# The Approach of Induced Generalized Neutrosophic Cubic Shapley Choquet Integral Aggregation Operators via the CODAS Method to Solve Distance-Based Multicriteria Decision-Making Problems

Authors: Majid Khan; Muhammad Gulistan; Mohammed M. Al-Shamiri
Journal: Journal of Mathematics
 (2022)

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## Abstract

This study aims to define a conjecture that can handle complex frames of work more efficiently that occurs in daily life problems. In decision-making theory inter-relation of criteria, weights and choice decision-making method subject to the given circumstances which are an important component for appropriate decisions. For this, we define neutrosophic cubic Shapley–Choquet integral (NCSCI) measure; combinative distance-based assessment selection (CODAS) is accomplished over NCSCI and is implemented over a numerical example of a company foreign investment model as an application in decision-making (DM) theory. The neutrosophic cubic set (NCS) is a hybrid of the neutrosophic set (NS) and interval neutrosophic set (INS), which provides a better plate form to handle inconsistent and vague data more conveniently. The novel CODAS method is based on Shapley–Choquet integral and Minkowski distance which contain more information measures than usual criteria weights and distances. The weights of criteria are measured by Shapley–Choquet integral and distance is evaluated by Minkowski distance. The Choquet integral considers the interaction among the criteria, and Shapley considers the overall weight criteria. Motivated by these characteristics NCSCI, we defined two aggregation operators’ induced-generalized neutrosophic cubic Shapley–Choquet integral arithmetic (IGNCSCIA) and operators’ induced-generalized neutrosophic cubic Shapley–Choquet integral geometric (IGNCSCIG) operators. To find the distance between two NC values, Minkowski distance is defined to evaluate neutrosophic cubic combinative distance-based assessment selection (NCCODAS). To examine the feasibility of the proposed method, an example of company investment in a foreign country is considered. To check, the validity of the method, the comparative analysis of the proposed method with other methods is conducted.


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## Body

## 1. Introduction

Increasing uncertainty and complexity in decision-making (DM) theory, the representation of data is no longer the real number. The researcher developed different theories that can handle such data appropriately. Among these, Zadeh initiated the fuzzy set (FS) [1] to deal with the uncertainty. A fuzzy set consists of a crisp value from [0,1] referred to as a membership degree. FS was further extended into an interval-valued fuzzy set (IVFS) [2, 3], in which the membership degree is a subinterval of [0,1]. Atnassove instigated a nonmembership function to FS and named it as an intuitionistic fuzzy set (IFS) [4]. Both membership and nonmembership are dependent. IFS was generalized into an interval-valued intuitionistic fuzzy set (IVIFS) [5]. Jun combined FS and IVFS to form a cubic set (CS) [6]. These generalizations of FS handle vague and inconsistent data in the form of membership, and nonmembership degrees can be assigned crisp and interval values. In a complex frame of work, the situation often arises in which one is unable to completely specify the data by assigning an argument membership grade and nonmembership grades only. This limitation can be overcome by Smrandache neutrosophic set (NS) [7]. An NS consists of three independent components, truth, indeterminancy, and falsity grades. The NS provides a wide range of choosing so that the data can easily be associated according to the complex frame of the environment. NS was further extended into the interval neutrosophic set (INS) [8]. INS provides the choice of choosing in the form of interval values. The problem arises that whether these components can be assigned with both the interval value and the crisp value at the same time. This problem can be tackled by neutrosophic cubic set (NCS) [9] and the hybrid of NS and INS. NCS provides the plate form to choose the value in the form of a crisp value along with the interval value at the same time. This makes NCS a useful tool to represent the fuzziness of acceptance, neutral, and rejection in the complex frame of the environment more conveniently. These characteristics attract the researcher to apply in the field of DM theory. Majid et al. defined novel operational laws on NCS [10].The aggregation operator is an important component of DM theory. MCDM problem involves conflicting criteria and aggregation operators are used to aggregate the conflicting criteria to conclude problems [11–16]. Most of the aggregation operators deal with the criteria independently; interaction among the criteria and overall criteria is not considered by such aggregation operators. These limitations can be overcome by Coquet integral [17, 18] that considers the interaction among two adjacent criteria. To consider the overall interaction of criteria, Sugano defined Shapley fuzzy measure [19–23] that considers the overall interaction and importance of criteria. It can also be used to establish the weights and distribution of criteria [24]. Shapley measure is more flexible than probability by its additive property [25]. Combining the idea of Shapley measure and Choquet integral will tackle the overall and partial information of input argument [26, 27].


### 1.1. Motivation

The motivation of this research is to generalize Shapley measure and Choquet integral operator in the NCS plate form. That is aggregation operators that tackle the interaction among the criteria and overall interaction of criteria become a handy tool to handle complex frames of the environment. The Shapley measure will handle the overall interaction of criteria and weightage of criteria. Choquet integral will look after the interaction of amongst criteria, and NCS will provide a platform for data to handle complex frames of the environment.


### 1.2. Contribution

This study contributes the following work:(i)
The induced generalized Shapley–Choquet integral is defined(ii)
The IGNCSCIA operator is defined(iii)
The IGNCSCIG operator is defined(iv)
Some significant properties are investigated(v)
Minkowski distance is defined(vi)
NCCODAS method is defined to handle distance-based DM problemsTo check the validity of the proposed method, the comparative analysis is investigated with some existing methods.


### 1.3. Organization

The organization process of research is shown in Figure1.Figure 1
Research organization process.The research paper has been divided into four sections. Section1 comprises of introduction. Section 2 comprises of preliminaries, definition, and results. The section will help to work out the proposed research. Section 3 consists of IGNCSCI, IGNCSCIA, and IGNCSCIG aggregation operators along with some important properties and neutrosophic cubic Minkowski distance. Section 4 consists of NCCODAS method, numerical example as an application, and comparative analysis.


## 1.1. Motivation

The motivation of this research is to generalize Shapley measure and Choquet integral operator in the NCS plate form. That is aggregation operators that tackle the interaction among the criteria and overall interaction of criteria become a handy tool to handle complex frames of the environment. The Shapley measure will handle the overall interaction of criteria and weightage of criteria. Choquet integral will look after the interaction of amongst criteria, and NCS will provide a platform for data to handle complex frames of the environment.


## 1.2. Contribution

This study contributes the following work:(i)
The induced generalized Shapley–Choquet integral is defined(ii)
The IGNCSCIA operator is defined(iii)
The IGNCSCIG operator is defined(iv)
Some significant properties are investigated(v)
Minkowski distance is defined(vi)
NCCODAS method is defined to handle distance-based DM problemsTo check the validity of the proposed method, the comparative analysis is investigated with some existing methods.


## 1.3. Organization

The organization process of research is shown in Figure1.Figure 1
Research organization process.The research paper has been divided into four sections. Section1 comprises of introduction. Section 2 comprises of preliminaries, definition, and results. The section will help to work out the proposed research. Section 3 consists of IGNCSCI, IGNCSCIA, and IGNCSCIG aggregation operators along with some important properties and neutrosophic cubic Minkowski distance. Section 4 consists of NCCODAS method, numerical example as an application, and comparative analysis.


## 2. Preliminaries

This section consists of two section developments in NCS to neutrosophic cubic set and fuzzy preferences.


### 2.1. Development of NCS

Definition 1 (see [1]).
A mappingψ: U⟶0,1 is called a fuzzy set, and ψu is called a membership function, simply denoted by ψ.Definition 2 (see [2]).
A mappingψ˜:U⟶D0,1, where D0,1 is the interval value of 0,1, called the interval-valued fuzzy set(IVF). For all u∈U, ψ˜u=ψLu,ψUu|ψLu,ψUu∈0,1andψLu≤ψUu is membership degree of u in ψ˜. This is simply denoted by ψ˜=ψL,ψU.Definition 3 (see [6]).
A structureC=u,ψ˜u,ψu|u∈U is a cubic set in U in which ψ˜u is IVFS in U, that is, ψ˜=ψL,ψU and ψ˜ is a fuzzy set in U. This can be simply denoted by C=ψ˜,ψ.Definition 4 (see [7]).
The neutrosophic set is defined as(1)N=TNγ¨,INγ¨,FNγ¨|γ¨∈Γ,where TNγ¨,INγ¨,andFNγ¨ are truth, indeterminacy, and falsity fuzzy functions.Definition 5 (see [8]).
An INS is an extension of NS defined by(2)N˜=T˜Nγ¨,I˜Nγ¨,F˜Nγ¨|γ¨∈Γ,where T˜Nγ¨,I˜Nγ¨,andF˜Nγ¨ are interval-valued fuzzy truth, indeterminacy, and falsity function.Definition 6 (see [9]).
A NCS is hybrid of NS and INS and defined as(3)N=T˜Nγ¨,I˜Nγ¨,F˜Nγ¨,TNγ¨,INγ¨,FNγ¨/γ¨∈Γ,where T˜Nγ¨=TNLγ¨,TNUγ¨,I˜Nγ¨=INLγ¨,INUγ¨,F˜Nγ¨=FNLγ¨,FNUγ¨ is an INS, and TNγ¨,INγ¨,FNγ¨ is a NS, where 0˜≤T˜N+I˜N+F˜N≤3˜,0≤TN+IN+FN≤3.
For the sake of convenience, the NCS are written asN=TNL,TNU,INL,INU,FNL,FNU,TN,IN,FN.Definition 7 (see [10]).
The sum of two NCS,N=TNL,TNU,INL,INU,FNL,FNU,TN,IN,FN and M=TML,TMU,IML,IMU,FML,FMU,TM,IM,FM, is defined as(4)N⊕M=TNL+TML−TNLTML,TNU+TMU−TNUTMU,INL+IML−INLIML,INU+IMU−INUIMU,FNLFML,FNUFMU,TNTM,INIM,FN+FM−FNFM.Definition 8 (see [10]).
The product of two NCS,N=TNL,TNU,INL,INU,FNL,FNU,TN,IN,FN and M=TML,TMU,IML,IMU,FML,FMU,TM,IM,FM, is defined as(5)N⊗M=TNLTML,TNUTMU,INLIML,INUIMU,FNL+FML−FNLFML,FNU+FMU−FNUFMU,TN+TM−TNTM,IN+IM−INIM,FNFM,Definition 9 (see [10]).
The scalar multiplication on a NCSN=TNL,TNU,INL,INU,FNL,FNU,TN,IN,FN and a scalar η is defined:(6)ηN=1−1−TNLη,1−1−TNUη,1−1−INLη,1−1−INUη,FNLη,FNUη,TNη,INη,1−1−FNη.


### 2.2. Developments in Fuzzy Measure

In decision-making (DM) process, the valuel⌢ is weighted by wi weight and then aggregated using weighted averaging and weighted geometric aggregation operators, where wi∈0,1 such that ∑i=1n¨wi=1. In real-life problems, there exist interactive phenomena amongst the elements. The overall significance of an element not only specified by itself, but by all the other elements in process.Sangeno [20] established the notion of fuzzy measure, which not only determines weight of an element and each combination of elements as well, and sum of weights need not to be equal to one. Murofushi and Saneno [21] proposed Choquet integral as an extension of Lebesgue integral. It is a significant aggregation operator for MCDM by considering significance of element by fuzzy measure.Definition 10 (see [23]).
LetT⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ be a set. A fuzzy measure θ on T⌢ is defined as a function θ:T⌢⟶0,1 fulfilling the following properties:(i)
θφ=0andθT⌢=1(ii)
ForU,V∈PT⌢,U⊆V⟹θU≤θV, where PT⌢ is power set of T⌢Definition 11 (see [23]).
In MCDM, forU,V∈PT⌢ such that U∩V=φ, three types of interactive relation are possible, that is,(i)
Additive measure: if U and V are independent (no interaction), then(7)θU∪V=θU+θV.(ii)
Super additive measure: if U and V are positive synergetic interaction, then(8)θU∪V>θU+θV,(iii)
Subadditive measure: if U and V are negative synergetic interaction, then(9)θU∪V<θU+θV.Definition 12 (see [23]).
Letk be a function on T⌢ and θ be a fuzzy measure on T⌢. Then, discrete Choquet with respect to θ is defined by(10)Cθkl⌢ρi=∑i=1n¨kl⌢ρiθT⌢ρi−θT⌢ρi+1,for ρi as ρ1,ρ2,…,ρi,…,ρn¨ present the permutations of 1,2,…,i,…,n¨ such that kl⌢1≤kl⌢2≤kl⌢3≤⋯≤kl⌢n¨ and T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ with T⌢ρn¨+1=φ. From definition, it is observed that the Choquet integral handles the interaction between two consecutive values; it is unable to handle the overall all interaction. This limitation is overcome by the Shapley index [18]:(11)ϖSShθ,N=∑T⊆N\Sn¨−u−v!t!n¨−u+1!t!θU∪V−θV,∀U⊆N,where θ is a fuzzy function of fuzzy measure λ, on N, and the cardinality of U,N, and V is, respectively, u,n¨, and v.
Meng [25] generalized the Shapley index to generalized Shapley index as λ− fuzzy measure θλ on N by(12)ϖSShθλ,N=∑T⊆N\Sn¨−u−v!v!n¨−u+1!v!θλU∪V−θλV,∀U⊆N,where θλis fuzzy measure expressed as(13)θλA=1λ∏k∈T⌢λθλi+1−1ifλ≠0∑k∈T⌢λθλkifλ=0,where ∏c∈N1+θλck=1+θλ is used to measure λ.Thus, if in S=k, then(14)ϖkShθλ,N=∑T⊆N\St!n¨−1−t!t!n¨!θλk∏j∈T1+λθλj,∀k⊆N.Definition 13.
Based on these definitions, Meng [25] defined arithmetic λ−Shapley–Choquet integral operator as(15)CϖSShθλ,Nkl⌢ρi=∑i=1n¨kl⌢ρiΘϒ⌢ρT⌢i,whereϒ⌢ρT⌢i=ϖρT⌢iShθλ,N−ϖρT⌢i+1Shθλ,N,for ρi as ρ1,ρ2,…,ρi,…,ρn¨ present the permutations of 1,2,…,i,…,n¨ such that kl⌢1≤kl⌢2≤kl⌢3≤⋯≤kl⌢n¨ and T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ with T⌢ρn¨+1=φ.Definition 14 (see [25]).
Geometricλ−Shapley–Choquet integral operator is as(16)CϖSShηλ,Nkl⌢ρi=∏i=1n¨kl⌢ρiΘϒ⌢ρT⌢iwhereΘϒ⌢ρT⌢i=ϖρT⌢iShθλ,N−ϖρT⌢i+1Shθλ,Nfor ρi as ρ1,ρ2,…,ρi,…,ρn¨ present the permutations of 1,2,…,i,…,n¨ such that kl⌢1≤kl⌢2≤kl⌢3≤⋯≤kl⌢n¨ and T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ with T⌢ρn¨+1=φ.


### 2.3. Induced Generalized Shapley–Choquet Integral

Aggregation operator is an important component of DM theory. The suitable aggregation operator may reduce the challenges that are present in vague and inconsistent data. Different operators are defined to meet these challenges. IGNCSCIA and IGNCSCIG aggregation operators will be defined to meet the challenges of criterion weights and interaction.Definition 15.
Letl⌢i=TLl⌢i,TUl⌢i,ILl⌢i,IUl⌢i,FLl⌢i,FUl⌢i,Tl⌢i,Il⌢i,Fl⌢i, where i=1,2,…,n¨ be a collection of NC values, and θ be a fuzzy measure on T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n such that θl⌢i=u⌢i; then, the IGNCSCIA operator is defined as(17)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n,l⌢n¨=⊕i=1n¨kl⌢iqΘϒ⌢ρT⌢i1/q,where q∈0,∞ and ρi as ρ1,ρ2,…,ρi,…,ρn¨ present the permutations of 1,2,…,i,…,n¨ such that kl⌢1≤kl⌢2≤kl⌢3≤⋯≤kl⌢n¨ and T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ with T⌢ρn¨+1=φ.Theorem 1.
Letl⌢i=TLl⌢i,TUl⌢i,ILl⌢i,IUl⌢i,FLl⌢i,FUl⌢i,Tl⌢i,Il⌢i,Fl⌢i, where 1,2,…,i,…,n¨ be a collection of NC values, and θ be a fuzzy measure on T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ such that θl⌢i=u⌢i; then, the IGNCSCIA operator is an NC value:(18)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n,l⌢n¨=⊕i=1n¨kl⌢iqΘϒ⌢ρT⌢i1/q=1−∏i=1n¨1−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=1n¨1−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−∏i=1n¨1−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=1n¨1−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=1n¨1−1−Fl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−1−∏i=1n¨1−1−Fl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=1n¨1−1−Tl⌢ρiqΘϒ⌢ρT⌢i1/q,1−1−∏i=1n¨1−1−Il⌢ρiqΘϒ⌢ρT⌢i1/q,1−∏i=1n¨1−Fl⌢ρiqΘϒ⌢ρT⌢i1/q,where q∈0,∞ and ρ1,ρ2,…,ρi,…,ρn¨ present the permutations of 1,2,…,i,…,n¨ such that kl⌢1≤kl⌢2≤kl⌢3≤⋯≤kl⌢n¨ and T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ with T⌢ρn¨+1=φ.Proof.
Forn¨=1, (17) reduces to the NC value by operational laws and equations (15). For n¨=2,(19)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2=⊕i=12kl⌢ρiqΘϒ⌢ρT⌢i1/q=1−∏i=121−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=121−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−∏i=121−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=121−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=121−1−Fl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−1−∏i=121−1−Fl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=121−1−Tl⌢ρiqΘϒ⌢ρT⌢i1/q,1−1−∏i=121−1−Il⌢ρiqΘϒ⌢ρT⌢i1/q,1−∏i=121−Fl⌢ρiqΘϒ⌢ρT⌢i1/q,(20)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2=⊕i=12kl⌢ρiqΘϒ⌢ρT⌢i1/q=1−1−Tl⌢ρ1LqΘϒ⌢ρT⌢11−Tl⌢ρ2LqΘϒ⌢ρT⌢21/q,1−1−Tl⌢ρ1UqΘϒ⌢ρT⌢11−Tl⌢ρ2UqΘϒ⌢ρT⌢21/q,1−1−Il⌢ρ1LqΘϒ⌢ρT⌢11−Il⌢ρ2LqΘϒ⌢ρT⌢21/q,1−1−Il⌢ρ1UqΘϒ⌢ρT⌢11−Il⌢ρ2UqΘϒ⌢ρT⌢21/q,1−1−1−1−Fl⌢ρ1LqΘϒ⌢ρT⌢11−1−Fl⌢ρ2LqΘϒ⌢ρT⌢21/q,1−1−1−1−Fl⌢ρ1UqΘϒ⌢ρT⌢11−1−Fl⌢ρ2UqΘϒ⌢ρT⌢21/q,1−1−1−1−Tl⌢ρ1qΘϒ⌢ρT⌢11−1−Tl⌢ρ2qΘϒ⌢ρT⌢21/q,1−1−1−1−Il⌢ρ1qΘϒ⌢ρT⌢11−1−Il⌢ρ2qΘϒ⌢ρT⌢21/q,1−1−Fl⌢ρ1qΘϒ⌢ρT⌢11−Fl⌢ρ2qΘϒ⌢ρT⌢21/q.
By operational laws equation, letn¨=j result holds good:(21)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,...,u⌢j,l⌢j=⊕i=1jkl⌢ρiqΘϒ⌢ρT⌢i1/q=1−∏i=1j1−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=1j1−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−∏i=1j1−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=1j1−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j1−1−Fl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j1−1−Fl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j1−1−Tl⌢ρiqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j1−1−Il⌢ρiqΘϒ⌢ρT⌢i1/q,1−∏i=1j1−Fl⌢ρiqΘϒ⌢ρT⌢i1/q.
Forn¨=j+1,(22)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢j+1,l⌢j+1=⊕i=1j+1kl⌢ρiqΘϒ⌢ρT⌢i1/q=1−∏i=1j+11−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=1j+11−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−∏i=1j+11−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=1j+11−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j+11−1−Fl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j+11−1−Fl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j+11−1−Tl⌢ρiqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j+11−1−Il⌢ρiqΘϒ⌢ρT⌢i1/q,1−∏i=1j+11−Fl⌢ρiqΘϒ⌢ρT⌢i1/q,which in the form of(23)IGNCSCIAηλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢j,l⌢j⊕IGNCSCIAηu⌢j+1,l⌢j+1=⊕i=1jkl⌢iqΘϒ⌢ρT⌢i1/q⊕kl⌢j+1qΘϒ⌢ρT⌢j+11/q=1−∏i=1j1−Tl⌢ρiLqΘϒ⌢ρT⌢i1−1−Tl⌢ρj+1LqΘϒ⌢ρT⌢j+11/q,1−∏i=1j1−Tl⌢ρiUqΘϒ⌢ρT⌢i1−1−Tl⌢ρj+1UqΘϒ⌢ρT⌢j+11/q,1−∏i=1j1−Il⌢ρiLqΘϒ⌢ρT⌢i1−1−Il⌢ρj+1LqΘϒ⌢ρT⌢j+11/q,1−∏i=1j1−Il⌢ρiUqΘϒ⌢ρT⌢i1−1−Il⌢ρj+1UqΘϒ⌢ρT⌢j+11/q,1−1−∏i=1j1−1−Fl⌢ρiLqΘϒ⌢ρT⌢i1−1−Fl⌢ρj+1LqΘϒ⌢ρT⌢j+11/q,1−1−∏i=1j1−1−Fl⌢ρiUqΘϒ⌢ρT⌢i1−1−Fl⌢ρj+1UqΘϒ⌢ρT⌢j+11/q,1−1−∏i=1j1−1−Tl⌢ρiqΘϒ⌢ρT⌢i1−1−Tl⌢ρj+1qΘϒ⌢ρT⌢j+11/q,1−1−∏i=1j1−1−Tl⌢ρiqΘϒ⌢ρT⌢i1−1−Tl⌢ρj+1qΘϒ⌢ρT⌢j+11/q1−∏i=1j1−Fl⌢ρiqΘϒ⌢ρT⌢i1−Fl⌢ρj+1qΘϒ⌢ρT⌢j+11/q=1−∏i=1j+11−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=1j+11−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−∏i=1j+11−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=1j+11−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j+11−1−Fl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j+11−1−Fl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j+11−1−Tl⌢ρiqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j+11−1−Il⌢ρiqΘϒ⌢ρT⌢i1/q,1−∏i=1j+11−Fl⌢ρiqΘϒ⌢ρT⌢i1/q.is a NC value by assumption hypothesis n¨=k and n¨=2. Hence, n¨=j+1 is NC, which completes the proof.Definition 16.
Letl⌢i=TLl⌢i,TUl⌢i,ILl⌢i,IUl⌢i,FLl⌢i,FUl⌢i,Tl⌢i,Il⌢i,Fl⌢i, where 1,2,…,i,…,n¨ be a collection of NC values, and θ be a fuzzy measure on T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ such that θl⌢i=u⌢i; then, the IGNCSCIG operator is defined as(24)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨=1q⊗i=1n¨qkl⌢ρiΘϒ⌢ρT⌢ρi,where q∈0,∞ and ρ1,ρ2,…,ρi,…,ρn¨ present the permutations of 1,2,…,i,…,n¨ such that kl⌢1≤kl⌢2≤kl⌢3≤…≤kl⌢n¨ and T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ with T⌢ρn¨+1=φ.Theorem 2.
Letl⌢i=TLl⌢i,TUl⌢i,ILl⌢i,IUl⌢i,FLl⌢i,FUl⌢i,Tl⌢i,Il⌢i,Fl⌢i, where i=1,2,…,n¨ be a collection of NC values, and θ be a fuzzy measure on T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ such that θl⌢i=u⌢i; then, the aggregated result obtained by the IGNCGSCIG operator is an NC value:(25)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨=1q⊗i=1n¨qkl⌢ρiΘϒ⌢ρT⌢ρi=1−1−∏i=1n¨1−1−Tl⌢ρiLqΘϒ⌢ρT⌢ρi1/q,1−1−∏i=1n¨1−1−Tl⌢ρiUqΘϒ⌢ρT⌢ρi1/q,1−1−∏i=1n¨1−1−Il⌢ρiLqΘϒ⌢ρT⌢ρi1/q,1−1−∏i=1n¨1−1−Il⌢ρiUqΘϒ⌢ρT⌢ρi1/q,1−∏i=1n¨1−Fl⌢ρiLqΘϒ⌢ρT⌢ρi1/q,1−∏i=1n¨1−Fl⌢ρiUqΘϒ⌢ρT⌢ρi1/q,1−∏i=1n¨1−Tl⌢ρiqΘϒ⌢ρT⌢ρi1/q,1−∏i=1n¨1−Il⌢ρiqΘϒ⌢ρT⌢ρi1/q,1−1−∏i=1n¨1−1−Fl⌢ρiqΘϒ⌢ρT⌢ρi1/q,where q∈0,∞ and ρ1,ρ2,…,ρi,…,ρn¨ present the permutations of 1,2,…,i,…,n¨ such that kl⌢1≤kl⌢2≤kl⌢3≤…≤kl⌢n¨ and T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ with T⌢ρn¨+1=φ.Proof.
The proof is analogy of Theorem1.


### 2.4. Properties of IGNCSCIA and IGNCSCIG

The IGNCSCIA and IGNCSCIG satisfy the following properties.Proposition 1 (idempotency).
Letl⌢i=TLl⌢i,TUl⌢i,ILl⌢i,IUl⌢i,FLl⌢i,FUl⌢i,Tl⌢i,Il⌢i,Fl⌢i, where i=1,2,…,n¨ be a collection of NC values and θ be a fuzzy measure on T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ such that θl⌢i=u⌢i:(26)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n,l⌢n¨=⊕i=1n¨kl⌢iqΘϒ⌢T⌢ρi1/q,(27)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨=1q⊗i=1n¨qkl⌢ρiΘϒ⌢T⌢ρi,where q∈0,∞ and ρ1,ρ2,…,ρi,…,ρn¨ present the permutations of 1,2,…,i,…,n¨ such that kl⌢1≤kl⌢2≤kl⌢3≤…≤kl⌢n¨ and T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ with T⌢ρn¨+1=φ. For l⌢i=l⌢,(28)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨=l⌢,IGNCSCIGθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨=l⌢.Proposition 2 (monotonicity).
Letl⌢i=TLl⌢i,TUl⌢i,ILl⌢i,IUl⌢i,FLl⌢i,FUl⌢i,Tl⌢i,Il⌢i,Fl⌢i, where i=1,2,…,n¨ be a collection of NC values and θ be a fuzzy measure on T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ such that θl⌢i=u⌢i, B⌢=b⌢1,b⌢2,b⌢3,…,b⌢n¨, and θb⌢i=v⌢i with Tl⌢iL≤Tb⌢iL,Tl⌢iU≤Tb⌢iU,Il⌢iL≤Ib⌢iL,Il⌢iU≤Ib⌢iU,Tl⌢iL≥Tb⌢iL,Tl⌢iU≥Tb⌢iU,Tl⌢i≥Tb⌢i,Il⌢i≥Ib⌢i,Fl⌢i≤Fb⌢i; then,(29)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨≤IGNCSCIAθλv⌢1,b⌢1,v⌢2,b⌢2,…,v⌢n¨,b⌢n¨,IGNCSCIGθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨≤IGNCSCIGθλv⌢1,b⌢1,v⌢2,b⌢2,…,v⌢n¨,b⌢n¨.Proof.
To prove this proposition, the following inequality is proved:(30)x,y∈0,1such thatxk≤yk⟹1−xk≥1−yk⟹1−1−xk≤1−1−yki,forxk≥yk⟹1−xk≤1−yk⟹1−1−xk≤1−1−ykii.
Forx=y=0 and x=y=1, the result is trivial.
SinceΘϒ⌢T⌢ρi=ϖT⌢ρiShηλ,N−ϖT⌢ρi+1Shηλ,N=ϖB⌢ρiShηλ,N−ϖB⌢ρi+1Shηλ,N=Θϒ⌢B⌢ρi, putting x=Tl⌢iL,y=Tb⌢iL,andk=q and using equation i,(31)1−∏i=1n¨1−Tl⌢ρiLqΘϒ⌢T⌢ρi1/q≤1−∏i=1n¨1−Tb⌢ρiLqΘϒ⌢B⌢ρi1/q,1−∏i=1n¨1−Tl⌢ρiUqΘϒ⌢T⌢ρi1/q≤1−∏i=1n¨1−Tb⌢ρiUqΘϒ⌢B⌢ρi1/q1−∏i=1n¨1−Il⌢ρiLqΘϒ⌢T⌢ρi1/q≤1−∏i=1n¨1−Ib⌢ρiLqΘϒ⌢B⌢ρi1/q,1−∏i=1n¨1−Il⌢ρiUqΘϒ⌢T⌢ρi1/q≤1−∏i=1n¨1−Ib⌢ρiUqΘϒ⌢B⌢ρi1/q1−∏i=1n¨1−Fl⌢ρiqΘϒ⌢T⌢ρi1/q≤1−∏i=1n¨1−Fb⌢ρiqΘϒ⌢B⌢ρi1/q,
Puttingx=Fl⌢ρiL,y=Fb⌢ρiL,andk=q and using equation (ii),(32)1−1−∏i=1n¨1−1−Fl⌢ρiLqΘϒ⌢T⌢ρi1/q≥1−1−∏i=1n¨1−1−Fb⌢ρiLqΘϒ⌢B⌢ρi1/q,1−1−∏i=1n¨1−1−Fl⌢ρiUqΘϒ⌢T⌢ρi1/q≥1−1−∏i=1n¨1−1−Fb⌢ρiUqΘϒ⌢B⌢ρi1/q,1−1−∏i=1n¨1−1−Tl⌢ρiqΘϒ⌢T⌢ρi1/q≥1−1−∏i=1n¨1−1−Tb⌢ρiqΘϒ⌢B⌢ρi1/q,1−1−∏i=1n¨1−1−Il⌢ρiqΘϒ⌢T⌢ρi1/q≥1−1−∏i=1n¨1−1−Ib⌢ρiqΘϒ⌢B⌢ρi1/q,IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨≤IGNCSCIAθλv⌢1,b⌢1,v⌢2,b⌢2,…,v⌢n¨,b⌢n¨,IGNCSCIGθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨≤IGNCSCIGθλv⌢1,b⌢1,v⌢2,b⌢2,…,v⌢n¨,b⌢n¨.Proposition 3 (boundedness).
Letl⌢i=TLl⌢i,TUl⌢i,ILl⌢i,IUl⌢i,FLl⌢i,FUl⌢i,Tl⌢i,Il⌢i,Fl⌢i, where i=1,2,…,n¨ be a collection of NC values and θ be a fuzzy measure on T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ such that θl⌢i=u⌢i, such that(33)l⌢−=minTLl⌢i,minTUl⌢i,minILl⌢i,minIUl⌢i,1−maxFLl⌢i,1−maxFUl⌢i,minTl⌢i,minIl⌢i,1−maxFl⌢i,l⌢i+=maxTLl⌢i,maxTUl⌢i,maxILl⌢i,maxIUl⌢i,1−minFLl⌢i,1−minFUl⌢i,maxTl⌢i,maxIl⌢i,1−minFl⌢i.
Then,(34)l⌢−≤IGNCSCIAθλl⌢1,l⌢2,l⌢3,…,l⌢n¨≤l⌢+,l⌢−≤IGNCSCIGθλl⌢1,l⌢2,l⌢3,…,l⌢n¨≤l⌢+.
The proof is followed by idempotency.Remark 1.
When different values are assigned top, IGNCSCA and IGNCSG will deduce into different neutrosophic cubic aggregation operators as follows.Proof.
Whenq=1, IGNCSCIA deduce into induced neutrosophic cubic Shapley–Choquet integral average (INCSCIA) operator, and IGNCSCIG deduced into induced neutrosophic cubic Shapley–Choquet integral geometric (INCSCIG) operator.
Whenu⌢i=l⌢i, IGNCSCIA deduce into generalized neutrosophic cubic Shapley–Choquet integral average (GNCSCIA) operator, and IGNCSCIG deduce into generalized neutrosophic cubic Shapley–Choquet integral geometric (GNCSCIG) operator.
Whenu⌢i=l⌢i and q=1, IGNCSCIA deduce into neutrosophic cubic Shapley–Choquet integral average (NCSCIA) operator, and IGNCSCIG deduce into neutrosophic cubic Shapley–Choquet integral geometric (NCSCG) operator.


## 2.1. Development of NCS

Definition 1 (see [1]).
A mappingψ: U⟶0,1 is called a fuzzy set, and ψu is called a membership function, simply denoted by ψ.Definition 2 (see [2]).
A mappingψ˜:U⟶D0,1, where D0,1 is the interval value of 0,1, called the interval-valued fuzzy set(IVF). For all u∈U, ψ˜u=ψLu,ψUu|ψLu,ψUu∈0,1andψLu≤ψUu is membership degree of u in ψ˜. This is simply denoted by ψ˜=ψL,ψU.Definition 3 (see [6]).
A structureC=u,ψ˜u,ψu|u∈U is a cubic set in U in which ψ˜u is IVFS in U, that is, ψ˜=ψL,ψU and ψ˜ is a fuzzy set in U. This can be simply denoted by C=ψ˜,ψ.Definition 4 (see [7]).
The neutrosophic set is defined as(1)N=TNγ¨,INγ¨,FNγ¨|γ¨∈Γ,where TNγ¨,INγ¨,andFNγ¨ are truth, indeterminacy, and falsity fuzzy functions.Definition 5 (see [8]).
An INS is an extension of NS defined by(2)N˜=T˜Nγ¨,I˜Nγ¨,F˜Nγ¨|γ¨∈Γ,where T˜Nγ¨,I˜Nγ¨,andF˜Nγ¨ are interval-valued fuzzy truth, indeterminacy, and falsity function.Definition 6 (see [9]).
A NCS is hybrid of NS and INS and defined as(3)N=T˜Nγ¨,I˜Nγ¨,F˜Nγ¨,TNγ¨,INγ¨,FNγ¨/γ¨∈Γ,where T˜Nγ¨=TNLγ¨,TNUγ¨,I˜Nγ¨=INLγ¨,INUγ¨,F˜Nγ¨=FNLγ¨,FNUγ¨ is an INS, and TNγ¨,INγ¨,FNγ¨ is a NS, where 0˜≤T˜N+I˜N+F˜N≤3˜,0≤TN+IN+FN≤3.
For the sake of convenience, the NCS are written asN=TNL,TNU,INL,INU,FNL,FNU,TN,IN,FN.Definition 7 (see [10]).
The sum of two NCS,N=TNL,TNU,INL,INU,FNL,FNU,TN,IN,FN and M=TML,TMU,IML,IMU,FML,FMU,TM,IM,FM, is defined as(4)N⊕M=TNL+TML−TNLTML,TNU+TMU−TNUTMU,INL+IML−INLIML,INU+IMU−INUIMU,FNLFML,FNUFMU,TNTM,INIM,FN+FM−FNFM.Definition 8 (see [10]).
The product of two NCS,N=TNL,TNU,INL,INU,FNL,FNU,TN,IN,FN and M=TML,TMU,IML,IMU,FML,FMU,TM,IM,FM, is defined as(5)N⊗M=TNLTML,TNUTMU,INLIML,INUIMU,FNL+FML−FNLFML,FNU+FMU−FNUFMU,TN+TM−TNTM,IN+IM−INIM,FNFM,Definition 9 (see [10]).
The scalar multiplication on a NCSN=TNL,TNU,INL,INU,FNL,FNU,TN,IN,FN and a scalar η is defined:(6)ηN=1−1−TNLη,1−1−TNUη,1−1−INLη,1−1−INUη,FNLη,FNUη,TNη,INη,1−1−FNη.


## 2.2. Developments in Fuzzy Measure

In decision-making (DM) process, the valuel⌢ is weighted by wi weight and then aggregated using weighted averaging and weighted geometric aggregation operators, where wi∈0,1 such that ∑i=1n¨wi=1. In real-life problems, there exist interactive phenomena amongst the elements. The overall significance of an element not only specified by itself, but by all the other elements in process.Sangeno [20] established the notion of fuzzy measure, which not only determines weight of an element and each combination of elements as well, and sum of weights need not to be equal to one. Murofushi and Saneno [21] proposed Choquet integral as an extension of Lebesgue integral. It is a significant aggregation operator for MCDM by considering significance of element by fuzzy measure.Definition 10 (see [23]).
LetT⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ be a set. A fuzzy measure θ on T⌢ is defined as a function θ:T⌢⟶0,1 fulfilling the following properties:(i)
θφ=0andθT⌢=1(ii)
ForU,V∈PT⌢,U⊆V⟹θU≤θV, where PT⌢ is power set of T⌢Definition 11 (see [23]).
In MCDM, forU,V∈PT⌢ such that U∩V=φ, three types of interactive relation are possible, that is,(i)
Additive measure: if U and V are independent (no interaction), then(7)θU∪V=θU+θV.(ii)
Super additive measure: if U and V are positive synergetic interaction, then(8)θU∪V>θU+θV,(iii)
Subadditive measure: if U and V are negative synergetic interaction, then(9)θU∪V<θU+θV.Definition 12 (see [23]).
Letk be a function on T⌢ and θ be a fuzzy measure on T⌢. Then, discrete Choquet with respect to θ is defined by(10)Cθkl⌢ρi=∑i=1n¨kl⌢ρiθT⌢ρi−θT⌢ρi+1,for ρi as ρ1,ρ2,…,ρi,…,ρn¨ present the permutations of 1,2,…,i,…,n¨ such that kl⌢1≤kl⌢2≤kl⌢3≤⋯≤kl⌢n¨ and T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ with T⌢ρn¨+1=φ. From definition, it is observed that the Choquet integral handles the interaction between two consecutive values; it is unable to handle the overall all interaction. This limitation is overcome by the Shapley index [18]:(11)ϖSShθ,N=∑T⊆N\Sn¨−u−v!t!n¨−u+1!t!θU∪V−θV,∀U⊆N,where θ is a fuzzy function of fuzzy measure λ, on N, and the cardinality of U,N, and V is, respectively, u,n¨, and v.
Meng [25] generalized the Shapley index to generalized Shapley index as λ− fuzzy measure θλ on N by(12)ϖSShθλ,N=∑T⊆N\Sn¨−u−v!v!n¨−u+1!v!θλU∪V−θλV,∀U⊆N,where θλis fuzzy measure expressed as(13)θλA=1λ∏k∈T⌢λθλi+1−1ifλ≠0∑k∈T⌢λθλkifλ=0,where ∏c∈N1+θλck=1+θλ is used to measure λ.Thus, if in S=k, then(14)ϖkShθλ,N=∑T⊆N\St!n¨−1−t!t!n¨!θλk∏j∈T1+λθλj,∀k⊆N.Definition 13.
Based on these definitions, Meng [25] defined arithmetic λ−Shapley–Choquet integral operator as(15)CϖSShθλ,Nkl⌢ρi=∑i=1n¨kl⌢ρiΘϒ⌢ρT⌢i,whereϒ⌢ρT⌢i=ϖρT⌢iShθλ,N−ϖρT⌢i+1Shθλ,N,for ρi as ρ1,ρ2,…,ρi,…,ρn¨ present the permutations of 1,2,…,i,…,n¨ such that kl⌢1≤kl⌢2≤kl⌢3≤⋯≤kl⌢n¨ and T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ with T⌢ρn¨+1=φ.Definition 14 (see [25]).
Geometricλ−Shapley–Choquet integral operator is as(16)CϖSShηλ,Nkl⌢ρi=∏i=1n¨kl⌢ρiΘϒ⌢ρT⌢iwhereΘϒ⌢ρT⌢i=ϖρT⌢iShθλ,N−ϖρT⌢i+1Shθλ,Nfor ρi as ρ1,ρ2,…,ρi,…,ρn¨ present the permutations of 1,2,…,i,…,n¨ such that kl⌢1≤kl⌢2≤kl⌢3≤⋯≤kl⌢n¨ and T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ with T⌢ρn¨+1=φ.


## 2.3. Induced Generalized Shapley–Choquet Integral

Aggregation operator is an important component of DM theory. The suitable aggregation operator may reduce the challenges that are present in vague and inconsistent data. Different operators are defined to meet these challenges. IGNCSCIA and IGNCSCIG aggregation operators will be defined to meet the challenges of criterion weights and interaction.Definition 15.
Letl⌢i=TLl⌢i,TUl⌢i,ILl⌢i,IUl⌢i,FLl⌢i,FUl⌢i,Tl⌢i,Il⌢i,Fl⌢i, where i=1,2,…,n¨ be a collection of NC values, and θ be a fuzzy measure on T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n such that θl⌢i=u⌢i; then, the IGNCSCIA operator is defined as(17)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n,l⌢n¨=⊕i=1n¨kl⌢iqΘϒ⌢ρT⌢i1/q,where q∈0,∞ and ρi as ρ1,ρ2,…,ρi,…,ρn¨ present the permutations of 1,2,…,i,…,n¨ such that kl⌢1≤kl⌢2≤kl⌢3≤⋯≤kl⌢n¨ and T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ with T⌢ρn¨+1=φ.Theorem 1.
Letl⌢i=TLl⌢i,TUl⌢i,ILl⌢i,IUl⌢i,FLl⌢i,FUl⌢i,Tl⌢i,Il⌢i,Fl⌢i, where 1,2,…,i,…,n¨ be a collection of NC values, and θ be a fuzzy measure on T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ such that θl⌢i=u⌢i; then, the IGNCSCIA operator is an NC value:(18)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n,l⌢n¨=⊕i=1n¨kl⌢iqΘϒ⌢ρT⌢i1/q=1−∏i=1n¨1−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=1n¨1−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−∏i=1n¨1−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=1n¨1−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=1n¨1−1−Fl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−1−∏i=1n¨1−1−Fl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=1n¨1−1−Tl⌢ρiqΘϒ⌢ρT⌢i1/q,1−1−∏i=1n¨1−1−Il⌢ρiqΘϒ⌢ρT⌢i1/q,1−∏i=1n¨1−Fl⌢ρiqΘϒ⌢ρT⌢i1/q,where q∈0,∞ and ρ1,ρ2,…,ρi,…,ρn¨ present the permutations of 1,2,…,i,…,n¨ such that kl⌢1≤kl⌢2≤kl⌢3≤⋯≤kl⌢n¨ and T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ with T⌢ρn¨+1=φ.Proof.
Forn¨=1, (17) reduces to the NC value by operational laws and equations (15). For n¨=2,(19)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2=⊕i=12kl⌢ρiqΘϒ⌢ρT⌢i1/q=1−∏i=121−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=121−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−∏i=121−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=121−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=121−1−Fl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−1−∏i=121−1−Fl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=121−1−Tl⌢ρiqΘϒ⌢ρT⌢i1/q,1−1−∏i=121−1−Il⌢ρiqΘϒ⌢ρT⌢i1/q,1−∏i=121−Fl⌢ρiqΘϒ⌢ρT⌢i1/q,(20)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2=⊕i=12kl⌢ρiqΘϒ⌢ρT⌢i1/q=1−1−Tl⌢ρ1LqΘϒ⌢ρT⌢11−Tl⌢ρ2LqΘϒ⌢ρT⌢21/q,1−1−Tl⌢ρ1UqΘϒ⌢ρT⌢11−Tl⌢ρ2UqΘϒ⌢ρT⌢21/q,1−1−Il⌢ρ1LqΘϒ⌢ρT⌢11−Il⌢ρ2LqΘϒ⌢ρT⌢21/q,1−1−Il⌢ρ1UqΘϒ⌢ρT⌢11−Il⌢ρ2UqΘϒ⌢ρT⌢21/q,1−1−1−1−Fl⌢ρ1LqΘϒ⌢ρT⌢11−1−Fl⌢ρ2LqΘϒ⌢ρT⌢21/q,1−1−1−1−Fl⌢ρ1UqΘϒ⌢ρT⌢11−1−Fl⌢ρ2UqΘϒ⌢ρT⌢21/q,1−1−1−1−Tl⌢ρ1qΘϒ⌢ρT⌢11−1−Tl⌢ρ2qΘϒ⌢ρT⌢21/q,1−1−1−1−Il⌢ρ1qΘϒ⌢ρT⌢11−1−Il⌢ρ2qΘϒ⌢ρT⌢21/q,1−1−Fl⌢ρ1qΘϒ⌢ρT⌢11−Fl⌢ρ2qΘϒ⌢ρT⌢21/q.
By operational laws equation, letn¨=j result holds good:(21)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,...,u⌢j,l⌢j=⊕i=1jkl⌢ρiqΘϒ⌢ρT⌢i1/q=1−∏i=1j1−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=1j1−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−∏i=1j1−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=1j1−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j1−1−Fl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j1−1−Fl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j1−1−Tl⌢ρiqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j1−1−Il⌢ρiqΘϒ⌢ρT⌢i1/q,1−∏i=1j1−Fl⌢ρiqΘϒ⌢ρT⌢i1/q.
Forn¨=j+1,(22)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢j+1,l⌢j+1=⊕i=1j+1kl⌢ρiqΘϒ⌢ρT⌢i1/q=1−∏i=1j+11−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=1j+11−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−∏i=1j+11−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=1j+11−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j+11−1−Fl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j+11−1−Fl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j+11−1−Tl⌢ρiqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j+11−1−Il⌢ρiqΘϒ⌢ρT⌢i1/q,1−∏i=1j+11−Fl⌢ρiqΘϒ⌢ρT⌢i1/q,which in the form of(23)IGNCSCIAηλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢j,l⌢j⊕IGNCSCIAηu⌢j+1,l⌢j+1=⊕i=1jkl⌢iqΘϒ⌢ρT⌢i1/q⊕kl⌢j+1qΘϒ⌢ρT⌢j+11/q=1−∏i=1j1−Tl⌢ρiLqΘϒ⌢ρT⌢i1−1−Tl⌢ρj+1LqΘϒ⌢ρT⌢j+11/q,1−∏i=1j1−Tl⌢ρiUqΘϒ⌢ρT⌢i1−1−Tl⌢ρj+1UqΘϒ⌢ρT⌢j+11/q,1−∏i=1j1−Il⌢ρiLqΘϒ⌢ρT⌢i1−1−Il⌢ρj+1LqΘϒ⌢ρT⌢j+11/q,1−∏i=1j1−Il⌢ρiUqΘϒ⌢ρT⌢i1−1−Il⌢ρj+1UqΘϒ⌢ρT⌢j+11/q,1−1−∏i=1j1−1−Fl⌢ρiLqΘϒ⌢ρT⌢i1−1−Fl⌢ρj+1LqΘϒ⌢ρT⌢j+11/q,1−1−∏i=1j1−1−Fl⌢ρiUqΘϒ⌢ρT⌢i1−1−Fl⌢ρj+1UqΘϒ⌢ρT⌢j+11/q,1−1−∏i=1j1−1−Tl⌢ρiqΘϒ⌢ρT⌢i1−1−Tl⌢ρj+1qΘϒ⌢ρT⌢j+11/q,1−1−∏i=1j1−1−Tl⌢ρiqΘϒ⌢ρT⌢i1−1−Tl⌢ρj+1qΘϒ⌢ρT⌢j+11/q1−∏i=1j1−Fl⌢ρiqΘϒ⌢ρT⌢i1−Fl⌢ρj+1qΘϒ⌢ρT⌢j+11/q=1−∏i=1j+11−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=1j+11−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−∏i=1j+11−Tl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−∏i=1j+11−Tl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j+11−1−Fl⌢ρiLqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j+11−1−Fl⌢ρiUqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j+11−1−Tl⌢ρiqΘϒ⌢ρT⌢i1/q,1−1−∏i=1j+11−1−Il⌢ρiqΘϒ⌢ρT⌢i1/q,1−∏i=1j+11−Fl⌢ρiqΘϒ⌢ρT⌢i1/q.is a NC value by assumption hypothesis n¨=k and n¨=2. Hence, n¨=j+1 is NC, which completes the proof.Definition 16.
Letl⌢i=TLl⌢i,TUl⌢i,ILl⌢i,IUl⌢i,FLl⌢i,FUl⌢i,Tl⌢i,Il⌢i,Fl⌢i, where 1,2,…,i,…,n¨ be a collection of NC values, and θ be a fuzzy measure on T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ such that θl⌢i=u⌢i; then, the IGNCSCIG operator is defined as(24)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨=1q⊗i=1n¨qkl⌢ρiΘϒ⌢ρT⌢ρi,where q∈0,∞ and ρ1,ρ2,…,ρi,…,ρn¨ present the permutations of 1,2,…,i,…,n¨ such that kl⌢1≤kl⌢2≤kl⌢3≤…≤kl⌢n¨ and T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ with T⌢ρn¨+1=φ.Theorem 2.
Letl⌢i=TLl⌢i,TUl⌢i,ILl⌢i,IUl⌢i,FLl⌢i,FUl⌢i,Tl⌢i,Il⌢i,Fl⌢i, where i=1,2,…,n¨ be a collection of NC values, and θ be a fuzzy measure on T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ such that θl⌢i=u⌢i; then, the aggregated result obtained by the IGNCGSCIG operator is an NC value:(25)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨=1q⊗i=1n¨qkl⌢ρiΘϒ⌢ρT⌢ρi=1−1−∏i=1n¨1−1−Tl⌢ρiLqΘϒ⌢ρT⌢ρi1/q,1−1−∏i=1n¨1−1−Tl⌢ρiUqΘϒ⌢ρT⌢ρi1/q,1−1−∏i=1n¨1−1−Il⌢ρiLqΘϒ⌢ρT⌢ρi1/q,1−1−∏i=1n¨1−1−Il⌢ρiUqΘϒ⌢ρT⌢ρi1/q,1−∏i=1n¨1−Fl⌢ρiLqΘϒ⌢ρT⌢ρi1/q,1−∏i=1n¨1−Fl⌢ρiUqΘϒ⌢ρT⌢ρi1/q,1−∏i=1n¨1−Tl⌢ρiqΘϒ⌢ρT⌢ρi1/q,1−∏i=1n¨1−Il⌢ρiqΘϒ⌢ρT⌢ρi1/q,1−1−∏i=1n¨1−1−Fl⌢ρiqΘϒ⌢ρT⌢ρi1/q,where q∈0,∞ and ρ1,ρ2,…,ρi,…,ρn¨ present the permutations of 1,2,…,i,…,n¨ such that kl⌢1≤kl⌢2≤kl⌢3≤…≤kl⌢n¨ and T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ with T⌢ρn¨+1=φ.Proof.
The proof is analogy of Theorem1.


## 2.4. Properties of IGNCSCIA and IGNCSCIG

The IGNCSCIA and IGNCSCIG satisfy the following properties.Proposition 1 (idempotency).
Letl⌢i=TLl⌢i,TUl⌢i,ILl⌢i,IUl⌢i,FLl⌢i,FUl⌢i,Tl⌢i,Il⌢i,Fl⌢i, where i=1,2,…,n¨ be a collection of NC values and θ be a fuzzy measure on T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ such that θl⌢i=u⌢i:(26)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n,l⌢n¨=⊕i=1n¨kl⌢iqΘϒ⌢T⌢ρi1/q,(27)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨=1q⊗i=1n¨qkl⌢ρiΘϒ⌢T⌢ρi,where q∈0,∞ and ρ1,ρ2,…,ρi,…,ρn¨ present the permutations of 1,2,…,i,…,n¨ such that kl⌢1≤kl⌢2≤kl⌢3≤…≤kl⌢n¨ and T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ with T⌢ρn¨+1=φ. For l⌢i=l⌢,(28)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨=l⌢,IGNCSCIGθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨=l⌢.Proposition 2 (monotonicity).
Letl⌢i=TLl⌢i,TUl⌢i,ILl⌢i,IUl⌢i,FLl⌢i,FUl⌢i,Tl⌢i,Il⌢i,Fl⌢i, where i=1,2,…,n¨ be a collection of NC values and θ be a fuzzy measure on T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ such that θl⌢i=u⌢i, B⌢=b⌢1,b⌢2,b⌢3,…,b⌢n¨, and θb⌢i=v⌢i with Tl⌢iL≤Tb⌢iL,Tl⌢iU≤Tb⌢iU,Il⌢iL≤Ib⌢iL,Il⌢iU≤Ib⌢iU,Tl⌢iL≥Tb⌢iL,Tl⌢iU≥Tb⌢iU,Tl⌢i≥Tb⌢i,Il⌢i≥Ib⌢i,Fl⌢i≤Fb⌢i; then,(29)IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨≤IGNCSCIAθλv⌢1,b⌢1,v⌢2,b⌢2,…,v⌢n¨,b⌢n¨,IGNCSCIGθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨≤IGNCSCIGθλv⌢1,b⌢1,v⌢2,b⌢2,…,v⌢n¨,b⌢n¨.Proof.
To prove this proposition, the following inequality is proved:(30)x,y∈0,1such thatxk≤yk⟹1−xk≥1−yk⟹1−1−xk≤1−1−yki,forxk≥yk⟹1−xk≤1−yk⟹1−1−xk≤1−1−ykii.
Forx=y=0 and x=y=1, the result is trivial.
SinceΘϒ⌢T⌢ρi=ϖT⌢ρiShηλ,N−ϖT⌢ρi+1Shηλ,N=ϖB⌢ρiShηλ,N−ϖB⌢ρi+1Shηλ,N=Θϒ⌢B⌢ρi, putting x=Tl⌢iL,y=Tb⌢iL,andk=q and using equation i,(31)1−∏i=1n¨1−Tl⌢ρiLqΘϒ⌢T⌢ρi1/q≤1−∏i=1n¨1−Tb⌢ρiLqΘϒ⌢B⌢ρi1/q,1−∏i=1n¨1−Tl⌢ρiUqΘϒ⌢T⌢ρi1/q≤1−∏i=1n¨1−Tb⌢ρiUqΘϒ⌢B⌢ρi1/q1−∏i=1n¨1−Il⌢ρiLqΘϒ⌢T⌢ρi1/q≤1−∏i=1n¨1−Ib⌢ρiLqΘϒ⌢B⌢ρi1/q,1−∏i=1n¨1−Il⌢ρiUqΘϒ⌢T⌢ρi1/q≤1−∏i=1n¨1−Ib⌢ρiUqΘϒ⌢B⌢ρi1/q1−∏i=1n¨1−Fl⌢ρiqΘϒ⌢T⌢ρi1/q≤1−∏i=1n¨1−Fb⌢ρiqΘϒ⌢B⌢ρi1/q,
Puttingx=Fl⌢ρiL,y=Fb⌢ρiL,andk=q and using equation (ii),(32)1−1−∏i=1n¨1−1−Fl⌢ρiLqΘϒ⌢T⌢ρi1/q≥1−1−∏i=1n¨1−1−Fb⌢ρiLqΘϒ⌢B⌢ρi1/q,1−1−∏i=1n¨1−1−Fl⌢ρiUqΘϒ⌢T⌢ρi1/q≥1−1−∏i=1n¨1−1−Fb⌢ρiUqΘϒ⌢B⌢ρi1/q,1−1−∏i=1n¨1−1−Tl⌢ρiqΘϒ⌢T⌢ρi1/q≥1−1−∏i=1n¨1−1−Tb⌢ρiqΘϒ⌢B⌢ρi1/q,1−1−∏i=1n¨1−1−Il⌢ρiqΘϒ⌢T⌢ρi1/q≥1−1−∏i=1n¨1−1−Ib⌢ρiqΘϒ⌢B⌢ρi1/q,IGNCSCIAθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨≤IGNCSCIAθλv⌢1,b⌢1,v⌢2,b⌢2,…,v⌢n¨,b⌢n¨,IGNCSCIGθλu⌢1,l⌢1,u⌢2,l⌢2,…,u⌢n¨,l⌢n¨≤IGNCSCIGθλv⌢1,b⌢1,v⌢2,b⌢2,…,v⌢n¨,b⌢n¨.Proposition 3 (boundedness).
Letl⌢i=TLl⌢i,TUl⌢i,ILl⌢i,IUl⌢i,FLl⌢i,FUl⌢i,Tl⌢i,Il⌢i,Fl⌢i, where i=1,2,…,n¨ be a collection of NC values and θ be a fuzzy measure on T⌢=l⌢1,l⌢2,l⌢3,…,l⌢n¨ such that θl⌢i=u⌢i, such that(33)l⌢−=minTLl⌢i,minTUl⌢i,minILl⌢i,minIUl⌢i,1−maxFLl⌢i,1−maxFUl⌢i,minTl⌢i,minIl⌢i,1−maxFl⌢i,l⌢i+=maxTLl⌢i,maxTUl⌢i,maxILl⌢i,maxIUl⌢i,1−minFLl⌢i,1−minFUl⌢i,maxTl⌢i,maxIl⌢i,1−minFl⌢i.
Then,(34)l⌢−≤IGNCSCIAθλl⌢1,l⌢2,l⌢3,…,l⌢n¨≤l⌢+,l⌢−≤IGNCSCIGθλl⌢1,l⌢2,l⌢3,…,l⌢n¨≤l⌢+.
The proof is followed by idempotency.Remark 1.
When different values are assigned top, IGNCSCA and IGNCSG will deduce into different neutrosophic cubic aggregation operators as follows.Proof.
Whenq=1, IGNCSCIA deduce into induced neutrosophic cubic Shapley–Choquet integral average (INCSCIA) operator, and IGNCSCIG deduced into induced neutrosophic cubic Shapley–Choquet integral geometric (INCSCIG) operator.
Whenu⌢i=l⌢i, IGNCSCIA deduce into generalized neutrosophic cubic Shapley–Choquet integral average (GNCSCIA) operator, and IGNCSCIG deduce into generalized neutrosophic cubic Shapley–Choquet integral geometric (GNCSCIG) operator.
Whenu⌢i=l⌢i and q=1, IGNCSCIA deduce into neutrosophic cubic Shapley–Choquet integral average (NCSCIA) operator, and IGNCSCIG deduce into neutrosophic cubic Shapley–Choquet integral geometric (NCSCG) operator.


## 3. Application

Example 1.
In order to evaluate the IGNCSCIA and IGNCSCIG operator, the following example is presented. A company wants to expand its foreign country investment, to choose the best country out of five alternatives (countries)T⌢=l⌢1,l⌢2,l⌢3,l⌢4,l⌢5 to make investment. The four main factors (criteria) E to decide are resources, policies, economy, and infrastructure, E=e1,e2,e3,e4. First, the data are presented in the form of NC values in Table 1.
The weights are obtained by fuzzy Shapley measures presented in Table2.
With the help of these values, the following weight are measured:ϖ1Shμλ,N=0.0892,ϖ2Shμλ,N=0.0056,ϖ3Shμλ,N=0.5939,andϖ4Shμλ,N=0.2624.
Using IGNCSCIA and IGNCSCIG operator, the following values are obtained Table3.
The alternatives are ranked, and graphical representation is given below for both IGNCSCIA and IGNCSCIG operators (Figures2 and 3).
The ranking of alternatives in the IGNCSCIA operator for different values ofq is tabulated in Table 4.
The ranking of alternatives in the IGNCSCIG operator for different values ofq is tabulated in Table 5.
From graphs and tabulated data, it is observed that the value ofq affects the rank of alternatives in the IGNCSCIG operator.Table 1
The data in NC form of given data.
e1e2e3e4a˙11=0.7,0.8,0.5,0.7,0.1,0.2,0.9,0.7,0.2a˙12=0.6,0.8,0.4,0.6,0.4,0.6,0.2,0.3,0.7a˙13=0.4,0.5,0.5,0.6,0.4,0.6,0.3,0.3,0.3a˙14=0.6,0.8,0.5,0.6,0.4,0.5,0.5,0.4,0.5a˙15=0.6,0.7,0.4,0.5,0.4,0.5,0.5,0.4,0.5a˙21=0.6,0.8,0.4,0.5,0.3,0.3,0.9,0.8,0.2a˙22=0.5,0.7,0.4,0.6,0.1,0.3,0.6,0.4,0.2a˙23=0.6,0.7,0.4,0.6,0.3,0.4,0.8,0.7,0.5a˙24=0.5,0.6,0.3,0.4,0.4,0.5,0.4,0.4,0.3a˙25=0.8,0.9,0.3,0.4,0.1,0.2,0.7,0.4,0.3a˙31=0.8,0.8,0.4,0.6,0.1,0.2,0.8,0.9,0.1a˙32=0.6,0.6,0.2,0.3,0.4,0.5,0.5,0.1,0.3a˙33=0.7,0.8,0.6,0.7,0.1,0.2,0.5,0.4,0.4a˙34=0.6,0.8,0.5,0.6,0.1,0.2,0.5,0.7,0.3a˙35=0.7,0.8,0.5,0.6,0.1,0.2,0.5,0.3,0.4a˙31=0.7,0.7,0.3,0.4,0.2,0.2,0.8,0.6,0.2a˙32=0.6,0.8,0.4,0.4,0.2,0.4,0.7,0.3,0.5a˙33=0.5,0.6,0.5,0.6,0.2,0.3,0.6,0.6,0.4a˙34=0.8,0.9,0.3,0.4,0.1,0.2,0.7,0.5,0.4a˙35=0.5,0.7,0.5,0.5,0.2,0.3,0.6,0.5,0.3Table 2
The fuzzy measure of elements of power set ofA.
AμλAAμλAAμλAϕ0e1,e20.1441e1,e2,e30.7378e10.664e1,e30.6596e1,e2,e40.4050e20.0777e1,e40.3271e1,e3,e40.9217e30.5929e2,e30.6710e2,e3,e40.9931e40.2605e2,e40.33840E1e3,e40.8584Table 3
The neutrosophic cubic values using IGNCSCIA and IGNCSCIG operator.
IGNCSCAu˜31=0.735,0.862,0.412,0.557,0.158,0.204,0.172,0.267,0.191u˜32=0.594,0.746,0.383,0.468,0.137,0.329,0.531,0.207,0.375u˜33=0.648,0.750,0.528,0.670,0.165,0.270,0.588,0.516,0.480u˜34=0.662,0.801,0.406,0.511,0.156,0.267,0.444,0.449,0.358u˜35=0.761,0.872,0.451,0.528,0.093,0.188,0.573,0.348,0.369IGNCSCGu˜31=0.644,0.803,0.339,0.468,0.251,0.271,0.888,0.843,0.869u˜32=0.518,0.663,0.286,0.372,0.268,0.421,0.634,0.343,0.757u˜33=0.560,0.664,0.436,0.597,0.261,0.370,0.731,0.646,0.598u˜34=0.540,0.664,0.311,0.414,0.317,0.415,0.560,0.573,0.714u˜35=0.645,0.786,0.350,0.436,0.158,0.260,0.665,0.432,0.708Figure 2
Graphical view of IGNCSCIA.Figure 3
Graphical view of IGNCSCIG.Table 4
The ranking of alternatives subject to the different values ofq using the IGNCSCIA operator.
Tabulated view of IGNCSCIA for different values ofqqRanksq=0.1l⌢1>l⌢3>l⌢5>l⌢4>l⌢2q=0.5l⌢1>l⌢3>l⌢5>l⌢4>l⌢2q=1l⌢1>l⌢3>l⌢5>l⌢4>l⌢2q=2l⌢1>l⌢3>l⌢5>l⌢4>l⌢2q=3l⌢1>l⌢3>l⌢5>l⌢4>l⌢2q=5l⌢1>l⌢3>l⌢5>l⌢4>l⌢2Table 5
The ranking of alternatives subject to the different values ofq using the IGNCSCIG operator.
Tabulated view of IGNCSCIG for different values ofqqRanksq=0.1l⌢1>l⌢3>l⌢5>l⌢4>l⌢2q=0.5l⌢1>l⌢3>l⌢5>l⌢4>l⌢2q=1l⌢1>l⌢3>l⌢4>l⌢5>l⌢2q=2l⌢1>l⌢3>l⌢4>l⌢5>l⌢2q=3l⌢3>l⌢1>l⌢5>l⌢4>l⌢2q=5l⌢3>l⌢1>l⌢5>l⌢4>l⌢2


### 3.1. Sensitivity Analysis

From the data, it is observed that the ranking of alternative changes with change value ofq. For 0<q<1, the ranking is l⌢1>l⌢3>l⌢5>l⌢4>l⌢2. For q∈1,2, the ranking changes to l⌢1>l⌢3>l⌢4>l⌢5>l⌢2. For q=3, the ranking changes to l⌢3>l⌢1>l⌢5>l⌢4>l⌢2, and for q≥5, the ranking l⌢3>l⌢1>l⌢5>l⌢4>l⌢2. So, the decision of best alternative changes with the value q. For such situation, the distance decision-making methods are the best choices to conclude the best ranking. A number of distance-based decision-making methods are proposed by researchers. Among these, the CODAS method is one of the user-friendly methods that evaluate the data more precisely. This limitation is overcome by proposing novel distance-based decision method and CODAS method in NC environment. Since the CODAS method is the distance-based method, so first the Minkwoski distance formula is proposed for NC values.


### 3.2. Minkowski Distance

In this section, Minkowski distance formula is introduced to determine the distance of two NC values. Minkowski distance is the generalized distance formula which can deduced to hamming, Euclidian, and Chebyshev formula. This provides a plate form for expressing more information.Definition 17.
LetN=TNL,TNU,INL,INU,FNL,FNU,TN,IN,FN and M=TML,TMU,IML,IMU,FML,FMU,TM,IM,FM; the Minkowski distance is defined as(35)dN,M=19TNL−TMLq+TNU−TMUq+INL−IMLq+INU−IMUq+FNL−FMLq+FNU−FMUq+TN−TMq+IN−IMq+FN−FMq1/q.
Assigning different values toq Minkowski distance generates different distances:(i)
q=1 will reduce it into hamming distance(ii)
q=2 will reduce it into Euclidean distance(iii)
q⟶∞ will reduce it into Chebyshev distance


### 3.3. The NCCODAS Method

This section proposes the neutrosophic cubic combinative distance-based assessment (NCCODAS) method, and the CODAS method was proposed by Keshavarz et al. [28], which is a systemized method to handle MCDM problems. The CODAS method predominantly uses the combinative form of distances for computation of alternative rating.


#### 3.3.1. The Presentation of Methodology

The demonstration of the proposed method is shown in Figure4.Figure 4
Flowchart.The steps of the NCCODAS method for MCDM problems are as follows:Step 1: construction of DM asA=l⌢ij,i=1,2,…,mandj=1,2,…,n, and l⌢ij=TLl⌢ij,TUl⌢ij,ILl⌢ij,IUl⌢ij,FLl⌢ij,FUl⌢ij,Tl⌢ij,Il⌢ij,Fl⌢ij is neutrosophic cubic values assigned to alternative a˙i subject to criteria ej by decision makers.Step 2: the neutrosophic cubic normalized weighted matrix is accomplished byU=u˜ij,i=1,2,…,mandj=1,2,…,n, where(36)u˜ij=wj⊗l⌢ij,such that∑j=1nwj=1 and. 0<wj<1Step 3: determine the neutrosophic cubic negative ideals by score/ accuracy function:(37)ri=miniu˜ij|j=1,2,…,n.Step 4: calculate neutrosophic cubic Minkwoski distances of alternatives from negative ideals obtained by equation (37).Step 5: construct relative assessment matrix on neutrosophic cubic distances by(38)Ra=hikmnbyhik=MDi−MDk+tMDi+MDkMDi−MDk,wherek=1,2,…,n and t expresses a threshold function in accordance with the parameter θ assigned by the decision makers determined as tθ=1,ifθ≥1,0,ifθ<1.Note: for better result, different values ofq are used in Minkwoski distance in MDi−MDk which is used twice. In general, (38) is considered asRa=hikmnbyhik=EDi−EDk+tEDi+EDkHDi−HDk, where HD is hamming and ED is Euclidean distance.Step 6: calculate the assessment of each alternative by(39)ASi=∑k=1mhik.Step 7: rank the alternative in the ascending order. The alternative with highest rank is desirable alternative.


#### 3.3.2. A Case Study of Foreign Country Investment

A company wants to expand its foreign country investment to choose the best country out of five alternatives (countries)T⌢=l⌢1,l⌢2,l⌢3,l⌢4,l⌢5 and to make investment. The four main factors (criteria) E to decide are resources, policies, economy, and infrastructure. E=e1,e2,e3,e4. Jiao et al. [26] used different techniques to evaluate the results. The proposed method will be applied to the given data, and results will be compared with some existing methods.


### 3.4. Application of NCCODAS

The proposed method includes IGNCSCIA, IGNCSCIG aggregation operators, and NCCODAS decision-making techniques. The steps defined in NCCODAS are follows:Step 1: the data are organized in the form of NC values considered.It shows that a firm is interested to finance in a country out of five countries (attributes). The countries are represented byT⌢=l⌢1,l⌢2,l⌢3,l⌢4,l⌢5 subject to the criteria resources, politics and policies, economy, and infrastructure of countries represented by E=e1,e2,e3,e4 tabulated in Table 1.Step 2: to obtain the weighted matrix, the Shapley–Choquet measure is used to find the weight with the help of equations (10) and (11). Then, the aggregated values of IGNCSCA and IGNCSCG are tabulated in Tables 3 and 4. The measured values are tabulated in Table 2.Step 3: the minimum value is calculated using equation (23).The minimum value in IGNCSCA isminiu˜ij=0.594,0.746,0.383,0.468,0.165,0.329,0.588,0.516,0.191.The minimum value in IGNCSCG isminiu˜ij=0.518,0.663,0.286,0.372,0.317,0.421,0.888,0.843,0.598.Step 4: calculate Minkowski distance from negative ideals by equation (19).Distance from IGNCSCAminiu˜ij is tabulated in Table 4.Step 5: construction of relative assistance matrix with the help of equation (23) is tabulated in Table 4.Step 6: calculation of assessment is tabulated in Table4 by equation (20).Step 7: ranks of alternative in the ascending order are to get the desired alternative in Table4.In Table2, the data calculated in Steps 4–7 are tabulated for different values of q (Tables 6 and 7):(40)q=0.1Table 6
The aggregated neutrosophic cubic value data by IGNCSCA and IGNCSCG.
IGNCSCAu˜31=0.735,0.862,0.412,0.557,0.158,0.204,0.172,0.267,0.191u˜32=0.594,0.746,0.383,0.468,0.137,0.329,0.531,0.207,0.375u˜33=0.648,0.750,0.528,0.670,0.165,0.270,0.588,0.516,0.480u˜34=0.662,0.801,0.406,0.511,0.156,0.267,0.444,0.449,0.358u˜35=0.761,0.872,0.451,0.528,0.093,0.188,0.573,0.348,0.369IGNCSCGu˜31=0.644,0.803,0.339,0.468,0.251,0.271,0.888,0.843,0.869u˜32=0.518,0.663,0.286,0.372,0.268,0.421,0.634,0.343,0.757u˜33=0.560,0.664,0.436,0.597,0.261,0.370,0.731,0.646,0.598u˜34=0.540,0.664,0.311,0.414,0.317,0.415,0.560,0.573,0.714u˜35=0.645,0.786,0.350,0.436,0.158,0.260,0.665,0.432,0.708Table 7
The table represents the numerical data calculated by steps 4–7 for different values ofq.
AlternativesEDiHDiRaASiR10.19550.48560.184441R20.15910.3847−0.17663R30.18400.46610.0885462R40.16740.4143−0.086394R50.17860.4263−0.009993Rankingl⌢1>l⌢3>l⌢5>l⌢4>l⌢2Different values are assigned toq and alternatives ranked are as shown in Table 8.Table 8
The graphical interpretation of all alternatives subject to the different values ofq.
Tabulated view of NCCODAS for different values ofqqRanksp=0.1l⌢1>l⌢3>l⌢5>l⌢4>l⌢2p=0.5l⌢1>l⌢3>l⌢5>l⌢4>l⌢2p=1l⌢1>l⌢3>l⌢5>l⌢4>l⌢2p=2l⌢1>l⌢3>l⌢5>l⌢4>l⌢2p=3l⌢1>l⌢3>l⌢5>l⌢4>l⌢2p=5l⌢1>l⌢3>l⌢5>l⌢4>l⌢2The graphical interpretation of tabulated values is shown in Figure5.Figure 5
NCCODAS graphical view.It is observed that changing values of firm do not change the overall ranking of alternatives. To check the validity if method is under NC environment, it is compared with different methods.


### 3.5. Comparative Analysis

In order to validate the method, comparative analysis is provided with some existing MCDM methods. The analysis is tabulated in Table9.Table 9
The tabulated view of comparative analysis.
MethodologyRankingNC Einstein geometric aggregation operator in MCDM [10]l⌢1>l⌢3>l⌢2>l⌢5>l⌢4Generalized NC aggregation operators with application MEDM [13]l⌢1>l⌢3>l⌢5>l⌢4>l⌢2GRA MADM in NCS [14]l⌢1>l⌢5>l⌢3>l⌢2>l⌢4Cosine measure of NCS for MCDM [15]l⌢1>l⌢2>l⌢5>l⌢3>l⌢4Proposed NCCODASl⌢1>l⌢3>l⌢5>l⌢4>l⌢2From Table9, it is observed that the NCCODAS method l⌢1 is the best alternative and l⌢2 is the worst alternative evaluated by all the methods considered. Furthermore, it is also observed that the best alternative also obtained by other methods is l⌢1. This validates the validity of the NCCODAS method under neutrosophic cubic environment.


## 3.1. Sensitivity Analysis

From the data, it is observed that the ranking of alternative changes with change value ofq. For 0<q<1, the ranking is l⌢1>l⌢3>l⌢5>l⌢4>l⌢2. For q∈1,2, the ranking changes to l⌢1>l⌢3>l⌢4>l⌢5>l⌢2. For q=3, the ranking changes to l⌢3>l⌢1>l⌢5>l⌢4>l⌢2, and for q≥5, the ranking l⌢3>l⌢1>l⌢5>l⌢4>l⌢2. So, the decision of best alternative changes with the value q. For such situation, the distance decision-making methods are the best choices to conclude the best ranking. A number of distance-based decision-making methods are proposed by researchers. Among these, the CODAS method is one of the user-friendly methods that evaluate the data more precisely. This limitation is overcome by proposing novel distance-based decision method and CODAS method in NC environment. Since the CODAS method is the distance-based method, so first the Minkwoski distance formula is proposed for NC values.


## 3.2. Minkowski Distance

In this section, Minkowski distance formula is introduced to determine the distance of two NC values. Minkowski distance is the generalized distance formula which can deduced to hamming, Euclidian, and Chebyshev formula. This provides a plate form for expressing more information.Definition 17.
LetN=TNL,TNU,INL,INU,FNL,FNU,TN,IN,FN and M=TML,TMU,IML,IMU,FML,FMU,TM,IM,FM; the Minkowski distance is defined as(35)dN,M=19TNL−TMLq+TNU−TMUq+INL−IMLq+INU−IMUq+FNL−FMLq+FNU−FMUq+TN−TMq+IN−IMq+FN−FMq1/q.
Assigning different values toq Minkowski distance generates different distances:(i)
q=1 will reduce it into hamming distance(ii)
q=2 will reduce it into Euclidean distance(iii)
q⟶∞ will reduce it into Chebyshev distance


## 3.3. The NCCODAS Method

This section proposes the neutrosophic cubic combinative distance-based assessment (NCCODAS) method, and the CODAS method was proposed by Keshavarz et al. [28], which is a systemized method to handle MCDM problems. The CODAS method predominantly uses the combinative form of distances for computation of alternative rating.


### 3.3.1. The Presentation of Methodology

The demonstration of the proposed method is shown in Figure4.Figure 4
Flowchart.The steps of the NCCODAS method for MCDM problems are as follows:Step 1: construction of DM asA=l⌢ij,i=1,2,…,mandj=1,2,…,n, and l⌢ij=TLl⌢ij,TUl⌢ij,ILl⌢ij,IUl⌢ij,FLl⌢ij,FUl⌢ij,Tl⌢ij,Il⌢ij,Fl⌢ij is neutrosophic cubic values assigned to alternative a˙i subject to criteria ej by decision makers.Step 2: the neutrosophic cubic normalized weighted matrix is accomplished byU=u˜ij,i=1,2,…,mandj=1,2,…,n, where(36)u˜ij=wj⊗l⌢ij,such that∑j=1nwj=1 and. 0<wj<1Step 3: determine the neutrosophic cubic negative ideals by score/ accuracy function:(37)ri=miniu˜ij|j=1,2,…,n.Step 4: calculate neutrosophic cubic Minkwoski distances of alternatives from negative ideals obtained by equation (37).Step 5: construct relative assessment matrix on neutrosophic cubic distances by(38)Ra=hikmnbyhik=MDi−MDk+tMDi+MDkMDi−MDk,wherek=1,2,…,n and t expresses a threshold function in accordance with the parameter θ assigned by the decision makers determined as tθ=1,ifθ≥1,0,ifθ<1.Note: for better result, different values ofq are used in Minkwoski distance in MDi−MDk which is used twice. In general, (38) is considered asRa=hikmnbyhik=EDi−EDk+tEDi+EDkHDi−HDk, where HD is hamming and ED is Euclidean distance.Step 6: calculate the assessment of each alternative by(39)ASi=∑k=1mhik.Step 7: rank the alternative in the ascending order. The alternative with highest rank is desirable alternative.


### 3.3.2. A Case Study of Foreign Country Investment

A company wants to expand its foreign country investment to choose the best country out of five alternatives (countries)T⌢=l⌢1,l⌢2,l⌢3,l⌢4,l⌢5 and to make investment. The four main factors (criteria) E to decide are resources, policies, economy, and infrastructure. E=e1,e2,e3,e4. Jiao et al. [26] used different techniques to evaluate the results. The proposed method will be applied to the given data, and results will be compared with some existing methods.


## 3.3.1. The Presentation of Methodology

The demonstration of the proposed method is shown in Figure4.Figure 4
Flowchart.The steps of the NCCODAS method for MCDM problems are as follows:Step 1: construction of DM asA=l⌢ij,i=1,2,…,mandj=1,2,…,n, and l⌢ij=TLl⌢ij,TUl⌢ij,ILl⌢ij,IUl⌢ij,FLl⌢ij,FUl⌢ij,Tl⌢ij,Il⌢ij,Fl⌢ij is neutrosophic cubic values assigned to alternative a˙i subject to criteria ej by decision makers.Step 2: the neutrosophic cubic normalized weighted matrix is accomplished byU=u˜ij,i=1,2,…,mandj=1,2,…,n, where(36)u˜ij=wj⊗l⌢ij,such that∑j=1nwj=1 and. 0<wj<1Step 3: determine the neutrosophic cubic negative ideals by score/ accuracy function:(37)ri=miniu˜ij|j=1,2,…,n.Step 4: calculate neutrosophic cubic Minkwoski distances of alternatives from negative ideals obtained by equation (37).Step 5: construct relative assessment matrix on neutrosophic cubic distances by(38)Ra=hikmnbyhik=MDi−MDk+tMDi+MDkMDi−MDk,wherek=1,2,…,n and t expresses a threshold function in accordance with the parameter θ assigned by the decision makers determined as tθ=1,ifθ≥1,0,ifθ<1.Note: for better result, different values ofq are used in Minkwoski distance in MDi−MDk which is used twice. In general, (38) is considered asRa=hikmnbyhik=EDi−EDk+tEDi+EDkHDi−HDk, where HD is hamming and ED is Euclidean distance.Step 6: calculate the assessment of each alternative by(39)ASi=∑k=1mhik.Step 7: rank the alternative in the ascending order. The alternative with highest rank is desirable alternative.


## 3.3.2. A Case Study of Foreign Country Investment

A company wants to expand its foreign country investment to choose the best country out of five alternatives (countries)T⌢=l⌢1,l⌢2,l⌢3,l⌢4,l⌢5 and to make investment. The four main factors (criteria) E to decide are resources, policies, economy, and infrastructure. E=e1,e2,e3,e4. Jiao et al. [26] used different techniques to evaluate the results. The proposed method will be applied to the given data, and results will be compared with some existing methods.


## 3.4. Application of NCCODAS

The proposed method includes IGNCSCIA, IGNCSCIG aggregation operators, and NCCODAS decision-making techniques. The steps defined in NCCODAS are follows:Step 1: the data are organized in the form of NC values considered.It shows that a firm is interested to finance in a country out of five countries (attributes). The countries are represented byT⌢=l⌢1,l⌢2,l⌢3,l⌢4,l⌢5 subject to the criteria resources, politics and policies, economy, and infrastructure of countries represented by E=e1,e2,e3,e4 tabulated in Table 1.Step 2: to obtain the weighted matrix, the Shapley–Choquet measure is used to find the weight with the help of equations (10) and (11). Then, the aggregated values of IGNCSCA and IGNCSCG are tabulated in Tables 3 and 4. The measured values are tabulated in Table 2.Step 3: the minimum value is calculated using equation (23).The minimum value in IGNCSCA isminiu˜ij=0.594,0.746,0.383,0.468,0.165,0.329,0.588,0.516,0.191.The minimum value in IGNCSCG isminiu˜ij=0.518,0.663,0.286,0.372,0.317,0.421,0.888,0.843,0.598.Step 4: calculate Minkowski distance from negative ideals by equation (19).Distance from IGNCSCAminiu˜ij is tabulated in Table 4.Step 5: construction of relative assistance matrix with the help of equation (23) is tabulated in Table 4.Step 6: calculation of assessment is tabulated in Table4 by equation (20).Step 7: ranks of alternative in the ascending order are to get the desired alternative in Table4.In Table2, the data calculated in Steps 4–7 are tabulated for different values of q (Tables 6 and 7):(40)q=0.1Table 6
The aggregated neutrosophic cubic value data by IGNCSCA and IGNCSCG.
IGNCSCAu˜31=0.735,0.862,0.412,0.557,0.158,0.204,0.172,0.267,0.191u˜32=0.594,0.746,0.383,0.468,0.137,0.329,0.531,0.207,0.375u˜33=0.648,0.750,0.528,0.670,0.165,0.270,0.588,0.516,0.480u˜34=0.662,0.801,0.406,0.511,0.156,0.267,0.444,0.449,0.358u˜35=0.761,0.872,0.451,0.528,0.093,0.188,0.573,0.348,0.369IGNCSCGu˜31=0.644,0.803,0.339,0.468,0.251,0.271,0.888,0.843,0.869u˜32=0.518,0.663,0.286,0.372,0.268,0.421,0.634,0.343,0.757u˜33=0.560,0.664,0.436,0.597,0.261,0.370,0.731,0.646,0.598u˜34=0.540,0.664,0.311,0.414,0.317,0.415,0.560,0.573,0.714u˜35=0.645,0.786,0.350,0.436,0.158,0.260,0.665,0.432,0.708Table 7
The table represents the numerical data calculated by steps 4–7 for different values ofq.
AlternativesEDiHDiRaASiR10.19550.48560.184441R20.15910.3847−0.17663R30.18400.46610.0885462R40.16740.4143−0.086394R50.17860.4263−0.009993Rankingl⌢1>l⌢3>l⌢5>l⌢4>l⌢2Different values are assigned toq and alternatives ranked are as shown in Table 8.Table 8
The graphical interpretation of all alternatives subject to the different values ofq.
Tabulated view of NCCODAS for different values ofqqRanksp=0.1l⌢1>l⌢3>l⌢5>l⌢4>l⌢2p=0.5l⌢1>l⌢3>l⌢5>l⌢4>l⌢2p=1l⌢1>l⌢3>l⌢5>l⌢4>l⌢2p=2l⌢1>l⌢3>l⌢5>l⌢4>l⌢2p=3l⌢1>l⌢3>l⌢5>l⌢4>l⌢2p=5l⌢1>l⌢3>l⌢5>l⌢4>l⌢2The graphical interpretation of tabulated values is shown in Figure5.Figure 5
NCCODAS graphical view.It is observed that changing values of firm do not change the overall ranking of alternatives. To check the validity if method is under NC environment, it is compared with different methods.


## 3.5. Comparative Analysis

In order to validate the method, comparative analysis is provided with some existing MCDM methods. The analysis is tabulated in Table9.Table 9
The tabulated view of comparative analysis.
MethodologyRankingNC Einstein geometric aggregation operator in MCDM [10]l⌢1>l⌢3>l⌢2>l⌢5>l⌢4Generalized NC aggregation operators with application MEDM [13]l⌢1>l⌢3>l⌢5>l⌢4>l⌢2GRA MADM in NCS [14]l⌢1>l⌢5>l⌢3>l⌢2>l⌢4Cosine measure of NCS for MCDM [15]l⌢1>l⌢2>l⌢5>l⌢3>l⌢4Proposed NCCODASl⌢1>l⌢3>l⌢5>l⌢4>l⌢2From Table9, it is observed that the NCCODAS method l⌢1 is the best alternative and l⌢2 is the worst alternative evaluated by all the methods considered. Furthermore, it is also observed that the best alternative also obtained by other methods is l⌢1. This validates the validity of the NCCODAS method under neutrosophic cubic environment.


## 4. Conclusion

This study defined the NCSCI operator based on fuzzy Shapley and Choquet integration operators in NCS. Two aggregation operators IGNCSCIA and IGNCSCIG are defined and furnished upon numerical application of foreign investment of a company. A sensitive analysis is conducted over these investigated cases. A novel CODAS is proposed on IGNCSCIA and IGNCSCIG in NCS environment to handle complex frame of data that occurs in daily life. This method is also furnished upon the same application and observed that it yields the same ranking for different values ofq. The comparative analysis is conducted to investigate the validity of the proposed method. It is observed that the proposed method yields the best alternative is the same as existing methods. Comparative analysis is conducted. In future, some extension of this work will be explored in the field of Bonferroni mean operators and unified generalized aggregation operators, and these operators will be furnished upon daily life problems.



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