Interval and Fuzzy-Valued Functions: Generalized Differentiability and Applications

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (1 May 2022) | Viewed by 4692

Special Issue Editors


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Department of Statistics and Operational Research, University of Cadiz, Cádiz, Spain
Interests: optimization methods under uncertainty; interval and fuzzy numbers; multi-objective programming; generalized convexity
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Guest Editor
Associate Professor, Department of Statistical Sciences “Paolo Fortunati”, University of Bologna, Bologna, Italy
Interests: financial mathematics; interval and fuzzy mathematics; uncertainty modeling
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Guest Editor
Department of Economics, Society and Politics - University of Urbino “Carlo Bo”, Italy
Interests: fuzzy numbers; optimization models; computational mathematics

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Guest Editor
Department of Economics, Society, Politics – University of Urbino “Carlo Bo”, Urbino, Italy
Interests: interval and fuzzy mathematics; uncertainty modeling and decision making; soft computing; fuzzy numbers and algorithms

Special Issue Information

Dear Colleagues,

Interval and fuzzy-valued functions have been a field of intensive research during several decades and received an increasing interest after the introduction of concepts of generalized differentiability. The purpose of this Special Issue is to bring together research coming from different approaches and perspectives and used to formulate or revisit several topics in generalized differentiability, with a particular focus on applications. We strongly believe that the Special Issue will offer an occasion to encourage cross-fertilization and to enhance this newly emerging research area.

A partial list of topics:

  • Interval and fuzzy-valued functions of a single and multiple variables;
  • Generalized derivative and differentiability of fuzzy-valued functions;
  • Comparison of different definitions and forms of generalized differentiability;
  • Properties of interval-valued, set-valued and fuzzy-valued functions;
  • Calculus and algebra for interval and fuzzy generalized derivative;
  • Characterization of generalized differentiable functions;
  • Generalized fuzzy differentiability on time scales and applications;
  • Fractional differentiability of interval and fuzzy-valued functions and applications;
  • Partial derivatives of interval and fuzzy-valued functions of multiple variables;
  • Fuzzy differential equations: theory, numerical solution, algorithms, applications;
  • Optimization with nonlinear fuzzy-valued objective functions;
  • Global optimization with fuzzy-valued objective;
  • Vector-valued optimization with interval and fuzzy objectives;
  • Linear interval and fuzzy optimization: theory, algorithms, applications;

Prof. Dr. Manuel Arana-Jimenez
Prof. Dr. Maria Letizia Guerra
Prof. Dr. Laerte Sorini
Prof. Dr. Luciano Stefanini
Guest Editors

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Keywords

  • Interval-valued functions
  • Set-valued function
  • Fuzzy-valued function
  • Generalized derivative
  • Generalized differentiability
  • Interval optimization
  • Fuzzy optimization

Published Papers (2 papers)

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Research

23 pages, 392 KiB  
Article
Interval-Valued Pseudo Overlap Functions and Application
by Rong Liang and Xiaohong Zhang
Axioms 2022, 11(5), 216; https://doi.org/10.3390/axioms11050216 - 6 May 2022
Cited by 17 | Viewed by 1878
Abstract
A class of interval-valued OWA operators can be constructed from interval-valued overlap functions with interval-valued weights, which plays an important role in solving multi-attribute decision making (MADM) problems considering interval numbers as attribute values. Among them, when the importance of multiple attributes is [...] Read more.
A class of interval-valued OWA operators can be constructed from interval-valued overlap functions with interval-valued weights, which plays an important role in solving multi-attribute decision making (MADM) problems considering interval numbers as attribute values. Among them, when the importance of multiple attributes is different, it can only be calculated by changing the interval-valued weights. In fact, we can directly abandon the commutativity and extend the interval-valued overlap functions (IO) to interval-valued pseudo overlap functions (IPO) so that function itself implies the weights of the attributes, thus there is no need to calculate the OWA operator, which is more flexible in applications. In addition, the similar generalization on interval-valued pseudo t-norms obtained from interval-valued t-norms further enhances the feasibility of our study. In this paper, we mainly present the notion of interval-valued pseudo overlap functions and a few their qualities, including migrativity and homogeneity, and give some construction theorems and specific examples. Then, we propose the definitions of residuated implications induced by interval-valued pseudo overlap functions, give their equivalent forms, and prove some properties satisfied by them. Finally, two application examples about IPO to interval-valued multi-attribute decision making (I-MADM) are described. The results show that interval-valued pseudo overlap functions can not only be used to obtain the same rankings, but also be more flexible, simple and widely used. Full article
14 pages, 1712 KiB  
Article
Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System
by Ji-Eun Kim
Axioms 2021, 10(3), 206; https://doi.org/10.3390/axioms10030206 - 30 Aug 2021
Cited by 2 | Viewed by 1665
Abstract
The step derivative of a complex function can be defined with various methods. The step direction defines a basis that is distinct from that of a complex number; the derivative can then be treated by using Taylor series expansion in this direction. In [...] Read more.
The step derivative of a complex function can be defined with various methods. The step direction defines a basis that is distinct from that of a complex number; the derivative can then be treated by using Taylor series expansion in this direction. In this study, we define step derivatives based on complex numbers and quaternions that are orthogonal to the complex basis while simultaneously being distinct from it. Considering previous studies, the step derivative defined using quaternions was insufficient for applying the properties of quaternions by setting a quaternion basis distinct from the complex basis or setting the step direction to which only a part of the quaternion basis was applied. Therefore, in this study, we examine the definition of quaternions and define the step derivative in the direction of a generalized quaternion basis including a complex basis. We find that the step derivative based on the definition of a quaternion has a relative error in some domains; however, it can be used as a substitute derivative in specific domains. Full article
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