The most useful optimisation software feature you have probably never heard of

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Struggling with which pit shell to select? In this post Snowden Global Manager, Tarrant Elkington, describes a software feature you can use to help you make the decision quickly and consistently.

The selection of an ultimate pit shell has always been a key part of the mine planning process. I have seen engineers pore over the decision for days, completing umpteen scenario schedules and contorting themselves into spreadsheet knots before making a decision.

This has always bothered me because it is a decision that usually has a very small impact on project value. Why? Because this decision usually only impacts on your last cutback, often many years into the future when discounting erodes the present value of cash flows. Call me crazy if you like, but I would rather spend my time (and my client’s money) focusing on aspects that will have a big impact on value.

Perhaps the greatest significance of the ultimate pit shell selection is in the number of ounces (or equivalent) you want featured in your Ore Reserve announcement. This might sound a little facetious, but the reality is that there are often a wide range of pit sizes (and ounce totals) that produce similar value, meaning you can pick any of them and get a similar outcome. However, to an investor, 800 koz doesn’t have the same ring as 1 Moz, does it?

Typical practice in selecting pit shells is to generate shells without any discounting and then evaluate them with “best” and “worst” case schedules. The “best case” is usually around the revenue factor (RF) 1 pit, but the “worst case” can be the shell at a much lower RF (sometimes as low as 0.5). This provides a range of pits that can be selected. The engineer can then apply analysis and ultimately judgement to select a pit for final design.

It is curious to some that the “optimal” pit can be at a RF much lower than 1. I have had cases of selecting a pit in the order of RF 0.6 produces the highest NPV. However for some pits, the RF 1 pit is most appropriate. After going through the process many times I took a deeper look and noticed the trend:

  • Shallow pits are typically “optimised” at close to RF 1 pits
  • Deeper pits are typically “optimised” at lower RF pits.

Why is this the case? Let’s get a couple of truths out there:

  1. By definition, pit shell increments (differences between sequential pit shells) are marginal when the RFs are near 1. Think about it. If you go from RF 0.95 to RF 1, the maximum return of that incremental material is 5%. So, if the economics deteriorated only slightly, these increments would no longer be economic. For fun (sad, I know), I looked at a random selection of 15 pit optimisations and normalised the pit sizes and values in order to quantify this. The optimisations cover a range of commodities and geometries. The characteristic curves below are remarkably similar (Figure 1). Worth noting is that the top of the hill in all cases is very flat. 95% of peak margin is achieved in some cases with a pit size ±50% and at the least ±20%. There is a very wide margin for error in ultimate pit shell selection! When you include discounting the effective curve would be even flatter at the top. Just make sure you are somewhere on the top of the curve (unless you have a strategic reason for picking a smaller pit).
  2. Typically, but not always, you have to mine through waste before you access revenue-bearing ore. If you cannot mine the overburden in the same period as the ore there is a difference in discounting of cost (waste) and revenue (ore). The revenue-generating ore material is more heavily discounted than the associated overburden removal costs.

Figure 1 Pit size versus operating margin curve (normalised) for 15 projects

Let’s go through an example. Say the cost of mining waste is $10 M and the value of the ore uncovered is $10.5 M. So, we make a profit of $0.5 M undiscounted. If we assume waste is mined in Year 1 and ore is mined in Year 2, and the discount rate is 10% we get a negative present value (-10/1.1 + 10.5/1.1^2 = -0.4). So, while this increment generates undiscounted profit, it reduces discounted profit and should be excluded. Thus, we end up at an optimised pit that is lower than RF1.

Now consider an example of a shallow pit, where we can access ore simultaneous to waste (no overburden). The discounting of cost and revenue is that same, thus the revenue factor 1 pit may be selected.

If you are looking at a very deep pit with a long pre-strip for that cutback, which can be 5 years, then the ore revenue is greatly discounted, making it more likely that pit increments that generate positive undiscounted profit will deliver negative discounted profit. Thus, we select lower optimal pit shells for deeper pits.

Now we understand the theory behind this, can’t we just figure out a way to go straight to the optimised pit without wasting all this time on analysis? Yes!

We want to assign greater discounting to deeper blocks. But how much? To determine this, we look to a key, but usually valid assumption: That when you mine a cutback you will typically look to commence as late as possible (to defer cost) without impacting on your ore schedule, and mine at your maximum vertical rate of advance. In this situation the relative timing between blocks is set based on each block’s depth. Using this we can apply discount factor to each block prior to optimisation:

Discount factor = 1 / (1+ discount rate) ^ (depth / maximum vertical rate of advance)

If we optimise the pit (using standard process, but with the augmented values applied) the new RF1 pit will be our optimal discounted pit and we don’t need to do this other analysis or make judgements before moving along in the mine planning workflow.

Now I can hear what you are saying already: The total measured value of a pit with this discounting applied will be incorrect, as you will not mine all of each bench in the same year. I thought the same too, at first. And technically, yes, you would be correct.

However, what we are looking for here is a relative difference in values between blocks being considered for each pit shell expansion. So, while a block at surface of a later cutback might not be mined until, say, year 10 and it is discounted as if it would be mined in year 1 (due to its shallow depth), the relative discounting between this block and a block that will take 5 years to get to, will be preserved. Thus, the decision to mine this increment of material would have fair reflection of the relative discounting between upper blocks and lower blocks.

Also, you may wonder about whether augmenting these values in the optimisation will impact on cut-off grades. However, the great characteristic here is that the revenue and cost of a block are discounted by the same factor, offsetting each other in the cut-off calculation.

Sound like a bit of work to implement? Luckily, there is a feature within at least one of the commercially available tools, Whittle, which does this for us, albeit buried in an area resigned to rarely used parameters. It is called “Discounted by depth” (DBD) and simply requires the input of a text file with a string of values corresponding to the discount factor for each bench. The software then does the rest.

An example of this in a real-world situation is shown in the image below (Figure 2). The image shows the cross section through the model showing both the undiscounted block values, and those values obtained after discounting by depth (noting that the section does not show all the way to surface. The black outline is the RF1 pit for the undiscounted case. The red outline is the RF1 pit for the discounted case. There is a significant (35%) decrease in total tonnes between the outlines; effectively contracting the outline in by 30 m. The discounting associated with having a number of years of costs upfront do not get repaid (on a discounted basis) by the revenue generated at the base of the cutback. It is worth knowing this before you start with your pit design!

Figure 2 Demonstration of discounting by depth mechanics

For shallow pits there is little discounting difference between waste and ore, and hence the optimised pit will be similar to that for the undiscounted method. For deep pits, there may be a large difference in the results.

To test this hypothesis, I used the same 15 cases and run with and without the DBD feature. For each I used common parameters, with a 10% discount rate and 50 m maximum vertical advance rate. I measured the change in pit size at RF 1 between cases, and calculated the equivalent shell (based on pit size) that the DBD RF 1 pit shell would correspond to in an undiscounted case. In all cases the equivalent RF was less than 1 i.e. the DBD RF 1 was smaller than the undiscounted RF 1 pit. Plotting a scatter of depth (based on the undiscounted RF 1 pit) versus the equivalent RF shell we see a slight trend of decreasing equivalent RF shell with increasing depth (Figure 3).

Figure 3 Equivalent RF pit selected for a range of projects using DBD

Worth noting here is that the minimum equivalent RF shell was approximately 0.89, not that far off the undiscounted RF 1 pit. And they would all fit on the top of the undiscounted value plateau. Changing the method does not change the fact that there are many solutions with similar values.

Discounting by depth may also change ranking of nested shells as it penalises deeper blocks more than an undiscounted optimisation will. This may have important implications for staging of pits.

 You might be asking yourself, why only consider depth (or vertical rate of advance) in this decision, and not processing or mining rate. And this is a fair point. Let’s work through these:

  • If your mining rate (and pushback size selection) is limiting this is the same as having a lower rate of advance. You can adjust the discounting by depth to suit this. As an aside, if this is the case perhaps consider smaller stages.
  • If processing rate is limiting your advance (i.e. you slow down at the bottom) this should not make a big difference, particularly if you are stockpiling high grade for later.

While consulting workload has gotten in the way of me completing further scenario planning on these cases, the discounting depth will assuredly produce smaller optimal pits as either (a) the vertical advance rate is reduced, or (b) the discount rate increases.

SO WHAT?

I now use this feature routinely when completing pit optimisation. It saves time taken to figure out which shell to select. This is a scientific method that can be applied consistently, considering the main drivers of discounted value. And you only need to run it once, without having to run schedule after scheduling using pit shells which often bear only vague resemblance to the final pit staging that will be employed.

Perhaps most importantly, this time can be devoted to other, more value-accretive tasks such as initial pit staging, intelligent design and optimised scheduling. All things that add a lot more value to your project than ultimate pit shell selection.

I would guess that only 5% of mine planners have tested this out, and 95% of those have disregarded after one run, not understanding its underlying mechanics and usefulness. I have never reviewed a report that has described the use of this feature. It is truly unloved and underappreciated, in my opinion.

I am hoping that by highlighting the benefits of this feature that it will get the use it deserves, with the benefit being to the mine planner, and ultimately the mine owner, with better discounting trade-offs being made in the mine plan and more time to work on the important value-adding tasks.

Please give this feature a go and let us know what you find.

Look out for future posts where I will delve into the world of strategic mine planning and optimisation, apply some of these topics, provide some tools and tricks and identify some of the big unsolved problems you might never have thought of. Please provide feedback, ask questions, and share your own experiences.


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