Russia, Petrozavodsk, Petrozavodsk State University (PetrSU)

Rogov@mainpgu.karelia.ru

Abstract

The paper presents some approaches to solving the problems that arise in investigation of the problem of optimal control of a renewal system for finite time interval from the stage of statement and analysis of the problem to the stage of software programming.

Россия, Петрозаводск, Петрозаводский государственный университет (ПетрГУ)

Rogov@mainpgu.karelia.ru

Аннотация

В докладе рассказывается о некоторых подходах к решению задач, возникающих при полном цикле исследования (от постановки задачи и ее анализа до создания соответствующего программного обеспечения) по организации оптимального управления восстанавливаемыми системами, которые функционируют на конечном интервале времени.

The problem of renewal system control for finite time interval [1, 5, 6 and 8] is attracting increased interest due to its practical applications. Usually the renewal processes, alternating and semi-Markovian processes are used as a model of renewal systems operation. Some models are based on embedded processes [12, 14]. After a model has been developed, its adequacy has to be checked. In order to determine control parameter values a corresponding optimization problem has to be stated and solved. The solution found is analyzed and if this analysis proves that this solution is practically acceptable then the corresponding computer software should be developed. This software must be convenient enough for the managerial personnel who do not possess specialized mathematical and computer skills.

Some approaches to simulation of renewal system operation on the example of paper-machines (PM) with the use of embedded alternating processes as a model are presented in [1, 12 and 14]. In those papers five stages have been distinguished for describing the operation efficiency of a paper-machine. These are working, accident by mechanical cause, accident by technological cause, stopping by administrative and economical cause and planned preventive maintenance. The model is based on three embedded alternating processes.

Unfortunately, the data received from this model is censored. It satisfies the following testing plans [n, B, T], [n, T₁, …, T_n], [n, A, (F₁, T₁, …, T_n)], [n, A, (F₁, F₂, T₁, …, T_n)] etc. This fact complicates the checking of model adequacy and requires development of special statistical methods.

In author’s opinion the use of embedded processes as models is more prospective than the use of semi-Markovian processes.

We now present (perhaps incomplete) list of statistical problems, which arise when renewal processes are used as a model of a renewal system [6]:

1. Identification of both reliability function and renewal time distribution function usually on the basis of censored data.
2. Prediction identification of parameters both of the reliability function and of the renewal time distribution function on the basis of the information about supposed system functioning regimes and technical modernization of separate system units and technological function schemes.
3. Characterization of a class of renewal function for renewal processes.
4. Identification of renewal function and other reliability characteristics.

Statistical checking of the model described in the previous section and estimation of reliability parameters of the renewal system have required the development of special statistical methods. The results obtained by the author are presented in [4-6, 9-15].

The following problems arise during control system simulation:

1. Synthesis of mathematical models of renewal system functioning that are adequate to chosen optimization criteria and to real functioning conditions for accepted model assumption.
2. Investigation of complexity of these problems and stability of its optimal solution. Estimates of the number of quasioptimal solutions.

The real practice leads to a series of optimization problems for renewal systems functioning. Therefore, synthesis of optimization mathematical model for functioning of a renewal system is a very complicated problem. To get an idea of how to approach this problem, the author suggests taking a look at the results of development and investigation of optimization models for planning preventive measures (PPM) for a complex system of paper-machines on finite time interval [0, T]. These results can be found in [1-3, 6 and 8]. Optimization problems described there are nonlinear problems of separable continuous-discrete programming with constraints of “or-or” type. Besides that, these are multi-criterion problems. As a rule, the number of variables in these problems is too large to be able to solve them by straight item-by-item examination. The dynamic programming method often does not apply, too. Application of some heuristic methods with local optimization gives acceptable results.

The following problems arise when the information system of control support is being developed:

1. The use of computer for solving optimization problems occurring in renewal theory.
2. Computing of industrial, economical, technical and diagnostic characteristics of renewal systems to estimate the economical and industrial effectiveness of a functioning renewal system on a finite time interval.

Generalizing the results of development and applying the information and mathematical support for solving different optimization problems of control for some enterprises of North-Western region of Russia (Karelia, Arkhangelsk area), we come to the conclusion, that now there is a need in developing special software environment. It should be oriented on solution of practical problems in the following areas: information systems and networks, service of operative repair of technological equipment, robot-technical complex, transport systems, problems of calendar network planning etc [7].

Let formulate the main requirements of this environment. It must consist of the following units:

• Information unit;
• Mathematical unit;
• Service unit;
• User interface unit;
• Printing unit.

Information unit is intended for statistical data entry, lexicographic data entry and database entry. This information about control support system will be used to solve optimization problems.

Mathematical unit will

• solve problems of linear, nonlinear and stochastic programming by different exact and heuristic methods,
• allow addition of new problems and their solutions.

Mathematical unit has to have modular structure. Every user can use only modules he/she needs.

Service unit is intended to specify characteristics and parameters of the control support system. This unit is used with specialists having mathematical knowledge and experience of mathematical model development. Such specialists also apply user interface, as users are not supposed to have special mathematical and mathematical programming knowledge. If application is correct then specialist-mathematician does not take part in the environment exploitation. Thus the service includes choice of mathematical model of control support system (possibly, several models) and choice of user interface.

Printing unit is intended to prepare and print different reports. Unit has to have graphic means.

Existing software, which can be used for solving the optimization problems, may be divided into two groups. The first, such as MathLab, MathCad, Mathematica, are used by specialist-mathematician in laboratory conditions. The second type – Excel, Quattro Pro – is intended for economical calculation. It has limited mathematical means and can not be used for solving optimization problems. Besides, practically all software does not have convenient tools for customizing the environment for non-specialist users.

1. Rogov A.A., Tchernetskii V.I. (1990) On one method of calculating optimal schedule for prophylactic repair of paper machine. In: Mathematical modeling of economic processes, Petrozavodsk, pp. 42-56 (in Russian).
2. Rogov A.A. (1991) On a problem of optimal maintenance of complex of “n” machines with fixed number of planning preventive repairs. Dep. in VINITI, No. 3478-B91 (in Russian).
3. Rogov A.A. (1991) On some cases of solving constructing problem for schedule of planning repair of “n” machines. Dep. in VINITI, No. 3477-B91 (in Russian).
4. Rogov A.A. (1992) On a problem of identification of renewal function. Dep. in VINITI, No. 2979-B92 (in Russian).
5. Tchernetskii V.I., Rogov A.A. (1992) Mathematical problems of renewal of systems on finite time interval. In: Applied mathematics and informatics, No. 1, Petrozavodsk, pp. 21-31 (in Russian).
6. Rogov A.A., Tchernetskii V.I. (1993) On problems of renewal systems theory for finite time interval. In: Probabilistic methods in discrete mathematics, Utrecht/Moscow, pp. 386-394.
7. Rogov A.A., Polyakov V.V. (1995) Information support of optimization control of queueing systems. In: Investigation of communication and computer networks by queueing theory methods, Minsk, pp. 105-106 (in Russian).
8. Rogov A.A. (1995) Mathematical model of problem for schedule of PPM taking into account queueing system structure. In: Applied mathematics and informatics, No. 4, Petrozavodsk, pp. 13-19 (in Russian).
9. Rogov A.A., Shchyegoleva L.V. (1996) On recurrent estimation of renewal function on basis of censured sample. In: Applied mathematics and informatics, No. 5, Petrozavodsk, pp. 43-52 (in Russian).
10. Tchernetskii V.I., Rogov A.A., Shchyegoleva L.V. (1997) Statistical analysis of renewal processes. In: Probabilistic methods in discrete mathematics, Utrecht, pp. 137-144.
11. Rogov A.A., Shchyegoleva L.V. (1997) The statistical estimation of a renewal function. In: Queues: flows, systems, networks, Minsk, pp. 175-178.
12. Rogov A.A., Shchyegoleva L.V. (1997) Application of embedded processes for simulation of operation efficiency of paper-machine. In: Statistical and applied analysis of time series, Brest, pp. 159-165 (in Russian).
13. Rogov A.A., Shchyegoleva L.V. (1998) Approximation of statistical estimation of renewal function. In: Queues: flows, systems, networks, Minsk, pp. 157-159 (in Russian).
14. Rogov A.A., Shchyegoleva L.V. (1998) Modeling of the operation efficiency of a complex technical unit based on the embedded processes. In: Proceedings of the 3rd St.Petersburg workshop on simulation, St.Petersburg, pp. 340-342.
15. Rogov A.A., Shchyegoleva L.V., Shulgin A.V. (1999) Comparison of empirical estimators of a renewal function. In: Probabilistic analysis of rare events, Riga, pp. 209-211.