Харківський національний університет радіоелектроніки Державне підприємство "Південний державний проектно-конструкторський та науково-дослідний інститут авіаційної промисловості" Kharkov National University of Radio Electronics State Enterprise "Southern National Design & Research Institute of Aerospace Industries" СУЧАСНИЙ СТАН НАУКОВИХ ДОСЛІДЖЕНЬ ТА ТЕХНОЛОГІЙ В ПРОМИСЛОВОСТІ INNOVATIVE TECHNOLOGIES AND SCIENTIFIC SOLUTIONS FOR INDUSTRIES № 1 (1) 2017 No. 1 (1) 2017 Щоквартальний науковий журнал Quarterly scientific journal Харків 2017 Kharkiv 2017 No. 1 (1), 2017 Затверджений до друку Вченою Радою Харківського національного університету радіоелектроніки (Протокол № 13 від 01 вересня 2017 р.). Свідоцтво про державну реєстрацію журналу Серія КВ № 22696-12596Р від 04.05.2017 р. За достовірність викладених фактів, цитат та інших відомостей відповідальність несе автор. © Харківський національний університет радіоелектроніки, Державне підприємство "Південний державний проектно-конструкторський та науково-дослідний інститут авіаційної промисловості" EDITORIAL BOARD Editor in Chief Semenets Valery, Dr. Sc. (Engineering), Professor, Ukraine Deputy Editor in Chief Ruban Igor, Dr. Sc. (Engineering), Professor, Ukraine Deputy Editor in Chief Momot Tetiana, Dr. Sc. (Economics), Professor, Ukraine Assistant Editor Kovalenko Andrey, Ph. D. (Engineering Sciences), Associate Professor, Ukraine Editorial Board Members: Artiukh Roman, Ph. D. (Engineering Sciences) (Ukraine); Akhmetov Bahidzhan Srazhatdinovich, Dr. Sc. (Engineering), Professor (Kazakhstan); Bayramov Azad oglu Agalar, Dr. Sc. (Physical and Mathematical), Professor (Azerbaijan); Bezkorovainyi Volodymyr, Dr. Sc. (Engineering), Professor (Ukraine); Zaitseva Elena, Dr. Sc. (Engineering), Professor (Slovak Republic); Karpіnski Nicholas, Dr. Sc., Professor (Poland); Kononenko Igor, Dr. Sc. (Engineering), Professor (Ukraine); Kosenko Viktor, Ph. D. (Engineering Sciences), Associate Professor (Ukraine); Kostin Yuri, Dr. Sc. (Economics), Professor (Ukraine); Kuchuk Heorhii, Dr. Sc. (Engineering), Professor (Ukraine); Lepeyko Tetyana, Dr. Sc. (Economics), Professor (Ukraine); Malyeyeva Olga, Dr. Sc. (Engineering), Professor (Ukraine); Nazarova Galina, Dr. Sc. (Economics), Professor (Ukraine); Nevliudov Igor, Dr. Sc. (Engineering), Professor (Ukraine); Pushkar Olexandr, Dr. Sc. (Economics), Professor (Ukraine); Ramazanov Sultan, Dr. Sc. (Engineering), Dr. Sc. (Economics), Professor (Ukraine); Savchenko Olga, Ph. D. (Philosophic Sciences), Associate Professor (Ukraine); Sokolova Lyudmila, Dr. Sc. (Economics), Professor (Ukraine); Timofeyev Volodymyr, Dr. Sc. (Engineering), Professor (Ukraine); Todorov Kiril, Dr. Sc. (Economics), Professor (Bulgaria); Chumachenko Igor, Dr. Sc. Engineering), Professor (Ukraine); Chukhray Nataliya, Dr. Sc. (Economics), Professor (Ukraine); Iastremska Olena, Dr. Sc. (Economics), Professor (Ukraine). ESTABLISHERS Kharkiv National University of Radio Electronics, State Enterprise "National Design & Research Institute of Aerospace Industries" EDITORIAL OFFICE ADDRESS: Ukraine, 61166, Kharkiv, Nauka Ave, 14 Phone: +38 (057) 704-10-51 Information site: http://itssi-journal.com E-mail of the editorial board: journal.itssi@gmail.com РЕДАКЦІЙНА КОЛЕГІЯ Головний редактор Семенець Валерій Васильович, д-р. техн. наук, професор Заступник головного редактора Рубан Ігор Вікторович, д-р. техн. наук, професор Заступник головного редактора Момот Тетяна Валеріївна, д-р. екон. наук, професор Відповідальний секретар Коваленко Андрій Анатолійович, канд. техн. наук, доцент Члени редколегії: Артюх Роман Володимирович, канд. техн. наук; Ахмєтов Бахиджан Сражатдінович, д-р. техн. наук, професор (Казахстан); Байрамов Азад Агалар огли, д-р. фіз.-мат. наук, професор (Азербайджан); Безкоровайний Володимир Валентинович, д-р. техн. наук, професор; Зайцева Єлєна, д-р. техн. наук, професор (Словаччина); Карпінські Міколай, доктор наук, професор (Польща) Кононенко Ігор Володимирович, д-р. техн. наук, професор; Косенко Віктор Васильович, канд. техн. наук, доцент; Костін Юрій Дмитрович, д-р. екон. наук, професор; Кучук Георгій Анатольович, д-р. техн. наук, професор; Лепейко Тетяна Іванівна, д-р. екон. наук, професор; Малєєва Ольга Володимирівна, д-р. техн. наук, професор; Назарова Галина Валентинівна, д-р. екон. наук, професор; Невлюдов Ігор Шакирович, д-р. техн. наук, професор; Пушкар Олександр Іванович, д-р. екон. наук, професор; Рамазанов Султанахмед Курбанович, д-р. техн. наук, д-р. екон. наук, професор; Савченко Ольга Олександрівна, канд. філос. наук, доцент; Соколова Людмила Василівна, д-р. екон. наук, професор; Тімофєєв Володимир Олександрович, д-р. техн. наук, професор; Тодоров Кирил, д-р. екон. наук, професор (Болгарія); Чумаченко Ігор Володимирович, д-р. техн. наук, професор; Чухрай Наталія Іванівна, д-р. екон. наук, професор; Ястремська Олена Миколаївна, д-р. екон. наук, професор. ЗАСНОВНИКИ Харківський національний університет радіоелектроніки, Державне підприємство «Південний державний проектно-конструкторський та науково-дослідний інститут авіаційної промисловості» АДРЕСА РЕДАКЦІЇ: Україна, 61166, м. Харків, проспект Науки, 14 Телефон: +38 (057) 704-10-51 Інформаційний сайт: http://itssi-journal.com E-mail редколегії: journal.itssi@gmail.com javascript:openRTWindow('http://pm.khpi.edu.ua/about/editorialTeamBio/56305') http://itssi-journal.com/ mailto:journal.itssi@gmail.com http://itssi-journal.com/ mailto:journal.itssi@gmail.com № 1 (1), 2017 5 Семенець В. В. Вступне слово 6 Бабенко В. О., Алісейко О. В., Кочуєва З. А. Задача мінімаксного адаптивного управління інноваційними процесами на підприємстві з урахуванням ризиків (eng.) 14 Безкоровайний В. В., Березовський Г. В. Оцінка властивостей технологічних систем із використанням нечітких множин (eng.) 21 Гурин В. М., Персіянова О. Ю. Загальні принципи побудови моделі формування і функціонування неоднорідних команд для управління проектами (eng.) 28 Діденко Є. В. Модель оцінки ризиків виробничого травматизму 36 Ковалев В. Д., Шелковой А. Н., Клочко А. А. Технологические решения стационарной задачи рабочих жидкостей повышения долговечности зубчатых колес главных приводов тяжелых токарных станков 46 Косенко В. В. Принципи і структура методології ризик-адаптивного управління параметрами інформаційно- телекомунікаційних мереж систем критичного застосування (eng.) 53 Косенко Н. В., Кадикова І. М., Артюх Р. В. Формалізація завдання формування команди проекту на основі теорії корисності (eng.) 58 Мартиненко О. С., Гусєва Ю. Ю., Чумаченко І. В. Метод освоєних вимог для моніторингу виконання проекту (eng.) 64 Носова Я. В., Аврунін О. Г., Семенець В. В. Біотехнічна система для комплексної ольфактометричної діагностики (eng.) 69 Парфененко Ю. В. Модель підтримки прийняття рішень при управлінні централізованим теплозабезпеченням на стороні споживача (eng.) 75 Рубан І. В., Кучук Г. А., Коваленко А. А. Перерозподіл навантаження базових станцій в мобільних мережах зв'язку (eng.) 82 Стародубцев Н. Г., Фомовский Ф. В., Невлюдова В. В., Малая И. А., Демская Н. П. Математическое моделирование выбора информативных признаков для анализа состояния процессов жизненного цикла радиоэлектронных средств 90 Шулима О. В. Інформаційна технологія планування енергозабезпечення будівель з відновлювальними джерелами енергії 98 Аванесова Н. Е., Чупрін Є. С. Економічна безпека підприємства: сутнісна характеристика поняття (eng.) 103 Момот Т. В., Чжан Хаоюй, Момот Д. Т. Забезпечення кадрової безпеки підприємств промисловості в умовах конфліктного середовища (eng.) 110 Руженцев И. В., Луцкий С. В. Мера информации в экономических задачах 117 Солодовнік О. О., Докуніна К. І. Економічний механізм енергозбереження на підприємствах комунального господарства (eng.) 124 Чех Н. О., Вінник І. Ю. Регуляторне середовище в діяльності підприємницьких структур України (eng.) 130 Шаповал Г. М., Ващенко О. М. Вартісно-орієнтований підхід до управління оборотними активами корпоративних будівельних підприємств (eng.) 135 Алфавітний показчик No. 1 (1), 2017 5 Semenetz V. Introduction 6 Babenko V., Alisejko E., Kochuyeva Z. The task of minimax adaptive management of innovative processes at an enterprise with risk assessment 14 Beskorovainyi V., Berezovskyi G. Estimating the properties of technological systems based on fuzzy sets 21 Gurin V., Persiyanova E. General principles of building the model of development and operation of heterogeneous teams for project management 28 Didenko E. The model of estimating the risks of work-related fatalities, injuries, and illnesses 36 Kovalev V., Shelkovoy A., Klochko A. Technological solutions of the stationary task of working liquids for increasing the durability of main drive gears of high-power lathes 46 Kosenko V. Principles and structure of the methodology of risk-adaptive management of parameters of information and telecommunication networks of critical application systems 53 Kosenko N., Kadykova I., Artiukh R. Formalizing the problem of a project team bulding based on the utility theory 58 Martynenko O., Husieva Yu., Chumachenko I. The method of earned requirements for project monitoring 64 Nosova Y., Avrunin O., Semenets V. Biotechnical system for integrated olfactometry diagnostics 69 Parfenenko Yu. The model of decision support in centralized heating management on the consumer side 75 Ruban I., Kuchuk H., Kovalenko A. Redistribution of base stations load in mobile communication networks 82 Starodubtsev N., Fomovsky F., Nevliudova V., Malaja I., Demska N. Mathematical мodelling of selecting informative features for analyzing the life cycle processes of radio- electronic means 90 Shulyma О. Information technology of planning power supply of buildings with renewable energy sources 98 Avanesova N., Chuprin Y. Enterprise economic security: essential characteristics of the concept 103 Momot T., Zhang Haoyuy, Momot D. Ensuring personnel security of industrial enterprises under conditions of environmental conflicts 110 Ruzhentsev I., Lutsky S. Information measure in economic tasks 117 Solodovnik O., Dokunina K. Economic mechanism of energy saving at public utility companies 124 Chekh N., Vinnyk I. Regulatory environment of business activities in Ukraine 130 Shapoval G., Vashenko O. Value-based approach to managing current assets of corporate construction companies 135 Alphabetical index ISSN 2522-9818 (print) Сучасний стан наукових досліджень та технологій в промисловості. 2017. № 1 (1) 5 Шановні друзі! У сучасному світі значення промисловості складно переоцінити, адже саме цей сектор виробництва визначає рівень і якість життя людей. Промисловість становить невід’ємну частину світової та національної економіки, оскільки це сила, здатна привести в рух технологічні дослідження різного ступеня складності. Саме в цю сферу людської діяльності в першу чергу впроваджуються всі нові розробки і досягнення науково-технічного прогресу. Інноваційний розвиток все в більшій мірі перетворюється в найважливішу складову процесу розвитку виробництва, інновації перетворилися на визначальний фактор ефективного розвитку промисловості, а стійке економічне зростання і створення нових конкурентних переваг стає можливим тільки за умови переходу на інноваційну модель економічного розвитку. Специфіка і унікальність нинішнього етапу економічного розвитку полягає в тому, що промисловість дає напрямки і орієнтири для теоретичних досліджень і, навпаки, теоретичні дослідження інтегруються в промисловість. Таким чином, теоретичне знання виступає важливим чинником розвитку прикладної науки і подальшого впровадження її досягнень у виробництво. Тому сьогодні як ніколи важливим і цікавим стає обмін думками, обговорення оригінальних ідей та нетрадиційних підходів до вирішення теоретичних і практичних проблем. Саме цьому и присвячене нове наукове видання – науковий журнал "Сучасний стан наукових досліджень та технологій в промисловості". Спілкування на сторінках журналу дозволить науковцям та виробничникам різних напрямів обмінятися результатами досліджень, сприятиме успішному виконанню наукових робіт та зближенню науки та виробництва. Закликаю представників виробництва та науковців, що працюють як у технічній сфері, так й в галузі економіки, до активної участі в роботі нашого журналу та сподіваюсь на плідну співпрацю и бажаю успіхів, наснаги та творчої взаємоді. Dear colleague! In the modern world, it is impossible to overstress the importance of industry, because this sector of production determines the level and quality of human life. Industry is an integral part of the world and national economy because this is the power that can drive various technological researches. This is the sphere of human activity where all new developments and achievements of scientific and technological progress are implemented. Innovation development is increasingly becoming an important component of the process of production development, innovation has become a determining factor in the efficient development of industry, whilst sustainable economic growth and creation of new competitive advantages becomes possible only if the innovative model of economic development is actualized. The specific and unique feature of the current stage of economic development is emphasized by the fact that industry provides directions and benchmarks for theoretical studies, and vice versa, theoretical studies are integrated into industry. Thus, theoretical knowledge is an essential factor for the development of applied science and for the further introduction of its achievements in production Therefore, interchanging views, discussing original ideas and non-traditional approaches to solving theoretical and practical problems are becoming significant and interesting as never before. And these are the issues the scientific journal "Innovative technologies and scientific solution for industries" focuses on. Scientific discussions on the pages of the journal will allow scientists and practitioners working in various branches of science to exchange the results of their studies, promote successful research work and convergence of science and production. I call scientists and practitioners engaged both in engineering and economics for participation in the work of our journal and hope for effective cooperation and wish success, inspiration, and creative interaction. Головний редактор журналу, доктор технічних наук, професор Семенець Валерій Васильович Valery Semenetz Chief Editor, Doctor of Engineering, Professor ISSN 2522-9818 (print) Innovative technologies and scientific solutions for industries. 2017. No. 1 (1) 6 UDC 519.86 V. BABENKO, Е. ALISEJKO, Z. KOCHUYEVA THE TASK OF MINIMAX ADAPTIVE MANAGEMENT OF INNOVATIVE PROCESSES AT AN ENTERPRISE WITH RISK ASSESSMENT The subject matter of the article is a discrete dynamic system that consists of an object whose dynamics is described by a vector linear discrete recurrent relation and is affected by control parameters (managements) and uncontrolled parameters (the vector of risks or interference). It is supposed that the phase conditions of the object, management actions and the vector of risks of the considered dynamic system at any moment of time are constrained by given finite or convex polyhedral sets in corresponding finite-dimensional vector spaces. The objective of the article is to model a task of adaptive management of an enterprise innovative processes (EIP) under risks, which requires to complete the following tasks: to develop a software model of managing EIP under risks; to formalize the task of optimizing the EIP adaptive management and general paradigm of its solving as a guaranteed result based on minimax (optimizing a guaranteed result at a given final moment of time considering risks). In such a case, risks in the system of EIP management are thought of as factors that negatively or even catastrophically affect the results of the processes considered in it. In view of this, it is suggested to use the deterministic approach based of the methods of the theory of optimal management and dynamic optimization. The result of the research is a recurrent algorithm which reduces the initial multi-step task to the implementation of finite sequence of tasks of minimax software management of EIP. In turn, the implementation of each task is reduced to the implementation of finite sequence of only one-step optimizing operations as the tasks of linear convex mathematical and discrete optimization. The following conclusions are made: the suggested method makes it possible to work out efficient numerical procedures that enable computer modelling the dynamics of the target task, developing adaptive minimax management of EIP and obtaining an optimal guaranteed result. The results demonstrated in the work can be used for economic and mathematical modelling and solving other tasks of optimizing processes of data prediction and management under the lack of information and under risks as well as for developing corresponding software and hardware complexes to support efficient managerial decisions in practice. Keywords: innovative process, economic and mathematical model, risks, dynamical model, optimization, process of management, minimax adaptive management, guaranteed result. Introduction To achieve the set tasks under increasing competition among Ukrainian enterprises leads to an increase in the amount and complexity of production processes, analysis, planning, management, internal and external relations with suppliers, intermediaries, etc. Effective implementation of tasks linked with these processes is impossible without the appropriate economic and mathematical modelling of managing an enterprise innovative processes (EIP) as a computer information system. However, innovative activity in the process of dynamic development of production relations cannot be considered fully justified and adapted without using modern approaches of economic and mathematical modelling as an effective tool for theoretical processing and practical generalization of mechanisms and tools for managing the innovative activity of an enterprise that is a complex, open, capable to self-organization and self- development economic system with dynamically changing nondeterministic and conflicting characteristics. Modelling in EIP management provides for the solution of tasks of software and adaptive control. The result of the EIP software management is forecast values for a certain prospective period of time. But when an innovation process is introduced in each period of time, the model parameters can change (technological processes are compromised, financial indicators, types and suppliers of raw materials change and so on). In addition, in order to obtain a guaranteed result, the task of optimizing the EIP software management under risks takes into account risks that can lean to maximum losses. But risks that lead to a maximum damage can affect a real process. In this case, in order to take into account changes in the economic environment and the current state of an innovation process, the procedure for adapting the model to the current conditions should be specified on the basis of the results of the EIP software management. Thus, in order to take into account the instability of the innovation process, which is characterized by various "disturbances" as changes in the current state of the production process and the economic environment, an adaptive control model should be developed, which enables correcting and considering the dynamics of the main production characteristics within the innovation process It should be noted that the problem of economic and mathematical modelling of the EIP adaptive management under uncertainty and risks at enterprises have not been solved yet by scientific researches that deal with the problems of enterprise management, by various economic and mathematical models and techniques for finding out optimal solutions as well as by methods of business process modelling; so this problem remains a burning topic for researching. Analysis of literature sources A number of reputed Ukrainian and foreign scientists deal with the problems of economic and mathematical modelling of production and financial processes, among them are: N.N. Krasovsky [1], A.F. Shorikov [2], A.V. Lotov [3], A.I. Propoy [4], A.V. Ter-Krikorov [5] and others. However, some issues require further elaboration. So, at this stage, there are practically no economic and mathematical models that consider the specificity of production process dynamics, and take into account the impact of risks while managing innovative processes at enterprises and optimizing these processes. Research and solving the task of the EIP managing requires the development of a dynamic economic and © V. Babenko, Е. Alisejko, Z. Kochuyeva, 2017 ISSN 2522-9818 (print) Сучасний стан наукових досліджень та технологій в промисловості. 2017. № 1 (1) 7 mathematical model that takes into account control actions, uncontrolled parameters (risks, modelling errors, etc.) and the lack of information. At the same time, available approaches to solving similar problems are based mainly on static models and stochastic modelling apparatus to use which it is necessary to know the probabilistic characteristics of the model main parameters and special conditions for performing the process under consideration. It should be noted that very severe conditions, which usually are unachievable beforehand, are necessary for using the stochastic modelling apparatus. Economic and mathematical models of such problems are presented, for example, in [68]. This article continues the studies presented in [9], the concepts and notation that are introduced in it are used in this paper without additional explanations. Developing a generalized EIP management model under risks Let a multi-step dynamic system be considered for a given integer time, 0, {0,1, , } ( 0)T T T  ; this system consists of one controlled object – I (the subject of management as it is managed by player P), whose motion is described by a linear discrete recurrent vector equation:                  1 ,x t A t x t B t u t c t w t D t v t       00 .x x (1) Here 0, 1t T  , nx R is the phase vector of object I, which consists of 2n n m   coordinates for the model of the EIP management [2], that is 1 2 1 2( ) ( ( ), ( ),..., ( ), ( ), ( ),..., ( ),n mx t x t x t x t y t y t y t ( ), ( )) nZ t k t R , where, according to the notions in [2], 1 2( ) ( ( ), ( ),..., ( ))nx t x t x t x t  nR is the vector of amounts of production residues stored at the warehouses of the enterprise over the period of time t; 1 2( ) ( ( ), ( ),..., ( ))my t y t y t y t  mR is the vector of amounts of residues of production resources stored at the warehouses of the enterprise over the period of time t; ( )Z t is the enterprise total costs over the period of time t; ( )k t is the amount of available financial resources accumulated before the beginning of period t; ,n mN ; N is the set of all natural numbers; for kN , k R is K- dimensional Euclidean vector space of column vectors, even if they are written as a string for saving the space);         1 2, ,..., n nu t u t u t u t   R is the vector of innovative management of the intensity of production over the period of time t ( 0, 1)t T  , where each j-th coordinate ( )ju t is the value of the j-th production amount  1,j n constrained by the given restriction: 1( ) ( ) ( ) p N t u t t U t  U R ( : )p p n N , (2)   tNU t is the finite set of vectors for each 0, 1t T  , i.e. the finite set consisting of tN ( tN N ) of vectors in n R , defining all possible implementations of different management scenarios at the moment of time t; 1 2( ) ( ( ), ( ),..., ( )) ( )m m w t w t w t w t m m  R is the vector of intensity of replenishment of storage resources over the period of time t ( 0, 1)t T  , which depends on the permissible implementation of management 1( ) ( )u t tU and must meet the following specified limit: 1( ) ( ( )) ( ( )) m M t w t u t W u t  W R ( : )m m m N ; (3) ( ( )) tMW u t is the finite set of vectors for every moment of time 0 1t ,T  and management ( ) ( ) tNu t U t , i.e. the finite set consisting of tM (i) ( tM (i)N , 1, t i N ) vectors in space m R , defining all possible realizations of various scenarios of replenishment of the warehouse resources at the moment of time t. It is also assumed that for every 0, 1t T  , each permissible realization of the phase vector 1 2 1 2( ) ( ( ), ( ),..., ( ), ( ), ( ),..., ( ), ( ), ( )) ,n n mx t x t x t x t y t y t y t Z t k t R meets the following phase constraint 1 2 1 2 1( ) ( ( ), ( ),..., ( ), ( ), ( ),..., ( ), ( ), ( )) ( )n mx t x t x t x t y t y t y t Z t k t t  X = 0 ( ) 0, (0) 0, 1 ; ( ) 0, (0) , 1 ; ( ) 0, (0) 0; ( ) 0, Z(0) 0. j j i i i x t x j ,n y t y b i ,m k t k G G Z t                  (4) where G is the amount of financial resources of a bank loan intended for investments to the expansion of production within the initial period of management (when t = 0); 0G is the amount of own financial resources, deducted from net profit and directed to the expansion of production (when t = 0); ( ) ( ( ), ( ), ( )) q l rv t v t v t v t     R R R is the generalized vector of risks ( 1 2( ) ( ( ), ( ), , ( )) q qv t v t v t v t  R is the vector of risks describing possible unfavorable implementations of the vector of priori indefinite factors or the vector combining the modeling error of the considered process, affecting the release of a product unit over the period of time t; 1 2( ) ( ( ), ( ), , ( )) l lv t v t v t v t     R is the vector of risks that affects the state of a unit of available resources over the period of time t; 1 2( ) ( ( ), ( ), , ( )) r rv t v t v t v t     R is the vector of financial risks affecting a unit of total costs of the enterprise over the period of time t; , , q l r N ) which depends on the permissible implementation of the management 1( ) ( )u t tU during the EIP management over the period of time t ( 0, 1)t T  and must meet the following specified limit: 1( ) ( ( )) q v t u t V R ( : )q q q l r   N . (5) ISSN 2522-9818 (print) Innovative technologies and scientific solutions for industries. 2017. No. 1 (1) 8 Matrices ( )A t , ( )B t , ( )C t and ( )D t in the vector equation (1) for the economic and mathematical model describing the dynamics of the EIP management are real matrices of orders ( )n n , ( )n p , ( )n m and ( )n q respectively, and such ones that for all 0, 1t T  matrix ( )A t is nondegenerate, i.e. there exists an inverse matrix 1( )A t corresponding to it, and the rank of matrix ( )B t equals p (the dimension of vector ( )u t ). For the considered EIP management process [2], these matrices have the following specific form: 11 22 11 22 1 2 1 2 ( ) 0 0 0 0 0 0 0 0 ( ) 0 0 0 0 0 0 0 0 ( ) 0 0 0 0 0 0 0 0 ( ) 0 0 0 0 ( ) 0 0 0 0 ( ) 0 0 0 0 0 0 0 0 ( ) 0 0 ( ) ( ) ( ) ( ) ( ) ( ) 1 0 0 0 . 0 0 0 0 0 1 nn mm n m a t a t a t r t A t r t r t z t z t z t p t p t p t                                 ; 11 12 (1,( 1)) 1 21 22 (2,( 1)) 2 1 2 ( ,( 1)) 1 0 0 0 0 1 0 0 0 0 0 1 ( ) 0 0 1 0 0 0 0 1 n n n n m m m n mn b b b b B t b b b b b b b b                                             ; 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ( ) 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 C t                                 ; 11 12 1 11 12 1 1 2 11 12 1 21 22 2 1 2 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 q q n n nq l l m m ml r c c c c c c c c c c c c D t c c c c c c c c c                                                               . It should be pointed out that for all 0, 1t T  set 1( )tU in the restriction (2) is not empty and is the finite set consisting of tN ( tN N ) vectors of space p R ; for all 0, 1t T  and vectors 1( ) ( )u t tU , set 1( ( ))u tW in the restriction (3) is not empty and is the finite set consisting of tM (i) ( tM (i)N , 1, ti N ) vectors of space m R ; set 1( )tX , according to its definition (4) is not empty and is a convex, closed and bounded polyhedron (with the finite number of vertices) in space n R ; it is assumed that set 1( ( ))u tV in the restriction (5) is not empty and is a convex, closed, and bounded polyhedron (with the finite number of vertices) in space q R . Let us describe the information capabilities of player P in the process of minimax adaptive (according to the feedback principle) management of EIP for a discrete dynamical system (1) − (5). It is assumed that while managing EIP for any moment of time 1,T  and the corresponding integer time interval 0, 0,T  ( 0  ) up to the moment of time  player P has measured and stored the following values: 0(0)x x , i.e. the initial phase state of object I; 0, 1 ( ) { ( )} t u u t     , i.e. the history of the implementation ISSN 2522-9818 (print) Сучасний стан наукових досліджень та технологій в промисловості. 2017. № 1 (1) 9 of player P management over the period 0, ; 0, 1 ( ) { ( )} t w w t     , i.e. the history of implementation of the vector of the intensity of replenishment of the warehouse resources over the period of time 0, ; 0, 1 ( ) { ( )} t v v t     , i.e. the history of the implementation of the vector of risks over the period of time 0, . Equation (1) and constraints for it (2) – (5) are also known. The considered process of the EIP management is estimated by the value of the convex functional 1: n F R R defined at possible implementations of the phase vector ( ) nx T R of the system (1) − (5) at the final moment of time T. Then, for system (1) − (5) from the point of view of player P the goal of optimal adaptive management can be formulated as follows: for a given time interval 0,T player P should organize management 0, 1 ( ) { ( )} t T u u t     (for all 10, 1: ( ) ( )t T u t t  U ) according to the feedback principle (as the implementation of the minimax adaptive strategy [1], [3], [4] from the selected class of admissible adaptive strategies), using all the available information about this process in such a way that possible maximum value of the functional F defined on vector ( ) nx T R (where ( )x T is the implementation of the phase vector of object I at the moment of time T which corresponds to management ( )u  ) was minimal. In this case it is assumed that the worst (largest) values of functional F can be implemented with respect to possible unfavorable realizations 0, 1 ( ) { ( )} t T v v t     (for all 10, 1: ( ) ( ( ))t T v t u t  V of generalized risk vector; while the implementations 0, 1 ( ) { ( )} t T w w t     (for all 10, 1: ( ) ( ( ))t T w t u t  W ) of the vector of the intensity of the replenishment of the warehouse resources further P player’s goals, i.e. their selection (according to player P) is aimed at minimizing the functional F according to the strategy he has selected. Formalizing the task of optimization of the EIP adaptive management It should be noted that definitions and notations which were introduced in work [2] are straightly used in this section while formalizing and solving the task of the EIP minimax software management since the considered dynamic model (1) – (5) coincides with the model for this task in [2]. To assess the quality of the EIP management by player P under adaptive management in the dynamic system (1) − (5) over a time interval, a vector terminal functional (the process quality index) (1) (2) ( ) , , , ( , , , )r T T T T F F F      , F is introduced , 0,T T  similarly to the formalization described in work [2]. This functional is a collection of r convex functionals of ( ) ,T k  F : ˆ ( ) ( , ) ( , ; ( )) ( , ; ( ))T T u T u        G U W V 1 R ( 1, )k r such as. to implement the set ˆ( ( ), ( ), ( ), ( )) ( ) ( , ) ( , ; ( ))g u w v T T u          G U W ( , ; ( ))T u V , where ( )g   { , ( )}x   ˆ ( )G , their values are determined by the following relation: ( ) ( ) , T , ( ( ), ( ), ( ), ( )) ( ( ; ( ), ( ), ( ), ( )))k k T , T g u w v F x T x u w v            F ( ) ( ( )), 1, T k = F x T k r ,  , (6) where ( ) 1:k n ,T F  R R is the convex functional for each 1,k r ; , ( ) ( ; ( ), ( ), ( ), ( )) T x T x T x u w v      . On the basis of the vector functional (1) (2) ( ) , , , , ( , , , )r T T T T F F F     F introduced by relation (6), to assess the quality of the process of optimizing the considered EIP management, the scalar objective function ( ( ), ( ), ( ), ( )) T g u w v      , F is introduced; its values for all admissible implementations of ˆ( ( ), ( ), ( ), ( )) ( ) ( , ) ( , ; ( ))g u w v T T u          G U W ( , ; ( ))T u V over the time interval ,Т , where ( )g   ˆ{ , ( )} ( )x  G , , 1 ( ) { ( )} (0, ) t T u u t T     U , , 1 , ( ) { ( )} ( , ; ( )), ( ) { ( )} (0, ; ( )), t T t T w w t T u v v t T u             W V are determined according to the following relation: ( ) , 1 ( ( ), ( ), ( ), ( )) ( ( ), ( ), ( ), ( )) r k kT T ki g u w v F g u w v                , F ( ) , 1 ( ( ; ( ), ( ), ( ), ( )) r k k ,T T k F x T x τ u w v           ( ) 1 ( ( )) ( ( )). r i k ,T k F x T x T       F 1 1 : 0, 1 r k k k k ,r        , (7) where , ( ) ( ; ( ), ( ), ( ), ( )) T x T x T x u w v      , and F is the convex functional introduced earlier. The objective function (functional) ( ( ), ( ), ( ), ( )) T g u w v      , F is a convex scalar convolution of the vector functional (1) (2) ( ) , , , , ( , , , )r T T T T F F F     F , i.e. it is formed according to the method of scalarization of vector objective functions (e.g.[6]), with nonnegative weighting factors k , 1k ,r , which can be determined, for example, by expertise or on the basis of statistical information on the history of the implementation of the main parameters of the considered EIP management. Assume that player P having selected management 1( ) ( ), 0, 1u t t t T  U in the dynamic system (1) − (5) for a given period of time 0, ( 0)T T  is under agreed awareness conditions. Then, on the basis of stated above, from the position of player P his goal in the task of the ISSN 2522-9818 (print) Innovative technologies and scientific solutions for industries. 2017. No. 1 (1) 10 EIP minimax adaptive management for the dynamic system (1) − (5) can be formulated as follows. Player Р over the period of time 0,T is supposed to arrange the selection of his management 0, 1 ( ) { ( )} t T u u t     (for all х 10, 1: ( ) ( )t T u t t  U ) of object I in the adaptive mode (according to the feedback principle) knowing his   position ˆ( ) { , ( )} ( )g t t x t t G at every moment of time 0, 1t T  so that functional 0,T F , determined by relation (7) when 0  has the smallest possible value when the implementation of the EIP management is completed. It should be taken into consideration that the worst values of the vector function ( ) (0, ; ( ))v T u  V can be realized, i.e. maximizing the given functional, and the realization of the vector function ( ) (0, ; ( ))w T u  W furthers player's P goal. Then, using the above arguments and similarly to [3], [4] the achievement of this goal of player P can be formalized in the following way. The permissible strategy of the EIP adaptive management aU of player Р for a discrete dynamical system (1) − (5) over the time interval 0,T can be mapping 1 ˆ: ( ) ( )a  U G U that assigns set 1( ( )) ( )a g t U U of 1( ) ( )u t U management of player Р to each moment of time 0, 1T   and to possible realization of   position 0 ˆ( ) { , ( )} ( ) ( (0) )g x g g     G . The set of all admissible management strategies for player P for the process considered through a  U is denoted. Further, the group of motions of object I over the time interval 0,T , corresponding to the equation of motion (1), the initial position P 0 0 0 ˆ{0, }g x G of player Р, the permissible strategy ( ( ))a a ag t  U U U , 0, 1t T  , ˆ( ) { , ( )} ( )g t t x t t  G , and the admissible software implementation of the intensity of the replenishment of the warehouse resources ( ) (0, ; ( ))aw T u  W , where , 1 ( ) { ( )} (0, )a a t T u u t T     U any admissible management of player P over the time interval 0,T generated by strategy aU will be called as follows: * * * 0( ;0, , , , ( )) { ( ): ( ) (0, ), ( ) (0, ),nT g w x x T u Ta        X U S U *( ) (0, ; ( )),v T u   V * * * 00, 0, , ( ) ( ; , ( ), ( ), ( )),t t tT t T x t x t x u w v      0 0 ˆ( ) { , ( )} (0, , , ( ), ( )) ( ), (0) ,t tg t t x t g t u w t g g         G G 0, 1 ( ) { ( )} , 0, 1, ( ) ( ( )),t at u u t T u t g t            U 0, 1 0, 1 ( ) { ( )} , ( ) { ( )} }t tt t w w v v              . (8) Then the following nonlinear multi-step problem of the EIP minimax adaptive management for the dynamical system (1) − (5) can be formulated. Task 1. For the given time interval 0,T (Т>0) and initial position 0 0 0 ˆ{0, }g x G of player Р in the discrete dynamical system (1) − (5), the strategy of the EIP minimax adaptive management ( ) ( ) ˆ( ( )) , ( ) { , ( )} ( ),e e a a ag t g t t x t t   U U U G 0, 1t T  , 0( (0) )g g , should be found, which meets the relation ( ) ( ) 00, 0,( ) ( ) ( ) (0, ; ( )) ( ) (0, ; ( )) ( , , ( ), ( ))e,a e aT Te e w T u v T u a a min max g w v           W V F F U 00, ( ) (0, ; ( )) ( ) (0, ; ( )) ( , , ( ), ( )) =aT* w T u v T u a aa a min min max g w v          W VU U F U ( ) (0, ; ( )) ( ) ( ;0, , , , ( )) 0 ( ( )) = * w T u x T T T g w a aa a min min max x T       W X UU U F ( , ) 0 ( ) ( ) ( ) ( ;0, , , , ( )) 0 ( ( )) = (0, , )e a e e x T T T g w a max x T c T g    F X U F , (9) as the realization of the finite sequence of only one-step operations. Here functional 0,T F is determined according to the relation (7); , 1 ( ) { ( )} (0, )a a t T u u t T     U is any admissible management of player P over the time interval 0,T generated by strategy aU ; ( ) ( ) , 1 ( ) { ( )} (0, )e e a a t T u u t T     U is any admissible management of player Р over the time interval 0,T generated by the strategy ( )e aU . The number ( , ) 0(0, , )e ac T g F = ( ) 0 e,a T, F will be called the optimal guaranteed (minimax) result of the minimax adaptive management of P player's EIP over the time interval 0,T for the discrete dynamical system (1) − (5) concerning its initial position 0g and functional 0,T F . It should be noted that the above conditions for the parameters of the system (1) − (5) and the results of works [3], [4] demonstrate that there is the solution for this task. Further, for any realizations of management ( ) ( ) 0,T 1 ( ) { ( )}e e a a t u u t     , ( ) ( ) ( )0, 1: ( ) ( ( ))e e e a at T u t w t   U of player Р generated by strategy ( ) ,e a a U U for vector functions ( ) ( )( ) (0, ; ( ))e e a aw T u  W and ( )( ) (0, ; ( ))e a av T u  V , for the group of motions ( ) ( ) ( ) 00, ( ) ( ; , ( ), ( ), ( ))e e e a a aT x x x u w v       ( ) 0( ;0, , , , ( ))e a aT g w X U corresponding to it, on the basis of relations (6) − (9), it is not difficult to show the validity of the following equation: ( ) ( ) 00, ( , ( ), ( ), ( ))e e a a aT g u w v   F ( ) ( ) ( ) 00, ( ( ; , ( ), ( ), ( )) ( ( ))e e e a a a aT x T x u w v x T    = F F ( , ) 0(0, , )e ac T g F = ( ) 0 e,a T  , F ( ) 0 e, T, F = ( ) 0(0, , )ec T g F , (10) ISSN 2522-9818 (print) Сучасний стан наукових досліджень та технологій в промисловості. 2017. № 1 (1) 11 where 0 0 0 ˆ{0, }g x G ; ( ) 0(0, , )ec T g F is the optimal guaranteed (minimax) result of solving the problem of the EIP minimax software management. It should be noted that the relations (6) demonstrate that the result of solving task 1 can only improve the result of solving the task of the EIP minimax software management, i.e. the EIP minimax adaptive management is more promising in comparison with the minimax software management for the considered process. Thus, in this section we formalize the task of the EIP minimax adaptive management for the dynamical system (1) − (5). It should be noted that task 1 is the main one in this chapter, but its formalization and solution are based on the task of the EIP minimax software management [2]. General pattern for solving task 1 The general pattern for solving task 1 on the basis of the results of [2] − [4] is suggested. Using the solution of the task of the EIP minimax software management considered in the previous chapter, for all the moments of time 0, 1T   and all  -positions ( ) ( ) ˆ( ) { , ( )} ( )e eg x    G ( ) 0 0 0 ˆ( (0) {0, } )eg g x  G of player Р, where ( ) ( ) ( ) 00, ( ) ( ; , ( ), ( ), ( ))e e e T x x x u w v     , ( ) ( ) ( )( ) (0, , (0))e e eu T g  F U , ( ) ( )ew  ( )(0, ; ( ) )eT u W the following sets can be developed: ( ) ( ) ( ) ( ) ( ) ( ) 1( ( )) { ( ): ( ) ( ), ( ) ( ),e e e e e eg u u u u       U U ( ) ( ) ( )( ) ( , , ( ))e e eu T g   F U , 0, 1}T   , (11) where ( ) ( )( , , ( ))e eT g  F U is the set of minimax software managements developed from the solution of the corresponding task of the EIP minimax software management considered in the previous chapter. Then the management strategy ( ) ( ) ( ( ))e e a a ag   U U U , 0, 1T   , 0 ˆ( ) ( ) ( (0) )g g g  G of player Р for the considered EIP minimax adaptive management in a discrete dynamical system over the time interval 0,T from all admissible management strategies a  U is determined; it is formally described by the following relations: For all 0, 1T   and  - positions ( ) ( ) ( ) ( ) 0( ) { , ( )} (0, , , ( ), ( ))e e e eg t x g u w      G ( ) 0( (0) )eg g assume that ( ) ( ) ( ) ( ) 1( ( )) ( ( )) ( )e e e e a g g   U U U . (12) For all 0,T 1   and  -positions ( ) ( ) 0 ˆ{ ( )\ (0, , , ( ), ( ))} ( (0) ) 0 e eg u w g g      G G assume that ( ) 1( ( )) ( )e a g   U U , (13) where ( ) ( ) 0, 1 ( ) { ( )}e e t u u t     , ( ) ( ) 0( ) (0, , )e eu T g  F U ; ( ) ( ) 0, 1 ( ) { ( )}e e t w w t     , ( ) ( )( ) (0, ; ( ) )e ew T u  W . Let ( ) ( ) 0, 1 ( ) { ( )} (0, )e e a a t T u u t T     U be the realization of the management of player P over the time interval 0,T , which is developed as a result of using strategy ( )e a a U U over this interval of time, and the realization of vector functions ( ) ( )( ) (0, ; ( ))e e a aw T u  W and ( ) (0, )v T V . Then, for ( 1)T  -position ( ) ( ) ˆ( 1) { 1, ( 1)} ( 1)e e a ag T T x T T     G of player Р (here ( ) ( 1)e ax T  )= ( ) 00, ( ; , ( ),e aT x x u  ( ) ( ), ( ))e a aw v  , which corresponds to these realizations, the following relations are true ( , ) ( ) ( ) 0 00, (0, , ) ( , ( ), ( ), ( )) =e a e e a aT c T g g u w v    F F ( ) ( ) ( ) ( ) 1, ( ( 1), ( 1), ( 1), ( 1)) =e e e e a a aT - T g T u T w T v T    F ( ) ( ) ( ) 1,( ) ( 1) ( 1, ; ( )) ( ( 1), ( 1), ( 1), ( 1))e e e a a aT - Te v T T T u a max g T u T w T v T           V F ( ) ( ) ( , ) ( ) 0 0 00,( ) ( ) (0, ; ( )) ( , ( ), ( ), ( )) (0, , ) (0, , )e e e a e a aTe v T u a max g u w v c T g c T g          F F V F . (14) On the basis of the results of [2] − [4] and relations (11) − (14), the following statement, which is the main result of this paper, can be justified. Statement 1. For the given initial position 0 0 0 ˆ(0) {0, }g g x  G of player Р in the discrete dynamical system (1) − (5) the strategy of the EIP management ( )e a a U U over the period of time 0,T which is determined by relations (11) − (13), is the minimax adaptive management strategy for task 1, i.e. ( ) ( )e e a a a  U U U and number ( , ) 0(0, , )e ac T g F are the optimal guaranteed (minimax) result for this task, i.e. ( , ) 0(0, , )e ac T g F ( , ) 0(0, , )e ac T g F , which corresponds to the implementation of this strategy over the period of time 0,T for the considered EIP management, and both these elements are developed by implementing the finite sequence of only one step operations. Thus, for the organization of the EIP minimax adaptive management, i.e. the solution of task 1 in the ( )g   { , ( )}x   ( ) ( ) ( ) 1, ( ( 1), ( 1), ( 1), ( 1))e e e a a aT - T g T u T w T v T    = F ISSN 2522-9818 (print) Innovative technologies and scientific solutions for industries. 2017. No. 1 (1) 12 selected class of admissible adaptive management strategies, the recurrent algorithm that reduces the initial multistage problem to the realization of the finite sequence of tasks of the EIP minimax software management is suggested. In turn, the solution of each of these tasks is reduced to the realization of the finite sequence of only one-step optimization operations as solving the tasks of linear and convex mathematical programming, as well as discrete optimization (e.g. [3], [4]). Then it can be stated that the solution of the considered task 1 is reduced to the solution of the finite sequence of tasks of linear convex mathematical programming and discrete optimization. References 1. Krasovskiy, N. (1968), Theory of motion control [Teorija upravlenija dvizheniem], Moscow, Nauka, 476 p. 2. Shorikov, A. (2005), "Algorithm for solving the problem of optimal terminal control in linear discrete dynamical systems" ["Algoritm reshenija zadachi optimal'nogo terminal'nogo upravlenija v linejnyh diskretnyh dinamicheskih sistemah"], Information Technologies in Economics: Theory, Models and Methods: a collection of scientific works, Publishing house Ural State University of Economics, Ekaterinburg, pp. 119-138. 3. Lotov, A. (1984), Introduction to economic and mathematical modeling [Vvedenie v jekonomiko-matematicheskoe modelirovanie], Nauka, Home Edition of Physical and Mathematical Literature, Moscow, 332 p. 4. Propoy, A. (1973), Elements of the theory of optimal discrete processes [Jelementy teorii optimal'nyh diskretnyh processov], Science, Home Edition of Physical and Mathematical Literature, Moscow, 368 p. 5. Ter-Krikorov, A. (1977), Optimal control and mathematical economics [Optimal'noe upravlenie i matematicheskaja jekonomika], Nauka, Moscow, 216 p. 6. Babenko, V. (2014), Management of innovation processes of processing enterprises of agrarian and industrial complex (mathematical modeling and information technologies): monograph [Upravlinnya innovatsiynymy protsesamy pererobnykh pidpryyemstv APK (matematychne modelyuvannya ta informatsiyni tekhnolohiyi): monohrafiya], V. Dokuchaev Kharkiv National Agrarian University, Machulin, 380 p. 7. Babenko V. O. (2013), Formation of an economic-mathematical model of dynamics of process of management by innovative technologies at the enterprises of AIC [Formirovaniye ekonomiko-matematicheskoy modeli dinamiki protsessa upravleniya innovatsionnymi tekhnologiyami na predpriyatiyakh APK], Scientific Economic Journal "Actual Problems of Economics", No. 1 (139), University "National Academy of Management", Kyiv, pp. 182-186. EID: 2-s2.0-84929991982 8. Babenko V. O. (2017), Modelling of factors affecting innovational agricultural activity of enterprises AIC in Ukraine, Scientific bulletin Polesie, No. 1 (9), pp. 115-121. DOI: 10.25140/2410-9576-2017-2-2(10) 9. Shorikov, A. F., Babenko V. A. (2014), Optimization of the guaranteed result in the dynamic control model of innovative process on the enterprise, Scientific Information and Analytical Economic Journal "The Economy of the Region" ["Optimizatsiya garantirovannogo rezul'tata v dinamicheskoy modeli upravleniya innovatsionnym protsessom na predpriyatii", Nauchnyy informatsionno-analiticheskiy ekonomicheskiy zhurnal "Ekonomika regiona"], No. 1 (37). Institute of Economics, Ural Branch of RAS, pp. 196–202. EID: 2-s2.0-84979807246 10. Johnson, M. (2010), Seizing the white space, Business Model Innovation for growth and renewal, Harvard Business Press: Boston, Massachussetts. Receive 02.06.2017 Відомості про авторів / Сведения об авторах / About the Authors Бабенко Віталіна Олексіївна – доктор економічних наук, кандидат технічних наук, доцент, Харківський національний університет імені В.Н. Каразіна, професор кафедри міжнародного бізнесу та економічної теорії, м. Харків, Україна; e-mail: vitalinababenko@karazin.ua; ORCID: 0000-0002-4816-4579. Бабенко Виталина Алексеевна – доктор экономических наук, кандидат технических наук, доцент, Харьковский национальный университет имени В.Н. Каразина, профессор кафедры международного бизнеса и экономической теории, г. Харьков, Украина; e-mail: vitalinababenko@karazin.ua; ORCID: 0000-0002-4816-4579. Babenko Vitalina – Doctor of Sciences (Economics), PhD. (Engineering Sciences), Associate professor, V.N. Karazin Kharkiv National University, Professor of the Department of International Business and Economic Theory, Kharkiv, Ukraine; e-mail: vitalinababenko@karazin.ua; ORCID: 0000-0002-4816-4579. Алісейко Олена В’ячеславівна – кандидат технічних наук, старший науковий співробітник, доцент, Харківський торгівельно-економічний інститут Київського національного торгівельно-економічного університету, доцент кафедри вищої математики та інформатики, м. Харків, Україна; e-mail: alisejkoev@gmail.com, ORCID: 0000-0001-6917-164X. Алисейко Елена Вячеславовна – кандидат технических наук, старший научный сотрудник, доцент, Харьковский торгово-экономический институт Киевского национального торгово-экономического университета, доцент кафедры высшей математики и информатики, г. Харьков, Украина; e-mail: alisejkoev@gmail.com, ORCID: 0000-0001-6917-164X. Alisejko Elena – Candidate of Sciences (Engineering), Assistant Professor, Kharkiv Trade and Economic Institute of Kyiv National Trade and Economic University, Associate Professor of the Department of Higher Mathematics and Informatics, Kharkiv, Ukraine; e-mail: alisejkoev@gmail.com, ORCID: 0000-0001-6917-164X. Кочуєва Зоя Анатоліївна – кандидат технічних наук, Національний технічний університет "Харківський політехнічний інститут", доцент кафедри інтелектуальних комп’ютерних систем, м. Харків, Україна; e-mail: kochueva@kochuev.com; ORCID: 0000-0002-4300-3370. Кочуева Зоя Анатольевна – кандидат технических наук, Национальный технический университет "Харьковский политехнический институт", доцент кафедры интеллектуальных компьютерных систем, г. Харьков, Украина; e-mail: kochueva@kochuev.com; ORCID: 0000-0002-4300-3370. Kochuyeva Zoya – Ph. D. (Engineering Sciences), National Technical University "Kharkiv Polytechnic Institute", Associate Professor at the Department of Intellectual Computer Systems, Kharkiv, Ukraine; e-mail: kochueva@kochuev.com; ORCID: 0000-0002-4300-3370. mailto:vitalinababenko@karazin.ua mailto:vitalinababenko@karazin.ua mailto:vitalinababenko@karazin.ua mailto:alisejkoev@gmail.com mailto:alisejkoev@gmail.com mailto:alisejkoev@gmail.com mailto:kochueva@kochuev.com mailto:kochueva@kochuev.com mailto:kochueva@kochuev.com ISSN 2522-9818 (print) Сучасний стан наукових досліджень та технологій в промисловості. 2017. № 1 (1) 13 ЗАДАЧА МІНІМАКСНОГО АДАПТИВНОГО УПРАВЛІННЯ ІННОВАЦІЙНИМИ ПРОЦЕСАМИ НА ПІДПРИЄМСТВІ З УРАХУВАННЯМ РИЗИКІВ Предметом дослідження статті є дискретна динамічна система, що складається з об’єкта, динаміка якого описується векторним лінійним дискретним рекурентним співвідношенням і схильна до впливу керованих параметрів (управлінь) і неконтрольованого параметра (вектору ризиків або перешкоди). Передбачається, що фазові стани об’єкта, керуючі впливи та вектор ризиків динамічної системи, що розглядається, в кожен момент часу обмежені заданими кінцевими або опуклими багатогранними множинами в відповідних скінченновимірних векторних просторах. Ціллю статті є моделювання адаптивного управління інноваційними процесами підприємства (ІПП) при наявності ризиків що вимагає виконання наступних завдань: формування моделі програмного управління ІПП при наявності ризиків; формалізація задачі оптимізації адаптивного управління ІПП та загальної схеми її вирішення у вигляді гарантованого результату на основі мінімакса (оптимізації гарантованого результату) на заданий фінальний момент часу з урахуванням наявності ризиків. При цьому під ризиками в системі управління ІПП будемо розуміти фактори, які впливають негативно або катастрофічно на результати розглянутих в ній процесів. З цією метою пропонується використовувати детермінований підхід на основі методів теорії оптимального управління та динамічної оптимізації. Результатом дослідження є рекурентний алгоритм, який зводить вихідне багатокрокове завдання до реалізації кінцевої послідовності завдань мінімаксного програмного управління ІПП. У свою чергу, рішення кожного з таких завдань зводиться до реалізації кінцевої послідовності тільки однокрокових оптимізаційних операцій в формі вирішення завдань лінійного опуклого математичного програмування та дискретної оптимізації. Висновки: пропонований метод дає можливість розробляти ефективні чисельні процедури, що дозволяють реалізувати комп’ютерне моделювання динаміки розглянутої задачі, сформувати адаптивне мінімаксне управління ІПП та отримати оптимальний гарантований результат. Представлені в роботі результати можуть бути використані для економіко- математичного моделювання та вирішення інших завдань оптимізації процесів прогнозування даних і управління в умовах дефіциту інформації та наявності ризиків, а також для розробки відповідних програмно-технічних комплексів для підтримки прийняття ефективних управлінських рішень на практиці. Ключові слова: інноваційний процес, економіко-математична модель, ризики, динамічна модель, оптимізація, процес управління, мінімаксний адаптивний менеджмент, гарантований результат. ЗАДАЧА МИНИМАКСНОГО АДАПТИВНОГО УПРАВЛЕНИЯ ИННОВАЦИОННЫМИ ПРОЦЕССАМИ НА ПРЕДПРИЯТИИ С УЧЕТОМ РИСКОВ Предметом исследования статьи является дискретная динамическая система, состоящая из объекта, динамика которого описывается векторным линейным дискретным рекуррентным соотношением и подвержена влиянию управляемых параметров (управлений) и неконтролируемого параметра (вектора рисков или помехи). Предполагается, что фазовые состояния объекта, управляющие воздействия и вектор рисков рассматриваемой динамической системы в каждый момент времени стеснены заданными конечными или выпуклыми многогранными множествами в соответствующих конечномерных векторных пространствах. Целью статьи является моделирование задачи адаптивного управления инновационными процессами предприятия (ИПП) при наличии рисков, что требует выполнение следующих задач: формирование модели программного управления ИПП при наличии рисков; формализация задачи оптимизации адаптивного управления ИПП и общей схемы ее решения в виде гарантированного результата на основе минимакса (оптимизации гарантированного результата на заданный финальный момент времени с учетом наличия рисков. При этом под рисками в системе управления ИПП будем понимать факторы, которые влияют негативно или катастрофически на результаты рассматриваемых в ней процессов. С этой целью предлагается использовать детерминированный подход на основе методов теории оптимального управления и динамической оптимизации. Результатом исследования является рекуррентный алгоритм, который сводит исходную многошаговую задачу к реализации конечной последовательности задач минимаксного программного управления ИПП. В свою очередь, решение каждой из таких задач сводится к реализации конечной последовательности только одношаговых оптимизационных операций в форме решения задач линейного выпуклого математического программирования и дискретной оптимизации. Выводы: предлагаемый метод дает возможность разрабатывать эффективные численные процедуры, позволяющие реализовать компьютерное моделирование динамики рассматриваемой задачи, сформировать адаптивное минимаксное управление ИПП и получить оптимальный гарантированный результат. Представленные в работе результаты могут быть использованы для экономико-математического моделирования и решения других задач оптимизации процессов прогнозирования данных и управления в условиях дефицита информации и наличия рисков, а также для разработки соответствующих программно-технических комплексов для поддержки принятия эффективных управленческих решений на практике. Ключевые слова: инновационный процесс, экономико-математическая модель, риски, динамическая модель, оптимизация, процесс управления, минимаксное адаптивное управление, гарантированный результат. Бібліографічні описи / Библиографические описания / Bibliographic descriptions Бабенко В. О., Алісейко О. В., Кочуєва З. А. Задача мінімаксного адаптивного управління інноваційними процесами на підприємстві з урахуванням ризиків. Сучасний стан наукових досліджень та технологій в промисловості. Харків. 2017. № 1 (1). С. 6–13. Бабенко В. А., Алисейко Е. В., Кочуева З. А. Задача минимаксного адаптивного управления инновационными процессами на предприятии с учетом рисков. Сучасний стан наукових досліджень та технологій в промисловості. Харків. 2017. № 1 (1). С. 6–13. Babenko V., Alisejko E., Kochuyeva Z. The task of minimax adaptive management of innovative processes at an enterprise with risk assessment. Innovative technologies and scientific solutions for industries. Kharkiv. 2017. No. 1 (1). P. 6–13. ISSN 2522-9818 (print) Innovative technologies and scientific solutions for industries. 2017. No. 1 (1) 14 UDC 004.9 V. BESKOROVAINYI, G. BEREZOVSKYI ESTIMATING THE PROPERTIES OF TECHNOLOGICAL SYSTEMS BASED ON FUZZY SETS The subject of the research in the article is the process of evaluating the properties of technological systems at the stages of their design and reengineering. The goal – to improve the efficiency of procedures for multi-criteria evaluation of options for constructing technological systems using the apparatus of fuzzy sets. Objectives: to search for new or modification of known functions of belonging to fuzzy sets "the best variant of building a technological system" by particular criteria in the direction of reducing the complexity of procedures for calculating their values; perform a comparative analysis of the temporal complexity and accuracy of approximation of the preferences of the decision maker with the help of monotonic membership functions; give recommendations on the practical use of monotonous membership functions in decision support systems. Common scientific methods are used, such as: decision making, utility theory, fuzzy sets and identification. The following results are obtained. The article presents the model of preferences of the decision-maker developed by the authors for evaluating individual properties of technological systems using the membership function of fuzzy sets, which allows to realize both linear and nonlinear (convex, concave, S-shaped and Z-shaped) criteria. The carried out experimental research has revealed its advantages in terms of accuracy and time complexity in comparison with the functions of Gauss, Harrington, logistic function, gluing of power functions and their modifications. Methods are proposed that reduce the time complexity of procedures for calculating the values of membership functions. Conclusions. As a result of the analysis of known membership functions for fuzzy sets, it has been established that they do not adequately reflect the preferences of the decision-maker for the characteristics of systems close to extreme values and have a relatively high computational complexity. The proposed membership function and its calculation method make it possible to increase the adequacy of multifactorial estimation models and significantly reduce the time complexity of procedures for calculating its values. Practical use of the proposed membership function in the support systems for the adoption of design and management solutions will make it possible to obtain solutions of the problems of multifactor estimation and choice of a much larger dimension or with less expenditure of computing resources practically without loss of accuracy. Keywords: technological system, quality criteria, multicriteria optimization, fuzzy set, membership function. Introduction Modern technological systems are difficult complexes of functionally interconnected means of technological equipment, items of production and actors for conducting specified technological processes or operations under regulated production conditions. Their efficiency is largely determined by decisions made in the process of their designing or reengineering [1]. Processes of designing and reengineering of technological systems assume the solution of a set of interrelated problems of structural, topological and parametric optimization. These tasks belong to the NP- complex class and are solved in conditions of incomplete information according to a set of functional and cost indicators (particular criteria of efficiency). The best solution from a set of efficient ones can be selected by a decision-maker only in the simplest situations [2]. Otherwise, to automate the procedures for estimating project solutions, additional information about the utility of different values of formalized properties of decisions is necessary (particular criteria) [3]. Analysis of the current state of the problem The methodology of solving tasks of optimization of technological systems as well other anthropogenic systems is based on the theory of multicriteria optimization [46]. It is based on mathematical models and methods of utility theory. Two main approaches to estimating the quality of system variants are used, that is the ordinal and cardinal ones. Within the ordinal approach a decision-maker establishes the procedure "better-worse" on the subset  X x of admissible X X * or efficient variants CX X for developing the system; for example, a decision maker can determine the binary relations of strict preference SR ( X ) , the preference-indifference relation NSR ( X ) or equivalence ER ( X ) : ( ) { , : , , }S i j i j i jR X x x x x X x x    ; O S k l nR ( X ) : x x ... x ; (1) NS i j i j i jR ( X ) { x ,x : x ,x X , x x }    ; O NS k l nR ( X ) : x x ... x ; (2) E i j i j i jR ( X ) { x ,x : x ,x X , x x }    ; O E k l nR ( X ) : x x ... x . (3) Within the cardinal approach, preferences are determined by attributing some value  P x , that is interpreted as its utility or value, to each of the alternatives x X . The utility function that is common according to a set of indicators  P x determines the appropriate order (preferences of a decision maker)  R X (1), (2) or (3). The value of the argument ox X , corresponding to the maximum of general utility function  P x , corresponds to the best variant of the technological system development. The pattern for solving the task of multicriteria selection within the cardinal approach is presented as: 0S A Opt G x   , (4) © V. Beskorovainyi, G. Berezovskyi, 2017 ISSN 2522-9818 (print) Сучасний стан наукових досліджень та технологій в промисловості. 2017. № 1 (1) 15 where S is the situation of multicriteria selection; A is the axiomatics of multicriteria selection, which is a set of axioms that determine the pattern of compromise (the principle of ordering solutions); Opt G is compromise pattern  P x ); 0x is the best variant for developing a technological system [3]. A decision-maker or a group of experts determine the principle of ordering decisions A grounding on heuristic considerations. Currently, the apparatus of fuzzy sets theory is used to develop a scalar estimation on a set of partial criteria ( ), 1,ik x i m (where m is a number of partial criteria), representing the most important properties of the technological system [3, 7]. In this case, the fuzzy set adjective "the best variant of developing a technological system" is used as the general utility function P(x). The fuzzy set adjective "the best variant of developing a technological system" can be represented as a set of ordered pairs:    '' The best variant of developing a technological system'' x, P x ,   where x X is a variant of system development;  P x is a degree of the fuzzy set adjective "the best variant of developing a technological system". The most universal function among the ones for multifactor estimation is the function developed on the basis of the Kolmogorov-Gabor polynomial [3]: m m m i i ij i i i 1 i 1 j i P( x ) ( x ) ( x ) ( x )           m m m ijl i j l i 1 j i l j ( x ) ( x ) ( x ) ...         , (5) where i ij ijl, ,   – weight coefficients of partial criteria and their products; i j l( x ), ( x ), ( x )   – utility functions of partial criteria i jk ( x ), k ( x ),..., lk ( x ) . The utility functions of partial criteria i j l( x ), ( x ), ( x )   in this case are considered as the fuzzy set adjective "the best variant of developing a technological system" according to particular criteria i jk ( x ), k ( x ),..., lk ( x ) . They map 1 i i: k ( x ) E , i 1,m   , and should be universal and well-adapted to considering the peculiarities of specific situations of multicriteria selection [3]: they should be monotonous and dimensionless; have a common change interval (from 0 to 1); be invariant to the extremum of a particular criterion (min or max); enable implementing both linear and non-linear dependencies on the characteristics of options for developing systems. In the practice of multicriteria optimization the mostly wide-spread functions are membership functions [3, 8–10]:        i i i i i i i i k x k x k x k k               , (6) where i i ik ( x ), k , k  is the value of the i-th partial criterion for the variant x, the best and worst values of the i-th criterion, i 1,m ; i is a parameter determining the variant of the dependence ( 1i  is linear, 0 1i  is concave, 1i  is convex). The disadvantage of the function (6) is the impossibility of implementing S- and Z-shaped dependencies on the values of the particular criterion, which more adequately represent the situations of making project optimization solutions. In particular, power functions sewing from [7] and [11], the Gaussian function, the Harrington function, the logistic function, and their modifications do not have such drawback [12]. The experimental study of the utility functions of particular criteria used in practice [12] showed that: the procedures for selecting their parameters have a linear or quadratic time complexity with respect to a number of approximation nodes and take from several hundredths to tens of seconds; power functions sewing from [11] has several times greater accuracy of approximation of a decision maker’s estimations than other functions. The main disadvantage of these functions is a various degree of approximation of their values to the boundary values (0 and 1) with the approximation of the normalized values of the particular criterion to the extremal values 1, 0, 1,i ik k i m    . This reduces their capability to differentiate the utility of different values of the particular criterion. To avoid this disadvantage the ordinates ik ( x ), i 1,m should be scaled, which, in turn, increases the computation time of the membership function i( x ), i 1,m  . The conducted analysis and the review of literature sources on the topic of the study [112] showed that nowadays there remain some unexplored problems of evaluating the time complexity of procedures for calculating the values of utility functions of partial criteria. Their preliminary analysis shows that the computation time basing on them can differ by several times. Taking into account the fact that the tasks of structural, topological and parametric optimization of technological systems suggest, in the worst case, the analysis of the variants of the n2 order (where n is a variable that determines a number of structural elements, variants of topologies or parameters of the technological system), the search for new functions or the modification of available ones are required in order to reduce the complexity of calculating their values. The goal and objectives of the study The goal of the research is to increase the efficiency of multicriteria estimation of the properties of technological systems at the stages of their designing and reengineering using the apparatus of fuzzy sets. To achieve this goal, it is necessary: - to search new monotonous fuzzy set adjectives "the best variant of developing a technological system" or their ISSN 2522-9818 (print) Innovative technologies and scientific solutions for industries. 2017. No. 1 (1) 16 modification according to particular criteria for reducing the complexity of their values calculation; - to make a comparative analysis of time complexity and accuracy of approximation of a decision maker’s preferences for available and suggested monotonous membership functions; - to give recommendations about the practical use of monotonous fuzzy set adjectives "the best variant of developing a technological system" according to private criteria. Study materials Dependencies representing the change in the utility of the values of certain technological properties, like other anthropogenic systems, on the values of their criterion estimates are monotonous: S-shaped for particular criteria ik ( x ) max and Z-shaped for particular criteria jk ( x ) min ,  i, j 1,...,m (where m is a number of partial criteria). In this case, by normalizing the values of particular criteria (regardless their type), all membership functions can be reduced to S-type [3]: k( x ) k k( x ) k k       , i 1,m , (7) where k( x ), k , k  – the value of the particular criterion for the variant x, the best and worst values of the criterion k( x ) . Power functions sewing from [7] and functions (6) [11], the Gaussian function, the Harrington functions, logistic function, and their modifications belong to the utility functions of partial criteria (as a fuzzy set adjective "the best variant"), that describe a decision-maker’s estimations in the most accurate way and are widely accepted in practice [12]. Let us represent the listed functions using the normalization of the form (6): - the Gaussian function [12]: 2( k( x ) 1) ( x ) exp c         , (8) where c 0 is the parameter, defining specific type of dependencies; - the logistics function [12]: 1 ( x ) ( k( x ) a ) 1 exp b          , (9) where a is the abscissa of the inflection point; b is the parameter, defining the type of dependencies; - the Harington function [12]:  ( x ) exp exp ( g k( x ) a )        , (10) where g is the parameter of nonlinearity; a / g determines the inflection point; - the modified Gaussian function [12]: 2( k( x ) 1) ( x ) exp c          , (11) where c 0 is the parameter defining the form of dependence;  is the parameter determining the type of nonlinearity; - power functions sewing [11]: 1 2 a a a a a k( x ) a , 0 k( x ) k ; k ( x ) k( x ) k a (1 a ) , k k( x ) 1, 1 k                             (12) where ak , a – are normalized values of the coordinates of the point of sewing function, a0 k 1  , 0 a 1  ; 1 2,  – are the coefficients that determine the form of the dependence on the initial and final segments of the function; - power functions sewing based on the function [7]: p p 1 p p 1 2 k( x ) , 0 k( x ) 0.5; ( x ) 0.5 k( x ) 1 2 , 0.5 k( x ) 1, 0.5                         (13) where p is the parameter defining the form of dependence. Functions (8) (13) greatly change their values at the entrance to the dead bands (when the partial characteristics of the system approach to the worst and best values, i.e., k( x ) 0 and k( x ) 1 . This can lead to significant errors in determining the properties of technological systems according to specific criteria and have a significant effect on the error in calculating the quality of options for their development as a whole  P x (5). To overcome these drawbacks, sewing function (12) should be modified by using fractional-linear functions instead of power functions: a1 1 1 a a a2 2 2 a k( x ) a ( b 1) 1 b / b , 0 k( x ) k ; k ( x ) k( x ) k a (1 a ) ( b 1) 1 b / b , k k( x ) 1, 1 k                                                     (14) where 1b , 2b are the coefficients that determine the form of the dependence on the initial and final segments of the function. ISSN 2522-9818 (print) Сучасний стан наукових досліджень та технологій в промисловості. 2017. № 1 (1) 17 The suggested modification of sewing function (14) substantially reduces the dead band of membership function, thereby widening the region of the estimation model adequacy (fig. 1). а – function (12) b – function (14) Fig. 1. The components of sewing the utility functions of particular criteria for different values of 1 2,  and 1 2,b b parameters To reduce a number of operations that are required while calculating the values of function ( x ) , it is suggested to use a single preliminary calculation of their parts that do not change when the values of the particular criterion change k( x ) . The values of the functions (8) - (14) transformed to reduce the computation time are represented as: - the Gaussian function (8): 2( x ) exp z ( k( x ) 1)       , (16) where z 1 / c  ; - the logistics function (9): 1 2 1 ( x ) 1 exp z z k( x )         , (17) where 1z a / b ; 2z 1 / b ; - the Harrington function (10):  ( x ) exp exp ( a g k( x ))        , (18) where g – is the nonlinearity parameter; a / g – determines the inflection point; - the modified Gaussian function (11): 2z 1( x ) exp z ( k( x ) 1)      , (19) where z 1 / c  ; 2z 2   ; - power functions sewing (12): 1 2 a1 a a2 z [ k( x )] , 0 k( x ) k ; ( x ) a z [ k( x ) k ] , k k( x ) 1,                (20) where 1 1 a 1 z a k          , 2 2 a 1 z (1 a ) 1 k          ; - power functions sewing (13) based on the function [7]: p p z k( x ) , 0 k( x ) 0.5; ( x ) 0.5 k( x ) 1 z , 0.5 k( x ) 1, 0.5                       ( 21) where p 1z 2  ; - the suggested modification of sewing function (14): 12 a11 13 22 a21 23 z z , 0 k( x ) k ; z k( x ) ( x ) z z , k k( x ) 1, z k( x )               (22) where 11 1z a (b 1)   ; a12 1 11z b z k   ; a13 1z b k  ; 21 2z b (1 a ) 1    ; a22 21 2z ( z 1) (b 1) (1 k )      ; a a22 2z b (1 k ) k    . The results of the study The software was developed and the series of experiments were carried out for a comparative analysis of the accuracy of approximation of a decision maker’s preferences and the time complexity of calculating the values of utility functions of particular criteria ( x ) (8) – (14). ISSN 2522-9818 (print) Innovative technologies and scientific solutions for industries. 2017. No. 1 (1) 18 The values of the particular criterion ik( x ) and the corresponding values of their significance i( x ), i 1,20  were developed while simulating the work of experts using a random number generator. The selection of the best values of the parameters of the functions q was carried out by the method of the golden section according to the criterion of the least squares: 220 i i q Q i 1 K ( x ,q ) ( x ) min         , (15) where Q – set of admissible values of the parameters of the functions (8)  (14). The average time for calculating the value of the original functions was estimated (8) – (14) 1ct ; as well as the average time for calculating the value of the transformed functions (16)  (22) the average error in approximating a decision maker’s preferences cK (15); the maximum error of one decision maker’s estimation, the value maxK according to the series of experiments (tab. 1). Table 1. Results of the experimental study of functions Function type 1ct , Ns 2ct , Ns cK maxK Gauss (8) 2.523 2.23 1.83101 0.34394 Logistic (9) 2.471 2.361 0.08763 0.02251 Harrington (10) 4.658 4.611 0.07969 0.01285 Modified Gaussian (11) 7.74 7.501 0.40765 0.09471 Gluing (12) 5.739 5.431 0.03707 0.00655 Gluing (13) 0.876 0.722 1.02613 0.25744 Proposed function (14) 0.786 0.624 0.04151 0.01131 The results of the experiments justified that the accuracy of approximating a decision maker’s preferences using sewing function (12) and the suggested modification (14) is several times higher than with the help of other functions. At the same time, all functions except for the Gaussian function (8) and power functions sewing (13) have an error of approximation of preferences according to the maximum error maxK , that is satisfactory for practice. A much shorter time for calculating the values is required for the suggested modification of sewing function (14) and power functions sewing (13). At the same time, according to this indicator, they are 1.1512.02 times higher than all other membership functions. The suggested method for simplifying the algorithms for calculating the functions (8)  (14) made it possible to reduce the time for estimating by 1.0125.96% more. A number of compromises according to the accuracy of approximation of a decision maker’s preferences the calculation time include sewing function (12) and the suggested modifications (14). The suggested modification of sewing function (14) is the most efficient function according to the complex parameter "accuracy – computational complexity". Conclusions The analysis of the problem of estimating the properties of technological systems in the process of their multicriteria optimization resulted in the study of available monotonous fuzzy set adjectives "the best variant". It was established experimentally that power functions sewing (12) has much higher accuracy of approximation of expert estimates in comparison with the Gaussian function, the Harrington function, the logistic function and sewing function from (13) among the functions that are used in decision making support systems and enable implementing S (Z)-like dependencies on the values of particular criteria. In this case, the available membership functions greatly change their values when the system’s partial characteristics approach to the worst and best values, which can lead to significant errors in determining the properties of technological systems according to specific indicators and, consequently, to the error of their complex multicriteria estimation. To use the methods of solving combinatorial tasks of structural, topological, and parametric optimization of technological systems, membership functions with little time complexity are required. To overcome the mentioned drawbacks, the modification (14) of sewing function (12) is suggested by using fractional-linear functions instead of power functions. The suggested modification of the membership function significantly reduces the dead band practically without any loss of accuracy, thereby increasing the adequacy of the model of multifactor estimation and the selection of design solutions. In this case, the time for calculating the values of the suggested modification of the function is 8.7 times less than for the basic sewing function. Its practical use in decision making support systems of design and management solutions enables solving multifactor estimation tasks practically without any loss of accuracy and selecting solutions of much larger dimension with less computational resources. Directions for further research in this area can be the development of mathematical models, methods and software tools for selecting the parameters of the fuzzy set adjective "the best variant of developing a technological system" according both to specific indicators and to a set of quality indicators simultaneously. ISSN 2522-9818 (print) Сучасний стан наукових досліджень та технологій в промисловості. 2017. № 1 (1) 19 References 1. Averchenkov, V. I., Kazakov, Yu. M. (2011), Automation of the design of technological processes, FLINTA, Moscow, 229 p. 2. Kryuchkovsky, V., Petrov, E., Sokolova, N., Khodakov, V., Petrova, E. (ed.) (2011), Introspective analysis. Methods and means of expert evaluation, Grin DS, Kherson, 168 p. 3. Ovezgeldyev O., Petrov E., Petrov K. (2002), Synthesis and identification of models of multifactor estimation and optimization, Naukova Dumka, Kyiv, 161 p. 4. Greco, S., Ehrgott, M., Figueira, J. (2016), Multiple Criteria Decision Analysis - State of the Art Surveys, Springer, New York, 1346 p. 5. Kaliszewski, I., Miroforidis, I., Podkopaev, D. (2016), Multiple Criteria Decision Making by Multiobjective Optimization - A Toolbox, Springer, New York, 142 p. 6. Kaliszewski, I., Kiczkowiak, T., Miroforidis, J. (2016), "Mechanical design, Multiple Criteria Decision Making and Pareto optimality gap", Engineering Computations, Vol. 33 (3), pp. 876-895. 7. Ruskin, L. G., Gray, O. V. (2008), Fuzzy Mathematics. Fundamentals of the theory. Applications, Parus, Kharkov, 352 p. 8. Petrov, K. E, Kryuchkovsky,V. V. (2009), Comparative structural-parametric identification of models of scalar multivariate estimation, Oldi-plus, Kherson, 294 p. 9. Kryuchkovsky, V., Petrov, E., Sokolova, N., Khodakov, V. (2013), Introduction to the normative theory of decision-making, Grin DS, Kherson, 284 p. 10. Petrov, E., Brynza, N., Kolesnik, L., Pisklakova, O. (2014), Methods and models of decision-making under conditions of multi- criteria and uncertainty, Grin DS, Kherson, 192 p. 11. Petrov, E. G, Beskorovainy, V., Pisklakova, V. (1997), "Formation of utility functions of particular criteria in multicriteria estimation problems", Radioelectronics and Informatics, No. 1, pp. 71-73. 12. Beskorovayny, V. V, Soboleva, E. V (2010), "Identification of the partial utility of multifactorial alternatives using S-shaped functions", Bionics of Intellect, No. 10, pp. 50-54. 13. Beskorovainy, V. V, Trofimenko, I. V (2006), "Structural-parametric identification of models of the bug-factoring estimation", Armament systems and military equipment, No. 3 (7), pp. 56-59. Receive 05.06.2017 Відомості про авторів / Сведения об авторах / About the Authors Безкоровайний Володимир Валентинович – доктор технічних наук, професор, Харківський національний університет радіоелектроніки, професор кафедри системотехніки, м. Харків, Україна; e-mail: vladimir.beskorovainyi@nure.ua, ORCID: 0000-0001-7930-3984. Бескоровайный Владимир Валентинович – доктор технических наук, профессор, Харьковский национальный университет радиоэлектроники, профессор кафедры системотехники, г. Харьков, Украина; e-mail: vladimir.beskorovainyi@nure.ua, ORCID: 0000-0001-7930-3984. Beskorovainyi Vladimir – Doctor of Sciences (Engineering), Professor, Kharkiv National University of Radioelectronics, Professor of the Department of System Engineering, Kharkiv, Ukraine; e-mail: vladimir.beskorovainyi@nure.ua, ORCID: 0000-0001-7930-3984. Березовський Георгій Вячеславович – Харківський національний університет радіоелектроніки, студент кафедри штучного інтелекту, м. Харків, Україна; e-mail: heorhii.berezovskyi@nure.ua, ORCID: 0000-0001-7277-1531. Березовский Георгий Вячеславович – Харьковский национальный университет радиоэлектроники, студент кафедры искусственного интеллекта, г. Харьков, Украина; e-mail: heorhii.berezovskyi@nure.ua, ORCID: 0000-0001-7277-1531. Berezovskyi Heorhii – Kharkiv National University of Radioelectronics, Student of the Department of Artificial Intelligence, Kharkiv, Ukraine; e-mail: heorhii.berezovskyi@nure.ua, ORCID: 0000-0001-7277-1531. ОЦІНКА ВЛАСТИВОСТЕЙ ТЕХНОЛОГІЧНИХ СИСТЕМ ІЗ ВИКОРИСТАННЯМ НЕЧІТКИХ МНОЖИН Предметом дослідження в статті є процес оцінки властивостей технологічних систем на етапах їхнього проектування та реінжинірингу. Мета – підвищення ефективності процедур багатокритеріальної оцінки варіантів побудови технологічних систем з використанням апарату нечітких множин. Завдання: провести пошук нових чи внесення змін до відомих функцій приналежності нечітких множин «кращий варіант побудови технологічної системи» за частковими критеріями в напрямку зниження складності процедур обчислення їх значень; виконати порівняльний аналіз часової складності та точності апроксимації переваг особи, що приймає рішення, за допомогою монотонних функцій приналежності; дати рекомендації щодо практичного використання монотонних функцій приналежності в системах підтримки прийняття рішень. Використовуються загальнонаукові методи: прийняття рішень, теорії корисності, нечітких множин, ідентифікації. Отримані такі результати. У статті подана розроблена авторами модель переваг особи, що приймає рішення, для оцінки окремих властивостей технологічних систем з використанням функції приналежності нечітким множинам, що дозволяє реалізувати як лінійні, так і нелінійні (опуклі, увігнуті, S-образні і Z-образні) залежно від значень часткових критеріїв. Проведене експериментальне дослідження виявило її переваги за показниками точності та часової складності в порівнянні з функціями Гауса, Харрінгтона, логістичною функцією, склейками ступеневих функцій і їх модифікаціями. Запропоновано прийоми, що знижують часову складність процедур обчислення значень функцій приналежності. Висновки. У результаті аналізу відомих функцій приналежності нечітких множин встановлено, що вони недостатньо адекватно відображають переваги особи, що приймає рішення, для характеристик систем близьких до екстремальних значень і мають відносно високу обчислювальну складність. Запропонована функція приналежності та спосіб її обчислення дозволяють підвищити адекватність моделей багатофакторного оцінювання та суттєво знизити часову складність процедур обчислення її значень. Практичне http://orcid.org/0000-0001-7277-1531 http://orcid.org/0000-0001-7277-1531 http://orcid.org/0000-0001-7277-1531 ISSN 2522-9818 (print) Innovative technologies and scientific solutions for industries. 2017. No. 1 (1) 20 використання запропонованої функції приналежності в системах підтримки прийняття проектних і управлінських рішень дозволить практично без втрати точності отримувати розв’язки задач багатофакторного оцінювання та вибору набагато більшої розмірності або з меншими витратами обчислювальних ресурсів. Ключові слова: технологічна система, критерії якості, багатокритеріальна оптимізація, нечітка множина, функція приналежності. ОЦЕНКА СВОЙСТВ ТЕХНОЛОГИЧЕСКИХ СИСТЕМ С ИСПОЛЬЗОВАНИЕМ НЕЧЕТКИХ МНОЖЕСТВ Предметом исследования в статье является процесс оценки свойств технологических систем на этапах их проектирования и реинжиниринга. Цель – повышение эффективности процедур многокритериальной оценки вариантов построения технологических систем с использованием аппарата нечетких множеств. Задачи: провести поиск новых или модификацию известных функций принадлежности нечетким множествам «лучший вариант построения технологической системы» по частным критериям в направлении снижения сложности процедур вычисления их значений; выполнить сравнительный анализ временной сложности и точности аппроксимации предпочтений лица, принимающего решения, с помощью монотонных функций принадлежности; дать рекомендации по практическому использованию монотонных функций принадлежности в системах поддержки принятия решений. Используются общенаучные методы: принятия решений, теории полезности, нечетких множеств, идентификации. Получены следующие результаты. В статье представлена разработанная авторами модель предпочтений лица, принимающего решения, для оценки отдельных свойств технологических систем с использованием функции принадлежности нечетким множествам, позволяющая реализовать как линейные, так и нелинейные (выпуклые, вогнутые, S-образные и Z-образные) зависимости от значений частных критериев. Проведенное экспериментальное исследование выявило ее преимущества по показателям точности и временной сложности в сравнении с функциями Гаусса, Харрингтона, логистической функцией, склейками степенных функций и их модификациями. Предложены приемы, снижающие временную сложность процедур вычисления значений функций принадлежности. Выводы. В результате анализа известных функций принадлежности нечетким множествам установлено, что они недостаточно адекватно отображают предпочтения лица, принимающего решения, для характеристик систем близких к экстремальным значениям и имеют относительно высокую вычислительную сложность. Предложенная функция принадлежности и способ ее вычисления позволяют повысить адекватность моделей многофакторного оценивания и существенно снизить временную сложность процедур вычисления ее значений. Практическое использование предложенной функции принадлежности в системах поддержки принятия проектных и управленческих решений позволит практически без потери точности получать решения задач многофакторного оценивания и выбора гораздо большей размерности или с меньшими затратами вычислительных ресурсов. Ключевые слова: технологическая система, критерии качества, многокритериальная оптимизация, нечеткое множество, функция принадлежности. Бібліографічні описи / Библиографические описания / Bibliographic descriptions Безкоровайний В. В., Березовський Г. В. Оцінка властивостей технологічних систем із використанням нечітких множин. Сучасний стан наукових досліджень та технологій в промисловості. Харків. 2017. № 1 (1). С. 14–20. Бескоровайный В. В., Березовский Г. В. Оценка свойств технологических систем с использованием нечетких множеств. Сучасний стан наукових досліджень та технологій в промисловості. Харків. 2017. № 1 (1). С. 1420. Beskorovainyi V., Berezovskyi H. Estimating the properties of technological systems based on fuzzy sets. Innovative technologies and scientific solutions for industries. Kharkiv. 2017. No. 1 (1). P. 14–20. ISSN 2522-9818 (print) Сучасний стан наукових досліджень та технологій в промисловості. 2017. № 1 (1) 21 UDC 004:519.711.3 V. GURIN, E. PERSIYANOVA GENERAL PRINCIPLES OF BUILDING THE MODEL OF DEVELOPMENT AND OPERATION OF HETEROGENEOUS TEAMS FOR PROJECT MANAGEMENT The subject of the research is the basic principles of making the model of managing the production of software and models of development and operation of heterogeneous teams for project management. The objective of the research is to develop the mathematical model representing the operation of heterogeneous teams for project management. The sociometry is used for psychodiagnostic procedures in the course of social and psychological analysis of group relations. This method is directed at determining the structure of interpersonal relations by identifying mutual feelings of frienliness and unfrienliness among the members of groups. The mathematical methods of processing data and information obtained during the sociometric survey lie in calculating mathematical indicators which can be subdivided into group and individual indexes. In the course of the research the following tasks were solved: the requirements for team building and development of models for software implementation were analyzed; the method for analyzing the cohesion of team members was selected; the options for developing the mathematical model representing the principles of building the team which works on the project were considered. Groups different in structure were compared with the help of mathematical processing of statistical data; and correlation procedures were conducted. Individual indexes were defined; among them are: the index of sociometric status which indicates the advantage of any member of the group over other participants; the indexes of positive and negative emotional expansivity; the index of group cohesion; the index of sociometric coherence. The methods used are: statistical and correlation analysis, sociological, Hungarian, mathematical. As a result of the conducted researches the basic principles of making the model of managing software development are shown, mathematical methods of development and operation of heterogeneous teams for project management are suggested. Thus, the goals and objectives of the research are carried out. Keywords: software, the model of management, sociometry, individual indexes, Hungarian method. Introduction The key object of studying software engineering is the process of software development. However, nowadays, there is no universal software development process, that is a set of techniques, rules and regulations that are suitable for any software, for any company, for teams of any nationality. Each current development process, carried out by any team within a particular project, has a large number of features and individualities. Before starting a project, it would make sense to create a process template, for example, as in the Microsoft Visual Team System that is created or adapted (if a standard one is used) before developing. In VSTS, there are workpieces for specific processes based on CMMI, Scrum, and others. These features of software development process require a model that would enable managing this process. Problem analysis and task setting Generally, the development of a model is considered as a process of purposeful "creation" of a special way of interaction among the people in a group (called team), which makes it possible to realize their professional, intellectual and creative potential efficiently according to the strategic objectives of this model of management (team). The model in this case is defined as a group of people who are mutually reinforcing and interchanging one another in the course of achieving the goals [1]. Nominally, four types of models (groups) that are often formed in the course of practical activities of enterprises and are classified in terms of their work can be determined. 1. Teams that create something new for the organization or do work that has been done earlier. Project teams fall into this group. They are temporary in nature, as the content of a project is determined as a temporary specific organizational form for achieving goals and solving unique tasks. 2. Teams (groups) that deal with problems, goals and tasks of the enterprise using analysis, control and recommendations, e.g. auditing and controlling teams, quality assessment teams. 3. Teams (groups) that are not special, but a permanent part of the organizational structure and which carry out the process of production and performance of repetitive work, e.g. production teams (groups), sales teams and service teams (brigades, groups). 4. Teams that are of multifunctional executive management nature. These teams are usually formed at higher levels of enterprise management and act as executive committees, management teams or top managers of the enterprise. The following conditions determine the efficient work of a team: - each member of a team should clearly understand their role, which enables performing their tasks without disturbing the work of others; - project specification and schedule of work should be coordinated with all the members of a team; - all the team members should interact with one another and respect the professional qualitie