Solution of Some Transportation Problems with Relaxed or Additional Constraints

The authors consider some modifications of the usual transportation problem by allowing bounds for the admissible supply—respectively, demand—distributions. In particular, the case that the marginal distribution function of the supply is bounded below by a $df$$F_1 $, while the marginal $df$ of the demand is bounded above by a $df$ is considered. For the case that the difference of the marginals is fixed—this is an extension of the well-known Kantorovich–Rubinstein problem—the authors obtain new and general explicit results and bounds, even without the assumption that the cost function is of Monge type. The multivariate case is also treated. In the last section, theauthors study Monge–Kantorovich problems with constraints of a local type, that is, on the densities of the marginals. In particular, the classical Dobrushin theorem on optimal couplings is extended with respect to total variation.