Mining: Which Pits to Work and When to Close Them Down

A mining company is going to continue operating in a certain area for the next five years.^[8] There are four mines in this area but it can operate at most three in any one year. Although a mine may not operate in a certain year it is still necessary to keep it ‘open’, in the sense that royalties are payable, should it be operated in a future year. Clearly if a mine is not going to be worked again it can be closed down permanently and no more royalties need be paid. The yearly royalties payable on each mine kept ‘open’ are

There is an upper limit to the amount of ore which can be extracted from each mine in a year. These upper limits are:

The ore from the different mines is of varying quality. This quality is measured on a scale so that blending ores together results in a linear combination of the quality measurements, e.g. if equal quantities of two ores were combined the resultant ore would have a quality measurement half way between that of the ingredient ores. Measured in these units the qualities of the ores from the mines are given below:

In each year it is necessary to combine the total outputs from each mine to produce a blended ore of exactly some stipulated quality. For each year these qualities are

The final blended ore sells for £10 per ton each year. Revenue and expenditure for future years must be discounted at a rate of 10% per annum.

Which mines should be operated each year and how much should they produce?