Consider a minimum cost network flow problem instance with five nodes.

Suppose node $1$ has a supply of $32$ units, node $4$ has a demand of $22$ units

and node $5$ has a demand of $10$ units. The remaining nodes $($nodes $2$ and

are transshipment nodes. The arcs of the network with their capacities

and unit cost values are provided in the below table.

a$.$ Formulate the problem as a linear program.

b$.$ Solve the problem using network simplex. Start with the feasible

basic solution $x_{12}=20,x_{25}=10,x_{34}=12,x_{24}=10,(x_{13}=12).$