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

1. atlas 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

2. atlas 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

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

4. Activities and travel requests from OpenAMOS Google GTFS and/or transit line information Transit and intermodal trips Routes, stops,schedules Auto trips Auto skims Auto part of intermodal trips Passenger arrival from auto Transit vehicleapproach Need to stop Stop Vehicle Pax 4 Pax 1 … Pax 8 … Pax 3 Transit vehiclearrival Pax 12 Pax 6 … … Dwell time atlas Structure of FAST-TrIPs FAST-TrIPs MALTA Transit Passenger Assignment Simulation of Vehicle Movements Passenger arrival time, stop, boarding behavior Passenger Simulation Passenger experience Transit Skims, Operating Statistics

5. atlas 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.

6. atlas 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)

7. atlas 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)

8. atlas Number of destinations: N • Number of P & R: M • Number of parking actions: i 1 2 3 4 Intermodal Shortest Tour Problem Specification • N = 3 • M = 27 Origin • Tucson • 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 = 54,081 • Number of possible tours: = 214,866 Number of auto legs: = 323,028 • Number of Transit legs:

9. atlas 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&R’s Bousquet, Constans, and Faouzi(2009)

10. atlas Mathematical Formulation Minimize Z = Σ d∈{1,…,Nd+1} Σ(i,j,t)∈Exijtd ∙ (cijt+wijtd) Subject to 1- Σj,t:(i,j,t)∈AUxijtd+ Σj,t:(i,j,t)∈MTxijtd= Σj,t:(j,i,t)∈AUxjitd+Σj,t:(j,i,t)∈MTxjitd; ∀i∈V\D; ∀d∈{1, … , Nd+1} 2- Σj,t:(i,j,t)∈TRxijtd+ Σj,t:(i,j,t)∈MTxijtd= Σj,t:(j,i,t)∈TRxjitd+Σj,t:(j,i,t)∈MTxjitd;∀i∈V\D; ∀d∈{1, … , Nd+1} 3- Σj,t:(o,j,t)∈AU xojt1=1;o=origin 4- Σj,t:(a,j,t)∈Exajtd=1;∀d∈{1, … , Nd+1}; a=Dest(d-1) 5- Σi,t:(i,b,t)∈Exibtd=1;∀d∈{1, … , Nd+1}; b=Dest(d) 6- Σj,t:(b,j,t)∈AU xbjtd+1= Σj,t:(j,b,t)∈AUxjbtd; ∀d∈{1, … , Nd}; b=Dest(d) 7- Σj,t:(b,j,t)∈TR xbjtd+1= Σj,t:(j,b,t)∈TRxjbtd; ∀d∈{1, … , Nd}; b=Dest(d) 8- Σ d∈{1,…,Nd+1} Σ t:(i,j,t)∈MTxijtd≤1; ∀i,j,∈V 9- Σ d∈{1,…,Nd+1} [(Σ t:(i,j,t)∈MTt∙xijtd)∙(Σa,t:(a,i,t)∈AUxaitd)]≤ Σ d∈{1,…,Nd+1} Σ t:(j,i,t)∈MTt∙xjitd; ∀i,j,∈V 10- To1=Start_time;o=origin 11- (Tjd-Tid)∙xijtd= (cijt+wijtd)∙xijtd; ∀(i,j,t)∈E; ∀d∈{1, … , Nd+1} 12- (Tid+wijtd)∙xijtd= t∙xijtd; ∀(i,j,t)∈E; ∀d∈{1, … , Nd+1} 13- Tad+1-Tad=Add; ∀d∈{1, … , Nd}; a=Dest(d) 14- xijtd∈{0,1}; 15- wijtd, Tid, cijt≥0;

11. atlas Methodology • 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

12. atlas Methodology- Network Expansion D3 D2 D1 P2 P1 Origin

13. atlas Methodology- Network Expansion D3 D2 D32 D22 P22 D12 D1 P2 D21 D31 P1 D11 P11 Origin D30 D20 SPTP D10 P20 P10 Origin

14. atlas Methodology- Shortest Path Tour Problem (SPTP) 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.

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

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

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

18. atlas Methodology- Iterative Labeling (SPTP) Activity 2 candidates Activity 3 candidates D23 D33 D22 D32 D21 D31 D13 D12 D11 Origin Activity 1 candidates

19. atlas Methodology- Iterative Labeling (SPTP) • Iterative Labeling : • Based on Dijkstra labeling method • One iteration per trip leg • One layer per iteration • Multi-source shortest path runs • Steps: • Starts from origin, finds the SP tree, labels the network in layer 0. • Picks the labels of candidates nodes for 1st destination from layer 0,and takes to layer 1. • Finds the SP tree from candidates nodes for 1st destination, labels the network in layer 1. • Continues until all the layers are labeled. • Label of origin in the last layer is the shortest travel time.

20. atlas One Iteration of Iterative Labeling in Intermodal Networks D1-2 D1-1 D1 (a)

21. atlas One iteration of Iterative Labeling in intermodal network D1-2 D1-1 D1 (b)

22. atlas One iteration of Iterative Labeling in intermodal network D1-2 D1-1 D1 (c)

23. atlas One iteration of Iterative Labeling in intermodal network D1-2 D1-1 D1 (d)

24. atlas One iteration of Iterative Labeling in intermodal network D1-2 D1-1 D1 (e)

25. atlas One iteration of Iterative Labeling in intermodal network D1-2 D1-1 D1 (f)

26. atlas Efficiency of the Algorithm D1-2 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&R’s and N is number of destination) D1-1 D1

27. atlas Efficiency of the Algorithm D1-2 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&R’s and N is number of destination) D1-1 D1

28. atlas Real Network Application Rancho Cordova, CA 447 nodes 850 links 163 bus stops 6 bus routes P2 P1 Origin D2 D1

29. atlas Tour using P1: 71 min Tour using P2: 78 min Tour using auto: 62 min First leg using P1: 29 min First leg using P2: 22 min First leg using Auto: 29 min Real Network Application P2 P1 Origin D2 D1 Computation time: 0.6 sec

30. atlas Conclusions • 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.

31. atlas References 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.

32. atlas References 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. 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.

33. Questions?