# A Multi-Criteria Group Decision Making Model for Green Supplier Selection under the Ordered Weighted Hesitant Fuzzy Environment

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

- (1)
- ${}^{\omega}h{}^{\lambda}=\underset{1\le j\le L}{\cup}\left\{\langle {({h}^{\delta (j)})}^{\lambda},{w}^{\delta (j)}\rangle \right\}$;
- (2)
- $\lambda {}^{\omega}h=\underset{1\le j\le L}{\cup}\left\{\langle 1-{(1-{h}^{\delta (j)})}^{\lambda},{w}^{\delta (j)}\rangle \right\}$;
- (3)
- ${}^{\omega}h{}_{1}\oplus {}^{\omega}h{}_{2}=\underset{1\le j\le L}{\cup}\left\{\langle {h}_{1}^{\delta (j)}+{h}_{2}^{\delta (j)}-{h}_{1}^{\delta (j)}{h}_{2}^{\delta (j)},\overline{({w}_{1}^{\delta (j)}+{w}_{2}^{\delta (j)})}\rangle \right\}$;

**Definition**

**3.**

- (1)
- If$\Delta {(}^{\omega}{h}_{1})>\Delta {(}^{\omega}{h}_{2})$, then${}^{\omega}{h}_{1}{>}^{\omega}{h}_{2}$
- (2)
- If$\Delta {(}^{\omega}{h}_{1})<\Delta {(}^{\omega}{h}_{2})$, then${}^{\omega}{h}_{1}{<}^{\omega}{h}_{2}$
- (3)
- If$\Delta {(}^{\omega}{h}_{1})=\Delta {(}^{\omega}{h}_{2})$, then$\{\begin{array}{c}\nabla {(}^{\omega}{h}_{1})>\nabla {(}^{\omega}{h}_{2}){\Rightarrow}^{\omega}{h}_{1}{<}^{\omega}{h}_{2}\\ \nabla {(}^{\omega}{h}_{1})=\nabla {(}^{\omega}{h}_{2}){\Rightarrow}^{\omega}{h}_{1}{=}^{\omega}{h}_{2}\\ \nabla {(}^{\omega}{h}_{1})<\nabla {(}^{\omega}{h}_{2}){\Rightarrow}^{\omega}{h}_{1}{>}^{\omega}{h}_{2}\end{array}$

**Definition**

**4.**

## 3. GOWHFPWA Operator and Its Properties

**Definition**

**5.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Theorem**

**6.**

**Theorem**

**7.**

**Theorem**

**8.**

**Proof.**

## 4. The MCGDM Approach with Order Weighted Hesitant Fuzzy Information

**Step 1**. Calculate the values of ${T}_{pq}(p=1,2,\dots m;q=1,2,\dots ,n)$ based on Equation (11).

**Step 2**. Aggregate the OWHFEs ${}^{\omega}h{}_{pq}$ for each supplier ${x}_{p}(p=1,2,\dots ,m)$ by the GOWHFPWA operator, then we can get the overall OWHFE ${}^{\omega}h{}_{p}(p=1,2,\dots ,m)$ for the supplier ${x}_{p}(p=1,2,\dots ,m)$ as follows:

**Step 3**. Calculate the score functions $\Delta ({}^{\omega}h{}_{p})(p=1,2,\dots ,m)$ of the OWHFE ${}^{\omega}h{}_{p}(p=1,2,\dots ,m)$ for the supplier ${x}_{p}(p=1,2,\dots ,m)$, that is,

**Step 4**. Rank the score functions $\Delta ({}^{\omega}h{}_{p})$ in ascending order. Then, the supplier with the highest priority is the most desirable green supplier.

## 5. Numerical Example

_{i}(i = 1, 2, 3, 4, 5) have been determined for further assessment. In order to choose the most suitable supplier, the company established a team of six DMs ${e}_{u}(u=1,2,\dots ,6)$ from the department of purchasing, quality, and production who have abundant knowledge and experience in GSCM. Finally, four criteria are chosen from the Table 1 criteria list by experts to evaluate possible green suppliers. The four selected criteria are quality (c

_{1}), technology(c

_{2}), environment(c

_{3}), cost(c

_{4}), and the priority relationship among the criteria is ${c}_{1}\succ {c}_{2}\succ {c}_{3}\succ {c}_{4}$ in the evaluation process. For a supplier under a criterion, six DMs need to give their evaluation values. As an instance, for the supplier x

_{1}under the criterion c

_{1}, the evaluation values 0.3, 0.5, and 0.8 are provided by two, one and three DMs, respectively, and then an OWHFE ${}^{\omega}h{}_{11}$ can be represented by {<0.3,2/6>,<0.5,1/6>,<0.8,3/6>}.

**Step 1**. According to Equation (11), ${T}_{pq}(p=1,2,\dots ,5,q=1,2,3,4)$ are calculated as follows:

**Step 2**. Aggregate ${}^{\omega}h{}_{pq}(p=1,2,\dots ,5,q=1,2,3,4)$ by using a GOWHFPWA ($\alpha =1$) operator to derive the overall OWHFEs ${}^{\omega}h{}_{p}(p=1,2,\dots ,5)$ for the supplier ${x}_{p}(p=1,2,\dots ,5)$.

**Step 3**. Calculate the score functions $\Delta ({}^{\omega}h{}_{p})(p=1,2,\dots ,5)$ of the OWHFEs ${}^{\omega}h{}_{p}(p=1,2,\dots ,5)$ for the supplier ${x}_{p}(p=1,2,\dots ,5)$, that is,

**Step 4**. Rank all the suppliers ${x}_{p}(p=1,2,\dots ,5)$ in accordance with the score functions $\Delta ({}^{\omega}h{}_{p})(p=1,2,\dots ,5)$ and the priority relationship of five suppliers can be obtained, that is,

## 6. Performance Analysis and Comparation Analysis

_{1}as an example, we have ${h}_{1}=HFPWA({h}_{11},{h}_{12},{h}_{13},{h}_{14})$ = {0.3083,0.3179,0.3295,0.3620,0.3709,0.3816,0.3879,0.3965,0.4067,0.4055,0.4138,0.4238,0.4517,0.4593,0.4685,0.4740,0.4813,0.4902,0.4501,0.4578,0.4670,0.4928,0.4998,0.5084,0.5134,0.5202,0.5284,0.4169,0.4250,0.4348,0.4622,0.4697,0.4787,0.4841,0.4912,0.4999,0.4989,0.5059,0.5143,0.5378,0.5442,0.5520,0.5566,0.5628,0.5702,0.5365,0.5429,0.5507,0.5724,0.5784,0.5856,0.5898,0.5956,0.6024,0.6338,0.6389,0.6451,0.6623,0.6670,0.6727,0.6760,0.6805,0.6860,0.6853,0.6897,0.6950,0.7098,0.7138,0.7187,0.7216,0.7255,0.7301,0.7089,0.7130,0.7179,0.7315,0.7353,0.7398,0.7424,0.7460,0.7504}.

## 7. Conclusions and Further Directions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Guo, S.; Shen, B.; Choi, T.M.; Jung, S. A review on supply chain contracts in reverse logistics: Supply chain structures and channel leaderships. J. Clean. Prod.
**2017**, 144, 387–402. [Google Scholar] [CrossRef] - Blome, C.; Hollos, D.; Paulraj, A. Green procurement and green supplier development: Antecedents and effects on supplier performance. Int. J. Prod. Econ.
**2014**, 124, 252–264. [Google Scholar] [CrossRef] - Ho, W.; Xu, X.; Dey, P.K. Multi-criteria decision making approaches for supplier evaluation and selection: A literature review. Eur. J. Oper. Res.
**2010**, 202, 16–24. [Google Scholar] [CrossRef] - Chai, J.; Liu, J.N.; Ngai, E.W. Application of decision-making techniques in supplier selection: A systematic review of literature. Expert. Syst. Appl.
**2013**, 40, 3872–3885. [Google Scholar] [CrossRef] - Deng, X.; Hu, Y.; Deng, Y.; Mahadevan, S. Supplier selection using AHP methodology extended by D numbers. Expert. Syst. Appl.
**2014**, 41, 156–167. [Google Scholar] [CrossRef] - Govindan, K.; Rajendran, S.; Sarkis, J.; Murugesan, P. Multi criteria decision making approaches for green supplier evaluation and selection: A literature review. J. Clean. Prod.
**2015**, 98, 66–83. [Google Scholar] [CrossRef] - Handfield, R.; Walton, S.V.; Sroufe, R.; Melnyk, S.A. Applying environmental criteria to supplier assessment: A study in the application of the analytical hierarchy process. Eur. J. Oper. Res.
**2002**, 141, 70–87. [Google Scholar] [CrossRef] - Lu, L.Y.Y.; Wu, C.H.; Kuo, T.C. Environmental principles applicable to green supplier evaluation by using multi-objective decision analysis. Int. J. Prod. Res.
**2007**, 45, 4317–4331. [Google Scholar] [CrossRef] - Hsu, C.W.; Hu, A.H. Applying hazardous substance management to supplier selection using analytic network process. J. Clean. Prod.
**2009**, 17, 255–264. [Google Scholar] [CrossRef] - Kuo, R.J.; Wang, Y.C.; Tien, F.C. Integration of artificial neural network and MAMD methods for green supplier selection. J. Clean. Prod.
**2010**, 18, 1161–1170. [Google Scholar] [CrossRef] - Bai, C.; Sarkis, J. Green supplier development: Analytical evaluation using rough set theory. J. Clean. Prod.
**2010**, 18, 1200–1210. [Google Scholar] [CrossRef] - Yeh, W.C.; Chuang, M.C. Using multi-objective genetic algorithm for partner selection in green supply chain problems. Expert. Syst. Appl.
**2011**, 38, 4244–4253. [Google Scholar] [CrossRef] - Zhou, R.; Ma, X.; Li, S.; Li, J. The green supplier selection method for chemical industry with analytic network process and radial basis function neural network. Adv. Inf. Sci. Serv. Sci.
**2012**, 4, 147–158. [Google Scholar] - Kuo, R.J.; Lin, Y.J. Supplier selection using analytic network process and data envelopment analysis. Int. J. Prod. Res.
**2012**, 50, 2852–2863. [Google Scholar] [CrossRef] - Jauhar, S.K.; Pant, M.; Deep, A. An approach to solve multi-criteria supplier selection while considering environmental aspects using differential evolution. In Proceedings of Swarm, Evolutionary, and Memetic Computing; Springer International Publishing: Switzerland, 2013; Volume 8297, pp. 199–208. [Google Scholar]
- Dobos, I.; Vörösmarty, G. Green supplier selection and evaluation using DEA-type composite indicators. Int. J. Prod. Econ.
**2014**, 157, 273–278. [Google Scholar] [CrossRef] - Freeman, J.; Chen, T. Green supplier selection using an AHP-Entropy-TOPSIS framework. Supply Chain Manag. An Int. J.
**2015**, 20, 327–340. [Google Scholar] [CrossRef] - Hashemi, S.H.; Karimi, A.; Tavana, M. An integrated green supplier selection approach with analytic network process and improved Grey relational analysis. Int. J. Prod. Econ.
**2015**, 159, 178–191. [Google Scholar] [CrossRef] - Yazdani, M.; Chatterjee, P.; Zavadskas, E.K.; Zolfani, S.H. Integrated QFD-MCDM framework for green supplier selection. J. Clean. Prod.
**2017**, 142, 3728–3740. [Google Scholar] [CrossRef] - Liu, B.; Yang, X.; Huo, T.; Shen, G.Q.; Wang, X. A linguistic group decision making framework for bid evaluation in mega public projects considering carbon dioxide emissions reduction. J. Clean. Prod.
**2017**, 148, 811–825. [Google Scholar] [CrossRef] - Zadeh, L.A. Fuzzy sets. Inf. Control.
**1965**, 8, 338–356. [Google Scholar] [CrossRef] - Akman, G. Evaluating suppliers to include green supplier development programs via Fuzzy c-means and VIKOR methods. Comput. Ind. Eng.
**2015**, 86, 69–82. [Google Scholar] [CrossRef] - Lourenzutti, R.; Krohling, R.A. The hellinger distance in multicriteria decision making: An illustration to the TOPSIS and TODIM methods. Expert. Syst. Appl.
**2014**, 41, 4414–4421. [Google Scholar] [CrossRef] - Tseng, M.L.; Lin, Y.H.; Tan, K.; Chen, R.H.; Chen, Y.H. Using TODIM to evaluate green supply chain practices under uncertainty. Appl. Math. Model.
**2014**, 38, 2983–2995. [Google Scholar] [CrossRef] - Govindan, K.; Khodaverdi, R.; Jafarian, A. A fuzzy multi criteria approach for measuring sustainability performance of a supplier based on triple bottom line approach. J. Clean. Prod.
**2013**, 47, 345–354. [Google Scholar] [CrossRef] - Chiou, C.Y.; Hsu, C.W.; Hwang, W.Y. Comparative investigation on green supplier selection of the American, Japanese and Taiwanese electronics industry in China. In Proceedings of the IEEE International Conference on IE&EM, Singapore, 8–11 December 2008; pp. 1909–1914. [Google Scholar]
- Lee, A.H.I.; Kang, H.Y.; Hsu, C.F.; Hung, H.C. A green supplier selection model for high-tech industry. Expert. Syst. Appl.
**2009**, 36, 7917–7927. [Google Scholar] [CrossRef] - Tsai, W.H.; Hung, S.J. A fuzzy goal programming approach for green supply chain optimization under activity-based costing and performance evaluation with a value-chain structure. Int. J. Prod. Res.
**2009**, 47, 4991–5017. [Google Scholar] [CrossRef] - Tuzkaya, G.; Ozgen, A.; Ozgen, D.; Tuzkaya, U.R. Environmental performance evaluation of suppliers: A hybrid fuzzy multi-criteria decision approach. Int. J. Environ. Sci. Technol.
**2009**, 6, 477–490. [Google Scholar] [CrossRef] [Green Version] - Büyüközkan, G.; Çifçi, G. A novel hybrid MCDM approach based on fuzzy DEMATEL, fuzzy ANP and fuzzy TOPSIS to evaluate green suppliers. Expert. Syst. Appl.
**2012**, 39, 3000–3011. [Google Scholar] [CrossRef] - Datta, S.; Samantra, C.; Mahapatra, S.S.; Banerjee, S.; Bandyopadhya, A. Green supplier evaluation and selection using VIKOR method embedded in fuzzy expert system with interval-valued fuzzy numbers. Int. J. Procure. Manag.
**2012**, 5, 647–678. [Google Scholar] [CrossRef] - Shen, L.; Olfat, L.; Govindan, K.; Khodaverdi, R.; Diabat, A. A fuzzy multi criteria approach for evaluating green supplier’s performance in green supply chain with linguistic preferences. Resour. Conserv. Recycl.
**2013**, 74, 170–179. [Google Scholar] [CrossRef] - Wang, X.; Chan, H.K. A hierarchical fuzzy TOPSIS approach to assess improvement areas when implementing green supply chain initiatives. Int. J. Prod. Res.
**2013**, 51, 3117–3130. [Google Scholar] [CrossRef] - Cao, Q.W.; Wu, J.; Liang, C.Q. An intuitionsitic fuzzy judgment matrix and TOPSIS integrated multi-criteria decision making method for green supplier selection. J. Intell. Fuzzy Syst.
**2014**, 28, 117–126. [Google Scholar] - Kannan, D.; Govindan, K.; Rajendran, S. Fuzzy axiomatic design approach based green supplier selection: A case study from Singapore. J. Clean. Prod.
**2015**, 96, 194–208. [Google Scholar] [CrossRef] - Hamdan, S.; Cheaitou, A. Supplier selection and order allocation with green criteria: An MCDM and multi-objective optimization approach. Comput. Oper. Res.
**2016**, 81, 282–304. [Google Scholar] [CrossRef] - Guo, Z.X.; Liu, H.T.; Zhang, D.Q.; Yang, J. Green supplier evaluation and selection in apparel manufacturing using a fuzzy multi-criteria decision-making approach. Sustainability
**2017**, 9, 650–663. [Google Scholar] - Xu, Y.; Cabrerizo, F.J.; Herrera-Viedma, E. A consensus model for hesitant fuzzy preference relations and its application in water allocation management. Appl. Soft Comput.
**2017**, 58, 265–284. [Google Scholar] [CrossRef] - Tsui, C.W.; Tzeng, G.H.; Wen, P.U. A hybrid MCDM approach for improving the performance of green suppliers in the TFT-LCD Industry. Int. J. Prod. Res.
**2015**, 53, 6436–6454. [Google Scholar] [CrossRef] - Darabi, S.; Heydari, J. An interval-valued hesitant fuzzy ranking method based on group decision analysis for green supplier selection. IFAC PapersOnLine
**2016**, 49, 12–17. [Google Scholar] [CrossRef] - Gitinavard, H.; Ghaderi, H.; Pishvaeeet, S.M. Green supplier evaluation in manufacturing systems: A novel interval-valued hesitant fuzzy group outranking approach. Soft Comput.
**2017**, 3, 1–20. [Google Scholar] [CrossRef] - Qin, J.; Liu, X.; Pedrycz, W. An extended todim multi-criteria group decision making method for green supplier selection in interval type-2 fuzzy environment. Eur. J. Oper. Res.
**2017**, 258, 626–638. [Google Scholar] [CrossRef] - Tang, S.L. Green supplier selection model with hesitant fuzzy information. J. Intell Fuzzy. Syst.
**2017**, 32, 189–195. [Google Scholar] [CrossRef] - Krohling, R.A.; Pacheco, A.G.C.; Siviero, A.L.T. IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making. Knowl. Base Syst.
**2013**, 53, 142–146. [Google Scholar] [CrossRef] - Torra, V.; Narukawa, Y. On hesitant fuzzy sets and decision. In Proceedings of the IEEE International Conference on Fuzzy Systems Fuzzy Systems, Jeju Island, South Korea, 20–24 August 2009; pp. 1378–1382. [Google Scholar]
- Torra, V. Hesitant fuzzy sets. Int. J. Intell. Syst.
**2010**, 25, 529–539. [Google Scholar] [CrossRef] - Rodríguez, R.M.; Martínez, L.; Torra, V.; Xu, Z.S.; Herrera, F. Hesitant fuzzy sets: State of the art and future directions. Int. J. Intell. Syst.
**2014**, 29, 495–524. [Google Scholar] [CrossRef] - Chen, N.; Xu, Z.S.; Xia, M.M. Interval-valued hesitant preference relations and their applications to group decision making. Knowl. Based. Syst.
**2013**, 37, 528–540. [Google Scholar] [CrossRef] - Yu, D.J.; Zhang, W.Y.; Xu, Y.J. Group decision making under hesitant fuzzy environment with application to personnel evaluation. Knowl. Based Syst.
**2013**, 52, 1–10. [Google Scholar] [CrossRef] - Peng, D.H.; Gao, C.Y.; Gao, Z.F. Generalized hesitant fuzzy synergetic weighted distance measures and their application to multiple criteria decision making. Appl. Math. Model.
**2013**, 37, 5837–5850. [Google Scholar] [CrossRef] - Farhadinia, B. Correlation for dual hesitant fuzzy sets and dual interval-valued hesitant fuzzy sets. Int. J. Intell. Syst.
**2014**, 29, 184–205. [Google Scholar] [CrossRef] - Rodríguez, R.M.; Martínez, L.; Herrera, F. Hesitant fuzzy linguistic term sets for decision making. IEEE Trans. Syst.
**2012**, 20, 109–119. [Google Scholar] [CrossRef] - Dong, Y.C.; Chen, X.; Herrera, F. Minimizing adjusted simple terms in the consensus reaching process with hesitant linguistic assessments in group decision making. Inf. Sci.
**2015**, 297, 95–117. [Google Scholar] [CrossRef] - Farhadinia, B. Distance and similarity measures for higher order hesitant fuzzy sets. Knowl. Based Syst.
**2014**, 55, 43–48. [Google Scholar] [CrossRef] - Morente-Molinera, J.A.; Kou, G.; González-Crespo, R.; Corchado, J.M.; Herrera-Viedma, E. Solving multi-criteria group decision making problems under environments with a high number of alternatives using fuzzy ontologies and multi-granular linguistic modelling methods. Knowl. Syst.
**2017**, 137, 54–64. [Google Scholar] [CrossRef] - Alcantud, J.C.R.; Giarlotta, A. Necessary and possible hesitant fuzzy sets: A novel model for group Decision making. Inf. Fusion
**2018**, 46, 63–76. [Google Scholar] [CrossRef] - Zhang, Z.; Wu, C. Weighted hesitant fuzzy sets and their application to multi-criteria decision making. British J. Math. Comput. Sci.
**2014**, 4, 1091–1123. [Google Scholar] [CrossRef] - Farhadinia, B.; Xu, Z.S. Distance and aggregation-based methodologies for hesitant fuzzy decision making. Cogn. Comput.
**2017**, 9, 81–94. [Google Scholar] [CrossRef] - Yager, R.R. Prioritized aggregation operators. Int. J. Approx. Reason.
**2008**, 48, 263–274. [Google Scholar] [CrossRef] - Xia, M.; Xu, Z.S. Hesitant fuzzy information aggregation in decision making. Int. J. Approx. Reason.
**2011**, 52, 395–407. [Google Scholar] [CrossRef] [Green Version] - Wei, G.W. Hesitant fuzzy prioritized operators and their application to multiple attribute decision making. Knowl. Based Syst.
**2012**, 31, 176–182. [Google Scholar] [CrossRef] - Qua, G.H.; Zhang, H.P.; Qua, W.H.; Zhang, Z.H. Induced generalized dual hesitant fuzzy Shapley hybrid operators and their application in multi-attributes decision making. J. Intell. Fuzzy Syst.
**2016**, 31, 633–650. [Google Scholar] [CrossRef] - Wei, G.W.; Lu, M.; Tang, X.Y.; Wei, Y. Pythagorean hesitant fuzzy Hamacher aggregation operators and their application to multiple attribute decision making. Int. J. Intell. Syst.
**2018**, 33, 1197–1233. [Google Scholar] [CrossRef] - Galankashi, M.R.; Chegeni, A.; Soleimanynanadegany, A.; Memari, A.; Anjomshoae, A.; Helmi, S.A.; Dargi, A. Prioritizing green supplier selection criteria using fuzzy analytical network process. Procedia CIRP
**2015**, 26, 689–694. [Google Scholar] [CrossRef] - Mousakhani, S.; Nazari-Shirkouhi, S.; Bozorgi-Amiri, A. A novel interval type-2 fuzzy evaluation model based group decision analysis for green supplier selection problems: A case study of battery industry. J. Clean. Prod.
**2017**, 168, 205–218. [Google Scholar] [CrossRef] - Omurca, S.I. An intelligent supplier evaluation, selection and development system. Appl. Soft Comput.
**2013**, 13, 690–697. [Google Scholar] [CrossRef] - Alcantud, J.C.R.; Santos-García, G. Expanded hesitant fuzzy sets and group decision making. In Proceedings of the IEEE International Conference on Fuzzy Systems, Naples, Italy, 9–12 July 2017. [Google Scholar]

Variable | Criterion | Definition | Authors |
---|---|---|---|

c_{1} | Cost | Total cost of product and service | Yeh and Chuang [12], Govindan et al. [6], Mousakhani et al. [65] |

c_{2} | Quality | The quality of product and service | Omurca [66], Govindan et al. [6], Mousakhani et al. [65] |

c_{3} | Service | Performance in terms of product service and social service | Omurca [66], Kannan et al. [35], Govindan et al. [6] |

c_{4} | Environment | Environmental protection; certification and materials recycling capacity | Govindan et al. [6], Mousakhani et al. [65], Lee et al. [27] |

c_{5} | Technology | Ability to facilitate the development of green products | Lee et al. [27], Govindan et al. [6], Mousakhani et al. [65] |

c_{6} | Management | Capcity for environmental management | Kuo et al. [12], Tseng et al. [24], Mousakhani et al. [65] |

c_{7} | Responsibility | Including safety production, social morality and public interest | Galankashi, et al. [6], Mousakhani et al. [65] |

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

x_{1} | {<0.3,2/6>,<0.5,1/6>,<0.8,3/6>} | {<0.3,2/6>,<0.6,1/6>,<0.7,3/6>} | {<0.3,2/6>,<0.6,1/6>,<0.7,3/6>} | {<0.4,2/6>,<0.5,1/6>,<0.6,3/6>} |

x_{2} | {<0.1,2/6>,<0.4,1/6>,<0.5,3/6>} | {<0.2,2/6>,<0.3,1/6>,<0.5,3/6>} | {<0.1,2/6>,<0.4,1/6>,<0.5,3/6>} | {<0.2,2/6>,<0.3,1/6>,<0.4,3/6>} |

x_{3} | {<0.1,2/6>,<0.2,1/6>,<0.3,3/6>} | {<0.1,2/6>,<0.2,1/6>,<0.4,3/6>} | {<0.1,2/6>,<0.2,1/6>,<0.3,3/6>} | {<0.1,2/6>,<0.2,1/6>,<0.4,3/6>} |

x_{4} | {<0.3,2/6>,<0.4,1/6>,<0.7,3/6>} | {<0.2,2/6>,<0.3,1/6>,<0.6,3/6>} | {<0.1,2/6>,<0.5,1/6>,<0.7,3/6>} | {<0.3,2/6>,<0.4,1/6>,<0.5,3/6>} |

x_{5} | {<0.7,2/6>,<0.8,1/6>,<0.9,3/6>} | {<0.5,2/6>,<0.7,1/6>,<0.8,3/6>} | {<0.4,2/6>,<0.6,1/6>,<0.7,3/6>} | {<0.5,2/6>,<0.6,1/6>,<0.7,3/6>} |

**Table 3.**The results of the generalized ordered weighted hesitant fuzzy prioritized weighted average operator (GOWHFPWA) operator with different α.

α | x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | Rankings |
---|---|---|---|---|---|---|

0.1 | 0.5666 | 0.3527 | 0.2255 | 0.4777 | 0.7176 | ${x}_{5}\succ {x}_{1}\succ {x}_{4}\succ {x}_{2}\succ {x}_{3}$ |

0.2 | 0.5667 | 0.3529 | 0.2256 | 0.4780 | 0.7180 | ${x}_{5}\succ {x}_{1}\succ {x}_{4}\succ {x}_{2}\succ {x}_{3}$ |

0.5 | 0.5670 | 0.3535 | 0.2258 | 0.4791 | 0.7192 | ${x}_{5}\succ {x}_{1}\succ {x}_{4}\succ {x}_{2}\succ {x}_{3}$ |

1 | 0.5676 | 0.3547 | 0.2262 | 0.4809 | 0.7211 | ${x}_{5}\succ {x}_{1}\succ {x}_{4}\succ {x}_{2}\succ {x}_{3}$ |

2 | 0.5689 | 0.3581 | 0.2270 | 0.4847 | 0.7254 | ${x}_{5}\succ {x}_{1}\succ {x}_{4}\succ {x}_{2}\succ {x}_{3}$ |

5 | 0.5737 | 0.3703 | 0.2309 | 0.4941 | 0.7389 | ${x}_{5}\succ {x}_{1}\succ {x}_{4}\succ {x}_{2}\succ {x}_{3}$ |

10 | 0.5836 | 0.3854 | 0.2389 | 0.5045 | 0.7581 | ${x}_{5}\succ {x}_{1}\succ {x}_{4}\succ {x}_{2}\succ {x}_{3}$ |

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

x_{1} | {0.3,0.5,0.8} | {0.3,0.6,0.7} | {0.3,0.6,0.7} | {0.4,0.5,0.6} |

x_{2} | {0.1,0.4,0.5} | {0.2,0.3,0.5} | {0.1,0.4,0.5} | {0.2,0.3,0.4} |

x_{3} | {0.1,0.2,0.3} | {0.1,0.2,0.4} | {0.1,0.2,0.3} | {0.1,0.2,0.4} |

x_{4} | {0.3,0.4,0.7} | {0.2,0.3,0.6} | {0.1,0.5,0.7} | {0.3,0.4,0.5} |

x_{5} | {0.7,0.8,0.9} | {0.5,0.7,0.8} | {0,4,0.6,0.7} | {0.5,0.6,0.7} |

Methods | x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | Ranking Order |
---|---|---|---|---|---|---|

GOWHFPWA | 0.5676 | 0.3547 | 0.2262 | 0.4809 | 0.7211 | ${x}_{5}\succ {x}_{1}\succ {x}_{4}\succ {x}_{2}\succ {x}_{3}$ |

HFPWA | 0.5689 | 0.3581 | 0.2270 | 0.4847 | 0.7254 | ${x}_{5}\succ {x}_{1}\succ {x}_{4}\succ {x}_{2}\succ {x}_{3}$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, Y.; Jin, L.; Zhu, F.
A Multi-Criteria Group Decision Making Model for Green Supplier Selection under the Ordered Weighted Hesitant Fuzzy Environment. *Symmetry* **2019**, *11*, 17.
https://doi.org/10.3390/sym11010017

**AMA Style**

Liu Y, Jin L, Zhu F.
A Multi-Criteria Group Decision Making Model for Green Supplier Selection under the Ordered Weighted Hesitant Fuzzy Environment. *Symmetry*. 2019; 11(1):17.
https://doi.org/10.3390/sym11010017

**Chicago/Turabian Style**

Liu, Yumin, Linlin Jin, and Feng Zhu.
2019. "A Multi-Criteria Group Decision Making Model for Green Supplier Selection under the Ordered Weighted Hesitant Fuzzy Environment" *Symmetry* 11, no. 1: 17.
https://doi.org/10.3390/sym11010017