Why Mine Plans Based On Average Assumptions Are Wrong On Average

By Dr. Luis A. Martínez Tipe, PhD Director General & Principal Researcher, CAIDTech Originally published: June 4, 2017

This article explains, in a simple fashion, the concept of the “flaw of averages in mine planning” displayed in Fig. 1. It states that, in mining, when mine planners and mine managers plan and evaluate the future of a mine operation, uncertain variables are often replaced with single average values. The result of doing this is the generation of wrong/spurious outcomes for the project economic and operational indicators throughout its operating life, which could be fatal for the success of the mine operation. Following professor Sam Savage’s suggestions (Savage, S., 2009), these systematic errors are the “flaw of averages in mine planning”, which explain why forecasts based on average numbers are always wrong (For more information about the Flaw of Averages in Mine Project evaluation follow the link: https://www.linkedin.com/pulse/flaw-averages-mine-project-evaluation-dr-luis-martinez).

Figure 1. The Flaw of Averages in Mine Planning. Indeed, as stated by many authors and research studies (see for example Savage, 2009 and de Neufville and Scholtes, 2011), the practice of replacing uncertain numbers by single averages when evaluating the future economical and operational performance of a project, in our case a mine project, is wrong except in exceptional cases when all the relevant relationships in the mine evaluation process are linear; although there are some mining activities characterised by simple linear processes, in general the mine evaluation process is a complex non-linear process. To understand the flaw of averages in mine planning we present a simple example in Figure 2. In the top part of the figure there is an ore block, of let’s say copper (with T tonnes), which is sent to the mill to recover the copper metal. Despite the fact that resource engineers are aware that the copper grade, g_cu, in the block is uncertain, it is common practice to use Kriging techniques (Rossi and Deutsch, 2014) to interpolate and estimate the copper grade of the block, which is then used as input data to the mill process to estimate the effective copper grade (after the mill recovery process) that can be used to estimate the copper content that can be recovered from the block.  In our simple example, the estimated average copper grade of the block is calculated as, g_cu=0.8% per tonne. To recover the copper content in the block, it is sent to the mill plant where the block passes throughout a metallurgical process to finally render the recovered copper content, Q. The metallurgical process normally has a yield or recovery function, R, which is commonly a non-linear function of the copper grade (a more general definition of recovery function is provided by geo-metallurgical concepts); in our example the recovery function, R, is defined as:

Then, if we follow traditional techniques that use the single estimated copper grade of g_cu=0.8 as input to estimate the recovered copper from the block, the result will be an effective copper grade of, g_cu^R=g_cu×84%=0.67%, which is used to estimate the recovered copper content as Q_cu^trad=T×0.67%.  In this case, mine planners and mine managers will assume that because they used an average copper grade of, g_cu=0.8%, as input to the mill process then the recovered metal quantity, Q_cu^trad, is also an expected value of the copper the mine will recover from the block, which is wrong and consequently will lead them to make unrealistic investment decisions – we show next why this practice provides a wrong estimate of the expected metal recovered. If we now use advanced mine planning optimisation techniques that use the copper grade distribution of probabilities, via conditional simulation techniques (in this case we have 6 simulations), as input to the mill process, the output will be a distribution of probabilities of effective copper grades, f(g_cu ) (see Figure 2, bottom part), which provides information about the expected effective copper grade value, E{f(g_cu )}=0.64%, and its variance, 0.21%. The expected recovered metal quantity is then calculated as Q_cu^Advanced=T×0.64%. Despite in this simple example the difference in effective copper grades seems to be minimal, the effect of “the flaw of averages” can be better observed when estimating annual project cash flow. For example, if ore production is assumed to be 80m tonnes per year, and the copper price is assumed to be US$5,500/t, then the difference in annual cash flow is around $118.8 million, which could be decisive in deciding among two or more operational strategies. In summary, “the flaw of averages in mine planning” can be explained by the Jensen’s inequality, summarized in Figure 3, which shows the danger of using average (or expected) values as input when dealing with nonlinear (either concave or convex) functions, i.e., E{f(g_cu )}≠f(E{g_cu }).

Figure 3. Simple explanation of the Jensen’s inequality that explain the flaw of averages. In general, although we have shown a simple example to demonstrate the “flaw of averages in mine planning”, the problem is more complex when dealing with a real mine project as it involves the input of several operational and economic uncertainty variables which need to pass through several non-linear stages before rendering the value of the project over time. Since the vital importance of running further risk and option analysis “IN” your projects (as opposed to ON your projects) to minimize the effect of the flaw of averages. We hope that this article raises awareness about the problem of using single averages in mine project evaluation by stating: “Mine plans based on average assumptions are wrong on average”.  If you want to find out how we can assist you in managing the “flaw of averages” in your mine projects contact us at: contact@randoanalytics.com  *Thanks goes to Richard Wooller for having the time to read the paper and check the numbers in Figure 2, suggesting us to include more details in the paper in order to make it more transparent. References

• De Neufville, R. and Scholtes, S., 2011. Flexibility in Engineering Design. The MIT Press, Cambridge, Massachusetts, London, p 293. 

• Rossi, M. E., and Deutsch, C. V., 2014. Mineral Resource Estimation. Springer Dordrecht Heilderberg New York, London, p 332.

• Savage, S., 2009. The Flaw of Averages. Wiley, New York, p 392.

Editor's note: This article was written in 2017. Since then, CAIDTech has developed and applied the probabilistic frameworks described here across multiple mine projects in Latin America and Australia, integrating geological variability, operational dynamics and economic uncertainty into a single quantitative model. Learn more at [caidtechnology.com]