Theory and Applications of Ordered Fuzzy Numbers

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 t-norms, s-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.

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 \(\mathfrak {B}\) 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.

The chapter presents a novel way of describing changes in the stock index and the identification of potential trends. The authors already used a similar approach to describe the stock exchange index [16]; this chapter is a continuation and another application of work on this issue. The method for detecting patterns in a trend by means of linguistic variables is described. The use of computational operations on numbers in the Ordered Fuzzy Number (OFN) notation [40‐42] enables us to set the values of linguistic variables and thus conduct fuzzification of the input. By using one OFN number it is possible to store five parameters of index quotations (open, high, low, and close values as well as a change direction). The OFN numbers are conveyed into a linguistic form. In order to find trend sequence similarity the following applies: sequence identity with the input frame expressed as a percentage, frame size, the level of threshold conformity with the frame (threshold), and how often the pattern is present (frequency). A dedicated computer program to detect patterns is implemented. The program used data from the index Nasdaq Composite from the years 2006-2016. The results represent a further step to develop effective methods of rule-based forecasting.

This chapter presents a hybrid method of swarm intelligence current. Intelligence represented by ant colonies has been enriched with fuzzy logic arithmetics. In this case Kosiński’s Ordered Fuzzy Numbers were specifically used. Apart from a fuzzy decision model of a single ant used earlier by other researchers, the author used the order as a trend support. By associating the direction of a number in Ordered Fuzzy Numbers (OFNs) with the trend observed in the ant colony it is possible to provide a unique description of a fuzzy observation of a colony behavior. The experiments were carried out in the area of searching for the optimal connecting route in the field. The experiment covered 10 complex issues of searching for the optimal route. All are benchmarks from the TSPlib repository which are well known among researchers. They represent the actual problems of route selection such as transport connections depending on geographic conditions and optimizing the machining process or the layout of the power networks. The complexity level of optimal solutions for problems to be solved amounted from several hundred to several thousand connections. Each of them was solved using six swarm intelligence methods and five well-known classical methods dedicated to the traveling salesman problem (TSP). The results were presented in the form of tables and graphs, and some of the routes were shown in graphical form. Final conclusions of the experiment indicate the superiority of methods based on ant colony optimization as regards closeness to optimal solutions. The results achieved by the OFNAnt method are generally better (in 92% of cases) than those achieved by classic methods and are in the forefront of solutions from the swarm intelligence group.

The chapter includes a hybrid concept combining bee colony optimization with the application of Ordered Fuzzy Numbers. This is another research, after the OFNAnt method, prepared in AIRlab - Artificial Intelligence and Robotics Laboratory at Kazimierz Wielki University in Bydgoszcz, in which authors enriched metaheuristics by implementing the arithmetics of Ordered Fuzzy Numbers (OFNs). Applied fuzzy observation enabled very faithful modeling of the navigation mechanism used by bees when orienting with reference to the position of the sun. Experiments aimed at verification of the developed concept have been carried out on a set of several commonly known benchmarks. The preliminary results of experiments allow us to nurture grounded hope that further modifications of the metaheuristics using OFN arithmetics shall enable smooth control of the optimization criteria of the tested phenomena.

DDoS attacks are able to block Web servers. Such attacks could be started from anywhere in the network. This chapter presents the possibility of using Ordered Fuzzy Numbers (OFNs) for observation of a DDoS attack. The proposed algorithm could be implemented on routers and predict the moment of the attack. Such prediction gives a possibility for the network administrators to protect server resources. In the chapter the author presents the real test results made on a prepared IP network. The presented results prove that OFNs have a huge potential for usage in observation of DDoS attacks.

This chapter presents the potential use of fuzzy observance implementation for detecting transmission problems that could appear in the near future. Using quick detection, appropriate action could be taken and the security and reliability of data transfer could be maintained at a high level. As a result the authors present a proposed solution for dividing a data stream between different data links and predicting transmission problems.

This chapter presents a trend phenomenon and application of the fuzzy controller for Ordered Fuzzy Numbers (OFNs). The authors propose to use a trend in a combustion process for a simplified model of a solid fuel fired furnace. Better control over the process translates into reduced CO \(_2\) emission as well as optimal use of the furnace. When carrying out the fuzzy observation of the efficiency of the furnace, the authors apply the OFN notation by connecting the trend of furnace temperature changes with the order appropriate for this notation. Thanks to this approach it is possible to enhance information without the additional need to multiply the transmitted data. It is particularly effective in the multidimensional fuzzy observation when monitoring not only the condition of the temperature in the furnace but also the ambient temperature and the temperatures in several rooms of the heated building. The chapter is a continuation of a series of papers published by the authors on multidimensional fuzzy observation using OFN notation. A controller in the conventional fuzzy logic approach is also presented in the chapter. The controller was built using jFuzzyLogic software. The fact that there are more and more OFN applications seems to be a good predictor of the development of this generalization, an example of which is the problem analyzed in this chapter.

Locomotion in post-stroke patients may be severely compromised. Assessment and treatment of gait disorders after stroke are crucial. Scientists and clinicians still look for more effective diagnostic and therapeutic tools. The aim of the study was to assess a new fuzzy-based tool for measurement of observed gait parameters (velocity, cadence, and stride length, and their normalized values), both in healthy people and post-stroke patients.

A modeling approach may significantly help to explore the problem of weak understanding of the physiological and pathological central nervous system function in the most noninvasive and comprehensive way. The aim of this chapter is to assess and discuss the extent to which possible opportunities concerning computational brain models based on fuzzy logic techniques may be exploited.