1 Introduction

The emergence in 2019 of a new coronavirus, SARS-CoV-2 (COVID-19), has made devastating and huge consequences worldwide. Although some control mechanisms like hand hygiene, use of masks, and physical distancing [22] have helped decrease the transmission, unfortunately, these measures have failed to prevent the spread of COVID-19. As of 23 September 2021, as per Johns Hopkins University report (Johns Hopkins University and Medicine, 2021), 230,898,440 individuals have been infected, and 4,733,154 persons passed away due to this outbreak globally. Vaccines are required to reduce the complications and deaths associated with COVID-19, as can be expected, thus some vaccine candidates have started to emerge recently thanks to the extraordinary efforts of scholars [12].

The World Health Organization (WHO) has set the lower limit of the efficacy rate of a coronavirus vaccine at 50 percent [88]. In other words, in the era of COVID-19, each vaccine that is declared to provide protection above this rate is considered successful. Accordingly, to date, some coronavirus vaccines such as Comirnaty (BNT162b2), mRNA-1273, AZD1222, CoronaVac, and Sputnik V managed to obtain emergency use approval from the European Medicines Agency (EMA), the American Food and Drug Administration (FDA), and the WHO.

Obviously, the demand for vaccines will soar as long as the COVID-19 pandemic remains harsh. Questions about which vaccine is better are at present on the world agenda. Nevertheless, the available information on both existing coronavirus vaccines and COVID-19 is insufficient to respond to such a question. In parallel to the increase in vaccine production in the near future, should it be easier to reach different vaccines, it is no doubt that policymakers and customers will ask to choose the best vaccine against the COVID-19 illness. However, selecting a coronavirus vaccine is a challenging problem that encompasses numerous quantitative and qualitative attributes and alternatives together. Fortunately, the MCDM terminology has unique methods that allow to reliably solve such problems. Despite the fact that the Multi-Attribute Ideal-Real Comparative Analysis (MAIRCA) method was recently introduced by Pamucar et al. [65], it has attracted great attention from researchers. It is based on defining the difference between theoretical and actual outcomes. Thanks to a unique linear normalization algorithm, the main advantage of MAIRCA, one can obtain very reliable results. Further, MAIRCA is an effective mathematical tool and solution method that allows combining with other methods. However, assessments via exact numbers are not very sufficient in expressing human thoughts and opinions. Put it differently, the MAIRCA method cannot express experts' vagueness and ambiguity appropriately. To overcome such shortcomings, the fuzzy set theory proposed by Zadeh [99] presents rather useful tools [27]. After Zadeh's fuzzy sets, on the other hand, there have been some attempts to model uncertainty in a more realistic way [30]. As a result of the intense efforts of the researchers, some extensions of fuzzy sets have been developed [19, 63]. Intuitionistic fuzzy sets (IFSs), as an extension of the theory of fuzzy sets, were introduced by Atanassov [9] both to perform more flexible evaluations and to model imprecision properly. IFSs are also helpful to depict the ambiguous reasoning of experts. They yield a proper approach for coping with decision-making procedures containing membership and non-membership degrees as well as hesitation degrees. In a nutshell, with IFSs, the opinions of individuals are stated more inclusively since vagueness and imprecisions are identified by both the membership and non-membership degrees, simultaneously. According to existing literature, we discovered that no study had focused on the development of multi-criteria field integrated with IFS concept and aimed to assess coronavirus vaccines. For instance, Yener and Can [95] proposed the IF Multi-Attribute Border Approximation Area (IF-MABAC) approach for risk assessment. For assessing sustainable development indicators, Alrasheedi et al. [5] suggested the Combined Compromise Solution (CoCoSo) method with IF concept. IF best–worst analytic hierarchy process was recently introduced by Majumder et al. [52] for efficiency analysis. Rani et al. used an IF-gray relational analysis (IF-GRA) approach for selecting service providers. A combination of IFs and the combinative distance-based assessment (CODAS) technique was introduced by Mishra et al. [55] to evaluate sustainable suppliers. Ziquan et al. [103] employed the stepwise weight assessment ratio analysis (SWARA) and the complex proportional assessment (COPRAS) method under IF environment for determining the finest supplier. In this work, dissimilar to other studies, a framework based on group decision-making and developing from the integration of the MAIRCA method and IFSs is proposed, named the intuitionistic fuzzy MAIRCA (IF-MAIRCA) methodology for coronavirus vaccine selection problem for a real-life case study. Consequently, the IF-MAIRCA framework can cope with vague and indeterminate information in a satisfactory way in multi-criteria real-life problems.

Application of MCDM in COVID-19 outbreak is still at a very early stage. Hezam et al. [37] determined the priority group for the COVID-19 vaccine via an integrated neutrosophic analytical hierarch process (AHP) and technique for order of preference by similarity to ideal solution (TOPSIS) approach. By using measurement of alternatives and ranking according to the compromise solution under IF environment (IF-MARCOS), Ecer and Dragan (2021) evaluated healthcare services of insurance companies during the COVID-19 epidemic. Maqbool and Khan [53] analyzed barriers to prevent the transmission of COVID-19 illness through decision-making trial and evaluation laboratory (DEMATEL). In spite of these efforts, the research questions below remain unanswered:

Q1. Which methodology is most proper for a comprehensive evaluation of coronavirus vaccines during the COVID-19 epidemic?

Q2. How can policymakers and authorities as well as customers choose their coronavirus vaccines reliably and scientifically?

Q3. Which criteria are more appropriate for coronavirus vaccine selection?

Below are the main motivations for this paper.

  • Because IFSs include a non-membership function, vagueness and uncertainty can be coped with more conveniently. Besides, IFSs are suitable vehicles to be utilized for describing ambiguous or vague decision information.

  • The modifications of the MAIRCA technique and operators of IFSs are performed to introduce the fused model.

  • A multi-criteria problem regarding the coronavirus vaccine selection is given and solved by employing the suggested MAIRCA approach, which discloses the properness of the methodology proposed in this work.

Motivated by the above reasons, this research has the following goals:

  • To identify evaluation criteria of coronavirus vaccine assessment;

  • To weigh the criteria of coronavirus vaccine selection;

  • To evaluate coronavirus vaccines approved in the age of COVID-19.

The research is structured as follows. An extensive literature survey is presented in Sect. 2. Section 3 turns to the methodological subject. Particularly, preliminary knowledge on IFSs is discussed briefly. However, the introduced framework is explained in detail. A case study is introduced in the next section. Section 5 presents a comprehensive sensitivity analysis. The penultimate section conducts managerial implications and limitations. Conclusions are drawn in Sect. 7.

2 Review of earlier work

The literature section is grouped into two subsections. The first part of the literature survey aims to summarize the papers that employed the MAIRCA technique, while its second part focuses on the research related to any MCDM technique under the IFS.

2.1 Studies using the MAIRCA method

Given that the enormous potential of MCDM, lots of MCDM techniques have been posited so far to solve multi-criteria problems like MAIRCA. Due to the following superiorities of the MAIRCA technique, this work prefers MAIRCA: (i) it can be employed problems where there are numerous evaluation criteria and alternatives, (ii) it is capable of solving problems having both qualitative and quantitative evaluation criteria, (iii) it is easy to understand and apply, and (iv) it produces consistent solutions due to its own algorithm.

After the first study conducted by Pamucar et al. [65], till now, several studies have presented in the MCDM literature that utilize the MAIRCA method's superiorities, such as for selecting ammunition depots [33], assessing financial performance [11, 36], neighborhood selection [104], supplier selection [29, 59], selecting catering firm [85], site selection [61], etc.

However, the traditional MAIRCA technique is limited by its utilize of solely numerical values when prioritizing the alternatives according to the evaluation criteria; thus, extending MAIRCA to deal with the fuzziness and uncertainty of human thought is of practical importance. Pamucar et al. [64], for example, developed a model including DEMATEL, ANP, and MAIRCA techniques in the context of interval rough numbers to cope with a bidder evaluation problem. To evaluate suppliers, Chatterjee et al. [24] applied rough DEMATEL and rough ANP as well as rough MAIRCA. Pamucar et al. [60] suggested an interval-valued fuzzy-rough MAIRCA methodology for the treatment of vagueness. Boral et al. [15] introduced an AHP and MAIRCA methodology under a fuzzy environment. A combined fuzzy BWM and fuzzy MAIRCA approach developed by Gul and Ak [34] for the evaluation of occupational risks in terms of human health and the environment.

2.2 MCDM techniques under IF environment

On the basis of integration, several decision-making methodologies have been provided to solve real-world problems in an IF environment, such as third-party logistics evaluation [86], cost optimization [84], supplier selection [87, 89], module selection [51], sustainable performance evaluation [45], machine selection [4], job appointment [83], partner selection [97], technology evaluation [17], etc. To help waste managers, Karagoz et al. [40] introduced the intuitionistic combinative distance-based assessment (IF-CODAS) approach. By combining the AHP technique and the axiomatic design, Büyüközkan and Göçer [20] provided a methodology for solving supplier selection problem in IF concept. Using quality function deployment (QFD) and TOPSIS for knowledge management system evaluation in an IF environment, Li et al. [47] suggested a novel fuzzy MCDM model. Table 1 demonstrates a summary of researches in relation to MCDM techniques in IF context recently.

Table 1 MCDM studies based on IFSs
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With respect to the detailed literature survey, therefore, the following gaps are revealed:

  • To the best of the authors’ knowledge, there is no earlier work that has used the IF-MAIRCA approach. Some extensions of MAIRCA intending to contain more ambiguity are available in the literature [3, 13]. However, the IF extension of the method has not been performed yet.

  • There is no work on the performance evaluation of coronavirus vaccines according to various qualitative and quantitative attributes.

Keeping in mind the matters discussed above, this paper focuses on the coronavirus vaccine evaluation and selection problem. Since this problem has multiple vagueness, in this work, the MAIRCA method under an intuitionistic fuzzy environment is introduced for the coronavirus vaccine problem.

3 Research methodology

3.1 IFSs and related knowledge

Zadeh’s fuzzy sets [99] copes with vagueness successfully in a scientifically style. Yet, it can solely focus on the membership degree of undefined cases [28, 62]. Besides, it fails in coping with the non-membership degree of uncertain conditions. To eliminate this problem, Atanassov [9] presented IFSs as a useful way of employing uncertainty and fuzziness by means of the degree of hesitation in the system. To better understand the following part, IFS is clarified below [10].

Let a set X be a fixed universe of discourse and its subset \(A=\left\{\left.x,{\mu }_{A}\left(x\right),{\vartheta }_{A}(x)\right|x\epsilon X\right\}\) which is allocated by the membership function \({\mu }_{A}\left(x\right)=[\mathrm{0,1}]\) and non-membership function \({\vartheta }_{A}\left(x\right)=[\mathrm{0,1}]\), satisfying \(0\le {\mu }_{A}\left(x\right)+{\vartheta }_{A}\left(x\right)\le 1\). Additionally, for each IFS A, \({{\pi }_{A}=1-\mu }_{A}\left(x\right)-{\vartheta }_{A}\left(x\right)\) which corresponds to the degree of hesitancy. It is obvious that \(0\le {\pi }_{A}\left(x\right)\le 1\).

3.2 Intuitionistic fuzzy MAIRCA (IF-MAIRCA)

In the present work, it is applied to the MCDM technique to derive the weight value of each attribute as per the IF method and IF weighted averaging (IFWA) operator proposed by Xu [92], and the linguistic phrases are transformed into intuitionistic fuzzy numbers (IFNs). The description IF-MAIRCA framework can be carried out as follows.

(i) e = {1,2,…,k} is the set of experts and their weights are \(\varphi =\left[{\varphi }_{1},{\varphi }_{2},\dots ,{\varphi }_{k}\right]\) and \(\sum_{e=1}^{k}{\varphi }_{e}=1\). The importance weight of each expert is computed by Eq. (1).

$${\varphi }_{e}=\frac{\left({\mu }_{e}+{\pi }_{e}.\left(\frac{{\mu }_{e}}{{\mu }_{e}+{\vartheta }_{e}}\right)\right)}{\sum_{e=1}^{k}\left({\mu }_{e}+{\pi }_{k}.\left(\frac{{\mu }_{e}}{{\mu }_{e}+{\vartheta }_{e}}\right)\right)}$$
(1)

In Eq. (1), \(\mu , \vartheta\), and \(\pi\) represent membership function, non-membership function, and hesitancy degree, respectively.

(ii) \({C}_{m}\) is the number of criteria and their weights are W = [w1, w2,…,wn],

where \(\sum_{m=1}^{n}{w}_{m}=1\); \(m=\mathrm{1,2},\dots ,n.\)

Step 1. Determining importance weights of experts.

The linguistic evaluations for both the experts and evaluation criteria are carried out via Table 2.

Table 2 Linguistic phrases for a rating of experts and evaluation criteria [73]
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Step 2. Construct the aggregated IF decision matrix.

Let \(D = \left[ {P_{{{\text{me}}}} } \right]_{n * k}\) \(\left( {e = 1,2, \ldots ,k;m = 1,2, \ldots ,n} \right)\) be the IF decision matrix of experts. Herein, Pme expresses the assessment of dth about expert the jth criteria. Pme is used by IFN, and it can be stated that \(P_{{{\text{me}}}} = \left( {\mu_{{P_{{{\text{me}}}} }} , \vartheta_{{P_{{{\text{me}}}} }} ,\pi_{{P_{{{\text{me}}}} }} } \right)\) where \(\pi_{{S_{jd} }}\) is the hesitation degree of \(\pi_{{{\text{me}}}}\) is performed by Eq. (2).

$$\pi_{{{\text{me}}}} = 1 - \mu_{{P_{{{\text{me}}}} }} - \vartheta_{{P_{{{\text{me}}}} }}$$
(2)

The aggregated IF decision matrix is depicted as \(\hat{R} = \left[ {\hat{P}_{{{\text{me}}}} } \right]_{n*k}\)

$$\hat{P}_{m} = \;{\text{IFWA}}_{w} \left( {P_{{m1}} ,P_{{m2}} , \ldots ,P_{{{\text{me}}}} } \right) = {\text{ }}\left[ {1 - \prod\limits_{{e = 1}}^{k} {\left( {1 - \mu _{{P_{{{\text{me}}}} }} } \right)^{{\varphi _{e} }} } ,\prod\limits_{{e = 1}}^{k} {\left( {\nu _{{P_{{{\text{me}}}} }} } \right)^{{\varphi _{e} }} } ,\prod\limits_{{e = 1}}^{k} {\left( {1 - \mu _{{P_{{{\text{me}}}} }} } \right)^{{w_{d} }} } - \prod\limits_{{e = 1}}^{k} {\left( {1 - \vartheta _{{P_{{{\text{me}}}} }} } \right)^{{\varphi _{e} }} } } \right]{\text{ }}$$
(3)

where \(\hat{P}_{m} = \left( {\mu_{{\hat{P}_{m} }} , \vartheta_{{\hat{P}_{m} }} ,\pi_{{\hat{P}_{m} }} } \right)\).

By using Table 3, the linguistic assessments for the alternatives are realized.

Table 3 Linguistic phrases for a rating of alternatives [73]
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Step 3. Define the IF ideal solutions.

An IF has a positive ideal solution (IFPIS) and a negative ideal solution (IFNIS), which take values \({\tau }^{+}=(\mathrm{1,0},0)\) and \({\tau }^{-}=(\mathrm{0,1},0)\), respectively. Whereas IFNIS and IFPIS are characterized by max and min operators, it is noted that there is no significant gap in their outcomes [8].

Step 4. Calculate the distance measures.

To compute the distance measure, a fuzzy normalized Euclidean distance equation is applied [80]. \({\delta }_{m}^{+}\) and \({\delta }_{m}^{-}\) are handled in Eqs. (4)–(5) to illustrate positive and negative distance measures, respectively.

$${\delta }_{m}^{+}=\sqrt{{({\mu }_{{\widehat{P}}_{m}}-{\tau }^{+})}^{2}+{({\vartheta }_{{\widehat{P}}_{m}}-{\tau }^{+})}^{2}+{({\pi }_{{\widehat{P}}_{m}}-{\tau }^{+})}^{2}}$$
(4)
$${\delta }_{m}^{-}=\sqrt{{({\mu }_{{\widehat{P}}_{m}}-{\tau }^{-})}^{2}+{({\vartheta }_{{\widehat{P}}_{m}}-{\tau }^{-})}^{2}+{({\pi }_{{\widehat{P}}_{m}}-{\tau }^{-})}^{2}}$$
(5)

Step 5. Determine the closeness coefficient (CC) values and calculate the criteria weights.

As \({\mathrm{CW}}_{m}\) is the CC of the mth criterion, it is computed by using Eq. (6).

$${\mathrm{CW}}_{m}=\frac{{\delta }_{m}^{-}}{{\delta }_{m}^{-}+{\delta }_{m}^{+}}$$
(6)

After the whole CC values are computed, the criteria weights could be found by normalization.

Step 6. Establish the initial IF decision matrix.

It is the matrix of CC values obtained in Step 5. This matrix also specifies the type of optimization (benefit or cost) of each criterion.

Step 7. Construct the normalized IF decision matrix.

Evaluation criteria are normalized by using the linear max–min normalization approach. To do so, Eq. (7) is used for benefit-based criteria, while Eq. (8) is applied for cost-based criteria.

$$n_{ij} = \frac{{x_{ij} - x_{\min } }}{{x_{\max } - x_{\min } }}\;\;\;\;{\text{if}}\;x_{ij} \in B$$
(7)
$$n_{ij} = \frac{{x_{\max } - x_{ij} }}{{x_{\max } - x_{\min } }}\;\;\;\;{\text{if}}\;x_{ij} \in C$$
(8)

Step 8. Determine the theoretical IF decision matrix.

Via Eq. (9), first, the preference for any of the f possible alternatives is decided.

$${P}_{{A}_{i}}=\frac{1}{f}$$
(9)

where \({\sum }_{i=1}^{f}{P}_{{A}_{i}}=1\).

Thereafter, preferences for the selection of alternatives are multiplied with criteria weights, thus the theoretical IF matrix (Tp) is derived (Eq. 10).

$${T}_{p}={P}_{{A}_{i}}.[{t}_{p1} {t}_{p2}\dots {t}_{pn}]$$
(10)

where n is the total number of criteria and \({t}_{pi}\) is the theoretical rating.

Step 9. Establish the real IF evaluation matrix.

To form the real IF evaluation matrix, the elements of the normalized IF decision matrix are multiplied by the elements of the theoretical IF decision matrix (Eq. (11)).

$${r}_{ij}={n}_{ij}. {t}_{pi}$$
(11)

Step 10. Build the IF gap matrix.

By using Eq. (12), the IF gap matrix is gathered by subtracting the real IF evaluation matrix from the theoretical IF decision matrix.

$${g}_{ij}= {t}_{pi}-{r}_{ij}$$
(12)

Step 11. Calculate the utility scores of alternatives.

The utility scores of alternatives can be found by summing the elements of the IF gap matrix by rows, Eq. (13).

$${q}_{ij}= \sum_{j=1}^{n}{g}_{ij}, i=\mathrm{1,2},\dots ,f$$
(13)

where n is the total number of criteria and f is the total number of the alternatives.

Step 12. Rank the alternatives and choose the finest one.

Once all utility scores are calculated, finally, alternatives are ranked from smallest to largest according to their utility scores. It should be stated that the alternative with the smallest utility score is the best among the others since it is very close to the theoretically best positioning alternative.

4 The IF-MAIRCA framework for coronavirus vaccine selection

In this research, an IF-MAIRCA framework for aiding to determine the best coronavirus vaccine is introduced, as depicted in Fig. 1.

Fig. 1
figure 1

Flow diagram of the introduced coronavirus vaccine selection framework

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The decision-makers committee consists of four experts with doctorate degrees in medicine from infectious diseases (E1), internal medicine (E2), virology (E3), and chest diseases (E4). The experts who have sufficient knowledge about coronavirus vaccines have working experience of no less than eighteen years and work still in various universities in Turkey.

In the study, five coronavirus vaccines approved by the EMA, FDA, and WHO are included for analysis. That is, Comirnaty (A1), mRNA-1273 (A2), AZD1222 (A3), CoronaVac (A4), and Sputnik V (A5) are taken as alternatives into account. Detailed information about the alternatives is given in Appendix A.

Furthermore, the following eight quantitative and qualitative criteria are determined for the analysis based on both the knowledge of experts. Hundred percent ambiguities are accepted to data to which uncertainty information is not ensured [60]. To better define these criteria, this research addresses linguistic terms, which are closer to human thinking than exact numbers.

4.1 Duration of protection (C1)

It refers to the duration in which vaccinated people are expected to become immune to coronavirus. After receiving the second dose of vaccine for all coronavirus vaccines, meanwhile, the protection begins 2–3 weeks later. Experts are still trying to learn how long vaccines are protective in real life. However, clinical trials conducted by vaccine manufacturers give some clues about the protection period of the vaccines. For example, in people who received two doses of the mRNA-1273 vaccine, it was reported that the high level of protection of the vaccine was increased for up to 6 months [12].

4.2 Success against the mutations (C2)

It refers to the vaccine's ability to protect against different genetic variants of the coronavirus. Recently, it has been observed that the coronavirus has changed in different countries like England, South Africa, Brazil, and Finland and different variants have emerged. Such variants make people worry and, as a result, the reliability of vaccines developed is questioned. A variant remains susceptible to one potential mechanism of vaccine-mediated protection but cannot escape from that protection [50]. Very recently, Liu et al. [50] proved that the Comirnaty vaccine is also effective against recombinant (mutant) viruses.

4.3 Storage conditions (C3)

It means to the conditions such as temperature and storage environment necessary for the vaccines to not lose their unique properties. Storage conditions for vaccines are a significant factor because disadvantageous storage conditions are a barrier to the distribution of vaccines to the rest of the world. From this point of view, conventional vaccines (AZD1222, Sputnik V, and CoronaVac) offer storage advantages [43].

4.4 Effectiveness of the vaccine (C4)

It refers to the success in preventing death and serious illness. Effectiveness is a vital parameter for vaccines [43]. It is claimed that the effectiveness of the current vaccines in preventing serious illness and death is close to 100 percent. Khurana et al. [43] stated that the primary data revealed Comirnaty and mRNA-1273 have an efficacy of 95% and 94.5% against COVID-19, respectively. However, it has been noted that the effectiveness in preventing disease development in the vaccinated person differs from vaccine to vaccine.

4.5 Logistics (C5)

It refers to the ease or difficulty experienced in transporting vaccines. For instance, Pfizer has developed its own packaging that allows the doses to be preserved for 10 days without special freezers, but the doses still need to be flown from Belgium and then sent to vaccination centers in trucks with thermosensors and GPS trackers. This situation constitutes one of the biggest obstacles to its distribution to poor countries that do not have the necessary technological infrastructure. On the other hand, traditional vaccines have important superiorities like easy shipping [72].

4.6 Number of vaccine doses (C6)

It expresses the number of vaccine doses required for the vaccine to be effective against coronavirus. The current coronavirus vaccines are usually received in two doses with a time interval of a few weeks to ensure adequate immunity in the body [58]. The reason for this is to increase the amount of antibody, which decreased after receiving the first dose, to a certain level with the second dose. According to Bouazzaoui et al. [18], one of the most significant disadvantages of conventional vaccines is the need for multiple doses to achieve immunity.

4.7 Side effects (C7)

It depicts undesirable conditions such as headache, fever, and fatigue that occur after vaccination. According to Deutsche [25], since the introduction of the Comirnaty vaccine, some people have been shown to have allergic reactions immediately after vaccination. In the mRNA-1273, allergic reactions were observed in very few of the volunteers who were administered the vaccine, while exhaustion was observed in 9.7 percent. So far, there has been no report of serious side effects of the Sputnik V vaccine. However, complaints such as headache and fever, which are considered as usual side effects, were reported. According to CoronaVac volunteers, mild side effects occurred and these side effects disappeared within two days. It was stated that the most important side effect was a pain in the area where the vaccine was given. On the other hand, Oldenburg et al. [58] highlighted that 13 cases of thrombosis with more than 1.6 million AZD1222 vaccine doses administered.

4.8 Price (C8)

It refers to the price of one dose of coronavirus vaccine. It can be stated that another difference between available coronavirus vaccines is their price [43]. At present, the AZD1222 vaccine attracts attention as the cheapest vaccine (only $6/per dose). Prices per dose for Comirnaty, mRNA-1273, CoronaVac, and Sputnik V vaccines are $37, $39, $60, and $20, respectively [25]. Thus, the low price of mRNA vaccines is seen as an important advantage [18].

In sum, an evaluation system for the coronavirus vaccine selection is presented in Fig. 2. The solution stages of the introduced framework are clarified in detail below.

Fig. 2
figure 2

An assessment system for coronavirus vaccine selection

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Step 1. As the proposed IF-MAIRCA model is a group decision-based model, at first, the importance weights of decision-makers are determined via Eq. (1). The importance weights of experts are presented in Table 4.

Table 4 The importance weights of experts
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For instance, the importance weight of E1 is computed as follows:

$${\varphi }_{{E}_{1}}=\frac{0.88+0.04\left(\frac{0.88}{0.88+0.08}\right)}{0.88+0.04\left(\frac{0.88}{0.88+0.08}\right)+0.75+0.05\left(\frac{0.75}{0.75+0.2}\right)+0.88+0.04\left(\frac{0.88}{0.88+0.08}\right)+0.75+0.05\left(\frac{0.75}{0.75+0.2}\right)}=0.269$$

Step 2. The linguistic assessments of eight criteria are carried out by the experts by helping of Table 2, which evaluate the five coronavirus vaccine options according to the eight attributes utilizing Table 3. The linguistic expressions for the evaluation criteria and options are given in Tables 5 and 6, respectively.

Table 5 Linguistic evaluations of criteria by experts
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Table 6 Linguistic evaluations of alternatives by experts
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By using Eqs. (2) and (3), the aggregated IF decision matrix shown in Table 7 is established. For example, aggregated \(\mu , \vartheta ,\) and \(\pi\) of C1 could be found as follows:

Table 7 The aggregated IF decision matrix
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$$\mu =1-\left({\left(1-0.88\right)}^{0.269}\times {\left(1-0.88\right)}^{0.231}\times {\left(1-0.88\right)}^{0.269}\times {\left(1-0.88\right)}^{0.231}\right)=0.88$$
$$\vartheta ={0.08}^{0.269}\times {0.08}^{0.231}\times {0.08}^{0.269}\times {0.08}^{0.231}=0.08$$
$$\pi =1-\left(0.88+0.08\right)=0.04$$

Step 3. In this step, as mentioned above, the IFPIS and IFNIS are \({\tau }^{+}=(\mathrm{1,0},0)\) and \({\tau }^{-}=(\mathrm{0,1},0)\), respectively.

Step 4. Compute the distance measures.

Taking Table 7 into consideration, \({\delta }_{m}^{+}\), \({\delta }_{m}^{-}\), and the IF weights of each criterion are calculated by using Eqs. (4)–(5) as presented in Table 8. For example, \({\delta }_{1}^{+}\) and \({\delta }_{1}^{-}\) for C1 are calculated as follows:

Table 8 The \({\delta }_{1}^{+}\) and \({\delta }_{1}^{-}\) values of the criteria
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$${\delta }_{1}^{+}=\sqrt{{(0.88-1)}^{2}+{(0.08-0)}^{2}+{(0.04-0)}^{2}}=0.15$$
$${\delta }_{1}^{-}=\sqrt{{(0.88-0)}^{2}+{(0.08-1)}^{2}+{(0.04-0)}^{2}}=1.274$$

Step 5. Determine the CC values and compute the criteria weights.

CC values of criteria are calculated by Eq. (6). For instance, the CC value of C1 is found as follows:

$${\mathrm{CW}}_{1}=\frac{1.274}{1.274+0.15}=0.895$$

Based on the obtained CC values shown in Table 9, thereafter, normalization is performed and the criteria weights are determined.

Table 9 CC values and weights of evaluation criteria
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Thereby, C1 is specified as the most significant criterion.

Step 6. Form the initial IF decision matrix.

To determine elements of the IF decision matrix, firstly, the values \(\mu , \vartheta ,\) and \(\pi\) are computed in this stage based on Tables 6, 10.

Table 10 Aggregated \(\mu , \vartheta ,\) and \(\pi\) values of alternatives subject to each criterion
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Thereafter, through Table 10 and Eqs. (4)–(6), Table 11 is obtained.

Table 11 Aggregated IF decision matrix for alternatives
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Last but not least, the initial IF decision matrix can be built as shown in Table 12.

Table 12 The initial IF decision matrix
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Step 7. Construct the normalized IF decision matrix.

By both utilizing Eqs. (7)–(8) and Table 12, elements of the normalized IF decision matrix are identified (Table 13).

Table 13 The normalized IF decision matrix
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For instance, the normalized value of C1 (as a benefit-type criterion) subject to A1 is found as follows:

$${n}_{11}=\frac{1-0.7844}{1-0.7844}=1$$

On the other hand, the normalized value of C6 (as a cost-type criterion) as per A1 is determined as follows:

$${n}_{16}=\frac{0.7214-0.7213}{0.7214-0.7181}=0.031$$

Step 8. Determine the theoretical IF decision matrix.

As pointed out previously, by using Eq. (10), the theoretical IF decision matrix (Table 14) is identified by multiplying preferences for the selection of alternatives and criteria weights. Since the case study in this work is made according to the evaluation of five alternatives, via Eq. (9), the preference for the selection of alternatives is 0.2 (1/5 = 0.2).

Table 14 The theoretical IF decision matrix
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Subject to C1, for instance, the theoretical value of A1 is:

$${P}_{{A}_{i}}.{t}_{p1}=0.1422\times 0.2=0.0284$$

Step 9. Establish the real IF evaluation matrix.

Via Eq. (11), it is obtained the real IF evaluation matrix as presented in Table 15.

Table15 The real IF evaluation matrix
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According to C1, the real value of A1 is:

$${r}_{11}=1\times 0.0284=0.0284$$

Step 10. Build the IF gap matrix.

The IF gap matrix shown in Table 16 is constructed through Eq. (12).

Table 16 The IF gap matrix
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For instance,

$${g}_{11}= 0.0284-0.0284=0.$$

Steps 11 and 12. Calculate the utility scores and the alternatives.

As presented in Table 17, by using Eq. (13), the utility scores of alternatives are calculated by summing the elements of the IF gap matrix by rows.

Table 17 The utility scores and rank orders
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For instance, the utility score of A1 is found as follows:

$${q}_{1}= 0+0.0241+0.0249+0+0.0269+0.0188+0+0=0.0947$$

Once alternatives are ranked in ascending order as per their benefit scores, the most preferred alternative is A3. In a nutshell, AZD1222 is the best coronavirus vaccine among others with regard to the proposed framework.

5 Comparison and sensitivity assessment

The suitability of employing the proposed methodology for the goal of coronavirus vaccine selection could be shown by a detailed sensitivity check. The sensitivity check consists of four parts. First, the effect of the change of the most significant criterion (C1) on the ranking outcomes is examined. The modification of w1 is made by 40 scenarios, whereas the weights of the rest are proportionally changed to satisfy the condition \(\sum\nolimits_{j = 1}^{n} {w_{j} } = 1\) simultaneously. Second, the effect of changes in the weight of experts on ranking outcomes is investigated. Third, it is the comparison of the rankings derived performing some IF extensions of MCDM methods. Last but not least, the effect of a change in linguistic assessments is discovered. The sensitivity check is performed in MS Excel.

5.1 Changing the weight values of the criteria

In this subsection, the effect of the change in the weight of the most significant criterion (w1) on the ranking outcomes is examined. Firstly, it is formed 40 new vectors of weight coefficients, which are categorized into 40 scenarios. The novel vectors of weight coefficients are created by reducing the weight coefficient w1 by 1% in each scenario. So, the change of w1 in the interval \({w}_{1}\in [\mathrm{0.0839,0.1408}]\) is created, where w1 = 0.1408 means the value of the weighting factor in Scenario 1 (Sc1), while the value w1 = 0.0839 means the value of the weighting factor in Scenario 40 (Sc40). After each modification of w1, by performing Eq. 14, the weights of the rest criteria are obtained.

$${\omega }_{m\gamma }=\left(1-{\omega }_{m\pi }\right).\frac{{\omega }_{\gamma }}{\left(1-{\omega }_{m}\right)}$$
(14)

where m shows any criterion, \({\omega }_{m\gamma }\) shows the modified value of m, \({\omega }_{m\pi }\) shows the reduced value of the finest criterion (C1), \({\omega }_{\gamma }\) shows the original value of m, and \({\omega }_{m}\) shows the original value of finest criterion (C1). For example, the calculations for the first scenario (Sc1) are as follows. The Sc1 value of C1 becomes 0.1408 as a result of a 1% reduction of its real weight value of 0.1422 (\(0.1422\times 0.99=0.1408\)). Thereafter, by Eq. 14, the weights of the remaining criteria are as follows:

$${\omega }_{2}=\left(1-0.1408\right).\frac{0.1344}{\left(1-0.1422\right)}=0.1347$$
$${\omega }_{3}=\left(1-0.1408\right).\frac{0.1243}{\left(1-0.1422\right)}=0.1245$$
$${\omega }_{4}=\left(1-0.1408\right).\frac{0.1390}{\left(1-0.1422\right)}=0.1392$$
$${\omega }_{5}=\left(1-0.1408\right).\frac{0.1344}{\left(1-0.1422\right)}=0.1347$$
$${\omega }_{6}=\left(1-0.1408\right).\frac{0.0970}{\left(1-0.1422\right)}=0.0972$$
$${\omega }_{7}=\left(1-0.1408\right).\frac{0.1232}{\left(1-0.1422\right)}=0.1234$$
$${\omega }_{8}=\left(1-0.1408\right).\frac{0.1053}{\left(1-0.1422\right)}=0.1055$$

Consequently, the new weight values of criteria derived as per 40 scenarios are given in Table 18 and depicted in Fig. 3.

Table 18 New weight scenarios achieved with 1% reduction of w1
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Fig. 3
figure 3

Changes in criteria weights according to various scenarios

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After the use of new criteria weights in the proposed approach, as presented in Fig. 4, the updated rankings of alternatives are provided via different scenarios. With respect to Fig. 4, it is obvious that alternatives A3, A2, and A4 kept their ranks through all 40 scenarios. On the other hand, differentiation in the weight values of C1 has the most critical impact on the change in the rank of A1 and A5. A1 is third, while A5 is fourth if w1 is \({w}_{1}\in [\mathrm{0.1067,0.1408}]\). Should w1 is \({w}_{1}\in [\mathrm{0.0839,0.1053}]\), however, A5 is third, whereas A1 is fourth. In a nutshell, the rank order of alternatives A1 and A5 changes with each other. In that their score functions are very close (A1 = 0.0947, A5 = 0.1027), the change of ranking is no surprise. As a result, A1 and A5 could be thought equally good alternatives. Nonetheless, the weight change of C1 does not impact the ranking of the rest. According to the findings, therefore, alternative A3 remains the best, followed by alternative A2. Nevertheless, it is found that A4 is the worst alternative through all 40 scenarios. In spite of a little change in the ranking results, it can be concluded that the initial ranking (i.e., A3 \(\succ\) A2 \(\succ\) A1 \(\succ\) A5 \(\succ\) A4) is approved and accepted.

Fig. 4
figure 4

Ranking of alternatives in the light of different scenarios

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5.2 Changing the weight values of the experts

Because the importance weights of experts directly affect the determination of alternative rankings, it is crucial to make an analysis of the influence of changes in their importance on the rank orders. The importance weights of the experts were calculated above as follows: \({w}_{E1}={w}_{E3}=0.269\) and \({w}_{E2}={w}_{E4}=0.231\). On the basis of the values obtained, we can conclude that the experts E1 and E3 have the most meaningful impact on the introduced frameworks' final outcomes.

To demonstrate the influences of several importance weights of all experts on the outcomes found by the IF-MAIRCA framework in this research, six various cases are formed and evaluated the results as depicted in Table 19. In Case 1, the current importance weights of experts are addressed. All experts are handled to have the equal importance weight (0.25) in Case 2. In the cases from 3 to 6, a high weight (0.7) is allocated to each expert, respectively, while a small weight (0.1) is assigned to the remaining ones to satisfy the condition \(\sum {w}_{Ej}=1\). According to the sensitivity analysis, it is noticed that different importance weights of experts influence the ranking order of alternatives. For example, as depicted in Fig. 5, A3 is the alternative with the best performance in both Case 1 and Case 5. Similarly, A1 is the most preferred alternative in two cases: 2 and 4. In both Cases 3 and 6, A2 is the best alternative. These results emphasize that there is an effect of experts’ importance weights on determining a suitable alternative.

Table 19 Sensitivity analysis outcomes as per various importance weights of experts
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Fig. 5
figure 5

Final ranking changes of alternatives

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5.3 Benchmarking with the results of the IF-MARCOS approach

In the last decade, it is noticeable that extended MCDM techniques with IFSs have been employed in making decisions. However, each of these approaches has its unique processes. To execute a comparison, MARCOS extensions of IFSs [31] are employed in this work to tackle the coronavirus vaccine selection problem. The detailed information regarding the IF-MARCOS can be found in Ecer and Pamucar’s [31] study. Ranking results of the application of these approaches are presented in Table 20.

Table 20 Results of the IF-MARCOS approach
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As per the findings derived, it is confirmed that alternative A3 demonstrates the most preferred option according to IF-MARCOS. Besides, the alternative A4 is placed last in the final ranking, whereas A2 is the second-best alternative. Obviously, the application of the introduced framework and IF-MARCOS give the same ranking order. Based on the above analysis and Fig. 6, it can be concluded that the final ranking result obtained with IF-MAIRCA is as reliable and useful as the outcomes obtained from the IF-MARCOS model. It should note that the symmetrical structure in Fig. 6 is due to the fact that IF-MAIRCA identifies the smallest result as the best alternative, while IF-MARCOS emphasizes the highest result as the most preferred. So, the proposed IF-MAIRCA model is as robust as the approach compared and can be used confidently in challenging real-world decision-making problems such as those managed by Kaya and Ertugrul [42], Precup et al. [66], and Yuhana et al. [98], among others.

Fig. 6
figure 6

Utility scores of IF-MAIRCA and IF-MARCOS

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5.4 The effect of a change in linguistic assessments

Besides, it is examined the effect of a change in linguistic assessments on the final decision. To achieve this, the linguistic assessments of E1 as per the alternative A3 in Table 6 were changed from EG, G, VVG, VG, EG, G, VVG, and VVG to G, G, G, G, G, G, G, and G. Such a change produces the following utility scores: \({q}_{1}=0.0945, {q}_{2}=0.0776, {q}_{3}=0.0802, {q}_{4}=0.1281, {q}_{5}=0.1036\). A2 becomes the most acceptable alternative in this case, while A3 is the second best. Thus, the proposed model is fairly sensitive to linguistic ratings.

6 Managerial implications and limitations

This work develops an MCGDM framework called IF-MAIRCA, which goals to select the most preferable coronavirus vaccine among various vaccine options considering some conflicting criteria. This approach allows experts to express their thoughts, experiences, and knowledge with linguistic values and allows diverse persons to join by a strong method to come to a final decision. Because the MAIRCA method defines the gap between real and empirical outcomes successfully, a high performance could be accomplished in an uncertain environment by using it. Additionally, comparison and sensitivity check confirm the authority of the proposed framework.

With respect to the outcomes found in this research, thus, the most significant attribute for coronavirus vaccine selection is “duration of protection” with a relative weight of 0.1422. The next attribute is “effectiveness of the vaccine” (0.1390). The next attributes for effective selection are both “success against the mutations” and “logistics” (0.1344). Further, the next attribute is “storage conditions” (0.1243) and followed by “side effects” (0.1232). The findings as well state that “price” (0.1053) is ranked seventh. Finally, “number of vaccine doses” is the least crucial attribute. In relation to the findings, apart from this, coronavirus vaccines are prioritized from best to worst as AZD1222, mRNA-1273, Comirnaty, Sputnik V, and CoronaVac. The reason behind being the best coronavirus vaccine of AZD1222 over the rest could be its satisfactory protection duration, storage conditions, easy logistics, and low price.

The main contributions of the research are as follows.

  • To model and analyze information with a high degree of fuzziness, for the first time in the literature, this work posits a new MCGDM methodology, named IF-MAIRCA.

  • IF-MAIRCA allows the preferred coronavirus vaccine to meet the necessities of not only authorities and experts but for consumers willing to purchase coronavirus vaccine.

  • Bearing in mind the importance of experts, the proposed methodology determines the weight values of evaluation criteria and alternative rankings simultaneously, which makes it a useful decision tool.

  • As far as the authors' knowledge, in the era of COVID-19, integrating the MAIRCA method and IFS in the context of coronavirus vaccine selection has not been studied.

  • One of the unique properties of the introduced framework is that it takes the importance weight of each expert into account.

This research holds certain limitations. As there are still many uncertainties regarding the coronavirus, the criteria set is confined to the experts’ knowledge. Further, a simple form of IFSs is addressed. Hezam et al. [37] argued that the decision-making team should be involved experts from different medical fields to get benefit from their knowledge in determining priorities and ratings. The proposed model, therefore, makes calculations based on the subjective evaluations of a team of experts in their fields. An assessment based on those regions can be made if experts are selected from different countries or continents. Children and older patient groups are generally not included from clinical trials. Therefore, there are no published data on the safety and efficacy of vaccines in children and the elderly [77]. As a result, this study considers people over 18 years old but younger than 65 years old in Turkey, which is another limitation. When we are conducting this work, no country had access to all vaccines approved. As a result of the admirable intensive vaccine development and production efforts, it is hoped that all countries will have access to various vaccines in the near future. So, it will be likely to test the effectiveness of the proposed model with actual applications for different age groups. Further work could be performed on interval-valued IFNs or trapezoidal IFNs.

7 Conclusions

Vaccines work together with the natural defenses of the human body to protect health and prevent disease so that when anybody is exposed, s/he is ready to fight the virus. Because the MAIRCA method is more effective in detecting vagueness with combined IFSs, this work introduces IF-MAIRCA framework in order to choose the finest coronavirus vaccine during the COVID-19 outbreak. The IF-MAIRCA methodology, which has not been proposed before, has a great chance of success in solving multi-criteria problems as it can be operated easily. The introduced IF-MAIRCA methodology is effective in overcoming the ambiguity and uncertainty in experts' perceptions. Finally, the reliability and practicability of the introduced model are also checked with a three-phase sensitivity analysis. The primary goals of governments are to reduce deaths during the pandemic, remove obstacles to the speedy recovery of patients infected with COVID-19, and well-being for their people. So, the proposed methodology may give governments selecting the finest vaccine for their country.

As expected, this work also has some its own limitations. It can be noted that the importance weights of the evaluation criteria may vary for different countries and continents. Whereas storage conditions are less critical for a country in the Arctic, this criterion may be the most significant in a country in Africa. As a result, the criteria weights and vaccine rankings obtained in this study may vary from country to country. Since the beginning of the COVID-19 in China, many scholars around the world have declared predictions for the pandemic [69, 90] though the modeling results often differed from each other [70]. Because the methodology proposed in this paper is an assessment framework, not a prediction model, it is not possible to use the model to predict any stage of the epidemic (peak time, ending, etc.).

MAIRCA multi-criteria technique integrated with the IFS has a tremendous chance of success for MCGDM problems because it also takes uncertain and incomplete information of experts into account. Thus, in the future, this framework can be addressed for coping with fuzziness in MCDM problems like personnel selection, site selection, renewable energy source evaluation, and many other fields of engineering, management, agriculture, etc. problems.