Abstract-The aim of this paper is to distinguish between some of the more intrinsic differences that exist between grey system theory (GST) and fuzzy system theory (FST). There are several aspects of both paradigms that are closely related, it is precisely these close relations that will often result in a misunderstanding or misinterpretation. The subtly of the differences in some cases are difficult to perceive, hence why a definitive explanation is needed. This paper discusses the divergences and similarities between the interval-valued fuzzy set and grey set, interval and grey number; for both the standard and the generalised interpretation. A preference based analysis example is also put forward to demonstrate the alternative in perspectives that each approach adopts. It is believed that a better understanding of the differences will ultimately allow for a greater understanding of the ideology and mantras that the concepts themselves are built upon. By proxy, describing the divergences will also put forward the similarities. We believe that by providing an overview of the facets that each approach employs where confusion may arise, a thorough and more detailed explanation is the result. This paper places particular emphasis on grey system theory, describing the more intrinsic differences that sets it apart from the more established paradigm of fuzzy system theory.
The main aim of this paper is to connect R-Fuzzy sets and type-2 fuzzy sets, so as to provide a practical means to express complex uncertainty without the associated difficulty of a type-2 fuzzy set. The paper puts forward a significance measure, to provide a means for understanding the importance of the membership values contained within an R-fuzzy set. The pairing of an R-fuzzy set and the significance measure allows for an intermediary approach to that of a type-2 fuzzy set. By inspecting the returned significance degree of a particular membership value, one is able to ascertain its true significance in relation, relative to other encapsulated membership values. An R-fuzzy set coupled with the proposed significance measure allows for a type-2 fuzzy equivalence, an intermediary, all the while retaining the underlying sentiment of individual and general perspectives, and with the adage of a significantly reduced computational burden. Several human based perception examples are presented, wherein the significance degree is implemented, from which a higher level of detail can be garnered. The results demonstrate that the proposed research method combines the high capacity in uncertainty representation of type-2 fuzzy sets, together with the simplicity and objectiveness of type-1 fuzzy sets. This in turn provides a practical means for problem domains where a type-2 fuzzy set is preferred but difficult to construct due to the subjective type-2 fuzzy membership.
This paper presents a newly created significance measure based on a variation of Bayes' theorem, one which quantifies the significance of any value contained within an R-fuzzy set. An R-fuzzy set is a relatively new concept and an extension to fuzzy sets. By utilising the lower and upper approximations from rough set theory, an R-fuzzy approach allows for uncertain fuzzy membership values to be encapsulated. The membership values associated with the lower approximation are regarded as absolute truths, whereas the values associated with the upper approximation maybe be the result of a single voter, or the vast majority, but definitely not all. By making use of the significance measure one can inspect each and every encapsulated membership value. The significance value itself is a coefficient, this value will indicate how strongly it was agreed upon by the populace for a specific R-fuzzy descriptor. There has been no recent effort made in order to make sense of the significance of any of the values contained within an R-fuzzy set, hence the motivation for this paper. Also presented is a worked example, demonstrating the coupling together of an Rfuzzy approach and the significance measure.
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