# Some Novel Geometric Aggregation Operators of Spherical Cubic Fuzzy Information with Application

Authors: Zafar Ullah; Huma Bashir; Rukhshanda Anjum; Mabrook Al-Rakhami; Suheer Al-Hadhrami; Abdul Ghaffar
Journal: Security and Communication Networks
 (2021)

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## Abstract

Technology is quickly evolving and becoming part of our lives. Life has become better and easier due to the technologies. Although it has lots of benefits, it also brings serious risks and threats, known as cyberattacks, which are neutralized by cybersecurities. Since spherical fuzzy sets (SFSs) and interval-valued SFS (IVSFS) are an excellent tool in coping with uncertainty and fuzziness, the current study discusses the idea of spherical cubic FSs (SCFSs). These sets are characterized by three mappings known as membership degree, neutral degree, and nonmembership degree. Each of these degrees is spherical cubic fuzzy numbers (SCFNs) such that the summation of their squares does not exceed one. The score function and accuracy function are presented for the comparison of SCFNs. Moreover, the spherical cubic fuzzy weighted geometric (SCFWG) operators and SCF ordered weighted geometric (SCFOWG) operators are established for determining the distance between two SCFNs. Furthermore, some operational rules of the proposed operators are analyzed and multiattribute decision-making (MADM) approach based on these operators is presented. These methods are applied to make the best decision on the basis of risks factors as a numerical illustration. Additionally, the comparison of the proposed method with the existing methods is carried out; since the proposed methods and operators are the generalizations of existing methods, they provide more general, exact, and accurate results. Finally, for the legitimacy, practicality, and usefulness of the decision-making processes, a detailed illustration is given.


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## Body

## 1. Introduction

Multiple-attribute decision-making (MADM) means that, from the restricted alternatives set according to multiple attribute, the best alternative is selected that could be called cognitive processing. MADM is a key subdivision of the theory of decision-making (DM), commonly used in human activities [1]. The fuzzy knowledge usually is of two forms: qualitative and quantitative. The quantitative fuzzy knowledge is determined by fuzzy set (FS) [2], intuitionistic FS (IFS) [3], Pythagorean FS (PyFS) [4], and so forth. Zadeh’s FS theory was utilized to characterize fuzzy quantitative knowledge comprising of only membership degree. In light of this, Atanassov proposed IFS, consisting of two degrees, namely, membership and nonmembership. The summation of two grades should be less than or equal to one in an IFS. However, the two degrees often do not fulfill the constraints, but the sum of their squares does. Yager et al. [5] laid down the PyFS, which contains that the square sum of both the degrees is less than or equal to 1. In certain cases, the PyFSs will convey the details more effectively than the IFSs. For instance, if the membership, representing the support of an expert, is 0.8 and nonmembership representing the opposition, is 0.6, then surely, this information cannot be represented by IFS, but it can be effectively described by PyFS.Now, the IFS and PyFS do not provide a satisfactory result, since the neutral degree calculates the real-world problems independently. In order to handle this situation, Cuong and Kreinovich [6] originated the notion of a picture fuzzy set (PFS). A PFS is able to use three indexes, namely, membership degree Pˇx, neutral degree Iˇx, and nonmembership degree Nˇx on condition that 0≤Pˇx+Iˇx+Nˇx≤1 . Of course, PFSs are more appropriate for managing the fuzziness and ambiguity than IFS or PyFS. Garg [7] presented the picture fuzzy weighted average operator (PFWAO) and picture fuzzy ordered weighted averaging aggregation operators. Since the last few decades, several researchers have investigated PyFS and have successfully applied it to a wide range of fields such as strategic decision-making, decision-making qualities, and design recognition [8–11]. In real life, we often have a lot of problems which cannot be solved with PFS; for instance, Pˇx+Iˇx+Nˇx>1. In these conditions, PFS cannot produce an acceptable result. To make this clear, an example is given: to support and to oppose the extent of an alternative membership, they are, respectively, 1/5,3/5, and 3/5. This is based on their number exceeding 1 and not being presented for PFS. In light of these conditions, a generalization of PFS is introduced as the concept of spherical fuzzy sets (SFSs). The degrees of membership, neutral, and nonmembership in an SFS has the following condition:(1)0≤P⌄2x+Iˇ2x+Nˇ2x≤1.In PyFS, Peng and Yang [12] introduced some new properties that are division, subtraction, and other important characteristics. The authors addressed the superiority and dependence ranking methodologies in the Pythagorean fuzzy setting to clarify the multiattribute decision-making problems. We et al. [13] implemented a maximizing variance protocol in order to clarify decisions based on Pythagorean fuzzy interval conditions. Garg [14] presented the IVPyF average (IVPyFWA) and IVPyF geometric (IVPFWG) and provided a concept of the new precision function based on PyF’s interval-evaluated setting. Liang et al. [10] proposed the concepts of the medium and weighted PyF geometric Bonferroni mean (WPFGBM) operator. Many researchers proposed the idea of IVSF and its applications in DM problems [15–17]. Ayaz et al. [18] introduced the idea of SCFSs and applied it to the problems of multiattribute decision-making.Many researchers presented the various applications of SFSs, IVSFSs. Kutlu Gündoğdu and Kahraman [19] presented the new approach of SFSs by using the TOPSIS methodology. Gündoğdu [20] presented the principals of SFSs and their applications in DM theory. Zeng et al. [21] gave the new concept by defining the probabilistic interactive aggregation operator of T-SFSs and their applications in solar cell selection. Liu et al. [22] gave the concept of multiattribute decision-making approach for Baiyao’s R&D project selection problem. Mathew et al. [23] presented the novel approach under SFSs for the selection of advanced manufacturing system. Gong et al. [24] gave a new approach of spherical distance for IFSs and its applications in DM. Kutlu Gundogdu and Kahraman [25] presented the WASPAS extension with spherical fuzzy sets. After that, Gundogdu and Kahraman [26] gave a new concept and related the TOPSIS methods to IVSFSs. Zeng et al. [27] introduced the TOPSIS methodology with covering-based SF rough set model for MADM. Liu et al. [28] gave a linguistic SFSs approach with applications for evaluation of shared bicycles in China. In 2019, Kutlu Gundogdu and Kahraman [29] showed a novel approach of VIKOR method using the SFSs and their applications in selection of warehouse site. We et al. [30] presented the similarity measures of SFSs based on cosine function. After that, in 2020, Khan et al. [31] introduced a new approach and related the distance and similarity measures for SFSs and their application in selection of mega project selection. Shishavan et al. [32] extended the idea of similarity measure in the environment of SF information. Recently, Mahmood et al. [33] further enhanced the idea of similarity measure and discussed it applications in pattern recognition and medical diagnosis.The objectives of this paper include the following: (1) to determine an SCFS, (2) to define SCFNs and the associated operating identity; (3) to propose comparison functions for score, accuracy, or certainty; (4) to propose SCF geometric aggregation operators and some discussion on their properties.Convincing accretion is one of the commanding tools of decision-making. The values are normalized by the collection operators. Additionally, these operators represent a wide range of data values. The weighted geometric aggregation operators are used for the position values of the required weight. The problems arise whenever the load segments of the weight vectors comprise segments that have a vital difference in parts of the weight vectors. This issue inspired proposing the idea of spherical cubic geometric aggregation operators. Henceforth, we present the notion of spherical cubic geometric aggregation operators. Moreover, the target of the DM techniques is to select the best choice among the available choices. In certain situations, the available choices are arranged in order to select the most suitable choice. These circumstances motivated presenting a method for the ranking of the available choices. We aim to combine the Einstein product and introduce the concepts of SCF Einstein weighted averaging (SCFEWA) operator, SCF Einstein ordered weighted averaging (SCFEOWA) operator, SCF Einstein weighted geometric (SCFEWG) operator, SCF Einstein ordered weighted geometric (SCFEOWG) operator, and some more generalized operators in MADM processes.This article describes the concept of the SCFS, which is the extension of IVSFS based on the constraints of the fact that the square sum of the supremum of its degrees of membership is less than one. Here, the concept of the SCFS is introduced, that is, the generalization of the IVSFS. We analyze certain SCFS properties. For comparison of SCFNs, the score and degree of deviation are described. The distance between SCFNs is defined. SCFS states that the square of the supremacy to its membership is less than or equal to 1. Based on this information, aggregation operators are, for example, SCF weighted geometric (SCFWG), SCF ordered weighted geometric (SCFOWG), and SCF hybrid weighted geometric (SCFHWG). In addition, the proposed operators are utilized in problems of decision-making where experts give their preferences in the SCF details to illustrate the practicality of the new method and its efficiency.The remaining of the paper is organized as follows: Section2 presents some basic definitions and important properties. Section 3 proposes some geometric aggregation operators for SCFNs and their properties. Section 4 applies the spherical cubic geometric aggregation operators for MADM process. Section 5 presents the numerical illustration for the application and the final section concludes the study.


## 2. Preliminaries

This section presents a few elementary definitions with their key properties.Definition 1 (see [2]).
SupposingŢˆ be a universal set, the fuzzy set (FS) F˜ formulates as follows:(2)F˜=ţˆ,μ˜F˜ţˆ|ţˆ∈Ţˆ,where μ˜F˜:Ţˆ⟶0,1 is the membership degree of ţˆ∈Ţˆ.Definition 2 (see [3]).
SupposingŢˆ be a universal set, the fuzzy set (FS) I˜ formulates as follows:(3)I˜=ţˆ,μ˜I˜ţˆ,ϑ˜I˜ţˆ|ţˆ∈Ţˆ,where μ˜I˜:Ţˆ⟶0,1 is the membership degree and ϑ˜I˜:Ţˆ⟶0,1 is the nonmembership degree of ţˆ∈Ţˆ, under the specified condition(4)0≤μ˜I˜+ϑ˜I˜≤1.Definition 3 (see [18]).
SupposeŢˆ be a universal set. A cubic set (CS) C˜ formulates as follows:(5)C˜=ţˆ,μ˜C˜ţˆ,ϑ˜C˜ţˆ|ţˆ∈Ţˆ,where μ˜C˜ is an IVFS in Ţˆ and ϑ˜C˜ is a simple FS.Definition 4 (see [18]).
SupposingŢˆ be a universal set, then a spherical fuzzy set (SFS) Q˜ is of the following form:(6)Q˜=ţˆ,μ˜Q˜ţˆ,η˜Q˜ţˆ,ϑ˜Q˜ţˆ|ţˆ∈Ţˆ,where μ˜Q˜:Ţˆ⟶0,1 is the membership degree, η˜Q˜:Ţˆ⟶0,1 is the neutral degree, and ϑ˜Q˜:Ţˆ⟶0,1 is the nonmembership degree of ţˆ∈Ţˆ, under the specified condition:(7)0≤μ˜Q˜2+η˜Q˜2+ϑ˜Q˜2≤1.
The indeterminacy degree for SFSQ˜ is defined as follows:(8)π˜Q˜=1−μ˜Q˜2−η˜Q˜2−ϑ˜Q˜2.
For uncomplicatedness, we represent the SFN asQ˜=ΓQ˜,ΥQ˜,ΛQ˜.Definition 5 (see [26]).
SupposingŢˆ be a universal set, an interval-valued SFS IVSFSA˜Q˜ is defined as follows:(9)A˜Q˜=ţˆ,μ˜A˜Q˜Lţˆ,μ˜A˜Q˜Uţˆ,η˜A˜Q˜Lţˆ,η˜A˜Q˜Uţˆ,ϑ˜A˜Q˜Lţˆ,ϑ˜A˜Q˜Uţˆ|ţˆ∈Ţˆ,where 0≤μ˜A˜Q˜Lţˆ≤μ˜A˜Q˜Uţˆ≤1,0≤η˜A˜Q˜Lţˆ≤η˜A˜Q˜Uţˆ≤1,0≤ϑ˜A˜Q˜Lţˆ≤ϑ˜A˜Q˜Uţˆ≤1, and 0≤μ˜A˜Q˜Uţˆ2+η˜A˜Q˜Uţˆ2+ϑ˜A˜Q˜Uţˆ2≤1. If μ˜A˜Q˜Lţˆ=μ˜A˜Q˜Uţˆ,η˜A˜Q˜Lţˆ=η˜A˜Q˜Uţˆ,ϑ˜A˜Q˜Lţˆ=ϑ˜A˜Q˜Uţˆ, then IVSFS reduces to SFS.Definition 6 (see27).
LetA˜Q˜=μ˜A˜Q˜Lţˆ,μ˜A˜Q˜Uţˆ,η˜A˜Q˜Lţˆ,η˜A˜Q˜Uţˆ,ϑ˜A˜Q˜Lţˆ,ϑ˜A˜Q˜Uţˆ be IVSFS, the score function of A˜Q˜ can be described in the following way:(10)ScoreA˜Q˜=12μ˜A˜Q˜Lţˆ2+μ˜A˜Q˜Uţˆ2−η˜A˜Q˜Lţˆ2−η˜A˜Q˜Uţˆ2−ϑ˜A˜Q˜Lţˆ2−ϑ˜A˜Q˜Uţˆ2,where score A˜Q˜∈−1,1.Definition 7 (see27).
LetA˜Q˜=μ˜A˜Q˜Lţˆ,μ˜A˜Q˜Uţˆ,η˜A˜Q˜Lţˆ,η˜A˜Q˜Uţˆ,ϑ˜A˜Q˜Lţˆ,ϑ˜A˜Q˜Uţˆ be IVSFS, the accuracy function of A˜Q˜ can be described in the following way:(11)accuracyA˜Q˜=12μ˜A˜Q˜Lţˆ2+μ˜A˜Q˜Uţˆ2+η˜A˜Q˜Lţˆ2+η˜A˜Q˜Uţˆ2+ϑ˜A˜Q˜Lţˆ2+ϑ˜A˜Q˜Uţˆţˆ2,where accuracy A˜Q˜∈0,1.Definition 8 (see [27]).
LetA˜Q˜1=μ˜A˜Q˜1Lţˆ,μ˜A˜Q˜1Uţˆ,η˜A˜Q˜1Lţˆ,η˜A˜Q˜1Uţˆ,ϑ˜A˜Q˜1Lţˆ,ϑ˜A˜Q˜1Uţˆ and A˜Q˜2=μ˜A˜Q˜2Lţˆ,μ˜A˜Q˜2Uţˆ,η˜A˜Q˜2Lţˆ,η˜A˜Q˜2Uţˆ,ϑ˜A˜Q˜2Lţˆ,ϑ˜A˜Q˜2Uţˆ be two IVSFNs; then,(12)SA˜Q˜1=ScoreA˜Q˜1=12μ˜A˜Q˜1Lţˆ2+μ˜A˜Q˜1Uţˆ2−η˜A˜Q˜1Lţˆ2−η˜A˜Q˜1Uţˆ2−ϑ˜A˜Q˜1Lţˆ2−ϑ˜A˜Q˜1Uţˆ2,SA˜Q˜2=ScoreA˜Q˜2=12μ˜A˜Q˜2Lţˆ2+μ˜A˜Q˜2Uţˆ2−η˜A˜Q˜2Lţˆ2−η˜A˜Q˜2Uţˆ2−ϑ˜A˜Q˜2Lţˆ2−ϑ˜A˜Q˜2Uţˆ2,are the score functions of A˜Q˜1 and A˜Q˜2, respectively. And(13)HA˜Q˜1=accuracyA˜Q˜1=12μ˜A˜Q˜1Lţˆ2+μ˜A˜Q˜1Uţˆ2+η˜A˜Q˜1Lţˆ2+η˜A˜Q˜1Uţˆ2+ϑ˜A˜Q˜1Lţˆ2+ϑ˜A˜Q˜1Uţˆ2,HA˜Q˜2=accuracyA˜Q˜2==12μ˜A˜Q˜2Lţˆ2+μ˜A˜Q˜2Uţˆ2+η˜A˜Q˜2Lţˆ2+η˜A˜Q˜2Uţˆ2+ϑ˜A˜Q˜2Lţˆ2+ϑ˜A˜Q˜2Uţˆ2,are the accuracy functions of A˜Q˜1 and A˜Q˜2, respectively. Then, we have the following properties:(1)
IfSA˜Q˜1<SA˜Q˜2, then A˜Q˜1<A˜Q˜2,(2)
IfSA˜Q˜1>SA˜Q˜2, then A˜Q˜1>A˜Q˜2,(3)
IfSA˜Q˜1=SA˜Q˜2, then we have(a)
IfHA˜Q˜1<HA˜Q˜2, then A˜Q˜1<A˜Q˜2,(b)
IfHA˜Q˜1>HA˜Q˜2, then A˜Q˜1>A˜Q˜2,(c)
IfHA˜Q˜1=HA˜Q˜2, then A˜Q˜1=A˜Q˜2.Definition 9 (see [27]).
LetA˜Q˜1=μ˜A˜Q˜1Lţˆ,μ˜A˜Q˜1Uţˆ,η˜A˜Q˜1Lţˆ,η˜A˜Q˜1Uţˆ,ϑ˜A˜Q˜1Lţˆ,ϑ˜A˜Q˜1Uţˆ and A˜Q˜2=μ˜A˜Q˜2Lţˆ,μ˜A˜Q˜2Uţˆ,η˜A˜Q˜2Lţˆ,η˜A˜Q˜2Uţˆ,ϑ˜A˜Q˜2Lţˆ,ϑ˜A˜Q˜2Uţˆ be two IVSFNs; then,(1)
λ˜A˜Q˜1=μ˜A˜Q˜1Lţˆ2+μ˜A˜Q˜2Lţˆ2−μ˜A˜Q˜1Lţˆ2μ˜A˜Q˜2Lţˆ2,μ˜A˜Q˜1Uţˆ2+μ˜A˜Q˜2Uţˆ2−μ˜A˜Q˜2Uţˆ2μ˜A˜Q˜1Uţˆ2,η˜A˜Q˜1Lţˆ⋅η˜A˜Q˜2Lţˆ,η˜A˜Q˜1Lţˆ⋅η˜A˜Q˜2Lţˆ,1−μ˜A˜Q˜1Lţˆ2μ˜A˜Q˜1Lţˆ2+1−μ˜A˜Q˜2Lţˆ2ϑ˜A˜Q˜2Lţˆ2−ϑ˜A˜Q˜1Lţˆ2ϑ˜A˜Q˜2Lţˆ21−μ˜A˜Q˜2Lţˆ2ϑ˜A˜Q˜1Uţˆ2+1−μ˜A˜Q˜1Uţˆ2ϑ˜A˜Q˜2Uţˆ2−ϑ˜A˜Q˜1Uţˆ2ϑ˜A˜Q˜2Uţˆ2(2)
μ˜A˜Q˜1Lţˆ⋅μ˜A˜Q˜2Lţˆ,μ˜A˜Q˜1Uţˆ⋅μ˜A˜Q˜2Uţˆη˜A˜Q˜1Lţˆ2+η˜A˜Q˜2Lţˆ2−η˜A˜Q˜1Lţˆ2η˜A˜Q˜2Lţˆ2,η˜A˜Q˜1Uţˆ2+η˜A˜Q˜2Lţˆ2−η˜A˜�˜1Lţˆ2η˜A˜Q˜2Lţˆ2μ˜A˜Q˜1Lţˆ⋅μ˜A˜Q˜2Lţˆ,μ˜A˜Q˜1Uţˆ⋅μ˜A˜Q˜2Uţˆ1−η˜A˜Q˜1Lţˆ2η˜A˜Q˜1Lţˆ2+1−η˜A˜Q˜1Lţˆ2η˜A˜Q˜2Lţˆ2−ϑ˜A˜Q˜2Lţˆ2ϑ˜A˜Q˜2Lţˆ21−η˜A˜Q˜1Uţˆ2η˜A˜Q˜1Uţˆ2+1−η˜A˜Q˜1Uţˆ2η˜A˜Q˜2Uţˆ2−ϑ˜A˜Q˜2Uţˆ2ϑ˜A˜Q˜2Uţˆ2(3)
λ˜A˜Q˜1=1−1−μ˜A˜Q˜1Lţˆ2λ˜,1−1−μ˜A˜Q˜1Uţˆ2λ˜,η˜A˜Q˜1Lţˆλ˜,η˜A˜Q˜1Uţˆλ˜,1−μ˜A˜Q˜1Lţˆ2λ˜−1−μ˜A˜Q˜1Lţˆ2−ϑ˜A˜Q˜1Lţˆ2λ˜,1−μ˜A˜Q˜1Uţˆ2λ˜−1−μ˜A˜Q˜1Uţˆ2−ϑ˜A˜Q˜1Uţˆ2λ˜(4)
A˜Q˜1λ˜=μ˜A˜Q˜1Lţˆλ˜,μ˜A˜Q˜1Uţˆλ˜1−1−η˜A˜Q˜1Lţˆ2λ˜,1−1−η˜A˜Q˜1Uţˆ2λ˜,η˜A˜Q˜1Lţˆλ˜,η˜A˜Q˜1Uţˆλ˜,1−η˜A˜Q˜1Lţˆ2λ˜−1−η˜A˜Q˜1Lţˆ2−ϑ˜A˜Q˜1Lţˆ2λ˜,1−η˜A˜Q˜1Uţˆ2λ˜−1−η˜A˜Q˜1Uţˆ2−ϑ˜A˜Q˜1Uţˆ2λ˜.Definition 10 (see [27]).
LetA˜Q˜1=μ˜A˜Q˜1Lţˆ,μ˜A˜Q˜1Uţˆ,η˜A˜Q˜1Lţˆ,η˜A˜Q˜1Uţˆ,ϑ˜A˜Q˜1Lţˆ,ϑ˜A˜Q˜1Uţˆţˆ and A˜Q˜2=μ˜A˜Q˜2Lţˆ,μ˜A˜Q˜2Uţˆ,η˜A˜Q˜2Lţˆ,η˜A˜Q˜2Uţˆ,ϑ˜A˜Q˜2Lţˆ,ϑ˜A˜Q˜2Uţˆ be the set comprised of IVSFNs; then, IVSF weighted geometric (IVSFWG) operator is defined as follows:(14)IVSFWGA˜Q˜1,A˜Q˜2,…,A˜Q˜N=∏i=1nμ˜A˜Q˜1Lţˆωi,∏i=1nμ˜A˜Q˜1Uţˆωi,1−∏i=1n1−η˜A˜Q˜1Lţˆ2ωi,1−∏i=1n1−η˜A˜Q˜1Uţˆ2ωi,η˜A˜Q˜1Lţˆλ˜,η˜A˜Q˜1Uţˆλ˜,∏i=1n1−η˜A˜Q˜1Lţˆ2ωi−∏i=1n1−η˜A˜Q˜1Lţˆ2−ϑ˜A˜Q˜1Lţˆ2ωi,∏i=1n1−η˜A˜Q˜1Uţˆ2ωi−∏i=1n1−η˜A˜Q˜1Uţˆ2−ϑ˜A˜Q˜1Uţˆ2ωi,where ϖ=ϖ1,ϖ2,…,ϖnT represent the weight vector of A˜Q˜ii=1,2,…,n and 0≤ϖi≤1,∑i=1nϖi=1.Definition 11 (see [18]).
SupposingŢˆ be a universal set, then a spherical cubic FS (SCFS) Q˜C˜ is defined as follows:(15)Q˜C˜=ţˆ,ΓQ˜C˜ţˆ,ΥQ˜C˜ţˆ,ΛQ˜C˜ţˆ∣ţˆ∈Ţˆ,where ΓQ˜C˜ţˆ=aQ˜C˜−,aQ˜C˜+,μ˜Q˜C˜ represents the membership function, ΥQ˜C˜ţˆ=nQ˜C˜−,nQ˜C˜+,η˜Q˜C˜ represents the neutral degree, and ΛQ˜C˜ţˆ=bQ˜C˜−,bQ˜C˜+,ϑ˜Q˜C˜ represents the nonmembership function, in which μ˜Q˜C˜:Ţˆ⟶0,1,η˜Q˜C˜:Ţˆ⟶0,1, and ϑ˜Q˜C˜:Ţˆ⟶0,1,ţˆ∈Ţˆ, under the specified condition:(16)0≤supaQ˜C˜−,aQ˜C˜+2+supnQ˜C˜−,nQ˜C˜+2+supbQ˜C˜−,bQ˜C˜+2≤1,0≤μ˜Q˜C˜2+η˜Q˜C˜2+ϑ˜Q˜C˜2≤1.
The indeterminacy degree for (SCFS)Q˜C˜ is defined as follows:(17)π˜Q˜=|1−supaQ˜C˜−,aQ˜C˜+2−supnQ˜C˜−,nQ˜C˜+2−supbQ˜C˜−,bQ˜C˜+2,1−μ˜Q˜C˜2−η˜Q˜C˜2−ϑ˜Q˜C˜2.
For simplicity, we represent the spherical numberSCFN as Q˜C˜=ΓQ˜C˜,ΥQ˜C˜,ΛQ˜C˜.Definition 12 (see [19]).
LetQ˜C˜=aQ˜C˜−,aQ˜C˜+,μ˜Q˜C˜,nQ˜C˜−,nQ˜C˜+,η˜Q˜C˜,bQ˜C˜−,bQ˜C˜+,ϑ˜Q˜C˜ be SCFS; then, the score function of Q˜C˜ is defined as follows:(18)ScoreQ˜C˜=19aQ˜C˜−+aQ˜C˜++μ˜Q˜C˜2+nQ˜C˜−+nQ˜C˜++η˜Q˜C˜2−bQ˜C˜−+bQ˜C˜++ϑ˜Q˜C˜2,where Score Q˜C˜∈−1,1.Definition 13 (see [19]).
SupposingQ˜C˜=aQ˜C˜−,aQ˜C˜+,μ˜Q˜C˜,nQ˜C˜−,nQ˜C˜+,η˜Q˜C˜,bQ˜C˜−,bQ˜C˜+,ϑ˜Q˜C˜ be SCFS, then the accuracy function of Q˜C˜ is defined as follows:(19)accuracyQ˜C˜=19aQ˜C˜−+aQ˜C˜++μ˜Q˜C˜2+nQ˜C˜+nQ˜C˜++η˜Q˜C˜2+bQ˜C˜−+bQ˜C˜++ϑ˜Q˜C˜2,where accuracy Q˜C˜∈0,1.Definition 14 (see [19]).
LetQ˜C˜1=ΓQ˜C˜1,ΥQ˜C˜1,ΛQ˜C˜1 and Q˜C˜2=ΓQ˜C˜2,ΥQ˜C˜2,ΛQ˜C˜2 be two SCFNs; then, we have the following:(1)
If scoreQ˜C˜1<scoreQ˜C˜2, then Q˜C˜1<Q˜C˜2,(2)
If scoreQ˜C˜1> score Q˜C˜2, then Q˜C˜1>Q˜C˜2,(3)
If scoreQ˜C˜1=scoreQ˜C˜2, then(a)
If accuracyQ˜C˜1< accuracy Q˜C˜2, then(b)
If accuracyQ˜C˜1> accuracy Q˜C˜2, then iQ˜C˜1>Q˜C˜2,(c)
If accuracyQ˜C˜1= accuracy Q˜C˜2, then Q˜C˜1=Q˜C˜2.Definition 15 (see [19]).
LetQ˜C˜1=ΓQ˜C˜1,ΥQ˜C˜1,ΛQ˜C˜1 and Q˜C˜2=ΓQ˜C˜2,ΥQ˜C˜2,ΛQ˜C˜2 be two SCFNs; then, the distance function among Q˜C˜2 and Q˜C˜2 is defined as follows:(20)D˜Q˜C˜1,Q˜C˜2=19aQ˜C˜1−2−aQ˜C˜2−2+aQ˜C˜1+2−aQ˜C˜2+2+μ˜Q˜C˜12−μ˜Q˜C˜22+nQ˜C˜1−2−nQ˜C˜2−2+nQ˜C˜1+2−nQ˜C˜2+2+η˜Q˜C˜12−η˜Q˜C˜22+bQ˜C˜1−2−bQ˜C˜2−2+bQ˜C˜1+2−bQ˜C˜2+2+ϑ˜Q˜C˜12−ϑ˜Q˜C˜22.Definition 16 (see [19]).
LetQ˜C˜1=ΓQ˜C˜1,ΥQ˜C˜1,ΛQ˜C˜1 and Q˜C˜2=ΓQ˜C˜2,ΥQ˜C˜2,ΛQ˜C˜2 be two spherical cubic fuzzy numbers; then, the below operational laws hold:(1)
Q˜C˜1⊕Q˜C˜2=aQ˜C˜1−2+aQ˜C˜2−2−aQ˜C˜1−2aQ˜C˜2−2,aQ˜C˜1+2+aQ˜C˜2+2−aQ˜C˜1+2aQ˜C˜2+2,μ˜Q˜C˜12+μ˜Q˜C˜22−μ˜Q˜C˜12μ˜Q˜C˜22,nQ˜C˜1−.nQ˜C˜2−,nQ˜C˜1+.nQ˜C˜2+,η˜Q˜C˜1.η˜Q˜C˜2,bQ˜C˜1−.bQ˜C˜2−,bQ˜C˜1+.bQ˜C˜2+,ϑ˜Q˜C˜1.ϑ˜Q˜C˜2;(2)
Q˜C˜1⊗Q˜C˜2=aQ˜C˜1−.aQ˜C˜2−,aQ˜C˜1+.aQ˜C˜2+,μ˜Q˜C˜1.μ˜Q˜C˜2,nQ˜C˜1−.nQ˜C˜2−,nQ˜C˜1+.nQ˜C˜2+,η˜Q˜C˜1.η˜Q˜C˜2bQ˜C˜1−2+bQ˜C˜2−2−bQ˜C˜1−2bQ˜C˜2−2,bQ˜C˜1+2+bQ˜C˜2+2−bQ˜C˜1+2bQ˜C˜2+2,ϑ˜Q˜C˜12+ϑ˜Q˜C˜22−ϑ˜Q˜C˜12ϑ˜Q˜C˜22,;(3)
τ.Q˜C˜1=1−1−aQ˜C˜1−2τ,1−1−aQ˜C˜1+2τ,1−1−μ˜Q˜C˜12τ,nQ˜C˜1−τ,nQ˜C˜1+τ,η˜Q˜C˜1τ,bQ˜C˜1−τ,bQ˜C˜1+τ,ϑ˜Q˜C˜1τ;(4)
Q˜C˜1τ=aQ˜C˜1−τ,aQ˜C˜1+τ,μ˜Q˜C˜1τ,nQ˜C˜1−τ,nQ˜C˜1+τ,η˜Q˜C˜1τ1−1−bQ˜C˜1−2τ,1−1−bQ˜C˜1+2τ,1−1−ϑ˜Q˜C˜12τ.


## 3. Spherical Cubic Fuzzy Geometric Aggregation Operators

This section proposes a few geometric aggregation operators for spherical cubic fuzzy numbers and discusses their properties.Definition 17.
LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜ii=1,2,3,…,n be a collection of SCFNs; then, SCF weighted geometric SCFWG operator is a mapping SCFWG: SCFNn⟶SCFN defined as follows:(21)SCFWGQ˜C˜1,Q˜C˜2,Q˜C˜3…,Q˜C˜n=Q˜C˜1ϖ1⊗Q˜C˜2ϖ2⊗,…,Q˜C˜nϖn,where ϖ=ϖ1,ϖ2,…,ϖnT is the weight vector of Q˜C˜i, with 0≤ϖi≤1 and ∑i=1nϖi=1.Theorem 1.
LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜ii=1,2,3,…,n be a collection of SCFNs, and let ϖ=ϖ1,ϖ2,…,ϖnT be the weight vector of Q˜C˜i, with 0≤ϖi≤1 and ∑i=1nϖi=1. Then, the aggregation of SCFWG operator is also an SCFN:(22)SCFWGQ˜C˜1,Q˜C˜2,…,Q˜C˜n=∏i=1naQ˜C˜i−ϖi,∏i=1naQ˜C˜i+ϖi,∏i=1nμ˜Q˜C˜iϖi,∏i=1nnQ˜C˜i−ϖi,∏i=1nnQ˜C˜i+ϖi,∏i=1nη˜Q˜C˜iϖi,1−∏i=1n1−bQ˜C˜i−2ϖi,1−∏i=1n1−bQ˜C˜i+2ϖi,1−∏i=1n1−ϑ˜Q˜C˜i2ϖi,.Proof.
Proof of the theorem follows from theorem [7] of [18].


### 3.1. Properties

LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜ii=1,2,3…,n be a set comprising of SCFNs, and the weight vector of Q˜C˜i is ϖ=ϖ1,ϖ2,…,ϖnT with 0≤ϖi≤1 and ∑i=1nϖi=1 Then, there are some properties that the SCFWG operator has clearly fulfilled.


#### 3.1.1. Idempotency

If allQ˜C˜i are identical, that is, Q˜C˜i=Q˜C˜, then(23)SCFWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n=Q˜C˜.


#### 3.1.2. Boundary

(24)Q˜C˜min≤SCFWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n≤Q˜C˜max,where(25)Q˜C˜min=minΓQ˜C˜i,minΥQ˜C˜i,maxΛQ˜C˜i,Q˜C˜max=maxΓQ˜C˜i,minΥQ˜C˜i,minΛQ˜C˜i.


#### 3.1.3. Monotonicity

LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜i and Q˜C˜i∗=ΓQ˜C˜i∗,ΥC˜i∗,ΛC˜i∗i=1,2,…,n be collections of SCFNs; if Q˜C˜i⊆Q˜C˜i∗ for all, then(26)SCFWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n⊆SCFWGQ˜C˜1∗,Q˜C˜2∗,Q˜C˜3∗,…,Q˜C˜n∗.Definition 18.
LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜ii=1,2,3,…,n be a set comprising of SCFNs; then, SCF ordered weighted geometric (SCFOWG) operator is a mapping SCFOWG: SCFNn⟶SCFN defined as follows:(27)SCFOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n=Q˜C˜1ϖ1⊗Q˜C˜2ϖ2⊗,…,Q˜C˜nϖn,where the ith largest weighted value is Q˜σC˜i by the total order Q˜σC˜1≥Q˜σC˜2≥⋯≥Q˜σC˜n and ϖ=ϖ1,ϖ2,…,ϖnT is the weight vector of Q˜C˜i, with 0≤ϖi≤1 and ∑i=1nϖi=1.Theorem 2.
LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜ii=1,2,3…,n be a set comprising of SCFNs, and let ϖ=ϖ1,ϖ2,…,ϖnT be the weight vector of Q˜C˜i, with 0≤ϖi≤1 and ∑i=1nϖi=1. Then, the aggregation of the SCFOWG operator is also a SCFN:(28)SCFOWGQ˜C˜1,Q˜C˜2,…,Q˜C˜n=∏i=1naQ˜σC˜i−ϖi,∏i=1naQ˜σC˜i+ϖi,∏i=1nμ˜Q˜σC˜iϖi,∏i=1nnQ˜σC˜i−ϖi,∏i=1nnQ˜σC˜i+ϖi,∏i=1nη˜Q˜σC˜iϖi,1−∏i=1n1−bQ˜σC˜i−2ϖi,1−∏i=1n1−bQ˜σC˜i+2ϖi,1−∏i=1n1−ϑ˜Q˜σC˜i2ϖi,.Proof.
Proof of the theorem follows from theorem [8] of [18].


### 3.2. Properties

LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜ii=1,2,3…,n be a set comprising of SCFNs, and the weight vector of Q˜C˜i is ϖ=ϖ1,ϖ2,…,ϖnT with 0≤ϖi≤1 and ∑i=1nϖi=1. Then, there are some properties that the SCFOWG operator has clearly fulfilled.


#### 3.2.1. Idempotency

If allQ˜C˜i are identical, that is, Q˜C˜i=Q˜C˜, then(29)SCFOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n=Q˜C˜.


#### 3.2.2. Boundary

(30)Q˜C˜min≤SCFOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n≤Q˜C˜max,where(31)Q˜C˜min=minσQ˜C˜i,minβQ˜C˜i,maxγQ˜C˜i,Q˜C˜max=maxσQ˜C˜i,minβQ˜C˜i,minγQ˜C˜i.


#### 3.2.3. Monotonicity

LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜i and Q˜C˜i∗=ΓQ˜C˜i∗,ΥC˜i∗,ΛC˜i∗i=1,2,…,n be collections of SCFNs; if Q˜C˜i⊆Q˜C˜i∗ for all i, then(32)SCFOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n⊆SCFOWGQ˜C˜1∗,Q˜C˜2∗,Q˜C˜3∗,…,Q˜C˜n∗.Definition 19.
Leti be a set comprised of SCFNs; then, SCF hybrid ordered weighted geometric (SCFHOWG) operator is a mapping SCFHOWG: SCFNn⟶SCFN defined as follows:(33)SCFHOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n=Q˜C˜1ϖ1⊗Q˜C˜2ϖ2⊗,…,Q˜C˜nϖn,where the ithi largest weighted value is Q˜σC˜i and Q˜C˜i′Q˜C˜i′=nϖiQ˜C˜i and ϖ=ϖ1,ϖ2,…,ϖnT is the weight vector of Q˜C˜i, with 0≤ϖi≤1 and ∑i=1nϖi=1. Also, wˆ=wˆ1,wˆ2,wˆ3,…,wˆn is the associated weight vector such that 0≤wˆj≤1 and ∑j=1nwˆj=1.Theorem 4.
LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜ii=1,2,3,…,n be a set comprised of SCFNs, and let ϖ=ϖ1,ϖ2,…,ϖnT be the weight vector of Q˜C˜i, with 0≤ϖi≤1 and ∑i=1nϖi=1. Then, the aggregation of SCFHOWG operator is an SCFN as well, and(34)SCFHOWGQ˜C˜1,Q˜C˜2,…,Q˜C˜n=∏i=1naQ˜σC˜i−ϖi,∏i=1naQ˜σC˜i+ϖi,∏i=1nμ˜Q˜σC˜iϖi,∏i=1nnQ˜σC˜i−ϖi,∏i=1nnQ˜σC˜i+ϖi,∏i=1nη˜Q˜σC˜iϖi,1−∏i=1n1−bQ˜σC˜i−2ϖi,1−∏i=1n1−bQ˜σC˜i+2ϖi,1−∏i=1n1−ϑ˜Q˜σC˜i2ϖi,.Proof.
Proof of the theorem follows from theorem [9] of [18].


### 3.3. Properties

LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜ii=1,2,3,…,n be a set comprised of SCFNs, and the weight vector of Q˜C˜i is ϖ=ϖ1,ϖ2,…,ϖnT with 0≤ϖi≤1 and ∑i=1nϖi=1. Then, there are some properties that the SCFHOWG operator has clearly fulfilled.


#### 3.3.1. Idempotency

If allQ˜C˜i are identical, that is, Q˜C˜i=Q˜C˜, then(35)SCFHOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n=Q˜C˜.


#### 3.3.2. Boundary

(36)Q˜C˜min≤SCFHOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n≤Q˜C˜max,where(37)Q˜C˜min=minσQ˜C˜i,minβQ˜C˜i,maxγQ˜C˜i,Q˜C˜max=maxσQ˜C˜i,minβQ˜C˜i,minγQ˜C˜i.


#### 3.3.3. Monotonicity

LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜i and Q˜C˜i∗=ΓQ˜C˜i∗,ΥC˜i∗,ΛC˜i∗i=1,2,…,n be collections of SCFNs; if Q˜C˜i⊆Q˜C˜i∗ for all i, then(38)SCFHOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n⊆SCFHOWGQ˜C˜1∗,Q˜C˜2∗,Q˜C˜3∗,…,Q˜C˜n∗.


## 3.1. Properties

LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜ii=1,2,3…,n be a set comprising of SCFNs, and the weight vector of Q˜C˜i is ϖ=ϖ1,ϖ2,…,ϖnT with 0≤ϖi≤1 and ∑i=1nϖi=1 Then, there are some properties that the SCFWG operator has clearly fulfilled.


### 3.1.1. Idempotency

If allQ˜C˜i are identical, that is, Q˜C˜i=Q˜C˜, then(23)SCFWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n=Q˜C˜.


### 3.1.2. Boundary

(24)Q˜C˜min≤SCFWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n≤Q˜C˜max,where(25)Q˜C˜min=minΓQ˜C˜i,minΥQ˜C˜i,maxΛQ˜C˜i,Q˜C˜max=maxΓQ˜C˜i,minΥQ˜C˜i,minΛQ˜C˜i.


### 3.1.3. Monotonicity

LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜i and Q˜C˜i∗=ΓQ˜C˜i∗,ΥC˜i∗,ΛC˜i∗i=1,2,…,n be collections of SCFNs; if Q˜C˜i⊆Q˜C˜i∗ for all, then(26)SCFWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n⊆SCFWGQ˜C˜1∗,Q˜C˜2∗,Q˜C˜3∗,…,Q˜C˜n∗.Definition 18.
LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜ii=1,2,3,…,n be a set comprising of SCFNs; then, SCF ordered weighted geometric (SCFOWG) operator is a mapping SCFOWG: SCFNn⟶SCFN defined as follows:(27)SCFOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n=Q˜C˜1ϖ1⊗Q˜C˜2ϖ2⊗,…,Q˜C˜nϖn,where the ith largest weighted value is Q˜σC˜i by the total order Q˜σC˜1≥Q˜σC˜2≥⋯≥Q˜σC˜n and ϖ=ϖ1,ϖ2,…,ϖnT is the weight vector of Q˜C˜i, with 0≤ϖi≤1 and ∑i=1nϖi=1.Theorem 2.
LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜ii=1,2,3…,n be a set comprising of SCFNs, and let ϖ=ϖ1,ϖ2,…,ϖnT be the weight vector of Q˜C˜i, with 0≤ϖi≤1 and ∑i=1nϖi=1. Then, the aggregation of the SCFOWG operator is also a SCFN:(28)SCFOWGQ˜C˜1,Q˜C˜2,…,Q˜C˜n=∏i=1naQ˜σC˜i−ϖi,∏i=1naQ˜σC˜i+ϖi,∏i=1nμ˜Q˜σC˜iϖi,∏i=1nnQ˜σC˜i−ϖi,∏i=1nnQ˜σC˜i+ϖi,∏i=1nη˜Q˜σC˜iϖi,1−∏i=1n1−bQ˜σC˜i−2ϖi,1−∏i=1n1−bQ˜σC˜i+2ϖi,1−∏i=1n1−ϑ˜Q˜σC˜i2ϖi,.Proof.
Proof of the theorem follows from theorem [8] of [18].


## 3.1.1. Idempotency

If allQ˜C˜i are identical, that is, Q˜C˜i=Q˜C˜, then(23)SCFWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n=Q˜C˜.


## 3.1.2. Boundary

(24)Q˜C˜min≤SCFWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n≤Q˜C˜max,where(25)Q˜C˜min=minΓQ˜C˜i,minΥQ˜C˜i,maxΛQ˜C˜i,Q˜C˜max=maxΓQ˜C˜i,minΥQ˜C˜i,minΛQ˜C˜i.


## 3.1.3. Monotonicity

LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜i and Q˜C˜i∗=ΓQ˜C˜i∗,ΥC˜i∗,ΛC˜i∗i=1,2,…,n be collections of SCFNs; if Q˜C˜i⊆Q˜C˜i∗ for all, then(26)SCFWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n⊆SCFWGQ˜C˜1∗,Q˜C˜2∗,Q˜C˜3∗,…,Q˜C˜n∗.Definition 18.
LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜ii=1,2,3,…,n be a set comprising of SCFNs; then, SCF ordered weighted geometric (SCFOWG) operator is a mapping SCFOWG: SCFNn⟶SCFN defined as follows:(27)SCFOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n=Q˜C˜1ϖ1⊗Q˜C˜2ϖ2⊗,…,Q˜C˜nϖn,where the ith largest weighted value is Q˜σC˜i by the total order Q˜σC˜1≥Q˜σC˜2≥⋯≥Q˜σC˜n and ϖ=ϖ1,ϖ2,…,ϖnT is the weight vector of Q˜C˜i, with 0≤ϖi≤1 and ∑i=1nϖi=1.Theorem 2.
LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜ii=1,2,3…,n be a set comprising of SCFNs, and let ϖ=ϖ1,ϖ2,…,ϖnT be the weight vector of Q˜C˜i, with 0≤ϖi≤1 and ∑i=1nϖi=1. Then, the aggregation of the SCFOWG operator is also a SCFN:(28)SCFOWGQ˜C˜1,Q˜C˜2,…,Q˜C˜n=∏i=1naQ˜σC˜i−ϖi,∏i=1naQ˜σC˜i+ϖi,∏i=1nμ˜Q˜σC˜iϖi,∏i=1nnQ˜σC˜i−ϖi,∏i=1nnQ˜σC˜i+ϖi,∏i=1nη˜Q˜σC˜iϖi,1−∏i=1n1−bQ˜σC˜i−2ϖi,1−∏i=1n1−bQ˜σC˜i+2ϖi,1−∏i=1n1−ϑ˜Q˜σC˜i2ϖi,.Proof.
Proof of the theorem follows from theorem [8] of [18].


## 3.2. Properties

LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜ii=1,2,3…,n be a set comprising of SCFNs, and the weight vector of Q˜C˜i is ϖ=ϖ1,ϖ2,…,ϖnT with 0≤ϖi≤1 and ∑i=1nϖi=1. Then, there are some properties that the SCFOWG operator has clearly fulfilled.


### 3.2.1. Idempotency

If allQ˜C˜i are identical, that is, Q˜C˜i=Q˜C˜, then(29)SCFOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n=Q˜C˜.


### 3.2.2. Boundary

(30)Q˜C˜min≤SCFOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n≤Q˜C˜max,where(31)Q˜C˜min=minσQ˜C˜i,minβQ˜C˜i,maxγQ˜C˜i,Q˜C˜max=maxσQ˜C˜i,minβQ˜C˜i,minγQ˜C˜i.


### 3.2.3. Monotonicity

LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜i and Q˜C˜i∗=ΓQ˜C˜i∗,ΥC˜i∗,ΛC˜i∗i=1,2,…,n be collections of SCFNs; if Q˜C˜i⊆Q˜C˜i∗ for all i, then(32)SCFOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n⊆SCFOWGQ˜C˜1∗,Q˜C˜2∗,Q˜C˜3∗,…,Q˜C˜n∗.Definition 19.
Leti be a set comprised of SCFNs; then, SCF hybrid ordered weighted geometric (SCFHOWG) operator is a mapping SCFHOWG: SCFNn⟶SCFN defined as follows:(33)SCFHOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n=Q˜C˜1ϖ1⊗Q˜C˜2ϖ2⊗,…,Q˜C˜nϖn,where the ithi largest weighted value is Q˜σC˜i and Q˜C˜i′Q˜C˜i′=nϖiQ˜C˜i and ϖ=ϖ1,ϖ2,…,ϖnT is the weight vector of Q˜C˜i, with 0≤ϖi≤1 and ∑i=1nϖi=1. Also, wˆ=wˆ1,wˆ2,wˆ3,…,wˆn is the associated weight vector such that 0≤wˆj≤1 and ∑j=1nwˆj=1.Theorem 4.
LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜ii=1,2,3,…,n be a set comprised of SCFNs, and let ϖ=ϖ1,ϖ2,…,ϖnT be the weight vector of Q˜C˜i, with 0≤ϖi≤1 and ∑i=1nϖi=1. Then, the aggregation of SCFHOWG operator is an SCFN as well, and(34)SCFHOWGQ˜C˜1,Q˜C˜2,…,Q˜C˜n=∏i=1naQ˜σC˜i−ϖi,∏i=1naQ˜σC˜i+ϖi,∏i=1nμ˜Q˜σC˜iϖi,∏i=1nnQ˜σC˜i−ϖi,∏i=1nnQ˜σC˜i+ϖi,∏i=1nη˜Q˜σC˜iϖi,1−∏i=1n1−bQ˜σC˜i−2ϖi,1−∏i=1n1−bQ˜σC˜i+2ϖi,1−∏i=1n1−ϑ˜Q˜σC˜i2ϖi,.Proof.
Proof of the theorem follows from theorem [9] of [18].


## 3.2.1. Idempotency

If allQ˜C˜i are identical, that is, Q˜C˜i=Q˜C˜, then(29)SCFOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n=Q˜C˜.


## 3.2.2. Boundary

(30)Q˜C˜min≤SCFOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n≤Q˜C˜max,where(31)Q˜C˜min=minσQ˜C˜i,minβQ˜C˜i,maxγQ˜C˜i,Q˜C˜max=maxσQ˜C˜i,minβQ˜C˜i,minγQ˜C˜i.


## 3.2.3. Monotonicity

LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜i and Q˜C˜i∗=ΓQ˜C˜i∗,ΥC˜i∗,ΛC˜i∗i=1,2,…,n be collections of SCFNs; if Q˜C˜i⊆Q˜C˜i∗ for all i, then(32)SCFOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n⊆SCFOWGQ˜C˜1∗,Q˜C˜2∗,Q˜C˜3∗,…,Q˜C˜n∗.Definition 19.
Leti be a set comprised of SCFNs; then, SCF hybrid ordered weighted geometric (SCFHOWG) operator is a mapping SCFHOWG: SCFNn⟶SCFN defined as follows:(33)SCFHOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n=Q˜C˜1ϖ1⊗Q˜C˜2ϖ2⊗,…,Q˜C˜nϖn,where the ithi largest weighted value is Q˜σC˜i and Q˜C˜i′Q˜C˜i′=nϖiQ˜C˜i and ϖ=ϖ1,ϖ2,…,ϖnT is the weight vector of Q˜C˜i, with 0≤ϖi≤1 and ∑i=1nϖi=1. Also, wˆ=wˆ1,wˆ2,wˆ3,…,wˆn is the associated weight vector such that 0≤wˆj≤1 and ∑j=1nwˆj=1.Theorem 4.
LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜ii=1,2,3,…,n be a set comprised of SCFNs, and let ϖ=ϖ1,ϖ2,…,ϖnT be the weight vector of Q˜C˜i, with 0≤ϖi≤1 and ∑i=1nϖi=1. Then, the aggregation of SCFHOWG operator is an SCFN as well, and(34)SCFHOWGQ˜C˜1,Q˜C˜2,…,Q˜C˜n=∏i=1naQ˜σC˜i−ϖi,∏i=1naQ˜σC˜i+ϖi,∏i=1nμ˜Q˜σC˜iϖi,∏i=1nnQ˜σC˜i−ϖi,∏i=1nnQ˜σC˜i+ϖi,∏i=1nη˜Q˜σC˜iϖi,1−∏i=1n1−bQ˜σC˜i−2ϖi,1−∏i=1n1−bQ˜σC˜i+2ϖi,1−∏i=1n1−ϑ˜Q˜σC˜i2ϖi,.Proof.
Proof of the theorem follows from theorem [9] of [18].


## 3.3. Properties

LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜ii=1,2,3,…,n be a set comprised of SCFNs, and the weight vector of Q˜C˜i is ϖ=ϖ1,ϖ2,…,ϖnT with 0≤ϖi≤1 and ∑i=1nϖi=1. Then, there are some properties that the SCFHOWG operator has clearly fulfilled.


### 3.3.1. Idempotency

If allQ˜C˜i are identical, that is, Q˜C˜i=Q˜C˜, then(35)SCFHOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n=Q˜C˜.


### 3.3.2. Boundary

(36)Q˜C˜min≤SCFHOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n≤Q˜C˜max,where(37)Q˜C˜min=minσQ˜C˜i,minβQ˜C˜i,maxγQ˜C˜i,Q˜C˜max=maxσQ˜C˜i,minβQ˜C˜i,minγQ˜C˜i.


### 3.3.3. Monotonicity

LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜i and Q˜C˜i∗=ΓQ˜C˜i∗,ΥC˜i∗,ΛC˜i∗i=1,2,…,n be collections of SCFNs; if Q˜C˜i⊆Q˜C˜i∗ for all i, then(38)SCFHOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n⊆SCFHOWGQ˜C˜1∗,Q˜C˜2∗,Q˜C˜3∗,…,Q˜C˜n∗.


## 3.3.1. Idempotency

If allQ˜C˜i are identical, that is, Q˜C˜i=Q˜C˜, then(35)SCFHOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n=Q˜C˜.


## 3.3.2. Boundary

(36)Q˜C˜min≤SCFHOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n≤Q˜C˜max,where(37)Q˜C˜min=minσQ˜C˜i,minβQ˜C˜i,maxγQ˜C˜i,Q˜C˜max=maxσQ˜C˜i,minβQ˜C˜i,minγQ˜C˜i.


## 3.3.3. Monotonicity

LetQ˜C˜i=ΓQ˜C˜i,ΥQ˜C˜i,ΛQ˜C˜i and Q˜C˜i∗=ΓQ˜C˜i∗,ΥC˜i∗,ΛC˜i∗i=1,2,…,n be collections of SCFNs; if Q˜C˜i⊆Q˜C˜i∗ for all i, then(38)SCFHOWGQ˜C˜1,Q˜C˜2,Q˜C˜3,…,Q˜C˜n⊆SCFHOWGQ˜C˜1∗,Q˜C˜2∗,Q˜C˜3∗,…,Q˜C˜n∗.


## 4. Application for MADM Based on Spherical Cubic Fuzzy Geometric Aggregation Operators

This section utilizes the SCF geometric aggregation operators for the MADM process. Suppose we haven choices Yˆ=yˆ1,yˆ2,…,yˆn and m criteria Aˆ=Aˆ1,Aˆ2,Aˆ3,…,Aˆm to be determined with weight vector ϖ=ϖ1,ϖ2,…,ϖnT with 0≤ϖj≤1 and ∑j=1nϖj=1. In order to assess the proficiency of the alternative yˆi according to Aˆj criteria, the decision-maker must not simply provide the details where yˆi fulfills the criteria Aˆj, but the alternative yˆi does not fulfill the criteria Aˆj or the alternative yˆi remains unchanged. These three components can be defined by ΓQ˜C˜,ΥQ˜C˜,ΛQ˜C˜ where ΓQ˜C˜=aQ˜C˜−,aQ˜C˜+,μ˜Q˜C˜ represent the degree where the alternative yˆi fulfills the criteria Aˆj and Υ=nQ˜C˜−,nQ˜C˜+,η˜Q˜C˜ represents the degree where the alternative yˆi remains unchanged and ΛQ˜C˜=bQ˜C˜−,bQ˜C˜+,ϑ˜Q˜C˜ represents the degree where the alternative yˆi does not fulfill the criteria Aˆj; then, the efficiency of yˆi under the criteria Aˆj can be defined by SCFN, Q˜C˜=aQ˜C˜−,aQ˜C˜+,μ˜Q˜C˜,nQ˜C˜−,nQ˜C˜+,η˜Q˜C˜,bQ˜C˜−,bQ˜C˜+,ϑ˜Q˜C˜ with the specified condition 0≤supaQ˜C˜−,aQ˜C˜+2+supaQ˜C˜−,aQ˜C˜+2+supbQ˜C˜−,bQ˜C˜+2≤ and 0≤μ˜Q˜C˜2+η˜Q˜C˜2+ϑ˜Q˜C˜2≤1. The following steps are specified to obtain the ranking of the alternatives.Step 1: Firstly, we make a spherical cubic fuzzy matrix:(39)DN˜=aQ˜C˜,aQ˜C˜+,μ˜Q˜C˜,nQ˜C˜,nQ˜C˜+,η˜Q˜C˜,bQ˜C˜−,bQ˜C˜+,ϑ˜Q˜C˜m×n.If there are two types of criteria, for instance, cost-benefit criteria, the SC decision matrix could be transformed into a normalized one:(40)α˜Q˜C˜ij=Q˜C˜ijif  the benefit−type attributeQ˜C˜ijcif the cost−type attribute,whereQ˜C˜ijc represents the complement of Q˜C˜ij if every parameter is of the same form as they are not necessary to normalize the decision matrix.Step 2: To obtain an SCFNQ˜C˜ii=1,2,…,n, use the suggested aggregation operators for the alternatives yˆi. To be precise, the established operators stem the Q˜C˜ii=1,2,3,…,n of alternative yˆi where the weighted vector for the criteria is ϖ=ϖ1,ϖ2,…,ϖnT.Step 3: Determine the scores, scoreQ˜C˜ii=1,2,3,…,n , and the degree of accuracy Q˜C˜i of each Q˜C˜i.Step 4: The best alternative is selected by the ranking of all the alternatives.


## 5. Numerical Illustration

Suppose an investment group wants to spend money on the best option (alternative). A board with four possibilities options (alternative) for investing money:yˆ1,yˆ2,yˆ3,yˆ4⋅yˆ1 is a vehicle manufacturer, yˆ2 a food company, yˆ3 a washing machine company, and yˆ4 a television company. Four criteria, including a risk analysis Aˆ1, a growth and development analysis Aˆ2, a social impact analysis Aˆ3, and an environment impact analysis Aˆ4, are taken into account. ϖ=0.1,0.2,0.25,0.45T is the Aˆj criterion weight vector j=1,2,…,n. A three-decision-maker committee explores four possible alternatives yˆii=1,2,3,4 in the four Aˆjj=1,2,3,4 criteria above. Then, the ranking is needed to assess the investment company yˆ1,yˆ2,yˆ3,yˆ4. The following matrices represent these experts in the form of spherical cubic fuzzy numbers:Step 1.
The decision-makers make their decisions given in Table31–3.Table 1
First spherical cubic fuzzy decision matrix.
Aˆ1Aˆ2Aˆ3Aˆ4yˆ1.1,.2,.3,.4,.5,.4,.1,.3,.1.1,.3,.3,.1,.4,.3,.1,.3,.1.3,.5,.3,.1,.5,.5,.1,.3,.1.1,.4,.4,.1,.4,.2,.3,.5,.5yˆ2.2,.4,.1,.1,.4,.3,.1,.5,.6.3,.6,.5,.1,.2,.2,.1,.4,.2.1,.2,.1,.1,.4,.2,.1,.3,.3.3,.5,.5,.1,.4,.5,.1,.4,.4yˆ3.1,.2,.1,.1,.4,.2,.3,.6,.5.2,.5,.5,.4,.5,.2,.2,.3,.1.2,.4,.1,.1,.4,.3,.2,.4,.7.1,.2,.2,.1,.3,.4,.3,.4,.4yˆ4.1,.2,.3,.1,.3,.7,.1,.5,.2.1,.1,.3,.2,.4,.6,.3,.7,.2.4,.5,.8,.1,.2,.2,.1,.2,.1.1,.2,.1,.1,.1,.3,.2,.3,.6Table 2
Second spherical cubic fuzzy decision matrix.
Aˆ1Aˆ2Aˆ3Aˆ4yˆ1.1,.2,.5,.4,.4,.2,.2,.3,.5.1,.3,.2,.1,.4,.1,.1,.4,.3.1,.5,.6,.2,.3,.1,.1,.3,.2.1,.2,.5,.3,.6,.5,.1,.4,.2yˆ2.1,.4,.3,.1,.3,.4,.1,.7,.6.1,.4,.2,.1,.2,.1,.3,.7,.2.1,.3,.2,.2,.4,.5,.1,.5,.3.1,.6,.2,.1,.2,.6,.2,.5,.2yˆ3.2,.1,.3,.1,.5,.5,.1,.3,.2.2,.4,.5,.6,.7,.1,.1,.2,.5.1,.3,.2,.1,.5,.5,.1,.2,.3.1,.3,.2,.1,.4,.5,.1,.4,.5yˆ4.2,.4,.3,.4,.5,.3,.2,.4,.2.6,.7,.5,.1,.1,.2,.1,.4,.3.1,.4,.2,.3,.2,.7,.2,.6,.4.1,.3,.2,.1,.2,.3,.3,.7,.2Table 3
Third spherical cubic fuzzy decision matrix.
Aˆ1Aˆ2Aˆ3Aˆ4yˆ1.1,.3,.2,.1,.2,.3,.2,.1,.5.1,.2,.1,.2,.4,.2,.2,.3,.5.3,.5,.6,.1,.3,.1,.1,.3,.2.1,.2,.5,.2,.4,.2,.3,.6,.5yˆ2.1,.2,.3,.1,.2,.3,.2,.4,.1.1,.3,.2,.1,.4,.3,.3,.6,.2.1,.2,.1,.1,.2,.1,.4,.5,.3.2,.3,.1,.3,.4,.5,.1,.2,.1yˆ3.1,.5,.3,.2,.3,.1,.1,.2,.5.4,.3,.1,.2,.4,.5,.2,.3,.2.4,.3,.6,.1,.4,.1,.5,.6,.4.1,.4,.2,.3,.4,.1,.1,.3,.2yˆ4.1,.2,.4,.4,.5,.2,.2,.3,.2.1,.2,.4,.5,.7,.1,.1,.3,.2.1,.5,.2,.2,.3,.5,.1,.2,.4.1,.3,.2,.1,.2,.3,.1,.2,.3Step 2.
Using the SCFWG aggregation operator, we get the collection of SCFN foryˆi alternative given in Table 4. Table 5 contains the decision-makers’ SCF ordered weighted geometric (SCFOWG) aggregation information. And the rowwise aggregated (SCFDWG) decision-makers are given in Table 6. In Table 7, the rowwise aggregation of (SCFOWG) operators is given.Table 4
Spherical cubic fuzzy weighted geometric aggregation information of decision-makers.
Aˆ1Aˆ2Aˆ3Aˆ4yˆ1.10,.22,.33,.16,.36,.28,.17,.27,.42.10,.27,.19,.12,.40,.17,.13,.34,.33.19,.50,.47,.13,.36,.18,.10,.30,.17.10,.25,.46,.18,.47,.29,.24,.50,.41yˆ2.13,.34,.20,.10,.30,.34,.13,.58,.54.15,.43,.28,.10,.24,.17,.25,.60,.20.10,.24,.13,.13,.34,.24,.22,.44,.30.17,.47,.23,.13,.30,.45,.15,.41,.28yˆ3.13,.19,.20,.12,.41,.24,.20,.43,.41.24,.40,.33,.40,.54,.19,.17,.27,.35.18,.33,.21,.10,.44,.28,.29,.41,.52.10,.28,.20,.13,.36,.31,.20,.38,.41yˆ4.13,.26,.32,.25,.42,.36,.17,.42,.20.20,.26,.40,.19,.26,.25,.20,.53,.25.16,.46,.32,.18,.22,.42,.15,.43,.33⌈.10,.26,.16,⌉.10,.16,.30,.23,.52,.42Table 5
SCF ordered weighted geometric aggregation information of decision-makers.
Aˆ1Aˆ2Aˆ3Aˆ4yˆ1.19,.45,.62,.26,.58,.46,.20,.44,.30.17,.40,.42,.28,.49,.31,.23,.40,.39.15,.36,.33,.13,.44,.36,.20,.38,.51.13,.40,.43,.16,.50,.26,.32,.60,.54yˆ2.25,.51,.33,.14,.34,.50,.15,.46,.27.16,.49,.38,.16,.50,.50,.29,.56,.39.10,.40,.26,.17,.36,.40,.20,.58,.48.23,.47,.24,.13,.32,.21,.34,.64,.51yˆ3.30,.50,.46,.50,.64,.30,.21,.36,.51.26,.50,.26,.20,.51,.40,.21,.39,.62.13,.29,.27,.24,.50,.43,.26,.47,.44.10,.28,.20,.13,.36,.31,.20,.38,.41yˆ4.13,.26,.32,.25,.42,.36,.17,.42,.20.20,.26,.40,.19,.26,.25,.20,.53,.25.10,.32,.30,.29,.48,.60,.34,.70,.42.24,.36,.19,.10,.24,.42,.35,.61,.52Table 6
Rowwise aggregated (SCFDWG) decision-makers.
yˆ1.16,.11,.25,.20,.29,.31,.19,.41,.35yˆ2.21,.38,.31,.17,.20,.33,.19,.49,.31yˆ3.18,.19,.26,.24,.21,.28,.22,.38,.43yˆ4.19,.13,.25,.19,.18,.18,.20,.49,.35Table 7
Rowwise aggregated (SCFDOWG) decision-makers.
yˆ1.16,.11,.25,.20,.29,.31,.19,.41,.35yˆ2.21,.38,.31,.17,.20,.33,.19,.49,.31yˆ3.18,.19,.26,.24,.21,.28,.22,.38,.43yˆ4.19,.13,.25,.19,.18,.18,.20,.49,.35Step 3.
Using equation (22) to determine the scores of Q˜C˜ii=1,2,3,…,n, the ranking criteria of alternatives using the spherical cubic geometric aggregation operators are given in Table 8.Table 8
Ranking criteria of alternatives.
OperatorQ˜C˜iAˆ1Q˜C˜iAˆ2Q˜C˜iAˆ3Q˜C˜iAˆ4RankingSCFWG0.060.020.040.01Aˆ1>Aˆ3>Aˆ2>Aˆ4SCFOWG0.110.070.080.03Aˆ1>Aˆ3>Aˆ2>Aˆ4SCFHWG0.050.020.040.01Aˆ1>Aˆ3>A2>Aˆ4Step 4.
The scores are arranged in a descending order. Choose the highest alternative. Figure1 represents the graphical representation of alternative by using the spherical cubic geometric aggregation operators. By using the proposed aggregation operator, the ranking of the four possible alternatives achieved is more accurate in comparison with Table 9. The best choice for these operators is Aˆ1. These approaches are ideal for addressing circumstances in which the inputs, viewpoints, and the interaction of experts and criteria may be considered which are more likely to handle such problems. Figure 2 shows the graphical representation of alternative by using the spherical geometric aggregation operators.Figure 1
Ranking of alternatives using SCFSs.Table 9
Comparison analysis using spherical fuzzy sets.
OperatorQ˜C˜iAˆ1Q˜C˜iAˆ2Q˜C˜iAˆ3Q˜C˜iAˆ4RankingSFWG0.080.070.020.05Aˆ1>Aˆ2>Aˆ4>Aˆ2SFOWG0.020.030.050.01Aˆ3>Aˆ2>Aˆ1>Aˆ4SFHWG0.060.090.030.02Aˆ2>A1>A3>Aˆ2Figure 2
Ranking of alternatives using spherical fuzzy sets.


## 6. Conclusion

In this paper, we have discussed the spherical cubic fuzzy set (SCFS) which is the generalization of the interval-valued SFS (IVSFS). Some operational laws of SCFS were presented. For the comparison of SCFNs, we have established accuracy and score functions. We also described SCF distances between SCFNs. Further, we proposed various aggregation operators in the SCF environment, SCFWG, SCFOWG, and SCFHWG operators.Some interesting properties such as idempotency, boundary, and monotonicity were also discussed. Furthermore, a relationship was formed among the proposed operators. Additionally, we suggested a MADM to demonstrate the prominence and the strength of the proposed operators. Moreover, by applying the developed aggregation operators, we explained the problems of decision-making. A numerical illustration is presented to indicate an alternative way of addressing the decision-making process more effectively by using the proposed operators. Finally, we compared the practicality, efficiency, and validity of the new approach with existing operators.We would combine other SCFS approaches such as Einstein product and introduce the concept of SCF Einstein weighted averaging (SCFEWA) operator, SCF Einstein ordered weighted averaging (SCFEOWA) operator, SCF Einstein weighted geometric (SCFEWG) operator, SCF Einstein ordered weighted geometric (SCFEOWG) operator, and some more generalized operators in MADM processes.



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