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

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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 2

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

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

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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. 5

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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 6

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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 7

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

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

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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.) 10

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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 11

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

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

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

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

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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. 16

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Infeasibility in SODTA Solution- Relaxed Network 17 Feasible Infeasible Relax

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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. 18 Warm start SODTA

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Small Network Example- Single Lane Links 19

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Cell-Based Network 20

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

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

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1 st Scenario D 2 =15 at time 0 D 5 =15 at time 0 D 4 =15 at time 0 D 1 =100 at time 0 D 3 =15 at time 0 23

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Optimal Flow in Relaxed Network 1 st Scenario D 2 =15 at time 0 D 5 =15 at time 0 D 4 =15 at time 0 D 1 =100 at time 0 D 3 =15 at time 0 24

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Algorithm Solution 1 st Scenario D 2 =15 at time 0 D 5 =15 at time 0 D 4 =15 at time 0 D 1 =100 at time 0 D 3 =15 at time 0 Original Network Optimal Flow = 3083 Relaxed Network Optimal Flow = 3083 No Capacity Violations Feasible! No Link Reversals 25

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2 nd Scenario D 2 =15 at time 0 D 5 = 200 at time 0 D 4 =15 at time 0 D 1 = 15 at time 0 D 3 =15 at time 0 26

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Optimal Flow in Relaxed Network 2 nd Scenario D 2 =15 at time 0 D 5 = 200 at time 0 D 4 =15 at time 0 D 1 =15 at time 0 D 3 =15 at time 0 27

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Algorithm solution 2 nd Scenario D 2 =15 at time 0 D 5 = 200 at time 0 D 4 =15 at time 0 D 1 =15 at time 0 D 3 =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% 28

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D 2 =15 at time 0 D 5 = 200 at time 5 D 4 =15 at time 0 D 1 =15 at time 0 D 3 =15 at time 0 Optimal Flow in Relaxed Network 3 rd Scenario 29

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D 2 =15 at time 0 D 5 = 200 at time 5 D 4 =15 at time 0 D 1 =15 at time 0 D 3 =15 at time 0 Optimal Flow in Relaxed Network 3 rd Scenario ? ? 30

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D 2 =15 at time 0 D 5 = 200 at time 5 D 4 =15 at time 0 D 1 =15 at time 0 D 3 =15 at time 0 Algorithm Solution 3 rd Scenario ? ? 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 31

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D 2 =15 at time 0 D 5 = 200 at time 5 D 4 =15 at time 0 D 1 =15 at time 0 D 3 =15 at time 0 Algorithm Solution 3 rd Scenario 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% 32

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Conclusions The relaxed network SODTA : 1. Gives an insight to the pattern of evacuation flow 2. Largely confines the feasible set 3. 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. 33

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