Decision making based on intuitionistic fuzzy preference relations with additive approximate consistency

2.1 Basic notations

The theory of fuzzy sets considers that everything has the flexibility to a certain degree [1, 31]. The vagueness or imprecision of human-originated information can be modeled by using the fuzzy sets. The definition of fuzzy sets is recalled as follows:

Definition 1. [31] Let U be a non-empty universe set. A fuzzy set is defined as the following mapping: F : U ↦ [ 0 , 1 ] . The membership grade of u ∈ U to a fuzzy set F is denoted as F ( u ) .

One can see from Definition 1 that when the membership grade F ( u ) is given, it implies that the non-membership grade of u to a fuzzy set is 1 - F ( u ) . However, for some practical cases, the non-balanced bipolarity of the events and objects exhibits [4, 5]. The theory of fuzzy sets cannot work, then the definition of intuitionistic fuzzy sets is proposed. That is, we have the following definition:

Definition 2. [6] Assume that U is a non-empty universe set. An intuitionistic fuzzy set is represented as the following form: A = { 〈 u , μ A ( u ) , ν A ( u ) 〉 | u ∈ U } , where 0 ≤ μ A ( u ) ≤ 1 , 0 ≤ ν A ( u ) ≤ 1 , 0 ≤ μ A ( u ) + ν A ( u ) ≤ 1 . The symbols μ A ( u ) and ν A ( u ) are called as the membership degree and non-membership degree of u to A , respectively. The term π A ( u ) = 1 - μ A ( u ) - ν A ( u ) stands for the non-determinacy degree.

It is obvious that when the non-determinacy degree π A ( u ) equals to zero, the intuitionistic fuzzy sets degenerate to fuzzy sets. For the sake of simplicity, hereafter the intuitionistic fuzzy set A is denoted as 〈 μ A , ν A 〉 [5]. In addition, the basic operations of intuitionistic fuzzy sets have been studied [5, 32]. For example, for any two intuitionistic fuzzy sets A 1 and A 2 , we have the following definition:

Definition 3. [5] Suppose that A 1 = 〈 μ A 1 , ν A 1 〉 and A 2 = 〈 μ A 2 , ν A 2 〉 are two intuitionistic fuzzy numbers. Some basic operations are given as follows: A ¯ 1 = 〈 ν A 1 , μ A 1 〉 , A 1 + A 2 = 〈 μ A 1 + μ A 2 - μ A 1 μ A 2 , ν A 1 ν A 2 〉 , A 1 · A 2 = 〈 μ A 1 μ A 2 , ν A 1 + ν A 2 - ν A 1 ν A 2 〉 , λ A 1 = 〈 1 - ( 1 - μ A 1 ) λ , ν A 1 λ 〉 , λ > 0 , A 1 λ = 〈 μ A 1 λ , 1 - ( 1 - ν A 1 ) λ 〉 , λ > 0 .

In what follows, let us consider a decision making problem with a finite set of alternatives expressed as X = {x₁, x₂, …, x _n } and I = {1, 2, ⋯ , n} . The decision maker compares the alternatives in pairs and a binary relation on X is constructed as: r ij = R ( x i , x j ) , i , j ∈ I ,(1) where r _ij stands for a fuzzy set or an intuitionistic fuzzy set. When r _ij  ∈ [0, 1] , an additive reciprocal matrix B = (b _ij ) _n×n is formed as:

Definition 4. [3] An additive reciprocal preference relation B = (b _ij ) _n×n satisfies the relations of b _ij  + b _ji  = 1, 0 ≤ b _ij  ≤ 1, and b _ii  = 0.5 for ∀i, j ∈ I . The value of b _ij is the positive preference intensity of the alternative x _i over the alternative x _j  .

Then by considering the multiplicative and additive transitivity, we have the following definitions:

Definition 5. [3] The additive reciprocal preference relation B = (b _ij ) _n×n is of multiplicative consistency, if b _ij  · b _jk  · b _ki  = b _ik  · b _kj  · b _ji for ∀i, j, k ∈ I .

Definition 6. [3] The additive reciprocal preference relation B = (b _ij ) _n×n is additively consistent, if b _ij  = b _ik  - b _jk  + 0.5, for ∀i, j, k ∈ I .

When the additive reciprocity of b _ij  + b _ji  = 1 is applied, the conditions in Definitions 5 and 6 can be rewritten. For example, the condition of b _ij  = b _ik  - b _jk  + 0.5 can be reexpressed as [33]: b ij + b jk + b ki = b ji + b kj + b ik ,(2) In addition, it is assumed that the importance of alternatives is expressed as the weight vector ω = (ω₁, ω₂, ⋯ , ω _n ) . Then the matrix B = (b _ij ) _n×n is of multiplicative consistency when [33] b ij = ω i ω i + ω j , ∀ i , j ∈ I .(3) And the matrix B = (b _ij ) _n×n with additive consistency satisfies the following relation [34]: b ij = 0.5 ( ω i - ω j + 1 ) , ∀ i , j ∈ I .(4)

Moreover, when the preference degree r _ij is expressed as an intuitionistic fuzzy set [35], the IFPR is defined as follows:

Definition 7. [7] An IFPR R ¯ = ( r ¯ ij ) n × n is represented as R ¯ = ( 〈 0.5 , 0.5 〉 〈 μ 12 , ν 12 〉 ⋯ 〈 μ 1 n , ν 1 n 〉 〈 μ 21 , ν 21 〉 〈 0.5 , 0.5 〉 ⋯ 〈 μ 2 n , ν 2 n 〉 ⋮ ⋮ ⋮ ⋮ 〈 μ n 1 , ν n 1 〉 〈 μ n 2 , ν n 2 〉 ⋯ 〈 0.5 , 0.5 〉 ) , where r ¯ ij = 〈 μ ij , ν ij 〉 with μ _ij , ν _ij  ∈ [0, 1] , 0 ≤ μ _ij  + ν _ij  ≤ 1, μ _ij  = ν _ji , μ _ji  = ν _ij , and μ _ii  = ν _ii  = 0.5 for ∀i, j ∈ I . The values of μ _ij and ν _ij stand for the positive and negative preference degrees of the alternative x _i over the alternative x _j , respectively.

It is worth noting that the consistency of R ¯ = ( r ¯ ij ) n × n is of much importance and it has been investigated widely. In what follows, let us give some reviews on the existing consistency definitions of R ¯ = ( r ¯ ij ) n × n and analyze the underlying ideas.

The consistency of IFPRs has been defined from three viewpoints: multiplicative consistency, additive consistency and consistency definitions based on the weights of alternatives. In what follows, we give an analysis in detail, respectively.

(I) Multiplicative consistency.

The first consistency definition of IFPRs was given in [17] and it is recalled as follows:

Definition 8. [17] Assume that R ¯ = ( r ¯ ij ) n × n is an IFPR. When the entries of R ¯ satisfy the following relation: r ¯ ij = r ¯ ik · r ¯ kj , i , j , k ∈ I ,(5) the matrix R ¯ is a consistent IFPR.

Some existing shortcoming in Definition 8 has been pointed out such as those in [7, 20]. Here we further consider the underlying idea and the necessary condition of the relation (5). It is seen that the multiplicative transitivity (5) stems from the condition of consistent pairwise comparison matrices arising in the AHP method [15]. That is, in the relative measurements, when we have x _i  = a _ik x _k and x _k  = a _kj x _j , it is consistent to obtain x _i  = a _ik a _kj x _j , then a _ij  = a _ik a _kj  . In other words, the consistent relation (5) is based on the comparison ratios of alternatives generated in the relative measurements. However, the entries of R ¯ = ( r ¯ ij ) n × n stand for the preference degrees in the absolute measurements. This means that the consistent relation (5) is unsuitable to characterize the consistency of R ¯ = ( r ¯ ij ) n × n . In addition, the relative measurement means the reciprocal property a _ij  = 1/a _ji owing to x _i  = a _ij x _j and x _j  = a _ji x _i  . The reciprocal property has been considered as one of the axiomatic properties of the AHP model [36] and the consistent relation (5) [25], respectively.

In order to improve Definition 8, the following consistency definition of IFPRs has been proposed:

Definition 9. [18] Let R ¯ = ( r ¯ ij ) n × n with r ¯ ij = 〈 μ ij , ν ij 〉 be an IFPR. The multiplicative consistency of R ¯ is attributed to the following relation: μ ij = { 0 , ( μ ik , μ kj ) = ( 0 , 1 ) , or ( 1 , 0 ) , μ ik μ kj μ ik μ kj + ( 1 - μ ik ) ( 1 - μ kj ) , otherwise , ν ij = { 0 , ( ν ik , ν kj ) = ( 0 , 1 ) , or ( 1 , 0 ) , ν ik ν kj ν ik ν kj + ( 1 - ν ik ) ( 1 - ν kj ) , otherwise .

The idea behind Definition 9 is based on the multiplicative consistency of additive reciprocal matrices in Definition 5. That is, one constructs the following two matrices: U ¯ = ( μ ij ) n × n = ( 0.5 μ 12 ⋯ μ 1 n μ 21 0.5 ⋯ μ 2 n ⋮ ⋮ ⋯ ⋮ μ n 1 μ n 2 ⋯ 0.5 ) ,(6) V ¯ = ( ν ij ) n × n = ( 0.5 ν 12 ⋯ ν 1 n ν 21 0.5 ⋯ ν 2 n ⋮ ⋮ ⋯ ⋮ ν n 1 ν n 2 ⋯ 0.5 ) .(7) When the entries of U ¯ and V ¯ satisfy the derived conditions in Definition 9 from the consistent condition in Definition 5, the IFPR R ¯ is of multiplicative consistency. It is noted that the existing disadvantage of Definition (9) has been analyzed in [20]. Here we should point out that the matrices U ¯ and V ¯ usually do not satisfy the additive reciprocity in Definition 4, meaning μ _ij  + μ _ji  ≠ 1 and ν _ij  + ν _ji  ≠ 1 . The serious issue implies that the extension of Definition 5 to Definition 9 lacks a reasonable ground.

Moreover, Definition 9 has been modified as the following form:

Definition 10. [20] Suppose that R ¯ = ( r ¯ ij ) n × n = ( 〈 μ ij , ν ij 〉 ) n × n is an IFPR. When the following multiplicative transitivity is satisfied: μ ij · μ ik · μ kj = ν ij · ν ik · ν kj ,(8) for ∀i, j, k ∈ I, the matrix R ¯ is of multiplicative consistency.

By considering the relations of μ _ij  = ν _ji and μ _ji  = ν _ij , the equality (8) means that μ ij · μ ik · μ kj = μ ji · μ ki · μ jk ,(9) ν ij · ν ik · ν kj = ν ji · ν ki · ν jk ,(10) for ∀i, j, k ∈ I . This means that when the two matrices U ¯ and V ¯ directly satisfy Definition 5, the IFPR R ¯ is multiplicatively consistent in Definition 10. The strict conditions (9) and (10) mean that the positive and negative judgements of decision makers should be all consistent. The shortcoming is that the non-determinacy degrees π _ij  = 1 - μ _ij  - ν _ij for ∀i, j, k ∈ I have been neglected and the important property of IFPRs has been not considered.

(II) Additive consistency.

By extending the additive consistency of additive reciprocal matrices in Definition 6, the additive consistency of IFPRs has been investigated. For example, one has the following definition [19]:

Definition 11. [19] An IFPR R ¯ = ( r ¯ ij ) n × n = ( 〈 μ ij , ν ij 〉 ) n × n is of additive consistency when the following relation hold: μ ij + μ jk + μ ki = μ ji + μ kj + μ ik ,(11) for ∀i, j, k ∈ I .

Clearly, the condition (11) can be used to obtain ν ij + ν jk + ν ki = ν ji + ν kj + ν ik , and it is the direct extension of (5a). It is noted that the relation (5a) is based on the reciprocal property of b _ij  + b _ji  = 1 . However, in general, the entries of U ¯ and V ¯ do not own the similar reciprocal property. Hence, the reason behind Definition 11 is unclear, which is similar to Definition 10.

(III) Consistency definitions using the weights of alternatives.

Multiplicative and additive consistency of IFPRs have been defined according to the priority vector of alternatives. For example, by extending the relations (3) and (4) to IFPRs, the following consistency definitions have been given:

Definition 12. [37] An IFPR R ¯ = ( r ¯ ij ) n × n = ( 〈 μ ij , ν ij 〉 ) n × n is of multiplicative consistency if there is a priority vector ω = (ω₁, ω₂, ⋯ , ω _n ) such that μ ij ≤ ω i ω i + ω j ≤ 1 - ν ij ,(12) for ∀i, j ∈ I .

Definition 13. [37] An IFPR R ¯ = ( r ¯ ij ) n × n = ( 〈 μ ij , ν ij 〉 ) n × n is additively consistent if there is a priority vector ω = (ω₁, ω₂, ⋯ , ω _n ) such that μ ij ≤ 0.5 ( ω i - ω j + 1 ) ≤ 1 - ν ij ,(13) for ∀i, j ∈ I .

The underlying idea behind Definitions 12 and 13 is that the IFPR R ¯ has been transformed into an interval-valued preference relation as follows: P = ( [ 0.5 , 0.5 ] [ μ 12 , p 12 ] ⋯ [ μ 1 n , p 1 n ] [ μ 21 , p 21 ] [ 0.5 , 0.5 ] ⋯ [ μ 2 n , p 2 n ] ⋮ ⋮ ⋱ ⋮ [ μ n 1 , p n 1 ] [ μ n 2 , p n 2 ] ⋯ [ 0.5 , 0.5 ] ) , where p ij = 1 - ν ij , ∀ i , j ∈ I . One can see that the inherent characteristic with non-determinacy degrees has been lost in Definitions 12 and 13. Furthermore, the interval weights have been introduced to define the consistency of IFPRs. For instance, one has the following definition:

Definition 14. [22] It is assumed that R ¯ = ( r ¯ ij ) n × n = ( 〈 μ ij , ν ij 〉 ) n × n is an IFPR. If there exists a normalized priority vector: ω ˜ = ( ( ω 1 l , ω 1 u ) , ( ω 2 l , ω 2 u ) , ⋯ , ( ω n l , ω n u ) ) , such that μ ij = ω i l ω i l + ω j u , ν ij = ω j l ω j l + ω i u , π ij = ω i u ω j l + ω i u - ω i l ω i l + ω j u , for ∀i, j ∈ I, the matrix R ¯ is of multiplicative consistency.

It is seen that Definition 14 is based on P together with the following constructed preference relations: Q = ( [ 0.5 , 0.5 ] [ ν 12 , q 12 ] ⋯ [ ν 1 n , q 1 n ] [ ν 21 , q 21 ] [ 0.5 , 0.5 ] ⋯ [ ν 2 n , q 2 n ] ⋮ ⋮ ⋱ ⋮ [ ν n 1 , q n 1 ] [ ν n 2 , q n 2 ] ⋯ [ 0.5 , 0.5 ] ) , Π = ( 0 π 12 ⋯ π 1 n π 21 0 ⋯ π 2 n ⋮ ⋮ ⋱ ⋮ π n 1 π n 2 ⋯ 0 ) , where q ij = 1 - μ ij , π ij = 1 - μ ij - ν ij , ∀ i , j ∈ I . In a sense, Definition 14 has reflected some characteristic of IFPRs. However, since the non-determinacy degrees π _ij are equally allocated to the matrices P and Q, Definition 14 should be further developed by considering the unequally allocation of the non-determinacy degrees π _ij  .

3.1 Additive approximate consistency

In what follows, in order to reasonably allocate the non-determinacy degree π _ij , we introduce an attitude factor 0 ≤ α ≤ 1 of the decision maker to form the following preference relations: P ˜ ( α ) = ( [ 0.5 , 0.5 ] [ μ 12 , p ˜ 12 ] ⋯ [ μ 1 n , p ˜ 1 n ] [ μ 21 , p ˜ 21 ] [ 0.5 , 0.5 ] ⋯ [ μ 2 n , p ˜ 2 n ] ⋮ ⋮ ⋱ ⋮ [ μ n 1 , p ˜ n 1 ] [ μ n 2 , p ˜ n 2 ] ⋯ [ 0.5 , 0.5 ] ) , Q ˜ ( α ) = ( [ 0.5 , 0.5 ] [ ν 12 , q ˜ 12 ] ⋯ [ ν 1 n , q ˜ 1 n ] [ ν 21 , q ˜ 21 ] [ 0.5 , 0.5 ] ⋯ [ ν 2 n , q ˜ 2 n ] ⋮ ⋮ ⋱ ⋮ [ ν n 1 , q ˜ n 1 ] [ ν n 2 , q ˜ n 2 ] ⋯ [ 0.5 , 0.5 ] ) , where p ˜ ij = μ ij + α π ij , q ˜ ij = ν ij + ( 1 - α ) π ij , for ∀i, j ∈ I . This means that when the decision maker evaluates her/his opinions on the importance of the alternative x _i over the alternative x _j to give an intuitionistic fuzzy number 〈μ _ij , ν _ij 〉, the positive or negative attitude can affect the allocation of the non-determinacy degree π _ij  . When α = 0, the matrices P ˜ ( α ) and Q ˜ ( α ) degenerate to Q and U ¯ , respectively. When α = 1, the matrices P and V ¯ can be retrieved, respectively. When we investigate the consistency property of IFPRs, the two matrices P ˜ ( α ) and Q ˜ ( α ) should be for all concerned.

Based on the above idea, when one part of the non-determinacy degree απ _ij is allocated to μ _ij , the other part (1 - α) π _ij is allocated simultaneously to ν _ij  . This implies that when the decision maker expresses the positive preference intensity of the alternatives x _i over x _j as p ˜ ij = μ ij + α π ij , the positive preference intensity of the alternatives x _j over x _i is obtained as q ˜ ij = ν ij + ( 1 - α ) π ij . When the decision maker gives the negative preference intensity of the alternatives x _i over x _j as q ˜ ij = ν ij + ( 1 - α ) π ij , the negative preference intensity of the alternatives x _j over x _i is determined as p ˜ ij = μ ij + α π ij . In other words, it is natural to determine the following real-valued preference relations: S ( α ) = ( x 1 x 2 ⋯ x n x 1 0.5 s 12 ( α ) ⋯ s 1 n ( α ) x 2 s 21 ( α ) 0.5 ⋯ s 2 n ( α ) ⋮ ⋮ ⋮ ⋱ ⋮ x n s n 1 ( α ) s n 2 ( α ) ⋯ 0.5 ) ,(14) T ( α ) = ( x 1 x 2 ⋯ x n x 1 0.5 t 12 ( α ) ⋯ t 1 n ( α ) x 2 t 21 ( α ) 0.5 ⋯ t 2 n ( α ) ⋮ ⋮ ⋮ ⋱ ⋮ x n t n 1 ( α ) t n 2 ( α ) ⋯ 0.5 ) ,(15) where s ij ( α ) = { p ˜ ij , i < j , q ˜ ji , i > j , t ij ( α ) = { q ˜ ji , i < j , p ˜ ij , i > j , for ∀i, j ∈ I . Moreover, since p ˜ ij + q ˜ ij = 1 (∀ i, j ∈ I) , the two matrices S (α) and T (α) have the additive reciprocity of s _ij  + s _ji  = 1 and t _ij  + t _ji  = 1 (∀ α ∈ [0, 1]) , respectively. It is seen from the expressions of S (α) and T (α) that we have constructed the additive reciprocal matrixes with positive decision information of the decision maker from the views of i > j and i < j, respectively. Moreover, the transpositions of S (α) and T (α) can be used to characterize the negative decision information of the decision maker.

In particular, for α = 0 and 1, we have the following results: S ( 0 ) = ( x 1 x 2 ⋯ x n x 1 0.5 μ 12 ⋯ μ 1 n x 2 1 - μ 12 0.5 ⋯ μ 2 n ⋮ ⋮ ⋮ ⋱ ⋮ x n 1 - μ 1 n 1 - μ 2 n ⋯ 0.5 ) , T ( 0 ) = ( x 1 x 2 ⋯ x n x 1 0.5 1 - μ 21 ⋯ 1 - μ n 1 x 2 μ 21 0.5 ⋯ 1 - μ n 2 ⋮ ⋮ ⋮ ⋱ ⋮ x n μ n 1 μ n 2 ⋯ 0.5 ) , S ( 1 ) = ( x 1 x 2 ⋯ x n x 1 0.5 1 - ν 12 ⋯ 1 - ν 1 n x 2 ν 12 0.5 ⋯ 1 - ν 2 n ⋮ ⋮ ⋮ ⋱ ⋮ x n ν 1 n ν 2 n ⋯ 0.5 ) , and T ( 1 ) = ( x 1 x 2 ⋯ x n x 1 0.5 ν 21 ⋯ ν n 1 x 2 1 - ν 21 0.5 ⋯ ν n 2 ⋮ ⋮ ⋮ ⋱ ⋮ x n 1 - ν n 1 1 - ν n 2 ⋯ 0.5 ) . By considering the conditions of μ _ij  = ν _ji and μ _ji  = ν _ij for ∀i, j ∈ I, it is found that S (0) = T (1) and S (1) = T (0) . In general, the following result is observed:

Theorem 1. Let R ¯ = ( r ¯ ij ) n × n = ( 〈 μ ij , ν ij 〉 ) n × n be an IFPR with μ _ij  = ν _ji and μ _ji  = ν _ij for ∀i, j ∈ I . The constructed matrices S (α) and T (α) in (14) and (15) satisfy the following relation: S ( α ) = T ( 1 - α ) , α ∈ [ 0 , 1 ] .(16)

Proof. We discuss the two situations with i < j and i > j, respectively. When i < j, it follows s ij ( α ) = p ˜ ij ( α ) = μ ij + α π ij , t ij ( 1 - α ) = q ˜ ji ( 1 - α ) = ν ji + α π ji . The application of μ _ij  = ν _ji and π _ij  = π _ji yields s _ij  (α) = t _ij  (1 - α) . On the other hand, when i > j, one has s ij ( α ) = q ˜ ji ( α ) = ν ji + ( 1 - α ) π ji , t ij ( 1 - α ) = p ˜ ij ( 1 - α ) = μ ij + ( 1 - α ) π ij . According to μ _ij  = ν _ji and π _ij  = π _ji , we have s _ij  (α) = t _ij  (1 - α) . The proof is completed.

Theorem 1 reveals the relation of the two constructed matrices S (α) and T (α) . In particular, when α = 0.5, we have S (α) = T (α) , implying that the two matrices are identical. In other words, the case of α = 0.5 means that the half non-determinacy degree is offered equally to the positive and negative membership degrees, respectively. This ideal case can be used to characterize the consistency property of IFPRs. Hence, following the methods of defining multiplicative and additive consistency of additive reciprocal matrices presented in Definitions 5 and 6, the multiplicative and additive approximate consistency of IFPRs can be defined. In the study, we mainly consider the additive approximate consistency of IFPRs and propose the following definition:

Definition 15. An IFPR expressed as R ¯ = ( r ¯ ij ) n × n = ( 〈 μ ij , ν ij 〉 ) n × n is of additive approximate consistency, when the entries of S (0.5) satisfy the relation of s _ij  = s _ik  - s _jk  + 0.5 for ∀i, j, k ∈ I .

For example, we consider the following IFPR: R ¯ 1 = ( 〈 0.5 , 0.5 〉 〈 0.2 , 0.6 〉 〈 0.2 , 0.7 〉 〈 0.6 , 0.2 〉 〈 0.5 , 0.5 〉 〈 0.3 , 0.4 〉 〈 0.7 , 0.2 〉 〈 0.4 , 0.3 〉 〈 0.5 , 0.5 〉 ) . The matrices S (α) and T (α) can be constructed as follows: S ( α ) = ( 0.5 0.2 + 0.2 α 0.2 + 0.1 α 0.8 - 0.2 α 0.5 0.3 + 0.3 α 0.8 - 0.1 α 0.7 - 0.3 α 0.5 ) , T ( α ) = ( 0.5 0.4 - 0.2 α 0.3 - 0.1 α 0.6 + 0.2 α 0.5 0.6 - 0.3 α 0.7 + 0.1 α 0.4 + 0.3 α 0.5 ) . When α = 0.5, we obtain the additively consistent matrix as S ( 0.5 ) = T ( 0.5 ) = ( 0.5 0.3 0.25 0.7 0.5 0.45 0.75 0.55 0.5 ) . This means that R ¯ 1 is of additive approximate consistency.

In order to illustrate the definitions of additive approximate consistency of IFPRs, the properties should be addressed. First, the following result is given:

Theorem 2. Assume that R ¯ = ( 〈 μ ij , ν ij 〉 ) n × n is an IFPR. When n = 2, the matrix R ¯ must be of additive approximate consistency according to Definition 15.

Proof. When n = 2, the matrix R ¯ is written as R ¯ = ( 〈 0.5 , 0.5 〉 〈 μ 12 , ν 12 〉 〈 μ 21 , ν 21 〉 〈 0.5 , 0.5 〉 ) . The matrices S (α) and T (α) can be determined as: S ( α ) = ( 0.5 μ 12 + α π 12 ν 12 + ( 1 - α ) π 12 0.5 ) , T ( α ) = ( 0.5 ν 21 + ( 1 - α ) π 21 μ 21 + α π 21 0.5 ) . By using the proposition in [33], it is sufficient to consider the case of i ≤ j ≤ k . Since 0.5 + ( μ 12 + α π 12 ) + ( ν 12 + ( 1 - α ) π 12 ) = 0.5 + μ 12 + ν 12 + π 12 = 1.5 , and 0.5 + ( ν 21 + ( 1 - α ) π 21 ) + ( μ 21 + α π 21 ) = 0.5 + ν 21 + μ 21 + π 21 = 1.5 , the matrix R ¯ is of additive approximate consistency for n = 2 and ∀α ∈ [0, 1] .

The result in Theorem 2 is in agreement with the common knowledge about the consistency of a preference relation. Second, we consider S (0) , S (1) , T (0) , and T (1) to give the following result:

Theorem 3. Suppose that R ¯ = ( 〈 μ ij , ν ij 〉 ) n × n is an IFPR. When S (0) = T (1) and S (1) = T (0) are of additive consistency (Definition 4), the IFPR R ¯ is with additive approximate consistency.

Proof. Following the idea in [33], the case of i < j < k only needs to be dealt with. In virtue of additive consistency of S (0) and S (1) , we have μ ij + μ jk + ( 1 - μ ik ) = 1.5 , and ( 1 - ν ij ) + ( 1 - ν jk ) + ν ik = 1.5 . Then it follows: s ij + s jk + s ki(17) = μ ij + α π ij + μ jk + α π jk + v ik + ( 1 - α ) π ik = μ ij + μ jk + 1 - μ ik + α ( π ij + π jk - π ik ) = 1.5 , t ij + t jk + t ki = ν ji + ( 1 - α ) π ji + ν kj + ( 1 - α ) π kj + μ ik + α π ik = ν ji + ν kj + μ ik + π ji + π kj + α ( π ik - π ji - π kj ) = μ ik + 2 - ν ji - ν kj = 1.5 . This means that S (α) and T (α) are additively consistent for ∀α ∈ [0, 1] . According to Definition 15, the IFPR R ¯ is with additive approximate consistency.

Based on Theorem 3, it is easy to construct an IFPR with additive approximate consistency. For instance, we can construct an IFPR with additive approximate consistency as follows: R ¯ 2 = ( x 1 x 2 x 3 x 1 〈 0.5 , 0.5 〉 〈 0.3 , 0.6 〉 〈 0.15 , 0.7 〉 x 2 〈 0.6 , 0.3 〉 〈 0.5 , 0.5 〉 〈 0.35 , 0.6 〉 x 3 〈 0.7 , 0.15 〉 〈 0.6 , 0.35 〉 〈 0.5 , 0.5 〉 ) .(18) It is seen that Theorem 3 shows a sufficient condition to generate an IFPR with additive approximate consistency. However, the condition is not necessary to obtain an IFPR with additive approximate consistency. For example, R ¯ 1 is with additive approximate consistency. But the constructed matrices S (0) and S (1) are not additively consistent by Definition 6.

On the other hand, the permutation of alternatives reflects the random behavior of decision makers in choosing alternatives to pairwisely compare [25, 27]. It is of interest to investigate the effect of the permutations of alternatives on additive approximate consistency of IFPRs. It is convenient to let σ be a one-to-one mapping from I to I . Applying σ to X, one gives the permutation of alternatives as (x_σ(1), x_σ(2), ⋯ , x _σ ) . Under the permutation of σ, the constructed matrices S (α) and T (α) are rewritten as S σ ( α ) = ( x σ ( 1 ) x σ ( 2 ) ⋯ x σ ( n ) x σ ( 1 ) 0.5 s σ ( 1 ) σ ( 2 ) ⋯ s σ ( 1 ) σ ( n ) x σ ( 2 ) s σ ( 2 ) σ ( 1 ) 0.5 ⋯ s σ ( 2 ) σ ( n ) ⋮ ⋮ ⋮ ⋱ ⋮ x σ ( n ) s σ ( n ) σ ( 1 ) s σ ( n ) σ ( 2 ) ⋯ 0.5 ) , T σ ( α ) = ( x σ ( 1 ) x σ ( 2 ) ⋯ x σ ( n ) x σ ( 1 ) 0.5 t σ ( 1 ) σ ( 2 ) ⋯ t σ ( 1 ) σ ( n ) x σ ( 2 ) t σ ( 2 ) σ ( 1 ) 0.5 ⋯ t σ ( 2 ) σ ( n ) ⋮ ⋮ ⋮ ⋱ ⋮ x σ ( n ) t σ ( n ) σ ( 1 ) t σ ( n ) σ ( 2 ) ⋯ 0.5 ) , where s σ ( i ) σ ( j ) ( α ) = { p ˜ σ ( i ) σ ( j ) , i < j , q ˜ σ ( j ) σ ( i ) , i > j , t σ ( i ) σ ( j ) ( α ) = { q ˜ σ ( j ) σ ( i ) , i < j , p ˜ σ ( i ) σ ( j ) , i > j , for ∀i, j ∈ I . Following the observations in [25, 26], the permutations of alternatives have no influence on the consistency property of S _σ  (0.5) and T _σ  (0.5) . That is, the following result is obtained:

Theorem 4. Let R ¯ = ( 〈 μ ij , ν ij 〉 ) n × n be an IFPR and σ be a permutation of alternatives. The application of σ to R ¯ leads to R ¯ σ . The matrices S (α) and S _σ  (α) are determined from R ¯ and R ¯ σ , respectively. If S (0.5) is additively consistent (Definition 4), S _σ  (0.5) is of additive consistency regardless of the permutation σ .

Proof. It is sufficient to prove that S _σ  (0.5) can be derived from S (0.5) under the application of the permutation σ . In fact, the values of the entries in S _σ  (0.5) are the same as those in S (0.5) according to the methods of defining S (0.5) and S _σ  (0.5) , respectively. The existing difference between S (0.5) and S _σ  (0.5) is the locations of entries, which are dependent on the permutation σ . That is, S _σ  (0.5) is determined by applying the permutation σ to S (0.5) . Since a permutation cannot affect the consistency property of an additive reciprocal matrix [26], the additive consistency of S (0.5) leads to the additive consistency of S _σ  (0.5) . The proof is completed.

According to Theorem 4, the additive approximate consistency of IFPRs in virtue of Definition 15 is independent of the permutations of alternatives. However, the permutations of alternatives affect the performance of S _σ  (0) and S _σ  (1) . For example, applying σ = (1, 3, 2) to R ¯ 2 , it gives R ¯ 2 σ = ( x 1 x 3 x 2 x 1 〈 0.5 , 0.5 〉 〈 0.15 , 0.7 〉 〈 0.3 , 0.6 〉 x 3 〈 0.7 , 0.15 〉 〈 0.5 , 0.5 〉 〈 0.6 , 0.35 〉 x 2 〈 0.6 , 0.3 〉 〈 0.35 , 0.6 〉 〈 0.5 , 0.5 〉 ) . Then the matrices S _σ  (0) and S _σ  (1) are determined as follows: S σ ( 0 ) = ( x 1 x 3 x 2 x 1 0.5 0.15 0.3 x 3 0.85 0.5 0.6 x 2 0.7 0.4 0.5 ) , S σ ( 1 ) = ( x 1 x 3 x 2 x 1 0.5 0.3 0.4 x 3 0.7 0.5 0.65 x 2 0.6 0.35 0.5 ) . It is found that S _σ  (0) and S _σ  (1) are not additively consistent. Therefore, in order to consider the effect of the permutations of alternatives, Theorem 3 should be rewritten as the following one:

Theorem 5. Suppose that R ¯ = ( 〈 μ ij , ν ij 〉 ) n × n is an IFPR and σ is a permutation of alternatives. R ¯ σ is obtained by applying σ to R ¯ . If there is a permutation of alternatives σ such that S _σ  (0) and S _σ  (1) are additively consistent, the IFPR R ¯ is of additive approximate consistency.

Proof. The proof is the straightforward of Theorem 3 and the detail procedure is omitted.

In the next section, we will propose a consistency improving method and the approach to properties of alternatives.

4.1 The method of improving additive approximate consistency

One can see that the two matrices S _σ  (α) and T _σ  (α) are with additive reciprocal property. When S _σ  (0.5) = T _σ  (0.5) are not additively consistent, the known improving consistency methods can be developed such as those in [38, 39] to adjust S _σ  (0.5) = T _σ  (0.5) to a new one with additive consistency. With the knowledge of Theorem 4, the matrix S _σ  (0.5) under a permutation only needs to be adjusted.

In what follows, we adjust the matrix S _σ  (0.5) to a new one with additive consistency. As shown in [39], it is assumed that the weights derived from S _σ  (0.5) are expressed as ω σ = { ω 1 σ , ω 2 σ , ⋯ , ω n σ } , with ω i σ = ∑ j = 1 n s σ ( i ) σ ( j ) ( 0.5 ) n , i ∈ I . Then based on (17), we can obtain the matrix with additive consistency. By using (4), we establish the following equality: s ij σ ( 0.5 ) = ω i σ - ω j σ + 1 2 , ∀ i , j ∈ I . Then a new additive reciprocal matrix with additive consistency is constructed as S σ a ( 0.5 ) = ( s ij σ ( 0.5 ) ) n × n , and it is further used to construct an IFPR with additive approximate consistency.

As an example, let us adjust the following IFPR to a new matrix R ¯ 3 a such that S ^a  (0.5) is additively consistent: R ¯ 3 = ( x 1 x 2 x 3 x 1 〈 0.5 , 0.5 〉 〈 0.15 , 0.30 〉 〈 0.36 , 0.34 〉 x 2 〈 0.30 , 0.15 〉 〈 0.5 , 0.5 〉 〈 0.45 , 0.30 〉 x 3 〈 0.34 , 0.36 〉 〈 0.30 , 0.45 〉 〈 0.5 , 0.5 〉 ) .

Step 1: In virtue of R ¯ 3 , we obtain the following matrix S ( 0.5 ) = ( x 1 x 2 x 3 x 1 0.5 0.425 0.51 x 2 0.575 0.5 0.525 x 3 0.49 0.475 0.5 ) .

Step 2: Using (17) and (18), the additively consistent matrix S ^a  (0.5) is determined as follows: S a ( 0.5 ) = ( x 1 x 2 x 3 x 1 0.5 0.4725 0.495 x 2 0.5275 0.5 0.5225 x 3 0.505 0.4775 0.5 ) .

Step 3: Based on the following formulae: μ σ ( i ) σ ( j ) = p ˜ σ ( i ) σ ( j ) - 0.5 π σ ( i ) σ ( j ) , and ν σ ( i ) σ ( j ) = q ˜ σ ( i ) σ ( j ) - 0.5 π σ ( i ) σ ( j ) , the new IFPR with additive approximate consistency is constructed as the following form:

( 〈 0.5 , 0.5 〉 〈 0.1975 , 0.2525 〉 〈 0.345 , 0.355 〉 〈 0.2525 , 0.1975 〉 〈 0.5 , 0.5 〉 〈 0.3975 , 0.3525 〉 〈 0.355 , 0.345 〉 〈 0.3525 , 0.3975 〉 〈 0.5 , 0.5 〉 ) .

4.2 The priorities of alternatives

Now let us derive the priorities of alternatives from an IFPR R ¯ with additive approximate consistency. It is seen from the above analysis that the constructed matrices S _σ  (α) and T _σ  (α) reveal the positive preference information of decision makers by allocating the non-determinacy degrees. The two matrices S _σ  (α) and T _σ  (α) can be used to derive the priorities of alternatives. It is worth noting that the methods of deriving priorities from an additive reciprocal matrix have been studied widely [40, 41]. Here the simple method proposed in [41] is used to derive the priorities from S _σ  (α) and T _σ  (α) as: ω S σ ( α ) = ( ω 1 s σ ( α ) , ω 2 s σ ( α ) , ⋯ , ω n s σ ( α ) ) ,(19) ω T σ ( α ) = ( ω 1 t σ ( α ) , ω 2 t σ ( α ) , ⋯ , ω n t σ ( α ) ) ,(20) where ω i s σ ( α ) = 2 ∑ j = 1 n s i σ ( j ) ( α ) n , i ∈ I ,(21) ω i t σ ( α ) = 2 ∑ j = 1 n t i σ ( j ) ( α ) n , i ∈ I .(22) The symbols ω i s σ ( α ) and ω i t σ ( α ) stand for the weights of the alternative x _i under the permutation σ derived from S _σ  (α) and T _σ  (α) , respectively. In order to normalize the obtained weights, let us suppose N 1 σ = ∑ i = 1 n ω i s σ ( α ) , N 2 σ = ∑ i = 1 n ω i t σ ( α ) .

For convenience, it is assumed that the weights of alternatives are expressed as ω ( α ) = ( ω 1 ( α ) , ω 2 ( α ) , ⋯ , ω n ( α ) ) .(23) Then by considering the effects of permutations, we have ω i ( α ) = 1 n ! ∑ σ ω i s σ ( α ) N 1 σ , i ∈ I ,(24) or ω i ( α ) = 1 n ! ∑ σ ω i t σ ( α ) N 2 σ , i ∈ I .(25) According to the methods of constructing S _σ  (α) and T _σ  (α) , we have the following property:

Theorem 6. Let R ¯ = ( 〈 μ ij , ν ij 〉 ) n × n be an IFPR and R ¯ σ be determined by applying σ to R ¯ . The two matrices S _σ  (α) and T _σ  (α) are constructed from R ¯ σ . Then the derived priorities in (24) and (25) are equivalent and independent of the parameter α .

Proof. According to the process of deriving (24) and (25), it follows N 1 σ = N 2 σ = n . Moreover, since S _σ  (α) and T _σ  (α) are the positive preference information of the decision maker by considering i < j and i > j, respectively, one has the following relations for all permutations: ∑ σ s i σ ( j ) ( α ) = ∑ σ t i σ ( j ) ( α ) ,(26) and s i σ ( j ) ( α ) + t i σ ( j ) ( α ) = μ i σ ( j ) + 1 - ν i σ ( j ) ,(27) for i ∈ I . Then using (21), (22), (24) and (25), it gives ∑ σ ω i s σ ( α ) N 1 σ = ∑ σ ω i t σ ( α ) N 2 σ ,(28) and ω i ( α ) = 1 n 2 · n ! ∑ σ ∑ j = 1 n ( μ i σ ( j ) + 1 - ν i σ ( j ) ) .(29) This means that the derived priorities in (24) and (25) are equivalent and independent of the parameter α .