A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem

Abstract

The objective of the present work is divided into two folds. Firstly, an interval-valued Pythagorean fuzzy set (IVPFS) has been introduced along with their two aggregation operators, namely, interval-valued Pythagorean fuzzy weighted average and weighted geometric operators for different IVPFS. Secondly, an improved accuracy function under IVPFS environment has been developed by taking the account of the unknown hesitation degree. The proposed function has been applied to decision making problems to show the validity, practicality and effectiveness of the new approach. A systematic comparison between the existing work and the proposed work has also been given.

Keywords

Pythagorean fuzzy set interval-valued Pythagorean fuzzy set MCDM accuracy function aggregated operators

1 Introduction

Multi-criteria decision making (MCDM) is one of the process for finding the optimal alternative from all the feasible alternatives according to some criteria or attributes. Traditionally, it has been generally assumed that all the information which access the alternative in terms of criteria and their corresponding weights are expressed in the form of crisp numbers. But most of the decisions in the real-life situations are taken in the environment where the goals and constraints are generally imprecise or vague in nature. For this, Bellman and Zadeh [1] introduced the theory of fuzzy sets [2] in which the information relative to the degree of satisfaction of a particular attribute are taken in the form of membership functions and simultaneously considered that the degree of unsatisfactory is just the complement of it. After their successful application of the fuzzy set theory, researchers are engaged in their advancement. Out of that, intuitionistic fuzzy set (IFS) theory [3, 4] is one of the most successful extension which is characterized by the degrees of membership and non-membership satisfaction of the particular alternative with respect to the criteria, such that their sum is equal to or less than 1. Under this environment, various researchers pay more attention on IFSs for aggregating the different alternatives using different aggregation operators [5 –14]. However, there may be a situation where the decision maker may provide the degree of membership and non-membership of a particular attribute in such a way that their sum is greater than 1. For instance, a person expresses his preference about the degree of alternative satisfies the criteria is $\frac{\sqrt{3}}{2}$ while dissatisfies the criteria is $\frac{1}{2}$ . Then, it is clearly seen that $\frac{\sqrt{3}}{2} + \frac{1}{2} ≰ 1$ . Therefore, this situation is not properly handled in the IFS. To overcome this shortcoming, Yager [15, 16] introduced a concept of Pythagorean fuzzy set (PFS), generalization of the IFS, under the restriction that square sum of its membership degrees are less than or equal to 1. For instance, corresponding to above considered example, we see that $(\frac{\sqrt{3}}{2})^{2} + (\frac{1}{2})^{2} \leq 1$ . Later on, Yager and Abbasov [17] studied the relationship between the PFNs and the complex numbers and concluded that Pythagorean degrees are a subclass of complex numbers. Zhang and Xu [18] presented a technique for finding the best alternative based on its ideal solution under the Pythagorean fuzzy environment. Yager [16] proposed some new fuzzy weighted average and geometric aggregated operators using the different Pythagorean fuzzy numbers (PFNs) for solving MCDM problems. Peng and Yang [19] defined the new operations such as division; subtraction and their corresponding properties on PFNs. Recently, Garg [14] proposed a new generalized Pythagorean fuzzy information aggregation operators using Einstein operations.

Following the pioneering work of Yager and their co-authors [15 –17], we will present some novel concepts and theorems under the interval-valued Pythagorean fuzzy environment. To the best of our knowledge, interval-valued Pythagorean fuzzy sets have not been defined so far. Thus, the objective of this work is to extend the theory of Pythagorean fuzzy set to the interval-valued Pythagorean fuzzy sets (IVPFSs) and hence some aggregated operators for aggregating the different interval-valued Pythagorean fuzzy numbers are introduced. Some desirable properties of these operators are also investigated. Finally, an improved accuracy function has been proposed by taking into account the unknown degree (hesitancy degree) for comparing the two different IVPFNs and a decision-making method based on it.

The rest of the paper is organized as follows. Section 2 describe some basic definitions of PFS and their corresponding interval-valued PFSs. Based on these definitions, some weighted aggregated average and geometric operators on interval-valued Pythagorean fuzzy numbers (IVPFNs) has been proposed. In Section 3, an improved accuracy function has been proposed for ranking the IVPFNs by taking the degree of hesitation of IVPFSs. An illustrative example has been taken for showing the superiority of the proposed function. In Section 4, we proposed a method for solving the MCDM problems using these aggregation operators and an improved accuracy function. In Section 5, some illustrative examples are pointed out. Finally, some concrete conclusion about the paper has been summarized and give some remarks in Section 6.

2 Interval-valued Pythagorean fuzzy sets

In this section, firstly some basic concepts related to IFS and PFS have been defined and then interval-valued Pythagorean fuzzy set are introduced.

2.1 Pythagorean fuzzy set

Definition 2.1. An intuitionistic fuzzy set (IFS) A in a finite universe of discourse X = {x₁, x₂, …, x_n} is given by [3, 4] $\begin{matrix} A = {〈 x, μ_{A} (x), ν_{A} (x) 〉 ∣ x \in X} \end{matrix}$ where μ_A (x) and ν_A (x) are the membership and non-membership functions of A. The numbers μ_A (x) and ν_A (x) describes respectively the degree of belonging and rejection of x ∈ X in A. For a given x, the pair 〈μ, ν〉 is called intuitionistic fuzzy value (IFV) or intuitionistic fuzzy number (IFN) where μ ∈ [0, 1] , ν ∈ [0, 1] and μ + ν ∈ [0, 1]. The degree of hesitation between the membership functions is given by π_A (x) =1 - μ_A (x) - ν_A (x).

Definition 2.2. A Pythagorean fuzzy set (PFS) P in a finite universe of discourse X is given by [15] $\begin{matrix} P = {〈 x, μ_{P} (x), ν_{P} (x) 〉 ∣ x \in X} \end{matrix}$ where μ_P, ν_P : X ⟶ [0, 1] be the degree of membership and nonmembership of the element x ∈ X to the set P, respectively with the condition that 0 ≤ (μ_P (x))²+ (ν_P (x))²≤ 1. The degree of indeterminacy (hesitation) between the membership function is given by $π_{P} (x) = \sqrt{1 - (μ_{P} (x))^{2} - (ν_{P} (x))^{2}}$ .

For convenience, Zhang and Xu [18] Zhang2014a called 〈μ_P (x) , ν_P (x) 〉 a Pythagorean fuzzy number (PFN) denoted by P = 〈μ_P, ν_P〉. For any PFN P = 〈μ_P, ν_P〉, the score function of P is defined as follows: $\begin{matrix} S (P) = (μ_{P})^{2} - (ν_{P})^{2} \end{matrix}$ where S (P) ∈ [-1, 1]. For any two PFNs P₁, P₂, if S (P₁) < S (P₂) then P₁ ≺ P₂. If S (P₁) > S (P₂), then P₁ ≻ P₂. If S (P₁) = S (P₂), then P₁ ∼ P₂. On the other hand, the accuracy function of P be defined as follows $\begin{matrix} H (P) = (μ_{P})^{2} + (ν_{P})^{2} \end{matrix}$ where H (P) ∈ [0, 1].

As for the PFNs, Yager [15, 17] defined the operations for three PFNs α = 〈μ, ν〉, α₁ = 〈μ₁, ν₁〉 and α₂ = 〈μ₂, ν₂〉, λ > 0, as follows

$α_{1} \oplus α_{2} = 〈 \sqrt{μ_{1}^{2} + μ_{2}^{2} - μ_{1}^{2} μ_{2}^{2}}, ν_{1} ν_{2} 〉$

$α_{1} \otimes α_{2} = 〈 μ_{1} μ_{2}, \sqrt{ν_{1}^{2} + ν_{2}^{2} - ν_{1}^{2} ν_{2}^{2}} 〉$

$λ \cdot α = 〈 \sqrt{1 - (1 - μ^{2})^{λ}}, ν^{λ} 〉$

$α^{λ} = 〈 μ^{λ}, \sqrt{1 - (1 - ν^{2})^{λ}} 〉$

2.2 Interval-valued Pythagorean fuzzy set

Definition 2.3. Let D [0, 1] be the set of all closed intervals of [0, 1], an interval-valued Pythagorean fuzzy set (IVPFS) A in X is defined as $\begin{matrix} A = {〈 x, [μ_{A}^{L} (x), μ_{A}^{U} (x)], [ν_{A}^{L} (x), ν_{A}^{U} (x)] 〉 ∣ x \in X} \end{matrix}$ where $0 \leq μ_{A}^{L} (x) \leq μ_{A}^{U} \leq 1$ , $0 \leq ν_{A}^{L} (x) \leq ν_{A}^{U} \leq 1$ and $(μ_{A}^{U} (x))^{2} + (ν_{A}^{U} (x))^{2} \leq 1$ .

Similar to PFSs, for each element x ∈ X, its hesitation interval relative to A is given as $\begin{matrix} π_{A} (x) & = & [π_{A}^{L} (x), π_{A}^{U} (x)] \\ = & [\sqrt{1 - (μ_{A}^{U} (x))^{2} - (ν_{A}^{U} (x))^{2}}, \\ \sqrt{1 - (μ_{A}^{L} (x))^{2} - (ν_{A}^{L} (x))^{2}}] \end{matrix}$ Especially, for every x ∈ X, if $\begin{matrix} μ_{A} (x) = μ_{A}^{L} (x) = μ_{A}^{U} (x), ν_{A} (x) = ν_{A}^{L} (x) = ν_{A}^{U} (x) \end{matrix}$ then, IVPFS A reduces to an ordinary PFS.

For an IVPFS A, the pair $〈 [μ_{A}^{L} (x), μ_{A}^{U} (x)]$ , $[ν_{A}^{L} (x)$ , $ν_{A}^{U} (x)] 〉$ is called an interval-valued Pythagorean fuzzy number (IVPFN). For convenience, the pair $〈 [μ_{A}^{L} (x), μ_{A}^{U} (x)]$ , $[ν_{A}^{L} (x), ν_{A}^{U} (x)] 〉$ is often denoted by 〈 [a, b] , [c, d] 〉 where

$[a, b] \subset [0, 1], [c, d] \subset [0, 1], and b^{2} + d^{2} \leq 1$ (1)

Obviously, α⁺= 〈[1, 1] , [0, 0] 〉 is the largest IVPFN, and α^-= 〈[0, 0] , [1, 1] 〉 is the smallest IVPFN.

In particular, if α₁ = 〈 [a₁, b₁] , [c₁, d₁] 〉 and α₂ = 〈 [a₂, b₂] , [c₂, d₂] 〉 are IVPFNs then α₁ = α₂ if and only if a₁ = a₂, b₁ = b₂, c₁ = c₂ and d₁ = d₂.

Definition 2.4. Let α = 〈 [a, b] , [c, d] 〉, α₁ = 〈 [a₁, b₁] , [c₁, d₁] 〉 and α₂ = 〈 [a₂, b₂], [c₂, d₂] 〉 be IVPFNs then

$\bar{α} = 〈 [c, d], [a, b] 〉$

α₁ ∧ α₂ = 〈 [min {a₁, a₂} , min {b₁, b₂}],

[max {c₁, c₂} , max {d₁, d₂}] 〉

α₁ ∨ α₂ = 〈 [max {a₁, a₂} , max {b₁, b₂}],

[min {c₁, c₂} , min {d₁, d₂}] 〉

$α_{1} \oplus α_{2} = 〈 [\sqrt{a_{1}^{2} + a_{2}^{2} - a_{1}^{2} a_{2}^{2}},$

$\sqrt{b_{1}^{2} + b_{2}^{2} - b_{1}^{2} b_{2}^{2}}]$ ,

[c₁c₂, d₁d₂] 〉

α₁ ⊗ α₂ = 〈 [a₁a₂, b₁b₂],

$[\sqrt{c_{1}^{2} + c_{2}^{2} - c_{1}^{2} c_{2}^{2}},$ $\sqrt{d_{1}^{2} + d_{2}^{2} - d_{1}^{2} d_{2}^{2}}] 〉$

$λ \cdot α = 〈 [\sqrt{1 - (1 - a^{2})^{λ}}, \sqrt{1 - (1 - b^{2})^{λ}}]$ ,

[c^λ, d^λ] 〉, λ > 0

$α^{λ} = 〈 [a^{λ}, b^{λ}], [\sqrt{1 - (1 - c^{2})^{λ}}$ ,

$\sqrt{1 - (1 - d^{2})^{λ}}] 〉, λ > 0$

Theorem 2.1. All the operational results in Definition 2.4 are IVPFNs

Proof. (i) Since α = 〈 [a, b] , [c, d] 〉 is an IVPFN, so it satisfies the Equation 1 and hence $\bar{α} = 〈 [c, d], [a, b] 〉$ also satisfies this condition. Thus $\bar{α}$ is an IVPFN.

(ii) Since both α₁ = 〈 [a₁, b₁] , [c₁, d₁] 〉 and α₂ = 〈 [a₂, b₂] , [c₂, d₂] 〉 are IVPFNs, so α₁ and α₂ satisfy the condition (1), i.e., $\begin{matrix} [a_{1}, b_{1}] \subset [0, 1], [c_{1}, d_{1}] \subset [0, 1], b_{1}^{2} + d_{1}^{2} \leq 1 \end{matrix}$ Thus, we have [min {a₁, a₂} , min {b₁, b₂}] ⊂ [0, 1], [max {c₁, c₂} , max {d₁, d₂}] ⊂ [0, 1] and (min {b₁, b₂})²+ (max {d₁, d₂})²≤ 1. Then α₁ ∧ α₂ satisfies the condition (1) i.e. α₁ ∧ α₂ is an IVPFN.

(iii) Similar to (ii), we can prove that α₁ ∨ α₂ is an IVPFN.

(iv) Since both α₁ and α₂ satisfies the condition (1), it follows that $\begin{matrix} \sqrt{a_{1}^{2} + a_{2}^{2} - a_{1}^{2} a_{2}^{2}} = \sqrt{a_{1}^{2} (1 - a_{2}^{2}) + a_{2}^{2}} \\ \geq \sqrt{a_{2}^{2}} \geq a_{2} \geq 0, and c_{1} c_{2} \geq 0 \end{matrix}$

Also, $b_{1}^{2} + b_{2}^{2} - b_{1}^{2} b_{2}^{2} + d_{1}^{2} d_{2}^{2} \leq b_{1}^{2} + b_{2}^{2} - b_{1}^{2} b_{2}^{2}$ $+ (1 - b_{1}^{2}) (1 - b_{2}^{2}) = 1$

Therefore, the value of α₁ ⊕ α₂ satisfies the condition of Equation 1 and hence it is an IVPFN. In the similar way, (v) can be proven.

(vi) Since $\sqrt{1 - (1 - a^{2})^{λ}} \geq 0, \sqrt{1 - (1 - b^{2})^{λ}}$ ≥0, c^λ≥ 0, d^λ≥ 0 and 1 - (1 - b²)^λ+ (d²)^λ≤ 1 - (1 - b²)^λ+ (1 - b²)^λ= 1. Thus, the value of λ · α is an IVPFN.

(vii) Can be proven similarly. This completes the proof.

Remark 2.1. In the following, let us look at λ · α and α^λ for some special cases of λ and α.

If α = 〈 [a, b] , [c, d] 〉 = 〈[1, 1] , [0, 0] 〉 then $\begin{matrix} λ \cdot α = 〈 [\sqrt{1 - (1 - a^{2})^{λ}}, \sqrt{1 - (1 - b^{2})^{λ}}], [c^{λ}, d^{λ}] 〉 \\ = 〈 [1, 1], [0, 0] 〉 \\ α^{λ} = 〈 [a^{λ}, b^{λ}], [\sqrt{1 - (1 - c^{2})^{λ}}, \sqrt{1 - (1 - d^{2})^{λ}}] 〉 \\ = 〈 [1, 1], [0, 0] 〉 \\ i . e . λ \cdot 〈 [1, 1], [0, 0] 〉 = 〈 [1, 1], [0, 0] 〉; \\ 〈 [1, 1], [0, 0] 〉^{λ} = 〈 [1, 1], [0, 0] 〉 \end{matrix}$

If α = 〈 [a, b] , [c, d] 〉 = 〈[0, 0] , [1, 1] 〉 then $\begin{matrix} λ \cdot α = 〈 [\sqrt{1 - (1 - a^{2})^{λ}}, \sqrt{1 - (1 - b^{2})^{λ}}], [c^{λ}, d^{λ}] 〉 \\ = 〈 [0, 0], [1, 1] 〉 \\ α^{λ} = 〈 [a^{λ}, b^{λ}], [\sqrt{1 - (1 - c^{2})^{λ}}, \sqrt{1 - (1 - d^{2})^{λ}}] 〉 \\ = 〈 [0, 0], [1, 1] 〉 \\ i . e . λ \cdot 〈 [0, 0], [1, 1] 〉 = 〈 [0, 0], [1, 1] 〉; \\ 〈 [0, 0], [1, 1] 〉^{λ} = 〈 [0, 0], [1, 1] 〉 \end{matrix}$

If λ → 0 and 0 < a, b, c, d < 1 then $\begin{matrix} λ \cdot α = 〈 [\sqrt{1 - (1 - a^{2})^{λ}}, \sqrt{1 - (1 - b^{2})^{λ}}], [c^{λ}, d^{λ}] 〉 \\ ⟶ 〈 [0, 0], [1, 1] 〉 \\ α^{λ} = 〈 [a^{λ}, b^{λ}], [\sqrt{1 - (1 - c^{2})^{λ}}, \sqrt{1 - (1 - d^{2})^{λ}}] 〉 \\ ⟶ 〈 [1, 1], [0, 0] 〉 \\ i . e . λ \cdot α \to 〈 [0, 0], [1, 1] 〉, as λ \to 0 \\ α^{λ} \to 〈 [0, 0], [1, 1] 〉, as λ \to 0 \end{matrix}$

If λ→ + ∞ and 0 < a, b, c, d < 1 then $\begin{matrix} λ \cdot α = 〈 [\sqrt{1 - (1 - a^{2})^{λ}}, \sqrt{1 - (1 - b^{2})^{λ}}], [c^{λ}, d^{λ}] 〉 \\ \to 〈 [1, 1], [0, 0] 〉 \\ α^{λ} = 〈 [a^{λ}, b^{λ}], [\sqrt{1 - (1 - c^{2})^{λ}}, \sqrt{1 - (1 - c^{2})^{λ}}] 〉 \\ \to 〈 [0, 0], [1, 1] 〉 \\ i . e . λ \cdot α = 〈 [1, 1], [0, 0] 〉 and α^{λ} = 〈 [0, 0], [1, 1] 〉 \end{matrix}$

If λ = 1 then $\begin{matrix} λ \cdot α = 〈 [\sqrt{1 - (1 - a^{2})^{λ}}, \sqrt{1 - (1 - b^{2})^{λ}}], [c^{λ}, d^{λ}] 〉 \\ = 〈 [a, b], [c, d] 〉 = α \\ α^{λ} = 〈 [a^{λ}, b^{λ}], [\sqrt{1 - (1 - c^{2})^{λ}}, \sqrt{1 - (1 - d^{2})^{λ}}] 〉 \\ = 〈 [a, b], [c, d] 〉 = α \\ i . e . λ \cdot α = α and α^{λ} = α \end{matrix}$

Theorem 2.2. Let λ, λ₁, λ₂ ≥ 0, then

α₁ ⊕ α₂ = α₂ ⊕ α₁

α₁ ⊗ α₂ = α₂ ⊗ α₁

λ · (α₁ ⊕ α₂) = λ · α₁ ⊕ λ · α₂

$(α_{1} \otimes α_{2})^{λ} = α_{1}^{λ} \otimes α_{2}^{λ}$

λ₁ · α ⊕ λ₂ · α = (λ₁ + λ₂) · α

α^{λ
₁}⊗ α^{λ
₂}= α^λ₁+λ₂

Proof. (i) By (iv) of Definition 2.4, we have $\begin{matrix} α_{1} \oplus α_{2} = 〈 [\sqrt{a_{1}^{2} + a^{2} - a_{1}^{2} a_{2}^{2}}, \sqrt{b_{1}^{2} + b_{2}^{2} - b_{1}^{2} b_{2}^{2}}], \\ [c_{1} c_{2}, d_{1} d_{2}] 〉 \\ = 〈 [\sqrt{a_{2}^{2} + a^{1} - a_{2}^{2} a_{1}^{2}}, \sqrt{b_{2}^{2} + b_{1}^{2} - b_{2}^{2} b_{1}^{2}}], \\ [c_{2} c_{1}, d_{2} d_{1}] 〉 \\ = α_{2} \oplus α_{1} \end{matrix}$

(ii) can be proven similarly, so we omit here.

(iii) It follows from (iv) in definition 2.4 that $\begin{matrix} α_{1} \oplus α_{2} = 〈 [\sqrt{a_{1}^{2} + a_{2}^{2} - a_{1}^{2} a_{2}^{2}}, \sqrt{b_{1}^{2} + b_{2}^{2} - b_{1}^{2} b_{2}^{2}}], \\ [c_{1} c_{2}, d_{1} d_{2}] 〉 \end{matrix}$

According to (vi) in Definition 2.4, we get $\begin{matrix} λ \cdot (α_{1} \oplus α_{2}) = 〈 [\sqrt{1 - (1 - (a_{1}^{2} + a_{2}^{2} - a_{1}^{2} a_{2}^{2})^{λ})}, \\ \sqrt{1 - (1 - (b_{1}^{2} + b_{2}^{2} - b_{1}^{2} b_{2}^{2})^{λ})}], \\ [(c_{1} c_{2})^{λ}, (d_{1} d_{2})^{λ}] 〉 \\ = 〈 [\sqrt{1 - (1 - a_{1}^{2})^{λ} (1 - a_{2}^{2})^{λ}}, \\ \sqrt{1 - (1 - b_{1}^{2})^{λ} (1 - b_{2}^{2})^{λ}}], \\ [(c_{1} c_{2})^{λ}, (d_{1} d_{2})^{λ}] 〉 \end{matrix}$

Also since,

\begin{matrix} λ \cdot α_{1} \\ = 〈 [\sqrt{1 - (1 - a_{1}^{2})^{λ}}, \sqrt{1 - (1 - b_{1}^{2})^{λ}}], [c_{1}^{λ}, d_{1}^{λ}] 〉 \\ λ \cdot α_{2} \\ = 〈 [\sqrt{1 - (1 - a_{2}^{2})^{λ}}, \sqrt{1 - (1 - b_{2}^{2})^{λ}}], [c_{2}^{λ}, d_{2}^{λ}] 〉 \end{matrix}

we have $\begin{matrix} λ α_{1} \oplus λ α_{2} & = & 〈 [\sqrt{1 - (1 - a_{1}^{2})^{λ} (1 - a_{2}^{2})^{λ}}, \\ \sqrt{1 - (1 - b_{1}^{2})^{λ} (1 - b_{2}^{2})^{λ}}], \\ [(c_{1} c_{2})^{λ}, (d_{1} d_{2})^{λ}] 〉 \end{matrix}$ Thus, $λ \cdot (α_{1} \oplus α_{2}) = λ \cdot α_{1} \oplus λ \cdot α_{2}$

Similarly, we can prove (iv)

(v) Since, $\begin{matrix} λ_{1} \cdot α_{1} \\ = 〈 [\sqrt{1 - (1 - a_{1}^{2})^{λ_{1}}}, \sqrt{1 - (1 - b_{1}^{2})^{λ_{1}}}], [c_{1}^{λ_{1}}, d_{1}^{λ_{1}}] 〉 \\ λ_{2} \cdot α_{1} \\ = 〈 [\sqrt{1 - (1 - a_{1}^{2})^{λ_{2}}}, \sqrt{1 - (1 - b_{1}^{2})^{λ_{2}}}], [c_{1}^{λ_{2}}, d_{1}^{λ_{2}}] 〉 \end{matrix}$

we can obtain, $\begin{matrix} λ_{1} α_{1} \oplus λ_{2} α_{1} = 〈 [\sqrt{1 - (1 - a_{1}^{2})^{λ_{1}} + 1 - (1 - a_{1}^{2})^{λ_{2}} - (1 - (1 - a_{1}^{2})^{λ_{1}}) (1 - (1 - a_{1}^{2})^{λ_{2}})}, \\ \sqrt{1 - (1 - b_{1}^{2})^{λ_{1}} + 1 - (1 - b_{1}^{2})^{λ_{2}} - (1 - (1 - b_{1}^{2})^{λ_{1}}) (1 - (1 - b_{1}^{2})^{λ_{2}})}], \\ [c_{1}^{λ_{1}} c_{1}^{λ_{2}}, d_{1}^{λ_{1}} d_{1}^{λ_{2}}] 〉 \\ = 〈 [\sqrt{1 - (1 - a_{1}^{2})^{λ_{1}} (1 - a_{1}^{2})^{λ_{2}}}, \sqrt{1 - (1 - b_{1}^{2})^{λ_{1}} (1 - b_{1}^{2})^{λ_{2}}}], \\ [c_{1}^{λ_{1} + λ_{2}}, d_{1}^{λ_{1} + λ_{2}}] 〉 \\ = 〈 [\sqrt{1 - (1 - a_{1}^{2})^{λ_{1} + λ_{2}}}, \sqrt{1 - (1 - b_{1}^{2})^{λ_{1} + λ_{2}}}], \\ [c_{1}^{λ_{1} + λ_{2}}, d_{1}^{λ_{1} + λ_{2}}] 〉 \\ = (λ_{1} + λ_{2}) \cdot α_{1} \end{matrix}$

(vi) can be proven similarly. This complete the proof.

Theorem 2.3. Let α₁ = 〈 [a₁, b₁] , [c₁, d₁] 〉 and α₂ = 〈 [a₂, b₂], [c₂, d₂] 〉 be two IVPFNs then

$α_{1}^{c} \land α_{2}^{c} = (α_{1} \lor α_{2})^{c}$

$α_{1}^{c} \lor α_{2}^{c} = (α_{1} \land α_{2})^{c}$

$α_{1}^{c} \oplus α_{2}^{c} = (α_{1} \otimes α_{2})^{c}$

$α_{1}^{c} \otimes α_{2}^{c} = (α_{1} \oplus α_{2})^{c}$

(α₁ ∨ α₂) ⊕ (α₁ ∧ α₂) = α₁ ⊕ α₂

(α₁ ∨ α₂) ⊗ (α₁ ∧ α₂) = α₁ ⊗ α₂

Proof 2.3. The proof is trivial, so it is omitted here.

Theorem 2.4. Let α₁ = 〈 [a₁, b₁] , [c₁, d₁] 〉, α₂ = 〈 [a₂, b₂], [c₂, d₂] 〉 and α₃ = 〈 [a₃, b₃] , [c₃, d₃] 〉 be three IVPFNs then

(α₁ ∨ α₂) ∧ α₃ = (α₁ ∧ α₃) ∨ (α₂ ∧ α₃)

(α₁ ∧ α₂) ∨ α₃ = (α₁ ∨ α₃) ∧ (α₂ ∨ α₃)

(α₁ ∨ α₂) ⊕ α₃ = (α₁ ⊕ α₃) ∨ (α₂ ⊕ α₃)

(α₁ ∧ α₂) ⊕ α₃ = (α₁ ⊕ α₃) ∧ (α₂ ⊕ α₃)

(α₁ ∨ α₂) ⊗ α₃ = (α₁ ⊗ α₃) ∨ (α₂ ⊗ α₃)

(α₁ ∧ α₂) ⊗ α₃ = (α₁ ⊗ α₃) ∧ (α₂ ⊗ α₃)

Proof 2.4. The proof is trivial, so it is omitted here.

2.3 Interval-valued Pythagorean Fuzzy aggregation operators

We now introduce, based on Definition 2.4, some operators for aggregating IVPFNs.

Definition 2.5. Let α_j = 〈 [a_j, b_j] , [c_j, d_j] 〉 (j = 1, 2, …, n) be a collection of IVPFNs and let IPFWA_ω : Ωⁿ⟶ Ω. If

$\begin{matrix} {IPFWA}_{ω} (α_{1}, α_{2}, \dots, α_{n}) \\ = ω_{1} \cdot α_{1} \oplus ω_{2} \cdot α_{2} \oplus \dots \oplus ω_{n} \cdot α_{n} \end{matrix}$ (2) where Ω is the collection of all IVPFNs, ω_j is the weight of α_j (j = 1, 2, …, n), ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ , then the function IPFWA_ω is called an interval-valued Pythagorean fuzzy weighted averaging operator. In particular, if ω_j = 1/n, ∀ j then IPFWA_ω operator reduces to interval-valued Pythagorean fuzzy averaging (IPFA) operator $\begin{matrix} IPFA (α_{1}, α_{2}, \dots, α_{n}) = \frac{1}{n} (α_{1} \oplus α_{2} \oplus \dots \oplus α_{n}) \end{matrix}$

Theorem 2.5. Let α_j = 〈 [a_j, b_j] , [c_j, d_j] 〉, (j = 1, 2, …, n) be a collection of IPFNs then the aggregated value by using Equation 2 is also an IVPFN and

$\begin{matrix} {IPFWA}_{ω} (α_{1}, α_{2}, \dots, α_{n}) \\ = 〈 [\sqrt{1 - \prod_{j = 1}^{n} (1 - a_{j}^{2})^{ω_{j}}}, \sqrt{1 - \prod_{j = 1}^{n} (1 - b_{j}^{2})^{ω_{j}}}], \\ [\prod_{j = 1}^{n} c_{j}^{ω_{j}}, \prod_{j = 1}^{n} d_{j}^{ω_{j}}] 〉 \end{matrix}$ (3) where ω_j is the weight of α_j (j = 1, 2, …, n), ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$

Proof 2.5. We prove Equation 3 by using mathematical induction on n:

When n = 2, ${IPFWA}_{ω} (α_{1}, α_{2}) = ω_{1} \cdot α_{1} \oplus ω_{2} \cdot α_{2}$

According to Theorem 2.1, we can see that both ω₁α₁ and ω₂α₂ are IVPFNs, and the value of ω₁α₁ ⊕ ω₂α₂ is an IVPFN. By the operational law (vi) in Definition 2.4, we have $\begin{matrix} ω_{1} \cdot α_{1} \\ = 〈 [\sqrt{1 - (1 - a_{1}^{2})^{ω_{1}}}, \sqrt{1 - (1 - b_{1}^{2})^{ω_{1}}}], [c_{1}^{ω_{1}}, d_{1}^{ω_{1}}] 〉 \\ ω_{2} \cdot α_{2} \\ = 〈 [\sqrt{1 - (1 - a_{2}^{2})^{ω_{2}}}, \sqrt{1 - (1 - b_{2}^{2})^{ω_{2}}}], [c_{2}^{ω_{2}}, d_{2}^{ω_{2}}] 〉 \end{matrix}$ Then $\begin{matrix} {IPFWA}_{ω} (α_{1}, α_{2}) = ω_{1} \cdot α_{1} \oplus ω_{2} \cdot α_{2} \\ = 〈 [\sqrt{1 - (1 - a_{1}^{2})^{ω_{1}} + 1 - (1 - a_{2}^{2})^{ω_{2}} - (1 - (1 - a_{1}^{2})^{ω_{1}}) (1 - (1 - a_{2}^{2})^{ω_{2}})}, \\ \sqrt{1 - (1 - b_{1}^{2})^{ω_{1}} + 1 - (1 - b_{2}^{2})^{ω_{2}} - (1 - (1 - b_{1}^{2})^{ω_{1}}) (1 - (1 - b_{2}^{2})^{ω_{2}})}], \\ [c_{1}^{ω_{1}} c_{2}^{ω_{2}}, d_{1}^{ω_{1}} d_{2}^{ω_{2}}] 〉 \\ = 〈 [\sqrt{1 - (1 - a_{1}^{2})^{ω_{1}} (1 - a_{2}^{2})^{ω_{2}}}, \sqrt{1 - (1 - b_{1}^{2})^{ω_{1}} (1 - b_{2}^{2})^{ω_{2}}}], \\ [c_{1}^{ω_{1}} c_{2}^{ω_{2}}, d_{1}^{ω_{1}} d_{2}^{ω_{2}}] 〉 \end{matrix}$

Thus, result is true for n = 2.

Assume that result is true for n = k, Equation 3 holds, i.e. $\begin{matrix} {IPFWA}_{ω} (α_{1}, α_{2}, \dots, α_{k}) \\ = < [\sqrt{1 - \prod_{j = 1}^{k} (1 - a_{j}^{2})^{ω_{j}}}, \sqrt{1 - \prod_{j = 1}^{k} (1 - b_{j}^{2})^{ω_{j}}}], \\ [\prod_{j = 1}^{k} c_{j}^{ω_{j}}, \prod_{j = 1}^{k} d_{j}^{ω_{j}}] > \end{matrix}$ Then, when n = k + 1, by (iv) and (vi) in Definition 2.4, we get $\begin{matrix} {IPFWA}_{ω} (α_{1}, α_{2}, \dots, α_{k + 1}) \\ = {IPFWA}_{ω} (α_{1}, α_{2}, \dots, α_{k}) \oplus ω_{k + 1} \cdot α_{k + 1} \\ = < [\sqrt{1 - \prod_{j = 1}^{k} (1 - a_{j}^{2})^{ω_{j}}}, \sqrt{1 - \prod_{j = 1}^{k} (1 - b_{j}^{2})^{ω_{j}}}], \\ [\prod_{j = 1}^{k} c_{j}^{ω_{j}}, \prod_{j = 1}^{k} d_{j}^{ω_{j}}] > \\ \oplus 〈 [\sqrt{1 - (1 - a_{k + 1}^{2})^{ω_{k + 1}}}, \sqrt{1 - (1 - b_{k + 1}^{2})^{ω_{k + 1}}}], \\ [c_{k + 1}^{ω_{k + 1}}, d_{k + 1}^{ω_{k + 1}}] 〉 \\ = 〈 [\sqrt{1 - \prod_{j = 1}^{k + 1} (1 - a_{j}^{2})^{ω_{j}}}, \sqrt{1 - \prod_{j = 1}^{k + 1} (1 - b_{j}^{2})^{ω_{j}}}], \\ [\prod_{j = 1}^{k + 1} c_{j}^{ω_{j}}, \prod_{j = 1}^{k + 1} d_{j}^{ω_{j}}] 〉 \end{matrix}$ i.e. when n = k + 1, Equation 3 also holds.

Hence, Equation 3 holds for any n.

Next, in order to show IPFWA_ω is an IVPFN.

As α_j = 〈 [a_j, b_j] , [c_j, d_j] 〉 for all j is an IVPFN, thus 0 ≤ a_j, b_j, c_j, d_j ≤ 1 and $b_{j}^{2} + d_{j}^{2} \leq 1$ . Thus $0 \leq \prod_{j = 1}^{n} (1 - a_{j}^{2})^{ω_{j}} \leq 1$ and hence 0≤ $\sqrt{1 - \prod_{j = 1}^{n} (1 - a_{j}^{2})^{ω_{j}}} \leq 1$ and $0 \leq \prod_{j = 1}^{n} c_{j}^{ω_{j}} \leq 1$ . Similarly, $0 \leq \sqrt{1 - \prod_{j = 1}^{n} (1 - b_{j}^{2})^{ω_{j}}} \leq 1$ and 0≤ $\prod_{j = 1}^{n} d_{j}^{ω_{j}} \leq 1$ .

Again, $\begin{matrix} (\sqrt{1 - \prod_{j = 1}^{n} (1 - b_{j}^{2})^{ω_{j}}})^{2} + (\prod_{j = 1}^{n} d_{j}^{ω_{j}})^{2} \\ = & 1 - \prod_{j = 1}^{n} (1 - b_{j}^{2})^{ω_{j}} + \prod_{j = 1}^{n} d_{j}^{2 ω_{j}} \\ \leq & 1 - \prod_{j = 1}^{n} d_{j}^{2 ω_{j}} + \prod_{j = 1}^{n} d_{j}^{2 ω_{j}} = 1 \end{matrix}$ Hence, IPFWA_ω is an IVPFN and therefore proof is completed.

Definition 2.6. Let IPFWG_ω : Ωⁿ⟶ Ω. If

$\begin{matrix} {IPFWG}_{ω} (α_{1}, α_{2}, \dots, α_{n}) \\ = α_{1}^{ω_{1}} \otimes α_{2}^{ω_{2}} \otimes \dots \otimes α_{n}^{ω_{n}} \end{matrix}$ (4) then the function IPFWG_ω is called an interval-valued Pythagorean fuzzy weighted geometric operator. In particular, if ω = (1/n, 1/n, …, 1/n)^T then the IPFWG_ω operator reduces to an interval-valued Pythagorean fuzzy geometric operator

$\begin{matrix} {IPFWG}_{ω} (α_{1}, α_{2}, \dots, α_{n}) \\ = (α_{1} \otimes α_{2} \otimes \dots \otimes α_{n})^{1 / n} \end{matrix}$ (5)

Theorem 2.6. The aggregated value by using Equation 4 is also an IVPFN, and

$\begin{matrix} {IPFWG}_{ω} (α_{1}, α_{2}, \dots, α_{n}) = 〈 [\prod_{j = 1}^{n} a_{j}^{ω_{j}}, \prod_{j = 1}^{n} b_{j}^{ω_{j}}], \\ [\sqrt{1 - \prod_{j = 1}^{n} (1 - c_{j}^{2})^{ω_{j}}}, \sqrt{1 - \prod_{j = 1}^{n} (1 - d_{j}^{2})^{ω_{j}}}] 〉 \end{matrix}$ (6)

Proof 2.6. The proof of this theorem is similar to Theorem 2.5, so we omit here.

Example 2.1. Suppose that α₁ = 〈[0.7, 0.9] , [0.2, 0.3] 〉, α₂ = 〈[0.7, 0.8] , [0.1, 0.2] 〉, α₃ = 〈[0.2, 0.3] , [0.8, 0.9] 〉 and α₄ = 〈[0.5, 0.6] , [0.35, 0.45] 〉 and ω = (0.2, 0.3, 0.1, 0.4)^T is the weight vector of α_j (j = 1, 2, 3, 4). Then $\begin{matrix} {IPFWA}_{ω} (α_{1}, α_{2}, α_{3}, α_{4}) \\ = 〈 [\sqrt{1 - (1 - 0 . 7^{2})^{0.2} (1 - 0 . 7^{2})^{0.3} (1 - 0 . 2^{2})^{0.1} (1 - 0 . 5^{2})^{0.4}}, \\ \sqrt{1 - (1 - 0 . 9^{2})^{0.2} (1 - 0 . 8^{2})^{0.3} (1 - 0 . 3^{2})^{0.1} (1 - 0 . 6^{2})^{0.4}}], \\ [(0.2)^{0.2} (0.1)^{0.3} (0.8)^{0.1} (0.35)^{0.4}, (0.3)^{0.2} (0.2)^{0.3} (0.9)^{0.1} (0.45)^{0.4}] 〉 \\ = 〈 [0.6050, 0.7500], [0.2334, 0.3487] 〉 \\ {IPFWG}_{ω} (α_{1}, α_{2}, α_{3}, α_{4}) = \\ 〈 [0 . 7^{0.2} (0.7)^{0.3} (0.2)^{0.1} (0.5)^{0.4}, 0 . 9^{0.2} (0.8)^{0.3} (0.3)^{0.1} (0.6)^{0.4}], \\ [\sqrt{1 - (1 - 0 . 2^{2})^{0.2} (1 - 0 . 1^{2})^{0.3} (1 - 0 . 8^{2})^{0.1} (1 - 0 . 35^{2})^{0.4}}, \\ \sqrt{1 - (1 - 0 . 3^{2})^{0.2} (1 - 0 . 2^{2})^{0.3} (1 - 0 . 9^{2})^{0.1} (1 - 0 . 45^{2})^{0.4}}] 〉 \\ = 〈 [0.5398, 0.6618], [0.3907, 0.5000] 〉 \end{matrix}$

In order to rank the IVPFNs, we now introduce the score function and accuracy function of IVPFNs.

Definition 2.7. Let α = 〈 [a, b] , [c, d] 〉 be an IVPFN then we call $S (α) = \frac{a^{2} + b^{2} - c^{2} - d^{2}}{2}$ be the score of α, S (α) ∈ [-1, 1]. Clearly, the greater the S (α), the larger the α. In particular, if S (α) =1 then α = 〈[1, 1] , [0, 0] 〉; if S (α) = -1, then α is the smallest IVPFN 〈[0, 0] , [1, 1] 〉.

However, if we take α₁ = 〈[0.2, 0.4] , [0.2, 0.4] 〉 and α₂ = 〈[0.5, 0.6] , [0.5, 0.6] 〉 then S (α₁) = S (α₂) =0. In this case, score function cannot distinguish between the IVPFNs α₁ and α₂.

Definition 2.8. The accuracy function of IVPFN α is defined as $H (α) = \frac{a^{2} + b^{2} + c^{2} + d^{2}}{2}$ , where H (α) ∈ [0, 1]

3 Proposed novel accuracy function

Let α = 〈 [a, b] , [c, d] 〉 be an interval-valued Pythagorean fuzzy number, a novel accuracy function M of an interval-valued Pythagorean fuzzy value, based on unknown degree (hesitant degree), is proposed by the following formula

$\begin{matrix} M (α) \\ = \frac{a^{2} - \sqrt{1 - a^{2} - c^{2}} + b^{2} - \sqrt{1 - b^{2} - d^{2}}}{2} \end{matrix}$ (7) where M (α) ∈ [-1, 1].

Based on these functions, a prioritized comparison method for any two IVPFNs α = 〈 [a₁, b₁] , [c₁, d₁] 〉 and β = 〈 [a₂, b₂] , [c₂, d₂] 〉 is defined as follows

Definition 3.1. Let α and β be any two IVPFNs. Then

If S (α) < S (β), then α ≺ β;

If S (α) > S (β), then α ≻ β;

If S (α) = S (β),

If H (α) < H (β), then α ≺ β.

If H (α) > H (β), then α ≻ β.

If H (α) = H (β),

If M (α) > M (β), then α ≻ β.

If M (α) < M (β), then α ≺ β.

If M (α) = M (β), then α ∼ β.

Let us consider the following examples.

Example 3.1. If interval-valued Pythagorean fuzzy values for two alternatives are α₁ = $〈 [\frac{1}{10}, \frac{1}{4} \sqrt{\frac{1}{5}}]$ , $[\frac{\sqrt{3}}{10}, \frac{1}{4} \sqrt{\frac{3}{5}}] 〉$ and $α_{2} = 〈 [0, \frac{1}{10} \sqrt{\frac{3}{2}}], [\frac{1}{10} \sqrt{\frac{7}{2}}, \frac{1}{5}] 〉$ , the desirable alternative is selected in accordance with their accuracy functions.

By applying the accuracy function defined in definition 2.8, we can obtain $\begin{matrix} H (α_{1}) = 0.0450 H (α_{2}) = 0.0450 \end{matrix}$ Hence, we do not compare these two alternative based on its accuracy function. On the other hand, if we apply Equation (7), we obtain $\begin{matrix} M (α_{1}) = - 0.9660 M (α_{2}) = - 0.9697 \end{matrix}$ Thus, the alternative α₁ is better than the alternative α₂.

Example 3.2. Let $α_{1} = 〈 [\frac{1}{2}, \frac{\sqrt{6}}{4}], [\frac{1}{2 \sqrt{2}}, \frac{1}{\sqrt{2}}] 〉$ and $α_{2} = 〈 [\frac{1}{8}, \frac{\sqrt{39}}{8}], [\frac{\sqrt{15}}{8}, \frac{5}{8}] 〉$ be two IVPFNs then according to Definition 1, we have H (α₁) =0.6250 and H (α₂) =0.6250, hence we are unable to select the best alternative among these. Now, if we apply Equation 7 on these two IVPFNs then we have M (α₂) = -0.2596 and M (α₂) = -0.1205. Therefore, the alternative α₂ is better than the alternative α₁.

From the above examples, we can see that the proposed function is reasonable in some cases. It should be noted that the proposed method for ranking interval-valued Pythagorean fuzzy values has the following properties.

Property 3.1. (Zero Property) If interval-valued Pythagorean fuzzy value α = 〈 [a, b] , [c, d] 〉 = 〈[0, 0] , [1, 1] 〉 then M (α) =0

Proof 3.1. If α = 〈[0, 0] , [1, 1] 〉 be an IVPFN then $M (α) = \frac{0 - \sqrt{1 - 0 - 1} + 0 - \sqrt{1 - 0 - 1}}{2} = 0$

Property 3.2. (One Property) If interval-valued Pythagorean fuzzy value α = 〈 [a, b] , [c, d] 〉 = 〈[1, 1] , [0, 0] 〉 then M (α) =1

Proof 3.2. If α = 〈[1, 1] , [0, 0] 〉 be an IVPFN then $M (α) = \frac{1 - \sqrt{1 - 1 - 0} + 1 - \sqrt{1 - 1 - 0}}{2} = 1$

Property 3.3. For a subset $α = 〈 [a, b], [\sqrt{1 - a^{2}}, \sqrt{1 - b^{2}}] 〉$ the proposed accuracy function is $M (α) = \frac{a^{2} + b^{2}}{2}$ . In particular, if a = b then M (α) = a².

Proof 3.3. As $α = 〈 [a, b], [\sqrt{1 - a^{2}}, \sqrt{1 - b^{2}}] 〉$ be an IVPFN then from Equation 7, we have $\begin{matrix} M (α) \\ = \frac{a^{2} - \sqrt{1 - a^{2} - (1 - a^{2})} + b^{2} - \sqrt{1 - b^{2} - (1 - b^{2})}}{2} \\ = \frac{a^{2} + b^{2}}{2} \end{matrix}$

4 Multi-criteria fuzzy decision making method based on the new accuracy function

Let A = {A₁, A₂, …, A_m} be a set of m alternatives which are evaluated under the set of different criteria G = {G₁, G₂, …, G_n} in which characteristic of each alternative is represented in the form of interval-valued Pythagorean fuzzynumber as $\begin{matrix} A_{i} = {〈 G_{j}, [μ_{A_{i}}^{L} (G_{j}), μ_{A_{i}}^{U} (G_{j})], \\ [ν_{A_{i}}^{L} (G_{j}), ν_{A_{i}}^{U} (G_{j})] 〉 ∣ G_{j} \in G} \end{matrix}$ where $0 \leq μ_{A_{i}}^{L} (G_{j}) \leq μ_{A_{i}}^{U} (G_{j}) \leq 1$ and $0 \leq ν_{A_{i}}^{L}$ $(G_{j}) \leq ν_{A_{i}}^{U} (G_{j}) \leq 1$ , for i = 1, 2, …, m andj = 1, 2, …, n with the condition that $(μ_{A_{i}}^{U})^{2} + (ν_{A_{i}}^{U})^{2} \leq 1$ . The IVPFN value is denoted by α_ij = 〈 [a_ij, b_ij] , [c_ij, d_ij] 〉, where [a_ij, b_ij] indicates the degree that the alternative A_i satisfies the criteria C_j given by the decision maker, [c_ij, d_ij] indicates the degree that the alternative A_i does not satisfies the criteria C_j given by the decision maker, [a_ij, b_ij] ⊂ [0, 1], [c_ij, d_ij] ⊂ [0, 1]. Therefore, we can elicit the decision matrix D = (α_ij) _m×n. Based on this decision matrix, an aggregating IVPFN α_i for A_i (i = 1, 2, …, n) is α_i = 〈 [a_i, b_i] , [c_i, d_i] 〉 = IPFWA_ω (α_i1, α_i2, …, α_in) or IPFWG_ω (α_i1, α_i2, …, α_in). Finally, utilize the Equation 7 to calculate the accuracy M (α_i) of interval-valued Pythagorean fuzzy number α_i (i = 1, 2, …, m) to rank the alternative A_i (i = 1, 2, …, m) and then to select the bestone(s).

In the nutshells, the procedure for computing the MCDM is summarized in the followingsteps.

(Step 1:) Construction of Pythagorean fuzzy decision-making matrix: Let A = {A₁, A₂, …, A_m} be a set of alternatives and G = {G₁, G₂, …, G_n} be set of attributes whose weights are known. Suppose that D_m×n (x_ij) = 〈 [a_ij, b_ij] , [c_ij, d_ij] 〉 be the intuitionistic fuzzy decision matrix, where [a_ij, b_ij] indicates the degree that the alternative A_i satisfies the attribute G_j given by the decision maker, [c_ij, d_ij] indicates the degree that the alternative A_i doesn’t satisfy the attribute G_j given by the decision maker, [a_ij, b_ij] ⊂ [0, 1], [c_ij, d_ij] ⊂ [0, 1], $b_{ij}^{2} + d_{ij}^{2} \leq 1$ , i = 1, 2, …, m ; j = 1, 2, …, n. Therefore, an interval-valued Pythagorean fuzzy decision matrix is expressed as

$\begin{matrix} D_{m \times n} (x_{ij}) \\ = [\begin{matrix} 〈 [a_{11}, b_{11}], [c_{11}, d_{11}] 〉 & 〈 [a_{12}, b_{12}], [c_{12}, d_{12}] 〉 & \dots & 〈 [a_{1 n}, b_{1 n}], [c_{1 n}, d_{1 n}] 〉 \\ 〈 [a_{21}, b_{21}], [c_{21}, d_{21}] 〉 & 〈 [a_{22}, b_{22}], [c_{22}, d_{22}] 〉 & \dots & 〈 [a_{2 n}, b_{2 n}], [c_{2 n}, d_{2 n}] 〉 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 〈 [a_{m 1}, b_{m 1}], [c_{m 1}, d_{m 1}] 〉 & 〈 [a_{m 2}, b_{m 2}], [c_{m 2}, d_{m 2}] 〉 & \dots & 〈 [a_{mn}, b_{mn}], [c_{mn}, d_{mn}] 〉 \end{matrix}] \end{matrix}$ (8)

(Step 2:) Obtain the normalized Pythagorean fuzzy decision matrix. Classify the attributes set G into two categories, namely as benefit and cost denoted by B and C respectively. If all the attributes are of the same type, then the rating values do not need normalization, whereas if there are benefit as well as cost attributes in MCDM, in such cases, we may transform the rating values of the benefit types into the rating values of the cost type by the following normalization formula: $r_{ij} = 〈 ξ_{ij}, φ_{ij} 〉 = {\begin{matrix} (α_{ij})^{c} & ; j \in B \\ α_{ij} & ; j \in C \end{matrix}$ (9) where $α_{ij}^{c}$ is the complement of α_ij. Hence, we obtain the normalized Pythagorean fuzzy decision matrix R = (r_ij) _m×n = 〈ξ_ij, φ_ij〉_m×n.

(Step 3:) Compute the overall assessments of alternatives. Utilize the IPFWA_ω or IPFWG_ω operator (given in Equation 3 and Equation 6) to aggregate all the rating values r_ij (j = 1, 2, …, n) of the i^th line and get the overall rating value r_i corresponding to the alternative A_i (i = 1, 2, …, m).

(Step 4:) Compute accuracy values: Calculate the accuracy value of the overall collective overall values α_i, i = 1, 2, …, m by using Equation (7).

(Step 5:) Ranking the alternative: Rank all the alternatives A_i (i = 1, 2, …, m) according to degree of accuracy function M (α_i) and thus select the most desirable alternative(s).

5 Illustrative example

The example for decision-making problem has been taken from [5, 7]. The aim of this problem is to make a decision for a panel who wants to invest the money in the four possible alternatives namely as car, food, company and arm company denoted by A₁, A₂, A₃ and A₄ respectively. The panel takes the decision according to the three criteria given by G₁ is risk analysis; G₂ is the growth analysis and G₃ is the environmental impact analysis. Assume that weight of G₁, G₂ and G₃ are 0.35, 0.25 and 0.40, respectively.

5.1 By proposed approach

We utilize the approach developed to get the most desirable alternative(s) as follows.

The information by the decision-maker under the above three criteria of these four possible alternative A_i, (i = 1, 2, 3, 4) are to be evaluated under the interval-valued intuitionistic Pythagorean fuzzy environment, as listed in the following decision matrix D_4×3 (x_ij). $\begin{matrix} D_{4 \times 3} (x_{ij}) \\ = [\begin{matrix} 〈 [0.4, 0.5], [0.3, 0.4] 〉 & 〈 [0.4, 0.6], [0.2, 0.4] 〉 & 〈 [0.1, 0.3], [0.5, 0.6] 〉 \\ 〈 [0.6, 0.7], [0.2, 0.3] 〉 & 〈 [0.6, 0.7], [0.2, 0.3] 〉 & 〈 [0.4, 0.7], [0.1, 0.2] 〉 \\ 〈 [0.3, 0.6], [0.3, 0.4] 〉 & 〈 [0.5, 0.6], [0.3, 0.4] 〉 & 〈 [0.5, 0.6], [0.1, 0.3] 〉 \\ 〈 [0.7, 0.8], [0.1, 0.2] 〉 & 〈 [0.6, 0.7], [0.1, 0.3] 〉 & 〈 [0.3, 0.4], [0.1, 0.2] 〉 \end{matrix}] \end{matrix}$

Since G₁, G₃ are cost criteria while G₂ is of benefit criteria, so normalized intuitionistic Pythagorean fuzzy decision matrix becomes $\begin{matrix} R_{4 \times 3} (r_{ij}) \\ = [\begin{matrix} 〈 [0.4, 0.5], [0.3, 0.4] 〉 & 〈 [0.2, 0.4], [0.4, 0.6] 〉 & 〈 [0.1, 0.3], [0.5, 0.6] 〉 \\ 〈 [0.6, 0.7], [0.2, 0.3] 〉 & 〈 [0.2, 0.3], [0.6, 0.7] 〉 & 〈 [0.4, 0.7], [0.1, 0.2] 〉 \\ 〈 [0.3, 0.6], [0.3, 0.4] 〉 & 〈 [0.3, 0.4], [0.5, 0.6] 〉 & 〈 [0.5, 0.6], [0.1, 0.3] 〉 \\ 〈 [0.7, 0.8], [0.1, 0.2] 〉 & 〈 [0.1, 0.3], [0.6, 0.7] 〉 & 〈 [0.3, 0.4], [0.1, 0.2] 〉 \end{matrix}] \end{matrix}$

Utilize the IPFWA_ω operator to aggregate all the performance values r_ij (j = 1, 2, 3) and get the overall performance value α_i corresponding to the alternative A_i as below $\begin{matrix} α_{1} = 〈 [0.2692, 0.4079], [0.3955, 0.5206] 〉; \\ α_{2} = 〈 [0.4586, 0.6408], [0.1995, 0.3153] 〉; \\ α_{3} = 〈 [0.3972, 0.5612], [0.2197, 0.3946] 〉; \\ α_{4} = 〈 [0.4910, 0.6024], [0.1565, 0.2736] 〉 \end{matrix}$

By applying Equation (7), we compute M (α_i) (i = 1, 2, 3, 4) as $\begin{matrix} M (α_{1}) = - 0.6946, M (α_{2}) = - 0.4725, \\ M (α_{3}) = - 0.5730, M (α_{4}) = - 0.5014 \end{matrix}$

Rank all alternatives in accordance with the accuracy degree of M (α_i) (i = 1, 2, 3, 4): A₂ ≻ A₄ ≻ A₃ ≻ A₁, and thus, the most desirable alternative is A₂.

5.2 By classical accuracy function

If we use the accuracy function as defined in Definition 2.8 then their corresponding values are $\begin{matrix} H (α_{1}) & = & 0.3332,; H (α_{2}) = 0.3800, \\ H (α_{3}) & = & 0.3383,; H (α_{4}) = 0.3517 \end{matrix}$ and hence the most desirable alternative is A₂.

5.3 By Ye [5] approach

If we apply Ye [5] approach for aggregating these different alternatives by using interval-valued intuitionistic fuzzy aggregating operator then we get $\begin{matrix} α_{1} & = & 〈 [0.2417, 0.4013], [0.3955, 0.5206] 〉, \\ α_{2} & = & 〈 [0.4406, 0.6292], [0.1995, 0.3153] 〉, \\ α_{3} & = & 〈 [0.3881, 0.5573], [0.2197, 0.3946] 〉, \\ α_{4} & = & 〈 [0.4459, 0.5755], [0.1565, 0.2736] 〉 \end{matrix}$

and hence by the accuracy function as proposed by [5], we have $\begin{matrix} M (α_{1}) & = & 0.1011,; M (α_{2}) = 0.3272, \\ M (α_{3}) & = & 0.2526,; M (α_{4}) = 0.2364 \end{matrix}$ Therefore, ranking of the alternative is A₂ ≻ A₄ ≻ A₃ ≻ A₁, and thus, the most desirable alternative is A₂.

5.4 By Chen et al. [6] approach

If we apply Chen et al. [6] approach on the considered data and the ranking value (RV) (For detail, we refer to [6]) corresponding to each alternative are obtained as follow $\begin{matrix} RV (α_{1}) & = & 0.5763,; RV (α_{2}) = 0.7220, \\ RV (α_{3}) & = & 0.6621,; RV (α_{4}) = 0.7075 \end{matrix}$ Therefore, ranking of the alternative is A₂ ≻ A₄ ≻ A₃ ≻ A₁, and thus, the most desirable alternative is A₂.

5.5 By Bai [7] approach

If we apply an improved score function, as proposed by Bai [7], for the effective ranking order of the intuitionistic interval-valued numbers approach on the considered data then we get a relative closeness coefficient (CC) corresponding to each alternative as follow $\begin{matrix} CC (A_{1}) & = & 0.4130,; CC (A_{2}) = 0.6983, \\ CC (A_{3}) & = & 0.6026,; CC (A_{4}) = 0.6532 \end{matrix}$ Therefore, the ranking order of the four alternatives is A₂ ≻ A₄ ≻ A₃ ≻ A₁ and hence the most desirable alternative is A₂.

6 Conclusion

In this paper, Pythagorean fuzzy sets have been extended to the interval-valued Pythagorean fuzzy sets and hence their corresponding operators have been defined to explore the multi-criteria decision making problems under an interval-valued Pythagorean fuzzy environment. For this, we introduce the two aggregation operators on it, namely, interval-valued Pythagorean fuzzy weighted averaging (IPFWA_ω) operator and interval-valued Pythagorean fuzzy weighted geometric (IPFWG_ω) operator for aggregating the different IVPFNs. A new accuracy function has been introduced to evaluate the degree of interval-valued Pythagorean fuzzy information. A multi-criteria decision making method based on the novel accuracy function has been established for IVPFSs by taking the account of degree of hesitation. Based on this function, a ranking of the alternative has been done for finding the most desirable alternative out of the given alternative under the given criteria. Finally, an illustrative example is proving to show the efficiency of the proposed approach. By comparison with the existing paper’s results with the proposed results, it has been obtained that the decision making method proposed in this paper is more stable and practical and also found that the proposed results coincides with the ones shown in existing approaches. Therefore, it has been concluded from the aforementioned results that the proposed decision making method can be suitably utilized to solve the multiple and decision making problems. In further research, we may apply these operators in the field of other domains such as pattern recognition, fuzzy cluster analysis and uncertain programming.

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