An Introduction to the Theory of Pit Optimisation


Open pit mining "Pit Optimisation" involved juggling a number competing goals. Though many things need to optimised, "Pit Optimisation" here means one clearly defined problem. This problem is as follows:
  1. A body of rock, where a mine is considered, can be considered as a finite set of 'blocks'. This set must include not only potentially valuable ore but all possible overburden.
  2. For each block a value must be ascribed representing the net profit from mining it. Overburden blocks will have a negative value since they cost money to mine but do not, in themselves, yield a profit.
  3. For each block, it must be possible to determine its overburden i.e a list of blocks that must be mined first, if the final pit shape is to have slopes that the rock slope can support.
  4. We will use the term 'feasible pit' to mean a subset of blocks that obey geotechnical constraints. In other words for any block in this subset, all the blocks in its overburden are also in the subset.
  5. The optimum pit is the subset of blocks that gives a feasible pit and has the greatest value.

Discretisation -a horrible word

I use the term "discretisation" and "discretising" to mean making a problem discrete. We start with the problem of mine design, for which involves rational numbers, continuous variables and an infinite possible mine designs. We convert the problem into one of a finite number of fixed chunks (blocks) of rock, with therefore a finite, if large number of possible mines. Blocks are 'discrete' and are either mined or not mined.

Simple Examples and Illustrations

There are, of course, simple geometries which can be solved analytically and do not require discretising the problem into blocks. I have chosen not to consider this here. On the other hand I have been deliberately vague about the shape of the blocks. Throughout this site, I will tend to draw two dimensional problems, when I intend three dimensional ones. Unless otherwise stated, you can assume that the problem extends backwards into the screen.


pit diagram
This image (above) is intended to depict a simple block model. The mineralised (valuable) blocks have been shaded warm colours and have positive values. The blocks that cannot be mined at a profit are shaded blue.


pit diagram
If we assume that slopes of up to 45 degrees are possible, then the above is what we are calling a feasible pit. (i.e. feasible geotechnically). We can't slice through blocks to get our 45 degree slope (blocks are either mined or not) We arrange our removal of blocks so that the block to block angles do not exceed 45 degrees.


pit diagram
By contrast this is not a feasible pit. The pit walls are vertical. (i.e. not feasible geotechnically). This violates out chosen criterion of 45 degrees.


pit diagram
Here the green line indicates the optimal pit. Note that it is feasible and that no alternative could have a greater value (this pit has a value of 7).

Summary Points

When we talk about pit optimisation in this paper we mean the following:
  1. The problem is decribed in terms of a finite number of blocks that will either be mined or not mined.
  2. The blocks have positive or negative values.
  3. There is a slope criterion which determines maximum possible slopes and determines which blocks are in the overburden of which other blocks.
  4. The optimum pit is the set of blocks that is feasible and has the greatest possible value. For some problems the optimum pit is the empty set (nothing can be mined profitably)
It is not the case that:
  1. Blocks need to be cubes
  2. The slope angle needs to be 45 degrees.
  3. The problem must be two dimensional.
However, problems will generally be drawn that way for convenience.