Approaches to multi-attribute decision making with risk preference under extended Pythagorean fuzzy environment

1 Introduction

Due to high complexity of socioeconomics it is very difficult to obtain appropriate and accurate data for real world decision making. To overcome this situation in 1965 Zadeh [24] initiated the notion of fuzzy set and widely has been in many fields in our modern society [3, 26]. The theory of fuzzy set has been extended and generalized by many others to different forms of fuzzy set. Some of these extensions are intuitionistic fuzzy set (IFS) [1], interval-valued intuitionistic fuzzy set (IVIFS) [2], hesitant fuzzy set (HFS) [18, 19], dual hesitant fuzzy set (DHFS) [29], Pythagorean fuzzy set (PFS) [22], interval-valued Pythagorean fuzzy set (IVPFS) (15), Pythagorean hesitant fuzzy set (PHFS) (10), extended intuitionistic fuzzy set (EIFS)(28) to further know the fuzziness and hesitation of the real data by giving objective mathematical symbols.

In the above-mentioned fuzzy sets, the IFS is an intuitionistic fuzzy set also known as Atanassov-IFS is a widely used extension of fuzzy set is the immediate consideration of membership and non-membership degrees [1]. The IFS fulfill the condition that the sum of membership degree and non-membership degree is less than or equal to 1. The IFS is a better tool to deal with uncertainty and hesitant. Moreover the concept of HFS has been proposed to permit the degree of membership consisting of an element set to be represented as several possible values between 0 and 1 [18, 19]. The HFS is very powerful tool in decision making environment in which people are hesitant to provide their preferences. Then, the DHFS has been developed by combining the IFS and the HFS and consists of the hesitancy membership function and hesitancy non-membership function [29]. Yager [22] developed the concept of PFS. The PFS fulfill the conditions that the square sum of the membership degree and non-membership degree is less than or equal to 1. PFS is a more suitable to deal with uncertainty and fuzziness. Since PFS is a powerful tool to deal with vagueness and hesitation therefore many authors have applied the concept to multi-attribute decision making (MADM) problems [5, 27]. Further the concept of IVPFS [15] has been developed and applied to practical decision making problems [12, 13]. Recently, researchers made some progresses in decision-making with risk preferences. Adding risk preferences can affect the decision-maker’s psychological factors into the decision-making process. That can reduce the error of decision results and improve the quality of the decision-making. Zhou and Xu [28] developed the concept of EIFS based on hesitant fuzzy membership function and applied the concept to decision making with risk preferences. The EIFS is the generalization of IFS with hesitant fuzzy membership function. However PFS is more suitable than IFS to deal with vagueness and uncertainty. The motivation of presenting PFSs is that in the real-life decision process, the sum of the support (membership) degree and the against (nonmembership) degree to which an alternative satisfying a criterion provided by the decision maker may be bigger than 1 but their square sum is equal to or less than 1. Proposing a considerable more generalized fuzzy set by combining PFS and a HFS is therefore a reasonable notion; this generalized fuzzy set can have all the characters, such as the concurrent consideration of both membership and non-membership degrees and the general appearance of the membership degree with a set of possible values. Based on this idea the concept of PHFS has been developed and investigated, but the drawback of calculations in the HFS has also been presented in the PHFS and has even been enlarged for its two hesitant fuzzy parts [18]. Therefore in this study generalized the concept of PFS with hesitant fuzzy membership function and introduce the concept of extended PFS (EPFS). The EPFS includes the hesitant membership function and real number non-membership function and reduce the multi calculation involved in the PHFS. To do this the remainder of the paper is organized as under.

In section 2 we briefly review some basic concepts of HFS, DHFS, PFS, EIFS and PHFS. In section 3, we define EPFSs, C-EPFSs and P-EPFSs, provides the corresponding EPFNs, C-EPFNs and P-EPFNs. We develop some basic operational laws and the comparison method of the proposed concepts. In Section 4, based on the proposed operations we develop the aggregation operators for EPFSs, C-EPFSs and P-EPFSs and discuss their properties in detail. In section 5 we present a numerical example of risk decision making to demonstrate the applications of the preceding concepts and approaches. Concluding remark is in section 6.

2 Preliminaries

This section reviews some basic concept and operation of IFS, HFS, PFS, PHFS and EIFS to contextualize the concept of EPFSs.

Definition 1. [18] Let X = (x₁, x₂,. . . , x _n ) be a fixed set. Then hesitant fuzzy set (HFS) H in X can be define as follows. H = { 〈 x i , h H ( x i ) 〉 | x i ∈ X }(1) where h _H  (x _i ) denotes the set of some values belonging to [0, 1], that is the possible membership degree of the element x _i  ∈ X to the set H. For convenience we denote a hesitant fuzzy number (HFN) by h = h _H  (x _i ) and HFN the set of all HFNs.

Definition 2. [22] Let X = (x₁, x₂,. . . , x _n ) be a fixed set. Then a Pythagorean fuzzy set (PFS) P ⌢ in X can be defined as follows: P = { 〈 x i , γ P ( x i ) , μ P ( x i ) 〉 | x i ∈ X }(2) where γ _P  (x _i ) and μ _P  (x _i ) are mappings from X to [0, 1], such that 0 ≤ γ _P  (x _i ) ≤1, 0 ≤ μ _P  (x _i ) ≤1 and also 0 ≤ γ P 2 ( x i ) + μ P 2 ( x i ) ≤ 1 , for all x _i  ∈ X, here γ _P  (x _i ) and μ _P  (x _i ) denotes the membership degree and non-membership degree of element x ∈ X to set P, respectively. Let π P ( x i ) = 1 - γ P ( x i ) - μ P ( x i ) . Then it is commonly called the Pythagorean fuzzy index of element x _i  ∈ X to set P, representing the degree of indeterminacy of x _i to P. Also 0 ≤ π _P  (x _i ) ≤1, for every x _i  ∈ X. We denote the Pythagorean fuzzy number (PFN) byp =〈 γ _p , μ _p  〉.

Definition 3. [10] Let X = (x₁, x₂,. . . , x _n ) be a fixed set. A Pythagorean hesitant fuzzy set abbreviated as PHFS P _H in X is an object with the following notion. P H = { 〈 x i , γ P H ( x i ) , μ P H ( x i ) | x i ∈ X 〉 }(3) where γ _{P H}  (x _i ) and μ _{P H}  (x _i ) are mappings from X to [0, 1], denoting a possible degree of membership and non-membership degree of element x _i  ∈ X in P _H respectively, and for each element x _i  ∈ X, ∀ h _{P H}  (x _i ) ∈ γ _{P H}  (x _i ), ∃ h P H ' ( x i ) ∈ μ P H ( x i ) such that 0 ≤ h P H 2 ( x i ) + h P H ' 2 ( x i ) ≤ 1 , and ∀ h P H ′ ( x i ) ∈ μ P H ( x i ) , ∃h _{P H}  (x _i ) ∈ γ _{P H}  (x _i ) such that 0 ≤ h P H 2 ( x i ) + h P H ' 2 ( x i ) ≤ 1 . Moreover, PHES(X) denotes the set of all elements of PHFSs. If X has only one element〈x _i , γ _{P H}  (x _i ) , μ _{P H}  (x _i ) 〉, then it is said to be Pythagorean hesitant fuzzy number and is denoted by p _H  = 〈γ _{p H} , μ _{p H} 〉 for convenience. We denote the set of all PHFNs by PHFNS. For all x _i  ∈ X if γ _{P H}  (x _i ) and μ _{P H}  (x _i ) have only one element. Then the PHFS become a PFS. If the non-membership degree is{0}, then PHFS become a HFS.

In This section presents the concept of extended Pythagorean fuzzy set, which include the hesitant fuzzy membership and real number non-membership degree, which fulfill the conditions that the square sum of the maximum (max) of hesitant fuzzy membership and the real number non-membership degree is less than or equal to 1. This section further develop the credible EPFS (C-EPFS) and possible EPFS (P-EPFS) in which non-membership degrees are obtained based on the conversion formula and the max membership degree, and provides the corresponding credible extended Pythagorean fuzzy number (C-EPFN) and possible EPFN (P-EPFN), to mine the hesitant fuzzy membership information in the EPFN and to avoid the logical difficulty of simultaneously providing membership and non-membership degrees in the EPFN. Throughout in this paper a extended Pythagorean fuzzy set will be denoted by P _H and extended Pythagorean hesitant fuzzy number by P ⌢ .

Definition 4. Let X = (x₁, x₂,. . . , x _n ) be a fixed set. An extended Pythagorean fuzzy set abbreviated as EPHFS P _E in X is an object with the following notion. P E = { 〈 x i , h P E ( x i ) , μ P E ( x i ) | x i ∈ X 〉 }(4) where h _{P E}  (x _i ) is a hesitant fuzzy set and μ _{P E}  (x _i ) is a fuzzy number both are mappings from X to [0, 1], denoting a possible degree of membership and non-membership degree of element x _i  ∈ X in P _E respectively, and for each element x ∈ X, such that 0 ≤ ( max { h P E ( x i ) } ) 2 + μ P E 2 ( x i ) ≤ 1 x _i  ∈ X.

Here, h _{P E}  (x _i ) ⊆ [0, 1] and μ _{P E}  (x _i ) is a fuzzy number. Moreover, EPFS(X) denotes the set of all elements of EPFSs. Also the pair P ⌢ = 〈 h P ⌢ , μ P ⌢ 〉 is called the Extended Pythagorean fuzzy number (EPFN).

In the following we present some demonstration and comparison between four simplified forms of fuzzy numbers.

1) PFN: (γ _P , μ _P )

2) HFN: h ={ γ₁, γ₂,. . . , γ _n  }

3) PHFN: 〈f _P , g _P 〉 = 〈 {f₁, f₂,. . . , f _n } , {g₁, g₂,. . . , g _n } 〉

4) EPFN: 〈 h P ⌢ , μ P ⌢ 〉 = 〈 { γ 1 , γ 2 , , … , γ n } μ p ⌢ 〉

In EPFN 〈 h P ⌢ , μ P ⌢ 〉 = 〈 { γ 1 , γ 2 , , … , γ n } μ p ⌢ 〉 , the membership is denoted by h P ⌢ , the HFN, and μ P ⌢ is the non-membership degree. Hence this new developed fuzzy number can effectively synthesize the characteristics of the PFN and HFN that are similar to PHFN. The EPFN has only a HFN part, which is more simplified than of the PHFN when applied to decision making Theory.

On the bases of operational laws defined in [21], [22] for HFNs and PFNs respectively, we develop some new operational laws as follows:

Definition 5. Let p ⌢ i = 〈 h p ⌢ i , μ p ⌢ i 〉 ( i = 1 , 2 , 3 ) , be three EPFNs, and λ > 0, then their operations are defined as follows. The result obtain by these operations is also Extended Pythagorean fuzzy numbers (EPFNs). ( 1 ) p ⌢ 1 ∪ p ⌢ 2 = ( max ⁡ { h p ⌢ 1 , h p ⌢ 2 } , min ⁡ { μ p ⌢ 1 , μ p ⌢ 2 } ) , ( 2 ) p ⌢ 1 ∩ p ⌢ 2 = ( min ⁡ { h p ⌢ 1 , h p ⌢ 2 } , max ⁡ { μ p ⌢ 1 , μ p ⌢ 2 } ) , ( 3 ) p ⌢ c = ( μ p ⌢ , h p ⌢ ) , ( 4 ) p ⌢ 1 ⊕ p ⌢ 2 = ( ∪ γ p ⌢ 1 ∈ h γ p ⌢ 1 , γ p ⌢ 2 ∈ h γ p ⌢ 2 { γ p ⌢ 1 2 + γ p ⌢ 2 2 − γ p ⌢ 1 2 γ p ⌢ 2 2 } , μ p ⌢ 1 μ p ⌢ 2 )

( 5 ) p ⌢ 1 ⊗ p ⌢ 2 = ( ∪ γ p ⌢ 1 ∈ h γ p ⌢ 1 , γ p ⌢ 2 ∈ h γ p ⌢ 2 { γ p ⌢ 1 γ p ⌢ 2 } , { μ p ⌢ 1 2 + μ p ⌢ 2 2 − μ p ⌢ 1 2 μ p ⌢ 1 2 } ) ( 6 ) λ p ⌢ = ( ∪ γ p ⌢ ∈ h p ⌢ { 1 − ( 1 − ( γ p ⌢ ) 2 ) λ } , ( μ p ⌢ ) λ ) , λ > 0 ( 7 ) p ⌢ λ = ( ∪ γ p ⌢ ∈ h p ⌢ { γ p ⌢ λ } , 1 − ( 1 − ( μ p ⌢ ) 2 ) λ ) , λ > 0

Following example shows that for any three EPFNs p ⌢ 1 , p ⌢ 2 , and λ > 0 p ⌢ 1 ⊕ p ⌢ 2 , p ⌢ 1 ⊗ p ⌢ 2 , λ p ⌢ 1 and p ⌢ 1 λ are also EPFNs.

To compare two EPFNs in following we define score function and some basic laws on the basis of the score function.

Definition 6. Let p ⌢ = 〈 h p ⌢ , μ p ⌢ 〉 is an EPFN. Then we define the score function of p ⌢ as follows: S ( p ⌢ ) = ( 1 l γ p ⌢ ∈ h p ⌢ ∑ γ p ⌢ ∈ h p ⌢ γ p ⌢ ) 2 - ( μ p ⌢ ) 2(5)

Where S ( p ⌢ ) ∈ [ - 1 , 1 ] , l γ p ⌢ denotes the number of elements in h p ⌢ and μ p ⌢ denotes the non-membership.

Definition 7. Let p ⌢ = 〈 h p ⌢ , μ p ⌢ 〉 is EPFNs. Then the accuracy degree of p ⌢ is denoted by α ¯ ( p ⌢ ) and can be defined as follows: α ¯ ( p ⌢ ) = ( 1 l γ p ⌢ ∑ γ p ⌢ ∈ h p ⌢ γ p ⌢ - S ( p ⌢ ) ) 2 + ( μ p ⌢ ) 2(6)

Here we can see that α ¯ ( p ⌢ ) is just the mean value in statistics, and α ¯ ( p ⌢ ) is just the standard variance, which reflects the accuracy degree between all values in the EPFN p ⌢ and their mean value. Inspired by this idea, based on the score S ( p ⌢ ) and the accuracy degree α ¯ ( p ⌢ ) , we can compare and rank, two EPFNs in the following as:

Definition 8. let p ⌢ 1 and p ⌢ 2 be two EPFNs, S ( p ⌢ 1 ) be the score of p ⌢ 1 , S ( p ⌢ 2 ) be the score of p ⌢ 2 , and α ¯ ( p ⌢ 1 ) be the deviation degree of p ⌢ 1 , α ¯ ( p ⌢ 2 ) be the deviation degree of p ⌢ 2 Then

1) If S ( p ⌢ 1 ) < S ( p ⌢ 2 ) , then p ⌢ 1 < p ⌢ 2 .

2) If S ( p ⌢ 1 ) > S ( p ⌢ 2 ) , then p ⌢ 1 > p ⌢ 2 .

3) If S ( p ⌢ 1 ) = S ( p ⌢ 2 ) , then p ⌢ 1 ∼ p ⌢ 2 .

i) If S ( p ⌢ 1 ) < S ( p ⌢ 2 ) , then p ⌢ 1 < p ⌢ 2 .

ii) If α ¯ ( p ⌢ 1 ) > α ¯ ( p ⌢ 2 ) , then p ⌢ 1 > p ⌢ 2 .

ii) If α ¯ ( p ⌢ 1 ) = α ¯ ( p ⌢ 2 ) , then p ⌢ 1 ∼ p ⌢ 2 .

Definition 9. If X = (x₁, x₂,. . . , x _n ) is a fixed set, then a C-EPFS C on X can be defined as follows: C ⌢ 2 = { ( x i , h C ⌢ 2 ( x i ) , ( 1 - ( max { h C ⌢ 2 ( x i ) } ) 2 ) | x i ∈ X }(7)

Where h C ⌢ 2 ( x i ) and 1 - ( max { h C ⌢ 2 ( x i ) } ) 2 represent the membership and non-membership degree of the element x _i to C h C ⌢ 2 ( x i ) is HFN with h C ⌢ 2 ( x i ) ⊆ [ 0 , 1 ] and 1 - ( max { h C ⌢ 2 ( x i ) } ) 2 is a real number. Furthermore the pair ( x i , h C ⌢ 2 ( x i ) , ( 1 - ( max { h C ⌢ 2 ( x i ) } ) 2 ) represents the credible extended Pythagorean fuzzy number (C-EPFN).

Definition 10. If X = (x₁, x₂,. . . . . . , x _n ) is a fixed set then an P-EPFS p ⌢ on X can be defined as follows: p ⌢ = { 〈 x i , h p ( x i ) , [ 0 , 1 - ( { max h p ( x i ) } ) 2 ] | x i ∈ X 〉 }(8)

Where h p ⌢ ( x i ) and [ 0 , 1 - ( { max γ p ⌢ ( x i ) } ) 2 ] represent the membership and non-membership degrees of the element x _i to p ⌢ , h p ⌢ ( x i ) is a HFN with h p ⌢ ( x i ) ⊆ [ 0 , 1 ] and [ 0 , 1 - ( max h p ⌢ ( x i ) } ) 2 ] ⊆ [ 0 , 1 ] . The pair 〈 h p ⌢ ( x i ) , [ 0 , 1 - ( { max γ p ⌢ ( x i ) } ) 2 ] 〉 represent a possible extended Pythagorean fuzzy number (P-EPFN).

For computational convenience, the C-EPFN and the P-EPFN may be denoted by C ⌢ 2 = 〈 h i , 1 - ( max { h i } ) 2 〉 and ρ ⌢ = 〈 h i , [ 0 , 1 - ( { max ( h i ) } ) 2 ] 〉 respectively, where h _i is a HFN. The first non-membership degree (1 - (max {h _i }) ²) is the maximum number of {1 - h _i |h _i ∈ H } and a credible non-membership degree. The second non-membership degree [0, 1 - ({max h _i }) ²] is an interval-valued non-membership degree.

Any real number or HFE can be transformed into C-EPFN and P-EPFN through the following Formulas: μ ¯ → ( ( μ ¯ ) , ( 1 - ( μ ¯ ) 2 ) ) and μ ¯ → ( ( μ ¯ ) , [ 0 , ( 1 - ( μ ¯ ) 2 ) ] ) γ i → ( γ i , ( 1 - { max γ i } 2 ) ) and γ i → ( γ i , [ 0 , ( 1 - { max γ i } 2 ) ] )

Theorem 1. If ρ ⌢ i = 〈 h i , [ 0 , 1 - ( max { h i } ) 2 ] 〉 (i = 1, 2) and ρ = 〈h, [0, 1 - (max {h}) ²] 〉 are three P-EPFN with h₁, h₂ and h being three HFNs and λ > 0, then we the following operations and the aggregated values derived by these values are also the P-EPFNs: ρ ⌢ ' = 〈 [ 0 , 1 - ( max { h } ) 2 ] , h 〉 ( 2 ) p ⌢ 1 ∩ p ⌢ 2 = { h ∈ h 1 ∩ h 2 | h ≤ min { ( max { h 1 } ) 2 , ( max { h 2 } ) 2 } , [ 0 , max { 1 - ( max { h 1 } ) 2 , ( max { h 2 } ) 2 } ] } ( 3 ) p ⌢ 1 ∪ p ⌢ 2 = { h ∈ h 1 ∪ h 2 | h ≥ max { ( min { h 1 } ) 2 , ( min { h 2 } ) 2 } , [ 0 , min { 1 - ( max { h 1 } ) 2 , 1 - ( max { h 2 } ) 2 } ] } ( 4 ) p ⌢ 1 ⊕ p ⌢ 2 = 〈 ∪ γ 1 ∈ h 1 , γ 2 ∈ h 2 { γ 1 2 + γ 2 2 - γ 1 2 γ 2 2 } , [ 0 , ( 1 - ( max { h 1 } ) 2 ) . ( 1 - ( max { h 2 } ) 2 ) ] 〉 ( 5 ) p ⌢ 1 ⊗ p ⌢ 2 = 〈 ∪ γ 1 ∈ h 1 , γ ∈ h 2 { γ 1 γ 2 } , [ 0 , 1 - ( max { h 1 h 2 } ) 2 ] 〉 ( 6 ) λ ρ ⌢ = 〈 ∪ γ ∈ h { 1 - ( 1 - γ 2 ) λ } , [ 0 , 1 - ( max { h } ) 2 ) λ ] 〉 , λ > 0 ; ( 7 ) ρ ⌢ λ = 〈 ∪ γ ′ ∈ γ { γ λ } , [ 0 , 1 - ( ( max { γ } ) 2 ) λ ] 〉 , λ > 0 Proof: Proof is easy. Moreover on the bases of score and accuracy functions for EPFNs in the following we propose the score and accuracy functions for C-EPHFNs and P-EPHFNs as follows respectively. S ( c ⌢ i ) = ( 1 l γ c ⌢ ∈ h c ⌢ ∑ γ c ⌢ ∈ h c ⌢ γ c ⌢ ) 2 − ( 1 − ( max ⁡ { ( h c ⌢ ) } ) 2 ) 2 ,(9) α ( c ⌢ i ) = ( 1 l γ C ⌢ 2 ∈ h C ⌢ 2 ∑ γ C ⌢ 2 ∈ h C ⌢ 2 γ C ⌢ 2 ) 2 + ( 1 - ( max { ( h C ⌢ 2 ) } ) 2 ) 2(10) and S ( ρ ⌢ i ) = ( 1 l γ ρ ⌢ ∈ h ρ ⌢ ∑ γ ρ ⌢ ∈ h ρ ⌢ γ ρ ⌢ ) 2 − 1 2 ( 1 − ( max ⁡ { ( h ρ ⌢ ) } ) 2 ) 2 ,(11) α ( ρ ⌢ i ) = ( 1 l γ ρ ⌢ ∈ h ρ ⌢ ∑ γ ρ ⌢ ∈ h ρ ⌢ γ ρ ⌢ ) 2 + 1 2 ( 1 − ( max ⁡ { ( h ρ ⌢ ) } ) 2 ) 2(12) □

Yager [23] and Xu [21] developed the aggregation operators for PFS and HFS respectively. Based on these aggregation operators and the operations for EPFS, we propose a series of aggregation operators and investigate some of their desirable properties.

Definition 11. Let ρ ⌢ i = 〈 h ρ ⌢ i , μ ρ ⌢ i 〉 (i=1,2, ... ,n) be a collection of all EPFNs, and w = (w₁, w₂,. . . , w _n )  ^T be the weight vector of ρ ⌢ i (i = 1,2, ... ,n) with w _i  ≥ 0 (i = 1,2, ... ,n) where w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 . Then Extended Pythagorean fuzzy weighted averaging (EPFWA) operator is a mapping EPFWA : EPFN ⁿ  → EPFN can be defined as

E P F W A ( p ⌢ 1 , p ⌢ 2 , … , p ⌢ n ) = ω 1 p ⌢ 1 ⊕ ω 2 p ⌢ 2 ⊕ … ⊕ ω n p ⌢ n . ∪ γ p ⌢ 1 ∈ h p ⌢ 1 , γ p ⌢ 2 ∈ h p ⌢ 2 , … , γ p ⌢ n ∈ h p ⌢ n = 〈 { 1 − ∏ i = 1 n ( 1 − γ p ⌢ i 2 ) ω i } , 〉 ∏ i = 1 n ( μ p ⌢ i ω i )(13) and the EPFWA operator is said to be an extended Pythagorean fuzzy weighted averaging operator. In particular if w = ( 1 n , 1 n , . . . , 1 n ) T then the EPFWA operator reduces to extended Pythagorean fuzzy averaging (EPFA) operator.

Definition 12. Let ρ ⌢ i = 〈 h ρ ⌢ i , μ ρ ⌢ i 〉 (i = 1,2, ... ,n) be a collection of all EPFNs, and w = (w₁, w₂,. . . , w _n ) be the weight vector of ρ ⌢ i (i = 1,2, ... ,n) with w _i  ≥ 0 (i = 1,2, ... ,n) such that w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 . Then, Extended Pythagorean fuzzy weighted geometric (EPFWG) operator is a mapping EPFWG : EPFN ⁿ  → EPFN can be defined as

E P F W G ( p ⌢ 1 , p ⌢ 2 , … , p ⌢ n ) = p ⌢ 1 ω 1 ⊗ p ⌢ 2 ω 2 ⊗ , … , ⊗ p ⌢ n ω n ∪ γ p ⌢ 1 ∈ h p ⌢ 1 1 , γ p ⌢ 2 ∈ h p ⌢ 2 , … , = 〈 γ p ⌢ n ∈ h p ⌢ { ∏ i = 1 n ( γ p ⌢ i ) ω i } , 〉 ( 1 − ∏ i = 1 n ( 1 − μ p ⌢ i 2 ) ω i )(14) and the EPFWG operator is said to be an Extended Pythagorean fuzzy weighted geometric operator. In particular if w = ( 1 n , 1 n , . . . , 1 n ) T then the EPFWG operator reduces to extended Pythagorean fuzzy geometric (EPFG) operator.

Definition 13. Let ρ ⌢ i = 〈 h ρ ⌢ i , μ ρ ⌢ i 〉 (i = 1,2, ... ,n) be a collection of all EPFNs, and w = (w₁, w₂,. . . , w _n )  ^T be the weight vector of ρ ⌢ i (i = 1,2, ... ,n) with w _i  ≥ 0 (i = 1,2, ... ,n) where w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 . Then generalized extended Pythagorean fuzzy weighted averaging (GEPFWA) operator is a mapping GEPFWA : EPFN ⁿ  → EPFN can be defined as

GEPFWA ( ρ ⌢ 1 , ρ ⌢ 2 , . . . , ρ ⌢ n ) = 〈 ∪ γ ρ ⌢ 1 ∈ η ρ ⌢ 1 , γ ρ ⌢ 2 ∈ η ρ ⌢ 2 , . . . , γ ρ ⌢ n ∈ η ρ ⌢ n { ( 1 - ∏ i = 1 n ( 1 - γ p i ⌢ 2 λ ) w i ) 1 λ } , ( 1 - ( 1 - ∏ i = 1 n ( 1 - ( 1 - μ p i ⌢ 2 ) λ ) w i ) 1 λ 〉(15) and the GEPFWA operator is said to be a generalized extended Pythagorean fuzzy weighted averaging operator. In particular if λ = 1 then the GEPFWA operator reduces to EPFWA operator.

Definition 14. Let ρ ⌢ i = 〈 h ρ ⌢ i , μ ρ ⌢ i 〉 (i = 1,2, ... ,n) be a collection of all EPFNs, and w = (w₁, w₂,. . . , w _n ) be the weight vector of (i = 1,2, ... ,n) with w _i  ≥ 0 (i = 1,2, ... ,n) such that w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 . Then generalized extended Pythagorean fuzzy weighted geometric (GEPFWG) operator is a mapping GEPFWG : EPFN ⁿ  → EPFN can be defined as GEPFWG ( ρ ⌢ 1 , ρ ⌢ 2 , . . . , ρ ⌢ n ) = 〈 ∪ γ ρ ⌢ 1 ∈ h ρ ⌢ 1 , γ ρ ⌢ 2 ∈ h ρ ⌢ 2 , . . . , γ ρ ⌢ n ∈ h p ⌢ { ( 1 - ( 1 - ∏ i = 1 n ( 1 - ( 1 - γ p i ⌢ 2 ) λ ) w i ) 1 λ } , ( 1 - ∏ i = 1 n ( 1 - μ p i ⌢ 2 λ ) w i ) 1 λ 〉(16) and the GEPFWG operator is said to be a generalized extended Pythagorean fuzzy weighted geometric operator. In particular if λ = 1 then the GEPFWG operator reduces to EPFWG operator.

In the following we present some relationship of the EPFWA operators, EPFWG operator and GEPFWA operators, GEPFWG operator respectively.

Lemma 1. Let ρ ⌢ i > 0 , w _i  > 0 (i = 1,2, ... ,n) and ∑ i = 1 n w i = 1 . Then, ∏ i = 1 n ( ρ ⌢ i ) w i ≤ ∑ i = 1 n w i ρ ⌢ i where the equality holds if and only if p ⌢ 1 = p ⌢ 2 = p ⌢ 3 = . . . = p ⌢ n .

Theorem 2. Let ρ ⌢ i = 〈 h ρ ⌢ i , μ ρ ⌢ i 〉 (i = 1,2, ... ,n) be a collection of EPHFNs. Then

EPFWG ( ρ ⌢ 1 , ρ ⌢ 2 , . . . , ρ ⌢ n ) ≤ EPFWA ( ρ ⌢ 1 , ρ ⌢ 2 , . . . , ρ ⌢ n ) ,(17)

GEPFWG ( ρ ⌢ 1 , ρ ⌢ 2 , . . . , ρ ⌢ n ) ≤ GEPFWA ( ρ ⌢ 1 , ρ ⌢ 2 , . . . , ρ ⌢ n ) ,(18) where w = (w₁, w₂,. . . w _n )  ^T is the weighted vector of ρ ⌢ i (i = 1,2, ... ,n) such that w _i  ∈ [0, 1] (i = 1,2, ... ,n) and ∑ i = 1 n w i = 1 .

Proof. Based on Lemma 1, we can easily prove the Theorem.

Based on the credible extended Pythagorean fuzzy operations and possible extended Pythagorean fuzzy operations in the following we develop credible extended Pythagorean fuzzy weighted averaging (CEPFWA) operator, generalized credible extended Pythagorean fuzzy weighted averaging (GCEPFWA) operator, credible extended Pythagorean fuzzy weighted geometric (CEPFWG) operator and generalized credible extended Pythagorean fuzzy weighted geometric (GCEPFWG) operator.

Definition 15. For a collection of C-EPFNs c i ⌢ = ( h i , ( 1 - ( max { h i } ) 2 ) ) (i = 1,2, ... ,n) w = (w₁, w₂,. . . , w _n )  ^T is the weight vector of c i ⌢ (i = 1,2, ... ,n) with w i ∈ [ 0 , 1 ] , ∑ i = 1 n w i = 1 and λ > 0 . Then,

(1) The credible EPFWA (CEPFWA) operator: CEPFWA ( c 1 ⌢ , c 2 ⌢ , , . . . , c n ⌢ ) = ⊕ i = 1 n ( w i c ⌢ i ) = 〈 ∪ γ 1 ∈ h ( c 1 ⌢ ) , γ 2 ∈ h ( c 2 ⌢ ) , . . . , γ n ∈ h ( c n ⌢ ) { 1 - ∏ i = 1 n ( 1 - γ i 2 ) w i } , ∏ i = 1 n ( 1 - ( max { h i } ) 2 ) w i 〉

(2) The generalized CEPFWA (GCEPFWA) operator: G C E P F W A ( c ⌢ 1 , c ⌢ 2 , , … , c ⌢ n ) = ( ⊕ i = 1 n ( ω i c ⌢ i λ ) ) 1 λ ∪ γ 1 ∈ h ( c ⌢ 1 ) , γ 2 ∈ h ( c ⌢ 2 ) , … , γ n ∈ h ( c ⌢ n ) = 〈 { ( 1 − ∏ i = 1 n ( ( 1 − ( γ i 2 ) λ ) ω i ) 1 λ } , 〉 ( 1 − ( 1 − ∏ i = 1 n ( 1 − ( max ⁡ { h i } ) 2 ) λ ) ω i ) 1 λ

(3) The credible EPFWG (CEPFWG) operator: C E P F W G ( c ⌢ 1 , c ⌢ 2 , , … , c ⌢ n ) = ⊗ i = 1 n ( c ⌢ i ) ω i ∪ γ 1 ∈ h ( c ⌢ 1 ) , γ 2 ∈ h ( c ⌢ 2 ) , , , , , , , , γ n ∈ h ( c ⌢ n ) = 〈 { ∏ i = 1 n ( γ i ) ω i } , ( 1 − ∏ i = 1 n ( max ⁡ ( h i 2 ) ) ) ω i 〉 (4) The generalized CEPFWG (GCEPFWG) operator: G C E P F W G ( c ⌢ 1 , c ⌢ 2 , , … , c ⌢ n ) = 1 λ ( ⊗ i = 1 n ( λ i c ⌢ i ) ω i ) ∪ γ 1 ∈ h ( c ⌢ 1 ) , γ 2 ∈ h ( c ⌢ 2 ) , … , γ n ∈ h ( c ⌢ n ) = 〈 { 1 − ( 1 − ∏ i = 1 n ( 1 − ( 1 − γ i 2 ) λ ) ω i ) 1 λ } , 〉 ( ∏ i = 1 n ( 1 − ( 1 − ( 1 − max ⁡ ( h i 2 ) ) λ ) ω i ) 1 λ

Definition 16. For a collection of P-EPFNs ρ ⌢ i = 〈 h i , [ 1 - ( max { h i } ) 2 ] 〉 (i = 1,2, ... ,n), w = (w₁, w₂,. . . , w _n )  ^T is the weight vector of k _i (i = 1,2, ... ,n) with w i ∈ [ 0 , 1 ] , ∑ i = 1 n w i = 1 and λ > 0 . Then,

(1) The possible EPFWA (PEPFWA): PEPFWA ( p ⌢ 1 , p ⌢ 2 , . . . , ρ n ⌢ ) = ⊕ i = 1 n ( w i ρ i ⌢ ) = 〈 ∪ γ 1 ∈ h ( p ⌢ 1 ) , γ 2 ∈ h ( p ⌢ 2 ) , . . . , γ n ∈ h ( ρ n ⌢ ) { 1 - ∏ i = 1 n ( 1 - γ i 2 ) w i } , [ 0 , ∏ i = 1 n ( 1 - ( max { h i } ) 2 ) w i ] 〉 (2) The possible EPFWG (PEPFWG) operator: PEPFWG ( p ⌢ 1 , p ⌢ 2 , . . . , ρ n ⌢ ) = ⊗ i = 1 n ( w i ρ i ⌢ ) = 〈 ∪ γ 1 ∈ h ( p ⌢ 1 ) , γ 2 ∈ h ( p ⌢ 2 ) , . . . , γ n ∈ h ( ρ n ⌢ ) { ∏ i = 1 n ( γ i ) w i } , [ 0 , 1 - ∏ i = 1 n ( max { h i } ) 2 ) w i ] 〉 (3) The generalized possible EPFWG (GPEPFWG) operator: G P E P F W G ( ρ ⌢ 1 , ρ ⌢ 2 , … , ρ ⌢ n ) = 1 λ ( ⊗ i = 1 n ( λ p i ) ω i ) ∪ γ 1 ∈ h ( c ⌢ 1 ) , γ 2 ∈ h ( c ⌢ 2 ) , … , γ n ∈ h ( c ⌢ n ) = 〈 { 1 − ( 1 − ∏ i = 1 n ( 1 − ( 1 − γ i 2 ) λ ) ω i ) 1 λ } , 〉 [ 0 , ( ∏ i = 1 n ( 1 − ( 1 − ( 1 − max ⁡ ( h i 2 ) ) λ ) ω i ) 1 λ ]

This section presents a multi-attribute decision making with anonymity based on extended Pythagorean fuzzy aggregation information. Suppose that there are n alternatives X ={ x_1,x₂,. . . , x _n  } and m attributes A ={ A_1,A₂,. . . , A _m  } to be evaluated having weight vector w = (w₁, w₂,. . . , w _m )  ^T such that w _j  ∈ [0, 1], j = 1, 2,. . . , m and ∑ j = 1 m w j = 1 .

To evaluate the performance of the alternative x _i under the attributes A _j , the decision maker is required to provide not only the information that the alternative x _i satisfies the attributes A _j , but also the information that the alternative x _i does not satisfy the attributes A _j . These two part information can be expressed by h _ij and μ _ij which denote the degrees that the alternative x _i satisfy the attribute A _j and does not satisfy the attribute A _j , then the performance of the alternative x _i under the criteria A _j can be expressed by an EPFN p ij ⌢ = 〈 h ij , μ ij 〉 . To obtain the ranking of the alternatives, the following steps are given:

Step 1. Construct a hesitant fuzzy decision matrix and convert the hesitant fuzzy information to credible extended Pythagorean fuzzy information and possible extended Pythagorean fuzzy information. If the attribute have two types, such as cost and benefit attributes. Then the credible extended Pythagorean fuzzy decision matrix and possible extended Pythagorean fuzzy decision matrix can be converted into the normalized credible extended Pythagorean fuzzy decision matrix and possible extended Pythagorean fuzzy decision matrix D _N  = (γ _ij ) _m×n, where γ i j = { p ⌢ i j if the attribute is of benefit type p ⌢ i j c if the attribute is of cost type where p ⌢ i j c = 〈 μ i j , h i j 〉 (i = 1, 2,. . . , n ; j = 1, 2,. . . , m). If all the attributes have the same type than there is no need to normalized the decision matrix.

Step 2. Utilize the developed aggregation operators to obtain the CEPFN ρ ⌢ i (i = 1, 2,. . . , n) and the PEPFN ρ ⌢ i (i = 1, 2,. . . , n) for the alternativesX _i . That is the developed operators to obtain the collective overall preference values ρ ⌢ i (i = 1, 2,. . . , n) of the alternativeX _i , wherew = (w₁, w₂,. . . w _n )  ^T is the weighting vector of the attributes.

Step 3. By using the definition of score and accuracy functions for C-EPFS and P-EPFS, calculate the scores $( ρ ⌢ i ) (i = 1, 2, …, n) and the accuracy @ ( ρ ⌢ i ) ( i = 1 , 2 , … , n ) of overall values ρ ⌢ i (i = 1, 2, …, n).

Step 4. Rank the alternatives X _i (i = 1,2, ... ,n) and then select the best one.

5 Illustrative example

In In this section, we presents two examples of decision making with risk preference to demonstrate the C-EIFNs and the P-EIFNs and applies two aggregation operators (i.e., GCEIFWA and GPEIFWA) to obtain the optimal alternative.

Example 1. An investor plans to place the idle funds on the Growth Enterprise Market (GEM) board of the Shenzhen Stock Exchange in Shenzhen, China, as his/her long-term investment. There are five listed companies Xi (i = 1,2, ... 5) that are impressive and promising, which represent five growing prospective industries. There are a 3-D print technology company X₁, a hybrid energy company X₂, a composite material company X₃, a waste gas treatment company X₄, and an e-business company X5. The investor just wants to select the most desirable and suitable company as his/her investment target for the limited time and money available. Meanwhile, as these companies belong to five new industries, their financial data are limited and their share prices are volatile. Therefore, the decision-making approach based on the hesitant fuzzy information provided by the investor is suitable and feasible. Furthermore, if the risk appetite of the investor is considered, such as the appetites for preference risk and neutral risk, then the credible and possible decision-making processes and the corresponding Extended Pythagorean fuzzy aggregation operators are exercisable.

Here, this investor makes a decision according to five attributes: (1) A₁ is the sustainability analysis; (2) A₂ is the risk analysis; (3) A₃ is the profitability analysis, and (4) A₄ is the stock prices increase analysis. It is easily found that these attributes are the benefit type. The five listed companies Xi (i = 1,2, ... ,5) are to be evaluated by this investor using the hesitant fuzzy information x _ij (i = 1, 2, …, 5; j= 1, 2,…, 5), (see Table 1) and he/she considers the weight vector of five attributes Aj (j = 1, 2, 3, 4, 5) as w=(0.20, 0.25, 0.35, 0.20)T to be suitable. Then, we construct the following fuzzy decision matrix C = ( p ij ⌢ ) 4 × 5 .

The HFE information in Table 1 can be directly aggregated, and the optimal alternative can be obtained. However, as a risk preference investor, the DMs will pay more attention to the probability of the maximum rise of the stock price. Moreover, the non-membership should be also introduced to synthesize the negative information implied in the HFEs. Thus, in this decision, we think that using C-EIFNs is more suitable and reasonable than using HFEs and P-EIFNs. In this study, the C-EIFNs focus on the boundary values that can represent the investor’s preference and whose credible non-membership can reflect the possible negative information. Below, the GCEIFWA operator and GCEIFWA operator are used to aggregate the preceding extended fuzzy information to obtain the optimal alternative.

Case 1: If the risk appetite is consider as a preference risk. Then the GCEPFWA/GCEPFWG operators will be used to aggregate previous extended Pythagorean fuzzy information to get the best alternative. In order to avoid manipulating each other, the decision makers are required to provide their preferences in anonymity and the decision matrix C = ( p ij ⌢ ) m × n is presented in Table 1, where p ij ⌢ ( i = 1 , 2 , 3 , 4 , 5 ; j = 1 , 2 , 3 ) are in the form of EPHFNs.

Step 1. Transfer the hesitant fuzzy decision matrix to the corresponding credible extended Pythagorean fuzzy matrix as shown in Table 2.

Step 2. Utilize the GCEPFWA operator and GCEPFWG operator to get the overall C-EPFNs c i ⌢ (i = 1, 2, 3, 4, 5). By Definition 21 as the parameter λ. λ changes we can get different results for each alternative, here we will not list them for vast amounts of data.

Step 3. The score values S( c i ⌢ )(i = 1,2,3,4,5) are calculated as shown in Tables 4 and 5.

Step 4. By ranking S( c i ⌢ )(i = 1,2,3,4,5) the most desirable alternatives are obtained whereas the parameter changes, as shown in Table 4 and Table 5. The optimal alternative is X₂. Therefore for the preference risk the most desirable investment option is X₂.

Case 2: If the risk appetite is consider as a neutral risk. Then the GPEPFWA/GPEPFWG operators will be used to aggregate previous extended Pythagorean fuzzy information to get the best alternative.

Step 1. Transfer the hesitant fuzzy decision matrix to the corresponding possible extended Pythagorean fuzzy matrix shown in Table 3.

Step 2. Utilize the GPEPFWA operator and GPEPFWG operator to get the overall C-EPFNs c i ⌢ (i = 1, 2, 3, 4, 5) and consider λ = 1. By Definition 22 as the parameter λ changes we can get different results for each alternative, here we will not list them for vast amounts of data.

Step 3. The score values S( ρ ⌢ i )(i = 1,2,3,4,5) are calculated as shown in Tables 6 and 7.