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: [email protected].
2Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, PR China.E-mail:[email protected].
3School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026,P.R. China. E-mail: [email protected].
4Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail:[email protected].
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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
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& 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
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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 =
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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)
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• |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).
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• 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)
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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
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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.
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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.
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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)
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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)
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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)
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⇒ 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.
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