A Survey of Interval-Valued Fuzzy Sets

Information Sciences, 2009

This paper, which is tutorial in nature, demonstrates how the Embedded Sets Representation Theorem (RT) for a general type-2 fuzzy set (T2 FS), when specialized to an interval (I)T2 FS, can be used as the starting point to solve many diverse problems that involve IT2 FSs. The problems considered are: set theoretic operations, centroid, uncertainty measures, similarity, inference engine computations for Mamdani IT2 fuzzy logic systems, linguistic weighted average, person membership function approach to type-2 fuzzistics, and Interval Approach to type-2 fuzzistics. Each solution obtained from the RT is a structural solution but is not a practical computational solution, however, the latter are always found from the former. It is this author's recommendation that one should use the RT as a starting point whenever solving a new problem involving IT2 FSs because it has had such great success in solving so many such problems in the past, and it answers the question ''Where do I start in order to solve a new problem involving IT2 FSs?" Ó

Studies in Fuzziness and Soft Computing, 2019

The series "Studies in Fuzziness and Soft Computing" contains publications on various topics in the area of soft computing, which include fuzzy sets, rough sets, neural networks, evolutionary computation, probabilistic and evidential reasoning, multi-valued logic, and related fields. The publications within "Studies in Fuzziness and Soft Computing" are primarily monographs and edited volumes. They cover significant recent developments in the field, both of a foundational and applicable character. An important feature of the series is its short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results.

Fuzzy Sets and Systems, 2007

In this paper we stress the relevance of a particular family of fuzzy sets, where each element can be viewed as the result of a classification problem. In particular, we assume that fuzzy sets are defined from a well-defined universe of objects into a valuation space where a particular graph is being defined, in such a way that each element of the considered universe has a degree of membership with respect to each state in the valuation space. The associated graph defines the structure of such a valuation space, where an ignorance state represents the beginning of a necessary learning procedure. Hence, every single state needs a positive definition, and possible queries are limited by such an associated graph. We then allocate this family of fuzzy sets with respect to other relevant families of fuzzy sets, and in particular with respect to Atanassov's intuitionistic fuzzy sets. We postulate that introducing this graph allows a natural explanation of the different visions underlying Atanassov's model and interval valued fuzzy sets, despite both models have been proven equivalent when such a structure in the valuation space is not assumed.

In this paper, we introduce specific types of intuitionistic fuzzy sets, inspired by the multi-dimensional intuitionistic fuzzy sets and general type-2 fuzzy sets. The newly proposed sets extend the opportunities of the general type-2 fuzzy sets when modeling particular types of uncertainty. Short comparison between concepts of interval type-2 fuzzy sets and intuitionistic fuzzy sets is presented. In addition, new future directions of research are outlined.

Bulletin of the American Mathematical Society

2006

This work considers an interval extension of fuzzy implication based on the best interval representation of continuous t-norms. Some related properties can be naturally extended and that extension preserves the behaviors of the implications in the interval endpoints.

12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006), 2006

We analyze sufficient and necessary conditions for these three classes of implications as inclusion-monotonic functions in both arguments satisfying the minimal properties of fuzzy implication.

Fuzzy Sets and Systems, 2006

Among the various extensions to the common [0, 1]-valued truth degrees of "traditional" fuzzy set theory, closed intervals of [0, 1] stand out as a particularly appealing and promising choice for representing imperfect information, nicely accommodating and combining the facets of vagueness and uncertainty without paying too much in terms of computational complexity. From a logical point of view, due to the failure of the omnipresent prelinearity condition, the underlying algebraic structure L I falls outside the mainstream of the research on formal fuzzy logics (including MV-, BL-and MTL-algebras), and consequently so far has received only marginal attention. This comparative lack of interest for interval-valued fuzzy logic has been further strengthened, perhaps, by taking for granted that its algebraic operations amount to a twofold application of corresponding operations on the unit interval. Abandoning that simplifying assumption, however, we may find that L I reveals itself as a very rich and noteworthy structure allowing the construction of complex and surprisingly well-behaved logical systems. Reviewing the main advances on the algebraic characterization of logical operations on L I , and relating these results to the familiar completeness questions (which remain as major challenges) for the associated formal fuzzy logics, this paper paves the way for a systematic study of interval-valued fuzzy logic in the narrow sense.

IEEE Transactions on Fuzzy Systems, 2000

This paper explains how to compute normalized interval type-2 fuzzy sets in closed form and explains how the results reduce to well-known results for type-1 fuzzy sets and interval sets. Such normalized interval type-2 fuzzy sets may be needed in linguistic probability computations or multiple criteria decision analysis under uncertainty. Index Terms-Fuzzy weighted average (FWA), interval type-2 fuzzy sets (IT2 FSs), interval weighted average (IWA), linguistic probability, linguistic weighted average (LWA), normalized interval type-2 fuzzy sets. I. INTRODUCTION This paper is devoted to calculation of normalized interval type-2 fuzzy numbers, which, as we shall see, may be useful in dealing with linguistic probabilities and multicriteria decision making problems. In this section, we first introduce some notations and preliminary concepts and then elaborate on the problem description. A. Preliminaries and Notations An interval type-2 fuzzy set (IT2 FS) [2] A over a universe of discourse U is characterized by the membership function μ A (x), which assigns a closed subinterval of [0,1], J x ⊆ [0, 1], to each x ∈ U μ A (x) = J x = [μ A (x), μ A (x)]. (1) In other words, A is characterized by two type-1 fuzzy sets (T1 FSs) A and A, whose membership functions μ A (x) and μ A (x) are called the lower membership function (LMF) and the upper membership function (UMF) of A, respectively. Recall that an IT2 FS is completely described by its footprint of uncertainty (FOU) [3], [16], [17]; hence, IT2 FS A is described by FOU(A). In addition, FOU(A) is completely described by its LMF and UMF, μ A (x) and μ A (x), respectively, both of which are membership functions of T1 FSs, i.e., FOU(A) = ∀x ∈X μ A (x), μ A (x) (2) where ∪ denotes set-theoretic union. A nine-parameter FOU of a trapezoidal IT2 FS is shown in Fig. 1. Note that the UMF is determined by the parameters (a, b, c, d), and LMF is determined by (a, b, c, d, h), where h is the height of the LMF. Recall that the T1 FS W is called a type-1 fuzzy number if [6, Ch. 2] it is normal, convex, and has an upper semicontinuous membership function and a bounded support. The IT2 FS W i is called an interval type-2 fuzzy number if both its UMF and LMF, W and W ,

IEEE Transactions on Fuzzy Systems, 2020

In many contexts, type-2 fuzzy sets (T2 FS) are obtained from a type-1 fuzzy set to which we wish to add uncertainty. However, in the current type-2 representation, there is no restriction on the shape of the footprint of uncertainty and the embedded sets (ESs) that can be considered acceptable. This leads, usually, to the loss of the semantic relationship between the T2 FS and the concept it models. As a consequence, the interpretability of some of the ESs and the explainability of the uncertainty measures obtained from them can decrease. To overcome these issues, constrained type-2 (CT2) fuzzy sets have been proposed. However, no formal definitions for some of their key components [e.g., acceptable ESs (AESs)] and constrained operations have been given. In this article, we provide some theoretical underpinning for the definition of CT2 sets, their inferencing and defuzzification method. To conclude, the constrained inference framework is presented, applied to two real-world cases and briefly compared to the standard interval type-2 inference and defuzzification method. Index Terms-Constrained type-2 (CT2) fuzzy sets (CT2 FS), embedded sets (ESs), type-2 fuzzy logic, explainable artificial intelligence (XAI).