Optimization Modeling for Pandemic Vaccine Supply Chain Management: A Review and Future Research Opportunities

Abstract

During various stages of the COVID-19 pandemic, countries implemented diverse vaccine management approaches, influenced by variations in infrastructure and socio-economic conditions. This article provides a comprehensive overview of optimization models developed by the research community throughout the COVID-19 era, aimed at enhancing vaccine distribution and establishing a standardized framework for future pandemic preparedness. These models address critical issues such as site selection, inventory management, allocation strategies, distribution logistics, and route optimization encountered during the COVID-19 crisis. A unified framework is employed to describe the models, emphasizing their integration with epidemiological models to facilitate a holistic understanding. This article also summarizes evolving nature of literature, relevant research gaps, and authors’ perspectives for model selection. Finally, future research scopes are detailed both in the context of modeling and solutions approaches.

1. Introduction

The COVID-19 pandemic emerged as the largest global health crisis in recent history, with an staggering incidence of over 750 million reported cases and death toll approaching seven million individuals world-wide (World Health Organization 2023). Beyond the profound health implications, the pandemic has presented an array of multifaceted challenges encompassing social, technological, economic, environmental, political, legal, and ethical domains (Sampath et al. 2021, Taskinsoy 2020, Klemeš et al. 2020, Shah et al. 2021, Saberi Isfeedvajani et al. 2020). In response, numerous researchers have delved into crucial issues such as resource allocation for essential care resources like ventilators and hospital beds/capacity (Mehrotra et al. 2020, Bozkir et al. 2023, Yang et al. 2021), as well as investigating the impact of community mitigation strategies like social distancing and mask-wearing (Keskinocak et al. 2020, Gel et al. 2020) during the initial stages of the pandemic. However, with the rapid development of highly effective vaccines, the research efforts have progressively shifted towards studying operational dilemmas related to vaccine allocation and distribution.

The COVID-19 pandemic underscored the significance of timely and widespread vaccine deployment, particularly when highly effective vaccines are shown to prevent infections and hospitalizations to achieve herd immunity against novel pathogens (MacIntyre et al. 2022). However, the operational challenges associated with rapidly developed vaccines, such as those created and approved for COVID-19, necessitate careful consideration to ensure efficient, effective, and equitable deployment (Huang et al. 2012). Primarily, vaccine availability is typically limited, especially in the early stages of production, requiring prioritized allocation of scarce doses among various subgroups and geographical regions. Additionally, the diverse range of vaccines developed by different manufacturers offer varying levels and types of immunity against different pathogen strains. These vaccines also present different storage requirements (shelf life, doses per vial, cold storage, etc.) and dosing protocols (single or multiple shots), thereby complicating transportation, allocation, and distribution logistics. From a global perspective, addressing these challenges requires tailored solutions in different regions due to factors such as varying vaccine availability, healthcare infrastructure, socioeconomic pressures, education levels, and risk perceptions among citizens. Moreover, different countries may prioritize subpopulations and employ conflicting economic measures, leading to the adoption of diverse vaccine deployment and delivery strategies (Saadi et al. 2021).

In response to the various operational challenges encountered during the COVID-19 pandemic, the Operations Research/Management Science (OR/MS) community has generated a significant body of research comprising modeling and optimization studies, introducing novel dimensions and capabilities to address operational problems in this domain. Some of these optimization models incorporate dynamic considerations of disease transmission and infection stages, enabling accurate estimation of population-level vaccination requirements. Considering this diverse range of work on optimization models for pandemic vaccine management, this literature review offers a comprehensive overview of these papers using a systematic framework for the first time in COVID-19 literature. The primary objectives of this review are to (i) advocate for the adoption of such models in decision-making processes by public health officials in future pandemics and (ii) identify research gaps that hinder ongoing pandemic preparedness efforts undertaken by our nation and various global agencies. Furthermore, we categorize solution methodologies used in literature, encompassing heuristics, global and local optimization techniques, followed by research questions for operations research/management scientists applicable beyond pandemic vaccine deployment. Real life interpretation and importance of such models are also demonstrated. Thus, this review work provides a comprehensive view of pandemic vaccine supply chain models, spanning from their conceptualization to execution and fostering connections among interdisciplinary communities.

1.1. Search Methodology and Organization

Our search methodology involved identifying relevant articles published or made available online between the years 2020 and 2023, employing various combinations of keywords listed in the first, second, and third columns of Table 1. Through this approach, we identified a total of 109 articles that met the inclusion criteria. Additionally, to ensure comprehensive coverage of pertinent topics, we incorporated a selection of highly cited articles published between 2011 and 2019, as they offered valuable insights into issues of interest such as equitable vaccine allocation, vial size, and the vaccine supply chain. Figure 3 in the Appendix outlines the paper selection and classification process as a flow chart.

Table 1.

List of keywords used in the initial search.

Keyword Set 1	Keyword Set 2	Keyword Set 3
coronavirus vaccine	allocation	mathematical modelling
COVID-19 vaccine	distribution	mathematical programming
influenza vaccine	equity	network
pandemic vaccine	facility location problem	optimization
vaccination	inventory	optimization model
	supply chain management	simulation

Figure 3.

Flowchart of the process used to select and classify papers on pandemic vaccine supply chain management.

The articles covered in this review can be categorized into three main groups (i) Optimization articles, which encompass resource allocation, location-allocation, inventory management, supply chain management, and routing problems, employing at least one mathematical optimization model. Some of these articles incorporate simulation and compartmental epidemiological dynamics models, such as Susceptible-Infected-Recovered (SIR) models, integrated with the optimization models. (ii) Simulation-based articles, which focus on prioritization, management of multiple vaccine, and inventory control, utilizing simulation modeling to compare various scenarios or interventions of interest. These articles do not include an optimization model but rely on simulation-based approaches. (iii) Other articles, involving descriptive analysis, conceptualization, vaccine hesitancy, and waste management. These articles do not involve optimization or simulation models but offer valuable insights to guide future research in this field. In total, our review covers 77 optimization model-based articles, 17 simulation-based articles, and 15 articles employing alternative approaches. Notably, 76 out of the 94 articles involving optimization and simulation modeling specifically pertain to COVID-19 vaccine studies, highlighting the significant recent interest in pandemic vaccine management prompted by the COVID-19 pandemic.

The subsequent sections of this paper are organized as follows: Section §2 presents the vaccine supply chain framework along with the nomenclature adopted throughout the paper. In Sections §3, and §4, we propose a comprehensive optimization modeling framework for COVID-19 vaccine allocation and distribution management. These sections respectively address supply chain and location problems using mixed-integer linear programming techniques. Section §5 reviews the optimization technique found in vaccine logistics literature in presence of uncertainties. Additionally, Section §6 provides an overview of how equity and sustainability are incorporated into the models. In Section §7, we outline a generalized epidemiological modeling framework utilizing an extended version of the compartmental SIR model called DELPHI-V and present a framework for integrating optimization and epidemiological models. In Section §8, we provide a classification of methodological considerations and highlight the important insights from case studies. In Section §9, we offer an overall discussion of the literature as well as our own perspectives for model selection. In Section §10, we provide a critique and suggestions for future research directions. Appendix §A covers routing problems. Finally, Appendix §B covers simulation-based and other relevant articles in alignment with the optimization-based articles.

2. Vaccine Supply Chain Framework and Model Notations

The vaccine supply chain comprises of four primary echelons: manufacturers (M), distribution centers (DC), vaccination centers (VC), and population demand (PS), as depicted in Figure 1. Production facilities can be located domestically or internationally, catering to multiple distribution centers (DCs) or warehouses at various levels, such as national, state, city, regional, or district levels. The fundamental unit of inventory is the vaccine vial, which is typically transported in refrigerated trucks to numerous distribution centers (DCs) that serve geographically dispersed vaccination centers (VCs). These vaccination centers (VCs) encompass hospitals, community health centers, pharmacies, and mobile vaccination clinics. Depending on vaccine characteristics and availability, certain subpopulations may be prioritized for vaccination based on factors like age, occupation, or risk level. It is important to note that vaccine vials can contain multiple doses, necessitating the consideration of shelf-life and the duration for which an open vial remains viable. In this paper, we assume that each vial contains a single vaccine dose. Furthermore, the paper employs four distinct time frames: ordering time, receiving time, opening time, and vaccination time. The primary types of decisions addressed in the surveyed papers are as follows:

Figure 1.

Vaccine Supply Chain

• Location: Decisions regarding locating facilities, such as mass vaccination centers.

• Inventory: Decisions on managing various aspects of vaccine inventory.

• Allocation: Joint decisions on vaccine allocation and assignment of demand nodes to locations.

• Distribution: Decisions regarding distribution of vaccines to demand nodes.

• Routing: Decisions regarding the management of vehicles/mobile vaccination clinics visiting depots, vaccination centers (VCs), etc.

Tables 3, 4, and 5 provide the index sets, parameters, and decision variables employed in our overarching optimization model. We strongly suggest the reader to refer to these tables when reading the technical expressions appearing in this review, since they are not necessarily described in the main text. Note that, in the above notation convention, the subscripts refer to the indexes. Letters $M,D,m,d,v$, etc. are used as superscripts in many variables or parameters to highlight their association with the manufacturer (M), distribution center (DC), and vaccination center (VC), respectively. The manufacturer set $ℳ$ can also be defined as $\{\mathrm{1,\; 2},\cdots ,|ℳ\left|\right\}$ where $|\cdot |$ is its cardinality. All sets are defined using calligraphic letters. As necessary, we also interchangeably use $[\cdot ]$ to represent an index set with the entry ‘·’ representing the last entry of that set starting from 1. Decision variable types are indicated with letters B (binary), C (continuous) and I (integer). I/C refers to decision variables that should have been integer but were treated as continuous to reduce complexity.

Table 3.

Sets

Notation	Description
$ℳ$	Set of manufacturers (M)/suppliers/vendors, indexed by $i$.
$𝒩$	Set of distribution centers (DC), indexed by $j$.
$𝒱$	Set of vaccination centers (VC), indexed by $k$.
${𝒱}_j$	Set of vaccination centers (VC) that can be served by distribution center (DC) $j$, indexed by $k$.
$𝒢$	Set of risk-based population groups, indexed by $g$.
$P$	Set of different types of vaccines, indexed by $p$.
${𝒫}_i$	Set of vaccine types that can only come from manufacturer (M) $i$, indexed by $p$.
$𝒯$	Set of discrete time periods, indexed by $t$.
$\mathscr{H}$	Set of vehicles, indexed by $h$.
$ℛ$	Set of regions, indexed by $r$ or $r^{\prime }$.
$𝒮$	Set of population sites (PS), indexed by $s$.
${𝒬}_s$	Set of coverage levels to serve population sites (PS)/villages $s$, indexed by $q$.
$𝒪$	Set of outreach centers/teams, indexed by $\ell$.
$\mathrm{\Omega }$	Set of scenarios representing uncertainty, indexed by $\omega$.
$ℬ$	Ambiguity set in distributionally robust optimization (DRO).

Table 4.

Parameters

Notation	Description	Equations
Capacity
$C_{ipt}^M$	Production capacity of manufacturer $i$ at time period $t$ for type $p$ of vaccine.	1
$C_i^D$	Capacity of distribution center (DC) j.	8
$C_k^V\left(C^V\right)$	Capacity of vaccination center (VC) $k$ (Overall capacity/availability over all VCs).	34,63
$C^F$	Fleet size availability.	15
$S_j^D$	Safety stock level of distribution center (DC) $j$.	8
$C_{\ell }^O$	Capacity of outreach center (OC) $\ell$.	37
$C_h^H$	Capacity of vehicle $h$.	64
$C_j^{D(c)}$	Cold refrigeration capacity of distribution center (DC) $j$.	10
$C_j^{D(vc)}$	Very cold refrigeration capacity of distribution center (DC) $j$.	11
$C_j^{D(uc)}$	Ultra cold refrigeration capacity of distribution center (DC) $j$.	11,12
Cost
$C_{kp}^{SH}$	Shortage cost for vaccine $p$ at vaccination center (VC) k.	In Text
$C^{EI}$	Environmental impact cost for establishing a facility.	48
$C^{CR}$	Amount of carbon emmission by transshipping a unit of vaccine.	48
$C_k$	Cost of locating vaccination center (VC) at $k$.	35,36
b	Total budget.	35,36
Demand
$d_{gkpt}$	Demand of vaccine type $p$ among population group $g$ at location $k$ in time period $t$.	28,29,30,31,32,33,44,45
$d_{kpt}$	Demand for vaccine type $p$ at vaccination center (VC) $k$ at time period $t$.	16,17
$d_{rg}$	Demand of region $r$ for risk group $g$.	64,46,47
$d_j^D$	Demand (average demand rate) of distribution center (DC) $j$.	38
$\gamma$	Minimum demand satisfaction ratio.	44,45
$d_s^P$	Demand of population site (PS) $s$.	34
$P_s$	Coverage coefficient at population site (PS) $s$.	35,36
${\theta }_q$	Coverage fraction attainable when the nearest serving center is between ${\mathfrak{y}}_{q-1}$ and ${\mathfrak{y}}_q$.	36
${\hat{P}}_r\left({\hat{P}}_s\right)$	Population size of region $r$ (of population site (PS) $s$).	63
Inventory
$l_j$	Lead time to deliver order to distribution center $(DC)j$.	38
$c_{ij}^{OC}$	Ordering cost to order from manufacturer (M) $i$ by distribution center (DC) $j$.	38
$c_j^{HC}$	Holding cost to hold inventory at facility $j$.	38
Logistics
${\mathrm{\delta }}_i^{(vc)}\left({\mathrm{\delta }}_p^{(vc)}\right)$	1, if vaccine coming from manufacturer $i$ (type p) needs very cold or ultra cold refrigeration.	10,11,12,13
${\mathrm{\delta }}_i^{(uc)}\left({\mathrm{\delta }}_p^{(uc)}\right)$	1, if vaccine coming from manufacturer $i$ (type p) needs ultra cold refrigeration.	11,12,13
$a_{kj}$	1, if vaccination center (VC) $k$ is feasible to be assigned to distribution center (DC) $j$.	18
$a_{sk}$	1, if population site (PS) $s$ is feasible to be assigned to vaccination center (VC) $k$.	34
${\eta }_{ab}$	Distance from node $a$ to node $b$ (general index notation $a$ and $b$).	34,35,36,37,48
${\mathrm{\tau }}_{r{r}^{\prime }}$	Travel time from region $r$ to region $r^{\prime }$.	64
${\mathrm{\tau }}_r^S$	Service time of vehicle (mobile vaccination clinic) at region $r$.	64
${\eta }_{\mathrm{m}\mathrm{a}\mathrm{x}}$	Maximum allowable distance.	35,37
${\mathfrak{y}}_q$	Distance at coverage level $q$	37
$e_{rg}^v$	Estimated equitable quantity that should be allocated to region $r$ for priority group $g$.	47
Others
$\varsigma$	Weight such that $0\le \varsigma \le 1$.	46
M	Big-M value.
$\check{\mathbf{M}}$	Context specific penalty term (not necessarily a large value) or importance factor.	41,46
Vaccine
${\mathrm{\lambda }}^{sl}$	Vaccine vial shelf life.	20,21,22,23,24
$\mathrm{\tau }$	Vaccine safe-use-life after opening the vial.	26,27,28,29
${\iota }_v$	Number of vaccines contained in a vial size $v\in 𝒱$.	25
${\iota }_{vp}$	Number of vaccines contained in a vial size $v\in 𝒱$, for group $p$.	26,27
Workforce
$C^{HW}$	Number of people a health worker can serve within time t.	19
$C_{kt}^{EH}$	Existing workforce in vaccination center (VC) $k$ in time $t$.	19
Uncertainty Modeling
$\alpha$	Significance level such that $0\le \alpha \le 1$.	38,43
${\mathrm{Z}}_{1-\alpha }$	Z score of standard normal distribution at $1-\alpha$ confidence level.	38,43
$\mu (\sigma )$	Mean (standard deviation) estimate of an uncertain parameter.	38,42,43
$\epsilon$	Ambiguity in empirical estimate of a parameter vaccination center (VC) $k$ in time $t$.	42
$\mathrm{\epsilon },\overline{\mathrm{\epsilon }}$	Factor to scale an ambiguously estimated parameter such that $0\le \mathrm{\epsilon }\le 1\le \epsilon$.	42
${\mathrm{p}}_{\omega }$	Probability that a scenario $\omega \in \mathrm{\Omega }$ would happen.	41

Table 5.

Decision Variables

Notation	Description	Type	Equations
Allocation
$x_{jk}$	1, if vaccination center (VC) $k$ is assigned to distribution center (DC) $j$.	B	18,45
$x_{sk}$	Assignment variable (or percentage, Continuous). Assigning population site (PS) $s$ to vaccination center (VC) $k$.	$\mathrm{B}/\mathrm{C}$	34
${\mathfrak{z}}_s$	1, if population site (PS) $s$ is covered by a facility for vaccinating its population.	B	35
${\mathfrak{z}}_{sq}$	1, if population site (PS) $s$ is covered by a facility at its distance based level $q\in Q_s$.	B	36,37
$y_{jkt}^{dv}$	1, if supply happens to vaccination center (VC) $k$ by distribution center (DC) $j$ at time period $t$.	B	In Text
$x_{gkpt}^v$	Number of people of group $g$ vaccinated by vaccine type $p$ at vaccination center (VC) $k$ during time period $t$.	$\mathrm{I}/\mathrm{C}$	16,19,25,44
$x_{kpt}^v$	Number of people vaccinated by vaccine type $p$, at vaccination center (VC) $k$ during time $t$.	$\mathrm{I}/\mathrm{C}$	17
$x_{gkpt{t}^{\prime }}^v$	Number of people of group $g$ vaccinated by vaccine type $p$ at vaccination center (VC) $k$ during time period $t^{\prime }$ from a vial opened at time period $t$.	$\mathrm{I}/\mathrm{C}$	26,27,28,29
$y_{gkpt}^v$	Number of vaccine type $p$ available to fulfill demand or exactly meeting demand of priority group $g$ in vaccination center (VC) $k$ at time period $t$.	$\mathrm{I}/\mathrm{C}$	30,31,32,33
$s_{gkpt}^v$	Amount of shortages of vaccine type $p$ in vaccination center $(\mathrm{V}\mathrm{C})k$ at time period $t$ by group $g$.	$\mathrm{I}/\mathrm{C}$	29,48
$s_{kpt}^v$	Amount of shortages of vaccine type $p$ in vaccination center $(\mathrm{V}\mathrm{C})k$ at time period $t$.	$\mathrm{I}/\mathrm{C}$	17
$L_{kpt{t}^{\prime }}^v$	Number of type $p$ vaccine vials allocated to vaccination center (VC) $k$ at time $t$ and opened in time $t^{\prime }$.	$\mathrm{I}/\mathrm{C}$	20,21,22,23
$N_{kp{t}^{\prime }}^v$	Number of type $p$ vaccine vials opened at vaccination center $(\mathrm{V}\mathrm{C})k$ in time $t^{\prime }$.	$\mathrm{I}/\mathrm{C}$	22,23
$N_{vkp{t}^{\prime }}^{v{r}^{\prime }}$	Number of type $p$ vaccine vial of size $v$ opened at vaccination center (VC) $k$ in time $t^{\prime }$.	$\mathrm{I}/\mathrm{C}$	25,26,27
${\mathfrak{x}}_{\ell k}$	1, if vaccination center $(\mathrm{V}\mathrm{C})k$ served by an outreach center/team $\ell \in O$.	B	37
$A_{kp0t^{\prime }}^v$	Number of type $p$ vaccine vials available in vaccination center (VC) $k$ at initial time period and opened in time $t^{\prime }$.	$\mathrm{I}/\mathrm{C}$	23,24
${\mathrm{\Delta }}_{rg}^{+},{\mathrm{\Delta }}_{rg}^{-}$	Positive and negative deviation from a quantity particular to a region $r$ for group $g$.	$\mathrm{C}$	46,47
${\mathrm{\Pi }}_{rg}$	Positive difference between minimum percentage of demand to be met and allocation in region $r$ for group $g$.	$\mathrm{C}$	46
$f_r\left(f_g\right)$	Fraction of total population in region $r$ (of risk group g).	C	63
Inventory
$I_j^t$	Inventory available at distribution center (DC) $j$ at the end of time period $t$.	$\mathrm{I}/\mathrm{C}$	3,4,8,9
$I_k^t$	Inventory available at vaccination center $(\mathrm{V}\mathrm{C})k$ at the end of time period $t$.	$\mathrm{I}/\mathrm{C}$	16,30,31,32,33
$I_{kp}^0$	Number of type $p$ vaccine vials available at initial time period at vaccination center (VC) $k$.	$\mathrm{I}/\mathrm{C}$	24
$Q_j$	Order quantity of distribution center (DC) $j$.	$\mathrm{I}/\mathrm{C}$	38
Location
$\left.Y_j^{D(c)}\right)$	1, if distribution center (DC) at location $j$ with usual cold storage facility $c$ is launched.	$\mathrm{B}$	10,13,14,48
$Y_j^{D(vc)}$	$1$, if distribution center (DC) at location $j$ with very cold storage facility $vc$ is launched.	B	13,14,48
$Y_j^{D(uc)}$	$1$, if ultra cold storage facility $uc$ is added to an existing distribution center (DC) $j$.	B	11,12,13,48
$Y_j^D$	$1$, if distribution center (DC) $j$ having only usual cold storage facility $j$ is launched.	B	5,6,7,8,48
$Z_k^V$	$1$, if vaccination center $(\mathrm{V}\mathrm{C})$ is established.	B	34,35,36,48
Routing
$y_{r{r}^{\prime }}$	1, if region $r$ and $r^{\prime }$ are served in order in a vehicle routing problem.	B	64
$z_{rh}$	$1$, if region $r$ is served by vehicle $h$.	B	64
$T_r$	Service start time of region $r$.	$\mathrm{C}$	64
$Z_r^R$	$1$, if region $r$ is visited by a vehicle.	B	In Text
Transshipment
$x{x}_{ijpt}^{md}$	Number of vaccine-type $p$ arriving from manufacturer $i$ to distribution center (DC) $j$ at the beginning of time period $t$.	$\mathrm{I}/\mathrm{C}$	1,9,48
$x_{ijpt-l}^{mm}$	Number of vaccine-type $p$ arriving from manufacturer $i$ to distribution center (DC) $j$ at the beginning of time period $t$, ordered at $t-l$.	$\mathrm{I}/\mathrm{C}$	3,6
$x_{ij{t}^{\prime }t}^{mb}$	Amount of vaccines shipped from manufacturer $i$ to distribution center (DC) $j$, ordered at time $t^{\prime }$, received at time $t$.	$\mathrm{I}/\mathrm{C}$	10,11,12,14
$x_{jkpt}^{dv}\left(x_{jkpth}^{dv}\right)$	Number of vaccine type $p$ transferred from distribution center (DC) $j$ to vaccination center (VC) $k$ at time $t$ (using vehicle $h$).	$\mathrm{I}/\mathrm{C}$	3,4,,5,9,10,11,12,13,15,16,18,20,21,30,31,32,33,43,48
$x_{j{j}^{\prime }pt}^{d{d}^{\prime }}$	Number of vaccine type $p$ transshipped from distribution center (DC) $j$ to another distribution center (DC) $j^{\prime }$ at time period $t$.	$\mathrm{I}/\mathrm{C}$	3,4,5,48
$x_{j^{\prime }jpt}^{d^{\prime }d}$	Number of vaccine type $p$ arriving to distribution center (DC) $j$ from another distribution center (DC) $j^{\prime }$ via transshipment at time period $t$.	$\mathrm{I}/\mathrm{C}$	3,4,6,7
$x_{k{k}^{\prime }pt}^{v{v}^{″}}$	Number of vaccine type $p$ transshipped from vaccination center $(\mathrm{V}\mathrm{C})k$ to another vaccination center $(\mathrm{V}\mathrm{C})j^{\prime }$ at time period $t$.	$\mathrm{I}/\mathrm{C}$	30,31,48
	Number of vaccine type $p$ arriving to vaccination center (VC) $k$ from another vaccination center (VC) $k^{\prime }$ via transshipment at time period $t$.	$\mathrm{I}/\mathrm{C}$	30,31
$X_{it}^m$	$1$, if selection of manufacturer $i$ at time $t$ for vaccine production occurs.	B	1
$X_{ip{t}^{\prime }t}^m$	1, if vaccine type $p$ ordered from manufacturer $i$ at time $t^{\prime }$ to be delivered to distribution center (DC) at the beginning of time $t$.	B	2
Vaccine Wastage
$w_{jpt}$	Wasted amount of type $p$ vaccine vial at distribution center (DC) $j$ at time $t$.	$\mathrm{I}/\mathrm{C}$	3,4,5,9
$w_{kpt}$	Wasted amount of type $p$ vaccine vial at vaccination center (VC) $k$ at time $t$.	$\mathrm{I}/\mathrm{C}$	16,20,21,30,31,32,33
$W_{kp0t^{\prime }}^v$	Amount of type $p$ vaccine vials available in vaccination center (VC) $k$ at initial time period and wasted in time $t^{\prime }$ due to time expiration.	$\mathrm{I}/\mathrm{C}$	24
$w_{kpt}^{ov}$	Wasted dosage of vaccine type $p$ in vaccination center (VC) $k$ at time period $t$ from opened vial $ov$.	$\mathrm{I}/\mathrm{C}$	25,28
Workforce
$F_{kt}^R$	Required number of health care worker in vaccination center $(\mathrm{V})k$ at time period $t$.	$\text{I}$	19
$F_{kt}^A$	Additional healthcare workers required in vaccination center (VC) $k$ at time period $t$.	I	19
Uncertainty Modeling
${\mathrm{p}}_{\omega }$	Probability that a scenario $\omega \in \mathrm{\Omega }$ would happen.	$\mathrm{C}$	42

Table 6 classifies the articles based on the specific type of decision being modeled, such as location, inventory, allocation, distribution, or routing. It also indicates whether a compartmental Susceptible-Infected-Recovered (SIR) type epidemiological model was utilized to capture the dynamic nature of the infectious disease being studied. It is worth noting that papers incorporating a SIR-type model often consider demand uncertainty within the modeling framework.

Table 6.

Decisions considered in each optimization article.

	Type of Decision
Article	Location	Inventory	Allocation	Distribution	Routing	SIR
Supply Chain Models (See Section §3)
Resource Allocation
Abbasi et al. (2020)			✓	✓
Anahideh et al. (2022)			✓
Balcik et al. (2022)			✓	✓
Bandi et al. (2021)			✓	✓
Barth et al. (2023)			✓			✓
Bennouna et al. (2022)			✓	✓		✓
Bertsimas et al. (2020)			✓			✓
Enayati and Özaltın (2020)			✓	✓		✓
Hu et al. (2023)			✓	✓		✓
Huang et al. (2017)			✓
Jadidi et al. (2021)			✓			✓
Jarumaneeroj et al. (2022)			✓			✓
Li and Aprahamian (2023)			✓	✓		✓
Minoza et al. (2021)			✓	✓		✓
Munguía-López and Ponce-Ortega (2021)			✓
Orgut et al. (2023)			✓
Rao and Brandeau (2021)			✓			✓
Roy et al. (2021)			✓			✓
Shukla et al. (2022)			✓	✓
Thul and Powell (2023)			✓			✓
Yang et al. (2022)			✓			✓
Yarmand et al. (2014)			✓
Inventory Management
Azadi et al. (2020)		✓		✓
Azadi et al. (2020)		✓
Bonney and Jaber (2011)		✓
Fadaki et al. (2022)		✓	✓	✓
Georgiadis and Georgiadis (2021)		✓	✓	✓
Hovav and Tsadikovich (2015)		✓	✓	✓
Işik and Yildiz (2023)		✓	✓	✓
Jahani et al. (2022)		✓	✓	✓
Karakaya and Balcik (2023)		✓	✓
Mak et al. (2022)		✓	✓			✓
Mohammadi et al. (2022)		✓	✓	✓
Shah et al. (2022)		✓
Sinha et al. (2021)		✓	✓
Supply Chain Management
Abbasi et al. (2023)	✓		✓	✓
Basciftci et al. (2023)	✓	✓	✓
Chowdhury et al. (2022)	✓	✓	✓	✓	✓
Gilani and Sahebi (2022)	✓	✓	✓	✓
Goodarzian et al. (2022)	✓	✓	✓	✓
Goodarzian et al. (2022)	✓		✓	✓
Habibi et al. (2023)	✓	✓	✓	✓	✓
Kohneh et al. (2023)	✓	✓	✓	✓
Lai et al. (2021)	✓	✓	✓
Li et al. (2021)	✓	✓	✓
Lim et al. (2022)	✓	✓	✓	✓
Manupati et al. (2021)	✓	✓	✓	✓
Rahman et al. (2023)	✓	✓	✓	✓
Rastegar et al. (2021)	✓	✓	✓	✓
Sazvar et al. (2021)	✓	✓	✓	✓
Shiri and Ahmadizar (2022)	✓	✓	✓	✓
Tang et al. (2022)	✓	✓	✓
Tavana et al. (2021)	✓	✓	✓	✓
Wang et al. (2023)	✓	✓	✓	✓
Xu et al. (2021)	✓		✓	✓
Yang et al. (2021)	✓	✓	✓	✓
Location and Routing Models (See Section §4 and Appendix §A)
Location-Allocation
Bertsimas et al. (2022)	✓		✓			✓
Bravo et al. (2022)	✓		✓
Cabezas et al. (2021)	✓		✓
Cao et al. (2023)	✓		✓
Dastgoshade et al. (2022)	✓		✓
Emu et al. (2021)	✓			✓
Enayati et al. (2023)	✓			✓
Enayati et al. (2023)	✓			✓
Kumar et al. (2022)	✓		✓
Leithäuser et al. (2021)	✓		✓
Lim et al. (2016)	✓		✓
Luo and Mehrotra (2023)	✓		✓
Lusiantoro et al. (2022)	✓		✓
Polo et al. (2015)	✓		✓
Soria-Arguello et al. (2021)	✓			✓
Srivastava et al. (2021)	✓		✓
Zhang et al. (2022)	✓		✓
Routing
Shamsi Gamchi et al. (2021)				✓	✓	✓
Yang and Rajgopal (2021)	✓			✓	✓
Yucesoy et al. (2022)	✓			✓	✓

Optimization articles are further grouped into five categories based on the decisions studied: (i) resource allocation, (ii) location-allocation, (iii) inventory management, (iv) supply chain management, and (v) routing. Further details on each paper in each category (model type, uncertainty type, commodity, period, echelons, objective, case study, and solution methodology) are compiled in Tables 9 through 13 in Appendix. Table 2 provides an overall summary for all of the optimization articles.

Table 2.

Summary of articles based on decision levels, model type, commodity, period and case study for each category.

	Decision Levels							Model Type		Commodity Type			Period		Case Study		Solution Method
Category	Total	Location	Inventory	Allocation	Distribution	Routing	SIR	Deterministic	Uncertain	Single	Multiple	Multiple Dose	Single	Multiple	Real Life	Synthetic	Commercial Solver
Supply Chain Models
Resource Allocation	23	0 (0%)	0 (0%)	23 (100%)	10 (43%)	0 (0%)	13 (57%)	9 (39%)	14 (61%)	17 (74%)	6 (26%)	2 (9%)	10 (43%)	13 (57%)	18 (78%)	5 (22%)	4 (17%)
Inventory Management	13	0 (0%)	13 (100%)	9 (69%)	7 (54%)	0 (0%)	1 (8%)	4 (31%)	9 (69%)	8 (62%)	5 (38%)	4 (31%)	2 (15%)	11 (85%)	10 (77%)	3 (23%)	1 (8%)
Supply Chain Management	20	20 (100%)	18 (90%)	20 (100%)	16 (80%)	2 (10%)	0 (0%)	9 (45%)	11 (55%)	13 (65%)	8 (40%)	0 (0%)	7 (35%)	14 (70%)	19 (95%)	1 (5%)	4 (20%)
Location and Routing Models
Location-Allocation	18	18 (100%)	0 (0%)	16 (89%)	5 (28%)	0 (0%)	1 (6%)	12 (67%)	6 (33%)	15 (83%)	2 (11%)	0 (0%)	14 (78%)	3 (17%)	16 (89%)	2 (11%)	7 (39%)
Routing	3	2 (67%)	0 (0%)	0 (0%)	3 (100%)	3 (100%)	1 (33%)	1 (33%)	2 (67%)	3 (100%)	0 (0%)	1 (33%)	2 (67%)	1 (33%)	3 (100%)	0 (0%)	2 (67%)
Total	77	40 (52%)	31 (40%)	68 (88%)	41 (53%)	5 (6%)	16 (21%)	35 (45%)	42 (55%)	56 (73%)	21 (27%)	7 (9%)	35 (45%)	42 (55%)	66 (86%)	11 (14%)	18 (23%)

In the next five sections of mathematical models for pandemic vaccine supply chain, the categorization can be seen into two broad classes: (i) traditional supply chain model, with specific considerations for handling vaccines (Sections §3 - §6), and (ii) models in epidemiological dynamics while building upon traditional vaccine supply chain models (Section §7). This is important for endemic/pandemic management. If the focus of a work is on location (e.g., coverage model) or routing decision without much consideration for other details in the entire supply chain, we position them in a separate section (Section §4). In vaccine supply chain model, it is important to take uncertainty into account in practice. This uncertainty can be modeled in various ways, and hence we delve into these diverse approaches in Section §5, offering some details and illustrating their broad applicability. Finally, to highlight the importance - equity and multi-criteria decision-making in pandemic vaccine distribution, especially in the early stage, we separately discuss them in Section §6.

Note that we first present core mathematical models pertinent to the scope of each section/subsection. If some of the models are unique to the vaccine supply chain, we also present them separately. Models dealing with cold storage refrigeration requirements, vaccine vial shelf life, relationship between vaccine vials, dosages and open vial wastage, priority group, equity, epidemiological dynamics, and herd immunity are some of those considerations. Given the scope of this review some of the advanced models are only discussed to ensure greater completeness.

3. Supply Chain Models

In this section, we present a comprehensive perspective on the diverse factors considered in the existing literature concerning the COVID-19 vaccine supply chain across multiple echelons. We achieve this by examining various objective functions and constraints that have been utilized for different components of the supply chain. Specifically, we emphasize key constraints derived from mathematical models documented in relevant literature, while also highlighting the potential challenges associated with modeling the intricate network of vaccine allocation and distribution for manufacturers (Section §3.1), distribution centers (Section §3.2), and vaccination centers (Section §3.3).

3.1. Manufacturers

In this section, we focus on the production capacity constraint applicable to manufacturers in the vaccine supply chain. Several articles have addressed inventory management (Georgiadis and Georgiadis 2021, Hovav and Tsadikovich 2015, Jahani et al. 2022, Shah et al. 2022) and supply chain management (Chowdhury et al. 2022, Gilani and Sahebi 2022, Goodarzian et al. 2022, Manupati et al. 2021, Sazvar et al. 2021, Tavana et al. 2021) aspects related to manufacturers’ requirements. In the vaccine supply chain, capacity constraints can exist between the manufacturer and the distribution center (DC), as well as between the distribution center (DC) and the vaccination center (VC). These capacity constraints can pertain to various resources, such as the number of vaccines, the number of vaccine lots, fleet size, workforce, time, budget, and other restricted resources necessary for production. Additionally, the capacity constraints can be specific to commodities, time, or facilities. Moreover, lead time considerations may arise due to the time required for production. In its simplest form, the capacity constraint can be expressed as in (1).

$$\underset{\begin{array}{c}\text{Product-and time-specific production capacity}\\ \text{constraint of a manufacturer.}\end{array}}{\underbrace{\sum _{j\in 𝒩}{x}_{ijpt}^{md}\le C_{ipt}^M{X}_{it}^m,\forall i,p,t}}$$ (1)

The capacity constraint can take various forms depending on the specific problem context. To achieve a more aggregated representation of the constraints in terms of vaccine type, time, and facility, the decision variables, parameters, summations, and set indexes in (1) can be redefined. This allows for a more generalized formulation of the capacity constraint, accommodating different variations based on the problem.

In practice, it is common for multiple distribution centers (DCs) to request vaccines from the same manufacturer. However, due to the manufacturer’s capacity limitations, it can only fulfill the requests of a distribution center (DC) once its current order is processed. To address this consideration in the optimization model, two time indices ($t^{\prime }$ representing the ordering time and $t$ representing the receiving time $\left.t^{\prime }\le t\right)$ are used (e.g., see Tavana et al. (2021)).

$$\underset{\begin{array}{c}\text{Manufacturer can receieve at most one order}\\ \text{for the same delivery time}t\text{over all DCs.}\end{array}}{\underbrace{\sum _{t^{\prime }=1}^{|𝒯|}{X}_{ip{t}^{\prime }t}^m\le 1,\forall i,p,t}};\underset{\begin{array}{c}\text{Manufacturer can not accept a new order}\\ \text{from a DC unless its previous order is served.}\end{array}}{\underbrace{\sum _{t^{\prime }<{\hat{t}}^{\prime }<t}\sum _{{\hat{t}}^{\prime }\le \hat{t}}{X}_{ip{\hat{t}}^{\prime }\hat{t}}^m\le \mathbf{M}\cdot \left(1-X_{ip{t}^{\prime }t}^m\right),\forall i,p,t,t^{\prime }\le t}}$$ (2)

While the majority of research on COVID-19 focuses on shipments from manufacturers to distribution centers (DCs), it is worth noting that manufacturers can also directly transship vaccines to vaccination centers (VCs). Additionally, in the context of COVID-19, all manufacturers typically produce a single type of vaccine. For example, Johnson & Johnson plants exclusively manufacture the Janssen Vaccine, while Pfizer plants exclusively produce the BioNTech Vaccine. However, in cases where a manufacturing plant produces multiple types of vaccines, the set $p\in {𝒫}_i$ can be incorporated into (1) and (2) to account for this variation. Furthermore, some articles consider the flow of raw materials from suppliers to manufacturers (Chowdhury et al. 2022, Gilani and Sahebi 2022, Sazvar et al. 2021), the manufacturing process itself (Shah et al. 2022), and the location decisions of manufacturers with different technology levels (Sazvar et al. 2021). These aspects contribute to a more comprehensive understanding of the COVID-19 vaccine supply chain.

3.2. Distribution Center

In this section, we present the considerations related to distribution centers (DCs) in terms of flow conservation (Section §3.2.1), cold chain requirements (Section §3.2.2), and fleet size (Section §3.2.3). These considerations are based on optimization models proposed in recent COVID-19 literature addressing resource allocation (Abbasi et al. 2020, Anahideh et al. 2022, Balcik et al. 2022, Bennouna et al. 2022, Bertsimas et al. 2020, Enayati and Özaltın 2020, Huang et al. 2017, Jadidi et al. 2021, Jarumaneeroj et al. 2022, Minoza et al. 2021, Munguía-López and Ponce-Ortega 2021, Rao and Brandeau 2021, Roy et al. 2021, Shukla et al. 2022, Thul and Powell 2023, Yang et al. 2022, Yarmand et al. 2014), inventory management (Azadi et al. 2020, Bonney and Jaber 2011, Azadi et al. 2020, Fadaki et al. 2022, Georgiadis and Georgiadis 2021, Hovav and Tsadikovich 2015, Karakaya and Balcik 2023, Mak et al. 2022, Mohammadi et al. 2022, Shah et al. 2022, Sinha et al. 2021), and supply chain management (Basciftci et al. 2023, Chowdhury et al. 2022, Gilani and Sahebi 2022, Goodarzian et al. 2022, Habibi et al. 2023, Goodarzian et al. 2022, Kohneh et al. 2023, Lim et al. 2022, Manupati et al. 2021, Rahman et al. 2023, Rastegar et al. 2021, Sazvar et al. 2021, Shiri and Ahmadizar 2022, Tavana et al. 2021, Wang et al. 2023, Xu et al. 2021, Yang et al. 2021). These models contribute to a comprehensive understanding of the distribution center (DC) aspects within the COVID-19 vaccine supply chain.

3.2.1. Flow Conservation

Distribution centers (DCs) play a critical role in the vaccine supply chain, facilitating both inflows and outflows of vaccines. The inflows to distribution centers (DCs) include procurement from manufacturers, their own inventory carried over from previous time periods, and shipments received from other distribution centers (DCs). Conversely, the outflows involve transshipments to vaccination centers (VCs), maintaining inventory for future time periods, shipments to other distribution centers (DCs), and addressing vaccine waste resulting from expiration, damage (e.g., physical or temperature-related), or mishandling. If distribution centers (DCs) can also function as vaccination centers (VCs), an additional vaccine outflow can be the administration of vaccines to individuals.

To maintain flow conservation in distribution centers (DCs) (3) to (8) are utilized. Equations (3) and (4) represent the inventory flow conservation constraints for distribution centers (DCs). Equation (5) ensures that if there is an outflow from a distribution center (DC), it must correspond to an existing outflow location. Similarly, (6) and (7) ensure that if there is an inflow to a distribution center (DC), it must originate from an existing inflow location. It should be noted that not all distribution centers (DCs) may serve all vaccination centers (VCs). In such cases, the flow balance constraints can be modified by incorporating the index set ${𝒱}_j$, which represents the set of vaccination centers (VCs) exclusively served by distribution center (DC) $j$.

Furthermore, in practical scenarios, there may exist a lead time $(l)$ between the ordering and receiving of vaccines from one location to another. This lead time can be either deterministic or uncertain. In the current formulation, let us consider the case where there is only a deterministic lead time between the manufacturer and the distribution center (DC). For $l=0$, the formulation has no lead time. In such a case, (4) and (7) become redundant and the decision variable $x_{ijpt-l}^{md}$ becomes $x_{ijpt}^{md}$ in (3) and (6).

$${I^t}_j=\underset{\text{Vaccine inflow}.}{\underbrace{I_j^{t-1}+\sum _{i=1}^{|ℳ|}\sum _{p=1}^{|𝒫|}{x}_{ijpt-l}^{md}+\sum _{\begin{array}{c}{j}^{\prime }\in 𝒩\\ j^{\prime }\ne j\end{array}}\sum _{p=1}^{|𝒫|}{x}_{j^{\prime }jpt}^{d^{\prime }d}}}\underset{\text{Vaccine outflow}.}{\underbrace{-\sum _{k=1}^{|𝒱|}\sum _{p=1}^{|𝒫|}{x}_{jkpt}^{dv}-\sum _{\begin{array}{c}{j}^{\prime }\in 𝒩\\ j^{\prime }\ne j\end{array}}\sum _{p=1}^{|𝒫|}{x}_{j{j}^{\prime }pt}^{d{d}^{\prime }}-\sum _{p=1}^{|𝒫|}{w}_{jpt},}}\forall j,t>l$$ (3)

$${I^t}_j=\underset{\text{Vaccine inflow}.}{\underbrace{I_j^{t-1}+\sum _{\begin{array}{c}{j}^{\prime }\in 𝒩\\ j^{\prime }\ne j\end{array}}\sum _{p=1}^{|𝒫|}{x}_{j^{\prime }jpt}^{d^{\prime }d}}}\underset{\text{Vaccine outflow}.}{\underbrace{-\sum _{k=1}^{|𝒱|}\sum _{p=1}^{|𝒫|}{x}_{jkpt}^{dv}-\sum _{\begin{array}{c}{j}^{\prime }\in 𝒩\\ j^{\prime }\ne j\end{array}}\sum _{p=1}^{|𝒫|}{x}_{j{j}^{\prime }pt}^{d{d}^{\prime }}-\sum _{p=1}^{|𝒫|}{w}_{jpt}}},\forall j,t\le l$$ (4)

$$\underset{\text{Outflow to VCs}.}{\underbrace{\sum _{p=1}^{|𝒫|}\sum _{k=1}^{|𝒱|}{x}_{jkpt}^{dv}}}+\underset{\text{Outflow to other DCs}.}{\underbrace{\sum _{\begin{array}{c}{j}^{\prime }\in 𝒩\\ j^{\prime }\ne j\end{array}}\sum _{p=1}^{|𝒫|}{x}_{j{j}^{\prime }pt}^{d{d}^{\prime }}}}+\underset{\text{Wasted vaccines}.}{\underbrace{\sum _{p=1}^{|𝒫|}{w}_{jpt}}}\le M\cdot Y_j^D,\forall j,t$$ (5)

$$\underset{\text{Inflow from manufacturers}.}{\underbrace{\sum _{i=1}^{|ℳ|}\sum _{p=1}^{|𝒫|}{x}_{ijpt-l}^{md}}}+\underset{\text{Inflow from other DCs}.}{\underbrace{\sum _{\begin{array}{c}{j}^{\prime }\in 𝒩\\ j^{\prime }\ne j\end{array}}\sum _{p=1}^{|𝒫|}{x}_{j^{\prime }jpt}^{d^{\prime }d}}}\le M\cdot Y_j^D,\forall j,t>l$$ (6)

$$\underset{\text{Inflow from other DCs}.}{\underbrace{\sum _{\begin{array}{l}{j}^{\prime }\in 𝒩\\ j^{\prime }\ne j\end{array}}\sum _{p=1}^{|𝒫|}{x}_{j^{\prime }jpt}^{d^{\prime }d}}}\le \mathbf{M}.Y_j^D,\forall j,t\le l$$ (7)

$$\underset{\text{Inventory capacity constraint}.}{\underbrace{{I^t}_j\le C_j^D{Y}_j^d,\forall j,t;}}\underset{\text{Inventory safety stock constraint}.}{\underbrace{{I^t}_j\ge S_j^D{Y}_j^d,\forall j,t}}$$ (8)

Georgiadis and Georgiadis (2021) formulate an aggregated version of the inventory balancing constraint with lead time ($l$) using (9):

$$\underset{\begin{array}{c}\text{Outflow from a DC}\\ \text{until time}t.\end{array}}{\underbrace{\sum _{t^{\prime }\le t}\sum _{k=1}^{|𝒱|}{x}_{jkp{t}^{\prime }}^{dv}}}\le \underset{\text{Initial inventory.}}{\underbrace{I_j^0}}+\underset{\begin{array}{c}\text{Quantity of vaccines ordered at or before}\\ \text{time}(t-l{)}^{+}\text{, and arrived at or before time}t.\end{array}}{\underbrace{\sum _{t^{″}\le (t-l{)}^{+}}\sum _{i=1}^{|ℳ|}{x}_{ijp{t}^{″}}^{md}}}-\underset{\text{Vaccine wastage.}}{\underbrace{\sum _{t^{″}\le (t-l{)}^{+}}{w}_{j{t}^{″}p}}},\forall j,t,p$$ (9)

In Fadaki et al. (2022), flow units of different decision variables or parameters differ. For example, if inflow is in terms of vaccine lots and outflow is in terms of vaccine vials, corresponding variables must be multiplied by the appropriate conversion coefficient. Moreover, Hinojosa et al. (2000) consider fractional flow (i.e., using continuous decision variables instead of discrete decision variables).

3.2.2. Cold Storage Refrigeration Requirement

Vaccines may have varying storage requirements in terms of temperature, e.g., cold, very cold, or ultra cold. Ultra cold storage satisfies very cold storage requirements, but not the other way around. Additionally, distribution centers (DCs) can have different capacities for each level of refrigeration, which can vary over time. For example, Tavana et al. (2021) employ (10) to maintain the capacity restriction for very cold storage, (11) to sustain the capacity restriction for very cold storage, and (12) to ensure the capacity restriction for ultra cold storage at distribution center (DC) $j$ during time period $t$. These equations assume that everything received by a distribution center (DC) will be distributed to vaccination centers (VCs) during the same time duration.

$$\sum _{i=1}^{|ℳ|}\sum _{t^{\prime }}{x}_{ij{t}^{\prime }t}^{md}(1-{\mathrm{\delta }}_i^{(vc)})=\sum _{p=1}^{|𝒫|}\sum _{k=1}^{|𝒱|}{x}_{jkpt}^{dv}(1-{\mathrm{\delta }}_p^{(vc)})\le C_j^{D(c)}{Y}_j^{D(c)},\forall j,t$$ (10)

$$\sum _{i=1}^{|ℳ|}\sum _{t^{\prime }}{x}_{ij{t}^{\prime }t}^{md}{\mathrm{\delta }}_i^{(vc)}(1-{\mathrm{\delta }}_i^{(uc)})=\sum _{p=1}^{|𝒫|}\sum _{k=1}^{|𝒱|}{x}_{jkpt}^{dv}{\mathrm{\delta }}_p^{(vc)}(1-{\mathrm{\delta }}_p^{(uc)})\le C_j^{D(vc)}-C_j^{D(uc)}{Y}_j^{D(uc)},\forall j,t$$ (11)

$$\sum _{i=1}^{|ℳ|}\sum _{t^{\prime }}{x}_{ij{t}^{\prime }t}^{md}{\mathrm{\delta }}_i^{(vc)}{\mathrm{\delta }}_i^{(uc)}=\sum _{p=1}^{|𝒫|}\sum _{k=1}^{|𝒱|}{x}_{jkpt}^{dv}{\mathrm{\delta }}_p^{(vc)}{\mathrm{\delta }}_p^{(uc)}\le C_j^{D(uc)}{Y}_j^{D(uc)},\forall j,t$$ (12)

The binary location decisions also need to be linked to the outflow and inflow from a distribution center (DC). Equation (13) links the outflow to the location decision variable, based on the required refrigeration type.

$$\begin{gathered} x_{jkpt}^{dv}\left(1-{\mathrm{\delta }}_p^{(vc)}\right)\le \mathbf{M}\cdot Y_j^{D(c)},\forall j,k,p,t;x_{jkpt}^{dv}{\mathrm{\delta }}_p^{(vc)}\left(1-{\mathrm{\delta }}_p^{(uc)}\right)\le \mathbf{M}\cdot Y_j^{D(vc)},\forall j,k,p,t \\ {x}_{jkpt}^{dv}{\mathrm{\delta }}_p^{(vc)}{\mathrm{\delta }}_p^{(uc)}\le \mathbf{M}\cdot Y_j^{D(uc)},\forall j,k,p,t \end{gathered}$$ (13)

Similarly, (14) ensures the same relation for inflow respecting the assumption that unlike cold and very cold storage type, no new distribution centers (DCs) will be set up for ultra cold storage; ultra cold storage facility can only be added to an existing distribution center (DC).

$$x_{ij{t}^{\prime }t}^{md}\le \mathbf{M}\cdot Y_j^{D(c)},\forall i,j,t^{\prime },t;x_{ij{t}^{\prime }t}^{md}\le \mathbf{M}\cdot Y_j^{D(vc)},\forall i,j,t^{\prime },t$$ (14)

3.2.3. Fleet and Drones

Vaccine vials can be transported between different echelons using various refrigerated vehicles, each with its own capacity for different levels of refrigeration (Goodarzian et al. 2022, Dastgoshade et al. 2022, Chowdhury et al. 2022, Habibi et al. 2023, Hovav and Tsadikovich 2015, Lim et al. 2022). However, it is worth noting that the majority of existing studies in the literature do not consider modeling of vehicles and exclude them from their scope. Among the works that do incorporate vehicles, refrigerated trucks are commonly utilized as the primary transportation mode. Similarly, Lim et al. (2022) explore the use of motorbikes.

In practice, each distribution center (DC) may have its own dedicated fleet of vehicles, which can be employed to serve different vaccination centers (VCs). Detailed considerations regarding vehicle routing are presented in Section §A. Equation (15) provides an aggregated representation of fleet availability across all distribution centers (DCs), capturing the overall capacity of the available vehicles.

$$\sum _{j=1}^{|𝒱|}\sum _{p=1}^{|𝒫|} \sum _{k=1}^{|K|} x_{jkpt}^{dv}\le C^F,\forall t$$ (15)

Lim et al. (2016) use vehicle trips required to transport vaccines in a replenishment as a decision variable for each specific inventory replenishment frequency. Moreover, Yang et al. (2021) consider transportation capacity-based required trips for two different annual inventory replenishment frequencies. Travel time to complete a trip is an important parameter in this setting.

Notable studies by Wang et al. (2023), Enayati et al. (2023), and Enayati et al. (2023) explore the use of drones for vaccine distribution. While Wang et al. (2023) optimizes drone capacity with vaccination scheduling in facility location problems, models in the latter two works involve location decision of drone base, drone relay station, and distribution centers (DCs) layering each possible origin-destination path for drone delivery with controlled drone stops, drone range, and cold-time.

3.3. Vaccination Center

In this section, we discuss the vaccination center (VC) considerations in terms of flow conservation (Section §3.3.1), DC-VC assignment (Section §3.3.2), healthcare personnel capacity/scheduling (Section §3.3.3), vaccine shelf life (Section §3.3.4), vaccine vials (Section §3.3.5), and priority groups (Section §3.3.6), based on the optimization models presented in the recent COVID-19 literature.

3.3.1. Inventory and Flow Balancing

Vaccination centers (VCs), similar to distribution centers (DCs), experience both inflows and outflows of vaccines. The inflows of vaccines at vaccination centers (VCs) can originate from several sources, including procurement from manufacturers, their own inventory carried over from previous time periods, and shipment from other vaccination centers (VCs) (lateral transshipment) or distribution centers (DCs). Conversely, the outflows from vaccination centers (VCs) involve shipment to other vaccination centers (VCs), maintaining inventory for future periods, returning leftover vaccines to distribution centers (DCs), managing vaccine waste (such as open vial wastage or improper administration), and administering vaccines to individuals. In the context of this article, we assume that a vaccination center (VC) can only have inventory, vaccination, and waste as outflows. Additionally, the center can only receive inflows from its own inventory from the previous periods and distribution centers (DCs). It is important to note that vaccine demand can be either deterministic or stochastic. Furthermore, when managing lateral shipment to minimize open vial wastage, the flow should be considered at both the dosage and vial levels. In this formulation, the demand is expressed in terms of vaccine vials. If the demand were stated in terms of vaccine doses, appropriate conversion coefficients would need to be included for accurate calculations.

Equation (16) is the inventory flow conservation constraint for vaccination centers (VCs), assuming there is no supply shortage. Similar to flow conservation constraints at distribution center (DC) given in constraints (5) to (8), constraints for location-flow binding, inventory capacity and safety stock level can be formulated for vaccination centers (VCs). Section §4 further elaborates on facility location decision for vaccination centers (VCs) and the routing dynamics when a vaccination center (VC) is mobile.

$$I_k^t=I_k^{t-1}+\underset{\text{Inflow from DCs.}}{\underbrace{\sum _{j=1}^{|𝒩|}\sum _{p=1}^{|𝒫|}{x}_{jkpt}^{dv}}}-\underset{\text{Vaccine demand.}}{\underbrace{\sum _{p=1}^{|𝒫|}{d}_{kpt}}}-\underset{\text{Wasted vaccines.}}{\underbrace{\sum _{p=1}^{|𝒫|}{w}_{kpt}}},\forall k,t$$ (16)

Under supply scarcity, (16) would create infeasibility, therefore $d_{kpt}$ must be replaced by decision variable $x_{kpt}^v$ that denotes allocation decisions. In addition, (17) must be added to ensure that shortages are accounted for.

$$\underset{\text{Vaccination.}}{\underbrace{x_{kpt}^v}}+\underset{\text{Shortage.}}{\underbrace{s_{kpt}^v}}=\underset{\text{Demand.}}{\underbrace{d_{kpt}}},\forall k,p,t$$ (17)

Alternatively, the constraint $x_{kpt}^v\le d_{kpt}\forall k,p,t$ can be used if tracking shortages is not necessary. Hovav and Tsadikovich (2015) penalize the vaccine shortages using a shortage cost $c_{kp}^{SH}$ in the objective function. This practice is very common in works that consider uncertain demand.

3.3.2. Assigning a Vaccination Center to a Distribution Center

The linkage between distribution centers (DCs) and vaccination centers (VCs) can be established in multiple ways. Several articles in the literature explore different assignment methods, including one-to-one assignment, set covering assignment, and set packing assignment. In the case of the flow of vaccines from a distribution center (DC) to its corresponding vaccination center (VC), this can be modeled using (18). It is worth noting that similar types of assignments can also be considered for the linkage between manufacturers and distribution centers (DCs) (Manupati et al. 2021) or between vaccination centers (VCs) themselves (Fadaki et al. 2022).

$$\underset{\text{Direct pairwise assignment.}}{\underbrace{\sum _{p,t}{x}_{jkpt}^{dv}\le \mathbf{M}\cdot x_{jk},\forall j,k}};\underset{\text{Set cover type constraint.}}{\underbrace{\sum _j{a}_{kj}{x}_{jk}\ge 1,\forall k}};\underset{\text{Set packing type constraint.}}{\underbrace{\sum _j{a}_{kj}{x}_{jk}\le 1,\forall k}}$$ (18)

Once the distribution center (DC) assignment is specified, the next step is to allocate vaccines from the distribution center (DC) to various destinations, such as regional depots (Balcik et al. 2022, Huang et al. 2017, Munguía-López and Ponce-Ortega 2021), vaccination centers (VCs) (Abbasi et al. 2020, Anahideh et al. 2022, Bertsimas et al. 2020, Roy et al. 2021, Shukla et al. 2022, Thul and Powell 2023, Yang et al. 2022, Yarmand et al. 2014), or specific sub-population groups (Bennouna et al. 2022, Enayati and Özaltın 2020, Jadidi et al. 2021, Jarumaneeroj et al. 2022, Minoza et al. 2021, Rao and Brandeau 2021). Different allocation rules and prioritization strategies are used in these models to determine the amount of vaccine to be allocated, with the objective of minimizing the spread of infection while considering the limited availability of vaccines. In addition to efficiency concerns, equity among the served sub-populations is often a key objective or constraint in these models. For more detailed information on resource allocation articles, refer to Table 9 in Appendix.

3.3.3. Healthcare Personnel Capacity and Vaccination Scheduling

The healthcare workforce plays a crucial role in pandemic management, as skilled personnel are needed to administer vaccines to individuals. The allocation of healthcare workforce is addressed in several COVID-19 articles (Leithäuser et al. 2021, Georgiadis and Georgiadis 2021, Kumar et al. 2022, Lai et al. 2021, Rahman et al. 2023). The availability and sustainability of healthcare personnel in vaccination centers (VCs) are captured by (19).

$$\underset{\text{Healthcare workforce availability constraint.}}{\underbrace{\sum _{g,p}{x}_{gkpt}^v\le F_{kt}^R{C}^{HW},\forall k,t}};\underset{\text{Additional healthcare workforce constraint.}}{\underbrace{F_{kt}^R\le F_{kt}^A+C_{kt}^{EH},\forall k,t}}$$ (19)

Wang et al. (2023), Zhang et al. (2022) focus on vaccination scheduling. Zhang et al. (2022) propose a mass vaccination appointment scheduling model that has the potential to be extended to healthcare personnel scheduling, including overtime requirements. Their linear ordering formulation considers location, population size, and time window inputs to determine optimal vaccination center (VC) location, appointment acceptance, assignment of appointments to selected vaccination centers (VCs), and the scheduling of the vaccination timetable while accounting for possible tardiness. The formulation is further enhanced using dominance rules and valid inequalities, and is solved using a logic-based Benders approach. Bandi et al. (2021) emphasize the importance of efficient appointment slot management, second-dose reserves, and flexible vaccine preferences for optimizing vaccination rates during a two-dose roll-out.

3.3.4. Vaccine Vial Shelf Life

Vaccine vials are considered perishable and have a limited shelf life (denoted by ${\mathrm{\lambda }}^{sl}$). Once the shelf life of a vaccine vial expires, the vaccine doses within the vial become unsafe for use, resulting in waste. Many articles do not explicitly account for the shelf life of vaccine vials and instead define the problem time horizon to be shorter than the shelf life. However, Georgiadis and Georgiadis (2021) address this issue by introducing (20) to (24) to ensure the proper flow of vaccine vial shelf life. These equations aim to balance the forward flow of vaccine vial shelf life by fixing the receiving time $(t)$ (see (20) and (21)) and the backward flow by fixing the opening time ($t^{\prime }$) (see (23) and (22)). It is assumed that there is no lead time in transhipped vaccines between distribution centers (DCs) and vaccination centers (VCs). Other key assumptions include the shelf life applying to a vial immediately after its receiving time and a vial being opened one day after its receiving time. Thus, the relationship $t+1\le t^{\prime }\le t+{\mathrm{\lambda }}^{sl}$ holds, where ${\left(t^{\prime }-{\mathrm{\lambda }}^{sl}\right)}^{+}\le$ $t\le t^{\prime }-1$, and all times are bounded by the time horizon $|𝒯|$. Additionally, in (20), it is assumed that all vial openings occur within the planning time horizon.

$$\underset{\begin{array}{c}\text{Vaccines can be opened at}{t}^{\prime }\\ \text{where}t+1\le t^{\prime }\le t+{\mathrm{\lambda }}^{\mathit{sl}}.\end{array}}{\underbrace{\sum _{t^{\prime }=t+1}^{t+{\mathrm{\lambda }}^{sl}}{L}_{kpt{t}^{\prime }}^v}}=\underset{\text{Vaccines received at}t.}{\underbrace{\sum _{j\in 𝒩}{x}_{jkpt}^{dv}}}-\underset{\text{Wasted vaccines.}}{\underbrace{w_{kpt}}},\forall k,p,t\in \left[|𝒯|-{\mathrm{\lambda }}^{sl}\right]$$ (20)

$$\underset{\begin{array}{l}\text{Similar to Constraint (20), by not considering}\\ \text{the vaccines opened after time horizon}|𝒯|.\end{array}}{\underbrace{\sum _{t^{\prime }=t+1}^{|𝒯|}{L}_{kpt{t}^{\prime }}^v\le \sum _{j\in 𝒩}{x}_{jkpt}^{dv}-w_{kpt}}},\forall k,p,|𝒯|-{\lambda }^{\mathit{sl}}<t\le |𝒯|$$ (21)

$$\underset{\begin{array}{c}\text{Number of vaccines}\\ \text{opened at}{t}^{\prime }.\end{array}}{\underbrace{N_{kp{t}^{\prime }}^v}}=\underset{\begin{array}{c}\text{Number of vaccines received}\\ \mathrm{in}{t}^{\prime }-{\mathrm{\lambda }}^{sl}\le t\le t^{\prime }-1.\end{array}}{\underbrace{\sum _{t^{\prime }-{\mathrm{\lambda }}^{sl}\le t}^{t^{\prime }-1}{L}_{kpt{t}^{\prime }}^v}},\forall k,p,t^{\prime }>{\mathrm{\lambda }}^{sl}$$ (22)

$$\underset{\begin{array}{c}\text{Similar to Constraint (22), by considering}\\ \text{initial net vaccine allocation}.\end{array}}{\underbrace{N_{kp{t}^{\prime }}^v=A_{kp0t^{\prime }}^v+\sum _{(t^{\prime }-{\mathrm{\lambda }}^{sl})+\le t}^{t^{\prime }-1}{L}_{kpt{t}^{\prime }}^v}},\forall k,p,t^{\prime }\le {\mathrm{\lambda }}^{sl}$$ (23)

$$\sum _{1\le t^{\prime }\le {\mathrm{\lambda }}^{sl}}(\underset{\begin{array}{c}\text{Initial net vaccine}\\ \text{availability}.\end{array}}{\underbrace{A_{kp0t^{\prime }}^v}}+\underset{\begin{array}{c}\text{Initial vaccine}\\ \text{waste}.\end{array}}{\underbrace{W_{kp0t^{\prime }}^v}})=\underset{\begin{array}{c}\text{Initial gross vaccine}\\ \text{inventory}.\end{array}}{\underbrace{I_{kp}^0}},\forall k,p$$ (24)

3.3.5. Relationship Between Vaccine Vials, Dosages and Open Vial Wastage

Several articles employ inventory replenishment policies in multi-echelon systems (Fadaki et al. 2022, Georgiadis and Georgiadis 2021, Hovav and Tsadikovich 2015) to optimize vaccination delivery and minimize waste, considering the unique requirements of considered vaccines. These requirements encompass aspects such as cold chain management, storage conditions, vaccine-specific vial sizes at distribution (DCs) and vaccination centers (VCs), and expiration timelines. It is important to note that a single vaccine vial can contain multiple vaccine doses. In general, when a vaccine vial is opened, all doses within the vial should be administered to individuals before any doses expire and go to waste. It is worth mentioning that the open vial shelf life is typically shorter than the shelf life of the unopened vial. For instance, the Pfizer COVID-19 vaccine is packaged in vials containing five doses. While the vial shelf life is 30 days, the open vial shelf life is only 24 hours. This consideration underscores the need to efficiently utilize the doses within an opened vial to prevent wastage. Equation (25) ensures vial-dosage balancing, by equating the total number of opened doses to the summation of vaccinations among all priority groups and open vial wastage.

$$\underset{\text{Opened vaccine dosages.}}{\underbrace{\sum _{v\in 𝒱}{\mathrm{ı}}_{\mathrm{v}}{N}_{vkpt}^v}}=\underset{\text{Vaccinated people.}}{\underbrace{\sum _{g\in 𝒢}{x}_{gkpt}^v}}+\underset{\text{Open vial wastage.}}{\underbrace{w_{kpt}^{ov}}},\forall k,p,t$$ (25)

In their works, Azadi et al. (2020) and Azadi et al. (2020) address the impact of vials on inventory management. They propose an integrated vial opening-inventory replenishment policy and develop a framework that takes into account the specific vaccine type and vial size. To mitigate the issue of open vial wastage, Azadi et al. (2020) focus on childhood vaccines with different vial sizes within a two-stage stochastic model that considers demand uncertainty. The first stage of their model involves determining the timing, quantity, and size of vials to be ordered. The model aims to achieve a balance between the number of vaccines in terms of vials at the first stage and the number of doses at the second stage. Equations (26) to (29) illustrate the process of dose balancing through the opening time $(t)$ and vaccination time $\left(t^{\prime }\right)$. It is important to note that vaccinations can be safely administered before $t+\mathrm{\tau }$, but not at $t+\mathrm{\tau }$.

$$\underset{\begin{array}{c}\text{Vaccinated people in terms of doses,}\\ \text{vaccination time in}t\le t^{\prime }\le t+\mathrm{\tau }-1.\end{array}}{\underbrace{\sum _{t^{\prime }=t}^{t^{\prime }+\mathrm{\tau }-1}\sum _{g\in 𝒢}{x}_{gkpt{t}^{\prime }}^v}}+\underset{\text{Expired vaccine}.}{\underbrace{x_{gkptt+\mathrm{\tau }}^v}}=\underset{\begin{array}{c}\text{Amount of doses in}\\ \text{the opened vials}.\end{array}}{\underbrace{\sum _{v\in 𝒱}{\iota }_{vp}{N}_{vkpt}^v}},\forall k,p,t\le |\mathcal{T}|-\mathrm{\tau }+1$$ (26)

$$\underset{\begin{array}{l}\text{Similar to Constraint (26), by not considering}\\ \text{the vaccination after time horizon}|𝒯|.\end{array}}{\underbrace{\sum _{t^{\prime }=t}^{|\mathcal{T}|}\sum _{g\in 𝒢}{x}_{gkpt{t}^{\prime }}^v\le \sum _{\text{v}\in 𝒱}{\iota }_{vp}{N}_{\text{v}kpt}^v}},\forall k,p,t>|𝒯|-\mathrm{\tau }+1$$ (27)

$$\sum _{t\ge {(t^{\prime }-\mathrm{\tau }+1)}^{+}}^{t^{\prime }}\sum _{g\in 𝒢}{x}_{gkpt{t}^{\prime }}^v+\underset{\text{Open vial wastage}.}{\underbrace{w_{kp{t}^{\prime }}^{ov}}}\le \sum _{g\in 𝒢}{d}_{gkp{t}^{\prime }},\forall p,k,t^{\prime }\le \mathrm{\tau }$$ (28)

$$\underset{\begin{array}{c}\text{Vaccinated people in terms of doses,}\\ \text{opening time in}{t}^{\prime }-\mathrm{\tau }+1\le t\le t^{\prime }.\end{array}}{\underbrace{\sum _{t\ge (t^{\prime }-\mathrm{\tau }+1)}^{t^{\prime }}\sum _{g\in 𝒢}{x}_{gkpt{t}^{\prime }}^v}}+\underset{\text{Vaccine dose shortage}.}{\underbrace{\sum _{g\in 𝒢}{s}_{gpk{t}^{\prime }}^v}}=\underset{\text{Vaccine demand}.}{\underbrace{\sum _{g\in 𝒢}{d}_{gkp{t}^{\prime }}}},\forall p,k,t^{\prime }>\mathrm{\tau }$$ (29)

Finally, Mak et al. (2022) assess the inventory dynamics of the rollout process for three strategies: (i) holding back second doses, (ii) releasing second doses, and (iii) stretching the lead time between doses.

3.3.6. Priority Group Focus

In order to effectively address the pandemic, vaccine prioritization among different subgroups is crucial, particularly when faced with limited supply. Various factors can be taken into account for prioritization, such as age, profession, social contact, health condition, income, and race. Optimization modeling can be employed to incorporate vaccination strategies. For instance, Fadaki et al. (2022) allocate COVID-19 vaccines based on a “community transmission” parameter, which reflects the number of unvaccinated individuals at a specific location at a given time. On the other hand, Shiri and Ahmadizar (2022) introduce a formulation that divides the population into a high-priority group $(g=1)$ and a low-priority group $(g=2)$. Equations (30) to (33) are used to define this grouping. The formulation includes lateral vaccine shipment between vaccination centers (VCs) exclusively during the first time period. With these constraints in place, the low-priority group receives vaccinations only after the high-priority group’s needs have been met. Any remaining vaccines at the end of the time period are carried over to the subsequent period through inventory management.

$$y_{1kp1}^v=\text{min}\{d_{1kp1},{\mathit{I}}_k^0+\sum _{j\in 𝒩}{x}_{jkp1}^{dv}+\sum _{\begin{array}{c}{k}^{\prime }\in 𝒱\\ k^{\prime }\ne k\end{array}}{x}_{k^{\prime }kp1}^{v^{\prime }v}-\sum _{\begin{array}{c}{k}^{\prime }\in 𝒱\\ k^{\prime }\ne k\end{array}}{x}_{k{k}^{\prime }p1}^{v{v}^{\prime }}-w_{kp1}\},\forall k,p$$ (30)

$$y_{2kp1}^v=\text{min}\{d_{2kp1},{\mathit{I}}_k^0+\sum _{j\in 𝒩}{x}_{jkp1}^{dv}+\sum _{\begin{array}{c}{k}^{\prime }\in 𝒱\\ k^{\prime }\ne k\end{array}}{x}_{k^{\prime }kp1}^{v^{\prime }v}-\sum _{\begin{array}{c}{k}^{\prime }\in 𝒱\\ k^{\prime }\ne k\end{array}}{x}_{k{k}^{\prime }p1}^{v{v}^{\prime }}-y_{1kp1}^v-w_{kp1}\},\forall k,p$$ (31)

$$y_{1kpt}^v=\text{min}\{d_{1kpt},{\mathit{I}}_k^{t-1}+\sum _{j\in 𝒩}{x}_{jkpt}^{dv}-w_{kpt}\},\forall k,p,t\ge 2$$ (32)

$$y_{2kpt}^v=\text{min}\{d_{2kpt},{\mathit{I}}_k^{t-1}+\sum _{j\in 𝒩}{x}_{jkpt}^{dv}-y_{1kpt}^v-w_{kpt}\},\forall k,p,t\ge 2$$ (33)

4. Location and Routing Models

Decisions on distribution center (DC) (Dastgoshade et al. 2022, Emu et al. 2021, Soria-Arguello et al. 2021, Srivastava et al. 2021) and vaccination center (VC) locations (Bertsimas et al. 2022, Bravo et al. 2022, Cabezas et al. 2021, Leithäuser et al. 2021, Lim et al. 2016, Lusiantoro et al. 2022, Polo et al. 2015, Kumar et al. 2022, Luo and Mehrotra 2023, Zhang et al. 2022) are important for vaccine supply-demand matching in the context of COVID-19. Models are based on demand, distance, coverage, priority, required service levels, and fairness. A recent article considers locating waste disposal center (Cao et al. 2023). In this section, we present studies that focus on a variant of the assignment type (Section §4.1) or coverage type (Section §4.2) facility location models. In the formulations used in this section, vaccination center (VC) is the facility and population site (PS) is the demand source. Similar formulations can be used for the case where distribution center (DC) is the facility and vaccination center (VC) is the demand source. We present vehicle routing in Appendix §A.

4.1. Facility Location Models

Lusiantoro et al. (2022) consider a multi-objective facility location model (see (34)), where the first objective function ensures effectiveness by maximizing demand coverage and the second objective function ensures efficiency by minimizing demand weighted distance, simultaneously (Gralla et al. 2014).

$$\begin{gathered} \text{(Obj. 1)}\mathrm{m}\mathrm{a}\mathrm{x}\underset{\text{Demand coverage.}}{\underbrace{\sum _{s\in 𝒮}\sum _{k\in 𝒱}{d}_s^P{x}_{sk}}};\text{(Obj. 2) min}\underset{\text{Demand weighted distance.}}{\underbrace{\sum _{s\in 𝒮}\sum _{k\in 𝒱}{d}_s^P{\mathrm{\eta }}_{sk}{x}_{sk}}} \\ \text{s.t.}\underset{\text{Assigning a PS to at most one VC.}}{\underbrace{\sum _{k\in 𝒱}{x}_{sk}\le 1,\forall s}};\underset{\text{There can be at most}|\overline{𝒱}|\text{VCs.}}{\underbrace{\sum _{k\in 𝒱}{Z}_k^V\le |\overline{𝒱}|}};\underset{\text{VC capacity constraint.}}{\underbrace{\sum _{s\in S}{d}_s^P{x}_{sk}\le C_k^V,\forall k}} \\ \underset{\text{Can not assign to a non-existing VC.}}{\underbrace{0\le x_{sk}\le Z_k^V,\forall s,k}};\underset{\text{Viability constraint.}}{\underbrace{x_{sk}\le a_{sk},\forall s,k}};Z_k^V\in \{\mathrm{0,1}\},\forall k \end{gathered}$$ (34)

Bravo et al. (2022) consider a formulation similar to the one given in (34), where the second objective function is not considered and the decision variable $x_{sk}$ is a binary assignment variable, which captures percent coverage in the current formulation. In addition to the classical facility location factors, Polo et al. (2015) consider healthcare workforce planning and spatial accessibility using GIS data (Radke and Mu 2000). Cabezas et al. (2021) use stochastic optimization to incorporate patient preference of vaccination centers (VCs) in terms of distance. Bertsimas et al. (2022) is the only paper that considers an epidemiological model within a location model. Dastgoshade et al. (2022) develop an allocation optimization framework using Rawls’ theory, Sadr’s theory, and utilitarianism fairness theories. Unlike other studies, Soria-Arguello et al. (2021) locate intermediate distribution centers (DCs) based on production plants and existing distribution centers (DCs). Kumar et al. (2022) compute number of optimal days a vaccination center (VC) should operate. Luo and Mehrotra (2023) integrate linear utility function with COVID-19 service center location decision that depends on travel distance, waiting time and location based service features. Zhang et al. (2022) merge appointment scheduling with vaccination center (VC) location problem. The location decisions are dependent on the accepted appointments and vaccination sequencing in each site. Cao et al. (2023) find the best locations for COVID-19 waste disposal centers by considering stochastic infection risk, while carrying the waste and the population around candidate disposal centers.

4.2. Coverage Models

Lim et al. (2016), Polo et al. (2015) and Srivastava et al. (2021) consider the maximum coverage problem (see (35)).

$$\begin{gathered} \mathrm{m}\mathrm{a}\mathrm{x}\underset{\text{Total coverage.}}{\underbrace{\sum _{s\in 𝒮}{\bar{P}}_s{\mathbf{𝔷}}_s}} \\ \text{s.t.}\underset{\begin{array}{c}\text{A PS cannot be covered by}\\ \text{a VC not within}{\mathrm{\eta }}_{\max }.\end{array}}{\underbrace{{𝔷}_s\le \sum _{k\in 𝒱\mid {\mathrm{\eta }}_{sk}\le {\mathrm{\eta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{Z}_k^V,\forall s}};\underset{\text{There can be at most}|\stackrel{-}{𝒱}|\text{VCs.}}{\underbrace{\sum _{k\in 𝒱}{Z}_k^V\le |\stackrel{-}{𝒱}|}};\underset{\text{Budget constraint.}}{\underbrace{\sum _{k\in 𝒱} c_k{Z}_k^V\le \mathrm{b}}} \\ {Z}_k^V,{𝔷}_s\in \{\mathrm{0,1}\},\forall k,s \end{gathered}$$ (35)

There are multiple ways to calculate the coverage coefficient $\left({\bar{P}}_s\right)$. The most common is to equate it to the population size in a population site (PS), assuming that in a covered site all population is covered. In practice, coverage can be affected by factors like accessibility and willingness to be vaccinated Xu et al. (2021). develop two different ${\bar{P}}_s$ measures based on the number of public transportation spots and the social vulnerability index. Lim et al. (2016) provide a step-wise distance based coverage formulation, where coverage percentage and distance are inversely proportional (see (36)).

$$\begin{gathered} \mathrm{m}\mathrm{a}\mathrm{x}\underset{\text{Step-wise coverage.}}{\underbrace{\sum _{s\in 𝒮}{\bar{P}}_s\sum _{q\in Q_s}{\mathrm{\theta }}_q{𝔷}_{sq}}} \\ \text{s.t.}\underset{\text{A PS cannot be covered by a VC outside allowed level.}}{\underbrace{{\mathbf{𝔷}}_{sq}\le \sum _{k\in {𝒱}_s}{Z}_k^V,{𝒱}_s=\left\{k\in 𝒱\mid {\mathrm{𝔶}}_{q-1}\le {\mathrm{\eta }}_{sk}\le {\mathrm{\eta }}_q\right\},\forall q,s}};\underset{\text{A PS can be covered in at most one level}q.}{\underbrace{\sum _{q\in Q_s}{\mathbf{𝔷}}_{sq}\le 1,\forall s}} \\ \sum _{k\in 𝒱}{Z}_k^V\le |\stackrel{-}{𝒱}|;\sum _{k\in 𝒱}{c}_k{Z}_k^V\le \mathrm{b};Z_k^V,{\mathbf{𝔷}}_{sq}\in \{\mathrm{0,\; 1}\},\forall k,q,s \end{gathered}$$ (36)

Lim et al. (2016) consider outreach centers (OC) for covering vaccination center (VC). Outreach happens when healthcare workers take vaccines to remote, temporary locations according to accessibility needs. Generally, these outreach centers (OCs) cover multiple rural vaccination centers (VCs) (see (37)).

$$\underset{\text{OC coverage constraint.}}{\underbrace{{\mathbf{𝔷}}_{sq}\le \sum _{k\in {𝒱}_s}\sum _{\ell \in O}{\mathrm{𝔵}}_{\ell k},{𝒱}_s=\left\{k\in 𝒱\mid {\mathrm{𝔶}}_{q-1}\le {\mathrm{\eta }}_{sk}\le {\mathrm{𝔶}}_q\right\},\forall q,s}};\sum _{q\in {𝒬}_s}{\mathbf{𝔷}}_{sq}\le 1,\forall s\underset{\text{OC capacity constraint.}}{\underbrace{\sum _{k\in 𝒱}{\mathfrak{x}}_{\ell k}\le C_{\ell }^O,\forall \ell }};\underset{\text{OC distance constraint.}}{\underbrace{{\mathrm{\eta }}_{\ell k}{\mathfrak{x}}_{\ell k}\le {\eta }_{\text{max}},\forall \ell ,k}};\underset{\text{A VC can be covered by one OC.}}{\underbrace{\sum _{\ell \in O}{\mathfrak{x}}_{\ell k}\le 1,\forall k}}$$ (37)

These types of location models support regional decision making by considering the specific requirements that may arise due to factors such as population density, existing pharmacy stores, and other infrastructure related factors. Further information on location-allocation articles is given in Table 10 in Appendix §A.

5. Uncertainty Modeling in COVID-19 Vaccine Supply Chain Models

In COVID-19 vaccine supply chain literature, demand uncertainty is the predominant form of uncertainty (Bennouna et al. 2022, Bertsimas et al. 2020, Enayati and Özaltın 2020, Jadidi et al. 2021, Jarumaneeroj et al. 2022, Mehrotra et al. 2020, Minoza et al. 2021, Rao and Brandeau 2021, Thul and Powell 2023, Yang et al. 2022, Bertsimas et al. 2022, Dastgoshade et al. 2022, Azadi et al. 2020, Mak et al. 2022, Azadi et al. 2020, Sinha et al. 2021, Basciftci et al. 2023, Chowdhury et al. 2022, Goodarzian et al. 2022, Habibi et al. 2023, Kohneh et al. 2023, Lai et al. 2021, Manupati et al. 2021, Sazvar et al. 2021, Shiri and Ahmadizar 2022, Shamsi Gamchi et al. 2021, Yang and Rajgopal 2021). However, research has also considered uncertainty in supply (Dastgoshade et al. 2022, Jahani et al. 2022, Karakaya and Balcik 2023, Mohammadi et al. 2022, Chowdhury et al. 2022, Wang et al. 2023), lead time (Jahani et al. 2022, Karakaya and Balcik 2023, Habibi et al. 2023, Manupati et al. 2021, Yang and Rajgopal 2021), infection (Roy et al. 2021, Cao et al. 2023, Basciftci et al. 2023), vaccination location choice (Cabezas et al. 2021, Luo and Mehrotra 2023), vaccine accessibility (Yarmand et al. 2014, Gilani and Sahebi 2022) and vaccine wastage (Shah et al. 2022, Rahman et al. 2023). While demand from a distibution center (DC) to manufacturer (Gilani and Sahebi 2022, Goodarzian et al. 2022, Manupati et al. 2021, Shah et al. 2022) and manufacturing lead time (Manupati et al. 2021) impact the DC-manufacturer level uncertainty, vaccination center (VC) level demand uncertainty and their variation across regions is influenced by numerous factors including gender, education, employment, exposure to misinformation, and hesitancy (Seboka et al. 2021). These uncertainties affect both DC-VC and VC-PS related echelons.

In terms of mathematical modeling of such uncertainties, stochastic inventory model (Manupati et al. 2021, Shah et al. 2022, Sinha et al. 2021, Mak et al. 2022, Yarmand et al. 2014, Chick et al. 2008), two-stage stochastic programming (Lai et al. 2021, Basciftci et al. 2023, Mohammadi et al. 2022, Karakaya and Balcik 2023, Shiri and Ahmadizar 2022, Azadi et al. 2020), robust and distributionally robust optimization approaches (Basciftci et al. 2023, Luo and Mehrotra 2023, Mohammadi et al. 2022, Gilani and Sahebi 2022, Wang et al. 2023), chance-constraint modeling (Goodarzian et al. 2022, Rahman et al. 2023, Azadi et al. 2020) are noteworthy. Forecasting, predictive analytics, machine learning methods, simulation tools as well as their ensembles (Bennouna et al. 2022) are also used to obtain estimates of the uncertain parameters and aid optimization under various types of uncertainties. Other practices of modeling uncertainty found in COVID-19 vaccine literature include queuing (Jahani et al. 2022), fuzzy programming (Kohneh et al. 2023), game theory (Sinha et al. 2021), decision trees (Manupati et al. 2021), epidemiological modeling (Bertsimas et al. 2020, Enayati and Özaltın 2020, Jadidi et al. 2021, Jarumaneeroj et al. 2022, Minoza et al. 2021, Rao and Brandeau 2021, Thul and Powell 2023, Yang et al. 2022), etc. The latter is separately explored in Section §7. In this section, we present stochastic inventory models (Section §5.1), two stage stochastic programming approaches (Section §5.2), robust/distributional robust approaches (Section §5.3), and chance constrained modeling (Section §5.4).

5.1. Stochastic Inventory Models

In response to supply chain uncertainties, vaccine literature, besides leveraging the inventory balancing equations discussed earlier, also uses to continuous or periodic inventory review policies. Mak et al. (2022) focus on the scenario of very limited supply during the early period of COVID-19, examine varied two-dose vaccine rollout policies by analyzing the related inventory dynamics.

Under a continuous review policy, the reorder point and reorder quantity are the key decisions to make such that the remaining inventory $\left(d_{\mathrm{m}\mathrm{a}\mathrm{x}}^{@l}\right)$ satisfies the lead time demand $(d(l))$ at the desired service level $(1-\mathrm{\alpha })$ (i.e., $\mathbb{P}\left(d(l)\le d_{\mathrm{m}\mathrm{a}\mathrm{x}}^{@l}\right)\ge 1-\mathrm{\alpha }$). Manupati et al. (2021) use a continuous review policy and EOQ structure to model this uncertainty assuming Normally distributed demand. In their work, resulting total cost for a distribution center (DC) and reorder point is expressed in terms of the order quantity as shown in (38). Shah et al. (2022) also use similar type of ensembles of continuous review and EOQ policy and investigate carbon emission from a vaccine inventory system.

$$\underset{\text{Total cost for DC}j.}{\underbrace{\left(\sum _i{c}_{ij}^{OC}\right)\left(\frac{d_j^D}{Q_j}\right)+c_j^{HC}\left(\frac{Q_j}{2}+{\mathrm{Z}}_{1-\mathrm{\alpha }}\sqrt{\left(l_j\right)\left({\mathrm{\sigma }}_{d_j^D}^2\right)}\right)}};\underset{\text{Reorder point of DC}j.}{\underbrace{\left(d_j^D\right)\left(l_j\right)+{\mathrm{Z}}_{1-\mathrm{\alpha }}\sqrt{{\left(d_j^D\right)}^2\left({\mathrm{\sigma }}_{l_j}^2\right)+{\left(l_j\right)}^2\left({\mathrm{\sigma }}_{d_j^D}^2\right)}}}$$ (38)

Periodic review policy-based inventory decisions that review stocks periodically are considered using a newsvendor modeling framework. Yarmand et al. (2014) propose a closed-form solution to Phase-1 allocation of a two-phase vaccine allocation problem using the newsvendor approach. This work uses the ratio of underage cost to summation of underage and overage cost and finds a closed-form solution for vaccine allocation. The newsvendor structure is also utilized to model contracts for supply chain coordination (Chick et al. 2008).

Sinha et al. (2021) use inventory balancing equations similar to those described in the earlier sections to model uncertainties. For instance, using forecasts, Sinha et al. (2021) replace the ${\sum }_p{d}_{kpt}$ term by $\mathrm{\gamma }{\sum }_p{d}_{kpt}$ in (16) for critical nodes of a supply chain network. Due to budget and infrastructure inefficiencies (especially in developing countries), vaccination centers (VCs) may not get sufficient number of vaccines from distribution centers (DCs) on a regular basis. A model in such a context adds binary variables $y_{jkt}^{dv}$ to decide at which time period a distribution center (DC) should supply vaccine to a vaccination center (VC). For a similar reason, lead time can be uncertain as well. Sinha et al. (2021) analyze major disruption scenarios and store additional vaccine supplies in strategic higher-echelon nodes to prevent transshipment lead time uncertainty.

5.2. Two-stage Stochastic Programming Approach

In a two-stage stochastic programming (TSSP) approach, the first stage makes here-and-now strategic decisions while, given the first stage decisions, the second stage makes wait-and-see decisions. Below, we outline a COVID-19 motivated TSS mixed-integer linear programming model from a number of papers (Mehrotra et al. 2020, Lai et al. 2021, Basciftci et al. 2023, Mohammadi et al. 2022, Karakaya and Balcik 2023).

$$\min \underset{\text{First-stage cost}.}{\underbrace{{\mathrm{c}}^{\mathrm{\top }}\mathrm{x}}}+\underset{\text{Expected optimal cost of second-stage problem}.}{\underbrace{{𝔼}_{\mathrm{\xi }\sim \mathbb{P}}[𝔓(\mathit{x},\mathrm{\xi })]}}\text{s.t.}\underset{\text{First-stage constraints}.}{\underbrace{\mathrm{A}\mathrm{x}\ge \mathrm{a},\mathrm{x\; is\; Integer}}}$$ (39)

$$\begin{gathered} \underset{\begin{array}{c}\text{Recourse problem parameterized by}\mathbf{x}\text{and}\\ \text{and random parameters}{\mathrm{\xi }}^{\mathrm{\omega }}\text{under scenario}\mathrm{\omega }\in \mathrm{\Omega }.\end{array}}{\underbrace{𝔓(\mathrm{x},{\mathrm{\xi }}^{\mathrm{\omega }})}}:=\min \underset{\text{Second-stage cost for scenario}\mathrm{\omega }.}{\underbrace{{\mathrm{e}}^{{\mathrm{\omega }}^{\mathrm{\top }}}{\text{y}}^{\mathrm{\omega }}}} \\ \mathrm{s.t.}\underset{\begin{array}{c}\text{Second-stage coupling constraint}\\ \text{with technology matrix}{\mathrm{T}}^{\mathrm{\omega }}.\end{array}}{\underbrace{{\mathrm{B}}^{\mathrm{\omega }}{\mathrm{y}}^{\mathrm{\omega }}\ge {\mathrm{b}}^{\mathrm{\omega }}-{\mathrm{T}}^{\mathrm{\omega }}{\mathrm{x}}^{\mathrm{\omega }}}};\underset{\begin{array}{c}\text{Second stage constraint}\\ \text{free of first-stage variable}\mathrm{x}.\end{array}}{\underbrace{{\mathit{D}}^{\mathrm{\omega }}{\mathrm{y}}^{\mathrm{\omega }}\ge {\mathrm{d}}^{\mathrm{\omega }}}} \\ {\mathrm{y}}^{\mathrm{\omega }}\text{is continuous or mixed-integer} \end{gathered}$$ (40)

Mehrotra et al. (2020) were the earliest to use a TSSP framework in the context of ventilator allocation. The TSSP model proposed by Lai et al. (2021) minimizes setup costs by considering the opening of vaccination centers (VCs), vaccination center (VC) assignment to population sites, and required healthcare personnel as first-stage variables. Each discrete scenario of the second stage model incorporates multiperiod planning and considers vaccine transportation, inventory, shortages, and satisfied demand at each location within the planning horizon as decision variables. The number of personnel required at each vaccination center (VC) in each time period is also considered, ensuring that the minimum average realized demand threshold is met based on their service rate. Additionally, a coupling constraint prevents the number of healthcare personnel needed to meet demand from exceeding the total workers assigned in the strategic stage at any given time period. Similarly, demand of a population site is satisfied by a vaccination center (VC) if only if the first stage establishes a one-to-one assignment between them.

In the work of Basciftci et al. (2023), the TSSP model minimizes setup and operational costs of different distribution centers (DCs), while considering inventory balancing and lead time. However, it does not account for the connection between facility requirements and healthcare workers. Karakaya and Balcik (2023), on the other hand, propose a TSSP model that focuses on supply level uncertainty and aims to minimize the expected weighted sum of deviations from the vaccination timeline for each priority group made in the first stage. In each scenario, their second stage model calculates earliness and lateness deviation, taking into account factors such as scenario-specific vaccination starting and completion time, dosage scheme, inventory availability, and other factors with necessary coupling of the first stage decisions. Mohammadi et al. (2022) present a TSSP model that incorporates uncertainty in both supply and demand, including parameters related to pandemic progression, vaccination effectiveness, social contacts, and uncertain death rates. Their objective is to minimize cost and the number of deaths, with the added consideration of vaccination center (VC) capacity determined through a queuing system.

5.3. Robustness and Distributionally Robust Approaches

In addition to their TSSP approach, Mohammadi et al. (2022) test a robust approach that minimizes deaths using a scenario-dependent second objective value $\left({\mathrm{\vartheta }}_2^{\mathrm{\omega }}\right)$ of a bi-objective model. The resulting objective given in (41) is subject to the same constraints as their TSSP model. To solve the problem, the non-linearities due to the death minimization objective and robustification are linearized.

$$\mathrm{m}\mathrm{i}\mathrm{n}\underset{\text{Expected number of deaths.}}{\underbrace{\sum _{\mathrm{\omega }\in \mathrm{\Omega }}{\mathrm{p}}^{\mathrm{\omega }}{\mathrm{\vartheta }}_2^{\mathrm{\omega }}}}+\underset{\text{Weighted mean deviation from mean objective value.}}{\underbrace{\check{\mathbf{M}}\sum _{\mathrm{\omega }\in \mathrm{\Omega }}{\mathrm{p}}^{\mathrm{\omega }}\left|{\mathrm{\vartheta }}_2^{\mathrm{\omega }}-\sum _{{\mathrm{\omega }}^{\prime }\in \mathrm{\Omega }}{\mathrm{p}}^{{\mathrm{\omega }}^{\prime }}{\mathrm{\vartheta }}_2^{{\mathrm{\omega }}^{\prime }}\right|}}$$ (41)

Motivated by Bertsimas et al. (2016), Gilani and Sahebi (2022) propose a dynamic robust optimization approach based on uncertainty sets and cutting hyperplanes, to reduce conservatism compared to its static counterpart while maintaining robustness. Wang et al. (2023) develop a two-stage robust optimization model for facility location and scheduling under supply uncertainty, with the goal of distributing vaccines to remote areas using drones.

Besides a TSSP model, Basciftci et al. (2023) also utilize a distributinally robust optimization (DRO) technique for robustification (Rahimian and Mehrotra 2019). The DRO approach selects the worst-case distribution from an ambiguity set $(ℬ)$ defined on some support set for random parameters, without assuming any specific probability distribution for the expected cost calculation in the TSSP model. Basciftci et al. (2023) assume finite support set and moment-based ambiguity set as defined in (42). Luo and Mehrotra (2023) propose a utility-robust service center location model with a pandemic case study. The ambiguity set used for uncertain linear utility function parameter are a generalized version of a moment-based ambiguity set (42) because of its additional dependence on the location decision.

$$\underset{\text{Finitely}\text{supported}\text{moment}\text{based}\text{ambiguity}\text{set}\text{under}\text{imperfect}\text{moment}\text{information}.}{\underbrace{ℬ=\{\text{p}\in {ℝ}_{+}^{|\text{Ω}|}\mid {\mu }_{{\tilde{d}}_{kt}}-{\mathrm{\epsilon }}_{kt}^{\mu }\le \sum _{\mathrm{\omega }\in \text{Ω}}{\text{p}}_{\mathrm{\omega }}{\tilde{d}}_{\mathrm{\omega }kt}\le {\mu }_{{\tilde{d}}_{kt}}+{\mathrm{\epsilon }}_{kt}^{\mu },{\mathrm{\sigma }}_{{\tilde{d}}_{kt}}{\underset{\_}{\mathrm{\epsilon }}}_{kt}^{\mathrm{\sigma }}\le \sum _{\mathrm{\omega }\in \text{Ω}}{\text{p}}_{\mathrm{\omega }}{\tilde{d}}_{\mathrm{\omega }kt}^2\le {\mathrm{\sigma }}_{{\tilde{d}}_{kt}}{\mathrm{\epsilon }}_{kt}^{-\mathrm{\sigma }},\forall k,t,{\text{p}}^{\top }1=1\}}}$$ (42)

5.4. Chance-constrained Modeling

In stochastic optimization, handling chance-constrained (CC) problems can be challenging. However, in the context of the vaccine literature, the normality assumption of random parameters simplifies the problem and is commonly used. For example, Azadi et al. (2020) use chance constraints to ensure that supplied amounts of pediatric vaccine at every vaccination center (VC) in every time period $\left(x_{jkpt}^{dv}\right)$, meet the random demand $\left({\tilde{d}}_{kpt}\right)$ at least $1-\mathrm{\alpha }$ percent of time (i.e., $\left.\mathbb{P}\left({\sum }_j x_{jkpt}^{dv}\ge {\tilde{d}}_{kpt}\right)\ge 1-\mathrm{\alpha },\forall k,p,t\right)$. Similarly, Rahman et al. (2023) impose chance constraints for total cost and make span of the vaccination program to remain within their random budget with at least a probability of $1-\mathrm{\alpha }$. If normality assumption remains in place, such presence of a random parameter only in the right hand side of the expression (not as a coefficient of variables) is very easy to deal with since resultant deterministic equivalent of CC is linear. Other than Azadi et al. (2020), Goodarzian et al. (2022) also leverage this technique to impose chance constraints for every vehicle responsible to distribute vaccine between each of the possible supply chain players: manufacturer (main and local plant) to distribution center (DC), or warehouse, warehouse to vaccination center (VC) (hospital, pharmacy), according to the respective uncertain demands. If ${\tilde{d}}_{th}$ is the random demand at time $t$ to be transported by vehicle $h$, a linear reformulation of $\mathbb{P}\left({\sum }_{j,k,p} x_{jkpth}^{dv}\ge {\tilde{d}}_{th}\right)\forall t,h$ is

$$\underset{\text{Each vehicle}h\text{should at least transport}1-\mathrm{\alpha }\text{confidence level (upper) of the demand at time}t.}{\underbrace{\sum _{j,k,p}{x}_{jkpth}^{dv}\ge {\mu }_{{\tilde{d}}_{th}}+{\mathrm{Z}}_{1-\mathrm{\alpha }}{\mathrm{\sigma }}_{{\tilde{d}}_{th}},\forall t,h}}$$ (43)

6. Equity, Sustainability and Multi-objective Modeling

6.1. Equity Considerations

In public health decision-making, equity plays a significant role. A significant number of papers has studied this concern in the context of COVID-19 (Balcik et al. 2022, Bertsimas et al. 2020, Enayati and Özaltın 2020, Munguía-López and Ponce-Ortega 2021, Dastgoshade et al. 2022, Lim et al. 2016, Fadaki et al. 2022, Rastegar et al. 2021, Shiri and Ahmadizar 2022, Tavana et al. 2021, Wang et al. 2023, Bennouna et al. 2022, Basciftci et al. 2023, Mohammadi et al. 2022). Equity can be integrated into decision models by incorporating it into the objective function and/or the constraints. Within the COVID-19 literature, two main dimensions of equity are examined: priority group-based (Fadaki et al. 2022, Enayati and Özaltın 2020, Shiri and Ahmadizar 2022, Basciftci et al. 2023, Mohammadi et al. 2022), geography-based (Lim et al. 2016, Wang et al. 2023, Bertsimas et al. 2020, Munguía-López and Ponce-Ortega 2021) or both (Rastegar et al. 2021, Tavana et al. 2021, Balcik et al. 2022, Dastgoshade et al. 2022, Bennouna et al. 2022). Priority group equity involves distributing vaccines equitably among diverse sub-populations, taking into account factors like risk, age, occupation, and other demographic characteristics. This approach ensures that individuals with higher vulnerability or essential roles have equitable access to vaccines. Geography-based equity aims to ensure that vaccines are distributed equitably across different geographical locations. Decision makers consider factors such as population density, healthcare infrastructure, and disease prevalence to achieve an equitable allocation of vaccines.

The commonly used equity measures are: equitable demand satisfaction (Tavana et al. 2021, Rastegar et al. 2021, Shiri and Ahmadizar 2022), deviation from fair coverage level (Balcik et al. 2022), prohibiting daily variations in the number of vaccinations distributed to each location (Bertsimas et al. 2020) and using equity frameworks (e.g., Social Welfare, Nash, Rawlsian Justice) (Munguía-López and Ponce-Ortega 2021). For example, Tavana et al. (2021), Rastegar et al. (2021) and Shiri and Ahmadizar (2022) use (44) and (45) to model equity in the form of demand satisfaction:

$$\underset{\begin{array}{c}\text{Maximizing}\text{the}\text{minimum}\\ \text{vaccination}\text{percentage}.\end{array}}{\underbrace{\max \underset{k,g}{\min }\left\{\frac{{\sum }_{p,t}{x}_{gkpt}^v}{{\sum }_{p,t}{d}_{gkpt}}\right\};}}\underset{\begin{array}{c}\text{Minimum}\text{demand}\text{satisfaction}\\ \text{rate}\text{constraint}.\end{array}}{\underbrace{\sum _{p,t}{x}_{gkpt}^v\ge \gamma \sum _{p,t}{d}_{gkpt}\forall g,k}}$$ (44)

$$\underset{\begin{array}{c}\text{Minimum demand satisfaction rate}\\ \text{constraint with assignment}.\end{array}}{\underbrace{\frac{{\sum }_{i,t}{x}_{ijpt}^{md}}{{\sum }_{g,k,t}{d}_{gkpt}{x}_{jk}}\ge \mathrm{\gamma },\forall j,p}}$$ (45)

Balcik et al. (2022) incorporate equity across different regions and priority groups using (46) and (47) based on the deviation of actual allocation from equitable quantity estimated. Negative deviation with respect to demand (first term of (46)) is discouraged by using a penalty term $\check{\mathbf{M}}$.

$$\underset{\begin{array}{c}\text{Weighted sum of deviation from}\\ \text{fair amount to demand ratio.}\end{array}}{\underbrace{\mathrm{m}\mathrm{i}\mathrm{n}\check{\mathbf{M}}\sum _{r=1}^{|ℛ|}\sum _{g=1}^{|𝒢|}\frac{\varsigma {\mathrm{\Delta }}_{rg}^{-}}{d_{rg}}+\sum _{r=1}^{|ℛ|}\sum _{g=1}^{|𝒢|}\frac{(1-\varsigma ){\mathrm{\Delta }}_{rg}^{+}}{d_{rg}}}}+\underset{\begin{array}{c}\text{Penalizing gap between minimum}\\ \text{percentage demand and allocation.}\end{array}}{\underbrace{\sum _{r=1}^{|𝒱|}\sum _{g=1}^{|𝒢|}\mathbf{M}\cdot {\mathrm{\Pi }}_{rg}}}$$ (46)

$$\begin{array}{c}\\ \underset{\begin{array}{c}\text{Deviation of allocation}\\ \text{from fair amount.}\end{array}}{\underbrace{\sum _{p=1}^{|𝒫|}{x}_{grp}^v=e_{rg}^v+{\mathrm{\Delta }}_{rg}^{+}-{\mathrm{\Delta }}_{rg}^{-}}};\underset{\begin{array}{c}\text{Allocation should not be}\\ \text{more than the demand.}\end{array}}{\underbrace{e_{rg}^v+{\mathrm{\Delta }}_{rg}^{+}-{\mathrm{\Delta }}_{rg}^{-}\le d_{rg}}},\forall r,g\end{array}$$ (47)

The approaches discussed in Munguía-López and Ponce-Ortega (2021) include Rawlsian justice and the Social Welfare II approaches. Rawlsian justice principle maximizes the minimum allocated vaccine over all regions. Social Welfare II ensures vaccine allocation per region by maximizing the sum of their squared deviations from the allocation quantity possible in each region without equity consideration to achieve the overall maximum allocation. Bertsimas et al. (2022) consider equity in deciding the location of vaccination centers (VCs) at a strategic level. The number of sites opened in each state is determined based on the total target facility setup weighted by the state’s population share, with efficiency-fairness trade-offs controlled through model parameters. Enayati and Özaltın (2020) incorporate equity using the well-known Gini coefficient (or index) expressed as $GI(f)=\frac{{\sum }_{g^{\prime }\in 𝒢} {\sum }_{g\in 𝒢} \left|f_g-f_{g^{\prime }}\right|}{2|𝒢|{\mathrm{\Sigma }}_{g^{\prime }\in 𝒢}{f}_{g^{\prime }}}$, where $f_{g^{\prime }}$ is a decision variable representing the fraction of population covered from population class $g^{\prime }$.

6.2. Sustainability Considerations

Sustainability focused models consider the environmental cost and social benefit, with the aim of minimizing carbon emissions and adhering to carbon emission budgets. The quantification of environmental cost typically involves measuring carbon emissions from activities such as truck usage, inventory holding, and waste generation or disposal (Gilani and Sahebi 2022, Goodarzian et al. 2022, Chowdhury et al. 2022, Goodarzian et al. 2022, Sazvar et al. 2021). On the other hand, the social cost is assessed by considering factors such as waiting times for vaccination, transmission risk at workplaces, minimizing unmet demand, delivery time reduction, and staff balancing. Meanwhile, social benefits are in the form of job creation (Gilani and Sahebi 2022, Goodarzian et al. 2022, Shamsi Gamchi et al. 2021, Goodarzian et al. 2022, Chowdhury et al. 2022, Sazvar et al. 2021). An illustrative example can be found in (48) where the two equations are two objective functions, based on the work of Goodarzian et al. (2022).

$$\mathrm{m}\mathrm{i}\mathrm{n}\underset{\text{Carbon emission cost of establishing DCs and VCs.}}{\underbrace{c^{EI}\left(\sum _j\left(Y_j^{D(c)}+Y_j^{D(vc)}+Y_j^{D(uc)}\right)+\sum _k{Z}_k^V\right)}}+\underset{\text{Carbon emission cost of vaccine transshipments.}}{\underbrace{c^{CR}\sum _{i,j,j^{\prime },k,k^{\prime },p,t}\left({\mathrm{\eta }}_{ij}{x}_{ijpt}^{md}+{\mathrm{\eta }}_{jk}{x}_{jkpt}^{dv}+{\mathrm{\eta }}_{j{j}^{\prime }}{x}_{j{j}^{\prime }pt}^{d{d}^{\prime }}+{\mathrm{\eta }}_{k{k}^{\prime }}{x}_{k{k}^{\prime }pt}^{v{v}^{\prime }}\right)}}\mathrm{m}\mathrm{i}\mathrm{n}\underset{\text{Total vaccine shortages.}}{\underbrace{\sum _{g,k,p,t}{s}_{gkpt}^v}}$$ (48)

Bonney and Jaber (2011) use carbon emissions, waste disposal cost and social cost, in addition to ordering cost per replenishment cycle, to derive the optimal order quantity of an environmentally-friendly EOQ model. Moreover, Shah et al. (2021) consider carbon tax and amount of carbon emissions during manufacturing, holding, preparation, purchase and inspection processes to build an inventory model that minimizes total cost of an inventory system per unit time.

6.2.1. Waste Management

There are two optimization papers that consider waste management in the context of COVID-19 vaccine supply chain. Abbasi et al. (2023) emphasize the importance of building a green SCM that not only addresses the environmental impact of biomedical and plastic waste but also considers the hygiene costs associated with waste management. In parallel, Işık and Yildiz (2023) delves into provincial-level waste management in Turkey, with a specific focus on medical waste generated during the vaccine distribution process.

6.3. Multi-objective Considerations

Equity, sustainability, and economic objectives often conflict. Several papers consider conflicting multiple objective functions in resource allocation (Anahideh et al. 2022, Minoza et al. 2021), location-allocation (Bertsimas et al. 2022, Cao et al. 2023, Dastgoshade et al. 2022, Leithäuser et al. 2021, Lusiantoro et al. 2022, Polo et al. 2015), inventory management (Fadaki et al. 2022, Jahani et al. 2022, Mohammadi et al. 2022), supply chain management (Chowdhury et al. 2022, Goodarzian et al. 2022, Gilani and Sahebi 2022, Goodarzian et al. 2022, Kohneh et al. 2023, Li et al. 2021, Sazvar et al. 2021, Tang et al. 2022, Wang et al. 2023, Mohammadi et al. 2022) and routing (Shamsi Gamchi et al. 2021). For example, expressions in (48) are conflicting, since, in order to prevent vaccine shortage, more vaccine should be shipped and more facilities should be located. In the bi-objective TSSP and robust optimization model of Mohammadi et al. (2022), death and cost minimization are also conflicting to each other. To handle the conflicting nature of the models, different multi-objective approaches have been employed. Regardless of the number of objectives in these multi-objective models, most literature eventually works with a single objective after using other objective(s) as $\epsilon$—constraints.

7. Models Considering Epidemiological Dynamics

Demand estimation in vaccine management is accomplished using epidemiological models. Several articles (especially in resource allocation category) unify mathematical optimization with compartmental epidemiological modeling (Bertsimas et al. 2020, Enayati and Özaltın 2020, Jadidi et al. 2021, Jarumaneeroj et al. 2022, Minoza et al. 2021, Rao and Brandeau 2021, Roy et al. 2021, Thul and Powell 2023, Yang et al. 2022, Bertsimas et al. 2022, Mak et al. 2022, Shamsi Gamchi et al. 2021, Bennouna et al. 2022), for decision making under demand uncertainty. SIR (Susceptible-Infectious-Recovered) model is the most commonly used compartmental epidemiological model, which is represented by a set of ordinary differential equations (ODE). The SIR model has many extensions (i.e., SEIR, SEIQR, SIRD, DELPHI). Bertsimas et al. (2020) and Bertsimas et al. (2022) consider the DELPHI-V model, representing the rate of susceptible $(S)$, exposed $(E)$, infectious $(I)$, undetected $(U)$, quarantined $(Q)$ and dead $(D)$ population in a certain region, using compartment-wise flow rates. The compartmental flows for a region are illustrated in Figure 2. Table 7 outlines the nomenclature used in the DELPHI-V model and the differential equations given in (49) through (56) provide the ODE representation of the system dynamics model.

$$\underset{\begin{array}{c}\text{Rate at which susceptible population under risk}\\ \text{class}g\text{flows out to other compartments}.\end{array}}{\underbrace{\frac{d{S}_g}{dt}}}=-\mathrm{\beta }{V}_g(t)-\mathrm{\alpha }\mathrm{\gamma }(t)(S_g(t)-\mathrm{\beta }{V}_g(t))\underset{\begin{array}{c}\text{Currently infected}\\ \text{population}.\end{array}}{\underbrace{\left(\sum _{g\in 𝒢}{I}_g(t)+I^{\prime }(t)\right)}}$$ (49)

$$\frac{d{S}^{\prime }}{dt}=+\mathrm{\beta }\sum _{g\in 𝒢}{V}_g(t)-\mathrm{\alpha }\mathrm{\gamma }(t)\left(S^{\prime }(t)+\mathrm{\beta }\sum _{g\in 𝒢}{V}_g(t)\right)\left(\sum _{g\in 𝒢}{I}_g(t)+I^{\prime }(t)\right)$$ (50)

$$\frac{d{E}_g}{dt}=\mathrm{\alpha }\mathrm{\gamma }(t)(S_g(t)-\mathrm{\beta }{V}_g(t))\left(\sum _{g\in 𝒢}{I}_g(t)+I^{\prime }(t)\right)-r^I{E}_g(t)$$ (51)

$$\frac{d{E}^{\prime }}{dt}=\mathrm{\alpha }\mathrm{\gamma }(t)\left(S^{\prime }(t)+\mathrm{\beta }\sum _{g\in 𝒢}{V}_g(t)\right)\left(\sum _{g\in 𝒢}{I}_g(t)+I^{\prime }(t)\right)-r^I{E}^{\prime }(t)$$ (52)

$$\frac{d{I}_g}{dt}=r^I{E}_g(t)-r^d{I}_g(t);\frac{d{I}^{\prime }}{dt}=r^I{E}^{\prime }(t)-r^d{I}^{\prime }(t)$$ (53)

$$\frac{d{U}_g}{dt}=r_g^U(t)I_g(t)-r^D{U}_g(t);\frac{d{H}_g}{dt}=r_g^H(t)I_g(t)-r^D{H}_g(t);\frac{d{Q}_g}{dt}=r_g^Q(t)I_g(t)-r^D{Q}_g(t)$$ (54)

$$\frac{d{D}_g}{dt}=r^D(U_g(t)+H_g(t)+Q_g(t))$$ (55)

$$\frac{dM}{dt}=r^d{I}^{\prime }(t)$$ (56)

Figure 2.

DELPHI-V Compartmental Model (Adapted from (Bertsimas et al. 2022))

Table 7.

Parameter Description of Compartmental Epidemiological Model

Notation	Description
$\mathrm{\beta }$	Vaccine effectiveness percentage.
$\mathrm{\alpha }$	Nominal infection rate.
$\mathrm{\gamma }(t)$	Governmental and societal response function modulating infection rate over different time.
$r^I$	Infection progression rate.
$r^d$	Detection rate.
$r^D$	Death rate.
$r_g^U(t)$	Transition rate to undetected compartment for risk group $g$.
$r_g^H(t)$	Hospitalization rate for risk-based population risk group $g$.
$r_g^Q(t)$	Quarantine rate for risk-based population risk group $g$.
$S_{rgt}\left(S_{rt}^{\prime }\right)$	Number of susceptible people in region $r$, risk group $g$ (vaccinated class, see Fig. 2) and time $t$.
${\overline{S}}_{rgt}$	Number of eligible people in region $r$, risk group $g$ and time $t$.
$V_{rgt}$	Number of vaccinated people in region $r$, risk group $g$ and time $t$.
$E_{rgt}\left(E_{rt}^{\prime }\right)$	Number of exposed people in region $r$, risk group $g$ (vaccinated class) and time $t$.
$I_{rgt}\left(I_{rt}^{\prime }\right)$	Number of infectious people in region $r$, risk group $g$ (vaccinated class) and time $t$.
$U_{rgt}$	Number of undetected people in region $r$, risk group $g$ and time $t$.
$D_{rgt}$	Number of dead people in region $r$, risk group $g$ and time $t$.
$M_{rgt}$	Number of immunized people in region $r$, risk group $g$ and time $t$.
$Q_{rgt}$	Number of quarantined people in region $r$, risk group $g$ and time $t$.
$H_{rgt}$	Number of hospitalized people in region $r$, risk group $g$ and time $t$.

7.1. Embedding SIR Dynamics into Optimization Modeling

In order to embed a compartmental model into an optimization model, ODEs are discretized within a time horizon $𝒯$ with discretization unit $\mathrm{\Delta }t$ (generally articles segment $𝒯$ into $|𝒯|$ many days). Equations (57) to (61) represent a unified epidemiological-optimization framework for region and compartment-based, multi-period vaccine allocation. The key decision variable $x^{\mathrm{\kappa }}$ is related to vaccine supply chain management while $y^{\mathrm{\kappa }}$ is related to the number of people in each compartment at each time period for each region (see (59) to (61)).

$$\min \underset{\mathrm{Cost}}{\underbrace{c^{\mathrm{\kappa }}{x}^{\mathrm{\kappa }}}}+\underset{\begin{array}{c}\text{Epidemiological spread and}\\ \text{deaths over time period}.\end{array}}{\underbrace{d^{\mathrm{\kappa }}{y}^{\mathrm{\kappa }}}}$$ (57)

$$\mathrm{s.t.}\underset{\begin{array}{c}\text{Manufacturer, DC and VC}\\ \text{spesific constrains}.\end{array}}{\underbrace{A^{\mathrm{\kappa }}{x}^{\mathrm{\kappa }}\le a^{\mathrm{\kappa }}}}$$ (58)

$$\underset{\begin{array}{c}\text{Discretized version of}\\ \text{ODEs (49) to}(56).\end{array}}{\underbrace{B^{\mathrm{\kappa }}{y}^{\mathrm{\kappa }}\ge b^{\mathrm{\kappa }};f(y^{\mathrm{\kappa }})\ge 0}}$$ (59)

$$\underset{\begin{array}{c}\text{Linking constraint sets (58) and (59) for meeting}\\ \text{vaccination target, schedule, etc}.\end{array}}{\underbrace{C^{\mathrm{\kappa }}{x}^{\mathrm{\kappa }}\le D^{\mathrm{K}}{y}^{\mathrm{\kappa }}+e^{\mathrm{\kappa }}}}$$ (60)

$$\begin{array}{l}{x}^{\mathrm{\kappa }}\text{is mixed binary or integer,}{y}^{\mathrm{\kappa }}\text{is continuous or pure integer}\\ \{\overline{S},S,V,E,I,U,D,M,Q\}\subseteq y^{\mathrm{\kappa }}\end{array}$$ (61)

Equation (59) builds the population upper bounds in each compartment. For example, (49) can be used in the form $S_{rgt+1}\ge S_{rgt}-\mathrm{\beta }{V}_{rgt}-{\mathrm{\alpha }}_r{\mathrm{\gamma }}_{rt}\left(S_{rgt}-\mathrm{\beta }{V}_{rgt}\right)\left({\sum }_{g\in 𝒢}{I}_{rgt}+I_{rt}^{\prime }\right)\mathrm{\Delta }t$ where, $\mathrm{\Delta }t=1$. Embedding $|𝒯|$ many forward difference based representative equations for the system of ODEs given in (49) to (56) into this optimization model for each region of interest gives an estimation of the proportion of population in each compartment in each time period, alongside of decisions on vaccine distribution. Note that, non-linearity arises from the multiplication of $S_{rg},I_{rgt}$ and $V_{rgt},I_{rgt}$ pairs. Finally, in (60), not all the elements are required to be linked.

It is common to write constraints in terms of epidemiological compartments. For instance, Bertsimas et al. (2022) use (62) to limit vaccination (V) by eligible population $(\stackrel{-}{\mathbf{S}}$), since number of susceptible individuals $\left(S_{rgt}\right)$ is highly dependent on vaccine availability. Note that eligible population $(\stackrel{-}{\mathbf{S}})$ is determined by susceptible population $(\mathbf{S})$ and vaccine effectiveness $(\beta )$.

$${\bar{S}}_{rgt+1}\le {\bar{S}}_{rgt}-\mathrm{\beta }{V}_{rgt}-\left(S_{rgt}-S_{rgt+1}\right)\forall r,g,t;V_{rgt}\le {\bar{S}}_{rgt}\forall r,g,t$$ (62)

Bertsimas et al. (2022) also decide where to locate mass vaccination centers (VCs) and how to assign each state to a vaccination center (VC) by embedding an epidemiological model into a facility location optimization model. The model aims to minimize total mortality and hospitalizations. Finally, their analytical approach determines the quantity of vaccines allocated to each vaccination center (VC) and the eligibility of individuals within a particular risk category in a state for receiving the vaccination. Several SIR related literature can be considered in the context of the general structure sketched above. The optimization model in Jarumaneeroj et al. (2022), for instance, is similar but considers an infectious case minimization objective. Barth et al. (2023) incorporated epidemiological dynamics with time-varying contact rate. The corresponding model is thus able to reflect multiple waves during pandemic. Note that other than an SIR type structure, a vaccine allocation model can reflect infection spread dynamics by incorporating subject-specific risk and contact information accounting social network, such as the model presented by Li and Aprahamian (2023) to underscore the benefits of customized vaccine distribution strategies.

7.2. Graph Laplacian-based Dynamics Representation

Epidemiological dynamics are also represented by graph Laplacian-based dynamic models. Enayati and Özaltın (2020) compute the minimum number of required vaccines to prevent the virus spread, while ensuring sufficient vaccine coverage. They model the infection growth in terms of generation instead of time (Hill and Longini Jr 2003). Using the current population in each compartment, their model gives the number of new infections due to contact with exposed or infectious individuals.

7.3. Model Parameterization and Herd Immunity

Embedding a discretized SIR model results in a large optimization model that can be computationally expensive to solve. Consequently, several papers incorporate SIR dynamics indirectly into their model. For example, Rao and Brandeau (2021) use a SIR model-generated patient population over time in each compartment as parameters in their optimization model. Their models examine different objectives separately including minimization of infections, deaths, life years lost, and quality adjusted life years. The model requires vaccination of a minimum fraction of population in each region. Minoza et al. (2021) also use the output from a SIR model as parameters in an optimization model that focuses on risk-class based prioritization for front-line workers and the elderly. As an alternative approach to incorporate an SIR model into optimization framework, Shamsi Gamchi et al. (2021) leverage optimal control theory to determine the proportion of susceptible population targeted for vaccination in a region.

On the other hand, the model proposed in Jadidi et al. (2021), Westerink-Duijzer (2017) aim to maximize the vaccinated population size and population achieving herd immunity using a herd immunity function (see (63)). The herd immunity function, $\mathrm{H}\mathrm{e}\mathrm{r}\mathrm{d}\left(f_r\right)$ (e.g., Lambert herd function (Westerink-Duijzer 2017)) is described as a variable of the fraction of total vaccinated population in region $r$.

$$\begin{gathered} \max \underset{\begin{array}{c}\text{Population size vaccinated}\\ \text{over all regions}.\end{array}}{\underbrace{\sum _r{\hat{P}}_r{f}_r}}+\underset{\begin{array}{c}\text{Population size benefiting from}\\ \text{herd immunity over all regions}.\end{array}}{\underbrace{\sum _r{\hat{P}}_r\mathrm{Herd}(f_r)}} \\ \mathrm{s.t.}\underset{\begin{array}{c}\text{Vaccinated Population cannot}\\ \text{outnumber vaccine availability}.\end{array}}{\underbrace{\sum _r{\hat{P}}_r{f}_r\le C^V}}\underset{\begin{array}{c}\text{At most susceptible}\\ \text{population fraction}\\ \text{can be vaccinated}.\end{array}}{\underbrace{0\le f_r\le S_r/{\hat{P}}_r,\forall r}} \end{gathered}$$ (63)

Instead of using a herd immunity function, MacIntyre et al. (2022), Rahman et al. (2023) use vaccine coverage coefficient, $\frac{1}{\beta }\left(1-\frac{1}{\alpha }\right)$, as a measure for herd immunity. When vaccine effectiveness $\beta$ is small and nominal infection rate $\alpha$ is large, vaccine coverage coefficient becomes larger than 1, implying that we need to vaccinate more people to achieve the desired herd-immunity level. Vaccine effectiveness is also addressed in Mohammadi et al. (2022) while minimizing expected number of deaths resulting from unvaccinated, single-dose and double-dose vaccinated groups.

7.4. Vaccine Roll-out Policy Evaluation With SIR Dynamics

Hu et al. (2023) propose a vaccine allocation model motivated by the trade-offs between partially vaccinating a segment of population versus fully vaccinating half of that segment by splitting each full vaccine dose into two under supply scarcity. Their work considers an SIR model with two vaccinated compartments, containing individuals with fractional and full dosage. Mak et al. (2022) evaluated vaccine roll-out policies such as hold-back, release, and single-dose-first strategies using SEIR dynamics, focusing on symptomatic cases, hospitalizations, and deaths.

8. Methodological Considerations and Case Studies

8.1. Methodological Considerations

Initial studies for COVID-19 vaccine supply chain management literature predominantly used commercial solvers to solve the analytical models developed. As the pandemic evolved, researchers integrated more realistic features into existing models or introduced new representations of previously studied features while adapting existing solution approaches. The following sections outline decomposition-based global optimization approaches (Section §8.1.1), decomposition-based heuristic approaches (Section §8.1.2), other heuristic and meta-heuristic approaches (Section §8.1.3), non-linear optimization approaches (Section §8.1.4) and multi-objective optimization algorithms (Section §8.1.5).

8.1.1. Decomposition and Global Optimization Approaches

These methods guarantee solution quality using a defined value for the optimality gap. Azadi et al. (2020) use the L-shaped method to solve their two-stage problem. Motivated by their problem structure, Zhang et al. (2022) utilize logic-based Benders to decompose location-assignment and appointment scheduling decisions. Wang et al. (2023) use a column-and-cut generation (C&CG) algorithm to solve their two-stage robust optimization problem, working with both primal and dual forms of C&CG. The primal approach requires exploring vertices of a specially structured uncertainty set, while a vertex-traversing algorithm helps obtain the worst-case scenario among all vertices. The dual approach dualizes the innermost problem, and the resultant bilinear problem is reformulated as a mixed-binary linear program by exploiting a vertex-traversal algorithm. This approach avoids the traditional Big-M approach. Enayati and Özaltın (2020) use the multi-parametric-disaggregation technique, a global optimization approach, to solve their proposed bilinear problem. This method introduces two new sets of variables (continuous and binary) for each bilinear term, in addition to a variable representing each bi-linear term. These representative variables are then expressed as a summation of a continuous set of variables with varying parameters as coefficients.

8.1.2. Decomposition-based Heuristic Approaches

These methods, while not assuring solution quality like global optimization, can still efficiently achieve good solutions depending on the problem and decomposition type. Lai et al. (2021) utilize a Benders decomposition-based heuristic to solve a two-stage stochastic model. Their method solves the LP relaxation of a mixed-integer linear subproblem to generate an optimality cut. Once a targeted optimality gap is reached, the fractional solution of the second stage is rounded up for integer components providing the heuristic solution. Georgiadis and Georgiadis (2021) treat hub/DC-VC level allocation as a subproblem in a vaccine supply chain network from manufacturer to VC. Their method first allocates VCs to nearby hubs and then condenses the model by clustering VCs within existing political boundaries, subsequently addressing the smaller hub-VC assignment problems with aggregated parameters. By treating binary solutions from this assignment as incumbent, original large-scale MILP subproblems become LPs, consequently reducing computational burden.

The heuristic decomposition method by Yang et al. (2021) first breaks down the main network using a hierarchical clustering technique, ensuring the resulting sub-network can handle a MILP problem. After solving the sub-problem corresponding to a sub-network, the method sequentially aggregates other sub-networks, keeping certain optimal hub-opening decisions fixed, while allowing adjustments to others. The nodes that are reconsidered as decision variables in the merged sub-networks depend on their positions. The aggregation-based algorithm by Karakaya and Balcik (2023), on the other hand, performs aggregation on the priority groups. The algorithm for priority group based vaccination timeline decision simultaneously deals with two groups: the highest priority group and the group of all others post-aggregation. The timeline established for the first group serves as a parameter in the subsequent iteration, which focuses on the second priority class and the rest in an aggregated form. To prevent potential delays in vaccinating the aggregated group due to its high demand (due to aggregation), the minimum coverage threshold is also suitably adjusted, guaranteeing an optimal timeline for each class at the end. Karakaya and Balcik (2023) also offer comprehensive guidance on modifying the initial vaccination calendar in response to unforeseen events, such as severe vaccine side effects during the planned period.

8.1.3. Other Heuristics and Meta-Heuristics

Heuristics are problem-specific strategies or rules of thumb that guide the search for scalable solutions but offer no performance guarantee and sometimes result in very poor solution quality. In comparison to heuristic approaches, meta-heuristics often yield better solutions, but they are generally computationally more intensive for exploring a large search space.

A commonly-practiced algorithm by local and state governments during the COVID-19 pandemic was the pro-rata heuristic, which divides the available vaccines equally based on the population size. The vaccine allocation heuristic by Orgut et al. (2023) distributes vaccines to population sites following capacity-to-demand ratio in ascending order. If a site’s demand is unmet, the heuristic checks neighboring sites in the order of proximity, adjusting available capacity with each allocation. After fulfilling a region’s entire demand, the process repeats for the next region, using the updated capacity-to-demand sequence. The greedy approach outlined in Li and Aprahamian (2023) ranks vaccination candidates based on a value derived from multiplying the associated risk measure, the likelihood of post-infection fatality, and the reduction in fatality rate after vaccination. Vaccine needs of a selected candidate are then fully met unless constrained by vaccine availability or budget. Although such an approach is optimal when restricted to a single period design, due to its myopic nature, it can be always far from an optimal allocation in a multi-period framework.

Enayati et al. (2023), Enayati et al. (2023) in their layered feasible path network formulation leverages depth-first-search algorithm to construct all viable paths between each origin-destination pair. Subsequent heuristic reduces layered network size through parameters that control the number of transshipments, and drone launch-landings in a vaccine delivery path. Thus, the computational cost to solve the resultant model is reduced. In Zhang et al. (2022), a meta-heuristic approach is proposed for a combined location-assignment and scheduling problem, which mitigates NP-hardness of the scheduling subproblem by using a heuristic sequencing policy. This policy allows for the elimination of binary variables that determine task sequences. The heuristic prioritizes tasks based on the smallest values of the earliest start and latest completion times. Consequently creating an approximation of the original MILP is created, which, in addition to other feasibility constraints, includes constraints for the start times of each vaccination task utilizing the predetermined sequences.

8.1.4. Nonlinear Optimization Approaches

A two-stage stochastic formulation of the facility location problem with location preference order by Cabezas et al. (2021) uses the accelerated dual ascent (ADA) algorithm. ADA in the first step solves the Lagrangian dual problem by a subgradient algorithm while the semi-Lagrangian-dual problem in the second step is solved by a dual ascent algorithm using the solution from the first step, followed by a heuristic for accelerating the entire procedure. Iterative coordinate descent algorithm, which takes a gradient descent step with respect to only one variable at a time, is used in Bertsimas et al. 2020, 2022) to solve the bilinear model, resulting from epidemiological dynamics considerations. Li and Aprahamian (2023) also employed the coordinate descent method, alternating between assignment variable and the product term associated to risk over multiple time periods, exploiting their multiplicative relationship. Among other methods, Barth et al. (2023) used sequential quadratic programming (SQP) to solve a discretized version of continuous time model that, alongside bilinearity also involves non-linearity resulting from time-dependent contact-rate considerations. We note that these non-linear optimization approaches for non-convex problems typically converges to a local solution, but the quality of the solution can be enhanced by exploring different initial solutions.

8.1.5. Algorithms for Multi-Objective Optimization

Among the types of algorithms used in multi-objective vaccine supply chain literature, especially including the sustainability criteria, weighted-sum approach, and epsilon-constraint approach are the two simplest ones used (e.g., Tang et al. (2022)). Nature-inspired and evolutionary-type algorithms, also known as meta-heuristics, appear next in the list of most common algorithm types. Under this category, grey-wolf optimization and its modifications (e.g., Goodarzian et al. (2022)), particle swarm optimization (PSO) (e.g., Chowdhury et al. (2022)), teaching-learning-based optimization (TLBO) and its hybrid with PSO and genetic algorithm (GA) (e.g., Goodarzian et al. (2022)), non-dominated sorting genetic algorithm II (NSGA-II) (e.g., Tang et al. (2022)), evolutionary strategies (ES) (e.g., Lim et al. (2022)), can be found in the literature. Other methods include, but not limited to, variable neighborhood search and LP-metric method (e.g., Goodarzian et al. (2022)). We note that constructing a Pareto frontier is an integral part of these works. We also note that evolutionary type algorithms are better regarded for building a Pareto frontier with a diverse set of non-dominated solutions compared to other multi-objective optimization approaches used.

8.2. Insights from Numerical Studies

In this section, we outline key highlights from representative numerical or case studies in the literature. While some studies use synthetic data to emphasize their main points, others integrate real data (e.g., Hajibabai et al. (2022)), supplemented with rough approximations. These studies either provide an artificial view of national vaccine supply chains, like those in the U.S. or India, or focus on specific regional details. In deterministic modeling, some papers validate models using a range of artificially generated parameter data and assessing their interpretational consistency, whereas under uncertainty, model outcomes are assessed under various uncertainty scenarios. Another aspect of model validation, observed in a few studies, involves stating modeling assumptions and evaluating their reasonableness in reflecting real-world conditions without oversimplification or misrepresentation. Regardless, sensitivity analysis is often performed with respect to central parameters such as budget in facility number, cost budget, vaccine availability, supplier/DC/VCs capacity, healthcare personnel service rate, minimum coverage threshold, minimum service ratio, cost-coefficient, parameters defining uncertainty/ambiguity set, etc.

Case studies also highlight the importance of using an optimal allocation policy compared to the pro-rata policy by using metrics such as region or risk class-wise number of active cases, deaths, (e.g., Bertsimas et al. (2020)), infected days (infection rate multiplied by length of time horizon in days) (e.g., Barth et al. (2023), percentage of unused vaccines, percentage of unmet demand (e.g., Orgut et al. (2023)), etc. For example, Bertsimas et al. (2020) showed that optimized vaccine allocation following their model can reduce $10-25\mathrm{\%}$ of the fatalities over the 90-day planning horizon. Balcik et al. (2022) compared several coverage-based key performance indicators (KPIs) across allocation policies systematically, factoring in priority groups and/or socio-economic differences. Barth et al. (2023) additionally assessed the efficiency of different heuristic vaccine allocation strategies, remarking on the consistency of the pro-rata policy compared to others. For more on such policies, see Section 4.1 of Barth et al. (2023).

8.2.1. Insights from Parameter Variation Effect

Mohammadi et al. (2022) demonstrated how decisions such as the number of vaccine centers, vaccine orders, transfers, trans-shipments, and first-dose and second-dose vaccinations per risk class vary across different uncertainty scenarios. Their study revealed that as vaccine campaign parameters (like warehouse capacity, dose time-lag, and manufacturer trust rate) enhance, deaths decrease and first-dose vaccinations increase. Longer vaccine delivery lead times, however, have the opposite effect. Mortality also increases with increased contact and death rates but reduces with vaccine effectiveness. The authors suggest prioritizing high-risk individuals for vaccination when supplies are limited but high-contact individuals otherwise. Vaccination strategy and dosing intervals depend on the situation: in the face of a lethal and highly contagious virus variant, it is important to fully vaccinate the high-risk group rather than just giving one dose to many. Boosting vaccination capacity can combat such variants, and resource sharing offers flexible capacity planning.

Sensitivity analyses in the literature explicitly show that equity can be improved via a change in facility capacity (e.g., Balcik et al. (2022), Orgut et al. (2023)). In addition to evaluating fairness and efficiency through case studies, many studies assess other performance metrics as indicated earlier in this section. When optimization models introduce or emphasize specific features, assessing their outcomes numerically is standard to highlight their importance. For instance, Balcik et al. (2022) evaluated the fairness measure they use against other known equity measures. Barth et al. (2023) analyzed how the ratio of available vaccine percentage to population size affects infected days, Gini-coefficients of vaccinations as well as Gini-coefficients of infected days.

9. Discussion

This section provides a comprehensive discussion of the above discussed models and their usefulness in a future pandemic. The discussion includes unique features of a vaccine supply chain during a pandemic (§9.1), trends in the literature (§9.2), and authors’ perspectives for model selection (§9.3).

9.1. Unique Challenges in Pandemic Vaccine Supply Chain

The vaccine supply chain presents unique challenges that distinguish it from other supply chains. First, it requires rigorous cold chain management, especially for vaccines necessitating ultra-cold storage. Secondly, it necessitates the development of adaptable distribution strategies that take into account limited manufacturing sites and unpredictable epidemiological patterns. Third, complex inventory management is needed due to diverse factors such as vaccine dose policies, vial sizes, vial shelf life, open vial shelf life, wastage, and the need for specialized healthcare staff. Additionally, the supply chain should maintain a delicate balance between equity and efficiency while addressing vaccine hesitancy and priority groups. Extending vaccine access to remote areas poses another hurdle, wherein drone delivery does not fully resolve the issue of healthcare personnel availability. Moreover, the integration of demand into allocation models is critical, considering vaccine preferences and epidemiological dynamics in quantifying demand.

9.2. Temporal Trends in the Literature

In consideration of the above mentioned distinctions, several temporal trends in the literature can be noted. In the initial phases of the pandemic, models considered are similar to those of conventional supply chains, albeit with some added emphasis on cold chain management. Additionally, real-life case studies were rare, most probably due to limitations in the availability of data. However, as the pandemic progressed, the number of real-life case studies increased, providing a rich repository of insights and experiences. In this stage, researchers also produced more realistic models that consider different sources of uncertainty and epidemiological dynamics, enhancing the relevance of these tools in public health decision-making.

A later wave of studies has aimed to advance equity and efficiency in vaccine deployment beyond that enabled by prior work. Recent work also features models that address vaccine efficacy in preventing transmission and infections, as well as vaccine hesitancy to take the vaccine among different segments of the population. Hence, we see a strong interest in modeling characteristics pertaining to human behavior and social dynamics as well as clinical performance of the developed vaccines against mutations of the vaccine.

Despite these advances, however, there are still important issues to study. For example, the more general question of how to prioritize the inoculation of various segments of the population (e.g., fractional dosing, prioritizing individuals of high-risk occupations/social interaction) while at the same time considering realistic models of vaccine characteristics (e.g., type, dosing, storage requirements) population mobility, social determinants of health, and access to healthcare are lacking.

Regardless of the specific circumstances, it is clear that vaccine allocation models that incorporate epidemiological dynamics and realistic mobility behavior can greatly enhance the effectiveness of public health decision-making. However, even without the explicit consideration of these aspects of the problem, optimization models that facilitate resource sharing among different geographical areas and possess the flexibility to adapt to changing environments can still prove to be highly valuable.

9.3. Perspectives for Model Selection

Models that primarily address location selection and opening decisions, as these are strategic choices that are not easily altered once made, should incorporate different practical factors. In parallel, there should be solution approaches that provide solutions of high quality. On the other hand, operational-level decision-making models, which prioritize enhancing coverage in hard-to-reach regions or fortifying vulnerable supply chains to mitigate the aftermath of a major impact, warrant attention for enhancing global vaccination efforts. Furthermore, it is advisable to develop models that considers epidemiological dynamics, either via compartment or network structure, to account for long-term effects. These models should focus on the time-varying nature of a pandemic and should be capable of optimizing resource allocations, while also considering factors such as vaccine availability, quarantine, lockdown measures, and other preventive strategies. Although such models may pose computational challenges, they are deemed to be ideal, even when epidemiological dynamics is represented in approximate or simulated forms.

Effective utilization of Operations Research often demands a high level of expertise, and without it, there exists a risk of improperly applying models and techniques. Hence, models should be carefully developed, providing clear and simple guidelines for public health officers to facilitate implementation and execution. Additionally, caution is required as recommendations from models based on unreliable or insufficient data can deviate significantly from reality, particularly in scenarios with rapidly changing dynamics such as the COVID-19 pandemic. It is also worth noting that vaccine supply chains can be influenced by cultural, social, or behavioral factors, which may prove challenging to quantify within a model’s framework. Moreover, if the chosen solution approach to address the modeled problem lacks quality or fails to align with the specific purpose, the effort and expenses invested in the model may outweigh the benefits it provides.

10. Concluding Remarks on Future Research Directions

A pandemic vaccine supply chain can be viewed in three phases: Phase-I represents the initial period following vaccine development, characterized by a pronounced gap between vaccine supply and demand for inoculation. During Phase-II, the supply of vaccines increases, meeting a substantial portion of the demand but still falling short of full coverage. Phase-III defines the state in which the vaccine supply matches or even surpasses the demand. Note that the pandemic vaccine deployment literature, including the planning of distribution center (DC) and vaccination center (VC) locations, is predominantly centered on problems pertaining to Phase-II. Below, we include some concluding remarks on fruitful research directions in modeling (§10.1 and §10.2) and solution methods (§10.3).

10.1. Models with Explicit Consideration of Epidemiological Dynamics

Traditionally, an estimated size of the susceptible population is regarded as the demand in vaccine allocation models. However, several factors, such as the administration of multiple vaccine doses, variations in vaccine efficacy, and fundamental epidemiological dynamics strongly influence the size of the population susceptible to infection over time. Hence, it is imperative that such decision making models incorporate effective epidemiological dynamics models to estimate the demand for vaccines over time in different geographical regions that may be going through different phases of the pandemic.

Although several papers incorporate compartmentalized epidemiological models (e.g., SIR-type models) into vaccine allocation models, present studies exhibit several limitations. As discussed in Section 7, directly integrating SIR dynamics into optimization models or employing separate models with diverse SIR parameters results in computational complexity or challenges in achieving interpretability. This leads to a critical question: is it possible to develop a computationally efficient SIR-surrogate that preserves essential epidemiological dynamics?

Moreover, the ever-evolving nature of virus challenges the accuracy of models incorporating epidemiological dynamics, even using advanced machine-learning models. Addressing this uncertainty underscores the need for using multiple prediction models. However, current studies lack a unified approach for integrating these models into optimization algorithms. Additionally, concerning the dynamic environment of a pandemic, it is worth exploring whether prediction and optimization should be performed sequentially or concurrently in a decision-making framework. Furthermore, in a sequential decision-making framework, the potential utilization of an integrated approach that combines offline and online learning for a fast-yet-more informed decision-making warrants new studies. The question of how contextual optimization model outcomes, modeled either with real or simulated data, respond to contextual changes should also be examined. The OR/MS community needs to address these critical issues to advance the field beyond superficial adjustments to carefully-studied vaccine supply chain models with potential for real-world impact.

10.2. Additional Modeling Considerations

Several additional gaps exist in the modeling of vaccine supply chain models. The uncertainty-driven dynamism of these problems poses the question of whether distribution networks aided by novel comprehensive topological metrics and graph representation learning can yield decisions that are more adaptable. Research examining the interplay between quarantine/isolation policies and vaccine allocation is notably scarce, as is the development of models that account for time-varying contact and mortality rates. Recognizing that vaccination does not uniformly reduce the risks of infection and mortality across various subpopulations in different risk categories, equity metrics based on demand or coverage may prove inadequate, prompting the exploration of alternative methods. Exploring vaccine rollout strategies that consider time-varying contact rates and the occurrence of multiple infection spikes, and utility-driven location models that factor in diverse determinants such as distance, regional income, and education, present promising avenues for further research. Modeling regulated by risk-neutral or risk-aversion type utility can further serve as a viable alternative to conventional vaccine hesitancy models.

Current discrete scenario-based approaches to two-stage stochastic programming fall short of addressing multiple spikes of infections during a pandemic. Multi-stage stochastic programming, while computationally intensive, could more accurately capture epidemiological dynamics over time. The challenge lies in devising effective surrogates for this approach and rigorously assessing their performance.

Finally, further developments on model validation approaches are required for ascertaining the adaptability of a vaccine supply chain network to rapid shifts in infection dynamics. Defining criteria for allocating limited vaccine supplies in a prioritized order during the initial vaccination phase (Phase-I) necessitates systematic analysis, streamlining long-term risks and benefits, while considering access to healthcare, social determinants of health, human behaviour, mobility and adherence to non-pharmaceutical interventions such as social distancing and vaccine hesitancy.

10.3. Solution Methodologies

The majority of reviewed work directly adopts solution methods from similar literature, which, regrettably, hampers the advancement of comprehensive models equipped with practical features. To address this gap, the OR/MS community needs to innovate effective solution strategies that leverage the inherent problem structure, decomposition techniques (e.g., graph-based methodologies), and valid inequalities synchronized with model condensation techniques, and harness combinatorial heuristic algorithms to solve realistic problems with large network structures. Such methods will be particularly crucial to efficiently solve multi-objective models with conflicting objectives as well as bilinear/biconvex models resulting from robust optimization formulations that are needed to represent the plethora of sources of uncertainties in this problem domain.

Appendix. Detailed Features of Optimization and Simulation Articles in Pandemic Vaccine Management

Table 8.

List of abbreviations used in Appendix.

Model
CCP: Chance Constraint Programming	CoP: Copositive Programming	CSP: Constraint Satisfaction Programming	DT: Decision Trees
FP: Fuzzy Programming	GT: Game Theory	I: Inventory Modeling	LP: Linear Programming
LRP: Location Routing Problem	MC: Markov Chains	MILP: Mixed Integer Linear Programming	MINLP: Mixed Integer Non-linear Programming
NLP: Non-linear Programming	OC: Optimal Control	Q: Queuing	RL: Reinforcement Learning
RO: Robust Optimization	S: Simulation	SCP: Set Covering Problem	SN: Social Networks
SP: Stochastic Programming	U: Utility Function	VRP: Vehicle Routing Problem
Uncertainty
D: Demand	DR: Deterioration Rate	I: Infection	VLC: Vaccination Location Choice
LT: Lead Time	S: Supply	VA: Vaccine Accessibility	VW: Vaccine Wastage
Commodity
Quantity:	M: Multiple	S: Single
Type:	ChV: Childhood Vaccine	CP: COVID-19 Pills	CT: COVID-19 Test
	CV: COVID-19 Vaccine	CW: COVID-19 Waste	GHS: General Health Service
	GV: General Vaccine	IV: Influenza Vaccine
Characteristics:	MD: Multiple Doses	P: Perishable	SD: Single Dose
Period
M: Multiple	S: Single
Echelon
DC: Distribution Center	GHSC: General Health Service Center	M: Manufacturer	MVC: Mobile Vaccination Clinic
OC: Outreach Center	PS: Population Sub-group	RT: Refrigerated Truck	VC: Vaccination Center
WDC: Waste Disposal Center
Objective Function
Quantity:	[Max]: Maximize	[Min]: Minimize	[n]: n Different Separate Objective Functions
Operator:	M: Multiple Objectives	S: Single Objective
Vehicle
Type:	MVC: Mobile Vaccination Clinic	RT: Refrigerated Truck
Characteristics:	He: Heterogeneous	Ho: Homogeneous

Figure 4.

Number of real-life case studies used by optimization-based articles according to countries.

Table 9.

Detailed features of the optimization articles studying resource allocation problems.

Article	Model	Prioritization	Uncertainty	Commodity	Period	Echelon: From	Echelon: To	Objective	Case Study	Solution Method
Abbasi et al. (2020)	MILP	✓	×	S,CV,SD	S	DC	PS	S, [Min] Vaccine oversupply, transshipment time.	Victoria, Australia	Risk-based heuristic algorithm.
Anahideh et al. (2022)	MILP	×	×	S,GV,SD	S	DC	VC	M, [Min] Geographical diversity, [Min] social fairness gap.	Baltimore, Chicago, New York, USA	LP Relaxation, Penalty Method, Binary Search
Balcik et al. (2022)	MILP	✓	×	M,CV,SD	S	DC	DC	S, [Min] Deviation from the fair coverage levels.	Turkey	Commercial Solver
Bandi et al. (2021)	CoP	✓	D	M,CV,MD,P	M	VC	PS	M, [Max] Vaccination rates, [Min] appointment waiting times.	Singapore	Copositive Program Reformulation
Barth et al. (2023)	OC	✓	D	S,CV,SD	M	VC	PS	S, [Min] Total number of infected people.	Synthetic	NLP reformulation by discretizing the time dimension.
Bennouna et al. (2022)	SP	✓	D	M,CV,MD	M	DC	PS	S, [Min] Expected number of deaths.	Massachusetts, USA	ML-based aggregation method.
Bertsimas et al. (2020)	NLP	✓	D	S,CV,SD	M	DC	PS	S, [Min] Number of deaths.	New York, USA	Iterative Coordinate Descent
Enayati and Özaltin (2020)	NLP	✓	D	S,IV,SD	M	DC	PS	S, [Min] Number of vaccine doses distributed.	Synthetic (Asian Flu Pandemic)	Discretization with Multi-parametric Disaggregation Algorithm
Hu et al. (2023)	OC	✓	D	S,CV,SD	M	VC	PS	S, [Min] Total number of infected people.	World	Optimal Control with Binary Search
Huang et al. (2017)	MILP	✓	×	M,IF,SD	S	DC	DC	S, [Max] Equal access.	Texas, USA	Commercial Solver
Jadidi et al. (2021)	NLP,S	✓	D	S,CV,SD	S	DC	PS	S, [Max] Total immunity among populations.	Synthetic (Another Paper)	Graph Theory Algorithm
Jarumaneeroj et al. (2022)	NLP	✓	D	M,CV,SD	M	DC	PS	S, [Min] Stress on healthcare system.	Thailand	Iterative Algorithm
Li and Aprahamian (2023)	NLP,SN,S	✓	×	S,CV,SD	M	VC	PS	S, [Min] Number of total fatalities.	Texas	Greedy solution scheme, solution algorithm using coordinate descent.
Minoza et al. (2021)	LP,S	✓	D	S,CV,SD	S	DC	PS	M, [Max] Vaccine to be allocated for each location, [Max] priority factor.	Quezon City, Philippines	Simulating Scenarios
Munguía-López and Ponce-Ortega (2021)	LP,NLP	✓	×	S,CV,SD	S	DC	DC	[3], [Max] Total vaccines allocated, [Max] smallest allocated vaccines, [Max] social welfare.	Mexico	Commercial Solver
Orgut et al. (2023)	MILP	✓	×	S,CV,SD	S	DC	VC	S, [Min] Maximum travel duration of patients.	Alabama	Pro Rata Vaccine Allocation Heuristic
Rao and Brandeau (2021)	MILP	✓	D	S,CV,SD	S	DC	PS	[4], [Min] New infections, [Min] deaths, [Min] life years lost, [Min] quality-adjusted life years.	New York, USA	Exact Analytical Conditions
Roy et al. (2021)	LP	×	I	S,CV,SD	M	DC	PS	S, [Min] Transportation cost.	New York, USA	Exact Solution Algorithm
Shukla et al. (2022)	MILP	✓	×	M,CV,SD	M	DC	VC,MVC	S, [Max] Vaccine distribution efficiency.	USA	Python NetworkX, Minimum Cost Flow Problem
Thul and Powell (2023)	SP,RL	✓	D	S,CV,SD,CT	M	DC	PS	S, [Max] Expected sum of cumulative rewards.	USA	Partially-Observable Markov Decision Process
Yang et al. (2022)	SP	✓	D	S,CV,SD	S	DC	PS	S, [Min] Total number of confirmed cases.	Synthetic	Commercial Solver
Yarmand et al. (2014)	SP	×	VA	S,GV,SD	M	DC	PS	S, [Min] Phase-I vaccination cost, expected Phase-II vaccination cost.	North Carolina	Newsvendor Closed Form Solution, Heuristic Algorithm

Table 10.

Detailed features of the optimization articles studying location-allocation problems.

Article	Model	Uncertainty	Commodity	Period	Facility Echelon	Demand Echelon	Objective	Case Study	Solution Method
Bertsimas et al. (2022)	NLP	D	S,CV,SD	M	VC	PS	M, [Min] Deaths, [Min] exposed, [Min] distance.	New York, USA	Iterative Coordinate Descent
Bravo et al. (2022)	MILP	×	S,CV,SD	S	VC	PS	S, [Max] Aggregate vaccinations.	California, USA	Commercial Solver
Cabezas et al. (2021)	SP	VLC	M,CV,SD	S	VC	PS	S, [Min] Demand coverage and facility cost.	Synthetic	Lagrangian Relaxation Heuristic
Cao et al. (2023)	RO	I	S,CW	M	WDC	VC	M, [Min] Transportation and waste disposal cost, [Min] infectious risk.	Maharashtra, India	KKT, Branch and Bound, Polyhedral Uncertainty Set
Dastgoshade et al. (2022)	SP	D,S	S,CV,SD	M	VC	PS	M, [Min] Network cost, [Max] social equity.	Iran	Sample Average Approximation, Lexicographic Goal Programming
Emu et al. (2021)	CSP	×	S,CV,SD	S	DC	PS	[4], [Max] Overall vaccine distribution, [Max] vaccine distribution among the higher priority groups, [Max] vaccine distribution among people located closer to DCs, [Max] vaccine distribution among the higher priority groups and people located closer to DCs.	Chennai, India	K-medoids Algorithm
Enayati et al. (2023)	MILP	×	S,CV,SD,P	S	DC	DC	S, [Min] Total operating costs.	Synthetic	Commercial Solver
Enayati et al. (2023)	MILP	×	S,CV,SD	S	DC	VC	S, [Min] Total operating costs.	Vanatu Islands	Commercial Solver
Kumar et al. (2022)	MINLP,I	×	S,CV,SD	S	VC	PS	S, [Min] Operational (transportation, location, salary) costs and travel distance.	Chandigarh, India	Two-stage MILP
Leithäuser et al. (2021)	MILP	×	S,CV,SD	S	VC	PS	M, [Min] Number of vaccination centers, [Min] number of physicans, [Min] sum of patient travel distance.	Germany	Different Assignment Heuristics (Greedy, Same State)
Lim et al. (2016)	MILP	×	S,ChV,SD	S	OC	PS	S, [Max] Demand coverage.	Bihar, India	Commercial Solver
Luo and Mehrotra (2023)	RO,U	VLC	S,CT	S	VC	PS	S, [Max] Total utility of the service received by the customers.	San Diego, USA	Mixed 0-1 second-order cone (MISOCP) formulation and cutting-plane algorithm.
Lusiantoro et al. (2022)	MILP	×	S,CV,SD	S	VC	PS	M, [Max] Coverage, [Min] weighted distance.	Yogyakarta, Indonesia	Commercial Solver
Polo et al. (2015)	MILP	×	S,GHS	S	GHSC	PS	M, [Max] Coverage, [Min] total distance.	Bogota, Colombia	Commercial Solver
Soria-Arguello et al. (2021)	MILP	×	M,CV,SD	S	M,DC	DC	S, [Min] Total operational cost.	Mexico	Commercial Solver
Srivastava et al. (2021)	MILP	×	S,CV,SD	S	DC	VC	S, [Max] Coverage of demand points, within a maximum time.	Madhya Prades, India	Greedy Adding Algorithm-Based Optimization
Zhang et al. (2022)	MILP	×	S,CV,SD	S	VC	PS	S, [Min] Operational cost (location, traveling distance, appointment rejection, tardiness).	Yiwu, China	Logic Based Benders Decomposition and Metaheuristic

Table 11.

Detailed features of the optimization articles studying inventory management problems.

Article	Model	Uncertainty	Commodity	Period	Echelons	Objective	Case Study	Solution Method
Azadi et al. (2020)	CCP	D	M,ChV,SD	M	DC,VC	S, [Max] Number of fully immunized children, vaccine availability.	Niger	Sample Average Approximation
Azadi et al. (2020)	SP	D	M,ChV,SD	M	DC,VC	S, [Min] Cost and open vial waste.	Bangladesh	Stochastic Benders Decomposition, L-Shaped Method
Bonney and Jaber (2011)	I	×	S,GHS	S	DC	S, [Min] Environmental cost.	×	Closed Form
Fadaki et al. (2022)	MINLP	×	S,CV,SD	M	DC,VC	M, [Min] Weighted risk of unvaccinated population, transshipment mechanism and return mechanism.	Victoria, Australia	Augmented Multi-period Algorithm
Georgiadis and Georgiadis (2021)	MILP	×	M,CV,SD,P	M	M,DC,VC	S, [Min] Distribution, inventory and wastage cost.	Greece	Two-Step Decomposition Algorithm
Hovav and Tsadikovich (2015)	LP	×	S,IV,SD	M	M,DC,VC	S, [Min] Operational and shortage cost.	Israel	Commercial Solver
Işık and Yildiz (2023)	RO	D,DR,S	S,CV,MD,P	M	DC,VC,WDC	M, [Max] Total vaccinations, [Min] holding, transhipment, waste management cost.	Izmir, Turkey	ε-constraint Method
Jahani et al. (2022)	NLP,Q	S,LT	M,CV,SD	M	M,VC	M, [Min] Expected vaccine waiting time, [Min] holding and ordering cost.	Melbourne, Australia	Goal Attainment Algorithm, Non-dominated Sorting Genetic Algorithm
Karakaya and Balcik (2023)	SP	S,LT	M,CV,MD	M	DC,PS	S, [Min] Deviations from the announced COVID-19 calendar.	Norway	Aggregation-based algorithm.
Mak et al. (2022)	I	D	S,CV,MD	M	DC,VC	Three roll-out scenarios are compared: (i) holding back second doses, (ii) releasing second doses, and (iii) stretching the lead time between doses.	Synthetic	Proof Based
Mohammadi et al. (2022)	RO,Q	S	S,CV,MD	S	DC,VC,PS	M, [Min] Total expected number of deaths among the population, [Min] total distribution cost.	France	Scenario-based robust-stochastic optimization approach.
Shah et al. (2022)	I	VW	S,CV,SD,P	M	M,DC	S, [Min] Ordering, manufacturing, holding, inspection and carbon emission cost.	Synthetic	Exact Solution Algorithm
Sinha et al. (2021)	I,GT	D	S,CV,SD,P	M	DC,VC	[3], [Min] Total cost, under no disruption scenario, [Min] total cost, under disruption, [Min] excess inventory cost.	Gorakhpur, India (Japanese Encephalitis Vaccine)	Heuristic Algorithm

Table 12.

Detailed features of the optimization articles studying supply chain management problems.

Article	Model	Uncertainty	Commodity	Period	Echelons	Objective	Case Study	Solution Method
Abbasi et al. (2023)	MILP	×	S,CV,SD	S	M,DC,VC,PS,WDC	M, [Min] Economic cost, [Min] environmental cost.	Iran	Multi-objective grey wolf optimizer algorithm.
Basciftci et al. (2023)	SP,RO	D,I	S,CV,SD	M	DC,PS	S, [Min] Total cost of locating DCs, installing their capacities, expected cost of shipping, holding inventory, back-orders.	Michigan, USA	Moment-based ambiguity set approach.
Chowdhury et al. (2022)	MILP	D,S	S,CV,SD,P	M	M,DC,VC	M, [Min] Operational cost, [Min] environmental cost, [Max] job opportunities.	Bangladesh	Two heuristic algorithms: multi-objective social engineering optimizer and multi-objective feasibility enhanced particle swarm optimization.
Gilani and Sahebi (2022)	RO	VA	M,CV,SD	M	M,DC,VC	M, [Min] Economical cost, [Min] environmental cost, [Min] social cost.	Iran	Cutting Planes Approach
Goodarzian et al. (2022)	MILP,CCP,S	D	M,CP	M	M,DC,VC	M, [Min] Operational cost [Min] environmental cost, [Max] social responsibility.	USA	Multi-Objective Teaching-learning-based Optimization Heuristic, Particle Swarm Optimization Heuristic, Genetic Algorithm Heuristic
Goodarzian et al. (2022)	MILP	×	M,CV,SD	M	DC,VC	M, [Min] Operational, shortage, surplus cost, [Min] unmet demand, [Min] delivery time, [Min] environmental impact.	Iran	Gray Wolf Optimization Heuristic, Variable Neighborhood Search Heuristic
Habibi et al. (2023)	SP,MC	D,LT	S,CV,SD	S	DC,VC	S, [Min] Operational cost.	Iran	Two heuristics: population-based simulated annealing and improved grey wolf optimizer.
Kohneh et al. (2023)	FP	D	M,CV,SD,P	M	DC,VC,PS	M, [Min] System cost, [Max] vaccination service level.	Mashhad, Iran	Model Linearization, Defuzzification
Lai et al. (2021)	SP	D	M,CV,SD	M	VC,PS	S, [Min] Operational cost.	Synthetic	Benders decomposition-based heuristic algorithm.
Li et al. (2021)	MINLP	×	S,CV,SD	S	VC,PS	M, [Min] Travel distance, cost, [Max] number of fully opened vaccination stations.	Shenzhen, China	Two-stage MILP Reformulation, ε-Constraint Approach
Lim et al. (2022)	MILP	×	S,CV,SD	S	DC,VC	S, [Min] Operational, logistical cost.	Sub-Saharan Africa	Commercial Solver, Hybrid Heuristics for Large Scale
Manupati et al. (2021)	MILP,DT	D,LT	S,CV,SD	M	M,DC,VC	S, [Min] Overall cost.	India	Commercial Solver
Rahman et al. (2023)	MILP,CCP	VW	M,CV,SD	M	DC,VC,PS	S, [Min] Operational cost.	Canberra, Australia	Heuristic Genetic Algorithm
Rastegar et al. (2021)	MILP	×	S,IV,SD	M	DC	S, [Max] Minimum vaccine distributed-to-demand ratio.	Iran	Commercial Solver, Linearization
Sazvar et al. (2021)	RO,FP	D	S,IV,SD	M	M,DC,PS	M, [Min] Economic cost, [Min] environmental cost, [Max] social responsibility, [Max] resilience.	Iran	Multi-choice goal programming with a utility function approach.
Shiri and Ahmadizar (2022)	SP	D	M,CV,SD	M	DC,VC	S, [Min] Transportation, operation, shortage cost.	Iran	Scenario Reduction Algorithm
Tang et al. (2022)	MILP	×	S,CV,SD	S	VC,PS	M, [Min] Operational, inventory cost, [Min] total travel distance.	Beijing City, China	Tailored Genetic Algorithm, DP Based Heuristic
Tavana et al. (2021)	MILP	×	M,CV,SD	M	M,DC	S, [Max] Minimum delivery-to-demand ratio.	India	Commercial Solver
Wang et al. (2023)	RO	S	S,CV,MD	M	DC,VC	M, [Max] vaccine benefits, [Min] cost.	Linshui, China	Column-and-Constraint Generation, Vertex Traversal and Dual Methods
Xu et al. (2021)	MILP	×	S,CV,SD	S	DC,OC	[3], [Max] Total coverage using three different coverage measures.	New Jersey, USA	Commercial Solver
Yang et al. (2021)	MILP	×	S,ChV,SD	S	DC,VC	S, [Min] Annual hub facility, transportation costs.	Sub-Saharan Africa	Disaggregation and Merging Algorithm

A. Appendix: Vehicle Routing Considerations for Vaccine Distribution

For the last-mile delivery of pandemic vaccines, routing of refrigerated trucks (Shamsi Gamchi et al. 2021) or mobile vaccine clinics (Yang and Rajgopal 2021, Yucesoy et al. (2022)) plays an important role. Routing problems are considered for the VC-PS part of the supply chain, under scenarios where vaccination center (VC) is mobile. All of the studies consider a single commodity, a single central depot and assume that the vehicles are equipped with all of the required equipment (i.e., vaccines, refrigerated storage units, and nurses to administer the vaccine). Shamsi Gamchi et al. (2021) propose a routing problem under an epidemiological model, after a disaster. Given the quantity of available vaccines and information about heterogeneous vehicles, the model distributes vaccines according to prioritization schemes by determining the time at which a vaccine is dropped at a population site (PS) (see (64)):

$$\begin{gathered} \underset{\begin{array}{l}\text{A region can be}\\ \text{served by one vehicle}.\end{array}}{\underbrace{\sum _h{z}_{rh}=1,\forall r;}}\underset{\begin{array}{c}\text{All vehicles that depart the}\\ \text{depot should return back.}\end{array}}{\underbrace{\sum _r{y}_{r0}\le |\mathscr{H}|;\sum _r{y}_{0r}\le |\mathscr{H}|;}}\underset{\text{Sequencing constraints.}}{\underbrace{\sum _{r^{\prime }}{y}_{r{r}^{\prime }}=1,\forall r;\sum _{r^{\prime }}{y}_{r^{\prime }r}=1,\forall r}} \\ \underset{\text{Sub-tour elimination constraint}.}{\underbrace{\sum _{r,r^{\prime }}{y}_{r^{\prime }r}\le |{ℛ}^{-}|-1,\forall {ℛ}^{-}\subseteq ℛ\backslash \{0\},|{ℛ}^{-}|\ge 2}} \\ \underset{\text{Vehicle capacity loading constraint.}}{\underbrace{\sum _r{d}_{rg}{z}_{rh}\le C_h^H,\forall h}};\underset{\begin{array}{c}\text{Consecutive vaccination start time}\\ \text{considering order}.\end{array}}{\underbrace{T_{r^{\prime }}\ge y_{r{r}^{\prime }}{(T}_r+{\mathrm{\tau }}_{r{r}^{\prime }}+{\mathrm{\tau }}_r^S),\forall r,r^{\prime },r\ne r^{\prime }}} \end{gathered}$$ (64)

This article uses time to serve a region and waiting time in each region to approximate the social cost per region in their objective function. On the other hand, Yang and Rajgopal (2021) and Yucesoy et al. (2022) assume homogeneous vehicles for routing vaccines to different priority groups (i.e., pregnant, elderly and disabled people) in different locations. Yucesoy et al. (2022) assume that the demands at each node are known whereas Yang and Rajgopal (2021) use optimization under uncertainty techniques for handling demand uncertainty.

One major feature in both Yang and Rajgopal (2021) and Yucesoy et al. (2022) is that the model selects population sites (PSs) to visit under demand information at hand, with the binary decision variable $Z_r^R$. Subsequently, the routing constraints are expressed as ${\sum }_{r^{\prime }} y_{r{r}^{\prime }}=Z_r^R,\forall r$ and ${\sum }_{r^{\prime }}{y}_{r^{\prime }r}=Z_r^R,\forall r$. Service time in each population site (PS) is treated as a decision variable and the total time budget constraint is formulated. Moreover, some articles consider routing within the entire supply chain design (Chowdhury et al. 2022, Habibi et al. 2023). Further information on routing articles can be found in Table 13.

Table 13.

Detailed features of the optimization articles studying routing problems.

Article	Model	Uncertainty	Commodity	Period	Vehicle	Echelons	Objective	Case Study	Solution Method
Shamsi Gamchi et al. (2021)	VRP	D	S,GV,SD	S	RT,He	DC,PS	M, [Min] Social cost, [Min] routing cost.	Tehran, Iran	Dynamic Programming
Yang and Rajgopal (2021)	VRP,SCP	D,LT	S,ChV,SD	M	MVC,Ho	DC,PS	S, [Min] Vehicle and outreach trip cost.	Sub-Saharan Africa	MILP Commercial Solver, Multi-period Stochastic Modeling Approach
Yucesoy et al. (2022)	LRP	×	S,CV,SD	S	MVC,Ho	DC,PS	S, [Max] Vaccination service levels.	Van, Turkey	Commercial Solver

B. Appendix: Additional Literature

We now review papers that do not use mathematical optimization models but use simulation approaches (§B.1) to support decisions (§B.2) in vaccine management using descriptive statistics, observation, or other relevant methods in order to identify and conceptualize crucial steps. These papers highlight some additional, less studied aspects of vaccine management. Typically, these studies involve testing different scenarios using a simulation model and offering managerial insights for public health authorities, by observing the trends in the identified Key Performance Indicators (KPIs).

B.1. Simulation-based Articles

Table 14 contains information about the decision levels, type of the simulation model, commodity, duration of the simulation, priority groups, and case studies used in simulation-based articles.

Prioritization:

A large number of the simulation papers analyzed questions on the prioritization of different population subgroups, due to vaccine scarcity at the early stages. The most common KPIs studied under a number of alternative prioritization schemes (e.g., by age groups, by profession, or by infection risk/immunity) are the number of infections, hospitalizations, and mortality. Chen et al. (2020) and Rosenstrom et al. (2022) also consider fairness by using the Gini index and race-based sub-population grouping, respectively. Papers that employ system dynamics models (i.e., SIR-type) addressed prioritization issues through the use of additional compartments that reflect differences between different population subgroups, but typically agent-based simulation models are deployed when analyzing the impact of interactions that may result in infections (Chen et al. 2021, Oruc et al. 2021).

The simulation based studies resulted in valuable insights shaping the COVID-19 vaccine deployment, such as prioritization of the elderly to minimize mortality, prioritization of younger and working individuals to minimize spread, benefits of dynamic prioritization (first vaccinate elderly, then switch to young), importance of equity objectives in reducing health disparities, etc. In addition, simulation approaches provided timelines for the duration of the pandemic as well as required number of vaccine doses to reach herd immunity (Chaturvedi and Chakravarty 2021).

Managing multiple vaccines:

COVID-19 vaccines have different, possibly time-dependent, efficacy, resource, storage and dosing requirements, and became available at different times during the pandemic. While most papers have considered a single type of vaccine, a few focused on issues related to managing multiple vaccines. For example, Kim et al. (2021) provide the insight that a vaccine with fewer resource requirements and becoming available later can reduce the virus spread as much as a vaccine that becomes available earlier but requires more resources. In subsequent work, Kim et al. (2022) showed that if a vaccine with lower initial efficacy can be distributed fast, then it would be as effective as a vaccine with higher initial efficacy with a slower distribution rate, assuming that vaccine efficacy drops by time. In a similar vein, Shim (2021) show that to minimize deaths under scarce supply of vaccines, COVID-19 vaccines with lower efficacy should be given to younger people and COVID-19 vaccines with higher efficacy should be given to older people, highlighting an important principle in prioritization with multiple vaccines. Finally in Romero-Brufau et al. (2021) a simulation model was constructed to compare standard COVID-19 vaccination with a delayed second dose strategy, prioritizing the first dose, for vaccines with varying degrees of effectiveness. The results indicate that implementing a delayed second dose vaccination plan, specifically for individuals under the age of 65, could potentially lower the overall number of cumulative deaths under specific circumstances.

Management of vaccine inventory:

Vaccine vial size, recommended dosing, and population targeted for deployment are parameters that affect vaccine inventory management. For example, there are significant operational differences between storing and administering vaccines with 10-dose vials versus those with 2-dose vials. Assi et al (2011, 2012) build discrete-event simulation models where every vaccine with its storage location, refrigerator, freezer, and transport device is modeled as an entity to study the impact of vial size, vaccination period duration, and target vaccination population on supply chain bottlenecks.

B.2. Additional Considerations

Conceptual articles:

Dai and Song (2021) outline the COVID-19 supply chain research agenda under supply, demand, and supply-demand matching and highlight important areas such as cold chain management, demand eligibility, and vaccine hesitancy. Forman et al. (2021) outline the obstacles to successful and coordinated international vaccine initiatives to prevent COVID-19, as well as proposed strategies to overcome them. Schmidt et al. (2021) use descriptive statistics to assess the equity of COVID-19 vaccine allocation within the US, and conclude that policymakers at the federal, state, and municipal levels should widely adopt the use of disadvantage indices and associated place-based indicators to ensure that fairness plays a prominent role in allocation strategies. Alam et al. (2021) use the decision-making trial and evaluation laboratory framework to examine the main challenges of the COVID-19 vaccine supply chain and come up with practical policy guidelines for the decision-makers. The article identifies the main bottlenecks of the COVID-19 vaccine supply chain as (i) limited vaccine producers, (ii) improper collaboration with local organizations, (iii) lack of vaccine monitoring (iv) difficulties in regulating vaccine temperature, and (v) high cost of vaccinations. Li et al. (2021) describe the main methods used by a sizable hospital pharmacy department to overcome the challenges of preparing a large number of COVID-19 vaccine doses in a short period of time. Cano-Marin et al. (2023) utilize social media data and natural language processing to demonstrate that vaccine supply chain challenges, especially in developing countries, significantly affect COVID-19 healthcare responses, emphasizing the critical need for improved coordination in vaccine supply chains to ensure equitable access.

Vaccine hesitancy:

As mentioned in Dai and Song (2021), a major obstacle to vaccination adoption is vaccine hesitancy, which the World Health Organization named as of the top threats to world health. Vaccine hesitancy is quantified using surveys. Bogart et al. (2021) performed a questionnaire among HIV-infected African Americans about COVID-19 vaccine hesitancy and found that more than half of the participants supported at least one COVID-19 treatment reluctance or vaccination stance. Silva et al. (2021) deployed a comparative questionnaire at a college campus to quantify COVID-19 and influenza vaccine hesitancy among college students. The results show that if the vaccinations are shown to be both safe and effective, college students are generally willing to get the vaccine. Sallam (2021) reviewed the vaccine hesitancy surveys and came up with a vaccine hesitancy percentage for each country. Troiano and Nardi (2021) performed a narrative review to find the underlying factors of vaccine hesitancy (i.e., age, gender, race, belief, education, work status) in the context of COVID-19. Finally, Hegde et al. (2023) proposed a framework with two distinct operating modes to examine service rates, which signify the capacity of the system to administer vaccinations under the first mode and the throughput affected by vaccine hesitancy under the second mode.

Waste management:

Mass COVID-19 vaccination practices increase medical waste dramatically. The typical waste is vaccine vials, needles, syringes, and plastic vaccine administration equipment. Hasija et al. (2022) conceptualize and classify the types of vaccine waste and provide guidelines for vaccine waste management and treatment. Similarly, Rayhan et al. (2022) assess the COVID-19 vaccine-related waste management practices in Bangladesh.

Table 14.

Detailed features of the simulation-based articles.

Article Prioritization	Decision Level					Model Characteristics
	Inventory	Allocation	Distribution	Type	SIR	Commodity	Period	Priority Groups	Case Study
	Bubar et al. (2021)	×	✓	×	SEIR Simulation	✓	Single COVID-19 vaccine.	One year.	✓ Age based.	USA
Chaturvedi and Chakravarty (2021)	×	×	✓	SEIR Simulation	✓	Single COVID-19 vaccine.	Nine months.	×	India, Brazil, USA
Chen et al. (2020)	×	✓	×	SAPHIRE Simulation	✓	Single COVID-19 vaccine.	Three months.	✓ Age based.	New York, USA
Chen et al. (2021)	×	✓	×	Agent Based Simulation	✓	Single COVID-19 vaccine.	Four months.	✓ Social contact based.	Virginia, USA
Foy et al. (2021)	×	✓	×	SEIR Simulation	✓	Single COVID-19 vaccine.	Five years.	✓ Age based.	India
Fujimoto et al. (2021)	×	✓	×	SEIR Simulation	✓	Single COVID-19 vaccine.	550 days.	✓ Antibody testing based.	Synthetic
MacIntyre et al. (2022)	×	✓	✓	SEIR Simulation	✓	Single COVID-19 vaccine.	700 days.	✓ Age, health worker based.	New South Wales, Australia
Oruc et al. (2021)	×	✓	×	Agent Based Simulation	✓	Single COVID-19 vaccine.	16 months.	✓ Age, risk based.	Georgia, USA
Rosenstrom et al. (2022)	×	✓	×	Agent Based Simulation	✓	Single COVID-19 vaccine.	17 months.	✓ Income, race based.	North Carolina, USA
Walker et al. (2022)	×	✓	×	SEPIR Simulation	✓	Single COVID-19 vaccine.	731 days.	✓ Age, health worker based.	USA
Multiple Vaccine Management
Kim et al. (2021)	×	✓	×	SIRD Simulation	✓	Two different COVID-19 vaccines.	One year.	×	Synthetic
Kim et al. (2022)	×	×	✓	SIRD Simulation	✓	COVID-19 vaccines with different efficacy.	One year.	×	Synthetic
Romero-Brufau et al. (2021)	×	✓	×	Agent Based Simulation	✓	COVID-19 vaccines with different efficacy.	180 days.	✓ Age, health worker based.	USA
Shim (2021)	×	✓	×	SEIR Simulation	✓	COVID-19 vaccines with different efficacy.	One year.	✓ Age based.	South Korea
Management of Vaccine Inventory
Assi et al. (2011)	✓	✓	✓	Discrete Event Simulation	×	Multiple vial sized measles vaccines.	One year.	×	Niger
Assi et al. (2012)	✓	✓	✓	Discrete Event Simulation	×	Ten different influenza vaccines.	One-six months.	✓ Age based.	Trang Province, Thailand
Govindan et al. (2020)	×	×	×	Fuzzy Inference System	×	General demand management.	N/A.	✓ Age, disease, risk based.	Synthetic