The set of constraints is generally called a *program*. If the constraints are expressed by linear relationships, then it is a linear programming situation.

The expression *function to be optimized* is also a synonym for *economic function* or *objective function*.

### Example

At least 150 litres of paint in total will be used to paint the classrooms in a school whose total surface area is less than 500 m². White and green paint will be used to match the school colours. According to the data provided by the decorator, the amount of green paint used should be at most 3 times the amount of white paint used. Given the characteristics of the paint, one litre of white paint covers 2 m² and costs $12, while one litre of green paint covers 3 m² and costs $15.

How many litres of each colour should the decorator use to keep costs as low as possible?

The number of litres of white paint is represented by*x* and the number of litres of green paint is represented by *y*.

Constraints (*program*) :

x ≥ 0 |
The number of litres of white paint is positive. |

y ≥ 0 |
The number of litres of green paint is positive. |

y ≤ 3x |
The quantity of green paint will be at most 3 times greater than the quantity of white paint. |

2x + 3y ≤ 500 |
Constraint related to the surface area to be painted. |

x + y ≥ 150 |
Constraint related to the total quantity of paint to be applied. |

The function to be optimized, that is, the rule that will provide the lowest cost is : *f*(*x*, *y*) = 12*x* + 15*y*.

This is the graph of the ** polygon of constraints **:

The table below provides the value of the function to be optimized at each of the vertices of the polygon of constraints :

Vertices (values rounded to the nearest whole number) |
Function to be optimized f(x, y) |
Values (in dollars) |

A(45, 136) | 540 + 2040 | 2580 |

B(38, 113) | 456 + 1695 | 2151 |

C(150, 0) | 1800 + 0 | 1800 |

D(250, 0) | 3000 + 0 | 3000 |

Based on the results, the lowest cost is obtained by using 150 litres of white paint and 0 litres of green paint. Therefore, the painting project will cost $1800.