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Article

Confidence Levels-Based Cubic Fermatean Fuzzy Aggregation Operators and Their Application to MCDM Problems

1
School of Mathematics, Thapar Institute of Engineering & Technology, Deemed University, Patiala 147004, Punjab, India
2
Department of Mathematics, Graphic Era Deemed to be University, Dehradun 248002, Uttarakhand, India
3
Applied Science Research Center, Applied Science Private University, Amman 11931, Jordan
4
Department of Mathematics, Hazara University Mansehra, Mansehra 21120, Pakistan
5
College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, Denmark
6
Department of Statistics & Operations Research, College of Sciences King Saud University, Riyadh 11451, Saudi Arabia
*
Author to whom correspondence should be addressed.
Symmetry 2023, 15(2), 260; https://doi.org/10.3390/sym15020260
Submission received: 25 December 2022 / Revised: 7 January 2023 / Accepted: 13 January 2023 / Published: 17 January 2023
(This article belongs to the Special Issue Research on Fuzzy Logic and Mathematics with Applications II)

Abstract

:
Assessment specialists (experts) are sometimes expected to provide two types of information: knowledge of rating domains and the performance of rating objects (called confidence levels). Unfortunately, the results of previous information aggregation studies cannot be properly used to combine the two categories of data covered above. Additionally, a significant range of symmetric/asymmetric events and structures are frequently included in the implementation process or practical use of fuzzy systems. The primary goal of the current study was to use cubic Fermatean fuzzy set features to address such situations. To deal with the ambiguous information of the aggregated arguments, we defined information aggregation operators with confidence degrees. Two of the aggregation operators we initially proposed were the confidence cubic Fermatean fuzzy weighted averaging (CCFFWA) operator and the confidence cubic Fermatean fuzzy weighted geometric (CCFFWG) operator. They were used as a framework to create an MCDM process, which was supported by an example to show how effective and applicable it is. The comparison of computed results was carried out with the help of existing approaches.

1. Introduction

Researchers working in the general area of fuzzy decision-making have drawn inspiration from the Bellman–Zadeh conception of a symmetrical decision model in an uncertain environment, with complete symmetry between constraints and decision variables. A significant range of symmetric/asymmetric events and structures is frequently included in the implementation process or practical use of fuzzy systems. Multi-criteria decision-making (MCDM) is one of the fast-developing active research problems for obtaining conclusive results in a reasonable time. However, due to different restrictions, it is not always possible to express the requirements precisely, hence the corresponding solutions are not always optimal. The intuitionistic fuzzy set (IFS) [1] theory is one of the most effective and promising strategies scholars usually apply to manage the ambiguity and imprecision of information. In this context, different scholars focus more on IFSs for integrating the different alternatives using various aggregation algorithms. The performances of the criteria for alternatives are aggregated throughout the data synthesis process using weighted and ordered weighted aggregation operators (AOs) [2,3]. In an IFS environment, Xu and Yager [4] presented a geometric aggregation operator (GAO) while Xu [5] proposed a weighted averaging aggregation operator (AAO). Wang and Liu [6] proposed Einstein aggregation operators by using Einstein norm operations in the IFS context. Lai et al. [7] presented a matching algorithm based on similarity measures and adaptive weights. Ye [8] proposed an accuracy function (ac) for interval-valued IFS to compare them to interval-valued intuitionistic fuzzy numbers. Garg [9] presented a series of communicating AOs for IFSs. Garg [10,11] introduced interacting geometric operators employing Einstein t-norm and Einstein t-conorm operations to aggregate intuitionistic fuzzy data. Xu et al. [12] introduced the intuitionistic fuzzy Einstein–Choquet integral-based operators for decision-making (DM) problems. According to the results of the research mentioned above, they are legitimate as long as the sum of the membership grades does not exceed one. However, in real life, it is not always possible to communicate one’s preferences within this restriction. For instance, if someone were to review an option according to their preferences, they would give it a satisfaction rating of 0.7 and an unsatisfaction rating of 0.6. As a result, the review would be unable to meet the IFS condition, as 0.7 + 0.6 > 1 . Because the effectiveness cannot be tested under these circumstances, the IFS theory has some limits and disadvantages. To address these issues, Yager [13,14] introduced Pythagorean fuzzy sets (PFSs) as an extension of the IFS theory. PFSs relax the limitations of IFS. Furthermore, it has been demonstrated that all intuitionistic fuzzy values are part of Pythagorean fuzzy values, which specifies that PFSs have superior ability to manage ambiguous issues (See Figure 1). Following his pioneering work, scholars are continually attempting to improve PFSs. According to Yager and Abbasov [15], Pythagorean fuzzy grades are subclasses of complex numbers. Moreover, Zhang and Xu [16] provided a method for determining the optimal alternative based on an ideal solution in a Pythagorean fuzzy environment. Yager [14] presented a series of aggregation operators in a PFS environment. Peng and Yang [17] defined some basic operational laws and their related properties for Pythagorean fuzzy numbers. Garg presented correlation and correlation coefficients for PFSs. Geo and Deng [18] proposed a Pythagorean fuzzy generation technique based on probability negativity to handle MCDM problems. Zhang [19] presented the notions of interval-valued PFSs (IVPFSs) by extending PFSs. Some important properties of IVPFSs were presented by Peng and Yang [20]. To relax the limitations of PFSs, Senapati and Yager [21] proposed Fermatean fuzzy sets (FFSs) and some operational laws of FFSs. Senapati and Yager [22] proposed weighted averaging and weighted geometric aggregation operators under an FFS environment. Rani and Mishra proposed interval-valued FFSs.
According to the available research, fuzzy sets, IFS, PFS, and their corresponding implementations are the main topics of all current research. Later on, Jun et al. proposed cubic sets (CSs) by integrating fuzzy sets and interval-valued fuzzy sets. Kaur and Garg [23] presented cubic IFSs and a series of AOs based on t-norm operations. Khan et al. [24,25] suggested CS operations and their characteristics. Abbas et al. [26] proposed cubic PFSs (CPFSs) by combining PFSs and IVPFSs for solving MCDM problems. The flaws and ambiguities of CPFSs were investigated by Amin et al. [27]. Rahim et al. [28] proposed Bonferroni mean aggregation operators under a CPFS environment. Rong and Mishra [29] proposed cubic FFSs and their application in MCDM problems.
Despite the popularity of the aforementioned work, the level of confidence in the criteria was not assessed in any of the studies described above. To put it another way, every researcher has approached the studies with the premise that decision-makers are unquestionably competent in the subjects being investigated. However, these types of prerequisites are only partially accomplished in real-world situations. To compensate for this limitation, decision-makers may examine the alternatives in terms of cubic Fermatean fuzzy numbers (CFFNs) and their associated confidence levels based on their familiarity with the evaluation. As a result, during the evaluation of the alternative in terms of CFFNs, the present study proposes the concept of confidence levels in the optimization processes. First, some basic operations such as P-union (rep. P-intersection), R-union (rep. R-intersection) and so on are defined. Based on these investigations a series of weighted and geometric operators and are proposed in this paper. Additionally, a method to address MCDM issues is suggested. The following is a summary of the study’s primary goals:
(1)
Define some basic operations of CFFSs and their properties.
(2)
Based on these operational laws, propose a series of aggregation operators with confidence levels in a CFFS environment.
(3)
Develop a new approach to solve MCDM problems under CFFSs.
(4)
Provide an example to evaluate the accuracy and reliability of the proposed approach.
(5)
Compare the results of the proposed framework with some existing approaches.

2. Preliminaries

In this section, we briefly present some concepts of PFS, IVPFS, and others to understand the paper.

2.1. PFSs, IVPFSs, and CPFSs

Definition 1
Ref. [13]. Let F be a non-empty finite set. A PFS over element t F is defined as
A = { t ,   φ A ( t ) ,   ψ A ( t ) | t F } ,
where φ A ( t ) [ 0 ,   1 ] and ψ A ( t ) [ 0 , 1 ] are the membership and non-membership function of an element t F such that  ( φ A ( t ) ) 2 + ( ψ A ( t ) ) 2 1 .
For convenience, Zhang and Xu [16] called φ A ( t ) ,   ψ A ( t ) a PFN denoted by φ A , ψ A . The score function of A can be calculated as s c ( A ) = φ A 2 ψ A 2 .
Definition 2
Ref. [19]. For a non-empty set F , an IVPFS over an element t F is defined as follows:
B = { t , φ ˜ B ( t ) ,   ψ ˜ B ( t ) | t F } ,
where φ ˜ B ( t ) and ψ ˜ B ( t ) are interval-valued fuzzy numbers representing the interval membership and non-membership grades of set B , respectively. Let φ ˜ B ( t ) = [ φ ˜ B L ( t ) , φ ˜ B U ( t ) ] and ψ ˜ B ( t ) = [ ψ ˜ B L ( t ) , ψ ˜ B U ( t ) ] then IVPFS can be written as B = { t , [ φ ˜ B L ( t ) , φ ˜ B U ( t ) ] , [ ψ ˜ B L ( t ) , ψ ˜ B U ( t ) ] | t F } .
For convenience, we denote these pairs as [ φ ˜ B L , φ ˜ B U ] , [ ψ ˜ B L , ψ ˜ B U ] and call this an interval-valued PFN (IVPFN). We also set, 0 φ ˜ B L , φ ˜ B U , ψ ˜ B L , ψ ˜ B U 1 such that ( φ ˜ B U ) 2 + ( ψ ˜ B U ) 2 1 . The score function of B can be calculated as s c ( B ) = 1 2 ( ( φ ˜ B L ) 2 + ( φ ˜ B U ) 2 ( ψ ˜ B L ) 2 ( φ ˜ B U ) 2 ) .
Definition 3
Refs. [26,27]. Let F be a non-empty finite set. A CPFS over an element t F is defined as
C = { t , ˜ C ( t ) , A C ( t ) | t F } ,
where ˜ C ( t ) = ( [ φ ˜ ˜ c L ( t ) , φ ˜ ˜ c U ( t ) ] , [ ψ ˜ ˜ c L ( t ) , ψ ˜ ˜ c U ( t ) ] ) represents an IVPFS while A C ( t ) = ( φ ˜ C ( t ) , ψ ˜ C ( t ) ) represents a PFS. We also set, 0 φ ˜ ˜ c L ( t ) , φ ˜ ˜ c U ( t ) , ψ ˜ ˜ c L ( t ) , ψ ˜ ˜ c U ( t ) 1 such that ( φ ˜ ˜ c U ( t ) ) 2 + ( ψ ˜ ˜ c U ( t ) ) 2 1 .
For convenience, we denote the pairs as [ φ ˜ ˜ c L , φ ˜ ˜ c U ] , [ ψ ˜ ˜ c L , ψ ˜ ˜ c U ] , φ ˜ C , ψ ˜ C and call this a CPFN.
Definition 4
Ref. [28]. Let C 1 = ( [ φ C 1 L ,   φ C 1 U ] ,   [ ψ C 1 L , ψ C 1 U ] ,   φ C 1 ,   ψ C 1 ) be a CPFN, then the score function is defined under R-order as
s c ( C 1 ) = ( φ C 1 L ) 2 + ( φ C 1 L ) 2 ( ψ C 1 L ) 2 ( ψ C 1 L ) 2 2 + ( ψ C 1 ) 2 ( φ C 1 ) 2 ,
and for P-order as
s c ( C 1 ) = ( φ C 1 L ) 2 + ( φ C 1 L ) 2 ( ψ C 1 L ) 2 ( ψ C 1 L ) 2 2 + ( φ C 1 ) 2 ( ψ C 1 ) 2 ,
where  2 s c ( β 1 ) 2 .
Definition 5
Ref. [28]. Let C 1 = ( [ φ C 1 L ,   φ C 1 U ] ,   [ ψ C 1 L , ψ C 1 U ] ,   φ C 1 ,   ψ C 1 ) be a CPFN, then the accuracy function is defined as
a c ( C 1 ) = ( φ C 1 L ) 2 + ( φ C 1 L ) 2 + ( ψ C 1 L ) 2 + ( ψ C 1 L ) 2 2 + ( φ C 1 ) 2 + ( ψ C 1 ) 2 ,
where  0 a c ( C 1 ) 2 .
Definition 6
Ref. [28]. Let  C 1 = ( [ φ C 1 L ,   φ C 1 U ] ,   [ ψ C 1 L , ψ C 1 U ] ,   φ C 1 ,   ψ C 1 )  and C 2 = ( [ φ C 2 L ,   φ C 2 U ] ,   [ ψ C 2 L , ψ C 2 U ] ,   φ C 2 ,   ψ C 2 )  be two CPFSs in F . Then:
  • (Equality): C 1 = C 2 , if and only if  [ φ C 1 L ,   φ C 1 U ] = [ φ C 2 L ,   φ C 2 U ] , ψ C 1 L , ψ C 1 U = ψ C 2 L , ψ C 2 U , φ C 1 = φ C 2  and  ψ C 1 = ψ C 2 ;
  • (P-order): C 1 P C 2 if [ φ C 1 L ,   φ C 1 U ] [ φ C 2 L ,   φ C 2 U ] , [ ψ C 1 L ,   ψ C 1 U ] [ ψ C 2 L ,   ψ C 2 U ] ,   φ C 1 φ C 2  and  ψ C 1 ψ C 2 ;
  • (R-order): C 1 R C 2 if [ φ C 1 L ,   φ C 1 U ] [ φ C 2 L ,   φ C 2 U ] , [ ψ C 1 L ,   ψ C 1 U ] [ ψ C 2 L ,   ψ C 2 U ] ,   φ C 1 φ C 2  and  ψ C 1 ψ C 2 .
Definition 7
Ref. [27]. For the CPFNs  C i = ( [ φ C i L ,   φ C i U ] ,   [ ψ C i L , ψ C i U ] ,   φ C i ,   ψ C i )   ( 1 ,   2 ,   3 ,   4 )  we have:
(a)
If C 1 P C 2 and C 2 P C 3 then C 1 P C 3 ;
(b)
If C 1 P C 2 then C 2 c P C 1 c :
(c)
If C 1 P C 2   and C 1 P C 3 then C 1 P C 2 C 3 ;
(d)
If C 1 P C 2 and C 3 P C 4 then C 1 C 3 P C 2 C 4 and C 1 C 3 P C 2 C 4 ;
(e)
If C 1 P C 2 and C 3 P C 2 then C 1 C 3 P C 2 ;
(f)
If C 1 R C 2 and C 2 R C 3 then C 1 R C 3 ;
(g)
If C 1 R C 2 then C 2 c R C 1 c ;
(h)
If C 1 R C 2   and C 1 R C 3 then C 1 R C C 3 ;
(i)
If C 1 R C 2 and C 3 R C 4 then C 1 C 3 R C 2 C 4 and C 1 C 3 R C 2 C 4 ;
(j)
If C 1 R C 2 and C 3 R C 2 then C 1 C 3 R C 2 .

2.2. FFSs, IVFFSs, and CFFSs

Definition 8
Ref. [21]. Let F be a non-empty set and t F . The FFS over element t is defined as
= { t , φ ( t ) , ψ ( t ) | t F } ,
where φ ( t ) [ 0 ,   1 ] and ψ ( t ) [ 0 , 1 ] are the membership and non-membership function of an element t F such that ( φ ( t ) ) 3 + ( ψ ( t ) ) 3 1 .
For convenience, Senapati and Yager [21] called φ ( t ) ,   ψ ( t ) an FFN denoted by φ , ψ . The score function of A can be calculated as s c ( A ) = φ 3 ψ 3 .
Definition 9
Ref. [30]. For a non-empty set F , an interval-valued FFS (IVFFS) over an element t F is defined as follows:
G = { t , φ ˜ G ( t ) ,   ψ ˜ G ( t ) | t F } ,
where φ ˜ G ( t ) and ψ ˜ G ( t ) are interval-valued fuzzy numbers representing the interval membership and non-membership grades of the set G repectively. Let φ ˜ G ( t ) = [ φ ˜ G L ( t ) , φ ˜ G U ( t ) ] and ψ ˜ G ( t ) = [ ψ ˜ G L ( t ) , ψ ˜ G U ( t ) ] then IVPFS can be written as G = { t , [ φ ˜ G L ( t ) , φ ˜ G U ( t ) ] , [ ψ ˜ G L ( t ) , ψ ˜ G U ( t ) ] | t F } .
For convenience, we denote these pairs as [ φ ˜ G L , φ ˜ G U ] , [ ψ ˜ G L , ψ ˜ G U ] and call this an interval-valued FFN (IVFFN). We also set, 0 φ ˜ G L , φ ˜ G U , ψ ˜ G L , ψ ˜ G U 1 such that ( φ ˜ G U ) 3 + ( ψ ˜ G U ) 3 1 . The score function of G can be calculated as s c ( B ) = 1 2 ( ( φ ˜ B L ) 3 + ( φ ˜ B U ) 3 ( ψ ˜ B L ) 3 ( φ ˜ B U ) 3 ) .
Definition 10
Ref. [29]. Let F be a non-empty finite set. A CPFS over an element t F is defined as
= { t , G ( t ) , ( t ) | t F } ,
where  G ( t ) = ( [ φ ˜ L ( t ) , φ ˜ U ( t ) ] , [ ψ ˜ L ( t ) , ψ ˜ U ( t ) ] )  represents an IVFFS while  ( t ) = ( φ ( t ) , ψ ( t ) )  represents PFS. We also set,  0 φ ˜ L ( t ) , φ ˜ U ( t ) , ψ ˜ L ( t ) , ψ ˜ U ( t ) 1  such that  ( φ ˜ U ( t ) ) 2 + ( ψ ˜ U ( t ) ) 2 1 .
For convenience, we denote the pairs as [ φ ˜ L , φ ˜ U ] , [ ψ ˜ L , ψ ˜ U ] , φ , ψ and call this a CPFN.

3. New Operational Laws and Aggregation Operators under CFFSs with Confidence Levels

In this section, the existing operations defined by Rong et al. [29] are modified. Furthermore, the order relations such as P-order and R-order of CFFNs are presented. Finally, based on these modified operations some series aggregation operators with confidence levels are proposed.

3.1. Modified Operations of CFFSs

Definition 11.
For a family of CFFS { i ,   i Δ } , it follows that
(a)
(P-union): i Δ P i = ( [ m a x i Δ ( φ i L ) ,   m a x i Δ ( φ i U ) ] ,   [ m i n i Δ ( ψ i L ) ,   m i n i Δ ( ψ i U ) ] , m a x i Δ μ i , m i n i Δ ν i ) ;
(b)
(P-intersection): i Δ R i = ( [ m i n i Δ ( φ i L ) ,   m i n i Δ ( φ i U ) ] ,   [ m a x i Δ ( ψ i L ) ,   m a x i Δ ( ψ i U ) ] , m i n i Δ μ i , m a x i Δ ν i ) ;
(c)
(R-union): i Δ R i = ( [ m a x i Δ ( φ i L ) ,   m a x i Δ ( φ i U ) ] ,   [ m i n i Δ ( ψ i L ) , m i n i Δ ( ψ i U ) ] , m i n i Δ μ i , m a x i Δ ν i ) ;
(d)
(R-intersection): i Δ P i = ( [ m a x i Δ ( φ i L ) ,   m a x i Δ ( φ i U ) ] ,   [ m i n i Δ ( ψ i L ) ,   m i n i Δ ( ψ i U ) ] , m i n i Δ φ i , m a x i Δ ψ i ) .
Definition 12.
Let  1 = ( [ φ 1 L ,   φ 1 U ] ,   [ ψ 1 L , ψ 1 U ] ,   φ 1 ,   ψ 1 )  and 2 = ( [ φ 2 L ,   φ 2 U ] ,   [ ψ 2 L , ψ 2 U ] ,   φ 2 ,   ψ 2 )  be two CFFSs in F . Then
(a)
(Equality): 1 = 2 , if and only if  [ φ 1 L ,   φ 1 U ] = [ φ 2 L ,   φ 2 U ] , [ ψ 1 L ,   ψ 1 U ] = [ ψ 2 L ,   ψ 2 U ] , φ 1 = φ 2  and  ψ 1 = ψ 2 :
(b)
(P-order): 1 P 2  if  [ φ 1 L ,   φ 1 U ] [ φ 2 L ,   φ 2 U ] , [ ψ 1 L ,   ψ 1 U ] [ ψ 2 L ,   ψ 2 U ] , φ 1 φ 2  and  ψ 1 ψ 2 ;
(c)
(R-order): 1 R 2  if  [ φ 1 L ,   φ 1 U ] [ φ 2 L ,   φ 2 U ] , [ ψ 1 L ,   ψ 1 U ] [ ψ 2 L ,   ψ 2 U ] , φ 1 φ 2  and  ψ 1 ψ 2 .
Definition 13.
Let  1 = ( [ φ 1 L ,   φ 1 U ] ,   [ ψ 1 L , ψ 1 U ] ,   φ 1 ,   ψ 1 )  be a CFFN, then the score function is defined under R-order as
s c ( 1 ) = ( φ 1 L ) 3 + ( φ 1 U ) 3 ( ψ 1 L ) 3 ( ψ 1 U ) 3 2 + ( ψ 1 3 φ 1 3 ) ,
and for P-order as
s c ( 1 ) = ( φ 1 L ) 3 + ( φ 1 U ) 3 ( ψ 1 L ) 3 ( ψ 1 U ) 3 2 + ( φ 1 3 ψ 1 3 ) ,
where  2 s c ( 1 ) 2 .
Definition 14.
Let  1 = ( [ φ 1 L ,   φ 1 U ] ,   [ ψ 1 L , ψ 1 U ] ,   φ 1 ,   ψ 1 )  be a CFFN, then the accuracy function is defined under R-order as
a c ( 1 ) = ( φ 1 L ) 3 + ( φ 1 U ) 3 + ( ψ 1 L ) 3 + ( ψ 1 U ) 3 2 + ( φ 1 3 + ψ 1 3 ) ,
where  0 a c ( 1 ) 2 .
Theorem 1.
For the CPFNs i = ( [ φ i L ,   φ i U ] ,   [ ψ i L , ψ i U ] ,   φ i ,   ψ i )   ( 1 ,   2 ,   3 ,   4 ) we have:
(a)
If 1 P 2 and 2 P 3 then 1 P 3 ;
(b)
If 1 P 2 then 2 c P 1 c ;
(c)
If 1 P 2   and 1 P 3 then 1 P 2 3 ;
(d)
If 1 P 2 and 3 P 4 then 1 3 P 2 4 and 1 3 P 2 4 ;
(e)
If 1 P 2 and 3 P 2 then 1 3 P 2 ;
(f)
If 1 R 2 and 2 R 3 then 1 R 3 ;
(g)
If 1 R 2 then 2 c R 1 c ;
(h)
If 1 R 2   and 1 R 3 then 1 R 2 3 ;
(i)
If 1 R 2 and 3 R 4 then 1 3 R 2 4 and 1 3 R 2 4 ;
(j)
If 1 R 2 and 3 R 2 then 1 3 R 2 .
Proof. 
(a) Since 1 = ( [ φ 1 L ,   φ 1 U ] ,   [ ψ 1 L , ψ 1 U ] ,   φ 1 ,   ψ 1 ) , 2 = ( [ φ 2 L ,   φ 2 U ] ,   [ ψ 2 L , ψ 2 U ] ,   φ 2 ,   ψ 2 ) , and 3 = ( [ φ 3 L ,   φ 3 U ] ,   [ ψ 3 L , ψ 3 U ] ,   φ 3 ,   ψ 3 ) be CPFNs. Using Definition 12, if 1 P 2 then [ φ 1 L ,   φ 1 U ] [ φ 2 L ,   φ 2 U ] , [ ψ 1 L ,   ψ 1 U ] [ ψ 2 L ,   ψ 2 U ] , φ 1 φ 2 , and ψ 1 ψ 2 . Similarly, if 2 P 3 , then [ φ 2 L ,   φ 2 U ] [ φ 3 L ,   φ 3 U ] , [ ψ 2 L ,   ψ 2 U ] [ ψ 3 L ,   ψ 3 U ] , φ 2 φ 3 , and ψ 2 ψ 3 which implies that [ φ 1 L ,   φ 1 U ] [ φ 2 L ,   φ 2 U ] [ φ 3 L ,   φ 3 U ] ; [ ψ 1 L ,   ψ 1 U ] [ ψ 2 L ,   ψ 2 U ] [ ψ 3 L ,   ψ 3 U ] ; φ 1 φ 2 φ 3 ; and ψ 1 ψ 2 ψ 3 and hence [ φ 1 L ,   φ 1 U ] [ φ 3 L ,   φ 3 U ] ; [ ψ 1 L ,   ψ 1 U ] [ ψ 3 L ,   ψ 3 U ] ; φ 1 φ 3 ; and ψ 1 ψ 3 . Therefore, if 1 P 2 and 2 P 3 , then 1 P 3 . Similarly, for the others. □
Definition 15.
Let  = ( [ φ L , φ U   ] ,   [ ψ L ,   ψ U ] ,   φ , ψ ) and  i = ( [ φ i L ,   φ i U ] ,   [ ψ i L , ψ i U ] ,   φ i ,   ψ i )   ( i = 1 , 2 ) be the collections of CFFNs, and ζ 0 be a real number then
(a)
1 2 = ( [ 1 i = 1 2 ( 1 ( φ i L ) 3 ) 3 , 1 i = 1 2 ( 1 ( φ i U ) 3 ) 3 ] ,   [ i = 1 2 ψ i L ,   i 2 ψ i U ]   , i = 1 2 φ i , 1 i = 1 2 ( 1 ( ψ i ) 3 )   3 ) ;
(b)
1 2 = ( [ i = 1 2 φ i L ,   i 2 φ i U ] ,   [ 1 i = 1 2 ( 1 ( ψ i L ) 3 ) 3 , 1 i = 1 2 ( 1 ( ψ i U ) 3 ) 3 ] , 1 i = 1 2 ( 1 ( φ i ) 3 ) 3 , i = 1 2 ψ i ) ;
(c)
ζ = ( [ 1 ( 1 ( φ L ) 3 ) ζ 3 ,   1 ( 1 ( φ U ) 3 ) ζ 3 ] ,   [ ( ψ L ) ζ ,   ( ψ U ) ζ ] , φ ζ , 1 ( 1 ψ 3 ) ζ 3 ) ;
(d)
ζ = ( [ ( φ L ) ζ ,   ( φ U ) ζ ] ,   [ 1 ( 1 ( ψ L ) 3 ) ζ 3 ,   1 ( 1 ( ψ U ) 3 ) ζ 3 ] , 1 ( 1 φ 3 ) ζ 3 , φ ζ ) .
Theorem 2.
For two CFFNs 1 = ( [ φ 1 L ,   φ 1 U ] ,   [ ψ 1 L , ψ 1 U ] ,   φ 1 ,   ψ 1 )  and 2 = ( [ φ 2 L ,   φ 2 U ] ,   [ ψ 2 L , ψ 2 U ] ,   φ 2 ,   ψ 2 ) , provided ζ 0 is a real number, then 1 2 , 1 2 , ζ , and ζ 1 are also CFFNs.
Proof. 
Since 1 = ( [ φ 1 L ,   φ 1 U ] ,   [ ψ 1 L , ψ 1 U ] ,   φ 1 ,   ψ 1 ) and 2 = ( [ φ 2 L ,   φ 2 U ] ,   [ ψ 2 L , ψ 2 U ] ,   φ 2 ,   ψ 2 ) are two CFFNs such that 0 φ 1 L ,   φ 1 U , ψ 1 L , ψ i U ,   φ 2 L , φ 2 U ,   ψ 2 L , ψ 2 U 1 and ( φ 1 U ) 3 + ( ψ 1 U ) 3 1 this implies that 0 ( 1 ( φ 1 L ) 3 ) ( 1 ( φ 1 L ) 3 ) 1 and hence 0 ( φ 1 L ) 3 + ( φ 2 L ) 3 ( φ 1 L ) 3 ( φ i L ) 3 3 1 . Similarly, we can prove that 0 ( φ 2 L ) 3 + ( φ 2 L ) 3 ( φ 2 L ) 3 ( φ 2 L ) 3 3 1 , 0 ψ 1 L ψ 2 L 1 and 0 ψ 1 U ψ 2 U 1 . We also set, 0 φ 1 ,   ψ 2 ,   φ 1 , ψ 2 1 and φ 1 3 + ψ 1 3 1 , φ 2 3 + ψ 2 3 1 , which implies that φ 1 φ 2 1 and φ 1 3 + φ 2 3 φ 1 3 μ 2 3 3 1 .
Finally, we have
( φ 1 U ) 3 + ( φ 2 U ) 3 ( φ 1 U ) 3 ( φ 2 U ) 3 + ( φ 1 U ) 3 ( φ 2 U ) 3 3 = 1 ( 1 ( φ 1 U ) 3 ) ( 1 ( φ 2 U ) 3 ) + ( φ 1 U ) 3 ( φ 2 U ) 3 3 1 ( φ 1 U ) 3 ( φ 2 U ) 3 + ( φ 1 U ) 3 ( φ 2 U ) 3 3 1 ,
and
φ 1 φ 2 + ψ 1 3 + ψ 2 3 ( ψ 1 U ) 3 ( ψ 1 U ) 3 3 = φ 1 φ 2 + 1 ( 1 ( φ 1 U ) 3 ) ( 1 ( φ 2 U ) 3 ) 3 φ 1 φ 2 + 1 φ 1 φ 2 3 1 .
Therefore, 1 2 is a CFFN. Furthermore, for any positive real number ψ and CFFN β = ( [ φ L ,   φ U ] ,   [ ψ L ,   ψ U ] ,   φ , ψ ) , we have 0 φ ζ 1 , 0 1 ( 1 ( φ ) 3 ) ζ 3 1 , 0 ( ψ L ) ζ ( ψ U ) ζ 1 and 0 1 ( 1 ( φ L ) 3 ) ζ ( 1 ( φ U ) 3 ) ζ 3 1 . Hence ζ is also a CFFN. Similarly, we can prove that 1 2 and ζ are also CFFNs. □

3.2. Cubic Fermatean AOs with Confidence Levels

In the available studies, all scholars have approached the studies by taking the postulation that decision-makers are confident in using the estimated objects. However, these kinds of prerequisites are only partially met in real-world situations. To address this problem, in this section we propose a set of averaging and geometric operators with different confidence levels in a cubic Fermatean fuzzy environment. Those are named confidence cubic Fermatean fuzzy weighted averaging (CCFFWA) operator and confidence cubic Fermatean fuzzy weighted geometric (CCFFWG) operator.

3.2.1. Weighted Averaging Operators

Definition 16.
A CCFFWA operator is a mapping  CCFFWA : Γ p Γ  defined as
CCFFWA ( 1 ,   2 , , p ) = σ 1 ( ξ 1 1 ) σ 2 ( ξ 2 2 ) σ p ( ξ p p )
where Γ is the collection of CPFNs with confidence level i = ( [ φ i L ,   φ i U ] ,   [ ψ i L , ψ i U ] ,   φ i ,   ψ i , σ i )   for i = 1 , 2 , , p ; ξ = ( ξ 1 ,   ξ 2 , , ξ p ) T is the weight vector of ξ i such that ξ i > 0 and i = 1 n ξ i = 1 ; and σ i are the confidence levels of the CFFNs i .
Theorem 3.
For the group of CCFFNs 1 ,   2 , , n , the value obtained via CCFFWA is a CFFN, which can be calculated as
CCFFWA ( 1 ,   2 , , p ) = ( [ 1 i = 1 p ( 1 ( φ i L ) 3 ) ξ i σ i 3 , 1 i = 1 p ( 1 ( φ i U ) 3 ) ξ i σ i 3 ] , [ i = 1 p ( ψ i L ) ξ i σ i , i = 1 p ( ψ i U ) ξ i σ i ] , i = 1 p ( φ i ) ξ i σ i , 1 i = 1 p ( 1 ( ψ i ) 3 ) ξ i σ i 3 ) .
Proof. 
We apply induction principle on 1 ,   2 , , P
Step 1 For p = 2 , using Definition 15, we get
CCFFWA ( 1 , 2 ) = σ 1 ξ 1 1 σ 2 ξ 2 2
= ( [ 1 ( 1 ( φ 1 L ) 3 ) σ 1 ξ 1 ( 1 ( φ 2 L ) 3 ) σ 2 ξ 2 3 , 1 ( 1 ( φ 1 U ) 3 ) σ 1 ξ 1 ( 1 ( φ 2 L ) 3 ) σ 2 ξ 2 3   ] , [ ( ψ 1 L ) σ 1 ξ 1 ( ψ 2 L ) σ 2 ξ 2 , ( ψ 1 U ) σ 1 ξ 1 ( ψ 2 U ) σ 2 ξ 2 ] , ( φ 1 ) σ 1 ξ 1 ( φ 2 ) σ 2 ξ 2 , 1 ( 1 ψ 1 3 ) σ 1 ξ 1 ( 1 ψ 1 3 ) σ 1 ξ 1 3 )
= ( [ 1 i = 1 2 ( 1 ( φ i L ) 3 ) ξ i σ i 3 , 1 i = 1 2 ( 1 ( φ i U ) 3 ) ξ i σ i 3 ] , [ i = 1 2 ( ψ i L ) ξ i σ i , i = 1 2 ( ψ i U ) ξ i σ i ] , i = 1 2 ( φ i ) ξ i σ i , 1 i = 1 2 ( 1 ( ψ i ) 3 ) ξ i σ i 3 ) .
Hence, it holds for P = 2 .
Step 2 Assume Equation (14) holds for p = κ , then
CCFFWA ( 1 ,   2 , , k ) = ( [ 1 i = 1 k ( 1 ( φ i L ) 3 ) ξ i σ i 3 , 1 i = 1 k ( 1 ( φ i U ) 3 ) ξ i σ i 3 ] , [ i = 1 k ( ψ i L ) ξ i σ i , i = 1 k ( ψ i U ) ξ i σ i ] , i = 1 k ( φ i ) ξ i σ i , 1 i = 1 k ( 1 ( ψ i ) 3 ) ξ i σ i 3 ) .
Step 3 For p = κ + 1 , we have
CCFFWA ( 1 ,   2 , , κ , κ + 1 ) = 1 2 . κ κ + 1
= ( [ 1 i = 1 κ ( 1 ( φ i L ) 3 ) ξ i σ i 3 , 1 i = 1 κ ( 1 ( φ i U ) 3 ) ξ i σ i 3 ] , [ i = 1 κ ( ψ i L ) ξ i σ i , i = 1 κ ( ψ i U ) ξ i σ i ] , i = 1 κ ( φ i ) ξ i σ i , 1 i = 1 κ ( 1 ( ψ i ) 3 ) ξ i σ i 3 ) ( [ 1 ( 1 ( φ κ + 1 L ) 3 ) σ κ + 1 ξ κ + 1 3 , 1 ( 1 ( φ κ + 1 U ) 3 ) σ κ + 1 ξ κ + 1 3   ] , [ ( ψ κ + 1 L ) σ κ + 1 ξ κ + 1 , ( ψ κ + 1 U ) σ κ + 1 ξ κ + 1 ] , ( φ κ + 1 ) σ κ + 1 ξ κ + 1 , 1 ( 1 ( ψ κ + 1 ) 3 ) σ κ + 1 ξ κ + 1 3 ) = ( [ 1 i = 1 κ + 1 ( 1 ( φ i 3 ) 3 ) ξ i σ i 3 , 1 i = 1 κ + 1 ( 1 ( φ i U ) 3 ) ξ i σ i 3 ] , [ i = 1 κ + 1 ( ψ i U ) ξ i σ i , i = 1 κ + 1 ( ψ i U ) ξ i σ i ] , i = 1 κ + 1 ( φ i ) ξ i σ i , 1 i = 1 κ + 1 ( 1 ( ψ i ) 3 ) ξ i σ i 3 ) .
As a result, the result is valid for p = κ + 1 , and hence
CCFFWA ( 1 ,   2 , , p ) = ( [ 1 i = 1 p ( 1 ( φ i L ) 3 ) ξ i σ i 3 , 1 i = 1 p ( 1 ( φ i U ) 3 ) ξ i σ i 3 ] , [ i = 1 p ( ψ i L ) ξ i σ i , i = 1 p ( ψ i U ) ξ i σ i ] , i = 1 p ( φ i ) ξ i σ i , 1 i = 1 p ( 1 ( ψ i ) 3 ) ξ i σ i 3 ) .
The proof is completed. □
Example 1.
Let  1 = ( [ 0.4 , 0.6 ] , [ 0.3.0.7 ] , ( 0.3 , 0.5 ) ; 0.8 ) , 2 = ( [ 0.5 , 0.6 ] , [ 0.4.0.5 ] , ( 0.2 , 0.4 ) ; 0.7 ) ,  and  3 = ( [ 0.2 , 0.3 ] , [ 0.4.0.5 ] , ( 0.7 , 0.2 ) ; 0.6 )  be three CFFNs with confidence levels and  ξ = ( 0.25 , 0.35 , 0.4 )  be their corresponding weight vector then
1 i = 1 3 ( 1 ( φ i L ) 3 ) ξ i σ i 3 = ( 1 ( ( 1 ( 0.4 ) 3 ) ( 0.25 ) ( 0.8 ) × ( 1 ( 0.5 ) 3 ) ( 0.35 ) ( 0.7 ) × ( 1 ( 0.2 ) 3 ) ( 0.4 ) ( 0.6 ) ) ) 1 3 = 0.3602 ;
1 i = 1 3 ( 1 ( φ i U ) 3 ) ξ i σ i 3 = ( 1 ( ( 1 ( 0.6 ) 3 ) ( 0.25 ) ( 0.8 ) × ( 1 ( 0.6 ) 3 ) ( 0.35 ) ( 0.7 ) × ( 1 ( 0.3 ) 3 ) ( 0.4 ) ( 0.6 ) ) ) 1 3 = 0.477 ;
i = 1 3 ( ψ i L ) ξ i σ i = ( 0.3 ) ( 0.25 ) ( 0.8 ) × ( 0.4 ) ( 0.35 ) ( 0.7 ) × ( 0.4 ) ( 0.4 ) ( 0.6 ) = 0.5040 ;
i = 1 3 ( ψ i L ) ξ i σ i = ( 0.7 ) ( 0.25 ) ( 0.8 ) × ( 0.5 ) ( 0.35 ) ( 0.7 ) × ( 0.6 ) ( 0.4 ) ( 0.6 ) = 0.6653 ;
i = 1 p ( φ i ) ξ i σ i = ( 0.3 ) ( 0.25 ) ( 0.8 ) × ( 0.2 ) ( 0.35 ) ( 0.7 ) × ( 0.7 ) ( 0.4 ) ( 0.6 ) = 0.4864 ;
1 i = 1 p ( 1 ( ψ i ) 3 ) ξ i σ i 3 = ( 1 ( ( 1 ( 0.5 ) 3 ) ( 0.25 ) ( 0.8 ) × ( 1 ( 0.4 ) 3 ) ( 0.35 ) ( 0.7 ) × ( 1 ( 0.2 ) 3 ) ( 0.4 ) ( 0.6 ) ) ) 1 / 3 = 0.3526 .
Thus, using Equation (14) we get
CCFFWA ( 1 ,   2 , 3 ) = ( [ 1 i = 1 3 ( 1 ( φ i L ) 3 ) ξ i σ i 3 , 1 i = 1 3 ( 1 ( φ i U ) 3 ) ξ i σ i 3 ] , [ i = 1 3 ( ψ i L ) ξ i σ i , i = 1 3 ( ψ i U ) ξ i σ i ] , i = 1 3 ( φ i ) ξ i σ i , 1 i = 1 3 ( 1 ( ψ i ) 3 ) ξ i σ i 3 ) = ( [ 0.3602 , 0.4770 ] , [ 0.5040.0.6653 ] , 0.4864 ,   0.3526 ) .
According to Theorem 3, the CCFFWA operator fulfils the certain properties listed below.
Property 1.
For  i =   i = 1 , 2 , , p , where  = ( [ φ L ,   φ u ] ,   [ ψ L ,   ψ U ] ,   φ , ψ ) , it follows that  CCFFWA ( 1 ,   2 , , n ) = . This property is called idempotency.
Proof. 
As ξ i 0 , I = 1 n ξ i = 1 and ξ i = ξ for all i , then
CCFFWA ( ,   , , ) = ( [ 1 i = 1 p ( 1 ( φ L ) 3 ) ξ i σ i 3 , 1 i = 1 p ( 1 ( φ U ) 3 ) ξ i σ i 3 ] , [ i = 1 p ( ψ L ) ξ i σ i , i = 1 p ( ψ U ) ξ i σ i ] , i = 1 p ( φ ) ξ i σ i , 1 i = 1 p ( 1 ( ψ ) 3 ) ξ i σ i 3 ) = ( [ 1 ( 1 ( φ L ) 3 ) ξ i σ i ,   1 ( 1 ( φ U ) 3 ) ξ i σ i ] ,   [ ( ψ L ) ξ i σ i , ( ψ U ) ξ i σ i   ] , ( φ ) ξ i σ i , 1 ( 1 ( ψ ) 3 ) ξ i σ i ) = ( [ φ L ,   φ u ] ,   [ ψ L ,   ψ U ] ,   φ , ψ ) = .
Property 2.
Let  i = ( [ φ i L ,   φ i U ] ,   [ ψ i L , ψ i U ] ,   φ i ,   ψ i )  and  ˜ i = ( [ φ ˜ i L ,   φ ˜ i U ] ,   [ ψ ˜ i L , ψ ˜ i U ] ,   φ ˜ i ,   ψ ˜ i )  be CCFFNs where ( i = 1 , 2 , , p ) , such that  i ˜ i , then
CPFWA ( 1 ,   2 , , n ) CCFFWA ( ˜ 1 ,   ˜ 2 , , ˜ p ) .
This property is called monotonicity.
Proof. 
First let us express the term of CCFFN as follows:
1 i = 1 p ( 1 ( φ i L ) 3 ) ξ i σ i 3 = α ,   1 i = 1 p ( 1 ( φ i U ) 3 ) ξ i σ i 3 = β , i = 1 p ( ψ i L ) ξ i σ i = γ ,   i = 1 p ( ψ i U ) ξ i σ i = δ , i = 1 p ( φ i ) ξ i σ i = ε ,   1 i = 1 p ( 1 ( ψ i ) 3 ) ξ i σ i 3 = ζ , 1 i = 1 p ( 1 ( φ ˜ i L ) 3 ) ξ i σ i 3 = α ˜ ,   1 i = 1 p ( 1 ( φ ˜ i U ) 3 ) ξ i σ i 3 = β ˜ , i = 1 p ( ψ ˜ i L ) ξ i σ i = γ ˜ ,   i = 1 p ( ψ ˜ i L ) ξ i σ i = δ ˜ , i = 1 p ( φ ˜ i ) ξ i σ i = ε ˜   and   1 i = 1 p ( 1 ( ψ ˜ i ) 3 ) ξ i σ i 3 = ζ ˜ .
Also, i ˜ i for all i , then we have φ i L φ ˜ i L , φ i U φ ˜ i U , ψ i L ψ ˜ i L , ψ i U ψ ˜ i U ; φ i φ ˜ i and ψ i ψ ˜ i , then we have α α ˜ , β β ˜ , γ γ ˜ , δ δ ˜ , ε ε ˜ , and ζ ζ ˜ . Therefore, using the score function defined in Definition 10 and 11, we have
s c ( CCFFWA ( 1 ,   2 , , p ) ) = α 3 + β 3 γ 3 δ 3 2 + ( ζ 3 ε 3 ) α ˜ 3 + β ˜ 3 γ ˜ 3 δ ˜ 3 2 + ( ζ ˜ 3 ε ˜ 3 ) = s c ( CCFFWA ( ˜ 1 ,   ˜ 2 , , ˜ p ) ) .
Thus, CCFFWA ( 1 ,   2 , , p ) CCFFWA ( ˜ 1 ,   ˜ 2 , , ˜ p ) . □
Property 3.
For any group of CCFFNs  i   ( i = 1 ,   2 , ,   p ) . If
= ( [ m i n i ( φ i L ) ,   m i n i ( φ i U ) ] ,   [ m a x i ( ψ i L ) ,   m a x i ( ψ i U ) ] ,   m a x i ( φ i ) ,   m i n i ( ψ i ) ) and + = ( [ m a x i ( φ i L ) , m a x i ( φ i U ) ] ,   [ m i n i ( ψ i L ) ,   m i n i ( ψ i U ) ] ,   m i n i ( φ i ) ,   m a x i ( ψ i ) )
then  CCFFWA ( 1 ,   2 , , n ) + . This property is called Boundedness.
Proof. 
As m i n i ( φ i L ) φ i L m a x i ( φ i L ) , m i n i ( φ i U ) φ i U m a x i ( φ i U ) , m i n i ( ψ i L ) ψ i L m a x i ( ψ i L ) , m i n i ( ψ i U ) ψ i U m a x i ( ψ i U ) , m i n i ( φ i ) φ i m a x i ( φ i ) , and m i n i ( ψ i ) ψ i m a x i ( ψ i ) it follows that
1 i = 1 n ( 1 m i n i ( φ i L ) 3 ) ξ i σ i 3 1 i = 1 n ( 1 ( φ i L ) 3 ) ξ i σ i 3 1 i = 1 n ( 1 m a x i ( φ i L ) 3 ) ξ i σ i 3 ; 1 i = 1 n ( 1 m i n i ( φ i U ) 3 ) ξ i σ i 3 1 i = 1 n ( 1 ( φ i U ) 3 ) ξ i σ i 3 1 i = 1 n ( 1 m a x i ( φ i U ) 3 ) ξ i σ i ; 3 i = 1 n m a x i ( ψ i L ) ξ i σ i i = 1 n ( ψ i L ) ξ i σ i i = 1 n m i n i ( ψ i L ) ξ i σ i ; i = 1 n m a x i ( ψ i U ) ξ i σ i i = 1 n ( ψ i U ) ξ i σ i i = 1 n m i n i ( ψ i U ) ξ i σ i ; i = 1 n m a x i ( φ i ) ξ i σ i i = 1 n ( φ i ) ξ i σ i i = 1 n m i n i ( φ i ) ξ i σ i ; 1 i = 1 n ( 1 m i n i ( ψ i ) 3 ) ξ i σ i 3 1 i = 1 n ( 1 ( ψ i ) 3 ) ξ i σ i 3 1 i = 1 n ( 1 m a x i ( ψ i ) 3 ) ξ i σ i 3
which implies that
m i n i ( φ i L ) 3 1 i = 1 n ( 1 ( φ i L ) 3 ) ξ i σ i 3 m a x i ( φ i L ) 3 ; m i n i ( φ i U ) 3 1 i = 1 n ( 1 ( φ i U ) 3 ) ξ i σ i 3 m a x i ( φ i U ) 3 ; m a x i ( ψ i L ) i = 1 n ( ψ i L ) ξ i σ i m i n i ( ψ i L ) ; m a x i ( ψ i U ) i = 1 n ( ψ i U ) ξ i σ i m i n i ( ψ i U ) ; m a x i ( φ i ) i = 1 n ( φ i ) ξ i σ i m i n i ( φ i ) ; m i n i ( ψ i ) 3 1 i = 1 n ( 1 ( ψ i ) 3 ) ξ i σ i 3 m a x i ( ψ i ) 3 .
Thus, CCFFWA ( 1 ,   2 , , n ) + . □
Property 4.
For the CCFFNs  1 ,   2 , ,   p  and  ˜ = ( [ φ ˜ ˜ L ,   φ ˜ ˜ U ] ,   [ ψ ˜ ˜ L , ψ ˜ ˜ U ] ,   φ ˜ ˜ ,   ψ ˜ ˜ ) , we have
CCFFWA ( 1 ˜ 2 ˜ p ˜ ) = CCFFWA ( 1 , 2 , , p ) ˜ .
Proof. 
Straightforward. □
Property. 5.
For a positive real number  ζ , we have
CCFFWA ( ζ 1 , ζ   2 , , ζ p ) = ζ CCFFWA ( 1 ,   2 , , p ) .
Proof. 
Straightforward. □

3.2.2. Ordered weighted Averaging Operator

Definition. 17.
A CCFFOWA is a mapping defined as  CCFFOWA :   Γ n Γ  on a collection of CPFNs  i , ( i = 1 , 2 , p )  as follows
CCFFOWA ( 1 ,   2 , ,   p ) = ξ 1 σ 1 δ ( 1 ) ξ 2 σ 2 β δ ( 2 )   ξ p σ p σ ( p )
where δ is a permutation of ( 1 ,   2 , , n ) , such that δ ( i 1 ) i for i = 1 , 2 , , p and ξ = ( ξ 1 , ξ 2 , ξ n ) T is its weight vector, such that ξ 0 and i = 1 p ξ i = 1 with confidence levels 0 σ i 1 . Furthermore, theith largest CFFN among i ' s is δ ( i ) .
Theorem. 4.
The value obtained by using the CCFFOWA operator for CFFNs i   ( i = 1 , 2 , , p ) is again a CFFN and given by
CCFFWA ( 1 ,   2 , , n ) = ( [ 1 i = 1 p ( 1 ( φ δ ( i ) L ) 3 ) ξ i σ i 3 , 1 i = 1 p ( 1 ( φ δ ( i ) U ) 3 ) ξ i σ i 3 ] , [ i = 1 p ( ψ δ ( i ) L ) ξ i σ i , i = 1 p ( ψ δ ( i ) U ) ξ i σ i ] , i = 1 p ( φ δ ( i ) ) ξ i σ i , 1 i = 1 p ( 1 ( ψ δ ( i ) ) 3 ) ξ i σ i 3 ) .
Proof. 
Similar proof as Theorem 3. □
Example 2.
Let  1 = ( [ 0.3 , 0.4 ] , [ 0.2.0.3 ] , ( 0.2 , 0.6 ) ; 0.5 ) , 2 = ( [ 0.4 , 0.5 ] , [ 0.3.0.4 ] , ( 0.6 , 0.2 ) ; 0.4 ) ,  and  3 = ( [ 0.6 , 0.7 ] , [ 0.5.0.6 ] , ( 0.4 , 0.3 ) ; 0.7 )  be three CFFNs with confidence levels, and  ξ = ( 0.5 , 0.3 , 0.2 )  be their corresponding weight vector. By using Equations (10) and (11) to calculate the score values of each CFFN it follows that
s c ( 1 ) = ( 0.3 ) 3 + ( 0.4 ) 3 ( 0.2 ) 3 ( 0.3 ) 3 2 + ( ( 0.6 ) 3 ( 0.2 ) 3 ) = 0.2360 ; s c ( 2 ) = ( 0.4 ) 3 + ( 0.5 ) 3 ( 0.3 ) 3 ( 0.4 ) 3 2 + ( ( 0.2 ) 3 ( 0.6 ) 3 ) = 0.1590 ; s c ( 3 ) = ( 0.6 ) 3 + ( 0.7 ) 3 ( 0.5 ) 3 ( 0.6 ) 3 2 + ( ( 0.3 ) 3 ( 0.4 ) 3 ) = 0.0720 .
The order of these CFFNs with respect to score values is 1 3 2 .
Arrange these CFFNs with respect to score values, i.e.,
1 = ( [ 0.3 , 0.4 ] , [ 0.2.0.3 ] , ( 0.2 , 0.6 ) ; 0.5 ) , 3 = ( [ 0.6 , 0.7 ] , [ 0.5.0.6 ] , ( 0.4 , 0.3 ) ; 0.7 ) ;
and
2 = ( [ 0.4 , 0.5 ] , [ 0.3.0.4 ] , ( 0.6 , 0.2 ) ; 0.4 ) .
Therefore, δ ( 1 ) = 1 , δ ( 2 ) = 3 , and δ ( 3 ) = 2 .
Now, we have
1 i = 1 3 ( 1 ( φ δ ( i ) L ) 3 ) ξ i σ i 3 = ( 1 ( ( 1 ( 0.3 ) 3 ) ( 0.5 ) ( 0.5 ) × ( 1 ( 0.6 ) 3 ) ( 0.3 ) ( 0.7 ) × ( 1 ( 0.4 ) 3 ) ( 0.2 ) ( 0.4 ) ) ) 1 / 3 = 0.3942 ; 1 i = 1 3 ( 1 ( φ δ ( i ) U ) 3 ) ξ i σ i 3 = ( 1 ( ( 1 ( 0.4 ) 3 ) ( 0.5 ) ( 0.5 ) × ( 1 ( 0.7 ) 3 ) ( 0.3 ) ( 0.7 ) × ( 1 ( 0.5 ) 3 ) ( 0.2 ) ( 0.4 ) ) ) 1 / 3 = 0.4777 ; i = 1 3 ( ψ δ ( i ) L ) ξ i σ i = ( 0.2 ) ( 0.5 ) ( 0.5 ) × ( 0.5 ) ( 0.3 ) ( 0.7 ) × ( 0.3 ) ( 0.2 ) ( 0.4 ) = 0.5251 ; i = 1 3 ( ψ δ ( i ) L ) ξ i σ i = ( 0.3 ) ( 0.5 ) ( 0.5 ) × ( 0.6 ) ( 0.3 ) ( 0.7 ) × ( 0.4 ) ( 0.2 ) ( 0.4 ) = 0.6178 ; i = 1 p ( φ δ ( i ) ) ξ i σ i = ( 0.2 ) ( 0.5 ) ( 0.5 ) × ( 0.4 ) ( 0.3 ) ( 0.7 ) × ( 0.6 ) ( 0.2 ) ( 0.4 ) = 0.5296 ; 1 i = 1 p ( 1 ( ψ δ ( i ) ) 3 ) ξ i σ i 3 = ( 1 ( ( 1 ( 0.6 ) 3 ) ( 0.5 ) ( 0.5 ) × ( 1 ( 0.3 ) 3 ) ( 0.3 ) ( 0.7 ) × ( 1 ( 0.2 ) 3 ) ( 0.2 ) ( 0.4 ) ) ) 1 / 3 = 0.4021 .
Hence,
CCFFOWA ( 1 ,   2 , 3 ) = ( [ 0.3942 , 0.4777 ] , [ 0.5251.0.6178 ] , 0.5296 ,   0.4021 ) .

3.2.3. Geometric Operator

Definition 18.
A CCFFWG operator is a mapping  CCFFWG : Γ n Γ  defined as
CCFFWG ( 1 ,   2 , , p ) = 𝓌 1 1 𝓌 2 2 𝓌 p p
where  Ω  is the collection of CPFNs  i ( i = 1 , 2 , , p ) , and  ξ = ( ξ 1 ,   ξ 2 , , ξ p ) T  is the weight vector of i such that  ξ i 0  and  i = 1 n ξ i = 1 . We also set, σ p be the confidence levels of CFFN p .
Theorem 5.
For  1 ,   2 , , n , the value obtained by CCFFWG is a CFFN, which is determined by
CCFFWG ( 1 ,   2 , , n ) = ( [ i = 1 p ( φ i L ) ξ i σ i , i = 1 p ( φ i U ) ξ i σ i ] , [ 1 i = 1 p ( 1 ( ψ i L ) 3 ) ξ i σ i 3 , 1 i = 1 p ( 1 ( ψ i U ) 3 ) ξ i σ i 3 ] , 1 i = 1 p ( 1 ( φ i ) 3 ) ξ i σ i 3 , i = 1 p ( ψ i ) ξ i σ i ) .
Proof. 
Similar to Theorem 3, therefore omitted here. □
Example 3.
Let  1 = ( [ 0.4 , 0.6 ] , [ 0.3.0.7 ] , ( 0.3 , 0.5 ) ; 0.8 ) , 2 = ( [ 0.5 , 0.6 ] , [ 0.4.0.5 ] , ( 0.2 , 0.4 ) ; 0.7 ) ,  and  3 = ( [ 0.2 , 0.3 ] , [ 0.4.0.5 ] , ( 0.7 , 0.2 ) ; 0.6 )  be three CFFNs with confidence levels and  ξ = ( 0.25 , 0.35 , 0.4 )  be their corresponding weight vector then
i = 1 3 ( φ i L ) ξ i σ i = ( 0.4 ) ( 0.25 ) ( 0.8 ) × ( 0.5 ) ( 0.35 ) ( 0.7 ) × ( 0.2 ) ( 0.4 ) ( 0.6 ) = 0.4774 ; i = 1 3 ( φ i L ) ξ i σ i = ( 0.6 ) ( 0.25 ) ( 0.8 ) × ( 0.6 ) ( 0.35 ) ( 0.7 ) × ( 0.3 ) ( 0.4 ) ( 0.6 ) = 0.5967 ; 1 i = 1 3 ( 1 ( ψ i L ) 3 ) ξ i σ i 3 = ( 1 ( ( 1 ( 0.3 ) 3 ) ( 0.25 ) ( 0.8 ) × ( 1 ( 0.4 ) 3 ) ( 0.35 ) ( 0.7 ) × ( 1 ( 0.4 ) 3 ) ( 0.4 ) ( 0.6 ) ) ) 1 3 = 0.3328 ; 1 i = 1 3 ( 1 ( ψ i U ) 3 ) ξ i σ i 3 = ( 1 ( ( 1 ( 0.7 ) 3 ) ( 0.25 ) ( 0.8 ) × ( 1 ( 0.5 ) 3 ) ( 0.35 ) ( 0.7 ) × ( 1 ( 0.5 ) 3 ) ( 0.4 ) ( 0.6 ) ) ) 1 3 = 0.5171 ; 1 i = 1 p ( 1 ( φ i ) 3 ) ξ i σ i 3 = ( 1 ( ( 1 ( 0.3 ) 3 ) ( 0.25 ) ( 0.8 ) × ( 1 ( 0.2 ) 3 ) ( 0.35 ) ( 0.7 ) × ( 1 ( 0.7 ) 3 ) ( 0.4 ) ( 0.6 ) ) ) 1 3 = 0.4682 : i = 1 p ( ψ i ) ξ i σ i = ( 0.5 ) ( 0.25 ) ( 0.8 ) × ( 0.4 ) ( 0.35 ) ( 0.7 ) × ( 0.2 ) ( 0.4 ) ( 0.6 ) = 0.4727 .
Hence, we have
CCFFWG ( 1 ,   2 , 3 ) = ( [ 0.4774 , 0.5967 ] , [ 0.3328 ,   0.5171 ] , 0.4682 ,   0.4727 ) .

3.2.4. Ordered Weighted Geometric Operator

Definition 19.
A CPFOWG is a mapping  CPFOWG :   Γ n Γ  defined over a collection of CCFFNs  i  with confidence levels  σ i   ( i = 1 , 2 , p )  as follows
CCFFOWG ( 1 ,   2 , ,   p ) = σ I ξ 1 σ ( 1 ) σ 2 ξ 2 σ ( 2 ) σ p ξ p δ ( p )
where  δ  is a permutation of ( 1 ,   2 , , p ) , such that δ ( i 1 ) i for i = 1 , 2 , , n and ξ = ( ξ 1 , ξ 2 , ξ p ) T is its weight vector, such that ξ 0 and i = 1 n ξ i = 1 . Moreover, theith largest CFFN among i s is δ ( i ) .
Theorem 6.
The value obtained by using the CPFOWG operator for CFFNs i   ( i = 1 , 2 , , p ) is again a CFFN and given by
CCFFWG ( 1 ,   2 , , n ) = ( [ i = 1 p ( φ δ ( i ) L ) ξ i σ i , i = 1 p ( φ δ ( i ) U ) ξ i σ i ] , [ 1 i = 1 p ( 1 ( φ δ ( i ) L ) 3 ) ξ i σ i 3 , 1 i = 1 p ( 1 ( φ δ ( i ) U ) 3 ) ξ i σ i 3 ] , 1 i = 1 p ( 1 ( φ δ ( i ) ) 3 ) ξ i σ i 3 , i = 1 p ( ψ δ ( i ) ) ξ i σ i ) .
Theorem 7.
Let  i ( i = 1 , 2 , , p ) , a n d   ξ = ( ξ 1 ,   ξ 2 , , ξ p ) T  be the weight vector of i such that  ξ i 0  and  i = 1 p ξ i = 1 , then we have
  • CCFFWA ( 1 c ,   2 c , , p c ) = ( CPFWG ( 1 ,   2 , , p ) ) c ;
  • CCFFWG ( 1 c ,   2 c , , p c ) = ( CPFWA ( 1 ,   2 , , p ) ) c .
Proof. 
Since i = ( [ φ i L ,   φ i U ] ,   [ ψ i L , ψ i U ] ,   φ i ,   ψ i ) and i c = (   [ ψ i L , ψ i U ] ,   [ φ i L ,   φ i U ] ,   ψ i ,   φ i ) , then using Equation (17), we have
CCFFWA ( 1 c ,   2 c , , p c ) = ( [ i = 1 n ( ψ i L ) ξ i σ i , i = 1 n ( ψ i L ) ξ i σ i ] ,   [ 1 i = 1 n ( 1 ( φ i L ) 2 ) ξ i σ i 3 , 1 i = 1 n ( 1 ( φ i L ) 2 ) ξ i σ i 3 ] , 1 i = 1 n ( 1 ( ψ i ) 2 ) ξ i σ i 3 , i = 1 n ( φ i ) ξ i σ i ) = ( CCFFWG ( 1 ,   2 , , p ) ) c .
Similarly, we can prove that CCFFWG ( 1 c ,   2 c , , p c ) = ( CCPFWA ( 1 ,   2 , , p ) ) c . □
Theorem 8.
Let  i ( i = 1 , 2 , , p ) , and ξ = ( ξ 1 ,   ξ 2 , , ξ p ) T  be the weight vector of i such that  ξ i 0  and  i = 1 p ξ i = 1 , then we have
CCFFWG ( 1 ,   2 , , p ) CCFFWA ( 1 ,   2 , , p )
Proof. 
Easy to prove. □
Definition. 20.
For the CFFNs  i   ( i = 1 ,   2 , , p )  the operator  CCFFHA :   Γ n Γ  is given as
CCFFHA ( 1 ,   2 , , p ) = σ 1 ξ 1 ˙ σ ( 1 ) σ 2 ξ 2 ˙ σ ( 2 ) σ p ξ p ˙ σ ( p )
where, ξ = ( ξ 1 ,   ξ 2 , , ξ p ) T  be the weight vector, such that  ξ i 0  and  i = 1 n ξ i = 1  and  ˙ i s   ( ˙ i = n ζ i i ) is ˙ σ ( i ) , where  n  is the number of CPFNs and  η = ( η 1 ,   η 2 , , η p ) T  is the vector corresponding to  i  with  ζ i 0  and  i = 1 p ζ i = 1 .
Theorem. 9.
The value obtained using the CCFFHA operator for the CFFNs  i   ( i = 1 , 2 , , p )  is again a CFFN and given by
CCFFHA ( 1 ,   2 , , n ) = ( [ 1 i = 1 p ( 1 ( φ ˙ δ ( i ) L ) 2 ) ξ i σ i 3 , 1 i = 1 p ( 1 ( μ ˙ δ ( i ) U ) 2 ) ξ i σ i 3 ] , [ i = 1 n