# Multiple Criteria Group Decision-Making Considering Symmetry with Regards to the Positive and Negative Ideal Solutions via the Pythagorean Normal Cloud Model for Application to Economic Decisions

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Intuitionistic Fuzzy Sets and Pythagorean Fuzzy Sets

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

#### 2.2. NC and the Backward Cloud Generator

**Definition**

**6.**

## 3. Pythagorean Normal Cloud

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Theorem**

**1.**

**Proof.**

#### 3.1. Backward Cloud Generator and Aggregation Operators for PNCs

**Definition**

**10.**

**Theorem**

**2.**

**Proof.**

**Definition**

**11.**

**Theorem**

**3.**

**Proof.**

#### 3.2. Distance Measures for PNCs

**Definition**

**12.**

**Property**

**1.**

## 4. PNC-Based MCGDM Method

## 5. Results

#### 5.1. Empirical Application

#### 5.2. Comparative Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Changes in the normalized distances ${d}_{i}^{\ast}$ due to changes in the weighting vector. Note: the cases represent those given in Table 2.

Case No. | ${\mathit{\u25b5}}_{3}$ | ${\mathit{w}}^{\prime}$ | The Final Ranking |
---|---|---|---|

1 | −0.499 | $(0.5994,0.3996,0.0010)$ | ${x}_{2}\succ {x}_{3}\succ {x}_{4}\succ {x}_{1}$ |

2 | −0.3 | $(0.4800,0.3200,0.2000)$ | ${x}_{2}\succ {x}_{4}\succ {x}_{3}\succ {x}_{1}$ |

3 | −0.1 | $(0.3600,0.2400,0.4000)$ | ${x}_{4}\succ {x}_{2}\succ {x}_{3}\succ {x}_{1}$ |

4 | 0.2 | $(0.2400,0.1600,0.6000)$ | ${x}_{4}\succ {x}_{2}\succ {x}_{3}\succ {x}_{1}$ |

5 | 0.4 | $(0.1200,0.0800,0.8000)$ | ${x}_{4}\succ {x}_{3}\succ {x}_{2}\succ {x}_{1}$ |

6 | 0.499 | $(0.0006,0.0004,0.9990)$ | ${x}_{4}\succ {x}_{3}\succ {x}_{1}\succ {x}_{2}$ |

**Table 2.**The ranking orders rendered by different methods. UPLC, pure linguistic information cloud; UPLHAA, pure linguistic hybrid harmonic averaging.

Method | The Final Ranking | The Best Camera | The Worst Camera |
---|---|---|---|

The method in [26] | ${x}_{4}\succ {x}_{3}\succ {x}_{2}\succ {x}_{1}$ | ${x}_{4}$ | ${x}_{1}$ |

The method in [41] | ${x}_{2}\succ {x}_{4}\succ {x}_{3}\succ {x}_{1}$ | ${x}_{2}$ | ${x}_{1}$ |

UPLC [42] | ${x}_{4}\succ {x}_{2}\succ {x}_{3}\succ {x}_{1}$ | ${x}_{4}$ | ${x}_{1}$ |

UPLHAA [43] | ${x}_{4}\succ {x}_{3}\succ {x}_{1}\succ {x}_{2}$ | ${x}_{4}$ | ${x}_{2}$ |

The proposed method | ${x}_{4}\succ {x}_{2}\succ {x}_{3}\succ {x}_{1}$ | ${x}_{4}$ | ${x}_{1}$ |

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**MDPI and ACS Style**

Zhou, J.; Su, W.; Baležentis, T.; Streimikiene, D.
Multiple Criteria Group Decision-Making Considering Symmetry with Regards to the Positive and Negative Ideal Solutions via the Pythagorean Normal Cloud Model for Application to Economic Decisions. *Symmetry* **2018**, *10*, 140.
https://doi.org/10.3390/sym10050140

**AMA Style**

Zhou J, Su W, Baležentis T, Streimikiene D.
Multiple Criteria Group Decision-Making Considering Symmetry with Regards to the Positive and Negative Ideal Solutions via the Pythagorean Normal Cloud Model for Application to Economic Decisions. *Symmetry*. 2018; 10(5):140.
https://doi.org/10.3390/sym10050140

**Chicago/Turabian Style**

Zhou, Jinming, Weihua Su, Tomas Baležentis, and Dalia Streimikiene.
2018. "Multiple Criteria Group Decision-Making Considering Symmetry with Regards to the Positive and Negative Ideal Solutions via the Pythagorean Normal Cloud Model for Application to Economic Decisions" *Symmetry* 10, no. 5: 140.
https://doi.org/10.3390/sym10050140