1. Minimum Cost Network Flow Problems

2. Some extensions to the Max Flow problem
If we have multiple sources or multiple sinks, we need to create additional nodes to have a single start or end point for the network flows.
A supersink can be created which has an edge leading to each source node, and a supersink would have a single edge leading form each sink to it.
This augmentation to a problem could be used to find the maximum capacity of a transportation network for example.
When we looked at transportation and transshipment problems in , we did not limit the amount from a supply point to a demand point.

3. Enter the Minimum Cost Network Flow Problem
The transportation and transshipment problems are just simplifications of the MCNFP.
The solution techniques for these problems are extensions of the work we did to solve those problems.
The transportation simplex is expanded on, to become the network simplex method.
We’ll look at the LP formulation of MCNFPs first.

4. Our simple network
We will use the network below again to illustrate the new class of problems

5. Capacities and costs
In MCNFP scenarios we will deal with the costs cij associated with edges of the network.
Ultimately, it is these costs that will determine the amount of flow along particular edges.
It is common for MCNFPs to have lower and upper bounds on flows. A set of bounds lij · xij · uij is given in a table in the notes.

6. Starting with the Max Flow problem
The objective function of the max flow problem is
max a0 =  xs. =  x.t
And there are only a few constraints.

7. Constraints There are two sets of constraints to worry about:
First, what goes into a node must leave it. These are called flow conservation constraints.
 xi. -  x.i = 0
And all edge flows must be within the allowable range.
lij · xij · uij

8. Treatment of undirected edges
Note that the undirected edge in our network between A and B must be dealt with.
We could say that the flow from A to B could take negative values to allow flow from B to A.
or,
We could use a pair of directed edges.
It will prove much easier to take this step yourself, but note that when we allow a variable to be unrestricted in sign, the simplex method will create the additional variable anyway.

9. The change to the MCNFP
The only alterations we must allow for to turn the max flow problem into an MCNFP is to alter the objective function and to ensure that a certain amount is transferred from the source to the sink.
This is an example of goal programming. We start with one goal and having found the answer we want (usually feasibility) we alter the objective function.
The new objective function is
min z = i j cijxij

10. The new constraint
The new constraint can be based on the product leaving the source or the product arriving at the sink.
We would then have the option of choosing between
 xs. ¸ k Or
 x.t ¸ k