Pythagorean hesitant fuzzy sets and their application to group decision making with incomplete weight information

1 Introduction

Zadeh in his important paper [33], initiated the concept of fuzzy sets to handle uncertainty. A method of finding the best decision among a set of reasonable alternatives is a group decision making method. The problem is that in what way to aggregate various inputs into a single output [27, 32]. In [4], the concept of fuzzy set used by Bellman and Zadeh in decision making for the solution of uncertainty in information became from human preferences. In [7], Dong et al. defined the concept of the ranking range of an alternative in the MADM, and propose a series of mixed 0-1 linear programming. In [8], Dong and Herrera-Viedma proposed a consistency-driven automatic methodology to set interval numerical scales of 2-tuple linguistic term sets in the decision making problems with linguistic preference relations models (MLPMs) to show the process of designing a strategic attribute weight vector. In [35], Zhang et al. proposed 2-rank MAGDM problem under the multi-granular linguistic context, and proposes a 2-rank consensus reaching framework with the minimum adjustments. For fuzzy decision analysis Dubois compares old and new procedures [9]. Atanassov was the first who defined intuitionistic fuzzy set which is the generalization of fuzzy set and characterized by a membership function and a non-membership function [1, 2]. The notion of intuitionistic fuzzy set is more appropriate to deal with uncertainty and fuzziness than that of fuzzy set. Intuitionistic fuzzy set is very suitable to show uncertainty and vagueness of an object, and hence intuitionistic fuzzy set can be used as a powerful tool to precise data information under different fuzzy environments which have great attentions. In decision making problems the concept of intuitionistic fuzzy set is broadly applied [3, 15]. The concept of fuzzy set further extended by Torra in [25], and introduced the notion of hesitant fuzzy sets. Hesitant fuzzy set permits that situation of the membership having a set of possible values. Using the concept of hesitant fuzzy set many researchers solve Group decision making problems and with aggregation operators in [16, 36]. In [23], Rodríguez et al. briefly studied the necessity of HFSs and provides a discussion about current proposals including a guideline that the proposals should follow and some challenges of HFSs. Extended the notion of intuitionistic fuzzy set Yager in [29, 30] initiated the notion of Pythagorean fuzzy set (PFS), under the restriction that the sum of square of membership degree and non-membership degree is less than or equal to 1. Many researchers have paid attention to the group decision making problems by using the concept of Pythagorean fuzzy. In [31] the relation between Pythagorean membership degrees and complex numbers has been discussed. The authors showed that Pythagorean degrees are a subclass of complex numbers and is said to be Π - i numbers. Zhang and Xu in [34], introduced a method for Order Preference by Similarity to an best Solution to solve MCDM problem with Pythagorean fuzzy information. In [30], Yager proposed a series of aggregation operators which are Pythagorean fuzzy weighted average (PFWA) operator, Pythagorean fuzzy weighted geometric average operator (PFAG), Pythagorean fuzzy weighted power average (PFWPA) operator and Pythagorean fuzzy weighted power geometric average (PFWPG) operator to aggregate the different Pythagorean fuzzy numbers. These proposed operators have been proved with an application to MCDM problem. Peng and Yang in [17], introduced some new operations in Pythagorean fuzzy set which are division, subtraction and discussed their corresponding properties. The authors also deal with the superiority and inferiority ranking method to solve the multi-attribute group decision-making problems with Pythagorean fuzzy information. Peng and Yong [20] developed the Pythagorean fuzzy linguistic sets (PFLSs), some operational laws and score function of Pythagorean fuzzy linguistic numbers were presented. Wei and Zhang [13] developed a maximizing deviation method to solve decision-making problems under interval-valued Pythagorean fuzzy circumstances. The subtraction and division operations of Pythagorean fuzzy numbers (PFNS) were proposed by Gou et al. [11]. Pend and Dai [21], introduced the concept of axiomatic definition of Pythagorean fuzzy distance measure, which is expressed by Pythagorean fuzzy number that will reduce the information loss and remain more original information. Also the defined the concept of novel score function. Liang et al. [14], initiated the concept of Pythagorean fuzzy geometric Bonferroni mean and weighted Pythagorean fuzzy geometric Bonferroni mean (WPFGBM) operators. In [10], Garg developed interval-valued Pythagorean fuzzy weighted average (IVPFWA) Operator, interval-valued Pythagorean fuzzy geometric (IVPFWG) operator, and introduced the concept of new accuracy function under interval-valued Pythagorean fuzzy environment. In [22], Qian et al. generalized the notion of hesitant fuzzy sets (HFSs) with intuitionistic fuzzy sets (IFSs) and referred to them as generalized hesitant fuzzy set HFSs, which in essence extended the element of HFSs from a real number to intuitionistic fuzzy numbers (IFNs). Zhu, Xu, and Xia in [36] developed the concept of dual hesitant fuzzy set (HFS), and also discussed their basic operations and properties. Peng, Wang, Wang, and Chen [18] introduced a MCDM approach with hesitant interval-valued intuitionistic fuzzy sets (HIVIFSs), which are an extension of dual IVHFSs. However, dual HFSs are defined in terms of sets of values as opposed to precise numbers for the membership degrees and non-membership degrees of IFSs. In [19], the authors apply the concept of Intuitionistic hesitant fuzzy set (IHFS) to group decision making problems using fuzzy cross-entropy.

Intuitionistic hesitant fuzzy set (IHFS) introduced in [19], as a generalization of IFS and HFS. The basic idea was to model the case in which instead of a single membership and non-membership degrees, human beings hesitate among a set of membership degree and non-membership degree and they need to represent such a hesitation. Theses sets of membership degree and non-membership degree are typically considered finite. IHFS satisfies the condition that the sum of membership and non-membership degree is less than or equal to 1. However, there may be a situation where the decision maker may provide the degree of membership and nonmembership of a particular attribute in such a way that their sum is greater than 1. To overcome this shortcoming, in this paper we introduce the concept of Pythagorean hesitant fuzzy set (PHFS) which is the generalization of intuitionistic hesitant fuzzy set under the restriction that the square sum of its membership degrees is less than or equal to 1.

In this paper we introduce the concept of Pythagorean hesitant fuzzy set (PHFS) which is the generalization of the notion of intuitionistic hesitant fuzzy set. We discuss some properties of PHFS. We define score and deviation degree of the Pythagorean hesitant fuzzy numbers (PHFNs) for the comparison of Pythagorean hesitant fuzzy numbers. We also define distance measure between Pythagorean fuzzy numbers. Pythagorean hesitant fuzzy set satisfies the condition that the sum of its membership’s degrees is less than one. Under these environments, aggregation operators, namely, Pythagorean hesitant fuzzy weighted averaging (PHFWA), are proposed in this paper. Furthermore, these operators are applied to decision-making problems in which experts provide their preferences in the Pythagorean hesitant fuzzy environment to show the validity, practicality, and effectiveness of the new approach.

2 Preliminaries

In this section, we introduce some basic definition and properties.

Definition 2.3. [29] Let X be a fixed set. Then a Pythagorean fuzzy set (PFS), P in X can be defined as follows: P = { 〈 x , h P ( x ) , h P ′ ( x ) 〉 | x ∈ X } ,(3) where h _P  (x) and h P ′ ( x ) are mappings from X to [0, 1], such that 0 ≤ h _P  (x) ≤1, 0 ≤ h P ′ ( x ) ≤ 1 and also 0 ≤ h P 2 ( x ) + h P ′ 2 ( x ) ≤ 1 , for all x ∈  X , here h _P  (x) and h′ (x) denotes the membership degree and nonmembership degree of element x ∈  X to set P , respectively. Let π P ( x ) = 1 - h P 2 ( x ) - h P ′ 2 ( x ) . Then it is commonly called the Pythagorean fuzzy index of element x ∈  X to set P , representing the degree of indeterminacy of x to P. Also, 0 ≤ π _P  (x) ≤1, for every x ∈  X . We denote the Pythagorean fuzzy number (PFN) by h ˆ = 〈 Λ h ˆ , Γ h ˆ 〉 .

To compare two Pythagorean fuzzy numbers in [34] the authors introduced the concept of score function and accuracy degree. They also discuss some relation between them.

Definition 2.4. [34] Let h ˆ 1 = 〈 Λ h ˆ 1 , Γ h ˆ 1 〉 and h ˆ 2 = 〈 Λ h ˆ 2 , Γ h ˆ 2 〉 be the two PFN. Then S ( h ˆ 1 ) = Λ h ˆ 1 2 - Γ h ˆ 1 2 and S ( h ˆ 2 ) = Λ h ˆ 2 2 - Γ h ˆ 2 2 are the scores of h ˆ 1 and h ˆ 2 respectively, and H ( h ˆ 1 ) = Λ h ˆ 1 2 + Γ h ˆ 1 2 , H ( h ˆ 2 ) = Λ h ˆ 2 2 + Γ h ˆ 2 2 are the accuracy degrees of h ˆ 1 , h ˆ 2 respectively. Then we have

If S ( h ˆ 1 ) < S ( h ˆ 2 ) , then h ˆ 1 is smaller than h ˆ 2 , denoted by h ˆ 1 < h ˆ 2 ,

Definition 2.5. [25] Let X be a fixed set. Then hesitant fuzzy set (HFS) H in X can be define as follows. H = { 〈 x , h H ( x ) 〉 | x ∈ X }(4)

Where h _H  (x) denotes the set of some values belonging to [0, 1], that is the possible membership degree of the element x ∈  X to the set H . For convenience we denote a hesitant fuzzy number (HFN) by h = h _H  (x) and HFNs the set of all hesitant fuzzy numbers.

Definition 2.6. [25] Let h,  h₁ and h₂ be three hesitant fuzzy numbers. Then some basic operations on HFNs can be defined as follows:

h ^c  =∪ _δ∈h {1 - δ} ;

Definition 2.7. [19] Let X be a fixed set. Then intuitionistic hesitant fuzzy set (IHFS), I _H in X can be defined as follows. I H = { 〈 x , Λ I H ( x ) , Γ I H ( x ) | x ∈ X 〉 }(5) where Λ _{I H}  (x) and Γ _{I H}  (x) are mappings from X to [0, 1], denoting a possible degree of membership and non-membership degree of element x ∈  X in I _H respectively, and for every element x ∈  X , for all h _{I H}  (x) ∈ Λ _{I H}  (x), ∃ h _{I H}  (x) ∈ Γ _{I H}  (x) such that 0 ≤ h I H ( x ) + h I H ′ ( x ) ≤ 1 , and for all h I H ′ ( x ) ∈ Γ I H ( x ) , ∃h _{I H}  (x) ∈ Λ _{I H}  (x) such that 0 ≤ h I H ( x ) + h I H ′ ( x ) ≤ 1 . If X has only one element 〈x,  Λ _{I H}  (x),  Γ _{I H}  (x)〉 is said to be intuitionistic hesitant fuzzy number and is denoted by h ˆ = 〈 Λ h ˆ , Γ h ˆ 〉 . The set of all IHFN’s is denoted by IHFNS.

In this section we initiate the concept of Pythagorean hesitant fuzzy set, which is the generalization of intuitionistic hesitant fuzzy set. We give example for Pythagorean hesitant fuzzy set which is not an intuitionistic hesitant fuzzy set. We discuss some properties. Throughout in this paper a Pythagorean fuzzy set will be denoted by P _H and Pythagorean hesitant fuzzy number by h ˆ .

Definition 3.1. Let X 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 , Λ P H ( x ) , Γ P H ( x ) | x ∈ X 〉 }(6) where Λ _{P H}  (x) and Γ _{P H}  (x) are mappings from X to [0, 1], denoting a possible degree of membership and non-membership degree of element x ∈  X in P _H respectively, and for each element x ∈  X , ∀ h P H ( x ) ∈ Λ P H ( x ) , ∃ h P H ′ ( x ) ∈ Γ P H ( x ) , such that 0 ≤ h P H 2 ( x ) + h P H ′ 2 ( x ) ≤ 1 , and ∀ h P H ′ ( x ) ∈ Γ P H ( x ) ∃ h P H ( x ) ∈ Λ P H ( x ) , such that 0 ≤ h P H 2 ( x ) + h P H ′ 2 ( x ) ≤ 1 .

Moreover, PHFS (X) denotes the set of all elements of PHFS’s. If X has only one element 〈x,  Λ _{P H}  (x),  Γ _{P H}  (x)〉, then it is said to be Pythagorean hesitant fuzzy number and is denoted by h ˆ = 〈 Λ h ˆ , Γ h ˆ 〉 for convenience. We denote the set of all PHFN’s by PHFNS.

Example 3.2. Let X ={ x₁,  x₂,  x₃ } be a fixed set and consider a set P _H in X given by P H = { 〈 x 1 , { 0.7 , 0.8 , 0.9 , 1 } , { 0.0 , 0.4 , 0.5 } 〉 , 〈 x 2 , { 0.04 , 0.7 , 0.8 } , { 0.2 , 0.8 , 0.9 } 〉 , 〈 x 3 , { 0.0 , 0.2 , 0.5 , 0.6 } , { 0.6 , 0.9 , 1 } 〉 }.

Then, (0.7)² + (0.0)² = 0.49, (0.8)² + (0.0)² = 0.64, (0.9)² + (0.0)² = 0.81, (1)² + (0.0)² = 1. Moreover, (0.04)² + (0.0)² = 0.0416, (0.7)² + (0.2)² = 0.53, (0.8)² + (0.2)² = 0.68. Finally, (0.0)² + (0.6)² = 0.36, (0.2)² + (0.6)² = 0.4, (0.5)² + (0.6)² = 0.61, (0.6)² + (0.6)² = 0.72. Similarly we can calculate the other case. Thus, ( Λ h ˆ ( x 1 ) , Γ h ˆ ( x 1 ) ) , ( Λ h ˆ ( x 2 ) , Γ h ˆ ( x 2 ) ) and ( Λ h ˆ ( x 3 ) , Γ h ˆ ( x 3 ) ) are PHFNs. Therefore, P _H is a PHFS.

Remark For all x ∈  X if Λ _{P H}  (x) and Γ _{P H}  (x) have only one element. Then the PHFS become a PFS. If the non-membership degree is {0}, then PHFS become a HFS.

Definition 3.3. For any PHFS P _H  = {〈x,  Λ _{P H}  (x), Γ _{P H}  (x)〉|x ∈ X} and for all x ∈  X , Π P H ( x ) = ∪ h P H ∈ Λ P H ( x ) , h P H ′ ( x ) ∈ Γ P H ( x ) 1 - h P H 2 - h P H ′ 2 is said to be the degree of indeterminacy of x to P _H , where 1 - h P H 2 - h P H ′ 2 ≥ 0 .

Example 3.4. From Example 3.2, we have Π _{P H}  (x₁) = {0.05, 0.1, 0.19, 0.2, 0.33, 0.36, 0.51, 0.76, 0.98}, Π _{P H}  (x₂) = {0.18, 0.32, 0.35, 0.47, 0.95} and Π _{P H}  (x₃) = {0.0, 0.19, 0.28, 0.39, 0.64}.

Definition 3.5. Let h ˆ = 〈 Λ h ˆ , Γ h ˆ 〉 , h ˆ = 〈 Λ h ˆ 1 , Γ h ˆ 1 〉 , h ˆ 2 = 〈 Λ h ˆ 2 , Γ h ˆ 2 〉 are three PHFNs and λ > 0, then their operations are defined as follows. The result obtain by these operations is also a Pythagorean hesitant fuzzy number. ( 1 ) h ˆ 1 ∪ h ˆ 2 = 〈 max { Λ h ˆ 1 , Λ h ˆ 2 } , min { Γ h ˆ 1 , Γ h ˆ 2 } 〉 , ( 2 ) h ˆ 1 ∩ h ˆ 2 = 〈 min { Λ h ˆ 1 , Λ h ˆ 2 } , max { Γ h ˆ 1 , Γ h ˆ 2 } 〉 , ( 3 ) h ˆ c = 〈 Γ h ˆ , Λ h ˆ 〉 , ( 4 ) h ˆ 1 ⊕ h ˆ 2 = 〈 ∪ h h ˆ 1 ∈ Λ h ˆ 1 , h h ˆ 2 ∈ Λ h ˆ 2 { h h ˆ 1 2 + h h ˆ 2 2 - h h ˆ 1 2 h h ˆ 2 2 } , ∪ h h ˆ 1 ′ ∈ Γ h ˆ 1 , h h ˆ 2 ′ ∈ Γ h ˆ 2 { h h ˆ 1 ′ h h ˆ 2 ′ } 〉 ( 5 ) h ˆ 1 ⊗ h ˆ 2 = 〈 ∪ h h ˆ 1 ∈ Λ h ˆ 1 , h h ˆ 2 ∈ Λ h ˆ 2 { h h ˆ 1 h h ˆ 2 } , ∪ h h ˆ 1 ′ ∈ Γ h ˆ 1 , h h ˆ 2 ′ ∈ Γ h ˆ 2 { h h ˆ 1 ′ 2 + h h ˆ 2 ′ 2 - h h ˆ 1 ′ 2 h h ˆ 2 ′ 2 } 〉 ( 6 ) λ h ˆ = 〈 ∪ h h ˆ ∈ Λ h ˆ { 1 - ( 1 - ( h h ˆ ) 2 ) λ } , ∪ h h ˆ ′ ∈ Γ h ˆ { ( h h ˆ ′ ) λ } 〉 , λ > 0 . ( 7 ) h ˆ λ = 〈 ∪ h h ˆ ∈ Λ h ˆ { h h ˆ λ } , ∪ h h ˆ ′ ∈ Γ h ˆ { 1 - ( 1 - ( h h ˆ ′ ) 2 ) λ } 〉 , λ > 0 .

Following example shows that for any three PHFN’s h ˆ 1 , h ˆ 2 , h ˆ 3 , and λ > 0 h ˆ 1 ⊕ h ˆ 2 , h ˆ 1 ⊗ h ˆ 2 , λ h ˆ and h ˆ λ are also PHFN’s.

Theorem 3.7.For three PHFNs h ˆ = 〈 Λ h ˆ , Γ h ˆ 〉 , h ˆ 1 = 〈 Λ h ˆ 1 , Γ h ˆ 1 〉 and h ˆ 2 = 〈 Λ h ˆ 2 , Γ h ˆ 2 〉 , the following hold:

h ˆ 1 ⊕ h ˆ 2 = h ˆ 2 ⊕ h ˆ 1

Proof. Proof is simple.

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

Definition 3.8. Let h ˆ = 〈 Λ h ˆ , Γ h ˆ 〉 be a PHFN. Then we define the score function of h ˆ as follows: S ( h ˆ ) = ( 1 l h h ˆ ∈ Λ h ˆ ∑ h h ˆ ∈ Λ h ˆ h h ˆ ) 2 - ( 1 l h h ˆ ′ ∈ Γ h ˆ ∑ h h ˆ ′ ∈ Γ h ˆ h h ˆ ′ ) 2(7) where S ( h ˆ ) ∈ [ - 1 , 1 ] , l h h ˆ denotes the number of elements in Λ h ˆ and l h h ˆ ′ denotes the number of elements in Γ h ˆ .

Definition 3.9. Let h ˆ 1 = 〈 Λ h ˆ 1 , Γ h ˆ 1 〉 and h ˆ 2 = 〈 Λ h ˆ 2 , Γ h ˆ 2 〉 be two PHFN’s, S ( h ˆ 1 ) be the score function of h ˆ 1 and S ( h ˆ 2 ) be the score function of h ˆ 2 . Then

If S ( h ˆ 1 ) < S ( h ˆ 2 ) , then h ˆ 1 < h ˆ 2 .

Example 3.10. Let h ˆ 1 = 〈 { 0.5 , 0.7 , 0.9 } , { 0.1 , 0.5 , 0.6 } 〉 , h ˆ 2 = 〈 { 0.1 , 0.7 , 0.9 } , { 0.0 , 0.8 , 0.9 } 〉 , h ˆ 3 = 〈 { 0.0 , 0.3 , 0.7 , 0.9 } , { 0.03 , 0.8 , 0.9 } 〉 be three PHFN’s. Then by Definition 3.8, we have, S ( h ˆ 1 ) = 0.33 , S ( h ˆ 2 ) = 0.0 , S ( h ˆ 3 ) = - 0.107 . Thus, S ( h ˆ 1 ) > S ( h ˆ 2 ) > S ( h ˆ 3 ) .

Example 3.11. Let h ˆ 1 = 〈 { 0.3 , 0.9 } , { 0.3 , 0.4 , 0.5 } 〉 , h ˆ 2 = 〈 { 0.3 , 0.7 , 0.8 } , { 0.3 , 0.5 } 〉 be two PHFN’s. Then by Definition 3.8, we have S ( h ˆ 1 ) = 0.20 , S ( h ˆ 2 ) = 0.20 . Thus, S ( h ˆ 1 ) = S ( h ˆ 2 ) .

Since, S ( h ˆ 1 ) = S ( h ˆ 2 ) . Therefore, by Definition (3.8) we cannot differentiate between h ˆ 1 and h ˆ 2 . Actually such a case is usually common in practice. It is clear from Definition (3.8), that it does not take into account the situation that two PHFN’s h ˆ 1 and h ˆ 2 have the same score. But their deviation degrees may be different from each other. The accuracy degree of all elements with respect to the average value in a PHFN’s reflects how these elements agree with each other, that is, they have a higher consistency. To better represent this issue, we initiate the notion of accuracy degree for the compression of two PHFN’s as follows.

Definition 3.12. Let h ˆ = 〈 Λ h ˆ , Γ h ˆ 〉 be a PHFN. Then the accuracy degree of h ˆ is denoted by σ ¯ ( h ˆ ) and can be defined as follows:

σ ¯ ( h ˆ ) = ( 1 l h h ˆ ∈ Λ h ˆ ∑ h h ˆ ∈ Λ h ˆ h h ˆ - S ( h ˆ ) ) 2 + ( 1 l h h ˆ ′ ∈ Γ h ˆ ∑ h h ˆ ′ ∈ Γ h ˆ h h ˆ ′ - S ( h ˆ ) ) 2(8)

Here we can see that S ( h ˆ ) is just the mean value in statistics, and σ ¯ ( h ˆ ) is just the standard variance, which reflects the accuracy degree between all values in the PHFN h ˆ and their mean value. Inspired by this idea, based on the score S ( h ˆ ) and the accuracy degree σ ¯ ( h ˆ ) , we can compare and rank, two PHFN’s in the following as:

Definition 3.13. Let h ˆ 1 and h ˆ 2 be two PHFN’s, S ( h ˆ 1 ) be the score of h ˆ 1 , S ( h ˆ 2 ) be the score of h ˆ 2 , and σ ¯ ( h ˆ 1 ) be the deviation degree of h ˆ 1 , σ ¯ ( h ˆ 2 ) be the deviation degree of h ˆ 2 . Then

If S ( h ˆ 1 ) < S ( h ˆ 2 ) , then h ˆ 1 < h ˆ 2 .

Example 3.14. From Example 3.11, since S ( h ˆ 1 ) = S ( h ˆ 2 ) = 0.2 ; so we have σ ¯ ( h ˆ 1 ) = 0.399 . σ ¯ ( h ˆ 2 ) = 0.2 . Thus, σ ¯ ( h ˆ 1 ) > σ ¯ ( h ˆ 2 ) . Hence, h ˆ 1 > h ˆ 2 .

Definition 3.15. Let h ˆ 1 , h ˆ 2 and h ˆ 3 be three PHFSs in X. A distance measure D ( h ˆ 1 , h ˆ 2 ) is a mapping D   :   PHFS × PHFS → [0, 1], satisfies the following properties:

0 ≤ D ( h ˆ 1 , h ˆ 2 ) ≤ 1 ;

Definition 3.16. Let h ˆ 1 and h ˆ 2 be two PHFNs on a set X = { x₁,  x₂, …,  x _n  } . Then we define the distance measure between h ˆ 1 and h ˆ 2 as follows:

D ( h ^ 1 , h ^ 2 ) = 1 2 [ 1 l x i ∑ i = 1 l x i | ( h h ^ 1 σ ( k ) ( x i ) ) 2 − ( h h ^ 2 σ ( k ) ( x i ) ) | + 1 l x i ∑ i = 1 l x i | ( h h ^ 1 ′ σ ( k ) ( x i ) ) 2 − ( h h ^ 2 ′ σ ( k ) ( x i ) ) 2 | ](9)

where, h h ˆ 1 σ ( k ) ( x i ) , h h ˆ 1 ′ σ ( k ) , h h ˆ 2 σ ( k ) , h h ˆ 2 ′ σ ( k ) , be the i ^th largest value in h ˆ 1 and h ˆ 2 respectively.

Since the number of elements for different PHFNs could be different, we can make them equivalent through adding elements to the PHFN that has a less number of elements. We can add the smallest element in terms of pessimistic principle while the opposite case will be adopted in optimistic principle.

Example 3.17. Let h ˆ 1 = 〈 { 0.6 , 0.7 , 0.9 } , { 0.5 , 0.7 , 0.8 } 〉 and h ˆ 2 = 〈 { 0.5 , 0.6 , 0.9 } , { 0.4 , 0.7 , 0.8 } 〉 be two PHFNs. Then D ( h ˆ 1 , h ˆ 2 ) = 1 2 [ 1 3 ( | 0 . 6 2 - 0 . 5 2 | + | 0 . 7 2 - 0 . 6 2 | + | 0 . 9 2 - 0 . 9 2 | ) + 1 3 ( | 0 . 5 2 - 0 . 4 2 | + | 0 . 7 2 - 0 . 8 2 | + | 0 . 8 2 - 0 . 9 2 | ) ] D ( h ˆ 1 , h ˆ 2 ) = 1 2 [ 0.08 + 0.14 ] = 0.19

In this section we develop some aggregation operators for Pythagorean hesitant fuzzy numbers, and investigate some of its properties.

Definition 4.1. Let h ˆ i = 〈 Λ h ˆ i , Γ h ˆ i 〉 (i = 1,  2, …,  n) be a collection of all PHFN’s, and w = (w₁,  w₂, …,  w _n )  ^T be the weight vector of h ˆ 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 Pythagorean hesitant fuzzy weighted averaging (PHFWA) operator is a mapping PHFWA   :   PHFN ⁿ  → PHFN can be defined as

PHFWA ( h ˆ 1 , h ˆ 2 , … , h ˆ n ) = w 1 h ˆ 1 ⊕ w 2 h ˆ 2 ⊕ … ⊕ w n h ˆ n .(10) and the PHFWA operator is said to be a Pythagorean hesitant fuzzy weighted averaging operator.

Theorem 4.2.Let h ˆ i = 〈 Λ h ˆ i , Γ h ˆ i 〉 (i = 1,  2, …,  n) be a collection of all PHFN’s, andw = (w₁,  w₂, …,  w _n )  ^T be the weight vector of h ˆ i (i = 1,  2, …,  n) withw _i  ≥ 0(i = 1,  2, …,  n) wherew _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 . Then the aggregation result usingPHFWAoperator is also aPHFNand

PHFWA ( h ˆ 1 , h ˆ 2 , … , h ˆ n ) = 〈 ∪ h h ˆ 1 ∈ Λ h ˆ 1 , h h ˆ 2 ∈ Λ h ˆ 2 , … , h h ˆ n ∈ Λ h ˆ n 1 - ∏ i = 1 n ( 1 - ( 1 - h h ˆ i 2 ) w i ) , ∪ h h ˆ 1 ′ ∈ Γ h ˆ 1 , h h ˆ 1 ′ ∈ Γ h ˆ 2 , … h h ˆ n ′ ∈ Γ h ˆ n ( ∏ i = 1 n ( h h ˆ i ′ ) w i ) 〉(11)

Proof. Straightforward.

Definition 4.4. Let h ˆ i = 〈 Λ h ˆ i , Γ h ˆ i 〉 (i = 1,  2, …,  n) be a collection of all PHFN’s, and w = (w₁,  w₂, …,  w _n )  ^T be the weight vector of h ˆ i (i = 1,  2, …,  n) with w _i  ≥ 0 (i = 1,  2,  3, …,  n) such that w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 . Then, Pythagorean hesitant fuzzy weighted geometric (PHFWG) operator is a mapping

PHFWG : PHFN ⁿ  → PHFN can be defined as

PHFWG ( h ˆ 1 , h ˆ 2 , … , h ˆ n ) = h ˆ 1 w 1 ⊗ h ˆ 2 w 2 ⊗ , … , ⊗ h ˆ n w n(12) and the PHFWG operator is said to be a Pythagorean hesitant fuzzy weighted geometric operator.

Theorem 4.5.Let h ˆ i = 〈 Λ h ˆ i , Γ h ˆ i 〉 (i = 1,  2, …,  n) be a collection of all PHFN’s, andw = (w₁,  w₂, …,  w _n )  ^T be the weight vector of h ˆ i (i = 1,  2, …,  n) withw _i  ≥ 0 (i = 1,  2, …,  n) wherew _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 . Then the aggregation result using PHFWG operator is also a PHFN, and

PHFWG ( h ˆ 1 , h ˆ 2 , … , h ˆ n ) = 〈 ∪ h h ˆ 1 ∈ Λ h ˆ 1 , h h ˆ 2 ∈ Λ h ˆ 2 , … , h h ˆ n ∈ Λ h ˆ n { ∏ i = 1 n ( h h ˆ i ) w i } , ∪ h h ˆ 1 ′ ∈ Γ h ˆ 1 , h h ˆ 2 ′ ∈ Γ h ˆ 2 , … , h h ˆ n ′ ∈ Γ h ˆ n { 1 - ∏ i = 1 n ( 1 - h h ˆ i ′ 2 ) w i } 〉(13)

Proof. Straightforward

Lemma 4.7.Let h ˆ i > 0 , w _i  > 0 (i = 1,  2, …,  n) and ∑ i = 1 n w i = 1 . Then, ∏ i = 1 n ( h ˆ i ) w i ≤ ∑ i = 1 n w i h ˆ i ,(14) where the equality holds if and only if h ˆ 1 = h ˆ 2 = h ˆ 3 = … = h ˆ n .

Theorem 4.8.Let h ˆ i = 〈 Λ h ˆ i , Γ h ˆ i 〉 (i = 1,  2, …,  n) be a collection of PHFN’s. Then

PHFWG ( h ˆ 1 , h ˆ 2 , h ˆ 3 , … , h ˆ n ) ≤ PHFWA ( h ˆ 1 , h ˆ 2 , h ˆ 3 , . . . , h ˆ n ) ,(15) where w = (w₁,  w₂,  w₃, … w _n )  ^T is the weighted vector of h ˆ i (i = 1,  2, …,  n) such that w _i  ∈ [0, 1] (i = 1,  2,  3, …,  n) and ∑ i = 1 n w i = 1 .

Proof Based on Lemma 4.7, we can easily prove the Theorem.

In this section, we forward a framework for framework for determining attribute weights and the ranking orders for all the alternatives with incomplete weight information under Pythagorean hesitant fuzzy environment.