Combined models for public and private travel modes

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TNK084 Traffic Theory series Vol.4, number.1
May 2008
Combined models for public and private travel modes
Magnus Edgren
Abstract – This report analyses a combined network model consisting of a user equilibrium car
network model, a model for modeling public transport and a nested logit model for modeling the
choices between the travel modes. First a derivation is presented that shows that the combined
model do consists of the three sub models. Then some results from a simulation using the model is
presented and analyzed. It is shown that monetary costs for different transportation modes play a
role for all people but most for people with low income. It is also shown that the dispersion in the
logit model must be chosen with great accuracy in order to get valid results.
Keywords: Combined models, nested logit functions, public transport, user equilibrium
I.
model complies with reality these things will be shown
in the results.
Introduction
I.4.
I.1.
Background
The network on which the model is applied is small
and not valid for a real case. The results shall only be
seen as indications on what happens when certain
parameters are changed.
The model is also simplified in the way that public
transport is assumed to have their own network. No
pedestrian or similar transportation is modeled either.
Modelling a car network or public transport can be
done separately in a relatively easy manner. But what is
really interesting to know is when the two models are
modelled together with a choice model for which travel
mode to use. Then we can see the interactions between
the models due to their individual attractiveness and
how different user groups react on the available choices.
A combined model is often quite advanced
mathematically and it is generally easier to apply a
model and prove that it is correct than to derive it from
scratch.
I.2.
II.
Theoretical background
An optimization problem is given together with
models for the car network, public transport and the
choice between travel modes. Here they are presented
together with a description of the notations.
Scope and objectives
The combined model will be analyzed and proved to
be made up by a car network model, a model for public
transport and a logit model for choosing among the
travel modes. The analysis consists of a theoretical part
and a implementation in AMPL. The AMPL model will
be used to se how the model behaves while changing
some of the parameters used as input to the model. For
instance monetary cost will be varied in order to see
how the model output is varying.
I.3.
Limitations
II.1.
Notations
Sets
C = set of all OD-pairs
M = set of user groups
L = set of links in the car network
Rpqm = set of all routes in OD-pair pq
For all notations below we have
lL
m
r  R pq
m M
 p, q  C
Aim
When changing the monetary cost, i.e. the actual cost
that people perceive in order to choose a certain travel
mode, it is expected that the travelling changes
accordingly. If the price for a ticket to the public
transport decreases the travelling will most certainly
increase. Similarily if the fuel price for private cars
increases, the travelling with cars will decrease. If the
Demand
Number of potential travellers
1
m
d pq
Magnus Edgren
Number of bus travellers
d bm
pq
Number of car travellers
cm
d pq
Number of travellers
dm
d pq
Number not making a trip
nm
d pq
User groups
Value of time
m
Car network model
Total flow on link l
vl
Link flow on link l by user group m
wlm
Route flow, user group m
hrm
hrm
 rl
Route cost, user group m
Route r uses link l
With a Wardrop user equilibrium mind we then get
the following model for the car network
t lc  l , vl 
Travel time on link l
l
l
Link capacity
Monetary cost
Public transport

Travel time from p to q
bm
t bm
pq  pq , d pq
Bus frequency
 pq
Free flow travel time p to q
 pq
Monetary cost
 pq
Choice model
Dispersion logit model, level i
 i , i  1,2

Other
t nm
pq
Cost of not making a trip
II.2.
Car network model
We assume link costs as follows
t lc  l , vl  
l
m
with tl being convex function for the travel time due
to link flows, road capacity and others, τl being the
monetary cost, e.g. the cost for petrol while traveling on
the link, and βm indicating the value of time for a certain
user group, m. We then get the cost for a certain route in
OD-pair “pq”
crm    rl t lc  l , vl  
lL
l
m
2
Magnus Edgren
Objective function
 vl


min z c     t lc  l , s ds   l wlm 


lL  0
mM  m

II.4.
The choice model is assumed to be a nested logit
model as shown in fig 2.
subject to
h
cm
 d pq
m
r
rR m
pq
vl 
pq  C , m  M
wlm 
lL
m
l

m
d pq
β1
w
mM
Choice model for travel modes
  rlm hrm
nm
No trip d pq
l  L, m  M
( p , q )C rR m
pq
dm
d pq
Make trip
β2
Un
Ud
r  R , pq  C, m  M
h 0
m
pq
m
r
cm Car
d pq
Bus d bm
pq
II.3.
Public transport model
The public transport is assumed to travel on its own
network and also with direct links between the ODpairs, i.e. links and routes are identical. It is also
assumed that the general travel time function is
t
bm
pq

pq
,d
bm
pq
 a
pq
 d bm

pq

 b pq 
  pq 


Fig. 2. The nested logit model.
The utilities for each level is marked Ux, with x
indicating travel mode. The probability of selecting one
alternative is given by the logit model as
n pq
e  1U n
e  1U n  e  1U d
e  1U d
p d   1U n
e
 e  1U d
e   2U c
pc  pd   2U c
e
 e   2U b
e   2U b
pb  pd   2U c
e
 e   2U b
pn 
where apq and bpq are constant parameters describing
the link, and αpq describing the bus frequency. The
demand for bus travel is denoted
d bm
pq .
Beside the travel time function we have the term
 pq 
Uc
Ub
 pq
m
with ρpq being a parameter for modelling changes in
travel time and τpq being the ticket price. The cost of
time for user group m is βm.
Then the demands for different travel modes are
calculated as
nm
m
m
d pq
 d pq
pn  d pq
The resulting model will then be
d
 pq  
bm 


min z b      t bm
pq  pq , s ds  d pq   pq 

 m  
pqC mM

0


bm
pq
1
1 e
1 U n U d 
1
dm
m
d pq
 d pq
1  e 1 U d U n 
1
cm
dm
d pq
 d pq
 2 U c U b 
1 e
1
dm
d bm
pq  d pq
 2 U b U c 
1 e
The composite utility for making a trip is also known
as
Ud 
3
1
2

ln e  2U c  e  2U b

Magnus Edgren
II.5.
Optimization problem
The optimization problem is given as follows
 vl c

 t l  l , s ds    l wlm  




lL  0
m M  m

bm
 d pq

 t bm  , s ds  d bm     pq   


pq
pq
pq
pq

 
 m  
( p , q ) C m  M

 0

  d pqnmt nmpq 
min T 
( p , q ) C m  M
1
1
  d ln d
nm
pq
dm
dm
 1  d pq
ln d pq
1 
  d ln d
dm
pq
1 
  d ln d
cm
pq
bm
 1  d bm
pq ln d pq  1
( p , q ) C m  M
1
2
( p , q ) C m  M
1
2
( p , q ) C m  M
nm
pq
dm
pq
cm
pq







subject to
cm
nm
d pq
 d bm
pq  d pq ,
nm
dm
m
d pq
 d pq
 d pq
,
h
m
r
cm
 d pq
,
 p, q   C ,
 p, q   C ,
 p, q   C ,
mM
mM
mM
rR m
pq
 
( p , q )C rR m
pq
m
rl
hrm  wlm ,
hrm  0,
 wlm  vl ,
mM
dm
d pq
nm
d pq
cm
d pq
bm
d pq
 0,
 0,
 0,
 0,
l  L, m  M
m
r  R pq
,
lL
 p, q   C ,
 p, q   C ,
 p, q   C ,
 p, q   C ,
 p, q   C ,
mM
mM
mM
mM
mM
4
Magnus Edgren
II.6.
L
0
hrm
Test network
The network used for test of the model is depicted in
fig 1 below. The circles in bold denotes nodes in the
OD-matrix, whereas the other circles are intersections.
The solid lines are links in the car network and the
dashed lines for the public transport. The public
transport is assumed to not intersect with the car
network.
hrm
Doing that we have

  t
c
l

l  m
m
   pq
0
 m  rl

 

 
m
0
hrm    tlc  l  rlm   pq

m 
 lL 

lL
1
2
4
L
0
hrm
This describes a model with route costs
5

 
crm    tlc  l  rlm
m 
lL 
3
In equilibrium either the route flows hrm=0 or the
route costs takes the minimum values πpqm, hence it is a
user equilibrium model.
6
In order to show that the solution is unique it has to
be shown that the objective function, T, is convex. This
can be done by showing that the Hessian matrix, H, is
positive definite.
7
  2T2
 v1
H  
  2T
 v v
 1 K
Fig. 1. The test network.
III. Verification of the model
Formulate the Lagrangean equation as follows
   d
   d  d
L T
( p ,q )C mM
m
pq
( p ,q )C mM
 
( p ,q )C mM
where
m
m
 pq
,  pq
m
pq
m
pq
m
pq
dm
pq

T
 t ac  a , va 
va

dm
 d pq

 cm

 d pq   hrm 
mM


and
m
 pq




 

 2T 
2
vK 
 2T
vK v1
By deriving T once we have
cm
 d pq
 d bm
pq 
nm
pq

which we know is convex. Further derivations will
not change this, i.e. T is convex.
are Lagrangean
III.2. Public transport model
multipliers for the first three constraints in the
optimization problem.
The public transport model consist only of a
costfunction, hence no proof is needed in order to see
that it is a part of the optimization problem.
III.1. Car network model
It can be shown that the car traffic route choice is
made according to a user equilibrium model by using the
following conditions
5
Magnus Edgren
Un Ud  t
III.3. Choice model
The following equations will be used to show that the
used choice model is the nestled logit model as
described in II.4.
nm
pq
 pq 



  2  t bm
pq   pq 
  2 m
 m  

pq

 ln e
e

2 


1
From this equation we see that Ud is the composite
m
cost of car (  p q ) and bus ( t pq
L
0
xm
d pq
d
xm
pq




L


0
xm

d pq

d
dm
pq
e
x  n, d , c, b
d bm
pq  e
cm
d pq
d

1  mpq t nm
pq



d
bm
pq
d
1

1 e
 pq 


 2   mpq t bm
pq   pq 
 m 


1
1 e
 2 U c U b 

 p q  
m



U c  U b   p q   t bpmq    p q 
 m  



m
which is the minimum cost for car,  p q , minus the
 pq 


 2   mpq t bm
pq   pq 
 m 

nm
d pq
d
). πpqm is the
and from this we see that

 2  mpq  mpq
cm
pq
cost for the public transport alternative.
IV. Analysis
Constructing the quota on the first logit level
nm
pq
 pq
m
On the second level in the choice model we have
 2 1 m
 pq  mpq
 2  1
cm
d pq
e
  pq 
minimum route cost for car trips from p to q as shown
next.
This will result in the following
nm
d pq
e
bm
1

dm
pq
1 e

1  mpq t nm
pq
   
2 1
2
m
m
pq  pq
IV.1. Method

The analysis is made with respect to variations in
some of the input parameters. The parameters analyzed
is as follows
1
and comparing it with the following from the logit
theory
l
Monetary cost for bus,  pq
 Monetary cost for car,
d
d
nm
pq
nm
pq
d
dm
pq


1
1 e
1 U n U d 
 Bus frequency,
 pq
 Dispersion in the choice model,
it is clear that
Un Ud

 
t
nm
pq
m
pq
t
nm
pq
 
2
2
 1

m
pq

m
pq
For each parameter change, the output travel
patterns, d pqxm , x  n, d , c, b, will be analysed. We will look

at the following
m
m
1 pq
  2 pq

1   2
 How the decisions on making the trip or not
changes for each user group.
 How the travel mode is changed for those making
the trip
To clear this expression up we look at the sum of trip
makers
dm
cm
d pq
 d pq
 d bm
pq
The user groups here used are “poor” (low income)
and “rich” high income.

m
m

1 pq
  2 pq
1    

ln  e
e
 2  1
2 
2

 i , i  1,2
m
pq

 bm
 t pq   pq  pq
2
m






 


6
Magnus Edgren
TABLE I
CHANGED NUMBER OF TRAVELERS DUE TO PARAMETER VARIATIONS
IV.2. Results and discussion
Parameter
In fig 3 the total flow in the network before changing
parameters is indicated. The main flow in the car
network goes in OD-pairs (1,5), (1,7), (5,1) and (7,1)
via node 3 and in OD-pairs (5,7) and (7,5) via node 6.
The biggest movement in the public transport mode is
between nodes 7 and 1.
Mon. cost car
Mon. cost bus
Bus frequency
Dispersion choice
model level 1
Dispersion choice
model level 2
1
5
“Poor”
“Rich”
+20%
-20%
+20%
-20%
+20%
-20%
Mult 5
Div 5
Mult 5
Div 5
-2.0%
+2.0%
-1.4%
+1.4%
+0.2%
-0.3%
+40%
-22%
-14%
+40%
-0.8%
+0.8%
-0.6%
+0.6%
+0.2%
-0.2%
+32%
-26%
-12%
+30%
By examining a change in the monetary cost for cars
in a wider range we get the result in fig 4. The y-axis
shows the ratio between making the trip and the total
demand for each user group and in total. We see that in
order to reduce the number of travelers from 98% to
95% we need to increase the cost to 16 kr per link for
poor people. But for the rich people to reduce the
traveling as much as the poor the cost needs to be
increased to 40 kr, which is 250% more.
2
4
Variation
3
1
0,98
6
0,96
0,94
0,92
Ratio
7
Fig. 3. Total flow in the network.
Poor
0,9
Rich
Total
0,88
0,86
IV.3. Change in no of travelers due to param changes
0,84
The change in number of travelers due to parameter
changes are presented in table I. It can be seen that
“Rich” people are less sensitive to changes in the
monetary cost for both the car and the public transport
travel mode. However, a change in the bus frequency
generates the same result for the two user groups, which
is reasonable due to the priceless nature of this
parameter.
The increased dispersion on level 1 yields a bigger
raise in number of poor travelers than the rich ones. As a
increased dispersion makes people more tend to follow
their true wish, the interpretation of this is that the
number of poor people wanting to do a trip but don’t
actually do it is bigger than the corresponding subset
among the rich people.
The change in dispersion on level 2 is not intuitively
easy to interpret. Raised dispersion yields more
tendency to choose the true wish alternative and maybe
it is so that people in this model would want to go by
car, but as they don’t afford it they rather stay at home
than go with the alternative they can afford (i.e. public
transport).
0,82
47
42
37
32
27
22
17
12
7
2
0,8
Mon. cost car [kr/link]
Fig. 4. Change of the monetary cost for car.
We continue by looking at the dispersion in the
choice model (level 1), as shown in fig 5. With a very
low dispersion all groups have the same travel habits but
for high dispersion the values differ a lot. We also see
that for high dispersion the values are stable. This
complies with the theory that says that people tend to
select more and more according to their utility as the
dispersion increases.
7
Magnus Edgren
tell us that all groups are similarly tend to use the car.
0,9
0,6
0,8
0,55
0,7
0,5
Poor
0,5
0,45
Rich
0,4
Ratio
Ratio
0,6
Total
Poor
0,4
Rich
Total
0,3
0,35
0,2
0,3
0,1
0,25
45
47
42
37
32
27
22
17
7
Dispersion, logit level 1
12
0,2
2
40
35
30
25
20
15
10
5
0
0
Mon. cost car [kr/link]
Fig. 5. Change of the dispersion, choice model level 1.
Fig. 6. Change of the monetary cost for car.
IV.4. Change in no of car users due to param changes
With the dispersion for “no trip” or ”make trip” set
low we have equally car usage for the user groups. With
a raised dispersion we get quite low shares of car usage
for all groups. The values go to stable values as the
dispersion increases.
Table II presents the change in number of car users
due to changes in the parameters. We see that raised fuel
price for car gives less car users as they switch to public
transport, while a raised fee for the public transport
gives more car users. As before the rich group is less
sensitive to changes in the monetary cost. A change in
the bus frequency is independent of user groups.
Changes in the dispersion on level 1 in the logit
model do not make a big difference on the ratios that is
presented here. The reason is that the number of people
actually doing the trip (which is changed due to the
dispersion change) still chooses the travel mode as they
did before.
The dispersion on level 2 makes a big difference
however. A raised dispersion makes people choose the
car alternative in a bigger extent which also reduces the
number of travelers in total as discussed above.
0,9
0,8
0,7
Ratio
0,6
0,2
0,1
45
40
35
30
25
-1.9%
+1.9%
+2.4%
-2.5%
-0.6%
+0.9%
+0.1%
+0.2%
+42%
-16%
20
-3.7%
+3.6%
+4.6%
-4.8%
-0.6%
+0.9%
0.0%
+0.2%
+42%
-18%
15
+20%
-20%
+20%
-20%
+20%
-20%
Mult 5
Div 5
Mult 5
Div 5
0
10
“Rich”
5
“Poor”
Total
0
Variation
Rich
0,4
0,3
TABLE II
CHANGED NUMBER OF CAR USERS DUE TO PARAMETER VARIATIONS
Parameter
Poor
0,5
Dispersion, logit level 1
Mon. cost car
Mon. cost bus
Bus frequency
Dispersion choice
model level 1
Dispersion choice
model level 2
Fig. 7. Change of the dispersion for choice mode level 1.
V.
Conclusion
The given model is shown to be a combination of the
given models for the car network, public transport and
choice. It is also, due to the tests performed within this
work, likely that the model is valid in reality. However
the parameters must be chosen carefully as discussed
below.
The results indicate that people are more sensitive to
variations in monetary cost than for variations in other
costs such as bus frequency. However there are
important parameters that have not been analyzed here,
such as changes in travel time. Between user groups we
With a look on the monetary cost we get the graph in
fig 6, which has the share of car users among all
travelers for each user group on the y-axis. High cost
yields that poor people uses the bus instead so the ratio
decreases rapidly for this group. A low cost, however,
8
Magnus Edgren
see that rich people are less sensitive to changes in cost
than poor people are.
We also see that the results are very sensitive to the
dispersion parameters. With the dispersions being wrong
it doesn’t matter how good the model and parameter is
in other respects.
Time spent on the project
The time spent on the project is shown in table III.
TABLE III
TIME SPENT ON THE PROJECT
Parameter
Theory
Implementation
Tests
Analysis
Documentation
45 h
20 h
14 h
9h
35 h
Appendix
The AMPL code is attached last in the report. The
following source files are attached:
prob_def.txt
network.txt
prob_solve.txt
defines the optimization
problem
defines all parameters
AMPL script for running a
calculation
References
Journal Papers:
[1] C. Rydergren, Short text on the combined mode and route
choice model, (2005)
Books:
[2] Y. Sheffi, Urban Transportation Networks: Equilibrium
Analysis with Mathematical Programming Methods, (1985),
Prentice-Hall, Englewood Cliffs, NJ.
9
Magnus Edgren
prob_def.txt
var
var
var
var
var
var
var
var
var
var
var
var
#-----------------------------------------------------------# TNK084 Traffic theory
# Project - Combined models for private and public
transport
# Optimization problem
# Magnus Edgren
# 2008-04-27
#------------------------------------------------------------
#-------------------# Sets
#-------------------set
set
set
set
set
set
set
set
set
LINKS;
ODPAIR;
GROUPS;
R15;
R17;
R51;
R57;
R71;
R75;
v
{LINKS};
# link flows
w
{LINKS, GROUPS}; # link flows per group
h15 {R15, GROUPS}; # route flows OD15
h17 {R17, GROUPS}; # route flows OD17
h51 {R51, GROUPS}; # route flows OD51
h57 {R57, GROUPS}; # route flows OD57
h71 {R71, GROUPS}; # route flows OD71
h75 {R75, GROUPS}; # route flows OD75
dn
{ODPAIR, GROUPS}; # demand no trip
dd
{ODPAIR, GROUPS}; # demand trip
db
{ODPAIR, GROUPS}; # demand bus
dc
{ODPAIR, GROUPS}; # demand car
#-------------------# Objective function
#--------------------
#
#
#
#
#
#
#
#
#
minimize T:
sum{l
in
LINKS}
(ffc[l]*v[l]
+
A[l]*((1/crho[l])^n[l])*((v[l])^(n[l]+1))/(n[l]+1)
+
sum{m in GROUPS} (ctao[l]*w[l,m]/beta[m])) +
sum{pq
in
ODPAIR,
m
in
GROUPS}
(db[pq,m]/alfa[pq]
+
db[pq,m]*(brho[pq]+(btao[pq]/beta[m]))) +
sum{pq
in
ODPAIR,
m
in
GROUPS}
(dn[pq,m]*tn[pq,m]) +
(1/beta1) * (sum{pq in ODPAIR, m in GROUPS}
(dn[pq,m]*(log(dn[pq,m])-1)
+
dd[pq,m]*(log(dd[pq,m])-1))) (1/beta2) * (sum{pq in ODPAIR, m in GROUPS}
(dd[pq,m]*(log(dd[pq,m])-1)
)) +
(1/beta2) * (sum{pq in ODPAIR, m in GROUPS}
(dc[pq,m]*(log(dc[pq,m])-1)
+
db[pq,m]*(log(db[pq,m])-1)));
#-------------------# Parameters
#-------------------param ffc {LINKS};
# Free flow
param A
{LINKS};
#
param crho {LINKS};
#
param n
{LINKS};
#
param ctao {LINKS};
#
param alfa {ODPAIR};
#
param d
{ODPAIR, GROUPS}; #
param brho {ODPAIR};
#
param btao {ODPAIR};
#
param beta {GROUPS};
#
param beta1;
#
param beta2;
#
param delta15 {R15, LINKS}; #
param delta17 {R17, LINKS}; #
param delta51 {R51, LINKS}; #
param delta57 {R57, LINKS}; #
param delta71 {R71, LINKS}; #
param delta75 {R75, LINKS}; #
param tn
{ODPAIR, GROUPS}; #
#-------------------# Constraints
#-------------------subject to Constraint1
{pq in ODPAIR, m in GROUPS}:
dc[pq,m] + db[pq,m] = dd[pq,m];
subject to Constraint2
{pq in ODPAIR, m in GROUPS}:
dn[pq,m] + dd[pq,m] = d[pq,m];
subject to Constraint3a
{m in GROUPS}:
sum{r in R15} h15[r,m] = dc['OD15',m];
#-------------------# Variables
#--------------------
subject to Constraint3b
10
Magnus Edgren
{m in GROUPS}:
sum{r in R17} h17[r,m] = dc['OD17',m];
subject to Constraint7
{pq in ODPAIR, m in GROUPS}:
dn[pq,m] >= 1;
subject to Constraint3c
{m in GROUPS}:
sum{r in R51} h51[r,m] = dc['OD51',m];
subject to Constraint8
{pq in ODPAIR, m in GROUPS}:
dd[pq,m] >= 1;
subject to Constraint3d
{m in GROUPS}:
sum{r in R57} h57[r,m] = dc['OD57',m];
subject to Constraint9
{pq in ODPAIR, m in GROUPS}:
dc[pq,m] >= 1;
subject to Constraint3e
{m in GROUPS}:
sum{r in R71} h71[r,m] = dc['OD71',m];
subject to Constraint10
{pq in ODPAIR, m in GROUPS}:
db[pq,m] >= 1;
subject to Constraint3f
{m in GROUPS}:
sum{r in R75} h75[r,m] = dc['OD75',m];
subject to Constraint11
{l in LINKS}:
v[l] >= 0;
subject to Constraint4
{l in LINKS, m in GROUPS}:
sum{r in R15} (delta15[r,l]*h15[r,m]) +
sum{r in R17} (delta17[r,l]*h17[r,m]) +
sum{r in R51} (delta51[r,l]*h51[r,m]) +
sum{r in R57} (delta57[r,l]*h57[r,m]) +
sum{r in R71} (delta71[r,l]*h71[r,m]) +
sum{r in R75} (delta75[r,l]*h75[r,m]) = w[l,m];
subject to Constraint12
{l in LINKS, m in GROUPS}:
w[l,m] >= 0;
network.txt
subject to Constraint5a
{m in GROUPS, r in R15}:
h15[r,m] >= 0;
#-----------------------------------------------------------# TNK084 Traffic theory
# Project - Combined models for private and public
transport
# Input data
# Magnus Edgren
# 2008-04-25
#------------------------------------------------------------
subject to Constraint5b
{m in GROUPS, r in R17}:
h17[r,m] >= 0;
subject to Constraint5c
{m in GROUPS, r in R51}:
h51[r,m] >= 0;
set LINKS := 1 4 5 6 7 8 9 10 11 12 14 15 16 18
19 20 21 24;
set ODPAIR := OD15 OD17 OD51 OD57 OD71 OD75;
set GROUPS := POOR RICH;
set R15 := R1 R2 R3 R4 R5 R6;
set R17 := R7 R8 R9 R10 R11 R12;
set R51 := R13 R14 R15 R16 R17 R18;
set R57 := R19 R20 R21 R22 R23;
set R71 := R24 R25 R26 R27 R28 R29;
set R75 := R30 R31 R32 R33 R34;
subject to Constraint5d
{m in GROUPS, r in R57}:
h57[r,m] >= 0;
subject to Constraint5e
{m in GROUPS, r in R71}:
h71[r,m] >= 0;
subject to Constraint5f
{m in GROUPS, r in R75}:
h75[r,m] >= 0;
#------------------------# Car model parameters
#------------------------param ffc:=
1 0.0125
4 0.0125
subject to Constraint6
{l in LINKS}:
sum{m in GROUPS} w[l,m] = v[l];
11
Magnus Edgren
5 0.03
6 0.033333
7 0.03
8 0.025
9 0.075
10 0.033333
11 0.026667
12 0.07625
14 0.025
15 0.026667
16 0.02
18 0.075
19 0.07625
20 0.02
21 0.0125
24 0.0125;
24 1800;
param n:=
1 4.5
4 4.5
5 3
6 3.1
7 3
8 3.2
9 3.5
10 3.1
11 3.1
12 3
14 3.2
15 3.1
16 3.1
18 3.5
19 3
20 3.1
21 4.5
24 4.5;
param A:=
1 0.0026515
4 0.0026515
5 0.03
6 0.033333
7 0.03
8 0.025
9 0.015909
10 0.033333
11 0.026667
12 0.016174
14 0.025
15 0.026667
16 0.02
18 0.015909
19 0.016174
20 0.02
21 0.0026515
24 0.0026515;
param ctao:= # Car fee / link [kr]
1 5
4 5
5 5
6 5
7 5
8 5
9 5
10 5
11 5
12 5
14 5
15 5
16 5
18 5
19 5
20 5
21 5
24 5;
param crho:=
1 1800
4 1800
5 1100
6 1100
7 1100
8 1100
9 1100
10 1100
11 1100
12 1100
14 1100
15 1100
16 1100
18 1100
19 1100
20 1100
21 1800
param delta15: 1 4 5 6 7 8 9 10 11 12 14 15 16 18
19 20 21 24 :=
R1
0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0
R2
0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0
R3
0 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0
R4
0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0
R5
0 1 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0
R6
0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 0 0
0;
param delta17: 1 4 5 6 7 8 9 10 11 12 14 15 16 18
19 20 21 24 :=
R7
0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1
R8
0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1
12
Magnus Edgren
R9
0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1
R10
0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1
R11
0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1
R12
0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1;
param delta51: 1 4 5 6 7 8 9 10 11 12 14 15 16 18
19 20 21 24 :=
R13
1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
R14
1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0
R15
1 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 0 0
R16
1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0
R17
1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
R18
1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0;
param delta57: 1 4 5 6 7 8 9 10 11 12 14 15 16 18
19 20 21 24 :=
R19
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1
R20
0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1
R21
0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1
R22
0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1
R23
0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 1;
param delta71: 1 4 5 6 7 8 9 10 11 12 14 15 16 18
19 20 21 24 :=
R24
1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0
R25
1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0
R26
1 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0
R27
1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0
R28
1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0
R29
1 0 1 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0;
param delta75: 1 4 5 6 7 8 9 10 11 12 14 15 16 18
19 20 21 24 :=
R30
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0
R31
0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0
R32
0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0
R33
0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0
R34
0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1 0;
#------------------------# Other parameters
#------------------------param d(tr): OD15 OD17 OD51 OD57 OD71
OD75 :=
POOR
562.5 525.0 337.5 300.0 525.0 425.0
RICH
562.5 525.0 337.5 300.0 525.0 425.0;
param tn(tr): OD15 OD17 OD51 OD57
OD75 :=
POOR
1
1
1
1
1
1
RICH
1
1
1
1
1
1;
param beta:=
POOR 30
RICH 60;
OD71
# Value of time [kr/hour]
prob_solve.txt
#-----------------------------------------------------------# TNK084 Traffic theory
# Project - Combined models for private and public
transport
# Problem solver script
# Magnus Edgren
# 2008-04-27
#-----------------------------------------------------------reset;
option solver minos;
model prob_def.txt;
data network.txt
solve > prob_result.txt;
display v > prob_result.txt;
display w > prob_result.txt;
display h15 > prob_result.txt;
display h17 > prob_result.txt;
display h51 > prob_result.txt;
display h57 > prob_result.txt;
display h71 > prob_result.txt;
display h75 > prob_result.txt;
#------------------------# Bus model parameters
#------------------------param brho := OD15 0.8 OD17 0.8 OD51 0.16 OD57
0.2 OD71 0.25 OD75 0.2;
param alfa := OD15 10 OD17 10 OD51 10 OD57 10
OD71 10 OD75 10; # Bus frequency
exit;
param btao := OD15 20 OD17 20 OD51 20 OD57
20 OD71 20 OD75 20; # Bus fee [kr]
#------------------------# Choice model parameters
#------------------------param beta1:= 1;
# Dispersion logit model level
1
param beta2:= 0.2;
# Dispersion logit model
level 1
13
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