Interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging aggregation operator and their application to group decision making

Interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging aggregation operator and... The objective of the present work is divided into two folds. Firstly, interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging aggregation operator has been introduced along with their several properties, namely idempotency, boundedness and monotonicity. Secondly, we apply the proposed operator to deal with multi-attribute group decision-making problem under Pythagorean fuzzy information. For this, we construct an algorithm for multi-attribute group decision making. At the last, we construct a numerical example for multi-attribute group decision making. The main advantage of using the proposed operator is that this operator provides more accurate and precise results is compared to the existing methods. Keywords IVPFS · IVPFEHWA averaging operator · MAGDM problems Introduction successful and positive applications of intuitionistic fuzzy set, aggregation operators become more interesting topic for Multi-criteria group decision making is one of the success- research. Thus, many scholars in [3–16] developed several ful processes for finding the optimal alternative from all the aggregation operators for group decision making using intu- feasible alternatives according to some criteria or attributes. itionistic fuzzy information. Traditionally, it has been generally assumed that all the infor- However, there are many cases where the decision maker mation that access the alternative in terms of criteria and their may provide the degree of membership and nonmembership corresponding weights are expressed in the form of crisp of a particular attribute in such a way that their sum is greater numbers. But most of the decisions in the real-life situations than one. To solve these types of problems, Yager [17,18] are taken in the environment where the goals and constraints introduced the concept of another set called Pythagorean are generally imprecise or vague in nature. In order to han- fuzzy set. Pythagorean fuzzy set is more powerful tool to dle the uncertainties and fuzziness intuitionistic fuzzy set [1] solve uncertain problems. Like intuitionistic fuzzy aggrega- theory is one of the successful extensions of the fuzzy set the- tion operators, Pythagorean fuzzy aggregation operators are ory [2], which is characterized by the degree of membership also become an interesting and important area for research, and degree of non-membership has been presented. After the after the advent of Pythagorean fuzzy set theory. Several researchers in [19–28] introduced many aggregation oper- ators for decision using Pythagorean fuzzy information. B Khaista Rahman But, in some real decision-making problems, due to insuf- khaista355@yahoo.com ficiency in available information, it may be difficult for Saleem Abdullah decision makers to exactly quantify their opinions with a saleemabdullah81@yahoo.com crisp number, but they can be represented by an interval num- Asad Ali ber within [0, 1]. Therefore, it is so important to present the asad_maths@hu.edu.pk idea of interval-valued Pythagorean fuzzy sets, which permit Fazli Amin the membership degrees and non- membership degrees to a fazliamin@hu.edu.pk given set to have an interval value. Thus in [29] Peng and Department of Mathematics, Hazara University, Mansehra, Yang introduced the concept of interval-valued Pythagorean KPK, Pakistan fuzzy set. Rahman et al. [30–33] introduced many aggre- Department of Mathematics, Abdul Wali Khan University gation operators using interval-valued Pythagorean fuzzy Mardan, Mardan, KPK, Pakistan 123 Complex & Intelligent Systems numbers and applied them to multi-attribute group decision and making. a b v (k) =[v (k), v (k)]⊂[0, 1]. (5) Thus, keeping the advantages of these operators, in this I I I paper, we introduce the notion of interval-valued Pythagorean Also fuzzy Einstein hybrid weighted averaging operator. More- over, we introduce some of their basic properties such as u (k) = inf(u (k)), (6) idempotency, boundedness and monotonicity. This motiva- I tion comes from [32], in which the authors introduced the u (k) = sup(u (k)), (7) notion of IVPFEWA operator and IVPFEOWA operator and a v (k) = inf(v (k)), (8) applied them to group decision making. But in this paper we v (k) = sup(v (k)), (9) introduce the notion of IVPFEHWA operator, which is the generalization of the above mention operators. and The remainder of this paper is structured as follows. 2 2 In Sect. "Preliminaries", we give some basic definitions b b 0 ≤ u (k) + v (k) ≤ 1. (10) I I and results which will be used in our later sections. In Sect. "Interval-valued Pythagorean fuzzy Einstein hybrid If weighted averaging aggregation operator", we introduce the notion of interval-valued Pythagorean fuzzy Einstein hybrid a b π (k) = π (k), π (k) , for all k ∈ K . (11) I I weighted averaging operator. In Sect. "An approach to mul- tiple attribute group decision-making problems based on Then, it is called the interval-valued Pythagorean fuzzy intervalvalued Pythagorean fuzzy information", we apply index of k to I , where the proposed operator to multi-attribute group decision- making problem with Pythagorean fuzzy information. In 2 2 a b b π (k) = 1 − u (k) − v (k) , (12) Sect. "Illustrative example", we develop a numerical exam- I I I ple. In Sect. "Conclusion", we have conclusion. and 2 2 a a π (k) = 1 − u (k) − v (k) . (13) Preliminaries I I Definition 3 [29]Let λ = ([u ,v ], [x , y ]) be an IVPFN, Definition 1 [17,18]Let K be a fixed set, then a Pythagorean λ λ λ λ then the score function and accuracy function of λ can be fuzzy set can be defined as: defined as follows, respectively: P ={k, u (k), v (k)|k ∈ K }, (1) P P 2 2 2 2 s(λ) = (u ) + (v ) − (x ) − (y ) , (14) λ λ λ λ where u (k) : P →[0, 1],v (k) : K →[0, 1] are called P P membership function and non-membership function, respec- and 2 2 tively, with condition 0 ≤ (u (k)) + (v (k)) ≤ 1, for all P P k ∈ K . 2 2 2 2 h(λ) = u + v + x + y . (15) ( ) ( ) ( ) ( ) λ λ λ λ Let If λ and λ are two IVPFNs, then 1 2 2 2 π (k) = 1 − u (k) − v (k). (2) P P 1. If s(λ ) ≺ s(λ ), then λ ≺ λ . 1 2 1 2 Then, it is called the Pythagorean fuzzy index of k ∈ K , 2. If s(λ ) = s(λ ), then we have the following three con- 1 2 with condition 0 ≤ π (k) ≤ 1, for every k ∈ K . ditions. Definition 2 [29]Let K be a fixed set, then an interval-valued 1) If h(λ ) = h(λ ), then λ = λ . 1 2 1 2 Pythagorean fuzzy set can be defined as: 2) If h(λ ) ≺ h(λ ), then λ ≺ λ . 1 2 1 2 3) If h(λ ) h(λ ), then λ λ . 1 2 1 2 I ={k, u (k), v (k)|k ∈ K }, (3) I I Definition 4 [32]Let λ = ([u,v], [x , y]), λ = ([u ,v ], 1 1 1 where [x , y ]), λ = ([u ,v ], [x , y ]) are three IVPFNs, and 1 1 2 2 2 2 2 δ 0, then some Einstein operations for λ, λ ,λ can be 1 2 a b u (k) =[u (k), u (k)]⊂[0, 1], (4) defined as follows: I I 123 Complex & Intelligent Systems 1. Definition 5 [32]Let λ = ([u ,v ], [x , y ])( j = 1, 2, 3, j j j j j ..., n) be the collection of IVPFVs, then IVPFEWA operator ⎛ ⎡ ⎤ 2 2 2 2 u + u v + v can be defined as: 1 2 1 2 ⎝ ⎣ ⎦ λ ⊕ λ =  ,  , 1 ε 2 IVPFEWA (λ ,λ ,λ , ..., λ ) w 1 2 3 n 2 2 2 2 1 + u u 1 + v v 1 2 1 2 ⎡   ⎤ ⎛ ⎞ n w n w n w n w j j j j 2 2 2 2 1+u − 1−u 1+v − 1−v ⎡ λ λ λ λ ⎢ j j j j ⎥ ⎜ j =1 j =1 j =1 j =1 ⎟ ⎢ ⎥ , , ⎜ ⎟ ⎣ n n n n ⎦ x x  w  w  w  w ⎜ j j j j ⎟ 1 2 2 2 2 2 ⎣ 1+u + 1−u 1+v + 1−v ⎜ ⎟ , λ λ λ λ j j j j ⎜ j =1 j =1 j =1 j =1 ⎟ 2 2 ⎜ ⎟ 1 + 1 − x 1 − x = ⎜ ⎟ , 1 2 ⎡   ⎤ ⎜ ⎟ n n w  w ⎜ j j ⎟ 2 2 2 x 2 y ⎜ ⎟ λ λ ⎤ ⎞ ⎢ j j ⎥ ⎜ j =1 j =1 ⎟ ⎢ ⎥ ⎝ , ⎠ n   n   n   n ⎣ w w w w ⎦ j  j  j  j y y 2 2 2 2 1 2 2−x + x 2−y + y λ λ λ λ ⎦ ⎠ j j j j j =1 j =1 j =1 j =1 2 2 1 + 1 − y 1 − y (16) 1 2 where w = (w ,w ,w , ..., w ) is the weighted vector of 1 2 3 n ⎛ ⎡ λ ( j = 1, 2, 3, ..., n), such that w ∈[0, 1] and w = j j j j =1 u u 1. 1 2 ⎝ ⎣ λ ⊗ λ =  , 1 ε 2 2 2 1 + 1 − u 1 − u Definition 6 [32]Let λ ( j = 1, 2, 3, ..., n) be a collection of 1 2 j IVPFVs, then IVPFEOWA operator can be defined as: v v 1 2 IVPFEOWA (λ ,λ ,λ , ..., λ ) w 1 2 3 n ⎛ ⎡   ⎤ ⎞ 2 2 n   n   n   n w  w  w  w 1 + 1 − v 1 − v j j j j 2 2 2 2 1 2 1+u − 1−u 1+v − 1−v λ λ λ λ ⎢ σ( j ) σ( j ) σ( j ) σ( j ) ⎥ ⎜ ⎟ j =1 j =1 j =1 j =1 ⎢ ⎥ ⎜   ⎟ , , ⎜ ⎣ n w n w n w n w ⎦ ⎟ ⎡ ⎤ ⎞ j j j j 2 2 2 2 ⎜ 1+u + 1−u 1+v + 1−v ⎟ λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) 2 2 2 2 ⎜ ⎟ j =1 j =1 j =1 j =1 x + x y + y ⎜ ⎟ 1 2 1 2 ⎜ ⎟ = , ⎣ ⎦ ⎠ ,  ⎜ ⎟ ⎡   ⎤ ⎜ n   n   ⎟ w w j  j 2 2 2 2 ⎜ 2 2 ⎟ 2 x 2 y λ λ 1 + x x 1 + y y ⎜ ⎟ ⎢ σ( j ) σ( j ) ⎥ 1 2 1 2 j =1 j =1 ⎜ ⎢ ⎥ ⎟ ⎝         ⎠ ⎣ n w n w n w n w ⎦ j j j j 2 2 2 2 2−x + x 2−y + y λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) j =1 j =1 j =1 j =1 (17) 2 δ 2 δ (1 + u ) − (1 − u ) δλ =  , where (σ (1), σ (2), σ (3), ..., σ (n)) is a permutation of (1, 2, 2 δ 2 δ (1 + u ) + (1 − u ) 3, ..., n) such that σ( j ) ≤ σ( j − 1) for all jand w = (w ,w ,w , ..., w ) is the weighted vector of λ ( j = 1 2 3 n σ( j ) 2 δ 2 δ (1 + v ) − (1 − v ) , 1, 2, 3, ..., n) such that w ∈[0, 1] and w = 1. j j j =1 2 δ 2 δ (1 + v ) + (1 − v ) 2 δ 2 δ 2(x ) 2(y ) Interval-valued Pythagorean fuzzy Einstein 2 δ 2 δ 2 δ 2 δ (2 − y ) + (y ) (2 − x ) + (x ) hybrid weighted averaging aggregation operator In this section, we introduce the notion of interval-valued 2 δ 2 δ 2(u ) 2(v ) Pythagorean fuzzy Einstein hybrid weighted averaging aggre- λ =  ,  , 2 δ 2 δ 2 δ 2 δ (2 − u ) + (u ) (2 − v ) + (v ) gation operator. We also discuss some desirable properties such as idempotency, boundedness and monotonicity. 2 δ 2 δ (1 + x ) − (1 − x ) Definition 7 An interval-valued Pythagorean fuzzy Einstein 2 δ 2 δ (1 + x ) + (1 − x ) hybrid weighted averaging operator of dimension n is a mapping IVPFEHWA :  → ,which has associated 2 δ 2 δ (1 + y ) − (1 − y ) vectorw = (w ,w ,w , ..., w ) , such that w ∈[0, 1] and 1 2 3 n j 2 δ 2 δ (1 + y ) + (1 − y ) n w = 1. Furthermore j =1 123 Complex & Intelligent Systems IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 n is the balancing coefficient, which plays a role of bal- ⎡ ⎤ ⎛ w w w w ⎞ n n n n T j  j  j  j ance. If the vector w = (w ,w ,w ,...,w ) approaches 2 2 2 2 1 2 2 n 1+u − 1−u 1+v − 1−v ˙ ˙ ˙ ˙ λ λ λ λ ⎢ ⎥ σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ T ⎢ ⎥ 1 1 1 1 ⎜     ,     ,⎟ w w w w , , ,..., , then the vector (nwωλ , ⎣ n j n j n j n j ⎦ ⎜ ⎟ 2 2 2 2 n n n n 1+u + 1−u 1+v + 1−v ⎜ ⎟ ˙ ˙ ˙ ˙ λ λ λ λ T T σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ nω λ ,..., nω λ ) approaches(λ ,λ ,λ ,...,λ ) . 2 2 n n 1 2 3 n ⎜ ⎟ = , ⎡   ⎤ ⎜ ⎟ w w n j n j ⎜ ⎟ 2 2 2 x 2 y ⎜ ⎟ ˙ ˙ ⎢ λ λ ⎥ σ( j ) σ( j ) ⎜ j =1 j =1 ⎟ Proof We can prove this theorem by mathematical induction ⎢   ⎥ ⎝         ⎠ w w w w ⎣ n j n j n j n j⎦ 2 2 2 2 2−x + x 2−y + y ˙ ˙ ˙ ˙ λ λ λ λ on n. σ( j ) σ( j ) σ( j ) σ( j ) j =1 j =1 j =1 j =1 For n = 2 (18) ⎛ ⎡ w w 1 1 ˙ 2 2 where λ is the jth largest of the weighted interval-valued σ( j ) 1 + u − 1 − u ˙ ˙ ⎜ ⎢ λ λ 1 1 ˙ ˙ ⎜ ⎢ Pythagorean fuzzy values, λ (λ = nω λ ). ω = ˙ σ( j ) σ( j ) j j w λ =  , 1 1 ⎝ ⎣ w w T 1 1 (ω ,ω ,ω , ..., ω ) is the weighted vector of λ ( j = 1, 2, 2 2 1 2 3 n j 1 + u + 1 − u n ˙ ˙ λ λ 1 1 3, ..., n) such that ω ∈[0, 1], ω = 1, and n j j j =1 is the balancing coefficient, which plays a role of bal- ⎤ w w 1 1 2 2 ance. If the vector w = (w ,w ,w , ..., w ) approaches 1 + v − 1 − v 1 2 3 n ˙ ˙ λ λ ⎥ 1 1 1 1 1 1 to , , , ..., , then the vector (nω λ , nω λ , ..., 1 1 2 2 n n n n w w 1 1 T T 2 2 nω λ ) approaches to (λ ,λ ,λ , ..., λ ) . n n 1 2 3 n 1 + v + 1 − v ˙ ˙ λ λ 1 1 Theorem 1 Let λ, λ ,λ be the three interval-valued 1 2 ⎡ Pythagorean fuzzy numbers and δ, δ ,δ 0, then the fol- 1 2 2 x lowing conditions always hold: w w 1 1 2 2 2 − x + x ˙ ˙ λ λ 1 1 1. λ ⊕ λ = λ ⊕ λ , 1 ε 2 2 ε 1 2. λ ⊗ λ = λ ⊗ λ , ⎤ ⎞ 1 ε 2 2 ε 1 3. δ(λ ⊕ λ ) = δλ ⊕ δλ , 1 ε 2 1 ε 2 2 y λ ⎥ ⎟ δ δ δ ⎥ ⎟ 4. (λ ⊗ λ ) = (λ ) ⊗ (λ ) , 1 ε 2 1 ε 2 ⎦ ⎠ w w 1 1 5. δ (λ) ⊕ δ (λ) = (δ ⊕ δ )λ, 2 2 1 ε 2 1 ε 2 2 − y + y ˙ ˙ δ δ (δ ⊗ δ ) λ λ 1 1 1 2 1 ε 2 6. (λ) ⊗ (λ) = λ . and Proof The proof is trivial, so it is omitted here. ⎛ ⎡ w w 2 2 2 2 Theorem 2 Let λ = ([u ,v ], [x , y ])( j = 1, 2, 3, ..., n) j j j j j 1 + u − 1 − u ˙ ˙ ⎜ ⎢ λ λ 2 2 ⎜ ⎢ be a collection of IVPFVs, then their aggregated value using ˙ w λ =  , 2 2 ⎝ ⎣ w w 2 2 the IVPFEHWA operator is also an IVPFV, and 2 2 1 + u + 1 − u ˙ ˙ λ λ 2 2 IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 n ⎡ ⎤ ⎛ w w w w ⎞ n n n n ⎤ j  j  j  j 2 2 2 2 w w 1+u − 1−u 1+v − 1−v 2 2 ˙ ˙ ˙ ˙ λ λ λ λ ⎢ σ( j ) σ( j ) σ( j ) σ( j ) ⎥ 2 2 ⎜ j =1 j =1 j =1 j =1 ⎟ ⎢ ⎥ 1 + v − 1 − v ⎜     ,     ,⎟ ˙ ˙ w w w w λ λ ⎥ ⎣ n j n j n j n j ⎦ ⎜     ⎟ 2 2 2 2 2 2 1+u + 1−u 1+v + 1−v ⎥ ⎜ ⎟ λ ˙ λ ˙ λ ˙ λ ˙ σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ ⎜ ⎟ w w 2 2 = , ⎡   ⎤ ⎜     ⎟ 2 2 w w n j n j 1 + v + 1 − v ⎜ ⎟ 2 2 2 x 2 y ˙ ˙ ⎜ ⎟ λ λ ˙ ˙ 2 2 ⎢ λ λ ⎥ σ( j ) σ( j ) ⎜ j =1 j =1 ⎟ ⎢ ⎥ ⎝         ⎠ w w w w ⎣ n j n j n j n j ⎦ 2 2 2 2 2−x + x 2−y + y ⎡ ˙ ˙ ˙ ˙ λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) j =1 j =1 j =1 j =1 w 2 x (19) ⎢ λ w w 2 2 2 2 2 − x + x where λ is the jth largest of the weighted interval-valued σ( j ) ˙ ˙ λ λ 2 2 ˙ ˙ Pythagorean fuzzy values, λ (λ = nω λ ), w = σ( j ) σ( j ) j j ⎤ ⎞ (w ,w ,w ,...,w ) is the weighted vector of IVPFE- 1 2 2 n n 2 y ⎥ ⎟ HWA, such that w ∈[0, 1], w = 1.ω = λ j j 2 j =1 ⎥ ⎟ ⎦ ⎠ (ω ,ω ,ω ,...,ω ) is the weighted vector of λ ( j = 1, 2, w w 1 2 2 n j 2 2 2 2 n 2 − y + y 3,..., n) such that ω ∈[0, 1], ω = 1, and n ˙ ˙ λ λ j j 2 2 j =1 123 Complex & Intelligent Systems w w k+1 k+1 Then 2 2 m = 1 + u − 1 − u ˙ ˙ λ λ k+1 k+1 IVPFEHWA (λ ,λ ) ω,w 1 2 ⎡ ⎤ ⎛         ⎞ w w w w 2 2 2 2 j j j j w w k+1 k+1 2 2 2 2 1+u − 1−u 1+v − 1−v 2 2 ˙ ˙ ˙ ˙ λ λ λ λ ⎢ ⎥ m = 1 + u + 1 − u σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ 2 ˙ ˙ ⎢ ⎥ λ λ ⎜     ,     ,⎟ k+1 k+1 ⎣ w w w w ⎦ 2 j 2 j 2 j 2 j ⎜     ⎟ 2 2 2 2 1+u + 1−u 1+v + 1−v ⎜ ⎟ ˙ ˙ ˙ ˙ λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ w w k+1 k+1 ⎜ ⎟ = . 2 2 ⎡   ⎤ ⎜     ⎟ w w a = 1 + v − 1 − v 2 j 2 j 1 ⎜ ⎟ ˙ ˙ 2 2 λ λ 2 x 2 y k+1 k+1 ⎜ ⎟ ˙ ˙ ⎢ λ λ ⎥ σ( j ) σ( j ) ⎜ j =1 j =1 ⎟ ⎢ ⎥ ⎝         ⎠ w w w w ⎣ 2 2 2 2 ⎦ j  j  j  j w wk+1 2 2 2 2 k+1 2−x + x 2−y + y ˙ ˙ ˙ ˙ 2 2 λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) j =1 j =1 j =1 j =1 a = 1 + v + 1 − v ˙ ˙ λ λ k+1 k+1 k k !# # w w j j Thus, the result is true for n = 2, now we assume that Eq. " 2 2 r = 2 − x + x ˙ ˙ λ λ σ( j ) σ( j ) (19) holds for n = k. Thus j =1 j =1 IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 k ! ! k k ⎡   ⎤ # # ! w ! w ⎛         ⎞ j j w w w w k k k k j j j j 2 2 " " 2 2 2 2 1+u − 1−u 1+v − 1−v r = 2 x , s = 2 y ˙ ˙ ˙ ˙ 1 1 λ λ λ λ ⎢ ⎥ ˙ ˙ σ( j ) σ( j ) σ( j ) σ( j ) λ λ ⎜ j =1 j =1 j =1 j =1 ⎟ σ( j ) σ( j ) ⎢ ⎥ ⎜     ,     ,⎟ ⎣ w w w w ⎦ k j n j k j k j j =1 j =1 ⎜     ⎟ 2 2 2 2 1+u + 1−u 1+v + 1−v ⎜ ⎟ ˙ ˙ ˙ ˙ λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ ⎜ ⎟ k k = . ⎡   ⎤ ⎜     ⎟ # # ! w w w w j j k j k j ⎜ ⎟ 2 2 2 2 2 x 2 y ⎜ ⎟ ˙ ˙ s = 2 − y + y ⎢ λ λ ⎥ 2 σ( j ) σ( j ) ⎜ j =1 j =1 ⎟ ˙ ˙ λ λ ⎢ ⎥ σ( j ) σ( j ) ⎝         ⎠ w w w w ⎣ k j k j k j k j ⎦ j =1 j =1 2 2 2 2 2−x + x 2−y + y ˙ ˙ ˙ ˙ λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) j =1 j =1 j =1 j =1 w w k+1 k+1 2 2 b = 2 − x + x ˙ ˙ λ λ k+1 k+1 If Eq. (19) holds for n = k, then we show that Eq. (19) w w k+1 k+1 2 2 holds for n = k + 1. Thus b = 2 x , c = 2 y 1 1 ˙ ˙ λ λ k+1 k+1 IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 k+1 w w ⎡   ⎤ k+1 k+1 ⎛         ⎞ w w w w k k k k 2 2 j  j  j  j 2 2 2 2 c = 2 − y + y 1+u − 1−u 1+v − 1−v 2 ˙ ˙ ˙ ˙ ˙ ˙ ⎢ λ λ λ λ ⎥ λ λ σ( j ) σ( j ) σ( j ) σ( j ) k+1 k+1 ⎜ j =1 j =1 j =1 j =1 ⎟ ⎢ ⎥ ,  , ⎜         ⎟ ⎣ w w w w ⎦ k j n j k j k j ⎜     ⎟ 2 2 2 2 ⎜ 1+u + 1−u 1+v + 1−v ⎟ λ ˙ λ ˙ λ ˙ λ ˙ σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ ⎜ ⎟ ⎡   ⎤ ⎜     ⎟ w w k j k j ⎜   ⎟ 2 2 Now putting these values in Eq. (20), we have 2 x 2 y ⎜ ⎟ ˙ ˙ ⎢ λ λ ⎥ σ( j ) σ( j ) ⎜ j =1 j =1 ⎟ ⎢ ⎥ ⎝ , ⎠ w w w w ⎣ k j k j k j k j⎦ 2 2 2 2 2−x + x 2−y + y ˙ ˙ ˙ ˙ λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) j =1 j =1 j =1 j =1 IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 k+1 $ % $ % $ % $ % t p r s m a b c 1 1 1 1 1 1 1 1 ⎡   ⎤ ⎛         ⎞ w w w w k+1 k+1 k+1 k+1 = , , , ⊕ , , , 2 2 2 2 1+u − 1−u 1+v − 1−v ˙ ˙ ˙ ˙ ⎢ λ λ λ λ ⎥ t p r s m a b c 2 2 2 2 2 2 2 2 ⎜ k+1 k+1 k+1 k+1 ⎟ ⎢ ⎥ , , ⎜     ⎟ ⎣ w w w wk+1 ⎦ ⎛ ⎡ ⎤ k+1 k+1 ⎜ k+1 ⎟ 2 2 2 2 2 2 2 2 ⎜ 1+u + 1−u 1+v + 1−v ⎟ ˙ ˙ ˙ ˙ λ λ λ λ t m p a ⎜ k+1 k+1 k+1 k+1 ⎟ 1 1 1 1 + + ⎜ ⎟ ⊕ . ε ⎡ ⎤ ⎜ ⎢ t m p a ⎥ 2 2 2 2 ⎜ ⎟ w w k+1 k+1 ⎜ ⎟ ⎜ ⎢ ⎥ 2 2 =  ,  , ⎜ 2 x 2 y ⎟ ˙ ˙ ⎢ λ λ ⎥ ⎝ ⎣         ⎦ k+1 k+1 ⎜ ⎟ 2 2 2 2 ⎢ ⎥ ⎝ , ⎠ t m p a ⎣ w w w w ⎦ 1 1 1 1 k+1 k+1 k+1 k+1 1 + 1 + 2 2 2 2 2−x + x 2−y + y t m p a ˙ ˙ ˙ ˙ 2 2 2 2 λ λ λ λ k+1 k+1 k+1 k+1 (20) ⎢ r b 1 1 r b 2 2 Let ⎣ 2 2 r b 1 1 1 + 1 − 1 − k k r b 2 2 # # ! w w j j 2 2 t = 1 + u − 1 − u ⎤ ⎞ ˙ ˙ λ λ σ( j ) σ( j ) s c j =1 j =1 1 1 ⎥ ⎟ s c 2 2 ⎥ ⎟ k n ⎦ ⎠ !# # w w j j 2 2 2 2 s c 1 1 t = 1 + u + 1 − u 2 1 + 1 − 1 − ˙ ˙ λ λ s c σ( j ) σ( j ) 2 2 j =1 j =1 ⎛ ⎡ ⎤ 2 2 ! 2 2 (p a ) + (a p ) k k (t m ) + (t m ) 1 2 1 2 !# # 1 2 2 1 w w j j ⎝ ⎣ ⎦ =  ,  , " 2 2 p = 1 + v − 1 − v 1 2 2 ˙ ˙ λ λ 2 2 σ( j ) σ( j ) (t m ) + (t m ) 2 2 1 1 (p a ) + (p a ) 2 2 1 1 j =1 j =1 k k # # ! w w j j 2 2 p = 1 + v + 1 − v ˙ ˙ λ λ σ( j ) σ( j ) j =1 j =1 123 Complex & Intelligent Systems ⎛ ⎡ ⎤ δ δ r b 2 2 1 1 2 u 2 v ⎝ ⎣ ⎦ λ =  ,  , 2 2 2 2 2 2 2 2 δ δ δ δ 2r b + r b − r b − r b 2 2 2 2 2 2 1 1 2 1 1 2 2 − u + u 2 − v + v ⎤ ⎞ δ δ s c 2 2 1 1 1 + x − 1 − x ⎦ ⎠ . (21) 2 2 2 2 2 2 2 2 δ δ 2s c + s c − s c − s c 2 2 2 2 1 1 2 1 1 2 1 + x + 1 − x ⎤ ⎞ δ δ 2 2 1 + y − 1 − y ⎦ ⎠ 2 2 2 Again putting the values of (t m ) + (t m ) ,(t m ) + 1 2 2 1 2 2 δ δ 2 2 2 2 2 2 2 2 2 1 + y + 1 − y (t m ) ,(p a ) +(a p ) ,(p a ) +(p a ) , r b ,2r b + 1 1 1 2 1 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 r b − r b − r b , s c , 2s c + s c − s c − s c , in δ δ 1 1 2(0) 2(0) 1 1 2 1 1 2 2 2 1 1 2 1 1 2 =  ,  , Eq. (21), then δ δ δ δ (2 − 0) + (0) (2 − 0) + (0) δ δ δ δ (1 + 1) − (1 − 1) (1 + 1) − (1 − 1) IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 k+1 ⎡   ⎤ ⎛ ⎞ w w w w δ δ δ δ k+1 k+1 j k+1 j k+1 j j    (1 + 1) + (1 − 1) (1 + 1) + (1 − 1) 2 2 2 2 1+u − 1−u 1+v − 1−v ˙ ˙ ˙ ˙ ⎢ λ λ λ λ ⎥ σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ ⎢ ⎥ , , ⎜         ⎟ = ([0, 0], [1, 1]). w w w w ⎣ k+1 j k+1 j k+1 j k+1 j ⎦ ⎜ ⎟ 2 2 2 2 1+u + 1−u 1+v + 1−v ⎜ ⎟ ˙ ˙ ˙ ˙ λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ ⎜ ⎟ = . ⎡   ⎤ ⎜     ⎟ δ w w k+1 k+1 j  j Thus λ = ([0, 0], [1, 1]) and δλ = ([1, 1], [0, 0]). ⎜ ⎟ 2 2 2 x 2 y ⎜ ⎟ ˙ ˙ λ λ ⎢ σ( j ) σ( j ) ⎥ ⎜ j =1 j =1 ⎟ ⎢ ⎥ 3. If λ = ([u,v], [x , y]) = ([0, 0], [0, 0]) i. e,. u = v = 0 ⎝     ,     ⎠ ⎣ w w w w ⎦ k+1 j k+1 j k+1 j k+1 j 2 2 2 2 2−x + x 2−y + y λ ˙ λ ˙ λ ˙ λ ˙ and x = y = 0, then σ( j ) σ( j ) σ( j ) σ( j ) j =1 j =1 j =1 j =1 ⎛ ⎡ ⎤ δ δ 2 2 2 u 2 v ⎝ ⎣ ⎦ λ =  ,  , δ δ δ δ 2 2 2 2 Hence, Eq. (19) holds for n = k + 1. Thus, Eq. (19) holds 2 − u + u 2 − v + v for all n. δ δ 2 2 1 + x − 1 − x δ δ 2 2 Remark 1 In the following, let us look δλ and λ some special 1 + x + 1 − x ⎤ ⎞ cases of δ and λ. δ δ 2 2 1 + y − 1 − y ⎦ ⎠ δ δ 2 2 1 + y + 1 − y 1. If λ = ([u,v], [x , y]) = ([1, 1], [0, 0]) i. e,. u = v = 1 δ δ and u = v = 1, then 2(0) 2(0) =  ,  , δ δ δ δ (2 − 0) + (0) (2 − 0) + (0) ⎛ ⎡ ⎤ δ δ 2 2 δ δ δ δ 2 u 2 v (1 + 0) − (1 − 0) (1 + 0) − (1 − 0) ⎝ ⎣ ⎦ λ = , , δ δ δ δ δ δ δ δ 2 2 2 2 (1 + 0) + (1 − 0) (1 + 0) + (1 − 0) 2 − u + u 2 − v + v = ([0, 0], [0, 0]). δ δ 2 2 1 + x − 1 − x Thus λ = ([0, 0], [0, 0]) and δλ = ([0, 0], [0, 0]). δ δ 2 2 1 + x + 1 − x 4. If δ → 0 and 0 ≤ u,v, x , y ≤ 1, then ⎤ ⎞ δ δ 2 2 1 + y − 1 − y ⎛ ⎡ ⎤ ⎦ ⎠ δ δ 2 2 2 u 2 v δ δ 2 2 1 + y + 1 − y ⎝ ⎣ ⎦ λ =  ,  , δ δ δ δ 2 2 2 2 2 − u + u 2 − v + v δ δ 2(1) 2(1) =  ,  , δ δ δ δ δ δ (2 − 1) + (1) (2 − 1) + (1) 2 2 1 + x − 1 − x δ δ  , δ δ (1 + 0) − (1 − 0) (1 + 0) − (1 − 0) δ δ 2 2 1 + x + 1 − x δ δ δ δ (1 + 0) + (1 − 0) (1 + 0) + (1 − 0) ⎤ ⎞ δ δ = ([1, 1], [0, 0]). 2 2 1 + y − 1 − y ⎦ ⎠ δ δ 2 2 1 + y + 1 − y Thus λ = ([1, 1], [0, 0]) and δλ = ([0, 0], [1, 1]). 2 0 2 0 2(u ) 2(v ) 2. If λ = ([u,v], [x , y]) = ([0, 0], [1, 1]) i. e,. u = v = 0 =  ,  , 2 0 2 0 2 0 2 0 and x = y = 1, then (2 − u ) + (u ) (2 − v ) + (v ) 123 Complex & Intelligent Systems 2 0 2 0 Lemma 1 [6] Let λ 0,w 0( j = 1, 2, 3, ..., n) and j j (1 + x ) − (1 − x ) , w = 1, then j =1 2 0 2 0 (1 + x ) + (1 − x ) n n 2 0 2 0 # & (1 + y ) − (1 − y ) (λ ) ≤ w λ , (22) j j j 2 0 2 0 (1 + y ) + (1 − y ) j =1 j =1 = ([1, 1], [0, 0]). where the equality holds if and only if λ = λ = ··· = λ . 1 2 n Thus λ = ([1, 1], [0, 0]) and δλ = ([0, 0], [1, 1]). 5. If δ →+∞ and 0 ≤ u,v, x , y ≤ 1, then Theorem 3 Let λ = ([u ,v ], [x , y ])( j = 1, 2, 3, ..., n) j j j j j ⎛ ⎡ ⎤ be a collection of IVPFVs, where the w = (w ,w ,w , ..., 1 2 3 δ δ 2 2 2 u 2 v w ) is the weighted vector of IVPFEHWA and IVPFHWA, ⎝ ⎣ ⎦ λ =  ,  , δ δ δ δ such that w ∈[0, 1] and w = 1.ω = (ω ,ω ,ω , ..., j j 1 2 2 2 2 2 2 j =1 2 − u + u 2 − v + v ω ) is the weighted vector of λ ( j = 1, 2, 3, ..., n) such ⎡ n j δ δ 2 2 1 + x − 1 − x thatω ∈[0, 1], ω = 1, then j j j =1 δ δ 2 2 1 + x + 1 − x IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 n ⎤ ⎞ δ δ 2 2 1 + y − 1 − y ≤ IVPFHWA (λ ,λ ,λ , ..., λ ). (23) ω,w 1 2 3 n ⎦ ⎠ δ δ 2 2 1 + y + 1 − y Proof Straight forward. 2 ∞ 2 ∞ 2(u ) 2(v ) =  ,  , 2 ∞ 2 ∞ 2 ∞ 2 ∞ ˙ ˙ (2 − u ) + (u ) (2 − v ) + (u ) Theorem 4 Idempotency: If λ = λ for all j ( j = 1, 2, σ( j ) 3, ..., n), where λ = ([u,v], [x , y]), then 2 ∞ 2 ∞ (1 + x ) − (1 − x ) 2 ∞ 2 ∞ (1 + x ) + (1 − x ) IVPFEHWA (λ ,λ ,λ , ..., λ ) = λ. (24) ω,w 1 2 3 n 2 ∞ 2 ∞ (1 + y ) − (1 − y ) = ([[0, 0], 1, 1]). 2 ∞ 2 ∞ (1 + y ) + (1 − y ) ˙ ˙ Proof Since λ = λ for all j, then we have σ( j ) IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 n Thus, λ = ([0, 0], [1, 1]) and δλ = ([1, 1], [0, 0]). ⎡ ⎤ ⎛ w w w w ⎞ n j n j n j n j 2 2 2 2 6. If δ = 1 and 0 ≤ u,v, x , y ≤ 1, then 1+u − 1−u 1+v − 1−v ˙ ˙ ˙ ˙ λ λ λ λ ⎢ σ ( j ) σ ( j ) σ ( j ) σ ( j ) ⎥ ⎜ j =1 j =1 j =1 j =1 ⎟ ⎢ ⎥ ⎜     ,     ,⎟ ⎣ w w w w ⎦ n j n j n j n j ⎛ ⎡ ⎤ ⎜ ⎟ 2 2 2 2 1+u + 1−u 1+v + 1−v δ δ ⎜ ⎟ λ ˙ λ ˙ λ ˙ λ ˙ 2 2 σ ( j ) σ ( j ) σ ( j ) σ ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ 2 u 2 v ⎜ ⎟ ⎡   ⎤ ⎝ ⎣ ⎦ = ⎜ ⎟ λ =  ,  , w w n j n j ⎜ ⎟ 2 2 δ δ δ δ 2 x 2 y ⎜ ⎟ λ ˙ λ ˙ 2 2 2 2 ⎢ ⎥ σ ( j ) σ ( j ) 2 − u + u 2 − v + v ⎜ j =1 j =1 ⎟ ⎢ ⎥ ⎜ , ⎟ w w w w ⎣ n j n j n j n j ⎦ ⎝ ⎠ 2 2 2 2 2−x + x 2−y + y ˙ ˙ ˙ ˙ λ λ λ λ δ δ σ ( j ) σ ( j ) σ ( j ) σ ( j ) j =1 j =1 j =1 j =1 2 2 1 + x − 1 − x ⎛ ⎞ ⎡ ⎤ δ δ ! ! n n n n 2 2 ! ! 1 + x + 1 − x w   w   w   w " j j " j j 2 j =1 2 j =1 2 j =1 2 j =1 ⎜ ⎢ ⎥ ⎟ 1+u − 1−u 1+v − 1−v ˙ ˙ ˙ ˙ ⎜ ⎢ λ λ λ λ ⎥ ⎟ ⎤ ⎞ ⎜ , ,⎟ ⎢ ⎥ δ δ ! ! n n n n ⎜     ⎟ 2 2 ⎣ ⎦ ! ! 1 + y − 1 − y w w w w ⎜ " j j " j j ⎟ 2 j =1 2 j =1 2 j =1 2 j =1 ⎜ ⎟ ⎦ ⎠ 1+u + 1−u 1+v + 1−v ˙ ˙ ˙ ˙ ⎜ λ λ λ λ ⎟ = ⎜ ⎡ ⎤ ⎟ δ δ ! ! n n 2 2 ⎜ ⎟ 1 + y + 1 − y ! ! w w ⎜ " j " j ⎟ 2 j =1 2 j =1 ⎜ ⎢ ⎥ ⎟ 2 x 2 y ˙ ˙ ⎛ ⎡ ⎤ ⎜ ⎢ λ λ ⎥ ⎟ ⎢ , ⎥ 1 1 ⎜ ⎟ ! ! n n n n 2 2 ⎝ ⎣ ! ! ⎦ ⎠ 2 u 2 v w   w   w   w " j j " j j 2 j =1 2 j =1 2 j =1 2 j =1 ⎝ ⎣ ⎦ 2−x + x 2−y + y = , , ˙ ˙ ˙ ˙ λ λ λ λ 1 1 1 1 2 2 2 2 2 − u + u 2 − v + u ⎛ ⎡ ⎤ ⎞ 2 2 2 2 1+u − 1−u 1+v − 1−v ˙ ˙ ˙ ˙ λ λ λ λ 1 1 ⎜ ⎣   ⎦ ⎟ , , 2 2 ⎜ ⎟ 1 + x − 1 − x 2 2 2 2 ⎜ 1+u + 1−u 1+v + 1−v ⎟ ˙ ˙ ˙ ˙ λ λ λ λ ⎜ ⎟ ⎜ ⎟ ˙ = = λ. 1 1 ⎜   ⎟ 2 2 ⎡ ⎤ 1 + x + 1 − x ⎜ ⎟ 2 2 2 x 2 y ⎜ ⎟ ˙ ˙ λ λ ⎝ ⎠ ⎣   ⎦ ⎤ ⎞ , 1 1 2 2 2 2 2 2 2−x + x 2−y + y ˙ ˙ ˙ ˙ 1 + y − 1 − y λ λ λ λ ⎦ ⎠ = λ. 1 1 2 2 1 + y + 1 − y The proof is completed. Thus, λ = λ and δλ = λ. 123 Complex & Intelligent Systems 1 1 1 1 Theorem 5 Boundedness: Let λ = ([u ,v ], [x , y ]) j λ λ λ λ Proof Let w = , , , ..., , , and λ = λ , then j j j j σ( j ) σ( j ) n n n n ( j = 1, 2, 3, ..., n) be a collection of IVPFNs, then we have ˙ ˙ λ ≤ IVPFEHWA (λ ,λ ,λ , ..., λ ) ≤ λ , (25) min ω,w 1 2 3 n max IVPFEHWA (λ ,λ ,λ ,...,λ ) ω,w 1 2 3 n ˙ ˙ λ = max(λ ), (26) max σ( j ) ˙ ˙ ˙ = w λ ⊕ w λ ⊕ ··· ⊕ w λ 1 σ(1) ε 2 σ(2) ε ε n σ(n) ˙ ˙ λ = min(λ ). (27) min σ( j ) = w λ ⊕ w λ ⊕ ··· ⊕ w λ 1 σ(1) ε 2 σ(2) ε ε n σ(n) = IVPFEOWA (λ ,λ ,λ ,...,λ ). w 1 2 3 n Proof Proof is easy so it is omitted here. Theorem 6 Monotonicity: If λ ≤ λ for all j ( j = The proof completed. 1, 2, 3, ..., n), then IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 n An approach to multiple attribute group ∗ ∗ ∗ ∗ ≤ IVPFEHWA (λ ,λ ,λ , ..., λ ). (28) decision-making problems based on ω,w 1 2 3 n interval-valued Pythagorean fuzzy Proof As we know that. information IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 n Algorithm Let X ={X , X , X , ..., X } be a finite set of m 1 2 3 m ˙ ˙ ˙ ˙ alternatives and C ={C , C , C , ..., C } be a finite set of = w λ ⊕ w λ ⊕ w λ ⊕ ··· ⊕ w λ , 1 2 3 n 1 σ(1) ε 2 σ(2) ε 3 σ(3) ε ε n σ(n) n attributes. Suppose the grade of the alternativesX (i = (29) 1, 2, 3, ..., m)on attributeC ( j = 1, 2, 3, ..., n) given by decision makers is interval-valued Pythagorean fuzzy num- and bers. Let D ={D , D , D , ..., D } be the set of k 1 2 3 k ∗ ∗ ∗ ∗ decision makers, and let w = (w ,w ,w , ..., w ) be 1 2 3 n IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 n the weighted vector of the attributes C ( j = 1, 2, 3, ..., n), ∗ ∗ ∗ ∗ j ˙ ˙ ˙ ˙ = w λ ⊕ w λ ⊕ w λ ⊕ ··· ⊕ w λ . 1 ε 2 ε 3 ε ε n σ(1) σ(2) σ(3) σ(n) such that w ∈[0, 1], w = 1, and let ω = j j j =1 (30) (ω ,ω ,ω , ..., ω ) be the weighted vector of the deci- 1 2 3 k sion makers D (s = 1, 2, 3, ..., k), such that ω ∈[0, 1] and Since λ ≤ λ for all j,thus Eq.(28) always holds. j k ω = 1. Let D = (a ) =[u ,v ], [x , y ](i = s ji ji ji ji ji s=1 1, 2, 3, ..., m, j = 1, 2, 3, ..., n) where [u ,v ] indicates Theorem 7 Interval-valued Pythagorean fuzzy Einstein ji ji the interval degree that the alternative X (i = 1, 2, 3, ..., m) weighted averaging operator is a special case of the interval- i satisfies the attribute C ( j = 1, 2, 3, ..., n) and [x , y ] valued Pythagorean fuzzy Einstein hybrid weighted averag- j ji ji indicates the interval degree that the alternative X (i = ing operator. i 1, 2, 3, ..., m) does not satisfy the attribute C ( j = 1, 2, 3, 1 1 1 1 Proof Let ω = , , , ..., , , then we have ..., n), And also [u ,v ]∈[0, 1], [x , y ]∈[0, 1] with n n n n ji ji ji ji 2 2 condition 0 ≤ (v ) + (y ) ≤ 1,(i = 1, 2, 3, ..., m, j = ji ji IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 n 1, 2, 3, ..., n). This method has the following steps. ˙ ˙ ˙ = w λ ⊕ w λ ⊕ ··· ⊕ w λ 1 σ(1) ε 2 σ(2) ε ε n σ(n) ˙ ˙ ˙ = (λ ⊕ λ ⊕ ··· ⊕ λ ) σ(1) ε σ(2) ε ε σ(n) Step 1 Utilize the given information in the form of matrices, (s) D = a (s = 1, 2, 3, ..., k). ji n×m = (nω λ ⊕ nω λ ⊕ ··· ⊕ nω λ ) 1 1 ε 2 2 ε ε n n Step 2 If the criteria have two types, such as benefit = ω λ ⊕ ω λ ⊕ ··· ⊕ ω λ 1 1 ε 2 2 ε ε n n criteria and cost criteria, then the interval-valued (s) = IVPFEWA (λ ,λ ,λ , ..., λ ). Pythagorean fuzzy decision matrices, D = a w 1 2 3 n ji n×m (s = 1, 2, 3, ..., k) can be converted into the nor- The proof is completed. malized interval-valued Pythagorean fuzzy decision (s) matrices, R = r (s = 1, 2, 3, ..., n), where Theorem 8 Interval-valued Pythagorean fuzzy Einstein ji n×m ordered weighted averaging operator is a special case of the (s) interval-valued Pythagorean fuzzy Einstein hybrid weighted a , for benefit criteria C j j = 1, 2, 3, ..., n (s) ji r = , (s) ji averaging operator. i = 1, 2, 3, ..., m a ¯ , for cost criteria C , ji 123 Complex & Intelligent Systems (s) s T and a ¯ is the complement of α . If all the criteria makers, whose weight vector is ω = (0.2, 0.3, 0.5) . There ji ji have the same type, then there is no need of normal- are many factors that must be considered while selecting ization. the most suitable system, but here, we have consider only Step 3 Utilize the IVPFEWA operator to aggregate all the the following four criteria, whose weighted vector is w = individual normalized interval-valued Pythagorean (0.1, 0.2, 0.3, 0.4) (s) fuzzy decision matrices, R = r (s = 1, 2, ji n×m 1. C : Costs of hardware. 3, ..., k) into a single interval-valued Pythagorean 1 2. C : Support of the organization. fuzzy decision-matrix, R =[r ] , where r = 2 ji n×m ji 3. C : Effort to transform from current systems. [u ,v ], [x , y ]. 3 ji ji ji ji 4. C : Outsourcing software developer reliability, Step 4 In this step, we calculate r˙ = nw r . 4 ji j ji Step 5 Calculate the scores function of r˙ (i = 1, 2, 3, ..., ji m, j = 1, 2, 3, ..., n). If there is no difference where C , C , are cost type criteria and C , C are benefit 1 3 2 4 between two or more than two scores, then we must type criteria, i.e., the attributes have two types of criteria; find out the accuracy degrees of the collective overall thus, we must change the cost type criteria into benefit type preference values. criteria. Step 6 Utilize the IVPFEHWA operator to aggregate all preference values. Step 1 Construct the decision-making matrices (Tables 1, 2 Step 7 Arrange the scores of the all alternatives in the form and 3). of descending order and select that alternative which has the highest score function. Step 2 Construct the normalized decision making matrices (Tables 4, 5 and 6). Illustrative example Step 3 Utilize the IVPFEWA operator to aggregate all the Suppose in Hazara University, the IT department wants to individual normalized interval-valued Pythagorean select a new information system for the purpose of the best (s) fuzzy decision matrices, R = r into a ji productivity. After the first selection, there are only three n×m single interval-valued Pythagorean fuzzy decision X (i = 1, 2, 3) alternatives have been short listed. There are s matrix, R =[r ] (Table 7). ji n×m three experts D (s = 1, 2, 3) from a group to act as decision Table 1 Interval-valued X X X 1 2 3 Pythagorean fuzzy decision matrix of D C ([0.5, 0.8], [0.3, 0.4])([0.6, 0.7], [0.3, 0.6])([0.3, 0.7], [0.3, 0.5]) C ([0.3, 0.5], [0.6, 0.7])([0.3, 0.7], [0.2, 0.6])([0.3, 0.6], [0.4, 0.7]) C ([0.5, 0.7], [0.3, 0.7])([0.5, 0.6], [0.3, 0.7])([0.2, 0.6], [0.3, 0.7]) C ([0.3, 0.6], [0.6, 0.7])([0.6, 0.5], [0.2, 0.7])([0.3, 0.4], [0.5, 0.6]) Table 2 Interval-valued X X X 1 2 3 Pythagorean fuzzy decision matrix of D C ([0.5, 0.6], [0.3, 0.5])([0.5, 0.7], [0.3, 0.6])([0.2, 0.8], [0.3, 0.4]) C ([0.3, 0.4], [0.6, 0.8])([0.3, 0.8], [0.2, 0.6])([0.3, 0.6], [0.3, 0.7]) C ([0.4, 0.5], [0.3, 0.8])([0.5, 0.7], [0.3, 0.6])([0.2, 0.6], [0.3, 0.8]) C ([0.3, 0.6], [0.5, 0.7])([0.3, 0.4], [0.2, 0.8])([0.3, 0.5], [0.5, 0.7]) Table 3 Interval-valued X X X 1 2 3 Pythagorean fuzzy decision matrix of D C ([0.3, 0.8], [0.5, 0.6])([0.3, 0.5], [0.5, 0.7])([0.2, 0.4], [0.5, 0.7]) C ([0.5, 0.7], [0.3, 0.4])([0.4, 0.6], [0.5, 0.8])([0.5, 0.7], [0.2, 0.5]) C ([0.3, 0.6], [0.4, 0.6])([0.3, 0.5], [0.5, 0.6])([0.2, 0.8], [0.4, 0.6]) C ([0.5, 0.7], [0.3, 0.4])([0.5, 0.7], [0.2, 0.4])([0.5, 0.6], [0.3, 0.5]) 123 Complex & Intelligent Systems Table 4 Normalized X X X 1 2 3 Pythagorean fuzzy decision matrix R C ([0.3, 0.4], [0.5, 0.8])([0.3, 0.6], [0.6, 0.7])([0.3, 0.5], [0.3, 0.7]) C ([0.3, 0.5], [0.6, 0.7])([0.3, 0.7], [0.2, 0.6])([0.3, 0.6], [0.4, 0.7]) C ([0.3, 0.7], [0.5, 0.7])([0.3, 0.7], [0.5, 0.6])([0.3, 0.7], [0.2, 0.6]) C ([0.3, 0.6], [0.6, 0.7])([0.6, 0.5], [0.2, 0.7])([0.3, 0.4], [0.5, 0.6]) Table 5 Normalized X X X 1 2 3 Pythagorean fuzzy decision matrix R C ([0.3, 0.5], [0.5, 0.6])([0.3, 0.6], [0.5, 0.7])([0.3, 0.4], [0.2, 0.8]) C ([0.3, 0.4], [0.6, 0.8])([0.3, 0.8], [0.2, 0.6])([0.3, 0.6], [0.3, 0.7]) C ([0.3, 0.8], [0.4, 0.5])([0.3, 0.6], [0.5, 0.7])([0.3, 0.8], [0.2, 0.6]) C ([0.3, 0.6], [0.5, 0.7])([0.3, 0.4], [0.2, 0.8])([0.3, 0.5], [0.5, 0.7]) Table 6 Normalized X X X 1 2 3 Pythagorean fuzzy decision matrix R C ([0.5, 0.6], [0.3, 0.8])([0.5, 0.7], [0.3, 0.5])([0.5, 0.7], [0.2, 0.4]) C ([0.5, 0.7], [0.3, 0.4])([0.4, 0.6], [0.5, 0.8])([0.5, 0.7], [0.2, 0.5]) C ([0.4, 0.6], [0.3, 0.6])([0.5, 0.6], [0.3, 0.5])([0.4, 0.6], [0.2, 0.8]) C ([0.5, 0.7], [0.3, 0.4])([0.5, 0.7], [0.2, 0.4])([0.5, 0.6], [0.3, 0.5]) Table 7 Collective interval-valued Pythagorean fuzzy decision matrix R X X X 1 2 3 C ([0.413, 0.537], [0.389, 0.738])([0.413, 0.653], [0.405, 0.595])([0.413, 0.593], [0.216, 0.562]) C ([0.413, 0.593], [0.429, 0.563])([0.352, 0.692], [0.320, 0.697])([0.413, 0.653], [0.260, 0.595]) C ([0.352, 0.692], [0.363, 0.587])([0.413, 0.622], [0.389, 0.576])([0.352, 0.692], [0.200, 0.697]) C ([0.413, 0.653], [0.405, 0.536])([0.475, 0.593], [0.200, 0.563])([0.413, 0.537], [0.389, 0.576]) Step 4 Calculate λ = nwλ . Step 5 Calculate the score functions (Table 8). ji ji λ = ([0.262, 0.343], [0.733, 0.897]), ˙ ˙ ˙ s(λ ) =−0.57, s(λ ) =−0.13, s(λ ) 11 21 31 λ = ([0.370, 0.534], [0.523, 0.645]) = 0.18, s(λ ) = 0.37 λ = ([0.385, 0.745], [0.281, 0.513]), ˙ ˙ ˙ s(λ ) =−0.50, s(λ ) =−0.12, s(λ ) 12 22 32 λ = ([0.518, 0.788], [0.201, 0.329]) = 0.15, s(λ ) = 0.37 λ = ([0.262, 0.424], [0.742, 0.837]), ˙ ˙ ˙ s(λ ) =−0.41, s(λ ) =−0.04, s(λ ) 13 23 33 λ = ([0.315, 0.628], [0.420, 0.757]) = 0.13, s(λ ) = 0.26. λ = ([0.452, 0.665], [0.307, 0.501]), λ = ([0.593, 0.726], [0.061, 0.359]) λ = ([0.262, 0.382], [0.605, 0.823]), Step 6 Utilize the IVPFEHWA aggregation operator to λ = ([0.370, 0.590], [0.357, 0.672]) aggregate all preference values. λ = ([0.385, 0.745], [0.136, 0.638]), λ = ([0.518, 0.664], [0.188, 0.374]). r = ([0.354, 0.567], [0.550, 0.674]) 123 Complex & Intelligent Systems Table 8 Pythagorean fuzzy hybrid decision matrix R X X X 1 2 3 C ([0.518, 0.788], [0.201, 0.329])([0.593, 0.726], [0.061, 0.359])([0.518, 0.664], [0.188, 0.374]), C ([0.385, 0.745], [0.281, 0.513])([0.525, 0.665], [0.307, 0.501])([0.585, 0.745], [0.136, 0.638]), C ([0.370, 0.534], [0.523, 0.645])([0.315, 0.628], [0.420, 0.757])([0.370, 0.590], [0.357, 0.672]), C ([0.262, 0.343], [0.733, 0.897])([0.262, 0.424], [0.742, 0.837])([0.262, 0.382], [0.605, 0.823]). r = ([0.367, 0.581], [0.422, 0.686]) References r = ([0.354, 0.571], [0.347, 0.695]). 1. Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96 2. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353 3. Xu ZS, Yager RR (2006) Some geometric aggregation operators Step 7 Calculate the score functions. based on intuitionistic fuzzy sets. Int J Gen Syst 35(4):417–433 4. Xu ZS (2007) Intuitionistic fuzzy aggregation operators. IEEE s(r ) =−0.154, s(r ) =−0.088, s(r ) =−0.076. 1 1 1 Trans Fuzzy Syst 15(6):1179–1187 5. Wang W, Liu X (2011) Intuitionistic fuzzy geometric aggrega- tion operators based on Einstein operations. Int J Intell Syst Step 8 Arrange the scores of the all alternatives in the form 26(11):1049–1075 of descending order and select that alternative which 6. Wang WZ, Liu XW (2012) Intuitionistic fuzzy information has the highest score function. aggregation using Einstein operations. IEEE Trans Fuzzy Syst 20(5):923–938 7. 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Yager RR (2014) Pythagorean membership grades in multi-criteria decision making. IEEE Trans Fuzzy Syst 22(4):958–965 19. Yager RR, Abbasov AM (2013) Pythagorean membership grades, Open Access This article is distributed under the terms of the Creative complex numbers and decision making. Int J Intell Syst 28(5):436– Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, 20. Peng X, Yang Y (2015) Some results for Pythagorean fuzzy sets. and reproduction in any medium, provided you give appropriate credit Int J Intell Syst 30(11):1133–1160 to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 123 Complex & Intelligent Systems 21. Garg H (2016) A new generalized Pythagorean fuzzy information 29. Peng X, Yang Y (2016) Fundamental properties of interval- aggregation using Einstein operations and its application to deci- valued Pythagorean fuzzy aggregation operators. Int J Intell Syst sion making. Int J Intell Syst 31(9):886–920 31(5):444–487 22. Garg H (2017) Generalized Pythagorean fuzzy geometric aggreg- 30. Rahman K, Ali Asad, Khan MSA (2018) Some interval-valued tion operators using Einstein t-Norm and t-Conorm for multicrite- Pythagorean fuzzy weighted averaging aggregation operators and ria decision-making process. Int J Intell Syst 32:597–630. https:// their application to multiple attribute decision making, Punjab Uni- doi.org/10.1002/int.21860 versity. J Math 50(2):113–129 23. Rahman K, Abdullah S, Husain F, Ali Khan MS (2016) Approaches 31. Rahman K, Abdullah S, Shakeel M, Khan MSA, Ullah Murad to Pythagorean fuzzy geometric aggregation operators. Int J Com- (2017) Interval-valued Pythagorean fuzzy geometric aggregation put Sci Inf Secur IJCSIS 4(9):174–200 operators and their application to decision making. Cogent Math 24. Rahman K, Khan MSA, Ullah M, Fahmi A (2017) Multiple 4(1):1338638. https://doi.org/10.1515/jisys-2017-0212 attribute group decision making for plant location selection with 32. Rahman K, Abdullah S, Khan MSA (2018) Some interval-valued Pythagorean fuzzy weighted geometric aggregation. Operator Pythagorean fuzzy Einstein weighted averaging aggregation oper- Nucleus 54(1):66–74 ator and their application to group decision making. J Intell Syst. 25. Rahman K, Abdullah S, Husain F, Ali Khan MS, Shakeel M https://doi.org/10.1515/jisys-2017-0212 (2017) Pythagorean fuzzy ordered weighted geometric aggregation 33. Rahman K, Abdullah S (2018) Generalized interval-valued operator and their application to multiple attribute group decision Pythagorean fuzzy aggregation operators and their application to making. J Appl Environ Biol Sci 7(4):67–83 group decision making. Granul Comput. https://doi.org/10.1007/ 26. Rahman K, Abdullah S, Ali Khan MS, Shakeel M (2016) s41066-018-0082-9 Pythagorean fuzzy hybrid geometric aggregation operator and their applications to multiple attribute decision making. Int J Comput Sci Inf Secur IJCSIS 837–854 Publisher’s Note Springer Nature remains neutral with regard to juris- 27. Rahman K, Ali A, Shakeel M, Khan MSA, Ullah Murad (2017) dictional claims in published maps and institutional affiliations. Pythagorean fuzzy weighted averaging aggregation operator and its application to decision making theory. Nucleus 54(3):190–196 28. Rahman K, Abdullah S, Ahmed R, Ullah Murad (2017) Pythagorean fuzzy Einstein weighted geometric aggregation oper- ator and their application to multiple attribute group decision making. J Intell Fuzzy Syst 33(1):635–647 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Complex & Intelligent Systems Springer Journals

Interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging aggregation operator and their application to group decision making

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Abstract

The objective of the present work is divided into two folds. Firstly, interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging aggregation operator has been introduced along with their several properties, namely idempotency, boundedness and monotonicity. Secondly, we apply the proposed operator to deal with multi-attribute group decision-making problem under Pythagorean fuzzy information. For this, we construct an algorithm for multi-attribute group decision making. At the last, we construct a numerical example for multi-attribute group decision making. The main advantage of using the proposed operator is that this operator provides more accurate and precise results is compared to the existing methods. Keywords IVPFS · IVPFEHWA averaging operator · MAGDM problems Introduction successful and positive applications of intuitionistic fuzzy set, aggregation operators become more interesting topic for Multi-criteria group decision making is one of the success- research. Thus, many scholars in [3–16] developed several ful processes for finding the optimal alternative from all the aggregation operators for group decision making using intu- feasible alternatives according to some criteria or attributes. itionistic fuzzy information. Traditionally, it has been generally assumed that all the infor- However, there are many cases where the decision maker mation that access the alternative in terms of criteria and their may provide the degree of membership and nonmembership corresponding weights are expressed in the form of crisp of a particular attribute in such a way that their sum is greater numbers. But most of the decisions in the real-life situations than one. To solve these types of problems, Yager [17,18] are taken in the environment where the goals and constraints introduced the concept of another set called Pythagorean are generally imprecise or vague in nature. In order to han- fuzzy set. Pythagorean fuzzy set is more powerful tool to dle the uncertainties and fuzziness intuitionistic fuzzy set [1] solve uncertain problems. Like intuitionistic fuzzy aggrega- theory is one of the successful extensions of the fuzzy set the- tion operators, Pythagorean fuzzy aggregation operators are ory [2], which is characterized by the degree of membership also become an interesting and important area for research, and degree of non-membership has been presented. After the after the advent of Pythagorean fuzzy set theory. Several researchers in [19–28] introduced many aggregation oper- ators for decision using Pythagorean fuzzy information. B Khaista Rahman But, in some real decision-making problems, due to insuf- khaista355@yahoo.com ficiency in available information, it may be difficult for Saleem Abdullah decision makers to exactly quantify their opinions with a saleemabdullah81@yahoo.com crisp number, but they can be represented by an interval num- Asad Ali ber within [0, 1]. Therefore, it is so important to present the asad_maths@hu.edu.pk idea of interval-valued Pythagorean fuzzy sets, which permit Fazli Amin the membership degrees and non- membership degrees to a fazliamin@hu.edu.pk given set to have an interval value. Thus in [29] Peng and Department of Mathematics, Hazara University, Mansehra, Yang introduced the concept of interval-valued Pythagorean KPK, Pakistan fuzzy set. Rahman et al. [30–33] introduced many aggre- Department of Mathematics, Abdul Wali Khan University gation operators using interval-valued Pythagorean fuzzy Mardan, Mardan, KPK, Pakistan 123 Complex & Intelligent Systems numbers and applied them to multi-attribute group decision and making. a b v (k) =[v (k), v (k)]⊂[0, 1]. (5) Thus, keeping the advantages of these operators, in this I I I paper, we introduce the notion of interval-valued Pythagorean Also fuzzy Einstein hybrid weighted averaging operator. More- over, we introduce some of their basic properties such as u (k) = inf(u (k)), (6) idempotency, boundedness and monotonicity. This motiva- I tion comes from [32], in which the authors introduced the u (k) = sup(u (k)), (7) notion of IVPFEWA operator and IVPFEOWA operator and a v (k) = inf(v (k)), (8) applied them to group decision making. But in this paper we v (k) = sup(v (k)), (9) introduce the notion of IVPFEHWA operator, which is the generalization of the above mention operators. and The remainder of this paper is structured as follows. 2 2 In Sect. "Preliminaries", we give some basic definitions b b 0 ≤ u (k) + v (k) ≤ 1. (10) I I and results which will be used in our later sections. In Sect. "Interval-valued Pythagorean fuzzy Einstein hybrid If weighted averaging aggregation operator", we introduce the notion of interval-valued Pythagorean fuzzy Einstein hybrid a b π (k) = π (k), π (k) , for all k ∈ K . (11) I I weighted averaging operator. In Sect. "An approach to mul- tiple attribute group decision-making problems based on Then, it is called the interval-valued Pythagorean fuzzy intervalvalued Pythagorean fuzzy information", we apply index of k to I , where the proposed operator to multi-attribute group decision- making problem with Pythagorean fuzzy information. In 2 2 a b b π (k) = 1 − u (k) − v (k) , (12) Sect. "Illustrative example", we develop a numerical exam- I I I ple. In Sect. "Conclusion", we have conclusion. and 2 2 a a π (k) = 1 − u (k) − v (k) . (13) Preliminaries I I Definition 3 [29]Let λ = ([u ,v ], [x , y ]) be an IVPFN, Definition 1 [17,18]Let K be a fixed set, then a Pythagorean λ λ λ λ then the score function and accuracy function of λ can be fuzzy set can be defined as: defined as follows, respectively: P ={k, u (k), v (k)|k ∈ K }, (1) P P 2 2 2 2 s(λ) = (u ) + (v ) − (x ) − (y ) , (14) λ λ λ λ where u (k) : P →[0, 1],v (k) : K →[0, 1] are called P P membership function and non-membership function, respec- and 2 2 tively, with condition 0 ≤ (u (k)) + (v (k)) ≤ 1, for all P P k ∈ K . 2 2 2 2 h(λ) = u + v + x + y . (15) ( ) ( ) ( ) ( ) λ λ λ λ Let If λ and λ are two IVPFNs, then 1 2 2 2 π (k) = 1 − u (k) − v (k). (2) P P 1. If s(λ ) ≺ s(λ ), then λ ≺ λ . 1 2 1 2 Then, it is called the Pythagorean fuzzy index of k ∈ K , 2. If s(λ ) = s(λ ), then we have the following three con- 1 2 with condition 0 ≤ π (k) ≤ 1, for every k ∈ K . ditions. Definition 2 [29]Let K be a fixed set, then an interval-valued 1) If h(λ ) = h(λ ), then λ = λ . 1 2 1 2 Pythagorean fuzzy set can be defined as: 2) If h(λ ) ≺ h(λ ), then λ ≺ λ . 1 2 1 2 3) If h(λ ) h(λ ), then λ λ . 1 2 1 2 I ={k, u (k), v (k)|k ∈ K }, (3) I I Definition 4 [32]Let λ = ([u,v], [x , y]), λ = ([u ,v ], 1 1 1 where [x , y ]), λ = ([u ,v ], [x , y ]) are three IVPFNs, and 1 1 2 2 2 2 2 δ 0, then some Einstein operations for λ, λ ,λ can be 1 2 a b u (k) =[u (k), u (k)]⊂[0, 1], (4) defined as follows: I I 123 Complex & Intelligent Systems 1. Definition 5 [32]Let λ = ([u ,v ], [x , y ])( j = 1, 2, 3, j j j j j ..., n) be the collection of IVPFVs, then IVPFEWA operator ⎛ ⎡ ⎤ 2 2 2 2 u + u v + v can be defined as: 1 2 1 2 ⎝ ⎣ ⎦ λ ⊕ λ =  ,  , 1 ε 2 IVPFEWA (λ ,λ ,λ , ..., λ ) w 1 2 3 n 2 2 2 2 1 + u u 1 + v v 1 2 1 2 ⎡   ⎤ ⎛ ⎞ n w n w n w n w j j j j 2 2 2 2 1+u − 1−u 1+v − 1−v ⎡ λ λ λ λ ⎢ j j j j ⎥ ⎜ j =1 j =1 j =1 j =1 ⎟ ⎢ ⎥ , , ⎜ ⎟ ⎣ n n n n ⎦ x x  w  w  w  w ⎜ j j j j ⎟ 1 2 2 2 2 2 ⎣ 1+u + 1−u 1+v + 1−v ⎜ ⎟ , λ λ λ λ j j j j ⎜ j =1 j =1 j =1 j =1 ⎟ 2 2 ⎜ ⎟ 1 + 1 − x 1 − x = ⎜ ⎟ , 1 2 ⎡   ⎤ ⎜ ⎟ n n w  w ⎜ j j ⎟ 2 2 2 x 2 y ⎜ ⎟ λ λ ⎤ ⎞ ⎢ j j ⎥ ⎜ j =1 j =1 ⎟ ⎢ ⎥ ⎝ , ⎠ n   n   n   n ⎣ w w w w ⎦ j  j  j  j y y 2 2 2 2 1 2 2−x + x 2−y + y λ λ λ λ ⎦ ⎠ j j j j j =1 j =1 j =1 j =1 2 2 1 + 1 − y 1 − y (16) 1 2 where w = (w ,w ,w , ..., w ) is the weighted vector of 1 2 3 n ⎛ ⎡ λ ( j = 1, 2, 3, ..., n), such that w ∈[0, 1] and w = j j j j =1 u u 1. 1 2 ⎝ ⎣ λ ⊗ λ =  , 1 ε 2 2 2 1 + 1 − u 1 − u Definition 6 [32]Let λ ( j = 1, 2, 3, ..., n) be a collection of 1 2 j IVPFVs, then IVPFEOWA operator can be defined as: v v 1 2 IVPFEOWA (λ ,λ ,λ , ..., λ ) w 1 2 3 n ⎛ ⎡   ⎤ ⎞ 2 2 n   n   n   n w  w  w  w 1 + 1 − v 1 − v j j j j 2 2 2 2 1 2 1+u − 1−u 1+v − 1−v λ λ λ λ ⎢ σ( j ) σ( j ) σ( j ) σ( j ) ⎥ ⎜ ⎟ j =1 j =1 j =1 j =1 ⎢ ⎥ ⎜   ⎟ , , ⎜ ⎣ n w n w n w n w ⎦ ⎟ ⎡ ⎤ ⎞ j j j j 2 2 2 2 ⎜ 1+u + 1−u 1+v + 1−v ⎟ λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) 2 2 2 2 ⎜ ⎟ j =1 j =1 j =1 j =1 x + x y + y ⎜ ⎟ 1 2 1 2 ⎜ ⎟ = , ⎣ ⎦ ⎠ ,  ⎜ ⎟ ⎡   ⎤ ⎜ n   n   ⎟ w w j  j 2 2 2 2 ⎜ 2 2 ⎟ 2 x 2 y λ λ 1 + x x 1 + y y ⎜ ⎟ ⎢ σ( j ) σ( j ) ⎥ 1 2 1 2 j =1 j =1 ⎜ ⎢ ⎥ ⎟ ⎝         ⎠ ⎣ n w n w n w n w ⎦ j j j j 2 2 2 2 2−x + x 2−y + y λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) j =1 j =1 j =1 j =1 (17) 2 δ 2 δ (1 + u ) − (1 − u ) δλ =  , where (σ (1), σ (2), σ (3), ..., σ (n)) is a permutation of (1, 2, 2 δ 2 δ (1 + u ) + (1 − u ) 3, ..., n) such that σ( j ) ≤ σ( j − 1) for all jand w = (w ,w ,w , ..., w ) is the weighted vector of λ ( j = 1 2 3 n σ( j ) 2 δ 2 δ (1 + v ) − (1 − v ) , 1, 2, 3, ..., n) such that w ∈[0, 1] and w = 1. j j j =1 2 δ 2 δ (1 + v ) + (1 − v ) 2 δ 2 δ 2(x ) 2(y ) Interval-valued Pythagorean fuzzy Einstein 2 δ 2 δ 2 δ 2 δ (2 − y ) + (y ) (2 − x ) + (x ) hybrid weighted averaging aggregation operator In this section, we introduce the notion of interval-valued 2 δ 2 δ 2(u ) 2(v ) Pythagorean fuzzy Einstein hybrid weighted averaging aggre- λ =  ,  , 2 δ 2 δ 2 δ 2 δ (2 − u ) + (u ) (2 − v ) + (v ) gation operator. We also discuss some desirable properties such as idempotency, boundedness and monotonicity. 2 δ 2 δ (1 + x ) − (1 − x ) Definition 7 An interval-valued Pythagorean fuzzy Einstein 2 δ 2 δ (1 + x ) + (1 − x ) hybrid weighted averaging operator of dimension n is a mapping IVPFEHWA :  → ,which has associated 2 δ 2 δ (1 + y ) − (1 − y ) vectorw = (w ,w ,w , ..., w ) , such that w ∈[0, 1] and 1 2 3 n j 2 δ 2 δ (1 + y ) + (1 − y ) n w = 1. Furthermore j =1 123 Complex & Intelligent Systems IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 n is the balancing coefficient, which plays a role of bal- ⎡ ⎤ ⎛ w w w w ⎞ n n n n T j  j  j  j ance. If the vector w = (w ,w ,w ,...,w ) approaches 2 2 2 2 1 2 2 n 1+u − 1−u 1+v − 1−v ˙ ˙ ˙ ˙ λ λ λ λ ⎢ ⎥ σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ T ⎢ ⎥ 1 1 1 1 ⎜     ,     ,⎟ w w w w , , ,..., , then the vector (nwωλ , ⎣ n j n j n j n j ⎦ ⎜ ⎟ 2 2 2 2 n n n n 1+u + 1−u 1+v + 1−v ⎜ ⎟ ˙ ˙ ˙ ˙ λ λ λ λ T T σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ nω λ ,..., nω λ ) approaches(λ ,λ ,λ ,...,λ ) . 2 2 n n 1 2 3 n ⎜ ⎟ = , ⎡   ⎤ ⎜ ⎟ w w n j n j ⎜ ⎟ 2 2 2 x 2 y ⎜ ⎟ ˙ ˙ ⎢ λ λ ⎥ σ( j ) σ( j ) ⎜ j =1 j =1 ⎟ Proof We can prove this theorem by mathematical induction ⎢   ⎥ ⎝         ⎠ w w w w ⎣ n j n j n j n j⎦ 2 2 2 2 2−x + x 2−y + y ˙ ˙ ˙ ˙ λ λ λ λ on n. σ( j ) σ( j ) σ( j ) σ( j ) j =1 j =1 j =1 j =1 For n = 2 (18) ⎛ ⎡ w w 1 1 ˙ 2 2 where λ is the jth largest of the weighted interval-valued σ( j ) 1 + u − 1 − u ˙ ˙ ⎜ ⎢ λ λ 1 1 ˙ ˙ ⎜ ⎢ Pythagorean fuzzy values, λ (λ = nω λ ). ω = ˙ σ( j ) σ( j ) j j w λ =  , 1 1 ⎝ ⎣ w w T 1 1 (ω ,ω ,ω , ..., ω ) is the weighted vector of λ ( j = 1, 2, 2 2 1 2 3 n j 1 + u + 1 − u n ˙ ˙ λ λ 1 1 3, ..., n) such that ω ∈[0, 1], ω = 1, and n j j j =1 is the balancing coefficient, which plays a role of bal- ⎤ w w 1 1 2 2 ance. If the vector w = (w ,w ,w , ..., w ) approaches 1 + v − 1 − v 1 2 3 n ˙ ˙ λ λ ⎥ 1 1 1 1 1 1 to , , , ..., , then the vector (nω λ , nω λ , ..., 1 1 2 2 n n n n w w 1 1 T T 2 2 nω λ ) approaches to (λ ,λ ,λ , ..., λ ) . n n 1 2 3 n 1 + v + 1 − v ˙ ˙ λ λ 1 1 Theorem 1 Let λ, λ ,λ be the three interval-valued 1 2 ⎡ Pythagorean fuzzy numbers and δ, δ ,δ 0, then the fol- 1 2 2 x lowing conditions always hold: w w 1 1 2 2 2 − x + x ˙ ˙ λ λ 1 1 1. λ ⊕ λ = λ ⊕ λ , 1 ε 2 2 ε 1 2. λ ⊗ λ = λ ⊗ λ , ⎤ ⎞ 1 ε 2 2 ε 1 3. δ(λ ⊕ λ ) = δλ ⊕ δλ , 1 ε 2 1 ε 2 2 y λ ⎥ ⎟ δ δ δ ⎥ ⎟ 4. (λ ⊗ λ ) = (λ ) ⊗ (λ ) , 1 ε 2 1 ε 2 ⎦ ⎠ w w 1 1 5. δ (λ) ⊕ δ (λ) = (δ ⊕ δ )λ, 2 2 1 ε 2 1 ε 2 2 − y + y ˙ ˙ δ δ (δ ⊗ δ ) λ λ 1 1 1 2 1 ε 2 6. (λ) ⊗ (λ) = λ . and Proof The proof is trivial, so it is omitted here. ⎛ ⎡ w w 2 2 2 2 Theorem 2 Let λ = ([u ,v ], [x , y ])( j = 1, 2, 3, ..., n) j j j j j 1 + u − 1 − u ˙ ˙ ⎜ ⎢ λ λ 2 2 ⎜ ⎢ be a collection of IVPFVs, then their aggregated value using ˙ w λ =  , 2 2 ⎝ ⎣ w w 2 2 the IVPFEHWA operator is also an IVPFV, and 2 2 1 + u + 1 − u ˙ ˙ λ λ 2 2 IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 n ⎡ ⎤ ⎛ w w w w ⎞ n n n n ⎤ j  j  j  j 2 2 2 2 w w 1+u − 1−u 1+v − 1−v 2 2 ˙ ˙ ˙ ˙ λ λ λ λ ⎢ σ( j ) σ( j ) σ( j ) σ( j ) ⎥ 2 2 ⎜ j =1 j =1 j =1 j =1 ⎟ ⎢ ⎥ 1 + v − 1 − v ⎜     ,     ,⎟ ˙ ˙ w w w w λ λ ⎥ ⎣ n j n j n j n j ⎦ ⎜     ⎟ 2 2 2 2 2 2 1+u + 1−u 1+v + 1−v ⎥ ⎜ ⎟ λ ˙ λ ˙ λ ˙ λ ˙ σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ ⎜ ⎟ w w 2 2 = , ⎡   ⎤ ⎜     ⎟ 2 2 w w n j n j 1 + v + 1 − v ⎜ ⎟ 2 2 2 x 2 y ˙ ˙ ⎜ ⎟ λ λ ˙ ˙ 2 2 ⎢ λ λ ⎥ σ( j ) σ( j ) ⎜ j =1 j =1 ⎟ ⎢ ⎥ ⎝         ⎠ w w w w ⎣ n j n j n j n j ⎦ 2 2 2 2 2−x + x 2−y + y ⎡ ˙ ˙ ˙ ˙ λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) j =1 j =1 j =1 j =1 w 2 x (19) ⎢ λ w w 2 2 2 2 2 − x + x where λ is the jth largest of the weighted interval-valued σ( j ) ˙ ˙ λ λ 2 2 ˙ ˙ Pythagorean fuzzy values, λ (λ = nω λ ), w = σ( j ) σ( j ) j j ⎤ ⎞ (w ,w ,w ,...,w ) is the weighted vector of IVPFE- 1 2 2 n n 2 y ⎥ ⎟ HWA, such that w ∈[0, 1], w = 1.ω = λ j j 2 j =1 ⎥ ⎟ ⎦ ⎠ (ω ,ω ,ω ,...,ω ) is the weighted vector of λ ( j = 1, 2, w w 1 2 2 n j 2 2 2 2 n 2 − y + y 3,..., n) such that ω ∈[0, 1], ω = 1, and n ˙ ˙ λ λ j j 2 2 j =1 123 Complex & Intelligent Systems w w k+1 k+1 Then 2 2 m = 1 + u − 1 − u ˙ ˙ λ λ k+1 k+1 IVPFEHWA (λ ,λ ) ω,w 1 2 ⎡ ⎤ ⎛         ⎞ w w w w 2 2 2 2 j j j j w w k+1 k+1 2 2 2 2 1+u − 1−u 1+v − 1−v 2 2 ˙ ˙ ˙ ˙ λ λ λ λ ⎢ ⎥ m = 1 + u + 1 − u σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ 2 ˙ ˙ ⎢ ⎥ λ λ ⎜     ,     ,⎟ k+1 k+1 ⎣ w w w w ⎦ 2 j 2 j 2 j 2 j ⎜     ⎟ 2 2 2 2 1+u + 1−u 1+v + 1−v ⎜ ⎟ ˙ ˙ ˙ ˙ λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ w w k+1 k+1 ⎜ ⎟ = . 2 2 ⎡   ⎤ ⎜     ⎟ w w a = 1 + v − 1 − v 2 j 2 j 1 ⎜ ⎟ ˙ ˙ 2 2 λ λ 2 x 2 y k+1 k+1 ⎜ ⎟ ˙ ˙ ⎢ λ λ ⎥ σ( j ) σ( j ) ⎜ j =1 j =1 ⎟ ⎢ ⎥ ⎝         ⎠ w w w w ⎣ 2 2 2 2 ⎦ j  j  j  j w wk+1 2 2 2 2 k+1 2−x + x 2−y + y ˙ ˙ ˙ ˙ 2 2 λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) j =1 j =1 j =1 j =1 a = 1 + v + 1 − v ˙ ˙ λ λ k+1 k+1 k k !# # w w j j Thus, the result is true for n = 2, now we assume that Eq. " 2 2 r = 2 − x + x ˙ ˙ λ λ σ( j ) σ( j ) (19) holds for n = k. Thus j =1 j =1 IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 k ! ! k k ⎡   ⎤ # # ! w ! w ⎛         ⎞ j j w w w w k k k k j j j j 2 2 " " 2 2 2 2 1+u − 1−u 1+v − 1−v r = 2 x , s = 2 y ˙ ˙ ˙ ˙ 1 1 λ λ λ λ ⎢ ⎥ ˙ ˙ σ( j ) σ( j ) σ( j ) σ( j ) λ λ ⎜ j =1 j =1 j =1 j =1 ⎟ σ( j ) σ( j ) ⎢ ⎥ ⎜     ,     ,⎟ ⎣ w w w w ⎦ k j n j k j k j j =1 j =1 ⎜     ⎟ 2 2 2 2 1+u + 1−u 1+v + 1−v ⎜ ⎟ ˙ ˙ ˙ ˙ λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ ⎜ ⎟ k k = . ⎡   ⎤ ⎜     ⎟ # # ! w w w w j j k j k j ⎜ ⎟ 2 2 2 2 2 x 2 y ⎜ ⎟ ˙ ˙ s = 2 − y + y ⎢ λ λ ⎥ 2 σ( j ) σ( j ) ⎜ j =1 j =1 ⎟ ˙ ˙ λ λ ⎢ ⎥ σ( j ) σ( j ) ⎝         ⎠ w w w w ⎣ k j k j k j k j ⎦ j =1 j =1 2 2 2 2 2−x + x 2−y + y ˙ ˙ ˙ ˙ λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) j =1 j =1 j =1 j =1 w w k+1 k+1 2 2 b = 2 − x + x ˙ ˙ λ λ k+1 k+1 If Eq. (19) holds for n = k, then we show that Eq. (19) w w k+1 k+1 2 2 holds for n = k + 1. Thus b = 2 x , c = 2 y 1 1 ˙ ˙ λ λ k+1 k+1 IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 k+1 w w ⎡   ⎤ k+1 k+1 ⎛         ⎞ w w w w k k k k 2 2 j  j  j  j 2 2 2 2 c = 2 − y + y 1+u − 1−u 1+v − 1−v 2 ˙ ˙ ˙ ˙ ˙ ˙ ⎢ λ λ λ λ ⎥ λ λ σ( j ) σ( j ) σ( j ) σ( j ) k+1 k+1 ⎜ j =1 j =1 j =1 j =1 ⎟ ⎢ ⎥ ,  , ⎜         ⎟ ⎣ w w w w ⎦ k j n j k j k j ⎜     ⎟ 2 2 2 2 ⎜ 1+u + 1−u 1+v + 1−v ⎟ λ ˙ λ ˙ λ ˙ λ ˙ σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ ⎜ ⎟ ⎡   ⎤ ⎜     ⎟ w w k j k j ⎜   ⎟ 2 2 Now putting these values in Eq. (20), we have 2 x 2 y ⎜ ⎟ ˙ ˙ ⎢ λ λ ⎥ σ( j ) σ( j ) ⎜ j =1 j =1 ⎟ ⎢ ⎥ ⎝ , ⎠ w w w w ⎣ k j k j k j k j⎦ 2 2 2 2 2−x + x 2−y + y ˙ ˙ ˙ ˙ λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) j =1 j =1 j =1 j =1 IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 k+1 $ % $ % $ % $ % t p r s m a b c 1 1 1 1 1 1 1 1 ⎡   ⎤ ⎛         ⎞ w w w w k+1 k+1 k+1 k+1 = , , , ⊕ , , , 2 2 2 2 1+u − 1−u 1+v − 1−v ˙ ˙ ˙ ˙ ⎢ λ λ λ λ ⎥ t p r s m a b c 2 2 2 2 2 2 2 2 ⎜ k+1 k+1 k+1 k+1 ⎟ ⎢ ⎥ , , ⎜     ⎟ ⎣ w w w wk+1 ⎦ ⎛ ⎡ ⎤ k+1 k+1 ⎜ k+1 ⎟ 2 2 2 2 2 2 2 2 ⎜ 1+u + 1−u 1+v + 1−v ⎟ ˙ ˙ ˙ ˙ λ λ λ λ t m p a ⎜ k+1 k+1 k+1 k+1 ⎟ 1 1 1 1 + + ⎜ ⎟ ⊕ . ε ⎡ ⎤ ⎜ ⎢ t m p a ⎥ 2 2 2 2 ⎜ ⎟ w w k+1 k+1 ⎜ ⎟ ⎜ ⎢ ⎥ 2 2 =  ,  , ⎜ 2 x 2 y ⎟ ˙ ˙ ⎢ λ λ ⎥ ⎝ ⎣         ⎦ k+1 k+1 ⎜ ⎟ 2 2 2 2 ⎢ ⎥ ⎝ , ⎠ t m p a ⎣ w w w w ⎦ 1 1 1 1 k+1 k+1 k+1 k+1 1 + 1 + 2 2 2 2 2−x + x 2−y + y t m p a ˙ ˙ ˙ ˙ 2 2 2 2 λ λ λ λ k+1 k+1 k+1 k+1 (20) ⎢ r b 1 1 r b 2 2 Let ⎣ 2 2 r b 1 1 1 + 1 − 1 − k k r b 2 2 # # ! w w j j 2 2 t = 1 + u − 1 − u ⎤ ⎞ ˙ ˙ λ λ σ( j ) σ( j ) s c j =1 j =1 1 1 ⎥ ⎟ s c 2 2 ⎥ ⎟ k n ⎦ ⎠ !# # w w j j 2 2 2 2 s c 1 1 t = 1 + u + 1 − u 2 1 + 1 − 1 − ˙ ˙ λ λ s c σ( j ) σ( j ) 2 2 j =1 j =1 ⎛ ⎡ ⎤ 2 2 ! 2 2 (p a ) + (a p ) k k (t m ) + (t m ) 1 2 1 2 !# # 1 2 2 1 w w j j ⎝ ⎣ ⎦ =  ,  , " 2 2 p = 1 + v − 1 − v 1 2 2 ˙ ˙ λ λ 2 2 σ( j ) σ( j ) (t m ) + (t m ) 2 2 1 1 (p a ) + (p a ) 2 2 1 1 j =1 j =1 k k # # ! w w j j 2 2 p = 1 + v + 1 − v ˙ ˙ λ λ σ( j ) σ( j ) j =1 j =1 123 Complex & Intelligent Systems ⎛ ⎡ ⎤ δ δ r b 2 2 1 1 2 u 2 v ⎝ ⎣ ⎦ λ =  ,  , 2 2 2 2 2 2 2 2 δ δ δ δ 2r b + r b − r b − r b 2 2 2 2 2 2 1 1 2 1 1 2 2 − u + u 2 − v + v ⎤ ⎞ δ δ s c 2 2 1 1 1 + x − 1 − x ⎦ ⎠ . (21) 2 2 2 2 2 2 2 2 δ δ 2s c + s c − s c − s c 2 2 2 2 1 1 2 1 1 2 1 + x + 1 − x ⎤ ⎞ δ δ 2 2 1 + y − 1 − y ⎦ ⎠ 2 2 2 Again putting the values of (t m ) + (t m ) ,(t m ) + 1 2 2 1 2 2 δ δ 2 2 2 2 2 2 2 2 2 1 + y + 1 − y (t m ) ,(p a ) +(a p ) ,(p a ) +(p a ) , r b ,2r b + 1 1 1 2 1 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 r b − r b − r b , s c , 2s c + s c − s c − s c , in δ δ 1 1 2(0) 2(0) 1 1 2 1 1 2 2 2 1 1 2 1 1 2 =  ,  , Eq. (21), then δ δ δ δ (2 − 0) + (0) (2 − 0) + (0) δ δ δ δ (1 + 1) − (1 − 1) (1 + 1) − (1 − 1) IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 k+1 ⎡   ⎤ ⎛ ⎞ w w w w δ δ δ δ k+1 k+1 j k+1 j k+1 j j    (1 + 1) + (1 − 1) (1 + 1) + (1 − 1) 2 2 2 2 1+u − 1−u 1+v − 1−v ˙ ˙ ˙ ˙ ⎢ λ λ λ λ ⎥ σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ ⎢ ⎥ , , ⎜         ⎟ = ([0, 0], [1, 1]). w w w w ⎣ k+1 j k+1 j k+1 j k+1 j ⎦ ⎜ ⎟ 2 2 2 2 1+u + 1−u 1+v + 1−v ⎜ ⎟ ˙ ˙ ˙ ˙ λ λ λ λ σ( j ) σ( j ) σ( j ) σ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ ⎜ ⎟ = . ⎡   ⎤ ⎜     ⎟ δ w w k+1 k+1 j  j Thus λ = ([0, 0], [1, 1]) and δλ = ([1, 1], [0, 0]). ⎜ ⎟ 2 2 2 x 2 y ⎜ ⎟ ˙ ˙ λ λ ⎢ σ( j ) σ( j ) ⎥ ⎜ j =1 j =1 ⎟ ⎢ ⎥ 3. If λ = ([u,v], [x , y]) = ([0, 0], [0, 0]) i. e,. u = v = 0 ⎝     ,     ⎠ ⎣ w w w w ⎦ k+1 j k+1 j k+1 j k+1 j 2 2 2 2 2−x + x 2−y + y λ ˙ λ ˙ λ ˙ λ ˙ and x = y = 0, then σ( j ) σ( j ) σ( j ) σ( j ) j =1 j =1 j =1 j =1 ⎛ ⎡ ⎤ δ δ 2 2 2 u 2 v ⎝ ⎣ ⎦ λ =  ,  , δ δ δ δ 2 2 2 2 Hence, Eq. (19) holds for n = k + 1. Thus, Eq. (19) holds 2 − u + u 2 − v + v for all n. δ δ 2 2 1 + x − 1 − x δ δ 2 2 Remark 1 In the following, let us look δλ and λ some special 1 + x + 1 − x ⎤ ⎞ cases of δ and λ. δ δ 2 2 1 + y − 1 − y ⎦ ⎠ δ δ 2 2 1 + y + 1 − y 1. If λ = ([u,v], [x , y]) = ([1, 1], [0, 0]) i. e,. u = v = 1 δ δ and u = v = 1, then 2(0) 2(0) =  ,  , δ δ δ δ (2 − 0) + (0) (2 − 0) + (0) ⎛ ⎡ ⎤ δ δ 2 2 δ δ δ δ 2 u 2 v (1 + 0) − (1 − 0) (1 + 0) − (1 − 0) ⎝ ⎣ ⎦ λ = , , δ δ δ δ δ δ δ δ 2 2 2 2 (1 + 0) + (1 − 0) (1 + 0) + (1 − 0) 2 − u + u 2 − v + v = ([0, 0], [0, 0]). δ δ 2 2 1 + x − 1 − x Thus λ = ([0, 0], [0, 0]) and δλ = ([0, 0], [0, 0]). δ δ 2 2 1 + x + 1 − x 4. If δ → 0 and 0 ≤ u,v, x , y ≤ 1, then ⎤ ⎞ δ δ 2 2 1 + y − 1 − y ⎛ ⎡ ⎤ ⎦ ⎠ δ δ 2 2 2 u 2 v δ δ 2 2 1 + y + 1 − y ⎝ ⎣ ⎦ λ =  ,  , δ δ δ δ 2 2 2 2 2 − u + u 2 − v + v δ δ 2(1) 2(1) =  ,  , δ δ δ δ δ δ (2 − 1) + (1) (2 − 1) + (1) 2 2 1 + x − 1 − x δ δ  , δ δ (1 + 0) − (1 − 0) (1 + 0) − (1 − 0) δ δ 2 2 1 + x + 1 − x δ δ δ δ (1 + 0) + (1 − 0) (1 + 0) + (1 − 0) ⎤ ⎞ δ δ = ([1, 1], [0, 0]). 2 2 1 + y − 1 − y ⎦ ⎠ δ δ 2 2 1 + y + 1 − y Thus λ = ([1, 1], [0, 0]) and δλ = ([0, 0], [1, 1]). 2 0 2 0 2(u ) 2(v ) 2. If λ = ([u,v], [x , y]) = ([0, 0], [1, 1]) i. e,. u = v = 0 =  ,  , 2 0 2 0 2 0 2 0 and x = y = 1, then (2 − u ) + (u ) (2 − v ) + (v ) 123 Complex & Intelligent Systems 2 0 2 0 Lemma 1 [6] Let λ 0,w 0( j = 1, 2, 3, ..., n) and j j (1 + x ) − (1 − x ) , w = 1, then j =1 2 0 2 0 (1 + x ) + (1 − x ) n n 2 0 2 0 # & (1 + y ) − (1 − y ) (λ ) ≤ w λ , (22) j j j 2 0 2 0 (1 + y ) + (1 − y ) j =1 j =1 = ([1, 1], [0, 0]). where the equality holds if and only if λ = λ = ··· = λ . 1 2 n Thus λ = ([1, 1], [0, 0]) and δλ = ([0, 0], [1, 1]). 5. If δ →+∞ and 0 ≤ u,v, x , y ≤ 1, then Theorem 3 Let λ = ([u ,v ], [x , y ])( j = 1, 2, 3, ..., n) j j j j j ⎛ ⎡ ⎤ be a collection of IVPFVs, where the w = (w ,w ,w , ..., 1 2 3 δ δ 2 2 2 u 2 v w ) is the weighted vector of IVPFEHWA and IVPFHWA, ⎝ ⎣ ⎦ λ =  ,  , δ δ δ δ such that w ∈[0, 1] and w = 1.ω = (ω ,ω ,ω , ..., j j 1 2 2 2 2 2 2 j =1 2 − u + u 2 − v + v ω ) is the weighted vector of λ ( j = 1, 2, 3, ..., n) such ⎡ n j δ δ 2 2 1 + x − 1 − x thatω ∈[0, 1], ω = 1, then j j j =1 δ δ 2 2 1 + x + 1 − x IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 n ⎤ ⎞ δ δ 2 2 1 + y − 1 − y ≤ IVPFHWA (λ ,λ ,λ , ..., λ ). (23) ω,w 1 2 3 n ⎦ ⎠ δ δ 2 2 1 + y + 1 − y Proof Straight forward. 2 ∞ 2 ∞ 2(u ) 2(v ) =  ,  , 2 ∞ 2 ∞ 2 ∞ 2 ∞ ˙ ˙ (2 − u ) + (u ) (2 − v ) + (u ) Theorem 4 Idempotency: If λ = λ for all j ( j = 1, 2, σ( j ) 3, ..., n), where λ = ([u,v], [x , y]), then 2 ∞ 2 ∞ (1 + x ) − (1 − x ) 2 ∞ 2 ∞ (1 + x ) + (1 − x ) IVPFEHWA (λ ,λ ,λ , ..., λ ) = λ. (24) ω,w 1 2 3 n 2 ∞ 2 ∞ (1 + y ) − (1 − y ) = ([[0, 0], 1, 1]). 2 ∞ 2 ∞ (1 + y ) + (1 − y ) ˙ ˙ Proof Since λ = λ for all j, then we have σ( j ) IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 n Thus, λ = ([0, 0], [1, 1]) and δλ = ([1, 1], [0, 0]). ⎡ ⎤ ⎛ w w w w ⎞ n j n j n j n j 2 2 2 2 6. If δ = 1 and 0 ≤ u,v, x , y ≤ 1, then 1+u − 1−u 1+v − 1−v ˙ ˙ ˙ ˙ λ λ λ λ ⎢ σ ( j ) σ ( j ) σ ( j ) σ ( j ) ⎥ ⎜ j =1 j =1 j =1 j =1 ⎟ ⎢ ⎥ ⎜     ,     ,⎟ ⎣ w w w w ⎦ n j n j n j n j ⎛ ⎡ ⎤ ⎜ ⎟ 2 2 2 2 1+u + 1−u 1+v + 1−v δ δ ⎜ ⎟ λ ˙ λ ˙ λ ˙ λ ˙ 2 2 σ ( j ) σ ( j ) σ ( j ) σ ( j ) ⎜ j =1 j =1 j =1 j =1 ⎟ 2 u 2 v ⎜ ⎟ ⎡   ⎤ ⎝ ⎣ ⎦ = ⎜ ⎟ λ =  ,  , w w n j n j ⎜ ⎟ 2 2 δ δ δ δ 2 x 2 y ⎜ ⎟ λ ˙ λ ˙ 2 2 2 2 ⎢ ⎥ σ ( j ) σ ( j ) 2 − u + u 2 − v + v ⎜ j =1 j =1 ⎟ ⎢ ⎥ ⎜ , ⎟ w w w w ⎣ n j n j n j n j ⎦ ⎝ ⎠ 2 2 2 2 2−x + x 2−y + y ˙ ˙ ˙ ˙ λ λ λ λ δ δ σ ( j ) σ ( j ) σ ( j ) σ ( j ) j =1 j =1 j =1 j =1 2 2 1 + x − 1 − x ⎛ ⎞ ⎡ ⎤ δ δ ! ! n n n n 2 2 ! ! 1 + x + 1 − x w   w   w   w " j j " j j 2 j =1 2 j =1 2 j =1 2 j =1 ⎜ ⎢ ⎥ ⎟ 1+u − 1−u 1+v − 1−v ˙ ˙ ˙ ˙ ⎜ ⎢ λ λ λ λ ⎥ ⎟ ⎤ ⎞ ⎜ , ,⎟ ⎢ ⎥ δ δ ! ! n n n n ⎜     ⎟ 2 2 ⎣ ⎦ ! ! 1 + y − 1 − y w w w w ⎜ " j j " j j ⎟ 2 j =1 2 j =1 2 j =1 2 j =1 ⎜ ⎟ ⎦ ⎠ 1+u + 1−u 1+v + 1−v ˙ ˙ ˙ ˙ ⎜ λ λ λ λ ⎟ = ⎜ ⎡ ⎤ ⎟ δ δ ! ! n n 2 2 ⎜ ⎟ 1 + y + 1 − y ! ! w w ⎜ " j " j ⎟ 2 j =1 2 j =1 ⎜ ⎢ ⎥ ⎟ 2 x 2 y ˙ ˙ ⎛ ⎡ ⎤ ⎜ ⎢ λ λ ⎥ ⎟ ⎢ , ⎥ 1 1 ⎜ ⎟ ! ! n n n n 2 2 ⎝ ⎣ ! ! ⎦ ⎠ 2 u 2 v w   w   w   w " j j " j j 2 j =1 2 j =1 2 j =1 2 j =1 ⎝ ⎣ ⎦ 2−x + x 2−y + y = , , ˙ ˙ ˙ ˙ λ λ λ λ 1 1 1 1 2 2 2 2 2 − u + u 2 − v + u ⎛ ⎡ ⎤ ⎞ 2 2 2 2 1+u − 1−u 1+v − 1−v ˙ ˙ ˙ ˙ λ λ λ λ 1 1 ⎜ ⎣   ⎦ ⎟ , , 2 2 ⎜ ⎟ 1 + x − 1 − x 2 2 2 2 ⎜ 1+u + 1−u 1+v + 1−v ⎟ ˙ ˙ ˙ ˙ λ λ λ λ ⎜ ⎟ ⎜ ⎟ ˙ = = λ. 1 1 ⎜   ⎟ 2 2 ⎡ ⎤ 1 + x + 1 − x ⎜ ⎟ 2 2 2 x 2 y ⎜ ⎟ ˙ ˙ λ λ ⎝ ⎠ ⎣   ⎦ ⎤ ⎞ , 1 1 2 2 2 2 2 2 2−x + x 2−y + y ˙ ˙ ˙ ˙ 1 + y − 1 − y λ λ λ λ ⎦ ⎠ = λ. 1 1 2 2 1 + y + 1 − y The proof is completed. Thus, λ = λ and δλ = λ. 123 Complex & Intelligent Systems 1 1 1 1 Theorem 5 Boundedness: Let λ = ([u ,v ], [x , y ]) j λ λ λ λ Proof Let w = , , , ..., , , and λ = λ , then j j j j σ( j ) σ( j ) n n n n ( j = 1, 2, 3, ..., n) be a collection of IVPFNs, then we have ˙ ˙ λ ≤ IVPFEHWA (λ ,λ ,λ , ..., λ ) ≤ λ , (25) min ω,w 1 2 3 n max IVPFEHWA (λ ,λ ,λ ,...,λ ) ω,w 1 2 3 n ˙ ˙ λ = max(λ ), (26) max σ( j ) ˙ ˙ ˙ = w λ ⊕ w λ ⊕ ··· ⊕ w λ 1 σ(1) ε 2 σ(2) ε ε n σ(n) ˙ ˙ λ = min(λ ). (27) min σ( j ) = w λ ⊕ w λ ⊕ ··· ⊕ w λ 1 σ(1) ε 2 σ(2) ε ε n σ(n) = IVPFEOWA (λ ,λ ,λ ,...,λ ). w 1 2 3 n Proof Proof is easy so it is omitted here. Theorem 6 Monotonicity: If λ ≤ λ for all j ( j = The proof completed. 1, 2, 3, ..., n), then IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 n An approach to multiple attribute group ∗ ∗ ∗ ∗ ≤ IVPFEHWA (λ ,λ ,λ , ..., λ ). (28) decision-making problems based on ω,w 1 2 3 n interval-valued Pythagorean fuzzy Proof As we know that. information IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 n Algorithm Let X ={X , X , X , ..., X } be a finite set of m 1 2 3 m ˙ ˙ ˙ ˙ alternatives and C ={C , C , C , ..., C } be a finite set of = w λ ⊕ w λ ⊕ w λ ⊕ ··· ⊕ w λ , 1 2 3 n 1 σ(1) ε 2 σ(2) ε 3 σ(3) ε ε n σ(n) n attributes. Suppose the grade of the alternativesX (i = (29) 1, 2, 3, ..., m)on attributeC ( j = 1, 2, 3, ..., n) given by decision makers is interval-valued Pythagorean fuzzy num- and bers. Let D ={D , D , D , ..., D } be the set of k 1 2 3 k ∗ ∗ ∗ ∗ decision makers, and let w = (w ,w ,w , ..., w ) be 1 2 3 n IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 n the weighted vector of the attributes C ( j = 1, 2, 3, ..., n), ∗ ∗ ∗ ∗ j ˙ ˙ ˙ ˙ = w λ ⊕ w λ ⊕ w λ ⊕ ··· ⊕ w λ . 1 ε 2 ε 3 ε ε n σ(1) σ(2) σ(3) σ(n) such that w ∈[0, 1], w = 1, and let ω = j j j =1 (30) (ω ,ω ,ω , ..., ω ) be the weighted vector of the deci- 1 2 3 k sion makers D (s = 1, 2, 3, ..., k), such that ω ∈[0, 1] and Since λ ≤ λ for all j,thus Eq.(28) always holds. j k ω = 1. Let D = (a ) =[u ,v ], [x , y ](i = s ji ji ji ji ji s=1 1, 2, 3, ..., m, j = 1, 2, 3, ..., n) where [u ,v ] indicates Theorem 7 Interval-valued Pythagorean fuzzy Einstein ji ji the interval degree that the alternative X (i = 1, 2, 3, ..., m) weighted averaging operator is a special case of the interval- i satisfies the attribute C ( j = 1, 2, 3, ..., n) and [x , y ] valued Pythagorean fuzzy Einstein hybrid weighted averag- j ji ji indicates the interval degree that the alternative X (i = ing operator. i 1, 2, 3, ..., m) does not satisfy the attribute C ( j = 1, 2, 3, 1 1 1 1 Proof Let ω = , , , ..., , , then we have ..., n), And also [u ,v ]∈[0, 1], [x , y ]∈[0, 1] with n n n n ji ji ji ji 2 2 condition 0 ≤ (v ) + (y ) ≤ 1,(i = 1, 2, 3, ..., m, j = ji ji IVPFEHWA (λ ,λ ,λ , ..., λ ) ω,w 1 2 3 n 1, 2, 3, ..., n). This method has the following steps. ˙ ˙ ˙ = w λ ⊕ w λ ⊕ ··· ⊕ w λ 1 σ(1) ε 2 σ(2) ε ε n σ(n) ˙ ˙ ˙ = (λ ⊕ λ ⊕ ··· ⊕ λ ) σ(1) ε σ(2) ε ε σ(n) Step 1 Utilize the given information in the form of matrices, (s) D = a (s = 1, 2, 3, ..., k). ji n×m = (nω λ ⊕ nω λ ⊕ ··· ⊕ nω λ ) 1 1 ε 2 2 ε ε n n Step 2 If the criteria have two types, such as benefit = ω λ ⊕ ω λ ⊕ ··· ⊕ ω λ 1 1 ε 2 2 ε ε n n criteria and cost criteria, then the interval-valued (s) = IVPFEWA (λ ,λ ,λ , ..., λ ). Pythagorean fuzzy decision matrices, D = a w 1 2 3 n ji n×m (s = 1, 2, 3, ..., k) can be converted into the nor- The proof is completed. malized interval-valued Pythagorean fuzzy decision (s) matrices, R = r (s = 1, 2, 3, ..., n), where Theorem 8 Interval-valued Pythagorean fuzzy Einstein ji n×m ordered weighted averaging operator is a special case of the (s) interval-valued Pythagorean fuzzy Einstein hybrid weighted a , for benefit criteria C j j = 1, 2, 3, ..., n (s) ji r = , (s) ji averaging operator. i = 1, 2, 3, ..., m a ¯ , for cost criteria C , ji 123 Complex & Intelligent Systems (s) s T and a ¯ is the complement of α . If all the criteria makers, whose weight vector is ω = (0.2, 0.3, 0.5) . There ji ji have the same type, then there is no need of normal- are many factors that must be considered while selecting ization. the most suitable system, but here, we have consider only Step 3 Utilize the IVPFEWA operator to aggregate all the the following four criteria, whose weighted vector is w = individual normalized interval-valued Pythagorean (0.1, 0.2, 0.3, 0.4) (s) fuzzy decision matrices, R = r (s = 1, 2, ji n×m 1. C : Costs of hardware. 3, ..., k) into a single interval-valued Pythagorean 1 2. C : Support of the organization. fuzzy decision-matrix, R =[r ] , where r = 2 ji n×m ji 3. C : Effort to transform from current systems. [u ,v ], [x , y ]. 3 ji ji ji ji 4. C : Outsourcing software developer reliability, Step 4 In this step, we calculate r˙ = nw r . 4 ji j ji Step 5 Calculate the scores function of r˙ (i = 1, 2, 3, ..., ji m, j = 1, 2, 3, ..., n). If there is no difference where C , C , are cost type criteria and C , C are benefit 1 3 2 4 between two or more than two scores, then we must type criteria, i.e., the attributes have two types of criteria; find out the accuracy degrees of the collective overall thus, we must change the cost type criteria into benefit type preference values. criteria. Step 6 Utilize the IVPFEHWA operator to aggregate all preference values. Step 1 Construct the decision-making matrices (Tables 1, 2 Step 7 Arrange the scores of the all alternatives in the form and 3). of descending order and select that alternative which has the highest score function. Step 2 Construct the normalized decision making matrices (Tables 4, 5 and 6). Illustrative example Step 3 Utilize the IVPFEWA operator to aggregate all the Suppose in Hazara University, the IT department wants to individual normalized interval-valued Pythagorean select a new information system for the purpose of the best (s) fuzzy decision matrices, R = r into a ji productivity. After the first selection, there are only three n×m single interval-valued Pythagorean fuzzy decision X (i = 1, 2, 3) alternatives have been short listed. There are s matrix, R =[r ] (Table 7). ji n×m three experts D (s = 1, 2, 3) from a group to act as decision Table 1 Interval-valued X X X 1 2 3 Pythagorean fuzzy decision matrix of D C ([0.5, 0.8], [0.3, 0.4])([0.6, 0.7], [0.3, 0.6])([0.3, 0.7], [0.3, 0.5]) C ([0.3, 0.5], [0.6, 0.7])([0.3, 0.7], [0.2, 0.6])([0.3, 0.6], [0.4, 0.7]) C ([0.5, 0.7], [0.3, 0.7])([0.5, 0.6], [0.3, 0.7])([0.2, 0.6], [0.3, 0.7]) C ([0.3, 0.6], [0.6, 0.7])([0.6, 0.5], [0.2, 0.7])([0.3, 0.4], [0.5, 0.6]) Table 2 Interval-valued X X X 1 2 3 Pythagorean fuzzy decision matrix of D C ([0.5, 0.6], [0.3, 0.5])([0.5, 0.7], [0.3, 0.6])([0.2, 0.8], [0.3, 0.4]) C ([0.3, 0.4], [0.6, 0.8])([0.3, 0.8], [0.2, 0.6])([0.3, 0.6], [0.3, 0.7]) C ([0.4, 0.5], [0.3, 0.8])([0.5, 0.7], [0.3, 0.6])([0.2, 0.6], [0.3, 0.8]) C ([0.3, 0.6], [0.5, 0.7])([0.3, 0.4], [0.2, 0.8])([0.3, 0.5], [0.5, 0.7]) Table 3 Interval-valued X X X 1 2 3 Pythagorean fuzzy decision matrix of D C ([0.3, 0.8], [0.5, 0.6])([0.3, 0.5], [0.5, 0.7])([0.2, 0.4], [0.5, 0.7]) C ([0.5, 0.7], [0.3, 0.4])([0.4, 0.6], [0.5, 0.8])([0.5, 0.7], [0.2, 0.5]) C ([0.3, 0.6], [0.4, 0.6])([0.3, 0.5], [0.5, 0.6])([0.2, 0.8], [0.4, 0.6]) C ([0.5, 0.7], [0.3, 0.4])([0.5, 0.7], [0.2, 0.4])([0.5, 0.6], [0.3, 0.5]) 123 Complex & Intelligent Systems Table 4 Normalized X X X 1 2 3 Pythagorean fuzzy decision matrix R C ([0.3, 0.4], [0.5, 0.8])([0.3, 0.6], [0.6, 0.7])([0.3, 0.5], [0.3, 0.7]) C ([0.3, 0.5], [0.6, 0.7])([0.3, 0.7], [0.2, 0.6])([0.3, 0.6], [0.4, 0.7]) C ([0.3, 0.7], [0.5, 0.7])([0.3, 0.7], [0.5, 0.6])([0.3, 0.7], [0.2, 0.6]) C ([0.3, 0.6], [0.6, 0.7])([0.6, 0.5], [0.2, 0.7])([0.3, 0.4], [0.5, 0.6]) Table 5 Normalized X X X 1 2 3 Pythagorean fuzzy decision matrix R C ([0.3, 0.5], [0.5, 0.6])([0.3, 0.6], [0.5, 0.7])([0.3, 0.4], [0.2, 0.8]) C ([0.3, 0.4], [0.6, 0.8])([0.3, 0.8], [0.2, 0.6])([0.3, 0.6], [0.3, 0.7]) C ([0.3, 0.8], [0.4, 0.5])([0.3, 0.6], [0.5, 0.7])([0.3, 0.8], [0.2, 0.6]) C ([0.3, 0.6], [0.5, 0.7])([0.3, 0.4], [0.2, 0.8])([0.3, 0.5], [0.5, 0.7]) Table 6 Normalized X X X 1 2 3 Pythagorean fuzzy decision matrix R C ([0.5, 0.6], [0.3, 0.8])([0.5, 0.7], [0.3, 0.5])([0.5, 0.7], [0.2, 0.4]) C ([0.5, 0.7], [0.3, 0.4])([0.4, 0.6], [0.5, 0.8])([0.5, 0.7], [0.2, 0.5]) C ([0.4, 0.6], [0.3, 0.6])([0.5, 0.6], [0.3, 0.5])([0.4, 0.6], [0.2, 0.8]) C ([0.5, 0.7], [0.3, 0.4])([0.5, 0.7], [0.2, 0.4])([0.5, 0.6], [0.3, 0.5]) Table 7 Collective interval-valued Pythagorean fuzzy decision matrix R X X X 1 2 3 C ([0.413, 0.537], [0.389, 0.738])([0.413, 0.653], [0.405, 0.595])([0.413, 0.593], [0.216, 0.562]) C ([0.413, 0.593], [0.429, 0.563])([0.352, 0.692], [0.320, 0.697])([0.413, 0.653], [0.260, 0.595]) C ([0.352, 0.692], [0.363, 0.587])([0.413, 0.622], [0.389, 0.576])([0.352, 0.692], [0.200, 0.697]) C ([0.413, 0.653], [0.405, 0.536])([0.475, 0.593], [0.200, 0.563])([0.413, 0.537], [0.389, 0.576]) Step 4 Calculate λ = nwλ . Step 5 Calculate the score functions (Table 8). ji ji λ = ([0.262, 0.343], [0.733, 0.897]), ˙ ˙ ˙ s(λ ) =−0.57, s(λ ) =−0.13, s(λ ) 11 21 31 λ = ([0.370, 0.534], [0.523, 0.645]) = 0.18, s(λ ) = 0.37 λ = ([0.385, 0.745], [0.281, 0.513]), ˙ ˙ ˙ s(λ ) =−0.50, s(λ ) =−0.12, s(λ ) 12 22 32 λ = ([0.518, 0.788], [0.201, 0.329]) = 0.15, s(λ ) = 0.37 λ = ([0.262, 0.424], [0.742, 0.837]), ˙ ˙ ˙ s(λ ) =−0.41, s(λ ) =−0.04, s(λ ) 13 23 33 λ = ([0.315, 0.628], [0.420, 0.757]) = 0.13, s(λ ) = 0.26. λ = ([0.452, 0.665], [0.307, 0.501]), λ = ([0.593, 0.726], [0.061, 0.359]) λ = ([0.262, 0.382], [0.605, 0.823]), Step 6 Utilize the IVPFEHWA aggregation operator to λ = ([0.370, 0.590], [0.357, 0.672]) aggregate all preference values. λ = ([0.385, 0.745], [0.136, 0.638]), λ = ([0.518, 0.664], [0.188, 0.374]). r = ([0.354, 0.567], [0.550, 0.674]) 123 Complex & Intelligent Systems Table 8 Pythagorean fuzzy hybrid decision matrix R X X X 1 2 3 C ([0.518, 0.788], [0.201, 0.329])([0.593, 0.726], [0.061, 0.359])([0.518, 0.664], [0.188, 0.374]), C ([0.385, 0.745], [0.281, 0.513])([0.525, 0.665], [0.307, 0.501])([0.585, 0.745], [0.136, 0.638]), C ([0.370, 0.534], [0.523, 0.645])([0.315, 0.628], [0.420, 0.757])([0.370, 0.590], [0.357, 0.672]), C ([0.262, 0.343], [0.733, 0.897])([0.262, 0.424], [0.742, 0.837])([0.262, 0.382], [0.605, 0.823]). r = ([0.367, 0.581], [0.422, 0.686]) References r = ([0.354, 0.571], [0.347, 0.695]). 1. 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Yager RR (2014) Pythagorean membership grades in multi-criteria decision making. IEEE Trans Fuzzy Syst 22(4):958–965 19. Yager RR, Abbasov AM (2013) Pythagorean membership grades, Open Access This article is distributed under the terms of the Creative complex numbers and decision making. Int J Intell Syst 28(5):436– Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, 20. Peng X, Yang Y (2015) Some results for Pythagorean fuzzy sets. and reproduction in any medium, provided you give appropriate credit Int J Intell Syst 30(11):1133–1160 to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 123 Complex & Intelligent Systems 21. Garg H (2016) A new generalized Pythagorean fuzzy information 29. Peng X, Yang Y (2016) Fundamental properties of interval- aggregation using Einstein operations and its application to deci- valued Pythagorean fuzzy aggregation operators. Int J Intell Syst sion making. Int J Intell Syst 31(9):886–920 31(5):444–487 22. Garg H (2017) Generalized Pythagorean fuzzy geometric aggreg- 30. Rahman K, Ali Asad, Khan MSA (2018) Some interval-valued tion operators using Einstein t-Norm and t-Conorm for multicrite- Pythagorean fuzzy weighted averaging aggregation operators and ria decision-making process. Int J Intell Syst 32:597–630. https:// their application to multiple attribute decision making, Punjab Uni- doi.org/10.1002/int.21860 versity. J Math 50(2):113–129 23. Rahman K, Abdullah S, Husain F, Ali Khan MS (2016) Approaches 31. Rahman K, Abdullah S, Shakeel M, Khan MSA, Ullah Murad to Pythagorean fuzzy geometric aggregation operators. Int J Com- (2017) Interval-valued Pythagorean fuzzy geometric aggregation put Sci Inf Secur IJCSIS 4(9):174–200 operators and their application to decision making. Cogent Math 24. Rahman K, Khan MSA, Ullah M, Fahmi A (2017) Multiple 4(1):1338638. https://doi.org/10.1515/jisys-2017-0212 attribute group decision making for plant location selection with 32. Rahman K, Abdullah S, Khan MSA (2018) Some interval-valued Pythagorean fuzzy weighted geometric aggregation. Operator Pythagorean fuzzy Einstein weighted averaging aggregation oper- Nucleus 54(1):66–74 ator and their application to group decision making. J Intell Syst. 25. Rahman K, Abdullah S, Husain F, Ali Khan MS, Shakeel M https://doi.org/10.1515/jisys-2017-0212 (2017) Pythagorean fuzzy ordered weighted geometric aggregation 33. Rahman K, Abdullah S (2018) Generalized interval-valued operator and their application to multiple attribute group decision Pythagorean fuzzy aggregation operators and their application to making. 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Journal

Complex & Intelligent SystemsSpringer Journals

Published: Jun 5, 2018

References

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