# Dynamic Intuitionistic Fuzzy Multi-Attribute Group Decision-Making Based on Power Geometric Weighted Average Operator and Prediction Model

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Concepts and Operational Rules

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

#### 2.2. The Comparing Method of IFVs

**Definition**

**4.**

- If$s\left(\alpha \right)<s\left(\beta \right)$, then$\alpha \prec \beta $,
- If$s\left(\alpha \right)>s\left(\beta \right)$, then$\alpha \succ \beta $,

## 3. The Proposed IF-Operators and IFVs-GM(1,1)-PM

#### 3.1. The IFPGWA Operator

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Example**

**1.**

#### 3.2. The DIFPGWA Operator

**Definition**

**8.**

**Example**

**2.**

#### 3.3. The Improved IFVs-GM(1,1)-PM

**Definition**

**9.**

**Example**

**3.**

#### 3.4. Future Adjustment Coefficient

**Definition**

**10.**

**Example**

**4.**

## 4. Decision-Making Steps

**Description**

**1.**

**Step 1.**Determine expert weights and group decision matrix

**Step 2.**Determine attribute weights

**Step 3.**Determine all alternative’s comprehensive evaluation value by operator in each period

**Step 4.**Determine time weights

**Step 5.**Determine all alternative’s comprehensive evaluation value by operator

**Step 6.**Predicting the $g+1$ period comprehensive evaluation value

**Step 7.**Comprehensive evaluation

## 5. Example

**Step 1.**Based on the above evaluation information given by the experts in each period, the group mean evaluation matrix could be constructed. Then, using Equations (18)–(20), to get the expert weights. The group decision matrices for each period are calculated by Equation (21), which is shown in Table 6.**Step 2.**The closer the evaluation value of the attribute is, the smaller the weight should be given. Otherwise, the greater weight should be given. The attribute weights for each period are calculated by Equations (22)–(24), which are shown in Table 7.**Step 3.**The comprehensive evaluation values are calculated by Equation (25) in each period, which are shown in Table 8.**Step 4.**In determining the time weight, experts think the recent data is more important. The time weight vectors that are calculated by Equation (26) are $\theta \left(t\right)={\left(0.154,0.292,0.554\right)}^{T}$, where $\delta =0.3$.**Step 5.**The comprehensive evaluation values are calculated by Equation (27), which are shown in Table 9.**Step 6.**The predict comprehensive evaluation values are calculated by Equations (28)–(30), which are shown in Table 10.**Step 7.**The comprehensive evaluation values are calculated by Equation (31), which are shown in Table 11. Let $\psi =0.3$.All alternatives are sorted according to the ranking method of IFVs. When $\delta =0.3$, the final sort result is ${A}_{2}\succ {A}_{1}\succ {A}_{3}\succ {A}_{5}\succ {A}_{4}$.**Step 8.**Result Analysis

#### Result Analysis

## 6. Comparative Analysis

- (1)

- (2)

- (3)

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1986**, 20, 87–96. [Google Scholar] [CrossRef] - Bao, T.T.; Xie, X.L.; Meng, P.P. Intuitionistic fuzzy hybrid multi-criteria decision making based on prospect theory and evidential reasoning. Syst. Eng. Theory Pract.
**2017**, 37, 460–468. [Google Scholar] - Chen, Q.W. Application of Intuitionistic Fuzzy Entropy Weight Method to Evaluate Quality of Medical Care. J. Math. Med.
**2010**, 23, 85–486. [Google Scholar] - Yu, G.F.; Li, D.F.; Qiu, J.M. Solution to intuitionistic fuzzy linear programming and its application. Control Decis.
**2015**, 30, 640–644. [Google Scholar] - Li, B.P.; Chen, H.Y. Intuitionistic Fuzzy Area Set and Its Application in Pattern Recognition. Fuzzy Syst. Math.
**2015**, 29, 134–141. [Google Scholar] - Liu, P.D.; Li, Y.; Antuchevičienė, J. Multi-criteria decision-making method based on intuitionistic trapezoidal fuzzy prioritised OWA operatorm. Technol. Econ. Dev. Econ.
**2016**, 22, 453–469. [Google Scholar] [CrossRef] - Hashemi, H.; Mousavi, S.M.; Zavadskas, E.K.; Chalekaee, A.; Turskis, Z. A New Group Decision Model Based on Grey-Intuitionistic Fuzzy-ELECTRE and VIKOR for Contractor Assessment Problem. Sustainability
**2018**, 10, 1635. [Google Scholar] [CrossRef] - Kahraman, C.; Ghorabaee, M.; Zavadskas, E.; Onar, S.; Yazdani, M.; Oztaysi, B. Intuitionistic fuzzy EDAS method: An application to solid waste disposal site selection. J. Environ. Eng. Landsc. Manag.
**2017**, 25, 1–12. [Google Scholar] [CrossRef] - Yager, R.R. OWA aggregation of intuitionistic fuzzy sets. Int. J. Gen. Syst.
**2009**, 38, 617–641. [Google Scholar] [CrossRef] - Xu, Z.S. Intuitionistic Fuzzy Aggregation Operators. IEEE Trans. Fuzzy Syst.
**2007**, 15, 1179–1187. [Google Scholar] - Xu, Z.S.; Yager, R.R. Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gen. Syst.
**2006**, 35, 417–433. [Google Scholar] [CrossRef] - Tan, C.Q.; Chen, X.H. Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making. Int. J. Intell. Syst.
**2011**, 26, 659–686. [Google Scholar] [CrossRef] - Xu, Z.S. Approaches to Multiple Attribute Group Decision Making Based on Intuitionistic Fuzzy Power Aggregation Operators; Elsevier Science Publishers B. V.: New York, NY, USA, 2011. [Google Scholar]
- Wei, G.W. Gray Relational Analysis Method for Intuitionistic Fuzzy Multiple Attribute Decision Making. Expert Syst. Appl.
**2011**, 38, 11671–11677. [Google Scholar] [CrossRef] - Yue, Z.L. TOPSIS-based group decision-making methodology in intuitionistic fuzzy setting. Inf. Sci.
**2014**, 277, 141–153. [Google Scholar] [CrossRef] - Li, P.; Wu, J.M.; Zhu, J.J. Stochastic multi-criteria decision-making methods based on new intuitionistic fuzzy distance. Syst. Eng. Theory Pract.
**2014**, 34, 1517–1524. [Google Scholar] - Dymova, L.; Sevastjanov, P. A new approach to the rule-base evidential reasoning in the intuitionistic fuzzy setting. Knowl-Based Syst.
**2014**, 61, 109–117. [Google Scholar] [CrossRef] - Iancu, I. Intuitionistic fuzzy similarity measures based on Frank t-norms family. Pattern Recognit. Lett.
**2014**, 42, 128–136. [Google Scholar] [CrossRef] - Xu, Z.S.; Yager, R.R. Dynamic intuitionistic fuzzy multi-attribute decision making. Int. J. Approx. Reason.
**2008**, 48, 246–262. [Google Scholar] [CrossRef] - Wei, G.W. Some geometric aggregation functions and their application to dynamic multiple attribute decision making in the intuitionistic fuzzy setting. Int. J. Uncertain. Fuzziness Knowl-Based Syst.
**2009**, 17, 179–196. [Google Scholar] [CrossRef] - Su, Z.X.; Chen, M.Y.; Xia, G.P.; Wang, L. An interactive method for dynamic intuitionistic fuzzy multi-attribute group decision making. Expert Syst. Appl.
**2011**, 38, 15286–15295. [Google Scholar] [CrossRef] - Park, J.H.; Cho, H.J.; Kwun, Y.C. Extension of the VIKOR method to dynamic intuitionistic fuzzy multiple attribute decision making. Comput. Math. Appl.
**2013**, 65, 731–744. [Google Scholar] [CrossRef] - Chen, W.; Yang, Z.L.; Zhou, W.; Chen, H. Dynamic Intuitionistic Fuzzy Compromise Decision Making Method Based on Time Degrees. Oper. Res. Manag. Sci.
**2016**, 25, 83–89. [Google Scholar] - Li, P.; Liu, S.F.; Zhu, J.J. GM(1,1) PM based on intuitionistic fuzzy numbers. Control Decis.
**2013**, 28, 1583–1586. [Google Scholar] - Chen, Z.P.; Yang, W. A new multiple attribute group decision making method in intuitionistic fuzzy setting. Appl. Math. Model.
**2011**, 35, 4424–4437. [Google Scholar] [CrossRef] - Wang, Y.M. Using the method of maximizing deviation to make decision for multiindices. J. Syst. Eng. Electron.
**1997**, 8, 21–26. [Google Scholar] - Wu, Z.B.; Chen, Y. The Maximizing Deviation Method for Group Multiple Attribute Decision Making under Linguistic Environment; Elsevier North-Holland, Inc.: New York, NY, USA, 2007. [Google Scholar]
- Guo, Y.J.; Yao, Y.; Yi, P.T. A Method and Application of Dynamic Comprehensive Evaluation. Syst. Eng. Theory Pract.
**2007**, 27, 154–158. [Google Scholar] [CrossRef] - Xu, Z.S. Intuitionistic preference relations and their application in group decision making. Inf. Sci. Int. J.
**2007**, 177, 2363–2379. [Google Scholar] [CrossRef] - Chen, S.M.; Tan, J.M. Handling Multicriteria Fuzzy Decision-Making Problems Based on Vague Set Theory; Elsevier North-Holland, Inc.: New York, NY, USA, 1994. [Google Scholar]
- Hong, D.H.; Choi, C.H. Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst.
**2000**, 114, 103–113. [Google Scholar] [CrossRef] - Aczél, J.; Saaty, T.L. Procedures for synthesizing ratio judgements. J. Math. Psychol.
**1983**, 27, 93–102. [Google Scholar] [CrossRef] - Xu, Z.S.; Yager, R.R. Power-Geometric Operators and Their Use in Group Decision Making. IEEE Trans. Fuzzy Syst.
**2010**, 18, 94–105. [Google Scholar] - Zhang, Y.Y.; Feng, Q.; Zhou, D.Y.; Zhang, K. Multi-Attribute Dynamic Threat Assessment in Air Combat Based on Intuitionistic Fuzzy Sets. Electron. Opt. Control
**2015**, 22, 17–21. [Google Scholar] - Mei, X.L. Dynamic Intuitionistic Fuzzy Multi-attribute Decision Making Method Based on Similarity. Stat. Decis.
**2016**, 15, 22–24. [Google Scholar] - Du, Y.; Wu, Z.Q.; Li, D.F. Dynamic Intuitional Fuzzy Multiple Attribute Group Decision-making Method Based on DIFHA Operator. Fire Control Command Control
**2011**, 8, 15–18. [Google Scholar]

Time Series A | Scores Degree of A | Time Series B | Scores Degree of B |
---|---|---|---|

(0.4, 0.1) | 0.3 | (0.60, 0.10) | 0.50 |

(0.5, 0.1) | 0.4 | (0.55, 0.10) | 0.45 |

(0.6, 0.1) | 0.5 | (0.50, 0.10) | 0.40 |

Time Series | Type | Result | Scores Degree |
---|---|---|---|

A | Result of aggregation | (0.52, 0.10) | 0.42 |

Result of prediction | (0.50, 0.00) | 0.50 | |

comprehensive evaluation | (0.51, 0.00) | 0.51 | |

B | Result of aggregation | (0.53, 0.10) | 0.43 |

Result of prediction | (0.48, 0.12) | 0.36 | |

comprehensive evaluation | (0.506, 0.11) | 0.396 |

$\mathbf{Period}\text{}{\mathit{t}}_{1}$ | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | |
---|---|---|---|---|---|---|

Experts ${e}_{1}$ | ${C}_{1}$ | (0.56, 0.33) | (0.56, 0.16) | (0.78, 0.13) | (0.73, 0.27) | (0.88, 0.12) |

${C}_{2}$ | (0.57, 0.32) | (0.58, 0.16) | (0.74, 0.17) | (0.48, 0.52) | (0.82, 0.18) | |

${C}_{3}$ | (0.57, 0.31) | (0.61, 0.17) | (0.83, 0.07) | (0.02, 0.40) | (0.36, 0.24) | |

Experts ${e}_{2}$ | ${C}_{1}$ | (0.63, 0.34) | (0.65, 0.16) | (0.82, 0.07) | (0.72, 0.14) | (0.77, 0.16) |

${C}_{2}$ | (0.63, 0.35) | (0.64, 0.16) | (0.81, 0.08) | (0.70, 0.15) | (0.58, 0.42) | |

${C}_{3}$ | (0.57, 0.31) | (0.63, 0.17) | (0.78, 0.13) | (0.78, 0.22) | (0.16, 0.47) | |

Experts ${e}_{3}$ | ${C}_{1}$ | (0.66, 0.33) | (0.52, 0.17) | (0.80, 0.09) | (0.58, 0.42) | (0.70, 0.30) |

${C}_{2}$ | (0.63, 0.32) | (0.53, 0.16) | (0.79, 0.09) | (0.76, 0.11) | (0.28, 0.72) | |

${C}_{3}$ | (0.63, 0.34) | (0.48, 0.19) | (0.77, 0.10) | (0.65, 0.16) | (0.19, 0.25) |

$\mathbf{Period}\text{}{\mathit{t}}_{2}$ | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | |
---|---|---|---|---|---|---|

Experts ${e}_{1}$ | ${C}_{1}$ | (0.62, 0.33) | (0.51, 0.17) | (0.75, 0.12) | (0.50, 0.50) | (0.57, 0.11) |

${C}_{2}$ | (0.60, 0.35) | (0.51, 0.17) | (0.72, 0.14) | (0.85, 0.15) | (0.85, 0.15) | |

${C}_{3}$ | (0.66, 0.32) | (0.62, 0.17) | (0.70, 0.14) | (0.74, 0.13) | (0.41, 0.06) | |

Experts ${e}_{2}$ | ${C}_{1}$ | (0.61, 0.32) | (0.61, 0.32) | (0.68, 0.15) | (0.40, 0.39) | (0.40, 0.60) |

${C}_{2}$ | (0.55, 0.18) | (0.65, 0.16) | (0.67, 0.16) | (0.74, 0.26) | (0.24, 0.31) | |

${C}_{3}$ | (0.53, 0.16) | (0.61, 0.32) | (0.64, 0.16) | (0.87, 0.13) | (0.84, 0.16) | |

Experts ${e}_{3}$ | ${C}_{1}$ | (0.77, 0.10) | (0.71, 0.14) | (0.68, 0.15) | (0.44, 0.20) | (0.72, 0.28) |

${C}_{2}$ | (0.71, 0.14) | (0.74, 0.13) | (0.72, 0.13) | (0.51, 0.18) | (0.75, 0.25) | |

${C}_{3}$ | (0.70, 0.16) | (0.77, 0.10) | (0.63, 0.17) | (0.70, 0.30) | (0.43, 0.39) |

$\mathbf{Period}\text{}{\mathit{t}}_{3}$ | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | |
---|---|---|---|---|---|---|

Experts ${e}_{1}$ | ${C}_{1}$ | (0.62, 0.32) | (0.72, 0.14) | (0.56, 0.16) | (0.39, 0.61) | (0.91, 0.09) |

${C}_{2}$ | (0.72, 0.14) | (0.83, 0.07) | (0.53, 0.17) | (0.13, 0.33) | (0.15, 0.53) | |

${C}_{3}$ | (0.68, 0.25) | (0.62, 0.32) | (0.54, 0.16) | (0.63, 0.21) | (0.07, 0.16) | |

Experts ${e}_{2}$ | ${C}_{1}$ | (0.62, 0.32) | (0.67, 0.25) | (0.55, 0.16) | (0.47, 0.11) | (0.87, 0.02) |

${C}_{2}$ | (0.67, 0.25) | (0.61, 0.32) | (0.49, 0.18) | (0.41, 0.49) | (0.35, 0.30) | |

${C}_{3}$ | (0.61, 0.32) | (0.74, 0.17) | (0.40, 0.22) | (0.21, 0.44) | (0.92, 0.08) | |

Experts ${e}_{3}$ | ${C}_{1}$ | (0.82, 0.13) | (0.61, 0.32) | (0.38, 0.23) | (0.42, 0.50) | (0.51, 0.21) |

${C}_{2}$ | (0.83, 0.13) | (0.81, 0.13) | (0.36, 0.24) | (0.14, 0.59) | (0.81, 0.19) | |

${C}_{3}$ | (0.62, 0.32) | (0.82, 0.13) | (0.37, 0.23) | (0.23, 0.65) | (0.26, 0.74) |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | ||
---|---|---|---|---|---|---|

Period ${t}_{1}$ | ${C}_{1}$ | (0.62, 0.33) | (0.57, 0.16) | (0.80, 0.09) | (0.68, 0.28) | (0.78, 0.19) |

${C}_{2}$ | (0.61, 0.33) | (0.58, 0.16) | (0.78, 0.11) | (0.65, 0.25) | (0.53, 0.47) | |

${C}_{3}$ | (0.58, 0.31) | (0.58, 0.17) | (0.79, 0.10) | (0.29, 0.25) | (0.22, 0.32) | |

Period ${t}_{2}$ | ${C}_{1}$ | (0.65, 0.27) | (0.61, 0.21) | (0.70, 0.14) | (0.44, 0.37) | (0.56, 0.35) |

${C}_{2}$ | (0.62, 0.22) | (0.63, 0.15) | (0.70, 0.14) | (0.70, 0.20) | (0.58, 0.23) | |

${C}_{3}$ | (0.63, 0.21) | (0.66, 0.20) | (0.65, 0.16) | (0.77, 0.19) | (0.52, 0.23) | |

Period ${t}_{3}$ | ${C}_{1}$ | (0.66, 0.27) | (0.66, 0.24) | (0.50, 0.18) | (0.42, 0.47) | (0.77, 0.10) |

${C}_{2}$ | (0.73, 0.17) | (0.75, 0.16) | (0.46, 0.19) | (0.19, 0.49) | (0.33, 0.36) | |

${C}_{3}$ | (0.63, 0.30) | (0.73, 0.20) | (0.42, 0.21) | (0.30, 0.47) | (0.26, 0.43) |

${\mathit{w}}_{1}$ | ${\mathit{w}}_{2}$ | ${\mathit{w}}_{3}$ | |
---|---|---|---|

Period ${t}_{1}$ | 0.25888 | 0.29594 | 0.44518 |

Period ${t}_{2}$ | 0.43661 | 0.23824 | 0.32415 |

Period ${t}_{3}$ | 0.27301 | 0.37785 | 0.34914 |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | |
---|---|---|---|---|---|

Period ${t}_{1}$ | (0.599, 0.322) | (0.576, 0.166) | (0.789, 0.099) | (0.458, 0.258) | (0.404, 0.342) |

Period ${t}_{2}$ | (0.635, 0.242) | (0.634, 0.193) | (0.683, 0.146) | (0.588, 0.276) | (0.554, 0.284) |

Period ${t}_{3}$ | (0.675, 0.244) | (0.716, 0.196) | (0.458, 0.193) | (0.279, 0.478) | (0.383, 0.324) |

Comprehensive Evaluation Value by Operator | Scores Degree | Accuracy Degree | |
---|---|---|---|

${A}_{1}$ | (0.649, 0.257) | 0.392 | 0.906 |

${A}_{2}$ | (0.665, 0.191) | 0.474 | 0.856 |

${A}_{3}$ | (0.563, 0.164) | 0.399 | 0.727 |

${A}_{4}$ | (0.379, 0.389) | −0.01 | 0.768 |

${A}_{5}$ | (0.433, 0.315) | 0.118 | 0.748 |

$\mathit{a}$ | $\mathit{b}$ | Predict Comprehensive Evaluation Value | Scores Degree | Accuracy Degree | |
---|---|---|---|---|---|

${A}_{1}$ | −0.09 | 0.35 | (0.674, 0.204) | 0.470 | 0.878 |

${A}_{2}$ | −0.17 | 0.33 | (0.679, 0.064) | 0.615 | 0.743 |

${A}_{3}$ | 0.68 | 1.19 | (0.392, 0.258) | 0.134 | 0.650 |

${A}_{4}$ | 9.04 | 3.53 | (0.358, 0.358) | 0.000 | 0.716 |

${A}_{5}$ | 1.28 | 0.52 | (0.363, 0.344) | 0.019 | 0.707 |

Comprehensive Evaluation Value | Scores Degree | Accuracy Degree | |
---|---|---|---|

${A}_{1}$ | (0.657, 0.240) | 0.417 | 0.896 |

${A}_{2}$ | (0.669, 0.138) | 0.532 | 0.807 |

${A}_{3}$ | (0.517, 0.188) | 0.330 | 0.705 |

${A}_{4}$ | (0.373, 0.379) | −0.007 | 0.752 |

${A}_{5}$ | (0.413, 0.323) | 0.089 | 0.736 |

**Table 12.**The final sort results of the case in [34].

Decision Method | Weight | Result |
---|---|---|

Decision method in [34] | Weights used in [34] | ${A}_{2}\succ {A}_{3}\succ {A}_{1}\succ {A}_{4}$ |

the proposed DIFPGWA operator | Weights used in [34] | ${A}_{2}\succ {A}_{3}\succ {A}_{1}\succ {A}_{4}$ |

The TFVs-GM(1,1)-PM | Weights used in [34] | ${A}_{2}\succ {A}_{1}\succ {A}_{3}\succ {A}_{4}$ |

Comprehensive evaluation results | Weights used in [34] | ${A}_{2}\succ {A}_{3}\succ {A}_{1}\succ {A}_{4}$ |

the proposed DIFPGWA operator | Weights are unknown | ${A}_{2}\succ {A}_{3}\succ {A}_{4}\succ {A}_{1}$ |

The TFVs-GM(1,1)-PM | Weights are unknown | ${A}_{2}\succ {A}_{1}\succ {A}_{4}\succ {A}_{3}$ |

Comprehensive evaluation results | Weights are unknown | ${A}_{2}\succ {A}_{3}\succ {A}_{4}\succ {A}_{1}$ |

**Table 13.**The final sort results of the case in [35].

Decision Method | Weight | Result |
---|---|---|

Decision method in [35] | Weights used in [35] | ${A}_{4}\succ {A}_{3}\succ {A}_{1}\succ {A}_{2}$ |

the proposed DIFPGWA operator | Weights used in [35] | ${A}_{4}\succ {A}_{3}\succ {A}_{1}\succ {A}_{2}$ |

The TFVs-GM(1,1)-PM | Weights used in [35] | ${A}_{1}\succ {A}_{3}\succ {A}_{4}\succ {A}_{2}$ |

Comprehensive evaluation results | Weights used in [35] | ${A}_{1}\succ {A}_{3}\succ {A}_{4}\succ {A}_{2}$ |

the proposed DIFPGWA operator | Weights are unknown | ${A}_{3}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ |

The TFVs-GM(1,1)-PM | Weights are unknown | ${A}_{3}\succ {A}_{1}\succ {A}_{4}\succ {A}_{2}$ |

Comprehensive evaluation results | Weights are unknown | ${A}_{3}\succ {A}_{1}\succ {A}_{4}\succ {A}_{2}$ |

**Table 14.**The final sort results of the case in [36].

Decision Method | Weight | Result |
---|---|---|

Decision method in [36] | Weights used in [36] | ${A}_{1}\succ {A}_{2}\succ {A}_{3}$ |

the proposed DIFPGWA operator | Weights used in [36] | ${A}_{1}\succ {A}_{2}\succ {A}_{3}$ |

The TFVs-GM(1,1)-PM | Weights used in [36] | ${A}_{2}\succ {A}_{1}\succ {A}_{3}$ |

Comprehensive evaluation results | Weights used in [36] | ${A}_{2}\succ {A}_{1}\succ {A}_{3}$ |

the proposed DIFPGWA operator | Weights are unknown | ${A}_{1}\succ {A}_{2}\succ {A}_{3}$ |

The TFVs-GM(1,1)-PM | Weights are unknown | ${A}_{1}\succ {A}_{2}\succ {A}_{3}$ |

Comprehensive evaluation results | Weights are unknown | ${A}_{1}\succ {A}_{2}\succ {A}_{3}$ |

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**MDPI and ACS Style**

Yin, K.; Wang, P.; Jin, X.
Dynamic Intuitionistic Fuzzy Multi-Attribute Group Decision-Making Based on Power Geometric Weighted Average Operator and Prediction Model. *Symmetry* **2018**, *10*, 536.
https://doi.org/10.3390/sym10110536

**AMA Style**

Yin K, Wang P, Jin X.
Dynamic Intuitionistic Fuzzy Multi-Attribute Group Decision-Making Based on Power Geometric Weighted Average Operator and Prediction Model. *Symmetry*. 2018; 10(11):536.
https://doi.org/10.3390/sym10110536

**Chicago/Turabian Style**

Yin, Kedong, Pengyu Wang, and Xue Jin.
2018. "Dynamic Intuitionistic Fuzzy Multi-Attribute Group Decision-Making Based on Power Geometric Weighted Average Operator and Prediction Model" *Symmetry* 10, no. 11: 536.
https://doi.org/10.3390/sym10110536