Optimal Models for Distributing Vaccines in a Pandemic

Md Yeakub Hassan

1,2

, Mahzabeen Emu

, Zubair Md Fadlullah

1,2

and Salimur Choudhury

Thunder Bay Regional Health Research Institute, Thunder Bay, Ontario, Canada

Department of Computer Science, Lakehead University, Thunder Bay, Ontario, Canada

Keywords:

Pandemic, ILP, Vaccine Distribution, Operational Research, Policy Making.

Abstract:

Distributing vaccines among a massive population is one of the challenging tasks in a pandemic. Therefore,

health care organizations need to optimize the assignment of vaccination appointments for people while con-

sidering their priorities and preferences. In this paper, we propose two optimal vaccine distribution models

as Integer Linear Programming (ILP) models; namely, Priority-based Model (PM) and Priority & Preference-

based Model (PPM), to maximize the distribution of vaccines among a given population. In PM, we divide

the people among several priority groups and ensure maximum vaccine distribution among the higher priority

groups. However, along with the priority groups, PPM also considers a list of preferred vaccine distribution

centers and time slots for each person. Thus, this model maximizes vaccine distribution among the higher

priority groups by assigning appointments in their desired location and time. We analyzed the performance

of our proposed models on a randomly generated dataset. In addition, we also performed a case study for

our proposed models on the COVID-19 vaccination dataset from Thunder Bay, Canada. In both experiments,

we show that PPM outperforms PM in full-ﬁlling people’s preferences while maximizing the distribution of

vaccines among the higher priority groups.

1 INTRODUCTION

A pandemic creates massive challenges for health

care organizations, including inadequate capacity, ﬁ-

nancial loss, and resource management. Recently,

the COVID-19 pandemic shows us how a pandemic

can cause an array of acute challenges, from provid-

ing enough protective equipment to adjusting hospi-

tal’s capacity by reducing ﬁnancial loss (Ali and Al-

harbi, 2020; Xiong et al., 2020; Nicole et al., 2020).

As of September 2021, more than 230 million cases

are reported worldwide, along with 4.7 million deaths

(Worldometers, 2021). Health care specialists are

urging people to get vaccinated to get over the pan-

demic. However, as most countries have already

started to vaccinate people, it becomes challenging to

distribute vaccines among a massive population with

limited resources. People can have different prefer-

ences of locations and timing for getting their vac-

cines; moreover, our vaccine distribution centers have

limited capacity. Therefore, assigning appointments

for vaccination becomes challenging for health care

organizations. Mahzabeen et al. (Emu et al., 2021)

propose several vaccines distributing models using

Constraint Satisfaction Programming (CSP). These

models do not consider people’s preferences for set-

ting up appointments in their preferred vaccine distri-

bution centers at their preferred time.

In this paper, we propose two optimal vaccine dis-

tribution models, namely,

• Priority based Model (PM)

• Priority & Preference based Model (PPM),

to optimize the distribution of vaccines in a given

geographical region. We formulate both models as

Integer Linear Programming (ILP) models and show

that these problems are NP-hard. In PM, our goal is

to maximize the distribution of vaccines among the

people with higher priority and maximize the mini-

mum resource utilization of any distribution centers

among all vaccine distribution centers. In PPM, we

aim to maximize the number of vaccinated people

with higher priority assigned to one of their preferred

vaccine distribution centers and time slots. We also

aim to maximize the minimum resource utilization of

any distribution center among all distribution centers.

We perform a random experiment and a case study

on these models and show how these models provide

the most optimal vaccine distribution. For the case

study, we use COVID-19 vaccination data obtained

from Thunder Bay, Ontario, Canada. Our experimen-

tal analysis shows that PPM assigns most people to

Hassan, M., Emu, M., Fadlullah, Z. and Choudhury, S.

Optimal Models for Distributing Vaccines in a Pandemic.

DOI: 10.5220/0010819600003117

In Proceedings of the 11th International Conference on Operations Research and Enterprise Systems (ICORES 2022), pages 337-344

ISBN: 978-989-758-548-7; ISSN: 2184-4372

337

their preferred distribution centers and time slot. Our

models also maximize the minimum resource utiliza-

tion of any distribution centers among all distribution

centers. Our vaccine distribution models can opti-

mally distribute vaccines or any medical equipment

among a massive population.

The rest of the paper is organized the follows. In

Section 2, we discuss the related works in the litera-

ture. The system model and optimal vaccine distribu-

tion models are discussed in Section 3 and 4, respec-

tively. Section 5 discusses the experimental analysis

on our proposed models. Finally, we conclude the pa-

per in Section 6.

2 RELATED WORKS

Vaccines are one of the most effective ways to pre-

vent a sudden outbreak and develop immunity against

certain infectious diseases (Tan et al., 2012). The

overall supply chain of vaccines can be categorized

into four broad categories: product, production, al-

location, and distribution (Duijzer et al., 2018). The

importance of strategic spatiotemporal vaccine distri-

bution to control the spread of highly infectious dis-

eases has been proven undeniable through existing lit-

erature (Grauer et al., 2020). The ﬁnal step of vaccine

distribution involves various decisions such as inven-

tory control, location of vaccine stockpiles, logistics

related to the point of dispensing, stafﬁng levels, rout-

ing, and scheduling, etc. (Duijzer et al., 2018). Of-

ten, the operations research perspective is adopted to

achieve optimal vaccine distribution schemes. Most

of the operations research-based models have been

developed using Quadratic Programming (QP), Inte-

ger Linear Programming (ILP), Mixed Integer Linear

Programming (MILP), Constraint Optimization (CO)

frameworks (Emu et al., 2021).

Sharon et al. (Hovav and Tsadikovich, 2015)

propose a mathematical model to improve the over-

all supply chain by optimizing inventory manage-

ment of inﬂuenza vaccines. With the help of the La-

grangian approach and branch-and-bound techniques,

a research study has been conducted to factor in en-

vironmental considerations for the cold supply chain

of vaccines (Saif and Elhedhli, 2016). The authors

propose a hybrid optimization-simulation tool to re-

duce the effects of refrigerant gases and carbon emis-

sions caused by the preservation of vaccines as much

as possible (Saif and Elhedhli, 2016). Lin et al. have

developed a policy-based model for taking intermedi-

ary decisions on the transportation of vaccines from

distributors to retailers (Lin et al., 2020).

An equity constraints-based framework using the

Gini index has been studied to distribute Inﬂuenza

vaccines optimally (Enayati and

Ozaltın, 2020). The

authors justify the model implications and the scala-

bility of the model on larger instances through exten-

sive simulation studies (Enayati and

Ozaltın, 2020).

Another research study has developed a simulation

tool to optimize the average waiting time of individ-

uals to expedite mass vaccination rate (Gupta et al.,

2013). Furthermore, Rajan et al. formulated a

stochastic genetic algorithm for deriving optimal vac-

cine distribution strategies that have been proven to

demonstrate 85% more efﬁcacy compared to random

vaccination schemes (Patel et al., 2005). Recently,

some of the research studies have made an effort to

maintain transparency, data integrity, and immutabil-

ity using blockchain framework for vaccine rollouts

(Antal et al., 2021). This research study highly em-

phasized the employment of smart contracts to enable

awareness among network peers (Antal et al., 2021).

To the best of our knowledge, existing literature

studies ignore the preferences of individuals for the

vaccine distribution decision-making process. In this

paper, we propose an ILP based vaccine distribution

model that simultaneously prioritizes individual pref-

erences and resource utilization of vaccine distribu-

tion centers. Such a model can be generalized and

adapted for sudden pandemic and epidemic urgency

situations that may arise in the future. Moreover, the

convenience caused by incorporating the preferences

of the people alongside demographics may even fur-

ther diminish vaccine hesitancy and accelerate vacci-

nation rates.

3 SYSTEM MODEL

In this section, we discuss the system model for the

vaccine distribution problem. We are given a set E =

,...,e

} of n people required to be vaccinated.

We use a set P = {p

, p

,..., p

} of n non-zero posi-

tive integers to specify the priority of people for vacci-

nation purpose, where p

deﬁnes the priority level of a

person e

∈ E. The higher values of p

indicate higher

priority. A person with higher priority is desired to

get faster vaccination service. Let N represent the to-

tal number of available vaccines. We are given a set

H = {h

,...,h

} of m vaccine distribution cen-

tres (DCs), i.e., hospitals. Let T be a set of time slots.

We denote T

as a set of available time slots provided

by the DC h

where T

⊆ T . Therefore, we can in-

fer that T = ∪

j=1

. Let B = {b

,...,b

} be a

set of m positive integers where b

speciﬁes the num-

ber of people can be vaccinated in each time slot in

. Each person e

∈ E provides a list L

of preferred

ICORES 2022 - 11th International Conference on Operations Research and Enterprise Systems

338

DCs including a set of preferred time slots from the

corresponding DCs. Hence, each element in L

in-

cludes a DC h

∈ H and a set T

⊆ T

of preferred

time slots from h

, i.e., each element of L

is deﬁned

as {h

∈ H ,{T

⊆ T

}}. In this paper, we consider

the solution of the vaccine distribution problem in a

binary decision variable as x

i, j,k

∈ {0,1} where,

i, j,k

(

1, a person e

is assigned to h

at t

∈ T

;

0, otherwise.

4 OPTIMAL VACCINE

DISTRIBUTION MODELS

In this paper, we formulate the vaccine distribution

research problem as an Integer Linear Programming

(ILP) model. A DC h

can give vaccine to at most

| T

| ×b

number of people. We deﬁne the resource

utilization of a DC as the utilization rate of its total

capacity. Therefore, the resource utilization R

of a

DC h

∑

i=1

∑

∈T

i, j,k

| T

| ×b

. (1)

Our primary goal is to maximize the number of over-

all vaccinated people and maximize the minimum re-

source utilization of any DC among all DCs. How-

ever, in many cases, we aim to maximize the distribu-

tion of vaccines among the people with higher prior-

ity. Moreover, we may want to maximize the number

of vaccinated people assigned to one of their preferred

DCs and time slots. Thus, we propose two differ-

ent vaccine distribution ILP models: Priority-based

Model (PM) and Priority & Preference-based Model

(PPM).

4.1 Priority based Model (PM)

In this model, our goal is to maximize the distribu-

tion of vaccines among the people with higher prior-

ity and maximize the minimum resource utilization of

any DC among all DCs. Therefore, we formulate the

following optimization problem:

MAX





min

j=1

∑

i=1

∑

j=1

∑

∈T

× x

i, j,k





(2)

subject to,

∑

j=1

∑

∈T

i, j,k

≤ 1 ; ∀e

∈ E (3)

∑

i=1

i, j,k

≤ b

; ∀h

∈ H and ∀t

∈ T

(4)

∑

i=1

∑

∈T

i, j,k

≤| T

| ×b

; ∀h

∈ H (5)

∑

i=1

∑

j=1

∑

∈T

i, j,k

≤ N (6)

i, j,k

= {0,1}; ∀e

∈ E,∀h

∈ H and ∀t

∈ T

(7)

The constraint in equation (3) ensures that each per-

son can take at most one vaccine. The constraint in

equation (4) ensures that in a DC h

, at most b

peo-

ple can be vaccinated in each time slot. In equation

(5), we ensure that each DC h

can give vaccine to at

most | T

| ×b

people. The constraint in equation (6)

ensures that the total vaccine distribution should not

be more than the total number of available vaccines.

4.2 Priority & Preference based Model

(PPM)

In this model, our goal is to maximize the number

of vaccinated people with higher priority assigned to

one of their preferred DCs and time slots and max-

imize the minimum resource utilization of any DC

among all DCs. We deﬁne a preference score func-

tion f which takes a person e

, a DC h

and a time

slot t

as input and calculates the preference score for

assigning the person e

to the DC h

at the time slot

. The deﬁnition of the function f is

f (e

) =











2 + p

, if h

is in L

and t

∈ T

;

1 + p

, if h

is in L

and t

/∈ T

;

, otherwise.

We formulate the following optimization problem

MAX





min

j=1

∑

i=1

∑

j=1

∑

∈T

f (e

) × x

i, j,k





(8)

subject to the equations (3)-(7).

4.3 The Complexity of the PM and

PPM Optimization Problems

The PPM optimization problem is NP-hard, and we

can obtain the hardness result by reduction from the

Generalized Assignment Problem (GAP), which is

NP-hard (

Ozbakir et al., 2010). In the generalized

assignment problem, we have given a set of n items

and m bins with a ﬁxed capacity. For each bin, each

item has a proﬁt and weight. The goal is to ﬁnd the

maximum possible proﬁt from a feasible item-bin as-

signment. In a viable solution, the total weight of

Optimal Models for Distributing Vaccines in a Pandemic

339

assigned items in a bin must not exceed its capac-

ity. If the proﬁt and weight of an item do not vary

between different bins, then the generalized assign-

ment problem is reduced to the multiple knapsack

problem. Therefore, our proposed PM problem can

also be shown NP-hard by reduction from the mul-

tiple knapsack problem, which is NP-hard (Chekuri

and Khanna, 2005). Consequently, we can derive the

following theorem.

Theorem 1. The PM and PPM optimization problems

are NP-hard.

5 EXPERIMENTAL EVALUATION

In this section, we demonstrate the experimental re-

sults of our proposed models in two different sce-

narios. In the following, we show the performances

of our models on randomly generated datasets and

a real-world dataset from Thunder Bay, Ontario,

Canada. In all cases, we compare the results between

our two proposed models and show how PPM op-

timizes both priority and preference in each assign-

ment. We have used Gurobi (Gurobi Optimization,

LLC, 2021) optimization solver to solve the proposed

ILP models.

5.1 Random Simulation

In this experiment, we consider 320 people (| E |) for

vaccination in 5 different vaccine DCs and the total

number of available vaccines (N ) is 150. The details

of each vaccine DC are given in Table 1. We consider

ﬁve different priority groups P = {1, 2, 3, 4, 5} where

5 being the highest priority group and 1 being the low-

est priority group. To show the impact of the popula-

tion distribution and preference distribution, we use

two different distribution models in our random sim-

ulation; namely,

• Random-Simulation-1 (RS-1): Population and

preference lists of vaccine DCs are distributed us-

ing uniform random distribution.

• Random-Simulation-2 (RS-2): Population is

distributed into the ﬁve groups using the normal

distribution. Preference lists of vaccine DCs are

generated using a poisson like distribution where

and h

are likely to be chosen by most people.

In both models, the preference of time slots is chosen

using a uniform random distribution. Figure 1 shows

the distribution of 300 people among 5 priority

groups for RS-1 and RS-2.

Table 1: No of time slots and maximum no of vaccination

in each time slot of vaccine DCs in the random simulation.

H : h

| T

| 20 15 20 25 10

2 1 3 2 1

(a) RS-1

(b) RS-2

Figure 1: Distribution of 300 people among 5 priority

groups in the random simulation.

Figure 2 and 3 show the distribution of assigned

people to their preferred DC and time slot by ﬁnding

an optimal solution for both PPM and PM for RS-1

and RS-2. In both random simulations, PPM assigns

around 80% of people to their preferred DC and time

slot because the model considers each person’s prior-

ity level and preference list in the objective function.

However, the priority-based model (PM) only consid-

ers the priority level of each people in the objective

function. Therefore, in this model, nearly 3% of peo-

ple are assigned to their preferred DC and time slot,

and most of the people are assigned to DCs and time

ICORES 2022 - 11th International Conference on Operations Research and Enterprise Systems

340

(a) RS-1 (PM)

(b) RS-1 (PPM)

Figure 2: No of people assigned to their preferred DC and

time slot in RS-1 (total 150 vaccinated people).

slots that are not in their preference list.

In Figure 4, we show the comparison between av-

erage and minimum resource utilization (percentage)

of the vaccine DCs in each experiment for the ran-

dom simulation. In both random simulations, the av-

erage resource utilization in PPM is slightly higher

than the average resource utilization in PM. More-

over, the minimum and the average resource utiliza-

tion rates are higher than 80% in all cases. Since the

objective functions maximize the minimum resource

utilization of any DC among all DCs in both PPM

and PM, both the models perform similarly in terms

of maximizing the minimum resource utilization.

Figure 5 shows the distribution of vaccinated peo-

ple among the 5 priority groups. The average per-

centage of vaccinated people in each group for both

PM and PPM is equal. In our objective functions, the

weight of priority is much higher than the weight of

the preference list. Therefore, in both models, people

(a) RS-2 (PM)

(b) RS-2 (PPM)

Figure 3: No of people assigned to their preferred DC and

time slot in RS-2 (total 150 vaccinated people).

in higher priority groups will always be vaccinated

before those in lower priority groups.

5.2 Case Study - Thunder Bay

In this experiment, we consider the COVID-19 vac-

cine distribution dataset from Thunder Bay, Ontario,

Canada. First, we use the 2016 census proﬁle to get

the number of people in different age ranges (Census,

2019). Then, we divide the people into ﬁve priority

groups according to their age. In Table 2, we show

the distribution of the Thunder Bay population into

ﬁve priority groups. We consider 30 vaccine distribu-

tion centers where ﬁve of them are primary vaccina-

tion dispatch centers and others are local pharmacies

(cov, 2020). The primary distribution centers provide

vaccines to 8-9 people in each time slot, whereas local

pharmacies offer 1-2 people. We assume that a total

of 4000 vaccines are available for vaccination. The

preference list of DCs and time slots for each person

are generated using uniform random distribution.

Optimal Models for Distributing Vaccines in a Pandemic

341

(a) RS-1

(b) RS-2

Figure 4: Comparison of average and minimum resource

utilization of the vaccine DC’s in random simulation.

Table 2: Distribution of Thunder Bay population into ﬁve

priority groups.

Group no Age range No of people

1 20-29 13160

2 30-39 10905

3 40-49 10990

4 50-59 14505

5 60 and above 26195

In Figure 6, we show the distribution of assigned

people to their preferred DC and time slot in opti-

mal solutions for both models. In our priority and

preference-based model, around 65% of people on

average are assigned to their preferred DC and time

slot. However, in only the priority-based model, only

7.5% of people are assigned to their preferred DC and

time slots. In this experiment, all the vaccine distri-

(a) RS-1

(b) RS-2

Figure 5: Distribution of vaccinated people among the 5

priority groups.

bution centres are ﬁlled to 100% capacity. Therefore,

their minimum and average resource utilization be-

come 100%. Since the no of people in group 5 is

much higher than the number of available vaccines,

only people from that group get vaccinated.

6 CONCLUSION

This paper proposes two optimization models (PM

and PPM) based on Integer Linear Programming

(ILP) framework to ﬁnd optimal solutions for assign-

ing vaccination appointments for people while con-

sidering their priorities and preferences. In PM, we

aim to maximize the distribution of vaccines among

the people with higher priority. In PPM, our goal is

to maximize the number of vaccinated people with

higher priority groups by assigning appointments in

their desired locations and time slots. Moreover, in

both models, we maximize the minimum resource uti-

ICORES 2022 - 11th International Conference on Operations Research and Enterprise Systems

342

(a) PM

(b) PPM

Figure 6: No of people assigned to their preferred DC and

time slot in Thunder Bay dataset.

lization of any vaccine distribution center. Experi-

mental analysis on the random dataset and COVID-

19 vaccination dataset from Thunder Bay shows that

our proposed PPM outperforms PM in full-ﬁlling peo-

ple’s preferences while maximizing the distribution of

vaccines among the higher priority groups. Although

our proposed models can be adapted across various

scenarios for distributing any medicine among a mas-

sive population, we have shown a minimal number of

constraints in our models due to resource constraints.

In the future, depending on any speciﬁc scenario, our

models can be expanded by adding more real-world

constraints.

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