There is another way. As noted, the typical value equivalence formula, which is a linear function of the metal grades describing a value plane in multidimensional space, is:
Value = a × Metal A grade + b × Metal B grade + c × Metal C grade + …
In the same way that we require two points to define a line in two dimensions, and three points to define a plane in three dimensions, so we require n points to define a plane in n dimensions. If we were to investigate the relationship between two variables, we would obtain many more than two points to perform a statistical regression and define the straight-line relationship (the line of best fit) between variables. If we have many more than n points in an n-dimensional problem, we can perform a multiple linear
cHapter 9 | Specifying the Grade Descriptor regression to define the plane of best fit. In a two-dimensional analysis, a simple linear regression will generate the best fit values for the y axis intercept and the coefficient of the independent variable x. In the same way, a multiple linear regression will calculate the best fit values of the dependent variable axis intercept and the coefficients of the independent variables in n dimensions. If we have a number of sets of ore grades and their corresponding dollar values, a multiple linear regression can be performed, and the best fit values of the multipliers for the constituent grades will be determined. There is no need to make arbitrary allocations of costs. The maths automatically derives the best relationship between value and metal grades.
Mines usually have a model for deriving the final net revenue from the mine head grade, accounting for a large number of metallurgical performance and product sales terms parameters, which have an effect on the ultimate value received by the mine for a parcel of ore treated. If the processes are too complex to derive an NSR-type grade value for each geological model block, a data set of grades for a representative sample of blocks can be extracted from the block model. Assuming that each sample represents the grades and other characteristics of the mill feed, the physical, economic and financial model can calculate the net revenue generated in each case. Each combination of input grades and the resulting value will be the coordinates of a point in multidimensional space. With a number of data points greater than the number of variables, and covering the full range of individual metal grades and ratios of grades (to avoid problems of collinearity), the equation of the multidimensional plane can then be determined by multiple linear regression. Software such as Microsoft Excel™ provides a simple way of doing this.
If all the points lie close to a multidimensional plane, the linear equivalence formula model is as accurate as can be. The multipliers derived by regression will be the theoretically correct multipliers, in that the value predicted by the simple equivalence formula model will be the same as that resulting from the more complex economic and financial model. The coefficient of correlation (R-squared) calculated by the regression process will help to identify whether this is so. The closer R-squared is to 1, the better the linear equivalence formula fits the data points. This will be the case where the net payable grades are linear functions of head grade.
As well as generating linear multipliers to apply to each metal grade to derive an equivalent value, the multiple linear regression process will generate a constant term. It is unusual for equivalent grade formulas to include a constant term, but there is no reason why they should not. There are two options regarding the constant term. Since it will be applied to all blocks, it will not affect the ranking of blocks relative to each other. If simple ranking of blocks is the purpose of the grade descriptor, the constant term may be ignored, and only the multipliers for each metal applied to the metal grades. If it is desired that a dollar value grade descriptor should represent the true value as accurately as possible, the constant term should be included in the equivalence formula. This will need to be the case if, for example, sets of multipliers need to be derived for different rock types and different constants are derived from the regressions. Alternatively, the regression can be conducted in such a way as to force the constant term to be zero. Multipliers for metal grades will differ from those obtained with a constant term, and the correlation will not be as good. Consideration of the resulting change in the correlation coefficient or of values derived for actual grades, using a process similar to that illustrated in Figures 6.14 and 7.1, will indicate whether eliminating the constant term is appropriate.
Where the net payable grades are not linear functions of head grade, or for some other reason the value versus grades relationship is non-linear, one could potentially identify a more complex but accurate mathematical formula relating value and the metal grades. Alternatively, if the main purpose of the equivalent value is to define blocks above and below cut-off, a linear formula chosen to be accurate in the region of the cut-off could be a good solution.3
DoLLar VaLueS VerSuS MetaL eQuiVaLentS
Many polymetallic mines use a dollar value as a grade descriptor. Common practice at such operations is to calculate the cost per tonne and then use that unit cost as the cut- off. This has the appearance of both rigour and simplicity, and is effectively applying a break-even cut-off.4 As noted in the preceding subsection, there is no guarantee that
the dollar value assigned to a block accurately represents its true value. The application of break-even cut-offs to dollar values of material can potentially lead to under or over- stated cut-offs.
When price predictions change (often when new forecasts are issued annually as part of the company’s planning cycle), typical practice is to derive and apply new multipliers to block grades to produce new dollar values, and then re-create ore boundaries by applying the existing cut-off value to the new block dollar values. If break-even cut-offs are being used, this process is logical, but again there is no guarantee that the new dollar values accurately represent the true value, nor that the errors, if any, are consistent between the old and new values.
However, there are problems with this approach when optimum rather than break-even cut-offs are used. Optimum cut-offs are intended to achieve some particular optimised goal, such as maximising NPV, whereas break-even cut-offs merely ensure that every tonne pays for itself. As indicated in previous chapters, NPV-maximising cut-offs are typically higher than break-even cut-offs. Conventional wisdom regarding changes in cut-offs when prices change is based on the behaviour of break-evens; changes in cut- offs are expected to be inversely proportional to changes in prices. Yet the outcomes of optimisation studies show that, for a single metal deposit using the metal grade as the grade descriptor, the NPV-maximising optimum cut-off will change by a significantly smaller proportion than the inverse proportion of the price change. Figure 9.3 illustrates the problems that can arise when using dollar values.
The figure shows value versus cut-off curves for two price scenarios, the higher prices being 20 per cent higher than the lower prices. There are two cut-off axes for dollar values derived from each price set. Since all prices are assumed to increase by 20 per
3. the regression would be conducted using only the grades of sample blocks, for which the true values lie within a small range above and below the cut-off value.
4. there is an operation with a number of polymetallic orebodies that are mined by separate mines feeding a single concentrator. Staff in a centralised section are responsible for deriving NSr-type grade descriptors for the blocks in the block models of all orebodies. these are derived to represent the net value of the ore at the point of delivery to surface at each mine. Junior technical staff are tasked with determining the cost per tonne at their mines. these costs are then applied as break-even cut-offs to the NSr-type payable value grades. By virtue of these procedures, the company effectively allocates all responsibility for defining what is and is not its ore first to the market, which sets prices and hence ore values; and second, to its junior technical staff, who then determine what will be classified as ore. the company in question is not alone in doing this.