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An event driven single game solution for resource allocation in a multi-crisis environment

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An event driven single game solution for resource allocation in a multi-crisis environment
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Shetty, Rashmi S
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game theory
normal form game
N player game
Nash equilibrium
crisis management
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ABSTRACT: The problem of resource allocation and management in the context of multiple crises occurring in an urban environment is challenging. In this thesis, the problem is formulated using game theory and a solution is developed based on the Nash equilibrium to optimize the allocation of resources to the different crisis events in a fair manner considering several constraints such as the availability of resources, the criticality of the events, the amount of resources requested etc. The proposed approach is targeted at managing small to medium level crisis events occurring simultaneously within a specific pre-defined perimeter with the resource allocation centers being located within the same fixed region. The objective is to maximize the utilization of the emergency response units while minimizing the response times. In the proposed model, players represent the crisis events and the strategies correspond to possible allocations.The choice of strategies by each player impacts the decisions of the other players. The Nash equilibrium condition will correspond to the set of strategies chosen by all the players such that the resource allocation optimal for a given player also corresponds to the optimal allocations of the other players. The implementation of the Nash equilibrium condition is based on the Hansen's combinatorial theorem based approximation algorithm. The proposed solution has been implemented using C++ and experimental results are presented for various test cases. Further, metrics are developed for establishing the quality and fairness of the obtained results.
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Thesis (M.S.C.S.)--University of South Florida, 2004.
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Includes bibliographical references.
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by Rashmi S. Shetty.
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An Event Driven Single Game Solution For Resource Allocation In A Multi-Crisi s Environment by Rashmi S. Shetty A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Computer Science Department of Computer Science and Engineering College of Engineering University of South Florida Major Professor: Nagarajan Ranganathan, Ph.D. Sudeep Sarkar, Ph.D. Soontae Kim, Ph.D. Date of Approval November 9, 2004 Keywords: crisis management, nash equilibrium, n player game, normal form gam e, game theory Copyright 2004, Rashmi S. Shetty

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DEDICATION To my Parents and my Sister

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ACKNOWLEDGEMENTS I would like to express my gratitude to my major professor, Dr. N. Ranganathan, for his guidance and support through every step of my work. I would also lik e to thank Narender Hanchate for his valuable insights and advise. Last but not the least, I would like to thank my family and friends who have always been a constant source of support.

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i TABLE OF CONTENTS LIST OF TABLES iii LIST OF FIGURES iv ABSTRACT vi CHAPTER 1 INTRODUCTION 1 1.1 Crisis Management 1 1.2 Crisis Management Systems/Agencies 4 1.2.1 FEMA Federal Emergency Management Agency 4 1.2.2 INFOSPHERE – Sense and Respond Systems 7 1.2.3 CMS (Crisis Management System) 8 1.3 Motivation and Contributions 8 1.4 Thesis Outline 9 CHAPTER 2 RELATED WORK 10 2.1 Types of Algorithmic Approaches to Resource Allocation 12 2.1.1 Dynamic Programming 12 2.1.2 Integer Programming 14 2.1.3 Lagrange Multiplier Method 15 2.1.4 Simulated Annealing 16 2.1.5 Genetic Algorithms 18 2.1.6 Branch and Bound 19 2.1.7 Greedy Algorithm 20 2.1.8 Tabu Search 21 2.2 Why Game Theory? 21 2.3 Game Theoretic Concepts 22 CHAPTER 3 FORMULATION OF A GAME 26 3.1 Crisis Scenario 26 3.2 Modeling of Crisis Scenario as a Noncooperative Strategic Game 28

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ii 3.3 Notations 29 CHAPTER 4 OPTIMAL ALLOCATION MODEL USING NASH EQUILIBRIUM 31 4.1 Structure of a strategy 31 4.2 Generation of Strategies 33 4.2.1 Pruning of Strategies 34 4.2.2 Cost Function 36 4.3 Payoff Modeling 40 4.3.1 Creation of Payoff Matrices 40 4.3.2 The Payoff Function 41 4.4 Algorithm to Approximate Nash Equilibrium 43 4.5 Software Implementation 45 4.5.1 System Input and Output 46 4.5.2 Object-Oriented Design 47 4.5.3 Overview of Classes and Functions 48 CHAPTER 5 EXPERIMENTAL RESULTS 52 5.1 Fairness 53 5.2 Execution Time 57 5.3 Statistical Significance of Experimental Results for Execution T ime 60 CHAPTER 6 CONCLUSIONS AND FUTURE WORK 62 6.1 Conclusions 62 6.2 Future Work 63 REFERENCES 64

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iii LIST OF TABLES Table 3.1 Resource Types and Availability 27 Table 3.2 Crisis Types and Requests 27 Table 3.3 Crisis Priorities 27 Table 3.4 Time (in minutes) Taken to Reach Crises 28 Table 4.1 Overview of Classes Used 48 Table 5.1 Fairness Measures 55 Table 5.2 Regression Analysis Results 60

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iv LIST OF FIGURES Figure 1.1 Crisis Scenario 2 Figure 1.2 FEMA – SLG 101 The Planning Process 5 Figure 1.3 FEMA – SLG 101 Resource Management Organization 6 Figure 1.4 Infosphere – An Overview 7 Figure 2.1 Criteria for Resource Allocation 11 Figure 2.2 Generic Algorithm for Dynamic Programming 13 Figure 2.3 Structure of Simulated Annealing Algorithm 17 Figure 2.4 Structure of a genetic algorithm 18 Figure 2.5 Flow of Branch and Bound Algorithm 20 Figure 4.1 Formulation of a Strategy 32 Figure 4.2 Generation of Strategies 33 Figure 4.3 Algorithm to Generate Strategies 34 Figure 4.4 Recursive Algorithm to Generate Strategy Set for each Cri sis 35 Figure 4.3 Flowchart Illustrating Ordering of Strategies 26 Figure 4.4 Normal Form Game Representation 27 Figure 4.5 Flowchart Illustrating Ordering of Strategies 37 Figure 4.6 Algorithm to Add a Strategy to a Strategy Set 38 Figure 4.7 Normal Form Game Representation 39 Figure 4.8 Structure of Payoff Matrices for Crises 41 Figure 4.9 Algorithm to Approximate Nash Equilibrium 44 Figure 4.10 Overview of Crisis Management System 45

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v Figure 4.11 Workflow Model of the Proposed System 51 Figure 5.1 Effect on Execution Time Due to Increase in Number of Resource Cent ers 58 Figure 5.2 Effect of Increasing Number of Crises on Execution Time 58 Figure 5.3 Percentage Increase in Computation time of Nash Equilibrium 59

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vi AN EVENT DRIVEN SINGLE GAME SOLUTION FOR RESOURCE ALLOCATON IN A MULTI-CRISIS ENVIRONMENT Rashmi S. Shetty ABSTRACT The problem of resource allocation and management in the conte xt of multiple crises occurring in an urban environment is challenging. In this thesis, the problem is formulated using game theory and a solution is developed based on the Nash equilibrium to opt imize the allocation of resources to the different crisis events in a fair manne r considering several constraints such as the availability of resources, the criticality of the eve nts, the amount of resources requested etc. The proposed approach is targeted at managing small to medium lev el crisis events occurring simultaneously within a specific pre-defined perimeter with t he resource allocation centers being located within the same fixed region. The objective is to maxim ize the utilization of the emergency response units while minimizing the response times. In the proposed model, players represent the crisis events and the strategies correspond to pos sible allocations. The choice of strategies by each player impacts the decisions of the other players. The Nash equilibrium condition will correspond to the set of strategies chosen by all the players such that the resource allocation optimal for a given player also corresponds to the opt imal allocations of the other players. The implementation of the Nash equilibrium condition is base d on the Hansen’s combinatorial theorem based approximation algorithm. The proposed so lution has been

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vii implemented using C++ and experimental results are presented for various test cases. Further, metrics are developed for establishing the quality and fairness of the obtained results.

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1 CHAPTER 1 INTRODUCTION Over the decades, crisis management has developed into a complex a nd multifaceted issue. The nature of a crisis or a disaster ranges from natural disasters like hurricanes and earthquakes to man-made crises like plane crashes, terrorist attacks, willful acts of mass destruction, industrial accidents, etc. With the development in infr astructure, the impact on a community in terms of damage to property and loss of lives due to t he occurrence of a crisis necessitates the need for an organized and effective crisis ma nagement system. The scope of a crisis management system includes but is not limited to ris k analysis, sensing, responding, monitoring and mitigating the effects of a crisis. 1.1 Crisis Management Every community is equipped with an array of emergency respons e units to cater to varied crisis scenarios. Effective recovery from a crisi s requires immediate deployment of requested units to the crisis locations. The complexity of this a rises from the heterogeneity of emergency response units, e.g., fire engines, ambulances and police ca rs. Furthermore, these units are distributed over a wide area and control-led by multiple org anizations. Each crisis is unique in its severity, request for number and type of resources, location and potential growth. In the event of multiple simultaneous crises, it is critical to ascertain the severi ty of each crisis and allocate the optimum number and appropriate type of emergency units to each location. Depen ding on the

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2 Fig 1.1 Crisis Scenario

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3 location and nature of the crisis, there may be additional requests for emergency units to prevent further deterioration of a situation. Although, some crises may be more critical than others, all of them need to be serviced immediately to prevent spawning of additi onal crises and further damage. The main aim of allocation of emergency services is maximization of the utility of existing and available emergency response units and minimization of response time to mitigate the effects of one or many crises. Fig 1.1 illustrates a scena rio to facilitate further understanding of the situation. Let us assume a hypothetical crisis scenario in a city. A plane crashes as it lands in the airport. It is a small pas senger plane with 30-35 people including the crew. The crash has affected 2-3 planes parked at their terminals at the airport. A fire breaks out in an apartment on the sixth floor of a high -rise building trapping the people within the apartment. The fire is slowly spreading to other apartments on the same floor and the one above. A demonstration in front of the town hall has turned into a riot. A few members of the riot have become violent and are throwing inflammable objects all ar ound the area. Each of the above incidents qualifies as a crisis although eac h varies in its degree of criticality and the number of resources it requires. The plane crash has the highest priority because of the casualties and the possibility of worsening. The airplanes have fuel in them which is highly inflammable and as other planes have been affected, the explosion has to be controlled before it spreads to the airport and causes further damage. The fire is next in priority and has to be controlled before it consumes the entire building and causes furt her damage to life and property. The riot comes next in terms of criticality. In the scenario, we consider three types of resources available – ambulances from the two hospitals, police cars from the polic e station and fire engines from the fire department. The plane crash and the fire scene would require fire engines and

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4 ambulances. Both the locations might require a few police cars to regulate the population and direct people to safety. The riot scene would require a lot o f police cars to control the crowd, maintain law and order and prevent stampedes. It would also need a fe w ambulances for riot casualties and fire engines to control fires started by rio ters. A lot of factors like availability, distance, traffic conditions, etc determine how many resources are dispatched to the crisis locations and from which resource center. For example, there are tw o hospitals, Hospital A and Hospital B. It is important to note that Hospital A is closer t o the riot than Hospital B. Hence, it is more practical to send more ambulances from Hospital A to the r iot than Hospital B. However, the plane crash would receive more ambulances in total than the riot due to its higher criticality. It is highly possible that there may not be sufficient number of ambulances and fire engines as compared to the requests made by the crisis locations. Hence it is critical to make an optimal allocation of resources as under or over utilization of resources could cost lives. The allocation has to weigh in factors such as criticality of crises, reques t and availability of resources, distance and number of resource centers. 1.2 Crisis Management Systems/Agencies Several agencies have been set up and systems have been des igned to monitor and mitigate the effects of crises. We will review some of the salient features of these agencies and systems. 1.2.1 FEMA Federal Emergency Management Agency The Federal Emergency Management Agency a former independent agency that became part of the new Department of Homeland Security in March 2003 – h as the task of responding to, planning for, recovering from and mitigating against disasters. F EMA can trace its beginnings to the Congressional Act of 1803, a piece of disaster management legislation, which provided assistance to a New Hampshire town following an extensive fir e. In the century that followed, ad hoc legislation was passed more than 100 times in response to hurricanes, earthquake s, floods and

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5 other natural disasters. Over the course of the last few ye ars, nuclear power plants, transportation of hazardous substances and civil defense responsibilities were added to this agency further compounding the complexity of emergency management. According to Federal Emergency Management Agency (FEMA) Str ategic Plan Fiscal Years 2003 – 2008: Crisis/Disaster: Broadly defined to include disasters and emerg encies that may be caused by any natural or man-made event Response: Conducting emergency operations to save lives and property including positioning emergency equipment and supplies; evacuating potential v ictims; providing food, water, shelter, and medical care to those in need; and restori ng critical public services Research Development Validation Maintenance Review Law, Plans, Mutual Aid Agreements and Guidance Conduct Hazard/Risk Analysis Determine Resource B ase Note Special Facets of the Planning Environment Fig 1.2 FEMA – SLG 101 Planning Process (Chapter 2)

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6 Recovery: Rebuilding communities so individuals, businesses, and gover nmental infrastructure can function on their own, return to normalcy, and are protected against future hazards Fig 1.2 shows an excerpt from the State and Local Guide (SLG) 101: Guide for All-Hazard Emergency Operations Planning – Chapter 2. These are guidelines to emergency preparedness. The grey boxes indicate those tasks which are relevant to resourc e allocation. Any proposed solution would require complete information about the resource base, topogra phy, jurisdiction and classification of priority. After collecting information abou t resources and possible hazards, a system needs to have guidelines on distribution of resources. Fig 1.3 g ives a brief outline of the resource management organization as proposed by Chapter 6 of FEMA – SLG 101. The shaded region has relevance to our work in terms of acquiring real-t ime resource updates, prioritization of events and dispatching resources. Resource Manager Needs Analysis Receive requests Prioritize events Pass requests Track request status Report resource status to RM Supply Coordination Procure personnel and units Coordinate transport Coordination Coordinate routing, reception, storage and handling of stocks Fig 1.3 FEMA – SLG 101 Resource Management Organization (Chapter 6)

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7 1.2.2 INFOSPHERE – Sense and Respond Systems The Caltech Infospheres Project [72] is devoted to research on com positional systems or, systems built from interacting components. One of the applications of this project includes a “sense and response” system. The basic purpose of the system is to hold a repository of data from multiple sources and normalize it to a standardized vocabulary Certain conditions are specified that generate system alerts. The system monitors data from v arious institutions and when the specified condition is met, alerts are sent securely to the destination. The system can be programmed for specific applications, for e.g., an airplane switc hes to a different mode when the system detects an equipment malfunction. The data sources monitor ed for specific applications are immense in terms of volume, speed, heterogeneity and distrib uted nature. The proposed system is evaluated on the basis of – frequency of errors, respons e time, computational resources consumed, scalability and ease of adaptation. Fig 1.4 gives an over view of the control flow of the proposed system. Configure a set of sense-respond conditions for an application Monitor data from agencies to detect significant ch anges in data values Detect when specific (alert) conditions are met Disseminate secure alerts to destinations Fig 1.4 Infosphere – An Overview

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8 1.2.3 CMS (Crisis Management System) CMS [71] is a powerful crisis management software develope d by Applied Science Associates, Inc. It has been designed as a tool for marine emer gency response managers to model the impacts and biological effects of spill or a disaster It can be used for training and simulation exercises, cost-benefit analysis and facilitate real-time response to marine disasters. Its interface has been designed for an oil spill, chemical spill, search a nd rescue mission, marine emergency and nuclear disaster. All the machines part of CMS are connect ed to a single resource database and enable immediate access to resource information. It is e quipped with a Geographical Information System (GIS) to obtain real-time environmental inf ormation. It is used for management and distribution of information for an organization handling response and recovery. 1.3 Motivation and Contributions Multi-crisis management is a complex problem encompassing sensi ng, responding and recovering from crises. It is important to set good precedents f or future occurrences to enable a better level of preparedness and response to crises to minimize destruction of life and property. In the previous section, we observe systems and agencies designed to i mprove and enhance the ability of emergency response managers to make effective crisis management decisions. They are primarily protocols or guidelines (like FEMA) for risk anal ysis, resource base analysis, and coordination of personnel and resource units. Some others are ale rting mechanisms (like INFOSPHERE) used to monitor the activities of systems (a house, an airplane, city, etc). Systems like CMS are tools to monitor crises and provide real-time updates on them and the resources sent to mitigate their effects. While ample work has been done in collecting data regarding resources and classification of crises, there has not been significant wor k to automate the allocation of resources to crisis situations. The actual allocation of res ources is made manually based on predefined guidelines and protocols and real-time information gathere d from the scenes and is susceptible to human error.

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9 This work proposes a model that uses the same information and enume rates feasible allocation strategies and determines an optimal set of stra tegies for managing each crisis. The proposed application is based on an optimization algorithm to obtain the be st possible allocation that benefits all crisis locations. The multi-crisis scenari o is modeled as a strategic game [22, 23] where the crisis locations are considered as players and each p ossesses a set of strategies that correspond to allocations from different resource centers. All the crises compete for the same set of resources and their adversarial nature is used to model them as players in a noncooperative game where each player tries to maximize his own utility. Ea ch strategy of a crisis has an associated cost which is a function of resources contributed and the time taken to service a crisis’ request. The strategy sets are inputs to the optimization alg orithm. The algorithm uses an objective function that associates a payoff with each allocat ion based on priority of a crisis, its request, number of resources available and time taken to reac h the crisis. The optimization algorithm produces as an output an allocation of resources from eac h resource center to each of the crises. It is based on the principle of the Nash Equilibrium [22, 23, 69, 70] which provides a socially viable solution whereby any player which deviates f rom the equilibrium strategy will earn less than if it remained in its current strategy. 1.4 Thesis Outline The thesis is organized as follows: Chapter 2 briefly look s at other algorithmic approaches to solving resource allocation problems and the approach us ed in this work. We explain the reasons for adopting game theoretic concepts to our probl em. Chapter 3 introduces us to the formulation of the problem and the algorithmic design. We expl ain the constraints of the problem and the objective function used. Chapter 4 presents the expe rimental results followed by conclusions and an overview of future work in Chapter 5.

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10 CHAPTER 2 RELATED WORK Technological advances over the last few decades have drasti cally decreased the time delay between theoretical progress and its practical impact. O ne such subject which has been widely researched is resource allocation and it has found numerous applications in load distribution [10], production planning [11], computer scheduling [12] and many ot her areas. Since Koopman’s work [5] on the optimal distribution of effort in 1953, a significant number of papers [2, 3, 4, 6, 7] have been published on the subject. The allocation of resources is an optimization problem with the constraint – given a fixed quantity of a resource type, determine its allocation to a set of activities, such that the objective function or ( in our case) the payoff function is optimized. Formally, this can be stated as follows [1]: Resource: minimize ( ) n x x x f ,..., 2 1 Subject to = = n j j N x 1 where 0 j x j = 1, 2, …, n Here, j x represents the amount of the resource that is allocated to act ivity j and f represents the objective function. N represents the total amount of the resource available and n, the total number of activities. The objective value ( ) n x x x f ,..., 2 1 could be a cost, a reward, profit or loss as a result of the allocation. If the resource is divisible, it can be re presented by any nonnegative value

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11 Fig 2.1 Criteria for Resource Allocation and j x is a continuous variable. The resource may not be divisible if it repre sents persons, j j j u x l £ £ j = 1, 2, …, n vehicles, parts and so on. In the context of this work, j x is a discrete variable that can be represented only in nonnegative integer values. Sometimes, lower bounds (other than/greater than 0) and/or upper bounds are imposed such that, it is required to allocat e atleast j l and at most j u resources to activity j [1]. The choices of algorithms for resource allocation depend on how effic iently an objective function can be exploited. Some of the typical forms of objective functions are [1]: Constraints Resource Allocation Objective Function Algorithmic Approaches Discrete/ Continu ous Lower/upper bound Separable Convex Concave Fair Minimax Dynamic Programming [51,52,53] Integer Programming [59,60,61] Lagrange Multiplier [56,57,58] Simulated Annealing [29,30,31] Genetic Algorithm [38,39,40] Branch and Bound [43,44,45] Miscellaneous Algorithms Game theory [73,74,75] Greedy [48,49,50] Tabu Search [65,66,67]

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12 Separable: ( ) = n j j j x f 1 where each j f is a function of one variable Convex [9]: A function f(x) is convex if, on an interval [x, y], for any points 1 a and 2 a in the interval [x, y], ( ) [ ] ( ) ( ) [ ] 2 1 2 1 2 1 2 1 x f x f x x f + £ + Concave [9]: A function f(x) is concave if, on an interval [x, y], for any points 1 a and 2 a in [x, y], the function –f(x) is convex on that interval. Minimax: Minimize ( ) j j j x f max ; and Maximin: maximize ( ) j j j x f min Fair: Minimize g ( ( ) j j j x f max ( ) j j j x f min ), where g ( u, v ) is nondecreasing (respectively, nonincreasing) with respect to u (respectively, v ) 2.1 Types of Algorithmic Approaches to Resource Allocation Several algorithmic solutions and their generalizations have bee n proposed to obtain optimal solutions to the resource allocation problem. In this chapter, we will examine some of the widely used algorithms to solve the resource allocation problem and examine how game theory is well suited for modeling the solution in our case. Although we exami ne algorithmic approaches to solving discrete and continuous resource allocation problems, we em phasize discrete algorithms due to its relevance in this work. 2.1.1 Dynamic Programming Dynamic programming typically applies to optimization problems where we make a set of choices to obtain an optimal solution. There may be several solutions to obtain the optimal value. A dynamic-programming algorithm can be categorized into four steps: Define the structure of an optimal solution Recursively define an optimal solution Compute an optimal solution in a bottom-up manner

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13 Build an optimal solution from the information obtained in the previous steps In [13], a dynamic programming formulation is used for a time-optim al multi-agent task assignment problem. Here, m tasks are to be assigned to n agents, with m n and one agent can perform only one task. A task assignment algorithm for a global optimal task assignment is obtained based on a problem specific recurrence relation derive d using the Principal of Optimality [14]. Next, a dynamic programming styled time-optimal t ask assignment algorithm is constructed since each stage of the algorithm is based on the re currence relation derived earlier. Dynamic programming is similar to the divide-and-conquer problem. H owever, the latter approach is not suitable for cases when there are common subproble ms as it solves them repeatedly. Dynamic programming solves each subproblem just once a nd saves it in a table and thereby avoids recomputation every time the subproblem is encounter ed. However, the disadvantage of dynamic programming is that when it is applie d to any multistage optimization problem, the dimensionality explodes when there are several state variables and each of them has large discretization Fig 2.2 Generic Algorithm for Dynamic Programming [15] Define a non-empty state space X with a finite set of states Define a finite set of actions, U( x ) that can be applied from a state x X Let k {1 …K +1} where k is a stage. We assume that K is larger than the lo ngest optimal path between any two states X Let F = K + 1 where F is the final stage ) ( 1 k k k u x f x = + for k x X and k u U( k x ) where f denotes the state transition equation Let 1 x denote the initial state and g x denote the state we want to reach or the goal state Define an additive loss function L, = + = K k F F k k x l u x l L 1 ) ( ) ( where ) ( F F x l = 0, if G f x x = and ) ( F F x l = otherwise Define a termination action T u U( x ), such that if T u is applied to k x then the action is repeatedly applied until stage K and the state remains in k x until the final stage. Also, ) ( T k u x l = 0 for any k and k x Find K u u ......... 1 that minimizes L

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14 levels. In our case, the state variables would correspond to the number of units in each resource center. Also, the computational cost of the proposed algorithm grows ra pidly with the number of agents making it infeasible for a large number of agents. Inspi te of the curse of dimensionality, we find dynamic programming being applied to a variety of resource a llocation scenarios as described in [51, 52, 53, 54, 55]. 2.1.2 Integer Programming Many optimization problems can be expressed as linear or nonli near programming problems. A linear program is a problem which can be expressed as follows: Minimize cx Subject to Ax = b x 0 where x is a vector of variables to be solved and A is the matrix of coefficients, and c and b are vectors of known coefficients. “ cx ” is referred to as the objective function, and the expression “ Ax = b ” is called a constraint. A nonlinear program is a problem of the form, Minimize f(x) Subject to gi(x) = 0 for i = 1, …, m1 where m1 0 hj(x) 0 for j = m1 + 1, …, m f(x) is a an objective function consisting of several variables and the other two functions are constraints. If the unknown variables are required to be integers, as is the case with this work, then the problem is referred to as integer programming. If t he problem requires only some of the variables to take on integer values, it is called mixed intege r programming. Although this is more realistic, it is harder to solve. Integer programming techni ques can be applied over a substantial range of problem sizes and applications. The work in [16] integer linear programming is used to improve bandwidth efficiency in networks using a segment protection a lgorithm. An active path (AP) in a network is divided into several active segments ( AS) which are protected by backup segments (BS). In case of a failure, the traffic is reroute d through a BS. Integer linear

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15 programming is used to determine an optimal partition of a given AP into ASs and find the corresponding BSs. The work in [17] uses integer linear programmi ng to obtain an optimal allocation of registers for general purpose processors and embe dded systems. Although integer programming techniques are known to provide optimal solutions, in both the works above it has been found that they can be used for medium sized and not large netwo rks (in the first case) and solution times are slow (in the second case). Integer programs are undecidable in the worst case and in some cases found to be NP-Hard. [59, 60, 61, 62, 63, 64] are some works whi ch use pure integer and mixed integer programming techniques for their resource all ocation problems. 2.1.3 Lagrange Multiplier Method In mathematical optimization problems, Lagrange multipliers a re used to deal with problems with constraints. They are used to find the maxima or minim a of a multivariate function subject to a constraint [1]. Optimize f(x, y) Subject to g(x, y) = 0 The Lagrangian is written as, L = f(x, y) + g(x, y) where is a constant called the Lagrange multiplier. According to Lagrange’s multiplier method, the simultaneous c onditions are, ;0 = y x L ;0 = x y L The goal is to find the maximum and minimum values taken on by f along the curve with the constraint on the points, g(x, y) = 0 The Lagrangian approach treats all variables and constraint s in a symmetrical fashion so that problems involving numerous vari ables and constraints can be neatly organized. [18] uses Lagrangian methodology to schedule and al locate resources in a manufacturing unit. The objective is to efficiently use lim ited resources to meet dynamic customer requirements. Factories use a flexible manufactur ing system by using production layouts to simplify production flow lines and increase productivity. S cheduling is used to decide

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16 when to set up a cell for a production lot and the quantity of machines to allocate to the cell. The machine capacities, processing time and the machine type are quantified as constraints. The model uses Lagrange relaxation and forms a dual function by rela xing complicating constraints with Lagrangian multipliers. The original problem is divided into subproblem s which are easier to solve. The model produces schedules which are 16%-29% within optimal. [ 19] is another work using a similar approach to optimally allocate resources in a distributed computing environment. Lagrangian relaxation and subgradient methods are applied to solv e this problem. It was observed that an optimal solution would occur in an earlier iteration without converging. It has been observed that in the works above solutions to subproblems are not f easible and additional heuristics are applied to arrive at feasible schedules. [56, 57, 58] are some other examples of the application of the Lagrange multiplier method to the resource allocation probl em. 2.1.4 Simulated Annealing This algorithm is based on that of Metropolis et al. [9] to f ind an equilibrium configuration of a collection of atoms at a given temperature. In 1973, Pincus et al. [26] drew an analogy between this algorithm and mathematical minimization. It was proposed as a n optimization technique for combinatorial problems by [27]. Simulated annealing is a random sear ch technique which has the advantage of not getting trapped in local minima. It accepts change s that increase and decrease an objective function f An increase in the change is accepted with the probability p [28], where = T f e pd. t is the increase in f and T is a control parameter referred to as the “system temperature”. [41] suggests that the initial temperature 0 T has a significant impact on the performance of the algorithm. The algorithm requires a problem-s pecific annealing schedule, i.e. an initial temperature and the rules for lowering it as the sea rch proceeds. Figure 2.2 shows the structure of a simulated annealing algorithm. One of the major problems in the implementation of simulated annealing lies in the difficulty of drawing an analogy between T and a free parameter in

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17 a real-life problem. Furthermore, staying out of local minima is dependent on the choice of an annealing schedule, number of iterations performed for each temperat ure and the decrements of temperature towards the cooling process. [29, 30, 31, 32, 33, 34] are some recent examples of the practical applications of simulated annealing as an optimization algor ithm. Fig 2.3 Structure of Simulated Annealing Algorithm [28] Input and examine initial solution Determine initial value of T = 0 T Generate and examine new solution Accept new solution? Update Stores Adjust temperature Terminate search? Start Stop No No Yes Yes

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18 2.1.5 Genetic Algorithms The framework for genetic algorithms lies in the natural evol ution of species searching for beneficial adaptations in a changing environment. Although the in formation encoded in the chromosomes of individual members changes by random mutation, it is es sentially a combination of chromosomal material during breeding. In 1975, the incorporation of the pr inciples of evolution into optimization routines was formally established in [36] To use a genetic algorithm, we need to represent the solution as a genome (or chromosome). T he algorithm takes as an input an initial population of solutions and applies the genetic operators (mutation, crossover) to evolve and find an optimum solution. The main aspects of applying genetic algorithms to rea l-life Fig 2.4 Structure of a genetic algorithm Start Generate random population of n chromosomes Create a new population by performing selection, crossover, mutation and accepting Replace the current population with the generated p opulation Is end condition satisfied? Return best solution of the population Stop Evaluate the fitness f(x) of each chromosome Yes No

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19 problems are: (i) defining an objective function; (ii) obtaining a genetic representation of the solution; (iii) defining the genetic operators. The efficienc y of a genetic algorithm is dependent on the control parameters: (i) initial population, (ii) size of the population, N iii) crossover probability, Pc, (iv) mutation probability, Pm Genetic algorithms are similar to simulated annealing as both use probabilistic transition rules and use obje ctive function information and not derivatives [28]. Genetic algorithms, although computationally expens ive, can be easily parallelized as the evaluation of an objective function and co nstraints can be done simultaneously for a whole population. The figure 2.3 shows the structure of a geneti c algorithm. [38, 39, 40, 41, 42] are some of the recent works which use genetic algorithms to obtain optimal solutions for the resource allocation problem. 2.1.6 Branch and Bound Branch and bound is another algorithm used to solve optimization problem s. It searches the entire solution space for the best solution. Howeve r, as the number of possible solutions increases exponentially, it becomes infeasible to enum erate all of them and hence we use bounds for the function to be optimized. The algorithm consists of three main pa rts [37]: A bounding function to determine the lower bound for the best solution i n the given subspace of the solution A strategy for determining the next solution subspace to be ana lyzed. A branching rule to be applied if a subspace after analysis cannot be excluded and furt her subdivide the subspace into two or more subspaces The performance of the branch and bound algorithm depends to a great degr ee on the initial search space fed to the algorithm. Convergence is ensured if the size of each generated subspace is smaller than the original one. [43, 44, 45, 46] are works which employ the branch and bound algorithm to find op timal solutions to their versions of the resource allocation problem. Although br anch and bound

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20 Fig 2.5 Flow of Branch and Bound Algorithm algorithms are favorable for discrete and continuous global optimiz ation problems, they have high memory requirements. 2.1.7 Greedy Algorithm A greedy algorithm is one which follows a problem solving heuris tic of making a locally optimum choice in the hope of obtain a globally optimum solution. Greedy algorithms do not always yield optimal solutions as they do not exhaustively examine all the possible solutions of a search space. The basic elements of a greedy algorithm are [47]: A solution space from which the solution is created A selection function to choose the next best element to be added to the solution A feasibility function to examine an element’s eligibility to be added to t he solution An objective function to calculate the value of the (full or partial) s olution obtained A solution function to determine if the complete solution has been reached Greedy algorithms are rarely used to obtain optimal solutions and usually form the basis of a heuristic approach. Even though there maybe problems which can be sol ved using the greedy S S3 S2 S1 S13 S31 S22 S21 S1 2 S11 ……… ……… ………. ……….. ……….. …………

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21 approach, establishing their optimality is non-trivial. [48, 49, 50] are examples of works use the greedy approach to solve resource allocation problems. Typically, greedy approache s are not used because of their unreliability in providing optimal solutions as is proved in t he case of the work in [50]. 2.1.8 Tabu Search A Tabu search is a global optimization algorithm which is a me ta-heuristic imposed on another heuristic. In 1977, Fred Glover introduced this approach of movi ng through a solution space and using memory techniques to avoid cycling. The algorithm re cords moves in a Tabu list and penalizes it if it takes the solution to a point in the sol ution space that has been previously visited. Hence, the algorithm avoids getting trapped in cycles. Tabu search is still an evolving and highly researched technique of optimization. Although Tabu searc h provides comparable or superior solutions to optimization problems, it does not guarante e optimality. Also, the construction of a Tabu list to keep a record of the moves is heurist ic. [65, 66, 67, 68] are some of the recent applications of Tabu search to practical problems. 2.2 Why Game Theory? Game theory is a tool for modeling and analyzing conflict and cooperation between decision makers called players [20]. Such a situation occurs when multiple decision makers with different objectives act on a system or share resources. Gam e theory is considered to have been formalized with the publishing of von Neumann and Morgenstern’s The Theory of Games and Economic Behaviour in 1944. Game theory provides a natural framework for the modeling of the crisis scenario in this thesis. In the context of this work, we use noncooperative games due to the competitive nature of the players (or, crises in our case). Tw o or more crises compete for a limited number of emergency units from various centers. They have a finite set of actions or allocation strategies available to them, the choice of which leads to a well defined numerical

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22 payoff associated with the each combination of strategies and for each crisis. [21]The strategy selected by a crisis depends on three parameters: Strategy space which consists of the set of strategies (allocati ons) available to the crisis Information about the other crises (priority, distance from resource cen ter) Payoff or utility function which quantifies the satisfaction a user can g et from an outcome Every player attempts to maximize his gain in the game. A si gnificant aspect of game theory is that each player’s decision is based on the decision of every othe r player and hence, each player can optimize his gain with respect to every other player. This is quite useful in modeling the crisis scenario where the overall optimization is feasible only if each crisis has been satisfied with respect to all other crises The adversarial nature of the game and the interdependence of each player’s objective function on the decisions of the other players require the resource allocation strategy to take into account all the other players’ objec tive functions. The Nash solution in game theory provides an equilibrium solution taking into account the objectiv e functions of all players. A significant aspect of game theory is that it has been prov en that a finite noncooperative game has at least one Nash equilibrium [22]. This is motivation for us to use the Nash equilibrium as we are guaranteed an equilibrium strategy set for a crisis scenario. 2.3 Game Theoretic Concepts Games, as represented in game theory, consist of four essent ial elements – players, actions, payoffs, and information. These constitute the rules of t he game. Depending on the information available, players try to maximize their payoffs by choosing strategies Players are the individuals/ entities who make decisions. Each player’s goal is to maximize his utility by a choice of actions An action or strategy of player i denoted i s is a decision he can make. Player i ’s action i S = { i s }, is the entire set of actions available to a player.

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23 By player i ’s payoff ) ,..., ( 1 n i s sp, we mean the utility received by player i after all players have decided their strategies and the game has been played; or The expected utility received as a function of the strategies c hosen by the player i and the other players There are two kinds of strategies: pure and mixed [22] A pure strategy maps each of a player’s possible information sets to one a ction. i i i a s w: A mixed strategy maps each of a player’s possible information sets to a probability distribution over actions ( ) i i i a m s w: Where m 0 and ( ) = i A i i da a m 1 Here iwrefers to the information set. A strategy combination ( ) n s s s ,..., 1 = is an ordered set consisting of one strategy from each of the n players. Every player in a game maximizes its payoff and arr ives at an equilibrium state. An equilibrium ) ,..., ( * 1 n s s s = is a strategy combination of the best strategy for each player in an nplayer game. The strategy combination s (a set of strategies) is a Nash equilibrium is, if any player which deviates from its strategy will earn less t han if it remained in its current strategy. Formally, this can be stated as, ( ) ( ) i i i i i i s s s s s i ,' , * -p p The inputs required to formulate Nash equilibrium are: Strategies available for each player Number of players in a game

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24 Some of the major categories of classifying games are as follows [25] : Number n of players – A game can consist of one, two or n players Number of strategies of players – This may be finite, a discrete infi nite or a continuum Zero-sum, constant-sum or general-sum – In a zero-sum game, t he numerical payoffs to the player after any possible play of the game sum up to zero. A constant-sum game is one where the sum of all payoffs to the player is the same for any outcome. A generalsum game includes the other two. Cooperative and noncooperative In cooperative games, players communi cate with one another and can make binding commitments as opposed to noncooperative games Complete and incomplete information Single-stage or multi-stage games Non cooperative games can be further classified into strategic ( normal form) games and extensive form games. A normal form game is a game of complete information played be tween n players, each having a strategy set, i S and a payoff function i p where £ n i i i S p : If there are a finite number of strategies, we can define a normal form g ame as a matrix. In such a game, each player simultaneously selects a move i i S s and receives a payoff ( ) n i s s u ,..., 1 An extensive form game is one which can be represented as a connec ted tree with no cycles and a distinguished node where each node represents a deci sion made by a player. A function specifies which player moves at a node, what actions are available, and which node comes next for each action [24]. For the modeling of the problem stated in this work, our definition of a finite game is a noncooperative nplayer game, with each player associated with a finite set of pure strategies; and

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25 corresponding to each player, i a payoff function i p which maps all the n -tuples of pure strategies into real numbers. A tuple is a set of n strategies with each strategy associated with a different player.

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26 CHAPTER 3 FORMULATION OF A GAME The automation of the allocation of emergency response units is a logical step in crisis management to minimize if not completely eliminate human errors in decision making. In this work, we propose a game theoretic framework to address the proble m of allocation of resources to multiple crises. In drawing upon the concepts of game theory and con sequently Nash Equilibrium we obtain a framework in which we can address the i ssue of minimization of response time, maximization of utility and fairness of the all ocation. The idea of using a Nash solution in the context of resource allocation is not new and has been implemented in several areas [73, 74, 75, 76, 77, 78]. In this chapter, we illustrate the transform ation of a multi-crisis environment into a noncooperative strategic game. 3.1 Crisis Scenario We revisit the example presented in Chapter 1, Figure 1.1. Consi der a hypothetical crisis scenario as described in the figure – a plane crash, a fire a nd a riot. Tables 3.1 to 3.4 contain information about resource types and availability, crisis re quests, crisis priorities and time taken to reach crises from each of the resource centers. In the gi ven scenario, we note that the requirement of crises exceeds the capacity of the resource c enters. Typically, if the number of resources available for dispatch were less than or equal to the requests made by crises, the resource manager is left with the task of determining the distributi on of resources from resource

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27 Table 3.1 Resource Types and Availability Resource Centers Resource Types Hospital A Hospital B Police Station Fire Department Total Availability Ambulances 8 10 18 Police Cars 16 16 Fire Engines 14 14 Table 3.2 Crisis Types and Requests Resource Types Crisis Ambulances Police Cars Fire Engines Plane Crash 11 6 10 Fire 8 3 6 Riot 3 9 1 Total Request 22 18 17 Table 3.3 Crisis Priorities Crisis Priority (1 10) Plane Crash 9 Fire 7 Riot 3

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28 Table 3.4 Time (in minutes) Taken to Reach Crises Resource Centers Crisis Hospital A Hospital B Police Station Fire Department Plane Crash 30 7 31 24 Fire 18 15 33 8 Riot 8 11 13 3 centers between crises based on time taken to reach each cris is. Shortage of resources is not an issue and every crisis is guaranteed satisfaction of its request. However, in the current scenario, the resource manager has the additional task of determining the optimal allocation in the face of shortage and possible starvation of lesser priority crises. For e xample, there are 10 + 8 = 18 ambulances. However, the total request for ambulances is 11 + 8 + 3 = 22 > 18 ambulances. There are two hospitals, A and B and it is important to note tha t A is closer to the plane crash B while allocating resources. Not only does a resource manager ha ve to decide how many ambulances have to be dispatched from Hospital A and B, he has to decide on an optimal distribution based on severity and distance from crisis due to the sho rtage of ambulances. In our methodology, we obtain an optimal solution for each type of resource separately, i.e., we determine an allocation for ambulances, police cars and fire engines separ ately. 3.2 Modeling of Crisis Scenario as a Noncooperative Strategic Game We apply game theoretic concepts to the crisis scenario res ource allocation problem. As mentioned in Section 2.3 of Chapter 2, the main elements of every game are players, actions, payoffs and information. In our model, each crisis is modeled as a player. The objective of each player is to maximize its utility by choosing actions most beneficial to i t The actions, in this case, will be the

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29 allocations to resource centers. For example, the plane crash, f ire and riot have requested for 6, 3 and 9 police cars respectively. The action of sending 6 police car s to the plane crash, 2 police cars to the fire and 8 police cars to the riot constitutes an all ocation. In the context of this work, we refer to the actions as strategies. The utilities are the payoffs received as a function of actions chosen by other players. The framework rests on the modeling of t he payoff function as it captures the effect on other crises when one crisis chooses a particular strategy. The information available to the players corresponds to requests, availabiliti es, priority and response time. The conflicting objectives of the crises (players) contribut e to the noncooperative nature of the game. None of the crises make prior commitments to sha re or lend resources before the game is played. This is a game of complete information as we play the game on the assumption that we have the latest information updates on the resource avai lability and crisis requests. Also, the information regarding time taken to reach a crisis locat ion is assumed to be accurate. All the crises will make their selections simultaneously in a gam e and the game can be represented in a matrix where each cell represents a payoff value. Hence the game is a strategic or normal form game where all crises have finite number of strategies in their sets and the players’ actions are mapped to a probability distribution. 3.3 Notations The following notations are used in this work: n Number of crises, n N C Set of crises, C = { C 1 C 2 C 3 ,.....C n } m Number of resource centers, m N R Set of resource centers, R = { R 1 R 2 R 3 .....R m } Q Set of requirements of all crises, Q = { q 1 q 2 .....q i .....q n } q i Number of resources requested by crisis C i where q i N O Set of resources available at resource centers, O = { o 1 o 2 ......o i ....o m }

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30 o i Number of resources available at resource center R i where o i N S Set of strategies of all crises, S = { S 1 S 2 ,....S n } S i Set of strategies of crisis C i S i = { s i,1 s i,2 ,.....,s i,gi } g i Total number of strategies in S i s i,j j th strategy of the i th crisis, s i,j S i r k Number of resources contributed by k th resource center, r k o k L i Priority of i th crisis, i = 1, 2, .... 10 where 1 is the lowest and 10 is the highest level T Set of response times of resources from resource center to crises, w here T = { t i,j } t i,j Time to by j th resource center to reach i th crisis NS i Set of strategies constituting the Nash Equilibrium solution

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31 CHAPTER 4 OPTIMAL ALLOCATION MODEL USING NASH EQUILIBRIUM The optimal allocation of resources in the crisis scenario is dependent on the modeling of the problem. In the previous chapter, we observed the transforma tion of such a scenario into a noncooperative game as it provides a framework for analyzing s trategic interactions. In this chapter, the details of this transformation are presented. W e propose a definition of a strategy in the context of this thesis and elaborate the process of the fo rmulation and pruning of the strategy space. The basic idea behind this approach is to apply heuristics to eliminate strategies which contribute to infeasible and “poor” solutions. Another goal is t o control the explosion of the strategy space as the dimension of the problem increases. The computati on of a Nash Equilibrium increases in complexity as the strategy space grows and hence i t becomes necessary to prune the strategy space. After the strategy spaces are constructed, we play a noncooperative strategic game and obtain a probability distribution over the strategy set of eac h crisis which is used to determine the allocations. 4.1 Structure of a Strategy A game is modeled around its information set and a strategy captures the essence of this information in formulating an “action” for a player. Each playe r can play the game by selecting an action from its own set of “actions” or strategies, S i. The definition of an “action” or a strategy

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32 Fig 4.1 Formulation of a Strategy plays a significant role in determining the outcome of any game. In our crisis scenario model, the information available consists of, Number of crisis locations, Number of resource centers, Time taken to travel from the resource centers to each crisis, Priority of each crisis Number of resources requested by each crisis The crises are modeled as players and the amount of resources allocated to them constitutes a strategy. Let C = { C i } for i n be a finite set of crises and R = { R j } for j m be a finite set of resource centers, where n N is the number of crises and m N is the number of resource centers. Let i S be the finite set of pure strategies available to crisis i C In the context of this work, we define a strategy as an n -tuple consisting of non-negative integers, one for each of the m resource centers. We write, ) ... ,.. ( 2 1 m k j i r r r r s = N r k and i j i S s where j i s denotes the j th strategy of the i th crisis and r k is number of resources contributed by resource center R k from its pool of resources. R 1 R 3 R 2 R 4 C 1 s 1, j = (r 1 r 2 r 3 r 4 ) Resource Center Crisis r 1 r 2 r 3 r 4

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33 4.2 Generation of Strategies The concept of a noncooperative, normal form game is used in this w ork. All the players move simultaneously in a game which can be represented in a matrix form. Hence, it becomes necessary to enumerate all possible and feasible allocation st rategies before the game is played. With each resource center equipped with r k resources, there are o 1 *o 2 *…o k *…o m different possibilities of allocating resources from the different center s to a crisis. Every crisis has its set of allocation strategies, { } ) ( 2 1 ,.... i g i s s s S = where g(i) o 1 *o 2 *…o k *…o m g(i) N, i = 1,2, …, n We use a recursive algorithm to generate the combinations of a llocations that each crisis can select from. This is repeated for each of the crises. Figur e 3.4 shows the algorithm used to generate these combinations. During the process of generating the strategies, we apply certain constraints on them to eliminate infeasible strategies. Fig 4.2 Generation of Strategies 5 4 2 R 1 R 2 R 3 0 0 0 0 0 0 0 1 1 0 1 0 0 2 1 R 1 R 2 R 3 2 4 5 5 4 1 5 5 4 0 3 2 . . . Resource Pool Strategy Set

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34 Fig 4.3 Algorithm to Generate Strategies 4.2.1 Pruning of Strategies Since this is a game of complete information, the number of resources requested by a crisis location is known. During the allocation of a strategy to a cri sis, it is imperative that a crisis is not allocated more resources than it needs. This is to avoid unf air allocation and wastage of resources. Resources that were allocated in excess of the requ irement could have been used for crises whose needs are more urgent or for other crises which might occur. Let Q = { q i } be the set of requirements of all the crises where i = 1, 2, ... n and q i N Similarly, let O = { o j } be the set of the resources available at the resource ce nters where j = 1, 2, …, m and o j N We apply our first constraint to the strategy generation process. For any strategy, s i, j S i r k q i where i = 1, 2, …, n and k = 1, 2, …, m ---------(1) The above statement implies that in any strategy s i, j belonging to crisis C i the sum of all the entries in the strategy tuple should not exceed the requirem ent q i of that crisis. For e.g., if crisis C 1 requires 4 resources and it has two strategies – s 1,1 ( 1,2,1 ) and s 1,2 ( 1,2,2 ), the strategy s 1,2 2 nd strategy of crisis C 1 is invalid as 1+2+2 = 5 > 4. While generating strategies, it should be kept in mind that the individual entries in a strategy tuple should not exceed the availability of the corre sponding resource center. Hence, we arrive at the second constraint on the strategies. Algorithm : GenStrat Generation of strategies for each cris is Input : Resource array containing number of resource unit s in each center R[m] Number of resource centers m Crisis array containing requests of each crisis – C[n] ,Number of crises n Output: Set of strategies S i for each crisis Declare T[m] Temporary array Declare head Pointer to ordered list of strategies for crises in C i i 1 to n Strategy Set S i Call RecStrat ( R, m, C, n, T, 0, head ) end

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35 For any strategy, s i, j S i r k o k where i = 1, 2, …, n and k = 1, 2, …, m ------------(2) The above condition indicates that in any strategy s i,j belonging to crisis C i the individual entries in the strategy tuple i.e. s i, j = ( r 1 r 2 r 3 ,… r k ,…, r m ), should not exceed the corresponding resource center’s total capacity i.e. r 1 p 1 r 2 p 2 r 3 p 3 ,… r k p k ,…, r m p m For e.g., Let there be three resource centers R 1 R 2 and R 3 with capacities 3, 4 and 2 units respectively. A strategy s i, j = (4, 3, 2) ( j th strategy of i th crisis) is invalid as the first entry in the tuple corre sponding to the contribution of resource center R 1 is 4 whereas the capacity of R 1 is 3. Fig 4.4 Recursive Algorithm to Generate Strategy Set for each Crisis 4.2.2 Cost Function In modeling this problem, we provide the user with an option to sele ct n best strategies from each crisis’s set and play the game. These strategi es are ordered in ascending order of cost to the crisis. Here cost is not tangible and is used as a meas ure of practicality The objective Algorithm : RecursiveStrat Generation of strategies for eac h crisis Input : Resource array containing number of resource unit s in each center R[m] Number of resource centers m Crisis array containing requests of each crisis – C[n] ,Number of crises n, Temporary array T[m] Variables curr Pointer to ordered list of strategies head Output: Strategy set S i temp = curr + 1 if temp == m then last = 1; else last = 0; end for resource units in R i i 1 to curr T[curr] = i ; if last == 1 then head Call AddStrat else head Call RecursiveStrat ( R, m, C, n, T, curr+1, head ) end end return head

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36 behind the ordering is to sort the strategies by those that provide the greatest number of resources in the shortest possible time In a strategy, we compare the time taken to reach a crisi s from each resource center and the number of resources contributed by that r esource center as the factors in determining the cost of that strategy. Let T = { t i, j } be the set of measures of time taken to reach each of the c rises from the resource centers. t i, j is the time taken to reach the i th crisis from the j th resource center. Here i = 1, 2, …, n and j = 1, 2, …, m where n is the number of crises and m is the number of resource centers. For any strategy, Cost ( s i, j ) = min n r m m i i i r t r t r t 2 2, 1 1, ,......., for any strategy ) ... ,.. ( 2 1 m k j i r r r r s = …….. (3) Cost ( s i, j ) measures the ratio between the time taken to reach a cris is i from each of the resource centers in a strategy tuple s i, j and the number of units contributed by each of the resource centers. It is not uncommon for two strategies to have the same cost. In s uch a case, we examine the two strategies and look for the next lowest value of the ra tio in the tuples. The strategy with the next lower value is ranked higher among the two strategie s being compared. Let Cost ( s i, j ) = min n r m m i i i r t r t r t 2 2, 1 1, ,......., = a where a = p p i r t and Cost ( s i, k ) = min n r m m i i i r t r t r t 2 2, 1 1, ,......., = b where b = q q i r t where ( s i, j ) and ( s i, k ) are j th and k th strategies of crisis i If a = b then compare min n r + + m m i p p i p p i i i r t r t r t r t r t 1 1 1 1 2 2, 1 1, ...., ,... , and min n r + + m m i q q i q q i i i r t r t r t r t r t 1 1 1 1 2 2, 1 1, ...., ,... ………………. (4)

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37 Fig 4.5 Flowchart Illustrating Ordering of Strategies Yes Generate Strategy Calculate Cost Is there another strategy (s) in the ordered list with the same cost? Is the next lowest cost in the tuple different from next lowest values in the other strategy (s) with the same cost? Compare the sum of the resources contributed by the strategy with the other strategy (s) with the same cost. Add strategy to the ordered list of strategies in t he appropriate place in ascending order of cost The strategy with the next lowest value is ranked higher The strategy contributing the higher number of resources is ranked higher No No Yes

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38 Fig 4.6 Algorithm to Add a Strategy to a Strategy Set The process above (4) is repeated until we find two unequal costs. If two strategies being compared have identical k k i r t values in their tuples, we compare the two strategies for the one which contributes the higher sum total of resources to the cris is. The strategy contributing the higher number of resources is ranked higher among the two strategies being compar ed, i.e. compare r k where k = 1, 2, ..., m in ( s i, j ) and ( s i, k ) for the higher value ....................... (5) We perform the ordering of the strategies at the time of ge neration to avoid the additional overhead of sorting a huge list of strategies at the end th ereby adding the time taken to sort strategies of each of the crises to the overall computation ti me. Figure 3.5 shows the process of ordering of strategies. Algorithm : AddStrat Add strategy to strategy set of each c risis Input: Strategy S[m] Number of resource centers m Pointer to ordered list of strategies head, Crisis array containing requests of each crisis – C[n] ,Number of crises n Output: Pointer to ordered list of strategies head Declare sum Temporary Variable for resource units in S i i 1 to m sum sum + S i end if sum C[present crisis] then Compute cost of strategy S Determine order strategy in list of strategies bas ed on cost Add strategy to strategy set end return head

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39 Figure 4.7 Normal Form Game Representation s 1,1 s 1,2 s 2,1 s 2,2 1, 1 2, 3 3, 2 4, 1 s 2 ,1 s 1,2 s 1,1 s 2 ,2 s 1,1 s 1,2 s 2,1 s 2 ,2 1 3 2 4 1 1 3 2 Player 2 Player 1 Payoffs to Player 2 a) Typical 2-player Normal form game representation b) Our 2-player Normal form game representation s 1,1 s 1,2 s 2,1 s 2,2 s 2,1 s 2,2 s 3 ,1 s 3, 2 s 1,1 s 1,2 1,1,2 2,3,4 3,2,1 4,1,3 2,3,4 5,2,1 2,1,3 2,2,1 Payoffs to Player 1 Payoffs to Player 2 Payoffs to Player 3 s 1,1 s 1,2 s 2,1 s 2,2 s 2,1 s 3,1 s 2,1 s 3,2 s 2, 2 s 3,1 s 2, 2 s 3,2 s 1 ,1 s 3,1 s 1 ,1 s 3,2 s 1 2 s 3,1 s 1 2 s 3,2 s 2,1 s 1 ,1 s 2,1 s 1,2 s 2 ,2 s 1 ,1 s 2 ,2 s 1,2 s 3,1 s 3,2 1 1 2 3 3 3 2 2 2 2 4 5 2 2 1 1 2 1 4 3 4 1 3 1 c) Typical 3-player Normal form game representation d) Our 3-player Normal form game representation Payoffs to Player 1

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40 4.3 Payoff Modeling A normal form representation of an n -player game is specified with, A finite set of n players, A finite set of strategies S = { S 1 S 2 ..., S n }for each player A utility/payoff function u i to be applied on the set S Figure 3.7 shows the typical representation and our representati on of a 2-player game. In (a), the strategies of player 1 are vertical and the strategies of player 2 a re horizontal. Each cell has two entries – the first entry is the payoff to player 1 and the se cond entry is the payoff to player 2 when player 1 and player 2 choose the strategies at the far left and top r espectively. Similarly, in (c) the matrix is divided into two parts. Each part is simila r to (a) but represents a payoff when a third player chooses its first strategy (left half) and its second strategy (second half). In (b) and (d), we divide (a) into two matrices and (c) into three matr ices respectively, one for each player’s payoffs for choosing a particular combination. 4.3.1 Creation of Payoff Matrices A normal form game can be represented as a matrix. For the purp ose of our implementation we create n payoff matrices, one for each crisis. In our representation of th e payoff matrices in Figure 3.8, the first column of each crisis’ s matrix holds its strategies, one on each row starting from the second row. The first row of the crisis’ s matrix holds the combinations of strategies selected by all the other crises, one on each column starting from the second column. The ‘X’ in the first matrix is for the payoff for the first crisis when C 1 selects s 1,1 and C 2 selects s 2,1 and C 3 selects s 3,1

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41 Fig 4.8 Structure of Payoff Matrices for Crises 4.3.2 The Payoff Function In any game, the best action for any player depends on the act ions of the other players. A payoff to a crisis for choosing a particular strategy when the other crises make their selection can be represented as a gain to the crisis or a loss to the other crises. In our model, we depict it as a summation of the losses to the other players and each player tr ies to maximize this loss to other players. The payoff to a crisis is representative of the loss incurred by the other crises on the allocation of a particular strategy to the crisis in question and the remaining crises are allocated their strategies. Essentially, the possible combinations of stra tegies remain the same. Each matrix captures the payoff to a particular crisis for a particular combination. Crisis C 1 s 1, 1 s 1, 2 s 1, 3 Crisis C 2 s 2, 1 s 2, 2 s 2, 3 Crisis C 3 s 3, 1 s 3, 2 s 3, 3 s 1, 1 s 1, 2 s 1, 3 s 2, 1 s 2, 2 s 2, 3 s 3, 1 s 3, 2 s 3, 3 s 2, 1 s 3, 1 s 2, 1 s 3, 2 s 2, 1 s 3, 3 s 2, 2 s 3, 1 s 2, 2 s 3, 2 s 2, 2 s 3, 3 s 2, 3 s 3, 1 s 2, 3 s 3, 2 s 2, 3 s 3, 3 s 1, 1 s 3, 1 s 1, 1 s 3, 2 s 1, 1 s 3, 3 s 1, 2 s 3, 1 s 1, 2 s 3, 2 s 1, 2 s 3, 3 s 1, 3 s 3, 1 s 1, 3 s 3, 2 s 1, 3 s 3, 3 s 1, 1 s 2, 1 s 1, 1 s 2, 2 s 1, 1 s 2, 3 s 1, 2 s 2, 1 s 1, 2 s 2, 2 s 1, 2 s 2, 3 s 1, 3 s 2, 1 s 1, 3 s 2, 2 s 1, 3 s 2, 3 X Payoff matrix of crisis C 1 Payoff matrix of crisis C 2 Payoff matrix of crisis C 3 Strategy Sets for Crises – C 1 C 2 and C 3

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42 In our model, every crisis is assigned a priority L on a scale of 1 to 10 to indicate its severity. This priority is used as a weight in a payoff funct ion to facilitate the calculation of the loss to a crisis(s). Payoff to k th strategy of i th crisis when the crises t i choose s m i, l { S 1 S 2 ... S i-1 Si +1 Sn } is given by, () () () () == = n t m j i t j li St j k i St j j li St j L o r r o r 11 , , , constant li St j r , Resources contributed by the j th resource center of l th strategy of crisis C t i k i St j r , = Resources contributed by the j th resource center of k th strategy of crisis C t=i o j Total number of resources available at resource center R j L t i Priority of crisis C t i M Number of resource centers n Number of crises The term ( ) ( ) k i St j j li St j r o r , , = refers to the ratio between the resources contributed by a strategy and number of resources available from that resource center after allocating to the crisis in question, C i The term ( ) j li St j o r , refers to the ratio between the resources contributed by a strategy and number of resources available from that resource c enter without allocating to the crisis in question, C i The difference captures the loss to the crises C t i and the priority of the crises adds a weight to the loss. The last term of the expres sion, constant, is value which is either 0 or If the combination of strategies is feasible, then constant = 0, else constant = (or a very high number). Consider a combination of strategies, { s 1, p s 2, q ...., s n, r }. Let { r 1 r 2 ..., r m } be the sum of resources.

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43 ( r 1 r 2 ..., r m ) belonging to s 1, p + ( r 1 r 2 ..., r m ) belonging to s 2, q + ...... .................... + ( r 1 r 2 ..., r m ) belonging to s n, r ___________ ( r 1 r 2 ..., r m ) If r i o i for any i = 1, 2, ... m the combination of strategies becomes infeasible. 4.4 Algorithm to Approximate Nash Equilibrium We apply a variant of the Scarf-Hansen fixed-point algorithm [ 69, 70] to approximate a Nash Equilibrium point in our noncooperative game. The algorithm is base d on a combinatorial theorem [69] which is expressed in terms of a primitive set Consider a collection of h dimensional vectors X = ( x 1 x 2 ...., x h ) of the form ( m 1 /D ..., m h /D ) with each value greater than or equal to -1 and summing up to D which is a very large number, typically a multiple of the number of crises. Let the numerators of the vectors in X be [69], m 11 m 12 .... m 1h m 21 m 22 .... m 2h M = . . . m h1 m h2 .... m hh The above matrix is a primitive set if and only if and there is a rearrangement of the columns and a permutation of the labels of the columns, I ( l ) such that[69], 1. The lth column is identical to column l-1, except for the two rows I ( l )-1 and I ( l )

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44 2. m k,l = m k,l-1 + 1 for k = I ( l )-1 m k,l = m k,l-1 – 1 for k = I ( l ) Note: For l = 1, l -1 = h Similarly, I ( l ) = 1, I ( l ) – 1 = h Fig 3.9 shows the steps of the algorithm used to compute the Nash Equil ibrium solution. As shown in the figure the output is a probability vector p = ( p 1 p 2 ,..... p i ... p n ). p i is a probability distribution over the strategy set S i of crisis C i p i = { p i,j } where j n and i g j total number of strategies in S j Fig 4.9 Algorithm to Approximate Nash Equilibrium Algorithm : NashSolve Approximate Nash Equilibrium Input : n – number of crises, X, Payoff matrices Output : Probability vector p Calculate h = Total number of strategies of all crises – n +1 While label of x j 1 Let x j be an arbitrary nonnegative vector from X We associate it with the probability vectors ( j n j j p p p ,.... 2 1 ) as follows: t 0 = 0 and t i = s i -1 where i = 1, 2,..., n and s i is the number of strategies of crisis Ci -= + + = 1 0 1 i v v ki k t r where k = 1, ...., t i and i = 1,...... n j r j i k ki x n p = where k = 1, ...., t i and i = 1,...... n = = i i t i j i k j i s p p 1 , 1 where j i k p 0 i f j i s i p 0 then Let B ki = B ki ( j n j i j i j p p p p .... ,.., 1 1 1 + ) be the expected payoff to a crisis it uses its k th strategy and k i is the lowest index for which i k i B ( j n j i j i j p p p p .... ,.., 1 1 1 + ) B ki ( j n j i j i j p p p p .... ,.., 1 1 1 + ) The vector x j is labeled = + + 1 0 1 i u i u k t where i = min{ j l s l p l | > 0 and k l s i } Perform replacement step end if j i s i p < 0 or j i s i p = 0 for all i or k i = s i for all i with j i s i p > 0 then x j is labeled 1 Terminate algorithm end end

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45 We compute max j ( p i,j ) for each i The Nash Equilibrium solution is given by, () { } j i j p i i s NS max = max j ( p i,j ) corresponds to the strategy with maximum value of probability among strategies of a crisis. In the event that there are two strategies with ident ical probabilities, we pick the one with the lower cost to the crisis. 4.5 Software Implementation Crisis management encompasses a whole range of activities from “s ensing” a crisis to deployment of resources to monitoring of crisis development. The objective of our work is to provide an automated mechanism for determining the number of units assigned to each crisis location based on priority and requirement. In this work, we have implemented a tool tha t Fig 4.10 Overview of Crisis Management System Crisis Alert Mechanism eg. “Sense and Respond System” Crisis Damage Assessment Mitigation of crisis and deployment of resources Monitor crisis development and other possible crises eg. CMS Recovery

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46 determines an optimal allocation of resources in a multi-crisi s environment. The user provides a set of inputs to the system, namely, resource capacities, cr isis requests, crisis priorities and response time information. The software evaluates the information u sing the underlying algorithm and presents the user with an optimal allocation of resources from each center. In Fig 4.11, the grey box indicates where our tool for allocation of resources to crises will fit into a crisis management model. 4.5.1 System Input and Output Every system is unique in the nature of inputs fed into it and the output prese nted to the user. Below are the specifications of the input and the output of our system: Input: Number of crises (2 5 crises) Number of resource centers (2 – 10 centers) Resources requested by each crisis (8 – 15 resources) Resources available at each resource center (2 – 10 resources) Priority of each crisis (1 – 10; 1 being the lowest and 10 being the highest) Time taken to travel between crisis and resource center (in minutes) Output: Allocation consisting of combination of resources contributed by each resour ce center

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47 4.5.2 Object-Oriented Design We have used an object oriented design to implement our solution because of the two main benefits: Maintainability – This is achieved through a simplified mapping to the prob lem domain, which results in less analysis effort, less complexity in system design and easier user verification. Reusability – Segments of the structured code can be reused by adding new functiona lities with slight or no modification. This reduces implementation time, localizes t he modifications in code when a change in implementation is required and also increases the possibi lity that prior testing has removed bugs. The programming language used to implement it is C++. It was c hosen because of its ability to program in a C-like style, or an object-oriented style or both and also its ability to utilize the predefined classes and be able to create user-defined clas ses to characterize the features of the input. Object oriented design allows us to organize data into discre te, distinguishable entities called objects. A single object has a state and behavior ass ociated with it. For example, in our work, crises and resource centers are objects. Each has its own characteristics and behavior. A crisis is described in terms of it severity, requests and loc ation. A resource center is described in terms of its capacity. Furthermore, each object has its ow n characteristic behavior. For example, each crisis generates its strategy set.

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48 4.5.3 Overview of Classes and Functions We have the following four classes with object functions in our implementa tion: Table 4.1 Overview of Classes Used Class Attributes Behavior Resource Center Name, Capacity Crisis Name, Priority, Number of Strategies, Pointer to Strategy Set, Response time from each Resource center Recursively generates all possible allocations from resource centers Adds a strategy to its set Sorts its strategies based on cost Payoff Matrix Name, Number of Crises, Number of combinations, Structure for each Combination Generates rows and columns of a matrix Computes payoffs Nash Primitive Set, X Vector, Probability Vector, D, n, k array and t array Computes the Nash Equilibrium The following functions have been used: RecNum() Class : Crisis Input : Number of resource centers, resource center capacities Function : Generate strategies for a crisis from the available res ources from each center AddStrat() Class : Crisis

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49 Input : Strategy of a crisis Function : Add strategy to a crisis’ strategy set costSort() Class : Crisis Input : Strategy of a crisis Function : Locate the ordering of a strategy based on its cost compareStrategy() Class : Crisis Input : Strategies having the same cost Function : Determine ordering of two strategies with same cost GenerateComIndex() Class : PayoffMatrix Input : Number of crises Function : Generate indexes for combination of strategies belonging to diff erent crises generateComIndex() Class : PayoffMatrix Input : Number of crises Function : Generate indexes for combination of strategies belonging to diff erent crises getRowStrategy() Class : PayoffMatrix Input : Number of crises

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50 Function : Generate strategies in each row of payoff matrix getColumnStrategy() Class : PayoffMatrix Input : Number of crises, Strategy set of each crisis Function : Generate combinations of strategies in each column of payoff matrix getPayoff() Class : PayoffMatrix Input : Number of crises Function : Compute payoffs in the payoff matrix of each crisis selectedStrategy() Class : Nash Input : Crises, resource centers, payoff matrices Function : Determine the optimal allocation for each crisis

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51 Fig 4.11 Workflow Model of the Proposed System Optimal Allocation of Resources Crisis Request Resource Center Capacity Priority Re sponse Time Generate Strategy Set for each Crisis Apply Constraints on the Strategies in each Set bas ed on Crisis Request Resource Center Capacity Compute Cost of each Strategy and place it in Ascending order Generate Combinations of Strategies and Construct Payoff Matrices Calculate Payoffs for each Strategy Combination in the Payoff Matrices Apply Hansen’s Algorithm to Compute Nash Equilibrium Obtain Probability Vector and determine Strategies that form Nash Equilibrium

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52 CHAPTER 5 EXPERIMENTAL RESULTS This chapter presents a set of experimental results to v erify the efficacy of the solution obtained using our implementation. The implementation has been divided int o three phases – generation of strategies, computation of payoff matrices and f inally, the generation of the Nash Equilibrium solution. For the integration of the three phases, the output of each phase is made compatible for the next step. The user inputs information regard ing number of crises, requests, response times and criticality levels. C++ is the program ming language chosen for implementing our tool to obtain an optimal solution. The object-oriented characte ristic of C++ facilitates modeling of the characteristics of players and strategies, and the modularization of the implementation. In this chapter, we observe the performance of our approach in var ious crisis scenarios by varying the inputs and noting its effects on execution time and quali ty of the solution. It is logical to assume that as the dimensionality of an input set increases so does the execution time of an implementation. In our implementation, the process of generating st rategies and the computation a Nash Equilibrium are time intensive tasks and we examine the effect of increasing number of crises on the execution time. Also, the total availability of resources in a resource pool versus the total demand or requests made by crises are contributing factors in determining the quality of the solution. We have performed experiments on various test cases by alter ing inputs such as crisis requests and resource center capacities. The test cases are used to study the effects of inputs on the time taken to reach a solution and also to determine the qualit y of a solution. Furthermore, we

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53 apply the fairness measures as described in [79] to quantify the f easibility and fairness of the implementation. All experiments were run on a Sun Fire V880 Ultr aSPARC III server that features eight 900 MHz processors and 32 GB of memory running the SunOS. 5.1 Fairness Typically, in a priority based system, a high priority event w ould receive its request and the lower priority event would receive less than its fair share and possibly starve due to lack of resources. This is not very different from a first-come-first -serve principle where the more critical event is sent ahead in line to claim its share of resource s. Although, it would seem logical to allow a more critical event to satisfy its request, such a scheme suffers from a high degree of unfairness. A priority-based system would perform reasonably wel l when there is sufficient number of resources in the resource pool. However, the problem arise s when the total demand exceeds the total supply. Higher priority crises satisfy their requests, while lower priority crises events suffer from starvation due to inevitable shortage. I n a crisis environment, this is unacceptable as starvation of a lesser priority crisis co uld lead to further loss in life and property and increase the possibility of worsening of a crisis scenario. Generally, a system is deemed to be fair or unfair based on whether or not it me ets certain requirements or not. Generally, a system is considered fair i f it meets certain criteria on throughput or delay, and it is considered unfair if these criteria are not met. For example, a scheduling algorithm is fair or unfair depending on whether any use r receives a throughput of x bits/sec or not. In our system, fairness would be determined by whe ther or not a crisis receives its share of resources. We use the fairness concepts of [79] to det ermine the fairness of our implementation. In our approach, we compare the best-cost strategy of a crisis with the actualcost strategy, i.e. the cost of the strategy that is assigned as the Nash Equilibrium solution. A measure of fairness of the system can be measured in terms of the self-fairness of the individual crises. The proportion of the cost associated with crisis j that is deemed to be fair is given by,

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54 lowest i i i lowest j j fair t t p 1 , cos cos = = T j r r = Here, lowest j t cos is the lowest or best cost strategy of a crisis. In Chapter 4, we have described the ordering of strategies based on cost. We have used the term r j for brevity. = = n i actual i actual j j actual t t p 1 , cos cos Here, actual j t cos is the cost of the strategy assigned to a crisis as part of the Nash S olution. The definition of the self-fairness of a given user is, ( ) () T j j actual j r r p f log log = We also define, () () T k T k r r r r C log 1 1 log 1 1 1 + + =

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55 The values C k are normalization constants. Their objective is to ensure that the maxi mum value of the weighted average fairness is unity and the maximum occur s when each user consumes its fair share of the resources. The average fairness of the system is given by = = = n k k k n k k k k T r C f p C r F 1 1 The average fairness value of F ranges between 0 and 1. The value of unity results when a cris is consumes resources using a strategy with the lowest possible cost. Tables 5.1 (a-d) show the fairness measure for test cases with varying differences between demand and supply. Every crisis attempts to select a strategy from its set of strategies that costs the least. We observe a slight decrease in fairness as the total request for resources from all the crises exceeds the total resources available from all th e centers. Table 5.1 Fairness Measures Demand vs. Supply Degree of Fairness Total demand 21%-50% less than total supply 0.83730 9 Total demand 1%-20% less than total supply 0.82711 7 Total demand = Total availability 0.833342 Total demand 1%-20% greater than total supply 0.803 526 Total demand 21%-50% greater than total supply 0.80 5155 (a) Crisis Scenario with 2 Crises

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56 Table 5.1 Continued Demand vs. Supply Degree of Fairness Total demand 21%-50% less than total supply 0.81152 3 Total demand 1%-20% less than total supply 0.84878 8 Total demand = Total availability 0.891354 Total demand 1%-20% greater than total supply 0.71144 Total demand 21%-50% greater than total supply 0.78 1591 (b) Crisis Scenario with 3 Crises Demand vs. Supply Degree of Fairness Total demand 21%-50% less than total supply 0.82297 8 Total demand 1%-20% less than total supply 0.81860 3 Total demand = Total availability 0.845423 Total demand 1%-20% greater than total supply 0.710 171 Total demand 21%-50% greater than total supply 0.69 1537 (c) Crisis Scenario with 4 Crises Demand vs. Supply Degree of Fairness Total demand 21%-50% less than total supply 0.78929 2 Total demand 1%-20% less than total supply 0.76923 1 Total demand = Total availability 0.769231 Total demand 1%-20% greater than total supply 0.769 858 Total demand 21%-50% greater than total supply 0.76 9231 (d) Crisis Scenario with 5 Crises

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57 5.2 Execution Time Although the optimality of the solution is important, the time taken to arrive a t an optimal solution is an important factor in determining the feasibility of our approach. For example, on the occurrence of 3-4 crises simultaneously, a formulation which took m ore than 1-3 minutes would not be tolerated. Every second utilized to construct a solution has a direct effect on the response time of the emergency units and consequently on their ability to pre vent damage. The nature of the problem is such that we are limited in scope in terms of comparison with other works. We analyze the implementation by varying the inputs and observing its effect s on the runtime. The graphs in Fig 5.1 reveal three important observations rega rding the effect of inputs on execution time. Firstly, all three graphs show the sudden increas e in execution time as the number of resource centers increases. As the number of resource centers reaches 8-9, there is an exponential increase in the time. In Fig 5.1 (a) and (b), the number of resources in each center is increased to 3 and 7 respectively. The trend in exponential increase persists over an increase in resources per center with a sudden surge observed around 7-9 cent ers. The second observation is that as the number of resources per center increases, the ra nge of execution time increases significantly. The range of execution time when there are 3 re sources per center is 0 -14 seconds in Fig. 5.1 (b) as compared to a range of 0 – 250 seconds when there a re 7 resources per center in Fig. 5.1 (a). The third observation is that as the number of crises increases, the execution time for the same number of resource centers is higher. For example, in Fi g. 5.1 (b), when the number of resource centers is 9, a 2-crisis scenario takes around 110 sec onds and a 4-crisis scenario takes about 225 seconds. These observations typically aid a user in determi ning the range of an input set used to produce an optimal solution within an acceptable time frame.

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58 Effect on Execution Time Due To Increase In Number of Resource Centers 0 2 4 6 8 10 12 14 01234567891011 Number of Resource Centers (Resources per Center = 3) Execution Time (seconds) Number of Crises = 2 (Request of each crisis = 5) Number of Crises = 3 (Request of each crisis = 5) Number of Crises = 4 (Request of each crisis = 5) (a) Effect on Execution Time Due To Increase In Number of Resource Centers 0 50 100 150 200 250 012345678910 Number of Resource Centers (Resources per Center = 7) Execution Time (seconds) Number of Crises = 2 (Request of each crisis = 5) Number of Crises = 3 (Request of each crisis = 5) Number of Crises = 4 (Request of each crisis = 5) (b) Fig 5.1 Effect on Execution Time Due to Increase in Number of Resource Centers Fig 5.2 Effect of Increasing Number of Crises on Execution Time Effect of increasing number of crises on execution time 0 100 200 300 400 500 0123456Number of crises (Request of each crisis = 8) Execution Time (seconds) Number of resource centers = 7; Number of resources per center = 5 Number of resource centers = 5; Number of resources per center = 7

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59 Another factor which determines the execution time is the number of crisis locations. Fig 5.2 is a graph with a fixed number of total resources. Both the series show a n increase in execution time as the number of crises increases. In the pink series, the number of resources centers is 5 with 7 resources per center and the blue series is vice versa. Alt hough the total number of resources is constant, we observe that the series with 7 resource centers and 5 resources per center has a higher execution time with the same number of crises. For eas e of plotting the graph, we have used the logarithm function to plot execution time. For a fixed set of resource centers, capacities and request as the number of crises increases, the size of a game increases thereby increasing t he time taken to arrive at the Nash solution. Fig 5.3 shows this increase in the percentage of time taken to compute Na sh Equilibrium as the dimension of the problem increases. Percentage Increase in Computation Time for Nash Equilibrium with Increase in Number of Crises0 10 20 30 40 50 2345 Number of Crises Increase in Percentage of Computation Time (%) Fig 5.3 Percentage Increase in Computation time of Nash Equilibrium

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60 5.3 Statistical Significance of Experimental Results for Execu tion Time We perform regression analysis on our test data in order to det ermine the statistical significance of our results. We perform linear regression an alysis by using the "least squares" method to fit a line through a set of observations. By doing so, we will be able to analyze how a single dependent variable is affected by the values of one or more independe nt variables. Table 5.2 Regression Analysis Results Independent Variables Coefficient of independence PValue (Confidence) Number of crises 3.59 2.64E-12 Number of resource centers 2.10 5.76E-13 Number of resources per center 0.61 0.25 In the linear regression model, the dependent variable is assum ed to be a linear function of one or more independent variables plus an error introduced to acc ount for all other factors. Consider a dependent variable y and an independent variable x the coefficient of y on x is given by, 2 x xy C = In our case, we examine execution time as the dependent variable and number of crises, number of resource centers and number of resources per center as the independent variables. We observe a significant impact of number of crises on the execution time. F or every unit change in Log (Number of Crises), Log (Execution Time) increases by 3.592021. Simil arly, for every unit change in Log (Number of resource centers) and Log (Number of r esources per center), Log (Execution Time) increases by 2.10 and 0.61 respectively.

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61 The p-value is the probability of finding a value as extreme or more extreme is a chance given that the null hypothesis is true. A null hypothesis basically assumes that none of the variables have any effect on the execution time. In our case, t he p-values for number of crises, number of resource centers and number of resources per center are 2.64E-12, 5.76E-13 a nd 0.25. In this chapter, we have examined the various aspects of our solution. We have quant ified the fairness of our implementation and found fairness measures as high as 0.89. We derived significant inferences regarding the relationship between the various input parameters – crisis requests, resource availabilities and number of resource center s. Finally, we evaluated our test cases to understand the dependent and independent variables in the s ystem and also verified the confidence of our test cases.

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62 CHAPTER 6 CONCLUSIONS AND FUTURE WORK 6.1 Conclusions Crisis management has gained considerable importance over the last few years and the automation of allocation of emergency services is a logical step towa rd erasing human error. Each crisis in a multi-crisis scenario makes a request for a certain number of resources. A conflicting situation occurs when there is a shortage of resources or compe tition for resources from the same center. Although, each crisis has varying degrees of severity i t is highly essential to cater to each crisis’ request in the best possible manner. We have proposed an a pproach using game theory to allocate an optimal number of resources in a multi-crisis environme nt. Our method is a novel way of modeling a crisis scenario in a game theoretic framework and obtaining an allocation of resources that benefits all the crises in the game. We have examined the effects of various input parameters l ike number of crises, resource capacities etc on the execution time. We have examined the effec t of increasing crisis on the overall execution time of the system. Although, the implementation is affected significantly by the dimensions of the inputs, it has shown a degree of fairness o f up to 0.89 in its results and can be used as a basis for modeling other resource allocation problems and obtaining feasible solutions. We have performed a linear regression analysis on our test case s and found the degrees of dependence between the variables and verified the confidence of our test cases.

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63 6.2 Future Work In the experiments that we performed, we restricted ourselves in the number of resource centers and the number of resources per center due to constraint s on execution time. We need to investigate additional schemes to prune strategy spaces more effectively and improve the definition of a strategy in order to enhance the performance of the algorithm with increased dimensionality of the input set. The process of generation of s trategies provides scope for parallelism which could improve execution time. Also, we need to e xplore additional factors that can be incorporated to enrich the payoff function like traffic de lays, crisis growth probability, etc in order to obtain the best possible representation of a real life crisis env ironment and improve the quality of the solution.

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An event driven single game solution for resource allocation in a multi-crisis environment
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ABSTRACT: The problem of resource allocation and management in the context of multiple crises occurring in an urban environment is challenging. In this thesis, the problem is formulated using game theory and a solution is developed based on the Nash equilibrium to optimize the allocation of resources to the different crisis events in a fair manner considering several constraints such as the availability of resources, the criticality of the events, the amount of resources requested etc. The proposed approach is targeted at managing small to medium level crisis events occurring simultaneously within a specific pre-defined perimeter with the resource allocation centers being located within the same fixed region. The objective is to maximize the utilization of the emergency response units while minimizing the response times. In the proposed model, players represent the crisis events and the strategies correspond to possible allocations.The choice of strategies by each player impacts the decisions of the other players. The Nash equilibrium condition will correspond to the set of strategies chosen by all the players such that the resource allocation optimal for a given player also corresponds to the optimal allocations of the other players. The implementation of the Nash equilibrium condition is based on the Hansen's combinatorial theorem based approximation algorithm. The proposed solution has been implemented using C++ and experimental results are presented for various test cases. Further, metrics are developed for establishing the quality and fairness of the obtained results.
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