In the paper some results of investigations of two intelligent information systems: a feedforward neural network and an adaptive fuzzy expert system, are presented. The systems can be used for example in approximation and control problems or in diagnostics. The adaptive fuzzy expert system is constructed as a hybrid in which a fuzzy inference system is combined with a neural network. In the learning process for given set of training points an optimal value of the so-called generalized weight vector is searched. The Lapunov theory is used to examine the non-sensitivity of the optimal value of a generalized weight vector to initial conditions and training data. Some necessary and sufficient conditions are formulated in terms of the Hessian matrix of the error function.

An adaptive information system is constructed in order to approximate a set of multidimensional data. To get better approximation properties a pre-processing stage of data is proposed in which the set of points, forming the multidimensional data base and called a training set TRE, undergoes a clustering analysis. In the analysis two independent clustering algorithms are used; on each cluster a feed-forward neural network is trained and a membership function of a fuzzy set is constructed. The constructed system contains a module of two-conditional fuzzy rules consequent parts of which are of the functional type. Each rule is designed on a pair of clusters.

Markovian operators, non-negative linear operators and its subgroups play a significant role for the description of phenomena observed in the nature. Research on asymptotic stability is one of the main issues in this respect. A. Lasota and J. A. Yorke proved in 1982 that the necessary and sufficient condition of the asymptotic stability of a Markovian operator is the existence of a non-trivial lower-bound function. In the present paper it is shown how the method of lower-bound function can be applied to the investigation of genetic algorithms. Genetic algorithms considered used for solving of non-smooth optimization problems are compositions of two random operators: selection and mutation. The compositions are Markovian matrices.

Dnia 15 maja 2009 roku zmarla przedwcześnie w Warszawie, po przegranej walce z nieuleczalną chorobą, dr Izabella Czochralska – Sekretarz Komitetu Redakcyjnego naszego pisma, mocno zaangazowana w dzialalnośc na rzecz Polskiego Towarzystwa Matematycznego. Byla wieloletnim czlonkiem Zarządu Oddzialu Warszawskiego PTM i Komisji Rewizyjnej, a takze delegatem na liczne Zjazdy Towarzystwa. W roku 2000 byla glowną organizatorką Zjazdu PTM w Warszawie. Od 2007 r. pelnila funkcje Sekretarza Komitetu Redakcyjnego Matematyki Stosowanej. Wszyscy, ktorzy Ją znali, stracili bliskiego przyjaciela, czlowieka o niezwyklej osobowości i umyśle.

SYMBAD is the structure of Artificial Neural Network (ANN) developed on the basis of experimentally gained data from the military aircraft testing. These tests were done in Air Force Institute of Technology for many years and for the different types of aircraft. Potential usage of this programme for the managing of aircraft prototypes testing has been formulated as well. This case is presented in this article. The presented program will be improved based on the knowledge of researchers.The SYMBAD program has been developed under the Borland C++ Builder programming environment. It uses an innovative approach to any issues of the aeronautical testing.The SYMBAD's structure can be easily adjusted depending on the type of test equipment, eg. for unmanned air systems, helicopters and combat aircrafts. Moreover, programme modification can include type of research being conducted: for a new-designed and - developed aircraft as well as for an upgraded one, for those in production, and for the aircraft equipment/systems built in the aircraft.Input programme data take into consideration requirements listed in technical specifications, rules and regulations of aircraft building, standards, and Customer's requirements.Paper presented programme uses neutral network to optimize the flight test programme as well, according to the Customer's requirements, at a proposed cost, with safety of both the staff and the aircraft provided.

The paper concerns two methods of inverse problem solution for the equation of heat conduction, what allows to determine the thermal diffusivity of the materials using pulsed infrared thermography.Both methods are related to finding the time dependence of the temperature of an infinite plate surface, when opposite surface of the plate was heated by a short heat pulse.This dependence is compared with the time evolution of the temperature of the rear plate surface measured by the means of an infrared (IR) camera.Such comparison allows to extract, from experimental data, the information about thermal diffusivity of the tested material.

The aim of the article is to investigate defuzzification functionals in the theory of Ordered Fuzzy Numbers (OFN). The model of OFN was introduced in 2002 to overcome drawbacks of classical (convex) fuzzy numbers. Each OFN is equipped with an additional feature – the orientation. New forms of defuzzification functionals are proposed which are sensitive to the orientation change.

The aim of the article is to propose some tools of inventory management. One is based on the so-called fixed order quantity model which takes into account several elements of inventory cost, such as ordering cost, transportation and storing costs, frozen capital cost, as well as extra discounts. The tool deals with fuzzy concepts represented by Ordered Fuzzy Numbers. The second tool takes into account the dynamics and works on the base of replenishment system. This tool can be regarded as a kind of controller.

The fixed order quantity model of inventory management system is used in the deterministic part. Several elements of inventory cost, such as ordering cost, transportation and storing costs, frozen capital cost, as well as extra rebates, are taken into account in the model. Then the fuzzy optimization problem for the total cost function is formulated within the space of Ordered Fuzzy Numbers when all variables of the model are fuzzy. After the choice of a particular defuzzification functional an appropriate theorem is formulated which gives the solution of the problem.

The purpose of this article is to suggest tools of inventory management which would determine economically optimal order quantities.One of them is based on the so-called fixed order quantity model which takes into account several elements of inventory cost, such as ordering cost, transportation and storing cost, frozen capital cost, as well as extra discounts.The tool is based on fuzzy concepts represented by Ordered Fuzzy Numbers.The second tool takes into account the dynamics and works on the basis of replenishment system.This tool can be treated as a kind of controller.Examples of using this tools are presented.

Ordered fuzzy numbers (OFN) were introduced by Kosiński, Prokopowicz and Ślȩzak in 2002. The definition of OFN uses the extension of the parametric representation of convex fuzzy numbers. So far, they were applied to deal with optimization problems when data are fuzzy. In 2011 Kacprzak and Kosiński observed that a subspace of OFN called step ordered fuzzy numbers (SOFN) may be equipped with a lattice structure. In consequence, a Boolean operations like conjunction, disjunction and, what is more important, diverse types of implications can be defined on SOFN. In this paper we show how OFN can be applied in multi-agent systems for modelling agents’ beliefs about fuzzy expressions. Then we present preliminary version of a logic based on SOFN and study how this logic can be helpful in evaluating features of multi-agent systems concerning agents’ fuzzy beliefs.

Applicability of the fuzzy approach to problems in economics and finance is presented. One problem arrives when a comparison between mutually exclusive investment alternatives is performed using the internal rate of return (IRR). Another problem originating from administrating accounting, deals with the determination of an economic order quantity (EOQ) with imprecise and vague data. Financial stock time series are presented together with the so-called Japanese candlesticks. The model of Ordered Fuzzy Numbers, developed by the first author (W.K.) and his two co-workers is used. The present approach generalizes that developed within the model of Convex Fuzzy Numbers and stays outside the probabilistic one.

Defuzzification functionals, which play the main role when dealing with fuzzy controllers and fuzzy inference systems, for convex as well for ordered fuzzy numbers, are discussed. Three characteristic conditions for them are formulated. It is shown that most of the known defuzzification functionals satisfy them. Motivations for introducing the extended class of convex fuzzy numbers are presented, together with operations on them.

Financial stock time series are presented together with the so-called Japanese candlesticks. Model of Ordered Fuzzy Numbers is shortly presented and its use in presentation of Japanese candlesticks. Then the ogive, the graphical representation of the cumulative relative frequency of transactions is introduced, as the next characteristic of price time series. Linear operations on ogive curves are defined. It is shown that ogive is reflecting some properties of stock time series additional to the Japanese candlestick.

Aim of the paper is to propose a new tool for a decision supporting system concerning the financial project evaluation. It is based on the determination of the internal rate of return (IRR) of a investment project in which all expenditure and anticipated incomes are vague, and described by Ordered Fuzzy Numbers (OFNs). It means that the probabilistic approach is neglected in this paper and the use of the well developed arithmetics of OFNs is made to find a positive fuzzy root of a fuzzy polynomial representing the fuzzy net present value of the project. Since in the space of OFNs a partial order relation is defined together with a number of defuzzification functionals, the authors can construct a decision support system for investors helping them in acceptance procedure of most profitable investment projects.

Metaset is a new concept of set with partial membership relation. It is directed towards computer implementations and applications. The degrees of membership for metasets are expressed as binary sequences and they may be evaluated as real numbers too. The forcing mechanism discussed in this paper is used to assign certainty values to sentences involving metasets. It turns out, that for a sentence involving finite first order metasets only its certainty value complements the certainty value of its negation. This is not true in general: sentences expressing properties of metasets may have positive uncertainty value. We supply an example of a sentence which is totally uncertain.

Diversity of opinion is an empirical fact often appearing in social networks. In the paper known statistical methods of evaluation is substituted by fuzzy concepts, namely Step Ordered Fuzzy Numbers (SOFN). SOFN are extentions of Ordered Fuzzy Numbers (OFN) introduced by Kosiński, Prokopowicz and Ślęzak in 2002. In 2011 Kacprzak and Kosiński observed that SOFN may be equipped with a lattice structure. In consequence, Boolean operations like conjunction, disjunction and, what is more important, diverse types of implications can be defined on SOFN. In this paper we show how SOFN can be applied for modelling diversity of beliefs even in fuzzy expressions.