Minimize $w=9y_1+4y_2$ subject to linear inequalities

Question

Minimize $w=9y_1+4y_2$

subject to : $4y_1+9y_2\geq 360$

$y_1+4y_2\geq 40$

$y_1\geq 0,~y_2\geq 0$

Answer 1

As SiXUlm said in comments, the minimum you're looking for will be found at a vertex where two of the boundary conditions meet. This means it is possible to find the minimum using the following (fairly naive) algorithm:

1) take each pair of boundary conditions

2) for each pair, calculate the point at which the two boundaries meet

3) check the meeting point fulfils the remaining conditions

4) if so, calculate the value of the function you're minimising

5) see which meeting point gave you the minimal value in (4).

EDIT: we also need to consider whether the function actually does have a finite minimum by seeing how it behaves (and whether the boundary conditions are still met) as $y_1$ and $y_2$ get very large in the positive or negative direction. But in this case we can see they are both constrained to be positive and that the function we're trying to minimise only grows as the variables increase.