Distance measures for higher order dual hesitant fuzzy sets
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Distance measures for higher order dual hesitant fuzzy sets Jianjian Chen1 · Xianjiu Huang1 · Jing Tang1
Received: 31 July 2016 / Revised: 25 January 2017 / Accepted: 30 January 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract In this study, we propose new distance measures for dual hesitant fuzzy sets (DHFSs) in terms of the mean, standard deviation of dual hesitant fuzzy elements (DHFEs), respectively, which overcome some drawbacks of the existing distance measures. Meanwhile, we extend DHFS to its higher order type and refer to it as the higher order dual hesitant fuzzy set (HODHFS). HODHFS is the actual extension of DHFS that enables us to define the membership and non-membership of a given element in terms of several possible generalized type of fuzzy sets (G-Type FSs). The rationale behind HODHFS can be seen in the case that the decision makers are not satisfied by providing exact values for the membership degrees and the non-membership degrees. To indicate HODHFSs have a good performance in decision making, we introduce several distance measures for HODHFSs based on our proposed new distance for dual hesitant fuzzy sets. Finally, we practice our proposed measures for HODHFSs in multi-attribute decision making illustrating their applicability and availability. Keywords DHFS · Mean · Standard deviation · HODHFS · Distance measure Mathematics Subject Classification 03E72 · 90B50
1 Introduction When people make a decision, they are usually hesitant and irresolute for one thing or another which makes it difficult to reach a final agreement, that is, there usually exists a hesitation or uncertainty about the degree. Zhu et al. (2012) introduced the definition of DHFS, which is an extension of hesitant fuzzy set (Torra 2010; Torra and Narukawa 2009). DHFSs can better deal Communicated by Rosana Sueli da Motta Jafelice.
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Xianjiu Huang [email protected] Jianjian Chen [email protected]
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Department of Mathematics, Nanchang University, Nanchang 330031, People’s Republic of China
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J. Chen et al.
with the situations that permit both the membership and the non-membership of an element to a given set having a few different values, which can reflect the human’s hesitance of not only membership degrees, but also non-membership degrees. Then, a growing number of studies focus on DHFSs. Ye (2014) proposed a correlation coefficient between DHFSs as a new extension of existing correlation coefficients for hesitant fuzzy sets and intuitionistic fuzzy sets and apply it to multiple attribute decision making under dual hesitant fuzzy environments. Wang et al. (2014) first investigated a variety of distance measures and the corresponding similarity measures for dual hesitant fuzzy sets, based on which they presented a TOPSIS approach for the weapon selection problem. After that, Singh (2015) proposed some distance measures based on the geometric distance model, the set-theoretic approach, and the matching functions. However, these existing distance measures for dual hesitant
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