All of the above models are special types of network flow problems: they each have a specialized algorithm that can find solutions hundreds of times faster than plain linear programming.
They can all also be seen as examples of a much broader model, the minimum cost network flow model. This model represents the broadest class of problem that can be solved much faster than linear programming while still retaining such nice properties as integrality of solution and appeal of concept.
Like the maximum flow problem, it considers flows in networks with capacities. Like the shortest path problem, it considers a cost for flow through an arc. Like the transportation problem, it allows multiple sources and destinations. In fact, all of these problems can be seen as special cases of the minimum cost flow problem.
Consider a directed network with n nodes. The decision variables are , the flow through arc (i,j). The given information includes:
Note that for this model, it is not necessary that every arc exists. We will use the convention that summations are only taken over arcs that exist. The linear programming formulation for this problem is:
Again, we will assume that the network is balanced, so , since dummies can be added as needed. We also still have a nice integrality property. If all the and are integral, then the resulting solution to the linear program is also integral.
Minimum cost network flows are solved by a variation of the simplex algorithm and can be solved more than 100 times faster than equivalently sized linear programs. From a modeling point of view, it is most important to know the sort of things that can and cannot be modeled in a single network flow problem:
Note that although these cannot be done in a single network, it may be possible to use the solutions to multiple networks to give you an answer. For instance, if there is only one arc with a fixed cost, you can solve both with and without the arc to see if it is advantageous to pay the fixed cost.
Here is an example of a problem that doesn't look like a network flow problem, but it really is:
A company must meet the following demands for cash at the beginning of each of the next six months:
At the beginning of month 1, the company has $150 in cash and $200 worth of bond 1, $100 worth of bond 2 and $400 worth of bond 3. Of course, the company will have to sell some bonds to meet demands, but a penalty will be charged for any bonds sold before the end of month 6. The penalties for selling $1 worth of each bond are shown in the table below.
[ Hint: Transshipment points are needed, in the form cash available at beginning of month t after bonds for month t have been sold, but before month t demand is met. Shipments into occur from bond sales and . Shipments out of occur to and demands for months .]