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

Intuitionistic fuzzy logic is the main tool in the recently developed step-wise “cross-evaluation” procedure that aims at the assessment of different optimization algorithms. In this investigation, the procedure previously applied to compare the effectiveness of two or three algorithms has been significantly upgraded to evaluate the performance of a set of four algorithms. For the first time, the procedure applied here has been tested in the evaluation of the effectiveness of genetic algorithms (GAs), which are proven as very promising and successful optimization techniques for solving hard non-linear optimization tasks. As a case study exemplified with the parameter identification of a S. cerevisiae fed-batch fermentation process model, the cross-evaluation procedure has been executed to compare four different types of GAs, and more specifically, multi-population genetic algorithms (MGAs), which differ in the order of application of the three genetic operators: Selection, crossover and mutation. The results obtained from the implementation of the upgraded intuitionistic fuzzy logic-based procedure for MGA performance assessment have been analyzed, and the standard MGA has been outlined as the fastest and most reliable one among the four investigated algorithms.

How to Assess Different Algorithms Using Intuitionistic Fuzzy Logic

Model of Big Data Map/Reduce Processing

The Urals and Siberian vacuum residues are considered difficult to process in the ebullated bed hydrocracking because of their increased tendency to form sediments. Their achievable conversion rate reported in the literature is 60%. Intercriteria analysis was used to assess data from a commercial vacuum residue hydrocracker during processing blends from three vacuum residues: Urals, Siberian Light, and Basra Heavy. The analysis revealed that the main contributors to conversion enhancement is hydrodemetallization (HDM) and the first reactor ΔT augmentation. The increase of HDM from 40 to 98% and the first reactor ΔT (ΔT(R1)) from 49 to 91 °C were associated with a vacuum residue conversion enhancement of 62.0 to 82.7 wt.%. The developed nonlinear regression prediction of conversion from HDM and ΔT(R1) suggests a bigger influence of ΔT(R1) enhancement on conversion augmentation than the HDM increase. The intercriteria analysis evaluation revealed that the higher first reactor ΔT suppresses the sediment formation rate to a greater extent than the higher HDM. During processing Basrah Heavy vacuum residue, a reduction in hydrodeasphaltization (HDAs) from 73.6 to 55.2% and HDM from 88 to 81% was observed. It was confirmed that HDM and HDAs are interrelated. It was found that the attainment of conversion of 80 wt.% and higher during processing Urals and Siberian Light vacuum residues is possible when the HDM is about 90% and LHSV ≤ 0.19 h−1.

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Commercial Ebullated Bed Vacuum Residue Hydrocracking Performance Improvement during Processing Difficult Feeds

F. Smarandache discussed the following particular cases of the well-know characteristic functions.

On Four Pr.ime and Coprime Functions

The human body is an open stationary system and has different adaptation mechanisms in changing conditions of the external and internal environment. In this aspect there is no exception to the circulatory system, respectively the arteries of the lower limbs. The narrowed or blocked arteries reduce blood flow to the lower limbs. In this case, the blood flows through the smaller blood vessels and the anastomoses (connections) between them above and below the thrombosis of the main vessel. In this research, an approach using an intelligent tool for analyzing data through fuzzy logic is presented. The intuitionistic fuzzy assessments for the blood supply evaluation of the lower limbs with peripheral artery disease are suggested. The permeability of the artery is within a certain range, limited by the minimum and maximum thresholds. The intuitionistic fuzzy evaluations determine the degree of permeability of the artery. They are formed on the basis of a set of intuitionistic fuzzy estimations \( \left\langle {\mu ,\nu } \right\rangle \) of real numbers from the set [0, 1] × [0, 1].

Krassimir T. Atanassov

Papers

How to Assess Different Algorithms Using Intuitionistic Fuzzy Logic

Model of Big Data Map/Reduce Processing

Commercial Ebullated Bed Vacuum Residue Hydrocracking Performance Improvement during Processing Difficult Feeds

On Four Pr.ime and Coprime Functions

Intuitionistic Fuzzy Assessments of the Abdominal Aorta and Its Branches