OR-Notes

J E Beasley

OR-Notes are a series of introductory notes on topics that fall under the broad heading of the field of operations research (OR). They were originally used by me in an introductory OR course I give at Imperial College. They are now available for use by any students and teachers interested in OR subject to the following conditions.

A full list of the topics available in OR-Notes can be found here.

Integer programming tutorial class question

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, 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. Once a mine has been closed it cannot be reopened. The yearly royalties payable on each mine kept 'open' are:

• Mine 1 £5m
• Mine 2 £4m
• Mine 3 £4m
• Mine 4 £5m

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

• Mine 1 2m tons
• Mine 2 2.5m tons
• Mine 3 1.3m tons
• Mine 4 3m tons

The ore from the different mines is of varying quality. This quality is measured on a scale such 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 two ingredient ores. Measured in these units the qualities of the ores from the mines are given below:

• Mine 1 1.0
• Mine 2 0.7
• Mine 3 1.5
• Mine 4 0.5

To make this blending clearer if we blend a quantity a from mine 1, b from mine 2, c from mine 3 and d from mine 4 the resulting blend has quality (1.0a+0.7b+1.5c+0.5d)/(a+b+c+d).

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

• Year 1 0.9
• Year 2 0.8
• Year 3 1.2
• Year 4 0.6
• Year 5 1.0

The final blended ore sells for £10 per ton each year. Formulate a mixed integer programming problem to determine which mines should be operated each year and how much they should produce.