The PIPRECIA method integrates two procedures for determining criteria weight, the first known as the PIPRECIA method and the second known as the inverse PIPRECIA method, which are together integrated into the Extended PIPRECIA (PIPRECIA-E) method (Stanujkic et al., 2017).

In the first approach, each criterion, except the first one, the significance of each criterion is compared with the significance of the previous criterion, whereby the comparison starts from the second criterion in the list of criteria, and ends with the last one in the list of criteria. This weighting procedure is often used independently and is known as the PIPRECIA method.

The second procedure for determining criteria weights is somewhat like the procedure used in the PIPRECIA method. However, in the inverse procedure (PIPRECIA-I) method, the comparison of the relative criteria importance begins from the n − 1 criterion and ends with the first criterion in the list of criteria. The computational procedure for determining criteria weights using the PIPRECIA-I procedure is very similar to the procedure of the PIPRECIA procedure, but this pairwise comparison procedure can sometimes be much more complex for the respondent, which is why there are often certain deviations in the criteria weights obtained using the PIPRECIA and PIPRECIA-I procedures. However, combining the above two approaches can be used to more realistically determine the significance of the criteria.

The fuzzy set (FS) theory was proposed by Zadeh (1965), as an extension of a classical set theory. In contrast to the classical set theory, FS theory allows for partial membership to a set.

Definition 1

A fuzzy subset A ˜ of a non-empty set X is defined as follows: (4) A ˜ = { 〈 x , μ A ˜ ( x ) 〉 | x ∈ X } ,

where μ A ˜ ( x ) : X → [ 0 , 1 ] is the membership function of FS A ˜ , μ A ˜ ( x ) ∈ [ 0 , 1 ] is the degree of membership of the element x to the A ˜ .

Membership functions can have different shapes such as bell-shaped, Gaussian, trapezoidal, and triangular.

Definition 2

Triangular fuzzy number (TFN) A ˜ can be represented as a triplet A ˜ = ( l , m , u ) where membership function is defined as follows: (5) μ A ˜ ( x ) = ( x − l ) / ( m − l ) , l ⩽ x < m , 1 , x = m , ( u − x ) / ( u − m ) , m < x ⩽ u , 0 , otherwise ,

where l, m, and u denote left endpoint, mode, and right endpoint, respectively, of TFN A ˜ , and l < m < u.

Definition 3

Let A ˜ = ( a l , a m , a u ) and B ˜ = ( b l , b m , b u ) be two positive TFNs, and k is a positive crisp number. The arithmetic operations of these TFNs, necessary for the development of the fuzzy extension of the PIPRECIA method, are as follows: (6) A ˜ ⊕ B ˜ = ( a l + b l , a m + b m , a u + b u ) , (7) A ˜ ⊘ B ˜ = ( a l b u , a m b m , a u b l ) , (8) A ˜ · k = ( a l · k , a m · k , a u · k ) , (9) A ˜ k = ( a l · 1 k , a m · 1 k , a u · 1 k ) .

Definition 4.

Let F i = ( l i , m i , u i ) be a collection of n fuzzy numbers. The fuzzy weighted arithmetic (FWA) operator of F i is as follows: (10) FW A ( F j ) = ∑ j = 1 n w j F j = ( ∑ j = 1 n w j l j , ∑ j = 1 n w j m j , ∑ j = 1 n w j u j ) ,

where: w j denotes the weight of element j of the collection A j , w j ∈ [ 0 , 1 ], and ∑ j = 1 n w j = 1.

This subsection presents an adaptation of the PIPRECIA-E method, which enables all values of variables used in the calculation procedure of the PIPRECIA-E method to be kept in the interval [ 0 , 1 ].

The computational procedure of the modified fuzzy PIPRECIA (MF PIPRECIA) method for determining criteria weights of n criteria can be presented through the following steps:

Step 1. Assigning the relative significance s j of each criterion, excepting the first criterion, using triangular fuzzy numbers, as follows: (21) s ˜ j ( l j , m j , u j ) = s j = ( ⩾ 0.5 , ⩾ 0.5 ∧ ⩽ 0.95 , ⩽ 0.95 ) , when C j ≻ C j − 1 , s j = ( 0.5 , 0.5 , 0.5 ) , when C j = C j − 1 , s j ∈ ( ⩾ 0.05 , ⩾ 0.05 ∧ ⩽ 0.5 , ⩽ 0.5 ) , when C j ≺ C j − 1 ,

where j ≠ 1.

Step 2. Transform the value of s ˜ j as follows: (22) sk ˜ j ( l j , m j , u j ) = ( 0.5 , 0.5 , 0.5 ) , when j = 1 , 1 ˜ − s ˜ j , when j ≠ 1 , (23) sq ˜ j ( l j , m j , u j ) = ( 0.5 , 0.5 , 0.5 ) , when j = 1 , sq ˜ j − 1 2 · sk ˜ j , when j ≠ 1 , (24) sn ˜ j ( l j , m j , u j ) = 1 max j u j sq ˜ j ( l j , m j , u j ) .

Step 3. Determine the relative weights of the evaluation criteria as follows: (25) w ˜ j ( l j , m j , u u ) = sn ˜ j ∑ k = 1 n sn ˜ k .

The weights of the criteria obtained by applying the previously presented procedure are fuzzy or triangular fuzzy numbers. Depending on the further calculation procedure, these weights can be used in their existing form, or if necessary, they can be transformed into crisp values, which is a more common approach.

The simplest way to transform fuzzy weight w ˜ j ( l j , m j , u j ) into crisp values w j is as follows: (26) w j c = l j + m j + u j 3 .

However, Eq. (25) does not ensure the fulfillment of the condition ∑ j = 1 n w j c = 1, which is why it is necessary to determine the final weights as follows: (27) w j = w j c ∑ k = 1 n w k c .

In addition to the previously presented approach for transforming fuzzy to crisp weights, the following transformation approaches, which enable giving different weights to the left and right boundaries of the triangular fuzzy number, or the mode of the fuzzy numbers, can also be used: (28) w j c = α · l j + m j + ( 1 − α ) · u j 2 , (29) w j c = α · l j + β · m j + ( 1 − α ) · u j 3 ,

where coefficient α ∈ [ 0 , 1 ], and coefficient β ∈ [ 0 , 2 ]. By varying the values of coefficients α and β, decision-makers can change the influence, of the left and right bounds, and/or mode, of triangular fuzzy numbers, and in this way consider different analyses, or scenarios.

As in the case of using Eq. (26), it is also necessary to apply Eq. (27) in these cases to ensure that the sum of criteria weights is equal to 1.

The computational procedure of the modified inverse fuzzy PIPRECIA (MFI PIPRECIA) method for determining criteria weights of n criteria can be represented through the following steps:

Step 1. Assigning the relative significance s ˜ j of each criterion, excepting the first criterion, using triangular fuzzy numbers, as follows: (30) s ˜ j ( l j , m j , u j ) = s j = ( ⩾ 0.5 , ⩾ 0.5 ∧ ⩽ 0.95 , ⩽ 0.95 ) , when C j ≻ C j + 1 , s j = ( 0.5 , 0.5 , 0.5 ) , when C j = C j + 1 , s j ∈ ( ⩾ 0.05 , ⩾ 0.05 ∧ ⩽ 0.5 , ⩽ 0.5 ) , when C j ≺ C j + 1 ,

where j ≠ n.

Step 2. Transform the value of s j as follows: (31) sk ˜ j ( l j , m j , u j ) = ( 0.5 , 0.5 , 0.5 ) , when j = n , 1 ˜ − s ˜ j , when j ≠ n , (32) sq ˜ j ( l j , m j , u j ) = ( 0.5 , 0.5 , 0.5 ) , when j = n , sq ˜ j + 1 2 · sk ˜ j , when j ≠ n , (33) sn ˜ j ( l j , m j , u j ) = 1 max j u j sq ˜ j ( l j , m j , u j ) .

Step 3. Determine the relative weights of the evaluation criteria as follows: (34) w ˜ j ( l j , m j , u u ) = sn ˜ j ∑ k = 1 n sn ˜ k .

As a result of using the MF PIPRECIA-E method, two fuzzy criterion weights are actually obtained, which are realistically different to some extent, especially if we consider that the inverse criteria evaluation procedure can be significantly more difficult for the examinee. In the case of transforming fuzzy weights into crisp values, consistency and weights can be determined as in the PIPRECIA-E method, that is using Eq. (1).

However, the weights obtained by the MF PIPRECIA-E method do not always have to be transformed into crisp values, because they can still be used in a fuzzy form for further calculations. In this case, the consistency of the performed pairwise comparison in the direct and indirect form can be determined as follows: (35) c c = 1 1 + 1 3 n ∑ j = 1 n ( ⌊ l j ′ − l j ′ ′ ⌋ + ⌊ m j ′ − m j ′ ′ ⌋ + ⌊ u j ′ − u j ′ ′ ⌋ ) .

In the case of applying the direct and inverse procedure of determining the weights of the fuzzy PIPRECIA-E method, two fuzzy weight vectors, w j ′ and w j ′ ′ , are formed.

In the case of further application of the crisp weight criteria, the fuzzy weights w j ′ and w j ′ ′ can be transformed into crisp weights, and the resulting weight can be determined by applying Eq. (2) and Eq. (3). However, in the case of further calculations based on the application of fuzzy criteria weights, the resulting fuzzy weights of the criteria can be determined in the simplest way as follows: (36) w ˜ j ( l j , m j , u u ) = l j ′ + l j ′ ′ 2 , m j ′ + m j ′ ′ 2 , u j ′ + u j ′ ′ 2 .

Certainly, in the case of the need for carrying out additional analyses, the resulting fuzzy weights can be determined by applying significantly more complex approaches, which enable giving different meanings to the left and right boundaries of the fuzzy number, i.e. the mode of the fuzzy number.

The use of triangular fuzzy numbers, or more precisely setting values of three parameters that define triangular fuzzy numbers, i.e. lower bound, mode, and upper bound, for expressing preferences about relative importance of criteria, can be too complex for less experienced respondents, that is, respondents who are not familiar with the use of fuzzy numbers.

For this reason, many fuzzy extensions of MCDM methods also suggest the use of appropriate linguistic variables, such as linguistic scales, to express respondents’ attitudes. However, the precise expression of respondents’ preferences can be limited using linguistic variables to some extent. For this reason, in the proposed fuzzy extension of PIPRECIA-E method usage of a linguistic scale whose linguistic variables can be transformed into one of the two offered fuzzy numbers, with different spreads, as shown in Table 1 and Table 2, is also proposed. Table 1

Fuzzy linguistic scale proposed for use with MF PIPRECIA-E method.

Depending on their preferences, respondents can choose one of the two offered values of triangular fuzzy numbers, and if necessary, they can also modify the proposed values.

In the case of solving multiple-criteria decision-making problems, i.e., determining the criteria, using a group decision-making approach enables problem-solving in two basic ways:

–

The first, and certainly simpler, way is to calculate the crisp values of criteria weights for each respondent and then determine the resulting group weight of the criteria.

In this approach, emphasis is placed on the calculation procedure based on the application of the group decision matrix, which is formed by applying the FWA operator, that is, applying Eq. (11). After the formation of the group decision-making matrix, the further calculation procedure is like the procedure carried out in the case of calculation based on the attitudes of single respondent.

As the weakness of this approach, the difficulty of determining the consistency of the pairwise comparisons can be mentioned. In the case of calculations made based on attitudes of each respondent separately, the consistency of the comparisons can be easily checked, and respondents with inconsistent attitudes can be excluded from further calculation, or they may be asked to re-evaluate comparisons.

However, even in the case of applying group approach for determining criteria weights, the consistency of the ”group comparisons” can be determined based on the weights of the criteria obtained by the using direct and indirect comparisons, that is using Eq. (35).

As mentioned previously, the first numerical illustration presents the use of the proposed approach for determining criteria weights for evaluating customer satisfaction in traditional Serbian restaurants, based on the following criteria (Stanujkic et al., 2019):

–

The building’s interior and hospitable atmosphere ( C 1 ),

The attitudes of respondents obtained by applying the direct procedure for comprising the relative importance of the MF PIPRECIA method are shown in Table 3, while the details of the calculation are presented in Table 4. Table 3

Attitudes of respondent obtained using MF PIPRECIA.

The previously presented example of calculating criteria weights presents the rarer case of calculations in which the upper limits of the triangular fuzzy numbers are slightly greater than 1.

The values in columns sk ˜ j , sq ˜ j , sn ˜ j and w ˜ j were calculated using Eqs. (21) to (24). Crisp values of criteria weights obtained using Eqs. (25) and (26) are shown in Table 5. Table 5

Fuzzy and crisp values of criteria weights.

A similar process of transforming fuzzy to crisp criterion weights could be done by applying Eqs. (27) or (28). The results of transforming fuzzy into crisp criteria weights using Eq. (27), and different values of coefficient α, is presented in Table 6. Table 6

Transformation of fuzzy into crisp criteria weights using Eq. (27).

The results of transforming fuzzy into crisp criteria weights applying Eq. (28) are presented in Table 7. To ensure a simpler presentation, in this case, only the values of coefficient β were varied, while the values of coefficient α were constant, and it was 0.5. Table 7

Transformation of fuzzy into crisp criteria weights using Eq. (28).

From the previous two tables, the application of Eqs. (27) and (28) has a certain influence on the criteria weights, which can enable decision makers to explore different sce narios, especially in cases when the fuzzy weights of the criteria are used in combination with the fuzzy expansion of various MCDM methods.

The attitudes of the respondent obtained by applying the inverse procedure of the MF PIPRECIA method (MFI PIPRECIA) are shown in Table 8, while the details of the calculations are presented in Table 9. Table 8

Attitudes of respondent obtained applying MFI PIPRECIA.

As in the previous case, i.e. in the case of direct relative criteria importance comparisons, transformation of fuzzy criteria weights into crisp values was performed using equations (27) and (28), as is presented in Table 10. Table 10

Fuzzy and crisp values of criteria weights obtained using MFI PIPRECIA.

As a result of using the fuzzy MF PIPRECIA-E method, two vectors of fuzzy criteria weights are obtained, which are realistically different to some extent, especially if it is considered that the inverse criteria importance comparisons procedure can be significantly more difficult for the examinee. In the case of transforming fuzzy weights into crisp values, consistency and criteria weights can be determined as in the PIPRECIA-E method, that is using Eq. (1), for determining consistency of the comparisons, and using Eq. (2) or Eq. (3) for determining resulting criteria weights.

Crisp criteria weights calculated using Eq. (2) are presented in Table 11, with achieved consistency of 0.966. Table 11

Resulting criteria weights obtained using Eq. (2).

However, weights obtained by using the MF PIPRECIA-E method do not always have to be transformed into crisp values. In these cases, transformation of the fuzzy weights obtained by the direct and indirect procedure of comparing the relative importance of the criteria into resulting fuzzy criteria weights can be determined by applying Eq. (35), while consistency can be determined using Eq. (34).

Fuzzy criteria weights, calculated by using Eq. (35) are shown in Table 12, where the consistency of the evaluation was 0.994. Table 12

Resulting fuzzy criteria weights.

The second numerical illustration demonstrates the application of the MF PIPRECIA-E method in a group decision-making environment. In this case, the determination of criteria weight for determining customer satisfaction in traditional Serbian restaurants is also considered, based on the attitudes of three respondents.

The attitudes of the first of the three respondents were previously shown in Tables 3 and 5, while the attitudes of the remaining two respondents are shown in Tables 13, 14, 15 and 16. Table 13

Attitudes of the second respondent obtained using MF PIPRECIA.

It should be emphasized here that the second respondent transformed the linguistic variables into TFNs using the values from the TFN II column. Table 14

Attitudes of the third respondent obtained using MF PIPRECIA.

In this case, the second respondent also performed the transformation of linguistic variables into TFNs using the values from the TFN II column. Table 16

Attitudes of the third respondent obtained applying MFI PIPRECIA.

The group decision matrices, shown in Tables 17 and 18, were formed by applying the FWA operator, that is, by applying Eq. (10), where the same importance is assigned to all respondents, i.e. w j = 0.333. Table 17

Group decision matrix formed using MF PIPRECIA.

The calculation details and fuzzy criteria weights obtained using MF PIPRECIA and MFI PIPRECIA methods are summarized in Tables 19 and 20.

The fuzzy criteria weights obtained by applying the MF PIPRECIA-E method are summarized again in Table 21. Based on the specified fuzzy criteria weights, the resulting fuzzy or crisp criteria weights can be calculated, as well as the consistency of the group pairwise comparisons. Table 19

Calculation details and fuzzy criteria weights obtained using group MF PIPRECIA method.

The consistency of group pairwise comparisons, calculated using Eq. (35) and data from Table 21, is very high and it is 0.9732.

Resulting fuzzy criteria weights can be calculated applying Eq. (36), as shown in Table 22. Table 22

Resulting fuzzy criteria weights obtained using group MF PIPRECIA-E method.

Crisp criteria weights can be calculated using one of the following Eqs. (26), (28) or (29), and certainly using Eq. (26), separately for w ˜ j ′ and w ˜ j ′ ′ . After that, using Eq. (2), or Eq. (3), resulting crisp criteria weights can be calculated, as shown in Table 23. Table 23

Resulting crisp criteria weights obtained using group MF PIPRECIA-E method.

The values of criteria weights w j ′ and w j ′ ′ are calculated using Eq. (26), while the values in column ( w j ′ + w j ′ ′ ) / 2 were calculated using Eq. (2). The resulting crisp criteria weights calculated using Eq. (27) are shown in column w j .

A similar calculation of criteria weights applying Eq. (28) and different values of the coefficient α is summarized in Table 24. Table 24

Resulting criteria weights obtained using Eq. (28) and different values of coefficient α.

From the previous table, it can be noticed that the changes in values of the coefficient α affect the criteria weights. For example, when the value of the coefficient α increases, the weight of criterion C 1 increases, while the weight of criterion C 2 decreases.

Similar calculations of criterion weights can be carried out by varying the value of coefficients α and β in Eq. (29), i.e. coefficient α in Eq. (3), which enables decision makers to examine different decision-making scenarios.