### Example of Linear Programming including Maximising and Minimising.

### Question

A firm exports two types of
machine P and Q. Type P occupies 2m3 of space and type Q, 4 m3.
Type P weighs 9kgs and type Q 6kgs. The available shipping space is
1,600m3 and the total weight of the machines cannot exceed 3,600 kgs.
The profit on type P is €100 and the profit on type Q is €80. How many
of each machine must be exported to bring a maximum profit and what is
the profit.

### Answer

Let x represent the number of type P exported and y represent the number of Q exported.

therefore 2x + 4y<= 1,600 which on dividing across by 2 gives x+ 2y<=800

9x + 6y<=3,600 which on dividing across by 3 gives 3x + 2y <= 1200

The other two inequalities will be x>=0 and y>=0 as you cannot have a negative no of machines.

The four inequalities are

- x>=0
- y>=0
- x + 2y<=800 substituting x=0 and y=0 we get the points (0,400) and (800,0)
- 3x + 2y<=1200 substituting x=0 and y=0 we get the points (0,600) and (400,0)

When 3 and 4 are solved
simultaneously we get the point (200,300). Now we draw the lines and use
the value (0,0) to indicate the region represented by each inequality.Click here to see the two steps again.

The following is the required
diagram.

The profit on P is €100 and the profit on Q is €80 therefore the profit is 100x + 80y.

Following Step 4 of finding the maximum and minimun values we construct a table for simplicity.

Vertex | 100x | 80y | 100x +80y |

(0,0) | 0 | 0 | 0 |

(0,400) | 0 | 32,000 | 32,000 |

(200,300) |
20,000 |
24,000 |
44,000 |

(400,0) | 40,000 | 0 | 40,000 |

From the table we can see that the company should export 200 machines of type P and 300 machines of type Q to get a Maximum value of €44,000.