# Selecting Cooking Methods to Decrease Persistent Organic Pollutant Concentrations in Food of Animal Origin Using a Consensus Decision-Making Model

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## Abstract

**:**

## 1. Introduction

## 2. Definition of Two Consistent Interval Fuzzy Preference Relations

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

## 3. Decision Rules for Ideal Interval Fuzzy Preference Relations

- Form I: Element ${r}_{ij}^{*}$ is a crisp number, i.e., ${r}_{ij}^{*}\in [0,1]$.
- Form II: Element ${r}_{ij}^{*}$ is an interval number, i.e., ${r}_{ij}^{*}\subseteq [0,1]$.

## 4. Construction of Ideal Fuzzy Preference Relations with $\mathbf{I}$-Consistency Using Different Social Choice Modes

#### 4.1. Construction of Ideal Fuzzy Preference Relations with I-Consistency Using the Freedom Principle

#### 4.2. Construction of Ideal Fuzzy Preference Relations with I-Consistency Using the Fraternity Principle

#### 4.3. Construction of Ideal Fuzzy Preference Relations with I-Consistency Using the Equality Principle

#### 4.4. Construction of Ideal Fuzzy Preference Relations with I-Consistency Using the Mixed Principles

## 5. Construction of Ideal Fuzzy Preference Relations with II-Consistency Using Different Social Choice Modes

#### 5.1. Construction of Ideal Fuzzy Preference Relations with II-Consistency Using the Freedom Principle

#### 5.2. Construction of Ideal Fuzzy Preference Relations with II-Consistency Using the Fraternity Principle

#### 5.3. Construction of Ideal Fuzzy Preference Relations with II-Consistency Using the Equality Principle

#### 5.4. Construction of Ideal Fuzzy Preference Relations with II-Consistency Using Mixed Principles

## 6. Case Study of a Method for Selecting a Cooking Method to Decrease the POP Concentrations in Food

#### 6.1. I-Consistency

#### 6.2. II-Consistency

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Freedom | Fraternity | Equality | Mixture | |
---|---|---|---|---|

$min$ | 0.2010 | 0.096 | 0 | 0.0481 |

D | 0.096 | 0.0956 | ||

${\omega}_{1}$ | 0.1768 | 0.1699 | 0.1909 | 0.1805 |

${\omega}_{2}$ | 0.4188 | 0.4194 | 0.3704 | 0.4136 |

${\omega}_{3}$ | 0.2573 | 0.2603 | 0.2653 | 0.2453 |

${\omega}_{4}$ | 0.1471 | 0.1504 | 0.1734 | 0.1605 |

${\alpha}_{12}^{1}$ | 0.4792 | 0.5262 | 0.5917 | 0.5274 |

${\alpha}_{13}^{1}$ | 1 | 0.6243 | 0.5697 | 0.6237 |

${\alpha}_{14}^{1}$ | 0.1 | 0.2549 | 0.4276 | 0.4099 |

${\alpha}_{23}^{1}$ | 0.1559 | 0.4842 | 0.5285 | 0.2584 |

${\alpha}_{24}^{1}$ | 0.2279 | 0.3931 | 0.4644 | 0.1297 |

${\alpha}_{34}^{1}$ | 0.1 | 0.4034 | 0.5105 | 0.1 |

${\alpha}_{12}^{2}$ | 0.9792 | 0.9361 | 0.6361 | 1 |

${\alpha}_{13}^{2}$ | 0.1 | 0.1123 | 0.5288 | 0.1 |

${\alpha}_{14}^{2}$ | 0.1 | 0.1112 | 0.4961 | 0.1 |

${\alpha}_{23}^{2}$ | 0.2890 | 0.2815 | 0.4384 | 0.3146 |

${\alpha}_{24}^{2}$ | 0.7279 | 0.7058 | 0.5026 | 0.6297 |

${\alpha}_{34}^{2}$ | 0.65 | 0.6404 | 0.5273 | 0.5564 |

${\alpha}_{12}^{3}$ | 0.1 | 0.1 | 0.5573 | 0.1 |

${\alpha}_{13}^{3}$ | 1 | 0.9982 | 0.6737 | 1 |

${\alpha}_{14}^{3}$ | 1 | 1 | 0.6842 | 1 |

${\alpha}_{23}^{3}$ | 0.1 | 0.1 | 0.4649 | 0.1292 |

${\alpha}_{24}^{3}$ | 0.8186 | 0.8030 | 0.5197 | 0.7531 |

${\alpha}_{34}^{3}$ | 0.1 | 0.1 | 0.4081 | 0.1 |

$\mathbf{min}$ | ${\mathit{\omega}}_{1}$ | ${\mathit{\omega}}_{2}$ | ${\mathit{\omega}}_{3}$ | ${\mathit{\omega}}_{4}$ | |
---|---|---|---|---|---|

$\mathrm{freedom}\phantom{\rule{4.pt}{0ex}}\mathrm{principle}$ | 0.2369 | 0.1714 | 0.4 | 0.2571 | 0.1714 |

$\mathrm{fraternity}\phantom{\rule{4.pt}{0ex}}\mathrm{principle}$ | 0.12 | 0.2001 | 0.3997 | 0.2001 | 0.2001 |

$\mathrm{equality}\phantom{\rule{4.pt}{0ex}}\mathrm{principle}$ | 0 | 0.2339 | 0.2891 | 0.2547 | 0.2222 |

$\mathrm{mixed}\phantom{\rule{4.pt}{0ex}}\mathrm{principles}$ | 0.1137 | 0.2025 | 0.3347 | 0.2603 | 0.2025 |

$\mathbf{Principle}$ | $\mathbf{Weight}$ | $\mathbf{Change}{\mathit{R}}^{1}$ | $\mathbf{Change}{\mathit{R}}^{2}$ | $\mathbf{Change}{\mathit{R}}^{3}$ |
---|---|---|---|---|

freedom | ${\omega}_{1}$ | 0.1780 | 0.1698 | 0.1803 |

${\omega}_{2}$ | 0.4135 | 0.3731 | 0.3920 | |

${\omega}_{3}$ | 0.2596 | 0.2573 | 0.2414 | |

${\omega}_{4}$ | 0.1488 | 0.1998 | 0.1858 | |

fraternity | ${\omega}_{1}$ | 0.1709 | 0.1623 | 0.1868 |

${\omega}_{2}$ | 0.4169 | 0.3520 | 0.4177 | |

${\omega}_{3}$ | 0.2618 | 0.2347 | 0.2397 | |

${\omega}_{4}$ | 0.1504 | 0.2510 | 0.1555 | |

eequality | ${\omega}_{1}$ | 0.2008 | 0.2053 | 0.2271 |

${\omega}_{2}$ | 0.3505 | 0.3201 | 0.3604 | |

${\omega}_{3}$ | 0.2545 | 0.2740 | 0.2460 | |

${\omega}_{4}$ | 0.1942 | 0.2005 | 0.1665 | |

mixture | ${\omega}_{1}$ | 0.1804 | 0.1616 | 0.2023 |

${\omega}_{2}$ | 0.4055 | 0.4045 | 0.4121 | |

${\omega}_{3}$ | 0.2447 | 0.2517 | 0.2388 | |

${\omega}_{4}$ | 0.1693 | 0.1820 | 0.1439 |

$\mathbf{Principle}$ | $\mathbf{Weight}$ | $\mathbf{Change}{\mathit{R}}^{1}$ | $\mathbf{Change}{\mathit{R}}^{2}$ | $\mathbf{Change}{\mathit{R}}^{3}$ |
---|---|---|---|---|

freedom | ${\omega}_{1}$ | 0.1875 | 0.25 | 0.25 |

${\omega}_{2}$ | 0.4375 | 0.25 | 0.25 | |

${\omega}_{3}$ | 0.1875 | 0.25 | 0.25 | |

${\omega}_{4}$ | 0.1875 | 0.25 | 0.25 | |

fraternity | ${\omega}_{1}$ | 0.2037 | 0.25 | 0.25 |

${\omega}_{2}$ | 0.3899 | 0.25 | 0.25 | |

${\omega}_{3}$ | 0.2037 | 0.25 | 0.25 | |

${\omega}_{4}$ | 0.2037 | 0.25 | 0.25 | |

equality | ${\omega}_{1}$ | 0.2364 | 0.2465 | 0.2550 |

${\omega}_{2}$ | 0.2891 | 0.2725 | 0.2790 | |

${\omega}_{3}$ | 0.2504 | 0.2524 | 0.2443 | |

${\omega}_{4}$ | 0.2241 | 0.2286 | 0.2216 | |

mixture | ${\omega}_{1}$ | 0.2118 | 0.2286 | 0.2283 |

${\omega}_{2}$ | 0.3294 | 0.2571 | 0.3302 | |

${\omega}_{3}$ | 0.2471 | 0.2571 | 0.2568 | |

${\omega}_{4}$ | 0.2118 | 0.2571 | 0.1848 |

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## Share and Cite

**MDPI and ACS Style**

Tan, X.; Gong, Z.; Huang, M.; Wang, Z.-J.
Selecting Cooking Methods to Decrease Persistent Organic Pollutant Concentrations in Food of Animal Origin Using a Consensus Decision-Making Model. *Int. J. Environ. Res. Public Health* **2017**, *14*, 187.
https://doi.org/10.3390/ijerph14020187

**AMA Style**

Tan X, Gong Z, Huang M, Wang Z-J.
Selecting Cooking Methods to Decrease Persistent Organic Pollutant Concentrations in Food of Animal Origin Using a Consensus Decision-Making Model. *International Journal of Environmental Research and Public Health*. 2017; 14(2):187.
https://doi.org/10.3390/ijerph14020187

**Chicago/Turabian Style**

Tan, Xiao, Zaiwu Gong, Minji Huang, and Zhou-Jing Wang.
2017. "Selecting Cooking Methods to Decrease Persistent Organic Pollutant Concentrations in Food of Animal Origin Using a Consensus Decision-Making Model" *International Journal of Environmental Research and Public Health* 14, no. 2: 187.
https://doi.org/10.3390/ijerph14020187