1

Location and Capacity Modeling of Multimodal Networks Interchanges

By:

Aldo Fabregas

Committee Members:

Dr. Grisselle Centeno, IMSE (Advisor)Dr. Tapas Das, IMSE

Dr. Qiang Huang, IMSEDr. Jianguo Ma, Geography

Dr. Pei-Sung Lin, CUTR

University of South Florida2009

Outline

• Problem overview and motivation• Problem Description• Previous Research• Example• General Solution Strategy• Ongoing research• Computational Resources• Expected Contributions

2

3

Problem Overview

An urban multimodal interchange is the place where it is possible to switch from one transportation mode to another

An example of an interchange could be a station that communicates the street network (mode: Car) with a bus network (mode: Transit)

Destination

Origin

Interchange

Network A

Network B

4

Motivation

Need for improved transportation alternatives and infrastructure

Increased demandUrban sprawlFuel pricesReduction of emissions

Transit capital expenses in 2005 were 12.8 Billion26.3% corresponded to Bus projects(APTA , Public Transportation Fact book , May 2007)

Capital cost decisions:

Land acquisition…Number, size, and length of stationsNumber of tracks or lanesNumber and size of parking lots or garages…

5

Intellectual Merit

The design of urban multimodal interchange involves the optimization of investments in infrastructure.

Mathematical modeling of these decisions have to take into account:

Central authority objectiveUsers objectivesCapacity constraintsGeographically referenced dataDemand uncertaintyBudget constraints

The resulting models are large scale mixed-integer non-linear problems requiring efficient solution .

6

Problem Description

Formulation and solution methodology for the problem of communicating two or more networks subject to user behavior and uncertain demand.

Application in transportation: Urban multimodal interchange design. Decision of intermediate locations where roadway users can switch to bus or train to continue to their final destination.

Keywords: Equilibrium, Traffic Assignment, Mode Choice, Stackelberg Games, Complementarity Problems, Variational Inequalities, MPEC, Discrete Network Design, Integer programming, Non-linear programming, Global Optimization

7

Problem Description (2)

Leader

Departure time, Mode, Routes, Interchange

Topology,Capacity,

Fares

NetworkAttributes

Followers

Performance,Utilization

System Optimum

Budget, User Behavior, Demand Uncertainty

System Optimum

Budget, User Behavior, Demand Uncertainty

Self benefitOrigin-Destination pairs,Perceived cost,Socio-econ. Attributes

Self benefitOrigin-Destination pairs,Perceived cost,Socio-econ. Attributes

Stackelberg Game

User Equilibrium

Network Design Problem

8

Previous Research

Route Choice

TransportationModes

DesignProblem

DemandType

DesignProblem

DemandType

DemandUncertainty

DemandUncertainty

Traffic Assignment

TransportationModes

9

Previous Research

Route Choice

TransportationModes

DesignProblem

DemandType

DesignProblem

DemandType

DemandUncertainty

DemandUncertainty

Traffic Assignment

TransportationModes

10

Proposed ResearchDemand

UncertaintyDesignProblem

Traffic Assignment

DemandType

DemandType

TransportationModes

TransportationModes

DesignProblem

DemandUncertainty

11

Selected Literature

GARCÍA, R. and MARÍN, A., 2001. Urban Multimodal Interchange Design Methodology. Mathematical Methods on Optimization in Transportation Systems, 48, pp. 49–79.

GARCÍA, R. and MARÍN, A.,2002. Parking Capacity and Pricing in Park'n Ride Trips: A Continuous Equilibrium Network Design Problem. Annals of Operations Research, 116(1), pp. 153-178.

FAGHRI, A., LANG, A., HAMAD, K., & HENCK, H.,2002. Integrated knowledge-based geographic information system for determining optimal location of park-and-ride facilities.Journal of Urban Planning and Development, 128, 18-41.

WANG, J.Y.T., YANG, H. and LINDSEY, R., 2004. Locating and pricing park-and-ride facilities in a linear monocentric city with deterministic mode choice. Transportation Research Part B: Methodological, 38(8), pp. 709-731.

LAM, W.H.K., LI, Z., HUANG, H. and WONG, S.C., 2006. Modeling time-dependent travel choice problems in road networks with multiple user classes and multiple parking facilities. Transportation Research Part B: Methodological, 40, pp. 368-395.

LAM, W.H.K., LI, Z., WONG, S.C. and ZHU, D.,2007. Modeling an Elastic-Demand Bimodal Transport Network with Park-and-Ride Trips. Tsinghua Science & Technology, 12(2), pp. 158-166.

FARHAN, B., & MURRAY, A. T.,2008. Siting park-and-ride facilities using a multi-objective spatial optimization model. Computers & Operations Research, 35(2), 445-456.

Example ( Adapted from Marin et. al. 2002)

Notation:

a : Index for park and ride trips (car-transit)b : Index for transit-transit tripsc : Other modes not using park and rideW: Set of O-D pairs ωgω:Demand for O-D pairt: Index for park and ride tTω: Set of all the park and ride locations feasible

for route ω

12

13

Mathematical Model (1)

{ }),()(),()( vgBuPzyRyLMin −++

mZz Ζ⊂∈

Location cost (L)

Cost of providing transit service z to location y)

Cost related to the parking capacity u

Performance measure as function of the

demand g and fare v

{ } mmSzy ℜ⊂×⊂∈ 1,0),(

{ }mYy 1,0⊂∈

mUu ℜ⊂∈

m is the number of locations (existing + proposed) for the interchanges

mVv ℜ⊂∈

Location

Transit design

Parking design

Parking Fare

Combination parking lots-transit

Design variables x

Upper Level Problem UPL(g)

14

Mathematical Model (2)

),(min gxULPXx∈

Design variables x

Upper Level Problem UPL(g)

Subject to:

),(min )( xqLLPArgg xq Ω∈=14

Leader

Topology,Capacity,

Fares

NetworkAttributes

Followers

Performance,Utilization (g)

g

Lower Level Problem LLP(x)

15

Mathematical Model (3)Demand modeling

Interchanges

t : 1,2 ,3…

∑∈′

+−

+−

′′

=

ω

ω

ω

βα

βα

ωω

Tt

U

Uaa

ta

tt

att

e

eUG)(

)(

,,2

,2

)(

{ }∑

∈′

+−

+−

′′

=

cbak

U

Ukk

kk

kk

e

eUG

,,

)(

)(

1

1

)(ω

ω

βα

βα

ωω

O-D pairs ω

[ ] Wg ∈ωω

ωωωωωω gUGUGg aaat

at )()(,, =

a b ccar other

transit

⎥⎦

⎤⎢⎣

⎡−= ∑

∈′

+− ′

ω

ωβαω β Tt

Ua atteU )(

2

,2log1

Parking locations…

Mode

Interchange

Utility

Demand

16

Mathematical Model (4)

n

t

tttt u

gBvvugc ⎟⎟⎠

⎞⎜⎜⎝

⎛+=),,(

Capacity modeling(1)

Capacity modeling (2)

Fare at parking

lot tCapacity at parking lot t

tt ug ≤

ParameterParameter

τω

ω∑∈= tW

at

t

gg

,

Car occupancy

Number of users in the parking lot

Mathematical Model (5)

17

{ })(),(

,0,, gGgUgCdsxsc

W cbk

kk

Tt

ga

ta

tt

t

+⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎟⎠

⎞⎜⎜⎝

⎛+∑ ∑∑ ∫

∈ ∈∈ωωωωω

ω

=),(min xgLLPg

{ }∑

∈

=cbak

k gg,,

ωω

∑∈

=)(

,xTt

aat gg

ω

ωω

tW

at gg

t

=∑∈ω

ω ,

Demand for OD pairs

Demand by mode choice

Demand by interchange

General Solution Strategy(1)

18

Equilibrium problemsVariational inequalities problems (VIP)

Fixed-point problem

Existing of solution by Brouwer’s fixed point

theorem

Reformulation as nonlinear programming

problems

Unconstrained nonlinear problems

Newton’s Method

BFGS

Trust region methods

Constrained nonlinear problems

Simplicial decomposition

Penalty and barrier

Fank-Wolfe

General Solution Strategy(2)

19

{ }∑∑ ∑∈∈ ∈

−⎟⎟⎠

⎞⎜⎜⎝

⎛−+−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

W

aa

cbak Ww

kkk gggggGω

ωωωω βα

β)1(ln1)1(ln1)(

,, 21

∑ ∑∑ ∑∈ ∈∈ ∈

−⎟⎟⎠

⎞⎜⎜⎝

⎛−+−⎟⎟

⎠

⎞⎜⎜⎝

⎛+

W Tt

aat

Ww Ttt

at

ay gggg

ωωωωω

ωωβ

αβ

)1(ln1)1(ln1,

3,,

2

{ })(),(

,0,, gGgUgCdsxsc

W cbk

kk

Tt

ga

ta

tt

t

+⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎟⎠

⎞⎜⎜⎝

⎛+∑ ∑∑ ∫

∈ ∈∈ωωωωω

ω

321 βββ << Ensures the convexity of the objective function

20

Solution Approaches

The linear BPP is NP-Hard (Jeroslow,1985), the checking of a BPP is also an NP-Hard problem (Vicente, Savard and Judice, 1994).

Numerical approaches

Cost approximation algorithm and variations (Patriksson,1999) which includes:

Simplicial decomposition: García, R., & Marín, A. (2005). Gauss-Seidel methods: Lam, W.H.K., Li, Z., Huang, H. and Wong, S.C.(2006)Stochastic programming (Patil, Ukkusuri, 2007)

Artificial intelligence approaches

Simulated annealing (García, R. and Marín, A., 2001)Genetic algorithms (Dimitriou, L., Tsekeris, T., & Stathopoulos,A.,2008)

The problem structure will be studied to devise special solutions algorithms that can be compared with the solution of state-of the-art solvers.

21

Current Research

Detailed decision process: include decisions on whether or not to make a trip, mode choice, parking choice

Capacity modeling: implicit or explicit demand

Incorporate congestion effects: Consider the interaction between the modes throughout the network

Combined equilibrium: consider passenger assignment model using hyperpath formulations

System vs User optimum: Evaluate the tradeoffs between these two objectives

Incorporate uncertainty: Create a scenario generation tool to account for variations in the demand

22

Computational Resources

The resulting mathematical models (medium size) will be coded in AMPL and tested using COIN-OR open source solvers through the NEOS website for benchmarking purposes.

http://www-neos.mcs.anl.gov/

http://www.coin-or.org/index.html

23

Expected Contributions

Extend the application of operations research in the transportation field by developing a series of models to address emerging issues in transportation planning.

Post new models for capacity expansion in multimodal network systems under uncertainty and subject to user behavior.

The outcome of this research can be implemented in a transportation planning software (or in GIS) helping to make well-informed decisions in transportation planning.

The modeling framework and solution algorithm can be used for topology, pricing and capacity planning of highly utilized networks subject to equilibrium constraints.