1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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: 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

9. 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

10. 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

11. 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

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

13. 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

14. 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

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

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

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

18. 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

19. 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 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

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

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

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

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

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

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

26. 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)

27. 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)

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

29. 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

30. 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

31. 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, 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. 31

32. 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. 240–282. 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 and optimization. In Proceedings of INOC 2009, Pisa, April. 14- DynusT online user manual, http://dynust.net/wikibin/doku.php. Accessed July 2011.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.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. 32

33. Questions?