Given the problem:
maximize $a_1x_1+a_2x_2$
s.t.
$2x_1-x_2\ge 2$
$x_1+x_2\le 4$
$x_1,x_2\ge 0$a) Given that $a_1=1,a_2=2$, solve the problem graphically.
b) Find all the basic feasible solutions of the problem.
c) Using the basic feasible solutions you found, find a range for the values of $a_1,a_2$ such that the solution you've found is still optimal.
I have solved (a) and (b), I've found that the optimal solution is $(2,2)$ with optimal value $6$, and found all the basic feasible solutions, and saw that they are the vertices of the feasible area in my graph in (a).
What I don't understand is how to solve (c), I can understand graphically that I could change the parameters until $(2,2)$ is not an optimal solution anymore, but I can't understand how do I use the basic feasible solutions I've found to do that?
What does knowing the basic feasible solutions tell me about the parameters? Is it just them being the vertices?
I feel lost, and would appreciate any help.
Thanks in advance.
Best Answer
For any values of $a_1$ and $a_2$, you know that the optimal solution will be a basic feasible solution (BFS). So one way to answer the question is to write the object value of each BFS as a function of $a_1$ and $a_2$, compare those expressions, and figure out for what values of $a_1$ and $a_2$ the solution you found has objective value at least as large as that of any other BFS.