A dynamic continuum modeling approach to the spatial analysis of air quality and

housing location choice in a polycentric city

Liangze Yang1, S.C. Wong2, Mengping Zhang3 and Chi-Wang Shu4

AbstractThis study develops a model that integrates land use, transport, and environment factors.

Specifically, a continuum modeling approach is used to access how air quality, among otherfactors influence people’s housing location choices. We assume that pollutants generated bythe transport sector are dispersed through turbulent diffusion and advection by the wind andthat they affect air quality. Air quality affects people’s housing choices, which in turn changestheir travel behavior. A polycentric urban city with multiple central business districts (CBDs)is considered, and the road network within the modeled city is assumed to be sufficientlydense that it can be viewed as a continuum. The predictive continuum dynamic user-optimal(PDUO-C) model is used to describe the traffic flow for a given traffic demand distribution.The dispersion of the vehicle exhaust is modeled with an advection-diffusion equation. In thisstudy, we incorporate departure time choice into the PDUO-C model to describe the traffic flowand show that the departure time choice problem is equivalent to a variational inequality (VI)problem in which housing location choice is determined by travel cost, air quality and rent.The coupled system can be treated as a fixed-point problem. A projection method is adopted tosolve the VI problem, and a self-adaptive version of the method of successive averages (MSA)is proposed to solve the whole coupled system. A numerical example is given to illustrate theeffectiveness of the proposed model.

Key Words: predictive dynamic user-optimal model; departure time; variational inequali-ty; continuum model; housing location choice.

1School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026,P.R. China. E-mail: lzeyang@mail.ustc.edu.cn.

2Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, PR China.E-mail:hhecwsc@hku.hk.

3School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026,P.R. China. E-mail: mpzhang@ustc.edu.cn.

4Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail:shu@dam.brown.edu.

1

1 Introduction

As Vaughan (1987) discussed, traffic is the interaction between land use and transport,

and this interaction is a two-way process. After Hansen (1959) demonstrated this reciprocal

relationship, models that consider the interaction between land-use and transport have been ex-

tensively studied. Lowry (1964) was one of the first to analytically and operationally consider

the urban land-use transport feedback cycle .

Advanced computation systems have led to the establishment of many land-use and trans-

port models over the last two decades. Chang (2006) and Chang & Mackett (2006) classi-

fied existing land-use and transport models into four groups: spatial interaction, mathematical

programming, random utility, and bid-rent models. However, few studies have incorporat-

ed environmental problems (e.g., noise pollution, vehicle emissions, and pollutant dispersion

and concentration) into these models. Most studies have developed models based on discrete

choice theory and utility maximization and have aggregated data in zone levels. More recently,

Yin et al. (2013, 2017) used a continuum modeling approach to develop a spatial analysis of

the interaction between air quality and housing location choice in a polycentric city.

Many studies have attempted to establish an integrated land-use and transport model, in

which housing location choice is integrated into land use and transport models, and ultimately

into travel demand models. The ability to predict housing location choices is important for

people making decisions on the allocation of housing developments and commercial activities.

Traditionally, housing location choice has been viewed as a tradeoff between transport

costs and housing prices (Giuliano, 1989). Wheaton (1977) found that when the elasticity of

housing prices exceeded the travel costs, people might choose to live further from their work

locations. Rosen (1974) formulated hedonic prices theory to explain the spatial equilibrium

between provider and buyer. The idea of hedonic prices was later widely adopted in studies

of housing prices (Huh & Kwak, 1997; Orford, 2000). However, Ellickson (1981) pointed out

that hedonic prices could not predict consumer behavior. Later, logit models have been used to

predict housing location choices (Ben-Akiva & Bowman, 1998; Bhat & Guo, 2004; Zongdag

2

& Pieters, 2005). These studies have considered various kinds of detailed information about

housing and households, including the sizes of houses, neighborhood, family structures, and

the types or times of travel. Based on the interactions between land use and transport, discrete

modeling techniques such as logit models have been used by several researchers to develop

models that integrate land use, transport, and environmental factors (Wegener & Fuerst, 2004;

Wegener, 2004; Wagner & Wegener, 2007). These models consider how land use influences

transport type and how transport-related pollutants and noise affect land-use patterns.

In situations where many variables are continuous, a continuous modeling approach may

be more appropriate than the discrete modeling approach often applied to land use, transport,

and environment models. Ho and Wong (2007, 2005) incorporated a continuous logit model

into a continuum modeling framework to study travel patterns and housing location choic-

es in an urban city. They assumed that within the modeled region, all of the variables were

continuous, and that the differences between adjacent areas were small. Thus, the transporta-

tion system in the city could be described by smooth mathematical functions (Vaughan, 1987;

Huang et al., 2009; Du et al., 2013).

The traffic equilibrium problem was initially studied as a static point problem, however, as

the temporal variation of flow and cost cannot be examined in static models, they cannot be

used to consider elements such as travelers’ departure/arrival time choices or dynamic traffic

management and control. the dynamic traffic assignment (DTA) problems has received much

attention in recent decades. The DTA problem includes two fundamental components: traffic

flow and travel choice (Szeto & Lo, 2006). DTA models can be divided into three types based

on available route and departure time choices (Szeto & Wong, 2012): pure departure time

choice models (Small 1982), pure route choice models (Tong & Wong, 2000; Huang et al.,

2009; Du et al., 2013), and simultaneous departure time and route choice models (Friesz et al.,

1993; Wie et al., 1995; Han et al., 2013).

For the transportation emission problem, emission models are used to calculate the emis-

sion rate and fuel consumption of different kinds of vehicles in different situations. Gaussian

3

dispersion models are extensively used to assess the impacts of different sources of air pol-

lution on local and urban areas. The use of Gaussian dispersion can be traced back to the

1920s, when gradient transport theory and statistical theory of dispersion were proposed (Tay-

lor, 1921; Roberts, 1923; Richardson, 1926). Roberts (1923) study of how smoke scattered in a

turbulent atmosphere identified a solution for the mean diffusion equation with constant-eddy

diffusivities for different source configurations, such as point and line sources. Specifically,

the solution is the ensemble-averaged smoke puffs from instantaneous point sources.

In this study, we apply the continuum modeling approach to study the interactions between

land use, transport, and the transportation emissions in an urban city with multiple central

business districts (multi-CBDs). We adopt the continuum modeling approach that was applied

to the static analysis of air pollution in Yin et al. (2013, 2017). For this study, we extend

this continuum modeling approach to the dynamic analysis of air pollution and transport pat-

terns. We integrate departure time choice into the predictive continuum dynamic user-optimal

(PDUO-C) model to describe the traffic flow. We principally define the simultaneous user-

optimal and departure time principle mathematically; that is, for each origin-destination (OD)

pair, the actual total cost incurred by travelers departing at any given time is equal and mini-

mized. Then, we prove that the principle is equivalent to a variational inequality (VI) problem.

In our model, there are two problems that need to be solved; housing location choice and de-

parture time. The first problem is a fixed-point problem, and the second can be solved using

the Goldstein-Levitin-Polyak (GLP) projection method. In fact, the whole coupled system is

also a fixed-point problem, which can be solved by iteration methods such as the method of

successive averages (MSA). We use the numerical results to evaluate the effectiveness of the

proposed model.

2 Model formulation

As shown in Figure 1, the modeled region is an urban city with multiple CBDs. We denote

the whole area of the city as Ω, Then, we let Γo be the outer boundary of the city and Γmc (m =

4

1, ...,M) be the boundary of the m-th compact CBD. Thus, the boundary of Ω is Γ = Γo ∪

(∪mΓmc ). The travelers are classified into M groups based on the different CBDs. CBDs other

than the m-th CBD are viewed as obstacles for the travelers of Group m. The road network

outside the CBDs is assumed to be relatively dense and can be approximated as a continuum.

Fig. 1: Modelling domain

We denote the variables as follow:

• ρm(x, y, t) (in veh/km2) is the density of Group m at location (x, y) at time t.

• vm = (um1 (x, y, t), um

2 (x, y, t)) is the velocity vector of Group m at location (x, y) at time t.

• Um(x, y, t) (in km/h) is the speed of Group m, which is the norm of the velocity vector,

i.e., Um = |vm|, and is determined by the density as

Um(x, y, t) = Umf e−β(

M∑m=1

ρm)2

, ∀(x, y) ∈ Ω, t ∈ T j, (1)

where Umf (x, y) (in km/h) is the free-flow speed of Group m and β(x, y) (in km4/veh2) is

a positive scalar related to the road condition.

• F m = ( f m1 (x, y, t), f m

2 (x, y, t)) is the flow vector of Group m at location (x, y) at time t,

which is defined as

F m = ρmvm, ∀(x, y) ∈ Ω, t ∈ T j (2)

5

• |F m| is the flow intensity, which is the norm of the flow vector F m, and is defined as

|F m| = ρmUm, ∀(x, y) ∈ Ω, t ∈ T j (3)

• qm(x, y, t) (in veh/km2/h) is the travel demand of Group m at location (x, y) at time t,

which is a time-varying function.

• qm(x, y) (in veh/km2/h) is the total travel demand of Group m at location (x, y) .

• q(x, y) (in veh/km2/h) is the total travel demand of all of a group’s travelers at location

(x, y).

• cm(x, y, t) (in $/km) is the local travel cost per unit distance of Group m at location (x, y)

at time t, and is defined as

cm(x, y, t) = κ(1

Um + π(M∑

m=1

ρm)), ∀(x, y) ∈ Ω, t ∈ T j (4)

where κ is the value of time, κUm represents the cost associated with the travel time, and

κπ(ρ) represents other costs that are dependent on the density .

• φm1 (x, y, t) is the actual travel cost potential of Group m who departs from location (x, y)

at time t to travel to the m-th CBD using the constructed path-choice strategy (i.e., going

to work).

• φm2 (x, y, t) is the actual travel cost potential of the traveler who departs from the m-th

CBD travel to location (x, y) at time t using the constructed path-choice strategy (i.e.,

returning home).

• Im(x, y, t) is the travel time of Group m from location (x, y) to the m-th CBD at time t.

• pm(x, y, t) is the schedule delay cost of Group m departing from location (x, y) for the

m-th CBD at time t, which is a kind of penalty for late or early arrival. It is determined

by the arrival time t + Im(x, y, t).

6

• lm(x, y, t) is the total cost of the travel of Group m from location (x, y) to the m-th CBD,

and

lm(x, y, t) = pm(x, y, t) + φm1 (x, y, t) (5)

• C(x, y, z, t) (in kg/km3) is the concentration of the pollution at location (x, y, z) at time t,

where z represents the height about groud.

• u f (x, y, z, t) (in km/h) is the wind velocity vector at location (x, y, z) at time t.

• Kx,Ky, andKz (in km2/h) are the eddy diffusivities in the x, y and z directions, respec-

tively.

• Ψ(x, y) (in kg/(veh h)) is the average amount of pollutants generated per vehicle, per

hour at location (x, y).

• Pm(x, y) is the total perceived cost for the vehicles traveling to the m−th CBD at location

(x, y), and is defined as

Pm(x, y) = Pm0 + Φm(x, y) (6)

where Pm0 is the origin perceived cost of the vehicle travel to the m-th CBD, denoted as

Pm0 = θm + S m(Vm), (7)

where θm is the biased component that represents the preference of travelers for the m-

th CBD, and S m is the internal operating cost of traffic, such as the parking cost and

local circulation cost, within the m−th CBD, which is specified as a function of the total

demand attracted to the m−th CBD.

Vm =

∫T 1

∫Ω

qm(x, y, t)dΩ.dt (8)

Φm(x, y) is the average transportation cost between location (x, y) and the m-th CBD,

which is defined as

Φm(x, y) =1|T 1|

∫T 1φm

1 (x, y, t)dt +1|T 2|

∫T 2φm

2 (x, y, t)dt. (9)

7

Finally, T = T 1 ∪ T 2, for j ∈ 1, 2, T j = [t jbeginning, t

jend] is the modeling period of the j-th

part of the traffic. T 1 refers to the time taken to go to the CBD, and T 2 refers to the time taken

to return home from the CBD, m = 1, ...,M.

2.1 Predictive continuum dynamic user-optimal model

In this section, we briefly review the predictive continuum dynamic user-optimal model.

For a more detailed discussion, we refer readers to Yang et al. (2018) and Du et al. (2013). As

we consider the traffic behavior for a complete day, we describe the model in two parts: for

vehicles traveling to the CBDs and for vehicles traveling from CBDs.

Traffic model for vehicles traveling to the CBD

For this part,the conservation law is

ρmt + ∇ · F m(x, y, t) = qm(x, y, t), ∀(x, y) ∈ Ω, t ∈ T 1

F m = ρmυm = ρmUm (−∇φm1 (x, y, t))

‖ ∇φm1 (x, y, t) ‖

, ∀(x, y) ∈ Ω, t ∈ T 1,

F m(x, y, t) · n(x, y) = 0, ∀(x, y) ∈ Γ \ Γmc , t ∈ T 1

ρm(x, y, t1beginning) = 0, ∀(x, y) ∈ Ω

(10)

and the Hamilton-Jacobi equation is

1Umφ

m1,t − |∇φ

m1 | = −cm, ∀(x, y) ∈ Ω, t ∈ T 1

φm1 (x, y, t) = 0, ∀(x, y) ∈ Γm

c , t ∈ T 1,

φm1 (x, y, t1

end) = φm0 (x, y), ∀(x, y) ∈ Ω.

(11)

where the initial value φm0 (x, y) is computed by the following 2D Eikonal equation:‖ ∇φm

0 (x, y) ‖= cm(x, y, t1end), ∀(x, y) ∈ Ω, t ∈ T 1

φm0 (x, y) = φm

CBD, ∀(x, y) ∈ Γmc .

(12)

Traffic model of the vehicles returning from the CBD

8

For this part,the conservation law is

ρmt + ∇ · F m(x, y, t) = qm(x, y, t), ∀(x, y) ∈ Ω, t ∈ T 2

F m = ρmυm = ρmUm (−∇φm2 (x, y, t))

‖ ∇φm2 (x, y, t) ‖

, ∀(x, y) ∈ Ω, t ∈ T 2,

F m(x, y, t) · n(x, y) = 0, ∀(x, y) ∈ Γ \ Γmc , t ∈ T 2

ρm(x, y, t2end) = 0, ∀(x, y) ∈ Ω

(13)

and the Hamilton-Jacobi equation is

1Umφ

m2,t − |∇φ

m2 | = −cm, ∀(x, y) ∈ Ω, t ∈ T 2

φm2 (x, y, t) = 0, ∀(x, y) ∈ Γm

c , t ∈ T 2,

φm2 (x, y, t2

beginning) = φm0 (x, y), ∀(x, y) ∈ Ω.

(14)

where the initial value φm0 (x, y) is computed by the following 2D Eikonal equation:

‖ ∇φm0 (x, y) ‖= cm(x, y, t2

beginning), ∀(x, y) ∈ Ω, t ∈ T 2

φm0 (x, y) = φm

CBD, ∀(x, y) ∈ Γmc .

(15)

Here, ρm(x, y, t) is governed by the conservation law, and φm1 (x, y, t), φm

2 (x, y, t) are computed

using the Hamilton-Jacobi equation.

2.2 Dispersion model

In this subsection, we consider the dispersion of vehicle exhaust through turbulent diffusion

and the wind advection. The concentration C(x, y, z, t) is governed by the following three-

dimensional advection-diffusion equation

∂C∂t

+ ∇ · (Cu f ) =∂

∂x(Kx

∂C∂x

) +∂

∂y(Ky

∂C∂y

) +∂

∂z(Kz

∂C∂z

) + S , (16)

where S (x, y, z, t) [kg/(km3 h)] is a source term. In this study, we only consider the dispersion

of vehicle exhaust, so the source term can be written as

S (x, y, z, t) = ρΨδ(z), (17)

where δ(·) is the Dirac delta function and Ψ (in kg/(veh ·h)) is the average amount of pollutants

generated per vehicle, per hour, which we discuss in the next subsection.

9

2.3 Emission model

Several models can estimate transport-related pollutant emissions. Here, an acceleration

and speed-based model is used, in which the emission rate is defined as a function of vehicle

type, instantaneous speed, and acceleration. Specifically we apply the model proposed by

Ahn et al. (1999), in which the emission rate is defined as a function of the instantaneous

acceleration and speed as follows:

ψ = exp(3∑

i=0

3∑j=0

wi, jvis j), (18)

where ψ (in kg/(veh · h)) is the amount of pollutants generated per vehicle, per hour, s is

the instantaneous acceleration (km/h2) of a individual vehicle, v is the instantaneous speed

(km/h) of a individual vehicle, wi, j is the model regression coefficient for speed power i and

acceleration power j, and the coefficient may vary according to the kind of emission, such as

HC, CO, or NOx (mg/s).

The above emission model is a microcosmic model for an individual vehicle. In our contin-

uum model, as we can easily derive the average speed U and average acceleration a in a local

region, we develop a macroscopic emission model from the above microcosmic emission mod-

el based on the average speed and acceleration, Firstly, we make the following assumptions.

In a local region, a vehicle’s instantaneous speed and instantaneous acceleration satisfy the

normal distribution with respect to the average speed and average acceleration respectively,

and v and s are independent random variables, i.e. v ∼ N(U, σ2U) and s ∼ N(a, σ2

a), where U

and a are the expected distributions and σU and σa are the the standard deviations.

Therefore, we have

E(v − U) = 0, E(s − a) = 0

E((v − U)2) = σ2U , E((s − a)2) = σ2

a, and (19)

E((v − U)(s − a)) = E(v − U)E(s − a) = 0.

where E(·) is the excepted value.

10

We first denote ψ(v, s) = ψ = exp(3∑

i=0

3∑j=0

wi, jvis j), then do a Taylor series expansion for the

emission rate with respect to speed v and acceleration s at (U, a):

ψ(v, s) = ψ(U, a) +∂ψ(U, a)∂v

(v − U) +∂ψ(U, a)∂s

(s − a)

+12∂2ψ(U, a)∂v2 (v − U)2 +

12∂2ψ(U, a)∂s2 (s − a)2 (20)

+∂2ψ(U, a)∂v∂s

(v − U)(s − a) + R(v, s)

where R(v, s) is the residual of the Taylor expansion. We should note that where ψ(U, a), ∂ψ(U,a)∂v , ∂ψ(U,a)

∂s ,

∂2ψ(U,a)∂v2 , ∂

2ψ(U,a)∂s2 , and ∂2ψ(U,a)

∂v∂s are fixed constants, the expectation of ψ(v, s) is

E[ψ(v, s)] = E[ψ(U, a)] + E[∂ψ(U, a)∂v

(v − U)] + E[∂ψ(U, a)∂s

(s − a)]

+ E[12∂2ψ(U, a)∂v2 (v − U)2] + E[

12∂2ψ(U, a)∂s2 (s − a)2] (21)

+ E[∂2ψ(U, a)∂v∂s

(v − U)(s − a)] + E[R(v, s)]

= ψ(U, a) +12∂2ψ(U, a)∂v2 σ2

U +12∂2ψ(U, a)∂s2 σ2

a + E[R(v, s)]

We denote

Ψ = ψ(U, a) +12∂2ψ(U, a)∂v2 σ2

U +12∂2ψ(U, a)∂s2 σ2

a (22)

where Ψ ≈ E[ψ(v, s)] is the average emission rate, which is a macroscopic emission model.

Under the route choice governed by user-optimal conditions, vehicles may accelerate or

decelerate along their trajectory, based on the spatial variation in traffic conditions in the neigh-

boring areas. Acceleration in the direction of movement is determined by

a =a1φ1,x + a2φ1,y√

(φ21,x + φ2

1,y), (23)

where a1 and a2 represent accelerations in the x and y directions, respectively, and the follow-

ing equations should be satisfied:

a1 = (u1)t + u1(u1)x + u2(u1)y, and a2 = (u2)t + u1(u2)x + u2(u2)y. (24)

11

There is no single representative measure of traffic-related air pollution. However, as sug-

gested by Wardman & Bristow (2004), NOx levels can be useful indicators, as this pollutant

clearly has adverse health effects. Here, we consider only the NOx emissions. We use the

parameters given in Ahn et al. (1999) and Yang et al. (2018), other pollutants can be easi-

ly considered in the modeling framework by choosing appropriate coefficients in Ahn et al.

(1999).

2.4 Travel time and the schedule delay function

In this subsection, we consider travel time Im(x, y, t) and the schedule delay cost pm(x, y, t).

According to the traffic model introduced in the previous subsection, travel time satisfies

‖ ∇Im(x, y, t) ‖=1

Um(x, y, t),∀(x, y) ∈ Ω, t ∈ T j, (25)

Let the time interval [tm∗− M, tm∗+ M] be the desired arrival time period for the m-th group

of travelers, where M≥ 0, tm∗ denotes the center of the period and M is a measure of work

start time flexibility. We introduce the schedule delay function pm(x, y, t), which describes the

penalty for early or late arrival. The function is defined as

pm(x, y, t) =

γ1((tm∗ − ∆) − (t + Im(x, y, t))), t + Im(x, y, t) < tm∗ − ∆,

0, tm∗ − ∆ ≤ t + Im(x, y, t) ≤ tm∗ + ∆,

γ2((t + Im(x, y, t)) − (tm∗ − ∆)), t + Im(x, y, t) > tm∗ − ∆,

(26)

where γ1, γ2 > 0 are the parameters. In accordance with previous empirical results, we assume

that γ2 > κ > γ1; thus, the total cost can be calculated by Equation (5).

2.5 Simultaneous dynamic user-optimal and departure time principle

Given the traffic demand distribution qm(x, y, t), we can model the traffic flow in the city

and obtain the total cost using the PDUO-C model. As the total cost depends on traffic demand,

we denote lm(x, y, t) = lm(x, y, t, q), where q(x, y, t) = (q1(x, y, t), ..., qM(x, y, t)). Next we define

m = 1, ...,M (where we take the period T 1, for example) as

lm(x, y, q∗) = ess inflm(x, y, t, q∗) : t ∈ T 1 (27)

12

Next, we give the following simultaneous dynamic user-optimal and departure time prin-

ciple definition.

Definition 2.1. The simultaneous dynamic user-optimal and departure time principle is satis-

fied if the following two conditions are satisfied for all (x, y) ∈ Ω and m = 1, ...,M:

lm(x, y, t, q) = lm(x, y, q), if qm(x, y, t) > 0. (28)

and

lm(x, y, t, q) ≥ lm(x, y, q), if qm(x, y, t) = 0. (29)

where q ∈∨

and

∨=q : qm(x, y, t) ≥ 0,

∫T 1

qm(x, y, t)dt = qm(x, y),

m = 1, ...,M,∀(x, y) ∈ Ω,∀t ∈ T 1, (30)

The user-optimal conditions (28) and (29) ensure that the total cost incurred by travelers

departing at any time are equal and minimized, and no traveler in the system can change their

total cost by changing his departure time.

Then we develop a variational inequality formulation of the simultaneous dynamic user-

optimal and departure time principle.

Theorem 2.1. If Definition 2.1 is equivalent to the following variational inequality problem:

find q∗ = (q1∗, ..., qM∗) ∈∨

so that for all q ∈∨

, the user-optimal problem is

∑1≤m≤M

"Ω

∫T 1

lm(x, y, t, q∗)(qm(x, y, t) − q∗m(x, y, t))dtdΩ ≥ 0. (31)

Proof. (Necessity) First, according to the definition of lm(x, y, q∗), we have lm(x, y, t, q∗) ≥

lm(x, y, q∗). Next, if qm(x, y, t) − qm∗(x, y, t) < 0; then as q∗ satisfies the simultaneous dynamic

user-optimal and departure time principle, we have

qm(x, y, t) < qm∗(x, y, t)

13

⇒ qm∗(x, y, t) > 0 ⇒ lm(x, y, t, q∗) − lm(x, y, q∗) = 0. (32)

Hence, for all q ∈∨, ∀(x, y) ∈ Ω, m = 1, ...,M and ∀t ∈ T 1,

(lm(x, y, t, q∗) − lm(x, y, q∗))(qm(x, y, t) − qm∗(x, y, t)) ≥ 0. (33)

Hence, it is obvious that∫T 1

(lm(x, y, t, q∗) − lm(x, y, q∗))(qm(x, y, t) − qm∗(x, y, t))dt ≥ 0. (34)

Moreover, according to the definition of q, q∗ ∈∨

, we have∫T 1

(qm(x, y, t) − qm∗(x, y, t))dt = 0. (35)

It follows that (34) is equivalent to the following condition;∫T 1

lm(x, y, t, q∗)(qm(x, y, t) − qm∗(x, y, t))dt ≥ 0. (36)

Hence, (31) is proven.

(sufficiency) Next,suppose that q∗ satisfies (31) for all q. Observe first that according to

the definition of lm(x, y, q∗) for ∀(x, y) ∈ Ω, m = 1, ...,M and all t ∈ T 1,

lm(x, y, t, q∗) ≥ lm(x, y, q∗). (37)

Hence, q∗ satisfies (29) by construction.Next, we must establish (28). To do so, suppose that

(28) fails, and for some m0, (x0, y0) ∈ Ω, t0 ∈ T 1, we have

qm0∗(x0, y0, t0) > 0, lm0(x0, y0, t0, q∗) − lm0(x0, y0, q

∗) > 0. (38)

Then, by the continuity of qm0∗(x, y, t), lm0(x, y, t, q∗) and lm0(x, y, q∗), there exist positive valves

δ > 0, ε > 0, and ∃ Ω0 × T0 in the neighborhood of (x0, y0) × t0, for ∀(x, y) ∈ Ω0, t ∈ T0, such

that

qm0∗(x, y, t) > δ, lm0(x, y, t, q∗) − lm0(x, y, q∗) > 2ε. (39)

14

Note that the set Ω0,T0 has a positive measure, i.e., |Ω0| > 0, |T0| > 0. Then, according to the

definition

lm(x, y, q∗) = ess inflm(x, y, t, q∗) : t ∈ T 1, (40)

there exists a non-empty set T1 ⊂ T 1, such that for ∀ t ∈ T1,

lm0(x0, y0, t, q∗) < lm0(x0, y0, q∗) + ε (41)

Again using the continuity of lm0(x, y, t, q∗) and lm0(x, y, q∗), there exists Ω1, a neighborhood of

(x0, y0), such that for ∀(x, y) ∈ Ω1, t ∈ T1,

lm0(x, y, t, q∗) < lm0(x, y, q∗) + ε, (42)

Note that the set Ω1,T1 also has a positive measure. Without loss of generality, we assume that

|T0| = |T1|,Ω0 = Ω1, and T0 ∩ T1 = ∅. Then, we define

qm(x, y, t) =

qm∗(x, y, t) − δ, (x, y) ∈ Ω0, t ∈ T0,m = m0

qm∗(x, y, t) + δ, (x, y) ∈ Ω0, t ∈ T1,m = m0

qm∗(x, y, t), otherwise

(43)

Next, we show that for q ∈∨

, if m = m0, (x, y) ∈ Ω0, t ∈ T0, then

qm∗(x, y, t) > δ⇒ qm(x, y, t) = qm∗(x, y, t) − δ ≥ 0.

If m = m0, (x, y) ∈ Ω0, t ∈ T1, then

qm∗(x, y, t) ≥ 0⇒ qm(x, y, t) = qm∗(x, y, t) + δ ≥ 0.

Otherwise,

qm(x, y, t) = qm∗(x, y, t) ≥ 0.

Moreover, for m = m0∫T 1

qm(x, y, t)dt

15

=

∫T 1\T0\T1

qm(x, y, t)dt +

∫T0

qm(x, y, t)dt +

∫T1

qm(x, y, t)dt

=

∫T 1\T0\T1

qm∗(x, y, t)dt +

∫T0

(qm∗(x, y, t) − δ)dt +

∫T1

(qm∗(x, y, t) + δ)dt

= qm(x, y) (44)

and this is obvious also for m , m0, thus q ∈∨

. Then∑1≤m≤M

"Ω

∫T 1

lm(x, y, t, q∗)(qm(x, y, t) − qm∗(x, y, t))dtdΩ

=

"Ω0

∫T0∪T1

lm0(x, y, t, q∗)(qm0(x, y, t) − qm0∗(x, y, t))dtdΩ

=

"Ω0

∫T0

lm0(x, y, t, q∗)(qm0(x, y, t) − qm0∗(x, y, t))dtdΩ

+

"Ω0

∫T1

lm0(x, y, t, q∗)(qm0(x, y, t) − qm0∗(x, y, t))dtdΩ

=

"Ω0

(−∫

T0

δlm0(x, y, t, q∗)dt +

∫T1

δlm0(x, y, t, q∗)dt)dΩ

≤

"Ω0

(−δ|T0|(lm0(x, y, q∗) + 2ε) + δ|T1|(lm0(x, y, q∗) + ε))dΩ

=

"Ω0

(−δ|T0|ε)dΩ

= −δ|T0||Ω0|ε < 0 (45)

which contradicts (31) for this choice of q ∈∨

. Thus, we can conclude that q∗ satisfies the

simultaneous dynamic user-optimal and departure time principle.

Gap functions have been used in many studies to evaluate the quality of the numerical

solutions to traffic equilibrium problems (Lin et al., 2016). We define the gap function as

GAP =∑

1≤m≤M

"Ω

∫T 1

qm(x, y, t)(lm(x, y, t, q) − lm(x, y, q))dtdΩ. (46)

The gap function has the following properties.

• GAP ≥ 0, because qm(x, y, t) ≥ 0 and lm(x, y, t, q) ≥ lm(x, y, q) for all (x, y) ∈ Ω, m =

1, ...,M, t ∈ T 1.

• GAP = 0⇐⇒ q is a solution to the VI problem (equivalent to the user-optimal problem).

If q is a solution to the VI problem, then for all (x, y) ∈ Ω, m = 1, ...,M, t ∈ T 1, either

16

qm(x, y, t) = 0 or lm(x, y, t, q) − l(x, y, q) = 0, thus GAP = 0. If GAP = 0, then for all

(x, y) ∈ Ω, m = 1, ...,M, t ∈ T 1, we have qm(x, y, t)(lm(x, y, t, q) − lm(x, y, q)) = 0. Thus,

if qm(x, y, t) > 0, then lm(x, y, t, q) = lm(x, y, q). Hence GAP = 0 is equivalent to the fact

that q is a solution to the VI problem.

The gap function provides a measure of convergence in the VI problem (user-optimal prob-

lem). We use the relative gap function

RGAP =

∑1≤m≤M

!Ω

∫T 1 qm(x, y, t)(lm(x, y, t, q) − lm(x, y, q))dtdΩ∑

1≤m≤M

!Ω

qm(x, y)l(x, y, q)dΩ(47)

as a stopping criterion of the numerical algorithm.

2.6 Housing location choice

In this subsection, we integrate land use, transport, and environment factors into a con-

tinuum model in which transport-related pollutants are assumed to influence peoples housing

location. Therefore, travel demand depends on travel cost, air quality and the externalities of

the CBDs. Next, we describe travel demand in detail.

In the traffic model, qm(x, y, t) is travel demand. As it depends on the housing location

choice and the departure time distribution at location (x, y), we can write travel demand as

qm(x, y, t) = qm(x, y)gm(x, y, t), (48)

where gm(x, y, t) is the departure time distribution at (x, y), and∫

T j gm(x, y, t)dt = 1. In the pre-

vious subsection, we fixed qm(x, y) to find a gm(x, y, t) that satisfies the simultaneous dynamic

user-optimal and departure time principle. In contrast, in this subsection we fix gm(x, y, t) to

find a desired qm(x, y).

For a particular user, the probability of choosing a CBD as his or her destination depends

on the total perceived cost from his/her home to his or her destination. This probability is

governed by a logit-type distribution:

qm(x, y) = q(x, y)exp(−χPm(x, y))∑Mi=1 exp(−χPi(x, y))

, (49)

17

where q(x, y) =∑M

m=1 qm(x, y) and χ is a sensitivity parameter of the commuters. Then we

define Π(x, y) as a function of the log-sum cost of the travel between location (x, y) and the

CBD:

Π(x, y) = −1χ

log(M∑

m=1

exp(−χPm(x, y))) (50)

Next we define the utility function, σ(x, y) = Π(x, y) + τ(x, y) + r(x, y), which consists

of three components. Π(x, y) is the log-sum cost, as obtained from Equation (50). τ(x, y)

is the group m commuters’ perception of air quality, which is linear to the local pollutant

concentration τ(x, y) = ξC(x, y, 0), defined as

C(x, y, 0) =1|T |

∫T

C(x, y, 0, t)dt, (51)

where ξ is a parameter that measures the sensitivity of group m commuters to air quality,

and C(x, y, 0) is the average pollutant concentration at location (x, y). The housing rent r(x, y)

depends on the total demand density q(x, y) and the total housing supply density H(x, y), which

are estimated as follows:

r(x, y) = α(x, y)(1 +β1(x, y)q(x, y)

H(x, y) − q(x, y)) (52)

where α(x, y) represents the perceptions of housing rents, β1(x, y) are scalar parameters that

represent the demand-dependent components of the rent function at location (x, y), and H(x, y) >

q(x, y).

The interaction between housing location choice and traffic equilibrium is governed by the

demand distribution function, which is used to describe the way in which road users choose

their home locations in the city. Many studies have identified housing rent and travel cost as

the basic variables that affect commuters’ choices of where to live. In the case examined in this

study, the externalities of the CBDs and the local air quality are also considered. The following

equation is used to incorporate the housing location choice problem into the transportation

equilibrium problem:

q(x, y) = Qexp(−γσ(x, y))!

Ωexp(−γσ(x, y))dΩ

. (53)

18

where Q is the total demand, which is fixed in this model, and γ is a positive scalar parameter

that measures the model’s sensitivity.

Combining Equations (49) and (53) reveals that for ∀(x, y) ∈ Ω,m = 1, ...,M,, we get

qm(x, y) = Qexp(−γσ(x, y))!

Ωexp(−γσ(x, y))dΩ

exp(−χPm(x, y))∑Mi=1 exp(−χPi(x, y))

(54)

3 Solution algorithm

In this section, we describe the solution algorithm that can be used to solve the whole

model, including the Lax-Friedrichs scheme used for the conservation law equation, the fast

sweeping method used for the Eikonal equation, the projection method used for the finite vari-

ational inequality problem, and the self-adaptive MSA used to solve the fixed-point problem.

3.1 The Lax-Friedrichs scheme used to solve the conservation law

In this subsection, we focus on the numerical method to solve the conservation law. We

assume that the cost potential function φ(x, y, t) is known for all (x, y) ∈ Ω.

For the 2D mass conservation law equation of the system, we use the conservative differ-

ence scheme to approximate the point value ρni, j ≈ ρ(xi, y j, tn):

ρn+1i, j = ρn

i, j −∆t∆x

(( f1)i+ 12 , j− ( f1)i− 1

2 , j) −

∆t∆y

(( f2)i, j+ 12− ( f2)i, j− 1

2) + qi, j∆t. (55)

where qi, j = q(xi, y j, t) is the given demand at location (xi, y j) at time t. ∆x and ∆y are the mesh

sizes in x and y, respectively, which for simplicity are assumed to be uniform ∆x = ∆y = h.

( f1)i+ 12 , j

and ( f2)i, j+ 12

are numerical fluxes in the x and y directions, respectively. Here, we use

the Lax-Friedrichs flux, which is a monotone flux:

( f1)i+ 12 , j

=12

[ f1(ρni, j) + f1(ρn

i+1, j) − α f1(ρni+1, j − ρ

ni, j)] (56)

( f2)i+ 12 , j

=12

[ f2(ρni, j) + f1(ρn

i, j+1) − α f2(ρni, j+1 − ρ

ni, j)]. (57)

where α f1 = max | f′

1 | and α f2 = max | f′

2 |.

19

3.2 Fast sweeping method for the Eikonal equation

The Eikonal equation is a special type of steady-state Hamilton-Jacobi equations. We use

the first-order Godunov fast sweeping method (Zhao, 2005) to solve it. The fast sweeping

method starts with the following initialization. Based on the boundary condition φm0 (x, y) =

φmCBD for (x, y) ∈ Γm

c , we assign the exact boundary values on Γmc . Large values (for example

1012) are assigned as initial guess at all of the other grid points.

The following Gauss-Seidel iterations with four alternating direction sweepings are per-

formed after initialization

(1) i = 1 : Nx, j = 1 : Ny; (2) i = Nx : 1, j = 1 : Ny;

(3) i = Nx : 1, j = Ny : 1; (4) i = 1 : Nx, j = Ny : 1,(58)

where (i, j) is the grid index pair in (x, y) and Nx and Ny are the number of grid points in x and

y, respectively. When we loop to a point (i, j), the solution is updated as follows, using the

Godunov Hamiltonian:

φnewi, j =

min(φxmin

i, j , φymini, j ) + ci, jh, if |φxmin

i, j − φymini, j | ≤ ci, jh,

φxmini, j + φ

ymini, j + (2c2

i, jh2 − (φxmin

i, j − φymini, j )2)

12

2, otherwise,

(59)

where ci, j = c(xi, y j, t).

For a first-order fast sweeping method, we define φxmini, j and φymin

i, j asφxmin

i, j = min(φi−1, j, φi+1, j),

φymini, j = min(φi, j−1, φi, j+1).

(60)

Convergence is declared if

‖φnew − φold‖ ≤ δ, (61)

where δ is a given convergence threshold value. δ = 10−9 and L1 norm are used in our compu-

tation.

20

3.3 Lax-Friedrichs scheme used to solve the time-dependent Hamilton-Jacobi equation

In this subsection, we suppose that the density ρm(x, y, t) is known for all (x, y) ∈ Ω and

t ∈ T 1 (or T 2), and thus focus on the numerical method to solve the Hamilton-Jacobi equation:

1Umφ

m1,t − |∇φ

m1 | = −cm, ∀(x, y) ∈ Ω, t ∈ T 1

φm1 (x, y, t) = 0, ∀(x, y) ∈ Γm

c , t ∈ T 1,

φm1 (x, y, t1

end) = φm0 (x, y), ∀(x, y) ∈ Ω.

(62)

Note that the initial time is t = t1tend and that the initial value φm

0 (x, y) is computed by the

Eikonal equation. In this subsection we assume that φm0 (x, y) is known.

As the initial time is t1end, we define

τ = t1end − t, Φm(x, y, τ) = φm

1 (x, y, t1end − τ). (63)

and thus we rewrite the time-dependent HJ equation into the usual form:

1Um Φm

τ + |∇Φm| = cm, ∀(x, y) ∈ Ω, τ ∈ T 1

Φm(x, y, t) = 0, ∀(x, y) ∈ Γmc , τ ∈ T 1,

Φm(x, y, 0) = Φm0 (x, y), ∀(x, y) ∈ Ω.

(64)

When we define

H(Φmx ,Φ

my ) = U(|∇Φm| − cm) (65)

then the scheme to solve Φmτ + H(Φm

x ,Φmy ) = 0 is

Φm,n+1i, j = Φm,n

i, j − ∆tH((Φmx )−i, j, (Φ

mx )+

i, j, (Φmy )−i, j), (Φ

my )+

i, j (66)

with

(Φmx )−i, j =

Φmi, j − Φm

i−1, j

∆x, (Φm

x )+i, j =

Φmi+1, j − Φm

i, j

∆x(67)

(Φmy )−i, j =

Φmi, j − Φm

i, j−1

∆y, (Φm

y )+i, j =

Φmi, j+1 − Φm

i, j

∆y(68)

21

where H is a Lipschitz continuous monotone flux consistent with H. Here, we use the global

Lax-Friedrichs flux:

H(u−, u+, v−, v+) = H(u− + u+

2,

v− + v+

2) −

12αx(u+ − u−) −

12αy(v+ − v−) (69)

where αx and αy are the viscosity constants and are defined as

αx = maxA≤u≤B,C≤v≤D

|H1(u, v)|, αy = maxA≤u≤B,C≤v≤D

|H2(u, v)| (70)

where H1(H2) is the partial derivative of H in terms of Φmx (Φm

y ), [A, B] is the value range of u±

and [C,D] is the value range of v±.

3.4 Schemes for the advection-diffusion equation

In this subsection, we focus on the numerical method to solve the advection-diffusion e-

quation with the Dirac source term. For the first order derivative, we use the Lax-Friedrichs

scheme described above. The second-order standard central finite difference is used to approx-

imate the second derivatives.

The Dirac function is a singular term in a differential equation, which is difficult to approx-

imate in numerical computation. Based on the finite difference method, the most common and

effective technique is to find a more regular function to approximate the Dirac function. A

detailed description of this process can be found in Tornberg & Engquist (2004). Here, we use

the following approximation to replace the Dirac function:

δε(z) =

1

4∆zmin(

z∆z

+2, 2 −z

∆z), |z| ≤ 2∆z,

0, |z| > 2∆z.(71)

3.5 Finite dimensional variational inequality and the projection method

In this subsection, we first introduce the finite dimensional variational inequality, then we

introduce the projection method used to solve the finite dimensional variational inequality.

Finally, based on the spatial and time discretization, we transform our VI problem into a finite

variational inequality problem, and solve it using the projection method.

22

Definition 3.1. Let X be a nonempty closed convex subset of Rn and let F be a mapping from

Rn into itself. The variational inequality problem, denoted by VI(X, F), finds a vector x∗ ∈ X

such that

F(x∗)T (y − x∗) ≥ 0 for all y ∈ X. (72)

Theorem 3.1. Let β > 0, x∗ is a solution to the variational inequality problem (72) if and only

if

x∗ = PX(x∗ − βF(x∗)), (73)

where PX(z) is defined as

PX(z) = Argmin‖ y − z ‖: y ∈ X (74)

To prove Theorem 3.1, we need the following Lemma.

Lemma 3.1. Let X be a nonempty closed convex subset of Rn. We have

(y − PX(y))T (x − PX(y)) ≤ 0, ∀y ∈ Rn, ∀x ∈ X. (75)

Proof: According to the definition of PX(y), we have

‖ y − PX(y) ‖≤‖ y − z ‖, ∀z ∈ X. (76)

Note that PX(y) ∈ X and X is a closed convex set. We have for all x ∈ X and θ ∈ (0, 1)

z := θx + (1 − θ)PX(y) = PX(y) + θ(x − PX(y)). (77)

Considering Equations (76) and (77), we have

‖ y − PX(y) ‖2≤‖ y − PX(y) − θ(x − PX(y)) ‖2 (78)

Thus, for all x ∈ X and θ ∈ (0, 1), we have

(y − PX(y))T (x − PX(y)) ≤θ

2‖ x − PX(y) ‖2 . (79)

23

Let θ → 0+, Eq. (75) is satisfied.

Next we prove Theorem 3.1.

Proof: (Necessity.) If x∗ is a solution to the VI problem, then according to the Lemma 3.1,

we have

(y − PX(y))T (x∗ − PX(y)) ≤ 0, ∀y ∈ Rn. (80)

Let y = x∗ − βF(x∗), then we have

(x∗ − βF(x∗) − PX(x∗ − βF(x∗)))T (x∗ − PX(x∗ − βF(x∗))) ≤ 0. (81)

Hence,

‖ x∗ − PX(x∗ − βF(x∗)) ‖2≤ β(x∗ − PX(x∗ − βF(x∗)))T F(x∗). (82)

As PX(x∗ − βF(x∗)) ∈ X and x∗ ∈ X is a solution to the VI problem, we have

(PX(x∗ − βF(x∗)) − x∗)T F(x∗) ≥ 0. (83)

According to Eqs. (82) and (83), we have

(PX(x∗ − βF(x∗)) − x∗)T F(x∗) = 0. (84)

Hence either PX(x∗ − βF(x∗)) − x∗ = 0 or F(x∗) = 0, and if F(x∗) = 0, it is obvious to derive

x∗ = PX(x∗), so

x∗ = PX(x∗ − βF(x∗)). (85)

(Sufficiency.) Let y = x∗ − βF(x∗) in Eq. (75), we have

(x∗ − βF(x∗) − PX(x∗ − βF(x∗)))T (x∗ − PX(x∗ − βF(x∗))) ≤ 0, ∀x ∈ X (86)

As x∗ = PX(x∗ − βF(x∗)), we have

−βF(x∗)T (x − x∗) ≤ 0, ∀x ∈ X. (87)

24

As β ≥ 0, we have for all x ∈ X:

(x − x∗)T F(x∗) ≥ 0. (88)

Then we focus on our problem. First, we make a spatial discretization and time discretiza-

tion. Let Nx,Ny and Nt be the numbers of the grid points in x, y and t, respectively. We note

that Nt must be subject to the CFL condition. Based on the spatial and time discretization, we

transform the VI problem into the following finite dimensional VI problem.

We denote the discrete∨

as∧

, and∧=q : (qm)n

i, j ≥ 0,∑

1≤n≤Nt

(qm)ni, j = (qm)i, j, (89)

i = 1, ...,Nx, j = 1, ...,Ny, n = 1, ...,Nt,m = 1, ...,M

Therefore, we can write the VI problem in the discrete form given below.

Theorem 3.2. The user-optimal problem of Definition 3.1 is equivalent to the following dis-

crete variational inequality problem: find q∗ ∈∧

so that for all q ∈∧

∑1≤m≤M

∑1≤i≤Nx

∑1≤ j≤Ny

∑1≤n≤Nt

(lm)ni, j((q

m)ni, j − (qm∗)n

i, j)∆x∆y∆t ≥ 0, (90)

where ∆x and ∆y are the mesh sizes in the x and y directions,respectively, ∆t is the time step.

According to Theorem 3.1 and Theorem 3.2, we have

q∗ = P∧(q∗ − λl(q∗)), (91)

where l(q∗) = (l1(q∗), ..., lM(q∗)) and P∧ is the projection of x on the set∧

under Euclidean

norm.

Then we will use the GLP projection algorithm proposed by Goldstein (1964) and Levitin

& Polyak (1966) to solve our problem: given an initial q0, generate a sequence qk according

to the following equation:

qk+1 = P∧(qk − λkl(qk)), (92)

where λk is a given positive step size, which should be set according to the specific problem.

25

3.6 Solution procedure for the finite dimensional VI problem

To solve the finite dimensional VI problem using the GLP projection method, the projec-

tion P∧(q∗−λl(q∗)) should be known. By definition, this is equivalent to solving the following

convex quadratic program:

minq

z(q) =‖ q − p ‖

s.t.

q ∈∧

We use the Frank-Wolfe method (Frank & Wolfe, 1956) to solve the quadratic program. Given

a feasible solution xk, then another feasible solution yk, we have approximately

z(yk) = z(xk) + ∇z(yk)(yk − xk) = z(xk) + ∇z(xk)yk − ∇z(xk)xk. (93)

To find the maximum drop direction at xk, we solve the following linear optimization problem

to find the descent direction yk − xk:

min∇z(xk) · yk (94)

s.t.

yk ∈∧

. (95)

The solution procedure for solving the convex quadratic problem is as follows

Algorithm 1

1. Given an initial feasible solution xk and set k = 1.

2. Find yk by solving the linear optimization problem to get the descend direction yk − xk.

3. Find descend step size λ by solving min0≤λ≤1 z(xk + λ(yk − xk)).

4. Set xk+1 = xk + λ(yk − xk).

26

5. If z(xk)−z(xk+1)z(xk) ≤ ε, stop; otherwise go to step 2

Therefore, the complete solution algorithm for solving the discrete VI problem is as fol-

lows.

Algorithm 2

1. Given an initial arbitrary point q0 ∈∧

and set k = 1.

2. Compute lm(q): firstly, compute the travel cost φm and travel time cost T m by solving

the PDUO-C model, then compute the schedule delay pm(x, y, t) and derive the total cost

lm(q)

3. Compute qk+1 through Eq. (92) by using Algorithm 1.

4. Compute the relative gap function RGAP. If ‖qk+1−qk‖

‖qk‖< ε1 and RGAP < ε2, stop;

otherwise set k = k + 1 and go to step 2.

3.7 Fixed-point problem

We first introduce the method for solving the housing location choice problem. Note that

when computing the whole system, we must know qm(x, y) at every point, and as it depends

on the utility function σ(x, y), we must know σ(x, y) at every point. To compute the utility

function, we must know qm(x, y) at every point. As in the housing location choice problem,

there are two sub-problems that need to be considered when considering the whole problem.

The first is the housing location choice problem, and the second is the departure time choice

problem. Note that when computing the housing location choice problem, we must know

the departure time choice g(x, y, t). To compute the departure time problem, we need the

house location choice information. As mentioned in the introduction, the house location choice

problem and the whole model form a fixed-point problem, that we can solve it using a self-

adaptive MSA (Du et al., 2013). We illustrate this in detail in this subsection.

Define the vector of the numerical solutions at each grid point and each time level as (where

27

we only consider t ∈ T 1, and when t ∈ T 2 is similar)

~q = qmi, j, i = 1, ...,Nx, j = 1, ...,Ny,m = 1, ...,M, (96)

~g = gm,ni, j , i = 1, ...,Nx, j = 1, ...,Ny, n = 1, ...,Nt,m = 1, ...,M, (97)

~φ = φm,ni, j , i = 1, ...,Nx, j = 1, ...,Ny, n = 1, ...,Nt,m = 1, ...,M, (98)

~C = Cni, j,0, i = 1, ...,Nx, j = 1, ...,Ny, n = 1, ...,Nt, (99)

~σ = σi, j, i = 1, ...,Nx, j = 1, ...,Ny, (100)

where Nx,Ny, and Nt are the numbers of grid points in x, y, and t, respectively. First, let us

now give the definition for one iteration of the housing location choice problem.

Step 1.1. With a given ~qold, we solve the model from t = t1beginning to t = t1

end and thus obtain

the vector ~φ and ~C. From these, we obtain ~σ . We denote this step as

~σ = h1(~qold). (101)

Step 1.2. Using the vector ~σ, solve the equation (54) to obtain an updated vector ~qnew. We

denote this step as

~qnew = h2(~σ). (102)

We consider Step 1.1 and 1.2 as one iteration and denote it as

~qnew = h2(h1(~qold)) = f (~qold). (103)

With this definition of one iteration and the function f , the model translates to a fixed-point

problem:

~q = f (~q). (104)

28

Similarly, we now define one iteration for the whole model.

Step 2.1. Doe a given (~q)old, i.e. ~qold, ~gold is known. We solve the housing location choice

problem and obtain the vector ~qnew. We denote this step as

~qnew = h3((~q)old) = h3(~qold, ~gold). (105)

Step 2.2. Using the vector ~qnew, we solve the departure time problem to obtain an updated

vector ~gnew. We denote this step as

~gnew = h4(~qnew). (106)

We consider Step 2.1 and 2.2 as one iteration and denote it as

(~q)new = ~qnew~gnew = ~qnewh4(~qnew) = h3(~qold~gold)h4(h3(~qold, ~gold)) = f1((~q)old). (107)

With this definition of one iteration and the function f1, the whole model translates to a fixed-

point problem

~q = f1(~q). (108)

3.8 Solution procedure

We now summarize the complete solution procedure.

1. Given an initial travel demand (~qm)k and set k = 1.

2. Solve the housing location choice problem using the following steps:

(a) set the ~qk as ~qk,l and set l = 1;

(b) use the l-th solution vector ~qk,l to complete the l-th iteration, i.e. ~yk,l = f (~qk,l). (as

discussed in the previous subsection);

(c) compute the step sizes λl(l > 7) using the method described in Du et al. (2013)

(The step sizes λl, l = 1, ..., 7 are predetermined.);

29

(d) compute the (l + 1)-th solution vector using the formula ~qk,l+1 = (1− λl)~qk,l + λl~yk.l;

and

(e) if ‖ ~qk,l+1 − ~qk,l ‖≤ δ, stop; otherwise go to step (b).

3. Solve the departure time problem using the Algrithm 1 and Algrithm 2 to obtain ~gmk+1.

4. If ‖ (~q)k+1 − (~q)k ‖≤ δ, stop; otherwise go to step 2.

4 Numerical experiments

4.1 Problem description

Fig. 2: Modeling domain

As shown in Figure 2, in our numerical simulation, we consider a rectangular computation-

al domain that is 35 km long and 25 km wide with two CBDs. The center of CBD 1 located at

(6 km, 10 km). The center of CBD 2 located at (30 km, 15 km), and the center of power plant

located at (18.5 km, 4.5 km) has a size of 1 km × 1 km, where traffic is not allowed to enter or

leave. We assume that within the power plant region, the emission rate is 20 kg/(km2 · h), the

emission rate is 0.5 × Lm(t)Vm kg/(km2 · h) within the m−th CBD, where Lm(t) is the cumulative

number of vehicles in the m−th CBD. We simulate the pollution dispersion in three dimensions

30

1 km above the rectangular domain (i.e., Ω = [0, 35]× [0, 25] and Ω = Ω× [0, 1]). We assume

that there is no traffic at the beginning of the modeling period (i.e., ρm0 (x, y) = 0, ∀(x, y) ∈ Ω),

and φmCBD = 0, ∀(x, y) ∈ Γm

c , t ∈ T . We consider the period from 6:00 am on the first day

to 6:00 am the following day in the city, i.e., the modeling period is T = [0, 24h]. We set

T 1 = [0, 11h], T 2 = [11, 24h]. We assume that travelers heading for the CBD generally have

similar desired arrival times, regardless of their resident locations. For the vehicles returning

from the CBD, as we assume that this journey is just a reversal of the vehicle traveling to the

CBD, we set the desired arrival time backward in time. We set γ1 = 48 $/h, γ2 = 108 $/h,

and M= 0.2 h. When the vehicles travel to the CBD, the desired arrival times are t1∗ = 3.0,

and t2∗ = 2.8. As we assume that the vehicle returning from the CBD is just the reversed of a

vehicle traveling to the CBD, the desired arrival time can be viewed as the desired departure

time, and t1∗ = 12.5, and t2∗ = 13.

The free-flow speed of group m is defined as

Umf (x, y, t) = Umax[1 + γ3d(x, y)] (109)

where Umax = 56 km/h is the maximum speed and γ3 = 4×10−3 km−1. The factor [1+γ3d(x, y)]

is used to express the faster free-flow speed in the domain far from the CBD, where there are

fewer junctions. Here, we define d(x, y) = 34 min d1(x, y), d2(x, y) + 1

4 max d1(x, y), d2(x, y)

where dm(x, y) is the distance from location (x, y) to the center of CBD m, which shows that

the closer to CBD has a greater impact. The local travel cost per unit of distance is defined as

cm(x, y, t) = κ( 1Um + π(ρ1 + ρ2)), where κ = 90 $/h and π(ρ) = 10−8ρ2. We assume that the wind

velocity is constant, |u f | = 10 km/h, and Kx = Ky = Kz = 0.01 km2/h.

We set Q = 350000, θ1 = 12, θ2 = 15, S 1(V1) = 8 × 10−11(V1 − 150000)2, S 2(V2) =

10 × 10−11(V2 − 100000)2, and Vm =∫

Ωqm(x, y)dΩ, γ = 0.0015, χ = 0.012.

τ(x, y) is people’s perception of air quality, which is linear to the local pollutant concentra-

tion τ(x, y) = ξC(x, y, 0) where ξ = 10 is a parameter that measures the sensitivity of group m

commuters to air quality, and C(x, y, 0) is the average pollution concentration at location (x, y).

31

r(x, y) is the housing rent, where

r(x, y) = 5(1 + 12q(x, y)

H(x, y) − q(x, y)) (110)

where H(x, y) is the total housing supply, and

H(x, y) = hmax × (1 − exp(−(0.5d1(x, y))1.5)) × (1 − exp(−(0.5d2(x, y))1.5)) (111)

where hmax = 1000 is the maximum housing supply.

We now use the algorithm described in the previous section to perform the numerical sim-

ulation. A mesh with a Nx × Ny × Nz grid is used. The numerical boundary conditions are

summarized as below.

1. On the solid wall boundary, i.e., the outer boundary of the city, and CBDs other than the

m-th CBD are viewed as obstacles for the travelers of group m, Γic(i , m), and we let

the normal numerical flux be 0. We set ρm = 0 at the ghost points inside the wall. In the

Eikonal equation, we set φmi = 1012 at the ghost points.

2. On the boundary of the CBD, i.e., Γmc , we set φm

i = 0 in the Eikonal equation. The

boundary conditions for ρm inside the CBD are obtained by extrapolating from the grids

outside the CBD. To maintain the maximum flow intensity on the boundary of the CBD

under the congested condition, we set Um(x, y, t) = Umf inside the CBD.

4.2 Numerical results

We now present the numerical results. To verify the convergence of the composed algo-

rithm, we test three grids (grid 1: 35 × 25 × 50; grid 2: 70 × 50 × 100; and grid 3: 140 × 100

× 200). Note that there are two kinds of convergence to be verified: the convergence of the

self-adaptive MSA under each grid, and the convergence between different grids.

Let us consider the first kind of convergence. From Figure 3 we can see that during the

first several iterations, the error decreases very fast. After 12 iterations, the error reduction

becomes extremely slow, and we can consider these numerical results convergent.

32

Fig. 3: Convergence of the MSA method.

We now consider the second type of convergence. The housing location curves along

different cuts, plotted in Figure 4, show good grid convergence for the numerical solution. The

grid 140 × 100 × 200 is selected for further discussion.

Figure 5 shows the travel demand and the total cost for each group of vehicles at different

points on the vehicles’ route to the CBD between 6:00 am and 5:00 pm. We only plot the

figures from 6:00 am to 12:00 pm. As shown in Figure 5, all of the vehicles choose a depar-

ture time such that the total travel costs are equal and minimized; therefore, the simultaneous

dynamic user-optimal and departure time principle are satisfied. As the desired arrival time of

Group 1 is later than the desired time of Group 2, we can see the departure time of Group 1 is

later than the departure time of Group 2. As shown in each sub-figure, at the beginning of the

travel, the traffic in the city is in the non-congested condition, the travel cost of the vehicle trav-

eling to CBD is almost the same at this time, and the penalty for early arrival decreases. Thus

the total cost decreases at a linear rate of 48 $/h. As the vehicles gradually depart, the travel

cost gradually increases, thus the vehicles can arrive at the CBD at the desired time. There

is no penalty, but the total cost increases with the increase in travel cost. Vehicles that depart

late, arrive late, when the city is highly congested; the total cost increases with the increase in

33

(a) x=3 (b) x=10

(c) x=24 (d) x=33

(e) y=3 (f) y=10

(g) y=12 (h) y=20

Fig. 4: Grid convergence of the demand (unit: veh/km2).

34

(a) x=5, y=5 (b) x=10, y=5

(c) x=10, y=20 (d) x=15, y=10

(e) x=25, y=20 (f) x=34, y=24

Fig. 5: Travel demand and total cost for each group of vehicles when vehicles travel to theCBD.

35

(a) x=5, y=5 (b) x=10, y=5

(c) x=10, y=20 (d) x=15, y=10

(e) x=25, y=20 (f) x=34, y=24

Fig. 6: Travel demand and the total cost for each group of vehicles when the vehicles returnfrom the CBD.

36

(a) 8:00 am, Group 1 (b) 8:00 am, Group 2 (c) 8:00 am, Total density

(d) 8:24 am, Group 1 (e) 8:24 am, Group 2 (f) 8:24 am, Total density

(g) 8:48 am, Group 1 (h) 8:48 am, Group 2 (i) 8:48 am, Total density

(j) 9:12 am, Group 1 (k) 9:12 am, Group 2 (l) 9:12 am, Total density

Fig. 7: Density plot of multiple CBDs when the vehicles are traveling to the CBD (unit:veh/km2).

37

(a) 6:18 pm, Group 1 (b) 6:18 pm, Group 2 (c) 6:18 pm, Total density

(d) 6:30 pm, Group 1 (e) 6:30 pm, Group 2 (f) 6:30 pm, Total density

(g) 6:54 pm, Group 1 (h) 6:54 pm, Group 2 (i) 6:54 pm, Total density

(j) 7:18 pm, Group 1 (k) 7:18 pm, Group 2 (l) 7:18 pm, Total density

Fig. 8: Density plot of multiple CBDs when the vehicles are returning from the CBD (unit:veh/km2).

38

(a) Group 1

(b) Group 2

(c) total vehicles in CBD

Fig. 9: Total demand and total inflow plot.

both the travel cost and the schedule delay cost, and the speed of increase is very high. Finally,

once all of the vehicles have entered the CBD, the city returns to a non-congested condition,

and the total cost increases again at a linear rate of 108 $/h.

Figure 6 shows the travel demand and the total cost for each group of vehicles at different

points in the vehicles return journey from the CBD in the period from 5:00 pm to 6:00 am

the next day. We only plot the points from 6:00 am to 12:00 pm, when the travel demand

indicates the vehicles have arrived at their destination. As in the case of vehicles traveling to

the CBD, the simultaneous dynamic user-optimal and departure time principle is satisfied. As

the desired departure time of Group 2 is later than the desired departure time of Group 1, the

arrival time of Group 2 is later than the arrival time of Group 1. For the total cost, the trend for

vehicles traveling from the CBD is the opposite of the trend for the vehicles traveling to the

39

(a) Group 1 (b) Group 2 (c) The log-sum cost

Fig. 10: Travel cost plot.

CBD.

Figure 7 shows the temporal and spatial distributions of the density ρm within the model-

ing region when the vehicles are traveling to the CBD. As shown in Figures 7(a), 7(b), and

7(c), the density is low at the beginning of the period, and the traffic is in the non-congested

condition, especially for the density of the first group. As the traffic demand grows, more

vehicles gradually join the traffic system, as the desired arrival time of the first group is later

than that of the second group, the density of the second group becomes high as the region near

the CBD2 becomes congested (see Figure 7(e)), but the density of first group remains low (see

Figure 7(d)). As vehicles gradually enter the CBD2, and the travel demand of the Group 1

grows, the density of Group 1 becomes high (see Figure 7(g)). Finally, once all of the vehicles

have entered the CBDs, all parts of the city return to the non-congested condition (Figures 7(j),

7(k), and 7(l)). Figure 7(a), 7(d), 7(g), and 7(j) (or Figures 7(b), 7(e), 7(h), and 7(k)), show

that the high density regions of each group close to its related CBD gradually diminish over

time.

Figure 8 shows the temporal and spatial distributions of the density ρm within the modeling

region when the vehicles are returning from the CBD. The results are similar to those foe the

period when the vehicles are traveling to the CBD. As the travel demand of each group grows,

the density around each CBD increases, traffic becomes congested and the high density areas

away from the related CBD gradually become less dense. The second group always has a

40

Fig. 11: Average pollutant concentrations (unit: kg/km3).

delay in the travel pattern relative to the Group 1.

We consider the total flow to the CBD through Γmc , which measures the inflow when the

vehicles travel to the CBD and the outflow when the vehicles return from the CBD; if f mCBD > 0,

it represents the inflow, otherwise, it represents the outflow. defined as

f mCBD(t) =

∮Γc

(F · n)(x, y, t)ds. (112)

where n is the unit normal vector pointing toward the CBD, and the total demand over the

whole domain is defined as

qmΩ(t) =

"Ω

qm(x, y, t)dxdy. (113)

Figure 9 shows the relationship between f mCBD(t) and qm

Ω(t). The numerical data show that the

areas under these two curves are the same for the two groups, which demonstrates that for

Group 1, all of the vehicles have entered the CBD by 9:00 am, and have left the CBD and

41

Fig. 12: Housing rent (unit: $).

reached their homes by 8:00 pm, and for Group 2, all of the vehicles have entered the CBD

by 9:30 am and have left the CBD and reached their homes by 7:30 pm. The curve for the

total inflow, f mCBD(t), always lags behind the curve for the total demand, qm

Ω(t). In contrast, the

curve for the total demand, qmΩ

(t), always lags behind the curve for the total outflow, f mCBD(t).

The number of vehicles in the city increases when qmΩ

(t) is larger than f mCBD(t), and it decreases

when f mCBD(t) is larger than qm

Ω(t). Furthermore, Figure 9 (c) shows the distribution of the total

vehicles in each CBD about t, we can see that from 9:00 am to 6:00 pm, all vehicle stays in

the CBD, the emission rate within the CBD is most high.

Figure 10 shows the travel cost for each group of vehicles traveling to the CBDs and the

log-sum cost. The travel cost increases with the distance to the destination (see Figures 10 (a)

and 10(b)). The area between the two CBDs is convenient for travel to the CBDs. Hence, as

shown in Figure 10 (c), the log-sum cost is low.

42

Fig. 13: Travel demand plot (unit: veh/km2).

Figure 11 shows the temporal and spatial distribution of the average pollutant concentra-

tion, C, for NOx at ground level, given a wind direction aligned with the positive x-axis and

|u f | = 10 km/h. This figure clearly shows that the upwind locations are much less polluted

than the downwind locations. The region around the CBD2 is the most highly polluted, be-

cause it is located on the downwind side of the city and has a high traffic flow intensity. In

particular, there is a huge emission rate in the power plant region, and the locations downwind

of the power plant are the most polluted.

Figure 12 shows the distribution of the housing rent. The housing rent is extremely high

near the CBDs, because there is limited housing available in these places, which is reasonable,

as there is limited housing supply near the CBDs. However, we should note that in the area to

the right of the power plant, the rent is very low. For most parts of the modeled region, housing

rent is comparatively low and only varies slightly from location to location.

43

Figure 13 shows the total travel demand distribution across the city. Given the cost of

travel, people prefer to live closer to the CBDs, but the rents closer to the CBDs are relatively

high, and the air quality is relatively poor, especially around the CBD2. Hence, the actual

total travel demand distribution is a tradeoff between all of these factors. Figure 13 also shows

that in the area around the CBDs, the total travel demand is low, because housing rent is very

high in these places. Thus the total cost is low in areas around the CBDs. The housing rent

distribution indicates that the housing rent varies only slightly across the other regions, but the

total demand decreases as the distance to the CBD increases. This is reasonable, as housing

rent and total cost are the main factors in people’s choices of housing location. An examination

of the distribution of the pollutant concentrations in areas to the left of the CBD1 and the right

of the CBD2 (see Figure 11) shows that in the area to the left of CBD1 where the pollutant

concentration is high, the total demand is low; in locations downwind of the power plant, the

demand is very low, even though the housing rent is low (see Figure 12). This indicates that

air quality is as an important factor in housing location decisions.

Finally, we can define the health cost based on the average pollution distribution and the

demand distribution, defined as

Υ =

∫Ω

C(x, y, 0)q(x, y)dΩ. (114)

In our example, Υ = 202190. This quantity is a useful measure of residants’ exposure to

health risk in the city.

5 Conclusions

In this study, the continuum modeling approach is used to study how air quality, among

other factors affects people’s choices of residence location. We develop a model that inte-

grates land use, transport, and environment factors to solve this problem, in which we focus

on transport-related source of pollutants. In our model, we combine the departure time choice

with the PDUO-C model to describe a traffic pattern that satisfies the predictive dynamic user

equilibrium principle. That is, we assume a vehicle chooses a route that minimizes the total

44

travel cost to the destination, and the simultaneous dynamic user-optimal and departure time

principle in which the total cost incurred by the vehicles departing at any time is equal and

minimized. We use the advection-diffusion model to describe the dispersion of the vehicle

exhaust, and use this to derive the air quality in different parts of the city. We show that the

departure time problem is equivalent to a VI problem, and can be solved by the projection

method. The whole model and the housing location choice problem are fixed-point problems

that can be solved by a self-adaptive MSA. The numerical results show the effectiveness of the

proposed model.

Acknowledgements

The work described in this paper was supported by a grant from the Research Grants Coun-

cil of the Hong Kong Special Administrative Region, China (Project No. 17208614). The

second author was also supported by the Francis S Y Bong Professorship in Engineering. The

research of the third author is supported by NSFC Grant 11471305. The research of the fourth

author was supported by NSF grant DMS-1719410.

References

Ahn, K., Trani, A.A., Rakha, H. & van Aerde, M. (1999). Microscopic fuel consumption and

emission models. Transportation Research Board 78th Annual Meeting, Washington DC.

Ben-Akiva, M., & Bowman, J. L. (1998). Integration of an activity-based model system and a

residential location model. Urban Studies, 35, 1131-1153.

Bhat, C. R., & Guo, J. (2004). A mixed spatially correlated logit model: Formulation and

application to residential choice modeling. Transportation Research Part B: Methodological,

38, 147-168.

Chang, J. S. (2006). Models of the relationship between transport and land-use: A review.

Transport Reviews: A Transnational Transdisciplinary Journal, 26, 325-350.

45

Chang, J. S. & Mackett, R. L. (2006). A bi-level model of the relationship between transport

and residential location. Transportation Research Part B: Methodological, 40, 123-146.

Du, J., Wong, S. C., Shu, C. W., Xiong, T., Zhang, M. P., & Choi,K. (2013). Revisiting Jiang’s

dynamic continuum model for urban cities. Transportation Research Part B: Methodologi-

cal, 56, 96-119.

Ellickson, B. (1981). An alternative test of the hedonic theory of housing markets. Journal of

Urban Economics, 9, 56-79.

Frank, M., & Wolfe, P. (1956). An algorithm for quadratic programming. Naval Research

Logistics Quarterly, 3, 95-110.

Friesz, T. L., Bernstein, D., Smith, T. E., Tobin,R. L., & Wie, B. W. (1993). A variational

inequality formulation of the dynamic network user equilibrium problem. Operations Re-

search, 41, 179-191.

Giuliano, G. (1989). Research policy and review 27. New directions for understanding trans-

portation and land use. Environment and Planning A, 21, 145-159.

Goldstein, A. A. (1964). Convex programming in Hilbert space. Bulletin of the American

Mathematical Society, 70, 709-710.

Han, K., Friesz, T. L., & Yao, T. (2013). Existence of simultaneous route and departure choice

dynamic user equilibrium. Transportation Research Part B: Methodological, 53, 17-30.

Hansen, W. G. (1959). How accessibility shapes land use. Journal of American Institute of

Planners, 25, 73-76.

Ho, H. W., & Wong, S. C. (2005). Combined model of housing location and traffic equilibri-

um problems in a continuous transportation system. The 16th International Symposium on

Transportation and Traffic Theory (ISTTT16). Maryland, U. S. A.

46

Ho, H. W., & Wong, S. C. (2007). Housing allocation problem in a continuum transportation

system. Transportmetrica, 3, 21-39.

Huang, L., Wong, S. C., Zhang, M. P., Shu, C. W., & Lam, W. H. K. (2009) Revisiting Hughes’

dynamic continuum model for pedestrian flow and the development of an efficient solution

algorithm. Transportation Research Part B: Methodological, 43(1): 127-141.

Huh, S., & Kwak, S. J. (1997). The choice of functional form and variables in the hedonic

price model in Seoul. Urban Studies, 34, 989-998.

Levitin, E. S., & Polyak, B. T. (1966). Constrained minimization methods. Zhurnal Vychisli-

tel’noi Matematiki i Matematicheskoi Fiziki, 6, 787-823.

Lin Z.Y., Wong S.C., Zhang P. & Choi K. (2016) A predictive continuum dynamic user-optimal

model for the simultaneous departure time and route choice problem in a polycentric city.

Transportation Science, in press.

Lowry, I. (1964). A model of metropolis (No. RM-40535-RC). Rand Corporation, Santa Mon-

ica, California, USA.

Orford, S. (2000). Modeling spatial structures in local housing market dynamics: A multilevel

perspective. Urban Studies, 37, 1643-1671.

Richardson, L. F. (1926). Atmospheric diffusion shown on a distance-neighbour graph. Royal

Society of London Proceedings Series A, 110, 709-737.

Roberts, O.F.T. (1923). The theoretical scattering of smoke in a turbulent atmosphere. Pro-

ceedings of the Royal Society of London. Series A, 104(728): 640-654.

Rosen, S. (1974). Hedonic prices and implicit markets: Product differentiation in pure compe-

tition. Journal of Political Economy, 82, 34-55.

Small, K. A. (1982). The scheduling of consumer activities-work trips. The American Eco-

nomic Review, 72, 467-479.

47

Szeto, W. Y., & Lo, H. K. (2006). Dynamic traffic assignment: Properties and extensions.

Transportmetrica, 2, 31-52.

Szeto, W. Y., & Wong, S. C. (2012). Dynamic traffic assignment: Model classifications and

recent advances in travel choice principles. Central European Journal of Engineering, 2,

1-18.

Taylor, G. I. (1921). Diffusion by continuous movements. Proceedings of the London Mathe-

matical Society, 20, 196-211.

Tong, C. O., & Wong, S. C. (2000). A predictive dynamic traffic assignment model in congest-

ed capacityconstrained road networks. Transportation Research Part B: Methodological, 34,

625-644.

Tornberg A.K. & Engquist B. (2004) Numerical approximations of singular source terms in

differential equations. Journal of Computational Physics, 200(2): 462-488.

Vaughan, R. (1987). Urban Spatial Traffic Patterns, London, Pion.

Wagner, P., & Wegener, M. (2007). Urban land use, transport, and environment models. Ex-

periences with an integrated microscopic approach. DisP-The Planning Review, 43(170),

45-56.

Wardman, M. & Bristow, A.L. (2004). Traffic related noise and air quality valuations: Ev-

idence from stated preference residential choice models. Transportation Research Part D:

Transport and Environment, 9, 1-27.

Wegener, M. (2004). Overview of land-use transport models. Handbook of Transport Geogra-

phy and Spatial Systems, 5, 127-146.

Wegener, M., & Fuerst, F. (2004). Land-use transport interaction: State of the art. Deliver-

able D2a of the project TRANSLAND (Integration of Transport and Land use Planning).

48

Berichte aus den Insititut fur Raumplanung, Insititut fur Raumplanung, Universitat Dort-

mund.46.

Wheaton, W. C. (1977). Income and urban residence: An analysis of consumer demand for

location. American Economic Review, 67, 620-631.

Wie, B. W., Tobin, R. L., Friesz, T. L., & Bernstein, D. (1995). A discrete-time, nested cost op-

erator approach to the dynamic network user equilibrium problem. Transportation Science,

29, 79-92.

Yang L., Li, T., Wong, S.C., Shu, C.-W., & Zhang, M. (2018). Modeling and simulation of

urban air pollution from the dispersion of vehicle exhaust: A continuum modeling approach.

International Journal of Sustainable Transportation, in press.

Yin, J., Wong, S. C., Sze, N. N. & Ho, H.W. (2013). A continuum model for housing allo-

cation and transportation emission problems in a polycentric city. International Journal of

Sustainable Transportation, 7, 275-298.

Yin, J., Wong, S.C., Choi, K. & Du, Y.C. (2017). A continuum modeling approach to the spatial

analysis of air quality and housing location choice. International Journal of Sustainable

Transportation, 11, 319-329.

Zhao, H.K. (2005). A fast sweeping method for Eikonal equations. Mathematics of Computa-

tion, 74 (250), 603-627.

Zongdag, B., & Pieters, M. (2005). Influence of accessibility on residential location choice.

Transportation Research Record: Journal of the Transportation Research Board, 1992, 63-

70.

49