In this chapter, the basic definitions and properties of the interval valued intuitionistic fuzzy sets (IVIFSs) will be introduced. We will omit the majority of the proofs below, which are, in general, analogous to the proofs from Chapter 1.

Interval-Valued Intuitionistic Fuzzy Sets

Several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, Atanassov's intuitionistic fuzzy sets, type 2 fuzzy sets, and fuzzy multisets. In this paper, we propose hesitant fuzzy sets. Although from a formal point of view, they can be seen as fuzzy multisets, we will show that their interpretation differs from the two existing approaches for fuzzy multisets. Because of this, together with their definition, we also introduce some basic operations. In addition, we also study their relationship with intuitionistic fuzzy sets. We prove that the envelope of the hesitant fuzzy sets is an intuitionistic fuzzy set. We prove also that the operations we propose are consistent with the ones of intuitionistic fuzzy sets when applied to the envelope of the hesitant fuzzy sets. © 2010 Wiley Periodicals, Inc.

Hesitant fuzzy sets

An intuitionistic fuzzy set, characterized by a membership function and a non-membership function, is a generalization of fuzzy set. In this paper, based on score function and accuracy function, we introduce a method for the comparison between two intuitionistic fuzzy values and then develop some aggregation operators, such as the intuitionistic fuzzy weighted averaging operator, intuitionistic fuzzy ordered weighted averaging operator, and intuitionistic fuzzy hybrid aggregation operator, for aggregating intuitionistic fuzzy values and establish various properties of these operators.

Intuitionistic Fuzzy Aggregation Operators

In this paper, the authors study the theory of soft sets initiated by Molodtsov. The authors define equality of two soft sets, subset and super set of a soft set, complement of a soft set, null soft set, and absolute soft set with examples. Soft binary operations like AND, OR and also the operations of union, intersection are defined. De Morgan's laws and a number of results are verified in soft set theory.

Soft set theory

What is modal logic

In a series of research papers, the authors study some of the most important processes of functioning of universities and construct their Generalized Net (GN) models. The main focus in this paper is to analyse the process of the production of course timetables in a digital university and to evaluate their quality. The opportunity of using GNs as a tool for modeling such processes is also analyzed.

Towards a Model of the Digital University: A Generalized Net Model for Producing Course Timetables and for Evaluating the Quality of Subjects

During years of research, there have been defined 149 intuitionistic fuzzy implications. In the paper, it is checked which of these implications satisfy Klir-Yuan’s axioms, whether as (classical) tautologies or as intuitionistic fuzzy tautologies.

Intuitionistic Fuzzy Implications and Klir-Yuan’s Axioms

In the paper we describe a generalized net GACOA realizing an arbitrary algorithms for ant colony optimization. In this sense, this net is universal for all standard algorithms for ant colony optimization, since it describes the way of functioning and results of their work. Then, we discuss the way of constructing a GN that includes the GACOA as a subnet. In this way, we ensure the generalized net tokens' optimal transfer with regard to the results of GACOA . Thus, we construct a generalized net, featuring an optimization component and thus optimally functioning.

A generalized net with an ACO-Algorithm optimization component

As a continuation of the previous paper [8], here are presented two models of Generalized Nets (GN) — an n-bit digital magnitude comparator and an n-bit digital full adder. We can perform the measurements, if we have a set of several logical circuits that can be used to obtain identical output data. The aforementioned logical circuits must be composed of different logical elements. By using several measurement points and different schematics, we can suggest the best solution for the considered type of task.

Generalized net model of logic gates and circuits

This paper is a continuation of the previous two papers [7, 8], representing generalized net models of two of the basic sequential logic circuits - SR and JK flip-flop circuits. We can perform the measurements, if we have a set of several logical circuits that can be used to obtain identical output data. The aforementioned logical circuits must be composed of different logical elements. By using several measurement points and different schematics, we can infer the best solution for the considered type of task.

Krassimir T. Atanassov

Papers

Towards a Model of the Digital University: A Generalized Net Model for Producing Course Timetables and for Evaluating the Quality of Subjects

Intuitionistic Fuzzy Implications and Klir-Yuan’s Axioms

A generalized net with an ACO-Algorithm optimization component

Generalized net model of logic gates and circuits

Generalized net models of basic sequential logic circuits