# Dimensional Analysis under Pythagorean Fuzzy Approach for Supplier Selection

^{*}

## Abstract

**:**

## 1. Introduction

- Do not take into account the interrelationship among three or more arguments.
- Limited qualitative criteria.
- Imprecise preferences of decision makers.

## 2. Preliminaries

#### 2.1. Pythagorean Fuzzy Set (PFS)

**Definition**

**1.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

#### 2.2. Dimensional Analysis (DA)

**Definition**

**6.**

## 3. Methodology

**Definition**

**7.**

**Theorem**

**1.**

**Proof.**

- Step 1: Define the Pythagorean decision matrix.
- Step 2: Select the ideal solution in accordance with BN or C criteria values.
- Step 3: Establish criteria weights, use Equations (30) and (31).
- Step 4: Standardized matrix—use Equation (25) for BN criteria, Equation (26) for C criteria.
- Step 5: Standardized matrix elevated in accordance with criteria weights, use Equation (16).
- Step 6: Generate PFIS index, use Equation (21).
- Step 7: Establish the highest index of the index of similarity (IS), use Equation (4).
- Step 8: Establish the ranking, with highest values to lowest values.

## 4. Numerical Illustration

- Price $({X}_{1})$: the most minimum values are selected.
- Facility $({X}_{2})$: great assessments are selected.
- Lead time $({X}_{3})$: high assessments are selected.
- Quality $({X}_{4})$: great assessments are selected.

- Product Quality $({X}_{1})$: BN criteria.
- Delivery Compliance $({X}_{2})$: BN criteria.
- Price $({X}_{3})$: high assessment C criteria.
- Production Capability $({X}_{4})$: C criteria.
- Technological Capability $({X}_{5})$: C criteria.

#### 4.1. Sensitivity Analysis

#### 4.2. Comparative Analysis

#### 4.3. Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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C1 | C2 | C3 | C4 | Rankings |
---|---|---|---|---|

0.24 | 0.24 | 0.22 | 0.31 | A1 > A2 > A5 > A3 > A4 |

0.25 | 0.25 | 0.25 | 0.25 | A1 > A2 > A5 > A3 > A4 |

0.2 | 0.2 | 0.3 | 0.3 | A1 > A2 > A5 > A3 > A4 |

0.3 | 0.3 | 0.2 | 0.2 | A1 > A2 > A5 > A3 > A4 |

0.1 | 0.3 | 0.2 | 0.4 | A1 > A2 > A5 > A3 > A4 |

0.5 | 0.2 | 0.15 | 0.15 | A1 > A2 > A5 > A3 > A4 |

C1 | C2 | C3 | C4 | C5 | Rankings |
---|---|---|---|---|---|

0.25 | 0.1 | 0.5 | 0.1 | 0.05 | P4 > P1 > P3 > P2 |

0.1 | 0.1 | 0.3 | 0.4 | 0.1 | P4 > P2 > P3 > P1 |

0.15 | 0.15 | 0.3 | 0.2 | 0.2 | P4 > P3 > P2 > P1 |

0.1 | 0.1 | 0.4 | 0.2 | 0.2 | P4 > P2 > P3 > P1 |

0.3 | 0.1 | 0.15 | 0.15 | 0.3 | P4 > P3 > P1 > P2 |

0.2 | 0.2 | 0.2 | 0.2 | 0.2 | P3 > P4 > P2 > P1 |

$\mathit{D}({\mathit{X}}_{\mathit{i}},{\mathit{X}}^{-})$ | $\mathit{D}({\mathit{X}}_{\mathit{i}},{\mathit{X}}^{+})$ | $\mathbf{Closeness}\text{}({\mathit{X}}_{\mathit{i}})$ | Rank |
---|---|---|---|

0.7063 | 0.2062 | 0.43059021 | 1 |

0.3355 | 0.3955 | −1.23888967 | 2 |

0.4937 | 0.4743 | −1.30058103 | 3 |

0.2427 | 0.8114 | −3.4439003 | 5 |

0.3559 | 0.4981 | −1.69490467 | 4 |

Alternative | PF-DA Ranking | PF-MOORA Ranking | PF-TOPSIS Ranking | d | ${\mathit{d}}^{2}$ |
---|---|---|---|---|---|

A1 | 1 | 1 | 1 | 0 | 0 |

A2 | 2 | 2 | 2 | 0 | 0 |

A3 | 4 | 3 | 3 | 1 | 1 |

A4 | 5 | 5 | 5 | 0 | 0 |

A5 | 3 | 4 | 4 | 1 | 1 |

Cronbach’s Alpha | Cronbach’s Alpha Based on the Typified Elements | N of Elements |
---|---|---|

0.977 | 0.977 | 3 |

Alternative | PF-DA Ranking | PF CODAS Ranking | d | ${\mathit{d}}^{2}$ |
---|---|---|---|---|

P1 | 4 | 4 | 0 | 0 |

P2 | 1 | 1 | 0 | 0 |

P3 | 3 | 2 | 1 | 1 |

P4 | 2 | 3 | 1 | 1 |

Cronbach’s Alpha | N of Elements |
---|---|

0.889 | 2 |

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**MDPI and ACS Style**

Villa Silva, A.J.; Pérez Dominguez, L.A.; Martínez Gómez, E.; Alvarado-Iniesta, A.; Pérez Olguín, I.J.C.
Dimensional Analysis under Pythagorean Fuzzy Approach for Supplier Selection. *Symmetry* **2019**, *11*, 336.
https://doi.org/10.3390/sym11030336

**AMA Style**

Villa Silva AJ, Pérez Dominguez LA, Martínez Gómez E, Alvarado-Iniesta A, Pérez Olguín IJC.
Dimensional Analysis under Pythagorean Fuzzy Approach for Supplier Selection. *Symmetry*. 2019; 11(3):336.
https://doi.org/10.3390/sym11030336

**Chicago/Turabian Style**

Villa Silva, Aldo Joel, Luis Asunción Pérez Dominguez, Erwin Martínez Gómez, Alejandro Alvarado-Iniesta, and Iván Juan Carlos Pérez Olguín.
2019. "Dimensional Analysis under Pythagorean Fuzzy Approach for Supplier Selection" *Symmetry* 11, no. 3: 336.
https://doi.org/10.3390/sym11030336