UFDC Home  myUFDC Home  Help 



Full Text  
PAGE 1 1 TRANSIT BASED EVACUATION PLANNING UNDER DEMAND UNCERTAINTY By ASHISH KULSHRESTHA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF D OCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2011 PAGE 2 2 2011 Ashish Kulshrestha PAGE 3 3 To my parents PAGE 4 4 ACKNOWLEDGMENTS No research endeavor is ever carried out in solitude. I owe my gratitude to all those people who in one way or another contributed to wards this dissertation. First and foremost, I would like to thank my advisor Dr. Yafeng Yin for his valuable support, guidance and encouragement, throughout the course of my Ph.D. study. He has been an incredible source of inspiration and one simply could not wish for a better advisor. I would also like to thank my committee members Drs. Scott Washburn, Lily Elefteriadou, Sivaramakrishnan Srinivasan and Siriphong Lawphongpanich for their valuable comments, feedback and suggestions from various perspectives throughout the research. On a personal note, I want to thank my parents and family members for their love and support at each and every stage of my life. They have been very understanding and patient in all these years. I would also like to thank all my friends and fellow students at the Transportation Research Center for making my stay, a memorable one. Finally, special thanks go to Sonia, for the happiness and joy she brings to my life. She has been a huge support throughout this rese arch and h er motiva tion and encouragement was in the end what made this dissertation possible. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 ABSTRACT ................................ ................................ ................................ ..................... 9 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 12 1.1 Background ................................ ................................ ................................ ... 12 1.2 Dissertation Outline ................................ ................................ ....................... 16 2 LITERATURE REVIEW ................................ ................................ .......................... 17 2.1 Emergency Planning and Management ................................ ........................ 17 2.1.1 Evacuation Management Process ................................ ....................... 18 2.1.2 Emergency Response Plan ................................ ................................ 19 2.1.3 Evacuation Plann ing ................................ ................................ ........... 20 2.1.4 Emergency Management Roles in Evacuation ................................ ... 21 2.1.5 Authority and Command Structure ................................ ...................... 22 2.2 Role of Transportation in Evacuation Planning ................................ ............. 23 2.3 Transit Evacuation Planning ................................ ................................ .......... 25 2.3.1 Transit System Roles in Emergency Evacuation ................................ 26 2.3.2 Lack of Transit Planning in Emergency Response .............................. 27 2.3.3 Need for Transit Evacuation Planning ................................ ................. 29 2.3.4 Optimizing the Use of Transit in Evacuation ................................ ....... 30 2 .4 Evacuation Planning under Uncertainty ................................ ........................ 32 2.4.1 Sources of Uncertainty ................................ ................................ ........ 33 2.4.2 Optimization Under Uncertainty ................................ .......................... 33 2.4.2.1 Simulation optimization ................................ ................................ 34 2.4.2.2 Stochastic p rogramming ................................ .............................. 34 2.4.2.3 Robust optimization ................................ ................................ ..... 35 2.4.3 Incorporating Uncertainty in Evacuation Modeling .............................. 35 2.5 Contributions of This Dissertation ................................ ................................ 39 3 ROBUST PUBLIC SHELTER LOCATIONS WITH DEMAND UNCERTAINTY ....... 41 3.1 Background ................................ ................................ ................................ ... 41 3.2 Model Components ................................ ................................ ....................... 44 3.2.1 Representing Demand Uncertainty ................................ ..................... 44 3.2.2 Notations ................................ ................................ ............................. 46 PAGE 6 6 3.2.3 Combined Distribution and Assi gnment Model ................................ ... 48 3.2.4 List of Notations ................................ ................................ .................. 49 3.3 Model Formulation ................................ ................................ ........................ 50 3.4 Solution Algorithm ................................ ................................ ......................... 52 3.4.1 Solving Robust Shelter Location Problem ................................ ........... 52 3.4.2 Solving Restricted Robust Shelter Location Problem .......................... 55 3.4.3 Solving Worst Case Demand Problem ................................ ................ 58 3.5 Numerical Example ................................ ................................ ....................... 60 3.6 Summary ................................ ................................ ................................ ....... 68 4 TRANSIT PICK UP LOCATIONS AND BUS ALLCOATION WITH DEMAND UNCERTAINTY ................................ ................................ ................................ ...... 70 4.1 Background ................................ ................................ ................................ ... 70 4.2 Modeling Considerations ................................ ................................ ............... 71 4.2.1 Basic Assumptions ................................ ................................ .............. 71 4.2.2 Demand Uncertainty ................................ ................................ ........... 73 4.2.3 List of Notations ................................ ................................ .................. 73 4.3 Model Formulation ................................ ................................ ........................ 75 4.4 Solution Algorithm ................................ ................................ ......................... 78 4.5 Numerical Example ................................ ................................ ....................... 80 4.6 Summary ................................ ................................ ................................ ....... 86 5 INTEGRATED APPROACH FOR PUBLIC SHELTERS AND TRANSIT PICK UP LOCATIONS WITH DEMAND UNCERTAINTY ................................ ................ 88 5.1 Model Considerations ................................ ................................ ................... 89 5.1.1 Basic Assumptions ................................ ................................ .............. 89 5.1.2 List of Notations ................................ ................................ .................. 91 5.2 Robust Optimization Approach ................................ ................................ ...... 92 5.3 Two Stage Robust Optimization Approach ................................ ................... 95 5.3.1 General Modeling Approach for Two Stage Problem .......................... 96 5.3.2 Two Stage Robust Formulation ................................ .......................... 97 5.4 A Cutting Pl ane Algorithm for the Two Stage Problem ................................ 99 5.4.1 Solving Two Stage Robust Model ................................ ....................... 99 5.4.2 Solving Subproblem ................................ ................................ .......... 104 5.5 Numerical Example ................................ ................................ ..................... 108 5.6 Summary ................................ ................................ ................................ ..... 114 6 CONCLUSION ................................ ................................ ................................ ...... 116 LIST OF REFERENCES ................................ ................................ ............................. 120 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 128 PAGE 7 7 LIST OF TABLES Table page 3 1 Demand scenarios at each county ................................ ................................ ..... 62 3 2 Maximum travel time for each county to shelters ................................ ................ 64 3 3 Fixed cost associated with establishing the shelters at potential sites ................ 65 3 4 Comparison of nominal, robust and worst case plans ................................ ........ 65 3 5 Reliability of cost esti mate and travel time constraint with varying degree of pessimism ................................ ................................ ................................ ........... 67 4 1 Demand values at each demand point ................................ ............................... 83 4 2 Available capacity of public shelters ................................ ................................ ... 83 4 3 Pick up points under nominal, robust and worst case plans ............................... 84 4 4 Bus allocation under nominal, robust and worst case plans ............................... 85 4 5 Reliability of plans with varying degree of pessimism ................................ ......... 86 5 1 Demand values at each demand point ................................ ............................. 110 5 2 Pick up and shelter locations under nominal, one stage robust, two stage robust and worst case plans ................................ ................................ ............. 111 5 3 Evacuation time of one stage and two stage robust plans with varying degree of pessimism ................................ ................................ ................................ ..... 114 PAGE 8 8 LIST OF FIGUR ES Figure page 1 1 Evacuation frequency ba sed on hazard type (1990 2003) ................................ 13 1 2 Evacuation frequency based on evacua ting population size (1990 2003) .......... 13 2 1 Summary of transit evacuation modeling studies ................................ ............... 32 2 2 A simulation optimization model ................................ ................................ ......... 34 2 3 Summary of studies considering uncertainties in evacuation modeling .............. 39 3 1 Sioux Falls network with origin nodes and potential shelter sites ....................... 63 4 1 Sioux Falls network with demand points and known shelter locations ................ 82 5 1 Sioux Falls network for demand points and potential shelter locations ............. 109 5 2 Plot of total evacuation time for sampled demand vectors ............................... 113 PAGE 9 9 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy TRANSIT BASED EVACUATION PLANNING UNDER DEMAND UNCERTAINTY By Ashish Kulshrestha December 2011 Chair: Yafeng Yin Cochair: Scott S. W ashburn Major: Civil Engineering Evacuation planning has drawn a significant amount of attention over the past several decades. Previous studies have focused on different aspects of evacuation planning, however, m ost deal s with auto based evacuation. Unfo rtunately, many people in risk areas do not own or have access to private vehicles during evacuation. Recent experiences with Hurricane Katrina and Rita in 2005 have highlighted a need for a plan to evacuate those who depend on public transportation. These include individuals without access to private vehicles, the poor and elderly, tourist s and people with medical needs Despite a great deal of attention and focus towards transit evacuation planning since 2005 deficiencies still remain in the emergency re sponse plans using public transportation T here are very few attempts in the literature on modeling of transit based evacuation. On the other hand, evacuation planning is challenging due to the uncertainties involved. There can be numerous sources of uncer tainties in planning an evacuation and most of them are not easy to quan tify. T he majority of existing evacuation planning models does not account for the uncertainty during evacuation. Ignoring uncertainties PAGE 10 10 during evacuation can result in some serious un desirable difficulties during the entire process and the whole evacuation could end up with a failure. There is a pressing need for models that incorporate uncertainty and thus assure viable measures for the unanticipated The main purpose of this disserta tion is to develop a plan that evacuates those who depend on public transportation to safe areas reliably and as fast as possible during an emergency. In this dissertation, optimization model s are developed for transit based evacuation planning which are m ore efficient and reliable in the face of demand uncertainty. Demand uncertainty refers to the uncertainty associated with the number of transit dependent evacuees. Two critical decision making components of transit evacuation planning are studied, namely, public shelter locations and pick up locations ( staging areas ). Considering that it is practically difficult to obtain the probability distribution of the uncertain demand, a set based robust optimization approach is adopted to address the demand uncertai nty in the proposed models. T hree models considering different aspects of transit evacuation are developed in this dissertation. The first model determine s the optimal locations of public shelters and their capacities in the region from a given set of po tential sites for a large scale transit evacuation (i.e., regional level planning) The other two models focus more on transit evacuation planning on a local level, i.e., from the perspective of a transit agency in a particular county In the second model, a robust optimization model is proposed to locate the pick up points for evacuees to assemble, and allocate available transit vehicles to transport the assembled evacuees between the pick up locations and different public shelters. For this model, the loc ations of public shelters are assumed to PAGE 11 11 be given and known. T he last model consider s an integrated approach to model pick up and shelter locations simultaneously. In particular, it simultaneously determines the optimal shelter locations, pick up locations allocation of given transit vehicles and capacities of public shelters. Efficient solution algorithms based on cutting plane schemes are proposed to solve the models. The computational results illustrate the effectiveness of the proposed models. Robust p lans achieve nearly the same level of reliability ( in terms of meeting the evacuee demand) with a significant reduction in evacuation time as c ompared to a conservative plan, i.e., one based on the worst case demand. The i mportance of considering demand un certainty in evacuation planning is also demonstrated. The resulting plans obtained from the proposed model are robust with respect to different demand scenarios and eff e ctive in real world mass evacuations Overall, the proposed models are very useful for the planners and emergency managers for a more reliable and efficient transit based evacuation planning. PAGE 12 12 CHAPTER 1 INTRODUCTION 1.1 Background Natural and man made disasters occur frequently and have always been a major concern to humanity and a threat to lives and property. The most commonly encountered natural disasters include hurricanes, earthquakes, and floods, while nuclear meltdowns, hazardous material spills, and terrorist attacks comprise the primary man made threats. Despite their differences, sim ilar issues arise in the aftermath of these events. It is essential that we use the knowledge gained from the past disasters to develop effective plans and avoid loss of life and damage to infrastructure in the future ones. Aiming to reduce adverse consequ ences of these disasters, the field of Emergency Management has evolved substantially over the years (McLoughlin, 1985). Evacuation is often the most viable response action for protecting affected people. f people, of a temporary nature, (Quarantelli, 1998). Recent trends illustrate an increase in vulnerability towards these disasters. According to Federal Emergency Manageme nt Agency (FEMA), there are 45 to 75 major emergency incidents annually in the United States that require evacuation (FEMA, 2008). A nother recent study of evacuations by the Sandia National Laboratories (Jones et al., 2008) showed that between1990 and 2003 230 evacuations involved 1,000 or more evacuees. Figure 1 1 shows the evacuation frequency based on the hazard type and Figure 1 2 shows the evacuation frequency based on the evacuating population size. Some of these evacuations are large scale events (e .g., evacuations in response to hurricanes), potentially affecting millions of people. PAGE 13 13 Figure 1 1. Evacuation frequency based on hazard type (1990 2003) ( Source: Jones et al., 2008) Figure 1 2. Evacuation frequency based on evacuating population size (1990 2003) ( Source: Jones et al., 2008) Over the past several decade s interest in and awareness of the topic of e vacuation has grown enormously. There are various studies that focus on different PAGE 14 14 aspects of evacuation planning such as evacuee behaviors ( e.g., Johnson and Zeigler, 1986; Baker, 1991; Whitehead et al., 2000; Fu and Wilmot, 2004), traffic control strategies (e.g., Wolshon, 2001; Theodoulou and Wolshon, 2004; Chen et al., 2007), sheltering site selection (e.g., Sherali et al., 1991; Kongsomsak sakul et al., 2005), demand management (e.g., Sbyati and Mahmassani, 2006; Chen and Zhan, 2008), and real time route guidance (e.g., Chiu et al., 2006; Yuan et al., 2006; Liu et al., 2007) etc. However most of these studies are for auto based evacuation. U nfortunately, not all the people in risk areas will own or have access to personal vehicles during evacuation. Recent experience with hurricane Katrina and Rita in 2005 have highlighted the need of evacuation planning for transit dependent peoples (e.g., L itman, 2006; Renne et al., 2008; TRB, 2008). More specifically, during hurricane Katrina, there was no effective plan to evacuate those who were dependent on the public transportation. The plans lacked important details such as how to use the buses, availa bility of drivers, bus routes or the pick up locations (staging areas) to collect evacuees (USDOT & USDHS, 2006). E vacuation planning during Katrina was relatively effective for people with automobiles but failed for transit dependent residents (Litman, 20 06). State and local plans during hurricane Rita appeared to be better coordinated in providing public transportation during the evacuation. However, the plan still lacked many details concerning the evacuation of transit dependent r esidents with special n eeds (USDOT & USDHS, 2006). It is believed that the public transit could have played an important role in assisting the evacuation of carless residents in such emergency situations, if it had been properly integrated in the evacuation plans (TRB, 2008). Th e potential role of transit is even PAGE 15 15 more evident in states and cities that have high percentage of carless residents, as is the case with various cities in the United States (Renne et al, 2008). The extra risks faced by carless population during an evacuat ion are well recognized and thus transit evacuation planning has been recently viewed as an important component of evacuation planning and management. Interest in the topic of transit evacuation has increased significantly since 2005. Despite a great deal of attention and focus towards transit evacuation planning, deficiencies still remain in the state and local emergency response plans to evacuate using public transportation (Bailey et al., 2007; Hess and Gotham, 2007; TRB, 2008). Evacuation planning conti nues to focus more heavily on auto based strategies while only a little emphasis on transit based evacuation s In fact, t here are a limited number of studies on modeling of transit based evacuation. This dissertation aims to address this critical gap. On t he other hand, e vacuation planning is very challenging due to the uncertainties involved during evacuation There can be numerous sou rces of uncertainties and most of them are not easy to quantify. However, t he majority of existing evacuation planning mode ls does not account for the uncertainty during evacuation Deterministic models cannot reliably represent a real life evacuation under uncertain conditions. Therefore, for more effective evacuation planning, it is imperative that plans account for uncertai nties. There is a pressing need for models that incorporate uncertainty and thus assure viable measures for the unanticipated This dissertation approaches the transit based evacuation problem from a robust planning perspective and aims to facilitate the t ransit based evacuation in response to PAGE 16 16 various disasters and cata strophic events. The objective of this dissertation is to develop a plan that evacuates those who depend on public transportation to safe areas reliably and as fast as possible during an emer gency. In this dissertation, optimization model s are developed for transit based evacuation planning that are more efficient and reliable in the face of demand uncertainty. Two critical decision making components of transit evacuation planning are studied, namely, public shelter locations and pick up locations ( staging areas ) and demand uncertainty in the models refers to the uncertainty associated with number of transit dependent evacuees. 1.2 Dissertation Outline This dissertation is organized as follows. Cha pter 2 reviews the existing literature on evacuation planning, the role of transportation in evacuation planning, transit evacuation planning and evacuation planning under uncertainty. The next three chapters present different models for transit based evac uation planning. Chapter 3 develops a robust model for optimal shelter locations and their capacities for regional transit evacuation planning with demand uncertainty. Chapter 4 proposes a model for optimal transit pick up locations and allocation of trans it vehicles in a county or region Chapter 5 then presents an integrated approach to determine optimal shelter and pick up location for transit evacuation planning under demand uncertainty. Finally conclusions and suggestions of future research work are pr ovided in Chapter 6. PAGE 17 17 CHAPTER 2 LITERATURE REVIEW Evacuation planning has drawn a significant amount of attention over the past several decades. There is an extensive literature on evacuation planning and management spanning various disciplines. In this ch apter, we will start with the current understanding of the overall emergency planning and management process and practice Next, we will summarize the role played by transportation systems in supporting and assisting evacuation during an emergency response Different roles, practices and strategies will be discussed in brief We then turn our focus to transit evacuation planning, which aims at assisting transit dependent populations during evacuation. Given that there are substantial uncertainties involved in evacuation planning, we lastly review the studies on evacuation planning under uncertain conditions 2.1 Emergency Planning and Management In the event of a natural or man made disaster, emergency preparedness plays a vital role in ensuring the safety of re sidents and transportation system. Emergency Emergency preparedness for a state or locality i s often measured in terms of its ability to respond to an emergency in a timely and effective manner. Transportation officials have a very important role to play in preparing for emergencies and need to work closely with emergency management agencies and o ther stakeholders to promote a smooth functioning of transportation system during or following a n emergency. According to the U. S. Department of Transportation, planning to optimize PAGE 18 18 the operation and coordination of its tran sportation system during natural disasters or (U SDOT, 2003) In preparing transportation systems for emergencies it is helpful to understand the emergen cy management planning process. The m ajor components of an emergency response and management plan are mitigation, preparedness, response and recovery (Nakanishi et al., 2003; Sisiopiku et al., 2004). The m itigation step of the emergency planning process refers to the de velopment of measures to reduce or minimize the risk of damage in the event of an emergency situation. Preparedness refers to the development of emergency response plans and decision making guidelines in advance and it should focus on designating roles and responsibilities to the agencies, implementing communication systems and protocols; develop ing emergency training and drills to maintain readiness; and revis ing and updat ing plans. The r esponse phase can be defined as taking actions (e.g., allocati ng reso urces, implementing evacuation orders etc.) when an emergency situation takes place in order to save lives and reduce damage. The r ecovery phase consists of efforts to return evacuees to affected areas and restore the economic, physiological, and social li fe of a population back to its normal condition. 2.1.1 Evacuation Management Process This section introduces the general framework for the evacuation management process after the detection of any event that poses a disaster threat for a region. The first step af ter the detection of disaster is hazard analysis and forecasting to understand the type of disaster and its impact. Next, vulnerability analysis is conducted to assess how many people are at risk and what is the expected damage to the infrastructure. The PAGE 19 19 f irst decision is whether the situation needs disaster response or not. If the disaster is not going to affect people in the region and they can remain at their homes without harm, then no evacuation action may be needed other than sending people home. If t he event is characterized as a major disaster, then the disaster response needs to be invoked. Evacuation is often the most viable response action for handling emergency situation during many disasters. Appropriate actions for evacuation are determined at this stage. The main tasks required for evacuation operations are shelter analysis, traffic management, evacuation of people with special needs, continuous monitoring of the event and network conditions, and establishment of effective public communication for the sharing of necessary information. During the disaster response, evacuation management operations include a variety of tasks requiring different departments and agencies e.g., firefighting, medical services, department of transportation, police etc. to work together to facilitate the evacuation. During evacuation, specific government agencies are assigned different emergency support roles. Once the state of emergency is over, response actions are replaced mostly by recovery actions. 2.1.2 Emergency Respon se Plan The federal government, through FEMA, requires every state to have a comprehensive emergency response plan. These plans guide emergency operations for all types of hazards and their specifics depend on the regional characteristics of the geography and transportation system. Most states take a two tiered approach to emergency planning and response. Primarily, evacuation planning, response, and recovery activities are developed at the local (e.g. county or city) level. State level PAGE 20 20 emergency management (EM) agencies typically serve to coordinate local transportation agencies, EM agencies, and law enforcement. Federal, state and local officials have recognized the need for better evacuation planning after some recent disasters (e.g., Hurricane Katrina a nd Rita) in 2005 and they have taken initiatives and made changes to address the lessons learned. Some of the key elements of evacuation planning and implementation which should be addressed in a state emergency response plan includes, decision making and management; planning; public communication and preparedness; evacuation of people with special needs; evacuation operations; sheltering; training and exercises. For more detailed information about these key elements, please refer to USDOT & USDHS (2006). 2.1.3 E vacuation Planning Evacuation planning is a very challenging part of an emergency management plan. The visible part of the evacuation process occurs when people get to the road trying to reach safe destinations. However, in order to accomplish a successful evacuation, much planning is necessary before this last step is taken. The sequence of activities that leads up to an evacuation order is typically led and coordinated by state and local level emergency management officials. State and local government ag encies must properly coordinate the movement of evacuating vehicles. There must be better integration between evacuation plans at the state and local levels and greater real time coordination between state and local governments while managing an evacuation For instance, in order to avoid future massive traffic jams and coordination problems, state and local governments must be flow operations. Their plans must be seamlessly integrated so that when the need for a mass evacuation arises, these government PAGE 21 21 entities can work together based on the same pre existing, shared plan. Furthermore, as a mass evacuation usually affects a population dispersed over a wide geographical area, a number of governmental jurisdictions must work together in order to improve the mobility of different transportation modes across their boundaries during an evacuation. An evacuation plan is the outcome of a planning process and cannot be created by one agency. Plans for a complex, multi jurisdic tional disaster require coordination and integration of plans between federal, state, and local agencies as well as NGOs and the private sector. Evacuation plans that are integrally linked with the plans of supporting agencies must be updated regularly to reflect changes in capabilities and resources of the partners. The evacuation plans evaluate the risk to the community; provide the information that emergency managers need to execute an evacuation; determines what actions will be taken and who will carry out those actions; and discuss strategies and alternatives if events do not go as anticipated. 2.1.4 Emergency Management Roles in Evacuation Historically, evacuation planning has been the responsibility of emergency management and law enforcement agencies. In response to the problems experienced during some of the recent disaster events, many state departments of transportation have begun to take a more active role in evacuation planning. In current practice, evacuation planning involves not only emergency mana gers and law enforcement officials, but also transportation officials and officials from other agencies, because of their expertise in different aspects of evacuation planning. Under the emergency response plan, specific governments and agencies are assign ed different emergency PAGE 22 22 support roles. For the details on roles during an emergency and their primary responsible agencies, refer USDOT & USDHS (2006). Evacuations are typically ordered and managed by local officials. In case of a mass evacuation, where mul tiple counties (or states) are concerned, state level agencies become involved in the evacuation process. Although federal agencies like FEMA are involved in some aspects of evacuations, the role of federal agencies during an evacuation is typically limite d to providing assistance to state and local agencies as the conditions of the emergency surpass local resources and capabilities. Non government organizations, such as American Red Cross also play an important role in disaster response and provide shelter food, and health services to address basic needs. 2.1.5 Authority and Command Structure Command structure is one of the key elements that must be specified in emergency response plans. Command is defined as the authority to issue evacuation orders and every st ate has a different command structure. By law, g overnors in most states have the ultimate authority to order an evacuation in case of a disaster However, this authority is often delegated to local officials (e.g. mayor, city council, county law enforcemen t, county judge and county president). At the same time, local officials rely on county or municipal emergency managers for advice. This is primarily because emergency managers have better knowledge and tra ining in the evacuation process and are better inf ormed on current local conditions Details about major components of state and federal level command structure can be found in USDOT & USDHS (2006). PAGE 23 23 2.2 Role of Transportation in Evacuation Planning Although history has shown that evacuation planning has been addressed relatively less within the transportation community throughout the years, interest and involvement of transportation professionals in the field have grown considerably in the past decade. In response to the difficulties experienced during some of the past evacuations, many Department s of Transportation (DOT ) have begun to take a more active role in planning, management and operations of evacuations. Evacuations are dependent on the ability of transportation systems to move evacuees out of the a ffected region. The transportation system plays an important role in supporting and assisting evacuations during an emergency response. Transportation personnel are involved before, during and after evacuations by managing and maintaining transportation sy stems, including traffic control, monitoring, planning, and management. Among the best defined and well developed roles of transportation professionals in evacuations are in the area of direction and control of highway networks. Transportation network cong estion is one major problem encountered during an evacuation, and it can be catastrophic and life threatening. Therefore, better utilization of available resources and the development of effective strategies are essential to manage traffic and lead people to safety during evacuation. Numerous traffic control strategies have been developed to facilitate evacuations e.g., contraflow operations (e.g., FEMA, 2000a; PBS&J, 2000; Wolshon, 2001), phased evacuations (e.g., Chiu, 2004; Tuydes and Ziliaskopoulos, 20 05; Mitchell and Radwan, 2006; Sbayti and Mahmassani, 2006) and arterial signal control (e.g., Sisiopiku et al., 2004; Chen, 2005; Chen et al., 2007). These strategies help optimize the throughput of evacuees during evacuation. PAGE 24 24 Transportation professionals also focus on support activities such as providing information on open and closed roadways and detours; committing manpower and material resources for roadway closures; performing inspections, repairs, and debris removal on affected roadways; and using In telligent Transportation Systems (ITS) to provide en route guidance and communications and traffic flow monitoring. Transportation planners also play a key role in the process of evacuation demand modeling, which is an important and valuable aspect of evac uation planning in order to quantify the travel patterns of evacuees and to assess the transportation related proble m that may hinder an evacuation. Behavioral models for evacuation demand have received a significant amount of attention for many years. Beh avioral aspects of evacuees during evacuation include how and when they chose to evacuate and choice of their routes and destinations. The difficulty with these demand models lies in the complexity of human psychology. Researchers have examined a wide rang e of factors that affect evacuation behavior and demand (Baker, 1991 ; Dow and Cutter, 1997; Dash and Gladwin, 2007 for a detailed review). Current practice in evacuation planning to estimate time dependent travel demand is a two step process In the first step the total number of people expected to evacuate is estimated using participation rates. These rates are determined by the different characteristics of the disaster and the factors affecting evacuee behaviors. In the second step the time at which the evacuees are expected to evacuate is estimated. This is typically done using a response (loading) curve which estimates the percent of total evacuation demand which will evacuate in each time period. For a more detailed review on studies related to evacua tion demand modeling refer Yazici and Ozbay (2008) and Pel et al. (2011). PAGE 25 25 importance of and contributions made by transportation agencies in supporting and assisting evacuations. Over the past several decades, the role of transportation in evacuation has significantly expanded. In the United States the transportation system is dominated by the automobile. As a result, most of the evacuation planning is based on auto based strategi es. Unfortunately, not all the people in risk areas will own or have access to personal vehicles during evacuation. Evacuation planning for these carless and low mobility populations, who are dependent on public transportation to evacuate during disaster, has been largely ignor ed. Recently, there is increasing interest in the role of public transportation during evacuation and some efforts have been made towards the use of transit in assisting evacuation. In the follow ing section, transit based evacuation p lanning for transit dependent residents will be reviewed. 2.3 Transit Evacuation Planning Public transit agencies are capable of providing assistance during crisis situations, performing services such as evacuating victims and transport of emergency personnel etc. In the aftermath of major disasters, public transit has often maintained the mobility of residents faced with damaged or blocked roadway conditions. Successful application of public transit in some of the incidents from recent years highlights the cri tical role of transit systems: During the San Francisco earthquake of 1989, some primary arterials of the area were destroyed, which closed connection to the other roadways within San Francisco and connections with Oakland, making travel in the city diffic ult. Within a few hours of the earthquake, the BART subway system was running and providing reliable transport in the city. Later, other transit agencies in the area also joined in to help restore the transportation system (Public Transit, 1992). Harrisbur emergency conditions during the blizzard of 1996 and its aftermath. CAT vehicles PAGE 26 26 response and recovery by, for example, evacuating residents from flood zones and transporting firefighters during a four alarm fire (Harrisburg, 1996). tornado in February 1998. In April, a tornado in Nashv ille led to similar response Public Transportation played a critical role in the evacuation of people from Manhattan, NY during the terrorist attack of September 11, 2001. Within minutes of the attacks, New York City Transit went to emergency operations and dispatched hundreds of buses to successfully evacuate people from the World Trade Center area (Jenkins and Winslow, 2003). 2.3.1 Transit System Roles in Emergency Evacuation Historically, trains and buses have played an imp ortant role in the evacuation Zelinsky and Kosinski (1991) examined twenty seven major evacuations that have occurred in the past fifty years. According to this study, trains and buses were important modes in 20 of the 27 studied evacuations. In ten of these evacuations, the majority of people used trains and buses Public transit is an extremely versatile and flexible mode of transportation that can play a vital role in supporting evacuations during emergency situations. Transit can play multiple roles in an emergency evacuation, and these roles depend on the nature of the disaster and its location, availability of vehicles and operators at the time of disaster, and the extent of damage, if any, to transit equipment and facilities Th e case studies conducted in TRB ( 2008 ) illustrat e the role that transit can play in an emergency evacuation. Transit can play a critical role in evacuating people without access to private vehicles, moving them out of the affected area to safe shelter to an incident site. Finally, transit can also be involved in the recovery phase, bringing the evacuees back to the area and resuming normal transport services for the re sidents. PAGE 27 27 2.3.2 Lack of Transit Planning in Emergency Response Use of public transit in emergency situations is not well studied in the literature and the failure of evacuating residents without access to private transportation, has highlighted the need for urban evacuation and planning for transit dependent people. Transit dependent citizens include people without access to private vehicles, the poor and elderly, tourists and people with medical needs. The evacuation of New Orleans during hurricane Katrina was on e of the most significant evacuations in United States history and it was both a success and a miserable failure. Years of planning and coordination amongst transportation planners, emergency managers, and police led to an effective contraflow system that enabled everyone with a car to evacuate. Unfortunately, much of this hard work went unnoticed because national attention focused on the glaring failures in the evacuation effort, particularly the failure to evacuate a significant number of carless people w ho were unable to leave the city. State and local plans did not properly address the evacuation of those who were dependent on public transportation modes. Although various emergency preparedness groups including the Red Cross, the New Orleans Public Healt the use of Amtrak and city buses for the evacuation of public transit dependent people even before Katrina struck, yet these plans lacked important details such as how to use the buses, availability of drivers, bus routes or the pick up location ( staging areas ) to collect evacuees (USDOT& USDHS, 2006). State and local plans during hurricane Rita appeared to be better coordinated in providing public transit during the evacuation of residents. Houston METRO, which assisted greatly during the evacuation, possessed a robust fleet of buses, rail and lift PAGE 28 28 vehicles. However, the plan still lacked many details concerning the evacuation of special needs residents (USDOT& USDHS, 2006). It is evident that there are serious deficiencies in government plans to evacuate using public transportation. The Victoria Transport Policy Institute describes New implemented by of evacuation was relatively effective for people with automobiles but failed transit TRB (2008) evaluated the potential role of transit systems in acco mmodating the evacuation in times of emergency. As a part of the study, a literature review was conducted to address the question of the extent to which transit is incorporated into local emergency evacuation plans. According to the findings of this report the majority of emergency operation plans are only partially sufficient to conduct a successful evacuation and few focus on the role of transit. Transit has a unique role to play in evacuating the carless and people with special needs during an emergency However, these groups are inadequately addressed in most local emergen cy evacuation plans (TRB, 2008). In a national survey of hurricane evacuation plans (Wolshon et al., 2001); it was found that state departments of transportation (DOTs) ignored the spe cial needs population. This report also found that most of the cities do not have a sufficient number of buses to evacuate all special needs evacuees. Hess and Gotham (2007) studied counties in rural New York and found that multimodal evacuation planning was not seriously considered in most evacuation plans and that there are inadequate public transportation resources for mass evacuation. PAGE 29 29 Bailey et al. (2007) surveyed the emergency response and evacuation plans in 20 metropolitan areas with higher than ave rage proportions of minorities, low income levels, limited English proficiency, and households without vehicle access. It was found that only a few agencies had included transit dependent population in their emergency plans. 2.3.3 Need for Transit Evacuation Pla nning In the United States, New Orleans is not unique in terms of its high percentage of carless residents In fact, according to U.S. Census (2000), seven cities including New York (56 percent), Washington D.C. (37 percent), Baltimore (36 percent), Phila delphia (36 percent), Boston (35 percent), Chicago (29 percent) and San Francisco (29 percent) had carless population s higher than the 27 percent found in New Orleans. In addition, the people who own an automobile may also need to rely on other modes due to some other constraints for e.g., the elderly people who have cars may be reluctant to d rive them during an evacuation. These demographics face higher risk during disasters. For instance, 71% of those who died during Katrina in New Orleans were over the age of 60 and 47% over the ag e of 75 (AARP 2006a and 2006b). The extra risks faced by carless households during an evacuation are well recognized and have been documented in numerous reports ( e.g., Fischett, 2001; Bourne, 2004). Despite the attention, rela tively little has been done to improve the situation and some serious deficiencies in the evacuation planning involving public transit remain throughout the United States. Based on the current level of public transit emergency planning and preparedness, it is quite likely that unfortunate tragedies seen in New Orleans during and after hurricane Katrina are bound to repeat unless be tter practices can be developed and widely adopted (Jenkins et al., 2007). PAGE 30 30 For detailed information about the role of public tra nsportation during evacuation and evacuation planning for transit dependent residents, please refer to Higgins et al., 1999; Renne et al., 2008; and TRB, 2008. 2.3.4 Optimizing the Use of Transit in Evacuation Before 2005, public transportation operators in the United States did not play a strong role in evacuation planning, nor were they viewed as a viable option for evacuation. Now, there is increased national awareness and interest in the role of public transportation in evacuation. Ever since hurricanes Katr ina and Rita in 2005, the Federal G overnment has increased the efforts to assist s tate and local governments in improv ing the effectiveness of buses and trains to evacuate special needs population s from disaster areas. The Federal Transit Administration (F TA) has been working with the American Bus Association and the American Public Transportation Association (APTA) to identify and meet the resource requirement s of transit agencies in evacuating special needs population s APTA also created an Emergency Prep aredness Task Force to develop strategies for improving coordination between all levels of government in emergency response. Recent research shows some contributions by the res ear chers in the area of transit evacuation planning using simulation model. Howe ver, optimizing the use of public transit during evacuation events has not received as much attention in the literature. Interest began recently with Margulis et al. (2006 ) which developed a deterministic decision support model to maximize the number of p eople evacuated during a hurricane by determining the optimal bus allocation for specific pick up point PAGE 31 31 to shelter routes. The model execution was limited to a small scale evacuation because of its inability to deal with a large area. The assumption underl ying the model is that the pick up points, number of buses and shelter locations are known and from this the model outputs the number of bus trips for specific pick up point to shelter routes. Sayyady (2007) presented an optimization technique to develop a n evacuation plan for transit dependent people during no notice disasters. A mixed integer linear programming model is developed to generate a transit evacuation routing plan in order to minimize the evacuation time. The assumption underlying the model is that the location of pick up points and shelters are known and transit vehicles will perform only one trip considering the limitation on evacuation time during no notice disaster. Model outputs are the expected number of vehicles required, the expected num ber of citizens evacuated within a given time period, and the evacuation routes for transit vehicles. Song et al. (2009) presented an optimization modeling technique to develop an evacuation plan for transit dependent residents during hurricane evacuation. The proposed model is formulated as a location routing problem with uncertain demand. Pick up points, shelter locations and number of buses are known and the goal is to optimize the bus routing during evacuation in order to minimize the total evacuation t ime. Abdelgawad et al. (2009) presents an approach to optimally operate the available capacity of mass transit to evacuate transit dependent people during the no notice evacuation of urban areas. They proposed an extended vehicle routing problem incorporat ing constraints for the number of depots, evacuation time frame, and pick up and delivery locations of evacuees. A framewor k, using constraint programming was developed to model and solve the problem. A hypothetical evacuation event was PAGE 32 32 considered and the results demonstrate the optimal scheduling and routing for the buses. Figure 2 1 provides a summary of the reviewed modeling studies f or transit evacuation planning. Figure 2 1. Summary of transit evacuation modeling studies 2.4 Evacuation Planning under Un certainty Evacuation modeling is subject to uncertainties and al though a great deal of research has been dedicated towards evacuation management, there is a lack of modeling efforts in the literature addressing uncertainty during evacuation. Even after ex tensive evacuation planning, it is challenging to model evacuation s in real world transportation network s due to the associated uncertainties. Public officials face a great challenge during evacuations because of the unpredictability of different aspects o f evacuations. For example, studies in the literature show that the total number of people evacuat ing and their departure time s are hardly predictable PAGE 33 33 2.4.1 Sources of Uncertainty There are numerous sources of uncertainties during an evacuation and most of them are not easy to quantify or control. Uncertainties in the evacuation problem can be related to the demand side, to the supply side or to the disaster characteristics. On the demand side, uncertainty can be in terms of total number of evacuees and their lo cations. Demand uncertainty is associated with human behavior e.g., decision to evacuate or not, timing of evacuation etc. Furthermore, there can be behavioral uncertainties regarding compliance and driving behavior. On the supply side, uncertainty may be caused by impact of disaster on the infrastructure e.g., reduced link capacities, flooded links during hurricanes, limited network connectivity due to collapse of a bridge during earthquakes etc. The limited predictability of the severity of a disaster, e. g. the intensity and track of a hurricane, or the timing of any disaster occurrence, are also some of the factors causing uncertainty during evacuations. 2.4.2 Optimization U nder Uncertainty A large number of problems in production planning and scheduling, loca tion, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. Problems of optimization under uncertainty are characterized by the necessity of making decisions without knowing what their full effects w ill be. In addition, a key difficulty in optimization under uncertainty is in dealing with an uncertainty space that is huge and frequently leads to very large scale optimization models. Approaches to optimization under uncertainty have followed a variety of modeling techniques, and in this section we will review some of the main approaches. PAGE 34 34 2.4.2.1 Simulation optimization Simulation based optimization integrates optimization te chniques into simulation analysi s. Simulation optimization can be defined as the process of finding the optimal decision variable values from among all possibilities without explicitly evaluating each possibility. At each iteration, the simulation based optimizer provides values for the decision variables, followed by the simulation runs to e valuate the objective and constraints Simulation optimization is often slow and takes too much time even for a small problem. Figure 2 2 presents a general simulation optimization model (Carson and Maria, 199 7). For a more detailed review on simulation ba sed optimization methods, refer Andradottir (1998), Fu (1994), and Fu (2002). Figure 2 2 A simulation optimization model 2.4.2.2 Stochastic programming In stochastic programming, the uncertain elements in a problem can often be modeled as random variables to which the theory of probability can be applied. This approach requires known probability distributions of the uncertain data and it is assumed that the probability structure is unaffected by any of the decisions that are taken during the process (Rockaf ellar, 2001). The goal of stochastic programming approach is to find some policy which is feasible for all (or almost all) the possible data instances and the expected values is optimized. This approach is the most widely used Feed back on progress Input Output Simulation Model Optimization Strategy PAGE 35 35 approach for optimization und er uncertainty and has applications in broad range of areas. We refer the interested reader to several textbooks ( Infanger, 1994; Kall and Wallace, 1994; Prekopa, 1995; Birge and Louveaux, 1997 ) and the references therein for a more comprehensive understa nding of s tochastic programming. 2.4.2.3 Robust optimization Stochastic models of decision making under uncertainty assume perfect information and known probability distributions for the random variables. However such precise knowledge of probability distributions is rarely available. Robust optimization is a more recent approach to optimization under uncertainty, in which uncertainty model is not stochastic, but rather set based. In robust optimization, random variables are modeled as uncertain parameters belongin g to a bounded set and a decision maker usually protects the system against the worst case value within that set. This notion of modeling uncertainty is useful in many applications and has advantage in computational tractability depending on the problem st ructure. Interested readers are referred to Ben Tal and Nemirovski (2002) and Bertsimas et al. (200 8 ) for reviews of the robust optimization methods. 2.4.3 Incorporating Uncertainty in Evacuation Modeling Due to the associated uncertainties during an evacuation, it is very unlikely that the evacuation process would go (or even close) according to the predetermine d plan. There can be significant consequences of uncertainties during real life mass evacuations and the whole evacuation planning may result in a failur e. Past experience has shown that uncertainty might lead to overcrowded shelters, high network congestion and other failure situations. For instance, it was found that the actual number of evacuees during Hurricane Rita from the Galveston and Harris Counti es PAGE 36 36 equaled 1.8 million people, whereas the predicted number was only 686,000 (Lindell and Prater, 2007). Although the real life evacuation process shows a significant uncertainty, the general state of the practice ignores the real world uncertainties and a ssumes fixed demand and capacity. In order to avoid the undesirable surprises during the actual evacuation process, it is essential to develop evacuation plan and decisions which could guarantee a high probability of the actual evacuation to be as close as possible to the predicted one. Therefore, while planning for optimal evacuation measures, it is important to deal with these uncertainties and decisions must be made by accounting for different measures of uncertainties. In the literature, there are not many studies in evacuation planning focusing on uncertainty. The m ajority of existing evacuation planning model s assumes determinism. Clearly, there is an inconsistency between th is deterministic assumption and what is actually experienced in practice Tak ing a deterministic approach might lead to sub optimal evacuation strategies with potential ly significant consequences in the real world. A f ew studies have considered uncertainties in evacuation modeling. Yazici and Ozbay (2007) incorporated uncertainty in their C ell T ransmission M odel (CTM) based S ystem O ptimal D ynamic T raffic A ssignment (SODTA) model. They approached the problem from stochastic programming perspective and used a chance constraint approach to model the uncertainties in road capacity duri ng evacuation. Changes in clearance time and spatial shelter utilization were analyzed and it was demonstrated that introducing probabilistic link capacities can adjust the overall flow in the network as PAGE 37 37 well as shelter utilization. However, it was assumed that the total number of evacuees is deterministically known. Later, Yazici and Ozbay (2010) also included the demand uncertainty in their modeling approach and proposed a SODTA model with probabilistic demand and capacity constraints during evacuation. The problem was modeled based on two different stochastic modeling approaches, namely individual chance constraints and joint chance constraints. It was shown that the model can incorporate different user defined probability values and can be used to calcu late the change in clearance time, average travel time and risk exposure measures. While the work by Yazici and Ozbay is certainly an improvement over previous studies assuming determinism, they adopt the implicit assumption of chance constraint programmin g that the probability distributions of random demand and capacities are explicitly known. However, this might not be the case because of the very limited or non existent historical data on evacuation s Yao et al. (2009) consider evacuation in an uncertai n environment and proposed a robust linear programming model considering demand uncertainty. In their study, the demand for evacuation was explicitly considered to be a random variable. They realized the fact that it is not possible to obtain the exact pro bability distribution of number of evacuees. Road capacities were assumed to be deterministic. It was found that a robust solution improves both feasibility and quality compar ed to a deterministic solution. Ng and Waller (2010) presented a CTM based evacua tion route planning model that accounts for both evacuation demand uncertainty as well as road capacity uncertainty. Their model also does not require the assumption that probability PAGE 38 38 distributions are known explicitly. A novel distribution free approach is used to provide probabilistic guarantees on the resulting evacuation plan, i.e. they allow for infeasibilities with a pre specified tolerance level. It was demonstrated that the reliable evacuation plans were able to provide a good estimate of realized ev acuation time. Huibregtse et al. (2010) presented an approach to optimize evacuation measures under uncertainty. The uncertainty in the evacuation problem is related to the evacuees (the demand and the behavior of people) and the hazard (location, time, a nd intensity). They adopted a scenario based uncertainty approach (the uncertainty is translated into scenarios which could occur) and defined different criteria to evaluate the effectiveness of evacuation measures. Although the scenario based approach has some restrictions for real applications, this case study shows the usefulness of dealing with uncertainty in the evacuation problem. Pel et al. (2010) performed a sensitivity analysis to identify and quantify the impacts of variations in both travel deman d and network supply for evacuations. Sensitivity analysis was done using a macroscopic evacuation traffic simulation model EVAQ. The authors vary different aspects such as trip generation, departure rates, route flow rates, road capacities and maximum spe eds. It was found that the departure rates and route flow rates have a substantial non linear impact on the network conditions and arrival pattern, in particular when the network is highly congested, while the trip generation and road capacities have a sma ller quasi linear impact. Figure 2 3 provides a summary of the studies focusing on modeling unce rtainty in evacuation planning. PAGE 39 39 Figure 2 3. Summary of studies considering uncertainties in evacuation modeling 2.5 Contributions of This Dissertation The review of current trends in emergency planning showed that many of these areas are well studied and that there is a continuous exploration and enhancement of practices and plans. Recently, use of transit has been viewed as an important part of evacuation plannin g and management. However, there is still a need to address transit evacuation planning to assist transit dependent residents, in order to develop a comprehensive evacuation response plan. In the literature, there are a limited number of studies focusing on modeling and optimizing the use of public transit during evacuation. The studies mainly focused on optimizing transit routing plans. In this dissertation, two critical decision making components for transit evacuation planning are studied i.e., public shelter locations and pick up locations (staging areas). It has already been shown that the shelter location decisions are extremely important for auto evacuation planning and can greatly PAGE 40 40 influence the evacuation process. Likewise, public sheltering is an important component of transit evacuation planning and the location of public shelters can be critical to the efficiency of transit evacuation. In addition, all the previous research on modeling transit during evacuation assumes that the pick up locations (staging areas) for transit are already known, yet this is not the case in real life evacuations and exact pick up locations are not known To our best knowledge, no analytical approach is available in the literature that determines the optimal shelter loc ations and transit pick up locations during transit evacuation. In this dissertation, we try to fill this gap. On the other hand, d espite a great deal of research dedicated towards evacuation planning, there is a lack of modeling efforts in the literature addressing uncertainty during evacuation. The re is a pressing need for models that incorporate uncertainty and thus assure viable measures for the unanticipated. This dissertation aims to address this important gap. The main focus of this dissertation is to develop a plan that evacuates those who depend on public transportation to safe areas reliably and as fast as possible during an emergency. More specifically, optimization model s are developed for transit based evacuation planning under demand uncertain ty using robust optimization approach, which are more efficient and reliable in the face of demand uncertainty. Demand uncertainty refers to the uncertainty associated with the number of transit dependent evacuees. In particular, three different models are proposed to model public shelter and pick up locations for transit evacuation The resulting plans obtained from the proposed model s are expected to perform robustly and effectively in real world mass evacuations, thereby making the transit based evacuati on planni ng more reliable and efficient. PAGE 41 41 CHAPTER 3 ROBUST PUBLIC SHELTER LOCATIONS WITH DEMAND UNCERTAINTY 3.1 Background The evacuat ion planning and management has been studied from various management, shelter and route choice etc. One of the critical tasks in developing an evacuation plan is to determine where evacuees would seek shelter. FDOT (2000) reported poor information on the availability and location of shelters for the evacuees, as one of the weakness of evacuation process during Hurricane Floyd. As indicated by Sherali et al. (1991), the selection of shelter locations from among potential alternatives can greatly influence the evacuation process. However, only a few studies in the literature have explicitly considered the impact of shelter locations on evacuation time. Most of the studies related to shelters during evacuation focus on structural and operational issues rather than location (e.g., ARC, 1992; FEMA, 2000b; Coulbourne et al., 2002 and Yi and Ozdamar, 2007). Sherali et al. (1991) seemed to be the first to study the shelter location problem for evacuation planning under hurricane/flood conditions. A nonlinear mixed integer programming model was proposed to determine the she lter locations and prescribe paths for evacuees to minimize the system wide evacuation time. The assumption that under the disaster conditions. Recognizing this limitati on, Kongsomsaksakul et al. (2005) proposed a bi level programming model to determine the optimal location of shelters during flood evacuation. In their model, the planning authority determines the number and shelter locations (the upper level problem), whi le the evacuees choose PAGE 42 42 shelters and routes to evacuate (the lower level problem). The model assumed that the shelter capacities are known and the number of evacuees at each origin is also given. More recently, Ng et al. (2010) proposed a hybrid bi level mo del for optimal shelter assignment in emergency evacuations. The upper level problem specifies a system optimum assignment of the evacuee demand while evacuees choose their routes to the assigned shelters in the lower level problem. In their study, the she lter locations are assumed to be kno wn. Above studies indicate the importance of shelter location decision for auto based evacuation. Likewise, public sheltering is a n important component of transit ba sed evacuation planning and the selection of shelter lo cations can greatly influence the efficiency of transit operations during evacuation. Especially, for the emergency situation, where the significant portion of demand is transit dependent and transit can play an important role, it is critical that the shel ter location decision be made carefully, identifying the residential location of transit dependent evacuees in the affected area. However, there is no analytical approach in literature to determine the optimal location of shelters f or transit evacuation pl anning. In addition, in the planning stage, it is difficult, if not impossible, to estimate accurately the number of transit dependent evacuees and the public shelter usage during an evacuation. To develop a successful transit evacuation plan, it is critic al to proactively consider such a demand uncertainty in determining shelter locations and capacities. Otherwise, the capacities of certain shelters may not be sufficient to accommodate the evacuees while the other shelters remain underutilized. Moreover, t ravel times to certain shelters may become prohibitively large due to traffic congestion. PAGE 43 43 Capacity of shelters hereinafter refers to the optimal stocking of supplies at the facilities, and not the physical capacity of shelters. In developing a mass evacuat ion plan for transit operations, information regarding the extent to which evacuation plans are coordinated among different counties and neighboring States is essential. The inability to evacuate the transit dependent population during hurricane Katrina is assumed to be the result of poor coordination and communication of available resources, rather than the lack of transportation resources (USDOT, 2006; Renne 2006). After hurricane Katrina, the U.S. Department of Transportation (USDOT, 2006) stressed the c oordination of available resources as a critical issue for a successful mass evacuation plan. However, during large scale evacuation, problem often arises in trying to manage operations involving transit agencies across different counties and jurisdictions It is observed that the real time coordination between different state and local government while managing an evacuation is very limited. This chapter focuses on transit evacuation planning from a regional level planning perspective for a large scale eva cuation where transit agencies in different counties are trying to perform their own evacuation and transport the transit dependent evacuees to the public shelters in order to minimize their evacuation time. In this chapter a robust model is developed to dete rmine optimal shelter locations and capacities, from a given set of potential sites, under demand uncertainty for transit evacuation. Demand uncertainty refers to the uncertainty associated with the number of transit dependent evacuees originating fro m different counties or cities, seeking shelters to evacuate For the purpose of this dissertation, hereafter, we refer to the use of bus service when mentioning the public transit in the developed models. The proposed PAGE 44 44 model is formulated as a mathematical program with complementarity constraints, a class of problems difficult to solve. The problem has a bi level structure. At the upper level, planning authority determines the number and locations of shelters along with their capacitie s while at the lower l evel, different transit agencies choose shelters and route to evacuate in order to transport the transit dependent evacuees A cutting plane algorithm (Lawphongpanich and Hearn, 2004) is applied to solve the problem. For the remainder of this chapter, Sect ion 3.2 describes the model components and considerations. Section 3.3 formulates the shelter location model. Section 3.4 proposes an efficient solution algorithm to solve the model. Results from a numerical example based on Sioux Falls network to illustra te the effectiveness of proposed model are presented in Section 3.5. Finally, Section 3.6 provides the summary of this chapter. 3.2 Model Components 3.2.1 Representing Demand Uncertainty Quite a few techniques have been developed in the literature to accommodate un certainty in decision making as discussed in Section 2.4.2. Considering that it is practically difficult to obtain the probability distribution of the uncertain demand, a robust optimization approach (e.g. Ben Tal and Nemirovski, 2002) is adopted in this d issertation to model demand uncertainty. In this approach, the uncertain demand is assumed to be confined to an uncertainty set, and the shelter locations and capacities are determined against the worst case demand scenario realized in the set. More specif ically, it is assume d that the transit dependent demand for each county/ city can be one of the several possible values whose probabilities of occurrence are unknown. These possible demand values are not difficult to obtain and can be identified, e.g., with alternative forecasts of the magnitude and impact area of the PAGE 45 45 disaster. For each county let denote the transit dependent demand under scenario where corresponds to the nominal demand, i.e., the most likely forecast. To hedge against uncertainty, a shelter location plan can be designed against a worst case scenario where each county generates the maximal possible transit dependent demand. However, this will lead to an overly conservative plan. More specifically, it is rare for all the counties to reach their maximum transit dependent demand valu es simultaneously. Designing the location allocation plan against such a rare case would be overly protective. With a philosophy that the nature is neither our enemy nor friend, it is assume d that demands for at most counties can deviate from their nominal values simultaneously while the others must remain at their nominal values. is called as the degree of pessimism (also known as uncertainty budget) which is a parameter for controlling the degree of conservatism attitude toward risk. A larger implies that decision makers are more risk averse. This approach of accounting uncertainty (e.g., Bertsimas and Sim, 2003; Lou et al., 2009) leads to computationally tractable robust formulation and is flexible in controlling the Mathematically, let denote a binary variable that indicates whether scenario is realized at county Since, only one demand scenario can be realized for each county we have Also, since at most county demands are allowed to deviate from their nominal values and represents that nominal demand value is realized at county we have Additionally, the binary requirement for the variable can be written in the form of a continuous variable by adding constraints, PAGE 46 46 which ensures that is 0 or 1 for each Thus, the uncertainty set for the transit dependent demands can be characterized as follows: where is the demand vector realized for the network and is the demand realized for each county Hereinafter, the superscript denotes variables associated with a particular demand vector Note tha t the above uncertainty set does not necessarily include all the possible realizations of transit dependent demand, which would be really difficult due to limited information at the planning stage. The set is a purely mathematical construct to model demand uncertainty in the robust optimization approach and choice of affects the efficiency and robustness of the resulting robust plan (Ben Tal and Nemirovski, 2002). Compared with other types of configuration, the above set is easier to obtain because planning agencies are often able to produce a few different demand fo recasts as previously discussed. 3.2.2 Notations Let denote the network of roads, where and are the sets of nodes and links in the network respectively. We denote a link as or the pair of its starting and ending nodes, i.e., As sume that the demand during evacuation is originated from a subset of origin nodes Note that, each origin node here represents a county in the network. Next, let denote a set of potential sites to locate shelters in the network. A binary varia ble represents the shelter location decision, which is 1 if a shelter is located at node and 0 otherwise. PAGE 47 47 Let be the node link incidence matrix associated with the network and is a vector in with two non zero component: one has a value 1 in the component corresponding to county and other has a value 1 in the component corresponding to shelter For a given demand vector the number of buses between any county and shelter node are the results from tr destinations choices, and thus are de cision variables in the model. The set of feasible county shelter demand and link flow distributions for the network, denoted as can be expressed as follows: where is the aggregate link flow vector; is the link flow vector ( i.e., number of transit vehicles ) corresponding to the county shelter pair ; is the demand in terms of total number of transit vehicles going from county to shelter ; is the capacity of transit vehicles available with county ; is a sufficiently large number and the inequality, i.e., ensures that transit vehicles from each county only go to the sites where a shelter is located. Hereinafter, the subscript or superscript d denotes variables assoc iated with a particular demand vector Additionally, it is assumed that there exists background traffic in the network. This background traffic represents the automobile traffic within the network. Let denote the estimated background flow o n link Hence, the total flow on link will be Consequently, for calculating the travel time for link the following form of BPR function is used: PAGE 48 48 where is the free flow travel time for link and is the capacity of link Since is assumed to be known, we hereinafter denote the travel time function as 3.2.3 Combined Distribution and Assignment Model Given the location of shelters, a combined distribution and assignment (CDA) model is used to describe the choices of which shelters to evacuate and which routes to access the selected shelters. In the CDA model, the network traffic condition is assumed to be in user equilibrium and the selection of shelters is represented by a logit model with a dispersion parameter The CDA model for a specific demand scenario is written as follows: s.t. (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) where is the shelter bound aggregate link flow at lin k is the shelter bound flow between the county shelter pair at link is the total number of transit vehicles going from county to shelter Constraint (3.1) calculates the aggregate link flow. Constraint (3.2) ens ures the flow balance between county and shelter nodes in the network. Constraint (3.3) requires PAGE 49 49 that the total number of evacuees transported fro m any county to different shelters will be equal to the total demand at that particular county Const raint (3.4) ensures that transit vehicles only go to the sites where a shelter is located. Constraint (3.5) and (3.6) are the non negativity constraints for the link flows and demand. 3.2.4 List of Notations Below is the list of some useful notation s used for t he proposed model i n this chapter. index for origin/county nodes, index for potential shelter sites, set of nodes set of links capacity of transit vehicle available at origin/county fixed cost for establishing a shelter at site unit cost to operate shelter at site dispersion parameter for CDA model upper bound on travel time from origin to any open shelter degree of pessimism demand vector demand realized for origin demand for origin node under scenario demand from origin to shelter transit flow on link background traffic flow on link PAGE 50 50 transit link flow corresponding to county shelter pair travel time for link travel time from origin to shelter binary variable indicating which scenario is realized for origin binary variable indicating shelter location decision capacity of shelter Note that, although not all the notations used i n this chapter are listed here, they are defined appropriately wherever used. 3.3 Model Formulation In this section, the mathematical formulation for the shelter location problem is presented For computational efficiency, a static modeling fr amework is adopted which appears sufficient for such a macroscopic planning practice. Our future study will examine the impacts of traffic dynamics on the shelter deployment plan. Using the notations defined in previous section, a robust shelter location model or RSL can be formulated as follows: RSL s.t. (3.7) (3.8) (3.9) (3.10) PAGE 51 51 (3.11) (3.12) (3.13) (3.14) (3.15) (3.16) (3.17) (3.18) The decision variables in this model include shelter location shelter capacity county shelter flow distribution link flow distribution variables and and auxiliary variables which are the Lagrangian multipliers for each demand scenario associate d with Equation (3.1) (3.6) respectively in the Karush Kuhn Tucker (KKT) conditions (e.g., Bazaraa et al., 2006) of the CDA model. The objective of RSL is to minimize total cost to establish and operate the shelters to accommodate all the potential transi t dependent evacuees. The cost includes a fixed cost for establishing shelters with being the fixed cost for one shelter and a variable cost to operate shelters, which is proportional to the shelter capacity with being the unit cost. Constraint (3.7) ensures that the capacity of each shelter, denoted as K is always suff icient to accommodate its potential transit dependent demand under all demand scenarios realized from the uncertainty set D Constraint (3.8) suggests that the link flow and demand distribution should be feasible. Constraints (3.9) (3.15) are the KKT condi tions for the CDA problem for each demand scenario, ensuring that the distribution of link flow and county shelter demand is in user equilibrium for the PAGE 52 52 combined choices of shelters and routes. Constraints (3.16) (3.17) ensure the shelter location variable to be binary. Finally, constraint (3.18) requires that the travel time for each county to any open shelter, is less than an upper bound This requirement is to ensure transit dependent evacuees from all the counties to reach a safe location within a specified time. Note that since a logit model is used to describe shelter choices, an open shelter will be utilized by transit agencies from all counties even though the number from certain county could be very small. This is due t o the fact that a logit model will always lead to a strictly positive probability even for the least attractive option. As formulated, RSL is a mathematical program with complementarity constraints (MPCC), a class of problems difficult to solve (e.g., Sch eel and Scholtes, 2000). The formulation has four sets of complementarity constraints; three sets to ensure that the flow and demand distribution is in user equilibrium and the other one to make binary variables continuous. Standard stationarity co nditions such as the KKT conditions do not hold for MPCC (Scheel and Scholtes, 2000) and many (e.g., Luo et al., 1996, and references cited therein) have proposed special algorithms to solve them. However, many of these algorithms work well for small and m edium problems and others, especially those based on solving equivalent nonlinear programs (e.g., Fletcher and Leyffer, 2004), can handle larger problems. 3.4 Solution Algorithm 3.4.1 Solving Robust Shelter Location Problem In this section a cutting plane algorithm ( Lawphongpanich and Hearn, 2004) to solve RSL is presented considering that the number of demand vectors in the uncertainty set can be extremely large. The algorithm solves a sequence of restricted PAGE 53 53 RSL problems with respect to a subset of demand vector s and then solves another problem to generate demand vectors one at a time as needed to expand The restricted RSL problem (R RSL) is written as follows: R RSL s.t. (3.19) (3.20) (3.2 1) (3.22) (3.23) (3.24) (3.25) (3.26) (3.27) (3.28) and (3.16) (3.17) Let so lve the R RSL globally and let denote the corresponding set of locations with shelters. Then, the solution also solves the original RSL problem if the shelter capacity is sufficient and travel time between each county and op en shelter is within its threshold value, under all demand scenarios in For a given PAGE 54 54 location of shelters and demand vector the set of all feasible link flow and county shelter distributions for the network, denoted as is ex pressed as follows: To verify this mathematically, fol lowing worst case demand problem (WCD) is solved : WCD s.t. (3.29) and (3.9) (3.15) The objective of WCD is to find a demand vector the current design of shelters. More specifically, the demand scenario will make, to a largest possible extent, the shelter capacity insufficient and travel time larger th an the required threshold. If the optimal objective value of WCD problem is no greater than zero, then is an optimal solution to the RSL problem. Otherwise, an improved solution can be obtained by expanding the demand vector set where is the solution to above WCD. Because the number of demand vectors in the uncertainty set is finite, the cutting plane algorithm terminates after a finite number of iterations. Below is the formal solution procedure of the cutting plane a lgorithm outline d above. Cutting Plane Algorithm : Step 0: Choose an initial demand vector Set and PAGE 55 55 Step 1: Solve the R RSL problem with a subset of demand vectors (Section 3.4.2) and let denote the r esulting optimal solution. Step 2: Solve the WCD problem associated with the shelter location and shelter capacities (Section 3.4.3) and let denote the optimal solution and denote the corresponding objective value. Step 3: If stop and is an optimal solution to the RSL problem. Otherwise, set and go to step 1. The solution obtained from the above cutting plane algorithm will be global optimal to the RSL problem, if the R RSL and WCD problems can be solved optimally. However, both the sub problems (R RSL and WCD) are still MPCC, a class of non convex problems difficult to solve to the global optimality. 3.4.2 Solving Restricted Robust Shelter Location Problem A n active set algorithm (Zhang et al., 2009) is applied to solve the R RSL. The basic idea of the algorithm is to solve a sequence of relaxation of the original MPCC problem and strategically updating the feasible design (shelter locations in our case) using the dual i nformation associated with the relaxed problem. Given a feasible shelter locations to R RSL, consider the two index sets defined as and The index set based relaxed R RSL problem (R RSL I) can thus be wri tten as follows: R RSL I s.t. (3.30) PAGE 56 56 (3.31) and (3.19) (3.28) The solution to R RSL I is very easy to obtain wit hout directly solving the problem. For a known feasible shelter locations the flow and county shelter demand distribution can be easily obtained by solving the CDA model (Section 3.2.2) for each demand scenario in With the known O D demand dist ribution for each demand scenario, the minimum shelter capacities and total cost can be easily calculated. With the optimal solution to R RSL I as the initial solution, the problem is solved directly once again. The purpose is to obtain the Lagrangian mult ipliers associated with the constraints (3.30) and (3.31), which provide valuable information on the marginal impact of adjusting the index sets to obtain an updated shelter location plan. Let and denote the Lagrangian multipliers associated with (3.30) and (3.31) respectively. To update the index sets, the following knapsack problem is considered: KP RSL s.t. (3.32) (3.33) (3.34) (3.35) is binary where the binary variable indicates whether to flip the value of or not. Constraints (3.32) and (3.33) strategically update the value of based on the value of For PAGE 57 57 example, if (i.e. ), and is flipped (i.e. ) then the updated value of will become 1, otherwise the updated value of will remain 0 if is not flipped (i.e. ). Thus, the objective of the above knapsack problem is to minimize the estimated change to the total cost by adjusting the current shelter locations and a neg ative objective value indicates that a reduction in total cost is possible. Constraint (3.34) is introduced to help find suboptimal solutions. Initially, is and solving KP RSL would yield updated shelter locations, with the most negative objective va lue If the updated shelter location actually yields a lower total estimated cost (solving the R RSL I with updated locations to calculate the new total cost and comparing with the previous value of total estimated cost) and is feasible (i.e. satisfying the travel time requirement), then an improved shelter location design is found. However, because the Lagrangian multipliers are only estimate of marginal changes while the actual changes are either zero or one, the new design may not lead to an actual de crease in the total cost. If the new shelter location is not a better and feasible solution, then another design is generated by setting = where is a sufficiently small positive number. Constraint (3.35) ensures that there is at least one open shelter in the updated shelter location plan. If the optimal objective value of KP RSL is zero, then no adjustment in the shelt er locations leads to a better solution and the algorithm is terminated. The procedure to solve R RSL can be described as follows: Active Set Algorithm (R RSL) : Step 0: Locate shelters at all potential sites and set it as an initial feasible shelter locati on plan PAGE 58 58 Step 1: Solve the R RSL I problem with and let denote the resulting optimal objective value. Obtain the Lagrangian multipliers and Set Step 2: Solve the KP RSL problem. Let denote the resulting optimal soluti on and denote the corresponding objective value. If stop and is the optimal solution to R RSL. Otherwise go to step 3. Step 3: Solve the R RSL I with new updated solution Let denote the corresponding objective value. If and the new updated solution is feasible then an improved shelter location design is found, set update the active sets and and go to Step 1. Otherwise, set = and go to Step 2. 3.4.3 Solving Worst Case Demand Problem WCD is also an MP CC problem with a structure si milar to R RSL and therefore the same active set algorithm is applied to solve it. Given a demand vector and its corresponding consider two index sets defined as and The index set based relaxed WCD problem (WCD I) can thus be written as follows: WCD I s.t. (3.36) (3.37) (3.38) (3.39) PAGE 59 59 (3.40) (3.41) (3.42) (3.44) (3.45) (3.46) (3.47) (3.48) (3.49) Simil ar to R RSL I, the optimal solution to WCD I is also easy to obtain. For a given demand vector the corresponding O D and flow distribution can be easily obtained by solving the CDA model. In a similar manner, WCD I is then solved directly using the o ptimal solution as an initial solution. Let and denote the Lagrangian multipliers associated with (3.36) and (3.37) respectively. To update the index sets, the following knapsack problem is considered: KP WCD s.t. (3.50) (3.51) (3.52) PAGE 60 60 is binary and (3.38) (3.40) where binary variable indicates whether to flip the value of or not. The objective of the above knapsack problem is to maximize the estimated change to the worst case objectiv e value by adjusting and a positive objective value indicates that an increase in worst case objective is possible. The procedure to solve WCD can be described as follows: Active Set Algorithm (WCD) : Step 0: Choose an initial demand vector Step 1: Solve the WCD I problem with and let denote the resulting worst case objective value. Obtain the Lagrangian multipliers and Set Step 2: Solve the KP WCD problem. Let denote the resulting op timal solution and denote the corresponding objective value. If stop and is the worst case demand to WCD. Otherwise go to step 3. Step 3: Solve the WCD I with the new updated solution Let denote the corresponding objective value. If then a worse demand vector is found, set update the active sets and and go to Step 1. Otherwise, set and go to Step 2. 3.5 Numerical Example This section presents results from solving the robust shelter location mod el for the Sioux Falls network ( LeBlanc et al., 1975 ). T he cutting plane algorithm is implemented in GAMS (Brooke et al., 1992) in which CDA, R RSL I and WCD I problems are solved PAGE 61 61 using CONOPT (Drud, 1992) and the knapsack problems (KP RSL and KP WCD) usin g CPLEX (CPLEX, 2008). The Sioux Falls network contains 76 links and 24 nodes. Mass evacuation is considered because of a hypothetical disaster in the region. Eight potential sites to locate shelter in the network are assumed as nodes 1, 4, 8, 13, 14, 18, 20, and 22. The remaining 16 nodes are the origin nodes with nominal demands as shown in Table 3 1. Each origin node which represents a county is assumed to have three demand scenarios. The other two demand scenarios, low and high, are generated by multipl ying the nominal demand by a uniformly distributed number between (0.5, 1) and (1.5, 2). The demand values reported here are in terms of number of transit vehicles after considering the total number of transit dependent evacuees and the capacity of availab le buses. PAGE 62 62 Table 3 1. Demand scenarios at each county County Nominal Demand Low Demand High Demand Node 2 4.00 2.34 6.32 Node 3 3.00 2.76 4.88 Node 5 5.00 3.88 9.17 Node 6 7.00 4.55 12.02 Node 7 11.00 7.11 18.48 Node 9 14.00 8.57 23.46 Node 10 30.0 0 20.25 46.97 Node 11 20.00 18.56 31.50 Node 12 12.00 6.40 21.53 Node 15 16.00 12.00 30.65 Node 16 22.00 21.98 35.54 Node 17 18.00 14.21 32.99 Node 19 10.00 9.96 18.88 Node 21 8.00 7.05 13.21 Node 23 13.00 7.35 20.22 Node 24 7.00 5.74 12.26 Total 200.00 152.71 338.08 The locations of origins (counties) and potential shelter nodes are shown in Figure 3 1. The original data for each link in LeBlanc et al. (1975 ) consist of two parameters flow travel ti me and capacity. As shown in Table 3 2, the threshold of maximum travel time from each origin to any shelter is set to be slightly greater than the maximum travel time from the origin when all origins have the highest demand simultaneously and shelters are located at all potential sites, i.e., This is to ensure that at least a feasible shelter location plan exists. PAGE 63 63 Figure 3 1. Sioux Falls network with origin nodes and potential shelter sites PAGE 64 64 Table 3 2. Maximum travel time for each county to shelters County Maximum Travel Time (min) Node 2 44 Node 3 36 Node 5 48 Node 6 52 Node 7 38 Node 9 90 Node 10 90 Node 11 92 Node 12 36 Node 15 95 Node 16 65 Node 17 100 Node 19 100 Node 21 36 Node 23 36 Node 24 36 For simplification, it is assumed that the background traffic in the network is zero. The dispersion parameter in the CDA model is assumed to be 0.1. The fixed costs associated with establishing shelters at potential sites are shown in Table 3 3 and the variable cost to operate shelters is assumed to be 100 per unit capacity. It is further assume d that capacity of all the buses available in different counties is same and is assumed to be 30. PAGE 65 65 Table 3 3. Fixed cost associated with establishing the shelters at potential sites Potential Shelter Sites Fixed Cost Node 1 4100 Node 4 4200 Node 8 4600 Node 13 4200 Node 14 4400 Node 18 4500 Node 20 4100 Node 22 4300 Table 3 4 compares the results for the nominal, robust and worst case plans for the optimal shelter plans in the network. Robust plan is obtained using the degree of pessimism, while the nominal plan is obtained with (deterministic demands with nominal value). The worst case plan is the optimal plan against deterministic demand with all highest values. It is equivalent, in this particular example, to solve the robust form ulation with the degree of pessimism, Table 3 4. Comparison of nominal, robust and worst case plans Nominal Plan Robust Plan Worst Case Plan Shelter Locations Capacity Shelter Locations Capacity Shelter Locations Capacity Node 4 2098.8 Node 4 2400.9 Node 1 900.9 Node 18 2258.1 Node 14 1217.7 Node 4 1432.2 Node 22 1643.1 Node 18 2852.1 Node 8 1265.4 Node 22 1730.4 Node 13 1059.6 Node 14 1270.2 Node 18 2423.4 Node 22 1791.3 Total Capacity Estimate 6 0 00.00 8201.10 10143.00 Total Cost Estimate 613,0 00.00 837 510 .00 1 044,600.00 Reliability of Shelter Capacity Provision 4.20 % 90.40 % 100.00 % Probability of Meeting Travel Time Requirement 90.52 % 98.82 % 100.00 % PAGE 66 66 In order to evaluate the performa nce of these three different plans, travel demand vectors are randomly generated by assuming that there is an equal probability of occurrence of the nominal, low and high demand scenarios at each origin. For each sampled demand vector, the CDA model is sol ved to evaluate whether the shelter capacities are sufficient and the travel times to shelters are less than the thresholds under three different plans. The percent times that these two requirements are met are reported in Table 3 4. Note that a sufficient large number of demand vectors are drawn such that the variance of the estimate of the reliability of shelter capacity provision or the reliability of meeting travel time requirement is less than 0.1. D ifference in the total capacity and cost estimates fo r the nominal, robust and worst case plan can be observed from Table 3 4 The worst case plan achieves 100% reliability in the capacity provision and meeting the travel time requirement. However, it is expensive and may be too conservative. On the other ha nd, the robust plan, saving 20 % in the total cost, achieves nearly 90% and 99% chance in providing sufficient capacity and meeting the travel time requirements. The nominal plan saves 4 2 % in the total cost, but achieves only 4% reliability of sufficient c apacity provision. This underscores the importance of accommodating the demand uncertainty in evacuation planning. Table 3 5 compares the nominal and worst case plans against t he robust plans with varying the degree of pessimism In general, as the degree of pessimism increases, the capacity provision and travel time guarantee becomes more and more reliable while the total cost increases. It can be seen that robust plans with a large PAGE 67 67 enough degree of pessimism e.g., provides a good cost saving and a sup erior performance with the reliability of 90% or more. Note that the degree of pessimism reflects the risk taking attitude of the decision maker. However, without doing a posterior simulation as above, it is difficult to decide the appropriate value for t he degree of pessimism It is recommended to start with a half of the number of the origins with uncertain demand, i.e., for the numerical example. Our future research is to examine how to construct an uncertainty set to explicitly ensure the performance of the resulting robust plan. Table 3 5. Reliability of cost estimate and travel time constraint with varying degree of pessimism Degree of Pessimism Total Cost Estimate Savings in Cost (Compared to the Worst Case Plan) Total Shelter Capacity Estimate Reliability of Shelter Capacity Provision Probability of Meeting Travel Time Requirement 0 (nominal case) 613,000 .00 4 1.3 % 6000.00 4.17 % 90.52 % 1 672,970.10 35.6 % 6599.70 23.07 % 94.78 % 2 726,539.50 30 .4 % 7094.40 42.13 % 97.04 % 3 771,420.00 26.2 % 7543.20 64.37 % 97.25 % 4 808,109.80 22.6 % 7907.10 82.38 % 98.37 % 5 837,510.20 19.8 % 8201.10 90.40 % 98.82 % 6 880,099.90 15.7 % 8586.00 92.15 % 99.87 % 7 926,329.70 11.3 % 9048.30 97.22 % 99.92 % 8 938,360.50 10.2 % 9168.60 98.13 % 100.00 % 9 955,529.70 8.5 % 9294.30 98.63 % 100.00 % 10 975,390.00 6.6 % 9492.90 98.75 % 100.00 % 16 ( worst case) 1,044,600.00 10143.00 100.00 % 100.00 % PAGE 68 68 3.6 Summary The chapter focuses on regional level transit evacuation planning and develops robust shelter location model to determine optimal shelter locations and their capacities under demand uncertaint y. The proposed model is formulated as a mathematical program with complementarity constraints, in which the planning authority determines the shelter locations along with their capacities, while the transit agencies in different counties choose the shelte rs and routes to evacuate. A cutting plane algorithm was proposed to solve the problem. Numerical results from the Sioux Falls network verify the importance of considering demand uncertainty and using a robust approach for evacuation planning. The worst ca se plan can be 100% reliable in satisfying the transit dependent demand and meeting the travel time requirement under all demand scenarios, but is much more expensive and very conservative. On the other hand, a nominal plan may be less expensive, but is mu ch less reliable too. In contrast, the robust plan can be significantly economical and produces nearly the same level of performance. As the degree of pessimism increases, the robust plans become more reliable and expensive as well. Depending on decision m degree of pessimism can be chosen. Although only demand uncertainty is incorporated in this model, the proposed model framework is general enough to accommodate other types of uncertainty, such as c apacity uncertainty. As part of future work, we plan to implement the proposed model and solution algor ithm for larger scale networks. Regional coordination is an important aspect of mass evacuation transit planning. Consequently, it is important to create coordinated transit evacuation plan to evacuate transit dependent residents more PAGE 69 69 effectively. In future, the proposed model could be extended to incorporate the coordination among the transit agencies in different counties PAGE 70 70 CHAPTER 4 TRANSIT PICK UP LOC ATIONS AND BUS ALLCO ATION WITH DEMAND UNCERTAINTY 4.1 Background This chapter focuses on transit evacuation planning on a local level, i.e., from the perspective of a transit agency in a particular county or city In the previous chapter, we discussed tha t the public shelter location as a critical component for transit evacuation planning. Another critical component of transit evacuation planning is the location of transit pick up points, also known as staging areas, within the affected area. Transit pick up po ints refer to the locations where transit dependent evacuees will assemble and the transit agencies will allocate the available transit vehicles to pick up those evacuees and tra nsport them to public shelters. All the previous studies in the literature on modeling the transit based evacuation (Chapter 2) typically assume that the pick up location s for transit are always known and given, but it is not the case with real life evacuations. Renne et al. (2008) found that out of 26 studied cities that had publis hed their evacuation plans, 20 cities had some provision for transit dependent residents. However, most of the plans did not specify any specific locations for transit pick up points. Evacuation plans for t en cities mentioned pick up points during an evacu ation but did not specify any location, two plans advise residents to wait on main roads, four advise residents to wait along bus routes and only four (New York, Boston, Cleveland, Charlotte) show maps of exact pick up locations. Optimal locations of pick up points can influence the total evacuation time and the efficiency of the evaluation plans. However, no analytical model has been proposed to help local governments and transit agencies with this decision making. This chapter aims to fill this void PAGE 71 71 This chapter presents a robust model for determining the optimal pick up locations and bus allocation for transit based evacuation planning with demand uncertainty. More specifically, for tr ansit based evacuation planning, wil l assemble are determined and allocate transit vehicles to assembly locations to pick up the evacuees and transport them to public shelters. The location of public shelters is assumed to be known and given. We consider that the number of transit dependent evacuees is uncertain and attempt to design a location allocation plan that performs reasonably well in any realization of the uncertain demand. The proposed model is formulated as an integer program and is solved via a cutting plane algorithm. In Section 4.2, modeling considerations are discussed in detail. Section 4.3 then formulates the pick up location model, which is solved using a cutting plane scheme proposed in Section 4.4. Results from a numerical example based on Sioux Falls network to illustrate the effectiveness of proposed model are then presented in Section 4.5. Summary of this chapter is provided in Section 4.6. 4.2 Modeling Considerations 4.2.1 Basic Assumptions Consider a situation when residents are required to evacuate in the event of a disaster. I n response, many residents will use their own private vehicles to evacuate while transit dependent residents will rely on the public transportation system to take them to safe locations. When the evacuation orders are released, it is assumed that transit d ependent evacuees from different residential locations will assemble at designated pick up locations, and allocated transit vehicles will pick up and transport them to public shelters. In this model, we determine the optimal locations of pick up PAGE 72 72 points and allocation of a given number of transit vehicles to pick up evacuees in order to minimize the total evacuati on time. In our modeling framework, it is assumed that evacuees will go to t heir nearest pick up points. It is further assume d that evacuees will a ccumulate at the pick up locations before the start of the evacuation process and the proposed model thus does not take into account delays or times for evacuees to get to the pick up locations. The number of available transit vehicles and their capacitie s are assumed to be known in the model. For allocation, each vehicle is assigned to a particular pick up location. The vehicle can make trips to different public shelters, but it only serves its assigned one pick up point without going to another one. L oca tions of open public shelters and their capacities available for transit dependent evacuees are considered to be known. It is further assume d that transit vehicles, in a limited number, have negligible impact on overall congestion, and thus the travel time between pick up points and public shelters are given and constant. Evacuation planning is multifaceted with a variety of objectives. For this model the location allocation plan is optimized to minimize the maximum total evacuation time realized in the un certainty set. Other objectives can be used, such as minimizing the longest evacuation time, maximizing the total number of people evacuated, minimizing the total number of vehicles needed. According to a survey report (Zhang and He, 2008), the highest pri ority for transit evacuation planning is often to minimize the total evacuation time of all the transit dependent evacuees. PAGE 73 73 4.2.2 Demand Uncertainty N umbers of transit dependent evacuees at different residential locations are considered to be uncertain, and an optimal location allocation plan is designed against the uncertain demands. A similar approach as discussed in Section 3.2.1 is used to model demand uncertainty for transit dependent evacuees Let denote the possible demand value at the resident location or demand point with corresponding to the nominal demand value, i.e., the most likely forecast. Let be a binary variable that indicates whether de mand value is realized at demand point Since, only one demand value can be realized at each demand point, we have Also, since at most demand points are allowed to deviate from their nominal values and repre sents that the nominal demand value is realized at demand point we have Thus, the uncertainty set for the transit dependent evacuation demand at demand points can be characterized as follows: w here is the demand vector realized for the network and is the demand realized at demand point Hereinafter, the su perscript d denotes variables associated with a particular demand vector 4.2.3 List of Notations Below is the list of some useful notations used for the proposed model in this chapter. index for demand points index for shelter locations PAGE 74 74 set of nodes set of links degree of pessimism total number of available buses index for the buses capacity of bus round trip travel time from pick up location to shelter walking distance between the no de pair in the network longest walking distance allowed from any demand point to its nearest pick up location maximum running time allowed for each bus during evacuation capacity of shelter demand vector demand realized for demand point demand for demand point under scenario demand accumulated at the pick up location binary variable indicating which scenario is realized for demand point binary variable in dicating bus allocation decision binary variable indicating evacuees choice of pick up location distance of demand point to its nearest pick up locations binary variable indicating pick up location decision integer variable representing number of trips bus makes between pick up location and shelter PAGE 75 75 Note that, although not all the notations used in this chapter are listed here, they are defined appropriately wherever used. 4.3 Model Formula tion Let den ote the network, where and are the sets of nodes and links in the network respectively. Assume that the transit dependent demand during evacuation is originated from a subset of the nodes, i.e., the set of demand points Next, let denote a set of shelter locations in the network. A binary variable represents the pick up location decision, which is 1 if a pick up point is located at and 0 otherwise. It is assume d that buses are the only publication transportation available for evacuation. Let denote the number of buses available for the evacuation and denote the capacity of bus A binary variable represents the bus allocation decision, which is 1 if bus is assigned to pick up point and 0 otherwise We use a n integer variable to represent the number of trips bus needs to make between pick up point to shelter We further use a binary variable choice of pick up points. The variable is 1 if pick up point is nearest to the demand point and is thus selected. Otherwise, it is 0. Let denote the distance of demand point to its nearest pick up location and the demand accumulated at the pick up location Using the above notatio ns, a robust transit pick up location and bus allocation model or RTPL can be formulated as follows: RTPL PAGE 76 76 s.t. (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) (4.9) (4.10) (4.11) (4.12) integer where is the round trip travel time from a pick up point to a shelter ; is the capacity of public shelter available for transit dependent evacuees; is the distance between the node pair in the network and is an input to the model; is the longest walking distance allowed from any dem and point to its nearest pick up point; is the PAGE 77 77 maximum running time allowed for each bus during evacuation; and is a sufficiently large number. The objective of RTPL is to minimize the total evacuation time. Constraint (4.1) ensures all the evacuees under all demand scenarios realized from the uncertainty set can be transported by buses. Constraint (4.2) requires the total number of evacuees transported to a shelter to be less than the available capacity at that shelter. Constraint (4.3) ensu res that each bus is assigned to only one particular pick up point. Constraint (4.4) implies that the node a bus is assigned to is a pick up point. Constraint (4.5) specifies that a bus can make trips to different shelters but it only goes to its assigned pick up point. Constraints (4.6) (4.7) calculate the walking distance of a demand point to its nearest pick up point. Constraint (4.8) assumes that all the evacuees from a particular demand point go to the same nearest pick up point. Constraint (4.9) ensur es that the node where evacuees assemble is a pick up point. Constraint (4.10) is to estimate the total number of evacuees assembling at each pick up point. Constraint (4.11) requires that the walking distance from each demand point to its nearest pick up point should be less than the maximum allowed. Finally, constraint (4.12) imposes an upper bound on the total running time of each bus during evacuation. Note that constraints ( 4. 6) (4.9) and ( 4. 11) can be further simplified as the following equivalent set of constraints. We present as above to facilitate the understanding of the model structure. PAGE 78 78 where In summary, the output of the above model includes th e optimal locations of pick up points, i.e., allocation of buses to pick up points, i.e., the number of trips made by each bus, i.e., up points, i.e., and the total demand accumulated at each pick up point, i.e., 4.4 Solution Algorithm As formulated, RTPL is a mixed integer program. The number of demand vectors in the uncertainty set can be extremely large, which renders RTPL difficult to solve. A cutting plane scheme ( Lawphongpanich and Hearn, 2004) is develope d to solve RTPL. More specifically, a sequence of restricted RTPL problems are solved with respect to a subset of demand vectors that contains a much smaller number of demand vectors and then another problem is solved to generate demand vectors one at a time as needed to expand The restricted RTPL problem (R RTPL) can be written as follows: R RTPL s.t. (4.13) (4.14) integer (4.2) (4.9), (4.11) and (4.12) PAGE 79 79 Compared with RTPL, the above formulation contain s a less number of constraints, and is thus much easier to solve. Let solve the R RTPL globally with the optimal objective value as Then the solution also solves the original RTPL problem if the bus allocation pl an is sufficient to transport the demands at pick up points under all demand scenarios in Mathematically, this can be verified by solving the following worst case demand problem (WCD): WCD s.t. (4.15) (4.16) (4.17) (4.18) (4.18) (4.20) (4.21) where denote the excess demand at pick up point and is a sufficiently large number. The objective of WCD is to find a demand vector that maximizes the total excess demand, i.e., the number of evacuees not transported by the current bus allocation plan. Constraints (4.15) (4.18) are to calculate the excess demand at pick up point i. PAGE 80 80 Con straint (4.19) calculates the total number of evacuees at each pick up point. Constraints (4.20) (4.21) ensure that the demand vector belongs to the demand uncertainty set. If the optimal objective value of WCD is no greater than zero, then the current bus allocation plan is sufficient, i.e., is the optimal solution to the original RTPL problem. Otherwise, an improved solution may be obtained by expanding the demand vector set where is the solution to the above WCD. B elow is the procedure of the cutting plane scheme outlined above. Cutting Plane Algorithm : Step 0: Choose an initial demand vector Set and Step 1: Solve the R RTPL problem with a subset of demand vectors and let denote the resulting optimal solution. Step 2: Solve the WCD problem associated with the current location of pick up points and the bus allocation Let denote the optimal solutio n and denote the corresponding objective value. Step 3: If stop and ( is an optimal solution to the RTPL problem. Otherwise, set and go to step 1. Because t he number of demand vectors in the uncertainty set is finite, the cutting plane algorithm terminates after a finite number of iterations. 4.5 Numerical Example This section presents results from solving the proposed model on the Sioux Falls network (LeBlan c et al., 1975). T he cutting plane algorithm is implemented in GAMS PAGE 81 81 (Brooke et al., 1992) where the R RTPL and WCD problems are solved using CPLEX (CPLEX, 2008). For the Sioux Falls network, the original data for each link consist of two parameters and the flow travel time and capacity. Mass evacuation is considered for a hypothetical disaster in the region. As shown in Figure 4 1, we assume that nodes 1 12 and 16 18 are the demand points and public shelters are located at nodes 13, 20, 21 and 22. It is further assume d that the planning agency has estimated three possible demand values at each demand point i.e., nominal, low and high. The nominal demand is reported in Table 4 1 and the other two demand values, low and h igh, are generated by multiplying the nominal demand by a uniformly distributed number between (0.5, 1) and (1.5, 2). The shelter location and capacity are reported in Table 4 2. It is further assumed that there are 10 buses available for evacuation and ea ch has a capacity of 30. The maximum allowable running time of each bus is 180 min and the maximum allowable walking time from a demand point to a pick up point is 5 min. PAGE 82 82 Figure 4 1. Sioux Falls network with demand points and known shelter locations PAGE 83 83 Tab le 4 1. Demand values at each demand point Demand Points Nominal Demand Low Demand High Demand Node 1 60.00 35.15 109.20 Node 2 42.00 38.70 66.35 Node 3 40.00 31.00 65.00 Node 4 46.00 29.92 84.38 Node 5 38.00 24.55 65.28 Node 6 50.00 30.60 84.00 Nod e 7 34.00 22.95 56.98 Node 8 44.00 40.84 68.90 Node 9 44.00 23.48 69.30 Node 10 52.00 39.00 93.32 Node 11 48.00 47.96 91.95 Node 12 40.00 31.58 64.62 Node 16 36.00 35.84 65.98 Node 17 26.00 22.90 49.10 Node 18 30.00 16.96 49.55 Total Demand 630.00 471.43 1083.91 Table 4 2. Available capacity of public shelters Public Shelters Available Capacity Node 13 240.00 Node 20 333.00 Node 21 360.00 Node 22 300.00 Total Capacity 1230.00 Tables 4 3 and 4 4 present the optimal locations of pick up poin ts and allocation of bus trips for the nominal, robust and worst case plans. The r obust plan is obtained using the degree of pessimism, while the nominal plan is obtained with (deterministic demand with nominal value). The worst case plan is the op timal plan against deterministic high demand at each demand point. It is equivalent, in this particular example, to solving the proposed model with the degree of pessimism PAGE 84 84 In order to evaluate the performance of these three different plans, 1 000 dema nd vectors are randomly generated by assuming that for each demand point, there is an equal probability of occurrence for the nominal, low and high demand values For each sampled demand vector, it is then evaluate d that whether the bus allocation is suff icient to transport all the evacuees at pick up point s. The probability of meeting the demand is calculated accordingly and reported in Table 4 3 Table 4 3. Pick up points under nominal, robust and worst case plans Pick up Locations Nominal Plan Robu st Plan Worst Case Plan Node 3 Node 3 Node 3 Node 6 Node 6 Node 6 Node 10 Node 10 Node 10 Node 18 Node 17 Node 12 Node 18 Node 17 Node 18 Total Evacuation Time (min) 612.00 1128.80 1234.20 Probability of Meeting Demand with Allo cated Bus Trips 2.13 % 97.94 % 100.00 % PAGE 85 85 Table 4 4. Bus allocation under nominal, robust and worst case plans Nominal Plan Robust Plan Worst Case Plan Bus # Trips Pick up Location Shelter Bus # Trips Pick up Location Shelter Bus # Trips Pick up Location Shelter 1 5 Node 18 Node 20 1 2 Node 6 Node 20 1 5 Node 10 Node 22 2 7 Node 3 Node 13 2 4 Node 10 Node 22 2 3 Node 6 Node 21 3 5 Node 10 Node 22 2 1 Node 10 Node 21 3 4 Node 18 Node 20 4 2 Node 6 Node 20 3 4 Node 10 Node 22 4 3 Node 6 Node 20 5 4 Node 6 Node 20 4 7 Node 3 Node 13 5 1 Node 3 Node 13 5 4 Node 6 Node 20 5 1 Node 3 Node 21 6 1 Node 17 Node 20 6 4 Node 6 Node 20 6 2 Node 17 Node 22 7 7 Node 3 Node 13 7 4 Node 18 Node 20 8 4 Node 17 Node 22 8 3 Node 6 Node 21 9 3 Nod e 12 Node 21 9 1 Node 3 Node 13 10 3 Node 10 Node 21 9 2 Node 3 Node 21 10 1 Node 10 Node 22 10 1 Node 17 Node 21 From Table 4 3, the difference in total evacuation time and the reliability of the nominal, robust and worst case plans c an be observed The worst case plan is 100% reliable in meeting the demand realized under uncertainty but the estimated total evacuation time (and the required bus running time, and the associated resource) is very high. On the other hand, the robust plan requires less total evacuation time, saving approximately 9% of the evacuation time (and the associated resource) while achieving nearly the same level of reliability in meeting the realized demand. The nominal plan yields a lot of saving in total evacuati on time, but meets the evacuation demand with a probability of 2%. Such an exceptionally low reliability underscores the importance of accommodating the demand uncertainty in evacuation planning. Table 4 5 compares the nominal and worst case plans against the robust plans with varying the degree of pessimism In general, the probability of a robust plan in meeting the demand increases with the increase in the degree of pessimism but the total evacuation time estimate also increases. As previously mentioned, controls the PAGE 86 86 degree of conservatism in the solution. An appropriate value of degree of pessimism T able 4 5. Reliability of plans with varying degree of pessimism Degree of Pessimism Total Evacuation Time (min) Probability of Meeting Travel Demand 0 (nominal case) 612.00 2.13 % 1 782.00 34.88 % 2 965.60 76.42 % 3 1128.80 97.94 % 4 11 81.00 99.59 % 5 1227.40 100.00 % 6 1227.40 100.00 % 7 1227.40 100.00 % 8 1234.20 100.00 % 15(worst case) 1234.20 100.00 % 4.6 Summary This chapter proposed a robust optimization approach to determine the optimal locations of transit pick up points and t rip allocations for buses in transit based evacuation planning under demand uncertainty. The proposed model was formulated as a mixed integer program and solved by a cutting plane scheme. Numerical results from the Sioux Falls network illustrate that the p roposed robust model produced a less conservative but reliable plan, compared with the one optimized against the worst case demand. In addition, importance of incorporating the demand uncertainty in evacuation planning was also demonstrated. PAGE 87 87 Although only demand uncertainty is considered, the proposed model framework is general enough to accommodate other types of uncertainty. For future studies, we plan to implement the proposed model and solution algorithm for a large scale realistic network. In addition, model can be further extend ed to capture the time dependent nature of evacuation into the framework. PAGE 88 88 CHAPTER 5 INTEGRATED APPROACH FOR PUBLIC SHELTER S AND TRANSIT PICK UP LOCATIONS WITH DEMAND UNCERTAI NTY The previous chapter discussed a modeling approa ch to determine the optimal pick up locations for transit evacuation, where the locations of public shelters are assumed to be known and given However, along with pick up locations, it is critical that the shelter location decision be made carefully based on the different residential locations of transit dependent evacuees. For instance, studies have shown that households with higher income tend to evacuate to motels/hotels and households with higher education and/or, with pets and children are more likely to stay with families and friends (Whitehead et al., 2000). It is found that the low income and minority families or elderly people are the ones who are more likely to evacuate to shelters (Mileti et al., 1992; Whitehead et al., 2000; Willigen et al., 200 2). In general, it can be said that significant portion of evacuee population that are more likely to stay at public shelters are transit dependent residents. Therefore it would be more realistic to determine the optimal shelter locations based on the resi dential locations of transit dependent evacuees, which in turn could potentially affect the optimal locations of transit pick up points. In other words, optimal locations of shelters, residential locations of transit dependent evacuees, and optimal transit pick up locations could be mutually interconnected. Consequently, it is imperative to consider an integrated approach to model shelter locations and pick up locations simultaneously for transit evacuation Thus, in this chapter, the modeling approach for transit evacuation planning is further enhanced and extend ed it in two aspects: we determine both pick up and shelter locations under demand uncertainty and consider a two stage robust optimization approach to achieve flexible and less conservative results PAGE 89 89 For the remain der of this chapter, Section 5.1 describes the model considerations, which are fairly similar to those discussed in Chapter 4. Section 5. 2 presents the model formulation based on single stage robust optimization approach. Section 5.3 then p resents the two stage robust optimization approach. An efficient solution algorithm to solve two stage problem is proposed in Section 5. 4 A numerical example is given in Section 5.5 to illustrate the effectiveness of proposed model and compare the results from the two approaches. Summary of this chapter is provided in Section 5. 6 5.1 Model Considerations 5.1.1 Basic Assumptions For transit evacuation, it is assume d that the transit dependent evacuees will assemble at designated pick up locations and the transit veh icles will pick up and transport them to the public shelters. But, now the location of public shelters is also considered a decision variable, unlike the pick up location model discussed in Chapter 4. Thus our aim is to determine the optimal shelter locati ons, pick up locations, allocation of given transit vehicles and the capacities of public shelters. In our modeling framework, pick up locations are modeled in a way similar to the pick up location problem of Chapter 4. It is assume d that the evacuees from different demand points go to the nearest pick up locations and assemble there before the start of evacuation process. For modeling the public shelters, we consider that the potential sites where shelters can be located are given and known and there are a maximum number of shelters which can be opened out of all the potential sites based on severity of the disaster. The number of available transit vehicles and their capacities are assumed to be known and each transit vehicle is assigned to a particular pi ck up location and the PAGE 90 90 vehicle can make trips to different shelters but serves only its assigned pick up location. It is further assume d that transit vehicles, in a limited number, have negligible impact on overall congestion, and thus the travel time betw een pick up points and public shelters are given and constant. N umbers of transit dependent evacuees at different residential locations are considered to be uncertain. A similar approach as discussed earlier is used to model demand uncertainty and the same uncertainty set is used to characterize the tran sit dependent evacuation demand, i.e., where is the demand vector realized for the network and is the demand realized at demand poin t Let denote the network, where and are the sets of nodes and links in the network respectively. Assume that the transit dependent demand during evacuation is originated from a subset of the nodes, i.e., the set of demand points Next, let denote a set of potential sites to locate shelters in the network. A binary variable represents the pick up location decision, which is 1 if a pick up point is located at and 0 otherwise. A binary variable represents the shelter location decision, which is 1 if a shelter is located at node and 0 otherwise. Let denote the number of buses available for the evacuation and denote the capacity of bus A binary variable represents the bus allo cation decision, which is 1 if bus is assigned to pick up point and 0 otherwise. We use a n integer variable to represent the number of trips bus needs to make between pick up location to shelter We further use a binary variable up PAGE 91 91 points. The variable is 1 if pick up point is nearest to the demand point and is thus selected; otherwise, it is 0. Let denote the distance of demand point to its nearest pick up location Furthermore, is the demand accumulated at the pick up location and denote the capacity of shelter 5.1.2 List of Notations Below is the list of some useful notations used for the proposed model in this chapter. index for demand points index for shelter locations set of nodes set of links degree of pessimism total number of available buses index for the buses capacity of bus round trip travel time from pick up location to shelter walking distance between the node pair in the network longest walking distance allowed from any demand point to its nearest pick up location maximum running time allowed for each bus during evacuation demand vector demand realized for demand point demand for demand point under scenario PAGE 92 92 demand accumulated at the pick up location binary variable indicating which scenario is realized for demand point binary variable indicating bus allocation decision binary variable indicating evacuees choice of pick up location distance of demand point to its nearest pick up locations capacity of shelter binary variable indicating she lter location decision binary variable indicating pick up location decision integer variable representing number of trips bus makes between pick up location and shelter Note that, although not all the not ations used i n this cha pter are listed here, they are defined appropriately wherever used. 5.2 Robust Optimization Approach The robust pick up and shelter locations (RPSL) can be obtained by solving the following mathematical program: RPSL s.t. (5.1) (5.2) (5.3) (5.4) PAGE 93 93 (5.5) (5.6) (5.7) (5.8) (5.9) (5.10) (5.11) (5.12 ) (5.13 ) (5.14) integer where is the round trip travel time from a pick up point to a sh elter ; is the distance between the node pair in the network and is an input to the model; is the longest walking distance allowed from any demand point to its nearest pick up point; is the maximum running time allowed for each bus during evacuation; is the maximum number of shelter which can be opened; and are sufficiently large numbers. In the above RPSL, the objective function is to minimize the total evacuation time. Constraint (5.1) ensures that all the evacuees under each demand scenario realized from the uncertainty set can be transported by the allocated buses Constraint (5.2) requires the available capacity of public shelter to be greater than the total number of PAGE 94 94 evacuees transported to that particular shelter. Constraint (5.3) ensures that each bus is assigned to only one particular pick up point. Constraint (5.4) implies that the node a bus is assigned to is a pick up point. Constraint (5.5) specifies that a bus can make trips to different shelters but it only goes to its assigned pick up point. Constraint (5.6) specifies that a bus make trips to only open shelters. Constraint (5.7) imposes a bound on the maximum number of shelters which can be opened. Constraint s (5.8) (5.9) calculate the w alking distance of a demand point to its nearest pick up location. Constraint (5.10) assumes that all the evacuees from a particular demand point go to the same nearest pick up location. Constraint (5.11) ensures that the node where evacuees assemble is a pick up location. Constraint (5.1 2 ) requires that the walking distance from each demand point to its nearest pick up location should be less than the maximum required. Constraint (5.1 3 ) estimates the total number of evacuees assembling at each pick up loca tion. Finally, constraint (5.14) imposes a n upper bound on the total running of each bus during evacuation. The above model is a straightforward extension of the pick up location model proposed in Chapter 4. The model output includes the optimal locations of public shelters, i.e., capacit ies of public shelters, i.e., optimal locations of pick up points, i.e., allocation of buses to pick up points, i.e., the number of trips made by each bus, i.e., up points, i.e., and the total demand accumulated at each pick up point, i.e., The RPSL model can be solved using a similar cutting plane scheme as proposed in Section 4.4. One of the modeling limitations of the above single stage robust optimization is to make al l the decisions before the realization of uncertain parameters, and thus the PAGE 95 95 solution tends to be conservative. This is a disadvantage of single stage robust optimization in the situations when the decisions can be made sequentially and some of the decisio n variables can adjust themselves, to some extent, to the true value of the uncertain data. The proposed single stage robust optimization model (RSPL) does not consider the fact that in transit evacuation situation, decisions are made at different stages a nd consequently produces too conservative estimates. In order to address this issue and obtain a less conservative solution, a two stage robust optimization approach is proposed next for optimal shelter and pick up locations for transit evacuation planning 5.3 Two Stage Robust Optimization Approach Robust optimization is a recent approach to optimization under uncertainty. To hedge against uncertainty, a robust optimization approach constructs a single solution that is feasible for all the possible realization of the uncertain data. However, it is possible that because a robust (single stage) optimization approach tries to optimize over the worst case scenario, it may produce conservative solutions. In addition, in single stage robust optimization approaches, d ecision maker has to obtain a solution before the real values of uncertain parameters are observed and therefore do not allow for recourse actions that may be based on realized values of some of these parameters. This disadvantage prevents its application to the problems where decisions are made sequentially. To address this drawback of single stage robust optimization, two stage robust optimization (more general, multi stage) also known as robust adjustable or adaptable optimization, has been introduced an d studied (e.g., Ben Tal et al., 2004). Two stage robust optimization, an extension of robust optimization, aims to solve mathematical PAGE 96 96 programming problems where the data is uncertain and decisions can be made at different points in time, thus producing so lutions that are less conservative in nature than produced by one stage robust optimization. It is possible that in some cases, one stage and two stage robust optimization approaches are equivalent. However, the reason to explore two stage approach is to e nable more flexibility in cases when the one stage approach is unjustifiably conservative. The basic idea behind the two stage robust optimization approach is to consider two set of variables, such that the first set of variables are determined before the realization of uncertainty, and the second set of variables are derived after the uncertainty is revealed. Such a framework enables the decision makers to fully take advantage of the revealed information to make less conservative decisions. Two stage robus t optimization is becoming more popular for decision making under uncertainty and some of the recent applications include network/transportation problems (e.g., Atamturk and Zhang, 2007; Ordonez and Zha o, 2007; Gabrel et al., 2011; Z eng, 2011), portfolio o ptimization (Takeda, 2008) and power system control and scheduling (Zhao and Zeng, 2010; Bertsimas et al., 2011; Jiang et al., 2011). 5.3.1 General Modeling Approach for Two Stage Problem In this sub section, a general modeling approach for two stage robust opti mization problem (e.g., Zeng, 2011) is provided to facilitate the understanding of our proposed model. Let us consider a linear optimization problem formulated as: PAGE 97 97 where, is the first stage and is the second stage variables respectively, is the first stage cost, is the second stage cost, is the uncertain parameter belonging to uncer tainty set is a polyhedron described by a finite list of inequalities. In a two stage problem, the objective is to minimize the total cost, which includes first stage cost and worst case second stage cost over the uncertainty set. The general f orm of the two stage robust optimization formulation can thus be written as: Once the first stage decisions are obtained, an d the realization of the uncertainty is known, the second stage problem can be defined as: Therefore, the two stage problem can also be written as follows: 5.3.2 Two Stage Robust Formulation We now extend our single stage robust optimization formulation, i.e., RPSL, and propose a two stage robust optimization formulation. In the context of pick up and shelter location problem and w ith the uncertainty set for the demand, note that the whole decision process can be decomposed into two steps. In the first stage, optimal locations of pick up points and public shelters are determined and once the realized demands are known, bus allocatio n and shelter capac ities are decided in the second stage. PAGE 98 98 The model considerations are same as discussed above The only difference in the modeling framework is that rather than having one set of variables, now there are second stage variables correspondin g to each demand vector in the uncertainty set. Thus, the set of constraints which involves second stage variables are written for each demand vector in the set. Optimal locations of public shelters, i.e., optimal locations of pick up points, i.e., up points, i.e., are the first stage decision variables. Allocation of buses to pick up points, i.e., the number of trips made by each bus, i.e., capacity of public shelters, i.e., and the total demand accumulate d at each pick up point, i.e., are the second stage variables in the model. The superscript denotes variables associated with a particular demand vector The two stage robust pick up and shelter location problem or TS RPSL can be formulated as the following mathematical program: TS RPSL s.t. (5.15) (5.16) (5.17) (5.18) (5.19) (5.20) (5.21) PAGE 99 99 (5.22) (5.23) (5.24) (5.25) (5.26 ) (5.27 ) (5.28) integer The objective function for the above TS RPSL model is to minimize the worst case minimum evacuation time realized over the uncertainty set. Constraints are defined similar to the corresponding set of constraints in the RPSL model. Constraints (5.15) (5.2 0 ) and (5.27) (5.28) involves the second stage variables, which are now corresponding to each demand vector in the uncertainty set The s et of constraints (5.21 ) (5.26) whic h involves only the first stage variables, remains unchanged 5.4 A Cutting Plane Algorithm for the Two Stage Problem 5.4.1 Solving Two Stage Robust Model Although two stage robust optimization approach enables more flexibility compared to the single stage approach however there is a price to pay for this flexibility, i.e., it results in additional complexity. In general, two stage robust optimization leads to computationally intractable problems and are very difficult to solve As shown in Ben Tal et al. (2004), even a simple two stage robust optimization problem could be NP hard and become intractable. In the literature, t wo types of solution PAGE 100 100 approaches have been proposed to solve two stage robust optimization problem. The first one is approximation algorithm wh ich assumes the second stage variables to be simple functions (e.g., affine functions) of the uncertainty (e.g., Bertsimas et al., 2008). Howeve r, when the structure of second stage decision problem is not simple, such an approximation approach may not be applica ble. The second type of algorithms are various cutting plane based methods e.g., Benders dual cutting plane algorithms (Gabrel et al., 2011; Jiang et al., 2011), c olumn and constraint generation algorithm (Zhao and Zeng, 2010; Zeng, 2011), and Kell gorithm (Thiele et al., 2009 ), which seek to derive the exact solutions. In this section, a cutting plane scheme to solve two stage robust problem (TS RPSL) is presented The algorithm is implemented in a master subproblem framework, in which a sequ ence of master problem s are solved with respect to a subset of demand vectors and then another subproblem is solved to generate demand vectors one at a time as needed to expand The master probl em is to find the optimal first stage variables i.e., pick up locations and shelter locations, for a set of known demand vector s. The master problem (MP) is written as follows: MP s.t. (5.29) (5.30) (5.31) PAGE 101 101 (5.32) (5.33) (5.34) (5.35) (5.36 ) (5.37 ) integer and (5.21) (5.26 ) The objective of above MP model is to find a pick up and shelter location plan, which minimizes where is the maximum of minimum evacuation times corresponding to a set of known demand vectors in Constrain t (5.29) ensures that is the maximum of evacuation times for different demand vectors. The expression on the right hand side of constraint (5.29) is the total evacuation time value. S ince the objective is to minimize the value of the solution will for ce the total evacuation time to be minimum. More specifically, constraint (5.29) ensures the total evacuation time value is the minimum corresponding to the demand vector for which the constraint is binding but not necessarily the minimum evacuation time for the rest demand vectors in the set. Since we only require minimizing the maximum of minimum evacuation time, this formulation will give us the desired location plan. As formulated, the above master problem is a mixed integer program (MIP) and c an be so lved directly using CPLEX (CPLEX, 2008) PAGE 102 102 Let be the optimal solution to the above master problem with the objective function value being The objective function value is the minimum evacuation time for the given set of demand vecto rs The obtained solution will be an optimal solution to the original TS RPSL, if for the pick up ( and shelter location ( plan, there does not exists any other demand vector in for which the minimum evacuation time is higher than Mathemati cally, this can be verified by solving the following subproblem (SP). SP s.t. (5.38 ) (5.39 ) (5.40 ) (5.41 ) (5.42 ) (5.43 ) (5.44 ) (5.45 ) (5.46 ) (5.47 ) (5.48 ) PAGE 103 103 integer The purpose of solving the above subproblem is to find a demand vector which creates trouble with the curr ent pick up and shelter location plan. More specifically t he objective is to find a demand vector that maximizes the minimum evacuation time for the given pick up and shelter locations If the optimal objective value of above subproblem is less than (i.e., the objective value of master problem) then is the optimal pick up and shelter location plan to the TS RPSL problem. Otherwise, an improved location plan can be obtained by expanding the set of demand vectors where is the solution to the above subproblem. Because the number of demand vectors in the uncertainty set is finite, the cutting plane algorithm terminates after a finite number of iterations. Below is the formal solution procedure of the cutting plane al gorithm outlined above. Cutting Plane Algorithm : Step 0: Choose an initial demand vector Set and Step 1: Solve the master problem (MP) with a subset of demand vectors and let denote the r esulting optimal solution with objective function value as Step 2: Solve the subproblem associated with the pick up location and shelter location (Section 5.5.2) and let denote the optimal solution and denote the corresponding objective value. Step 3: If stop and is an optimal solution to the TS RPSL problem. Otherwise, set and go to step 1. Furthermore, once the optimal pick up ( a nd shelter locations ( are obtained for the above TS RSPL problem, the second stage variables for any particular realized PAGE 104 104 demand can be obtained by solving the following minimization problem (MinET). This problem can be solved directly and the s olution is very easy to obtain. MinET s.t. (5.49 ) integer and (5.38) (5.44) 5.4.2 Solving Subproblem O btaining the solution to the subproblem is the most computationally difficult part of the proposed algorithm. The objective of SP model has a max min structure which makes it difficult to solve. There are two levels in the SP formulation and it is not poss ible to obtain an equivalent one level formulation. Although it is difficult to calculate the objective function value itself, it is easier to obtain an approximation on the lower bound of the objective for the subproblem. More specifically, we make use of the dual problem associated with the subproblem to closely approximate the value of objective function. Since, subproblem is an integer program, we use the dual of the LP relaxation of the subproblem. In the current subproblem formulation, allocation of buses to pick up points, i.e., and the number of trips made by each bus, i.e., are both integer variables. Therefore, instead of writing the dual of the LP relaxation of the current subproblem, we will reformulate the subproblem such that the relaxation of integer constraints is minimized. PAGE 105 105 For given pick up and shelter locations, and a known demand vector the problem to comp ute the minimum evacuation time (MinET) is re formulated as: MinET II s.t. (5.50 ) (5.51 ) (5.52 ) (5.53 ) (5.54 ) (5.55 ) where is the number of trips from pick up location to shelter The objective is to minimize the evacuation time. Constraint (5.50 ) ensures that all the evacuees at each pick up location ca n be transported by t he total trips. Constraint (5.51 ) requires the available capacity of public shelter to be greater than the total number of evacuees transported to that part icular shelter. Constraint (5.52 ) specifies that the trips can be made from a l ocation only when it is assigned as pick up point. Constraint (5.53 ) specifies that trips are made to only open shelters. Constraint (5.54 ) estimates the total number of evacuees assembling at each pick up location. Finally, constrai nt (5.55 ) imposes a n up per bound on the average running time of each bus, where is the total number of available buses. Since the objective here is to find the minimum evacuation time, we do not exactly need the specific allocation of each bus. The information about PAGE 106 106 total num ber of trips between pick up locations and shelter is sufficient in this case, in order to calculate the minimum evacuation time. Now, the dual of the above LP relaxation of above problem can be written as: D MinET II s.t. (5.56 ) (5.57 ) (5.58 ) and unrestricted w here are the dual variable ass ociated with the equations (5.50) (5.55 ) respectively. Now, the dual problem of the relaxed sub problem ( D RSP) can thus be written as below : D RSP s.t. (5. 59 ) (5.60 ) (5.61 ) (5.62 ) (5.63 ) PAGE 107 107 (5.64 ) and unrestricted Note that, the max min formulation for subproblem is converted to max max formulation using the dual of the relaxed subprob lem. The above D RSP model is a mixed integer nonlinear program (MINLP), which is easier to solve and can be solved directly using DICOPT a framework for solving MINLP (e.g., Grossmann et al., 2008 ). By solving D RSP problem, we obtain a maximum objective function and a demand vector corresponding to that maximum value. N ote that t he solution to D RSP is a lower bound on the objective function value of the subproblem). In order to calculate the minimum evacuation time corresponding to obtained demand vect or, we solve the MinET problem This minimum evacuation time value is then compared with the current objective value of the master problem to check if we need to include this obtained dem and vector into the set or not. R elaxing the integer constraint s and then computing the dual objective value can produce weak bound for the original problem. Therefore after solving the D RSP, and obtaining an corresponding minimum evacuation time less than the current MP objective does not guarantee that there does not exi st any other demand vector that can lead to a higher minimum evacuation time. Thus, in order to be on the safer side, a sampling procedure is done to ensure a good solution. After solving the D RSP problem, if we get a demand vector for which the minimum e vacuation time is higher than the objective of master problem, we include that demand vector in the set and go to Step 2 of the cutting plane scheme. Otherwise, 100 demand vectors are randomly PAGE 108 108 generated from the uncertainty set. For each sampled demand ve c tor, MinET problem is solved to calculate the minimum evacuation time. If the minimum evacuation time for any of the sampled demand vector exceeds t he objective of master problem, this particular demand vector is included in the set. If none of the 100 ran dom demand vectors produces minimum evacuation time larger than MP objective value, we terminate the cutting plane algorithm. 5.5 Numerical Example This section presents results from solving the proposed model on the Sioux Falls network (LeBlanc et al., 197 5). T he cutting plane algorithm is implemented in GAMS (Brooke et al., 1992) where the master problem (MP) and dual of the relaxed subproblem (D RSP) are solved using CPLEX (CPLEX, 2008) and DICOPT (Grossmann et al., 2008) respectively. Figure 5 1 displays th e underlying network in which we assume that nodes 1 12, 16 and 18 are the demand points and eight potential sites to locate shelter in the network as nodes 13 15 and 20 24. Out of the eight potential sites, it is assume d that a maximum of six sites can be opened as shelters. It is further assume d that the planning agency has estimated three possible demand values at each demand point i.e., nominal, low and high. The nominal demand is reported in Table 5 1 and the other two demand values, low and high, are generated by multiplying the nominal demand by a uniformly distributed number between (0.5, 1) and (1.5, 2). It is further assumed that there are 10 buses available for evacuation and each has a capacity of 30. The maximum allowable running time of each b us is 180 min and the maximum allowable walking time from a demand point to a pick up point is 5 min. PAGE 109 109 Figure 5 1. Sioux Falls network for demand points and potential shelter locations PAGE 110 110 Table 5 1. Demand values at each demand point Demand Points Nominal Demand Low Demand High Demand Node 1 50.00 29.29 78.27 Node 2 42.00 38.71 76.43 Node 3 40.00 31.01 63.19 Node 4 46.00 29.93 74.75 Node 5 38.00 24.55 69.71 Node 6 50.00 30.60 85.88 Node 7 34.00 22.95 57.11 Node 8 44.00 40.84 73.73 Node 9 44.00 23.4 8 68.89 Node 10 52.00 39.00 81.90 Node 11 48.00 47.95 86.14 Node 12 40.00 31.57 76.62 Node 16 36.00 35.84 58.15 Node 18 30.00 26.43 54.98 Total Demand 594.00 452.15 1005.75 Table 5 2 present s the optimal pick up locations and shelter locations for the nominal, one stage robust, two stage robust and worst case plans. The one stage robust and two stage robust plans are obtained using the degree of pessimism while the nominal plan is obtained with (determinis tic demand with nominal value). The worst case plan is the optimal plan against deterministic high demand at each demand point. It is equivalent, in this particular example, to solving the proposed model with the degree of pessimism, PAGE 111 111 Table 5 2. Pick up and shelter locations under nominal, one stage robust two stage robust and worst case plans Nominal Plan One Stage Robust Plan Two Stage Robust Plan Worst Case Plan Pick up Locations S helter Locations Pick up Locations Shelter Locations Pick up Locations Shelter Locations Pick up Locations Shelter Locations Node 3 Node 13 Node 3 Node 13 Node 3 Node 13 Node 3 Node 13 Node 6 Node 14 Node 6 Node 14 Node 6 Node 14 Node 6 Node 14 No de 10 Node 15 Node 9 Node 15 Node 8 Node 15 Node 8 Node 15 Node 12 Node 20 Node 11 Node 20 Node 10 Node 20 Node 10 Node 20 Node 18 Node 18 Node 12 Node 24 Node 11 Node 24 Node 18 Node 12 Node 18 Total Evacuation Ti me (mins) 765.00 1150.00 965.00 1260.00 PAGE 112 112 The total evacuation time reported in Table 5 2 is the mini mum evacuation time for nominal, one stage robust and worst case plan, whereas for the two stage robust plan, it is the maximum value of minimum evacuati on time among all the realized demand vectors. We can observe the difference in total evacuation time of the different plans and the fact that two stage plan produce less conservative estimate as compared to the one stage robust plan. It can be noted that for an degree of pessimism, the two stage robust plan result in 16% or more reduction in total evacuation time compared to the one stage robust plan. Moreover, it has already been demonstrated that even though the nominal plan results in a lot of redu ction i n total evacuation time, it achieves very less reliability in terms of meeting the demand. Next, in order to further demonstrate the flexibility of the two stage robust plan compared to the one stage robust plan, 1000 travel demand vectors are rando mly generated from the uncertainty set for degree of pessimism, For the one stage robust approach, the total evacuation time will be same for all the generated demand vectors as reported in Table 5 2. On the other hand, according to the two stage robust plan w e have different allocation of bus trips and shelter c apacity estimate corresponding to each demand vector. Therefore, for the two stage plan the MinET problem is solved for each sampled demand vector, to compute the corresponding total evacuation time. Figure 5 2 shows the plot of the total minimum evacuati on time for the sampled demand vectors corresponding to the one stage and two stage robust plans. I t can be noted that the total evacuation time in the case of two stage robust plan is always less than the evac uation time for one stage plan. In addition, s ince the bus PAGE 113 113 allocation and shelter capacity estimate in two stage robust plan is corresponding to the particular demand vector, the reliability of two stage robust plan will always be 100%. Figure 5 2 Plot of total evacuation time for sampled demand v ectors Table 5 3 compares the one stage ( the min imum evacuation time) and two stage ( the maximum of the min imum evacuation time) robust plans with v arying the degree of pessimism In general, the total evacuation time estimate increases with increase in t he degree of pessimism, As p reviously mentioned, controls the degree of conservatism in the solution. An appropriate value of degree of pessimism can be chosen depending 600 700 800 900 1000 1100 1200 0 200 400 600 800 1000 Total Evacuation Time (Min) Demand Vectors TwoStage Robust Plan OneStage Robust Plan PAGE 114 11 4 Table 5 3 Evacuation time of one stage and two stage robust plans with varying degree of pessimism Degree of Pessimism Total Evacuation Time (min) One Stage Robust Plan Two Stage Robust Plan 1 915.00 820.00 2 1055.00 900.00 3 1150.00 965.00 4 1175.00 1010.00 5 1175.00 1055.00 6 1175.00 1095.00 7 1180.00 1120.00 8 1180.00 1150.00 5.6 Summary In this chapter, an integrated approach to model public shelters and pick up locations for transit based evacuation planning under demand uncertainty is proposed. The model is formulated as a mixed integer program to determine the optimal shelter locations, pick up locations, allocation of available transit vehicles and capacities of public shelters. Two approaches, robust (single stage) optimization and two stage robust optimization are pres ented. The first approach i.e., single stage robust optimization, determines all the decision variables before the realization of uncertain demand and constructs a solution that is feasible for all the possible realization of demands. In contrast, in the t wo stage robust optimization approach, some of the decision variables are determined before the uncertain demands are realized while the other are decided once the realized demands are known. A c utting plane scheme is proposed to solve both the approaches. Numerical results from Sioux Falls network demonstrate that the robust plans (one stage and two stage) yields better and reliable solutions. In addition, it was PAGE 115 115 demonstrated that two stage robust optimization approach offers more flexibility to the decis ion maker than the single stage approach and produce less conservative solutions. However, two stage robust optimization problems are difficult to solve compared to one stage robust optimization problems. It is believe d that the two stage robust optimizati on approach proposed in this chapter offers a new perspective on evacuation pla nning under demand uncertainty. Our future work will involve demonstrating the effectiveness and usefulness of proposed model and solution algorithm to a realistic large scale network Proposed model can be further refined and extended to be more realistic and practical. For example, we assume that transit vehicles can make trip s only from its assigned pick up location. In future, better allocation of transit vehicle can be considered and possibly allowing for trip chaining. To model pick up locations, we assume that evacuees will go the nearest pick up location, which might not be the case in real world. Evacuees may choose some other nearby pick up location which is not necessarily closest e.g., a more familiar location. In future, evacuee behavior in choosing the pick up location can be studied Although only demand uncertaint y is considered, the proposed model framework is general enough to accomm odate other types of uncertainties In addition, model can be further extended to capture the time dependent nature of evacuation into the framework. PAGE 116 116 CHAPTER 6 CONCLUSION This disser tation develops robust optimization models for effective and reliable transit based evacuation planning in order to evacuate transit dependent population during emergency situations. Two critical decision making components of transit evacuation planning ar e studied, namely, pick up locations ( staging areas) and public shelter locations. The proposed models take into account the demand uncertainty, which refers to the uncertainty in the number of transit dependent evacuees. In particular, three models are pr oposed in this dissertation. The first model focuses on transit evacuation planning on a regional level, with transit agencies in different counties performing their own evacuation. T he proposed model is to determine the optimal locations of public shelter s and their capacities, from a given set of potential sites for transit operations under demand uncertainty. The model is formulated as mathematical program with complementarity constraints and is solved by a cutting plane scheme. A n umerical example on th e Sioux Falls network demonstrates that robust plans are able to achieve nearly the same level of performance with a significant lower cost as compared to a conservative plan, which assumes the highest demand of each origin node Next, a decision support model is developed for a local level transit based evacuation planning under demand uncertainty A robust optimization model is proposed to locate the pick up points for evacuees to assemble, and allocate available transit vehicles to transport the assembl ed evacuees between the pick up locations and different public shelters. For this model, the location s of public shelters are assumed to be given and known. The model is formulated as a mixed integer linear program and is PAGE 117 117 solved via a cutting plane scheme. The numerical results based on the Sioux Falls network demonstrates that the robust plan yields lower total evacuation time and is reliable in serving the realized evacuee demand. In the last model, an integrated approach to model pick up and shelter loca tions simultaneously for transit evacuation planning under demand uncertainty is considered. The developed model determines the optimal shelter locations, pick up locations, allocation of given transit vehicles and capacities of public shelters. Two approa ches, single stage and two stage robust optimization are proposed. The former is believed to produce conservative solutions, while the latter enables decision makers to fully take advantage of the revealed information to make less conservative decisions. I n a two stage robust optimization approach, the whole decision process is decomposed into two stages. In the first stage, optimal locations of pick up points and public shelters are determined and once the realized demands are known, the allocation of give n transit vehicles and shelter capacities are decided in the second stage. A c utting plane scheme is used to solve both proposed approaches and results from the Sioux Falls network demonstrate that the robust plans (one stage and two stage) yield better an d reliable solutions. In addition, the two stage robust optimization approach offers more flexibility to the decision maker than the single stage approach and produce s less conservative solutions. In order to reduce the impact of disasters in the future, p lanners need to de vote at least as much attention to transit based evacuation as to auto based evacuation in emergency response plans, and develop ways to optimize the use of available resources during emergencies. In addition, it is important to understan d why many PAGE 118 118 people ignore evacuation orders, e.g., some face logistical or transportation problem to evacuate, some had nowhere to go and are fearful of shelter conditions etc. Addressing these kinds of issues in an emergency response plan, pertaining to es pecially transit dependent residents can increase evacuation rates and decrease damage to the society. To our best knowledge, no analytical approach is available in the literature that determines the optimal shelter locations and transit pick up locations during transit evacuation. The proposed models are very useful to planners, transit providers and emergency management officials for a better and effective transit evacuation planning. For emergency management and transit management agencies, how to effici ently operate the transit system while leaving no citizens behind is a key question. The results from this research provide strategies for efficient transit operation that address these issues. The proposed models have the capability to be applied in real world evacuation situations. In the future, the proposed models can be further refined and extended, to be more realistic and practical. Although only demand uncertainty is incorporated in the proposed models, the proposed framework is general enough to ac commodate other types of uncertainty as well. Our f uture work may expand the current models to address a variety of uncertainties from both the demand side and supply side. Another future direction is to extend the models to capture the time dependent natu re of evacuation and incorporate delays and waiting times for evacuees to get to the pick up locations. Future research may also examine the possibility of integrating both auto based and the transit based aspects of evacuation. PAGE 119 119 In summary, this dissertat ion develops transit based evacuation plans to evacuate those who depend on public transportation to safe areas reliably and as fast as possible during an emergency. Optimization model s are developed for transit based evacuation planning which are more eff icient and reliable in the face of demand uncertainty T he proposed modeling frameworks are feasible and promising for the robust and effective transit operations during evacuation PAGE 120 120 LIST OF REFERENCES AARP (2006a). AARP Offers Tips to Help Older American s Prepare for Emergencies. American Association of Retired Persons. AARP (2006b). We Can Do Better: Lessons Learned for Protecting Older Persons in Disasters. American Association of Retired Persons. Abdelgawad, H., & Abdulhai, B. (2010). Managing l arge sc ale multimodal e mergency e vacuations. Journal of T ransportation Safety and Securit y, 2(2), 122 151. Andradottir, S. (1998). Simulation optimization. In J. Banks, editor, Handbooks of Simulation: Principles, Methodology, Advances, Applications, and Practic e, chapter 9. John Wiley & Sons, New York. ARC (1992). Guidelines for Hurricane Evacuation Shelter Selection. American Red Cross, ARC 4496. Atamturk, A., & Zhang, M. (2007). Two stage robust network flow and design under demand uncertainty. Operations Rese arch, 55 (4), 662 673. Bailey, D., Swiacki, S., Byrnes, A., Buckley, J., King, D., Piper, P., Marino, M., Mundle, S., Pierlott, G., & Lynd, A. (2007). Transportation Equity and Emergency Preparedness: A Review of the Practices of State Departments of Transp ortation, Metropolitan Planning Organizations, and Transit Agencies in 20 Metropolitan Areas. Final Report FTA PA 26 8001 2007.U.S.Department of Transportation, Federal Transit Administration, Washington, D.C. Baker, E.J. ( 1991 ) Hurricane evacuation behav ior. International Journal of Mass Emergencies and Disasters, 9 (2), 287 310. Bazaraa, M. S., Sherali, H.D., & Shetty, C.M. (2006).Nonlinear Programming: Theory and Algorithms New York : John Wiley & Sons Ben Tal, A. & Nemirovski, A. (2002). Robust optimi zation methodology and applications. Mathematical Programming Series B, 92 453 480. Ben Tal, A., Goryashko A. Guslitzer E., & Nemirovski A. ( 2004 ) Adjustable robust solutions of uncertain linear programs. Mathematical Programming 99 351 376. Berl i n, S. (1998). Transit systems, personnel rise to e mergencies. Passenger Transport, 56 (50), p. 12. Bertsimas, D., Brown, D. B. and Caramanis, C. (2008). Theory and applications of robust optimization. PAGE 121 121 Bertismas, D., Litvinov, E., Sun, X. A., Zhao, J., & Zh eng, T. (2011). Adaptive robust optimization for the security constrained unit commitment problem Technical report, 2011. Bertsimas, D. & Sim, M. (2003). Robust discrete optimization and network flows. Mathematical Programming, 98, 49 71. Birge, J. R., & and Louveaux, F. (19 97). Introduction to Stochastic Programming. Springer Verlag. Bourne, J. K. (2004). Gone with the water. National Geographic. Brooke, A., Kendirck, D., & Meeraus, A. (1992). Press, South San Francis co, California. Carson, Y., & Maria, A. (1997). Simulation Optimization: Methods and Applications, Proceedings of the 1998 Winter Simulation Conference, 118 126. Chen, M. (2005). Traffic signal timing for urban evacuation. Master thesis, University of Mary land, College Park. Chen, M., Chen, L., & Hooks, M. (2007). Traffic signal timing for urban evacuation. Journal of Urban Planning and Development, American Society of Civil Engineers, 133(1), 30 42. Chen, X. & Zhan, F. ( 2008 ) Agent based modeling and sim ulation of urban evacuation: Relative effectiveness of simultaneous and staged evacuation strategies. Journal of the Operational Research Society, 59(1), 25 33. Chiu, Y.C. (2004). Traffic scheduling simulation and assignment for area wide evacuation. IEEE Intelligent Transportation Systems Conference, Washington, D.C. Chiu, Y.C., Villalobos, J., Gautam, B., &Zhen, H. (2006).Optimal scheduling of evacuation operations.CD ROM. Transportation Research Board of the National Academies, Washington D.C. Coulbourne W.L., Tezak, E.S., & McAllister, T.P. (2002). Design guidelines for community shelters for extreme wind events. Journal of Architectural Engineering, 8(2), 69 77. CPLEX (2008). ILOG Inc. Sunnyvale, California. Dash, N. & Gladwin, H. ( 2007 ) Evacuation d ecision making and behavioral responses: Individual and household. National Hazards Review, 8 (3), 69 77. Dow, K. & Cutter, S. L. ( 1997 ) Crying wolf: Repeat responses to hurricane evacuation orders. Coastal Management, 26 237 251. PAGE 122 122 Drud, A. (1992). CONOPT A Large Scale GRG Code. ORSA Journal on Computing, 6, 207 216. FDOT (2000). Summary. Florida Department of Transportation. FEMA (2000a). Southeast United States Hurricane Evacuation Traffic St udy Executive Summary. Final Report for Southeast United States Hurricane Evacuation Traffic Study, Federal Emergency Management Agency, Washington D.C. FEMA (2000b). Design and Construction Guidance for Community Shelters. Federal Emergency Management Age ncy, Washington D.C. FEMA (2008). Producing Emergency Plans: A Guide for All Hazard Emergency Operations Planning. Federal Emergency Management Agency, Washington D.C. Fischett, M. (2001). Drowning New Orleans. Scientific American. Fletcher, R., & Leyffer, S. (2004). Solving mathematical program with complementarity constraints for nonlinear programs. Optimization Methods and Software, 19 (1), 15 40. Fu, H., & Wilmot, C. G. (2004). A sequential logit dynamic travel demand model for hurricane evacuation. Tran sportation Research Record, 1882, 19 26. Fu, M. (1994). Optimization via simulation: A review. Annals of Operations Research, 53, 199 248. Fu, M. (2002). Optimization for simulation: Theory vs. practice. INFORMS Journal on Computing, 14(3), 192 215. Gabrel V., Lacroix, M., Murat, C., & Remli, N. (2011). Robust location transportation problems under uncertain demands Technical report submitted to Discrete Applied Mathematics, Lamsade, Universite Paris Dauphine, 2011. Grossmann, I. E., Viswanathan, J., Vec chietti, A., Raman, R., & Kalvelagen, E. (2008). DICOPT. In: GAMS The Solver Manuals. GAMS Corporation, Washington. Harri c tivates snow, flood contingency p lans. Passenger Transport, 54 (10), p. 4. Hess, D. B., & Gotham J. C. (2007). Multi modal mass evacuation in upstate New York: A review of disaster p lans. Journal of Homeland Security and Emergency Management, 4 (3), 1 19. Higgins, L. L., Hickman, M. D., & Weatherby, C. A. (1999).Role of P ublic T ransportation O perations in Emergency M anagement: Research R eport. Texas Transportation Institute Report No. 1834 2, College Station, TX, Texas A&M University. PAGE 123 123 Huibregtse, O.L., Hoogendoorn, S.P.,& Bliemer, M.C. J. (2010). Optimization of evacuation measures under uncertainty. Proce edings of the 89th Annual Meeting of the Transportation Research Board, Washington, D.C. Infanger, G. (1994). Planning under Uncertainty: Solving Large Scale Stochastic Linear Programs. Boyd and Fraser. Jenkins, B. M., & Winslow, F. E. (2003). Saving City Lifelines: Lessons Learned in the 9 11 Terrorist Attacks. Mineta Transportation Institute, San Jose University. Jenkins, P., Laska, S., & W illiamson, G. (2007). Connecting future evacuation to current recovery: Saving the lives of o lder people in the n ext c atastrophe. Generations, 31 (4), 49 52. Jiang, R., Zhang, M., Li, G., & Guan, Y. (2011). Two stage robust power grid optimization problem Technical report, 2011. Johnson, J., & Zeigler, D. (1986). Modeling evacuation behavior during the Three Mile Island r eactor crisis. Socio Economic Planning Sciences, 20(3), 165 171. Jones, J. A., Walton, F., Smith, J. D., & Wolshon, B. (2008). Assessment of Emergency Response Planning and Implementation in the Aftermath of Major Natural Disasters and Technological Accide nts. Sandia National Laboratories Report No. SAND2007 1776P, U.S. Nuclear Regulatory Commission Report No. NUREG/CR 6981, NRC Division of Preparedness and Response, Washington, D.C. Kall, P., & Wallace, S. (1994). Stochastic Programming. John Wiley & Sons. Kongsomsaksakul, S., Yang, C. & Chen, A. (2005). Shelter location allocation model for flood evacuation planning. Journal of the Eastern Asia Society for Transportation Studies, 6 4237 4252. Lawphongpanich, S. & Hearn, D. W. (2004). An MPEC approach to second best toll pricing. Mathematical Programming Series B, 7 33 55. LeBlanc, L.J., Morlok, E.K., & Pierskalla, W.P. (1975). An efficient approach to solving the road network equilibrium traffic assignment problem. Transport Research, 9, 309 318. Lindel l, M.K. & Prater, C. S. (2007). Critical behavioral assumptions in evacuation analysis for private vehicles: Examples from hurricane research and planning. Journal of Urban Planning and Development, 133 ( 1 ) 18 29. Litma n, T. ( 2006 ) Lessons from Katrina and Rita: W hat major disasters can teach transportation planners? Journal of Transportation Engineering, 132 11 18. PAGE 124 124 Liu, H. X., Ban J. X., Ma, W. & Mirchandani, P. (2007). Model reference adaptive control framework for real time tr affic management under emergency evacuation. ASCE Journal of Urban Planning and Development, 133 (1), 43 50. Lou, Y., Yin, Y., & Lawphongpanich, S. (2009). Robust approach to discrete network design with demand uncertainty. Transportation Research Record, 2 090 86 94. Luo, Z. Q., Pang, J. S., & Ralph, D. (1996). Mathematical Programs with Equilibrium Constraints. New York: Cambridge University Press. Margulis, L., Charosky P., & Fernandez J. (2003) Hurricane Evacuation Decision Support Model for Bus Dispa tch. Fourth International Latin American and Caribbean Conference on Engineering and Technology, Puerto Rico. McLoughlin, D. (1985). A framework for integrated emergency management. Public Administration Review, 45 165 172. Mileti, D. S., Sorensen, J. H., Toward an explanation of mass care shelter use in evacuations. International Journal of Mass Emergencies and Disasters, 10(1), 25 42. Mitchell, S.W. & Radwan, E. (2006). Heuristic prioritization of emergency evacuation staging to re duce clearance time. Presented at the 85th annual meeting of Transportation Research Board, Washington D.C. Nakanishi, Y., Kim, K., Ulusoy, Y., & Bata, A. (2003). Assessing emergency preparedness of transit agencies: A focus performance indicators. Transpo rtation Research Board, 82nd Annual Meeting, Washington D.C. Ng, M. W., Park, J., & Waller, S. T. (2010). A hybrid bi level model for the optimal shelter assignment in emergency evacuations. Computer Aided Civil and Infrastructure Engineering, 25 (8), 547 5 56. Ng, M. W. & Waller, S. T. (2010). Reliable evacuation p lanning via demand inflation and s upply d eflation Transportation Research Part E: Logi stics and Transportation Review, 46(6), 1086 1094. Ordonez, F., & Zhao, J. (2007). Robust capacity expansion of network flows. Networks, 50 (2), 136 145. PBS&J (2000). Reverse Lane Standards and ITS Strategies Southeast United States Hurricane Study. Technical Memorandum 3, Final Report, Federal Emergency Management Agency, Tallahassee, Florida. Pel, A.J., Bliemer M.C.J., & Hoogendoorn, S.P. (2011). A review on travel behavior modeling in dynamic traffic simulation models for evacuations. Transportation, 1 27. PAGE 125 125 Pel, A. J., Hoogendoorn, S. P., & Bliemer, M. C. J. (2010).Impact of variations in travel demand and n etw ork supply factors for evacuation s tudies. Transportation Research Record: Journal of the Transportation Research Board 2196, 45 55 Public Transit (1992). Public Transit Lifeline in Time of Community Crisis. Brochure published by the American Public Tr ansit Association, Washington, D.C. Prekopa, A. (1995). Stochastic Programming. Kluwer. Quarantelli, E. (1998). What is a Disaster? Perspectives on the Question New York: Routledge. Renne, J. (2006). Evacuation and equity: A post Katrina New Orleans diar y. Planning, 72(5), 33 42. Renne, J., Sanchez, T., & Litman, T. (2008). National Study on Carless and Special Needs Evacuation Planning: A Literature Review Prepared for the Federa l Transit Administration. Rockafellar, R.T., 2001. Optimization under Uncertaint y: Lecture Notes. University of Washington Sayyady, F. (2007). Optimizing the use of public transit system in no notice evacuations in urban areas. Master Thesis, Mississippi State University. Sbyati, H. & Mahmassani, H. S. ( 2006 ) .Optimal scheduling of evacuation operations. Transportation Research Record, 1964 238 246. Schee l, H., & Scholtes, S. (2000). Mathematical programs with complementarity constraints: S tationarity, op timality, and sensitivity. Mathematics of Operations Research, 25 1 22. Sherali, H. D., Carter, T. B. & Hobeika, A. G. (1991).A location allocation model and algorithm for evacuation planning under hurricane/flood conditions. Transportation Research Part B, 25(6), 439 452. Sisiopiku, V. P., Jones, S. L., Sullivan, A. J., Patharkar, S. S. & Tang, X. (2004). Regional Traffic Simulation for Emergency Preparedness. Technical report, Department of Civil and Environmental Engineering, University of Alabama. Song R., He, S ., & Zhang, L. (2009). Optimum transit operations during emergency e vacuations. Journal of Transportation Systems Engineering and Information Technology 9 (6), 154 160. Takeda A., Taguchi, S., & Tutuncu, R. H. (2008). Adjustable robust optimization models fora nonlinear two period system. Journal of Optimization Theory and Applications, 136 (2), 275 295 PAGE 126 126 Thiele, A., Terry, T., & Epelman M. (2009) Robust linear optimization wit h recourse Tech nical Report. Theodoulou, G., & Wolshon, B. (2004). Modeling and analyses of freeway contraflow to improve future evacuations. Proceedings of the 83rd Transportation Research Board, Washington, DC. TRB (2008).The Role of Transit in Emergenc y Evacuation. Transportation Research Board, Special Report 294. Tuydes, H. & Ziliaskopoulos, A. (2005). The network evacuation problem and solution algorithms. Presented at the INFORMS Annual Meeting, San Francisco. U.S. Census (2000). United States Cens us Bureau. Retrieved from http://www.census.gov/main/www/cen2000.html U.S. Department of Transportation and the U.S. Department of Homeland Security. (2006). Congress on Catastrophic Hurricane Evacuation Plan Evaluation: A Report to Congress. Retrieved fro m http://www.fhwa.dot.gov/reports/hurricanevacuation USDOT (2003). Public Safety and Security Brochure. U.S. Department of Transportation Whitehead, J. C., Edwards, B., Willigen, M. V. Maiolo, J. & Wilson, K. ( 2000 ) Heading for higher ground factors a ffecting real and hypothetical hurricane evacuation behavior. Environmental Hazards, 2 133 142. Willigen, M. V., Edwards, T., Edwards, B., & Hessee, S. (2002). Riding out the storm: Experiences of the physically disabled during hurricanes Bonnie, Dennis, and Floyd. Natural Hazards Review, 3, 98 106. Wolshon, B. (2001). One way out: contra flow freeway operation for hurricane evacuation. Natural Hazards Review, 2(3), 105 112. Wolshon, B., Urbina, E., & Levitan, M. (2001). National Review of Hurricane Evacua tion Plans and Policies. Research report, Louisiana State University Hurricane Center. Yao, T., Mandala, R. S., & Chung, B. D. (2009). Evacuation transportation planning u nder uncertainty: A robust o ptimization approach. Networks and Spatial Economics, 9 (2 ), 171 189. Yazici, M. A. & Ozbay K. (2007). Impact of p robabilistic road capacity c onstraints on the s patial distribution of h urricane e vacuation shelter capacities. T ransportation Research Record, 2022 55 62. Yazici, M. A. & Ozbay, K. (2010). Evacuat ion network modeling via dynamic traffic assignment with probabilistic demand and capacity constraints. Transportation Research Record: Journal of the Transportation Research Board, 2196, 11 20. PAGE 127 127 Yazici, M. A. & Ozbay K. ( 2008 ) Evacuation modeling in the United States: Does the demand model choice matter? Transport Reviews, 28(6), 757 779. Yi, W., & Ozdamar, L. (2007). A dynamic logistics coordination model for evacuation and support in disaster response activities. European Journal of Operational Researc h, 179(3), 1177 1193. Yuan, F., Han, L. D., Chin, S. M., & Hwang, H. (2006). A proposed framework for simultaneous optimization of evacuation traffic destination and route assignment. Transportation Research Record: Journal of the Transportation Research B oard, 1964 50 58. Zelinsky, W., & Kosinski L. A. (1991).The Emergency Evacuation of Cities: A Cross National Historical and Geographical Study. Lanham, MD: Rowman & Littlefield. Zeng, B. (2011). Solving two stage robust optimization problems by a constra int and column generation method. Optimization Online Submitted. Zhang, L., & He, S. (2008). Optimum transit operations during the emergency evacuations Technical Report, Institute for Multimodal Transportation. Zhang, L., Lawphongpanich, S., & Yin, Y. (2009). An active set algorithm for discrete network design problems. Transportation and Traffic Theory 2009: Golden Jubilee, 283 300. Zhao, L., & Zeng, B. (2010). Robust unit commitment problem with demand response and wind energy Technical report, Unive rsity of South Florida, 2010. PAGE 128 128 BIOGRAPHICAL SKETCH Ashis h Kulshrestha was born in India in 1985. He studied civil engineering at 2006. Afterwards he joined the Univers ity of Florida, Gainesville for the Ph.D. program in transportation engineering. During his Ph.D. study at the Transportation Research Center in University of Florida, Ashish Kulshrestha was a research assistant under the supervision of Dr. Yafeng Yin and worked on various projects. Ashish Kulshrestha has co authored six papers in transportation journals and conference proceedings and made presentations at various conferences. His primary research interests include transportation system modeling and optimi zation, managed lane operations, traffic control and operations, and travel demand modeling. He received his Doctor of Philosophy degree in December 20 1 1 