Determining the Economic Efficiency of Increasing the Service Lives of Machines Being Designed

V. N. Shukin

(Presented by the scientific seminar of the Department of Industrial Economics and Organization of Enterprises)

As is known, an increase in the service life of a machine being designed can be achieved by carrying out various measures available to the designer, such as, for example: changing parts, assemblies, and the machine as a whole; using higher-quality materials that ensure high wear resistance; using standard and unified parts and assemblies that have proved themselves positively in operation, etc.

If the implementation of measures that increase the service life is not accompanied by an increase in the cost of the machine and its operating expenses, then there can be no doubt as to the effectiveness of such measures. If, however, the measures are accompanied by an increase in the cost of the machine and by a change in the amount of operating expenses, then their effectiveness must be verified on the basis of appropriate calculations.

Unfortunately, in the specialized literature the questions of methodology for carrying out such calculations have not been reflected. In our opinion, such a calculation should be carried out as follows.

First it is necessary to determine the amount of savings that the national economy will obtain from the introduction of the indicated measures. For this, above all, it should be borne in mind that increasing the service life leads to savings in capital investments that would have been required for the reproduction of the machine upon expiration of the service life established without taking into account the measures under consideration. The amount of these investments is

\[ K = K_{\mathrm{v}} n, \tag{1} \]

where \(K\) is the total amount of capital investments that would have been required for the reproduction of the machine over a period of time equal to the increase in service life;

\(K_{\mathrm{v}}\) is the amount of capital investments that would have been required for the reproduction of the machine upon expiration of the original service life;

\(n\) is a number showing how many times the increase in service life exceeds its original value.

If \(n\) turns out to be fractional, it is rounded up to the larger integer. In turn, the quantity

\[ K_{\mathrm{v}} = C_1 + C_2 - C_3, \tag{2} \]

where \(C_1\) is the full cost of the machine at the time of its reproduction upon expiration of the initial service life;

\(C_2\) is the cost of installation work connected with reproduction of the machine;

\(C_3\) is the residual value of the machine upon expiration of its service life.

Capital investments released as a result of increasing the service life of the machine will be used in other sectors of the national economy.

\[ \mathcal{E}' = \mathcal{E}_1 \Delta t + \mathcal{E}_1(\Delta t - t_1) + \mathcal{E}_1(\Delta t - 2t_1) + \ldots \mathcal{E}_1[\Delta t - (n-1)t_1] = \]

\[ = \mathcal{E}_1\,[\Delta t\, n - 0.5\,t_1 n(n-1)], \tag{3} \]

where \(\mathcal{E}'\) is the total savings that can be obtained from the use of all released capital investments in other sectors of the national economy;

\(\mathcal{E}_1\) is the average annual savings that can be obtained from the use, in other sectors of the national economy, of capital investments;

\(\Delta t\) is the increase in service life, in years;

\(t\) is the initial service life, in years.

The quantities \(K_{\mathrm{v}}\) and \(\mathcal{E}_1\) are related by

\[ \frac{K_{\mathrm{v}}}{\mathcal{E}_1}=\tau, \tag{4} \]

where \(\tau\) is the payback period, in years, of the released capital investments through the average annual savings for the national economy.

Substituting the value of \(\mathcal{E}_1\) from expression (4) into expression (3), we obtain

\[ \mathcal{E}'= \frac{1}{\tau}K_{\mathrm{v}}\,[\Delta t\, n - 0.5\,t_1 n(n-1)]. \tag{5} \]

If the introduction of measures that increase the service life also affects the amount of the machine’s operating costs, then the total amount of savings from the introduction of such measures will be

\[ \mathcal{E}''= \frac{1}{\tau}K_{\mathrm{v}}\,[\Delta t\,n - 0.5t_1 n(n-1)] +\mathcal{E}_{\mathrm{r}}(t_1+\Delta t), \tag{6} \]

where \(\mathcal{E}''\) is the total amount of savings from the introduction of measures increasing the machine’s service life;

\(\mathcal{E}_{\mathrm{r}}\) is the amount of savings or increase in annual operating costs.

If there is a saving, then \(\mathcal{E}_{\mathrm{r}}\) is taken with the sign “+”; if there is an increase, then with the sign “−”.

On the other hand, in order to obtain this saving we are forced to make additional capital investments that increase the cost of the machine.

If these investments were used in other sectors of the national economy, the following saving would be obtained:

\[ \mathcal{E}'''=\mathcal{E}_2(t_1+\Delta t), \tag{7} \]

where \(\mathcal{E}'''\) is the saving that can be obtained in other sectors of the national economy through the use of additional capital investments during the service life of the machine;
\(\mathcal{E}_2\) is the average annual saving under the same conditions.

At the same time, these capital investments are related to the magnitude of the average annual saving \(\mathcal{E}_2\) by the relation

\[ \frac{K_{\mathrm{d}}}{\mathcal{E}_2}=\tau, \tag{8} \]

where \(K_{\mathrm{d}}\) is the additional capital investment.

Substituting the value of \(\mathcal{E}_2\) from expression (8) into expression (7), we obtain

\[ \mathcal{E}'''=\frac{1}{\tau}K_{\mathrm{v}}(t_1+\Delta t). \tag{9} \]

The difference \(\mathcal{E}''-\mathcal{E}'''\) will characterize the economic effectiveness of increasing the service life of the machine. The greater this difference, the greater the amount of saving ensured for the national economy:

\[ \mathcal{E}=\mathcal{E}''-\mathcal{E}'''= \frac{1}{\tau}K_{\mathrm{v}}\,[\Delta t\cdot n-0.5\,t_1 n(n-1)]- \]

\[ -\left(\frac{1}{\tau}K_{\mathrm{d}}-\mathcal{E}_{\mathrm{r}}\right)(t_1+\Delta t), \tag{10} \]

where \(\mathcal{E}\) is the economic effectiveness of increasing the service life of the machine.

It follows from this that increasing the service life is economically justified if the condition

\[ \mathcal{E}'' \geq \mathcal{E}''' \tag{11} \]

is satisfied.

Substituting the values of \(\mathcal{E}''\) and \(\mathcal{E}'''\) from expressions (6) and (9), and making certain transformations, we obtain the final condition for an economically expedient increase in service life:

\[ \frac{\Delta t}{t_1}\geq \frac{K_{\mathrm{d}}+0.5K_{\mathrm{v}}\cdot n(n-1)-\mathcal{E}_{\mathrm{r}}\tau} {K_{\mathrm{v}}\cdot n-K_{\mathrm{d}}+\mathcal{E}_{\mathrm{r}}\tau}. \tag{12} \]

In the particular case when an increase in service life is accompanied by an insignificant change in operating costs,

\[ \frac{\Delta t}{t_1}\geq \frac{K_{\mathrm{d}}+0.5K_{\mathrm{v}}n(n-1)} {K_{\mathrm{v}}\cdot n-K_{\mathrm{d}}}. \tag{13} \]

From this expression it is possible to establish easily the greatest limiting ratio \(\dfrac{K_{\text{д}}}{K_{\text{в}}}\), which must correspond to an increase in the machine’s service life,

\[ \frac{K_{\text{д}}}{K_{\text{в}}} = \frac{\dfrac{\Delta t}{t_1} n - 0.5 n(n-1)} {1 - \dfrac{\Delta t}{t_1}} . \tag{14} \]

For quickly finding the greatest limiting ratio \(\dfrac{K_{\text{д}}}{K_{\text{в}}}\), one may use the graph (Fig. 1), which reflects the dependence of this ratio on the relative increase in the machine’s service life.

In the graph: vertical axis—“Greatest limiting ratio \(\dfrac{K_{\text{д}}}{K_{\text{в}}}\)”; horizontal axis—“Relative increase in service life \(\dfrac{\Delta t}{t}\).” Curves are marked \(n:1\), \(n:2\), \(n:3\).

Fig. 1. Dependence of the greatest limiting ratio
\[ \frac{K_{\text{д}}}{K_{\text{в}}} \]
on the relative increase in service life.

Example. Suppose that, during the reconstruction of a mine fan with an initial cost of 40 thousand rubles and a service life of 15 years, it appears possible, through the introduction of a number of measures, to increase its service life by 10 years. At the same time, the cost of the fan increases by 10 thousand rubles. Operating expenses change very insignificantly. The cost of installing the fan will be 8 thousand rubles, and the residual value 3 thousand rubles.

It is required to establish whether these measures are profitable, and, if they are, what their economic efficiency is. We find that the increase in service life is 0.667 of its initial value. Therefore, \(n = 1\).

Substituting the values of the components into expression (14), we find that the measures will be profitable if the ratio \(\dfrac{K_\text{д}}{K_\text{в}}\) does not exceed 0.4. An analogous answer will be obtained if one uses the graph (Fig. 1).

To determine the actual ratio \(\dfrac{K_\text{д}}{K_\text{в}}\), we find the value of \(K_\text{в}\). Using expression (2), we find

\[ K_\text{в} = 40 + 8 - 3 = 45 \text{ thousand rubles}. \]

Then

\[ \frac{K_\text{д}}{K_\text{в}} = 0.22. \]

Since the actual ratio \(\dfrac{K_\text{д}}{K_\text{в}}\) is less than the limiting one, the measures are profitable.

We calculate the savings according to expression (10). Assuming that the payback period is \(\tau = 5\) years,

\[ \Э = \frac{1}{5} \, 45 \{10 \cdot 1 - 0.5 \cdot 15(1 - 1)\} - \]

\[ -\left(\frac{1}{5}\cdot 10 - 0\right)(15 + 10) = 40 \text{ thousand rubles}. \]

In conclusion, it should be noted that increasing the service life also depends on a whole series of factors related to the operation of the machine, such as intensity of use, quality of maintenance, timeliness of repairs, modernization, etc. Establishing the economic efficiency of increasing the service life through such measures lies beyond the scope of the present article and is subject to separate consideration.