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

OLAP query answering requirements for a knowledge based treatment of user requests led us to introduce the concept of hierarchical preferences over a universe that has a hierarchical structure. We introduce the automatic analysis of queries according to concepts defined as part of knowledge based hierarchy in order to guide the query answering as part of a data-warehouse environment with the aid of hierarchical Intuitionistic fuzzy sets, H-IFS. Based on the notion of H-IFS we propose an ad-hoc utility build on top of current OLAP tools like Oracle10g that allows us to enhance the query capabilities of by providing better and knowledgeable answers to user’s requests. The theoretical aspects as well the practical issues and achieved results are presented throughout the rest of the paper.

Expressing Hierarchical Preferences in OLAP Queries

A new approach for design and self-organization of complex hierarchical control systems is proposed. Using the constructive abilities of the Generalized Nets (GNs), a situation-based control is accepted in order to integrate business, operational and control functions. The main alternation is paid in organization and coordination of a big number of structured parallel procedures aiming choice of optimal, or at least rational, control strategy, current structure, algorithms and tuning parameters for each possible real life situation, covering the full range of technological, business environmental and safety conditions and requirements.

Design and Multiaspect Functionality Realization of Hybrid Intelligent Control Systems via Generalized Nets.

Generalized net model of a university's classes schedule

The Generalized Nets (GN), proposed in 1982, are graphically oriented means for modelling, including as special cases the Petri nets. By their means was described the diagnostic processes of gout and turbid urine. The autors intend to present a series of communications on GN-models of basic nephrological symptoms. This report is dealing with the diagnostics of renal colic. The autors’ final aim is to create a broader GN including these GN.

The Application of Generalized Nets in Medicine (Renal Colic)

A new formula for the n-th prime number is introduced. It is based on an arithmetic function, similar to the operation “differentiation”, but defined only over natural numbers.

http://doi.baspress.com/pdf-journal/Comptes%20Rendus_4_2013/4_2013_4.pdf

Krassimir T. Atanassov

Papers

Expressing Hierarchical Preferences in OLAP Queries

Design and Multiaspect Functionality Realization of Hybrid Intelligent Control Systems via Generalized Nets.

Generalized net model of a university's classes schedule

The Application of Generalized Nets in Medicine (Renal Colic)

A FORMULA FOR THE n-TH PRIME NUMBER