Comparing Fuzzy Numbers Using Defuzzificators on OFN Shapes

Abstract

This chapter concerns an issue of comparing fuzzy numbers. The relationship of similarity is probably the most widely used and most difficult to determine the measure of compliance precisely. Analysis of the similarity between two objects is an essential tool in biology, taxonomy, and psychology, and is the basis for reasoning by analogy. This chapter describes methods for determining the similarity used in fuzzy logic. Many of them were dedicated only to triangular or trapezoidal fuzzy numbers. This was a computing inconvenience and raised the question about the axiological basis for such comparisons. The authors have proposed two new approaches to comparing fuzzy numbers using one of the known fuzzy number extensions that are Ordered Fuzzy Numbers (OFNs). This has allowed us to simplify operations and eliminate said dualism. Two order-sensitive defuzzification methods are presented in the chapter. For OFN numbers with positive order (compliant with the direction of the OX axis increase) the results of defuzzifications are results for numbers of different notations, for example, L-R, whereas for numbers with negative orders, the defuzzification result changes. An important part of the chapter is a catalogue of the shapes of numbers in OFN notation. This is probably the first summary of basic shapes of those numbers with the results of defuzzifications using several methods.

Comparative Study: Defuzzification Functions and Their Effect on the Performance of the OFNbee Optimization Algorithm

“…Another operator used in the new method is the Mandala Factor operator [21] [22] proposed by J. Czerniak. Mandala is a painting composed of colorful grains of sand, arranged by Buddhist monks.…”

“…The crisp value of the number is obtained in the place where the last of the columns poured ended (Figure 8). Mathematically, the notation is as follows 9 [21]:…”

A Cellular Automata-Based Simulation Tool for Real Fire Accident Prevention

“…Thus, there are two possible sets of movements for this cell in question, depending on the determinant [27,49,50]. Since each of the sets is a fourelement, then the notation of fuzzy numbers called ordered fuzzy numbers introduced by Kosiń ski is suitable for its description [51,52]. After the death of the creator OFNs in some works are also called Kosiń ski's Fuzzy Numbers [23,47,48,53,54].…”