The Generalized Neutrosophic Cubic Aggregation Operators and Their Application to Multi-Expert Decision-Making Method

: In the modern world, the computation of vague data is a challenging job. Di ﬀ erent theories are presented to deal with such situations. Amongst them, fuzzy set theory and its extensions produced remarkable results. Samrandache extended the theory to a new horizon with the neutrosophic set (NS), which was further extended to interval neutrosophic set (INS). Neutrosophic cubic set (NCS) is the generalized version of NS and INS. This characteristic makes it an exceptional choice to deal with vague and imprecise data. Aggregation operators are key features of decision-making theory. In recent times several aggregation operators were deﬁned in NCS. The intent of this paper is to generalize these aggregation operators by presenting neutrosophic cubic generalized uniﬁed aggregation (NCGUA) and neutrosophic cubic quasi-generalized uniﬁed aggregation (NCQGUA) operators. The accuracy and precision are a vital tool to minimize the potential threat in decision making. Generally, in decision making methods, alternatives and criteria are considered to evaluate the better outcome. However, sometimes the decision making environment has more components to express the problem completely. These components are named as the state of nature corresponding to each criterion. This complex frame of work is dealt with by presenting the multi-expert decision-making method (MEDMM).


Introduction
In real-life problems complex phenomena occur. One of the complex phenomena is to deal with the vagueness and uncertainty in data. Because uncertainty is inevitable in problems in different areas of life, conventional methods have failed to cope with such problems. The big task was to deal with uncertain information for many years. Many models have been introduced to incorporate uncertainty into the description of the system. Zadeh presented their theory of Fuzzy sets [1]. The possibilistic nature of fuzzy set theory attracted researchers to apply it in different fields of sciences like artificial intelligence, decision making theory, information sciences, medical sciences and more. Due to its applicability in sciences and daily life problems, fuzzy set has been extended to interval valued fuzzy sets (IVFS) [2,3], intuitionistic fuzzy sets (IFS) [4], interval valued intuitionistic fuzzy sets (IVIFS) [5], cubic sets [6], etc. Over the last decades, researchers used it for decision making problems [7][8][9][10][11][12].

Preliminaries
This section consists of some work that provides the foundation for our work. Definition 1. [13] A structure N = (T N (u), I N (u), F N (u)) u ∈ U is NS, where T N (u), I N (u), F N (u) ∈ [0 − , 1 + ] and T N (u), I N (u), F N (u) are truth, indeterminacy and falsehood respectively. Definition 2. [15] A structure N = (T N (u), I N (u), F N (u)) u ∈ U is SVNS, where T N (u), I N (u), F N (u) ∈ [0, 1] respectively called truth, indeterminancy and falsehood are simply denoted by N = (T N , I N , F N ).
The scalar multiplication on a NCS, [0,0], 0, 0, 1) be maximum NCS, then the cosine measure (C m ) is defined as For comparison of two NCS cosine measure is used.

The Neutrosophic Cubic Generalized Unified Aggregation Operator
The NCGUA operators are the generalization of many aggregation operators. The NCGUA operators unify several aggregations operators consequent upon their importance to analyze the imprecise data according to their importance. Moreover, it allows to use arithmetic, quadratic and geometric aggregation operators. By including a wide range of systems, it can adopt different scenario without losing any information.
where C j is the relevance that each sub-aggregation has in the system with C j ∈ [0, 1] and m j=1 C j = 1;w j i is the ith weight of the jth weighing vector W with w j i ∈ [0, 1], j = 1,λ h is the parameter such that λ j ∈ R and A i is the argument value of neutrosophic cubic value. Definition 10. The further generalization can be expressed as, . Usually λ j remain same but for complex type of aggregation different values can be assumed.
The operation used on NC is defined by [28]. NCGUA operators accomplish properties like monotonicity, boundness and idempotency.

Families of NCGUA Operators
The main aspect of NCGUA operator is that it characterizes a variety of aggregation operators. The aim of this section is to analyze these sub aggregations operators. First generalized NCOWA, NCWA and NCPOWA operators are analyzed.
where A i is the ith largest of A n , and w 1 , w 2 , w 3 represent the weights corresponding to the NCOWA, NCWA and NCPOWA operators, λ j = δ = 1. Note that in NCOWA operator, an additional order is made of (A 1 , A 2 , . . . , A n ), then it is weighted.
Some other family of aggregation operators can be analyzed by assigning values to λ j and δ, these values depend on the type of problem under discussion.
The averaging aggregation operators have a most practical operator among their competitors, but in some situations, other operators like geometric, quadratic, cubic operators are in a much better position to evaluate the values.

•
If λ = δ = 1 and for all j, the aggregation operator is deduced to NCUA.
• If λ → 1, δ = 1 and for all j, the aggregation operator is deduced to NCUG.
• If λ = 2, δ = 1 and for all j, the aggregation operator is deduced to NCUQA.
• If λ → 0, δ → 0 and for all j, the aggregation operator is deduced to NCGUG.
• If λ = 2, δ = 2 and for all j, the aggregation operator is deduced to NCQUQA.
The particular cases of NCUA operators can be analyzed by different values of λ and δ as shown in Table 1.
λ and δ have different values in the sub-aggregation operators. Thus, the aggregation operators have listed aggregation operators above. For further analysis, assume that aggregation operators follow the weighted averaging aggregation approach. Observe that a different scenario may be constructed by assigning different values to λ and δ. Today's world has much more complex phenomena; to deal with such situation, complex aggregation operators become a vital tool. Note that the complex and simple aggregation operators can be studied by assigning different values not only to λ and δ but to the weight as well.

Neutrosophic Cubic Quasi-Generalized Unified Aggregation Operators
The NCGUA operator can further be generalized by quasi-arithmetic means by neutrosophic cubic quasi-generalized unified aggregation (NCQGUA) operator. The characteristic of NCQGUA operators is that they are a generalization of not only NCGUA operators but of some other aggregation operators by the function introduced in NCQGUA.
Definition 11. The NCQGUA operator can be defined as where f j and g j are strictly continuous monotone functions.
The NCQGUA operator can be deduced to a wide range of aggregation operators. The Table 2 illustrates the range generated by NCQGUA operators. Table 2.
In Table 1 it is assumed that g 1 = g 2 = . . . = g m for all j.

The Application of NCQGUA and NCGUA Operators to Multi-Expert Decision-Making Method
The NCQGUA and NCGUA aggregation operators are generalized forms of most of the aggregation operators that can easily be deduced under some special conditions. This characteristic offers a great advantage when applying them to different MCDM. For this purpose, the multi-expert decision-making method MEDMM is proposed. This method is specially designed for complex situations where need of more than one criterion has a further classification, which is the state of nature. In the modern world, the areal study becomes a vital tool to set foreign policy or investments to a country or region. The choice of country or region to invest is a risk-taking job. The countries and multinational companies hire experts for these regions to come up with profitable decisions. For such situation MEDMM is developed under a NC environment that is a neutrosophic cubic multi-expert decision-making method (NCMEDMM).

Algorithm
This decision-making technique, which is specially designed for the problem, has different criteria, and each criterion has different classifications named as state of nature. The choice of alternatives in such a situation becomes different from the group decision making studied until now. Due to this phenomenon the following method is proposed.
The problem consist of n alternatives Construction of expert's criteria matrices for each criterion corresponding to the given alternatives and finite state of nature. Transformation of expert criteria matrices to general group expert's matrix by aggregation operator. Transformation of all the general group experts' matrices to a single matrix by aggregation operator. Ranking of alternatives.

Model Formulation
European Monetary Union EMU is an organization working for the benefit of the EU. The EMU formation has a lot of micro-and macro-finance decisions to build a productive state. All the areas like economics, finance, politics, marketing, management and EU laws are taken into account for making any decision.
Consider an illustrative example in multi-person decision making in the EMU [35][36][37][38]. This type of problem in macroeconomics usually deals with huge amounts of capital or other variables. This makes it critical to find the accurate decision; otherwise, a small deviation may cause huge economic difference in the region. The two different criteria are considered to analyze MPDMM. It is very common that a decision of EMU is usually influenced by several experts and criteria.
The model consists of the following data.

Alternative
A 1 : Increase the rates 1%. The company collected the data regarding these alternatives. The assumption is that this decision has potential states of nature benefit corresponding to the two criteria.
Cr1 : Internal economic condition S 1 : Negative growth. S 2 : Growth close to 0. S 3 : Positive growth. Cr2 : Global economic condition. S 1 : Negative growth. S 2 : Growth close to 0. S 3 : Positive growth. The EMU nominates individuals responsible for this decision which is divided into three groups; each group provides their opinion regarding the outcome and the possible strategy. The data of expert 1 subject to criterion 1 is shown in Table 3. The data of expert 1 subject to criterion 2 is shown in Table 4. Subject to this information, the first criterion is weighted as 0.70 and the second criterion as 0.30. The NCGUA operators are applied to form the general matrix, which represents the matrix of the group of experts illustrated in Table 5.
The data of expert 2 subject to criterion 1 is shown in Table 6. The data of expert 2 subject to criterion 2 is shown in Table 7. Subject to this information, the first criterion is weighted as 0.70 and the second criterion as 0.30. The NCGUA operators are applied to form the general matrix, which represents the matrix of the group of experts illustrated in Table 8.
The data of expert 3 subject to criterion 1 is shown in Table 9.  The data of expert 3 subject to criterion 2 is shown in Table 10. Subject to this information, the first criterion is weighted as 0.70 and the second criterion as 0.30. The NCGUA operators are applied to form the general matrix, which represents the matrix of the group of experts illustrated in Table 11. Table 11. Expert 3-General result.             (0.45, 0.35, 0.20), the following matrix is obtained illustrated in Table 12.
Now applying NCGUA operator subject to the weight (0.33, 0.33, 0.34) the following matrix is obtained illustrated in Table 13.

From this data the ranking is
For comparison purposes, some other aggregation operators are considered, and their results are computed.
The aggregation operators like neutrosophic cubic weighted geometric (NCWG), neutrosophic cubic probabilistic (NCP), neutrosophic cubic maximum (NCmax) and neutrosophic cubic minimum (NCmin) operators are also applied to the data. The following results are obtained.
The graphical comparison of these operators is shown in the following figure. The following table also illustrates the comparison of these aggregation operators under the MPDMM.
From the above table with the different alternatives of each decision maker, the alternative in closest accordance to his interests will be selected; this data indicates that the optimal decision is 5 A . Thus, the optimal alternative is 5 A .

Discussion
Jose et al. [40] analyze the data on fuzzy sets and conclude with the following results. Comparison of Tables 14 and 15. It is observed that in both tables, the optimal alternative is 5 , although individually, some aggregation operators may have different results. The reason may be that NC has more components compared to the FS, like indeterminate and falsehood. Overall comparison has the same result, which ensures the validity of this method. But the advantage of NCS over table FS is that NCS is a generalized version of FS. That is, it enables the expert to deal with inconsistent and indeterminate data more efficiently than the FS.  The following table also illustrates the comparison of these aggregation operators under the MPDMM.
From the above table with the different alternatives of each decision maker, the alternative in closest accordance to his interests will be selected; this data indicates that the optimal decision is A 5 . Thus, the optimal alternative is A 5 .

Discussion
Jose et al. [39] analyze the data on fuzzy sets and conclude with the following results. Comparison of Tables 14 and 15. It is observed that in both tables, the optimal alternative is A 5 , although individually, some aggregation operators may have different results. The reason may be that NC has more components compared to the FS, like indeterminate and falsehood. Overall comparison has the same result, which ensures the validity of this method. But the advantage of NCS over table FS is that NCS is a generalized version of FS. That is, it enables the expert to deal with inconsistent and indeterminate data more efficiently than the FS.

Operators
Ranking Table 15. Ranking using different operators in fuzzy data.

Conclusions
This paper generalized aggregation operators, such as NCGUA operators, which provide a single plate form for researchers and experts to deal with different types of aggregation operators. This generalization helps to work in a complex frame of work where more than one operator is required. The data involves inconsistent and indeterminate factors and is furnished upon a daily life problem. The NCGUA operator is further generalized to NCQGUA aggregation operators. Moreover, it can include complex aggregation operators that deal with complex environments such as problems with a wide range of sub-aggregations. It is shown that some particular aggregation operators like NCWA, NCOWA, NCGA, NCOGA, NCmax, NCmin and other aggregation operators can easily be deduced from NCQGUA or NCGUA operator. A new decision-making method is formed to deal with the problem in which each criterion is further classified into nature of stats. A numeric example involving the EMU is provided as an application. It is concluded that the proposed method provides the best results in comparison to the method previously proposed due to the indeterminate factors involved in data.