Depending on the nature of the mineralisation, cut-offs may need to be applied to mining blocks rather than to mine-wide tonnage increments. For example, the geological conditions in some open stoping operations may mean that, once the stoping limits have been established, they cannot be changed without causing operational problems. The production schedule may then change to a number of mining blocks being mined in parallel, but starting and ending at different times. This type of requirement is similar to the stages used in open pit mine plans, but whereas the open pit plan is rearranged into sequential steps, this may be unfeasible in an underground operation. Parallel ore streams are not handled by the simple form of Lane’s methodology.
Stockpiles
Stockpiling of mineralisation with a grade below the current cut-off is often viable with an open pit,22 with stockpiled material becoming additional rock that can be reclassified
as ore along with the rock that is freshly mined at a later date. There are costs associated with building, maintaining and reclaiming a stockpile.
Difficulties in applying Lane’s methodology may arise when mining capacities need to be considered. Drilling and blasting capacity, and perhaps that of the in-pit loading fleet, will impose a limit on the amount of new rock mined; however, if the same trucks are used to haul both newly mined rock from the pit to waste dumps, stockpiles and treatment plant and reclaimed stockpile material to the treatment plant, the mining
22. Stockpiling of underground material is less common – we would not usually incur the costs of mining material simply to put it on a stockpile; however, where the mine can produce more potential ore than the mill can handle and there is a need to mine through lower-grade material to reach higher grades, production and stockpiling of that lower-grade material may be appropriate if separation of material of different grades is feasible.
ChApter 5 | Lane’s Methodology capacity may be a function of the cut-off. This may depend on the tonnage and grade versus cut-off relationships in both newly mined and stockpiled material and haulage rates associated with each. The problem is not insurmountable: there is no reason why the tonnage and grade versus cut-off curves or tables that would be used to find the two balancing cut-offs cannot include mining capacities that also vary with cut-off rather than remain constant, as described earlier.
Reclaimed stockpile material is ore, so the cost of reclaiming it is an ore-related treatment cost, even if the reclaim is being done by the rock-mining fleet and recorded as mining costs in the company’s cost reports.
Lane (1988, 1997) presents a formula for calculating the stockpile break-even grade. It is similar to Lane’s treatment-limited cut-off formula, but includes the cost of reclaim as part of the cost per tonne of ore. The recovery term (y) is for the stockpiled material, which might suffer some degradation during the years it is stockpiled. This grade is the lower boundary of material that should be stockpiled in any year, and represents the lowest grade that could be economically treated at the end of the operation’s life. The upper bound will, of course, be the cut-off applied to run-of-mine material. The construction of the stockpile is therefore easily defined.
However, the optimum stockpile withdrawal strategy cannot be determined in the same way as newly mined material using Lane’s methodology without stockpiling, as described earlier in this chapter. Part of the reason is that, as indicated previously, simple Lane assumes a single stream of mined material delivered in sequential steps, from which ore is separated. The details of the material in each mining step are known before the cut-off optimisation process begins. Stockpiling not only adds a second potential source of ore, but the tonnage is not fixed – it can fluctuate over the life of the mining operation – so the tonnage and its grade distribution are functions of the cut-off policy applied to the newly mined material. Lane describes briefly how an appropriate algorithm will draw material for treatment from both newly mined material and stockpiles, such that the net contribution from the lowest grade material from each is the same. The capacities for mining, treatment and marketing – that is, for handling rock, ore and product – must also be balanced when dealing with these components of both the newly mined material and stockpiled material. These more complex algorithms fall within the realm of complex Lane or strategy optimisation, though some of the issues identified may be handled by the iterative processes required for simple Lane.
SuMMAry
Lane’s methodology takes account of costs and prices, the grade distribution in the rock mass and the capacities of the various stages of the production process. In principle, Lane’s methodology can be applied to optimise any strategic decisions, including cut- off. To develop an understanding of the parameters that influence cut-off derivation, we refer to simple Lane, which seeks to optimise cut-offs given a specified size of mine, mining sequence and capacities for dealing with rock, ore and product. The explicit goal of a Lane-style cut-off optimisation is maximising NPV; however, there is no guarantee that the maximum NPV will be a positive value.
Six potential cut-offs are derived for each sequential increment of rock (or mining step). These increments are the constants in the analysis. The optimum cut-off policy will determine the duration of each mining step and hence the overall life of the operation.
Changing the assumptions about costs, prices or capacities may change the duration and overall life.
Three of the six cut-offs are limiting cut-offs, derived by assuming that the processes dealing with rock (identified by Lane as mining), ore (identified as treatment) and product (identified as marketing) are each independently the constraint on throughput. Limiting cut-offs are derived by break-even formulas. Since each increment of rock must be mined regardless of the amount of rock that is classified as ore, the mining variable costs will be the same for all possible cut-offs. Costs common to all options will not impact on the cut-off decision and do not need to be included in the analysis. Mining variable costs such as drilling, blasting, loading and hauling do not enter into any of the limiting cut-off calculations.
The mining-limited cut-off calculation includes only ore- and product-related variable costs. As well as these costs, the treatment- and marketing-limited cut-offs also include fixed costs and opportunity cost. The latter is the time value of money cost of deferring receipt of the NPV of the rest of the operation by treating additional material from the mining step under consideration as ore, as well as accounting for the impact of any changes in economic conditions in the period within which the displaced material would be produced. When the ore or product stream is assumed to be the capacity- limiting process, any additional material classed as ore from the rock increment will extend the mine life and hence incur additional fixed costs and opportunity cost, which must be paid for by the grade of the additional material processed.
The other three cut-offs are balancing cut-offs. These are determined for each of the three pairs of two of the production stages, and ensure that the two stages in each case are operating at their capacity limits. These cut-offs are physical functions of the grade distribution in the rock in each mining step and the capacities of the production stages considered. They are in no way defined by costs and prices.
Having generated all six potential cut-offs, there is a basic two-stage process to reduce the six, firstly to three and finally to the one optimum cut-off to be used.
For determining a short-term single-period optimum cut-off, the process derives the necessary cost data from mine cost accounting systems, though it may require effort to ensure that costs are attributed to the correct quantities. The opportunity cost is usually extracted from an existing long-term plan. For determining a long-term optimum cut-off policy – the sequence of optimum cut-offs over time – an iterative process uses NPVs remaining at the end of each mining step in one iteration to determine the opportunity costs for the next iteration. This process normally converges to a stable cut-off policy, though there is no guarantee.
It is likely that in a long-life operation with a consistent grade distribution over time, debottlenecking of the mining and treatment processes will result in a balance between at least two of the production processes, and the balancing cut-off created will then remain as the optimum cut-off regardless of price and cost changes over time. If constraints do not change, prices and costs can alter significantly, but the optimum (balancing) cut-off will not change. If the optimum cut-off is a balancing cut-off, it is not logical to change it (for example, because changes in costs and prices have changed the break-even grade) unless one of the capacities in balance is also changed. This must be justified by an evaluation of the change’s impact on value.
ChApter 5 | Lane’s Methodology In a single-metal open pit, Lane’s theories and methods permit analytical optimisation of the cut-off policy for a given mining sequence. There are no simple analytical techniques available to optimise concurrently both the schedule and the cut-off policy.23
These comments may also prove true underground, but only for some mining methods; for example, sublevel caving.
For such issues as multiple mines, ore sources or ore types; polymetallic orebodies; and complex stockpiling requirements, analytical techniques may be inadequate. Exhaustive search techniques or specialist software may be required to optimise the overall plan. This will be the key focus of the following chapter. In many cases, complex issues can be dealt with by the iterative processes used to determine an optimum cut-off policy. In the simplest analyses, only the opportunity cost is varied from iteration to iteration. In reality, every component accounted for may change from one iteration to another, with values in a later iteration being derived from the outcomes of the preceding iteration.
Lane’s methodology represents a major advance in cut-off theory. The insights it provides help to explain many of the sometimes counter-intuitive outcomes of a full strategy optimisation. In particular, the concept of opportunity cost – often referred to in discussions of strategy optimisation – is rigorously accounted for to indicate to what extent future production can be deferred to immediately treat additional material as ore. In addition, the concept of the balancing cut-off is introduced in cut-off theory. This shows that it is feasible for the optimum cut-off to be purely a function of the physical features of the resource and the operation’s infrastructure – the grade distribution and the capacities of the various stages of the mining and processing operations – and be unaffected by price and cost changes.
23. the key words here are simple and analytical, implying analyses that could be done by a competent engineer using spreadsheet software, for example. More complex techniques that could potentially optimise concurrently both the schedule and the cut-off policy, such as dynamic programming and linear and mixed integer programming, will be discussed in part 2.