Discussion: What is the best block model to run the mine plan and design optimisation process?

By Dr. Luis A. Martínez Tipe, PhD Director General & Principal Researcher, CAIDTech Originally published: June 27, 2022

I would like to comment about the fifth “true/false” statement I posted previously in https://lnkd.in/eW5d2_-8). 

Is the Kriging block grade model the best block model to run the mine plan and design optimisation process, where the (technical and economic) results, including the mine plan and design, are seen as expected (averages) results?

Even though this question seems to be a bit off the mark (because it is a normal practice in mining to use the Kriging block model for open pit mine plan and design optimisation), the objective of this question is to think a bit beyond traditional practices. 

So, in order to provide an answer to this question we need first to stablish the following facts from previous questions:

1. The processing of the drill hole data (and all available necessary data) results in the generation of: 

 I.         An average block model estimated using a best linear estimator (Kriging), and 

 II.         A set of simulated block models estimated using an algorithm for simulating block models conditional to the data.

Note all these models are conditional to the original available data (see Figure 1) – see comments on first question about kriging and simulation here: 

Figure 1. Block models - Kriging and Conditional Simulations – both of them are generated using the same initial data.

2. The mine plan and design optimisation process is a non-linear process (see Figure 2) – for information about the non-linearity of the optimisation process check previous question here.

Figure 2. Schematic showing the Non-Linearity of the Open Pit Mine Plan and Design Optimisation Process.

3. Jensen’s inequality states that when dealing with a complex nonlinear process, i.e. whose non-linear function is concave, convex, or combinations of both, serious trouble can arise when a single number is substituted for a distribution of probabilities. This is what we called the “flaw of averages in mine project evaluation” (read the article here)”. That is, if the expected value, E{X}, of the uncertainty variable, X, is input into the non-linear process F(.), the resulting output, F(E{X}), will not be the same as the expected value of the resulting outputs, E{F(X)}, generated by inputting the entire distribution of values, i.e., F(E{X}) ≠ E{F(X)} (Figure 3). As a matter of fact, depending of the structure of the non-linear process, the average of the function can be either greater or equal, or lower an equal, than the function of the average (See Figure 4).

Figure 3. The Flaw of Averages in Mine Plan and Design Optimisation Process.

Figure 4. The Jensen's Inequality Theory.

So, after reviewing the previous theory it seems that the kriging (average) block model is not the most suitable block model to use in the mine plan and design optimisation process; as a matter of fact, some researches, where conditional simulation were used in mine planning and design, suggest that the (economic and operational) values obtained from kriging ( F(E{X}) ) overestimate the average value of the results obtained from the simulations (no references provided trying to avoid misunderstandings). 

• Does this mean the mine plan and design optimisation process is a concave shape non-linear process?) 

• ...Can we prove this?

Note that, in this discussion, we did not deal with other sources of uncertainty such as the economic and operational uncertainties (modifying factors) assuming them to remain constants.

Open for discussion... Editor's note: This article was written in 2022. 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]