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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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.
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.
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.
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 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.
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.
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.
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.
Method for solving distributional problems of mathematical programming under conditions of fuzzy initial data
(Technology center PC, 2024) Raskin, Lev; Sira, Oksana; Hatunov, Artur; Riabokon, Roman; Sinitsyn, Rodyslav
The object of this study is a large class of mathematical programming problems under conditions of uncertainty of initial data. The formulated object generates a subclass of problems of rational distribution of a limited resource under conditions of initial data described in terms of fuzzy mathematics. The conventional, standardly used method for solving such problems is based on optimization on average. To obtain such a solution, it is sufficient to replace all fuzzy initial data with their modal values in the analytical description of the mathematical model of the corresponding problem. To solve the resulting deterministic problem, one can use known methods of mathematical programming. However, the results of such a solution can be used in practice if the carriers of fuzzy parameters are specified compactly, that is, the intervals of possible values of fuzzy parameters of the problem are small. Otherwise, the implementation of this solution may lead to unpredictably large losses. Other alternative approaches are based on the use of insufficiently informative estimates of the best or worst possible values of fuzzy parameters of the problem. These circumstances make the statement of the problem and the objective of the study relevant: devising a method for solving the problem of rational distribution of a limited resource under conditions of fuzzy initial data. To solve the stated problem of rational distribution of a limited resource, a productive idea of constructing the proposed general optimization method under conditions of uncertainty of the initial data has been constructively implemented. In this case, the initial problem is reduced to a clear problem of optimizing a complex criterion constructed on the basis of the objective function of the initial problem and a set of membership functions of its fuzzy parameters. An example of solving the problem has been considered, leading to a solution that is better than that obtained on the basis of the modal values of the fuzzy parameters of the problem.