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Abstract:
At the beginning of the article the authors describe a new trend in the artificial intelligence, associated with fuzzy sets and the accompanying derivative solutions which may include L-R numbers by Dubois and Prade. On their basis the redefined theory has started a new trend in the form of ordered fuzzy numbers (OFN). Main features of ordered fuzzy numbers further in this article. Due to the nature of this article, which is related to the proposed defuzzyfication method, the authors mentions a fuzzy controller model and in particular the defuzzyfication process. Criteria used for conventional solutions of fuzzy numbers and ordered fuzzy numbers were also presented. In the further part of the article the defuzzyfication method called Triangular Expanding was presented. The author compared it to the Geometrical Mean method introduced earlier, which inspired his solution. Results of comparison with other methods such as FOM, LOM, COG were presented in the paper as well. The summary including conclusions and directions of further research were provided at the end.

Keywords:
Fuzzy logic, Ordered fuzzy numbers

Abstract:
The article concerns the issue of seeking patterns in trends. In the study a method to detect patterns in trends recorded in a linguistic has been proposed. Linguistic variables take their values as a result of the calculations in Ordered Fuzzy Numbers notation. Thus, in a first stage fuzzyfication of the source data occurs. Transposition of the parameters was applied to daily quotations (min, max, the opening value, the closing value and the direction of change) which were interpreted as a single OFN number. This is the first usage of this notation to describe the stock index which allows to describe five different parameters in a single number. Then the data are converted into linguistic form. The level of trend sequence similarity is determined by following set parameters: the Frame size of the pattern, expressed as a percentage similarity of trend sequence to a frame set at the outset, threshold indicating how many trend fragments is consistent with the frame and the frequency of the pattern occurrences. Patterns detected in this way in the nature of things are characterized by various support, and coefficients of similarity in both the whole pattern and the individual elements. For the purposes of this study, we developed a dedicated computer program performing patterns search. As research material the main index of the Warsaw Stock Exchange, i.e. WIG20 from the years 2008-2016 was used as a data set. This preliminary study is beginning to develop Rule base forecasting methods, and in this direction further experiments will be carried out.

Keywords:
WIG20, OFN, Trend, Rule-based forecasting, Pattern recognition, Financial engineering

Abstract:
One of the many reasons why a human searches for new solutions is inspiration for innovation perceived in natural processes and phenomena. The authors of the paper present new algorithms for the defuzzyfication block which is the final process of the fuzzy controller (fuzzy control systems), for which a defuzzyfied value controls a given object. The presented new methods are: Golden Ratio (GR) and Mandala Factor (MF). The first of them uses the ancient Golden Ratio rule which is known, among others, from the Fibonacci sequence. The second proposal is based on the interpretation of drawing technique used in Asia, consisting in arranging pictures of color sand grains. In Tibetan Buddhism this technique is known as Mandala, a symbol of perfection and harmony. The interpretation of the perfection symbol and the Golden Ratio method in this paper has been referenced to other methods used in the defuzzyfication process, including weight averange method, centroid and mean of maxima. The scene for solutions presented here is provided by the ordered fuzzy numbers (OFN) theory which allows to use both the trend of a given phenomena, as well as more precisely wield mathematical methods. A special property of the proposed methods is their sensitivity to the OFN number order. This means that MF and GR operators applied to the numbers of the same shape but of opposite orders would result in different defuzzyfication values. The paragraph relating to discussion on the research includes a comparison of the existing defuzzyfication operators as regards the sensitivity to order.

Keywords:
Fuzzy logic, Ordered fuzzy numbers, OFN, Defuzzification

Abstract:
Energy efficiency is often a key optimization problem. Many control systems use fuzzy logic and as a result applying compare operators to fuzzy numbers. The article deals with the issue of comparing fuzzy numbers. The similarity relation is most probably the most frequently used and the most difficult to precisely determine the convergence measure. Analysis of the similarity of two objects is a basic assessment tool and constitutes the basis for reasoning by analogy. It also directly affects the energy effectiveness of the universe that it controls. This article presents the methods for determining the similarity used in fuzzy logic. Many of these methods were dedicated only to fuzzy triangular or trapezoidal numbers (Dobrosielski et al. 2017, Chi-Tsuen Yeh 2017, Abbasbandy and Hajjar 2009). This was a computational inconvenience and posed a question about the axiological basis of this type of comparison. The authors proposed two new approaches for comparing fuzzy numbers using one of the known extensions of fuzzy numbers (Kacprzyk and Wilbik 2009, 2005). This allowed to simplify the operation and eliminate the duality (Zadrożny, 2004).

Keywords:
fuzzy logic, comparison, OFN, GR, ML, TR

Abstract:
This article presents a proposal for a new method of defuzzification a fuzzy controller, which is based on the concept of the golden ratio, derived from the Fibonacci series [1]. The origin of the method was the observation of numerous instances of the golden ratio in such diverse fields as biology, architecture, medicine, and painting. A particular area of its occurrence is genetics, where we find the golden ratio in the very structure of the DNA molecule [2] (deoxyribonucleic acid molecules are 21 angstroms wide and 34 angstroms long for each full length of one double helix cycle). This fact makes the ratio in the Fibonacci series in some sense a universal design principle used by man and nature alike. In keeping with the requirements, the authors of the present study first explain the essential concepts of fuzzy logic, including in particular the notions of a fuzzy controller and a method of defuzzification. Then, they postulate the use of the golden ratio in the process of defuzzification and call the idea the Golden Ratio (GR) Method. In the subsequent part of the article, the proposed GR-based instrument is compared with the classical methods of defuzzification, including COG, FOM, and LOM. In the final part, the authors carry out numerous calculations and formulate conclusions which serve to classify the proposed method. At the end they present directions of further research.

Keywords:
Fuzzy logic, Fuzy sets, Fuzzy control system, Deffuzification, Fibonacci series