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Neema Nassir, Mark Hickman, and Hong Zheng

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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

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

Introduction- Motivation

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

Woodlands, TX. Sept. 11, 2008

Contraflow Reconfiguration

Introduction

- ADOT Project SPR-679:
- “Platform for Evaluating Emergency Evacuation Strategies – Phase II”

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

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

Mathematical Programming

CTM Based System Optimal DTA with Capacity Reversibility

r

Tuydes and Ziliaskopoulos

(2006)

Mathematical Programming

Single Destination System Optimal DTA with Capacity Reversibility

r

Tuydes and Ziliaskopoulos

(2006)

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.)

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

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)

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)

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

Network Transformation of

Cell-based SD SODTA

Zheng and Chiu, 2011

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.

Infeasibility in SODTA Solution- Relaxed Network

Feasible

Relax

Infeasible

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

Small Network Example- Single Lane Links

Cell-Based Network

Cell-Based Network

Cell-Based Network

Cell-Based Network

Original Cell Based Network

Number of Cells: 105

Number of Connectors: 164

Cell-Based Network

Cell-Based Network

Relaxed Cell Based Network

Number of Cells: 203

Number of Connectors: 430

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

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

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

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

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

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

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

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

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 1920.……………………………….obj=5795

Cut 20’19’..…………………………….obj=5800

Second Iteration:

Cut 2627………………..………………obj=5795

Cut 27’26’.………………………………obj=6072

Feasible!

Cut 1920 and 2627………….….obj=5795

D5=200

at time 5

?

D3=15

at time 0

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

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.

Comment, suggestions and questions?