Determination of Optimal Parameters of a Set of Machines for Excavation of Frozen Soils
Abstract Generated abstract
This paper examines how to select the number and productivity of ripping and excavating machines used together in winter excavation of frozen soils. It develops calculation schemes for sequential and parallel-sequential operations, relating set productivity to machine productivities, work volume, refreezing constraints, and periods of single-machine-type and joint operation. The analysis indicates that increasing the number of machines can raise reduced costs disproportionately and may fail to improve productivity when one operation becomes the limiting stage. The authors conclude that optimal machine sets should minimize the imbalance between total ripper productivity and total excavator productivity, since unequal capacities create intra-set losses.
Full Text
DETERMINATION OF THE OPTIMAL PARAMETERS OF A SET OF MACHINES FOR EXCAVATING FROZEN SOILS
I. G. BASOV, N. A. DUBROVSKII
(Presented by the scientific seminar of the Department of Industrial Economics and Enterprise Organization)
Winter earthworks usually include preliminary preparation of frozen soil for excavation, performed by ripping machines, and the actual excavation of the soil by earth-moving machines.
These operations may be carried out sequentially or in parallel-sequential fashion.
When the operations are performed sequentially, the duration of soil excavation \(T_{\mathrm{p}}\) is determined as the sum of the time expenditures for ripping \(T_1\) and for direct excavation \(T_2\), i.e.,
\[ T_{\mathrm{p}} = T_1 + T_2. \]
The indicators \(T_1\) and \(T_2\) may be determined from the relations:
\[ T_1 = \frac{V}{Q_{ij}}, \qquad T_2 = \frac{V}{Q_{\varphi j}}, \]
where \(Q_{ij}\) and \(Q_{\varphi j}\) are, respectively, the productivity of the ripping machine and of the excavating machine.
Knowing the volume \(V\) and the duration \(T_{\mathrm{p}}\) of soil excavation, one can determine the productivity of the set
\[ Q_{\mathrm{p}} = \frac{V}{T_{\mathrm{p}}}, \text{ in } m^3/\text{h}. \]
When the operations are performed sequentially it is necessary to satisfy the condition
\[ T_1 \leq T_{\mathrm{cm}} \geq T_2, \]
where \(T_{\mathrm{cm}}\) is the time required for the prepared soil to freeze again before excavation.
If ripping and excavation of the soil are carried out in parallel-sequential fashion, then in determining the productivity of the set it is necessary to take into account the ratio of the productivities of the machines included in the set and the time of their parallel operation. The sequence for determining the productivity of a set of machines for various ratios of the productivities of rippers and excavators is given in the table.
Using the formulas proposed above, a study was conducted of the nature of the change in the productivity of the set as a function of the type and number of rippers \((n)\) and excavating machines \((m)\).
The research methodology was as follows. Initial parameters of the set were specified and its productivity was determined. After changing one of the parameters, the productivity of the new set was determined and expressed through the parameters of the initial one. Such operations were carried out until the influence of one parameter or another on the character of the change in the productivity of the set became clear.
Table
| Name of indicators | Dependencies for determining indicators at productivity ratios $Q_{ij} > Q_{\varphi j}$ |
Dependencies for determining indicators at productivity ratios $Q_{ij} = Q_{\varphi j}$ |
Dependencies for determining indicators at productivity ratios $Q_{ij} < Q_{\varphi j}$ |
|---|---|---|---|
| Type of excavating machine selected | $\varphi_1$ | $\varphi_2$ | $\varphi_3$ |
| Productivity of machines | $Q_{\varphi j1}$ | $Q_{\varphi j2}$ | $Q_{\varphi j3}$ |
| Type of ripper selected | $i_1$ | $i_2$ | $i_3$ |
| Ratio of frozen soil loosened by the ripper working element to that prepared for excavation | $S_1$ | $S_2$ | $S_3$ |
| Productivity of the ripper | $Q_{ij1}$ | $Q_{ij2}$ | $Q_{ij3}$ |
| Minimum allowable value of the cut | $V_{3\min_1}$ | $V_{3\min_2}$ | $V_{3\min_3}$ |
| Operating time of only machines of one type | $T'_1=\dfrac{V_{3\min_1}}{Q_{ij1}}$ | $T'_1$ or $T'_2$ | $T'_2=\dfrac{V_{3\min_3}}{Q_{\varphi j3}}$ |
| Possibility of forming a set according to the condition | $T'_1 \leq T_{\mathrm{sm}}$ | $T'_1 \leq T_{\mathrm{sm}}$ and $T'_2 \leq T_{\mathrm{sm}}$ | $T'_2 = T_{\mathrm{sm}}$ |
| Soil excavation time | $T_2=\dfrac{V}{Q_{\varphi j}}$ | ||
| Time for preparing frozen soil for excavation | $T_1=\dfrac{V}{Q_{ij}}$ | ||
| Time of joint operation of the machines | $\tau=T_1-T'_1$ | $\tau$ | $\tau=T_2-T'_2$ |
| Operating time of only machines of one type | $T'_2=T_2-\tau$ | $T'_2$ or $T'_1$ | $T'_1=T_1-\tau$ |
| Possibility of forming sets according to the condition | $T'_2 \leq T_{\mathrm{sm}}$ | $T'_1 \leq T_{\mathrm{sm}}$ | |
| Time for developing the specified volume of soil | $T_{\mathrm{pp}}=T_2+T'_1$ | $T_{\mathrm{pp}}$ | $T_{\mathrm{pp}}=T_1+T'_2$ |
| Productivity of the set | $Q_{\mathrm{pp}}=\dfrac{V}{T_{\mathrm{pp}}}$ | $Q_{\mathrm{pp}}=\dfrac{V}{T_{\mathrm{pp}}}$ | $Q_{\mathrm{pp}}=\dfrac{V}{T_{\mathrm{pp}}}$ |
Analysis of the results obtained showed the following:
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For $nQ_{ij}<Q_{\varphi j}$ and small values of $n$, a significant change in the productivity of the set can be achieved by changing the number of ripper machines.
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If $nQ_{ij}>Q_{\varphi j}$, then it is impossible to achieve a substantial increase in the productivity of the set by increasing $n$.
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The productivity of the set for $Q_{ij}>mQ_{\varphi j}$ and an increase in $m$ changes as a result of a decrease in the soil excavation time and an increase in $T'_1$.
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In the case $Q_{ij}<mQ_{\varphi j}$, as the number of excavating machines increases, the productivity of the set does not change.
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With a change in the productivity of the ripping machine, the productivity of the set changes substantially when the condition \(Q_{ij}<Q_{\varphi j}\) is met, and only slightly when \(Q_{ij}>Q_{\varphi j}\).
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If the productivity of the excavating machine changes, then the productivity of the set changes both as a result of a decrease in the time of soil excavation and as a result of a change in the duration of its preparation for excavation.
Simultaneously with the study of the character of the change in the productivity of sets of machines under the influence of various factors, the question was considered of the change in the value of the reduced costs associated with the operation of the sets, which makes it possible to establish the following:
a) when the productivity of sets is changed by increasing the number of machines, the reduced costs increase;
b) if the productivity of a set changes due to the productivity of the machines included in it, then the magnitude of the reduced costs may remain constant, increase, or decrease.
In addition, an analysis of the influence of various factors on the productivity of the set and on the character of the change in the costs of excavating frozen soil showed that, with an increase in the number of identical machines included in the set, there is a disproportionate increase in its productivity with a proportional increase in reduced costs. Cases are also possible \((Q_{ij}<mQ_{\varphi j})\) in which increasing the number of excavating machines in the set does not increase its productivity, but does increase the reduced costs.
Thus, when composing sets of machines from identical rippers and identical excavating machines, by varying their number one can obtain sets that are equivalent in productivity but differ in reduced costs. In this case there are clearly expressed losses of the set—“intraset” losses. On this basis it may be assumed that intraset losses will occur in any set if the condition \(nQ_{ij}=mQ_{\varphi j}\) is not observed.
This proposition was checked in the following way. For a set of machines with \(nQ_{ij}=mQ_{\varphi j}\), the values of productivity \(Q\) and reduced costs \(Z_{\mathrm{pr}}\) were determined. Then it was determined how many of the same rippers and excavating machines had to be included in the set in order to achieve the same productivity \(Q\), while observing the condition \(nQ_{ij}=mQ_{\varphi j}\).
Determination of the number of machines began with calculation of the lag time in completion of the work of the excavating machines
\[ T'_2=\frac{V_3}{Q_{\varphi j}}. \]
But since for the set being calculated \(nQ_{ij}=mQ_{\varphi j}\), it follows that \(T'_{2r}=T'_{1r}\). By virtue of the equality of the productivities of the sets being compared, the time required for these sets to perform identical volumes of work will be equal, i.e., \(T=T_r\).
Then, for the known value \(T_r\), the duration of preparation of the frozen soil for excavation was determined as \(T_{1r}=T_r-T'_{2r}\), and the duration of its excavation as \(T_{2r}=T_r-T'_{1r}\). Finally, the required number of rippers and excavating machines was found from the expressions
\[ n=\frac{V}{T_{1r}Q_{ij}}, \qquad m=\frac{V}{T_{2r}Q_{\varphi j}}. \]
After determining the number of machines included in the set, the value of the reduced costs \((Z_{\mathrm{pr.r}})\) was determined, and these were compared with \(Z_{\mathrm{pr}}\).
For \(nQ_{ij} > mQ_{\varphi j}\), the following situation is obtained. Equalization of the time for preparing frozen soil for excavation with the excavation time occurs through a change in the number of machines included in the set, i.e., \(Q_{ij}\), \(Q_{\varphi j}\), and \(V\) are constant quantities. In this case the number of rippers decreases (\(T_1\) increases), while the number of excavating machines increases (\(T_2\) decreases).
In determining the reduced costs, it was found that, for a set in which the condition \(nQ_{ij} = mQ_{\varphi j}\) is satisfied, there is a saving in one-time costs and capital investments for ripper machines; but since the number of excavating machines increases, there is also an increase in the reduced costs associated with soil excavation.
Consequently, the amount of reduced costs for a set with different productivity of rippers and excavating machines will be less than the reduced costs of the set for which the condition \(nQ_{ij} > mQ_{\varphi j}\) is satisfied, so long as the saving provided by the rippers is greater than the excess expenditure provided by the excavating machines. This conclusion may be written by means of the expression
\[ Z_{\mathrm{pr}\cdot \mathrm{p}} - Z_{\mathrm{pr}\cdot \mathrm{pr}} \geq Z_{\mathrm{pr}\cdot \mathrm{vr}} - Z_{\mathrm{pr}\cdot \mathrm{v}}, \]
where \(Z_{\mathrm{pr}\cdot \mathrm{p}}\) and \(Z_{\mathrm{pr}\cdot \mathrm{v}}\) are, respectively, the values of the reduced costs for ripping and for excavating the soil under the condition \(nQ_{ij} \ne mQ_{\varphi j}\);
\(Z_{\mathrm{pr}\cdot \mathrm{pr}}\) and \(Z_{\mathrm{pr}\cdot \mathrm{vr}}\) are, respectively, the values of the reduced costs for ripping and for excavating the soil under the condition \(nQ_{ij} = mQ_{\varphi j}\).
In the case \(nQ_{ij} < mQ_{\varphi j}\), equalization of the productivities of rippers and excavating machines occurs only by decreasing the number of excavating machines (\(T_2\) increases). The reduced costs for ripping frozen soil for the sets being compared remain the same; for the excavating machines there is always a saving for the set in which the condition \(nQ_{ij} = mQ_{\varphi j}\) is satisfied.
In this case the inequality is always satisfied:
\[ Z_{\mathrm{pr}\cdot \mathrm{p}} + Z_{\mathrm{pr}\cdot \mathrm{v}} > Z_{\mathrm{pr}\cdot \mathrm{pr}} + Z_{\mathrm{pr}\cdot \mathrm{vr}}, \]
because \(Z_{\mathrm{pr}\cdot \mathrm{p}} = Z_{\mathrm{pr}\cdot \mathrm{pr}}\) and \(Z_{\mathrm{pr}\cdot \mathrm{v}} > Z_{\mathrm{pr}\cdot \mathrm{vr}}\).
From the foregoing it follows that in a set of machines with \(nQ_{ij} \ne mQ_{\varphi j}\), intra-set losses occur.
Consequently, when forming sets of machines for carrying out winter earthworks, it is necessary to strive to make the difference between the productivities of the rippers and the excavating machines minimal, i.e., \(nQ_{ij} - mQ_{\varphi j} \to \min\).