Abstract
The normal wiggly hesitant fuzzy set (NWHFS) is one of the latest extensions of hesitant fuzzy set (HFS). It can depict experts’ preference information consisting of several possible values and the potential information dug in the traditional hesitant fuzzy set (HFS) at the same time. However, traditional aggregation operators of NWHFSs always ignore the influence of psychological behaviors. Therefore, to make use of the potential information and manage the applications with incomplete rationality, this paper combines the advantages of TODIM (TOmada deDecisão Iterativa Multicritério) method and NWHFSs. Considering the structures and characteristics of NWHFSs, we also provide the definition of distance measure of NWHFSs. Based on this, we develop an improved TODIM method. The specific implementation process is also provided. Finally, we apply the improved TODIM method to the environmental quality evaluation. The comparison result with traditional aggregation operators indicates that the proposed method can enlarge the competitive relationship better. It also demonstrates the rationality and accuracy of our method.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00500-023-08155-3/MediaObjects/500_2023_8155_Fig1_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00500-023-08155-3/MediaObjects/500_2023_8155_Fig2_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00500-023-08155-3/MediaObjects/500_2023_8155_Fig3_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00500-023-08155-3/MediaObjects/500_2023_8155_Fig4_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00500-023-08155-3/MediaObjects/500_2023_8155_Fig5_HTML.png)
Similar content being viewed by others
Data availability
Enquiries about data availability should be directed to the authors.
References
Bogacz R, Gurney K (2007) The basal ganglia and cortex implement optimal decision making between alternative actions. Neural Comput 19:442–477
Chen TY (2013) An interval-valued intuitionistic fuzzy LINMAP method with inclusion comparison possibilities and hybrid averaging operations for multiple criteria group decision making. Knowl-Based Syst 45:134–146
Chen N, Xu ZS, **a MM (2013) Interval-valued hesitant preferece relations and their applications to group decision making. Knowl-Based Syst 37:528–540
Chhipi-Shrestha CK, Hewage K, Sadiq R (2017) Selecting sustainability indicators for small to medium sized urban water systems using fuzzy ELECTRE. Water Environ Res 89:238–249
Choi H, Han K, Choi K et al (2016) A fuzzy medical diagnosis based on quantiles of diagnostic measures. J Intell Fuzzy Syst 31:3197–3202
Couzin I, Krause J, Franks N et al (2005) Effective leadership and decision-making in animal groups on the move. Nature 433:513–516
De MB, Kumaran D, Seymour B et al (2006) Frames, biases and rational decision-making in the human brain. Science 313:684–687
Divsalar M, Ahmadi M, Ebrahimi E et al (2022) A probabilistic hesitant fuzzy Choquet integral-based TODIM method for multi-attribute group decision-making. Expert Syst Appl. https://doi.org/10.1016/j.eswa.2021.116266
Gold JI, Shadlen MN (2007) The neural basis of decision making. Annu Rev Neurosci 30:5350574
Gomes LFAM, Lima MMPP (1991) TODIM: Basic and application to multi-criteria ranking of projects with environmental impacts. Fund Comput Decis Sci 16:113–127
Hancock TO, Hess S, Choudhury CF (2018) Decision field theory: improvements to current methodology and comparisons with standard choice modelling techniques. Transp Res Part B: Methodol 107:18–40
Hao ZN, Xu ZS, Zhao H et al (2017) Probabilistic dual hesitant fuzzy set and its application in risk evaluation. Knowl-Based Syst 127:16–28
Hashemkhani Zolfani S, Pourhossein M, Yazdani M, Kazimieras Zavadskas E (2017) Evaluating construction projects of hotels based on environmental sustainability with MCDM framework. Alex Eng J. https://doi.org/10.1016/j.aej.2016.1011.1002
Li DF (2005) An approach to fuzzy multi-attribute decision making under uncertainty. Inf Sci 169:97–112
Liao SH (2000) Case-based decision support system: architecture for simulating military command and control. Eur J Oper Res 123:558–567
Liu PD, Pei Z (2020) Normal wiggly hesitant fuzzy TODIM approach for multiple attribute decision making. J Intell Fuzzy Syst. https://doi.org/10.3233/JIFS-191569
Liu F, Zhang WG (2013) TOPSIS-based consensus model for group decision-making with incomplete interval fuzzy preference relationships. IEEE Trans Cybern 44:1283–1294
Liu PD, Zhang P (2020) Normal wiggly hesitant fuzzy TODIM approach for multiple attribute decision making. J Intell Fuzzy Syst 39:1–18
Liu PD, Xu H, Pedrycz W (2020) A normal wiggly hesitant fuzzy linguistic projection-based multi-attributive border approximation area comparison method. Int J Intell Syst. https://doi.org/10.1002/int.22213
Manrai AK (1995) Mathematical models of brand choice behavior. Eur J Oper Res 82:1–17
Ming H, Bhatt M, Adolphs R et al (2006) Neural systems responding to degrees of uncertainty in human decision making. Science 310:1680–1683
Ren ZL, Xu ZS, Wang H (2018) Normal wiggly hesitant fuzzy sets and their application to environmental quality evaluation. Knowl-Based Syst 159:286–297
Sanfey AG, Rilling JK, Aronson JA et al (2003) The neural basis of economic decision-making in the ultimatum game. Science 300:1755–1758
Song CY, Zhao H, Xu ZS et al (2019) Interval-valued probabilistic hesitant fuzzy set and its application in the Arctic geopolitical risk evaluation. Int J Intell Syst 34:627–651
Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25:529–539
Wang D, Wei S, Luo H et al (2017) A novel hybrid model for air quality index forecasting based on two-phase decomposition technique and modified extreme learning machine. Sci Total Environ 580:719–733
**a MM, Xu ZS (2011) Hesitant fuzzy information aggregation in decision making. Int J Approx Reason 52:395–407
Yu DJ (2013) Triangular Hesitant fuzzy set and its application to teaching quality evaluation. J Inf Comput Sci 10:1925–1934
Zadeh LA (1965) Fuzzy sets. Information. Control 8:338–356
Zhang C, Li D, Yan Y (2015) A dual hesitant fuzzy multi-granulation rough set over two-universe model for medical diagnoses. Comput Math Methods Med 5:1–12
Zhang S, Xu ZS, He Y (2017) Operations and integrations of probabilistic hesitant fuzzy information in decision making. Inf Fus 38:1–11
Zhu B, Xu ZS (2014) Some results for dual hesitant fuzzy sets. J Intell Fuzzy Syst 26:1657–1668
Zindani D, Maity SR, Bhowmik S (2021) Extended TODIM method based on normal wiggly hesitant fuzzy sets for deducing optimal reinforcement condition of agro-waste fibers for green product development. J Clean Prod 1:126947
Acknowledgements
The work was supported by the National Natural Science Foundation of China (No. 72071135).
Funding
The funding was provided by National Natural Science Foundation of China (No. 72071135).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have not disclosed any competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Song, C., Xu, Z., Zhang, Y. et al. Environmental quality evaluation based on the TODIM method with normal wiggly hesitant fuzzy set. Soft Comput 27, 8161–8173 (2023). https://doi.org/10.1007/s00500-023-08155-3
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-023-08155-3