1. Process 1 has a 90 per cent recovery and costs $10/t of ore treated, as in the previous example
2. Process 2 has a 98 per cent recovery but incurs costs of $15/t of ore treated. The break-even cut-over occurs when:
Price × grade × recoveryprocess 2 - costprocess 2 = price × grade × recoveryprocess 1 - costprocess 1
Break-even grade = product rice (recovery recovery )
cost cost
p process 2 process 2process 1- process 1 - # $ ( . 8 .9 ) $( 1 ) $(5 10) 105 #50 9 0 0 = + - - + = $5.00/$0.80 = 6.25 gu/t
As determined previously, the ore–waste break-even for Process 1 is 1.11 gu/t. Applying the same calculation for Process 2 generates an ore–waste break-even of 1.53 gu/t but this is not used as a cut-off or cut-over. Combined with the derivation of the Process 1 - Process 2 cut-over of 6.25 gu/t, it can be seen that rock with a grade: • less than 1.11 should be classified as waste
• greater than 1.11 and less than 6.25 should be classified as ore and treated by Process 1 • greater than 6.25 should be classified as ore and treated by Process 2.
Figure 3.4 shows graphically the net benefits of classifying material as waste and as ore treated by both Process 1 and Process 2. The first two lines are the same as in Figure 3.3. The net benefit of waste is –$5/t at all grades of material. The benefit of ore treated by Process 1 increases linearly at the rate of $9/t/gu from –$15.00/t at zero grade. Similarly,
FIGURE 3.4
Graphical representation of ore–waste break-evens and cut-over between two processes.
1 0 2 3 4 5 6 7 8 9 10 –20 0 20 40 60 80 $/tonne
Grade (grade units) Contribution (Process 1) Contribution (Waste) Waste P1/P2 Cut-over (6.25 gu/t) Ore to
Process 1 Process 2Ore to
Contribution (Process 2) Ore/Waste Break-even (P1: 1.11 gu/t P2: 1.53 gu/t)
CHAPTER 3 | Break-even Analysis the benefit of ore treated by Process 2 increases linearly at the rate of $9.80/t/gu from –$20.00/t at zero grade, being the total cost of mining and treatment by Process 2. The Process 2 line crosses the Process 1 line at 6.25 gu/t.
Note that the ore–waste break-even for Process 2 (1.53 gu/t) does not enter into the specifications of how rock is classified. In the absence of Process 1, rock with a grade above 1.53 would be classed as ore and treated by Process 2, but Process 1 provides better returns than Process 2 for all grades up to the identified cut-over of 6.25.
As mining progresses through the deposit, there is no guarantee that the distribution of grades will be such that both processes will operate at full capacity. If one process has spare capacity while the other is overloaded, the simple 1D break-even calculation does not suggest what to do with the excess material which, according to the cut-over value, should be processed by the overloaded process. It could be stockpiled until it can be treated by that process or treated immediately through the less profitable (for that grade of material) process with spare capacity. The simple 1D break-even cut-off model does not do everything we might reasonably require in order to maximise the cash generated.
rock grades equal to cut-off
The preceding description of the split of rock between destinations does not address grades that are equal to the specified cut-off or cut-over. As with specifying costs to include in the calculation, there is no commonly agreed position on including or excluding material with grade equal to the cut-off.
In many cases this issue may be purely academic. For example, if grades are modelled to more significant figures than the cut-offs, the number of blocks in the geological model with an estimated grade the same as the cut-off is likely to be negligible and of no practical concern. The author would, however, suggest that it is pointless to classify material with the same grade as the ore–waste break-even grade as ore – it would then be processed for no net benefit.
If the cut-off and modelled grades have the same number of significant figures, both will probably have been calculated to a greater precision and then rounded. If the cut- off has been rounded up, it might be logical to say that rock with the same grade as the cut-off should be classified as ore. Conversely, if rounded down, you might specify the cut-off to be classified as waste. There will still be some potential for misclassification if the unrounded values for the rock grade and cut-off had been used. Again, this is likely to be negligible relative to other uncertainties implicit in the grade estimation and break- even calculation processes.
This rationale is applicable to the ore–waste cut-off, but the cut-over between the two processes is a different issue. As noted, the 1D break-even calculation takes no account of treatment plant capacities. Since the same net return is derived in either process, material whose grade is exactly equal to the cut-over would be sent to the process with spare capacity. If there is a significant amount of such material, it might be that some is sent to one process and the rest to the other. Again, the simple 1D break-even model does not account for all issues that need to be considered, and additional information must be brought into the decision-making process.
CoStS in ore–WASte BreAk-even CAlCulAtionS
As stated, there are no standard definitions for names to be used for break-even grades and the costs that are included in the calculations. Four examples are described below, but these should not be seen as any particular recommendation for names or definitions; rather, they illustrate the range of costs that might be included in a break-even calculation: • marginal break-even – costs to be covered = mining variable costs + milling variable
costs
• mine operating break-even – costs to be covered = total mining costs + total milling costs • site operating break-even – costs to be covered = total mining costs + total milling costs
+ total site administration and services costs
• full cost break-even – costs to be covered = total mining costs + total milling costs + total site administration and services costs + head office charges + allowance for capital. Many operations use at least two of these types of calculations for specifying cut-offs in a given situation. Companies tend to use a cut-off similar to the first of these as an operational cut-off for identifying marginal ore that can be treated opportunistically when there is spare capacity in the ore handling and treatment processes. Similarly, cut-offs similar to the third or fourth options are often used as the planning cut-off for identifying ore and waste in the long-term mine plan; however, the costs used in the break-even calculations are often in practice indicated with little understanding of what is being achieved.
As previously stated, many operations use the terms break-even and cut-off synonymously, so cut-off might be substituted for break-even in these definitions in company documentation. The author prefers in these circumstances to use the compound term break-even cut-off to indicate that the cut-off is to be specified by a break-even calculation. It should be clearly understood that in this book break-even is a generic term to describe the derivation methodology, with additional descriptors used to indicate the costs included in the calculation.
As can be seen from the formulas and example calculations, costs to be applied in break-even calculations ought to be the differential costs of classifying rock as ore rather than waste. A similar rationale applies when dealing with cut-overs between two processes. In the examples, costs that are common to both options (ore or waste, Process 1 or Process 2) effectively cancel out, leaving only the cost differential between the two options in the calculation.
What are the differential costs to be accounted for? If there is spare capacity in both the ore and product streams, the extra costs of classifying a piece of rock as ore are only the marginal variable costs of treating it as ore and dealing with the derived product. If either the ore or product stream is at capacity, choosing to treat a piece of rock as ore extends the operation’s life; the extra costs of classifying it as ore are the variable costs of treating it as ore and handling the derived product, plus the fixed costs associated with the extension of the mine life.
The following subsections discuss the variable and fixed costs that arise in different parts of the operation and how they apply in break-even cut-off derivation.
CHAPTER 3 | Break-even Analysis
open pit – in the mine
When a decision is made to mine a pit to a particular size, the variable costs of mining (typically drill, blast, load and haul) are incurred for all rock removed from the pit, irrespective of its classification as ore or waste. All rock must be transported to the point where ore and waste streams diverge. The location of the rock in the pit – whether from deep or shallow areas – is therefore not relevant to the ore–waste decision.1 Hence,
open pit mining variable costs should typically not go into an ore–waste break-even calculation. This was seen in the more complex example, where the mining costs were included in the costs of both ore and waste and cancel out in the resulting calculation. The exception, which follows directly from this rationale, is any differential cost of mining ore and waste. This could be the result of different:
• grade control procedures • drill and blast patterns • mining fleets
• haul distances.
The base mining variable cost should therefore be thought of as the cost of mining all the rock as waste, with any incremental cost of mining ore considered an extra cost that must be covered by the net revenue received from classing it as ore. The cost difference may be positive or negative. For example, drilling patterns for ore may be closer-spaced than for waste (generating a higher unit cost or positive incremental cost for ore), while haulage distances to waste dumps may be longer than for ore (resulting in a lower unit cost or negative incremental cost for ore).
Many readers may quite rightly raise an objection that, although the mining costs in general should not be included in the ore–waste break-even cut-off calculation, they nevertheless must be accounted for somehow or the operation might make a loss. In practice, this could lead to including these costs in the break-even cut-off grade, even though the preceding discussion indicates that they should not be. How is this resolved? That the problem arises at all indicates that the simple 1D break-even model of cut-off derivation is inadequate for generating a cut-off that can be used in long-term planning. The issue will be addressed and resolved in the more complex cut-off derivation models discussed in Chapters 4 and 6.