INTRODUCTION
Computer hardware, and to a lesser extent software, has for the last 20 years consistently advanced at a rate which has exceeded all expectations. As a result, calculations which were difficult or impossible to do only a few years ago can now easily be completed on a computer small enough to fit on a desk and costing only a few months’ salary. What is more, the calculations can be done by users with very little knowledge of computers.
Pit optimization is a field which has benefited greatly from this process in recent years, and we can now go far beyond simple optimization of a pit outline. Thorough sensitivity work, which has often only received lip service in the past, can now be carried out routinely on every ore body that is examined. Management can be offered the real possibility of trading profit for reduced corporate risk in an explicit manner.
Pit optimization was touched upon briefly in the previous section, but we will now go into it in much more detail and describe what can be done at the time of writing (early 1990). There will undoubtedly be further developments.
THE MEANING OF PIT OPTIMIZATION
The first thing to realize is that any feasible pit outline has a dollar value which can, in theory, be calculated.
By feasible, here, we mean that no wall slope is steeper than the rock can support after allowing for the insertion of haul roads and safety berms. That is, we are talking about overall pit slopes.
To calculate the dollar value we must decide on a mining sequence and then conceptually mine out the pit, progressively accumulating the revenues and costs as we go. If we wish to allow for the time value of money—that is the fact that a dollar we receive today is more valuable than one that we (might) receive next year—then we must discount the revenues and costs by a factor which increases with time.
The second thing to realize is that in doing this calculation we have, in effect, allocated a value to every cubic meter or to every block of rock. What is more, we have allocated these values without taking any account of the mining which has gone before, except that the value may depend on the position of the block and the effect that its position has on haulage distances.
Current computer optimization techniques attempt to find the feasible pit outline which has the maximum total dollar value. The good ones guarantee that there is no single block or combination of blocks which can be added to or subtracted from the outline to produce an increase in total outline value. That is, they guarantee the absolute mathematical maximum. They also exclude any block combinations which have a zero value.
Once we have fixed the block values and the slopes, we have fixed the optimal outline, and it is important to make the point that there is only one optimal outline. If we assume that there are two outlines of the same value, then it is easy to show that the two taken together would produce an outline of higher value. Consequently the assumption of the existence of two different optimal outlines of equal value is false.
If the block values increase then, in general, the optimal pit gets bigger. If the slopes increase then, in general, the optimal pit gets deeper.
Of course, we have to know the pit outline in order to calculate the values of the blocks, particularly if the time value of money is important. Conversely, we have to know the block values in order to find the optimal outline. We therefore have a chicken and egg situation, and we will return to this.
A SIMPLE EXAMPLE
Let us assume that we have a flat topography and a vertical rectangular ore body of constant grade as is shown in Fig. 5.3.1. Let us further assume that the ore body is sufficiently long in strike for end effects to be ignored. Under these circumstances, we only have to concern ourselves with a section.
In this simplified case there are eight possible pit outlines that we can consider, and the tonnages for these outlines are given in Table 5.3.1.
If we assume that ore is worth $2.00 per tonne after all mining and processing costs have been paid, and that waste costs $1.00 per tonne to remove, then we obtain the values shown in Table 5.3.2 for the possible pit outlines.
When plotted against pit tonnage, these values produce the graph in Fig. 5.3.2. With these very simple assumptions the outline with the highest value is number five.
There are other things that we can learn from this curve.
Firstly, outlines four and six have values which are close to that of outline five, and this is not just an artefact of this particular ore body. For any continuous ore body, as the pit is expanded towards optimality, the last shell which is added will have only a small positive value. If it had a large one, there would probably be another positive shell to follow. This means that in this case, and in the vast majority of real ore bodies, the curve of value against tonnage is smooth and surprisingly flat at the peak. It is common to find that a 10% range of pit tonnage covers only a 1% range of pit value. The trick is to find the peak, and good optimizers guarantee to do this.
Secondly, consider Fig. 5.3.3. If we are working without an optimizer and doing a detailed design for a realistically complex ore body, then we might be working away from the peak at ‘A,’ where changes in pit tonnage can have a significant effect on the value of the pit. In fact, generations of mining engineers have learned that a series of small adjustments, involving a great deal of work, can significantly affect the profitability of the mine. Contrast this with starting from an optimized outline at ‘B.’ From this point, providing that ore and waste are kept in step with each other, it is difficult to go wrong. Certainly there is no need to experiment with small adjustments. Since, with modern software, we can plot this graph for real ore bodies, we can actually find out how much freedom of movement we have before we start the detailed design. In other words, designs based on optimized outlines are very much easier to do.
THE EFFECTS OF SCHEDULING ON THE OPTIMAL OUTLINE
When we schedule a pit, we plan the sequence in which various parts of it will be mined and the time interval in which each is to be mined. This affects the value of the mine because it determines when various items of revenue and expenditure will occur. This is important because the dollar we have today is more valuable to us than the dollar that we are going to receive or spend in a year’s time. There are various reasons for this:
- Delayed revenue may increase our need to borrow funds and pay interest, thus reducing the effective revenue;
- Delayed revenue may not eventuate—one of the risk factors;
- Delayed expenditure may reduce our need to borrow funds and pay interest, thus reducing the effective expenditure;
- Something unexpected may go wrong with the operation—another risk factor; etc.
The standard way to allow for this is to discount next year’s dollar by a certain percentage and to apply that idea cumulatively into the future. Thus we discount future revenues and costs by a particular discount rate and reduce them all to a net present value.
There are two discount rates. The notional discount rate is applied to actual revenues and costs which are likely to occur. That is, revenues and costs which follow the inflation rate. Thus the notional rate (typically 20%) includes an allowance for inflation. It is correct to use this, provided that we inflate our revenues and costs for future years. However, we are then in the position of guessing at the future inflation rate and then guessing at a figure to correct for it! It is easier to work out revenues and costs in today’s dollars and then to use the real discount rate (typically 10%), which does not allow for inflation.
In what we will call worst case mining, each bench is mined completely before the next bench is started. Waste at the top of the outer shells is mined early, and the cost is discounted less than the revenue from the corresponding ore which is mined much later. This can make the outer shells uneconomic. The optimal pit for worst case mining is thus generally smaller than is indicated by simple optimization using today’s costs and revenues. This can easily be seen by referring to Fig. 5.3.1.
In what we will call best case mining, each shell is mined in turn and thus the related ore and waste is mined in approximately the same time period. In this case, the optimal pit is usually close to the one obtained by simple optimization. Unfortunately, if we try to mine each shell separately, mining costs usually increase and cancel out some of the gains.
In small pits, worst case mining may be the only possibility. The larger the pit, the more opportunity there is for creative sequencing, and the closer it is possible to get to best case mining.
PRODUCTION OF A DETAILED DESIGN FROM AN OPTIMAL OUTLINE
The precise method used in creating a detailed pit design depends on the tools which are available. It may be done entirely by hand, or with varying degrees of computer assistance.
Whatever the method, the aim is to produce a detailed design which deviates as little as possible from the outline provided by optimization. Where deviation is unavoidable, we try to balance extra tonnage in one place with reduced tonnage in another. The resultant design should in most cases contain ore and waste tonnages very similar to those contained by the optimal outline. If it is not possible to achieve this, then it may be that the slopes were not set correctly for the optimization. For example, insufficient allowance may have been made for the effect of haul roads.
While all reasonable steps should be made to follow the optimal outline, the shape of the graph shown in Fig. 5.3.2 should be borne in mind. Provided that waste is not included without the ore which it uncovers, small deviations from the outline have little or no effect on the pit value. A useful concept is to say that the spirit of the outline should be followed rather than the detail. Certainly the square edges of the blocks on the outer surface of the outline are irrelevant. As a starting point, a smooth line should be drawn through them as is shown in Fig. 5.3.4. Remember that the block edges are artefacts, they do not represent geological or grade boundaries.
The achievement of the necessary minimum mining widths at the bottom of the pit is often cited as a problem with pit optimization. This problem is more apparent than real in that, for large disseminated or near horizontal ore bodies, the necessary adjustments at the bottom of the pit are usually easy, whereas, for steeply dipping reef structures, it may be possible to put extra constraints into the optimization so as to ensure the necessary width. In the remaining cases, some loss of pit value will be involved in adjusting the bottom of the pit, but it should never exceed 1 or 2%.
THE AVAILABLE OPTIMIZATION METHODS
All currently available methods of optimization attempt to find the optimal outline in terms of a block model. That is, they try to find the list of blocks which has the maximum total value while still obeying the slope constraints.
The enormity of this problem is seldom appreciated.
Trial and Error
Consider a trivial model with only one section and 10 benches of 10 blocks. If we take a very simple-minded approach, each of the 100 blocks can either be mined or not, so there are 2100 or 1030 alternatives, many of them not feasible. Even if a computer could assess a million alternatives a second, it would still take three million times the current age of the universe to find the best one!
If the allowable slope is one block up or down at each column change, and we use this information to ensure that we try only feasible alternatives, the number of alternatives is reduced to 10 × 39 or 200,000. A computer could easily assess this number of alternatives. However, if we extend the model to 10 sections, the number of alternatives rises to 10 × 299 or about 1030 again, and we still have only 1,000 blocks, which is insufficient for serious work.
Put simply, trial and error is useless.
Floating Cone
The floating cone method has been popular because it is easy to program and easy to understand. It works by searching through the block model for ore blocks and then assessing the value of the inverted cones which have to be mined to expose them. If the value of a cone is positive, it is mined out and all the blocks it includes are changed to air blocks. The search then continues.
Unfortunately, this simple-minded approach rarely finds the optimal pit because of two distinct problems; one causes it to omit profitable ore from the pit and the other causes it to include non-profitable ore.
The first occurs because it cannot try all possible combinations of ore blocks, as that would be a trial and error process, and we have seen that that is computationally unreasonable. Most pits are viable in part at least because numbers of ore blocks combine to pay for the stripping of waste above them, when no individual block or even close group of blocks can do so. The floating cone method cannot detect this co-operation between different parts of the ore body if neither part is viable in its own right.
The second occurs for slightly more technical reasons. In Fig. 5.3.5 there are three small ore bodies and their corresponding waste volumes, with their values and costs shown. A floating cone program will examine A and will find that the corresponding cone has a total value of (40 - 20 - 30) = -10, and so is not worth mining. It will then examine B, will find a cone of value (200 - 80 - 30) = +90 and will convert it to air, leaving the values shown in Fig. 5.3.6
If a floating cone program is to work correctly, whenever it converts a cone to air, it should start searching again at the top of the model. However, this is computationally very expensive so that most programs continue their search downwards and would consider C next.
At this time the cone for C has a total value of (40 - 50 + 40 - 20) = +10, so that the program mines it. This should not happen, because some of the value of ore body A is being used to help pay for the mining of waste (the -50 region) which is below it. The true optimal pit in this case includes A and B, but not C.
Apart from being easy to understand and program, the one advantage that the floating cone method has over other methods is that, if instead of using just one block the program uses a disk of blocks as its starting point, then this can ensure a particular minimum mining width at the bottom of the pit.
Two-Dimensional Lerchs-Grossmann Method
In 1965 Lerchs and Grossmann gave two different methods for open pit optimization in the same paper. One works on a single section at a time. It only handles slopes which are one block up or down and one across, so that the block proportions have to be chosen so as to create the required slopes. This method is easy to program and is reliable in what it does, but, since sections are optimized independently, there is no guarantee that successive sections can be joined up in a feasible manner. Consequently a good deal of manual adjustment is usually required to produce a detailed design. The end result is erratic and unlikely to be truly optimal.
Two later variants of this method exist. One (Johnson, Sharp, 1971) uses the two-dimensional method both along sections and across them, in an attempt to join them up. The other (Koenigsberg, 1982) uses a similar idea but works in both directions at once. Both are restricted to slopes which are defined by the block proportions and neither honors even these slopes at 45ྻ to section. This last point is best illustrated by running the programs on a model which contains only one (very valuable) ore block. The resulting pit is diamond shaped rather than circular, with slopes correct in the E-W and N-S directions, but much too steep in between.
Three-Dimensional Lerchs-Grossmann and Network Flow
The second method given by Lerchs and Grossmann (1965) was based on a graph theory method, and Johnson (1968) published a network flow method of optimizing a pit. Both guarantee to find the optimum in three dimensions regardless of block proportions. Both, naturally, give the same result.
Both are difficult to program for a production environment where there are large numbers of blocks. Nevertheless this has been achieved and programs are now available which can run on any computer from a PC upwards. Most of these use the Lerchs-Grossmann method.
Because these programs guarantee to find the sub-set of blocks with the absolute maximum value consistent with the slope constraints, the alterations to the pit outline caused by small slope or block value changes are reliable indicators of the effect of such changes. This has opened up the field of real sensitivity analysis, where the effects of slope, price and cost changes can be measured accurately. With other methods, only the crudest sensitivity work is possible.
This has led to the development of programs which automate some aspects of sensitivity analysis to the point where graphs of net present value against, say, total pit tonnage, can easily be plotted. Further mention of this will be made later.
CALCULATING BLOCK VALUES
The correct calculation of block values is essential for any optimization. If the block values are wrong, the optimized pit outline will also be wrong.
For optimization purposes, there are two basic rules which must be followed when calculating the value of a block.
The First Rule
Calculate the block value on the assumption that it HAS been uncovered and that it WILL be mined.
No allowance for assumed stripping ratios should be made, because stripping is precisely what pit optimization works out. If a stripping ratio is assumed when calculating the block values, the result of the optimization is being prejudged.
Similarly, take no notice of any pre-conceived breakeven cutoff. The use of a breakeven cutoff can be helpful in manual pit design; it is inappropriate for optimized pit design. A consequence of this is that a block model in which only rock containing grades above a breakeven cutoff is designated as ore, is also inappropriate for pit optimization.
The only relevant cutoff in this context is that grade at which the revenue from recovered product will just pay for the cost of processing and any extra mining cost which is only applicable to ore.
Second Rule
Include any on-going cost which would stop if mining were stopped.
This is because, when the optimization program is adding a block to the pit outline, it is effectively extending the life of the mine. It must therefore pay for all the costs involved in extending the life of the mine.
Incremental costs such as fuel costs, wages, etc. must obviously be included in the cost of mining or processing, whichever is involved.
Overhead costs WHICH WILL STOP IF MINING STOPS must also be included. If the mine throughput is to be limited by the overall mining capacity, then these overheads should be included in the mining costs. If the throughput is to be limited by the processing capacity, then these overheads should be included in the processing cost, because only the addition of an ore block extends the life of the mine.
Nonrecoverable upfront costs, such as the cost of building access roads, should not be included in the costs used in optimization. Although these may be paid for with a loan which is to be repaid over a number of years, these repayments will be required whether mining continues or not. If the value of the optimized pit is less than the nonrecoverable upfront costs, then the mine should not be proceeded with.
BLOCK SIZES
There are four block sizes which are relevant in this work.
For Outlining the Ore Body
The size of the block that is needed for outlining the ore body depends on the shape and size of the ore body and on the particular computer modeling package that is being used. It may be quite small, which can lead to a model consisting of millions of blocks.
For Calculating Block Values
The value of blocks should be calculated with a block size which is similar to the selective mining size. That is, a parcel of rock should not be so small that it could not be mined separately, nor so large that grades are artificially smoothed. This block is sometimes bigger than that needed for outlining the ore body, requiring blocks to be combined and their grades averaged.
For Designing a Pit
There is now considerable experience in pit design using optimization techniques and, assuming that the pit occupies most of the width and length of the model and that the outline is not too convoluted, then a full model of 100,000 to 200,000 blocks is usually more than sufficient for pit design purposes. This leads to a block size which may be bigger than that for calculating values.
If it is necessary to re-block the value model, then it should be done by adding component block values and NOT by averaging grades.
For Sensitivity Work
If we want to do a series of optimizations using, say, different product prices so as to plot a graph of pit value against price, a model of 20,000 to 50,000 blocks will give just the same shape of graph with a very small shift of absolute value. Thus, most optimizations for sensitivity work can be done very quickly and this approach generally leads to a much more thorough sensitivity analysis.
Again, re-blocking should be done by adding values and not by averaging grades.
SENSITIVITY WORK
Although an optimized block outline and the corresponding detailed design are not the same, they do have a close relationship and, provided a good optimizer is used, are very similar in value. Consequently, when comparing two designs, the difference in value between the two optimal block outlines will be very similar to the difference in value between the two detailed designs. This means that sensitivity work can be carried out without doing any detailed designs at all.
Also, because a good optimizer produces a result which is objective and single-valued, it is quite reasonable to take note of small value differences due to, say, changing the slopes by a few degrees. This is not true when designs are done by hand, because an engineer will probably produce different designs on different days, without any change of slope.
During sensitivity work, we explore the economic and slope sensitivity of the mine. We sort out the general scale of mining and hence the operating costs. We decide approximately where the haul roads are to go and adjust the slopes in these regions to the average slope.
This requires a large number of quick optimization runs. However, it is probably the most valuable part of the whole design exercise because it inevitably leads to a much better understanding of the ore body and its economics. Graphs can be prepared which show how various characteristics of the mine, such as value or tonnage, are related to product price, costs, etc.
Probably the most significant graph is the one shown in Fig. 5.3.7. This relates net present value (NPV) to total pit tonnage for a given throughput and product price.
First, a set of optimal outlines is prepared, where each is optimal for a different product price. For some fixed product price, each of the outlines is then scheduled as though it was to be the limiting pit. If an automated practical scheduling scheme is available, it should be used. In producing Fig. 5.3.7, two limiting schedules have been used. Best case scheduling involves mining with many small pushbacks or cutbacks. Although in no sense a practical schedule, it indicates the highest possible NPV. Worst case scheduling involves completing the mining of each bench before starting the next. This is usually practical, but produces the lowest possible NPV.
The NPV for any practical mining schedule must lie somewhere between the two lower curves, with smaller pits tending towards the bottom curve and larger pits providing opportunities to get nearer to the middle curve.
This graph, which can be plotted for different product prices, is the single-most useful presentation known to the writer. It is meaningful to engineers, accountants, and management alike and can usefully be discussed in committee. It allows profit and corporate risk, in the form of mine life (pit tonnage), to be related and traded explicitly. Once a pit size has been chosen, it is easy to use the corresponding pit outline as a starting point for the detailed design.
This graph can be prepared by using any good optimizer and by doing a lot of work. However, software now exists which will produce the data for it automatically and quickly.
CONCLUSION
We have seen how good pit optimizers can be used not only to help design ultimate pit outlines, but also to carry out sensitivity analysis to an extent which is not possible without them.
Pit optimization is a tool which, used properly, can greatly speed and ease the process of pit design and can significantly increase the value of most pits. It can also be used to reduce the corporate risk involved in mining.
REFERENCE LIST
Johnson, T.B., 1968, “Optimum Open Pit Mine Scheduling,” Ph.D. Diss. University of California, Berkeley, CA, 120 pp.
Johnson, T.B., and Sharpe, R.W., 1971, “Three Dimensional Dynamic Programming Method for Optimal Ultimate Pit Design,” Report of Investigation 7553, US Bureau of Mines.
Koenigsberg, E., 1982, “The Optimum Contours of an Open Pit Mine: An Application of Dynamic Programming,” Proceedings, 17th APCOM Symposium, AIME, New York, pp. 274–287.
Lerchs, H., and Grossmann, I.F., 1965, “Optimum Design of Open Pit Mines,” CIM Bulletin, Canadian Institute of Mining and Metallurgy, Vol. 58, January.
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