Continuous interval-valued Pythagorean fuzzy aggregation operators for multiple attribute group decision making

Abstract

The interval-valued Pythagorean fuzzy set (IVPFS), as a new generalization of Pythagorean fuzzy set, which can be used to model complex uncertainty in the real multiple attribute group decision making (MAGDM) problems. This paper aims to develop some new operators for dealing with MAGDM issue with interval-valued Pythagorean fuzzy information, such as continuous interval-valued Pythagorean fuzzy ordered weighted quadratic averaging (C-IVPFOWQA) operator, weighted C-IVPFOWQA operator, ordered weighted C-IVPFOWQA operator and hybrid C-IVPFOWQA operator. The proposed operators not only improve the accuracy of decision making but also can reflect the decision makers’ risk preferences. Additionally, some desirable properties and special cases of these operators are investigated in detail. Later, we present a MAGDM method based on the hybrid C-IVPFOWQA operator. Finally, a numerical example is presented to illustrate the proposed method and comparison analysis is given to demonstrate the practicality and effectiveness.

Keywords

Continuous interval-valued Pythagorean fuzzy ordered weighted quadratic averaging (C-IVPFOWQA) operator weighted C-IVPFOWQA operator ordered weighted C-IVPFOWQA operator hybrid C-IVPFOWQA operator multiple attribute group decision making

1. Introduction

Atanassov [1] introduced the concept of intuitionistic fuzzy set (IFS), which is a successfully generalization of the concept of fuzzy set (FS) [2]. IFS is characterized by a membership degree and a nonmembership degree satisfying the condition that the sum of these two degrees is equal to or less than 1. Therefore, it can express the fuzzy character of data more comprehensively and specifically. Since IFS was first introduced in 1986, it has received widely attentions and has been successfully applied to the multiple attribute decision making (MADM) problems or multiple attribute group decision making (MAGDM) problems [3 –22]. Considering various types of score functions, Chen [23] gave a comparative analysis of score functions for MADM problems in intuitionistic fuzzy environment. Li et al. [24] developed a linear programming method for MAGDM problems with intuitionistic fuzzy number (IFS), where the information about attribute weights is unknown, and the group consistency and inconsistency indices, generalizations or specializations of the linear programming model are investigated. Xu and Yager [25] proposed dynamic intuitionistic fuzzy weighted averaging (DIFWA) operator and uncertain dynamic intuitionistic fuzzy weighted averaging (UDIFWA) operator for the intuitionistic fuzzy MADM problems at different period. Tan and Chen [26] introduced two new operational laws on IFSs and proposed intuitionistic fuzzy Choquet integral operator.

However, sometimes the sum of the membership degree and the nonmembership degree provided by the decision maker (DM) may be greater than 1,for instances, if DM gives the membership degree about the support to an alternative is 0.7 and his against membership degree is 0.5, it can be seen that 0.7 + 0.5 > 1 and 0 . 7² + 0 .5² < 1, so, IFS is invalid to handle this decision information. To overcome this shortcoming, Yager [27, 28] developed the concept of Pythagorean fuzzy set (PFS), which is an effective extension of IFS. Similar to the IFS, the PFS is also characterized by a membership degree and a nonmembership degree satisfying the condition that the square sum of these two degrees is equal to or less than 1, while the sum of these two degrees can be more than 1. Obviously, the PFS is more capable than IFS to represent and model complex uncertainty in the real decision making problems. Zhang and Xu [29] introduced the concept of Pythagorean fuzzy number (PFN) and extended the technique for order preference by similarity to ideal solution (TOPSIS) method to handle with PFN in MADM. Peng and Yang [30] developed a Pythagorean fuzzy superiority and inferiority ranking method for MAGDM problems. Zeng et al. [31] proposed a hybrid method for Pythagorean fuzzy MAGDM problems. Garg [32] proposed a correlation coefficient for PFSs and applied to decision making problems. Gou et al. [33] investigated the continuity, derivability and differentiability of PFNs. Based on the entropy theory, Xue et al. [34] proposed a linear programming technique for multidimensional analysis of preference (LINMAP) method to handle the railway project investment decision making with PFSs. Yager [27] developed the Pythagorean fuzzy weighted averaging (PFWA) operator and Pythagorean fuzzy weighted geometric (PFWG) operator, and utilized these to handle MADM problems. Ma and Xu [35] proposed the symmetric Pythagorean fuzzy weighted geometric/averaging operators. Garg [36] developed Pythagorean fuzzy Einstein weighted averaging (PFEWA), Pythagorean fuzzy Einstein ordered weighted averaging (PFEOWA), generalized Pythagorean fuzzy Einstein weighted averaging (GPFEWA) and generalized Pythagorean fuzzy Einstein ordered weighted averaging (GPFEOWA) operators, and applied them to Pythagorean fuzzy MADM problems. Zhang [37] presented new PFWA operator and Pythagorean fuzzy ordered weighted averaging (PFOWA) operator to aggregate PFSs. Peng and Yang [38] developed a Pythagorean fuzzy Choquet integral operator in MAGDM. Wei and Lu [39, 40] presented some Pythagorean fuzzy Maclaurin symmetric mean operators and Pythagorean fuzzy power aggregation operators in MADM, respectively. Additionally, Zeng et al. [41] proposed the Pythagorean fuzzy induced ordered weighted averaging weighted average (PFIOWAWA) operator and applied it to solve MAGDM problems with Pythagorean fuzzy information.

After the introduction of PFS, Zhang [42] extended the concept of PFS to interval-valued Pythagorean fuzzy set (IVPFS) whose membership degree and nonmembership degree can be represented by an interval value within [0, 1] instead of a crisp value. Since IVPFS appears, it has received increasing attention in MADM or MAGDM. For instance, Peng and Yang [43] proposed an interval-valued Pythagorean fuzzy elimination and choice translating reality (ELECTRE) method for interval-valued Pythagorean fuzzy MAGDM problems, meanwhile, some aggregation operators were developed, such as interval-valued Pythagorean fuzzy weighted average (IVPFWA) operator, interval-valued Pythagorean fuzzy weighted geometric (IVPFWG) operator, and interval-valued Pythagorean fuzzy point weighted averaging (IVPFPWA) operator. Liang [44] proposed a maximizing deviation method to handle the MAGDM problems with interval-valued Pythagorean fuzzy information. Garg [45] developed a novel accuracy function for ranking the interval-valued Pythagorean fuzzy numbers (IVPFNs), and an improved accuracy function was also presented by Garg [46]. Chen [47] developed a closeness-based assignment method for MADM problems with interval-valued Pythagorean fuzzy information. Based on the basic operational laws of IVPFNs [42], another type of interval-valued Pythagorean fuzzy weighted average (IVPF-WAA) operator was developed by Liang et al. [44], and another type of interval-valued Pythagorean fuzzy weighted geometric (IPFWG) operator was developed by Garg [45]. Based on the Bonferroni mean, Liang et al. [48] introduced interval-valued Pythagorean fuzzy extended Bonferroni mean (IVPFEBM), weighted interval-valued Pythagorean fuzzy extended Bonferroni mean (WIVPFEBM) operators, and applied WIVPFEBM operator to solve MADM problems. Garg [49] introduced new exponential operational laws for IVPFSs and presented some new weighted aggregation operators.

In the process of decision making, the aggregation operators is an interesting and important research topic. A variety of operators have been proposed in the past decades, among them the ordered weighted averaging (OWA) operator [50] is the well known one, which aggregates multi-dimension crisp evaluation values into collective crisp values. Based on the OWA operator, Merigó and Casanovas [51] introduced generalized OWA operator. In order to aggregate the continuous interval value, Yager [52] developed the continuous ordered weighted averaging (C-OWA) operator, which is an important extension of the OWA operator. The main advantage of the C-OWA operator is that it can aggregate interval value into different crisp value according to the attitudinal character of BUM function Q. Based on the C-OWA operator, some extensions are developed, for instance, the continuous ordered weighted geometric (C-OWG) operator [53], the continuous generalized OWA (C-GOWA) operator [54], the continuous Quasi-OWA operator [55]. Additionally, Zhou et al. [56] developed the continuous interval-valued intuitionistic fuzzy ordered weighted averaging (C-IVIFOWA) operator to handle MAGDM problems with interval-valued intuitionistic fuzzy information. Lin and Zhang [57] revised the flaws of C-IVIFOWA operator. Peng et al. [58] introduced the continuous hesitant fuzzy averaging/geometric aggregation operators, and applied them to solve MADM problems. Continuous ordered weighted quadratic averaging (C-OWQA) operator was developed by Liu et al. [55], which is a new form of the extension of OWA operator and the special cases of continuous Quasi-OWA operator.

From above analysis, we can see that the existing interval-valued Pythagorean fuzzy aggregation operators only focus on the endpoints of the closed interval, while all the values contained in IVPFSs are not full considered. Meanwhile, the C-OWQA operator can aggregate interval values into different crisp value. So it’s very necessary to utilize the C-OWQA operator to handle interval-valued Pythagorean fuzzy information. Therefore, in this paper, based on the C-OWQA operator, we developed the continuous interval-valued Pythagorean fuzzy aggregation operators and applied them to solve MAGDM problems.

The remainder of the paper is organized as follows: In Section 2, we briefly review some basic concepts of IFS, PFS, IVPFS, OWA operator, Pythagorean fuzzy OWA operator, C-OWA operator and C-OWQA operator. In Section 3, we develop the continuous interval-valued Pythagorean fuzzy ordered weighted quadratic averaging (C-IVPFOWQA) operator, new score function and accuracy function of the IVPFSs are defined based on the C-IVPFOWQA operator, meanwhile, some properties of the C-IVPFOWQA operator are investigated, such as, boundedness, identity and monotonicity. Some extended C-IVPFOWQA operators, such as the weighted C-IVPFOWQA (WC-IVPFOWQA) operator, the ordered weighted C-IVPFOWQA (OWC-IVPFOWQA) operator and the hybrid C-IVPFOWQA (HC-IVPFOWQA) operator are developed in Section 4, additionally, their desirable properties and special cases are investigated. In Section 5, we develop a method for solving the MAGDM problems using the HC-IVPFOWQA operator. In Section 6, a numerical example is given to demonstrate the applicability and effectiveness of the proposed methods and compare it with some of the existing operators. The paper is concluded in Section 7.

2. Preliminaries

Basic concepts of IFS [1], PFS [27, 28], IVPFS [42, 43], The OWA operator [50], the PFOWA operator [37], the C-OWA operator [52] and the C-OWQA operator [55] are briefly reviewed in this section.

2.1 Pythagorean fuzzy set

Definition 1. [1] Let X be a fixed set. An IFS A is defined as: $A = {〈 x, (μ_{A} (x), ν_{A} (x)) 〉 | x \in X} .$ (1) where the functions μ_A (x) , ν_A (x) : X → [0, 1] satisfy the condition: (μ_A (x) + ν_A (x)) ⊆ [0, 1] . μ_A (x) , v_A (x) denote, respectively, the degree of membership and the degree of non-membership of the element x ∈ X to the set A. The degree of indeterminacy π_A (x) =1 - μ_A (x) - ν_A (x).

Definition 2. [27, 28]. Let X be a fixed set. A PFS P is defined as: $P = {〈 x, (μ_{P} (x), ν_{P} (x)) 〉 | x \in X} .$ (2) where the functions μ_P (x) , ν_P (x) : X → [0, 1] satisfy the condition: $0 \leq μ_{P}^{2} (x) + ν_{P}^{2} (x) \leq 1 .$ μ_P (x) , v_P (x) denote, respectively, the degree of membership and the degree of non-membership of the element x ∈ X to the set P. The degree of indeterminacy $π_{P} (x) = \sqrt{1 - μ_{P}^{2} (x) - ν_{P}^{2} (x)}$ .

For convenience, Zhang and Xu [29] defined α = (μ_α, ν_α) as the Pythagorean fuzzy number (PFN), where μ_α ∈ [0, 1] , ν_α ∈ [0, 1] and $0 \leq μ_{α}^{2} + ν_{α}^{2} \leq 1$ . Some operational laws for PFNs were defined as follows:

Definition 3. [28, 29]. Let α = (μ_α, ν_α) , α₁ = (μ_{α
₁}, ν_{α
₁}) and α₂ = (μ_{α
₂}, ν_{α
₂}) be three PFNs, then $\begin{matrix} (1) α^{c} = (ν_{α}, μ_{α}); \\ (2) α_{1} \oplus α_{2} = (\sqrt{μ_{α_{1}}^{2} + μ_{α_{2}}^{2} - μ_{α_{1}}^{2} μ_{α_{2}}^{2}}, ν_{α_{1}} ν_{α_{2}}); \\ (3) α_{1} \otimes α_{2} = (μ_{α_{1}} μ_{α_{2}}, \sqrt{ν_{α_{1}}^{2} + ν_{α_{2}}^{2} - ν_{α_{1}}^{2} ν_{α_{2}}^{2}}); \\ (4) λ α = (\sqrt{1 - (1 - μ_{α}^{2})^{λ}}, (ν_{α})^{λ}), λ > 0; \\ (5) α^{λ} = ((μ_{α})^{λ}, \sqrt{1 - (1 - ν_{α}^{2})^{λ}}), λ > 0 . \end{matrix}$

Zhang and Xu [29] also defined the following laws of comparison.

Definition 4. Let α = (μ_α, ν_α) be a PFN, $s (α) = μ_{α}^{2} - ν_{α}^{2}$ and $h (α) = μ_{α}^{2} + ν_{α}^{2}$ denote the score and the accuracy degree of α, respectively. For two PFNs α₁ = (μ_{α
₁}, ν_{α
₁}) , α₂ = (μ_{α
₂}, ν_{α
₂}),

If s (α₁) > s (α₂), then α₁ > α₂;

If s (α₁) = s (α₂), then:

If h (α₁) > h (α₂), then α₁ > α₂;

If h (α₁) = h (α₂), then α₁ = α₂.

2.2 Interval-valued Pythagorean Fuzzy Set

Definition 5. [42, 43]. Let X be a fixed set. An IVPFS $\tilde{P}$ can be defined as: $\tilde{P} = {〈 x, (μ_{\tilde{P}} (x), ν_{\tilde{P}} (x)) 〉 | x \in X} .$ (3) where the functions $μ_{\tilde{P}} (x), ν_{\tilde{P}} (x) \subseteq [0, 1]$ satisfy $μ_{\tilde{P}} (x) = [μ_{\tilde{P}}^{L} (x), μ_{\tilde{P}}^{U} (x)],$ $ν_{\tilde{P}} (x) = [ν_{\tilde{P}}^{L} (x), ν_{\tilde{P}}^{U} (x)]$ , ${(μ_{\tilde{P}}^{U} (x))}^{2} + {(ν_{\tilde{P}}^{U} (x))}^{2} \leq 1 .$ The degree of indeterminacy, $π_{\tilde{P}} (x) = [π_{\tilde{P}}^{L} (x), π_{\tilde{P}}^{U} (x)]$ , where $π_{\tilde{P}}^{L} (x) = \sqrt{1 - {(μ_{\tilde{P}}^{U} (x))}^{2} - {(ν_{\tilde{P}}^{U} (x))}^{2}}$ and $π_{\tilde{P}}^{U} (x) =$ $\sqrt{1 - {(μ_{\tilde{P}}^{L} (x))}^{2} - {(ν_{\tilde{P}}^{L} (x))}^{2}} .$ If $μ_{\tilde{P}}^{L} (x) = μ_{\tilde{P}}^{U} (x)$ and $ν_{\tilde{P}}^{L} (x) = ν_{\tilde{P}}^{U} (x)$ , then the IVPFS reduces to the PFS.

Zhang [42] defined $\tilde{p} = (μ_{\tilde{p}}$ , $ν_{\tilde{p}}) = ([μ_{p}^{L}$ , $μ_{p}^{U}]$ , $[ν_{p}^{L}$ , $ν_{p}^{U}])$ as the interval-valued Pythagorean fuzzy number (IVPFN), where ${(μ_{p}^{U})}^{2} + {(ν_{p}^{U})}^{2} \leq 1 .$ For convenience, the pair $([μ_{p}^{L}, μ_{p}^{U}], [ν_{p}^{L}, ν_{p}^{U}])$ is often denoted by ([a, b] , [c, d]), where [a, b] ⊆ [0, 1] , [c, d] ⊆ [0, 1], and 0 ≤ b² + d² ≤ 1.

Definition 6. [42] Let $\tilde{α} = ([a, b], [c, d]), {\tilde{α}}_{1} = ([a_{1}, b_{1}], [c_{1}, d_{1}])$ and ${\tilde{α}}_{2} = ([a_{2}, b_{2}], [c_{2}, d_{2}])$ be three IVPFNs, then: $\begin{matrix} (1) {\tilde{α}}^{c} = ([c, d], [a, b]); \\ (2) {\tilde{α}}_{1} \oplus {\tilde{α}}_{2} = ([\sqrt{a_{1}^{2} + a_{2}^{2} - a_{1}^{2} a_{2}^{2}}, \\ \sqrt{b_{1}^{2} + b_{2}^{2} - b_{1}^{2} b_{2}^{2}}], [c_{1} c_{2}, d_{1} d_{2}]); \\ (3) {\tilde{α}}_{1} \otimes {\tilde{α}}_{2} = ([a_{1} a_{2}, b_{1} b_{2}], [\sqrt{c_{1}^{2} + c_{2}^{2} - c_{1}^{2} c_{2}^{2}}, \\ \sqrt{d_{1}^{2} + d_{2}^{2} - d_{1}^{2} d_{2}^{2}}]); \\ (4) λ \tilde{α} = ([\sqrt{1 - (1 - a^{2})^{λ}}, \sqrt{1 - (1 - b^{2})^{λ}}], \\ [c^{λ}, d^{λ}]), λ > 0; \\ (5) {\tilde{α}}^{λ} = ([a^{λ}, b^{λ}], [\sqrt{1 - (1 - c^{2})^{λ}}, \\ \sqrt{1 - (1 - d^{2})^{λ}}]), λ > 0 . \end{matrix}$ Definition 7. [42, 43] Let $\tilde{α} = ([a, b], [c, d])$ be an IVPFN, $S (\tilde{α}) = (a^{2} + b^{2} - c^{2} - d^{2}]) / 2$ and $H (\tilde{α}) = (a^{2} + b^{2} + c^{2} + d^{2}]) / 2$ denote the score and the ccuracy degree of $\tilde{α}$ , respectively. For two IVPFNs ${\tilde{α}}_{1} = ([a_{1}, b_{1}], [c_{1}, d_{1}]), {\tilde{α}}_{2} = ([a_{2}, b_{2}], [c_{2}, d_{2}])$ , the following comparison laws can be given:

If $S ({\tilde{α}}_{1}) > S ({\tilde{α}}_{2})$ , then ${\tilde{α}}_{1} > {\tilde{α}}_{2}$ ;

If $S ({\tilde{α}}_{1}) = S ({\tilde{α}}_{2})$ , then:

If $H ({\tilde{α}}_{1}) > H ({\tilde{α}}_{2})$ , then ${\tilde{α}}_{1} > {\tilde{α}}_{2}$ ;

If $H ({\tilde{α}}_{1}) = H ({\tilde{α}}_{2})$ , then ${\tilde{α}}_{1} = {\tilde{α}}_{2}$ .

An improved score function for IVPFNs was developed by Garg [59] and Garg [60].

For simplicity, we denote that Θ is the set of all Pythagorean fuzzy values and that Ω is the set of all interval-valued Pythagorean fuzzy values.

2.3 The OWA operator

Ordered weighted averaging (OWA) operator was introduced by Yager [50], which can be defined as follows:

Definition 8. An OWA operator of dimension n is a mapping OWA : Rⁿ → R that has an associated weighting ω = (ω₁, ω₂, …, ω_n) ^T with ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ such that $OWA (a_{1}, a_{2}, . . ., a_{n}) = \sum_{j = 1}^{n} ω_{j} b_{j} .$ (4) where b_j is jth largest of the arguments a_i and (a₁, a₂, . . . , a_n) is a finite collection of argument.

The OWA operator is bounded, idempotent, commutative and monotonic [50].

2.4 The Pythagorean fuzzy OWA operator

Based on the operation of PFNs and the OWA operator, Zhang [37] introduced the Pythagorean fuzzy ordered weighted averaging (PFOWA) operator as follows:

Definition 9. [37] Let α_j = (μ_{α
_j}, ν_{α
_j}) (j = 1, 2, ⋯, n) be a collection of PFNs, a PFOWA operator of dimension n is a mapping PFOWA : Θⁿ → Θ that has an associated weighting ω = (ω₁, ω₂, …, ω_n) ^T with ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ such that $\begin{matrix} (α_{1}, α_{2}, \dots, α_{n}) = \oplus_{j = 1}^{n} ω_{j} α_{σ (j)} = \\ (\sqrt{1 - \prod_{j = 1}^{n} (1 - μ_{α_{σ (j)}}^{2})^{ω_{j}}}, \prod_{j = 1}^{n} ν_{α_{σ (j)}}^{ω_{j}}) . \end{matrix}$ (5) where (σ (1) , σ (2) , …, σ (n)) is a permutation of (1, 2, …, n) such that α_σ(j-1) ≥ α_σ(j) for all j.

2.5 The C-OWA operator

A continuous ordered weighted averaging (C-OWA) operator was developed by Yager [52], which is the generalization of the OWA operator, the C-OWA operator can be defined as follows:

Definition 10. A C-OWA operator is a mapping φ : K → R⁺ associated with a BUM function Q such that $φ_{Q} ([a, b]) = \int_{0}^{1} [b + y (b - a)] dQ (y) .$ (6) where [a, b] ∈ K, K is the set of all nonnegative interval numbers, and the basic unit interval monotonic (BUM) function Q satisfies Q (0) =0, Q (1) =1 and Q (x) ≥ Q (y) for any x ≥ y.

2.6 The C-OWQA operator

Based on the Definition 10, Liu et al. [55] developed continuous ordered weighted quadratic averaging(C-OWQA) operator, which can be defined as follows:

Definition 11. A continuous ordered weighted quadratic averaging(C-OWQA) operator f : K → R⁺ associated with a BUM function Q such that $f_{Q} ([a, b]) = {(\int_{0}^{1} [b^{2} + y (b^{2} - a^{2})] dQ (y))}^{\frac{1}{2}} .$ (7) where the concepts K and Q are the same as in Definition 10.

If $λ = \int_{0}^{1} Q (y) dy$ is the attitudinal character of Q, then f_Q ([a, b]) can be expressed as $f_{Q} ([a, b]) = \sqrt{(1 - λ) a^{2} + λ b^{2}} .$ (8)

For simplicity, f_Q is denoted by f_λ.

3. Continuous interval-valued Pythagorean fuzzy ordered weighted quadratic averaging operator

In the following, we will introduce the continuous interval-valued Pythagorean fuzzy ordered weighted quadratic averaging (C-IVPFOWQA) operator based on the operations of IVPFNs and C-OWQA operator.

Definition 12. A C-IVPFOWQA operator is a mapping g : Ω → Θ, which has associated with it a BUM function Q : [0, 1] → [0, 1] having the properties: (1) Q (0) =0; (2) Q (1) =1; and (3) Q (x) ≥ Q (y) for any x ≥ y, such that $g_{Q} (\tilde{α}) = (μ_{g_{Q} (\tilde{α})}, ν_{g_{Q} (\tilde{α})}) = (f_{λ} [a, b], f_{1 - λ} [c, d]) .$ (9) where $\tilde{α} = (μ_{\tilde{α}}, ν_{\tilde{α}}) = ([a, b], [c, d]) \in Ω$ .

For convenience, g_Q is also denoted by g_λ. Based on Equation (9) and Definition 11, we can drive the following result:

Theorem 1. Let $\tilde{α} = (μ_{\tilde{α}}, ν_{\tilde{α}}) = ([a, b], [c, d])$ be an IVPFN, if λ is the attitudinal character of Q, then the aggregated value by using Equation (9) is a PFN and $g_{λ} (\tilde{α}) = (\sqrt{(1 - λ) a^{2} + λ b^{2}}, \sqrt{λ c^{2} + (1 - λ) d^{2}}) .$ (10)

Proof. Since $\tilde{α} = ([a, b], [c, d])$ is an IVPFN, by Definition of IVPFN, we have 0 ≤ a, b, c, d ≤ 1, a ≤ b, c ≤ d and 0 ≤ b² + d² ≤ 1. Therefore, for a given λ, we obtain $0 \leq \sqrt{(1 - λ) a^{2} + λ b^{2}} \leq 1$ , $0 \leq \sqrt{λ c^{2} + (1 - λ) d^{2}} \leq 1$ ,

Moreover, $\begin{matrix} {(\sqrt{(1 - λ) a^{2} + λ b^{2}})}^{2} + {(\sqrt{λ c^{2} + (1 - λ) d^{2}})}^{2} \\ = (1 - λ) a^{2} + λ b^{2} + λ c^{2} + (1 - λ) d^{2} \\ \leq (1 - λ) b^{2} + λ b^{2} + λ d^{2} + (1 - λ) d^{2} \\ = b^{2} + d^{2} \leq 1 . \end{matrix}$

Hence, the theorem is proved □.

Example 1. Let $\tilde{α} = ([0.6, 0.8], [0.3, 0.4])$ and Q (y) = y³, then $λ = \int_{0}^{1} y^{3} dy = 1 / 4$ . By Equation (10), we get $\begin{matrix} g_{λ} (\tilde{α}) = (\sqrt{(1 - 1 / 4) \times 0 . 6^{2} + (1 / 4) \times 0 . 8^{2}}, \\ \sqrt{(1 / 4) \times 0 . 3^{2} + (1 - 1 / 4) \times 0 . 4^{2}}) \\ = (0 . 6557, 0 . 3775) . \end{matrix}$

In order to compare any two IVPFNs, we also can introduce the new concepts of score function and accuracy function based on the C-IVPFOWQA operator, which are defined as follows:

Definition 13. Let $\tilde{α} = (μ_{\tilde{α}}, ν_{\tilde{α}}) = ([a, b], [c, d])$ be an IVPFN, for a given λ $\begin{matrix} S_{λ} (\tilde{α}) = μ_{g_{λ} (\tilde{α})}^{2} - ν_{g_{1 - λ} (\tilde{α})}^{2} = (f_{λ} ([a, b]))^{2} \\ - (f_{1 - λ} ([c, d]))^{2}, \end{matrix}$ (11) and $\begin{matrix} H_{λ} (\tilde{α}) = μ_{g_{λ} (\tilde{α})}^{2} + ν_{g_{1 - λ} (\tilde{α})}^{2} = (f_{λ} ([a, b]))^{2} \\ + (f_{1 - λ} ([c, d]))^{2} . \end{matrix}$ (12) are called, respectively, the score function and the accuracy function of an IVPFN $\tilde{α}$ with respect to λ, where λ is the attitudinal character of BUM function Q.

Specially, if λ = 1/2, then we get $\begin{matrix} S_{λ} (\tilde{α}) = (f_{λ} ([a, b]))^{2} - (f_{1 - λ} ([c, d]))^{2} \\ = (a^{2} + b^{2} - c^{2} - d^{2}) / 2, \end{matrix}$ (13) and $\begin{matrix} H_{λ} (\tilde{α}) = (f_{λ} ([a, b]))^{2} + (f_{1 - λ} ([c, d]))^{2} \\ = (a^{2} + b^{2} + c^{2} + d^{2}) / 2 . \end{matrix}$ (14)

It can be seen that the $S_{λ} (\tilde{α})$ in Equation (13) and $H_{λ} (\tilde{α})$ in Equation (14) reduce to the score function and accuracy function, respectively, developed by Peng and Yang [43].

Based on the attitudinal character λ, we can propose a method for the comparison between two IVPFNs, which can be defined as follows:

Definition 14. Let ${\tilde{α}}_{1} = ([a_{1}, b_{1}], [c_{1}, d_{1}])$ and ${\tilde{α}}_{2} = ([a_{2}, b_{2}], [c_{2}, d_{2}])$ be two IVPFNs, $S_{λ} ({\tilde{α}}_{1})$ and $S_{λ} ({\tilde{α}}_{2})$ be the scores of ${\tilde{α}}_{1}$ and ${\tilde{α}}_{2}$ with respect to λ, respectively, and let $H_{λ} ({\tilde{α}}_{1})$ and $H_{λ} ({\tilde{α}}_{2})$ be the accuracy degrees of ${\tilde{α}}_{1}$ and ${\tilde{α}}_{2}$ with respect to λ, respectively, then if $S_{λ} ({\tilde{α}}_{1}) < S_{λ} ({\tilde{α}}_{2})$ , then ${\tilde{α}}_{1} < {\tilde{α}}_{2}$ ; if $S_{λ} ({\tilde{α}}_{1}) = S_{λ} ({\tilde{α}}_{2})$ , then (1) If $H_{λ} ({\tilde{α}}_{1}) = H_{λ} ({\tilde{α}}_{2})$ , then ${\tilde{α}}_{1} = {\tilde{α}}_{2}$ , with respect to λ; (2) If $H_{λ} ({\tilde{α}}_{1}) < H_{λ} ({\tilde{α}}_{2})$ , then ${\tilde{α}}_{1} < {\tilde{α}}_{2}$ , with respect to λ.

If λ is the attitudinal character of Q, it satisfies 0 ≤ λ ≤ 1, then we can drive the following result:

Theorem 2. (Boundedness) Let $\tilde{α} = (μ_{\tilde{α}}, ν_{\tilde{α}}) = ([a, b], [c, d])$ be an IVPFN, and let g_λ be the C-IVPFOWQA operator and λ be the attitudinal character of BUM function Q, then for a given λ, $(a, d) \leq g_{λ} (\tilde{α}) \leq (b, c) .$ (15)

Proof. From Equation (10), $g_{λ} (\tilde{α}) = (\sqrt{(1 - λ) a^{2} + λ b^{2}}, \sqrt{λ c^{2} + (1 - λ) d^{2}})$ , since 0 ≤ a ≤ b ≤ 1, 0 ≤ c ≤ d ≤ 1, so. $\begin{matrix} S (g_{λ} (\tilde{α})) = {(\sqrt{(1 - λ) a^{2} + λ b^{2}})}^{2} \\ - {(\sqrt{λ c^{2} + (1 - λ) d^{2}})}^{2} \\ = (1 - λ) a^{2} + λ b^{2} - λ c^{2} - (1 - λ) d^{2} \\ \leq (1 - λ) b^{2} + λ b^{2} - λ c^{2} - (1 - λ) c^{2} \\ = b^{2} - c^{2} = S ((b, c)) . \end{matrix}$ and $\begin{matrix} S (g_{λ} (\tilde{α})) = {(\sqrt{(1 - λ) a^{2} + λ b^{2}})}^{2} \\ - {(\sqrt{λ c^{2} + (1 - λ) d^{2}})}^{2} \\ = (1 - λ) a^{2} + λ b^{2} - λ c^{2} - (1 - λ) d^{2} \\ \geq (1 - λ) a^{2} + λ a^{2} - {dc}^{2} - (1 - λ) d^{2} \\ = a^{2} - d^{2} = S ((a, d)) . \end{matrix}$

Thus, the proof is completed.

Theorem 3. (Identity) Let $\tilde{α}$ = $(μ_{\tilde{α}}$ , $ν_{\tilde{α}})$ = ([a, b] , [c, d]) be an IVPFN, and let g_λ be the C-IVPFOWQA operator and λ be the attitudinal character of BUM function Q. If a = b = μ and c = d = ν, then for a given λ, $g_{λ} (([a, b], [c, d])) = (μ, ν) .$ (16)

Proof. Since a = b = μ and c = d = ν, then we obtain $\begin{matrix} g_{λ} (([a, b], [c, d])) = (\sqrt{(1 - λ) a^{2} + λ b^{2}}, \\ \sqrt{λ c^{2} + (1 - λ) d^{2}}) \\ = (\sqrt{(1 - λ) μ^{2} + λ μ^{2}}, \sqrt{λ ν^{2} + (1 - λ) ν^{2}}) \\ = (μ, ν) . \end{matrix}$

Therefore, the proof is completed.

Theorem 4. (Monotonicity with respect to IVPFN $\tilde{α}$ ) Let ${\tilde{α}}_{1} = (μ_{{\tilde{α}}_{1}}, ν_{{\tilde{α}}_{1}}) = ([a_{1}, b_{1}], [c_{1}, d_{1}])$ and ${\tilde{α}}_{2} = (μ_{{\tilde{α}}_{2}}, ν_{{\tilde{α}}_{2}}) = ([a_{2}, b_{2}], [c_{2}, d_{2}])$ be two IVPFNs, and let g_λ be the C-IVPFOWQA operator and λ be the attitudinal character of BUM function Q. If ${\tilde{α}}_{1} < {\tilde{α}}_{2}$ with respect to λ, then $g_{λ} ({\tilde{α}}_{1}) < g_{λ} ({\tilde{α}}_{2}) .$ (17)

Proof. For a given attitudinal character λ, the proof can be divided into two cases:

Case 1: If $S_{λ} ({\tilde{α}}_{1})$ < $S_{λ} ({\tilde{α}}_{2})$ , from Equation (11), we have $(f_{λ} (μ_{{\tilde{α}}_{1}}))^{2}$ - $(f_{1 - λ} (ν_{{\tilde{α}}_{1}}))^{2}$ < $(f_{λ} (μ_{{\tilde{α}}_{2}}))^{2}$ - $(f_{1 - λ} (ν_{{\tilde{α}}_{2}}))^{2}$ , because $g_{λ} ({\tilde{α}}_{1})$ = $(f_{λ} (μ_{{\tilde{α}}_{1}})$ , $f_{1 - λ} (ν_{{\tilde{α}}_{1}}))$ and $g_{λ} ({\tilde{α}}_{2})$ = $(f_{λ} (μ_{{\tilde{α}}_{2}})$ , $f_{1 - λ} (ν_{{\tilde{α}}_{2}}))$ , by Definition 14, we obtain $g_{λ} ({\tilde{α}}_{1})$ < $g_{λ} ({\tilde{α}}_{2})$ ;

Case 2: If $S_{λ} (\tilde{α}) = S_{λ} (\tilde{β})$ but $H_{λ} (\tilde{α}) < H_{λ} (\tilde{β})$ , from Definition 13, we have $(f_{λ} (μ_{{\tilde{α}}_{1}}))^{2} - (f_{1 - λ} (ν_{{\tilde{α}}_{1}}))^{2} = (f_{λ} (μ_{{\tilde{α}}_{2}}))^{2} - (f_{1 - λ} (ν_{{\tilde{α}}_{2}}))^{2}$ and $(f_{λ} (μ_{{\tilde{α}}_{1}}))^{2} + (f_{1 - λ} (ν_{{\tilde{α}}_{1}}))^{2} < (f_{λ} (μ_{{\tilde{α}}_{2}}))^{2} + (f_{1 - λ} (ν_{{\tilde{α}}_{2}}))^{2}$ By Definition 14, we obtain $g_{λ} ({\tilde{α}}_{1}) < g_{λ} ({\tilde{α}}_{2})$ .

According to Case 1 and Case 2, the proof is completed.

Theorem 5. (Monotonicity with respect to attitudinal character λ) Let $\tilde{α} = (μ_{\tilde{α}}, ν_{\tilde{α}}) = ([a, b], [c, d])$ be an IVPFN and g_λ be the C-IVPFOWQA operator, and let $λ_{1} = \int_{0}^{1} Q_{1} (y) dy$ and $λ_{2} = \int_{0}^{1} Q_{2} (y) dy$ , if λ₁ ≤ λ₂, then $g_{λ_{1}} (\tilde{α}) \leq g_{λ_{2}} (\tilde{α})$ (18)

Proof. Since $\begin{matrix} g_{λ_{1}} (\tilde{α}) = (\sqrt{(1 - λ_{1}) a^{2} + λ_{1} b^{2}}, \\ \sqrt{λ_{1} c^{2} + (1 - λ_{1}) d^{2}}), \end{matrix}$ $\begin{matrix} g_{λ_{2}} (\tilde{α}) = (\sqrt{(1 - λ_{2}) a^{2} + λ_{2} b^{2}}, \\ \sqrt{λ_{2} c^{2} + (1 - λ_{2}) d^{2}}) . \end{matrix}$

From Definition 4, we have $\begin{matrix} s (g_{λ_{1}} (\tilde{α})) = (1 - λ_{1}) a^{2} + λ_{1} b^{2} - λ_{1} c^{2} - (1 - λ_{1}) \\ d^{2} = a^{2} - d^{2} + λ_{1} (b^{2} + d^{2} - a^{2} - c^{2}), \end{matrix}$ $\begin{matrix} s (g_{λ_{2}} (\tilde{α})) = (1 - λ_{2}) a^{2} + λ_{2} b^{2} - λ_{2} c^{2} - (1 - λ_{2}) \\ d^{2} = a^{2} - d^{2} + λ_{2} (b^{2} + d^{2} - a^{2} - c^{2}) . \end{matrix}$

When λ₁ ≤ λ₂, thus we can get $\begin{matrix} s (g_{λ_{1}} (\tilde{α})) - s (g_{λ_{2}} (\tilde{α})) = (λ_{1} - λ_{2}) \\ (b^{2} + d^{2} - a^{2} - c^{2}) \leq 0, \end{matrix}$

Therefore, $g_{λ_{1}} (\tilde{α}) \leq g_{λ_{2}} (\tilde{α})$ .

4. Extended C-IVPFOWQA operators

The C-IVPFOWQA operator can only aggregate single IVPFN, which can’t be suitable for dealing with group decision making. In order to aggregate multiple IVPFNs, we shall present some extend C-IVPFOWQA operators, such as the weighted C-IVPFOWQA (WC-IVPFOWQA) operator, the ordered weighted C-IVPFOWQA (OWC-IVPFOWQA) operator and the hybrid C-IVPFOWQA (HC-IVPFOWQA) operator.

4.1 Weighted C-IVPFOWQA operator

Definition 15. Let ${\tilde{α}}_{j}$ = $(μ_{{\tilde{α}}_{j}}$ , $ν_{{\tilde{α}}_{j}})$ = ([a_j, b_j] , [c_j, d_j]) (j = 1, 2, …, n) be a collection of IVPFNs, a WC-IVPFOWQA operator of dimension n is a mapping WC - IVPFOWQA : Ωⁿ → Θ that has an associated weighting ω = (ω₁, ω₂, …, ω_n) ^T with ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ such that $WC - IVPFOWQA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) = \oplus_{j = 1}^{n} ω_{j} g_{λ} ({\tilde{α}}_{j}) .$ (19) where λ is the attitudinal character of BUM function Q and g_λ is the C-IVPFOWQA operator.

Theorem 6. Let ${\tilde{α}}_{j} = (μ_{{\tilde{α}}_{j}}, ν_{{\tilde{α}}_{j}}) = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ (j = 1, 2, …, n) be a collection of IVPFNs, ω = (ω₁, ω₂, …, ω_n) ^T be the weight vector of ${\tilde{α}}_{j} (j = 1, 2, \dots, n)$ with ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ , then the aggregated value deduced from WC-IVPFOWQA operator is a PFN, and $\begin{matrix} WC - IVPFOWQA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = (\sqrt{1 - \prod_{j = 1}^{n} (1 - (f_{λ} (μ_{{\tilde{α}}_{j}}))^{2})^{ω_{j}}}, \prod_{j = 1}^{n} (f_{1 - λ} (ν_{{\tilde{α}}_{j}}))^{ω_{j}}) . \end{matrix}$ (20)

Proof. Based on the Equation (10) and the results of Ref.37, we know that Theorem 6 holds.

Example 2. Let ${\tilde{α}}_{1} = ([0.5, 0.7], [0.3, 0.4])$ , ${\tilde{α}}_{2} = ([0.3, 0.6], [0.2, 0.6])$ , ${\tilde{α}}_{3} = ([0.3, 0.5], [0.1, 0.6])$ , and ${\tilde{α}}_{4} = ([0.2, 0.6], [0.3, 0.8])$ be four IVPFNs and ω = (0.2, 0.3, 0.1, 0.4) ^T be the weight vector of ${\tilde{a}}_{j} (j = 1, 2, 3, 4)$ . Let the BUM function Q (y) = y², then $λ = \int_{0}^{1} y^{2} dy = \frac{1}{3}$ . Here we use the WC-IVPFOWQA operator to aggregate these four IVPFNs, from Equation (10), we have $\begin{matrix} g_{λ} ({\tilde{α}}_{1}) = (\sqrt{(1 - 1 / 3) \times 0 . 5^{2} + (1 / 3) \times 0 . 7^{2}}, \\ \sqrt{(1 / 3) \times 0 . 3^{2} + (1 - 1 / 3) \times 0 . 4^{2}}) \\ = (0 . 5745, 0 . 3697), \end{matrix}$ $\begin{matrix} g_{λ} ({\tilde{α}}_{2}) = (\sqrt{(1 - 1 / 3) \times 0 . 3^{2} + (1 / 3) \times 0 . 6^{2}}, \\ \sqrt{(1 / 3) \times 0 . 2^{2} + (1 - 1 / 3) \times 0 . 6^{2}}) \\ = (0 . 4243, 0 . 5033), \end{matrix}$ $\begin{matrix} g_{λ} ({\tilde{α}}_{3}) = (\sqrt{(1 - 1 / 3) \times 0 . 3^{2} + (1 / 3) \times 0 . 5^{2}}, \\ \sqrt{(1 / 3) \times 0 . 1^{2} + (1 - 1 / 3) \times 0 . 6^{2}}) \\ = (0 . 3786, 0 . 4933), \end{matrix}$ $\begin{matrix} g_{λ} ({\tilde{α}}_{4}) = (\sqrt{(1 - 1 / 3) \times 0 . 2^{2} + (1 / 3) \times 0 . 6^{2}}, \\ \sqrt{(1 / 3) \times 0 . 3^{2} + (1 - 1 / 3) \times 0 . 8^{2}}) \\ = (0 . 3830, 0 . 6758) . \end{matrix}$

By Equation (20), it follows that $\begin{matrix} WC - IVPFOWQA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, {\tilde{α}}_{3}, {\tilde{α}}_{4}) \\ = (\sqrt{1 - \prod_{j = 1}^{4} (1 - (f_{λ} (μ_{{\tilde{α}}_{j}}))^{2})^{ω_{j}}}, \prod_{j = 1}^{4} (f_{1 - λ} (ν_{{\tilde{α}}_{j}}))^{ω_{j}}) \\ = (0 . 4431, 0 . 5313) . \end{matrix}$

Based on the Theorem 6, the WC-IVPFOWQA operator satisfies the properties of the idempotency, boundedness and monotonicity, which can be demonstrated as the following theorems.

Theorem 7. (Idempotency) Let ${\tilde{α}}_{j} = (μ_{{\tilde{α}}_{j}}, ν_{{\tilde{α}}_{j}}) = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ (j = 1, 2, …, n) be a collection of IVPFNs and h_λ be the WC-IVPFOWQA operator. If ${\tilde{α}}_{j} = \tilde{α} = (μ_{\tilde{α}}, ν_{\tilde{α}}) = ([a, b], [c, d]) (j = 1, 2, \dots, n)$ for all j, then $\begin{matrix} h_{λ} ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) = g_{λ} (\tilde{α}) . \end{matrix}$ (21) where λ is the attitudinal character of BUM function Q and g_λ is the C-IVPFOWQA operator.

Proof. If ${\tilde{α}}_{i} = \tilde{α}$ for all j, then based on Theorem 6, we obtain $\begin{matrix} h_{λ} ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = (\sqrt{1 - \prod_{j = 1}^{n} (1 - (f_{λ} (μ_{{\tilde{α}}_{j}}))^{2})^{ω_{j}}}, \prod_{j = 1}^{n} (f_{1 - λ} (ν_{{\tilde{α}}_{j}}))^{ω_{j}}) \\ = (\sqrt{1 - \prod_{j = 1}^{n} (1 - (f_{λ} (μ_{\tilde{α}}))^{2})^{ω_{j}}}, \prod_{j = 1}^{n} (f_{1 - λ} (ν_{\tilde{α}}))^{ω_{j}}) \\ = (\sqrt{1 - (1 - (f_{λ} (μ_{\tilde{α}}))^{2})^{\sum_{j = 1}^{n} ω_{j}}}, (f_{1 - λ} (ν_{\tilde{α}}))^{\sum_{j = 1}^{n} ω_{j}}) \\ = (f_{λ} (μ_{\tilde{α}}), f_{1 - λ} (ν_{\tilde{α}})) = g_{λ} (\tilde{α}) . \end{matrix}$

The proof is completed.

Theorem 8. (Boundedness) Let ${\tilde{α}}_{j} = (μ_{{\tilde{α}}_{j}}, ν_{{\tilde{α}}_{j}}) = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ (j = 1, 2, …, n) be a collection of IVPFNs and h_λ be the WC-IVPFOWQA operator, then ${\tilde{α}}^{-} \leq h_{λ} ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \leq {\tilde{α}}^{+} .$ where ${\tilde{α}}^{-} = (min_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})}$ , $max_{j} {f_{1 - λ} (ν_{{\tilde{α}}_{j}})})$ , ${\tilde{α}}^{+} = (max_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})}$ , $min_{j} {f_{1 - λ} (ν_{{\tilde{α}}_{j}})})$ and λ is the attitudinal character of BUM function Q.

Theorem 9. (Monotonicity) Let ${\tilde{α}}_{j} = (μ_{{\tilde{α}}_{j}}, ν_{{\tilde{α}}_{j}}) = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ and ${\tilde{β}}_{j}$ = $(μ_{{\tilde{β}}_{j}}$ , $ν_{{\tilde{β}}_{j}})$ = $([a_{{\tilde{β}}_{j}}$ , $b_{{\tilde{β}}_{j}}]$ , $[c_{{\tilde{β}}_{j}}$ , $d_{{\tilde{β}}_{j}}])$ (j = 1, 2, …, n) be two collections of IVPFNs, h_λ be the WC-IVPFOWQA operator. If $f_{λ} (μ_{{\tilde{α}}_{j}}) \geq f_{λ} (μ_{{\tilde{β}}_{j}})$ and $f_{1 - λ} (ν_{{\tilde{α}}_{j}}) \leq f_{1 - λ} (ν_{{\tilde{β}}_{j}})$ for all j, then $h_{λ} ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \geq h_{λ} ({\tilde{β}}_{1}, {\tilde{β}}_{2}, \dots, {\tilde{β}}_{n}) .$

4.2 Ordered weighted C-IVPFOWQA operator

Ordered weighted C-IVPFOWQA (OWC-IVPFOWQA) operator is an extension of the WC-IVPFOWQA operator, which can be defined as follows:

Definition 16. Let ${\tilde{α}}_{j}$ = $(μ_{{\tilde{α}}_{j}}$ , $ν_{{\tilde{α}}_{j}})$ = ([a_j, b_j], [c_j, d_j]) (j = 1, 2, …, n) be a collection of IVPFNs, an OWC-IVPFOWQA operator of dimension n is a mapping OWC - IVPFOWQA : Ωⁿ → Θ that has an associated weighting ω = (ω₁, ω₂, …, ω_n) ^T with ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ such that $\begin{matrix} OWC - IVPFOWQA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = \oplus_{j = 1}^{n} ω_{j} g_{λ} ({\tilde{α}}_{σ (j)}) . \end{matrix}$ (24) where λ is the attitudinal character of BUM function Q and g_λ is the C-IVPFOWQA operator, (σ (1) , σ (2) , …, σ (n)) is a permutation of (1, 2, …, n) such that $g_{λ} ({\tilde{α}}_{σ (j - 1)}) \geq g_{λ} ({\tilde{α}}_{σ (j)})$ for all j = 1, 2, …, n.

Similarly to the Theorem 6, we have the following Theorem.

Theorem 10. Let ${\tilde{α}}_{j}$ = $(μ_{{\tilde{α}}_{j}}$ , $ν_{{\tilde{α}}_{j}})$ = ([a_j, b_j], [c_j, d_j]) (j = 1, 2, …, n) be a collection of IVPFNs, ω = (ω₁, ω₂, . . . , ω_n) ^T be the weight vector of ${\tilde{α}}_{j} (j = 1, 2, \dots, n)$ with ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ , then the aggregated value deduced from OWC-IVPFOWQA operator is a PFN, and $\begin{matrix} OWC - IVPFOWQA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = (\sqrt{1 - \prod_{j = 1}^{n} (1 - (f_{λ} (μ_{{\tilde{α}}_{σ (j)}}))^{2})^{ω_{j}}}, \\ \prod_{j = 1}^{n} (f_{1 - λ} (ν_{{\tilde{α}}_{σ (j)}}))^{ω_{j}}) . \end{matrix}$ (25)

Example 3. Let ${\tilde{α}}_{1}$ = ([0.2, 0.8], [0.4, 0.5]), ${\tilde{α}}_{2}$ = ([0.3, 0.7], [0.1, 0.6]), ${\tilde{α}}_{3}$ = ([0.1, 0.3], [0.2, 0.8]), and ${\tilde{α}}_{4}$ = ([0.4, 0.7], [0.2, 0.5]) be four IVPFNs and ω = (0.4, 0.2, 0.3, 0.1) ^T be the weight vector of ${\tilde{α}}_{j} (j = 1, 2$ , …, 4). Let the BUM function Q (y) = y², then $λ = \int_{0}^{1} y^{2} dy = \frac{1}{3}$ . Here we use the OWC-IVPFOWQA operator to aggregate these four IVPFNs, by Equation (9), we get

g_λ $({\tilde{α}}_{1})$ = (0.4899, 0.4690), g_λ $({\tilde{α}}_{2})$ = (0.4726, 0.4933), g_λ $({\tilde{α}}_{3})$ = (0.1915, 0.6633), g_λ $({\tilde{α}}_{4})$ = (0.5196, 0.4243),

Based on Definition 13, we have

S_λ $(g_{λ} ({\tilde{α}}_{1}))$ = 0 . 0200, S_λ $(g_{λ} ({\tilde{α}}_{2}))$ = -0 . 0200, S_λ $(g_{λ} ({\tilde{α}}_{3}))$ = -0 . 4033, S_λ $(g_{λ} ({\tilde{α}}_{4}))$ = 0 . 0900, Since, S_λ $(g_{λ} ({\tilde{α}}_{4}))$ > S_λ $(g_{λ} ({\tilde{α}}_{1}))$ > S_λ $(g_{λ} ({\tilde{α}}_{2}))$ > S_λ (g_λ $({\tilde{α}}_{3}))$ , then g_λ $({\tilde{α}}_{4})$ > g_λ $({\tilde{α}}_{1})$ > g_λ $({\tilde{α}}_{2})$ > g_λ $({\tilde{α}}_{3})$ and g_λ $(({\tilde{α}}_{(1)}))$ = $g_{λ} ({\tilde{α}}_{4})$ = (0.5196, 0.4243), g_λ $(({\tilde{α}}_{(2)}))$ = $g_{λ} ({\tilde{α}}_{1})$ = (0.4899, 0.4690), g_λ $(({\tilde{α}}_{(3)}))$ = g_λ $({\tilde{α}}_{2})$ = (0.4726, 0.4933) , g_λ $(({\tilde{α}}_{(4)}))$ = g_λ $({\tilde{α}}_{3})$ = (0.1915, 0.6633).

By Equation (25), it follows that $\begin{matrix} OWC - IVPFOWQA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, {\tilde{α}}_{3}, {\tilde{α}}_{4}) \\ = (\sqrt{1 - \prod_{j = 1}^{4} (1 - (f_{λ} (μ_{{\tilde{α}}_{σ (j)}}))^{2})^{ω_{j}}}, \\ \prod_{j = 1}^{4} (f_{1 - λ} (ν_{{\tilde{α}}_{σ (j)}}))^{ω_{j}}) = (0.4788, 0.4736) . \end{matrix}$

The OWC-IVPFOWQA operator has similar properties to the WC-IVPFOWQA operator.

Theorem 11. (Idempotency) Let ${\tilde{α}}_{j} = (μ_{{\tilde{α}}_{j}}, ν_{{\tilde{α}}_{j}}) = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ (j = 1, 2, …, n) be a collection of IVPFNs and g_λ be the C-IVPFOWQA operator. If ${\tilde{α}}_{j} = \tilde{α} = (μ_{\tilde{α}}, ν_{\tilde{α}}) = ([a, b], [c, d]) (j = 1, 2, \dots, n)$ for all j, then $OWC - IVPFOWQA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) = g_{λ} (\tilde{α}) .$ (26)

Theorem 12. (Boundedness) Let ${\tilde{α}}_{j} = (μ_{{\tilde{α}}_{j}}, ν_{{\tilde{α}}_{j}}) = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ (j = 1, 2, …, n) be a collection of IVPFNs,then ${\tilde{α}}^{-} \leq OWC - IVPFOWQA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \leq {\tilde{α}}^{+} .$ (27) where ${\tilde{α}}^{-} = (min_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})}$ , $max_{j} {f_{1 - λ} (ν_{{\tilde{α}}_{j}})})$ , ${\tilde{α}}^{+} = (max_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})}$ , $min_{j} {f_{1 - λ} (ν_{{\tilde{α}}_{j}})})$ .

Theorem 13. (Monotonicity) Let ${\tilde{α}}_{j} = (μ_{{\tilde{α}}_{j}}, ν_{{\tilde{α}}_{j}}) = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ and ${\tilde{β}}_{j} = (μ_{{\tilde{β}}_{j}}$ , $ν_{{\tilde{β}}_{j}})$ = $([a_{{\tilde{β}}_{j}}$ , $b_{{\tilde{β}}_{j}}]$ , $[c_{{\tilde{β}}_{j}}$ , $d_{{\tilde{β}}_{j}}])$ (j = 1, 2, …, n) be two collections of IVPFNs. If $f_{λ} (μ_{{\tilde{α}}_{j}})$ ≥ $f_{λ} (μ_{{\tilde{β}}_{j}})$ and $f_{1 - λ} (ν_{{\tilde{α}}_{j}})$ ≤ $f_{1 - λ} (ν_{{\tilde{β}}_{j}})$ for all j, then $\begin{matrix} OWC - IVPFOWQA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ \geq OWC - IVPFOWQA ({\tilde{β}}_{1}, {\tilde{β}}_{2}, \dots, {\tilde{β}}_{n}) . \end{matrix}$ (28)

Theorem 14. (Commutativity) Let ${\tilde{α}}_{j} = (μ_{{\tilde{α}}_{j}}, ν_{{\tilde{α}}_{j}}) = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ (j = 1, 2, …, n) be a collection of IVPFNs, then $\begin{matrix} OWC - IVPFOWQA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = OWC - IVPFOWQA ({\tilde{β}}_{1}, {\tilde{β}}_{2}, \dots, {\tilde{β}}_{n}) . \end{matrix}$ (29) where $({\tilde{β}}_{1}, {\tilde{β}}_{2}, \dots, {\tilde{β}}_{n})$ is any permutation of $({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n})$ .

Proof. Since $({\tilde{β}}_{1}, {\tilde{β}}_{2}, \dots, {\tilde{β}}_{n})$ is any permutation of $({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n})$ , then ${\tilde{α}}_{σ (j)} = {\tilde{β}}_{σ (j)}$ for all j, according to the Definition of OWC-IVPFOWQA operator in Equation (24), we obtain $\begin{matrix} OWC - IVPFOWQA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = \oplus_{j = 1}^{n} ω_{j} g_{λ} ({\tilde{α}}_{σ (j)}) = \oplus_{j = 1}^{n} ω_{j} g_{λ} ({\tilde{β}}_{σ (j)}) \\ = OWC - IVPFOWQA ({\tilde{β}}_{1}, {\tilde{β}}_{2}, \dots, {\tilde{β}}_{n}) . \end{matrix}$

Thus, the proof is completed.

Besides the above theorems, the following desirable property can be easily obtained.

Property 1. Let ${\tilde{α}}_{j} = (μ_{{\tilde{α}}_{j}}, ν_{{\tilde{α}}_{j}}) = [a_{j}, b_{j}], [c_{j}, d_{j}])$ (j = 1, 2, …, n) be a collection of IVPFNs, ω = (ω₁, ω₂, …, ω_n) ^T be the weight vector of ${\tilde{α}}_{j}$ (j = 1, 2, …, n) with ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ . Then

1) If ω = (1, 0, …, 0) ^T, then $OWC - IVPFOWQA ({\tilde{α}}_{1}$ , ${\tilde{α}}_{2}$ , …, ${\tilde{α}}_{n})$ = $max_{j} {g_{λ} ({\tilde{α}}_{j})};$

2) If ω = (0, 0, …, 1) ^T, then $OWC - IVPFOWQA ({\tilde{α}}_{1}$ , ${\tilde{α}}_{2}$ , …, ${\tilde{α}}_{n})$ = $min_{j} {g_{λ} ({\tilde{α}}_{j})};$

3) If ω_i = 1, ω_j = 0 and i ≠ j, then $OWC - IVPFOWQA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) = g_{λ} ({\tilde{α}}_{σ (i)}) .$

where $g_{λ} ({\tilde{α}}_{(i)})$ is the ith largest value of $g_{λ} ({\tilde{α}}_{j})$ .

4.3 Hybrid C-IVPFOWQA operator

Since WC-IVPFOWQA operator focuses on the IVPFNs only while OWC-IVPFOWQA operator considers the ordered positions of them, in order to weigh both the given argument and its ordered position, we propose the hybrid C-IVPFOWQA (HC-IVPFOWQA) operator.

Definition 17. Let ${\tilde{α}}_{j}$ = $(μ_{{\tilde{α}}_{j}}$ , $ν_{{\tilde{α}}_{j}})$ = ([a_j, b_j], [c_j, d_j]) (j = 1, 2, …, n) be a collection of IVPFNs, a HC-IVPFOWQA operator of dimension n is a mapping: HC - IVPFOWQA : Ωⁿ → Θ that has an associated weighting ω = (ω₁, ω₂, …, ω_n) ^T with ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ such that $\begin{matrix} HC - IVPFOWQA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = \oplus_{j = 1}^{n} ω_{j} g_{λ} ({\dot{\tilde{α}}}_{σ (j)}) . \end{matrix}$ (30) where λ is the attitudinal character of BUM function Q and g_λ is the C-IVPFOWQA operator, (σ (1) , σ (2) , …, σ (n)) is a permutation of (1, 2, …, n) such that $g_{λ} ({\dot{\tilde{α}}}_{σ (j - 1)}) \geq g_{λ} ({\dot{\tilde{α}}}_{σ (j)})$ for all j. $g_{λ} ({\dot{\tilde{α}}}_{(j)})$ is the jth largest value of the weighted PFNs ${{nw}_{1} g_{λ} ({\tilde{α}}_{1}), {nw}_{2} g_{λ} ({\tilde{α}}_{2}), \dots, {nw}_{n} g_{λ} ({\tilde{α}}_{n})}$ , n is the balancing coefficient, and w = (w₁, w₂, …, w_n) ^T is the weight vector of ${\tilde{α}}_{j}$ (j = 1, 2, …, n) with w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ .

Theorem 15. Let ${\tilde{α}}_{j}$ = $(μ_{{\tilde{α}}_{j}}$ , $ν_{{\tilde{α}}_{j}})$ = ([a_j, b_j], [c_j, d_j]) (j = 1, 2, …, n) be a collection of IVPFNs, ω = (ω₁, ω₂, . . . , ω_n) ^T be the weight vector with ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ , then $\begin{matrix} HC - IVPFOWQA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = (\sqrt{1 - \prod_{j = 1}^{n} (1 - (f_{λ} (μ_{{\dot{\tilde{α}}}_{σ (j)}}))^{2})^{ω_{j}}}, \\ \prod_{j = 1}^{n} (f_{1 - λ} (ν_{{\dot{\tilde{α}}}_{σ (j)}}))^{ω_{j}}) . \end{matrix}$ (31)

Proof. The proof is similar to Theorem 6, so it is omitted here.

From Equation (31), we know that the aggregated value by utilizing HC-IVPFOWQA operator is also a PFN.

Example 4. Utilizing the information in Example 3 and let w = (0.15, 0.34, 0.28, 0.23) ^T, based on Definition 3, we have $g_{λ} ({\dot{\tilde{α}}}_{1}) = {nw}_{1} g_{λ} ({\tilde{α}}_{1}) = 4 \times 0.15 \times (0.4899, 0.4690) = (0.3896, 0.6349),$ $\begin{matrix} g_{λ} ({\dot{\tilde{α}}}_{2}) = {nw}_{2} g_{λ} ({\tilde{α}}_{2}) = 4 \times 0.34 \times (0.4726, 0.4933) \\ = (0.5393, 0.3825), \end{matrix}$ $\begin{matrix} g_{λ} ({\dot{\tilde{α}}}_{3}) = {nw}_{3} g_{λ} ({\tilde{α}}_{3}) = 4 \times 0.28 \times (0.1915, 0.6633) \\ = (0.2024, 0.6314), \end{matrix}$ $\begin{matrix} g_{λ} ({\dot{\tilde{α}}}_{4}) = {nw}_{4} g_{λ} ({\tilde{α}}_{4}) = 4 \times 0.23 \times (0.5196, 0.4243) \\ = (0.5014, 0.4544), \end{matrix}$ and S_λ $(g_{λ} ({\tilde{α}}_{1}))$ = -0 . 2513, S_λ $(g_{λ} ({\tilde{α}}_{2}))$ = 0 . 1446, S_λ $(g_{λ} ({\tilde{α}}_{3})) =$ - 0 . 3577, S_λ $(g_{λ} ({\tilde{α}}_{4}))$ = 0 . 0449,

Since, S_λ $(g_{λ} ({\tilde{α}}_{2}))$ > S_λ $(g_{λ} ({\tilde{α}}_{4}))$ > S_λ $(g_{λ} ({\tilde{α}}_{1}))$ > S_λ $(g_{λ} ({\tilde{α}}_{3}))$ , then g_λ $({\dot{\tilde{α}}}_{2})$ > g_λ $({\dot{\tilde{α}}}_{4})$ > g_λ $({\dot{\tilde{α}}}_{1})$ > g_λ $({\dot{\tilde{α}}}_{3})$ and g_λ $(({\dot{\tilde{α}}}_{(1)}))$ = $g_{λ} ({\dot{\tilde{α}}}_{2})$ = (0.5393, 0.3825), g_λ $(({\dot{\tilde{α}}}_{(2)}))$ = g_λ $({\dot{\tilde{α}}}_{4})$ = (0.5014, 0.4544), g_λ $(({\dot{\tilde{α}}}_{(3)}))$ = $g_{λ} ({\dot{\tilde{α}}}_{1})$ = (0.3896, 0.6349), g_λ $(({\dot{\tilde{α}}}_{(4)}))$ = $g_{λ} ({\dot{\tilde{α}}}_{3})$ = (0.2024, 0.6314), By Equation (31), it follows that $\begin{matrix} HC - IVPFOWQA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, {\tilde{α}}_{3}, {\tilde{α}}_{4}) \\ = (\sqrt{1 - \prod_{j = 1}^{4} (1 - (f_{λ} (μ_{{\dot{\tilde{α}}}_{σ (j)}}))^{2})^{ω_{j}}}, \\ \prod_{j = 1}^{4} (f_{1 - λ} (ν_{{\dot{\tilde{α}}}_{σ (j)}}))^{ω_{j}}) = (0.4695, 0.4846) . \end{matrix}$

It can be easily proved that the HC-IVPFOWQA operator has the following properties.

Property 2. If the weight vector ω = (1/n, 1/n, …, 1/n), then the HC-IVPFOWQA operator is reduced to the WC-IVPFOWQA operator.

Property 3. If the weight vector w = (1/n, 1/n, …, 1/n), then the HC-IVPFOWQA operator is reduced to the OWC-IVPFOWQA operator.

5. An approach to MAGDM based on the extended C-IVPFOWQA operators

In this section, we will utilize the extended C-IVPFOWQA aggregation operators to MAGDM with interval-valued Pythagorean fuzzy information. Let A = {A₁, A₂, …, A_m} be a discrete set of alternatives, and C = {C₁, C₂, …, C_n} be the set of attributes, and ω = (ω₁, ω₂, …, ω_n) ^T be the weight vector of attribute C_j (j = 1, 2, …, n) with ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ . Let E = {e₁, e₂, …, e_l} be the set of decision makers whose weight vector is ρ = (ρ₁, ρ₂, …, ρ_l) ^T with ρ_k ∈ [0, 1] and $\sum_{k = 1}^{l} ρ_{k} = 1$ . Assume that ${\tilde{R}}^{(k)} = ({\tilde{r}}_{ij}^{(k)})_{m \times n}$ is an interval-valued Pythagorean fuzzy decision matrix, where ${\tilde{r}}_{ij}^{(k)} = (μ_{{\tilde{r}}_{ij}^{(k)}}, ν_{{\tilde{r}}_{ij}^{(k)}}) = ([a_{{\tilde{r}}_{ij}^{(k)}}, b_{{\tilde{r}}_{ij}^{(k)}}], [c_{{\tilde{r}}_{ij}^{(k)}}, d_{{\tilde{r}}_{ij}^{(k)}}])$ is attribute value given by decision maker e_k ∈ E for the alternative A_i ∈ A with respect to the attribute C_j ∈ C, $μ_{{\tilde{r}}_{ij}^{(k)}} = [a_{{\tilde{r}}_{ij}^{(k)}}, b_{{\tilde{r}}_{ij}^{(k)}}]$ indicates the degree that the alternative A_i satisfies the attribute C_j given by the decision maker e_k, $v_{{\tilde{r}}_{ij}^{(k)}} = [c_{{\tilde{r}}_{ij}^{(k)}}, d_{{\tilde{r}}_{ij}^{(k)}}]$ indicates the degree that the alternative A_i does not satisfy the attribute C_j given by the decision maker e_k, $[a_{{\tilde{r}}_{ij}^{(k)}}, b_{{\tilde{r}}_{ij}^{(k)}}], [c_{{\tilde{r}}_{ij}^{(k)}}, d_{{\tilde{r}}_{ij}^{(k)}}] \in [0, 1]$ , and $0 \leq b_{{\tilde{r}}_{ij}^{(k)}}^{2} + d_{{\tilde{r}}_{ij}^{(k)}}^{2} \leq 1$ .

In the following, a novel MAGDM approach is developed with interval-valued Pythagorean fuzzy information based on the extended C-IVPFOWQA aggregation operators.

Step 1. Aggregate interval-valued Pythagorean fuzzy decision matrix ${\tilde{r}}_{ij}^{(k)} = (μ_{{\tilde{r}}_{ij}^{(k)}}, ν_{{\tilde{r}}_{ij}^{(k)}}) = ([a_{{\tilde{r}}_{ij}^{(k)}}, b_{{\tilde{r}}_{ij}^{(k)}}], [c_{{\tilde{r}}_{ij}^{(k)}}, d_{{\tilde{r}}_{ij}^{(k)}}])$ into the overall decision matrix R = (r_ij) _m×n based on HC-IVPFOWQA operator $r_{i j} = h_{λ} ({\tilde{r}}_{i j}^{(1)}, {\tilde{r}}_{i j}^{(2)}, \dots, {\tilde{r}}_{i j}^{(l)}) = \oplus_{k = 1}^{l} ρ_{k} g_{λ} ({\dot{\tilde{r}}}_{i j}^{σ (k)})$ (32)

Step 2. Utilize the decision matrix R = (r_ij) _m×n and the PFOWA operator $r_{i} = PFOWA (r_{i 1}, r_{i 2}, \dots, r_{in}) = \oplus_{j = 1}^{n} ω_{j} r_{i σ (j)}$ (33) to drive the overall preference values r_i (i = 1, 2, …, m) for each alternative A_i (i = 1, 2, …, m).

Step 3. Calculate the score values S (r_i) (i = 1, 2, …, m) of the collective overall values r_i (i = 1, 2, …, m) to rank all the alternatives A_i (i = 1, 2, …, m). If there is no difference between two scores S (r_i) and S (r_j) , then we need to calculate the accuracy degrees H (r_i) and H (r_j) of the collective overall values r_i and r_j, respectively.

Step 4. Rank all the alternatives A_i (i = 1, 2, …, m) and select the most desirable one(s) in accordance with S (r_i) and H (r_i) (i = 1, 2, …, m).

Step 5. End.

6. Numerical example

In this section, we are going to present a numerical example to show the risk evaluation of technological innovation in high-tech enterprises with interval-valued Pythagorean fuzzy information (adapt from Ref.44).

A management committee wants to select the best high-tech enterprise with the lowest risk of technologic innovation. Three experts {e₁, e₂, e₃} form a committee to act as decision makers, whose weight vector is ρ = (0.4, 0.35, 0.25) ^T, after careful reviewing of the information, they choose four possible alternatives {A₁, A₂, A₃, A₄}. Management committee selects six attributes to evaluate the four possible high-tech enterprises (1) C₁ is policy risk; (2) C₂ is financial risk; (3) C₃ is technological risk; (4) C₄ is production risk; (5) C₅ is market risk and (6) C₆is managerial risk, and ω = (0.1894, 0.1841, 0.1361, 0.1257, 0.1753, 0.1894) ^T is the weight vector of them. As the environment is uncertain, the three experts e_k (k = 1, 2, 3) evaluate the high-tech enterprise A_i (i = 1, 2, …, 4) with respect to the attributes C_j (j = 1, 2, …, 6) and construct the following three interval-valued Pythagorean fuzzy decision matrixs

{\tilde{R}}^{(k)} = ({\tilde{r}}_{ij}^{(k)})_{4 \times 6}

in Tables 1–3.

Table 1
The interval-valued Pythagorean fuzzy decision matrix ${\tilde{R}}^{(1)}$ for e₁

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
A ₁	([0.8, 0.9],	([0.5, 0.7],	([0.7, 0.9],	([0.7, 0.8],	([0.8, 0.9],	([0.4, 0.6],
	[0.2, 0.3])	[0.4, 0.5])	[0.3, 0.4])	[0.4, 0.5])	[0.2, 0.3])	[0.5, 0.7])
A ₂	([0.6, 0.7],	([0.6, 0.8],	([0.7, 0.9],	([0.8, 0.9],	([0.5, 0.6],	([0.8, 0.9],
	[0.3, 0.5])	[0.4, 0.5])	[0.2, 0.3])	[0.2, 0.3])	[0.1, 0.3])	([0.1, 0.3],
A ₃	([0.5, 0.6],	([0.7, 0.8],	([0.6, 0.7],	([0.7, 0.9],	([0.4, 0.6],	([0.7, 0.8],
	[0.4, 0.5])	[0.4, 0.5])	[0.3, 0.4])	[0.1, 0.3])	[0.2, 0.3])	[0.4, 0.5])
A ₄	([0.4, 0.6],	([0.8, 0.9],	([0.7, 0.8],	([0.6, 0.8],	([0.5, 0.6],	([0.8, 0.9],
	[0.3, 0.5])	[0.2, 0.3])	[0.4, 0.5])	[0.4, 0.5])	[0.1, 0.3])	[0.2, 0.3])

Table 2

The interval-valued Pythagorean fuzzy decision matrix ${\tilde{R}}^{(2)}$ for e₂

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
A ₁	([0.6, 0.7],	([0.6, 0.9],	([0.4, 0.5],	([0.8, 0.9],	([0.5, 0.7],	([0.5, 0.7],
	[0.2, 0.3])	[0.1, 0.3])	[0.4, 0.6])	[0.2, 0.3])	[0.2, 0.4])	[0.4, 0.5])
A ₂	([0.5, 0.7],	([0.5, 0.6],	([0.5, 0.8],	([0.5, 0.6],	([0.6, 0.8],	([0.8, 0.9],
	[0.4, 0.5])	[0.3, 0.5])	[0.4, 0.6])	[0.1, 0.3])	[0.4, 0.5])	[0.2, 0.3])
A ₃	([0.5, 0.6],	([0.8, 0.9],	([0.5, 0.6],	([0.6, 0.7],	([0.8, 0.9],	([0.6, 0.7],
	[0.4, 0.5])	[0.2, 0.3])	[0.4, 0.5])	[0.4, 0.6])	[0.2, 0.3])	[0.4, 0.6])
A ₄	([0.7, 0.9],	([0.7, 0.8],	([0.6, 0.8],	([0.4, 0.6],	([0.7, 0.9],	([0.5, 0.7],
	[0.1, 0.3])	[0.4, 0.5])	[0.4, 0.5])	[0.2, 0.3])	[0.1, 0.3])	[0.4, 0.5])

Table 3

The interval-valued Pythagorean fuzzy decision matrix ${\tilde{R}}^{(3)}$ for e₃

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
A ₁	([0.8, 0.9],	([0.7, 0.9],	([0.5, 0.7],	([0.6, 0.8],	([0.6, 0.8],	([0.6, 0.7],
	[0.2, 0.3])	[0.1, 0.3])	[0.2, 0.5])	[0.3, 0.4])	[0.4, 0.5])	[0.4, 0.5])
A ₂	([0.6, 0.7],	([0.4, 0.6],	([0.8, 0.9],	([0.8, 0.9],	([0.7, 0.8],	([0.8, 0.9],
	[0.1, 0.3])	[0.1, 0.3])	[0.2, 0.3])	[0.2, 0.3])	[0.2, 0.5])	[0.2, 0.3])
A ₃	([0.6, 0.8],	([0.8, 0.9],	([0.7, 0.9],	([0.6, 0.8],	([0.6, 0.7],	([0.7, 0.8],
	[0.4, 0.5])	[0.2, 0.3])	[0.1, 0.3])	[0.4, 0.5])	[0.4, 0.6])	[0.4, 0.5])
A ₄	([0.8, 0.9],	([0.7, 0.8],	([0.6, 0.8],	([0.8, 0.9],	[0.4, 0.5])	([0.5, 0.6],
	[0.2, 0.3])	[0.4, 0.5])	[0.4, 0.5])	[0.2, 0.3])	[0.4, 0.6])	[0.4, 0.5])

6.1 Decision procedure based on HC-IVPFOWQA operator

To select the best high-tech enterprise, the computation steps are presented as follows:

Step 1. Utilize the HC-IVPFOWQA operator and the weight ρ = (0.4, 0.35, 0.25) ^T to aggregate interval-valued Pythagorean fuzzy decision matrix

{\tilde{R}}^{(k)} = {({\tilde{r}}_{ij}^{(k)})}_{4 \times 6} (k = 1, 2, 3)

into the overall decision matrix R = (r_ij) _4×6, and the aggregating results are shown in Table 4. Assume that w = (0.37, 0.33, 0.3) ^T and BUM function

Q (y) = \sqrt[4]{y}

in Equation (31).

Table 4
The Pythagorean fuzzy collective decision matrix R

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
A ₁	(0.8512,0.2116)	(0.8170,0.2027)	(0.7575,0.3214)	(0.8278,0.2987)	(0.8090,0.2583)	(0.6434,0.4457)
A ₂	(0.7958,0.2695)	(0.6185,0.2790)	(0.8497,0.2511)	(0.7968,0.1967)	(0.8316,0.2359)	(0.8308,0.1998)
A ₃	(0.6630,0.4105)	(0.8570,0.2588)	(0.7542,0.2525)	(0.8029,0.2739)	(0.7657,0.2534)	(0.7555,0.4167)
A ₄	(0.8213,0.2159)	(0.8361,0.3085)	(0.7755,0.4085)	(0.7891,0.2728)	(0.7305,0.1979)	(0.7849,0.3085)

Step 2. Utilize the PFOWA operator and ω = (0.1894, 0.1841, 0.1361, 0.1257, 0.1753, 0.1894) ^T to drive the overall preference values of each alternative: $\begin{matrix} r_{1} = (0 . 7924, 0 . 2795), r_{2} = (0 . 7979, 0 . 2391), \\ r_{3} = (0 . 7770, 0 . 3084), r_{4} = (0 . 7938, 0 . 2766) . \end{matrix}$

Step 3. Calculate the score values S (r_i) (i = 1, 2, …, 4) of the collective overall values r_i (i = 1, 2, …, 4). S (r₁) =0 . 5497, S (r₂) =0 . 5794, S (r₃) =0 . 5085, S (r₄) =0 . 5536.

Then we have S (r₂) > S (r₄) > S (r₁) > S (r₃).

Step 4. Rank all the alternatives A_i (i = 1, 2, …, 4) according to S (r_i) (i = 1, 2, …, 4) A₂ ≻ A₄ ≻ A₁ ≻ A₃.

Thus, the most desirable alternative is A₂. It is noted that “≻” means “preferred to”.

The attitudinal characterλplays an important role during the computation process of the HC-IVPFOWQA operator. In what follows, we study the impact on the scores and the ranking with the attitudinal character λ changes from 0 to 1 with step length 0.1. The scores of collective overall values of alternatives and corresponding ranking results are shown in Table 5.

Table 5

The scores and ranking results with different attitudinal character λ

λ	Score function				Ranking results
λ	S (r₁)	S (r₂)	S (r₃)	S (r₄)	Ranking results
0	0.2953	0.3476	0.2787	0.3133	A₂ ≻ A₄ ≻ A₁ ≻ A₃
0.1	0.3257	0.3750	0.3062	0.3422	A₂ ≻ A₄ ≻ A₁ ≻ A₃
0.2	0.3562	0.4027	0.3338	0.3712	A₂ ≻ A₄ ≻ A₁ ≻ A₃
0.3	0.3973	0.4400	0.3709	0.4102	A₂ ≻ A₄ ≻ A₁ ≻ A₃
0.4	0.4181	0.4594	0.3898	0.4301	A₂ ≻ A₄ ≻ A₁ ≻ A₃
0.5	0.4496	0.4887	0.4180	0.4600	A₂ ≻ A₄ ≻ A₁ ≻ A₃
0.6	0.4816	0.5184	0.4475	0.4903	A₂ ≻ A₄ ≻ A₁ ≻ A₃
0.7	0.5150	0.5485	0.4776	0.5215	A₂ ≻ A₄ ≻ A₁ ≻ A₃
0.8	0.5497	0.5794	0.5085	0.5536	A₂ ≻ A₄ ≻ A₁ ≻ A₃
0.9	0.5859	0.6113	0.5407	0.5868	A₂ ≻ A₄ ≻ A₁ ≻ A₃
1.0	0.6242	0.6451	0.5753	0.6224	A₂ ≻ A₁ ≻ A₄ ≻ A₃

The result in Table 5 indicates that all scores of collective overall values of alternatives are increasing with an increasing λ. For each attitudinal character λ in unit interval [0,1], the most desirable alternative is always A₂.

Furthermore, to provide a better view of the change results in regard to λ, the results of the scores of alternatives are described in Fig. 1. From Fig. 1, we obtain the ranking of the alternatives which is A₂ ≻ A₄ ≻ A₁ ≻ A₃ if λ ∈ [0, 0.9335], and A₂ ≻ A₁ ≻ A₄ ≻ A₃ if λ ∈ [0 . 9335, 1], the slightly difference is the ranking order between A₁ and A₄, the former order is A₄ ≻ A₁, whereas the latter order is A₁ ≻ A₄. Moreover, it can be seen from Fig. 1 that no matter what value the attitudinal character λ takes, A₂ is still the most desirable alternative.

Fig. 1

The results of HC-IVPFOWQA with different attitudinal character λ.

6.2 Comparison analysis

The prominent characteristic of the proposed operators is that they not only improve the accuracy of decision making but also can reflect the decision makers’ risk preferences. To demonstrate the advantages, we discussed some comparative analyses with some existing operators: IVPFWA operator [43], IVPFWG operator [43], IVPF-WAA operator [44], IPFWG operator [45], and proposed operators: WC-IVPFOWQA operator and OWC-IVPFOWQA operator. The aggregating results and ranking results by different methods are shown in Table 6.

Table 6
The aggregating results by different methods

	IVPFWA [43]	IVPFWG [43]	IVPF-WAA [44]	IPFWG [45]	WC-IV PFOWQA	OWC-IV PFOWQA	HC-IV PFOWQA
A ₁	([0.6143,0.7759],	([0.5991,0.7660],	([0.6487, 0.8123],	([0.5991,0.7660,],	(0.7780,	(0.7920,	(0.7924,
	[0.2861, 0.4224])	0.2583,0.4052])	[0.2583, 0.4052])	[0.3152, 0.4513])	0.3024)	0.2896)	0.2795)
A ₂	([0.6324,0.7721],	([0.6189,0.7629],	([0.6656, 0.8076],	([0.6189,0.7629],	(0.7860	(0.7955,	(0.7979,
	0.2355,0.3914])	[0.2067,0.3775])	[0.2067, 0.3775])	[0.2651,0.4116])	0.2533)	0.2456)	0.2391)
A ₃	([0.6295, 0.7528],	([0.6181, 0.7441],	([0.6547, 0.7863],	([0.6181,0.7441],	(0.7635,	(0.7801,	(0.7770,
	[0.3209, 0.4447])	[ 0.2963, 0.4301])	[0.2963, 0.4301])	[0.3413,0.4644])	0.3311)	0.3193)	0.3084)
A ₄	([0.6238, 0.7684],	([0.6066, 0.7566],	([0.6593, 0.8078],	([0.6066,0.7566],	(0.7787,	(0.7929,	(0.7938,
	[0.2787, 0.4104])	[0.2464, 0.3964])	[0.2464, 0.3964])	[0.3077,0.4294])	0.2941)	0.2887)	0.2766)
Ranking	A₂ ≻ A₄ ≻ A₁ ≻ A₃	A₂ ≻ A₄ ≻ A₁ ≻ A₃	A₂ ≻ A₄ ≻ A₁ ≻ A₃	A₂ ≻ A₄ ≻ A₁ ≻ A₃	A₂ ≻ A₄ ≻ A₁ ≻ A₃	A₂ ≻ A₄ ≻ A₁ ≻ A₃	A₂ ≻ A₄ ≻ A₁ ≻ A₃

From Table 6, we can see that although the aggregating results are different, the ranking results of alternatives derived from the IVPFWA operator [43], IVPFWG operator [43], IVPF-WAA operator [44] and the IPFWG operator [45] are the same as obtained by our proposed method, that is A₂ ≻ A₄ ≻ A₁ ≻ A₃, and A₂ is the most desirable alternative, which shows that the proposed method is reasonable and effective.

Fig. 2

The discrimination of different methods.

(1) Although many operators have been proposed to aggregate interval-valued Pythagorean fuzzy information [43–45 , 49], all these existing interval-valued Pythagorean fuzzy aggregation operators only focus on the endpoints of the closed interval, while all the values contained in IVPFSs are not full considered, which indicate that much useful information may be lost. Our aggregation operators have the ability to overcome this and improve the accuracy of decision making. Therefore, they are the more suitable operators to handle problems.

(2) From Table 5 and Fig. 1, it is concluded that the HC-IVPFOWQA operator can aggregate the IVPFNs into different the PFN according to attitudinal character λ, that is to say, our operators consider the DMs’ risk preferences by parameter λ, which are very suitable for the MAGDM problems. Zhou et al. [56] developed the combined continuous interval-valued intuitionistic fuzzy ordered weighted averaging (CC-IVIFOWA) operator, which can also reflect the DMs’ risk preferences, but the CC-IVIFOWA operator is invalid to handle interval-valued Pythagorean fuzzy information. From the managerial implication, parameter λ expresses the optimistic character of DMs. 0 ≤ λ < 0.5, λ = 0.5 and 0.5 < λ ≤ 1 denotes the DMs’ attitude in regard to pessimistic tendency, neutral and optimistic tendency, respectively. According to the practical application, the DMs can flexibly select the appropriate values of parameter λ respectively.

Moreover, it is possible to check the discrimination of these methods, and the results are shown in Fig. 2. From Fig. 2 we can see that results by IVPFWA operator [43], IVPFWG operator [43], IVPF-WAA operator [44] and IPFWG operator [45] are quite close, these results of decision values can’t clearly distinguish. On the contrary, our proposed method is clearly distinguished.

7. Conclusions

In this paper, by combining IVPFNs and C-OWQA operator, we have provided an interesting topic about continuous interval-valued Pythagorean fuzzy aggregation operators. The major contributions in this paper can be summarized as follows: (1) We proposed the C-IVPFOWQA operator; (2) We defined some new score function, accuracy function and comparison rules of IVPFSs based on C-IVPFOWQA operator; (3) We developed WC-IVPFOWQA operator, OWC-IVPFOWQA operator and HC-IVPFOWQA operator, and discussed their some desirable properties and special cases; (4) We introduced an effective interval-value Pythagorean fuzzy MAGDM method; and (5) The proposed method is applied to address the risk evaluation problem of high-tech enterprises, and comparison analysis is also given.

In the future, we will extend the proposed method to deal with large-scale group decision making problems [61, 62] and many other fields under uncertain environment, such as hesitant fuzzy linguistic, probabilistic interval-valued Pythagorean fuzzy set, interval-valued Pythagorean fuzzy linguistic variable set^, and so on [63 –69].

Footnotes

Acknowledgments

The authors are very grateful to the anonymous reviewers for their valuable comments and constructive suggestions that greatly improved the quality of this paper. The work was partly supported by the National Natural Science Foundation of China (No. 61304173).

Appendix

Proof. Assume that $α = h_{λ} ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) = (μ_{α}, ν_{α})$ . For all j, we have $min_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})} \leq f_{λ} (μ_{{\tilde{α}}_{j}}) \leq max_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})}$ , since $(min_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})})^{2} \leq (f_{λ} (μ_{{\tilde{α}}_{j}}))^{2} \leq (max_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})})^{2}$ , then we obtain $1 - \prod_{j = 1}^{n} {(1 - {(f_{λ} (μ_{{\tilde{α}}_{j}}))}^{2})}^{ω_{j}} \leq 1 - \prod_{j = 1}^{n} {(1 - {(\max_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})})}^{2})}^{ω_{j}} = 1 - {(1 - {(\max_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})})}^{2})}^{\sum_{j = 1}^{n} ω_{j}} = {(\max_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})})}^{2},$ and $1 - \prod_{j = 1}^{n} {(1 - {(f_{λ} (μ_{{\tilde{α}}_{j}}))}^{2})}^{ω_{j}} \geq 1 - \prod_{j = 1}^{n} {(1 - {(\min_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})})}^{2})}^{ω_{j}} = 1 - {(1 - {(\min_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})})}^{2})}^{\sum_{j = 1}^{n} ω_{j}} = {(\min_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})})}^{2},$

It follows that $min_{j} {f_{λ} (μ_{{\tilde{a}}_{j}})} \leq \sqrt{1 - \prod_{j = 1}^{n} (1 - (f_{λ} (μ_{{\tilde{a}}_{j}}))^{2})^{ω_{j}}} \leq max_{j} {f_{λ} (μ_{{\tilde{a}}_{j}})},$

Similarly, we obtain $min_{j} {f_{1 - λ} (ν_{{\tilde{α}}_{j}})} \leq \prod_{j = 1}^{n} (f_{1 - λ} (ν_{{\tilde{α}}_{j}}))^{ω_{j}} \leq max_{j} {f_{1 - λ} (ν_{{\tilde{α}}_{j}})},$

From Definition 14, we have $(min_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})})^{2} - (max_{j} {f_{1 - λ} (ν_{{\tilde{α}}_{j}})})^{2} \leq S (α) = μ_{α}^{2} - ν_{α}^{2} \leq (max_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})})^{2} - (min_{j} {f_{1 - λ} (ν_{{\tilde{α}}_{j}})})^{2},$

Then S (α^-) ≤ S (α) ≤ S (α⁺).

If S (α^-) < S (α) < S (α⁺), from Definition 14, we obtain ${\tilde{α}}^{-} < α < {\tilde{α}}^{+}$ , the result holds.

If S (α) = S (α^-), then $(min_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})})^{2} - (max_{j} {f_{1 - λ} (ν_{{\tilde{α}}_{j}})})^{2} = μ_{α}^{2} - ν_{α}^{2}$ , so $μ_{α}^{2} = (min_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})})^{2}$ , $ν_{α}^{2} = (max_{j} {f_{1 - λ} (ν_{{\tilde{α}}_{j}})})^{2}$ , from Definition 14, we obtain $H (α) = μ_{α}^{2} + ν_{α}^{2} = (min_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})})^{2} + (max_{j} {f_{1 - λ} (ν_{{\tilde{α}}_{j}})})^{2} = H (α^{-})$ , hence α = α^-, the result holds.

If S (α) = S (α⁺), then $μ_{α}^{2} = (max_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})})^{2}$ , $ν_{α}^{2} = (min_{j} {f_{1 - λ} (ν_{{\tilde{α}}_{j}})})^{2}$ , from Definition 14, we obtain $H (α) = μ_{α}^{2} + ν_{α}^{2} = (max_{j} {f_{λ} (μ_{{\tilde{α}}_{j}})})^{2} + (min_{j} {f_{1 - λ} (ν_{{\tilde{α}}_{j}})})^{2} = H (α^{+})$ , hence α = α⁺, the result holds.

Theorem 9. (Monotonicity) Let ${\tilde{α}}_{j} = (μ_{{\tilde{α}}_{j}}, ν_{{\tilde{α}}_{j}}) = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ and ${\tilde{β}}_{j} = (μ_{{\tilde{β}}_{j}}, ν_{{\tilde{β}}_{j}}) = ([a_{{\tilde{β}}_{j}}, b_{{\tilde{β}}_{j}}], [c_{{\tilde{β}}_{j}}, d_{{\tilde{β}}_{j}}])$ (j = 1, 2, …, n) be two collections of IVPFNs, h_λ be the WC-IVPFOWQA operator. If $f_{λ} (μ_{{\tilde{α}}_{j}}) \geq f_{λ} (μ_{{\tilde{β}}_{j}})$ and $f_{1 - λ} (ν_{{\tilde{α}}_{j}}) \leq f_{1 - λ} (ν_{{\tilde{β}}_{j}})$ for all j, then $h_{λ} ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \geq h_{λ} ({\tilde{β}}_{1}, {\tilde{β}}_{2}, \dots, {\tilde{β}}_{n}) .$

Proof. Since $f_{λ} (μ_{{\tilde{α}}_{j}}) \geq f_{λ} (μ_{{\tilde{β}}_{j}})$ and $f_{1 - λ} (ν_{{\tilde{α}}_{j}}) \leq f_{1 - λ} (ν_{{\tilde{β}}_{j}})$ . For all j, we have $(1 - (f_{λ} (μ_{{\tilde{α}}_{j}}))^{2})^{ω_{j}} \geq (1 - (f_{λ} (μ_{{\tilde{β}}_{j}}))^{2})^{ω_{j}}$ , $(f_{1 - λ} (ν_{{\tilde{α}}_{j}}))^{ω_{j}} \leq (f_{1 - λ} (ν_{{\tilde{β}}_{j}}))^{ω_{j}}$ ,Then $1 - \prod_{j = 1}^{n} (1 - (f_{λ} (μ_{{\tilde{α}}_{j}}))^{2})^{ω_{j}} \geq 1 - \prod_{j = 1}^{n} (1 - (f_{λ} (μ_{{\tilde{β}}_{j}}))^{2})^{ω_{j}}, {(\prod_{j = 1}^{n} (f_{1 - λ} (ν_{{\tilde{α}}_{j}}))^{ω_{j}})}^{2} \leq {(\prod_{j = 1}^{n} (f_{1 - λ} (ν_{{\tilde{β}}_{j}}))^{ω_{j}})}^{2},$

Furthermore, $1 - \prod_{j = 1}^{n} (1 - (f_{λ} (μ_{{\tilde{α}}_{j}}))^{2})^{ω_{j}} - {(\prod_{j = 1}^{n} (f_{1 - λ} (ν_{{\tilde{α}}_{j}}))^{ω_{j}})}^{2} \geq 1 - \prod_{j = 1}^{n} (1 - (f_{λ} (μ_{{\tilde{β}}_{j}}))^{2})^{ω_{j}} - {(\prod_{j = 1}^{n} (f_{1 - λ} (ν_{{\tilde{β}}_{j}}))^{ω_{j}})}^{2},$

Assume that $h_{λ} ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) = α$ , $h_{λ} ({\tilde{β}}_{1}, {\tilde{β}}_{2}, \dots, {\tilde{β}}_{n}) = β$ , from Definition 14, we have S (α) ≥ S (β). If S (α) > S (β), then α > β, the result holds.

If S (α) = S (β), then $1 - \prod_{j = 1}^{n} (1 - (f_{λ} (μ_{{\tilde{α}}_{j}}))^{2})^{ω_{j}} - {(\prod_{j = 1}^{n} (f_{1 - λ} (ν_{{\tilde{α}}_{j}}))^{ω_{j}})}^{2} = 1 - \prod_{j = 1}^{n} (1 - (f_{λ} (μ_{{\tilde{β}}_{j}}))^{2})^{ω_{j}} - {(\prod_{j = 1}^{n} (f_{1 - λ} (ν_{{\tilde{β}}_{j}}))^{ω_{j}})}^{2},$

Since $f_{λ} (μ_{{\tilde{α}}_{j}}) \geq f_{λ} (μ_{{\tilde{β}}_{j}})$ and $f_{1 - λ} (ν_{{\tilde{α}}_{j}}) \leq f_{1 - λ} (ν_{{\tilde{β}}_{j}})$ , we have $1 - \prod_{j = 1}^{n} (1 - (f_{λ} (μ_{{\tilde{α}}_{j}}))^{2})^{ω_{j}} = 1 - \prod_{j = 1}^{n} (1 - (f_{λ} (μ_{{\tilde{β}}_{j}}))^{2})^{ω_{j}},$ ${(\prod_{j = 1}^{n} (f_{1 - λ} (ν_{{\tilde{α}}_{j}}))^{ω_{j}})}^{2} = {(\prod_{j = 1}^{n} (f_{1 - λ} (ν_{{\tilde{β}}_{j}}))^{ω_{j}})}^{2},$

Thus $1 - \prod_{j = 1}^{n} (1 - (f_{λ} (μ_{{\tilde{α}}_{j}}))^{2})^{ω_{j}} + {(\prod_{j = 1}^{n} (f_{1 - λ} (ν_{{\tilde{α}}_{j}}))^{ω_{j}})}^{2} = 1 - \prod_{j = 1}^{n} (1 - (f_{λ} (μ_{{\tilde{β}}_{j}}))^{2})^{ω_{j}} + {(\prod_{j = 1}^{n} (f_{1 - λ} (ν_{{\tilde{β}}_{j}}))^{ω_{j}})}^{2},$

Therefore, H (α) = H (β), which implies that α = β, so the result holds.

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Continuous interval-valued Pythagorean fuzzy aggregation operators for multiple attribute group decision making

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1 Pythagorean fuzzy set

4.1 Weighted C-IVPFOWQA operator

Table 1 The interval-valued Pythagorean fuzzy decision matrix R ˜ ( 1 ) for e1

Table 4 The Pythagorean fuzzy collective decision matrix R

Table 6 The aggregating results by different methods

Footnotes

Acknowledgments

Appendix

References

Table 1
The interval-valued Pythagorean fuzzy decision matrix ${\tilde{R}}^{(1)}$ for e₁

Table 4
The Pythagorean fuzzy collective decision matrix R

Table 6
The aggregating results by different methods