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.

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 [1116]. 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 [1923] 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 problems

To 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 Figure 1.

The research paper has been divided into four sections. Section 1 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 mapping : is called a fuzzy set, and is called a membership function, simply denoted by .

Definition 2 (see [2]). A mapping , where is the interval value of , called the interval-valued fuzzy set(IVF). For all , is membership degree of in . This is simply denoted by .

Definition 3 (see [6]). A structure is a cubic set in in which is IVFS in , that is, and is a fuzzy set in . This can be simply denoted by .

Definition 4 (see [7]). The neutrosophic set is defined as where are truth, indeterminacy, and falsity fuzzy functions.

Definition 5 (see [8]). An INS is an extension of NS defined bywhere are interval-valued fuzzy truth, indeterminacy, and falsity function.

Definition 6 (see [9]). A NCS is hybrid of NS and INS and defined aswhere is an INS, and is a NS, where
For the sake of convenience, the NCS are written as .

Definition 7 (see [10]). The sum of two NCS, and , is defined as

Definition 8 (see [10]). The product of two NCS, and , is defined as

Definition 9 (see [10]). The scalar multiplication on a NCS and a scalar is defined:

2.2. Developments in Fuzzy Measure

In decision-making (DM) process, the value is weighted by weight and then aggregated using weighted averaging and weighted geometric aggregation operators, where such that . 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]). Let be a set. A fuzzy measure on is defined as a function fulfilling the following properties:(i)(ii), where is power set of

Definition 11 (see [23]). In MCDM, for such that , three types of interactive relation are possible, that is,(i)Additive measure: if and are independent (no interaction), then(ii)Super additive measure: if and are positive synergetic interaction, then(iii)Subadditive measure: if and are negative synergetic interaction, then

Definition 12 (see [23]). Let be a function on and be a fuzzy measure on . Then, discrete Choquet with respect to is defined byfor as present the permutations of such that and with . 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]:where is a fuzzy function of fuzzy measure , on , and the cardinality of , and is, respectively, , and .
Meng [25] generalized the Shapley index to generalized Shapley index as fuzzy measure on N bywhere fuzzy measure expressed aswhere is used to measure .Thus, if in then

Definition 13. Based on these definitions, Meng [25] defined arithmetic Shapley–Choquet integral operator asfor as present the permutations of such that and with .

Definition 14 (see [25]). Geometric Shapley–Choquet integral operator is asfor as present the permutations of such that and with .

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. Let where be a collection of NC values, and be a fuzzy measure on such that ; then, the IGNCSCIA operator is defined aswhere and as present the permutations of such that and with .

Theorem 1. Let where be a collection of NC values, and be a fuzzy measure on such that ; then, the IGNCSCIA operator is an NC value:where and present the permutations of such that and with .

Proof. For , (17) reduces to the NC value by operational laws and equations (15). For ,By operational laws equation, let result holds good:For ,which in the form ofis a NC value by assumption hypothesis and Hence, is NC, which completes the proof.

Definition 16. Let where be a collection of NC values, and be a fuzzy measure on such that ; then, the IGNCSCIG operator is defined aswhere and present the permutations of such that and with .

Theorem 2. Let where be a collection of NC values, and be a fuzzy measure on such that ; then, the aggregated result obtained by the IGNCGSCIG operator is an NC value:where and present the permutations of such that and with .

Proof. The proof is analogy of Theorem 1.

2.4. Properties of IGNCSCIA and IGNCSCIG

The IGNCSCIA and IGNCSCIG satisfy the following properties.

Proposition 1 (idempotency). Let where be a collection of NC values and be a fuzzy measure on such that :where and present the permutations of such that and with . For ,