On the Annual Productivity of a Coal Mine for a Section with Limited Reserves

N. G. Kapustin

(Presented by the scientific seminar of the departments of bedded-deposit mining, ore-deposit mining, and mining-enterprise construction)

The design determination of the annual productivity of a mine is a classical problem of Russian mining science. However, the present-day state of study of this problem still remains insufficient.

This provides grounds for continuing the search for methods for the computational determination of annual productivity and the service life of newly constructed mines.

It is well known that many sections of the richest coal regions of the country are coal fields with limited reserves, i.e., allowing the design of obviously no more than one mine. In such cases, the adoption of a decision on the magnitude of the production capacity of the mine directly indicates the corresponding term of its service. Therefore, for deposits with limited reserves, the problem of capacity and service life reduces to the computational determination only of the annual productivity of the mine.

The modern computational-analytical method for solving the problem of the capacity of a mine with limited reserves consists in the following.

As is known, Professors A. P. Sudoplatov, A. I. Smirnov, and others have expressed the function of the mine cost of coal production in the following form

\[ a = C_1 + C_2 A + C_3 : A = f(A), \tag{1} \]

where \(C_1, C_2, C_3\) are coefficients established individually for each region and having a physical meaning;

\(C_1\) is the element of cost not dependent on the magnitude of \(A\);

\(C_2\) is the same, directly proportional to the magnitude \(A\);

\(C_3\) is the same, inversely proportional to the magnitude \(A\);

\(A\) is the production capacity of the mine.

Analyzing the properties of function (1), Academician L. D. Shevyakov gives the important conclusion that for this function there exists an extremal value of the argument \(A_0\), at which the cost of coal will be minimal. At the same time, the optimal value of capacity should be expressed not by a specific number, but by a certain “from—to” interval, in which two values of mine capacity give cost values that differ from one another within the limits of the accuracy of the calculations.

If function (1) is interpreted graphically, then the optimal zone of the value \(A\), according to Sudoplatov, will appear as shown in Fig. 1 and will be calculated by the formulas

\[ A_n=(1+\Delta)A_0+\sqrt{(2+\Delta)\Delta}\cdot A_0; \tag{2} \]

\[ A_\ell=(1+\Delta)A_0-\sqrt{(2+\Delta)\Delta}\cdot A_0, \tag{3} \]

where \(\Delta\) is the adopted accuracy of calculating the value \(a\), in fractions of a unit.

If \(\Delta=0.09\) (i.e., 9%) is adopted, then the zone of optimal values of the quantity \(A\) will be bounded on the left by \(A_\ell=0.65A_0\) and on the right by \(A_n=1.52A_0\). The fluctuation of these limiting values will be \(A_n:A_\ell=1.56:0.64=2.34\), or 234%. Academician L. D. Shevyakov and Prof. A. P. Sudoplatov, in studying the causes of the emergence of a large range in the optimal zone, confined themselves only to pointing out that the range arises in connection with the slow variation of the function in the extremum zone. However, the slow variation of the function and the presence of a large range are readily explained if one turns to statistical data [1] on the amounts of depreciation of fixed assets (Table 1).

Table 1

Since the principle of attaining minimum cost is the most advantageous combination [2, 190] of depreciation and operating expenses in the cost of production, the coupling of the share of operating expenses at 97% with the share of depreciation at 3% cannot yield an exact value of the optimal capacity of a mine, giving rise to an extensive zone of its values. This is the shortcoming of the existing theory of the analytical calculation of a mine’s annual productivity.

In the literature, the opinion of Academician S. G. Strumilin was expressed [3] regarding the determination of the production capacity of an enterprise with account taken of the time factor. This thesis of Academician S. G. Strumilin, because of its extreme generality, could not be implemented by mining engineers, although in its profound scientific-economic content it is very promising.

The purpose of the present article is to set forth new principles for the design determination of the production capacity of a mine being laid out in a section with limited coal reserves.

When analyzing the operation of mines under equal conditions in one and the same coal deposit, it can be established that the size of accumulation may prove to be substantially different for two compared types of mines. Obviously, it is expedient to build the type of mine for which the share of accumulation will be greatest. The size of accumulation depends on the levels of cost and labor productivity. The decisive condition for reducing the cost of production is the growth of labor productivity; to ensure this, an increase in the relative share of workers in the principal production process—at the expense of underground, service, and auxiliary personnel—is of enormous importance.

The personnel structure of coal mines contains seven groups of workers: engineering and technical personnel, mine foremen, clerical employees, junior service personnel, apprentices, workers of the permanent co-

staff and workers of the variable staff. The first six groups are, as it were, fixed and do not depend, or almost do not depend, on the volume of production. For example, the administrative and managerial personnel for a mine with a capacity of 240 thousand tons per year amounts to 38 persons; for a mine with a capacity of 450 thousand tons—only 44 persons. The capacity of the mine has increased by almost 200%, while the office staff has increased by 15%.

Consequently, designers must increase the productive capacity of a new mine. However, an excessive increase in mine capacity entails an irrational character of the capital investments made in mine construction, representing the expenditures of past years. The struggle to raise labor productivity requires not only saving working time and materials, making maximum use of tools of labor, and improving the quality of output; this everyday struggle for the productivity of living labor cannot obscure another important goal—the effective use of, and careful attitude toward, the fixed assets of the enterprise, created by the labor of past years.

A large mine, compared with a small mine, has savings in auxiliary and subsidiary services; it is better equipped with machinery and therefore has greater labor productivity at the mine. Consequently, it gives an increase in accumulation.

But the larger fixed assets of a large mine, with fixed reserves of the mining allotment, tie up in production for many years larger resources of materialized labor and deprive the national economy of mobility at the necessary moment for introducing new, and then the newest, technology. At a large mine, with fixed reserves of the mineral resource, the fixed assets are amortized over a shorter period, causing a considerable share of depreciation, which can substantially raise the level of production cost, thereby causing unprofitable operation of the enterprise and, consequently, a decrease in accumulation. In addition, there arises the undesirable phenomenon of liquidating one mine after all the reserves of its field have been extracted and constructing another mine on a new site, although the mine workings, buildings, and structures of the mine being liquidated are still suitable for performing their functions. As a rule, such liquidation is irrational, for the working condition of the fixed assets of any mine, under the technical conditions of production, is maintained throughout its entire service life. This is achieved by means of systematic capital repairs, carried out at the expense of a certain share of depreciation charges.

The liquidation of a large mine because all the reserves of its mining allotment have been worked out is an undesirable phenomenon also because its service life may be such that its fixed assets do not become technically obsolete.

The considerations set forth inevitably lead to the conviction that cheaper mines should be built, i.e., mines of lower productivity. However, a small mine operates with lower labor productivity and a higher production cost, which follows at least from consideration of the properties of the mine-cost function $a = C_1 + C_2 A + C_3 : A$.

Obviously, there exists such a value of the productive capacity of a mine to be laid out on a site with fixed coal reserves that gives an advantageous combination of the amount of initial capital investment and the productivity of the workers at this mine.

There arises the necessity of considering jointly the expenditures of past labor materialized in fixed assets and of current living labor. Finding the optimal mine capacity according to the single principle of achieving the minimum production cost does not yet ensure, to the proper extent, a joint consideration of these two aspects of production. It is necess—

an additional criterion of the scale of production, making it possible to take such accounting fully into account. The required joint accounting of the labor of past years and of living labor is achieved by calculating the gross accumulation for the enterprise, representing normalized accumulation less the value of fixed assets at the moment the enterprise is closed.

For construction it is desirable to assign such a capacity to a mine that the extraction of the fixed reserves of the mining allotment will ensure maximum gross accumulation at the minimum cost of extracting the useful mineral. If the normalized accumulation over the service life of the mine is \(E\), and the value of the mine’s assets, evaluated according to the market conditions of the year of the mine’s liquidation, is \(K_0\), then the gross accumulation will be the function

\[ b = E - K_0. \tag{4} \]

Mathematically, the problem consists in finding such a capacity of the mine that would turn the function \(b = E - K_0\) into a maximum. At the same time, the same capacity of the mine must give the minimum value of the cost of coal extraction when substituted into the equation \(a = C_1 + C_2 A + C_3 : A\). Thus there arises the principle of a joint analysis of two functions: accumulation—\(b\) and cost—\(a\).

In assigning the production capacity of a mine, one must ensure the highest accumulation at the minimum cost of coal not in a single calendar year, but over the entire service life of the mine.

The principles set forth for the calculated substantiation of the design productivity of a mine may be formulated as follows: the designed capacity of a mine, while satisfying mining-geological and directive requirements, must give the greatest accumulation, after deducting the value of the fixed assets lost upon completion of mining operations within the boundaries of its field, while simultaneously ensuring the lowest cost of output and the highest labor productivity during the period of operation of this mine.

This general principle is subdivided into particular requirements:

1) ensuring the minimum cost of the useful mineral;

2) achieving such a combination of the scale of capital investments with an expedient structure of the mine’s labor force, under which the maximum national income and its share in the form of accumulation, less the value of fixed assets in the prices of the time of closure of the given mine, will be ensured;

3) correspondence of the mine to the natural conditions of the deposit and to the directive guidelines of the Party and the Government in the field of the regional distribution of coal extraction, the introduction of new technology and advanced experience.

The requirement of the greatest accumulation at the lowest production cost follows from the essence of the basic economic law of socialism; therefore this requirement must be fully satisfied when designing the production capacity of a mine as the principal parameter of its type.

Let us formulate and solve the problem of finding the function of gross accumulation \(b = F(A)\), where the quantity \(b\) would represent the accumulation for a mine of production capacity \(A\).

The number of man-shifts worked over the service life of the mine is expressed by the formula:

\[ N = 10^6 \cdot \frac{A \Theta}{p_0}, \tag{5} \]

where \(p_0\) is the productivity of a worker at the mine,

\(\Theta\) is a coefficient taking into account the systematic increase in labor productivity,

\[ \Theta = \frac{1+\mu}{\mu}\left[1 - \frac{1}{(1+\mu)^t}\right], \tag{6} \]

\(\mu\) is the annual percentage increase in labor productivity in the coal industry, adopted for the mine’s service life as an average value; the quantity \(\mu\) is expressed in hundredths of a unit.

Formulas (5) and (6) follow from the following reasoning. The number of man-shifts worked is:

\[ \begin{array}{rcl} \text{in the 1st year} & \ldots & 10^6 A : p_0,\\ \text{in the 2nd year} & \ldots & 10^6 A : p_0(1+\mu),\\ \text{in the 3rd year} & \ldots & 10^6 A : p_0(1+\mu)^2,\\ & \ldots & \\ \text{in year } t & \ldots & 10^6 A : p_0(1+\mu)^{t-1},\\ \hline \text{over the entire period of } t \text{ years} & \ldots & N . \end{array} \]

\[ N=10^6\frac{A}{p_0}\left[1+\left(\frac{1}{1+\mu}\right)+\left(\frac{1}{1+\mu}\right)^2+\ldots\left(\frac{1}{1+\mu}\right)^{t-1}\right] =10^6\cdot \frac{A\Theta}{p_0}\ \text{man-shifts}. \tag{7} \]

We shall denote the sum of the geometric progression in the square brackets of expression (7) by \(\Theta\). The wage fund of the mine’s workers over the period will be \(TN\), or \(10^6AT\Theta : p_0\), where \(T\) is the average daily earnings.

Let us find the accumulations \(E\) created by the workers according to the norm \(m\%\). If over the period \(t\) the mine yields national income \(U\), and the share of personal consumption in this income is

\[ \frac{100-m}{100}\cdot U, \]

then the accumulations will be

\[ E=\frac{m}{100}\cdot U. \]

Then we have

\[ E+10^6AT\Theta:p_0=U, \]

\[ \frac{100-m}{100}\cdot U=10^6AT\Theta:p_0. \tag{8} \]

Solving equations (8) jointly, we find

\[ E=10^6\frac{m}{100-m}\cdot \frac{AT}{p_0}\cdot \Theta\ \text{rubles}. \tag{9} \]

The factor \(AT:p_0\), which appears in formula (9), represents the annual wage fund of the mine’s workers at the level of the base year, i.e., the year in which the mine begins operation.

Mining statistics in the coal industry, organized on a scientific basis, can and should serve the purpose of revealing the functional relationship between the annual wage fund for a mine and the annual productivity of this mine.

As applied to Kuzbass, these statistical data for 1948–1953 for 88 mine-years, when processed by the method of least squares, can be interpreted analytically by the following formula:

\[ AT:p_0=700\log(1+0.1A)\pm\rho, \tag{10} \]

where \(A\) is in million tons, and \(\rho\) is the root-mean-square error, 0.5 million rubles.

Formula (10) gives deviations from the specific statistical data, as we have established, within no more than \(\pm 5\%\) for mines with a capacity of \(0.3\) million tons/year and for mines with a capacity of \(1.5\) million tons/year—\(1\%\); this makes it possible to adopt it for subsequent calculations.

Expression (4) contains the value of the mine’s fixed assets at the time of the enterprise’s closure, \(K_0\), which depends on the size of their initial value, i.e., the sum of the general estimate \(K\), and the mine’s service life \(t\). Let us find the function \(K=\psi(A)\).

As is known [5, 89], in general form

\[ K=E_1+E_2A+E_3\sqrt{A}. \tag{11} \]

If, for three specific mines for which the values \(K\) and \(A\) are known, expression (11) is written down, then, when considered jointly, they form a system of three equations with three as yet unknown coefficients \(E_i\).

Using the production statistics of Kuzbass, taking account of design data, in the course of the present study 30 such equations with three unknowns were compiled. As a result of solving them by the method of least squares, we obtain, in final form, formula (12).

\[ K = 91A + 22\sqrt{A},\ \text{million rubles}. \tag{12} \]

Next, let us denote the total cost of the mine’s housing construction by \(\varphi K\), and the cost of machines, apparatus, and mechanical devices by \(\beta K\), where \(\varphi, \beta\) are the shares of these types of expenditures [6] in the amount of the general estimate. These two types of expenditures must be considered separately, since the housing stock and mechanical equipment are realized in the national economy after the liquidation of the mine at a new, so-called replacement cost at the prices of the year of the mine’s liquidation.

As is known, some part of depreciation charges is systematically paid into the industrial bank. Then over the service life of the mine \(t\), the sum of initial capital investments that will enter the bank, i.e. will be restored, is equal to

\[ wKt, \tag{13} \]

where \(w\) is the annual percentage (in hundredths of a unit) paid into the industrial bank for national purposes while simple reproduction of the means of production continues.

In addition, which is very important, after the liquidation of the mine, in the realization of the housing stock, machines and equipment, and scrap metal, a certain part of the initial investments will be restored. This part can be calculated by formulas (14) and (15), whereby the factor of housing construction will be taken into account in solving the posed problem. The neglect in previous studies of the factor of housing construction, which accounts for 44% of the amount of the mine’s general estimate, is a substantial omission.

The replacement cost of the housing stock will be

\[ \frac{\varphi K}{(1+\delta)^t}, \tag{14} \]

where \(\delta\) is the average annual increase in labor productivity in the construction industry (hundredths of a unit) relative to the preceding year.

Formula (14) was recommended by Academician S. G. Strumilin [3] for determining the new value of fixed assets created in the past and subject to revaluation after \(t\) years. Analogously, the replacement cost of machines and equipment will be

\[ \frac{0.9\cdot \beta \cdot K}{(1+\delta)^t}, \tag{15} \]

where \(0.9\) is a coefficient accounting for dismantling costs.

In view of its small magnitude, we neglect the value of scrap metal. Thus, the total amount of return after liquidation of the mine will be

\[ D = wKt + \frac{K(\varphi + 0.9\beta)}{(1+\delta)^t}. \tag{16} \]

The difference between expressions (12) and (16) will give the desired remainder \(K_0=K-D\) of the value of the fixed assets, not restored during the service life of the mine and subject to deduction from the accumulations \(E\) created by living labor.

The difference \(E-K_0=b\) will already be accumulation in pure form, whose relative magnitude fully characterizes the comparative expediency of variants of the annual productivity of the mine.

On the basis of formulas (9), (10), (16), we write, in millions of rubles:

\[ b=E-K_0=\frac{m}{100-m}\,700\log(1+0.1A)\Theta-K+wKt+ \frac{K(\varphi+0.9\beta)}{(1+\delta)^t}. \]

\[ b=700\cdot \frac{m}{100-m}\log(1+0.1A)\Theta -K\left[1-wt-\frac{\varphi+0.9\beta}{(1+\delta)^t}\right]. \tag{17} \]

Introduce the notation

\[ 1-wt-\frac{\varphi+0.9\beta}{(1+\delta)^t}=W. \tag{18} \]

Then, in final form, for \(m=25\%\),

\[ b=F(A)=233\log(1+0.1A)\Theta-(91A+22\sqrt{A})W. \tag{19} \]

For further analysis of function (19), we shall apply the method of variants, combined with the analytical method.

Assigning variant values of the mine capacity and having at our disposal the industrial reserves of the section, we find the corresponding service lives of the mine \(t_i\). For the variant values \(t_i\), by formulas (6) and (18) we find the quantities \(\Theta_i\) and \(W_i\).

The system of dependent quantities: \(A_1,\Theta_1,W_1;\ A_2,\Theta_2,W_2;\ A_3,\Theta_3,W_3\) makes it possible, by formula (19), to find the corresponding quantities \(b_1,b_2\), and \(b_3\).

On the basis of the pairs of values \(A_1,b_1,\ A_2,b_2\), and \(A_3,b_3\), we construct the graph (Fig. 1) of the function \(b=F(A)\) and make sure that it is nothing other than a parabola of the form

\[ b=d_1+d_2A-d_3A^2=F(A), \tag{20} \]

turned with its convexity toward the ordinate axis.

Such a parabola has one maximum ordinate \(b_0\) at the abscissa

\[ A_{\mathrm{og}}=d_2:2d_3, \tag{21} \]

\[ b_0=d_1+d_2^2:4d_3. \tag{22} \]

The variable part \((d_2^2:4d_3)\) of the quantity \(b_0\) may be permissibly reduced to the quantity \((1-\Delta)d_2^2:4d_3\), where \(\Delta\) is the accuracy of the techno-economic calculations. Allowance for the inaccuracy of the calculations will lead to the emergence of a zone of values of \(A\) in which all ordinates from \(b_0\) to \((1-\Delta)b_0\) are sufficiently acceptable. On this basis we write

\[ d_2A-d_3A^2=(1-\Delta)d_2^2:4d_3. \tag{23} \]

We transform equation (23):

\[ A^2-\frac{d_2}{d_3}A+0.25\,\frac{d_2^2}{d_3^2}(1-\Delta)=0. \tag{24} \]

The roots of equation (24) represent (Fig. 1) the right and left abscissas of the range:

\[ A_n=(1+\sqrt{\Delta})A_{\mathrm{og}};\qquad A_\ell=(1-\sqrt{\Delta})A_{\mathrm{og}}. \tag{25} \]

Taking \(\Delta = 0.09\), we obtain \(A_n = 1.3 A_{OF}\), \(A_l = 0.7 A_{OF}\), which gives a fluctuation of the optimal values of \(1.30 : 0.70 = 1.86\), or \(186\%\).

Taking into account the meaning of the graph of the functions \(f(A)\) and \(F(A)\) in Fig. 1, and pursuing the intention of simultaneously satisfying the two principles (\(a\)—minimum, \(b\)—maximum), it is logical to assign for construction such a standard value of the production capacity of the mine \(A_T\), according to Order No. 270 of the Ministry of the Coal Industry of May 31, 1952, that satisfies the condition \(A_{OF} \leq A_T \leq A_{Of}\).

Fig. 1. Graphs of the functions of the cost of mining 1 ton of coal and of the gross accumulations of the mine.

It turns out that such a condition readily makes it possible to find one standard value \(A_T\). Along with this, the value \(A_T\) must also satisfy other requirements:

1) the technical feasibility of mining operations with a level height \(h\), obtained on the basis of the formula \(h = A_T : 2 L \sigma c\), where \(L\) is the advance of the longwalls, \(\sigma\) is the productivity of the seams possible for simultaneous extraction, and \(c\) is the coefficient of extraction of reserves;

2) sufficient correspondence of the height \(h\) to its economically most advantageous certain value \(h_0\), established by the method of variants [2, 184]. In determining the magnitude \(h_0\), the length of the field and the mine capacity \(A_T\) are considered given;

3) satisfaction of the state plan for the development of coal production in the given district;

4) assignment of the mine capacity in accordance with the reliability of the reserves, the structure of the deposit, the grade of coal, and the degree of difficulty of carrying out capital mining operations and construction on the surface.

The factors listed affect the final decision on the designed capacity of the mine in the following way.

1. If the analytical value \(A_T\) satisfies the above-listed requirements, then it should be adopted for construction.

2. If it is found that \(h_0\) is substantially greater or less than \(h\), then the designed capacity of the mine should accordingly be reduced or increased, adopting the adjacent standard value greater or less than \(A_T\), but without going outside the range of optimal values established by the formulas

2, 3, 25; in this case the range may be limited, for example, by values from \(A_l = 0.65 A_{of}\) to \(A_n = 1.30 A_{of}\), or from \(A_l = 0.70 A_{of}\) to \(A_n = 1.52 A_{of}\).

1. If the coal-production plan for the district is high, or if reserves of coal of industrial categories amount to no less than 50%, and also in the case of coking coals, the typical value of mine capacity should preferably be chosen closer to the right-hand boundary of the range. Conversely, when the reliability of the deposit is low, in the case of steam coals, and with a moderate depth of the deposit, preference should be given to the left-hand side of the range of optimal values.

The factors listed will be quite sufficient for making a well-argued decision on the production capacity of the mine being designed.

Using the method described, let us find the annual output of a mine for a section with dimensions: \(S = 3400\) m, \(\vartheta h = 1560\) m, \(\sigma = 4.2\) t/m\(^2\). The seams are gently dipping; the seams crop out beneath the overburden and are opened by two inclined shafts. We divide the mine field into two wings. Coal reserves are \(G = 20\) million t. Here \(\vartheta\) is the number of levels; \(h\) is the height of a level.

In order to determine the cost function for mining one ton of coal, we take into account seven types of production costs.

1. Amortization of the initial capital investments and capital repair of industrial fixed assets will amount, per 1 ton of coal, to

\[ a_1 = \frac{K}{G}\left[1 + 0.018\,t - \frac{0.44}{(1+\delta)^t}\right], \tag{26} \]

where \(0.44\) is the coefficient allowing for conjunctural losses in housing construction arising upon liquidation of the mine;

\(0.018\) is the coefficient allowing for capital repairs, calculated as 3.2% of the amount of the general estimate according to the industrial factor \(0.56K\);

\(K\) is the amount of the general estimate for the mine according to formula (12), \(= 91A + 22\sqrt{A}\) million rubles;

\(\delta, t, G\) are the previously adopted notation.

1. Amortization of systematic capital investments made in opening and preparing new levels will amount, per 1 t of coal, to

\[ a_2 = \frac{k_{\mathrm{sh}}}{hc\sigma} + \frac{D + k_c h}{G}\,\vartheta, \tag{27} \]

where \(k_{\mathrm{sh}}, k_c\) are the cost of driving 1 running meter of gallery and of all shafts;

\(D\) is the cost of driving and equipping the shaft station yard at each horizon;

\(h,\vartheta\) are the previously adopted notation; \(\sigma\) is the coal reserves extracted from 1 m\(^2\) of seam floor; \(c\) is their extraction coefficient.

1. The cost element associated with the factor of the staff of surface workers, as is known, stands in significant relation to the annual productivity of the mine. Therefore this expenditure must be taken into account in determining the function \(a = f(A)\).

In the general case, mine workers are divided into three categories. The first category consists of workers at extraction operations; their labor productivity does not depend on the size of production.^1 The second category consists of auxiliary underground workers; their number and labor productivity, depending on the profession, influence the formation of the cost of coal mining in different ways. One part of the staff of auxiliary underground workers is proportional to the amount of coal mined, and it may be neglected when forming and investigating the minimum of the cost function. However, another part of the workers servicing the maintenance of workings, haulage, and mine hoisting must and can be

^1 With the same mechanization of extraction operations (Ed.).

accounted for in the problem being solved by calculating the expenditures for these types of production costs. Such accounting is achieved by the following formulas (29–31).

Only the last, third, category of mine workers—surface workers, employees, and junior service personnel—is to be accounted for as a separate expense item. The wages of the mine-surface workers per \(1\ m\) of coal will be

\[ a_3=\frac{T_n}{P_n}, \tag{28} \]

where \(T_n\) is the average daily wage of a mine-surface worker, taken from column 17, section 1, of tabular report form No. 17-sh, approved by the Central Statistical Administration of the USSR on July 15, 1953, for a group of mines or a coal trust;

\(P_n\) is the shift productivity of a mine-surface worker, selected from Table 2 for the variant value of the mine’s daily output.

The statistical data of Table 2 for the Donbass are taken from the work of I. E. Atlas [7, 34]; the indicators of the analytical calculation are taken from the literary source [8, 372].

Table 2

The coefficient for converting from the productivity of surface workers to the productivity of surface personnel in 1953 was, for the Vakhrushev mine (Kuzbass), 0.67; for the Taibinskaya mine, 0.77; for mine No. 3 of the Kiselevsk district of Kuzbass, 0.79; for mine No. 5, 0.77; and for the Surtaikha mine of the same district, 0.67. On average for these small and large mines, this coefficient is equal to 0.75, which is what has been adopted in Table 2.

1. The cost of transport along the entries and their maintenance per \(1\ m\) of coal will be

\[ a_4=0.25\,q_{\text{ш}}S+\frac{r_{\text{ш}}St}{G}, \tag{29} \]

where \(q_{\text{ш}}\) is the unit cost of transportation per tonne-meter;
\(S\) is the length of the mine field; \(r_{\text{ш}}\) is the cost of maintaining an entry, in rubles per \(1\) linear meter per year; \(G\) is the industrial reserves of the mine field; \(t\) is the service life of the mine.

1. In view of the presence of “counter” movement of coal—down the dip in the extraction district, and upward along the mine shaft toward the rise—there will be production costs for the “overhaulage” of coal in the following amount per \(1\ m\):

\[ a_5=0.5\,(q_y+q_c)\,h. \tag{30} \]

where \(q_y\) is the cost of haulage in the section along the line of dip of the seams, in rubles per 1 ton-meter; \(q_c\) is the same along the mine shaft.

1. The lowest costs for drainage would occur in the case of lifting water directly from each running meter of depth of occurrence. In fact, water is concentrated at the haulage level, which entails the cost of “re-lifting” the water, amounting per 1 \(t\) of coal to

\[ a_6 = 0.5\, k_B q_B h, \tag{31} \]

where \(k_B\) is the water-abundance coefficient; \(q_B\) is the cost of lifting 1 \(m^3\) of water through 1 linear meter of height.

1. The cost of energy for moving air through the drifts and shafts in ventilating the mine will be, per 1 \(t\) of coal [2, 210],

\[ a_7=\left[C_{\!ш}\left(\frac{Q}{2}\right)^3 S + C_c h Q^3 \vartheta\right] : A, \tag{32} \]

where \(Q\) is the quantity of air supplied to the mine, in \(m^3/\mathrm{sec}\); \(C_{\!ш}, C_c\) are letter designations according to Academician L. D. Shevyakov [2, 210].

The cost of coal production according to the seven named types of expenditures will be

\[ a=f(A)=\sum_{1}^{7} a_i . \tag{33} \]

We shall obtain the cost function in explicit form by combining the method of variants with the analytical method. The initial parameters for the calculation, in accordance with the adopted notation, are summarized in Table 3.

In addition, for all variants we take \(L=300\,m/\mathrm{year}\), \(\sigma c=3.78\,t/m^2\), \(T_n=24\) rub/cm, \(\partial=0.04\), \(\psi=0.06\), \(\gamma=0.1\) rub/kWh, the unit rate for driving a drift as 120 rub/\(m^3\), for the shaft station as 150 rub/\(m^3\), and for the shaft as 250 rub/\(m^3\); \(\xi=0.44\), \(\beta=0.14\). The volume of the shaft station and its chambers is found by A. S. Popov’s formula [2, 93]: \(5\sqrt{A}+0.002A\).

On the basis of the data in Table 3, using formulas (27–32), we compute the elements of cost, as shown in Table 4.

On the basis of pairs of values of the quantities \(A, a\), selected from Table 4, we find by the method of least squares the function (34), which expresses in algebraic form the results of the variant calculations:

\[ a = 14.7 + 5.1 A + 1.74 : A . \tag{34} \]

The accuracy of this algebraic interpretation of the data of Table 4 will be characterized by the absolute error

\[ \rho=\sqrt{[\varepsilon\varepsilon] : n'(n-3)} =\sqrt{1.11:28} =\pm 0.17\ \text{rub}/t, \]

which amounts to a relative error of less than 1%. The squares of the deviations \([\varepsilon\varepsilon]\) of function (34) are presented in Table 5.

Thus, in order to satisfy the first requirement of minimum cost, the mine capacity may be taken in the project in the range from \(0.65\sqrt{1.74:5.1}=0.37\) to \(1.52\sqrt{1.74:5.1}=0.88\) million \(t\). Within this range lie three standard mine capacities \(A_T=0.45;\ 0.6;\ 0.9\), which give equal or nearly equal costs of coal production.

Next we find the system of dependent quantities \(A_1, \Theta_1, W_1, A_2, \Theta_2, W_2\), as shown in Table 6, on the basis of which, by formula (19), we compute the corresponding values of the quantities \(b_1, b_2, b_i\); the calculations are carried out according to the form of Table 6. Using pairs of values of the quantities \(A, b\), following—

Table 3

Technical and cost parameters as a function of the variant value of mine capacity

Table 4

Calculation of the cost of coal extraction for variants of mine capacity

Table 5

Calculation of the squares of deviations for function (34)

\[ [\varepsilon\varepsilon]=1{,}11 \]

Table 6

Calculation of the accumulation magnitude for variants of mine capacity

Table 7

Calculation of the squares of deviations for function (35)

\[ [\varepsilon\varepsilon]=48 \]

... from the calculations in Table 6, we find, by the method of least squares, the function (35), which will express in algebraic form the results of the variant calculations,

\[ b=14+92.5A-50A^2. \tag{35} \]

The replacement of the tabular data by function (35) will contain a root-mean-square error
\(\rho=\sqrt{[\varepsilon\varepsilon]:n(n-3)}=\sqrt{48:28}=\mp 1.3\) million rubles, or a relative error of less than \(2.5\%\). The squares of the deviations \([\varepsilon\varepsilon]\) of function (35) are presented in Table 7.

Consequently, in order to achieve the second requirement—maximum accumulations of the enterprise—it is possible to adopt in the design the mine capacity in the range, according to expression (25), from
\(0.7\,\dfrac{92.5}{2\cdot 50}=0.63\) to \(1.3\,\dfrac{92.5}{2\cdot 50}=1.17\) million tons/year. In this range there also lie three standard mine capacities, \(A_T=0.6;\ 0.9\), and \(1.2\) million tons/year.

In accordance with the proposition advanced concerning the designation in the design of a standard mine capacity that would yield maximum accumulations at minimum cost, we adopt the design capacity of the mine in the example under consideration in the range from \(A_{nf}=0.88\) to \(A_{AF}=0.63\) million tons/year, which constitutes a variation of \(140\%\), as against \(240\%\) according to formulas (2,3).

Proceeding from the principles of industrial methods of construction, which provide for the standardization of enterprises of the national economy, under the conditions of the example under consideration it proves advisable to adopt, within the limits of the established range \((0.63—0.88)\), a standard mine capacity equal to \(A_T=0.6\) million tons/year, with a service life of \(20:0.6+3=35\) years. Let us note that the right-hand boundary of the range, \(A_{nf}=0.88\), also comes close to the standard value \(A_T=0.9\); however, we reject a mine capacity of \(0.9\) million tons/year for two reasons:

a) a mine with a capacity of \(0.9\) million tons/year will have a service life of 25 years, which contradicts the advanced experience of coal-mine construction in the USSR and actual production data;

b) a mine with a capacity of \(0.9\) million tons/year lies beyond the boundary of the range of optimal productivities with respect to the minimum cost.

For the example under consideration, the figures obtained, \(A_T=0.6\) and \(t=35\), fully correspond to the technical directives of the Ministry of the Coal Industry in the field of mine construction, set forth in Order No. 270 of November 31, 1952.

Conclusions

The existing method of computationally establishing the capacity of a mine being designed must be further developed and improved by the introduction of the new principle of “achieving maximum accumulations at minimum extraction cost.”

The proposed method of calculation is consistent with the known propositions:

a) of Academician L. D. Shevyakov on the most advantageous combination of depreciation and operating expenses;

b) of Academician S. G. Strumilin on taking account of the time factor in the design of capital investments;

c) of Prof. P. I. Gorodetsky [10; 215], determining the choice of the productivity of a mining enterprise according to six conditions;

d) of Prof. Ya. V. Aleshkevich on ensuring the regional state coal-production plan by means of a type of mine that is expedient in techno-economic terms.

The studies set forth above do not contradict the preceding works of other authors, but achieve the satisfaction of diverse requirements by a single computational method and derive the solution of the classical problem of mine capacity from a wide range of equivalent answers for specific conditions.

REFERENCES

1. Sobol’. On the question of the effectiveness of capital investments. Planovoe khozyaistvo, No. 4, 1947.

2. Shevyakov L. D. Fundamentals of the theory of designing coal mines. Ugletekhizdat, 1950.

3. Strumilin S. G. Taking account of the time factor in the design of capital investments. Izvestiya AN SSSR (Department of Economics), No. 2, 1946.

4. Report of the Central Committee of the VKP(b) to the XIX Party Congress on the work of the Central Committee of the VKP(b). G. M. Malenkov, Gospolitizdat, 1952.

5. Popov A. S. Techno-economic analysis in mining. State Scientific-Technical Mining Publishing House, 1932.

6. Kabatyanskii Kh. S. Basic questions of mining statistics. Ugletekhizdat, 1953.

7. Atlas I. E. Face loading and labor productivity of workers in mines in the Donbas. Ugol’, No. 8, 1952.

8. Basic questions of mining engineering (analytical study). Scientific-Technical Council of the Coal Industry. State Scientific-Technical Mining Publishing House, 1933.

9. Gorodetskii P. I. Design of mining-ore enterprises. Metallurgizdat, 1949.