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I am trying to solve the following problem using Network Simplex method. But I have questions.

enter image description here

My attempt:

Basis Matrix$(B)$

Rows: 1, 2, 3, 4, 5

Column: (1,3) (1,4) (1,5) (2,3) (2,4) (2,5)

$$ \begin{bmatrix} 1&1&1&0&0&0 \\ 0&0&0&1&1&1\\ -1&0&0&-1&0&0\\ 0&-1&0&0&-1&0\\ 0&0&-1&0&0&-1 \\ \end{bmatrix} $$

Vector of basic variables$(x_B)$

$$ \begin{bmatrix} x_{13} \\ x_{14}\\ x_{15}\\ x_{23}\\ x_{24}\\ x_{25} \\ \end{bmatrix} $$

My questions:

I couldn't be sure where to add the root arc both on the matrix and the graph. I guess I need to add 2 root arcs, but where do I add them?

To solve the problem I need to solve the $B*x_B=b$ equality where $B$ is basis, $x_B$ is the vector of basis variables(consists of the arcs) and $b$ is the capacities. I also couldn't find a way to represent the basis variables.

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    $\begingroup$ Would you see this link? $\endgroup$
    – A.Omidi
    Jan 25 at 12:59

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