Neutrosophic Cubic Einstein Geometric Aggregation Operators with Application to Multi-Criteria Decision Making Method

Abstract

Neutrosophic cubic sets (NCs) are amore generalized version of neutrosophic sets(Ns) and interval neutrosophic sets (INs). Neutrosophic cubic setsare better placed to express consistent, indeterminate and inconsistent information, which provides a better platform to deal with incomplete, inconsistent and vague data. Aggregation operators play a key role in daily life, and in relation to science and engineering problems. In this paper we defined the algebraic and Einstein sum, multiplication and scalar multiplication, score and accuracy functions. Using these operations we defined geometric aggregation operators and Einstein geometric aggregation operators. First, we defined the algebraic and Einstein operators of addition, multiplication and scalar multiplication. We defined score and accuracy function to compare neutrosophic cubic values. Then we definedthe neutrosophic cubic weighted geometric operator (NCWG), neutrosophic cubic ordered weighted geometric operator (NCOWG), neutrosophic cubic Einstein weighted geometric operator (NCEWG), and neutrosophic cubic Einstein ordered weighted geometric operator (NCEOWG) over neutrosophic cubic sets. A multi-criteria decision making method is developed as an application to these operators. This method is then applied to a daily life problem.

1. Introduction

The theory of fuzzy sets was introduced by Zadeh [1].Soon after, it attracted experts of sciences and engineering due to its possibilistic behavior. The applicability of fuzzy sets extended it to interval valued fuzzy sets(IVFs) [2,3]. In 1986, K. Atnassov developed the theory of intuitionistic fuzzy sets [4], which were further extended to interval valued intuitionistic fuzzy sets in 1989 [5]. In 2012, Y.B. Jun generalized the idea of fuzzy sets and intuitionistic fuzzy sets to form cubic sets [6]. Smarandache presented his theory regarding the inconsistent and indeterminate behavior of data in 1999, and named it the neutrosophic set [7]. Neutrosophic sets consist of three components:Truth, indeterminate and falsehood, which provides a more general platform to deal with vague and insufficient data. In 2005, Wang et al. [8] presented the idea of interval valued neutrosophic sets. Interval valued neutrosophic sets provide a range to experts which makes them more comfortable with making the choice. Jun et al. defined the neutrosophic cubic set [9,10]. Neutrosophic cubic sets are a generalization of neutrosophic sets and interval neutrosophic sets. They enable us to choose both interval values and single value membership. This characteristic of neutrosophic cubic sets enables us to deal with uncertain and vague data more efficiently.

Decision making is one of the most important factors in scienceand day-to-day life as well. Aggregation operators are an imperative part of modern decision making. A lack of data or information makes it difficult for decision makers to take an appropriatedecision. This uncertain situation can be minimized using the vague nature neutrosophic cubic set and its extensions. Neutrosophic cubic set (NCs) are a more generalized version of neutrosophic sets (Ns) and interval neutrosophic sets (INs). Neutrosophic cubic setsare better placed to express consistent, indeterminate, and inconsistent information, which provides a better platform to deal with incomplete, inconsistent, and vague data. Aggregation operators have a key role in daily life, science and engineering problems. Zhan et al. [11] in their workapplications of neutrosophic cubic sets in multi-criteria decision making in 2017. Banerjee et al. [12] usedgrey rational analysis in their workGRA for multi attribute decision making in neutrosophic cubic set environment in 2017.Lu and Ye [13] definedcosine measure for neutrosophic cubic sets for multiple attribte decision making in 2017. Pramanik et al. [14] defined neutrosophic cubic MCGDM method based on similarity measurein 2017. Shi and Ye [15] defined Dombi aggregation operators of neutrosophic cubic set for multiple attribute deicision makingin 2018. Baolin et al. [16] applied Einstein aggregations onneutrosophic sets in a novel generalized simplified neutrosophic number Einstein aggregation operator 2018. Alot of work has been done and is being done by different researchers in decision making using neutrosophic cubic sets.

In this paper, we define algebraic and Einstein sum, multiplication and scalar multiplication, score and accuracy functions. Using these operations, we define geometric aggregation operators and Einstein geometric aggregation operators. First, we define algebraic and Einstein operators of addition, multiplication and scalar multiplication. We then define score and accuracy functions to compare neutrosophic cubic values. Following this, we propose a neutrosophic cubic ordered weighted geometric operator (NCOWG), neutrosophic cubic Einstein weighted geometric operator (NCEWG), and a neutrosophic cubic Einstein ordered weighted geometric operator (NCEOWG) over neutrosophic cubic sets. A multi-criteria decision making method is then developed as an application for these operators. This method is then applied to a daily life problem.

2. Preliminaries

This section consists of two parts: Notations, which consists of notations with their descriptions and some previous definitions; and results. We recommend the reader to see [1,2,3,6,7,8,9,16].

2.1. Notations

This section consists of some notations with their descriptions, as shown in Table 1.

2.2. Pre-Defined Definitions

This section consists of some predefined definitions and results.

Definition 1

[1].A mapping $\psi$:$U\to \left[0,1\right]$ is called a fuzzy set, and $\psi (u)$ is called a membership function, simply denoted by $\psi$.

Definition 2

[2,3].A mapping $\tilde{\Psi }:U\to D\left[0,1\right]$, where $D\left[0,1\right]$ is the interval value of $[0,1]$, called the interval valued fuzzy set(IVF). For all $u\in U$ $\tilde{\Psi }(u)=\left\{\left[{\psi }^L\left(u\right),{\psi }^U\left(u\right)\right]|{\psi }^L\left(u\right),{\psi }^U\left(u\right)\in [0,1]and{\psi }^L\left(u\right)\le {\psi }^U\left(u\right)\right\}$ is membership degree of $u$ in $\tilde{\Psi }$. This is simply denoted by $\tilde{\Psi }=\left[{\Psi }^L,{\Psi }^U\right]$.

Definition 3

[6].A structure$C=\left\{\left(u,\tilde{\Psi }\left(u\right),\Psi \left(u\right)\right)|u\in U\right\}$is a cubic set in$U$in which$\tilde{\Psi }\left(u\right)$is IVF in$U$, that is,$\tilde{\Psi }=\left[{\Psi }^L,{\Psi }^U\right]$and$\Psi$is a fuzzy set in$U$. This can be simply denoted by $C=\left(\tilde{\Psi },\Psi \right)$.$C^U$denotes the collection of cubic sets in $U$.

Definition 4

[7].A structure$N=\left\{\left(T_N(u),I_N(u),F_N(u)\right)|u\in U\right\}$is a neutrosophic set(Ns), where$\left\{T_N(u),I_N(u),F_N(u)\in [0,1]\right\}$are called truth, indeterminacy and falsity functions, respectively.This can be simply denoted by $N=\left(T_N,I_N,F_N\right)$.

Definition 5

[8].An interval neutrosophic set (INs) in$U$is a structure$N=\left\{\left({\tilde{T}}_N(u),{\tilde{I}}_N(u),{\tilde{F}}_N(u)\right)|u\in U\right\}$, where$\left\{{\tilde{T}}_N(u),{\tilde{I}}_N(u),{\tilde{F}}_N(u)\in D[0,1]\right\}$is calledtruth, indeterminacy an falsity functionin$U$, respectively. This can be simply denoted by$N=\left({\tilde{T}}_N,{\tilde{I}}_N,{\tilde{F}}_N\right)$. For convenience, we denote$N=\left({\tilde{T}}_N,{\tilde{I}}_N,{\tilde{F}}_N\right)$by $N=\left({\tilde{T}}_N=\left[T_N^L,T_N^U\right],{\tilde{I}}_N=\left[I_N^L,I_N^U\right],{\tilde{F}}_N=\left[F_N^L,F_N^U\right]\right)$.

Definition 6

[9].A structure N =$\left\{\left(u,{\tilde{T}}_N(u),{\tilde{I}}_N(u),{\tilde{F}}_N(u),T_N(u),I_N(u),F_N(u)\right)|u\in U\right\}$is neutrosophic cubic set(NCs) in$U$, in which$\left({\tilde{T}}_N=\left[T_N^L,T_N^U\right],{\tilde{I}}_N=\left[I_N^L,I_N^U\right],{\tilde{F}}_N=\left[F_N^L,F_N^U\right]\right)$is an interval neutrosophic set and$\left(T_N,I_N,F_N\right)$is a neutrosophic set in$U$. Simply denoted by $N=\left({\tilde{T}}_N,{\tilde{I}}_N,{\tilde{F}}_N,T_N,I_N,F_N\right)$,$[0,0]\le {\tilde{T}}_N+{\tilde{I}}_N+{\tilde{F}}_N\le [3,3]$and $0\le T_N+I_N+F_N\le 3$.$N^U$denotes the collection of neutrosophic cubic sets in$U$. Simply denoted by $N=\left({\tilde{T}}_N,{\tilde{I}}_N,{\tilde{F}}_N,T_N,I_N,F_N\right)$.

Definition 7

[16].The t-operators are basically union and intersection operators in the theory of fuzzy sets, which are denoted by t-conorm $\left({\Gamma }^{*}\right)$ and t-norm $(\Gamma )$, respectively. The role of t-operators is very important in fuzzy theory and its applications.

Definition 8

[16].${\Gamma }^{*}:\left[0,1\right]\times \left[0,1\right]\to \left[0,1\right]$ is called t-conorm if it satisfies the following axioms:

• Axiom 1.${\Gamma }^{*}\left(1,u\right)=1$ and ${\Gamma }^{*}\left(0,u\right)=0$;
• Axiom 2.${\Gamma }^{*}\left(u,v\right)={\Gamma }^{*}\left(v,u\right)$ for all a and b;
• Axiom 3.${\Gamma }^{*}\left(u,{\Gamma }^{*}\left(v,w\right)\right)={\Gamma }^{*}\left({\Gamma }^{*}\left(u,v\right),w\right)$ for all a, b and c;
• Axiom 4.If $u\le u^{\prime }$ and $v\le v^{\prime }$, then ${\Gamma }^{*}\left(u,v\right)\le {\Gamma }^{*}\left(u^{\prime },v^{\prime }\right)$.

Most known t-conorms are as follows:

• The default t-conorm: ${\Gamma }_{\max }^{*}\left(u,v\right)=\max \left(u,v\right)$.
• The bounded t-conorm: ${\Gamma }_{bounded}^{*}\left(u,v\right)=\min \left(1,u+v\right)$.
• The algebraic t-conorm: ${\Gamma }_{a\lg ebraic}^{*}\left(u,v\right)=u+v-uv$.

Definition 9

[16].$\Gamma :\left[0,1\right]\times \left[0,1\right]\to \left[0,1\right]$ is called t-norm if it satisfies the following axioms:

• Axiom 5.$\Gamma \left(1,u\right)=u$ and $\Gamma \left(0,u\right)=0$;
• Axiom 6.$\Gamma \left(u,v\right)=\Gamma \left(v,u\right)$ for all a and b;
• Axiom 7.$\Gamma \left(u,\Gamma \left(v,w\right)\right)=\Gamma \left(\Gamma \left(u,v\right),w\right)$ for all a, b and c;
• Axiom 8.If $u\le u^{\prime }$ and $v\le v^{\prime },$ then $\Gamma \left(u,v\right)\le \Gamma \left(u^{\prime },v^{\prime }\right)$.

Most well known t-norms are as follows:

• The default t-norm: ${\Gamma }_{\min }\left(u,v\right)=\min \left(u,v\right)$.
• The bounded t-norm: ${\Gamma }_{bounded}\left(u,v\right)=\max \left(0,u+v-1\right)$.
• The algebraic t-norm: ${\Gamma }_{a\lg ebraic}\left(u,v\right)=uv$.

If ${\Gamma }^{*}\left(u,v\right)$, $\Gamma \left(u,v\right)$ are continuous and ${\Gamma }^{*}\left(u,u\right)>u$, $\Gamma \left(u,u\right)<u$, then ${\Gamma }^{*}$ and $\Gamma$ are said to be Archimedes t-conorm and t-norm, respectively. Any pair of dual t-conorm (${\Gamma }^{*}$) and t-norm ($\Gamma$) is used. It is known that t-norms and t-conorms operators satisfy the condition of conjunction and disjunction operators, respectively. However, the algebraic operations, like algebraic sum and product, are not unique and may correspond to union and intersection. The t-conorms and t-norms families have a vast range, which corresponds to unions and intersections. Among these, the Einstein sum and Einstein product are good choices since they give the smooth approximation like algebraic sum and algebraic product, respectively. Einstein sum ${\oplus }_E$ and Einstein product ${\otimes }_E$ are examples of t-conorm and t-norm, respectively:

$${\Gamma }_E^{*}(u,v)=\frac{u+v}{1+uv}$$

$${\Gamma }_E(u,v)=\frac{uv}{1+\left(1-u\right)\left(1-v\right)}$$

Group decision making is an important aspect of decision making theory. We are often in situationsin which we have to deal with more then one expert, attribute and alternative. Motivated by such situations, a multi-attribute decision making method for more then one expert is proposed on neutrosophic cubic aggregation operators.This whole work consisted of six sections. In Section 3, we define some algebraicEinstein operations and score and accuracy functions, along with some important results and examples. On the basis of these definitions and results, we define geometric and Einstein geometric aggregation operators on neutrosophic cubic sets in Section 4. In Section 5, an algorithm is proposed based on neutrosophic cubic geometric and Einstein geometric aggregation operators to deal with multi-attribute decision making problems. In the final section, a numerical example from daily life is presented as an application of the work.

3. Operations on Neutrosophic Cubic Sets

In this section, we introduce some new operations on neutrosophic cubic sets which are further used in the article.

3.1. Algebraic Addition, Multiplication and Scalar Multiplication

We introduce the algebraic addition, multiplication, and scalar multiplication on neutrosophic cubic sets(NCs). An important result of exponential multlipliction is established on the basis of these defintions, which provides the basis to define neutrosophic cubic geometric aggregation operators.

Definition 10.

The sum of two neutrosophic cubic sets(NCs),$A=\left({\tilde{T}}_A,{\tilde{I}}_A,{\tilde{F}}_A,T_A,I_A,F_A\right)$, where${\tilde{T}}_A=\left[T_A^L,T_A^U\right],{\tilde{I}}_A=\left[I_A^L,I_A^U\right],{\tilde{F}}_A=\left[F_A^L,F_A^U\right],andB=\left({\tilde{T}}_B,{\tilde{I}}_B,{\tilde{F}}_B,T_B,I_B,F_B\right)$, where ${\tilde{T}}_B=\left[T_B^L,T_B^U\right],{\tilde{I}}_B=\left[I_B^L,I_B^U\right],{\tilde{F}}_B=\left[F_B^L,F_B^U\right]$ is defined as

$$A\oplus B=\left(\begin{array}{c}\left[T_A^L+T_B^L-T_A^L{T}_B^L,T_A^U+T_B^U-T_A^U{T}_B^U\right],\\ \left[I_A^L+I_B^L-I_A^L{I}_B^L,I_A^U+I_B^U-I_A^U{I}_B^U\right],\\ \left[F_A^L{F}_B^L,F_A^U{F}_B^U\right],\\ T_A{T}_B,I_A{I}_B,F_A+F_B-F_A{F}_B\end{array}\right)$$

Definition 11.

The product between two neutrosophic cubic sets (NCs),$A=\left({\tilde{T}}_A,{\tilde{I}}_A,{\tilde{F}}_A,T_A,I_A,F_A\right)$, where${\tilde{T}}_A=\left[T_A^L,T_A^U\right],{\tilde{I}}_A=\left[I_A^L,I_A^U\right],{\tilde{F}}_A=\left[F_A^L,F_A^U\right]$and$B=\left({\tilde{T}}_B,{\tilde{I}}_B,{\tilde{F}}_B,T_B,I_B,F_B\right)$, where${\tilde{T}}_B=\left[T_B^L,T_B^U\right],{\tilde{I}}_B=\left[I_B^L,I_B^U\right],{\tilde{F}}_B=\left[F_B^L,F_B^U\right]$is defined as

$$A\otimes B=\left(\begin{array}{c}\left[T_A^L{T}_B^L,T_A^U{T}_B^U\right],\\ \left[I_A^L{I}_B^L,I_A^U{I}_B^U\right],\\ \left[F_A^L+F_B^L-F_A^L{F}_B^L,F_A^U+F_B^U-F_A^U{F}_B^U\right],\\ T_A+T_B-T_A{T}_B,I_A+I_B-I_A{I}_B,F_A{F}_B\end{array}\right)$$

Definition 12.

The scalar multiplication on a neutrosophic cubic set (NCs),$A=\left({\tilde{T}}_A,{\tilde{I}}_A,{\tilde{F}}_A,T_A,I_A,F_A\right)$, where${\tilde{T}}_A=\left[T_A^L,T_A^U\right],{\tilde{I}}_A=\left[I_A^L,I_A^U\right],{\tilde{F}}_A=\left[F_A^L,F_A^U\right]$, and a Scalar$k$ is defined as

$$kA=\left(\begin{array}{c}\left[1-{(1-T_A^L)}^k,1-{(1-T_A^U)}^k\right],\\ \left[1-{(1-I_A^L)}^k,1-{(1-I_A^U)}^k\right],\\ \left[{\left(F_A^L\right)}^k,{\left(F_A^U\right)}^k\right],\\ {\left(T_A\right)}^k,{\left(I_A\right)}^k,1-{\left(1-F_A\right)}^k\end{array}\right)$$

The following result is established to deal with the exponential multiplication on neutrosophic cubic values. This result enables us to define geometric aggregation operators along some important results on neutrosophic cubic sets.

Theorem 1.

Let$A=\left({\tilde{T}}_A,{\tilde{I}}_A,{\tilde{F}}_A,T_A,I_A,F_A\right)$, where${\tilde{T}}_A=\left[T_A^L,T_A^U\right],{\tilde{I}}_A=\left[I_A^L,I_A^U\right],{\tilde{F}}_A=\left[F_A^L,F_A^U\right],$be a neutrosophic cubic value, then the exponential operation can be defined by

$$A^k=\left(\begin{array}{c}\left[{(T_A^L)}^k,(T_A^U{)}^k\right],\\ \left[{(I_A^L)}^k,(I_A^U{)}^k\right],\\ \left[1-{\left(1-F_A^L\right)}^k,1-{\left(1-F_A^U\right)}^k\right],\\ 1-{\left(1-T_A\right)}^k,1-{\left(1-I_A\right)}^k,{\left(F_A\right)}^k\end{array}\right)$$

where$A^k=A\otimes A\otimes ,\dots \otimes A(k-times)$, and$A^k$is a neutrosophic cubic value for every positive value of k.

Proof. 

We prove the theorem by mathematical induction, as the $k=1$, $A^1=A$ result holds.We assume that for $k=m$ the result is true:

$$A^m=\left(\begin{array}{c}\left[{(T_A^L)}^m,(T_A^U{)}^m\right],\\ \left[{(I_A^L)}^m,(I_A^U{)}^m\right],\\ \left[1-{\left(1-F_A^L\right)}^m,1-{\left(1-F_A^U\right)}^m\right],\\ 1-{\left(1-T_A\right)}^m,1-{\left(1-I_A\right)}^m,{\left(F_A\right)}^m\end{array}\right)$$

That is $A^m$ is neutrosophic cubic value. We prove that for $k=m+1$ is also neutrosophic cubic value.

Since

$$\begin{array}{c}{A}^m\otimes A=\left(\begin{array}{c}\left[{(T_A^L)}^m,(T_A^U{)}^m\right],\\ \left[{(I_A^L)}^m,(I_A^U{)}^m\right],\\ \left[1-{\left(1-F_A^L\right)}^m,1-{\left(1-F_A^U\right)}^m\right],\\ 1-{\left(1-T_A\right)}^m,1-{\left(1-I_A\right)}^m,{\left(F_A\right)}^m\end{array}\right)\otimes \left(\begin{array}{c}\left[(T_A^L),(T_A^U)\right],\\ \left[(I_A^L),(I_A^U)\right],\\ \left[F_A^L,F_A^U\right],\\ T_A,I_A,F_A\end{array}\right)\\ =\left(\begin{array}{c}\left[{(T_A^L)}^{m+1},(T_A^U{)}^{m+1}\right],\\ \left[{(I_A^L)}^{m+1},(I_A^U{)}^{m+1}\right],\\ \left[1-{\left(1-F_A^L\right)}^m+F_A^L-\left(1-{\left(1-F_A^L\right)}^m\right)F_A^L,1-{\left(1-F_A^U\right)}^m+F_A^U-\left(1-{\left(1-F_A^U\right)}^m\right)F_A^U\right],\\ 1-{\left(1-T_A\right)}^m+T_A-\left(1-{\left(1-T_A\right)}^m\right)T_A,1-{\left(1-I_A\right)}^m+I_A-\left(1-{\left(1-I_A\right)}^m{I}_A\right),{\left(F_A\right)}^{m+1}\end{array}\right)\\ =\left(\begin{array}{c}\left[{(T_A^L)}^{m+1},(T_A^U{)}^{m+1}\right],\\ \left[{(I_A^L)}^{m+1},(I_A^U{)}^{m+1}\right],\\ \left[1-{\left(1-F_A^L\right)}^m+F_A^L-F_A^L+{\left(1-F_A^L\right)}^m{F}_A^L,1-{\left(1-F_A^U\right)}^m+F_A^U-F_A^U+{\left(1-F_A^U\right)}^m{F}_A^U\right],\\ 1-{\left(1-T_A\right)}^m+T_A-T_A+{\left(1-T_A\right)}^m{T}_A,1-{\left(1-I_A\right)}^m+I_A-I_A+{\left(1-I_A\right)}^m{I}_A,{\left(F_A\right)}^{m+1}\end{array}\right)\\ =\left(\begin{array}{c}\left[{(T_A^L)}^{m+1},(T_A^U{)}^{m+1}\right],\\ \left[{(I_A^L)}^{m+1},(I_A^U{)}^{m+1}\right],\\ \left[1-{\left(1-F_A^L\right)}^m+{\left(1-F_A^L\right)}^m{F}_A^L,1-{\left(1-F_A^U\right)}^m+{\left(1-F_A^U\right)}^m{F}_A^U\right],\\ 1-{\left(1-T_A\right)}^m+{\left(1-T_A\right)}^m{T}_A,1-{\left(1-I_A\right)}^m+{\left(1-I_A\right)}^m{I}_A,{\left(F_A\right)}^{m+1}\end{array}\right)\\ =\left(\begin{array}{c}\left[{(T_A^L)}^{m+1},(T_A^U{)}^{m+1}\right],\\ \left[{(I_A^L)}^{m+1},(I_A^U{)}^{m+1}\right],\\ \left[1-{\left(1-F_A^L\right)}^m\left(1-F_A^L\right),1-{\left(1-F_A^U\right)}^m\left(1-F_A^U\right)\right],\\ 1-{\left(1-T_A\right)}^m\left(1-T_A\right),1-{\left(1-I_A\right)}^m\left(1-I_A\right),{\left(F_A\right)}^{m+1}\end{array}\right)\\ =\left(\begin{array}{c}\left[{(T_A^L)}^{m+1},(T_A^U{)}^{m+1}\right],\\ \left[{(I_A^L)}^{m+1},(I_A^U{)}^{m+1}\right],\\ \left[1-{\left(1-F_A^L\right)}^{m+1},1-{\left(1-F_A^U\right)}^{m+1}\right],\\ 1-{\left(1-T_A\right)}^{m+1},1-{\left(1-I_A\right)}^{m+1},{\left(F_A\right)}^{m+1}\end{array}\right)\\ =A^{m+1}.\end{array}$$

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3.2. Einstein Addition, Multiplication and Scalar Multiplication

Taking into account the dual t-conorm (${\Gamma }^{*}$) and t-norm ($\Gamma$), the Einstein operations of union, intersection, addition, multiplication and scalar multiplication are defined on the neutrosophic cubic sets.An important result of Einstein exponential multlipliction is established on the basis of these defintions, which provides the base with which to define neutrosophic cubic Einstein geometric aggregation operators.

Definition 13.

The Einstein union between two neutrosophic cubic sets (NCs),$A=\left({\tilde{T}}_A,{\tilde{I}}_A,{\tilde{F}}_A,T_A,I_A,F_A\right)$where${\tilde{T}}_A=\left[T_A^L,T_A^U\right],{\tilde{I}}_A=\left[I_A^L,I_A^U\right],{\tilde{F}}_A=\left[F_A^L,F_A^U\right]$, and$B=\left({\tilde{T}}_B,{\tilde{I}}_B,{\tilde{F}}_B,T_B,I_B,F_B\right)$where ${\tilde{T}}_B=\left[T_B^L,T_B^U\right],{\tilde{I}}_B=\left[I_B^L,I_B^U\right],{\tilde{F}}_B=\left[F_B^L,F_B^U\right]$ is defined as

$$A\vee B=\left(\Gamma \left\{{\tilde{T}}_A,{\tilde{T}}_B\right\},\Gamma \left\{{\tilde{I}}_A,{\tilde{I}}_B\right\},{\Gamma }^{*}\left\{{\tilde{F}}_A,{\tilde{F}}_B\right\},{\Gamma }^{*}\left\{T_A,T_B\right\},{\Gamma }^{*}\left\{I_A,I_B\right\},\Gamma \left\{F_A,F_B\right\}\right)$$

Definition 14.

The Einstein intersection between two neutrosophic cubic sets(NCS),$A=\left({\tilde{T}}_A,{\tilde{I}}_A,{\tilde{F}}_A,T_A,I_A,F_A\right)$, where${\tilde{T}}_A=\left[T_A^L,T_A^U\right],{\tilde{I}}_A=\left[I_A^L,I_A^U\right],{\tilde{F}}_A=\left[F_A^L,F_A^U\right]$and$B=\left({\tilde{T}}_B,{\tilde{I}}_B,{\tilde{F}}_B,T_B,I_B,F_B\right),$where ${\tilde{T}}_B=\left[T_B^L,T_B^U\right],{\tilde{I}}_B=\left[I_B^L,I_B^U\right],{\tilde{F}}_B=\left[F_B^L,F_B^U\right]$ is defined as

$$A\wedge B=\left({\Gamma }^{*}\left\{{\tilde{T}}_A,{\tilde{T}}_B\right\},{\Gamma }^{*}\left\{{\tilde{I}}_A,{\tilde{I}}_B\right\},\Gamma \left\{{\tilde{F}}_A,{\tilde{F}}_B\right\},\Gamma \left\{T_A,T_B\right\},\Gamma \left\{I_A,I_B\right\},{\Gamma }^{*}\left\{F_A,F_B\right\}\right).$$

On the basis of Einstein union and intersection the Einstein sum and product is defined over neutrosophic cubic values.

Definition 15.

The Einstein sum between two neutrosophic cubic sets (NCS),$A=\left({\tilde{T}}_A,{\tilde{I}}_A,{\tilde{F}}_A,T_A,I_A,F_A\right)$, where${\tilde{T}}_A=\left[T_A^L,T_A^U\right],{\tilde{I}}_A=\left[I_A^L,I_A^U\right],{\tilde{F}}_A=\left[F_A^L,F_A^U\right]$and$B=\left({\tilde{T}}_B,{\tilde{I}}_B,{\tilde{F}}_B,T_B,I_B,F_B\right),$where ${\tilde{T}}_B=\left[T_B^L,T_B^U\right],{\tilde{I}}_B=\left[I_B^L,I_B^U\right],{\tilde{F}}_B=\left[F_B^L,F_B^U\right]$ is defined as

$$A{\oplus }_{E}B=\left(\begin{array}{c}\left[\frac{T_A^L+T_B^L}{1+T_A^L{T}_B^L},\frac{T_A^U+T_B^U}{1+T_A^U{T}_B^U}\right],\\ \left[\frac{I_A^L+I_B^L}{1+I_A^L{I}_B^L},\frac{I_A^U+I_B^U}{1+I_A^U{I}_B^U}\right],\\ \left[\frac{F_A^L{F}_B^L}{1+(1-F_A^L)\left(1-F_B^L\right)},\frac{F_A^U{F}_B^U}{1+(1-F_A^U)\left(1-F_B^U\right)}\right]\\ \frac{T_A{T}_B}{1+(1-T_A)\left(1-T_B\right)},\frac{I_A{I}_B}{1+(1-I_A)\left(1-I_B\right)},\frac{F_A+F_B}{1+F_A{F}_B}\end{array}\right)$$

Definition 16.

The Einstein product between two neutrosophic cubic sets (NCS),$A=\left({\tilde{T}}_A,{\tilde{I}}_A,{\tilde{F}}_A,T_A,I_A,F_A\right)$, where${\tilde{T}}_A=\left[T_A^L,T_A^U\right],{\tilde{I}}_A=\left[I_A^L,I_A^U\right],{\tilde{F}}_A=\left[F_A^L,F_A^U\right]$and$B=\left({\tilde{T}}_B,{\tilde{I}}_B,{\tilde{F}}_B,T_B,I_B,F_B\right),$where ${\tilde{T}}_B=\left[T_B^L,T_B^U\right],{\tilde{I}}_B=\left[I_B^L,I_B^U\right],{\tilde{F}}_B=\left[F_B^L,F_B^U\right]$ is defined as

$$A{\otimes }_{E}B=\left(\begin{array}{c}\left[\frac{T_A^L{T}_B^L}{1+(1-T_A^L)\left(1-T_B^L\right)},\frac{T_A^U{T}_B^U}{1+(1-T_A^U)\left(1-T_B^U\right)}\right],\\ \left[\frac{I_A^L{I}_B^L}{1+(1-I_A^L)\left(1-I_B^L\right)},\frac{I_A^U{I}_B^U}{1+(1-I_A^U)\left(1-I_B^U\right)}\right],\\ \left[\frac{F_A^L+F_B^L}{1+F_A^L{F}_B^L},\frac{F_A^U+F_B^U}{1+F_A^U{F}_B^U}\right]\\ \frac{T_A+T_B}{1+T_A{T}_B},\frac{I_A+I_B}{1+I_A{I}_B},\frac{F_A{F}_B}{1+(1-F_A)\left(1-F_B\right)}\end{array}\right)$$

Definition 17.

The scalar multiplication on a neutrosophic cubic set(NCS),$A=\left({\tilde{T}}_A,{\tilde{I}}_A,{\tilde{F}}_A,T_A,I_A,F_A\right)$, where${\tilde{T}}_A=\left[T_A^L,T_A^U\right],{\tilde{I}}_A=\left[I_A^L,I_A^U\right],{\tilde{F}}_A=\left[F_A^L,F_A^U\right],$and scalar $k$ is defined as

$$k_{E}A=\left(\begin{array}{c}\left[\frac{{(1+T_A^L)}^k-{(1-T_A^L)}^k}{{(1+T_A^L)}^k+{(1-T_A^L)}^k},\frac{{(1+T_A^U)}^k-{(1-T_A^U)}^k}{{(1+T_A^U)}^k+{(1-T_A^U)}^k}\right],\\ \left[\frac{{(1+I_A^L)}^k-{(1-I_A^L)}^k}{{(1+I_A^L)}^k+{(1-I_A^L)}^k},\frac{{(1+I_A^U)}^k-{(1-I_A^U)}^k}{{(1+I_A^U)}^k+{(1-I_A^U)}^k}\right],\\ \left[\frac{2{\left(F_A^L\right)}^k}{{\left(2-F_A^L\right)}^k+{\left(F_A^L\right)}^k},\frac{2{\left(F_A^U\right)}^k}{{\left(2-F_A^U\right)}^k+{\left(F_A^U\right)}^k}\right],\\ \frac{2{\left(T_A\right)}^k}{{\left(2-T_A\right)}^k+{\left(T_A\right)}^k},\frac{2{\left(I_A\right)}^k}{{\left(2-I_A\right)}^k+{\left(I_A\right)}^k},\frac{{(1+F_A)}^k-{(1-F_A)}^k}{{(1+F_A)}^k+{(1-F_A)}^k}\end{array}\right)$$

After defining the scalar multiplication over the neutrosophic cubic set, we established the following result, which deals with the Einstein exponential multiplication on neutrosophic cubic values. This result enabled us to define Einstein geometric aggregation operators along with some important results on neutrosophic cubic sets.

Theorem 2.

Let$A=\left({\tilde{T}}_A,{\tilde{I}}_A,{\tilde{F}}_A,T_A,I_A,F_A\right)$, where${\tilde{T}}_A=\left[T_A^L,T_A^U\right],{\tilde{I}}_A=\left[I_A^L,I_A^U\right],{\tilde{F}}_A=\left[F_A^L,F_A^U\right]$, be a neutrosophic cubic value, then the exponential operation defined by

$$A^{E^k}=\left(\begin{array}{c}\left[\frac{2(T_A^L{)}^k}{{2(-T_A^L)}^k+{(T_A^L)}^k},\frac{2(T_A^U{)}^k}{{(2-T_A^U)}^k+{(T_A^U)}^k}\right],\\ \left[\frac{2(I_A^L{)}^k}{{(2-I_A^L)}^k+{(I_A^L)}^k},\frac{2(I_A^U{)}^k}{{(2-I_A^U)}^k+{(I_A^U)}^k}\right],\\ \left[\frac{{(1+F_A^L)}^k-{(1-F_A^L)}^k}{{(1+F_A^L)}^k+{(1-F_A^L)}^k},\frac{{(1+F_A^U)}^k-{(1-F_A^U)}^k}{{(1+F_A^U)}^k+{(1-F_A^U)}^k}\right],\\ \frac{{(1+T_A)}^k-{(1-T_A)}^k}{{(1+T_A)}^k+{(1-T_A)}^k},\frac{{(1+I_A)}^k-{(1-I_A)}^k}{{(1+I_A)}^k+{(1-I_A)}^k},\frac{2{\left(F_A\right)}^k}{{\left(2-F_A\right)}^k+{\left(F_A\right)}^k}\end{array}\right)$$

where$A^{E^k}=A{\otimes }_{E}A{\otimes }_E\dots {\otimes }_{E}A(k-times)$, moreover$A^{E^k}$is a neutrosophic cubic value for every positive value of k.

Proof. 

We prove the theorem by mathematical induction. For $k=1$

$$A^E=\left(\begin{array}{c}\left[\frac{2(T_A^L)}{(2-T_A^L)+(T_A^L)},\frac{2(T_A^U)}{(2-T_A^U)+(T_A^U)}\right],\\ \left[\frac{2(I_A^L)}{(2-I_A^L)+(I_A^L)},\frac{2(I_A^U)}{(2-I_A^U)+(I_A^U)}\right],\\ \left[\frac{(1+F_A^L)-(1-F_A^L)}{(1+F_A^L)+(1-F_A^L)},\frac{(1+F_A^U)-(1-F_A^U)}{(1+F_A^U)+(1-F_A^U)}\right],\\ \frac{(1+T_A)-(1-T_A)}{(1+T_A)+(1-T_A)},\frac{(1+I_A)-(1-I_A)}{(1+I_A)+(1-I_A)},\frac{2\left(F_A\right)}{\left(2-F_A\right)+\left(F_A\right)}\end{array}\right)$$

We observe that the components $T_A^L,T_A^U,I_A^L,I_A^U,F_A$ are of the form $\frac{2x}{(2-x)+x}$, and $F_A^L,F_A^U,T_A,I_A$ are of the form $\frac{(1+y)-(1-y)}{(1+y)+(1-y)}$,

For all $x,y\in [0,1]$, clearly $x=\frac{2x}{(2-x)+x}$ and $y=\frac{(1+y)-(1-y)}{(1+y)+(1-y)}$

Hence $A^E$ is neutrosophic cubic value.

Assuming $k=m$ is a neutrosophic cubic value i.e.,

$$A^{E^m}=\left(\begin{array}{c}\left[\frac{2(T_A^L{)}^m}{{(2-T_A^L)}^m+{(T_A^L)}^m},\frac{2(T_A^U{)}^m}{{(2-T_A^U)}^m+{(T_A^U)}^m}\right],\\ \left[\frac{2(I_A^L{)}^m}{{(2-I_A^L)}^m+{(I_A^L)}^m},\frac{2(I_A^U{)}^m}{{(2-I_A^U)}^m+{(I_A^U)}^m}\right],\\ \left[\frac{{(1+F_A^L)}^m-{(1-F_A^L)}^m}{{(1+F_A^L)}^m+{(1-F_A^L)}^m},\frac{{(1+F_A^U)}^m-{(1-F_A^U)}^m}{{(1+F_A^U)}^m+{(1-F_A^U)}^m}\right],\\ \frac{{(1+T_A)}^m-{(1-T_A)}^m}{{(1+T_A)}^m+{(1-T_A)}^m},\frac{{(1+I_A)}^m-{(1-I_A)}^m}{{(1+I_A)}^m+{(1-I_A)}^m},\frac{2{\left(F_A\right)}^m}{{\left(2-F_A\right)}^m+{\left(F_A\right)}^m}\end{array}\right)$$

is a neutrosophic cubic value. Then we prove $A^{E^{k+1}}$ is neutrosophic cubic value.

$$\begin{array}{c}\mathrm{Consider},\\ A^{E^m}{\otimes }_E{A}^E=\left(\begin{array}{c}\left[\frac{2(T_A^L{)}^m}{{(2-T_A^L)}^m+{(T_A^L)}^m},\frac{2(T_A^U{)}^m}{{(2-T_A^U)}^m+{(T_A^U)}^m}\right],\\ \left[\frac{2(I_A^L{)}^m}{{(2-I_A^L)}^m+{(I_A^L)}^m},\frac{2(I_A^U{)}^m}{{(2-I_A^U)}^m+{(I_A^U)}^m}\right],\\ \left[\frac{{(1+F_A^L)}^m-{(1-F_A^L)}^m}{{(1+F_A^L)}^m+{(1-F_A^L)}^m},\frac{{(1+F_A^U)}^m-{(1-F_A^U)}^m}{{(1+F_A^U)}^m+{(1-F_A^U)}^m}\right],\\ \frac{{(1+T_A)}^m-{(1-T_A)}^m}{{(1+T_A)}^m+{(1-T_A)}^m},\frac{{(1+I_A)}^m-{(1-I_A)}^m}{{(1+I_A)}^m+{(1-I_A)}^m},\frac{2{\left(F_A\right)}^m}{{\left(2-F_A\right)}^m+{\left(F_A\right)}^m}\end{array}\right){\otimes }_E\left(\begin{array}{c}\left[\frac{2(T_A^L)}{(2-T_A^L)+(T_A^L)},\frac{2(T_A^U)}{(2-T_A^U)+(T_A^U)}\right],\\ \left[\frac{2(I_A^L{)}^1}{(2-I_A^L)+(I_A^L)},\frac{2(I_A^U{)}^1}{(2-I_A^U)+(I_A^U)}\right],\\ \left[\frac{(1+F_A^L)-(1-F_A^L)}{(1+F_A^L)+(1-F_A^L)},\frac{(1+F_A^U)-(1-F_A^U)}{(1+F_A^U)+(1-F_A^U)}\right],\\ \frac{(1+T_A)-(1-T_A)}{(1+T_A)+(1-T_A)},\frac{(1+I_A)-(1-I_A)}{(1+I_A)+(1-I_A)},\frac{2\left(F_A\right)}{\left(2-F_A\right)+\left(F_A\right)}\end{array}\right)\end{array}$$

$$=\left(\begin{array}{c}\left[\frac{\frac{4{\left(T_A^L\right)}^{m+1}}{\left({(2-T_A^L)}^m+{\left(T_A^L\right)}^m\right)\left((2-T_A^L)+T_A^L\right)}}{1+\left(1-\frac{2{\left(T_A^L\right)}^m}{{(2-T_A^L)}^m+{\left(T_A^L\right)}^m}\right)\left(1-\frac{2T_A^L}{(2-T_A^L)+T_A^L}\right)},\frac{\frac{4{\left(T_A^U\right)}^{m+1}}{\left({(2-T_A^U)}^m+{\left(T_A^U\right)}^m\right)\left((2-T_A^U)+T_A^U\right)}}{1+\left(1-\frac{2{\left(T_A^U\right)}^m}{{(2-T_A^U)}^m+{\left(T_A^U\right)}^m}\right)\left(1-\frac{2T_A^U}{(2-T_A^U)+T_A^U}\right)}\right],\\ \left[\frac{\frac{4{\left(I_A^L\right)}^{m+1}}{\left({(2-I_A^L)}^m+{\left(I_A^L\right)}^m\right)\left((2-I_A^L)+I_A^L\right)}}{1+\left(1-\frac{2{\left(I_A^L\right)}^m}{{(2-I_A^L)}^m+{\left(I_A^L\right)}^m}\right)\left(1-\frac{2I_A^L}{(2-I_A^L)+I_A^L}\right)},\frac{\frac{4{\left(I_A^U\right)}^{m+1}}{\left({(2-I_A^U)}^m+{\left(I_A^U\right)}^m\right)\left((2-I_A^U)+I_A^U\right)}}{1+\left(1-\frac{2{\left(I_A^U\right)}^m}{{(2-I_A^U)}^m+{\left(I_A^U\right)}^m}\right)\left(1-\frac{2I_A^U}{(2-I_A^U)+I_A^U}\right)}\right],\\ \left[\frac{\left(\frac{{(1+F_A^L)}^m-{(1-F_A^L)}^m}{{(1+F_A^L)}^m+{(1-F_A^L)}^m}\right)+\left(\frac{(1+F_A^L)-(1-F_A^L)}{(1+F_A^L)+(1-F_A^L)}\right)}{1+\left(\frac{{(1+F_A^L)}^m-{(1-F_A^L)}^m}{{(1+F_A^L)}^m+{(1-F_A^L)}^m}\right)\left(\frac{(1+F_A^L)-(1-F_A^L)}{(1+F_A^L)+(1-F_A^L)}\right)},\frac{\left(\frac{{(1+F_A^U)}^m-{(1-F_A^U)}^m}{{(1+F_A^U)}^m+{(1-F_A^U)}^m}\right)+\left(\frac{(1+F_A^U)-(1-F_A^U)}{(1+F_A^U)+(1-F_A^U)}\right)}{1+\left(\frac{{(1+F_A^U)}^m-{(1-F_A^U)}^m}{{(1+F_A^U)}^m+{(1-F_A^U)}^m}\right)\left(\frac{(1+F_A^U)-(1-F_A^U)}{(1+F_A^U)+(1-F_A^U)}\right)}\right],\\ \frac{\left(\frac{{(1+T_A)}^m-{(1-T_A)}^m}{{(1+T_A)}^m+{(1-T_A)}^m}\right)+\left(\frac{(1+T_A)-(1-T_A)}{(1+T_A)+(1-T_A)}\right)}{1+\left(\frac{{(1+T_A)}^m-{(1-T_A)}^m}{{(1+T_A)}^m+{(1-T_A)}^m}\right)\left(\frac{(1+T_A)-(1-T_A)}{(1+T_A)+(1-T_A)}\right)},\frac{\left(\frac{{(1+I_A)}^m-{(1-I_A)}^m}{{(1+I_A)}^m+{(1-I_A)}^m}\right)+\left(\frac{(1+I_A)-(1-I_A)}{(1+I_A)+(1-I_A)}\right)}{1+\left(\frac{{(1+I_A)}^m-{(1-I_A)}^m}{{(1+I_A)}^m+{(1-I_A)}^m}\right)\left(\frac{(1+I_A)-(1-I_A)}{(1+I_A)+(1-I_A)}\right)},\\ \frac{\frac{4{\left(F_A\right)}^{m+1}}{\left({(2-F_A)}^m+{\left(F_A\right)}^m\right)\left((2-F_A)+F_A\right)}}{1+\left(1-\frac{2{\left(F_A\right)}^m}{{(2-F_A)}^m+{\left(F_A\right)}^m}\right)\left(1-\frac{2F_A}{(2-F_A)+F_A}\right)}\end{array}\right)$$

$$=\left(\begin{array}{c}\left[\frac{\frac{4{\left(T_A^L\right)}^{m+1}}{\left({(2-T_A^L)}^m+{\left(T_A^L\right)}^m\right)\left((2-T_A^L)+T_A^L\right)}}{1+\left(\frac{{(2-T_A^L)}^m+{\left(T_A^L\right)}^m-2{\left(T_A^L\right)}^m}{{(2-T_A^L)}^m+{\left(T_A^L\right)}^m}\right)\left(\frac{(2-T_A^L)+T_A^L-2T_A^L}{(2-T_A^L)+T_A^L}\right)},\frac{\frac{4{\left(T_A^U\right)}^{m+1}}{\left({(2-T_A^U)}^m+{\left(T_A^U\right)}^m\right)\left((2-T_A^U)+T_A^U\right)}}{1+\left(\frac{{(2-T_A^U)}^m+{\left(T_A^U\right)}^m-2{\left(T_A^U\right)}^m}{{(2-T_A^U)}^m+{\left(T_A^U\right)}^m}\right)\left(\frac{(2-T_A^U)+T_A^U-2T_A^U}{(2-T_A^U)+T_A^U}\right)}\right],\\ \left[\frac{\frac{4{\left(I_A^L\right)}^{m+1}}{\left({(2-I_A^L)}^m+{\left(I_A^L\right)}^m\right)\left((2-I_A^L)+I_A^L\right)}}{1+\left(\frac{{(2-I_A^L)}^m+{\left(I_A^L\right)}^m-2{\left(I_A^L\right)}^m}{{(2-I_A^L)}^m+{\left(I_A^L\right)}^m}\right)\left(\frac{(2-I_A^L)+I_A^L-2I_A^L}{(2-I_A^L)+I_A^L}\right)},\frac{\frac{4{\left(I_A^U\right)}^{m+1}}{\left({(2-I_A^U)}^m+{\left(I_A^U\right)}^m\right)\left((2-I_A^U)+I_A^U\right)}}{1+\left(\frac{{(2-I_A^U)}^m+{\left(I_A^U\right)}^m-2{\left(I_A^U\right)}^m}{{(2-I_A^U)}^m+{\left(I_A^U\right)}^m}\right)\left(\frac{(2-I_A^U)+I_A^U-2I_A^U}{(2-I_A^U)+I_A^U}\right)}\right],\\ \left[\begin{array}{c}\frac{\frac{\left({\left(1+F_A^L\right)}^m-{\left(1-F_A^L\right)}^m\right)\left(\left(1+F_A^L\right)+\left(1-F_A^L\right)\right)+\left({\left(1+F_A^L\right)}^m+{\left(1-F_A^L\right)}^m\right)\left((1+F_A^L)-(1-F_A^L)\right)}{\left({(1+F_A^L)}^m+{(1-F_A^L)}^m\right)\left((1+F_A^L)+(1-F_A^L)\right)}}{\frac{\left({\left(1+F_A^L\right)}^m+{\left(1-F_A^L\right)}^m\right)\left(\left(1+F_A^L\right)+\left(1-F_A^L\right)\right)+{\left(1+F_A^L\right)}^{m+1}-{\left(1+F_A^L\right)}^m\left(1-F_A^L\right)-{\left(1-F_A^L\right)}^m\left(1+F_A^L\right)+{\left(1-F_A^L\right)}^{m+1}}{\left({(1+F_A^L)}^m+{(1-F_A^L)}^m\right)\left((1+F_A^L)+(1-F_A^L)\right)}},\\ \frac{\frac{\left({\left(1+F_A^U\right)}^m-{\left(1-F_A^U\right)}^m\right)\left(\left(1+F_A^U\right)+\left(1-F_A^U\right)\right)+\left({\left(1+F_A^U\right)}^m+{\left(1-F_A^U\right)}^m\right)\left((1+F_A^U)-(1-F_A^U)\right)}{\left({(1+F_A^U)}^m+{(1-F_A^U)}^m\right)\left((1+F_A^U)+(1-F_A^U)\right)}}{\frac{\left({\left(1+F_A^U\right)}^m+{\left(1-F_A^U\right)}^m\right)\left(\left(1+F_A^U\right)+\left(1-F_A^U\right)\right)+{\left(1+F_A^U\right)}^{m+1}-{\left(1+F_A^U\right)}^m\left(1-F_A^U\right)-{\left(1-F_A^U\right)}^m\left(1+F_A^U\right)+{\left(1-F_A^U\right)}^{m+1}}{\left({(1+F_A^U)}^m+{(1-F_A^U)}^m\right)\left((1+F_A^U)+(1-F_A^U)\right)}}\end{array}\right],\\ \frac{\frac{\left({\left(1+T_A\right)}^m-{\left(1-T_A\right)}^m\right)\left(\left(1+T_A\right)+\left(1-T_A\right)\right)+\left({\left(1+T_A\right)}^m+{\left(1-T_A\right)}^m\right)\left((1+T_A)-(1-T_A)\right)}{\left({(1+T_A)}^m+{(1-T_A)}^m\right)\left((1+T_A)+(1-T_A)\right)}}{\frac{\left({\left(1+T_A\right)}^m+{\left(1-T_A\right)}^m\right)\left(\left(1+T_A\right)+\left(1-T_A\right)\right)+{\left(1+T_A\right)}^{m+1}-{\left(1+T_A\right)}^m\left(1-T_A\right)-{\left(1-T_A\right)}^m\left(1+T_A\right)+{\left(1-T_A\right)}^{m+1}}{\left({(1+T_A)}^m+{(1-T_A)}^m\right)\left((1+T_A)+(1-T_A)\right)}},\\ \frac{\frac{\left({\left(1+I_A\right)}^m-{\left(1-I_A\right)}^m\right)\left(\left(1+I_A\right)+\left(1-I_A\right)\right)+\left({\left(1+I_A\right)}^m+{\left(1-I_A\right)}^m\right)\left((1+I_A)-(1-I_A)\right)}{\left({(1+I_A)}^m+{(1-I_A)}^m\right)\left((1+I_A)+(1-I_A)\right)}}{\frac{\left({\left(1+I_A\right)}^m+{\left(1-I_A\right)}^m\right)\left(\left(1+I_A\right)+\left(1-I_A\right)\right)+{\left(1+I_A\right)}^{m+1}-{\left(1+I_A\right)}^m\left(1-I_A\right)-{\left(1-I_A\right)}^m\left(1+I_A\right)+{\left(1-I_A\right)}^{m+1}}{\left({(1+I_A)}^m+{(1-I_A)}^m\right)\left((1+I_A)+(1-I_A)\right)}},\\ \frac{\frac{4{\left(F_A\right)}^{m+1}}{\left({(2-F_A)}^m+{\left(F_A\right)}^m\right)\left((2-F_A)+F_A\right)}}{1+\left(\frac{{(2-F_A)}^m+{\left(F_A\right)}^m-2{\left(F_A\right)}^m}{{(2-F_A)}^m+{\left(F_A\right)}^m}\right)\left(\frac{(2-F_A)+F_A-2F_A}{(2-F_A)+F_A}\right)}\end{array}\right)$$

$$=\left(\begin{array}{c}\left[\begin{array}{c}\frac{\frac{4{\left(T_A^L\right)}^{m+1}}{\left({(2-T_A^L)}^m+{\left(T_A^L\right)}^m\right)\left((2-T_A^L)+T_A^L\right)}}{\frac{\left(\left({(2-T_A^L)}^m+{\left(T_A^L\right)}^m\right)\left((2-T_A^L)+T_A^L\right)\right)+\left(\left({(2-T_A^L)}^m-{\left(T_A^L\right)}^m\right)\left((2-T_A^L)-T_A^L\right)\right)}{\left({(2-T_A^L)}^m+{\left(T_A^L\right)}^m\right)\left((2-T_A^L)+T_A^L\right)}},\\ \frac{\frac{4{\left(T_A^U\right)}^{m+1}}{\left({(2-T_A^U)}^m+{\left(T_A^U\right)}^m\right)\left((2-T_A^U)+T_A^U\right)}}{\frac{\left(\left({(2-T_A^U)}^m+{\left(T_A^U\right)}^m\right)\left((2-T_A^U)+T_A^U\right)\right)+\left(\left({(2-T_A^U)}^m-{\left(T_A^U\right)}^m\right)\left((2-T_A^U)-T_A^U\right)\right)}{\left({(2-T_A^U)}^m+{\left(T_A^U\right)}^m\right)\left((2-T_A^U)+T_A^U\right)}}\end{array}\right],\\ \left[\begin{array}{c}\frac{\frac{4{\left(I_A^L\right)}^{m+1}}{\left({(2-I_A^L)}^m+{\left(I_A^L\right)}^m\right)\left((2-I_A^L)+I_A^L\right)}}{\frac{\left(\left({(2-I_A^L)}^m+{\left(I_A^L\right)}^m\right)\left((2-I_A^L)+I_A^L\right)\right)+\left(\left({(2-I_A^L)}^m-{\left(I_A^L\right)}^m\right)\left((2-I_A^L)-I_A^L\right)\right)}{\left({(2-I_A^L)}^m+{\left(I_A^L\right)}^m\right)\left((2-I_A^L)+I_A^L\right)}},\\ \frac{\frac{4{\left(I_A^U\right)}^{m+1}}{\left({(2-I_A^U)}^m+{\left(I_A^U\right)}^m\right)\left((2-I_A^U)+I_A^U\right)}}{\frac{\left(\left({(2-I_A^U)}^m+{\left(I_A^U\right)}^m\right)\left((2-I_A^U)+I_A^U\right)\right)+\left(\left({(2-I_A^U)}^m-{\left(I_A^U\right)}^m\right)\left((2-I_A^U)-I_A^U\right)\right)}{\left({(2-I_A^U)}^m+{\left(I_A^U\right)}^m\right)\left((2-I_A^U)+I_A^U\right)}}\end{array}\right],\\ \left[\begin{array}{c}\frac{\begin{array}{c}{\left(1+F_A^L\right)}^{m+1}+{\left(1+F_A^L\right)}^m\left(1-F_A^L\right)-{\left(1-F_A^L\right)}^m\left(1+F_A^L\right)-{\left(1-F_A^L\right)}^{m+1}+\\ {\left(1+F_A^L\right)}^{m+1}-{\left(1+F_A^L\right)}^m\left(1-F_A^L\right)+{\left(1-F_A^L\right)}^m\left(1+F_A^L\right)-{\left(1-F_A^L\right)}^{m+1}\end{array}}{\begin{array}{c}{\left(1+F_A^L\right)}^{m+1}+{\left(1+F_A^L\right)}^m\left(1-F_A^L\right)-{\left(1-F_A^L\right)}^m\left(1+F_A^L\right)+{\left(1-F_A^L\right)}^{m+1}+\\ {\left(1+F_A^L\right)}^{m+1}-{\left(1+F_A^L\right)}^m\left(1-F_A^L\right)+{\left(1-F_A^L\right)}^m\left(1+F_A^L\right)+{\left(1-F_A^L\right)}^{m+1}\end{array}},\\ \frac{\begin{array}{c}{\left(1+F_A^U\right)}^{m+1}+{\left(1+F_A^U\right)}^m\left(1-F_A^U\right)-{\left(1-F_A^U\right)}^m\left(1+F_A^U\right)-{\left(1-F_A^U\right)}^{m+1}+\\ {\left(1+F_A^U\right)}^{m+1}-{\left(1+F_A^U\right)}^m\left(1-F_A^U\right)+{\left(1-F_A^U\right)}^m\left(1+F_A^U\right)-{\left(1-F_A^U\right)}^{m+1}\end{array}}{\begin{array}{c}{\left(1+F_A^U\right)}^{m+1}+{\left(1+F_A^U\right)}^m\left(1-F_A^U\right)-{\left(1-F_A^U\right)}^m\left(1+F_A^U\right)+{\left(1-F_A^U\right)}^{m+1}+\\ {\left(1+F_A^U\right)}^{m+1}-{\left(1+F_A^U\right)}^m\left(1-F_A^U\right)+{\left(1-F_A^U\right)}^m\left(1+F_A^U\right)+{\left(1-F_A^U\right)}^{m+1}\end{array}}\end{array}\right],\\ \frac{\begin{array}{c}{\left(1+T_A\right)}^{m+1}+{\left(1+T_A\right)}^m\left(1-T_A\right)-{\left(1-T_A\right)}^m\left(1+T_A\right)-{\left(1-T_A\right)}^{m+1}+\\ {\left(1+T_A\right)}^{m+1}-{\left(1+T_A\right)}^m\left(1-T_A\right)+{\left(1-T_A\right)}^m\left(1+T_A\right)-{\left(1-T_A\right)}^{m+1}\end{array}}{\begin{array}{c}{\left(1+T_A\right)}^{m+1}+{\left(1+T_A\right)}^m\left(1-T_A\right)-{\left(1-T_A\right)}^m\left(1+T_A\right)+{\left(1-T_A\right)}^{m+1}+\\ {\left(1+T_A\right)}^{m+1}-{\left(1+T_A\right)}^m\left(1-T_A\right)+{\left(1-T_A\right)}^m\left(1+T_A\right)+{\left(1-T_A\right)}^{m+1}\end{array}},\\ \frac{\begin{array}{c}{\left(1+I_A\right)}^{m+1}+{\left(1+I_A\right)}^m\left(1-I_A\right)-{\left(1-I_A\right)}^m\left(1+I_A\right)-{\left(1-I_A\right)}^{m+1}+\\ {\left(1+I_A\right)}^{m+1}-{\left(1+I_A\right)}^m\left(1-I_A\right)+{\left(1-I_A\right)}^m\left(1+I_A\right)-{\left(1-I_A\right)}^{m+1}\end{array}}{\begin{array}{c}{\left(1+I_A\right)}^{m+1}+{\left(1+I_A\right)}^m\left(1-I_A\right)-{\left(1-I_A\right)}^m\left(1+I_A\right)+{\left(1-I_A\right)}^{m+1}+\\ {\left(1+I_A\right)}^{m+1}-{\left(1+I_A\right)}^m\left(1-I_A\right)+{\left(1-I_A\right)}^m\left(1+I_A\right)+{\left(1-I_A\right)}^{m+1}\end{array}},\\ \frac{\frac{4{\left(F_A\right)}^{m+1}}{\left({(2-F_A)}^m+{\left(F_A\right)}^m\right)\left((2-F_A)+F_A\right)}}{\frac{\left(\left({(2-F_A)}^m+{\left(F_A\right)}^m\right)\left((2-F_A)+F_A\right)\right)+\left(\left({(2-F_A)}^m-{\left(F_A\right)}^m\right)\left((2-F_A)-F_A\right)\right)}{\left({(2-F_A)}^m+{\left(F_A\right)}^m\right)\left((2-F_A)+F_A\right)}}\end{array}\right)$$

$$=\left(\begin{array}{c}\left[\begin{array}{c}\frac{4{\left(T_A^L\right)}^{m+1}}{{(2-T_A^L)}^{m+1}+T_A^L{(2-T_A^L)}^m+{\left(T_A^L\right)}^{m+1}+T_A^{Lm}\left(2-T_A^L\right)+\left({(2-T_A^L)}^{m+1}-T_A^L{(2-T_A^L)}^m+{\left(T_A^L\right)}^{m+1}-{\left(T_A^L\right)}^m\left(2-T_A^L\right)\right)},\\ \frac{4{\left(T_A^U\right)}^{m+1}}{{(2-T_A^U)}^{m+1}+T_A^U{(2-T_A^U)}^m+{\left(T_A^U\right)}^{m+1}+T_A^{Um}\left(2-T_A^U\right)+\left({(2-T_A^U)}^{m+1}-T_A^U{(2-T_A^U)}^m+{\left(T_A^U\right)}^{m+1}-{\left(T_A^U\right)}^m\left(2-T_A^U\right)\right)}\end{array}\right],\\ \left[\begin{array}{c}\frac{4{\left(I_A^L\right)}^{m+1}}{{(2-I_A^L)}^{m+1}+I_A^L{(2-I_A^L)}^m+{\left(I_A^L\right)}^{m+1}+I_A^{Lm}\left(2-I_A^L\right)+\left({(2-I_A^L)}^{m+1}-I_A^L{(2-I_A^L)}^m+{\left(I_A^L\right)}^{m+1}-{\left(I_A^L\right)}^m\left(2-I_A^L\right)\right)},\\ \frac{4{\left(I_A^U\right)}^{m+1}}{{(2-I_A^U)}^{m+1}+I_A^U{(2-I_A^U)}^m+{\left(I_A^U\right)}^{m+1}+I_A^{Um}\left(2-I_A^U\right)+\left({(2-I_A^U)}^{m+1}-I_A^U{(2-I_A^U)}^m+{\left(I_A^U\right)}^{m+1}-{\left(I_A^U\right)}^m\left(2-I_A^U\right)\right)}\end{array}\right],\\ \left[\frac{2\left({\left(1+F_A^L\right)}^{m+1}-{\left(1-F_A^L\right)}^{m+1}\right)}{2\left({\left(1+F_A^L\right)}^{m+1}+{\left(1-F_A^L\right)}^{m+1}\right)},\frac{2\left({\left(1+F_A^U\right)}^{m+1}-{\left(1-F_A^U\right)}^{m+1}\right)}{2\left({\left(1+F_A^U\right)}^{m+1}+{\left(1-F_A^U\right)}^{m+1}\right)}\right],\\ \frac{2\left({\left(1+T_A\right)}^{m+1}-{\left(1-T_A\right)}^{m+1}\right)}{2\left({\left(1+T_A\right)}^{m+1}+{\left(1-T_A\right)}^{m+1}\right)},\\ \frac{2\left({\left(1+I_A\right)}^{m+1}-{\left(1-I_A\right)}^{m+1}\right)}{2\left({\left(1+I_A\right)}^{m+1}+{\left(1-I_A\right)}^{m+1}\right)},\\ \frac{4{\left(F_A\right)}^{m+1}}{{(2-F_A)}^{m+1}+F_A{(2-F_A)}^m+{\left(F_A\right)}^{m+1}+F_A^m\left(2-F_A\right)+\left({(2-F_A)}^{m+1}-F_A{(2-F_A)}^m+{\left(F_A\right)}^{m+1}-{\left(F_A\right)}^m\left(2-F_A\right)\right)}\end{array}\right)$$

$$=(\begin{array}{c}\left[\frac{4{\left(T_A^L\right)}^{m+1}}{2\left({(2-T_A^L)}^{m+1}+{\left(T_A^L\right)}^{m+1}\right)},\frac{4{\left(T_A^U\right)}^{m+1}}{2\left({(2-T_A^U)}^{m+1}+{\left(T_A^U\right)}^{m+1}\right)}\right],\\ \left[\frac{4{\left(I_A^L\right)}^{m+1}}{2\left({(2-I_A^L)}^{m+1}+{\left(I_A^L\right)}^{m+1}\right)},\frac{4{\left(I_A^U\right)}^{m+1}}{2\left({(2-I_A^U)}^{m+1}+{\left(I_A^U\right)}^{m+1}\right)}\right]\end{array}$$