**Neutrosophic Cubic Einstein Hybrid Geometric Aggregation Operators with Application in Prioritization Using Multiple Attribute Decision-Making Method**

#### **Khaleed Alhazaymeh 1, Muhammad Gulistan 2,\*, Majid Khan <sup>2</sup> and Seifedine Kadry <sup>3</sup>**


Received: 5 January 2019; Accepted: 18 February 2019; Published: 10 April 2019

**Abstract:** Viable collection is one of the imperative instruments of decision-making hypothesis. Collection operators are not simply the operators that normalize the value; they represent progressively broad values that can underline the entire information. Geometric weighted operators weight the values only, and the ordered weighted geometric operators weight the ordering position only. Both of these operators tend to the value that relates to the biggest weight segment. Hybrid collection operators beat these impediments of weighted total and request total operators. Hybrid collection operators weight the incentive as well as the requesting position. Neutrosophic cubic sets (NCs) are a classification of interim neutrosophic set and neutrosophic set. This distinguishing of neutrosophic cubic set empowers the decision-maker to manage ambiguous and conflicting data even more productively. In this paper, we characterized neutrosophic cubic hybrid geometric accumulation operator (NCHG) and neutrosophic cubic Einstein hybrid geometric collection operator (NCEHG). At that point, we outfitted these operators upon an everyday life issue which empoweredus to organize the key objective to develop the industry.

**Keywords:** neutrosophic cubic set; neutrosophic cubic hybrid geometric operator; neutrosophic cubic Einstein hybrid geometric operator; multiattributedecision-making (MADM)

#### **1. Introduction**

Life is loaded with indeterminacy and vagueness, which makes it hard to get adequate and exact information. This uncertain and obscure information can be tended to by fuzzy set [1], interim-valued fuzzy set (IVFS) [2,3], intuitionistic fuzzy set (IFS) [4], interim-valued intuitionistic fuzzy set (IVIFS) [5], cubic sets [6], neutrosophic set (Ns) [7], single-valued neutrosophic set (SVNs) [8], interim neutrosophic set (INs) [9], and neutrosophic cubic set [10]. Smarandache first investigated the hypothesis of neutrosophic sets [7].

Not long after thisinvestigation, it became a vital tool to manage obscure and conflicting information. The neutrosophic set comprises of three segments: truth enrollment, indeterminant participation, and deception enrollment. These segments can, likewise, be alluded to as participation, aversion, andnon-membership, and these segments range from ]0−, 1+[. For science and designing issues, Wang et al. [8] proposed the idea of a single-valued neutrosophic set, which is a class of neutrosophic set, where the parts of single-valued neutrosophic set are in [0,1]. Wang et al. stretched it

outto the interim neutrosophic set [9]. Jun et al. [10] consolidated both of these structures to frame the neutrosophic cubic set, which is the speculation of single-valued neutrosophic set and interim neutrosophic set. These structures drew scientistsinto apply it to various fields of sciences, building day-by-day life issues.

Decision-making is a basic instrument of everyday life issues. Analysts connected distinctive collection operators to neutrosophic sets and its augmentations. Zhan et. al. [11] took a shot at multicriteria decision-making on neutrosophic cubic sets. Banerjee et al. [12] utilized GRA(Grey Rational Analysis) for multicriteria decision-making on neutrosophic cubic sets. Lu and Ye [13] characterized cosine measure inneutrosophic cubic sets. Pramanik et al. [14] utilized a likeness measure to neutrosophic cubic sets. Shi and Ye [15] characterized Dombi total operators on neutrosophic cubic sets. Baolin et al. [16] connected Einstein accumulations to neutrosophic sets. Majid et al. [17] proposed neutrosophic cubic geometric and Einstein geometric collection operators. Different applied aspects of different types of fuzzy sets can be seen in [18–27].

A compelling accumulation is one of the imperative instruments of decision-making. Collection operators are not simply the operators that normalize the value, theyrepresent progressively broad values that can underline the entire data. The geometric weighted operator weights the values just where the requested weighted geometric collection operators weight the requesting position of values. In any case, the issue emerges when the load segments of weight vectors are so that one segment is a lot bigger than the other in parts of the weight vector. Motivated by such a circumstance, the thought of neutrosophic cubic crossbreed geometric and neutrosophic cubic Einstein hybrid geometric total operators are proposed. That is the reason we present the idea of neutrosophic cubic hybrid geometric and neutrosophic cubic Einstein hybrid geometric (NCEHG) collection operators. More often than not, the decision-making strategies are produced to pick one fitting option among the given. Be that as it may, frequently, in certain circumstances, we instead organize the option to pick a suitable one. Roused by such a circumstance, a technique is being created toprioritize the options. A numerical model is outfitted upon these operators to organize the vital objective to develop the industry.

#### **2. Preliminaries**

This section consists of some predefined definitions and results. We recommend the reader to see [1–3,6–10,16].

**Definition 1.** [1] *Mapping* ψ*: U* → [0, 1] *is called fuzzy set,* ψ(*u*) *is called membership function. Simply denoted by* ψ.

**Definition 2.** [2,3] *Mapping* Ψ- : *U* → *D*[0, 1]*, D*[0, 1] *has interval value of* [0, 1]*, and is called interval-valued fuzzy set(IVF). For all u* ∈ *U* Ψ-(*u*) = ψ*L*(*u*),ψ*U*(*u*) ψ*L*(*u*),ψ*U*(*u*) <sup>∈</sup> [0, 1] *and*ψ*L*(*u*) <sup>≤</sup> <sup>ψ</sup>*U*(*u*) *is membership degree of u in* Ψ-*. Simply denoted by* Ψ- = <sup>Ψ</sup>*L*, <sup>Ψ</sup>*U* .

**Definition 3.** [6] *A structure <sup>C</sup>* = *u*, Ψ-(*u*), Ψ(*u*) *<sup>u</sup>* <sup>∈</sup> *<sup>U</sup> is cubic set in U, in which* Ψ-(*u*) *is IVF in U, i.e.,* Ψ- = <sup>Ψ</sup>*L*, <sup>Ψ</sup>*U , and* <sup>Ψ</sup> *is fuzzy set in U. Simply denoted by <sup>C</sup>* = Ψ-, Ψ *. CU denotes collection of cubic sets in U.*

**Definition 4.** [7] *A structure N* = (*TN*(*u*), *IN*(*u*), *FN*(*u*)) *<sup>u</sup>* <sup>∈</sup> *<sup>U</sup> is neutrosophic set (Ns), where TN*(*u*), *IN*(*u*), *FN*(*u*) <sup>∈</sup> ]0−, 1+[ *and TN*(*u*), *IN*(*u*), *FN*(*u*) *are truth, indeterminacy, andfalsity function.*

**Definition 5.** [8] *A structure N* = (*TN*(*u*), *IN*(*u*), *FN*(*u*)) *<sup>u</sup>* <sup>∈</sup> *<sup>U</sup> is single value neutrosophic set (SVNs), where TN*(*u*), *IN*(*u*), *FN*(*u*) ∈ [0, 1] *are called truth, indeterminacy, and falsity functions respectively. Simply denoted by N* = (*TN*, *IN*, *FN*)*.*

*.*

**Definition 6.** [9] *An interval neutrosophic set (INs) in <sup>U</sup> is a structure <sup>N</sup>* = -*TN*(*u*),-*IN*(*u*), -*FN*(*u*) *<sup>u</sup>* <sup>∈</sup> *<sup>U</sup> where* -*TN*(*u*),-*IN*(*u*), -*FN*(*u*) ∈ *D*[0, 1] *respectively called truth, indeterminacy, and falsity function in U. Simply denoted by <sup>N</sup>* = -*TN*,-*IN*, -*FN . For convenience, we denote <sup>N</sup>* = -*TN*,-*IN*, -*FN by N* = -*TN* <sup>=</sup> *TL <sup>N</sup>*, *<sup>T</sup><sup>U</sup> N* ,-*IN* <sup>=</sup> *I L <sup>N</sup>*, *I U N* , -*FN* <sup>=</sup> *FL <sup>N</sup>*, *<sup>F</sup><sup>U</sup> N* .

**Definition 7.** [10] *A structure N* = *u*, -*TN*(*u*),-*IN*(*u*), -*FN*(*u*), *TN*(*u*), *IN*(*u*), *FN*(*u*) *<sup>u</sup>* <sup>∈</sup> *<sup>U</sup> is neutrosophic cubic set in U, in which* -*TN* <sup>=</sup> *TL <sup>N</sup>*, *<sup>T</sup><sup>U</sup> N* ,-*IN* <sup>=</sup> *I L <sup>N</sup>*, *I U N* , -*FN* <sup>=</sup> *FL <sup>N</sup>*, *<sup>F</sup><sup>U</sup> N is an interval neutrosophic set and* (*TN*, *IN*, *FN*) *is neutrosophic set in U. Simply denoted by <sup>N</sup>* <sup>=</sup> -*TN*,-*IN*, -*FN*, *TN*, *IN*, *FN* , [0, 0] ≤ -*TN* +-*IN* + -*FN* ≤ [3, 3]*, and* 0 ≤ *TN* + *IN* + *FN* ≤ 3*. N<sup>U</sup> denotes the collection of neutrosophic cubic sets in U. Simply denoted by N* = -*TN*,-*IN*, -*FN*, *TN*, *IN*, *FN* .

**Definition 8.** [16] *The t-operators are basically Union and Intersection operators in the theory of fuzzy sets which are denoted by t-conorm* (Γ∗) *and t-norm* (Γ)*, respectively. The role of t-operators is very important in fuzzy theory and its applications.*

**Definition 9.** [16] Γ<sup>∗</sup> : [0, 1] × [0, 1] → [0, 1] *is called t-conorm if it satisfies the following axioms. Axiom 1* Γ∗(1, *u*) = 1 *and* Γ∗(0, *u*) = 0 *Axiom 2* Γ∗(*u*, *v*) = Γ∗(*v*, *u*) *for all a and b. Axiom 3* Γ∗(*u*, Γ∗(*v*, *w*)) = Γ∗(Γ∗(*u*, *v*), *w*) *for all a, b, and c. Axiom 4 If u* ≤ *u and v* ≤ *v* , *then* Γ∗(*u*, *v*) ≤ Γ∗(*u* , *v* )

**Definition 10.** [16] Γ : [0, 1] × [0, 1] → [0, 1] *is called t-norm if it satisfies the following axioms. Axiom 1* Γ(1, *u*) = *u and* Γ(0, *u*) = 0 *Axiom 2* Γ(*u*, *v*) = Γ(*v*, *u*) *for all a and b. Axiom 3* Γ(*u*, Γ(*v*, *w*)) = Γ(Γ(*u*, *v*), *w*) *for all a, b, and c. Axiom 4 If u* ≤ *u and v* ≤ *v* , *then* Γ(*u*, *v*) ≤ Γ(*u* , *v* )

The t-conorms and t-norms families have a vast range, which correspond to unions and intersections, among these, Einstein sum and Einstein product are good choices since they give smooth approximations, like algebraic sum and algebraic product, respectively. Einstein sums ⊕*<sup>E</sup>* and Einstein products ⊗*<sup>E</sup>* are respectively the examples of t-conorm and t-norm:

$$
\Gamma\_E^\*(\mu, \upsilon) = \frac{\mu + \upsilon}{1 + \iota \upsilon},
$$

$$
\Gamma\_E(\mu, \upsilon) = \frac{\iota \upsilon}{1 + (1 - \iota)(1 - \upsilon)}
$$

.

**Definition 11.** [17] *The sum of two neutrosophic cubic sets, <sup>A</sup>* = -*TA*,-*IA*, -*FA*, *TA*, *IA*, *FA* , *where* -*TA* = *TL <sup>A</sup>*, *<sup>T</sup><sup>U</sup> A* ,-*IA* <sup>=</sup> *I L A*, *I U A* , -*FA* <sup>=</sup> *FL <sup>A</sup>*, *FU A* , *and <sup>B</sup>* = -*TB*,-*IB*, -*FB*, *TB*, *IB*, *FB* , *where* -*TB* <sup>=</sup> *TL <sup>B</sup>*, *<sup>T</sup><sup>U</sup> B* ,-*IB* = *I L B*, *I U B* , -*FB* <sup>=</sup> *FL <sup>B</sup>*, *FU B is defined as*

$$A \oplus B = \begin{pmatrix} \begin{bmatrix} T\_A^L + T\_B^L - T\_A^L T\_{B'}^L \, ^{I\underline{I}} + T\_B^{\underline{I}I} - T\_A^{\underline{I}I} T\_B^{\underline{I}I} \end{bmatrix} \\ \begin{bmatrix} I\_A^L + I\_B^L - I\_A^L I\_{B'}^L \, I\_A^{\underline{I}I} + I\_B^{\underline{I}I} - I\_A^{\underline{I}I} I\_B^{\underline{I}I} \end{bmatrix} \\ \begin{bmatrix} F\_A^L F\_{B'}^L \, F\_A^{\underline{I}} F\_B^{\underline{I}} \\ T\_A T\_{B'} I\_A I\_{B'} F\_A + F\_B - F\_A F\_B \end{bmatrix} \end{pmatrix}$$

*.*

**Definition 12.** [17] *The product between two neutrosophic cubic sets, <sup>A</sup>* = -*TA*,-*IA*, -*FA*, *TA*, *IA*, *FA* , *where* -*TA* <sup>=</sup> *TL <sup>A</sup>*, *<sup>T</sup><sup>U</sup> A* ,-*IA* <sup>=</sup> *I L A*, *I U A* , -*FA* <sup>=</sup> *FL <sup>A</sup>*, *FU A* , *and <sup>B</sup>* = -*TB*,-*IB*, -*FB*, *TB*, *IB*, *FB* , *where* -*TB* <sup>=</sup> *TL <sup>B</sup>*, *<sup>T</sup><sup>U</sup> B* ,-*IB* = *I L B*, *I U B* , -*FB* <sup>=</sup> *FL <sup>B</sup>*, *FU B is defined as*

$$A \otimes B = \begin{pmatrix} \begin{bmatrix} T\_A^L T\_{B'}^L \ T\_A^{\iota I} T\_B^{\iota I} \\ \left[ I\_A^L I\_{B'}^L I\_A^{\iota I} I\_B^{\iota I} \right]\_{\iota} \end{bmatrix} \\\ \begin{bmatrix} F\_A^L + F\_B^L - F\_A^L F\_B^L, F\_A^{\iota I} + F\_B^{\iota I} - F\_A^{\iota I} F\_B^{\iota I} \\ T\_A + T\_B - T\_A T\_B, I\_A + I\_B - I\_A I\_B, F\_A F\_B \end{bmatrix} \end{pmatrix}$$

**Definition 13.** [17] *The scalar multiplication on a neutrosophic cubic set <sup>A</sup>* = -*TA*,-*IA*, -*FA*, *TA*, *IA*, *FA* , *where* -*TA* <sup>=</sup> *TL <sup>A</sup>*, *<sup>T</sup><sup>U</sup> A* ,-*IA* <sup>=</sup> *I L A*, *I U A* , -*FA* <sup>=</sup> *FL <sup>A</sup>*, *<sup>F</sup><sup>U</sup> A* , *and a scalar k is defined.*

$$kA = \begin{pmatrix} \left[1 - \left(1 - T\_A^L\right)^k, 1 - \left(1 - T\_A^{\mathrm{II}}\right)^k\right] \\ \left[1 - \left(1 - I\_A^L\right)^k, 1 - \left(1 - I\_A^{\mathrm{II}}\right)^k\right] \\ \left[\left(F\_A^{\mathrm{I}}\right)^k, \left(F\_A^{\mathrm{II}}\right)^k\right] \\ \left(T\_A\right)^k \left(I\_A\right)^k, 1 - \left(1 - F\_A\right)^k \end{pmatrix}$$

The exponential multiplication is followed by the following result.

**Theorem 1.** [17] *Let <sup>A</sup>* = -*TA*,-*IA*, -*FA*, *TA*, *IA*, *FA* , *where* -*TA* <sup>=</sup> *TL <sup>A</sup>*, *<sup>T</sup><sup>U</sup> A* ,-*IA* <sup>=</sup> *I L A*, *I U A* , -*FA* <sup>=</sup> *FL <sup>A</sup>*, *FU A* , *is a neutrosophic cubic value, then, the exponential operation defined by*

$$A^k = \begin{pmatrix} \left[ \left(T\_A^L\right)^k, \left(T\_A^{\mathcal{U}}\right)^k \right] \\ \left[ \left(I\_A^L\right)^k, \left(I\_A^{\mathcal{U}}\right)^k \right] \\ \left[1 - \left(1 - F\_A^L\right)^k, 1 - \left(1 - F\_A^{\mathcal{U}}\right)^k \right] \\ 1 - \left(1 - T\_A\right)^k, 1 - \left(1 - I\_A\right)^k, \left(F\_A\right)^k \end{pmatrix}.$$

*where A<sup>k</sup>* = *A* ⊗ *A*⊗, ....⊗*A*(*k* − *times*), *moreover, A<sup>k</sup> is a neutrosophic cubic value for every positive value of k.*

**Definition 14.** [17] *The Einstein sum between two neutrosophic cubic sets <sup>A</sup>* = -*TA*,-*IA*, -*FA*, *TA*, *IA*, *FA* , *where* -*TA* <sup>=</sup> *TL <sup>A</sup>*, *<sup>T</sup><sup>U</sup> A* ,-*IA* <sup>=</sup> *I L A*, *I U A* , -*FA* <sup>=</sup> *FL <sup>A</sup>*, *FU A* , *and <sup>B</sup>* = -*TB*,-*IB*, -*FB*, *TB*, *IB*, *FB* , *where* -*TB* <sup>=</sup> *TL <sup>B</sup>*, *<sup>T</sup><sup>U</sup> B* ,-*IB* = *I L B*, *I U B* , -*FB* <sup>=</sup> *FL <sup>B</sup>*, *FU B is defined as*

$$A \oplus\_{E} B = \begin{pmatrix} \begin{bmatrix} \frac{T\_{A}^{I} + T\_{B}^{I}}{1 + T\_{A}^{I} \frac{T\_{B}^{I}}{A} \cdot \frac{T\_{A}^{II} + T\_{B}^{II}}{1 + T\_{A}^{II} \frac{T\_{B}^{I}}{A}} \end{bmatrix} \end{pmatrix} \\\\ \begin{bmatrix} \frac{I\_{A}^{I} + I\_{B}^{I}}{1 + I\_{A}^{I} \frac{T\_{B}^{I}}{A} \cdot \frac{I\_{A}^{II} + I\_{B}^{I}}{1 + I\_{A}^{I} \frac{T\_{B}^{I}}{A}} \end{bmatrix} \end{pmatrix}$$

$$\begin{bmatrix} \frac{F\_{A}^{I} F\_{B}^{I}}{1 + (1 - F\_{A}^{I}) \left(1 - F\_{B}^{I}\right)} \prime \frac{F\_{A}^{II} F\_{B}^{I}}{1 + (1 - F\_{A}^{I}) \left(1 - F\_{B}^{I}\right)} \end{bmatrix}\_{\begin{bmatrix} I\_{A} I\_{B} \\ 1 + (1 - F\_{A}^{I}) \left(1 - F\_{B}^{I}\right) \end{bmatrix}}$$

.

**Definition 15.** [17] *The Einstein product between two neutrosophic cubic sets, <sup>A</sup>* = -*TA*,-*IA*, -*FA*, *TA*, *IA*, *FA* , *where* -*TA* <sup>=</sup> *TL <sup>A</sup>*, *<sup>T</sup><sup>U</sup> A* ,-*IA* <sup>=</sup> *I L A*, *I U A* , -*FA* <sup>=</sup> *FL <sup>A</sup>*, *FU A* , *and <sup>B</sup>* = -*TB*,-*IB*, -*FB*, *TB*, *IB*, *FB* , *where* -*TB* = *TL <sup>B</sup>*, *<sup>T</sup><sup>U</sup> B* ,-*IB* <sup>=</sup> *I L B*, *I U B* , -*FB* <sup>=</sup> *FL <sup>B</sup>*, *<sup>F</sup><sup>U</sup> B is defined as*

$$A \otimes\_{E} B = \begin{pmatrix} \begin{bmatrix} \frac{T\_{A}^{L}T\_{B}^{L}}{1 + \left(1 - T\_{A}^{L}\right)\left(1 - T\_{B}^{L}\right)} \prime \frac{T\_{A}^{LI}T\_{B}^{II}}{1 + \left(1 - T\_{A}^{L}\right)\left(1 - T\_{B}^{L}\right)} \prime \\\\ \begin{bmatrix} \frac{I\_{A}^{L}I\_{B}^{L}}{1 + \left(1 - I\_{A}^{L}\right)\left(1 - I\_{B}^{L}\right)} \prime \frac{I\_{A}^{I}I\_{B}^{I}}{1 + \left(1 - I\_{A}^{I}\right)\left(1 - I\_{B}^{I}\right)} \end{bmatrix} \prime \\\\ \begin{bmatrix} \frac{F\_{A}^{I} + F\_{B}^{I}}{1 + F\_{A}^{I}F\_{B}^{I}} \prime \frac{F\_{A}^{I} + F\_{B}^{I}}{1 + F\_{A}^{I}F\_{B}^{I}} \\\ \frac{T\_{A} + T\_{B}}{1 + T\_{A}T\_{B}} \prime \frac{I\_{A} + I\_{B}}{1 + I\_{A}I\_{B}} \prime \frac{F\_{A}F\_{B}}{1 + \left(1 - F\_{A}\right)\left(1 - F\_{B}\right)} \end{bmatrix} \end{pmatrix}$$

.

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ .

,

**Definition 16.** [17] *The scalar multiplication on a neutrosophic cubic set, <sup>A</sup>* = -*TA*,-*IA*, -*FA*, *TA*, *IA*, *FA* , *where* -*TA* <sup>=</sup> *TL <sup>A</sup>*, *<sup>T</sup><sup>U</sup> A* ,-*IA* <sup>=</sup> *I L A*, *I U A* , -*FA* <sup>=</sup> *FL <sup>A</sup>*, *<sup>F</sup><sup>U</sup> A* , *and a scalar k is defined*

$$k\_{E}A = \begin{pmatrix} \begin{bmatrix} \left(1+T\_{A}^{L}\right)^{k}-\left(1-T\_{A}^{L}\right)^{k} & \left(1+T\_{A}^{L}\right)^{k}-\left(1-T\_{A}^{L}\right)^{k} \\ \left(1+T\_{A}^{L}\right)^{k}+\left(1-T\_{A}^{L}\right)^{k} & \left(1+T\_{A}^{L}\right)^{k}+\left(1-T\_{A}^{L}\right)^{k} \end{bmatrix}, \\ \begin{bmatrix} \left(1+I\_{A}^{L}\right)^{k}-\left(1-I\_{A}^{L}\right)^{k} & \left(1+I\_{A}^{L}\right)^{k}-\left(1-I\_{A}^{L}\right)^{k} \\ \left(1+I\_{A}^{L}\right)^{k}+\left(1-I\_{A}^{L}\right)^{k} & \left(1+I\_{A}^{L}\right)^{k}+\left(1-I\_{A}^{L}\right)^{k} \end{bmatrix}, \\ \begin{bmatrix} 2\left(F\_{A}^{L}\right)^{k} & 2\left(F\_{A}^{L}\right)^{k} \\ \left(2-F\_{A}^{L}\right)^{k}+\left(F\_{A}^{L}\right)^{k} & \left(2-F\_{A}^{L}\right)^{k}+\left(F\_{A}^{L}\right)^{k} \end{bmatrix}, \\ \hline 2\left(T\_{A}\right)^{k}+\left(T\_{A}\right)^{k}+\left(T\_{A}\right)^{k}+\left(1-T\_{A}\right)^{k} & \left(1+F\_{A}\right)^{k}+\left(1-F\_{A}\right)^{k} \\ \hline \end{bmatrix}$$

The Einstein exponential multiplication is followed by the following result.

**Theorem 2.** [17] *Let <sup>A</sup>* = -*TA*,-*IA*, -*FA*, *TA*, *IA*, *FA* , *where* -*TA* <sup>=</sup> *TL <sup>A</sup>*, *<sup>T</sup><sup>U</sup> A* ,-*IA* <sup>=</sup> *I L A*, *I U A* , -*FA* <sup>=</sup> *FL <sup>A</sup>*, *FU A* , *is a neutrosophic cubic value, then, the exponential operation defined by*

*AEk* = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ <sup>2</sup>(*T<sup>L</sup> A*)*k* (2−*T<sup>L</sup> A*) *k* +(*T<sup>L</sup> A*) *<sup>k</sup>* , <sup>2</sup>(*T<sup>U</sup> A* )*k* (2−*T<sup>U</sup> A* ) *k* +(*T<sup>U</sup> A* ) *k* , <sup>2</sup>(*<sup>I</sup> L A*)*k* (2−*I L A*) *k* +(*I L A*) *<sup>k</sup>* , <sup>2</sup>(*<sup>I</sup> U A* )*k* (2−*I U A* ) *k* +(*I U A* ) *k* , (1+*FL A*) *k* <sup>−</sup>(1−*FL A*) *k* (1+*FL A*) *k* +(1−*FL A*) *k* , (1+*F<sup>U</sup> A* ) *k* <sup>−</sup>(1−*FU A* ) *k* (1+*FU A* ) *k* +(1−*FU A* ) *k* , (1+*TA*) *k* −(1−*TA*) *k* (1+*TA*) *k* +(1−*TA*) *<sup>k</sup>* , (1+*IA*) *k* −(1−*IA*) *k* (1+*IA*) *k* +(1−*IA*) *<sup>k</sup>* , <sup>2</sup>(*FA*) *k* (2−*FA*) *k* +(*FA*) *k* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

*where AE<sup>k</sup>* = *A* ⊗*<sup>E</sup> A* ⊗*<sup>E</sup>* ... ⊗*<sup>E</sup> A*(*k* − *times*), *moreover, AE<sup>k</sup> is a neutrosophic cubic value for every positive value of k.*

To compare two neutrosophic cubic values the score function is defined.

**Definition 17.** [17] *Let <sup>N</sup>* = -*TN*,-*IN*, -*FN*, *TN*, *IN*, *FN* , *where* -*TN* <sup>=</sup> *TL <sup>N</sup>*, *<sup>T</sup><sup>U</sup> N* ,-*IN* <sup>=</sup> *I L <sup>N</sup>*, *I U N* , -*FN* <sup>=</sup> *FL <sup>N</sup>*, *FU N is a neutrosophic cubic value, and the score function is defined as*

$$S(N) = \left[T\_N^L - F\_N^L + T\_N^{II} - F\_N^{II} + T\_N - F\_N\right].$$

If the score function of two values are equal, the accuracy function is used.

**Definition 18.** [17] *Let <sup>N</sup>* = -*TN*,-*IN*, -*FN*, *TN*, *IN*, *FN* , *where* -*TN* <sup>=</sup> *TL <sup>N</sup>*, *<sup>T</sup><sup>U</sup> N* ,-*IN* <sup>=</sup> *I L <sup>N</sup>*, *I U N* , -*FN* <sup>=</sup> *FL <sup>N</sup>*, *FU N is a neutrosophic cubic value, and theaccuracy function is defined as*

$$H(u) = \frac{1}{9} \Big\{ T\_N^L + I\_N^L + F\_N^L + T\_N^{II} + I\_N^{II} + F\_N^{II} + T\_N + I\_N + F\_N \Big\}.$$

The following definition describes the comparison relation between two neutrosophic cubic values.

**Definition 19.** [17] *Let N*1*, N*<sup>2</sup> *be two neutrosophic cubic values, with core functions SN*<sup>1</sup> , *SN*<sup>2</sup> *, and accuracy function HN*<sup>1</sup> , *HN*<sup>2</sup> *. Then,*

	- *(i) HN*<sup>1</sup> > *HN*<sup>2</sup> ⇒ *N*<sup>1</sup> > *N*<sup>2</sup>
	- *(ii) HN*<sup>1</sup> = *HN*<sup>2</sup> ⇒ *N*<sup>1</sup> = *N*<sup>2</sup>

**Definition 20.** [17] *The neutrosophic cubic weighted geometric operator (NCWG) is defined as*

$$\text{NCWG} : \mathbb{R}^m \to \mathbb{R} \text{ defined by } \text{NCWG}\_w(N\_1, N\_2, \dots, N\_m) = \bigotimes\_{j=1}^m N\_j^{w\_j}.$$

*where the weight W* = (*w*1, *<sup>w</sup>*2, ..., *wm*)*<sup>T</sup> of Nj*(*<sup>j</sup>* <sup>=</sup> 1, 2, 3, ..., *<sup>m</sup>*), *such that wj* <sup>∈</sup> [0, 1] *and <sup>m</sup> j*=1 = 1.

**Definition 21.** [17] *The neutrosophic cubic ordered weighted geometric operator(NCOWG) is defined as*

$$\text{C}\ \text{NCOWG}:\ \text{R}^{\text{m}} \to \text{R defined by NCOWG}\_{\text{w}}(\text{N}\_{1}, \text{N}\_{2}, ..., \text{N}\_{\text{m}}) = \stackrel{\text{m}}{\underset{j=1}{\otimes}} \text{N}\_{(\text{\textdegree{'}})^{j}\_{j}}^{\text{w}\_{j}}$$

*where N*(γ)*<sup>j</sup> is the descending ordered neutrosophic cubic values, W* = (*w*1, *w*2, ..., *wm*)*<sup>T</sup> of Nj*(*j* = 1, 2, 3, ..., *m*), *such that wj* <sup>∈</sup> [0, 1] *and <sup>m</sup> j*=1 = 1.

**Definition 22.** [17] *The neutrosophic cubic Einstein weighted geometric operator(NCEWG) is defined as*

$$\text{NCCEWG}: \mathbb{R}^m \to \mathbb{R} \text{ defined by } \text{NCECWG}\_w(\text{N}\_1, \text{N}\_2, \dots, \text{N}\_m) = \stackrel{\text{m}}{\otimes} \left(\text{N}\_j\right)^{E^{\nu\_j}},$$

*where W* = (*w*1, *<sup>w</sup>*2, ..., *wm*)*<sup>T</sup> is weight of Nj*(*<sup>j</sup>* <sup>=</sup> 1, 2, 3, ..., *<sup>m</sup>*),*such that wj* <sup>∈</sup> [0, 1] *and <sup>m</sup> j*=1 = 1.

**Definition 23.** [17] *Order neutrosophic cubic Einstein weighted geometric operator(NCEOWG) is defined as*

$$\text{NCEOWG}: \mathbb{R}^m \to \mathbb{R} \text{ by } \text{NCEOWG}\_w(\text{N}\_1, \text{N}\_2, ..., \text{N}\_m) = \stackrel{\text{m}}{\otimes} \begin{pmatrix} \text{B}\_j \end{pmatrix}^{E^{\nu\_j}}.$$

*where Bj is the jth largest neutrosophic cubic value, and W* = (*w*1, *w*2, ..., *wm*)*<sup>T</sup> is weight of Nj*(*j* = 1, 2, 3, ..., *m*), *such that wj* <sup>∈</sup> [0, 1] *and <sup>m</sup> j*=1 = 1.

The neutrosophic cubic geometric aggregation operators weight only the neutrosophic cubic values, whereas neutrosophic cubic order geometric aggregation operators weight the orders of the values first then weight them. In the two cases, the amassed values that focused on the value relate to the biggest weight. The accompanying precedents represent the impediments of the NCWG and NCEWG.

Let *W* = (0.7, 0.2, 0.1) be the weight corresponding to the neutrosophic cubic values

$$\begin{array}{l} N\_1 = ([0.2, 0.7], [0.2, 0.4], [0.2, 0.5], 0.8, 0.5, 0.8) \\ N\_2 = ([0.4, 0.6], [0.4, 0.7], [0.1, 0.3], 0.2, 0.6, 0.5) \\ N\_3 = ([0.5, 0.8], [0.3, 0.6], [0.4, 0.9], 0.5, 0.8, 0.9) \end{array}$$

Then *S*(*N*1) = 0.2, *S*(*N*2) = 0.3, and *S*(*N*3) = −0.4.

Therefore, *NCWG* = ([0.251, 0.688], [0.239, 0.465], [0.204, 0.524], 0.711, 0.563, 0.737) and *NCOWG* = ([0.356, 0.636], [0.338, 0.616], [0.155, 0.544], 0.421, 0.609, 0.737).

We observe that the higher the weight component, the aggregated value will tend to the corresponding neutrosophic cubic value of that vector. In NCWG, the value tendsto *N*1, as the weight that corresponds to *N*<sup>1</sup> is highest, and in NCOWG, the highest component of weight corresponds to *N*2. This situation often arises in aggregation problems. Motivated by such a situation, the idea of neutrosophic cubic hybrid geometric and neutrosophic cubic Einstein hybrid geometric operators are proposed.

#### **3. Neutrosophic Cubic Hybrid Geometric and Neutrosophic Cubic Einstein Geometric Operators**

This segment comprises of the following subsections. In Section 3.1 neutrosophic cubic crossbreed, the geometric operator is characterized. In Section 3.2 neutrosophic cubic Einstein crossbreed, the geometric operator is characterized. In Section 3.3, a calculation is characterized to organize the neutrosophic cubic values utilizing these tasks. In Section 3.4, a numerical model is outfitted upon Section 3.3.

#### *3.1. Neutrosophic Cubic Hybrid Geometric Operator*

NCWG operator weights only the neutrosophic cubic values, where NCOWG weights only the ordering positions. The idea of neutrosophic cubic hybrid geometric aggregation operators is developed to overcome these limitations. NCHG weights both the neutrosophic cubic values and its order positioning as well.

**Definition 24.** *NCHG* : Ω*<sup>m</sup>* → Ω *is a mapping from m-dimenion, which has associated weight W* = (*w*1, *<sup>w</sup>*2, ..., *wm*)*T,such that wj* <sup>∈</sup> [0, 1] *and <sup>m</sup> j*=1 *wj* = 1, *such that*

$$\text{NCECOWG:} \, R''' \to R \text{ by } \text{NCECOWG}\_{\bowtie}(N\_1, N\_2, ..., N\_m) = \stackrel{m}{\otimes} \left(B\_{\slash}\right)^{E\_{\ll}},$$

*where N*∼ *<sup>j</sup> jth largest of the weighted neutrosophic cubic values N*∼ (*j*) *N*∼ (*j*) <sup>=</sup> *<sup>N</sup>mwj j* , *j* = 1, 2, 3, , , *m*), *W* = (*w*1, *w*2, ..., *wm*)*<sup>T</sup> , such that wj* <sup>∈</sup> [0, 1] *and <sup>m</sup> j*=1 *wj* = 1*, and m is the balancing coe*ffi*cient.*

**Theorem 3.** *Let Nj* <sup>=</sup> -*TNj* ,-*INj* , -*FNj* , *TNj* , *INj* , *FNj* , *where* -*TNj* = *TL Nj* , *T<sup>U</sup> Nj* ,-*INj* = *I L Nj* , *I U Nj* , -*FNj* = *FL Nj* , *FU Nj* , (*j* = 1, 2, ..., *m*) *be collection of neutrosophic cubic values, then the aggregated value (NCHWG) is also a cubic value and*

$$\text{NCHG}(N\_{j}) = \begin{pmatrix} \begin{bmatrix} \frac{m}{\otimes} \left( T\_{\circ \circ (j)}^{\sim L} \right)^{w\_{j}} \, \Big|\limits\_{j=1}^{w\_{j}} \left( T\_{\circ (j)}^{\sim L} \right)^{w\_{j}} \Big| \begin{bmatrix} \frac{m}{\otimes} \left( I^{\sim L}\_{\circ (j)} \right)^{w\_{j}} \left( I^{\sim L}\_{\circ (j)} \right)^{w\_{j}} \\ \frac{m}{\otimes} \left( 1 - F^{\sim L}\_{\circ (j)} \right)^{w\_{j}} \, 1 - \mathop{\otimes} \Big( 1 - F^{\sim L}\_{\circ (j)} \Big)^{w\_{j}} \end{bmatrix} \\\ \begin{array}{ll} \begin{array}{ll} -\frac{m}{\otimes} \left( 1 - T\_{\circ (j)}^{\sim L} \right)^{w\_{j}} , 1 - \mathop{\otimes} \Big( 1 - F^{\sim L}\_{\circ (j)} \Big)^{w\_{j}} \end{array} \Big| \\\ 1 - \frac{m}{j=1} \left( 1 - T\_{\circ (j)}^{\sim L} \right)^{w\_{j}} \, 1 - \mathop{\otimes} \Big( 1 - \mathop{I}\_{\circ (j)}^{\sim} \Big)^{w\_{j}} \Big( \mathbb{P}\_{\circ (j)}^{\sim} \Big)^{w\_{j}} \end{array} \end{pmatrix}$$

,

*the weight W* = (*w*1, *<sup>w</sup>*2, ..., *wm*)*T, such that wj* <sup>∈</sup> [0, 1] *and <sup>m</sup> j*=1 = 1.

**Proof.** By mathematical induction for *m* = 2, using

$$\mathop{\otimes}\_{j=1}^{2} \mathcal{N}\_{j}^{\text{w}\_{j}} = \mathcal{N}\_{1}^{\text{w}\_{1}} \otimes \mathcal{N}\_{2}^{\text{w}\_{2}}.$$

= ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ (*T<sup>L</sup> N*σ(*j*) ) *w*<sup>1</sup> ,(*T<sup>U</sup> N*σ(*j*) )*w*<sup>1</sup> , (*I L N*σ(*j*) ) *w*<sup>1</sup> ,(*I U N*σ(*j*) )*w*<sup>1</sup> , 1 − <sup>1</sup> <sup>−</sup> *<sup>F</sup><sup>L</sup> N*σ(*j*) *w*<sup>1</sup> , 1 − <sup>1</sup> <sup>−</sup> *<sup>F</sup><sup>U</sup> N*σ(*j*) *w*<sup>1</sup> , 1 − 1 − *TN*σ(*j*) *<sup>w</sup>*<sup>1</sup> , 1 − 1 − *IN*σ(*j*) *<sup>w</sup>*<sup>1</sup> , *FN*σ(*j*) *<sup>w</sup>*<sup>1</sup> ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⊗ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ (*T<sup>L</sup> N*σ(*j*) ) *w*<sup>2</sup> ,(*T<sup>U</sup> N*σ(*j*) )*w*<sup>2</sup> , (*I L N*σ(*j*) ) *w*<sup>2</sup> ,(*I U N*σ(*j*) )*w*<sup>2</sup> , 1 − <sup>1</sup> <sup>−</sup> *FL N*σ(*j*) *w*<sup>2</sup> , 1 − <sup>1</sup> <sup>−</sup> *<sup>F</sup><sup>U</sup> N*σ(*j*) *w*<sup>2</sup> , 1 − 1 − *TN*σ(*j*) *<sup>w</sup>*<sup>2</sup> , 1 − 1 − *IN*σ(*j*) *<sup>w</sup>*<sup>2</sup> , *FN*σ(*j*) *<sup>w</sup>*<sup>2</sup> ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ <sup>2</sup> ⊗ *j*=1 (*T<sup>L</sup> N*σ(*j*) ) *wj* , 2 ⊗ *j*=1 (*T<sup>U</sup> N*σ(*j*) ) *wj* , <sup>2</sup> ⊗ *j*=1 (*I L N*σ(*j*) ) *wj* , 2 ⊗ *j*=1 (*I U N*σ(*j*) ) *wj* , <sup>1</sup> <sup>−</sup> <sup>2</sup> ⊗ *j*=1 1 − *F<sup>L</sup> N*σ(*j*) *wj* , 1 <sup>−</sup> <sup>2</sup> ⊗ *j*=1 <sup>1</sup> <sup>−</sup> *<sup>F</sup><sup>U</sup> N*σ(*j*) *wj* , <sup>1</sup> <sup>−</sup> <sup>2</sup> ⊗ *j*=1 1 − *TN*σ(*j*) *wj* , 1 <sup>−</sup> <sup>2</sup> ⊗ *j*=1 1 − *IN*σ(*j*) *wj* , 2 ⊗ *j*=1 *FN*σ(*j*) *wj* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ .

Let the results hold for *m*.

$$\mathop{\otimes}\_{j=1}^{m}\mathop{\mathcal{N}}\_{j}^{w\_{j}} = \begin{pmatrix} \left[\mathop{\otimes}\limits\_{j=1}^{m} \left(T\_{N\_{\sigma(j)}}^{\mathcal{L}}\right)^{w\_{j}} \mathop{\otimes}\_{j=1}^{m} \left(T\_{N\_{\sigma(j)}}^{\mathcal{U}}\right)^{w\_{j}}\right] \left[\mathop{\otimes}\_{j=1}^{m} \left(I\_{N\_{\sigma(j)}}^{\mathcal{L}}\right)^{w\_{j}} \mathop{\otimes}\_{j=1}^{m} \left(I\_{N\_{\sigma(j)}}^{\mathcal{U}}\right)^{w\_{j}}\right],\\ \left[\mathbbm{1} - \mathop{\otimes}\limits\_{j=1}^{m} \left(1 - F\_{N\_{\sigma(j)}}^{\mathcal{L}}\right)^{w\_{j}}, \mathbbm{1} - \mathop{\otimes}\limits\_{j=1}^{m} \left(1 - F\_{N\_{\sigma(j)}}^{\mathcal{U}}\right)^{w\_{j}}\right],\\ \mathbbm{1} - \mathop{\otimes}\limits\_{j=1}^{m} \left(1 - T\_{N\_{\sigma(j)}}\right)^{w\_{j}}, \mathbbm{1} - \mathop{\otimes}\limits\_{j=1}^{m} \left(1 - I\_{N\_{\sigma(j)}}\right)^{w\_{j}} \mathop{\otimes}\_{j=1}^{m} \left(F\_{N\_{\sigma(j)}}\right)^{w\_{j}} \end{pmatrix}$$

We prove the result for *m* + 1,

$$\mathbf{as}\left(\mathbf{N}\_{j+1}\right)^{\mathbf{w}\_{j+1}} = \begin{pmatrix} \left[\left(\boldsymbol{T}\_{\boldsymbol{N}\_{j+1}}^{\mathrm{L}}\right)^{\boldsymbol{w}\_{j+1}}, \left(\boldsymbol{T}\_{\boldsymbol{N}\_{j+1}}^{\mathrm{LI}}\right)^{\boldsymbol{w}\_{j+1}}\right] \left[\left(\boldsymbol{I}\_{\boldsymbol{N}\_{j+1}}^{\mathrm{L}}\right)^{\boldsymbol{w}\_{j+1}}, \left(\boldsymbol{I}\_{\boldsymbol{N}\_{j+1}}^{\mathrm{LI}}\right)^{\boldsymbol{w}\_{j+1}}\right] \\\left[1-\left(\boldsymbol{1}-\boldsymbol{F}\_{\boldsymbol{N}\_{j+1}}^{\mathrm{L}}\right)^{\boldsymbol{w}\_{j+1}}, \left\boldsymbol{1}-\left(\boldsymbol{1}-\boldsymbol{F}\_{\boldsymbol{N}\_{j+1}}^{\mathrm{LI}}\right)^{\boldsymbol{w}\_{j+1}}\right] \boldsymbol{f}\_{\boldsymbol{N}\_{j+1}} \\\-\left(\boldsymbol{1}-\boldsymbol{T}\_{\boldsymbol{N}\_{j+1}}\right)^{\boldsymbol{w}\_{j+1}}, \left\boldsymbol{1-\left(\boldsymbol{1}-\boldsymbol{I}\_{\boldsymbol{N}\_{j+1}}\right)^{\boldsymbol{w}\_{j+1}}, \left\boldsymbol{\left$$

*m* ⊗ *j*=1 *Nwj <sup>j</sup>* <sup>⊕</sup> *<sup>N</sup>wj*+<sup>1</sup> *<sup>j</sup>*+<sup>1</sup> = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *<sup>m</sup>* ⊗ *j*=1 (*T<sup>L</sup> <sup>N</sup>*σ(*j*) ) *wj* , *m* ⊗ *j*=1 (*T<sup>U</sup> <sup>N</sup>*σ(*j*) ) *wj* , *<sup>m</sup>* ⊗ *j*=1 (*I L <sup>N</sup>*σ(*j*) ) *wj* , *m* ⊗ *j*=1 (*I U <sup>N</sup>*σ(*j*) ) *wj* , <sup>1</sup> <sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 1 − *FL <sup>N</sup>*σ(*j*) *wj* , 1 <sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 1 − *F<sup>U</sup> <sup>N</sup>*σ(*j*) *wj* , <sup>1</sup> <sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 1 − *TN*σ(*j*) *wj* , 1 <sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 1 − *IN*σ(*j*) *wj* , *m* ⊗ *j*=1 *FN*σ(*j*) *wj* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⊕ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ (*T<sup>L</sup> Nj*+1 ) *wj*+<sup>1</sup> ,(*T<sup>U</sup> Nj*+1 ) *wj*+1 , (*I L Nj*+1 ) *wj*+<sup>1</sup> ,(*I U Nj*+1 ) *wj*+1 , 1 − 1 − *F<sup>L</sup> Nj*+1 *wj*+<sup>1</sup> , 1 − 1 − *F<sup>U</sup> Nj*+1 *wj*+<sup>1</sup> , 1 − 1 − *TNj*<sup>+</sup><sup>1</sup> *wj*+<sup>1</sup> , 1 − 1 − *INj*<sup>+</sup><sup>1</sup> *wj*+<sup>1</sup> *I* ,(*FNj*<sup>+</sup><sup>1</sup> ) *wj*+1 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

*m*+1 ⊗ *j*=1 *<sup>N</sup>wj <sup>j</sup>* = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *<sup>m</sup>* ⊗ *j*=1 *TL N*σ(*j*) *wj TL Nm*+1 *wm*+<sup>1</sup> , *m* ⊗ *j*=1 *T<sup>U</sup> N*σ(*j*) *wj T<sup>U</sup> Nm*+1 *wm*+<sup>1</sup> , *<sup>m</sup>* ⊗ *j*=1 *I L N*σ(*j*) *wj I L Nm*+1 *wm*+<sup>1</sup> , *m* ⊗ *j*=1 *I U N*σ(*j*) *wj I U Nm*+1 *wm*+<sup>1</sup> , ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ <sup>1</sup> <sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1 − *FL N*σ(*j*) ) *wj* <sup>+</sup> <sup>1</sup> <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>F</sup><sup>L</sup> Nm*+1 ) *wm*+<sup>1</sup> <sup>−</sup> <sup>1</sup> <sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1 − *FL N*σ(*j*) ) *wj* 1 − (1 − *FL Nm*+1 ) *wm*+1 , <sup>1</sup> <sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1 − *FU N*σ(*j*) ) *wj* <sup>+</sup> <sup>1</sup> <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>F</sup><sup>U</sup> Nm*+1 ) *wm*+<sup>1</sup> <sup>−</sup> <sup>1</sup> <sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1 − *F<sup>U</sup> N*σ(*j*) ) *wj* 1 − (1 − *FU Nm*+1 ) *wm*+1 , ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ , <sup>1</sup> <sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1 − *TN*σ(*j*) ) *wj* + 1 − (1 − *TNm*<sup>+</sup><sup>1</sup> ) *wm*+<sup>1</sup> <sup>−</sup> <sup>1</sup> <sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1 − *TN*σ(*j*) ) *wj* 1 − (1 − *TNm*<sup>+</sup><sup>1</sup> ) *wm*+<sup>1</sup> <sup>1</sup> <sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1 − *IN*σ(*j*) ) *wj* + 1 − (1 − *INm*<sup>+</sup><sup>1</sup> ) *wm*+<sup>1</sup> <sup>−</sup> <sup>1</sup> <sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1 − *IN*σ(*j*) ) *wj* 1 − (1 − *INm*<sup>+</sup><sup>1</sup> ) *wm*+<sup>1</sup> *m* ⊗ *j*=1 *FN*σ(*j*) *wj FNm*<sup>+</sup><sup>1</sup> *wm*+<sup>1</sup> = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *m*+1 ⊗ *j*=1 *TL N*σ(*j*) *wj* , *m*+1 ⊗ *j*=1 *T<sup>U</sup> N*σ(*j*) *wj* , *m*+1 ⊗ *j*=1 *I L N*σ(*j*) *wj* , *m*+1 ⊗ *j*=1 *I U N*σ(*j*) *wj* , ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ <sup>2</sup> <sup>−</sup> *<sup>m</sup>*+<sup>1</sup> ⊗ *j*=1 (1 − *FL N*σ(*j*) ) *wj* <sup>−</sup> <sup>1</sup> <sup>+</sup> *<sup>m</sup>*+<sup>1</sup> ⊗ *j*=1 (1 − *F<sup>L</sup> N*σ(*j*) ) *wj* + (<sup>1</sup> <sup>−</sup> *<sup>F</sup><sup>L</sup> Nm*+1 ) *wm*+<sup>1</sup> <sup>−</sup> *m*+1 ⊗ *j*=1 (1 − *FL N*σ(*j*) ) *wj* (1 − *FL Nm*+1 ) *wm*+<sup>1</sup> , <sup>2</sup> <sup>−</sup> *<sup>m</sup>*+<sup>1</sup> ⊗ *j*=1 (1 − *F<sup>U</sup> N*σ(*j*) ) *wj* <sup>−</sup> <sup>1</sup> <sup>+</sup> *<sup>m</sup>*+<sup>1</sup> ⊗ *j*=1 (1 − *FU N*σ(*j*) ) *wj* + (<sup>1</sup> <sup>−</sup> *FU Nm*+1 ) *wm*+<sup>1</sup> <sup>−</sup> *m*+1 ⊗ *j*=1 (1 − *F<sup>U</sup> N*σ(*j*) ) *wj* (1 − *FU Nm*+1 ) *wm*+1 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ , <sup>2</sup> <sup>−</sup> *<sup>m</sup>*+<sup>1</sup> ⊗ *j*=1 (1 − *TN*σ(*j*) ) *wj* <sup>−</sup> <sup>1</sup> <sup>+</sup> *<sup>m</sup>*+<sup>1</sup> ⊗ *j*=1 (1 − *TN*σ(*j*) ) *wj* + (<sup>1</sup> <sup>−</sup> *TNm*<sup>+</sup><sup>1</sup> ) *wm*+<sup>1</sup> <sup>−</sup> *m*+1 ⊗ *j*=1 (1 − *T wj N*σ(*j*) (1 − *TNm*<sup>+</sup><sup>1</sup> ) *wm*+<sup>1</sup> , <sup>2</sup> <sup>−</sup> *<sup>m</sup>*+<sup>1</sup> ⊗ *j*=1 (1 − *IN*σ(*j*) ) *wj* <sup>−</sup> <sup>1</sup> <sup>+</sup> *<sup>m</sup>*+<sup>1</sup> ⊗ *j*=1 (1 − *IN*σ(*j*) ) *wj* + (<sup>1</sup> <sup>−</sup> *INm*<sup>+</sup><sup>1</sup> ) *wm*+<sup>1</sup> <sup>−</sup> *m*+1 ⊗ *j*=1 (1 − *I wj N*σ(*j*) (1 − *INm*<sup>+</sup><sup>1</sup> ) *wm*+1 , *m*+1 ⊗ *j*=1 *FNj wj* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *m*+1 ⊗ *j*=1 *TL N*σ(*j*) *wj* , *m*+1 ⊗ *j*=1 *T<sup>U</sup> N*σ(*j*) *wj* , *m*+1 ⊗ *j*=1 *I L N*σ(*j*) *wj* , *m*+1 ⊗ *j*=1 *I U N*σ(*j*) *wj* , <sup>1</sup> <sup>−</sup> *<sup>m</sup>*+<sup>1</sup> ⊗ *j*=1 (1 − *F<sup>L</sup> N*σ(*j*) ) *wj* , 1 <sup>−</sup> *<sup>m</sup>*+<sup>1</sup> ⊗ *j*=1 (<sup>1</sup> <sup>−</sup> *FU N*σ(*j*) ) *wj* , <sup>1</sup> <sup>−</sup> *<sup>m</sup>*+<sup>1</sup> ⊗ *j*=1 1 − *TN*σ(*j*) *wj* , 1 <sup>−</sup> *<sup>m</sup>*+<sup>1</sup> ⊗ *j*=1 1 − *IN*σ(*j*) *wj* , *m*+1 ⊗ *j*=1 *FN*σ(*j*) *wj* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

which completes the proof. -

**Theorem 4.** *The NCWG is a special case of NCHG operator.*

$$\begin{array}{l} \textbf{Proof. Let } W = \left(\frac{1}{m}, \frac{1}{m}, \dots, \frac{1}{m}\right)^{T}. \text{ Then,} \\ \textbf{NCHG}(N\_{1}, N\_{2}, \dots, N\_{m})^{w} = \left(\textbf{N^{\smcorner}}\_{\boldsymbol{\sigma}(1)}\right)^{w\_{1}} \otimes \left(\textbf{N^{\smcorner}}\_{\boldsymbol{\sigma}(2)}\right)^{w\_{2}} \otimes \dots \otimes \left(\textbf{N^{\smcorner}}\_{\boldsymbol{\sigma}(m)}\right)^{w\_{m}} \\ = \left(\textbf{N^{\smcorner}}\_{\boldsymbol{\sigma}(1)}\right)^{\dagger} \otimes \left(\textbf{N^{\smcorner}}\_{\boldsymbol{\sigma}(2)}\right)^{\dagger} \otimes \dots \otimes \left(\textbf{N^{\smcorner}}\_{\boldsymbol{\sigma}(m)}\right)^{\dagger} \\ = \left(N\_{1}, N\_{2}, \dots, N\_{m}\right)^{\frac{1}{m}} \\ = \left(N\_{1}\right)^{w\_{1}}\left(N\_{2}\right)^{w\_{2}}, \dots, \left(N\_{m}\right)^{w\_{m}} \\ = \textbf{NCWG}(N\_{1}, N\_{2}, \dots, N\_{m}). \quad \square \end{array}$$

**Theorem 5.** *The NCOWG is a special case of NCHG.*

$$\begin{aligned} \textbf{Proof. Let } W &= \left(\frac{1}{m}, \frac{1}{m}, \dots, \frac{1}{m}\right)^{T}. \text{ Then,} \\ &N! \textbf{CHG}(N\_{1}, N\_{2}, \dots, N\_{m})^{\varpi} = \left(\mathrm{N}^{\sim}\_{\sigma(1)}\right)^{w\_{1}} \otimes \left(\mathrm{N}^{\sim}\_{\sigma(2)}\right)^{w\_{2}} \otimes \dots \otimes \left(\mathrm{N}^{\sim}\_{\sigma(m)}\right)^{w\_{m}} \\ &= \left(\mathrm{N}^{\sim}\_{\sigma(1)}\right)^{\frac{1}{\pi}} \otimes \left(\mathrm{N}^{\sim}\_{\sigma(2)}\right)^{\frac{1}{\pi}} \otimes \dots \otimes \left(\mathrm{N}^{\sim}\_{\sigma(m)}\right)^{\frac{1}{\pi}} \\ &= \left(N\_{1}, N\_{2}, \dots, N\_{m}\right)^{\frac{1}{\pi}} \end{aligned}$$

$$\begin{array}{l} = (N\_1)^{\text{w}\_1} \text{ (}N\_2)^{\text{w}\_2} \dots \text{ (}N\_m\text{)}^{\text{w}\_m} \\ = \text{NCOWG}(N\_1, N\_2, \dots, N\_m) . \quad \square \end{array}$$

#### *3.2. Neutrosophic Cubic Einstein Hybrid Geometric Operator*

NCEWG operator weights only the neutrosophic cubic values, where NCEOWGA weights only the ordering positions. The idea of neutrosophic cubic Einstein hybrid aggregation operators (NCEHG) is developed to overcome these limitations, which weights both the given neutrosophic cubic value and its order position as well.

**Definition 25.** *NCEHG* : Ω*<sup>m</sup>* → Ω *is a map from m-dimension which has an associated vector W* = (*w*1, *<sup>w</sup>*2, ..., *wm*)*T, where wj* <sup>∈</sup> [0, 1] *and <sup>m</sup> j*=1 = 1*, such that*

$$\text{NCEHG}\_{\text{w}}(\text{N}\_{1}, \text{N}\_{2}, \dots, \text{N}\_{m}) = \left(\text{N}\_{\sigma(1)}^{\smile}\right)^{\text{w}\_{1}} \otimes\_{\text{E}} \left(\text{N}\_{\sigma(2)}^{\smile}\right)^{\text{w}\_{2}} \otimes\_{\text{E}} \dots \otimes\_{\text{E}} \left(\text{N}\_{\sigma(m)}^{\smile}\right)^{\text{w}\_{m}}.$$

*where N*∼ *<sup>j</sup> is the jth largest of the weighted neutrosophic cubic values N*∼ (*j*) *N*∼ (*j*) <sup>=</sup> *<sup>N</sup>mwj j* , *j* = 1, 2, 3, , , *m*), *W* = (*w*1, *w*2, ..., *wm*)*<sup>T</sup> , with wj* <sup>∈</sup> [0, 1] *and <sup>m</sup> j*=1 = 1*, and m is the balancing coe*ffi*cient.*

**Theorem 6.** *Let Nj* <sup>=</sup> -*TNj* ,-*INj* , -*FNj* , *TNj* , *INj* , *FNj* , *where* -*TNj* = *TL Nj* , *T<sup>U</sup> Nj* ,-*INj* = *I L Nj* , *I U Nj* , -*FNj* = *FL Nj* , *FU Nj* , *<sup>N</sup>* = -*TNj* ,-*INj* , -*FNj* , *TNj* , *INj* , *FNj* , *where* -*TNj* = *TL Nj* , *T<sup>U</sup> Nj* ,-*INj* = *I L Nj* , *I U Nj* , -*FNj* = *FL Nj* , *F<sup>U</sup> Nj* (*j* = 1, 2, ..., *m*) *is acollection of neutrosophic cubic values, then, their aggregated value by NCEWG operator is also a cubic value and*

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ,

*NCEHG*(*Nj*) = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ 2 *m* ⊗ *j*=1 *TL <sup>N</sup>*σ(*j*) *wj m* ⊗ *j*=1 <sup>2</sup>−*T<sup>L</sup> <sup>N</sup>*σ(*j*) *wj* <sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 *TL <sup>N</sup>*σ(*j*) *wj* , 2 *m* ⊗ *j*=1 *T<sup>U</sup> <sup>N</sup>*σ(*j*) *wj m* ⊗ *j*=1 <sup>2</sup>−*T<sup>U</sup> <sup>N</sup>*σ(*j*) *wj* <sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 *T<sup>U</sup> <sup>N</sup>*σ(*j*) *wj* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ , ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ 2 *m* ⊗ *j*=1 *I L <sup>N</sup>*σ(*j*) *wj m* ⊗ *j*=1 2−*I L <sup>N</sup>*σ(*j*) *wj* <sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 *I L <sup>N</sup>*σ(*j*) *wj* , 2 *m* ⊗ *j*=1 *I U <sup>N</sup>*σ(*j*) *wj m* ⊗ *j*=1 2−*I U <sup>N</sup>*σ(*j*) *wj* <sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 *I U <sup>N</sup>*σ(*j*) *wj* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ , ⎡ ⎢⎢⎢⎢⎢⎢⎢⎣ *m* ⊗ *j*=1 (1+*FL <sup>N</sup>*σ(*j*) ) *wj*<sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*FL <sup>N</sup>*σ(*j*) ) *wj m* ⊗ *j*=1 (1+*FL <sup>N</sup>*σ(*j*) ) *wj*<sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*F<sup>L</sup> <sup>N</sup>*σ(*j*) ) *wj* , *m* ⊗ *j*=1 (1+*F<sup>U</sup> <sup>N</sup>*σ(*j*) ) *wj*<sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*F<sup>U</sup> <sup>N</sup>*σ(*j*) ) *wj m* ⊗ *j*=1 (1+*FU <sup>N</sup>*σ(*j*) ) *wj*<sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*FU <sup>N</sup>*σ(*j*) ) *wj* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎦ *m* ⊗ *j*=1 (1+*TN*σ(*j*) ) *wj*<sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*TN*σ(*j*) ) *wj m* ⊗ *j*=1 (1+*TN*σ(*j*) ) *wj*<sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*TN*σ(*j*) ) *wj* , *m* ⊗ *j*=1 (1+*IN*σ(*j*) ) *wj*<sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*IN*σ(*j*) ) *wj m* ⊗ *j*=1 (1+*IN*σ(*j*) ) *wj*<sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*IN*σ(*j*) ) *wj* , 2 *m* ⊗ *j*=1 *FN*σ(*j*) *wj m* ⊗ *j*=1 <sup>2</sup>−*FN*σ(*j*) *wj* <sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 *FN*σ(*j*) *wj*

*where W* = (*w*1, *<sup>w</sup>*2, ..., *wm*)*<sup>T</sup> is weight of Nj*(*<sup>j</sup>* <sup>=</sup> 1, 2, 3, ..., *<sup>m</sup>*), *with wj* <sup>∈</sup> [0, 1] *and <sup>m</sup> j*=1 = 1. **Proof.** We use mathematical induction to prove this result for *k* = 2, using definition

 *N<sup>E</sup>* 1 *<sup>w</sup>*<sup>1</sup> <sup>=</sup> ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎡ ⎢⎢⎢⎢⎢⎣ 2 *TL N*1 *w*1 <sup>2</sup>−*T<sup>L</sup> N*1 *w*1 +*T<sup>L</sup> N*1 , <sup>2</sup> *T<sup>U</sup> N*1 *w*1 <sup>2</sup>−*T<sup>U</sup> N*1 *w*1 +*T<sup>U</sup> N*1 ⎤ ⎥⎥⎥⎥⎥⎦ , ⎡ ⎢⎢⎢⎢⎢⎣ 2 *I L N*1 *w*1 2−*I L N*1 *w*1 +*I L N*1 , <sup>2</sup> *I U N*1 *w*1 2−*I U N*1 *w*1 +*I U N*1 ⎤ ⎥⎥⎥⎥⎥⎦ , (1+*F<sup>L</sup> N*1 ) *<sup>w</sup>*1−(1−*F<sup>L</sup> N*1 ) *w*1 (1+*FL N*1 ) *<sup>w</sup>*1+(1−*FL N*1 ) *w*1 , (1+*F<sup>U</sup> N*1 ) *<sup>w</sup>*1−(1−*FU N*1 ) *w*1 (1+*F<sup>U</sup> N*1 ) *<sup>w</sup>*1+(1−*FU N*1 ) *w*1 , (1+*TN*<sup>1</sup> ) *<sup>w</sup>*1−(1−*TN*<sup>1</sup> ) *w*1 (1+*TN*<sup>1</sup> ) *<sup>w</sup>*1+(1−*TN*<sup>1</sup> ) *w*1 , (1+*IN*<sup>1</sup> ) *<sup>w</sup>*1−(1−*IN*<sup>1</sup> ) *w*1 (1+*IN*<sup>1</sup> ) *<sup>w</sup>*1+(1−*IN*<sup>1</sup> ) *<sup>w</sup>*<sup>1</sup> , <sup>2</sup>(*FN*<sup>1</sup> ) *w*1 (2−*FN*<sup>1</sup> ) *<sup>w</sup>*1+*FN*<sup>1</sup> ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ *N<sup>E</sup>* 2 *<sup>w</sup>*<sup>2</sup> <sup>=</sup> ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎡ ⎢⎢⎢⎢⎢⎣ 2 *TL N*2 *w*2 <sup>2</sup>−*T<sup>L</sup> N*2 *w*2 +*T<sup>L</sup> N*2 , <sup>2</sup> *T<sup>U</sup> N*2 *w*2 <sup>2</sup>−*T<sup>U</sup> N*2 *w*2 +*T<sup>U</sup> N*2 ⎤ ⎥⎥⎥⎥⎥⎦ , ⎡ ⎢⎢⎢⎢⎢⎣ 2 *I L N*2 *w*2 2−*I L N*2 *w*2 +*I L N*2 , <sup>2</sup> *I U N*2 *w*2 2−*I U N*2 *w*2 +*I U N*2 ⎤ ⎥⎥⎥⎥⎥⎦ , (1+*F<sup>L</sup> N*2 ) *<sup>w</sup>*2−(1−*F<sup>L</sup> N*2 ) *w*2 (1+*FL N*2 ) *<sup>w</sup>*2+(1−*FL N*2 ) *<sup>w</sup>*<sup>2</sup> , (1+*F<sup>U</sup> N*2 ) *<sup>w</sup>*2−(1−*FU N*2 ) *w*2 (1+*F<sup>U</sup> N*2 ) *<sup>w</sup>*2+(1−*FU N*2 ) *w*2 , (1+*TN*<sup>2</sup> ) *<sup>w</sup>*2−(1−*TN*<sup>2</sup> ) *w*2 (1+*TN*<sup>2</sup> ) *<sup>w</sup>*2+(1−*TN*<sup>2</sup> ) *<sup>w</sup>*<sup>2</sup> , (1+*IN*<sup>2</sup> ) *<sup>w</sup>*2−(1−*IN*<sup>2</sup> ) *w*2 (1+*IN*<sup>2</sup> ) *<sup>w</sup>*2+(1−*IN*<sup>2</sup> ) *<sup>w</sup>*<sup>2</sup> , <sup>2</sup>(*FN*<sup>2</sup> ) *w*2 (2−*FN*<sup>2</sup> ) *<sup>w</sup>*2+*FN*<sup>2</sup> ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ 2 ⊗ *j*=1 *N<sup>E</sup> j wj* = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ 2 2 ⊗ *j*=1 *TL <sup>N</sup>*σ(*j*) *wj* 2 ⊗ *j*=1 <sup>2</sup>−*T<sup>L</sup> <sup>N</sup>*σ(*j*) *wj* <sup>+</sup> <sup>2</sup> ⊗ *j*=1 *TL <sup>N</sup>*σ(*j*) *wj* , 2 2 ⊗ *j*=1 *T<sup>U</sup> <sup>N</sup>*σ(*j*) *wj* 2 ⊗ *j*=1 <sup>2</sup>−*T<sup>U</sup> <sup>N</sup>*σ(*j*) *wj* <sup>+</sup> <sup>2</sup> ⊗ *j*=1 *T<sup>U</sup> <sup>N</sup>*σ(*j*) *wj* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ , ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ 2 2 ⊗ *j*=1 *I L <sup>N</sup>*σ(*j*) *wj* 2 ⊗ *j*=1 2−*I L <sup>N</sup>*σ(*j*) *wj* <sup>+</sup> <sup>2</sup> ⊗ *j*=1 *I L <sup>N</sup>*σ(*j*) *wj* , 2 2 ⊗ *j*=1 *I U <sup>N</sup>*σ(*j*) *wj* 2 ⊗ *j*=1 2−*I U <sup>N</sup>*σ(*j*) *wj* <sup>+</sup> <sup>2</sup> ⊗ *j*=1 *I U <sup>N</sup>*σ(*j*) *wj* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ , ⎡ ⎢⎢⎢⎢⎢⎢⎢⎣ 2 ⊗ *j*=1 (1+*FL <sup>N</sup>*σ(*j*) ) *wj*<sup>−</sup> <sup>2</sup> ⊗ *j*=1 (1−*FL <sup>N</sup>*σ(*j*) ) *wj* 2 ⊗ *j*=1 (1+*FL <sup>N</sup>*σ(*j*) ) *wj*<sup>+</sup> <sup>2</sup> ⊗ *j*=1 (1−*F<sup>L</sup> <sup>N</sup>*σ(*j*) ) *wj* , 2 ⊗ *j*=1 (1+*F<sup>U</sup> <sup>N</sup>*σ(*j*) ) *wj*<sup>−</sup> <sup>2</sup> ⊗ *j*=1 (1−*FU <sup>N</sup>*σ(*j*) ) *wj* 2 ⊗ *j*=1 (1+*F<sup>U</sup> <sup>N</sup>*σ(*j*) ) *wj*<sup>+</sup> <sup>2</sup> ⊗ *j*=1 (1−*F<sup>U</sup> <sup>N</sup>*σ(*j*) ) *wj* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎦ , 2 ⊗ *j*=1 (1+*TN*σ(*j*) ) *wj*<sup>−</sup> <sup>2</sup> ⊗ *j*=1 (1−*TN*σ(*j*) ) *wj* 2 ⊗ *j*=1 (1+*TN*σ(*j*) ) *wj*<sup>+</sup> <sup>2</sup> ⊗ *j*=1 (1−*TN*σ(*j*) ) *wj* , 2 ⊗ *j*=1 (1+*IN*σ(*j*) ) *wj*<sup>−</sup> <sup>2</sup> ⊗ *j*=1 (1−*IN*σ(*j*) ) *wj* 2 ⊗ *j*=1 (1+*IN*σ(*j*) ) *wj*<sup>+</sup> <sup>2</sup> ⊗ *j*=1 (1−*IN*σ(*j*) ) *wj* , 2 2 ⊗ *j*=1 *FN*σ(*j*) *wj* 2 ⊗ *j*=1 <sup>2</sup>−*FN*σ(*j*) *wj* <sup>+</sup> <sup>2</sup> ⊗ *j*=1 *FN*σ(*j*) *wj*

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ .

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ .

Let the result holds for *m*.

*m* ⊗ *j*=1 *N<sup>E</sup> j wj* = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ 2 *m* ⊗ *j*=1 *TL <sup>N</sup>*σ(*j*) *wj m* ⊗ *j*=1 <sup>2</sup>−*T<sup>L</sup> <sup>N</sup>*σ(*j*) *wj* <sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 *TL <sup>N</sup>*σ(*j*) *wj* , 2 *m* ⊗ *j*=1 *T<sup>U</sup> <sup>N</sup>*σ(*j*) *wj m* ⊗ *j*=1 <sup>2</sup>−*T<sup>U</sup> <sup>N</sup>*σ(*j*) *wj* <sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 *T<sup>U</sup> <sup>N</sup>*σ(*j*) *wj* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ , ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ 2 *m* ⊗ *j*=1 *I L <sup>N</sup>*σ(*j*) *wj m* ⊗ *j*=1 2−*I L <sup>N</sup>*σ(*j*) *wj* <sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 *I L <sup>N</sup>*σ(*j*) *wj* , 2 *m* ⊗ *j*=1 *I U <sup>N</sup>*σ(*j*) *wj m* ⊗ *j*=1 2−*I U <sup>N</sup>*σ(*j*) *wj* <sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 *I U <sup>N</sup>*σ(*j*) *wj* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ , ⎡ ⎢⎢⎢⎢⎢⎢⎢⎣ *m* ⊗ *j*=1 (1+*FL <sup>N</sup>*σ(*j*) ) *wj*<sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*FL <sup>N</sup>*σ(*j*) ) *wj m* ⊗ *j*=1 (1+*FL <sup>N</sup>*σ(*j*) ) *wj*<sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*F<sup>L</sup> <sup>N</sup>*σ(*j*) ) *wj* , *m* ⊗ *j*=1 (1+*F<sup>U</sup> <sup>N</sup>*σ(*j*) ) *wj*<sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*FU <sup>N</sup>*σ(*j*) ) *wj m* ⊗ *j*=1 (1+*F<sup>U</sup> <sup>N</sup>*σ(*j*) ) *wj*<sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*F<sup>U</sup> <sup>N</sup>*σ(*j*) ) *wj* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎦ , *m* ⊗ *j*=1 (1+*TN*σ(*j*) ) *wj*<sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*TN*σ(*j*) ) *wj m* ⊗ *j*=1 (1+*TN*σ(*j*) ) *wj*<sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*TN*σ(*j*) ) *wj* , *m* ⊗ *j*=1 (1+*IN*σ(*j*) ) *wj*<sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*IN*σ(*j*) ) *wj m* ⊗ *j*=1 (1+*IN*σ(*j*) ) *wj*<sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*IN*σ(*j*) ) *wj* , 2 *m* ⊗ *j*=1 *FN*σ(*j*) *wj m* ⊗ *j*=1 <sup>2</sup>−*FN*σ(*j*) *wj* <sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 *FN*σ(*j*) *wj*

We prove the result holds for *m* + 1.

*as N<sup>E</sup> m*+1 *wm*+<sup>1</sup> <sup>=</sup> ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎡ ⎢⎢⎢⎢⎢⎢⎣ 2 *TL Nm*+1 *wm*+1 <sup>2</sup>−*T<sup>L</sup> Nm*+1 *wm*+1 +(*T<sup>L</sup> N*) *wm*+1 , <sup>2</sup> *T<sup>U</sup> Nm*+1 *wm*+1 (2−*T<sup>U</sup> N*) *wm*+1+(*T<sup>U</sup> N*) *wm*+1 ⎤ ⎥⎥⎥⎥⎥⎥⎦ , ⎡ ⎢⎢⎢⎢⎢⎢⎣ 2 *I L Nm*+1 *wm*+1 2−*I L Nm*+1 *wm*+1 +(*I L N*) *wm*+1 , <sup>2</sup> *I U Nm*+1 *wm*+1 2−*I U Nm*+1 *wm*+1 +(*I U N*) *wm*+1 ⎤ ⎥⎥⎥⎥⎥⎥⎦ , ⎡ ⎢⎢⎢⎢⎣ (1+*FL Nm*+1 ) *wm*+1−(1−*F<sup>L</sup> Nm*+1 ) *wm*+1 (1+*FL Nm*+1 ) *wm*+1+(1−*F<sup>L</sup> Nm*+1 ) *wm*+<sup>1</sup> , (1+*FU Nm*+1 ) *wm*+1−(1−*FU Nm*+1 ) *wm*+1 (1+*FU Nm*+1 ) *wm*+1+(1−*FU Nm*+1 ) *wm*+1 ⎤ ⎥⎥⎥⎥⎦, (1+*TNm*<sup>+</sup><sup>1</sup> ) *wm*+1−(1−*TNm*<sup>+</sup><sup>1</sup> ) *wm*+1 (1+*TNm*<sup>+</sup><sup>1</sup> ) *wm*+1+(1−*TNm*<sup>+</sup><sup>1</sup> ) *wm*+<sup>1</sup> , (1+*INm*<sup>+</sup><sup>1</sup> ) *wm*+1−(1−*INm*<sup>+</sup><sup>1</sup> ) *wm*+1 (1+*INm*<sup>+</sup><sup>1</sup> ) *wm*+1+(1−*INm*<sup>+</sup><sup>1</sup> ) *wm*+<sup>1</sup> , <sup>2</sup> *FNm*<sup>+</sup><sup>1</sup> *wm*+1 2−*FNm*<sup>+</sup><sup>1</sup> *wm*+1+(*FN*) *wm*+1 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ *so <sup>m</sup>* ⊗ *j*=1 *N<sup>E</sup> j wj* ⊗*E N<sup>E</sup> m*+1 *wm*+<sup>1</sup> <sup>=</sup> ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ 2 *m* ⊗ *j*=1 *TL <sup>N</sup>*σ(*j*) *wj m* ⊗ *j*=1 <sup>2</sup>−*T<sup>L</sup> <sup>N</sup>*σ(*j*) *wj* <sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 *TL <sup>N</sup>*σ(*j*) *wj* , 2 *m* ⊗ *j*=1 *T<sup>U</sup> <sup>N</sup>*σ(*j*) *wj m* ⊗ *j*=1 <sup>2</sup>−*T<sup>U</sup> <sup>N</sup>*σ(*j*) *wj* <sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 *T<sup>U</sup> <sup>N</sup>*σ(*j*) *wj* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ , ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ 2 *m* ⊗ *j*=1 *I L <sup>N</sup>*σ(*j*) *wj m* ⊗ *j*=1 2−*I L <sup>N</sup>*σ(*j*) *wj* <sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 *I L <sup>N</sup>*σ(*j*) *wj* , 2 *m* ⊗ *j*=1 *I U <sup>N</sup>*σ(*j*) *wj m* ⊗ *j*=1 2−*I U <sup>N</sup>*σ(*j*) *wj* <sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 *I U <sup>N</sup>*σ(*j*) *wj* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ , ⎡ ⎢⎢⎢⎢⎢⎢⎢⎣ *m* ⊗ *j*=1 (1+*F<sup>L</sup> <sup>N</sup>*σ(*j*) ) *wj*<sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*FL <sup>N</sup>*σ(*j*) ) *wj m* ⊗ *j*=1 (1+*FL <sup>N</sup>*σ(*j*) ) *wj*<sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*FL <sup>N</sup>*σ(*j*) ) *wj* , *m* ⊗ *j*=1 (1+*FU <sup>N</sup>*σ(*j*) ) *wj*<sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*FU <sup>N</sup>*σ(*j*) ) *wj m* ⊗ *j*=1 (1+*FU <sup>N</sup>*σ(*j*) ) *wj*<sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*FU <sup>N</sup>*σ(*j*) ) *wj* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎦ , *m* ⊗ *j*=1 (1+*TN*σ(*j*) ) *wj*<sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*TN*σ(*j*) ) *wj m* ⊗ *j*=1 (1+*TN*σ(*j*) ) *wj*<sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*TN*σ(*j*) ) *wj* , *m* ⊗ *j*=1 (1+*IN*σ(*j*) ) *wj*<sup>−</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*IN*σ(*j*) ) *wj m* ⊗ *j*=1 (1+*IN*σ(*j*) ) *wj*<sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 (1−*IN*σ(*j*) ) *wj* , 2 *m* ⊗ *j*=1 *FN*σ(*j*) *wj m* ⊗ *j*=1 <sup>2</sup>−*FN*σ(*j*) *wj* <sup>+</sup> *<sup>m</sup>* ⊗ *j*=1 *FN*σ(*j*) *wj* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⊕*E* ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎡ ⎢⎢⎢⎢⎢⎢⎣ 2 *TL Nm*+1 *wm*+1 <sup>2</sup>−*T<sup>L</sup> Nm*+1 *wm*+1 +(*T<sup>L</sup> N*) *wm*+1 , <sup>2</sup> *T<sup>U</sup> Nm*+1 *wm*+1 (2−*T<sup>U</sup> N*) *wm*+1+(*T<sup>U</sup> N*) *wm*+1 ⎤ ⎥⎥⎥⎥⎥⎥⎦ , ⎡ ⎢⎢⎢⎢⎢⎢⎣ 2 *I L Nm*+1 *wm*+1 2−*I L Nm*+1 *wm*+1 +(*I L N*) *wm*+1 , <sup>2</sup> *I U Nm*+1 *wm*+1 2−*I U Nm*+1 *wm*+1 +(*I U N*) *wm*+1 ⎤ ⎥⎥⎥⎥⎥⎥⎦ , ⎡ ⎢⎢⎢⎢⎣ (1+*FL Nm*+1 ) *wm*+1−(1−*F<sup>L</sup> Nm*+1 ) *wm*+1 (1+*F<sup>L</sup> Nm*+1 ) *wm*+1+(1−*FL Nm*+1 ) *wm*+<sup>1</sup> , (1+*F<sup>U</sup> Nm*+1 ) *wm*+1−(1−*FU Nm*+1 ) *wm*+1 (1+*F<sup>U</sup> Nm*+1 ) *wm*+1+(1−*F<sup>U</sup> Nm*+1 ) *wm*+1 ⎤ ⎥⎥⎥⎥⎦, (1+*TNm*<sup>+</sup><sup>1</sup> ) *wm*+1−(1−*TNm*<sup>+</sup><sup>1</sup> ) *wm*+1 (1+*TNm*<sup>+</sup><sup>1</sup> ) *wm*+1+(1−*TNm*<sup>+</sup><sup>1</sup> ) *wm*+<sup>1</sup> , (1+*INm*<sup>+</sup><sup>1</sup> ) *wm*+1−(1−*INm*<sup>+</sup><sup>1</sup> ) *wm*+1 (1+*INm*<sup>+</sup><sup>1</sup> ) *wm*+1+(1−*INm*<sup>+</sup><sup>1</sup> ) *wm*+<sup>1</sup> , 2 *FNm*<sup>+</sup><sup>1</sup> *wm*+1 2−*FNm*<sup>+</sup><sup>1</sup> *wm*+1+(*FN*) *wm*+1 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

*m*+1 ⊗ *j*=1 *N<sup>E</sup> j wj* = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ 2 *m*+1 ⊗ *j*=1 *TL <sup>N</sup>*σ(*j*) *wj m*+1 ⊗ *j*=1 <sup>2</sup>−*T<sup>L</sup> <sup>N</sup>*σ(*j*) *wj* +*m*+<sup>1</sup> ⊗ *j*=1 *TL <sup>N</sup>*σ(*j*) *wj* , 2 *m*+1 ⊗ *j*=1 *T<sup>U</sup> <sup>N</sup>*σ(*j*) *wj m*+1 ⊗ *j*=1 <sup>2</sup>−*T<sup>U</sup> <sup>N</sup>*σ(*j*) *wj* +*m*+<sup>1</sup> ⊗ *j*=1 *T<sup>U</sup> <sup>N</sup>*σ(*j*) *wj* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ , ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ 2 *m*+1 ⊗ *j*=1 *I L <sup>N</sup>*σ(*j*) *wj m*+1 ⊗ *j*=1 2−*I L <sup>N</sup>*σ(*j*) *wj* +*m*+<sup>1</sup> ⊗ *j*=1 *I L <sup>N</sup>*σ(*j*) *wj* , 2 *m*+1 ⊗ *j*=1 *I U <sup>N</sup>*σ(*j*) *wj m*+1 ⊗ *j*=1 2−*I U <sup>N</sup>*σ(*j*) *wj* +*m*+<sup>1</sup> ⊗ *j*=1 *I U <sup>N</sup>*σ(*j*) *wj* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ , ⎡ ⎢⎢⎢⎢⎢⎢⎢⎣ *m*+1 ⊗ *j*=1 (1+*FL <sup>N</sup>*σ(*j*) ) *wj*− *m*+1 ⊗ *j*=1 (1−*F<sup>L</sup> <sup>N</sup>*σ(*j*) ) *wj m*+1 ⊗ *j*=1 (1+*F<sup>L</sup> <sup>N</sup>*σ(*j*) ) *wj*+*m*+<sup>1</sup> ⊗ *j*=1 (1−*F<sup>L</sup> <sup>N</sup>*σ(*j*) ) *wj* , *m*+1 ⊗ *j*=1 (1+*FU <sup>N</sup>*σ(*j*) ) *wj*− *m*+1 ⊗ *j*=1 (1−*F<sup>U</sup> <sup>N</sup>*σ(*j*) ) *wj m*+1 ⊗ *j*=1 (1+*FU <sup>N</sup>*σ(*j*) ) *wj*+*m*+<sup>1</sup> ⊗ *j*=1 (1−*FU <sup>N</sup>*σ(*j*) ) *wj* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎦ , *m*+1 ⊗ *j*=1 (1+*TN*σ(*j*) ) *wj*− *m*+1 ⊗ *j*=1 (1−*TN*σ(*j*) ) *wj m*+1 ⊗ *j*=1 (1+*TN*σ(*j*) ) *wj*+*m*+<sup>1</sup> ⊗ *j*=1 (1−*TN*σ(*j*) ) *wj* , *m*+1 ⊗ *j*=1 (1+*IN*σ(*j*) ) *wj*− *m*+1 ⊗ *j*=1 (1−*IN*σ(*j*) ) *wj m*+1 ⊗ *j*=1 (1+*IN*σ(*j*) ) *wj*+*m*+<sup>1</sup> ⊗ *j*=1 (1−*IN*σ(*j*) ) *wj* , 2 *m*+1 ⊗ *j*=1 *FN*σ(*j*) *wj m*+1 ⊗ *j*=1 <sup>2</sup>−*FN*σ(*j*) *wj* +*m*+<sup>1</sup> ⊗ *j*=1 *FN*σ(*j*) *wj*

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ,

so the result holds for all values of *m*. -

**Theorem 7.** *The NCEWG is special case of the NCEHG operator.*

**Proof.** Followed by Theorem 4. -

**Theorem 8.** *The NCOWG is a special case of NCEHG.*

**Proof.** Followed by Theorem 5. -

#### *3.3. An Application of Neutrosophic Cubic Hybrid Geometric and Einstein Hybrid Geometric Aggregation Operator to Group Decision-Making Problems*

In this section, we develop an algorithm for group decision-making problems using the neutrosophic cubichybrid geometric and Einstein hybrid geometric aggregation (NCHWG and NCEHWG).

**Algorithm 1.** Let *F* = {*F*1, *F*2, ..., *Fn*} be the set of n alternatives, *H* = {*H*1, *H*2, ..., *Hm*} be the m attributes subject to their corresponding weight *<sup>W</sup>* <sup>=</sup> {*w*1, *<sup>w</sup>*2, ..., *wm*} , such that *wj* <sup>∈</sup> [0, 1] and *<sup>m</sup> j*=1 = 1. The

method has the following steps.

**Step 1:** First of all, we construct neutrosophic cubic decision matrix *<sup>D</sup>* = *Nij <sup>n</sup>*×*m*.

**Step 2:** The attributes *H* = {*H*1, *H*2, ..., *Hm*} are weighted to their corresponding weight *W* = {*w*1, *w*2, ..., *wm*}, and these values multipliedby the balancing coefficient *m*.

**Step 3:** The new weights are calculated using [18] so that we get new weights *V* = {*v*1, *v*2, ..., *vm*}. **Step 4:** By using aggregation operators like (NCHG, NCEHG), the decision matrix is aggregated by the new weightsassigned to the *m* attributes.

**Step 5:** The *n* alternatives are ranked according to their scores and arranged in descending order to select the alternative with highest score.

#### *3.4. NumericalApplication*

A steering committee is interested in prioritizingthe set of information for improvement of the project using a multiple attribute decision-making method. The committee must prioritize the development and implementation of a set of six information technology improvement projects *Aj* (*j* = 1, 2, ..., 6). The three factors, *B*<sup>1</sup> productivity, to increase the effectiveness and efficiency, *B*<sup>2</sup> differentiation, from products and services of competitors, and *B*<sup>3</sup> management, to assist the management in improving their planning, are considered to assess the potential contribution of each project. The list of proposed information systems are *A*<sup>1</sup> Quality Assurance, (2) *A*<sup>2</sup> Budget Analysis, (3) *A*<sup>3</sup> Itemization, (4) *A*<sup>4</sup> Employee Skills Tracking, (5) *A*<sup>5</sup> Customer Returns and Complaints, and (6) *A*<sup>6</sup> Materials Acquisition. Suppose the weight *W* = (0.5, 0.3, 0.2) corresponds to the *Bj*, (*j* = 1, 2, 3,) factors and characteristics of projects *Ai* (*i* = 1, 2, ..., 10) by the neutrosophic cubic value *Nij*. **Step 1:** Construction of neutrosophic cubic decision matrix *<sup>D</sup>* = *Nij* 6×3

*D* = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *B*<sup>1</sup> *B*<sup>2</sup> *B*<sup>3</sup> *A*1 [0.5, 0.6], [0.2, 0.5], [0.4, 0.8], 0.7, 0.8, 0.4 [0.3, 0.7], [0.1, 0.6], [0.3, 0.7], 0.4, 0.7, 0.2 [0.4, 0.8], [0.3, 0.6], [0.5, 0.7], 0.4, 0.2, 0.6 *A*<sup>2</sup> [0.2, 0.5], [0.7, 0.9], [0.3, 0.7], 0.8, 0.5, 0.3 [0.1, 0.6], [0.4, 0.7], [0.2, 0.5], 0.6, 0.4, 0.7 [0.2, 0.6], [0.1, 0.7], [0.3, 0.7], 0.6, 0.4, 0.8 *A*<sup>3</sup> [0.4, 0.7], [0.2, 0.5], [0.5, 0.7], 0.3, 0.6, 0.2 [0.3, 0.8], [0.1, 0.4], [0.6, 0.7], 0.6, 0.2, 0.6 [0.3, 0.5], [0.5, 0.9], [0.2, 0.7], 0.7, 0.5, 0.6 *A*4 [0.3, 0.6], [0.4, 0.7], [0.2, 0.5], 0.6, 0.4, 0.7 [0.5, 0.8], [0.1, 0.5], [0.3, 0.8], 0.4, 0.8, 0.6 [0.1, 0.7], [0.2, 0.6], [0.4, 0.7], 0.5, 0.6, 0.8 *A*<sup>5</sup> [0.2, 0.5], [0.3, 0.7], [0.2, 0.6], 0.5, 0.3, 0.8 [0.4, 0.8], [0.3, 0.7], [0.1, 0.6], 0.4, 0.6, 0.7 [0.6, 0.8], [0.5, 0.9], [0.4, 0.9], 0.6, 0.8, 0.3 *A*<sup>6</sup> [0.1, 0.6], [0.3, 0.6], [0.4, 0.8], 0.6, 0.9, 0.4 [0.2, 0.7], [0.6, 0.9], [0.3, 0.6], 0.4, 0.8, 0.3 [0.4, 0.7], [0.3, 0.5], [0.1, 0.6], 0.4, 0.6, 0.7 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

**Step 2:** The attributes are weighted *W* = (0.5, 0.3, 0.2) and multiplied by balancing coefficient 3.

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

*D* = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *B*<sup>1</sup> *B*<sup>2</sup> *B*<sup>3</sup> *A*1 ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.3535, 0.4647], [0.0894, 0.3535], [0.5352, 0.9105], 0.8356, 0.9105, 0.2529 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.3383, 0.7254], [0.1258, 0.6314], [0.27453, 0.6616], 0.3685, 0.6616, 0.2349 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.5770, 0.8746], [0.4855, 0.7360], [0.4229, 0.5144], 0.2639, 0.1253, 0.7360 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ *A*<sup>2</sup> ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.0894, 0.3535], [0.5856, 0.8538], [0.4143, 0.8356], 0.9105, 0.6464, 0.1643 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.1258, 0.6314], [0.4383, 0.7254], [0.1819, 0.4641], 0.5616, 0.3685, 0.7254 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.3807, 0.7360], [0.2511, 0.8073], [0.1926, 0.5144], 0.4229, 0.2639, 0.8073 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ *A*<sup>3</sup> ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.2529, 0.5856], [0.0894, 0.3535], [0.6464, 0.8356], 0.4143, 0.7470, 0.0894 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.3383, 0.8180], [0.1258, 0.4383], [0.5616, 0.6616], 0.5616, 0.1819, 0.6314 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.4855, 0.6597], [0.6597, 0.9387], [0.1253, 0.5144], 0.5144, 0.3402, 0.7360 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ *A*4 ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.1643, 0.4647], [0.2529, 0.5856], [0.2844, 0.6464], 0.7470, 0.5352, 0.5856 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.5358, 0.8180], [0.1258, 0.5358], [0.2745, 0.7650], 0.3685, 0.7650, 0.6314 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.2511, 0.8073], [0.3807, 0.7360], [0.2639, 0.5144], 0.6402, 0.4229, 0.8073 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ *A*<sup>5</sup> ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.0894, 0.5353], [0.1643, 0.5856], [0.2844, 0.7470], 0.6464, 0.4143, 0.7155 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.4383, 0.8180], [0.3383, 0.7254], [0.0904, 0.5616], 0.3685, 0.5616, 0.7254 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.7360, 0.8073], [0.6597, 0.9387], [0.2639, 0.7488], 0.4229, 0.6192, 0.4855 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ *A*<sup>6</sup> ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.0316, 0.4647], [0.1643, 0.4647], [0.5352, 0.9105], 0.7470, 0.9683, 0.2529 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.2349, 0.7254], [0.6314, 0.9035], [0.2745, 0.5616], 0.3685, 0.7650, 0.3383 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ [0.5770, 0.8073], [0.4855, 0.6597], [0.0612, 0.4229], 0.2639, 0.4229, 0.8073 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

**Step 3:** The new weights are calculated using the normal distribution method. Let *W* = (0.2429, 0.5142, 0.2429) be its weighting vector derived by the normal distribution-based method [18]. **Step 4:** By neutrosophic cubic weighted geometric aggregation operator (NCWG), the decision matrix is aggregated by the new weights assigned to the *m* attributes.

$$D = \begin{pmatrix} A\_1 & \begin{pmatrix} [0.3892, 0.6812], [0.1606, 0.5692] \\ [0.3840, 0.7325], 0.5273, 0.6914, 0.3270 \\ [0.1852, 0.5692], [0.4107, 0.7745] \\ [0.2480, 0.4946], 0.6813, 0.4306, 0.5190 \\ A\_3 & \begin{pmatrix} [0.3441, 0.7158], [0.1731, 0.5005] \\ [0.7078, 0.6899], 0.5178, 0.4161, 0.4076 \end{pmatrix} \\ A\_4 & \begin{pmatrix} [0.3344, 0.7107], [0.1950, 0.5913] \\ [0.2743, 0.6904], 0.7010, 0.6550, 0.6580 \\ A\_5 & \begin{pmatrix} [0.3378, 0.7375], [0.3388, 0.7331] \\ [0.1848, 0.6649], 0.4633, 0.5454, 0.6557 \end{pmatrix} \\ A\_6 & \begin{pmatrix} [0.1795, 0.6681], [0.4271, 0.7122] \\ [0.3068, 0.6813], 0.4751, 0.8937, 0.3893 \end{pmatrix} \end{pmatrix}$$

**Step 5:** The scores are

$$S(A\_1) = 0.1542, S(A\_2) = 0.1741, S(A\_3) = -0.2276, S(A\_4) = 0.1297, S(A\_5) = 0.0332, S(A\_6) = -0.0547, S(A\_7) = 0.0332, S(A\_8) = 0.0332$$

$$S(A\_2) > S(A\_1) > S(A\_4) > S(A\_5) > S(A\_6) > S(A\_3).$$

List of priorities are as follows.

$$A\_2 > A\_1 > A\_4 > A\_5 > A\_6 > A\_3.$$

Hence, the project *A*<sup>1</sup> has the highest potential contribution to the firm's strategic goal of gaining competitive advantage in the industry.

#### **4. Conclusions**

This paper was influenced by the impediment of neutrosophic cubic geometric and Einstein geometric collection operators as preliminarily discussed, that is, we observed that the higher the weight component, the aggregated value tended to the corresponding neutrosophic cubic value of that vector. Consequent upon such circumstances, we characterized neutrosophic cubic hybrid and neutrosophic cubic Einstein hybrid aggregation operators. At that point, these operators are outfitted upon a day-by-day life precedent structure industry to organize the potential contributions that serve to achieve the strategic objective of getting favorable circumstances in industry.

**Author Contributions:** All authors contributed equally.

**Acknowledgments:** The authors gratefully acknowledged the financial from the Deanship of Scientific research and Graduate Studies at Philadelphia University in Jordan.

**Conflicts of Interest:** There is no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Refined Neutrosophy and Lattices vs. Pair Structures and YinYang Bipolar Fuzzy Set**

#### **Florentin Smarandache**

Mathematics Department, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA; smarand@unm.edu

Received: 4 March 2019; Accepted: 1 April 2019; Published: 16 April 2019

**Abstract:** In this paper, we present the lattice structures of neutrosophic theories. We prove that Zhang-Zhang's YinYang bipolar fuzzy set is a subclass of the Single-Valued bipolar neutrosophic set. Then we show that the pair structure is a particular case of refined neutrosophy, and the number of types of neutralities (sub-indeterminacies) may be any finite or infinite number.

**Keywords:** neutrosophic set; Zhang-Zhang's YinYang bipolar fuzzy set; single-valued bipolar neutrosophic set; bipolar fuzzy set; YinYang bipolar fuzzy set

#### **1. Introduction**

First, we prove that Klement Dand Mesiar's lattices [1] do not fit the general definition of neutrosophic set, and we construct the appropriate nonstandard neutrosophic lattices of the first type (as neutrosophically ordered set) [2], and of the second type (as neutrosophic algebraic structure, endowed with two binary neutrosophic laws, inf*<sup>N</sup>* and sup*N*) [2].

We also present the novelties that neutrosophy, neutrosophic logic, set, and probability and statistics, with respect to the previous classical and multi-valued logics and sets, and with the classical and imprecise probability and statistics, respectively.

Second, we prove that Zhang-Zhang's YinYang bipolar fuzzy set [3,4] is not equivalent with but a subclass of the Single-Valued bipolar neutrosophic set.

Third, we show that Montero, Bustince, Franco, Rodríguez, Gómez, Pagola, Fernández, and Barrenechea's paired structure of the knowledge representation model [5] is a particular case of Refined Neutrosophy (a branch of philosophy that generalized dialectics) and of the Refined Neutrosophic Set [6]. We disprove again the claim that the bipolar fuzzy set (renamed as YinYang bipolar fuzzy set) is the same of neutrosophic set as asserted by Montero et al [5].

About the three types of neutralities presented by Montero et al., we show, by examples and formally, that there may be any finite number or an infinite number of types of neutralities *n*, or that indeterminacy (*I*), as neutrosophic component, can be refined (split) into 1 ≤ *n* ≤ ∞ number of sub-indeterminacies (not only 3 as Montero et al. said) as needed to each application to solve.

Also, we show, besides numerous neutrosophic applications, many innovatory contributions to science were brought on by the neutrosophic theories, such as: generalization of Yin Yang Chinese philosophy and dialectics to neutrosophy [7], a new branch of philosophy that is based on the dynamics of opposites and their neutralities, the sum of the neutrosophic components *T*, *I*, *F* up to 3, the degrees of dependence/independence between the neutrosophic components [8,9]; the distinction between absolute truth and relative truth in the neutrosophic logic [10], the introduction of nonstandard neutrosophic logic, set, and probability after we have extended the nonstandard analysis [11,12], the refinement of neutrosophic components into subcomponents [6]; the ability to express incomplete information, complete information, paraconsistent (conflicting) information [13,14]; and the extension of the middle principle to the multiple-included middle principle [15], introduction of neutrosophic crisp set and topology [16], and so on.

#### **2. Answers to Erich Peter Klement and Radko Mesiar**

*2.1. Oversimplification of the Neutrosophic Set*

At [1], page 10 (Section 3.3) in their paper, related to neutrosophic sets, they wrote:

"*As a straightforward generalization of the product lattice* <sup>I</sup> <sup>×</sup> <sup>I</sup>,≤*comp , for each n* ∈ *N, the n-dimensional unit cube* <sup>I</sup>*n*,≤*comp , i.e., the n-dimensional product of the lattice (*I*,* ≤*comp), can be defined by means of (1) and (2).*

*The so-called "neutrosophic" sets introduced by F. Smarandache [93] (see also [94–97], which are based on the bounded lattices* I3,≤I<sup>3</sup> *and* <sup>I</sup>3,≤I<sup>3</sup> *, where the orders* ≤*I*<sup>3</sup> *and* ≤*I*<sup>3</sup> *on the unit cube I3 are defined by the Equations below.*

$$\mathbb{P}\left(\mathbf{x}\_{1},\mathbf{x}\_{2},\mathbf{x}\_{3}\right) \leq\_{\mathbb{P}} \mathbb{P}\left(y\_{1},y\_{2},y\_{3}\right) \Leftrightarrow \mathbf{x}\_{1} \leq y\_{1} \text{ AND } \mathbf{x}\_{2} \leq y\_{2} \text{ AND } \mathbf{x}\_{3} \geq y\_{3} \tag{-13}$$

$$\mathbf{x}\_1(\mathbf{x}\_2, \mathbf{x}\_2, \mathbf{x}\_3) \preceq^{l^3} (y\_1, y\_2, y\_3) \Leftrightarrow \mathbf{x}\_1 \le y\_1. \text{ AND } \mathbf{x}\_2 \ge y\_2. \text{ AND } \mathbf{x}\_3 \ge y\_3 \tag{-14}$$

The authors have defined Equations *(1)* and *(2)* as follows:

$$\left(\prod\_{i=1}^{n} L\_{i\prime} \preceq\_{\text{comp}}\right) \\ \text{where } \left(L\_{i\prime} \preceq\_{\text{L}}\right) \text{ are } f \\ \text{uzzy lattices, } \text{ for all } 1 \le i \le n \tag{1}$$

$$(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) \leq\_{\text{comp}} (y\_1, y\_2, \dots, y\_n) \Leftrightarrow \mathbf{x}\_1 \leq y\_1 \text{ AND } \mathbf{x}\_2 \leq y\_2 \text{ AND } \dots \text{ AND } \mathbf{x}\_n \leq y\_n \tag{2}$$

The authors did not specify what type of lattices they employ: of the first type (lattice, as a partially ordered set), or the second type (lattice, as an algebraic structure). Since their lattices are endowed with some inequality (referring to the neutrosophic case), we assume it is as the first type.

The authors have used the notations:

$$\begin{aligned} \mathbb{I} &= [0,1]\_{\prime} \\ \mathbb{I}^2 &= [0,1]^2 \\ \mathbb{I}^3 &= [0,1]^3 \end{aligned}$$

The order relationship <sup>≤</sup>comp on I<sup>3</sup> can be defined as:

$$(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3) \preceq\_{\text{comp}} (y\_1, y\_2, y\_3) \Leftrightarrow \mathbf{x}\_1 \preceq y\_1 \text{ and } \mathbf{x}\_2 \preceq y\_2 \text{ and } \mathbf{x}\_3 \preceq y\_3)$$

The three lattices they constructed are denoted by *KL*1, *KL*2, *KL*3, respectively.

$$KL\_1 = (\mathbb{I}^3 \urcorner \preceq\_{\text{comp}}) , \ KL\_2 = (\mathbb{I}^3 \urcorner \preceq\_{\mathbb{I}^3}) , \ KL\_3 = (\mathbb{I}^3 \urcorner \preceq^{\mathbb{I}^3})$$

Contain only the very particular case of standard single-valued neutrosophic set, i.e., when the neutrosophic components *T* (truth-membership), *I* (indeterminacy-membership), and *F* (false-membership) of the generic element *x*(*T*, *I*, *F*), of a neutrosophic set *N* are single-valued (crisp) numbers from the unit interval [0, 1].

The authors have *oversimplified* the neutrosophic set. Neutrosophic is much more complex. Their lattices do not characterize the *initial definition of the neutrosophic set* ([10], 1998): a set whose elements have the degrees of appurtenance *T*, *I*, *F*, where *T*, *I*, *F* are standard or nonstandard subsets of the nonstandard unit interval: ] <sup>−</sup>0, 1+[, where ] <sup>−</sup>0, 1+[ overpasses the classical real unit interval [0, 1] to the left and to the right.

#### *2.2. Neutrosophic Cube vs. Unit Cube*

Clearly, their I<sup>3</sup> = [0, 1] <sup>3</sup> - ] <sup>−</sup>0, 1+[ <sup>3</sup> that is our neutrosophic cube (Figure 1), where ] −0 = μ(−0) is the left nonstandard monad of number 0, and 1<sup>+</sup> = μ(1+) is the right nonstandard monad of number 1.

**Figure 1.** Neutrosophic cube.

The unit cube I<sup>3</sup> used by the authors does not equal the above neutrosophic cube. The neutrosophic cube *A'B'C'D'E'F'G'H'* was introduced by Dezert [17] in 2002.

#### *2.3. The Most General Neutrosophic Lattices*

The authors' lattices are far from catching the *most general definition of the neutrosophic set*.

Let U be a universe of discourse, and *M* ⊂ U be a set. Then an element *x*(*T*(*x*), *I*(*x*), *F*(*x*)) ∈ *M*, where *T*(*x*), *I*(*x*), *F*(*x*) are standard or nonstandard subsets of nonstandard interval: ] <sup>−</sup>Ω, Ψ+[, where Ω ≤ 0 < 1 ≤ Ψ, with Ω, Ψ ∈ R, whose values Ω and Ψ depend on each application, and

$$[\cdot,\cdot]^-\Omega,\ \Psi^+\left[=\!\_N\left\{\varepsilon,a,a^-,a^{-0},\ a^+,\ a^{+0},\ a^{\mp},\ a^{-0+}\right\}\varepsilon,a\in[\Omega,\ \Psi],\ \varepsilon\text{ is infinitesimal}\right\}\_{\prime\prime}$$

where *<sup>m</sup> a*, *m* ∈ −, <sup>−</sup><sup>0</sup> , + , <sup>+</sup><sup>0</sup> , <sup>−</sup><sup>+</sup> , <sup>−</sup>0<sup>+</sup> are monads or binads [12].

It follows that the nonstandard neutrosophic mobinad real offsets lattices ] <sup>−</sup>Ω, <sup>Ψ</sup>+[,≤*nonS N* and ] <sup>−</sup>Ω, <sup>Ψ</sup>+[, inf*N*, sup*N*, <sup>−</sup> <sup>Ω</sup>, <sup>Ψ</sup><sup>+</sup> of the first type and, respectively, of the second type are the most general (non-refined) neutrosophic lattices.

While the most general refined neutrosophic lattices of the first type is: ] <sup>−</sup>Ω, <sup>Ψ</sup>+[,≤*nonS nN* , where <sup>≤</sup>*nonS nN* is the n-tuple nonstandard neutrosophic inequality dealing with nonstandard subsets, defined as:

$$\begin{pmatrix} T\_1(\mathbf{x}), \ T\_2(\mathbf{x}), \dots, \ T\_p(\mathbf{x}); \ I\_1(\mathbf{x}), \ I\_2(\mathbf{x}), \dots, \ I\_r(\mathbf{x}); \ F\_1(\mathbf{x}), \ F\_2(\mathbf{x}), \dots, \dots, \ F\_s(\mathbf{x}) \end{pmatrix} \leq\_{\text{nNS}}^{\text{nNS}} \left( T\_1(y), \dots, T\_p(y) \right)$$
 
$$T\_2(y), \dots, \dots, T\_p(y); \ I\_1(y), \ I\_2(y), \dots, \dots, \ I\_r(y); \ F\_1(y), \ F\_2(y), \dots, \dots, \ F\_s(y) \right) \text{ iff }$$
 
$$\begin{aligned} T\_1(\mathbf{x}) \leq\_{\text{nNS}}^{\text{nNS}} & T\_1(y), \; T\_2(\mathbf{x}) \leq\_{\text{nNS}}^{\text{nNS}} \; T\_2(y), \; \dots, \; T\_p(\mathbf{x}) \leq\_{\text{nNS}}^{\text{nNS}} \; T\_p(y) \\\ I\_1(\mathbf{x}) \geq\_{\text{nNS}}^{\text{nNS}} I\_1(y), \; I\_2(\mathbf{x}) \geq\_{\text{nNS}}^{\text{nNS}} I\_2(y), \; \dots, \; I\_r(\mathbf{x}) \geq\_{\text{nNS}}^{\text{nNS}} I\_r(y) \end{aligned} \\\begin{aligned} T\_1(y) \leq\_{\text{nNS}}^{\text{nNS}} \; T\_1(y) \end{aligned}$$

#### *2.4. Distinction between Absolute Truth and Relative Truth*

The authors' lattices are incapable of making distinctions between absolute truth (when *T* = 1<sup>+</sup> >*<sup>N</sup>* 1) and relative truth (when *T* = 1) in the sense of Leibniz, which is the essence of nonstandard neutrosophic logic.

#### *2.5. Neutrosophic Standard Subset Lattices*

Their three lattices are not even able to deal with *standard subsets* [including intervals [8], and hesitant (discrete finite) subsets] *T*, *I*, *F* ⊆ [0, 1], since they have defined the 3D-inequalities with respect to single-valued (crisp) numbers: *x*1, *x*2, *x*<sup>3</sup> ∈ [0, 1] and *y*1, *y*2, *y*<sup>3</sup> ∈ [0, 1].

In order to deal with standard subsets, they should use *inf*/*sup*, i.e.,

(*T*1, *I*1, *F*1) ≤ (*T*2, *I*2, *F*2) ⇔ inf*T*<sup>1</sup> ≤ inf*T*<sup>2</sup> and sup*T*<sup>1</sup> ≤ sup*T*2, inf*I*<sup>1</sup> ≥ inf*I*<sup>2</sup> and sup*I*<sup>1</sup> ≥ sup*I*2, and inf*F*<sup>1</sup> ≥ inf*F*<sup>2</sup> and sup*F*<sup>1</sup> ≥ sup*F*<sup>2</sup>

[I have displayed the most used 3D-inequality by the neutrosophic community.]

#### *2.6. Nonstandard and Standard Refined Neutrosophic Lattices*

The *Nonstandard Refined Neutrosophic Set* [2,6,12], defined on ] <sup>−</sup>0, 1+[ *<sup>n</sup>*, strictly includes their n-dimensional unit cube (I*n*), and we use a nonstandard neutrosophic inequality, not the classical inequalities, to deal with inequalities of monads and binads, such as <sup>≤</sup>*nonS nN* and <sup>≤</sup>*nonS <sup>N</sup>* .

Not even the Standard Refined Single-Valued Neutrosophic Set [6] (2013) may be characterized with *KL*1, *KL*2, and *KL*<sup>3</sup> nor with <sup>I</sup>*n*, <sup>≤</sup>comp , since the *n*-D neutrosophic inequality is different from *n*-D ≤comp, and from *n*-D extensions of ≤*I*<sup>3</sup> or ≤*I*<sup>3</sup> respectively, as follows:

Let *T* be refined into *T*1, *T*2, ... , *Tp*;

*I* be refined into *I*1, *I*2, ... , *Ir*;

and *F* be refined into *F*1, *F*2, ... , *Fs*;

with *p*, *r*, *s* ≥ 1 are integers, and *p* + *r* + *s* = *n* ≥ 4, produced the following *n*-D neutrosophic inequality.

Let *x Tx* <sup>1</sup>, *<sup>T</sup><sup>x</sup>* <sup>2</sup>, ... , *<sup>T</sup><sup>x</sup> <sup>p</sup>*; *I x* 1, *I x* <sup>2</sup>, ... , *I x <sup>r</sup>* ; *Fx* <sup>1</sup>, *<sup>F</sup><sup>x</sup>* <sup>2</sup>, ... , *Fx s* , and *y Ty* <sup>1</sup> , *<sup>T</sup><sup>y</sup>* <sup>2</sup> , ... , *<sup>T</sup><sup>y</sup> <sup>p</sup>* ; *I y* 1 , *I y* <sup>2</sup> , ... , *I y <sup>r</sup>* ; *<sup>F</sup><sup>y</sup>* <sup>1</sup>, *<sup>F</sup><sup>y</sup>* <sup>2</sup>, ... , *<sup>F</sup><sup>y</sup> s* . Then:

$$\propto \le\_N \mathcal{Y} \Leftrightarrow \left( \begin{array}{l} T\_1^{\ge} \le T\_{1'}^{\mathcal{Y}}, T\_2^{\ge} \le T\_{2'}^{\mathcal{Y}}, \dots, T\_p^{\mathcal{X}} \le T\_p^{\mathcal{Y}}; \\\ I\_1^{\ge} \ge I\_{1'}^{\mathcal{Y}}, I\_2^{\ge} \ge I\_{2'}^{\mathcal{Y}}, \dots, I\_r^{\mathcal{X}} \ge I\_r^{\mathcal{Y}}; \\\ F\_1^{\ge} \ge F\_{1'}^{\mathcal{Y}}, F\_2^{\ge} \ge F\_{\prime}, \dots, F\_s^{\mathcal{X}} \ge F\_s^{\mathcal{Y}}. \end{array} \right).$$

#### *2.7. Neutrosophic Standard Overset*/*Underset*/*O*ff*set Lattice*

Their three lattices *KL*1, *KL*<sup>2</sup> and *KL*<sup>3</sup> are no match for neutrosophic overset (when the neutrosophic components *T*, *I*, *F* > 1), nor for neutrosophic underset (when the neutrosophic components *T*, *I*, *F* < 0), and, in general, no match for the neutrosophic offset (when the neutrosophic components *T*, *I*, *F* take values outside the unit interval [0, 1] as needed in real life applications [13,14,18–20] (2006–2018): [Ω, Ψ] with Ω ≤ 0 < 1 ≤ Ψ.)

Therefore, a lattice may similarly be built on the *non-unitary neutrosophic cube* [ϕ, ψ] 3 .

#### *2.8. Sum of Neutrosophic Components up to 3*

The authors do not mention the novelty of neutrosophic theories regarding the sum of single-valued neutrosophic components *T* + *I* + *F* ≤ 3, extended up to 3, and, similarly, the corresponding inequality when *T*, *I*, *F* are subsets of [0, 1]: sup*T* + sup*I* + sup*F* ≤ 3, for neutrosophic set, neutrosophic logic, and neutrosophic probability never done before in the previous classic logic and multiple-valued logics and set theories, nor in the classical or imprecise probabilities.

This makes a big difference, since, for a single-valued neutrosophic set *S*, all unit cubes [0, 1] <sup>3</sup> are fulfilled with points, each point *P*(*a*, *b*, *c*) into the unit cube may represent the neutrosophic coordinates (*a*, *b*, *c*) of an element *x*(*a*, *b*, *c*) ∈ *S*, which was not the case for previous logics, sets, and probabilities.

This is not the case for the Picture Fuzzy Set (Cuong [21], 2013) whose domain is <sup>1</sup> <sup>6</sup> of the unit cube (a cube corner):

$$\mathbb{D}^\* = \left\{ (\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3) \in \mathbb{I}^3 \middle| \mathbf{x}\_1 + \mathbf{x}\_2 + \mathbf{x}\_3 \le 1 \right\}$$

For Intuitionistic Fuzzy Set (Atanassov [22], 1986), the following is true.

$$\mathbb{D}\_A = \left\{ (\mathfrak{x}\_1, \mathfrak{x}\_2, \mathfrak{x}\_3) \in \mathbb{I}^3 \Big| \mathfrak{x}\_1 + \mathfrak{x}\_2 + \mathfrak{x}\_3 = 1 \right\}$$

where *x*<sup>1</sup> = membership degree, *x*<sup>2</sup> = hesitant degree, and *x*<sup>3</sup> = nonmembership degree, whose domain is the main cubic diagonal triangle that connects the vertices: (1, 0, 0), (0, 1, 0), and (0, 0, 1), i.e., triangle BDE (its sides and its interior) in Figure 1.

#### *2.9. Etymology of Neutrosophy and Neutrosophic*

The authors [1] write ironically twice, in between quotations, "neutrosophic" because they did not read the etymology [10] of the word published into my first book (1998), etymology, which also appears into Denis Howe's 1999 *The Free Online Dictionary of Computing* [23], and, afterwards, repeated by many researchers from the neutrosophic community in their published papers:

**Neutrosophy** [23]: <*philosophy*> *(From Latin "neuter"—neutral, Greek "sophia"—skill*/*wisdom). A branch of philosophy, introduced by Florentin Smarandache in 1980, which studies the origin, nature, and scope of neutralities, as well as their interactions with di*ff*erent ideational spectra. Neutrosophy considers a proposition, theory, event, concept, or entity, "A"in relation to its opposite, "Anti-A" and that which is not A, "Non-A", and that which is neither "A" nor "Anti-A", denoted by "Neut-A". Neutrosophy is the basis of neutrosophic logic, neutrosophic probability, neutrosophic set, and neutrosophic statistics.*

While **neutrosophic** means what is derived/resulted from *neutrosophy*.

Unlike the "intuitionistic|" and "picture fuzzy" notions, the notion of *neutrosophic* was carefully and meaningfully chosen, coming from **neutral** (or indeterminate, denoted by <neutA>) between two opposites, *A* and anti*A*, which made the main distinction between neutrosophic logic/set/probability, and the previous fuzzy, intuitionistic fuzzy logics and sets, i.e.,


Their irony is malicious and ungrounded.

#### *2.10. Neutrosophy as Extension of Dialectics*

Let *A* be a concept, notion, idea, or theory.

Then anti*A* is the opposite of *A*, while neut*A* is the neutral (or indeterminate) part between them.

While in philosophy, Dialectics is the dynamics of opposites (*A* and anti*A*), Neutrosophy is an extension of dialectics. In other words, neutrosophy is the dynamics of opposites and their neutrals (*A*, anti*A*, neut*A*), because the neutrals play an important role in our world, interfering in one side or the other of the opposites.

Refined Neutrosophy is an extension of Neutrosophy, and it is the dynamics of the refined-items <*A1*>*,* <*A2*>*,* ... *,* <*An*>, their refined-opposites <*antiA1*>*,* <*antiA2*>*,* ... *,* <*antiAn*>, and their refined-neutrals <*neutA1*>*,* <*neutA2*>*,* ... *,* <*neutAn*>*.*

As an extension of Refined Neutrosophy one has the Plithogeny [24–27].

#### *2.11. Refined Neutrosophic Set and Lattice*

At page 11, Klement and Mesiar ([1], 2018) assert that: Considering, for *n* > 3, lattices which are isomorphic to *Ln*(I), <sup>≤</sup>*comp* , further generalizations of "neutrosophic" sets can be introduced.

The authors are uninformed so that a generalization was done in 2013 when we have published a paper [6] that introduced, for the first time, the refined neutrosophic set/logic/probability, where *T*, *I*, *F* were refined into *n* neutrosophic subcomponents:

*T*1, *T*2, ... , *Tp*; *I*1, *I*2, ... , *Ir*; *F*1, *F*2, ... , *Fs*,

With *p*, *r*, *s* ≥ 1 are integers and *p* + *r* + *s* = *n* ≥ 4.

But in our lattice (I*n*, ≤*nN*), the neutrosophic inequality is adjusted to the categories of sub-truths, sub-indeterminacies, and sub-falsehood, respectively.

$$(T\_1(\mathbf{x}), T\_2(\mathbf{x}), \dots, T\_p(\mathbf{x}); I\_1(\mathbf{x}), I\_2(\mathbf{x}), \dots, I\_r(\mathbf{x}); F\_1(\mathbf{x}), F\_2(\mathbf{x}), \dots, F\_s(\mathbf{x})) \le\_{\text{nN}} (T\_1(\mathbf{y}), T\_2(\mathbf{y}), \dots, T\_s(\mathbf{x})),$$

$$\dots, T\_p(\mathbf{y}); I\_1(\mathbf{y}), I\_2(\mathbf{y}), \dots, I\_r(\mathbf{y}); F\_1(\mathbf{y}), F\_2(\mathbf{y}), \dots, F\_s(\mathbf{y})) \text{ if and only if}$$

$$T\_1(\mathbf{x}) \le T\_1(\mathbf{y}), T\_2(\mathbf{x}) \le T\_2(\mathbf{y}), \dots, T\_p(\mathbf{x}) \le T\_p(\mathbf{y})$$

$$I\_1(\mathbf{x}) \ge I\_1(\mathbf{y}), \ I\_2(\mathbf{x}) \ge I\_2(\mathbf{y}), \dots, I\_r(\mathbf{x}) \ge I\_r(\mathbf{y})$$

$$F\_1(\mathbf{x}) \ge F\_1(\mathbf{y}), \ F\_2(\mathbf{x}) \ge F\_2(\mathbf{y}), \dots, \ F\_s(\mathbf{x}) \ge F\_s(\mathbf{y})$$

Therefore, <sup>≤</sup>*nN* is different from the n-D inequalities <sup>≤</sup>comp, and from <sup>≤</sup>I*<sup>n</sup>* and <sup>≤</sup>I*<sup>n</sup>* (extending from authors inequalities <sup>≤</sup>I<sup>3</sup> and <sup>≤</sup>I<sup>3</sup> , respectively).

#### *2.12. Nonstandard Refined Neutrosophic Set and Lattice*

Even more, Nonstandard Refined Neutrosophic Set/Logic/Probability (which include infinitesimals, monads, and closed monads, binads and closed binads) has no connection and no isomorphism whatsoever with any of the authors' lattices or extensions of their lattices for *2D* and *3D* to *nD*.

#### *2.13. Nonstandard Neutrosophic Mobinad Real Lattice*

We have built ([2], 2018) a more complex Nonstandard Neutrosophic Mobinad Real Lattice, on the nonstandard mobinad unit interval ] <sup>−</sup>0, 1+[ defined as:

$$[1^-0, 1^+] = \left\{ \varepsilon, a, a^-, a^{-0}, \ a^+, \ a^{+0}, \ a^{-+}, \ a^{-0+} \, \middle| \, \text{with } 0 \le a \le 1, \ a \in \mathbb{R}, \text{ and } \varepsilon > 0, \ \varepsilon \text{ infinitesimal, } \varepsilon \in \mathbb{R}^\* \right\}.$$

which is both **nonstandard neutrosophic lattice of the first type** (as partially ordered set, under neutrosophic inequality ≤*N*) and lattice of the second type (as algebraic structure, endowed with two binary nonstandard neutrosophic laws: inf*<sup>N</sup>* and sup*N*).

Now, ] <sup>−</sup>0, 1+[ <sup>3</sup> is a nonstandard unit cube, with much higher density than [0, 1] <sup>3</sup> and which comprise not only real numbers *a* ∈ [0, 1] but also infinitesimals ε > 0 and monads and binads neutrosophically included in ] <sup>−</sup>0, 1+[.

#### *2.14. New Ideas Brought by the Neutrosophic Theories and Never Done Before*


For example, when T, I, and F are totally dependent with each other, then *T* + *I* + *F* ≤ 1. Therefore, we obtain the particular cases of intuitionistic fuzzy set (when *T* + *I* + *F* = 1) and picture set when *T* + *I* + *F* ≤ 1.

— Nonstandard analysis used in order to distinguish between absolute and relative (truth, membership, chance).

— Refinement of the components into sub-components:

$$\left(T\_1, T\_2, \dots, T\_p; \; I\_1, I\_2, \dots, \; I\_r; \; F\_1, F\_2, \dots, F\_s\right)$$

with the newly introduced Refined Neutrosophic Logic/Set/Probability.


Klement's and Mesiar's claim that the neutrosophic set (I do not talk herein about intuitionistic fuzzy set, picture fuzzy set, and Pythagorean fuzzy set that they criticized) is not a new result is far from the truth.

#### **3. Neutrosophy vs. Yin Yang Philosophy**

Ying Han, Zhengu Lu, Zhenguang Du, Gi Luo, and Sheng Chen [3] have defined the "YinYang bipolar fuzzy set" (2018).

However, the "YinYang bipolar" is already a pleonasm, because, in Taoist Chinese philosophy, from the 6th century BC, Yin and Yang was already a bipolarity, between negative (Yin)/positive (Yang), or feminine (Yin)/masculine (Yang).

Dialectics was derived, much later in time, from Yin Yang.

Neutrosophy, as the dynamicity and harmony between opposites (Yin <A> and Yang (antiA>) together with their neutralities (things which are neither Yin nor Yang, or things which are blends of both: <neutA>) is an extension of Yin Yang Chinese philosophy. Neutrosophy came naturally since, into the dynamicity, conflict, cooperation, and even ignorance between opposites, the neutrals are attracted and play an important role.

#### *3.1. YinYang Bipolar Fuzzy Set Is the Bipolar Fuzzy Set*

The authors sincerely recognize that: "*In the existing papers, YinYang bipolar fuzzy set also was called bipolar fuzzy set [5] and bipolar-valued fuzzy set [13,16]*."

These papers are cited as References [31–33].

We prove that the YinYang bipolar fuzzy set is not equivalent with the neutrosophic set, but a particular case of the bipolar neutrosophic set.

The authors [3] say that: "Denote *I <sup>P</sup>* = [0, 1] and *I <sup>N</sup>* <sup>=</sup> [−1, 0], and *<sup>L</sup>* <sup>=</sup> ∼ α = (<sup>∼</sup> α *P* , ∼ α *N* ) ∼ α *P* ∈ *I P*, ∼ α *N* ∈ *I N* , then <sup>∼</sup> α is called the YinYang bipolar fuzzy number. (YinYang bipolar fuzzy set) *X* = {*x*1, ···, *xn*} represents the finite discourse. YinYang bipolar fuzzy set in *X* is defined by the mapping below.

$$\stackrel{\sim}{A} : X \to L, \mathfrak{x} \to \left( \stackrel{\sim}{A}^P(\mathfrak{x}), \stackrel{\sim}{A}^N(\mathfrak{x}) \right) \forall \mathfrak{x} \in X.$$

where the functions <sup>∼</sup> *A P* : *X* → *I <sup>P</sup>*, *<sup>x</sup>* <sup>→</sup> <sup>∼</sup> *A P* (*x*) ∈ *I <sup>P</sup>* and <sup>∼</sup> *A N* : *X* → *I <sup>N</sup>*, *<sup>x</sup>* <sup>→</sup> <sup>∼</sup> *A N* (*x*) ∈ *I <sup>N</sup>* define the satisfaction degree of the element *x* ∈ *X* to the property, and the implicit counter-property to the YinYang bipolar fuzzy set <sup>∼</sup>

*A* in *X*, respectively (see [3], page 2).

With simpler notations, the above set *L* is equivalent to:

*L* = {(*a*, *b*), *with a* ∈ [0, 1], *b* ∈ [−1, 0]}, and the authors denote (*a*, *b*) as the YinYang bipolar fuzzy number.

Further on, again with simpler notations, the so-called *YinYang bipolar fuzzy set* in

*X* = {*x*1, ... , *xn*} is equivalent to:

*X* = {*x*1(*a*1, *b*1), ... , *xn*(*an*, *bn*)}, where all *a*1, ... ,*an* ∈ [0, 1], and all *b*1, ... , *bn* ∈ [−1, 0]}. Clearly, this is the bipolar fuzzy set and there is no need to call it the "YinYang bipolar fuzzy set." The authors added that: "Montero et al. pointed out that the neutrosophic set is equivalent to the YinYang bipolar fuzzy set in syntax." However, the bipolar fuzzy set is not equivalent to the neutrosophic set at all. The bipolar fuzzy set is actually a particular case of the bipolar neutrosophic set, defined as (keeping the previous notations):

$$X = \langle \mathbf{x}\_1(\begin{pmatrix} a\_1, b\_1 \end{pmatrix}, \begin{pmatrix} c\_1, d\_1 \end{pmatrix}, \begin{pmatrix} c\_1, f\_1 \end{pmatrix} \rangle, \dots, \begin{pmatrix} \mathbf{x}\_n(\begin{pmatrix} a\_n, b\_n \end{pmatrix}, \begin{pmatrix} c\_n, d\_n \end{pmatrix}, \begin{pmatrix} c\_n, f\_n \end{pmatrix}) \end{pmatrix} \rangle$$

where

all *a*1, ... ,*an*, *c*1, ... , *cn*, *e*1, ... ,*en* ∈ [0, 1], and all *b*1, ... , *bn*, *d*1, ... , *dn*, *f* 1, ... , *fn* ∈ [−1, 0]}; for a generic *xj*((*aj*, *bj*),(*cj*, *dj*), (*ej*, *fj*)) ∈ *X*, 1 ≤ *j* ≤ *n*,

*ai* = *positive membership degree of xi*, and *bi* = *negative membership degree of xi*;

*ci* = *positive indeterminate-membership degree of xi*, and *di* = *negative indeterminate membership degree of xi*;

*ei* = *positive non-membership degree of xi*, and *fi* = *negative non-membership degree of xi*.

Using notations adequate to the neutrosophic environment, one found the following.

Let U be a universe of discourse, and *M* ⊂ U be a set. *M* is a **single-valued bipolar fuzzy set** (that authors call *YinYang bipolar fuzzy set*) if, for any element, *x*(*T*<sup>+</sup> (*x*) , *T*− (*x*) ) <sup>∈</sup> *<sup>M</sup>*, *<sup>T</sup>*<sup>+</sup> (*x*) <sup>∈</sup> [0, 1], and *T*− (*x*) <sup>∈</sup> [−1, 0], where *<sup>T</sup>*<sup>+</sup> (*x*) is the positive membership of *x*, and *T*<sup>−</sup> (*x*) is the negative membership of *x*. (BFS).

The authors write that: "*Montero et al. pointed that the neutrosophic set [22] is equivalent to the YinYang bipolar fuzzy set in syntax [17]*".

Montero et al.'s paper is cited below as Reference [5].

If somebody says something, it does not mean it is true. They have to verify. Actually, it is *untrue*, since the neutrosophic set is totally different from the so-called YinYang bipolar fuzzy set.

Let U be a universe of discourse, and *M* ⊂ U be a set, if for any element.

$$\mathbf{x}(T(\mathbf{x}), I(\mathbf{x}), F(\mathbf{x})) \in M$$

*T*(*x*),*I*(*x*), *F*(*x*) are standard or nonstandard real subsets of the nonstandard real subsets of the nonstandard real unit interval ] <sup>−</sup>0, 1+[. (NS).

Clearly, the definitions (BFS) and (NS) are totally different. In the so-called YinYang bipolar fuzzy set, there is no indeterminacy *I*(*x*), no nonstandard analysis involved, and the neutrosophic components may be subsets as well.

#### *3.2. Single-Valued Bipolar Fuzzy Set as a Particular Case of the Single-Valued Bipolar Neutrosophic Set*

The Single-Valued bipolar fuzzy set (alias YinYang bipolar fuzzy set) is a particular case of the Single-Valued bipolar neutrosophic set, employed by the neutrosophic community, and defined as follows:

Let U be a universe of discourse, and *M* ⊂ U be a set. *M* is a single-valued bipolar neutrosophic set, if for any element:

$$x(T^+\_{(x)}, T^-\_{(x)}; I^+\_{(x)}, I^-\_{(x)}; F^+\_{(x)}, F^-\_{(x)}) \in M$$

$$T^+\_{(x)}, I^+\_{(x)}, F^+\_{(x)} \in [0, \ 1]$$

$$T^-\_{(x)}, I^-\_{(x)}, F^-\_{(x)} \in [-1, \ 0]$$

#### *3.3. Dependent Indeterminacy vs. Independent Indeterminacy*

The authors say: "*Attanassov's intuitionistic fuzzy set [4] perfectly reflects indeterminacy but not bipolarity.*"

We disagree, since Atanassov's intuitionistic fuzzy set [22] perfectly reflects **hesitancy** between membership and non-membership not **indeterminacy**, since **hesitancy is dependent** on membership and non-membership: *H* = 1 − *T* − *F*, where *H* = hesitancy, *T* = membership, and *F* = non-membership.

It is the single-valued neutrosophic set that "perfectly reflects indeterminacy" since indeterminacy (*I*) in the neutrosophic set is **independent** from membership (*T*) and from nonmembership (*F*).

On the other hand, the neutrosophic set perfectly reflects the bipolarity membership/non-membership as well, since the membership (*T*) and nonmembership (*F*) are independent of each other.

#### *3.4. Dependent Bipolarity vs. Independent Bipolarity*

The *bipolarity* in the single-valued fuzzy set and intuitionistic fuzzy set is **dependent** (restrictive) in the sense that, if the truth-membership is *T*, then it involves the falsehood-nonmembership *F* ≤ 1 − *T* while the *bipolarity* in a single-valued neutrosophic set is independent (nonrestrictive): if the truth-membership *T* ∈ [0, 1], the falsehood-nonmebership is not influenced at all, then *F* ∈ [0, 1].

#### *3.5. Equilibriums and Neutralities*

Again: "While, in semantics, the YinYang bipolar fuzzy set suggests *equilibrium*, and neutrosophic set suggests a general *neutrality*. While the neutrosophic set has been successfully applied to a medical diagnosis [9,27], from the above analysis and the conclusion in [31], we see that the YinYang bipolar fuzzy set is clearly the suitable model to a bipolar disorder diagnosis and will be adopted in this paper."

I'd like to add that the single-valued bipolar neutrosophic set suggests:


Therefore, the single-valued bipolar neutrosophic set is 3 × 2 = 6 times more complex and more flexible than the YinYang bipolar fuzzy set. Due to higher complexity, flexibility, and capability of catching more details (such as falsehood-nonmembership, and indeterminacy), the single-valued bipolar neutrosophic set is more suitable than the YinYang bipolar fuzzy set to be used in a bipolar disorder diagnosis.

#### *3.6. Zhang-Zhang's Bipolar Model is not Equivalent with the Neutrosophic Set*

Montero et al. [5] wrote: "*Zhang-Zhang's bipolar model is, therefore, equivalent to the neutrosophic sets proposed by Smarandache [70]*" *(p. 56)*.

This sentence is false and we proved previously that what Zhang & Zhang proposed in 2004 is a subclass of the single-valued bipolar neutrosophic set.

#### *3.7. Tripolar and Multipolar Neutrosophic Sets*

Not talking about the fact that, in 2016, we have extended our bipolar neutrosophic set to tripolar and even multipolar neutrosophic sets [18], the sets have become more general than the bipolar fuzzy model.

#### *3.8. Neutrosophic Overset*/*Underset*/*O*ff*set*

Not talking that the unit interval [0, 1] was extended in 2006 below 0 and above 1 into the neutrosophic overset/underset/offset: [Ω, Ψ] with Ω ≤ 0 < 1 ≤ Ψ (as explained above).

#### *3.9. Neutrosophic Algebraic Structures*

The Montero et al. [5] continue: "*Notice that none of these two equivalent models include any formal structure, as claimed in [48]*".

First, we have proved that these two models (Zhang-Zhang's bipolar fuzzy set, and neutrosophic logic) are not equivalent at all. Zhang-Zhang's bipolar fuzzy set is a subclass of a particular type of neutrosophic set, called the single-valued bipolar neutrosophic set.

Second, since 2013, Kandasamy and Smarandache have developed various algebraic structures (such as neutrosophic semigroup, neutrosophic group, neutrosophic ring, neutrosophic field, neutrosophic vector space, etc.) [28] on the set of neutrosophic numbers:

*SR* <sup>=</sup> *a* + *bI*|, where *a*, *b* ∈ R, and *I* = indeterminacy, *I* <sup>2</sup> = *I* , where R is the set of real numbers. And extended on:

*SC* <sup>=</sup> *a* + *bI*|, where *a*, *b* ∈ *C*, and *I* = indeterminacy, *I* <sup>2</sup> = *I* , where *C* is the set of complex numbers.

However, until 2016 [year of Montero et al.'s published paper], I did not develop a formal structure on the neutrosophic set. Montero et al. are right.

Yet, in 2018, and, consequently at the beginning of 2019, we [2] developed, then generalized, and proved that the neutrosophic set has a structure of the lattice of the first type (as the neutrosophically partially ordered set): ( ] <sup>−</sup>0, 1+[,≤*N*), where ] <sup>−</sup>0, 1+[ is the nonstandard neutrosophic mobinad (monads and binads) real unit interval, and ≤*<sup>N</sup>* is the nonstandard neutrosophic inequality. Moreover, ] <sup>−</sup>0, 1+[, inf*N*, sup*N*, <sup>−</sup> 0, 1<sup>+</sup> has the structure of the bound lattice of the second type (as algebraic structure), under two binary laws inf*<sup>N</sup>* (nonstandard neutrosophic infimum) and sup*<sup>N</sup>* (nontandard neutrosophic supremum).

#### *3.10. Neutrality (*<*neutA*>*)*

Montero et al. [5] continue: " ... *the selected denominations within each model might suggest di*ff*erent underlying structures: while the model proposed by Zhang and Zhang suggests conflict between categories (a specific type of neutrality di*ff*erent from Atanassov's indeterminacy), Smarandache suggests a general neutrality that should, perhaps jointly, cover some of the specific types of neutrality considered in our paired approach.*"

In neutrosophy and neutrosophic set/logic/probability, the neutrality <neutA> means everything in between <A> and <antiA>, everything which is neither <A> nor <antiA>, or everything which is a blending of <A> and <antiA>.

Further on, in Refined Neutrosophy and Refined Neutrosophic Set/Logic/Probability [9], the neutrality <neutA> was split (refined) in 2013 into sub-neutralities (or sub-indeterminacies), such as: <neutA1>, <neutA2>, ... , <neutAn> whose number could be finite or infinite depending on each application that needs to be solved.

Thus, the paired structure becomes a particular case of refined neutrosophy (see next).

#### **4. The Pair Structure as a Particular Case of Refined Neutrosophy**

Montero et al. [5] in 2016 have defined a **paired structure**: "*composed by a pair of opposite concepts and three types of neutrality as primary valuations: L* = *{concept, opposite, indeterminacy, ambivalence, conflict}.*"

Therefore, each element *x* ∈ *X*, where *X* is a universe of discourse, is characterized by a degree function, with respect to each attribute value from *L*:

$$
\mu: X \to [0,1]^5
$$

$$\mu(\mathfrak{x}) = (\mu\_1(\mathfrak{x}), \mu\_2(\mathfrak{x}), \mu\_3(\mathfrak{x}), \mu\_4(\mathfrak{x}), \mu\_5(\mathfrak{x})) $$

where μ1(*x*) represents the degree of *x* with respect to the concept;

μ2(*x*) represents the degree of *x* with respect to the opposite (of the concept);

μ3(*x*) represents the degree of *x* with respect to 'indeterminacy';

μ4(*x*) represents the degree of *x* with respect to 'ambivalence';

μ5(*x*) represents the degree of *x* with respect to 'conflict'.

However, this paired structure is a particular case of Refined Neutrosophy.

#### *4.1. Antonym vs. Negation*

First, Dialectics is the dynamics of opposites. Denote them by *A* and anti*A*, where *A* may be an item, a concept, attribute, idea, theory, and so on while anti*A* is the opposite of *A*.

Secondly, Neutrosophy ([10], 1998), as a generalization of Dialectics, and a new branch of philosophy, is the dynamics of opposites and their neutralities (denoted by neut*A*). Therefore, Neutrosophy is the dynamics of *A*, anti*A*, and neut*A*.

neut*A* means everything, which is neither *A* nor anti*A*, or which is a mixture of them, or which is indeterminate, vague, or unknown.

The **antonym** of *A* is anti*A*.

The **negation** of *A* (which we denote by non*A*) is what is not *A*, therefore:

$$\neg\_N \langle A \rangle = \langle n \omicron n A \rangle =\_N \langle n \textit{nut} A \rangle \cup\_N \langle a \textit{nti} A \rangle$$

We preferred to use the lower index <sup>N</sup> (neutrosophic) because we deal with items, concepts, attributes, ideas, and theories such as *A* and, in consequence, its derivates anti*A*, neut*A*, and non*A*, whose borders are ambiguous, vague, and not clearly delimited.

#### *4.2. Refined Neutrosophy as an Extension of Neutrosophy*

Thirdly, Refined Neutrosophy ([6], 2013), as an extension of Neutrosophy, and a refined branch of philosophy, is the dynamics of refined opposites: *A*1, *A*2, ... , *Ap* with anti*A*1, anti*A*2, ... , anti*As*, and their refined neutralities: neut*A*1, neut*A*2, ... , neut*Ar*, for integers p, r, *s* ≥ 1, and *p* + *r* + *s* = *n* ≥ 4. Therefore, the item *A* has been split into sub-items *Aj*, 1 ≤ *j* ≤ *p*, the anti*A* into sub-(anti-items) anti*Ak*, 1 ≤ *l* ≤ *s*, and the neut*A* into sub-(neutral-items) neut*Al*, 1 ≤ *k* ≤ *r*.

#### *4.3. Qualitative Scale as a Particular Case of Refined Neutrosophy*

Montero et al.'s qualitative scale [5] is a particular case of Refined Neutrosophy where the neutralities are split into three parts.

*L* = *{concept, opposite, indeterminacy, ambivalence, conflict}* = *{*<*A*>*,* <*antiA*>*,* <*neutA1*>*,* <*neutA2*>*,* <*neutA3*>*}*

where: <*A*> = *concept*, <*antiA*> = *opposite*, <*neutA1*> = *indeterminacy*, <*neutA2*> = *ambivalence*, <*neutA3*> = *conflict*.

Yin Yang, Dialectics, Neutrosophy, and Refined Neutrosophy (the last one having only neut*A* as refined component), are bipolar: *A* and anti*A* are the poles.

Montero et al.'s qualitative scale is bipolar ('concept', and its 'opposite').

#### *4.4. Multi-Subpolar Refined Neutrosophy*

However, the Refined Neutrosophy, whose at least one of *A* or anti*A* is refined, is *multi-subpolar*.

#### *4.5. Multidimensional Fuzzy Set as a Particular Case of the Refined Neutrosophic Set*

Montero et al. [5] defined the *Multidimensional Fuzzy Set AL* as: *At* <sup>=</sup> <sup>&</sup>lt; *<sup>x</sup>*;(μ*s*(*x*))*s*∈*<sup>L</sup>* <sup>&</sup>gt; *<sup>x</sup>* <sup>∈</sup> *<sup>X</sup>* , where X is the universe of discourse, *L* = the previous qualitative scale, and μ*s*(*x*) ∈ *S*, where *S* is a valuation scale (in most cases *S* = [0, 1]), μ*s*(*x*) is the degree of *x* with respect to *s* ∈ *L*.

A *Single-Valued Neutrosophic Set* is defined as follows. Let U be a universe of discourse, and *M* ⊂ U a set. For each element *x*(*T*(*x*), *I*(*x*), *F*(*x*)) ∈ *M*, *T*(*x*) ∈ [0, 1] is the degree of truth-membership of element *x* with respect to the set *M*, *I*(*x*) ∈ [0, 1] is the degree of indeterminacy-membership of element *x* with respect to the set *M*, and *F*(*x*) ∈ [0, 1] is the degree of falsehood-nonmembership of element *x* with respect to the set *M*.

Let's refine *I*(*x*) as *I*1(*x*), *I*2(*x*), and *I*3(*x*) ∈ [0, 1] sub-indeterminacies. Then we get a single-valued refined neutrosophic set.

μ*concept*(*x*) = *T*(*x*) (truth-membership); μ*opposite*(*x*) = *F*(*x*) (falsehood-non-membership); μindeterminacy(*x*) = *I*1(*x*) (first sub-indeterminacy); μambivalence(*x*) = *I*2(*x*) (second sub-indeterminacy); μconflict(*x*) = *I*3(*x*) (third sub-indeterminacy).

The *Single-Valued Refined Neutrosophic Set* is defined as follows. Let U be a universe of discourse, and *M* ⊂ U a set. For each element:

$$\text{tr}\left(T\_1(\mathbf{x}), T\_2(\mathbf{x}), \dots, T\_p(\mathbf{x}); I\_1(\mathbf{x}), I\_2(\mathbf{x}), \dots, I\_r(\mathbf{x}); F\_1(\mathbf{x}), F\_2(\mathbf{x}), \dots, F\_s(\mathbf{x})\right) \in M$$

*Tj*(*x*), 1 ≤ *j* ≤ *p*, are degrees of subtruth-submembership of element *x* with respect to the set *M*.

*Ik*(*x*), 1 ≤ *k* ≤ *r*, are degrees of subindeterminacy-membership of element *x* with respect to the set *M*.

Lastly, *Fl*(*x*), 1 ≤ *l* ≤ *s*, are degrees of sub-falsehood-sub-non-membership of element *x* with respect to the set *M*, where integers p, r, *s* ≥ 1, and *p* + *r* + *s* = *n* ≥ 4.

Therefore, Montero et al.'s **multidimensional fuzzy set** is a particular case of the **refined neutrosophic set**, when *p* = 1, *r* = 3, and *s* = 1, where *n* = 1 + 3 + 1 = 5.

#### *4.6. Plithogeny and Plithogenic Set*

Fourthly, in 2017 and in 2018 [24–27], the Neutrosophy was extended to Plithogeny, which is multipolar, being the dynamics and hermeneutics [methodological study and interpretation] of many opposites and/or their neutrals, together with non-opposites.

*A*, neut*A*, anti*A*;

*B*, neut*B*, anti*B*; etc.

*C*, *D*, etc.

In addition, the **Plithogenic Set** was introduced, as a generalization of **Crisp**, **Fuzzy**, **Intuitionistic Fuzzy**, and **Neutrosophic Sets**.

Unlike previous sets defined, whose elements were characterized by the attribute 'appurtenance' (to the set), which has only one (membership), or two (membership, nonmembership), or three (membership, nonmembership, indeterminacy) attribute values, respectively. For the Plithogenic Set, each element may be characterized by a multi-attribute, with any number of attribute values.

*4.7. Refined Neutrosophic Set as a Unifying View of Opposite Concepts*

Montero et al.'s statement [5] from their paper Abstract: "*we propose a consistent and unifying view to all those basic knowledge representation models that are based on the existence of two somehow opposite fuzzy concepts.*"

With respect to the "unifying" claim, their statement is not true, since, as we proved before, their **paired structure** together with three types on neutralities (**indeterminacy**, **ambivalence**, and **conflict**) is a simple, particular case of the refined neutrosophic set.

The real unifying view currently is the **Refined Neutrosophic Set**.

{I was notified about this paired structure article [5] by Dr. Said Broumi, who forwarded it to me.}

#### *4.8. Counter-Example to the Paired Structure*

As a counter example to the paired structure [5], it cannot catch a simple voting scenario.

The election for the United States President from 2016: Donald Trump vs. Hillary Clinton. USA has 50 states and since, in the country, there is an **Electoral vote**, not a **Popular vote**, it is required to know the winner of each state.

There were two opposite candidates.

The candidate that receives more votes than the other candidate in a state gets all the points of that state.

As in the neutrosophic set, there are three possibilities:

*T* = percentage of USA people voting for Mr. Trump;

*I* = percentage of USA people not voting, or voting but giving either a blank vote (not selecting any candidate) or a black vote (cutting all candidates);

*F* = percentage of USA people voting against Mr. Trump.

The opposite concepts, using Montero et al.'s knowledge representation, are T (voting for, or truth-membership) and F (voting against, or false-membership). However, *T* > *F*, or *T* = *F*, or *T* < *F*, that the Paired Structure can catch, mean only the Popular vote, which does not count in the United States.

Actually, it happened that *T* < *F* in the US 2016 presidential election, or Mr. Trump lost the Popular vote, but he won the Presidency using the Electoral vote.

The paired structure is not capable of refining the opposite concepts (*T* and *F*), while the indeterminate (*I*) could be refined by the paired structure only in three parts.

Therefore, the paired structure is not a unifying view of all basic knowledge that uses opposite fuzzy concepts. However, the refined neutrosophic set/logic/probability do.

Using the refined neutrosophic set and logic, and splits (refines) *T*, *I*, and *F* as:

*Tj* = percentage of American state *Sj* people voting for Mr. Trump;

*Ij* = percentage of American state *Sj* people not voting, or casting a blank vote or a black vote;

*Fj* = percentage of American state *Sj* people voting against Mr. Trump, with *Tj*, *Ij*, *Fj* ∈ [0, 1] and *Tj* + *Ij* + *Fj* = 1, for all *j* ∈ {1, 2, ... , 50}.

Therefore, one has:

(*T*1, *T*2, ... , *T*50; *I*1, *I*2, ... , *I*50; *F*1, *F*2, ... , *F*50).

On the other hand, due to the fact that the sub-indeterminacies *I*1, *I*2, ... , *I*<sup>50</sup> did not count towards the winner or looser (only for indeterminate voting statistics), it is not mandatory to refine *I*. We could simply refine it as:

(*T*1, *T*2, ... , *T*50; *I*; *F*1, *F*2, ... , *F*50).

#### *4.9. Finite Number and Infinite Number of Neutralities*

Montero et al. [5]: "*(* ... *) we emphasize the key role of certain neutralities in our knowledge representation models, as pointed out by Atanassov [4], Smarandache [70], and others. However, we notice that our notion of neutrality should not be confused with the neutral value in a traditional sense (see [22–24,36,54], among others).* *Instead, we will stress the existence of di*ff*erent kinds of neutrality that emerge (in the sense of Reference [11]) from the semantic relation between two opposite concepts (and notice that we refer to a neutral category that does not entail linearity between opposites).*"

In neutrosophy, and, consequently, in the neutrosophic set, logic, and probability, between the opposite items (concepts, attributes, ideas, etc.) *A* and anti*A*, there may be a large number of neutralities/indeterminacies (all together denoted by neut*A* even an infinite spectrum—depending on the application to solve.

We agree with different kinds of neutralities and indeterminacies (vague, ambiguous, unknown, incomplete, contradictory, linear and non-linear information, and so on), but the authors display only three neutralities.

In our everyday life and in practical applications, there are more neutralities and indeterminacies.

In another example (besides the previous one about Electoral voting), there may be any number of sub indeterminacies/sub neutralities.

The opposite concepts attributes are: *A* = white, anti*A* = black, while neutral concepts in between may be: neut*A*1 = yellow, neut*A*2 = orange, neut*A*3 = red, neut*A*4 = violet, neut*A*5 = green, and neut*A*6 = blue. Therefore, we have six neutralities. Example with infinitely many neutralities:

— The opposite concepts: *A* = white, anti*A* = black;

— The neutralities: neut*A*1, 2, ..., ∞ = the whole light spectrum between white and black, measured in nanometers (*nn*) [a nanometer is a billionth part of a meter].

#### **5. Conclusions**

The neutrosophic community thank the authors for their criticism and interest in the neutrosophic environment, and we wait for new comments and criticism, since, as Winston Churchill had said, *the eagles fly higher against the wind*.

**Funding:** The author received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **Notations**


#### **References**


© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Linguistic Neutrosophic Numbers Einstein Operator and Its Application in Decision Making**

**Changxing Fan, Sheng Feng and Keli Hu \***

Department of Computer Science, Shaoxing University, Shaoxing 312000, China; fcxjszj@usx.edu.cn (C.F.); fengsheng\_13@aliyun.com (S.F.)

**\*** Correspondence: ancimoon@gmail.com

Received: 28 March 2019; Accepted: 25 April 2019; Published: 28 April 2019

**Abstract:** Linguistic neutrosophic numbers (LNNs) include single-value neutrosophic numbers and linguistic variable numbers, which have been proposed by Fang and Ye. In this paper, we define the linguistic neutrosophic number Einstein sum, linguistic neutrosophic number Einstein product, and linguistic neutrosophic number Einstein exponentiation operations based on the Einstein operation. Then, we analyze some of the relationships between these operations. For LNN aggregation problems, we put forward two kinds of LNN aggregation operators, one is the LNN Einstein weighted average operator and the other is the LNN Einstein geometry (LNNEWG) operator. Then we present a method for solving decision-making problems based on LNNEWA and LNNEWG operators in the linguistic neutrosophic environment. Finally, we apply an example to verify the feasibility of these two methods.

**Keywords:** multiple attribute group decision making (MAGDM); Linguistic neutrosophic; LNN Einstein weighted-average operator; LNN Einstein weighted-geometry (LNNEWG) operator

#### **1. Introduction**

Smarandache [1] proposed the neutrosophic set (NS) in 1998. Compared with the intuitionistic fuzzy sets (IFSs), the NS increases the uncertainty measurement, from which decision makers can use the truth, uncertainty and falsity degrees to describe evaluation, respectively. In the NS, the degree of uncertainty is quantified, and these three degrees are completely independent of each other, so, the NS is a generalization set with more capacity to express and deal with the fuzzy data. At present, the study of NS theory has been a part of research that mainly includes the research of the basic theory of NS, the fuzzy decision of NS, and the extension of NS, etc. [2–14]. Recently, Fang and Ye [15] presented the linguistic neutrosophic number (LNN). Soon afterwards, many research topics about LNN were proposed [16–18].

Information aggregation operators have become an important research topic and obtained a wide range of research results. Yager [19] put forward the ordered weighted average (OWA) operator considering the data sorting position. Xu [20] presented the arithmetic aggregation (AA) of IFS. Xu and Yager [21] presented the geometry aggregation (GA) operator of IFS. Zhao [22] proposed generalized aggregation operators based on IFS and proved that AA and GA were special cases of generalized aggregation operator. The operators mentioned above are established based on the algebraic sum and the algebraic product of number sets. They are respectively referred to as a special case of Archimedes t-conorm and t-norm to establish union or intersection operation of the number set. The union and intersection of Einstein operation is a kind of Archimedes t-conorm and t-norm with good smooth characteristics [23]. Wang and Liu [24] built some IF Einstein aggregation operators and proved that the Einstein aggregation operator has better smoothness than the arithmetic aggregation operator. Zhao and Wei [25] put forward the IF Einstein hybrid-average (IFEHA) operator and IF

Einstein hybrid-geometry (IFEHG) operator. Further, Guo etc. [26] applied the Einstein operation to a hesitate fuzzy set. Lihua Yang etc. [27] put forward novel power aggregation operators based on Einstein operations for interval neutrosophic linguistic sets. However, neutrosophic linguistic sets are different from linguistic neutrosophic sets. The former still use two values to describe the evaluation value, while the latter can use a pure language value to describe the evaluation value. As far as we know, this is the first work on Einstein aggregation operators for LNN. It must be noticed that the aggregation operators in References [15–18] are almost based on the most commonly used algebraic product and algebraic sum of LNNs for carrying the combination process, which is not the only operation law that can be chosen to model the intersection and union on LNNs. Thus, we establish the operation rules of LNN based on Einstein operation and put forward the LNN Einstein weighted-average (LNNEWA) operator and LNN Einstein weighted-geometry (LNNEWG) operator. These operators are finally utilized to solve some relevant problems.

The other organizations: in Section 2, concepts of LNN and Einstein are described, operational laws of LNNs based on Einstein operation are defined, and their performance is analyzed. In Section 3, LNNEWA and LNNEWG operators are proposed. In Section 4, multiple attribute group decision making (MAGDM) methods are built based on LNNEWA and LNNEWG operators. In Section 5, an instance is given. In Section 6, conclusions and future research are given.

#### **2. Basic Theories**

#### *2.1. LNN and Its Operational Laws*

**Definition 1.** [15] *Set a finite language set* <sup>Ψ</sup> = ψ*t <sup>t</sup>* <sup>∈</sup> [0, *<sup>k</sup>*] *, where* ψ*<sup>t</sup> is a linguistic variable, k* +*1 is the cardinality of* Ψ*. Then, we define u* = ψβ,ψγ,ψδ*, in which* ψβ,ψγ,ψδ ∈ Ψ *and* β, γ, δ ∈ *[0, k],* ψβ,ψδ *and* ψγ *expresse truth, falsity and indeterminacy degree, respectively, we call u an LNN.*

**Definition 2.** [15] *Set three LNNs u* = ψβ,ψγ,ψδ*, u*<sup>1</sup> = ψβ<sup>1</sup> ,ψγ<sup>1</sup> ,ψδ<sup>1</sup> *and u*<sup>2</sup> = ψβ<sup>2</sup> ,ψγ<sup>2</sup> ,ψδ<sup>2</sup> *in* Ψ *and* λ ≥ 0*, then, the operational rules are as following:*

$$\mathbf{u} \oplus \mathbf{u}\_2 = \langle \psi\_{\beta\_1}, \psi\_{\gamma\_1}, \psi\_{\delta\_1} \rangle \oplus \langle \psi\_{\beta\_2}, \psi\_{\gamma\_2}, \psi\_{\delta\_2} \rangle = \langle \psi\_{\beta\_1 + \beta\_2 - \frac{\beta\_1 \beta\_2}{k}}, \psi\_{\frac{\gamma\_1 \gamma\_2}{k}}, \psi\_{\frac{\delta\_1 \delta\_2}{k}} \rangle;\tag{1}$$

$$\mathbf{u}\_1 \otimes \mathbf{u}\_2 = \langle \psi\_{\beta\_1}, \psi\_{\gamma\_1}, \psi\_{\delta\_1} \rangle \otimes \langle \psi\_{\beta\_2}, \psi\_{\gamma\_2}, \psi\_{\delta\_2} \rangle = \langle \psi\_{\frac{\delta\_1 \delta\_2}{k}}, \psi\_{\gamma\_1 + \gamma\_2 - \frac{\gamma\_1 \gamma\_2}{k}}, \psi\_{\delta\_1 + \delta\_2 - \frac{\delta\_1 \delta\_2}{k}} \rangle;\tag{2}$$

$$
\lambda \mu = \lambda \langle \psi\_{\not\rhd\_1}, \psi\_{\not\rhd\_1}, \psi\_{\not\rhd\_1} \rangle = \langle \psi\_{\not\rhd\_{k-\left(1-\frac{\theta}{k}\right)}\lambda}, \psi\_{\not\rhd\_{k}\left(\frac{\theta}{k}\right)}, \psi\_{\not\rhd\_{k}\left(\frac{\theta}{k}\right)} \rangle; \tag{3}
$$

$$
\mu^{\lambda} = \langle \psi\_{\mathbb{R}^1}, \psi\_{\mathbb{M}^\lambda}, \psi\_{\mathbb{S}\_1} \rangle^{\lambda} = \langle \psi\_{\mathbb{A}(\frac{\mathbb{R}}{k})^{\lambda}}, \psi\_{k - k(1 - \frac{\mathbb{Y}}{k})^{\lambda}}, \psi\_{k - k(1 - \frac{\mathbb{Y}}{k})^{\lambda}} \rangle. \tag{4}
$$

**Definition 3.** [15] *Set an LNN u* = ψβ,ψγ,ψδ *in* Ψ*, we define* ζ(*u*) *as the expectation and* η(*u*) *as the accuracy:*

$$\mathcal{L}(\mathbf{u}) = (2\mathbf{k} + \beta - \gamma - \delta) / 3\mathbf{k} \tag{5}$$

$$
\eta(u) = (\beta - \delta) / k \tag{6}
$$

**Definition 4.** [15]: *Set two LNNs u*<sup>1</sup> = ψβ<sup>1</sup> ,ψγ<sup>1</sup> ,ψδ<sup>1</sup> *and u*<sup>2</sup> = ψβ<sup>2</sup> ,ψγ<sup>2</sup> ,ψδ<sup>2</sup> *in* Ψ*, then*

*If* ζ(*u*1) > ζ(*u*2)*, then u*<sup>1</sup> *u*2*; If* ζ(*u*1) = ζ(*u*2) *then If* η(*u*1) > η(*u*2)*, then u*<sup>1</sup> *u*2*; If* η(*u*1) = η(*u*2)*, then u*<sup>1</sup> ∼ *u*2*.*

#### *2.2. Einstein Operation*

**Definition 5.** [28,29] *For any two real Numbers a, b*∈ [0, 1]*, Einstein* ⊕*<sup>e</sup> is an Archimedes t-conorms, Einstein* ⊗*<sup>e</sup> is an Archimedes t-norms, then*

$$\mathbf{a} \circledast \mathbf{b} = \frac{a+b}{1+ab'} \quad \mathbf{a} \circledast \mathbf{b} = \frac{ab}{1+(1-a)(1-b)}.\tag{7}$$

#### *2.3. Einstein Operation Under the Linguistic Neutrosophic Number*

**Definition 6.** *Set u* = ψβ,ψγ,ψδ*, u*<sup>1</sup> = ψβ<sup>1</sup> ,ψγ<sup>1</sup> ,ψδ<sup>1</sup> *and u*<sup>2</sup> = ψβ<sup>2</sup> ,ψγ<sup>2</sup> ,ψδ<sup>2</sup> *as three LNNs in* Ψ*,* λ ≥ 0, *the operation of Einstein* ⊕*<sup>e</sup> and Einstein* ⊗*<sup>e</sup> under the linguistic neutrosophic number are defined as follows:*

$$\Psi\_1 \Psi\_t \,\upmu\_2 = \langle \psi\_{\frac{k^2(\beta\_1+\beta\_2)}{k^2+\beta\_1\beta\_2}}, \psi\_{\frac{k\gamma\_1\gamma\_2}{k^2+(k-\gamma\_1)(k-\gamma\_2)}}, \Psi\_{\frac{k\delta\_1\delta\_2}{k^2+(k-\delta\_1)(k-\delta\_2)}} \rangle; \tag{8}$$

$$\mu\_1 \otimes \mu\_2 = \langle \psi\_{\frac{k\beta\_1\beta\_2}{k^2 + (k-\beta\_1)(k-\beta\_2)}} \prime \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!$$

$$\lambda \mu = \langle \psi\_{k\*\frac{(k+\beta)^{\lambda}-(k-\beta)^{\lambda}}{(k+\beta)^{\lambda}+(k-\beta)^{\lambda}}} \cdot \psi\_{k\*\frac{2\gamma^{\lambda}}{(2k-\gamma)^{\lambda}+\gamma^{\lambda}}} \cdot \psi\_{k\*\frac{2\delta\cdot\lambda}{(2k-\delta)^{\lambda}+\delta^{\lambda}}} \rangle;\tag{10}$$

$$\boldsymbol{\mu}^{\boldsymbol{\lambda}} = \langle \boldsymbol{\psi}\_{\boldsymbol{k} \* \frac{2\boldsymbol{p}^{\boldsymbol{\lambda}}}{\left(2\boldsymbol{k} - \boldsymbol{\beta}\right)^{\boldsymbol{\lambda}} + \boldsymbol{p}^{\boldsymbol{\lambda}}}}, \boldsymbol{\psi}\_{\boldsymbol{k} \* \frac{\left(\boldsymbol{k} + \boldsymbol{\gamma}\right)^{\boldsymbol{\lambda}} - \left(\boldsymbol{k} - \boldsymbol{\gamma}\right)^{\boldsymbol{\lambda}} \boldsymbol{\lambda}^{\boldsymbol{\lambda}}}{\left(\boldsymbol{k} + \boldsymbol{\gamma}\right)^{\boldsymbol{\lambda}} + \left(\boldsymbol{k} - \boldsymbol{\gamma}\right)^{\boldsymbol{\lambda}}}}, \boldsymbol{\psi}\_{\boldsymbol{k} \* \frac{\left(\boldsymbol{k} + \boldsymbol{\delta}\right)^{\boldsymbol{\lambda}} - \left(\boldsymbol{k} - \boldsymbol{\delta}\right)^{\boldsymbol{\lambda}}}{\left(\boldsymbol{k} + \boldsymbol{\delta}\right)^{\boldsymbol{\lambda}} + \left(\boldsymbol{k} - \boldsymbol{\delta}\right)^{\boldsymbol{\lambda}}}}\rangle. \tag{11}$$

**Theorem 1.** *Set u* = ψβ,ψγ,ψδ*, u*<sup>1</sup> = ψβ<sup>1</sup> ,ψγ<sup>1</sup> ,ψδ<sup>1</sup> *and u*<sup>2</sup> = ψβ<sup>2</sup> ,ψγ<sup>2</sup> ,ψδ<sup>2</sup> *as three LNNs in* Ψ*,* λ ≥ 0, *then, the operation of Einstein* ⊕*<sup>e</sup> and Einstein* ⊗*<sup>e</sup> have the following performance:*

$$
u\_1 \oplus\_{\mathfrak{k}} \mathfrak{u}\_{\mathfrak{k}} = \mathfrak{u}\_{\mathfrak{k}} \oplus\_{\mathfrak{k}} \mathfrak{u}\_{\mathfrak{k}}.\tag{12}$$

$$
\mu\_1 \otimes\_\mathfrak{e} \mu\_2 = \mu\_2 \otimes\_\mathfrak{e} \mu\_1;\tag{13}
$$

$$
\lambda(\mu\_1 \oplus\_\mathfrak{e} \mu\_2) = \lambda \mu\_1 \oplus\_\mathfrak{e} \lambda \mu\_2;\tag{14}
$$

$$(\mu\_1 \otimes\_{\mathfrak{e}} \mu\_2)^{\lambda} = \mu\_1^{\lambda} \otimes\_{\mathfrak{e}} \mu\_2^{\lambda};\tag{15}$$

**Proof.** Performance (1) and (2) are easy to be obtained, so we omit it; Now we prove the performance (3): According to Definition 6, we can get

$$\begin{aligned} \left\langle \right\rangle\_{\mathfrak{l}} \, \upharpoonright \, \upharpoonright \, \upharpoonright \, \nu\_2 &= \left\langle \psi\_{\frac{k^2(\beta\_1+\beta\_2)}{k^2+\beta\_1\beta\_2}} \, \!\!\psi\_{\frac{k\gamma\_1\gamma\_2}{k^2+(k-\gamma\_1)(k-\gamma\_2)}} \, \n~\!\!\!\psi\_{\frac{k\delta\_1\delta\_2}{k^2+(k-\delta\_1)(k-\delta\_2)}} \right\rangle; \end{aligned}$$

② λ(*u*<sup>1</sup> ⊕*<sup>e</sup> u*2) = ψ *k*∗ (*k*<sup>+</sup> *<sup>k</sup>*2(β1+β2) *<sup>k</sup>*2+β1β<sup>2</sup> ) λ <sup>−</sup>(*k*<sup>−</sup> *<sup>k</sup>*2(β1+β2) *<sup>k</sup>*2+β1β<sup>2</sup> ) λ (*k*<sup>+</sup> *<sup>k</sup>*2(β1+β2) *<sup>k</sup>*2+β1β<sup>2</sup> ) λ +(*k*<sup>−</sup> *<sup>k</sup>*2(β1+β2) *<sup>k</sup>*2+β1β<sup>2</sup> ) λ ,ψ *k*∗ <sup>2</sup>( *<sup>k</sup>*γ1γ<sup>2</sup> *<sup>k</sup>*2+(*k*−γ1)(*k*−γ2) ) λ (2*k*<sup>−</sup> *<sup>k</sup>*γ1γ<sup>2</sup> *<sup>k</sup>*2+(*k*−γ1)(*k*−γ2) ) λ +( *<sup>k</sup>*γ1γ<sup>2</sup> *<sup>k</sup>*2+(*k*−γ1)(*k*−γ2) ) λ ,ψ *k*∗ <sup>2</sup>( *<sup>k</sup>*δ1δ<sup>2</sup> *<sup>k</sup>*2+(*k*−δ1)(*k*−δ2) ) λ (2*k*<sup>−</sup> *<sup>k</sup>*δ1δ<sup>2</sup> *<sup>k</sup>*2+(*k*−δ1)(*k*−δ2) ) λ +( *<sup>k</sup>*δ1δ<sup>2</sup> *<sup>k</sup>*2+(*k*−δ1)(*k*−δ2) ) λ = ψ *k*∗ (*k*+β1)λ(*k*+β2)λ−(*k*−β1)λ(*k*−β2)<sup>λ</sup> (*k*+β1)λ(*k*+β2)λ+(*k*−β1)λ(*k*−β2)<sup>λ</sup> ,ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup>(γ1γ2)<sup>λ</sup> ((2*k*−γ1)λ(2*k*−γ2)λ)+(γ1γ2)<sup>λ</sup> ,ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup>(δ1δ2)<sup>λ</sup> ((2*k*−δ1)λ(2*k*−δ2)λ)+(δ1δ2)<sup>λ</sup> ; ③ λ*u*<sup>1</sup> = ψ *k*∗ (*k*+β1)λ−(*k*−β1)<sup>λ</sup> (*k*+β1)λ+(*k*−β1)<sup>λ</sup> ,ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup>γ<sup>1</sup> λ (2*k*−γ1)λ+γ<sup>1</sup> λ ,ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup>δ<sup>1</sup> λ (2*k*−δ1)λ+δ<sup>1</sup> λ ; ④ λ*u*<sup>2</sup> = ψ *k*∗ (*k*+β2)λ−(*k*−β2)<sup>λ</sup> (*k*+β2)λ+(*k*−β2)<sup>λ</sup> ,ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup>γ<sup>2</sup> λ (2*k*−γ2)λ+γ<sup>2</sup> λ ,ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup>δ<sup>2</sup> λ (2*k*−δ2)λ+δ<sup>2</sup> λ ;

⑤ λ*u*<sup>1</sup> ⊕*<sup>e</sup>* λ*u*<sup>2</sup>

<sup>=</sup> <sup>ψ</sup> *<sup>k</sup>*2(*k*<sup>∗</sup> (*k*+β1)λ−(*k*−β1)<sup>λ</sup> (*k*+β1)λ+(*k*−β1)<sup>λ</sup> <sup>+</sup>*k*<sup>∗</sup> (*k*+β2)λ−(*k*−β2)<sup>λ</sup> (*k*+β2)λ+(*k*−β2)<sup>λ</sup> ) *<sup>k</sup>*2+((*k*<sup>∗</sup> (*k*+β1)λ−(*k*−β1)<sup>λ</sup> (*k*+β1)λ+(*k*−β1)<sup>λ</sup> )(*k*<sup>∗</sup> (*k*+β2)λ−(*k*−β2)<sup>λ</sup> (*k*+β2)λ+(*k*−β2)<sup>λ</sup> )) ,<sup>ψ</sup> *<sup>k</sup>*(*k*<sup>∗</sup> <sup>2</sup>γ<sup>1</sup> λ (2*k*−γ1)λ+γ<sup>1</sup> <sup>λ</sup> )(*k*<sup>∗</sup> <sup>2</sup>γ<sup>2</sup> λ (2*k*−γ2)λ+γ<sup>2</sup> λ ) *<sup>k</sup>*2+(*k*−(*k*<sup>∗</sup> <sup>2</sup>γ<sup>1</sup> λ (2*k*−γ1)λ+γ<sup>1</sup> <sup>λ</sup> ))(*k*−(*k*<sup>∗</sup> <sup>2</sup>γ<sup>2</sup> λ (2*k*−γ2)λ+γ<sup>2</sup> <sup>λ</sup> )) ,<sup>ψ</sup> *<sup>k</sup>*(*k*<sup>∗</sup> <sup>2</sup>δ<sup>1</sup> λ (2*k*−δ1)λ+δ<sup>1</sup> <sup>λ</sup> )(*k*<sup>∗</sup> <sup>2</sup>δ<sup>2</sup> λ (2*k*−δ2)λ+δ<sup>2</sup> λ ) *<sup>k</sup>*2+(*k*−(*k*<sup>∗</sup> <sup>2</sup>δ<sup>1</sup> λ (2*k*−δ1)λ+δ<sup>1</sup> <sup>λ</sup> ))(*k*−(*k*<sup>∗</sup> <sup>2</sup>δ<sup>2</sup> λ (2*k*−δ2)λ+δ<sup>2</sup> <sup>λ</sup> )) = ψ *k*∗ (*k*+β1)λ(*k*+β2)λ−(*k*−β1)λ(*k*−β2)<sup>λ</sup> (*k*+β1)λ(*k*+β2)λ+(*k*−β1)λ(*k*−β2)<sup>λ</sup> ,ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup>(γ1γ2)<sup>λ</sup> ((2*k*−γ1)λ(2*k*−γ2)λ)+(γ1γ2)<sup>λ</sup> ,ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup>(γ1δ2)<sup>λ</sup> ((2*k*−δ1)λ(2*k*−δ2)λ)+(δ1δ2)<sup>λ</sup> 

So, we can get λ(u1 ⊕<sup>e</sup> u2) = λu1 ⊕<sup>e</sup> λu2. Now, we prove the performance (4):

① *u*<sup>1</sup> <sup>λ</sup> <sup>=</sup> <sup>ψ</sup> *<sup>k</sup>*<sup>∗</sup> <sup>2</sup>β<sup>1</sup> λ (2*k*−β1)λ+β<sup>1</sup> λ ,ψ *k*∗ (*k*+γ1)λ−(*k*−γ1)<sup>λ</sup> (*k*+γ1)λ+(*k*−γ1)<sup>λ</sup> ,ψ *k*∗ (*k*+δ1)λ−(*k*−δ1)<sup>λ</sup> (*k*+δ1)λ+(*k*−δ1)<sup>λ</sup> ; ② *u*<sup>2</sup> <sup>λ</sup> <sup>=</sup> <sup>ψ</sup> *<sup>k</sup>*<sup>∗</sup> <sup>2</sup>β<sup>2</sup> λ (2*k*−β2)λ+β<sup>2</sup> λ ,ψ *k*∗ (*k*+γ2)λ−(*k*−γ2)<sup>λ</sup> (*k*+γ2)λ+(*k*−γ2)<sup>λ</sup> ,ψ *k*∗ (*k*+δ2)λ−(*k*−δ2)<sup>λ</sup> (*k*+δ2)λ+(*k*−δ2)<sup>λ</sup> ; ③ *u*<sup>1</sup> <sup>λ</sup> <sup>⊕</sup>*<sup>e</sup> <sup>u</sup>*<sup>2</sup> <sup>λ</sup> = <sup>ψ</sup> *<sup>k</sup>*(*k*<sup>∗</sup> <sup>2</sup>β<sup>1</sup> λ (2*k*−β1)λ+β<sup>1</sup> <sup>λ</sup> )(*k*<sup>∗</sup> <sup>2</sup>β<sup>2</sup> λ (2*k*−β2)λ+β<sup>2</sup> λ ) *<sup>k</sup>*2+(*k*−(*k*<sup>∗</sup> <sup>2</sup>β<sup>1</sup> λ (2*k*−β1)λ+β<sup>1</sup> <sup>λ</sup> ))(*k*−(*k*<sup>∗</sup> <sup>2</sup>β<sup>2</sup> λ (2*k*−β2)λ+β<sup>2</sup> <sup>λ</sup> )) , ψ*<sup>k</sup>*2((*k*<sup>∗</sup> (*k*+γ1)λ−(*k*−γ1)<sup>λ</sup> (*k*+γ1)λ+(*k*−γ1)<sup>λ</sup> )+(*k*<sup>∗</sup> <sup>2</sup>β<sup>2</sup> λ (2*k*−β2)λ+β<sup>2</sup> <sup>λ</sup> )) *<sup>k</sup>*2+(*k*<sup>∗</sup> (*k*+γ1)λ−(*k*−γ1)<sup>λ</sup> (*k*+γ1)λ+(*k*−γ1)<sup>λ</sup> )(*k*<sup>∗</sup> <sup>2</sup>β<sup>2</sup> λ (2*k*−β2)λ+β<sup>2</sup> λ ) , ψ*<sup>k</sup>*2((*k*<sup>∗</sup> (*k*+δ1)λ−(*k*−δ1)<sup>λ</sup> (*k*+δ1)λ+(*k*−δ1)<sup>λ</sup> )+(*k*<sup>∗</sup> (*k*+δ2)λ−(*k*−δ2)<sup>λ</sup> (*k*+δ2)λ+(*k*−δ2)<sup>λ</sup> )) *<sup>k</sup>*2+(*k*<sup>∗</sup> (*k*+δ1)λ−(*k*−δ1)<sup>λ</sup> (*k*+δ1)λ+(*k*−δ1)<sup>λ</sup> )(*k*<sup>∗</sup> (*k*+δ2)λ−(*k*−δ2)<sup>λ</sup> (*k*+δ2)λ+(*k*−δ2)<sup>λ</sup> ) = ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup>(β1β2)<sup>λ</sup> ((2*k*−β1)λ(2*k*−β2)λ)+(β1β2)<sup>λ</sup> ,ψ *k*∗ (*k*+γ1)λ(*k*+γ2)λ−(*k*−γ1)λ(*k*−γ2)<sup>λ</sup> (*k*+γ1)λ(*k*+γ2)λ+(*k*−γ1)λ(*k*−γ2)<sup>λ</sup> ,ψ *k*∗ (*k*+δ1)λ(*k*+δ2)λ−(*k*−δ1)λ(*k*−δ2)<sup>λ</sup> (*k*+δ1)λ(*k*+δ2)λ+(*k*−δ1)λ(*k*−δ2)<sup>λ</sup> ; ④ *u*<sup>1</sup> ⊗*<sup>e</sup> u*<sup>2</sup> = ψ *<sup>k</sup>*β1β<sup>2</sup> *<sup>k</sup>*2+(*k*−β1)(*k*−β2) ,ψ*k*2(γ1+γ2) *<sup>k</sup>*2+γ1γ<sup>2</sup> ,ψ*k*2(δ1+δ2) *<sup>k</sup>*2+δ1δ<sup>2</sup> ; ⑤ (*u*<sup>1</sup> ⊗*<sup>e</sup> u*2) <sup>λ</sup> <sup>=</sup> <sup>ψ</sup> *k*∗ <sup>2</sup>( *<sup>k</sup>*β1β<sup>2</sup> *<sup>k</sup>*2+(*k*−β1)(*k*−β2) ) λ (2*k*<sup>−</sup> *<sup>k</sup>*β1β<sup>2</sup> *<sup>k</sup>*2+(*k*−β1)(*k*−β2) ) λ +( *<sup>k</sup>*β1β<sup>2</sup> *<sup>k</sup>*2+(*k*−β1)(*k*−β2) ) λ ,ψ *k*∗ (*k*<sup>+</sup> *<sup>k</sup>*2(γ1+γ2) *<sup>k</sup>*2+γ1γ<sup>2</sup> ) λ <sup>−</sup>(*k*<sup>−</sup> *<sup>k</sup>*2(γ1+γ2) *<sup>k</sup>*2+γ1γ<sup>2</sup> ) λ (*k*<sup>+</sup> *<sup>k</sup>*2(γ1+γ2) *<sup>k</sup>*2+γ1γ<sup>2</sup> ) λ +(*k*<sup>−</sup> *<sup>k</sup>*2(γ1+γ2) *<sup>k</sup>*+γ1γ<sup>2</sup> ) λ ,ψ *k*∗ (*k*<sup>+</sup> *<sup>k</sup>*2(δ1+δ2) *<sup>k</sup>*2+δ1δ<sup>2</sup> ) λ <sup>−</sup>(*k*<sup>−</sup> *<sup>k</sup>*2(δ1+δ2) *<sup>k</sup>*2+δ1δ<sup>2</sup> ) λ (*k*<sup>+</sup> *<sup>k</sup>*2(δ1+δ2) *<sup>k</sup>*2+δ1δ<sup>2</sup> ) λ +(*k*<sup>−</sup> *<sup>k</sup>*2(δ1+δ2) *<sup>k</sup>*2+δ1δ<sup>2</sup> ) λ = ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup>(β1β2)<sup>λ</sup> ,ψ *k*∗ (*k*+γ1)λ(*k*+γ2)λ−(*k*−γ1)λ(*k*−γ2)<sup>λ</sup> ,ψ *k*∗ (*k*+δ1)λ(*k*+δ2)λ−(*k*−δ1)λ(*k*−δ2)<sup>λ</sup> ;

So, we can get (*u*<sup>1</sup> ⊕*<sup>e</sup> u*2) <sup>λ</sup> = *<sup>u</sup>*<sup>1</sup> <sup>λ</sup> <sup>⊕</sup>*<sup>e</sup> <sup>u</sup>*<sup>2</sup> <sup>λ</sup>. -

((2*k*−β1)λ(2*k*−β2)λ)+(β1β2)<sup>λ</sup>

#### **3. Einstein Aggregation Operators**

#### *3.1. LNNEWA Operator*

**Definition 7.** *Set a LNN ui* = ψβ*<sup>i</sup>* ,ψγ*<sup>i</sup>* ,ψδ*<sup>i</sup> in* Ψ*, for i* = *1,2,* ... *, z, we define the LNNEWA operator:*

(*k*+γ1)λ(*k*+γ2)λ+(*k*−γ1)λ(*k*−γ2)<sup>λ</sup>

$$\text{LNNEWA}(\mu\_1, \mu\_2, \dots, \mu\_z) = \underset{i=1}{\stackrel{z}{\oplus}}\_{i=1} \varepsilon\_i \mu\_{i\star} \tag{16}$$

(*k*+δ1)λ(*k*+δ2)λ+(*k*−δ1)λ(*k*−δ2)<sup>λ</sup>

*with the relative weight vector* = ( 1, 2, ... , *<sup>z</sup>*) *T, z <sup>i</sup>*=<sup>1</sup> *<sup>i</sup>* = 1 *and <sup>i</sup>* ∈ [0, 1].

**Theorem 2.** *Set a collection ui* = ψβ*<sup>i</sup>* ,ψγ*<sup>i</sup>* ,ψδ*<sup>i</sup> in* Ψ*, for i* = *1,2,* ... *,z, then according to the LNNEWA aggregation operator, we can get the following result:*

$$\begin{split} \text{LINENW}A(u\_1, u\_2, \dots, u\_{\mathbb{Z}}) &= \bigoplus\_{i=1}^{z} \varepsilon\_i \mathbb{1}\_{i} \\ &= \langle \boldsymbol{\psi} \sum\_{\begin{subarray}{c} \Pi\_{i=1}^{\pm}(k + \boldsymbol{\beta}\_{i})^{\varepsilon\_{i}^{\pm}} - \Pi\_{i=1}^{\pm}(k - \boldsymbol{\beta}\_{i})^{\varepsilon\_{i}^{\pm}} \\ \Pi\_{i=1}^{\pm}(k + \boldsymbol{\beta}\_{i})^{\varepsilon\_{i}^{\pm}} + \Pi\_{i=1}^{\pm}(k - \boldsymbol{\beta}\_{i})^{\varepsilon\_{i}^{\pm}} \end{subarray}} \, \, \, \forall \, \frac{\,\_{2}\Pi\_{i=1}^{\pm}\gamma^{\varepsilon\_{i}}}{\Pi\_{i=1}^{\pm}(2k - \gamma\_{i})^{\varepsilon\_{i}^{\pm}} + \Pi\_{i=1}^{\pm}\gamma^{\varepsilon\_{i}^{\pm}}} \, \, \forall \, \frac{\,\_{2}\Pi\_{i=1}^{\pm}\gamma^{\varepsilon\_{i}^{\pm}}}{\Pi\_{i=1}^{\pm}(2k - \boldsymbol{\beta}\_{i})^{\varepsilon\_{i}^{\pm}} + \Pi\_{i=1}^{\pm}\gamma^{\varepsilon\_{i}^{\pm}}} \, \, \} \end{split} \tag{17}$$

*with the relative weight vector* = ( 1, 2, ... , *<sup>z</sup>*) *T , z <sup>i</sup>*=<sup>1</sup> *<sup>i</sup>* = 1 *and <sup>i</sup>* ∈ [0, 1]*.*

**Proof.**

$$\begin{array}{rcl} \bigotimes & \mathfrak{e}\_{i}u\_{i} = & \langle \psi\_{k\*\frac{(k+\beta\_{i})^{\mathfrak{s}\_{i}^{\mathfrak{s}\_{i}}-(k-\beta\_{i})^{\mathfrak{s}\_{i}^{\mathfrak{s}\_{i}}}}}{(k+\beta\_{i})^{\mathfrak{s}\_{i}^{\mathfrak{s}\_{i}}+(k-\beta\_{i})^{\mathfrak{s}\_{i}^{\mathfrak{s}\_{i}}}}} \ \ \ \ \mathsf{l}^{\mathfrak{l}} \star & \frac{2\gamma\_{i}^{\mathfrak{s}\_{i}^{\mathfrak{s}\_{i}}}}{(2k-\gamma\_{i})^{\mathfrak{s}\_{i}^{\mathfrak{s}\_{i}}}+\gamma\_{i}^{\mathfrak{s}\_{i}^{\mathfrak{s}\_{i}}}} \ \ \mathsf{l}^{\mathfrak{l}} \star & \frac{2\delta\_{i}^{\mathfrak{s}\_{i}^{\mathfrak{s}\_{i}}}}{(2k-\mathfrak{s}\_{i})^{\mathfrak{s}\_{i}^{\mathfrak{s}\_{i}}}+\delta\_{i}^{\mathfrak{s}\_{i}^{\mathfrak{s}\_{i}}}} \end{array} \end{array}$$

<sup>②</sup> *<sup>z</sup>* <sup>=</sup> 2 , *LNNEWA*(*u*1, *<sup>u</sup>*2) = <sup>2</sup> ⊕*e i*=1 *iui* <sup>=</sup> <sup>ψ</sup> *<sup>k</sup>*2(*k*<sup>∗</sup> (*k*+β1) <sup>1</sup>−(*k*−β1) 1 (*k*+β1) <sup>1</sup> +(*k*−β1) <sup>1</sup> <sup>+</sup>*k*<sup>∗</sup> (*k*+β2) <sup>2</sup>−(*k*−β2) 2 (*k*+β2) <sup>2</sup> +(*k*−β2) <sup>2</sup> ) *<sup>k</sup>*2+((*k*<sup>∗</sup> (*k*+β1) <sup>1</sup>−(*k*−β1) 1 (*k*+β1) <sup>1</sup> +(*k*−β1) <sup>1</sup> )(*k*<sup>∗</sup> (*k*+β2) <sup>2</sup>−(*k*−β2) 2 (*k*+β2) <sup>2</sup> +(*k*−β2) <sup>2</sup> ) ,<sup>ψ</sup> *<sup>k</sup>*(*k*<sup>∗</sup> <sup>2</sup>γ<sup>1</sup> 1 (2*k*−γ1) <sup>1</sup> <sup>+</sup>γ<sup>1</sup> <sup>1</sup> )(*k*<sup>∗</sup> <sup>2</sup>γ<sup>2</sup> 2 (2*k*−γ2) <sup>2</sup> <sup>+</sup>γ<sup>2</sup> <sup>2</sup> ) *<sup>k</sup>*2+(*k*−(*k*<sup>∗</sup> <sup>2</sup>γ<sup>1</sup> 1 (2*k*−γ1) <sup>1</sup> <sup>+</sup>γ<sup>1</sup> <sup>1</sup> ))(*k*−(*k*<sup>∗</sup> <sup>2</sup>γ<sup>2</sup> 2 (2*k*−γ2) <sup>2</sup> <sup>+</sup>γ<sup>2</sup> <sup>2</sup> )) ,<sup>ψ</sup> *<sup>k</sup>*(*k*<sup>∗</sup> <sup>2</sup>δ<sup>1</sup> 1 (2*k*−δ1) <sup>1</sup> <sup>+</sup>δ<sup>1</sup> <sup>1</sup> )(*k*<sup>∗</sup> <sup>2</sup>δ<sup>2</sup> 2 (2*k*−δ2) <sup>2</sup> <sup>+</sup>δ<sup>2</sup> <sup>2</sup> ) *<sup>k</sup>*2+(*k*−(*k*<sup>∗</sup> <sup>2</sup>δ<sup>1</sup> 1 (2*k*−δ1) <sup>1</sup> <sup>+</sup>δ<sup>1</sup> <sup>1</sup> ))(*k*−(*k*<sup>∗</sup> <sup>2</sup>δ<sup>2</sup> 2 (2*k*−δ2) <sup>2</sup> <sup>+</sup>δ<sup>2</sup> <sup>2</sup> )) = ψ *k*∗ (*k*+β1) <sup>1</sup> (*k*+β2) <sup>2</sup>−(*k*−β1) <sup>1</sup> (*k*−β2) 2 (*k*+β1) <sup>1</sup> (*k*+β2) <sup>2</sup> +(*k*−β1) <sup>1</sup> (*k*−β2) 2 ,ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup>γ<sup>1</sup> <sup>1</sup> <sup>γ</sup><sup>2</sup> 2 (2*k*−γ1) <sup>1</sup> (2*k*−γ2) <sup>2</sup> <sup>+</sup>γ<sup>1</sup> <sup>1</sup> <sup>γ</sup><sup>2</sup> 2 ,ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup>δ<sup>1</sup> <sup>1</sup> <sup>δ</sup><sup>2</sup> 2 (2*k*−δ1) <sup>1</sup> (2*k*−δ1) <sup>2</sup> <sup>+</sup>δ<sup>1</sup> <sup>1</sup> <sup>δ</sup><sup>2</sup> 2 = ψ *k*∗ \$2 *<sup>i</sup>*=<sup>1</sup> (*k*+β*i*) *i*− \$2 *<sup>i</sup>*=<sup>1</sup> (*k*−β*i*) *i* \$2 *<sup>i</sup>*=<sup>1</sup> (*k*+β*i*) *<sup>i</sup>* +\$<sup>2</sup> *<sup>i</sup>*=<sup>1</sup> (*k*−β*i*) *i* ,ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup> \$2 *<sup>i</sup>*=<sup>1</sup> <sup>γ</sup>*<sup>i</sup> i* \$2 *<sup>i</sup>*=<sup>1</sup> (2*k*−γ*i*) *<sup>i</sup>* +\$<sup>2</sup> *<sup>i</sup>*=<sup>1</sup> <sup>γ</sup>*<sup>i</sup> i* ,ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup> \$2 *<sup>i</sup>*=<sup>1</sup> <sup>δ</sup>*<sup>i</sup> i* \$2 *<sup>i</sup>*=<sup>1</sup> (2*k*−δ*i*) *<sup>i</sup>* +\$<sup>2</sup> *<sup>i</sup>*=<sup>1</sup> <sup>δ</sup>*<sup>i</sup> i* .

Suppose *z* = *m*, according t formula (17), we can get

$$\begin{array}{rclcrcl}\text{LNNEWA}(\boldsymbol{\mu}\_{1},\boldsymbol{\mu}\_{2},\ldots,\boldsymbol{\mu}\_{m}) & = & \stackrel{\text{in}}{\oplus}\_{\mathbb{R}^{c}}\boldsymbol{e}\_{i}\boldsymbol{\mu}\_{i} \\ & = & \langle\boldsymbol{\psi},\underset{\begin{subarray}{c}\prod\_{i=1}^{m}(3+\boldsymbol{\beta}\_{i})^{\ell\_{i}}=\prod\_{i=1}^{m}(3-\boldsymbol{\beta}\_{i})^{\ell\_{i}} \\ \end{subarray}}{\prod\_{i=1}^{m}(3+\boldsymbol{\beta}\_{i})^{\ell\_{i}}+\prod\_{i=1}^{m}(3-\boldsymbol{\beta}\_{i})^{\ell\_{i}}} \times & \stackrel{\text{\textquotedblleft}}{\prod\_{i=1}^{m}(3-\boldsymbol{\beta}\_{i})^{\ell\_{i}}+\prod\_{i=1}^{m}\boldsymbol{\gamma}\_{i}^{\ell\_{i}}},\ \boldsymbol{\forall}\underset{\begin{subarray}{c}\prod\_{i=1}^{m}(3-\boldsymbol{\beta}\_{i})^{\ell\_{i}}=\prod\_{i=1}^{m}\boldsymbol{\gamma}\_{i}^{\ell\_{i}}\end{subarray}}{\prod\_{i=1}^{m}(3-\boldsymbol{\beta}\_{i})^{\ell\_{i}}+\prod\_{i=1}^{m}\boldsymbol{\gamma}\_{i}^{\ell\_{i}}} \end{array} \tag{18}$$

Then z = m + 1, the following can be found:

*LNNEWA*(*u*1, *<sup>u</sup>*2, ... *um*, *um*+1)=( *<sup>m</sup>* ⊕*e i*=1 *iui*) ⊕*<sup>e</sup> <sup>m</sup>*+1*um*+<sup>1</sup> = ψ *k*∗ \$*m <sup>i</sup>*=<sup>1</sup> (*k*+β*i*) *i*− \$*m <sup>i</sup>*=<sup>1</sup> (*k*−β*i*) *i* \$*m <sup>i</sup>*=<sup>1</sup> (*k*+β*i*) *<sup>i</sup>* +\$*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> (*k*−β*i*) *i* ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup> \$*m <sup>i</sup>*=<sup>1</sup> <sup>γ</sup>*<sup>i</sup> i* \$*m <sup>i</sup>*=<sup>1</sup> (2*k*−γ*i*) *<sup>i</sup>* +\$*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> <sup>γ</sup>*<sup>i</sup> i* , ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup> \$*m <sup>i</sup>*=<sup>1</sup> <sup>δ</sup>*<sup>i</sup> i* \$*m <sup>i</sup>*=<sup>1</sup> (2*k*−δ*i*) *<sup>i</sup>* +\$*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> <sup>δ</sup>*<sup>i</sup> i* ⊕*e* ψ *k*∗ (*k*+β*m*+1) *<sup>m</sup>*+1−(*k*−β*m*+1) *m*+1 (*k*+β*m*+1) *<sup>m</sup>*+<sup>1</sup> +(*k*−β*m*+1) *m*+1 , ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup>γ*m*+<sup>1</sup> *m*+1 (2*k*−γ*m*+1) *<sup>m</sup>*+<sup>1</sup> <sup>+</sup>γ*m*+<sup>1</sup> *m*+1 , ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup>δ*m*+<sup>1</sup> *m*+1 (2*k*−δ*m*+1) *<sup>m</sup>*+<sup>1</sup> <sup>+</sup>δ*m*+<sup>1</sup> *m*+1 = ψ*<sup>k</sup>*2((*k*<sup>∗</sup> \$*m <sup>i</sup>*=<sup>1</sup> (*k*+β*i*) *i*− \$*m <sup>i</sup>*=<sup>1</sup> (*k*−β*i*) *i* \$*k <sup>i</sup>*=<sup>1</sup> (*k*+β*i*) *<sup>i</sup>* +\$*<sup>k</sup> <sup>i</sup>*=<sup>1</sup> (*k*−β*i*) *i* )+(*k*∗ (*k*+β*m*+1) *<sup>m</sup>*+1−(*k*−β*m*+1) *m*+1 (*k*+β*m*+1) *<sup>m</sup>*+<sup>1</sup> +(*k*−β*m*+1) *<sup>m</sup>*+<sup>1</sup> )) *<sup>k</sup>*2+(*k*<sup>∗</sup> \$*m <sup>i</sup>*=<sup>1</sup> (*k*+β*i*) *i*− \$*m <sup>i</sup>*=<sup>1</sup> (*k*−β*i*) *i* \$*m <sup>i</sup>*=<sup>1</sup> (*k*+β*i*) *<sup>i</sup>* +\$*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> (*k*−β*i*) *<sup>i</sup>* )(*k*<sup>∗</sup> (*k*+β*m*+1) *<sup>m</sup>*+1−(*k*−β*m*+1) *m*+1 (*k*+β*m*+1) *<sup>m</sup>*+<sup>1</sup> +(*k*−β*m*+1) *<sup>m</sup>*+<sup>1</sup> ) , ψ *<sup>k</sup>*(*k*<sup>∗</sup> <sup>2</sup> \$*m <sup>i</sup>*=<sup>1</sup> <sup>γ</sup>*<sup>i</sup> i* \$*m <sup>i</sup>*=<sup>1</sup> (2*k*−γ*i*) *<sup>i</sup>* +\$*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> <sup>γ</sup>*<sup>i</sup> <sup>i</sup>* )(*k*<sup>∗</sup> <sup>2</sup>γ*m*+<sup>1</sup> *m*+1 (2*k*−γ*m*+1) *<sup>m</sup>*+<sup>1</sup> <sup>+</sup>γ*m*+<sup>1</sup> *<sup>m</sup>*+<sup>1</sup> ) *<sup>k</sup>*<sup>2</sup> + (*<sup>k</sup>* <sup>−</sup> (*<sup>k</sup>* <sup>∗</sup> <sup>2</sup> \$*m <sup>i</sup>*=<sup>1</sup> γ*<sup>i</sup> i* \$*m <sup>i</sup>*=<sup>1</sup> (2*k*−γ*i*) *<sup>i</sup>*+\$*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> γ*<sup>i</sup> <sup>i</sup>* ))(*<sup>k</sup>* <sup>−</sup> (*<sup>k</sup>* <sup>∗</sup> <sup>2</sup>γ*m*+<sup>1</sup> *m*+1 (2*k*−γ*m*+1) *<sup>m</sup>*+1+γ*m*+<sup>1</sup> *<sup>m</sup>*+<sup>1</sup> )) ψ *<sup>k</sup>*(*k*<sup>∗</sup> <sup>2</sup> \$*m <sup>i</sup>*=<sup>1</sup> <sup>δ</sup>*<sup>i</sup> i* \$*m <sup>i</sup>*=<sup>1</sup> (2*k*−δ*i*) *<sup>i</sup>* +\$*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> <sup>δ</sup>*<sup>i</sup> <sup>i</sup>* )(*k*<sup>∗</sup> <sup>2</sup>δ*m*+<sup>1</sup> *m*+1 (2*k*−δ*m*+1) *<sup>m</sup>*+<sup>1</sup> <sup>+</sup>δ*m*+<sup>1</sup> *<sup>m</sup>*+<sup>1</sup> ) *<sup>k</sup>*2+(*k*−(*k*<sup>∗</sup> <sup>2</sup> \$*m <sup>i</sup>*=<sup>1</sup> <sup>δ</sup>*<sup>i</sup> i* \$*m <sup>i</sup>*=<sup>1</sup> (2*k*−δ*i*) *<sup>i</sup>* +\$*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> <sup>δ</sup>*<sup>i</sup> <sup>i</sup>* ))(*k*−(*k*<sup>∗</sup> <sup>2</sup>δ*m*+<sup>1</sup> *m*+1 (2*k*−δ*m*+1) *<sup>m</sup>*+<sup>1</sup> <sup>+</sup>δ*m*+<sup>1</sup> *<sup>m</sup>*+<sup>1</sup> )) , = ψ *k*∗ \$*m*+1 *<sup>i</sup>*=<sup>1</sup> (*k*+β*i*) *i*− \$*m*+1 *<sup>i</sup>*=<sup>1</sup> (*k*−β*i*) *i* \$*m*+1 *<sup>i</sup>*=<sup>1</sup> (*k*+β*i*) *<sup>i</sup>* +\$*m*+<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> (*k*−β*i*) *i* ,ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup> \$*m*+1 *<sup>i</sup>*=<sup>1</sup> <sup>γ</sup>*<sup>i</sup> i* \$*m*+1 *<sup>i</sup>*=<sup>1</sup> (2*k*−γ*i*) *<sup>i</sup>* +\$*m*+<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> <sup>γ</sup>*<sup>i</sup> i* ,ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup> \$*m*+1 *<sup>i</sup>*=<sup>1</sup> <sup>δ</sup>*<sup>i</sup> i* \$*m*+1 *<sup>i</sup>*=<sup>1</sup> (2*k*−δ*i*) *<sup>i</sup>* +\$*m*+<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> <sup>δ</sup>*<sup>i</sup> i* .

So, Equation (17) is satisfied for any z according to the above results. This proves Theorem 1. -

**Theorem 3.** *(Idempotency). Set an LNN u* = ψβ,ψγ,ψδ *in* Ψ*, for every ui in u is equal to u, we can get:*

*LNNEWA*(*u*1, *u*2, ... *uz*) = *LNNEWA*(*u*, *u* ... *u*) = *u*.

**Proof.** For *ui* = *u*, *then* β*<sup>i</sup>* = β; γ*<sup>i</sup>* = γ; δ*<sup>i</sup>* = δ = *(i* = *1, 2,* ...,z), the following result can be found:

*LNNEWA*(*u*1, *<sup>u</sup>*2, ... *uz*) = *LNNEWA* (*u*, *<sup>u</sup>* ... *<sup>u</sup>*) =( *<sup>z</sup>* ⊕*e i*=1 *iu*) = ψ *k*∗ \$*z <sup>i</sup>*=<sup>1</sup> (*k*+β) *i*− \$*z <sup>i</sup>*=<sup>1</sup> (*k*−β) *i* \$*z <sup>i</sup>*=<sup>1</sup> (*k*+β) *<sup>i</sup>* +\$*<sup>z</sup> <sup>i</sup>*=<sup>1</sup> (*k*−β) *i* ,ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup> \$*z <sup>i</sup>*=<sup>1</sup> γ *<sup>i</sup>* \$*z <sup>i</sup>*=<sup>1</sup> (2*k*−γ) *<sup>i</sup>* +\$*<sup>z</sup> <sup>i</sup>*=<sup>1</sup> γ *<sup>i</sup>* ,ψ *<sup>k</sup>*<sup>∗</sup> <sup>2</sup> \$*z <sup>i</sup>*=<sup>1</sup> δ *<sup>i</sup>* \$*z <sup>i</sup>*=<sup>1</sup> (2*k*−δ) *<sup>i</sup>* +\$*<sup>z</sup> <sup>i</sup>*=<sup>1</sup> <sup>δ</sup> *<sup>i</sup>* <sup>=</sup> ψ*k*<sup>∗</sup> (*k*+β)−(*k*−β) (*k*+β)+(*k*−β) ,ψ*k*<sup>∗</sup> <sup>2</sup><sup>γ</sup> (2*k*−γ)+γ ,ψ*k*<sup>∗</sup> <sup>2</sup><sup>δ</sup> (2*k*−δ)+δ = ψβ,ψγ,ψδ = *u*

**Theorem 4.** *(Monotonicity) set two collections of LNNs ui* = ψβ*<sup>i</sup>* ,ψγ*<sup>i</sup>* ,ψδ*<sup>i</sup> and ui* = ψβ*<sup>i</sup>* ,ψγ*<sup>i</sup>* ,ψδ*<sup>i</sup> (i* = *1, 2,* ... *, z) in* Ψ*, if ui* ≤ *ui then*

$$
\text{LNNEVA}(\mu\_1, \mu\_2, \dots, \mu\_z) \le \text{LNNEVA}(\mu\_1', \mu\_2', \dots, \mu\_z').
$$

**Proof.** For *ui* ≤ *ui* , then *iui* ≤ *iui* 

So, we can easily obtain:

$$\bigoplus\_{i=1}^{z} \mathfrak{e}\_i u\_i \le \bigoplus\_{i=1}^{z} \mathfrak{e}\_i u\_i{'} $$

For *LNNEWA*(*u*1, *<sup>u</sup>*2, ... *uz*) = *<sup>z</sup>* ⊕*e i*=1 *iui* and *LNNEWA*(*u*<sup>1</sup> , *u*<sup>2</sup> , ... *uz* ) = *<sup>z</sup>* ⊕*e i*=1 *iui* , then we can get: *LNNEWA*(*u*1, *u*2, ... *uz*) ≤ *LNNEWA*(*u*<sup>1</sup> , *u*<sup>2</sup> , ... *uz* ). -

**Theorem 5.** *(Boundedness) Let a collection ui* = ψβ*<sup>i</sup>* ,ψγ*<sup>i</sup>* ,ψδ*<sup>i</sup> in* Ψ*, u*<sup>−</sup> = *min*(ψβ*<sup>i</sup>* ), *max*(ψγ*<sup>i</sup>* ), *max*(ψδ*<sup>i</sup>* ) *and u*<sup>+</sup> = *max*(ψβ*<sup>i</sup>* ), *min*(ψγ*<sup>i</sup>* ), *min*(ψδ*<sup>i</sup>* ), *we can get:*

*<sup>u</sup>*<sup>−</sup> <sup>≤</sup> *LNNEWA*(*u*1, *<sup>u</sup>*2, ... *uz*) <sup>≤</sup> *<sup>u</sup>*+.

**Proof.** The following can be obtained by using Theorem 3:

$$\mathbf{u}^{\top} = \text{LNNEWA}(\mathbf{u}^{\top}, \mathbf{u}^{\top} \dots \mathbf{u}^{\top}), \\ \mathbf{u}^{+} = \text{LNNEWA}(\mathbf{u}^{+}, \mathbf{u}^{+} \dots \mathbf{u}^{+}).$$

The following can be obtained by using Theorem 4:

$$\text{LNNEWA}\left(\mathbf{u}^-, \mathbf{u}^- \dots \mathbf{u}^-\right) \le \text{LNNEWA}\left(\mathbf{u}\_1, \mathbf{u}\_2, \dots \mathbf{u}\_\mathbf{z}\right) \le \text{LNNEWA}\left(\mathbf{u}^+, \mathbf{u}^+ \dots \mathbf{u}^+\right).$$

Above all, we can get:

$$\mathbf{u}^- \le \text{LNNEWA}(\mathbf{u}\_1, \mathbf{u}\_2, \dots, \mathbf{u}\_\mathbf{z}) \le \mathbf{u}^+.$$


#### *3.2. LNNEWG Operators*

**Definition 8.** *Set a collection ui* = ψβ*<sup>i</sup>* ,ψγ*<sup>i</sup>* ,ψδ*<sup>i</sup> in* Ψ*, for i* = *1, 2,* ... *, z, we define the LNNEWG operator:*

$$\text{LNNEWG}(u\_1, u\_2, \dots, u\_z) = \underset{i=1}{\stackrel{z}{\otimes}} (u\_i)^{\mathfrak{e}\_i} \tag{19}$$

*with the relative weight vector* = ( 1, 2, ... , *<sup>z</sup>*) *T , z <sup>i</sup>*=<sup>1</sup> *<sup>i</sup>* = 1 *and <sup>i</sup>* ∈ [0, 1]*.* **Theorem 6.** *Set a collection ui* = ψβ*<sup>i</sup>* ,ψγ*<sup>i</sup>* ,ψδ*<sup>i</sup> in* Ψ*, for i* = *1,2,* ... *,z, then according to the LNNEWG aggregation operator, we can get the following result:*

$$\begin{split} \text{LINENWG}(\boldsymbol{u}\_{1},\boldsymbol{u}\_{2},\ldots \quad \boldsymbol{u}\_{\boldsymbol{\varepsilon}}) &= \underset{\boldsymbol{\delta} = 1}{\text{ $\dot{\Phi}\_{\boldsymbol{\varepsilon}}$ }} (\boldsymbol{u}\_{i})^{\text{4}{}} \\ &= \underset{\boldsymbol{k} = \frac{2\prod\_{i=1}^{2}\boldsymbol{\theta}^{\boldsymbol{x}\_{i}}\boldsymbol{\theta}^{\boldsymbol{x}\_{i}}}{2\prod\_{i=1}^{2}(3-\boldsymbol{\theta}\_{1})^{\boldsymbol{\theta}\_{i}^{\boldsymbol{x}}+\prod\_{i=1}^{2}\boldsymbol{\theta}^{\boldsymbol{x}\_{i}}}} \; \text{ $\boldsymbol{\Psi}\_{\boldsymbol{\varepsilon}}$ } \underset{\boldsymbol{\Pi}\_{\boldsymbol{\varepsilon}\text{in1}}^{\boldsymbol{x}}(\boldsymbol{k}+\boldsymbol{\gamma})^{\text{4}\boldsymbol{\varepsilon}}=\boldsymbol{\Pi}\_{\boldsymbol{\varepsilon}\text{in1}}^{2}(\boldsymbol{k}-\boldsymbol{\gamma})^{\text{4}} $}{$ \boldsymbol{\varepsilon} $} \; \text{$ \boldsymbol{\varepsilon} $} \underset{\boldsymbol{\Pi}\_{\boldsymbol{\varepsilon}\text{in}}^{\boldsymbol{x}}(\boldsymbol{k}+\boldsymbol{\gamma})^{\text{4}\boldsymbol{\varepsilon}}=\boldsymbol{\Pi}\_{\boldsymbol{\varepsilon}\text{in}}^{2}(\boldsymbol{k}-\boldsymbol{\gamma})^{\text{4}\boldsymbol{\varepsilon}}$ }{ $\boldsymbol{\varepsilon}$ } \end{split} \tag{20}$$

*with the relative weight vector* = ( 1, 2, ... , *<sup>z</sup>*) *T, z <sup>i</sup>*=<sup>1</sup> *<sup>i</sup>* = 1 *and <sup>i</sup>* ∈ [0, 1]*.*

**Theorem 7.** *(Idempotency) Set a collection ui* = ψβ*<sup>i</sup>* ,ψγ*<sup>i</sup>* ,ψδ*<sup>i</sup> in* Ψ*, for i* = *1,2,* ... *,z, for every ui in u is equal to u, we can get*

$$LNNEWG(\mu\_1, \mu\_2, \dots, \mu\_z) = LNNEWG(\mu, \mu, \dots, \mu) = \mu.$$

**Theorem 8.** *(Monotonicity). Set two collections of LNNs ui* = ψβ*<sup>i</sup>* ,ψγ*<sup>i</sup>* ,ψδ*<sup>i</sup> and ui* = ψβ*<sup>i</sup>* ,ψγ*<sup>i</sup>* ,ψδ*<sup>i</sup> (i* = *1, 2,* ... *, z) in* Ψ*, if ui* ≤ *ui then*

$$LNNEWG(\mu\_1, \mu\_2, \dots, \mu\_z) \le LNNEWG(\mu\_1', \mu\_2', \dots, \mu\_z').$$

**Theorem 9.** *(Boundedness) Let a collection ui* = ψβ*<sup>i</sup>* ,ψγ*<sup>i</sup>* ,ψδ*<sup>i</sup> in* Ψ*, u*<sup>−</sup> = *min*(ψβ*<sup>i</sup>* ), *max*(ψγ*<sup>i</sup>* ), *max*(ψδ*<sup>i</sup>* ) *and u*<sup>+</sup> = *max*(ψβ*<sup>i</sup>* ), *min*(ψγ*<sup>i</sup>* ), *min*(ψδ*<sup>i</sup>* ), *we can get:*

$$
\mu^- \le L \\
NNEWG(\mu\_1, \mu\_2, \dots, \mu\_z) \le \mu^+ + 1
$$

We omit the proof here because it is similar to Theorems 2–5.

#### **4. Methods with LNNEWA or LNNEWG Operator**

We introduce two MAGDM methods with the LNNEWA or LNNEWG operator in LNN information.

Now, we suppose that a collection of alternatives is expressed Θ = {Θ1, Θ2, ... , Θ*m*} and a collection of attributes is expressed *E* = {*E*1, *E*2, ... , *En*}. Then, = ( 1, 2, ... , <sup>n</sup>) <sup>T</sup> with n <sup>i</sup>=<sup>1</sup> <sup>i</sup> = 1 *and* <sup>i</sup> ∈ [0, 1] is the weight vector of *Ei*(*i* = 1, 2, ... , *n*). Establishing a set of experts *D* = {*D*1, *D*2, ... , *Dt*} , μ = (μ1,μ2, ... , μ*t*) *<sup>T</sup>* with 1 <sup>≥</sup> <sup>μ</sup>*<sup>j</sup>* <sup>≥</sup> 0 and *<sup>t</sup> <sup>j</sup>*=<sup>1</sup> μ*<sup>j</sup>* = 1 is the weight vector of *Di*(*i* = 1, 2, ... , *t*). Assuming that the expert *Dy*(*y* = 1, 2, ... , *t*) uses the LNNs to give out the assessed value <sup>θ</sup>(*y*) *ij* for alternative <sup>Θ</sup>*<sup>i</sup>* with the attribute <sup>E</sup>*j*, the value <sup>θ</sup>(*y*) *ij* can be written as <sup>θ</sup>(*y*) *ij* <sup>=</sup> ψ*<sup>y</sup>* β*ij* ,ψ*<sup>y</sup>* <sup>γ</sup>*ij* ,ψ*<sup>y</sup>* δ*ij* (*<sup>y</sup>* <sup>=</sup> 1, 2, ... , *<sup>t</sup>* ; *<sup>i</sup>* <sup>=</sup> 1, 2, ... , *<sup>m</sup>*; *<sup>j</sup>* <sup>=</sup> 1, 2, ... , *<sup>n</sup>*),ψ*<sup>y</sup>* β*ij* ,ψ*<sup>y</sup>* <sup>γ</sup>*ij* ,ψ*<sup>y</sup>* <sup>δ</sup>*ij* <sup>∈</sup> <sup>Ψ</sup>. Then, the decision evaluation matrix can be found. Table 1 is the decision matrix.

**Table 1.** The decision matrix using linguistic neutrosophic numbers (LNN).


The decision steps are described as follows:

Step 1: the integrated matrix can be obtained by the *LNNEWA* operator:

$$\begin{split} \mathcal{O}\_{ij} &= \langle \psi\_{\mathbb{B}\_{ij'}}, \psi\_{\mathbb{V}\_{ij}'}, \psi\_{\mathbb{S}\_{ij}} \rangle = \text{LNNEWA}(\mathcal{O}\_{ij'}^{1}, \mathcal{O}\_{ij'}^{2}, \dots, \mathcal{O}\_{ij}^{t}) = \mathop{\frac{t}{\mathbb{B}\_{\ell}}}\_{l=1} \mathcal{O}\_{l} \mathcal{O}\_{ij}^{l} \\ &= \langle \psi\_{\mathbb{A}\* \frac{\prod\_{l=1}^{t}(k+\vartheta\_{ij}^{l})^{\mathbb{A}\_{j}} - \prod\_{l=1}^{t}(k-\vartheta\_{ij}^{l})^{\mathbb{A}\_{l}}}^{\prod\_{l=1}^{t}(k+\vartheta\_{ij}^{l})^{\mathbb{A}\_{l}} + \prod\_{l=1}^{t}(k-\vartheta\_{ij}^{l})^{\mathbb{A}\_{l}}}^{\prod\_{l=1}^{t}(l^{\mathbb{A}\_{l}})^{\mathbb{A}\_{l}}} k^{\star} \frac{2 \operatorname{\prod}\_{l=1}^{t} \operatorname{s}\_{ij}^{l} \mu\_{l}}{\prod\_{l=1}^{t}(2k-\vartheta\_{ij}^{l})^{\mathbb{A}\_{l}} + \prod\_{l=1}^{t} \operatorname{s}\_{ij}^{l} \mu\_{l}} \frac{2 \operatorname{\prod}\_{l=1}^{t} \operatorname{s}\_{ij}^{l} \mu\_{l}}{\prod\_{l=1}^{t}(2k-\vartheta\_{ij}^{l})^{\mathbb{A}\_{l}} + \prod\_{l=1}^{t} \operatorname{s}\_{ij}^{l} \mu\_{l}} \end{split} \tag{21}$$

Step 2: the total collective LNN θ*<sup>i</sup>* (*i* = 1, 2, ... , *m*) can be obtained by the *LNNWEA* or *LNNEWG* operator.

$$\begin{array}{lcl}\theta\_{i} &= \text{LNNEWA}(\theta\_{i1}, \theta\_{i2}, \dots, \theta\_{i\bar{n}}) &= \sum\_{j=1}^{n} \epsilon\_{ij} \theta\_{i\bar{j}} \\ &= \langle \psi\_{k\*} \frac{\prod\_{j=1}^{n} (k + \theta\_{i\bar{j}})^{e\_{ij}} \prod\_{j=1}^{n} (k - \theta\_{i\bar{j}})^{e\_{ij}}}{\prod\_{j=1}^{n} (k + \theta\_{i\bar{j}})^{e\_{ij}} + \prod\_{j=1}^{n} (k - \theta\_{i\bar{j}})^{e\_{ij}}} \times \frac{{}\_{2}^{\mathsf{I}} \Pi\_{j=1}^{\mathsf{T}} \iota\_{i\bar{j}}^{e\_{ij}} }{\prod\_{j=1}^{n} (2k - \varprojlim\_{j \to 1} \iota\_{j\bar{j}}^{e\_{ij}})} \mathsf{array} \frac{{}\_{2}^{\mathsf{I}} \Pi\_{j=1}^{\mathsf{T}} \iota\_{i\bar{j}}^{e\_{ij}}}{\prod\_{j=1}^{n} (2k - \varprojlim\_{j \to 1} \iota\_{j\bar{j}}^{e\_{ij}})} \end{array} \rangle{}$$

Or

$$\begin{array}{lcl}\theta\_{\mathtt{i}} &= \mathtt{LNNEWG}(\theta\_{\mathtt{i}1}, \theta\_{\mathtt{i}2}, \ldots, \theta\_{\mathtt{in}}) = \bigwedge\_{j=1}^{\mathtt{n}} (\theta\_{\mathtt{i}j})^{\mathtt{e}\_{\mathtt{i}j}^{\mathtt{e}}} \\ &= \{\psi \brace{\begin{subarray}{c} 2\,\prod\_{j=1}^{\mathtt{n}} \theta\_{ij}^{\mathtt{e}\_{\mathtt{i}j}^{\mathtt{e}}} \\ \prod\_{j=1}^{\mathtt{n}} (2\boldsymbol{\upmu} - \boldsymbol{\upmu}\_{ij})^{\mathtt{e}\_{\mathtt{i}j}^{\mathtt{e}} + \prod\_{j=1}^{\mathtt{n}} \theta\_{ij}^{\mathtt{e}\_{\mathtt{i}j}^{\mathtt{e}}} \end{subarray}} \; \forall \begin{subarray}{c} \forall \psi \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \} \quad \begin{subarray}{c} \forall \begin{subarray}{c} \Pi^{\mathtt{n}}\_{j} \ (\boldsymbol{\upmu}\_{\mathtt{i}} - \boldsymbol{\upmu}\_{i})^{\mathtt{e}\_{\mathtt{i}j}^{\mathtt{e}}} \end{subarray} \; \forall \end{subarray} \mid \} \end{array} \tag{23}$$

Step 3: according to Definition 3, we can calculate ζ(θ*i*) and η(θ*i*) of every LNN Θ*i*(*i* = 1, 2, ... , *m*). Step 4: According to ζ(θ*i*), then we can rank the alternatives and the best one can be chosen out. Step 5: End.

#### **5. Illustrative Examples**

#### *5.1. Numerical Example*

Now, we adopt illustrative examples of the MAGDM problems to verify the proposed decision methods. An investment company wants to find a company to invest. Now, there are four companies Θ = {Θ1, Θ2, Θ3, Θ4} to be considered as candidates, the first is for selling cars (Θ1), the second is for selling food (Θ2), the third is for selling computers (Θ3), and the last is for selling arms (Θ4). Next, three experts *<sup>D</sup>* <sup>=</sup> *D*1,*D*2,*D*<sup>3</sup> are invited to evaluate these companies, their weight vector is μ = (0.37, 0.33, 0.3) *<sup>T</sup>*. The experts make evaluations of the alternatives according to three attributes *E* = {*E*1, *E*2, *E*3}, *E*<sup>1</sup> is the ability of risk, *E*<sup>2</sup> is the ability of growth, and *E*<sup>3</sup> is the ability of environmental impact, the weight vector of them is = (0.35, 0.25, 0.4) *<sup>T</sup>*. Then, the experts use LNNs to make the evaluation values with a linguistic set Ψ = {ψ<sup>0</sup> = extremely poor, ψ<sup>1</sup> = very poor, ψ<sup>2</sup> = poor, ψ<sup>3</sup> = slightly poor, ψ<sup>4</sup> = medium , ψ<sup>5</sup> = slightlygood, ψ<sup>6</sup> = good, ψ<sup>7</sup> = very good, ψ<sup>8</sup> = extremely good}.

Then, the decision evaluation matrix can be established, Tables 2–4 show them.

**Table 2.** The decision matrix based on the data of *D*1.



**Table 3.** The decision matrix based on the data of *D*2.


**Table 4.** The decision matrix based on the data of *D*3.

Now, the proposed method is applied to manage this MAGDM problem and the computational procedures are as follows:

Step 1: the overall decision matrix can be obtained by the *LNNEWA* operator in Table 5.

**Table 5.** The overall decision matrix.


Step 2: the total collective LNN θ*i*(*i* = 1, 2, ... , *m*) can be obtained by the *LNNWEA* operator:

θ<sup>1</sup> = ψ6.0661,ψ1.7313,ψ2.3644, θ<sup>2</sup> = ψ6.0961,ψ1.7929,ψ1.9840, θ<sup>3</sup> = ψ5.7523,ψ1.7260,ψ2.2199, and θ<sup>4</sup> = ψ6.4198,ψ1.4753,ψ1.5957.

Step 3: according to Definition 3, the expected values of ζ(θ*i*) for θ*i*(*i* = 1, 2, 3, 4) can be calculated:

ζ(θ1) = 0.7488, ζ(θ2) = 0.7633, ζ(θ3) = 0.7419, and ζ(θ4) = 0.8062.

Based on the expected values, four alternatives can be ranked Θ<sup>4</sup> Θ<sup>2</sup> Θ<sup>1</sup> Θ3, thus, company Θ<sup>4</sup> is the optimal choice.

Now, the *LNNEWG* operator was used to manage this MAGDM problem:

Step 1 : the overall decision matrix can be obtained by the *LNNEWA* operator;

Step 2 : the total collective LNN θ*<sup>i</sup>* (*i* = 1, 2, ... , *m*) can be obtained by the *LNNEWG* operator, which are as following:

θ<sup>1</sup> = ψ5.9491,ψ1.7507,ψ2.4660, θ<sup>2</sup> = ψ6.5864,ψ1.8026,ψ2.0000, θ<sup>3</sup> = ψ6.8354,ψ1.8390,ψ2.2614, and θ<sup>4</sup> = ψ6.3950,ψ1.4868,ψ1.6033.

Step 3 : according to Definition 3, the expected values of ζ(θ*i*) for θ*i*(*i* = 1, 2, 3, 4) can be calculated:

ζ(θ1) = 0.7389, ζ(θ2) = 0.7827,(θ3) = 0.7806, and ζ(θ4) = 0.8043.

Based on the expected values, four alternatives can be ranked Θ<sup>4</sup> Θ<sup>2</sup> Θ<sup>3</sup> Θ1, thus, company Θ<sup>4</sup> is still the optimal choice.

Clearly, there exists a small difference in sorting between these two kinds of methods. However, we can get the same optimal choice by using the LNNEWA and LNNEWG operators. The proposed methods are effective ranking methods for the MCDM problem.

#### *5.2. Comparative Analysis*

Now, we do some comparisons with other related methods for LNN, all the results are shown in Table 6.


**Table 6.** The ranking orders by utilizing three different methods.

As shown in Table 6, we can see that company θ<sup>4</sup> is the best choice for investing by using four methods. Many methods such as arithmetic averaging, geometric averaging, and Bonferroni mean can all be used in LNN to handle the multiple attribute decision-making problems and can get similar results. Additionally, The Einstein aggregation operator is smoother than the algebra aggregation operator, which is used in the literature [15,16]. Compared to the existing literature [2–14], LNNs can express and manage pure linguistic evaluation values, while other literature [2–14] cannot do that. In this paper, a new MAGDM method was presented by using the LNNEWA or LNNEWG operator based on LNN environment.

#### **6. Conclusions**

A new approach for solving MAGDM problems was proposed in this paper. First, we applied the Einstein operation to a linguistic neutrosophic set and established the new operation rules of this linguistic neutrosophic set based on the Einstein operator. Second, we combined some aggregation operators with the linguistic neutrosophic set and defined the linguistic neutrosophic number Einstein weight average operator and the linguistic neutrosophic number Einstein weight geometric (LNNEWG) operator according the new operation rules. Finally, by using the LNNEWA and LNNEWG operator, two methods for handling MADGM problem were presented. In addition, these two methods were introduced into a concrete example to show the practicality and advantages of the proposed approach. In future, we will further study the Einstein operation in other neutrosophic environment just like the refined neutrosophic set [30]. At the same time, we will use these aggregation operators in many actual fields, such as campaign management, decision making and clustering analysis and so on [31–33].

**Author Contributions:** C.F. originally proposed the LNNEWA and LNNEWG operators and their properties; C.F., S.F. and K.H. wrote the paper together.

**Acknowledgments:** This research was funded by the National Natural Science Foundation of China grant number [61603258], [61703280]; General Research Project of Zhejiang Provincial Department of Education grant number [Y201839944]; Public Welfare Technology Research Project of Zhejiang Province grant number [LGG19F020007]; Public Welfare Technology Application Research Project of Shaoxing City grant number [2018C10013].

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Semi-Idempotents in Neutrosophic Rings**

#### **Vasantha Kandasamy W.B. 1, Ilanthenral Kandasamy 1,\* and Florentin Smarandache <sup>2</sup>**


Received: 13 April 2019; Accepted: 27 May 2019; Published: 3 June 2019

**Abstract:** In complex rings or complex fields, the notion of imaginary element *i* with *i* <sup>2</sup> <sup>=</sup> <sup>−</sup>1 or the complex number *i* is included, while, in the neutrosophic rings, the indeterminate element *I* where *<sup>I</sup>*<sup>2</sup> <sup>=</sup> *<sup>I</sup>* is included. The neutrosophic ring *<sup>R</sup>* <sup>∪</sup> *<sup>I</sup>* is also a ring generated by *<sup>R</sup>* and *<sup>I</sup>* under the operations of *R*. In this paper we obtain a characterization theorem for a semi-idempotent to be in *Zp* ∪ *I*, the neutrosophic ring of modulo integers, where *p* a prime. Here, we discuss only about neutrosophic semi-idempotents in these neutrosophic rings. Several interesting properties about them are also derived and some open problems are suggested.

**Keywords:** semi-idempotent; neutrosophic rings; modulo neutrosophic rings; neutrosophic semi-idempotent

**MSC:** 16-XX; 17C27

#### **1. Introduction**

According to Gray [1], an element *α* = 0 of a ring *R* is called a semi-idempotent if and only if *α* is not in the proper two-sided ideal of *<sup>R</sup>* generated by *<sup>α</sup>*<sup>2</sup> <sup>−</sup> *<sup>α</sup>*, that is *<sup>α</sup>* <sup>∈</sup>/ *<sup>R</sup>*(*α*<sup>2</sup> <sup>−</sup> *<sup>α</sup>*)*<sup>R</sup>* or *<sup>R</sup>* <sup>=</sup> *<sup>R</sup>*(*α*<sup>2</sup> <sup>−</sup> *<sup>α</sup>*)*R*. Here, 0 is a semi-idempotent, which we may term as trivial semi-idempotent. Semi-idempotents have been studied for group rings, semigroup rings and near rings [2–9].

An element *I* was defined by Smarandache [10] as an indeterminate element. Neutrosophic rings were defined by Vasantha and Smarandache [11]. The neutrosophic ring *R* ∪ *I* is also a ring generated by *R* and the indeterminate element *I* (*I*<sup>2</sup> = *I*) under the operations of *R* [11]. The concept of neutrosophic rings is further developed and studied in [12–16]. As the newly introduced notions of neutrosophic triplet groups [17,18] and neutrosophic triplet rings [19], neutrosophic triplets in neutrosophic rings [20] and their relations to neutrosophic refined sets [21,22] depend on idempotents, thus the relative study of semi-idempotents will be an innovative research for any researcher interested in these fields. Finding idempotents is discussed in [18,23–25]. One can also characterize and study neutrosophic idempotents in these situations as basically neutrosophic idempotents are trivial neutrosophic semi-idempotents. A new angle to this research can be made by studying quaternion valued functions [26].

We call a semi-idempotents *x* in *R* ∪ *I* as neutrosophic semi-idempotents if *x* = *a* + *bI* and *b* = 0; *a*, *b* ∈ *R* ∪ *I*. Several interesting results about semi-idempotents are derived for neutrosophic rings in this paper. As the study pivots on idempotents it has much significance for the recent studies on neutrosophic triplets, duplets and refined sets.

Here, the notion of semi-idempotents in the case of neutrosophic rings is introduced and several interesting properties associated with them are analyzed. We discuss only about neutrosophic semi-idempotents in these neutrosophic rings. This paper is organized into three sections. Section 1 is introductory in nature. In Section 2, the notion of semi-idempotents in the case of

$$\langle Z\_n \cup I \rangle = \{ a + bI | a, b \in Z\_n; n < \infty; I^2 = I \}$$

is considered. Section 3 gives conclusions and proposes some conjectures based on our study.

#### **2. Semi-Idempotents in the Modulo Neutrosophic Rings** *-Zn ∪ I*

Throughout this paper, *Zn* <sup>∪</sup> *<sup>I</sup>* <sup>=</sup> {*<sup>a</sup>* <sup>+</sup> *bI*/*a*, *<sup>b</sup>* <sup>∈</sup> *Zn*, 2 <sup>≤</sup> *<sup>n</sup>* <sup>&</sup>lt; <sup>∞</sup>; *<sup>I</sup>*<sup>2</sup> <sup>=</sup> *<sup>I</sup>*} denotes the neutrosophic ring of modulo integers. We illustrate some semi-idempotents of *Zn* ∪ *I* by examples and derive some interesting results related with them.

**Example 1.** *Let <sup>S</sup>* <sup>=</sup> *Z*<sup>2</sup> <sup>∪</sup> *<sup>I</sup>* <sup>=</sup> {*<sup>a</sup>* <sup>+</sup> *bI*/*a*, *<sup>b</sup>* <sup>∈</sup> *<sup>Z</sup>*2, *<sup>I</sup>*<sup>2</sup> <sup>=</sup> *<sup>I</sup>*} *be the neutrosophic ring of modulo integers. Clearly, I*<sup>2</sup> = *I and* (1 + *I*)<sup>2</sup> = 1 + *I are the two non-trivial idempotents of S. Here, 0 and 1 are trivial idempotents of S. Thus, S has no non-trivial semi-idempotents as all idempotents are trivial semi-idempotents of S.*

#### **Example 2.** *Let*

$$R = \langle Z\_3 \cup I \rangle = \{a + bI | a, b \in Z^3, I^2 = I\} = \{0, 1, 2, I, 2I, 1 + I, 2 + I, 1 + 2I, 2 + 2I\}$$

*be the neutrosophic ring of modulo integers. The trivial idempotents of R are 0 and 1. The non-trivial neutrosophic idempotents are I and* 1 + 2*I. Thus, the idempotents I and* 1 + 2*I are trivial neutrosophic semi-idempotents of R. Clearly, 2 and* 2 + 2*I are units of R as* 2 × 2 *= 1(mod 3) and* 2 + 2*I* × 2 + 2*I = 1(mod 3).* 1 + *I* ∈ *R is such that*

$$(1+I)^2 - (1+I) = 1 + 2I + I - (1+I) = 1 + 2 + 2I = 2I.$$

*Thus,* <sup>1</sup> <sup>+</sup> *<sup>I</sup> is a semi-idempotent as the ideal generated by* <sup>1</sup> <sup>+</sup> *<sup>I</sup> is* (<sup>1</sup> <sup>+</sup> *<sup>I</sup>*)<sup>2</sup> <sup>−</sup> (<sup>1</sup> <sup>+</sup> *<sup>I</sup>*) <sup>=</sup> 2*I is such that* <sup>1</sup> <sup>+</sup> *<sup>I</sup>* <sup>∈</sup>/ *R. However, it is important to note that* (<sup>1</sup> <sup>+</sup> *<sup>I</sup>*) <sup>∈</sup> *<sup>R</sup> is a unit as* (<sup>1</sup> <sup>+</sup> *<sup>I</sup>*)<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>+</sup> <sup>2</sup>*<sup>I</sup>* <sup>+</sup> *<sup>I</sup>* <sup>=</sup> <sup>1</sup>*, hence* 1 + *I is a unit in R but it is also a non-trivial semi-idempotent of R.* 2 + *I is not a semi-idempotent as*

$$\left(\left(2+I\right)^2-\left(2+I\right)=1+4I+I-\left(2+I\right)=2+I;I\right)$$

*hence the claim.* <sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>I</sup>* <sup>∈</sup> *<sup>R</sup> is a unit, now* (<sup>2</sup> <sup>+</sup> <sup>2</sup>*I*)<sup>2</sup> <sup>=</sup> <sup>4</sup> <sup>+</sup> <sup>8</sup>*<sup>I</sup>* <sup>+</sup> <sup>4</sup>*I*<sup>2</sup> <sup>=</sup> <sup>1</sup>*, thus* <sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>I</sup> is a unit. However,* (<sup>2</sup> <sup>+</sup> <sup>2</sup>*I*)<sup>2</sup> <sup>−</sup> (<sup>2</sup> <sup>+</sup> <sup>2</sup>*I*) = <sup>1</sup> <sup>+</sup> <sup>1</sup> <sup>+</sup> *<sup>I</sup>* <sup>=</sup> <sup>2</sup> <sup>+</sup> *<sup>I</sup>*.

*Now, the ideal generated by* 2 + *I does not contain* 2 + 2*I as* 2 + *I* = {0, 2 + *I*, 1 + 2*I*}*, thus* 2 + 2*I is also a non-trivial semi-idempotent even though* 2 + 2*I is a unit of R. Thus, it is important to note that units in modulo neutrosophic rings contribute to non-trivial semi-idempotents. Let P* = {0, 2 + 2*I*, 2 + *I*, 1 + 2*I*, *I*, 1 + *I*, 1} *be the collection of trivial and non-trivial semi-idempotents.* 2*I is not a semi-idempotent as* (2*I*)<sup>2</sup> <sup>−</sup> <sup>2</sup>*<sup>I</sup>* <sup>=</sup> *<sup>I</sup>* <sup>+</sup> *<sup>I</sup>* <sup>=</sup> <sup>2</sup>*I, hence the claim. Thus, P is not closed under sum or product.*

**Theorem 1.** *Let S* = {*Zp* ∪ *I*, +, ×} *be the ring of neutrosophic modulo integers where p is a prime. x is semi-idempotent if and only if x* ∈ *Zp* ∪ *I*\{*Zp I*, 0, 1, *a* + *bI with a* + *b* = 0}*.*

**Proof.** The elements *<sup>x</sup>* <sup>=</sup> *<sup>a</sup>* <sup>+</sup> *bI* <sup>∈</sup> *<sup>S</sup>* with *<sup>b</sup>* <sup>=</sup> 0 are such that *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>x</sup>* generates the ideal, which is *<sup>S</sup>*, thus *x* is a semi-idempotent. Let *y* = *a* + *bI*; if *a* = 0, the ideal generated by *y* is *Zp I*, thus *y* ∈ *Zp I* ⊂ *S*, hence *y* ∈ *Zp I*, therefore *y* is not a semi-idempotent.

Consider *<sup>z</sup>* <sup>=</sup> *<sup>a</sup>* <sup>+</sup> *bI* <sup>∈</sup> *<sup>S</sup>* with *<sup>a</sup>* <sup>+</sup> *<sup>b</sup>* <sup>=</sup> <sup>0</sup>(*mod p*); then, *<sup>z</sup>*<sup>2</sup> <sup>−</sup> *<sup>z</sup>* generates an ideal *<sup>M</sup>* of *<sup>S</sup>* such that every element *x* = *d* + *cI* in *M* is such that *d* + *c* ≡ 0(*mod p*), thus *z* is not a semi-idempotent of *S*. Let *x* = *a* + *bI* ∈ *S*(*a* = 0, *b* = 0 and *a* + *b* = 0).

$$\mathbf{x}^2 - \mathbf{x} = \begin{cases} m & m \in Z\_p \text{ or} \\ nI & n \in Z\_P \text{ or} \\ n + mI & m + n \neq 0 \end{cases}$$

If *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>x</sup>* <sup>=</sup> *<sup>m</sup>*, then the ideal generated by *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>x</sup>* is *<sup>S</sup>*, thus *<sup>x</sup>* is a semi-idempotent. If *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>x</sup>* <sup>=</sup> *nI*, then the ideal generated by *nI* is *Zp <sup>I</sup>*, thus *<sup>x</sup>* <sup>∈</sup>/ *Zp <sup>I</sup>*, hence again *<sup>x</sup>* is a semi-idempotent. If *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>x</sup>* <sup>=</sup> *n* + *mI*(*m* + *n* = 0), then the ideal generated by *n* + *mI* is *S*, thus *x* is a semi-idempotent by using properties of *Zp*, *p* a prime. Hence, the theorem is proved.

If we take *S* = {*Zn* ∪ *I*, +, ×} as a neutrosophic ring where *n* is not a prime, it is difficult to find all semi-idempotents.

**Example 3.** *Let S* = {*Z*<sup>15</sup> ∪ *I*, +, ×} *be the neutrosophic ring. How can the non-trivial semi-idempotents of S be found? Some of the neutrosophic idempotents of S are* {1 + 9*I*, 6 + 4*I*, 1 + 5*I*, 1 + 14*I*, 6 + 5*I*, *6 + 9I*, *I*, 6*I*, 10*I*, 10, 6, 6 + 10*I*, 10 + 11*I*, 10 + 6*I*, 10 + 5*I*}.

*The semi-idempotents are* {1 + *I*, 1 + 2*I*, 1 + 3*I*, 1 + 4*I*, 1 + 6*I*, 1 + 7*I*, 1 + 8*I*, *1 + 10I*, *1 + 11I*, *1 + 12I*, *1 + 13I*, 6 + *I*, 6 + 2*I*, 6 + 3*I*, 6 + 6*I*, 6 + 7*I*, 6 + 8*I*, 6 + 11*I*, 6 + 12*I*, 6 + 13*I*, 6 + 14*I*, *10 + I*, 10 + 2*I*, 10 + 3*I*, 10 + 4*I*, 10 + 7*I*, 10 + 8*I*, 10 + 9*I*, 10 + 10*I*, 10 + 12*I*, 10 + 13*I*, 10 + 14*I*}*.*

Are there more non-trivial neutrosophic idempotents and semi-idempotents?

However, we are able to find all idempotents and semi-idempotents of *S* other than the once given. In view of all these, we have the following theorem.

**Theorem 2.** *Let S* = {*Zpq* ∪ *I*; ×, +} *where p and q are two distinct primes:*


**Proof.** Given *S* = {*Zpq* ∪ *I*, +, ×} is a neutrosophic ring where *p* and *q* are primes, we know from [12,17,18,20,23–25] that *Zpq* has two idempotents *r* and *s* to prove *A* = {*r*,*s*,*rIsI*, *I*,*r* + *tI* and *s* + *tI*/*t* ∈ *Zpq* \ {0}} are idempotents or semi-idempotents of *S*.{*r*,*s*,*r I*,*sI*, *I*} are non-trivial idempotents of *<sup>S</sup>*. Now, *<sup>r</sup>* <sup>+</sup> *tI* <sup>∈</sup> *<sup>A</sup>* and (*<sup>r</sup>* <sup>+</sup> *tI*)<sup>2</sup> <sup>−</sup> (*<sup>r</sup>* <sup>+</sup> *tI*) = *mI* as*r*<sup>2</sup> <sup>=</sup> *<sup>r</sup>*, thus the ideal generated by *mI* does not contain *rtI*. Therefore, *rtI* is a non-trivial semi-idempotent. Similarly, *s* + *tI* is a non-trivial semi-idempotent. Hence, the theorem is proved.

We in addition to this theorem propose the following problem.

**Problem 1.** *Let S* = {*Zpq* ∪ *I*, *I*, ×}*, where p and q are two distinct primes, be the neutrosophic ring. Can S have non-trivial idempotents and non-trivial semi-idempotents other than the ones mentioned in (b) of the above theorem?*

**Problem 2.** *Can the collection of all trivial and non-trivial semi-idempotents have any algebraic structure defined on them?*

We give an example of *Zpqr*, where *p*, *q* and *r* are three distinct primes, for which we find all the neutrosophic idempotents.

**Example 4.** *Let S* = {*Z*<sup>30</sup> ∪ *I*, +, ×}*, be the neutrosophic ring. The idempotents of Z*<sup>30</sup> *are 6, 10, 15, 16, 21 and 25. The non-trivial semi-idempotents of S are* {1 + *I*, 1 + 2*I*, 1 + 3*I*, *1 + 4I*, *1 + 6I*, *1 + 7I*, *1 + 8I, 1 + 10I*, *1 + 11I*, 1 + 13*I*, 1 + 12*I*, 1 + 16*I*, 1 + 17*I*, 1 + 18*I*, 1 + 19*I*, 1 + 21*I*, *1 + 22I*, 1 + 23*I*, 1 + 25*I*, 1 + 26*I*, 1 + 27*I*, 1 + 28*I*}*.*

*P*<sup>1</sup> = {1 + 5*I*, 1 + 9*I*, 1 + 14*I*, 1 + 15*I*, 1 + 20*I*, 1 + 24*I*, 1 + 29*I*} *are non-trivial idempotents of S. J*<sup>2</sup> = {6 + *I*, 6 + 2*I*, 6 + 3*I*, 6 + 5*I*, 6 + 6*I*, 6 + 7*I*, 6 + 8*I*, 6 + 11*I*, 6 + 12*I*, 6 + 13*I*, 6 + 14*I*, 6 + 16*I*, 6 + 17*I*, *6 + 18I,* 6 + 20*I*, 6 + 21*I*, 6 + 22*I*, 6 + 23*I*, 6 + 26*I*, 6 + 27*I*, 6 + 28*I*, 6 + 29*I*} *are non-trivial neutrosophic semi-idempotents of S. P*<sup>2</sup> = {6 + 4*I*, 6 + 9*I*, 6 + 10*I*, 6 + 15*I*, 6 + 24*I*, 6 + 19*I*, 6 + 25*I*} *are non-idempotents of S.*

*Now, we list the non-trivial semi-idempotents associated with 10 of Z*30*. J*<sup>3</sup> = {10 + *I*, 10 + 2*I*, *10 + 3I*, 10+4*I*, 10+7*I*, 10+8*I*, 10+9*I*, 10+10*I*, 10+11*I*, 10+12*I*, 10+13*I*, 10+14*I*, 10+16*I*, 10+17*I*, 10 + 18*I*, 10 + 19*I*, 10 + 22*I*, 10 + 23*I*, 10 + 24*I*, 10 + 25*I*, 10 + 27*I*, 10 + 28*I*, *10 + 29I*}

*P*<sup>3</sup> = {10 + 5, 10 + 6*I*, 10 + 15*I*, 10 + 20*I*, 10 + 21*I*, 10 + 26*I*, 10 + 11*I*} *are the collection of non-trivial idempotent related with the idempotents. Now, we find the non-trivial idempotents associated with 15: J*<sup>4</sup> = {15 + 2*I*, 15 + 3*I*, 15 + 4*I*, 15 + 7*I*, 15 + 8*I*, 15 + 9*I*, 15 + 11*I*, 15 + 12*I*, 15 + 13*I*, *15 + 14I,* 15 + 17*I*, 15 + 18*I*, 15 + 19*I*, 15 + 20*I*, 15 + 22*I*, 15 + 23*I*, 15 + 24*I*, 15 + 25*I*, 15 + 26*I*, 15 + 27*I*, *15 + 28I, 15 + 29I*}.

*P*<sup>4</sup> = {15 + *I*, 15 + 5*I*, 15 + 6*I*, 15 + 10*I*, 15 + 15*I*, 15 + 16*I*, 15 + 21*I*} *are the non-trivial idempotents associated with 15. The collection of non-trivial semi-idempotents associated with 16 are: J*<sup>5</sup> = {*16 + I*, 16 + 2*I*, 16 + 3*I*, 16 + 4*I*, 16 + 6*I*, 16 + 7*I*, 16 + 8*I*, 16 + 10*I*, 16 + 19*I*, 16 + 27*I*, 16 + 21*I*, 16 + 22*I*, 16 + 23*I*, 16 + 25*I*, 16 + 11*I*, 16 + 12*I*, 16 + 13*I*, 16 + 17*I*, 16 + 18*I*, *16 + 28I*. *P*<sup>5</sup> = {16 + 14*I*, 16 + 15*I*, 16 + 20*I*, 16 + 24*I*, 16 + 29*I*, 16 + 5*I*, 16 + 9*I*} *are the set of non-trivial idempotents related with the idempotent. We find the non-trivial semi-idempotents associated with the idempotent 21: J*<sup>6</sup> = {21 + *I*, 21 + 2*I*, 21 + 3*I*, 21+5*I*, 21+6*I*, 21+7*I*, 21+8*I*, 21+12*I*, 21+11*I*, 21+13*I*, 21+14*I*, *21 + 16I*, 21+17*I*, 21+18*I*, 21+ 20*I*, 21 + 21*I*, 21 + 22*I*, 21 + 23*I*, 21 + 26*I*, 21 + 27*I*, 21 + 28*I*, 21 + 29*I*}. *P*<sup>6</sup> = {21 + 4*I*, 21 + 9*I*, 21 + 10*I*, 21 + 15*I*, 21 + 19*I*, 21 + 24*I*, 21 + 25*I*} *is the collection of non-trivial idempotents related with the real idempotent 21. The collection of all non-trivial semi-idempotents associated with the idempotent 25. J*<sup>7</sup> = {25 + *I*, 25 + 2*I*, 25 + 3*I*, 25 + 4*I*, 25 + 7*I*, 25 + 8*I*, 25 + 9*I*, *25 + 10I,* 25 + 12*I*, 25 + 13*I*, 25 + 14*I*, 25 + 16*I*, 25 + 24*I*, 25 + 17*I*, 25 + 18*I*, 25 + 19*I*, 25 + 22*I*, *25 + 23I,* 25 + 27*I*, 25 + 28*I*, 25 + 29*I*} *P*<sup>7</sup> = {25 + 5*I*, 25 + 6*I*, 25 + 11*I*, 25 + 15*I*, 25 + 20*I*, 25 + 21*I*, 25 + 26*I*} *are the non-trivial collection of neutrosophic semi-idempotents related with the idempotent 25.*

We tabulate the neutrosophic idempotents associated with the real idempotents in Table 1. Based on that table, we propose some open problems.


**Table 1.** Idempotents.


**Table 1.** *Cont*.

We see there are eight idempotents including 0 and 1. It is obvious that using 0 we get only idempotents or trivial semi-idempotents.

In view of all these, we conjecture the following.

**Conjecture 1.** *Let S* = {*Zn* ∪ *I*, +, ×} *be the neutrosophic ring n* = *pqr, where p*, *q and r are three distinct primes.*


This has been verified for large values of *p*, *q* and *r*, where *p*, *q* and *r* are three distinct primes.

#### **3. Conjectures, Discussion and Conclusions**

We have characterized the neutrosophic semi-idempotents in *Zp* ∪ *I*, with *p* a prime. However, it is interesting to find neutrosophic semi-idempotents of *Zn* ∪ *I*, with *n* a non-prime composite number. Here, we propose a few new open conjectures about idempotents in *Zn* and semi-idempotents in *Zn* ∪ *I*.

**Conjecture 2.** *Given Zn* ∪ *I, where n* = *p*1, *p*2,... *pt*; *t* > 2 *and pis are all distinct primes, find:*


**Conjecture 3.** *Prove if Zn* <sup>∪</sup> *<sup>I</sup> and Zm* <sup>∪</sup> *<sup>I</sup> are two neutrosophic rings where <sup>n</sup>* <sup>&</sup>gt; *<sup>m</sup> and <sup>n</sup>* <sup>=</sup> *<sup>p</sup><sup>t</sup> q (t* > 2*, and p and q two distinct primes) and m* = *p*<sup>1</sup> *p*<sup>2</sup> ... *ps where pis are distinct primes.* 1 ≤ *i* ≤ *s, then*


Finding idempotents in the case of *Zn* has been discussed and problems are proposed in [18,23,24]. Further, the neutrosophic triplets in *Zn* are contributed by *Zn*. In the case of neutrosophic duplets, we see units in *Zn* contribute to them. Both units and idempotents contribute in general to semi-idempotents.

**Author Contributions:** The contributions of the authors are roughly equal.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors would like to thank the reviewers for their reading of the manuscript and many insightful comments and suggestions.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Neutrosophic Triplets in Neutrosophic Rings**

#### **Vasantha Kandasamy W. B. 1, Ilanthenral Kandasamy 1,\* and Florentin Smarandache <sup>2</sup>**


Received: 25 May 2019; Accepted: 18 June 2019; Published: 20 June 2019

**Abstract:** The neutrosophic triplets in neutrosophic rings *Q* ∪ *I* and *R* ∪ *I* are investigated in this paper. However, non-trivial neutrosophic triplets are not found in *Z* ∪ *I*. In the neutrosophic ring of integers *Z* \ {0, 1}, no element has inverse in *Z*. It is proved that these rings can contain only three types of neutrosophic triplets, these collections are distinct, and these collections form a torsion free abelian group as triplets under component wise product. However, these collections are not even closed under component wise addition.

**Keywords:** neutrosophic ring; neutrosophic triplets; idemponents; special neutrosophic triplets

#### **1. Introduction**

Handling of indeterminacy present in real world data is introduced in [1,2] as neutrosophy. Neutralities and indeterminacies represented by Neutrosophic logic has been used in analysis of real world and engineering problems [3–5].

Neutrosophic algebraic structures such as neutrosophic rings, groups and semigroups are presented and analyzed and their application to fuzzy and neutrosophic models are developed in [6]. Subsequently, researchers have been studying in this direction by defining neutrosophic rings of Types I and II and generalization of neutrosophic rings and fields [7–12]. Neutrosophic rings [9] and other neutrosophic algebraic structures are elaborately studied in [6–8,10,13–17]. Related theories of neutrosophic triplet, duplet, and duplet set were developed by Smarandache [18]. Neutrosophic duplets and triplets have fascinated several researchers who have developed concepts such as neutrosophic triplet normed space, fields, rings and their applications; triplets cosets; quotient groups and their application to mathematical modeling; triplet groups; singleton neutrosophic triplet group and generalization; and so on [19–36]. Computational and combinatorial aspects of algebraic structures are analyzed in [37].

Neutrosophic duplet semigroup [23], classical group of neutrosophic triplet groups [27], the neutrosophic triplet group [12], and neutrosophic duplets of {*Zpn*, ×} and {*Zpq*, ×} have been analyzed [28]. Thus, Neutrosophic triplets in case of the modulo integers *Zn*(2 < *n* < ∞) have been extensively researched [27].

Neutrosophic duplets in neutrosophic rings are characterized in [29]. However, neutrosophic triplets in the case of neutrosophic rings have not yet been researched. In this paper, we for the first time completely characterize neutrosophic triplets in neutrosophic rings. In fact, we prove this collection of neutrosophic triplets using neutrosophic rings are not even closed under addition. We also prove that they form a torsion free abelian group under component wise multiplication.

#### **2. Basic Concepts**

In this section, we recall some of the basic concepts and properties associated with both neutrosophic rings and neutrosophic triplets in neutrosophic rings. We first give the following notations: *<sup>I</sup>* denotes the indeterminate and it is such that *<sup>I</sup>* <sup>×</sup> *<sup>I</sup>* <sup>=</sup> *<sup>I</sup>* <sup>=</sup> *<sup>I</sup>*2. *<sup>I</sup>* is called as the neutrosophic value. *Z*, *Q* and *R* denote the ring of integers, field of rationals and field of reals, respectively. *<sup>Z</sup>* <sup>∪</sup> *<sup>I</sup>* <sup>=</sup> {*<sup>a</sup>* <sup>+</sup> *bI*|*a*, *<sup>b</sup>* <sup>∈</sup> *<sup>Z</sup>*, *<sup>I</sup>*<sup>2</sup> <sup>=</sup> *<sup>I</sup>*} is the neutrosophic ring of integers, *<sup>Q</sup>* <sup>∪</sup> *<sup>I</sup>* <sup>=</sup> {*<sup>a</sup>* <sup>+</sup> *bI*|*a*, *<sup>b</sup>* <sup>∈</sup> *<sup>Q</sup>*, *<sup>I</sup>*<sup>2</sup> <sup>=</sup> *<sup>I</sup>*} is the neutrosophic ring of rationals and *<sup>R</sup>* <sup>∪</sup> *<sup>I</sup>* <sup>=</sup> {*<sup>a</sup>* <sup>+</sup> *bI*|*a*, *<sup>b</sup>* <sup>∈</sup> *<sup>R</sup>*, *<sup>I</sup>*<sup>2</sup> <sup>=</sup> *<sup>I</sup>*} is the neutrosophic ring of reals with usual addition and multiplication in all the three rings.

### **3. Neutrosophic Triplets in** *Q* ∪ *I* **and** *R* ∪ *I*

In this section, we prove that the neutrosophic rings *Q* ∪ *I* and *R* ∪ *I* have infinite collection of neutrosophic triplets of three types. Both collections enjoy strong algebraic structures. We explore the algebraic structures enjoyed by these collections of neutrosophic triplets. Further, the neutrosophic ring of integers *Z* ∪ *I* has no nontrivial neutrosophic triplets. An example of neutrosophic triplets in *Q* ∪ *I* is provided before proving the related results.

**Example 1.** *Let S* = *Q* ∪ *I*, +, × *(or R* ∪ *I*, +, ×*) be the neutrosophic ring. If x* = *a* − *aI* ∈ *S*(*a* = 0)*, then*

$$y = \frac{1}{a} - \frac{I}{a} \in \mathcal{S}$$

*is such that*

$$\mathbf{x} \times \mathbf{y} = (a - aI) \times \left(\frac{1}{a} - \frac{I}{a}\right) = 1 - I - I + I = 1 - I.$$

*Thus, for every x* = *a* − *aI, of this form in S we have a unique y of the form*

$$\frac{1}{a} - \frac{I}{a}$$

*such that x* × *y* = 1 − *I. Further,* 1 − *I* ∈ *S is such that* 1 − *I* × 1 − *I* = 1 − *I* + *I* − *I* = 1 − *I* ∈ *S*. *Thus, these triplets*

$$\left\{a - aI, 1 - I, \frac{1}{a} - \frac{I}{a}\right\} \text{ and } \left\{\frac{1}{a} - \frac{I}{a}, 1 - I, a - aI\right\}.$$

*form neutrosophic triplets with* 1 − *I as a neutral element.*

*Similarly, for aI* ∈ *S*(*a* = 0)*, we have a unique*

$$\frac{I}{a} \in \mathcal{S} \text{ such that } aI \times \frac{I}{a} = I$$

*and I* × *I* = *I is an idempotent. Thus,*

$$\left\{ aI, I, \frac{I}{a} \right\} \text{ and } \left\{ \frac{I}{a}, I, aI \right\}.$$

*are neutrosophic triplets with I as the neutral element.*

First, we prove *Q* ∪ *I* and *R* ∪ *I* have only *I* and 1 − *I* as nontrivial idempotents as invariably one idempotents serve as neutrals of neutrosophic triplets.

**Theorem 1.** *Let S* = *Q* ∪ *I*, +, × *(or* {*R* ∪ *I*, +, ×} *) be a neutrosophic ring. The only non-trivial idempotents in S are I and* 1 − *I.*

**Proof.** We call 0 and 1 ∈ *S* as trivial idempotents. Suppose *x* ∈ *S* is a non-trivial idempotent, then *<sup>x</sup>* <sup>=</sup> *aI* or *<sup>x</sup>* <sup>=</sup> *<sup>a</sup>* <sup>+</sup> *bI* <sup>∈</sup> *<sup>S</sup>*(*<sup>a</sup>* <sup>=</sup> 0, *<sup>b</sup>* <sup>=</sup> <sup>0</sup>). Now, *<sup>x</sup>* <sup>×</sup> *<sup>x</sup>* <sup>=</sup> *aI* <sup>×</sup> *aI* <sup>=</sup> *<sup>a</sup>*<sup>2</sup> *<sup>I</sup>* (as *<sup>I</sup>*<sup>2</sup> <sup>=</sup> *<sup>I</sup>*); if *<sup>x</sup>* is to be an idempotent, we must have *aI* <sup>=</sup> *<sup>a</sup>*<sup>2</sup> *<sup>I</sup>*; that is, (*<sup>a</sup>* <sup>−</sup> *<sup>a</sup>*2)*<sup>I</sup>* <sup>=</sup> <sup>0</sup>(*<sup>I</sup>* <sup>=</sup> <sup>0</sup>), thus *<sup>a</sup>*<sup>2</sup> <sup>=</sup> *<sup>a</sup>*. However, in *<sup>Q</sup>* or *<sup>R</sup>*,

*<sup>a</sup>*<sup>2</sup> <sup>=</sup> *<sup>a</sup>* implies *<sup>a</sup>* <sup>=</sup> 0 or *<sup>a</sup>* <sup>=</sup> 1; as *<sup>a</sup>* <sup>=</sup> 0, we have *<sup>a</sup>* <sup>=</sup> 1; thus, *<sup>x</sup>* <sup>=</sup> *<sup>I</sup>* and *<sup>x</sup>* is a nontrivial idempotent in *S*. Now, let *y* = *a* + *bI*; *a* = 0 and *b* = 0 for *a* = 0 will reduce to case *y* = *I* is an idempotent.

$$y^2 = (a+bI) \times (a+bI) = a^2 + b^2I + 2abI$$

That is, *<sup>y</sup>*<sup>2</sup> <sup>=</sup> *<sup>a</sup>* <sup>+</sup> *bI* <sup>×</sup> *<sup>a</sup>* <sup>−</sup> *bI* <sup>=</sup> *<sup>a</sup>*<sup>2</sup> <sup>+</sup> *abI* <sup>+</sup> *abI* <sup>+</sup> *<sup>b</sup>*<sup>2</sup> *<sup>I</sup>* <sup>=</sup> *<sup>a</sup>* <sup>+</sup> *bI*, equating the real and neutrosophic parts.

$$a^2 = a \text{ i.e., } a(a-1) = 0 \Rightarrow a = 1 \text{ as } a \neq 0 \text{ and } 2ab + b^2 - b = 0$$

*<sup>b</sup>*(2*<sup>a</sup>* <sup>+</sup> *<sup>b</sup>* <sup>−</sup> <sup>1</sup>) = 0; *<sup>b</sup>* <sup>=</sup> 0, thus 2*<sup>a</sup>* <sup>+</sup> *<sup>b</sup>* <sup>−</sup> <sup>1</sup> <sup>=</sup> 0; further, *<sup>a</sup>* <sup>=</sup> 0 as *<sup>a</sup>* <sup>=</sup> 0 will reduce to the case *<sup>I</sup>*<sup>2</sup> <sup>=</sup> *<sup>I</sup>*, thus *a* = 1. Hence, 2 + *b* − 1 = 0, thus *b* = −1. Hence, *a* = 1 and *b* = −1 leading to *y* = 1 − *I*. Thus, only the non-trivial idempotents of *S* are *I* and 1 − *I*.

We next find the form of the triplets in S.

**Theorem 2.** *Let S* = {*Q* ∪ *I*, +, ×} *(or R* ∪ *I*, +, ×*) be the neutrosophic ring. The neutrosophic triplets in S are only of the following form for a*, *b* ∈ *Q or R.*

$$\boldsymbol{a}$$

$$\left(a - aI, 1 - I, \frac{1}{a} - \frac{I}{a}\right) \text{ and } \left(\frac{1}{a} - \frac{I}{a}, 1 - I, a - aI\right); a \neq 0.$$

*(ii)*

$$\left(bI, I, \frac{I}{b}\right) \text{ and } \left(\frac{I}{b}, I, b\right); b \neq 0.$$

*(iii)*

$$\left(a+bI, 1, \frac{1}{a} - \frac{bI}{a(a+b)}\right); a+b \neq 0 \; and \; \left(\frac{1}{a} - \frac{bI}{a(a+b)}, 1, a+bI\right).$$

**Proof.** Let *S* be the neutrosophic ring. Let *x* = {*a* + *bI*,*e* + *f I*, *c* + *dI*} be a neutrosophic triplet in *S*; *a*, *b*, *c*, *d*,*e*, *f* ∈ *Q* or *R*. We prove the neutrosophic triplets of *S* are in one of the forms. If *x* is a neutrosophic triplet, then we have

$$a + bI \times c + fI = a + bI \tag{1}$$

$$
\mathcal{c} + fI \times \mathcal{c} + dI = \mathcal{c} + dI \tag{2}
$$

and

$$a + bI \times \mathfrak{c} + dI = \mathfrak{c} + fI \tag{3}$$

Now, solving Equation (1), we get

*ae* + (*bfI* + *beI* + *afI*) = *a* + *bI*

Equating the real and neutrosophic parts, we get

$$a\mathbf{e} = a\tag{4}$$

$$bf + be + af = b\tag{5}$$

Expanding Equation (2), we get

$$ce + fcI + deI + fdI = c + dI.$$

Equating the real and neutrosophic parts, we get

$$cc = c \tag{6}$$

$$f\mathbf{c} + de + fd = d.\tag{7}$$

Solving Equation (3), we get

$$ac + bcI + bdI + adI = e + fI$$

Equating the real and neutrosophic parts, we get

$$ac = \varepsilon \tag{8}$$

$$abc + bd + ad = f\tag{9}$$

We find conditions so that Equations (4) and (5) are true.

Now, *ae* = *a* and *b f* + *be* + *a f* = *b*; *ae* = *a* gives *a*(*e* − 1) = 0 if *a* = 0 and *e* = 1 using in Equation (4), thus if *a* = 0, we get *e* = 0 and using *e* = 0 in Equation (6), we get *c* = 0. Thus, *a* = *c* = *e* = 0. This forces *b* = 0, *d* = 0 and *f* = 0. We solve for *b*, *d* and *f* using Equations (5), (7) and (9). Equations (5) and (7) gives *b f* = *b* as *b* = 0, *f* = 1. Now, *f d* = *d* as *f* = 1; *d* = *d*. Equation (9) gives *bd* = *f* or *bd* = 1, thus

$$d = \frac{1}{b}(b \neq 0).$$

Thus, we get

$$\left(bI\_{\prime\prime}I\_{\prime\prime}\frac{I}{b}\right)$$

to be neutrosophic triplet then

$$\left(\frac{I}{b'}, I, bI\right)$$

is also a neutrosophic triplet. Thus, we have proved (ii) of the theorem.

Assume in Equation (4) *ae* = *a*; *a* = 0, which forces *e* = 1. Now, using Equation (8), we get *ac* = 1, thus

$$c = \frac{1}{a}.$$

Using Equation (5), we get *b f* + *b* + *a f* = *b*, thus (*a* + *b*)*f* = 0. If *f* = 0, then we have

$$\left(a+bI, 1, \frac{1}{a}+dI\right)$$

should be a neutrosophic triplet. That is,

$$\begin{aligned} \left(a+bI\right) \times \left(\frac{1}{a}+dI\right) &= 1 \\\\ 1 + \frac{b}{a}I + daI + dbI &= 1 \\\\ \frac{b}{a} + da + db &= 0 \\\\ b + a^2d + abd &= 0 \\\\ b(ad+1) + a^2d &= 0 \\\\ d(a^2+ab) &= -b. \end{aligned}$$

$$d = \frac{-b}{a^2 + ab} = \frac{-b}{a(a+b)}$$

*a* = 0 and *a* + *b* = 0. *a* + *b* = 0 for if *a* + *b* = 0, then *b* = 0 we get *d* = 0. Thus, the trivial triplet

$$(a, 1, \frac{1}{a})$$

will be obtained. Thus, *a* + *b* = 0 and

$$\left(a+bI, 1, \frac{1}{a} - \frac{bI}{a(a+b)}\right) \text{ and } \left(\frac{1}{a} - \frac{bI}{a(a+b)}, 1, a+bI\right).$$

are neutrosophic triplets so that Condition (iii) of theorem is proved.

Now, let *f* = 0, thus *a* + *b* = 0 and *c* + *d* = 0. We get *a* = −*b* or *b* = −*a* and *d* = −*c*. We have already proved *c* = <sup>1</sup> *<sup>a</sup>* . Using Equations (8) and (9) and conditions *a* = −*b* and *c* = −*d*, we get *f* = −1. Hence, the neutrosophic triplets are

$$\left(a - aI, 1 - I, \frac{1}{a} - \frac{I}{a}\right) \text{ and } \left(\frac{1}{a} - \frac{I}{a'}, 1 - I, a - aI\right).$$

which is Condition (i) of the theorem.

**Theorem 3.** *Let S* = {*Q* ∪ *I*, +, ×} *(or R* ∪ *I*, +, ×}*) be the neutrosophic ring.*

$$M = \left\{ \left( a - aI\_\prime 1 - I\_\prime \frac{1}{a} - \frac{I}{a} \right) \, | \, a \in \mathcal{Q} \backslash \{ 0 \} \right\}.$$

*be the collection of neutrosophic triplets of S with neutral* 1 − *I is commutative group of infinite order with* (1 − *I*, 1 − *I*, 1 − *I*) *as the multiplicative identity.*

**Proof.** To prove *M* is a group of infinite order, we have to prove *M* is closed under component-wise product and has an identity with respect to which every element has an inverse.

Let

$$\mathbf{x} = \left( a - aI, 1 - I, \frac{1}{a} - \frac{I}{a} \right) \text{ and } y = \left( c - cI, 1 - I, \frac{1}{c} - \frac{I}{c} \right) \in M.$$

$$\mathbf{x} \times \mathbf{y} = \left( a - aI, 1 - I, \frac{1}{a} - \frac{I}{a} \right) \times \left( c - cI, 1 - I, \frac{1}{c} - \frac{I}{c} \right)$$

$$= \left( ac - acI - acI, 1 - 2I + I, \frac{1}{ac} - \frac{I}{ac} - \frac{I}{ac} + \frac{I}{ac} \right)$$

$$= \left( ac - acI, 1 - I, \frac{1}{ac} - \frac{I}{ac} \right) \in M.$$

Thus, *M* is closed under component wise product.

We see that, when *a* = 1, we get *e* = (1 − *I*, 1 − *I*, 1 − *I*) ∈ *M* is the identity of *M* under component wise multiplication. Clearly, *e* × *x* = *x* × *e* = *x* for all *x* ∈ *M*, thus *e* is the identity of *M*. For every

$$\mathbf{x} = \left(a - aI\_\prime \mathbf{1} - I\_\prime \frac{\mathbf{1}}{a} - \frac{I}{a}\right)\mathbf{x}$$

we have a unique

$$\mathbf{x}^{-1} = \left(\frac{1}{a} - \frac{I}{a'}, 1 - I, a - aI\right) \in M$$

*Mathematics* **2019**, *7*, 563

such that

$$\mathbf{x} \times \mathbf{x}^{-1} = \mathbf{x}^{-1} \times \mathbf{x} = \mathbf{c} = (1 - I, 1 - I, 1 - I)$$

$$\mathbf{x} \times \mathbf{x}^{-1} = \left( a - aI, 1 - I, \frac{1}{a} - \frac{I}{a} \right) \times \left( \frac{1}{a} - \frac{I}{a} \right) - \left( \frac{1}{a} - \frac{I}{a}, 1 - I, a - aI \right)$$

$$= \left( \frac{a}{a} - \frac{aI}{a} - \frac{aI}{a} + \frac{aI}{a}, 1 - 2I + I, \frac{a}{a} - \frac{aI}{a} - \frac{aI}{a} + \frac{aI}{a} \right)$$

$$= (1 - I, 1 - I, 1 - I)$$

as *a* = 0. Thus, (*M*, ×) is a group under component wise product, which is known as the neutrosophic triplet group.

**Theorem 4.** *Let S* = {*Q* ∪ *I*, +, ×} *(or* {*R* ∪ *I*, +, ×}*) be the neutrosophic ring. The collection of neutrosophic triplets*

$$N = \left\{ \left( aI, I, \frac{I}{a} \right) \, | \, a \in Q \backslash \{ 0 \} \right\}.$$

*(or R*\{0}) *forms a commutative group of infinite order under component wise multiplication with (I, I, I) as the multiplicative identity.*

**Proof.** Let

$$N = \left\{ \left( aI, I, \frac{I}{a} \right) \, | \, a \neq 0 \in Q \text{ or } R \right\}.$$

be a collection of neutrosophic triplets. To prove *N* is commutative group under component wise product, let

$$x = \left(aI\_\prime I\_\prime \frac{I}{a}\right)^2$$

and

$$y = \left(bI\_\prime I\_\prime \frac{I}{b}\right) \in M.$$

To show *x* × *y* ∈ *N*.

$$\mathbf{x} \times \mathbf{y} = \left( aI\_\prime I\_\prime \frac{I}{a} \right) \times \left( bI\_\prime I\_\prime \frac{I}{b} \right) = \left( abI\_\prime I\_\prime \frac{I}{ab} \right) \,\prime$$

using the fact *<sup>I</sup>*<sup>2</sup> <sup>=</sup> *<sup>I</sup>*. Hence, (*N*, <sup>×</sup>) is a semigroup under product.

Considering *e* = (*I*, *I*, *I*) ∈ *N*, we see that *e* × *e* = *x* × *e* = *x* for all *x* ∈ *N*.

$$\mathbf{x} \times \mathbf{x} = (I, I, I) \times \left( aI, I, \frac{I}{a} \right) = \left( aI, I, \frac{I}{a} \right) = \mathbf{x} (\text{ using } I^2 = I).$$

Thus, (*I*, *I*, *I*) is the identity element of (*N*, ×). For every

$$\mathbf{x} = \left(aI\_\prime I\_\prime \frac{I}{a}\right)\prime$$

we have a unique

$$\mathbf{x}^{-1} = \left(\frac{I}{a}, I, a\right) \in N$$

is such that

$$\mathbf{x} \times \mathbf{x}^{-1} = \left( aI, I, \frac{I}{a} \right) = \left( I, I, I \right)$$

as *<sup>a</sup>* <sup>=</sup> 0 and *<sup>I</sup>*<sup>2</sup> <sup>=</sup> *<sup>I</sup>*.

Thus, {*N*, ×} is a commutative group of infinite order.

It is interesting to note both the sets M and N are not even closed under addition. Next, let

$$P = \left\{ a + bI, 1, \frac{1}{a} - \frac{bI}{a(a+b)}; a \neq b; a+b \neq 0, a \neq 0. \right\}$$

We get

$$a + bI \times \frac{1}{a} - \frac{bI}{a(a+b)} = 1.$$

We call these neutrosophic triplets as special neutrosophic triplets contributed by the unity 1 of the ring which is the trivial idempotent of *S*; however, where it is mandatory, *x* and *anti*(*x*) are nontrivial neutrosophic numbers with *neut*(*x*) = 1.

**Theorem 5.** *Let S* = *Q* ∪ *I*, +, × *(or R* ∪ *I*, +, ×*) be the neutrosophic ring. Let*

$$P = \left\{ (a+bI, 1, \frac{1}{a} - \frac{bI}{a(a+b)}; a \neq b, \text{ where } a, b \in Q \backslash \{0\} \backslash (or \ R \backslash 0) \text{ and } a+b \neq 0 \right\}$$

*be the collection of special neutrosophic triplets with 1 as the neutral. P is a torsion free abelian group of infinite order with* (1, 1, 1) *as its identity under component wise product.*

**Proof.** It is easily verified *P* is closed under the component wise product and (1, 1, 1) acts as the identity for component wise product. For every

$$\infty = \left( a - bI, 1, \frac{1}{a} + \frac{bI}{a(a-b)} \right) \in P\_{\prime \prime}$$

we have a unique

$$y = \left(\frac{1}{a} + \frac{bI}{a(a-b)}, 1, a-bI\right) \in P$$

such that *<sup>x</sup>* <sup>×</sup> *<sup>y</sup>* = (1, 1, 1). We also see *<sup>x</sup><sup>n</sup>* = (1, 1, 1) for any *<sup>x</sup>* <sup>∈</sup> *<sup>P</sup>* and *<sup>n</sup>* <sup>=</sup> <sup>0</sup>(*<sup>n</sup>* <sup>&</sup>gt; <sup>0</sup>); *<sup>x</sup>* = (1, 1, 1), hence *P* is a torsion free abelian group.

#### **4. Discussion and Conclusions**

We show that, in the case of neutrosophic duplets in *Z* ∪ *I*, *Q* ∪ *I* or *R* ∪ *I*, the collection of duplets {*a* − *aI*} forms a neutrosophic subring. However, in the case of neutrosophic triplets, we show that *Z* ∪ *I* has no nontrivial triplets and we have shown there are three distinct collection of neutrosophic triplets in *R* ∪ *I* and *Q* ∪ *I*. We have proved there are only three types of neutrosophic triplets in these neutrosophic rings and all three of them form abelian groups that are torsion free under component wise product. For future research, we would apply these neutrosophic triplets to concepts akin to SVNS and obtain some mathematical models.

**Author Contributions:** Conceptualization, V.K.W.B. and F.S.; writing—original draft preparation, V.K.W.B. and I.K.; writing—review and editing, I.K.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors would like to thank the reviewers for their reading of the manuscript and many insightful comments and suggestions.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

1. Smarandache, F. *A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics*; American Research Press: Rehoboth, DE, USA, 2005; ISBN 978-1-59973-080-6.


c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Inspection Plan Based on the Process Capability Index Using the Neutrosophic Statistical Method**

#### **Muhammad Aslam \* and Mohammed Albassam**

Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah 21551, Saudi Arabia **\*** Correspondence: aslam\_ravian@hotmail.com or magmuhammad@kau.edu.sa; Tel.: +966-59-3329841

Received: 25 June 2019; Accepted: 11 July 2019; Published: 16 July 2019

**Abstract:** The Process Capability Index (PCI) has been widely used in industry to advance the quality of a product. Neutrosophic statistics is the more generalized form of classical statistics and is applied when the data from the production process or a product lot is incomplete, incredible, and indeterminate. In this paper, we will originally propose a variable sampling plan for the PCI using neutrosophic statistics. The neutrosophic operating function will be given. The neutrosophic plan parameters will be determined using the neutrosophic optimization solution. A comparison between plans based on neutrosophic statistics and classical statistics is given. The application of the proposed neutrosophic sampling plan will be given using company data.

**Keywords:** acceptance number; neutrosophic approach; operating characteristics; risks; sample size

#### **1. Introduction**

Acceptance sampling is the most widely used tool for the inspection of the raw material, semi-finished product, and finished product. But, the presence of the indeterminacy in the observations or parameters may affect the performance of the sampling plan. A well-designed sampling plan used for the inspection of the product under the uncertainty and determinacy environment is needed at each stage to check that the finished product meets either the customer's upper specification limit (USL) and lower specification limit (LSL) before sending it to market. The quality of interest beyond the LSL and USL creates a non-conforming item. At the time of inspection, a random sample is taken and lot sentencing is made on the basis of this primary information about the lot. Thus, the sample information may mislead the experimenters in making the decision about the submitted product lot. There is a chance of rejecting a good lot and accepting a bad lot on the basis of the sample information. Thus, the sampling schemes are developed with the aim of reducing the cost of the inspection, non-conforming items, and minimizes the risk of the sampling. The acceptance sampling plan has two major types, known as attribute sampling plans and variable sampling plans. Attribute sampling plans are easier to apply but are more costly than the variable sampling plans. On the other hand, the variable sampling plans are more informative than attribute sampling plans [1]. A number of authors designed variable and attribute sampling plans: Jun et al. [2] studied variable sampling plans for sudden death testing; Balamurali and Jun [3] studied skip-lot sampling for the normal distribution; Fallah Nezhad et al. [4] designed a sampling plan using cumulative sums of conforming run-lengths; Pepelyshev et al. [5] applied a variable sampling plan in photovoltaic modules; Gui and Aslam [6] designed a time truncated plan for weighted exponential distribution; and Balamurali et al. [7] designed a mixed variable sampling plan.

The Process Capability Index (PCI) has been widely used in industry for quality improvement purposes and to make a relation between specification limits and process quality. Kane [8] originally proposed the PCI for classical statistics. Boyles [9] provided the bounds on the process yield for the normally distributed process. Kotz and Johnson [10] provided a detailed review of PCIs. More

details on PCIs can be seen in [11]. Pearn et al. [12] discussed an effective decision method for product inspection; Montgomery [1] mentioned the applications of PCIs. Boyles [13] studied PCIs for an asymmetric tolerances case and Ebadi [14] studied a simple linear profile using PCIs. Due to the importance of the PCIs in industry, several authors focused on the development of inspection schemes using classical statistics based on PCIs for various situations including for example, and Chen et al. [15] studied PCIs for entire product inspection. Pearn et al. [12] presented an effective decision method for the inspection. Aslam et al. [16] designed various sampling plans using PCIs. Seifi and Nezhad [17] studied resubmitted sampling using PCI and Arif et al. [18] worked on a sampling plan using PCI for multiple manufacturing lines.

Fuzzy sampling plans have been widely used in the industry when the proportion of the non-conforming product is a fuzzy number [19]. Kanagawa and Ohta [20] introduced an attribute plan using fuzzy sets. Sadeghpour Gildeh et al. [19] designed a single sampling plan using fuzzy parameters. Kahraman et al. [21] designed single and double sampling plans using fuzzy approach. The PCIs using fuzzy logic can be seen in [22–24].

Smarandache [25] defined the neutrosophic logic in 1998 as the generalization of fuzzy logic. Smarandache [26] gave the idea of descriptive neutrosophic statistics. The neutrosophic statistics is the more generalized form of classical statistics and applied when the data from the production process or a product lot is incomplete, incredible, and indeterminate [26]. Chen et al. [27,28] studied the rock joint roughness coefficient using neutrosophic statistics. According to [29] "All observations and measurements of continuous variables are not precise numbers but more or less non-precise. This imprecision is different from variability and errors. Therefore also lifetime data are not precise numbers but more or less fuzzy. The best up-to-date mathematical model for this imprecision is so-called non-precise numbers".

Recently, Aslam [30] introduced the neutrosophic statistics in the area of the acceptance sampling plan. Aslam [30] proposed an acceptance sampling plan using the neutrosophic process loss function. The sampling plan for multiple manufacturing lines using the neutrosophic statistics is proposed by [31]. The sampling plan for the exponential distribution under the uncertainty is proposed by [32]. Some more details about the sampling plan using the neutrosophic plans can be seen in [33–37].

The existing sampling plans using PCIs cannot apply when the data is indeterminate or incomplete. Also, the available sampling plans using the neutrosophic statistics do not consider the PCIs for the inspection of the product. By exploring the literature and best of the author knows there is no work on the sampling plan for PCIs using the neutrosophic statistics. In this paper, we will originally propose a variable sampling plan for the PCIs using the neutrosophic statistics. The neutrosophic operating function will be given. The neutrosophic plan parameters will be determined using the neutrosophic optimization solution. A comparison between plans based on neutrosophic statistics and classical statistics is given. We expect that the proposed plan will be more effective to be applied in an uncertain environment. The application of the proposed sampling plan using neutrosophic statistics will be given using the company data.

#### **2. Design of a Neutrosophic Plan Based on PCI**

Let *nN* {*nL*, *nU*} be a random sample selected from the population having some uncertain observations, where *nL* and *nU* are the lower and upper sample size of the indeterminacy interval, respectively. Suppose that a neutrosophic quality of interest, *XNi*, is expressed in the indeterminacy interval, say, *XNi* {*XL*, *XU*}; *i* =1,2,3, ... , *nN* having indeterminate observations follow the neutrosophic normal distribution, where *XL* and *XU* are the lower and the upper values, respectively, with the neutrosophic population mean μ*<sup>N</sup>* μ*L*, μ*<sup>U</sup>* and neutrosophic population standard deviation (NSD) σ*<sup>N</sup>* {σ*L*, σ*U*} (see [26]). The neutrosophic process capability index process (NPCI), say, *c*ˆ*Npk* , is defined as:

$$\mathbb{C}\_{N\_{pk}} = \mathrm{Min}\{\frac{\mathrm{LSL} - \mu\_N}{3\sigma\_N}, \frac{\mu\_N - \mathrm{LSL}}{3\sigma\_N}\}; \,\mu\_N \mathrm{e}\{\mu\_{L,\prime}\mu\_{\mathrm{II}}\}, \,\sigma\_N \mathrm{e}\{\sigma\_{L,\prime}\sigma\_{\mathrm{II}}\} \tag{1}$$

where *USL* and *LSL* are the upper specification limit and lower specification limit, respectively.

Note that *CNpk* reduces to PCI for classical statistics when no indeterminate observations are recorded in *XN*. Usually, μ*<sup>N</sup>* μ*L*, μ*<sup>U</sup>* and σ*<sup>N</sup>* {σ*L*, σ*U*} are unknown in practice and the best linear unbiased estimate (BLUE) of μ*<sup>N</sup>* μ*L*, μ*<sup>U</sup>* is the neutrosophic sample mean *XN XL*, *XU* and a BLUE of σ*<sup>N</sup>* {σ*L*, σ*U*} is the neutrosophic sample standard deviation *sN* {*sL*,*sU*} which can be used to evaluate *CNpk* . The *<sup>C</sup>*ˆ*Npk* based on sample estimate is given as by:

$$\mathbf{C}\_{N\_{pk}} = \text{Min}\{\frac{\text{LSL} - \overline{\mathbf{X}}\_N}{\text{3s}\_N}, \frac{\overline{\mathbf{X}}\_N - \text{LSL}}{\text{3s}\_N}\}; \ \overline{\mathbf{X}}\_N \mathbf{e} \{\overline{\mathbf{X}}\_L, \overline{\mathbf{X}}\_{\text{ll}}\}\_{\text{\textdegree}} s\_N = \{s\_{\text{L}}, s\_{\text{ll}}\} \tag{2}$$

where *XL* = *<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *xL <sup>i</sup>* /*nL*,

$$\overline{X}\_{\text{II}} = \sum\_{i=1}^{n} \mathbf{x}\_{i}^{\text{II}} / n\_{\text{II}} \text{ s}\_{\text{L}} = \sqrt{\sum\_{i=1}^{n} \left( \mathbf{x}\_{i}^{\text{L}} - \overline{X}\_{\text{L}} \right)^{2} / n\_{\text{L}}} \text{ and } \mathbf{s}\_{\text{II}} = \sqrt{\sum\_{i=1}^{n} \left( \mathbf{x}\_{i}^{\text{II}} - \overline{X}\_{\text{L}} \right)^{2} / n\_{\text{L}}}.$$

To design the proposed sampling plan, it is assumed that there is uncertainty in the selection of a random sample from the submitted product lot. Thus, a random sample will be selected from a neutrosophic interval. The proposed sampling plan is stated as follows:

**Step 1:** Select a random sample of size *nN* {*nL*, *nU*} from the product lot. Compute the statistic *<sup>C</sup>*ˆ*Npk Min USL*−*XN* <sup>3</sup>*sN* , *XN*−*LSL* 3*sN* ; *XN XL*, *XU* , *sN* {*sL*,*sU*}.

**Step 2:** Accept a product lot of *<sup>C</sup>*ˆ*Npk* <sup>≥</sup> *kN*; *kN* {*kaL*, *kaU*}, otherwise reject a product lot, where *kN* {*kaL*, *kaU*} is the neutrosophic acceptance number. An acceptance number is also called the action number/boundary number. A product lot is rejected if the statistic *<sup>C</sup>*ˆ*Npk* is smaller than *kN*, otherwise, the product lot is accepted.

The evaluation of the proposed sampling plan will be used on two parameters, namely *nN* = {*nL*, *nU*} and *kN* {*kaL*, *kaU*}. The neutrosophic operating characteristic (NOC) for the proposed plan is derived as follows:

$$\begin{array}{l} L(p) = P\{\mathbf{C}\_{N\_{pk}} \ge k\_{N}\} = P\{LSL + 3k\_{N}\mathbf{s}\_{N} \le \overline{\mathbf{X}}\_{N} \le \text{LSL} - 3k\_{N}\mathbf{s}\_{N}\} = P\{\overline{\mathbf{X}}\_{N} + 3k\_{N}\mathbf{s}\_{N} \le \text{LSL}\} \\ \quad - P\{\overline{\mathbf{X}}\_{N} - 3k\_{N}\mathbf{s}\_{N} \le \text{LSL}\}; \; n\_{N}\mathfrak{s}\mathfrak{c}\{n\_{L}, n\_{\mathrm{II}}\}, \; \overline{\mathbf{X}}\_{N}\mathfrak{c}\{\overline{\mathbf{X}}\_{L}, \overline{\mathbf{X}}\_{\mathrm{II}}\} \mathfrak{s}\_{N}\mathfrak{c}\{\mathbf{s}\_{L}, \mathbf{s}\_{\mathrm{II}}\} \text{ and } k\_{N}\mathfrak{c}\{k\_{a\overline{L},} k\_{a\overline{L}}\}. \end{array} \tag{3}$$

Duncan [38] suggested *XN* ± *kNsN*; *XN XL*, *XU* and *sN* = {*sL*,*sU*} is distributed as an approximately neutrosophic normal distribution, that is *XN* ± *kNsN* ∼ *NN* μ*<sup>N</sup>* ± *c*σ*N*, σ2 *N nN* <sup>+</sup> *<sup>c</sup>*2σ<sup>2</sup> *N* 2*nN* . where *NN*(.) shows neutrosophic normal distribution.

Suppose that quality of interest *XN* beyond the USL or LSL is labeled as the defective item and this probability is defined as *pU* = *P XN* <sup>&</sup>gt; *USL* μ*N* and *pU* = *P XNLSL*|μ*<sup>N</sup>* ; μ*<sup>N</sup>* = μ*L*, μ*<sup>U</sup>* . Thus, the probability of acceptance is given by the following [39]:

$$L(p) = \Phi\left(\frac{LSL - \mu\_N - 3k\_N\sigma\_N}{\left(\sigma\_N/m\_N\right)\sqrt{1 + 9k\_N^2/2}}\right) - \Phi\left(\frac{LSL - \mu\_N + 3k\_N\sigma\_N}{\left(\sigma\_N/m\_N\right)\sqrt{1 + 9k\_N^2/2}}\right) \tag{4}$$

Let us define the neutrosophic standard normal random variable as:

$$Z\_{N\_{pll}} = \frac{USL - \mu\_N}{\sigma\_N} \text{ and } -Z\_{N\_{pl}} = \frac{LSL - \mu\_N}{\sigma\_N} \tag{5}$$

Now, the final form of NFOC is given by:

$$L(p) = \Phi\left\{ \left( \left( Z\_{Np\perp} - 3k\_N \right) \sqrt{\frac{n\_N}{1 + \left( 9k\_N^2/2 \right)}} \right) \right\} - \Phi\left\{ \left( -\left( Z\_{Np\perp} - 3k\_N \right) \sqrt{\frac{n\_N}{1 + \left( 9k\_N^2/2 \right)}} \right) \right\} \tag{6}$$

where Φ(.) is the neutrosophic cumulative standard normal distribution.

#### *Research Methodology*

To meet the given producer's risk, say, α, and the custumer's risk, say, β, the plan parameters of the proposed sampling plan will be determined in such a way that NFOC passes through the two points (*p*1, 1 − α) and (*p*2, β), where *p*<sup>1</sup> is the acceptable quality limit (AQL) and *p*<sup>2</sup> is the limiting quality limit (LQL). The plan parameters of the proposed sampling plans will be determined through the following non-linear solution under the neutrosophic statistical interval method:

Minimize:

$$m\_N \varepsilon \{ n\_{L, \prime} n\_{\ell I} \} \tag{7}$$

subject to:

$$L\_{N}(p\_{1}) = \Phi\left\{ \left( \left( Z\_{Np\_{\mathrm{LI}}} - 3k\_{\mathrm{N}} \right) \sqrt{\frac{n\_{\mathrm{I}}}{1 + \left( 9k\_{\mathrm{N}}^{2}/2 \right)}} \right) \right\} - \Phi\left\{ \left( - \left( Z\_{Np\_{\mathrm{LI}}} - 3k\_{\mathrm{N}} \right) \sqrt{\frac{n\_{\mathrm{I}}}{1 + \left( 9k\_{\mathrm{N}}^{2}/2 \right)}} \right) \right\} \geq 1-\tag{8}$$
  $n$ ;  $k\_{\mathrm{N}} \in [k\_{\mathrm{all}}, k\_{\mathrm{d}} \sqcup]$ ;  $m \in \{ n\_{\mathrm{L}}, n\_{\mathrm{L}} \}$ 

and:

$$L\_{N}(p\_{2}) = \Phi\left\{ \left( \left( Z\_{Np\_{12}} - 3k\_{N} \right) \sqrt{\frac{n\_{N}}{1 + \left( 9k\_{N}^{2}/2 \right)}} \right) \right\} - \Phi\left\{ \left( -\left( Z\_{Np\_{12}} - 3k\_{N} \right) \sqrt{\frac{n\_{N}}{1 + \left( 9k\_{N}^{2}/2 \right)}} \right) \right\} \leq \beta; \tag{9}$$
  $\Phi \in \{k\_{\mathrm{all}}, k\_{\mathrm{all}}\};$   $m\_{N} \in \{n\_{L}, n\_{L}\}$ 

The plan parameters of the proposed plan are determined through Equations (7)–(9) using the search grid method for the various combinations of AQL and LQL. Several combinations of plan parameters in the indeterminacy interval satisfy Equations (7)–(9). The plan parameters having the smallest range in indeterminacy interval are chosen and placed in Table 1. To save the space, we present Table 1 when α = 0.05 and β = 0.10. Similar tables for other values of α and β can be prepared. The neutrosophic lot acceptance probabilities, *LN*(*p*1) and *LN*(*p*2) at the consumer's risk and producer's risk are also reported in Table 1.

From Table 1, we note that, for the fixed values of all other parameters, the values of *kN* {*kaL*, *kaU*}; *nN* {*nL*, *nU*} decrease as LQL increases. This means the indeterminacy in the sample size and acceptance number reduces. For example, under the uncertainty, when AQL = 0.001 and LQL = 0.02, the sample size will be in the interval [18,20]. This means the industrial engineers should select a sample size between 18 and 20. Furthermore, for the smaller values of AQL and LQL, larger the values of *nN* {*nL*, *nU*} are required. Note here that the appropriate sample size is decided on the basis of pre-defined parameters, such as AQL, LQL, α, and β. The following algorithm is used to determine the neutrosophic plan parameters:



**Table 1.** The plan parameters of the plan when α = **0.05**, β = **0.10**.

#### **3. Comparison Study**

In this section, we will compare the efficiency of the proposed plan with the sampling plan using classical statistics in terms of the sample size required for the inspection of the submitted product lot. For a fair comparison, we will consider the same values of all the specified parameters. The sample size *nN* along with range (R = *nU* − *nL*) in the indeterminacy interval of the proposed plan and sample size *n* using classical statistics when α = 0.05, β = 0.10 are placed in Table 2. From Table 2, it can be noted that the proposed plan provides a smaller indeterminacy interval in the sample size as compared to the plan using classical statistics. For example, when AQL = 0.001 and LQL = 0.002, the proposed plan has *nN* ∈ [602, 643] while the existing plan has *n* = 1134. Therefore, the proposed plan needs a smaller sample size and range in the indeterminacy interval for the inspection of a product lot. From this comparison, it is quite clear that the proposed plan using neutrosophic statistics is more efficient than the existing sampling plan under classical statistics in terms of sample size. In addition, the proposed plan is quite suitable, effective, and informative to be used in uncertainty than the existing plan.




**Table 2.** *Cont.*

#### **4. Application of the Proposed Plan**

In this section, we will give the application of the proposed plan using the data of the amplified pressure sensor that came from industry. Viertl [29] commented that the observations obtained from the measurements are not usually precise. According to [40] "For this amplified pressure sensor process, the span is the focused characteristic". As the observations for the quality of interest are measured, some observations in the data may be indeterminate or imprecise. Under the uncertainty, the experimenter is not sure about the sample size for the inspection of a product lot when some indeterminate or imprecise observations are recorded. For this data, LSL = 1.9 V, USL 2.1. Suppose that AQL = 0.001, LQL = 0.04,α = 0.05, and β = 0.10. The neutrosophic plan parameters from Table 1 are *nN* {128, 133}. Thus, the experimenter should select a random sample between 128 and 133. Suppose that the industrial engineers decided to select a random sample size of 128 for the inspection of a product lot. The amplified pressure sensor data of *n* = 128 having some indeterminate observations are reported in Table 3. Based on the given data, the neutrosophic average and standard deviation (SD) are computed as follows:

$$\begin{aligned} \left[1.9422, 1.9422\right] + \left[1.9651, 1.9651\right] + \left[2.0230, 2.0230\right] + \dots + \left[1.9994, 1.9994\right], \\ \left[\text{X}\_{N} = \frac{\left[1.9422, 1.9422\right] + \left[1.9651, 1.9651\right] + \left[2.0435, 2.0435\right] + \dots + \left[2.0512, 2.0512\right]}{128} \right] = \left[1.9805, 1.9827\right] \end{aligned}$$

and, similarly, *sN* = {0.0193, 0.0225}.

The NPCI is computed as follows: *<sup>C</sup>*ˆ*Npk* <sup>=</sup> *Min USL*−*XN* <sup>3</sup>*sN* , *XN*−*LSL* 3*sN* , *<sup>C</sup>*ˆ*Npk* [1.7377, 2.0639] for *XN* = [1.9805, 1.9827] and *sN* = {0.0193, 0.0225}.

The proposed plan will be implemented as follows:


The application of the proposed sampling plan shows that the proposed sampling plan is quite effective, adequate, and flexible to be used under the uncertainty environment than the plan based on classical statistics which provide the determined values of the plan parameters.



#### **5. Concluding Remarks**

In this paper, we originally proposed a variable sampling plan for the PCI under the neutrosophic logic. We presented the NPCI in the paper and used it to design the sampling plan. The proposed plan is the extension of the plan using classical statistics which can be applied where data is indeterminate or unclear. The plan parameters are presented for practical use in industry. A real example from industry is also added to show the application of the proposed sampling plan. The proposed plan is designed under the assumption that the data follow the neutrosophic normal distribution which can be tested using some statistical test or graphical depictions. For non-normal data, a suitable transformation can be applied to transfer non-normal data to normal data. From the comparison study, it is concluded that the proposed plan is more efficient than the plan based on classical statistics in terms of sample size. It is recommended to use the proposed plan in the industry where the data came from the complex situation or where there is a chance of some unclear data in the sampling. The proposed sampling plan using a double sampling scheme will be considered as a future research. The proposed plan using big data can be considered as future research.

**Author Contributions:** Conceived and designed the experiments: M.A. (Muhammad Aslam), M.A. (Mohammed Albassam); performed the experiments: M.A. (Muhammad Aslam); analyzed the data: M.A. (Muhammad Aslam); contributed reagents/materials/analysis tools: M.A. (Muhammad Aslam); wrote the paper: M.A. (Muhammad Aslam).

**Funding:** This article was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.

**Acknowledgments:** The authors are deeply thankful to the editor and reviewers for their valuable suggestions to improve the quality of this manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest regarding this paper.

#### **References**


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