Partially-linear transportation problems

The nonlinear relationship between quantities and the real cost for their transportation causes interest in some nonlinear transportation problems. Partially-linear transportation problems may be successfully applied in economics because every nonlinear function may be approximated by partially-linear functions. In this work we consider transportation problems with partially-linear dependence of the transportation costs and give the necessary and sufficient conditions for local optimality. A constructive method for solving such problems is suggested.