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The subject of this chapter is fuzzy sets and the basic issues related to them. The first section discusses concepts of sets: classic and fuzzy, and presents various ways of describing fuzzy sets. The second section is dedicated to -norms, -norms, and other terms associated with fuzzy sets. Subsequent sections describe the extension principle, fuzzy relations and their compositions, cylindrical extension and projection of a fuzzy set. The sixth section discusses fuzzy numbers and basic arithmetic operations on them. Finally, the last section summarizes the chapter.

The following chapter describes the basic concepts of fuzzy systems and approximate reasoning. The study focuses mainly on fuzzy models based on Zadeh’s compositional rule of inference. The presentation begins with an introduction of fundamental ideas of fuzzy conditional (if-then) rules. A collection of fuzzy if-then rules formulates the so-called knowledge base, which formally represents the knowledge to be processed during approximate reasoning. The subsequent sections present formal definitions related to the compositional rule of inference and approximate reasoning using a knowledge base. Theoretical considerations are supplemented with practical examples of fuzzy systems as the foundation of many modern structures. The description includes fuzzy systems proposed by Mamdani and Assilan, Takagi, Sugeno and Kang, and Tsukamoto.

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.

We outline basic notions and assumptions related to the Ordered Fuzzy Number (OFN) model. Definitions of mathematical operations, several interpretations of their results, as well as additional OFN parameters are presented. Some of them, such as inclination or order, are specific to OFNs, whereas others are equivalent to those present in the well-known convex fuzzy number model. An important aspect of this part is also a discussion of algebraic properties of the OFN model.

It was already mentioned in previous sections that the Ordered Fuzzy Number (OFN) model can represent a kind of tendency or direction. However, for a real practical use of this feature the tools for processing it are also needed. Of course some kind of quantitative processing is provided by the definitions of calculations, but there is much more potential for this feature apart from arithmetic operations. This part presents the idea of a property of processing data called . The main focus here is placed on the proposition of a direction determinant parameter that can be understood as a kind of measure of a direction. This determinant is a basis for the definition of such elements as the compatibility between two OFNs and also for an inference operator for a rule where the OFNs were used. The propositions of such operations are the important part of these sections of the book.

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.

We discuss construction of fuzzy implication and also correlation between negation and implication operators defined on fuzzy values. Two structures for fuzzy implications are studied: the lattice of Step-Ordered Fuzzy Numbers (SOFNs) and the Boolean algebra of membership degrees for metasets. Even though these two approaches stem from completely different areas it turned out that they lead to similar applications and results. Both of them emerged from research conducted by W. Kosiński and can be applied not only in the most popular application field which is approximate reasoning but also for designing decision-support systems, enriching methods and techniques of opinion mining, or modeling fuzzy beliefs in multiagent systems.

The aim of this chapter is to propose a new approach to incorporating uncertainty into capital budgeting. The chapter presents methods that can be used by an investor when the decision maker wants to be able to make an investment decision where there are alternative investment projects. This kind of problem is undertaken under the conditions of uncertainty and risk using Ordered Fuzzy Numbers (OFN). The starting point is the concept of Ordered Fuzzy Numbers. The chapter illustrates the implementation of the proposed approach with an example where two alternative investment projects are analyzed. The authors present the capital budgeting problem using a numerical example. The described methods dedicated to investment project selection lay the foundations for a fuzzy decision-making system. We utilize computer software such as MATLAB to demonstrate how the proposed methods can be applied to assessing the profitability of alternative investment projects.

The chapter presents the application of Ordered Fuzzy Numbers (OFNs) to the economic model. These numbers are used for input-output analysis (the Leontief model), which is a basic method of quantitative economics that presents macroeconomic activity as a system of interrelated goods and services. OFNs allow us not only to apply mathematical modeling of imprecise or ambiguous data but also simultaneously portray more information than could be presented by real numbers. It is shown based on the Leontief model, where at the same time the current level, the forecast level, and the level of change of the final demand or the production level can be determined. The example shows that use of OFNs in economic modeling can simplify and deepen the economic analyses.

The purpose of this chapter is to present how Ordered Fuzzy Numbers (OFNs) can be used with financial high-frequency time series. Considering this approach the financial data are modeled using OFNs called further ordered fuzzy candlesticks. Their use allows modeling uncertainty associated with financial data and maintaining more information about price movement at an assumed time interval than compared to commonly used price charts (e.g., Japanese Candlestick chart). Furthermore, in a simple way, it is possible to include the information about volume and the bid-ask spread. Thanks to the well-defined arithmetic of OFNs, one can be used in technical analysis or to construct models of fuzzy time series in the form of classical equations. Examples of an ordered fuzzy moving average indicator and ordered fuzzy autoregressive process are presented.