8. A company makes three sizes of lamps, small, medium and large. The company is trying to determine how many of each size to make in a day, in order to maximise its profit. As part of the process the lamps need to be sanded, painted, dried and polished. A single machine carries out these tasks and is available 24 hours per day. A small lamp requires one hour on this machine, a medium lamp 2 hours and a large lamp 4 hours.
Let \(x =\) number of small lamps made per day,
$$\begin{aligned}
& y = \text { number of medium lamps made per day, } \\
& z = \text { number of large lamps made per day, }
\end{aligned}$$
where \(x \geq 0 , y \geq 0\) and \(z \geq 0\).
- Write the information about this machine as a constraint.
- Re-write your constraint from part (a) using a slack variable \(s\).
- Explain what \(s\) means in practical terms.
Another constraint and the objective function give the following Simplex tableau. The profit \(P\) is stated in euros.
| Basic variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | Value |
| \(r\) | 3 | 5 | 6 | 1 | 0 | 50 |
| \(s\) | 1 | 2 | 4 | 0 | 1 | 24 |
| \(P\) | - 1 | - 3 | - 4 | 0 | 0 | 0 |
- Write down the profit on each small lamp.
- Use the Simplex algorithm to solve this linear programming problem.
- Explain why the solution to part (d) is not practical.
- Find a practical solution which gives a profit of 30 euros. Verify that it is feasible.