The fuzzy set theory was introduced by Zadeh in 1965 for better reflecting on human judgments and assessment in the decision-making process. Real case decision-making problems include fuzziness and uncertainty, as decision, goals, constraints, decision-maker opinions are not completely known. For that reason, group decision maker problems practically have used fuzzy numbers (Zadeh, 1965). In this study, we prefer to use a triangular fuzzy number that can be defined as A ˜ = ( a 1 , b 1 , c 1 ), where, a 1 , b 1 and c 1 denote its lower, medium, and upper number. Definition 1.

Let A ˜ = ( a 1 , b 1 , c 1 ), a 1 < b 1 < c 1 be a fuzzy set on R = ( − ∞ , ∞ ). It is called a triangular fuzzy number (Carlsson and Fullér, 2001), if its membership function is illustrated as follows (see in Fig. 2).

Fig. 2

Triangular fuzzy numbers.

Fuzzy BWM method have been applied successfully in various areas such as evaluating the sustainable supplier selection criteria (Ecer and Pamucar, 2020; Pamučar and Savin, 2020; Amiri et al., 2021), identifying challenges and barriers for development of solar energy (Mostafaeipour et al., 2021), evaluating driver behaviour factors (Muravev and Mijic, 2020; Malakoutikhah et al., 2021), weighting the risk parameters of FMEA (Tian et al., 2018) plant site selection process (Luo et al., 2020), environmental performance evaluation (Liu et al., 2021; Dwivedi et al., 2021), evaluating the green supplier selection criteria (Wu et al., 2019), evaluating traffic parameters (Subotić et al., 2020) and hospital performance evaluation (Liao et al., 2019).

In addition, BWM has been incorporated with a different type of fuzzy sets such as interval type-2 fuzzy number (Wu et al., 2019; Qin and Liu, 2019), intuitionistic fuzzy sets (Tian et al., 2018; Mou et al., 2017), triangular fuzzy numbers (Guo and Zhao, 2017; Hafezalkotob and Hafezalkotob, 2017; Ecer and Pamucar, 2020), Z-numbers (Aboutorab et al., 2018), hesitant fuzzy numbers (Mi and Liao, 2019; Yazdani et al., 2021; Liao et al., 2019), rough-fuzzy approach (Chen and Ming, 2020), Pythagorean hesitant fuzzy sets (Liu et al., 2019).

The fuzzy pairwise comparisons are applied based on the linguistic terms given in Table 1. Then, the linguistic evaluations are transformed into triangular fuzzy numbers. The fuzzy comparison matrix ( A ˜ ) can be obtained as follows, where x ˜ i j denotes the relative fuzzy preference of criterion i to criterion j, which is a triangular fuzzy number; x ˜ i j = ( 1 , 1 , 1 ) when i = j. In this study, we prefer to use the steps of F-BWM in Guo and Zhao, 2017. The steps of F-BWM are shown as follows:

Step 1. Build the decision criteria system. The decision criteria system consists of a set of decision criteria. n decision criteria set is presented as follows: { c 1 , c 2 , … , c n }.

Step 2. Decide the best and the worst criterion. In this step, the best and the worst criterion is decided by experts based on the constructed criteria set in Step 1. The best and the worst criterion are denoted as c Best and c Worst for each expert’s team.

Step 3. Implement the fuzzy reference comparisons for the best criterion ( c Best ). In this step, the fuzzy preferences of the best criterion over all the other criteria are decided by experts. Then, the fuzzy comparisons in the linguistic format are converted to triangular fuzzy numbers. The fuzzy Best-to-Other’s vector can be obtained as follows: A ˜ BO = { x ˜ B 1 , x ˜ B 2 , … , x ˜ B n } , where x ˜ B j denotes fuzzy comparison of the best criterion c Best over criterion j, j = { 1 , 2 , … , n }.

Step 4. Do the fuzzy reference comparisons for the worst criterion ( c Worst ). In this step, the fuzzy preferences of all the criteria over the worst criterion are determined. The fuzzy Others-to-Worst vector can be obtained as: A ˜ OW = { x ˜ 1 W , x ˜ 2 W , … , x ˜ n W } , where x ˜ j W denotes the fuzzy comparison of the worst criterion c Worst , i = { 1 , 2 , … , n }.

Step 5. Determine the optimal fuzzy weights ( w ˜ 1 ∗ , w ˜ 2 ∗ , … , w ˜ n ∗ ). The optimal fuzzy weight for each criterion is determined for each fuzzy pair w ˜ B / w ˜ j and w ˜ j / w ˜ W . It should have w ˜ B / w ˜ j = x ˜ B j and w ˜ j / w ˜ W = x ˜ j W . A solution is obtained that the maximum absolute gaps | w ˜ B w ˜ j − x ˜ B j | and | w ˜ j w ˜ W − x ˜ j W | for all j are minimized to satisfy these conditions for all j. w ˜ B , w ˜ j and w ˜ W in fuzzy BWM are triangular fuzzy numbers. In some cases, we prefer to use w ˜ j = ( a j w , b j w , c j w ) for optimal criteria selection. The triangular fuzzy weight of the criterion w ˜ j = ( a j w , b j w , c j w ) is transformed into a crisp value using the graded mean integration representation (GMIR) equation in Table 4. Consequently, the constrained optimization problem is constructed for obtaining the optimal fuzzy weights ( w ˜ 1 ∗ , w ˜ 2 ∗ , … , w ˜ n ∗ ) as follows Guo and Zhao (2017). (6) min max j { | w ˜ B w ˜ j − x ˜ B j | , | w ˜ j w ˜ W − x ˜ j W | } s.t. ∑ j = 1 n R ( w ˜ i ) = 1 , a j w ⩽ b j w ⩽ c j w , a j w ⩾ 0 , j = 1 , 2 , … , n , where w ˜ B = ( a B w , b B w , c B w ), w ˜ j = ( a j w , b j w , c j w ), w ˜ W = ( a W w , b W w , c W w ), x ˜ B j = ( a B j w , b B j w , c B j w ) and Eq. (6) is transformed to the nonlinearly constrained optimization problem. (7) min θ s.t. | w ˜ B w ˜ j − x ˜ B j | ⩽ θ , | w ˜ j w ˜ W − x ˜ j W | ⩽ θ , ∑ j = 1 n R ( w ˜ i ) = 1 , a j w ⩽ b j w ⩽ c j w , a j w ⩾ 0 , j = 1 , 2 , … , n , where θ = ( a θ , b θ , c θ ).

Considering a ξ ⩽ b ξ ⩽ c ξ , it is supposed that θ ∗ = ( k ∗ , k ∗ , k ∗ ), k ∗ ⩽ a ξ then Eq. (7) can be transferred as (8) min θ ∗ s.t. | ( a B w , b B w , c B w ) ( a j w , b j w n , c j w ) − ( a B j , b B j , c B j ) | ⩽ ( k ∗ , k ∗ , k ∗ ) , | ( a j w , b j w , c j w ) ( a W w n , b W w , c W w ) − ( a j W , b j W , c j W ) | ⩽ ( k ∗ , k ∗ , k ∗ ) , ∑ j = 1 n R ( w ˜ i ) = 1 , a j w ⩽ b j w ⩽ c j w , a j w ⩾ 0 , j = 1 , 2 , … , n . Step 6. Compute the crisp weights. After obtaining fuzzy weights, the GMIR is used to alter the fuzzy weight of criterion to crisp weights. Where a ˜ indicates the ranking of triangular fuzzy number (Omrani et al., 2018).

The GMIR formula is as follows: (9) R ( a ˜ i ) = a i + 4 b i + c i 6 . Step 7. Check the consistency level. The consistency ratio is checked in the same way as BWM and through computing the consistency ratio (CR) from the following equation. In this step, the consistency index (CI) for F-BWM is used that is listed in Table 2. (10) CR = Q ∗ C I .

Ranking of Alternatives through Functional mapping of criterion subintervals Into a Single Interval (RAFSI) method (Žižović et al., 2020) is based on defining ideal and anti-ideal reference points and defining the relationship between alternatives concerning defined reference points. The relationships between criterion values and reference points are defined using criterion functions that map criterion sub-intervals into a single criterion interval. This achieves two key advantages of the RAFSI method (Alosta et al., 2021; Božanić et al., 2021): i) A data standardization algorithm that allows the translation of data from the initial decision matrix into an interval that is suitable for rational decision making; and ii) The mathematical formulation of the RAFSI method eliminates the rank reversal problem, as one of the significant shortcomings of many traditional MCDM methods. The following section shows the extension of the RAFSI method to a fuzzy environment (RAFSI-F). By applying fuzzy sets, the RAFSI algorithm has been improved and adapted to handle the inaccuracies and uncertainties that arise when solving real-world problems. The algorithm of the RAFSI-F method is realized through four steps:

Step 1: Formation of an aggregated fuzzy initial decision matrix. Suppose that the evaluation of alternatives from the set A i ( i = 1 , 2 , … , m) is carried out by k experts. Experts evaluate alternatives in relation to a defined set of criteria C j ( j = 1 , 2 , … , n) using a predefined fuzzy linguistic scale. Then we can present the judgment of k expert as a matrix X ( e ) = [ ξ ˜ i j ( e ) ] m × n , where 1 ⩽ e ⩽ k. (11) X ( e ) = ξ ˜ i j ( e ) ξ ˜ 12 ( e ) ⋯ ξ ˜ 1 n ( e ) ξ ˜ 21 ( e ) ξ ˜ 22 ( e ) ⋯ ξ ˜ 2 n ( e ) ⋮ ⋮ ⋱ ⋮ ξ ˜ m 1 ( e ) ξ ˜ m 2 ( e ) ⋯ ξ ˜ m n ( e ) m × n ; 1 ⩽ i ⩽ m ; 1 ⩽ j ⩽ n ; 1 ⩽ e ⩽ k , where ξ ˜ i j ( e ) = ( ξ i j l ( e ) , ξ i j s ( e ) , ξ i j u ( e ) ); ( i = 1 , … , m; j = 1 , … , n) represents the fuzzy value from the fuzzy linguistic scale.

Since we have a group decision-making model, we obtain k experts’ initial decision-making matrices X ( 1 ) , X ( 2 ) , … , X ( e ) , … , X ( k ) , ( 1 ⩽ e ⩽ k ). For each expert matrix X ( e ) = [ ξ ˜ i j ( e ) ] m × n at position ( i , j ) we obtain the fuzzy sequence ξ ˜ i j ( e ) = ( ξ i j l ( e ) , ξ i j s ( e ) , ξ i j u ( e ) ). Using the fuzzy Heronian operator (Yu, 2013), Eq. (12), we obtain the averaged fuzzy number ξ ˜ i j = ( ξ i j l , ξ i j s , ξ i j u ), where ξ i j l and ξ i j u respectively represent the lower and upper limits of the fuzzy number interval, while ξ i j s represents the value in which the fuzzy number ξ ˜ i j has the maximum value. The Heronian mean (HM) operator (Yu, 2013) was used to aggregate the values as it allows the representation of the interrelationships between the elements being aggregated. (12) ξ ˜ i j = ( ξ i j l , ξ i j s , ξ i j u ) = ξ i j l = ( 2 k ( k + 1 ) ∑ i = 1 n ∑ j = i n ξ i l p ξ j l q ) 1 p + q , ξ i j s = ( 2 k ( k + 1 ) ∑ i = 1 n ∑ j = i n ξ i s p ξ j s q ) 1 p + q , ξ i j u = ( 2 k ( k + 1 ) ∑ i = 1 n ∑ j = i n ξ i u p ξ j u q ) 1 p + q , where k represents the number of experts participating in the research, while p, q ⩾ 0 is a set of non-negative numbers. By applying Eq. (12) we obtain an averaged fuzzy initial decision-matrix X = [ ξ ˜ i j ] m × n .

Step 2: Mapping the elements of the initial decision matrix into criterion intervals. For each criterion C j ( j = 1 , 2 , … , n ), the decision-maker defines ξ ˜ I j and ξ ˜ N j , where ξ ˜ I j represents the ideal value according to the criterion C j , while ξ ˜ N j represents the anti-ideal value according to the criterion C j . For each alternative from the set A i ( i = 1 , 2 , … , m), we define a function fi that maps the criterion intervals from the aggregated initial decision matrix (9) to the criterion interval [ n 1 , n b ], Eq. (13): (13) f ˜ A i ( C j ) = n b − n 1 ξ ˜ I j − ξ ˜ N j ξ ˜ i j + ξ ˜ I j · n 1 − ξ ˜ N j · n b ξ ˜ I j − ξ ˜ N j , where n b and n 1 represent a ratio that shows how much the ideal value is better than the anti-ideal value, while ξ ˜ i j denotes the value of the i-th alternative for the j-th criterion from the aggregated initial decision matrix. It is recommended that the ideal value is at least six times better than the anti-ideal (barely acceptable value), i.e. that n 1 = 1 and n b = 6.

Thus, we obtain a standardized decision matrix T = [ φ ˜ i j ] m × n ( i = 1 , 2 , … , m, j = 1 , 2 , … , n) in which all elements of the matrix are translated into the interval φ ˜ i j ∈ [ n 1 , n b ]. The elements of the matrix T are obtained by applying expression (10), i.e. φ ˜ i j = f A i ( C j ).

Step 3: Formation of a normalized decision matrix N = [ φ ˆ i j ] m × n ( i = 1 , 2 , … , m, j = 1 , 2 , … , n). By applying Eq. (14), the normalization of the element of the matrix T is performed. (14) φ ˆ i j = φ ˜ i j 2 A , for max criteria , H 2 φ ˜ i j , for min criteria , where A and H represent the arithmetic and harmonic mean of the elements n 1 and n b , respectively.

Step 4: Calculation of fuzzy criterion functions of alternatives Q ˜ ( A i ) and ranking of alternatives. By applying Eq. (15), the criterion functions of alternatives Q ˜ ( A i ) are calculated and the ranking of alternatives is performed. (15) Q ˜ ( A i ) = ∑ j = 1 n w j φ ˆ i j . From the considered set of alternatives, the alternative that has a higher value of the fuzzy criterion function Q ˜ ( A i ) is chosen.

Step 1: Eight criteria for medical selection are shown in Table 3.

Step 2: According to F-BWM, evaluations of experts in linguistic terms are used to obtain the importance weights of antivirus mask criteria, the best criteria, the worst criteria, the best to other comparison matrix, and the other to worst comparison matrix of each expert (EX) are given in Table 6 and Table 7 by using linguistic terms, respectively.

Step 3: To take evaluations of Expert 1, as an example, the fuzzy preferences of the best criterion over all the criteria can be obtained with respect to Table 3. A ˜ B O = [ ( 0.67 , 1 , 1.5 ) , ( 3.5 , 4 , 4.5 ) , ( 1.5 , 2 , 2.5 ) , ( 2.5 , 3 , 3.5 ) , ( 3.5 , 4 , 4.5 ) , ( 3.5 , 4 , 4.5 ) , ( 1 , 1 , 1 ) , ( 2.5 , 3 , 3.5 ) ] .

Step 4: The fuzzy preferences of all the criteria over the worst criterion can be presented in Table 4. A ˜ OW = [ ( 1.5 , 2 , 2.5 ) , ( 0.67 , 1 , 1.5 ) , ( 1.5 , 2 , 2.5 ) , ( 0.67 , 1 , 1.5 ) , ( 0.67 , 1 , 1.5 ) , ( 1 , 1 , 1 ) , ( 3.5 , 4 , 4.5 ) , ( 0.67 , 1 , 1.5 ) ] .

Step 5: Then, for obtaining the optimal fuzzy weights of all the criteria, the nonlinearly constrained model is established as follows in Eq. (7).

Step 6: The following nonlinearly constrained optimization problem is obtained using represented by crisp numbers as in Eq. (8). (16)

Solving above model by using LINGO 18.0 software, the optimal fuzzy weights with regards to EX1 can be calculated, which are: w C 1 ∗ = ( 0.156 , 0.179 , 0.215 ) ; w C 2 ∗ = ( 0.064 , 0.073 , 0.079 ) ; w C 3 ∗ = ( 0.154 , 0.167 , 0.174 ) ; w C 4 ∗ = ( 0.080 , 0.082 , 0.100 ) ; w C 5 ∗ = ( 0.063 , 0.073 , 0.079 ) ; w C 6 ∗ = ( 0.073 , 0.073 , 0.079 ) ; w C 7 ∗ = ( 0.241 , 0.259 , 0.298 ) ; w C 8 ∗ = ( 0.080 , 0.082 , 0.100 ) . θ ∗ is obtained 0.4494 and the consistency ratio can be computed as: CR = 0.4494 8.04 = 0.0559. The CR is lower than 10%, therefore the obtained result is acceptable.

Then, all F-BWM steps have been implemented for each expert. Results of all optimal fuzzy weights and average optimal fuzzy weights (AOFW) of eight criteria are given in Table 8. The average crisp weights of eight criteria are illustrated in Fig. 3, respectively. Heronian function, Eq. (12) was used for aggregation of fuzzy weight coefficients.

According to the results of the F-BWM model, among the antivirus mask criteria, “Filtration rate ( w C 7 )” were found to be the most critical criteria related to antivirus mask selection and the next important criteria are “Leakage rate ( w C 1 )” and “Tear and deformation-resistant ( w C 8 )”, respectively. On the other hand, “Easy to wear and take off ( w C 6 )” is the least important criteria to the experts. Ranking from the most important criteria to the least important criteria is as follows: w C 7 ≻ w C 1 ≻ w C 8 ≻ w C 4 ≻ w C 3 ≻ w C 5 ≻ w C 2 ≻ w C 6 .

Step 7: The consistency ratio is an important indicator for calculating the consistency of pairwise comparisons for all experts’ opinions. Its closeness to zero indicates its higher consistency. The consistency ratio is computed for pairwise comparisons that indicate high consistency in paired comparisons, as shown in Table 9. The consistency ratio for each expert is close to zero. Therefore, the weights obtained for the criteria are confirmed.

After defining the weight coefficients of the criteria, five experts evaluated the alternatives A i ( i = 1 , 2 , … , 6 ) in relation to the eight criteria C j ( j = 1 , 2 , … , 8 ) that were defined in the previous part of the paper. Criteria belongs to the group of max criteria, while the criterion C 1 belongs to the group of min criteria. To evaluate the evaluation of alternatives, the experts used the fuzzy linguistic scale shown in Table 10.

After evaluating the alternatives, the experts’ correspondence matrices were obtained and are shown in Table 11.

By applying expression (12) we get an aggregated initial decision matrix, Table 12. When calculating the initial rank of alternatives, it is recommended that decision-makers choose the values p = q = 1, since the adoption of the value p = q = 1 simplifies the decision-making process.

The element at position A 1 – C 1 , by applying expression (12), we obtain as follows: ξ ˜ 11 = ( 2.57 , 3.40 , 4.24 ) = ξ 11 l = 2 6 ( 6 + 1 ) 1 1 · 1 1 + 1 1 · 1 1 + 1 1 · 3 1 + 1 1 · 1 1 + 1 1 · 4 1 + 1 1 · 5 1 + ⋯ + 1 1 · 4 1 + 1 1 · 5 1 + 4 1 · 4 1 + 4 1 · 5 1 + 5 1 · 5 1 1 1 + 1 = 2.57 ξ 11 s = 2 6 ( 6 + 1 ) 1 1 · 1 1 + 1 1 · 2 1 + 1 1 · 4 1 + 1 1 · 2 1 + 1 1 · 5 1 + 1 1 · 6 1 + · + 2 1 · 5 1 + 2 1 · 6 1 + 5 1 · 5 1 + 5 1 · 6 1 + 6 1 · 6 1 1 1 + 1 = 3.40 ξ i j u = 2 6 ( 6 + 1 ) 1 1 · 1 1 + 1 1 · 3 1 + 1 1 · 5 1 + 1 1 · 3 1 + 1 1 · 6 1 + 1 1 · 7 1 + ⋯ + 3 1 · 6 1 + 3 1 · 7 1 + 6 1 · 6 1 + 6 1 · 7 1 + 7 1 · 7 1 1 1 + 1 = 4.24. The remaining elements of the aggregated initial decision matrix (Table 12) are aggregated similarly.

Step 2: The experts defined the ideal and anti-ideal points, ξ ˜ I j = ( 10 , 10 , 10 ) and ξ ˜ N j = ( 0.5 , 0.5 , 0.5 ), by consensus for each criterion C j ( i = 1 , 2 , … , 8 ). Based on the defined ideal and anti-ideal points, criterion intervals are formed. Using expression (13), the functions for standardization of criteria are defined. Since all the values of the criteria in the initial decision matrix are defined using the same linguistic scale, the same function f ˜ A i ( C j ) was used to map all the criteria C j ( j = 1 , 2 , … , 8): f A i ( C j ) = 9 − 1 10 − 0.5 · ξ ˜ i j + 10 · 1 − 0.5 · 9 10 − 0.5 = 0.84 · ξ ˜ i j + 0.58. By applying the function f ˜ A i ( C j ), we obtain a standardized initial decision matrix ( T = [ φ ˜ i j ] 6 × 8 , i = 1 , 2 , … , 6, j = 1 , 2 , … , 8), given in Table 13.