This presentation outlines some aspects of interval analysis from the constraint interval point of view, focusing on some issues associated with mathematical analysis , especially those that are fundamental to fuzzy mathematical analysis. We begin by reviewing interval representations from the standard interval point of view and the study of intervals as functions of parameters varying in hypercubes, called constraint interval representation (CI). These concepts are used to define arithmetic operations between intervals, as well, interval-valued and interval functions for rational and transcedental functions. Issues of semantics as they apply to interval analysis and fuzzy interval analysis are discussed. The definition of constrained interval function extension is used to prove that each CI arithmetic operation is an isotonic inclusion.

In this article, we analyze the properties of automorphisms in$\mathbb {R}^{2}$. Then, we consider a subclass of these automorphisms which generate a preference order relation and asymmetric distances between bounded and closed intervals. Symmetrizing these distances, we generate a new family of metrics in this space of intervals which depends on the automorphisms. We study its properties and give a characterization that allows us to calculate the distance between intervals in a simple way. We provide many examples to illustrate our results as well as an application in multicriteria decision-making methods with interval data.

Abstract In this paper we present a review of the most important notions and characterizations of differentiability and necessary optimality conditions for a fuzzy multiobjective problem. As basis of this review, we first study the fundamental aspects of the notions of differentiability for interval valued functions, since the fuzzy environment and the interval environment are closely related. Those aspects are related to the different definitions of difference for intervals and their drawbacks, the different definitions and characterizations of the differentiability for interval-valued functions and their drawbacks and how they have been solved in the literature. Based on the most important and meaning results on interval valued functions you can find in the literature, a review on notions of differentiability in fuzzy context is given, both in the case of functions of one variable, and several variables. And finally we present the review results of the necessary optimality conditions for fuzzy multiobjective problems and the main conclusions.

This paper presents experimental results about the relationship between the Footprint Of Uncertainty (FOU) and the centroid of Interval Type-2 Fuzzy Numbers (IT2FN) which are a subclass of Type-2 fuzzy numbers. Four types of IT2FNs are analyzed: singleton-core triangular, interval-core triangular, singleton-core Gaussian and interval-core Gaussian IT2FNs. A quadratic relationship was detected, characterized, and their results are presented and discussed using numerical experiments.

This paper presents a ranking method, two distances, and some definitions for ordering interval type-2 fuzzy numbers based on the Yager index for type-1 fuzzy numbers. The proposed Yager index provides closed equations for the central tendency of well-defined interval type-2 fuzzy numbers. Some numerical examples are given, a comparison to the centroid of an interval type-2 fuzzy number is performed, and some interpretation issues are discussed.

In this work we study optimization problems where both objective and constraints are given by fuzzy functions. In order to solve them, we first prove that such problems are equivalent to fuzzy optimization problems where the constraints are non-fuzzy (crisp) functions. Besides we prove a new and appropriate Karush-Kuhn-Tucker type necessary optimality condition based on the gH-differentiability and that has many computational advantages that we describe. The gH-derivative for fuzzy functions is a more general notion than Hukuhara and level-wise derivatives ones that are usually used in fuzzy optimization so far, in the sense that it can be applied to a wider number of fuzzy function classes than above concepts. With this new differentiability concept, we prove a KKT-type necessary optimality condition for fuzzy mathematical programming problems that is more operational and less restrictive that the few ones we can find in the literature so far.

In this paper, a two-dimensional Hardy type inequality for fuzzy integrals is proved. Also, some illustrative examples are presented.

In this paper we show an approximation for [Formula: see text], where u is a fuzzy interval and [Formula: see text] is obtained from a continuous function f applying the Zadeh's extension principle. In this process, we used a decomposition for u and the linear spline function for the approximation of f. A bound for the error in the approximation is given. Also, some examples are presented.

In this work, we prove a Prékopa-Leindler type inequality for the Sugeno integral. More precisely, if 0 < λ < 1 and h((1-λ)x+λy) ≥ f(x)1-λg(y)λ, ∀ x,y ∈ ℝn, where h, f and g are nonnegative μ-measurable functions on ℝn, then [Formula: see text], for any concave fuzzy measure μ. Also, we derive a general Brunn-Minkowski inequality (standard form) for any homogeneous quasiconcave fuzzy measure μ on ℝn.