Motorway Interchange Location Model (10min Presentation) Wctr

Optimum Location of Motorway Interchanges: Concessionaires’ Perspective
Hugo Repolho 12th WCTR Lisboa
12th WCTR Conference
Lisboa, Portugal
July 11-15, 2010
Hugo M. Repolho Contact: repolho@dec.uc.pt
Richard L. Church
António P. Antunes
Optimum Location of Motorway
Interchanges: Concessionaires’ Perspective

Summary
1. Introduction
3. Deterministic Motorway Interchange Location Model
4. Portuguese Case Study
• Results for the deterministic model
2. Route Choice Model
5. Stochastic Motorway Interchange Location Models
• Results for the stochastic models
6. Final Considerations

Many times the construction of a motorway takes place within the framework of built-operate-
transfer (BOT) contracts:
OBJECTIVE: develop an optimization model for assisting toll-motorway concessionaires in the
analysis of the most profitable solutions for Motorway Interchanges Location Problem.
Defines the corridor of the motorway
Defines the detailed design for the motorway
• motorway interchanges location
The Government
The Concessionaire
The location of the interchanges strongly impacts the amount of traffic that the motorway can
capture from the existing road network.
Introduction [1]

OBJECTIVE: develop an optimization model for assisting toll-motorway concessionaires in the
analysis of the most profitable solutions for Motorway Interchanges Location Problem.
The location of the interchanges strongly impacts the amount of traffic that the motorway
can capture from the existing road network.
In Europe most motorways are owned by the State but operated by private concessionaires.
The concessionaires may define certain design details for the motorway, namely the motorway
interchanges location (access and exit points).
Their profit comes from the application of a certain toll fee per mile to the motorway users.
Introduction [1/2]

1 n m M
i
j
i
There are two types of routes to consider:
1. Routes through the existing road network (choice 1);
2. Routes through a combination of existing roadway segments and new motorway
segments (choice 2).
• People will travel through the least cost route;
• The proportion of people using the motorway increases as the travel costs decrease;
Choice 1
Choice 2
Route choice model [1/2]
ASSUMPTIONS:

ij
ij
ij
ij
ij
ij
ij
q
c
c
c
c
c
c
q
ij
0
1
2
0
2
1
2
0
2
)
(
)
( 2
0
2














The traffic flow between i and j travelling through a
combination of existing roadways and new
motorway segments is as follows:
We present a route choice model to predict the traffic flow on the new motorway based
upon interchange locations.
Route choice model [2/2]
 The new connections may generate additional traffic flows if travel costs decrease.
 Some users may travel through the existing roadways even when these routes are less
cost efficient than using the new motorway.

DMILM [1/2]

    
 
  





M
m
m
J
i j
i
J
j M
m a
m
n
M
n
ijmn
mn
ijmn w
fy
x
d
q
ta
Max
ijmn
ij
: 0
and
:
0
2

OBJECTIVE FUNCTION: expresses the profit for the concessionaire, given as the
difference between total toll fee revenue and fixed charges for installing and
operating the interchanges and constructing the motorway.
Total toll fee revenue Fixed charges
The MILM can be seen as a particular case of the p-hub median problem, which was
formulated by Campbell, 1994.
OUTPUTS (decision variables):
• interchange locations -
• trips assigned to motorway routes -
m
y
ijmn
x
t = toll fee value/km, defined by the decision maker

j
i
J
j
i
x
M
m a
n
m
M
n
ijmn
ijmn

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:
,
1
0
and
:
0
:
,
,
,
,
0 
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m
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j
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  
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 


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j a
M
n
m
a
m
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ijmn
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m
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x
: 0
:
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
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J
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m
n
e
n
ijmn
ijmn
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n
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1
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1
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j
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

 ,
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  M
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ym 

 1
,
0
1.Assignment constraints.
2.Elimination of all non cost
efficient routes.
3.Trips are assigned only if the
motorway segment mn is limited
by two motorway interchanges.
4.Each trip is assigned to the least
cost route available.
5.Interchanges located by default
at the extremities of the
motorway.
6.Nonnegative constraints.
7.Location decision variables are
binary.
0
:
,
,
,
2 







 
 
ijmn
R
v R
b
n
m
ijvb a
M
n
m
J
j
i
y
y
x
ijmn ijmn
CONSTRAINTS
1
5
8
10
j
i
1
5
8
10
j
i
4. Each trip is assigned to the least cost route available
DMILM [2/2]

•Motorway A25 located in the center of Portugal
•Dataset : 55 centers and 33 candidate motorway interchanges;
ArcMap 9.2 Image
Portuguese Case Study [1/2]

Portuguese Case Study [2/2]
Routes’ attractiveness is measured by the costs borne by users.
The probability of users choosing a given route is a function of the route’s relative
attractiveness.
TC
UTC
AC
VOC
RUC 



The Road User Costs (RUC) expression is as
follows:
VOC, AC and TC are expressed in €/km/vehicle
TUC is expressed in €/hour/vehicle
FUEL COST (€/LITRE) SCENARIOS
VOC (€/km)
Fuel type SCN1 SCN2 SCN3 SCN4 SCN5
Diesel 0.498 0.663 0.995 1.493 1.990
Gas 0.610 0.813 1.219 1.829 2.438

DMILM – results [1]
DETERMINISTIC MILM
•Fuel cost scenario SCN3
Toll fee (€/Km) Interchanges location Routes CPU (sec)
0.030 1, 3, 4, 6, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 25, 26, 27, 28, 31, 33 15929 19
0.040 1, 3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 22, 23, 24, 27, 28, 31, 33 13277 13
0.045 1, 3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 20, 22, 23, 24, 26, 28, 29, 31, 33 12059 13
0.049 1, 3, 4, 6, 7, 9, 10, 11, 13, 14, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33 11161 11
0.050 1, 3, 4, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 22, 24, 26, 28, 29, 31, 33 10890 11
0.051 1, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 22, 24, 26, 28, 29, 31, 33 10598 10
0.055 1, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 18, 19, 20, 22, 24, 26, 28, 29, 31, 33 9771 9
0.060 1, 3, 4, 5, 6, 7, 8, 10, 11, 14, 15, 19, 20, 22, 24, 26, 28, 29, 31, 33 8786 8
0.065 1, 3, 5, 6, 7, 8, 9, 10, 11, 14, 15, 20, 22, 24, 26, 28, 29, 31, 33 7789 7
0.070 1, 3, 5, 6, 7, 8, 9, 10, 11, 14, 15, 20, 24, 26, 28, 29, 31, 33 6804 6
0.080 1, 3, 5, 6, 7, 8, 9, 10, 11, 15, 16, 17, 19, 20, 21, 24, 26, 28, 29, 31, 33 5179 6
0.081 1, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 19, 21, 24, 26, 28, 29, 31, 33 5046 4
0.090 1, 3, 5, 6, 7, 8, 9, 10, 11, 15, 16, 19, 21, 24, 26, 28, 29, 31, 33 4011 5
0.100 1, 3, 5, 6, 7, 8, 9, 10, 11, 15, 16, 19, 24, 26, 28, 29, 31, 33 3073 1
19
13 13
11 11
10
9 9 9
14
8
7
6
6 4
5 5
1
0.03 0.04 0.045 0.049 0.05 0.051 0.053 0.054 0.055 0.056 0.06 0.065 0.07 0.08 0.081 0.085 0.09 0.1
Toll fee (€/km)
CPU time (sec.)
15929
13277
12059
11161 1089010598 10156 9919 9771
9530
8786
7789
6804
5179 5046
4476 4011
3073
Number of routes
4729
22844
32414
39642
41611
35552
39143
41382
39632
23071
20844
18783
23474
17344 16342
17840
13157
11590
0
10000
20000
30000
40000
50000
0.03 0.04 0.045 0.049 0.05 0.051 0.053 0.054 0.055 0.056 0.06 0.065 0.07 0.08 0.081 0.085 0.09 0.1
Profit
(€/day)
Toll fee (€/km)
ArcMap 9.2 Image
1
3
4
6 7
9
11
13
18 26
24
28
33
10
14 15 17
29
22 31

Reality isn’t stationary and the future isn’t entirely predictable.
ROBUSTNESS: solutions that perform well under any realization of the uncertain
parameters .
We use scenarios to represent evolution trends or potential changes. Each scenario is
characterized by an occurrence probability.
Based on the stochastic optimization model in
Weaver and Church, 1983.
SMILM
Based on the Stochastic r-robust
uncapacitated fixed-charge location problem
in Snyder and Daskin, 2006.
r-SMILM
Fuel costs uncertainty
Traffic flow uncertainty
Stochastic Models [1]

SMILM OBJECTIVE FUNCTION:

    
  
  




M
m
m
S
s J
i j
i
J
j M
m a
m
n
M
n
ijmn
mn
ijmn
s fy
x
d
q
a
tp
Max
ijmn
ijs
: 0
and
:
0
2

OBJECTIVE FUNCTION: expresses the expected profit for the toll-motorway
concessionaire over all scenarios, considering the corresponding probabilities.
SMILM
• Fuel cost scenario SCN3
•50 traffic flow scenarios
Toll fee
(€/Km)
π
(€/day) Interchanges location Routes CPU (sec)
0.030 3066 1, 3, 4, 6, 9, 10, 11, 12, 13, 15, 16, 20, 26, 28, 29, 31, 33 15929 18
0.040 23731 1, 3, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 23, 24, 28, 29, 31, 33 13277 17
0.045 33203 1, 3, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 17, 20, 22, 23, 24, 28, 29, 31, 33 12059 11
0.050 41480 1, 3, 4, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 22, 24, 26, 28, 29, 31, 33 10890 9
0.051 32210 1, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 20, 22, 24, 26, 28, 29, 31, 33 10598 9
0.065 9457 1, 3, 5, 6, 7, 9, 10, 11, 14, 15, 20, 22, 24, 26, 28, 29, 31, 33 7789 7
0.070 14001 1, 3, 5, 6, 7, 9, 10, 11, 14, 15, 20, 24, 26, 28, 29, 31, 33 6804 6
0.081 -1804 1, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 19, 21, 24, 26, 28, 29, 31, 33 5046 7
Adds Interchange 20 and
substitutes interchange 18
by interchange 19.
SMILM [1]
Set of scenario Scenario probability Different traffic flows

Additional constraints to enforce the r-robustness condition.
r-SMILM OBJECTIVE FUNCTION:
w
fy
x
d
q
a
tp
Max
M
m
m
S
s J
i j
i
J
j M
m a
m
n
M
n
ijmns
mn
ijmns
s
ijmns
ijs


 
    
  
  


: 0
and
:
0
2

OBJECTIVE: find the solution that maximizes the expected profit for the toll-motorway
concessionaire over all scenarios, considering the corresponding probabilities, and
simultaneously is r-robust, i.e. whose relative regret in each scenario is no more than r.
S
s
V
r
f
y
t
d
x
a
q s
M
m
m
S
s J
i j
i
J
j M
m a
m
n
M
n
mn
ijmns
ijmns
ij
ijmns




 
    
  
  


)
1
(
2
: 0
and
:
r-SMILM [1/2]
Desirable robustness level Best OF value for scenario s

r-SMILM
•Five fuel cost scenario
• P = [0.05; 0.225; 0.45; 0.225; 0.05]
• t = 0.05 €/km
•Vs= [42064; 40998; 41611; 20240; 2250]
r(%) π (€/day) Interchanges location CPU (sec)
Maximum
regret (%)
100.0 33646 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 19, 20, 22, 24, 26, 28, 29, 31, 33 269 4.19
4.0 33626 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 19, 20, 22, 24, 26, 28, 31, 33 223 3.89
< 3.9 infeasible - - -
Profit losses are
only 0.06%
r-SMILM [2/2]
0.00%
0.50%
1.00%
1.50%
2.00%
2.50%
3.00%
3.50%
4.00%
4.50%
SCN 1 SCN 2 SCN 3 SCN 4 SCN 5
Relative Regret
r=100%
r=4%

The optimum location of motorway interchanges under the concessionaires’
perspective follows a previous work done under the users’ perspective, which aimed
to minimize travel costs.
We believe that the concessionaires’ perspective is the most relevant in real-world
applications.
The models presented are useful in toll-motorway concessionaires’ cost-benefit
analysis.
The route choice model turns the motorway travel demand elastic and thus more
approximated to users’ choices.
Final considerations

Users’ perspective…

19
13 13
11 11
10
9 9 9
14
8
7
6
6 4
5 5
1
0.03 0.04 0.045 0.049 0.05 0.051 0.053 0.054 0.055 0.056 0.06 0.065 0.07 0.08 0.081 0.085 0.09 0.1
Toll fee (€/km)
CPU time (sec.)
15929
13277
12059
11161 1089010598 10156 9919 9771
9530
8786
7789
6804
5179 5046
4476 4011
3073
Number of routes

4729
22844
32414
39642
41611
35552
39143
41382
39632
23071
20844
18783
23474
17344 16342
17840
13157
11590
0
10000
20000
30000
40000
50000
0.03 0.04 0.045 0.049 0.05 0.051 0.053 0.054 0.055 0.056 0.06 0.065 0.07 0.08 0.081 0.085 0.09 0.1
Profit
(€/day)
Toll fee (€/km)

ArcMap 9.2 Image
1
3
4
6 7
9
11
13
18 26
24
28
33
10
14 15 17
29
22 31

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