Milm Panam Novo

16th PANAM Conference
Lisboa, Portugal / July 15-18, 2010
The Optimal Location for Motorway Interchanges – A25 case study, 0
Hugo Repolho
Lisboa, Portugal
July 15-18, 2010
Hugo M. Repolho Contact: repolho@dec.uc.pt
António P. Antunes
Richard L. Church
The Optimal Location for Motorway
Interchanges – A25 case study

Hugo Repolho
Summary
1. Motivation
2. Problem
3. Case study
4. Model Formulation
a) MILM - C
b) MILM - EK
c) MILM - L
5. Application results
6. Conclusions

Hugo Repolho
Motivation [1/2]
“Easy access to markets, customers or clients”
“Transport links with other cities and internationally”
2nd and 4th positions (out of 12) in a ranking by Cushman and Wakefield, 2007.
“For a large sample of Scottish road links was the No. 1 location factor (…)”
(out of 18) in a ranking by Button, 2007.
Absolutely essential location factors :
• The fastest road trips take place through motorways.
• Interchanges locations have important implications upon the geographic
pattern of economic development.
Kawamura, 2001; DeBok and Sanders, 2005.
Motorway interchanges location (access/exit points):

Hugo Repolho
•Complex process due to the diversity of factors involved;
•Obeys to pre-established regional and national development plans;
•Rationalizes the existing traffic and promotes economic development;
•Suffers the noxious influence of political and economical pressures.
Supporting choices in a rigorous and exempt technical analysis is very important.
Motivation [2/2]
Defining the motorway network:
OBJECTIVE: Determine the locations for a given number of interchanges such that the
total cost incurred by road users is minimized.

Hugo Repolho
Problem [1/2]
1. Road users travel through the least cost routes;
2. There is and will be no traffic congestion;
3. Travel cost function has mainly to do with travel distances and
design speeds;
4. Can be formulated as a p-hub median problem;
5. The interchanges are the hubs and the motorway segments
are the inter-hub links;
6. The discount in the motorway segments is due to the fastest
travel speeds;
7. Interchanges location influences the network travel costs;
8. It’s a non-strict problem:

Hugo Repolho
There are 2 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).
1 n m M
i
j
i
Choice 1
Choice 2
Problem [2/2]

Hugo Repolho
Case study [1]
• 73 traffic generation centers
• 33 interchanges
• Toll-free motorway
• Average annual daily traffic
ranged between 5000 and
23000 pcu – Level of service A
• O/D matrix is known
• Motorways - 120 kph
• National roads – 90 kph
• Municipal roads – 50 kph
• Other roads – 70 kph
FACT:
Each interchange can cost more than 2 million Euros.
QUESTIONS:
A. Is it justified to build all the 33 interchanges?
B. If not:
• How many interchanges should we build?
• How much money would we save?
• What would be the lost in the net travel cost savings?

Hugo Repolho
Model Formulation [1/7]
MILM – basic
MILM – EK
MILM – L
Based on p-hub median problem – Campbell (1994)
and Skorin-Kapov (1996)
Based on Ernst and Krishnamoorthy (1998) and
Marín et al. (2006)
New model

Hugo Repolho
Sets
Centers (J)
Interchanges (M)
Traffic assignment (xijmn)
Interchange location (ym)
Travel Costs
Between centers (cij)
Between interchanges (cmn)
Between center/interchange (cim)
Traffic flows
Demand O/D (qij)
Model Outputs
Network travel cost without the
motorway
C0=∑∑ qij cij
Other parameters
N0. of interchanges (p)
a) MILM – basic

Hugo Repolho
Objective function – minimizes aggregated
travel cost
Assignment constraints – each route is assigned to
no more than one route
Facility constraint – only p or less facilities are
located
Bounding constraints – prevent a trip to be assigned
to a segment not limited by 2 interchanges
Default locations – locate interchanges by default at
the extremities of the motorway
Nonnegative and binary constraints
       

Ji jiJj Mm nmMn
ijmnijjnmnimij xqccccCCMin
: :
0
)'(
 
 

Mm nmMn
ijmn ijJjix :,1
:



Mm
m py
Mmygx
mnMn Ji ijJj
m
a
mijmn   
  : :
Mnygx
nmMm Ji ijJj
n
e
nijmn   
  : :
11 y1My
MnmJjixijmn  ,,,0   Mmym  1,0
Could not find a solution. The computer ran out of memory.
a) MILM – basic

Hugo Repolho
      
      









Ji Mm Mm nmMn Mn jiJj jiJj
ijijijnjnimnmnimim wcxchczcCMin
: : :
'
JiOzw
Mm
iim
Jj
ij  

jiJjiqxw
Mn
ijijnij  

:,
MmJixhhz
jiJj
ijm
nmMn
imn
nmMn
inmim   
,0
:::
MmJiyOz miim  ,
jiMnJjiyqx nijijn  :,,
 
n
miimn MmJiyOy ,
 
m
niimn MnJiyOy ,
 
n
imimn MmJizy ,
MnmJjihzx imnimijn  ,,,0,,



Mm
m py
11 y1My
  Mmym  1,0
ijjnijn ccMnJjix  :,,0
Marín et al. (2006)
New constraints
Considered more efficient than the Campbell model
Uses 2 and 3 subscripts assignment variables (instead of 4 subscripts)
 Fewer number of variables.
The solution for the application is found in at most 70 min.
The final formulation is up to 80% faster than the model proposed by Marín et al. (2006)
b) MILM – EK

Hugo Repolho
Model Formulation [5/7] c) MILM – L
The MILM-L is based on the concept of lists and has only 1 subscript decision variables.
Parameters acquired in a data pre-analysis process:
k = number of cost efficient routes through the motorway (cim+ cmn+ cjn< cij ).
R(K_values,4) = matrix containing the definition of the routes.
i j m n
1 2 9 19
1 2 9 20
. . . .
. . . .
67 72 12 15
68 70 24 25
# rows = k
# columns = 4 xijmn
xk
Is replaced by

Hugo Repolho
Model Formulation [6/7] c) MILM – L



Mm
m py
11 y 1My
  Mmym  1,0


Kk
kk xsCCMin 0
jiJjix
jkRikRKk
k 

:,1
)2,(and)1,(:
Mmygx
mkRKk
m
a
mk 
 )3,(:
Mnygx
nkRKk
n
e
nk 
 )4,(:
Kkxk  0
Objective function – minimizes aggregated
travel cost
Assignment constraints – each route is assigned to
no more than one route
Facility constraint – only p or less facilities are
located
Bounding constraints – prevent a trip to be assigned
to a segment not limited by 2 interchanges
Default locations – locate interchanges by default at
the extremities of the motorway
Nonnegative and binary constraints
The data pre-analysis identified 61 183 cost efficient routes through the motorway.
The solution for the application is found in at most 21 min.

Hugo Repolho
0
500
1000
1500
2000
2500
3000
3500
4000
4500
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
CPUtime(sec.)
Number of interchanges
MILM-EK MILM-L
The MILM-L is faster than MILM-EK for p<10, p=13, p=14 and p>17.

Hugo Repolho
Application Results [1/2] Number of
interchanges
Travel cost
savings
Hours/day
%
2 7 0.13
3 124 2.46
4 1710 34.02
5 2337 46.48
6 2802 55.73
7 3241 64.47
8 3573 71.07
9 3875 77.09
10 4086 81.27
11 4270 84.95
12 4428 88.08
13 4516 89.84
14 4596 91.43
15 4673 92.96
16 4736 94.20
17 4795 95.39
18 4844 96.36
19 4881 97.09
20 4916 97.79
21 4943 98.32
22 4968 98.83
23 4990 99.26
24 5006 99.58
25 5009 99.64
26 5024 99.95
27 5027 100.00
28 5027 100.00
0.00
1000.00
2000.00
3000.00
4000.00
5000.00
6000.00
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Travelcostssavings
Number of interchanges
p=13  90% of the maximum total travel costs savings
 Save 40 million Euros
p=17  95% of the maximum total travel costs savings
 Save 32 million Euros
Interchanges 2, 18, 21, 23, 30 and 32 are never used.

Hugo Repolho
Number of
interchanges
Location of interchanges
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
2 x x
3 x x x
4 x x x x
5 x x x x x
6 x x x x x x
7 x x x x x x x
8 x x x x x x x x
9 x x x x x x x x x
10 x x x x x x x x x x
11 x x x x x x x x x x x
12 x x x x x x x x x x x x
13 x x x x x x x x x x x x x
14 x x x x x x x x x x x x x x
15 x x x x x x x x x x x x x x x
16 x x x x x x x x x x x x x x x x
17 x x x x x x x x x x x x x x x x x
18 x x x x x x x x x x x x x x x x x x
19 x x x x x x x x x x x x x x x x x x x
20 x x x x x x x x x x x x x x x x x x x x
21 x x x x x x x x x x x x x x x x x x x x x
22 x x x x x x x x x x x x x x x x x x x x x x
23 x x x x x x x x x x x x x x x x x x x x x x x
24 x x x x x x x x x x x x x x x x x x x x x x x x
25 x x x x x x x x x x x x x x x x x x x x x x x x x
26 x x x x x x x x x x x x x x x x x x x x x x x x x x
27 x x x x x x x x x x x x x x x x x x x x x x x x x x X
Application Results [2/2]

Hugo Repolho
Conclusions [1]
The motorway interchanges location problem is an good example for hub theory.
To the best of our knowledge the models presented were never used to help deciding on
motorway interchange locations.
The MILM-EK formulation is up to 80% faster than the original one.
The new model, MILM-L, is a valid formulation that, in most cases, performed better than
the other two formulations.
Results have to be carefully analyzed:
• travel demand is more disperse across the region than we assume
• We assumed that roads are uncongested
• Travel demand was considered inelastic

Hugo Repolho
Lisboa, Portugal
July 15-18, 2010
Hugo M. Repolho Contact: repolho@dec.uc.pt
António P. Antunes
Richard L. Church
The Optimal Location for Motorway
Interchanges – A25 case study

Milm Panam Novo

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