# Solving transportation problem by the Network Simplex

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

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.

• Would you see this link? Jan 25 at 12:59