Fuzzy parameterized fuzzy soft sets and decision making
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ORIGINAL ARTICLE
Fuzzy parameterized fuzzy soft sets and decision making Kuanyun Zhu1 • Jianming Zhan1
Received: 14 December 2014 / Accepted: 19 October 2015 Ó Springer-Verlag Berlin Heidelberg 2015
Abstract The aim of this paper is to define t-norms and t-conorms products of fuzzy parameterized fuzzy soft sets (briefly, fpfs-sets). By using these products, AND–fpfs decision making and OR–fpfs decision making methods are constructed, respectively. Finally, the decision making methods are applied to solve problems which contain uncertainties. Keywords Soft set fpfs-set t-norm t-conorm t-norm product t-conorm product decision making
1 Introduction We know that the real world is full of indeterminacy, inaccuracy and vagueness. In fact, most of the problems we dealt with are vague rather than precise. Facing so many uncertain data, classical methods are not always successful, the reason is that various types of uncertainties present in these problems. It is well known that the theories of probabilities, fuzzy sets [24], rough sets [23] and other mathematical theories are often useful approaches to describe uncertainties. However, Molodtsov [21] pointed out all these theories have their own difficulties. To overcome these difficulties, Molodtsov proposed a new approach for modelling uncertainty, which is called a soft set. Now, this theory has been applied in many areas, such & Jianming Zhan [email protected] Kuanyun Zhu [email protected] 1
Department of Mathematics, Hubei University for Nationalities, Enshi 445000Hubei, China
as information sciences, intelligent systems, machine learning, cybernetics, the smoothness of functions, game theory, operations research, measurement theory, probability theory and so on. Most of these applications have already been demonstrated in Molodtsov’s book. For more details, the reader is referred to [22]. Up to now, the research on soft sets is progressing rapidly. Some basic operations on soft sets were defined by Maji [18]. In 2009, Ali [1] gave some new operations on soft sets. Also, Maji et al. [20] defined a fuzzy soft set and they gave the application of fuzzy soft sets in decision problem in [19]. By using these definitions, the applications of soft set theory have been studied increasingly. C¸ag˘man and Maji [9, 10] applied soft set theory to decision making. Chen [11] discussed the parameterization reduction of soft sets and its applications. At the same time, Feng [16] introduced the application of level soft sets in decision making based on interval-valued fuzzy soft sets. Jiang [17] introduced a novel approach to interval-valued intuitionistic fuzzy sets in decision making. Later on, more general properties and applications of soft set theory have been studied by Feng, Maji and others, for example, see [14, 19]. It is pointed out that soft set theory has been expanded by fuzzy soft sets, which are refereed to [5, 15, 20]. Recently, Zhan [25] firstly applied rough soft sets to hemirings, and described some characterizations of rough soft hemirings.
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