9. T42 Co. Ltd produces three different blends of tea, Morning, Afternoon and Evening. The teas must be processed, blended and then packed for distribution. The table below shows the time taken, in hours, for each stage of the production of a tonne of tea. It also shows the profit, in hundreds of pounds, on each tonne.
| \cline { 2 - 5 }
\multicolumn{1}{c|}{} | Processing | Blending | Packing | Profit (£100) |
| Morning blend | 3 | 1 | 2 | 4 |
| Afternoon blend | 2 | 3 | 4 | 5 |
| Evening blend | 4 | 2 | 3 | 3 |
The total times available each week for processing, blending and packing are 35, 20 and 24 hours respectively. T42 Co. Ltd wishes to maximise the weekly profit.
Let \(x , y\) and \(z\) be the number of tonnes of Morning, Afternoon and Evening blend produced each week.
- Formulate the above situation as a linear programming problem, listing clearly the objective function, and the constraints as inequalities.
An initial Simplex tableau for the above situation is
| \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | 3 | 2 | 4 | 1 | 0 | 0 | 35 |
| \(s\) | 1 | 3 | 2 | 0 | 1 | 0 | 20 |
| \(t\) | 2 | 4 | 3 | 0 | 0 | 1 | 24 |
| \(P\) | - 4 | - 5 | - 3 | 0 | 0 | 0 | 0 |
- Solve this linear programming problem using the Simplex algorithm. Take the most negative number in the profit row to indicate the pivot column at each stage.
(11)
T42 Co. Ltd wishes to increase its profit further and is prepared to increase the time available for processing or blending or packing or any two of these three. - Use your answer to part (b) to advise the company as to which stage(s) it should increase the time available.
(2)