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Optimizing the selection of COVID-19 vaccine distribution centers and allocation quantities: a case study for the county of Los Angeles

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Optimizing the selection of COVID-19 vaccine distribution centers and allocation quantities: a case study for the county of Los Angeles

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Optimizing the Selection of COVID-19 Vaccine Distribution Centers
and Allocation Quantities:
A Case Study for the County of Los Angeles
By
Basim Masad Aljohani

A Thesis Presented to the
FACULTY OF THE USC VITERBI SCHOOL OF ENGINEERING
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF SCIENCE
(INDUSTRIAL AND SYSTEMS ENGINEERING)

August 2023

ii
Epigraph

[[And certainly did WE create man from an extract of clay ۞ Then
WE placed him as a sperm-drop in a firm lodging ۞Then WE made
the sperm-drop into a clinging clot, and WE made the clot into a lump
[of flesh], and WE made [from] the lump, bones and WE covered the
bones with flesh; then WE developed him into another creation. So
blessed be Allah, the best of creators]]
- Holy Quran, Chapter 23, verses 12 through 14

iii
Dedication

A dedication to my inspiration and role model, my father, Masad Aljohani:
I would not be the man I am today without you and the lessons you taught me. Thank you for
everything you have done to make my life better and easier. Your absence has left me with an
unfillable void. I miss you deeply, and I pray to Almighty Allah to grant you the highest levels of
paradise.

iv
Acknowledgments
First, all praise is due to Allah the Almighty, the most gracious and the most merciful, for his
blessing and guidance throughout my life, during my study, and in completing my thesis.
I would like to express my deepest gratitude and sincere thanks to my advisor, Dr. Randolph Hall,
for his valuable support, guidance, and deep knowledge, which have been extremely valuable and
helpful throughout my research journey. The skills and knowledge I have acquired working with
you are invaluable assets that will undoubtedly help me in my future. Thanks are also extended to
King Fahd University of Petroleum and Minerals for providing me with a scholarship to continue
my graduate education.
I would like to extend my sincere gratitude to my thesis committee members: Dr. Sze-chuan Suen
and Dr.Andres Gomez. I am very appreciative and grateful for the time and effort you devoted to
reviewing my thesis and for being on my committee.
Thanks are also extended to Mingdong Lyu for his valuable feedback and support throughout my
thesis journey.
Last, but certainly not least, I would like to express my heartfelt gratitude to my family, whose
continuous support has always been a source of strength throughout my life. To my dear mother,
whose endless love and encouragement have inspired me to strive for excellence and helped make
me the man I am today. To my loving wife, Mawaddah, thank you for always believing in me and
for your support during times of stress. Your patience during the long times I was away busy
working is sincerely appreciated. And to my siblings, Nahla, Mohammed, Khalid, Tawfiq, Fahad,
and Norah, thank you for always being there when I needed someone to lean on. Your support and
love have always been a source of constant inspiration.

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Table of Contents

Epigraph .......................................................................................................................................... ii
Dedication ...................................................................................................................................... iii
Acknowledgments.......................................................................................................................... iv
List of Tables ................................................................................................................................ vii
List of Figures ............................................................................................................................... vii
List of Maps ................................................................................................................................. viii
Abstract .......................................................................................................................................... ix
Chapter One: Introduction .............................................................................................................. 1
1.1 Background and Context of the Study ......................................................................................... 1
1.2 Research Objectives and Question ............................................................................................... 3
Chapter Two: Literature Review .................................................................................................... 5
2.1 Review on the Special Analysis of Vaccine Distribution ............................................................ 5
2.2 Review on Optimization of the COVID-19 Vaccine Centers Locations ..................................... 7
2.3 Review on Allocation of the COVID-19 Vaccines .................................................................... 10
Chapter Three: Methodology ........................................................................................................ 16
3.1 Travel Time and Distance. ......................................................................................................... 16
3.2 Optimization Model. .................................................................................................................. 17
3.2.1 Model Sets, Parameters, and Decision Variables. ...................................................................... 18
3.2.2 Objective Function and Constraints ........................................................................................... 19
3.3 Modeling The Different Scenarios ............................................................................................. 22
3.3.1 Scenario One: Prioritizing Areas Based on Population. ............................................................ 22
3.3.2 Scenario Two: Prioritizing Areas Based on the Healthy Places Index. ..................................... 22
3.3.3 Scenario Three: Prioritizing Areas Based on the Vulnerability to Covid-19. ............................ 23
Chapter Four: Study Area and Datasets ........................................................................................ 27
4.1 Los Angeles County Metropolitan Area. ................................................................................... 27
4.2 Datasets ...................................................................................................................................... 29
4.2.1 Demographic Data...................................................................................................................... 29
4.2.2 Healthy Places Index .................................................................................................................. 29
4.2.3 Covid-19 Data ............................................................................................................................ 30
4.2.4 Vulnerability to Covid-19 .......................................................................................................... 30
4.2.5 Car Ownership Data ................................................................................................................... 30

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4.2.6 Geographical Data ...................................................................................................................... 31
Chapter Five: Results and Discussion........................................................................................... 32
5.1 Model Output for Optimal Vaccine Distribution Sites. ............................................................. 32
5.1.1 Scenario One: ............................................................................................................................. 32
5.1.2 Scenario Two: ............................................................................................................................ 35
5.1.3 Scenario Three: .......................................................................................................................... 36
5.2 Model Output Optimal Vaccine Distribution Sites: ................................................................... 38
5.2.1 Scenario One: ............................................................................................................................. 38
5.2.2 Scenario Two: ............................................................................................................................ 39
5.2.3 Scenario Three: .......................................................................................................................... 39
5.3 Sensitivity Analysis. ................................................................................................................... 41
5.3.1 Changing the Maximum Number of Sites.................................................................................. 41
5.3.2 Changing the Number of Sites Residents Can Choose From (F). .............................................. 44
5.4 Comparison Between the Optimal Sites vs the Actual LA County Vaccine Sites..................... 47
5.5 Limitations ................................................................................................................................. 52
Chapter Six: Conclusion. .............................................................................................................. 53
6.1 Summary of Findings. ................................................................................................................ 53
6.2 Contributions to the Field. .......................................................................................................... 55
6.3 Future Research. ......................................................................................................................... 56
References ..................................................................................................................................... 57
Appendix ....................................................................................................................................... 61

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List of Tables
Table 1: Car travel time matrix in minutes ................................................................................... 17
Table 2: Wi calculations using HPI .............................................................................................. 23
Table 3: Racial vulnerability to Covid-19 by CDC ...................................................................... 24
Table 4: Risks scores calculations example for zip code 90029 ................................................... 25
Table 5: Weights for risks measures ............................................................................................. 26
Table 6: Optimal costs breakdown ............................................................................................... 32
Table 7: Optimal costs breakdown for scenario 2 ........................................................................ 35
Table 8: Optimal costs breakdown for scenario 3 ........................................................................ 36
Table 9: Summary of cost types for all scenarios ......................................................................... 37
Table 10: average travel times and distance values for all scenarios ........................................... 40
Table 11: Total cost for different max sites for all scenarios ....................................................... 43
Table 12: Total cost weighted by HPI for different max sites for all scenarios ........................... 43
Table 13: Total cost weighted by COVID-19 Vulnerability Index for different max sites .......... 44
Table 14: Total weighted costs for different F values for all scenarios ........................................ 46
Table 15: Loop output for scenario 1 ............................................................................................ 66
Table 16: Loop output for scenario 2 ............................................................................................ 68
Table 17: Loop output for scenario 3 ............................................................................................ 70

List of Figures
Figure 1: Vaccination uptake rates in LA County according to LA Public Health[1] ................... 2
Figure 2: Scatterplot of average travel time by transit vs population of zip codes ....................... 33
Figure 3: Scatterplot of average travel time by car vs the population of zip codes ...................... 34
Figure 4: Scatterplot of average distance vs the population of zip codes ..................................... 34
Figure 5: Box plot of HPI percentile for selected sites ................................................................. 36
Figure 6: Cost types vs number of maximum sites ....................................................................... 41
Figure 7: Average travel times and distance vs max sites ............................................................ 42
Figure 8: Average travel times and distance comparison ............................................................. 49
Figure 9: Cost types comparison................................................................................................... 50
Figure 10: Average travel times and distance comparison ........................................................... 51
Figure 11: Cost types comparison................................................................................................. 51
Figure 12: Cost types vs max sites for scenario 2 ......................................................................... 61
Figure 13: Average travel times and distance vs max sites for scenario 2 ................................... 61
Figure 14: Cost types vs max sites for scenario 3 ......................................................................... 63
Figure 15: Average travel times and distance vs max sites for scenario 2 ................................... 63

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List of Maps
Map 1: LA County population heatmap ....................................................................................... 28
Map 2: Map of optimal sites for the first scenario ........................................................................ 33
Map 3: Map of optimal sites for the second scenario ................................................................... 35
Map 4: Map of optimal sites for third scenario............................................................................. 37
Map 5: Vaccine allocation for the first scenario ........................................................................... 38
Map 6: Vaccine allocation for the second scenario ...................................................................... 39
Map 7: Vaccine allocation for the third scenario .......................................................................... 40
Map 8: Map of optimal sites when F=1 ........................................................................................ 45
Map 9: Map of optimal sites when F=5 ........................................................................................ 45
Map 10: Map of the actual vaccine sites in LA county ................................................................ 48
Map 11: Map of selected sites when F=1, scenario 2 ................................................................... 62
Map 12: Map of selected sites when F=5, scenario 2 ................................................................... 62
Map 13: Optimal sites for F=1, scenario 3 ................................................................................... 64
Map 14: Optimal sites for F=5, scenario 3 ................................................................................... 64

ix
Abstract
This thesis aims to provide a tool to help policy and decision-makers establish well-informed plans
about the selection and distribution of locations and quantities of vaccines. Optimization of vaccine
distribution centers' locations plays a crucial role in providing communities with easy access to
vaccines, which will help in controlling global pandemics, such as the recent COVID-19, and
mitigate the risks of losing lives and economic losses. As studies have proven, strategic planning
of the vaccine site locations and the allocated quantities of the vaccines could help boost the
amount of vaccine uptake. This thesis utilizes mathematical modeling techniques to develop a
mixed integer program that aims to minimize travel time, distance, and associated costs in one of
the largest counties in the United States, Los Angeles County. The developed model takes into
account the diverse demographics and socioeconomic factors of the County and plans for the
selection and allocations accordingly. The model explores 277 zip codes within Los Angeles and
analyzes them as potential vaccine distribution centers. It also incorporates the two different and
most common means of transportation, cars, and public transit, to account for all users. Three
scenarios are explored where each zip code of the 277 is assigned priority based on the following
factors: population, Healthy Places Index, and a Vulnerability to COVID-19 index. The output
showed significant improvements in reducing average travel times and distances as well as savings
in costs when compared to the actual selected sites within the County.

1
Chapter One: Introduction
1.1 Background and Context of the Study
The coronavirus disease 2019 (COVID-19) emerged as a global threat in late 2019, quickly
evolving into a pandemic that has deeply impacted every aspect of modern life. The rapid spread
of the virus across the globe has placed massive pressure on healthcare systems, disrupted
economies, and altered social norms. Governments, organizations, and individuals have been
forced to adapt to new challenges and find innovative solutions in order to save lives and minimize
any economic losses.
In response to the crisis, researchers and pharmaceutical companies worldwide have worked
tirelessly to develop vaccines and medicine to combat the virus. The rapid development and
distribution of COVID-19 vaccines have been critical in mitigating the pandemic's impact and
making way for a gradual return to everyday life. Strategically choosing the sites to administer the
vaccines and planning the allocation of the quantities have become a top priority to ensure smooth
delivery to vulnerable populations and regions with limited resources in order to receive the
protection they need.
In this context, the study of sites selection and vaccines distribution, particularly in densely
populated and diverse regions like the Los Angeles (LA) County metropolitan area, is of major
importance. By optimizing the location of vaccine distribution centers while considering important
factors such as travel time, distances, costs, and the demographics of the populations, this research
aims to contribute to more efficient and equitable vaccine accessibility, ultimately improving
public health outcomes in the face of ongoing global challenges.

2
The Los Angeles County metropolitan area is among the most densely populated and diverse
regions in the United States. It comprises many urban, suburban, and rural areas, each exhibiting
distinct demographic and health characteristics. With a population of over 10 million people
dispersed across roughly 4,000 square miles, LA County confronts substantial logistical obstacles
in vaccine distribution. According to LA County Department of Public Health [1] data, as of June
11
th,
2023, around 8.33 million residents (81% of total LA county population) are vaccinated with
at least a single dose, and 7.56 million (74%) are fully vaccinated. While these numbers might
appear high, higher vaccination rates are desired in order to contain the severe risks of the virus.

Figure 1: Vaccination uptake rates in LA County according to LA Public Health[1]

These challenges are further aggravated by inequalities in health outcomes and access to healthcare
services across different communities within the County. Socioeconomic aspects, environmental
factors, and healthcare infrastructure contribute to an uneven distribution of health risks and
resources in the region. For instance, areas with lower socioeconomic status might face a higher
prevalence of chronic diseases or limited access to healthcare facilities, amplifying the importance
of strategically choosing the locations of the sites and efficiently distributing the vaccine.

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Moreover, incorporating health indices like the Healthy Places Index (HPI) allows for a more
targeted approach, prioritizing areas with greater health disparities and vulnerabilities. This
ensures that those most in need receive vital immunizations in a timely manner, contributing to a
more equitable distribution of resources and improved public health outcomes.
In summary, this study aims to identify optimal locations for vaccine distribution centers in the
LA County metropolitan area by considering a range of factors, including travel time, costs, and
health indices. By doing so, it seeks to promote more cost-efficient and equitable access to
vaccines, ultimately leading to improved public health outcomes for the diverse communities
within the region while maintaining the minimum possible costs.

1.2 Research Objectives and Question
The primary aim of this study is to formulate a well-informed strategy for identifying the most
suitable locations for vaccine distribution centers within the Los Angeles County metropolitan
area. By considering a range of factors such as travel duration, costs, health indicators, and ease of
access, the study aspires to advance the effectiveness and fairness of vaccine distribution across
the region's varied communities as well as keeping the costs to a minimum.
An essential part of this research is identifying which critical variables should be taken into account
when deciding on the best locations for vaccine distribution centers. These variables need to be
integrated into a comprehensive framework that strikes a balance between efficiency, equity, and
accessibility, thereby enhancing the vaccine distribution process for all residents. The investigation
will incorporate health indicators, such as the HPI index, into the location strategy. This inclusion
aims to give priority to areas with higher health disparities and susceptibilities.

4
Through addressing these research focus areas, the study hopes to offer valuable insights and
strategic recommendations for public health authorities, policymakers, and other parties involved
in organizing and executing vaccine distribution efforts. The wider implications of the proposed
strategy concerning public health and emergency preparedness in urban, suburban, and rural
settings will also be assessed. Ultimately, the research seeks to foster more cost-efficient and
equitable vaccine distribution, improving public health outcomes for the diverse communities
within the Los Angeles County metropolitan area and beyond.

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Chapter Two: Literature Review
Optimization models have been increasingly utilized in the healthcare sector to improve health
outcomes and increase efficiency and effectiveness. Applications include resource allocations and
strategic planning of healthcare facilities’ locations, especially in times of global pandemics such
as the recent COVID-19. In the case of vaccine distribution and optimal selection of vaccine
centers, optimization models can help determine the most effective ways to distribute the vaccines
to achieve the maximum converge and reduce the impact of the pandemic. In the context of this
thesis, the following sections will explore the progress recent research has achieved in the analysis
of accessibility, optimization of facility and vaccine administration locations, and allocations of
vaccines using mathematical modeling.
2.1 Review on the Special Analysis of Vaccine Distribution
Spatial analysis and accessibility have been a major part of the research conducted on vaccine
distribution as it helps identify disparities in vaccination rates and areas with highly vulnerable
population groups. One comprehensive spatial analysis was conducted by Mollalo, Mohammadi,
Mavaddati, and Kiani (2021) [2] in their paper “Spatial Analysis of COVID-19 Vaccination: A
Scoping Review”, which summarized previous findings and highlighted research gaps. They also
concluded that significant geographic disparities in COVID-19 vaccination rates are present, with
some regions and countries having higher rates, which they attribute to factors such as having
access to more supply and healthcare services, as well as the presence of vaccine hesitancy. This
comprehensive study provided an excellent overview on the topic and facilitated the search for
previous studies relevant to this thesis. In another study, "Spatial Modeling of COVID-19 Vaccine
Hesitancy in the United States", by Mollalo and Tatar (2021) [3], the authors presented a GIS-
based study to analyze the spatial heterogeneity of COVID-19 vaccination rates across all counties

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in the United States using Social Vulnerability Index (SVI) data. The study utilized regression
models, such as the Ordinary Least Squares and Geographically Weighted Regression models, to
study the relationship between the socioeconomic factors of the counties and their achieved
vaccination rates. Such application of regression tools could be utilized to study the relationships
between COVID-19 deaths and the demographics of the zip codes within the scope of this thesis,
to enhance establishing more targeted vaccine distribution strategies. Moreover, the paper “The
Use of GIS Technology to Optimize COVID-19 Vaccine Distribution: A Case Study of the City
of Warsaw, Poland" by Krzysztofowicz and Osinska-Skotak (2021) [4], the authors utilized GIS
technology to visualize the spatial distributions and patterns, which could be crucial for
understanding and determining the sites that have the maximum reach and efficiency. The tool
also assisted them in identifying the areas with higher populations of age groups eligible to receive
the vaccines, which helped achieve a more targeted vaccination campaign.
In another study, by Alemdar, Kaya, Çodur, Campisi, and Tesoriere (2021) [5], the authors
highlighted the importance of the logistics of the vaccines and the selection process of the sites for
dispensing vaccines. They proposed a three-step solution for site selection, where they defined
eight evaluation criteria based on the suggestion of advisory boards, assigned a weight to each
criterion, and finally assigned potential sites and determined service areas by obtaining a suitability
map. Building upon these previous studies, this thesis extends the strategic selection of vaccination
sites by the inclusion of additional factors such as travel times, populations, and costs into the
optimization model, which leads to a more comprehensive tool for decision-makers. Lastly, spatial
accessibility to urban medical facilities in China was assessed by Cheng, Tao, Lian, and Huang
(2021) [6], where they used a comprehensive transportation network composed of ground
transportation and rail transit and calculated the spatial accessibility of small units. They adopted

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empirical Kriging interpolation methods as well as cluster and outlier analysis, concluding that
poor accessibility was presented in most of their study area and that residential areas along the
subway had better access to the facilities. In the end, the authors proposed that improving the
transportation network leads to overall better medical facility accessibility.
2.2 Review on Optimization of the COVID-19 Vaccine Centers Locations
The topic of COVID-19 vaccine center locations has emerged as significant area of study since the
global pandemic emerged in 2020. Numerous studies have employed mathematical modeling
techniques, incorporating various factors such as travel times, distances, and operational costs to
identify optimal locations that would help reduce the impact of the pandemic. A study by Bravo,
Hu, and Long (2022) [7], [8] considered travel distance when optimizing the selection of vaccines.
The authors argue that reducing the travel distances can improve vaccination rates, where they
proposed using retail pharmacies and dollar stores as vaccination sites to reduce the disparities
across demographic groups. This reduction allows for an additional 25% of the population to be
within walking distance (<1 Km) of a vaccine site. Moreover, they concluded that an increase of
5% in vaccinations would be achieved by applying the pharmacy and dollar stores strategy.
However, this paper incorporated Euclidian distance between populations centroid and locations
in their model whereas this thesis explores traveled distance in a more accurate approach using
Maps API, which yields real life like travel times and distances. Similarly, using a facility location
optimization model and travel estimates from the US National Household Travel Survey data, the
authors Risanger, Singh, Morton, and Meyers (2021) [9] introduced a function calculating the
fractions of populations willing to travel, where it showed different behaviors for trip distances
above and below 5 miles. Such application of functions could be helpful when deciding on the
locations of public health facilities and vaccination sites. Their model is formulated to maximize

8
the population willing to travel to its closest site based on their introduced travel willingness
function with the assumption that there is a maximum number of locations that can be selected,
unlike what the introduced model in this thesis tris to achieve which is to minimize the travel
distance and time between populations and locations. In another study, Bertsimas, Digalakis Jr,
Jacquillat, Li, and Previero (2021) [10] proposed an epidemiological model called DELPHI and
integrated it into an optimization model that plans strategic vaccine distributions. Their approach
starts by forecasting the dynamic of the pandemic using an SEIR model and then it accounts for
the vaccination impact on the populations based on the vaccine effectiveness. Lastly, these inputs
are then integrated into a model used to optimize the vaccine distribution strategy, which increases
the effectiveness of the vaccination campaign by an estimated 20%, which is projected to save an
extra 4000 lives in the United States within three months. Unlike the SEIR model, this thesis takes
another approach and explores a distribution strategy based on prioritization criteria that can be
selected by decision makers.
Points of Dispensing (PODs) were also proposed in a study by Alghanmi, Alotaibi, Alshammari,
Alhothali, Bamasag, and Faisal (2022) [11], where the authors discussed the importance of PODs
during public health emergencies. They presented a survey of PODs location-allocation models,
where various optimization models were discussed which included minimizing maximum travel
times, minimizing total weighted travel time and distance, and the associated costs. This thesis
extends this work by analyzing adding more flexibility to the model, where demand points could
have more than one option of facilities to receive the vaccines. Furthermore, minimizing the
traveled distance is also discussed by Lusiantoro, Mara, and Rifai (2022) [12], which they
incorporated in their bi-objective linear programming model. They aimed at maximizing the
coverage for vaccine recipients while also minimizing the weighted distance travelled. They

9
applied the model to Yogyakarta of Indonesia, and their results suggested that prioritizing areas
with high COVID-19 cases caused less efficient coverage, which suggests that other factors
beyond the cases might need to be considered, which is considered in the developed model in this
thesis, where decision makers have the choice to select from multiple prioritization criteria.
Another multi-objective optimization model was proposed by Tang, Li, Bai, Liu, and Coelho
(2022) [13], where the authors tried to optimize the operational costs of vaccine sites and the total
travel distance for multi-period COVID-19 vaccination planning. A decision framework was also
proposed to help decision-makers choose based on real-life limitations or preferences while
optimizing the service level. The results of the model showed a 9.3% decrease in the operational
cost and a 36.6% decrease in the total traveled distance. Our work builds on this by considering
other costs such as the travel time, cost of public transit, and cost of traveled distance into the
objective function. At a municipality level, Cabanilla, Enriquez, Mendoza, and Mendoza (2022)
[14] presented optimal locations of vaccine sites, where they considered existing public facilities,
such as hospitals and schools, as potential sites. They divided the town into several smaller areas
and assigned weights to densely populated and highly contagious areas with higher case counts.
After that, they incorporated those weighting factors into a distance minimization problem between
the sites and the areas of the town. This thesis extends this work by considering every area in the
study as a potential site and by incorporating different means of transportation into the model.
Furthermore, a Location-Allocation model was developed by Faisal, Alshammari, Alotaibi,
Alghanmi, Bamsagm, and Bin Yamin (2022) [15] to improve the distribution of COVID-19
vaccine centers in Jeddah, Saudi Arabia. The authors introduced a maximal coverage model with
and without facility capacity constraints. They applied the model with different impedance cutoffs,
which are the maximum travel times required from demand points to vaccine centers. Moreover,

10
the authors explored the minimum number of facilities needed to satisfy all the demand points
within the city by minimizing the overall transportation time and distance.
2.3 Review on Allocation of the COVID-19 Vaccines
As we dive into the third section of the literature review, we shift our focus to vaccine allocation,
particularly through the lens of optimization techniques. During times of emergencies, like the
COVID-19 pandemic, equitable distribution of the vaccines is essential to ensure that all
populations, especially the most vulnerable, have access. This is a complex task that requires the
use of advanced mathematical modeling and optimization techniques to mitigate pandemics.
We first need to understand the relationship between the different characteristics of communities
and COVID-19 vulnerability and death rates. This was explored by Karmakar, Lantz, and
Tipirneni (2021) [16], where the authors analyzed the association between the US county-level
sociodemographic factors and COVID-19 incidence and mortality using a mixed-effects negative
binomial regression. They used the SVI as a measure of community vulnerability to the virus and
found a significant correlation between the two. Moreover, Ong, Pech, Gutierrez, and Mays (2021)
[17] presented in their paper "COVID-19 Medical Vulnerability Indicators: A Predictive, Local
Data Model for Equity in Public Health Decision Making", a study that aimed to develop a
predictive model of the vulnerability indicators for COVID-19 in Los Angeles County using four
key indicators to capture the dimensions of COVID-19 related vulnerabilities. The indicators were
pre-existing health conditions, barriers to accessing healthcare, built environment risks, and the
SVI. The study concluded that the racial groups of LatinX, Blacks and segments of the Asian
populations were the most vulnerable to COVID-19. Building on this, this thesis uses similar social

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characteristics in the optimization of sites locations, where it ensures equitability by addressing
the communities that are most vulnerable.
Many research papers have addressed fairness and equality in allocations of vaccines, as in Balcik,
Yucesoy, Akca, Karakaya, Gevsek, Baharmand and Sgarbossa (2022) [18]. The authors presented
a mathematical model that is designed to find an effective and equitable allocation plan while
prioritizing vulnerable groups and regions. Their objective function is to minimize the weighted
sum of deviations (both positive and negative) from the fair coverage levels while also minimizing
the deviation for the coverage threshold for certain groups by penalizing this deviation. The model
achieved a 74% coverage, which is very close to the ratio of total supply to total demand (74.93%).
In another paper by Anaahideh, Kang, and Nezami (2022) [19], the authors tried to achieve a fair
allocation plan in terms of demographic disparity by maximizing geographical diversity while also
maximizing fairness among certain groups of the population. They designed a multi-objective
optimization model that minimizes the weighted sum of deviation from the fair coverage levels to
ensure that the allocation is fair and diverse across different geographical and social groups. They
defined geographical diversity by a standard that all centers of vaccine should have the same
average amount of resources per capita regardless of location, which helps achieve an evenly
spread-out distribution of resources. The social fairness in their paper is defined by the standard
that for every social group, the average amount of resources allocated per capita should not vary
depending on the socioeconomic attributes of the groups. In addition, Wang (2021) [20] analyzed
data visualization and optimization to achieve an equitable distribution of COVID-19 vaccines.
He analyzed demographic data, such as population, race, age, and socioeconomic data, such as
poverty and unemployment, using business analytics tools to identify trends, such as regions with
high probability needs for hospitalization. He then integrated his optimization model to determine

12
the optimal cost-efficient method of allocations using the trends and patterns identified. His results
suggested an allocation plan that leans towards the areas in most need to ensure equitability.
One more contribution to the literature on the field of fair allocation of resources was made by
Singh (2020) [21]. The author presented a math model to analyze the allocation of resources based
on a set of user groups and a set of resource types while also considering the abundance of those
resources. The study concluded that if a resource is abundant, all users will achieve the maximum
possible coverage, whereas if it was scarce, the allocation will be equitable across all the user
groups. Taking inspiration from these studies, this thesis incorporates giving priority to vulnerable
groups and demographics of each zip code within the study area, which helps achieve a more
efficient health outcomes.
Similarly, other contributions to the field of vaccine allocation focused on specific goals, such as
minimizing the death counts caused by the virus, as studied by the authors of [22]. The study
utilized a constrained optimization model that aimed to minimize the projected number of
additional deaths in the Philippines by considering factors such as the current percentage of
unvaccinated populations, the number of susceptible people, the maximum outbreak size, and the
effectiveness and cost of vaccines. The authors also accounted for the population density and
specific interventions like social distancing by incorporating a scaling contact coefficient into the
model, which computes the rate of contact based on the population density. The developed model
ensured that priority groups, which were selected to be the front-line health workers and old
people, were all given priority to receive the vaccines. Furthermore, a model that prioritized groups
based on their occupations and age was developed in a study by Babus, Das, and Lee (2020) [23].
The model assumed different scenarios where a stay-at-home order was effective and where there
was no order. The policy minimized both the number of deaths and the loss due to the suspension

13
of the economic activities carried out by those occupations. Their findings suggested that age is
more important than occupation.
The authors of [24] developed a multi-period decision-making model to design an allocation
strategy by incorporating susceptibility rate and exposure risk while also taking into account
operational costs like transshipment between medical centers, capacity, and return mechanism.
Their results suggested that superior performance can be achieved in minimizing the risk of
susceptibility and integrating the operational costs aspects. However, unlike these studies, this
thesis incorporates other factors such as travel time and distance costs, as well as vulnerability
indices into the optimization process of costs and allocations.
In the context of forecasting and predicting the death outcomes of pandemics, the author of [25]
presented a predictive model named DELPHI-V-OPT that forecasts the number of detected cases,
hospitalizations, and deaths for each US state. This model also dynamically captures the effects of
vaccination on the pandemic in the sense that vaccinated individuals will be immune to the virus
and that reducing the infections will lead to a slower spread of the disease. The model then
optimizes the vaccine allocation using the epidemiological data of each state as well as clinical
data and availability of the vaccine data to minimize the number of deaths. The output of the model
suggested that a decrease of 10 to 25% in the death toll could be achieved, which could save up to
20,000 deaths over a period of three months in the United States. Moreover, authors of [26]
formulated a quadratic program that minimizes the sum of quadratic errors between the reported
cumulative number of deaths and the predicted number. The model is then formulated into a global
optimization problem that is solved using particle swarm optimization to find the parameters that
lead to the minimization of the quadratic programming model. The output of the model is then
tested in two scenarios, and the results turned out to be fairly close to prior estimates in the

14
literature. Furthermore, reference [27] discussed the uncertainty associated with the real-time
predictions of COVID-19 infection rates, where the authors utilized statistical modeling techniques
to predict the spread of the virus in China and Italy. They used a framework to estimate uncertainty
as more information becomes available. They noticed that there is an observed shift from
exponential growth at the beginning of the pandemic to a sigmoid pattern once lockdown
restriction measures are introduced, indicating the effective impact of these measures on the spread
of the disease. These studies provide inspiration for how the model could include other important
objectives such as reducing deaths from COVID-19.
Finally, reference [28] studied the optimization of the vaccine, where the authors proposed four
models to optimize the allocation: minimizing the travel distance and maximizing the vaccination
of high-priority groups. The models are developed using a Constraint Satisfaction Problem (CSP)
formulation, which is a powerful mechanism to address the logistics of resource allocations. Their
four models were: (1) a basic vaccine distribution model, (2) a priority-based distribution model,
(3) a distance-based distribution model, and (4) a combination of priority and distance distribution
models. They concluded that the last model of the four achieved the best results.
In summary, the fields of vaccine distribution centers optimization and allocation of vaccines have
seen significant attention in the past years, particularly in the context of COVID-19. In terms of
optimizing the locations of vaccine centers, studies aimed to ensure that the selected sites are in
areas that are easily accessible to the populations in general and priority groups in particular. They
achieved this by considering factors such as geographical distribution, travel time metrics, and the
use of existing retail locations such as dollar stores. The allocation of vaccines has also been a
focus of research. Studies have used various optimization techniques along with factors such as

15
the vulnerability of different groups, fairness, and equity to achieve the most effective allocation
plans that minimize the spread of the virus and reduce the number of cases and deaths.
Building on these foundations, this thesis introduces several novel aspects to the field. Firstly, it
covers a large metropolitan area, a study area that only a few papers have explored, with an
emphasis on the early stages of the pandemic when the number of sites was limited. Secondly, my
optimization model considers accurate values of travel times by car and transit and travel distance
between the proposed optimal locations and the population centroids of the zip codes of a large
metropolitan area. This enhanced accuracy was achieved by using Bing Maps API. The costs
associated with those travel times and distances were also incorporated in the model as the
objective to be minimized. Lastly, multiple prioritization assignment methods within zones were
considered, such as the Healthy Places Index, a vulnerability index based on CDC data, and
population-based prioritization. These methods allow exploring different scenarios and their
results, providing a more comprehensive study.

16
Chapter Three: Methodology
The designed optimization model in this thesis is a Mixed Integer Programming (MIP) model that
aims to find the optimal locations of vaccine centers and their associated costs in a large
metropolitan area. The model assumes that each zip code in the studied area represents a potential
site for COVID-19 vaccines and represents a population that needs to be served by these sites.
Therefore, the model depends on specific types of data for its computations, such as the actual
travel times and distances data between zip codes, the demographics of zip codes, and COVID-19
related data.
3.1 Travel Time and Distance.
Travel times by car and transit, along with the distance matrices, play a significant role in the
optimization model and the selection of vaccination sites, as they determined the accessibility of
all parts of large metropolitan areas and their residents. To generate the travel times and distances,
N by N matrices are created for the travel time by car, travel time by transit, and the distance
between the weighted population centroids, where N represents the number of areas/districts/ zip
codes in a large city. Each area N in the study serves as a potential site and demand point for the
vaccines. These matrices can be obtained in several methods using available software and APIs,
each having its advantages and disadvantage for this study. Bing Maps API was selected for the
calculations of those matrices due to its accessibility, ease of use, integration with Excel, and
accurate results. Excel is utilized in the process where a database with all coordinates were
organized. Three new Excel functions are developed using Excel Visual Basic macros, where
Excel is connected with the Bing API to generate the results. The matrices can then be computed
where the rows are considered as the source points and the columns as destinations, with every
cell representing a pair of both. Every cell in the matrix represented the value of either the travel

17
time in minutes or distance in miles according to the formula generated by Excel macros. The
following is a snapshot of a car travel time matrix.

Table 1: Car travel time matrix in minutes

3.2 Optimization Model.
The optimization model is designed to select the most effective vaccination sites and provide an
allocation plan for any major metropolitan city around the globe. The model incorporates a priority
assignment parameter that provides the option to assign importance levels to the zip codes based
on chosen criteria. In this thesis, the developed model is specifically applied to Los Angeles
County, taking into account its diverse demographics and socioeconomic factors. To evaluate the
model’s robustness and understand how the outcomes vary under different circumstances, we
implement three unique scenarios. These scenarios are differentiated based on the following
prioritization parameters: using population, the HPI, and a specially developed COVID-19
vulnerability index. The developed MIP model allows both continuous and integer decision
variables and relies on solid and robust data sources, which will provide the necessary inputs for a
comprehensive study that fits LA County's needs and constraints. The model utilizes travel times
by car and transit and the distance between the zip codes in order to find the optimal selection of
sites. The objective of the model is to minimize the total costs, which include the cost of opening
vaccination sites, the time cost of travel, the distance cost, and the bus cost for those who do not
own cars. Additionally, the model aims to maximize the total quantity of vaccines allocated to
each zip code based on its priority assignment. The model introduces several constraints to ensure

18
a feasible and practical solution. It sets the maximum and minimum number of sites to specific
numbers, which can be set by decision-makers according to the specifications of each study area.
the model also introduces a parameter, denoted as F, which represents the number of vaccine
centers people can choose from when receiving their vaccine. This parameter can be adjusted to
provide people with more flexibility and convenience. The model also assumes a restriction on the
budget, which the cost of opening sites cannot exceed.
3.2.1 Model Sets, Parameters, and Decision Variables.
Sets:

T: Set of time periods t
A: Set of all zip codes areas i
S: Set of all potential sites j
Parameters:

TBij : Travel time by transit between area i and site j
TCij : Travel time by car between area i and site j
DDij : Distance between area i and site j.
Oi : Percentage of car ownership at area i
Wi : Priority assigned for area i.
Pi : Population of area i.
K : Available Budget for opening sites.
Qt : Available quantity of the vaccine at time t.
TP : Total population of LA county
C : the cost of opening a site.

19
F : A flexibility parameter indicating the maximum number of sites that people can select
from.
M : Big M.
MS : the maximum number of allowable sites.
LS : the minimum number of allowable sites.
VT : the cost value of time.
VD: the cost of traveled distance.
VB: the cost of a public transit ticket.
Decision Variables:
Di : 1 if area i is selected as a site, 0 otherwise.
Xij : 1 if area i is assigned to site j, 0 otherwise.
Vijt : Allocated vaccines from site j to area i at time t
3.2.2 Objective Function and Constraints
The following objective function is formulated in order to achieve the goal of minimizing the
overall costs of administering the vaccines from the selected sites as well as to provide a plan for
allocations of vaccines between areas.

𝑀𝑖𝑛 ∑ 𝐷 j
C
𝑆 𝑗 + ∑ ∑(
1
𝐹 𝑋 𝑖𝑗
𝑃 𝑖 (𝑂 𝑖 𝑇𝐶
𝑖𝑗
+ (1 − 𝑂 𝑖 )
𝑆 𝑗 𝐴 𝑖 𝑇𝐵
𝑖𝑗
) × 2 × VT)
+ ∑ ∑(
1
𝐹 𝑋 𝑖𝑗
𝑃 𝑖 𝐷𝐷
𝑖𝑗
𝑆 𝑗 𝑂 𝑖 𝐴 𝑖 × 2 × VD) + ∑ ∑(
1
𝐹 𝑋 𝑖𝑗
𝑃 𝑖 𝑆 𝑗 (1 − 𝑂 𝑖 )
𝐴 𝑖 × VB)

(1a)

20
Subject to:
∑ 𝑉 𝑖𝑗𝑡
𝑆 𝑗 =
𝑃 𝑖 𝑇𝑃
× 𝑄 𝑡 , for i A, t T (2a)
∑ 𝐷 𝑗 𝑆 𝑗 ≤ 𝑀𝑆 (3)
∑ 𝐷 𝑗 𝑆 𝑗 ≥ 𝐿𝑆 (4)
∑ 𝑋 𝑖𝑗
𝑆 𝑗 ≥ 𝐹 , for i A (5)
𝑋 𝑖𝑗
≤ 𝐷 𝑗 , for i A, j S (6)
∑ 𝐷 𝑗 𝑆 𝑗 𝐶 ≤ 𝐾 (7)
𝑉 𝑖𝑗𝑡
≤ 𝑋 𝑖𝑗
𝑀 , for i A, j S, t T (8)
𝐷 𝑗 ≥ 0 , for j S (9)
𝑋 𝑖𝑗
= {0,1} , for i A, j S (10)
𝑉 𝑖𝑗𝑡
≥ 0 , for i A, j S, t T. (11)

The objective function (1a) minimizes the total cost that results from opening the sites, cost of
travel time to the vaccination sites, cost of miles traveled, and the cost of bus tickets as follows:
• The first term in the objective function (∑𝑫 𝐣 𝐂 𝑺 𝒋 ), calculates the direct cost resulted from
opening the sites.
• The second term of the function (∑∑ (
𝟏 𝐅 𝐗 𝐢𝐣
𝐏 𝐢 (𝐎 𝐢 𝐓𝐂
𝐢𝐣
+ (𝟏 − 𝐎 𝐢 )
𝐒 𝐣 𝐀 𝐢 𝐓𝐁
𝐢𝐣
) × 𝟐 × 𝐕𝐓 )),
calculates the cost of time spent by the populations traveling to their F closest sites when they

21
are given the option to choose from F locations. F is added here to simulate real-life scenarios
where the individuals have flexibility in choosing their preferred location to receive the
vaccine. Means of travel are accounted for by multiplying the percentage of people who own
cars by the TCij and by multiplying the remaining percentage, which will have to use public
transit, by TBij. The whole term is then multiplied by 2 to account for the round trips and by
the travel time cost VT. The whole term is divided by F to account for the average of the F
possible trips that could be made by the population, as only one trip is going to take place.
• The term (∑∑ (
𝟏 𝑭 𝑿 𝒊𝒋
𝑷 𝒊 𝑫𝑫 𝒊𝒋
𝑺 𝒋 𝑶 𝒊 𝑨 𝒊 × 𝟐 × 𝐕𝐃 )), finds the mile costs of the distance traveled by
car users. It is multiplied by 2 to account for the round trip and by VD, which is the cost per
mile.
• The fourth term (∑∑ (
𝟏 𝑭 𝑿 𝒊𝒋
𝑷 𝒊 𝑺 𝒋 (𝟏 − 𝑶 𝒊 )
𝑨 𝒊 × 𝐕𝐁 )), finds the total cost spent by transit users by
multiplying the number of trips by the cost of transit ticket VB.
Constraint (2a) provides an allocation strategy that is based on the ratio of the population of area
i to the total population in all areas multiplied by the available vaccine quantities Qt for the given
period t. Constraint (3) sets the maximum allowable sites to MS. Constraint (4) sets the minimum
possible sites to LS. Constraint (5) sets the number of options the populations of each area can
have to a minimum of F. Constraint (6) is an upper bound constraint to link decision variables Xij
and Dj. Constraint (7) sets the maximum budget for the costs of opening sites to K, which can be
adjusted by policymakers. In constraint (8), the amount of vaccines allocated is linked to the
assignment between areas and sites so that vaccines are only allocated when there is an
assignment, where M is a large positive number that acts as an upper bound on Vijt. M value could
be as low as the amount of vaccines available at t. Constraints (9), (10), and (11) regulate the
value of the decision variables.

22
3.3 Modeling The Different Scenarios
In this section, three scenarios will be discussed and applied to the introduced model to provide a
more comprehensive analysis of the model from different perspectives. The following scenarios
were not intended to represent the best strategies, but rather to demonstrate the model’s ability to
accommodate different prioritization strategies based on the specific needs and constraints of the
given situations and area of study. By doing so, we can gain insights in how different strategies
impact the outcomes and effectiveness of the optimization problem, which in return, will provide
guidance to decision-makers. The implementation of these scenarios is built on the assumption
than comprehensive data and indices about zip codes is readily available for the intended area of
study. The feasibility of these scenarios is directly tied to the extent and quality of available data.
3.3.1 Scenario One: Prioritizing Areas Based on Population.
In the first scenario, the population will serve as the weighting factor when deciding on the
importance and assigning the priorities of the areas. Therefore, the model in this case, will favor
the areas with higher populations and select the sites and allocation plan accordingly.
The objective function and constraints remain the same, as described and explained previously in
this chapter.
3.3.2 Scenario Two: Prioritizing Areas Based on the Healthy Places Index.
In scenario two, we utilize the HPI, a comprehensive tool that calculates an overall health score
for regions based on multiple factors such as socioeconomic conditions, environmental
characteristics, and accessibility to healthcare [29] (more discussion on HPI can be found in section
4.2.2) . HPI data is used as the weighting factor. We will rank all areas based on their HPI
percentile, as provided by The Public Health Alliance of Southern California. Five categories, 1

23
(most important) thru 5 (least important), are then introduced, and areas are categorized based on
their percentile. The population of each area will be multiplied by a multiplier based on the area’s
category to yield Wi for each area i that will be used in the model, as explained in the following
table:
Percentile Range Category Weighted Population (Wi)
0.19 - 0 1 population x 1.5
0.39 - 0.2 2 population x 1.25
0.59 - 0.4 3 population x 1
0.79 - 0.6 4 population x 0.75
1 - 0.8 5 population x 0.5
Table 2: Wi calculations using HPI
The new Wi will only change the objective function and the first constraint. The rest of the model
will remain unchanged. The modified objective function and constraint are as follows:
𝑀𝑖𝑛 ∑ 𝐷 j
C
𝑆 𝑗 + ∑ ∑(
1
𝐹 𝑋 𝑖𝑗
𝑊 𝑖 (𝑂 𝑖 𝑇𝐶
𝑖𝑗
+ (1 − 𝑂 𝑖 )
𝑆 𝑗 𝐴 𝑖 𝑇𝐵
𝑖𝑗
) × 2 × VT)
+ ∑ ∑(
1
𝐹 𝑋 𝑖𝑗
𝑊 𝑖 𝐷𝐷
𝑖𝑗
𝑆 𝑗 𝑂 𝑖 𝐴 𝑖 × 2 × VD) + ∑ ∑(
1
𝐹 𝑋 𝑖𝑗
𝑊 𝑖 𝑆 𝑗 (1 − 𝑂 𝑖 )
𝐴 𝑖 × VB)

(1b)
∑ 𝑉 𝑖𝑗𝑡
𝑆 𝑗 =
𝑊 𝑖 ∑ 𝑊𝑖
× 𝑄 𝑡 , for i A, t T (2b)

3.3.3 Scenario Three: Prioritizing Areas Based on the Vulnerability to
Covid-19.
In this scenario, the weighting factor Wi will be calculated based on how different ethnicities are
vulnerable to Covid-19. The following table is provided by the Centers for Disease Control and

24
Prevention (CDC) [30], and it outlines the risks of getting infection, hospitalized, and death by
Covid-19 by race and ethnicity.
Race/Ethnicity
American Indian or
Alaska Native, Non-
Hispanic
Asian, Non-
Hispanic
Black or
African
American, Non-
Hispanic
Hispanic or
Latino
Cases 1.6x 0.8x 1.1x 1.5x
Hospitalization 2.4x 0.7x 2.0x 1.8x
Death 2.0x 0.7x 1.6x 1.7x
Table 3: Racial vulnerability to Covid-19 by CDC
The “x” in these values shows the rate ratios compared to White, Non-Hispanic race. For instance,
a Black Non-Hispanic individual is 1.1 times more likely to get infected, two times more likely to
get hospitalized, and 1.6 times more likely to die from the virus compared to a White individual.
These rates, combined with the racial compositions for the targeted areas of study, are used for the
calculation of the Covid-19 racial vulnerability index. A weighted score for each of the three risks
(cases, hospitalization, death) measures is computed by multiplying the rates from the above table
by the proportion of each race for each area. The output of the above calculations will result in a
score for each of the above three risk measures. The following table is an example of these
calculations for an area for demonstration:

25

Race Population
% of Total
Population
Risk
Ratio
(Cases)
Weighted
Cases
Risk
Risk Ratio
(Hospitalization)
Weighted
Hospitalization
Risk
Risk
Ratio
(Death)
Weighted
Death
Risk
American Indian
and Alaska
Native, Non-
Hispanic
88 0.0025 1.6 0.004 2.4 0.006 2 0.005
Asian, Non-
Hispanic
5471 0.1546 0.8 0.1237 0.7 0.1082 0.7 0.1082
Black or African
American, Non-
Hispanic
1301 0.0368 1.1 0.0405 2 0.0736 1.6 0.0589
Hispanic or
Latino
19899 0.5623 1.5 0.8435 1.8 1.0114 1.7 0.9559
White only 8629 0.2438 1.0 0.2438 1.0 0.2438 1.0 0.2438
Total /
Composite Risk
Score
35488 1 - 1.2555 - 1.443 - 1.3618
Table 4: Risks scores calculations example for zip code 90029

26
Following the above calculations, we combine all three risk scores into a single score for all three
by assigning a weighting factor for each of the risk measures. We assigned a weight of 0.15, 0.3,
and 0.55 to cases, hospitalization, and death, respectively as shown in the following table:
Risk Measure Weight
Cases 0.15
Hospitalization 0.3
Death 0.55
Table 5: Weights for risks measures

The selection of these weights is subjective and was chosen based on the severity of the risks, with
death being the most serious risk. Finally, by implementing the previous steps in all areas of study,
we end up with a final score for each. The percentiles are calculated for all areas, and then the
areas are categorized into five categories, similar to the earlier categorization method used in
scenario two. Each percentile is multiplied by a multiplier based on the category to yield Wi, which
will be reflected on the objective function and the first constraint as described before.

27
Chapter Four: Study Area and Datasets
4.1 Los Angeles County Metropolitan Area.
Los Angeles County is the most populous County in the United States, with an estimated
population of over 10 million residents of mixed and diverse ethnicities and backgrounds. The
County is composed of a varied mix of urban, suburban, and rural areas, offering a comprehensive
view of various community settings, from the Pacific Ocean in the west to the San Gabriel
Mountains in the east, which could pose challenges and impact the distribution and administration
process of the vaccines. The demographic composition of LA County is extremely diverse.
According to the US Census Bureau [31], the County consists of 49% Latinos, 25.5% White, 7.6%
Black, 14% Asians, and less than 1% American Indian and Alaska Native. The County is also
linguistically diverse, with many distinct languages spoken in households, with the English and
Spanish being the most common ones. In addition, socioeconomic factors, such as income level,
education, occupations, and accessibility to cars, widely range in the area which could eventually
lead to health outcome disparities. Moreover, Los Angeles County consists of around 295 zip
codes, of which 277 are considered for the scope of the thesis after excluding some of the
incorporated areas in the County due to the lack of sufficient data needed to conduct the study.
Catalina Island zip codes were also excluded because they are accessible only by ferries and
airplanes. Each zip code has its own unique demographics and geographic characteristics, which
will result in more informed, tailored, and effective vaccine distribution centers and allocations.
The following map shows the populations of the zip codes in LA County:

28

Map 1: LA County population heatmap

The application within this thesis covers 12 biweekly periods (approximately five months and a
half) starting from the day the first vaccine was approved by the Food and Drug Administration
(FDA) on 12/14/2020 and continuing until 05/30/202. This timeframe was chosen to focus on the
initial phase of the pandemic when the sites and vaccine availability were limited.
A 277 by 277 matrices were generated by Bing Maps for car travel time, transit travel time, and
distance between the zip codes. Random values were chosen and compared with actual times, and
all were within 10% margin of error.
The parameter F in the model is assumed to be 3, this is to ensure people are given more than one
option and to simulate real-life flexibility. The value of parameter VT for LA County is set at
$0.264167. This number represents 50% of the average wage per minute in the County. According
to the US Department of Transportation [32], 50% is the recommended percent of wages for the
travel time valuation when the trip purpose is personal. The value of parameter VD is $0.615,
which is the suggested cost per mile in Southern California as per the American Automobile

29
Association [33]. Parameter VB represents the cost of public transit tickets, which is approximately
$1.75. The values of parameters C and K, which represent the cost of opening a site and the total
available budget, are assumed to be $500,000 and $10 million, respectively. These numbers are
hypothetical and were chosen to provide a framework for the model. Actual costs that are based
on real-life data on the availability of budgets should be used in the application of the model.
4.2 Datasets
The selection and use of appropriate datasets are crucial in carrying out an effective study. Several
large datasets were utilized in this thesis to enhance and inform the process of optimally locating
the vaccination sites and the quantity allocated to each one.
4.2.1 Demographic Data
Demographic data was taken from the U.S. Census Bureau [29] and utilized throughout the thesis.
It provides a complete breakdown of the racial and age groups composition of the County on the
zip code level. This level of detail is essential in order to make tailored and informed decisions
and outcomes.
4.2.2 Healthy Places Index
The Healthy Places Index (HPI), as described by [34], is a tool that provides data on numerous
elements that influence the areas’ health. It calculates an overall health score for regions using a
variety of factors, such as socioeconomic factors, environmental characteristics, neighborhood
conditions, healthcare accessibility, and more. The HPI is computed by standardizing 25 indicators
across 8 domains, calculating domain scores, and weighting them using a regression model against
life expectancy. The final score for each zip code is then calculated and validated to inform
decisions about the communities’ health. The scores are ranked from 1 to 99, with higher scores

30
indicating better healthy areas. This index was developed by The Public Health Alliance of
Southern California [29] for multiple counties within the state of California. This index provides
a powerful tool that could be used by policymakers in various fields and help them make more
informed decisions. The HPI will be utilized in this thesis to analyze different scenarios of the
optimization model.
4.2.3 Covid-19 Data
Data on the quantity of vaccines administered is sourced from the LA County Department of Public
Health [35]. It also displays the number of vaccines and the date of when they were delivered and
administered to each neighborhood within the LA county, which will serve as the assumed
available quantities in this thesis.
4.2.4 Vulnerability to Covid-19
In order to simulate different scenarios and analyze how the output behave, we will be introducing
a COVID-19 vulnerability index that takes into calculations the data provided by the Center for
Disease Control and Prevention (CDC) [30] on how certain racial groups are vulnerable to get
infected, hospitalized, and potentially die from the virus. This, when paired with the demographics
of each zip code, provides insights and helps design a better and more strategic solution.
4.2.5 Car Ownership Data
In order to capture more accurate results when it comes to the calculations of the travel times and
the trips made by the population of each unit of study (zip codes), the number of households
owning a car is needed and is sourced from the 5-Year American Community Survey Data
provided by the U.S Census Bureau [36]. This data consists of several house characteristics that
will help in estimating how many people in each zip code would use public transportation or their

31
private cars to travel to vaccination sites. For this thesis, we are interested in the percentage of
households owning at least one car.
4.2.6 Geographical Data
LA County shapefiles and geographical data will be utilized in this thesis to develop different types
of maps, which help analyze and visualize the areas of the County. These data were sourced from
the Los Angeles GeoHub [37]. Moreover, Travel times by both bus and cars, as well as the distance
between the zip codes in LA County, were generated using Bing API. Furthermore, the zip codes’
population-weighted centroid coordinates were retrieved from the Office of Police Development
and Research and then were utilized as potential facilities for the administration of the vaccines.
They were used during the computations of the travel times and distances matrices as well.

32
Chapter Five: Results and Discussion.
In this chapter, we look at the results of applying our optimization model to the three defined
scenarios, where we focus on the total costs, selection of the vaccine distribution centers, and the
allocations plan of the limited vaccine quantities, and then a sensitivity analysis will be conducted
to examine the robustness of our results.
5.1 Model Output for Optimal Vaccine Distribution Sites.
5.1.1 Scenario One:
The optimal objective value for the first scenario is Z= $194,815,903. The following table
showcases the breakdown of the costs:
Cost Type Cost ($)
Construction Cost 10,000,000.00
Travel Time Cost 89,647,526.18
Travel Distance Cost 93,682,480.82
Bus Ticket Cost 1,485,895.97
Total 194,815,902.98
Table 6: Optimal costs breakdown
As shown in the table, the travel time and distance costs contribute the most to the overall costs
since the number of trips to be made is very large. Moreover, the following LA map shows the
selected optimal sites when the population is the weighting factor.

33

Map 2: Map of optimal sites for the first scenario
The following scatterplots show the relationship between the average travel times for bus and
cars as well as the average traveled distance with the populations of the areas.

Figure 2: Scatterplot of average travel time by transit vs population of zip codes

34

Figure 3: Scatterplot of average travel time by car vs the population of zip codes

Figure 4: Scatterplot of average distance vs the population of zip codes

As shown in the previous charts, the relationship between the populations and the average travel
and distance is inversely proportional, which is reasonable considering that this scenario aims to
prioritize more populated areas.

35
5.1.2 Scenario Two:
The optimal objective value when applying the second scenario on the model is Z =
$204,038,094.2, and the following table shows the breakdown of the costs:
Cost Type Cost ($)
Construction Cost 10,000,000.00
Travel Time Cost 90,351,438.72
Travel Distance Cost 94,565,654.05
Bus Ticket Cost 1,485,895.97
Total 196,402,988.73
Table 7: Optimal costs breakdown for scenario 2
Please note that Z is different from the total costs since the costs are calculated based on the true
populations making the trips without Wi affecting the actual costs. The following is the map of the
selected sites:

Map 3: Map of optimal sites for the second scenario

36
Figure 5 shows the box plot of the HPI percentile for the selected sites and compares it to the non-
selected sites. As expected, the plot shows a notable difference in the median of the selected sites
vs the non-selected, where the sites appear to be in zip codes within the lower side of percentiles.

Figure 5: Box plot of HPI percentile for selected sites

5.1.3 Scenario Three:
When applying the third scenario with the developed Covid-19 vulnerability index, the optimal
objective becomes Z = $182,327,956.5, and the following table shows the breakdown of the costs:
Cost Type Cost ($)
Construction Cost 10,000,000.00
Travel Time Cost 92,633,900.93
Travel Distance Cost 95,914,698.13
Bus Ticket Cost 1,485,895.97
Total 200,034,495.03
Table 8: Optimal costs breakdown for scenario 3

37
Please note that Z is different from the total costs since the costs are calculated based on the true
populations making the trips without Wi affecting the actual costs. The following is the map of the
selected sites:

Map 4: Map of optimal sites for third scenario
The following table summarizes the travel time cost, distance cost, and total weighted cost for all
scenarios:

Cost type
Scenario Travel time cost ($) Distance cost ($) Total weighted cost ($)
1 89,647,526.18 93,682,480.82 194,815,902.98
2 90,351,438.72 94,565,654.05 196,402,988.73
3 92,633,900.93 95,914,698.13 200,034,495.03
Table 9: Summary of cost types for all scenarios
In scenario 1, the total weighted cost, which includes both the travel time and distance cost as well
as cost of opening sites and the bus ticket costs, is approximately $194.8 million. The second

38
scenario yielded a higher total weighted cost of around $196.4 million, which is due to the increase
of both the travel time cost and distance cost from the first scenario. In scenario 3, we observe a
further increase to approximately $200 million, which is also a result of the travel time and distance
increase. These results highlight the trade-offs involved in the optimization model. Different
prioritization techniques will result in different objective functions but could also mean better
public health outcomes.
5.2 Model Output Optimal Vaccine Distribution Sites:
5.2.1 Scenario One:
The following heatmap demonstrates the allocation quantities to all selected areas, which will then
be serving all zip codes around them based on the assignment output of the model.

Map 5: Vaccine allocation for the first scenario

39
As can be observed from the above map, most of the vaccine quantities are allocated towards and
near the areas where the population is very high (map 1 shows the population for reference).
5.2.2 Scenario Two:
The following heatmap demonstrates the allocation quantities to all selected areas when the second
scenario is applied, which will then be serving all zip codes around them based on the assignment
output of the model:

Map 6: Vaccine allocation for the second scenario

5.2.3 Scenario Three:
The resulted allocations of the vaccines for the third scenario are demonstrated in the following
heatmap:

40

Map 7: Vaccine allocation for the third scenario
The following table shows the average travel times both for cars and transit as well as the average
distance for all scenarios:

Scenario 1 Scenario 2 Scenario 3
Average Travel Time (Bus) (Min) 47.4 48.9 48.5
Average Travel Time (Car) (Min) 16 16.3 16.2
Average Distance (Mile) 9 9.3 9
Table 10: average travel times and distance values for all scenarios
After running the three scenarios, we could observe that the first scenario yielded the least total
costs, as well as better average values for times and distances. However, the higher costs and
slightly longer times and distances could be traded for the purpose of making the locations more
accessible to the people in need the most.

41
5.3 Sensitivity Analysis.
For sensitivity analysis, two types of analysis are conducted as follows:
• Changing the maximum number of sites between 7 and 40.
• Changing the flexibility for the number of sites people can choose from.
These two are done for the first scenario in this section, and the results for the second and third
scenarios can be found in the appendix.
5.3.1 Changing the Maximum Number of Sites.
An important part of the analysis is testing the sensitivity of the model results to changing the
maximum number of allowed vaccine distribution sites. This is implemented by changing the
range of the number of allowed sites between 7 and 40. The output for all iterations for all three
scenarios can be found in the appendix in Table 13 through 15, and it is displayed in the following
charts, showcasing the objective function (total costs), breakdowns of the costs, and the average
travel times in car and bus as well as the average traveled distance for some of the iterations:

Figure 6: Cost types vs number of maximum sites

42
As can be observed in Figure 6, starting from a baseline of 7 sites, the total cost decreases as the
number of sites increases. The cost of opening sites increases linearly as more sites are added,
which is expected since each additional site involves a fixed cost. The bus ticket cost is constant
as the number of trips is unchanged and reflects the number of people who do not own cars
regardless of the number of sites. The travel time cost and the distance cost decrease as more sites
are open, indicating the cost-effectiveness of having more sites that are spread out across the
County. The decrease in total cost, as well as the travel time and distance costs, become less
significant with each additional vaccination site.

Figure 7: Average travel times and distance vs max sites

Similarly, figure 7 shows a trend of reduction in the average travel time by bus and car and the
average distance traveled as the number of sites increases. A significant reduction is observed in
the average travel time for bus users, which emphasizes the importance of opening more sites to
increase accessibility for people who rely on public transit. A less steep decrease is seen for car
users, which is expected since these individuals are generally more mobile and less influenced by

43
the number of available centers. Overall, the results demonstrate the advantage of having more
vaccination sites, showcasing their benefits in terms of time and cost savings for the public.
Figures 12 through 15 shows these charts for sensitivity analysis on scenario 2 and scenario 3,
which can be found in the appendix.
Table 11 shows a comparison of the total costs for different maximum sites inputs for all scenarios
when Pi is the weighting factor for all . As can be observed across all the values of maximum sites,
scenario 1 almost always yields the minimum total costs. In reality, prioritization based on
population might be the most straightforward approach, which ensures maximum number of
people have more access to vaccination centers.

Total Cost Using Pi as the weghiting factor
Maximum Sites Number
Scenario 7 15 20 25 35
1 293,530,630 215,072,181 194,815,902 181,461,824 164,154,071
2 293,530,631 215,873,640 196,402,989 184,380,838 166,859,124
3 296,432,268 219,795,157 200,034,495 186,274,012 168,092,380
Table 11: Total cost for different max sites for all scenarios
Table 12 shows the same comparison when the HPI is used as the weighting factor for the
calculations of the total cost for the solutions. The second scenario in this case always yielded the
minimum total costs. In the real-world context, this approach ensures that areas with less healthy
measures have more access to the vaccines.

Total Cost Using HPI as the weghiting factor

Maximum Sites Number
Scenario 7 15 20 25 35
1 314,211,669 224,258,711 206,463,780 192,624,605 175,583,854
2 314,211,669 223,104,384 204,038,094 190,008,979 172,629,281
3 320,952,051 235,975,067 218,327,102 204,744,962 185,368,606
Table 12: Total cost weighted by HPI for different max sites for all scenarios

44
Table 13 shows the total cost when the COVID-19 vulnerability index is used as the weighting
factor when calculating the total cost for the solutions of each scenario. The table below shows
that scenario 3 yielded the minimum costs for all different maximum site numbers. This approach
would ensure that areas most at risk from the virus are prioritized, which potentially will reduce
the spread and impact of the disease.

Total Cost Using COVID-19 Vulnerqability Index as the weghiting factor

Maximum Sites Number
Scenario 7 15 20 25 35
1 274,597,515 206,453,642 185,640,461 172,862,364 153,968,630
2 274,597,515 208,648,060 190,659,797 179,711,961 161,123,811
3 272,518,937 203,221,875 182,327,957 168,479,526 151,182,940
Table 13: Total cost weighted by COVID-19 Vulnerability Index for different max sites
This sensitivity analysis shows that as the number of sites increases, the total costs decrease,
primarily due to the significant contribution of time and distances costs to the overall cost. These
outputs underscore the trade-offs involved in selecting different priority assignments when
optimizing the vaccine sites, which could justify the less optimal solutions in some instances.
5.3.2 Changing the Number of Sites Residents Can Choose From (F).
The second analysis that is conducted on the model is to change the number of sites each resident
can choose from by changing the parameter F in the model. The model is run with F=1 and F=5 to
test how this change affects the selection of the sites. In map 8 below, we can see the selected
optimal sites when F=1.

45

Map 8: Map of optimal sites when F=1

And the following map shows the sites when F=5.

Map 9: Map of optimal sites when F=5

46
We notice that as the number of options (F) given to the populations increases, the model tends to
select sites that are adjacent to each other, and as the number decreases, the selected locations are
far spread from each other. This trend could be explained as that the model is trying to select as
fewer sites as it could and potentially select a single site if it was possible. Another explanation for
this would be that areas with higher populations are more favorable to be near the optimal
locations, and therefore the model tends to select the number of sites equal to F it was given, all
next to areas that are highly populated.
Similar sensitivity analysis was conducted for scenario 2 and scenario 3, and their resulted maps
(Map 11 through 14) can be found in the appendix section.
Table 12 shows the total weighted costs when F is equal to 1, 3, and 5 for all three scenarios.

Total weighted cost ($)
Flexibility parameter (F)
Scenario 1 3 5
1 122,738,270.51 194,815,903.00 245,332,618.45
2 125,869,410.61 196,402,988.73 248,242,439.57
3 127,534,927.50 200,034,495.03 247,077,458.39
Table 14: Total weighted costs for different F values for all scenarios

As can be observed from the table, the value of the objective function gets worse as the parameter
F increases. Providing more choices might be more convenient for individuals, but it comes with
a cost. The model incorporates this trade-off by allowing the F value to be adjusted, which helps
capture the impact of levels of choice on the overall cost.

47
5.4 Comparison Between the Optimal Sites vs the Actual LA County
Vaccine Sites.
In this section, we will compare the model’s optimal costs for the selected sites and then compare
it to the actual sites that were operational in Los Angeles County. During the early phase of
vaccination, there were 9 locations established to provide the residents of the County with the
vaccine, according to NBC Los Angeles [38]. Those locations were:
1. Dignity Health Sports Park in Carson.
2. Cal State LA.
3. Cal Poly Pomona.
4. Dodger Stadium.
5. Pomona Fairplex.
6. The Forum.
7. Cal State Northridge.
8. Los Angeles County Office of Education.
9. Six Flags Magic Mountain.
The following map shows the locations for these 9 sites:

48

Map 10: Map of the actual vaccine sites in LA county

As can be observed from the map, the distribution sites are scattered across the region. However,
some highly populated areas, particularly in the northern and northeastern parts of LA County,
appear to be situated at long distances from these sites.
To ensure a fair comparison, the proposed model will be applied with a maximum limit of 9 sites,
mirroring the actual number of sites. The model will be executed for both the existing sites as well
as the three proposed scenarios, allowing us to compare the output effectively.
• When F=3:
The following chart shows the values for the average travel times and distance for all scenarios vs
the actual sites, along with the average improvement percentage:

49

Figure 8: Average travel times and distance comparison

As observed from the chart, we can see that the second scenario resulted in the lowest average
times with an average improvement of 17.9% from the actual sites travel times and distance.
Similarly, the three scenarios’ optimal costs were an improvement from the total cost of the actual
sites, with a 17% cost reduction achieved by the first scenario. The following chart shows the total
cost and its breakdown for all the scenarios and the actual sites, along with the savings percentage:

50

Figure 9: Cost types comparison

• When F=1:
Similarly, we compared the three scenarios to the actual sites when F=1. This yielded an even
larger gap between the proposed sites by the model than the true sites selected in LA County,
as can be seen in the following charts that show the comparisons in average travel times and
distance as well as the total costs:

51

Figure 10: Average travel times and distance comparison

Figure 11: Cost types comparison

52
5.5 Limitations

This thesis provided valuable insights into the optimization of vaccine distribution centers and
vaccine allocation. However, it is important to acknowledge certain limitations that may have
limited the work, such as in the estimation of site opening costs, and COVID-19 data availability
on zip codes level.
One of the key assumptions in the model is the cost of opening a vaccination site, which is roughly
estimated to be $500,000. This number is general approximation and does not consider the
variability in actual real estate’s costs across different areas within the LA County. The actual cost
of establishing vaccination sites can vary significantly depending on the location, size, and other
factors such as rental, renovation and operational costs. Therefore, the interpretation of results of
the model should consider this limitation in mind. Future research could build on this aspect by
conducting a more precise cost analysis.
Another limitation in the study is the lack of certain COVID-19 data on the zip code level. Data
on COVID-19 cases and deaths are only available and consolidated on district/community level,
which could not be accurately converted to the zip code level. The model could potentially yield
better results and explore the problem from different angles given the availability of such data.

53
Chapter Six: Conclusion.
6.1 Summary of Findings.
This thesis studied the selection of COVID-19 vaccine distribution centers, allocation quantities,
and associated costs within the County of Los Angeles. The study, using a Mixed Integer
Programming model, explored three different scenarios based on the following prioritization
techniques:
• Priority assignment based on population.
• Priority assignment based on the California Healthy Places Index (HPI).
• Priority assignment based on a COVID-19 Vulnerability index.
The model assumed that up to 20 sites could be selected, with a limited budget of $10M.
The first scenario, using the population as a weighting factor, yielded the lowest costs, which came
to a total of $194,815,902. Moreover, the average travel time by car and transit, as well as the
distance traveled between the selected sites and the 277 zip code, were also the lowest for this
scenario. The average travel times were 47.4 min and 16 min for transit and car users, respectively.
The average distance was 9 miles. Since priorities were given based on populations, the selected
sites appear to be close to the highly populated sides of the County. This suggests that prioritizing
areas with higher populations can lead to be more cost-efficient.
For the second scenario, the optimal weighted costs were $196,402,988. The average travel times
and distance were the highest among the three, with 48.9 min for transit, 16.3 min for car, and 9.3
miles for distance. The selected sites were very close to zip codes with a wide range of HPI
percentiles, from as low as 0.032 to 0.55. This range suggests that the model selected a diverse set
of sites, with the majority being on the lower side of percentiles. This indicates while prioritizing

54
areas based on the HPI can lead to a more diverse locations, it might also result in higher costs and
travel times due to the broader spread of these areas.
The last scenario yielded the highest weighted total cost of all, with a total cost of $200,034,495,
while resulting in average times and distances of 48.5 min for transit, 16.2 min for car, and 9 miles.
This highlights the potential trade-off between prioritizing areas with high vulnerability and the
increase in the associated costs and travel times.
After comparing the results of the three scenarios, it becomes clear that different prioritization
techniques have a significant impact on the outcomes, which highlights the importance of careful
consideration during the deciding-making and planning process.
The sensitivity analysis explored changing the maximum number of sites to up to 40 sites and
changing the number of choices people have for taking the vaccines from a single site to 5 sites.
The findings showed that the total costs decreased as the number of sites increased, primarily due
to the time and distance expenses constituting the bulk of costs. The results also showed that as
the number of site choices (F) decreases, the geographic spread of the sites is increased.
In Conclusion, the developed model in this thesis introduces innovative approaches to solving the
problem of optimizing vaccination centers. The model is designed to account for multiple factors,
such as minimizing the travel time and distance between residential areas and supply points of
limited resources and their associated costs. It also takes into account socioeconomic and
demographic factors, allowing for the assignment of priority to areas based on chosen criteria.
Furthermore, the model assumes that individuals will travel to the closest vaccination site, but in
reality, they may select from multiple sites based on preference. The model gives this flexibility
to the people. However, our findings suggest that this might lead to a less optimal solution

55
accompanied by a decrease in the spread of the selected sites. Finally, the application of the model
to Los Angeles County, which is one of the largest and most diverse counties in the United States,
demonstrates its flexibility and effectiveness.
Overall, our findings highlight the trade-offs involved in the selection of vaccination sites and the
benefits of using an optimization model to inform this process.
6.2 Contributions to the Field.
This research contributes to the ongoing efforts to manage and battle rapid evolving global
pandemics, particularly in managing the selection of distribution centers of scarce resources. By
taking into account various factors for assigning priority, policy and decision-makers can choose
the option that best fits the specific characteristics of the pandemic, the study area and its
populations.
The findings of the research have practical implications that could improve public health
outcomes. They offer valuable insights that can give guidance and recommendations to the
authorities in making informed decisions about the strategic selection of distribution centers and
how to allocate resources. This would lead to better use of resources, reduced travel times for
individuals, and consequently, higher vaccination rates and reduced deaths.
By optimizing the locations of the COVID-19 centers, we hope to save not only resources, but also
lives. Efficient distribution leads to better control of the pandemic, more protected communities,
and a quicker return to normal life and economic activities.

56
6.3 Future Research.
Future research could build on this thesis by incorporating additional factors into the optimization
process of vaccine centers. This could include an in-depth analysis of the staffing requirements,
the process of administering the vaccines and optimizing the capacity of each distribution center.
Additionally, the cost analysis associated with establishing these sites is by itself a substantial area
of study. Future studies could explore this aspect deeper, developing a dynamic approach that
considers all factors involved in estimating the costs of opening health facilities. Examples of those
factors could be real estate costs, construction costs, equipment, and supply costs.
In conclusion, the potential for research in this field is vast, and by exploring those ideas, we can
continue to refine our approaches and improve the efficiency and effectiveness of public health
outcomes.

57
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Appendix
• Sensitivity Analysis Plots for Scenario 2

Figure 12: Cost types vs max sites for scenario 2

Figure 13: Average travel times and distance vs max sites for scenario 2

62

Map 11: Map of selected sites when F=1, scenario 2

Map 12: Map of selected sites when F=5, scenario 2

63
• Sensitivity Analysis Plots for Scenario 3

Figure 14: Cost types vs max sites for scenario 3

Figure 15: Average travel times and distance vs max sites for scenario 2

64

Map 13: Optimal sites for F=1, scenario 3

Map 14: Optimal sites for F=5, scenario 3

65

max_sites Z Total cost
cost of
opening
sites
travel time
cost
distance cost
bus tickets
cost
Avg
TT
Bus
Avg
TT
Car
Avg
Distance
7 293530630 293530630.6 3500000 121517011.4 167027723.2 1485895.97 66.99 21.14 15.35
8 279588541 279588541.3 4000000 116979361.3 157123284.1 1485895.97 67.8 20.11 14.6
9 268525699 268525700 4500000 113075508 149464296 1485895.97 64.55 19.61 13.93
10 257748481 257748481.7 5000000 109408872.6 141853713.1 1485895.97 61.36 19.01 13.24
11 248222650 248222650.9 5500000 106806783 134429972 1485895.97 60.53 18.58 12.67
12 238917314 238917314.4 6000000 104010059.8 127421358.6 1485895.97 57.94 18.12 12.05
13 230139102 230139102.5 6500000 100606998.1 121546208.5 1485895.97 55.47 17.75 11.56
14 221572843 221572843.9 7000000 97661968.69 115424979.3 1485895.97 53.34 17.35 11.09
15 215072181 215072181.7 7500000 96398286.08 109687999.7 1485895.97 52.67 17.28 10.59
16 209845679 209845679.3 8000000 94493547.33 105866236 1485895.97 52.04 16.92 10.27
17 205491821 205491821.5 8500000 93139869.8 102366055.8 1485895.97 50.17 16.69 9.85
18 201617936 201617936.5 9000000 91694275.72 99437764.76 1485895.97 49.11 16.46 9.63
19 198000598 198000598.7 9500000 90670525.99 96344176.73 1485895.97 48 16.25 9.27
20 194815902 194815903 10000000 89647526.18 93682480.82 1485895.97 47.44 15.99 9.01
21 191716863 191716863.9 10500000 88152370.94 91578597.01 1485895.97 47.09 15.75 8.84
22 188998579 188998579.3 11000000 86978129.04 89534554.3 1485895.97 46.12 15.59 8.68
23 186298834 186298834.6 11500000 86232898.41 87080040.25 1485895.97 45.38 15.51 8.47

66

max_sites Z Total cost
cost of
opening
sites
travel time
cost
distance cost
bus tickets
cost
Avg
TT
Bus
Avg
TT
Car
Avg
Distance
24 183851856 183851856.2 12000000 85259681.59 85106278.59 1485895.97 44.71 15.34 8.28
25 181461824 181461824.2 12500000 84437526.85 83038401.34 1485895.97 43.62 15.18 8.04
26 179428133 179428133.1 13000000 83514212.29 81428024.81 1485895.97 43.26 15.1 7.95
27 177369678 177369678.9 13499999.98 82952355.99 79431426.92 1485895.97 42.69 15.03 7.7
28 175430677 175430677.2 14000000 82443665.9 77501115.34 1485895.97 42.26 14.97 7.53
29 173550073 173550073.4 14500000 81753489.44 75810688.03 1485895.97 41.8 14.89 7.41
30 171864627 171864627.8 15000000 80967791.81 74410940.06 1485895.97 41.43 14.77 7.29
31 170209903 170209903.3 15500000 79818782.47 73405224.83 1485895.97 40.71 14.46 7.17
32 168613105 168613105.8 16000000 79126310.24 72000899.56 1485895.97 39.53 14.35 7
33 167027225 167027225.7 16499999.96 79231554.4 69809775.37 1485895.97 39.54 14.41 6.83
34 165564549 165564549.6 17000000 78637194.97 68441458.63 1485895.97 38.54 14.35 6.7
35 164154071 164154071.6 17500000 78089564.11 67078611.5 1485895.97 38.21 14.19 6.54
36 162787398 162787398.8 18000000 76850483.09 66451019.7 1485895.97 37.22 14.02 6.48
37 161525057 161525057.2 18500000 76299826.73 65239334.46 1485895.97 37.13 13.9 6.37
38 160303115 160303115 19000000 75864339.31 63952879.72 1485895.97 36.53 13.82 6.25
39 159110460 159110460.9 19500000 74885096.98 63239467.94 1485895.97 36.5 13.68 6.21
40 158010146 158010146.9 20000000 73951317.96 62572932.97 1485895.97 36.28 13.58 6.17
Table 15: Loop output for scenario 1

67

max_sites Z Total cost
cost of
opening
sites
travel time
cost
distance cost
bus tickets
cost
Avg
TT
Bus
Avg
TT
Car
Avg
Distance
7 314211669 293530630.6 3500000 121517011.4 167027723.2 1485895.97 66.99 21.14 15.35
8 299398750 279985724.3 4000000 117398384.2 157101444.2 1485895.97 67.41 20.34 14.6
9 286479444 269083845.3 4500000 113650807.8 149447141.6 1485895.97 64.37 19.61 13.88
10 274343273 258398982.1 5000000 109708996.7 142204089.4 1485895.97 61.11 19.15 13.31
11 262678555 249923842.2 5500000 106862097 136075849.2 1485895.97 58.83 18.76 12.81
12 251048330 241153704.3 6000000 103915245.4 129752563 1485895.97 56.58 18.35 12.32
13 240835696 231661498.8 6500000 101346781 122328821.8 1485895.97 55.86 17.93 11.75
14 231018316 221820172.4 7000000 98190615.63 115143660.8 1485895.97 54.17 17.44 11.09
15 223104384 215873639.5 7500000 96078846.63 110808896.9 1485895.97 52.62 17.31 10.74
16 218298607 212382499.6 8000000 95281388.9 107615214.7 1485895.97 52.19 17.28 10.56
17 214149173 207591078.3 8500000 93908006.21 103697176.1 1485895.97 50.94 17.04 10.22
18 210609335 203609589.5 9000000 92950486.92 100173206.6 1485895.97 50.32 16.87 9.86
19 207305505 200106839.5 9500000 91642610.01 97478333.56 1485895.97 49.6 16.66 9.6
20 204038094 196402988.7 10000000 90351438.72 94565654.05 1485895.97 48.86 16.34 9.29
21 200899577 194043569.9 10500000 89475241.72 92582432.23 1485895.97 48.08 16.24 9.13
22 197902306 192108864.1 11000000 88993072.58 90629895.53 1485895.97 47.9 16.28 9
23 195081310 189017913.2 11500000 87610028.22 88421989.02 1485895.97 46.66 16.07 8.74

68

max_sites Z Total cost
cost of
opening
sites
travel time
cost
distance cost
bus tickets
cost
Avg
TT
Bus
Avg
TT
Car
Avg
Distance
24 192375526 186415764.6 12000000 86576423.75 86353444.82 1485895.97 45.89 15.91 8.56
25 190008978 184380838.4 12500000 85953573.89 84441368.56 1485895.97 45.55 15.84 8.4
26 187780472 181994637.6 13000000 84994226.71 82514514.94 1485895.97 45.09 15.68 8.22
27 185755249 179345970.7 13500000 83255004.9 81105069.84 1485895.97 44.04 15.33 8.05
28 183844659 176846056.5 14000000 82155139.58 79205020.99 1485895.97 43.55 15.13 7.88
29 181987018 175677568.6 14500000 81277567.87 78414104.71 1485895.97 43.21 15.02 7.84
30 180132736 173645993.6 15000000 80264633.03 76895464.56 1485895.97 42.25 14.77 7.63
31 178398604 172102178.9 15500000 79686707.96 75429574.95 1485895.97 42.21 14.71 7.52
32 176821580 170302270.2 16000000 78866442.07 73949932.14 1485895.97 41.65 14.56 7.37
33 175309097 168925657.8 16500000 78436800.24 72502961.55 1485895.97 41.63 14.48 7.24
34 173969220 168091427.8 17000000 77888860.64 71716671.22 1485895.97 40.97 14.31 7.1
35 172629280 166859124 17500000 77638292.11 70234935.94 1485895.97 40.37 14.23 6.96
36 171310679 165676178.3 18000000 77093154.74 69097127.61 1485895.97 39.89 14.15 6.85
37 169991849 164653428.8 18499999.97 76568110.01 68099422.8 1485895.97 39.52 14.1 6.78
38 168689571 163130837.9 19000000 75584774.95 67060166.99 1485895.97 38.9 13.92 6.66
39 167464712 161621325.6 19500000 74774834.74 65860594.87 1485895.97 38.46 13.78 6.53
40 166326531 160656541.6 20000000 74223757.9 64946887.74 1485895.97 37.86 13.71 6.46
Table 16: Loop output for scenario 2

69

max_sites Z Total cost
cost of
opening
sites
travel time
cost
distance cost
bus tickets
cost
Avg
TT
Bus
Avg
TT
Car
Avg
Distance
7 272518936 296432268.1 3500000 123239473.6 168206898.5 1485895.97 66.04 21.35 15.33
8 259432668 282696790.6 4000000 119137094.2 158073800.5 1485895.97 66.62 20.44 14.52
9 247834993 270909716 4500000 114733249.1 150190570.9 1485895.97 67.25 19.57 13.89
10 239132641 261339460.2 5000000 112140280.9 142713283.3 1485895.97 66.51 19.13 13.32
11 231007504 250628829.2 5500000 108569289.8 135073643.5 1485895.97 63.5 18.6 12.67
12 222888181 242435503.4 6000000 107018225.1 127931382.3 1485895.97 60.96 18.44 11.99
13 215949395 234976532.5 6500000 104491506.8 122499129.8 1485895.97 58.6 18.02 11.47
14 209284336 226204524.5 7000000 101553045.7 116165582.8 1485895.97 56.33 17.62 10.98
15 203221875 219795156.9 7500000 99168651.23 111640609.7 1485895.97 54.22 17.31 10.53
16 198213948 215842409.4 8000000 97524969.6 108831543.8 1485895.97 52.53 16.96 10.16
17 193748988 210566254.3 8500000 95457957.96 105122400.4 1485895.97 50.79 16.71 9.88
18 189552500 206187462.2 9000000 94045187.68 101656378.6 1485895.97 50.2 16.5 9.61
19 185816922 202458300.1 9500000 93563661.08 97908743.01 1485895.97 49.9 16.41 9.27
20 182327956 200034495 10000000 92633900.93 95914698.13 1485895.97 48.53 16.2 9.01
21 179071687 195771582.1 10500000 91241544.25 92544141.92 1485895.97 47.1 16.04 8.77
22 176161853 193190691 11000000 90272135.57 90432659.47 1485895.97 46.34 15.88 8.55
23 173470468 191917763.9 11500000 89914408.69 89017459.24 1485895.97 45.93 15.77 8.42

70

max_sites Z Total cost
cost of
opening
sites
travel time
cost
distance cost
bus tickets
cost
Avg
TT
Bus
Avg
TT
Car
Avg
Distance
24 170843273 190025850.9 12000000 88981869.2 87558085.73 1485895.97 45.58 15.55 8.25
25 168479525 186274011.9 12500000 87048394.54 85239721.36 1485895.97 45.1 15.23 8.08
26 166322155 184806913.4 13000000 86492379.48 83828637.9 1485895.97 44.34 15.23 7.92
27 164269358 183621917.9 13500000 86173194.03 82462827.9 1485895.97 44.15 15.14 7.77
28 162363575 181269222.8 14000000 85160921.11 80622405.74 1485895.97 42.88 15 7.63
29 160533065 179882639.4 14500000 84231276.77 79665466.65 1485895.97 42.42 14.79 7.51
30 158789947 178072999.3 15000000 83562419.15 78024684.19 1485895.97 41.75 14.78 7.36
31 157065371 176182611.6 15500000 82719288.68 76477426.95 1485895.97 41.28 14.62 7.21
32 155461116 173905567.7 16000000 81815308.17 74604363.58 1485895.97 40.8 14.54 7.08
33 153979662 171141715.7 16500000 80499124.44 72656695.28 1485895.97 40.27 14.39 6.97
34 152495413 169354502.4 17000000 79723248.13 71145358.29 1485895.97 39.76 14.26 6.85
35 151182940 168092379.8 17500000 79289602.92 69816880.91 1485895.97 38.95 14.19 6.73
36 149871223 166508626.5 18000000 78681406.59 68341323.9 1485895.97 38.23 14.16 6.59
37 148704289 165331589.6 18500000 77707086.49 67638607.16 1485895.97 37.86 13.99 6.52
38 147571611 163438406.2 19000000 76710982.62 66241527.57 1485895.97 37.51 13.84 6.42
39 146468145 161467432.8 19500000 75633591.22 64847945.63 1485895.97 37 13.72 6.32
40 145406585 160710235.1 20000000 75163378.84 64060960.29 1485895.97 36.33 13.62 6.22
Table 17: Loop output for scenario 3

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Asset Metadata

Creator Aljohani, Basim (author)

Core Title Optimizing the selection of COVID-19 vaccine distribution centers and allocation quantities: a case study for the county of Los Angeles

School Viterbi School of Engineering

Degree Master of Science

Degree Program Industrial and Systems Engineering

Degree Conferral Date 2023-08

Publication Date 08/14/2023

Defense Date 07/25/2023

Publisher University of Southern California. Libraries (digital)

Tag allocation,COVID-19,location,OAI-PMH Harvest,optimization,transportation,travel time,vaccines

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Hall, Randolph (committee chair), Gomez, Andres (committee member), Suen, Sze-chuan (committee member)

Creator Email baljohan@usc.edu;basim.aljohani93@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC113298004

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Legacy Identifier etd-AljohaniBa-12248

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Rights Aljohani, Basim

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