Кафедра "Комп'ютерна математика і аналіз даних"

Постійне посилання колекціїhttps://repository.kpi.kharkov.ua/handle/KhPI-Press/7570

Офіційний сайт кафедри http://web.kpi.kharkov.ua/kmmm

Кафедра "Комп'ютерна математика і аналіз даних" заснована в 2002 році.

Кафедра входить до складу Навчально-наукового інституту комп'ютерних наук та інформаційних технологій Національного технічного університету "Харківський політехнічний інститут", забезпечує підготовку бакалаврів і магістрів за проектно-орієнтованою освітньою програмою за напрямом науки про дані "DataScience".

У складі науково-педагогічного колективу кафедри працюють: 3 доктора наук: 1 – технічних, 1 – фізико-математичних, 1 – педагогічних; 15 кандидатів наук: 10 – технічних, 4 – фізико-математичних, 1 – педагогічних; 3 співробітників мають звання професора, 9 – доцента.

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Analysis and development of compromise solutions in multicriteria transport tasks
(Technology center PC, 2017) Raskin, Lev; Sira, Oksana; Parfeniuk, Yurii
The object of research is the multicriteria transport problem of linear programming. Simultaneous consideration of several criteria is a problematic problem, since the optimal solutions for different criteria do not coincide. The possible solution of the problem is investigated – finding a way to obtain a compromise solution. Based on the results of the analysis of known methods for solving multicriteria problems (Pareto-set formation, scalarization of the vector criterion, concessions method), the last is justified. To implement the method, an iterative procedure is suggested, in which the initial plan is optimal according to the main criterion. At subsequent iterations, an assignment is made to the main criterion in order to improve the values of the additional criteria. The solution of the problem is continued until a compromise solution is obtained, ensuring the best value for the main criterion, provided that the values for the remaining criteria are no worse than those given. Important advantages of the proposed method: the simplicity of the computational procedure, the grounded technology of forming a new solution at each iteration, realizing the concept of assignment, quality control of the solution obtained at each step. The application of the proposed method opens the prospect of its generalization to the case when the initial data for the solution of the problem contain uncertainty.
Analysis of efficiency of the bioinspired method for decoding algebraic convolutional codes
(Технологический центр, 2019) Panchenko, Sergii; Prykhodko, Sergii; Kozelkov, Sergii; Shtompel, Mykola; Kosenko, Viktor; Shefer, Oleksandr; Dunaievska, О. I.
It has been shown that convolutional codes are widely used, along with various decoding methods, to improve the reliability of information transmission in wireless telecommunication systems. The general principles of synthesis and the parameters and algebraic non-systematic convolutional codes with arbitrary coding rate and maximum achievable code distance have been shown. The basic stages of the bioinspired method for decoding algebraic convolutional codes using a random shift mechanism have been presented. It has been shown that the essence of the presented decoding method implies applying the procedure of differential evolution with the heuristically determined parameters. In addition, this method uses information about the reliability of the adopted symbols to find the most reliable basis for the generalized generator matrix. The mechanism of random shift for the modification of the accepted sequence is additionally applied for the bioinspired search based on various most reliable bases of a generalized generator matrix. The research results established that the bioinspired method for decoding algebraic convolutional codes ensures greater efficiency compared with the algebraic decoding method in the communication channel with additive white Gaussian noise. Depending on the parameters of the algebraic convolutional code and the necessary error coefficient, the energy gain from encoding ranges from 1.6 dB to 3 dB. It was shown that the presented bioinspired decoding method can be used for convolutional codes with a large code constraint length. In doing so, the presented method for decoding algebraic convolutional codes is less efficient than the Viterbi decoding method and turbo codes at a sufficient number of decoding iterations.
Analysis of operating modes of single-phase current source rectifier with rectangular-stepped pulse-width modulation
(Технологический центр, 2018) Krasnov, Oleksii; Liubarskyi, Borys; Bozhko, Vladimir; Petrenko, Оleksandr; Dubinina, Oksana; Nuriiev, Ramil
The operating modes of the single-phase active current-source rectifier in the case of rectangular-stepped pulse-width modulation and load in the form of DC traction motor are investigated. The single-phase bridge rectifier circuit with a discharge diode is considered. The mathematical model of the rectifier is developed and the main ratios for pulse-width modulation with rectangular-stepped modulating signal are determined. On the computer model, electromagnetic processes at three modulation frequencies (900, 1,200, 1,800 Hz) are studied. The features of the effect of the modulation depth and frequency on the rectifier power factor and the total harmonic voltage and current distortion in the mains supply are determined. The DC motor for today remains the main type of the traction motor of the 25 kV, 50 Hz AC mainline electric locomotives of alternating current in Ukraine and in some other countries. To power such motors, diode or thyristor rectifiers are used. At the same time, it is known that converters on fully controlled semiconductor devices provide higher power efficiency. The studies allow determining what values of modulation frequency and depth provide a high power factor (more than 0.9) and minimum total harmonic voltage and current distortion distortions in the mains supply. This allows finding rational approaches to the selection of power circuits and control algorithms for active converters in the traction electric drive of electric locomotives. The efficiency of increasing the power factor and reducing the total harmonic voltage and current distortion can be achieved, first of all, by reducing the power consumption for traction of trains. According to the set of selected comparison criteria, the active current-source rectifier with a modulation frequency of 1,200 Hz is the most suitable for implementation in the traction electric drive of the electric locomotive. Provision of high power characteristics in a wide range of traction loads can be achieved in the multi-zone circuit of such a converter.
Analysis of semi-Markov systems with fuzzy initial data
(Scientific Route OÜ, Estonia, 2022) Raskin, Lev; Sira, Oksana; Sukhomlyn, Larysa; Korsun, Roman
In real operating conditions of complex systems, random changes in their possible states occur in the course of their operation. The traditional approach to describing such systems uses Markov models. However, the real non-deterministic mechanism that con trols the duration of the system’s stay in each of its possible states predetermines the insufficient adequacy of the models obtained in this case. This circumstance makes it expedient to consider models that are more general than Markov ones. In addition, when choosing such models, one should take into account the fundamental often manifested feature of the statistical material actually used in the processing of an array of observations, their small sample. All this, taken together, makes it relevant to study the possibility of developing less demanding, tolerant models of the behavior of complex systems. A method for the analysis of systems described under conditions of initial data uncertainty by semi-Markov models is proposed. The main approaches to the description of this uncertainty are considered: probabilistic, fuzzy, and bi-fuzzy. A procedure has been developed for determining the membership functions of fuzzy numbers based on the results of real data processing. Next, the following tasks are solved sequentially. First, the vector of stationary state probabilities of the Markov chain embedded in the semi-Markov process is found. Then, a set of expected values for the duration of the system’s stay in each state before leaving it is determined, after which the required probability distribution of the system states is calculated. The proposed method has been developed to solve the problem in the case when the parameters of the membership functions of fuzzy initial data cannot be clearly estimated under conditions of a small sample.
Comparator identification in the conditions of bifuzzy initial data
(Scientific Route, Estonia, 2021) Raskin, Lev; Sira, Oksana; Katkova, Tetiana
When solving a large number of problems in the study of complex systems, it becomes necessary to establish a relationship between a variable that sets the level of efficiency of the system’s functioning and a set of other variables that determine the state of the system or the conditions of its operation. To solve this problem, the methods of regression analysis are traditionally used, the application of which in many real situations turns out to be impossible due to the lack of the possibility of direct measurement of the explained variable. However, if the totality of the results of the experiments performed can be ranked, for example, in descending order, thus forming a system of inequalities, the problem can be presented in such a way as to determine the coefficients of the regression equation in accordance with the following requirement. It is necessary that the results of calculating the explained variable using the resulting regression equation satisfy the formed system of inequalities. This task is called the comparator identification task. The paper proposes a method for solving the problem of comparator identification in conditions of fuzzy initial data. A mathematical model is introduced to describe the membership functions of fuzzy parameters of the problem based on functions (L–R)-type. The problem is reduced to a system of linear algebraic equations with fuzzy variables. The analytical relationships required for the formation of a quality criterion for solving the problem of comparator identification in conditions of fuzzy initial data are obtained. As a result, a criterion for the effectiveness of the solution is proposed, based on the calculation of membership functions of the results of experiments, and the transformation of the problem to a standard problem of linear programming is shown. The desired result is achieved by solving a quadratic mathematical programming problem with a linear constraint. The proposed method is generalized to the case when the fuzzy initial data are given bifuzzy.
Construction of the fractional-nonlinear optimization method
(Technology center PC, 2019) Raskin, Lev; Sira, Oksana
A method for solving the fractional nonlinear optimization problem has been proposed. It is shown that numerous inventory management tasks, on the rational allocation of limited resources, on finding the optimal paths in a graph, on the rational organization of transportation, on control over dynamical systems, as well as other tasks, are reduced exactly to such a problem in cases when the source data of a problem are described in terms of a probability theory or fuzzy math. We have analyzed known methods for solving the fractional nonlinear optimization problems. The most efficient among them is based on the iterative procedure that sequentially improves the original solution to a problem. In this case, every step involves solving the problem of mathematical programming. The method converges if the region of permissible solutions is compact. The obvious disadvantage of the method is the uncontrolled rate of convergence. The current paper has proposed a method to solve the problem, whose concept echoes the known method of fractional-linear optimization. The proposed technique transforms an original problem with a fractional-rational criterion to the typical problem of mathematical programming. The main advantage of the method, as well its difference from known ones, is the fact that the method is implemented using a single-step procedure for obtaining a solution. In this case, the dimensionality of a problem is not a limiting factor. The requirements to a mathematical model of the problem, which narrow the region of possible applications of the devised procedure, imply: 1) the components of the objective function must be separable functions; 2) the indicators for the power of all nonlinear terms of component functions should be the same. Another important advantage of the method is the possibility of using it to solve the problem on unconditional and conditional optimization. The examples have been considered.
Determining parameters of electromagnetic radiation for energoinformational disinfection of wool in its pretreatment
(Технологический центр, 2017) Kosulina, N.; Cherenkov, A.; Pirotti, E.; Moroz, S.; Chorna, M.
Угнетение инфекционных микроорганизмов в кипах шерсти возможно при помощи электромагнитного излучения (ЭМИ) с параметрами: частота 36 ГГц, мощность 0,5 кВт, экспозиция 3 мин. Энергоинформационные параметры были определены в результате теоретических и экспериментальных исследований процесс взаимодействия ЭМИ с инфекционными микроорганизмами.
Development of a model for the dynamics of probabilities of states of semi-Markov systems
(Kharkiv National University of Radio Electronics, 2021) Raskin, Lev; Sira, Oksana; Sukhomlyn, Larysa; Korsun, Roman
The subject is the study of the dynamics of probability distribution of the states of the semi-Markov system during the transition process before establishing a stationary distribution. The goal is to develop a technology for finding analytical relationships that describe the dynamics of the probabilities of states of a semi-Markov system. The task is to develop a mathematical model that adequately describes the dynamics of the probabilities of the states of the system. The initial data for solving the problem is a matrix of conditional distribution laws of the random duration of the system's stay in each of its possible states before the transition to some other state. Method. The traditional method for analyzing semi-Markov systems is limited to obtaining a stationary distribution of the probabilities of its states, which does not solve the problem. A well-known approach to solving this problem is based on the formation and solution of a system of integral equations. However, in the general case, for arbitrary laws of distribution of the durations of the stay of the system in its possible states, this approach is not realizable. The desired result can only be obtained numerically, which does not satisfy the needs of practice. To obtain the required analytical relationships, the Erlang approximation of the original distribution laws is used. This technique significantly increases the adequacy of the resulting mathematical models of the functioning of the system, since it allows one to move away from overly obligatory exponential descriptions of the original distribution laws. The formal basis of the proposed method for constructing a model of the dynamics of state probabilities is the Kolmogorov system of differential equations for the desired probabilities. The solution of the system of equations is achieved using the Laplace transform, which is easily performed for Erlang distributions of arbitrary order. Results. Analytical relations are obtained that specify the desired distribution of the probabilities of the states of the system at any moment of time. The method is based on the approximation of the distribution laws for the durations of the stay of the system in each of its possible states by Erlang distributions of the proper order. A fundamental motivating factor for choosing distributions of this type for approximation is the ease of their use to obtain adequate models of the functioning of probabilistic systems. Conclusions. A solution is given to the problem of analyzing a semi-Markov system for a specific particular case, when the initial distribution laws for the duration of its sojourn in possible states are approximated by second-order Erlang distributions. Analytical relations are obtained for calculating the probability distribution at any time.
Development of a model of the service system of batch arrivals in the passengers flow of public transport
(Technology center PC, 2019) Raskin, Lev; Sira, Oksana; Palant, Oleksii; Vodovozov, Yevgeniy
A mathematical model of the queuing system for the passenger flow of urban public transport is proposed. The resulting model differs from canonical models of queuing theory by taking into account the fundamental features of real systems. Firstly, the service process is divided into different successive service sessions. Secondly, arrival and departures are batch. Thirdly, the arrival rates vary in different service sessions. Fourthly, the laws of distribution of the number of jobs in batch arrivals for different sessions are different. Fifth, the laws of distribution of the number of batch arrivals and departures are also different. A criterion of efficiency of the service system is developed. The criterion is based on the calculation of the probability distribution of the service system states at the input and similar distribution at the output. These distributions are determined independently for each service session, into which the entire service cycle is divided. The numerical value of the criterion is set by the ratio of the average number of service rejections to the average number of jobs in the batch arrival for the entire service cycle. It can be used to assess the efficiency of the service system at any selected time interval during the day, because the value of the proposed criterion depends on the length of the interval between sessions, determined by the number of vehicles on the route. The resulting models adequately reflect the functioning of the system, which makes it possible to predict many different situations and evaluate the consequences of proposed solutions. Thus, it becomes possible to predict the provision of the population with public transport and determine quantitative values of efficiency of the urban public transport system.
Development of graphic-analytical models for the software security testing algorithm
(Technology center PC, 2018) Semenov, Serhii; Sira, Oksana; Kuchuk, Nina
An analysis of one of the main types of software testing, namely security testing has been made. It was established that there are a number of specific features associated with the possibility of negative manipulation with software products. A graphic-analytical model of the algorithm of testing software security was developed. The model based on the theory of semi-Markov processes provides an adequate structural description of the actual testing process. However, accuracy of this model essentially depends on accuracy of reproduction of densities of distribution of duration of the system residence in each of the possible states. An alternative model that uses the method of probability-time graphs is less demanding. For its implementation, it is sufficient to know the mean values of duration of residence in each of the states and the probability of transitions from one state to another. Correlations were obtained for calculating statistical characteristics and density of distribution of the mean time of execution of the software security testing algorithm. The model can be used to study basic stages of software security testing. Application of this model will reduce software vulnerability and improve security of the IT project as a whole. Also, the model is applicable when developing new methods, algorithms, and procedures for managing the IT projects.
Development of methods for extension of the conceptual and analytical framework of the fuzzy set theory
(Technology center PC, 2020) Raskin, Lev; Sira, Oksana
Fuzzy set theory is an effective alternative to probability theory in solving many problems of studying processes and systems under conditions of uncertainty. The application of this theory is especially in demand in situations where the system under study operates under conditions of rapidly changing influencing parameters or characteristics of the environment. In these cases, the use of solutions obtained by standard methods of the probability theory is not quite correct. At the same time, the conceptual, methodological and hardware base of the alternative fuzzy set theory is not sufficiently developed. The paper attempts to fill existing gaps in the fuzzy set theory in some important areas. For continuous fuzzy quantities, the concept of distribution density of these quantities is introduced. Using this concept, a method for calculating the main numerical characteristics of fuzzy quantities, as well as a technology for calculating membership functions for fuzzy values of functions from these fuzzy quantities and their moments is proposed. The introduction of these formalisms significantly extends the capabilities of the fuzzy set theory for solving many real problems of computational mathematics. Using these formalisms, a large number of practical problems can be solved: fuzzy regression and clustering, fuzzy multivariate discriminant analysis, differentiation and integration of functions of fuzzy arguments, state diagnostics in a situation where the initial data are fuzzy, methods for solving problems of unconditional and conditional optimization, etc. The proof of the central limit theorem for the sum of a large number of fuzzy quantities is obtained. This proof is based on the characteristic functions of fuzzy quantities introduced in the work and described at the formal level. The concepts of independence and dependence for fuzzy quantities are introduced. The method for calculating the correlation coefficient for fuzzy numbers is proposed. Examples of problem solving are considered.
Development of methods for supply management in transportation networks under conditions of uncertainty of transportation cost values
(Scientific Route, Estonia, 2021) Raskin, Lev; Sira, Oksana; Parfeniuk, Yurii; Bazilevych, Kseniia
The problem of transport management in a distributed logistics system «suppliers – consumers» is considered. Under the assumption of a random nature of transportation costs, an exact algorithm for solving this problem by a probabilistic criterion has been developed. This algorithm is implemented by an iterative procedure for sequential improvement of the transportation plan. The rate of convergence of a computational procedure to an exact solution depends significantly on the dimension of the problem and is unacceptably low in real problems. In this regard, an alternative method is proposed, based on reducing the original problem to solving a nontrivial problem of fractional-nonlinear programming. A method for solving this problem has been developed and substantiated. The corresponding computational algorithm reduces the fractional-nonlinear model to the quadratic one. The resulting problem is solved by known methods. Further, the original problem is supplemented by considering a situation that is important for practice, when in the conditions of a small sample of initial data there is no possibility of obtaining adequate analytical descriptions for the distribution densities of the random costs of transportation. In this case, the available volume of statistical material is sufficient only to estimate the first two moments of unknown distribution densities. For this marginal case, a minimax method for finding the transportation plan is proposed. The first step is to solve the problem of determining the worst distribution density with the given values of the first two moments. In the second step, the transportation plan is found, which is the best in this most unfavorable situation, when the distribution densities of the random cost of transportation are the worst. To find such densities, let’s use the modern mathematical apparatus of continuous linear programming.
Development of modern models and methods of the theory of statistical hypothesis testing
(Technology center PC, 2020) Raskin, Lev; Sira, Oksana
Typical problems of the theory of statistical hypothesis testing are considered. All these problems belong to the same object area and are formulated in a single system of axioms and assumptions using a common linguistic thesaurus. However, different approaches are used to solve each of these problems and a unique solution method is developed. In this regard, the work proposes a unified methodological approach for formulating and solving these problems. The mathematical basis of the approach is the theory of continuous linear programming (CLP), which generalizes the known mathematical apparatus of linear programming for the continuous case. The mathematical apparatus of CLP allows passing from a two-point description of the solution of the problem in the form {0; 1} to a continuous one on the segment [0; 1]. Theorems justifying the solution of problems in terms of CLP are proved. The problems of testing a simple hypothesis against several equivalent or unequal alternatives are considered. To solve all these problems, a continuous function is introduced that specifies a randomized decision rule leading to continuous linear programming models. As a result, it becomes possible to expand the range of analytically solved problems of the theory of statistical hypothesis testing. In particular, the problem of making a decision based on the maximum power criterion with a fixed type I error probability, with a constraint on the average risk, the problem of testing a simple hypothesis against several alternatives for given type II error probabilities. The method for solving problems of statistical hypothesis testing for the case when more than one observed controlled parameter is used to identify the state is proposed.
Devising a method for finding a family of membership functions to bifuzzy quantities
(Technology center PC, 2021) Raskin, Lev; Sira, Oksana; Sukhomlyn, Larysa; Korsun, Roman
This paper has considered a task to expand the scope of application of fuzzy mathematics methods, which is important from a theoretical and practical point of view. A case was examined where the parameters of fuzzy numbers’ membership functions are also fuzzy numbers with their membership functions. The resulting bifuzziness does not make it possible to implement the standard procedure of building a membership function. At the same time, there are difficulties in performing arithmetic and other operations on fuzzy numbers of the second order, which practically excludes the possibility of solving many practical problems. A computational procedure for calculating the membership functions of such bifuzzy numbers has been proposed, based on the universal principle of generalization and rules for operating on fuzzy numbers. A particular case was tackled where the original fuzzy number’s membership function contains a single fuzzy parameter. It is this particular case that more often occurs in practice. It has been shown that the correct description of the original fuzzy number, in this case, involves a family of membership functions, rather than one. The simplicity of the proposed and reported analytical method for calculating a family of membership functions of a bifuzzy quantity significantly expands the range of adequate analytical description of the behavior of systems under the conditions of multi-level uncertainty. A procedure of constructing the membership functions of bifuzzy numbers with the finite and infinite carrier has been considered. The method is illustrated by solving the examples of using the developed method for fuzzy numbers with the finite and infinite carrier. It is clear from these examples that the complexity of analytic description of membership functions with hierarchical uncertainty is growing rapidly with the increasing number of parameters for the original fuzzy number’s membership function, which are also set in a fuzzy fashion. Possible approaches to overcoming emerging difficulties have been described.
Devising methods for planning a multifactorial multilevel experiment with high dimensionality
(Technology center PC, 2021) Raskin, Lev; Sira, Oksana
This paper considers the task of planning a multifactorial multilevel experiment for problems with high dimensionality. Planning an experiment is a combinatorial task. At the same time, the catastrophically rapid growth in the number of possible variants of experiment plans with an increase in the dimensionality of the problem excludes the possibility of solving it using accurate algorithms. On the other hand, approximate methods of finding the optimal plan have fundamental drawbacks. Of these, the main one is the lack of the capability to assess the proximity of the resulting solution to the optimal one. In these circumstances, searching for methods to obtain an accurate solution to the problem remains a relevant task. Two different approaches to obtaining the optimal plan for a multifactorial multilevel experiment have been considered. The first of these is based on the idea of decomposition. In this case, the initial problem with high dimensionality is reduced to a sequence of problems of smaller dimensionality, solving each of which is possible by using precise algorithms. The decomposition procedure, which is usually implemented empirically, in the considered problem of planning the experiment is solved by employing a strictly formally justified technique. The exact solutions to the problems obtained during the decomposition are combined into the desired solution to the original problem. The second approach directly leads to an accurate solution to the task of planning a multifactorial multilevel experiment for an important special case where the costs of implementing the experiment plan are proportional to the total number of single-level transitions performed by all factors. At the same time, it has been proven that the proposed procedure for forming a route that implements the experiment plan minimizes the total number of one-level changes in the values of factors. Examples of problem solving are given.
Dynamic problem of formation of securities portfolio under uncertainty conditions
(Scientific Route, Estonia, 2019) Raskin, Lev; Sira, Oksana; Katkova, Tetiana
The analysis of known methods for solving the problem of forming a portfolio of securities in the face of uncertainty is carried out. Traditionally, the problem is solved under the assumption that for each type of asset, the values of the main statistical characteristics of the random value of their profitability (mathematical expectation and variance) are known. At the same time, the variance of portfolio returns, which is minimized, is used as a criterion for portfolio optimization. Two alternative approaches to solving the formulated problem are proposed. The first of them provides a decision on the criterion of the probability that the random total portfolio return will not be lower than the given. It is assumed that the random return for each type of asset is distributed normally and the statistical characteristics of the respective densities are known. The original problem is reduced to the problem of maximizing the quadratic fractional criterion in the presence of linear constraints. To solve this non-standard optimization problem, a special iterative algorithm is proposed that implements the procedure for sequential improvement of the plan. The method converges and the computational procedure for obtaining a solution can be stopped by any of the standard criteria. The second approach considers the possibility of solving the problem under the assumption that the distribution densities of random asset returns are not known, however, based on the results of preliminary statistical processing of the initial data, estimates of the values of the main numerical characteristics for each of the assets are obtained. To solve the problem, a new mathematical apparatus is used – continuous linear programming, which is a generalization of ordinary linear programming to the case when the task variables are continuous. This method, in the considered problem, is based on solving an auxiliary problem: finding the worst-case distribution density of a random total portfolio return at which this total return does not reach an acceptable threshold with maximum probability. Now the main minimax problem is being solved: the formation of the best portfolio in the worst conditions. The resulting computational scheme leads to the problem of quadratic mathematical programming in the presence of linear constraints. Next, a method is proposed for solving the problem of forming a portfolio of securities, taking into account the real dynamics of the value of assets. The problem that arises in this case is formulated and solved in terms of the general theory of control, using the Riccati equation.
Effect of deformation degree under quasihydroextrusion at 77K on the structure and properties of CuCrZr alloy
(Харківський національний університет імені В. Н. Каразіна, 2019) Belyaeva, A. I.; Khaimovich, P. A.; Galuza, A. A.; Kolenov, I. V.; Savchenko, Alla Aleksandrovna; Shul`gin, N. A.
Execution of arithmetic operations involving the second-order fuzzy numbers
(Technology center PC, 2020) Raskin, Lev; Sira, Oksana
The need to improve the adequacy of conventional models of the source data uncertainty in order to undertake research using fuzzy mathematics methods has led to the development of natural improvement in the analytical description of the fuzzy numbers' membership functions. Given this, in particular, in order to describe the membership functions of the three-parametric fuzzy numbers of the (L-R)-type, the modification implies the following. It is accepted that these functions' parameters (a modal value, the left and right fuzzy factors) are not set clearly by their membership functions. The numbers obtained in this way are termed the second-order fuzzy numbers (bi-fuzzy). The issue, in this case, is that there are no rules for operating on such fuzzy numbers. This paper has proposed and substantiated a system of operating rules for a widely used and effective class of fuzzy numbers of the (L-R)-type whose membership functions' parameters are not clearly defined. These rules have been built as a result of the generalization of known rules for operating on regular fuzzy numbers. We have derived analytical ratios to compute the numerical values of the membership functions of the fuzzy results from executing arithmetic operations (addition, subtraction, multiplication, division) over the second-order fuzzy numbers. It is noted that the resulting system of rules is generalized for the case when the numbers-operands' fuzziness order exceeds the second order. The examples of operations execution over the second-order fuzzy numbers of the (L-R)-type have been given.
Finding the probability distribution of states in the fuzzy markov systems
(Technology center PC, 2017) Raskin, Lev; Sira, Oksana; Katkova, Tetiana
A problem on finding the stationary distributions of probabilities of states for the Markov systems under conditions of uncertainty is solved. It is assumed that parameters of the analyzed Markov and semi-Markov systems (matrix of transition intensities, analytical description of distribution functions of the durations of being in states of the system before exiting, as well as a matrix of transition probabilities) are not clearly assigned. In order to describe the fuzziness, we employ the Gaussian membership functions, as well as functions of the type. The appropriate procedure of systems analysis is based on the developed technology for solving the systems of linear algebraic equations with fuzzy coefficients. In the problem on analysis of a semi-Markov system, the estimation of components of the stationary distribution of probabilities of states of the system is obtained by the minimization of a complex criterion. The criterion considers the measure of deviation of the desired distribution from the modal one, as well as the level of compactness of membership functions of the fuzzy result of solution. In this case, we apply the rule introduced for the calculation of expected value of fuzzy numbers. The criterion proposed is modified through the introduction of weight coefficients, which consider possible differences in the levels of requirements to different components of the criterion.
Formation of securities portfolio under conditions of uncertainty
(Technology center PC, 2017) Sira, Oksana; Katkova, Tetiana
We examined a problem on the formation of securities portfolio. A criterion for portfolio effectiveness is determined – a probability that the total portfolio profitability exceeds a threshold. In connection with real shortage of the volume of initial data, we substantiated the rejection of hypothesis about the normality of their distribution law and the problem is solved under assumption about the worst distribution density of these data. In this case, it is accepted that mathematical expectation and the dispersion of values for the cost of assets are the fuzzy numbers. The form of membership function of the fuzzy parameters in the problem is selected. We constructed an analytical expression to describe the criterion in the terms of fuzzy mathematics. In this case, a problem on the maximization of fractional-quadratic functional with linear constraints is obtained. We devised a method for solving the obtained fuzzy problem of mathematical programming, which reduces this problem to the conventional problem of nonlinear programming. In order to solve this problem, it is proposed to employ the optimization method of zero order. It is demonstrated that the portfolio risk depends quadratically on the mathematical expectation of its profitability. Recommendations are given regarding the choice of numerical value for the mathematical expectation of portfolio profitability depending on the acceptable portfolio risk.