Methodology of the process design of gross product output of an enterprise

1* Department of Applied Mathematics and Information Technologies. Pridniprovsk State Academy of Cyvil Engineering and Architecture. 24-a Chernishevskogo st. 49600, Dnipropetrovsk, Ukraine, tel. +38 (0567) 56-34-10, email: prmat@mail.pgasa.dp.ua ORCID ID: 0000-0001-5894-0642 2* Department of Applied Mathematics and Information Technologies. Pridniprovsk State Academy of Cyvil Engineering and Architecture. 24-a Chernishevskogo st. 49600, Dnipropetrovsk, Ukraine, tel. +38 (0567) 56-34-10, email: nersova107@gmail.com, ORCID ID: 0000-0003-0198-0883 3* Department of Applied Mathematics and Information Technologies. Pridniprovsk State Academy of Cyvil Engineering and Architecture. 24-a Chernishevskogo st. 49600, Dnipropetrovsk, Ukraine, tel. +38 (0567) 56-34-10, email: velmagina24@gmail.com ORCID ID: 0000-0002-5584-3748 Annotation. The methods for designing the process of gross product output of an enterprise, based on the stochastic method of dynamic programming for continuous deterministic systems was created. The selection of weight factors of quadratic quality functional was substantiated, the dependence between parameters of design and weight factors was established, and the region of optimum values of design parameters was determined, the optimal filter of the process of gross product output of an enterprise was obtained. Modeling of the process of gross product output with optimum values of coefficients of growth and retirement of basic production assets. Production capacity is determined by the output at the achieved level of organization and production technology. The shape of production capacity graphs changes noticeably even with minor changes in parameters, i.e., capacity is sensitive to changes in the state of a firm. Design parameters are determined by the matrix method of dynamic programming. Based on the concepts of observability and controllability, it has been established that the process of producing a gross product of an enterprise is subject to management. By modeling and calculations, it was proved that using the matrix method of dynamic programming, one can obtain analytical dependencies for growth and retirement coefficients, as well as calculate the regions of their optimal values, i.e. The production capacity of an enterprise can be the main indicator of the characteristics of the life cycle of an enterprise. The technique should be used when designing the process of producing a gross product of enterprises, its implementation will reduce the design time and ensure the stability of the process of producing a gross product.


Introduction
The process of gross product manufacturing is the basis of any enterprise. It must be continuous in time, i. e. it can be described by an ordinary differential equation. Production of gross product is impossible without investment of the basic production assets (BPA). Therefore, parameters of the process of manufacturing of gross product must include coefficients of growth and retirement of BPA. This model is constructed in [2]. The problem arises: how to determine the values of these parameters so that the process of gross product manufacturing could be sustainable. This problem has not been solved, although it is rather urgent. This paper proposes the procedure for designing the process of gross product output of an enterprise, based on stochastic method of dynamic programming for continuous deterministic systems.

Research goal and objectives
The goal of the present research is to create a technology of design of the process of gross product manufacturing by an enterprise, the parameters of which are determined using the method of dynamic programming for continuous stochastic systems.
To accomplish the set goal, the following tasks were to be solved:  to prove that the process of gross product manufacturing by the company is a control object;  to determine analytical dependences for design parameters using the matrix method of dynamic programming;  to substantiate the choice of weight factors of the quadratic functional of quality;  to determine the region of optimal values of design parameters;  to obtain the optimal filter of the process of gross product manufacturing;  to perform mathematical modeling of the process of gross product manufacturing with optimum values of coefficients of BPA growth and retirement.

Development of mathematical models and methods of design
1. Determining of analytical dependences for design parameters with the help of the matrix method of dynamic programming The result of production activity of an enterprise is the gross product. Mathematical model of the process of manufacturing of gross product is [2]: Here m is the capital ratio of BPA for manufacturing of given products; c is the coefficient of BPA growth; Y is the gross product; () Vt is the instantaneous value of basic assets;  , q are the coefficients of depreciation and proportionality (parameter of the model). Model (1) can be used to analyze the process of gross product manufacturing at different values of its parameters. The model includes a large number of parameters. There arises the problem how in the process of design to determine a range of a change in parameters, in which the process of manufacturing gross product will be sustainable. Sustainability is an internal property of a process independent of instantaneous values of basic assets. An enterprise and a process are only created, so it is not possible to judge about their level of scientific and technological development. As an assumption, we will take capital ratio as constant. Then the mathematical model of the process of gross product manufacturing will take the form: This model can be accepted as the model-analogue in optimization of parameters by the methods of the optimal control theory [8].
We set the goal to define calculation formulas for design parameters: coefficients of BPA retirement and growth at constant capital ratio.
In the matrix method of dynamic programming the differential equation of the designed process is written down as where u is the unknown synthesizing function, representing acceleration of the process of gross product manufacturing. Let us introduce designations: 12 ; x Y x Y  , then we have ; As the optimality criterion, we will accept quadratic functional of quality: where ,,    are the weight factors of the functional.

Matrices
, PG of functional (6) have the form: The problem is stated: to determine synthesizing function of equation (3), which will minimize the functional of quality. The physical sense of the functional is monetary costs to maintain stability of the process of gross product manufacturing [4].
A necessary optimality condition is solution of the nonlinear algebraic Riccati equation ' 1 ' We will write down equation (7) Hence we will obtain the system of non-linear equations for determining the elements of the symmetric matrix S.
If we substitute synthesizing function (10) into equation (3), we will obtain where 11 22 We will write characteristic equation down for (11) (13), we obtain analytical dependences for determining of design parameterscoefficients of BPA growth and retirement; As it is seen, design parameters are determined through weight coefficients of the quadratic functional of quality.
2. Substantiation of selection of weight factors of quadratic functional of quality [9].
Selection of weight factors of quadratic functional of quality is a complex problem, the elements of matrices P and G are often selected by trial and error method, which significantly impedes the synthesis of optimal systems by this criterion.
In economic systems, aperiodic law of development is close to optimal. Synthesizing function must be physically realizable. To verify this, it is necessary to have dependences between weight factors of the quadratic functional of quality and the range of their changes. Dependences are established from formulas (14) in view of the fact that coefficient of growth of BPA is normally by two orders of magnitude lower than the coefficient of retirement.    Table 1 shows calculation of optimal values of coefficient of BPA growth from formulas (14), Table 2 shows the values of coefficient of BPA retirement. Original data include: capital ratio m =1,5;  =10000; 15000; 20000;  =10; 30; 50; 70.

Таble 2 Region of optimal values of coefficient of BPA retirement
The data of Table 2 indicate that the values of the coefficient of BPA retirement are less than 0.3. The value of the coefficient of BPA growth (Table 1) is by two orders of magnitude less than the values of the coefficient of retirement, i.e., optimal parameters provide for physical implementation of the process of gross product output.

Algorithm of optimal filters of Kalman-Bucy
The state of the studied process is described by equations: Kalman proved that the solution of the problem of optimal filtration is assessment ˆ( ) Xtoutput vector of the dynamic process, which is described by differential equation Expression (17) is called a differential equation of the filter, which is «disturbed» by observed signal () Yt and at the output gives the best linear assessment ˆ( ) Xt of vector of state () Xt of the dynamic process. Matrix () Ct is called matrix coefficient of strengthening of the optimal filter and is determined by expression Kalman obtained the nonlinear differential equation to which is called a correlation or a dispersion equation.
It is system ( 1) 2 nn of differential equations of the Riccati type. Thus, solution of the optimal filtration problem is derived from the system of equations (18) Matrix of parameters of the optimal filter is formed from equation (17) When implementing the Kalman filter, it is important to provide for its stability. R. Kalman and R. Bucy showed that the optimal filter is a uniformly asymptotically stable dynamic process. In this case, solution of a correlation equation represents a stable computation process, not sensitive to insignificant errors.

The process of gross product output of an enterprise as a control object
To prove that the process of gross product manufacturing by an enterprise is a control object, the concepts of controllability and observability of dynamic systems are used. In paper [7], the methods of obtaining controllability and observability criteria of linear dynamic systems are clearly presented.
Research into observability of the process of gross product manufacturing was conducted on the mathematical model '' ; ' , Controllability matrix of the conjugate process is referred to as the matrix of observability of the original process, i.
The highest order of the minor of this matrix that is different from zero is equal to the size of vector * X .
Therefore, the conjugate process is fully controllable, and the original process is fully observable, i. e. the studied process of gross product manufacturing by an enterprise is a control object [5].
2. Determining the parameters of the process of gross product output of an enterprise Analytical dependences for determining of design parameterscoefficients of growth and retirement BPA: Dependences between weight factors of quadratic functional of quality and the range of their change: Selection of design parameters from the region of optimal values is performed by analysis of the steady law of the process development [10].
Let us verify the stability of the process of gross product manufacturing by the roots of the characteristic equation. The process is steady under the aperiodic law of development. In this case, the roots of the characteristic equation must be negative and different.
We select  =0.1086 from Table 2, c =0.00106 at m =1,5 from Table 1 Therefore, the process of gross product manufacturing of the enterprise is steady.
3. Obtaining of the optimal filter of the process of gross product manufacturing We will determine the matrix of the parameters of the optimal filter for gross product manufacturing by an enterprise .
Let us assume that during the process of gross product manufacturing, motions and velocities are measured, that is, all components of the vector of the process state are "observed". In this case, the dynamic process will be presented by differential equations The expression for the filter misalignment factor is 11 1 12 2 1 Them the matrix of parameters of the optimal filter will take the form 11 1 12 2 Analyzing the matrix of parameters of the optimal filter, we conclude that in this case coefficients of BPA growth and retirement must change [5]. If the values of the correlation functions 11 12 22 ,, K K K approximate or are equal to zero, the original process will be close to optimal. We will write down the matrix of parameters of the optimal filter as Therefore, coefficients of growth and retirement of BPA can be determined form formulas To obtain numerical values for parameters, it is necessary to solve the system of Riccatti equations on a computer and substitute values 11 12 22 ,, corresponding to the established process, in (25).
Analysis [7] of the matrix of the optimal filter parameters (25) shows that the values of coefficients of growth and retirement of BPA must change over time, i.e. coefficients of retirement and growth of BPA should not be constant at the stages of the lifecycle and an enterprise must develop intensively.

Conclusions
Thus, as a result of the conducted research: 1. Based on the concepts of observability and controllability, it was proved that the process of gross product manufacturing by an enterprise is a control object. 2. Analytical dependences for coefficients of growth and retirement of BPA were determined using the matrix method of dynamic programming [6] 3. Selection of weight coefficients of quadratic quality functional was performed. The selection was based on the requirement to get aperiodic law of development of the process of gross product output and the physically implemented synthesizing function [11]. 4. The regions of optimal values of design parameters were determined (Table 1 and Table 2). The data of Table 2 indicate that the value of the coefficient of BPA retirement is less than 0.3. The values of the coefficient of BPA growth (Table 1) are by two orders of magnitude less than the values of the coefficient of retirement, i.e. the optimal values of parameters provide for physical feasibility of the process of gross product output. 5. Selection of design parameters from the region of optimal values, which is provided by the steady law of process development, was substantiated. The growth coefficient is 0.00106 and BPA retirement coefficient is 0.1086 at fund capacity of 1.5. 6. Optimal filter of Kalman-Bucy for the process of gross product manufacturing by an enterprise was constructed. The values of coefficients of BPA growth and retirement of the filter matrix can change over time, i.e., these coefficients should not be constant at the stages of the lifecycle and an enterprise must develop intensively. 7. Modeling of the process of gross product output with optimal values of coefficients of BPA growth and retirement was performed in modeling system MVTU 3.7. The diagram of the process of gross product output without external investments (Fig. 5) and transitional characteristic (Fig. 7) prove the sustainability of the process. .