I got stuck with the solution of an optimization problem which I have to solve. I read a lot about how to use the simplex method and I can successfully apply it when the constraints are of the type "<= " or "<" , but I don't manage to understand how to use the method when I have equations instead.
Could someone explain to me how to make my Simplex table and how to solve it step by step? Thank you very much !
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Another way of looking at the problem is by noticing that the constraints are equivalent to \begin{cases} -2x_4+4x_5 &\le 1 \\ x_4 &\le 1 \\ 3x_4+2x_5 &\le 4 \end{cases} In other words, you can consider that $x_1,x_2,x_3$ are your slack variables (with non null coefficients in the objective function).
In this case, the initial solution consists in setting variables $x_4$ and $x_5$ to $0$, which yields $X_0 =(1,1,4,0,0)$ which is the initial solution that you proposed.
Now, start your table simplex like you usually would. In the first iteration, $x_4$ is the entering variable (with reduced cost $10$), and $x_2$ exits. Can you take it from there?
You can transform equality constraints to inequality constraints. A constraint like $x=5$ can be written as two constraints: $$x \leq 5 \\ x\geq5 $$. These two constraints will force x to equal 5. Your first constraint can thus be written as the following two inequality constraints: $$ x_1 -2x_4 +4x_5 \leq 1 \\ x_1 -2x_4 +4 x_5 \geq 1$$.
Multiplying the second constraint by -1 will bring it into the form $\leq$.