A Novel Comparison of Probabilistic Hesitant Fuzzy Elements in Multi-Criteria Decision Making
Abstract
:1. Introduction
2. Preliminaries
2.1. Concept of P-HFE
- (1)
- ;
- (2)
- ;
- (3)
- ;
- (4)
- .
2.2. The Ranking Method of the P-HFEs
- (1)
- If , then ;
- (2)
- If , then ;
- (3)
- If and , then ;
- (4)
- If and , then ;
- (5)
- If and , then we define that is equivalent to , denoted as .
3. Possibility Degree Formula for Ranking P-HFEs
3.1. The Different Methods for Ranking Fuzzy Numbers
3.2. A Concrete Formula for Ranking P-HFEs
- (1)
- With Equation (1), if two P-HFEs and have no common values in hesitant fuzzy sets, then or ; if , then we get .
- (2)
- The main innovations of our new method for the P-HFEs’ ranking are as follows:
- It is based on the structure of the P-HFEs and it considers their full information so that it can avoid the loss of information.
- The comparison result can show the relationship between different P-HFEs.
4. The Novel Ranking Method Based on the Possibility Degree Formula for P-HFEs
- Step 1
- If the weighting vector is given to us, with and , then we could use the probabilistic hesitant fuzzy weighted averaging (PHFWA) operator [15] to aggregate the P-HFEs of the alternatives :
- Step 2
- Construct a possibility degree matrix by computing using Equation (1):
- Step 3
- Derive the priorities from for its complementary judgment by employing the exact solution [27]:
- Step 4
- Let be the descending order of , thus we can obtain the ranking results of , :
- Step 5
- Based on the equation above, the ranking results of the alternatives are shown as follows:
5. A Case Study
- Step 1
- As the weighting vector has already been provided to us, we can use the PHFWA operator [15] to aggregate the evaluation information of the hospitals :
- Step 2
- Build the possibility degree matrix by contrasting each pair of P-HFEs based on Equation (1).
- Step 3
- According to the possibility degree matrix P above, we get the rank of :
6. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
References
- Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–356. [Google Scholar] [CrossRef]
- Atanassov, K. Intuitionistic Fuzzy Sets. Fuzzy Sets Syst. 1986, 20, 87–96. [Google Scholar] [CrossRef]
- Pang, Q.; Wang, H.; Xu, Z.S. Probabilistic linguistic term sets in multi-attribute group decision making. Inf. Sci. 2016, 369, 128–143. [Google Scholar] [CrossRef]
- Torra, V. Hesitant fuzzy sets. Int. J. Intell. Syst. 2010, 25, 529–539. [Google Scholar] [CrossRef]
- Gul, M.; Celik, E.; Gumus, A.T.; Guneri, A.F. A fuzzy logic based PROMETHEE method for material selection problems. Beni-Suef Univ. J. Basic Appl. Sci. 2018, 1, 68–79. [Google Scholar] [CrossRef]
- Tadić, S.; Zečević, S.; Krstić, M. A novel hybrid MCDM model based on fuzzy DEMATEL, fuzzy ANP and fuzzy VIKOR for city logistics concept selection. Expert Syst. Appl. 2014, 18, 8112–8128. [Google Scholar] [CrossRef]
- Meng, F.Y.; Chen, X.H. Correlation coefficients of hesitant fuzzy sets and their application based on fuzzy measures. Cognit. Comput. 2015, 7, 445–463. [Google Scholar] [CrossRef]
- Li, D.Q.; Zeng, W.Y.; Zhao, Y.B. Note on distance measure of hesitant fuzzy sets. Inf. Sci. 2015, 321, 103–115. [Google Scholar] [CrossRef]
- Bedregal, B.; Reiser, R.; Bustince, H.; Lopez-Molina, C.; Torra, V. Aggregation functions for typical hesitant fuzzy elements and the cation of automorphisms. Inf. Sci. 2014, 255, 82–99. [Google Scholar] [CrossRef]
- Xu, Z.S.; Xia, M.M. Distance and similarity measures for hesitant fuzzy sets. Inf. Sci. 2011, 181, 2128–2138. [Google Scholar] [CrossRef]
- Beliakov, G.; Bustince, H.; Calvo, T.; Mesiar, R.; Paternain, D. A class of fuzzy multisets with a fixed number of memberships. Inf. Sci. 2012, 189, 1–17. [Google Scholar]
- Bustince, H.; Barrenechea, E.; Pagola, M.; Fernandez, J.; Xu, Z.S.; Bedregal, B.; Montero, J.; Hagras, H.; Herrera, F.; de Baets, B. A historical account of types of fuzzy sets and their relationships. IEEE Trans. Fuzzy Syst. 2016, 24, 179–194. [Google Scholar] [CrossRef]
- Zhu, B. Decision Method for Research and Application Based on Preference Relation; Southeast University: Nanjing, China, 2014. [Google Scholar]
- Zhang, Z.; Wu, C. Weighted hesitant fuzzy sets and their application to multi-criteria decision making. Br. J. Math. Comput. Sci. 2014, 4, 1091–1123. [Google Scholar] [CrossRef]
- Zhang, S.; Xu, Z.S.; He, Y. Operations and integrations of probabilistic hesitant fuzzy information in decision making. Inf. Fusion 2017, 38, 1–11. [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]
- Dožić, S.; Lutovac, T.; Kalić, M. Fuzzy AHP approach to passenger aircraft type selection. J. Air Transp. Manag. 2018, 68, 165–175. [Google Scholar] [CrossRef]
- Eghbali-Zarch, M.; Tavakkoli-Moghaddam, R.; Esfahanian, F.; Sepehri, M.M.; Azaron, A. Pharmacological therapy selection of type 2 diabetes based on the SWARA and modified MULTIMOORA methods under a fuzzy environment. Artif. Intell. Med. 2018. [Google Scholar] [CrossRef] [PubMed]
- Bai, C.Z.; Zhang, R.; Qian, L.X.; Wu, Y.N. Comparisons of probabilistic linguistic term sets for multi-criteria decision making. Knowl. Based Syst. 2017, 119, 284–291. [Google Scholar] [CrossRef]
- Chen, S.H. Ranking fuzzy numbers with maximizing set and minimizing set. Fuzzy Sets Syst. 1985, 17, 113–129. [Google Scholar] [CrossRef]
- Abbasbandy, S.; Asady, B. Ranking of fuzzy numbers by sign distance. Inf. Sci. 2006, 176, 2405–2416. [Google Scholar] [CrossRef]
- Hao, M.; Kang, L. A method for ranking fuzzy numbers based on possibility degree. Math. Pract. Theory 2011, 21, 209–213. [Google Scholar]
- Dat, L.Q.; Yu, V.F.; Chou, S.Y. An improved ranking method for fuzzy numbers based on the centroid-index. Int. J. Fuzzy Syst. 2012, 14, 413–419. [Google Scholar]
- Chu, T.C.; Tsao, C.T. Ranking fuzzy numbers with an area between the centroid point and original point. Comput. Math. Appl. 2002, 43, 111–117. [Google Scholar] [CrossRef]
- Chai, K.C.; Kai, M.T.; Lim, C.P. A new method to rank fuzzy numbers using Dempster-Shafer theory with fuzzy targets. Inf. Sci. 2016, 346, 302–317. [Google Scholar] [CrossRef]
- Xu, Z.S. An overview of methods for determining OWA weights. Int. J. Intell. Syst. 2005, 20, 843–865. [Google Scholar] [CrossRef]
- Xu, Z.S. Two methods for priorities of complementary matrices-weighted least-square method and eigenvector method. Syst. Eng. Theory Pract. 2002, 7, 71–75. [Google Scholar]
Different Models | DMs Have more Choices | Retain most Decision-Making Information | DMs Have more Space to Hesitate |
---|---|---|---|
The HFS [4] | Yes | No | No |
The original P-HFS [13] | Yes | Yes | No |
The improved P-HFS [15] | Yes | Yes | Yes |
c1 | c2 | c3 | |
---|---|---|---|
h1 | |||
h2 | |||
h3 | |||
h4 |
Ranking Order | The Optimal Alternative | |
---|---|---|
The score and deviation method | ||
The novel possibility degree method |
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Song, C.; Xu, Z.; Zhao, H. A Novel Comparison of Probabilistic Hesitant Fuzzy Elements in Multi-Criteria Decision Making. Symmetry 2018, 10, 177. https://doi.org/10.3390/sym10050177
Song C, Xu Z, Zhao H. A Novel Comparison of Probabilistic Hesitant Fuzzy Elements in Multi-Criteria Decision Making. Symmetry. 2018; 10(5):177. https://doi.org/10.3390/sym10050177
Chicago/Turabian StyleSong, Chenyang, Zeshui Xu, and Hua Zhao. 2018. "A Novel Comparison of Probabilistic Hesitant Fuzzy Elements in Multi-Criteria Decision Making" Symmetry 10, no. 5: 177. https://doi.org/10.3390/sym10050177
APA StyleSong, C., Xu, Z., & Zhao, H. (2018). A Novel Comparison of Probabilistic Hesitant Fuzzy Elements in Multi-Criteria Decision Making. Symmetry, 10(5), 177. https://doi.org/10.3390/sym10050177