## No 1 (2022)

DETERMINATION, ANALYSIS AND SYNTHESIS OF THE ROOTS OF VALID ALGEBRAIC EQUATIONS USING LOGARITHMIC AMPLITUDE-PHASE FREQUENCY CHARACTERISTICS

#### Abstract

The article considers the method of approximate determination, analysis and synthesis of the roots of real algebraic equations of high order. The solution of such problems is relevant in the case of designing information-measuring and control systems, studying the dynamics of movement of various mechanisms (industrial robots, quadrocopters, etc.), determining the trajectories of aircraft, etc. The analytical solution of such problems is limited to equations of the third (sometimes fourth) degree, in other cases it is necessary to use either special sequential algorithms or packages of applied computer programs such as "Wolfram.Matematica", which allow only to find the roots of the equations, but not to synthesize them. The proposed method is based on the application for the decomposition of the studied polyomial (corresponding to the equation) into the simplest multipliers corresponding to aperiodic and/or oscillatory links, asymptotic logarithmic amplitude and phase-frequency characteristics. The form and values of the roots of the equation are proposed to be judged by the slopes at the fracture points of the logarithmic amplitude and phase-frequency characteristics of the polyparticle under study. The construction of logarithmic amplitude and phase-frequency characteristics is carried out by discarding the "small" terms of the polygamy at separate frequency intervals. A feature of the method is the possibility of its use both in conjunction with the computer and without it. Manual use of the method assumes that the user has a calculator and a ruler. The method allows to determine not only the roots of real algebraic equations (both real and complex), but also to establish a visual relationship between the coefficients for the terms of the equations with the type and values of the roots and purposefully change the necessary coefficients to change the parameters and type of roots. The possibilities of the method are not limited to solving real algebraic equations with positive coefficients and integer powers, it shows quite satisfactory results for equations with mixed coefficients and fractional powers. The method is quite simple, clear, has a small error in the case of far spaced roots, but in the case of closely spaced roots, its error increases, although it remains quite acceptable. The article presents the substantiation of the method, shows numerous examples of its capabilities, compares the results obtained with the results obtained with the help of the package of applied computer programs "Wolfram.Matematica".

**Applied Mathematics and Control Sciences**. 2022;(1):9-34

FORMALISING A RAIL PLANNING TASK FOR A MINING COMPANY

#### Abstract

The formalisation of the task of railway planning, namely, the formation of freight trains and their routes along the railway network of the mining company is considered. The statement of the problem including parameters of the problem, variables, system of constraints and target function is presented. An integer formulation of the problem, taking into account the constraints of a particular mining company, is proposed. The problem of railroad planning is relevant despite the various approaches and solutions available, since each case encounters different constraints. This formalisation differs from the others because there are different types of locomotives, and hence different capacities, and there are different types of materials to be transported. Four different categories of restrictions are presented. In addition, the existence of an extensive network of stations, a huge number of non-stationary constraints and other factors significantly increase the dimensionality of such problems, which increases the interest of researchers to them and contributes to the emergence of new and the development of existing methods and approaches to their solution. In the introduction, a description and features of the considered railway network of a mining company are presented. It then presents the formulation of the problem to be formalised, including parameters, variables, constraint system and target function. Then a numerical example with a solution is given. The integer linear programming method is used as a method for solving this problem. An example of scheduling freight trains for a network including four stations connected by double-track runs and having a star form is considered. The task of constructing routes is not considered in this example, as it is a separate complex task and is not required for this example, as there is a direct path from each departure station to each destination station.

**Applied Mathematics and Control Sciences**. 2022;(1):37-51

OPTIMIZATION OF A MULTICOMPONENT POLYMER COMPOSITION USING A FUZZY MATHEMATICAL MODEL

#### Abstract

Some aspects of the analysis of polymer composite materials as complex systems are considered. In this case, the system is presented as a combination of one polymer matrix and several active additives. Within the framework of this work, the composition is assumed to be unchanged, and the properties of the composition are controlled by changing the concentration of the ingredients. In work, a mathematical model was developed that calculates the optimal content of components to improve specific properties of the polymer composition. Obtaining such a model is partly hampered by complex interactions between components, but a solution was obtained within the framework of one specific composition. Nevertheless, it was impossible to transfer these results to other compositions in this case, and it was impossible to obtain a general mathematical model for an arbitrary composition. Therefore, to solve this problem, a black-box model was used in this work. The main methods for studying polymer compositions are presented; their systematization is considered according to the principle of controlling properties at different stages of material synthesis. In this work, a variant of controlling the properties of the polymer composition using active additives was used. The urgency of the problem related to the development of methods for assessing the properties and control of the latter by ranking the concentrations of the ingredients of the polymer matrix has been substantiated. As a result, a mathematical model for optimizing the composition of the polymer composition was obtained. It takes into account the positive and the negative influence of the ingredients on the entire composition of the polymer matrix. Also, computational experiments were carried out to find the optimal concentration of active additives in the composition of the polymer composition under conditions of pair interaction of additives. The model is presented and solved using a quadratic programming problem using a specific example. Different cut-off values were used for the content of the ingredients. The results obtained clearly demonstrate the dependence of the properties of a chemical system on the concentration of specific ingredients. Based on the results of two computational experiments under different boundary conditions, the optimal concentration was calculated for the full manifestation of two properties. The paper also presents a vector of further actions, prospects for improving the model, and possible areas of application of this model.

**Applied Mathematics and Control Sciences**. 2022;(1):52-71

ASSESSMENT SYSTEM FOR SPORTS EXERCISES BY NEURAL NETWORK VIDEO ANALYSIS

#### Abstract

This article describes the initial stage of developing an information system for evaluating a sports exercise based on the use of neural networks. The current approach to the evaluation of sports exercises and its shortcomings, as well as the advantages of introducing computer technology, were considered. The analysis of initial data is carried out. Next, the approaches that are used to analyze the position of the athlete's body during the exercise are considered, and an approach is chosen to obtain the position of the athlete's body in space and time. The definition of the concept of key points is given and their location, number on the human body are described. A number of pre-trained neural networks are considered that determine the location of key points on the human body, and the best option for solving the problem posed in this article is selected. The data sets that are used in the training of neural networks, in tasks of determining the position of the human body, are studied. The problem of image analysis is posed and an algorithm for solving the problem is introduced. As a result of the work, a prototype of an information system was developed that is capable of receiving frames from the original video sequence, processing frames using a neural network, recording the position of key points in the image to a text file, and processing data from the file for analysis.

**Applied Mathematics and Control Sciences**. 2022;(1):75-86

AN ADAPTIVE SOFT SENSOR DESIGN BASED ON A NEURAL NETWORK WITH A PREDICTIVE FILTER IN THE FEEDBACK LOOP FOR A REACTION DISTILATION TECHNOLOGICAL PROCESS

#### Abstract

Rectification and reactive distillation columns are the main of all units in the petrochemical and refining industry. Soft sensors consist of mathematical models that estimate of the quality of an output product in real time are used for technological processes control. In general, changes in the composition of raw materials, catalyst deactivation, etc. result in inconsistency between obtained data and the current state of the technological process. Soft sensor design obtained on such data will loss of accuracy in estimating the necessary parameters of the output product. An adaptive soft sensor design based on a neural network with a predictive filter in the feedback loop for solve the mismatching obtained data and the current state of the technological process problem is proposed. “Moving window” conception is used for size window adapting to the actual state of the technological process. Parameter estimation based on a neural network using data matching to the technological process. A predictive filter in the feedback loop for improve the estimation accuracy of the quality parameter at the cost of predicting the error of the soft sensor designed is proposed. A comparative analysis of several adaptive soft sensors based on neural networks using the "moving window" conception for estimation a by-product concentration of in the output product of the reaction-distillation column and the effectiveness of the proposed approach are shown. Application of the predictive filter in the feedback loop allows to improve the accuracy of the soft sensor based on a neural network by 12.94 % (coefficient of determination) and by 39.81 % (mean absolute error) in comparison with that of without predictive filter.

**Applied Mathematics and Control Sciences**. 2022;(1):89-108

ALGORITHMS FOR SEARCHING FOR AN ACCEPTABLE OPERATING MODE OF A CABLE CHANNEL BASED ON AN ASSESSMENT OF ITS TEMPERATURE STATE

#### Abstract

The great majority of currently used cable products contain polymer materials in their construction, which tend to degrade their physical and electrical properties over time. This process is aggravated when the temperature of the product increases. The design of cable lines and the conditions of their collective laying, for example, in cable channels, complicate the thermal control of their operation, while the conditions of an isolated channel contribute to the heating of individual lines at the expense of neighboring ones. The existing engineering methods for calculating the cable temperature specified in IEC 60287-1-1-2009 do not allow taking into account all the features of a particular cable structure. Therefore, the task of a quick and accurate assessment of the thermal state of all polymer elements of the lines laid in the channel is very urgent. The paper considers the possibility of replacing direct observation of the thermal field of a cable channel with its equivalent mathematical model capable of determining stable stationary temperature distributions at given current loads of lines. Such a model, in addition to the speed of finding the desired parameters, has such an advantage over a real line as the ability to evaluate the internal temperature field of the cable without violating the integrity of its design. Using a mathematical model, it is proposed to analyze the thermal state of the channel by comparing the temperatures of the elements of cable lines with a predetermined setpoint. When the fact of exceeding the setpoint is detected, an algorithm is started for selecting such a combination of current loads at which the temperatures of the line elements do not exceed the set value. An additional criterion for selecting the mode is the minimum reduction of currents in comparison with their initial value. The algorithms proposed in this paper are universal and can be used for channels of various designs and with different numbers of lines, as well as for analyzing the operation of similar engineering structures, for example, ventilation ducts with transit lines of air conditioning systems, pipelines for hot liquids and gases. The developed algorithms are functionally suitable for creating specialized software operating in the mode of an adviser to the operator of an electricity distribution node. The software is designed to pre-evaluate any change in line loads, checking for possible overheating and giving the operator either approval of the mode, or offering a safe option for long-term operation.

**Applied Mathematics and Control Sciences**. 2022;(1):109-122

IMPROVEMENT OF THE INFORMATION MANAGEMENT SYSTEM OF AN INDUSTRIAL ENTERPRISE

#### Abstract

This article is devoted to the development and research of models and methods for evaluating the information management system of an industrial enterprise (IOPP) as an indicator of the effectiveness of the management of the organizational and economic system. The importance and significance of the factor of improving the information support system to ensure the competitiveness of modern industrial enterprises and organizations in all spheres of the economy is shown. The industry specifics and the nature of the organization of production processes are analyzed on the basis of expert evaluation. The sequence of stages of application of the proposed methodological approach to improving the information management system of an industrial enterprise is highlighted. Models and methods are proposed that allow to establish and assess compliance with the requirements for information support and the level of competitive stability, external and internal regulatory documents. The synthesis of dominant analysis and factor analysis methods was applied to systematize the most significant factors of IOPP that have developed in the information environment, grouped into four groups that make up the content of the corresponding cells of the DETA matrix: I cell - a group of organizational factors (Development); II cell - a group of economic factors (Economic); III cell - a group of technical and technological factors (Technology); IV cell - a group of managerial factors (Administration). The DETA matrix presents significant factors that enable the management of an industrial enterprise to develop an appropriate methodological approach to improving the IOPP system by evaluating the achieved and expected indicators, together with modern trends in the development of the information environment. The matrix method of NDV analysis and the method of point estimates are presented and tested, which allowed: to determine the subsystems of the IOPP subject to evaluation procedures and criteria for their evaluation; to distribute evaluation indicators into groups according to NDV criteria (availability, sufficiency, availability, demand); to identify "bottlenecks" on which the managerial impact of the management of an industrial enterprise should be concentrated; to select criteria and indicators based on the specifics of a particular industrial enterprise. The methodical approach has been consistently tested at JSC "Lysvensky Enameled Ware Factory (LZEP)". Based on the industry specifics and the nature of the organization of production processes on the basis of expert evaluation, the parameters of the NDV were adopted as criteria, which made it possible to form an array including 16 indicators and build an evaluation classifier. The evaluation classifier made it possible to construct a conditional diagram of the IOPP management system in the criteria and subsystem sections Illustrative examples of the use of the proposed models and methods are given. Based on the results obtained, the author has developed an integral efficiency coefficient of the IOPP management system according to the criteria of the NDV. According to the information support management system of JSC "LZEP", an imbalance is stated with respect to the subsystem and criteria sections.

**Applied Mathematics and Control Sciences**. 2022;(1):125-150

VALUATION OF MACHINERY AND EQUIPMENT WITH RANDOMLY DEGRADED OPERATING CHARACTERISTICS

#### Abstract

Traditionally used methods of machinery and equipment valuation do not take into account the probabilistic nature of the process of their operation. Meanwhile, stochastic models of degradation of technical systems are widely used to solve various problems of the theory of reliability. This article refers to the direction of research in which the models of technical systems degradation used in the reliability theory are applied to the problems of machinery and equipment valuation. The objects of our research are equipment items subject to degradation and random failures. The subject of research is the service life of equipment and their market value. We define the (pre-tax) benefits from the use of equipment as the market value of the work they perform minus operating costs. The market value of equipment is determined by the sum of discounted benefits from its future use. The process of equipment degradation (deterioration of its technical and economic characteristics) is described by a Poisson random process of failures, and for each failure, the performance of the equipment is multiplied by a random coefficient. Operating costs are divided into a variable part depending on the amount of work performed by the equipment and a fixed part. We accept that equipment that has begun to bring negative benefits is decommissioned. The constructed model makes it possible to obtain explicit expressions for the market value of a unit of work performed by equipment, the average value and the coefficient of variation of its service life, and also to find the dependence of the equipment market value on its performance. In some cases, appraiser only knows the age of the equipment being assessed and has no information about its performance. However, equipment of the same age may have different performance. In this situation, he can only estimate the average market value of equipment of a given age.We got an integro-differential equation to calculate this average value. It turned out that in this model it can be expressed as a function of the relative age of the value (the ratio of its age to the average service life) and the coefficient of variation of the service life. The results obtained are presented as graphs that can be directly used to estimate the market value of used machinery and equipment.

**Applied Mathematics and Control Sciences**. 2022;(1):153-175

FEATURES SELECTION FOR USE IN PREDICTIVE ANALYTICS MODELS OF REGIONS FOREIGN ECONOMIC ACTIVITY

#### Abstract

The work is devoted to the actual problem of designing and selecting features in the tasks of constructing predictive models of indicators of foreign economic activity of regions. The aim of the work is to develop a methodology for the use of graph models and dimensionality reduction methods for the selection of features in the construction of predictive analytics models of foreign economic activity of regions. As an approach to describe the structure of foreign economic relations, graph models were used that implement the possibility of building on the basis of the UMAP dimension reduction algorithm. To build an optimal graph based on the UMAP algorithm, it is necessary to vary the number of nearest neighbors for each vertex and the minimum metric distance to establish a connection between the vertices. It is shown that the clique symmetry coefficient of the graph makes it possible to estimate the local connectivity of points in the constructed graph, forming a generalized idea of the network structure from the position of the presence of clusters in it. The Gini index of the graph allows us to assess the correspondence of the global structure of the graph to real networks. The selection of features is based on the analysis of the graph clicks, which provides maximum mutual information MI with a minimum of features, which maximally reduces distortions in describing the structure of regional foreign economic relations. The application of the described approach made it possible to eliminate the multicollinearity of features, to select indicators, to expand the possibilities of using the existing data set by including new indicators that introduce new useful information about the subject area into the model. The method of feature selection considered in the paper can be rationally used to construct interpreted predictive models of foreign economic activity indicators and as one of the ways to reduce the dimension of the model feature space. The results obtained allow us to conclude about the advantages of the considered approach to the implementation of the selection of features in the construction of predictive models of indicators of foreign economic activity of regions.

**Applied Mathematics and Control Sciences**. 2022;(1):176-195

ANALYSIS OF FACTORS INFLUENCING THE PRICING OF AIR TICKETS

#### Abstract

The article is devoted to discussing topical issues about the formation of the price of plane tickets. The paper analyzes and evaluates the degree of influence of factors on the pricing process using regression analysis methods. At the first stage, a review of the available research sources of foreign and Russian authors was carried out, according to the results of which the main characteristics determining the price level of air tickets were identified and considered. In the course of studying the issue, the experience and conclusions, based on previously performed studies of foreign segments of air transportation, are extended to the selection and analysis of current data on the Russian market. The assessment of the degree of a possible influence of the considered price-forming indicators on air ticket fares was carried out using four econometric models of the dependence of ticket prices on exchange rates, fare/air route characteristics, socio-economic and secondary factors. They are checked for lowcost and economy segments based on information on 15 thousand of air tickets of Russian air carriers for the period from 19.01.2021 to 12.02.2021, presented in the form of a single data set. The data was collected independently by daily monitoring of information from the official websites of airlines, federal websites of Rosstat, and Rosaviation. As a result of the research conducted within the framework of this article, it was revealed that with an increase in the distance of the flight, the cost of the ticket increases proportionally. The average income of the population negatively affects prices in the economy and budget classes. A similar trend can be traced concerning the cost of jet fuel at the airport of departure and nonstop flights. Reducing the number of days between the dates of ticket purchase and flight departure will also affect the increase in ticket prices. With a decrease in the exchange rate of foreign currency, the cost of economy class air tickets changes significantly compared to the budget. The reverse trend is observed in the case of the IATA exchange rate, where the effect falls on the last group of shipments. The listed characteristics are identified as the most significant relative to the considered classes of tariff groups in the Russian aviation industry. The results of the work can be recommended to air carriers for use in the process of forming the cost of tickets to achieve maximum efficiency and profitability of the business, used by consumers of the airline services as a tool for finding and buying air tickets with the best conditions and offers.

**Applied Mathematics and Control Sciences**. 2022;(1):196-213