Some new operations on Atanassov’s intuitionistic fuzzy sets in decision-making problems

Abstract

The arithmetic operations on intuitionistic fuzzy sets defined by Atanassov are the most popular in the intuitionistic fuzzy set theory. Based on these operations, many commonly used methods for solving the decision-making (DM) problems under intuitionistic fuzzy environment have been developed and various aggregation operators have been proposed. However, there have been revealed some undesirable properties of these operators, such as the inconsistency with the operations on the ordinary fuzzy sets (FSs), the non-monotonicity of the addition and multiplication operations or non-monotonicity under multiplication by a scalar. We show in this paper that these drawbacks pertain to also the recently proposed combined aggregations operators such as intuitionistic fuzzy Heronian mean (IFHM), intuitionistic fuzzy interaction partitioned Bonferroni mean (IFIPBM), intuitionistic fuzzy Dombi Bonferroni mean (IFDBM), intuitionistic fuzzy Maclaurin symmetric mean (IFMSM), Pythagorean fuzzy Maclaurin symmetric mean (PFMSM), q-rung orthopair fuzzy power Maclaurin symmetric mean (q-ROFPMSM), Muirhead mean (IFWMM) and intuitionistic fuzzy hybrid weighted arithmetic and geometric aggregation operators (IFHWAGA).This paper proposes some new arithmetic operations on Atanassov’s intuitionistic fuzzy sets that have good algebraic properties, such as idempotency, commutativity, monotonicity and monotonicity under multiplication by a scalar. Based on the proposed operations, the intuitionistic fuzzy weighted arithmetic mean and intuitionistic fuzzy weighted geometric mean operators with the acceptable properties are developed. Some illustrative examples are performed to demonstrate effectiveness and reliability of our method. Finally, in order to verify the validity of the proposed method in solving real-life DM problems, an application example is conducted with a comparative analysis with other existing methods.

Keywords

Intuitionistic fuzzy sets operations aggregation operator decision making

1 Introduction

The term fuzzy logic was introduced with the proposal of fuzzy set theory by Zadeh [35] as a form of many-valued logic in which the truth values (membership function) of variables may be any real number between 0 and 1 (between completely true and completely false). They are mathematical representation of vagueness of information related to such human activities as recognizing, representing, manipulating, interpreting, and utilizing data and information. The conception of intuitionistic fuzzy set proposed by Atanassov [1], abbreviated here as A-IFS, can be viewed as an alternative approach of ordinary fuzzy set to deal with vague and imperfect information. In the A-IFSs therefore, information is represented by three terms: membership degree, non-membership degree and hesitancy degree. In order to combine information in a single datum, different aggregation operators are used dependently on the assumptions on the data (data types) and about the type of information incorporated. The A-IFS theory and aggregation operator theory have become important tools to solve decision problems with vague and imprecise information available in many real-life decision making situations. Especially, in the multi-attribute decision making (MADM) and the multi-attribute group decision making (MAGDM) problems, they have become powerful tools for describing the uncertain information. In order to get a decision result, an important step is the aggregation of information expressed in terms of Atanassov’s intuitionistic fuzzy values (A-IFVs). Various families of aggregation functions have been put forward in the literature, such as the family of intuitionistic fuzzy averaging aggregation (IFAA) operators [29 , 37], the family of intuitionistic fuzzy ordered weighted averaging (IFOWA) operators [11 , 33], the family of induced intuitionistic fuzzy ordered weighted averaging (I-IFOWA) operators [36], the family of intuitionistic fuzzy hybrid aggregation (IFHA) operators [34], the family of intuitionistic fuzzy interaction Maclaurin symmetric means [18 , 27], the family of intuitionistic fuzzy power aggregation (IFPA) operators [10 , 31], the family of intuitionistic fuzzy power Heronian aggregation (IFPHA) operators [14], the family of intuitionistic fuzzy power Muirhead mean (IFPMM) operators [17] and the family of intuitionistic fuzzy Bonferroni mean (IFBM) aggregation operators [5 , 32]. Among them, the IFPHA, IFPMM and IFBM aggregation operators have been regarded as useful due to its ability to incorporate the interrelationships among the given arguments in decision-making contexts. For an overview, see [24].

The most important applications of A-IFS are MAGDM problems when the attribute values of alternatives and their aggregated results are in term of the A-IFVs. Therefore, appropriate operations on A-IFVs used for aggregation should be properly defined. Moreover, a reliable method for their comparison is needed to select the best alternative. Different definitions of operations on A-IFVs and their aggregation have been proposed in the literature. However, some drawbacks in the framework of A-IFS theory have been observed. For example, Beliakov, et al. [3] pointed out that existing averaging operators are not consistent with the operations on the ordinary FSs. However, their proposed averaging operator based on Łukasiewicz t-norms and t-conorms cannot be obtained using definition of operations on A-IFS defined by [1]. Dymova and Sevastjanov [6] also revealed some undesirable properties of the existing addition and multiplication operations on A-IFSs and concluded that they lead to the undesirable properties of commonly used aggregation operators on A-IFSs, i.e. intuitionistic fuzzy weighted arithmetic mean (IFWAM) and intuitionistic fuzzy weighted geometric mean (IFWGM) operators. Further, [7] claimed that there are methodological problems in the framework of the A-IFS theory and the operations on A-IFS defined by Atanassov [1] cannot be used to obtain the IFWAM and IFWGM operators with acceptable properties.

Inspired of the above mentioned limitations of the existing aggregation operators, this paper proposes some new operations on A-IFSs. The main originality of the paper is that the proposed operations on A-IFSs provide good algebraic properties, such as idempotency, commutativity, monotonicity and monotonicity under multiplication by a scalar in contrary to the widely used operations defined by Atanassov [1]. Based on the proposed operations, the IFWAM and IFWGM operators with the acceptable properties are developed. Some numerical examples are performed to demonstrate the effectiveness of the proposed approach. In solving real-life DM problems, an application example is conducted with a comparative analysis with other existing methods. Finally, an investigation of the sensitivity of the used methods by changing arbitrary an input evaluation A-IFV to a crisp value is conducted to show the reliability of the proposed method.

The rest of this paper is organized as follows. In Section 2, we briefly review some commonly used aggregation operators on A-IFSs. In Section 3, we define some new operations with proven properties. In Section 4, we compare and discuss these properties of the proposed operations with the existing ones in some numerical examples. In Section 5, we adopt the proposed aggregation operators in two experimental applications and discuss the results in comparison with that of the existing ones. The paper ends with a conclusion.

2 Operations on Atanassov’s intuitionistic fuzzy sets

An A-IFS is described in [1] as:

Definition 2.1. Let X be a finite universal set. Then an A-IFS A in X is an object having the form: $A = {〈 x, μ_{A} (x), v_{A} (x) 〉 | x \in X}$ (1) where μ_A (x) denotes a degree of membership and v_A (x) denotes a degree of non-membership of x to A, μ_A : X → [0, 1] and v_A : X → [0, 1] such that 0 ≤ μ_A (x) + v_A (x) ≤1, ∀ ∈ X.

The hesitancy degree of an element to an A-IFS is given by: $π_{A} (x) = 1 - μ_{A} (x) - v_{A} (x), where 0 \leq π_{A} (x) \leq 1, \forall x \in X$ (2) which is also called the intuitionistic fuzzy index or the hesitation margin of x to A.

If π_A (x) =0, ∀ x ∈ X, then μ_A (x) + v_A (x) =1 and the A-IFS A is reduced to an ordinary FS.

The concept of a complement of an A-IFS A, denoted by A^c is defined as: $A^{c} = {〈 x, v_{A} (x), μ_{A} (x) 〉 | x \in X}$ (3)

Hereinafter, for convenience we will use the following notation for A-IFV A as A =〈 μ_A, v_A 〉.

The operations of addition ⊕ and multiplication ⊗ on A-IFVs were defined by [2] as follows. Let A =〈 μ_A, v_A 〉 and B =〈 μ_B, v_B 〉 be A-IFVs, then: $A \oplus B = 〈 μ_{A} + μ_{B} - μ_{A} μ_{B}, v_{A} v_{B} 〉,$ (4) $A \otimes B = 〈 μ_{A} μ_{B}, v_{A} + v_{B} - v_{A} v_{B} 〉,$ (5)

Using operations (1) and (2) the following expressions were obtained for all real values α > 0: $α A = 〈 1 - (1 - μ_{A})^{α}, v_{A}^{α} 〉$ (6) $A^{α} = 〈 μ_{A}^{α}, 1 - (1 - v_{A})^{α} 〉$ (7)

The operations (4)–(7) have the following algebraic properties [29]: $A \oplus B = B \oplus A,$ (8) $A \otimes B = B \otimes A,$ (9) $α (A \otimes B) = α A + α B,$ (10) $(A \otimes B)^{α} = A^{α} \otimes B^{α},$ (11) $α_{1} A \oplus α_{2} A = (α_{1} \oplus α_{2}) A, α_{1}, α_{2} > 0,$ (12) $A^{α_{1}} \otimes A^{α_{2}} = A^{(α_{1} + α_{2})}, α_{1}, α_{2} > 0 .$ (13)

Let w = (w₁, w₂, . . . , w_n) be the standardized weight vector of a family of A-IFVs (A₁, A₂, . . . , A_n). Then based on the operations (4)–(7), the following aggregation operators called intuitionistic fuzzy weighted arithmetic mean (IFWAM) and intuitionistic fuzzy weighted geometric mean (IFWGM), were presented respectively:

$\begin{matrix} {IFWAM}_{w} (A_{1}, A_{2}, . . ., A_{n}) \\ = 〈 1 - \prod_{i = 1}^{n} (1 - μ_{A_{i}})^{w_{i}}, \prod_{i = 1}^{n} v_{A_{i}}^{w_{i}} 〉 \end{matrix}$ (14)

$\begin{matrix} {IFWGM}_{w} (A_{1}, A_{2}, . . ., A_{n}) \\ = 〈 \prod_{i = 1}^{n} μ_{A_{i}}^{w_{i}}, 1 - \prod_{i = 1}^{n} (1 - v_{A_{i}})^{w_{i}} 〉 \end{matrix}$ (15)

These aggregation operators are currently the most popular in MCDM problems under intuitionistic fuzzy environment. However, there have been noted some problems with these operators. For instance, in [26] Wan, et al. demonstrated that applying weights many times by the operators (14) or (15) leads the aggregated A-IFVs to the same results, i.e. 〈 [0, 0] , [1, 1]〉 or 〈 [1, 1] , [0, 0]〉, respectively. Recently, in order to overcome these flaws, there have been proposed such aggregation operators as intuitionistic fuzzy Heronian mean (IFHM) [23], intuitionistic fuzzy Bonferroni mean (IFBM) [32], intuitionistic fuzzy interaction partitioned Bonferroni mean (IFIPBM) [15], intuitionistic fuzzy Dombi Bonferroni mean (IFDBM) [19], intuitionistic fuzzy Maclaurin symmetric Mean (IFMSM) [25], Pythagorean fuzzy Maclaurin symmetric Mean (PFMSM) [27], q-rung orthopair fuzzy power Maclaurin symmetric mean (q-ROFPMSM) [16], Muirhead mean (IFWMM) [17] and intuitionistic fuzzy hybrid weighted arithmetic and geometric aggregation operators (IFHWAGA) [34]. These operators have the common feature that the interrelationship between arguments could be adjusted by combination of the arithmetic and geometric aggregation operators with the parameter vector P. However, they all have the same counterintuitive cases as considered below. Let us consider the called intuitionistic fuzzy Muirhead mean (IFWMM) aggregation operator [17], which is regarded as a generalization of such operators as power Heronian mean, Bonferroni mean, Maclaurin mean or hybrid arithmetic and geometric mean as follows.

Definition 2.2 [17]. Let α_i =〈 μ_i, v_i 〉 (i = 1,2,...,n) be a collection of A-IFVs, and P = (p₁, p₂, . . . , p_n) ∈ Rⁿ be a vector of parameters. If $\begin{matrix} {IFMM}^{p} (α_{1}, α_{2}, . . ., α_{n}) \\ = {(\frac{1}{n!} \sum_{ϑ \in s_{n}} \prod_{j = 1}^{n} α_{ϑ (j)}^{p_{j}})}^{\frac{1}{\sum_{j = 1}^{n} p_{j}}} = \\ 〈 {(1 - {(\prod_{ϑ \in S_{n}} (1 - \prod_{j = 1}^{n} μ_{ϑ (j)}^{p_{j}}))}^{\frac{1}{n!}})}^{\frac{1}{\sum_{j = 1}^{n} p_{j}}}, \\ 1 - {(1 - {(\prod_{ϑ \in S_{n}} (1 - \prod_{j = 1}^{n} (1 - v_{ϑ (j)})^{p_{j}}))}^{\frac{1}{n!}})}^{\frac{1}{\sum_{j = 1}^{n} p_{j}}} 〉, \end{matrix}$ (15a) then we call IFMM the intuitionistic fuzzy MM (IFMM), where ϑ (j), (j = 1, 2, …, n) is any a permutation of (1,2, . . . ,n), and S_n is the collection of all permutations of (1,2, . . . ,n).

According to the parameter vector P, the defined IFMM operator reduces to the following cases [17]:

If = (1, 0, . . . , 0), the IFMM reduces to intuitionistic fuzzy arithmetic mean (IFAM) operator a $\begin{matrix} {IFMM}^{(1, 0, . . ., 0)} (α_{1}, α_{2}, . . ., α_{n}) = \frac{1}{n} \sum_{i = 1}^{n} α_{i} \\ = 〈 1 - \prod_{i = 1}^{n} (1 - μ_{A_{i}})^{1 / n}, \prod_{i = 1}^{n} v_{A_{i}}^{1 / n} 〉 \end{matrix}$ (15b)

If = (1, 1, . . . , 1), the IFMM reduces to intuitionistic fuzzy geometric mean (IFGM) operator $\begin{matrix} {IFMM}^{(1, 1, . . ., 1)} (α_{1}, α_{2}, . . ., α_{n}) = {(\prod_{i = 1}^{n} α_{i})}^{1 / n} \\ = 〈 \prod_{i = 1}^{n} μ_{A_{i}}^{1 / n}, 1 - \prod_{i = 1}^{n} (1 - v_{A_{i}})^{1 / n} 〉 \end{matrix}$ (15c)

If = (1, 1, 0 . . . , 0), the IFMM reduces to intuitionistic fuzzy Bonferroni mean (IFBM) operator

\begin{matrix} {IFMM}^{(1, 1, 0 . . ., 0)} (α_{1}, α_{2}, . . ., α_{n}) \\ = (\frac{1}{n (n - 1)} \sum_{i, j = 1 i \neq j}^{n} α_{i} α_{j})^{1 / 2} \end{matrix}

(15d)

If $= ({\overset{︷}{1 . . ., 1}}^{k}, {\overset{︷}{0 . . ., 0}}^{n - k})$ , the IFMM reduces to intuitionistic fuzzy Bonferroni mean (IFBM) operator: $\begin{matrix} {IFMM}^{({\overset{︷}{1 . . ., 1}}^{k} {\overset{︷}{0 . . ., 0}}^{n - k})} (α_{1}, α_{2}, . . ., α_{n}) \\ = {(\frac{\sum_{1 \leq i_{1} < . . . < i_{k} \leq n} \prod_{j = 1}^{k} α_{ij}}{C_{n}^{k}})}^{1 / k} \end{matrix}$ (15e) where $C_{n}^{k}$ is the binomial coefficient, (i₁, i₂, . . . , i_k) traverses all the k-tuple combinations of (1,2, . . . , n)

Remark 1. It is easily seen that if any of the arguments in the IFMM operator (15b), i.e. α_i =〈 0, 1 〉, then resulting A-IFV equals to 〈0, 1〉, which is counterintuitive. It pertains also to such aggregation operators, as geometric mean, Bonferroni mean, Heronian mean, Maclaurin mean or hybrid arithmetic and geometric mean.

Example 2.1. Adopting example 1 from [17] instead of 〈0.5, 0.3〉, let 〈0, 1〉, 〈0.7, 0.1〉 and 〈0.8, 0.2〉be three A-IFVs, and P=(1.0,0.5,0.4), then according to (15b), we have $\begin{matrix} {IFMM}^{p} (α_{1}, α_{2}, . . ., α_{n}) = 〈 (1 - {(\prod_{ϑ \in S_{n}} (1 - 0))}^{\frac{1}{3!}}), \\ 1 - {(1 - {(\prod_{ϑ \in S_{n}} (1 - 0))}^{\frac{1}{3!}})}^{\frac{1}{1.9}} 〉 = 〈 0, 1 〉 . \end{matrix}$

Remark 2. The use of IFMM in [17] leads to the counterintuitive results as the aggregated IF values approach to 〈0, 1〉 regardless input values. The more input arguments, the faster decreased aggregated value.

Remark 3. The IFAM and IFGM operators are the special cases of the IFMM operator. So the drawbacks of the IFAM and IFGM operators imply the drawbacks of the IFMM operator as it is shown in examples 4.3 and 4.4.

In order to compare the A-IFVs of aggregation results, the score function S and accuracy function H of an A-IFV A =〈 μ_A, v_A 〉 were introduced by [29] as: $S (A) = μ_{A} - v_{A},$ $H (A) = μ_{A} + v_{A},$ in the algorithm of the order relation between any two A-IFVs A and B described as follows: $If S (A) > S (B), then B is smaller than A;$ (16) If S (A) = S (B), then

if H (A) > H (B), then B is smaller than A

if H (A) = H (B), then B=A.

They are rather of heuristic nature, but some different definitions of score function proposed in the literature as presented and analysed in [4], i.e. S₂ (A) = μ_A–v_Aπ_A, S₃ (A) = μ_A–0.5 (μ_A + π_A), S₄ (A) =0.5 (μ_A + v_A) –π_A, S₅ (A) = γμ_A + (1–γ) (1–v_A),are based on the same manner. The aggregation operators (14) and (15) as well as the score (16) and accuracy functions (17) have been shown by many researchers, that they have some undesirable properties. For example, Beliakov, et al. [3] revealed that the ordering (16) is not preserved under multiplication by a scalar, i.e. A ≻ B does not necessarily imply λA ≻ λB, λ > 0. Moreover, they concluded that if any of arguments in the aggregation operator (14) is A_i =〈 1, 0 〉 and its weight is not 0, then resulting A-IFV equals to 〈1, 0〉, which is rather counterintuitive. They proposed a new construction method of the aggregation operator, which is consistent with operation on ordinary fuzzy sets, based on the Łukasiewicz triangular norm as follows: $IWAM (A_{1}, A_{2}, . . ., A_{n}) = 〈 \sum_{i = 1}^{n} w_{i} v_{A_{i}} 〉$ (17)

Nevertheless, the methodological problem is that this operator cannot be obtained using the Atanassov’s basing definition of A-IFV arithmetic operations (4) and (6). In [8] and [6], it was pointed out, that the operators (14) and (15) are not monotonic with respect to the ordering (16). They concluded that the undesirable properties of the intuitionistic fuzzy weighted arithmetic mean operator are resulted in violation of the usual weighted averaging structure by the power operator. Finally, [7] have found that operations (8) and (9) on A-IFVs cannot be used to obtain the intuitionistic weighted arithmetic mean operator with acceptable properties and they constitute an important methodological problem in the framework of the A-IFS theory. However, the newly proposed aggregation operators are also counterintuitive as they do not fulfill such properties as idempotency or monotonicity under multiplication by a scalar.

3 New operations on A-IFSs

Definition 3.1. For any two A-IFVs A =〈 μ_A, v_A 〉 and B =〈 μ_B, v_B 〉 the following operations are proposed: $A \oplus B = 〈 \frac{μ_{A} + μ_{B}}{2}, 1 - \frac{(1 - v_{A}) + (1 - v_{B})}{2} 〉,$ (18) $A \otimes B = 〈 1 - (1 - μ_{A}) (1 - μ_{B}), v_{A} v_{B} 〉,$ (19) $λ A = 〈 λ μ_{A}, 1 - λ (1 - v_{A}) 〉 λ \in [0, 1],$ (20) $A^{λ} = 〈 1 - (1 - μ_{A})^{λ}, v_{A}^{λ} 〉, λ \in [0, 1],$ (21) $A \cup B = 〈 max {μ_{A} (x), μ_{B} (x)}, min {v_{A} (x), v_{B} (x)} 〉,$ (22) $A \cap B = 〈 min {μ_{A} (x), μ_{B} (x)}, max {v_{A} (x), v_{B} (x)} 〉,$ (23) $A \cup A = A \cap A = A,$ (23a) $A \subset B iff (\forall x \in X, μ_{A} (x) \leq μ_{B} (x) v_{A} (x) \geq v_{B} (x)),$ (24) $A = B iff A \subset B and B \subset A$ (24a)

It is easy to verify that they provide A-IFVs.

The formulas (18) and (19) can be generalized as follows: $\begin{matrix} A_{1} \oplus . . . \oplus A_{n} = 〈 \frac{μ_{A_{1}} + . . . + μ_{A_{n}}}{n}, \\ 1 - \frac{(1 - v_{A_{1}}) + . . . + (1 - v_{A_{n}})}{n} 〉 \\ = 〈 \frac{1}{n} \sum_{i = 1}^{n} μ_{A_{i}}, \frac{1}{n} \sum_{i = 1}^{n} v_{A_{i}} 〉 \end{matrix}$ (25) $A_{1} \otimes . . . \otimes A_{n} = 〈 1 - \prod_{i = 1}^{n} (1 - μ_{A_{i}}), \prod_{i = 1}^{n} (v_{A_{i}}) 〉$ (26)

The multiplier 1/n ensures conditions of the operation resulting A-IFV, i.e. μ, v ∈ [0, 1], 1 ≥ μ + v ≥ 0 for any x ∈ X.

Theorem 1. With the Definition 3.1, operations ⊕ and ⊗ provide A-IFVs, fulfilling the following properties:

Commutativity: A ⊕ B = B ⊕ A, $A \otimes B = B \otimes A$ (27)

Idempotency: A ⊕ A = A, A ⊗ A = A (28).

Monotonicity: For any three A-IFVs A, B and C, $if A ≽ B then A \oplus C ≻ B \oplus C A \oplus C ≽ B \oplus C$ (29)

Monotonicity under multiplication by a scalar: For any λ ∈ [0, 1],

if A ≻ B then λ A ≻ λ B .

(30)

Proof:

It is easily to prove the correctness of the first two properties (a and b follow directly from definition 3.1). For any two A-IFVs A =〈 μ_A, v_A 〉 and B =〈 μ_B, v_B 〉, we get μ_A, v_A, μ_B, v_B ∈ [0, 1] and 0 ≤ μ_A + v_A ≤ 1 and 0 ≤ μ_B + v_B ≤ 1. For example, regarding the addition operation (and similarly with the multiplication), we have $μ_{A \oplus B} = \frac{μ_{A} + μ_{B}}{2} \in [0, 1]$ , $v_{A \oplus B} = 1 - \frac{(1 - v_{A}) + (1 - v_{B})}{2} = \frac{v_{A} + v_{B}}{2} \in [0, 1]$ and $μ_{A \oplus B} + v_{A \oplus B} = \frac{μ_{A} + μ_{B} + v_{A} + v_{B}}{2} \in [0, 1]$ , showing that A ⊕ B is an A-IFV.

The proof is similar as above.

For any A-IFV C =〈 μ_C, v_C 〉, μ_C, v_C ∈ [0, 1], we have $\begin{matrix} A ≽ B \Leftrightarrow μ_{A} \geq μ_{B} v_{A} \leq v_{B} \Leftrightarrow μ_{A} + μ_{C} \\ \geq μ_{B} + μ_{C} v_{A} + v_{C} \leq v_{B} + v_{C} \\ \Leftrightarrow A \oplus C ≽ B \oplus C \end{matrix}$

For any λ ∈ [0, 1], A ≽ B ⇔ μ_A ≥ μ_B v_A ≤ v_B ⇔ λμ_A ≥ λμ_B λv_A ≤ λv_B ⇔ λA ≥ λB.

Lemma 1. From formula (28) one can obtain following equations:

$\begin{matrix} nA & = & A \oplus . . . \oplus A = A and A^{n} = A \otimes . . . \otimes A \\ = & A or n = 1, 2, \dots . \end{matrix}$ (31)

Indeed, operations on the same subsets, i.e. union, intersection, sum or multiplication should give the same subset. The subsets A of a finite universal set X can be treated as an imprecise statement or information. Repeating the same statement many times gives us the same information.

Lemma 2. The operations defined by (18) and (19) are consistent with the operations on ordinary fuzzy sets. Indeed, for ordinary fuzzy values (FVs) A and B, μ_A = 1–v_A and μ_B = 1–v_B that resulting values of the defined operations ⊕ and ⊗ are always FVs, i.e. μ_A⊕B = 1–v_A⊕B and μ_A⊗B = 1–v_A⊗B, respectively.

Theorem 2. The operations (18)–(21) have the following algebraic properties. Let A and B be any A-IFVs, if any λ₁, λ₂, λ ∈ [0, 1], then

λA ⊕ λB = λ (A ⊕ B), (32)

$λ_{1} A \oplus λ_{2} A = (\frac{λ_{1} + λ_{2}}{2}) A$ , 1 ≥ λ₁, λ₂ > 0,

(33)

A^λ ⊗ B^λ = (A ⊗ B) ^λ,(34)

A^{λ
₁} ⊗ A^{λ
₂} = A^λ₁+λ₂, 1 ≥ λ₁, λ₂ > 0.(35)

Proof:

λA ⊕ λB = 〈 λμ_A, 1–λ (1–v_A) 〉 ⊕ 〈λμ_B, 1–λ $(1 - v_{B}) 〉 = 〈 λ \frac{μ_{A} + μ_{B}}{2}, 1 - \frac{λ (1 - v_{A}) + λ (1 - v_{B})}{2} 〉 = λ (A \oplus B)$

$λ_{1} A \oplus λ_{2} A = 〈 λ_{1} μ_{A}, 1 - λ_{1} (1 - v_{A}) 〉 \oplus 〈 λ_{2} μ_{A}, 1 - λ_{2} (1 - v_{A}) 〉 = 〈 \begin{matrix} \frac{λ_{1} + λ_{2}}{2} μ_{A}, 1 - \\ \frac{λ_{1} (1 - v_{A}) + λ_{2} (1 - v_{A})}{2} \end{matrix} 〉 = (\frac{λ_{1} + λ_{2}}{2}) A .$

$A^{λ} \otimes B^{λ} = 〈 1 - (1 - μ_{A})^{λ}, v_{A}^{λ} 〉 \otimes 〈 1 - (1 - μ_{B})^{λ},$ $v_{B}^{λ} 〉 = 〈 1 - (1 - μ_{A})^{λ} (1 - μ_{B})^{λ}, v_{A}^{λ} v_{B}^{λ} 〉 = (A \otimes B)^{λ}$

$A^{λ_{1}} \otimes A^{λ_{2}} = 〈 1 - (1 - μ_{A})^{λ_{1}}, v_{A}^{λ_{1}} 〉 \otimes 〈 1 - (1 - μ_{A})^{λ_{2}}, v_{A}^{λ_{2}} 〉 = 〈 1 - (1 - μ_{A})^{λ_{1}} (1 - μ_{A})^{λ_{2}}$ , $v_{A}^{λ_{1}} v_{A}^{λ_{2}} 〉 = A^{λ_{1} + λ_{2}} .$

This completes the proof.□

Then, based on the above operations, the following aggregation operators are obtained.

Definition 3.2. Let A₁, A₂, . . . , A_n be A-IFVs representing the values of n attributes for some alternatives and z₄ = 〈0.535, 0.254〉 be real-valued weights of attributes, such that $ω_{i} ≻ 0, \sum_{i = 1}^{n} ω_{i} = 1$ , the intuitionistic weighted arithmetic mean (IFWAM_K) and intuitionistic weighted geometric mean (IFWGM_K) operators are defined respectively: $\begin{matrix} {IFWAM}_{K} (A_{1}, A_{2}, . . ., A_{n}) \\ = ω_{1} A_{1} \oplus ω_{2} A_{2} \oplus, . . ., ω_{n} A_{n}, \end{matrix}$ (36) ${IFWGM}_{K} (A_{1}, A_{2}, . . ., A_{n}) = A_{1}^{ω_{1}} \otimes A_{2}^{ω_{2}} \otimes . . . A_{n}^{ω_{n}},$ (37)

Theorem 3. With the Definition 3.2, the IFWAM_Kand IFWGM_Koperators based on the proposed operations are expressed as: $\begin{matrix} {IFWAM}_{K} (A_{1}, A_{2}, . . ., A_{n}) = \\ 〈 \sum_{i = 1}^{n} ω_{i} μ_{A_{i}}, \sum_{i = 1}^{n} ω_{i} ν_{A_{i}} 〉 \end{matrix}$ (38a) $\begin{matrix} {IFWGM}_{K} (A_{1}, A_{2}, . . ., A_{n}) = \\ 〈 1 - Π_{i = 1}^{n} (1 - μ_{A_{i}})^{ω_{i}}, Π_{i = 1}^{n} (1 - ν_{A_{i}})^{ω_{i}} 〉 \end{matrix}$ (38b)

Proof: For $ω_{i} > 0, \sum_{i = 1}^{n} ω_{i} = 1$ , on the basic of formulas (20), (25) and (18) we get $\begin{matrix} {IFWAM}_{K} (A_{1}, A_{2}, . . ., A_{n}) = ω_{1} A_{1} \oplus \\ ω_{2} A_{2} \oplus . . . \oplus ω_{n} A_{n} = \\ \sum_{i = 1}^{n} ω_{i} (1 - ν_{A_{i}}) 〉 = \\ 〈 \sum_{i = 1}^{n} ω_{i} μ_{A_{i}}, \sum_{i = 1}^{n} ω_{i} ν_{A_{i}} 〉, \end{matrix}$ and on the basic of formulas (21), (26) and (19) we get $\begin{matrix} {IFWGM}_{K} (A_{1}, A_{2}, . . ., A_{n}) = A_{1}^{ω_{1}} \otimes \\ A_{2}^{ω_{2}} \otimes . . . \oplus A_{n}^{ω_{n}} = \\ 〈 1 - Π_{i = 1}^{n} (1 - μ_{A_{i}})^{ω_{i}}, Π_{i = 1}^{n} (1 - ν_{A_{i}})^{ω_{i}} 〉, \end{matrix}$ which hold the formula (38a) and (38b).

Lemma 3. The proposed IFWAM_K and IFWAM_K are consistent with aggregation operation on the ordinary fuzzy sets. Indeed, when A_i reduce to FVs, i.e. μ_{A
_i} = 1–ν_{A
_i}, then z₃ = 〈0.270, 0.567〉 reduces to weighted arithmetic mean of the ordinary fuzzy sets.

Theorem 4. With the Definition 3.2, the aggregation IFWAM_Kand IFWGM_Koperators fulfill the following properties: Let {A₁, A₂, . . . , A_n}be a collection of A-IFVs and {ω₁, ω₂, . . . , ω_n}be any weighting vector, such that $ω_{i} > 0, \sum_{i = 1}^{n} ω_{i} = 1 .$

Idempotency:

If A₁ = A₂ = . . . A_n = A = 〈μ_A, ν_A〉 then ${IFWAM}_{K} (A_{1}, A_{2}, . . ., A_{n}) = A,$ (39) If A₁ = A₂ = . . . A_n = A = 〈μ_A, ν_A〉 then ${IFWGM}_{K} (A_{1}, A_{2}, . . ., A_{n}) = A .$ (40)

The proof follows directly from (38a) and (38b).

Commutativity:

If ${{\hat{A}}_{1}, {\hat{A}}_{2}, . . ., {\hat{A}}_{n}}$ is any permutation of {A₁, A₂, . . . , A_n}, then $\begin{matrix} {IFWAM}_{K} (A_{1}, A_{2}, . . ., A_{n}) = \\ {IFWAM}_{K} ({\hat{A}}_{1}, {\hat{A}}_{2}, . . ., {\hat{A}}_{n}) \end{matrix}$ (41) $\begin{matrix} {IFWGM}_{K} (A_{1}, A_{2}, . . ., A_{n}) = \\ {IFWGM}_{K} ({\hat{A}}_{1}, {\hat{A}}_{2}, . . ., {\hat{A}}_{n}) \end{matrix}$ (42)

The proof is straightforward from (38a) and (38b).

Boundedness:

Let A_L = min {A₁, A₂, . . . , A_n} and let A^U = max {A₁, A₂, . . . , A_n}. Then, $\begin{matrix} {IFWAM}_{K} (A_{1}, A_{2}, . . ., A_{n}) ≼ A^{U}, \end{matrix}$ (43) $\begin{matrix} A_{L} \leq {IFWAM}_{K} (A_{1}, A_{2}, . . ., A_{n}) ≼ A^{U} . \end{matrix}$ (44)

Proof:

We prove firstly the left side (lower bound) of the inequalities (43) and (44):

According to (38a), we have A_L = min {A₁, A₂, . . . , A_n}⇔ 〈μ_{A
_L}, ν_{A
_L}〉 ≼ 〈 μ_{A
_i}, ν_{A
_i}〉, (i = 1, 2, . . . , n) .

Then, we obtain $μ_{A_{L}} = \sum_{i = 1}^{n} ω_{i} μ_{A_{L}}$ $\leq \sum_{i = 1}^{n}$ ω_iμ_{A
_i} and $ν_{A_{L}} = \sum_{i = 1}^{n} ω_{i} ν_{A_{L}}$ $\geq \sum_{i = 1}^{n} ω_{i} ν_{A_{i}}$ .

Thus, 〈μ_{A
_L}, ν_{L
_A}〉 = A_L ≼ IFWAM_K (A₁, A₂, . . . , A_n) = 〈 $\sum_{i = 1}^{n} ω_{i} μ_{A_{i}}$ , $\sum_{i = 1}^{n} ω_{i} ν_{A_{i}} 〉$ .

Similarly we have $μ_{A_{L}} = \prod_{i = 1}^{n} (μ_{A_{L}})^{ω_{i}} \leq$ $\prod_{i = 1}^{n} (μ_{A_{i}})^{ω_{i}}$ and $ν_{A_{L}} = \prod_{i = 1}^{n}$ (ν_{A
_L}) ^{ω
_i}≥.

Thus, 〈μ_{A
_L}, v_{A
_L}〉 = A_L ≼ IFWGM_K (A₁, A₂, . . . , A_n) = 〈 $\prod_{i = 1}^{n} {(μ_{A_{i}})}^{ω_{i}}$ , $\prod_{i = 1}^{n} {(v_{A_{i}})}^{ω_{i}} 〉$ .

Analogically, we have for the upper bound.

Monotonicity:

Let A_i = 〈μ_{A
_i}, v_{A
_i}〉 and B_i = 〈μ_{B
_i}, v_{B
_i}〉, (i = 1, 2, . . . , n) be two collections of A-IFVs and ω₁, ω₂, . . . , ω_n be a weight vector, such that $ω_{i} > 0, \sum_{i = 1}^{n} ω_{i}$ . If A_i ≽ B_i for all i = 1, 2, . . . , n, then $\begin{matrix} {IFWAM}_{K} (A_{1}, A_{2}, . . ., A_{n}) \\ ⪢ {IFWAM}_{K} (B_{1}, B_{2}, . . ., B_{n}), \end{matrix}$ (45)

\begin{matrix} {IFWGM}_{K} (A_{1}, A_{2}, . . ., A_{n}) \\ ⪢ {IFWGM}_{K} (B_{1}, B_{2}, . . ., B_{n}) . \end{matrix}

(46)

Proof:

For, $ω_{i} > 0, \sum_{i = 1}^{n} ω_{i},$ i = 1, 2, . . . , n, we get A_i ≽ B_i ⇔ μ_{A
_i} ≥ μ_{B
_i}, v_{A
_i} ≤ v_{B
_i} ⇔ ω_iμ_{A
_i} ≤ ω_i $μ_{B_{i}}, ω_{i} v_{A_{i}} \leq ω_{i} v_{B_{i}} \Leftrightarrow 〈 \sum_{i = 1}^{n} ω_{i} μ_{A_{i}}, \sum_{i = 1}^{n} ω_{i} v_{A_{i}} 〉 ≽$ $〈 \sum_{i = 1}^{n} ω_{i} μ_{B_{i}}, \sum_{i = 1}^{n} ω_{i} v_{B_{i}} 〉 .$

This holds the proof according to (38a).

4 Illustrative examples

In the following, we show the effectiveness of the proposed methods by the numerical examples, where the existing ones fail as revealed in the literature. In [3, 6] and [7] some limitations of the existing operations (4) –(8) on A-IFVs were analyzed. To show the supervisor of the newly proposed methods, consider the following examples.

Example 4.1. In this example, we show the monotonicity of the proposed addition operation on A-IFVs under multiplication by a scalar. Adopting example 1 from [3], let a =〈 0.5, 0.4 〉, b =〈 0.3, 0.2 〉 be A-IFVs and λ = 0.6. Using the score function we obtain S (a) = S (b) =0.1 and H (a) =0.9, H (b) =0.5 indicating a ≻ b. Using (20) we get λa =〈 0.3, 0.64 〉, λb =〈 0.18, 0.52 〉, so S (λa) = S (λb) = –0.34, H (λa) =0.94 and H (λb) =0.7, indicating λa ≻ λb. Thus, a ≻ b implies λa ≻ λb showing the monotonicity of the proposed addition operation on A-IFVs under multiplication by a scalar.

Example 4.2. In this example, we show the monotonicity of the proposed multiplication operation on A-IFVs. Adopting example 1 from [7] let a =〈 0.1, 0.3 〉, b =〈 0.4, 0.5 〉 and c =〈 0.2, 0.1 〉 be three A-IFVs. The score functions (16) of a and b are computed respectively: S (a) = –0.2, S (b) = –0.1, which imply that b ≻ a. Besides, using operation (19), we obtain the following multiplications of these A-IFVs: a⊗ c = 〈 0.28, 0.03 〉, b⊗ c = 〈 0.52, 0.05 〉. Thus, we get S (a ⊗ c) =0.25, S (b ⊗ c) =0.47, which implies b ⊗ c ≻ a ⊗ c meaning the monotonicity of multiplication on the A-IFVs, whereas it is not monotonic with respect to the ordering (16) as it was shown in [7].

Example 4.3. In this example, we show the consistency of the proposed aggregation operator (38a) on A-IFVs with aggregation operator on ordinary fuzzy sets. Adopting example 6 from [6], let a₁ =〈 0.1, 0.9 〉, a₂ =〈 0.9, 0.1 〉 be A-IFVs (and FVs as well) and ω₁ = ω₂ = 0.5. In the framework of ordinary fuzzy sets, we get aggregation by fuzzy weighted arithmetic mean $FWAM = \sum_{i = 1}^{2} ω_{i} μ_{ai} = 0.1 + 0.5 + 0.9 + 0.5 + 0.5$ . Using our IFWAM_K operator (38), we get IFWAM_K (a₁, a₂) =〈 0.1 0.5 + 0.9 0.5, 0.9 0.5 + 0.1 0.5 〉 = 〈 0.5, 0.5 〉, which is exactly the same as in the framework of ordinary fuzzy sets. It is worth noticing that the aggregation operator based on Dempster-Shafer theory proposed in [6] was shown to be not always consistent with aggregation operations on the ordinary fuzzy sets. Using the IFWMM operator (15b) from [17] with parameter vector P=1, 0 we get z₃ = 〈0.379, 0.405〉 and with parameter vector P = 1, 1 we get IFWAM (a₁, a₂) =〈 0.7, 0.3 〉, which are rather not consistent with aggregation operation on ordinary FSs.

Example 4.4. In this example, we show the monotonicity of the aggregation IFWAM_K operator (38a) on A-IFVs. Adopting example 4 from [6], let a =〈 0.1 〉, b =〈 0.5, 0.4 〉, a =〈 0.3, 0.2 〉 and ω₁ = ω₂ = 0.5. According to score and accuracy functions, i.e. S (b) = S (c) =0.1 and H (b) =0.9, H (c) =0.5 we get b ≻ c. Using our IFWAM_K operator (36), we get IFWAM_K (a, b) =〈 0.25, 0.7 〉, IFWAM_K (a, c) =〈 0.15, 0.6 〉. Since S (IFWAM_K (a, b)) = S (IFWAM_K (a, c)) = –0.45 and H (IFWAM_K (a, b)) =0.95, H (IFWAM_K (a, c)) = 0.75, we conclude IFWAM_K (a, b) ≻ IFWAM_K (a, c) showing the monotonicity of the aggregation operator IFWAM_K (36) on A-IFVs, which was regarded as an important problem of the widely used IFWAM operator (14). Using the IFWMM operator (15b) from [17] with parameter vector P=1, 1 we get IFWMM (a, b) =〈 0, 0.265 〉 and IFWMM (a, c) =〈 0, 0.151 〉. Since S (IFWMM (a, b) = –0.265 and S (IFWMM (a, c) = –0.152 we conclude that IFWMM (a, c) ≻ IFWMM (a, b) and operator IFWMM is not monotone on A-IFVs.

Example 4.5. In this example, we show the monotonicity of the aggregation operator IFWAM_K (38b) on A-IFVs. Adopting example 2 from [7], let a =〈 0.4, 0.5 〉, b =〈 0.35, 0.448 〉, c =〈 0.5, 0.5 〉, w₁ = 0.5, x₄. According to (16) we have S (a) = –0.1, S (b) = –0.098, then b ≻ a. Besides, using the IFWAM_K operator (38b) with the weighting vector w = (w₁, w₂), we obtain the following aggregations: IFWGM_K (a, c) =〈 0.45, 0.5 〉 and IFWGM_K (b, c) =〈 0.43, 0.47 〉. Therefore, we have S (a ⊗ c) = –0.047, S (b ⊗ c) = –0.043, which implies IFWGM_K (b, c) ≻ IFWGM_K (a, c) meaning that the aggregation operator IFWGM_K (38b) on the A-IFVs is monotonic, whereas the aggregation operator IFWGM (15) is not monotonic with respect to the ordering (16) as it was shown in [7].

5 Applications of the proposed operators in MAGDM under intuitionistic fuzzy environment

In this section, we apply the new operations on A-IFSs and aggregation operators in solving the multi-attribute group decision making (MAGDM) problem. The MAGDM problem under consideration is described as follows. Let E = {e₁, e₂, . . . , e_l} be a set of l experts, who evaluate n alternatives from the set X = {x₁, x₂, . . . , x_m} regarding to a set of m attributes A = {a₁, a₂, . . . , a_m}. Let $R^{(k)} = (r_{ij}^{(k)})_{n \times m} (i = 1, 2, . . ., n; j = 1, 2, . . ., m; k = 1, 2, . . ., l)$ be intuitionistic fuzzy decision matrices, where $r_{ij}^{(k)} = 〈 μ_{r_{ij}^{(k)}}, μ_{r_{ij}^{(k)}} 〉$ is an evaluation in term of A-IFVs provided by the expert e_k ∈ E, related to the alternative x_i ∈ X with respect to the attribute a_j ∈ A. The goal is ranking the alternatives and finding the best one according to the information provided by a group of experts. In this example, we propose a new objective weighting method to determine attribute weights according to the objective information given in the decision matrix using the knowledge measures. Intuitively, the attribute with a larger knowledge measure should be assigned a higher weight. The whole decision making (DM) process is developed in the following steps:

Step 1. Normalize decision making matrices by converting the rates of cost attribute into the rates of benefit one as follows: For cost attributes a_j, $r_{ij}^{(k)} = r_{ij}^{(k) c} 〈 v_{r_{ij}^{k}}, μ_{r_{ij}^{(k)}} 〉 .$ (47)

Step 2. Aggregate all individual decision matrices with their weights λ_k ∈ [0, 1] $\sum_{k = 1}^{1} λ_{k} = 0$ using the IFWAM operator (18) into a group collective decision matrix denoted by Z = (z_ij) _n×m = (〈 v_{z
_ij}, μ_{z
_ij} 〉) _n×m: $\begin{matrix} z_{ij} = {IFWAM}_{k} (r_{ij}^{(1)}, r_{ij}^{(2)}, . . ., r_{ij}^{(l)}) \\ = 〈 \sum_{k = 1}^{l} λ_{k} μ_{r_{ij}^{(k)}}, \sum_{k = 1}^{l} λ_{k} v_{A_{r_{ij}^{(k)}}} 〉 . \end{matrix}$ (48)

Step 3. Aggregate all attribute values in the group collective decision matrix z_ij with the attribute weighting vector w = (w₁, w₂, . . . , w_m) ^T, w_j ∈ [0, 1] $\sum_{j = 1}^{m} w_{j} = 0$ , into overall evaluation values denoted by z_i =〈 v_zi, μ_zi 〉for the alternatives x_i ∈ X, (i = 1, 2, . . . , n), by the operator (19). $\begin{matrix} z_{i} = {IFWAM}_{w} (z_{i 1}, z_{i 2}, . . ., z_{im}) \\ = 〈 \sum_{j = 1}^{m} w_{j} μ_{{Ar}_{ij}}, \sum_{j = 1}^{m} w_{j} v_{A_{r_{ij}}} 〉 . \end{matrix}$ (49)

Step 4. Compute the score and accuracy functions of alternatives from their overall evaluation values z_i and rank the alternatives in the descending order of their score functions. The most desirable alternative is the one with the biggest score value.

In order to show the applicability of the proposed approach, we consider from literature an example of MAGDM problem. To verify it, we utilize some methods to solve the same MAGDM problem. Then a comparative analysis of obtained results will show the convergence and divergence between them. Moreover, a sensitivity investigation will be conducted to show their reliability in solving various MAGDM problems.

5.1 Experimental application

We adopt an example of the investment selection problem presented in [17]. Suppose that the investment company has to select one from five candidate companies {x₁, x₂, x₃, x₄, x₅}to invest the money. They must take a decision according to the candidate evaluations made by three invited experts {e₁, e₂, e₃}with respect to following four attributes: a₁ – the risk evaluation; a₂ – the growth evaluation; a₃ – the growth evaluation and a₄ – the environmental impact evaluation. They assumed the attribute weighting vector w = (0.2, 0.1, 0.3, 0.4) ^Tand the expert’s weighting vector λ = (0.35, 0.4, 0.25) ^T. The evaluation matrices provided by experts {e₁, e₂, e₃}are expressed in the intuitionistic fuzzy values as listed in Tables 5.1 –5.3. Our goal is to choose the best candidate using the proposed methods.

Table 5.1
The decision matrix $R_{ij}^{(1)}$ of the expert e₁

a ₁ a ₂ a ₃ a ₄

x ₁ <0.5, 0.4> <0.5, 0.3> <0.2, 0.6> <0.4, 0.4>

x ₂ <0.7, 0.3> <0.7, 0.3> <0.6, 0.2> <0.6, 0.2>

x ₃ <0.5, 0.4> <0.6, 0.4> <0.6, 0.2> <0.5, 0.3>

x ₄ <0.8, 0.2> <0.7, 0.2> <0.4, 0.2> <0.5, 0.2>

x ₅ <0.4, 0.3> <0.4, 0.2> <0.4, 0.5> <0.4, 0.6>

	a ₁	a ₂	a ₃	a ₄
x ₁	<0.5, 0.4>	<0.5, 0.3>	<0.2, 0.6>	<0.4, 0.4>
x ₂	<0.7, 0.3>	<0.7, 0.3>	<0.6, 0.2>	<0.6, 0.2>
x ₃	<0.5, 0.4>	<0.6, 0.4>	<0.6, 0.2>	<0.5, 0.3>
x ₄	<0.8, 0.2>	<0.7, 0.2>	<0.4, 0.2>	<0.5, 0.2>
x ₅	<0.4, 0.3>	<0.4, 0.2>	<0.4, 0.5>	<0.4, 0.6>

Table 5.2

The decision matrix $R_{ij}^{(2)}$ of the expert e₂

	a ₁	a ₂	a ₃	a ₄
x ₁	<0.4, 0.5>	<0.6, 0.2>	<0.5, 0.4>	<0.5, 03>
x ₂	<0.5, 0.4>	<0.6, 0.2>	<0.6, 0.3>	<0.7, 0.3>
x ₃	<0.4, 0.5>	<0.3, 0.5>	<0.4, 0.4>	<0.2, 0.6>
x ₄	<0.5, 0.4>	<0.7, 0.2>	<0.4, 0.4>	<0.6, 0.2>
x ₅	<0.6, 0.3>	<0.7, 0.2>	<0.4, 0.2>	<0.7, 0.2>

Table 5.3

The decision matrix $R_{ij}^{(3)}$ of the expert e₃

	a ₁	a ₂	a ₃	a ₄
x ₁	<0.4, 0.2>	<0.5, 0.2>	<0.5, 0.3>	<0.5, 0.2>
x ₂	<0.5, 0.3>	<0.5, 0.3>	<0.6, 0.2>	<0.7, 0.2>
x ₃	<0.4, 0.4>	<0.3, 0.4>	<0.4, 0.3>	<0.3, 0.3>
x ₄	<0.5, 0.3>	<0.5, 0.3>	<0.3, 0.5>	<0.5, 0.2>
x ₅	<0.6, 0.2>	<0.6, 0.4>	<0.4, 0.4>	<0.6, 0.3>

Step 1. All the attributes a_j (j = 1,2,3,4) are of the benefit type and the decision matrices $R_{ij}^{(k)}$ do not need normalization.

Step 2. Aggregating all individual decision matrices $R_{ij}^{(k)}$ with the expert’s weight vector λ = (0.35, 0.4, 0.25) ^Tinto the group decision matrix as listed in Table 5.4.

Table 5.4

The collective matrix of the expert group

	a ₁	a ₂	a ₃	a ₄
x ₁	<0.43, 0.39>	<0.54, 0.23>	<0.39, 0.44>	<0.46, 0.31>
x ₂	<0.57, 0.34>	<0.61, 0.26>	<0.60, 0.24>	<0.67, 0.24>
x ₃	<0.43, 0.44>	<0.41, 0.44>	<0.47, 0.31>	<0.33, 0.42>
x ₄	<0.61, 0.31>	<0.65, 0.23>	<0.38, 0.36>	<0.54, 0.20>
x ₅	<0.53, 0.28>	<0.57, 0.25>	<0.40, 0.36>	<0.57, 0.37>

Step 3. Aggregating the group decision matrix values z_ijinto the overall collective evaluation values z_i of alternatives x_i (i = 1, 2, 3, 4, 5)as:

z₁ =〈 0.446, 0.359 〉, z₂ =〈 0.621, 0.262 〉, z₃ =〈 0.401, 0.392 〉, z₄ =〈 0.515, 0.27 〉and z₅ =〈 0.511, 0.333 〉.

Step 4. Computing the score functions S of alternatives from their overall collective evaluation values z_i as:

S (z₁) =0.0865, S (z₂) =0.359, S (z₃) =0.009, S (z₄) =0.2445and S (z₅) =0.1785.

We have S (z₂) > S (z₄) > S (z₅) > S (z₁) > S (z₃). Accordingly, the ranking order of alternatives is x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃and the best candidate company is x₂, which coincides with that obtained in [17].

It is worth noticing that the obtained results in [17] by using IFWMM (or IFDWMM) do not follow the boundedness property, i.e. they exceed the minimum (or maximum) value of the input data set and approach to < 0, 1> (or < 1, 0>, respectively). Therefore, the called comprehensive values of alternatives obtained in [17] by using IFWMM, all have the membership degrees less than non-membership degrees in spite the input values are positive (μ > v). Similarly, but only the IFDWMM provides the inverse results. For example, from Table 5 in [17] the collective values of alternative S₁ under attributes A_iare as follows: A₁ =〈 0.445, 0.331 〉 A₂ =〈 0.542, 0.224 〉, A₃ =〈 0.428, 0.405 〉 and A₄ =〈 0.479, 0.281 〉. By using IFWMM operator, the aggregated value of alternative S₁shown in Table 6 of [17] is 〈0.313, 0.507〉, which is rather unreasonable.

Then a comparison of the proposed approach with existing related methods based on IFWAM, IFWGM, IFWMM and IFHWAGA operators, whose all results are given in Table 5.5. It can be seen that all considered methods provide the same ranking results in this MAGDM problem. However, looking closer at aggregated results of alternatives, just only the operator IFWMM [17] provides resulting A-IFVs with membership degrees less than non-membership degrees, i.e. they are of opposite significance (information) against to the input values. It means that for the almost positive information the operator IFWMM implies a negative information, which is for sure unreasonable case. Besides, [17] stated that with parameter vector P=1 . . . 1, the operator IFWMM reduces to the geometric weighted mean IFWGM operator, whereas their aggregated results are definitely different.

Table 5.5

Decision results of ranking order based on related aggregation operators

Aggregation operator	Aggregated results	Score function	Ranking order
IFWAM	z₁ = 〈0.453, 0.339〉	S (z₁) =0.114	x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃
	z₂ = 〈0.627, 0.255〉	S (z₂) =0.372
	z₃ = 〈0.414, 0.372〉	S (z₃) =0.042
	z₄ = 〈0.535, 0.254〉	S (z₄) =0.281
	z₅ = 〈0.529, 0.305〉	S (z₅) =0.224
IFWGM	z₁ = 〈0.431, 0.372〉	S (z₁) =0.059	x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃
	z₂ = 〈0.617, 0.265〉	S (z₂) =0.352
	z₃ = 〈0.379, 0.405〉	S (z₃) = –0.02
	z₄ = 〈0.499, 0.278〉	S (z₄) =0.221
	z₅ = 〈0.496, 0.351〉	S (z₅) =0.146
IFWMM
P=(1,1,1,1)	z₁ = 〈0.313, 0.507〉	S (z₁) =0.19	x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃
	z₂ = 〈0.431, 0.456〉	S (z₂) = –0.02
	z₃ = 〈0.270, 0.567〉	S (z₃) = –0.29
	z₄ = 〈0.369, 0.467〉	S (z₄) = –0.09
	z₅ = 〈0.356, 0.502〉	S (z₅) = –0.14
IFHWAGA
λ = 0.5	z₁ = 〈0.442, 0.355〉	S (z₁) =0.087	x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃
	z₂ = 〈0.622, 0.260〉	S (z₂) =0.362
	z₃ = 〈0.397, 0.389〉	S (z₃) =0.008
	z₄ = 〈0.516, 0.265〉	S (z₄) =0.251
	z₅ = 〈0.511, 0.327〉	S (z₅) =0.184
IFWAM_K
	z₁ = 〈0.446, 0.359〉	S (z₁) =0.087	x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃
	z₂ = 〈0.621, 0.262〉	S (z₂) =0.359
	z₃ = 〈0.401, 0.392〉	S (z₃) =0.009
	z₄ = 〈0.515, 0.27〉	S (z₄) =0.245
	z₅ = 〈0.511, 0.333〉	S (z₅) =0.178

5.2 Sensitivity investigation

In order to highlight the advantages of the proposed method, we assume that the experts may give a crisp value for any attribute evaluations. For instance, we modify a data in the experimental application 5.1 to assume the evaluating value of the alternative x₂ under attribute a₁, made by the expert e₁ of $r_{21}^{(1)} = 〈 0.7, 0.3 〉$ to $r_{21}^{(1)} = 〈 0, 1 〉$ . The other evaluation values are the same as the decision matrices shown in Tables 5.1, 5.3. The ranking results of the five alternatives for different methods are shown in Table 5.6. It is easily seen that the aggregated results of the alternative x₂ are obtained as z₂ = 〈0, 1〉 using the such operators as IFWGM, IFWMM or IFHWAGA, regardless other evaluating attributes values. Obviously, they give counterintuitive ranking results as another involved methods as Heronian mean, Bonferroni mean or symmetric Maclaurin mean. Only the our proposed operator and IFWAM operator overcome this drawback giving the intuitive aggregated results and ranking order of alternatives. It is worth mentioning that the IFWAM operator has some other drawbacks revealed in [3, 6] and [7].

Table 5.6
Decision results of the MAGDM methods in the sensitivity investigation based on related aggregation operators

Aggregation operator Aggregated results Score function Ranking order

IFWAM z₁ = 〈0.453, 0.339〉 S (z₁) =0.114 x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃

z₂ = 〈0.594, 0.277〉 S (z₂) =0.317

z₃ = 〈0.414, 0.372〉 S (z₃) =0.042

z₄ = 〈0.535, 0.254〉 S (z₄) =0.281

z₅ = 〈0.529, 0.305〉 S (z₅) =0.224

IFWGM z₁ = 〈0.431, 0.372〉 S (z₁) =0.059 x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃

z₂ = 〈0, 1〉 S (z₂) = –1

z₃ = 〈0.379, 0.405〉 S (z₃) = –0.02

z₄ = 〈0.499, 0.278〉 S (z₄) =0.221

z₅ = 〈0.496, 0.351〉 S (z₅) =0.146

IFWMM

P=(1,1,1,1) z₁ = 〈0.313, 0.507〉 S (z₁) = –0.19 x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃

z₂ = 〈0, 1〉 S (z₂) = –1

z₃ = 〈0.270, 0.567〉 S (z₃) = –0.29

z₄ = 〈0.369, 0.467〉 S (z₄) = –0.09

z₅ = 〈0.356, 0.502〉 S (z₅) = –0.14

IFHWAGA

λ = 0.5 z₁ = 〈0.442, 0.355〉 S (z₁) =0.087 x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃

z₂ = 〈0, 1〉 S (z₂) = –1

z₃ = 〈0.397, 0.389〉 S (z₃) =0.008

z₄ = 〈0.516, 0.265〉 S (z₄) =0.251

z₅ = 〈0.511, 0.327〉 S (z₅) =0.184

IFWAM_K

z₁ = 〈0.446, 0.359〉 S (z₁) =0.087 x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃

z₂ = 〈0.572, 0.311〉 S (z₂) =0.261

Aggregation operator	Aggregated results	Score function	Ranking order
IFWAM	z₁ = 〈0.453, 0.339〉	S (z₁) =0.114	x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃
	z₂ = 〈0.594, 0.277〉	S (z₂) =0.317
	z₃ = 〈0.414, 0.372〉	S (z₃) =0.042
	z₄ = 〈0.535, 0.254〉	S (z₄) =0.281
	z₅ = 〈0.529, 0.305〉	S (z₅) =0.224
IFWGM	z₁ = 〈0.431, 0.372〉	S (z₁) =0.059	x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃
	z₂ = 〈0, 1〉	S (z₂) = –1
	z₃ = 〈0.379, 0.405〉	S (z₃) = –0.02
	z₄ = 〈0.499, 0.278〉	S (z₄) =0.221
	z₅ = 〈0.496, 0.351〉	S (z₅) =0.146
IFWMM
P=(1,1,1,1)	z₁ = 〈0.313, 0.507〉	S (z₁) = –0.19	x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃
	z₂ = 〈0, 1〉	S (z₂) = –1
	z₃ = 〈0.270, 0.567〉	S (z₃) = –0.29
	z₄ = 〈0.369, 0.467〉	S (z₄) = –0.09
	z₅ = 〈0.356, 0.502〉	S (z₅) = –0.14
IFHWAGA
λ = 0.5	z₁ = 〈0.442, 0.355〉	S (z₁) =0.087	x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃
	z₂ = 〈0, 1〉	S (z₂) = –1
	z₃ = 〈0.397, 0.389〉	S (z₃) =0.008
	z₄ = 〈0.516, 0.265〉	S (z₄) =0.251
	z₅ = 〈0.511, 0.327〉	S (z₅) =0.184
IFWAM_K
	z₁ = 〈0.446, 0.359〉	S (z₁) =0.087	x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃
	z₂ = 〈0.572, 0.311〉	S (z₂) =0.261

6 Conclusion

In this study, we have proposed some new operations on A-IFVs with such desired properties as commutativity, idempotency, boundedness and monotonicity. Some numerical examples have been conducted to reveal the limitations of the existing operations on A-IFVs and the supervisor of the proposed operations in overcoming these drawbacks. Based on newly defined operations on A-IFVs, two popular aggregation operators for solving MAGDM problems under IF environment have been proposed, i.e. intuitionistic fuzzy weighted arithmetic mean and geometric mean, called IFWAM_K and IFWGM_K respectively. A practical application was conducted to show the consistency of the proposed method compared to the existing ones in solving the MAGDM problems. Then a sensitivity investigation was conducted to show their reliability in solving various MAGDM problems.

The obtained results have shown the consistency of the proposed method with the others in solving the same MAGDM problem. They have also revealed that the recent aggregation operators for A-IFVs as IFWGM, IFWMM or IFHWAGA, as well other involved methods based on such operators as Heronian mean, Bonferroni mean or symmetric Maclaurin mean have some flaws and may give counterintuitive ranking results, in spite they are good at incorporating interrelationship between the arguments.

The main contributions of the paper are as follows:

the proposed operations and aggregation operators have desired properties,

they overcome the limitations of the existing operations on A-IFVs,

they provide results consistent with that of the existing MAGDM methods in practical applications,

they are more robust and reliable in solving various MAGDM problems.

Future researches could be focused on utilizing the combined aggregation operators in the framework of the proposed methods to improve the aggregation feature in solving the more comprehensive MAGDM problems.

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