Fundamentals of Designing the Production Capacity of a Mine for Coal Deposits with Unlimited Reserves

N. G. Kapustin

(Presented by the scientific seminar of the departments of bedded-deposit mining, ore-deposit mining, mine construction, safety engineering, and mine ventilation)

In our work [1], a solution was set forth for the problem of selecting the design capacity of a mine under the conditions of a deposit with limited reserves. In doing so, it was assumed that the proper production capacity should be understood as its technically attainable magnitude, ensuring, over the service life, the maximum profitability of the mine at the minimum cost of extraction, established with allowance for the most expedient planning of mining operations, the effective use of new mining equipment, and the time factor in capital investments. Proceeding from this definition, two indicators must be established in the course of variant calculations: the cost of extraction and the income of the mine. Further joint consideration of these indicators becomes possible because both simultaneously depend on a single parameter—the production capacity of the mine.

In full accordance with this formulation of the question, the present article sets forth a solution to the problem of the design justification of mine capacity under the conditions of a suite of steeply dipping seams with unlimited reserves. We present the method as applied to the specific conditions of the Zenkovo deposit (Kuzbass), whose geological structure is shown in Fig. 1.

Fig. 1. Geological section along exploration line VIII.
Scale 1:5000

In the course of the study it is necessary first to establish the scheme and parameters for planning mining operations as the technical basis for solving the problem of mine capacity. This follows from the proposition that extraction operations account for [3, 147] about \(1/3\) of the production costs of coal mining. Therefore, the proposal [2] to base the calculation of mine capacity on the scheme and parameters for planning mining operations, including the optimal length of the longwall, is important and should be taken into account when designing a mine type. Bearing in mind the goal of achieving maximum labor productivity in the mine’s principal technological link—the extraction face—we establish the optimal length of the longwall face and the height of the sublevel; in our example it is equal to 40–50 m. The subsequent selection of a technically safe and economically most advantageous scheme for preparing the mine field and its parts will likewise serve the purpose of achieving maximum labor productivity for the auxiliary and service workers as well.

We establish the scheme for opening up the deposit on the basis of the geological structure of the deposit, the capacity of the mine, and the best practices of mine construction. In the example considered, on the grounds set forth, we establish: a) opening of the mine field by vertical shafts with level crosscuts; b) preparation of the level by group drifts under a flank ventilation scheme; c) preparation of the extraction field according to the scheme in Fig. 2, characterized by haulage to the rear crosscut.

Fig. 2. Basic scheme for the development of mining operations on the wing of a level
(plan of workings)

Optimal mine capacity with respect to the cost of mining

The capacity of a mine must be optimal with respect to the cost of mining 1 t of coal. This proposition is indisputable for any situation in mine construction. The point is that designing

according to the principle of achieving the minimum cost does not even pose the question of whether the mine is “to be or not to be”; a different problem is solved—to assign its capacity in such a variant as will ensure the lowest cost in comparison with other variants of the design under otherwise identical conditions. The success of solving the problem follows from the fact that, however high the absolute level of the cost of mining at a given deposit may be, there will always be, among a wide range of variants, a type of mine with a minimum cost.

If the mine capacity is \(A\), and the cost of mining \(1\ t\) of coal is \(a\), then the cost function will be [1]:

\[ a = E_1 + E_2 A + \frac{E_3}{A}, \tag{1} \]

where \(E_i\) are specified numerical coefficients established separately for each coal region. For example, for operating mines of the Kuzbass, we have identified function (1) in the following form (Fig. 3):

Fig. 3. Cost curve for coal mining at Kuzbass mines in 1948–1953 (costs for sorting and stoping extraction excluded)

\[ a = 3.53 + 20.95 A + \frac{10.6}{A}, \tag{2} \]

where \(A\) is in million \(t/\text{year}\).

For the mines of the Prokopyevsk-Kiselevsk district the coefficients \(E_i\) are different, namely:

\[ a=1.5+23.1\,A+\frac{9.95}{A}. \tag{3} \]

In this way one may verify that, regardless of whether the cost function has been obtained theoretically or empirically, its algebraic structure will remain unchanged in the form of equation (1).

Finding the specific form of function (1) for deposits with unlimited reserves is complicated by the absence of the dimensions of the mine field; it is complicated, but not excluded, since the dimensions of the field should be found from the same condition of attaining the minimum cost of extraction for each variant of mine capacity, with a previously established scheme for opening up and preparing the deposit.

For the subsequent calculations we shall adopt the following letter designations: \(h\)—height of a mine level, m; \(M\)—number of levels in the mine field; \(N_{\text{e}}\)—number of levels being worked simultaneously; \(S\)—length of the mine field along the strike, m; \(\sum p\)—total productivity of the seams, t/m\(^2\); \(c\)—coefficient of reserve extraction; \(m\)—number of sublevels in a level; \(x\)—length of an extraction field, m; \(L\)—annual advance of the total line of faces on a wing of a level, m/year; \(j\)—coefficient of simultaneity of seam working; \(\varphi\)—share of reserves extracted with stowing; \(n\)—number of extraction fields forming a continuous working front over time; \(\vartheta\)—number of seams in the mine take; \(G\)—industrial reserves of the mine field, t; \(\Theta\)—number of groups of seams actually being worked simultaneously, as a quantity reciprocal to the simultaneity coefficient; \(N_k\)—number of wings of the mine field in simultaneous operation; \(U\)—distance required for putting into operation a specified number of seams, \(U=nx\), m; \(B\)—distance along the strike that would be required in the case of simultaneous extraction of all seams of the mine take, m; \(l_k\)—length of a crosscut, m; \(l_t\)—haulage distance, m; \(H_o\)—depth of the mine shafts to the upper boundary of the mine field, m; \(m\)—height of the interlevel pillar, m; \(\tau_{\text{e}}\)—period of development of mining operations on a level, years; \(\tau\)—period of attaining the design capacity of the mine, years; \(r\)—cost of maintaining 1 linear m of working per year, rub./linear m; \(\varepsilon\)—sum of the costs of maintenance and of the energy for air movement per 1 linear m of working per year, rub./linear m; \(q_{\text{sh}}\)—unit cost of haulage work, rub./t·m; \(q_t\)—unit cost of delivery work, rub./t·km; \(q_{\text{z}}\)—the same for stowing; \(\sum k_c\)—cost of driving 1 linear m of all mine shafts, rub./linear m; \(K_d\)—cost of the pit-bottom yard with chambers, rub.; \(\lambda\)—capital investments in district facilities and geological exploration, rub./t; \(f\)—annual wage fund of the personnel servicing the horizon, rub./year; \(D\)—cost of buildings and structures on the mine surface, rub.; \(R\)—sum of capital repairs of buildings, structures, and mine equipment, rub./year; \(F\)—sum of expenditures for maintaining the conditionally fixed staff of workers on the mine surface, rub./year; \(\sum k_p\)—cost of preparing the horizon per linear m, rub.

In considering the types of production costs for extraction, one should distinguish the costs that depend on the capacity of the mine and on the dimensions of the mine field. Let us establish their algebraic expression, recalculated per 1 t of industrial reserves, taking as a guide the formulas for cost parameters recommended by Tsentrgiproshakht [4].

The construction of buildings and structures will amount, per 1 t of industrial reserves, to the value:

\[ m_1=\frac{D}{\sum pch}\cdot \frac{1}{MS}, \tag{4} \]

where, for mines with a capacity up to 0.75 million \(t\)/year, \(D=30A\); for mines with a capacity of more than 0.75 million \(t\)/year, \(D=18\cdot 10^6+7.2A\).

The maintenance of the staff of workers on the mine surface will amount, per \(1\,t\) of coal, to the expenditure:

\[ m_2=4.9+\frac{3.0}{A}. \tag{5} \]

The expenditure for maintaining drifts within the limits of the extraction field, prepared according to the scheme of Fig. 2, referred to \(1\,t\) of coal, will amount to the expenditure:

\[ m_3=\frac{N_zN_k\vartheta Jx}{Aj}\left[\frac{r_o+r_v}{2}+(m-1)r_p\right]. \tag{6} \]

The crosscut of the stowing on the ventilation horizon will amount, per \(1\,t\), to the expenditure:

\[ m_4=\varphi q_z x. \tag{7} \]

The costs for maintaining intermediate crosscuts and for the energy of air movement through them are determined by formula (8). Such costs per \(1\) running \(m\) per year will be:

\[ \varepsilon_k=r_k+40\alpha Q^3\cdot s^{2.5}, \]

where \(r_k\) is the cost of maintaining \(1\) running \(m\) per year; \(40\alpha Q^3\cdot s^{2.5}\) is the cost of the energy of movement of \(Q\,m^3\) of air per \(1\) running \(m\) per year in a working of cross section \(s\,m^2\) with aerodynamic coefficient \(\alpha\). The average length of crosscut maintenance is equal to \(0.5\,nl_k\); the maintenance time during the period of extraction of reserves \(nx\sum pch\), in the general case, will be \(\Theta xn:L+B:\Theta L\). The gross costs for maintenance and for the energy of air movement will be:

\[ W=\left(\frac{\Theta xn}{L}+\frac{B}{\Theta L}\right)0.5\,nl_k\varepsilon_k. \]

Then the specific costs, taking into account the formula for mine capacity \(jA=N_zN_k\sum pchLJ\), will be:

\[ m_5=W:nx\sum pch=0.5\,nl_k\varepsilon_kN_zN_k\frac{1+J}{jA}. \tag{8} \]

The cost of haulage of \(1\,t\) of coal will amount to the expenditure:

\[ m_6=0.25\,q_w S. \tag{9} \]

The costs for preparing a horizon, referred to \(1\,t\) of coal, will be:

\[ m_7=\sum K_p:\sum pch. \tag{10} \]

The costs for delivering \(1\,t\) of coal along intermediate drifts:

\[ m_8=\frac{m-1}{m}q_m l_m. \tag{11} \]

The specific costs for maintaining mine shafts and the cost of the energy of air movement through them will be:

\[ m_q=\frac{H_0}{A}\sum \varepsilon_c+\frac{h\sin\alpha^\circ}{2A}\sum \varepsilon_c M. \tag{12} \]

The cost of driving mine shafts, charged to 1 metric ton of coal:

\[ m_{10}=\frac{H_0\sum K_c}{\sum pch}\cdot\frac{1}{MS} +\frac{\sin\alpha\sum K_c}{\sum pc}\cdot\frac{1}{S}. \tag{13} \]

Expenditures per 1 metric ton of coal for mine hoisting and drainage:

\[ m_{11}=(H_0+0{,}5\,h\sin\alpha^\circ)q_c+0{,}5h\sin\alpha^\circ q_cM. \tag{14} \]

The cost of driving the shaft-yard workings, charged to 1 metric ton of coal:

\[ m_{12}=\frac{K_d}{\sum pch}\cdot\frac{1}{S}. \tag{15} \]

The maintenance of the conditionally fixed staff of workers serving a level during the period of its preparation will determine the expenditure per 1 metric ton of coal:

\[ m_{13}=\frac{10^6 f c_{\mathrm{э}}}{\sum pch}\cdot\frac{1}{S}. \tag{16} \]

We establish the expenditure for maintaining group drifts and for the energy of air movement through them as follows. With the direct and reverse order of extraction and the central ventilation scheme (Fig. 4), each group drift is maintained with an average length of

Fig. 4. Scheme of central ventilation

Labels in the figure: shafts; group ventilation drift; ventilation breakthrough; longwall; group haulage drift; group haulage drift; \(\varepsilon_v\); \(\varepsilon_o\); \(\varepsilon_o\).

\(0{,}25S\). Then the total expenditures over all operating levels in one year will be \(0{,}25SN_k(\varepsilon_v+N_{\mathrm{э}}\varepsilon_o)\), where \(\varepsilon_v,\varepsilon_o\) are the expenditures per 1 linear meter for the haulage and ventilation drifts.

With the flank ventilation scheme shown in Fig. 5,

Fig. 5. Scheme of flank ventilation

Labels in the figure: shafts; group ventilation drift; ventilation breakthrough; longwall; group haulage drift; group haulage drift; \(\varepsilon_v\); \(\varepsilon_o\); \(\varepsilon_o\).

annual costs per wing of the mine field for maintaining the ventilation entry will be \(0.5 S r_v\), and for the other entries—\(0.25 S N_e r_o\); the costs of air movement through the ventilation entry will be:

\[ 0.25 S \cdot 40 \alpha Q_k^3 : s^{2.5} \]

and through the other entries—

\[ 0.25 S N_e \cdot 40 \alpha \left(\frac{Q_k}{N_e}\right)^3 : s^{2.5} \]

Unit costs per 1 t of coal:

\[ m_{14}=\frac{1}{0.5 A}\left[0.5S(r_v+0.5N_e r_o)+0.25\,\frac{40\alpha Q_k^3}{s^{2.5}}\,S\left(1+\frac{1}{N_e^2}\right)\right], \]

where \(Q_k\) is the air flow rate in the wing of the mine field, in \(\mathrm{m^3/sec}\), passing through the ventilation entry to the flank. Assuming the cross sections of the haulage and ventilation entries to be equal (field entries), we obtain the final expression for the unit costs of maintaining and ventilating the group entries:

\[ m_{15}=\frac{1}{A}\left[r_v+0.5N_e r_o+\frac{40\alpha Q_k^3}{s^{2.5}}\left(\frac{1+N_e^2}{2N_e^2}\right)\right]S . \tag{17} \]

The repair of fixed assets and the maintenance of the conditionally permanent staff of surface workers during the period in which the mine reaches its design capacity will amount, per 1 t of coal, to:

\[ m_{16}=\frac{10^6 \tau (R+F)}{\Sigma pch}\cdot \frac{1}{MS}, \tag{18} \]

where \(F\) is the cost of maintaining the “surface,” and \(R\) is the cost of repairing the principal production assets of the mine, established as a function of mine capacity in the following amount:

Coal losses in protective pillars give rise to the costs:

\[ m_{17}=\frac{\lambda(50-w)}{ch}\cdot\frac{1}{M}+\frac{100\lambda}{c}\cdot\frac{1}{S}+\frac{\lambda w}{ch}. \tag{19} \]

After summing formulas (4–19), we obtain the algebraic expression, already known in the literature [3], for the cost of coal mining as a function of the dimensions of the mine field \(M\) and \(S\):

\[ a=\Sigma m_i=C_1+C_2S+\frac{C_3}{S}+\frac{C_4}{MS}+C_5M+\frac{C_6}{M}, \tag{20} \]

where the coefficients \(C_i\), for the previously selected schemes of opening and development of the specified group of seams, are determined by the expressions:

\[ C_1=4{,}9+3\cdot 10^6:A+\frac{\sum K_n}{\sum pch}+\frac{m-1}{m}l_m q_m+0{,}5\,n l_k \varepsilon_k N_\mathrm{e}N_k\frac{1+J}{jA}+ \]

\[ +\frac{\lambda u}{ch}+\frac{N_\mathrm{e}N_k \vartheta Jx}{jA} \left[0{,}5(r_0+r_v)+(m-1)r_n\right]+\varphi q_\mathrm{e}x+(H_0+0{,}5h\sin\alpha)q_c . \]

This coefficient \(C_1\) is recommended to be established after determining the dimensions of the mine field, and, consequently, the parameters \(r, q\).

\[ \left. \begin{aligned} C_2={}&0{,}25q_u+(\varepsilon_v+N_\mathrm{e}\varepsilon_o):2A+\\ &+\left[r_v+0{,}5N_\mathrm{e}r_o+\frac{40\alpha Q_k^3}{s^{2{,}5}} \left(\frac{1+N_\mathrm{e}^2}{2N_\mathrm{e}^2}\right)\right]:A;\\ C_3={}&\frac{100\lambda}{c}+(10^6 f\tau_\mathrm{e}+K_d+\sum K_c\sin\alpha):\sum pch;\\ C_4={}&[D+H_0\sum K_c+10^6\tau(F+R)]:\sum pch;\\ C_5={}&\frac{h}{2A}\sin\alpha\,\varepsilon_c+0{,}5h\sin\alpha\,q_c;\qquad C_6=\lambda(50-u):ch. \end{aligned} \right\} \tag{20a} \]

In order to use formulas (20) and (20a), it is necessary first to establish the story height \(h\), the length of the extraction field \(x\), and the development periods of the mining operations \(\tau_\mathrm{e}\) and \(\tau\). Such parameters are calculated by the method of variants, as shown in Table 2 for a story height of \(120\) m. Analogously, calculations are also made for other variants of the story height—60, 90, and 180 m. In doing so, the data of Table 1 are used, as well as the formula for mine capacity

\[ jA=N_\mathrm{e}N_k\sum pch\,LJ, \tag{21} \]

where the quantities \(A\) and \(h\) are given in their variant values. Then

\[ J=\frac{jA}{N_\mathrm{e}N_k\sum pch\,L}, \tag{22} \]

where \(j\) is the coefficient accounting for incidentally mined coal. For mines of the Kuzbass it may be calculated by the empirical formula:

\[ j=0{,}94-\frac{0{,}057}{A}\cdot 10^6 \]

Table 1

Technical parameters for the conditions of the example being solved

The distance for bringing seams into development is determined by a known graphical method [3]. For example, in order to bring all seams into development with a level height of 120 m, a distance of development of mine workings \(B = 1610\) m is required; with a level height of 60 m—\(B = 1030\), with a level height of 90 m—\(B = 1440\), and with a height \(h = 180\) m we have \(B = 1960\) m.

By specifying the level height, we thereby predetermine not only the coefficient \(J\), but also the necessary working front

\[ U = JB, \tag{23} \]

at which the simultaneous development of the required part of the exposed seams is ensured (for a mine of one capacity or another).

Table 2 contains 4 variants of calculations, and the total number of such tables corresponds to the number of level heights being tested. Thanks to such calculations, it becomes possible, already in the course of solving the problem of mine capacity, to investigate the question of the level height, the coefficient of simultaneity of seam development, and the extent of the front of development of mine workings on the level. Consequently, in finding the design capacity of the mine, the rational dimensions of the preparation scheme will be determined, as was stipulated as a condition for establishing this capacity.

When determining cost parameters from the data of Tsentrgiproshakht [4], which are included in the calculations according to formulas (20a), it is necessary first to establish the economically most advantageous cross section of the workings. This is quite possible to do, since the mine capacity is specified as a variant value.

The identified parameters of the level-preparation scheme for the mine-capacity variants make it possible to use formula (20) and calculate the cost of extraction, as shown by the graphs in Fig. 6.

Table 2

Parameters of the extraction-field preparation scheme with a level height of 120 m

As the basis for constructing graphs of the cost of extraction as a function of the mine-capacity variants and the level height, we take the optimum dimensions of the mine field, summarized in Table 3.

Table 3

Optimal dimensions of the mine field by variants of mine capacity and level heights, calculated by mathematical analysis of the cost function (20):

\[ f(M,S)=a-C_1=C_2S+C_3:S+C_4:MS+C_5M+C_6:M \]

From the data in Fig. 6 it can be seen that, owing to a change in level height at a constant mine capacity, the cost changes substantially. If the mine capacity and the level height are varied, then the cost may increase by 50% relative to its identified minimum in a number of the variants studied.

Next, we assign the level height for each mine capacity, taking into account mining-engineering possibilities and safety conditions, the mining systems used, and the relative economy of each level height.

For example, for a mine capacity of 2.4 million t/year, level heights of 60 and 90 m are rejected not only because of their high cost, but also in connection with the excess of the time required to prepare the next extraction field relative to the period of extraction of the preceding one.

A level height of 120 m is also eliminated, since its inherent cost remains relatively high. However, a level height of 180 m also cannot be accepted, because dangerous conditions would then arise for the movement of people, the delivery of timber, and the ventilation of four sublevels; excessive crushing of the coal occurs, which cannot fail to reduce its market value. A level height of 180 m for steeply dipping seams has not yet been sufficiently tested in practice; there is limited experience with a level height of 150 m in the mines of Kuzbass and the Ruhr.

It remains to adopt the panel height in the range from 120 to 180 m, where its single value is contained, equal to three optimal longwall lengths of steep dip plus the structural dimensions of the pillars, in all \(40 \cdot 3 + 25 = 145\) m. For this panel height we select, according to the graph in Fig. 6, the corresponding cost of coal production, which turns out to be 16.2 rubles/t, which differs by only 7% from the cost corresponding to a panel height of 180 m. As we see, in refining the panel height, the decisive significance was the optimal longwall length and the requirement for an integral number of sublevels. In an analogous way we analyze the panel height for the other variants and establish that the optimal panel height is the same for mine capacities from 0.6 to 2.4 million t/year and is equal to 145 m.

Fig. 6. Changes in the cost of coal according to variants of panel height

Having the panel height, we select from the graphs in Fig. 6 the magnitude of the cost of producing 1 t of coal for each variant of mine capacity. Substituting this cost into formula (1), we obtain the following equations:

\[ 17{,}5 = E_1 + 0{,}6 E_2 + \frac{E_3}{0{,}6}; \qquad 16{,}4 = E_1 + 1{,}2 E_2 + \frac{E_3}{1{,}2}; \]

\[ 15{,}6 = E_1 + 1{,}8 E_2 + \frac{E_3}{1{,}8}; \qquad 16{,}2 = E_1 + 2{,}4 E_2 + \frac{E_3}{2{,}4}. \]

We solve this system of equations by the method of least squares and obtain the numerical values of the coefficients: \(E_1 = 13{,}35\); \(E_2 = 0{,}763\); \(E_3 = 2{,}263\). Then the desired cost function (1) will be:

\[ a = 13{,}35 + 0{,}763 A + \frac{2{,}263}{A}. \tag{24} \]

The optimal mine capacity is determined from the condition:

\[ \frac{da}{dA} = E_2 - \frac{E_3}{A_{of}^{2}} = 0 \quad \text{or} \quad A_{of} = \sqrt{\frac{E_3}{E_2}} = 1{,}71 \text{ million t/year}. \tag{25} \]

Optimal mine capacity with respect to profitability from its operation

The main part of depreciation deductions is spent on capital repairs of fixed assets. Only 25% [7,246] of the amount

depreciation—the so-called renovation sum—goes to the Industrial Bank as repayment of the initial capital investments. For this reason, the completeness of repayment of the value of fixed assets depends on the service life of the mine in one or another variant of its capacity.

As is known, the calculation of the estimated production cost in any technical design for a mine is carried out taking into account definite average progressive depreciation rates by agency and by types of assets. Therefore, the systematic transfer of the value of assets to output cannot restore their original value by 100% if the service life of the mine has been chosen insufficient for this purpose.

Then the remaining part of the value of the assets will constitute a “loss” for the agency and a one-time expense for society, caused by the liquidation of the mine, as provided for in the design, after the reserves of the mining allotment have been extracted.

The housing stock can be realized, but at replacement value, which will likewise correspondingly increase the loss. The peculiarities of the location of mining settlements are such that, without taking them into account, questions of the efficiency of capital investments in mine construction cannot be correctly resolved.

If, over the service life of a mine, a certain national income \(W\) has been created, then the aforementioned “loss” \(\Omega\) should be deducted from the amount of income. Then capital investments will be fully accounted for under the item of society’s expenditures in developing the reserves of the given mining allotment, as is done when determining cost, for example, by formula (4). The proposed deduction of the “loss” from the amount of national income will make it possible to capture the net income from the activity of the mine as a socialist enterprise. The magnitude of such net income will show the relative efficiency of the mine-capacity variants studied in the design. We note that here the discussion concerns the unprofitability of those variants that are characterized by a mine capacity greater or smaller than in the optimal variant, which is unknown before the problem is solved. It is characteristic that such an optimal mine capacity will exist for any deposit. This is due to the fact that each design variant has its own individual mine service life. It is precisely this circumstance, under the existing procedure for charging depreciation (according to the uniform industry rate for each type of asset), that will determine a different degree of completeness in the repayment of fixed assets.

By decision of the Government, the USSR Gosplan may revise depreciation rates; however, the very principle of uniform industry rates for the repayment of individual types of fixed assets of the national economy remains unshakable, since it corresponds to the essence of socialist planning. Therefore the proposal [8; 9] to calculate the depreciation rate by dividing the value of fixed assets by the design service life of the mine has nothing in common with the planned conduct of the economy and would introduce into this matter an element of subjectivity on the part of designers, who are not competent to decide fundamental questions of the development of socialist industry, including the problem of accounting, planning, and collecting the renovation sum.

Academician S. G. Strumilin [5] showed the possibility of attaining in a design the maximum net income of an enterprise as a function of its capacity.

With respect to coal mines, the investigations carried out [1] permit the following general conclusion to be drawn.

As the capacity of a mine is increased up to a certain limit, its service life decreases, while net income increases. With further—

with a further increase in capacity, both the service life and the net income, referred to 1 m of coal reserves, fall substantially. Therefore, in order to economize social labor in construction and operation taken together, it is necessary to specify in the design such a mine capacity as would ensure its maximum income as a socialist enterprise.

But the calculation of income does not reveal the expediency of variants with respect to all expenditures in the sphere of production. For example, the wages of auxiliary workers, administrative and managerial expenses, capital repairs of fixed assets, etc., occupy different shares in the cost of production depending on the capacity of the mine; thereby a different level of this cost is conditioned among the variants. The overexpenditure or saving of these costs of the production sphere is not revealed in calculating income. Thus, the principle of maximum income, taken separately, is not capable of identifying the expedient variant of mine capacity.

On the other hand, the principle of achieving minimum cost is also insufficient, for it does not take into account the effectiveness of capital investments and is limited to the sphere of current production. In this case, in pursuit of an insignificant reduction in the cost of extraction, gigantic initial capital investments may arise that cannot be effectively used within the calculated service life of the mine.

Only by means of a double calculation—of net income, \(w\), and of cost, \(a\)—can the correct design capacity of the mine be found.

The calculation of the amount of income \(w\) is carried out for each variant of mine capacity on the following bases.

At the basis of price formation, as the principal components, lie material expenditures—\(M_3\), expenditures on payment for living labor—\(T\), and planned accumulations—\(H\). In this, the sum of planned accumulation and the value of living labor form the national income—\(W\), 75% of which, as stated in the report of the Central Committee of the CPSU to the Twentieth Party Congress, goes to satisfy the personal needs of the population.

Thus, the values circulating in production and formed in the production process are in definite relationships, which are illustrated by the diagram in Fig. 7.

Fig. 7. Diagram of income formation

In this diagram attention is drawn to the magnitude of the “loss” \(\Omega'\) arising at enterprises of the coal-mine type for the reasons set forth above. Let us find the amount of the value \(\Omega\), which has not been repaid in the usual manner by depreciation and therefore is subject to deduction from the sum of accumulation \(H\).

A distinction is made between the initial value and the value of assets with wear. The value with wear is the initial value minus the amount of depreciation for the period under consideration plus the cost of capital repairs for the same period. On the basis of these concepts one may assert that, under conditions in which the service life is insufficient, a one-time loss arises of the unliquidated part of the fixed assets and of their value with allowance for wear, which (loss) it is logical, in a comparative evaluation, to consider as

subtract from the accumulations for the variants and thereby determine the mine’s net income for one or another variant of its capacity.

We shall divide the mine’s fixed assets into four categories: buildings and structures; mine workings for opening up the deposit—shafts and shaft stations; machinery and equipment; and the residential settlement [7, 233].

Systematic capital repair of equipment ensures that, at the moment of the mine’s liquidation, it is in working condition. Therefore it can and should be dismantled and sold. In such a sale, the cost of dismantling will amount to a certain sum \(P\), which should be deducted from the mine’s accumulations.

The mine’s residential settlement, with initial cost \(K_{\mathrm{zh}}\), having existed for the service life of the mine \(t\), will likewise be sold at its replacement cost, which Academician S. G. Strumilin [5] determines by the formula:

\[ \frac{K_{\mathrm{zh}}}{(1+\delta)^t}, \tag{26} \]

where \(\delta\) is the average annual percentage increase in labor productivity in the construction industry (as a fraction of unity).

Buildings and structures with initial cost \(K_{\mathrm{c}}\) are lost at their full value, taking wear into account. This will constitute a “loss” from the liquidation of the mine in the amount

\[ K_{\mathrm{c}}[1+(p-n)t], \tag{27} \]

where \(p\) is the prescribed annual percentage (as a fraction of unity) for capital repair, and \(n\) is the annual depreciation percentage for the given group of structures.

The value, including wear, of the mine workings for opening up the deposit is determined as follows. The total cost of the workings for all levels will be:

\[ M K_{\mathrm{e}} = M(\Sigma K_{\mathrm{c}} h \sin \alpha + K_{\mathrm{d}}). \]

Their capital repair over the entire service life of the mine:

\[ p t_{\mathrm{e}} K_{\mathrm{e}} + 2 p t_{\mathrm{e}} K_{\mathrm{e}} + 3 p t_{\mathrm{e}} K_{\mathrm{e}} \ldots + M p t_{\mathrm{e}} K_{\mathrm{e}} = 0.5 p t (M+1) K_{\mathrm{e}}, \]

where \(t_{\mathrm{e}}\) is the service life of a level.

Depreciation deductions over the period of the mine’s operation:

\[ n t_{\mathrm{e}} K_{\mathrm{e}} + 2 n t_{\mathrm{e}} K_{\mathrm{e}} + \ldots M n t_{\mathrm{e}} K_{\mathrm{e}} = 0.5 n t (M+1) K_{\mathrm{e}}, \]

where \(K_{\mathrm{e}}\) is the cost of the capital workings for opening up a horizon.

Then the cost of the mine works, taking wear into account, will be:

\[ M K_{\mathrm{e}} + 0.5 p t (M+1) K_{\mathrm{e}} - 0.5 n t (M+1) K_{\mathrm{e}}. \tag{28} \]

On the basis of the considerations set forth, the total losses in the value of fixed assets will be:

\[ \Omega = K_{\mathrm{zh}} + p t K_{\mathrm{zh}} - n t K_{\mathrm{zh}} - \frac{K_{\mathrm{zh}}}{(1+\delta)^t} + \left[ M + 0.5(p-n)(M+1)t \right] K_{\mathrm{e}} + \]

\[ + \left[ 1+(p-n) \right] K_{\mathrm{c}} + P; \]

\[ \Omega = \left[ 1+(p-n)t - \frac{1}{(1+\delta)^t} \right] K_{\mathrm{zh}} + \left[ M + 0.5(p-n)(M+1)t \right] K_{\mathrm{e}} + \]

\[ + \left[1 + (p - n)t\right] K_c + \eta K_m , \tag{29} \]

where \(P = \eta K_m\), \(K_m\) is the initial cost of machinery and equipment; \(\eta\) is the coefficient allowing for the cost of dismantling.

Each variant of mine capacity has its own service life and initial amounts of fixed assets. Therefore the value \(\Omega\) may vary substantially among the variants, as can be seen from the data in Table 4. In compiling Table 4, the following were adopted: \(p = 3.1\%\); \(n = 4.2\%\); \(\eta = 15\%\).

According to the reporting data for 1955 [6], in the coal industry of the USSR wages \(T\) amounted to 57.2% of national income, while the sum of the funds of public consumption and accumulation amounted to 42.8% of national income \(W\).

In production design, this ratio must be taken according to the differentiated rate of accumulation in the particular branch of the economy. Taking into account the methodological aims of the present article and the relative evaluation of mine-capacity variants, as shown in the final line of Table 5, for a coal mine the ratio found above, \(H : T = 1/3\), is adopted approximately at the average level. The annual wage fund of the mine personnel is a function of its capacity. This function, according to production statistics, may be expressed [1], for example, for the Kuzbass mines by the formula:

\[ C = 700 \log(1 + 0.1 A). \tag{30} \]

Then

\[ H = \frac{Ct}{3} \quad \text{or} \quad w = \frac{H - \Omega}{G} = \left(\frac{Ct}{3} - \Omega\right)\frac{1}{G}. \tag{31} \]

Substituting in formula (31) the values of \(C\), \(t\), and \(\Omega\) for the mine-capacity variants under study, we obtain the amount of income, as shown in Table 5. Having four pairs of values \(w, A\), and regarding them as coordinates of a certain function \(w = F(A)\), we find the formula of the smoothing curve by the method of least squares:

\[ w = 5.49 + \frac{6.2}{A} - \frac{2.68}{A^2}. \tag{32} \]

The optimal mine capacity with respect to its profitability will be:

\[ \frac{dw}{dA} = -\frac{6.2}{A_{oF}^{2}} + \frac{2 \cdot 2.68}{A_{oF}^{3}} = 0 \quad \text{or} \quad A_{oF} = \frac{5.36}{6.2} = 0.87 \text{ million } t/\text{year}. \]

Design capacity of a mine satisfying the aggregate requirements

Thus, for the conditions of the example being solved, the functions (24 and 32) were identified, the graphs of which are presented in Fig. 8. The region of extre-

the sum of these functions is distinguished by an insignificant change in ordinates with a substantial increase or decrease in abscissas. However, with a considerable departure from the extreme values of mine capacity, the cost rises sharply and income decreases. A feature of the functions found is also their different points of extremum. The minimum cost is reached at a mine capacity of \(A_{of}=1.71\) million \(t/\text{year}\), and the maximum unit income at \(A_{of}=0.87\) million \(t/\text{year}\).

This difference in extrema contains an indication for the choice of mine capacity.

Fig. 8. Range for selecting the design capacity of a mine

Assuming the permissible deviation of the design cost from its minimum value to be \(10\%\), we find the boundaries of the range of optimal values of mine capacity by the formulas [1] of A. P. Sudoplatov:

\[ A_{lf}=\left(1+\Delta-\sqrt{2\Delta+\Delta^{2}}\right)A_{of} =0.64A_{of}=0.64\cdot1.71=1.09\ \text{million }t/\text{year}; \]

\[ A_{nf}=\left(1+\Delta+\sqrt{2\Delta+\Delta^{2}}\right)A_{of} =1.56A_{of}=1.56\cdot1.71=2.66\ \text{million }t/\text{year}, \]

where \(\Delta=0.1\) is the error of the cost calculation.

We shall find the boundaries of the range of mine-capacity values optimal with respect to income in the following way. Let us denote \(1/A=Z\). Then expression (32) will take the form: \(w=d_1+d_2Z-d_3Z^2\);

\[ \frac{dw}{dZ}=d_2-2d_3Z=0;\qquad Z_0=\frac{d_2}{2d_3}. \]

Table 4

Calculation of the amount of lost value of fixed assets
(million rubles)

Table 5

Amount of income by variants of mine capacity

The maximum income will be equal at the value $Z_0$:

$$ w_0=d_1+\frac{d_2^2}{4d_3}. $$

The variable part of the income

\[ \frac{d_2^2}{4d_3}, \]

which depends on \(A\), may permissibly be reduced to the value

\[ (1-\Delta)\frac{d_2^2}{4d_3}, \]

where \(\Delta\) is the error of its calculation. Such an assumption will lead to the emergence of a region of optimal values of the quantity \(Z\), and consequently also of \(A\). In this region all ordinates from \(w_0\) to \((1-\Delta)\dfrac{d_2^2}{4d_3}\) are acceptable. On this basis we write:

\[ d_2 Z - d_3 Z^2 = (1-\Delta)\frac{d_2^2}{4d_3}. \]

After transforming this expression we obtain:

\[ Z^2 - \frac{d_2}{d_3}Z + (1-\Delta)\frac{d_2^2}{4d_3^2}=0. \]

The roots of this equation represent the right and left boundaries of the region of variation of the auxiliary quantity \(Z\):

\[ \frac{1}{A_{nF}}=\frac{1-\sqrt{\Delta}}{A_{oF}} \quad \text{or} \quad A_{nF}=\frac{A_{oF}}{1-\sqrt{\Delta}}=\frac{0.87}{0.68}=1.27 \text{ million } t/\text{year}; \]

\[ \frac{1}{A_{nF}}=\frac{1+\sqrt{\Delta}}{A_{oF}} \quad \text{or} \quad A_{nF}=\frac{A_{oF}}{1+\sqrt{\Delta}}=\frac{0.87}{1.31}=0.66 \text{ million } t/\text{year}. \]

Proceeding from the adopted concept of the correct design capacity of a mine, it is necessary to assign such a standard capacity \(A_T\) as lies between the values \(A_{nF}=1.09\) and \(A_{nF}=1.27\) million \(t/\text{year}\), as shown in Fig. 8. Within this range there is one standard capacity, 1.2 million \(t/\text{year}\), for which the calculations yielded the following technical indicators: \(h=145\) m, \(N_e=1\), \(m=3\), \(U=625\) m, \(J=0.33\), \(M=4\), \(S=2500\) m, \(G=55.5\) million \(t\), \(t=47\) years.

In the postwar years, the average capacity of mines designed in the Kuzbass has been 1.3 million \(t/\text{year}\), and is distributed as follows:

Thus, the type of mine established in the example being solved is in sufficient agreement with production data.

REFERENCES

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