A mining company is going to continue operating in a certain area for the next five years. 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, if it be operated in a future year. Clearly, if a mine is not going to be worked again, it can be permanently closed down and no more royalties need be paid. The yearly royalties payable on each mine kept 'open' are as follows:

$\backslash$table$[[$Mine $1,5$ million$],[$Mine $2,4$ million$],[$Mine $3,4$ million$],[$Mine $4,5$ million$]]$

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

$\backslash$table$[[$Mine $1,2{10}^6$ tons$],[$Mine $2,2\mathrm{.}5{10}^6$ tons$],[$Mine $3,1\mathrm{.}3{10}^6$ tons$],[$Mine $4,3{10}^6$ tons$]]$

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, for example, 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 as follows:

$\backslash$table$[[$Mine $1,1\mathrm{.}0],[$Mine $2,0\mathrm{.}7],[$Mine $3,1\mathrm{.}5],[$Mine $4,0\mathrm{.}5]]$

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 as follows:

$\backslash$table$[[$Year $1,0\mathrm{.}9],[$Year $2,0\mathrm{.}8],[$Year $3,1\mathrm{.}2],[$Year $4,0\mathrm{.}6],[$Year $5,1\mathrm{.}0]]$

The final blended ore sells for $10$ 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?

This problem is based on a larger one arising in deciding which pits to work in the firm of English China Clays. In that problem $($in the $1970$s$),$ it was wished to work up to $4$ mines out of $20$ in each year. The model proved very difficult to solve.