My linear program instead of inequations also contains one equation. I do not understand how to handle this in every tutorial I searched the procedure is to add slack variables to convert the inequations to equations. My lp is the following:
Minimize x4
Subject to:
3x1+7x2+8x3<=x4
9x1+5x2+7x3<=x4
5x1+6x2+7x3<=x4
x1+x2+x3=1
I tried to add slack variables w1,w2,w3 to convert the in-equations to equations but then I do not understand how to find an Initial feasible solution. I am aware of the 2-phase simplex but I do not understand how to use it here.
Do I have the right to add a slack variable w4 to cope with the last equation? As far as I understand If I do that I will change the LP. How should I start to cope with this LP? Can I use as an initial feasible solution the vector (0,0,1,0) for example?
This is not a homework question (Preparatory question for exams)! I do not ask for a complete solution, just for a hint to get unstuck from the equation problem.
Edit: I am not able to solve this. And I am not able to prove that it is infeasible. The fact that I have so many zeros in the $b_i$ creates to me problems!
Best Answer
I have to be honest, my simplex is rusty. But perhaps you could split the equation into two inequalities: $$x_1+x_2+x_3\leq 1$$ $$-x_1-x_2-x_3\leq -1$$ This is exactly what some solvers do that can't handle mixtures of inequalities and equations.