., 2010. Global solution of nonlinear mixed-integer bilevel programs, Journal of Global Optimization, 47(4), 557-582.

[35] Perederieieva, O., Ehrgott, M., Raith, A., Wang, J. Y. T., 2015. A framework for and empirical study of algorithms for traffic assignment. Computers & Operations Research, 54, 90-107. [36] Saharidis, G.K., Ierapetritou, M.G., 2009. Resolution method for mixed integer bi-level linear problems based on decomposition technique. Journal of Global Optimization, 44(1), 29-51. [37] Sahin, K.H., Ciric, A.R., 1998. A dual temperature simulated annealing approach for solving bilevel programming prblems. Computers & Chemical Engineering, 23(1), 11-25. 35

ACCEPTED MANUSCRIPT

[38] Szeto, W. Y., Jaber, X, Margaret, O., 2010. Time-dependent discrete network design frameworks considering land use. Computer-Aided Civil and Infrastructure Engineering, 25(6), 411-426. [39] Suwansirikul, C., Friesz, T.L., Tobin, R.L., 1987. Equilibrium decomposed optimization: a heuristic for the continuous equilibrium network design problem. Transportation Science, 21(4),254-263. [40] US Bureau of Public Roads, 1964. Traffic Assignment Manual. US Department of Commerce,

CR IP T

Washington, DC.

[41] Verhoef, E., Koh, A., Shepherd, S., 2010. Pricing, capacity and long-run cost functions for first-best and second-best network problems. Transportation Research Part B, 44(7), 870-885. [42] Viegas, J.M., 2001. Making urban road pricing acceptable and effective: searching for quality and equity in urban mobility. Transport Policy 8 (4), 289–294.

AN US

[43] Wang, D. Z. W., Liu, H., Szeto, W. Y., 2015a. A novel discrete network design problem formulation and its global optimization solution algorithm. Transportation Research Part E, 79, 213-230.

[44] Wang, G.M., Gao Z.Y., Xu, M., Sun, H.J., 2014b. Models and a relaxation algorithm for continuous network design problem with a tradable credit scheme and equity constraints.

M

Computers & Operations Research 41(44), 252-261.

[45] Wang, G., Gao, Z., Xu, M., Sun, H., 2014c. Joint link-based credit charging and road capacity

ED

improvement in continuous network design problem. Transportation Research Part A, 67, 1– 14.

[46] Wang, S., Meng, Q., Yang, H., 2013. Global optimization methods for the discrete network

PT

design problem. Transportation Research Part B, 50(4), 42-60. [47] Wang, X.L., Yang, H., Zhu, D.L., Li, C., 2012. Tradable travel credits for congestion

CE

management with heterogeneous users. Transportation Research Part E, 48(2), 426-437. [48] Xu, M., Wang, G., Grant-Muller, S., Gao, Z., 2017. Joint road toll pricing and capacity development in discrete transport network design problem. Transportation, 44(4), 731-752.

AC

[49] Yang, H., Bell, M.G.H., 2001. Transport bilevel programming problems: recent methodological advances. Transportation Research Part B, 35(1), 1-4.

[50] Yang, H., Meng, Q., 2002. Benefit distribution and equity inroad network design. Transportation Research Part B 36(1), 19-35. [51] Yang, H., Wang, X.L., 2011. Managing network mobility with tradable credits. Transportation Research Part B, 45(3), 580-594. [52] Zhang, L., Yang, H., Wu, D., Wang, D., 2014. Solving a discrete multimodal transportation network design problem. Transportation Research Part C, 49(49), 73-86. 36

ACCEPTED MANUSCRIPT

[53] Zhang, H.X, Zhou, X.Z., 2009. Congestion pricing location problem of multi-class network with social and spatial equity constraints. Journal of Systems Engineering, 24(2), 184-189. (In Chinese) [54] Zhang, X.N, Wee, B. V., 2012. Enhancing transportation network capacity by congestion pricing with simultaneous toll location and toll level optimization. Engineering Optimization, 44(4): 477-488. [55] Zhu, D.L., Yang, H., Li, C.M., Wang, X.L., 2015. Properties of the multiclass traffic network

AC

CE

PT

ED

M

AN US

CR IP T

equilibria under a tradable credit scheme. Transportation Science, 49(3), 519-534.

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