DEPENDENCE OF THE COST LEVEL ON THE MAIN TECHNICAL AND ECONOMIC INDICATORS OF ELECTRICAL INSTALLATION WORKS

I. A. OMELCHENKO

(Presented by the scientific seminar of the Department of Electric Power Stations and Electrical Networks and Systems)

Existing methods for determining cost amount to comparing direct costs with the estimated cost of the work. Analysis of the cost level is limited to comparison of planned and actual indicators. At the same time, no account is taken of the influence on cost of factors associated with the technical growth of production. Such factors may include: growth of labor productivity, which affects the reduction of cost by decreasing the share of wages; an increase in the volume of work, which leads to a reduction of overhead costs per unit of output. But the volume of work and labor productivity, in turn, themselves depend on many conditions: the qualifications of engineering and technical personnel and workers, the volume of work of a single project, the distance to it, the pace of construction, the state of production preparation, provision with materials and equipment, climatic conditions, and so forth.

All this affects the results of organizations’ activity. Under such circumstances, the method of mathematical statistics with the application of probability theory becomes the most effective, and sometimes the only possible, method.

This article presents the results of work on investigating the influence on cost of technical and economic indicators on the basis of the results of the work of the installation departments of the Sibelektromontazh trust in 1966.

The following characteristics and conventional notations are adopted in the article:

$Y$ — cost level, the ratio of actual cost to the estimated cost of the work;

$X_1$ — annual volume of construction and installation work, in million rubles;

$X_2$ — output per worker, in thousand rubles;

$X_3$ — average annual wage of one worker, in thousand rubles.

As can be seen from Table 1, there is no functional relationship between the indicators under study. To identify the correlation relationship, we shall determine the correlation coefficients between the dependent quantities.

Let us find the mean values of $Y$ and $X_1$:

\[ m_y=\frac{\Sigma Y}{n}=\frac{10{,}1286}{12}=0{,}84405, \]

Table 1

\[ m_{x1}=\frac{\Sigma X_1}{n}=\frac{36,951}{12}=3,07925. \tag{1} \]

Determine the statistical second initial moments:

\[ a_2^*[Y]=\frac{\Sigma Y^2}{n}=\frac{8,5617}{12}=0,713475, \]

\[ a_2^*[X_1]=\frac{\Sigma X_1^2}{n}=\frac{128,94}{12}=10,745. \tag{2} \]

Compute the statistical variances:

\[ \begin{aligned} D_y^*&=a_2^*[Y]-m_y^2=0,001055,\\ D_{x1}^*&=a_2^*[X_1]-m_{x1}^2=1,25323. \end{aligned} \tag{3} \]

Multiplying the statistical variances by \(\dfrac{12}{n-1}\), we find the unbiased estimate of the variances

\[ D_y=0,001151,\quad D_{x1}=1,367274. \tag{4} \]

Then the root-mean-square deviation is

\[ \sigma_y=\sqrt{D_y}=0,034,\quad \sigma_{x1}=\sqrt{D_{x1}}=1,17. \tag{5} \]

Find the statistical initial moment:

\[ a_{1,1}^*[YX_1]=\frac{\Sigma YX_1}{n}=\frac{31,17}{12}=2,5975. \tag{6} \]

The statistical correlation moment:

\[ K_{yx1}^*=a_{1,1}[YX_1]-m_y m_{x1}=-0,185. \tag{7} \]

The unbiased estimate of the statistical correlation moment is

\[ K_{yx1}=K_{yx1}^*\cdot \frac{n}{n-1}=-0,201. \tag{8} \]

Then the correlation coefficient is

\[ r_{yx1}=\frac{K_{yx1}}{\sigma_y\cdot\sigma_{x1}}=\frac{-0,201}{0.034\cdot 1,17}=-0,505. \tag{9} \]

In the same way we find all the correlation coefficients and enter the results in Table 2.

The correlation coefficients between \(Y\) and \(X_1\), \(X_2\), \(X_3\) are quite significant. This makes it possible to conclude that there is a substantial relationship between the technical and economic indicators under study. Negative

Table 2

The signs of some coefficients indicate a negative relationship between the factors.

The values found for the correlation coefficients make it possible to obtain the equation of linear regression between \(Y\) and \(X_1\) [3].

\[ Y - m_y = r_{yx_1}\frac{\sigma_y}{\sigma_{x_1}}(X_1 - m_{x_1}), \]

\[ Y - 0.84405 = -0.505 \frac{0.034}{1.17}(X_1 - 3.07925), \tag{10} \]

\[ Y = 0.89005 - 0.0149X_1. \]

The other dependencies are calculated similarly.

\[ Y = 1.0782 - 0.0227X_2, \tag{11} \]

\[ Y = 0.76987 + 0.0401X_3. \tag{12} \]

Figure 1 shows graphs of the dependence of \(Y\) on \(X_1, X_2, X_3\).

The level of cost depends on several arguments \(X_k\) acting simultaneously. To obtain the coefficients of a linear dependence of the form

\[ Y = a_0 + a_1X_1 + a_2X_2 + a_3X_3 \tag{13} \]

it is necessary to solve a system of linear normal equations.

\[ \begin{aligned} S_{00}a_0 + S_{01}a_1 + S_{02}a_2 + S_{03}a_3 &= b_0,\\ S_{10}a_0 + S_{11}a_1 + S_{12}a_2 + S_{13}a_3 &= b_1,\\ S_{20}a_0 + S_{21}a_1 + S_{22}a_2 + S_{23}a_3 &= b_2,\\ S_{30}a_0 + S_{31}a_1 + S_{32}a_2 + S_{33}a_3 &= b_3. \end{aligned} \tag{14} \]

Here:

\[ \begin{aligned} &S_{00}=n;\quad S_{01}=\sum X_1;\quad S_{02}=\sum X_2;\quad S_{03}=\sum X_3;\\ &S_{11}=\sum X_1^2;\quad S_{12}=S_{21}=\sum X_1\cdot X_2;\quad S_{13}=S_{31}=\sum X_1\cdot X_3;\\ &S_{22}=\sum X_2^2;\quad S_{23}=S_{32}=\sum X_2\cdot X_3;\quad S_{33}=\sum X_3^2;\\ &b_0=\sum Y;\quad b_1=\sum Y\cdot X_1;\quad b_2=\sum Y\cdot X_2;\quad b_3=\sum Y\cdot X_3. \end{aligned} \]

Then the system of equations (14) will take the form:

\[ \begin{aligned} 12a_0 + 36.951a_1 + 123.787a_2 + 22.207a_3 &= 10.1286,\\ 36.951a_0 + 128.94a_1 + 389.63a_2 + 67.52a_3 &= 31.17,\\ 123.787a_0 + 389.63a_1 + 1293.43a_2 + 230.2a_3 &= 104.55,\\ 22.207a_0 + 67.52a_1 + 230.2a_2 + 42.58a_3 &= 18.77. \end{aligned} \tag{15} \]

There are several methods for solving equations of this kind. In our example, the coefficients of the linear dependence were calculated using the standard program for the “Razdan-2” electronic digital computer in the educational computing laboratory of the Omsk Polytechnic Institute. They have the values:

\[ a_c = 0.7761598, \]

\[ a_1 = 0.0033938, \]

\[ a_2 = 0.005372191, \tag{16} \]

\[ a_3 = 0.01236006. \]

Fig. 1.

Thus, the linear equation for the dependence of the cost level on the technical-and-economic indicators has the form

\[ Y = 0.7761598 - 0.0033938 X_1 + 0.00537219 X_2 + 0.01236006 X_3. \tag{17} \]

Comparison of the actual data with the theoretical cost level shows a high degree of agreement.

In order that the derived equation may be applied to the entire aggregate of installation organizations, let us determine the confidence intervals for the coefficients.

On the basis of [2], we determine the confidence intervals for the coefficients of the linear equation at reliability \(0.9\) and \(K-8\):

\[ \begin{aligned} 0.7626098 &< a_0 < 0.7897098,\\ 0.0173938 &< a_1 < 0.0106062,\\ 0.004017191 &< a_2 < 0.006727191,\\ 0.00687006 &< a_3 < 0.01785006. \end{aligned} \tag{18} \]

Substituting the values \(X_1, X_2, X_3\) into equation (17) determines the cost level of electrical installation works. From this same equation, if information on the cost level is available, one can determine any technical-and-economic indicator included in the formula. The greatest percentage reduction in cost will correspond-

correspond to the lowest level of cost. The cost model has been determined for installation administrations of the Siberian zone, but it can also be applied to other installation organizations.

For this purpose, within the confidence intervals for the coefficients of the linear dependence, suitable values are selected and the cost level is calculated.

REFERENCES

1. Edited by A. A. Sveshnikov. A Guide for Engineers on Solving Problems in Probability Theory. Sudpromgiz, 1962.

2. V. Yu. Linnik. The Method of Least Squares and the Foundations of the Theory of Processing Observations. Publishing House of Physical and Mathematical Literature, 1962.

3. E. S. Venttsel. Probability Theory, 1958.