# Fuzzy Group Consensus Decision Making and Its Use in Selecting Energy-Saving and Low-Carbon Technology Schemes in Star Hotels

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Triangular Fuzzy Numbers and Triangular Fuzzy Preference Relations

#### 2.2. Consistency and Fuzzy Weights of Triangular Fuzzy Preference Relations

**Definition**

**1.**

**Lemma**

**1.**

## 3. Inconsistency Measurement for Triangular Fuzzy Preference Relations

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Definition**

**2.**

**Definition**

**3.**

## 4. A Group Decision Making Consensus Model Based on Triangular Fuzzy Preference Relations

**Definition**

**4.**

**Definition**

**5.**

## 5. A Case Study of Selecting Energy-Saving and Low-Carbon Technology Schemes in Star Hotels

- (1)
- ${c}_{1}$: Energy efficiency. Efficiencies of the considered energy equipment and the overall technical system are two important factors in selecting energy-saving and low-carbon technology schemes for star hotels. Energy efficiency has been widely acknowledged as a promising approach for tackling environmental issues, and thus improving energy efficiency in star hotels is becoming increasingly significant. Energy efficiency programs offer a development prospect of renewable energy requirements. The energy efficient equipment in star hotels includes energy saving light bulbs, boilers and cooling equipment with high efficiency, recovery systems, and so on.
- (2)
- ${c}_{2}$: Capacity of energy-saving and carbon emission reduction. This capacity indicates the suitable performance of a technology scheme. The stronger the capacity, the better the technology scheme. Moreover, this criterion could be divided to two sub-criteria below.
- (i)
- ${c}_{21}$: Energy-saving capacity. This sub-criterion reflects the energy-saving performance and indicates how much energy is saved from the technology scheme.
- (ii)
- ${c}_{22}$: Low-carbon capacity. This sub-criterion reflects the low-carbon performance and shows how much carbon emission is reduced by the technology scheme.

- (3)
- ${c}_{3}$: Economic effectiveness. To rank energy-saving and low-carbon technology schemes, the investment cost plays an important role. The main goal of this criterion is lower investment cost with better performance. Therefore, this criterion is often measured and reflected by investment payback periods of per unit energy-saving and per unit carbon emission reduction.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Consistency Ratio (CR) | 0.01 | 0.05 | 0.1 | 0.15 |
---|---|---|---|---|

Threshold value (n = 3) | 0.0314 | 0.1573 | 0.3147 | 0.4720 |

Threshold value (n = 4) | 0.0352 | 0.1763 | 0.3562 | 0.5289 |

Threshold value (n > 4) | ~0.037 | ~0.185 | ~0.370 | ~0.555 |

TFPR | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ | |
---|---|---|---|---|---|

${\tilde{A}}_{(1)}^{{c}_{1}}$ | ${x}_{1}$ | 1 | (1/4, 1/3, 1/2) | (1/4, 1/3, 1/2) | (1/3, 1/2, 1) |

${x}_{2}$ | (2, 3, 4) | 1 | (3/2, 3/2, 2) | (1, 2, 3) | |

${x}_{3}$ | (2, 3, 4) | (1/2, 2/3, 2/3) | 1 | (1, 2, 3) | |

${x}_{4}$ | (1, 2, 3) | (1/3, 1/2, 1) | (1/3, 1/2, 1) | 1 | |

${\tilde{A}}_{(1)}^{{c}_{21}}$ | ${x}_{1}$ | 1 | (2/3, 5/6, 1) | (4/7, 5/7, 6/7) | (1, 3/2, 2) |

${x}_{2}$ | (1, 6/5, 3/2) | 1 | (4/7, 6/7, 1) | (1, 2, 5/2) | |

${x}_{3}$ | (7/6, 7/5, 7/4) | (1, 7/6, 7/4) | 1 | (3/2, 7/4, 5/2) | |

${x}_{4}$ | (1/2, 2/3, 1) | (2/5, 1/2, 1) | (2/5, 4/7, 2/3) | 1 | |

${\tilde{A}}_{(1)}^{{c}_{22}}$ | ${x}_{1}$ | 1 | (1/3, 1/2, 1) | (1/4, 1/3, 1/2) | (5/4, 2, 3) |

${x}_{2}$ | (1, 2, 3) | 1 | (1/2, 2/3, 1) | (3, 4, 5) | |

${x}_{3}$ | (2, 3, 4) | (1, 3/2, 2) | 1 | (5, 6, 7) | |

${x}_{4}$ | (1/3, 1/2, 4/5) | (1/5, 1/4, 1/3) | (1/7, 1/6, 1/5) | 1 | |

${\tilde{A}}_{(1)}^{{c}_{3}}$ | ${x}_{1}$ | 1 | (1/5, 1/4, 1/3) | (1/5, 1/4, 1/3) | (1/6, 1/5, 1/4) |

${x}_{2}$ | (3, 4, 5) | 1 | (3/2, 3/2, 2) | (1, 2, 3) | |

${x}_{3}$ | (3, 4, 5) | (1/2, 2/3, 2/3) | 1 | (1, 2, 3) | |

${x}_{4}$ | (4, 5, 6) | (1/3, 1/2, 1) | (1/3, 1/2, 1) | 1 |

TFPR | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ | |
---|---|---|---|---|---|

${\tilde{A}}_{(2)}^{{c}_{1}}$ | ${x}_{1}$ | 1 | (1/3, 1/2, 1) | (1/3, 2/3, 1) | (1, 4/3, 3/2) |

${x}_{2}$ | (1, 2, 3) | 1 | (1, 3/2, 2) | (2, 3, 7/2) | |

${x}_{3}$ | (1, 3/2, 3) | (1/2, 2/3, 1) | 1 | (1, 2, 3) | |

${x}_{4}$ | (2/3, 3/4, 1) | (2/7, 1/3, 1/2) | (1/3, 1/2, 1) | 1 | |

${\tilde{A}}_{(2)}^{{c}_{21}}$ | ${x}_{1}$ | 1 | (1/4, 2/7, 3/7) | (1/5, 2/5, 3/5) | (1/9, 2/9, 1/3) |

${x}_{2}$ | (7/3, 7/2, 4) | 1 | (6/5, 7/5, 8/5) | (2/3, 7/9, 8/9) | |

${x}_{3}$ | (5/3, 5/2, 5) | (5/8, 5/7, 5/6) | 1 | (4/9, 2/3, 1) | |

${x}_{4}$ | (3, 9/2, 9) | (9/8, 9/7, 3/2) | (1, 3/2, 9/4) | 1 | |

${\tilde{A}}_{(2)}^{{c}_{22}}$ | ${x}_{1}$ | 1 | (5/3, 2, 3) | (1/2, 5/8, 2/3) | (1, 5/4, 5/3) |

${x}_{2}$ | (1/3, 1/2, 3/5) | 1 | (2/7, 1/3, 1/2) | (1/2, 3/4, 1) | |

${x}_{3}$ | (3/2, 8/5, 2) | (2, 3, 7/2) | 1 | (1, 2, 3) | |

${x}_{4}$ | (3/5, 4/5, 1) | (1, 4/3, 2) | (1/3, 1/2, 1) | 1 | |

${\tilde{A}}_{(2)}^{{c}_{3}}$ | ${x}_{1}$ | 1 | (1/5, 2/7, 3/7) | (3/2, 2, 3) | (1/4, 1/3, 1/2) |

${x}_{2}$ | (7/3, 7/2, 5) | 1 | (6, 7, 8) | (1, 3/2, 2) | |

${x}_{3}$ | (1/3, 1/2, 2/3) | (1/8, 1/7, 1/6) | 1 | (1/7, 1/6, 1/5) | |

${x}_{4}$ | (2, 3, 4) | (1/2, 2/3, 1) | (5, 6, 7) | 1 |

TFPR | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ | |
---|---|---|---|---|---|

${\tilde{A}}_{(3)}^{{c}_{1}}$ | ${x}_{1}$ | 1 | (1/7, 1/6, 1/5) | (1/6, 1/5, 1/4) | (1/2, 1, 5/4) |

${x}_{2}$ | (5, 6, 7) | 1 | (1/2, 1, 6/5) | (5, 6, 7) | |

${x}_{3}$ | (4, 5, 6) | (5/6, 1, 2) | 1 | (4, 5, 6) | |

${x}_{4}$ | (4/5, 1, 2) | (1/7, 1/6, 1/5) | (1/6, 1/5, 1/4) | 1 | |

${\tilde{A}}_{(3)}^{{c}_{21}}$ | ${x}_{1}$ | 1 | (1/4, 2/7, 3/7) | (1/4, 1/3, 1/2) | (1, 6/5, 3/2) |

${x}_{2}$ | (7/3, 7/2, 4) | 1 | (1, 7/6, 3/2) | (3, 7/2, 4) | |

${x}_{3}$ | (2, 3, 4) | (2/3, 6/7, 1) | 1 | (2, 3, 4) | |

${x}_{4}$ | (2/3, 5/6, 1) | (1/4, 2/7, 1) | (1/4, 1/3, 1/2) | 1 | |

${\tilde{A}}_{(3)}^{{c}_{22}}$ | ${x}_{1}$ | 1 | (1/4, 1/3, 1/2) | (1/3, 3/7, 1/2) | (2/3, 2/3, 4/3) |

${x}_{2}$ | (2, 3, 4) | 1 | (1, 8/7, 4/3) | (2, 8/3, 4) | |

${x}_{3}$ | (2, 7/3, 3) | (3/4, 7/8, 1) | 1 | (2, 7/3, 7/2) | |

${x}_{4}$ | (3/4, 3/2, 3/2) | (1/4, 3/8, 1/2) | (2/7, 3/7, 1/2) | 1 | |

${\tilde{A}}_{(3)}^{{c}_{3}}$ | ${x}_{1}$ | 1 | (2/7, 1/3, 3/5) | (1/3, 1/2, 4/5) | (2/3, 1, 3/2) |

${x}_{2}$ | (5/3, 3, 7/2) | 1 | (3/2, 2, 3) | (2, 3, 4) | |

${x}_{3}$ | (5/4, 2, 3) | (1/3, 1/2, 2/3) | 1 | (3/2, 2, 3) | |

${x}_{4}$ | (2/3, 1, 3/2) | (1/4, 1/3, 1/2) | (1/3, 1/2, 2/3) | 1 |

TFPR | Inconsistency Index | Consensus Index | ${\tilde{\mathit{w}}}_{1\mathit{H}}$ | ${\tilde{\mathit{w}}}_{2\mathit{H}}$ | ${\tilde{\mathit{w}}}_{3\mathit{H}}$ | ${\tilde{\mathit{w}}}_{4\mathit{H}}$ |
---|---|---|---|---|---|---|

${\tilde{A}}_{(1)}^{{c}_{1}}$ | 0.0507 | 0.9758 | (0.421, 0.485, 0.638) | (1.536, 1.730, 1.897) | (1.167, 1.410, 1.441) | (0.579, 0.841, 1.313) |

${\tilde{A}}_{(1)}^{{c}_{21}}$ | 0.0079 | 0.7028 | (0.871, 0.972, 1.032) | (0.919, 1.198, 1.316) | (1.279, 1.300, 1.496) | (0.546, 0.661, 0.881) |

${\tilde{A}}_{(1)}^{{c}_{22}}$ | 0.0049 | 1.000 | (0.578, 0.760, 1.088) | (1.178, 1.520, 1.848) | (2.036, 2.280, 2.389) | (0.358, 0.380, 0.420) |

${\tilde{A}}_{(1)}^{{c}_{3}}$ | 0.1732 | 0.9167 | (0.331, 0.334, 0.352) | (1.592, 1.860, 2.141) | (1.210, 1.520, 1.627) | (0.817, 1.057, 1.564) |

${\tilde{A}}_{(2)}^{{c}_{1}}$ | 0.0295 | 0.8576 | (0.630, 0.816, 1.014) | (1.339, 1.730, 1.901) | (0.885, 1.189, 1.645) | (0.586, 0.595, 0.720) |

${\tilde{A}}_{(2)}^{{c}_{21}}$ | 0.0249 | 0.7972 | (0.273, 0.399, 0.540) | (1.259, 1.397, 1.434) | (0.884, 1.045, 1.334) | (1.452, 1.716, 2.191) |

${\tilde{A}}_{(2)}^{{c}_{22}}$ | 0.0239 | 0.6667 | (1.097, 1.118, 1.177) | (0.507, 0.595, 0.682) | (1.409, 1.760, 1.999) | (0.685, 0.855, 1.161) |

${\tilde{A}}_{(2)}^{{c}_{3}}$ | 0.0094 | 0.7500 | (0.534, 0.661, 0.877) | (2.077, 2.462, 2.786) | (0.314, 0.330, 0.341) | (1.609, 1.860, 2.137) |

${\tilde{A}}_{(3)}^{{c}_{1}}$ | 0.0503 | 0.9103 | (0.353, 0.427, 0.468) | (2.036, 2.450, 2.558) | (2.033, 2.240, 2.737) | (0.397, 0.427, 0.527) |

${\tilde{A}}_{(3)}^{{c}_{21}}$ | 0.0778 | 0.8641 | (0.549, 0.567, 0.686) | (1.878, 1.897, 1.917) | (1.376, 1.626, 1.858) | (0.446, 0.518, 0.852) |

${\tilde{A}}_{(3)}^{{c}_{22}}$ | 0.0098 | 0.6968 | (0.512, 0.579, 0.720) | (1.514, 1.811, 2.007) | (1.485, 1.539, 1.595) | (0.498, 0.730, 0.756) |

${\tilde{A}}_{(3)}^{{c}_{3}}$ | 0.0343 | 0.8204 | (0.540, 0.639, 0.857) | (1.669, 2.060, 2.281) | (0.976, 1.190, 1.426) | (0.537, 0.639, 0.760) |

TFPR | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ | |
---|---|---|---|---|---|

${\tilde{A}}_{(G)}^{{c}_{1}}$ | ${x}_{1}$ | 1 | (0.230, 0.306, 0.468) | (0.241, 0.352, 0.500) | (0.523, 0.826, 1.208) |

${x}_{2}$ | (2.137, 3.268, 4.348) | 1 | (0.955, 1.328, 1.716) | (1.995, 3.140, 4.051) | |

${x}_{3}$ | (2.000, 2.841, 4.419) | (0.583, 0.753, 1.047) | 1 | (1.516, 2.633, 3.693) | |

${x}_{4}$ | (0.828, 1.211, 1.912) | (0.247, 0.318, 0.501) | (0.271, 0.380, 0.660) | 1 | |

${\tilde{A}}_{(G)}^{{c}_{21}}$ | ${x}_{1}$ | 1 | (0.370, 0.438, 0.601) | (0.325, 0.478, 0.655) | (0.517, 0.791, 1.072) |

${x}_{2}$ | (1.664, 2.283, 2.703) | 1 | (0.844, 1.089, 1.300) | (1.231, 1.782, 2.111) | |

${x}_{3}$ | (1.527, 2.092, 3.077) | (0.769, 0.918, 1.185) | 1 | (1.135, 1.540, 2.187) | |

${x}_{4}$ | (0.933, 1.264, 1.934) | (0.474, 0.561, 0.812) | (0.457, 0.649, 0.881) | 1 | |

${\tilde{A}}_{(G)}^{{c}_{22}}$ | ${x}_{1}$ | 1 | (0.496, 0.671, 1.130) | (0.336, 0.434, 0.545) | (0.968, 1.250, 1.972) |

${x}_{2}$ | (0.885, 1.490, 2.018) | 1 | (0.521, 0.637, 0.886) | (1.553, 2.141, 2.882) | |

${x}_{3}$ | (1.835, 2.304, 2.980) | (1.129, 1.571, 1.921) | 1 | (2.342, 3.247, 4.405) | |

${x}_{4}$ | (0.507, 0.800, 1.033) | (0.347, 0.467, 0.644) | (0.227, 0.308, 0.427) | 1 | |

${\tilde{A}}_{(G)}^{{c}_{3}}$ | ${x}_{1}$ | 1 | (0.223, 0.284, 0.429) | (0.427, 0.574, 0.838) | (0.285, 0.378, 0.527) |

${x}_{2}$ | (2.331, 3.521, 4.484) | 1 | (2.274, 2.596, 3.424) | (1.231, 2.072, 2.896) | |

${x}_{3}$ | (1.193, 1.742, 2.342) | (0.292, 0.385, 0.440) | 1 | (0.630, 0.949, 1.331) | |

${x}_{4}$ | (1.898, 2.646, 3.509) | (0.345, 0.483, 0.812) | (0.751, 1.054, 1.587) | 1 |

TFPR | ${\tilde{\mathit{w}}}_{1\mathit{H}}^{(\mathit{G})}$ | ${\tilde{\mathit{w}}}_{2\mathit{H}}^{(\mathit{G})}$ | ${\tilde{\mathit{w}}}_{3\mathit{H}}^{(\mathit{G})}$ | ${\tilde{\mathit{w}}}_{4\mathit{H}}^{(\mathit{G})}$ | Ranking |
---|---|---|---|---|---|

${\tilde{A}}_{(G)}^{{c}_{1}}$ | (0.451, 0.546, 0.667) | (1.608, 1.921, 2.072) | (1.263, 1.541, 1.857) | (0.520, 0.618, 0.831) | ${x}_{2}\stackrel{100\%}{\underset{\_}{\succ}}{x}_{3}\stackrel{100\%}{\underset{\_}{\succ}}{x}_{4}\stackrel{85.5\%}{\underset{\_}{\succ}}{x}_{1}$ |

${\tilde{A}}_{(G)}^{{c}_{21}}$ | (0.528, 0.638, 0.762) | (1.284, 1.451, 1.474) | (1.154, 1.311, 1.564) | (0.708, 0.824, 1.027) | ${x}_{2}\stackrel{78.3\%}{\underset{\_}{\succ}}{x}_{3}\stackrel{100\%}{\underset{\_}{\succ}}{x}_{4}\stackrel{100\%}{\underset{\_}{\succ}}{x}_{1}$ |

${\tilde{A}}_{(G)}^{{c}_{22}}$ | (0.676, 0.777, 0.984) | (0.990, 1.194, 1.405) | (1.658, 1.852, 2.006) | (0.480, 0.582, 0.680) | ${x}_{3}\stackrel{100\%}{\underset{\_}{\succ}}{x}_{2}\stackrel{100\%}{\underset{\_}{\succ}}{x}_{1}\stackrel{100\%}{\underset{\_}{\succ}}{x}_{4}$ |

${\tilde{A}}_{(G)}^{{c}_{3}}$ | (0.443, 0.498, 0.605) | (1.749, 2.086, 2.360) | (0.757, 0.893, 0.979) | (0.882, 1.077, 1.384) | ${x}_{2}\stackrel{100\%}{\underset{\_}{\succ}}{x}_{4}\stackrel{100\%}{\underset{\_}{\succ}}{x}_{3}\stackrel{100\%}{\underset{\_}{\succ}}{x}_{1}$ |

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## Share and Cite

**MDPI and ACS Style**

Lu, P.; Yang, X.; Wang, Z.-J.
Fuzzy Group Consensus Decision Making and Its Use in Selecting Energy-Saving and Low-Carbon Technology Schemes in Star Hotels. *Int. J. Environ. Res. Public Health* **2018**, *15*, 2057.
https://doi.org/10.3390/ijerph15092057

**AMA Style**

Lu P, Yang X, Wang Z-J.
Fuzzy Group Consensus Decision Making and Its Use in Selecting Energy-Saving and Low-Carbon Technology Schemes in Star Hotels. *International Journal of Environmental Research and Public Health*. 2018; 15(9):2057.
https://doi.org/10.3390/ijerph15092057

**Chicago/Turabian Style**

Lu, Ping, Xuan Yang, and Zhou-Jing Wang.
2018. "Fuzzy Group Consensus Decision Making and Its Use in Selecting Energy-Saving and Low-Carbon Technology Schemes in Star Hotels" *International Journal of Environmental Research and Public Health* 15, no. 9: 2057.
https://doi.org/10.3390/ijerph15092057