Order-αCQ Divergence Measures and Aggregation Operators Based on Complex q-Rung Orthopair Normal Fuzzy Sets and Their Application to Multi-Attribute Decision-Making

Zeeshan Ali; Tahir Mahmood; Abdu Gumaei

doi:10.2991/ijcis.d.210622.004

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Volume 14, Issue 1, 2021, Pages 1895 - 1922

Order-α_CQ Divergence Measures and Aggregation Operators Based on Complex q-Rung Orthopair Normal Fuzzy Sets and Their Application to Multi-Attribute Decision-Making

Authors

Zeeshan Ali¹, Tahir Mahmood¹^,, Abdu Gumaei²^{, *}^,

¹Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan

²STC's Artificial Intelligence Chair, Department of Information Systems, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia

^*Corresponding author: Email: agumaei.c@ksu.edu.sa

Corresponding Author

Abdu Gumaei

Received 30 May 2020, Accepted 8 June 2021, Available Online 1 July 2021.

DOI: 10.2991/ijcis.d.210622.004 How to use a DOI?
Keywords: Complex q-Rung orthopair normal fuzzy sets; Order-αCQ divergence measures; Aggregation operators; Multi-attribute decision-making
Abstract: Complex q-rung orthopair fuzzy set (CQROFS) contains the grade of supporting and the grade of supporting against in the form of polar coordinates belonging to unit disc in a complex plane and is a proficient technique to address awkward information, although the normal fuzzy number (NFN) examines normal distribution information in anthropogenic action and a realistic environment. Based on the advantages of both notions, in this manuscript, we explored the novel concept of a complex q-rung orthopair normal fuzzy set (CQRONFS) as an imperative technique to evaluate unreliable and complicated information. Some operational laws based on CQRONFSs are also explored. Additionally, some distance measures, called complex q-rung orthopair normal fuzzy generalized distance measure (CQRONFGDM), complex q-rung orthopair normal fuzzy symmetric distance measure (CQROFNFSDM), two types of complex q-rung orthopair normal fuzzy order- divergence measures (CQRONFODMs), and their special cases are discussed. Moreover, weighted averaging, weighted geometric, generalized weighted averaging, and generalized weighted geometric operators based on CQRONFSs are also presented. In last, we solved a numerical example of a multi-attribute decision-making (MADM) problem is shown to justify the proficiency of the presented operators. The advantages, comparative and sensitive analyses are used to express the efficiency and flexibility of the explored approach.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

In our everyday life, most people are regularly confronted with MADM issues, which may include various other options and numerous assessment components. Because of the multifaceted nature of human social exercises and the vulnerability of indigenous habitats, the way to manage such dubious data has gotten the key to taking care of the MADM issues. Zadeh [1] explored a supporting-based fuzzy set (FS), which viably portrayed the fuzzy data and dubious condition, and hence the advantage to suggest a superior choice. Additionally, Atanassove [2] modified the theory of FS to intuitionistic FS (IFS) containing three components, that is, participation degree, nonenrollment degree, and hesitation degree. IFS has been broadly considered, and various scholars have extended into different kinds of notions [3–6]. Additionally, Yager [7] situated the modified version of IFS is called Pythagorean FS (PFS) and meet the conditions that the square sum of its supporting grade and supporting against grade is not exceeded from unit interval. Several scholars have utilized it in different fields [8–12]. Further, Yager [13] situated the modified version of PFS is called q-rung orthopair FS (QROFS) and meet the conditions that the q-power sum of its supporting grade and supporting against grade is not exceeded from unit interval. Several scholars have utilized it in different fields [14–18].

From the above winning investigations, it has been breaking down that the above examination has been led under the uncertainties or their expansions which can just arrangement with the vulnerability that exists in the data. None of these models can speak to the incomplete numbness of the information and its variances at a given period. Nonetheless, in complex informational indexes, for example, information from the clinical research, database for biometric and facial acknowledgment, and so on. Vulnerability and dubiousness in the information happen simultaneously with changes to the stage (periodicity) of the information. To deal with the periodicity of the information into the decision-making problems, Ramot et al. [19] presented the idea of the complex fuzzy set (CFS), a modified version of the FS, displayed by complex-valued supporting grade with a co-domain unit circle in an unpredictable plane. Additionally, Alkouri and Salleh [20] modified the theory of CFS to complex IFS (CIFS) containing three components, that is, complex-valued participation degree, complex-valued nonenrollment degree, and complex-valued hesitation degree. CIFS has been broadly considered and various scholars have extended it into different kinds of notions [21–23]. Additionally, Ullah et al. [24] situated the modified version of CIFS is called complex PFS (CPFS) and meet the conditions that the square sum of its real part (also for the imaginary part) of the supporting grade and real part (also for the imaginary part) of the supporting against grade is not exceeded form unit interval. Several scholars have utilized it in different fields [25]. Further, Liu et al. [26,27] situated the modified version of CPFS is called complex QROFS (CQROFS) and meet the conditions that the q-power sum of its real part (also for the imaginary part) of the supporting grade and real part (also for the imaginary part) of the supporting against grade is not exceeded form unit interval. Several scholars have utilized it in different fields [28].

The point of this exploration is to introduce a novel decision-making technique to take care of the MADM issues utilizing CQRONFSs which is a mixture of CQROFSs and normal fuzzy numbers (NFNs) [29] with powerful averaging and geometric operators. The CQRONFSs is a speculation of the QRONFS thinking about the supporting grade and supporting against are complex-valued and are expressed in polar coordinates with NFNs. The amplitude term gives the degree of belongingness of an item in a CQRONFS and the stage terms are commonly identified with periodicity. These stage terms recognize the CQRONFS and customary QRONFS theories. Uncertainties hypothesis manages just each measurement in turn which brings about data misfortune in certain examples. Be that as it may, all things considered, we run over complex characteristic marvels where it gets basic to add the second measurement to the declaration of participation and non-enrollment grades. By presenting this subsequent measurement, the total data can be expert projected in one set, and henceforth, loss of data can stay away from. To delineate the essentialness of the stage term, we give a model. “Consider XYZ organization chooses to set up biometric-based participation gadgets (BBPGs) in the entirety of its workplaces spread everywhere throughout the nation. For this, the organization counsels a specialist who gives the data regarding (i) demonstrates of BBPGs and (ii) creation dates of BBPGs. The organization needs to choose the most ideal model of BBPGs with its creation date all the while. Here, the issue is two-dimensional, which cannot be demonstrated at the same time utilizing customary QRONFS theories. The most ideal approach to speak to the entirety of the data gave by the master is by utilizing CQRONFS theories. The amplitude terms in CQRONFS might be utilized to give the organization's choice concerning the model of BBPGs, and the stage terms might be utilized to speak to the organization's judgment concerning creation date of BBPGs.”

When a decision-maker gives 0.0834eι2π0.1005 for truth grade, 0.1487eι2π0.304 for falsity grade, and 0.6845,0.1718 for a NFN, then the existing notions are cannot be able to cope with it. In the writing, there are a couple of techniques qualified for dealing with the MADM issues by utilizing the idea of CQRONFS. Because the explored idea of CQRONFS is more powerful and more general than existing notions, whose detail is discussed below:

If we choose the value of the imaginary part will be zero in CQRONFS, then the CQRONFS is converted for q-rung orthopair normal FS.
If we choose the value of the imaginary part will be zero in CQRONFS for q = 2, then the CQRONFS is converted for the Pythagorean normal FS.
If we choose the value of the imaginary part will be zero in CQRONFS for q = 1, then the CQRONFS is converted for an intuitionistic normal FS.
If we choose the value of the q = 2 in CQRONFS, then the CQRONFS is converted for a complex Pythagorean normal FS.
If we choose the value of the q = 1 in CQRONFS, then the CQRONFS is converted for a complex intuitionistic normal FS.
If we choose the value of normal FS will be zero in CQRONFS, then the CQRONFS is converted for CQROFS.
If we choose the value of the normal FS will be zero in CQRONFS for q = 2, then the CQRONFS is converted for a complex PFS.
If we choose the value of the normal FS will be zero in CQRONFS for q = 1, then the CQRONFS is converted for a complex IFS. The geometrical expression of the unit disc is discussed in the form of Figure 1.

In this way, inspired by the attributes of the CQRONFS model and the significance of data aggregation, this paper centers around investigating the basic qualities of CQRONFSs and their aggregation operators (AOs) for dealing with the multidimensional complex informational collections. The primary accomplishments of this examination are:

To handle the uncertainties in a more precise environment using CQRONFSs and their fundamental properties are explored.
To explore some new Order-αCQ divergence measures to combine the preferences. We have explored the TOPSIS Method (see Section 6), and the AO and distance measures are part of the TOPSIS method, so this is the reason, why we have explored two different concepts in one work.
To explore some new AOs to combine the preferences.
To explore an efficient algorithm based on explored measures and operators to solve MADM problems.
To illustrate the approach with a numerical example for evaluating the proficiency and reliability of the presented approaches.

To achieve these objectives, we provide more flexibility to the decision-maker to provide their preferences in terms of CQRONFSs to achieve the first objective. The second objective is done by proposing some new Order-αCQ divergence measures based on CQRONFSs. The third objective is done by proposing some new AOs, namely, weighted averaging, weighted geometric, generalized weighted averaging, and generalized weighted geometric operators based on CQRONFSs are also presented to aggregate the different CQRONFNs. Various relations and properties are also studied in it. The fourth objective is completed by developing an efficient algorithm based on explored measures and operators to solve MADM problems for finding to rank the alternatives by using the proposed approaches. Finally, the feasibility and the comparative analysis have been done to fulfill the fourth objective with several existing studies.

The remaining text is outlined as follows: in Section 2, we review some notions like NFN, CQROFS, and their operational laws. In Section 3, we explored the novel concept of CQRONFS is an important technique to evaluate unreliable and complicated information. Some operational laws based on CQRONFSs are also explored. In Section 4, some distance measures are called CQRONFGDM, CQROFNFSDM, two types of CQRONFODMs, and their special cases are discussed. In Section 5, some weighted averaging, weighted geometric, generalized weighted averaging, and generalized weighted geometric operators based on CQRONFSs are also presented. In Section 6, we solve a numerical example on multi-attribute decision-making (MADM) problem is shown to justify the proficiency of the presented operators. The advantages, comparative and sensitive analysis are used to express the efficiency and flexibility of the explored approach. The conclusion of this article is discussed in Section 7.

2. PRELIMINARIES

The purpose of this communication is to review some notions like NFN, CQROFS, and their operational laws. Throughout, this manuscript, the symbol XUNI denotes the universal set.

Definition 1.

[29] A NFN NNF=Ψ,Ω on a finite universal set XUNI based on truth function of fuzzy number is stated by

NNFx=e−x−ΨΩ2(1)

where ℛRN denotes the real number set with Ω,γSC>0.

Definition 2.

[29] For any two NFNs NNF−1=Ψ1,Ω1 and NNF−2=Ψ2,Ω2, then

γSCNNF−1=γSCΨ1,γSCΩ1(2)

NNF−1+NNF−2=Ψ1+Ψ2,Ω1+Ω2(3)

The distance measure based on NFNs is stated by

dDMNNF−1,NNF−2=Ψ1−Ψ22+12Ω1−Ω22(4)

Definition 3.

[26,27] The CQROFS NCQ is stated by

NCQ=ϕNCQ′xi,ψNCQ′xi:x∈XUNI(5)

where ϕNCQ′xi=ϕNCQxieι2πηϕNCQxi denotes the complex-valued truth degree and ψNCQ′xi=ψNCQxieι2πηψNCQxi denotes the complex-valued falsity degree, which holds the following conditions: 0≤ϕNCQqSCxi+ψNCQqSCxi≤1 and 0≤ηϕNCQqSCxi+ηψNCQqSCxi≤1. Further, μNCQxi=1−ϕNCQqSCxi−ψNCQqSCxi1qSCeι2π1−ηϕNCQqSCxi−ηψNCQqSCxi1qSC represents the hesitancy degree. The complex q-rung orthopair fuzzy number (CQROFNs) is followed as NCQ=ϕNCQeι2πηϕNCQ,ψNCQeι2πηψNCQ.

Definition 4.

[26,27] For any two CQROFNs NCQ−1=ϕNCQ−1eι2πηϕNCQ−1,ψNCQ−1eι2πηψNCQ−1 and NCQ−2=ϕNCQ−2eι2πηϕNCQ−2,ψNCQ−2eι2πηψNCQ−2, then

NCQ−1⊕CQNCQ−2=ϕNCQ−1qSC+ϕNCQ−2qSC−ϕNCQ−1qSCϕNCQ−2qSC1qSCeι2πηϕNCQ−1qSC+ηϕNCQ−2qSC−ηϕNCQ−1qSCηϕNCQ−2qSC1qSC,ψNCQ−1ψNCQ−2eι2πηψNCQ−1ηψNCQ−2(6)

NCQ−1⊗CQNCQ−2=ϕNCQ−1ϕNCQ−2eι2πηϕNCQ−1ηϕNCQ−2,ψNCQ−1qSC+ψNCQ−2qSC−ψNCQ−1qSCψNCQ−2qSC1qSCeι2πηψNCQ−1qSC+ηψNCQ−2qSC−ηψNCQ−1qSCηψNCQ−2qSC1qSC(7)

γSCNCQ−1=1−1−ϕNCQ−1qSCγSC1qSCeι2π1−1−ηϕNCQ−1qSCγSC1qSC,ψNCQ−1γSCeι2πηψNCQ−1γSC(8)

NCQ−1γSC=ϕNCQ−1γSCeι2πηϕNCQ−1γSC,1−1−ψNCQ−1qSCγSC1qSCeι2π1−1−ηψNCQ−1qSCγSC1qSC(9)

Definition 5.

[26,27] For any CQROFN NCQ−1=ϕNCQ−1eι2πηϕNCQ−1,ψNCQ−1eι2πηψNCQ−1, then the score function SSF and accuracy function ℋAF are stated by

SSFNCQ−1=ϕNCQ−1qSC−ψNCQ−1qSC+ηϕNCQ−1qSC−ηψNCQ−1qSC2(10)

ℋAFNCQ−1=ϕNCQ−1qSC+ψNCQ−1qSC+ηϕNCQ−1qSC+ηψNCQ−1qSC2(11)

For finding the relationships between any two CQROFNs, we use the following inequalities:

If SSFNCQ−1>SSFNCQ−2⇒NCQ−1>NCQ−2;
If SSFNCQ−1=SSFNCQ−2⇒
1. If ℋAFNCQ−1>ℋAFNCQ−2⇒NCQ−1>NCQ−2;
2. If ℋAFNCQ−1=ℋAFNCQ−2⇒NCQ−1=NCQ−2.

3. COMPLEX Q-RUNG ORTHOPAIR NF N

The purpose of this communication is to present the notion of CQRONFN, which is the mixture of CQROFS and NFN to cope with uncertain and awkward information in realistic decision theory. Some basic operational laws for CQRONFN are also explored.

Definition 6.

The CQRONFN NCQN is stated by

NCQN=ΨNCQN,ΩNCQN,ϕNCQN′xi,ψNNCQ′xi:x∈XUNI(12)

where ϕNCQ′xi=e−x−ΨΩ2ϕNCQxieι2πηϕNCQxi=e−x−ΨΩ2∗ϕNCQxieι2πe−x−ΨΩ2∗ηϕNCQxi=ϕRPxieι2πηϕIPxi denotes the complex-valued truth degree and ψNCQ′xi=e−x−ΨΩ21−1−ψNCQxieι2π1−1−ηψNCQxi=e−x−ΨΩ2∗1−1−ψNCQxieι2πe−x−ΨΩ2∗1−1−ηψNCQxi=ψRPxieι2πηψIPxi denotes the complex-valued falsity degree, which holds the following conditions: 0≤ϕRPqSCxi+ψRPqSCxi≤1 and 0≤ηϕIPqSCxi+ηψIPqSCxi≤1. The complex q-rung orthopair NFN (CQRONFNs) is followed as NCQN=ΨNCQN,ΩNCQN,ϕRPeι2πηϕIP,ψRPeι2πηψIP.

Definition 7.

For any two CQRONFNs NCQN−1=ΨNCQN−1,ΩNCQN−1,ϕRP−1eι2πηϕIP−1,ψRP−1eι2πηψIP−1 and NCQN−2=ΨNCQN−2,ΩNCQN−2,ϕRP−2eι2πηϕIP−2,ψRP−2eι2πηψIP−2, then

NCQN−1⊕CQNNCQN−2=ΨNCQN−1+ΨNCQN−2,ΩNCQN−1+ΩNCQN−2,ϕRP−1qSC+ϕRP−2qSC−ϕRP−1qSCϕRP−2qSC1qSCeι2πηϕIP−1qSC+ηϕIP−2qSC−ηϕIP−1qSCηϕIP−2qSC1qSC,ψRP−1ψRP−2eι2πηψIP−1ηψIP−2(13)

NCQN−1⊗CQNNCQN−2=ΨNCQN−1∗ΨNCQN−2,ΩNCQN−1∗ΩNCQN−2,ϕRP−1ϕRP−2eι2πηϕIP−1ηϕIP−2,ψRP−1qSC+ψRP−2qSC−ψRP−1qSCψRP−2qSC1qSCeι2πηψIP−1qSC+ηψIP−2qSC−ηψIP−1qSCηψIP−2qSC1qSC(14)

γSCNCQN−1=γSCΨNCQN−1,γSCΩNCQN−1,1−1−ϕRP−1qSCγSC1qSCeι2π1−1−ηϕIP−1qSCγSC1qSC,ψRP−1γSCeι2πηψIP−1γSC(15)

NCQN−1γSC=ΨNCQN−1γSC,ΩNCQN−1γSC,ϕRP−1γSCeι2πηϕIP−1γSC,1−1−ψRP−1qSCγSC1qSCeι2π1−1−ηψIP−1qSCγSC1qSC(16)

Definition 8.

For any CQRONFN NCQN−1=ΨNCQN−1,ΩNCQN−1,ϕRP−1eι2πηϕIP−1,ψRP−1eι2πηψIP−1, then the score function and accuracy function are stated by

SSFNCQN−1=ΨNCQN−1+ΩNCQN−12+ϕRP−1qSC−ψRP−1qSC+ηϕIP−1qSC−ηψIP−1qSC3(17)

ℋAFNCQN−1=ΨNCQN−1+ΩNCQN−12+ϕRP−1qSC−ψRP−1qSC+ηϕIP−1qSC−ηψIP−1qSC3(18)

For finding the relationships between any two CQRONFNs, we use the following inequalities:

If SSFNCQN−1>SSFNCQN−2⇒NCQN−1>NCQN−2;
If SSFNCQN−1=SSFNCQN−2⇒
1. If ℋAFNCQN−1>ℋAFNCQN−2⇒NCQN−1>NCQN−2;
2. If ℋAFNCQN−1=ℋAFNCQN−2⇒NCQN−1=NCQN−2.

Theorem 1.

For any two CQRONFNs NCQN−1=ΨNCQN−1,ΩNCQN−1,ϕRP−1eι2πηϕIP−1,ψRP−1eι2πηψIP−1 and NCQN−2=ΨNCQN−2,ΩNCQN−2,ϕRP−2eι2πηϕIP−2,ψRP−2eι2πηψIP−2, then

NCQN−1⊕CQNNCQN−2=NCQN−2⊕CQNNCQN−1(19)

NCQN−1⊕CQNNCQN−2⊕CQNNCQN−3=NCQN−1⊕CQNNCQN−2⊕CQNNCQN−3(20)

NCQN−1⊗CQNNCQN−2=NCQN−2⊗CQNNCQN−1(21)

NCQN−1⊗CQNNCQN−2⊗CQNNCQN−3=NCQN−1⊗CQNNCQN−2⊗CQNNCQN−3(22)

γSCNCQN−1⊕CQNNCQN−2=γSCNCQN−1⊕CQNγSCNCQN−2(23)

γSC−1+γSC−2NCQN−1=γSC−1NCQN−1+γSC−1NCQN−1(24)

NCQN−1⊗CQNNCQN−2γSC=NCQN−1γSC⊗CQNNCQN−2γSC(25)

Proof.

Straightforward.

4. DISTANCE MEASURES BASED ON CQRONFNs

The purpose of this section is to explore some distance measures are called CQRONFGDM, CQROFNFSDM, complex q-rung orthopair normal fuzzy Order-αCQ divergence measures dGDMSDM−1NCQN−1,NCQN−2, and dGDMSDM−2NCQN−1,NCQN−2 and their special cases are discussed. Further, these all measures are also converted into similarity measures.

Definition 9.

For any two CQRONFNs NCQN−1=ΨNCQN−1,ΩNCQN−1,ϕRP−1eι2πηϕIP−1,ψRP−1eι2πηψIP−1 and NCQN−2=ΨNCQN−2,ΩNCQN−2,ϕRP−2eι2πηϕIP−2,ψRP−2eι2πηψIP−2, the CQRONFGDM is stated by

dGDMNCQN−1,NCQN−2=121+ϕRP−1qSC−ψRP−1qSC+ηϕIP−1qSC−ηψIP−1qSC2∗ΨNCQN−1qSC−1+ϕRP−2qSC−ψRP−2qSC+ηϕIP−2qSC−ηψIP−2qSC2∗ΩNCQN−1qSC+12ΨNCQN−2+ΨNCQN−21qSC(26)

Eq. (26), must hold the following conditions:

dGDMNCQN−1,NCQN−2≥0;
dGDMNCQN−1,NCQN−2=dGDMNCQN−2,NCQN−1;
dGDMNCQN−1,NCQN−2=0⇔NCQN−1=NCQN−2.

In the CQRONFS hypothesis, enrollment and nonmembership degrees are intricate esteemed and are spoken to in polar directions. The abundance term comparing to the participation (nonmembership) degree gives the degree of things (not‐belongings) of an item in a CQRONFS, and the stage term related to enrollment (nonmembership) degree gives the extra data, by and large, related with periodicity. The stage terms are novel boundaries of the participation and nonmembership degrees, and these are the boundaries that recognize the customary QRONFS and CQRONFS hypothesis. QRONFS hypothesis manages just each measurement in turn, which brings about data misfortune on certain occasions. In any case, in day‐to‐day life, we run over complex normal marvels where it gets fundamental to add the second measurement to the statement of participation and nonmembership grades. By presenting this subsequent measurement, the total data can be extended in one set, and henceforth, loss of data can be maintained a strategic distance from. To show the hugeness of the stage term, consider an illustration of a specific organization that chooses to put in new information handling and examination programming. For this, the organization counsels a specialist who gives the data concerning (a) alternate choices of programing (b) relating programing variant. The organization needs to choose the most ideal alternative(s) of programing with its most recent form all the while. Here, the issue is two-dimensional, to be specific, to choose the ideal option of programing and its most recent form. This issue cannot be displayed precisely utilizing the conventional QRONFS hypothesis. Along these lines, the most ideal approach to speak to all the data gave by the master is by utilizing the CQRONFS hypothesis. The adequacy terms in CQRONFS might be utilized to give an organization's choice concerning the option of programing and the stage terms might be utilized to speak to the organization's choice regarding programing adaptation. Various researchers have used various sorts of measures in the fields of FS hypothesis and their expansions. However, cutting-edge nobody investigated the veers estimates dependent on proposed thoughts because the proposed thoughts are more summed up than existing thoughts.

Definition 10.

For any two CQRONFNs NCQN−1=ΨNCQN−1,ΩNCQN−1,ϕRP−1eι2πηϕIP−1,ψRP−1eι2πηψIP−1 and NCQN−2=ΨNCQN−2,ΩNCQN−2,ϕRP−2eι2πηϕIP−2,ψRP−2eι2πηψIP−2, the complex q-rung orthopair normal fuzzy symmetric divergence measure dGDMSDMNCQN−1,NCQN−2 is given by

dGDMSDMNCQN−1,NCQN−2=12dGDMSDM−1NCQN−1,NCQN−2+dGDMSDM−2NCQN−1,NCQN−2(27)

Definition 11.

For any two CQRONFNs NCQN−1=ΨNCQN−1,ΩNCQN−1,ϕRP−1eι2πηϕIP−1,ψRP−1eι2πηψIP−1 and NCQN−2=ΨNCQN−2,ΩNCQN−2,ϕRP−2eι2πηϕIP−2,ψRP−2eι2πηψIP−2 with XUNI=x, the complex q-rung orthopair normal fuzzy Order-αCQ divergence measures dGDMSDM−1NCQN−1,NCQN−2 and dGDMSDM−2NCQN−1,NCQN−2 are given by

dGDMSDM−1NCQN−1,NCQN−2=1αCQ−1log2ϕRP−1qSC+ηϕIP−1qSC2∗ΨNCQN−1αCQϕRP−1qSC+ηϕIP−1qSC+ϕRP−2qSC+ηϕIP−2qSC4∗ΩNCQN−11−αCQ+ψRP−1qSC+ηψIP−1qSC2∗ΨNCQN−2αCQψRP−1qSC+ηψIP−1qSC+ψRP−2qSC+ηψIP−2qSC4∗ΩNCQN−21−αCQ(28)

dGDMSDM−2NCQN−1,NCQN−2=1e2αCQ−1−eeϕRP−1qSC+ηϕIP−1qSC2∗ΨNCQN−1αCQϕRP−1qSC+ηϕIP−1qSC+ϕRP−2qSC+ηϕIP−2qSC4∗ΩNCQN−11−αCQ+ψRP−1qSC+ηψIP−1qSC2∗ΨNCQN−2αCQψRP−1qSC+ηψIP−1qSC+ψRP−2qSC+ηψIP−2qSC4∗ΩNCQN−21−αCQ−e(29)

where αCQ∈0,1 and e represents the exponential functions.

Eqs. (27–29) are also satisfied the three conditions of Definition 9. Further, we have discussed some special cases of the explored measures which are discussed below.

SGSMNCQN−1,NCQN−2=1−dGDMNCQN−1,NCQN−2(30)

Is called complex q-rung orthopair normal fuzzy generalized similarity measure. Similarly, we can find more similarity measures from Eqs. (27–29).

5. AOs BASED ON CQRONFNs

The purpose of this communication is to present some AOs based on CQRONFNs is called complex q-rung orthopair normal fuzzy weighted averaging (CQRONFWA), complex q-rung orthopair normal fuzzy weighted geometric (CQRONFWG), complex q-rung orthopair normal fuzzy generalized weighted averaging (CQRONFGWA), complex q-rung orthopair normal fuzzy generalized weighted geometric (CQRONFGWG) operators, and their special cases.

Definition 12.

For any family of CQRONFNs NCQN−j=ΨNCQN−j,ΩNCQN−j,ϕRP−jeι2πηϕIP−j,ψRP−jeι2πηψIP−j, j=1,2,…,n, then the CQRONFWA operator is stated by

CQRONFWANCQN−1,NCQN−2,…,NCQN−n=∑j=1nωW−jNCQN−j(31)

where ωW=ωW−1,ωW−2,…,ωW−n,ωW−j∈0,1 with a condition that is ∑j=1nωW−j=1.

Theorem 2.

For any family of CQRONFNs NCQN−j=ΨNCQN−j,ΩNCQN−j,ϕRP−jeι2πηϕIP−j,ψRP−jeι2πηψIP−j, j=1,2,…,n, then by using the Eq. (31), we get

CQRONFWANCQN−1,NCQN−2,…,NCQN−n=∑j=1nωW−jΨNCQN−j,∑j=1nωW−jΩNCQN−j,1−∏j=1n1−ϕRP−jqSCωW−j1qSCeι2π1−∏j=1n1−ηϕIP−jqSCωW−j1qSC,∏j=1nψRP−jeι2π∏j=1nηψIP−j(32)

Proof.

By using the mathematical induction, we have proven the Eq. (32), if n=2, then we have

CQRONFWANCQN−1,NCQN−2=ωW−1NCQN−1⊕CQNωW−2NCQN−2=ωW−1ΨNCQN−1,ωW−1ΩNCQN−1,1−1−ϕRP−1qSCωW−11qSCeι2π1−1−ηϕIP−1qSCωW−11qSC,ψRP−1eι2πηψIP−1⊕CQNωW−2ΨNCQN−2,ωW−2ΩNCQN−2,1−1−ϕRP−2qSCωW−21qSCeι2π1−1−ηϕIP−2qSCωW−21qSC,ψRP−2eι2πηψIP−2=ωW−1ΨNCQN−1+ωW−2ΨNCQN−2,ωW−1ΩNCQN−1+ωW−2ΩNCQN−2,1−1−ϕRP−1qSCωW−11qSC+1−1−ϕRP−2qSCωW−21qSC−1−1−ϕRP−1qSCωW−11qSC1−1−ϕRP−2qSCωW−21qSCeι2π1−1−ηϕIP−1qSCωW−11qSC+1−1−ηϕIP−2qSCωW−21qSC−1−1−ηϕIP−1qSCωW−11qSC1−1−ηϕIP−2qSCωW−21qSC,ψRP−1ψRP−2eι2πηψIP−1ηψIP−2

=∑j=12ωW−jΨNCQN−j,∑j=12ωW−jΩNCQN−j,1−∏j=121−ϕRP−jqSCωW−j1qSCeι2π1−∏j=121−ηϕIP−jqSCωW−j1qSC,∏j=12ψRP−jeι2π∏j=12ηψIP−j

Further, we choose for n=k, then we have

CQRONFWANCQN−1,NCQN−2,…,NCQN−k=∑j=1kωW−jΨNCQN−j,∑j=1kωW−jΩNCQN−j,1−∏j=1k1−ϕRP−jqSCωW−j1qSCeι2π1−∏j=1k1−ηϕIP−jqSCωW−j1qSC,∏j=1kψRP−jeι2π∏j=1kηψIP−j

We have proved that for n=k+1, such that

CQRONFWANCQN−1,NCQN−2,…,NCQN−k+1=∑j=1kωW−jΨNCQN−j+ωW−k+1ΨNCQN−k+1,∑j=1kωW−jΩNCQN−j+ωW−k+1ΩNCQN−k+1,1−∏j=1k1−ϕRP−jqSCωW−j1qSC+1−1−ϕRP−k+1qSCωW−k+11qSC−1−∏j=1k1−ϕRP−jqSCωW−j1qSC1−1−ϕRP−k+1qSCωW−k+11qSCeι2π1−∏j=1k1−ηϕIP−jqSCωW−j1qSC+1−1−ηϕIP−k+1qSCωW−k+11qSC−1−∏j=1k1−ηϕIP−jqSCωW−j1qSC1−1−ηϕIP−k+1qSCωW−k+11qSC,∏j=1kψRP−j∗ψRP−k+1eι2π∏j=1kηψIP−j∗ηψIP−k+1=∑j=1nωW−jΨNCQN−j,∑j=1nωW−jΩNCQN−j,1−∏j=1n1−ϕRP−jqSCωW−j1qSCeι2π1−∏j=1n1−ηϕIP−jqSCωW−j1qSC,∏j=1nψRP−jeι2π∏j=1nηψIP−j=CQRONFWANCQN−1,NCQN−2,…,NCQN−n

Hence the result is completed.

Further, we have discussed some properties based on CQRONFNs are called idempotent, boundedness, and monotonicity.

Theorem 3.

For any family of CQRONFNs NCQN−j=ΨNCQN−j,ΩNCQN−j,ϕRP−jeι2πηϕIP−j,ψRP−jeι2πηψIP−j, j=1,2,…,n, if NCQN−j=NCQN, then

CQRONFWANCQN−1,NCQN−2,…,NCQN−n=NCQN(33)

Proof.

By hypothesis, it's clear that NCQN−j=NCQN, then we get

CQRONFWANCQN−1,NCQN−2,…,NCQN−n=∑j=1nωW−jΨNCQN−j,∑j=1nωW−jΩNCQN−j,1−∏j=1n1−ϕRP−jqSCωW−j1qSCeι2π1−∏j=1n1−ηϕIP−jqSCωW−j1qSC,∏j=1nψRP−jeι2π∏j=1nηψIP−j=∑j=1nωW−jΨNCQN,∑j=1nωW−jΩNCQN,1−∏j=1n1−ϕRPqSCωW−j1qSCeι2π1−∏j=1n1−ηϕIPqSCωW−j1qSC,∏j=1nψRPeι2π∏j=1nηψIP=ΨNCQN,ΩNCQN,1−1−ϕRPqSC1qSCeι2π1−1−ηϕIPqSC1qSC,ψRPeι2πηψIP,∑j=1nωW−j=1=NCQN.

Theorem 4.

For any family of CQRONFNs NCQN−j=ΨNCQN−j,ΩNCQN−j,ϕRP−jeι2πηϕIP−j,ψRP−jeι2πηψIP−j, j=1,2,…,n, if NCQN−j−=minΨNCQN−j,minΩNCQN−j,minϕRP−jeι2πminηϕIP−j,maxψRP−jeι2πaxηψIP−j and NCQN−j+=maxΨNCQN−j,maxΩNCQN−j,maxϕRP−jeι2πmaxηϕIP−j,minψRP−jeι2πminηψIP−j, then

NCQN−j−≤CQRONFWANCQN−1,NCQN−2,…,NCQN−n≤NCQN−j+(34)

Proof.

From the above analysis it is clear that ∑j=1nωW−jΨNCQN−j−≤∑j=1nωW−jΨNCQN−j+ and ∑j=1nωW−jΩNCQN−j−≤∑j=1nωW−jΩNCQN−j+. Further, we can prove that for complex-valued supporting grade and also for complex-valued supporting against, we have

Case 1: We have considered the real part of the supporting grade, such that

1−∏j=1n1−min1≤j≤nϕRP−j qSCωW−j1qSC≤1−∏j=1n1−ϕRP−jqSCωW−j1qSC≤1−∏j=1n1−max1≤j≤nϕRP−jqSCωW−j1qSC

⇒1−1−min1≤j≤nϕRP−j qSC∑j=1nωW−j1qSC≤1−∏j=1n1−ϕRP−jqSCωW−j1qSC≤1−1−max1≤j≤nϕRP−j qSC∑j=1nωW−j1qSC

Because ∑j=1nωW−j=1, so

⇒min1≤j≤nϕRP−j≤1−∏j=1n1−ϕRP−jqSCωW−j1qSC≤max1≤j≤nϕRP−j

Similarly, we can find for the imaginary part of the complex-valued supporting grade, we have

⇒min1≤j≤nηϕIP−j≤1−∏j=1n1−ηϕIPqSCωW−j1qSC≤max1≤j≤nηϕIP−j

Case 2: We have considered the real part of the supporting against the grade, such that

∏j=1nmin1≤j≤nψRP−jωW−j≤∏j=1nψRP−jωW−j≤∏j=1nmax1≤j≤nψRP−jωW−j⇒min1≤j≤nψRP−j ∑j=1nωW−j≤∏j=1nψRP−jωW−j≤max1≤j≤nψRP−j∑j=1nωW−j

And because ∑j=1nωW−j=1 so

min1≤j≤nψRP−j≤∏j=1nψRP−jωW−j≤max1≤j≤nψRP−j

Then combined the above two cases, which is discussed for real parts, we have

min1≤j≤nϕRP−j−max1≤j≤nψRP−j≤1−∏j=1n1−ϕRP−jqSCωW−j1qSC−∏j=1nψRP−jωW−j≤max1≤j≤nϕRP−j−min1≤j≤nψRP−j

By the Eqs. (10) and (11), we get

SSFNCQ−j−≤SSFNCQ−j≤SSFNCQ−j+

Similarly, we will find imaginary parts. So based on cases (1) and (2) and Eq. (10), we get

NCQN−j−≤CQRONFWANCQN−1,NCQN−2,…,NCQN−n≤NCQN−j+.

Theorem 5.

For any family of CQRONFNs NCQN−j=ΨNCQN−j,ΩNCQN−j,ϕRP−jeι2πηϕIP−j,ψRP−jeι2πηψIP−j, j=1,2,…,n, and NCQN−∗j=ΨNCQN−∗j,ΩNCQN−∗j,ϕRP−∗jeι2πηϕIP−∗j,ψRP−∗jeι2πηψIP−∗j if ΨNCQN−j≤ΨNCQN−∗j,ϕRP−j≤ϕRP−∗j,ηϕIP−j≤ηϕIP−∗j,ψRP−j≥ψRP−∗j and ηψIP−j≥ηψIP−∗j, then

CQRONFWANCQN−1,NCQN−2,…,NCQN−n≤CQRONFWANCQN−∗1,NCQN−∗2,…,NCQN−∗n(35)

Proof.

Consider that CQRONFWANCQN−1,NCQN−2,…,NCQN−n=X¨,Y˙ and CQRONFWANCQN−∗1,NCQN−∗2,…,NCQN−∗n=X¨′,Y˙′, then for proving we consider the supporting grade, whose real part follows: X¨≤X¨′ and Y˙≥Y˙′. If ΨNCQN−j≤ΨNCQN−∗j,ϕRP−j≤ϕRP−∗j,ηϕIP−j≤ηϕIP−∗j,ψRP−j≥ψRP−∗j and ηψIP−j≥ηψIP−∗j, then we obtain

∑j=1nωW−jΨNCQN−j≤∑j=1nωW−jΨNCQN−∗j

And the real part of the supporting grade is following as

1−∏j=1n1−ϕRP−jqSCωW−j1qSC≤1−∏j=1n1−ϕRP−∗jqSCωW−j1qSC

Similarly, for the imaginary part of the supporting grade, we have

1−∏j=1n1−ηϕIP−jqSCωW−j1qSC≤1−∏j=1n1−ηϕIP−∗jqSCωW−j1qSC

And the real part of the supporting against grade is following as

∏j=1nψRP−j≥∏j=1nψRP−∗j

Similarly, for the imaginary part of the supporting against the grade, we have

∏j=1nηψIP−j≥∏j=1nηψIP−∗j

By combing these all, we get

∑j=1nωW−jΨNCQN−j,∑j=1nωW−jΩNCQN−j,1−∏j=1n1−ϕRP−jqSCωW−j1qSCeι2π1−∏j=1n1−ηϕIP−jqSCωW−j1qSC,∏j=1nψRP−jeι2π∏j=1nηψIP−j≤∑j=1nωW−jΨNCQN−∗j,∑j=1nωW−jΩNCQN−∗j,1−∏j=1n1−ϕRP−∗jqSCωW−j1qSCeι2π1−∏j=1n1−ηϕIP−∗jqSCωW−j1qSC,∏j=1nψRP−∗jeι2π∏j=1nηψIP−∗j

Hence

CQRONFWANCQN−1,NCQN−2,…,NCQN−n≤CQRONFWANCQN−∗1,NCQN−∗2,…,NCQN−∗n.

Definition 13.

For any family of CQRONFNs NCQN−j=ΨNCQN−j,ΩNCQN−j,ϕRP−jeι2πηϕIP−j,ψRP−jeι2πηψIP−j, j=1,2,…,n, then the CQRONFWG operator is stated by

CQRONFWGNCQN−1,NCQN−2,…,NCQN−n=∑j=1nNCQN−jωW−j(36)

where ωW=ωW−1,ωW−2,…,ωW−n,ωW−j∈0,1 with a condition that is ∑j=1nωW−j=1.

Theorem 6.

For any family of CQRONFNs NCQN−j=ΨNCQN−j,ΩNCQN−j,ϕRP−jeι2πηϕIP−j,ψRP−jeι2πηψIP−j, j=1,2,…,n, then by using the Eq. (36), we get

CQRONFWGNCQN−1,NCQN−2,…,NCQN−n=∑j=1nΨNCQN−jωW−j,∑j=1nΩNCQN−jωW−j,∏j=1nϕRP−jeι2π∏j=1nηϕIP−j,1−∏j=1n1−ψRP−jqSCωW−j1qSCeι2π1−∏j=1n1−ηψIP−jqSCωW−j1qSC(37)

Proof.

Straightforward.

Definition 14.

For any family of CQRONFNs NCQN−j=ΨNCQN−j,ΩNCQN−j,ϕRP−jeι2πηϕIP−j,ψRP−jeι2πηψIP−j, j=1,2,…,n, then the CQRONFGWA operator is stated by

CQRONFGWANCQN−1,NCQN−2,…,NCQN−n=∑j=1nωW−jNCQN−jγSC1γSC(38)

where ωW=ωW−1,ωW−2,…,ωW−n,ωW−j∈0,1 with a condition that is ∑j=1nωW−j=1 and γSC∈−∞,0U0,∞.

Theorem 7.

For any family of CQRONFNs NCQN−j=ΨNCQN−j,ΩNCQN−j,ϕRP−jeι2πηϕIP−j,ψRP−jeι2πηψIP−j, j=1,2,…,n, then by using the Eq. (38), we get

CQRONFGWANCQN−1,NCQN−2,…,NCQN−n=∑j=1nωW−jΨNCQN−jγSC1γSC,∑j=1nωW−jΩNCQN−jγSC1γSC,1−∏j=1n1−ϕRP−jγSCqSCωW−j1γSCqSCeι2π1−∏j=1n1−ηϕIP−jγSCqSCωW−j1γSCqSC,1−1−∏j=1n1−1−ψRP−jqSCγSC1/qSCωW−jqSC1γSC1qSCeι2π1−1−∏j=1n1−1−ηψIP−jγSCqSC1/qSCωW−jqSC1γSC1qSC(39)

Proof.

Straightforward.

Theorems 3–5 are the same for Definitions 13 and 14.

Definition 15.

For any family of CQRONFNs NCQN−j=ΨNCQN−j,ΩNCQN−j,ϕRP−jeι2πηϕIP−j,ψRP−jeι2πηψIP−j, j=1,2,…,n, then the CQRONFGWG operator is stated by

CQRONFGWGNCQN−1,NCQN−2,…,NCQN−n=1γSC∑j=1nγSCNCQN−jωW−j(40)

where ωW=ωW−1,ωW−2,…,ωW−n,ωW−j∈0,1 with a condition that is ∑j=1nωW−j=1 and γSC∈−∞,0U0,∞.

Theorem.

For any family of CQRONFNs NCQN−j=ΨNCQN−j,ΩNCQN−j,ϕRP−jeι2πηϕIP−j,ψRP−jeι2πηψIP−j, j=1,2,…,n, then by using the Eq. (40), we get

CQRONFGWGNCQN−1,NCQN−2,…,NCQN−n=∑j=1nωW−jΨNCQN−jγSC1γSC,∑j=1nωW−jΩNCQN−jγSC1γSC,1−1−∏j=1n1−1−ϕRP−jqSCγSC1/qSCωW−jqSC1γSC1qSCeι2π1−1−∏j=1n1−1−ηϕIP−jγSCqSC1/qSCωW−jqSC1γSC1qSC,1−∏j=1n1−ψRP−jγSCqSCωW−j1γSCqSCeι2π1−∏j=1n1−ηψIP−jγSCqSCωW−j1γSCqSC(41)

Proof.

Straightforward.

6. MADM METHOD BASED ON CQRONFNs

In this study, we present the proficiency and reliability of the explored approach, we develop a MADM technique based on CQRONFSs. For solving these issues, we choose the family of alternatives and their attributes concerning weight vector, whose representations are followed as AAL=AAL−1,AAL−2,…,AAL−n, CAT=CAT−1,CAT−2,…,CAT−m and ωW=ωW−1,ωW−2,…,ωW−nT. Further, we choose the CQRONFNs as alternatives AAL−j and their attributes CAT−j is followed as NCQN−j=ΨNCQN−jk,ΩNCQN−jk,ϕRP−jkeι2πηϕIP−jk,ψRP−jkeι2πηψIP−jk, then the steps of the MADM technique is summarized as follows:

TOPSIS Method

Step 1: By using Eq. (42), we construct the decision matrix, whose entry in the form of CQRONFNs.

DDM=NCQN−jkn×m(42)

Step 2: By using Eq. (43), we normalize the decision matrix, which is given in step 1, if needed, we have

DDM=ΨNCQN−jk,ΩNCQN−jk,ϕRP−jkeι2πηϕIP−jk,ψRP−jkeι2πηψIP−jkfor benefit typesΨNCQN−jk,ΩNCQN−jk,ψRP−jkeι2πηψIP−jk,ϕRP−jkeι2πηϕIP−jkfor cost types(43)

Step 3: By using Eq. (41), we aggregate the values, which are normalized in step 2.

Step 4: By using Eqs. (44) and (45), we evaluate the positive and negative ideas, such that

NCQN−j1+=maxΨNCQN−j1,maxΩNCQN−j1,maxϕRP−j1eι2πmaxηϕIP−j1,minψRP−j1eι2πminηψIP−j1,maxΨNCQN−j2,maxΩNCQN−j2,maxϕRP−j2eι2πmaxηϕIP−j2,minψRP−j2eι2πminηψIP−j2,…,maxΨNCQN−jm,maxΩNCQN−jm,maxϕRP−jmeι2πmaxηϕIP−jm,minψRP−jmeι2πminηψIP−jm(44)

NCQN−j1−=minΨNCQN−j1,minΩNCQN−j1,minϕRP−j1eι2πminηϕIP−j1,maxψRP−j1eι2πmaxηψIP−j1,minΨNCQN−j2,minΩNCQN−j2,minϕRP−j2eι2πminηϕIP−j2,maxψRP−j2eι2πmaxηψIP−j2,…,minΨNCQN−jm,minΩNCQN−jm,minϕRP−jmeι2πminηϕIP−jm,maxψRP−jmeι2πmaxηψIP−jm(45)

Step 5: By using the Eq. (28), we examine the Order-αCQ divergence measures, by using positive and negative ideals, we have dGDMSDM−1NCQN−j,NCQN−j1+ and dGDMSDM−1NCQN−j,NCQN−j1−.

Step 6: By using Eq. (46), we examine the overall distance measure based on the dGDMSDM−1NCQN−1,NCQN−j1+ and dGDMSDM−1NCQN−1,NCQN−j1−, we have

DOD−j=dGDMSDM−1NCQN−j,NCQN−j1+dGDMSDM−1NCQN−j,NCQN−j1++dGDMSDM−1NCQN−j,NCQN−j1+(46)

Step 7: Rank all alternatives, which we get in step 6, and examine the best alternative from the family of alternatives.

Step 8: The end. The graphical representation of the explored algorithm is summarized in the form of Figure 2.

Example 1.

With the improvement of internet business stages, web-based shopping has become a typical utilization propensity for buyers. A customer plans to purchase a cell phone on a web-based business stage. The mobile phones are considered as alternatives, whose representations are followed as AAL=AAL−1,AAL−2,AAL−3,AAL−4,AAL−5 and their attributes whose representation with details is followed as

CAT−1: Creditability of merchant;
CAT−2: Online satisfaction rate;
CAT−3: The preference of mobile phone system;
CAT−4: Price preference.

For alternatives AAL−j and their attributes CAT−j, the weight vector is followed as ωW=0.2,0.25,0.35,0.0.2T. Based on the above analysis, the procedure of the MADM technique is summarized in the following ways:

Step 1: By using Eq. (42), we construct the decision matrix, whose entries in the form of CQRONFNs.

Step 2: By using Eq. (43), we normalize the decision matrix, which is given in step 1, if needed. The matrix, which is mention in the Table 1 is not needed to normalize it.

Symbols	CAT−1	CAT−2	CAT−3	CAT−4
AAL−1	0.6,0.2,0.4eι2π0.5,0.2eι2π0.3	0.7,0.2,0.5eι2π0.6,0.1eι2π0.2	0.8,0.1,0.3eι2π0.4,0.1eι2π0.3	0.5,0.2,0.5eι2π0.3,0.2eι2π0.4
AAL−2	0.61,0.21,0.41eι2π0.51,0.21eι2π0.31	0.71,0.21,0.51eι2π0.61,0.11eι2π0.21	0.81,0.11,0.31eι2π0.41,0.11eι2π0.31	0.51,0.21,0.51eι2π0.31,0.21eι2π0.41
AAL−3	0.62,0.22,0.42eι2π0.52,0.22eι2π0.32	0.72,0.22,0.52eι2π0.62,0.12eι2π0.22	0.82,0.12,0.32eι2π0.42,0.12eι2π0.32	0.52,0.22,0.52eι2π0.32,0.22eι2π0.42
AAL−4	0.63,0.23,0.43eι2π0.53,0.23eι2π0.33	0.73,0.23,0.53eι2π0.63,0.13eι2π0.23	0.83,0.13,0.33eι2π0.43,0.13eι2π0.33	0.53,0.23,0.53eι2π0.33,0.23eι2π0.43
AAL−5	0.64,0.24,0.44eι2π0.54,0.24eι2π0.34	0.74,0.24,0.54eι2π0.64,0.14eι2π0.24	0.84,0.14,0.34eι2π0.44,0.14eι2π0.34	0.54,0.24,0.54eι2π0.34,0.24eι2π0.44

Table 1

Original decision matrix, whose every entry in the form of complex q-rung orthopair normal fuzzy numbers.

Step 3: By using Eq. (41), we aggregate the values, which are normalized in step 2, for γSC=2.

NCQN−1=0.6845,0.1718,0.0834eι2π0.1005,0.1487eι2π0.304

NCQN−2=0.6943,0.1814,0.088eι2π0.1056,0.1582eι2π0.3138

NCQN−3=0.7042,0.191,0.0927eι2π0.1108,0.1678eι2π0.3236

NCQN−4=0.7141,0.2007,0.0975eι2π0.1162,0.1774eι2π0.3335

NCQN−5=0.7239,0.2105,0.1024eι2π0.1217,0.187eι2π0.3433

Step 4: By using Eqs. (44) and (45), we evaluate the positive and negative ideas, such that

NCQN+=0.7239,0.2105,0.1024eι2π0.1217,0.1487eι2π0.304

NCQN−=0.6845,0.1718,0.0834eι2π0.1005,0.187eι2π0.3433

Step 5: By using the Eq. (28), we examine the Order-αCQ divergence measures, by using positive and negative ideals, we have dGDMSDM−1NCQN−j,NCQN−j1+ and dGDMSDM−1NCQN−j,NCQN−j1−.

Step 6: By using Eq. (46), we examine the overall distance measure based on the dGDMSDM−1NCQN−1,NCQN−j1+ and dGDMSDM−1NCQN−1,NCQN−j1−, we have

DOD−1=0.4820.482+0.562=0.4616

DOD−2=0.4617

DOD−3=0.4618

DOD−4=0.4619

DOD−5=0.4620

Step 7: Rank all alternatives, which we get in step 6 and we examine the best alternative from the family of alternatives, such that

DOD−5≥DOD−4≥DOD−3≥DOD−2≥DOD−1

The best alternative is DOD−5.

Step 8: The end.

6.1. Comparative Analysis

Keeping the advantages of the explored notions is called CQRONFSs and their AOs, we solve some numerical examples to examine the reliability and effectiveness of the presented work. For coping with such kind of issues, we considered different kinds of information and resolved it by using explored and existing operators, whose information's are discussed below.

Additionally, the compassion among presented work and existing works are discussed to observe the proficiency and expertise of the explored approach. The existing technique of intuitionistic normal fuzzy AOs was proposed by Wang and Li [30], and the q-rung orthopair normal fuzzy AOs were presented by Yang et al. [31].

From the above analysis, it is clear that the existing approaches [30,31] and explored approach in these manuscripts give the same ranking results, which is in the form of AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1. If, we can check the values, which are obtained in Example 1, which is following as: DOD−5≥DOD−4≥DOD−3≥DOD−2≥DOD−1 is also the same as the ranking values of Table 2. From the above discussion, we obtained with the help of explored work and existing works, the best alternative is AAL−5. The graphical representation of the information, which is discussed in Table 4, is explained with the help of Figure 3. The aggregated values for proposed work and existing works are illustrated in Table 4, for γSC=2.

Symbols	dGDMSDM−1NCQN−j,NCQN−j1+	dGDMSDM−1NCQN−j,NCQN−j1−
NCQN−1	0.482	0.562
NCQN−2	0.476	0.555
NCQN−3	0.469	0.547
NCQN−4	0.462	0.539
NCQN−5	0.455	0.53

Table 2

By using Eq. (28), we examine the distance measures based on positive and negative ideals.

Methods	Operators	Score Values	Ranking
Wang and Li [30]	WA	SSFAAL−1=0.3523,SSFAAL−2=0.3571,SSFAAL−3=0.362,SSFAAL−4=0.3668,SSFAAL−5=0.3717	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	WG	SSFAAL−1=0.3392,SSFAAL−2=0.3444,SSFAAL−3=0.3495,SSFAAL−4=0.3546,SSFAAL−5=0.3597	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWA	SSFAAL−1=0.4213,SSFAAL−2=0.4304,SSFAAL−3=0.4394,SSFAAL−4=0.4485,SSFAAL−5=0.4575	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWG	SSFAAL−1=0.1814,SSFAAL−2=0.1838,SSFAAL−3=0.1863,SSFAAL−4=0.1888,SSFAAL−5=0.1913	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
Yang et al. [31]	WA	SSFAAL−1=0.2498,SSFAAL−2=0.2575,SSFAAL−3=0.2654,SSFAAL−4=0.2734,SSFAAL−5=0.2816	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
Yang et al. [31]	WG	SSFAAL−1=0.2081,SSFAAL−2=0.2128,SSFAAL−3=0.2174,SSFAAL−4=0.2219,SSFAAL−5=0.2264	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWA	SSFAAL−1=0.2591,SSFAAL−2=0.2669,SSFAAL−3=0.2748,SSFAAL−4=0.2829,SSFAAL−5=0.2911	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWG	SSFAAL−1=0.2115,SSFAAL−2=0.216,SSFAAL−3=0.2204,SSFAAL−4=0.2248,SSFAAL−5=0.2291	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
Proposed method for q = 1	WA	SSFAAL−1=0.2928,SSFAAL−2=0.296,SSFAAL−3=0.2992,SSFAAL−4=0.3024,SSFAAL−5=0.3057	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	WG	SSFAAL−1=0.2724,SSFAAL−2=0.2759,SSFAAL−3=0.2794,SSFAAL−4=0.2829,SSFAAL−5=0.2864	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWA	SSFAAL−1=0.4233,SSFAAL−2=0.4317,SSFAAL−3=0.4399,SSFAAL−4=0.4482,SSFAAL−5=0.4564	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWG	SSFAAL−1=0.0531,SSFAAL−2=0.0532,SSFAAL−3=0.0532,SSFAAL−4=0.0534,SSFAAL−5=0.0537	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
Proposed method for q = 2	WA	SSFAAL−1=0.2725,SSFAAL−2=0.2818,SSFAAL−3=0.2911,SSFAAL−4=0.3006,SSFAAL−5=0.3102	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	WG	SSFAAL−1=0.1076,SSFAAL−2=0.1086,SSFAAL−3=0.1094,SSFAAL−4=0.1102,SSFAAL−5=0.1109	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWA	SSFAAL−1=0.2856,SSFAAL−2=0.2948,SSFAAL−3=0.3042,SSFAAL−4=0.3137,SSFAAL−5=0.3233	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWG	SSFAAL−1=0.1006,SSFAAL−2=0.1009,SSFAAL−3=0.101,SSFAAL−4=0.101,SSFAAL−5=0.101	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
Proposed method for q = 3	WA	SSFAAL−1=0.2031,SSFAAL−2=0.2105,SSFAAL−3=0.2182,SSFAAL−4=0.226,SSFAAL−5=0.234	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	WG	SSFAAL−1=0.1288,SSFAAL−2=0.1309,SSFAAL−3=0.133,SSFAAL−4=0.135,SSFAAL−5=0.1368	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWA	SSFAAL−1=0.2156,SSFAAL−2=0.2232,SSFAAL−3=0.231,SSFAAL−4=0.239,SSFAAL−5=0.2471	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWG	SSFAAL−1=0.1293,SSFAAL−2=0.1312,SSFAAL−3=0.1331,SSFAAL−4=0.1349,SSFAAL−5=0.1366	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1

Table 3

Comparison between explored work with some existing operators.

Symbols	CAT−1	CAT−2	CAT−3	CAT−4
AAL−1	0.6,0.2,0.6eι2π0.7,0.5eι2π0.4	0.7,0.2,0.7eι2π0.8,0.4eι2π0.3	0.8,0.1,0.9eι2π0.7,0.11eι2π0.3	0.5,0.2,0.5eι2π0.6,0.5eι2π0.4
AAL−2	0.61,0.21,0.61eι2π0.71,0.51eι2π0.41	0.71,0.21,0.71eι2π0.81,0.41eι2π0.31	0.81,0.11,0.92eι2π0.72,0.12eι2π0.32	0.51,0.21,0.51eι2π0.61,0.51eι2π0.41
AAL−3	0.62,0.22,0.62eι2π0.72,0.52eι2π0.42	0.72,0.22,0.72eι2π0.82,0.42eι2π0.32	0.82,0.12,0.93eι2π0.73,0.13eι2π0.33	0.52,0.22,0.52eι2π0.62,0.52eι2π0.42
AAL−4	0.63,0.23,0.63eι2π0.73,0.53eι2π0.43	0.73,0.23,0.73eι2π0.83,0.43eι2π0.33	0.83,0.13,0.94eι2π0.74,0.14eι2π0.34	0.53,0.23,0.53eι2π0.63,0.53eι2π0.43
AAL−5	0.64,0.24,0.64eι2π0.74,0.54eι2π0.44	0.74,0.24,0.74eι2π0.84,0.44eι2π0.34	0.84,0.14,0.95eι2π0.75,0.15eι2π0.35	0.54,0.24,0.54eι2π0.64,0.54eι2π0.44

Table 4

Original decision matrix, whose information in the form of complex Pythagorean normal fuzzy numbers.

From Table 2 we considered the complex intuitionistic normal fuzzy information and resolved it by using the explored and existing operators [30,31] to examine the proficiency and expertise of the presented approach. Further, to find the reliability of the explored operator, we choose the complex Pythagorean normal fuzzy information and solve it by using the explored and existing operators [30,31].

Form Figure 3 there are mentions five kinds of series, which are denoted the graph of alternatives in different colors. From Figure 3, we easily obtained that which one is the best alternative, see the above figure the series five is moved on the top in all series, so series five is the best alternative from the set of alternatives. The information's are discussed in Table 5 for γSC=2 and the weight vectors are discussed at the beginning of Example 1.

Methods	Operators	Score Values	Ranking
Wang and Li [30]	WA	Failed	Failed
	WG	Failed	Failed
	GWA	Failed	Failed
	GWG	Failed	Failed
Yang et al. [31]	WA	SSFAAL−1=0.4457,SSFAAL−2=0.473,SSFAAL−3=0.4908,SSFAAL−4=0.5097,SSFAAL−5=0.5298	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	WG	SSFAAL−1=0.1772,SSFAAL−2=0.1819,SSFAAL−3=0.1848,SSFAAL−4=0.1876,SSFAAL−5=0.1903	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWA	SSFAAL−1=0.472,SSFAAL−2=0.5,SSFAAL−3=0.5176,SSFAAL−4=0.536,SSFAAL−5=0.5557	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWG	SSFAAL−1=0.1714,SSFAAL−2=0.1751,SSFAAL−3=0.1772,SSFAAL−4=0.1791,SSFAAL−5=0.181	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
Proposed method for q = 1	WA	Failed	Failed
	WG	Failed	Failed
	GWA	Failed	Failed
	GWG	Failed	Failed
Proposed method for q = 2	WA	SSFAAL−1=0.5059,SSFAAL−2=0.5281,SSFAAL−3=0.5433,SSFAAL−4=0.559,SSFAAL−5=0.5754	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	WG	SSFAAL−1=0.1063,SSFAAL−2=0.1111,SSFAAL−3=0.1141,SSFAAL−4=0.1174,SSFAAL−5=0.1209	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWA	SSFAAL−1=0.5212,SSFAAL−2=0.5437,SSFAAL−3=0.5589,SSFAAL−4=0.5747,SSFAAL−5=0.591	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWG	SSFAAL−1=0.0478,SSFAAL−2=0.0478,SSFAAL−3=0.0469,SSFAAL−4=0.046,SSFAAL−5=0.0452	AAL−1≥AAL−2≥AAL−3≥AAL−4≥AAL−5
Proposed method for q = 3	WA	SSFAAL−1=0.4198,SSFAAL−2=0.4451,SSFAAL−3=0.4625,SSFAAL−4=0.4808,SSFAAL−5=0.5001	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
Proposed method for q = 3	WG	SSFAAL−1=0.1049,SSFAAL−2=0.1067,SSFAAL−3=0.1074,SSFAAL−4=0.1081,SSFAAL−5=0.1087	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
Proposed method for q = 3	GWA	SSFAAL−1=0.4407,SSFAAL−2=0.4664,SSFAAL−3=04836,SSFAAL−4=0.5016,SSFAAL−5=0.5207	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
Proposed method for q = 3	GWG	SSFAAL−1=0.0991,SSFAAL−2=0.1001,SSFAAL−3=0.1001,SSFAAL−4=0.1,SSFAAL−5=0.11	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1

Table 5

Comparison between explored work with some existing operators.

Based on the explored operators are called WA, WG, GWA, GWG operators based on complex Pythagorean normal fuzzy information's, the compassion among presented work and existing works are discussed to observe the proficiency and expertise of the explored approach. The existing technique of intuitionistic normal fuzzy AOs was proposed by Wang and Li [30], and the q-rung orthopair normal fuzzy AOs were presented by Yang et al. [31]. The aggregated values for proposed work and existing works are illustrated in Table 6.

Symbols	CAT−1	CAT−2	CAT−3	CAT−4
AAL−1	0.6,0.2,0.9eι2π0.8,0.8eι2π0.7	0.7,0.2,0.8eι2π0.9,0.7eι2π0.8	0.8,0.1,0.85eι2π0.75,0.75eι2π0.75	0.5,0.2,0.75eι2π0.56,0.85eι2π0.46
AAL−2	0.61,0.21,0.91eι2π0.81,0.81eι2π0.71	0.71,0.21,0.81eι2π0.91,0.71eι2π0.81	0.81,0.11,0.86eι2π0.76,0.76eι2π0.76	0.51,0.21,0.76eι2π0.56,0.86eι2π0.46
AAL−3	0.62,0.22,0.92eι2π0.82,0.82eι2π0.72	0.72,0.22,0.82eι2π0.92,0.72eι2π0.82	0.82,0.12,0.87eι2π0.77,0.77eι2π0.77	0.52,0.22,0.77eι2π0.57,0.87eι2π0.47
AAL−4	0.63,0.23,0.93eι2π0.83,0.83eι2π0.73	0.73,0.23,0.83eι2π0.93,0.73eι2π0.83	0.83,0.13,0.88eι2π0.78,0.78eι2π0.78	0.53,0.23,0.78eι2π0.58,0.88eι2π0.48
AAL−5	0.64,0.24,0.94eι2π0.84,0.84eι2π0.74	0.74,0.24,0.84eι2π0.94,0.74eι2π0.84	0.84,0.14,0.89eι2π0.79,0.79eι2π0.79	0.54,0.24,0.79eι2π0.59,0.89eι2π0.49

Table 6

Original decision matrix, whose information in the form of complex q-rung orthopair normal fuzzy numbers.

From the above analysis, it is clear that the existing approaches [30,31] and explored approach in this manuscripts give the different ranking results, which are in the form of AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1 and AAL−1≥AAL−2≥AAL−3≥AAL−4≥AAL−5. From the above discussion, we obtained with the help of explored work and existing works, the best alternatives are AAL−5 and AAL−1. The graphical representation of the information, which is discussed in Table 6, is explained with the help of Figure 4. The positive and negative ideals are discussed in the form of Table 2.

Form Figure 4 there are mentions five kinds of series, which are denoted the graph of alternatives in different colors. From Figure 4, we easily obtained that which one is the best alternative, see the above figure the series five is moved on the top in all series, so series five is the best alternative from the set of alternatives. From Tables 1 and 5 we considered the complex intuitionistic normal fuzzy information, complex Pythagorean normal fuzzy information and resolved it by using the explored and existing operators [30,31] to examine the proficiency and expertise of the presented approach. Further, to find the reliability of the explored operator, we choose the complex q-rung orthopair normal fuzzy information and solve it by using the explored and existing operators [30,31]. The information is discussed in Table 7.

Methods	Operators	Score Values	Ranking
Wang and Li [30]	WA	Failed	Failed
	WG	Failed	Failed
	GWA	Failed	Failed
	GWG	Failed	Failed
Yang et al. [31]	WA	Failed	Failed
	WG	Failed	Failed
	GWA	Failed	Failed
	GWG	Failed	Failed
Proposed method for q=1	WA	Failed	Failed
	WG	Failed	Failed
	GWA	Failed	Failed
	GWG	Failed	Failed
Proposed method for q = 2	WA	Failed	Failed
	WG	Failed	Failed
	GWA	Failed	Failed
	GWG	Failed	Failed
Proposed method for q = 8	WA	SSFAAL−1=0.289,SSFAAL−2=0.3092,SSFAAL−3=0.3301,SSFAAL−4=0.3529,SSFAAL−5=0.378	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	WG	SSFAAL−1=0.0627,SSFAAL−2=0.0588,SSFAAL−3=0.0528,SSFAAL−4=0.0458,SSFAAL−5=0.0379	AAL−1≥AAL−2≥AAL−3≥AAL−4≥AAL−5
	GWA	SSFAAL−1=0.3143,SSFAAL−2=0.3358,SSFAAL−3=0.358,SSFAAL−4=0.3822,SSFAAL−5=0.4087	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWG	SSFAAL−1=0.0544,SSFAAL−2=0.0494,SSFAAL−3=0.0425,SSFAAL−4=0.0346,SSFAAL−5=0.0256	AAL−1≥AAL−2≥AAL−3≥AAL−4≥AAL−5

Table 7

Comparison between explored work with some existing operators.

Based on the explored operators are called WA, WG, GWA, GWG operators based on complex Pythagorean normal fuzzy information's, the compassion among presented work and existing works are discussed to observe the proficiency and expertise of the explored approach. The existing technique of intuitionistic normal fuzzy AOs was proposed by Wang and Li [30], and the q-rung orthopair normal fuzzy AOs were presented by Yang et al. [31]. The aggregated values for proposed work and existing works are illustrated in Table 8.

Symbols	CAT−1	CAT−2	CAT−3	CAT−4
AAL−1	0.8,0.7,0.6,0.7	0.4,0.5,0.2,0.4	0.8,0.5,0.1,0.7	0.4,0.3,0.6,0.2
AAL−2	0.6,0.4,0.7,0.9	0.5,0.6,0.4,0.3	0.4,0.4,0.7,0.2	0.8,0.5,0.6,0.4
AAL−3	0.9,0.8,0.5,0.4	0.7,0.8,0.7,0.2	0.4,0.3,0.8,0.8	0.6,0.6,0.3,0.5
AAL−4	0.7,0.6,0.7,0.8	0.8,0.7,0.8,0.6	0.5,0.4,0.6,0.9	0.6,0.4,0.7,0.3
AAL−5	0.5,0.3,0.8,0.4	0.7,0.5,0.5,0.5	0.5,0.3,0.6,0.6	0.8,0.6,0.8,0.7

Table 8

Original decision matrix.

From the above analysis, it is clear that the existing approaches [30,31] and explored approach in this manuscripts give different ranking results, which are in the form of AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1 and AAL−1≥AAL−2≥AAL−3≥AAL−4≥AAL−5. From the above discussion, we obtained with the help of explored work and existing works, the best alternatives are AAL−5 and AAL−1. The graphical representation of the information, which is discussed in Table 8, is explained with the help of Figure 5.

Form Figure 5 there are mentions five kinds of series, which are denoted the graph of alternatives in different colors. From Figure 5, we easily obtained that which one is the best alternative, see the above figure the series five is moved on the top in all series, so series five is the best alternative from the set of alternatives. Further, we examine the reliability and proficiency of the explored measures and operators based on CQRONFSs. We illustrate a numerical example which is discussed below is taken from Ref. [31].

Example 2.

As economic globalization makes enterprises face a more complex internal and external environment, finding an appropriate partner is an important way to maintain their competitiveness, which is affected by many factors. To select a suitable global partner, an enterprise has selected five candidate enterprises in the global scope. The set of alternative enterprises is AAL={AAL−1,AAL−2,AAL−3,AAL−4,AAL−5}, and four attributes are considered, namely, R& D capability CAT−1, business operation level CAT−2, international cooperation level CAT−3 and credit level CAT−4. The set of attributes CAT=CAT−1,CAT−2,CAT−3,CAT−4 is formed, and they are all benefit-oriented indicators. The corresponding weight is ωW=0.3,0.2,0.2,0.3T, and the decision information matrix as shown in Table 8 is constructed according to the decision information. Besides, considering the problem of q-RONF information aggregation based on WA, WG, GWA, and GWG operators.

Based on the explored operators are called WA, WG, GWA, GWG operators based on complex Pythagorean normal fuzzy information's, the compassion among presented work and existing works are discussed to observe the proficiency and expertise of the explored approach. The existing technique of intuitionistic normal fuzzy AOs was proposed by Wang and Li [30], and the q-rung orthopair normal fuzzy AOs were presented by Yang et al. [31]. The aggregated values for proposed work and existing works are illustrated in Table 9.

Methods	Operators	Score Values	Ranking
Wang and Li [30]	WA	Failed	Failed
	WG	Failed	Failed
	GWA	Failed	Failed
	GWG	Failed	Failed
Yang et al. [31]	WA	SSFAAL−1=0.042,SSFAAL−2=0.132,SSFAAL−3=0.13,SSFAAL−4=0.131,SSFAAL−5=0.186	AAL−5≥AAL−2≥AAL−4≥AAL−3≥AAL−1
	WG	SSFAAL−1=0.142,SSFAAL−2=0.232,SSFAAL−3=0.23,SSFAAL−4=0.231,SSFAAL−5=0.286	AAL−5≥AAL−2≥AAL−4≥AAL−3≥AAL−1
	GWA	SSFAAL−1=0.049,SSFAAL−2=0.139,SSFAAL−3=0.19,SSFAAL−4=0.137,SSFAAL−5=0.189	AAL−5≥AAL−2≥AAL−4≥AAL−3≥AAL−1
	GWG	SSFAAL−1=0.112,SSFAAL−2=0.163,SSFAAL−3=0.133,SSFAAL−4=0.154,SSFAAL−5=0.192	AAL−5≥AAL−2≥AAL−4≥AAL−3≥AAL−1
Proposed method for q = 1	WA	Failed	Failed
	WG	Failed	Failed
	GWA	Failed	Failed
	GWG	Failed	Failed
Proposed method for q = 2	WA	Failed	Failed
	WG	Failed	Failed
	GWA	Failed	Failed
	GWG	Failed	Failed
Proposed method for q = 8	WA	SSFAAL−1=0.042,SSFAAL−2=0.132,SSFAAL−3=0.13,SSFAAL−4=0.131,SSFAAL−5=0.186	AAL−5≥AAL−2≥AAL−4≥AAL−3≥AAL−1
	WG	SSFAAL−1=0.142,SSFAAL−2=0.232,SSFAAL−3=0.23,SSFAAL−4=0.231,SSFAAL−5=0.286	AAL−5≥AAL−2≥AAL−4≥AAL−3≥AAL−1
	GWA	SSFAAL−1=0.049,SSFAAL−2=0.139,SSFAAL−3=0.19,SSFAAL−4=0.137,SSFAAL−5=0.189	AAL−5≥AAL−2≥AAL−4≥AAL−3≥AAL−1
	GWG	SSFAAL−1=0.112,SSFAAL−2=0.163,SSFAAL−3=0.133,SSFAAL−4=0.154,SSFAAL−5=0.192	AAL−5≥AAL−2≥AAL−4≥AAL−3≥AAL−1

Table 9

Comparison between explored work with some existing operators.

From the above analysis, the ranking results of the proposed and existing notions are the same which is in the form of AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1. The best alternative is AAL−5. Additionally, we choose the intuitionistic normal fuzzy information and examine the best alternative by using the explored and existing operators. The intuitionistic normal fuzzy information is stated in the form of Table 10.

	CAT−1	CAT−2	CAT−3	CAT−4
AAL−1	0.2,0.1,0.4,0.3	0.12,0.11,0.14,0.13	0.22,0.21,0.24,0.23	0.32,0.1,0.34,0.3
AAL−2	0.3,0.2,0.5,0.4	0.13,0.12,0.15,0.14	0.23,0.22,0.25,0.24	0.33,0.2,0.35,0.4
AAL−3	0.4,0.3,0.6,0.3	0.14,0.13,0.16,0.13	0.24,0.23,0.26,0.23	0.34,0.3,0.36,0.3
AAL−4	0.5,0.2,0.7,0.2	0.15,0.12,0.17,0.12	0.25,0.22,0.27,0.22	0.35,0.2,0.37,0.2
AAL−5	0.6,0.1,0.8,0.1	0.16,0.11,0.18,0.11	0.26,0.21,0.28,0.21	0.36,0.1,0.38,0.1

Table 10

Original decision matrix.

Based on the explored operators are called WA, WG, GWA, GWG operators based on complex Pythagorean normal fuzzy information's, the compassion among presented work and existing works are discussed to observe the proficiency and expertise of the explored approach. The existing technique of intuitionistic normal fuzzy AOs was proposed by Wang and Li [30], and the q-rung orthopair normal fuzzy AOs were presented by Yang et al. [31]. The aggregated values for proposed work and existing works are illustrated in Table 11.

Methods	Operators	Score Values	Ranking
Wang and Li [30]	WA	SSFAAL−1=0.2234,SSFAAL−2=0.2378,SSFAAL−3=0.2435,SSFAAL−4=0.2547,SSFAAL−5=0.26457	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	WG	SSFAAL−1=0.2246,SSFAAL−2=0.2358,SSFAAL−3=0.2416,SSFAAL−4=0.2556,SSFAAL−5=0.2678	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWA	SSFAAL−1=0.2278,SSFAAL−2=0.2317,SSFAAL−3=0.2419,SSFAAL−4=0.2511,SSFAAL−5=0.2615	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWG	SSFAAL−1=0.2242,SSFAAL−2=0.2347,SSFAAL−3=0.2455,SSFAAL−4=0.2557,SSFAAL−5=0.2611	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
Yang et al. [31]	WA	SSFAAL−1=0.1234,SSFAAL−2=0.1378,SSFAAL−3=0.1435,SSFAAL−4=0.1547,SSFAAL−5=0.16457	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	WG	SSFAAL−1=0.1246,SSFAAL−2=0.1358,SSFAAL−3=0.1416,SSFAAL−4=0.1556,SSFAAL−5=0.1678	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWA	SSFAAL−1=0.1278,SSFAAL−2=0.1317,SSFAAL−3=0.1419,SSFAAL−4=0.1511,SSFAAL−5=0.1615	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWG	SSFAAL−1=0.1242,SSFAAL−2=0.1347,SSFAAL−3=0.1455,SSFAAL−4=0.1557,SSFAAL−5=0.1611	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
Proposed method for q = 1	WA	SSFAAL−1=0.2234,SSFAAL−2=0.2378,SSFAAL−3=0.2435,SSFAAL−4=0.2547,SSFAAL−5=0.26457	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	WG	SSFAAL−1=0.2234,SSFAAL−2=0.2378,SSFAAL−3=0.2435,SSFAAL−4=0.2547,SSFAAL−5=0.26457	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWA	SSFAAL−1=0.2234,SSFAAL−2=0.2378,SSFAAL−3=0.2435,SSFAAL−4=0.2547,SSFAAL−5=0.26457	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWG	SSFAAL−1=0.2234,SSFAAL−2=0.2378,SSFAAL−3=0.2435,SSFAAL−4=0.2547,SSFAAL−5=0.26457	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
Proposed method for q = 2	WA	SSFAAL−1=0.112,SSFAAL−2=0.188,SSFAAL−3=0.197,SSFAAL−4=0.201,SSFAAL−5=0.217	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	WG	SSFAAL−1=0.177,SSFAAL−2=0.193,SSFAAL−3=0.202,SSFAAL−4=0.211,SSFAAL−5=0.228	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWA	SSFAAL−1=0.114,SSFAAL−2=0.189,SSFAAL−3=0.199,SSFAAL−4=0.207,SSFAAL−5=0.214	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWG	SSFAAL−1=0.117,SSFAAL−2=0.192,SSFAAL−3=0.197,SSFAAL−4=0.204,SSFAAL−5=0.218	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
Proposed method for q = 3	WA	SSFAAL−1=0.1234,SSFAAL−2=0.1378,SSFAAL−3=0.1435,SSFAAL−4=0.1547,SSFAAL−5=0.16457	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	WG	SSFAAL−1=0.1246,SSFAAL−2=0.1358,SSFAAL−3=0.1416,SSFAAL−4=0.1556,SSFAAL−5=0.1678	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWA	SSFAAL−1=0.1278,SSFAAL−2=0.1317,SSFAAL−3=0.1419,SSFAAL−4=0.1511,SSFAAL−5=0.1615	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
	GWG	SSFAAL−1=0.1242,SSFAAL−2=0.1347,SSFAAL−3=0.1455,SSFAAL−4=0.1557,SSFAAL−5=0.1611	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1

Table 11

Comparison between explored works with some existing operators.

From the above analysis, the ranking results of the proposed and existing notions are the same which is in the form of AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1. The best alternative is AAL−5. To determine the consistency of the parameters γSC by using the information of Example 1, which is discussed in the form of Table 12.

Parameter	Score Values	Ranking Values
γSC=2	SSFAAL−1=0.2031,SSFAAL−2=0.2105,SSFAAL−3=0.2182,SSFAAL−4=0.226,SSFAAL−5=0.234	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
γSC=3	SSFAAL−1=0.2017,SSFAAL−2=0.2077,SSFAAL−3=0.2152,SSFAAL−4=0.2253,SSFAAL−5=0.2336	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
γSC=4	SSFAAL−1=0.2011,SSFAAL−2=0.2075,SSFAAL−3=0.2149,SSFAAL−4=0.2248,SSFAAL−5=0.2331	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
γSC=5	SSFAAL−1=0.2008,SSFAAL−2=0.2066,SSFAAL−3=0.2145,SSFAAL−4=0.2244,SSFAAL−5=0.2329	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1
γSC=10	SSFAAL−1=0.1912,SSFAAL−2=0.1927,SSFAAL−3=0.2002,SSFAAL−4=0.2011,SSFAAL−5=0.2024	AAL−5≥AAL−4≥AAL−3≥AAL−2≥AAL−1

Table 12

Discussed for different values of the parameter γSC.

Therefore, the explored operators based on CQRONFSs are more proficient and more flexible than existing operators, which is discussed in Ref. [30,31]. Hence, the presented approach is extensively powerful and more general than complex intuitionistic normal fuzzy setFSs and complex Pythagorean normal FSs.

7. CONCLUSION

One of the most proficient and beneficial theories is called CQROFS, containing the grade of supporting and the grade of supporting against in the form of polar coordinates belonging to unit disc in a complex plane. CQROFS is a proficient technique to address awkward information, although the NFN is examining normal distribution information in anthropogenic action and realistic environment. Based on the advantages of both notions, in this manuscript, we explored the novel concept of CQRONFS as an important technique to evaluate unreliable and complicated information. Some operational laws based on CQRONFSs are also explored. Additionally, some distance measures are called CQRONFGDM, CQROFNFSDM, two types of CQRONFODMs, and their special cases are discussed. Moreover, weighted averaging, weighted geometric, generalized weighted averaging, and generalized weighted geometric operators based on CQRONFSs are also presented. In last, we solve a numerical example of the MADM problem is shown to justify the proficiency of the presented operators. The advantages, comparative and sensitive analysis are used to express the efficiency and flexibility of the explored approach.

In further research, considering the superiority of new CQRONFSs, we can extend them to some other work based on FSs [32,33], picture FSs [34], PFSs [35], hesitant fuzzy setFSs [36], and so on [37–42].

CONFLICTS OF INTEREST

The authors declare no conflict of interest.

AUTHORS’ CONTRIBUTIONS

All authors have equally contributed to this manuscript.

ACKNOWLEDGMENTS

The authors are grateful to the Deanship of Scientific Research, King Saud University for funding through Vice Deanship of Scientific Research Chairs.

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 14 - 1
Pages: 1895 - 1922
Publication Date: 2021/07/01
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
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TY  - JOUR
AU  - Zeeshan Ali
AU  - Tahir Mahmood
AU  - Abdu Gumaei
PY  - 2021
DA  - 2021/07/01
TI  - Order-αCQ Divergence Measures and Aggregation Operators Based on Complex q-Rung Orthopair Normal Fuzzy Sets and Their Application to Multi-Attribute Decision-Making
JO  - International Journal of Computational Intelligence Systems
SP  - 1895
EP  - 1922
VL  - 14
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.210622.004
DO  - 10.2991/ijcis.d.210622.004
ID  - Ali2021
ER  -

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