Optimization of the transportation expense of a firm with contractual supplies

We consider nonlinear nonconvex capacitated transportation problems where the nonlinearity occurs only in the last row of the transportation tableau. This transportation model can be successfully applied to economics representing the nonlinearity caused by penalties for unsatisfied contractual quantities or by changing in price. An algorithm for local optimization, based on the algorithms for solving linear transportation problems, is suggested. It consists of three phases – initial, linear and nonlinear. The nonlinear phase uses auxiliary linear transportation problems. The algorithm is illustrated by proper examples. Sufficient conditions for satisfying contractual supplies are given.