D1 June 2011 Q8
8. A firm is planning to produce two types of radio, type A and type B.
Market research suggests that, each week:
- At least 50 type A radios should be produced.
- The number of type A radios should be between 20% and 40% of the total number of radios produced.
Each type A radio requires 3 switches and each type B radio requires 2 switches. The firm can only buy 200 switches each week.
The profit on each type A radio is £15.
The profit on each type B radio is £12.
The firm wishes to maximise its weekly profit.
Formulate this situation as a linear programming problem, defining your variables.
| Scheme | Marks |
|---|---|
| Let \(x\) be the number of type A radios and y be the number of type B radios. | B1 |
| (Maximise P =) \(15x + 12y\) | B1 |
| Subject to \(x \geqslant 50\) | B1 |
| \(\tfrac{1}{5}(x + y) \lt x \quad (\text{accept } \leqslant)\ [y \lt 4x]\) | B1 |
| \(\tfrac{2}{5}(x + y) \gt x \quad (\text{accept } \geqslant)\ [2y \gt 3x]\) | B1 |
| \(3x + 2y \leqslant 200\) | B1 |
| \(y \geqslant 0\) | B1 |
| (7 marks) |
Notes
1B1 Defining \(x\) and \(y\); Must see ‘number of’
2B1 CAO objective function \(15x + 12y\)
3B1 CAO \(x \geqslant 50\)
4B1 CAO o.e \(\tfrac{1}{5}(x + y) \lt x \Rightarrow y \lt 4x\)
5B1 CAO o.e \(\tfrac{2}{5}(x + y) \gt x \Rightarrow 2y \gt 3x\) (corrected from the printed mark scheme: the notes print \(\tfrac{2}{3}(x + y)\); the scheme above has \(\tfrac{2}{5}(x + y)\), i.e. 40%)
6B1 CAO o.e \(3x + 2y \leqslant 200\)
7B1 CAO \(y \geqslant 0\)