# Dimensional Analysis under Linguistic Pythagorean Fuzzy Set

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- There is no consensus among experts that is widely accepted of generalized application, which is the best methodology to treat supplier selection, as diverse MCDM methods have been applied for supplier selection, furthermore, is a MCDM problem that has not been solved [45].

- DA is used to treat the interrelationship among the multi-input arguments, due to the advantages mentioned above.
- LPFS to overcome limitations concerning qualitative (fuzzy) criteria using linguistic assessments instead of numerical ones, in order to make DM’s opinions reliable.
- Get a method using DA under LPFS environment to solve a supplier problem, where index similarity (IS) equation of DA is replaced by linguistic Pythagorean fuzzy equations in order to make operative the decision matrix and obtain the ranking of alternatives.

## 2. Preliminaries

#### 2.1. Supplier Selection

#### 2.2. Dimensional Analysis (DA)

- Create a cluster of experts and define the importance of each one.
- Define the importance of each criteria in evaluation and join the opinions given by DMs.
- Create the aggregate decision matrix that represents the evaluations given by DMs for each of the alternatives
- Calculate index similarity using the following equation:

**Definition**

**1.**

- Once the index of similarity it is obtained, rank the alternatives.

#### 2.3. Intuitionistic Fuzzy Set (IFS)

**Definition**

**2.**

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

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

#### 2.5. Linguistic Pythagorean Fuzzy Set (LPFS)

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

## 3. The Proposed Model

- Step 1: Using the LPFS terms, construct decision matrix that represents the evaluations based on the opinions of the DMs for each of the alternatives.
- Step 2: In order to aggregate the opinions of DMs in a single decision matrix we utilize the LPFOWA Equation (21) defined in Section 2.4.
- Step 3: Select ideal solution in accordance with benefit (BN) or cost (C) criteria values.
- Step 4: Standardized matrix. In this step we take equation (22).
- Step 5: Standardized matrix elevated in accordance with criteria weights. In this step we take the Equation (19) in Section 2.4.
- Step 6: Generate LPFIS index, use the product Equation (18) in Section 2.4.
- Step 7: Establish the highest index of IS, use score Equation (15) in Section 2.4.
- Step 8: Arrange the score function of all the alternatives in descending order and select the alternative that has the highest score function value.

## 4. Illustrative Example

#### 4.1. An Illustrative Experiment with DA-LPFS

- CNC router construction (C1);
- Process design (C2);
- Performance (C3);
- CNC computer system (C4);
- Adaptation of workers (C5);
- Worker technical capability needed (C6);
- Initial cost (C7);
- Running cost (C8);
- Vendor’s CNC experience (C9);
- Reputation of vendor/brand (C11).

**Step 1:**The preferences of two experts over each alternative are summarized in the form of the decision matrix Table 1a,b and Table 2a,b:

**Step 2:**The weight vector concerning with each DM: W = (0.51, 0.49) was calculated using the entropy measures [46]. Then, the LPFOWA operator was utilized to develop the aggregate decision matrix in Table 3a,b:

**Step 3:**Establish the ideal solution in accordance with criteria values in Table 4a,b:

**Step 4:**Standardize the aggregate decision matrix (Table 5a,b).

**Step 5**: Standardized matrix elevated with criteria weights using entropy (Table 6a,b).

**Step 6:**Generate an index of similarity, LPFIS (Table 7).

**Step 7:**Obtain the highest index of similarity (Table 8).

#### 4.2. Sensitivity Analysis

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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(a) | ||||||

Alt | C1 | C2 | C3 | C4 | C5 | |

A1 | {S6, S1} | {S3, S1} | {S3, S3} | {S1, S6} | {S3, S4} | |

A2 | {S6, S1} | {S6, S2} | {S4, S3} | {S5, S1} | {S7, S1} | |

A3 | {S1, S6} | {S3, S4} | {S3, S4} | {S3, S2} | {S6, S1} | |

(b) | ||||||

Alt | C6 | C7 | C8 | C9 | C10 | C11 |

A1 | {S3, S4} | {S2, S5} | {S2, S4} | {S1, S3} | {S2, S3} | {S3, S2} |

A2 | {S1, S3} | {S2, S3} | {S3, S2} | {S3, S4} | {S2, S5} | {S2, S4} |

A3 | {S3, S1} | {S3, S3} | {S2, S4} | {S1, S3} | {S1, S6} | {S1, S3} |

(a) | ||||||

Alt | C1 | C2 | C3 | C4 | C5 | |

A1 | {S2, S3} | {S6, S7} | {S3, S4} | {S2, S3} | {S5, S2} | |

A2 | {S5, S3} | {S5, S2} | {S3, S3} | {S5, S2} | {S4, S1} | |

A3 | {S5, S2} | {S2, S1} | {S3, S4} | {S1, S2} | {S4, S1} | |

(b) | ||||||

Alt | C6 | C7 | C8 | C9 | C10 | C11 |

A1 | {S2, S1} | {S3, S4} | {S2, S5} | {S2, S3} | {S3, S3} | {S1, S2} |

A2 | {S2, S3} | {S3, S3} | {S6, S7} | {S3, S3} | {S5, S2} | {S3, S4} |

A3 | {S3, S4} | {S2, S3} | {S5, S2} | {S5, S2} | {S3, S3} | {S5, S2} |

(a) | ||||||

Alt | C1 | C2 | C3 | C4 | C5 | |

A1 | {3.58, 1.71} | {6.00, 2.59} | {3.00, 3.45} | {1.58, 4.27} | {4.31, 2.85} | |

A2 | {4.31, 1.71} | {4.59, 2.00} | {3.58, 3.00} | {5.00, 1.40} | {2.31, 1.00} | |

A3 | {4.17, 3.50} | {2.57, 2.03} | {3.00, 4.00} | {2.31, 2.00} | {2.58, 1.00} | |

(b) | ||||||

Alt | C6 | C7 | C8 | C9 | C10 | C11 |

A1 | {2.58, 2.02} | {2.56, 4.48} | {2.00, 4.46} | {1.58, 3.00} | {2.56, 3.00} | {2.31, 2.00} |

A2 | {1.58, 3.00} | {2.56, 3.00} | {6.00, 3.69} | {3.00, 3.47} | {4.21, 3.19} | {2.56, 4.00} |

A3 | {3.00, 1.97} | {2.58, 3.00} | {4.12, 2.84} | {4.01, 2.45} | {2.27, 4.27} | {4.01, 2.45} |

(a) | |||||

C1 | C2 | C3 | C4 | C5 | |

{4.18, 3.50} | {5.26, 2.02} | {4.84, 4.00} | {5.62, 4.27} | {3.11, 2.84} | |

(b) | |||||

C6 | C7 | C8 | C9 | C10 | C11 |

{5.62, 3.00} | {5.05, 3.00} | {3.25, 2.85} | {4.84, 3.47} | {5.17, 4.27} | {5.05, 4.00} |

(a) | ||||||

Alt | C1 | C2 | C3 | C4 | C5 | |

A1 | {0.49, 0.49} | {1.00, 1.28} | {0.00, 0.86} | {1.00, 0.00} | {0.00, 1.00} | |

A2 | {0.25, 0.49} | {0.70, 0.99} | {0.37, 0.75} | {0.81, 0.33} | {0.88, 0.35} | |

A3 | {0.00, 1.00} | {0.00, 1.00} | {0.00, 1.00} | {0.29, 0.47} | {0.83, 0.35} | |

(b) | ||||||

Alt | C6 | C7 | C8 | C9 | C10 | C11 |

A1 | {0.35, 0.68} | {0.00, 0.67} | {0.64, 0.64} | {0.49, 0.86} | {0.21, 0.70} | {0.20, 0.50} |

A2 | {0.00, 1.00} | {0.00, 1.00} | {1.00, 0.77} | {0.00, 1.00} | {0.62, 0.75} | {0.00, 1.00} |

A3 | {0.44, 0.66} | {0.05, 1.00} | {1.00, 1.00} | {0.51, 0.71} | {0.00, 1.00} | {0.57, 0.61} |

(a) | ||||||

Alt | C1 | C2 | C3 | C4 | C5 | |

A1 | {0.90, 0.28} | {0.97, 0.18} | {0.00, 0.56} | {0.00, 1.00} | {0.00, 1.00} | |

A2 | {0.88, 0.28} | {0.96, 0.28} | {0.72, 0.51} | {0.60, 0.62} | {0.85, 0.38} | |

A3 | {0.00, 1.00} | {0.00, 1.00} | {0.00, 1.00} | {0.46, 0.64} | {0.85, 0.38} | |

(b) | ||||||

Alt | C6 | C7 | C8 | C9 | C10 | C11 |

A1 | {0.65, 0.55} | {0.00, 0.44} | {0.85, 0.38} | {0.82, 0.46} | {0.90, 0.27} | {0.79, 0.37} |

A2 | {0.00, 1.00} | {0.00, 1.00} | {0.00, 0.41} | {0.00, 1.00} | {0.93, 0.28} | {0.00, 1.00} |

A3 | {0.67, 0.55} | {0.65, 1.00} | {0.00, 1.00} | {0.83, 0.42} | {0.00, 1.00} | {0.85, 0.38} |

IS |
---|

0, 0.010554 |

0, 0.028211 |

0, 0.011850 |

IS | Rank |
---|---|

4.242634 | 1 |

4.242594 | 3 |

4.242632 | 2 |

IS | Rank |
---|---|

4.24264 | 1 |

4.24259 | 3 |

4.24263 | 2 |

IS | Rank |
---|---|

4.24634 | 1 |

4.24594 | 3 |

4.24632 | 2 |

IS | Rank |
---|---|

4.242628 | 1 |

4.242594 | 3 |

4.242626 | 2 |

IS | Rank |
---|---|

4.242628 | 1 |

4.242594 | 3 |

4.242626 | 2 |

ALT | DA-LPFS Ranking | TOPSIS-LPFS Ranking | d | d^{2} |
---|---|---|---|---|

A1 | 3 | 5 | 2 | 4 |

A2 | 1 | 3 | 2 | 4 |

A3 | 4 | 1 | 3 | 9 |

A4 | 3 | 4 | 1 | 1 |

A5 | 2 | 2 | 0 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Villa Silva, A.J.; Pérez-Domínguez, L.; Martínez Gómez, E.; Luviano-Cruz, D.; Valles-Rosales, D.
Dimensional Analysis under Linguistic Pythagorean Fuzzy Set. *Symmetry* **2021**, *13*, 440.
https://doi.org/10.3390/sym13030440

**AMA Style**

Villa Silva AJ, Pérez-Domínguez L, Martínez Gómez E, Luviano-Cruz D, Valles-Rosales D.
Dimensional Analysis under Linguistic Pythagorean Fuzzy Set. *Symmetry*. 2021; 13(3):440.
https://doi.org/10.3390/sym13030440

**Chicago/Turabian Style**

Villa Silva, Aldo Joel, Luis Pérez-Domínguez, Erwin Martínez Gómez, David Luviano-Cruz, and Delia Valles-Rosales.
2021. "Dimensional Analysis under Linguistic Pythagorean Fuzzy Set" *Symmetry* 13, no. 3: 440.
https://doi.org/10.3390/sym13030440