Journal of Transportation Technologies, 2012, 2, 260-266
http://dx.doi.org/10.4236/jtts.2012.23028 Published Online July 2012 (http://www.SciRP.org/journal/jtts)
Modeling the Parking Pricing of Multiple Parking
Facilities under Different Operation Regimes
Wenbo Fan1,2, Muhammad Babar Khan3
1School of Transportation & Logistics, Southwest Jiaotong University, Chengdu, China
2Key Laboratory of Comprehensive Transportation of Sichuan Province, Chengdu, China
3National Institute of Transportation, School of Civil & Environmental Engineering,
National University of Sciences & Technology, Islamabad, Pakistan
Received March 20, 2012; revised April 22, 2012; accepted May 16, 2012
ABSTRACT
To explore the parking pricing of multiple parking facilities, this paper proposes a bi-level programming model, in
which the interactions between parking operators and travelers are explicitly considered. The upper-level sub-model
simulates the price decision-making behaviors of the parking operators whose objectives may vary under different
operation regimes, such as monopoly market, oligopoly competition, and social optimum. The lower level represents a
network equilibrium model that simulates how travelers choose modes, routes, and parking facilities. The proposed
model is solved by a sensitivity based algorithm, and applied to a numerical experiment, in which three types of parking
facilities are studied, i.e., the off-road parking lot, the curb parking lot, and the parking-and-ride (P & R) facility. The
results show in oligopoly market that the level of parking price reaches the lowest point, nonetheless the social welfare
decreases to the lowest simultaneously; and the share of P & R mode goes to the highest value, however the total net-
work costs rise also to the highest. While the monopoly market and the social optimum regimes result in solutions of
which P & R facilities suffer negative profits and have to be subsidized.
Keywords: Parking Pricing; Operation Regimes; Bi-Level Programming; Multiple Parking Facilities; Network
Equilibrium
1. Introduction
As one of the effective instruments of traffic demand
management (TDM), parking pricing has been widely
adopted by urban decision-makers around the world. The
price level determines not only the profits of the parking
operators, but also substantially influence travelers’ cho-
ices on modes, routes, and parking facilities [1,2]. Tradi-
tional methods, which generally determine the optimal
pricing by marginal cost pricing model [3,4], are not suf-
ficient to assess these impacts on travelers’ choices and
subsequent variants of parking demand and network per-
formance. Therefore, researchers applied network equi-
librium model to study the parking pricing problem [5,6].
In addition, the emergence of park and ride (P & R) fa-
cilities as a new type of parking facility contributes more
complexity to the situation, of which parking lots are lo-
cated on the edge of urban area to encourage car drivers
park and enter the city by public transport. Noticing the
difference of the P & R facility from pure parking facility,
R. Garcia and A. Marin [7] firstly studied the parking
pricing problem of the P & R facility, and established a
continuous network design model. J. Y. T. Wang et al.
[8] proposed an integrated model for a linear city to de-
terminate the pricing and siting issues of P & R facilities.
Nonetheless, these works neglected the impacts of other
types of parking facilities, e.g., off-road parking lot and
curb parking in central business district (CBD), thus could
not capture travelers’ choices between various types of
parking facilities. In addition, parking operators’ beha-
viors are highly dependent on the operation regimes,
such as monopolistic franchise, oligopolistic franchise,
and government operation. Their operating objectives
vary within different operation regimes, and lead to vari-
ous competing behaviors. Such phenomenon have re-
ceived some attentions, for instance, Z. C. Li et al. [9]
studied the optimal fare structure of public transport un-
der three operation regimes (i.e., monopoly market, oli-
gopoly competition, and social optimum). They found
that the operation regimes had significant influences on
the optimal solutions. With respect to studies on parking
issues, S.P. Anderson [10] made a comparative analysis
on the paring pricing of government operation and pri-
vate operation. The results showed that the private opera-
tion under free competition could lead to the lowest so-
C
opyright © 2012 SciRes. JTTs

W. B. FAN, M. B. KHAN 261
cial costs in total. J. F. Tsai and C. P. Chu [11] consi-
dered in parking pricing decision three types of players,
i.e., the government, the operation corporation, and the
customers, and established a three-stage Stachelberg game
model to describe the interactions between these players.
These existing valuable studies enriched our understand-
ing about the parking pricing under different operation
regimes, but had not yet explored the impacts of multiple
parking facilities.
The objective of this paper is to model the parking
pricing of multiple parking facilities under three opera-
tion regimes (i.e., the monopoly market, the oligopoly
competition, and the social optimum). The following sec-
tion presents several assumptions to facilitate the model-
ing. Section 3 formulates the pricing model with opera-
tors’ objective functions under the three operation re-
gimes. Section 4 discusses multimodal equilibrium mo-
del to represent the travelers’ minimum travel disutility.
A comprehensive solution algorithm is illustrated step-
wise in Section 5, followed by numerical experiments.
Finally, conclusive remarks are made along with recom-
mendations for future researches.
2. Assumptions and Variable Descriptions
Given a multimodal network G = (N, L), where N is the
set of all notes; L is the set of all links. G includes two
sub-networks, the auto network and the
metro network . Without loss of genera-
lity, the following basic assumptions are made in this
paper.

,
aaa
GNL



,
bbb
GNL
A1. There are two types of players in the network:
parking operators and auto travelers. The operators make
their pricing decisions based on the parking demand,
which in turn varies as a result of travelers’ adjustment to
their journey costs.
A2. Consider three operation regimes: monopoly mar-
ket, oligopoly competition, and social optimum; and three
types of parking facilities: off-road parking lot, curb par-
king lot, and P & R facility.
A3. There are two modes for travelers to accomplish
their journey, the automobile and the P & R. For conve-
nience of the following formulation, let “a” and “b” de-
note the two modes, and use the multinomial logit model
to describe travelers’ mode choice behaviors, which
could fit the diversity in people’s preferences.
A4. Suppose travelers are very familiar with the net-
work, and make their travel decisions in a deterministic
manner to minimize the travel disutility. Therefore, by
competition the user equilibrium (UE) would reach.
A5. Introduce an elastic demand function to depict
travelers’ responses to various level of parking charges,
such as switching the departure time, or even not making
the journey.
Tables 1 and 2 present the parameters and variables
that are used in the following model formulations.
3. Parking Pricing Model
3.1. Profit Function
The net profit k

of operator k can be formulated as
the total revenue minus the operation costs,


 



,
kk
kjjjj
jJ jJ
zvHCE C

 

zvzjj
(1)
where, the bold symbols represent the vectors of the cor-
responding variables. It is assumed that
j
H
and
j
E
are linear functions of the parking capacity
j
C, in forms
of


j
j
j
C
H
C

 and


01
j
j
j
C
EC   .
3.2. Objective Functions
3.2.1. Mon op o l y Market
In the monopoly market, there should be a single autho-
rized agent who is responsible for operating all parking
facilities. The objective of the agent would be to maxi-
mize its total net profit by guiding all operators in the
market. Given an elastic demand, the maximization is to
find the optimal parking fees for each parking facilities,
and can be expressed as follows.


(U1) max ,
k
k


zzvz

(2)
where the parking demand can be obtained from the
lower-level equilibrium model (given in Section 3), and
operator k’s net profit k

can be calculated by Equa-
tion (1).
3.2.2. Oligop o l y Comp eti tion
The oligopoly competition in this paper refers to the situ-
ation, in which operators act independently and compete
for their own profits. When the competitive equilibrium
reaches, no operator could earn more profits by his own
adjustment of the parking pricing. Thus, the oligopolistic
equilibrium can be formulated as a Cournot-Nash game
problem,



(U2) max ,,,,
k
kk kk
kkK



zzzvzz (3)
where, represents operator k’s pricing strategy for its
parking facilities, and
k
zk

z is the pricing strategy of
other operators.
3.2.3. Social Optimum
The social optimum represents the common situation
where all parking facilities are operated by the govern-
ment, who provides the parking as a public service.
Therefore, the operational objective is to maximize the
total social welfare (SW), which is defined here as a sum
of the consumer surplus and the producer surplus. The

W. B. FAN, M. B. KHAN 262
Table 1. Subscripts and parameters used in mathematical
formulations.
Symbol Description
K = set of parking operators
k = parking operator index
k
J
= set of parking facilities (i.e., parking lots and P & R
facilities) of operator k

, 1

= construction costs and operation costs of unit parking
space, respectively
0

= fixed operation costs of parking facility j
r, s = origin and destination (OD) of a trip
p = index of a path between OD pair (r, s)
m = set of travel modes (i.e., auto mode a and P & R mode b)
14

  = parameters converting travel costs by auto mode into the
same unit (e.g., monetary unit)
13

  = parameters converting travel costs of P & R mode
1
 4 = parameters converting the transferring costs between
auto to metro into the same unit

= a statistical calibrating parameter dependent on the
interval of trains’ arrivals at station and also the
distribution of passengers’ arrival times
m

= parameter reflecting traveler’s preference to mode m

= traveler’s perception variation on travel disutility

= parameter reflecting the elasticity of the demand to the
travel disutility
n = iteration counter in the solving algorithm
la = index of a link on roadway network
consumer surplus equals consumers’ total utility minus
their all disutility; and the producer surplus is the net
profits of all operators. The objective function under the
regime of social optimum can be given as,

 
1
0
,,
(U3) max SWd
,
rs
Q
rsrs rs
rs rs
k
kK
Duu Q 












zvz


(4)
where, the first part on the right-hand of Equation (4) is
the consumer surplus; and the second part is the total net
profits of all operators.
4. Multimodal Equilibrium Model
4.1. Equilibrium Conditions
According to the A4, travelers’ behaviors satisfy the UE
condition. That is to say, in the equilibrium state, the
travel alternatives chosen by travelers have the minimum
travel disutility. Such situation can be expressed as fol-
lows.
,
,
,
, if, 0
, if, 0
mm
rsrs jp m
rs jpmm
rsrs jp
f
Uf









(5)
Table 2. Variables used in mathematical formulations.
Symbol Description
k

= net profit of operator k
,
jj
zv = parking fee and demand of parking facility j
j
H
,
j
E= construction costs and the operation costs of
parking facility j, respectively
j
C = parking capacity of parking facility j
rs
Q = demand accommodated between (r, s)
rs

= travel disutility between (r, s)
1
rs
D

= inverse function of the elastic demand Q
rs
rs
Q = potential demand between (r, s)
m
rs
q = travel demand by mode m between (r, s)
,
m
rs jp
U,,
m
rs jp
f
= travel disutility and flow of mode m (i.e ., a and b)
on route p via parking facility j, between OD pair (r,
s)
,rj p
T = actual travel time by auto from origin r via path p
to parking facility j nearby the destination s
j

= parking search time within parking facility j

= average occupancy converting the parking fee into
per person
js
w = walking time from parking lot j to destination s
1
, rj p
T = auto travel time from the origin r to the P & R
facility j via path p
jj


= costs of transferring from automobile to metro
2
, js p
T = in-vehicle time by metro
b

= additional penalty for the transferring
jj
w

= walking time from P & R facility j to the adjacent
metro stationj

;
T
 jw
= waiting time at station j
F
b
= dispatch frequency of the metro trains

= a pre-specified precision
a
l
t,
a
l
v= travel time and traffic flow on auto link
a
l
0
a
l
t,
a
l
C= free-flow travel time and capacity on link la
0
j
d = free-flow parking search time of parking facility j
where, the travel disutility functions of mode a and b are
given as the following Equations (6) and (7), respec-
tively.

,1,2 34
a
rs jprj pjjjs
UT z   w
(6)
When the traveler chosen mode b to travel from origin
r and park at j, and continue the journey by taking metro
at station

j to destination s. Thus, his/her actual travel
disutility can be given formulated as,
12
,1,2 3,
b
rsjprjpjjj spb
UT T 


  (7)
where, the transferring costs
j
j

 can be given as,


123 4
w
j
jj jjj
zw 
j
T


 

(8)
where, the waiting time at station w
j
jT
 can be calcu-

W. B. FAN, M. B. KHAN 263
lated by,
w
j
T 
b
F (9)
It is to be noted that passengers’ arrival times is as-
sumed to follow the normal distribution and the metro
trains’ interval time is constant. Then, following refer-
ence [1], it can be derived that 0.5

.
The traveler’s choice model can be formulated ac-
cording to the assumption A2 as,
   
exp exp
m
rs
mm mm
rs rsrs
m
q
Q   
 (10)
where, rs is supposed to be a continuous monotone
decreasing function of the travel disutility between (r, s),
and is given in the following form:
Q

exp
rs rsrs
QQ


  (11)
where, the travel disutility rs

, according to the nature
of logit model, can be expressed by
 

ln exp mm
rs rs
m

  
.
4.2. Variational Inequality Model
Following the work of W. H. K. Lam et al. [2] on net-
work with multiple parking facilities, the aforementioned
equilibrium conditions (Equations (5)-(11)) can be ex-
pressed as a variational inequality problem (of which the
proof can be done by using Karush-Kuhn-Tucker con-
ditions but omitted for brevity of this paper) as follows.




*
,, ,
,
*
*
*
1* *
(L)
1ln
() 0
mm m
rs jprs jprs jp
rsmp j
mmmm
rs rs rs
rs mrs
rs rsrsrs
rs
Uff
qqq
Q
DQ QQ














q
a
b
(12)
Subject to:
{,}
m
rs rs
mab
Q

 (13)
,
,
a
rs j
a
rsrs jp
pP
qf

 (14)
,
,
b
rs j
b
rsrs jp
pP
qf

 (15)
,
,, 0
mm
rsrsrs jp
Qq f (16)
where, *
,
m
rs jp
f
, , and rs represent the optimal solu-
tions; the Equations (12)-(15) are demand conservation
constraints for network and modes; Equation (16) is non-
negativity constraints for the OD demand, parking de-
mand, and route flows.
*m
rs
q* Q
5. Solution Algorithm
A sensitivity analysis based algorithm [12] is proposed in
this section to solve equilibrium model. The fundamental
logic is to convert the intractable non-linear problem into
a quadratic programming problem by computing the de-
cision variable’s gradient information. The detailed step-
wise procedure is given as follows:
Step 1. Initialization. Set an initial parking fee , and
let the iteration counter
0
z
0n

.
Step 2. Lower-level assignment. Substitute the given
into the lower-level problem, and calculate the equi-
librium solution, and get the parking demand , the
OD travel disutility
n
zn
v
n

, and the accommodated demand
.
n
QStep 3. Sensitivity Analysis. Compute the equilibrium
solution’s gradient information with respect to the deci-
sion variable z, and get



vzz
,

z, and

Qz.
Step 4. Linearization. Linearize the parking demand
, the OD travel disutility
v

, and the accommodated
demand , and get the Equations (17)-(19). Q



()
n
kjj
jn
n

j
jj
jJ jzz
v
vv zz
z

 
 




z
vzj (17)




,
()
n
jj
rs n
n
rsrsj j
rs jzz
Q
QQ zz
z


 




z
zz

(18)




,
()
n
jj
rs n
n
rsrsj j
rs jzz
zz
z





 




z
zz

(19)
Step 5. Substitute the above Equations (17)-(19) in to
the upper-level objective function, and get a quadratic
programming problem of variable , which can be so-
lved by Newton methods. Yield an auxiliary solution
.
z
n
yStep 6. Update the parking fee by


1nnnn
n
 zzyz.
Step 7. Check convergence. If 1 max nn
jj
zz




stands, then stop and report the solution; otherwise, let
1nn

, and go to step 2. The

is a pre-specified
precision.
Remark. In step 2, the logit assignment can be under-
taken by the Dial approach [13], and the deterministic
network assignment can be preceded by the all-or-noth-
ing approach [14] combined with Moore’s shortest route
algorithm [15].
6. Numerical Experiments
6.1. Data Inputs
This section presents a numerical experiment that is car-
ried out with an illustrated network as shown in Figure 1.

W. B. FAN, M. B. KHAN 264
CBD
P&R
8
5
7
3
B
4
6
A
1
Auto linkWalking linkMe
t
ro link
1
2
3
4
5
2
6
Figure 1. Illustration of the hypothetical network.
Node 1 and 2 represent two residential communities at
suburban area; and note 3 denotes the central business
district (CBD). There are three different-type parking fa-
cilities in the network: nodes A and B are an off-road
parking lot and a curb parking lot within CBD, respec-
tively; near by node 4 is a P & R facility.
Functions of the auto link travel time and the parking
search time are given in the Bureau of Public Roads
(BPR) form,
 

4
010.15
aa
aa
ll
ll
vv
tt a
l
C
and
 
4.03
00.31
jjjj j
vd vC   . The involved parameters
are specified with values as shown in Table 3.
To make the experiment representative, the values of
all parameters are carefully designed. For this study, the
metro fare is taken as $0.3/km, the length of metro link is
30 km, and the average speed on metro line is 60 km/h;
The capacity of metro vehicle assumed is 300 pas-
sengers/carriage at the dispatching frequency of 6 car-
riages/hour. Other parameters include:
11.0

, 21.4

, 30.1


,41.8

; 11.0


,
22.0

, 31.0

; , 2,
10.7
 0.1  30.9


;
1.0

, 0.7

, 10

, 0.09


, and 1.0

. The
transferring penalty for P & R mode is set to be 0.1. The
potential demand of OD pair (1, 3) and (2, 3) are given to
be 2000 persons/hour and 1000 persons/hour, respe-
ctively.
6.2. Numerical Results
The proposed algorithms are coded and implemented in
Matlab on the Windows XP operating system, and the
numerical experiments are conducted on a laptop with a
Core2 Duo processor 2.4 GHz processor and 2.0 GB
RAM. Figures 2 and 3 reveal slice plots of the total pro-
fits and the social welfare with varying parking fees un-
der the monopoly market regime and the social optimum
regime, respectively. It is shown that the parking pricing
could lead to positive, neutral, and negative objectives.
The highest profit ($3403.6, vid. Table 4) is reached in
the monopoly market, when the parking fees are $15, $13,
and $7 for the curb parking, off-road parking lot, and P &
R facility, respectively (vid. Table 5). The highest social
welfare ($4012.4, vid. Table 4) is realized in the social
optimum regime with parking fees being $13, $11, and
$6 (vid. Table 5).
Figure 2. The total profit of the monopoly market.
Figure 3. The social welfare of the social optimum regime.
Table 3. Parameters of the link travel time function and the
parking search function.
Auto links 0
a
l
t (h) a
l
C (Vehicles/h)
1 0.15 800
2 0.30 600
3 0.65 800
4 0.60 800
5 0.15 800
Parking facilities0
j
d (h) j C (Vehicles/h)
A 0.10 1350
B 0.01 300
P & R 0.05 700
Walking links Walking time (h)
(4, 5) 0.05
(8, 3) 0.20
(A, 3) 0.10
(B, 3) 0.05

W. B. FAN, M. B. KHAN 265
Table 4. The networ k per f or mance under three re gime s ($).
Regimes Total profits Social welfare Network costs
Monopoly
market 3403.6 3967.7 1252.8
Oligopoly
competition 2095.9 3155.7 1734.7
Social optimum 3351.5 4012.4 1362.8
Table 5. The optimal pricing and corresponding profits of
three parking facilities ($).
Curb parking Off-road parking P & R facilities
Regimes Parking
fees Profits Parking fees Profits Parking fees Profits
Monopoly
market 15 2877 13 872.6 7 −345
Oligopoly
competition 6 1189 5 854.6 5 52.5
Social
optimum 13 2449 11 1245 6 −342
Table 5 shows the optimal pricing solution and the
corresponding profits of three parking facilities. It can be
found that the monopoly market yields the highest level
of parking fee, while the oligopoly competition results in
the lowest level. In addition, both the monopoly market
and the social optimum maximize the overall objective at
cost of the P & R operator’s loss, which means subsidies
for P & R operation should be necessary.
Table 4 presents the comparison of the three regimes
in terms of total profits, social welfare, and network costs.
In the view of parking operators, the monopoly market
allow them obtain the highest profits; from the point of
public administration, the social optimum regime results
in the highest social welfare. Interestingly, the oligopoly
competition yields positive profits for each operator,
nonetheless the lowest profits and social welfare in total.
Furthermore, the highest network costs emerge in the
oligopoly competition in spite of more auto travelers are
induced to choose the P & R mode (17.74%, vid. Table
6). To this result, the explanation can be found in Table
6, which shows that the most demand is accommodated
on the network under the oligopoly competition regime.
Consequently, although the average travel costs decrease
to the least of all ($2.05/person), the total network costs
rise ($1734.7).
The aforementioned analyses show that operation re-
gimes have significant influence on the optimal pricing
solution, the operators’ profits, the demand split, and also
the overall network performance. It is believed in this
paper that the least level of pricing is due to interactive
competitions between the three parking facilities.
7. Conclusions
This paper explores the optimal pricing of three-type
Table 6. The equilibrium solution under three regim es.
Regimes Curb
parking
Off-road
parking P & R facility Accommodated demand (person)
Monopoly
market 47.71%37.92% 14.37% 451
Oligopoly
competition 30.25%52.01% 17.74% 848
Social
optimum 40.81%44.71% 14.48% 528
parking facilities under three operation regimes (i.e.,
monopoly market, oligopoly competition, and social op-
timum). To handle the interactions between pricing deci-
sion and parking demand, we propose a bi-level pro-
gramming model simulating the decision-making of
parking operators and auto travelers, simultaneously. In
the upper level, operators optimize the parking pricing to
achieve certain objectives, which may differ with the
operation regimes; the lower-level sub-model is the net-
work equilibrium, where travelers choose the best travel
alternative according to parking fee, metro fare, travel
time, walking time, etc. In the light of the complexity of
the proposed non-linear model, a sensitivity analysis based
algorithm is adopted. A numerical experiment is de-
signed to assess the proposed model. The results verify
that operation regimes play an influential role on the
pricing decisions and the final outcomes. It is to be men-
tioned that parking operators’ competition under the oli-
gopoly regime decreases the overall level of pricing, at-
tracts more demand, and promotes the share of the P & R
mode. These phenomena remind our policy-makers should
carefully design the operation regimes to adjust the in-
terests of multiple parking operators and mass travelers.
Our further researches will address important issues to
relax the constraints made in this study. For instance,
travelers’ information about the network is not perfect,
and the network is uncertain due to some random inci-
dents, such as signal failure, road construction, and acci-
dents. Interesting results are expected in evaluating the
impacts of network uncertainty under different regimes.
8. Acknowledgements
The research is supported by National Natural Science
Foundation of China (No. 51178403, 51108391).
REFERENCES
[1] Z. C. Li, H. J. Huang, W. H. K. Lam and S. C. Wong,
“Optimization Of Time-Varying Parking Charges and
Parking Supply in Networks with Multiple User Classes
and Multiple Parking Facilities,” Tsinghua Science and
Technology, Vol. 12, No. 2, 2007, pp. 167-177.
doi:10.1016/S1007-0214(07)70024-X
[2] W. H. K. Lam, Z. C. Li, H. J. Huang and S. C. Wong,

W. B. FAN, M. B. KHAN 266
“Modeling Time-Dependent Travel Choice Problems in
Road Networks with Multiple User Classes and Multiple
Parking Facilities,” Transportation Research Part B, Vol.
40, No. 5, 2006, pp. 368-395.
doi:10.1016/j.trb.2005.05.003
[3] A. Glazer and E. Niskanen, “Parking Fees and Conges-
tion,” Regional Science and Urban Economics, Vol. 22,
No. 2, 1992, pp. 123-132.
doi:10.1016/0166-0462(92)90028-Y
[4] E. Verhoef, P. Nijkamp and P. Rietveld, “The Economics
of Regulatory Parking Polices: The (IM) Posssibilities of
Parking Policies in Traffic Regulation,” Transportation
Research Part A, Vol. 29, No. 2, 1995, pp. 141-156.
doi:10.1016/0965-8564(94)E0014-Z
[5] Z. Y. Mei, J. Chen and W. Wang, “Curb Parking Pricing
Method Based on Parking Choice Behavior,” Journal of
Southeast University (English Edition), Vol. 22, No. 4,
2006, pp. 559-563.
[6] S. Carrese, S. Gori and T. Picano, “A Parking Equilib-
rium Model for Park Pricing and Park & Ride Planning,”
Transportation Systems, Vol. 3, No. 6, 1997, pp. 1235-
1239.
[7] R. Garcia and A. Marin, “Parking Capacity and Pricing in
Park’n Ride Trips: A Continuous Equilibrium Network
Design Problem,” Annals of Operations Research, Vol.
116, No. 1-4, 2002, pp. 153-178.
doi:10.1023/A:1021332414941
[8] J. Y. T. Wang, H. Yang and R. Lindsey, “Locating and
Pricing Park-and-Ride Facilities in a Linear Monocentric
City with Deterministic Mode Choice,” Transportation
Research Part B, Vol. 38, No. 8, 2004, pp. 709-731.
doi:10.1016/j.trb.2003.10.002
[9] Z. C. Li, W. H. K. Lam and S. C. Wong, “The Optimal
Transit Fare Structure under Different Market Regimes
with Uncertainty in the Network,” Networks and Spatial
Economics, Vol. 9, No. 2, 2008, pp. 191-216.
doi:10.1007/s11067-007-9058-z
[10] S. P. Anderson and A. D. Palma, “The Economics of
Pricing Parking,” Journal of Urban Economics, Vol. 55,
2004, pp. 1-20. doi:10.1016/j.jue.2003.06.004
[11] J. F. Tsai and C. P. Chu, “Economic Analysis of Collect-
ing Parking Fees by a Private Firm,” Transportation Re-
search Part A, Vol. 40, No. 8, 2006, pp. 690-697.
doi:10.1016/j.tra.2005.12.001
[12] H. Yang, “Sensitivity Analysis for the Elastic-Demand
Network Equilibrium Problem with Applications,” Trans-
portation Research B, Vol. 31, No. 2, 1997, pp. 55-70.
doi:10.1016/S0191-2615(96)00015-X
[13] R. B. Dial, “A Probabilistic Multipath Traffic Assignment
Algorithm Which Obviates Path Enumeration,” Trans-
portation Research, Vol. 5, 1971, pp. 83-111.
doi:10.1016/0041-1647(71)90012-8
[14] Y. Sheffi, “Urban Transportation Networks: Equilibrium
Analysis with Mathematical Programming Methods,”
Prentice-Hall, Englewood Cliffs, 1985.
[15] E. F. Moore, “The Shortest Path through a Maze,” Har-
vard University Press, Cambridge, 1957.