[I don't know what is the direct translation to English but it should be duplex simplex method]
we have a problem:
$$\min(z_k)=-2x_1+7x_2$$ $$x_1+x_2-x_4=2$$ $$2x_1+7x_2=3$$ $$x_i \geq 0$$ After some computation I get it to the final form:
$$\max(W)=2\Pi + 4\Pi + 3\Pi$$ $$x_{1} \rightarrow \Pi_{1}+\Pi_{3} \leq -2$$ $$x_{2} \rightarrow \Pi_{1}+\Pi_{2} -2 \Pi_{3} \leq 7$$ $$x^{-}_{3} \rightarrow -\Pi_{3} \leq 3$$ $$x^{+}_{3} \rightarrow +\Pi_{3} \leq -3$$ $$x^{-}_{4} \rightarrow -\Pi_{1} \leq 0$$ $$x^{-}_{5} \rightarrow -\Pi_{2} \leq 0$$ So after that there is a table solution: \begin{array}{c|c} x_{B}&c_{b} b& x_{1} & x_{2} & x^{+}_{3} & x^{-}_{3} & x_{4}& x_{5} & x_{6} & \\ \hline x_{6} & M & & 1 & 0 & 0 & -1 & 0 & 1 &2 \\ \hline x_{5} & 0 & 0 & 1 & 0 & 0 &0 & 1 & 0 & 4 \\ \hline x^{-}_{3} & -3 & 1 & -2 & -1 & 1 & 0 & 0 & 0 & 3 \\ \hline && 1 & 1 & 0 & 0 & 0 & 0 & 0 & 9 \\ \hline & &-1 & -1 & 0 & 0 & 1 & 0 & 0 & -2 \end{array}
Questions:
What is the proper name of this simplex method?
In which cases do we use the $x^{-}_{3}$ and $x^{+}_{3}$? What is the rule?
How do we calculate the last two rows, the formula should be $\Pi^{*}=c^{T}_{B}*B^{-1}$ but I don't understand what should actually be done.