# Different Forms of Triangular Neutrosophic Numbers, De-Neutrosophication Techniques, and their Applications

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## Abstract

**:**

## 1. Introduction

#### 1.1. Theory of Uncertainty and Uncertainty Quantification

- The information belongs to a certain interval
- There is no concept of membership function

- The concept of belongingness of the elements comes
- The use of membership function is present

- The concept of belongingness and non-belongingness of the elementscomes
- The use of membership and non-membership function is present

- The concept of truthiness, falsity, and indeterminacy of the elements comes
- The use of membership function for truthiness, falsity, and indeterminacy is present

#### 1.2. Neutrosophic Number

#### 1.3. Ranking and De-Impreciseness

#### 1.4. Structure of the Paper

## 2. Neutrosophic Number

**Definition**

**1.**

**Definition**

**2.**

- ${\pi}_{\tilde{{S}_{neu}}}\langle \rho {a}_{1}+(1-\rho ){a}_{2}\rangle \ge min\langle {\pi}_{\tilde{{S}_{neu}}}({a}_{1}),{\pi}_{\tilde{{S}_{neu}}}({a}_{2})\rangle $
- ${\mu}_{\tilde{{S}_{neu}}}\langle \rho {a}_{1}+(1-\rho ){a}_{2}\rangle \le max\langle {\mu}_{\tilde{{S}_{neu}}}({a}_{1}),{\mu}_{\tilde{{S}_{neu}}}({a}_{2})\rangle $
- ${\vartheta}_{\tilde{{S}_{neu}}}\langle \rho {a}_{1}+(1-\rho ){a}_{2}\rangle \le max\langle {\vartheta}_{\tilde{{S}_{neu}}}({a}_{1}),{\vartheta}_{\tilde{{S}_{neu}}}({a}_{2})\rangle $

**Definition**

**3.**

## 3. Single Valued Linear Neutrosophic Number

- Triangular Single Valued Neutrosophic number of Type 1: The quantity of the truth, indeterminacy and falsity are not dependent: A Triangular Single Valued Neutrosophic number of Type 1 is defined as ${\tilde{\mathrm{A}}}_{\mathrm{Neu}}=({\mathrm{p}}_{1},{\mathrm{p}}_{2},{\mathrm{p}}_{3};{\mathrm{q}}_{1},{\mathrm{q}}_{2},{\mathrm{q}}_{3};{\mathrm{r}}_{1},{\mathrm{r}}_{2},{\mathrm{r}}_{3})$ whose truth membership, indeterminacy and falsity membership is defined as follows:$${T}_{{\tilde{A}}_{Neu}}(x)=\{\begin{array}{c}\begin{array}{cc}\frac{x-{p}_{1}}{{p}_{2}-{p}_{1}}& \mathrm{when}\text{}{p}_{1}\le x{p}_{2}\end{array}\\ \begin{array}{cc}1& \mathrm{when}\text{}x={p}_{2}\end{array}\\ \begin{array}{c}\begin{array}{cc}\frac{{p}_{3}-x}{{p}_{3}-{p}_{2}}& \mathrm{when}\text{}{p}_{2}x\le {p}_{3}\end{array}\\ \begin{array}{cc}0& \mathrm{otherwise}\end{array}\end{array}\end{array}$$$${T}_{{\tilde{A}}_{Neu}}(x)=\{\begin{array}{c}\begin{array}{cc}\frac{x-{p}_{1}}{{p}_{2}-{p}_{1}}& \mathrm{when}\text{}{p}_{1}\le x{p}_{2}\end{array}\\ \begin{array}{cc}1& \mathrm{when}\text{}x={p}_{2}\end{array}\\ \begin{array}{c}\begin{array}{cc}\frac{{p}_{3}-x}{{p}_{3}-{p}_{2}}& \mathrm{when}\text{}{p}_{2}x\le {p}_{3}\end{array}\\ \begin{array}{cc}0& \mathrm{otherwise}\end{array}\end{array}\end{array}$$$${I}_{{\tilde{A}}_{Neu}}(x)=\{\begin{array}{c}\begin{array}{cc}\frac{{q}_{2}-x}{{q}_{2}-{q}_{1}}& \mathrm{when}\text{}{q}_{1}\le x{q}_{2}\end{array}\\ \begin{array}{cc}0& \mathrm{when}\text{}x={q}_{2}\end{array}\\ \begin{array}{c}\begin{array}{cc}\frac{x-{q}_{2}}{{q}_{3}-{q}_{2}}& \mathrm{when}\text{}{q}_{2}x\le {q}_{3}\end{array}\\ \begin{array}{cc}1& \mathrm{otherwise}\end{array}\end{array}\end{array}$$$${T}_{{\tilde{A}}_{Neu}}(x)=\{\begin{array}{c}\begin{array}{cc}\frac{x-{p}_{1}}{{p}_{2}-{p}_{1}}& \mathrm{when}\text{}{p}_{1}\le x{p}_{2}\end{array}\\ \begin{array}{cc}1& \mathrm{when}\text{}x={p}_{2}\end{array}\\ \begin{array}{c}\begin{array}{cc}\frac{{p}_{3}-x}{{p}_{3}-{p}_{2}}& \mathrm{when}\text{}{p}_{2}x\le {p}_{3}\end{array}\\ \begin{array}{cc}0& \mathrm{otherwise}\end{array}\end{array}\end{array}$$

**Example**

**1.**

- 2.
- Triangular Single Valued Neutrosophic Number of Type 2: The quantity of indeterminacy and falsity are dependent: A triangular single valued neutrosophic number (TrSVNN) of Type 2 is defined as ${\tilde{A}}_{Neu}=({p}_{1},{p}_{2},{p}_{3};{q}_{1},{q}_{2},{q}_{3};{u}_{Neu},{y}_{Neu})$ whose truth membership, indeterminacy, and falsity membership are defined as follows:$${I}_{{\tilde{A}}_{Neu}}(x)=\{\begin{array}{c}\begin{array}{cc}\frac{{q}_{2}-x+{u}_{Neu}(x-{q}_{1})}{{q}_{2}-{q}_{1}}& \mathrm{when}\text{}{q}_{1}\le x{q}_{2}\end{array}\\ \begin{array}{cc}{u}_{Neu}& \mathrm{when}\text{}x={q}_{2}\end{array}\\ \begin{array}{c}\begin{array}{cc}\frac{x-{q}_{2}+{u}_{Neu}({q}_{3}-x)}{{q}_{3}-{q}_{2}}& \mathrm{when}\text{}{q}_{2}x\le {q}_{3}\end{array}\\ \begin{array}{cc}1& \mathrm{otherwise}\end{array}\end{array}\end{array}$$$${F}_{{\tilde{A}}_{Neu}}(x)=\{\begin{array}{c}\begin{array}{cc}\frac{{q}_{2}-x+{y}_{Neu}(x-{q}_{1})}{{q}_{2}-{q}_{1}}& \mathrm{when}\text{}{q}_{1}\le x{q}_{2}\end{array}\\ \begin{array}{cc}{y}_{Neu}& \mathrm{when}\text{}x={q}_{2}\end{array}\\ \begin{array}{c}\begin{array}{cc}\frac{x-{q}_{2}+{y}_{Neu}({q}_{3}-x)}{{q}_{3}-{q}_{2}}& \mathrm{when}\text{}{q}_{2}x\le {q}_{3}\end{array}\\ \begin{array}{cc}1& \mathrm{otherwise}\end{array}\end{array}\end{array}$$

**Example**

**2.**

- 3.
- Triangular Single Valued Neutrosophic number of Type 3: The quantity of the truth, indeterminacy, and falsity are dependent: A TrSVNN of Type 3 is defined as ${\tilde{A}}_{Neu}=({p}_{1},{p}_{2},{p}_{3};{w}_{Ne},{u}_{Neu},{y}_{Neu})$, whose truth membership, indeterminacy, and falsity membership are defined as follows:$${T}_{{\tilde{A}}_{Neu}}(x)=\{\begin{array}{c}{w}_{Neu}\frac{x-{p}_{1}}{{p}_{2}-{p}_{1}}\text{}\mathrm{when}\text{}{p}_{1}\le x{p}_{2}\\ {w}_{Neu}\text{}\mathrm{when}\text{}x={p}_{2}\\ \begin{array}{c}{w}_{Neu}\frac{{p}_{3}-x}{{p}_{3}-{p}_{2}}\\ 0\text{}\mathrm{otherwise}\end{array}\text{}\mathrm{when}\text{}{p}_{2}x\le {p}_{3}\end{array}$$$${I}_{{\tilde{A}}_{Neu}}(x)=\{\begin{array}{c}\frac{{p}_{2}-x+{u}_{Neu}(x-{p}_{1})}{{p}_{2}-{p}_{1}}\text{}\mathrm{when}\text{}{p}_{1}\le x{p}_{2}\\ {u}_{Neu}\text{}\mathrm{when}\text{}x={p}_{2}\\ \begin{array}{c}\frac{x-{p}_{2}+{u}_{Neu}({p}_{3}-x)}{{p}_{3}-{p}_{2}}\\ 1\text{}\mathrm{otherwise}\end{array}\text{}\mathrm{when}\text{}{p}_{2}x\le {p}_{3}\end{array}$$$${I}_{{\tilde{A}}_{Neu}}(x)=\{\begin{array}{c}\frac{{p}_{2}-x+{u}_{Neu}(x-{p}_{1})}{{p}_{2}-{p}_{1}}\mathrm{when}\text{}{p}_{1}\le x{p}_{2}\\ {u}_{Neu}\text{}\mathrm{when}\text{}x={p}_{2}\\ \begin{array}{c}\frac{x-{p}_{2}+{u}_{Neu}({p}_{3}-x)}{{p}_{3}-{p}_{2}}\\ 1\text{}\mathrm{otherwise}\end{array}\mathrm{when}\text{}{p}_{2}x\le {p}_{3}\end{array}$$

**Example**

**3.**

_{l}to upper limit U

_{l}.

- Addition$$\begin{array}{l}{\tilde{C}}_{Neu}={\tilde{A}}_{Neu}+{\tilde{B}}_{Neu}\\ =\langle \begin{array}{c}\left\{min({a}_{1}+{a}_{4},{U}_{l}),min({a}_{2}+{a}_{5},{U}_{l}),min({a}_{3}+raphical\text{}representation\text{}of\text{}type\text{}3\text{}TrSVNNs{a}_{6},{U}_{l})\right\};\\ \left\{min({b}_{1}+{b}_{4},{U}_{l}),min({b}_{2}+{b}_{5},{U}_{l}),min({b}_{3}+{b}_{6},{U}_{l})\right\};\left\{min({c}_{1}+{c}_{4},{U}_{l}),min({c}_{2}+{c}_{5},{U}_{l}),min({c}_{3}+{c}_{6},{U}_{l})\right\}\end{array}\rangle \end{array}$$
- Negative of SVNNs$$\begin{array}{l}{\tilde{S}}_{Neu}=-{\tilde{A}}_{Neu}\\ =\langle -{a}_{3},-{a}_{2},-{a}_{1};-{b}_{3},-{b}_{2},-{b}_{1};-{c}_{3},-{c}_{2},-{c}_{1}\rangle \end{array}$$
- Subtraction$$\begin{array}{l}{\tilde{D}}_{Neu}={\tilde{A}}_{Neu}-{\tilde{B}}_{Neu}\\ ={\tilde{A}}_{Neu}+(-{\tilde{B}}_{Neu})\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}}=\langle \begin{array}{c}\left\{max({a}_{1}-{a}_{6},{L}_{l}),max({a}_{2}-{a}_{5},{L}_{l}),max({a}_{3}-{a}_{4},{L}_{l})\right\};\\ \left\{max({b}_{1}-{b}_{6},{L}_{l}),max({b}_{2}-{b}_{5},{L}_{l}),max({b}_{3}-{b}_{4},{L}_{l})\right\};\\ \left\{max({c}_{1}-{c}_{6},{L}_{l}),max({c}_{2}-{c}_{5},{L}_{l}),max({c}_{3}-{c}_{4},{L}_{l})\right\}\end{array}\rangle \end{array}$$
- Multiplications$$\begin{array}{l}{\tilde{D}}_{Neu}={\tilde{A}}_{Neu}-{\tilde{B}}_{Neu}\\ ={\tilde{A}}_{Neu}+(-{\tilde{B}}_{Neu})\\ =\langle \begin{array}{c}\left\{max({a}_{1}-{a}_{6},{L}_{l}),max({a}_{2}-{a}_{5},{L}_{l}),max({a}_{3}-{a}_{4},{L}_{l})\right\};\\ \left\{max({b}_{1}-{b}_{6},{L}_{l}),max({b}_{2}-{b}_{5},{L}_{l}),max({b}_{3}-{b}_{4},{L}_{l})\right\};\\ \left\{max({c}_{1}-{c}_{6},{L}_{l}),max({c}_{2}-{c}_{5},{L}_{l}),max({c}_{3}-{c}_{4},{L}_{l})\right\}\end{array}\rangle \end{array}$$
- Multiplication by a constant$$\begin{array}{l}{\tilde{E}}_{Neu}=k\left[{\tilde{A}}_{Neu}\right]\\ =k\times \langle {a}_{1},{a}_{2},{a}_{3};{b}_{1},{b}_{2},{b}_{3};{c}_{1},{c}_{2},{c}_{3}\rangle \\ =\langle k{a}_{1},k{a}_{2},k{a}_{3};k{b}_{1},k{b}_{2},k{b}_{3};k{c}_{1},k{c}_{2},k{c}_{3}\rangle \end{array}$$
- Inverse of SVNNs$$\begin{array}{l}{\tilde{F}}_{Neu}={\tilde{A}}_{Neu}^{-1}=\frac{1}{\langle {a}_{1},{a}_{2},{a}_{3};{b}_{1},{b}_{2},{b}_{3};{c}_{1},{c}_{2},{c}_{3}\rangle}\\ =\langle \frac{1}{{a}_{3}},\frac{1}{{a}_{2}},\frac{1}{{a}_{1}};\frac{1}{{b}_{3}},\frac{1}{{b}_{2}},\frac{1}{{b}_{1}};\frac{1}{{c}_{3}},\frac{1}{{c}_{2}},\frac{1}{{c}_{1}}\rangle \text{}for\text{}(a,b,c)0\\ =\langle \frac{1}{{a}_{1}},\frac{1}{{a}_{2}},\frac{1}{{a}_{3}};\frac{1}{{b}_{1}},\frac{1}{{b}_{2}},\frac{1}{{b}_{3}};\frac{1}{{c}_{1}},\frac{1}{{c}_{2}},\frac{1}{{c}_{3}}\rangle \text{}for\text{}(a,b,c)0\end{array}$$
- Divisions$$\begin{array}{l}{\tilde{G}}_{Neu}{\tilde{A}}_{Neu}\xf7{\tilde{B}}_{Neu}\\ ={\tilde{A}}_{Neu}\xf7{\tilde{B}}_{Neu}\\ =\langle {a}_{1},{a}_{2},{a}_{3};{b}_{1},{b}_{2},{b}_{3};{c}_{1},{c}_{2},{c}_{3}\rangle \times \langle \frac{1}{{a}_{6}},\frac{1}{{a}_{5}},\frac{1}{{a}_{4}};\frac{1}{{b}_{6}},\frac{1}{{b}_{5}},\frac{1}{{b}_{4}};\frac{1}{{c}_{6}},\frac{1}{{c}_{5}},\frac{1}{{c}_{4}}\rangle \\ =\langle \begin{array}{c}\left\{\begin{array}{c}min(\frac{{a}_{1}}{{a}_{4}},\frac{{a}_{1}}{{a}_{5}},\frac{{a}_{1}}{{a}_{6}},\frac{{a}_{2}}{{a}_{4}},\frac{{a}_{2}}{{a}_{5}},\frac{{a}_{2}}{{a}_{6}}=vision\text{}of\text{}SVNNs,\frac{{a}_{3}}{{a}_{4}},\frac{{a}_{3}}{{a}_{5}},\frac{{a}_{3}}{{a}_{6}}),\\ mean(\frac{{a}_{1}}{{a}_{4}},\frac{{a}_{1}}{{a}_{5}},\frac{{a}_{1}}{{a}_{6}},\frac{{a}_{2}}{{a}_{4}},\frac{{a}_{2}}{{a}_{5}},\frac{{a}_{2}}{{a}_{6}}=vision\text{}of\text{}SVNNs,\frac{{a}_{3}}{{a}_{4}},\frac{{a}_{3}}{{a}_{5}},\frac{{a}_{3}}{{a}_{6}}),\\ \mathrm{max}(\frac{{a}_{1}}{{a}_{4}},\frac{{a}_{1}}{{a}_{5}},\frac{{a}_{1}}{{a}_{6}},\frac{{a}_{2}}{{a}_{4}},\frac{{a}_{2}}{{a}_{5}},\frac{{a}_{2}}{{a}_{6}}=vision\text{}of\text{}SVNNs,\frac{{a}_{3}}{{a}_{4}},\frac{{a}_{3}}{{a}_{5}},\frac{{a}_{3}}{{a}_{6}})\end{array}\right\};\\ \left\{\begin{array}{c}min(\frac{{b}_{1}}{{b}_{4}},\frac{{b}_{1}}{{b}_{5}},\frac{{b}_{1}}{{b}_{6}},\frac{{b}_{2}}{{b}_{4}},\frac{{b}_{2}}{{b}_{5}},\frac{{b}_{2}}{{b}_{6}},\frac{{b}_{3}}{{b}_{4}},\frac{{b}_{3}}{{b}_{5}},\frac{{b}_{3}}{{b}_{6}}),\\ mean(\frac{{b}_{1}}{{b}_{4}},\frac{{b}_{1}}{{b}_{5}},\frac{{b}_{1}}{{b}_{6}},\frac{{b}_{2}}{{b}_{4}},\frac{{b}_{2}}{{b}_{5}},\frac{{b}_{2}}{{b}_{6}},\frac{{b}_{3}}{{b}_{4}},\frac{{b}_{3}}{{b}_{5}},\frac{{b}_{3}}{{b}_{6}}),\\ \mathrm{max}(\frac{{b}_{1}}{{b}_{4}},\frac{{b}_{1}}{{b}_{5}},\frac{{b}_{1}}{{b}_{6}},\frac{{b}_{2}}{{b}_{4}},\frac{{b}_{2}}{{b}_{5}},\frac{{b}_{2}}{{b}_{6}},\frac{{b}_{3}}{{b}_{4}},\frac{{b}_{3}}{{b}_{5}},\frac{{b}_{3}}{{b}_{6}}),\end{array}\right\};\\ \left\{\begin{array}{c}min(\frac{{c}_{1}}{{c}_{4}},\frac{{c}_{1}}{{c}_{5}},\frac{{c}_{1}}{{c}_{6}},\frac{{c}_{2}}{{c}_{4}},\frac{{c}_{2}}{{c}_{5}},\frac{{c}_{2}}{{c}_{6}},\frac{{c}_{3}}{{c}_{4}},\frac{{c}_{3}}{{c}_{5}},\frac{{c}_{3}}{{c}_{6}}),\\ mean(\frac{{c}_{1}}{{c}_{4}},\frac{{c}_{1}}{{c}_{5}},\frac{{c}_{1}}{{c}_{6}},\frac{{c}_{2}}{{c}_{4}},\frac{{c}_{2}}{{c}_{5}},\frac{{c}_{2}}{{c}_{6}},\frac{{c}_{3}}{{c}_{4}},\frac{{c}_{3}}{{c}_{5}},\frac{{c}_{3}}{{c}_{6}}),\\ max(\frac{{c}_{1}}{{c}_{4}},\frac{{c}_{1}}{{c}_{5}},\frac{{c}_{1}}{{c}_{6}},\frac{{c}_{2}}{{c}_{4}},\frac{{c}_{2}}{{c}_{5}},\frac{{c}_{2}}{{c}_{6}},\frac{{c}_{3}}{{c}_{4}},\frac{{c}_{3}}{{c}_{5}},\frac{{c}_{3}}{{c}_{6}}),\end{array}\right\}\end{array}\rangle \end{array}$$

**Example**

**4.**

- Addition$${\tilde{A}}_{Neu}+{\tilde{B}}_{Neu}=\langle 9,16,23;5.5,11,16.5;11,19.25,25\rangle ,$$
- Subtraction$${\tilde{A}}_{Neu}-{\tilde{B}}_{Neu}=\langle 0,4,11;0,0,4.5;7.5,15.75,24\rangle $$
- Multiplication$${\tilde{A}}_{Neu}\times {\tilde{B}}_{Neu}=\langle 20,60,120;7.5,30,67.5;10,30.625,62.5\rangle $$
- Division$$\frac{{\tilde{A}}_{Neu}}{{\tilde{B}}_{Neu}}=\langle 0.625,1.806,3.75;0.278,1.0185,2.5;4,11.5,25\rangle ,$$
- Multiplication by a constant$$k{\tilde{B}}_{Neu}=\langle 12,18,24;9,18,27;3,5.25,7.5\rangle $$

## 4. Neutrosophic Non-Linear Number and Generalized Neutrosophic Number

#### 4.1. Single Valued Non-Linear Triangular Neutrosophic Number with Nine Components

**Note.**If ${a}_{1},{a}_{2},{b}_{1},{b}_{2},{c}_{1,}{c}_{2}=1$, then single valued non-linear triangular neutrosophic number with nine components will be converted into single valued linear triangular neutrosophic number with nine components.

#### 4.2. Single Valued Generalized Triangular Neutrosophic Number with Nine Components

#### 4.3. Single Valued Generalized Non-Linear Triangular Neutrosophic Number with Nine Components

**Note.**if ${a}_{1},{a}_{2},{b}_{1},{b}_{2},{c}_{1,}{c}_{2}=1$, then single valued generalized non-linear triangular neutrosophic number with nine components will be converted into single valued generalized linear triangular neutrosophic number with nine components.

## 5. De-Neutrosophication of Linear Neutrosophic Triangular Fuzzy Number

#### De-Neutrosophication Using Removal Area Method

**Example**

**5.**

## 6. PERT in Triangular Neutrosophic Environment and the Proposed Model

- Identify the specific activities and milestones.
- Determine the proper sequence of the activities.
- Construct a network diagram.
- Estimate the time required for each activity.
- Determine the critical path.
- Update the PERT chart as the project progresses.

**Note 2.**In Ref. [22], the authors introduced the concept of score and accuracy function to compute the crisp value of a trapezoidal neutrosophic number. In our proposed model, we choose all the three different times (optimistic, pessimistic, most likely) as triangular neutrosophic number.

**Solution.**Now, we solve the problem by the following steps, as shown in Table 6, Figure 9 and Figure 10.

**Step-1.**

## 7. Application of Triangular Neutrosophic Fuzzy Number in Assignment Problem Using De-Neutrosophic Value

- (i)
- if $S(\stackrel{\u02c7}{{A}_{1}})>S(\stackrel{\u02c7}{{A}_{2}})$, then $\stackrel{\u02c7}{{A}_{1}}>\stackrel{\u02c7}{{A}_{2}}$
- (ii)
- if $S(\stackrel{\u02c7}{{A}_{1}})=S(\stackrel{\u02c7}{{A}_{2}})$ and $H(\stackrel{\u02c7}{{A}_{1}})>H(\stackrel{\u02c7}{{A}_{2}})$, then $\stackrel{\u02c7}{{A}_{1}}>\stackrel{\u02c7}{{A}_{2}}$

**Note**: Since, using de-neutrosophic value, we observe that min cost is 8.55 units of dollar, whereas using score function, we get min cost in negative quantity that is loss, hence de-neutrosophication gives us a better result than the score function.

## 8. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 7.**Pictorial representation of de-neutrosophication. (

**a**) Area of trapezium OABR; (

**b**) Area of trapezium OABR; (

**c**) Area of trapezium OEDR;(

**d**) Area of trapezium OEFR; (

**e**) Area of trapezium OHGR: (

**f**) Area of trapezium OHKR.

**Table 1.**Value of ${T}_{Ne1}(\alpha )$, ${T}_{Ne2}(\alpha )$, ${I}_{Ne1}(\beta )$, ${I}_{Ne1}(\beta )$, ${F}_{Ne1}(\gamma )$, and ${F}_{Ne2}(\gamma )$.

$\mathit{\alpha}\mathbf{,}\mathit{\beta}\mathbf{,}\mathit{\gamma}$ | ${\mathit{T}}_{\mathit{N}\mathit{e}\mathbf{1}}\mathbf{\left(}\mathit{\alpha}\mathbf{\right)}$ | ${\mathit{T}}_{\mathit{N}\mathit{e}\mathbf{2}}\mathbf{\left(}\mathit{\alpha}\mathbf{\right)}$ | ${\mathit{I}}_{\mathit{N}\mathit{e}\mathbf{1}}\mathbf{\left(}\mathit{\beta}\mathbf{\right)}$ | ${\mathit{I}}_{\mathit{N}\mathit{e}\mathbf{1}}\mathbf{\left(}\mathit{\beta}\mathbf{\right)}$ | ${\mathit{F}}_{\mathit{N}\mathit{e}\mathbf{1}}\mathbf{\left(}\mathit{\gamma}\mathbf{\right)}$ | ${\mathit{F}}_{\mathit{N}\mathit{e}\mathbf{2}}\mathbf{\left(}\mathit{\gamma}\mathbf{\right)}$ |
---|---|---|---|---|---|---|

0 | 10 | 20 | 16 | 16 | 15 | 15 |

0.1 | 10 | 19.5 | 15.8 | 16.6 | 14.7 | 15.4 |

0.2 | 11 | 19 | 15.6 | 17.2 | 14.4 | 15.8 |

0.3 | 11.5 | 18.5 | 15.4 | 17.8 | 14.1 | 16.2 |

0.4 | 12 | 18 | 15.2 | 18.4 | 13.8 | 16.6 |

0.5 | 12.5 | 17.5 | 15 | 19 | 13.5 | 17 |

0.6 | 13 | 17 | 14.8 | 19.6 | 13.2 | 17.4 |

0.7 | 13.5 | 16.5 | 14.6 | 20.2 | 12.9 | 17.8 |

0.8 | 14 | 16 | 14.4 | 20.8 | 12.6 | 18.2 |

0.9 | 14.5 | 15.5 | 14.2 | 21.4 | 12.3 | 18.6 |

1 | 15 | 15 | 14 | 22 | 12 | 19 |

**Table 2.**Value of ${T}_{Ne1}(\alpha )$, ${T}_{Ne2}(\alpha )$, ${I}_{Ne1}(\beta )$, ${I}_{Ne1}(\beta )$, ${F}_{Ne1}(\gamma )$, and ${F}_{Ne2}(\gamma )$.

$\mathit{\alpha}\mathbf{,}\mathit{\beta}\mathbf{,}\mathit{\gamma}$ | ${\mathit{T}}_{\mathit{N}\mathit{e}\mathbf{1}}\mathbf{\left(}\mathit{\alpha}\mathbf{\right)}$ | ${\mathit{T}}_{\mathit{N}\mathit{e}\mathbf{2}}\mathbf{\left(}\mathit{\alpha}\mathbf{\right)}$ | ${\mathit{I}}_{\mathit{N}\mathit{e}\mathbf{1}}\mathbf{\left(}\mathit{\beta}\mathbf{\right)}$ | ${\mathit{I}}_{\mathit{N}\mathit{e}\mathbf{1}}\mathbf{\left(}\mathit{\beta}\mathbf{\right)}$ | ${\mathit{F}}_{\mathit{N}\mathit{e}\mathbf{1}}\mathbf{\left(}\mathit{\gamma}\mathbf{\right)}$ | ${\mathit{F}}_{\mathit{N}\mathit{e}\mathbf{2}}\mathbf{\left(}\mathit{\gamma}\mathbf{\right)}$ |
---|---|---|---|---|---|---|

0 | 10 | 20 | – | – | – | – |

0.1 | 10.5 | 19.5 | – | – | – | – |

0.2 | 11 | 19 | – | – | – | – |

0.3 | 11.5 | 18.5 | – | – | – | – |

0.4 | 12 | 18 | 16 | 16 | – | – |

0.5 | 12.5 | 17.5 | 15.6667 | 17 | 16 | 16 |

0.6 | 13 | 17 | 15.3333 | 18 | 15.6 | 17.2 |

0.7 | 13.5 | 16.5 | 15. | 19 | 15.2 | 18.4 |

0.8 | 14 | 16 | 14.6667 | 20 | 14.8 | 19.6 |

0.9 | 14.5 | 15.5 | 14.3333 | 21 | 14.4 | 20.8 |

1 | 15 | 15 | 14 | 22 | 14 | 22 |

**Table 3.**Value of ${T}_{Ne1}(\alpha )$, ${T}_{Ne2}(\alpha )$, ${I}_{Ne1}(\beta )$, ${I}_{Ne1}(\beta )$, ${F}_{Ne1}(\gamma )$ and ${F}_{Ne2}(\gamma )$.

$\mathit{\alpha}\mathbf{,}\mathit{\beta}\mathbf{,}\mathit{\gamma}$ | ${\mathit{T}}_{\mathit{N}\mathit{e}\mathbf{1}}\mathbf{\left(}\mathit{\alpha}\mathbf{\right)}$ | ${\mathit{T}}_{\mathit{N}\mathit{e}\mathbf{2}}\mathbf{\left(}\mathit{\alpha}\mathbf{\right)}$ | ${\mathit{I}}_{\mathit{N}\mathit{e}\mathbf{1}}\mathbf{\left(}\mathit{\beta}\mathbf{\right)}$ | ${\mathit{I}}_{\mathit{N}\mathit{e}\mathbf{1}}\mathbf{\left(}\mathit{\beta}\mathbf{\right)}$ | ${\mathit{F}}_{\mathit{N}\mathit{e}\mathbf{1}}\mathbf{\left(}\mathit{\gamma}\mathbf{\right)}$ | ${\mathit{F}}_{\mathit{N}\mathit{e}\mathbf{2}}\mathbf{\left(}\mathit{\gamma}\mathbf{\right)}$ |
---|---|---|---|---|---|---|

0 | 14 | 22 | ||||

0.1 | 14.4 | 20.8 | ||||

0.2 | 14.8 | 19.6 | ||||

0.3 | 15.2 | 18.4 | ||||

0.4 | 15.6 | 17.2 | ||||

0.5 | 16 | 16 | ||||

0.6 | ||||||

0.7 | 16 | 16 | ||||

0.8 | 16 | 16 | 16.2857 | 15.1429 | ||

0.9 | 15.75 | 16.75 | 16.5714 | 14.2857 | ||

1 | 15.5 | 17.5 | 16.8571 | 13.4286 |

Experiment No. | Neutrosophic Number | De-Neutrosophication Value |
---|---|---|

Set 1 | $\stackrel{\u02c7}{\mathit{A}}=(\mathbf{1},\mathbf{2},\mathbf{3};\mathbf{0.5},\mathbf{1.5},\mathbf{2.5};\mathbf{1.2},\mathbf{2.7},\mathbf{3.5})$ | 2.0083 |

Set 2 | $\stackrel{\u02c7}{\mathit{B}}=(\mathbf{0.5},\mathbf{1.5},\mathbf{2.5};\mathbf{0.3},\mathbf{1.3},\mathbf{2.2};\mathbf{0.7},\mathbf{1.7},\mathbf{2.2})$ | 1.45 |

Set 3 | $\stackrel{\u02c7}{\mathit{C}}=(\mathbf{0.3},\mathbf{1.2},\mathbf{2.8};\mathbf{0.5},\mathbf{1.5},\mathbf{2.5};\mathbf{0.8},\mathbf{1.7},\mathbf{2.7})$ | 1.533 |

Set 4 | $\stackrel{\u02c7}{\mathit{D}}=(\mathbf{1},\mathbf{3},\mathbf{5};\mathbf{0.5},\mathbf{1.5},\mathbf{2.5};\mathbf{1.2},\mathbf{2.7},\mathbf{4.5})$ | 2.425 |

Description | Predecessors | Optimistic Time | Pessimistic Time | Most Likely Time | |
---|---|---|---|---|---|

A | Selection of Officer and Force Member | – | (1,2,3;0.5,1.5,2.5;1.2,2.7,3.5) | (1,5,9;1.5,4.5,6.5;4,7,10) | (1.5,3.5,5.5;1,2,3;3,4.5,6) |

B | Selection of Site and do Site Survey | – | $(\mathbf{1},\mathbf{5},\mathbf{8};\mathbf{1},\mathbf{3},\mathbf{6};\mathbf{4},\mathbf{7},\mathbf{9})$ | (1,2,3;0.5,1.5,2.5;1.5,2.5,3.5) | (1,5,8;1.5,3,6.5;4,7,9) |

C | Selection of Arms | A | (1,4,7;1,3,5;3.5,6,7.5) | (1,1.5,4;0.5,1,2.5;1.25,3,4.25) | (1,5,9;1.5,4.5,6.5;4,7,10) |

D | Final Plan and Blueprint | B | (1,3,5;0.5,2.5,3.5;2.5,4,6) | (1.5,3.5,5.5;1,2,3;3,4.5,6) | (0.5,2.5,4.5;1,2,3;1.5,3.5,5.5) |

E | Bring Utilities to the Site | B | (0.5,2.5,4.5;0.5,1.5,3.5;2,4,6) | (1,5,9;1.5,4.5,6.5;4,7.5,10.5) | (1.5,2.5,3.5;1,1.5,3;2,3,4) |

F | Interview | A | (2,4,6;1.5,2.5,3.5;3,5,7) | (1,2,3;0.5,1.5,2.5;1.2,2.7,3.5) | (1,4,7;1,3,5;3.5,6,7.5) |

G | Acquisition and take delivery of arms | C | (0.5,2.5,4.5;1,2,3;2,4,6) | (1,5,8;1.5,3,6.5;4,7,9) | (1,5,9;1.5,4.5,6.5;4,7,10) |

H | Construct the battlefield | D | (1.5,3.5,5.5;1,2,3;3,4.5,6) | (0.5,3.5,6.5;0.5,2.5,4.5;3,5,7) | (1,2,3;0.5,1.5,2.5;1.5,2.5,3.5) |

I | Developed networking system | A | (1,5,9;1.5,4.5,6.5;4,7,10) | (0.5,2.5,4.5;1,2,3;1.5,3.5,5.5) | $(\mathbf{1},\mathbf{5},\mathbf{8};\mathbf{1},\mathbf{3},\mathbf{6};\mathbf{4},\mathbf{7},\mathbf{9})$ |

J | Run the system | E,G,H | (0.5,3.5,6.5;0.5,2.5,4.5;3,5,7) | (1.5,2.5,3.5;1,1.5,3;2,3,4) | (1,2,3;0.5,1.5,2.5;1.2,2.7,3.5) |

K | Training for all | F,I,J | (1,5,8;1.5,3.5,6.5;4,6,8.5) | (1,4,7;1,3,5;3.5,6,7.5) | (1,1.5,4;0.5,1,2.5;1.25,3,4.25) |

Optimistic Time (o) | Pessimistic Time (p) | Most Likely Time (m) | ${\mathit{E}}_{\mathit{j}\mathit{k}}\mathbf{=}\frac{\mathit{o}\mathbf{+}\mathbf{4}\mathit{m}\mathbf{+}\mathit{p}}{\mathbf{6}}$ | ${\mathit{\sigma}}_{\mathit{j}\mathit{k}}{}^{\mathbf{2}}\mathbf{=}{\mathbf{(}\frac{\mathit{p}\mathbf{-}\mathit{o}}{\mathbf{6}}\mathbf{)}}^{\mathbf{2}}$ |
---|---|---|---|---|

2.26 | 5.42 | 3.33 | 3.50 | 0.277 |

4.92 | 2.00 | 4.92 | 4.43 | 0.244 |

4.67 | 1.71 | 5.42 | 4.68 | 0.243 |

2.96 | 3.33 | 2.67 | 2.83 | 0.004 |

2.75 | 5.54 | 2.42 | 3.00 | 0.216 |

3.83 | 2.26 | 4.67 | 4.13 | 0.068 |

2.83 | 4.92 | 2.00 | 2.63 | 0.121 |

3.33 | 3.50 | 2.26 | 2.65 | 0.001 |

5.42 | 2.67 | 4.92 | 4.63 | 0.210 |

3.50 | 2.42 | 2.00 | 2.32 | 0.032 |

4.88 | 4.67 | 1.71 | 2.73 | 0.001 |

Destination-1 | Destination-2 | Destination-3 | |
---|---|---|---|

Truck 1 | (1,4,7;1,3,5;3.5,6,7.5) | (0.5,2.5,4.5;1,2,3;1.5,3.5,5.5) | (1,3,5;0.5,1.5,3.5;2,4,6) |

Truck 2 | (1,2,3;0.5,1.5,2.5;1.5,2.5,3.5) | (1,1.5,4;0.5,1,2.5;1.25,3,4.25) | (1.5,2.5,3.5;1,1.5,3;2,3,4) |

Truck 3 | (2,4,6;1.5,2.5,4.5;3,5,7) | (1,5,8;1.5,4.5,7.5;4,6.5,9) | (1,5,8;1.5,3,6.5;4,7,9) |

Destination-1 | Destination-2 | Destination-3 | |
---|---|---|---|

Truck 1 | 4.25 | 2.67 | 2.92 |

Truck 2 | 2.00 | 1.71 | 2.75 |

Truck 3 | 3.92 | 5.25 | 5.08 |

Destination-1 | Destination-2 | Destination-3 | |
---|---|---|---|

Truck 1 | 1.58 | 0 | 0.25 |

Truck 2 | 0.29 | 0 | 1.04 |

Truck 3 | 0 | 1.33 | 1.16 |

Destination-1 | Destination-2 | Destination-3 | |
---|---|---|---|

Truck 1 | 1.58 | 0 | 0 |

Truck 2 | 0.29 | 0 | 0.79 |

Truck 3 | 0 | 1.33 | 0.91 |

Destination-1 | Destination-2 | Destination-3 | |
---|---|---|---|

Truck 1 | 1.58 | 0 | 0 |

Truck 2 | 0.29 | 0 | 0.79 |

Truck 3 | 0 | 1.33 | 0.91 |

Destination-1 | Destination-2 | Destination-3 | |
---|---|---|---|

Truck 1 | 1.58 | 0 | $\left[\mathbf{0}\right]$ |

Truck 2 | 0.29 | $\left[\mathbf{0}\right]$ | 0.79 |

Truck 3 | $\left[\mathbf{0}\right]$ | 1.33 | 0.91 |

Destination-1 | Destination-2 | Destination-3 | |
---|---|---|---|

Truck 1 | (1,4,7;1,3,5;3.5,6,7.5) | (0.5,2.5,4.5;1,2,3;1.5,3.5,5.5) | (1,3,5;0.5,1.5,3.5;2,4,6) |

Truck 2 | (1,2,3;0.5,1.5,2.5;1.5,2.5,3.5) | (1,1.5,4;0.5,1,2.5;1.25,3,4.25) | (1.5,2.5,3.5;1,1.5,3;2,3,4) |

Truck 3 | (2,4,6;1.5,2.5,4.5;3,5,7) | (1,5,8;1.5,4.5,7.5;4,6.5,9) | (1,5,8;1.5,3,6.5;4,7,9) |

Destination-1 | Destination-2 | Destination-3 | |
---|---|---|---|

Truck 1 | −0.92 | −0.33 | −0.25 |

Truck 2 | 0.00 | −0.04 | −0.08 |

Truck 3 | −0.58 | −1.42 | −1.17 |

Destination-1 | Destination-2 | Destination-3 | |
---|---|---|---|

Truck 1 | 0.50 | 1.09 | 1.17 |

Truck 2 | 1.42 | 1.38 | 1.34 |

Truck 3 | 0.84 | 0.00 | 0.25 |

Destination-1 | Destination-2 | Destination-3 | |
---|---|---|---|

Truck 1 | 0 | 0.59 | 0.67 |

Truck 2 | 0.08 | 0.04 | 0 |

Truck 3 | 0.84 | 0 | 0.25 |

Destination-1 | Destination-2 | Destination-3 | |
---|---|---|---|

Truck 1 | 0 | 0.59 | 0.67 |

Truck 2 | 0.08 | 0.04 | 0 |

Truck 3 | 0.84 | 0 | 0.25 |

Destination-1 | Destination-2 | Destination-3 | |
---|---|---|---|

Truck 1 | 0 | 0.59 | 0.67 |

Truck 2 | 0.08 | 0.04 | 0 |

Truck 3 | 0.84 | 0 | 0.25 |

Destination-1 | Destination-2 | Destination-3 | |
---|---|---|---|

Truck 1 | [0] | 0.59 | 0.67 |

Truck 2 | 0.08 | 0.04 | [0] |

Truck 3 | 0.84 | [0] | 0.25 |

© 2018 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/).

## Share and Cite

**MDPI and ACS Style**

Chakraborty, A.; Mondal, S.P.; Ahmadian, A.; Senu, N.; Alam, S.; Salahshour, S.
Different Forms of Triangular Neutrosophic Numbers, De-Neutrosophication Techniques, and their Applications. *Symmetry* **2018**, *10*, 327.
https://doi.org/10.3390/sym10080327

**AMA Style**

Chakraborty A, Mondal SP, Ahmadian A, Senu N, Alam S, Salahshour S.
Different Forms of Triangular Neutrosophic Numbers, De-Neutrosophication Techniques, and their Applications. *Symmetry*. 2018; 10(8):327.
https://doi.org/10.3390/sym10080327

**Chicago/Turabian Style**

Chakraborty, Avishek, Sankar Prasad Mondal, Ali Ahmadian, Norazak Senu, Shariful Alam, and Soheil Salahshour.
2018. "Different Forms of Triangular Neutrosophic Numbers, De-Neutrosophication Techniques, and their Applications" *Symmetry* 10, no. 8: 327.
https://doi.org/10.3390/sym10080327