## No 4 (2022)

**Year:**2022**Articles:**11**URL:**https://ered.pstu.ru/index.php/amcs/issue/view/313**DOI:**https://doi.org/10.15593/2499-9873/2022.4

APPROXIMATE METHOD OF OPTIMIZATION OF NONLINEAR PROGRAMMING PROBLEMS

#### Abstract

Currently, the problem of choosing the optimal solution is one of the most important and urgent problems in industry, economy, agriculture and the military sphere. Methods and approaches of the theory of nonlinear programming are used to solve many applied optimization problems. The main difficulty of nonlinear optimization is the lack of a universal method for solving this class of problems. To solve this problem, special methods are being developed for solving particular nonlinear programming problems, for example, for positive or limited initial data The paper investigates the problem of analytical optimization of nonlinear programming problems. The purpose of this work is to develop a new approximate method for solving optimization problems of a nonlinear function under nonlinear constraints in the form of equalities. To do this, an approximation (expansion in a series) of the objective function and constraints is performed. All variables are considered bounded at the top and bottom. The objective function and constraints are considered infinitely differentiable by the set of arguments, and all their derivatives are assumed to be limited in absolute value by a given number. In this article, the theorem on the conditional maximum of the objective function under given constraints is proved. the results of which are the justification of the developed method. Since the developed optimization method is approximate, the error of the proposed representation of the objective function and constraint functions is estimated. In problems of an applied nature, the boundaries of variable changes are often set approximately and they can be adjusted. In addition, it is possible to adjust the point relative to which the functions are decomposed into series. Therefore, the article analyzes the sensitivity of the optimal solution of the problem when changing the decomposition point into a series of functions for different values of the coordinates of the left boundaries when searching for the maximum of the objective function. To explain the operation of the method, a specific numerical example is analyzed in detail. Modeling in the MS Excel environment was used to solve it. Graphs of the sensitivity study of the solution of the problem when changing the initial data are constructed. Nonlinear programming models are used, for example, to solve the following practically important issues: minimizing costs in the sale of products, optimizing consumer choice, maximizing production volume, determining the rational behavior of an individual in a given situation, rational use of resources, forming an optimal portfolio of securities.

**Applied Mathematics and Control Sciences**. 2022;(4):9-25

NUMERICAL CALCULATION OF HEAT TRANSFER IN A MULTILAYER COMPOSITE STRUCTURE WITH HONEYCOMB FILLER DURING AUTOCLAVE MOLDING AT THE HEATING STAGE

#### Abstract

The production of any multilayer structure begins with the development of technological documentation, which describes, among other things, the temperature regimes during their manufacture. Conventionally, the temperature process of polymerization can be divided into three stages: preheating, temperature stabilization and cooling. The paper presents the results of a numerical calculation of temperature fields in a multilayer composite structure with a honeycomb filler during its manufacture by autoclave molding at the preheating stage. This method of manufacturing composite structures makes it possible to mold parts of varying complexity and dimensions, the demand for which is growing in such industries as mechanical engineering, aircraft building, and shipbuilding. The requirements for the quality of such products are increasing, which is greatly influenced by compliance with the temperature regime during molding. Conducting direct experiments requires large energy costs, therefore, to solve the problem of controlling heat exchange processes inside the structure, mathematical models were developed that describe these processes. A non-stationary heat conduction problem is formulated for a multilayer unbounded plate with a constant initial distribution, boundary conditions of the third kind on the outer boundaries, and boundary conditions of the fourth kind on the contact surfaces of the layers. Using the finite element method, the problem is reduced to three-point difference equations, the solution of which is found by the sweep method. The determination of the sweep coefficients is shown taking into account the thermal characteristics of the layers. The results of a numerical calculation of the temperature distribution for a nine-layer composite structure with a honeycomb core are presented. The numerical calculation was carried out using the developed program in the BorlandDelphi 7.0 object-oriented programming environment. The results obtained are presented in the form of graphic dependences of the temperature over the thickness of the sample at different points in time, as well as the dependence of temperature on time at various nodes of the sample in comparison with the theoretical curve. An analysis of these dependences was carried out, which showed that the heating of the sample occurs unevenly over its thickness. The deviation from the theoretical temperature values is observed in the layers located closer to the honeycomb layer. This can adversely affect the course of the polymerization step, which is characterized by the conversion of the binder into a polymer, and occurs at certain temperatures. Therefore, achieving the desired temperature values at the stage of heating the structure is important for the manufacture of reliable and durable structures that can withstand extreme operating conditions. The obtained temperature distributions make it possible to correct the technological process of manufacturing various multilayer structures at the stage of its development, which will reduce the economic costs of production.

**Applied Mathematics and Control Sciences**. 2022;(4):29-43

THE USE OF DATA MINING METHODS TO IDENTIFY RISKS OF FETAL PATHOLOGY

#### Abstract

Reducing child mortality is reflected in several United Nations Sustainable Development Goals and is a key indicator of human progress. The use of information technology makes it possible to process a large number of generated statistical data, or to form new ones over time based on the results of analyses and examinations of a huge number of patients, use the results to determine deviations in the early stages and identify the influence of certain indicators on the likelihood of pathologies. Based on the analysis, the functional requirements for the system are formulated and the task of developing a new information and analytical system is set. The article is devoted to the actual problem of determining the risks of fetal pathology during pregnancy. A variant of the classification of the presence of fetal abnormalities using multinomial logistic regression and a method for clustering patient risk groups using the k -means method was proposed, an optimization technique based on the Broyden - Fletcher - Goldfarb - Shanno method was applied, where memory (L-BFGS) was used as a limitation, the classification of fetal health occurs according to the specified three classes. An information system has been developed to visualize the results of analytical methods. When creating the software package, the client - server architecture was used, which allows the server device to support multi-user mode and provide simultaneous work with several clients. The system includes: an application for Windows, an application for Android. Server interaction module, database interaction module, data analysis module, patient data processing module, doctor data processing module, module, statistical sampling module, personal data protection module, rights differentiation module, visualization module. The modular architecture allows you to upgrade the system, add new modules as needed and make changes to the modules while maintaining the integrity of the system. The developed system has passed the stages of testing and performance evaluation.

**Applied Mathematics and Control Sciences**. 2022;(4):47-59

DIGITAL MODELS AND TOOLS FOR DIAGNOSING THREE-DIMENSIONAL SPINAL DEFORMITY BASED ON STEREOPHOTOGRAMMETRY DATA

#### Abstract

Scoliosis is the most common orthopedic disease in children, which most often leads to disability. An urgent task of the current stage of the development of spinal deformity diagnostics is the search and introduction of new available methods of non-invasive safe diagnosis of spinal deformity at early stages, since existing diagnostic methods can detect spinal column deformity in three planes during the development of scoliosis, however, they have a number of limitations for use, since either X-ray radiation or diagnostic installations are used for their implementation, requiring complex installation in a separate room and training of operating personnel. In this regard, there is an urgent need to develop digital tools for the diagnosis of spinal deformity, which would neutralize these difficulties in clinical practice. The aim of the study was to develop a mathematical algorithm for identifying the deformation of the spinal column in the horizontal plane according to the photogrammetric evaluation of a three-dimensional model of the surface of the human back, which was obtained from the results of the operation of the original software for a personal telecommunications device. The algorithm for assessing spinal column deformity is a sequence of four stages: reposition of the coordinate system, stratification of the spine line into segments, calculation of the rotation of V-segments, separation of the V-segment ratios to the physiological bends of the spine and visualization of spinal deformity in the horizontal plane. The developed algorithm has a number of advantages compared to those developed earlier: it allows to identify the rotation of individual vertebrae and torsion of the spine as a whole, is implemented in relation to all vertebrae of the thoracic and lumbar spine, thereby providing the possibility of differentiated diagnosis of forms of spinal deformity - from posture disorders to scoliosis. The described algorithm is a separate part for modular software that allows diagnostics of spinal column deformity using a personal telecommunication device. This algorithm can be used in a comprehensive assessment of posture disorders and scoliosis, monitoring the dynamics of spinal column deformity in the rehabilitation process.

**Applied Mathematics and Control Sciences**. 2022;(4):60-82

ANALYSIS OF SOCIO-ECONOMIC SYSTEMS USING QUANTUM-LIKE MATHEMATICAL MODELS BASED ON STATUS FUNCTIONS. PART II

#### Abstract

It is proposed to use quantum-like models with the use of status functions for mathematical modeling and subsequent analysis of complex socio-economic systems. The limitations of the methods of classical probability theory and mathematical statistics, as well as the theory of fuzzy sets, algorithms Mamdani, Suzuki and others for solving similar problems are described. A description of the main assumptions that are used in the mathematical modeling of socio-economic objects based on status functions is given. Examples are considered that describe the features of transition paths through intermediate states. An operator for the transition of a socio-economic system to various states, similar to the Hamiltonian, is presented. A spectrum of possible virtual trajectories is introduced to describe transitions to different states. A mathematical model based on status functions is proposed to describe the transition of the system to a measurable state. In the proposed Hamiltonian, the first term represents a subsystem of indica-tors, the second is an analog of the energy of indicators in the information environment. At the same time, terms are distinguished that are analogues of the energies of the system of two controlled indica-tors: interaction, kinetic and potential. The description of the results of mathematical modeling and the analysis of the interaction of two hypothetical indicators of the socio-economic system are given. The indicators are taken from the statistics of innovation indicators of one of the regions of the Russian Federation.

**Applied Mathematics and Control Sciences**. 2022;(4):85-107

INTEGRATED ASSESSMENT MODEL OF MEDICAL ORGANIZATION MANAGEMENT EFFICIENCY BASED ON MATHEMATICAL APPROACH

#### Abstract

The article considers the problem of forming an integral indicator for evaluating the effectiveness of management in terms of ranking factors by their informativeness based on mathematical methods of pattern recognition. Since in the modern realities of the crisis, pandemic and other socio-political challenges, medical issues are becoming more and more relevant, it is proposed to take a more qualitative approach to assessing the effectiveness of management in this area. Assumptions are made about an integrated approach to the development of such an assessment that would cover the entire pool of necessary indicators. However, there are difficulties on this path, because in this case there will be many factors; with such a volume of data, calculations are difficult even with the available computing power. It is assumed that some of the parameters can be discarded, but the question arises: which are really less important, and which are simply necessary. To do this, it is proposed to use an algorithm to identify the most informative factors, as well as to use an integral evaluation indicator. The whole procedure for constructing an integral indicator for evaluating the effectiveness of management is divided into several parts. The most important part is the choice of informative factors. It is proposed to consider factors as some vector, with a priori weights found in a certain way. From a mathematical point of view, informativeness is proposed to be understood as the average distance between certain quantities. Then, with the help of the majority committee, the problem of pattern recognition is directly solved. At the same time, a question arises regarding the construction of the majority committee itself, and the main drawback of this approach is also indicated. The Seniority Committee is not considered in this case because of its inefficiency. To eliminate all the vulnerabilities, it is proposed to solve the problem of constructing a separating majority committee for sets by modifying the N. Nilson linear correction algorithm, which is the novelty of this method. This modification makes it possible to build the necessary dividing majority committee for the set, which subsequently leads to the choice of informative factors. The algorithm for selecting the most informative indicators is carried out for each enlarged group of factors. After its implementation, we get a certain number of indicators, much smaller than they were at the beginning, while the question of assigning weight to each factor remains open. To solve this problem, the authors propose the use of expert assessments. Thus, within each enlarged group of indicators, we get the sum of indicators with weighting coefficients, from which an integral indicator of the effectiveness of the management of a medical organization is already obtained.

**Applied Mathematics and Control Sciences**. 2022;(4):108-122

MODELING OF THE LIBRARY COLLECTION ACQUISITION PROCESS

#### Abstract

Сonstantly increasing assortment of literature of various genres and directions offered by publishers sets a problem for the libraries when purchasing new literature in the conditions of limited budgeting. In this connection, the task of the qualitative acquisition of the library stock is urgent. The main conceptual idea for the solution of the problem is the maximum satisfaction of the information needs of various types of library resource users. To implement the users informational preferences, a general mathematical model of decision-making support has been developed in the task of optimizing library stock acquisition. It allows it to be replenished and renewed in order to meet the information needs of library resource users and takes into account the limited budgeting for acquiring new books. The demand for library resources is considered as the optimality criterion. A formalized approach to the process of completing the library fund with popular printed publications is proposed. To reduce the area dimension of possible solutions, a model approximation of the optimization problem has been constructed. The transition from the optimization problem to its model approximation is based on the ideas of identifying the most important features of the area of possible values and the segmentation of the readership. In the practical implementation of the constructed model, the method of data mining, the method of analyzing strategic alternatives and the theory of fuzzy sets in accordance with the specifics of the problem to be solved were used in determining each significant classification problem. The model, constructed and tested on the basis of the scientific library of the Orenburg State University (OSU), allows to take into account not only the influence of various factors on the quality of the library stock, but also the issues caused by the limited budgeting of libraries.

**Applied Mathematics and Control Sciences**. 2022;(4):123-139

ANALYSIS OF FACTORS AFFECTING THE NUMBER OF INSPECTORS OF THE SUPERVISORY ACTIVITY UNITS OF THE EMERCOM OF RUSSIA

#### Abstract

Constant changes in the number of inspectors of the supervisory activity units of the Ministry of the Russian Federation for Civil Defence, Emergencies and Elimination of Consequences of Natural Disasters, both by reducing and increasing, have led to the need to search for and assess the significance of factors affecting the staffing structure of the units. In addition, for personnel decisions, it is necessary to conduct a retrospective analysis of changes in the number of staff units of the supervisory activity units of the Ministry of Emergency Situations of Russia, as well as the level of correlation (Pearson correlation) of the number of units with the main indicators of the activities of supervisory authorities: the number of inspections, objects of supervision, fires, the number of people killed and injured in fires. Aim of the study: to identify statistically significant factors influencing the regulation of the staffing of the supervisory activity units of the Ministry of Emergency Situations of Russia, to determine the level of correlation between these parameters, as well as: the number of objects of supervision and the number of actually conducted scheduled inspections; between the optimal number of inspections and the number of actually conducted scheduled inspections. Research method: the assessment of the level of influence of factors was carried out by the method of correlation analysis of statistical data on the main indicators of the EMERCOM of Russia in the period from 2011 to 2021. Results: the calculation of the optimal (evenly distributed over the years) number of inspections was made; the conclusion was made that there was no statistically significant relationship between the main performance indicators and the number of inspectors; the level of correlation between: the number of objects of supervision and the number of actually conducted scheduled inspections; between the optimal number of inspections and the number of actually conducted scheduled inspections. Practical significance: the results can be used to study the personnel potential and develop a methodology for calculating the optimal number of inspections for a specific department of supervisory activities of the Ministry of Emergency Situations of Russia.

**Applied Mathematics and Control Sciences**. 2022;(4):140-158

APPLICATION OF THE RANDOM DICHOTOMOUS CLASSIFICATION METHOD TO DEVELOPER’S BRAND ASSESSMENT BASED ON CONSUMER’S REVIEWS

#### Abstract

The brand influences the success and sustainability of the builder. In this paper, the authors continue to address the problem of managing developer branding based on information on the Internet. The paper proposes a dichotomous classification that allows you to classify developer reviews based on randomly labeled data. he dichotomous classification method allows you to work with small databases, in contrast to methods based on the construction of artificial neural networks, which require large samples of data. The method was implemented in the "Eidos" system. The Eidos system is based on automated system cognitive analysis (ASC-analysis). The decision to change the recall marker was made on the basis of changing the credibility of the model by Van Riesbergen's F-measure. The application of the dichotomous classification method to classify developers' reviews was tested on the example of OJSC "Perm Silicate Panels Plant" (PZSP). The dichotomous classification method allowed us to distinguish 20 negative and 63 positive reviews in the collected sample. The performance and adequacy of the method used is shown. In particular, it is shown that it is possible to divide the collected textual information about the developer into two clusters without prior training. The dichotomous classification method reduces the developer's response time to a reference. This paper proposes an algorithm for reference clustering based on the dichotomous classification method. This algorithm can be used as a basis for software to collect and analyze developer data, as well as to manage developer branding.

**Applied Mathematics and Control Sciences**. 2022;(4):159-173

MATHEMATICAL MODELING OF PREEMPTIVE MANAGEMENT BY THE STAGES COMPLEX OF ADMINISTRATIVE PRODUCTION

#### Abstract

In line with the fundamental reform of the Russia administrative legislation, this article discusses the synthesis of a geoinformation system for preventive management of the full cycle of production on affairs about administrative offenses. According to the Anokhin - Sudakov's theory of functional systems, the tasks of forming a structural image and synthesizing a mathematical model to manage the administrative process stages, as well as the tasks of substantiating the mathematical criterion and its structural and functional implementation to prevent violations of a reasonable time in production on affairs about administrative offenses, were set. According to the law of management object integrity preservation, it is obtained that the opposing side of the administrative conflict counteracts by the joint use of protective (geoinformation), target (enforcement) and providing (geolocation) management subsystems, each of which generates a different contribution to efficiency at the next stage of the administrative process. To create a mathematical model for making a managerial decision, adequately formalized for each stage of the administrative process, a natural-scientific approach to the synthesis of management in conditions of limited resources was used. An analytical dependence is determined, that integrates the functioning regularities of law enforcement, geoinformation and geolocation components of the management system at the stage of administrative practice. When concretizing it, the possibility of representing administrative production by Poisson transitions of the affairs flow about administrative offenses between the states of the administrative process stages in a continuous Markov chain is used. Through modeling a Markov chain by the Kolmogorov - Chapman's equations system the criterion of preemptive management existence of the administrative process stages complex is revealed, allowing under proper efficiency of administrative production to optimize the intensity of identification and neutralization of threats to ensure a reasonable time in the target, protective or providing subsystems of management. The structure and functionality of probabilistic transitions of the affairs flow about administrative offenses, including management procedures, in a Markov chain of administrative production are established by their network modelling based on accumulated observations of administrative statistics. Numerical studies of a synthesized model of preemptive management of administrative production have clarified the effects of disruption of transition processes between states of administrative practice on its efficiency.

**Applied Mathematics and Control Sciences**. 2022;(4):174-197

DEVELOPMENT AND APPROBATION OF A SOFTWARE SOLUTION FOR THE INVESTMENT PORTFOLIOS FORMATION USING FRACTAL ANALYSIS AND PREDICTIVE MODELS

#### Abstract

This paper describes the development and approbation of the software solution, which implements the methodology of forming recommendations for the composition of investment portfolios using fractal analysis and long memory predictive models, which is the result of research conducted over several years at the Department of Information Systems and Mathematical Methods in Economics of PSU. The general algorithm of the program includes four main steps: 1) obtaining and preparing data; 2) sorting assets by the fractal dimension of their price series; 3) forecasting asset returns; 4) forming portfolios (determining asset shares). The described algorithm stages correspond to the structure of the program solution, expressed by the set of its subsystems. The features of the developed program are as follows: possibility to load data of share prices from the FINAM website; calculation of the fractal dimension of asset price series by the DFA and minimum coverage methods; asset returns forecasting using the ARFIMA and ARFIMA-GARCH models; selection of portfolio structures based both on the forecasted and historical data; possibility of multiple generation of portfolios based on random-selected assets with characteristics averaging; evaluation of portfolio characteristics using test data (if any); support for multi-core processors for multiple acceleration of calculations; windowed graphical interface. The results of program approbation on the Russian stock market data under crisis economic conditions are given. On the whole, these results are in line with those obtained earlier on the developed US market data. The developed program can be used by portfolio investors, carrying out investment activities in the international financial markets. Scientific application of the program is important: its users can be scientists, students and other researchers of innovative methods of investment portfolios formation.

**Applied Mathematics and Control Sciences**. 2022;(4):201-223