# Function to Be Optimized

## Function to Be Optimized

In a decision-making situation involving real variables and in which the constraints are represented by functions involving these variables, the function to be optimized has the same variables and translates the relationship on which the decision must be made.

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 byx 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(xy) = 12x + 15y.

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.