On the Transportation Problem with Mixed Constraints

The standard transportation problem in linear programming is well known and may be started. The first statement of this type of problem is usually ascribed to Hitchcock. Dantzig gave the standard form and applied the simplex method to this special linear programming problem. An important limitation to the standard transportation problem is that the availabilities at the sources and the requirements at the destinations are fixed quantities and met exactly by the solution. The only exception is this when there is unbalance between the total available and the total required, in this case one introduces a dummy source and a dummy destination with zero costs. In real situations, it is frequently desirable to be more flexible and specify minimum quantities which may be taken from/or received at some sources destinations whilst specifying the exact amount to be taken from received others.