Ordered Fuzzy Numbers: Sources and Intuitions

Abstract

Most emerging methodologies, before they become well settled, stem from careful analysis of previous solutions. In that respect, this chapter refers to the roots of the Ordered Fuzzy Number (OFN) model. First, we outline some drawbacks of the most popular fuzzy number representations, which inspired us to search for a new approach. Then we discuss the idea of looking at fuzzy numbers from an alternative viewpoint. This leads towards formulation of the OFN model comprising three conceptual steps: (1) representing membership functions of fuzzy numbers as the pairs of increasing/decreasing components; (2) for each of two components treated as a locally defined function, inverting the meanings of its domain and its set of values; and finally (3) treating the obtained pairs of components as the ordered pairs. By introducing arithmetic operations on such ordered pairs, we obtain the framework, which is in many cases equivalent to the previous approaches but it also enables the representation of new information aspects.

Teaching machines to optimizing machining parameters: using independent fuzzy logic controller and image data

“…The amount of wear has a trapezoidal membership function [ 70 ]; this is summarized in Eq. 5 , where x a is the measured value of wear on the cutting tool in micrometers and p,q,r,s are the boundary values of the membership function.…”

“…R is the label used from family . The Gaussian membership function [ 70 ] for the response variable ( ) is given in Eq. 7 , where c is the mean of the distribution, s is the standard deviation, and y is the output value.…”

“…This method is suitable for the application at hand as it can work with the conjunctive interpretation of fuzzy rules in the canonical form given in Eq. 8 [ 70 ]. The conjunctive “AND” is interpreted as the minimum ( ) [ 70 ].…”

“…8 [ 70 ]. The conjunctive “AND” is interpreted as the minimum ( ) [ 70 ]. The inference results from each rule are finally added using maximum ( ) operation [ 70 ].…”

Evolution strategies based coefficient of TSK fuzzy forecasting engine

“…The fuzzy logic variable has a set of values characterized by the linguistic expression [26]. This study divide input variable (Table 2) into three fuzzy sets, namely DESCENDING, CONSTANT, and ASCENDING [23] [27]. Expert opinions have adjusted the determination of fuzzy sets.…”