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Research Article

Optimum Location of Autonomous Vehicle Lanes: A

Model Considering Capacity Variation

Sara Movaghar,

1

Mahmoud Mesbah ,

1

,

2

and Meeghat Habibian

1

1

Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

2

e University of Queensland, Brisbane, Australia

Correspondence should be addressed to Mahmoud Mesbah; mmesbah@aut.ac.ir

Received 24 December 2019; Revised 26 February 2020; Accepted 18 March 2020; Published 12 May 2020

Academic Editor: Anna M. Gil-Lafuente

Copyright ©2020 Sara Movaghar et al. is is an open access article distributed under the Creative Commons Attribution License,

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

is paper proposes a model to ﬁnd the optimal location of autonomous vehicle lanes in a transportation network consisting of

both Autonomous Vehicles (AVs) and Human-Driven Vehicles (HDVs) while accounting for the roadway capacity variation. e

main contribution of the model is considering a generalized deﬁnition of capacity as a function of AV proportion on a link and

incorporating it into the network design problem. A bilevel optimization model is proposed with total travel time as the objective

function to be minimized. At the upper-level problem, the optimal locations of AV lanes are determined, and at the lower level

which is a multiclass equilibrium assignment, road users including both AVs and HDVs seek to minimize their individual travel

times. It is shown that if capacity variation is ignored, the eﬀect of AV lane deployment can be misleading. Since there will be a

long transition period during which both AVs and HDVs will coexist in the network, this model can help the network managers to

optimally reallocate the valuable road space and better understand the eﬀects of AV lane deployment at the planning horizon as

well as during the transition period. Employing this model as a planning tool presents how the proposed AV lane deployment plan

could consider the AV market penetration growth during the transition period. Numerical analysis based on the Sioux Falls

network is presented in two cases with and without variable capacity to illustrate the application of this model. At the 60%

penetration rate of AVs, the improvement in total travel time was 3.85% with a ﬁx capacity while this improvement was 9.88% with

a variable capacity.

1. Introduction

In recent years, Autonomous Vehicles (AVs) have received

considerable attention from diﬀerent research groups all

over the world. ey are expected to have potential beneﬁts

on safety, mobility, traﬃc eﬃciency, driver productivity,

road capacity, and environment [1–4].

Despite signiﬁcant advances in autonomous vehicle

technology, there are still many stages and many years left

to make driverless cars ready for broad adoption [5].

Considering the patterns of previous vehicle technology

deployments such as automatic transmissions, airbags, and

hybrid vehicles, it takes decades for autonomous vehicle

technology to reach market saturation [5]. Reviewing

studies about AV adoption [5–8] leads to the conclusion

that there will be a long transition period in which AVs and

human-driven vehicles (HDVs) share the road. erefore,

policymakers and practitioners should prepare strategies

for the time that AVs and HDVs coexist and share the

same transportation network. ese strategies need to

recognize diﬀerent sources of uncertainty in trans-

portation planning for AVs such as technology, adoption

rates, roadway capacity, land use changes, and travel de-

mand [9].

Due to the high uncertainty in the penetration rate of

AVs, ﬂexible and low-cost strategies are required. Among

available strategies, providing dedicated lanes to auton-

omous vehicles using existing infrastructure is a ﬂexible

and low-cost one. It is a tactical decision which can

beneﬁt the network through existing resources [10].

However, conversion of a general purpose lane to a

dedicated lane may lead to public dissatisfaction. A

Hindawi

Mathematical Problems in Engineering

Volume 2020, Article ID 5782072, 13 pages

https://doi.org/10.1155/2020/5782072

dedicated AV lane is one of the managed lane strategies

with a similar concept to High Occupancy Vehicle (HOV)

lanes, High Occupancy Toll (HOT) lanes, and express

lanes. Previous experiences with these strategies show if

they are designed and operated properly, they can im-

prove safety, eﬃciency, and the environment quality [11].

Despite the similarities between dedicated AV lanes and

other managed lane strategies, there are yet diﬀerences

such as interactions between AVs and HDVs, which make

it necessary to speciﬁcally examine the adaptation of

dedicated AV lanes.

Some studies suggest that many potential AV beneﬁts

require platooning, which can be achieved by dedicating

lanes to AVs [5]. On the other hand, eﬀective platooning and

consequently successful implementation of dedicated AV

lanes highly depends on their market penetration rate; for

example, at low market penetration rates beneﬁts may not be

signiﬁcant [12]. Several studies have used simulation models

to investigate the eﬀect of a dedicated AV lane on traﬃc ﬂow.

Microscopic simulation models are one of the tools to

consider the interactions between AVs and HDVs. Tale-

bpour et al. [12] developed a microscopic simulation model

to explore how congestion and travel time reliability are

aﬀected by reserving a lane for AVs on a hypothetical two-

lane segment and a four-lane highway segment. eir results

indicate that optional use of dedicated lanes by AVs can

improve link performance both in terms of congestion and

travel time reliability while limiting AVs to dedicated lanes

decreases the overall performance. Vander Laan and

Sadabadi [13] used macroscopic traﬃc simulation to address

eﬀects of a dedicated AV lane on the performance of a

congested corridor. ey applied it to a four-lane corridor

with one AV lane. eir results showed that the best system

performance is obtained at medium market penetration

rates (i.e., 30%, 40%, and 50%). Ye and Yamamoto [14]

reached a similar conclusion. ey proposed a microscopic

simulation model to evaluate the performance of overall

traﬃc throughput under diﬀerent AV market penetration

rates while allocating a lane to connected and autonomous

vehicles (CAVs) in a heterogeneous traﬃc ﬂow. ey found

that the positive eﬀects of CAV dedicated lanes emerge at

medium market penetration rates. Ghiasi et al. [15] for-

mulated the capacity of a one-lane highway shared by CAVs

and HDVs considering stochastic headways between dif-

ferent vehicle types. ey also presented a lane management

model that determines the optimal number of CAV dedi-

cated lanes on a highway segment while maximizing

throughput.

While most studies have explored the eﬀect of a ded-

icated AV lane in a highway segment, there are only a

limited number of studies investigating these eﬀects at the

network level. To understand the eﬀects of exclusive AV

lanes at the network level, Ivanchev et al. [16] designed an

agent-based simulation for the city of Singapore. ey

converted one regular lane of each highway segment to an

exclusive AV lane. e results indicate that the introduc-

tion of AV lanes to all highway segments worsens the entire

network performance compared to the network without

AV lanes for all AV penetration rates. e reason may be

that introducing AV lanes to all highway segments limits

the access of HDVs to a large part of the network.

erefore, the optimal location of AV lanes should be

determined in order to obtain beneﬁts at the network level.

Wu et al. [17] examined network ﬂow pattern under user

equilibrium (UE) and system optimum (SO) in a linear

traﬃc corridor with dedicated AV expressways and non-

autonomous local streets. ey also developed a problem of

AV expressway exit as an optimization bilevel problem and

explored the eﬀect of exit locations on UE and SO patterns.

Chen et al. [18] developed a mathematical framework

which determined the optimal location, the time to deploy,

and the number of AV lanes in a transportation network.

ey considered the capacity increase associated with AV

shorter headways in an exclusive AV lane; however, ca-

pacity impacts of the presence of AVs in the traﬃc ﬂow

have been ignored. In fact they did not consider that the

capacity of a roadway shared by AVs and HDVs is diﬀerent

from the capacity of the same roadway with homogeneous

traﬃc ﬂow of AVs or HDVs.

e approach used in previous studies [18] may lead to

inaccurate conclusions as we show in this study. To avoid

such inexact results, we have introduced the capacity

function of mixed traﬃc to the lower level of the network

design problem investigated in this paper.

To bridge this research gap, this study aims to develop a

mathematical model that determines the optimal location of

AV lanes in a network accounting for capacity changes of a

roadway shared by AVs and HDVs. We considered capacity

of each link as a function of link autonomy level which is the

proportion of AVs on a link to total link ﬂow. We call this

approach variable capacity case, which can be compared

against the ﬁxed capacity case (i.e., the problem solved by

[18]). is problem is designed as a network design problem

in which at the upper level, the location of AV lanes are

determined. e lower-level problem is a multiclass equi-

librium assignment in which both AVs and HDVs follow the

Wardrop equilibrium principle to minimize their individual

travel costs. No study to date has formulated a network

design problem for a network with mixed traﬃc of AVs and

HDVs while examining the eﬀects of capacity variations

associated with presence of AVs.

e main advantage of the proposed model is consid-

ering a more realistic capacity function which is already

developed in previous link level studies [15, 19] to the

network level study by applying it in traﬃc assignment and

the network design problem. e proposed change improves

the estimated travel time accuracy as well as link ﬂows in a

network. To better explain the concept and contribution of

this study, an analysis on a small network of Figure 1 is

provided below.

Consider the network of Figure 1. Let wbe the set of

origin-destination (OD) pairs. ere are two OD pairs,

w (1,3),(1,4)

{ }. e demand of these OD pairs are d13

3000 and d14 2000 vehicles per hour. Assume all links have

one lane and the AV penetration rate is 50% for both OD

pairs. AVs and HDVs are allowed on all links. e BPR

function of equation (1) is used to estimate the link travel

time:

2Mathematical Problems in Engineering

tat0

a1+0.15 Va

Ca

4

⎛

⎝⎞

⎠,(1)

where tais the link travel time, t0

ais the link free-ﬂow travel

time which is assumed to be 6 minutes for all links of this

example, and Vaand Caare link ﬂow and capacity, re-

spectively. We computed total travel time at equilibrium for

two cases: the ﬁxed capacity case and the variable capacity

case (i.e., the approach of this study). Per-lane capacity for

the ﬁxed capacity case is assumed to be 2400 vehicles and for

the variable capacity case is computed by equation (2) [16],

where hAV and hHDV are headway of AVs and HDVs to a

leading vehicle, respectively, and pais proportion of AVs on

a link:

Ca3600

hAVpa+hHDV 1−pa

.(2)

Let hAV 0.85 and hHDV 1.5, the total travel time is

equal to 15,231 vehicle minutes for the ﬁrst case and 61,341

for the second case. e results indicate that integrating

capacity variations into the link performance function im-

proves the accuracy of travel time estimation. Since traﬃc

assignment is the lower-level problem of a network design

problem, the travel time estimation aﬀects the optimal so-

lution too.

e remainder of this paper is divided into ﬁve sections.

Section 2 provides a brief review of studies related to our

work. Section 3 proposes the mathematical framework, and

Section 4 explains the solution method used to solve the

problem. In Section 5, results of numerical analysis are

presented. Section 6 concludes the paper.

2. Literature Review

In this section, studies related to this work is summarized in

three main topics: (a) network design problems solved for

AVs so far, (b) equilibrium assignment problem in trans-

portation networks with AVs and HDVs, and (c) capacity

impacts of AVs in a heterogeneous traﬃc ﬂow (mix of AVs

and HDVs). Little research has been conducted on the ﬁrst

and second topics. However, the last issue has drawn the

main attention from researchers.

2.1. Network Design Problem (NDP) in the Context of Au-

tonomous Vehicles. NDP is a well-known optimization

problem in transportation planning literature, which aims

to ﬁnd the optimal value of speciﬁc objectives (e.g., total

travel time) under limited resources. e ultimate goal of

the NDP is to determine optimal projects among a set of

alternatives while accounting for route choice behavior of

road users. NDP can be considered as a two-stage leader-

follower Stackelberg game which is usually formulated as a

bilevel problem [20]. At the upper level, the network

manager (leader) aims to optimize network performance

under resource constraints, and at the lower level, the

network users (followers) choose a route with the mini-

mum travel cost.

Research on a network design problem for diﬀerent

transportation modes with various objective functions,

decision variables, and a wide range of solution methods has

a long history in transportation [21, 22]. However, a few

network design problems have considered the context of

AVs.

Chen et al. [18] proposed an optimization model for

AV lane deployment on a transportation network in a

mixed AV and HDV environment. e upper-level

problem determined the deployment plan of AV lanes to

minimize the social cost while considering a diﬀusion

model to forecast the AV adoption rate. e lower-level

problem was a multiclass user equilibrium which followed

Wardrop’s ﬁrst equilibrium principle [23]. e objective

function was the total social cost, which included the value

of travel time for both modes and the cost related to the

unsafety factor for using HDVs in the planning horizon.

ey used the US Bureau of Public Roads (BPR) formula as

a volume-delay function while assuming a ﬁx per-lane

capacity for regular links in the network. e per-lane

capacity for AV links was assumed ﬁxed and greater than

the per-lane capacity of the adjacent regular link, given

that AVs can achieve shorter headways compared to

HDVs. Applying the proposed model to the South Florida

network showed that there was no considerable gain from

deploying AV lanes at low market penetrations (e.g., less

than 20%). Furthermore, the optimal deployment plan was

at the 25th year of a 40 year horizon where the social cost

was reduced by 3.91%.

Another related network design problem is the optimal

design of AV zones in transportation networks [24]. e

problem was expressed as a mixed integer bilevel model that

determined the optimal AV zones in a general network while

both AVs and HDVs followed the user equilibrium outside

the AV zones. AVs assumed to follow the system optimum

within the AV zones. In their study, the AV zone is deﬁned

as a zone with links dedicated to AVs. Similar to their

previous study, it was assumed that the capacity of links

within an AV zone was more than mixed AV and HDV links

and the capacity of links with mixed AV and HDV and HDV

only links were equal. Numerical analyses showed that

designing AV zones in a transportation network reduced the

total travel cost.

1 2

3

4

1

2

3

Figure 1: A simple network to illustrate the study contribution.

Mathematical Problems in Engineering 3

2.2. User Equilibrium Assignment Problem in Mixed Traﬃc.

Static equilibrium assignment is the most common problem

used as the lower level of the NDP. Although many re-

searchers have studied the problem properties, formulation,

extensions, and solution methods since the introduction of

Wardrop principles in 1952 [23], there is little literature on the

static equilibrium assignment with mixed autonomy to the best

of the authors’ knowledge. Of course, the main reason is the

emerging nature of the subject. It will be of particular im-

portance to formulate the equilibrium assignment problem

appropriately when solving an NDP where AVs and HDVs

share same roads, and all network users have a selﬁsh route

behavior. e mixed autonomy traﬃc assignment is a multi-

class equilibrium model with special properties. e multiclass

user model, which was formulated by Dafermos [25] is more

complex than a single user model. e single user model can be

formulated as a convex optimization model with a unique

solution under the assumption of separable and increasing link

performance function [26]. However, there exists no convex

formulation for a general network with multiple classes of

vehicles. In addition, class-speciﬁc link ﬂow solution is not

unique for a general network. However, the total ﬂow of each

link is unique in certain cases. For instance, with a BPR vol-

ume-delay function, ﬁxed capacity for all links, and same

capacity for all vehicle classes, uniqueness of total link ﬂows is

guaranteed [27].

Levin and Boyles [28] is the ﬁrst study that accounted for

roadway capacity changes in the presence of AVs in a traﬃc

assignment model. ey developed a four-stage travel de-

mand model to investigate the eﬀects of AV ownership on

travel behavior. ey used a static assignment model to

identify the route choice behavior of multiple classes of road

users, which are divided according to their value of time. e

travel time of their model was based on the BPR function

with capacity speciﬁed as a linear function of the jam density

as deﬁned in Greenshield’s model (equation (3). In this

study, jam density is deﬁned by equation (4) as a function of

the proportion of AVs on the link:

Caxa

ρkaxa

,(3)

kaxa

5280

lHyϵYxy

a1−Γy

AV

yϵYxy

a

⎛

⎝⎞

⎠

+5280

lAyϵYxy

aΓy

AV

yϵYxy

a

⎛

⎝⎞

⎠,

(4)

where Ca(xa)is the capacity, ka(xa)is the jam density of

link ain units of vehicles per mile, ρis the calibration

constant, and lHand lAare the maintained vehicle spacing

for HDVs and AVs, respectively. xy

ais the ﬂow of class yand

Γy

AV is a Boolean variable indicating if class yuses AVs

entirely or not. Under the assumption 2lA>lH, the resulting

volume-delay function satisﬁes the monotonicity condition

with respect to each single class which is the necessary

condition (but not suﬃcient) for the convexity of a multi-

class problem. e Frank Wolfe algorithm was used to solve

the equilibrium problem applied to Austin, Texas, and

downtown network. Although the results of the numerical

analysis showed the convergence of the algorithm, Levin and

Boyles mentioned that the algorithm may not converge, and

multiple equilibria are certainly possible.

A recent study of Mehr and Horowitz [27] on the

equilibrium state of traﬃc network in a mixed AV and HDV

environment showed that introducing capacity impacts of

autonomous vehicles in the mixed ﬂow makes the multiclass

equilibrium problem more complex. ey found that the

equilibrium problem does not have a unique solution in a

general network with both AVs and HDVs. In this problem,

neither class-speciﬁc link ﬂows nor total link ﬂows are

unique. e results are based on BPR volume-delay function

with the following form:

tafa

t0

a1+ca

fa

Caαa

βa

⎛

⎝⎞

⎠,(5)

where fais the total link ﬂow of link a,t0

ais the free ﬂow

travel time, caand βaare nonnegative parameters, and

Ca(αa)is the capacity of link awhich is a function of the

proportion of AVs on link a. Despite other available capacity

functions for estimating link capacity in a mixed condition,

which will be discussed later in Section 2.3, capacity was

deﬁned as follows:

Caαa

maMa

αama+1−αa

Ma

,(6)

where maand Maare the capacity of link awhen all vehicles

on the link are HDVs and AVs, respectively (ma<Ma).μais

deﬁned as the ratio of mato Maand is conceptually the

degree of capacity asymmetry of link a.

Mehr and Horowitz proved that using the volume-delay

function of form (5) and the capacity function of form (6), as

well as assuming a homogeneous degree of capacity asymmetry

(i.e., same μafor all links) in a general network with a ﬁxed AV

market penetration, the uniqueness of total travel time is

guaranteed for all Wardrop equilibrium ﬂow vectors.

Wang et al. [29] developed a static multiclass traﬃc as-

signment model for a mixed AV and HDV environment. eir

proposed model oﬀers a diﬀerent type of route choice behavior

for each group of road users. AV users are assumed to follow

the user equilibrium principle while HDV users are assumed to

choose their route according to a crossnested logit model. ey

considered the fact that AVs could have lower reaction time

than HDVs which can lead to an increased link capacity. To

estimate link travel time, they used the BPR function with the

link capacity formulated as follows:

Ca1

va,H/va,H +va,A

1/Ca,H

+va,A/va,H +va,A

1/Ca,A

,

(7)

where Cais the capacity of link a,Ca,A and Ca,H are the

capacity of link awhen all vehicles on the link are AVs and

HDVs, respectively, and variables va,A and va,H are the AV

and HDV ﬂows on link a. ey have applied a new route-

swapping-based solution algorithm to solve the problem.

e suggested algorithm is path-based and can only ﬁnd one

local optimal depending on the initial path ﬂow pattern.

4Mathematical Problems in Engineering

2.3. Capacity Impacts of Autonomous Vehicles in the Het-

erogeneous Traﬃc Flow. Several studies suggest that AVs can

improve roadway capacity through the possibility of pla-

tooning and shorter safe-intervehicle distance compared to

HDVs. Tientrakool et al. [30] compared the highway ca-

pacity of three vehicle technologies: manual vehicles, ve-

hicles with sensors, and communicating vehicles. e results

indicated that when all vehicles were equipped with sensors,

the capacity was about 1.5 times of when all vehicles were

manual. Furthermore, communicating vehicles could lead to

a 237% increase in the highway capacity.

Chen et al. [31] proposed an analytical formulation for

capacity of a single-lane and a two-lane highway with

heterogeneous traﬃc of both AVs and HDVs. ey extend

their formulation to a general case of a multilane highway.

According to their study, the AV penetration rate, platoon

size, and spacing characteristics were the parameters that

aﬀect capacity. ey also proposed capacity functions under

diﬀerent lane policies including exclusive AV or HDV lanes

and mixed-used lanes. ey found that policies that seg-

regate AVs and HDVs (i.e., have exclusive HDV and AV

lanes), led to a lower capacity than policies including mixed-

used lanes.

Ghiasi et al. [15] formulated a mixed traﬃc capacity as a

function of the AV market penetration, platooning intensity,

and mixed traﬃc headway settings based on a Markov chain

model. ey showed that unlike the generally accepted as-

sumption, higher market penetration and platooning in-

tensity did not necessarily lead to higher capacity, since

headway setting is dependent on the AV technology and

plays an important role in determining the capacity.

Lazar et al. [19] proposed two capacity models under

two diﬀerent scenarios. e main assumption in the ﬁrst

scenario was that AVs are capable of maintaining reduced

headways regardless of the preceding vehicle. e second

scenario assumed that the headway depends on the pre-

ceding vehicle, and a short headway is possible when an

AV follows another AV. In this case, the distribution of

vehicles on the road is important. is study presented a

capacity formulation assuming that vehicle types are

determined as a result of the Bernoulli process. In both

scenarios, capacity was the function of proportion of AVs

on the road. e capacity calculated in the ﬁrst scenario

served as an upper bound for the capacity in the second

scenario [32].

3. Problem Statement and Formulation

In this study, we consider a network with two types of lanes:

regular lanes and AV lanes. Two modes use the trans-

portation network: AVs and HDVs. AVs can use all lanes in

the network and HDVs can only use regular lanes. e

demand for each origin-destination (O-D) pair is assumed

ﬁxed for each travel mode. e route choice behavior of both

modes is based on Wardrop’s ﬁrst principle.

We formulated AV lane location problem as an integer

bilevel programming (IBP). e IBP consists of optimally

selected lanes from the existing network to be dedicated to

AVs. Below we summarize the notations used in the model.

Sets:

N:set of nodes

A:set of links

A: set of candidate AV links

K:set of paired links

M:set of travel modes: mode 1 denotes HDVs and

mode 2 denotes AVs

W:set of origin-destination (OD) pairs

Parameters:

m:index of travel mode, m∈M

w:index of OD pair, w∈W

dw,m:demand of travel mode m∈Mbetween OD

pair w∈W

A: the node-link incidence matrix with elements αna,

n∈N, a ∈A.Aαna

, if link astarts in node n,

αna 1; if link aends in node n,αna −1; otherwise

αna 0

Dw,m: the demand vector with dw,m

n,

n∈N, w ∈W, m ∈M. Dw,m dw,m

n

, if node nis the

origin of w,dw,m

ndw,m; if node nis the destination of

w,dw,m

n −dw,m; otherwise dw,m

n0

La:number of lanes on link a∈A

ξ:capacity reduction factor in mixed traﬃc

hAV:headway of AVs to a leading vehicle

hHDV:headway of HDVs to a leading vehicle

dk

a:if link abelongs to the kth link pair, and it is an

AV link, dk

a1; if it is a regular link, dk

a −1; oth-

erwise dk

a0

Variables:

xw,m

a:ﬂow of travel mode m∈Mon link a∈A

between OD pair w∈W

xa:aggregate ﬂow on link a∈A

yk:if one lane of the kth link pair is converted to an

AV lane, yk1; otherwise yk0

Ca:total capacity of link a∈A

ca:per-lane capacity of link a∈A

pa:proportion of AVs on link a∈A

3.1. Network Representation. Let G(N, A)represent a general

transportation network with node set Nand link set A. Deﬁne

Aas a set of candidate links to introduce an AV lane (

A⊂A).

Each candidate AV link is paired with a regular link. Candidate

links are links with more than two lanes because there should

be at least one regular lane between each OD pair to maintain

network connectivity. Figure 2 illustrates a sample network

topology. In Figure 2, the solid line represents regular links, and

the dotted line denotes the candidate AV links. e set of

paired links is deﬁned as K. Speciﬁcally, in Figure 2,

A1,2,3,4,5

{ },

A4,5

{ }, and K (1,4),(2,5)

{ }. In this

example, links 1 and 2 have two lanes and link 3 has just one

lane, so no paired AV link is deﬁned for link 3.

3.2. Multiclass User Equilibrium Problem. An equilibrium

model for a general multiclass transportation network can be

Mathematical Problems in Engineering 5

written as a Variational Inequality (VI) problem of equation

(8) [33, 34]. e solution of the following VI formulation

with the feasible region in Ωis x∗. It can be shown that x∗

satisﬁes the multiclass user equilibrium condition as deﬁned

by Dafermos [25]:

w

m

a

tax∗

xw,m

a−xw,m∗

a

≥0,∀x∈ Ω.(8)

e feasible region Ωin a multiclass transportation

network with exclusive AV lanes is mathematically de-

scribed by equations (9)–(13). Constraint (9) simply ensures

the conservation of ﬂow in the network. Constraint (10) is

deﬁned on sets of regular links (i.e., A\

A). Nonnegativity of

ﬂows is expressed by (10) and (12). By (10) and (11), we allow

HDVs on regular links, and by (12) we allow AVs on both

lane types. Constraint (13) deﬁnes aggregate link ﬂow as the

sum of link ﬂows across all modes and O-D pairs:

Axw,m Dw,m,∀w∈W, ∀m∈M, (9)

xw,1

a≥0,∀a∈A\

A, ∀w∈W, (10)

xw,1

a0,∀a∈

A, ∀w∈W, (11)

xw,2

a≥0,∀a∈A, ∀w∈W, (12)

xa

m∈M

w∈W

xw,m

a,∀a∈A. (13)

e link performance function, which calculates travel

time, is the BPR function in the form of equation (14), where t0

a

is the free ﬂow travel time of link aand Cais the capacity of link

a. In this study, we assume that the capacity of each link is the

per-lane capacity times the number of lanes (equation (15). e

per-lane capacity is considered similar to that of Mehr and

Horowitz [27] (equation (16) with homogeneous degree of

capacity asymmetry (i.e., for all links (hAV/hHDV)is constant).

is function gives the lane capacity based on the share of AVs

in the ﬂow and also ensures that a unique total travel time is

resulted despite the fact that multiple Wardrop equilibria might

exist. According to Lazar et al. [19], the proposed capacity

function gives an upper bound for capacity; thus, we consider ξ

as a capacity reduction factor in mixed traﬃc. It is a realistic

assumption since it is unlikely that shared use lanes reach their

maximum capacity due to complex interactions between AVs

and HDVs. ξis a positive coeﬃcient less than 1.0 for shared use

lanes and is equal to 1.0 for exclusive AV lanes. Link autonomy

level which is deﬁned as the share of AVs on a link is calculated

by equation (17):

taxa, Ca

t0

a1+0.15 xa

Capa

4

⎛

⎝⎞

⎠,(14)

Capa

capa

La,(15)

capa

3600ξ

hAVpa+hHDV 1−pa

,(16)

paw∈Wxw,2

a

w∈Wxw,1

a+w∈Wxw,2

a

.(17)

3.3. AV Lane Location Problem. e problem of determining

the optimal location of AV lanes in the network is a bilevel

program in which the lower level is a multiclass user

equilibrium expressed by equations (8)–(13), and the upper-

level problem ﬁnds the AV lane location in the network to

minimize the total travel cost. e upper-level problem can

be formulated as follows.

Integer bilevel program:

min

x,y

a∈A

taxa

xa,(18)

s.t.

CacaLa+

k∈K

dk

ayk

⎛

⎝⎞

⎠,∀a∈A, (19)

yk∈0,1

{ },∀k∈K. (20)

Objective (18) is to minimize the total travel time of the

transportation network where xais determined by the lower

level user equilibrium model. A binary variable ykis in-

troduced to indicate whether an AV candidate link is

converted to AV link or not. dk

ais deﬁned as a parameter

which can take three values as below:

dk

a

1,

−1,

0,

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

if link a belongs to the kth link pair and it is an AV link,

if link a belongs to kth link pair and it is a regular link,

else.

(21)

1 2

3

4

1

2

3

4

5

Figure 2: An example of network topology.

6Mathematical Problems in Engineering

e capacity of link ais calculated by equation (19).

In brief, the problem is to ﬁnd the optimal location of AV

lanes in a network with mixed traﬃc ﬂow of AVs and HDVs

while minimizing the total travel time as the objective

function and accounting for capacity variation. us, given

the transportation network with a speciﬁed demand matrix,

the output of the model is the optimal location of AV lanes.

Below, we summarize basic assumptions of our model:

(i) A link with two lanes or more can be a candidate of

an AV lane

(ii) Both road users choose their routes based on the

user equilibrium principle

(iii) e link travel time is estimated by the BPR function

with the variable capacity formulated by equation

(16)

(iv) e ratio of hAV (i.e., headway of AVs to a leading

vehicle) to hHDV (i.e., headway of HDVs to a leading

vehicle) is the same for all network links

4. Solution Method

e transportation network design problem (TNDP) is NP-

hard even in the simplest form [35]. Several solution

methods, such as exact methods, heuristics, and meta-

heuristics have been developed to solve the TNDP

[22, 36, 37]. Hybrid heuristics and metaheuristics ap-

proaches are other methods that can be used to solve net-

work design problems. e exact methods such as the

branch and bound algorithm proposed by Leblanc [38] will

become computationally ineﬃcient in case of large-size

networks due to its high accuracy. On the contrary, heu-

ristics and metaheuristics are more eﬃcient but less accurate

methods. Hybrid methods which combine the two methods

are more eﬃcient than one heuristic or metaheuristic al-

gorithm, as shown by some studies [39–41]

To solve the bilevel problem proposed in this study, we

applied a heuristic method proposed by Bagloee et al. [42]

which is a hybrid machine learning and optimization. e

method employs a multivariate linear regression and an

integer linear program to solve the bilevel problem itera-

tively. e algorithm starts with a feasible solution. en, the

traﬃc assignment problem (i.e., the lower-level problem) is

solved based on this solution, and the objective function of

the upper-level problem is computed. e objective function

is regressed against the decision variables, and the coeﬃ-

cients of decision variables are calibrated. e next step is to

solve an integer linear program (ILP), which ﬁnds a new

feasible decision variable for the next iteration. e lower-

level problem is solved based on the new solution, and a new

regression is applied to the updated data. e termination

criterion is the number of iterations, which is a user-spec-

iﬁed parameter. e algorithm is applicable to large scale

networks and can quickly converge to a local optimum [42].

Let

i: iteration counter

imax: maximum number of iterations

yk

i: the feasible solution at iteration i

Zi: the objective function value at iteration i

e proposed algorithm is as follows [42]:

Step 0: specify imax; set i0; set yk

00 as the initial

feasible solution.

Step 1: solve the traﬃc assignment problem for the

given yk

iand compute the objective function Zi.

Step 2: estimate new regression coeﬃcients of equation

(18) using yk

iand the corresponding Zi:

Zi

k∈K

bkyk+b0.(22)

Step 3: update the ILP of equations (19)–(25) according

to the current binary solution yk

iand estimated

coeﬃcients:

min

yk

k

bkyk,(23)

s.t. yk∈0,1

{ },(24)

k∈Yj

1

yk−

k∈Yj

0

yk≤Yj

1

−1,where Yj

1k|yk

j1

;

Yj

0k|yk

j0

, j 1,..., i.

(25)

Step 4: if the termination criterion is met (i.e., iimax),

stop and report the best objective function (Zi) and

corresponding (yk

i) as the ﬁnal solution, else, solve the

updated ILP in step 3, introduce yk

i+1as a new feasible

solution, set ii+1, and go to Step 1.

5. Numerical Analysis

5.1. Basic Settings. e performance of the proposed model

on the Sioux Falls network is presented in this section. e

network has 76 links and 24 nodes and its data is available at

a repository maintained by Bar-Gera [43] including node

coordinates, start node, and end node of each link, capacity,

length, and free ﬂow travel time of each link. e given OD

ﬂows are 0.1 of the original daily ﬂows in LeBlanc et al. [43].

ere is no explicit data on the number of lanes of each link.

erefore, we have converted the given capacity to the

number of lanes assuming that capacity is deﬁned for the

same period as OD ﬂows and per-lane capacity of each link is

2400 vehicles per hour. Table 1 shows the results of the

calculation. Note that the values calculated for number of

lanes are rounded up to the nearest integer. Since the only

constraint for a link to be a candidate of introducing an AV

lane is to have more than one lane, there are 16 two way links

that make our alternative set. Dotted links in Figure 3 are the

candidates to get an AV lane. In Figure 3, both directions of a

link are considered as one alternative (e.g., (1, 2) and (2, 1) is

considered as one alternative to get an exclusive AV lane or

not). Note that each candidate for an AV lane is paired with a

Mathematical Problems in Engineering 7

regular link, as shown in Figure 2. e capacity reduction

factor (ξ) in mixed traﬃc is set to 0.8 for a mixed-used lane

and 1 for an exclusive AV lane. e headway of AVs and

HDVs from the preceding vehicle is 0.85 and 1.5 seconds,

respectively.

112

3 4 56

2

9 8 7

10 16 181112

17

191514

2223

24 21 2013

15

4

5

6

78

9

10

11

12

1

2

313

14

Candidate AV lane

Regular lane

Figure 3: e Sioux Falls network, location of AV lane candidates.

Table 1: Number of lanes on each link.

Start

node

End

node

Capacity (vehicle per

hour)

Number of

lanes

1 2 10791.75 5

1 3 9751.45 5

2 1 10791.75 5

2 6 2065.91 1

3 1 9751.45 5

3 4 7129.38 3

3 12 9751.45 5

4 3 7129.38 3

4 5 7409.50 4

4 11 2045.34 1

5 4 7409.50 4

5 6 2061.66 1

5 9 4166.67 2

6 2 2065.91 1

6 5 2061.66 1

6 8 2041.08 1

7 8 3267.42 2

7 18 9751.45 5

8 6 2041.08 1

8 7 3267.42 2

8 9 2104.25 1

8 16 2102.43 1

9 5 4166.67 2

9 8 2104.25 1

9 1 5798.25 3

10 9 5798.25 3

10 11 4166.67 2

10 15 5630.00 3

10 16 2022.88 1

10 17 2080.63 1

11 4 2045.34 1

11 1 4166.67 2

11 12 2045.34 1

11 14 2031.88 1

12 3 9751.45 5

12 11 2045.34 1

12 13 10791.75 5

13 12 10791.75 5

13 24 2121.36 1

14 11 2031.88 1

14 15 2136.47 1

14 23 2052.00 1

15 1 5630.00 3

15 14 2136.47 1

15 19 6068.65 3

15 22 3999.66 2

16 8 2102.43 1

16 1 2022.88 1

16 17 2179.13 1

16 18 8199.96 4

17 1 2080.63 1

17 16 2179.13 1

17 19 2009.98 1

18 7 9751.45 5

18 16 8199.96 4

18 2 9751.45 5

19 15 6068.65 3

19 17 2009.98 1

19 2 2084.42 1

Table 1: Continued.

Start

node

End

node

Capacity (vehicle per

hour)

Number of

lanes

20 18 9751.45 5

20 19 2084.42 1

20 21 2108.30 1

20 22 2114.87 1

21 2 2108.30 1

21 22 2179.13 1

21 24 2035.57 1

22 15 3999.66 2

22 2 2114.87 1

22 21 2179.13 1

22 23 2083.33 1

23 14 2052.00 1

23 22 2083.33 1

23 24 2116.05 1

24 13 2121.36 1

24 21 2035.57 1

24 23 2116.05 1

8Mathematical Problems in Engineering

To solve the proposed bilevel problem, a code in python

is developed which is synchronized to GAMS (the General

Algebraic Modeling System), which is used to formulate the

ILP problem as one part of the heuristic algorithm explained

in Section 4. e rest part of the algorithm is coded in

python.

5.2. Optimal Solution. During the transition period, market

penetration rate grows from very low values to relatively

high values until it reaches market saturation. Within the

ﬁrst stages of AV implementation, not much beneﬁt would

be obtained from the introduction of AV lanes to the

network; however, more beneﬁts are expected at higher

market penetration rates. To approve this hypothesis we

have conducted a sensitivity analysis on market penetration

rates, the results of which are shown in Table 2. e beneﬁts

began to be signiﬁcant at the 40% penetration rate, while

improvements at lower rates (i.e., 10%, 20%, and 30%) are so

marginal that a network manager may prefer not to im-

plement any AV lanes. According to the analysis, an im-

provement in the objective function value was maximized at

the market penetration rate of 60%. e beneﬁts of

deploying AV lanes were reduced at both lower and higher

rates. e optimal location plan at a market penetration rate

of 50% is shown in Figure 4 as an example. is solution

contains a selection of 12 alternatives out of 16 possible

alternatives. e total travel cost at the optimal scenario was

89371 vehicle hours, which was 3.26 percent better than the

base scenario (i.e., do nothing).

In addition to the optimal plan, a practical plan is also

proposed. If we want to present a practical AV lane de-

ployment plan to the network manager while accounting for

growth in the AV market penetration, we should consider

practical hassle of adding or removing an AV lane every year

during the transition period. Investigating the results shows

that there are common alternatives in the optimum plan of

most market penetration rates. For instance, alternatives 1

and 10 are selected in all penetration rates above 20%. Using

this observation and taking into account the fact that little

beneﬁt is gained when the market penetration rate was below

40%, it can be recommended to start the implementation of

AV lanes when 40% of all vehicles on the network are

autonomous. e optimal solution at 40% includes alter-

natives 1, 7, 8, 10, 11, 12, and 14. We add alternatives 3, 9, 14,

and 16 at 50%, and for 60%, 70%, and 80% alternative 2 is

added to the previous set of alternatives. When the AV

market size grows to 90%, fewer lanes need to be converted

to exclusive AV lanes; therefore, alternatives 3, 7, and 8 were

no longer selected to have an AV lane. Some alternatives that

were present in the optimal plan are not considered in the

practical plan to decrease the practical hassle of adding and

removing AV lanes every year during the transition period.

Running the traﬃc assignment and calculating the objective

function for this deployment plan (see Table 3) indicates that

there is a little diﬀerence between this plan and the optimal

plan of Table 2; therefore, the network beneﬁts from

implementation of AV lanes are maximized with minimum

disturbances to the network structure.

In order to show the computational eﬃcacy of the

applied algorithm, the number of iterations and the CPU

time to reach the optimal solution is presented in Table 4.

Table 2: e optimal plan for diﬀerent market penetration rates

(variable capacity).

Market

penetration rate Optimal plan

Improvement in the

objective function

(percent)

10 0000000000

0 0 0 0 0 0 0.00

20 0 0 1 0 0 1 0 0 0 0

0 0 1 1 1 0 0.11

30 1 0 1 0 0 0 1 1 0 1

0 0 1 0 0 0 0.67

40 1 0 0 0 0 0 1 1 0 1

1 1 0 1 0 0 1.87

50 1 0 1 0 0 1 1 1 1 1

1 1 1 1 0 1 3.26

60 1 1 1 0 0 0 1 1 1 1

1 1 1 1 0 1 3.85

70 1 1 1 0 0 1 1 1 1 1

1 1 0 1 0 1 2.83

80 1 1 1 0 0 0 1 1 1 1

1 1 1 1 0 1 2.19

90 1100000011

1 1 0 1 0 1 1.57

112

3456

2

987

10 16 18

11

12

17

19

15

14

22

23

24 21 20

13

15

4

5

6

78

9

10

11

12

1

2

313

14

Links that did not get an AV lane

Links that were not candidates for an AV lane

Links that were introduced with an AV lane

Figure 4: Optimal plan at market penetration of 50%.

Mathematical Problems in Engineering 9

We have used a laptop computer with Intel Corei7 of

2.70 GHz and 12 GB of RAM. It is worth noting that the

optimal solution in all cases is the global optimal solution

which is already identiﬁed by the exhaustive enumeration.

e results show that the algorithm converges to the

optimal solution in early iterations with a reasonable CPU

time.

5.3. e Variable Capacity Case Versus the Fixed Capacity

Case. Table 5 shows the optimal plan for diﬀerent market

penetration rates assuming a ﬁxed per-lane capacity. In this

case, the capacity of each regular lane is set to 2400 vehicles

per hour and the capacity of each AV lane is set to 4200

vehicles per hour, which equals to 1.75

((hAV/hHDV ) (1.5/0.85) 1.75) times the per-lane ca-

pacity of a regular lane. e comparison of the ﬁxed capacity

with variable capacity cases indicates that not only the

optimal plan varies at each market penetration rate but also

the trend of estimated beneﬁts obtained by the AV lane

deployment is signiﬁcantly overestimated. As shown in

Table 5, the reduction in the total travel time caused by an

AV lane deployment increased up to market penetration

rate of 60% and remained constant with the increase of

AV market size. Compared to its corresponding result in

the variable capacity case, the beneﬁts are overestimated

in the ﬁxed capacity case. e reason is that, in the variable

capacity case, the base scenario total travel time (TTTbs)

decreases as the AV market size grows. is decrease is

due to the dependence of capacity to market penetration

rate, while the ﬁxed capacity case ignores the change of

capacity and thus, takes the base scenario total travel time

as constant. It can be concluded that the introduction of

AVs to the network improved the TTTbs itself, as shown in

Figure 5 and, therefore, the estimated beneﬁt in the ﬁxed

capacity case is an overestimated value. is point can

have signiﬁcant implications in network planning and

should be considered in the process of decision making by

the network manager to have a reasonable expectation

from the network improvement obtained by an AV lane

deployment.

Table 3: e proposed practical plan.

Market penetration rate Proposed plan Improvement diﬀerence with the optimal plan (percent)

10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000

20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −0.110

30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −0.670

40 1 0 0 0 0 0 1 1 0 1 1 1 0 1 0 0 0.000

50 1 0 1 0 0 0 1 1 1 1 1 1 1 1 0 1 −0.003

60 1 0 1 0 0 0 1 1 1 1 1 1 1 1 0 1 −0.027

70 1 0 1 0 0 0 1 1 1 1 1 1 1 1 0 1 −0.003

80 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 0.000

90 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 1 −0.010

Table 4: Number of iterations and CPU time to reach the optimal solution.

Market penetration rate Number of iterations to reach the optimal solution CPU time (min) to reach the optimal solution

10 19 2.96

20 35 6.76

30 54 12.02

40 46 11.52

50 50 11.50

60 65 15.10

70 59 13.02

80 48 11.46

90 23 3.94

Table 5: e optimal plan for diﬀerent market penetration rates

(ﬁxed capacity).

Market

penetration rate Optimal plan Improvement in the

objective function (percent)

10 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0.00

20 0 0 1 1 0 01 1 01 0

0 1 1 0 0 0.39

30 1 0 0 0 0 0 1 1 01 1

0 0 1 0 0 2.36

40 1 0 1 0 0 0 1 1 1 1 1

1 1 1 0 1 5.99

50 1 1 1 0 0 1 1 1 1 1

1 1 1 1 0 1 9.04

60 1 1 1 0 1 1 1 1 1 1

1 1 1 1 0 1 9.88

70 1 1 1 0 1 1 1 1 1 1

1 1 1 1 0 1 9.90

80 1 1 1 1 1 1 1 1 1 1

1 1 1 1 0 1 9.89

90 1 1 1 1 1 1 1 1 1 1

1 1 1 1 0 1 9.91

10 Mathematical Problems in Engineering

6. Conclusion

is paper proposes a mathematical model that determines

the best scheme of road space reallocation between AVs and

HDVs by locating AV lanes in a transportation network. e

bilevel optimization problem is formulated as a discrete

network design problem which is an NP-hard problem and a

hybrid machine-learning and optimization method is

employed to solve it. e location of AV lanes is speciﬁed at

the upper-level problem. e multiclass assignment problem

at the lower level is formulated as a variational inequality

since it is not a convex problem. e main contribution of

the model is investigating the capacity impacts of autono-

mous vehicles at the network level. To achieve this goal, we

used the BPR volume-delay function as the travel time

function in the route choice procedure. e link capacity is

one of the parameters of BPR function which is usually

assumed as a constant value due to the link speciﬁcations. In

this study the link capacity is considered as a function of AV

proportion in a link, and, therefore, it can take diﬀerent

values according to the class-speciﬁc link ﬂows.

e model was applied on the Sioux Falls network and

the numerical results were presented at various market

penetration rates. It was found that the beneﬁts (i.e., re-

duction in total travel time) gained from AV lane de-

ployment is highly aﬀected by the market penetration rate.

To derive beneﬁts from the implementation of AV lanes

requires relatively wide adaptation of AVs (i.e., more than

30%). e results indicate that the beneﬁt is maximum at

medium ranges (i.e., 60%) of the market penetration rate.

At low market penetration rates (i.e., 10%, 20%, and 30%),

AV lane deployment does not lead to signiﬁcant network

improvements. e results also point out that ignoring the

capacity variation would lead to diﬀerent optimal plans as

well as overestimation in beneﬁts. A practical deployment

plan during transition period, when market penetration

rate grows from low levels to the saturation rate, is pre-

sented in this study. e practical plan considers the dis-

turbances that adding or removing an AV lane may cause in

the network.

is study can be followed up in diﬀerent directions. In

the proposed model, the total travel time serves as the

objective function, while it is possible to add other social

costs and account for diﬀerent values of time of each user

class. It is also interesting to examine the results using

diﬀerent capacity functions developed in various studies.

e model can be extended to a multistage problem in which

time is one of the dimensions. Since the introduction of AVs

can aﬀect travel demand and urban development, another

ﬁeld of research is to integrate the optimal AV lane location

problem with AV travel impacts such as spatial distribution

[44, 45], ridesharing [46], parking demand [47], and mode

choice [48], which are examined separately in some previous

studies.

Data Availability

e data used to support the ﬁndings of this study are

available from Leblanc et al. [43].

Conflicts of Interest

e authors declare that they have no conﬂicts of interest.

References

[1] L. M. Clements and K. M. Kockelman, “Economic eﬀects of

automated vehicles,” Transportation Research Record: Journal

of the Transportation Research Board, vol. 2606, no. 1,

pp. 106–114, 2017.

[2] D. J. Fagnant and K. Kockelman, “Preparing a nation for

autonomous vehicles: opportunities, barriers and policy

recommendations,” Transportation Research Part A: Policy

and Practice, vol. 77, pp. 167–181, 2015.

[3] T. Folsom, “Energy and autonomous urban land vehicles,”

IEEE Technology and Society Magazine, vol. 31, no. 2,

pp. 28–38, 2012.

[4] W. Gruel and K. F. Stanford, “Assessing the long-term eﬀects

of autonomous vehicles: a speculative approach,” Trans-

portation Research Procedia, vol. 13, pp. 18–29, 2016.

[5] T. Litman, Vehicle Implementation Predictions, Victoria

Transport Policy Institute, Victoria, Canada, 2018.

84000

86000

88000

90000

92000

94000

96000

0 102030405060708090100

AV market penetration rate

Base plan

Optimal plan

Total vehicle hours traveled

(a)

66000

76000

86000

96000

106000

116000

126000

136000

0 102030405060708090100

AV market penetration rate

Base plan

Optimal plan

Total vehicle hours traveled

(b)

Figure 5: Total vehicle hours traveled in the network for diﬀerent market penetration rates. (a) Variable capacity case. (b) Fixed capacity

case.

Mathematical Problems in Engineering 11

[6] P. Bansal and K. M. Kockelman, “Forecasting Americans’

long-term adoption of connected and autonomous vehicle

technologies,” Transportation Research Part A: Policy and

Practice, vol. 95, pp. 49–63, 2017.

[7] L. J. Lavasani, E. K. Jin, and W. P. Du, “Market penetration

model for autonomous vehicles on the basis of earlier tech-

nology adoption experience,” Transportation Research,

vol. 2597, no. 5, pp. 67–74, 2016.

[8] R. Shabanpour, A. Shamshiripour, and A. Mohammadian,

“Modeling adoption timing of autonomous vehicles: inno-

vation diﬀusion approach,” Transportation, vol. 45,

pp. 1607–1621, 2018.

[9] B. J. Cottam, “Transportation planning for connected au-

tonomous vehicles: how it all ﬁts together,” Transportation

Research Record: Journal of the Transportation Research

Board, vol. 2672, no. 51, pp. 12–19, 2018.

[10] G.-L. Jia, R.-G. Ma, and Z.-H. Hu, “Review of urban trans-

portation network design problems based on CiteSpace,”

Mathematical Problems in Engineering, vol. 2019, Article ID

5735702, 22 pages, 2019.

[11] B. Kuhn, G. Goodin, A. Ballard et al., “Managed lanes

handbook,” Texas Transportation Institute, College Station,

TX, USA, 2005.

[12] A. Talebpour, H. S. Mahmassani, and A. Elfar, “Investigating

the eﬀects of reserved lanes for autonomous vehicles on

congestion and travel time reliability,” Transportation Re-

search Record: Journal of the Transportation Research Board,

vol. 2622, no. 1, pp. 1–12, 2017.

[13] L. J. Vander Laan and K. F. Sadabadi, “Operational perfor-

mance of a congested corridor with lanes dedicated to au-

tonomous vehicle traﬃc,” International Journal of

Transportation Science and Technology, vol. 6, no. 1, pp. 42–52,

2017.

[14] L. Ye and T. Yamamoto, “Impact of dedicated lanes for

connected and autonomous vehicle on traﬃc ﬂow through-

put,” Physica A: Statistical Mechanics and its Applications,

vol. 512, pp. 588–597, 2018.

[15] A. Ghiasi, O. Hussain, Z. Qian, Sean), and X. Li, “A mixed

traﬃc capacity analysis and lane management model for

connected automated vehicles: a Markov chain method,”

Transportation Research Part B: Methodological, vol. 106,

pp. 266–292, 2017.

[16] J. Ivanchev, A. Knoll, D. Zehe, S. Nair, and D. Eckhoﬀ,

“Potentials and implications of dedicated highway lanes for

autonomous vehicles,” 2017, https://arxiv.org/abs/1709.

07658.

[17] W. Wu, F. Zhang, W. Liu, and G. Lodewijks, “Modelling the

traﬃc in a mixed network with autonomous-driving ex-

pressways and non-autonomous local streets,” Transportation

Research Part E: Logistics and Transportation Review, vol. 134,

Article ID 101855, 2020.

[18] Z. Chen, F. He, L. Zhang, and Y. Yin, “Optimal deployment of

autonomous vehicle lanes with endogenous market pene-

tration,” Transportation Research Part C: Emerging Technol-

ogies, vol. 72, pp. 143–156, 2016.

[19] D. A. Lazar, S. Coogan, and R. Pedarsani, “Capacity modeling

and routing for traﬃc networks with mixed autonomy,” in

Proceedings of the 2017 IEEE 56th Annual Conference on

Decision and Control (CDC), pp. 5678–5683, Melbourne,

Australia, December 2017.

[20] E. Miandoabchi, F. Daneshzand, R. Zanjirani Farahani, and

W. Y. Szeto, “Time-dependent discrete road network design

with both tactical and strategic decisions,” Journal of the

Operational Research Society, vol. 66, no. 6, pp. 894–913, 2015.

[21] M. Bagherian, M. Mesbah, and L. Ferreira, “Using delay

functions to evaluate transit priority at signals,” Public

Transport, vol. 7, no. 1, pp. 61–75, 2015.

[22] R. Z. Farahani, E. Miandoabchi, W. Y. Szeto, and H. Rashidi,

“A review of urban transportation network design problems,”

European Journal of Operational Research, vol. 229, no. 2,

pp. 281–302, 2013.

[23] M. J. Patriksson, e Traﬃc Assignment Problem: Models and

Methods, Dover Publications, Mineola, NY, USA, 2015.

[24] Z. Chen, F. He, Y. Yin, and Y. Du, “Optimal design of au-

tonomous vehicle zones in transportation networks,”

Transportation Research Part B: Methodological, vol. 99,

pp. 44–61, 2017.

[25] S. C. Dafermos, “e traﬃc assignment problem for multi-

class-user transportation networks,” Transportation Science,

vol. 6, no. 1, pp. 73–87, 1972.

[26] P. Marcotte and L. Wynter, “A new look at the multiclass

network equilibrium problem,” Transportation Science,

vol. 38, no. 3, pp. 282–292, 2004.

[27] A. T. Mehr and R. Horowitz, “How will the presence of

autonomous vehicles aﬀect the equilibrium state of traﬃc

networks?,” 2019, https://arxiv.org/abs/1901.05168.

[28] M. W. Levin and S. D. Boyles, “Eﬀects of autonomous vehicle

ownership on trip, mode, and route choice,” Transportation

Research Record: Journal of the Transportation Research

Board, vol. 2493, no. 1, pp. 29–38, 2015.

[29] J. Wang, S. Peeta, and X. He, “Multiclass traﬃc assignment

model for mixed traﬃc ﬂow of human-driven vehicles and

connected and autonomous vehicles,” Transportation Re-

search Part B: Methodological, vol. 126, pp. 139–168, 2019.

[30] P. Tientrakool, Y. Ho, and N. F. Maxemchuk, “Highway

capacity beneﬁts from using vehicle-to-vehicle communica-

tion and sensors for collision avoidance,” in Proceedings of the

2011 IEEE Vehicular Technology Conference (VTC Fall),

pp. 1–12, San Francisco, CA, USA, September 2011.

[31] D. Chen, S. Ahn, M. Chitturi, and D. A. Noyce, “Towards

vehicle automation: roadway capacity formulation for traﬃc

mixed with regular and automated vehicles,” Transportation

Research Part B: Methodological, vol. 100, pp. 196–221, 2017.

[32] M. W. Lazar, S. D. Chandrasekher, R. Pedarsani, and

D. Sadigh, “Maximizing road capacity using cars that inﬂu-

ence people,” 2018, https://arxiv.org/abs/1807.04414.

[33] S. Dafermos, “Traﬃc equilibrium and variational inequal-

ities,” Transportation Science, vol. 14, no. 1, pp. 42–54, 1980.

[34] M. J. Smith, “e existence, uniqueness and stability of traﬃc

equilibria,” Transportation Research Part B: Methodological,

vol. 13, no. 4, pp. 295–304, 1979.

[35] O. Ben-Ayed, D. E. Boyce, and C. E. Blair, “A general bilevel

linear programming formulation of the network design

problem,” Transportation Research Part B: Methodological,

vol. 22, no. 4, pp. 311–318, 1988.

[36] A. Ghaﬀari, M. Mesbah, and A. Khodaii, “Designing a transit

priority network under variable demand,” Transportation

Letters, pp. 1–14, 2019.

[37] M. Mesbah, R. ompson, and S. Moridpour, “Bilevel opti-

mization approach to design of network of bike lanes,”

Transportation Research Record: Journal of the Transportation

Research Board, vol. 2284, no. 1, pp. 21–28, 2012.

[38] L. J. Leblanc, “An algorithm for the discrete network design

problem,” Transportation Science, vol. 9, no. 3, pp. 183–199,

1975.

[39] H. Poorzahedy and H. S. Rouhani, “Hybrid meta-heuristic

algorithms for solving network design problem,” European

12 Mathematical Problems in Engineering

Journal of Operational Research, vol. 182, no. 2, pp. 578–596,

2007.

[40] S. Shiripour, N. Mahdavi-Amiri, and I. Mahdavi, “A trans-

portation network model with intelligent probabilistic travel

times and two hybrid algorithms,” Transportation Letters,

vol. 9, no. 2, pp. 90–122, 2017.

[41] T. Zhang, G. Ren, and Y. Yang, “Transit route network design

for low-mobility individuals using a hybrid metaheuristic

approach,” Journal of Advanced Transportation, vol. 2020,

Article ID 7059584, 12 pages, 2020.

[42] S. A. Bagloee, M. Asadi, M. Sarvi, and M. Patriksson, “A

hybrid machine-learning and optimization method to solve

bi-level problems,” Expert Systems with Applications, vol. 95,

pp. 142–152, 2018.

[43] L. J. LeBlanc, E. K. Morlok, and W. P. Pierskalla, “An eﬃcient

approach to solving the road network equilibrium traﬃc

assignment problem,” Transportation Research, vol. 9, no. 5,

pp. 309–318, 1975.

[44] S. Carrese, M. Nigro, S. M. Patella, and E. Toniolo, “A pre-

liminary study of the potential impact of autonomous vehicles

on residential location in Rome,” Research in Transportation

Economics, vol. 75, pp. 55–61, 2019.

[45] J. Meyer, O. M. Becker, P. M. B¨

osch, and K. W. Axhausen,

“Autonomous vehicles: the next jump in accessibilities?”

Research in Transportation Economics, vol. 62, no. 2,

pp. 80–91, 2017.

[46] A. T. Moreno, A. Michalski, C. Llorca, and R. Moeckel,

“Shared autonomous vehicles eﬀect on vehicle-Km traveled

and average trip duration,” Journal of Advanced Trans-

portation, vol. 2018, Article ID 8969353, 10 pages, 2018.

[47] W. Zhang and S. Guhathakurta, “Parking spaces in the age of

shared autonomous vehicles: how much parking will we need

and where?” Transportation Research Record: Journal of the

Transportation Research Board, vol. 2651, no. 1, pp. 80–91,

2017.

[48] P. Ashkrof, G. Homem de Almeida Correia, O. Cats, and

B. van Arem, “Impact of automated vehicles on travel mode

preference for diﬀerent trip purposes and distances,” Trans-

portation Research Record: Journal of the Transportation Re-

search Board, vol. 2673, no. 5, pp. 607–616, 2019.

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