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INFORMS 2011 Annual Meeting November 12-16, Charlotte, NC

Modeling Transit in Regional Dynamic Travel Models: FAST-TrIPs

Mark Hickman, Hyunsoo Noh, Neema Nassir, and Alireza KhaniThe University of Arizona Transit Research Unit

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Transit Modeling Requirements

Create a versatile tool for: Transit operations Transit assignment Inter-modal assignment

Capture operational dynamics for transit vehicles

Capture traveler assignment and network loading in a multi-modal context Within-day assignment Day-to-day adjustments to behavior

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Transit Modeling: FAST-TrIPs

Transit assignment Schedule-based Frequency-based Mix of schedule- and frequency-based

Intermodal assignment (P&R, K&R)

Simulation MALTA handles vehicle movements Transit vehicle hail behavior, dwell times, holding are real-time

inputs to MALTA from FAST-TrIPs Passenger behavior (access, boarding, riding, alighting, and egress)

handled within FAST-TrIPs

Feedback of skim information for next iteration of assignment

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FlexibleAssignment andSimulationTool forTransit andIntermodalPassengers

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Structure of FAST-TrIPs

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FAST-TrIPsMALTA

Simulation of Vehicle

Movements

Transit Passenger Assignment

Transit vehiclearrival

Dwell time

Passenger Simulation

VehiclePax 1Pax 3Pax 6

…

…

Passenger arrival time, stop, boarding behavior

Transit Skims, Operating Statistics

Passenger experience

Transit vehicleapproach

Need to stop

StopPax 4Pax 8Pax 12

…

…

Auto skims

Auto part of intermodal trips

Passenger arrival from auto

Activities and travel requests from OpenAMOS

Google GTFS and/or transit line information

Transit and intermodal trips

Routes, stops,schedules

Auto trips

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Intermodal Shortest Path Problem

Find the optimal path in intermodal (auto + transit) time-dependent network

Intermodal Path Viability Constraints:

Mode transfers are restricted to certain nodes, like “bus stop” and “P&R”.

Infeasible sequences of modes like “auto-bus-auto”.

Park-and-ride constraint : whichever park-and-ride facility is chosen for mode transfer, from auto to transit, must be used again when the immediate next mode transfer from transit back to auto takes place.

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Necessity of Tour-based Approaches

Due to park-and-ride constraint in intermodal trips, the route choices for the initial and return trips influence each other.

Baumann, Torday, and Dumont (2004)atlas

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Necessity of Tour-based Approaches

Due to park-and-ride constraint in intermodal trips, the route choices for both the initial and the return trips influence one another.

Bousquet, Constans, and Faouzi (2009)

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Intermodal Shortest Tour Problem Specification

Number of auto legs:

Number of Transit legs:

Number of destinations: N

Number of P & R: M

Number of parking actions: i

1 2 3 4

Origin

Number of possible tours:

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IMST: Find the best configuration/combination of P&R facilities, and the optimal path that serves sequence of destinations, AND satisfies the P&R constraint

N = 3M = 27

Tucson

= 54,081

= 214,866

= 323,028

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Existing Intermodal Tour-based Approach:

Bousquet, Constans, and Faouzi (2009)• Developed and tested a two-way optimal path (for a single destination)• Organized executions of the one-way shortest path algorithm• Extended their approach to optimal tours with multiple destinations

Performance of their approach:Number of Dijkstra one way iterations = M(M+1)(N-1) + 2M + 2N: Number of destinationsM: Number of P&R’s

Bousquet, Constans, and Faouzi(2009)atlas

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Mathematical Formulation

Minimize Z = Σ d {1,…,Nd+1} ∈ Σ(i,j,t) E∈ xijtd (c∙ ijt+wijt

d)Subject to1- Σj,t:(i,j,t) AU∈ xijt

d + Σ j,t:(i,j,t) MT∈ xijt

d = Σ j,t:(j,i,t) AU∈ xjit

d +Σ j,t:(j,i,t) MT∈ xjit

d; i V\D; d {1, … , N∀∈ ∀ ∈ d+1}2- Σ j,t:(i,j,t) TR∈ xijt

d + Σ j,t:(i,j,t) MT∈ xijt

d = Σ j,t:(j,i,t) TR∈ xjit

d +Σ j,t:(j,i,t) MT∈ xjit

d; i V\D; d {1, … , N∀∈ ∀ ∈ d+1}3- Σ j,t:(o,j,t) AU∈ xojt

1=1; o=origin4- Σ j,t:(a,j,t) E∈ xajt

d=1; d {1, … , N∀ ∈ d+1}; a=Dest(d-1) 5- Σ i,t:(i,b,t) E∈ xibt

d=1; d {1, … , N∀ ∈ d+1}; b=Dest(d)6- Σ j,t:(b,j,t) AU∈ xbjt

d+1= Σ j,t:(j,b,t) AU∈ xjbtd; d {1, … , N∀ ∈ d}; b=Dest(d)

7- Σ j,t:(b,j,t) TR∈ xbjtd+1= Σ j,t:(j,b,t) TR∈ xjbt

d; d {1, … , N∀ ∈ d}; b=Dest(d)8- Σ d {1,…,Nd+1}∈ Σ t:(i,j,t) MT∈ xijt

d ≤1; i,j, V∀ ∈

9- Σ d {1,…,Nd+1}∈ [(Σ t:(i,j,t) MT∈ t x∙ ijtd

) (Σ∙ a,t:(a,i,t) AU∈ xaitd)]≤ Σ d {1,…,Nd+1}∈ Σ t:(j,i,t) MT∈ t x∙ jit

d ; i,j, V∀ ∈

10- To1=Start_time; o=origin

11- (Tjd-Ti

d)∙ xijtd= (cijt+wijt

d) x∙ ijtd; (i,j,t) E; d {1, … , N∀ ∈ ∀ ∈ d+1}

12- (Tid+wijtd)∙ xijt

d= t x∙ ijtd; (i,j,t) E; d {1, … , N∀ ∈ ∀ ∈ d+1}

13- Tad+1-Ta

d=Add; d {1, … , N∀ ∈ d}; a=Dest(d)14- xijt

d {0,1}; ∈15- wijt

d, Tid, cijt≥0;atlas

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Methodology

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Network Expansion Technique Transforms the combinatorial optimization problem into a network

flow problem (Shortest Path Tour Problem, SPTP) Guarantees all the path flows satisfy the P&R constraint

Iterative Labeling Algorithm Solves SPTP in intermodal network Finds the optimal tour

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Methodology- Network Expansion

Origin

D1

D2

P1

P2

D3

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Methodology- Network Expansion

Origin

D1

D2

P1

P2

D3

Origin

D10

D20

P10

P20

D11

D12

D22

D21

P11

P22

D32

D31

D30

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SPTP

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Methodology- Shortest Path Tour Problem (SPTP)

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Festa (2009)SPTP is finding a shortest path from a given origin node s, to a given destination node d, in a directed graph with nonnegative arc lengths, with the constraint that the optimal path P should successively pass through at least one node from given node subsets A1, A2, … , AN.

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Methodology- Shortest Path Tour Problem (SPTP)

Festa (2009)atlas

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Methodology- Shortest Path Tour Problem (SPTP)

Festa (2009)atlas

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Methodology- Rivers Crossing Example

Origin-Start

Origin-End

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Methodology- Iterative Labeling (SPTP)

Origin

D11

D12

D13

D31

D32

D33

D21

D22

D23

Activity 1 candidates

Activity 2 candidates

Activity 3 candidates

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Iterative Labeling : Based on Dijkstra labeling method One iteration per trip leg One layer per iteration Multi-source shortest path runs

Steps: 1. Starts from origin, finds the SP tree, labels the network in layer 0.2. Picks the labels of candidates nodes for 1st destination from layer

0, and takes to layer 1.3. Finds the SP tree from candidates nodes for 1st destination, labels

the network in layer 1.4. Continues until all the layers are labeled.5. Label of origin in the last layer is the shortest travel time.

Methodology- Iterative Labeling (SPTP)

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One Iteration of Iterative Labeling in Intermodal Networks

D1-1

D1-2

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D1

(a)

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D1-1

D1-2

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D1

One iteration of Iterative Labeling in intermodal network

(b)

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D1-1

D1-2

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D1

One iteration of Iterative Labeling in intermodal network

(c)

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D1-1

D1-2

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D1

One iteration of Iterative Labeling in intermodal network

(d)

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D1-1

D1-2

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D1

One iteration of Iterative Labeling in intermodal network

(e)

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D1-1

D1-2

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D1

One iteration of Iterative Labeling in intermodal network

(f)

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Efficiency of the Algorithm

D1-1

D1-2

D1

In each iteration :Number of transit shortest path runs = M+1Number of auto shortest path runs = 1 Number of shortest path runs in Iterative labeling= N(M+2)

(M is number of P&R’s and N is number of destination)

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Efficiency of the Algorithm

D1-1

D1-2

D1

In each iteration :Number of transit shortest path runs = M+1Number of auto shortest path runs = 1 Number of shortest path runs in Iterative labeling= N(M+2)

Existing approach :

2M+2+(N-1)M(M+1)

(M is number of P&R’s and N is number of destination)

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Real Network Application

P1

P2

Origin

D2 D1

Rancho Cordova, CA 447 nodes850 links163 bus stops 6 bus routes

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Real Network Application

P1

P2

Origin

D2 D1

Tour using P1: 71 minTour using P2: 78 minTour using auto: 62 min

First leg using P1: 29 minFirst leg using P2: 22 minFirst leg using Auto: 29 min

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Computation time: 0.6 sec

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Conclusions

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Optimal intermodal tour algorithm is developed.

Network Expansion Technique is introduced that transforms the combinatorial optimization problem into a network flow problem.

Iterative Labeling Algorithm is introduced that solves SPTP in intermodal network.

Applied to real network.

Improved the efficiency.

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References

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1- Battista M.G., M. Lucertini and B. Simeone (1995) “Path composition and multiple choice in a bimodal transportation network,” In Proceedings of the 7th WCTR, Sydney, 1995. 2- Lozano, A., and G. Storchi (2001). “Shortest viable path algorithm in multimodal networks,” Transportation Research Part A 35, 225-241.3- Lozano, A., and G. Storchi (2002), “Shortest viable hyperpath in multimodal networks,” Transportation Research Part B 36(10), 853–874.4- Barrett C., K. Bisset, R. Jacob, G. Konjevod, and M. Marathe (2002). “Classical and contemporary shortest path problems in road networks: Implementation and experimental analysis of the TRANSIMS router”, In Proceedings of ESA 2002, 10th Annual European Symposium, 17-21 Sept., Springer-Verlag. 5- Ziliaskopoulos, A., and W. Wardell (2000). “An intermodal optimum path algorithm for multimodal networks with dynamic arc travel times and switching delays.” European Journal of Operational Research 125, 486–502.6- Barrett C. L., R. Jacob, and M. V. Marathe (2000).“Formal language constrained path problems.” Society for Industrial and Applied Mathematics, Vol. 30, No. 3, pp. 809–837.7-Baumann, D., A. Torday, and A. G. Dumont (2004). “The importance of computing intermodal round trips in multimodal guidance systems,” Swiss Transport Research Conference.8- Bousquet, A., S. Constans, and N. El Faouzi (2009). “On the adaptation of a label-setting shortest path algorithm for one-way and two-way routing in multimodal urban transport networks,” In Proceedings of International Network Optimization Conference, Pisa, Italy.9- Bousquet, A. (2009). “Routing strategies minimizing travel times within multimodal urban transport networks”, Young Researchers Seminar, Torino, Italy, June 2009.

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References

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10 - Pallottino, S., and M.G. Scutella (1998). “Shortest path algorithms in transportation models: Classical and innovative aspects.” In: Marcotte, P., Nguyen, S. (Eds.), Equilibrium and Advanced Transportation Modelling. Kluwer Academic Publishers, Dordrecht, pp. 240–282. 11- Jourquine, B., and M. Beuthe (1996). “Transportation policy analysis with a geographic information system: the virtual network of freight transportation in Europe.” Transportation Research Part C 4(6), 359–371.12- Bertsekas, D.P. (2005). Dynamic Programming and Optimal Control. 3rd Edition, Volume I. Athena Scientific.13- Festa, P. (2009). “The shortest path tour problem : Problem definition, modeling andoptimization.” In Proceedings of INOC 2009, Pisa, April.14- DynusT online user manual, http://dynust.net/wikibin/doku.php. Accessed July 2011.15- Khani, A., S. Lee, H. Noh, M. Hickman, and N. Nassir (2011). “An Intermodal Shortest and Optimal Path Algorithm Using a Transit Trip-Based Shortest Path (TBSP)”, 91st Annual Meeting of the Transportation Research Board, Washington D.C., Jan 2012.16- Tong, C. O., A. J. Richardson (1984). “A Computer Model for Finding the Time-Dependent Minimum Path in a Transit System with Fixed Schedule,” Journal of Advanced Transportation, 18.2, 145-161.17- Hamdouch, Y., S. Lawphongpanich, (2006). Schedule-based transit assignment model with travel strategies and capacity constraints. Transportation Research Part B 42 (2008) 663–684.18- Noh, H., M. Hickman, and A. Khani, (2011). “Hyperpaths in a Transit Schedule-based Network”, 91st Annual Meeting of the Transportation Research Board, Washington D.C., Jan 2012.19- General Transit Feed Specification. http://code.google.com/transit/spec/transit_feed_specification.html. Accessed July 2011.20- GTFS Data Exchange. www.gtfs-data-exchange.com. Accessed July 2011.

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