# N-Cubicq-Rung Orthopair Fuzzy Sets: Analysis of the Use of Mobile App in the Education Sector

Authors: Joseph David Madasi; Salma Khan; Nasreen Kausar; Dragan Pamucar; Muhammad Gulistan; Ben Sorowen
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

This study analyzes the description to examine the results of a new study and create the technique and also demonstrate the effectiveness of this technique. In this ever-changing world, students are increasingly encouraged to use mobile phones primarily to learn for educational purposes. The learning process is continuous and the goal has now been achieved. It has been replaced by online learning. Due to mobile phones as well as the many feature-oriented applications, students can study at their own place and use the application to spend their time understanding, because everything is accessible with a single click. To carry on the study we applied mobile applications for online education system. Now, because the traditional method is taken into consideration, it is normal to carry a bag full of books and copies and immerse yourself in the tradition of learning to write. However, it has been found that not all students learn when he takes notes. Therefore, we must make sure that the student focuses only on one thing at a time. To continue the research, we apply the N-cubic structure to q-rung orthopair fuzzy sets in multi-attribute group decision-making problems. This structure solves the problems of multi-attribute group decision-making techniques more generally.


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## Body

## 1. Introduction

Decision-making is an empathic process that allows the selection of alternatives from a set of possible attributes. In decision-making problems the data were ambiguous and uncertain and the representation of data is no longer in real number. For this purpose many researchers developed different theories to handle such type of data. Among these researchers, Zadeh [1] developed the theme of fuzzy set (FS) theory that could determine uncertainty and vagueness in classic sets which are based on only two values logic 0 and 1. In 1975, Zadeh [2–4] further expanded his ideas to interval-valued fuzzy sets (IVFS). Atanassov [5, 6] later came up with the idea that using intuitionistic fuzzy sets (IFS) to assist with the significance of the membership value as well as the nonmembership value. Wang et al. [7] defined some interval-valued intuitionistic fuzzy aggregation operators with basic operations and properties. Intuitionistic fuzzy set was generalized to the Pythagorean fuzzy set (PFS) [8] which described the value of membership and nonmembership with the condition that the square sum is less or equal to 1. PFS was generalized to q-rung orthopair fuzzy set [9]. In 2018 Ali [10] defined a new type of q-rung orthopair fuzzy sets where the domain of the function defining a q-ROF set is the region made up of orbits. To deal with the decision information, Liu and Wang [11] proposed the q-rung orthopair fuzzy weighted averaging operator and the q-rung orthopair fuzzy weighted geometric operator. Wei et al. [12] presented q-rung orthopair fuzzy Maclaurin symmetric mean operators and their applications to potential evaluation of emerging technology commercialization. Many researchers [13, 14] used the different versions of q-rung orthopair fuzzy sets in different field such as q-rung orthopair fuzzy soft sets, q-rung orthopair fuzzy hypersoft sets, and their operators. In 2012, Jun et al. [15] combined FS and IVFS and developed the theme of cubic set. In decision-making theory aggregation operators is an important component. The conflicting criteria are included in the multi-attribute decision-making (MADM) task, and the conflicting criteria are aggregated to solve the problem [13, 16]. Most aggregation operators treat criteria on an individual basis; they do not take into account how criteria interact with each other or with common criteria. Kaur and Garg [17, 18] developed cubic intuitionistic fuzzy aggregation operators, which includes two components at the same time. One component provides the degree of membership in the form of an interval value for cubic intuitionistic fuzzy numbers (CIFNs), as well as the second component, gives the degree of nonmembership in the form of fuzzy values. Abbas et al. [19] have described a modified version in CIFS that is known informally as cubic Pythagorean fuzzy sets (CPFS). Zang et al. [20] generalized CPFS into cubic q-rung orthopair fuzzy sets (CqROFSs). This allows decision-makers to explain their ideas better in the context of a fuzzy environment. In 2009, Jun et al. [21] defined negative-valued functions as well as the N-structure. This paper is on BCK/BCI algebra as well as subtraction algebra. Rashid et al. [22] used the concept of the N-structure and developed the theme of N-cubic sets, aggregate operators, and other concepts related to it. In 2020, Petrovic and Kankaras [23] developed a hybridized IT2FS-DEMATEL-AHP-TOPSIS multicriteria decision-making approach for the selection and evaluation of criteria for determination of air traffic control radar position. Agarwal et al. [24] discussed the development of management tools and techniques in decision-making for policy makers which are based on scientific evidence. Ali et al. [25] developed Einstein geometric aggregation operators using complex interval-valued pythagorean fuzzy set with application in green supplier chain management. We are currently employing the N-structure concept for q-ROFSs. The Cq-ROFS is a database that describes IVqROFS and q-ROFS in a way that is related to uncertainty in the information. In order to demonstrate how this structure might be used in decision-making, we shall examine issues relating to the N-structure of cubic q-rung orthopair fuzzy sets in this article. Although this study can manage decision-making more efficiently than fuzzy sets, using it manually is not simple. Therefore, we must create computer programming in order to overcome these constraints. By merging the N-structure with cubic q-ROF sets, this structure more specifically overcame the uncertainty issues. N-cubic q-rung orthopair fuzzy sets can effectively capture expert evaluation data and minimize fuzziness in decision-making outcomes.


## 2. Materials and Methods

In this section we recall some basic materials and methods.Definition 1 (see [6]).
LetG^≠∅ be universal set, then q-ROFS H^ be defined as(1)H^=g^,℧H^g^,ΩH^g^|g^∈G^,where ℧H^r´ and ΩH^r´ are a mapping from G^ to 0,1, also satisfy the condition as(2)0≤℧H^≤1,0≤ΩH^≤1,and(3)0≤℧H^g^q≤1,0≤ΩH^g^q≤1,where q≥1 for all g^∈G^ and represent the membership degree and the nonmembership degree to set H^.Definition 2 (see [6]).
LetG^≠∅ be universal set, then (IVq-ROFS) H^ be defined as(4)H^=g^,℧H^g^,ΩH^g^|g^∈G^,where ℧H^g^ and ΩH^g^ are a mapping from G^ to 0,1,(5)℧Pg^=℧H^Lg^,℧H^Ug^,and(6)ΩH^g^=ΩH^Lg^,ΩH^Ug^,also satisfy the condition as(7)0≤℧H^≤1,0≤ΩH^≤1,and(8)0≤℧H^Ug^q≤1,0≤ΩH^Ug^q≤1,where q≥1 for all g^∈G^ and represent the membership degree and the nonmembership degree to set H^.Definition 3 (see [10]).
LetX be the collection of some elements. A cubic-q-rung orthopair fuzzy set is represented as C=x,Hx,ϑx|x∈X, where Hx is an Interval-valued-q-rung orthopair fuzzy set and ϑx is a q-rung orthopair fuzzy set. Here Hx=℧L,℧UΩL,ΩU such that 0≤℧Uq+ΩUq≤1 and ϑx=℧,Ω with 0≤℧q+Ωq≤1 where q≥1. It can be described as C=H,ϑ, where H=℧L,℧UΩL,ΩU and ϑx=℧,Ω and it is known as the cubic-q-rung orthopair fuzzy set number.


## 3.N-Cubic q-Rung Orthopair Fuzzy Set and Hamy Mean Operators

This part develops the Nq-ROFS and NIVq-ROFS structures and introduces the innovative NCq-ROFS structure. The NCq-ROFS’s accuracy and score functions are defined. Both N-cubic q-rung orthopair fuzzy Hamy mean operator and N-cubic q-rung orthopair fuzzy power Hamy mean operator, as well as their characteristics and weighted forms, are covered in this section.Definition 4.
LetX be the collection of some elements. A Nq-ROFS define asNQRO=x,FNQROx,HNQROx:x∈X such that −1≤−12q˘+1FNQRO2q˘+HNQRO2q˘≤0, where FNQROx:X⟶−1,0 and HNQROx:X⟶−1,0.Definition 5.
A NIVq-ROFS in aϕ≠X is define as(9)NIV−QRO=x,℧˜NIV−QROx=℧NIV−QROL,℧NIV−QROU,Ω˜NIV−QROx=ΩNIV−QROL,ΩNIV−QROU:x∈x,with the condition(10)−1≤−12q˘+1℧NCQROU2q˘+ΩNCQROU2q˘≤0,where ℧NIV−QROL,℧NIV−QROU:X⟶−1,0 and ΩNIV−QROL,ΩNIV−QROU:X⟶−1,0.Definition 6.
An NCq-ROFS in aϕ≠X is define by the structure NCQRO = x,ΓNCQROx,ϜNCQROx:x∈X, where ΓNCQRO=x,℧˜NCQROx,Ω˜NCQROx:x∈X is an N−IVQROFS and ϜNCQRO=x,FNCQROx,HNCQROx:x∈X is an N−QROFS. Here ΓNCQRO=℧NCQROL,℧NCQROU,ΩNCQROL,ΩNCQROU:X⟶D−1,0 with the condition that−1≤−12q˘+1℧NCQROU2q˘+ΩNCQROU2q˘≤0 and ϜNCQRO=FNCQRO,HNCQRO:X⟶−1,0 with the condition−1≤−12q˘+1FNCQRO2q˘+HNCQRO2q˘≤0. For simplicity it is denoted by NCQRO = ΓNCQRO,ϜNCQRO.Definition 7.
An NCq-ROF setNCQRO = ΓNCQRO,ϜNCQRO in ϕ≠X is called internal NCq-ROF set if FNCQRO∈℧NCQROL,℧NCQROU and HNCQRO∈ΩNCQROL,ΩNCQROU for all x∈X, otherwise we called it an external NCq-ROF set.Definition 8.
The score functions under R-order of NCq-ROFNs(11)NiCQRO=℧NiCQROL,℧NiCQROU,ΩNiCQROL,ΩNiCQROU,FNiCQRO,HNiCQRO,is define as(12)SNiCQRO=1−12q˘+12141+℧NiCQROL2q˘−ΩNiCQROL2q˘+1+℧NiCQROU2q˘−ΩNiCQROU2q˘+FNiCQRO2q˘−HNiCQRO2q˘,now for P-order, we get(13)SNiCQRO=1−12q˘+12141+℧NiCQROL2q˘−ΩNiCQROL2q˘+1+℧NiCQROU2q˘−ΩNiCQROU2q˘+FNiCQRO2q˘−HNiCQRO2q˘,and accuracy function is defined as(14)HNiCQRO=1−12q˘+1212℧NiCQROL2q˘+℧NiCQROU2q˘+ΩNiCQROL2q˘+ΩNiCQROU2q˘+FNiCQRO2q˘+HNiCQRO2q˘,with the condition that(15)−1≤SNiCQRO≤1,0≤HNiCQRO≤1.Definition 9.
The comparison rule for two NCq-ROFNs(16)N1CQRO=℧N1CQROL,℧N1CQROU,ΩN1CQROL,ΩN1CQROU,FN1CQRO,HN1CQRO,and(17)N2CQRO=℧N2CQROL,℧N2CQROU,ΩN2CQROL,ΩN2CQROU,FN2CQRO,HN2CQRO,are defined as1 If SN1CQRO≻SN2CQRO, then N1CQRO≻N2CQRO. 2 If SN1CQRO=SN2CQROaHN1CQRO≻HN2CQRO, then N1CQRO≻N2CQRObHN1CQRO=HN2CQRO, then N1CQRO∼N2CQRO, where ″∼″ represent the “equivalent to.Example 1.
Assuming thatN1CQRO=−.7,−.2,−.2,−.1,−.3,−.5 andN2CQRO=−.5,−.4,−.4,−.3,−.2,−.7 are two NCq-ROFNs, the score function under R-order r are defined as(18)SNiCQRO=1−12q˘+12141+℧NiCQROL2q˘−ΩNiCQROL2q˘+1+℧NiCQROU2q˘−ΩNiCQROU2q˘+FNiCQRO2q˘−HNiCQRO2q˘,SN1CQRO=1−12q˘+12141+−.72q˘−−.22q˘+1+−.22q˘−−.12q˘+−.32q˘−−.52q˘,SN1CQRO=1−123+12141+−.723−−.223+1+−.223−−.123+−.323−−.523,SN1CQRO=1−172141+−.76−−.26+1+−.26−−.16+−.36−−.56,SN1CQRO=−12141.117585+1.000063+0.000729−0.15625,SN1CQRO=−12142.117648+−.155521,SN1CQRO=−120.529412−.155521,SN1CQRO=−120.3744202,SN1CQRO=−.1872101.For(19)SN2CQRO.SNiCQRO=1−12q˘+12141+℧NiCQROL2q˘−ΩNiCQROL2q˘+1+℧NiCQROU2q˘−ΩNiCQROU2q˘+FNiCQRO2q˘−HNiCQRO2q˘,SN2CQRO=1−12q˘+12141+−.52q˘−−.42q˘+1+−.42q˘−−.32q˘+−.22q˘−−.72q˘,where q = 3, then we get(20)SN2CQRO=1−123+12141+−.523−−.423+1+−.423−−.323+−.223−−.723,SN2CQRO=1−172141+−.56−−.46+1+−.46−−.36+−.26−−.76,SN2CQRO=−12141.011529+1.003367+0.000064−0.117649,SN2CQRO=−12142.014896+−0/117585,SN2CQRO=−120.503724−0117585,SN2CQRO=−120.386139,SN2CQRO=−.1930696.Now,(21)−.1872101≻−.1930696SN1CQRO≻SN2CQRON1CQRO≻N2CQRODefinition 10.
Considering the collection of NCq-ROFS to beNλλ=1,2,…n, j≥0, k≥0, if(22)NCq−ROFHMj,kN1,N2,…,Nn=2nn+1∑λ=1n∑s=1nNλjNsj1/j+k.It is then referred to as an NCq-ROFHM operator.Theorem 1.
Assuming thatj≥0, k≥0 and j+k≥0,Nλ=ΓNλ,ϜNλλ=1,2,…n are a set of NCq-ROFNs, the results of solving equation (22) are also NCq-ROFSs.(23)NCq−ROFHMλ1,λ2,…λn=−12q˘+11−℧NλLj℧NSLk2q˘2/nn+11/2q˘j+k,−12q˘+11−℧NλUj℧NSUk2q˘2/nn+11/2q˘j+k,−12q˘+11−1−Πλ=1nΠs=1n1−ΩNλL2q˘j1−ΩNsL2q˘k2/nn+11/j+k1/2q,−12q˘+11−1−Πλ=1nΠs=1n1−ΩNλU2q˘j1−ΩNsU2q˘k2/nn+11/j+k1/2q,−12q˘+11−℧Nλj℧NSk2q˘2/nn+11/2q˘j+k,−12q˘+11−1−Πλ=1nΠs=1n1−ΩNλ2q˘1−ΩNs2q˘k2/nn+11/j+k1/2q,.
(Idempotency) ConsiderNλ=NANλ,BNSλ=1,2,…n be a collection of NCq-ROFNS, if allNλ are identical, that is Nλ=N=ANλ,BNS for all λ, then NCq-ROFHMj,kN1,N2,..,Nn=N.Proof As,Nλ=N,∀λ we have(24)NCq−ROFHMj,kN1,N2,..,Nn=2nn+1∑λ=1n∑s=λnNλjNsk1/j+k=Nj+k1/j+k=N.(Monotonicity):Letαλ,βλλ=1,2,..n represent the two NCq-ROFN families, if αλ≤βλ∀λ=1,2,…,n then(25)NCq−ROFHMj,kα1,α2,…,αn≤NCq−ROFHMj,kβ1,β2,…βn.Proof.
Since,αλ≤βλ and αs≤βs for λ=1,2,..n and s=i,i+1,…,n, we have(26)αλjαsk≤βλjβsk.then(27)2nn+1∑λ=1n∑s=λnαλjαsk≤2nn+1∑λ=1n∑s=λnβλjβsk,so,(28)2nn+1∑λ=1n∑s=λnαλjαsk1/j+k≤2nn+1∑λ=1n∑s=λnβλjβsk1/j+k.And,(29)NCq−ROFHMj,kα1,α2,…,αn≤NCq−ROFHMj,kβ1,β2,…,βn.(Boundedness). Between the max and min operators is the NCq-ROFHM operator.(30)minN1,N2,…,Nn≤NCq−ROFHMj,kN1,N2,…,Nn≤maxN1,N2,…,Nn.Proof.
Letc=minN1,N2,…,Nn, d=maxN1,N2,…,Nn.
Using the aforementioned theorem, we obtain(31)NCq−ROFHMj,kc,c,…c≤NqQ−ROFHMj,kN1,N2,…,Nn≤NCq−ROFHMj,kd,d,…,d.And,(32)minN1,N2,…,Nn≤NCq−ROFHMj,kN1,N2,…,Nn≤maxN1,N2,…,Nn.Case 1.
The assertion that the recommended NCq-ROFHM operator transforms into the NCq-ROF basic HM operator ifj = k=(1/2).(33)NCq−ROFHM1/21/2N1,N2,…,Nn=−12q˘+11−Πλ=1nΠs=λn1−℧NλL℧NSL2q˘2/nn+11/2q˘,−12q˘+11−Πλ=1nΠs=λn1−℧NλU℧NSU2q˘2/nn+11/2q˘,−12q˘+1Πλ=1nΠs=λn1−1−ΩNλL2q˘1−ΩNλL2q˘2/nn+1∗1/2q˘,−12q˘+1Πλ=1nΠs=λn1−1−ΩNλU2q˘1−ΩNλU2q˘2/nn+1∗1/2q˘,,−12q˘+11−Πλ=1nΠs=λn1−℧Nλ℧NS2q˘2/nn+11/2q˘,−12q˘+1Πλ=1nΠs=λn1−1−ΩNλ2q˘1−ΩNs2q˘2/nn+1∗1/2q˘.Case 2.
Ifj=k=1 then (14) change into(34)NCQ−ROFHM1,1N1,N2,…,Nn=−12q˘+11−Πλ=1nΠs=λn1−℧NλL℧NSL2q˘2/nn+11/4q˘,−12q˘+11−Πλ=1nΠs=λn1−℧NλU℧NSU2q˘2/nn+11/4q˘,−12q˘+11−1−Πλ=1nΠs=λn1−1−ΩNλL2q˘1−ΩNsL2q˘2/nn+11/21/2q˘−12q˘+11−1−Πλ=1nΠs=λn1−1−ΩNλU2q˘1−ΩNsU2q˘2/nn+11/21/2q˘,−12q˘+11−Πλ=1nΠs=λn1−℧Nλ℧NS2q˘2/nn+11/2q˘,−12q˘+11−1−Πλ=1nΠs=λn1−1−ΩNλ2q˘1−ΩNs2q˘2/nn+11/21/2q˘.This means that it is also referred to as the N-cubic Q-rung orthopair fuzzy generalized interconnected square mean.Case 3.
Ifj⟶0, (34) is reduced to(35)limj⟶0NCQ−ROFHMj,kN1,N2,…,Nn=1n∑λ=1nNλj1/j=−12q˘+11−Πλ=1n1−℧NλLk2q˘1/n1/j+k,−12q˘+11−Πλ=1n1−℧NλUk2q˘1/n1/j+k,−12q˘+11−1−Πλ=1n1−1−ΩNλL2q˘k1/n1/k1/2q˘,−12q˘+11−1−Πλ=1n1−1−ΩNλU2q˘k1/n1/k1/2q˘,−12q˘+11−Πλ=1n1−℧Nλk2q˘1/n1/j+k,−12q˘+11−1−Πλ=1n1−1−ΩNλ2q˘k1/n1/k1/2q˘.It is sometimes referred to as the N-cubic q-rung s fuzzy generalized mean.Case 4.
Ifj=1 and k⟶0, (27) becomes an N-cubic q-rung orthopair fuzzy average mean.(36)limk⟶0NCQ−ROFHMj,kN1,N2,…,Nn=1n⊗λ=1nNλ=−12q˘+11−⊗λ=1n1−℧NλL2q˘2/n1/2q,−12q˘+11−⊗λ=1n1−℧NλU2q˘2/n1/2q,−12q˘+11−1−⊗λ=1n1−1−ΩNλL2q˘1/n1/2q,−12q˘+11−1−⊗λ=1n1−1−ΩNλU2q˘1/n1/2q,−12q˘+11−⊗λ=1n1−℧Nλ2q˘1/n12q˘,−12q˘+11−1−⊗λ=1n1−1−ΩNλ2q˘1/n1/2q.Case 5.
Ifj⟶0,k⟶0, then the existing NCq-ROFHM change into(37)limj⟶0NqQ−ROFHMj,0N1,N2,…,Nn=limk⟶01n⊗λ=1nNλj1/k=⊗λ=1nNλ1/n.Note that we can get a variety of orthopair fuzzy sets by varying the value of the parameter q. As an illustration, the N-cubic Pythagorean fuzzy set is renovated by NCq-ROFHM if j = 1 and k = 1. In MADM situations, different characteristics typically have significant advantages. Thus, it appears that the NCq-ROFHM operator is indifferent with this characteristic. The weighted version of the NCq-ROFHM operator is defined as follows to address this issue:Definition 11.
In this case,Nλ=ANλ,BNλλ=1,2,…,n be the NCq-ROFN family, the weight vector of NCq-ROFNs is indicated by j≥0,k≥0, j+k≥0, and w=w1,w2,…,wn for all wλ∈0,1 and ∑λ=1nw=1.Then NCq-ROFWHM: −1,0n⟶−1,0 such that(38)NCq−ROFWHMwj,kN1,N2,…,Nn=2nn+1∑λ=1n∑s=1nwλNλjwsNsk1/j+k.Theorem 2.
LetNλ=ANλ,BNλλ=1,2,…,n be the collection of NCq-ROFNs, j≥0,k≥0 and j+k≥0, and w=w1,w2,…,wn represents the weight vector of NCq-ROFNs, wλ∈0,1 and ∑λ=1nw=1. Then, NCq-ROFNs are also included in the resulting equation (38) as(39)NCq−ROFWHMwj,kN1,N2,…,Nn=−12q˘+11−Πλ=1nΠs=λn1−℧NλL2q˘2/nn+11/2q˘j+k,−12q˘+11−Πλ=1nΠs=λn1−℧NλU2q˘2/nn+11/2q˘j+k,−12q˘+11−1−Πλ=1nΠs=λnΩNλL4q/nn+11/j+k1/2q˘,−12q˘+11−1−Πλ=1nΠs=λnΩNλU4q/nn+11/j+k1/2q˘,−12q˘+11−Πλ=1nΠs=λn1−℧Nλ2q˘2/nn+11/2q˘j+k,−12q˘+11−1−Πλ=1nΠs=λnΩNλ4q/nn+11/j+k1/2q˘.where(40)℧λL,℧λU=−12q˘+11−1−℧λL2q˘wλj/2q˘1−1−℧sL2q˘wsk/2q˘,−12q˘+11−1−℧λU2q˘wλj/2q˘1−1−℧sU2q˘wsk/2q˘,ΩλL,ΩλU=−12q˘+11−1−ΩλL2q˘wλj1−ΩsL2q˘wsk1/2q˘,−12q˘+11−1−ΩλU2q˘wλj1−ΩsU2q˘wsk1/2q˘,℧λ=−12q˘+11−1−℧λ2q˘wλj/2q˘1−1−℧s2q˘wsk/2q˘,Ωλ=−12q˘+11−1−Ωλ2q˘wλj1−Ωs2q˘wsk1/2q˘.The relationship between the structure of the two attributes can be established through the HM operator. Each attribute is linked with other attributes of the HM operator. However, when it comes to decision-making issues, this condition is often not being met. To prevent the separation of characteristics we can use different partitions to solve decision-making problems because we remember the structure of attribute relationships. There is no link between attributes. When they are divided by two partitions, the same attributes present in partitions have a connection to each other. With the typical HM operator, the partitions do not solve these kinds of issues so we now provide the N-cubic q-rung orthopair fuzzy power Hamy mean operator with the ability to let us know the issue. The condition given above can be mathematically explained as: LetNλ=ANλ,BNλλ=1,2,..n be a collection of NCq-ROFNs, distributed into “g” different partitions FF1,F2,….,Fg with Fi∩Fı˜ and ∪i=1gFi=NiFi=Ni1,Ni2,…NiFi, where Fi denotes the cardinality of partitions Fi and ∑i=1gFi=n. By using above information, NCQ-ROFPHM operator is defined asDefinition 12.
LetNλ=ANλ,BNλλ=1,2,..n be a family of NCq-ROFNs,j≥0,k≥0 and j+k≥0. Then NCq-ROFPHM −1,0n⟶−1,0 and(41)NCq−ROFPHMj,kN1,N2,…,Nn=1g∑i=1g2FiFi+1∑λ=1Fi∑s=λFiNiλj⊗Nisk1/j+k.Theorem 3.
LetNλ=ANλ,BNλλ=1,2,..n be a family of NCq-ROFNs, j≥0,k≥0 and j+k≥0, then equation (41) is used to generate a consequent equation that is likewise an NCq-ROFN, as shown by(42)NCq−ROFPHMj,kN1,N2,…,Nn=−12q˘+11−Πi=1g1−1−1−℧iL2q˘2/FiFi+11/j+k1/g1/2q˘,−12q˘+11−Πi=1g1−1−1−℧iU2q˘2/FiFi+11/j+k1/g1/2q˘,−12q˘+1Πi=1g1−1−ΩiL4q/FiFi+11/j+k1/q∗g,−12q˘+1Πi=1g1−1−ΩiL4q/FiFi+11/j+k1/q∗g,−12q˘+11−Πi=1g1−1−1−℧i2q˘2/FiFi+11/j+k1/g1/2q˘,−12q˘+1Πi=1g1−1−Ωi4q/FiFi+11/j+k1/q∗g,Where(43)℧iL=−12q˘+11−Πλ=1FiΠs=λFi1−ANiλjBNiskq1/q˘,℧iU=−12q˘+11−Πλ=1FiΠs=λFi1−ANiλjBNiskq1/q˘,ΩiL=−12q˘+1Πλ=1FiΠs=λFi1−1−BNiλ2q˘j1−BNis2q˘k1/2q˘,ΩiU=−12q˘+1Πλ=1FiΠs=λFi1−1−BNiλ2q˘j1−BNis2q˘k1/2q˘,℧i=−12q˘+11−Πλ=1FiΠs=λFi1−ANiλjBNisk2q˘1/2q˘,Ωi=−12q˘+1Πλ=1FiΠs=λFi1−1−BNiλ2q˘j1−BNis2q˘k1/2q˘.Theorem 4.
Letj≥0,k≥0j+k≥0,(44)Nλ=ANλ,BNλλ=1,2,…,n.Be collection of NCq-ROFNs with g different subset Fλλ=1,2,…,n. Consequently, the NCq-ROFPHM operators have the following characteristics.(Idempotency) If allNλ are same that is, Nλ=N=AN,BN∀λ then(45)NCq−ROFPHMj,kN1,N2,…,Nn=N=AN,BN,Proof.
(46)NCq−ROFPHMj,kN1,N2,…,Nn=1g∑i=1g2FiFi+1∑λ=1Fi∑s=λFiNiλj⊗Nisk1/j+k=1g∑i=1g2FiFi+1∑λ=1Fi∑s=λFiNj⊗Nk1/j+k=1g∑i=1g2FiFi+1∑λ=1Fi∑s=λFiNj+k1/j+k1g∑λ=1gN=N.,
(Monotonicity) LetMλ=AMλ,BMλλ=1,2,…,n be a set NCq-ROFNs having the same partitioned structure as Nλ=ANλ,BNλλ=1,2,…,n,AMλ≥ANλand BMλ≤BNλ for all, then(47)NCq−ROFPHMj,kM1,M2,…,Mn¨≥NCq−ROFPHMj,kN1,N2,…,Nn¨.Proof.
Since,AMλ≥ANλand BMλ≤BNλ for all λ using Definition 6, we can obtain, Mλ≥Nλ for all, then AMiλjAMisk≥ANiλjAisk and(48)−12q˘+11−1−BMiλ2q˘j1−BMis2q˘k≥−12q˘+11−1−BNiλ2q˘j1−BNis2q˘k.Further,(49)℧Mi=−12q˘+11−Πλ=1FiΠs=λFi1−AMiλjBMisk2q˘1/2q≥−12q˘+11−Πλ=1FiΠs=λFi1−ANiλjBNisk2q˘1/2q=℧Ni,and(50)ΩMi=−12q˘+1Πλ=1FiΠs=λFi1−1−BMiλ2q˘j1−BMis2q˘k1/2q≥−12q˘+1Πλ=1FiΠs=λFi1−1−BNiλ2q˘j1−BNis2q˘k1/2q=ΩNi.Thus,(51)−12q˘+11−Πi=1g1−1−1−℧MiL2q˘2/FiFi+11/j+k1/g1/2q,−12q˘+11−Πi=1g1−1−1−℧MiU2q˘2/FiFi+11/j+k1/g1/2q≥−12q˘+11−Πi=1g1−1−1−℧NiL2q˘2/FiFi+11/j+k1/g1/2q,−12q˘+11−Πi=1g1−1−1−℧NiU2q˘2/FiFi+11/j+k1/g1/2q,and(52)−12q˘+1Πi=1g1−1−ΩMiL4q/FiFi+11/j+k1/2q∗g,−12q˘+1Πi=1g1−1−ΩMiU4q/FiFi+11/j+k1/2q∗g≤−12q˘+1Πi=1g1−1−ΩNiL4q/FiFi+11/j+k1/2q∗g,−12q˘+1Πi=1g1−1−ΩNiU4q/FiFi+11/j+k1/2q∗g.
Then we use (37), we get(53)NCq−ROFPHMj,kM1,M2,…,Mn≥NCq−ROFPHMj,kN1,N2,…,Nn.
(Boundedness) Letc = maxλAN,minλBN,d = minλAN,maxλBN,having a specific partition stricture Fλλ=1,2,…,n. Therefore,(54)c≤NCq−ROFPHMj,kN1,N2,…,Nn≤d.Proof.
Sincec = maxλAN,minλBN,d = minλAN,maxλBN,subsequently, based on the monotonicity, we have(55)NCq−ROFPHMj,kc,c,…c=c,and(56)NCq−ROFPHMj,kd,d,…,d=d.As a result,(57)c≤NCq−ROFPHMj,kN1,N2,…,Nn≤d,thus proved. Various particular examples of the NCq−ROFPHM operator can be obtained by altering the number of partitions and various values of the parameters “j,k.” The NCq−ROFPHM operator renovate into usual NCq−ROFPHM if g=1 as follows:(58)NCq−ROFPHMj,kN1,N2,…,Nn=2FiFi+1∑λ=1Fi∑s=λFiN1λjN1sk1/j+k=2nn+1∑λ=1n∑s=λnNλjNsk1/j+k.By giving varied values to the parameters “j, k” and g=1, we can clearly obtain the situations covered in equations (33)–(37).Definition 13.
LetNλ=ANλ,BNλλ=1,2,…,n be a set of NCq-ROFNs,j≥0,k≥0 and j+k≥0, and w=w1,w2,…,wn indicate the weight vector of NCq-ROFNs wλ∈1,0 and ∑λ=1nw=1. Then NCq-ROFWPHM: −1,0n⟶−1,0 such that(59)NCq−ROFWPHMwj,kN1,N2,…,Nn=1g⊕i=1g2FiFi+1⊕λ=1Fi⊕s=λFiwiλNiλj⊗wisNisk1/j+k.Theorem 5.
LetNλ=ANλ,BNλλ=1,2,…,n be a family of NCq-ROFNs where j≥0,k≥0and j+k≥0, and w=w1,w2,…,wn represents the weight vector of NCq-ROFNs, ∑λ=1nw=1. Then we get resultant equation by using equation (59) that is also a NCq-ROFNs given by(60)NCq−ROFWPHMwj,kN1,N2,…,Nn−12q˘+11−Πi=1g1−1−1−℧iL2q˘2/FiFi+11/j+k1/g1/2q,−12q˘+11−Πi=1g1−1−1−℧iU2q˘2/FiFi+11/j+k1/g1/2q,−12q˘+1Πi=1g1−1−ΩiL4q/FiFi+11/j+k1/q∗g,−12q˘+1Πi=1g1−1−ΩiL4q/FiFi+11/j+k1/q∗g,−12q˘+11−Πi=1g1−1−1−℧iU2q˘2/FiFi+11/j+k1/g1/2q,−12q˘+1Πi=1g1−1−ΩiL4q/FiFi+11/j+k1/q∗g.where(61)℧iL=−12q˘+11−Πλ=1FiΠs=λFi1−1−1−Niλ2q˘wiλj1−1−1−Nis2q˘wisk1/2q,℧iU=−12q˘+11−Πλ=1FiΠs=λFi1−1−1−Niλ2q˘wiλj1−1−1−Nis2q˘wisk1/2q,ΩiL=−12q˘+1Πλ=1FiΠs=λFi1−1−Ωiλwiλ2qj1−Ωiswit2qk1/2q,ΩiU=−12q˘+1Πλ=1FiΠs=λFi1−1−Ωiλwiλ2qj1−Ωiswit2qk1/2q,℧i=−12q˘+11−Πλ=1FiΠs=λFi1−1−1−Niλ2q˘wiλj1−1−1−Nis2q˘wisk1/2q,Ωi=−12q˘+1Πλ=1FiΠs=λFi1−1−Ωiλwiλ2qj1−Ωiswit2qk1/2q.


## 4. Multi Attribute Group Decision-Making Method as an Application

In this section we will use NCq-ROFWHM and NCq-ROFWPHM operators to examine MAGDM problems, and to show their applicability with the help of NCq-ROFNs. LetA¨=A1,A2,…,Am be a set of alternatives, C=C´1,C´2,…,C´m and attributes with weight vector w=w1,w2,…,wn, where wı˜∈0,1 and ∑ı˜=1nwı˜=1. Let ⊼=⊼`1,⊼`2,…,⊼`d be a group of experts with eight vector, ξ=ξ1,ξ2,…,ξdλ=where ξλ∈0,1 and ∑λ=1dξλ=1. Assume that the λth expert provides his opinion regarding the alternatives Ai=1,2,…,m with regard to the qualities C´ı˜=1,2,…,m as a NCq-ROFNs Niı˜λ=ANiı˜λ,BNiı˜λ.Using the expert’s preference, an NCq-ROF decision matrix is created as Tλ=Niı˜λm×n. Consider that there are ‘g’ divisions of the set F1,F2,F3,……,Fg and that there is a specified connection structure between the features while keeping in mind the natural relationship structure. There is no link between qualities from different partitions and those from the same partition. The established operators are then used to address these decision-making (DM) difficulties. Algorithm steps are provided byStep 1: To normalize the decision matrix and obtain the benefit and cost-type data.Ť=N´iı˜λ=AN´iı˜λ,BN´iı˜λm×n converting the value of the cost-type attributes first to the value of the benefit-type attributes, and then(62)N´iı˜λ=Niı˜λfor benefit−type attribute ofı˜CNiı˜λcfor cost−type attribute ofı˜C.whereNiı˜λc=BNiı˜λ,ANiı˜λ.Step 2: To aggregate all the normalized data.Ťλ= (λ=1,2,3,…,d) into a collective DM M=Viı˜m×n=AN´iı˜λ,BN´iı˜λm×n.(63)Vij=NCq−ROFWHMξj,kN´iı˜1,N´iı˜2,N´iı˜3,…,N´iı˜d−12q˘+11−Πλ=1nΠs=λn1−℧NλL2q˘2/nn+11/2q˘j+k,−12q˘+11−Πλ=1nΠs=λn1−℧NλU2q˘2/nn+11/2q˘j+k,−12q˘+11−1−Πλ=1nΠs=λnΩNλL2q˘/nn+11/j+k1/2q−12q˘+11−1−Πλ=1nΠs=λnΩNλU2q˘/nn+11/j+k1/2q,−12q˘+11−Πλ=1nΠs=λn1−℧Nλ2q˘2/nn+11/2q˘j+k,−12q˘+11−1−Πλ=1nΠs=λnΩNλ2q˘/nn+11/2j+k1/2q,,where(64)℧λL,℧λU=−12q˘+11−1−℧λL2q˘Bλj/2q˘1−1−℧sL2q˘Bsk/2q˘,−12q˘+11−1−℧λU2q˘Bλj/2q˘1−1−℧sU2q˘Bsk/2q˘,,ΩλL,ΩλU=−12q˘+11−1−ΩλL2q˘Bλj1−ΩsL2q˘Btkj/2q˘,−12q˘+11−1−ΩλU2q˘Bλj1−ΩsU2q˘Btkj/2q˘,℧λL=−12q˘+11−1−℧λ2q˘Bλj/2q˘1−1−℧s2q˘Bsk/2q˘,ΩλL=−12q˘+11−1−Ωλ2q˘Bλj1−Ωs2q˘Btkj/2q˘.Step 3: Assume a division form among the attributes to arrive at the collective assessment values.Vi=AN´iı˜λ,BN´iı˜λi=1,2,3,…m;ı˜=1,2,3,…n of alternatives Ai.(65)Vi=NCQ−ROFWHPwj,kVi1Vi2,…,Vin.Step 4: To find score valuesS^Vi of each alternative Ai=1,2,3,…,m.Example 2.
In this section we provide a brief overview of the outcomes of a brand-new technique and show its efficacy. Utilizing the full potential of mobile apps for online education, business administrators can check the effectiveness of these programs. Four possibilities have been suggested as possible options in the beginning stages. MoodleA1, LMS A2, Zoom A3, and NoonA4 are the four applications. There are four experts on the judgment board, E1,E2,E3,E4 each with a different area of competence. Take into account that λ represents the different expert weights, or λ=0.03,0.1,0.27,0.6. The five interconnected characteristics listed by the assessment committee are as follows: the app’s download, data storage speeds, data loading speed in and battery use (C1andC 2, C3 and C4, respectively). Assume represents different attribute weights, for example, w=0.17,0.2,0.23,0.4. The two subsets of the five qualities are separated based on how they relate to one another fundamentally. F1=C1,C3,C5,F1=C2,C4. Data in the form of NCq-ROFNs must be submitted by experts for examination. The expert assessment statistics are displayed in Tables 1–4, and Ei=i=1,2,3,4.Step 1: Given thatC3 is a cost-type attribute, we can normalize the decision-making data using equation (62). The normalized data is displayed in Tables 5–8.Step 2: To obtain the entire decision matrix, use equation (63). M=U˜iı˜4×5=AU˜iı˜,BU˜iı˜4×5. Additionally, we set the parameters j = 1, k = 1, and q = 3 to be true. This MAGDM seeks to identify the best choice. The complete NCq-ROF decision matrix M is shown in Table 9.STEP 3: Use (23) to calculate all of the evaluation values for each option, then use Ai and U˜i to obtain the values for each alternative’s Aii=1,2,3,4 collective evaluation.U˜1=−.0000754307,−.0039453,−.836279,−.827616,−.00345878,−.865973, U˜2=−.0000856423,−.0054665,−.839751,−.829231,−.00265612,−.860356, U˜3=−.0000346567,−.0039675,−.836134,−.826964,−.00467348,−.863867, U˜4=−.0000126569,−.0067831,−.830651,−.825320,−.00679601,−.865328.STEP 4: We compute score valuesSU˜iof U˜i as follows: SU˜1=−.57679527,SU˜2=−.5976312,SU˜3=−.56140327,SU˜4=0.5787565 as SU˜3>SU˜1>SU˜4>SU˜2.Hence A3>A1>A4>A2 and A3 is best alternative.The Influence of the parameter Values on the Ranking Results. In the following section, we will investigate how the parameters q, j, and k impact the findings of the alternatives. Put j=1,k=1 and q=3 in the previous computing technique for our convenience and without losing generality. From Table 10, it is clear that the ranking outcomes for the scenarios q=4,5,7,8 and A3>A1>A4>A2 are identical. Thus the ranking outcomes are shown as in Figure 1, and finally, we can say that the other top options remain the same when the parameter’s value changes.Table 1
For NCq-ROFDM ofQ1.
C1C2C3C4C5A1−6−3−5−4−6−1−7−3−5−4−9−4−7−3−5−4−9−4−9−6−5−5−4−1−2−2−9−5−9−7A2−8−5−7−6−4−3−6−2−6−5−6−1−8−5−6−3−6−3−6−2−8−6−6−3−7−2−4−5−7−1A3−3−2−5−5−9−6−6−5−7−4−6−1−7−6−7−4−9−8−8−6−5−5−6−4−9−8−6−5−7−1A4−4−2−8−5−8−7−9−4−7−5−6−6−8−7−6−5−7−4−6−2−9−5−5−4−7−5−6−4−9−3Figure 1
Ranking result forq = 4,5,7,8.Table 2
For NCq-ROFDM ofQ2.
C1C2C3C4C5A1−5−4−7−5−7−2−7−6−3−2−3−8−5−3−6−4−6−8−4−3−3−2−1−4−6−3−4−2−8−1A2−3−2−5−4−8−2−2−1−7−6−9−4−7−5−4−3−5−8−7−6−2−1−6−3−2−1−5−4−7−4A3−9−7−2−1−6−1−9−7−2−1−6−2−8−6−7−3−8−5−6−5−7−5−9−2−6−4−3−2−6−5A4−5−3−6−4−7−4−3−2−8−5−6−4−6−5−5−5−5−3−8−7−9−5−5−4−4−2−3−2−5−4Table 3
For NCq-ROFDM ofQ3.
C1C2C3C4C5A1−5−4−7−5−7−2−7−6−3−2−3−8−5−3−6−4−6−8−4−3−3−2−1−4−6−3−4−2−8−1A2−3−2−5−4−8−2−2−1−7−6−9−4−7−5−4−3−5−8−7−6−2−1−6−3−2−1−5−4−7−4A3−9−7−2−1−6−1−9−7−2−1−6−2−8−6−7−3−8−5−6−5−7−5−9−2−6−4−3−2−6−5A4−5−3−6−4−7−4−3−2−8−5−6−4−6−5−5−5−5−3−8−7−9−5−5−4−4−2−3−2−5−4Table 4
For NCq-ROFDM ofQ4.
C1C2C3C4C5A1−6−3−5−4−6−1−7−3−5−4−9−4−7−3−5−4−9−4−9−6−5−5−4−1−2−2−9−5−9−7A2−8−5−7−6−4−3−6−2−6−5−6−1−8−5−6−3−6−3−6−2−8−6−6−3−7−2−4−5−7−1A3−3−2−5−5−9−6−6−5−7−4−6−1−7−6−7−4−9−8−8−6−5−5−6−4−9−8−6−5−7−1A4−4−2−8−5−8−7−9−4−7−5−6−6−8−7−6−5−7−4−6−2−9−5−5−4−7−5−6−4−9−3Table 5
For normalized NCq-ROFDM ofQ1.
C1C2C3C4C5A1−5−4−7−5−7−2−7−6−3−2−3−8−5−3−6−4−6−8−4−3−3−2−1−4−6−3−4−2−8−1A2−3−2−5−4−8−2−2−1−7−6−9−4−7−5−4−3−5−8−7−6−2−1−6−3−2−1−5−4−7−4A3−9−7−2−1−6−1−9−7−2−1−6−2−8−6−7−3−8−5−6−5−7−5−9−2−6−4−3−2−6−5A4−5−3−6−4−7−4−3−2−8−5−6−4−6−5−5−5−5−3−8−7−9−5−5−4−4−2−3−2−5−4Table 6
Of normalized NCq-ROFDM ofQ2.
C1C2C3C4C5A1−6−3−5−4−6−1−7−3−5−4−9−4−7−3−5−4−9−4−9−6−5−5−4−1−2−2−9−5−9−7A2−8−5−7−6−4−3−6−2−6−5−6−1−8−5−6−3−6−3−6−2−8−6−6−3−7−2−4−5−7−1A3−3−2−5−5−9−6−6−5−7−4−6−1−7−6−7−4−9−8−8−6−5−5−6−4−9−8−6−5−7−1A4−4−2−8−5−8−7−9−4−7−5−6−6−8−7−6−5−7−4−6−2−9−5−5−4−7−5−6−4−9−3Table 7
Of normalized NCq-ROFDM ofQ3.
C1C2C3C4C5A1−4−2−5−3−7−1−8−7−4−2−9−4−4−1−5−4−2−6−5−3−4−2−1−7−6−3−4−2−6−3A2−6−4−2−1−3−5−6−5−4−3−9−4−4−2−6−5−7−7−9−6−5−3−2−8−2−1−4−2−7−2A3−2−1−4−3−9−6−5−4−3−2−7−4−2−1−7−5−3−8−6−5−7−5−9−2−7−4−2−1−6−5A4−5−3−6−4−5−5−3−2−8−5−6−4−5−3−4−2−4−5−8−6−8−5−5−4−5−2−2−2−5−6Table 8
Of normalized NCq-ROFDM ofQ4.
C1C2C3C4C5A1−5−4−7−5−7−2−7−6−3−2−3−8−5−3−6−4−6−8−4−3−3−2−1−4−6−3−4−2−8−1A2−3−2−5−4−8−2−2−1−7−6−9−4−7−5−4−3−5−8−7−6−2−1−6−3−2−1−5−4−7−4A3−9−7−2−1−6−1−9−7−2−1−6−2−8−6−7−3−8−5−6−5−7−5−9−2−6−4−3−2−6−5A4−5−3−6−4−7−4−3−2−8−5−6−4−6−5−5−5−5−3−8−7−9−5−5−4−4−2−3−2−5−4Table 9
For collective NCq-ROFDM of M.
C1C2C3C4C5A1−882357−0100657−722689−063629−437647−078382−677505−440853−094596−0287674−816469−176754−156341−056971−7804565−057936−709723−048765−980032−657461−745276−045512−832719−621549−786541−004215−967331−898736−481265−343255A2−0637383−420758−854671−542951−0764617−032604−976432−785421−6432515−443045−816469−176754−156341−056972−7804565−057936−709723−048765−844501−032167−726235−692963−564317−463246−854156−064874−905332−170456−567812−462467A3−96319−580327−494761−423218−998348−257343−049304−394847−768512−126789−346092−566718−998601−266534−599851−32192−908723−231168−304653−086215−374209−174129−444578−235075−946115−386430−753521−064251−987801−8521133A4−643012−074213−818596−254591−303785−032105−88506−873400−246543−194085−256375−019642−970547−075908−568652−181623−960782−567149−974534−375413−568786−293648−851636−166284−26301−143665−974624−356891−708643−226138Table 10
Ranking result for various values of parameterq.
QScore valuesRanking resultsq=4S1=−.5378,S2=−.5479,S3=−.5248,S4=−.5406A`3>A1>A4>A2q=5S1=−.5389,S2=−.5499,S3=−.5348,S4=−.5443A`3>A1>A4>A2q=7S1=−.5225,S2=−.5489,S3=−.5129,S4=−.5460A`3>A1>A4>A2q=8S1=−.5485,S2=−.5879,S3=−.5381,S4=−.5498A`3>A1>A4>A2These are different from the results obtained forj=0 and k=1 having ranking results A4>A1>A2>A3. As a result, it is possible to obtain varied ranking results by varying the values of the parameters j and k. If one parameter is fixed and the other is changed, the score and ranking results may change, as shown in Table 11. We can observe that the values of the parameters j and k affect the ranking outcomes, as shown in Figure 2.Table 11
Ranking result for different values of parametersj and k.
j And kScore valuesRanking resultsj=0,k=1S1=−.5286,S2=−.5759,S3=−.5148,S4=−.5463A3>A1>A4>A2j=3,k=1S1=−.5409,S2=−.5592,S3=−.5948,S4=−.5873A1>A2>A4>A3j=1,k=3S1=−.5825,S2=−.4489,S3=−.5879,S4=−.5260A2>A4>A1>A3j=0,k=5S1=−.5695,S2=−.5779,S3=−.5235,S4=−.5983A3>A1>A2>A4j=5,k=6S1=−.5805,S2=−.5949,S3=−.5621,S4=−.5498A4>A3>A1>A2j=7,k=1S1=−.5951,S2=−.5765,S3=−.5743,S4=−.5885A3>A2>A4>A1j=0,k=3S1=−.5195,S2=−.5669,S3=−.5930,S4=−.5573A1>A4>A2>A3j=1,k=0S1=−.4585,S2=−.5379,S3=−.3981,S4=−.5198A3>A1>A4>A2j=5,k=5S1=−.5578,S2=−.5678,S3=−.5421,S4=−.5950A3>A1>A2>A4j=6,k=4S1=−.5085,S2=−.3459,S3=−.4081,S4=−.4985A2>A3>A4>A1Figure 2
Ranking result for various values of parameters j and k.


## 5. Conclusion

In this study, we focus on the structure of N-cubic q-rung orthopair fuzzy sets. The score function under R-order and the comparison rule for two N-cubic q-rung orthopair fuzzy sets also define some aggregation operators, i.e., N-cubic q-rung orthopair fuzzy Hamy mean operator, N-cubic q-rung orthopair fuzzy weighted Hamy mean operator, N-cubic q-rung orthopair fuzzy power Hamy mean operator, and N-cubic q-rung orthopair fuzzy power weighted Hamy mean operator. N-structure can enhance decision-making performance. The recently discovered N-cubic q-ROFSs, which combine NQ-ROFSs and NIVqRFSs into a single structure, allow decision-makers greater space to work on multi-attribute group decision-making problems. As a result of the debate, we have discussed specific instances of the operators and created a method for solving MAGDM problems using NCq-ROFNs. In this study we analyze the use of mobile app in the education sector. Further research, problem-solving, and decision-making are possible to solve, and other operators may be able to be created through this method. In future someone can apply the N-cubic q-rung orthopair fuzzy sets in different decision-making technique.



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*Source: 9984314-2022-09-30.xml*