Distance and similarity measures for multiple-attribute decision making with dual hesitant fuzzy sets
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Distance and similarity measures for multiple-attribute decision making with dual hesitant fuzzy sets Pushpinder Singh
Received: 18 September 2013 / Revised: 25 February 2015 / Accepted: 26 February 2015 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015
Abstract Dual hesitant fuzzy set, first proposed by Zhu et al. (Dual hesitant fuzzy sets, J Appl Math, 1–13, 2012) as an extension of hesitant fuzzy sets, which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as a special case. Dual hesitant fuzzy sets consist of two parts, that is, the membership and nonmembership degrees, which are represented by two sets of possible values. Therefore, in accordance with the practical demand, these sets are more flexible than the existing fuzzy sets, and provide much more information about the situation. In this paper, the axiom definition of distance and similarity measures between dual hesitant fuzzy sets is introduced. Some new distance and similarity measures based on the geometric distance model, the set-theoretic approach, and the matching functions are proposed. The proposed distance measures are then applied to the multiple-attribute decision making under dual hesitant fuzzy environments. Through the distance measure between each alternative and the ideal alternative, the ranking order of all alternatives can be determined and the best alternative can be easily identified as well. Finally, a practical example of investment alternatives is given to demonstrate the effectiveness of the developed measures. The advantages of the proposed distance measure over existing measures have been presented. Keywords Fuzzy sets · Hesitant fuzzy sets · Dual hesitant fuzzy sets · Distance measures · Similarity measures · Multiple-attribute decision-making problems Mathematics Subject Classification
03E72 · 90B50
Communicated by Eduardo Souza de Cursi. P. Singh (B) Department of Computer Science, Palacky University, 17. listopadu 12, 77146 Olomouc, Czech Republic e-mail: [email protected]; [email protected]
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1 Introduction The theory of fuzzy sets, proposed by Zadeh (1965), has been successfully applied in various fields (Kumar et al. 2011a, b; Li et al. 2007; Singh 2012, 2014d; Verma et al. 2013). The concept of similarity is fundamentally important in almost every scientific field including pattern recognition, machine learning, decision making real and market prediction etc. (Gao et al. 2013; Liu et al. 2013; Singh 2014a, 2015). In literature, very large number of distance and similarity measures for fuzzy sets (FSs) and intuitionistic fuzzy set has been proposed (Atanassov 1986; Chen et al. 2013; Grzegorzewski 2004; Hung and Yang 2007; Liang and Shi 2003; Singh 2014c; Szmidt and Kacprzyk 2000; Wang 1997; Wang and Xin 2005). Very few authors discussed the similarity measures between type-2 fuzzy sets (Hung and Yang 2004; Hwang et al. 2011; Mitchell 2005; Singh 2014b; Yang and Lin 2009; Yang and Shih 2001). Torra and Narukawa (2009) and Tor
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