6. A company makes lighting sets to be sold to stores for use during the Christmas period. As the product is only required at this time of year, all manufacturing takes place during September, October and November.
The sets are delivered to stores at the end of each of these months. Any sets that have been made but do not need to be delivered at the end of each of September and October are put into storage which the company must pay for.
Let \(x , y\) and \(z\) be the number of sets manufactured in September, October and November respectively.
The demand for lighting sets and the relevant costs are shown in the table below.
| Month | September | October | November |
| Manufacturing costs per set during each month (£) | 500 | 800 | 600 |
| Demand for sets at the end of each month | 800 | 1000 | 700 |
| Cost of storing sets during each month ( £ ) | - | 100 | 150 |
The company must be able to meet the demand at the end of each month and there must be no unsold articles at the end of November.
- Express \(z\) in terms of \(x\) and \(y\).
- Hence, find an expression for the total costs incurred in terms of \(x\) and \(y\).
The company wishes to minimise its total costs by modelling this situation as a linear programming problem.
- Find as inequalities the constraints that apply in addition to \(x \geq 800\) and \(y \geq 0\).
(2 marks) - On graph paper, illustrate these inequalities and label clearly the feasible region.
(4 marks) - Use your graph to solve the problem. You must state how many sets should be produced in each month and the total costs incurred by the company.
(3 marks)