A new approach in decision making problems under the environment of neutrosophic cubic soft matrices

Abstract

There are many different techniques used so for in decision making problems. Some researchers used fuzzy sets, some used soft sets and some used combination of fuzzy sets with soft sets in order to handle the imprecise information. On the other hand, matrices play very important role in handling such problems. Different researchers used fuzzy matrices and fuzzy soft matrices in decision making problems. Some others generalized tools are Molodtsov’s soft sets, presented in 1999, Smarandache’s neutrosophic sets, in 1999, Jun’s cubic sets, in 2012 and Jun’s neutrosophic cubic sets in 2017. Our aim in this paper is to combine the above mentioned generalized tools and develop a new generalize approach namely neutrosophic cubic soft matrices. We first introduce the idea of Neutrosophic cubic soft aggregation operators with the help of neutrosophic cubic soft sets and then present the notion of neutrosophic cubic soft matrices and at the end, we present a max-min decision making problem using neutrosophic cubic soft matrices.

Keywords

Neutrosophic sets cubic sets soft sets neutrosophic cubic soft sets neutrosophic cubic soft matrices

1 Introduction

Introduction consists of four parts. First we give some historic background of different sets, their drawbacks, motivation and our current Model.

1.1 History of Different Sets

In many real world decision problems such as big data analysis, pattern recognition, expert systems and intelligent decision making, uncertainty and vagueness plays a fundamental role and it has been found that mathematical models based on classical sets might not be suitable for dealing with such problems. To overcome this situation, several types of theories were introduced like, fuzzy set [27] in 1965. Afterwards Molodtsov [16] initiated the concept of soft set which is a completely new approach for dealing with vagueness and uncertainty. Molodtsov pointed out several directions for the applications of soft set theory. Applications of soft set theory in algebraic structures was introduced by Aktas and Cagman [1]. Decision making always remain a main concerns for many researchers. Many researchers worked in decision making such as: Maji et al., [15], Perez et al., [20], studied consensus processes in group decision making problems recently and Dong et al., in [8] discussed the fusion process in opinion dynamics. Further in [9], Dong et al., studied, consensus reaching in social network group decision making: Research paradigms and challenges. Li et al., [13] studied, consistency of hesitant fuzzy linguistic preference relations. Group decision making based on linguistic distributions and hesitant assessments was discussed by Wu et al., in [24]. Xu et al., discussed the priority weights from incomplete hesitant fuzzy preference relations in group decision making [25]. Xu et al., provided a consensus model for hesitant fuzzy preference relations and its application in water allocation management [26]. [28] Zhang et al., discussed how to manage non-cooperative behaviors in consensus-based multiple attribute group decision making: An approach based on social network analysis. Further [29], Zhang et al., studied the 2-rank consensus reaching model in the multi-granular linguistic multiple-attribute group decision-making. Ref. [10], Dong et al., proceeded with the consensus reaching model in the complex and dynamic MAGDM-problem. Cagman et al. applied fuzzy soft matrix theory in decision making problems very effectively see [4], see also [2 , 18]. Being motivated by the realisms of physical life phenomenon i.e. different sports (win/ tie/ defeat), votes like yes/ NA/ NO and making a decision, Smarandache [21, 22] presented a fresh idea of a neutrosophic set (NS) and neutrosophic logic, which generalized fuzzy set and intuitionistic fuzzy set. NS is defined by (membership, indeterminacy and non-membership degrees). For applications in physical, technical and in different engineering regions, [23] Wang et al., in 2010 suggested the conception of a single valued neutrosophic sets. Now a days, neutrosophic sets are very actively used in applications and MCGM problems. Maji et al. [14] combined soft sets with neutrosophic sets. Deli and Said [7] gave the idea of neutrosophic soft matrices, see also [3, 6]. Jun et al. [11] gave the idea of cubic sets. Recently, Jun et al. [12] gave the idea of neutrosophic cubic sets, their different basic operations and applications. More applications can be seen in [5 , 30].

1.2 Drawbacks of the Developed sets/Methods

In order to present drawback of the developed sets we give a real life example as under; Let someone is watching a horror Hollywood Movie. Since fuzzy sets deal with the positive characteristics of a problem and cannot deal the negative characteristics so on the basis of it one can give the membership grade to that move without considering that what are the limitations in the Movie. This was handling through intutionistic fuzzy sets, where one can discuss about the positive and negative points of a Movie. Also it is not possible in many problems to give a certain fix membership grade to a certain thing on the basis of its positive characteristics so it was handled in cubic sets, which generalize the fuzzy sets and intutionistic fuzzy sets. On the other hand there may be some person who is not going to give any positive or negative point of view towards that certain Movie and this is handled in neutrosophic sets, which generalize the fuzzy sets and intutionistic fuzzy sets. Now if someone wants to include the different time of frames like Past, Present and Future in order to give its point of view then neutrosophic cubic sets are most suitable set, which generalize the intutionistic fuzzy sets, cubic sets and neutrosophic sets.

1.3 Motivation of Our Model

As Neutrosophic cubic sets are the most generalize version of all previously defined sets so this is the main advantage of it. So, the models presented in literature have different limitations in different situations. We mainly concern with the following tools:

a) Neutrosophic cubic sets are the more summed up class by which one can deal with uncertain data in a more successful manner when contrasted with fuzzy sets and all other versions of fuzzy sets. Neutrosophic cubic sets have the greater adaptability, accuracy and similarity to the framework when contrasted with past existing fuzzy models.

b) Soft set theory is an another general mathematical tool for dealing with uncertainty and vagueness. In soft set theory, the problem of setting the membership function does not arise, which makes the theory easily applied to many different fields including game theory, operations research, Riemann integration, and Perron integration.

c) Matrix mathematics applies to several branches of science, as well as different mathematical disciplines. Mathematicians, scientists and engineers represent groups of equations as matrices; then they have a systematic way of developing an algorithm. Computers have embedded matrix arithmetic in graphic processing algorithms, especially to render reflection and refraction. Some properties of matrix mathematics are important in math theory.

1.4 Current Proposal

We combine the idea of Jun’s cubic sets, Smarandache’s neutrosophic sets and Molodtsov’s soft sets with matrices in order to develop a new kind of theory known as neutrosophic cubic soft matrix theory. This paper consists of the seven sections. Section 1 is the introduction and in Section 2, we gather the required material from literature. In Section 3, we give the idea of neutrosophic cubic soft aggregation operator. In Section 4, we develop the idea of neutrosophic cubic soft matrices and different operations on them. Product of neutrosophic cubic soft matrices and some operators of neutrosophic cubic soft matrices are defined in this section. In Section 5, we give an application of the neutrosophic cubic soft matrices in decision making using the max-min operators. In Section 6, we provide a comparison analysis and in Section 7 we provide conclusions with some future research directions.

2 Preliminaries

Here we recall some of the existing ideas from the literature. More detail can be seen in [11 , 21].

Definition 2.1. [16] A pair (F, C) is called a soft set, over U, where F is a mapping given by F: C ⟶ P (U).

Definition 2.2. [21] Let X be a universe of discourse, then a set $A = {〈 x, T_{A} (x), I_{A} (x), F_{A} (x) 〉 : x \in X}$ is termed as neutrosophic set where T_A, I_A, F_A: X →]0^-, 1⁺ [ with 0 ≤ T_A (x) + I_A (x) + F_A (x) ≤3 and the functions T_A, I_A, F_A are truth, indeterminacy and falsity membership degrees respectively.

Definition 2.3. [23] Let X be a universe of discourse, then a set $A = {〈 x, T_{A} (x), I_{A} (x), F_{A} (x) 〉 : x \in X}$ is termed as single valued neutrosophic set where T_A, I_A, F_A: X → [0, 1] with 0 ≤ T_A (x) + I_A (x) + F_A (x) ≤3 and the functions T_A, I_A, F_A are truth, indeterminacy and falsity membership degrees respectively.

Definition 2.4. [11] Let X be any non-empty set. A cubic set in X is a structure $\underline{A} = {x, A (x), λ (x) | x \in X}$ where A is an interval valued fuzzy set in X and λ is a fuzzy set in X.

Definition 2.5. [12] A neutrosophic cubic set in X is a pair $ℑ = (\tilde{A}, ϑ)$ where $\tilde{A} = {〈 x, \tilde{T} (x), \tilde{I} (x), \tilde{F} (x) 〉 : x \in X}$ is an interval neutrosophic set in X where $\tilde{T}, \tilde{I}, \tilde{F} : X \to D [0, 1]$ and $ϑ = {〈 x, T (x), I (x), F (x) 〉 : x \in X}$ is a neutrosophic set in X where T, I, F: X → [0, 1]. Here D [0, 1] denotes set of all closed subintervals of [0, 1].

Definition 2.6. [12] Let X be a non-empty set. A neutrosophic cubic set (NCS), $ℑ = (\tilde{A}, ϑ)$ in X, is said to be internal neutrosophic cubic set (INCS) if ∀ x ∈ X, T (x) ∈ [T^- (x), T⁺ (x)], I (x) ∈ [I^- (x), I⁺ (x)], F (x) ∈ [F^- (x), F⁺ (x)].

Definition 2.7. [12] Let X be a non-empty set. A neutrosophic cubic set (NCS), $ℑ = (\tilde{A}, ϑ)$ in X, is said to be external neutrosophic cubic set (ENCS) if ∀x ∈ X, T (x) ∉ (T^- (x), T⁺ (x)), I (x) ∉ (I^- (x), I⁺ (x)), F (x) ∉ (F^- (x), F⁺ (x)).

3 Neutrosophic cubic soft aggregation operators

In this section as an application of neutrosophic cubic soft sets, we give the idea of neutrosophic cubic soft aggregation operator with an algorithm and numerical example in this section.

Definition 3.1. Let X be a universal set and NC (X) denotes the set of all neutrosophic cubic sets on X, E be the set of parameters that describes the elements of U, I ⊆ E. Then by neutrosophic cubic soft sets we mean a pair $(N, I)$ where $N$ is a mapping of I ⊆ E into the set of all neutrosophic cubic sets on X, i.e. $N : I \to NC (X) .$ It can be written as $(N, I) = {N (e_{i}), e_{i} \in I},$ where $N (e_{i}) = {〈 x, {\tilde{A}}_{N (e_{i})} (x), ϑ_{N (e_{i})} (x) 〉, x \in X}$ $\begin{matrix} {\tilde{A}}_{N (e_{i})} \\ = {〈 \begin{matrix} x, {\tilde{T}}_{{\tilde{A}}_{N (e_{i})}} (x) \\ {\tilde{I}}_{{\tilde{A}}_{N (e_{i})}} (x), \\ {\tilde{F}}_{{\tilde{A}}_{N (e_{i})}} (x) \end{matrix} 〉 : x \in X} \end{matrix}$ is an interval neutrosophic set in X and, $ϑ_{N (e_{i})} = {〈 x, T_{ϑ N (e_{i})} (x) I_{ϑ N (e_{i})} (x), F_{ϑ N (e_{i})} (x) 〉 : x \in X}$ is a neutrosophic set in X.

NCS (X)denotes the set of all neutrosophic cubic soft sets of X.

Definition 3.2. A neutrosophic cubic soft set $(N, I)$ in X is said to be

(i) Neutrosophic cubic soft truth-internal (briefly, NCST-internal) if for all e_i ∈ I ⊆ E, so that $T_{{\tilde{A}}_{N (e_{i})}}^{-} (x) \leq T_{ϑ_{N (e_{i})}} (x) \leq T_{{\tilde{A}}_{N (e_{i})}}^{+} (x),$ for all x ∈ X.

(ii) Neutrosophic cubic soft indeterminacy-internal (briefly, NCSI-internal) if for all e_i ∈ I ⊆ E, so that $I_{{\tilde{A}}_{N (e_{i})}}^{-} (x) \leq I_{ϑ_{N (e_{i})}} (x) \leq I_{{\tilde{A}}_{N (e_{i})}}^{+} (x),$ for all x ∈ X.

(iii) Neutrosophic cubic soft falsity-internal (briefly, NCSF-internal) if for all e_i ∈ I ⊆ E, so that $F_{{\tilde{A}}_{N (e_{i})}}^{-} (x) \leq F_{ϑ_{N (e_{i})}} (x) \leq F_{{\tilde{A}}_{N (e_{i})}}^{+} (x),$ for all x ∈ X.

If a neutrosophic cubic soft set $(N, I)$ in X, satisfy (i), (ii), (iii) it is known as neutrosophic cubic soft internal set in X, abbreviated as (NCSIS).

Definition 3.3. A neutrosophic cubic soft set $(N, I)$ in X is said to be

(i) Neutrosophic cubic soft truth-external (briefly, NCST-external) if for all e_i ∈ I ⊆ E, so that $T_{ϑ_{N (e_{i})}} (x) \notin (T_{{\tilde{A}}_{N (e_{i})}}^{-} (x), T_{{\tilde{A}}_{N (e_{i})}}^{+} (x)),$ for all x ∈ X.

(ii) Neutrosophic cubic soft indeterminacy-external (briefly, NCSI-external) if for all e_i ∈ I ⊆ E, so that $I_{ϑ_{N (e_{i})}} (x) \notin (I_{{\tilde{A}}_{N (e_{i})}}^{-} (x), I_{{\tilde{A}}_{N (e_{i})}}^{+} (x)),$ for all x ∈ X.

(iii) Neutrosophic cubic soft falsity-external (briefly, NCSF-external) if for all e_i ∈ I ⊆ E, so that $F_{ϑ_{N (e_{i})}} (x) \notin (F_{{\tilde{A}}_{N (e_{i})}}^{-} (x), F_{{\tilde{A}}_{N (e_{i})}}^{+} (x)),$ for all x ∈ X.

If a neutrosophic cubic soft set $(N, I)$ in X, satisfy (i), (ii), (iii) it is known as neutrosophic cubic soft external set in X, abbreviated as (NCSES).

Definition 3.4. Let $(N, I)$ be a neutrosophic cubic soft set in X. Then neutrosophic cubic soft set aggregation operator (NCSSAO), denoted by ${NCSSO}_{agg} : N (E) \times NCS (X) \to C (X),$ ${NCSSO}_{agg} (U, N_{1}) = N_{1}^{*}$ where $\begin{matrix} N_{1}^{*} & = {〈 {\tilde{A}}_{N_{1}^{*} (e_{i})} (x), ϑ_{N_{1}^{*} (e_{i})} (x) 〉 : x \in X} \\ = {〈 \begin{matrix} \frac{1}{| A |} \sum_{e_{i} \in A} (\begin{matrix} {\tilde{T}}_{{\tilde{A}}_{N (e_{i})}} (x) \\ + {\tilde{I}}_{{\tilde{A}}_{N (e_{i})}} (x) \\ - {\tilde{F}}_{{\tilde{A}}_{N (e_{i})}} (x) \end{matrix}), \\ \frac{1}{| A |} \sum_{e_{i} \in A} (\begin{matrix} T_{ϑ_{N (e_{i})}} (x) \\ + I_{ϑ_{N (e_{i})}} (x) \\ - F_{ϑ_{N (e_{i})}} (x) \end{matrix}) \end{matrix} 〉} \end{matrix}$ which is a cubic set over X and it is known as aggregate cubic set of the N₁ with |A| as a cardinality of A ⊆ E.

Algorithm 1: The algorithm for the solution is given below;

Step 1: Choose neutrosophic cubic soft set N₁ over X.

Step 2: Find aggregate cubic set of the $N_{1}^{*}$ of N₁.

Step 3: Find $max (N_{1}^{*} (x))$ which is the largest value of the membership and non-membership.

Example 3.5. Let X = {x₁, x₂, x₃} be the set under consideration, A = {e₁, e₂} ⊆ E = {e₁, e₂, e₃} be the set of attributes that best describe the elements in X with the neutrosophic cubic soft set as follows:

Step 1: Choose neutrosophic cubic soft set N₁over X

X	$N (e_{1}) = {\tilde{A}}_{N (e_{1})} (x), ϑ_{N (e_{1})} (x)$
x ₁	([0.2, 0.4], [0.2, 0.3], [0.1, 0.2]), (0.2, 0.7, 0.4)
x ₂	([0.2, 0.4], [0.2, 0.3], [0.3, 0.4]), (0.2, 0.7, 0.4)
x ₃	([0.1, 0.3], [0.3, 0.4], [0.1, 0.2]), (0.2, 0.7, 0.4)

and

X	$N (e_{2}) = {\tilde{A}}_{N (e_{2})} (x), ϑ_{N (e_{2})} (x)$
x ₁	([0.4, 0.5], [0.1, 0.3], [0.2, 0.4]), (0.5, 0.3, 0.4)
x ₂	([0.1, 0.2], [0.3, 0.4], [0.2, 0.25]), (0.1, 0.3, 0.4)
x ₃	([0.1, 0.2], [0.2, 0.3], [0.1, 0.2]), (0.2, 0.7, 0.4)

Step2: Aggregate cubic set of the

N_{1}^{*}

of N₁ is given by

N_{1}^{*} (x_{1}) = ([0.15, 0.45], 0.45)

N_{1}^{*} (x_{2}) = ([0.25, 0.4], 0.25)

N_{1}^{*} (x_{3}) = ([0.15, 0.5], 0.5)

Step3: $N_{1}^{*} (x_{3}) = ([0.15, 0.5], 0.5)$ is the $max (N_{1}^{*} (x_{3})) .$

Thus x₃ is selected for decision making.

4 Neutrosophic cubic soft matrices

This is the main section which provides a base for Section 5. In this section, we develop the idea of neutrosophic cubic soft matrices, neutrosophic cubic soft internal matrices and neutrosophic cubic soft external matrices with the help of neutrosophic cubic soft relations with examples. We also give the different operation on neutrosophic cubic matrices.

First we define the concept of neutrosophic cubic soft relations.

Definition 4.1. Let $(N, I) \in NCS (X) .$ Then a neutrosophic cubic soft relation is given by $R_{I} = {〈 (x_{i}, e_{i}), {\tilde{A}}_{R_{I}} (x_{i}, e_{i}), ϑ_{R_{I}} (x_{i}, e_{i}) 〉},$ for every (x_i, e_i) ∈ X × I, where ${\tilde{A}}_{R_{I}} : X \times I \to D [0, 1]$ , ϑ_{R
_I}: X × I → [0, 1]. If X = {x₁, x₂,..., x_m} and E = {e₁, e₂,..., e_n} and I ⊆ E, then R_I can be written in the Table "⊤".

Definition 4.2. Neutrosophic cubic soft matrix is defined as follows. If $a_{ij} = ({\tilde{A}}_{R_{I}} (x_{i}, e_{j}), ϑ_{R_{I}} (x_{i}, e_{j})),$ we define the matrix $[a_{ij}]_{m \times n} = [\begin{matrix} e_{11} & e_{12} & . & . & . & e_{1 n} \\ e_{21} & e_{22} & . & . & . & e_{2 n} \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ e_{m 1} & e_{m 2} & . & . & . & e_{mn} \end{matrix}]$ which is an m × n neutrosophic cubic soft matrix (NCSM) of the neutrosophic cubic soft set $(N, I)$ over X. Thus we can say that any neutrosophic cubic soft set $(N, I)$ is uniquely characterized by the neutrosophic cubic soft matrix (NCSM) and conversely. The set of all m × n neutrosophic cubic soft matrices denoted by NCSM_m×n.

Example 4.3. Let X = {x₁, x₂, x₃} be the set under consideration, A = {e₁, e₂} ⊆ E = {e₁, e₂, e₃} be the set of attributes that best describe the elements in X. Then the neutrosophic cubic soft set

X	$N (e_{1}) = {\tilde{A}}_{N (e_{1})} (x), ϑ_{N (e_{1})} (x)$
x ₁	([0.2, 0.4], [0.2, 0.3], [0.6, 0.7]), (0.2, 0.7, 0.4)
x ₂	([0.2, 0.4], [0.2, 0.3], [0.6, 0.7]), (0.2, 0.7, 0.4)
x ₃	([0.1, 0.3], [0.3, 0.4], [0.6, 0.7]), (0.2, 0.7, 0.4)

and

X	$N (e_{2}) = {\tilde{A}}_{N (e_{2})} (x), ϑ_{N (e_{2})} (x)$
x ₁	([0.4, 0.5], [0.1, 0.3], [0.3, 0.7]), (0.8, 0.3, 0.4)
x ₂	([0.1, 0.2], [0.3, 0.4], [0.3, 0.7]), (0.1, 0.3, 0.4)
x ₃	([0.1, 0.2], [0.2, 0.3], [0.6, 0.7]), (0.2, 0.7, 0.4)

we have the neutrosophic cubic soft matrix as

$\begin{matrix} [c] l [a_{ij}]_{3 \times 3} = \\ [\begin{matrix} {\begin{matrix} {\begin{matrix} ([0.2, 0.4], \\ [0.2, 0.3], \\ [0.6, 0.7]), \end{matrix} \\ {(0.2, 0.7, 0.4) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0.4, 0.5], \\ [0.1, 0.3], \\ [0.3, 0.7]), \end{matrix} \\ {(0.8, 0.3, 0.4) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0, 0], \\ [0, 0], \\ [0, 0]), \end{matrix} \\ {(0, 0, 0) \end{matrix} \\ {\begin{matrix} {\begin{matrix} ([0.2, 0.4], \\ [0.2, 0.3], \\ [0.6, 0.7]), \end{matrix} \\ {(0.2, 0.7, 0.4) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0.1, 0.2], \\ [0.8, 0.9], \\ [0.3, 0.7]), \end{matrix} \\ {(0.1, 0.3, 0.4) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0, 0], \\ [0, 0], \\ [0, 0]), \end{matrix} \\ {(0, 0, 0) \end{matrix} \\ {\begin{matrix} {\begin{matrix} ([0.2, 0.4], \\ [0.2, 0.3], \\ [0.6, 0.7]), \end{matrix} \\ {(0.2, 0.7, 0.4) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0.2, 0.4], \\ [0.2, 0.3], \\ [0.6, 0.7]), \end{matrix} \\ {(0.2, 0.7, 0.4) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0, 0], \\ [0, 0], \\ [0, 0]), \end{matrix} \\ {(0, 0, 0) \end{matrix} \end{matrix}] \end{matrix}$

Definition 4.4. A neutrosophic cubic soft matrix [a_ij] ∈ NCSM_m×n is called a neutrosophic cubic soft zero matrix if $[a_{ij}] = (([0, 0], [1, 1], [1, 1]), (0, 1, 1)) = [0]$ for all i and j.

Definition 4.5. A neutrosophic cubic soft matrix [a_ij] ∈ NCSM_m×n is called a neutrosophic cubic soft universal matrix if $[a_{ij}] = (([1, 1], [0, 0], [0, 0]), (1, 0, 0)) = [1]$ for all i and j.

Definition 4.6. A neutrosophic cubic soft matrix [a_ij] ∈ NCSM_m×n is said to be neutrosophic cubic soft square matrix if m = n.

Definition 4.7. The transpose of a neutrosophic cubic soft square matrix is denoted by $[a_{ij}]_{n \times m}^{t},$ which can be obtain by interchanging the rows and columns.

Definition 4.8. A neutrosophic cubic soft square matrix [a_ij] ∈ NCSM_m×n is called a neutrosophic cubic soft symmetric matrix if $[a_{ij}] = [a_{ij}]_{n \times m}^{t} .$

Definition 4.9. A neutrosophic cubic soft matrix [a_ij] ∈ NCSM_m×n is said to be (i) Neutrosophic cubic soft truth-internal matrix (briefly, NCST-internal matrix) if for all e_i ∈ I ⊆ E, so that $T_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}) \leq T_{ϑ_{R_{I}}} (x_{i}, e_{j}) \leq T_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j}),$ for all (x_i, e_j) ∈ X × I.

Table ¤
Table for Definition 4

R ₁ e ₁ e ₂ · · · e_n

x ₁ $({\tilde{A}}_{R_{1}} (x_{1}, e_{1}), ϑ_{R_{1}} (x_{1}, e_{1}))$ $({\tilde{A}}_{R_{1}} (x_{1}, e_{2}), ϑ_{R_{1}} (x_{1}, e_{2}))$ · · · $({\tilde{A}}_{R_{1}} (x_{1}, e_{n}), ϑ_{R_{1}} (x_{1}, e_{n}))$

x ₂ $({\tilde{A}}_{R_{1}} (x_{2}, e_{1}), ϑ_{R_{1}} (x_{2}, e_{1}))$ $({\tilde{A}}_{R_{1}} (x_{2}, e_{2}), ϑ_{R_{1}} (x_{2}, e_{2}))$ · · · $({\tilde{A}}_{R_{1}} (x_{2}, e_{n}), ϑ_{R_{1}} (x_{2}, e_{n}))$

· · · · · · ·

· · · · · · ·

· · · · · · ·

x_m $({\tilde{A}}_{R_{1}} (x_{m}, e_{1}), ϑ_{R_{1}} (x_{m}, e_{1}))$ $({\tilde{A}}_{R_{1}} (x_{m}, e_{2}), ϑ_{R_{1}} (x_{m}, e_{2}))$ · · · $({\tilde{A}}_{R_{1}} (x_{m}, e_{n}), ϑ_{R_{1}} (x_{m}, e_{n}))$

(ii) Neutrosophic cubic soft indeterminacy-internal matrix (briefly, NCSI-internal matrix) if for all e_i ∈ I ⊆ E, so that $I_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}) \leq I_{ϑ_{R_{I}}} (x_{i}, e_{j}) \leq I_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j}),$ for all (x_i, e_j) ∈ X × I.

(iii) Neutrosophic cubic soft falsity-internal matrix (briefly, NCSF-internal matrix) if for all e_i ∈ I ⊆ E, so that $F_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}) \leq F_{ϑ_{R_{I}}} (x_{i}, e_{j}) \leq F_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j}),$ for all (x_i, e_j) ∈ X × I.

If a neutrosophic cubic soft matrix $[a_{ij}] \in {NCSM}_{m \times n}$

satisfy (i), (ii), (iii) it is known as neutrosophic cubic soft internal matrix, abbreviated as (NCSIM).

Definition 4.10. A neutrosophic cubic soft matrix [a_ij] ∈ NCSM_m×n is said to be

(i) Neutrosophic cubic soft truth-external matrix (briefly, NCST-external matrix) if for all e_i ∈ I ⊆ E, so that $T_{ϑ_{R_{I}}} (x_{i}, e_{j}) \notin (T_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}), T_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j})),$ for all (x_i, e_j) ∈ X × I.

(ii) Neutrosophic cubic soft indeterminacy-external matrix (briefly, NCSI-external matrix) if for all e_i ∈ I ⊆ E, so that $I_{ϑ_{R_{I}}} (x_{i}, e_{j}) \notin (I_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}), I_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j})),$ for all (x_i, e_j) ∈ X × I.

(iii) Neutrosophic cubic soft falsity-external matrix (briefly, NCSF-external matrix) if for all e_i ∈ I ⊆ E, so that $F_{ϑ_{R_{I}}} (x_{i}, e_{j}) \notin (F_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}), F_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j})),$ for all (x_i, e_j) ∈ X × I.

If a neutrosophic cubic soft matrix $[a_{ij}] \in {NCSM}_{m \times n},$ satisfy (i), (ii), (iii) it is known as neutrosophic cubic soft external matrix, abbreviated as (NCSEM).

Remark: We may have a neutrosophic cubic soft matrix which is neither internal nor external.

We now define the P-union, P-intersection, R-union and R-intersection of neutrosophic cubic soft matrix with some properties as follows:

Definition 4.11. Let $[a_{ij}] = ({\tilde{A}}_{R_{I}} (x_{i}, e_{j}),$ ϑ_{R
_I} (x_i, e_j)) and $[b_{ij}] = ({\tilde{A}}_{R_{I}} (y_{i}, f_{j}), ϑ_{R_{I}} (y_{i}, f_{j}))$ be two neutrosophic cubic soft matrices in X. Then we define

(i) for (x_i, e_j) ∈ X × I, $\begin{matrix} [a_{i j}] \cup_{P} [b_{i j}] = [c i j] \\ = {〈 \begin{matrix} (x_{i}, e_{j}), {\tilde{A}}_{R_{I}} (x_{i}, e_{j}) \\ \lor {\tilde{A}}_{R_{I}} (x_{i}, e_{j}), ϑ_{R_{I}} (x_{i}, e_{j}) \\ \lor ϑ_{R_{I}} (x_{i}, e_{j}) \end{matrix} 〉} . \end{matrix}$ (ii) for (x_i, e_j) ∈ X × I, $\begin{matrix} [a_{i j}] \cap_{P} [b_{i j}] = [c_{i j}] \\ = {〈 \begin{matrix} (x_{i}, e_{j}), {\tilde{A}}_{R_{I}} (x_{i}, e_{j}) \\ \land {\tilde{A}}_{R_{I}} (x_{i}, e_{j}), ϑ_{R_{I}} (x_{i}, e_{j}) \\ \land ϑ_{R_{I}} (x_{i}, e_{j}) \end{matrix} 〉} . \end{matrix}$ (iii) for (x_i, e_j) ∈ X × I, $\begin{matrix} [a_{i j}] \cup_{R} [b_{i j}] = [c_{i j}] \\ = {〈 \begin{matrix} (x_{i}, e_{j}), {\tilde{A}}_{R_{I}} (x_{i}, e_{j}) \\ \lor {\tilde{A}}_{R_{I}} (x_{i}, e_{j}), ϑ_{R_{I}} (x_{i}, e_{j}) \\ \land ϑ_{R_{I}} (x_{i}, e_{j}) \end{matrix} 〉} . \end{matrix}$ (iv) for (x_i, e_j) ∈ X × I, $\begin{matrix} [a_{i j}] \cup_{R} [b_{i j}] = [c_{i j}] \\ = {〈 \begin{matrix} (x_{i}, e_{j}), {\tilde{A}}_{R_{I}} (x_{i}, e_{j}) \\ \land {\tilde{A}}_{R_{I}} (x_{i}, e_{j}), ϑ_{R_{I}} (x_{i}, e_{j}) \\ \lor ϑ_{R_{I}} (x_{i}, e_{j}) \end{matrix} 〉} . \end{matrix}$

(v) for (x_i, e_j) ∈ X × I, the complement of a neutrosophic cubic soft matrix ${[a_{i j}] = [a_{i j}]}^{c} = {〈 (x_{i}, e_{j}), {\tilde{A}}_{R_{I}}^{c} (x_{i}, e_{j}), ϑ_{R_{I}}^{c} (x_{i}, e_{j}) 〉} .$

Proposition 4.12. Let [a_ij], [b_ij], [c_ij], [d_ij] be neutrosophic cubic soft matrices in X. Then

(i) [a_ij] = [b_ij] if and only if the ${\tilde{A}}_{R_{I}} (x_{i}, e_{j}) = {\tilde{A}}_{R_{I}} (x_{i}, e_{j}), ϑ_{R_{I}} (x_{i}, e_{j}) = ϑ_{R_{I}} (x_{i}, e_{j}) .$

(ii) [a_ij] ⊆ _P [b_ij] if and only if ${\tilde{A}}_{R_{I}} (x_{i}, e_{j}) \leq {\tilde{A}}_{R_{I}} (x_{i}, e_{j}), ϑ_{R_{I}} (x_{i}, e_{j}) \leq ϑ_{R_{I}} (x_{i}, e_{j}) .$

(iii) [a_ij] ⊆ _R [b_ij] if and only if ${\tilde{A}}_{R_{I}} (x_{i}, e_{j}) \leq {\tilde{A}}_{R_{I}} (x_{i}, e_{j}), ϑ_{R_{I}} (x_{i}, e_{j}) \geq ϑ_{R_{I}} (x_{i}, e_{j}) .$

(iv) If [a_ij] ⊆ _P [b_ij] and [c_ij] ⊆ _P [d_ij] then [a_ij] ⊆ _P [d_ij].

(v) If [a_ij] ⊆ _P [b_ij] then [b_ij] ^c ⊆ _P [a_ij] ^c.

(vi) If [a_ij] ⊆ _P [b_ij] and [a_ij] ⊆ _P [d_ij] then [a_ij] ⊆ _P [b_ij] ∩ _P [d_ij].

(vii) If [a_ij] ⊆ _P [b_ij] and [c_ij] ⊆ _P [b_ij] then [a_ij] ∪ _P [c_ij] ⊆ _P [b_ij].

(viii) If [a_ij] ⊆ _P [b_ij] and [c_ij] ⊆ _P [d_ij] then [a_ij] ∪ _P [b_ij] ⊆ _P [c_ij] ∪ _P [d_ij] and [a_ij] ∩ _P [c_ij] ⊆ _P [b_ij] ∩ _P [d_ij].

(ix) If [a_ij] ⊆ _R [b_ij] and [c_ij] ⊆ _R [d_ij] then [a_ij] ⊆ _R [d_ij].

(x) If [a_ij] ⊆ _R [b_ij] then If [b_ij] ^c ⊆ _R [a_ij] ^c.

(xi) If [a_ij] ⊆ _R [b_ij] and [a_ij] ⊆ _R [d_ij] then [a_ij] ⊆ _R [b_ij] ∪ _R [d_ij].

(xii) If [a_ij] is neutrosophic cubic soft internal matrix in X then [a_ij] ^c is also neutrosophic cubic soft internal matrix in X.

(xiii) If [a_ij] is neutrosophic cubic soft external matrix in X then [a_ij] ^c is also neutrosophic cubic soft external matrix in X.

(xiv) The P-union and P-intersection of two neutrosophic cubic soft internal matrix in X is also neutrosophic cubic soft internal matrix in X.

(xv) The P-union and P-intersection of two neutrosophic cubic soft external matrix in X need not be neutrosophic cubic soft external matrix in X.

The follwoing Theorems gives us a condition under which the P- union and P- intersection of two neutrosophic cubic soft external matrix in X becomes neutrosophic cubic soft external matrix in X.

Theorem 4.13. Let $\begin{matrix} [a_{ij}] & = ({\tilde{A}}_{R_{I}} (x_{i}, e_{j}), ϑ_{R_{I}} (x_{i}, e_{j})) \\ and [b_{ij}] & = ({\tilde{A}}_{R_{I}} (y_{i}, f_{j}), ϑ_{R_{I}} (y_{i}, f_{j})) \end{matrix}$ be two neutrosophic cubic soft matrices in X. Then P-union of two neutrosophic cubic soft external matrix is neutrosophic cubic soft external matrix in X if it satisfies the following conditions for all (x_i, e_j) ∈ X × I:

(i) $\begin{matrix} max {\begin{matrix} min {T_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j}), T_{{\tilde{A}}_{R_{I}}}^{-} (y_{i}, f_{j})}, \\ min {T_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}), T_{{\tilde{A}}_{R_{I}}}^{+} (y_{i}, f_{j})} \end{matrix}} \\ \leq max {T_{ϑ_{R_{I}}} (x_{i}, e_{j}), T_{ϑ_{R_{I}}} (y_{i}, f_{j})} \\ < min {\begin{matrix} max {T_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j}), T_{{\tilde{A}}_{R_{I}}}^{-} (y_{i}, f_{j})}, \\ max {T_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}), T_{{\tilde{A}}_{R_{I}}}^{+} (y_{i}, f_{j})} \end{matrix}}, \end{matrix}$ (ii) $\begin{matrix} max {\begin{matrix} min {I_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j}), I_{{\tilde{A}}_{R_{I}}}^{-} (y_{i}, f_{j})}, \\ min {I_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}), I_{{\tilde{A}}_{R_{I}}}^{+} (y_{i}, f_{j})} \end{matrix}} \\ \leq max {I_{ϑ_{R_{I}}} (x_{i}, e_{j}), I_{ϑ_{R_{I}}} (y_{i}, f_{j})} \\ < min {\begin{matrix} max {I_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j}), I_{{\tilde{A}}_{R_{I}}}^{-} (y_{i}, f_{j})}, \\ max {I_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}), I_{{\tilde{A}}_{R_{I}}}^{+} (y_{i}, f_{j})} \end{matrix}}, \end{matrix}$ (iii) $\begin{matrix} max {\begin{matrix} min {F_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j}), F_{{\tilde{A}}_{R_{I}}}^{-} (y_{i}, f_{j})}, \\ min {F_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}), F_{{\tilde{A}}_{R_{I}}}^{+} (y_{i}, f_{j})} \end{matrix}} \\ \leq max {F_{ϑ_{R_{I}}} (x_{i}, e_{j}), F_{ϑ_{R_{I}}} (y_{i}, f_{j})} \\ < min {\begin{matrix} max {F_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j}), F_{{\tilde{A}}_{R_{I}}}^{-} (y_{i}, f_{j})}, \\ max {F_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}), F_{{\tilde{A}}_{R_{I}}}^{+} (y_{i}, f_{j})} \end{matrix}}, \end{matrix}$

Theorem 4.14. Let $[a_{ij}] = ({\tilde{A}}_{R_{I}} (x_{i}, e_{j}), ϑ_{R_{I}} (x_{i}, e_{j}))$ and $[b_{ij}] = ({\tilde{A}}_{R_{I}} (y_{i}, f_{j}), ϑ_{R_{I}} (y_{i}, f_{j}))$ be two neutrosophic cubic soft matrices in X. Then P-intersection of two neutrosophic cubic soft external matrix is neutrosophic cubic soft external matrix in X if it satisfies the following conditions for all (x_i, e_j) ∈ X × I:

(i) $\begin{matrix} min {\begin{matrix} max {T_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j}), T_{{\tilde{A}}_{R_{I}}}^{-} (y_{i}, f_{j})}, \\ max {T_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}), T_{{\tilde{A}}_{R_{I}}}^{+} (y_{i}, f_{j})} \end{matrix}} \\ \leq min {T_{ϑ_{R_{I}}} (x_{i}, e_{j}), T_{ϑ_{R_{I}}} (y_{i}, f_{j})} \\ < max {\begin{matrix} min {T_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j}), T_{{\tilde{A}}_{R_{I}}}^{-} (y_{i}, f_{j})}, \\ min {T_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}), T_{{\tilde{A}}_{R_{I}}}^{+} (y_{i}, f_{j})} \end{matrix}}, \end{matrix}$ (ii) $\begin{matrix} min {\begin{matrix} max {I_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j}), I_{{\tilde{A}}_{R_{I}}}^{-} (y_{i}, f_{j})}, \\ max {I_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}), I_{{\tilde{A}}_{R_{I}}}^{+} (y_{i}, f_{j})} \end{matrix}} \\ \leq min {I_{ϑ_{R_{I}}} (x_{i}, e_{j}), I_{ϑ_{R_{I}}} (y_{i}, f_{j})} \\ < max {\begin{matrix} min {I_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j}), I_{{\tilde{A}}_{R_{I}}}^{-} (y_{i}, f_{j})}, \\ min {I_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}), I_{{\tilde{A}}_{R_{I}}}^{+} (y_{i}, f_{j})} \end{matrix}}, \end{matrix}$ (iii) $\begin{matrix} min {\begin{matrix} max {F_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j}), F_{{\tilde{A}}_{R_{I}}}^{-} (y_{i}, f_{j})}, \\ max {F_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}), F_{{\tilde{A}}_{R_{I}}}^{+} (y_{i}, f_{j})} \end{matrix}} \\ \leq min {F_{ϑ_{R_{I}}} (x_{i}, e_{j}), F_{ϑ_{R_{I}}} (y_{i}, f_{j})} \\ < max {\begin{matrix} min {F_{{\tilde{A}}_{R_{I}}}^{+} (x_{i}, e_{j}), F_{{\tilde{A}}_{R_{I}}}^{-} (y_{i}, f_{j})}, \\ min {F_{{\tilde{A}}_{R_{I}}}^{-} (x_{i}, e_{j}), F_{{\tilde{A}}_{R_{I}}}^{+} (y_{i}, f_{j})} \end{matrix}}, \end{matrix}$

Definition 4.15. Two neutrosophic cubic soft matrix $\begin{array}{l} [a_{i j}] = ({\tilde{A}}_{R_{I}} (x_{i}, e_{j}), ϑ_{R_{I}} (x_{i}, e_{j})) \\ and [b_{i j}] = ({\tilde{A}}_{R_{I}} (y_{i}, f_{j}), ϑ_{R_{I}} (y_{i}, f_{j})) \end{array}$ in X are said to be P-disjoint (resp., R-disjoint) if [a_ij] ∩ _P [b_ij] = [0] ([a_ij] ∩ _R [b_ij] = [0]).

Proposition 4.16. Let [a_ij], [b_ij] be neutrosophic cubic soft matrices in X. Then

(i) (([a_ij]) ^c) ^c = [a_ij], (ii) [0] ^c = [1],

(iii) [0] ⊆ [a_ij] and [a_ij] ⊆ [1],

(iv) [a_ij] ∪ [b_ij] = [b_ij] ∪ [a_ij] and [a_ij] ∩ [b_ij] = [b_ij] ∩ [a_ij].

Proposition 4.17. Let [a_ij], [b_ij] be neutrosophic cubic soft matrices in X. Then De Morgan’s laws hold:

(i) ([a_ij] ∪ [b_ij]) ^c = ([a_ij]) ^c ∩ ([b_ij]) ^c,

(ii) ([a_ij] ∩ [b_ij]) ^c = ([a_ij]) ^c ∪ ([b_ij]) ^c.

4.1 Product of neutrosophic cubic soft matrices

In this subsection, we define different types of Product of neutrosophic cubic soft matrices.

Definition 4.18. Let [a_ij], [b_ik] be neutrosophic cubic soft matrices in X. Then And-product of [a_ij] and [b_ik] is denoted by [a_ij] ∧ [b_ik] is defined by $\land : {NCSM}_{m \times n} \times {NCSM}_{m \times n} \to {NCSM}_{m \times n^{2}}$ $\begin{matrix} [a_{ij}] \land [b_{ik}] \\ = [c_{ip}] = {〈 \begin{matrix} {\tilde{A}}_{R_{I}} (x_{i}, e_{j}) \land {\tilde{A}}_{R_{I}} (x_{i}, e_{k}), \\ ϑ_{R_{I}} (x_{i}, e_{i}) \land ϑ_{R_{I}} (x_{i}, e_{k}) \end{matrix} 〉} \\ = [\begin{matrix} {\begin{matrix} min (T_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{j}), T_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{k})), \\ max (I_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{j}), I_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{k})), \\ max (F_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{j}), F_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{k})) \end{matrix} \\ {\begin{matrix} min (T_{ϑ_{R_{I}}} (x_{i}, e_{j}), T_{ϑ_{R_{I}}} (x_{i}, e_{k})), \\ max (I_{ϑ_{R_{I}}} (x_{i}, e_{j}), I_{ϑ_{R_{I}}} (x_{i}, e_{k})), \\ max (F_{ϑ_{R_{I}}} (x_{i}, e_{j}), F_{ϑ_{R_{I}}} (x_{i}, e_{k})) \end{matrix} \end{matrix}] \end{matrix}$ such that p = n (j - 1) + k.

Definition 4.19. Let [a_ij], [b_ik] be neutrosophic cubic soft matrices in X. Then Or-product of [a_ij] and [b_ik] is denoted by [a_ij] ∨ [b_ik] is defined by $\lor : {NCSM}_{m \times n} \times {NCSM}_{m \times n} \to {NCSM}_{m \times n^{2}}$ $\begin{matrix} [a_{ij}] \lor [b_{ik}] \\ = [c_{ip}] = {〈 \begin{matrix} {\tilde{A}}_{R_{I}} (x_{i}, e_{j}) \lor {\tilde{A}}_{R_{I}} (x_{i}, e_{k}), \\ ϑ_{R_{I}} (x_{i}, e_{i}) \lor ϑ_{R_{I}} (x_{i}, e_{k}) \end{matrix} 〉} \\ = [\begin{matrix} {\begin{matrix} max (T_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{j}), T_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{k})), \\ min (I_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{j}), I_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{k})), \\ min (F_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{j}), F_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{k})) \end{matrix} \\ {\begin{matrix} max (T_{ϑ_{R_{I}}} (x_{i}, e_{j}), T_{ϑ_{R_{I}}} (x_{i}, e_{k})), \\ min (I_{ϑ_{R_{I}}} (x_{i}, e_{j}), I_{ϑ_{R_{I}}} (x_{i}, e_{k})), \\ min (F_{ϑ_{R_{I}}} (x_{i}, e_{j}), F_{ϑ_{R_{I}}} (x_{i}, e_{k})) \end{matrix} \end{matrix}] \end{matrix}$ such that p = n (j - 1) + k.

Definition 4.20. Let [a_ij], [b_ik] be neutrosophic cubic soft matrices in X. Then And Not-product of [a_ij] and [b_ik] is denoted by $[a_{ij}] \bar{\land} [b_{ik}]$ is defined by $\bar{\land} : {NCSM}_{m \times n} \times {NCSM}_{m \times n} \to {NCSM}_{m \times n^{2}}$ $\begin{matrix} [a_{ij}] \bar{\land} [b_{ik}] \\ = [c_{ip}] = {〈 \begin{matrix} {\tilde{A}}_{R_{I}} (x_{i}, e_{j}) \bar{\land} {\tilde{A}}_{R_{I}} (x_{i}, e_{k}), \\ ϑ_{R_{I}} (x_{i}, e_{i}) \bar{\land} ϑ_{R_{I}} (x_{i}, e_{k}) \end{matrix} 〉} \\ = [\begin{matrix} {\begin{matrix} min (F_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{j}), F_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{k})), \\ max (I_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{j}), I_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{k})), \\ max (T_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{j}), T_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{k})) \end{matrix} \\ {\begin{matrix} min (F_{ϑ_{R_{I}}} (x_{i}, e_{j}), F_{ϑ_{R_{I}}} (x_{i}, e_{k})), \\ max (I_{ϑ_{R_{I}}} (x_{i}, e_{j}), I_{ϑ_{R_{I}}} (x_{i}, e_{k})), \\ max (T_{ϑ_{R_{I}}} (x_{i}, e_{j}), T_{ϑ_{R_{I}}} (x_{i}, e_{k})) \end{matrix} \end{matrix}] \end{matrix}$ such that p = n (j - 1) + k.

Definition 4.21. Let [a_ij], [b_ik] be neutrosophic cubic soft matrices in X. Then Or Not-product of [a_ij] and [b_ik] is denoted by $[a_{ij}] \bar{\lor} [b_{ik}]$ is defined by $\bar{\lor} : {NCSM}_{m \times n} \times {NCSM}_{m \times n} \to {NCSM}_{m \times n^{2}}$ $\begin{matrix} [a_{ij}] \bar{\lor} [b_{ik}] \\ = [c_{ip}] = {〈 \begin{matrix} {\tilde{A}}_{R_{I}} (x_{i}, e_{j}) \bar{\lor} {\tilde{A}}_{R_{I}} (x_{i}, e_{k}), \\ ϑ_{R_{I}} (x_{i}, e_{i}) \bar{\lor} ϑ_{R_{I}} (x_{i}, e_{k}) \end{matrix} 〉} \\ = [\begin{matrix} {\begin{matrix} max (F_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{j}), F_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{k})), \\ min (I_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{j}), I_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{k})), \\ min (T_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{j}), T_{{\tilde{A}}_{R_{I}}} (x_{i}, e_{k})) \end{matrix} \\ {\begin{matrix} max (F_{ϑ_{R_{I}}} (x_{i}, e_{j}), F_{ϑ_{R_{I}}} (x_{i}, e_{k})), \\ min (I_{ϑ_{R_{I}}} (x_{i}, e_{j}), I_{ϑ_{R_{I}}} (x_{i}, e_{k})), \\ min (T_{ϑ_{R_{I}}} (x_{i}, e_{j}), T_{ϑ_{R_{I}}} (x_{i}, e_{k})) \end{matrix} \end{matrix}] \end{matrix}$ such that p = n (j - 1) + k.

Proposition 4.22. Let [a_ij], [b_ik] ∈ NCSM_m×n. Then the De Morgan’s Laws hold:

(i) ([a_ij] ∨ [b_ij]) ^c = ([a_ij]) ^c ∧ ([b_ij]) ^c,

(ii) ([a_ij] ∧ [b_ij]) ^c = ([a_ij]) ^c ∨ ([b_ij]) ^c,

(iii) ${([a_{ij}] \bar{\land} [b_{ij}])}^{c} = {([a_{ij}])}^{c} \bar{\lor} {([b_{ij}])}^{c}$ ,

(iv) ${([a_{ij}] \bar{\lor} [b_{ij}])}^{c} = {([a_{ij}])}^{c} \bar{\land} {([b_{ij}])}^{c} .$

5 Max-min decision making using neutrosophic cubic soft matrices

Decision matrix analysis is the simplest form of Multiple Criteria Decision Analysis (MCDA), also known as multiple criteria decision aid or multiple criteria decision management (MCDM). Decision matrix analysis is a useful technique used for making a decision. Being able to use decision matrix analysis means that we can take decisions confidently and rationally, at a time when other people might be struggling to make a decision. Decision matrix analysis helps you to decide between several options, where we need to take many different factors into account. Fuzzy matrix theory is a better approach which can cope with the uncertainty in a very affective way but it is considering only the positive effects of a problem while our presented model handles, positive, negative and situation where one is not sure in a more practical way. In this section as an application of the neutrosophic cubic soft matrix, we use the max-min decision making. One can may use (min-max,min-min, max-max decision neutrosophic cubic soft matrix), according to the type of the problem but max-min is the most popular version being used in literature so for. At the end, we provide a numerical example.

First we define the score function and then max-min decision function.

Definition 5.1. Let $(N_{1}, I_{1}) .$ Then the score function is defined as follows $\begin{matrix} \tilde{s} (N_{1}, I_{1}) \\ = {\begin{matrix} 〈 \frac{T_{{\tilde{A}}_{N (e_{i})}} + [1, 1] - {\tilde{I}}_{N (e_{i})} + [1, 1] - F_{{\tilde{A}}_{N (e_{i})}}}{3} 〉, \\ 〈 \frac{T_{ϑ_{N (e_{i})}} + 1 - I_{ϑ_{N (e_{i})}} + 1 - F_{ϑ_{N (e_{i})}}}{3} 〉 \end{matrix}} . \end{matrix}$

Definition 5.2. Let [c_ip] ∈ NCSM_m×n², I_k = {p: ∃ i, c_ip ≠ 0, (k - 1) n < p ≤ kn} for all k ∈ I = {1, 2,.. n}. Then Max-min decision function denoted by NCM_m: NCSM_m×n² → NCSM_m×1, ${NCM}_{m} [c_{ip}] = [D_{i 1}] = [max_{k} {{\tilde{t_{1}}}_{k}}, min_{k} {t_{2_{k}}}]$ where $\begin{array}{l} \tilde{t 1 k} = {\begin{cases} \begin{matrix} r \min_{p \in I_{k}} {{\tilde{A}}_{R I} (x_{i}, e_{j})}, & if I_{k} \neq ϕ, \end{matrix} \\ \begin{matrix} 0 & if I_{k} \neq ϕ . \end{matrix} \end{cases} \\ and t_{2} k = {\begin{matrix} \max_{p \in I_{k}} {ϑ_{R_{I}} (x_{i}, e_{j})} & if I_{k} \neq ϕ, \\ 1 & if I_{k} \neq ϕ . \end{matrix} \end{array}$ The one column NCM_m [c_ip] is called max-min decision neutrosophic cubic soft matrix.

Algorithm-2: Now we can construct an algorithm for max-min decision making using neutrosophic cubic soft matrices by using the Definition 5.

Step 1: Choose a set of parameters.

Step 2: Construct a neutrosophic cubic soft matrix for each chosen parameter for the set of parameters.

Step 3: Use the product of neutrosophic cubic soft matrices provided in Subsection 4.1 depending upon the set of parameters according to the situation.

Step 4: Find a max-min decision neutrosophic cubic soft matrix.

Step 5: Use the score function as in Definition 5, in order to get the suitable choice.

Example 5.3. Suppose we have a set U = {u₁, u₂, u₃} representing any three products of a company and suppose we have the E = {e₁, e₂, e₃, e₄} be the set of parameters that best describe the given products in the set U. Assume a couple is going to buy these products.

Step 1: Assume one member say A choose the parameters E = {e₁, e₂, e₃} and other member say B choose the parameters E = {e₂, e₃, e₄}.

Step 2: We now construct a neutrosophic cubic soft matrix for each chosen parameter for the member A and B as follows: $\begin{matrix} [c] l [a_{ij}]_{3 \times 3} = \\ [\begin{matrix} {\begin{matrix} {\begin{matrix} ([0.2, 0.4], \\ [0.2, 0.3], \\ [0.6, 0.7]), \end{matrix} \\ {(0.2, 0.7, 0.4) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0.4, 0.5], \\ [0.1, 0.3], \\ [0.3, 0.7]), \end{matrix} \\ {(0.8, 0.3, 0.4) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0, 0], \\ [0, 0], \\ [0, 0]), \end{matrix} \\ {(0, 0, 0) \end{matrix} \\ {\begin{matrix} {\begin{matrix} ([0.2, 0.4], \\ [0.2, 0.3], \\ [0.6, 0.7]), \end{matrix} \\ {(0.2, 0.7, 0.4) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0.1, 0.2], \\ [0.8, 0.9], \\ [0.3, 0.7]), \end{matrix} \\ {(0.1, 0.3, 0.4) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0, 0], \\ [0, 0], \\ [0, 0]), \end{matrix} \\ {(0, 0, 0) \end{matrix} \\ {\begin{matrix} {\begin{matrix} ([0.2, 0.4], \\ [0.2, 0.3], \\ [0.6, 0.7]), \end{matrix} \\ {(0.2, 0.7, 0.4) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0.2, 0.4], \\ [0.2, 0.3], \\ [0.6, 0.7]), \end{matrix} \\ {(0.2, 0.7, 0.4) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0, 0], \\ [0, 0], \\ [0, 0]), \end{matrix} \\ {(0, 0, 0) \end{matrix} \end{matrix}] \end{matrix}$ and $\begin{matrix} [c] l [b_{ik}]_{3 \times 3} = \\ [\begin{matrix} {\begin{matrix} {\begin{matrix} ([0.2, 0.4], \\ [0.2, 0.3], \\ [0.6, 0.7]), \end{matrix} \\ {(0.2, 0.7, 0.4) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0, 0], \\ [0, 0], \\ [0, 0]), \end{matrix} \\ {(0, 0, 0) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0.4, 0.5], \\ [0.1, 0.3], \\ [0.3, 0.7]), \end{matrix} \\ {(0.8, 0.3, 0.4) \end{matrix} \\ {\begin{matrix} {\begin{matrix} ([0.2, 0.4], \\ [0.2, 0.3], \\ [0.6, 0.7]), \end{matrix} \\ {(0.2, 0.7, 0.4) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0, 0], \\ [0, 0], \\ [0, 0]), \end{matrix} \\ {(0, 0, 0) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0.1, 0.2], \\ [0.8, 0.9], \\ [0.3, 0.7]), \end{matrix} \\ {(0.1, 0.3, 0.4) \end{matrix} \\ {\begin{matrix} {\begin{matrix} ([0.2, 0.4], \\ [0.2, 0.3], \\ [0.6, 0.7]), \end{matrix} \\ {(0.2, 0.7, 0.4) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0, 0], \\ [0, 0], \\ [0, 0]), \end{matrix} \\ {(0, 0, 0) \end{matrix} & {\begin{matrix} {\begin{matrix} ([0.2, 0.4], \\ [0.2, 0.3], \\ [0.6, 0.7]), \end{matrix} \\ {(0.2, 0.7, 0.4) \end{matrix} \end{matrix}] \end{matrix}$ Step 3: Now we find the product of neutrosophic cubic soft matrix using the And-product which is shown in Matrix "⊥".

Step 4: Find a max-min decision neutrosophic cubic soft matrix. Now ${NCM}_{m} [c_{ip}] = [D_{i 1}] = [max_{k} {{\tilde{t}}_{1_{k}}}, min_{k} {t_{2_{k}}}] .$

For i = 1 $\begin{matrix} [D_{11}] \\ = [\max_{k} {{\tilde{t}}_{1_{k}}}, \min_{k} {t_{1_{k}}}] \\ = [\begin{matrix} [c] c \max {{\tilde{t}}_{11}, {\tilde{t}}_{12}, {\tilde{t}}_{13}}, \\ \min {t_{11}, t_{12}, t_{13}} \end{matrix}] for k \in {1, 2, 3} \end{matrix}$ For i = 2 $\begin{matrix} [D_{21}] \\ = [\max_{k} {{\tilde{t}}_{1_{k}}}, \min_{k} {t_{1_{k}}}] \\ = [\begin{matrix} \max {{\tilde{t}}_{21}, {\tilde{t}}_{22}, {\tilde{t}}_{23}}, \\ \min {t_{21}, t_{22}, t_{23}} \end{matrix}] for k \in {1, 2, 3} \\ = [\begin{matrix} ([0, 0], [0.8, 0.9], [0.6, 0.7]), \\ {(0, 0.7, 0.4) \end{matrix}] \end{matrix}$ For i = 3 $\begin{matrix} [D_{31}] \\ = [\max_{k} {{\tilde{t}}_{1_{k}}}, \min_{k} {t_{1_{k}}}] \\ = [\begin{matrix} \max {{\tilde{t}}_{31}, {\tilde{t}}_{32}, {\tilde{t}}_{33}}, \\ \min {t_{31}, t_{32}, t_{33}} \end{matrix}] for k \in {1, 2, 3} \\ = [\begin{matrix} {([0, 0], [0.2, 0.3], [0.6, 0.7]), \\ {(0, 0.7, 0.4) \end{matrix}] \end{matrix}$ $\begin{matrix} Hence \\ {NCM}_{m} [c_{ip}] = [D_{i 1}] \\ = [\begin{matrix} {([0, 0], [0.1, 0.3], [0.3, 0.7]), {(0, 0.3, 0.4) \\ {([0, 0], [0.8, 0.9], [0.6, 0.7]), {(0, 0.7, 0.4) \\ {([0, 0], [0.2, 0.3], [0.6, 0.7]), {(0, 0.7, 0.4) \end{matrix}] . \end{matrix}$ Step 5: Now in order to get the most suitable choice we use the score function so that $s [D_{i 1}] = [\begin{matrix} ([0.33, 0.53], 0.433) \\ ([0.13, 0.2], 0.3) \\ ([0.33, 0.4], 0.3) \end{matrix}] .$ Thus ([0.33, 0.53], 0.433) is the most suitable choice.

Hence the product u₁ must be selected.

Remark: In this example, we have build an algorithm based on max-min product but we can use other products defined in Section 4.1 as well and it will not affect the conclusion.

6 Comparison analysis

This paper extends the idea provided in [4]. Mainly this paper is the extension of the paper of Deli and Said [7] where the authors combined the idea of neutrosophic sets with soft sets. After defining some basic operations on neutrosophic soft sets, they introduced the concept of neutrosophic soft matrix and their operators and finally NSM-decision making was developed to solve neutrosophic soft set based group decision making problems. We used neutrosophic cubic sets which are the generalization of neutrosophic sets and combined with soft sets. Neutrosophic cubic soft matrices have the ability to handle uncertain problems in a better way. Thus we developed a more general approach of neutrosophic cubic soft matrices and used it in MCDM-problems. Our model provide, more generalized information (internal or external) as compared to [4, 7]. Since our model of neutrosophic cubic soft matrices has the ability to handle the situation in the three intervals of the time period as, truth portion for past, indeterminacy portion for the present and falsity portion for the future time, which is main advantage of this model. If we restrict our model to one time frame then obviously it will reduces to previously defined models.

7 Conclusion

Since any neutrosophic cubic soft set $(N, I)$ is uniquely characterized by the neutrosophic cubic soft matrix (NCSM) and conversely, so we provide the generalized version of a fuzzy matrix in the form of neutrosophic cubic soft matrix. We discuss different types of products of neutrosophic cubic soft matrices and at the end we explain the use of neutrosophic cubic soft matrix theory in decision making. In future, our plan is to extract more results related with applications of neutrosophic cubic soft matrix theory. The main strength of our paper is that, it is the combination of different generalized theories so it would be more applicable in uncertain problems. As far as novelty and impact of this proposal is concerned, we develop a generalized approach by blending the three major concepts as mentioned above. So our model is more generalized and capable of handling many problems containing uncertainty. One limitation of our model is there is no pet rule to give certain membership to a certain element which could be handle in future in a more professional way. Other limitation which we feel is the use of different types of products and under which situation which product is most suitable, this could be handle in future. One of the main implication of our model is that it reduces the human’s biased nature and develop the consensus which is more reliable based on past, present and future framework of time which is not presented in previously defined models. In future, we aim to propose more applications of neutrosophic cubic soft matrices in different directions such as data storage in computers etc. We will define neutrosophic cubic soft eigen values, neutrosophic cubic soft eigen vectors and many more results. We also have a plan to combine different types of theories with matrix theory. Further our model can be used in many directions such as, (1) To construct the decision method for medical diagnosis by using neutrosophic cubic soft matrices. (2) In drug addiction effect in medical diagnosis by using neutrosophic cubic soft matrices. (3) The experimental results need to authenticate such as by comparing to other multi-criteria selection approaches. (4) In traffic control systems in order to controle the flow of traffic. (5) It can also be used in structures’s equation modeling. (6) Moreover, this approach will be useful to handle several realistic uncertainty problems such as problems in social, economic system, pattern recognition, feature extraction, image processing, game theory, coding theory, signature theory, dynamics consensus processes, opinion dynamics and in many social network group decision making problems in order to develop a reliable consensus by limiting the human’s biased nature.

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R ₁	e ₁	e ₂	·	·	·	e_n
x ₁	$({\tilde{A}}_{R_{1}} (x_{1}, e_{1}), ϑ_{R_{1}} (x_{1}, e_{1}))$	$({\tilde{A}}_{R_{1}} (x_{1}, e_{2}), ϑ_{R_{1}} (x_{1}, e_{2}))$	·	·	·	$({\tilde{A}}_{R_{1}} (x_{1}, e_{n}), ϑ_{R_{1}} (x_{1}, e_{n}))$
x ₂	$({\tilde{A}}_{R_{1}} (x_{2}, e_{1}), ϑ_{R_{1}} (x_{2}, e_{1}))$	$({\tilde{A}}_{R_{1}} (x_{2}, e_{2}), ϑ_{R_{1}} (x_{2}, e_{2}))$	·	·	·	$({\tilde{A}}_{R_{1}} (x_{2}, e_{n}), ϑ_{R_{1}} (x_{2}, e_{n}))$
·	·	·	·	·	·	·
·	·	·	·	·	·	·
·	·	·	·	·	·	·
x_m	$({\tilde{A}}_{R_{1}} (x_{m}, e_{1}), ϑ_{R_{1}} (x_{m}, e_{1}))$	$({\tilde{A}}_{R_{1}} (x_{m}, e_{2}), ϑ_{R_{1}} (x_{m}, e_{2}))$	·	·	·	$({\tilde{A}}_{R_{1}} (x_{m}, e_{n}), ϑ_{R_{1}} (x_{m}, e_{n}))$