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.
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.
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.
Multi-criteria optimization in terms of fuzzy criteria definitions
(Lviv Polytechnic National University, 2018) Raskin, Lev; Sira, Oksana; Sagaydachny, D.
The problems of multi-criteria optimization are considered. Known methods for solving these problems are generalized to the case when weights that take into account the relative importance of particular criteria are not clearly defined. The procedure for constructing membership functions of fuzzy numbers, given by sets of intervals of possible values, using a linearized computation of least squares methods is substantiated. In this case, for the description of fuzzy numbers, the membership functions of (L-R)-type were chosen. A method for solving a fuzzy multi-criteria optimization problem for a scalarized criterion is proposed. The technology of solving the problem reduces it to a linear fractional problem of mathematical programming. A converging iterative procedure for finding the optimal plan is described. An alternative method for solving the original fuzzy problem based on the formation of a Pareto-set of non-dominated options is considered. To solve this problem, a procedure has been proposed for comparing fuzzy numbers using a probability-theoretic approximation of their membership functions.
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.
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.
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.
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.
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.