# Simplex - Network flow problem : Arc from 1 to P with infinite capacity

The Network - Maximum flow problem below aims to find the maximum flow using simplex method :

With the LP as follow :

LP : $$\begin{Bmatrix} Z(Max) = \sum_{i=1}^{m} fi \\ Af =0 \end{Bmatrix}$$

Incidence Matrix :

With arcs indexed {1,2,3,4,5} in red.

Question part

As mentioned above, I need to get the Maximum Flow using the simplex method.
Can someone please solve it, and explain to me how to they did.

$$\begin{Bmatrix} f_{1} + f_{2} - f_{5} = 0\\ f_{1} + f_{3} = 0\\ -f_{2} + f_{4} = 0\\ -f_{3} -f_{4} + f_{5} = 0\\ \\ f_{1}\leq 4\\ f_{2}\leq 4\\ f_{3}\leq 2\\ f_{4}\leq 3 \end{Bmatrix}$$
• The last row of your incidence matrix is incorrect. Note that each column should one $1$ and one $-1$, with the other entries $0$. – RobPratt May 28 at 18:54