# Pythagorean 2-Tuple Linguistic Taxonomy Method for Supplier Selection in Medical Instrument Industries

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Pythagorean 2-Tuple Linguistic Sets

**Definition**

**1**

**Definition**

**2**

**Definition**

**3**

**Definition**

**4**

- (1)
- $if\text{\hspace{0.17em}}S\left({o}_{1}\right)<S\left({o}_{2}\right),{o}_{1}<{o}_{2};$
- (2)
- $if\text{\hspace{0.17em}}S\left({o}_{1}\right)>S\left({o}_{2}\right),{o}_{1}>{o}_{2};$
- (3)
- $if\text{\hspace{0.17em}}S\left({o}_{1}\right)=S\left({o}_{2}\right),H\left({o}_{1}\right)<H\left({o}_{2}\right),\text{\hspace{0.17em}}then\text{\hspace{0.17em}}{o}_{1}<{o}_{2};$
- (4)
- $if\text{\hspace{0.17em}}S\left({o}_{1}\right)=S\left({o}_{2}\right),H\left({o}_{1}\right)>H\left({o}_{2}\right),\text{\hspace{0.17em}}then\text{\hspace{0.17em}}{o}_{1}>{o}_{2};$
- (5)
- $if\text{\hspace{0.17em}}S\left({o}_{1}\right)=S\left({o}_{2}\right),H\left({o}_{1}\right)=H\left({o}_{2}\right),\text{\hspace{0.17em}}then\text{\hspace{0.17em}}{o}_{1}={o}_{2};$

**Definition**

**5**

**Definition**

**6**

**Theorem**

**1**

- (1)
- ${o}_{1}\oplus {o}_{2}={o}_{2}\oplus {o}_{1}$
- (2)
- ${o}_{1}\otimes {o}_{2}={o}_{2}\otimes {o}_{1}$
- (3)
- $k\left({o}_{1}\oplus {o}_{2}\right)=k{o}_{1}\oplus k{o}_{2},0\le k\le 1$
- (4)
- ${k}_{1}{o}_{1}\oplus {k}_{2}{o}_{1}=\left({k}_{1}\oplus {k}_{2}\right){o}_{1},0\le {k}_{1},{k}_{2},{k}_{1}+{k}_{2}\le 1$
- (5)
- ${o}_{1}{}^{{k}_{1}}\otimes {o}_{1}{}^{{k}_{2}}={\left({o}_{1}\right)}^{{k}_{1}+{k}_{2}},0\le {k}_{1},{k}_{2},{k}_{1}+{k}_{2}\le 1$
- (6)
- ${o}_{1}{}^{{k}_{1}}\otimes {o}_{2}{}^{{k}_{1}}={\left({o}_{1}\otimes {o}_{2}\right)}^{{k}_{1}},{k}_{1}\ge 0$
- (7)
- ${\left({\left({o}_{1}\right)}^{{k}_{1}}\right)}^{{k}_{2}}={\left({o}_{1}\right)}^{{k}_{1}{k}_{2}}$

#### 2.2. Some Operators with P2TLNs

**Definition**

**7**

**Definition**

**8**

## 3. Taxonomy Method for P2TL-MAGDM Issues

**Step 1.**Shift the cost attribute into the beneficial attribute.

**Step 2.**Set up a decision-making group composed of several experts, choose the best attributes to measure alternatives, and finally get a P2TL fuzzy decision matrix series ${\mathrm{X}}^{\left(t\right)}={\left({x}_{ij}^{\left(t\right)}\right)}_{m\times n}$ from each decision maker.

**Step 3.**Utilize the P2TLWA operator or the P2TLWG operator to fuse assessment information, then the P2TL fuzzy decision matrix $\mathrm{X}={\left({x}_{ij}\right)}_{m\times n}$ group can be obtained by the calculation.

**Step 4.**Equations (11) and (12) are used to calculate the mean and standard deviation of the attributes.

**Step 5.**The standard matrix:

**Remark.**

**Step 6:**The composite distances matrix:

**Remark.**

**Step 7:**Homogenizing the alternatives:

**Remark.**

**Step 8:**The development pattern:

**Remark.**

**Step 4**, the alternative development pattern is determined, where ${z}_{oj}$ represents the ideal value for the $j-th$ attribute. If ${\kappa}_{j}$ is a cost attribute, ${z}_{oj}$ is the minimum value of this column. Conversely, if ${\kappa}_{j}$ is a benefit attribute, ${z}_{oj}$ selects the maximum value of this column. ${z}_{ij}$ indicates the standardized value of the $j-th$ attribute for the $i-th$ alternative, and ${C}_{io}$ illustrates the development pattern for the $i-th$ attribute.

**Step 9:**The final ranking of alternatives:

**Remark.**

## 4. Numerical Example and Comparative Analysis

#### 4.1. Numerical Example

**Step 1:**Shift cost attribute ${\kappa}_{2}$ into beneficial attribute. If the cost attribute value is $\langle \left({s}_{\sigma},\psi \right),\left({u}_{o},{v}_{o}\right)\rangle $, then the corresponding beneficial attribute value is $\langle \left({s}_{\sigma},\psi \right),\left({v}_{o},{u}_{o}\right)\rangle $ (See Table 4, Table 5 and Table 6).

**Step 2:**Construct the evaluation matrix ${X}^{\left(3\right)}={\left({x}_{ij}^{3}\right)}_{5\times 4}\left(i=1,2,3,4,5,j=1,2,3,4\right)$ of each DM as in Table 4, Table 5 and Table 6. Based on Table 4, Table 5 and Table 6 and Equation (9), the group P2TLN decision matrix is computed and presented in Table 7.

**Step 3.**Equations (11) and (12) are used to calculate the mean ${\overline{x}}_{j}$ and standard deviation ${S}_{j}$ of the attributes.

**Step 4:**The standard matrix ${A}_{ij}$:

**Step 5:**The composite distances matrix ${C}_{ab}$:

**Step 6:**Homogenizing the alternatives:

**Step 7:**The development pattern:

**Step 8:**The final ranking of alternatives:

#### 4.2. Comparative Analyses

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Table 1.**The Pythagorean 2-tuple linguistic number (P2TLN) decision matrix by the first expert ${\mathsf{\Theta}}^{\left(1\right)}$.

Alternatives | ${\mathit{\kappa}}_{1}$ | ${\mathit{\kappa}}_{2}$ | ${\mathit{\kappa}}_{3}$ | ${\mathit{\kappa}}_{4}$ |
---|---|---|---|---|

${N}_{1}$ | <(s_{3}, 0), (0.6, 0.5)> | <(s_{6}, 0), (0.3, 0.3)> | <(s_{1}, 0), (0.6, 0.6)> | <(s_{4}, 0), (0.2, 0.3)> |

${N}_{2}$ | <(s_{1,} 0), (0.3, 0.8)> | <(s_{2}, 0), (0.6, 0.4)> | <(s_{4}, 0), (0.6, 0.3)> | <(s_{0}, 0), (0.1, 0.6)> |

${N}_{3}$ | <(s_{3}, 0), (0.4, 0.6)> | <(s_{1}, 0), (0.3, 0.4)> | <(s_{5}, 0), (0.6, 0.3)> | <(s_{3}, 0), (0.7, 0.4)> |

${N}_{4}$ | <(s_{4}, 0), (0.6, 0.7)> | <(s_{6}, 0), (0.4, 0.2)> | <(s_{6}, 0), (0.8, 0.4)> | <(s_{1}, 0), (0.4, 0.6)> |

${N}_{5}$ | <(s_{2}, 0), (0.5, 0.5)> | <(s_{4}, 0), (0.7, 0.4)> | <(s_{5}, 0), (0.6, 0.5)> | <(s_{6}, 0), (0.4, 0.5)> |

Alternatives | ${\mathit{\kappa}}_{1}$ | ${\mathit{\kappa}}_{2}$ | ${\mathit{\kappa}}_{3}$ | ${\mathit{\kappa}}_{4}$ |
---|---|---|---|---|

${N}_{1}$ | <(s_{6}, 0), (0.8, 0.7)> | <(s_{1}, 0), (0.5, 0.8)> | <(s_{6}, 0), (0.3, 0.3)> | <(s_{5}, 0), (0.1, 0.7)> |

${N}_{2}$ | <(s_{1}, 0), (0.6, 0.5)> | <(s_{0}, 0), (0.2, 0.2)> | <(s_{3}, 0), (0.8, 0.6)> | <(s_{3}, 0), (0.5, 0.3)> |

${N}_{3}$ | <(s_{3}, 0), (0.4, 0.2)> | <(s_{3}, 0), (0.1, 0.8)> | <(s_{1}, 0), (0.3, 0.6)> | <(s_{5}, 0), (0.1, 0.7)> |

${N}_{4}$ | <(s_{4}, 0), (0.7, 0.7)> | <(s_{5}, 0), (0.2, 0.6)> | <(s_{4}, 0), (0.7, 0.1)> | <(s_{2}, 0), (0.8, 0.5)> |

${N}_{5}$ | <(s_{4}, 0), (0.8, 0.3)> | <(s_{3}, 0), (0.3, 0.6)> | <(s_{5}, 0), (0.5, 0.6)> | <(s_{6}, 0), (0.4, 0.7)> |

Alternatives | ${\mathit{\kappa}}_{1}$ | ${\mathit{\kappa}}_{2}$ | ${\mathit{\kappa}}_{3}$ | ${\mathit{\kappa}}_{4}$ |
---|---|---|---|---|

${N}_{1}$ | <(s_{1}, 0), (0.4, 0.3)> | <(s_{0}, 0), (0.2, 0.6)> | <(s_{5}, 0), (0.8, 0.6)> | <(s_{3}, 0), (0.2, 0.1)> |

${N}_{2}$ | <(s_{4}, 0), (0.2, 0.6)> | <(s_{2}, 0), (0.5, 0.5)> | <(s_{3}, 0), (0.6, 0.6)> | <(s_{2}, 0), (0.7, 0.5)> |

${N}_{3}$ | <(s_{1}, 0), (0.3, 0.8)> | <(s_{5}, 0), (0.6, 0.5)> | <(s_{0}, 0), (0.1, 0.5)> | <(s_{4}, 0), (0.4, 0.7)> |

${N}_{4}$ | <(s_{6}, 0), (0.1, 0.5)> | <(s_{2}, 0), (0.7, 0.7)> | <(s_{4}, 0), (0.5, 0.3)> | <(s_{1}, 0), (0.6, 0.6)> |

${N}_{5}$ | <(s_{2}, 0), (0.3, 0.8)> | <(s_{4}, 0), (0.7, 0.6)> | <(s_{1}, 0), (0.4, 0.8)> | <(s_{2}, 0), (0.6, 0.3)> |

**Table 4.**The P2TLN normalized decision matrix by the first expert ${\mathsf{\Theta}}^{\left(1\right)}$.

Alternatives | ${\mathit{\kappa}}_{1}$ | ${\mathit{\kappa}}_{2}$ | ${\mathit{\kappa}}_{3}$ | ${\mathit{\kappa}}_{4}$ |
---|---|---|---|---|

${N}_{1}$ | <(s_{3}, 0), (0.6, 0.5)> | <(s_{6}, 0), (0.3, 0.3)> | <(s_{1}, 0), (0.6, 0.6)> | <(s_{4}, 0), (0.2, 0.3)> |

${N}_{2}$ | <(s_{1,} 0), (0.3, 0.8)> | <(s_{2}, 0), (0.4, 0.6)> | <(s_{4}, 0), (0.6, 0.3)> | <(s_{0}, 0), (0.1, 0.6)> |

${N}_{3}$ | <(s_{3}, 0), (0.4, 0.6)> | <(s_{1}, 0), (0.4, 0.3)> | <(s_{5}, 0), (0.6, 0.3)> | <(s_{3}, 0), (0.7, 0.4)> |

${N}_{4}$ | <(s_{4}, 0), (0.6, 0.7)> | <(s_{6}, 0), (0.2, 0.4)> | <(s_{6}, 0), (0.8, 0.4)> | <(s_{1}, 0), (0.4, 0.6)> |

${N}_{5}$ | <(s_{2}, 0), (0.5, 0.5)> | <(s_{4}, 0), (0.4, 0.7)> | <(s_{5}, 0), (0.6, 0.5)> | <(s_{6}, 0), (0.4, 0.5)> |

**Table 5.**The P2TLN normalized decision matrix by the second expert ${\mathsf{\Theta}}^{\left(2\right)}$.

Alternatives | ${\mathit{\kappa}}_{1}$ | ${\mathit{\kappa}}_{2}$ | ${\mathit{\kappa}}_{3}$ | ${\mathit{\kappa}}_{4}$ |
---|---|---|---|---|

${N}_{1}$ | <(s_{6}, 0), (0.8, 0.7)> | <(s_{1}, 0), (0.8, 0.5)> | <(s_{6}, 0), (0.3, 0.3)> | <(s_{5}, 0), (0.1, 0.7)> |

${N}_{2}$ | <(s_{1}, 0), (0.6, 0.5)> | <(s_{0}, 0), (0.2, 0.2)> | <(s_{3}, 0), (0.8, 0.6)> | <(s_{3}, 0), (0.5, 0.3)> |

${N}_{3}$ | <(s_{3}, 0), (0.4, 0.2)> | <(s_{3}, 0), (0.8, 0.1)> | <(s_{1}, 0), (0.3, 0.6)> | <(s_{5}, 0), (0.1, 0.7)> |

${N}_{4}$ | <(s_{4}, 0), (0.7, 0.7)> | <(s_{5}, 0), (0.6, 0.2)> | <(s_{4}, 0), (0.7, 0.1)> | <(s_{2}, 0), (0.8, 0.5)> |

${N}_{5}$ | <(s_{4}, 0), (0.8, 0.3)> | <(s_{3}, 0), (0.6, 0.3)> | <(s_{5}, 0), (0.5, 0.6)> | <(s_{6}, 0), (0.4, 0.7)> |

**Table 6.**The P2TLN normalized decision matrix by the third expert ${\mathsf{\Theta}}^{\left(3\right)}$.

Alternatives | ${\mathit{\kappa}}_{1}$ | ${\mathit{\kappa}}_{2}$ | ${\mathit{\kappa}}_{3}$ | ${\mathit{\kappa}}_{4}$ |
---|---|---|---|---|

${N}_{1}$ | <(s_{1}, 0), (0.4, 0.3)> | <(s_{0}, 0), (0.6, 0.2)> | <(s_{5}, 0), (0.8, 0.6)> | <(s_{3}, 0), (0.2, 0.1)> |

${N}_{2}$ | <(s_{4}, 0), (0.2, 0.6)> | <(s_{2}, 0), (0.5, 0.5)> | <(s_{3}, 0), (0.6, 0.6)> | <(s_{2}, 0), (0.7, 0.5)> |

${N}_{3}$ | <(s_{1}, 0), (0.3, 0.8)> | <(s_{5}, 0), (0.5, 0.6)> | <(s_{0}, 0), (0.1, 0.5)> | <(s_{4}, 0), (0.4, 0.7)> |

${N}_{4}$ | <(s_{6}, 0), (0.1, 0.5)> | <(s_{2}, 0), (0.7, 0.7)> | <(s_{4}, 0), (0.5, 0.3)> | <(s_{1}, 0), (0.6, 0.6)> |

${N}_{5}$ | <(s_{2}, 0), (0.3, 0.8)> | <(s_{4}, 0), (0.6, 0.7)> | <(s_{1}, 0), (0.4, 0.8)> | <(s_{2}, 0), (0.6, 0.3)> |

Alternatives | ${\mathit{\kappa}}_{1}$ | ${\mathit{\kappa}}_{2}$ |
---|---|---|

${N}_{1}$ | <(s_{4}, −0.13), (0.6875,0.638)> | <(s_{2}, 0.31), (0.6701,0.4975)> |

${N}_{2}$ | <(s_{2}, −0.28), (0.4618,0.6476)> | <(s_{1}, 0.1), (0.3632,0.4104)> |

${N}_{3}$ | <(s_{3}, −0.48),(0.3791,0.4594)> | <(s_{3}, −0.14), (0.6646,0.3139)> |

${N}_{4}$ | <(s_{4}, 0.48),(0.5987,0.7212)> | <(s_{5}, −0.41), (0.5593,0.4449)> |

${N}_{5}$ | <(s_{4}, −0.13), (0.6875,0.638)> | <(s_{2}, 0.31), (0.6701,0.4975)> |

Alternatives | ${\kappa}_{3}$ | ${\kappa}_{4}$ |

${N}_{1}$ | <(s_{4}, 0.21), (0.589, 0.5146)> | <(s_{4}, 0.21), (0.1631, 0.4901)> |

${N}_{2}$ | <(s_{3}, 0.31), (0.7113, 0.7029)> | <(s_{2}, −0.17), (0.5048, 0.4926)> |

${N}_{3}$ | <(s_{2}, 0), (0.4091, 0.6729)> | <(s_{4}, 0.14), (0.4745, 0.7818)> |

${N}_{4}$ | <(s_{5}, −0.38), (0.7056, 0.2658)> | <(s_{1}, 0.45), (0.6801, 0.6476)> |

${N}_{5}$ | <(s_{4}, 0.21), (0.589, 0.5146)> | <(s_{4}, 0.21), (0.1631, 0.4901)> |

Methods | Order |
---|---|

P2TLWA | ${N}_{4}>{N}_{1}>{N}_{5}>{N}_{3}>{N}_{2}$ |

P2TLWG | ${N}_{4}>{N}_{5}>{N}_{1}>{N}_{3}>{N}_{2}$ |

P2TL-TODIM | ${N}_{4}>{N}_{1}>{N}_{5}>{N}_{3}>{N}_{2}$ |

P2TL-EDAS | ${N}_{4}>{N}_{5}>{N}_{1}>{N}_{3}>{N}_{2}$ |

P2TL-CODAS | ${N}_{4}>{N}_{1}>{N}_{3}>{N}_{5}>{N}_{2}$ |

P2TL-Taxonomy | ${N}_{4}>{N}_{2}>{N}_{3}>{N}_{5}>{N}_{1}$ |

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## Share and Cite

**MDPI and ACS Style**

He, T.; Wei, G.; Lu, J.; Wei, C.; Lin, R. Pythagorean 2-Tuple Linguistic Taxonomy Method for Supplier Selection in Medical Instrument Industries. *Int. J. Environ. Res. Public Health* **2019**, *16*, 4875.
https://doi.org/10.3390/ijerph16234875

**AMA Style**

He T, Wei G, Lu J, Wei C, Lin R. Pythagorean 2-Tuple Linguistic Taxonomy Method for Supplier Selection in Medical Instrument Industries. *International Journal of Environmental Research and Public Health*. 2019; 16(23):4875.
https://doi.org/10.3390/ijerph16234875

**Chicago/Turabian Style**

He, Tingting, Guiwu Wei, Jianping Lu, Cun Wei, and Rui Lin. 2019. "Pythagorean 2-Tuple Linguistic Taxonomy Method for Supplier Selection in Medical Instrument Industries" *International Journal of Environmental Research and Public Health* 16, no. 23: 4875.
https://doi.org/10.3390/ijerph16234875