The formula at the heart (and mind) of mine plan optimisation

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This article was written by Snowden’s Global manager, Tarrant Elkington.

It is easy to forget, or take-for-granted, the foundation elements for your area of expertise. The blog looks at patient zero for mine optimisation.

Appreciating Simplicity

I was in a meeting many years ago working through some optimisation mathematics on the whiteboard with some colleagues. It looked something like this – please don’t worry about the detail.

\ IV_a=UGDV_a-OPDV_a

\ UGDV_a=\frac{r_a-h_u_g-m_u_g}{(1+d)^u^y}$

\ OPDV_a=max(\frac{r_a-h_o_p}{(1+d)^o^y},0)+\frac{v}{(1+d)^o^y}$

Midway through this review, my boss at the time saw one of our clients appear at a window and he jumped up, bringing him into the room. He pointed enthusiastically at the equations and said: “Look at this thing on the board…  this is going to be a paper!”

On the day, I smiled when considering the simple mathematics involved. Plus, minus, times and divide – trivial stuff that we all learnt in primary school. If this is enough to get people excited then, then I was pretty sure I’d found somewhere I could make a difference!

What I have come to realise in intervening years is that simple maths which everyone can understand is good maths and is much more likely to have impact. The ability to take old ideas and find new applications is a valuable skill.

While the equation on the board was reasonably simple, it was derived from the basic block value calculation (BVC) formula; a formula that looks something like this in its simplest form and has been around for decades without significant revision.

BV=[(P-S)RG-H]O-MT

Where:

BV = Block value in $

P = Commodity price in $ per recovered metal unit

S = “Selling cost”[1] in $ per recovered metal unit

R = overall metal recovery (fraction)

G = block grade

H = “Process cost” in $ per ore tonne

O = Ore tonnes

M = “Mining cost” in $ per rock tonne

T = Rock tonnes

The BVC is the basis of everything we do in mine plan optimisation, connecting our decisions to the objective function (see my earlier post). It’s the backbone of cut-off and cut-over grade analysis, metal equivalents, pit optimisation, and schedule optimisation.

Failure to appreciate the BVC and its simplicity can make your life unnecessarily difficult. Embrace it, vary it, really get to know it, and you have a powerful tool in your back pocket for solving even the most complex mine planning problem.

And yes, if you were wondering, the equation was the basis for a paper in 2008[2]. My boss of the day was a smart man and still is!

So, let’s get to know the BVC a little better. There were two seminal works in mine optimisation completed in the 1960s that presented the equation.

Lane’s Theory

The earliest was a foundation 1964 paper by Ken Lane, the father of cut-off grades. He, as what seems very simple now, asked the question “is this block worth more as ore or as waste”? Working through the maths you can derive the cut-off grade equation. Understanding this derivation (with middle school maths) is quite powerful:

Value_o_r_e=[(P-S)RG-H]O-MT

Value_w_a_s_t_e=-MT$as there is no revenue, or processing/selling costs associated with waste

We Set    Value_o_r_e=Value_w_a_s_t_e

Solve for G    G=H/[(P-S)R]

Lane introduced the concept of Throughput Accounting (“TA”), in contrast to the more popular but technically less useful Activity Based Costing (ABC). For simplicity the equations referred to mining (M), processing (P) and selling (S) costs. What he was actually referring to is the cost driven by

(a) the amount of Rock mined (T)

(b) the amount of ore processed (O) and,

(c) the amount of metal produced (OGR)

The focus was on the cost driver (TA) rather than the cost centre (ABC). Subtle, but quite significant on at least two fronts.

  1. There are some costs which “crossover” from one activity to another driver. The most common example is the “incremental ore cost” which is the additional expense of mining a block as ore, as opposed to waste. This can be additional haulage cost (when compared to a waste haul), grade control cost, or selective mining cost. While this cost is actually a mining activity, it is driven by ore tonnes. Hence, it’s recognised as a processing cost (H) for BVC purposes.
  2. Some costs are not driven by any of these three drivers. Instead they’re dependent on time and are known as fixed costs. In ABC, they’re by cost centre, however, in TA, they’re assigned to one cost driver and only one. This is because TA acknowledges that the duration of the mine is typically driven by one physical driver (rock mined, ore processed or metal sold) regardless from which cost centre it originated. The entire fixed cost is loaded onto this constraining physical driver (after dividing through by the capacity of that driver) and treated like the other variable costs thereafter. This was a very simple solution to a potentially intractable problem. Well done, Ken Lane!

Proper TA allows the cut-off analysis to work on a truly marginal analysis basis. This analysis holds as true today as when it was written. Subsequent adjustments have only fine- tuned the model to cater for variable cost and recovery complexities, multiple rock types or additional processing stages.

Lerchs and Grossman

One of the great features of the BVC is its additive nature. The calculation applies at block level, but accumulates over all blocks to give us the net cash flow of the mine (at least at a technical level, let’s leave tax to the accountants). Note that sometimes, annoyingly, this additive characteristic does not hold (a topic for a later post).

This feature was used by Lerchs and Grossman in 1965 to outline their algorithm for what later became known as “pit optimisation”. By leveraging the cut-off grade calculation to determine whether a block is ore or waste and introducing 3D modelling, a single dollar value can be assigned to each block in space. With graph theory, they could employ slope constraints to develop relationships between adjacent blocks and solve for the optimum pit shell by net cash flow for a given orebody. A brilliant contribution, and provided tremendous power to the mine planner to resolve the complex 3D problem of “what to mine?”.

Sure, more efficient algorithms have been developed over the years, but they still all rely on BVC mathematics.

So What?

Understanding the BVC, and concepts of TA and marginal analysis, are essential if you want identify opportunities to improve the value of a strategic mine plan. Why? Because it ties the technical aspects of a mining project together with the financial. It is the strategic mine plan on which a project is based. Those who truly understand the fundamental economics of any given project will be the most successful, hands-down!

While software may do most of the heavy lifting these days, these programs will not determine which parameters to apply. So, if you miss or even misclassify incremental ore costs or fixed costs, you are going to drive the wrong behaviours and get the wrong answer. Unsurprisingly, knowing what a realistic solution looks like before running the code lessens redundancy and can certainly lead you to the best outcome.

A good understanding of BVC can open you up to solving other more complex problems that software does not cover and here are just a couple of examples.

The first is a grade equivalent calculation. A grade equivalent is a means of simplifying a polymetallic deposit to a single equivalent grade by assigning credit for the minor elements to the major element. Using the BVC we can set the value of a block as ore, using, say, two elements to the same value as a block of ore with only a single element.

Value_{2 elements}=[(P_1-SS_1)R_1G_1 + (P_2-S_2)R_2G_2-H]O-MT

Value_{Equiv}=[(P_1-S_1)R_1G_{Eq}-H]O-MT

We set    Value_{2\,elements}=Value_{Equiv}

Solve for G_E_q    G_E_q=[(P_1-S_1)R_1G_1+(P_2-S_2)R_2G_2]/[(P_1-S_1)R_1]

The introductory equations in this post were used for a client contemplating an open pit cut back through an existing underground operation. The question posed was: “Should we continue mining underground if we are going to excavate an open pit later”? Now, if the answer was “yes”, which material should be mined by the underground operation?

So, what we did was develop a ranking parameter, that we called incremental value (or IV) that calculated the difference between the discounted value of the block mined from underground and the discounted value of the same block mined from open pit.

Applying Lane’s logic, we removed the open pit mining cost from the calculation since we were not deciding whether the material would be mined.

This was a given and the year was pre-determined, so we could simplify the decision to consider (a) the cost of underground mining, and (b) the timing difference between open pit and underground mining.

The upshot of the analysis is that underground material within the pit should only be mined if the pit was not going to mine in that area in the next 40 years. Now I thought this was pretty clever and interesting result.

How about you?

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.

References

Robert, B., Elkington, T. van Olden, K. and Maulen, M. (2009) Optimising a Combined Open Pit and Underground Strategic Plan, Project Evaluation 2009, Melbourne, 85-91.

Lane, K. (1964). Choosing the Optimum Cutoff Grade. Quarterly for the Colorado School of Mines , 59 (4), 811-829.

Lerchs, H., & Grossman, I. (1965). Optimum Design of Open-Pit Mines. Transactions of the CIM , LXVIII, 47-54.

[1] I put these in quotation marks as these terms are not technically correct and are the source of some confusion with practitioners. We will discuss this later.

[2] Optimising a combined open pit and underground strategic plan, 2008. This paper looked at the decision making around whether to extract blocks from underground from an area that will be subsequently mined by an open pit.


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