Redistribution of mass and probabilities

This paper is about optimal transportation, of mass (real) or probability mass (virtual). Not so much about the solution of such problems, but about their formulation.

Two non-negative vectors of the same size are given with the same totals. They can be considered supply and demand vectors for some good for a number of states (locations, sectors in the economy, etc.). The question that is considered in the present paper is how to redistribute the 'mass' in the supply vector to obtain the `mass' represented by the demand vector, in an optimal way.

What is to be considered optimal has to be defined. If we are dealing with a material good we want the transportation costs of this good to be minimized. Using a Hitchcock transportation model one only needs to specify the cost matrix to get a solution. However, if the vectors are not about a physical good, but about information, such as probability distributions, the problem is to find a joint distribution with the given distributions (vectors) as marginal distributions. This requires a different kind of model to find a solution. The domain (set of categories) needs to be a metric space. One can apply an iterative method, such as IPF, to compute a solution (if this exists).