# New Operators of Cubic Picture Fuzzy Information with Applications

Authors: Tehreem; Abdu Gumaei; Amjad Hussain
Journal: Journal of Mathematics
 (2021)

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## Abstract

The researcher has been facing problems while handling imprecise and vague information, i.e., the problems of networking, decision-making, etc. For encountering such complicated data, the notion of fuzzy sets (FS) has been considered an influential tool. The notion was extended to its generalizations by a number of researchers in different ways which helps to understand and assess even more complex issues. This article characterizes imprecision with four kinds of values of membership. In this work, we aim to define and examine cubic picture fuzzy sets and give an application on averaging aggregation operators. We first introduce the notion of a cubic picture fuzzy set, which is a pair of interval-valued picture fuzzy set and a picture fuzzy set by giving examples. Then, we define two kinds of ordering on these sets and also discuss some set-theoretical properties. Moreover, we introduce three kinds of averaging aggregation operators based on cubic picture fuzzy sets and, at the end, we illustrate the results with a decision-making problem by using one of the provided aggregation operators.


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## Body

## 1. Introduction

In 1965, Zadeh generalized the classical set and perceived the idea of fuzzy sets [1] to deal with uncertainty. This idea allows creating some new dimensions in the field of research and has been applied in many fields such as decision-making, medical diagnosis, and pattern recognition [1–6]. But in fuzzy set only the membership degree is considered. The limitation of fuzzy sets is that the nonmembership degree cannot be defined independently. To overcome this limitation, several extensions have been made by many researchers such as interval-valued fuzzy sets [7], intuitionistic fuzzy sets (IFSs) [8], cubic sets [9], and neutrosophic sets [10]. Among these various extensions of fuzzy sets, cubic set is one of the most prominent extensions. Jun [9] presented the idea of cubic sets in terms of interval-valued fuzzy set and fuzzy set in 2012. The very basic properties of cubic sets were studied, and some useful operations were defined successfully in his paper. Khir et al. [11] presented the idea of fuzzy sets and fuzzy logic and their application. Later on, the idea of cubic sets was applied to various fields by many authors (see [12–17]).In recent years, the notion of fuzzy sets was further generalized by Coung et al. and they proposed the concept of picture fuzzy sets [18, 19], and this idea gained more and more attention from the researchers. Several similarity measures, correlation coefficients, and entropy measures for picture fuzzy sets were defined by many authors and they applied these sets in various fields (see [20–28]). Recently, Coung et al. [29] have extended the picture fuzzy sets to the interval-valued picture fuzzy sets. For some works on picture fuzzy sets and several types of aggregation operators, we refer the reader to [24, 25, 30–35].Inspiring from the above study, we propose the concept of cubic picture fuzzy sets, which is an extension of cubic sets, picture fuzzy sets, and interval-valued picture fuzzy sets.The rest of the paper is organized as follows. In Section2, some basic definitions and results which are necessary for the main sections are discussed. In Section 3, the concept of cubic picture fuzzy sets which is a mixture of an interval-valued picture fuzzy set and a picture fuzzy set is introduced, and some basic operations on these sets were defined by giving several examples. Then the related theorems are studied. In Section 4, three types of aggregation operators in the environment of cubic picture fuzzy sets are discussed and, finally, one of them is applied in decision-making problem in the last section.


## 2. Preliminaries

Definition 1.
(see [1]). Let S⌣˙ be a nonempty set. Then U=s⌣˙,UUs⌣˙|s⌣˙∈S⌣˙ is called a fuzzy set, where UU is a membership function that maps each element of S˙˙ in0,1. Here we say that U is a fuzzy subset of S⌣˙.Definition 2.
(see [8]). Consider closed subinterval UU=UU−,UU+ of I where I=0,1 is called an interval number, where 0≤UU−≤UU+≤1. The set of all interval numbers is denoted by I. A function β:S⌣˙⟶I is said to be an interval-valued fuzzy (IVF) set of S⌣˙. The set of all IVF sets of S⌣˙ is denoted by Is⌣˙. For each UU∈Is⌣˙ ands⌣˙∈S⌣˙, UUs⌣˙=UU−s⌣˙,UU+s⌣˙ is called the degree of membership of an element s⌣˙ to UU; in this case UU−:S⌣˙⟶I and UU+:S⌣˙⟶I are fuzzy subsets of S⌣˙; these sets are known as lower fuzzy set and upper fuzzy subset of S⌣˙, respectively.Definition 3.
(see [9]). The cubic set of a nonempty set S⌣˙ is defined as follows: I^=s⌣˙,A^˙s⌣˙,ℬs⌣˙|s⌣˙∈S⌣˙, where A^˙ is an interval-valued fuzzy (IVF) set of S⌣˙ and ℬ is a fuzzy subset of S⌣˙. A cubic set is simply denoted by I^=A^˙,ℬ.Definition 4.
(see [9]). A cubic set I^=A^˙,ℬ is known to be(1)
an internal cubic set (briefly, ICS) ifA^˙−s⌣˙≤ℬs⌣˙≤A^˙+s⌣˙,∀s⌣˙∈S⌣˙(2)
an external cubic set (briefly, ECS) ifℬs⌣˙∉A^˙−s⌣˙,A^˙+s⌣˙,∀s⌣˙∈S⌣˙Example 1.
IfA^˙ is an IVF set of S⌣˙, then U=s⌣˙,A^˙s⌣˙,1s⌣˙|s⌣˙∈S⌣˙,V=s⌣˙,A^˙s⌣˙,0s⌣˙>|s⌣˙∈S⌣˙ and C=s⌣˙,A^˙s⌣˙,ℬs⌣˙|s⌣˙εS⌣˙, where λs⌣˙=A^˙−s⌣˙+A^˙+s⌣˙/2 are cubic sets of S⌣˙.Example 2.
LetU=s⌣˙,A^˙s⌣˙,1s⌣˙|s⌣˙∈S⌣˙ be a cubic set of S⌣˙ and A^˙s⌣˙=0.3,0.7 and ℬs⌣˙=0.4, for each s⌣˙∈S⌣˙. Then U is an ICS. If A^˙s⌣˙=0.3,0.7 and ℬs⌣˙=0.8, for each s⌣˙∈S⌣˙, then U is an ECS. If A^˙s⌣˙=0.3,0.7andℬs⌣˙=s⌣˙, for each s⌣˙∈S⌣˙, then U is neither an ICS nor an ECS.Definition 5.
(see [9]). Let S⌣˙ be a nonempty set and let U=A^˙,ℬ and V=J,K be two cubic sets of S⌣˙. Then the orderings are defined in the following way:(1)
(Equality)U=V⇔A^˙=J and ℬ=K(2)
(P-Order)U⊆PV⇔A^˙⊆J and ℬ≤K(3)
(R-Order)U⊆RV⇔A^˙⊆J and ℬ≥KDefinition 6.
(see [9]). For arbitrary indexed family of cubic sets Ui=s⌣˙,A^˙is⌣˙,ℬis⌣˙|s⌣˙εS⌣˙, where iε∧, we define the P-union, P-intersection, R-union, and R-intersection as follows:(1)
∪P,iε∧Ui=s⌣˙,∪iε∧Iis⌣˙,∨i∈Γαis⌣˙|s⌣˙∈S⌣˙P−union(2)
∩P,iε∧Ui=s⌣˙,∩iε∧Iis⌣˙,∧iε∧αis⌣˙|s⌣˙∈S⌣˙P−intersection(3)
∪R,iε∧Ui=s⌣˙,∩iε∧Iis⌣˙,∧iε∧αis⌣˙|s⌣˙∈S⌣˙R−union(4)
∩R,iε∧Ui=s⌣˙,∩iε∧Iis⌣˙,∨i∈Λαis⌣˙|s⌣˙∈S⌣˙R−intersection
The complement ofU=I,α is also a cubic set that is defined by Uc=s⌣˙,Ics⌣˙,1−αs⌣˙|s⌣˙εS⌣˙. Obviously, Ucc=U for any indexed family of cubic sets Ui=s⌣˙,Iis⌣˙,αis⌣˙|s⌣˙εS⌣˙iε∧:(1)
∪P,iε∧Uic=∩P,iε∧Uic and ∩P,iε∧Uic=∪P,iε∧Uic(2)
∪R,iε∧Uic=∩R,iε∧Uic and ∩R,iε∧Ui=∪R,iε∧UicDefinition 7.
(see [18, 19]). A picture fuzzy set (briefly, PFS) U of a universe S⌣˙ is an object in the form of s⌣˙,UUs⌣˙,ρUs⌣˙,ϑUs⌣˙|s⌣˙∈S⌣˙, where UU,ρU,ϑU:S⌣˙⟶0,1 are fuzzy sets that satisfy 0≤UUs⌣˙+ρUs⌣˙+ϑUs⌣˙≤1 for each s⌣˙∈S⌣˙. Then the values UUs⌣˙,ρUs⌣˙,ϑUs⌣˙ are called the degree of positive membership of s⌣˙ in U, the degree of neutral membership of s⌣˙ in U, and the degree of negative membership of s⌣˙ in U, respectively. Now 1−UUs⌣˙+ρUs⌣˙+ϑUs⌣˙ could be called the degree of refusal membership of s⌣˙ inU. Let PFSS⌣˙ represent the set of all picture fuzzy sets of a universeS⌣˙.Definition 8.
(see [18, 19]). Let U and V be the PFSs. Then the set of operations are defined as follows:(1)
U⊆Viff∀s⌣˙∈S⌣˙,UUs⌣˙≤UVs⌣˙andρUs⌣˙≤ρVs⌣˙andϑUs⌣˙≤ϑVs⌣˙(2)
U=ViffU⊆VandV⊆U(3)
U∪V=s⌣˙,MaxUUs⌣˙,UVs⌣˙,MinρUs⌣˙,ρVs⌣˙,MinϑUs⌣˙,ϑVs⌣˙|s⌣˙∈S⌣˙(4)
U∩V=s⌣˙,MinUUs⌣˙,UVs⌣˙,MinρUs⌣˙,ρVs⌣˙,MaxϑUs⌣˙,ϑVs⌣˙|s⌣˙∈S⌣˙
Now, a generalization of interval-valued fuzzy setU is proposed. Here int0,1 stands for the set of all closed subintervals of 0,1.Definition 9.
(see [29]). An interval-valued picture fuzzy set (briefly, IVPFS) U of a universe S⌣˙ is an object in the following form: U=MUs⌣˙,LUs⌣˙,NUs⌣˙|s⌣˙∈S⌣˙, where(1)MU:S⌣˙⟶int0,1,MUs⌣˙=MULs⌣˙,MUUs⌣˙∈int0,1,LU:S⌣˙⟶int0,1,LUs⌣˙=LULs⌣˙,LUUs⌣˙∈int0,1,NU:S⌣˙⟶int0,1,NUs⌣˙=NULs⌣˙,NUUs⌣˙∈int0,1.
The following condition is satisfied:supMUs⌣˙+supLUs⌣˙+supNUs⌣˙≤1,∀s⌣˙∈S⌣˙. The IVPFSS⌣˙ denotes the set of all interval-valued picture fuzzy sets of S⌣˙.


## 3. Cubic Picture Fuzzy Sets

In this section, we propose the notion of a cubic picture fuzzy set and investigate its set-theoretical operations and some basic properties by giving illustrative examples.Definition 10.
A cubic picture fuzzy set (briefly, CPFS) ofS⌣˙ is denoted and defined by CP=s⌣˙,A^˙s⌣˙,ℬs⌣˙|s⌣˙∈S⌣˙, where A^ is an interval-valued picture fuzzy set and ℬ is a picture fuzzy set of S⌣˙. A CPFS CP=s⌣˙,A˙^s⌣˙,ℬs⌣˙|s⌣˙∈S⌣˙ is simply denoted by CP=A^˙,ℬ.Definition 11.
A cubic picture fuzzy setCP=A^˙,ℬ of S⌣˙ is said to be(1)
positive internal CPFS ifℬ1−s⌣˙≤ℬ1s⌣˙≤ℬ1+s⌣˙, where ℬ1−s⌣˙,ℬ1+s⌣˙ are the lower and the upper positive degrees inS⌣˙, respectively(2)
negative internal CPFS ifℬ2−s⌣˙≤ℬ2s⌣˙≤ℬ2+s⌣˙, where ℬ2−s⌣˙,ℬ2+s⌣˙ are the lower and the upper negative degrees inS⌣˙, respectively(3)
indeterminacy internal CPFS ifℬ3−s⌣˙≤ℬ3s⌣˙≤ℬ3+s⌣˙, where ℬ3−s⌣˙,ℬ3+s⌣˙ are the lower and the upper indeterminacy degrees inS⌣˙, respectively
When conditions (1), (2), and (3) hold, then it is called an internal cubic picture fuzzy set (ICPFS) inS⌣˙.Definition 12.
A cubic picture fuzzy setCP=A^˙,ℬ is said to be an external cubic picture fuzzy set (ECPFS) if ℬ1s⌣˙∉ℬ1−s⌣˙,ℬ1+s⌣˙,ℬ2s⌣˙∉ℬ2−s⌣˙,ℬ2+s⌣˙andℬ3s⌣˙∉ℬ3−s⌣˙,ℬ3+s⌣˙ for all s⌣˙∈S⌣˙.Example 3.
LetS⌣˙=s⌣˙1,s⌣˙2,s⌣˙3 be given. Then, the CPFS,(2)U=s⌣˙1,0.1,0.3,0.3,0.4,0.1,0.3,0.2,0.3,0.2,s⌣˙2,0.1,0.3,0.3,0.5,0.0,0.2,0.2,0.4,0.1,s⌣˙3,0.1,0.3,0.1,0.4,0.0,0.2,0.2,0.3,0.1,is an internal cubic picture fuzzy set of S⌣˙.Example 4.
LetS⌣˙=s⌣˙1,s⌣˙2,s⌣˙3 be given; then the CPFS,(3)U=s⌣˙1,0.2,0.4,0.0,0.2,0.1,0.3,0.4,0.3,0.0,s⌣˙2,0.1,0.3,0.1,0.4,0.0,0.3,0.0,0.5,0.4,s⌣˙3,0.1,0.3,0.2,0.5,0.0,0.2,0.4,0.1,0.3,is an external cubic picture fuzzy set of S⌣˙.Theorem 1.
IfCP=A^˙,ℬ is a cubic picture fuzzy set, which is not an ECPFS, then there exists s⌣˙∈S⌣˙ such that ℬ1s⌣˙∈ℬ1−s⌣˙,ℬ1+s⌣˙,ℬ2s⌣˙∈ℬ2−s⌣˙,ℬ2+s⌣˙, and ℬ3s⌣˙∈ℬ3−s⌣˙∩ℬ3+s⌣˙.Proof.
The proof is straightforward and therefore is omitted.Theorem 2.
IfCP=A^,ℬℝ is both ICPFS and ECPFS, then the following is satisfied for each s⌣˙∈S⌣˙: ℬ1s⌣˙∈ℬ1−s⌣˙∪ℬ1+s⌣˙, ℬ2s⌣˙∈ℬ2−s⌣˙∪ℬ2+s⌣˙, and ℬ3s⌣˙∈ℬ3−s⌣˙∪ℬ3+s⌣˙.Proof.
Assume thatCP is both ICPFS and ECPFS. Then, by using the definitions of ICPFS and ECPFS, we have ℬ1−s⌣˙≤ℬ1s⌣˙≤ℬ1+s⌣˙ and ℬ1s⌣˙∉ℬ1−s⌣˙,ℬ1+s⌣˙,ℬ2−s⌣˙≤ℬ2s⌣˙≤ℬ2+s⌣˙ and ℬ2s⌣˙∉ℬ2−s⌣˙,ℬ2+s⌣˙,ℬ3−s⌣˙≤ℬ3s⌣˙≤ℬ3+s⌣˙ and ℬ3s⌣˙,∉ℬ3−s⌣˙,ℬ3+s⌣˙ for all s⌣˙∈S⌣˙. Thus ℬ1s⌣˙=ℬ1−s⌣˙ or ℬ1s⌣˙=ℬ1+s⌣˙, implying that ℬ1s⌣˙∈ℬ1−s⌣˙∪ℬ1+s⌣˙,ℬ2s⌣˙=ℬ2−s⌣˙ or ℬ2s⌣˙=ℬ1+s⌣˙ implying that ℬ2s⌣˙∈ℬ2−s⌣˙∪ℬ2+s⌣˙,ℬ3s⌣˙=ℬ3−s⌣˙ or ℬ3s⌣˙=ℬ3+s⌣˙ implying that ℬ3s⌣˙∈ℬ3−s⌣˙∪ℬ3+s⌣˙.Definition 13.
IfU=CP=A^,ℬ and V=Sp=J,K are the cubic picture fuzzy sets, then equality, P-order, and R-order are defined as follows:(1)
(Equality)U=V⇔A^=J and ℬ=K(2)
(P-order)U⊆PV⇔A^⊆J and ℬ≤K(3)
(R-order)U⊆RV⇔A^⊆J and ℬ≥KDefinition 14.
For any indexed family of CPFSsUi=s⌣˙,A^˙is⌣˙,ℬis⌣˙|s⌣˙∈S⌣˙i∈Λwe define the following:(1)
∪PUi=s⌣˙,∪i∈IA^˙is⌣˙,∨i∈Iℬis⌣˙/s⌣˙∈S⌣˙ (P-union)(2)
∩PUi=s⌣˙,∩i∈IA^˙is⌣˙,∧i∈Iℬis⌣˙/s⌣˙∈S⌣˙ (P-intersection)(3)
∪RUi=s⌣˙,∪i∈IA^˙is⌣˙,∧i∈Iℬis⌣˙/s⌣˙∈S⌣˙ (R-union)(4)
∩RUi=s⌣˙,∩i∈IA^˙is⌣˙,∨i∈Iℬis⌣˙/s⌣˙∈S⌣˙ (R-intersection)
The complement ofU=A^,ℬ is also a cubic picture fuzzy set which is defined by Uc=s⌣˙,A^cs⌣˙,ℬcs⌣˙|s⌣˙∈S⌣˙.Proposition 1.
For any CPFSU=A^,ℬ, V=J,K, C=L,M, and D=O,T, we have the following: (1)
IfU⊆PVandV⊆PCthenU⊆PC(2)
IfU⊆PVandU⊆PCthenU⊆PV∩PC(3)
IfU⊆PVandC⊆PVthenU∪PC⊆PV(4)
IfU⊆PVandC⊆PDthenU∪PC⊆PV∪PDandU∩PC⊆PV∩PD(5)
IfU⊆RVandV⊆RCthenU⊆RC(6)
IfU⊆RVandU⊆RCthenU⊆RV∩RC(7)
IfU⊆RVandC⊆RVthenU∪RC⊆RV(8)
IfU⊆RVandC⊆RDthenU∪RC⊆RV∪RDandU∩RC⊆RV∩RDProof.
The proof is straightforward and therefore is omitted.Remark 1.
Te following are noted:(1)
IfU⊆PV,thenVc⊈PUc(2)
IfU⊆RV,thenVc⊈RUcExample 5.
Let(4)U=0.1,0.2,0.2,0.25,0.2,0.4,0.1,0.2,0.2,V=0.1,0.2,0.2,0.3,0.2,0.5,0.1,0.2,0.3,and then U⊆pV.
Since(5)Uc=0.2,0.4,0.2,0.25,0.1,0.2,0.2,0.2,0.1,Vc=0.2,0.5,0.2,0.3,0.1,0.2,0.3,0.2,0.1,we obtain Vc⊈pUc.Example 6.
Let(6)U=0.1,0.2,0.2,0.25,0.2,0.4,0.1,0.2,0.3,V=0.1,0.2,0.2,0.3,0.2,0.5,0.1,0.2,0.2,and then U⊆RV.
Since(7)Uc=0.2,0.4,0.2,0.25,0.1,0.2,0.3,0.2,0.1,Vc=0.2,0.5,0.2,0.3,0.1,0.2,0.2,0.2,0.1,we have Vc⊈RUc.Theorem 3.
LetU=A^,ℬ be a cubic picture fuzzy set. If U is an ICPFS (resp., ECPFS), then Uc is an ICPFS (resp., ECPFS).Proof.
The proof is straightforward and therefore is omitted.Theorem 4.
P-union and P-intersection of arbitrary indexed family of ICPFSsUi=A^i,ℬi|i∈Λ are ICPFSs.Proof.
AsUi is an ICPFSs,(8)ℬ1i−s⌣˙≤ℬ1is⌣˙≤ℬ1i+s⌣˙,ℬ2i−s⌣˙≤ℬ2is⌣˙≤ℬ2i+s⌣˙,ℬ3i−s⌣˙≤ℬ3is⌣˙≤ℬ3i+s⌣˙,for each i∈Λ. This implies that(9)∪i∈Λℬ1i−s⌣˙≤∨i∈Λℬ1is⌣˙≤∪i∈Λℬ1i+s⌣˙,∪i∈Λℬ2i−s⌣˙≤∨i∈Λℬ2is⌣˙≤∪i∈Λℬ2i+s⌣˙,∪i∈Λℬ3i−s⌣˙≤∨i∈Λℬ3is⌣˙≤∪i∈Λℬ3i+s⌣˙,and, likewise,(10)∩i∈Λℬ1i−s⌣˙≤∧i∈Λℬ1is⌣˙≤∩i∈Λℬ1i+s⌣˙,∩i∈Λℬ2i−s⌣˙≤∧i∈Λℬ2is⌣˙≤∩i∈Λℬ2i+s⌣˙,∩i∈Λℬ3i−s⌣˙≤∧i∈Λℬ3is⌣˙≤∩i∈Λℬ3i+s⌣˙.
Hence, P-union and P-intersection ofUi are CPFSs.Remark 2.
P-union and P-intersection of ECPFSs need not be an ECPFS.Example 7.
LetU=A^,ℬ and V=J,K be the ECPFSs of I=0,1, where(11)A^˙s⌣˙=0.2,0.3,0.0,0.2,0.0,0.2,ℬs⌣˙=0.1,0.3,0.3,Js⌣˙=0.0,0.2,0.1,0.3,0.2,0.3,Ks⌣˙=0.3,0.0,0.1,for all s⌣˙∈I.(1)
We know thatU∪PV=s⌣˙,Js⌣˙,ℬs⌣˙|s⌣˙∈S⌣˙ and ℬ1s⌣˙∈ℬ1−s⌣˙,ℬ1+s⌣˙,ℬ2s⌣˙∈ℬ2−s⌣˙,ℬ2+s⌣˙,ℬ3s⌣˙∈ℬ3−s⌣˙,ℬ3+s⌣˙ for all s⌣˙∈S⌣˙. Hence, U∩PV is not an ECPFS.(2)
We know thatU∩PV=s⌣˙,A˙^s⌣˙,Ks⌣˙|s⌣˙∈S⌣˙ and K1s⌣˙∈K1−s⌣˙,K1+s⌣˙,K2s⌣˙∈K2−s⌣˙∪K2+s⌣˙,K3s⌣˙∈K3−s⌣˙,K3+s⌣˙ for all s⌣˙∈S⌣˙. Hence U∩PV is not an ECPFS.
The following example shows that the R-union and R-intersection of CPFSs need not be an CPFS.Example 8.
LetU=A^˙,ℬ and V=J,K be CPFSs in I=0,1, where A^˙s⌣˙=0.0,0.2,0.1,0.3,0.2,0.3,ℬs⌣˙=0.1,0.3,0.3 andJs⌣˙=0.2,0.3,0.0,0.2,0.0,0.2,Ks⌣˙=0.3,0.0,0.1 for alls⌣˙∈I.(1)
We know thatU∪RV=s⌣˙,Js⌣˙,Zs⌣˙|s⌣˙∈I and ℬ1s⌣˙∉ℬ1−s⌣˙,ℬ1+s⌣˙,ℬ2s⌣˙∉ℬ2−s⌣˙,ℬ2+s⌣˙,ℬ3s⌣˙∉ℬ3−s⌣˙,ℬ3+s⌣˙ for alls⌣˙∈I. Hence, U∪RV is not a CPFS.(2)
We know thatU∩RV=s⌣˙,A^˙s⌣˙,Ks⌣˙|s⌣˙∈I and K1s⌣˙∉K1−s⌣˙,K1+s⌣˙,K2s⌣˙∉K2−s⌣˙,K2+s⌣˙,K3s⌣˙∉K3−s⌣˙,K3+s⌣˙ for all s⌣˙∈I. Hence, U∩RV is not a CPFS.
The following example shows that “R-union” and “R-intersection” of ECPFS need not be an ECPFS.Example 9.
LetU=A^˙,ℬ and V=J,K be ECPFSs of I=0,1 in whichA^˙s⌣˙=0.0,0.2,0.1,0.4,0.2,0.3,ℬs⌣˙=0.3,0.0,0.1, andJs⌣˙=0.0,0.3,0.0,0.2,0.1,0.2, and Ks⌣˙=0.4,0.3,0.0 for all s⌣˙∈I.(1)
We know thatU∪RV=s⌣˙,Js⌣˙,ℬs⌣˙|s⌣˙∈I; clearly ℬ1s⌣˙∈ℬ1−s⌣˙,ℬ1+s⌣˙. Hence, U∩RV is not an ECPFS in I.(2)
We know thatU∩RV=s⌣˙,A^˙s⌣˙,Ks⌣˙|s⌣˙∈I; clearly K3A^˙∈K3−s⌣˙,K3+s⌣˙ for all s⌣˙∈I. Hence, U∩RV is not an ECPFS.Theorem 5.
LetU=A^˙,ℬ and V=J,K be the CPFSs, such that (12)Maxℬ1−s⌣˙,K1′−s⌣˙≤ℬ1ΛK1s⌣˙,Maxℬ2−s⌣˙,K2′−s⌣˙≤ℬ2ΛK2s⌣˙,Maxℬ3−s⌣˙,K3′−s⌣˙≤ℬ3ΛK3s⌣˙,
for eachs⌣˙∈S⌣˙. Then the “R-union” of U and V is a CPFS.Proof.
LetU=A^˙,ℬ and V=J,K be two CPFSs, which satisfy the conditions given in Theorem 5; then we have ℬ1−s⌣˙≤ℬ1s⌣˙≤ℬ1+s⌣˙,ℬ2−s⌣˙≤ℬ2s⌣˙≤ℬ2+s⌣˙,ℬ3−s⌣˙≤ℬ3s⌣˙≤ℬ3+s⌣˙, and K1′−s⌣˙≤K1s⌣˙≤K1′+s⌣˙,K2′−s⌣˙≤K2s⌣˙≤K2′+s⌣˙,K3′−s⌣˙≤K3s⌣˙≤K3′+s⌣˙.
These imply thatℬ1s⌣˙ΛK1s⌣˙≤ℬ1∪K1′−s⌣˙,ℬ2s⌣˙ΛK2s⌣˙≤ℬ2∪K2′−s⌣˙,ℬ3s⌣˙ΛK3s⌣˙≤ℬ3∪K3′−s⌣˙. It follows from the assumption that (13)ℬ1∪K1′−s⌣˙=Maxℬ1−s⌣˙,K1′−s⌣˙≤ℬ1ΛK1w≤ℬ1∪K1′+s⌣˙,ℬ2∪K2′−s⌣˙=Maxℬ2−s⌣˙,K2′−s⌣˙≤ℬ2ΛK2w≤ℬ2∪K2′+s⌣˙,ℬ3∪K3′−s⌣˙=Maxℬ3−s⌣˙,K3′−s⌣˙≤ℬ3ΛK3s⌣˙≤ℬ3∪K3′+s⌣˙,where U∪RV=s⌣˙,A^˙∪Js⌣˙,ℬVKs⌣˙|s⌣˙∈S⌣˙is a CPFS. For two ECPFSs U and V of S⌣˙, two CPFSs U∗ and V∗ derived from the given sets need not be CPFSs.Example 10.
LetU=A^˙,ℬ and V=J,K be ECPFSs of I=0,1, in which (14)A^˙s⌣˙=0.1,0.2,0.1,0.3,0.2,0.4,ℬs⌣˙=0.3,0.0,0.5,Js⌣˙=0.0,0.2,0.0,0.3,0.1,0.3,Ks⌣˙=0.3,0.4,0.0,for all s⌣˙∈I.
It is seen thatU∗=A^˙,K and V∗=J,ℬ are not CPFSs, because in U∗,ℬ1s⌣˙∉0.1,0.2,ℬ2s⌣˙∉0.1,0.3,ℬ3s⌣˙∉0.2,0.4 and in V∗,K1s⌣˙∉0.0,0.2,K3s⌣˙∉0.1,0.3.
The following example shows that the “P-union” of two ECPFSs need not be a CPFS.Example 11.
LetU=A^˙,ℬ and V=J,K be ECPFSs of I=0,1 in whichA^˙s⌣˙=0.2,0.4,0.1,0.2,0.0,0.3,ℬs⌣˙=0.1,0.3,0.4. Js⌣˙=0.0,0.2,0.1,0.4,0.2,0.3 and Ks⌣˙=0.3,0.0,0.1 for all s⌣˙∈I,U∪PV=s⌣˙,Js⌣˙,ℬs⌣˙|s⌣˙∈I; clearly, 0.4∉0.2,0.3. Hence U∪PV is not a CPFS of I.Theorem 6.
LetU=A^˙,ℬ and V=J,K be the CPFSs in W satisfying the following inequalities:(15)Minℬ1−s⌣˙,K1′−s⌣˙≥ℬ1∨K1s⌣˙,Minℬ2−s⌣˙,K2′−s⌣˙≥ℬ2∨K2s⌣˙,Minℬ3−s⌣˙,K3i′−s⌣˙≥ℬ3∨K3s⌣˙,
for alls⌣˙∈S⌣˙. Then the “R-intersection” of U and V is a CPFS.Proof.
The proof is straightforward and therefore is omitted.Theorem 7.
LetU=A^˙,ℬ and V=J,K be the ECPFSs. if U∗=A^˙,K and V∗=J,ℬ are CPFSs, then P-union U∪PV of U and V is a CPFS.Proof.
The proof is straightforward and therefore is omitted.Theorem 8.
LetU=A^˙,ℬ and V=J,K be the ECPFSs. If U∗=A^˙,K and V∗=J,ℬ are CPFSs, then P-intersection U∪PV of U=A^˙,ℬ and V=J,K is a CPFS.Proof.
The proof is straightforward and therefore is omitted.Remark 3.
For two ECPFSsU and V of W, the derived CPFSs U∗ and V∗ need not be ECPFSs.Example 12.
LetU=A^˙,ℬ and V=J,K be ECPFSs of I=0,1 in which(16)A^˙s⌣˙=0.0,0.3,0.1,0.2,0.2,0.4,ℬs⌣˙=0.4,0.0,0.1,Js⌣˙=0.1,0.2,0.0,0.3,0.2,0.3,Ks⌣˙=0.3,0.4,0.1,for all s⌣˙∈I. Now, from the above, we observe that U∗=A^˙,K and V∗=J,ℬ are not ECPFSs, because, in U∗,K1s⌣˙∈0.0,0.3, and, in V∗,ℬ2s⌣˙∉0.0,0.3.Theorem 9.
LetU=A^˙,ℬ and V=J,K be two ECPFSs. If U∗=A^˙,K and V∗=J,ℬ are ECPFSs, then P-union U∪PV of U=A^˙,ℬ and V=J,K is an ECPFS.Proof.
LetU=A^˙,ℬ and V=J,K be ECPFSs, such that U∗=A^˙,K and V∗=J,ℬ are ECPFSs. Then we obtain that ℬ1s⌣˙∉ℬ1−s⌣˙,ℬ1+s⌣˙,ℬ2s⌣˙∉ℬ2−s⌣˙,ℬ2+s⌣˙, and ℬ3s⌣˙∉ℬ3−s⌣˙,ℬ3+s⌣˙, and K1s⌣˙∉K1′−s⌣˙,K1′+s⌣˙, K2s⌣˙∉K2′−s⌣˙,K2′+s⌣˙, and K3s⌣˙∉K3′−s⌣˙,K3′+s⌣˙.
Hence,(17)ℬ1∨K1s⌣˙∉Maxℬ1−s⌣˙,K1′−s⌣˙,Maxℬ1+s⌣˙,K1′+s⌣˙,ℬ2∨K2s⌣˙∉Maxℬ2−s⌣˙,K2′−s⌣˙,Maxℬ2+s⌣˙,K2′+s⌣˙,ℬ3∨K3s⌣˙∉Maxℬ3−s⌣˙,K3′−s⌣˙,Maxℬ3+s⌣˙,K3′+s⌣˙.
This implies that(18)ℬ1∨K1w∉ℬ1∪K1′−s⌣˙,ℬ1∪K1′+s⌣˙,ℬ1∨K1w∉ℬ2∪K2′−s⌣˙,ℬ2∪K2′+s⌣˙,ℬ1∨K1w∉ℬ3∪K3′−s⌣˙,ℬ3∪K3′+s⌣˙.
Hence,U∪PV is an ECPFS.Theorem 10.
LetU=A^˙,ℬ and V=J,K be the ECPFSs of S⌣˙ such that(19)MinMaxℬ1+s⌣˙,K1′−s⌣˙,Maxℬ1−s⌣˙,K1′+s⌣˙≥ℬ1∨K1s⌣˙>MaxMinℬ1+s⌣˙,K1′−s⌣˙,Minℬ1−s⌣˙,K1′+s⌣˙,MinMaxℬ2+s⌣˙,K2′−s⌣˙,Maxℬ2−s⌣˙,K2′+s⌣˙≥ℬ1∨K2s⌣˙>MaxMinℬ2+s⌣˙,K1′−s⌣˙,Minℬ1−s⌣˙,K2′+s⌣˙,MinMaxℬ3+s⌣˙,K3′−s⌣˙,Maxℬ1−s⌣˙,K1′+s⌣˙≥ℬ3∨K3w>MaxMinℬ3+s⌣˙,K3′−s⌣˙,Minℬ3−s⌣˙,K3′+s⌣˙,for all s⌣˙∈S⌣˙.
Then, the P-intersection ofU and V is an ECPFS.Proof.
The proof is straightforward by the definitions in [11, 13].Theorem 11.
LetU=A^˙,ℬ and V=J,K be the CPFSs, such that the following implications are valid:(20)MinMaxℬ1+s⌣˙,K1′−s⌣˙,Maxℬ1−s⌣˙,K1′+s⌣˙=ℬ1∨K1s⌣˙=MaxMinℬ1+s⌣˙,K1′−s⌣˙,Minℬ1−s⌣˙,K1′+s⌣˙,MinMaxℬ2+s⌣˙,K2′−s⌣˙,Maxℬ2−s⌣˙,K2′+s⌣˙=ℬ2∨K2s⌣˙=MaxMinℬ2+s⌣˙,K2′−s⌣˙,Minℬ2−s⌣˙,K2′+s⌣˙,MinMaxℬ3+s⌣˙,K3′−s⌣˙,Maxℬ3−s⌣˙,K3′+s⌣˙≥ℬ3∨K3s⌣˙=MaxMinℬ3+s⌣˙,K3′−s⌣˙,Minℬ3−s⌣˙,K3′+s⌣˙,
for alls⌣˙∈S⌣˙. Then the P-intersection of U and V is both an ECPFS and a CPFS.Proof.
It is straightforward by the definitions in [11, 13].
The following example shows that the P-union of two ECPFSs needs not be an ECPFS.Example 13.
LetU=A^˙,ℬ and V=J,K be two ECPFSs of I=0,1 defined as follows:(21)A^˙s⌣˙=0.1,0.3,0.2,0.3,0.0,0.2,ℬs⌣˙=0.0,0.1,0.3,Js⌣˙=0.1,0.2,0.2,0.4,0.1,0.3,Ks⌣˙=0.3,0.1,0.4,for all s⌣˙∈I.
SinceU∪PV=s⌣˙,Js⌣˙,ℬs⌣˙s⌣˙∈I, clearly 0.3∈0.1,0.3. Hence U∪PV is not an ECPFS of I.Theorem 12.
LetU=A^˙,ℬ and V=J,K be two ECPFSs, such that the following are satisfied:(22)MinMaxℬ1+s⌣˙,K1′−s⌣˙,Maxℬ1−s⌣˙,K1′+s⌣˙>ℬ1∨K1s⌣˙≥MaxMinℬ1+s⌣˙,K1′−s⌣˙,Minℬ1−s⌣˙,K1′+s⌣˙,MinMaxℬ2+s⌣˙,K2′−s⌣˙,Maxℬ2−s⌣˙,K2′+s⌣˙>ℬ2∨K2s⌣˙≥MaxMinℬ2+s⌣˙,K2′−s⌣˙,Minℬ2−s⌣˙,K2′+s⌣˙,MinMaxℬ3+s⌣˙,K3′−s⌣˙,Maxℬ3−s⌣˙,K3′+s⌣˙>ℬ3∨K3s⌣˙≥MaxMinℬ3+s⌣˙,K3′−s⌣˙,Minℬ3−s⌣˙,K3′+s⌣˙,
for alls⌣˙∈S⌣˙. Then, P-union of U and V is an ECPFS.Proof.
The proof is straightforward and therefore is omitted.Theorem 13.
LetU=A^˙,ℬ and V=J,K be two ECPFSs, which satisfy the followings conditions:(23)MinMaxℬ1+s⌣˙,K1′−s⌣˙,Maxℬ1−s⌣˙,K1′+s⌣˙>ℬ1∨K1w≥MaxMinℬ1+s⌣˙,K1′−s⌣˙,Minℬ1−s⌣˙,K1′+s⌣˙,MinMaxℬ2+s⌣˙,K2′−s⌣˙,Maxℬ2−s⌣˙,K2′+s⌣˙>ℬ2∨K2s⌣˙≥MaxMinℬ2+s⌣˙,K2′−s⌣˙,Minℬ2−s⌣˙,K2′+s⌣˙,MinMaxℬ3+s⌣˙,K3′−s⌣˙,Maxℬ3−s⌣˙,K3′+s⌣˙>ℬ3∨K3s⌣˙≥MaxMinℬ3+s⌣˙,K3′−s⌣˙,Minℬ3−s⌣˙,K3′+s⌣˙,
for alls⌣˙∈S⌣˙. Then R-union of U and V is an ECPFS.Proof.
The proof is straightforward.Theorem 14.
LetU=A^˙,ℬ and V=J,K be the ECPFSs, such that the following are satisfied:(24)MinMaxℬ1+s⌣˙,K1′−s⌣˙,Maxℬ1−s⌣˙,K1′+s⌣˙≥ℬ1∨K1s⌣˙>MaxMinℬ1+s⌣˙,K1′−s⌣˙,Minℬ1−s⌣˙,K1′+s⌣˙,MinMaxℬ2+s⌣˙,K2′−s⌣˙,Maxℬ2−s⌣˙,K2′+s⌣˙≥ℬ2∨K2s⌣˙>MaxMinℬ2+s⌣˙,K2′−s⌣˙,Minℬ2−s⌣˙,K2′+s⌣˙,MinMaxℬ3+s⌣˙,K3′−s⌣˙,Maxℬ3−s⌣˙,K3′+s⌣˙≥ℬ3∨K3w>MaxMinℬ3+s⌣˙,K3′−s⌣˙,Minℬ3−s⌣˙,K3′+s⌣˙,
for alls⌣˙∈S⌣˙. Then R-intersection of U and V is an ECPFS.Proof.
The proof is straightforward.Remark 4.
LetU=A^˙,ℬ and V=J,K be two ECPFSs, such that the following are satisfied:(25)MinMaxℬ1+s⌣˙,K1′−s⌣˙,Maxℬ1−s⌣˙,K1′+s⌣˙>ℬ1∨K1s⌣˙=MaxMinℬ1+s⌣˙,K1′−s⌣˙,Minℬ1−s⌣˙,K1′+s⌣˙,MinMaxℬ2+s⌣˙,K2′−s⌣˙,Maxℬ2−s⌣˙,K2′+s⌣˙>ℬ2∨K2s⌣˙=MaxMinℬ2+s⌣˙,K2′−s⌣˙,Minℬ2−s⌣˙,K2′+s⌣˙,MinMaxℬ3+s⌣˙,K3′−s⌣˙,Maxℬ3−s⌣˙,K3′+s⌣˙>ℬ3∨K3w=MaxMinℬ3+s⌣˙,K3′−s⌣˙,Minℬ3−s⌣˙,K3′+s⌣˙,
for alls⌣˙∈S⌣˙. Then R-intersection of U and V may not be an ECPFS.Theorem 15.
LetU=A^˙,ℬ and V=J,K be two ECPFSs, such that the following are satisfied: (26)MinMaxℬ1+s⌣˙,K1′−s⌣˙,Maxℬ1−s⌣˙,K1′+s⌣˙=ℬ1∨K1s⌣˙=MaxMinℬ1+s⌣˙,K1′−s⌣˙,Minℬ1−s⌣˙,K1′+s⌣˙,MinMaxℬ2+s⌣˙,K2′−s⌣˙,Maxℬ2−s⌣˙,K2′+s⌣˙=ℬ2∨K2s⌣˙=MaxMinℬ2+s⌣˙,K2′−s⌣˙,Minℬ2−s⌣˙,K2′+s⌣˙,MinMaxℬ3+s⌣˙,K3′−s⌣˙,Maxℬ3−s⌣˙,K3′+s⌣˙>ℬ3∨K3w=MaxMinℬ3+s⌣˙,K3′−s⌣˙,Minℬ3−s⌣˙,K3′+s⌣˙,for all s⌣˙∈S⌣˙. Then R-intersection of U and V is both an ECPFS and a CPFS.Proof.
The proof is straightforward.Theorem 16.
LetU=A^˙,ℬ and V=J,K be two CPFSs. If the implications are satisfied, for all s⌣˙∈S⌣˙,(27)ℬ1∧K1s⌣˙≤Maxℬ1−s⌣˙,K1−s⌣˙,ℬ2∧K2s⌣˙≤Maxℬ2−s⌣˙,K2−s⌣˙,ℬ3∧K3s⌣˙≤Maxℬ3−s⌣˙,K3−s⌣˙,then the R-union of U and V is an EPCFS.Proof.
The proof is straightforward.Theorem 17.
LetU=A^˙,ℬ and V=J,K be two CPFSs. If the following implications are satisfied for all s⌣˙∈S⌣˙,ℬ1∨K1s⌣˙≥Minℬ1+s⌣˙,K1′+s⌣˙,ℬ2∨K2w≥Minℬ2+s⌣˙,K2′+s⌣˙, andℬ3∨K3w≥Minℬ3+s⌣˙,K3′+s⌣˙, then R-intersection of U and V is an ECPFS.Proof.
The proof is straightforward.Theorem 18.
LetU=A^˙,ℬ and V=J,K be two ECPFSs, such that the following implications hold:(28)MinMaxℬ1+s⌣˙,K1′−s⌣˙,Maxℬ1−s⌣˙,K1+s⌣˙≤ℬ1∧K1s⌣˙≤Maxℬ1+s⌣˙,K1′+s⌣˙,MinMaxℬ2+s⌣˙,K2′−s⌣˙,Maxℬ2−s⌣˙,K2+s⌣˙≤ℬ2∧K2s⌣˙≤MaxMinℬ2+s⌣˙,K2′+s⌣˙,MinMaxℬ3+s⌣˙,K3′−s⌣˙,Maxℬ3−s⌣˙,K3+s⌣˙≤ℬ3∧K3s⌣˙≤MaxMinℬ3+s⌣˙,K3′+s⌣˙,
for alls⌣˙∈S⌣˙. Then, R-union of U and V is a CPFS.Proof.
The proof is straightforward.


## 4. Averaging Aggregation Operators

In this section, we present three types of new aggregation operators called cubic picture fuzzy weighted averaging, cubic picture fuzzy ordered weighted averaging, and cubic picture fuzzy hybrid weighted averaging operators based on cubic picture fuzzy sets. LetCPF denote the collection of all CPFSs.Definition 15.
(see [11]). A function T: [0, 1] × [0, 1] −⟶ [0, 1] is said to be a t-norm which satisfies the following: (1)
Boundary:T (0, 0) = 0; T (x, 1) = T (1, x) = x for all x ∈ [0, 1](2)
Monotonicity: Ifx1≤y1 and x2≤y2, then Tx1,x2≤Ty1,y2(3)
Commutativity:Tx1,x2=Tx2,x1(4)
Associativity:Tx1,Tx2,x3=TTx1,x2,x3
A functionS defined by Sx,y=1−T1−x,1−y is called t-co-norm. A decreasing function g generates a t-norm as Tx,y=g−1gx+gy such that g1=0 and function h generates the t-co-norm as Sx,y=h−1hx+hy, where ht=g1−t. Based on these norms’ generators, g and h will be used in the next theorems.Definition 16.
LetC=M,L,N,U,ρ,ϑ,C1=M1,L1,N1,U1,ρ1,ϑ1, and C2=M2,L2,N2,U2,ρ2,ϑ2 be three CPFSs. Then the operations ⊕,⊗,λC,andCλ are defined as follows: (1)
C1⊕C2=h−1hM1+hM2,g−1gL1+gL2,g−1gN1+gN2,h−1hU1+hU2,g−1gρ1+gρ2,g−1gϑ1+gϑ2(2)
C1⊗C2=g−1gM1+gM2,h−1hL1+hL2,h−1hN1+hN2,g−1gU1+gU2,h−1hρ1+hρ2,h−1hϑ1+hϑ2(3)
λC=h−1λhM,g−1λgL,g−1λgN,h−1λhU,g−1λgρ,g−1λgϑ(4)
Cλ=g−1λgM,h−1λhL,h−1λhN,g−1λgU,h−1λhρ,h−1λhϑTheorem 19.
LetC1,C2, and C3 be three CPFNs and λ,λ1, and λ2>0. Then, we have the following:(1)
C1⊕C2=C2⊕C1(2)
C1⊗C2=C2⊗C1(3)
λC1⊕C2=λC1⊕λC2(4)
C1⊗C2λ=C1λ⊗C2λ(5)
λ1C⊕λ2C=λ1+λ2C(6)
Cλ1⊗Cλ1=Cλ1+λ2Proof.
It is easily obtained by the above definition.


### 4.1. Cubic Picture Fuzzy Weighted Averaging (CPFWA) Operators

Definition 17.
LetUii∈Λ be a collection of CPFSs. Then theCPFWACPℱn⟶CPF is defined as follows:(29)CPFWAU1,U2,…,Un=w¨1U1⊕w¨2U2⊕⋯⊕w¨nUn=⊕i=1nw¨iUi,where w¨=w¨1,w¨2,…,w¨nT is the weighted vector of Ui, s.t. w¨i>0 and ∑i=1nw¨i=1.Theorem 20.
fUi=A^˙is⌣˙,ℬis⌣˙,s⌣˙∈S⌣˙ is the collection of CPFSs, then the averaging value by using CPFWA operator is still CPFS and is given by(30)CPFWAU1,U2,…,Un=h−1∑i=1nw¨ihMi,g−1∑i=1nw¨igLi,g−1∑i=1nw¨igNi,h−1∑i=1nw¨ihUi,g−1∑i=1nw¨igρi,g−1∑i=1nw¨igϑi.Proof.
We shall prove the result by using the principle of mathematical induction on “n.”Step 1. For n=2, we have U1=A^˙1,ℬ1,U2=A^˙2,ℬ2; thus, by the operations of CPFSs, we get(31)w¨1U1=h−1w¨1hM1,g−1w¨1gL1,g−1w¨1gN1,h−1w¨1hU1,g−1w¨1gρ1,g−1w¨1gϑ1,w¨2U2=h−1w¨2hM2,g−1w¨2gL2,g−1w¨2gN2,h−1w¨2hU2,g−1w¨2gρ2,g−1w¨2gϑ2.Hence, by additive properties of CPFSs, we get(32)CPFWAU1,U2=w¨1U1⊕w¨2U2=h−1w¨1hM1,g−1w¨1gL1,g−1w¨1gN1,h−1w¨1hU1,g−1w¨1gρ1,g−1w¨1gϑ1⊕h−1w¨2hM2,g−1w¨2gL2,g−1w¨2gN2,h−1w¨2hU2,g−1w¨2gρ2,g−1w¨2gϑ2=h−1hh−1w¨1hM1+hh−1w¨2hM2,g−1gg−1w¨1gL1+gg−1w¨2gL2,g−1gg−1w¨1gN1+gg−1w¨2gN2,h−1hh−1w¨1hU1+hh−1w¨2hU2,g−1gg−1w¨1gρ1+gg−1w¨2gρ2,g−1gg−1w¨1gϑ1+gg−1w¨2gϑ2=h−1w¨1hM1+w¨2hM2,g−1w¨1gL1+w¨2gL2,g−1w¨1gN1+w¨2gN2,h−1w¨1hU1+w¨2hU2,g−1w¨1gρ1+w¨2gρ2,g−1w¨1gϑ1+w¨2gϑ2=h−1∑i=12w¨ihMi,g−1∑i=12w¨igLi,g−1∑i=12w¨igNi,h−1∑i=12w¨ihUi,g−1∑i=12w¨igρi,g−1∑i=12w¨igϑi.Then, the results hold forn=2.Step 2. If equation (30) holds for n=k, then, for n=k+1, we have(33)CPFWAU1,U2,…,UK+1=⊕i=1K+1w¨iUi=w¨1U1⊕w¨2U2⊕⋯⊕w¨K+1UK+1=⊕i=1Kw¨iUi⊕w¨K+1UK+1=h−1∑i=1Kw¨ihMi,g−1∑i=1Kw¨igLi,g−1∑i=1Kw¨igNi,h−1∑i=1Kw¨ihUi,g−1∑i=1Kw¨igρi,g−1∑i=1Kw¨igϑi⊕h−1w¨K+1hMK+1,g−1w¨K+1gLK+1,g−1w¨K+1gNK+1,h−1w¨K+1hUK+1,g−1w¨K+1gρK+1,g−1w¨K+1gϑK+1=h−1hh−1∑i=1Kw¨ihMi+h−1w¨K+1hMK+1,g−1gg−1∑i=1Kw¨igLi+g−1w¨K+1gLK+1,g−1gg−1∑i=1Kw¨igNi+g−1w¨K+1gNk+1,h−1hh−1∑i=1Kw¨ihUi+h−1w¨K+1hUK+1,g−1gg−1∑i=1Kw¨igρi+g−1w¨K+1gρK+1,g−1gg−1∑i=1Kw¨igϑi+g−1w¨K+1gϑK+1=h−1∑i=1Kw¨ihMi+w¨K+1hMK+1,g−1∑i=1Kw¨igLi+w¨K+1gLK+1,g−1∑i=1Kw¨igNi+vK+1gNk+1h−1∑i=1Kw¨ihUi+w¨K+1hUK+1,g−1∑i=1Kw¨igρi+w¨K+1gρK+1,g−1∑i=1Kw¨igϑi+w¨K+1gϑK+1=h−1∑i=1K+1w¨ihMi,g−1∑i=1K+1w¨igLi,g−1∑i=1K+1w¨igNi,h−1∑i=1K+1w¨ihUi,g−1∑i=1K+1w¨igρi,g−1∑i=1K+1w¨igϑi.
Since the results hold forn=k+1, hence, by the principle of mathematical induction, the result given in equation (30) holds for all positive integers n.Remark 5.
Ifgt is taken to be gt=−logt, then by equation (30) we have that (34)CPFWAU1,U2,…,Un=1−∏i=1n1−Miw¨i,∏i=1nLiw¨i,∏i=1nNiw¨i,1−∏i=1n1−Uiw¨i,∏i=1nρiw¨i,∏i=1nϑiw¨i,
which is called cubic picture fuzzy Archimedean weighted averaging operator.


### 4.2. Cubic Picture Fuzzy Ordered Weighted Averaging (CPFOWA) Operator

In this section, we intend to take the idea of OWA into CPFWA operator and propose a new operator which is defined as follows.Definition 18.
LetUii∈Λ be a collection of CPFSs. Then the CPFOWACPℱn⟶CPF is defined in the following way:(35)CPFOWAU1,U2,…,Un=w¨1Uo1⊕w¨2Uo2⊕⋯⊕w¨nUon=⊕i=1nw¨iUoi,where w¨=w¨1,w¨2,…,w¨nT is the weighted vector of Ui, such that w¨i>0 and ∑i=1nw¨i=1. Here o1,o2,…,on is the permutation of 1,2,…,n, such that Uoi−1≥Uoi, and Uoi is the ith largest of CPFSs Uiiε∧.Theorem 21.
LetUii∈Λ be a collection of CPFSs. Then, based on the CPFOWA operator, the aggregated CPFSs can be expressed as follows: (36)CPFOWAU1,U2,…,Un=h−1∑i=1nw¨ihMoi,g−1∑i=1nw¨igLoi,g−1∑i=1nw¨igNoi,h−1∑i=1nw¨ihUoi,g−1∑i=1nw¨igρoi,g−1∑i=1nw¨igϑoi.
In particular, ifLoi=ρoi=0 for all i, then equation (36) reduces to(37)CPFOWAU1,U2,…,Un=h−1∑i=1nw¨ihMoi,g−1∑i=1nw¨igNoi,h−1∑i=1nw¨ihUoi,g−1∑i=1nw¨igϑoi,which becomes cubic intuitionistic OWA operator.Proof.
The proof follows from Theorem19.


### 4.3. Cubic Picture Fuzzy Hybrid Averaging (CPFHA) Operator

CPFWA operator weighs the CPFSs only, while CPFOWA weighs the ordered positions of it. However, in order to combine these two aspects in one, we introduce CPFHA operator.Definition 19.
LetUii∈Λ be a collection of CPFSs. Then theCPFHACPℱn⟶CPF is defined as follows:(38)CPFHAU1,U2,…,Un=w¨1Uo1˙⊕w¨2Uo2˙⊕⋯⊕w¨nUon˙=⊕i=1nw¨iUoi˙,where w¨=w¨1,w¨2,…,w¨nT is the standard weight vector of Ui, such that w¨i>0 and ∑i=1nw¨i=1,Uoi˙ is the ith largest of the weighted CPFSs Ui˙Ui˙=nw¨˙iUi,i=1,2,…,n,wheren is the number of CPFSs. Then CPFHA is called cubic picture fuzzy hybrid averaging operator.Theorem 22.
LetUii∈Λ be a collection of CPFSs; then, based on CPFHA operator, the aggregated CPFSs can be expressed as(39)CPFHAU1,U2,…,Un=h−1∑i=1nw¨ihMoi˙,g−1∑i=1nw¨igLoi˙,g−1∑i=1nw¨igNoi˙,h−1∑i=1nw¨ihUoi˙,g−1∑i=1nw¨igρoi˙,g−1∑i=1nw¨igϑoi˙.Proof.
The proof is similar to Theorem20, so it is omitted here.


## 4.1. Cubic Picture Fuzzy Weighted Averaging (CPFWA) Operators

Definition 17.
LetUii∈Λ be a collection of CPFSs. Then theCPFWACPℱn⟶CPF is defined as follows:(29)CPFWAU1,U2,…,Un=w¨1U1⊕w¨2U2⊕⋯⊕w¨nUn=⊕i=1nw¨iUi,where w¨=w¨1,w¨2,…,w¨nT is the weighted vector of Ui, s.t. w¨i>0 and ∑i=1nw¨i=1.Theorem 20.
fUi=A^˙is⌣˙,ℬis⌣˙,s⌣˙∈S⌣˙ is the collection of CPFSs, then the averaging value by using CPFWA operator is still CPFS and is given by(30)CPFWAU1,U2,…,Un=h−1∑i=1nw¨ihMi,g−1∑i=1nw¨igLi,g−1∑i=1nw¨igNi,h−1∑i=1nw¨ihUi,g−1∑i=1nw¨igρi,g−1∑i=1nw¨igϑi.Proof.
We shall prove the result by using the principle of mathematical induction on “n.”Step 1. For n=2, we have U1=A^˙1,ℬ1,U2=A^˙2,ℬ2; thus, by the operations of CPFSs, we get(31)w¨1U1=h−1w¨1hM1,g−1w¨1gL1,g−1w¨1gN1,h−1w¨1hU1,g−1w¨1gρ1,g−1w¨1gϑ1,w¨2U2=h−1w¨2hM2,g−1w¨2gL2,g−1w¨2gN2,h−1w¨2hU2,g−1w¨2gρ2,g−1w¨2gϑ2.Hence, by additive properties of CPFSs, we get(32)CPFWAU1,U2=w¨1U1⊕w¨2U2=h−1w¨1hM1,g−1w¨1gL1,g−1w¨1gN1,h−1w¨1hU1,g−1w¨1gρ1,g−1w¨1gϑ1⊕h−1w¨2hM2,g−1w¨2gL2,g−1w¨2gN2,h−1w¨2hU2,g−1w¨2gρ2,g−1w¨2gϑ2=h−1hh−1w¨1hM1+hh−1w¨2hM2,g−1gg−1w¨1gL1+gg−1w¨2gL2,g−1gg−1w¨1gN1+gg−1w¨2gN2,h−1hh−1w¨1hU1+hh−1w¨2hU2,g−1gg−1w¨1gρ1+gg−1w¨2gρ2,g−1gg−1w¨1gϑ1+gg−1w¨2gϑ2=h−1w¨1hM1+w¨2hM2,g−1w¨1gL1+w¨2gL2,g−1w¨1gN1+w¨2gN2,h−1w¨1hU1+w¨2hU2,g−1w¨1gρ1+w¨2gρ2,g−1w¨1gϑ1+w¨2gϑ2=h−1∑i=12w¨ihMi,g−1∑i=12w¨igLi,g−1∑i=12w¨igNi,h−1∑i=12w¨ihUi,g−1∑i=12w¨igρi,g−1∑i=12w¨igϑi.Then, the results hold forn=2.Step 2. If equation (30) holds for n=k, then, for n=k+1, we have(33)CPFWAU1,U2,…,UK+1=⊕i=1K+1w¨iUi=w¨1U1⊕w¨2U2⊕⋯⊕w¨K+1UK+1=⊕i=1Kw¨iUi⊕w¨K+1UK+1=h−1∑i=1Kw¨ihMi,g−1∑i=1Kw¨igLi,g−1∑i=1Kw¨igNi,h−1∑i=1Kw¨ihUi,g−1∑i=1Kw¨igρi,g−1∑i=1Kw¨igϑi⊕h−1w¨K+1hMK+1,g−1w¨K+1gLK+1,g−1w¨K+1gNK+1,h−1w¨K+1hUK+1,g−1w¨K+1gρK+1,g−1w¨K+1gϑK+1=h−1hh−1∑i=1Kw¨ihMi+h−1w¨K+1hMK+1,g−1gg−1∑i=1Kw¨igLi+g−1w¨K+1gLK+1,g−1gg−1∑i=1Kw¨igNi+g−1w¨K+1gNk+1,h−1hh−1∑i=1Kw¨ihUi+h−1w¨K+1hUK+1,g−1gg−1∑i=1Kw¨igρi+g−1w¨K+1gρK+1,g−1gg−1∑i=1Kw¨igϑi+g−1w¨K+1gϑK+1=h−1∑i=1Kw¨ihMi+w¨K+1hMK+1,g−1∑i=1Kw¨igLi+w¨K+1gLK+1,g−1∑i=1Kw¨igNi+vK+1gNk+1h−1∑i=1Kw¨ihUi+w¨K+1hUK+1,g−1∑i=1Kw¨igρi+w¨K+1gρK+1,g−1∑i=1Kw¨igϑi+w¨K+1gϑK+1=h−1∑i=1K+1w¨ihMi,g−1∑i=1K+1w¨igLi,g−1∑i=1K+1w¨igNi,h−1∑i=1K+1w¨ihUi,g−1∑i=1K+1w¨igρi,g−1∑i=1K+1w¨igϑi.
Since the results hold forn=k+1, hence, by the principle of mathematical induction, the result given in equation (30) holds for all positive integers n.Remark 5.
Ifgt is taken to be gt=−logt, then by equation (30) we have that (34)CPFWAU1,U2,…,Un=1−∏i=1n1−Miw¨i,∏i=1nLiw¨i,∏i=1nNiw¨i,1−∏i=1n1−Uiw¨i,∏i=1nρiw¨i,∏i=1nϑiw¨i,
which is called cubic picture fuzzy Archimedean weighted averaging operator.


## 4.2. Cubic Picture Fuzzy Ordered Weighted Averaging (CPFOWA) Operator

In this section, we intend to take the idea of OWA into CPFWA operator and propose a new operator which is defined as follows.Definition 18.
LetUii∈Λ be a collection of CPFSs. Then the CPFOWACPℱn⟶CPF is defined in the following way:(35)CPFOWAU1,U2,…,Un=w¨1Uo1⊕w¨2Uo2⊕⋯⊕w¨nUon=⊕i=1nw¨iUoi,where w¨=w¨1,w¨2,…,w¨nT is the weighted vector of Ui, such that w¨i>0 and ∑i=1nw¨i=1. Here o1,o2,…,on is the permutation of 1,2,…,n, such that Uoi−1≥Uoi, and Uoi is the ith largest of CPFSs Uiiε∧.Theorem 21.
LetUii∈Λ be a collection of CPFSs. Then, based on the CPFOWA operator, the aggregated CPFSs can be expressed as follows: (36)CPFOWAU1,U2,…,Un=h−1∑i=1nw¨ihMoi,g−1∑i=1nw¨igLoi,g−1∑i=1nw¨igNoi,h−1∑i=1nw¨ihUoi,g−1∑i=1nw¨igρoi,g−1∑i=1nw¨igϑoi.
In particular, ifLoi=ρoi=0 for all i, then equation (36) reduces to(37)CPFOWAU1,U2,…,Un=h−1∑i=1nw¨ihMoi,g−1∑i=1nw¨igNoi,h−1∑i=1nw¨ihUoi,g−1∑i=1nw¨igϑoi,which becomes cubic intuitionistic OWA operator.Proof.
The proof follows from Theorem19.


## 4.3. Cubic Picture Fuzzy Hybrid Averaging (CPFHA) Operator

CPFWA operator weighs the CPFSs only, while CPFOWA weighs the ordered positions of it. However, in order to combine these two aspects in one, we introduce CPFHA operator.Definition 19.
LetUii∈Λ be a collection of CPFSs. Then theCPFHACPℱn⟶CPF is defined as follows:(38)CPFHAU1,U2,…,Un=w¨1Uo1˙⊕w¨2Uo2˙⊕⋯⊕w¨nUon˙=⊕i=1nw¨iUoi˙,where w¨=w¨1,w¨2,…,w¨nT is the standard weight vector of Ui, such that w¨i>0 and ∑i=1nw¨i=1,Uoi˙ is the ith largest of the weighted CPFSs Ui˙Ui˙=nw¨˙iUi,i=1,2,…,n,wheren is the number of CPFSs. Then CPFHA is called cubic picture fuzzy hybrid averaging operator.Theorem 22.
LetUii∈Λ be a collection of CPFSs; then, based on CPFHA operator, the aggregated CPFSs can be expressed as(39)CPFHAU1,U2,…,Un=h−1∑i=1nw¨ihMoi˙,g−1∑i=1nw¨igLoi˙,g−1∑i=1nw¨igNoi˙,h−1∑i=1nw¨ihUoi˙,g−1∑i=1nw¨igρoi˙,g−1∑i=1nw¨igϑoi˙.Proof.
The proof is similar to Theorem20, so it is omitted here.


## 5. MCDM Based on the Proposed Operation

In this section, we need the previous aggregation operators in a decision-making for CPFSs with illustrative example for evaluating the approach.Let a set ofm′ alternatives denoted by A=A1,A2,…,Am′ be found by the decision-maker under the set of the unlikely criteria Gˇ=Gˇ1,Gˇ2,…,Gˇn whose weight vector is w¨=w¨1,w¨2,…,w¨nT such that w¨i>0 and ∑i=1nw¨i=1.Suppose that the ranking of an alternativexj,j=1,2,…,m′ on the criteria Gˇi,i=1,2,…,m′ is assessed by the decision-maker in the form of CPFSs Uij=A^˙ij,ℬij,i,j∈1,2,….,n, where A^˙ij is the degree of VPFS and βij is the degree of PFS that the alternative Ai does not satisfy the attribute Gˇi. So we develop an approach for evaluating the best alternative based on the proposed operators for MCGDM problem whose steps are as follows:Step 1. Construct the decision matrix of CPFSs.Uij=A^˙ij,ℬij, where A^˙ij=Mij,Lij,Nij are VPFNs and βij=Uij,ρij,ϑij are the PFSs towards the alternative Ai and hence construct a cubic picture fuzzy decision matrix D=P¨ijm′×n.Step 2. Normalized decision matrix, namely, cost C and benefits iB, so we normalize(40)rij=P¨ijC,k∈B,P¨ij,k∈C,whereP¨ijC is the complement of P¨ij.Step 3. Aggregated assessment of alternative, based on the decision matrix, as taken from step 2, all the aggregated values of the alternatives Ai,i=1,2,…,m′ under the different criteria Gˇi are obtained by using either CPFWA or CPFOWA or CPFHA operator and we collect the value of ri for each alternative Ai,i=1,2,…,m′.Step 4. We compute the score values of rii=1,2,…,n.Step 5. At last, we find that the rank of the alternatives Ai,ii=1,2,…,m′ according to the descending value of the score value are most valuable.


## 6. Illustrative Example

In this section, we illustrate with the mathematical example for the decision-making studied as follows.Suppose few companies design their financial strategy for the next fiscal year, and according to their plan of strategy, they are picking three alternatives defined as follows:A1: to invest in the “Chinese markets”; A2: to invest in the “Indian markets”; and A3: to invest in “USA markets.” These proceed for finding the aspect as follows: Gˇ1: “the increases analysis,” Gˇ2: “the decreases analysis,” and Gˇ3: “the neutral analysis,” whose weight vector w=0.5,0.2,0.3T.


### 6.1. Example by the CPFWA Operator

The example is applied in CPFWA operator to calculate the best one.Step 1. These three alternatives Ai,i=1,2,3 are to be solved by an expert under the three aspects Gˇjj=1,2,3 by using cubic picture fuzzy decision matrix D=P¨ij3×3=A^˙ij,ℬij for i,j=1,2,3.Step 2. Since the criteria Gˇ2 and Gˇ3 are the porches criteria while Gˇ1 are losses criteria R˜=rij3×3, equation (40) is used as follows: (41)R˜=0.1,0.3,0.2,0.1,0.1,0.1,0.3,0.2,0.10.1,0.1,0.2,0.1,0.3,0.2,0.4,0.3,0.20.3,0.4,0.2,0.2,0.1,0.3,0.1,0.2,0.10.3,0.1,0.1,0.2,0.1,0.3,0.1,0.2,0.30.1,0.2,0.1,0.3,0.2,0.3,0.2,0.3,0.40.4,0.3,0.2,0.1,0.3,0.1,0.1,0.1,0.20.3,0.1,0.2,0.1,0.1,0.0,0.3,0.4,0.10.1,0.1,0.2,0.2,0.3,0.3,0.1,0.1,0.10.3,0.3,0.2,0.3,0.2,0.1,0.4,0.3,0.0.Step 3. By following the CPFWA given in equation (30) with generatorgt=−Logt, we obtain the overall rating value of each alternative Ai as(42)r1=CPFWAr11,r12,r13=0.15,0.28,0.20,0.12,0.14,0.165,0.32,0.19,0.11,r2=CPFWAr21,r22,r23=0.28,0.18,0.12,0.17,0.16,0.21,0.11,0.18,0.28,r3=CPFWAr21,r22,r23=0.25,0.16,0.19,0.16,0.15,0,0.29,0.27,0.Step 4. The definitions of the score functions of rii=1,2,3 are Sr1=−0.0054, Sr2=−0.19, and Sr3=−0.11.Step 5. Since Sr1>Sr3>Sr2, we have A1>A3>A2. Hence, the gorgeous financial strategy is A1, that is, to invest in the Chinese markets.


## 6.1. Example by the CPFWA Operator

The example is applied in CPFWA operator to calculate the best one.Step 1. These three alternatives Ai,i=1,2,3 are to be solved by an expert under the three aspects Gˇjj=1,2,3 by using cubic picture fuzzy decision matrix D=P¨ij3×3=A^˙ij,ℬij for i,j=1,2,3.Step 2. Since the criteria Gˇ2 and Gˇ3 are the porches criteria while Gˇ1 are losses criteria R˜=rij3×3, equation (40) is used as follows: (41)R˜=0.1,0.3,0.2,0.1,0.1,0.1,0.3,0.2,0.10.1,0.1,0.2,0.1,0.3,0.2,0.4,0.3,0.20.3,0.4,0.2,0.2,0.1,0.3,0.1,0.2,0.10.3,0.1,0.1,0.2,0.1,0.3,0.1,0.2,0.30.1,0.2,0.1,0.3,0.2,0.3,0.2,0.3,0.40.4,0.3,0.2,0.1,0.3,0.1,0.1,0.1,0.20.3,0.1,0.2,0.1,0.1,0.0,0.3,0.4,0.10.1,0.1,0.2,0.2,0.3,0.3,0.1,0.1,0.10.3,0.3,0.2,0.3,0.2,0.1,0.4,0.3,0.0.Step 3. By following the CPFWA given in equation (30) with generatorgt=−Logt, we obtain the overall rating value of each alternative Ai as(42)r1=CPFWAr11,r12,r13=0.15,0.28,0.20,0.12,0.14,0.165,0.32,0.19,0.11,r2=CPFWAr21,r22,r23=0.28,0.18,0.12,0.17,0.16,0.21,0.11,0.18,0.28,r3=CPFWAr21,r22,r23=0.25,0.16,0.19,0.16,0.15,0,0.29,0.27,0.Step 4. The definitions of the score functions of rii=1,2,3 are Sr1=−0.0054, Sr2=−0.19, and Sr3=−0.11.Step 5. Since Sr1>Sr3>Sr2, we have A1>A3>A2. Hence, the gorgeous financial strategy is A1, that is, to invest in the Chinese markets.


## 7. Conclusion

The article is based on a novel approach to CPFSs as a generalization of two new strong concepts of CSs and PFSs. The basic operations for CPFSs are developed and exemplified. Some related results based on proposed operations are discussed. Several aggregation operators are defined for CPFSs and their properties are investigated. The proposed aggregation operators are subjected to a decision-making problem and the results are discussed. Furthermore, we developed multicriteria decision-making (MCDM) to prove the effectiveness and validity of the proposed methodology. A numerical example showed that the proposed operators can resolve decision-making more accurately. We compared these with predefined operators to show the validity and effectiveness of the proposed methodology.In the future, some similarity measures for CPFS can be developed and can be applied in pattern recognition problems. We will define other methods with CPFS such as Dombi aggregation operators and introduce the idea of cubic picture fuzzy Dombi weighted average (CPFDWA), cubic picture fuzzy Dombi ordered weighted average (CPFDOWA), cubic picture fuzzy Dombi weighted geometric (CPFDWG), cubic picture fuzzy Dombi ordered weighted geometric (CPFDOWG), and generalized operators in multicriteria decision-making.



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*Source: 9938181-2021-05-20.xml*