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 Khani The University of Arizona Transit Research Unit atlas 1

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 atlas 2

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 atlas Flexible Assignment and Simulation Tool for Transit and Intermodal Passengers 3

Structure of FAST-TrIPs atlas FAST-TrIPs MALTA Simulation of Vehicle Movements Transit Passenger Assignment Transit vehicle arrival Dwell time Passenger Simulation Vehicle Pax 1 Pax 3 Pax 6 … … Passenger arrival time, stop, boarding behavior Transit Skims, Operating Statistics Passenger experience Transit vehicle approach Need to stop Stop Pax 4 Pax 8 Pax 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 4

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. atlas 5

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 6

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) atlas 7

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 Origin Number of possible tours: atlas 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 = 3 M = 27 Tucson = 54,081 = 214,866 = 323,028 8

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 + 2 N: Number of destinations M: Number of P&Rs Bousquet, Constans, and Faouzi(2009) atlas 9

Mathematical Formulation Minimize Z = Σ d {1,…,Nd+1} Σ (i,j,t) E x ijt d (c ijt +w ijt d ) Subject to 1- Σ j,t:(i,j,t) AU x ijt d + Σ j,t:(i,j,t) MT x ijt d = Σ j,t:(j,i,t) AU x jit d +Σ j,t:(j,i,t) MT x jit d ; i V\D; d {1, …, N d +1} 2- Σ j,t:(i,j,t) TR x ijt d + Σ j,t:(i,j,t) MT x ijt d = Σ j,t:(j,i,t) TR x jit d +Σ j,t:(j,i,t) MT x jit d ; i V\D; d {1, …, N d +1} 3- Σ j,t:(o,j,t) AU x ojt 1 =1; o=origin 4- Σ j,t:(a,j,t) E x ajt d =1; d {1, …, N d +1}; a=Dest(d-1) 5- Σ i,t:(i,b,t) E x ibt d =1; d {1, …, N d +1}; b=Dest(d) 6- Σ j,t:(b,j,t) AU x bjt d+1 = Σ j,t:(j,b,t) AU x jbt d ; d {1, …, N d }; b=Dest(d) 7- Σ j,t:(b,j,t) TR x bjt d+1 = Σ j,t:(j,b,t) TR x jbt d ; d {1, …, N d }; b=Dest(d) 8- Σ d {1,…,Nd+1} Σ t:(i,j,t) MT x ijt d 1; i,j, V 9- Σ d {1,…,Nd+1} [(Σ t:(i,j,t) MT tx ijt d )(Σ a,t:(a,i,t) AU x ait d )] Σ d {1,…,Nd+1} Σ t:(j,i,t) MT tx jit d ; i,j, V 10- T o 1 =Start_time; o=origin 11- (T j d -T i d ) x ijt d = (c ijt +w ijt d )x ijt d ; (i,j,t) E; d {1, …, N d+1 } 12- (T id +w ijt d ) x ijt d = tx ijt d ; (i,j,t) E; d {1, …, N d+1 } 13- T a d+1 -T a d =Ad d ; d {1, …, N d }; a=Dest(d) 14- x ijt d {0,1}; 15- w ijt d, T i d, c ijt 0; atlas 10

Methodology atlas 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 11

Methodology- Network Expansion Origin D1D1 D2D2 P1P1 P2P2 D3D3 atlas 12

Methodology- Network Expansion Origin D1D1 D2D2 P1P1 P2P2 D3D3 D 10 D 20 P 10 P 20 D 11 D 12 D 22 D 21 P 11 P 22 D 32 D 31 D 30 atlas SPTP 13

Methodology- Shortest Path Tour Problem (SPTP) atlas 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 A 1, A 2, …, A N. 14

Methodology- Shortest Path Tour Problem (SPTP) Festa (2009) atlas 15

Methodology- Shortest Path Tour Problem (SPTP) Festa (2009) atlas 16

Methodology- Rivers Crossing Example Origin-Start Origin-End atlas 17

Methodology- Iterative Labeling (SPTP) Origin D 11 D 12 D 13 D 31 D 32 D 33 D 21 D 22 D 23 Activity 1 candidates Activity 2 candidates Activity 3 candidates atlas 18

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 Picks the labels of candidates nodes for 1 st destination from layer 0, and takes to layer 1. 3.Finds the SP tree from candidates nodes for 1 st 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) atlas 19

One Iteration of Iterative Labeling in Intermodal Networks D 1-1 D 1-2 atlas D1D1 (a) 20

D 1-1 D 1-2 atlas D1D1 One iteration of Iterative Labeling in intermodal network (b) 21

D 1-1 D 1-2 atlas D1D1 One iteration of Iterative Labeling in intermodal network (c) 22

D 1-1 D 1-2 atlas D1D1 One iteration of Iterative Labeling in intermodal network (d) 23

D 1-1 D 1-2 atlas D1D1 One iteration of Iterative Labeling in intermodal network (e) 24

D 1-1 D 1-2 atlas D1D1 One iteration of Iterative Labeling in intermodal network (f) 25

atlas Efficiency of the Algorithm 26 D 1-1 D 1-2 D1D1 In each iteration : Number of transit shortest path runs = M+1 Number of auto shortest path runs = 1 Number of shortest path runs in Iterative labeling= N(M+2) (M is number of P&Rs and N is number of destination)

atlas Efficiency of the Algorithm 27 D 1-1 D 1-2 D1D1 In each iteration : Number of transit shortest path runs = M+1 Number 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&Rs and N is number of destination)

Real Network Application P1P1 P2P2 Origin D2D2 D1D1 Rancho Cordova, CA 447 nodes 850 links 163 bus stops 6 bus routes atlas 28

Real Network Application P1P1 P2P2 Origin D2D2 D1D1 Tour using P 1 : 71 min Tour using P 2 : 78 min Tour using auto: 62 min First leg using P 1 : 29 min First leg using P 2 : 22 min First leg using Auto: 29 min atlas Computation time: 0.6 sec 29

Conclusions atlas 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. 30

References atlas 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, Lozano, A., and G. Storchi (2001). Shortest viable path algorithm in multimodal networks, Transportation Research Part A 35, Lozano, A., and G. Storchi (2002), Shortest viable hyperpath in multimodal networks, Transportation Research Part B 36(10), 853– 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, 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– 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– 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

References atlas 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– 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– 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 and optimization. In Proceedings of INOC 2009, Pisa, April. 14- DynusT online user manual, Accessed July Accessed July 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 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, Hamdouch, Y., S. Lawphongpanich, (2006). Schedule-based transit assignment model with travel strategies and capacity constraints. Transportation Research Part B 42 (2008) 663– 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 General Transit Feed Specification. Accessed July Accessed July GTFS Data Exchange. Accessed July

Questions?