Models for Improving the Integration of Spontaneous Volunteers in Disaster Response

ABSTRACT

PARET, KYLE EDWARD. Models for Improving the Integration of Spontaneous Volunteers in Disaster Response. (Under the direction of Maria E. Mayorga).

In the wake of a disaster, people from nearby areas often converge to assist the affected community. Spontaneous volunteers (SVs) are not affiliated with relief agencies but are in a unique position to provide invaluable aid at a crucial point in the disaster cycle. Unfortunately, these volunteers are often ineffectively used or refused altogether. Existing emergency management plans often fail to take SVs into account due to negative perceptions and uncertainty around SV convergence. Despite SVs best intentions, they can cause more harm than good if volunteer sites are not prepared to integrate them. Disaster management plans can benefit from improved strategies to integrate the influx of SVs. This dissertation develops two models designed to improve the planning and integration of SVs in disaster response. First, a multi-server queueing model is formulated to represent the dynamics of assigning SVs to tasks in a post-disaster setting. In particular, we consider the stochastic arrival of demand tasks and volunteers, whose time in service is also stochastic. These assumptions mimic disaster relief tasks such as the distribution of relief items, where both donations and volunteers arrive randomly. We generate an optimal policy for assigning volunteers to tasks using a Markov Decision Process. Using simulation, we compare the optimal policy against several heuristic policies and discuss real-world implications. We then develop an agent-based model of SV convergence, designed to better represent the decision-making process of potential SVs during disaster response. We model individual agent motivation, communication, and site choice behaviors explicitly. Using the Theory of Planned Behavior, we evaluate an agent’s willingness to volunteer based on their behavioral intention in a given period. By implementing realistic agent behaviors, we can more effectively represent how and why volunteers appear at volunteer sites. We conduct verification and validation of the model through scenario evaluation, parameter calibration, and practitioner review. We conduct a statistical analysis of relevant model parameters to determine which internal and external factors most significantly affect SV responses. The model bridges a gap between operations research/management science, emergency management, and social science and provides disaster managers a new tool to evaluate volunteer integration plans. Finally, we conduct an analysis of strategic and operational level questions relevant to emergency management practitioners. Three unique scenarios are simulated in the agent-based model, using real-world data collected during Hurricane Florence. Results are summarized and implications related to disaster plans are discussed. The results can help communicate and educate community partners at the local level on the value of SV participation in disaster response. We close with a brief examination of how built-in simulation optimization capabilities can be used to create a decision support tool for messaging. With continued practitioner involvement, the SV Convergence model can be used to inform and improve existing disaster management plans. This work contributes to the field of Operations Research/Management Science in disaster response. © Copyright 2020 by Kyle Edward Paret

by Kyle Edward Paret

A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulﬁllment of the requirements for the Degree of Doctor of Philosophy

Industrial Engineering

Raleigh, North Carolina 2020

APPROVED BY:

Michael Kay Hong Wan

Sara Shashaani Maria E. Mayorga Chair of Advisory Committee DEDICATION

In memory of my grandmother, Mildred H. Paret, an engineer ahead of her time.

ii BIOGRAPHY

Kyle Paret is a doctoral candidate in the Edwards P. Fitts Department of Industrial and Systems Engineering at North Carolina State University. Born and raised in eastern Pennsylvania, Kyle received his bachelor’s degree in Industrial and Manufacturing Engineering (Magna Cum Laude and Honors) from the Pennsylvania State University in 2010. Prior to joining the Ph.D. program in 2014, Kyle worked as an Operations Research Analyst for the Department of Defense. His research interests include applying simulation models and stochastic processes to support complex real-world decision-making.

iii ACKNOWLEDGEMENTS

I want to thank my adviser, Dr. Mayorga, for her support and patience through good and bad times. Her guidance over the past four-plus years has helped me grow as a researcher and become a more disciplined writer. I would also like to thank Dr. Kay for providing advice and direction in my research and throughout my entire graduate journey. Thanks to my other committee members, Dr. Shashaani and Dr. Wan, for their feedback and suggestions, which helped improve and shape my dissertation research. I’ve always felt that there is more to graduate school than just conducting research. Thanks to all my friends in 444, 373, 375, and 458 for the conversations, suggestions, and non-research related distractions (Catan, Spikeball, Magic, Hearthstone, etc.). Special shout-out to Surfsol- jah13, Codex77, Drewpy, and Armnhammer7 for all the late-night “study sessions”. Thank you Eva, owner of the snack stand in 1911 Building, for the frequent chats and special order vanilla Coke sodas. These friendships made my graduate experience so enjoyable and something I will look back on fondly. Finally, I would like to thank my family - I couldn’t have done it without you. My parents, Cindy and Ron, who raised me to be so curious and questioning, and supported my academic pursuits. Jack Goggins and Olivia Wong for their encouragement, child care, and home-cooked meals during the ﬁnal year of my dissertation research. My wife Julia for putting up with (and encouraging) me through all the long hours and stressful times during my graduate journey. Last but certainly not least - thanks to my son Russell for constantly reminding me what is truly important in life.

This dissertation was made possible thanks to the funding support from the SMART Schol- arship Program, NSF Rapid Grants (#1760193 and #1901699), and the NCSU Dissertation Completion Grant.

iv TABLE OF CONTENTS

List of Tables ...... vii

List of Figures ...... ix

Chapter 1 Introduction ...... 1 1.1 Motivation...... 1 1.2 Spontaneous Volunteer Convergence and Integration...... 3 1.3 Models of Spontaneous Volunteers in Disaster Response ...... 4 1.4 Document Outline...... 6

Chapter 2 Assigning Spontaneous Volunteers to Relief Efforts under Uncertainty9 2.1 Introduction...... 9 2.2 Literature Review...... 11 2.2.1 Disaster Operations Management...... 11 2.2.2 Volunteer Scheduling...... 13 2.2.3 Queuing Theory...... 15 2.3 Model Development...... 17 2.3.1 Stability...... 19 2.3.2 Optimal Policy ...... 20 2.4 Heuristic Policies...... 23 2.5 Simulation Model...... 25 2.6 Computational Analysis and Discussion ...... 26 2.6.1 Sensitivity Analysis...... 30 2.6.2 Policy Recommendation ...... 36 2.7 Conclusions...... 37 2.7.1 Limitations ...... 38

Chapter 3 Agent-Based Model of Spontaneous Volunteer Convergence ...... 39 3.1 Introduction...... 39 3.2 Spontaneous Volunteer Convergence Model...... 41 3.2.1 Purpose...... 41 3.2.2 Entities, State Variables, and Scales ...... 42 3.2.3 Process Overview...... 46 3.2.4 Agent Decisions to Volunteer...... 48 3.2.5 Site Choice Behaviors ...... 55 3.2.6 Model Outputs at the System Level ...... 57 3.3 Veriﬁcation and Validation ...... 57 3.3.1 Veriﬁcation ...... 57 3.3.2 Validation...... 58 3.4 Analysis and Discussion ...... 65 3.5 Conclusions...... 70

v 3.5.1 Limitations ...... 71 3.6 Acknowledgements...... 72

Chapter 4 Application of SV Convergence Model to Disaster Planning Scenarios . 73 4.1 Introduction...... 73 4.2 Research Questions and Extensions...... 75 4.2.1 Volunteer Reception Centers...... 76 4.2.2 Local Agency Coordination ...... 78 4.2.3 Volunteer Messaging...... 80 4.3 Scenarios, Results, and Discussion...... 82 4.3.1 Importance of Volunteer Reception Centers in SV Assignment . . . . . 83 4.3.2 Value of Local Agency Coordination...... 87 4.3.3 Messaging Strategies and Site Capacity ...... 90 4.3.4 Optimal Size of Weekly Messages...... 93 4.4 Conclusions...... 97 4.4.1 Limitations ...... 98

Chapter 5 Conclusion ...... 100 5.1 Dissertation Summary and Contributions...... 100 5.2 Future Directions ...... 102

References ...... 106

Appendices ...... 114 Appendix A Full Computational Results for Chapter 2 ...... 115 Appendix B Veriﬁcation of the SV Convergence Model ...... 119 B.1 Summary ...... 119 B.2 Veriﬁcation Scenarios...... 121 B.2.1 Scenario 1 - Population Demographics ...... 121 B.2.2 Scenario 2 - Age-Based Demographic Differences ...... 122 B.2.3 Scenario 3 - Varying Messaging and Site Choice ...... 123 B.3 Results...... 124 Appendix C ODD+D Protocol Applied to the SV Convergence Model ...... 127 Appendix D Computational Results for Chapter 4 ...... 141

vi LIST OF TABLES

Table 2.1 Summary of ﬁxed and 2k factorial design parameters...... 27 Table 2.2 Summary of sensitivity analysis conducted (min:increment:max). For example, (2.9 : 0.1 : 3.9) in Case S1 indicates that λ was varied as (2.9, 3.0, ..., 3.8, 3.9)...... 30

Table 3.1 List of relevant fixed baseline parameter values and calibrated parameter ranges selected for parameter calibration...... 60 Table 3.2 Summary of parameters varied during model calibration using simulation optimization. Best fit values for each parameter are listed for each of the three different disasters...... 62 Table 3.3 List of relevant baseline parameter fixed values and range of new randomized values. Ranges were selected to test the robustness of the normalized convergence curve under the best fit calibrated parameters...... 63 Table 3.4 Comparison of recorded normalized convergence data and simulated results for both fixed and random baseline parameters across each scenario. Simulated convergence data is presented as mean ± 95% confidence interval over 100 replications...... 64 8−2 Table 3.5 Summary of parameters selected for fractional factorial design (2V ) and their corresponding low and high values...... 67

Table 4.1 List of relevant baseline parameter values selected for scenario evaluation. 83 Table 4.2 Summary of parameters varied during evaluation of research questions. Descretized range for each parameter is listed as (min : increment : max). 84 Table 4.3 Example of prescriptive messaging based on different disaster management objectives for synthetic example in New Hanover County...... 95

Table A.1 Input parameters for each of the 2k baseline cases...... 116 Table A.2 Computational results of the 2k factorial base cases...... 117 Table A.3 Computational results of the sensitivity cases...... 118

Table B.1 Full list of baseline model parameters and descriptions from original SV Convergence Model...... 121 Table B.2 Full list of varied parameters for scenario 1 experiments...... 122 Table B.3 Full list of varied parameters for scenario 2 experiments...... 123 Table B.4 Full list of varied parameters for scenario 3 experiments...... 123 Table B.5 Average remaining demand and SV count across all Sites on selected days.125 Table B.6 Average population level behavioral intention and PBC score across selected days...... 126 Table B.7 Average cumulative count of SV turned away and average number of SV possess full site knowledge for selected days...... 126

vii Table C.1 The ODD+D protocol responses for the SV Convergence model, including the guiding questions, to present a comprehensive model description. 128

Table D.1 Summary of average total demand completion for 30 replications across variations in VRC activation day and VRC enforcement levels. Data represents the mean value ± 95% confidence interval...... 142 Table D.2 Summary of average total SV rejections for 30 replications across variations in VRC activation day and VRC enforcement levels. Data represents the mean value ± 95% confidence interval...... 143 Table D.3 Summary of average total demand completion for 30 replications across variations in initial site message size and percent successful coordination. Data represents the mean value ± 95% confidence interval...... 144 Table D.4 Summary of average total SV rejection for 30 replications across variations in initial site message size and percent successful coordination. Data represents the mean value ± 95% confidence interval...... 145

viii LIST OF FIGURES

Figure 2.1 Pictorial representation of the system...... 18 Figure 2.2 Optimal policy assignments for a two queue system with parameters λ = 3, γ = 0.5, V1 = V2 = 5, µ1 = 2, µ2 = 4, α1 = 6, α2 = 8, h1 = h2 = 20, d1 = 10, and d2 = 5...... 23 Figure 2.3 Box plot of average deviation from MDP policy for each of the four heuristic policies. Mean values are represented as green diamonds, the red line represents the median value, the blue box represents the 1st and 3rd quartile range, and the pluses represent any outliers...... 28 Figure 2.4 Average daily holding cost versus volunteer arrival rate (λ) for Case S1. 31 Figure 2.5 Average daily holding cost versus maximum volunteer capacity (V2) for Case S2...... 33 Figure 2.6 Average daily holding cost versus volunteer work rate (µ1) for Case S3. 34 Figure 2.7 Average daily holding cost versus demand arrival rate (α2) for Case S4. 34 Figure 2.8 Average daily holding cost versus holding cost per item per unit time (h2) for Case S5...... 35 Figure 3.1 SV Convergence Model GIS user interface as shown in Anylogic Uni- versity Edition version 8.5.2 ...... 43 Figure 3.2 Visual representation of agent attributes proposed by Lindner et al. [2017]. The colored boxes indicate how they are captured within our modelling framework. Attributes are either directly captured (green), indirectly captured (yellow), ignored (red), or left for future work (no box). 44 Figure 3.3 Simplified UML diagram of the SV Convergence agent-based model. The diagram illustrates the attributes of each agent, their functions/behaviors, and relationship to each other...... 45 Figure 3.4 High level flow of events captured within the framework which occur following a natural disaster...... 46 Figure 3.5 Population agent state chart representing the daily flow of agents and decisions...... 48 Figure 3.6 Comparison of normalized attitude curves (attitude (Ai,t) versus day). . . 50 Figure 3.7 Graph of transformation from intention (BI) to attempting volunteer behavior (P[Bi,t = 1]) for Linear (blue), Sigmoid (red), and Threshold (green) functions...... 53 Figure 3.8 Recorded normalized percentage of volunteers over four weeks following the (1) 2011 Tornado in Tuscaloosa, Alabama, (2) 2018 Hurricane Florence response in North Carolina, (3) 2009 Victorian Bushfires. . . . 59 Figure 3.9 Simulated and actual normalized convergence over four weeks following: (A) 2011 Tornado in Tuscaloosa, (B) Hurricane Florence in 2018, (C) The 2009 Victorian Bushfires...... 61

ix Figure 3.10 Standardized least squares model for total demand completion. (Left) Plot of actual by predicted values. (Right) List of response with the corresponding logworth and p-value indicating signiﬁcance...... 68 Figure 3.11 Standardized least squares model for total volunteer rejections. (Left) Plot of actual by predicted values. (Right) List of response with the corresponding logworth and p-value indicating signiﬁcance...... 69 Figure 3.12 Main effects plot for (Top) demand completion (D) and (Bottom) volunteer rejection (R) when population agents follow the score based site choice policy...... 70

Figure 4.1 Updated state chart of spontaneous volunteer flow. Model extensions are identified by red borders and bold red text...... 77 Figure 4.2 New flow of volunteer agent upon arrival to volunteer site. Changes to allow for additional screening and reassignment of spontaneous volunteers are circled in red...... 80 Figure 4.3 Comparison of old (blue) and new (orange) attitude functions over time, highlighting the saw-tooth shape in the new function...... 82 Figure 4.4 Heat map of average demand completion across variations in VRC enforcement and VRC activation day...... 84 Figure 4.5 Heat map of average total volunteer rejections across variations in VRC enforcement and VRC activation day...... 86 Figure 4.6 Heat map of average demand completion across variations in percent successful coordination (PSR) and initial site message size...... 88 Figure 4.7 Heat map of average total volunteers rejected across variations in percent successful assignment and initial site message size...... 89 Figure 4.8 Summary of system level outputs for notional example evaluating site to agent messaging scenario. Conservative messaging is shown in blue and an aggressive strategy in red...... 91 Figure 4.9 Approximate distribution of PBC values across all population agents. The initial distribution of PBC values is shown in grey. The final distribution for the conservative and aggressive messaging strategies are shown in blue and red respectively...... 93 Figure 4.10 Pareto Front of simulated solutions for objectives: (1) demand completion and (2) volunteer rejection...... 97

x CHAPTER

INTRODUCTION

1.1 Motivation

The United Nations Office for Disaster Risk Reduction defines a disaster as a serious disruption of the functioning of a community or a society, leading to one or more of the following: human, material, economic and environmental losses and impacts1. Hurricanes, such as Harvey (2016) and Florence (2018), are among the most expensive and devastating natural disasters ever in the US. With the rising frequency of natural disasters, there is an increasing necessity to study disaster management. Rawls and Turnquist[2012] define Disaster Operations Management (DOM) as the sequence of operations that seek to prevent or reduce the injuries, fatalities, and damages resulting from a disaster; and facilitate the recovery from such an event. After major disasters, large numbers of people unaffiliated with traditional emergency response organizations converge on the scene to offer assistance [Fernandez et al. 2006]. These individuals respond on impulse immediately following disaster events and are referred to as

1https://www.unisdr.org/we/inform/terminology

1 spontaneous volunteers (SVs) [Lowe and Fothergill 2003]. Victims of disaster often have an enormous need for basic services such as food, relief supplies, and information. Spontaneous volunteers can be utilized to fulfill a significant portion of tasks that provide these needs during the early phases of disaster response [Whittaker et al. 2015]. Community residents, local organizations, and emergent groups constitute important resources that can mobilize rapidly when a disaster occurs [Goltz and Tierney 1997]. However, a 2016 study found that 22 of the 50 largest cities within the United States had no mention of volunteering in their disaster response plans [Rivera and Wood 2016]. The unpredictability, magnitude, uncertainty, and complexity of disasters make it difficult to develop efficient disaster management plans [Hoyos et al. 2015]. Although the fields of social sciences and humanities have provided an abundance of literature, there has been an increasing recognition of the need to study OR/MS DOM issues [Altay and Green 2006]. It has only been within the last 5-10 years that serious efforts have been made to bridge the gap between OR/MS and DOM [Altay and Green 2006, Galindo and Batta 2013, Hoyos et al. 2015]. However, SV convergence and integration have seen very little research in OR/MS DOM compared to other areas, such as material pre-positioning and evacuation planning. The current lack of research in this area of disaster management is problematic when one observes the increased frequency and intensity of disasters that are occurring within the United States in both historical hazard zones, but more importantly in places that have not traditionally experienced disasters [Rivera and Wood 2016]. By improving the planning, integration, and utilization of these spontaneous volunteers, traditional emergency response agencies are better able to address their central missions [Volunteer Florida 2005].

2 1.2 Spontaneous Volunteer Convergence and Integration

Following a disaster, affected communities receive an outpouring of support both in donations and offers of assistance. Fritz and Mathewson[1957] defines this mass movement of people and supplies to the affected area as volunteer convergence and material convergence, respectively. According to Shaskolsky[1965], volunteerism takes four forms in disaster situations (anticipated individual, anticipated organization, spontaneous individual, and spontaneous organization). Spontaneous individual volunteers often live within or near affected areas and can even be disaster survivors [Abualkhair et al. 2019]. Spontaneous offers to help during and following disasters are well documented and are strongly influenced by the amount of media coverage an event receives [Cottrell 2012]. Such coverage evokes viewers’ compassion for the victims and gratitude that their community was spared. Following a natural disaster, hundreds or thousands of people will feel compelled and energized to take action. For example, Lowe and Fothergill [2003] reports that 15,000 volunteers helped during the two-and-a-half week period following the September 11th attacks in New York City. The response to Hurricane Katrina in 2005 reportedly attracted 60,000 volunteers to New Orleans [Townsend et al. 2006]. Manpower planning, within the context of OR/MS DOM, has received minimal attention from an academic perspective [Simpson and Hancock 2009, Caunhye et al. 2012]. While traditional workforce scheduling problems have been studied extensively [e.g. Van den Bergh et al. 2013], there has been much less work that focuses on labor assignment from a DOM perspective. A comparison between traditional labor assignment and volunteer assignment is given by Sampson[2006]. Paciarotti et al.[2018] present an analysis of the management of spontaneous volunteers to identify strengths and weaknesses in current strategies. Falasca and Zobel[2012] identify several operational characteristics unique to volunteer planning and scheduling for disaster relief purposes that ought to be considered from an OR/MS modeling perspective. The need to address the implications of the rise in unaffiliated, spontaneous volunteering is one of the major takeaways from Emergency Volunteering 2030 [McLennan and Kruger 2019]. Many

3 organizations are not structured to incorporate SVs effectively. In fact, 58% of the organizations surveyed by Barraket et al.[2013] had turned away a SV during a past response. Many of the current management practices, as highlighted by FEMA and the Red Cross, focus on high-level guidelines [Points of Light Foundation 2004, Cross 2010]. Where operational guidelines do exist, they fail to elaborate on key points. For example, volunteer managers should "Refer unafﬁliated volunteers to appropriate response agencies after initial screening" [Points of Light Foundation 2004]. The referral is left to the best judgement of the volunteer manager at that volunteer reception center (VRC) or volunteer site. The manager must interpret the meaning of appropriate as it relates to the situation. Some of the major challenges for volunteer managers during disaster response include communicating with SVs, prioritizing the tasks of SVs, and incorporating SVs [Harris et al. 2017]. A quote from the Unafﬁliated Volunteers in Response and Recovery Handbook [Volunteer Florida 2005] captures this struggle well:

“As surely as disasters will happen, volunteers will come. Many emergency managers admit to being unnerved by the prospect of coping with convergent volunteers because there are so many unknowns. How many will come and when? How will they know where to go and what to do? Who will manage them?"

Management plans and more structured efforts to integrate SVs are necessary for effective disaster response [Sauer et al. 2014].

1.3 Models of Spontaneous Volunteers in Disaster Response

The popular research trends in OR/MS DOM focus primarily on mathematical programming models [Hoyos et al. 2015]. Many optimization models exist [e.g. Falasca and Zobel 2012, Lassiter et al. 2015, Rauchecker and Schryen 2018], seeking to balance skill matching with the needs of volunteer organizations. These models assume a deterministic pool of volunteers

4 with corresponding skill sets. Where research exists on the subject of disaster volunteerism, it tends to focus on the management of certified and affiliated volunteers and does not adequately address the important issue of spontaneous volunteers [Rivera and Wood 2016]. Although these models can be useful to optimally assign registered volunteers, it does not capture the randomness associated with truly spontaneous volunteers. Empirical studies have demonstrated that spontaneous volunteerism is characterized by various forms of uncertainty. For example, the times between volunteer arrivals, the amount of time volunteers contribute on a given day, and the sizes of volunteer groups who arrive and depart the relief effort together [Lodree and Davis 2016]. Work on the recovery stage, as well as on queuing theory methodologies, has not been given much attention [Hoyos et al. 2015]. Mayorga et al.[2017] were one of the first to attempt to capture the uncertainty seen in spontaneous volunteers. They proposed a Markov Decision Process (MDP) framework that captures the unique characteristics of spontaneous volunteer (SV) arrival and departure times. They model the SV assignment problem as a parallel queuing system with random server (i.e., volunteer) arrivals and abandonments, in which a deterministic amount of work known a priori is to be completed. In disaster response, the work remaining (demand) is often unknown until surveys of the affected area can be completed. Abualkhair et al.[2019] developed an extension to this problem by also considering uncertainty in beneficiary arrivals. They developed an agent-based model to compare different heuristic assignment policies. However, both of these models are limited to a single volunteer site. Much of the existing OR/MS DOM literature fails to account for the social and behavioral aspects when modeling spontaneous volunteers. Agent-based modeling (ABM) has been shown to be a valuable tool in capturing individual behavior in some areas of DOM, such as evacuation planning. Narzisi et al.[2007] develop an evacuation model that includes a heterogeneous population with unique attributes such as level of worry. Others consider uncertainty in human behavior by simulating different decisions taken by an agent involved in an evacuation [Massa-

5 guer et al. 2006]. ABM has also been used to model resource allocation during relief distribution [Das and Hanaoka 2014]. Simulations representing and evaluating spontaneous volunteer assignments are very limited. Dressler et al.[2016] evaluate the use of affiliated volunteers in response to flooding in Germany. Abualkhair et al.[2019] consider volunteer assignments within a single site. However, the volunteers are homogeneous and the number of available volunteers is known. Lindner et al. [2017] present a framework for the development of ABM of spontaneous volunteers. They identify twenty-five attributes necessary to capture the complexity of spontaneous volunteer agents. Lindner et al.[2018] develop a simple ABM implementation of SV convergence, as a proof of concept, based on their 2017 paper. The proof of concept model includes many assumptions and simplifications related to SV behaviors and heterogeneity of the agent population. Additional discussion of social science literature necessary to capture the complexities of individual behavior and motivation can be found in Chapter3.

1.4 Document Outline

This dissertation develops models to simulate the convergence and integration of spontaneous volunteers under uncertainty in disaster response. Results from the models are intended to inform SV policies and practices at both the strategic and operational level, with a goal to improve disaster resiliency. Volunteer managers are responsible for assigning incoming volunteers to available tasks within a volunteer site or reception center. They must make important decisions on assignment with very little information and often rely on "gut instincts". In Chapter2, we present a strategy for ﬁnding the optimal assignment of spontaneous volunteers within a single disaster relief location under uncertainty. An MDP is formulated and solved using value iteration to obtain the optimal policy. The optimal policy is compared against a variety of heuristic policies using

6 a discrete event simulation. Chapter2 extends the work done by Mayorga et al.[2017] by relaxing demand assumptions, specifically allowing for uncertainty in demand. We attempt to balance implementation complexity with available resources during a disaster to find the best implementable solution given the often hectic situations following a natural disaster. The chapter is an accepted research article in OMEGA: Journal of Management Science. With respect to assignment of SVs within a single volunteer site, the following research questions are defined:

• What is the shape of the optimal policy and is it implementable in disaster response?

• How do simple heuristic policies compare to the optimal policy?

• How robust are these policies across variations in volunteer arrival rates and other system changes?

• Do heuristics perform well under both deterministic and stochastic demand assumptions?

Through field work and practitioner interviews, we identified a need for a model that captures volunteer-volunteer and site-volunteer communications as they relate to volunteer convergence. In Chapter3, we develop an agent-based model framework to represent spontaneous volunteer convergence following a disaster. Our model framework bridges the gap between operations research/management science and social science. The framework considers a heterogeneous population of agents, each with unique attributes such as motivation, opinion, and site choice behavior. It provides the first step toward developing a tool that can be used to evaluate volunteer integration plans. We validate the model through calibration and external practitioner review. This chapter has been submitted to Production and Operations Management for consideration in the Disaster Management Department. With respect to modeling SV convergence and integration, we address the following research questions:

• Can we effectively model SV convergence while capturing individual motivation, site choice, and messaging strategies?

• Which parameters have a signiﬁcant effect on demand completion?

• Is the model robust enough to simulate different real world disaster scenarios?

7 Chapter4 provides an analysis of strategic and operational level questions using the SV convergence model. The research questions analyzed in this chapter were developed in collaboration with practitioners. The three scenarios evaluated represent areas of interest to emergency managers, based upon multiple round table discussions. The results of the initial scenarios will be used to help communicate the value of as a potential decision support tool and also to continue the discussion regarding SV integration and perceptions. With respect to disaster policies and plans, we address the following research questions:

• What are the beneﬁts/drawbacks of a Volunteer Reception Center?

• How does local agency coordination improve disaster response?

• What is the impact of different messaging strategies on convergence?

Finally, Chapter5 summarizes the dissertation, outlines contributions to the ﬁeld of OR/MS and DOM, and discusses directions for future research.

8 CHAPTER

ASSIGNING SPONTANEOUS VOLUNTEERS TO RELIEF EFFORTS UNDER UNCERTAINTY

2.1 Introduction

During and after major disasters, large numbers of people unafﬁliated with traditional emergency response organizations converge on the scene to offer assistance [Fernandez et al. 2006]. Motivated by the desire to do something for those in need, these individuals respond on impulse immediately following disaster events and are referred to as spontaneous volunteers [Lowe and Fothergill 2003]. Spontaneous offers to help during and following disasters are well documented, and are strongly inﬂuenced by the amount of media coverage an event receives [Cottrell 2012]. For example, Lowe and Fothergill[2003] report that 15,000 volunteers helped during the two- and-a-half week period following the September 11 attacks in New York City. The response to Hurricane Katrina in 2005 attracted 60,000 volunteers to New Orleans [Townsend et al. 2006].

9 Research shows that spontaneous volunteers are capable of positively contributing to relief efforts during the aftermath of disasters by performing a variety of services, including search and rescue, distribution of relief items, and the assessment of community needs [e.g., Whittaker et al. 2015]. While spontaneous volunteers can be a valuable resource, they are often ineffectively used and can potentially hinder emergency operations by creating health, safety, and security concerns. Furthermore, spontaneous volunteers require supervision, which can distract professional responders from their duties that directly serve disaster survivors [Fernandez et al. 2006]. Oftentimes, the services of spontaneous volunteers are refused purely because volunteer organizations are ill-equipped to manage them. A survey of non-governmental voluntary organizations (NVOs) found that the use of spontaneous volunteers is widespread, but NVOs are not necessarily structured to effectively engage them. Improved strategies for incorporating spontaneous volunteers into organized relief efforts are needed in order to achieve safe and responsive disaster management [Fernandez et al. 2006]. In this chapter, we consider the problem of assigning spontaneous volunteers arriving at a Volunteer Reception Center (VRC) to multiple disaster relief tasks. Such volunteers are commonly utilized to sort donations and distribute relief supplies (e.g. food, clothing, clean-up kits). One characteristic that distinguishes spontaneous volunteer assignment from all other forms of labor scheduling is that spontaneous volunteers randomly join and abandon the operation [Lodree and Davis 2016]. As such, we represent the spontaneous volunteer assignment problem as a multi-server parallel queuing system where the servers (spontaneous volunteers) randomly arrive and depart the system. Within this framework, optimal policies for assigning spontaneous volunteers to tasks are derived from a continuous time Markov Decision Process (MDP) model. In addition, we evaluate a variety of experimental cases using a discrete event simulation (DES) model in order to examine the performance of heuristic assignment policies relative to the optimal policy. We identify an effective assignment policy that is also easy to implement, and provide insights for individuals who may be in charge of managing spontaneous

10 volunteers in real-world disaster response environments. The remainder of this chapter is organized as follows. In Section 2.2, we review academic literature related to disaster operations management, volunteer scheduling, and server assignment in queuing theory. In Section 2.3, we formulate the spontaneous volunteer assignment problem as a Markov Decision Process (MDP), and discuss sufﬁcient conditions for the existence of steady-state solutions. In Section 2.4, we introduce and describe practical heuristic policies to be tested against the MDP policy. Section 2.5 provides a brief overview of the simulation model used to compare the performance of the MDP and heuristic policies. Section 2.6 details the experimental analysis generated for the simulation study and the results. Concluding remarks and implications for volunteer managers are provided in Section 2.7.

2.2 Literature Review

This study is related to three areas: (1) disaster operations management, (2) volunteer scheduling, and (3) server assignment in queuing systems.

2.2.1 Disaster Operations Management

The disaster operations management (DOM) literature has experienced rapid growth since the year 2001. For example, Gupta et al.[2016] surveyed 268 papers from among 25 Operations Research / Management Science (OR/MS) journals and reported an increase in the annual publication rate of 2.67 in 2001 to 33.67 in 2014 (these results are based on a three-year moving average). It is safe to say that DOM is a mainstream application area of OR/MS based on the numerous survey papers dedicated to the subject [e.g. Altay and Green 2006, Galindo and Batta 2013, Habib et al. 2016]. Moreover, multiple focus areas with critical mass have emerged from DOM literature, and review papers dedicated to these subjects have also been published. The most prevalent DOM topics include: inventory management [Ortuño et al. 2013], facility

11 location [Leiras et al. 2014], and relief distribution [Anaya-Arenas et al. 2014]. Manpower planning, however, is another important topic within the context of DOM that has received very limited attention from an academic perspective [Simpson and Hancock 2009, Caunhye et al. 2012]. Fritz and Mathewson[1957] first defined the mass movement of people and supplies to the affected area as volunteer and material convergence respectively. Volunteer management focuses on managing these volunteer resources through the entire process. Volunteer management plans cover the engagement, recruitment, placement, orientation, training, supervision, recognition, and evaluation of volunteers. Many software solutions exist (e.g. volgistics, EveryAction, MobileServe, etc.) to help volunteer managers track and schedule registered volunteers. Additionally, these software solutions are useful after disasters due to their reporting and database capabilities. However, they do not necessarily assist with the immediate assignment of incoming spontaneous volunteers following a natural disaster. Many of the current management practices related to spontaneous volunteers, as highlighted by FEMA [Points of Light Foundation 2004] and the Red Cross [Cross 2010], focus on high level guidelines. Where operational guidelines do exist, they fail to elaborate on key points. For example, one of FEMA’s concepts of operation suggests volunteer managers should “Refer unaffiliated volunteers to appropriate response agencies after initial screening" [Points of Light Foundation 2004]. However, the referral is left to the best judgement of the volunteer manager at that volunteer reception center (VRC) or volunteer site. The manager must interpret the meaning of appropriate as it relates to the situation. Prioritizing the tasks of SVs is considered to be one of the major challenges for volunteer managers during disaster response [Harris et al. 2017]. This chapter addresses a known gap in the DOM literature by considering the problem of assignment/placement for spontaneous volunteers after disaster events from an operational perspective.

12 2.2.2 Volunteer Scheduling

While traditional workforce scheduling problems have been studied extensively [e.g. Van den Bergh et al. 2013], there has been much less work that focuses on labor assignment from a DOM perspective. One of the most impactful distinctions between traditional and DOM manpower planning is the role of volunteer labor. In particular, volunteers complete a signiﬁcant portion of the tasks performed during the early phases of disaster response [e.g. Whittaker et al. 2015]. The labor in classical personnel scheduling is supported by a paid workforce, which tends to be more stable and predictable compared to volunteer labor [Sampson 2006]. Falasca and Zobel[2012] identify several operational characteristics that are unique to volunteer planning and scheduling for disaster relief purposes that ought to be considered from an OR/MS modeling perspective. One such characteristic is the importance of satisfying volunteers’ preferences, including the type of work, the times they want to work, how long they are willing to work, and with whom they prefer to work (volunteers often contribute to relief efforts in groups, e.g., churches, sports teams, families). The issue of prioritizing volunteer preferences applies to volunteer scheduling in general [e.g. Sampson 2006], not just volunteer scheduling for disaster relief. However, manpower planning for DOM may also require assigning volunteers across geographically dispersed locations, which for the most part, is irrelevant when it comes to labor scheduling in non-DOM contexts. Falasca and Zobel[2012] incorporate the above features into a bi-objective optimization model that seeks to balance the conﬂicting objectives of minimizing unmet task demands and maximizing volunteer preferences. Lassiter et al.[2015] also take unmet task demands and volunteer preferences into account within the context of humanitarian relief, but extend Falasca and Zobel[2012]’s deterministic model in an important way: they consider task uncertainty and propose a robust optimization model to handle this uncertainty. In both Falasca and Zobel[2012] and Lassiter et al.[2015], there is no uncertainty associated with labor capacity or capabilities. However, empirical studies have demonstrated that spontaneous volunteerism is characterized by various forms of uncertainty. Examples include the times

13 between volunteer arrivals, the amount of time volunteers contribute the relief efforts on a given day, and the sizes of volunteer groups who arrive and depart the relief effort together [Lodree and Davis 2016]. From this perspective, the studies by Falasca and Zobel[2012] and Lassiter et al.[2015] are appropriate for scheduling affiliated volunteers where uncertain labor capacity is less of an issue. However, assignment decisions that pertain to spontaneous volunteers require a different approach. Mayorga et al.[2017], the study closest to ours, proposes a framework that captures the unique characteristics of spontaneous volunteers, namely uncertainty in volunteer arrival and departure times. Specifically, they model the spontaneous volunteer assignment problem as a parallel queuing system with random server (i.e., volunteer) arrivals and abandonments. The control problem is formulated as a MDP, and a policy iteration algorithm is used to generate optimal policies for problem instances. Mayorga et al.[2017] also conduct a computational experiment in which the performance of several practical heuristic policies are examined through simulation. They find that simply assigning volunteers to the queue with fewest volunteers generally works well as an assignment policy. This chapter generalizes the deterministic demand model of Mayorga et al.[2017] by considering stochastic demand streams, allowing for the representation of more complex disaster relief tasks. By doing so, we now have to contend with deriving an appropriate stability condition for the queuing system (see Section 2.3.1), which was not necessary for the deterministic demand case described above. In summary, we have identified only three papers in the academic DOM literature that address manpower planning for disaster relief from an operational perspective: Falasca and Zobel[2012], Lassiter et al.[2015] and Mayorga et al.[2017]. Furthermore, only one of the three considers random volunteer arrival and abandonment processes [Mayorga et al. 2017], and is therefore relevant to the spontaneous volunteer assignment problem. We conclude our discussion here by noting that volunteer scheduling literature has occurred in contexts other than DOM. A general framework for volunteer scheduling is laid out by Sampson[2006], who

14 applies that framework to the problem of scheduling reviewer assignments for an academic conference. Other settings in which volunteer labor assignment research has been applied are an annual music festival [Gordon and Erkut 2004] and a bike sharing program [Kaspari 2010]. However, these applications do not consider labor uncertainty.

2.2.3 Queuing Theory

Optimal control of queues via server assignment is a widely studied class of problems in queuing theory with numerous variations. As such, a comprehensive review of this literature is beyond the scope of our discussion. Instead, we review a few representative studies that focus on the control of parallel queuing systems and highlight related areas of the queuing literature. A basic framework for server assignment in parallel queuing systems involves multiple customer classes, where each queue is dedicated to serving a speciﬁc class. The control problem entails dynamically allocating a ﬁxed pool of heterogeneous servers among the queues with the goal of minimizing waiting time. Squillante et al.[2001] consider this problem under the assumption of Poisson arrivals and exponential service times, and propose threshold-based priority policies

related to the well-known cµ rule. Variations of the cµ policy have since been applied to many different queuing systems. For example, a generalized version of the cµ rule is optimal when holding costs are nonlinear but convex [Van Mieghem 1995]. More recently, Cao and Xie[2016] derived conditions under which the cµ policy is optimal for a single server queuing network with two customer classes, where customers from the lower class are upgraded to the higher class after a random amount of

time. There are also some cases where the optimality of cµ is preserved for parallel queuing systems with multiple servers. Consider, for instance, the N-network that consists of two types of customers and servers; a dedicated server that can serve one customer type and a fully ﬂexible server that is capable of serving both customer types. Bell et al.[2001] and Saghaﬁan et al.

[2011] prove that the cµ rule is optimal under certain conditions in this setting, while Down and

15 Lewis[2010] extend the result to an N-Network with multiple servers of each type (dedicated and fully flexible) and customer upgrades. All of the above-mentioned studies involve the assignment of a fixed pool of servers where, as is the case in most papers, service rate is the only stochastic characteristic of the servers. However, the present study considers server assignment policies where not only service rates, but also server availability, are stochastic. Besides Mayorga et al.[2017], only a limited number of papers deal with optimal control of queues in the presence of unreliable servers. To our knowledge, only two papers do so within the context of assigning servers to parallel queues: Andradóttir et al.[2007] and Saghafian et al.[2011]. Both papers derive optimal policies for assigning unreliable servers to queues involving multiple customer classes. Here, servers are unreliable in the sense that they fail at random points in time and remain unavailable for random periods until they are repaired. Saghafian et al.[2011] establish conditions under which a version of the cµ rule is optimal for a generalization of the N-network that has three servers (instead of two): one dedicated, one fully flexible, and one partially flexible (the authors refer to this extension as the W-network). Andradóttir et al.[2007], on the other hand, investigate the prospect of using flexible servers to compensate for server unreliability under the long-run average throughput objective. Andradóttir et al.[2008] also consider the throughput objective, but within the context of a tandem (not parallel) queuing system. Wu et al.[2006] and Wu et al.[2008] also examine the assignment of unreliable servers in a tandem system, but with the objective of minimizing long-run average holding costs. The remaining papers that address the control of queues with unreliable servers do so by routing customers [Efrosinin 2013, Özkan and Kharoufeh 2014] or designing server repair policies [Wu et al. 2014, Peschansky and Kovalenko 2016] as opposed to server assignment policies. The present work differs from these in very fundamental ways in terms of the how server unreliability is represented. First, we consider servers which arrive randomly over time from an infinite population of spontaneous volunteers. This is unlike the queuing systems

16 mentioned above where the pool of servers is ﬁxed and ﬁnite. Additionally, after remaining in the system for random amounts of time, all volunteers eventually abandon the system forever. We assume they do not return after random periods of inactivity (due to repair) as in previous studies.

2.3 Model Development

To model this problem, we consider a queuing system where each queue i,(i = 1,...,N), represents a different job or task that volunteers may be assigned to work on, as shown in Figure 2.1. Spontaneous volunteers arrive to the system according to a Poisson process with

rate λ. As identified in the literature, it has been shown that there is typically an abundance of volunteers following a disaster [e.g. Cottrell 2012, Lowe and Fothergill 2003, Townsend et al. 2006], effectively representing an infinite population of potential servers. It is assumed that the volunteer manager is immediately able to assign volunteers to available queues according to a specified control policy. Available queues are all queues not currently at maximum volunteer capacity. Volunteer capacity refers to the number of volunteers who can actively contribute to the relief efforts. Capacity could be due to resource requirements (e.g. carts to transport items), limited space (e.g. a gymnasium being used as a distribution center), or safety concerns.

Volunteers’ times in system until abandonment are exponentially distributed with mean 1/γ. We assume work or demand for services (e.g. request for relief items, or donations to be sorted)

arrives to each queue separately, according to a Poisson process with rate αi.

The number of volunteers working in queue i can be represented as vi, with the maximum volunteer capacity represented as Vi. Work is completed at each queue according to an expo-

nential additive service rate of viµi; that is, volunteers are assumed to work on a single task

collaboratively. The work or demand for services at each queue i is represented by di. The hold-

ing cost per unit time for a job of type i is represented as hi, which for the purposes of this study

17 does not necessarily reﬂect the inventory holding cost to the relief organization. Alternatively,

hi can be thought of as a representation of the relative importance of speciﬁc goods or services during disaster response. For example, consider two tasks related to material convergence: (1) ofﬂoading truckloads of donations and (2) sorting/organizing donations. We expect volunteer

managers to assign importance such that h1 >> h2 in most cases. It is assumed that if all queues reach the maximum allowable capacity, volunteers will be turned away from the reception center or directed to another relief organization. Demand may continue to accumulate without being bounded. Volunteers remain idle or participate in ancillary tasks, such as cleaning up the work area, while they wait for additional work to arrive. Immediate reassignment of spontaneous volunteers is not considered.

Figure 2.1: Pictorial representation of the system.

We can deﬁne the feasible state space of the system described above as S = { v,d | vi ∈

(0,...,Vi), di ∈ (0,...,∞),i = 1 ... N }. The state of the system at time t can be deﬁned by

(v(t),d(t)), where v(t) = {v1(t),v2(t),...,vN(t)} and d(t) = {d1(t),d2(t),...,dN(t)} where

18 vi(t) and di(t) refer to the number of volunteers working and the number of unﬁnished units of work remaining at queue i at time t respectively. Feasible volunteer allocations include all

queues where vi < Vi. As described above, the state transitions are all Markovian and therefore this system can be formulated as a continuous time Markov decision process (MDP). The transition rates of the process are state and action dependent. Conditions to ensure stability of this system are provided in the following section.

2.3.1 Stability

In this section we derive conditions to guarantee that an assignment policy exists which maintains

system stability for a set of input parameters, i.e. a steady-state solution can be found (E[di] < ∞ as t → ∞). While the overall system is complex, we can break the model down into subsystems to more easily derive stability conditions. First, consider a random routing policy that thins

the volunteer arrival rate into N independent arrival streams with probability pi = 1/N, where

∑i pi = 1. The result is N subsystems, each with a volunteer arrival rate λ pi. The demand at each of the N queues can be represented as an M/M/1 queue with Markov modulated service rates. The Markov modulated service rates are determined by an underlying Birth Death (BD) process representing the number of volunteers. Births occur according to the rate volunteers arrive to the

queue, λ pi, and deaths occur according to volunteer abandonment, viγ. The expected number of volunteers in each queue is independent of the amount of work in that queue for this policy. Taking a closer look at the demand process, it has been shown by Núñez Queija[1997] that for M/M/1 queues with Markov Modulated service rates that follow a BD process, the system is stable if and only if

Vi αi < ∑ πiviµi, (2.1) i=0 where πi in Equation (2.1) is the steady state probability of being in state i of the BD system,

19 which represents the number of volunteers in queue i. Equation 2.1 can be rewritten as αi <

µiE[vi]. That is, the arrival rate must be less than the average service rate to maintain stability. Given that the volunteer process in queue i is a ﬁnite BD process, we can solve for the expected number of volunteers in queue i using known BD stability equations as

 Vi  ρi Vi Vi ρnn 1   i  Vi!  E[vi] = πin = n = ρi 1 − n . (2.2) ∑ ∑ n! Vi ρ  Vi ρ  n=0 n=0 ∑ i  ∑ i  n=0 n! n=0 n!

λ pi The queue intensity ρi in Equation (2.2) is represented as ρi = γ for volunteers in subsystem i. For the whole system to maintain stability in steady state, it follows that each subsystem

must satisfy αi < µiE[vi]. Given a set of input parameters (λ,α, µ,γ,V) and a random thinning policy such as the one discussed above, we meet sufﬁcient conditions for system stability if

 Vi  ρi    Vi!  αi < µiρi 1 − n for i = 1,...,N. (2.3)  Vi ρ   ∑ i  n=0 n! Essentially, the expected service rate for each relief task must be fast enough to keep up with task arrival rates for a given volunteer assignment policy. If the sufﬁcient conditions for stability are met, we have identiﬁed an assignment policy that maintains system stability. Thus, an optimal assignment policy is also guaranteed to result in a stable system. We discuss the process of developing an optimal assignment policy for a set of input parameters in the next section.

2.3.2 Optimal Policy

Given a stable system with a feasible assignment policy, we next ﬁnd the optimal assignment solution. The objective is to minimize the long-run average holding cost of the system. We note that solutions to steady state problems tend to be simpler and easier to implement than

20 ﬁnite horizon or non-stationary solutions, which is beneﬁcial in the case of relief efforts.

Additionally, note that even though the true arrival rate (λ) is likely to be time-varying, analyzing steady state models can indeed be helpful and tend to provide conservative results [Green 2004]. To ﬁnd the optimal policy, we solve an equivalent discrete time problem by employing uniformization, as detailed in Bertsekas[2011]. We deﬁne the maximum transition rate as:

N Γ = λ + ∑i=1 (αi +Viui +Viγi). Next, we deﬁne the recursive optimality equation (Bellman equation) for the discrete-time

equivalent ﬁnite horizon problem, Jk(v, d), as the value of being in state (v, d) with k < K

periods left-to-go out of K, and J0 represents the terminal cost.

( 1 N N Jk(v,d) = ∑ hidi + λ min(RiJk−1(v,d)) + ∑ αiJk−1(v,d + ei) Γ i=1 i=1 N N + ∑ µiviCiJk−1(v,d) + ∑ γiviJk−1(v − ei,d) (2.4) i=1 i=1 " N # ) + Γ − λ − ∑ (αi + µivi + γvi) Jk−1(v,d) i=1

The ﬁrst summation in Equation (2.4) corresponds to the total holding cost incurred at each queue. The second term denotes the allocation decision for the next arrival given that we are in state (v,d). The third term accounts for the arrival of work to each queue. The fourth term accounts for the completion of units of work, where the completion rate is proportional to the number of volunteers working at that queue. The ﬁfth term accounts for the abandonment of volunteers proportional to the number working at each queue. Finally the last term ensures that

all transition rates add up to Γ, which is required for uniformization. Transformation operators

Ri and Ci were included to simplify Equation (2.4). These operators ensure that the correct value

function is chosen based on the feasibility conditions. Speciﬁcally, Ri ensures that upon arrival,

the volunteer may be sent to queue i if that queue has not reached its maximum capacity; Ci

ensures that volunteers complete work with rate µi as long as there is work to complete. The

21 transformation operators are deﬁned in Equation 2.5.

  Jk(v + ei,d) if v_i 0 CiJk(v,d) = (2.5b)  Jk(v,d) otherwise

Next, we initialize the value function such that Jk(v,d)=0 for all (v,d) ∈ S , then using the

recursive optimality equation (2.4) we apply the value iteration algorithm until max(Jk(v,d)-

Jk−1(v,d))-min(Jk(v,d)-Jk−1(v,d))≤ ε. For tractability of the value iteration algorithm we

truncate the state space by limiting di ≤ Di. Here Di is chosen such that the probability of demand being turned away due to truncation in steady state is less than a small percentage (e.g. 3%). The value iteration algorithm was written and run in Matlab, on a 3.60 GHz Intel(R) Core(TM) i7-4790 CPU machine with 16.00 GB of RAM. The run times to convergence were approximately three minutes for cases with 100,000 states. The MDP policy is speciﬁed by the volunteer assignment to queue i, which minimizes the right hand side of Equation 2.4. We focus on minimizing the holding cost in the MDP equation as opposed to maximizing reward for completion to avoid scenarios in which a queue becomes completely inundated with demand. This would represent a severe material convergence issue, and would not be considered “optimal” in real world scenarios. An example of the MDP policy can be seen in Figure 2.2. When there are two volunteers at queue 1 and three volunteers at

queue 2 (v1 = 2,v2 = 3), the next volunteer should be assigned to queue 1. The MDP policy

reports an infeasible assignment when v1 = V1 and v2 = V2. In this case, the volunteer would not enter the system.

22 d1= 10, d2= 5 7 Queue 1 6 Queue 2 Infeasible

1 Number of Volunteers at Queue 2 0

-1 -1 0 1 2 3 4 5 6 7 Number of Volunteers at Queue 1

Figure 2.2: Optimal policy assignments for a two queue system with parameters λ = 3, γ = 0.5, V1 = V2 = 5, µ1 = 2, µ2 = 4, α1 = 6, α2 = 8, h1 = h2 = 20, d1 = 10, and d2 = 5.

The MDP policy generated using the recursive optimality equation is state dependent and thus is hard to characterize based solely upon the input parameters of the system. Additionally, implementing such a complex policy during disaster response is difﬁcult. Small to mid-size volunteer reception centers are often not equipped with technology capable of assessing the current system and producing complex optimal policies in real time. Instead, we propose the use of heuristic policies, which are easier to implement in practice and test their performance relative to the MDP policy.

2.4 Heuristic Policies

In this section we deﬁne four policies to compare to the MDP policy developed in the previous section. These heuristic policies act as alternatives to the MDP policy and are more easily implemented in practice. These policies come from both common sense assignment practices

23 and existing literature on queuing models, with the caveat that they need to implementable at a reception center during disaster response. Accordingly, we deﬁne the following heuristic policies:

Fewest Volunteers (FV): An arriving volunteer is assigned to the queue with the fewest number of volunteers. In the event of a tie between multiple feasible queues, the volunteer is assigned to one of the queues randomly with equal probability. This policy is very attractive for use in disaster relief because it is extremely easy for a volunteer manager to implement and only requires basic system knowledge. The FV policy is analogous to the “join the shortest queue policy" commonly used to minimize the total number of customers in the system at any time.

Largest Weighted Demand (LWD): An arriving volunteer is assigned to the queue with the

largest weighted demand from the subset of feasible queues which meet the condition vi < Vi and regardless of any other system parameters. Weighted demand for queue i is calculated as

dihi. This policy is similar to the largest demand (LD) policy and should also be easy to implement in practice. The benefit of LWD over LD is that the volunteer coordinator can impart the relative importance of each task during volunteer assignment. This policy can be thought of simply as trying to “put out the biggest fire first”.

Largest Queue Clearing Time (LQCT): An arriving volunteer is assigned to the queue with the largest weighted queue clearing time at the time of assignment. Note that this is not the actual queue clearing time for the queue as the number of volunteers is dynamic and service rates are stochastic. This policy is similar to the D/V policy evaluated by Mayorga et al.[2017]. These types of policies are a mix of FV and LWD in that they ensure each task is staffed, when possible, before balancing for workload. The expected queue clearing time is calculated as

(h d ) i i . (2.6) (µivi + ε)

24 Best Random (BR): In this case, an arriving server is assigned to queue i with probability pi, N such that ∑i=1 pi = 1. This heuristic is included only as a reference for comparison and is not intended to be implemented in actual disaster response. BR is provided in place of a true random policy (pi = 1/N) and serves as a tighter performance bound. The optimal thinning probabilities pi = (p1,..., pN), in terms of resulting average long-run holding costs, can be found by solving the mathematical program found in Equation 2.7.

N 2 φi min ∑ hi (2.7a) i=1 1 − φi  Vi  ρi    Vi!  s.t. αi < µiρi 1 − n for i = 1,...,N (2.7b)  Vi ρ   ∑ i  n=0 n! N ∑ pi = 1 (2.7c) i=1

pi > 0 for i = 1,...,N (2.7d)

λ pi Here ρi = γ , as deﬁned previously. The queue intensity of each M/M/1 queue with Markov

αi modulated service rates is represented as φi = . The objective function, shown in Equation E[vi]µi (2.7a) minimizes the expected weighted holding costs in steady state. Constraint (2.7b) represents the sufﬁcient condition for stability as discussed in Section 2.3. The ﬁnal constraints ensure that each queue has a positive weighted probability of volunteer assignment and that the total weight sums to one.

2.5 Simulation Model

In order to compare the heuristic policies deﬁned above to the MDP policy, a simulation model was developed in Matlab R2017a. The model simulates a system with two queues, but the

25 framework can easily be generalized to n > 2 queues. The code is separated into two distinct parts: (1) initialization and (2) main simulation. Model initiation creates and assigns a variety of variables for the case being simulated. These variables include case-specific system parameters, number of replications, warm-up period, simulation length, and simulation end time. The MDP optimal policy is stored in the Matlab Workspace or can also be read in from a separate file (for example using the xlsread function). Each run begins in state v = d = 0, i.e. zero volunteers and zero demand at all queues. The arrival times for the first volunteer and demand are sampled from the appropriate distribution. The main simulation tracks the time until next action over all possible actions and continu- ously compares it to the system time, t. Possible actions include: volunteer arrival, work arrival, volunteer departure, and work completion. The holding cost is updated at each discrete time epoch when a new action is triggered. When a volunteer arrival action is triggered, the simulation must decide how the incoming volunteer should be assigned. The logic for the heuristic policies is hard-coded into the simulation and the MDP policy uses a look-up function based on the current system state. Feasibility of the volunteer assignment decision is considered for all heuristics and no volunteer will be sent to a queue that is at maximum capacity. As discussed previously, if all queues are at capacity, the volunteer does not enter the system. When volunteer arrival or departure occurs, the time to next work completion must be updated preemptively, due to the additive service rate. The demand arrival, demand completion, and volunteer departure actions are modeled similarly. When an arrival or departure occurs, the respective count is updated and a new arrival or departure rate is sampled from the appropriate distribution.

2.6 Computational Analysis and Discussion

To evaluate the heuristic policies deﬁned in Section 2.4, a set of computational experiments was developed. The goal is to provide insight into which volunteer assignment policies perform

26 well over a robust set of scenarios. Experiments were designed for a system with two queues to more easily compare the relative performance of each heuristic policy to the MDP policy. A two queue system can be fully described by ten system parameters. We set the arrival and

departure of volunteers, λ and γ respectively, based on real world data collected by Lodree and Davis[2016]. The remaining system parameters are varied to create a robust set of system configurations that cover a broad range of potential real world scenarios (high intensity, low server capacity, varied service rates, etc). However, care must be taken when choosing these system parameters. We must ensure that an assignment policy exists which maintains system stability, as discussed in Section 2.3. We fix most parameters of queue 1, as shown in Table 2.1, to avoid duplicating cases. Next we assign high and low values for remaining parameters, following a 2k factorial design (k = 5), resulting in 32 unique cases (referred to as the baseline cases). A summary of the fixed and varied parameters is shown in Table 2.1.

Table 2.1: Summary of ﬁxed and 2k factorial design parameters.

V1 µ1 h1 λ γ 5 2.4 20 3.066 0.517

Case V2/V1 µ2/µ1 h2/h1 α1 α2 Low 0.6 0.5 0.5 3 1 High 2 1.5 3 5 2

We run each case and policy combination for 1000 replications in the simulation model. Holding cost was chosen as the primary metric for evaluation in the simulation. Holding cost can be thought of as a proxy for unmet need, with higher costs assigned to items/tasks of greater value to beneficiaries. It represents the cost of demand sitting at the volunteer site and not yet given to the beneficiaries. We choose to evaluate average holding cost (AHC) specifically

27 to allow equivalent comparison against the MDP policy, generated using value iteration. We find that the simulation reaches steady state performance in two model days, with a full run time of 24 days (representing approximately three weeks following a disaster event). For ease of comparison, we present percent deviation from the MDP policy. The percent deviation is AHC −AHC defined as, ∆ = Policy MDP . The full results of the experimental cases including: (1) the AHCMDP mean AHC, (2) percent deviation, and (3) statistical significance can be found in the Appendix. A summary of the baseline results is shown in Figure 2.3 below. The box plot represents the variation in percent deviation from the MDP policy for all policies across the baseline cases.

150

100

Deviation from MDP Policy (%) 0 LWD FV LQCT BR

Mean 13.2% 58.5% 13.3% 43.2%

Figure 2.3: Box plot of average deviation from MDP policy for each of the four heuristic policies. Mean values are represented as green diamonds, the red line represents the median value, the blue box represents the 1st and 3rd quartile range, and the pluses represent any outliers.

As expected, the MDP policy performs at least 13% better than all other policies on average. To further analyze the differences, we conduct a two sample t-test to compare the sample mean

28 AHC of each policy to the MDP policy across all cases. Using a p-value of 0.05, the MDP policy performs statistically significantly better than the FV and BR policy in all 32 cases. We can also see from Figure 2.3 that the LWD and LQCT policies are preferred over FV and BR given a system with stochastic demand tasks. However, it is difficult to identify a preferred policy just by evaluating the box plot. The LQCT policy has a slightly lower median value, but higher overall variation. A similar statistical analysis was conducted on the difference in sample mean AHC for the difference between LWD and LQCT. A summary of the results are included in Table A.2, found in the Appendix. In 9 cases, the LQCT policy performs significantly better than the LWD policy. Out of the remaining 23 policies, LWD outperforms LQCT in 5 of them. In the following sections, we conduct additional analysis to fully evaluate the performance differences between LWD and LQCT. Interestingly, the FV policy performs worse than both the LWD and LQCT policies, suggesting that the FV policy is not appropriate for cases with stochastic demand tasks. This warrants additional discussion as it is in complete contrast to the recommendation in the previous paper which features deterministic demand. There are two significant differences between this work and Mayorga et al.[2017] which contribute to the conflicting results. First and foremost is the underlying differences between deterministic and stochastic demand tasks. When a queue was cleared in the deterministic case, all volunteers would leave the system, as neither paper allows for reassignment. Given stochastic demand, it does not make sense to have volunteers leave immediately when the demand is zero, because additional demand may arrive. The FV policy likely performed well in the deterministic case because it minimized the volunteers lost after task completion. The second major difference between the two papers is the model objective. Mayorga et al.[2017] compared “time to completion” as the metric when determining optimality. This metric is inappropriate in the case with stochastic demand as there is no clear completion time for stochastic tasks. Additionally, differences in relative holding cost were not considered in the previous paper, as tasks were assumed to be similar or have the same level of importance.

29 Holding costs appear to be a signiﬁcant factor related to the performance of LWD and LQCT.

2.6.1 Sensitivity Analysis

We develop ﬁve different sensitivity analysis cases to further investigate the performance differences between the LWD and LQCT policies. Case S1 varies the arrival rate of spontaneous volunteers to determine the impact of varied volunteer arrival rates on AHC. Cases S2-S5 vary system parameters to create different types of queue imbalances. Cases S2 and S3 vary parameters on the volunteer side, maximum volunteer capacity and service rate respectively. The last two cases, S4 and S5, vary demand side parameters, namely arrival rate and relative holding cost, to determine the impact on AHC within the system. The arrival rate in case S4 is increased to allow for feasible solutions with higher demand arrival rates.

Table 2.2: Summary of sensitivity analysis conducted (min:increment:max). For example, (2.9 : 0.1 : 3.9) in Case S1 indicates that λ was varied as (2.9, 3.0, ..., 3.8, 3.9).

Case λ γ V1 V2 µ1 µ2 α1 α2 h1 h2 S1 2.85 : 0.1 : 3.95 0.5 5 5 2 4 6 8 20 20 S2 2.95 0.5 5 3 : 1 : 8 2 4 6 8 20 20 S3 2.95 0.5 5 5 2 : 1 : 8 4 6 8 20 20 S4 3.95 0.5 5 5 2 4 6 2 : 2 : 16 20 20 S5 2.95 0.5 5 5 2 4 6 8 20 10 : 10 : 80

A summary of the ﬁxed and varied parameters (min:increment:max) in each sensitivity case (S1-S5) can be found in Table 2.2. Results are provided as graphs of the sample mean AHC and a 95% half width over the 1000 replications for each of policies. Full results for the sensitivity analysis can be found in Table A.3.

30 Variations in Spontaneous Volunteer Arrival

It is assumed that in the immediate aftermath of a disaster, queues will experience extremely heavy demand and high spontaneous volunteer participation. We adjust the overall arrival rate of volunteers, λ, in Case S1 from moderate to very high and compare LWD and LQCT to the MDP solution. Figure 2.4 indicates that LWD and LQCT policies perform well over the majority of the cases within S1. When volunteer arrival rates are very high, LWD and LQCT perform nearly as well as the MDP solution. Table A.3 shows that for cases S1.11 and S1.12, there is no statistically signiﬁcant difference between MDP and LWD or LQCT policies. The LQCT policy does perform signiﬁcantly better than LWD for cases S1.8-S1.11, indicating that it may be more appropriate for use in cases with high volunteer arrival rates.

650 LWD 600 LQCT MDP 550

500

450

400

350

300

Average Holding Cost (per day) 250

200

150 2.8 3 3.2 3.4 3.6 3.8 4 Volunteer Arrival Rate Figure 2.4: Average daily holding cost versus volunteer arrival rate (λ) for Case S1.

At rates beyond λ=3.95 (S1.12), there is not a statistically signiﬁcant difference between LWD, LQCT, or MDP policy. If arrival rates are pushed to an extreme (e.g. λ > 20), SV positions will “always” be full, and so there is little to no difference between any policy

31 (including random assignment). As the volunteer arrival rates decrease, the average deviation grows for all policies. At moderate levels of spontaneous volunteer arrivals (S1.1-S1.4), all policies perform statistically significantly worse than the MDP policy. In these situations, volunteer assignment decisions become increasingly important to system performance. There is no significant difference between the LWD and LQCT policies in cases with moderate volunteer arrival rates. Overall, the LQCT policy performed significantly better than LWD in five of the twelve sub-cases tested.

Queue Imbalance Caused by Spontaneous Volunteers

Next we compare cases with queue imbalances in terms of spontaneous volunteers (S2 and S3).

We ﬁrst consider variations in maximum volunteer capacity (Vi) of one task. Within Case S2, all policies perform statistically worse than the MDP policy over all parameters tested. There is no signiﬁcant difference in mean AHC between the LWD and LQCT in any of the sub-cases. We

can see from Figure 2.5, that there exists some value of V2 at which additional work space does not translate to improved output. Clearly, the assignment policy alone is not always enough to take advantage of the increased floor space. Understanding when this bottleneck occurs is of practical importance for volunteer managers who need to make decisions on planned facility layout with limited floor space. Planners should consider volunteer participation levels when developing floor plans for VRCs. Other options to improve space utilization would be improved volunteer recruitment or initiatives to reduce departure rate of volunteers.

Queues with imbalance in volunteer service rates (αi) are compared in Case S3. Here, we

consider demand tasks with equal relative importance (h1 = h2) but large differences in service rates. Figure 2.6 shows LWD and LQCT performing well across the tested range, indicating they are fairly robust to changes in service rate. The LWD policy does not take into account the service rate differences directly, and therefore it is expected that the LQCT policy performs better. The LQCT performs signiﬁcantly better than the LWD policy in two of the seven sub-cases.

32 700 LWD LQCT 650 MDP

600

550

500

450 Average Holding Cost (per day) 400

350 3 4 5 6 7 8 maxVols 2

Figure 2.5: Average daily holding cost versus maximum volunteer capacity (V2) for Case S2.

Overall, the LQCT policy is the best performing heuristic policy, matching MDP performance in four of the seven sub-cases.

Queue Imbalances Caused by Demand Tasks

Next, we move to cases with a queue imbalance on the demand side (S4 and S5). Case S4 varies

the work arrival rate in queue 2 (α2) producing a queue imbalance that also indirectly affects system utilization. As the arrival rate of work in queue 2 increases, we see a nonlinear increase in holding costs across all policies in Figure 2.7. At higher demand arrival rates, all policies perform poorly when compared to the MDP policy. In fact, all policies perform signiﬁcantly worse than the MDP policy in all but one sub-case (S4.4). It is important to note that the LWD policy is the worst performing heuristic for cases with extremely imbalanced demand rates. In cases with extremely imbalanced queues, volunteer managers must exercise caution to ensure that volunteers are not all assigned to one task. For example, at high demand arrival rates, the LWD policy reacts by sending all volunteers to queue 2, effectively ignoring queue 1 for periods of time. The LQCT policy is less affected by this phenomenon due to the volunteer count

33 550 LWD 500 LQCT MDP 450

400

350

300

250

200

Average Holding Cost (per day) 150

100

50 2 3 4 5 6 7 8 mu 1

Figure 2.6: Average daily holding cost versus volunteer work rate (µ1) for Case S3. component within the heuristic. There is no signiﬁcant difference in the LWD and LQCT policy in seven of the eight sub-cases. However, the LQCT policy is more robust to large differences in demand arrival rates.

900 LWD 800 LQCT MDP 700

600

500

400

300

200 Average Holding Cost (per day)

100

0 2 4 6 8 10 12 14 16 Alpha 2

Figure 2.7: Average daily holding cost versus demand arrival rate (α2) for Case S4.

34 Finally, in case S5 we consider the scenario in which the relative importance of one task

is very large compared to another (e.g. h2 >> h1). For example, consider a case in which a relief organization is providing food/water to beneﬁciaries and also sorting incoming donations at a warehouse. The immediate importance of getting the food/water to beneﬁciaries likely outweighs the need to sort donations. For cases where the importance is relatively equal

(0.8h1 ≤ h2 ≤ 1.2h1), both LWD and LQCT perform well. Figure 2.8 identiﬁes a critical point

around h2 ≈ 1.5h1 where the trends change. The performance of LWD and LQCT diverges significantly to the right of this point. LQCT begins to approach the performance level of FV in cases with a large imbalances in relative importance. This is the only sensitivity case in which LWD performed statistically significantly better than LQCT (five of eight sub-cases). When extreme differences exist in task importance, LWD is the preferred heuristic assignment strategy.

1000 LWD LQCT 900 MDP

800

700

600

500

400 Average Holding Cost (per day)

300

200 10 20 30 40 50 60 70 80 HC 2

Figure 2.8: Average daily holding cost versus holding cost per item per unit time (h2) for Case S5.

35 2.6.2 Policy Recommendation

From the 32 original cases, there is little discernible difference between the LWD and LQCT policies. The LWD policy has a slightly lower mean average deviation (13.2 vs 13.3) and tighter variance when compared to LQCT. The LQCT policy is more robust than the LWD policy, as shown by outperforming LWD in three of the ﬁve sensitivity cases (S1, S3, S4). However, in cases with high spontaneous volunteer arrival rates (S1.12), there is no signiﬁcant difference between the LWD and LQCT policy. One major drawback to the LQCT policy is that it does not perform well given large differences in holding costs between demand tasks. In cases with a

high ratio of h2/h1, LWD is clearly the best performing policy. We must also consider difficulty of implementation when determining a policy recommendation. The FV policy is most attractive in volunteer management due to the ease of implementation and requiring limited system information. The LWD policy, an extension of the Largest Demand policy, requires the volunteer coordinators to keep track of demand and place value on the relative importance of each task. The LQCT policy requires complete system knowledge, which may be unknown during early disaster response efforts. In comparison to all other heuristic policies tested, FV requires the least amount of system knowledge. However, the FV policy is not robust, and performs poorly in a variety of system configurations. As shown in Figure 2.3, the FV policy is the worst performing on average and has an extremely large variance in AHC across all cases. Although LQCT performs slightly better than LWD over the majority of cases tested, it has two major drawbacks: (1) implementation difficulty and (2) performance in systems with large holding-cost differences. It is expected that volunteer coordinators may not have full system knowledge immediately following a disaster making LQCT difficult to implement. Throughout the disaster response, it is expected that tasks with largely different priorities require volunteers. For cases with a large difference in holding costs (or perceived costs), LWD should be used over LQCT to minimize average holding cost. Therefore, we recommend the use of the LWD

36 volunteer assignment policy for disaster response tasks with stochastic demand.

2.7 Conclusions

This chapter modeled volunteer assignment at a volunteer site or reception center in a post- disaster setting. The problem was formulated as a queuing model with stochastic arrival of demand, stochastic arrival and departure of volunteers, and stochastic and additive service rates. This is an expansion upon the spontaneous volunteer assignment model by Mayorga et al.[2017] by allowing for stochastic demand. The queuing system was formulated as a continuous time Markov Decision Process (MDP) and transformed to a discrete time MDP using uniformization. Solving the MDP using value iteration provided an optimal control policy. It was shown that the optimal policy is extremely complex and not suited for use in a disaster relief operation. Three different heuristic policies were introduced as implementable alternatives to the optimal MDP policy: (1) the Fewest Volunteer (FV) policy, (2) the Largest Weighted Demand (LWD) policy, (3) the Largest Queue Clearing Time (LQCT) policy. A fourth policy, (4) the Best Random (BR) policy was included as a reference. In order to test the performance of these heuristics compared to the MDP policy, a discrete event simulation model was developed in Matlab. A set of thirty two experiments were designed to test the robustness of each heuristic policy. Although the FV policy is easy to implement, it was found to not perform well in cases with stochastic demand. This is in contrast to the results found by Mayorga et al.[2017], which recommended FV in cases with deterministic demand. A short discussion of the underlying differences between the two papers was included for completeness. On average the LWD and LQCT policies performed well in relation to the MDP policy when minimizing average holding cost. An additional sensitivity analysis was conducted to further evaluate the effectiveness of LWD and LQCT. Again, both policies perform well, with LQCT performing better than LWD

37 in the majority of cases tested. The exception was for cases with large imbalances in relative holding costs, where LWD performed statistically signiﬁcantly better than LQCT. Ease of implementation and usability were also discussed. We recommend the use of the LWD policy for volunteer assignment given tasks that exhibit stochastic demand. While there are challenges with implementing any routing policy in a post-disaster response, the results show that value can be gained from planning for spontaneous volunteers. Volunteer organizations that are able to better manage spontaneous volunteers will be better prepared to serve the affected population.

2.7.1 Limitations

There are a few important limitations due to the formulation approach and model assumptions that should be discussed. With the Markov assumption, we cannot track volunteers day-to-day and therefore are unable to account for any learning that may occur across tasks. Our assumption is that tasks change frequently, dependent on the needs of the organization, and therefore the beneﬁt of learning is minimal. Similarly, we ignore training because we cannot track training across separate visits under the Markov assumptions. It is possible to include a training element within the model, through the development of additional states, but retraining would occur for each volunteer. This could be implemented by delaying spontaneous volunteers who enter the system and then allowing them to work on speciﬁc tasks at a faster rate.

38 CHAPTER

AGENT-BASED MODEL OF SPONTANEOUS VOLUNTEER CONVERGENCE

3.1 Introduction

Volunteer convergence presents signiﬁcant coordination, integration, communication, logistical, and health and safety challenges to emergency managers. This problem is magniﬁed due to the fact that emergency management plans rarely take emergent groups and spontaneous volunteers into account. A 2016 study found that 22 of the 50 largest cities within the United States had no mention of volunteering in their disaster response plans [Rivera and Wood 2016]. The UN Sustainable Development Goals seek to substantially increase the number of cities adopting and implementing integrated policies and plans towards resilience to disasters. The involvement of citizens into the emergency management process is inevitable and indispensable, from preparedness to recovery operations [Paciarotti and Cesaroni 2020]. It is likely to become

39 even more important in the future, with growing urban centers and populations, and in view of the high densities and proximity of urban people, buildings and infrastructure [Twigg and Mosel 2017]. Therefore, management plans and more structured efforts to integrate spontaneous volunteers (SVs) are necessary for effective disaster response [Sauer et al. 2014]. It is difficult for emergency managers to plan for and manage SVs due to the uncertainty surrounding when and where they will arrive and how long they will stay. Models inspired by historical trends can help authorities anticipate surges in the local population caused by volunteer convergence [Lodree and Davis 2016]. However, disaster events produce unique combinations of choices, actions, and reasoning that cannot be predicted in simple linear models [Comfort 1995]. Agent-based modeling (ABM) has been shown to be a valuable tool to model individual behaviors in disaster response. Hawe et al.[2012] present an overview of ABM usages in large scale emergency response. What is missing from the literature is a model of how and why individual citizens living in affected communities spontaneously volunteer to help when disaster strikes [Gunessee et al. 2018]. A model which considers the unique behaviors of spontaneous volunteers and how their response varies under different strategies can be used to communicate and improve disaster management plans. Lindner et al.[2017] present a framework for the development of an ABM of spontaneous volunteers consisting of a set of twenty-five attributes necessary to fully model a spontaneous volunteer agent. We expand upon the model developed by Lindner et al.[2018] by incorporating agent-agent communication, opinion sharing, site choice decision, and internal agent motivation. The inclusion of these behaviors allows us to better understand SV convergence and provides unique insight not found in other models. Analysis of model output, in conjunction with practitioner support, provides a platform to evaluate the benefit of different SV plans and integration strategies. Results from the model can be used to encourage a whole community approach to disaster planning and response. Communities that adopt a whole community approach to emergency management with a focus on community based volunteer activity are

40 observed to have more resilience after an emergency [Sobelson et al. 2015]. Our model is the first step toward developing a tool that can be used to evaluate and communicate the benefit of SV integration under different management plans. The remainder of this chapter is organized as follows: Section 3.2 describes the model purpose, scope, agents, attributes, and behaviors. Section 3.3 outlines verification and validation efforts including model calibration efforts. Section 3.4 includes statistical analysis of the model and relevant discussion. Section 3.5 summarizes the contribution of this model to disaster management literature.

3.2 Spontaneous Volunteer Convergence Model

The following model description is organized according to the structural elements found in the beginning of the Overview, Design Concepts, and Details + Decisions (ODD+D) protocol [Müller et al. 2013]. The full description can be found in AppendixC. The modelling methodology is supported where appropriate by qualitative data collected from a total of forty semi-structured interviews collected from 2017 to 2018. The interviews were collected both by phone and in person during ﬁeld work and were carried out with volunteer coordinators (26), volunteers (12) and beneﬁciaries (2) under IRB protocol # 12385, approved by North Carolina State University. Select quotes from these interviews are included in italics and serve as supplemental support for model assumptions where there is a lack of existing quantitative data.

3.2.1 Purpose

Perceptions surrounding the value of spontaneous volunteers may impact both how emergency managers plan for them and the value they eventually provide [Rivera and Wood 2016]. The purpose of the model is to provide emergency management practitioners a tool that can be used

41 to communicate, demonstrate, and comprehend the benefits of improved SV integration. Our model bridges an identified gap in existing literature by evaluating volunteer convergence from a macro perspective with an emphasis on information sharing between potential volunteers. The model simulates spontaneous volunteer convergence and the benefit that spontaneous volunteers provide to disaster response. The model allows users to analyze how spontaneous volunteer response is affected by varying (1) factors related to volunteer internal motivation and (2) communication and interactions with volunteer organizations. Disaster management practitioners at the city and state levels may use the model as a decision- support tool to evaluate changes to existing emergency management plans. The model can provide insight in the development of volunteer convergence response plans through numerical evaluation of alternative strategies. Value can also be found in the insightful discussions that come from evaluating different “what-if" scenarios. The model should encourage discussion regarding existing SV integration policies and can be used to enhance discussions around planning for and engaging with spontaneous volunteers.

3.2.2 Entities, State Variables, and Scales

The model framework has two primary agents: (1) Population Agents and (2) Volunteer Sites. Population agents, representing potential spontaneous volunteers, are a heterogeneous pool of agents living within a region of the affected area. Population agents act autonomously, communicate, and route themselves to volunteer sites to support relief efforts. Agents are connected through a scale-free network to mimic relationships and networks found on social media. Scale-free networks can be used to abstract social media networks with influencers or hub users [Urena et al. 2019]. Volunteer sites are stationary locations where population agents attempt to converge. These are locations where affiliated and unaffiliated volunteers work to complete tasks and satisfy demand created by a disaster. The status of volunteer sites transitions between activate, open, and closed as the model progresses through simulated time. A fixed

42 amount of population agents and volunteer sites found within the disaster area are modeled explicitly.

Figure 3.1: SV Convergence Model GIS user interface as shown in Anylogic University Edition version 8.5.2

A screenshot of the ABM graphic user interface is shown in Figure 3.1. It provides a visual representation and animation of the model, allowing for ease of communication when sharing with practitioners and community partners. Active volunteer sites are represented as green warehouses. Census block groups are illustrated with dashed lines, with houses representing the centroid. Specific volunteer assignment and demand completion at each volunteer site is abstracted for simplicity. Volunteer agents are represented in green, motivated agents are shown in yellow, and idle population agents are grey. Lindner et al.[2017] suggests that twenty-five attributes are necessary to fully model spontaneous volunteers in an agent-based model. As shown in Figure 3.2, our model directly adopts nine attributes (green) and indirectly includes another seven (yellow) of the recommended twenty-five. For ease of implementation, attributes

43 related to detailed levels of work within volunteer sites are not considered (red).

Figure 3.2: Visual representation of agent attributes proposed by Lindner et al.[2017]. The colored boxes indicate how they are captured within our modelling framework. Attributes are either directly captured (green), indirectly captured (yellow), ignored (red), or left for future work (no box).

A simpliﬁed Uniﬁed Modeling Language (UML) class diagram is shown in Figure 3.3. The UML Class diagram represents the structures and relationships found within the SV Convergence Model. It illustrates all agents, their attributes, and relationships to each other. For example, within a simulated affected area, there are one or more of each of the following: population agents, regions, and volunteer sites. Each region includes zero or more agents and volunteer sites.

44 Population agents have attributes related to age, volunteer site information, internal motivation, home region, and connections with other agents. Motivated agents inherit all attributes from population agents and also gain three new behaviors related to ﬁnding a suitable volunteer site. Similarly, Volunteers inherit attributes from motivated agents, including a choice of volunteer site and travel behaviors.

SV Convergence Model

Affected Area Volunteer Site agents : Agent 1..* volunteerCapacity sites : Site siteLocation regions : Region Region workRate 1..* Location 0..* demandRemaining activate() 1 requestVolunteers() close()

0..* 1..* 0..* Population Agent age 1..* Volunteer siteInformation siteChoice attitude Motivated Agent volunteerTime subjectiveNorm requestInfo() status pBC siteSearch() moveToSite() pReply chooseSite() updateInfo() homeRegion 1 changeStatus() networkConnections 1..* moveToHome() checkIntention()

Figure 3.3: Simpliﬁed UML diagram of the SV Convergence agent-based model. The diagram illustrates the attributes of each agent, their functions/behaviors, and relationship to each other.

45 3.2.3 Process Overview

The model is initialized immediately following a disaster at time t=0 with population agents located in their home region. Organizations respond to community need by activating volunteer sites throughout the community. Motivated population agents converge to these active volunteer sites. Demand is completed at each site at a base service rate, supplemented by any SVs accepted to the site. The model concludes after either: (1) All work is completed, or (2) System time reaches t=T. A summary of this ﬂow is shown in Figure 3.4.

Figure 3.4: High level ﬂow of events captured within the framework which occur following a natural disaster.

At initialization, all population agents start in an idle unmotivated state. When the disaster response begins, all population agents check internal motivation through a behavioral intention function, discussed in Section 3.2.4. Individual agent motivation is checked each morning, and population agents who fail this motivation check remain idle until the following day. Population agents who pass the motivation check become motivated agents and perform an internal check to see if they possess relevant site information, such as volunteer site status and location. Agents who do not have knowledge of an active volunteer site may send a request for site information to networked agents. Networked agents are those population agents connected to them through the scale-free network. Additionally, it is possible for motivated agents who lack site information to seek out volunteer opportunities independently. Agents may travel within the affected area and come across volunteer sites, as shown in the quote below.

“When these organizations set up ... they put up ﬂags, signs, and people just go by

46 and see those signs and ﬁnd out were they can help" - (Volunteer)

Motivated agents who posses site information choose a site based on their site decision policy discussed in detail in Section 3.2.5. Once a site choice is made, spatial information regarding region and site location is used to determine a travel route and travel time. Upon reaching the volunteer site, a determination is made by the site volunteer manager to accept or reject the volunteer. Volunteer rejections are assumed to only occur in the model if volunteer sites are at capacity or do not have the supervisory resources to handle additional volunteers.

“There were so many people who wanted to volunteer that they were in lines, and sometimes they were turned away ... because there were more volunteers than they could handle" - (Volunteer)

Volunteers who are turned away update their site opinion accordingly and return home. After multiple rejections, volunteer agents may become temporarily or permanently unmotivated and avoid future attempts to volunteer. Volunteer agents that are allowed to enter the site and provide assistance until there is no work left to complete or their internal abandonment time is met. In both cases, agents leave the site, update their site information, and return home. A summary of the population agent ﬂow can be seen in Figure 3.5.

47 Figure 3.5: Population agent state chart representing the daily ﬂow of agents and decisions.

3.2.4 Agent Decisions to Volunteer

Motivations behind volunteering is a topic discussed heavily in social sciences science and psychology [Clary et al. 1998, Gunessee et al. 2018, Veludo-de Oliveira et al. 2013, Greenslade and White 2005, Bourgeois 2010]. For example, Gunessee et al.[2018] argue that a citizen’s response to disaster situations is motivated by high degrees of empathy (and sympathy) and adherence to social and personal norms of equity and social responsibility. Cottrell[2012] found that media coverage was the main prompt for people to volunteer, followed by talking with other people about the event, and ﬁnally advertisements for volunteers. However, literature that includes a simulation of the decision to volunteer in a disaster setting is nearly nonexistent. Daimon and Atsumi[2018] develop a cellular autonoma model which parameterizes the decision to volunteer following an earthquake based primarily on normative behaviors. Due to the modelling methodology used, they are not able to incorporate heterogeneous agent characteristics or communication between agents. Lindner et al.[2018] use a Bernoulli random variable and acknowledge that modeling an individual’s decision to participate as a spontaneous volunteer after natural disasters is very complex.

48 We chose to parametrize motivation and the decision to volunteer using the Theory of Planned Behavior (TPB) [Ajzen and Fishbein 1980, Ajzen 1991]. The TPB proposes that the cause of behavior is behavioral intention, a prerequisite decision to engage in that behavior. The TPB can be used to effectively break down the reasoning process related to the development of a behavioral intention in the context of a decision to volunteer. Behavioral intention is made up

of the individual’s attitude (Ai) towards the behavior, their subjective norm (SNi) and perceived

behavioral control (PBCi). Mathematically, we can write the behavioral intention (BI) of agent i on model day t as

ATT SN PBC BIi,t = wi Ai,t + wi SNi,t + wi PBCi,t, (3.1)

ATT SN PBC where wi ,wi , and wi represent corresponding weights for each factor and are unique ATT for each agent. By accounting for age and considering unique values for wi we allow for additional heterogeneity of the agent population in behavioral intention. Weights for subjective

SN PBC norm (wi ) and perceived behavioral control (wi ) can be written as

SN ATT 18 PBC ATT SN ATT wi = (1 − wi ) , wi = 1 − wi − wi , 0 ≤ wi ≤ 1. (3.2) Agei

Attitude (A) is the function of beliefs about whether doing something will have particularly good or bad results. To determine the perception of this belief in regards to volunteering in disaster, we follow the methodology proposed by Daimon and Atsumi[2018]. The idea is the following: the more that the media is discussing a disaster, the more likely that population agents have a belief that volunteering is important or necessary for themselves or the community.

“I tried to ﬁgure out as much as I could from the news and websites about what relief efforts were going on" - (Volunteer)

To capture this, Daimon and Atsumi[2018] counted newspaper mentions of the earthquake following the natural disaster in Japan. They ﬁt the data to an exponential curve,

49 exp Ai,t = exp(−wi t), (3.3)

exp where wi is the exponential fit parameter and t is the model time in days. In Daimon and Atsumi[2018], it is assumed that all agents receive the same news and therefore have the exp same attitude value wi = 0.037, for all i . In our framework we can adjust the exponential fit exp ATT parameter wi and attitude weight wi to account for individual motivation to comply with this specific belief. Exponential functions have been used in other TPB equations to represent variations in motivation, such as the decision to purchase goods [Zhang and Zhang 2007].

Figure 3.6: Comparison of normalized attitude curves (attitude (Ai,t) versus day).

We perform a similar approximation using the number of mentions in tweets following Hurricane Sandy, collected by Wang and Zhuang[2017]. The normalized data on tweets also ﬁts an exponential distribution, however the curve drops much more quickly than with the exp newspaper mentions (wi = 0.3). The difference between the two exponential ﬁts is illustrated exp in Figure 3.6. An analysis of the impact of wi on agent attitude is conducted in Section

50 3.4. It should be noted that the model framework allows for any general function of attitude,

Ai,t = g(i,t), if an improved representation is identified. Subjective Norm (SN) is an individual’s perception about the particular behavior, which is influenced by the judgment of significant others (e.g., parents, spouse, friends, teachers). It represents the social pressure felt by an individual toward completing an action. The support for behavioral norms as a significant predictor of intention to volunteer indicates that individuals are influenced not only by the perceived views of others, but also by the behavior of others [Greenslade and White 2005].

“We learned very quickly that good work is about generally doing good to your neighbors, your friends" - (Volunteer Coordinator).

The volunteer behavior of friends and family may encourage individuals to feel that volunteering is an appropriate behavior for themselves [Warburton and Terry 2000]. We represent SN as

Si ∑ j=1 Vj,t−1 SNi,t = , (3.4) Si where the numerator is the number of agent i’s social network connections who volunteered in

the previous period. Si is the total number of network connections of agent i. In other words,

SNi,t represents the proportion of agent i’s connections that volunteered in the previous period. Perceived Behavioral Control (PBC) is an individual’s perceived ease or difﬁculty of performing the particular behavior. We abstracted this belief and represented as a single value

bounded by, 0 ≤ PBCi,t ≤ 1. PBC is unique for each volunteer and can be updated based on volunteer experience during the model execution. Higher initial PBC values represent a perceived ease in performing the volunteer work due to prior training or experience with the volunteer task. Alternatively PBC can be derived from a general distribution which represents the approximate PBC value expected for a target population. Equation 3.1 represents a convex combination of inputs, with individual weights based

51 on the agent’s own attributes. This summation, behavioral intention BIi,t, is bounded between [0,1] and effectively represents the likelihood agent i is motivated to volunteer in period t. We

represent an actual attempt to volunteer as an indicator variable Bi,t. To translate from BI to B, we consider a stochastic element. In the linear case, we can compute the probability of the indicator random variable as

P(Bi,t = 1) = BIi,t. (3.5)

Alternatively, there is support in the literature to transform B from BI using a Sigmoid function, which may better represent the unique elements of volunteer behavior [Daimon and Atsumi 2018]. In Daimon and Atsumi[2018], the parameters were calibrated using volunteer convergence output and found to ﬁt the function

1 P(Bi,t = 1) = . (3.6) 1 + exp(−12(BIi,t − 0.4))

We also consider the effect of fatigue on volunteer’s motivation decision. We impose a penalty

term Fi,t for each agent based upon the number of days previously volunteered. In general, this can be added to the translation from BI to B as

P(Bi,t = 1) = f (BIi,t,Fi,t), (3.7) with the penalty varying for each individual or based upon the type of volunteer work. A comparison of the two functions can be seen in Figure 3.7. The beneﬁt of the Sigmoid function is that it requires a larger ratio of BI to B initially. After a certain point, additional motivation is unlikely to increase the probability of actually completing the behavior. This acts as a pseudo threshold on motivation required to volunteer in a given period. Strict threshold values (P[Bi,t = 1] = 1 if BI > Threshold, else P[Bi,t = 1] = 0) are common in single event

52 behaviors such as evacuation (e.g. Du et al.[2017]) but are not appropriate for recurring tasks such as disaster volunteering. Again, the model framework is ﬂexible, and any general function

P(Bi,t = 1) = f (BIi,t) can be used to translate BI to B.

Figure 3.7: Graph of transformation from intention (BI) to attempting volunteer behavior (P[Bi,t = 1]) for Linear (blue), Sigmoid (red), and Threshold (green) functions.

Individuals who pass the motivation check (Bi,t = 1) next must consider whether or not they have actionable site information. Actionable site information means they know of one or more active volunteer sites. We separate the requirement of site knowledge and motivation such that population agents who are motivated but do not know site information, can request that information from network friends.

“I got on Facebook and started messaging friends that I knew were active in the community already and asked them [for information]" - (Volunteer)

Requests for volunteer site information occur when motivated individuals lack actionable site knowledge. An agent that receives a response will evaluate the response and update their site

53 opinion accordingly. Agents’ site opinion score represents the belief that an agent can impact disaster response at a given site (or a belief along the lines of “likelihood I can make an impact”) and is bounded [0,1]. Initial opinions are generated when agents learn about a site for the ﬁrst time which occurs in one of two ways: (1) An agent receives an unsolicited message from a volunteer site (or news outlet), as part of a one to many message, (2) An agent learns of a volunteer site through a one-one message in response to a request they sent for information. In both cases, the site knowledge is also updated to indicate the site is now known to the agent. An agent cannot have an opinion of an unknown site. Population agents who receive a site message generate an initial opinion based on a truncated normal distribution. Care should be taken when selecting the variance of the normal distribution as Anylogic re-samples for truncated distributions if the value selected is from outside the bound (e.g. [0,1]). However, the model framework allows for any distribution of initial opinions, allowing users to capture varying opinions across population agents. Population agents will directly adopt the site opinion of the agent who responded to their request for information. It is assumed that agents requesting information capture information being sent back to them (e.g. no chance at missing or ignoring the information) as long as they have not moved on to site selection. When agents receive information regarding a site they already know, the site status is updated if that information is newer to the receiving agent. That is, if the sender provides more current information about a site, that information will be updated. Site opinion is updated according to a simple weight-based approach when the receiving agent has an existing opinion [DeGroot 1974]. Agent i’s new opinion of site n can be written as

Oi,n = wiOi,n + w jO j,n wi + w j = 1 0 ≤ wi,w j ≤ 1, (3.8) where the agents opinion, Oi,n, and the friends opinion, O j,n, are combined. If wi = 1, then the

opinion remains unchanged, in the case of w j = 1, the new opinion is adopted completely. For

54 the purposes of our model, wi = w j = 0.5.

3.2.5 Site Choice Behaviors

Motivated agents from the population that are ready to volunteer (i.e. they have passed the motivation check and possess actionable site knowledge) must choose from available volunteer sites. We present four site choice behaviors; (1) Closest known site, (2) Site with highest score, (3) Site with most work, (4) Site most frequented by friends. Each behavior is discussed in more detail below and includes a supporting quote:

Closest Known Site (CLOSEST): This behavior is the simplest to implement, in that it only relies on the spatial distance between the population agent and other site agents. We choose the site with the minimum distance from the population agent’s location from a set of active sites. The development of this behavior came from practical consideration, but also from the idea that SVs want to give back within their immediate community. The assumption here is that the closest site provides the highest beneﬁt to their local community. There is no direct bound on the max distance an agent is willing to travel, but it is bounded by the geographic area modeled in Anylogic.

“There were numerous shelters that opened up and my ﬁrst thought was that I’m going to go to the [site] closest to me" - (Volunteer)

Site with Highest Score (SCORE): The choice here is the site with the highest score out of the set of known and active sites. Agent i’s score for site n is calculated as the following

di,n Sn = wD 1 − + wOOi,n wD + wO = 1 0 ≤ wD,wO ≤ 1, (3.9) dmax where di,n is the distance between agent i and site n, dmax is the maximum possible distance,

and Oi,n is agent i’s opinion score of site n. For the purposes of our model, we assign wO = 0.8

55 and wD = 0.2. With this policy, we attempt to balance the importance of giving back to the community with the idea that agents are volunteering at a site that they perceive needs help the most (has a high opinion score).

“I would be willing to volunteer anywhere if ... I knew that they had the need and I could get there" - (Volunteer)

By varying weights on distance and opinion we can create additional heterogeneity in site choice through the agent population.

Delayed Remaining Demand (DEMAND): Here the population agent chooses the site with the most demand remaining from feasible sites. This policy acts as an alternative to site opinion and distance, in that it is based on an ofﬁcial metric of actual “need". For example, population agents ﬁnd out information regarding remaining demand through the news.

“You can follow the news and you can ﬁnd out [the need]" - (Volunteer)

To add realism in the simulation model, demand information provided by this approach can different than the actual demand or delayed. For the purposes of our model, we provide correct demand information, lagged by two model days.

Site Most Frequented by Friends (FRIENDS): The population agent chooses the site with the highest friend score, from known sites. This choice is based on interview responses which indicated a desire to volunteer with friends.

“My best friend invited me to come volunteer with him, and because it was something that I want to do I started coming everyday" - (Volunteer)

Friend score is calculated in the model by tallying the last visited site of all agents connected to agent i and choosing the site with the highest value. Ties are broken based on distance.

56 3.2.6 Model Outputs at the System Level

The model captures a variety of performance indicators to measure SV convergence at the system level for a given set of model parameters and agent behaviors. Indicators are captured

each day and include: (1) demand remaining over time at each site Dn = (Dn,1,Dn,2,...,Dn,T ),

(2) active spontaneous volunteers at each site over time SVn = (SVn,1, SVn,2,...,SVn,T ), (3) total

number of SV rejections summed across all sites over time R = (R1,R2,...,RT ).

3.3 Veriﬁcation and Validation

Prior to the use of the model by practitioners for educational or evaluative purposes, it is important to conduct a thorough verification and validation. We first overview the model verification steps conducted and then move to validation. While validation tests typically begin after the model has been verified, they can supplement the model verification processes.

3.3.1 Veriﬁcation

To ease the verification process, the model was initially developed as three independent submodels: (1) communication/information sharing, (2) motivation and behavioral decisions, and (3) movement within the virtual space. Verification of the sub-models included traditional variable and parameter evaluation. We followed best practices identified in other disciplines such as observing model behavior when parameters are set to extreme values and when behavioral rules are modified [Cooley and Solano 2011]. After verification of the three individual submodels, we integrated all parts and began testing with homogeneous agent populations. We evaluated three specific scenarios to verify the effects of different parameters and behavioral changes on demand completion in the full model. For example, if the average initial perceived

behavioral control (PPBC) value is increased in all agents, we expect to see an increase in number

57 of volunteers and higher motivation initially. Similarly, if the probability of reply is decreased, there should be less volunteers who obtain site information through their social network. During all tests, behavioral intention should remain bounded by [0,1] for all agents, site demand should be monotonically decreasing, and no volunteer should be allowed to enter a site which is not active. A complete description of the veriﬁcation scenarios and the corresponding results can be found in AppendixB.

3.3.2 Validation

The complex nature of agent-based modelling and its emergent properties at a system level make validation of models challenging, and some argue inherently impossible due to the irreducibility of emergent properties [Heckbert 2009]. Agents’ motivation to volunteer are dependent on a multitude of internal and external factors and the decision to volunteer occurs in a stochastic manner. In addition, the information sharing and site choice behaviors can make the overall system highly nonlinear and noncontinuous. It is very difﬁcult, if not impossible, to obtain closed-form analytical solutions for such a system [Du et al. 2017]. Additionally, at the time of writing, there are no existing computational models which capture the convergence behavior of SVs that can be used to validate by direct comparison. A common approach to validate agent-based simulation models is to compare model output to historical data. This is typically done by calibrating the model on a training set and comparing the calibrated model outputs to the remaining portion of withheld test data. To an extent, calibration establishes the validity of the internal workings of the model and its results [Cooley and Solano 2011]. However, due to a lack of robust data on SV convergence, we are further limited in our validation approaches. To the best of our knowledge, only two papers exist which investigate and capture SV convergence data. We derive a new set of convergence data for a third point of comparison using open source information. A comparison of the three normalized convergence curves is shown in Figure 3.8.

58 Figure 3.8: Recorded normalized percentage of volunteers over four weeks following the (1) 2011 Tornado in Tuscaloosa, Alabama, (2) 2018 Hurricane Florence response in North Carolina, (3) 2009 Victorian Bushﬁres.

Lodree and Davis[2016] collected data from the main relief center in the month following the 2011 tornado disaster in Tuscaloosa, Alabama. They produced time-series plots of individual and group volunteer convergence by week. Cottrell[2012] collected data on the experiences of spontaneous volunteers who participated in the Victorian bushfires in February 2009 and/or the Queensland storms in November 2008. The data collection occurred post disaster in the form of online survey responses. It is important to note that the sample was self-selected and anonymous so it is not possible to identify which events the respondents participated in. Finally, we collected open source data from the volunteer response in the first four weeks following Hurricane Florence in 2018, as recorded by the North Carolina Voluntary Organizations Active In Disaster (NC-VOAD). The NC-VOAD data set includes all activities conducted by volunteer organizations across multiple geographic locations in North Carolina for the first ninety days following landfall. The goal is to calibrate select model parameters such that the simulated convergence output fits the real world convergence data. We define goodness of fit using the the mean squared error

59 (MSE) of the normalized weekly convergence. For our purposes, the MSE can be written as

W 2 1 SVj SVj ∑ W − W , (3.10) W j=1 ∑k=1 SVk ∑k=1 SVk where W is the total number of weeks of data analyzed (in our case, W = 4). The first term in the summation is the simulated normalized average number of SVs active across all sites for each week, j, and the second term is the actual normalized number of SVs reported. A summary of the fixed and calibrated parameters is shown in Table 3.1. The parameters selected for calibration and their relative ranges were chosen based upon results of the initial verification analysis and supplemented by practitioner input. All other parameters were fixed to baseline values during calibration. The values of the fixed parameters were also selected based upon literature, verification results, and expert input.

Table 3.1: List of relevant ﬁxed baseline parameter values and calibrated parameter ranges selected for parameter calibration.

Baseline Parameters Value Calibrated Parameters Range Max Info Age 2 P(Reply) [0.1, 0.9] Site Choice Behavior 2 Volunteer Request Day [1,14] P(Site Search) 0 Average Task Fatigue [0,0.3] Site Capacity 100 Message Size [5, 500] Network Parameter 5 Average wATT [0.2,0.8] Initial Demand 100,000 i Average wexp [-0.3,0.03] Max Requests 4 i Initial PBC Dist (α) [1,4] Adopt Opinion 0.5 f (BI) Linear or Sigmoid Rejection Penalty 1

Model calibration was conducted using the optimization experiment feature in Anylogic University 8.5.2 and the embedded OptQuest Optimization Engine. We allow OptQuest to search for 250 iterations, each with 10 replications. The run time is approximately one minute

60 per iteration on a machine with two Intel Xeon E5-2680 v2 (20 cores) and 128 GB of RAM. A visual comparison of the best found solutions to recorded normalized convergence data can be seen in Figure 3.9. Each ﬁgure shows the actual normalized convergence curve with the simulated average and 95% CI from 100 replications.

(a) (b)

(c)

Figure 3.9: Simulated and actual normalized convergence over four weeks following: (A) 2011 Tornado in Tuscaloosa, (B) Hurricane Florence in 2018, (C) The 2009 Victorian Bushﬁres.

The best found combination of parameter values for each scenario are shown in the right

61 hand column of Table 3.2. We observe similar results when comparing the calibrated parameters for the Tuscaloosa tornado and Hurricane Florence. However, the calibrated parameters for the Australian brush fires are significantly different than those of both U.S. based natural disasters. In the bushfire results we see a increase in average task fatigue and speed in which their attitude

ATT decreases (wi ) while also exhibiting a decrease in perceived ability to contribute (lower initial PBC). Recall that the data from the U.S. disasters do not distinguish between volunteer type while Cottrell[2012] specifically limited data collection to spontaneous volunteers. It is possible that the difference we see in calibrated parameters are due to the inclusion of affiliated volunteer data. For example, affiliated volunteers may trend the convergence data toward prolonged volunteer exp response (low wi ) and higher confidence (initial PBC). However, it is also possible that the differences seen in the calibrated parameters can be explained by differences in culture or disaster type. Additional practitioner feedback is required to fully analyze the differences in best fit calibrated parameters across disaster types.

Table 3.2: Summary of parameters varied during model calibration using simulation optimization. Best ﬁt values for each parameter are listed for each of the three different disasters.

Best Fit Simulated Value Parameter Range Hurricane Tornado Bushﬁres P(Reply) [0.1, 0.9] 0.154 0.181 0.245 f (BI) Linear or Sigmoid Linear Linear Sigmoid ATT Average wi [0.2,0.8] 0.68 0.631 0.613 exp Average wi [-0.3,0.03] -0.058 -0.07 -0.254 Initial PBC Dist β([1,4],8.5) β(3.135,8.5) β(2.604,8.5) β(1.575,8.5) Message Size [5, 500] 490 468 393 Volunteer Request Day [1,14] 5 5 2 Average Task Fatigue [0,0.3] 0.038 0.038 0.17

62 By calibrating the simulation to multiple data sources, we are able to highlight the flexibility of our model and showcase how volunteer convergence varies across different disaster types and locations. We see that the calibrated parameters provide a relatively good fit when all baseline parameters remain fixed. However, it is not likely that all baseline parameters will remain the same across different disaster types, locations, and population demographics. We are interested in evaluating how the robust the resultant convergence curves are across variations in the baseline parameters. Table 3.3 illustrates the changes to the baseline parameters. We assign bounds to each fixed parameter and uniformly sample each baseline parameter during the replications.

Table 3.3: List of relevant baseline parameter ﬁxed values and range of new randomized values. Ranges were selected to test the robustness of the normalized convergence curve under the best ﬁt calibrated parameters.

Baseline Parameters Fixed Value Random Value Max Info Age 2 U(2,7) Site Choice Behavior 2 {1, 2, 3, 4} P(Site Search) 0 U(0,0.05) Site Capacity 100 {50, 51, . . . , 199, 200} Network Parameter 5 U(3,20) Initial Demand 100,000 U(50000,200000) Max Requests 4 {2, 3, . . . , 9, 10} Adopt Opinion 0.5 U(0.2,0.8) Rejection Penalty 1 U(0,3)

We simulate 100 replications with the randomized baseline parameters, while keeping the best found calibrated parameters ﬁxed for each scenario. As noted previously, each data source captures convergence and response to different disaster types, sizes, response length, and locations. A comparison of the calibrated parameters with varied baseline parameters to recorded normalized convergence data can be seen in Table 3.4. We see that although the inclusion of

63 randomized baseline parameters does increase variability (larger 95% conﬁdence interval), the simulation is still able to ﬁt the normalized convergence curve well. This indicates that the calibrated parameters include the majority of factors affecting convergence. While the baseline parameters have minimal effect on SV convergence, they may impact other response variables such as volunteer rejections. We note that the model slightly underestimates the week 3 and over estimates the week 4 convergence for all scenarios.

Table 3.4: Comparison of recorded normalized convergence data and simulated results for both ﬁxed and random baseline parameters across each scenario. Simulated convergence data is presented as mean ± 95% conﬁdence interval over 100 replications.

2011 Tornado in Tuscaloosa, Alabama Source Week 1 Week 2 Week 3 Week 4 Recorded Data 0.1721 0.3443 0.3360 0.1476 Simulated w/ Fixed Base 0.1738 ± 0.0099 0.3749 ± 0.0100 0.2711 ± 0.0153 0.1802 ± 0.0157 Simulated w/ Random Base 0.1791 ± 0.0206 0.3744 ± 0.0163 0.2699 ± 0.0262 0.1766 ± 0.0159

2018 Hurricane Florence Response in NC Source Week 1 Week 2 Week 3 Week 4 Recorded Data 0.1943 0.3515 0.2741 0.1801 Simulated w/ Fixed Base 0.1653 ± 0.0093 0.3681 ± 0.0116 0.2729 ± 0.0163 0.1937 ± 0.0106 Simulated w/ Random Base 0.1692 ± 0.0291 0.3557 ± 0.0354 0.2465 ± 0.0356 0.2285 ± 0.0404

2009 Victorian Bushﬁres Source Week 1 Week 2 Week 3 Week 4 Recorded Data 0.7800 0.0940 0.0945 0.0315 Simulated w/ Fixed Base 0.7615 ± 0.0227 0.1131 ± 0.0239 0.0673 ± 0.0162 0.0581 ± 0.0160 Simulated w/ Random Base 0.7562 ± 0.0387 0.1155 ± 0.0247 0.0680 ± 0.0185 0.0603 ± 0.0146

We close the validation section by reviewing ongoing efforts to obtain feedback and support from emergency and volunteer managers. Face validity tests use the broad knowledge and

64 experience of subject matter experts as the source of model validation. Practitioners were presented the model and asked whether it is reasonably compatible with their experience with SV convergence. Participants had the opportunity to review the model, discuss its relevance, and provide feedback. This included a review of the conceptual model, agent behaviors, model assumptions and also graphical output from notional examples. Validation from practitioners is a paramount step toward the use of the model as both an educational and evaluation tool. External validation through practitioner input has already led to improvements to model assumptions. For example, practitioners provided guidance on speciﬁc ranges related to limits on hours and days worked by SVs. The meetings also help to foster ongoing collaboration with the emergency management community. Discussions regarding the problems they face allow us to identify areas in which this simulation model can be used to improve their understanding of SV convergence. Further analysis of operational level questions posed by the practitioners is discussed in Chapter4.

3.4 Analysis and Discussion

Quantitative sensitivity analysis can be used to strengthen trust in model realism and to eliminate model factors that have negligible inﬂuence on the variability of the output, allowing for a simpler, easier to understand model [Ligmann-Zielinska et al. 2014]. However, with over 20 input parameters and replication run times of 30-60 seconds, it is not feasible to conduct a full factorial design with a meaningful amount of replications. Fractional factorial designs provide a way to obtain good estimates of the main effects and two-factor interactions at a fraction of the computational effort required by a full 2k factorial design [Law 2017, Law et al. 2000].

8−2 We develop a resolution V fractional factorial design with 8 parameters (2V ), resulting in 64 iterations, to evaluate the model in computationally efﬁcient time. The goal of the analysis is to determine the statistically signiﬁcant factors affecting demand completion and also volunteer

65 rejection. A summary of the fractional factorial design parameters, and their high and low values can be found in Table 3.5. Parameters were selected for analysis based upon the calibrated parameters from the three examples and input from practitioners. The ﬁrst parameter f (BI) represents a

choice between two functions to translate intention into action (P(Bi,t = 1) = f (BIi,t,Fi,t)). The second parameter, PBCalpha defines the initial distribution of PBC values in population agents. The third parameter sets the minimum number of connections within the predefined social network. For example, a network parameter of 5 represents a network where population agents are connected with at least 5 other agents. Increasing the value to 20 represents a significantly

ATT exp more connected social network. The next parameters, wi and wi are both parameters related to the function to determine behavioral Intention (Equation 3.1). The high and low value of exp wi represent the two data sources, tweets [Wang and Zhuang 2017] and newspaper mentions [Daimon and Atsumi 2018] respectively. The fatigue parameter represents the physical and emotional fatigue which occurs when volunteering in disaster response. It acts as a penalty to motivation and encourages agents rest after prolonged volunteer experiences. The ﬁnal parameters, Messages and P(reply) impact how many individuals receive site information and the probability of successfully requesting information from others.

Each of the 64 iterations was run for 30 replications using Anylogic University 8.5.2. Using the fixed seed option in Anylogic Replication settings ensures equivalent replications through common random number generation. System level outputs total demand remaining and number of SV rejections were recorded for each replication. Results were averaged by replication and input into JMP Pro 14 for analysis. Using JMP Pro 14’s Fit function, we are able fit the 8 parameters to different models and estimate the response. Summary results are shown for a standardized least squares fit model for demand and rejections in Figure 3.10 and Figure 3.11, respectively. For the demand remaining response variable, the least squares fit model has an R2 = 0.81

66 8−2 Table 3.5: Summary of parameters selected for fractional factorial design (2V ) and their corresponding low and high values.

Parameter Description Low High f (BI) Choice of Parameter to Translate BI Linear Sigmoid PBCalpha Parameter to set initial distribution of PBC val- 1.5 3.5 ues in population agents β(α,8.5) Network Parameter Parameter that defines the shape of the scale-free 5 20 network ATT Average wi Weight of Attitude as defined in the TPB equa- 0.19 0.7 tion exp Average wi Exponential fit parameter inside the exponential -0.3 -0.037 function responsible for calculating individual Attitude Average Fatigue Parameter which defines how many days volun- 0.03 0.07 teers can participate before needing to rest Messages Amount of initial messages sent per site 50 200 P(reply) The probability that a connected agent will reply 0.4 0.8 to your request for site information

with p < 0.0001. The effect summary shows that 5 of the 8 factors are statistically significant. The most significant factor in predicting demand completion is the exponential fit parameter exp (wi ). The initial PBC distribution parameter (β(α,8)) is the second most significant factor of the 8 tested. This result is supported by Veludo-de Oliveira et al.[2013] who found that PBC was the most significant predictor of long term volunteer behavior. Additionally we see that the amount of initial messages significantly impacts demand completion. Messaging is one of the few methods that is currently considered in existing spontaneous volunteer management plans. FEMA states that consistent messaging from all sources is a vital tool in managing spontaneous volunteers [Points of Light Foundation 2014]. The final significant factor in the least squares model for demand completion is the choice of BI function. As identified by Lindner et al.[2017], the decision to volunteer is likely the most complex decision in a model of spontaneous volunteers. We see a high significance in the

67 Figure 3.10: Standardized least squares model for total demand completion. (Left) Plot of actual by predicted values. (Right) List of response with the corresponding logworth and p-value indicating signiﬁcance.

choice of function used to translate intention into action (P(Bi,t = 1) = f (BIi,t)). Interestingly,

ATT other factors related to motivation, wi , and the scale-free network parameter, do not have any statistical significance in first order effects. However, it is expected that these factors are influencing BI through second and third order effects. Finally, the factor P(reply), has no statistical significance. In a scenario in which sites were not able to disseminate large volunteer request messages (e.g. over news or social media), this would likely have a much more significant effect. Next we compare the least squares model for predicting SV rejections, shown in Figure 3.11. Although R2 = 0.68 with p < 0.0001, the residual plots indicate a non-linear trend in the rejection response. There appears to be very little middle ground in simulated rejection response, with scenarios having zero rejections or a very significant rejection count. We see

ATT the that the only statistically significant factors are the two related to motivation (wi and exp wi ). As suggested in the literature, media coverage and individual perception of need strongly impacts SV convergence. By educating the public to wait for official requests to volunteer prior to the disaster and maintaining a cohesive official message, it may be possible to further reduce SV rejections.

68 Figure 3.11: Standardized least squares model for total volunteer rejections. (Left) Plot of actual by predicted values. (Right) List of response with the corresponding logworth and p-value indicating signiﬁcance.

After removing the factors which were not statistically significant in predicting demand completion or volunteers rejection, we are left with six factors. We use a combined main effects plot to visualize statistically significant factors to both response variables. Figure 3.12 illustrates the conflicting nature of maximizing demand completion while minimizing the number of volunteers turned away. Most factors that improves demand completion negatively impacts volunteers turned away. Care must be taken when determining the messages sent to SVs so that a balance is obtained between demand completion and SV rejected. As stated by Cottrell [2012], authorities need to clearly articulate what they want the public to do: do they want to discourage spontaneous volunteers or call for them? Failing to consider the long term effects of SV rejection may result in fewer SVs attempting to volunteer in future disasters. The results of this analysis not only support the validation of the model, but also provide insight into directions for future data collection efforts. Through the identification of statistically significant parameters, we are able to refocus future data collection efforts. Targeted data collection of these parameters will not only improve the accuracy of our model, but it will also further the discussion of SV convergence with collaborating practitioners.

69 Figure 3.12: Main effects plot for (Top) demand completion (D) and (Bottom) volunteer rejection (R) when population agents follow the score based site choice policy.

3.5 Conclusions

This chapter presents a model to simulate spontaneous volunteer convergence during disaster response. We propose an implementable agent-based modeling framework which captures unique features of spontaneous volunteer convergence. This work was inspired by Lindner et al. [2017, 2018] but significantly expands the agent heterogeneity, behaviors, and communication. We model individual agent behavioral intention (motivation) to volunteer using the Theory of Planned Behavior. Messaging and information sharing between agents occurs through the use of a scale-free network. We propose four different behaviors for site choice selection, based on literature and interviews with spontaneous volunteers. We conduct verification and validation of the model through internal and external evaluation. The model was successfully calibrated against three unique data sources, showing the model is flexible enough to represent different disaster types and locations well. Face validation, through practitioner input, was conducted on the conceptual model and graphical output. Additionally, feedback from meetings with practitioners was used to improve the assumptions and validity

70 of the model. A sensitivity analysis was conducted to further evaluate selected parameters on demand completion and volunteers turned away. The function translating behavioral intention to behavior, along with the initial confidence agents felt toward volunteering significantly impacted the response. We found that the model had relatively high predictive accuracy for both response variables. The results, coupled with feedback from practitioners, suggest that further model development and support is warranted. Spontaneous volunteers represent a significant resource within communities. Planning for and effectively using them can improve resilience to disasters. Through cooperation with disaster practitioners, our model can be used to promote further development of standards, such as ISO 22319:2017 [International Organization for Standardization 2017], which supports integration of SV in disaster response. Additionally, the model can be used as an educational tool to help train the existing workforce on how to best manage SV arrivals, thus ensuring a better response in emergencies.

3.5.1 Limitations

While the model framework proposed is very flexible, many assumptions are drawn from practitioner interviews to fill gaps in existing data. For example, the decision to check motivation only once per day comes from our experience speaking with volunteer managers and even our own volunteer experiences. However, the actual behavior of SVs does not always follow this assumption. By enforcing the once per day check, we are significantly limiting the stochastic nature of SV arrivals. We are hopeful that through continued collaboration and data collection, we can begin to improve upon these assumptions and add additional realism to the model. Model development and validation is limited by the lack of available data surrounding spontaneous volunteer convergence. We note that the calibration scenarios are all evaluated using the same synthetic disaster scenario (based on a geographic area in New Hanover County, North Carolina). The agent and volunteer site locations are based upon regional census data and

71 site locations from a single disaster event. Therefore, we do not capture the impact of potential geographic differences on the calibrated parameters. Lindner et al.[2017] discuss the need to capture the kind of disaster and location to accurately model spontaneous volunteerism. It is possible that to improve calibration across the three unique scenarios, an element of geographic information needs to be included in the model. Finally, we note that there exists a practical limitation on the maximum size of the agent population such that the model is computationally efficient. Due to the complexity of decisions and interactions between agents, we artificially impose a bound (modeled population « actual population) to balance run time with information collection. We attempt to address this limitation by evaluating and comparing the normalized convergence data. With future improvements in coding efficiency and available computer hardware, this limitation should be able to be relaxed significantly.

3.6 Acknowledgements

We would like to thank the summer REU students, Connie Feinberg and Amrita Malur, for their support on this project and the National Science Foundation for their funding support (Abstract# 1901699).

72 CHAPTER

APPLICATION OF SV CONVERGENCE MODEL TO DISASTER PLANNING SCENARIOS

4.1 Introduction

Due to the uncertainty surrounding when and where SVs will arrive and how long they will stay, emergency managers struggle to plan for and integrate them into relief efforts. The problem is confounded by the lack of methods to evaluate disaster plans, as it is often not feasible or cost-effective to conduct live simulations of emergency response. To improve the use of and perceptions towards SVs and maximize the beneﬁt they provide, it is necessary to understand the unique nature of problems facing disaster managers. Thus, there exists a need for a tool to help practitioners evaluate different plans and communicate the results to a wide range of supporting organizations. By improving guidelines for using or managing spontaneous volunteers, we can increase the efﬁciency of volunteer management in disaster response activities [Rivera and

73 Wood 2016]. Improved use of SVs not only speeds up the transition to the recovery phase, it also has a direct economic impact on the community. FEMA’s Public Assistance (PA) Program allows donated resources to be credited toward the non-Federal share of grant costs under the PA program if documented and used appropriately. As discussed in the matching and cost-sharing regulations (44 CFR 13.24), credit may be given for volunteer labor in any field reasonably required for emergency work. For example, following the tornado in Osceola County, Florida, the initial estimate for cleanup alone was over $8 million and 90 days of work. According to Volunteer Florida[2005], the actual cleanup cost was reduced to $1.4 million and completed in 55 days, primarily due to SV support. Additionally, the value of volunteer labor covered $240,000 of the $300,000 county match for FEMA reimbursement [Volunteer Florida 2005]. We apply the SV Convergence Model developed in Chapter3 to answer strategic and operational questions identified by disaster practitioners. The research questions are based upon feedback gained through fieldwork, interviews, and panel discussions. They cover a variety of problems and highlight different ways the SV Convergence Model can support disaster planners. We develop tailored model extensions, use real demographic data, historical locations, and best fit model parameters to simulate the identified problems as closely as possible. This chapter aims to showcase the model’s ability to answer relevant strategic and operational questions while also highlighting its use as a tool to communicate the results to decision-makers and community partners.

By investing effort upfront to ensure that the model addresses exemplar questions appropriately, we ensure that the results are usable and improve practitioner conﬁdence in the model’s capabilities and results. The results of the model are translated into meaningful and easily understood insights to facilitate dissemination. We evaluate the practical implications and discuss future uses of the model as a tool to prescribe messaging strategies. Consistent, accurate, and timely messages about whether or not spontaneous volunteers are needed, and how offers of

74 help are being registered and coordinated can be the difference between volunteering efforts that are beneﬁcial or detrimental to recovery and the community [Leadbeater 2017]. The remainder of this chapter is organized as follows: Section 4.2 motivates the strategic and operational research questions and introduces necessary model extensions. Section 4.3 outlines computational scenarios, results, and discussion related to the operational questions. Section 4.4 summarizes the results, practical implications, and discusses limitations to adoption.

4.2 Research Questions and Extensions

Operations Research can strongly support humanitarian practitioners provided that it deals with pressing real problems [Besiou and Van Wassenhove 2020]. As part of the conceptual model validation in Chapter3, we engaged with disaster practitioners from a variety of governmental agencies (FEMA), non-governmental (Red Cross), and non-proﬁt relief organizations (FB-CENC, NC VOAD, Bread of Life, United Way, Samaritan’s Purse). Throughout these conversations and interviews, we asked participants what problems they faced and how they thought the SV Convergence Model could be best used to improve disaster response. We synthesized the most frequent responses and developed the following strategic and operational questions:

• RQ4.1: What are the beneﬁts/drawbacks of a Volunteer Reception Center?

• RQ4.2: How much does local agency coordination improve disaster response?

• RQ4.3: What is the impact of different messaging strategies on SV convergence?

These questions form the basis of the analysis conducted in Section 4.3. The takeaways of this analysis demonstrate the potential value of the SV convergence model to support disaster practitioners. Background, motivation, and necessary model extensions related to each question are discussed in the subsections below.

75 4.2.1 Volunteer Reception Centers

Volunteer reception centers (VRCs) provide officials with command and control over disaster response. However, large VRCs require a substantial amount of resources to activate and maintain. Once a major disaster has been declared, convergence will begin almost immediately. The sooner disaster management staff are able to activate the VRC, the faster and more efficiently volunteers will be utilized in response and recovery efforts [Points of Light Foundation 2004]. There are several practical reasons for incorporating a VRC process into an emergency operations plan. First, a VRC provides a specific location, staffed by skilled volunteer managers capable of screening, interviewing, and referring citizens in a professional manner. Second, the VRC encourages collaborative planning and implementation on a regional basis, contributing to a positive public perception [Ohio Citizen Corps 2006]. However, VRCs can become overrun with potential volunteers immediately following a disaster. When this occurs, it limits the effectiveness of a VRC and often hinders response activities.

“It discourages volunteers when they’re being turned away. It’s discouraging for them- it’s a difﬁcult process.” - (Volunteer Coordinator)

An alternative to a physical VRC location is a virtual VRC. Virtual VRCs can perform many functions without actually having volunteers on-site. By using volunteer websites and registration software, virtual VRCs can process and assign volunteers [Volunteers of America 2008]. However, virtual VRCs may act as a barrier to engagement for those without the means (hardware or connection) or ability (technological skills) to access them. Local agencies may also be hesitant to use virtual VRCs in favor of a more community based approach. Questions have also been raised about the necessity of managing spontaneous volunteers through the traditional command and control system. Attempts to integrate spontaneous volunteers into formal systems may prove counterproductive by limiting their adaptability, innovativeness, and responsiveness [Whittaker et al. 2015].

76 To study the beneﬁts and drawbacks of VRCs, it was necessary to introduce a new VRC agent to the model framework. The VRC agent has parameters such as VRC location, VRC activation day, and average VRC delay. Activation day refers to the model day in which the VRC is staffed and prepared to accept SVs. To avoid additional complexity we do not model detailed VRC operations, such as the number of servers, service rates, or capacity. Average VRC delay represents the expected service time for screening and assignment that each SV spends before traveling to a volunteer site. We also consider parameters such as VRC enforcement, value of training, and the VRC assignment policy. We summarize the model extensions in red on the SV state chart shown in Figure 4.1.

Figure 4.1: Updated state chart of spontaneous volunteer ﬂow. Model extensions are identiﬁed by red borders and bold red text.

VRC enforcement, represented by a value between [0,1], acts as a proxy for the level of SV integration into the Command and Control structure. A value of zero indicates the VRC is not used, such as in the case of a grass-roots or emergent organizational structure. An enforcement

77 value of one requires that all spontaneous volunteers MUST use the VRC. In practice, disaster managers have used colored bracelets to indicate that an SV has been screened and registered at the VRC. SVs arriving at volunteer sites without a bracelet were referred back to the VRC for registration. The enforcement policy is something that can be region-dependent and should be made known to population agents through volunteer messaging. To account for improved matching and training that often occurs at the VRC prior to volunteering, we add a new population agent attribute and site rejection criteria. For agents assigned through the VRC, the probability of rejection due to screening is zero (versus 0.1 for non-VRC arrivals). The VRC assignment policy considers delayed site information (demand and space available) when assigning a new volunteer. By evaluating multiple scenarios, varying the speed of activation and enforcement of use, we can begin to better understand the beneﬁts that VRCs provide to disaster response.

4.2.2 Local Agency Coordination

Despite high demand and resource limitations, humanitarian organizations typically do not share resources and/or coordinate in the field [Eftekhar et al. 2017]. The 2005 Hurricane Katrina disaster demonstrated a need for a mechanism to encourage collaboration among the local agencies as well as the hundreds of non-governmental organizations [Waugh Jr 2009]. Coordination through joint plans could help humanitarian organizations more efficiently use the available resources and improve service to beneficiaries [Moshtari and Gonçalves 2012]. Existing best practices suggest that disaster management plans include guidance on the communication of organizations’ volunteer needs. Volunteer organizations should develop memorandums of understanding (MOUs), outlining roles and responsibilities, including handling excess SVs. Based on interviews with volunteer managers, we have found that in practice, coordination at the local level is often lacking.

“It was super unorganized. It’s just a really tough situation... you can’t fully plan for

78 this sort of thing, and coordinating people is so difﬁcult.” - (Volunteer Coordinator)

Rejected spontaneous volunteers are not always provided alternate site locations [Augustine et al. 2019]. Inefficiencies occur due to lack of planning and coordination before the disaster and also from a lack of situational awareness during a disaster event. Additionally, volunteer managers of smaller organizations may not be equipped with a plan for dealing with or reassigning excess of spontaneous volunteers. Reports show that following the middle European flood catastrophe (2013) enormous support capacities could be activated. However, it also revealed that some disaster response sites were nearly overrun by SVs whereas other sites were in urgent need of additional manpower [Betke 2018]. Spontaneous, unaffiliated volunteer offers of assistance need to be effectively coordinated to ensure effective utilization of volunteers to support agencies active in an incident. Such efforts must involve inter-agency collaboration among governmental and tribal agencies, voluntary agencies, community-based organizations, faith-based groups, the private sector, and the media [Leadbeater 2017]. In the original model framework presented in Chapter3, there was a simplifying assumption that no reassignment occurred when an SV arrived at a site with no volunteer capacity. We generalize the original model by allowing for reassignment of volunteers, simulating a coordinated plan for volunteer reassignment across local volunteer sites. We allow for variations in the level

of coordination by creating a model parameter Probability of Successful Reassignment (PSR), which ranges from [0,1]. Low values of PSR are representative of the current state of disaster response, where rejected volunteers frequently do not receive an alternate site location. High values of PSR represent an optimistic future case where sites collaborate and volunteers are more likely to receive an alternate assignment.

79 Figure 4.2: New ﬂow of volunteer agent upon arrival to volunteer site. Changes to allow for additional screening and reassignment of spontaneous volunteers are circled in red.

Reassigned agents are redirected to an alternate volunteer site and incur a small travel time delay. Reassignment is only allowed in the case where there is excess capacity at another volunteer site. We allow one reassignment per volunteer attempt to avoid agents bouncing between multiple volunteer sites. The volunteer site ﬂow model was adapted to capture rejection and reassignment, as shown in Figure 4.2. Through simulation evaluation, we can quantify the relative value of coordination between local volunteer agencies.

4.2.3 Volunteer Messaging

Whereas SVs use popular tools such as Facebook or Twitter to organize themselves, official responders have separate command and control systems and communication channels. Typical communication between these two groups is conducted through mass media calls from officials or deployment of emergency call centers [Gerstmann et al. 2019]. Ideally, public messaging will regulate the flow of spontaneous volunteers into the affected area [Points of Light Foundation 2014]. However, research has shown that spontaneous volunteers respond when they perceive a

80 need, not necessarily when they are requested [Fernandez et al. 2006]. With the highest desire to volunteer coming in the ﬁrst week of the disaster, which coincides with the peak media coverage of disasters, this is hardly surprising [Cottrell 2012]. Consistent messaging from all sources is vital in managing spontaneous volunteers, but it is often one of the most difﬁcult aspects to manage.

“Communication [with SVs] was the biggest challenge we face. It’s always the biggest challenge we face. . . ” - (Volunteer Coordinator)

The collective messaging of the community disaster response network should be frequently updated. Messages need to reﬂect the conditions on the ground and include the need for volunteers [Volunteers of America 2008]. In the original model, site messaging was limited to an initial message at the start of the disaster (t=0). However, this assumption limits the ability to update population agents on volunteer needs as the disaster progresses. Therefore, we consider a modeling approach to simulate the beneﬁt of sending messages in the weeks following a disaster. While there are many possible approaches, we adopt a simple method to limit additional complexity in the model framework. As a reminder, the attitude function within the theory of planned behavior is represented as

exp Ai,t = exp(−wi t), (4.1) where wt is the exponential ﬁt parameter and t is the model time in days. In the new approach we slightly change the function to

exp Ai,t = exp −wi (t −tm) , (4.2) where tm is the most recent day an agent receives a request to volunteer (initialized as 0). In this way, we allow for a temporary increase in attitude in the population agents if they receive a speciﬁc request to help.

81 Figure 4.3: Comparison of old (blue) and new (orange) attitude functions over time, highlighting the saw-tooth shape in the new function.

By evaluating different messaging strategies (both size and schedule), we can identify the impact on SV convergence and perception of the disaster response. This analysis allows practitioners to improve policy around messaging and training of community partners.

4.3 Scenarios, Results, and Discussion

We apply the SV Convergence model to a synthetic disaster affecting Wilmington, North Car- olina, and surrounding New Hanover County. Population agents and their locations are generated using census block group data [U.S. Census Bureau 2013]. For computational purposes, we model only a small subset of the total population (∼2000 agents or 1% of New Hanover County) within the agent-based model. Three volunteer site locations are included, based upon actual locations used during Hurricane Florence (2018), as recorded by the NC-VOAD. We select a

82 set of baseline model parameters, based on a mix of the calibrated parameters found to ﬁt the NC-VOAD data of Hurricane Florence and the Victorian Bushﬁres, as shown in Table 4.1.

Table 4.1: List of relevant baseline parameter values selected for scenario evaluation.

Baseline Parameters Value Baseline Parameters Value Max Info Age 2 Rejection Penalty 1 Site Choice Behavior 2 Volunteer Request Day 1 P(Site Search) 0.05 Average Task Fatigue 0.03 ATT Network Parameter 5 Average wi 0.6 exp Initial Demand 100,000 Average wi 0.1 Max Requests 4 Initial PBC Dist (α) 3.135 Adopt Opinion 0.5 f (BI) Linear

In addition to the baseline parameters, we vary selected parameters in each scenario to capture the unique nature of each research question. A summary of the selected parameters and their values for each scenario are shown in Table 4.2. The model simulates 30 days following the synthetic disaster and each iteration includes 30 replications. Using the ﬁxed seed option for replications in the Experiment settings ensures equivalent replications through common random number generation.

4.3.1 Importance of Volunteer Reception Centers in SV Assignment

In this section, we evaluate the impact of volunteer reception centers on volunteer convergence, demand completion, and volunteer rejection. We consider the impact of VRC activation day and VRC enforcement, both of which can be controlled through planning and SV policy. VRC enforcement (percentage of SVs who use the VRC) refers to the regional policy and level of enforcement regarding whether or not volunteers are required to register before participating in the volunteer response. Summary level results of demand completion are shown as a heat map in

83 Table 4.2: Summary of parameters varied during evaluation of research questions. Descretized range for each parameter is listed as (min : increment : max).

Scenario Parameter RQ4.1 RQ4.2 RQ4.3a RQ4.3b Initial Message 75 (0:75:750) (75,600) (0:1:500) VRC Activation Day (0:1:10) 0 0 0 VRC Enforcement (0:0.1:1) 0 0 0 Reassignment 0 (0:0.1:1) 0 0 SV capacity 225 225 250 300 P(response) 0.8 0.2 0.4 0.4

Figure 4.4. The heat map shows the average demand completion for each iteration (representing pairs of VRC activation day and VRC enforcement). For demand completion, we are interested in pairings that result in higher values, indicated as yellow squares.

Figure 4.4: Heat map of average demand completion across variations in VRC enforcement and VRC activation day.

84 To maximize demand completion, it is important to activate the VRC early (< day 3), as the marginal benefit of activation decreases over time. Delayed activation of the VRC beyond day seven results in reduced demand completion. The best VRC strategies for our synthetic example activate on day zero and facilitate ∼50% increased demand completion when compared to no VRC use. The full numerical values including the 95% confidence intervals can be found in AppendixD. The results suggest that the benefit of requiring additional volunteers to register at the VRC (>0.7 enforcement) is offset by the additional delay (travel time to/from and wait time at VRC). While activation on day zero is optimal regardless of enforcement level, immediate activation of a physical VRC is not appropriate or feasible in all disaster scenarios. It is worth considering ways to adapt the VRC to fit the needs of the disaster or how an adapted VRC can support a larger area [Volunteers of America 2008]. Due to the size or nature of a disaster, it may be more effective to implement a virtual VRC. Rauchecker and Schryen[2018] propose an example of a virtual VRC integration plan. In that paper, volunteers can register via smartphones using the KUBAS volunteer application and submit offers for their help to an administrative component called KUBAS kernel. The command staff is connected to the KUBAS kernel via an interface to their command-and-control system, which allows them to submit and prioritize demands for volunteers [Rauchecker and Schryen 2018]. While virtual VRCs appear to be a good alternative because they can be activated quickly, there are additional challenges to consider. Preparation for immediate action of a virtual VRC requires a significant investment in technology and relies on internet/cellular services immediately following the disaster. Additionally, virtual VRCs lack the ability to provide pre- assignment training, improved assignment matching (confirmation of skills), and the capability to weed out volunteers who may not be suited for beneficiary facing tasks. Next, we investigate the number of volunteer rejections as they relate to VRC activation and enforcement. A similar heat map of volunteer rejections is shown in Figure 4.5. We note that the scale is inverted so that the best case (minimal rejections) is shown in yellow. We see that the

85 largest number of volunteer rejections actually occurs during full enforcement and immediate site activation. It appears that in this scenario, the overwhelming response to the VRC results in a signiﬁcant increase in SV rejections.

Figure 4.5: Heat map of average total volunteer rejections across variations in VRC enforcement and VRC activation day.

We can minimize the number of rejections by delaying VRC activation (e.g. beyond day seven). Delaying the activation allows the population level behavioral intent to decrease significantly before the volunteer requests are sent. However, this comes at the price of a substantial reduction in demand completion. When comparing VRC activation on day 10 to immediate activation, we see three times as many volunteer rejections on average. As first identified in Section 3.4 of Chapter3, there is a clear trade-off between demand completion and volunteer rejections. Regardless of activation day, there is a clear benefit of enforcing the VRC usage in almost

86 all cases tested. To maximize the use of the VRC, the enforcement of use should be at least 30%. Regardless of what the ofﬁcial policy states regarding VRCs, it is likely that some SVs will ignore those recommendations and attempt to volunteer directly with local agencies.

“Some people showed up and probably did not have that authorization.”- (Volunteer Coordinator).

Given this information, a strict enforcement policy appears to be optimal, assuming the VRC or virtual VRC is activated during the ﬁrst three days following a disaster. The VRC improves demand completion by controlling the assignment policy and minimizing site rejection when compared to individual choice. It is worth noting that strict enforcement of the VRC in late activation cases results in a decrease in demand completion. In these cases, motivated SVs have likely sought out their own volunteer opportunities. Enforcing a new site choice or forcing them to attend the VRC causes unnecessary delays and inefﬁciencies.

4.3.2 Value of Local Agency Coordination

To evaluate the effectiveness of local agency coordination and answer research question RQ4.2, we simulate a range of coordination values while also varying the number of initial site messages. Coordination here refers to the ability for volunteer sites to communicate with each other and reroute rejected volunteers. Coordination is represented as the probability of a successful volunteer reassignment [0,1]. For example, if a volunteer arrives but is rejected due to site

capacity, that volunteer is rerouted to another site with PSR. Like the previous section, we compare the total amount of work completed over 30 days and evaluate the number of SVs that are rejected during the same period. The average demand completion heat map can be seen in Figure 4.6. We note that the scale of demand completion is slightly lower than in the previous scenario, due to the decreased probability of response. The heat map shows that as coordination increases, completed demand

87 also increases for a fixed initial message size. The gradient suggests that improved coordination allows sites to handle a more significant influx of converging volunteers effectively. For example, given an initial message size of 100, full coordination increases demand completion by over 25%. This result may be useful to regional disaster managers to communicate the benefit and motivate collaboration among local agencies.

Figure 4.6: Heat map of average demand completion across variations in percent successful coordination (PSR) and initial site message size.

In cases with extensive requests for volunteers (relative to capacity), there is minimal beneﬁt of signiﬁcant coordination (e.g., beyond 0.3). There appears to be a threshold value for initial message size for a given site capacity in which additional messages are not effective in improving demand completion. This result provides some insight into why there is often a lack of coordination among local relief agencies. If all volunteer sites are inundated with offers of assistance, there is no need/capacity to reassign excess volunteers within the system. However,

88 if volunteers converge to a single site due to lack of knowledge or site selection, coordination is beneficial. Additionally, when volunteer resources are more scarce, such as three or more weeks after the disaster, we also expect a significant benefit from coordinating. Figure 4.7 provides similar results but instead depicts the number of volunteer rejections by model completion. Again, we note that the scale has been inverted, since it is best to minimize rejections. We see that for cases with lower volunteer coordination, there are more volunteers rejected. For the case of 100 messages, full coordination results in an approximately 20% decrease in rejections. This supports the development of local agency coordination plans and serves as motivation to encourage agency participation.

Figure 4.7: Heat map of average total volunteers rejected across variations in percent successful assignment and initial site message size.

We also see a signiﬁcant increase in rejected volunteers in cases where the message size is signiﬁcantly larger than available capacity. After coordination (percent successful reassignment)

89 reaches 0.3, there is minimal beneﬁt to additional coordination efforts for most cases. At initial message sizes above 200, site coordination has almost no impact on minimizing rejections. However, there are other practical beneﬁts to improved coordination that should be considered, such as avoiding duplication of work and improved sharing/distribution of donated resources. We discuss the limitations of messaging due to capacity constraints as they relate to demand completion and rejections in the next section.

4.3.3 Messaging Strategies and Site Capacity

In this section, we compare initial messaging strategies and investigate the impact on rejections and long term SV perception. The ﬁrst case represents a conservative messaging strategy, with an initial message size of 75 (∼ 3.6% of the population). The second case considers an aggressive call for volunteers, where organizations overestimate volunteer need. We assume the aggressive strategy results in 600 messages (∼ 28.8% of the population) sent immediately following the disaster. In addition to demand completion and volunteer rejection, we capture beginning and

ending intention to volunteer (BIi,t), for all agents i, t = 0,T. Figure 4.8 provides an overview of the average system-level outputs. The conservative messaging strategy is shown in blue and the aggressive strategy is shown in red. Figure 4.8

(a) shows the average normalized total demand remaining Dt/D0 and represents the progress of work conducted over time. Figure 4.8 (b) represents the average number of spontaneous volunteers participating in disaster response totaled across all three sites. The dashed line represents the maximum SV capacity of volunteer organizations within the model. We assume the SV capacity to be ﬁxed, although that assumption could be relaxed in future iterations.

Figure 4.8 (c) plots the number of rejections, Rt for both scenarios. Finally, Figure 4.8 (d) illustrates the change in average PBC value for population agents and changes due to individual agent experiences over time. By comparing messaging strategies, Figure 4.8 (a) shows that a strong initial request

90 allows more work to be completed during the first 30 days. The aggressive messaging strategy completes all initial demand by day 25. Recall that the demand is assumed to be deterministic and non-increasing which is often the case for debris clearing or mucking tasks. Alternatively, the conservative messaging strategy lags in demand completion by nearly a week. We see in Figure 4.8 (b) that this is due to the more significant number of SVs active at volunteer sites during the first two and a half weeks. While the amount of initial messages is small in the conservative case, site information travels across the population via the agent communication network.

(a) Total Demand Remaining (b) Total SV at Site by Day

Figure 4.8: Summary of system level outputs for notional example evaluating site to agent messaging scenario. Conservative messaging is shown in blue and an aggressive strategy in red.

While graphs (a) and (b) suggest that a policy of maximum messaging is best, it does not capture the full story. The aggressive site messaging is so successful that sites reached total site

91 capacity for almost a full week as indicated by the dashed line in Figure 4.8(b). During this period, there is a need to turn away many of the potential volunteers. SV convergence that is extreme and unplanned can overwhelm emergency managers and result in negative consequences for the current disaster. For example, turned away volunteers may start volunteering in an unofﬁcial capacity.

“All of the [sites] were swamped with volunteers. So people started just going out into the community and volunteering" - (Volunteer)

This, in turn, can result in further pressure on recovery agencies to manage duplication of effort or inappropriate activities [Leadbeater 2017]. Figure 4.8 (c) shows that the case with aggressive messaging results in almost double the number of volunteers turned away. Rejecting so many volunteers signiﬁcantly reduces average perception (PBC) of whether agents are able to complete the volunteering task, as shown in Figure 4.8 (d). The reduction in average perception is better illustrated by evaluating the density of PBC values across the population. Figure 4.9 overlays the initial PBC probability density function (PDF) in grey with the ﬁnal PBC densities for each messaging scenario at T=30. The initial

PBC PDF follows a beta distribution, β(3.5,8.5), which was selected to represent the target population in the synthetic example. The red and blue lines approximate the resulting PBC PDF for each case after 30 days. Both messaging strategies result in a subset of the population trending toward higher PBC values (>0.7), indicating an improved conﬁdence in their perceived ability to volunteer. However, overloading volunteers with requests and subsequently turning them away creates a signiﬁcant increase in permanently unmotivated agents (shown as agents with PBCi < 0.1). This result aligns with initial qualitative evidence that rejections can cause individuals to stop volunteering.

“A lot of them were being told, ‘hey, can you come back later? Or, there’s a huge line. We don’t need anybody right now.’ Then people stopped volunteering" - (Volunteer)

92 Figure 4.9: Approximate distribution of PBC values across all population agents. The initial distribution of PBC values is shown in grey. The ﬁnal distribution for the conservative and aggressive messaging strategies are shown in blue and red respectively.

The conservative messaging strategy improves the number of agents with high PBC values (0.9+), suggesting that more population agents are likely to convert to affiliated or longer-term role after the disaster. There is a clear trade-off between demand completion and motivation which can be impacted by messaging strategies. Due to communication between volunteers, volunteer messaging may result in a significantly larger volunteer response than expected. It is essential to consider the available capacity of local agencies to take in and manage SVs when developing communication plans. While aggressive requests for volunteers may result in additional work completion, there is a significant risk of overwhelming the system and causing a significant number of rejections.

4.3.4 Optimal Size of Weekly Messages

We have shown that disaster managers can influence volunteer convergence (and indirectly influence demand completion) through messaging, collaboration, and VRC activation policies. Volunteer messaging is one of the best intervention strategies to control the flow of spontaneous volunteers to disaster sites. As a proof of concept, we adapt the SV convergence model to provide prescriptive messaging strategies using simulation optimization. We focus on two objectives

93 of interest, one measuring cost (demand completion) and the other measuring performance (rejections). We are interested in determining a messaging schedule that maximizes objectives relevant to practitioners. These objectives are inversely related, and the resulting messaging decision is a trade-off between them. We can simplify the problem and formulate it as a single weighted objective, as shown in

Equation 4.3a. The value of weights w1 and w2 represent the trade-off between the importance of the two objectives. We constrain the total amount of messages sent during the response to a maximum value, M, set to 500 in this scenario. Total messages are capped to avoid donor fatigue, which refers to the feeling an agent has when they are being inundated with

requests to donate, or in our case, volunteer. The value of demand completion (Dr) and rejection

(Rr) for each replication (r) is solved using functions f and g, respectively. The simulation model acts as a function (whose explicit form is unknown) that evaluates the merit of a set of speciﬁcations, typically represented as a set of values [April et al. 2003]. Finally, we ensure that each weekly message is non-negative and integer. We solve for the schedule of weekly

messages, m = {m0,m1,...,mW }, which maximizes the expected objective function value over a set of N replications.

1 N max ∑ w1Dr − w2Rr (4.3a) N r=1 W s.t. ∑ mi ≤ M (4.3b) i=0

Dr = f (m) for r = 1,...,N (4.3c)

Rr = g(m) for r = 1,...,N (4.3d)

+ mi ∈ Z for i = 0,...,W (4.3e)

We use Anylogic’s built-in optimization functionality (OptQuest) to ﬁnd the optimal message sizes given different objective weights. OptQuest uses a variety of meta-heuristics (Scatter search,

94 tabu search, neural networks) to search for solutions. Utilizing the synthetic example of New Hanover County, we identify optimal messaging strategies under the following objectives: (1) Minimize SV rejections, (2) Maximize Demand Completion, and (3) Weighted Multi-Objective.

We consider a variety of weighted scenarios for w1 and w2, including setting them proportional

in magnitude. Proportional weights (w1=1, w2 ∈ [50,200]) were calculated by dividing average demand by average rejections, as reported in Sections 4.3.1 and 4.3.2. However, identifying “correct” weights for different disaster types requires signiﬁcant discussion with practitioners, which has not occurred at the time of publication. A summary of the results of the simulation optimization analysis is shown in Table 4.3. Solutions listed are the best found after 300 iterations, each with 10 replications.

Table 4.3: Example of prescriptive messaging based on different disaster management objectives for synthetic example in New Hanover County.

Objective (max) E[D] E[D − 50R] E[D − 100R] E[D − 150R] E[−R] Week 0 0.46M 0.70M 0.28M 0.16M 0 Week 1 0.54M 0 0 0 0 Week 2 0 0 0 0 0 Week 3 0 0 0.49M 0 0 Week 4 0 0.30M 0.23M 0.84M 0

The process serves as a decision-support tool that prescribes optimal weekly messaging strategies, given a chosen objective function. To interpret the results, we need to consider both the frequency and size of messages for each solution. To maximize demand completion, OptQuest suggests sending large proportion of requests (0.46 and 0.54 M) to SVs during the ﬁrst two weeks of the disaster. By getting the message out early and to as many population agents as possible, the information is able to propagate across the network and maximize the number of volunteers completing demand tasks. However, maximizing demand completion comes with a penalty of a signiﬁcant increase in volunteer rejections. In the single objective case, there is

95 an approximately 50% increase in demand completion but an over 400% increase in rejections. Given the potential effect of rejections on individuals’ sustained participation, it is important for disaster managers and local agency partners to consider that impact when developing messaging strategies. By evaluating the weighted cases which consider both objectives, we see solutions that are more in line with what was expected by practitioners. Volunteer interest typically decreases significantly in the weeks following the disaster event, and volunteer organizations must try significantly harder to encourage continued participation. In the case when the magnitudes of each objective are proportional (D-100R), we see this phenomena occurring. There is an initial request sent to population agents, representing less than half of available volunteer capacity, followed by requests in weeks three and four for significantly more volunteers. Because the messages are encouraging in nature (improves attitude, site opinion, and site knowledge), any positive value implies a real request for volunteers. However, the recommendation to send zero messages can be interpreted in two ways. For example, consider the scenario which seeks to minimize the number of rejections (R) as a single objective. We see that, as expected, the optimal strategy is to send no messages. Does the solution imply that sites are not in need of volunteers, or should sites actively discourage any volunteer participation? In practice, sites often send messages discouraging SVs, and asking them to delay participation until the area is safe or additional resources (such as capacity or housing) become available. When solving two objective simulation optimization problems, OptQuest first searches for optimal solutions to individual single objective problems. OptQuest then searches for solutions between the two, effectively generating an approximate Pareto Frontier. By capturing the results from each replication of the OptQuest search, we can visualize the approximate Pareto Frontier for our multi-objective problem. Figure 4.10 illustrates the trade-off that occurs between demand completion and volunteer rejection for the synthetic example in New Hanover County.

In practice, volunteer managers are not often prepared or equipped to handle large number of

96 Figure 4.10: Pareto Front of simulated solutions for objectives: (1) demand completion and (2) volunteer rejection.

spontaneous volunteers during the first few weeks of a disaster. The dominated solutions, shown in Figure 4.10, highlight the possible outcomes of poor volunteer management and integration. When this happens, it results in significant inefficiencies (low demand completion) and negative experiences (rejections). These results further highlight the need for improved management practices. Through the development of new tools and models, we can ensure that managers are better prepared to handle SVs and increase the speed of disaster response and eventual recovery.

4.4 Conclusions

In this chapter, we applied the SV Convergence Model to a synthetic example and answered relevant strategic and operational questions. The research questions were identiﬁed by disaster management practitioners and selected based on their potential for impact. We provided motivation for each research question and introduced the necessary model extensions for analysis.

97 Using targeted scenarios and comparative analysis, we evaluated each question and discussed policy implications related to SV convergence and integration . Results highlight how the SV Convergence Model can support the evaluation of disaster plans and help to communicate the benefit that spontaneous volunteers provide. We found that using volunteer reception centers provides a significant benefit if activated early enough in the response phase and use is enforced effectively. When physical VRC activation is not feasible, virtual VRCs can be used to obtain similar benefits. Next, we illustrated the benefit of local agency coordination in terms of demand completion and volunteers turned away. We show that minimal coordination efforts can improve demand completion and reduce rejections by approximately 25%. We conducted a detailed analysis of messaging to identify the potential long-term impacts of increased rejections. We showed that while aggressive messaging does maximize demand completion, there is a significant negative impact on volunteer perception which may hinder future volunteer efforts. Using Simulation Optimization, we identified strategies for weekly communication to maximize a variety of objectives. We also illustrated how mismanaging spontaneous volunteers can result in decreased work completion and/or an increased number of volunteer rejections. Results showcase the model’s ability to answer relevant strategic and operational questions and supports the development of additional “what-if” scenarios. We conclude by reiterating the need for continued practitioner engagement and an iterative review process. Collaboration with practitioners and community partners improves the long term likelihood that the model is adopted.

4.4.1 Limitations

While there was a signiﬁcant effort to identify relevant research questions, full interpretation of results also requires practitioners’ input. At the time of publication, we have not yet received feedback related to the results presented in this chapter. While there is value that can be gained

98 from the analysis provided, direct applicability is limited due to lack of available data. As discussed in Chapter 3, the evaluation scenarios are run using the same synthetic example based on a geographic area in New Hanover County, North Carolina. Therefore, caution should be exercised when attempting to generalize the results to different disaster locations or types.

99 CHAPTER

CONCLUSION

5.1 Dissertation Summary and Contributions

With fewer barriers to participation and greater access to real-time information about disasters, spontaneous volunteers are even more likely to respond [Leadbeater 2017]. Contemporary research and experience confirms the inevitability of spontaneous volunteering, and its potential to contribute to community resilience [Leadbeater 2017]. The need for improved planning to minimize the negative impacts of disasters is well documented: the Sendai Framework for Disaster Risk Reduction and the UN Sustainable Development Goals both seek to improve preparedness and risk reduction through the development of policies and plans. However, most cities in the US do not appear to plan for the use and management of spontaneous volunteers [Rivera and Wood 2016]. The unpredictability, magnitude, uncertainty, and complexity of disasters make it difficult to develop efficient disaster management plans [Hoyos et al. 2015]. Disaster and emergency management is more effective when community responses are anticipated, planned for and integrated within a formal system [Leadbeater 2017].

100 This dissertation develops mathematical and simulation models to better understand convergence and improve the integration of spontaneous volunteers in disaster response. In Chapter2, we present a strategy for finding the optimal assignment of spontaneous volunteers within a single disaster relief location under uncertainty. Volunteer managers typically are responsible for assigning incoming volunteers to available tasks within a volunteer site or VRC. They must make important decisions on assignment with very little information and often rely on “gut instincts”. We formulate a Markov Decision Process and solve for an optimal assignment policy using value iteration. The optimal policy is compared against a variety of heuristic policies using a discrete event simulation. A set of computational experiments are presented to test the robustness of each assignment policy. We attempt to balance implementation complexity with available resources during a disaster to find the best implementable solution given the often hectic situations following a natural disaster. In Chapter3, we develop an agent-based modeling framework to represent regional spontaneous volunteer convergence following a disaster. We blend concepts from social science, emergency management, and management science into the simulation model to evaluate different disaster management strategies. The framework considers a heterogeneous population of agents connected via a social network, each with unique attributes such as age, motivation, opinion, and site choice behavior. We conduct verification and validation of the model using both real world data and subject matter input. Through evaluation of a targeted design of experiment, we identify significant model parameters. The framework provides the necessary foundation toward developing a tool that can be used by disaster management practitioners to improve integration plans. Chapter4 presents an analysis of strategic and operational level questions identified by practitioners through panel discussions. We apply the SV Convergence Model to a synthetic disaster scenario in New Hanover County, North Carolina. We simulate actual population demographics and use historical volunteer site locations. Results from the models are intended

101 to inform SV policies and practices and encourage community participation to improve disaster resiliency. We close with a demonstration of how the model can be used as a decision support tool to identify optimal weekly messaging strategies. We present a variety of different site messaging strategies while varying the importance of demand completion and volunteer rejections. The analysis provides an overview of the various uses of the model in disaster planning and highlights the beneﬁt that can be gained from planning and integration. Research, if not properly informed by practice, may clearly lag instead of being timely and impactful [Besiou and Van Wassenhove 2020]. By conducting ﬁeld work and engaging practitioners throughout the research process, we ensure that the models developed in this dissertation address real problems. However, developing models that address the right questions is only part of the solution. Improving disaster response plans requires that practitioners trust the results and buy into the recommendations. We strive in each chapter to link computational results to real world implications in way that can be understood by practitioners. Where appropriate, we also suggest policy recommendations to improve current disaster plans. This dissertation serves as a foundation for the development of decision tools that can be used by practitioners to improve SV integration. We note that model development and implementation is an iterative process, and continued engagement with practitioners is necessary for us to expect any real improvements.

5.2 Future Directions

In this section, we brieﬂy identify some possible directions for future work. We include both ideas for model extensions and also suggest areas ripe for additional statistical analysis. The proposed model extensions focus on improving the validity, realism, and scope of the SV models. Regarding the single-site model developed in Chapter2, future work includes efforts to

102 relax assumptions and incorporate volunteer preferences in assignment decisions. In Section 2.7.1 we discuss limitations due to the Markov assumptions and suggest methods to allow task learning or training. Similarly, one could introduce volunteer preference by increasing the state space to represent different classes of volunteers. However, as the state space grows, the computational difficulty increases significantly. One possible extension would be to use reinforcement learning or other machine learning techniques instead of value iteration to solve for near-optimal assignment policies. However, the usefulness of any policy found may be limited due to the need for full system information and lack of interpretability. Alternatively, we propose developing a hybrid model that combines the models from Chapter 2 and Chapter 3. Using the SV Convergence Model to evaluate site assignment policies while considering individual agent preference introduces new set of research questions. Individual preference influences motivation, site opinion, site choice, probability of rejection, and length of stay. By incorporating task preference into volunteer assignment, practitioners can learn more about the long term impact of considering individual preference into assignment decisions. Another direct extension of the existing simulation developed in Chapter 3 would be to model the Volunteer Reception Center explicitly. Detailed representation, including queue capacity, service rates, and agent balking would allow practitioners to better identify size and space requirements for VRCs. This capability also allows for exciting analysis such as determining dynamic staffing requirements as the convergence rate is time dependent. Additionally, the optimal location and/or number of VRC facilities required for different types and sizes of disasters is an area of interest to practitioners. Due to the lack of existing data related to SV convergence, there exists a strong need for continued data collection and validation efforts. In Section 3.4 we identify some agent parameters which appear to have a significant effect on the response variables. Data for these parameters can be collected remotely using surveys or directly through continued partnership with local disaster agencies. We are hopeful that as electronic registration tools and volunteer tracking

103 software becomes more common, the ability to collect and utilize SV data improves. Similarly, there exists a need to collect convergence data across different geographic, demographic, and disaster types. All agent-based simulation runs from Chapter 3 and Chapter 4 were run using the same synthetic scenario of Wilmington, NC. Therefore, we are limited in our investigation of the geographic impacts that may be present in different disaster locations. We have the ability to capture some of the geographic changes indirectly through agent parameters (motivation, connectivity, social pressures, etc.) and site choice behaviors (distance and score). However, the results of the analysis conducted in Chapter 4 is not necessarily generalize to all disasters due to this limitation. One specific area of interest is in the development of a database of volunteer convergence, given different disaster types and locations. In conjunction with additional data collection efforts, input and output parameter modeling using bootstrapping, interpolation, or empirical analysis could be conducted to further improve the model results. By considering a wider range of input parameters instead of fixed values, we can evaluate scenarios in a way that is more adaptable. As the SV convergence model matures, there is also a need for additional statistical analysis. There is potential for research to evaluate model parameter selection (e.g. principal component analysis) and potentially remove redundant parameters or better capture parameter correlation and covariance. Similarly, there is an opportunity to develop more robust simulation optimization procedures to better identify calibrated parameter values and weekly messaging strategies. These simulation optimization methods may also be applied to finding a more exact Pareto frontier for the objectives demand completion and volunteer rejection. We are seeing an extreme downturn in the number of volunteers as the Coronavirus pandemic of 2020 continues to spread. A recent study by Volunteer Match found that 68% of non-profit organizations surveyed (463 total) reported heavy cancellations due to COVID-19 [Hughes 2020]. The model framework introduced in Chapter 3 could be adapted to evaluate the impact of COVID-19 on the decision to volunteer. By updating the agent behaviors to consider the risk

104 of COVID-19 when determining to converge to a volunteer site, queue at a VRC, or remain idle, we can help practitioners to understand how this pandemic is impacting response. We could use the agent-based model to consider different "what-if" scenarios to evaluate different strategies such as lower volunteer capacity at sites due to social distancing. The evaluation of behavioral decisions and the impact of COVID-19 is not limited to just volunteering. For example, we could develop new models to simulate the impact of COVID-19 on the decision to travel, using a variation of the BI framework presented in Chapter 3. Due to the modular and ﬂexible nature of the model framework, it could be adapted to evaluate any type of agent-system impacts in which decisions are made under uncertainty.

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113 APPENDICES

114 APPENDIX

FULL COMPUTATIONAL RESULTS FOR CHAPTER 2

The Tables in AppendixA provide the full results of the computational experiments. Results are shown for all policies (e.g. Fewest Volunteer, Largest Weighted Demand, Largest Queue Clearing Time, Markov Decision Process, and Best Random) over all cases. Table 2.1 summarizes the system parameters for each of the baseline cases. Tables A.2 and A.3 include the sample mean AHC from 1000 simulated replications for each policy. The difference in mean AHC between LWD and LQCT and percent deviation from the MDP policy are also included for ease of comparison. Bolded values indicate a statistically signiﬁcant difference (p < 0.05) in sample mean AHC between MDP and all other policies. Similarly, the difference between LWD and LQCT is bolded if the difference is statistically signiﬁcant (p < 0.05).

115 Table A.1: Input parameters for each of the 2k baseline cases.

Case λ γ V1 V2 µ1 µ2 α1 α2 h1 h2 1 3.066 0.517 5 3 2.4 1.2 3 1 20 10 2 3.066 0.517 5 3 2.4 1.2 3 1 20 60 3 3.066 0.517 5 3 2.4 1.2 3 2 20 10 4 3.066 0.517 5 3 2.4 1.2 3 2 20 60 5 3.066 0.517 5 3 2.4 1.2 5 1 20 10 6 3.066 0.517 5 3 2.4 1.2 5 1 20 60 7 3.066 0.517 5 3 2.4 1.2 5 2 20 10 8 3.066 0.517 5 3 2.4 1.2 5 2 20 60 9 3.066 0.517 5 3 2.4 3.6 3 1 20 10 10 3.066 0.517 5 3 2.4 3.6 3 1 20 60 11 3.066 0.517 5 3 2.4 3.6 3 2 20 10 12 3.066 0.517 5 3 2.4 3.6 3 2 20 60 13 3.066 0.517 5 3 2.4 3.6 5 1 20 10 14 3.066 0.517 5 3 2.4 3.6 5 1 20 60 15 3.066 0.517 5 3 2.4 3.6 5 2 20 10 16 3.066 0.517 5 3 2.4 3.6 5 2 20 60 17 3.066 0.517 5 8 2.4 1.2 3 1 20 10 18 3.066 0.517 5 8 2.4 1.2 3 1 20 60 19 3.066 0.517 5 8 2.4 1.2 3 2 20 10 20 3.066 0.517 5 8 2.4 1.2 3 2 20 60 21 3.066 0.517 5 8 2.4 1.2 5 1 20 10 22 3.066 0.517 5 8 2.4 1.2 5 1 20 60 23 3.066 0.517 5 8 2.4 1.2 5 2 20 10 24 3.066 0.517 5 8 2.4 1.2 5 2 20 60 25 3.066 0.517 5 8 2.4 3.6 3 1 20 10 26 3.066 0.517 5 8 2.4 3.6 3 1 20 60 27 3.066 0.517 5 8 2.4 3.6 3 2 20 10 28 3.066 0.517 5 8 2.4 3.6 3 2 20 60 29 3.066 0.517 5 8 2.4 3.6 5 1 20 10 30 3.066 0.517 5 8 2.4 3.6 5 1 20 60 31 3.066 0.517 5 8 2.4 3.6 5 2 20 10 32 3.066 0.517 5 8 2.4 3.6 5 2 20 60

116 Table A.2: Computational results of the 2k factorial base cases.

Mean Holding Cost per Day Average Deviation from MDP Case MDP LWD FV LQCT BR LWD-LQCT LWD FV LQCT BR 1 24.4 26.1 31.5 26.2 31.6 -0.1 6.8% 28.7% 7.4% 29.3% 2 60.5 61.4 65.1 61.4 80.7 0.0 1.5% 7.5% 1.5% 33.4% 3 52.8 55.1 58.5 52.8 76.6 2.3 4.3% 10.8% 0.0% 45.1% 4 181.8 178.5 222.9 189.9 317.0 -11.4 -1.8% 22.6% 4.4% 74.3% 5 52.7 55.4 110.3 61.6 68.3 -6.2 5.2% 109.3% 16.9% 29.7% 6 103.7 107.3 144.9 109.7 137.4 -2.4 3.4% 39.7% 5.7% 32.5% 7 103.1 110.9 134.9 109.4 140.4 1.5 7.6% 30.9% 6.1% 36.2% 8 270.9 275.8 295.5 264.4 420.8 11.5 1.8% 9.1% -2.4% 55.3% 9 17.0 19.7 27.5 20.0 20.7 -0.3 15.7% 61.7% 17.4% 21.6% 10 29.5 31.9 36.4 30.8 41.4 1.1 7.9% 23.2% 4.1% 40.0% 11 22.6 23.5 29.3 24.4 28.9 -0.9 4.3% 30.0% 8.1% 28.3% 12 46.9 48.6 51.0 47.6 68.8 1.0 3.6% 8.9% 1.6% 46.9% 13 39.8 45.6 104.9 47.5 49.5 -2.0 14.5% 163.6% 19.4% 24.3% 14 59.5 63.9 113.9 63.9 78.2 0.0 7.5% 91.6% 7.5% 31.5% 15 48.9 54.1 108.3 55.1 61.0 -0.9 10.8% 121.6% 12.7% 24.8% 16 85.5 91.6 130.2 87.7 121.8 3.9 7.1% 52.3% 2.5% 42.4% 17 22.9 28.6 30.8 29.0 34.7 -0.4 25.0% 34.4% 26.7% 51.5%

Baseline 18 50.7 57.8 57.6 54.8 76.8 3.0 13.9% 13.5% 8.0% 51.4% 19 38.7 44.7 46.6 43.6 67.2 1.1 15.5% 20.5% 12.7% 73.9% 20 108.4 116.6 152.5 111.2 171.2 5.4 7.6% 40.7% 2.6% 57.9% 21 47.9 57.3 122.4 64.4 73.9 -7.1 19.6% 155.6% 34.5% 54.4% 22 91.6 105.1 149.6 103.0 142.4 2.1 14.7% 63.2% 12.4% 55.3% 23 80.3 89.4 126.1 95.1 141.1 -5.7 11.3% 57.0% 18.3% 75.6% 24 194.7 218.1 225.2 206.1 307.3 12.0 12.0% 15.7% 5.8% 57.9% 25 16.5 23.5 27.1 24.0 20.7 -0.5 42.2% 64.0% 45.5% 25.1% 26 27.4 33.2 34.5 32.0 39.4 1.3 21.4% 26.0% 16.8% 44.0% 27 21.0 27.4 28.8 27.0 28.7 0.4 30.5% 37.4% 28.5% 36.6% 28 41.8 47.2 46.3 45.0 63.7 2.2 12.8% 10.8% 7.5% 52.3% 29 38.8 49.9 110.3 53.9 49.0 -4.1 28.5% 184.1% 38.9% 26.3% 30 55.3 68.0 117.6 65.3 74.8 2.7 23.0% 112.4% 18.0% 35.1% 31 47.0 57.8 117.6 57.9 62.8 -0.1 23.0% 150.0% 23.1% 33.4% 32 77.3 92.6 135.0 86.2 120.4 6.4 19.7% 74.6% 11.5% 55.7%

117 Table A.3: Computational results of the sensitivity cases.

Mean Holding Cost per Day Average Deviation from MDP Case MDP LWD FV LQCT BR LWD-LQCT LWD FV LQCT BR S1.1 476.3 591.3 636.8 568.9 619.6 22.4 24.1% 33.7% 19.4% 30.1% S1.2 420.7 513.4 595.7 497.3 547.3 16.1 22.0% 41.6% 18.2% 30.1%

λ S1.3 380.3 451.4 534.2 428.4 496.7 23.0 18.7% 40.5% 12.6% 30.6% S1.4 338.0 392.1 489.2 389.5 463.1 2.6 16.0% 44.7% 15.2% 37.0% S1.5 310.3 356.3 462.5 322.1 412.3 34.2 14.8% 49.0% 3.8% 32.8% S1.6 276.7 305.8 411.7 301.2 376.3 4.6 10.5% 48.8% 8.9% 36.0% S1.7 257.0 265.8 354.7 262.1 344.7 3.7 3.4% 38.0% 2.0% 34.1% S1.8 226.2 259.5 333.4 237.8 310.6 21.7 14.7% 47.4% 5.1% 37.3% S1.9 213.7 228.0 298.1 210.5 286.4 17.5 6.7% 39.5% -1.5% 34.0% S1.10 188.3 207.3 271.3 187.6 258.5 19.7 10.1% 44.1% -0.3% 37.3% S1: Vary Arrival Rate, S1.11 178.3 185.4 250.2 175.7 234.3 9.6 3.9% 40.3% -1.5% 31.3% S1.12 163.0 170.1 225.3 168.7 210.0 1.4 4.3% 38.2% 3.5% 28.8% S2.1 545.2 644.3 611.5 644.2 715.1 0.1 18.2% 12.2% 18.2% 31.1%

2 S2.2 450.0 531.0 585.7 524.6 572.6 6.4 18.0% 30.1% 16.6% 27.2% V S2.3 420.7 513.4 595.7 497.3 554.8 16.1 22.0% 41.6% 18.2% 31.9% S2.4 415.9 505.6 577.8 481.4 550.5 24.2 21.6% 38.9% 15.8% 32.4% S2.5 412.5 505.3 560.7 496.4 566.1 8.9 22.5% 35.9% 20.3% 37.2% S2: Vary S2.6 408.1 493.5 562.6 469.7 565.5 23.8 20.9% 37.9% 15.1% 38.6% S3.1 420.7 513.4 595.7 497.3 547.3 16.1 22.0% 41.6% 18.2% 30.1% S3.2 202.0 213.1 224.5 209.4 298.1 3.7 5.5% 11.1% 3.7% 47.6% 1

µ S3.3 139.7 146.2 167.7 140.4 209.7 5.8 4.6% 20.0% 0.5% 50.1% S3.4 112.1 121.3 145.4 114.4 179.0 6.9 8.2% 29.6% 2.0% 59.6% S3.5 94.1 102.7 145.1 93.8 148.8 8.9 9.1% 54.1% -0.3% 58.1%

S3: Vary S3.6 83.1 90.1 136.5 89.7 133.7 0.5 8.5% 64.3% 7.9% 60.9% S3.7 76.1 87.1 132.9 82.8 124.3 4.4 14.5% 74.6% 8.7% 63.4% S4.1 71.9 79.8 184.1 78.0 91.6 1.8 11.0% 156.1% 8.6% 27.4% S4.2 91.5 98.6 187.8 98.6 112.1 0.0 7.8% 105.2% 7.7% 22.4%

2 S4.3 118.4 125.9 208.0 123.6 156.2 2.3 6.4% 75.8% 4.4% 32.0% α S4.4 163.0 170.1 225.3 168.7 208.2 1.4 4.3% 38.2% 3.5% 27.7% S4.5 227.0 244.6 276.6 244.0 297.5 0.7 7.8% 21.9% 7.5% 31.1% S4.6 323.8 356.8 398.6 353.0 415.1 3.8 10.2% 23.1% 9.0% 28.2% S4: Vary S4.7 462.0 550.0 583.9 528.9 592.5 21.1 19.0% 26.4% 14.5% 28.3% S4.8 611.1 797.1 764.0 730.4 774.5 66.8 30.4% 25.0% 19.5% 26.7% S5.1 285.2 344.8 534.3 355.8 427.7 -11.1 20.9% 87.3% 24.7% 49.9% S5.2 420.7 513.4 595.7 497.3 554.8 16.1 22.0% 41.6% 18.2% 31.9%

2 S5.3 491.4 575.1 657.1 593.0 680.7 -17.9 17.0% 33.7% 20.7% 38.5% h S5.4 542.0 624.4 718.5 675.5 791.9 -51.1 15.2% 32.6% 24.6% 46.1% S5.5 573.9 657.9 779.9 713.9 908.6 -56.0 14.6% 35.9% 24.4% 58.3% S5.6 613.4 708.1 841.3 782.8 1028.7 -74.7 15.4% 37.1% 27.6% 67.7% S5: Vary S5.7 646.5 746.2 902.6 829.4 1133.6 -83.1 15.4% 39.6% 28.3% 75.3% S5.8 687.4 782.2 964.0 883.8 1299.7 -101.6 13.8% 40.2% 28.6% 89.1%

118 APPENDIX

VERIFICATION OF THE SV CONVERGENCE MODEL

Veriﬁcation of the original convergence model through scenario evaluation was supported by NCSU research undergraduate students Amrita Malur and Connie Feinberg. A summarized version of this work was submitted and presented at the IISE OR Division Undergraduate Dis- semination Competition (June 2020) where Connie took second place. All model development, examples, validation, and analysis in Chapter3 and extensions found in Chapter4 are my own work.

B.1 Summary

As part of the initial veriﬁcation of the SV convergence model, a variety of targeted scenarios were developed. Due to the large amount of input parameters, we targeted select parameters inﬂuencing primarily motivation and communication behaviors. Scenarios were designed to compare different combinations of spontaneous volunteer behavior dynamics in conjunction

119 with messaging strategies. All parameters for population agents and volunteer sites were set to baseline values unless explicitly changed. Table B.1 summarizes the full set of model parameters including name, baseline value, and description. In total, twenty-four experiments were run to fully evaluate agent behaviors and site communication methods. Each scenario was run for 100 replications in Anylogic PLE 8.4. We were successfully able to model a homogeneous population of spontaneous volunteers while allowing for uncertainty in arrival and abandonment. We found that there was no statistical

difference in the origin of social pressures (wN1,wN2) in regard to their effect on overall demand completion, leading to the removal of the neighbors in the Subjective Norm function. However, variations in weights on social pressures and perceived behavioral controls (wSN,wPBC) did have a statistically signiﬁcant effect on demand completion. Additionally, a pre-existing conﬁdence

in ability to volunteer (SPBC) signiﬁcantly increased intent to volunteer and improved overall disaster response. Evaluation of the results suggest that the model is working as intended and support model veriﬁcation. Details of each scenario are described in the subsections below.

120 Table B.1: Full list of baseline model parameters and descriptions from original SV Conver- gence Model.

Parameter Baseline Description populationSize 200 Size of SV agent population volunteerSitesSize 3 Number of volunteer sites homeSize 20 Number of neighborhoods networkParameter1 10 Parameter for scale free network probabilityOfReply 0.4 Probability that a connected agent responds to your request siteChoiceSelection 1 1 = DISTANCE, 2 = SCORE, 3 = DEMAND, 4 = FRIENDS

weightA (wA) 0.3 Weight of parameter A in TPB model

weightSN (wSN) 0.35 Weight of parameter SN in TPB model

weigthPBC (wPBC) 0.35 Weight of parameter PBC in TPB model

weightN1 (wN1) 10 Weight of parameter N1 in TPB model

weightN2 (wN2) 10 Weight of parameter N2 in TPB model

targetedMessaging (ITM) 1 Binary value 1 if targeted, 0 for blanket

useSigmoid (IS) 1 Binary value 1 if sigmoid, 0 if linear SVatSite T(6,8,10) Length of time each SV stays at the site initialDemand 10000 Initial demand at each site

B.2 Veriﬁcation Scenarios

B.2.1 Scenario 1 - Population Demographics

We begin with a comparison of urban and rural disaster response. The major assumption of this scenario is that rural areas are more tight-knit communities where face-face interactions are more likely to occur. We translate this to behavioral intention by assuming people are more likely to be inﬂuenced by their social network than their neighbors. By adjusting the weights for A, SN, and PBC, we can account for this difference in population demographic. In addition, we vary the initial values of Ppbc and Is. We evaluate the effect of disaster training through the implementation of initial Ppbc values. Finally, Is is an indicator variable representing whether or

121 not we use the Sigmoid or linear function when converting behavioral intention to behavior. A full 2k factorial design can be seen in Table B.2.

Table B.2: Full list of varied parameters for scenario 1 experiments.

Case wA wSN wPBC wN1 wN2 PPBC IS 1.1u 0.19 0.405 0.405 0.35 0.65 0 0 1.1r 0.19 0.405 0.405 0.65 0.35 0 0 1.2u 0.19 0.405 0.405 0.35 0.65 0 1 1.2r 0.19 0.405 0.405 0.65 0.35 0 1 1.3u 0.19 0.405 0.405 0.35 0.65 0.4 0 1.3r 0.19 0.405 0.405 0.65 0.35 0.4 0 1.4u 0.19 0.405 0.405 0.35 0.65 0.4 1 1.4r 0.19 0.405 0.405 0.65 0.35 0.4 1

B.2.2 Scenario 2 - Age-Based Demographic Differences

The second scenario also considers variations in base population motivation. Here we are interested in determining the effect of age on behavioral intention, and eventual response. It has been found (during long term volunteering) there is not a strong correlation between SN and volunteer behavior in older individuals Greenslade and White[2005]. Therefore we model

differences in young and older populations through a variation on the weights of wSN and wPBC. Here the assumption is that young people pay more attention to what their friends are doing and are less worried about their physical ability to help. On the contrary, as shown in literature,

k older populations put a much higher value on wPBC. A full 2 factorial design is shown in Table

B.3. In addition to varying the weights of SN and PBC, we also vary initial PBC and IS.

122 Table B.3: Full list of varied parameters for scenario 2 experiments.

Case wA wSN wPBC wN1 wN2 PPBC IS 2.1y 0.19 0.54 0.27 0.5 0.5 0 0 2.1o 0.19 0.27 0.54 0.5 0.5 0 0 2.2y 0.19 0.54 0.27 0.5 0.5 0 1 2.2o 0.19 0.27 0.54 0.5 0.5 0 1 2.3y 0.19 0.54 0.27 0.5 0.5 0.4 0 2.3o 0.19 0.27 0.54 0.5 0.5 0.4 0 2.4y 0.19 0.54 0.27 0.5 0.5 0.4 1 2.4o 0.19 0.27 0.54 0.5 0.5 0.4 1

B.2.3 Scenario 3 - Varying Messaging and Site Choice

The third scenario varies from the ﬁrst two in that we are no longer varying the weights related to behavioral intention. Instead we develop an experiment to explore the effect of two different types of population engagement. We compare the effect of initial site to agent messaging by sending targeted and blanket messages. We also vary the site choice policy for population agents to determine if there is any beneﬁt to alternate site choice policies.

Table B.4: Full list of varied parameters for scenario 3 experiments.

Case IT Initial Message Size Site Choice Policy 3.1t 1 10 1 3.2t 1 10 2 3.3t 1 10 3 3.4t 1 10 4 3.1b 0 40 1 3.2b 0 40 2 3.3b 0 40 3 3.4b 0 40 4

123 B.3 Results

Results are shown for all scenarios and cases over selected snapshots in simulated model time (t=6,12,18,24,30). Table B.5 provides the sample mean total demand and sample mean SV at site from 100 simulated replications for each case. The full results are not provided, but instead we provide the average of values at varying times. This was done in an effort to minimize the amount of data presented in the Appendix. Table B.6 presents additional detail for scenario 1 and scenario 2 related to population agent behavioral intention. Table B.7 provides additional insight into the effect of messaging and site choice behavior (scenario 3).

124 Table B.5: Average remaining demand and SV count across all Sites on selected days.

Average Demand Remaining on Day Average SV at Sites on Day Case 6 12 18 24 30 6 12 18 24 30 1.1u 27337.3 22311.3 16285.2 9614.6 4224.4 18.8 31.4 38.8 46.8 26.8 1.1r 27287.8 22216.4 16130.6 9457.8 4393.7 18.5 31.4 40.4 45.1 23.9 1.2u 27945.0 24956.1 22011.7 19146.5 16293.9 5.1 4.9 3.9 3.6 3.0 1.2r 27990.0 25004.8 22075.0 19196.2 16349.4 5.1 4.9 4.1 3.7 3.5 1.3u 26208.0 15939.7 5412.9 3425.9 2499.5 45.4 103.1 57.4 4.4 3.3 1.3r 26282.1 16235.3 6153.3 4083.0 3017.9 43.0 100.4 48.0 4.2 3.8 1.4u 25502.0 12894.2 5063.4 3517.9 2162.7 63.0 121.3 19.5 6.8 6.1 1.4r 25426.2 12327.7 4298.6 2912.4 1755.1 66.4 128.1 22.8 5.8 5.2 2.1y 27246.7 21824.9 15209.3 8114.7 3827.6 20.4 36.4 46.4 47.1 18.1 2.1o 27291.8 22433.4 16734.3 10361.2 5179.0 17.3 28.6 36.0 42.6 26.4 2.2y 27999.7 25003.5 22086.3 19231.0 16438.3 5.3 4.7 3.8 3.1 2.8 2.2o 27947.0 24967.0 22020.4 19111.6 16185.5 5.1 4.8 4.3 3.8 4.2 2.3y 26543.0 17421.8 6558.9 4023.9 3107.9 37.7 90.2 71.3 6.3 2.6 2.3o 25861.0 14573.1 4439.7 2836.9 1913.2 52.9 112.8 43.6 4.8 3.8 2.4y 26421.9 14447.7 4963.5 3885.6 2944.6 46.0 123.1 30.4 2.6 2.6 2.4o 24987.2 11625.5 3900.6 2507.0 1297.7 73.0 127.4 16.7 6.7 6.3 3.1t 26825.0 20745.7 14782.2 10109.0 6725.6 28.6 39.7 33.6 23.9 13.3 3.2t 26865.6 20842.8 14880.7 9930.8 6325.1 28.0 39.0 33.0 25.9 17.1 3.3t 26931.2 21072.1 15284.7 9948.8 6157.7 27.5 36.1 32.6 29.2 16.7 3.4t 26940.4 21165.8 16069.1 12266.2 9133.7 26.6 35.5 21.3 16.6 13.6 3.1b 24971.3 13948.7 6008.5 2594.4 962.5 68.4 96.3 46.1 18.4 8.3 3.2b 24779.6 13394.7 5278.4 2118.3 765.9 69.4 102.1 42.7 17.0 5.9 3.3b 24893.2 15185.6 5913.4 1530.9 639.0 65.9 74.7 69.7 14.0 2.1 3.4b 25043.9 16127.7 11468.5 7290.3 4318.3 63.0 62.0 24.4 28.9 15.5

125 Table B.6: Average population level behavioral intention and PBC score across selected days.

Average Behavioral Intention on Day Average PBC Value on Day Case 6 12 18 24 30 6 12 18 24 30 1.1u 0.191 0.200 0.218 0.248 0.242 0.022 0.075 0.142 0.229 0.324 1.1r 0.192 0.204 0.223 0.249 0.242 0.022 0.076 0.144 0.234 0.322 1.2u 0.160 0.128 0.104 0.085 0.072 0.008 0.022 0.033 0.041 0.049 1.2r 0.160 0.127 0.103 0.084 0.071 0.008 0.022 0.033 0.041 0.049 1.3u 0.447 0.622 0.624 0.450 0.430 0.556 0.775 0.905 0.916 0.916 1.3r 0.447 0.617 0.610 0.448 0.428 0.556 0.768 0.898 0.906 0.906 1.4u 0.481 0.678 0.553 0.464 0.450 0.574 0.796 0.884 0.885 0.885 1.4r 0.492 0.701 0.567 0.470 0.455 0.576 0.808 0.898 0.899 0.899 2.1y 0.204 0.226 0.249 0.269 0.200 0.023 0.085 0.161 0.261 0.343 2.1o 0.180 0.186 0.201 0.235 0.253 0.021 0.071 0.132 0.215 0.301 2.2y 0.162 0.128 0.101 0.081 0.066 0.008 0.023 0.034 0.042 0.049 2.2o 0.157 0.127 0.104 0.088 0.077 0.008 0.021 0.032 0.041 0.049 2.3y 0.379 0.542 0.585 0.342 0.306 0.547 0.750 0.901 0.917 0.918 2.3o 0.512 0.686 0.687 0.571 0.554 0.566 0.792 0.910 0.916 0.916 2.4y 0.388 0.653 0.525 0.342 0.328 0.547 0.783 0.892 0.895 0.895 2.4o 0.550 0.724 0.641 0.576 0.562 0.591 0.813 0.895 0.896 0.896

Table B.7: Average cumulative count of SV turned away and average number of SV possess full site knowledge for selected days.

Average Count of SV Turned Away Average Count of SV w/ Full Info Case 6 12 18 24 30 6 12 18 24 30 3.1t 1.10 1.17 9.43 41.38 83.64 2.42 12.68 17.78 26.66 35.56 3.2t 1.00 1.26 6.67 31.27 70.89 2.20 13.01 18.65 24.55 34.07 3.3t 1.36 2.52 5.19 16.35 52.84 1.68 11.26 16.64 19.25 24.83 3.4t 1.18 2.34 17.35 44.83 78.71 1.43 8.93 13.29 19.49 26.12 3.1b 4.76 28.27 149.06 273.34 333.09 9.95 24.90 36.39 60.31 72.50 3.2b 5.47 28.88 160.29 288.09 347.23 11.55 28.17 39.22 66.90 78.83 3.3b 8.54 46.83 106.59 246.42 328.95 10.71 25.84 29.12 39.32 57.22 3.4b 7.57 55.38 153.98 214.28 274.95 9.56 23.17 35.79 54.56 67.21

126 APPENDIX

ODD+D PROTOCOL APPLIED TO THE SV CONVERGENCE MODEL

In this appendix we provide a detailed description of the SV Convergence Model through the use of the ODD+D (Overview, Design Concepts and Details + Decisions) protocol. The ODD+D protocol is an expanded and reﬁned version of the ODD protocol proposed by Grimm et al. [2010], but designed to establish a standard for describing ABMs that include human decision making [Müller et al. 2013]. The ODD + D protocol allows for a concise and well-structured documentation of human decision-making in a more straightforward way than the original ODD protocol [Müller et al. 2013]. The descriptions provided in Table C.1 are intended to act as a quick reference guide and to supplement the model development discussion in Chapter 3.

127 Table C.1: The ODD+D protocol responses for the SV Convergence model, including the guiding questions, to present a comprehensive model description.

Structural Elements Guiding Questions SV Convergence Model I.i Purpose I.i.a What is the purpose of the Simulate the impact of individual agent motivation and informa- study? tion sharing on SV convergence to inform and improve disaster management policies related to volunteer integration. I.i.b For whom is the model Designed for volunteer managers in conjunction with academics designed to evaluate different "what-if" scenarios that may help to plan for SV convergence. Results of the model are also presented in a way that allows easy understanding and interpretation from non-technical community members. I.ii Entities, I.ii.a What kinds of entities are in Population agents, volunteer organizations/sites, and Volunteer state vari- the model? Reception Centers. ables and scales I) Overview I.ii.b By what attributes (i.e. state Population agents are characterized by a variety of internal variables and parameters) are these attributes primarily related to volunteering behaviors. A visual entities characterised? summary of the attributes and behaviors is provided in Figure 3.3. I.ii.c What are the exogenous fac- Disaster location which impacts demographic and census based tors/drivers of the model? information. Disaster type which impacts internal factors related to motivation such as agent fatigue. Continued on next page

128 Table C.1 – continued from previous page Structural Elements Guiding Questions SV Convergence Model I.ii.d If applicable, how is space The model is mapped on GIS space using internal Anylogic included in the model? functionality. All agents have lat/long locations based on census blkgroup data and follow map routing when traveling. A visual of the GIS space is shown in Figure 3.1.

I.ii.e What are the temporal and The model represents the ﬁrst 30 days following the a natural spatial resolutions and extents of disaster and highlights SV Convergence during the ﬁrst weeks of the model? disaster response. The simulated space is limited in our case to New Hanover County, North Carolina. The spatial resolution is modular and can be easily changed to any county or region based on the needs of practitioners.

I.iii Process I.iii.a What entity does what, and After the disaster occurs, volunteer sites and potential SV con- overview in what order? verge. Volunteer sites become active and start screening volun- and schedul- teers and completing demand tasks. Population agents follow a ing much more complex state chart, found in Figure 3.5. II.i II.i.a Which general concepts, the- The conceptual model of SV convergence following a disaster has Theoretical ories or hypotheses are underly- been discussed in the literature dating back to the late 1950s Fritz and ing the model’s design at the sys- and Mathewson[1957]. We also presented the conceptual model Empirical tem level or at the level(s) of the to disaster management practitioners for veriﬁcation and valida- Background submodel(s) (apart from the deci- tion. The model needs to accurately represent the realism found sion model)? What is the link to in disaster convergence while being computationally tractable. complexity and the purpose of the model? Continued on next page

129 Table C.1 – continued from previous page Structural Elements Guiding Questions SV Convergence Model II.i.b On what assumptions is/are The agent motivation to volunteer decisions are based upon the the agents’ decision model(s) Theory of Planned Behavior (TPB) and discussed heavily in exp based? Chapter 3. For the attitude function, we base the wi upon two data sources representing information sharing over time. The value of the TPB weights are based on literature of volunteerism Greenslade and White[2005], Warburton and Terry [2000], Veludo-de Oliveira et al.[2013]. The value of the population PBC is based upon SME input and is deﬁned using a generalized beta distribution. The four site choice decisions were based upon volunteer interviews conducted as part of the NSF Rapids Research. II.i.c Why is /are certain decision The TPB was found to be the model that had the strongest pre- model(s) chosen? dictive capability when predicting volunteer behavior Greenslade and White[2005], Warburton and Terry[2000], Veludo-de Oliveira et al.[2013]. We also considered the Volunteer Functions Inventory (VFI) and Social Impact Theory (SIT). The site choice decisions that were included in the model represented the majority of responses in our volunteer interviews. II) Design Concepts II.i.d If the model / submodel (e.g., Ideas for the behavioral intention equation came from Daimon the decision model) is based on and Atsumi[2018] and were updated to include data collected empirical data, where do the data by Wang and Zhuang[2017]. Additionally, data used to calibrate come from? the model was collected by Cottrell[2012] and Lodree and Davis [2016]. Continued on next page

130 Table C.1 – continued from previous page Structural Elements Guiding Questions SV Convergence Model II.i.e At which level of aggregation As discussed in Chapter 3, we are limited to weekly convergence were the data available? data. The lack of additional granularity signiﬁcantly limits our ability to evaluate the daily convergence trends and suggests that additional data collection is required. II.ii Individ- II.ii.a What are the subjects and Much of the decision making in the model takes place with the ual Decision objects of decision making? On individual agent as the subject and internal decisions as the object. Making which level of aggregation is deci- For example: (1) The decision to become motivated and seek out sion making modeled? Are mul- a volunteer opportunity, (2) request relevant site information, (3) tiple levels of decision making Choose a site, (4) Share updated information. included? Volunteer sites are also able to make decisions such as the number of initial site messages sent after activation. Another site decision is the decision to accept a volunteer agent (the object). Volunteer Reception Centers also can make site assignment decisions for the volunteer agents. While there are technically multiple levels of decision making occurring (at the site level and agent level), it is not clearly delin- eated into separate levels. Instead, there is a constant and complex amount of information sharing and decision making which occurs and impacts SV convergence.

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131 Table C.1 – continued from previous page Structural Elements Guiding Questions SV Convergence Model II.ii.b What is the basic rational- Sites make decisions with the typical goal of maximizing demand ity behind agent decision-making completion. In one of the model extensions we consider a dual in the model? Do agents pursue objective of maximizing demand while minimizing volunteer an explicit objective or have other rejections. success criteria? Agents make decision to volunteer based on internal motivating factors which are impacted by the environment within the model. The decision to volunteer is not necessarily made with an explicit objective, but the idea is that they feel motivated or compelled to volunteer. The site choice decision they make is based upon internal preference, with the overarching goal of trying to avoid potential site rejections.

II.ii.c How do agents make their Motivation decisions are made based upon TPB values and site decisions? choice is determined based on the prescribed policy. A variety of other decisions are abstracted to stochastic decision points due to a lack of available data and to limit the overall model complexity. The full description of agent decisions can be found in Chapter 3. II.ii.d Do the agents adapt their Agent’s opinions toward volunteers sites change over time based behaviour to changing endogenous upon their individual experiences. Additionally, they can request and exogenous state variables? information from other agents and update their opinions accord- And if yes, how? ingly. Internal motivation and the volunteer behavior changes over time based upon agent attitude, subjective norm, and perceived behavioral control. Continued on next page

132 Table C.1 – continued from previous page Structural Elements Guiding Questions SV Convergence Model II.ii.e Do social norms or cultural Yes, social pressure is represented as one aspect of the motivating values play a role in the decision- factors within the TPB. Again, see Chapter 3 for further details. making process? II.ii.f Do spatial aspects play a role Spatial aspects play a role in the site choice decision. Agents who in the decision process? choose a site based upon distance or score take into account the spatial distance between their home location and available sites when determining which site to attempt to volunteer. Additionally, travel times are calculated based upon GIS distances.

II.ii.g Do temporal aspects play a Yes, the attitude portion of the TPB function is represented by role in the decision process? a decreasing exponential function with a time component. As the time from disaster impact grows, agents are less motivated to contribute or volunteer. The exact impact is dependent on both ATT exp wi and wi , which is discussed in Chapter 3. II.ii.h To which extent and how is Uncertainty is included throughout the model in a variety of uncertainty included in the agents’ areas and decision points. The primary uncertainty related to the decision rules? motivation decision is the translation from attempted behavior and behavioral intention (P(Bi,t = 1) = BI). We choose two possible methods for translation but both are stochastic in nature. Continued on next page

133 Table C.1 – continued from previous page Structural Elements Guiding Questions SV Convergence Model II.iii Learn- II.iii.a Is individual learning Agent’s opinions toward volunteers sites change over time based ing included in the decision process? upon their individual experiences. An agent that has a negative How do individuals change experience at one volunteer site is less likely to attempt to volun- their decision rules over time as teer at that site again in the future. Additionally, after multiple consequence of their experience? negative experiences, a volunteer may decide to stop attempting to volunteer for the duration of the modeled period. II.iii.b Is collective learning imple- Information related to site opinions is shared when site informa- mented in the model? tion is requested, however collective learning is not otherwise directly considered. II.iv Individ- II.iv.a What endogenous and Population agents consider both endogenous (attitude and per- ual Sensing exogenous state variables are indi- ceived control) and exogenous (time since the disaster) when viduals assumed to sense and con- determining motivation. We do not consider partial or incorrect sider in their decisions? Is the sens- information in this evaluation. ing process erroneous? II.iv.b What state variables of Agents are able to perceive when connected agents volunteer in which other individuals can an the previous period. That is, agents feel pressure to volunteer if individual perceive? Is the sensing other connected agents are also volunteering. We do not consider process erroneous? partial or incorrect information in this evaluation. II.iv.c What is the spatial scale of There is no direct physical sensing which occurs in the model, sensing? however agents are able to identify volunteering action of other agents which are connected to them based through a social network (represented as a scale free network in the simulation) Continued on next page

134 Table C.1 – continued from previous page Structural Elements Guiding Questions SV Convergence Model II.iv.d Are the mechanisms by Yes, we model both one-one and one-many messaging mecha- which agents obtain information nisms which allow agents to obtain and update their information. modelled explicitly, or are individ- One-many information transfers occur when sites send messages uals simply assumed to know these to population agents requesting assistance. One-one messaging variables? occurs when agents request or receive actionable information regarding volunteer sites and site opinions. II.iv.e Are the costs for cognition There is no cognition cost or cost for gathering information in and the costs for gathering infor- a resource sense. However, both the decision to volunteer and mation explicitly included in the request information incurs some time delay for each agent. model? II.v Indi- II.v.a Which data do the agents use Agents use site opinion to determine the best action when choos- vidual to predict future conditions? ing a volunteer site location. Prediction II.v.b What internal models are Agents do not have forward thinking capabilities but do update agents assumed to use to estimate their site opinion based upon the experience they have at a given future conditions or consequences volunteer site. of their decisions? II.v.c Might agents be erroneous N/A in the prediction process, and how is it implemented? Continued on next page

135 Table C.1 – continued from previous page Structural Elements Guiding Questions SV Convergence Model II.vi Interac- II.vi.a Are interactions among We have both direct and indirect interactions between agents. tion agents and entities assumed as Examples of direct actions include messaging between agents direct or indirect? and site screening and rejection. Agents also have some indirect interaction related to the social pressure they feel toward volunteering. Another example of a very indirect interaction is when an individual arrives to a site that is already occupied and is therefore rejected. II.vi.b On what do the interactions Many of the direct interactions are dependent on each agents site depend? knowledge and motivation levels. Agents interact when they need actionable information or they are arriving to a volunteer site. II.vi.c If the interactions involve Agents are connected via a social network, represented in the communication, how are such model as a scale-free network. Communications are modeled communications represented? explicitly, with requests for information being either accepted or ignored. II.vi.d If a coordination network There exists coordination between local volunteer sites in one exists, how does it affect the agent example from Chapter 4. In that case the coordination is imposed behaviour? Is the structure of the and it improves both demand completion and reduces volunteer network imposed or emergent? rejections. Continued on next page

136 Table C.1 – continued from previous page Structural Elements Guiding Questions SV Convergence Model II.vii Collec- II.vii.a Do the individuals form No, we do not consider group dynamics or any prior afﬁliations tives or belong to aggregations that (like religious organizations, etc.) when modeling spontaneous affect and are affected by the indi- volunteers. viduals? Are these aggregations imposed by the modeller or do they emerge during the simulation? II.vii.b How are collectives repre- N/A sented? II.viii Het- II.viii.a Are the agents hetero- Yes, we allow heterogeneity through multiple variables such as erogeneity geneous? If yes, which state age, location, prior experience, and many of the motivation related variables and/or processes differ parameters. between the agents? II.viii.b Are the agents heteroge- Agents are heterogeneous in a variety of simulated decisions. The neous in their decision-making? primary motivation decision allows for extreme heterogeneity If yes, which decision models or through variable weights on the TPB terms along with addi- exp decision objects differ between the tional terms such as wi and the unique experiences impacting agents? their PBC values. Additionally, each agent location is unique and impacts their individual site choice decisions. Detailed information regarding these decisions can be found in Chapter 3.

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137 Table C.1 – continued from previous page Structural Elements Guiding Questions SV Convergence Model II.ix II.ix.a What processes (includ- A variety of model inputs and minor agent decisions are modeled Stochas- ing initialisation) are modelled randomly. For example, the initial PBC value for each agent is ticity by assuming they are random or drawn from a beta distribution. The age of each agent is ran- partly random? domly selected from a discrete distribution ﬁt to demographic information. Other decisions such as the decision to respond to an information request or seek out volunteer site information is based upon random chance (e.g. 5%). II.x Observa- II.x.a What data are collected from We track individual level and system level data. A summary is tion the ABM for testing, understand- provided at the end of the model discussion in Chapter 3. Individ- ing and analysing it, and how and ual level data includes values for each of the TPB components, when are they collected? the number of known sites, number of rejections, and total days volunteered. System level data is collected each model day for the number of volunteers active, demand completed, and volunteer rejections. The values are stored as arrays within Anylogic and are output to CSV ﬁles upon model completion.

II.x.b What key results, outputs We are interested in capturing the overall convergence behavior of or characteristics of the model are the group. We report the total number of volunteers at each site by emerging from the individuals? day, the demand completion by day, and the number of volunteer (Emergence) rejections which occur. Each of these values are metrics which can be helpful in evaluating the impact of different management and integration policies. Continued on next page

138 Table C.1 – continued from previous page Structural Elements Guiding Questions SV Convergence Model III.i Imple- III.i.a How has the model been The model was developed in Anylogic University Edition v8.5.2. mentation implemented? Each agent is represented within the main agent model. The Details main agent section includes high level inheritable attributes and the GIS view. Lower level agents such as population agents and volunteer site agents are represented by states and transitions with additional attributes as required. The volunteer site agent also includes a discrete event simulation element to track the ﬂow of SV through the submodel. III.i.b Is the model accessible, and The model is not currently publicly available. Interested parties if so where? can send requests for access to the dissertation author or the committee chair. III.ii Initiali- III.ii.a What is the initial state of In the initial state all agents are populated within the model in an sation the model world, i.e. at time t = 0 idle state. Populations wait for their ﬁrst daily motivation check of a simulation run? while volunteer sites and reception centers await activation. III.ii.b Is the initialisation always For the majority of parameter inputs, the initialization is similar III) Details the same, or is it allowed to vary across iterations and replications. We do pull some initial values among simulations? based upon input distributions (agent age, initial PBC value, etc.) which vary between replications, following a common random number scheme. Continued on next page

139 Table C.1 – continued from previous page Structural Elements Guiding Questions SV Convergence Model III.ii.c Are the initial values cho- Initial values are chosen based upon real data if available. When sen arbitrarily or based on data? that information is not available, due to the lack of data on SV convergence and SV motivation decisions, we do make a variety of assumptions in parameter values. These assumptions are based upon subject matter expertise and qualitative responses from our interview data. Where appropriate we also initialize the values using input modeling. III.iii Input III.iii.a Does the model use input No external data sources or input files are used to represent pro- Data from external sources such as data cesses that change over time. files or other models to represent processes that change over time? III.iv Sub- III.iv.a What, in detail, are the The majority of process scheduling is automated within the Any- models submodels that represent the pro- logic software. There are no submodels responsible for tracking cesses listed in ‘Process overview the overall process flow beyond the state chart representation that and scheduling’? agent’s follow during their daily process. III.iv.b What are the model param- The full description of model parameters is too expansive to eters, their dimensions and refer- include within the ODD+D table. However, a list of the model ence values? parameters, their baseline values, and varied values in each scenario can be found in Chapters 3 and 4. An additional exploration of the model parameters is conducted in AppendixB. III.iv.c How were the submod- Our model includes volunteer site and volunteer reception sub- els designed or chosen, and how models. Both of the submodels are designed to minimize the need were they parameterised and then for additional parameters, in an effort to limit the complexity of tested? the overall convergence model.

140 APPENDIX

COMPUTATIONAL RESULTS FOR CHAPTER 4

The Tables in AppendixD provide the full results of the scenarios for RQ4.1 and RQ4.2 as shown in the heat maps in Section 4.3. Results are separated into average total demand completion and average total spontaneous volunteer rejection for both scenarios. The values presented are the average values across 30 replications and include the 95% conﬁdence interval (±) for all cases.

141 Table D.1: Summary of average total demand completion for 30 replications across variations in VRC activation day and VRC enforcement levels. Data represents the mean value ± 95% conﬁdence interval.

VRC Activation Day 0 1 2 3 4 5 6 7 8 9 10 48355.6 46769 45356.6 43629.8 42106.9 40450.6 39258 37066.7 35376.5 33504.4 31647.9 1 ± 4533.7 ± 4077.8 ± 4308.1 ± 3123.8 ± 3173.2 ± 3322.9 ± 3462.0 ± 2918.7 ± 3188.8 ± 2697.6 ± 2644.2 48413.3 46810.7 45117.3 44074.6 43378 41518.9 39701.7 38635.5 36980.4 35808.3 34039.9 0.9 ± 4359.2 ± 5106.1 ± 4547.7 ± 4401.4 ± 3810.8 ± 4357.9 ± 3985.5 ± 3761.9 ± 2902.8 ± 2975.4 ± 2670.6 48343.6 46874.4 45793.4 44797.1 43791 42424.9 40754.4 39661 38178.1 37003.2 35792.2 0.8 ± 5336.1 ± 4371.9 ± 4256.6 ± 4466.2 ± 3056.0 ± 4447.5 ± 3815.3 ± 4249.0 ± 2975.4 ± 3615.9 ± 2659.9 48555.5 47465.3 46230 45563.8 44254.9 43037.8 41989.6 40735.7 39628.2 38665.4 37100.2 0.7 ± 4226.9 ± 4852.1 ± 4052.3 ± 3727.4 ± 4325.6 ± 3817.0 ± 3666.5 ± 3440.3 ± 3760.5 ± 3565.7 ± 3039.1 48162.9 47312.1 46554.9 45474.7 44796.1 44335.6 42645.7 41814.7 41090.5 39960.5 39288.9 0.6 ± 4657.8 ± 4017.7 ± 4935.5 ± 4945.3 ± 3549.1 ± 4456.8 ± 3744.4 ± 3270.2 ± 4840.4 ± 4225.6 ± 3857.5

VRC Enforcement Level 47368.3 47363.4 45883.3 45484.2 44973.9 44695.6 43571.5 43043.3 42353.6 41239 40501.6 0.5 ± 4362.9 ± 4681.0 ± 3869.3 ± 4815.8 ± 4320.3 ± 4352.2 ± 3699.3 ± 3940.9 ± 3865.1 ± 3750.3 ± 2671.3 46082 45807.8 45126.4 44452.8 44104.1 43762 43144 42827.4 42508.1 41396.7 40771.7 0.4 ± 4545.6 ± 4165.0 ± 3644.5 ± 3526.5 ± 2917.2 ± 3490.9 ± 3427.3 ± 3542.3 ± 4891.1 ± 3442.1 ± 2998.7 43212.1 42705.2 42589.5 42120.5 41886.8 41829.2 41452.5 40573.5 40043.8 39573 39099.3 0.3 ± 3181.8 ± 4154.2 ± 2454.7 ± 2621.6 ± 2696.7 ± 3534.1 ± 2858.4 ± 3183.4 ± 2565.6 ± 2864.3 ± 2809.2 39053.2 39022.7 38828.2 38561.9 38696.6 38260.8 38052.4 37474.1 37220.4 36505 36215.8 0.2 ± 2765.6 ± 2761.5 ± 3048.7 ± 2724.0 ± 2921.5 ± 2901.9 ± 2646.6 ± 3206.3 ± 2636.9 ± 2749.2 ± 3317.2 34955.4 34805.6 34606.2 34649.7 34436.5 34681.3 34233.4 34070.5 33868.3 33270.8 33106.1 0.1 ± 2919.3 ± 2111.1 ± 3012.5 ± 2244.0 ± 2480.8 ± 3271.0 ± 2988.0 ± 2723.9 ± 2481.9 ± 3351.2 ± 3025.2 30503.8 30619.7 30298.9 30356.8 30182.9 30311 30580.4 30393.4 30201.2 30174.6 30636.2 0 ± 3959.5 ± 3460.1 ± 3378.8 ± 3572.2 ± 3305.8 ± 3604.5 ± 3289.1 ± 3766.5 ± 3354.9 ± 3186.2 ± 3072.0

142 Table D.2: Summary of average total SV rejections for 30 replications across variations in VRC activation day and VRC enforcement levels. Data represents the mean value ± 95% conﬁdence interval.

VRC Activation Day 0 1 2 3 4 5 6 7 8 9 10 273.36 225.48 183.626 141.52 107.23 80.07 57.05 37.2 22.15 13.11 6.21 1 ± 57.4 ± 50.5 ± 54.7 ± 39.5 ± 34.2 ± 29.1 ± 26.1 ± 20.2 ± 15.2 ± 10.1 ± 5.2 285.88 247.76 207.5 176.86 142.35 118.01 92.45 67.47 49.86 33.7 23.77 0.9 ± 55.9 ± 47.8 ± 49.5 ± 39.9 ± 36.4 ± 39.0 ± 34.5 ± 28.5 ± 21.9 ± 19.7 ± 15.8 304.78 265.85 239.44 209.24 183.28 150.54 129.25 103.74 80.87 68.5 53.5 0.8 ± 52.8 ± 59.9 ± 45.4 ± 45.2 ± 51.3 ± 43.5 ± 35.3 ± 30.7 ± 28.9 ± 26.5 ± 21.5 312.54 280.38 252.66 228.72 208.34 185.4 157.28 135.9 119.9 98.77 85.09 0.7 ± 50.5 ± 53.4 ± 53.7 ± 54.7 ± 40.3 ± 37.4 ± 43.3 ± 48.9 ± 32.9 ± 31.8 ± 37.1 324.73 295.94 272.17 250.73 233.55 207.83 190.08 169.14 153.09 132.26 117.45 0.6 ± 68.7 ± 46.9 ± 56.2 ± 50.7 ± 50.6 ± 42.8 ± 42.1 ± 46.1 ± 34.1 ± 43.3 ± 35.5

VRC Enforcement Level 351.68 327.66 311.69 285.5 267.06 243.57 233.86 211.48 200.29 183.28 166.6 0.5 ± 65.7 ± 45.7 ± 60.7 ± 54.5 ± 45.1 ± 41.6 ± 48.9 ± 40.5 ± 42.3 ± 50.3 ± 35.6 395.47 376.33 362.37 343.99 324.28 309.39 289.79 280.01 262.55 254.43 238.79 0.4 ± 53.5 ± 47.3 ± 54.4 ± 51.0 ± 46.3 ± 51.4 ± 40.8 ± 50.0 ± 47.6 ± 62.4 ± 55.8 446.48 435.22 417.44 409.71 390.43 375.6 365.61 351.05 345.09 326.45 318.79 0.3 ± 61.5 ± 52.2 ± 55.4 ± 65.7 ± 49.4 ± 46.6 ± 47.6 ± 54.1 ± 46.1 ± 50.5 ± 46.8 500.09 488.09 473.83 464.48 452.14 445.39 439.02 425.32 420.26 410.56 402.41 0.2 ± 52.1 ± 43.1 ± 45.8 ± 53.5 ± 58.1 ± 57.4 ± 71.0 ± 47.3 ± 50.3 ± 39.6 ± 52.4 530.61 527.03 520.59 520.04 511.42 506.47 501.67 503.64 496.33 486.58 484.5 0.1 ± 61.6 ± 58.0 ± 59.6 ± 75.0 ± 67.4 ± 60.5 ± 60.7 ± 61.6 ± 66.3 ± 53.6 ± 59.5 568.9 557.79 561.12 566.2 560.92 562.88 563.23 567.03 563.6 564.21 563.58 0 ± 53.9 ± 55.8 ± 59.1 ± 58.2 ± 47.9 ± 68.9 ± 56.2 ± 58.0 ± 55.6 ± 60.2 ± 52.6

143 Table D.3: Summary of average total demand completion for 30 replications across variations in initial site message size and percent successful coordination. Data represents the mean value ± 95% conﬁdence interval.

Percent Successful Coordination (Level of Collaboration) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 44871.6 45490.2 46276.3 46458.3 46783.2 46901.3 47776 46982 47562.8 47488.4 47568 250 ± 3903.1 ± 3477.1 ± 3840.7 ± 4347.9 ± 4124.2 ± 4653.4 ± 5395.4 ± 3963.3 ± 5139.9 ± 5133.6 ± 4382.7 44432.3 45308.5 45972.5 46381.4 46909.6 47219.3 46722.3 47495.9 47118.4 47444.8 47776 225 ± 3791.6 ± 4446.2 ± 4177.2 ± 4037.8 ± 5283.6 ± 5205.4 ± 4712.2 ± 5626.3 ± 5065.8 ± 4838.0 ± 4269.3 43885.7 44872.9 46108 45922.4 46055.6 46864.8 46696 46675.9 47315.2 47323.8 47549.9 200 ± 3822.7 ± 4247.8 ± 4410.2 ± 4932.0 ± 4464.2 ± 4214.5 ± 4354.7 ± 4907.3 ± 5852.8 ± 4986.2 ± 4399.6 43057.1 44637.6 44822.7 44943.2 46086.4 45871.4 46371.1 46248.3 46736.4 47088.2 46592.1 175 ± 3789.9 ± 4212.1 ± 4221.7 ± 4134.0 ± 5003.3 ± 4219.8 ± 4818.7 ± 4563.1 ± 4490.3 ± 4531.0 ± 4539.0 42281.7 43327.8 44464 44610.3 44896.2 45369.7 45790.9 45847.1 46175.5 45968 45734.5 150 ± 3789.1 ± 3686.9 ± 4616.6 ± 3936.8 ± 4751.4 ± 5149.6 ± 4264.3 ± 4224.2 ± 5814.6 ± 4381.5 ± 5036.7 Initial Message Size 39928.2 41467.2 42683.9 43417.2 43989.4 43942.7 44361.7 44708.5 44900.7 45021.1 44877.5 125 ± 3862.3 ± 3781.2 ± 4634.9 ± 3108.4 ± 3861.8 ± 4367.4 ± 4570.7 ± 4509.3 ± 4288.0 ± 4899.5 ± 4994.7 38163.8 39348.4 40778.5 41926.2 42398.2 42415 42943.9 43585.5 43633.5 43389.7 44327.2 100 ± 4146.2 ± 3066.0 ± 3172.3 ± 3280.2 ± 4147.0 ± 4720.5 ± 4166.7 ± 4324.7 ± 4239.2 ± 5415.3 ± 4789.6 35392.2 36502.7 38050.2 39303.3 39963.6 40608.8 41260 41395.4 41607.8 42186.8 42537.1 75 ± 3358.9 ± 3473.9 ± 3585.0 ± 3385.5 ± 3350.8 ± 3308.6 ± 3683.8 ± 3942.3 ± 4259.9 ± 4543.2 ± 4075.9 32022.3 33231.4 35124.8 35865.1 37300.5 37560.6 38384.9 38992 39479.8 39855.4 39807.2 50 ± 2834.9 ± 3181.6 ± 3208.8 ± 3241.1 ± 3361.5 ± 3512.1 ± 3236.3 ± 3910.9 ± 3977.6 ± 3901.8 ± 3106.6 28014.3 29117.8 30509.8 32058.5 33220.7 34176.9 34728.8 35222.7 35723.5 36034.2 36173 25 ± 2461.3 ± 2884.3 ± 2890.6 ± 2946.3 ± 2563.2 ± 2853.8 ± 2900.3 ± 3039.8 ± 3980.7 ± 3473.6 ± 3802.8 23223.5 24278.3 25348.7 26547.3 28030.1 29164.6 29982.2 30496.1 30960.1 31357.3 31801.8 0 ± 2314.5 ± 2191.2 ± 2465.9 ± 2373.2 ± 2481.1 ± 1789.5 ± 3131.9 ± 2836.6 ± 3040.5 ± 3250.3 ± 3388.5

144 Table D.4: Summary of average total SV rejection for 30 replications across variations in initial site message size and percent successful coordination. Data represents the mean value ± 95% conﬁdence interval.

Percent Successful Coordination (Level of Collaboration) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 374.74 365.76 354.66 355.56 348.16 345.89 341.75 347.94 344.53 344.02 344.79 250 ± 55.7 ± 49.8 ± 63.7 ± 50.6 ± 66.2 ± 53.9 ± 55.8 ± 58.9 ± 56.5 ± 65.0 ± 47.8 357.88 339.99 329.7 327.72 324.83 319.21 325.89 320.78 324.24 321.6 316.29 225 ± 61.9 ± 58.0 ± 52.7 ± 58.7 ± 60.8 ± 68.2 ± 55.9 ± 53.8 ± 48.2 ± 59.6 ± 53.3 330.74 315.62 303.42 302.86 305.05 297.38 295.49 303.54 295.96 295.89 292.46 200 ± 64.7 ± 56.6 ± 55.4 ± 51.9 ± 50.1 ± 63.4 ± 74.5 ± 69.5 ± 62.0 ± 60.9 ± 47.5 301.67 286.12 282.31 276.78 276.03 275.5 267.21 272.34 268.04 268.1 272.98 175 ± 52.7 ± 53.1 ± 61.3 ± 60.8 ± 61.0 ± 59.5 ± 59.2 ± 56.3 ± 55.0 ± 61.1 ± 52.0 275.31 261.2 249.42 249.71 248.06 242.69 245.15 242.75 241.49 241.52 245.86 150 ± 58.3 ± 50.2 ± 45.4 ± 61.7 ± 58.1 ± 60.7 ± 50.2 ± 56.8 ± 60.5 ± 54.5 ± 60.9 Initial Message Size 255.36 236.18 222.23 219.89 214.19 213.3 214.59 212.58 213.53 214.91 215.06 125 ± 50.4 ± 53.2 ± 58.2 ± 50.9 ± 56.2 ± 60.3 ± 63.6 ± 50.6 ± 49.5 ± 51.9 ± 52.1 221.94 208.15 196.59 184.46 183.51 183.41 182.9 180.94 179.94 179.7 177.25 100 ± 49.9 ± 44.2 ± 52.6 ± 41.5 ± 49.5 ± 68.4 ± 43.9 ± 63.9 ± 59.9 ± 63.7 ± 57.3 185.65 178.91 163.18 152.53 151.24 150.29 147.27 144.32 145.02 146.18 145.24 75 ± 46.7 ± 43.9 ± 39.2 ± 48.5 ± 55.2 ± 49.3 ± 54.3 ± 52.3 ± 42.0 ± 47.2 ± 42.2 159.09 148.43 132.25 127.03 119.98 117.27 111.43 108.75 107.06 108.8 104.3 50 ± 57.1 ± 50.4 ± 36.3 ± 44.0 ± 41.0 ± 42.3 ± 42.4 ± 39.8 ± 43.1 ± 42.8 ± 44.3 130.53 119.65 105.22 97.12 84.25 80.97 78.48 76.39 72.06 73.47 71.38 25 ± 48.5 ± 41.7 ± 31.2 ± 41.1 ± 28.3 ± 28.0 ± 35.5 ± 38.4 ± 35.1 ± 35.5 ± 41.4 101.798 87.37 77.63 66.94 59.84 51.9 47.48 46.17 43.09 42.7 42.87 0 ± 34.8 ± 30.4 ± 25.6 ± 23.9 ± 21.8 ± 23.9 ± 24.5 ± 22.2 ± 23.1 ± 25.7 ± 24.9

145