To preparation of proposals for the target programs of development of housing and communal services

The mathematical model of the formation and implementation of strategies for the development of the park of control and measuring equipment used in construction and housing and communal services is considered. As control functions, purchases, repairs and write-offs are selected. The model is based on a linear system of ordinary differential equations with constant coefficients. The system describes the dynamics of changes in the number of control and measuring equipment samples with different levels of technical excellence and technical condition. The model mainly uses algorithms based on the properties of stationary solutions of dynamic systems. Three different strategies for the development of a measuring equipment park have been considered. For each strategy, analytical and semi-analytical solutions of the stationary model are obtained. A distinctive feature of the model is its visibility and ease of use. The results of calculations are given.


Introduction
The park of control and measuring equipment (CME), used in the field of construction and housing and communal services, includes tens and even hundreds of thousands of CME samples. Over time, CME samples may degrade, may break (become faulty), may become morally and physically obsolete. Therefore, there is the problem of managing such a park of CME. Indicator of modernity and the indicator of serviceability are generalized indicators of park efficiency. The problem of managing the indices of modernity and serviceability of the park of CME is an urgent practical task not only in the sphere of housing and public utilities [1][2][3][4][5], but in the construction industry [6], in the field of engineering, in the field of aircraft building and other activities [7][8][9]. For the solution of the problem of effective controlling the park of CME there used variety of methods, for example [10][11].

Mathematical model of transitions of state of samples of control and measuring equipment
In this paper, we assume that according to the levels of technical excellence and technical condition CME samples are subdivided into modern serviceable, modern faulty, outdated serviceable and outdated faults. Let us denote: 1 x -number of modern serviceable CME samples; 2 x -number of modern faulty CME samples; 3 x -number of obsolete serviceable CME samples; 4 x -number of obsolete faulty CME samples.
The total number of CME samples in the park is determined by directive standards, the needs of production and operation, as well as tactical and strategic tasks of management companies.
Suppose that the following statistical values of the probabilities of transitions between the states of the CME samples were determined as a result of statistical data processing for a sufficiently long period of time: 12 p -the probability of a transition from modern serviceable to modern faulty state; 13 p -the probability of a transition from modern serviceable to obsolete serviceable state; 24 p -the probability of transition from a modern faulty state to an obsolete faulty state; 34 p -the probability of a transition from the obsolete serviceable state to obsolete faulty state.
We describe other transitions of states, which in the present paper we consider as controls: k -the share of modern faulty samples to be repaired.
We will asume that after repair and verification faulty samples become serviceable.
The graph of state transitions for 21 21 Dynamic equations [5,10] describing the process of development of the CME park are: (1) We will assume that the initial conditions are given: If the coefficients of the variables ) will take the required predetermined values. A stationary solution can be found by equating the derivatives on the left side of the system (1) to zero. The stationary solution satisfies the following system of linear homogeneous algebraic equations for unknown variables The rank of system (2)

Development of strategies for the development of CME park
We describe three different strategies for the development of CME park.

Strategy 1
We consider the case , which corresponds to the strategy, when a certain proportion of modern faulty CME samples are subject to repair and verification, outdated faulty CME samples cannot be repaired, instead of them modern serviceable samples can be purchased. Then from (3)  (4) Restriction on total number of CME, as well as restrictions on the indices of modernity and serviceability, we write in the form of a system of equations: where c P -indicator of the modernity, и P -indicator of the serviceable of the CME park.
If in the left part of (5) we substitute the expression (4), then the left parts of relations (5) will depend on the unknown parameters x . Solving this system of nonlinear algebraic equations by the Newton method [11], we obtain the values 41 k and 21 k , providing the required level of modernity and serviceability of the CME park.
Note that the order of system (5) can be reduced to the second order, if we express the variable 2 x from the first relation (5) with regard to (4) and substitute (6) into the second and third relation (5): Restrictions on the total number of CME, as well as the limitations on the indicators of modernity and serviceability are (5). The left parts of (5) subject to condition (7) depend on the unknown parameters . If, as a result of solving a system of nonlinear equations, the values of the control parameters are obtained that do not satisfy the indicated constraints, then this indicates that the control resource is not sufficient for the simultaneous fulfillment of requirements for modernity and good condition. In this case, to obtain a result suitable for practical use, it is necessary to attract additional resources. For example, joint repair and verification of both modern faulty and obsolete faulty CME samples. One of these possible strategies is described below.
We also note that in some cases the simultaneous fulfillment of restrictions on modernity and serviceability indices in the form of two independent equalities turns out to be impossible. Then it is recommended to weaken one of the restrictions and take it into account in the form of inequality.

Strategy 3
Let us consider a special case , corresponding to the strategy, when all modern faulty samples are subject to repair and verification in the same planning interval in which they become faulty, and also part of the obsolete faulty CME samples is subject to repair and verification. In addition, instead of a portion of obsolete faulty samples, modern, serviceable samples are purchased.
The stationary solution satisfies the following system of linear homogeneous algebraic equations: . (12) The obtained analytical dependencies (12) allow us to calculate the values of the control parameters 41 k and 43 k , depending on the required values of the performance indicators с P and и P . Fig. 2a,b shows the corresponding two-dimensional dependences.