Literature DB >> 34099948

A mathematical programming approach for equitable COVID-19 vaccine distribution in developing countries.

Madjid Tavana1,2, Kannan Govindan3, Arash Khalili Nasr4, Mohammad Saeed Heidary5, Hassan Mina6.   

Abstract

Developing countries scramble to contain and mitigate the spread of coronavirus disease 2019 (COVID-19), and world leaders demand equitable distribution of vaccines to trigger economic recovery. Although numerous strategies, including education, quarantine, and immunization, have been used to control COVID-19, the best method to curb this disease is vaccination. Due to the high demand for COVID 19 vaccine, developing countries must carefully identify and prioritize vulnerable populations and rationalize the vaccine allocation process. This study presents a mixed-integer linear programming model for equitable COVID-19 vaccine distribution in developing countries. Vaccines are grouped into cold, very cold, and ultra-cold categories where specific refrigeration is required for their storage and distribution. The possibility of storage for future periods, facing a shortage, budgetary considerations, manufacturer selection, order allocation, time-dependent capacities, and grouping of the heterogeneous population are among the practical assumptions in the proposed approach. Real-world data is used to demonstrate the efficiency and effectiveness of the mathematical programming approach proposed in this study.
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021.

Entities:  

Keywords:  COVID-19; Coronavirus vaccine; Equitable distribution; Location-inventory problem; Mixed-integer linear programming model; Vaccine supply chain

Year:  2021        PMID： 34099948      PMCID： PMC8172366          DOI： 10.1007/s10479-021-04130-z

Source DB:  PubMed          Journal:  Ann Oper Res        ISSN： 0254-5330            Impact factor:   4.820


Introduction

The Developing Countries Vaccine Manufacturers Network (DCVMN) is a public-health agency representing vaccine manufacturers from emerging countries. The DCVMN is committed to protecting all people through research and development activities and manufacturing and distributing coronavirus disease 2019 (COVID-19) vaccines to developing countries (Pagliusi et al., 2020). COVID-19 has stressed the importance of preventing pulmonary infections and stopping the pandemic with vaccinations (Dinleyici et al., 2020). The World Health Organization (WHO) statistics show that nearly 84,000,000 cases of the virus have been identified worldwide by the end of 2020, and about 1,800,000 people have lost their lives (https://coronavirus.jhu.edu). The coronavirus outbreak has drastically changed global social norms and has brought about disruptions in health services provision (Chandir et al., 2020). Since the beginning of the disease outbreak, various strategies such as border closure, social distance, widespread testing, and homestay have been proposed by statesmen, the WHO, and other relevant centers to reduce the virus’s spread and cut its transmission chain (Coudeville et al., 2020). Some researchers, such as Govindan et al. (2020), have tried to reduce this disease’s prevalence by classifying people in the community and offering solutions to each class of people. Although these protective measures are crucial to managing this disease, vaccination is a critical defensive behavior to control and eradicate it (Reiter et al., 2020). More than 100 companies are developing the COVID-19 vaccine worldwide, some of which are placed in phase III trials (Degeling et al., 2020). With the mass production of the vaccine, demand is expected to outstrip supply in the early stages. Therefore, the vaccine distribution process will be of great importance. Government budget constraints, cold supply chain management, prioritization of people in the community, and waiting time for receiving vaccines are essential factors affecting vaccine distribution worldwide. Therefore, this study develops a mixed-integer linear programming (MILP) model for equitable COVID-19 vaccine distribution considering the factors mentioned above. Duijzer et al. (2018) examined the literature on the vaccine supply chain. They categorized these studies into the product, production, allocation, and distribution modeling groups by combining the WHO’s priorities for creating a robust and flexible vaccine supply chain with an operations research perspective. Similarly, De Boeck et al. (2019) studied the vaccine distribution studies in low- and middle-income countries. They identified several problems that had received little or no attention in the operations research literature. They found operations researchers heavily concentrate on the strategic decisions in vaccine distribution chains and, to a lesser extent, tactical and operational decisions. Hovav and Tsadikovich (2015) used mathematical programming and proposed an optimization model to design a healthcare supply chain to control influenza vaccines’ distribution and inventory control. Saif and Elhedhli (2016) developed a cold supply chain for the distribution and inventory management of vaccines considering environmental issues. For this purpose, they proposed a mixed-integer programming (MIP) model to minimize total costs and used a Lagrangian decomposition algorithm to solve the proposed model. Lim et al. (2019) proposed a MIP model for redesigning a cold supply chain for vaccine distribution by considering the location problem and limitations. They developed a hybrid heuristic algorithm for large-scale problem-solving and validated their model using several African countries’ data. A bi-objective model was developed by Zandkarimkhani et al. (2020) to design a perishable pharmaceutical supply chain network and minimize total costs and lost demands. They used an integrated inventory-location-routing problem and developed a chain to distribute Avonex (prefilled syringe for multiple sclerosis disease). One of the issues that should be considered in developing a vaccine distribution supply chain is the ambient temperature in which the vaccines are stored and transported. The transportation of vaccines at unfavorable temperatures has led to adverse events, especially in developing countries. Hence, Lin et al. (2020) proposed a model to solve this problem by implementing inspection strategies. Their intended chain includes distributors and retailers, and they analyze inspection policies of vaccines’ transportation in the cold supply chain. Gamchi et al. (2020) presented a novel bi-objective model using the susceptible-infected-recovered epidemic model and vehicle routing problem for vaccine distribution. Their model simultaneously minimizes social costs and the costs of operating vehicles and groups of people in the community. They validated their model using data from a cholera vaccine distribution chain. Yang et al. (2020) developed a MIP model for vaccine distribution in low- and middle-income countries in line with the WHO Expanded Program on Immunization (EPI). Their model aims to minimize total costs with validation using data from a vaccine distribution chain in African countries. Bulula et al. (2020) used a micro-costing approach to analyze costs in the vaccine supply chain. They showed the delegation of the vaccine supply chain’s responsibilities from the medical stores to the EPI would lead to a 27% reduction in the vaccine distribution and storage costs. The trade-offs between the equitable allocation of resources have been studied in energy (Sasse & Trutnevyte, 2019), bicycle-sharing (Conrow et al., 2018), and food donation (Fianu & Davis, 2018; Orgut et al., 2016, 2017). Equitable distribution of resources must consider a fair sharing of the resources among recipients; however, studies emphasizing equitable distribution of resources are limited (Fianu & Davis, 2018). Equity and fairness are among the most critical issues to be considered in the vaccine distribution chain (Abila et al., 2020). In this regard, Enayati and Özaltın (2020) proposed a mathematical programming model for equitable influenza vaccine distribution. They divided the population into several subgroups and prevented the epidemic outbreaks by allocating the necessary vaccines to each subgroup equitably. Rastegar et al. (2021) went one step further and developed a MILP model for equitable influenza vaccine distribution by considering the location-inventory problem under pandemic COVID-19 conditions. The possibility of storage for future periods, being faced with a shortage, and budget constraints are among the practical assumptions considered in this research. They proposed a novel objective function to consider the concept of equitable distribution. They then evaluated their proposed model’s performance using data from an influenza vaccine distribution chain in Iran. The remainder of this paper is organized as follows. In Sect. 2, we present our motivation and contributions. The proposed mathematical model is presented in Sect. 3. In Sect. 4, we present a case study to demonstrate the applicability of the method proposed in this study. A sensitivity analysis is conducted in Sect. 5 to exhibit the efficacy and robustness of our vaccine distribution method. We conclude the paper with our conclusions and future research directions in Sect. 6.

Motivation and contributions

A review of the vaccine distribution literature shows the research in this field is in its infancy. Some researchers, such as De Boeck et al. (2019) and Corey et al. (2020), have studied strategic and managerial approaches to vaccine distribution. Other researchers such as Gamchi et al. (2020), Yang et al. (2020), and Rastegar et al. (2021) have proposed mathematical models for vaccine distribution and supply chain network optimization. Supply chains for vaccine distribution require unique features. For example, some vaccine distribution requires a cold or very cold supply chain (i.e., Gamchi et al., 2020; Lim et al., 2019; Yang et al., 2020). Other vaccine distributions may require waiting time considerations in the supply chain (i.e., Gamchi et al., 2020). There is also the concern for fair and equitable access to vaccines. Sometimes it is impossible to provide vaccines for all members of society. Therefore, equitable distribution becomes a critical consideration and assumption in vaccine distribution networks (Enayati & Özaltın, 2020; Rastegar et al., 2021). Distribution of COVID-19 vaccines requires cold, very cold, and ultra-cold refrigeration. Waiting time to receive the vaccines from manufacturers also directly affects the delivery of vaccines to the public. This study presents the first mathematical model for equitable COVID-19 vaccine distribution considering time-dependent capacity and triple refrigeration requirement (i.e., cold, very cold, and ultra-cold). Rastegar et al. (2021) proposed a mathematical model for equitable influenza vaccine distribution. The current study presents a MILP model for equitable distribution of the COVID-19 vaccines. The model proposed by Rastegar et al. (2021) is a single-product model. However, the current study addresses the need for a multi-product model for COVID-19 vaccine distribution requiring cold, very cold, and ultra-cold refrigeration. Moreover, this study considers multiple cold supply chains with varying refrigeration requirements. This capability does not exist in the model proposed by Rastegar et al. (2021). The ordering and delivery times of vaccines can vary in the current study because of the potential waiting time between ordering and receiving the vaccines. However, the model proposed by Rastegar et al. (2021) does not consider waiting time because vaccines are provided in the same period they are ordered. In summary, the contributions of this study are to (i) introduce a location-inventory MILP model for a fair and equitable COVID-19 vaccine distribution in developing countries; (ii) propose a model to consider cold, very cold, and ultra-cold supply chain network design and; (iii) take into consideration an equitable vaccine distribution model capable of manufacturer selection, order allocation, capacity planning, and waiting time management; and (iv) validating the proposed model with real-world data.

Proposed model

Satisfying the global demand for COVID-19 vaccines is not a short-term problem due to limited production and supply. Consequently, vaccine delivery is subject to the waiting time. This study presents a MILP model for equitable COVID-19 vaccine distribution in developing countries by considering the location-inventory problem. In addition to limited production and supply constraints, the refrigeration requirement for the COVID-19 vaccine and the need for cold, very cold, and ultra-cold supply chains is a huge hurdle in developing countries. Furthermore, the need for grouping and prioritizing the population is another added complexity for the COVIS-19 vaccine distribution. The proposed model is considered a budget constraint and allows the applicant country to make the following customizations to the model: Which manufacturer should be selected? What period should the vaccine be ordered to the manufacturer, and how much? What is the waiting time for each manufacturer? Which distribution centers are needed? Which distribution centers need ultra-cold refrigeration equipment? How many vaccines should be transferred from distribution centers to warehouses in different states in each period? How many vaccines should be stored in the state warehouses in each period? How many vaccines should be allocated to each group in each state and period? To better understand the problem under study, the assumptions of the proposed model are given as follows: The proposed model is considered as multi-product and multi-period. The model determines the location of distribution centers. The established distribution centers can handle cold and very cold refrigeration. Ultra-cold refrigeration can only be installed in previously established distribution centers equipped with very cold refrigeration. An order can be placed after an existing order is received. One order can be placed with each manufacturer in each period. Time-dependent capacity is considered for the manufacturers. Distribution centers are capacitated. Each manufacturer produces only one type of vaccine. The possibility of storage in the state warehouses for future periods is considered. The possibility of facing a shortage is considered.

Mathematical model

Objective function

The proposed model’s objective function is derived from the model presented by Rastegar et al. (2021) that focuses on the equitable distribution of the vaccine. In this objective function, vaccines are distributed based on maximizing the minimum delivery-to-demand ratio. Constraint (2) ensures that each group receives the vaccine at least up to the coverage rate. Constraints (3) and (4) are related to the inventory balance in the state warehouses in period 1 and the periods greater than one, respectively. Not exceeding the capacity of vaccine manufacturers has been shown by constraint (5). Not exceeding the distribution centers’ capacity of cold, very cold, and ultra-cold refrigeration is guaranteed in constraints (6) to (88), respectively. The condition for the purchase of vaccines from manufacturers is that the order should be placed with the manufacturer. This condition is considered in constraint (9). According to constraint (10), if a distribution center with proper equipment for cold refrigeration vaccines has not been set up, it will not receive orders from manufacturers. Similarly, based on the constraint (11), if a distribution center has not been equipped with very cold and ultra-cold refrigeration, it will not receive orders from manufacturers. One order can be placed with each manufacturer in each period. This condition is satisfied by constraint (12). According to the location conditions, if a distribution center with proper equipment for cold refrigeration vaccines has not been set up, it will not be allowed to send cold refrigeration vaccines to the states’ warehouses. This condition is considered in constraint (13). Also, if a distribution center equipped with very cold refrigeration has not been set up, it will not be allowed to send very cold refrigeration vaccines to the states’ warehouses. Similarly, suppose a distribution center equipped with ultra-cold refrigeration has not been set up. In that case, that distribution center will not send ultra-cold vaccines to the states’ warehouses. These conditions are considered in constraints (14) and (15), respectively. Moreover, there is no possibility of setting up the distribution centers equipped with ultra-cold refrigeration until the distribution centers equipped with very cold refrigeration are set up. This condition is also expressed by constraint (16). The amount of vaccines delivered to each distribution center in each period is calculated by constraint (14). Constraint (15) is responsible for establishing the inventory balance in distribution centers. As long as the order is being processed (i.e., it has not yet been delivered to the distribution center), it will not be possible to place a new order with that manufacturer. This condition is satisfied by constraint (16). In all the constraints of the proposed model, the ordering time should always be less than or equal to the delivery time. Constraint (17) has been used to meet this condition. In the end, constraint (18) states that the supply chain’s total costs should not exceed the available budget. These costs include ordering cost to manufacturers, set-up cost of distribution centers for cold refrigeration, very cold refrigeration, and ultra-cold refrigeration vaccines, purchasing cost of vaccines, transportation cost from manufacturers’ location to distribution centers, transportation cost from distribution centers to state warehouses, and holding cost at the state warehouses. As can be seen, the objective function of the proposed model is nonlinear. To linearize it, we define a new free variable () and replace it with . Therefore, the following holds true: Based on Eq. (19), the following formula always holds true: Therefore, based on Eq. (19) and Eq. (20), the proposed nonlinear model is converted to a linear one as follows: Constraints (2) to (18).

Case study

This section demonstrates the applicability of the model proposed in this study with the data obtained from the Ministry of Health and Family Welfare (MOHFW) in India. COVID-19 vaccines have been discovered not so long ago, and, thereby, adequate information and data on their distribution are not yet fully available and accessible. As a result, some data, including transportation costs, vaccine prices, and manufacturers’ capacities, are simulated based on the MOHFW’s preliminary estimations. As in the study carried out by Rastegar et al. (2021), the heterogeneous population in this study is also divided into eight groups. The group numbers are not representative of priorities. These group numbers simply identify a segment of the population: Group (1): infants and toddlers ages 6–35 months, Group (2): pregnant women with pre-existing medical conditions, Group (3): adults aged 65 years and older with pre-existing medical conditions, Group (4): critical healthcare providers and first responders, Group (5): pregnant women without pre-existing conditions, Group (6): adults aged 65 years and older without pre-existing medical conditions, Group (7): people with pre-existing medical conditions, and Group (8): other people. This study considers five vaccine types. Vaccine types 1 and 2 require ultra-cold refrigeration (− 70 °C ± 10 °C), vaccine types 3 and 4 require very cold refrigeration (− 25 °C to − 15 °C), and vaccine type 5 requires cold refrigeration (− 8 °C to − 2 °C). The vaccines are purchased from manufacturers outside India and are then transported to distribution centers. Afterward, vaccines are shipped to different distribution centers and distributed equitably among multiple warehouses in multiple states according to demand (See Fig. 1).
Fig. 1

The vaccine distribution network

The vaccine distribution network We should note data such as the demand for each group in each state, holding cost for the vaccines, coverage rate, and transportation cost between distribution centers and states have been extracted from historical data proposed by the MOHFW. Furthermore, the MOHFW’s estimated budget for vaccine purchase and distribution is 4.5 billion dollars. Table 1 presents the vaccine manufacturers’ capacities. Additional data is provided in Appendix A (Tables 8, 9, 10, 11, 12, 13, 14, 15 and 16).
Table 1

The maximum delivery capacity of vaccine i for period t ordered in period w

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CP_{iwt}^{MN}$$\end{document}CPiwtMNt
iw12345678
11007,500,00010,000,00015,000,00019,000,00025,000,00030,000,000
12003,000,0007,000,00011,000,00015,000,00020,000,00025,000,000
130005,000,0009,000,00013,000,00018,000,00022,000,000
1400005,000,0009,000,00015,000,00020,000,000
15000007,500,00012,500,00017,500,000
160000006,000,00010,000,000
17000000015,000,000
1800000002,500,000
2106,000,00015,000,00020,000,00028,000,00032,000,00040,000,00060,000,000
220010,000,00014,000,00020,500,00027,000,00035,000,00042,000,000
230008,000,00013,500,00020,000,00027,500,00034,000,000
24000012,000,00018,000,00025,500,00032,000,000
25000009,500,00016,000,00030,000,000
2600000010,000,00025,000,000
27000000020,000,000
2800000000
3104,000,0007,000,00013,000,00020,000,00025,000,00030,000,00035,000,000
32004,000,0008,000,00015,000,00020,000,00025,000,00030,000,000
330005,000,0009,000,00015,000,00021,000,00025,000,000
3400004,000,0008,000,00014,000,00019,000,000
35000004,000,0008,000,00015,000,000
360000006,000,0009,000,000
3700000005,000,000
3800000000
4104,000,0007,000,00011,000,00017,500,00021,000,00027,500,00034,000,000
42001,500,0003,000,0006,000,0008,750,00011,250,00015,500,000
430002,000,0003,750,0006,000,0008,750,00011,500,000
4400001,500,0003,000,0004,500,0006,000,000
45000005,000,0008,000,00011,000,000
460000002,000,0005,250,000
4700000006,000,000
4800000000
5105,000,0009,000,00014,000,00018,000,00025,000,00030,000,00036,000,000
52004,000,0007,500,00012,000,00016,000,00022,000,00027,000,000
530004,000,0007,500,00013,000,00018,000,00024,000,000
5400004,000,0008,000,00013,000,00020,000,000
55000005,000,00010,000,00016,000,000
560000006,000,00012,000,000
5700000007,000,000
5800000000
Table 8

The demand for each group

StatesGroup (g)
12345678
Uttar Pradesh117,370,453136,9011,189,9451,165,6071,232,2332,210,133610,884213,966,569
Bihar27,997,20932,182331,682499,204607,2931,628,062177,995113,526,299
Maharashtra39,309,82438,385684,242480,298531,3812,047,686220,520109,831,887
West Bengal410,574,49647,384366,198547,857510,4321,682,009209,00985,671,917
Madhya Pradesh513,984,53335,530267,620495,085272,5921,021,796137,20969,144,601
Rajasthan67,122,23013,828243,591243,032291,3591,001,599176,54771,940,504
Tamil Nadu76,200,80636,331158,305233,510332,000849,269239,86769,791,179
Karnataka85,665,24623,41492,349405,377373,408978,112180,65459,844,125
Gujarat95,440,43623,251293,834185,263206,762957,655182,56856,582,632
Andhra Pradesh106,417,45829,164122,473204,846306,568703,711115,88346,003,290
Odisha113,999,59810,64165,970231,785249,602558,257113,15141,127,330
Telangana123,200,72019,29283,460177,136226,847616,78473,65534,964,837
Jharkhand133,781,11810,536159,971138,940246,098417,57368,42933,771,283
Kerala142,257,706999390,611142,803172,275421,08388,78832,516,184
Assam153,019,35219,51835,994121,062213,633572,96670,92431,553,590
Punjab163,790,53815,739179,62084,395203,320428,19057,74125,381,830
Chhattisgarh172,835,9436468103,989147,179123,582369,43464,75525,784,881
Haryana182,160,6318392154,933112,818148,955401,13983,54525,134,279
Delhi191,071,993696442,55484,195115,861205,25448,18617,135,916
Jammu & Kashmir201,189,383898918,02040,81948,076161,67042,38012,096,984
Uttarakhand211,050,549283015,25150,62840,805117,47827,2719,946,045
Himachal Pradesh22446,2583248848120,12229,58780,28590426,854,931
Tripura23334,328220522,48512,09317,28041,96378943,731,546
Meghalaya24300,049223715,60512,45816,08635,21867922,978,266
Manipur25327,429204115,33312,366956340,52597272,674,562
Nagaland26157,49049211,525674914,50121,82071722,029,945
Goa27128,9195454518682110,57015,93448491,414,093
Arunachal Pradesh28104,45124841457067891124,37436471,417,614
Puducherry29119,55067256824240590918,44620371,257,004
Mizoram30111,79418414214957552912,14437691,099,447
Chandigarh3174,24050333205213808711,36229741,052,775
Sikkim32113,08528721644003220482631110559,134
Dadra and Nagar Haveli and Daman and Diu3354,11810518511847221476111341546,637
Andaman and Nicobar Islands3433,2211958481251177945501285373,908
Ladakh3530,611191143311568943789909250,040
Lakshadweep366397489722025987022865,065
Table 9

The ordering cost to the manufacturer of vaccine i in period t

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$FX_{it}^{MN}$$\end{document}FXitMNt
i12345678
191009300820092009300880088008000
28200860091009900890010,00093009000
395008400820094009300920088009900
493009300970087009900960086009000
597009600900092008900840095008400
Table 10

The cost of setting up distribution centers equipped with cold refrigeration

Distribution center
PatnaBhopalNew DelhiHyderabad
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$FY_{d}^{DS}$$\end{document}FYdDS18,400,00018,800,00019,400,00019,200,000
Table 11

The cost of setting up distribution centers equipped with very cold refrigeration

Distribution center
PatnaBhopalNew DelhiHyderabad
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$FX_{d}^{DS}$$\end{document}FXdDS46,000,00047,000,00048,500,00048,000,000

Amn

Table 12

The coverage rate of each group

Group
12345678
Coverage rate (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi_{g}$$\end{document}ξg)0.750.90.8510.70.60.750.05
Table 13

The purchasing cost for two doses of vaccine i

Vaccine type
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 1$$\end{document}i=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 2$$\end{document}i=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 3$$\end{document}i=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 4$$\end{document}i=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 5$$\end{document}i=5
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$VP_{i}$$\end{document}VPi15.516.213.113.78.2
Table 14

Transportation cost for two doses of vaccine i from manufacturer’s location to distribution center d

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$TR_{id}^{MN}$$\end{document}TRidMNDistribution center
iPatnaBhopalNew DelhiHyderabad
18.259.088.919.8
27.68.368.219.03
38.329.158.999.89
47.698.468.319.14
56.897.587.448.18
Table 15

The holding cost for two doses of vaccine i in the state s warehouse

State (s)i = 1i = 2i = 3i = 4i = 5
11.71.70.680.680.58
21.631.630.650.650.56
31.51.50.60.60.52
41.751.750.70.70.6
51.781.780.710.710.61
61.131.130.450.450.39
70.980.980.390.390.34
82.052.050.820.820.71
91.051.050.420.420.36
100.830.830.330.330.28
111.531.530.610.610.52
121.41.40.560.560.48
131.151.150.460.460.4
141.381.380.550.550.47
151.31.30.520.520.45
160.80.80.320.320.28
171.551.550.620.620.53
181.531.530.610.610.52
191.451.450.580.580.5
201.231.230.490.490.42
211.31.30.520.520.45
221.11.10.440.440.38
230.950.950.380.380.33
241.21.20.480.480.41
251.281.280.510.510.44
261.081.080.430.430.37
271.281.280.510.510.44
281.331.330.530.530.46
291.131.130.450.450.39
301.231.230.490.490.42
311.31.30.520.520.45
321.091.090.930.930.81
331.011.010.860.860.75
340.980.980.830.830.72
350.940.940.80.80.7
360.840.840.710.710.62
Table 16

The distribution capacity of vaccines requiring cold, very cold, and ultra-cold refrigeration

Distribution center
PatnaBhopalNew DelhiHyderabad
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CP_{d}^{NR}$$\end{document}CPdNR36,000,00038,000,00040,000,00037,000,000
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CP_{d}^{SR}$$\end{document}CPdSR90,000,00090,000,00097,500,00087,000,000
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CP_{d}^{DFR}$$\end{document}CPdDFR45,000,00052,000,00045,000,00042,000,000
The maximum delivery capacity of vaccine i for period t ordered in period w The 7,500,000 in the first row of Table 1 indicates that if the order is placed to manufacturer 1 in period 1, the manufacturer will deliver a maximum of 7,500,000 doses of vaccines in period 3. The proposed model was run with the described data in GAMS software using CPLEX solver. The obtained results are as follows: Distribution centers in Patna, Bhopal, and Hyderabad were set up to distribute the vaccines. Patna and Bhopal distribution centers can distribute vaccines requiring cold, very cold, and ultra-cold refrigeration, whereas Hyderabad distribution center can distribute vaccines requiring cold and very cold refrigeration. Orders are placed with all five manufacturers. Patna and Bhopal distribution centers receive vaccines from all five manufacturers, but vaccines are purchased only from manufacturers 3, 4, and 5 for the Hyderabad distribution center. With the available budget of 4.5 billion dollars, 186,096,615 doses of COVID-19 vaccines were purchased. The optimal doses of vaccines ordered to each manufacturer and the optimal doses of vaccines delivered by each distribution center are shown in Tables 2 and 3, respectively.
Table 2

The optimal doses of vaccines delivered to distribution centers in each period

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{idwt}$$\end{document}Yidwtt
idw2345678
11107,500,00000000
116000006,000,0000
11700000011,123,260
123000010,488,40000
1270000003,876,740
211013,011,02800000
2130000016,736,1400
2170000009,756,236
22101,988,97200000
223000004,615,8390
3114,000,000000000
31203,812,26600000
313003,518,7820000
3140004,000,000000
31500001,289,24500
31600000338,1810
3170000003,389,863
32300286,4880000
32500002,710,75500
326000005,491,5950
3270000001,610,137
3420187,73400000
343001,194,7300000
34600000170,2240
41100000011,522,299
4210000004,080,327
44100000018,397,374
51203,690,68200000
513001,405,6150000
514000529,233000
51500004,492,53900
5170000005,553,466
5214,999,780000000
5220309,31800000
52300170,9800000
5240003,154,955000
5270000001,181,053
541220000000
543002,423,4050000
544000315,812000
5450000507,46100
546000006,000,0000
547000000265,481
Table 3

The optimal doses of vaccines delivered to distribution centers in each period

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{idt}$$\end{document}uidtt
id2345678
1107,500,0000006,000,00011,123,260
12000010,488,40003,876,740
21013,011,02800016,736,1409,756,236
2201,988,9720004,615,8390
314,000,0003,812,2663,518,7824,000,0001,289,245338,1813,389,863
3200286,48802,710,7555,491,5951,610,137
340187,7341,194,73000170,2240
4100000011,522,299
420000004,080,327
4400000018,397,374
5103,690,6821,405,615529,2334,492,53905,553,466
524,999,780309,318170,9803,154,955001,181,053
5422002,423,405315,812507,4616,000,000265,481
The optimal doses of vaccines delivered to distribution centers in each period The optimal doses of vaccines delivered to distribution centers in each period For example, 7,500,000 in the first row of Table 2 indicates that distribution center 1 (Patna) has ordered type 1 vaccine in period 1 and received 7.5 million doses of vaccines in period 3. The numbers included in Table 3 indicate the optimal doses of vaccines delivered to each distribution center in different periods. For example, 11,123,260 in the first row and last column of this table suggests that 11,123,260 doses of vaccine type 1 have been delivered to the Patna distribution center in period 8. Table 2 shows that this order has been placed in period 7.For example, 3,967,289 in the first row of Table 5 represents the number of type 1 vaccine doses shipped from Patna distribution center to state 1 (Uttar Pradesh) in period 3. Figure 2 presents the total doses of vaccines delivered to each state.
Table 5

The optimal vaccine doses shipped from distribution centers to warehouses in each period

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta_{idst}$$\end{document}θidstt
ids2345678
11103,967,2890006,000,0007,334,862
11130603,07500000
111502,264,51400000
11160000002,842,904
1121000000945,494
11320117,29200000
11330547,83000000
125000010,488,40000
1260000003,853,421
12900000022,785
1227000000534
21106,783,18200000
2120000014,198,0230
2140000008,478,729
211303,592,78100000
2115000001,863,8930
21160000001,269,092
211700000115,0900
211801,921,32800000
21210663,08300000
21250000008415
213200000559,1340
2133050,65400000
2250843,1470004,533,2600
22601,145,82500000
22180000082,5790
3111,165,60703,518,7820000
3140001,873,042000
31111,951,1973,812,266000338,1810
3115000000448,851
3117147,179002,126,9581,289,24500
3118000000688,132
31190000002,252,880
3124480,220000000
3130228,248000000
313527,549000000
323000005,491,5951,424,874
32900286,48802,710,7550185,263
34800409,19700170,2240
341000778,2230000
34230186,57800000
34280073100000
34350115600000
4140000004,458,437
4160000005,355,224
41130000001,085,490
4115000000121,062
4117000000436,095
412100000050,777
412400000015,214
4290000004,080,327
4470000004,686,209
4480000008,403,048
441000000028,580
44120000004,976,826
4427000000216,703
442800000086,008
516000000243,032
511002,300,1650529,2334,492,5390238,053
511300138,9400000
5116000000826,400
5118000000963,539
51200000001,765,361
5122000000870,248
5123000000390,773
51250496,78000000
51260297,21400000
51280112,28500000
513000110,3980000
5131001,146,8160000
51330094610000
51340412,19400000
5135000000256,060
5136072,04400000
5234,999,780309,318170,9803,154,955000
5290000001,181,053
547001,019,244004,000,7640
5412000000177,136
5414001,397,340315,812305,7331,999,23688,345
54270068210000
54290000201,72800
5436220000000
Fig. 2

The total doses of vaccines allocated to each state

The optimal doses of vaccines assigned to group 1 are given in Table 4. Similarly, in Appendix B (Tables 17, 18, 19, 20, 21, 22 and 23), the optimal doses of vaccines allocated to groups 2 to 8 are presented, respectively.
Table 4

The optimal doses of vaccines assigned to group 1 in each period

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu_{igst}$$\end{document}μigstt
igs2345678
1110000006,378,820
115000010,488,40000
111502,264,51400000
11160000002,842,904
1121000000945,494
11320000101,77600
21106,649,02000000
212000005,997,9070
2140000007,930,872
211302,835,83900000
2118001,620,4740000
21330048,7060000
311000778,2230000
31111,951,197000001,048,502
31170002,126,958000
3119000000964,793
312400000270,0440
313527,549000000
4160000005,341,673
4170000004,650,605
4180000004,248,935
4190000004,080,327
41120000002,400,540
4127000000116,027
5134,999,780000001,982,588
5110000004,034,8710
5114001,387,5470305,73300
5120000000892,038
5122000000401,632
5123000000300,895
5125000000294,686
5126000000141,741
5128094,00500000
5129000000107,595
5130000000100,614
5131000066,81600
5134000029,89800
51360000005757
Table 17

The optimal doses of vaccines allocated to group 2 in each period

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu_{igst}$$\end{document}μigstt
igs345678
1290000022,785
122700000534
123200002810
221134,16200000
222000031,5380
225034,8190000
221310,32500000
2217000063380
2218000008224
2221277300000
32110000010,428
32150000019,127
3219000006824
3224000002192
322800002430
4240000046,436
4260000013,551
4270000035,604
4280000022,945
42100000028,580
42120000018,906
523000037,6170
5214097930000
52160000015,424
5220000008809
5222000003183
5223000002160
5225000020000
522600000482
522900065800
523000000180
523104920000
523301020000
523400000191
523500000187
52360000047
Table 18

The optimal doses of vaccines allocated to group 3 in each period

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu_{igst}$$\end{document}μigstt
igs345678
1310001,011,45400
1332000210900
23200000323,389
235260,92900000
236237,50100000
2313155,97100000
23150000035,094
3380090,040000
3390286,4880000
33110000064,320
331800000151,059
33190000041,490
43400000357,043
43120000081,373
431700000101,389
43210000014,869
43240000015,214
4327000004405
4328000004041
53300000667,135
5370000154,347
531000000119,411
53140000088,345
531600000175,129
53200000017,569
5322000008268
53230000021,922
532500014,94900
53260000011,236
5329000553900
5330000013850
5331000003237
5333018040000
533400008260
5335000001397
53360000094
Table 19

The optimal doses of vaccines allocated to group 4 in each period

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu_{igst}$$\end{document}μigstt
igs2345678
14320000400300
24200000499,2040
244000000547,857
245000000495,085
242100050,628000
34101,165,60700000
349000000185,263
341100000231,7850
341700147,1790000
341900000084,195
342412,458000000
34280000007067
34300000004957
34350115600000
448000000405,377
4415000000121,062
54300480,2980000
546000000243,032
547000000233,510
54100000204,84600
5412000000177,136
5413000138,940000
541400000142,8030
541600000084,395
5418000000112,818
542000000040,819
542200000020,122
542300000012,093
54250012,3660000
54260000006749
54270000006821
54290000424000
54310000052130
54330001847000
54340000125100
5436220000000
Table 20

The optimal doses of vaccines allocated to group 5 in each period

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu_{igst}$$\end{document}μigstt
igs345678
15100000862,564
15600000256,395
1532000193900
25200000534,417
2550000239,8800
25130216,5660000
25150000187,9970
25170000108,7520
2525000008415
2533001948000
3580000328,5990
359000181,95000
351100000219,649
351900000101,957
352414,15500000
45400000449,180
451200000199,625
45210000035,908
4527000009301
4528000007841
55300467,615000
5570000292,1600
551000000269,779
55140000151,6020
551600000178,921
551800000131,080
55200000042,306
55220000026,036
55230000015,206
55260000012,760
5529000005199
5530004865000
5531000711600
5534000001565
553500000786
553600000227
Table 21

The optimal doses of vaccines allocated to group 6 in each period

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu_{igst}$$\end{document}μigstt
igs345678
16100001,326,0800
1632000619700
26200000976,838
26500000766,347
266751,19800000
2613313,17900000
2618300,85400000
2621088,1080000
363000001,228,612
361100000418,692
361500000429,724
361900000153,940
36240000026,413
3630000091080
464000001,009,206
46800000733,584
461200000462,588
461700000277,075
46270000011,950
5670000636,9510
56900000718,241
5610000527,78300
561400315,812000
561600000321,142
562000000121,252
56220000060,213
56230000031,472
56250000030,393
56260000016,365
562818,28000000
56290000013,834
5631000085210
5633005708000
5634000003412
5635000002841
563600000652
Table 22

The optimal doses of vaccines allocated to group 7 in each period

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu_{igst}$$\end{document}μigstt
igs35678
171000543,6860
17320098700
173311930000
2720000158,415
2750000122,116
276157,1260000
271360,9010000
271500063,1220
2718000074,355
272124,2710000
3730000196,262
3780160,782000
37900162,48500
37110000100,704
3719000042,885
372400060440
4740000186,018
4712000065,552
4717000057,631
472700004315
472800003245
577000213,4810
57100000103,135
571400079,0210
5716000051,389
5720000037,718
572200008047
572300007025
572500865700
572600006383
572900001812
573003354000
573100002646
573400011430
57350000809
57360000202
Table 23

The optimal doses of vaccines allocated to group 8 in each period

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu_{igst}$$\end{document}μigstt
igs345678
1812,955,83500004,223,712
186000003,597,026
1813603,07500000
1833546,63700000
28200005,676,3150
285000003,457,231
281500001,577,6800
2816000001,269,092
2821497,30300000
28320000559,1340
38103,518,7820000
38300005,491,5950
384001,873,042000
3890002,366,32000
3811000002,056,367
381700001,289,2450
381800000537,073
381900000856,796
38230000186,5780
382400000148,914
383000000214,183
484000002,410,554
488000002,992,207
4812000001,748,242
4813000001,085,490
48270000070,705
48280000070,881
587000003,489,559
58900000462,812
58102,300,16500000
581400001,625,8100
581800000719,641
582000000604,850
582200000342,747
5825000133,72900
582600000101,498
582900062,85100
5831000001,052,775
58340000373,9080
583500000250,040
583665,06500000
The optimal doses of vaccines shipped from distribution centers to the state warehouses in each period are shown in Table 5. Finally, Table 6 presents the number of vaccine doses stored in the state warehouses in each period.
Table 6

The doses of vaccines stored in the state warehouses in each period

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi_{ist}$$\end{document}ψistt
is234567
1101,011,4541,011,4541,011,45404,130,234
1320117,292117,292117,2922810
22000001,993,059
250582,218547,399547,399547,3994,840,779
2130216,5660000
2150000035,094
21801,620,47400082,579
2210138,73650,628000
233050,6541948000
311,165,60700000
3800409,197158,375158,3750
31103,812,2663,812,2663,812,2663,812,2663,918,662
317147,179147,179001,289,2450
3230186,578186,578186,578186,5780
324467,762453,607453,607453,607453,607177,519
328007310731073107067
330228,248228,248228,248228,248228,248219,140
530309,31802,687,3402,687,3402,649,723
57001,019,2441,019,2441,019,2443,723,069
510000529,2334,289,143254,272
51300138,940000
5250496,780484,414484,414327,079325,079
5260297,214297,214297,214297,214297,214
527006821682168216821
5290000128,440128,440
53000110,398102,179102,179100,794
531001,146,3241,146,3241,072,3921,058,658
533007555000
5340412,194412,194412,194381,0455168
536069796979697969796979
The optimal doses of vaccines assigned to group 1 in each period The optimal vaccine doses shipped from distribution centers to warehouses in each period The total doses of vaccines allocated to each state The doses of vaccines stored in the state warehouses in each period The following items have been depicted in Fig. 3 to provide a more accurate interpretation of the obtained result: the optimal doses of vaccines purchased from each manufacturer by the distribution centers, the optimal doses of vaccines assigned to the Kerala State to each group in each period, and the optimal doses of vaccines shipped from the Hyderabad distribution center to the Kerala State in each period.
Fig. 3

Assigned vaccines to each group in each period in Kerala state

Assigned vaccines to each group in each period in Kerala state The results presented in Fig. 3 show that a total of 186,096,615 double-dose vaccines have been purchased from each of the five manufacturers, where 85,096,615 vaccines required ultra-cold refrigeration, 66,000,000 vaccines required very cold refrigeration, and the remainder required cold refrigeration. From the total purchased vaccine, 111,668,835 and 44,965,339 double-dose vaccines have been allocated to the Patna and Bhopal distribution centers, respectively. The remaining 29,462,441 double-dose vaccines have been allocated to the Hyderabad distribution center. In addition, the equitable vaccine distribution in Kerala State is depicted in this figure. 1,397,340 double-dose vaccines are transferred to this state from the Hyderabad distribution center in period 4, out of which 1,387,340 vaccines were assigned to group 1, and the remaining 9793 vaccines were allocated to group 2. Similarly, vaccines assigned to each group in periods 5 to 8 are shown in this figure. It is worth noting again some parameters, including transportation costs, vaccine prices, and manufacturing capacities, have been simulated based on preliminary estimates at MOHFW. In summary, this study demonstrated a practical, structured, and yet flexible scientific approach for equitable COVID-19 vaccine distribution in developing countries. The obtained results confirm the efficiency and effectiveness of the proposed model.

Sensitivity analysis

In this section, we study the performance of the model proposed in this study according to various budgetary constraints. We begin the sensitivity analysis by increasing (decreasing) the total budget and expect that the total doses of the purchased vaccine will increase (decrease) accordingly. For this purpose, we consider nine budgeting scenarios and calculate the total doses of the purchased vaccine for each scenario. The total doses of purchased vaccines for these scenarios are presented in Table 7 and Fig. 4.
Table 7

Sensitivity analysis procedure using budget changes

ScenarioBudgetTotal doses of the purchased vaccine
14,100,000,000178,183,484
24,200,000,000180,073,849
34,300,000,000181,308,864
44,400,000,000184,706,426
5 (main problem)4,500,000,000186,096,615
64,600,000,000186,943,712
74,700,000,000188,607,348
84,800,000,000189,013,640
94,900,000,000189,956,071
Fig. 4

Total doses of the purchased vaccine for each scenario

Sensitivity analysis procedure using budget changes Total doses of the purchased vaccine for each scenario As shown in Table 7 and Fig. 4, as the budget amount increases (i.e., scenarios 6 to 9), the total doses of the purchased vaccine increase, and as the budget amount decreases (i.e., scenarios 4 to 1), the total doses of the purchased vaccine decreased. The results are logical and meet our expectations of the proposed model behavior. This sensitivity analysis confirms the applicability and logical performance of the model.

Conclusion

This study proposed a mathematical programming model for equitable COVID-19 vaccine distribution in developing countries in the context of a location-inventory problem, considering the concepts of equity and taking into account the needs for cold, very cold, and ultra-cold supply chains. This model is the general form of Rastegar et al.’s (2021) model that provides the possibility of distributing vaccines requiring cold, very cold, and ultra-cold refrigeration among heterogeneous populations. Budgetary considerations, manufacturer selection, and time-dependent capacities are considered some of the proposed model’s more general and practical assumptions. Data from a case study in India is used to validate the practical application of the proposed model. The results showed that with 4.5 billion dollars, the Indian government could purchase over 186 million double-dose COVID-19 vaccines, including over 85 million for ultra-cold, 66 million for very cold, and 35 million double-doses for cold supply chains. Finally, sensitivity analysis was used to confirm the applicability and logical performance of the model. This study has proposed an operational model with strategic consideration under certainty for equitable COVID-19 vaccine distribution. Life is full of uncertainty, and failing to fully consider operational uncertainties can have detrimental consequences in any operations, including vaccination efforts in developing countries. Future research is needed to study some of our operational assumptions under uncertain conditions. In addition, more complex models with additional objective functions (i.e., emission reduction) can improve the real-world applicability of the vaccine distribution model proposed in this study. Future advanced analytics research is needed to coordinate manufacturing and distribution with healthcare providers and pharmacies to deploy vaccines more effectively and efficiently through specialized supply chain networks.
Indices
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i$$\end{document}iVaccine type
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d$$\end{document}dDistribution center
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s$$\end{document}sState
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g$$\end{document}gGroup type
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w,\hat{w}$$\end{document}w,w^Period (ordering time)
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t,\hat{t}$$\end{document}t,t^Period (delivery time)
Parameters
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$DM_{gs}$$\end{document}DMgsThe total demand of group g for COVID-19 vaccines in state s
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$VR_{i} \left\{ {\begin{array}{l} 1 \\ 0 \\ \end{array} } \right.$$\end{document}VRi10If vaccine i requires very cold or ultra-cold refrigeration
Otherwise (vaccine i requires cold refrigeration)
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$VT_{i} \left\{ {\begin{array}{l} 1 \\ 0 \\ \end{array} } \right.$$\end{document}VTi10If vaccine i requires ultra-cold refrigeration
Otherwise (vaccine i requires very cold refrigeration)
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$FX_{it}^{MN}$$\end{document}FXitMNThe ordering cost to the manufacturer of vaccine i in period t
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$FY_{d}^{DS}$$\end{document}FYdDSThe set-up cost of the distribution center d equipped with cold refrigeration
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$FX_{d}^{DS}$$\end{document}FXdDSThe set-up cost of the distribution center d equipped with very cold refrigeration
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$EXC$$\end{document}EXCThe additional cost required to convert very cold refrigeration to ultra-cold refrigeration in a distribution center
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$VP_{i}$$\end{document}VPiThe purchasing cost of two-doses of vaccine i
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$TR_{id}^{MN}$$\end{document}TRidMNThe transportation cost of two-doses of vaccine i from manufacturer location to distribution center d
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$TR_{ids}^{DS}$$\end{document}TRidsDSThe transportation cost of two-doses of vaccine i from distribution center d to the warehouse is state s
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$HC_{is}$$\end{document}HCisThe holding cost for two-doses of vaccine i in the state warehouse s
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CP_{iwt}^{MN}$$\end{document}CPiwtMNThe maximum vaccine i production capacity, if ordering and delivery time are in periods w and t, respectively
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CP_{d}^{NR}$$\end{document}CPdNRThe maximum distribution center d capacity for cold refrigeration vaccines
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CP_{d}^{DFR}$$\end{document}CPdDFRThe maximum distribution center d capacity for vaccines requiring ultra-cold refrigeration
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CP_{d}^{SR}$$\end{document}CPdSRThe maximum distribution center d capacity for vaccines requiring very cold refrigeration
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi_{g}$$\end{document}ξgThe minimum percentage coverage rate for group g
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi$$\end{document}ΦAvailable budget
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$bigM$$\end{document}bigMA big number
Variables
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{iwt}^{MN} \left\{ {\begin{array}{l} 1 \\ 0 \\ \end{array} } \right.$$\end{document}XiwtMN10Binary\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ \begin{gathered} {\text{If the manufacturer of vaccine }}i{\text{ is ordered in period }}w{\text{ to receive the vaccine in period }}t \hfill \\ {\text{Otherwise}} \hfill \\ \end{gathered} \right.$$\end{document}If the manufacturer of vaccineiis ordered in periodwto receive the vaccine in periodtOtherwise
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{d}^{DS} \left\{ {\begin{array}{l} 1 \\ 0 \\ \end{array} } \right.$$\end{document}YdDS10Binary\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ \begin{gathered} {\text{If distribution center }}d{\text{ is capable of handling vaccines requiring cold refrigeration}} \hfill \\ {\text{Otherwise}} \hfill \\ \end{gathered} \right.$$\end{document}If distribution centerdis capable of handling vaccines requiring cold refrigerationOtherwise
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{d}^{DS} \left\{ {\begin{array}{l} 1 \\ 0 \\ \end{array} } \right.$$\end{document}XdDS10Binary\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ \begin{gathered} {\text{If distribution center }}d{\text{ is set up for vaccines requiring very cold refrigeration}} \hfill \\ {\text{Otherwise}} \hfill \\ \end{gathered} \right.$$\end{document}If distribution centerdis set up for vaccines requiring very cold refrigerationOtherwise
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{d}^{EX} \left\{ {\begin{array}{l} 1 \\ 0 \\ \end{array} } \right.$$\end{document}XdEX10Binary\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ \begin{gathered} {\text{If equipment of ultra - cold refrigeration is added to distribution center }}d \hfill \\ {\text{Otherwise}} \hfill \\ \end{gathered} \right.$$\end{document}If equipment of ultra - cold refrigeration is added to distribution centerdOtherwise
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{idwt}$$\end{document}YidwtIntegerThe doses of vaccines i ordered in period w by distribution center d received in period t
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{idt}$$\end{document}uidtIntegerThe doses of vaccines i allocated to distribution center d in period t
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu_{igst}$$\end{document}μigstIntegerThe doses of vaccines i allocated to group g in state s in period t
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi_{ist}$$\end{document}ψistIntegerThe doses of vaccines i stored in the warehouse of state s in period t
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta_{idst}$$\end{document}θidstIntegerThe doses of vaccines i shipped from distribution center d to warehouse of state s in period t
  11 in total

Review 1.  Vaccine storage and distribution between expanded program on immunization and medical store department in Tanzania: a cost-minimization analysis.

Authors:  Ngwegwe Bulula; Diana P Mwiru; Omary Swalehe; Amani Thomas Mori
Journal:  Vaccine       Date:  2020-11-06       Impact factor: 3.641

Review 2.  Vaccines and routine immunization strategies during the COVID-19 pandemic.

Authors:  Ener Cagri Dinleyici; Ray Borrow; Marco Aurélio Palazzi Safadi; Pierre van Damme; Flor M Munoz
Journal:  Hum Vaccin Immunother       Date:  2020-08-26       Impact factor: 3.452

3.  A strategic approach to COVID-19 vaccine R&D.

Authors:  Lawrence Corey; John R Mascola; Anthony S Fauci; Francis S Collins
Journal:  Science       Date:  2020-05-11       Impact factor: 47.728

4.  A decision support system for demand management in healthcare supply chains considering the epidemic outbreaks: A case study of coronavirus disease 2019 (COVID-19).

Authors:  Kannan Govindan; Hassan Mina; Behrouz Alavi
Journal:  Transp Res E Logist Transp Rev       Date:  2020-05-07       Impact factor: 6.875

5.  Impact of COVID-19 pandemic response on uptake of routine immunizations in Sindh, Pakistan: An analysis of provincial electronic immunization registry data.

Authors:  Subhash Chandir; Danya Arif Siddiqi; Mariam Mehmood; Hamidreza Setayesh; Muhammad Siddique; Amna Mirza; Riswana Soundardjee; Vijay Kumar Dharma; Mubarak Taighoon Shah; Sara Abdullah; Mohammed Adil Akhter; Anokhi Ali Khan; Aamir Javed Khan
Journal:  Vaccine       Date:  2020-08-15       Impact factor: 3.641

6.  Exploring uncertainty and risk in the accelerated response to a COVID-19 vaccine: Perspective from the pharmaceutical industry.

Authors:  L Coudeville; G B Gomez; O Jollivet; R C Harris; E Thommes; S Druelles; A Chit; S S Chaves; C Mahé
Journal:  Vaccine       Date:  2020-10-13       Impact factor: 3.641

7.  An inventory-location optimization model for equitable influenza vaccine distribution in developing countries during the COVID-19 pandemic.

Authors:  Mehdi Rastegar; Madjid Tavana; Afshin Meraj; Hassan Mina
Journal:  Vaccine       Date:  2020-12-09       Impact factor: 3.641

8.  Priority allocation of pandemic influenza vaccines in Australia - Recommendations of 3 community juries.

Authors:  C Degeling; J Williams; S M Carter; R Moss; P Massey; G L Gilbert; P Shih; A Braunack-Mayer; K Crooks; D Brown; J McVernon
Journal:  Vaccine       Date:  2020-12-13       Impact factor: 3.641

9.  Acceptability of a COVID-19 vaccine among adults in the United States: How many people would get vaccinated?

Authors:  Paul L Reiter; Michael L Pennell; Mira L Katz
Journal:  Vaccine       Date:  2020-08-20       Impact factor: 3.641

10.  We need to start thinking about promoting the demand, uptake, and equitable distribution of COVID-19 vaccines NOW!

Authors:  Derrick Bary Abila; Sharon D Dei-Tumi; Fabrice Humura; Godwin N Aja
Journal:  Public Health Pract (Oxf)       Date:  2020-12-22
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  10 in total

1.  Analyses on ICU and non-ICU capacity of government hospitals during the COVID-19 outbreak via multi-objective linear programming: An evidence from Istanbul.

Authors:  Nezir Aydin; Zeynep Cetinkale
Journal:  Comput Biol Med       Date:  2022-05-06       Impact factor: 6.698

2.  COVID-19 vaccine distribution planning using a congested queuing system-A real case from Australia.

Authors:  Hamed Jahani; Amir Eshaghi Chaleshtori; Seyed Mohammad Sadegh Khaksar; Abdollah Aghaie; Jiuh-Biing Sheu
Journal:  Transp Res E Logist Transp Rev       Date:  2022-05-30       Impact factor: 10.047

3.  Optimizing vaccine distribution via mobile clinics: a case study on COVID-19 vaccine distribution to long-term care facilities.

Authors:  Samta Shukla; Francois Fressin; Michelle Un; Henriette Coetzer; Sreekanth K Chaguturu
Journal:  Vaccine       Date:  2021-12-27       Impact factor: 3.641

4.  A sustainable-resilience healthcare network for handling COVID-19 pandemic.

Authors:  Fariba Goodarzian; Peiman Ghasemi; Angappa Gunasekaren; Ata Allah Taleizadeh; Ajith Abraham
Journal:  Ann Oper Res       Date:  2021-10-07       Impact factor: 4.820

Review 5.  COVID-19 in Southeast Asia: current status and perspectives.

Authors:  Dinh-Toi Chu; Suong-Mai Vu Ngoc; Hue Vu Thi; Yen-Vy Nguyen Thi; Thuy-Tien Ho; Van-Thuan Hoang; Vijai Singh; Jaffar A Al-Tawfiq
Journal:  Bioengineered       Date:  2022-02       Impact factor: 3.269

6.  Multi-period vaccine allocation model in a pandemic: A case study of COVID-19 in Australia.

Authors:  Masih Fadaki; Ahmad Abareshi; Shaghayegh Maleki Far; Paul Tae-Woo Lee
Journal:  Transp Res E Logist Transp Rev       Date:  2022-04-11       Impact factor: 10.047

7.  Student learning time analysis during COVID-19 using linear programming - Simplex method.

Authors:  Sujata Pardeshi; Sushopti Gawade; Palivela Hemant
Journal:  Soc Sci Humanit Open       Date:  2022-03-15

8.  A data-driven robust optimization model by cutting hyperplanes on vaccine access uncertainty in COVID-19 vaccine supply chain.

Authors:  Hani Gilani; Hadi Sahebi
Journal:  Omega       Date:  2022-03-11       Impact factor: 8.673

9.  A mathematical model for managing the multi-dimensional impacts of the COVID-19 pandemic in supply chain of a high-demand item.

Authors:  Sanjoy Kumar Paul; Priyabrata Chowdhury; Ripon Kumar Chakrabortty; Dmitry Ivanov; Karam Sallam
Journal:  Ann Oper Res       Date:  2022-04-11       Impact factor: 4.820

10.  Location-Allocation Model to Improve the Distribution of COVID-19 Vaccine Centers in Jeddah City, Saudi Arabia.

Authors:  Areej Alhothali; Budoor Alwated; Kamil Faisal; Sultanah Alshammari; Reem Alotaibi; Nusaybah Alghanmi; Omaimah Bamasag; Manal Bin Yamin
Journal:  Int J Environ Res Public Health       Date:  2022-07-19       Impact factor: 4.614
  10 in total

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