# Linguistic Multi-Attribute Group Decision Making with Risk Preferences and Its Use in Low-Carbon Tourism Destination Selection

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

_{0}is the neutral linguistic term in $S$, such as “middle”, “fair” and “indifference”. If ${\tau}_{1}={\tau}_{2}$, then $S$ satisfies the following characteristics [25]:

- (i)
- The set $S$ is ordered, i.e., ${s}_{i}{}^{}>{s}_{j}{}^{}$ if and only if $i>j$;
- (ii)
- A negation operator can be defined as $N\mathrm{e}g({s}_{i})={s}_{-i}$, where $N\text{eg}\left({s}_{0}\right)$$={s}_{0}$.

**Definition 1 [24].**

## 3. An Optimization Model for Determining a Group Generalized Linguistic Term Set

**Example 1.**

## 4. An Approach to Linguistic MAGDM with Risk Preferences and Incomplete Weight Information

- (i)
- A weak ranking: $\{{w}_{i}\ge {w}_{j}\},i\ne j$;
- (ii)
- A strict ranking: $\{{w}_{i}-{w}_{j}\ge {\epsilon}_{ij}\},i\ne j$, where ${\epsilon}_{ij}>0$;
- (iii)
- An interval form: $\{{\alpha}_{j}\le {w}_{j}\le {\alpha}_{j}+{\epsilon}_{j}\}$, where $0\le {\alpha}_{j}<{\alpha}_{j}+{\epsilon}_{j}\le 1$;
- (iv)
- A ranking with multiples: $\{{w}_{i}\ge {\beta}_{ij}{w}_{j}\}$, where $0\le {\beta}_{ij}\le 1,i\ne j;$ and
- (v)
- A ranking of deviations: $\{{w}_{i}-{w}_{j}\ge {w}_{k}-{w}_{l}\}$, where $i\ne j\ne k\ne l$.

#### Procedure

## 5. A Case Study of the Low-Carbon Tourism Destination Selection Problem

- (i)
- ${a}_{1}$: Low-carbon transportation, low-energy consumption vehicles and pick-up and drop-off services as reflected in connecting different scenic sites and reaching the destination.
- (ii)
- ${a}_{2}$: Food service including green food, a low-carbon environment and low-energy waste handling mechanisms.
- (iii)
- ${a}_{3}$: Hotels and accommodation, as reflected in green-material labels, low-carbon facilities and a low-carbon environment and education management.
- (iv)
- ${a}_{4}$: Consumption satisfaction, as reflected in the service cost of travel agencies, ticket prices for scenic sites and the cost of accommodation.
- (v)
- ${a}_{5}$: Attraction and impact of scenic sites, including low-carbon customer service and low-carbon management and control.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Dagoumas, A.S.; Barker, T.S. Pathways to a low-carbon economy for the UK with the macro-econometric E3MG model. Energy Policy
**2010**, 38, 3067–3077. [Google Scholar] [CrossRef] - Liu, X.; Liu, J. Measurement of low carbon economy efficiency with a three-stage data envelopment analysis: A comparison of the largest twenty CO
_{2}emitting countries. Int. J. Environ. Res. Public Health**2016**, 13, 1116. [Google Scholar] [CrossRef] [PubMed] - Cho, Y.J.; Wang, Y.; Hsu, L.L.I. Constructing Taiwan’s low-carbon tourism development suitability evaluation indicators. Asia Pac. J. Tour. Res.
**2016**, 21, 658–677. [Google Scholar] [CrossRef] - Cheng, Q.; Su, B.; Tan, J. Developing an evaluation index system for low-carbon tourist attractions in China—A case study examining the Xixi wetland. Tour. Manag.
**2013**, 36, 314–320. [Google Scholar] [CrossRef] - Herrera, F.; Martínez, L. A 2-Tuple fuzzy linguistic representation model for computing with words. IEEE Trans. Fuzzy Syst.
**2000**, 8, 746–752. [Google Scholar] - Wang, J.H.; Hao, J. A new version of 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans. Fuzzy Syst.
**2006**, 14, 435–445. [Google Scholar] [CrossRef] - Xu, Y.; Wang, H. Approaches based on 2-tuple linguistic power aggregation operators for multiple attribute group decision making under linguistic environment. Appl. Soft Comput.
**2011**, 11, 3988–3997. [Google Scholar] [CrossRef] - Park, J.H.; Park, J.M.; Kwun, Y.C. 2-Tuple linguistic harmonic operators and their applications in group decision making. Knowl.-Based Syst.
**2013**, 44, 10–19. [Google Scholar] [CrossRef] - Ju, Y.; Liu, X.; Wang, A. Some new Shapley 2-tuple linguistic Choquet aggregation operators and their applications to multiple attribute group decision making. Soft Comput.
**2016**, 20, 4037–4053. [Google Scholar] [CrossRef] - Sahin, R.; Liu, P.D. Maximizing deviation method for neutrosophic multiple attribute decision making with incomplete weight information. Neural Comput. Appl.
**2016**, 27, 2017–2029. [Google Scholar] [CrossRef] - Zhang, X.; Xu, Z.; Wang, H. Heterogeneous multiple criteria group decision making with incomplete weight information: A deviation modeling approach. Inf. Fusion
**2015**, 25, 49–62. [Google Scholar] [CrossRef] - Wei, G.W. Extension of TOPSIS method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information. Knowl. Inf. Syst.
**2010**, 25, 623–634. [Google Scholar] [CrossRef] - Hwang, C.L.; Yoon, K. Multiple Attributes Decision Making Methods and Applications; Springer: Berlin, Germany, 1981. [Google Scholar]
- Wei, G.W. Grey relational analysis method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information. Expert Syst. Appl.
**2011**, 38, 4824–4828. [Google Scholar] [CrossRef] - Zhang, Z.; Guo, C. A method for multi-granularity uncertain linguistic group decision making with incomplete weight information. Knowl.-Based Syst.
**2012**, 26, 111–119. [Google Scholar] [CrossRef] - Ju, Y. A new method for multiple criteria group decision making with incomplete weight information under linguistic environment. Appl. Math. Model.
**2014**, 38, 5258–5268. [Google Scholar] [CrossRef] - Andrés, R.; Espinilla, M.; Martínez, L. An extended hierarchical linguistic model for managing integral evaluation. Int. J. Comput. Intell. Syst.
**2010**, 3, 486–500. [Google Scholar] [CrossRef] - Zhai, Y.; Xu, Z.; Liao, H. Probabilistic linguistic vector-term set and its application in group decision making with multi-granular linguistic information. Appl. Soft Comput.
**2016**, 49, 801–816. [Google Scholar] [CrossRef] - Wan, S.P.; Dong, J.Y. Power geometric operators of trapezoidal intuitionistic fuzzy numbers and application to multi-attribute group decision making. Appl. Soft Comput.
**2015**, 29, 153–168. [Google Scholar] [CrossRef] - Tong, X.; Wang, Z.J. A group decision framework with intuitionistic preference relations and its application to low carbon supplier selection. Int. J. Environ. Res. Public Health
**2016**, 13, 923. [Google Scholar] [CrossRef] [PubMed] - Zhang, J. Evaluating regional low-carbon tourism strategies using the fuzzy delphi-analytic network process approach. J. Clean. Prod.
**2017**, 141, 409–419. [Google Scholar] [CrossRef] - Li, C.C.; Dong, Y. Unbalanced linguistic approach for venture investment evaluation with risk attitudes. Prog. Artif. Intell.
**2014**, 3, 1–13. [Google Scholar] [CrossRef] - Zhou, W.; Xu, Z. Generalized asymmetric linguistic term set and its application to qualitative decision making involving risk appetites. Eur. J. Oper. Res.
**2016**, 254, 610–621. [Google Scholar] [CrossRef] - Lin, H.; Wang, Z.J. Linguistic multi-attribute decision making with considering decision makers’ risk preferences. J. Intell. Fuzzy Syst.
**2017**, 33, 1775–1784. [Google Scholar] [CrossRef] - Xu, Z. Deviation measures of linguistic preference relations in group decision making. Omega
**2005**, 33, 249–254. [Google Scholar] [CrossRef] - Chen, C.T. Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Set Syst.
**2000**, 114, 1–9. [Google Scholar] [CrossRef] - Liou, T.S.; Wang, M.J.J. Ranking fuzzy numbers with integral value. Fuzzy Set Syst.
**1992**, 50, 247–255. [Google Scholar] [CrossRef]

**Figure 1.**Distribution of the semantic values of the generalized linguistic term sets (GLTS) ${\tilde{S}}_{1}^{*}$.

**Table 1.**Linguistic-term-based decision matrix ${R}_{1}={\left({s}_{{r}_{ij1}}\right)}_{4\times 5}$ given by ${e}_{1}$.

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

${x}_{1}$ | $P$ | $G$ | $VG$ | $M$ | $SG$ |

${x}_{2}$ | $SP$ | $M$ | $G$ | $P$ | $G$ |

${x}_{3}$ | $VG$ | $G$ | $SP$ | $SP$ | $M$ |

${x}_{4}$ | $SG$ | $SG$ | $M$ | $G$ | $M$ |

**Table 2.**Linguistic-term-based decision matrix ${R}_{2}={\left({s}_{{r}_{ij2}}\right)}_{4\times 5}$ given by ${e}_{2}$.

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

${x}_{1}$ | $SP$ | $VG$ | $SG$ | $G$ | $G$ |

${x}_{2}$ | $SG$ | $VG$ | $VG$ | $VP$ | $VG$ |

${x}_{3}$ | $G$ | $VG$ | $M$ | $G$ | $SG$ |

${x}_{4}$ | $SG$ | $G$ | $SP$ | $SG$ | $SG$ |

**Table 3.**Linguistic-term-based decision matrix ${R}_{3}={\left({s}_{{r}_{ij3}}\right)}_{4\times 5}$ given by ${e}_{3}$.

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

${x}_{1}$ | $M$ | $SP$ | $G$ | $SG$ | $VG$ |

${x}_{2}$ | $G$ | $SG$ | $SG$ | $M$ | $G$ |

${x}_{3}$ | $G$ | $SG$ | $G$ | $G$ | $SP$ |

${x}_{4}$ | $M$ | $VG$ | $SG$ | $VG$ | $G$ |

**Table 4.**Triangular fuzzy decision matrix ${\tilde{D}}_{1}={\left({\tilde{d}}_{ij1}\right)}_{4\times 5}$.

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

${x}_{1}$ | $(0.2100,0.2925,0.3913)$ | $(0.6380,0.7565,0.8456)$ | $(0.7565,0.8456,0.8456)$ | $(0.3913,0.5000,0.6380)$ | $(0.5000,0.6380,0.7565)$ |

${x}_{2}$ | $(0.2925,0.3913,0.5000)$ | $(0.3913,0.5000,0.6380)$ | $(0.6380,0.7565,0.8456)$ | $(0.2100,0.2925,0.3913)$ | $(0.6380,0.7565,0.8456)$ |

${x}_{3}$ | $(0.7565,0.8456,0.8456)$ | $(0.6380,0.7565,0.8456)$ | $(0.2925,0.3913,0.5000)$ | $(0.2925,0.3913,0.5000)$ | $(0.3913,0.5000,0.6380)$ |

${x}_{4}$ | $(0.5000,0.6380,0.7565)$ | $(0.5000,0.6380,0.7565)$ | $(0.3913,0.5000,0.6380)$ | $(0.6380,0.7565,0.8456)$ | $(0.3913,0.5000,0.6380)$ |

**Table 5.**Triangular fuzzy decision matrix ${\tilde{D}}_{2}={\left({\tilde{d}}_{ij2}\right)}_{4\times 5}$.

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

${x}_{1}$ | $(0.2925,0.3913,0.5000)$ | $(0.7565,0.8456,0.8456)$ | $(0.5000,0.6380,0.7565)$ | $(0.6380,0.7565,0.8456)$ | $(0.6380,0.7565,0.8456)$ |

${x}_{2}$ | $(0.5000,0.6380,0.7565)$ | $(0.7565,0.8456,0.8456)$ | $(0.7565,0.8456,0.8456)$ | $(0.2100,0.2100,0.2925)$ | $(0.7565,0.8456,0.8456)$ |

${x}_{3}$ | $(0.6380,0.7565,0.8456)$ | $(0.7565,0.8456,0.8456)$ | $(0.3913,0.5000,0.6380)$ | $(0.6380,0.7565,0.8456)$ | $(0.5000,0.6380,0.7565)$ |

${x}_{4}$ | $(0.5000,0.6380,0.7565)$ | $(0.6380,0.7565,0.8456)$ | $(0.2925,0.3913,0.5000)$ | $(0.5000,0.6380,0.7565)$ | $(0.5000,0.6380,0.7565)$ |

**Table 6.**Triangular fuzzy decision matrix ${\tilde{D}}_{3}={\left({\tilde{d}}_{ij3}\right)}_{4\times 5}$.

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

${x}_{1}$ | $(0.3913,0.5000,0.6380)$ | $(0.2925,0.3913,0.5000)$ | $(0.6380,0.7565,0.8456)$ | $(0.5000,0.6380,0.7565)$ | $(0.7565,0.8456,0.8456)$ |

${x}_{2}$ | $(0.6380,0.7565,0.8456)$ | $(0.5000,0.6380,0.7565)$ | $(0.5000,0.6380,0.7565)$ | $(0.3913,0.5000,0.6380)$ | $(0.6380,0.7565,0.8456)$ |

${x}_{3}$ | $(0.6380,0.7565,0.8456)$ | $(0.5000,0.6380,0.7565)$ | $(0.6380,0.7565,0.8456)$ | $(0.6380,0.7565,0.8456)$ | $(0.2925,0.3913,0.5000)$ |

${x}_{4}$ | $(0.3913,0.5000,0.6380)$ | $(0.7565,0.8456,0.8456)$ | $(0.5000,0.6380,0.7565)$ | $(0.7565,0.8456,0.8456)$ | $(0.6380,0.7565,0.8456)$ |

**Table 7.**Group triangular fuzzy decision matrix $\tilde{G}={\left({\tilde{g}}_{ij}\right)}_{4\times 5}={\left(({g}_{ij}^{L},{g}_{ij}^{M},{g}_{ij}^{U})\right)}_{4\times 5}$.

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

${X}_{1}$ | $(0.2793,0.3735,0.4841)$ | $(0.6163,0.7191,0.7765)$ | $(0.6302,0.7447,0.8100)$ | $(0.5117,0.6302,0.7447)$ | $(0.6065,0.7269,0.8100)$ |

${X}_{2}$ | $(0.4446,0.5630,0.6717)$ | $(0.5591,0.6658,0.7447)$ | $(0.6578,0.7684,0.8278)$ | $(0.2463,0.3010,0.4011)$ | $(0.6854,0.7921,0.8456)$ |

${X}_{3}$ | $(0.6854,0.7921,0.8456)$ | $(0.6578,0.7684,0.8278)$ | $(0.4011,0.5078,0.6243)$ | $(0.4998,0.6104,0.7074)$ | $(0.4150,0.5335,0.6578)$ |

${X}_{4}$ | $(0.4783,0.6104,0.7328)$ | $(0.6065,0.7269,0.8100)$ | $(0.3735,0.4841,0.6065)$ | $(0.6065,0.7269,0.8100)$ | $(0.4841,0.6065,0.7269)$ |

**Table 8.**A comparative study for attribute weight vectors and ranking results obtained from different models.

Model | Ref. | Attribute Weight Vector | Ranking Result |
---|---|---|---|

(M-3) and (20) | Wei [12] | $w={(0.3333,\text{}0,\text{}0.1,\text{}0.4167,\text{}0.15)}^{T}$ | ${x}_{3}\succ {x}_{4}\succ {x}_{1}\succ {x}_{2}$ |

(M-2) and (11)–(19) | Wei [14] | $w={(0.425,\text{}0,\text{}0.2,\text{}0.225,\text{}0.15)}^{T}$ | ${x}_{3}\succ {x}_{2}\succ {x}_{4}\succ {x}_{1}$ |

(M-5) and (8) | Ju [16] | $w={(0.425,\text{}0,\text{}0.1,\text{}0.325,\text{}0.15)}^{T}$ | ${x}_{3}\succ {x}_{4}\succ {x}_{1}\succ {x}_{2}$ |

(21) and (22) | This paper | $w={(0.14,\text{}0.235,\text{}0.2,\text{}0.175,\text{}0.25)}^{T}$ | ${x}_{1}\succ {x}_{2}\succ {x}_{3}\succ {x}_{4}$ |

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## Share and Cite

**MDPI and ACS Style**

Lin, H.; Wang, Z.-J.
Linguistic Multi-Attribute Group Decision Making with Risk Preferences and Its Use in Low-Carbon Tourism Destination Selection. *Int. J. Environ. Res. Public Health* **2017**, *14*, 1078.
https://doi.org/10.3390/ijerph14091078

**AMA Style**

Lin H, Wang Z-J.
Linguistic Multi-Attribute Group Decision Making with Risk Preferences and Its Use in Low-Carbon Tourism Destination Selection. *International Journal of Environmental Research and Public Health*. 2017; 14(9):1078.
https://doi.org/10.3390/ijerph14091078

**Chicago/Turabian Style**

Lin, Hui, and Zhou-Jing Wang.
2017. "Linguistic Multi-Attribute Group Decision Making with Risk Preferences and Its Use in Low-Carbon Tourism Destination Selection" *International Journal of Environmental Research and Public Health* 14, no. 9: 1078.
https://doi.org/10.3390/ijerph14091078