Neema Nassir, Mark Hickman, and Hong Zheng

1. atlas INFORMS 2011 Annual Meeting  November 12-16, Charlotte, NC A Heuristic for Solving the Evacuation Contraflow Problem Neema Nassir, Mark Hickman, and Hong Zheng Department of Civil Engineering and Engineering Mechanic The University of Arizona, Tucson, AZ

2. Contents • Introduction • Evacuation Control Strategies • Contraflow Design, Literature • - Mathematical Formulation • - Existing Heuristics • Proposing a Heuristic for Contraflow Design • - Network Flow Transformation of SD-SODTA • - Heuristic • - Small Network Application • Conclusions

3. Introduction- Motivation Traffic lines Interstate 45 leaving Houston as Hurricane Ike approaches the Texas Gulf Coast. September 11, 2008 in The Woodlands, Texas. 

4.  Woodlands, TX.  Sept. 11, 2008 Contraflow Reconfiguration

5. Introduction • ADOT Project SPR-679: • “Platform for Evaluating Emergency Evacuation Strategies – Phase II” • Develop a scalable integrated optimization platform of evacuation strategies in case of a disaster happening. • Propose optimal evacuation strategies for Tucson and Phoenix, AZ.

6. Evacuation Strategies • Optimal bus routing to assist carless evacuees • Contraflow design- lane closure • Staged evacuation (scheduling) • Signal control in emergency evacuation • Crossing elimination strategies • Destination choice

7. Contents • Introduction • Evacuation Control Strategies • Contraflow Design, Literature • - Mathematical Formulation • - Existing Heuristics • Proposing a Heuristic for Contraflow Design • - Network Flow Transformation of SD-SODTA • - Heuristic • - Small Network Application • Conclusions

8. Mathematical Programming CTM Based System Optimal DTA with Capacity Reversibility r Tuydes and Ziliaskopoulos (2006)

9. Mathematical Programming Single Destination System Optimal DTA with Capacity Reversibility r Tuydes and Ziliaskopoulos (2006)

10. Existing Heuristics for Contraflow Design Tuydes and Ziliaskopoulos (2006) Tabu Search Simulation-Based Heuristic. (VISTA for Evanston, IL) Basic idea: Heuristic is based on an insight into optimality conditions, by studying the dual problem and complementary slackness conditions. (If two coupled cells (or links) bear approximately the same level of congestion through the whole duration of the analysis, not necessarily at the same time, the capacity is distributed optimally. Otherwise, the system can be managed better by reversing some capacity from a less congested cell (link) to the more congested one.)

11. Contents • Introduction • Evacuation Control Strategies • Contraflow Design, Literature • - Mathematical Formulation • - Existing Heuristics • Proposing a Heuristic for Contraflow Design • - Network Flow Transformation of SD-SODTA • - Heuristic • - Small Network Application • Conclusions

12. SD-SODTA and Earliest Arrival Flow Number of vehicles exited the network from the beginning to t (cumulative) Number of vehicles exited the network in time interval t 9 vehicles t=1 t=2 t=3 t=0 t=1 t=2 t=3 Zheng and Chiu (2011)

13. SD-SODTA and Earliest Arrival Flow Number of vehicles existing in the network at time t Number of vehicles exited the network in time interval t 9 vehicles t=1 t=2 t=3 t=0 t=1 t=2 t=3 Zheng and Chiu (2011)

14. SD-SODTA and Earliest Arrival Flow max Z = Number of vehicles existing in the network at time t Number of vehicles exited the network in time interval t 9 vehicles t=1 t=2 t=3 t=0 t=1 t=2 t=3 SODTA = Minimize Red Boxes = Maximize Green Boxes = Earliest Arrival Flow Zheng and Chiu 2011

15. Network Transformation of Cell-based SD SODTA Zheng and Chiu, 2011

16. Proposing a Heuristic for SD-SODTA Contraflow Design • The basic idea is to: • Relax the capacities of each direction of the links to the total capacity of link, • Find the SO solution in the relaxed network, • Start from the infeasible solution and gradually move towards the feasible region, with least objective degradation.

17. Infeasibility in SODTA Solution- Relaxed Network Feasible Relax Infeasible

18. Proposing a Heuristic for SD-SODTA Contraflow Design Steps are: 1- For every link, relax the capacity of each direction to sum of the capacities in both directions, 2- Generate the network transformation, and find EAF in the relaxed network (traffic assignment), 3- Detect the streets which violate original capacities, choose the one with largest differential flow in two directions, 4- Cut back the capacity to the real capacity by closing the lanes with minimal degradation of objective function. Continue until feasibility is reached. Warm start SODTA

19. Small Network Example- Single Lane Links

20. Cell-Based Network Cell-Based Network

21. Cell-Based Network Cell-Based Network Original Cell Based Network Number of Cells: 105 Number of Connectors: 164

22. Cell-Based Network Cell-Based Network Relaxed Cell Based Network Number of Cells: 203 Number of Connectors: 430

23. 1st Scenario D2=15 at time 0 D4=15 at time 0 D1=100 at time 0 D5=15 at time 0 D3=15 at time 0

24. Optimal Flow in Relaxed Network 1st Scenario D2=15 at time 0 D4=15 at time 0 D1=100 at time 0 D5=15 at time 0 D3=15 at time 0

25. Algorithm Solution 1st Scenario D2=15 at time 0 Original Network Optimal Flow = 3083 Relaxed Network Optimal Flow = 3083 No Capacity Violations Feasible! No Link Reversals D4=15 at time 0 D1=100 at time 0 D5=15 at time 0 D3=15 at time 0

26. 2nd Scenario D2=15 at time 0 D4=15 at time 0 D1=15 at time 0 D5=200 at time 0 D3=15 at time 0

27. Optimal Flow in Relaxed Network 2nd Scenario D2=15 at time 0 D4=15 at time 0 D1=15 at time 0 D5=200 at time 0 D3=15 at time 0

28. Algorithm solution 2nd Scenario D2=15 at time 0 Original Network Optimal Flow = 5295 Relaxed Network Optimal Flow = 4906 No Capacity Violations Feasible! Two Link Reversals Needed Improvement= 7.3% D4=15 at time 0 D1=15 at time 0 D5=200 at time 0 D3=15 at time 0

29. Optimal Flow in Relaxed Network 3rd Scenario D2=15 at time 0 D4=15 at time 0 D1=15 at time 0 D5=200 at time 5 D3=15 at time 0

30. Optimal Flow in Relaxed Network 3rd Scenario D2=15 at time 0 D4=15 at time 0 D1=15 at time 0 ? D5=200 at time 5 ? D3=15 at time 0

31. Algorithm Solution 3rd Scenario D2=15 at time 0 D4=15 at time 0 D1=15 at time 0 ? Relaxed Network ……………..……….obj=5764 First iteration: Cut 1920.……………………………….obj=5795 Cut 20’19’..…………………………….obj=5800 Second Iteration: Cut 2627………………..………………obj=5795 Cut 27’26’.………………………………obj=6072 Feasible! Cut 1920 and 2627………….….obj=5795  D5=200 at time 5 ? D3=15 at time 0

32. Algorithm Solution 3rd Scenario D2=15 at time 0 D4=15 at time 0 D1=15 at time 0 Original Network Optimal Flow = 6224 Relaxed Network Optimal Flow = 5764 Reconfigured Network Optimal Flow = 5795 Two Links Capacity Violations Two Link Reversals Needed Improvement= 6.8% D5=200 at time 5 D3=15 at time 0

33. Conclusions • The relaxed network SODTA : • Gives an insight to the pattern of evacuation flow • Largely confines the feasible set • Smartly chooses the candidates for reversing • The warm start assignment estimate is used to find the move direction towards feasible set. • The warm start assignment estimate can be possible by utilizing the network flow approach to SODTA.

34. Comment, suggestions and questions?