The difficulty of establishing a common membership degree is not because there is a margin of error or some possibility distribution values, but because there is a set of possible values. Based on hesitant fuzzy sets and soft sets, a hesitant soft fuzzy rough set model is proposed in this paper. Basic properties of hesitant soft fuzzy rough sets are investigated in detail. We obtain a decomposition theorem for a hesitant fuzzy binary relation, which states that every typical hesitant fuzzy binary relation on a set can be represented by a well-structured family of fuzzy binary relations on that set. Indeed, a hesitant fuzzy soft set can induce a hesitant fuzzy binary relation. Then we give the relationship between hesitant fuzzy rough sets and hesitant soft fuzzy rough sets. In addition, we prove a characterization theorem for the hesitant soft fuzzy rough set model, which shows that the lower and upper hesitant soft fuzzy rough approximations can be equivalently defined by using level sets of the hesitant fuzzy soft set. Finally, by analyzing the limitations and advantages in the existing literatures, we establish an approach to decision making problem based on the hesitant soft fuzzy rough set model proposed in this paper and give a practical example to illustrate the validity of the novel method.

Introduction

The contemporary concern about knowledge representation and information systems has put forward useful extensions of classical set theory such as fuzzy set theory and rough set theory. The concept of rough set was originally proposed by Pawlak [21] in 1982 as a formal tool for studying information systems characterized by insufficient and incomplete information. The starting point of this theory is an observation that objects having the same description are indiscernible with respect to available information. While the fuzzy set theory, introduced by Zadeh [36] in 1965, offers a wide variety of techniques for analyzing imprecise data. It soon evoked a natural question concerning possible connections between rough sets and fuzzy sets. It is generally accepted that these two theories are related, but distinct and complementary, to each other. Generally speaking, both theories address the problem of information granulation: the theory of fuzzy sets is centred upon fuzzy information granulation, whereas the rough set theory is focused on crisp information granulation. Pawlak’s rough set can be described by a pair of crisp sets called the lower approximation and the upper approximation. The lower approximation is the greatest definable set contained in the given set of objects, while the upper approximation is the smallest definable set containing the given set. By using the concept of lower and upper approximations in rough set theory, knowledge hidden in information systems may be revealed and expressed in the form of decision rules.