5 Badgers is a small company that makes badges to customers' designs. Each badge must pass through four stages in its production: printing, stamping out, fixing pin and checking. The badges can be laminated, metallic or plastic.
The times taken for \(\mathbf { 1 0 0 }\) badges of each type to pass through each of the stages and the profits that Badgers makes on every 100 badges are shown in the table below. The table also shows the total time available for each of the production stages.
| Printing (seconds) | Stamping out (seconds) | Fixing pin (seconds) | Checking (seconds) | Profit (£) |
| Laminated | 15 | 5 | 50 | 100 | 4 |
| Metallic | 15 | 8 | 50 | 50 | 3 |
| Plastic | 30 | 10 | 50 | 20 | 1 |
| Total time available | 9000 | 3600 | 25000 | 10000 | |
Suppose that the company makes \(x\) hundred laminated badges, \(y\) hundred metallic badges and \(z\) hundred plastic badges.
- Show that the printing time leads to the constraint \(x + y + 2 z \leqslant 600\). Write down and simplify constraints for the time spent on each of the other production stages.
- What other constraint is there on the values of \(x , y\) and \(z\) ?
The company wants to maximise the profit from the sale of badges.
- Write down an appropriate objective function, to be maximised.
- Represent Badgers' problem as an initial Simplex tableau.
- Use the Simplex algorithm, pivoting first on a value chosen from the \(x\)-column and then on a value chosen from the \(y\)-column. Interpret your solution and the values of the slack variables in the context of the original problem.