② *z* = 2 , *LNNEWA*(*u*1, *u*2) = 2 ⊕*e i*=1 *iui* = ψ *k*2(*k*∗ (*k*+β1) 1−(*k*−β1) 1 (*k*+β1) 1 +(*k*−β1) 1 +*k*∗ (*k*+β2) 2−(*k*−β2) 2 (*k*+β2) 2 +(*k*−β2) 2 ) *k*2+((*k*∗ (*k*+β1) 1−(*k*−β1) 1 (*k*+β1) 1 +(*k*−β1) 1 )(*k*∗ (*k*+β2) 2−(*k*−β2) 2 (*k*+β2) 2 +(*k*−β2) 2 ) ,ψ *k*(*k*∗ 2γ1 1 (2*k*−γ1) 1 +γ1 1 )(*k*∗ 2γ2 2 (2*k*−γ2) 2 +γ2 2 ) *k*2+(*k*−(*k*∗ 2γ1 1 (2*k*−γ1) 1 +γ1 1 ))(*k*−(*k*∗ 2γ2 2 (2*k*−γ2) 2 +γ2 2 )) ,ψ *k*(*k*∗ 2δ1 1 (2*k*−δ1) 1 +δ1 1 )(*k*∗ 2δ2 2 (2*k*−δ2) 2 +δ2 2 ) *k*2+(*k*−(*k*∗ 2δ1 1 (2*k*−δ1) 1 +δ1 1 ))(*k*−(*k*∗ 2δ2 2 (2*k*−δ2) 2 +δ2 2 )) = ψ *k*∗ (*k*+β1) 1 (*k*+β2) 2−(*k*−β1) 1 (*k*−β2) 2 (*k*+β1) 1 (*k*+β2) 2 +(*k*−β1) 1 (*k*−β2) 2 ,ψ *k*∗ 2γ1 1 γ2 2 (2*k*−γ1) 1 (2*k*−γ2) 2 +γ1 1 γ2 2 ,ψ *k*∗ 2δ1 1 δ2 2 (2*k*−δ1) 1 (2*k*−δ1) 2 +δ1 1 δ2 2 = ψ *k*∗ \$2 *i*=1 (*k*+β*i*) *i*− \$2 *i*=1 (*k*−β*i*) *i* \$2 *i*=1 (*k*+β*i*) *i* +\$2 *i*=1 (*k*−β*i*) *i* ,ψ *k*∗ 2 \$2 *i*=1 γ*i i* \$2 *i*=1 (2*k*−γ*i*) *i* +\$2 *i*=1 γ*i i* ,ψ *k*∗ 2 \$2 *i*=1 δ*i i* \$2 *i*=1 (2*k*−δ*i*) *i* +\$2 *i*=1 δ*i 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, *u*2, ... *um*, *um*+1)=( *m* ⊕*e i*=1 *iui*) ⊕*e m*+1*um*+1 = ψ *k*∗ \$*m i*=1 (*k*+β*i*) *i*− \$*m i*=1 (*k*−β*i*) *i* \$*m i*=1 (*k*+β*i*) *i* +\$*m i*=1 (*k*−β*i*) *i* ψ *k*∗ 2 \$*m i*=1 γ*i i* \$*m i*=1 (2*k*−γ*i*) *i* +\$*m i*=1 γ*i i* , ψ *k*∗ 2 \$*m i*=1 δ*i i* \$*m i*=1 (2*k*−δ*i*) *i* +\$*m i*=1 δ*i i* ⊕*e* ψ *k*∗ (*k*+β*m*+1) *m*+1−(*k*−β*m*+1) *m*+1 (*k*+β*m*+1) *m*+1 +(*k*−β*m*+1) *m*+1 , ψ *k*∗ 2γ*m*+1 *m*+1 (2*k*−γ*m*+1) *m*+1 +γ*m*+1 *m*+1 , ψ *k*∗ 2δ*m*+1 *m*+1 (2*k*−δ*m*+1) *m*+1 +δ*m*+1 *m*+1 = ψ*k*2((*k*∗ \$*m i*=1 (*k*+β*i*) *i*− \$*m i*=1 (*k*−β*i*) *i* \$*k i*=1 (*k*+β*i*) *i* +\$*k i*=1 (*k*−β*i*) *i* )+(*k*∗ (*k*+β*m*+1) *m*+1−(*k*−β*m*+1) *m*+1 (*k*+β*m*+1) *m*+1 +(*k*−β*m*+1) *m*+1 )) *k*2+(*k*∗ \$*m i*=1 (*k*+β*i*) *i*− \$*m i*=1 (*k*−β*i*) *i* \$*m i*=1 (*k*+β*i*) *i* +\$*m i*=1 (*k*−β*i*) *i* )(*k*∗ (*k*+β*m*+1) *m*+1−(*k*−β*m*+1) *m*+1 (*k*+β*m*+1) *m*+1 +(*k*−β*m*+1) *m*+1 ) , ψ *k*(*k*∗ 2 \$*m i*=1 γ*i i* \$*m i*=1 (2*k*−γ*i*) *i* +\$*m i*=1 γ*i i* )(*k*∗ 2γ*m*+1 *m*+1 (2*k*−γ*m*+1) *m*+1 +γ*m*+1 *m*+1 ) *k*2 + (*k* − (*k* ∗ 2 \$*m i*=1 γ*i i* \$*m i*=1 (2*k*−γ*i*) *i*+\$*m i*=1 γ*i i* ))(*k* − (*k* ∗ 2γ*m*+1 *m*+1 (2*k*−γ*m*+1) *m*+1+γ*m*+1 *m*+1 )) ψ *k*(*k*∗ 2 \$*m i*=1 δ*i i* \$*m i*=1 (2*k*−δ*i*) *i* +\$*m i*=1 δ*i i* )(*k*∗ 2δ*m*+1 *m*+1 (2*k*−δ*m*+1) *m*+1 +δ*m*+1 *m*+1 ) *k*2+(*k*−(*k*∗ 2 \$*m i*=1 δ*i i* \$*m i*=1 (2*k*−δ*i*) *i* +\$*m i*=1 δ*i i* ))(*k*−(*k*∗ 2δ*m*+1 *m*+1 (2*k*−δ*m*+1) *m*+1 +δ*m*+1 *m*+1 )) , = ψ *k*∗ \$*m*+1 *i*=1 (*k*+β*i*) *i*− \$*m*+1 *i*=1 (*k*−β*i*) *i* \$*m*+1 *i*=1 (*k*+β*i*) *i* +\$*m*+1 *i*=1 (*k*−β*i*) *i* ,ψ *k*∗ 2 \$*m*+1 *i*=1 γ*i i* \$*m*+1 *i*=1 (2*k*−γ*i*) *i* +\$*m*+1 *i*=1 γ*i i* ,ψ *k*∗ 2 \$*m*+1 *i*=1 δ*i i* \$*m*+1 *i*=1 (2*k*−δ*i*) *i* +\$*m*+1 *i*=1 δ*i 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*i* = β; γ*i* = γ; δ*i* = δ = *(i* = *1, 2,* ...,z), the following result can be found:

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

**Theorem 4.** *(Monotonicity) set two collections of LNNs ui* = ψβ*i* ,ψγ*i* ,ψδ*i and ui* = ψβ*i* ,ψγ*i* ,ψδ*i (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, *u*2, ... *uz*) = *z* ⊕*e i*=1 *iui* and *LNNEWA*(*u*1 , *u*2 , ... *uz* ) = *z* ⊕*e i*=1 *iui* , then we can get: *LNNEWA*(*u*1, *u*2, ... *uz*) ≤ *LNNEWA*(*u*1 , *u*2 , ... *uz* ). -

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

*u*− ≤ *LNNEWA*(*u*1, *u*2, ... *uz*) ≤ *u*+.

**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*i* ,ψγ*i* ,ψδ*i 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, ... , *z*) *T , z i*=1 *i* = 1 *and i* ∈ [0, 1]*.* **Theorem 6.** *Set a collection ui* = ψβ*i* ,ψγ*i* ,ψδ*i 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, ... , *z*) *T, z i*=1 *i* = 1 *and i* ∈ [0, 1]*.*

**Theorem 7.** *(Idempotency) Set a collection ui* = ψβ*i* ,ψγ*i* ,ψδ*i 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*i* ,ψγ*i* ,ψδ*i and ui* = ψβ*i* ,ψγ*i* ,ψδ*i (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*i* ,ψγ*i* ,ψδ*i in* Ψ*, u*− = *min*(ψβ*i* ), *max*(ψγ*i* ), *max*(ψδ*i* ) *and u*+ = *max*(ψβ*i* ), *min*(ψγ*i* ), *min*(ψδ*i* ), *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, ... , n) T with n i=1 i = 1 *and* i ∈ [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*) *T* with 1 ≥ μ*j* ≥ 0 and *t j*=1 μ*j* = 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 θ(*y*) *ij* for alternative Θ*i* with the attribute E*j*, the value θ(*y*) *ij* can be written as θ(*y*) *ij* = ψ*y* β*ij* ,ψ*y* γ*ij* ,ψ*y* δ*ij* (*y* = 1, 2, ... , *t* ; *i* = 1, 2, ... , *m*; *j* = 1, 2, ... , *n*),ψ*y* β*ij* ,ψ*y* γ*ij* ,ψ*y* δ*ij* ∈ Ψ. 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*i* (*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 *D* = *D*1,*D*2,*D*3 are invited to evaluate these companies, their weight vector is μ = (0.37, 0.33, 0.3) *T*. The experts make evaluations of the alternatives according to three attributes *E* = {*E*1, *E*2, *E*3}, *E*1 is the ability of risk, *E*2 is the ability of growth, and *E*3 is the ability of environmental impact, the weight vector of them is = (0.35, 0.25, 0.4) *T*. Then, the experts use LNNs to make the evaluation values with a linguistic set Ψ = {ψ0 = extremely poor, ψ1 = very poor, ψ2 = poor, ψ3 = slightly poor, ψ4 = medium , ψ5 = slightlygood, ψ6 = good, ψ7 = very good, ψ8 = 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:

θ1 = ψ6.0661,ψ1.7313,ψ2.3644, θ2 = ψ6.0961,ψ1.7929,ψ1.9840, θ3 = ψ5.7523,ψ1.7260,ψ2.2199, and θ4 = 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 Θ4 Θ2 Θ1 Θ3, thus, company Θ4 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*i* (*i* = 1, 2, ... , *m*) can be obtained by the *LNNEWG* operator, which are as following:

θ1 = ψ5.9491,ψ1.7507,ψ2.4660, θ2 = ψ6.5864,ψ1.8026,ψ2.0000, θ3 = ψ6.8354,ψ1.8390,ψ2.2614, and θ4 = 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 Θ4 Θ2 Θ3 Θ1, thus, company Θ4 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 θ4 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 2**

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* 2 = −1 or the complex number *i* is included, while, in the neutrosophic rings, the indeterminate element *I* where *I*2 = *I* is included. The neutrosophic ring *R* ∪ *I* is also a ring generated by *R* and *I* 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 *R* generated by *α*2 − *α*, that is *α* ∈/ *R*(*α*2 − *α*)*R* or *R* = *R*(*α*2 − *α*)*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*2 = *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* ∪ *I* = {*a* + *bI*/*a*, *b* ∈ *Zn*, 2 ≤ *n* < ∞; *I*2 = *I*} 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 S* = *Z*2 ∪ *I* = {*a* + *bI*/*a*, *b* ∈ *Z*2, *I*2 = *I*} *be the neutrosophic ring of modulo integers. Clearly, I*2 = *I and* (1 + *I*)2 = 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,* 1 + *I is a semi-idempotent as the ideal generated by* 1 + *I is* (1 + *I*)2 − (1 + *I*) = 2*I is such that* 1 + *I* ∈/ *R. However, it is important to note that* (1 + *I*) ∈ *R is a unit as* (1 + *I*)2 = 1 + 2*I* + *I* = 1*, 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.* 2 + 2*I* ∈ *R is a unit, now* (2 + 2*I*)2 = 4 + 8*I* + 4*I*2 = 1*, thus* 2 + 2*I is a unit. However,* (2 + 2*I*)2 − (2 + 2*I*) = 1 + 1 + *I* = 2 + *I*.

*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*)2 − 2*I* = *I* + *I* = 2*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 *x* = *a* + *bI* ∈ *S* with *b* = 0 are such that *x*2 − *x* generates the ideal, which is *S*, 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 *z* = *a* + *bI* ∈ *S* with *a* + *b* = 0(*mod p*); then, *z*2 − *z* generates an ideal *M* of *S* 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 *x*2 − *x* = *m*, then the ideal generated by *x*2 − *x* is *S*, thus *x* is a semi-idempotent. If *x*2 − *x* = *nI*, then the ideal generated by *nI* is *Zp I*, thus *x* ∈/ *Zp I*, hence again *x* is a semi-idempotent. If *x*2 − *x* = *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*15 ∪ *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 *S*. Now, *r* + *tI* ∈ *A* and (*r* + *tI*)2 − (*r* + *tI*) = *mI* as*r*2 = *r*, 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*30 ∪ *I*, +, ×}*, be the neutrosophic ring. The idempotents of Z*30 *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*1 = {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*2 = {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*2 = {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*3 = {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*3 = {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*4 = {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*4 = {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*5 = {*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*5 = {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*6 = {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*6 = {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*7 = {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*7 = {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* ∪ *I and Zm* ∪ *I are two neutrosophic rings where n* > *m and n* = *pt q (t* > 2*, and p and q two distinct primes) and m* = *p*1 *p*2 ... *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 2**

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: *I* denotes the indeterminate and it is such that *I* × *I* = *I* = *I*2. *I* is called as the neutrosophic value. *Z*, *Q* and *R* denote the ring of integers, field of rationals and field of reals, respectively. *Z* ∪ *I* = {*a* + *bI*|*a*, *b* ∈ *Z*, *I*2 = *I*} is the neutrosophic ring of integers, *Q* ∪ *I* = {*a* + *bI*|*a*, *b* ∈ *Q*, *I*2 = *I*} is the neutrosophic ring of rationals and *R* ∪ *I* = {*a* + *bI*|*a*, *b* ∈ *R*, *I*2 = *I*} 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.  *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 *x* = *aI* or *x* = *a* + *bI* ∈ *S*(*a* = 0, *b* = 0). Now, *x* × *x* = *aI* × *aI* = *a*2 *I* (as *I*2 = *I*); if *x* is to be an idempotent, we must have *aI* = *a*2 *I*; that is, (*a* − *a*2)*I* = 0(*I* = 0), thus *a*2 = *a*. However, in *Q* or *R*,

*a*2 = *a* implies *a* = 0 or *a* = 1; as *a* = 0, we have *a* = 1; thus, *x* = *I* and *x* 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, *y*2 = *a* + *bI* × *a* − *bI* = *a*2 + *abI* + *abI* + *b*2 *I* = *a* + *bI*, equating the real and neutrosophic parts.

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

*b*(2*a* + *b* − 1) = 0; *b* = 0, thus 2*a* + *b* − 1 = 0; further, *a* = 0 as *a* = 0 will reduce to the case *I*2 = *I*, 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* = 1 *a* . 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 *I*2 = *I*. Hence, (*N*, ×) 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 *a* = 0 and *I*2 = *I*.

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 *x* × *y* = (1, 1, 1). We also see *xn* = (1, 1, 1) for any *x* ∈ *P* and *n* = 0(*n* > 0); *x* = (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<EMAIL_ADDRESS>or<EMAIL_ADDRESS>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*N* μ*L*, μ*U* and neutrosophic population standard deviation (NSD) σ*N* {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*N* μ*L*, μ*U* and σ*N* {σ*L*, σ*U*} are unknown in practice and the best linear unbiased estimate (BLUE) of μ*N* μ*L*, μ*U* is the neutrosophic sample mean *XN XL*, *XU* and a BLUE of σ*N* {σ*L*, σ*U*} is the neutrosophic sample standard deviation *sN* {*sL*,*sU*} which can be used to evaluate *CNpk* . The *C*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* = *n i*=1 *xL i* /*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 *C*ˆ*Npk Min USL*−*XN* 3*sN* , *XN*−*LSL* 3*sN* ; *XN XL*, *XU* , *sN* {*sL*,*sU*}.

**Step 2:** Accept a product lot of *C*ˆ*Npk* ≥ *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 *C*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*N* ± *c*σ*N*, σ2 *N nN* + *c*2σ2 *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* > *USL* μ*N* and *pU* = *P XNLSL*|μ*N* ; μ*N* = μ*L*, μ*U* . 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*1 is the acceptable quality limit (AQL) and *p*2 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: *C*ˆ*Npk* = *Min USL*−*XN* 3*sN* , *XN*−*LSL* 3*sN* , *C*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**

© 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*


IntechOpen Book Series Artificial Intelligence

### Volume 12

### Aims and Scope of the Series

Artificial Intelligence (AI) is a rapidly developing multidisciplinary research area that aims to solve increasingly complex problems. In today's highly integrated world, AI promises to become a robust and powerful means for obtaining solutions to previously unsolvable problems. This Series is intended for researchers and students alike interested in this fascinating field and its many applications.

## Meet the Series Editor

Andries Engelbrecht received the Masters and Ph.D. degrees in Computer Science from the University of Stellenbosch, South Africa, in 1994 and 1999 respectively. He is currently appointed as the Voigt Chair in Data Science in the Department of Industrial Engineering, with a joint appointment as Professor in the Computer Science Division, Stellenbosch University. Prior to his appointment at Stellenbosch University, he has been at the University of

Pretoria, Department of Computer Science (1998-2018), where he was appointed as South Africa Research Chair in Artifical Intelligence (2007-2018), the head of the Department of Computer Science (2008-2017), and Director of the Institute for Big Data and Data Science (2017-2018). In addition to a number of research articles, he has written two books, Computational Intelligence: An Introduction and Fundamentals of Computational Swarm Intelligence.

## Meet the Topic Editors

Dr. Marco Antonio Aceves Fernandez obtained his B.Sc. (Eng.) in Telematics from the Universidad de Colima, Mexico. He obtained both his M.Sc. and Ph.D. from the University of Liverpool, England, in the field of Intelligent Systems. He is a full professor at the Universidad Autonoma de Queretaro, Mexico, and a member of the National System of Researchers (SNI) since 2009. Dr. Aceves Fernandez has published more than 80 research papers as well as

a number of book chapters and congress papers. He has contributed to more than 20 funded research projects, both academic and industrial, in the area of artificial intelligence, ranging from environmental, biomedical, automotive, aviation, consumer, and robotics applications. He is also an Honorary President of the National Association of Embedded Systems (AMESE), a senior member of the IEEE, and a board member of many institutions. His research interests include intelligent and embedded systems.

Carlos M. Travieso-Gonzalez received his M.Sc. degree in Telecommunication Engineering at the Polytechnic University of Catalonia (UPC), Spain in 1997, and his Ph.D. degree in 2002 at the University of Las Palmas de Gran Canaria (ULPGC-Spain). He is a full professor of signal processing and pattern recognition and a Head of the Signals and Communications Department at ULPGC, teaching from 2001 on subjects on signal processing and learning

theory. His research lines are biometrics, biomedical signals and images, data mining, classification system, signal and image processing, machine learning, and environmental intelligence. He has participated in 52 international and Spanish research projects, some of them as a lead researcher. He is co-author of four books, co-editor of 27 proceedings books, guest editor for eight JCR-ISI international journals, and author of up to 24 book chapters. He has over 450 papers published in international journals and conferences (81 of them indexed in JCR – ISI - Web of Science) and has published seven patents in the Spanish Patent and Trademark Office. Dr. Travieso-Gonzalez has been a supervisor on eight Ph.D. theses (11 more under supervision), and 130 master theses. He is the founder of the IEEE IWOBI Conference Series and the President of its Steering Committee, as well as the founder of both the InnoEducaTIC and APPIS Conference Series. He is an evaluator of the project proposals for the European Union (H2020), Medical Research Council (MRC, UK), Spanish Government (ANECA, Spain), Research National Agency (ANR, France), DAAD (Germany), Argentinian Government, and the Colombian Institutions. He won the "Catedra Telefonica" Awards in Modality of Knowledge Transfer, 2017, 2018, and 2019 editions, and Award in Modality of COVID Research in 2020.

### Contents


**ANIMAL DOMESTICATION**

Edited by **Fabrice Teletchea**

#### **Animal Domestication**

http://dx.doi.org/10.5772/67910 Edited by Fabrice Teletchea

#### **Contributors**

Fabrice Teletchea, Pascal Fontaine, Alain Pasquet, Joao L Saraiva, Maria F. Castanheira, Pablo Arechavala-López, Jenny Volstorf, Billo Heinzpeter Studer, Thomas Lecocq, Mirte Bosse

#### **© The Editor(s) and the Author(s) 2019**

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#### **Notice**

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First published in London, United Kingdom, 2019 by IntechOpen IntechOpen is the global imprint of INTECHOPEN LIMITED, registered in England and Wales, registration number: 11086078, The Shard, 25th floor, 32 London Bridge Street London, SE19SG – United Kingdom Printed in Croatia

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*4.6. Total Cellular DNA Extraction and PCR Analyses*

Total cellular DNA was isolated from leaf tissues of antibiotic-resistant sugarcane clones as well as from nontransformed plants. PCR analysis was carried out to validate transgene integration. *Taq* DNA polymerase (Fermentas, Hanover, MD, USA) and platinum *Taq* polymerase (Fermentas, Hanover, MD, USA) were used with 100–300 ng of genomic DNA as a template. The *gfp* gene was amplified with primers P5 (5- -CCA TGG CTA GTA AAG GAG AA-3- ) and P6 (5- -TTA TTT GTA TAG TTC ATC CA-3- ). The left border fragment was amplified with primers P1 (5- -GAT ATC AAA ACC CGT CCT CAG TTC GGA TTG C-3- ) and P8 (5- -GGG CTG ATA CTG GGC CGG CAG G-3- ).


*4.1.3 Compressed air microwave distillation (CAMD)*

This method (**Figure 4**) uses the principle of steam entrainment, with compressed air instead of steam, to extract the essential oil. The extraction process consists of a compressor, a microwave oven and a refrigeration system. Compressed air is injected into the reactor, where the matrix is heated by microwaves and immersed in water. The steam, saturated with volatile molecules, is directed to a recovery container located outside the microwave oven and cooled by a refrigeration system. In just a few minutes, the water and aromatic molecules are condensed and recovered [30, 32].

A similar method using a condenser to cool the extraction gas (temperatures ranging from −20 to −15°C) has also been patented [33]. This extraction method is environmentally friendly, as no organic solvents or artificial chemical compounds are added.

**Figure 3.** *Experimental set-up for microwave solvent-assisted extraction.*

#### **Figure 4.**

*A schematic representation compressed air microwave distillation.*

### *4.1.4 Microwave hydrodistillation (MWHD)*

The MWHD (**Figure 5**) was developed by Stashenko et al., in 2004. It is based on the classic hydrodistillation principle. The process consists of a hydrodistillation unit placed inside a domestic microwave oven with a side port, through which an external glass condenser is connected to the round filter containing the matrix and water [34].

Microwave hydrodistillation is a widely used technique for extracting essential oils from various aromatic plants and spices, with examples such as *Thymus vulgaris* L., *Zataria multiflora* Boiss., and *Satureja montana*.

**Figure 5.** *Microwave hydrodistillation.*

### *Essential Oil Extraction Process DOI: http://dx.doi.org/10.5772/intechopen.113311*

An improved version of this technique was developed in 2007, which involves introducing a microwave coaxial antenna insulated inside a glass flask containing the matrix and water [35]. This in situ microwave heating offers advantages in terms of time and energy savings and can be useful for industrial applications.

Microwave Steam Distillation (MSD) (**Figure 6**) is another innovative technique that was developed. It is based on the conventional steam distillation principle and has been successfully used for the extraction of essential oil from Lavender flowers [30].

### *4.1.5 Solvent-free microwave extraction (SFME)*

This is one of the most recent techniques for the microwave-assisted extraction of essential oils, without solvents and using water at atmospheric pressure. The SFME process consists mainly of four parts: a reactor where the matrix to be treated is placed, a microwave oven, a cooling system, and an essential oil container where the oil is collected (**Figure 7**).

The process is based on a relatively simple principle, described as microwaveassisted dry distillation; the fresh matrix is placed in a microwave reactor without the addition of water or organic solvent. Heating the raw material with water breaks down the glands containing the essential oil. This phase releases the essential oil, which is then carried away by the steam produced by the water in the matrix. A cooling system located outside the microwave oven enables continuous condensation of the distillate, composed of water and essential oil, and the return of excess water

**Figure 6.** *Microwave steam distillation (MSD).*

**Figure 7.** *Improved solvent-free microwave extraction (improved SFME).*

to the bottle, thus maintaining the appropriate moisture content of the matrix. For example, Milestone's "DryDist" laboratory microwave oven makes it easy and efficient to extract high-quality essential oils.

Wang et al. in 2006 proposed an improved SFME extraction method. The method is based on the addition and mixing of carbonylated iron powder with the dry matrix. Spherical particles of carbonylated iron are capable of absorbing part of the energy emitted by microwaves and returning it to the medium in the form of heat. In this way, the matrix can be heated by simple conduction without any auxiliary energy. Various types of materials such as activated carbon, graphite powders, and ionic liquid (1-hexyl-3-methylimidazolium hexafluorophosphate) can absorb microwave radiation [36].


**2. Adsorption phenomenon**

Adsorption is a surface phenomenon with common mechanism for organic and inorganic pollutants removal. When a solution containing absorbable solute comes into contact with a solid with a highly porous surface structure, liquid–solid intermolecular forces of attraction cause some of the solute molecules from the solution to be concentrated or deposited at the solid surface. The solute retained (on the solid surface) in adsorption processes is called ad‐ sorbate, whereas, the solid on which it is retained is called as an adsorbent. This surface ac‐ cumulation of adsorbate on adsorbent is called adsorption. This creation of an adsorbed phase having a composition different from that of the bulk fluid phase forms the basis of separation by adsorption technology.

In a bulk material, all the bonding requirements (be they ionic, covalent, or metallic) of the constituent atoms of the material are filled by other atoms in the material. However, atoms on the surface of the adsorbent are not wholly surrounded by other adsorbent atoms and therefore can attract adsorbates. The exact nature of the bonding depends on the details of the species involved, but the adsorption process is generally classified as physicsorption (characteristic of weak Van Der Waals forces) or chemisorption (characteristic of covalent bonding). It may also occur due to electrostatic attraction.

As the adsorption progress, an equilibrium of adsorption of the solute between the solution and adsorbent is attained (where the adsorption of solute is from the bulk onto the adsorb‐ ent is minimum). The adsorption amount (qe, mmol g−1) of the molecules at the equilibrium step was determined according to the following equation:

$$\text{Age} = \text{V(Co-Ce)} \,\text{/M} \tag{1}$$

where V is the solution volume (L); M is the mass of monolithic adsorbents (g); and Co and Ce are the initial and equilibrium adsorbate concentrations, respectively.

Other definition of adsorption is a mass transfer process by which a substance is transferred from the liquid phase to the surface of a solid, and becomes bound by physical and/or chemical interactions. Large surface area leads to high adsorption capacity and surface reactivity [10].

#### **2.1. Adsorption isotherms and models**

Efficient techniques for the removal of highly toxic organic compounds from water have drawn significant interest. A number of methods such as coagulation, filtration with coagu‐ lation, precipitation, ozonation, adsorption, ion exchange, reverse osmosis and advanced ox‐ idation processes have been used for the removal of organic pollutants from polluted water and wastewater. These methods have been found to be limited, since they often involve high capital and operational costs. On the other hand ion exchange and reverse osmosis are more attractive processes because the pollutant values can be recovered along with their re‐ moval from the effluents. Reverse osmosis, ion exchange and advanced oxidation processes do not seem to be economically feasible because of their relatively high investment and op‐

Among the possible techniques for water treatments, the adsorption process by solid adsorb‐ ents shows potential as one of the most efficient methods for the treatment and removal of or‐ ganic contaminants in wastewater treatment. Adsorption has advantages over the other methods because of simple design and can involve low investment in term of both initial cost and land required. The adsorption process is widely used for treatment of industrial wastewa‐ ter from organic and inorganic pollutants and meet the great attention from the researchers. In recent years, the search for low-cost adsorbents that have pollutant –binding capacities has in‐ tensified. Materials locally available such as natural materials, agricultural wastes and industri‐ al wastes can be utilized as low-cost adsorbents. Activated carbon produced from these

Adsorption is a surface phenomenon with common mechanism for organic and inorganic pollutants removal. When a solution containing absorbable solute comes into contact with a solid with a highly porous surface structure, liquid–solid intermolecular forces of attraction cause some of the solute molecules from the solution to be concentrated or deposited at the solid surface. The solute retained (on the solid surface) in adsorption processes is called ad‐ sorbate, whereas, the solid on which it is retained is called as an adsorbent. This surface ac‐ cumulation of adsorbate on adsorbent is called adsorption. This creation of an adsorbed phase having a composition different from that of the bulk fluid phase forms the basis of

In a bulk material, all the bonding requirements (be they ionic, covalent, or metallic) of the constituent atoms of the material are filled by other atoms in the material. However, atoms on the surface of the adsorbent are not wholly surrounded by other adsorbent atoms and therefore can attract adsorbates. The exact nature of the bonding depends on the details of the species involved, but the adsorption process is generally classified as physicsorption (characteristic of weak Van Der Waals forces) or chemisorption (characteristic of covalent

As the adsorption progress, an equilibrium of adsorption of the solute between the solution and adsorbent is attained (where the adsorption of solute is from the bulk onto the adsorb‐

materials can be used as adsorbent for water and wastewater treatment [9].

erational cost.

**2. Adsorption phenomenon**

168 Organic Pollutants - Monitoring, Risk and Treatment

separation by adsorption technology.

bonding). It may also occur due to electrostatic attraction.

An adsorption isotherm is the presentation of the amount of solute adsorbed per unit weight of adsorbent as a function of the equilibrium concentration in the bulk solution at constant temperature. Langmuir and Freundlich adsorption isotherms are commonly used for the description of adsorption data.

The Langmuir equation is expressed as:

$$\text{Ce} \mid qe = \text{1} / b\text{X}m + \text{Ce} / \text{X}m,\tag{2}$$

Where *Ce* is the equilibrium concentration of solute (mmol L−1), *qe* is the amount of solute adsorbed per unit weight of adsorbent (mmol g−1 of clay), *Xm* is the adsorption capacity (mmol g−1), or monolayer capacity, and *b* is a constant (L mmol−1 ).

The Freundlich isotherm describes heterogeneous surface adsorption. The energy distribu‐ tion for adsorptive sites (in Freundlich isotherm) follows an exponential type function which is close to the real situation. The rate of adsorption/desorption varies with the strength of the energy at the adsorptive sites. The Freundlich equation is expressed as:

$$
\log qe = \log k + \text{ 1/ } n \log \text{Ce },\tag{3}
$$

Where *k* (mmol g−1) and 1*/n* are the constant characteristics of the system [11].


*2.2. In Vivo Analysis*

#### 2.2.1. Experimental Animals and Grouping

The protocol was approved by the Ethical Committee and the Laboratory Animal Center of China medical university, Shenyang, Liaoning, China. Eighteen 10-week-old specific pathogen-free (SPF) male mice from the Jackson Laboratory (Bar Harbor, ME, USA) were maintained at a constant temperature (22–25 ◦C) in a 12 h light/dark cycle and fed with a standard laboratory diet and water. Six non-diabetic BLKS/jdb/m male mice were selected as the control group (db/m, *n* = 6, weight: 18.4–10.7 g). Twelve spontaneous diabetic BLKS/jdb/db male mice were randomly separated into the diabetic group (db/db, *n* = 6, weight: 35.9–40.9 g) and the curcumin-treated diabetic group (db/db + C, *n* = 6, weight: 35.7–40.3 g). Each group was kept in the same cage. The curcumin-treated group was intragastrically given a dose of 200 mg/kg/d curcumin for 10 weeks, and the control group and diabetic group were given equivalent volumes of solution (normal saline + 0.1% DMSO) for 10 weeks. Curcumin was dissolved in DMSO, then diluted with normal saline to make the content of DMSO 0.1% and administered daily by gavage. All groups were sacrificed at 20 weeks, and mandibles (3 mm × 3 mm) were harvested and fixed in 4% paraformaldehyde solution for 7 days at 4 ◦C in the dark. The Block sections were sequentially dehydrated in ascending serial concentrations of ethanol, cleared with xylene twice, and embedded in poly(methyl methacrylate). Serial sections with a thickness of 70–80 μm were cut parallel to the coronal plane and prepared for fluorescence microscopy and histological staining. All analyses of in vivo experiments were made by an experienced pathologist (blind to the treatments of the mice) in order to characterize any changes.