Motorway Interchange Location Model (10min Presentation) Wctr

Optimum Location of Motorway Interchanges: Concessionaires’ Perspective

Hugo Repolho12th WCTR

Lisboa

12th WCTR ConferenceLisboa, PortugalJuly 11-15, 2010

Hugo M. Repolho Contact: [email protected]

Richard L. ChurchAntónio P. Antunes




Lisboa

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



Lisboa

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]



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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]



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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 1Choice 2

Route choice model [1/2]

ASSUMPTIONS:



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ij

ijijijij

ij

ijq

ccccc

cq

ij

01202

1

20

2 )()(20

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.



Lisboa

DMILM [1/2]

Mm

mJi jiJj Mm amnMn

ijmnmnijmn wfyxdqtaMaxijmn

ij: 0 and :

02

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 -

my

ijmnx

t = toll fee value/km, defined by the decision maker



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jiJjixMm anmMn

ijmn

ijmn

:, 10 and :

0:,,,, 0 ijmnijmn aMnmjiJjix

Ji jiJj aMn

mamijmn

ijmn

Mmygx: 0:

Ji jiJj aMm

nenijmn

ijmn

Mnygx: 0:

11 y1My

MnmJjixijmn ,,, 0

Mmym 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

ijmnRv Rb

nmijvb aMnmJjiyyxijmn 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]



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•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]



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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.

TCUTCACVOCRUC

The Road User Costs (RUC) expression is as follows:

VOC, AC and TC are expressed in €/km/vehicleTUC is expressed in €/hour/vehicle

FUEL COST (€/LITRE) SCENARIOS

VOC (€/km)

Fuel

type SCN1SCN2 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



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DMILM – results [1]

DETERMINISTIC MILM•Fuel cost scenario SCN3

Toll fee (€/Km)

Interchanges location RoutesCPU (sec)

0.0301, 3, 4, 6, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 25, 26, 27, 28, 31, 33

15929 19

0.0401, 3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 22, 23, 24, 27, 28, 31, 33

13277 13

0.0451, 3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 20, 22, 23, 24, 26, 28, 29, 31, 33

12059 13

0.0491, 3, 4, 6, 7, 9, 10, 11, 13, 14, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33

11161 11

0.0501, 3, 4, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 22, 24, 26, 28, 29, 31, 33

10890 11

0.0511, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 22, 24, 26, 28, 29, 31, 33

10598 10

0.0551, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 18, 19, 20, 22, 24, 26, 28, 29, 31, 33

9771 9

0.0601, 3, 4, 5, 6, 7, 8, 10, 11, 14, 15, 19, 20, 22, 24, 26, 28, 29, 31, 33

8786 8

0.0651, 3, 5, 6, 7, 8, 9, 10, 11, 14, 15, 20, 22, 24, 26, 28, 29, 31, 33

7789 7

0.0701, 3, 5, 6, 7, 8, 9, 10, 11, 14, 15, 20, 24, 26, 28, 29, 31, 33

6804 6

0.0801, 3, 5, 6, 7, 8, 9, 10, 11, 15, 16, 17, 19, 20, 21, 24, 26, 28, 29, 31, 33

5179 6

0.0811, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 19, 21, 24, 26, 28, 29, 31, 33

5046 4

0.0901, 3, 5, 6, 7, 8, 9, 10, 11, 15, 16, 19, 21, 24, 26, 28, 29, 31, 33

4011 5

0.1001, 3, 5, 6, 7, 8, 9, 10, 11, 15, 16, 19, 24, 26, 28, 29, 31, 33

3073 1

0.03

0.04

0.045

0.049 0.0

50.0

510.0

530.0

540.0

550.0

56 0.06

0.065 0.0

7

0.080

0000

0000

0...

0.081

0.085

0.090

0000

0000

0...

0.1

19

13 13

11 1110

9 9 9

14

87

6

6 45 5

1

CPU time ...

Toll fee (€/km)

15929

1327712059

1116110890105981015699199771

95308786

77896804

5179504644764011

3073

Number of routes

0.03

0.04

0.045

0.049

0000

0000

0...

0.05

0.051

0.053

0.054

0.055

0.056 0.0

60.0

65 0.07

0.08

0.081

0.085 0.0

9 0.10

10000

20000

30000

40000

50000

4729

22844

32414

3964241611

35552

39143 4138239632

2307120844

18783

23474

17344 16342

17840

13157

11590

Toll fee (€/km)

Pro

fit (

€/da

y)

ArcMap 9.2 Image

13

4

6 7

911

1318 2624

28

33

1014 15 17

29

22 31



Lisboa

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]



Lisboa

SMILM OBJECTIVE FUNCTION:

Mm

mSs Ji jiJj Mm amnMn

ijmnmnijmns fyxdqatpMaxijmn

ijs: 0 and :

02

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 (€/da

y) Interchanges locationRoute

sCPU (sec)

0.030 30661, 3, 4, 6, 9, 10, 11, 12, 13, 15, 16, 20, 26, 28, 29, 31, 33 15929 18

0.0402373

11, 3, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 23, 24, 28, 29, 31, 33 13277 17

0.0453320

31, 3, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 17, 20, 22, 23, 24, 28, 29, 31, 33 12059 11

0.0504148

01, 3, 4, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 22, 24, 26, 28, 29, 31, 33 10890 9

0.0513221

01, 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 94571, 3, 5, 6, 7, 9, 10, 11, 14, 15, 20, 22, 24, 26, 28, 29, 31, 33 7789 7

0.0701400

1 1, 3, 5, 6, 7, 9, 10, 11, 14, 15, 20, 24, 26, 28, 29, 31, 33 6804 6

0.081 -18041, 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



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Additional constraints to enforce the r-robustness condition.

r-SMILM OBJECTIVE FUNCTION:

wfyxdqatpMaxMm


ijmnsmnijmnss

ijmns

ijs

: 0 and :02

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.

SsVrfytdxaq sMm


mnijmnsijmnsij

ijmns

)1(2: 0 and :

r-SMILM [1/2]

Desirable robustness level

Best OF value for scenario s



Lisboa

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 locationCPU (sec)

Maximum regret (%)

100.0 336461, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 19, 20, 22, 24,

26, 28, 29, 31, 33269 4.19

4.0 336261, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 19, 20, 22, 24,

26, 28, 31, 33223 3.89

< 3.9infeasibl

e- - -

Profit losses are only 0.06%

r-SMILM [2/2]

SCN 1 SCN 2 SCN 3 SCN 4 SCN 50.00%0.50%1.00%1.50%2.00%2.50%3.00%3.50%4.00%4.50%

Relative Regret

r=100%r=4%



Lisboa

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



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Users’ perspective…



Lisboa

12th WCTR ConferenceLisboa, PortugalJuly 11-15, 2010

Hugo M. Repolho Contact: [email protected]

Richard L. ChurchAntónio P. Antunes




Lisboa

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

19

13 13

11 1110

9 9 9

14

87

6

6 45 5

1

CPU time ...Toll fee (€/km)

15929

1327712059

1116110890105981015699199771

95308786

77896804

5179504644764011

3073

Number of routes



Lisboa

0

10000

20000

30000

40000

50000

4729

22844

32414

3964241611

35552

39143 4138239632

2307120844

18783

23474

17344 16342

17840

13157

11590

Toll fee (€/km)

Pro

fit (

€/da

y)



Lisboa

ArcMap 9.2 Image

13

4

6 7

911

1318 2624

28

33

1014 15 17

29

22 31

Motorway Interchange Location Model (10min Presentation) Wctr

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