A Path-size Weibit Stochastic User
Equilibrium Model
Songyot Kitthamkesorn
Department of Civil & Environmental Engineering
Utah State University
Logan, UT 84322-4110, USA
Email: songyot.k@aggiemail.usu.edu
Adviser: Anthony Chen
Email: anthony.chen@usu.edu
Outline

Review of closed-form route choice/network
equilibrium models
 Weibit route choice model
 Weibit stochastic user equilibrium model
 Numerical results
 Concluding Remarks
2
Outline

Review of closed-form route choice/network
equilibrium models
 Weibit route choice model
 Weibit stochastic user equilibrium model
 Numerical results
 Concluding Remarks
3
Deterministic User Equilibrium (DUE)
Principle
Wardrop’s First Principle
“The journey costs on all used routes are equal, and less
than those which would be experienced by a single vehicle
on any unused route.”
Assumptions: All travelers have the same behavior and
perfect knowledge of network travel costs.
4
Stochastic User Equilibrium (SUE)
Principle and Conditions
Daganzo and Sheffi (1977)
“At stochastic user equilibrium, no travelers can improve his or
her perceived travel cost by unilaterally changing routes.”


Prij  gij   Pr Grij  Glij ,  l  Rij and r  l gij , r  Rij , ij  IJ
frij  Prij  g  qij ,  r  Rij , ij  IJ

rRij
Prij  g   1.0,  ij  IJ

rRij
f rij  qij ,  ij  IJ
Daganzo, C.F., Sheffi, Y., 1977. On stochastic models of traffic assignment.
Transportation Science, 11(3), 253-274.
5
Probabilistic Route Choice Models
Perceived travel cost
Gumbel
Normal
Multinomial logit (MNL) route
choice model Dial (1971)
Closed form
P 
ij
r
exp   g rij 
 exp   g
kRij
ij
k

Multinomial probit (MNP) route
choice model Daganzo and Sheffi
(1977)

Non-closed form

Prij   .. f t1ij ,.., t ijR dt1ij ..dt ijR
ij
ij
Dial, R., 1971. A probabilistic multipath traffic assignment model which obviates path enumeration. Transportation Research,
5(2), 83-111.
Daganzo, C.F. and Sheffi, Y., 1977. On stochastic models of traffic assignment. Transportation Science, 11(3), 253-274.
6
Gumbel Distribution
PDF
Gumbel
Perceived travel cost
CDF FG ij  t 
Location
parameter
r
Mean route travel cost g
1  exp e


 , t  , 



  ij
r
Euler constant
ij
r
ij
r
Route perception variance 

 rij t  rij

ij 2
r
2
6 rij
Scale
parameter
2
Variance is a function of scale
parameter only!!!
7
MNL Model and Closed-form Probability
Expression
Under the independently distributed assumption, we have the joint survival function:

 rij  trij  rij  

H  exp   e

r

R


ij
Then, the choice probability can be determined by


P   e
ij
r
ij ij
ij
ij  r tr  r
r
 exp 

 e
kij trij  kij
  dt ij



r

 kRij
Identically distributed
To obtain a closed-form,  is fixed for all routes
assumption


  trij kij  
  trij rij 

 ij
ij
Pr    e
exp   e
 dtr



 kRij

Finally, we have
Independently and
ij
exp  g r
Identically distributed (IID)
ij
Pr 
assumption
ij



 exp   g 
kRij
k
8
Independently Distributed Assumption:
Route Overlapping
(x)
(100-x)
(100-x)
j
i
Destination
P1  P  P 
ij
2
ij
3
e100
e100
1

 e100  e100 3
(100)
(Travel cost)
0.5
Prob. of choosing lower route
Origin
ij
0.46
MNL solution
MNP
0.42
Independently
distributed
0.38
MNL
0.34
0.3
0
20
40
60
80
100
x
Route overlapping
9
Identically Distributed Assumption:
Homogeneous Perception Variance
(125)
(10)
Origin
i
MNL (=0.1)
(5)
Destination
Origin
j
i
j
Destination
(Travel cost)
(Travel cost)
e0.15
1
Pl  0.15 0.110 
 0.62
0.1 5
e
e
1 e
ij
(120)
=
e0.1120
1
Pl  0.1120 0.1125 
 0.62
0.1 5
e
e
1 e
ij
Absolute cost
difference
MNP
Pl ij  0.85
>
Pl ij  0.54
2
Same perception variance of
6 2
PDF
5
10
120 125
Perceived travel cost
10
Existing Models
1. Gumbel
2. Normal
MNL
EXTENDED
LOGIT
Closed form
Closed form
MNP
11
Extended Logit Models
Gumbel
MNL
EXTENDED
LOGIT
Closed form
Closed form
Urij   grij  rij
Modification of the deterministic term
• C-logit (Cascetta et al., 1996)
• Path-size logit (PSL) (Ben-Akiva and Bierlaire, 1999)
Modification of the random error term
• Cross Nested logit (CNL) (Bekhor and Prashker, 1999)
• Paired Combinatorial logit (PCL) (Bekhor and Prashker, 1999)
• Generalized Nested logit (GNL) (Bekhor and Prashker, 2001)
Cascetta, E., Nuzzolo, A., Russo, F., Vitetta, A., 1996. A modified logit route choice model overcoming path overlapping problems: specification and some calibration results for interurban
networks. In Proceedings of the 13th International Symposium on Transportation and Traffic Theory, Leon, France, 697-711.
Ben-Akiva, M. and Bierlaire, M., 1999. Discrete choice methods and their applications to short term travel decisions. Handbook of Transportation Science, R.W. Halled, Kluwer Publishers.
12
Bekhor, S., Prashker, J.N., 1999. Formulations of extended logit stochastic user equilibrium assignments. Proceedings of the 14th International Symposium on Transportation and Traffic
Theory, Jerusalem, Israel, 351-372.
Bekhor S., Prashker, J.N., 2001. A stochastic user equilibrium formulation for the generalized nested logit model. Transportation Research Record 1752, 84-90.
Independently Distributed Assumption:
Route Overlapping
(x)
(100-x)
(100-x)
i
j
Destination
(100)
(Travel cost)
0.5
Prob. of choosing lower route
Origin
MNL
C-logit
PSL
PCL
CNL
GNL
0.46
0.42
MNP
0.38
MNL
0.34
0.3
0
20
40
60
x
80
100
13
Scaling Technique
(125)
(10)
Origin
(5)
i
Destination
Origin
j
(120)
i
CV = 0.5
j
Destination
(Travel cost)
(Travel cost)
(=0.51)
(=0.02)
e0.515
Pl  0.515 0.5110  0.93
e
e
ij
>
e0.04120
Pl  0.04120 0.04125  0.52
e
e
ij
Same
perception
variance
PDF
5
10
Same
perception
variance
120 125
Perceived travel cost
Chen, A., Pravinvongvuth, S., Xu, X., Ryu, S. and Chootinan, P., 2012. Examining the scaling effect and
overlapping problem in logit-based stochastic user equilibrium models. Transportation Research Part A, 46(8), 14
1343-1358.
3rd Alternative
1. Gumbel
2. Normal
3. Weibull
MNL
EXTENDED
LOGIT
Closed form
Closed form
MNP
Modification of the
deterministic term
MNW
PSW
Closed form
Closed form
Multinomial weibit model
Path-size weibit model
(Castillo et al., 2008)
Castillo et al. (2008) Closed form expressions for choice probabilities in the Weibull case. Transportation Research Part
15 B
42(4), 373-380
Outline

Review of closed-form route choice/network
equilibrium models
 Weibit route choice model
 Weibit stochastic user equilibrium model
 Numerical results
 Concluding Remarks
16
Weibull Distribution
PDF
Location
parameter
Shape
parameter
Weibull
Perceived travel cost
r
 
ij

 t   r 

1  exp  
ij
  r 



 , t   0,  
 Scale

parameter

1 
 rij  rij  1  ij 
 r 
ij
CDF FG ij  t 
r
Gamma function
Mean travel cost grij
Route perception variance  rij 
2

rij   1 
2
Variance is a function of route cost!!!

2 
ij
ij 2

g


  r
r 
 rij 
17
Multinomial Weibit (MNW) Model and
Closed-form Prob. Expression
Under the independently distributed assumption, we have the joint survival function:
rij 
  ij
ij
  tr   r   
 
H   exp  
ij
rRij

 
r
 


Then, the choice probability can be determined by
rij 1
rij 
lij 

  ij
ij
ij
ij

ij  t ij   ij 
r  r
r 
  tr   r   
   tr   l    ij
ij

   exp  
  dtr
Pr     ij 
exp  
ij
ij
ij
l

r
r  r

  lRij
 
  r
  l
 rij





Since the Weibull
To obtain a closed-form,  and  are fixed for all routes
variance is a function
 ij 1
 ij 

ij
ij
ij
ij

ij  t  



  exp   tr      dt ij
 r
of route cost, the
Prij   ij 



identically distributed
r  rij 
kij   r
kRij 

 ij




 ij

assumption does NOT


Finally, we have
ij
ij
apply
g


r
ij
Pr


 g
kRij

ij
k


ij
ij  
Castillo et al. (2008) Closed form expressions for choice probabilities in the
Weibull case. Transportation Research Part B 42(4), 373-380
18
Identically Distributed Assumption:
Homogeneous Perception Variance
(125)
(10)
Origin
i
MNW model
j
i
1
 10 
1  
5
(120)
j
Destination
(Travel cost)
(Travel cost)
52.1
Pl  2.1

5  102.1
ij
Origin
Destination
(5)
CV = 0.5
2.1
 0.81
>
1202.1
Pl 

2.1
2.1
120  125
1
ij
 125 
1 

 120 
2.1
 ij  2.1
 ij  0
 0.52
Relative cost
difference
Route-specific perception variance
PDF
2
 
ij 2
r
5
10
120
Perceived travel cost

  
g rij
2 
1
2



1



1

 

ij 
ij
ij
  1  1       
 



125
19
Path-Size Weibit (PSW) Model
MNW random utility maximization model
U rij   grij  

ij
ij 
 rij
Weibull distributed
random error term
To handle the route overlapping problem, a path-size
factor (Ben-Akiva and Bierlaire,
1999) is introduced,
ij
ij
ij 
i.e.,
gr  
ij
ij
Path-size factor
Ur 

r
ij


r
which gives the PSW model:
ij
r
P
g  


  g  

kRij
ij
r
ij
ij  
ij
r
ij
k
ij
k
 
ij
r

ar
la
Lijr
1

kRij
ij
ak
ij


ij
Ben-Akiva, M. and Bierlaire, M., 1999. Discrete choice methods and their applications to short term travel
20
decisions. Handbook of Transportation Science, R.W. Halled, Kluwer Publishers.
Independently Distributed Assumption:
Route Overlapping
(x)
(100-x)
(100-x)
Pl 
ij
Destination
Origin
(100)
Pl 
ij
(Travel cost)
1100 
2.1
1100 
2.1
 0.75 100 
2.1
1100 
2.1
 0.75 100 
1100 
2.1
 0.5 100 
2.1
2.1
 0.5 100 
2.1
 0.4
Pl 
ij
1100 
2.1
1100 
2.1
 1100 
2.1
 1100 
2.1
 0.33
Prob. of choosing lower route
0.5
PSW
0.46
MNL solution
0.42
MNP
0.38
MNL, MNW
0.34
0.3
0
20
40
60
x
80
100
21
 0.5
Outline

Review of closed-form route choice/network
equilibrium models
 Weibit route choice model
 Weibit stochastic user equilibrium model
 Numerical results
 Concluding Remarks
22
Comparison between MNL Model and
MNW Model
Extreme value distribution
Gumbel (type I)
Log
Weibull
Weibull (type III)
ij
Assume   0
IID
P 
ij
r
Independence
exp   g rij 
 exp   g 
kRij
ij
k
Log
Transformation
ij
r
P
g 


 g 
ij
ij  
r
kRij
ij
ij  
k
23
A Mathematical Programming (MP)
Formulation for the MNW-SUE model
Multiplicative Beckmann’s
transformation
(MBec)
Relative cost difference
under congestion
va
min Z    ln  a   d  
aA 0
ijIJ
1
 ij
 f  ln f
rRij
ij
r
ij
r
 1
s.t.

rRij
f rij  qij
ij
r
P
g 


 g 
ij
ij  
r
kRij
ij
ij  
k
f rij  0
24
A MP Formulation for the PSW-SUE Model
va
min Z    ln  a   d   
aA 0
ijIJ
1

ij

rRij
f rij  ln f rij  1  
ijIJ
1

f rij ln rij

ij
ij
r
g 


 g 
rRij
s.t.

rRij
f rij  qij
f rij  0
P

kRij
ij
ij  
r
ij
r
ij
k
ij
ij  
k
25
Equivalency Condition
By constructing the Lagrangian function, we have


ij
L  Z   ij  qij   f r 


ijIJ
r

R
ij


By setting the partial derivative w.r.t. route flow variable equal to zero, we have
f  exp   ij 
ij
r
ij
qij 
f
rRij
ij
r
ij
r
g 
ij
ij  
r
 exp   ij   
ij
rRij
ij
r
g 
ij
ij  
r
Then, we have the PSW route flow solution, i.e.,
 g 
f rij
P 

ij
ij
ij  
qij
  k  gk 
ij
r
ij
r
ij
ij  
r
kRij
26
Uniqueness Condition

The second derivative
2Z
f rij fl ij
1
 d b ij
 dv  br   f ij  0 , r  l
bA b
r

 d b  ij  ij ,
r l

br bl
bA dvb
By assuming d b dvb  0, b  A, the route flow solution
of PSW-SUE is unique.
27
Path-Based Partial Linearization
Algorithm
Initialization
·
·
n=0
f(0) = 0 à Free flow travel cost
n = n+1
Flow
Network
Search direction
PSW
probability
Update flow f(n)
yrij  n   qij Prij  g  n  
Route
travel cost
Line search
Solve
  n   arg min Z f  n     y  n   f  n   
0  1
Update
f  n  1  f  n     n    y  n   f  n  
Stopping
criteria

NO
n = n+1
Link flow
Link
travel cost
YES
Result
28
Outline

Review of closed-form route choice/network
equilibrium models
 Weibit route choice model
 Weibit stochastic user equilibrium model
 Numerical results
 Concluding Remarks
29
Real Network
0
2
4


6


Kilometers




































 






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
Winnipeg network, Canada
154 zones, 2,535 links, and
4,345 O-D pairs.
















0



2
4
6
Kilometers
30
Convergence Results
1.E+00
1.E-01 0
10
20
30
40
50
1.E-02

1.E-03
1.E-04
1.E-05
1.E-06
1.E-07
1.E-08
MNW-SUE
PSW-SUE
Iteration
31
Winnipeg Network Results
O-D (14, 100)
PSLs-SUE MNW-SUE PSW-SUE
0.140
0.191
0.149
0.233
0.120
0.226
0.139
0.176
0.148
0.154
0.129
0.138
0.193
0.196
0.196
0.141
0.188
0.142
Route
1
2
3
4
5
6
1
4
14
3
2
2
100
6
5
30
1
92
5
4
3
1
6
52
3
50
2
Route
1
2
3
4
5
6
O-D (92, 30)
PSLs-SUE MNW-SUE PSW-SUE
0.097
0.117
0.126
0.269
0.194
0.229
0.282
0.197
0.252
0.173
0.139
0.174
0.130
0.244
0.134
0.050
0.109
0.087
Route
1
2
3
O-D (50, 52)
PSLs-SUE MNW-SUE PSW-SUE
0.585
0.439
0.472
0.178
0.242
0.248
0.237
0.319
0.280
32
Link Flow Difference between MNW-SUE
and PSW-SUE Models
CBD
MNW - PSW
-500 to -300
PSLs
- PSW
-300
to -100
-500 to -300
-100 to 100
-300 to -100
100
to 300
-100
to 100
100
to 300
300
to 500
300 to 500
500
to 700
500
to 700
600
400
261
200
0
CBD
584
235
115
12
7
-500 to -300 to -100 to 100 to 300 to 500 to
-300 -100
100
300
500
700
Flow difference (MNW-SUE - PSW-SUE)
800
Number of links
Number of links
800
Non-CBD
600
503
400
200
0
2
31
37
7
0
-500 to -300 to -100 to 100 to 300 to 500 to
-300 -100
100
300
500
700
Flow difference (MNW-SUE - PSW-SUE)
33
Link Flow Difference between PSLs-SUE
and PSW-SUE Models
CBD
PSLs - PSW
-500 to -300
-300 to -100
-100 to 100
100 to 300
300 to 500
500 to 700
794
CBD
600
400
129
200
0
1
186
104
0
-500 to -300 to -100 to 100 to 300 to 500 to
-300 -100
100
300
500
700
Flow difference (PSLs-SUE - PSW-SUE)
800
Number of links
Number of links
800
Non-CBD
555
600
400
200
0
1
11
9
4
0
-500 to -300 to -100 to 100 to 300 to 500 to
-300 -100
100
300
500
700
Flow difference (PSLs-SUE - PSW-SUE)
34
Drawback: Insensitive to an Arbitrary
Multiplier Route Cost
(10)
Origin
(100)
Destination
(5)
i
Origin
j
i
j
(Travel cost)
(Travel cost)
 ij  2.1,  ij  0;
Destination
(50)
 ij  2.1,  ij  0;
52.1
1
Pl  2.1

 0.811
2.1
2.1
5  10
1 2
ij
 ij  2.1,  ij  4;
502.1
1
Pl  2.1

 0.811
2.1
2.1
50  100
1 2
ij
 ij  2.1,  ij  4;
5  4
Pl ij 
 0.977
2.1
2.1
5

4

10

4
 


2.1
a) Short network
 50  4 
Pl ij 
 0.824
2.1
2.1
50

4

100

4




2.1
b) Long network
35
Incorporating ij
Variational Inequality (VI)
f f 
* T


General route cost
 
f *  P g  f *  q  0, f 
g  
 g  
ij
ij  
ij
r
ij
k
kRij
Flow dependent
g  
  g  

kRij
ij
r
ij
ij  
ij
r
ij
k
MNW model
ij
ij  
ij
k
ij  
ij
PSW model
Zhou, Z., Chen, A. and Bekhor, S., 2012. C-logit stochastic user equilibrium model: formulations and
solution algorithm. Transportmetrica, 8(1), 17-41.
36
Concluding Remarks

Reviewed the probabilistic route choice/network
equilibrium models
 Presented a new closed-form route choice model
 Provided a PSW-SUE mathematical
programming formulation under congested
networks
 Developed a path-based algorithm for solving
the PSW-SUE model
 Demonstrated with a real network
37
Thank You
38