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

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 cir-cumstances 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.


Introduction
Increasing uncertainty and complexity in decision-making (DM) theory, the representation of data is no longer the real number. e researcher developed di erent 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]. ese 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. is limitation can be overcome by Smrandache neutrosophic set (NS) [7]. An NS consists of three independent components, truth, indeterminancy, and falsity grades. e 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. e problem arises that whether these components can be assigned with both the interval value and the crisp value at the same time. is 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. is makes NCS a useful tool to represent the fuzziness of acceptance, neutral, and rejection in the complex frame of the environment more conveniently. ese characteristics attract the researcher to apply in the field of DM theory. Majid et al. defined novel operational laws on NCS [10]. e 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][12][13][14][15][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. ese 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][20][21][22][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].

Motivation.
e motivation of this research is to generalize Shapley measure and Choquet integral operator in the NCS plate form. at 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. e 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.

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

Organization.
e organization process of research is shown in Figure 1.
e research paper has been divided into four sections. Section 1 comprises of introduction. Section 2 comprises of preliminaries, definition, and results. e 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.

Preliminaries
is section consists of two section developments in NCS to neutrosophic cubic set and fuzzy preferences.
Definition 4 (see [7]). e neutrosophic set is defined as Definition 6 (see [9]). A NCS is hybrid of NS and INS and defined as For the sake of convenience, the NCS are written as Definition 7 (see [10]). e sum of two NCS, N � ([T L N , Definition 8 (see [10]). e product of two NCS, Definition 9 (see [10]). e scalar multiplication on a NCS , T N , I N , F N ) and a scalar η is defined:

Developments in Fuzzy
Measure. In decision-making (DM) process, the value l ⌢ is weighted by w i weight and then aggregated using weighted averaging and weighted geometric aggregation operators, where w i ∈ [0, 1] such that € n i�1 w i � 1. In real-life problems, there exist interactive phenomena amongst the elements. e 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.

Induced Generalized Shapley-Choquet Integral.
Aggregation operator is an important component of DM theory. e suitable aggregation operator may reduce the challenges that are present in vague and inconsistent data.
Proof. For € n � 1, (17) reduces to the NC value by operational laws and equations (15). For € n � 2, Journal of Mathematics 5 Journal of Mathematics By operational laws equation, let € n � j result holds good: For € n � j + 1, which in the form of 8 Journal of Mathematics is a NC value by assumption hypothesis € n � k and € n � 2. Hence, € n � j + 1 is NC, which completes the proof.
, and k � q and using equation (i), , and k � q and using equation (ii),

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 to make investment. e four main factors (criteria) E to decide are resources, policies, economy, and infrastructure, E � e 1 , e 2 , e 3 , e 4 . First, the data are presented in the form of NC values in Table 1. e weights are obtained by fuzzy Shapley measures presented in Table 2.

Journal of Mathematics 13
With the help of these values, the following weight are measured: ϖ Sh 1 (μ λ , N) � 0.0892, ϖ Sh 2 (μ λ , N) � 0.0056, ϖ Sh 3 (μ λ , N) � 0.5939, and ϖ Sh 4 (μ λ , N) � 0.2624. Using IGNCSCIA and IGNCSCIG operator, the following values are obtained Table 3. e alternatives are ranked, and graphical representation is given below for both IGNCSCIA and IGNCSCIG operators (Figures 2 and 3). e ranking of alternatives in the IGNCSCIA operator for different values of q is tabulated in Table 4. e ranking of alternatives in the IGNCSCIG operator for different values of q is tabulated in Table 5.
From graphs and tabulated data, it is observed that the value of q affects the rank of alternatives in the IGNCSCIG operator.

Sensitivity Analysis.
From the data, it is observed that the ranking of alternative changes with change value of q . For 0 < q < 1, the ranking is l 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. is 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.
Journal of Mathematics where HD is hamming and ED is Euclidean distance.
Step 6: calculate the assessment of each alternative by Step 7: rank the alternative in the ascending order. e alternative with highest rank is desirable alternative. e proposed method will be applied to the given data, and results will be compared with some existing methods.

Application of NCCODAS.
e proposed method includes IGNCSCIA, IGNCSCIG aggregation operators, and NCCODAS decision-making techniques. e 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).  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). en, the aggregated values of IGNCSCA and IGNCSCG are tabulated in Tables 3 and 4. e measured values are tabulated in Table 2.
Step 3: the minimum value is calculated using equation (23   Step 4: calculate Minkowski distance from negative ideals by equation (19). Distance from IGNCSCA min i u 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 Table 4 by equation (20).
Step 7: ranks of alternative in the ascending order are to get the desired alternative in Table 4.
In Table 2, the data calculated in Steps 4-7 are tabulated for different values of q (Tables 6 and 7):     Different values are assigned to q and alternatives ranked are as shown in Table 8. e graphical interpretation of tabulated values is shown in Figure 5.
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.

Comparative Analysis.
In order to validate the method, comparative analysis is provided with some existing MCDM methods. e analysis is tabulated in Table 9.
From Table 9, 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 . is validates the validity of the NCCODAS method under neutrosophic cubic environment.

Conclusion
is 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. is method is also furnished upon the same application and observed that it yields the same ranking for different values of q. e 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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this article.