# The Three-Way Decisions Method Based on Theory of Reliability with SV-Triangular Neutrosophic Numbers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Three-Way Decisions

#### 2.2. Single-Valued Triangular Neutrosophic Numbers

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Example**

**1.**

#### 2.3. Theory of Reliability

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

## 3. Three-Way Decisions Based on the Theory of Reliability

#### 3.1. Loss Function Matrix with SVTNNs

**Example**

**2.**

**Definition**

**8.**

#### 3.2. The Calculation of Three Threshold Values with SVTNNs

#### 3.3. Decision-Making Rules Based on Theory of Reliability

## 4. Applications in the Overhaul of Factory’s Machines

#### The Decision-Making Process of Three-Way Decisions

**Step 1:**Based on the attributes of substitutability, quantity and the difficulty of maintenance, all machines are classified into three types. ${A}_{1}$: important, their substitutability is very low, they are few in number and their difficulty of maintenance is very high; ${A}_{2}$: medium, their substitutability is medium, they are medium in number and their difficulty of maintenance is medium; and ${A}_{3}$: general, their substitutability is high, there are a lot of them and their difficulty of maintenance is very low.

**Step 2:**For three different types of machines, different loss function matrices with SVTNNs are given by decision maker, respectively. The importance of three different types of machines is different, thus the loss cased by the same action in the same state is not the same.

**Step 3:**According to the stages of their life, we continue to subdivide the three types of machines further. In Definition 7, we define Stages I, $II$, and $III$ based on the malfunction’s rate of products. Therefore, we could divide Machines ${A}_{1}$ into three types, namely ${A}_{11}$, ${A}_{12}$, and ${A}_{13}$. ${A}_{11}$ refers to the important machines which are in the period of Stage I. Similarly, Machines ${A}_{2}$ could be divided into three types: ${A}_{21}$, ${A}_{22}$, and ${A}_{23}$. Machines ${A}_{3}$ could also be divided into three types: ${A}_{31}$, ${A}_{32}$, and ${A}_{33}$.

**Step 4:**Based on the historical data of the such machines, the function of malfunction’s rate of such machines is fitted. Then, we calculate the reliability of each machines based on Equations (14) and (15). The reliability of a type of machines could be calculated as follows:

**Step 5:**Based on the loss function matrices gained from Step 2 and Equation (12), we convert the loss function matrices with SVTNNs to the loss function matrices with real numbers. Then, we could calculate the threshold values ${\alpha}_{m}$, ${\beta}_{m}$, and ${\gamma}_{m}$ of three types of Machines ${A}_{1}$–${A}_{3}$ based on Equation (18).

**Step 6:**Based on the decision rules $({P}_{3})\sim ({N}_{3})$, we can determine the decision-making result for each object by comparing the reliability with the threshold values.

## 5. A Case Study about the Overhaul of the Factory’s Machines

**Step 1:**There are four kinds of machines in this glazed tile factory: blank extruder, slurry agitator, ball grinding mill and tile pressing machine. The blank extruder is the most important; all materials must be processed to form by it. It is the core machine for the production of glazed tiles, with a small number. Next, the ball grinding mill and the slurry agitator are the initial tools for processing raw materials; their substitutability is medium. Finally, the number of tile pressing machines is a little large, and we can replace these machines with manual operation. Therefore, we could divide the machines into three types:

- ${A}_{1}$: blank extruder;
- ${A}_{2}$: ball grinding mill, slurry agitator; and
- ${A}_{3}$: tile press.

**Step 2:**For three different types of machines, different loss function matrices with SVTNNs are given in the form of Table 2 by decision maker, respectively.

**Step 3:**In Machine ${A}_{1}$, the blank extruder is in the period of Stage $II$, therefore we put it in the ${A}_{12}$ category. In Machines ${A}_{2}$, the ball grinding mills are in the period of Stage I, while the slurry agitators are in the period of Stage $II$, therefore we put the ball grinding mills in the ${A}_{21}$ category, and the slurry agitators in the ${A}_{22}$ category. In Machines ${A}_{3}$, four tile press machines are in the period of Stage $II$, while four tile press machines are in the period of Stage $III$, therefore we put four tile press machines in the ${A}_{32}$ category, and the others in the ${A}_{33}$ category.

**Step 4:**Based on Step 4 of the decision-making process in Section 4, we can attain the reliability of equivalence classes shown as Table 6.

**Step 5:**Based on Equation (12) and the loss functions attained from Step 2, we convert the loss function matrices with SVTNNs to the loss function matrices with real numbers. For Machines ${A}_{1}$:

**Step 6:**At last, according to the decision rules ${P}_{3}\sim {N}_{3}$, we can compare the value of reliability to the threshold values. Then, the following results can be deduced: $P\phantom{\rule{-0.166667em}{0ex}}O\phantom{\rule{-0.166667em}{0ex}}S(X)=\{{A}_{12},{A}_{22},{A}_{32}\}$, $B\phantom{\rule{-0.166667em}{0ex}}N\phantom{\rule{-0.166667em}{0ex}}D(X)=\{{A}_{21}\}$, and $N\phantom{\rule{-0.166667em}{0ex}}E\phantom{\rule{-0.166667em}{0ex}}G(X)=\{{A}_{33}\}$. The results we attain imply that we should decide to make Machines ${A}_{12},{A}_{22},{A}_{32}$ keep working without overhaul and overhaul Machines ${A}_{33}$, while Machines ${A}_{21}$ need to be further investigated for their judgements.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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X(P) | ${\mathit{X}}^{\mathit{c}}$(N) | |
---|---|---|

${a}_{P}$ | ${\lambda}_{PP}$ | ${\lambda}_{PN}$ |

${a}_{B}$ | ${\lambda}_{BP}$ | ${\lambda}_{BN}$ |

${a}_{N}$ | ${\lambda}_{NP}$ | ${\lambda}_{NN}$ |

X(P) | ${\mathit{X}}^{\mathit{c}}$(N) | |
---|---|---|

${a}_{P}$ | ${\tilde{A}}_{PP}$ | ${\tilde{A}}_{PN}$ |

${a}_{B}$ | ${\tilde{A}}_{BP}$ | ${\tilde{A}}_{BN}$ |

${a}_{N}$ | ${\tilde{A}}_{NP}$ | ${\tilde{A}}_{NN}$ |

X(P) | ${\mathit{X}}^{\mathit{c}}$(N) | |
---|---|---|

${a}_{P}$ | {(0,0,0);1,0,0} | {(6.5,7,8);0.9,0.1,0.1} |

${a}_{B}$ | {(1,1.6,2);0.86,0.12,0.2} | {(3,3.5,4);0.82,0.4,0.3} |

${a}_{N}$ | {(3,3.5,4);0.9,0.2,0.24} | {(1.4,1.6,2);0.96,0.3,0.2} |

X(P) | ${\mathit{X}}^{\mathit{c}}$(N) | |
---|---|---|

${a}_{P}$ | {(0,0,0);1,0,0} | {(7,7.5,8.4);0.92,0.12,0.1} |

${a}_{B}$ | {(1.5,2,2.2);0.9,0.2,0.3} | {(3.6,4,4.4);0.86,0.2,0.2} |

${a}_{N}$ | {(4,4.6,5);0.88,0.3,0.1} | {(2,2.5,2.8);0.98,0.1,0.16} |

X(P) | ${\mathit{X}}^{\mathit{c}}$(N) | |
---|---|---|

${a}_{P}$ | {(0,0,0);1,0,0} | {(5,5.6,6.4);0.9,0.2,0.15} |

${a}_{B}$ | {(1.4,2,2.1);0.88,0.2,0.1} | {(2.8,3.2,3.6);0.88,0.1,0.12} |

${a}_{N}$ | {(2.4,2.8,3);0.92,0.1,0.3} | {(1.2,1.5,1.8);0.92,0.2,0.1} |

R(${\mathit{A}}_{12}$) | R(${\mathit{A}}_{21}$) | R(${\mathit{A}}_{22}$) | R(${\mathit{A}}_{32}$) | R(${\mathit{A}}_{33}$) |
---|---|---|---|---|

0.84 | 0.5 | 0.8 | 0.64 | 0.45 |

X(P) | ${\mathit{X}}^{\mathit{c}}$(N) | |
---|---|---|

${a}_{P}$ | 0 | 6.45 |

${a}_{B}$ | 1.30 | 2.47 |

${a}_{N}$ | 2.87 | 1.37 |

U | ${\mathit{\alpha}}_{\mathit{m}}$ | ${\mathit{\beta}}_{\mathit{m}}$ | ${\mathit{\gamma}}_{\mathit{m}}$ |
---|---|---|---|

${A}_{1}$ | 0.754 | 0.412 | 0.639 |

${A}_{2}$ | 0.703 | 0.324 | 0.554 |

${A}_{3}$ | 0.556 | 0.492 | 0.604 |

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**MDPI and ACS Style**

Li, X.; Huang, X.
The Three-Way Decisions Method Based on Theory of Reliability with SV-Triangular Neutrosophic Numbers. *Symmetry* **2019**, *11*, 888.
https://doi.org/10.3390/sym11070888

**AMA Style**

Li X, Huang X.
The Three-Way Decisions Method Based on Theory of Reliability with SV-Triangular Neutrosophic Numbers. *Symmetry*. 2019; 11(7):888.
https://doi.org/10.3390/sym11070888

**Chicago/Turabian Style**

Li, Xiang, and Xianjiu Huang.
2019. "The Three-Way Decisions Method Based on Theory of Reliability with SV-Triangular Neutrosophic Numbers" *Symmetry* 11, no. 7: 888.
https://doi.org/10.3390/sym11070888