Applications of probabilistic hesitant fuzzy rough set in decision support system
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METHODOLOGIES AND APPLICATION
Applications of probabilistic hesitant fuzzy rough set in decision support system Muhammad Ali Khan1 · Shahzaib Ashraf1 · Saleem Abdullah1 · Fazal Ghani1
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The objective of this manuscript is to present the notion of probabilistic hesitant fuzzy rough (PHFR) set and their basic operations. As a generalization of the sets, PHFR set is a more profitable way to express the uncertainties in the data. For it, firstly, we define the basic operational laws like, the union, intersection and the composition of probabilistic hesitant fuzzy approximation spaces with some basic properties are discussed in details. Secondly, presented the novel decision-making technique based on the PHFR sets over two nonempty fixed sets to deal with uncertainty in decision-making problems. Finally, two numerical examples are provided with some comparative study to validate the proposed approach. Keywords Probabilistic set · Hesitant fuzzy set · Rough set · Probabilistic hesitant fuzzy rough set · Decision-making technique
1 Introduction The theory of fuzzy set (FS) has been proposed by Zadeh (1965) for more than half a century. For its success in handling uncertainty, imprecision and ambiguity, fuzzy sets have been applied in many areas such as automatic control, decision making, and artificial intelligence. Fuzzy sets are generalized from classical sets by extending the characteristic function to the membership function. In a fuzzy set, values between 0 and 1 are used to depict the membership function. And the membership function and the non-membership function are summed to 1. Such constraint leads to the drawback that fuzzy sets cannot capture all kinds of uncertainty in reality.
Communicated by V. Loia.
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Shahzaib Ashraf [email protected] Saleem Abdullah [email protected] Muhammad Ali Khan [email protected] Fazal Ghani [email protected]
1
Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan
To represent the use of element in a quantitative value which is difficult task for the decision maker. For easy mode, we express in fuzzy sets or its novel extensions. In the daily life, we were facing different problems to take the decision of selection the best choice. To develop the technique for decision-making process, using the uncertain information is a difficult task. The uncertainty arised due to classical models and tools which cannot express the information through classical set. Many researchers got attraction to deal with uncertainty in decision-making problems to sort out the beat alternative according to given attributes. In this regard, many multi-attribute decision-making techniques are developed under fuzzy sets and their extension (Wang et al. 2014b; Xia and Xu 2011; Yager 1988; Zhang and Wei 2013). The concept of hesitant FS introduced by Torra (2010) to upgrade the form of fuzzy set, which has a set of values without having a single value in the form of membership. Hesitant
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