CC BY
3
MSC 97M40
DOI: 10.14529/mmp210210
MATHEMATICAL MODELLING OF INDUSTRIAL EQUIPMENT OPERATION BASED ON MARKOV PROCESSES
V.G. Mokhov1, G.S. Chebotareva2
1 South Ural State University, Chelyabinsk, Russian Federation 2Ural Federal University, Ekaterinburg, Russian Federation E-mails: mokhovvg@susu.ru, g.s.chebotareva@urfu.ru
The existence of an almost unlimited number of methods for evaluating the productivity of industrial equipment contributes to the uncertainty of choosing the most effective approach. At the same time, the presence of many possible states of equipment (from working to repair and other downtime) complicates the problem of modelling the operation of such a system. The problem of modelling does not lose its relevance, first of all, for large industrial companies. The article presents the methodological features of modelling the operation of industrial equipment based on the Markov method. This approach is used as a base for estimating the probabilities of equipment transitions between states, as well as for predicting the final state of operation of such a system. In terms of practical application, we consider an example of the functioning of the same type of industrial equipment in the framework of three possible states (functional, broken, and also in the mode of forced repair). Based on the results of calculations, we carry out assessment of the reality of the state transitions of equipment, designed rate of these transitions, as well as the predicted level of productivity equipment system after the period "t". The reliability of the research results is confirmed by their practical implementation. The obtained results are recommended to be used by the management and analysts of industrial companies in the process of making operational decisions and in the development of equipment repair strategies.
Keywords: Markov method; mathematical modelling; theory of probability; prediction; industrial equipment; equipment states; states transitions.
Introduction
The relevance of the problem of modelling the effective operation of industrial equipment for large companies is due to a number of significant reasons. The first reason is mandatory to implement the company production plans. The second one is the improving the reliability of equipment operation, improving its repair and maintenance systems [1, 2]. Thirdly, the need for accurate forecasting of equipment repair and maintenance costs. Together, these reasons are integral elements of the strategy of an industrial company development. Currently, this issue is updated due to the lack of uniform clear guidelines for the maintenance of equipment at industrial enterprises. The authors divide the solution of this problem into three stages.
1. Theoretical Foundations of Markov Modelling of Equipment Operation
Markov analysis is based on the concept of possible states of equipment and the transition between these states in time, assuming a constant probability of such transitions [3]. Based on the initial data on the functioning of the system under study, we construct a Markov transition matrix (Table), as well as its graphical representation in the form of a
diagram of states and transitions (Figure). They describe the probabilities of transitions between different states of equipment and model aspects of the reliability of the system behaviour over time.
Markov analysis is applicable in a situation where the future state of the system depends only on its current state. This method is usually used in industry companies to analyse maintainable systems that can operate in many modes.
The input data of the Markov analysis are [4]:
- a list of different system states (e.g. full or partial operation, failure, overhaul, scheduled maintenance, etc.);
- an accurate understanding of the possible transitions that need to be modeled;
- the rate of transition from one state to another, usually in the form of the transition probability (in the case of discrete analysis), the failure (A) or recovery rate (in the case of continuous analysis).
The output data are the values of the probabilities that the system is in various states, as well as the modelling of the system state after a specific period [4].
2. Estimation of Probabilities of Industrial Equipment Transitions between States Based on the Markov Method
In this section, we consider an example of the operation of the same type of industrial equipment for the case of discrete analysis. According to the initial data, there are three possible states of equipment:
- Si is a working state;
- S2 is a broken state;
- S3 is a state to be on a forced (unscheduled) repair.
The initial characteristic of this system is as follows: there are 1300, 40 and 12 pieces of equipment in the state S1, S2 and S3, respectively.
The matrix of the transitions from the initial state to the final state is presented in Table. The graphical interpretation of Table is shown in Figure. The indicator "t" means the number of periods when the object is in an unchanged state, and the unknowns X1, X2, X3, X4 are the probabilities of equipment transitions to different states.
Table
Matrix of the equipment states transitions
Final state
Si s2 s3
Initial state Si 0,95 X2
s2 0 t-»
S3 Xi 0,04 0,27
The values shown in Table and Figure interpret the probabilities of the equipment transition from the initial state to the final state, for example:
- 0,95 means that the probability that the equipment will remain in the working state S1 is 95%;
- 0 means that there is no possibility that the equipment will immediately move from the broken state S2 to the working state S1 without a prior repair;
Вестник ^ЭУрГУ. Серия «Математическое моделирование и программирование» (Вестник ЮУрГУ ММП). 2021. Т. 14, № 2. С. 94-99
0,95
Diagram of equipment states and transitions
- t-2 is a dependent variable that is equal to the probability of finding the equipment in the broken state S2, etc.
Initially, the estimation of all unknown equipment transition probabilities is based on the principles of probability theory. So, for each initial state, the sum of all the probabilities of equipment transitions to the final state is equal to one. Therefore, we construct the system
[ 0, 95 + X2 + X3 = 1,
I 0 + t-2 + X4 = 1, (1)
( Xi + 0, 04 + 0, 27= 1.
As a result, Xi = 0,69, X4 = (1 — t-2), the transition probability of X4 is dependent on the variable "t". The transitions of equipment X2 and X3 between states are interdependent, as shown in the equation
X2 + X3 = 0, 05. (2)
Therefore, to quantify the probabilities of these transitions, it is necessary to investigate their reality:
- X2 is the probability of transition of the equipment from the operating mode to the broken state;
- X3 is the probability of transition of the equipment from the operating mode to the repair state.
However, the equipment can not immediately move from a working state to a forced repair, since for this it must initially break down. Therefore, the probability of the transition X3 is zero, hence the probability of the transition X2 is 0,05.
3. Prediction of Final State of Operation of Industrial Equipment after Period "t" by Markov Method
To predict the final state of the equipment operation after the period "t" equal to 50 days, we construct the system of equations
Pi = 0, 95 ■ Pi + 0 ■ P2 + Xi ■ P3, P2 = X2 ■ Pi + t-2 ■ P2 + 0, 04 ■ P3, P3 = X3 ■ Pi + X4 ■ P2 + 0, 27 ■ P3, Pi + P2 + P3 = 1,
where Pi, P2, P3 are the probabilities of finding the system in each of the states S1, S2 and S3, respectively.
Suppose that the values X1, X2, X3, X4 are known, t = 50 days, then we specify the initial conditions:
System (4) has the following results:
- Pi = 0,8886, or 88,86%;
- P2 = 0,047, or 4,7%;
- P3 = 0,0644, or 6,44%.
This means that, after 50 days, this system of equipment is in working state, broken state, and in the state of repair with the probability 88.86%, 4.7%, and 6,44%, respectively. Suppose that the company has exactly 1 352 pieces of equipment at its disposal, when, after 50 days, the final state of the simulated system is as follows:
- 1 201 equipment work;
- 64 is in the broken state;
- 87 is in the state of repair.
Conclusions
1. A solution to the relevant problem of modelling the operation of industrial equipment using the Markov method is proposed.
2. All possible transitions between three states of the same type of equipment are modeled and their probabilities are quantified.
3. The forecast of the final state of operation of the equipment after the period "t" is made.
4. The results obtained are recommended for use in predicting the uninterrupted operation of industrial companies equipment and developing maintenance strategies.
Acknowledgements. The work was supported by grant of the President of the Russian Federation (MK-4549.2021.2).
References
1. Gitelman L.D., Kozhevnikov M.V., Chebotareva G.S., Kaimanova O.A. Asset Management of Energy Company Based on Risk-Oriented Strategy. WIT Transactions on Ecology and the Environment, 2020, no. 246, pp. 125-135. DOI: 10.2495/EPM200121
2. Aronson K.E., Murmansky B.E., Murmanskii I.B., Brodov Y.M. An Expert System for Diagnostics and Estimation of Steam Turbine Components' Condition. International Journal of Energy Production and Management, 2020, vol. 5, no. 1, pp. 70-81. DOI: 10.2495/EQ-V5-N1-70-81
3. Wentzel E.S. Introduction to Operations Research. Moscow, Soviet radio, 1964. (in Russian)
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Вестник !Ю"УрГ"У. Серия «Математическое моделирование и программирование» (Вестник ЮУрГУ ММП). 2021. Т. 14, № 2. С. 94-99
4. Haque S., Mengersen K., Stern S. Assessing the Accuracy of Record Linkages with Markov Chain Based Monte Carlo Simulation Approach. Journal of Big Data, 2021, vol. 8, no. 1, 8 p. DOI: 10.1186/s40537-020-00394-7
Received February 11, 2021
УДК 330.322.013+001.895 DOI: 10.14529/mmp210210
МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ РАБОТЫ ПРОМЫШЛЕННОГО ОБОРУДОВАНИЯ НА ОСНОВЕ МАРКОВСКИХ ПРОЦЕССОВ
В.Г. Мохов1, Г.С. Чеботарева2
1 Южно-Уральский государственный университет, г. Челябинск,
Российская Федерация
2Уральский федеральный университет, г. Екатеринбург, Российская Федерация
Существование практически неограниченного числа методов оценки работоспособности промышленного оборудования способствует возникновению неопределенности выбора наиболее эффективного подхода. Одновременно с этим наличие множества возможных состояний оборудования (от рабочего до ремонтного и иного простоя) усложняет задачу моделирования работы такой системы. Задача моделирования не теряет своей актуальности, в первую очередь, для крупных промышленных компаний. В статье представлены методологические особенности моделирования работы промышленного оборудования на основе Марковского метода. Данный подход использован в качестве базового при оценке вероятностей переходов оборудования между состояниями, а также при прогнозировании конечного состояния работы подобной системы. В части практической апробации рассмотрен пример функционирования однотипного промышленного оборудования в рамках трех возможных состояний: работоспособное, сломанное, а также находящееся в режиме вынужденного ремонта. По результатам расчетов проведена оценка реальности переходов между состояниями оборудования, рассчитана скорость подобных переходов, а также спрогнозирован уровень работоспособности системы оборудования через период «t». Достоверность результатов исследования подтверждена их практической реализацией. Полученные в рамках работы результаты рекомендуется использовать менеджментом и аналитикам промышленных компаний в процессе принятия операционных решений и при разработке стратегий ремонта оборудования.
Ключевые слова: Марковский метод; математическое моделирование; теория вероятностей; прогнозирование; промышленное оборудование; состояния оборудования; переходы между состояниями.
Литература
1. Gitelman, L.D. Asset Management of Energy Company Based on Risk-Oriented Strategy / L.D. Gitelman, M.V. Kozhevnikov, G.S. Chebotareva, O.A. Kaimanova // WIT Transactions on Ecology and the Environment. - 2020. - № 246. - P. 125-135.
2. Aronson, K.E. An Expert System for Diagnostics and Estimation of Steam Turbine Components' Condition / K.E. Aronson, B.E. Murmansky, I.B. Murmanskii, Y.M. Brodov // International Journal of Energy Production and Management. - 2020. - V. 5, № 1. -P. 70-81.
3. Вентцель, Е.С. Введение в исследование операций / Е.С. Вентцель. - М.: Советское радио, 1964.
4. Haque, S. Assessing the Accuracy of Record Linkages with Markov Chain Based Monte Carlo Simulation Approach / S. Haque, K. Mengersen, S. Stern // Journal of Big Data. - 2021. -V. 8, № 1. - P. 8.
Вениамин Геннадьевич Мохов, доктор экономических наук, профессор, ведущий научный сотрудник кафедры математического и компьютерного моделирования, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), mokhovvg@susu.ru.
Галина Сергеевна Чеботарева, кандидат экономических наук, доцент, старший научный сотрудник, кафедра систем управления энергетикой и промышленными предприятиями, Уральский федеральный университет (г. Екатеринбург, Российская Федерация) , g.s.chebotareva@urfu.ru.
Поступила в редакцию 11 февраля 2021 г.
Вестник !Ю"УрГ"У. Серия «Математическое моделирование и программирование» (Вестник ЮУрГУ ММП). 2021. Т. 14, № 2. С. 94-99
