Pythagorean cubic fuzzy aggregation operators and their application to multi-criteria decision making problems

1 Introduction

Multiple-criteria decision making (MCDM) means that the best alternative is selected from the limited alternatives set according to the multiple criteria, which can be regarded as cognitive processing. Multi criteria decision-making (MCDM) is an important branch of the decision-making theory which has been widely used in human activities [9]. Because the real decision making problems were frequently produced from the complicated environment, the evaluation information is usually fuzzy. In general, the fuzzy information takes two forms: one quantitative and one qualitative. The quantitative fuzzy information can be expressed by fuzzy set (FS) [39], intuitionistic fuzzy set (IFS) [1], Pythagorean fuzzy set (PFS) [32] and so on. FS theory proposed by Zadeh [39] has been used to describe fuzzy quantitative information which contains only a membership degree. Based on this, Atanassov [1] presented IFS, which consists of a member-ship and a non-membership degrees that fulfills the restriction form that the sum of two degrees is ⩽1. However, sometimes the two degrees don’t meet the restriction, but the sum of squares of the two degrees is ⩽1. Yager [32] introduced the PFS in which the square sum of membership and non-membership degrees is equal to or less than 1. In some situation, the PFS has the greater ability to express the fuzzy information than the IFS. For example, if an expert gives the membership about the support to a matter is 0.8 and the non-membership for against is 0.6. Obviously, IFS is unable to describe this decision information, but it can be effectively described by PFS. Peng et al. [20], presented some new properties in the Pythagorean fuzzy set which are division, subtraction and other important properties. Writers deal with the supremacy and dependency ranking methods to explain the multi-criteria decision-making problems in the Pythagorean fuzzy environment. In [11], khan et al. Developed prioritized aggregation operators for multi criteria decision-making established on Pythagorean fuzzy sets. Peng et al. [21] proposed Pythagorean fuzzy Linguistic sets (PFLSs) and there operating laws and score function of Pythagorean fuzzy linguistic numbers. Wei et al. [14] introduced a maximizing deviation method to explain decision-making problems based on interval-valued Pythagorean fuzzy environments. The subtraction and division concept of Pythagorean fuzzy numbers (PFNS) were proposed by Gou et al. [8]. Pend et al. [22], presented the notion of the obvious definition of Pythagorean fuzzy distance degree, which is communicated by a Pythagorean fuzzy number that will decrease the data loss and continue further creative evidence. Also the well-defined notion of novel score function. Liang et al. [15], presented the notation of Pythagorean fuzzy geometric Bonferroni mean and weighted Pythagorean fuzzy geometric Bonferroni (WPFGBM) operator. In [3], Garg proposed interval-valued Pythagorean fuzzy geometric (IVPFWA) operator, interval-valued Pythagorean fuzzy geometric (IVPFWG) operator and presented the notation of new accuracy function based on interval-valued Pythagorean fuzzy environment. Khan et al extended to concept of TOPSIS method for multi criteria decision-making and developed Choquet integral TOPSIS method based on IVPFNs [12]. Khan and Abdullah in [13], proposed gray relational analysis (GRA) scheme for multi criteria decision-making under interval valued Pythagorean fuzzy situation. The authors first developed interval valued Pythagorean fuzzy Choquet integral average (IVPFCI) operator, then advanced a method to multi criteria decision-making built on GRA method. Peng et al. [23], propose two algorithms to solve stochastic multi criteria decision-making problem under Interval-valued fuzzy soft set approaches. Peng and Liu in [24], presents three novel single-valued neutrosophic soft set methods and propose three algorithms to solve single-valued neutrosophic soft decision making problem. Peng and Selvachandran in [25], present an overview on Pythagorean fuzzy set with aim of offering a clear perspective on the different concepts, tools and trends related to their extension and algorithms in decision making. Peng [26], initiate some new interval-valued Pythagorean fuzzy operators and discuss their properties in relation with some existing operators in detail. It will promote the development of interval-valued Pythagorean fuzzy operators and propose an algorithm to deal with multi-criteria decision making (MCDM) problem based on proposed operator. Wei et al. [28], presents the generalized Heronian mean operator and geometric Heronian mean operator under the q-rung orthopair fuzzy sets is studied. Wei et al. [29], proposed Pythagorean hesitant fuzzy aggregation operators Pythagorean hesitant fuzzy Hamacher weighted average (PHFHWA) operator, Pythagorean hesitant fuzzy Hamacher weighted geometric (PHFHWG) operator, Pythagorean hesitant fuzzy Hamacher ordered weighted average (PHFHOWA) operator, Pythagorean hesitant fuzzy Hamacher ordered weighted geometric (PHFHOWG) operator, Pythagorean hesitant fuzzy Hamacher induced ordered weighted average (PHFHIOWA) operator, Pythagorean hesitant fuzzy Hamacher induced ordered weighted geometric (PHFHIOWG) operator, Pythagorean hesitant fuzzy Hamacher induced correlated aggregation operators, Pythagorean hesitant fuzzy Hamacher prioritized aggregation operators, and Pythagorean hesitant fuzzy Hamacher power aggregation operators based on Hamacher t-norm and t-conorm. Wu et al. in [30], developed 2-tuple linguistic neutrosophic Hamy mean (2TLNHM) operator, 2-tuple linguistic neutrosophic weighted Hamy mean (2TLNWHM) operator, 2-tuple linguistic neutrosophic dual Hamy mean (2TLNDHM) operator, and 2-tuple linguistic neutrosophic weighted dual Hamy mean (2TLNWDHM) operator under 2-tuple linguistic neutrosophic environment.

Wang et al. in [31], presented 2-tuple linguistic neutrosophic numbers weighted Bonferroni mean (2TLNNWBM) operator, 2-tuple linguistic neutrosophic numbers weighted geometric Bonferroni mean (2TLNNWGBM) operator, generalized 2-tuple linguistic neutrosophic numbers weighted Bonferroni mean (G2TLNNWBM) operator, generalized 2-tuple linguistic neutrosophic numbers weighted geometric Bonferroni mean (G2TLNNWGBM) operator, dual generalized 2-tuple linguistic neutrosophic numbers weighted Bonferroni mean (DG2TLNNWBM) operator, and dual generalized 2-tuple linguistic neutrosophic numbers weighted geometric Bonferroni mean (DG2TLNNWGBM) operator to deal with MCDM problems. Gao et al. in [7], investigated the dual hesitant bipolar fuzzy MCDM problems in which there exists a prioritization relationship over attributes and proposed dual hesitant bipolar fuzzy Hamacher prioritized geometric (DHBFHPG) operator, dual hesitant bipolar fuzzy Hamacher prioritized weighted average (DHBFHPWA) operator, dual hesiant bipolar fuzzy Hamacher prioritized weighted geometric (DHBFHPWG) operator.

In this article, we present the notion of the Pythagorean cubic fuzzy set (PCFS) which is the simplification of interval-valued Pythagorean fuzzy set based on the constraint that the square sum of supremum to its membership degrees is ⩽1. Here, we present the notion of the Pythagorean cubic fuzzy set (PCFS) which is the generality of the notion of interval valued Pythagorean fuzzy set. We study some properties of PCFS. We define score and deviation degree of the Pythagorean cubic fuzzy numbers (PCFNs) for the comparison of Pythagorean cubic fuzzy numbers. We also define distance measure between Pythagorean cubic fuzzy numbers. Pythagorean cubic fuzzy set hold the condition that the sum of square of supremum to its membership’s degrees is ⩽ 1. Based on this information, aggregation operators, namely, Pythagorean cubic fuzzy weighted averaging (PCFWA), Pythagorean cubic fuzzy weighted geometric (PCFWG), Pythagorean cubic fuzzy ordered weighted averaging (PCFOWA) and Pythagorean cubic fuzzy ordered weighted geometric (PCFOWG). Furthermore, these operators are then used for decision-making problems in which specialists offer their preferences in the Pythagorean cubic fuzzy information to show the practicality and effectiveness of the new approach.

2 Preliminaries

In this section, we introduce some elementary definition and their important properties.

Definition 1. [21] Let X be a universe of discourse, then a FS set F is an object having the following formulation: F = { 〈 x , μ F ˜ ( x ) 〉 | x ∈ X } ,(1)

where μ F ˜ is a function from X to [0, 1], and μ F ˜ ( x ) is called the membership degree of x in X.

Definition 2. [1] Let X be a universe of discourse, then a IFS set I is an object having the following form: I = { 〈 x , μ I ˜ ( x ) , ν I ˜ ( x ) 〉 | x ∈ X } ,(2)

where μ I ˜ ( x ) and ν I ˜ ( x ) is a function from X : → [0, 1], also hold 0 ⩽ μ I ˜ ( x ) ⩽ 1 , 0 ⩽ ν I ˜ ( x ) ⩽ 1 for all x in X, and denote the membership and nonmembership degree of element x in X to set I.

Definition 3. [42] Let U be a universal set. Then a cubic set can be defined as: C = { 〈 x , μ ¯ c ( x ) , ν c ( x ) 〉 | x ∈ X } .

Where μ ¯ c is an interval-valued fuzzy set in U and ν _c is a fuzzy set in U.

Definition 4. [37] Let X be a universe of discourse, then a PFS set P is an object having the following form: P = { 〈 x , μ P ˜ ( x ) , ν P ˜ ( x ) 〉 | x ∈ X } ,(3)

where μ P ˜ ( x ) and ν P ˜ ( x ) is a function from X : → [0, 1], such that 0 ⩽ μ P ˜ ( x ) ⩽ 1 , 0 ⩽ ν P ˜ ( x ) ⩽ 1 , and 0 ⩽ μ P ˜ 2 ( x ) ⩽ 1 , 0 ⩽ ν P ˜ 2 ( x ) ⩽ 1 for all x in X, and denote the membership and nonmembership degree of element x in X to set P. Let π P ˜ ( x ) = 1 - μ P ˜ 2 ( x ) - ν P ˜ 2 ( x ) , then it is called to be the Pythagorean fuzzy index of x in X to set P, denote the indeterminacy degree of x to P. Also 0 ⩽ π P ˜ ( x ) ⩽ 1 for every x ∈ X. We denote the Pythagorean fuzzy number (PFN) by P = 〈Λ _P , Γ _P 〉. To compare two Pythagorean fuzzy numbers, the authors presented the notion of score function, accuracy degree for them.

Definition 5. [37] Let P = 〈μ _P , λ _P 〉 P₁ = 〈μ _{P 1} , λ _{P 1} 〉, P₂ = 〈μ _{P 2} , λ _{P 2} 〉 be three (PFNs), then we have

P 1 ⊕ P 2 = 〈 μ P 1 2 + μ P 2 2 - μ P 1 2 μ P 2 2 , ν P 1 ν P 2 〉 ;

Definition 6. [38] Let X be a universe of discourse, then a IVPFS set R can be defined as: A = { 〈 x , a A ( x ) , b A ( x ) 〉 | x ∈ X } ,(4)

where a R ˜ ( x ) = [ a R ˜ L ( x ) , a R ˜ U ( x ) ] ⊂ [ 0 , 1 ] , b A ( x ) = [ b A L ( x ) , b A U ( x ) ] ⊂ [ 0 , 1 ] are the intervals, and a A L ( x ) = inf a A ( x ) and a A U ( x ) = sup a A ( x ) , similarly b A L ( x ) = inf b A ( x ) and b A U ( x ) = sup b A ( x ) for all x in X. And also 0 ⩽ ( a A U ( x ) ) 2 + ( b A U ( x ) ) 2 ⩽ 1 .

Let π A ( x ) = [ π A L ( x ) , π A U ( x ) ] , for all x ∈ X, then it is called the interval-valued Pythagorean fuzzy index of x to A, where π A L ( x ) = 1 - ( μ A U ( x ) ) 2 - ( ν A U ( x ) ) 2 , and π A U ( x ) = 1 - ( μ A L ( x ) ) 2 - ( ν A L ( x ) ) 2 .

Which satisfy the following relation.

If a A L ( x ) = a A U ( x ) and b A L ( x ) = b A U ( x ) , then an IVPFS set reduces to PFS.

Definition 7. [19] Let A = ( [ a A L , a A U ] , [ b A L , b A U ] ) are the IVPFN, we can defined the score functionof A in the following way: S ( A ) = 1 2 [ ( a A L ) 2 + ( a A U ) 2 - ( b A L ) 2 - ( b A U ) 2 ]

where S(A) ∈ [0,1].

Definition 8. [19] Let A = 〈 [ a A L , a A U ] , [ b A L , b A U ] 〉 are the IVPFN, we can defined the accurracy function of A in the following way: H ( A ) = 1 2 [ ( a A L ) 2 + ( a A U ) 2 + ( b A L ) 2 + ( b A U ) 2 ]

where H(A) ∈ [0,1].

Definition 9. [19] Let A = 〈 [ a A L , a A U ] , [ b A L , b A U ] 〉 and A 1 = 〈 [ a A 1 L , a A 1 U ] , [ b A 1 L , b A 1 U ] 〉 be two IVPFNs, then S ( A ) = 1 2 [ ( a A L ) 2 + ( a A U ) 2 - ( b A L ) 2 - ( b A U ) 2 ] , S ( A 1 ) = 1 2 [ ( a A 1 L ) 2 + ( a A 1 U ) 2 - ( b A 1 L ) 2 - ( b A 1 U ) 2 ]

Are the score of A and A₁, separately, while H ( A ) = 1 2 [ ( a A L ) 2 + ( a A U ) 2 + ( b A L ) 2 + ( b A U ) 2 ] , H ( A 1 ) = 1 2 [ ( a A 1 L ) 2 + ( a A 1 U ) 2 + ( b A 1 L ) 2 + ( b A 1 U ) 2 ]

Are the accuracy of A and A₁, separately, which satisfy the following properties:

If S (A) < S (A₁) then A < A₁

Definition 10. [4, 5] Let A _i  = 〈 [a _i , b _i ] , [c _i , d _i ] 〉 (i = 1, 2) be the collection of IVPFNs, and δ > 0, then the following oprational las are satisfied: δ A 1 = 〈 [ 1 - ( 1 - ( a 1 ) 2 ) δ , 1 - ( 1 - ( b 1 ) 2 ) δ ] , [ ( c 1 ) δ , ( d 1 ) δ ] 〉 , ( A 1 ) δ = 〈 [ ( a 1 ) δ , ( b 1 ) δ ] , [ 1 - ( 1 - ( c 1 ) 2 ) δ , 1 - ( 1 - ( d 1 ) 2 ) δ ] 〉 A 1 ⊗ A 2 = 〈 [ a 1 a 2 , b 1 b 2 ] , [ ( c 1 ) 2 + ( c 2 ) 2 - ( c 1 ) 2 ( c 2 ) 2 , ( d 1 ) 2 + ( d 2 ) 2 - ( d 1 ) 2 ( d 2 ) 2 ] 〉 , A 1 ⊕ A 2 = 〈 [ ( a 1 ) 2 + ( a 2 ) 2 - ( a 1 ) 2 ( a 2 ) 2 , ( b 1 ) 2 + ( b 2 ) 2 - ( b 1 ) 2 ( b 2 ) 2 ] , [ c 1 c 2 , d 1 d 2 ] 〉 .

Definition 11. [27] Let A _i  = 〈 [a _i , b _i ] , [c _i , d _i ] 〉 (i = 1, 2, …, n) be the collection of IVPFNs, then IVPFWA oprator is defined as: IVPFWA w ( A 1 , A 2 , … , A n ) = 〈 [ 1 - ∏ i = 1 n ( 1 - ( a i ) 2 ) w i , 1 - ∏ i = 1 n ( 1 - ( b i ) 2 ) w i ] , [ ∏ i = 1 n ( c i ) w i , ∏ i = 1 n ( d i ) w i ] 〉

where w = (w₁, w₂, …, w _n )  ^T be the weight vector of p _i (i = 1, 2, 3 … , n) and w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 .

Definition 12. [27] Let A _i  = 〈 [a _i , b _i ] , [c _i , d _i ] 〉 (i = 1, 2, …, n) be the collection of IVPFNs, then IVPFWG oprator is defined as: IVPFWG w ( A 1 , A 2 , … , A n ) = 〈 [ ∏ i = 1 n ( a i ) w i , ∏ i = 1 n ( b i ) w i ] , [ 1 - ∏ i = 1 n ( 1 - ( c i ) 2 ) w i , 1 - ∏ i = 1 n ( 1 - ( d i ) 2 ) w i ] 〉

where w = (w₁, w₂, …, w _n )  ^T be the weight vector of A _i (i = 1, 2, 3 … , n) and w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 .

In this section, we present the idea of some new concepts such as, Pythagorean cubic fuzzy set and discuss some properties with the help of an example for Pythagorean cubic fuzzy set which is not an intuitionistic cubic fuzzy set. Throughout in this paper a Pythagorean cubic fuzzy set will be denoted by P _C .

Definition 13. Let X be a fixed set, then a Pythagorean cubic fuzzy set can be defined as: P C = { 〈 x , ( Λ P C ( x ) , Γ P C ( x ) ) 〉 | x ∈ X } ,(5)

Where Λ _{P C}  (x) =〈A _{P C}  (x) ; λ _{P C}  (x) 〉 = 〈 [a _{P C}  (x) , |b _{P C}  (x)] ; λ _{P C}  (x) 〉 Γ P C ( x ) = 〈 A ˜ P C ( x ) ; μ P C ( x ) 〉 = 〈 [ a ˜ P C , b ˜ P C ] ; μ P C ( x ) 〉 and 0 ⩽ ( Λ P C ( x ) ) 2 + ( Γ P C ( x ) ) 2 ⩽ [ 1 ˜ , 1 ] .

The above condition can be also written in the following way: 0 ⩽ ( sup ( A P C ( x ) ) ) 2 + ( sup ( A ˜ P C ( x ) ) ) 2 ⩽ 1 , 0 ⩽ λ P C 2 ( x ) + μ P C 2 ( x ) ⩽ 1 .

The degree of indeterminacy for Pythagorean cubic set can be defined as: π C ˆ = 〈 1 - ( sup ( A P C ( x ) ) ) 2 - ( sup ( A ˜ P C ( x ) ) ) 2 ; 1 - λ P C 2 ( x ) - μ P C 2 ( x ) 〉 .

For simplicity we call ( 〈 A ; λ 〉 , 〈 A ˜ ; μ 〉 ) a Pythagorean cubic fuzzy number (PCFN) denoted by p c = ( 〈 A ; λ 〉 , 〈 A ˜ ; μ 〉 ) .

Example 1. Let X = {x₁, x₂, x₃} be a fixed set and consider a set in X by P C = { ( x 1 , 〈 [ 0.5 , 0.6 ] ; 0.7 ) , ( [ 0.6 , 0.7 ] ; 0.5 〉 ) , ( x 2 , 〈 [ 0.4 , 0.7 ] ; 0.6 〉 , 〈 [ 0.5 , 0.8 ] ; 0.6 〉 ) , ( x 3 , 〈 [ 0.4 , 0.6 ] , 0.7 〉 , 〈 [ 0.5 , 0.9 ] ; 0.6 〉 ) } .

Then, C ˆ is a PCFS.

Definition 14. Let p c = ( 〈 A ; λ 〉 , 〈 A ˜ ; μ 〉 ) , p c 1 = ( 〈 A 1 , λ 1 〉 , 〈 A ˜ 1 , μ 1 〉 ) and p c 2 = ( 〈 A 2 ; λ 2 〉 , 〈 A ˜ 2 ; μ 2 〉 ) be three PCFNs and δ > 0, where A₁ = [a₁, b₁], A ˜ 1 = [ a ˜ 1 , b ˜ 1 ] , A₂ = [a₂, b₂], A ˜ 2 = [ a ˜ 2 , b ˜ 2 ] , A = [a, b], A ˜ = [ a ˜ , b ˜ ] . Then the operations laws are: 1 ) p c 1 ⊕ p c 2 = ( 〈 [ a 1 2 + a 2 2 - a 1 2 a 2 2 , b 1 2 + b 2 2 - b 1 2 b 2 2 ] ; λ 1 2 + λ 2 2 - λ 1 2 λ 2 2 〉 , 〈 [ a ˜ 1 a ˜ 2 , b ˜ 1 b ˜ 2 ] ; μ 1 μ 2 〉 ) , 2 ) p c 1 ⊗ p c 2 = ( 〈 [ a 1 a 2 , b 1 b 2 ] ; λ 1 λ 2 〉 , 〈 [ a ˜ 1 2 + a ˜ 2 2 - a ˜ 1 2 a ˜ 2 2 , b ˜ 1 2 + b ˜ 2 2 - b ˜ 1 2 b ˜ 2 2 ] ; μ 1 2 + μ 2 2 - μ 1 2 μ 2 2 〉 ) , 3 ) δ p c 1 = ( 〈 [ 1 - ( 1 - a 1 2 ) δ , 1 - ( 1 - b 1 2 ) δ ] ; 1 - ( 1 - λ 1 2 ) δ 〉 , 〈 [ ( a ˜ 1 a ˜ 2 ) δ , ( b ˜ 1 b ˜ 2 ) δ ] ; ( μ 1 μ 2 ) δ 〉 ) , 4 ) p c 1 δ = ( 〈 [ ( a 1 a 2 ) δ , ( b 1 b 2 ) δ ] ; ( λ 1 λ 2 ) δ 〉 , 〈 [ 1 - ( 1 - a ˜ 1 2 ) δ , 1 - ( 1 - b ˜ 1 2 ) δ ] ; 1 - ( 1 - μ 1 2 ) δ 〉 ) , 5 ) p c ′ = 〈 A ˜ ; μ 〉 , 〈 A ; λ 〉 .

Theorem 1. Let p c = ( 〈 A ; λ 〉 , 〈 A ˜ ; μ 〉 ) , p c 1 = ( 〈 A 1 , λ 1 〉 , 〈 A ˜ 1 , μ 1 〉 ) and p c 2 = ( 〈 A 2 ; λ 2 〉 , 〈 A ˜ 2 ; μ 2 〉 ) be three PCFNs and δ > 0, δ₁ > 0andδ₂ > 0, where A₁ = [a₁, b₁], A ˜ 1 = [ a ˜ 1 , b ˜ 1 ] , A₂ = [a₂, b₂], A ˜ 2 = [ a ˜ 2 , b ˜ 2 ] , A = [a, b], A ˜ = [ a ˜ , b ˜ ] then the following will hold:

p _{c 1}  ⊕ p _{c 2}  = p _{c 2}  ⊕ p _{c 1}

Proof. Proof is very simple.

To compare two PCFNs, we define score function and there basic properties.

Definition 15. Let p c = ( 〈 A c ; λ c 〉 , 〈 A ˜ c ; μ c 〉 ) be a PCFN, where A _c  = [a _c , b _c ], A ˜ c = [ a ˜ c , b ˜ c ] . We can introduce the score function of p _c as: S ( p c ) = ( a + b - λ 3 ) 2 - ( a ˜ + b ˜ - μ 3 ) 2 .(6)

Where S (p _c ) ∈ [-1, 1].

Definition 16. Let p c 1 = ( 〈 A 1 ; λ 1 〉 , 〈 A ˜ 1 ; μ 1 〉 ) and p c 2 = ( 〈 A 2 ; λ 2 〉 , 〈 A ˜ 2 ; μ 2 〉 ) be two PCFNs, S (p _{c 1} ) be the score function of p _{c 1} and S (p _{c 2} ) be the score function of p _{c 2} . Then

If S (p _{c 1} ) < S (p _{c 2} ), then p _{c 1}  < p _{c 2} .

Example 2. Letp _{c 1}   =   (〈 [0.5, 0.7] ;0.9 〉 , 〈 [0.1, 0.5] ;   0.6 〉), p _{c 2}  = (〈 [0.4, 0.7] ;0.6 〉 , 〈 [0.2, 0.4] ;0.6 〉) and p _{c 3}  = (〈 [0.03, 0.8] ;0.9 〉 , 〈 [0.0, 0.3] ;0.7 〉) be three PCFNs. Then by Definition 20, we have, S (p _{c 1} ) =0.01, S (p _{c 2} ) =0.027 and S (p _{c 3} ) = -0.0173. Thus, S (p _{c 2} ) > S (p _{c 1} ) > S (p _{c 3} ).

Let p _{c 1}  = (〈 [0.5, 0.7] ;0.9 〉 , 〈 [0.1, 0.5] ;0.6 〉) and p _{c 2}  = (〈 [0.4, 0.7] ;0.6 〉 , 〈 [0.2, 0.4] ;0.7 〉) be two PCFNs. Then by Definition (15), we have, S (p _{c 1} ) =0.01 and S (p _{c 2} ) =0.01 Thus, S (p _{c 1} ) = S (p _{c 2} ). Therefore, by Definition (16) we cannot get information from p _{c 1} and p _{c 2} . Essentially, such a case has typically risen in preparation. From Definition (16), It is clear that it we cannot take the condition that two PCFNs in equal score. While the deviancy may be changed. The accuracy function of all the components to the average number in a PCFNs returns that they may be agree with each other. To improve this matter, we present the concept of accuracy degree for the compression of two PCFNs.

Definition 17. Let p c = ( 〈 A ; λ 〉 , 〈 A ˜ ; μ 〉 ) be a PCFN. Then we define the accuracy degree of p _c is denoted by α (p _c ), where A = [a, b], A ˜ = [ a ˜ , b ˜ ] , can be defined as: α ( p c ) = ( a + b - λ 3 ) 2 + ( a ˜ + b ˜ - μ 3 ) 2 .(7)

Where α (p _c ) ∈ [0, 1].

Definition 18. Let p c 1 = ( 〈 A 1 ; λ 1 〉 , 〈 A ˜ 1 ; μ 1 〉 ) and p c 2 = ( 〈 A 2 ; λ 2 〉 , 〈 A ˜ 2 ; μ 2 〉 ) be two PCFNs, α (p _{c 1} ) be the accuracy degree of p _{c 1} and α (p _{c 2} ) be the accuracy degree of p _{c 2} . Then

If α (p _{c 1} ) < α (p _{c 2} ), then p _{c 1}  < p _{c 2} .

Example 3. From Example 2, since S (p _{c 1} ) =0.01 and S (p _{c 2} ) =0.01 Thus, S (p _{c 1} ) = S (p _{c 2} ). So we have α (p _{c 1} ) =0.01 and α (p _{c 2} ) =0.044. Thus α (p _{c 1} ) > α (p _{c 2} ). Hence p _{c 1}  > p _{c 2} . Which clearing the situation that two PCFNs have the same score.

Definition 19. Let p _{c 1} and p _{c 2} be any two PCFNs on a set X. The distance measure between p _{c 1} and p _{c 2} can be defined in the following way: d ( p c 1 , p c 2 ) = 1 6 [ | a 1 2 - a 2 2 | + | b 1 2 - b 2 2 | + | a ˜ 1 2 - a ˜ 2 2 | + | b ˜ 1 2 - b ˜ 2 2 | + | λ 1 2 - λ 2 2 | + | μ 1 2 - μ 2 2 | ](8)

Example 4. Let p _{c 1}  = (〈[0.6, 0.7] ;0.3〉, 〈[0.5, 0.7] ; 0.8〉) and p _{c 2}  (〈[0.5, 0.6] ;0.4〉, 〈[0.4, 0.7] ;0.5〉) be two PCFNs. Then D ( p c 1 , p c 2 ) = 1 6 [ | 0.36 - 0.25 | + | 0.49 - 0.36 | + | 0.25 - 0.16 | + | 0.49 - 0.49 | + | 0.9 - 0.16 | + | 0.64 - 0.25 | ] D ( p c 1 , p c 2 ) = 1 6 [ 0.11 + 0.13 + 0.09 + 0.74 + 0.39 ] = 0.2433 .

In this section, we suggest some aggregation operators for Pythagorean cubic fuzzy numbers, and examine some of its properties.

Definition 20. Let p c i = ( 〈 A i ; λ i 〉 , 〈 A ˜ i ; μ i 〉 ) (i = 1, 2, …, n) be a collection of all PCFNs, while w = (w₁, w₂, …, w _n )  ^T be the weight vector of p _{c i} (i = 1, 2, …, n) with w _i  ⩾ 0 (i = 1, 2, 3 … , n) where w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 . Then Pythagorean cubic fuzzy weighted averaging (PCFWA) operator is a mapping PCFWA : PCFN ⁿ  → PCFN defined by PCFWA ( p c 1 , p c 2 , . . . , p c n ) = w 1 p c 1 ⊕ w 2 p c 2 ⊕ , . . . , ⊕ w n p c n(9)

where PCFWA is called Pythagorean cubic fuzzy weighted averaging (PCFWA) operator.

Definition 21. Let p c i = ( 〈 A i ; λ i 〉 , 〈 A ˜ i ; μ i 〉 ) (i = 1, 2, …, n) be a collection of all PCFNs, while w = (w₁, w₂, …, w _n )  ^T be the weight vector of p _{c i} (i = 1, 2, …, n) with w _i  ⩾ 0 (i = 1, 2, 3 … , n) where w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 . Then Pythagorean cubic fuzzy weighted geometric (PCFWG) operator is a mapping PCFWG : PCFN ⁿ  → PCFNPCFWG ( p c 1 , p c 2 , . . . , p c n ) = p c 1 w 1 ⊗ p c 2 w 2 ⊗ , . . . , ⊗ p c n w n(10)

where PCFWG is called Pythagorean cubic fuzzy weighted geometric (PCFWG) operator.

Theorem 2. Let p c i = ( 〈 A i ; λ i 〉 , 〈 A ˜ i ; μ i 〉 ) (i = 1, 2, …, n) be a collection of all PCFNs, and w = (w₁, w₂, …, w _n )  ^T be the weight vector of p _{c i} (i = 1, 2, …, n) with w _i  ⩾ 0 where w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 . Then the aggregation result using PCFWA/PCFWG operator is also a PCFN and PCFWA ( p c 1 , p c 2 , . . . , p c n ) = ( 〈 [ 1 - ∏ i = 1 n ( 1 - a i 2 ) w i , 1 - ∏ i = 1 n ( 1 - b i 2 ) w i ] ; 1 - ∏ i = 1 n ( 1 - λ i 2 ) w i 〉 , 〈 [ ∏ i = 1 n a ˜ i w i , ∏ i = 1 n b ˜ i w i ] ; ∏ i = 1 n μ i w i 〉 )(11)

and PCFWG ( p c 1 , p c 2 , . . . , p c n ) = ( 〈 [ ∏ i = 1 n a i w i , ∏ i = 1 n b i w i ] ; ∏ i = 1 n λ i w i 〉 , 〈 [ 1 - ∏ i = 1 n ( 1 - a ˜ i 2 ) w i , 1 - ∏ i = 1 n ( 1 - b ˜ i 2 ) w i ] ; 1 - ∏ i = 1 n ( 1 - μ i 2 ) w i 〉 )(12)

Proof. Straightforward similar proof.

Theorem 3. Let p c i = ( 〈 A i ; λ i 〉 , 〈 A ˜ i ; μ i 〉 ) (i = 1, 2, …, n) be a group of all PCFNs, and w = (w₁, w₂, …, w _n )  ^T be the weight vector of p _{c i} with where w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 . Then we have

(Idempotency): If all p _{c i} are equal, i.e., p _{c i}  = p _c , for all I. Then

PCFWA / PCFWG ( p c 1 , p c 2 , … , p c n ) = p c .(13)

(Boundary):

p c min ⩽ PCFWA / PCFWG ( p c 1 , p c 2 , … , p c n ) ⩽ p c max ,(14)

for all w. where p c min = ( min j ( μ i ) , max j ( ν i ) ) , p c max = ( max j ( μ i ) , min j ( ν i ) ) .

(Monotonicity): p c i = ( 〈 A i ; λ i 〉 , 〈 A ˜ i ; μ i 〉 ) and p c i * = ( 〈 A i * ; λ i * 〉 , 〈 A ˜ i * ; μ i * 〉 ) (i = 1, 2, 3, …, n) be the groups of pythagorean cubic fuzzy values,if A i ⩽ A i * and λ _i  ⩾ λ^{* _i} , A ˜ i ⩽ A ˜ i * and μ i ⩾ μ i * for all i. Then PCFWA / PCFWG ( p c 1 , p c 2 , … , p c n ) ⩽ PCFWA / PCFWG ( p c * 1 , p c * 2 , … , p c * n )(15)

for every w. Similarly for PCFWG operator, we have.

Proof. Proof of the Theorem follows from Theorm [2,3, 2,3] of [29], and Theorem [4] of [3].

Lemma 1. Let p _{c i}  > 0, w _i  > 0 (i = 1, 2, … ; , n) and ∑ i = 1 n w i = 1 . Then∏ i = 1 n ( p c i ) w i ⩽ ∑ i = 1 n w i p c i .(16)

Where the equality holds if and only if p _{c 1}  = p _{c 2}  = p _{c 3}  = ⋯ = p _{c n}  .

Theorem 4. Let p c i = ( 〈 A i ; λ i 〉 , 〈 A ˜ i ; μ i 〉 ) (i = 1, 2, …, n) be a collection of all PCFNs, then PCFWG ( p c 1 , p c 2 , … , p c n ) ⩽ PCFWA ( p c 1 , p c 2 , … , p c n ) .

Where w = (w₁, w₂, …, w _n )  ^T is the weighted vector of p _{c i} (i = 1, 2, 3, …, n) ∃ w _i  ∈ [0, 1] (i = 1, 2, 3, …, n) and ∑ i = 1 n w i = 1 .

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

Definition 22. Let p c i = ( 〈 A i ; λ i 〉 , 〈 A ˜ i ; μ i 〉 ) (i = 1, 2, …, n) be a group of all PCFNs, while w = (w₁, w₂, …, w _n )  ^T be the weight vector of p _{c 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 (PCFOWA) operator is a mapping PCFOWA : PCFN ⁿ  → PCFNPCFOWA ( p c 1 , p c 2 , … , p c n ) = w 1 p c 1 ⊕ w 2 p c 2 ⊕ , … , ⊕ w n p c n(17)

and the PCFOWA operator is said to be a Pythagorean cubic fuzzy ordered weighted averaging operator.

Definition 23. Let p c i = ( 〈 A i ; λ i 〉 , 〈 A ˜ i ; μ i 〉 ) (i = 1, 2, …, n) be a group of all PCFNs, while w = (w₁, w₂, …, w _n )  ^T be the weight vector of p _{c 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 the Pythagorean cubic fuzzy (PCFOWG) operator is a mapping PCFOWG : PCFN ⁿ  → PCFN can be defined as PCFOWG ( p c 1 , p c 2 , … , p c n ) = p c 1 w 1 ⊗ p c 2 w 2 ⊗ , … , ⊗ p c n w n(18)

and the PCFOWG operator is said to be a Pythagorean cubic fuzzy ordered weighted geometric operator.

Theorem 5. Let p c i = ( 〈 A i ; λ i 〉 , 〈 A ˜ i ; μ i 〉 ) (i = 1, 2, 3, …, n) be a collection of all PCFNs, and w = (w₁, w₂, …, w _n )  ^T be the weight vector of p _{c i} (i = 1, 2, 3, …, n) with w _i  ⩾ 0 where w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 . Then the aggregation result using PCFOWA operator is also a PCFN and PCFOWA ( p c 1 , p c 2 , … , p c n ) = ( 〈 [ 1 - ∏ i = 1 n ( 1 - a σ ( i ) 2 ) w i , 1 - ∏ i = 1 n ( 1 - b σ ( i ) 2 ) w i ] ; 1 - ∏ i = 1 n ( 1 - λ σ ( i ) 2 ) w i 〉 , 〈 [ ∏ i = 1 n a ˜ σ ( i ) w i , ∏ i = 1 n b ˜ σ ( i ) w i ] ; ∏ i = 1 n μ σ ( i ) w i 〉 )(19)

And the aggregation result using PCFOWG operator is also a PCFN and PCFOWG ( p c 1 , p c 2 , … , p c n ) = ( 〈 [ ∏ i = 1 n a σ ( i ) w i , ∏ i = 1 n b σ ( i ) w i ] , ∏ i = 1 n λ σ ( i ) w i 〉 , 〈 [ 1 - ∏ i = 1 n ( 1 - a ˜ σ ( i ) 2 ) w i , 1 - ∏ i = 1 n ( 1 - b ˜ σ ( i ) 2 ) w i ] , 1 - ∏ i = 1 n ( 1 - μ σ ( i ) 2 ) w i 〉 )(20)

Proof. Similar proof.

Theorem 6. Let p c i = ( 〈 A i ; λ i 〉 , 〈 A ˜ i ; μ i 〉 ) (i = 1, 2, 3, …, n) be a group of all PCFNs, and w = (w₁, w₂, …, w _n )  ^T be the weight vector of p _{c i} with w _i  ⩾ 0 where w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 . Then we have

(Idempotency): If all p _{c i} are equal, i.e., p _{c i}  = p _c , for all i. Then PCFOWA / PCFOWG ( p c 1 , p c 2 , … , p c n ) = p c .(21)

for every w.

Proof. Straightforwad.

Lemma 2. Let p _{c i}  > 0, w _i  > 0 (i = 1, 2, 3, …, n) and ∑ i = 1 n w i = 1 . Then∏ i = 1 n ( p c i ) w i ⩽ ∑ i = 1 n w i p c i .(23)

Where the equality holds if and only if p _{c 1}  = p _{c 2}  = p _{c 3}  = ⋯ = p _{c n} .

Theorem 7. Let p _{c i}  = 〈μ _{c ij} , ν _{c ij} 〉, (i = 1, 2, 3, …, n) be a collection of all PCFNs, thenPCFOWG ( p c 1 , p c 2 , … , p c n ) ⩽ PCFOWA ( p c 1 , p c 2 , … , p c n ) .(24)

Where w = (w₁, w₂, …, w _n )  ^T is the weighted vector of p _{c i} (i = 1, 2, 3, …, n) ∃ w _i  ∈ [0, 1] (i = 1, 2, 3 … , n) and ∑ i = 1 n w i = 1 .

Proof. Based on Lemma 2, we can easily prove the Theorem.

In this section, we use the Pythagorean cubic fuzzy aggregation operators to multi-criteria decision-making process. Assume, we have an n options X ={ x_1,x₂, …, x _n  } and m criteria A ={ A_1,A₂, …, A _m  } to be evaluated having weight vector w = (w₁, w₂, …, w _m )  ^T ∃ w _j  ∈ [0, 1], and ∑ j = 1 m w j = 1 . To assess the performance of the alternative x _i under the criteria A _j , the decision makers are required to convey not only the information that the alternative x _i satisfy the criteria A _j , but also the alternative x _i does not satisfy the criteria A _j . These two part information can be stated by 〈A _ij  ; λ _ij 〉 and 〈 A ˜ ij ; μ ij 〉 which represent the degrees that the alternative x _i satisfy the criterion A _j and does not satisfy the criterion A _j , then the performance of the alternative x _i under the criteria A _j can be stated by an PCFN p c ij = ( 〈 A ij ; λ ij 〉 , 〈 A ˜ ij ; μ ij 〉 ) with the condition that for all A _{c ij}  ∈ p _{c ij} , ∃ A ˜ c ij ∈ p c ij ′ such that 0 ⩽ ( A c ij ) 2 + ( A ˜ c ij ) 2 ⩽ 1 , i = 1, 2, …, n, j = 1, 2, …, m and k = 1, 2, …, t. To get the ranking of the alternatives, the following steps are stated: