A production-transportation problem with piecewise linear cost structures

In this paper, we study a multistage production-transportation problem for a make-to-order company with outsourcing options at each stage of production. We formulate the problem as a multi-commodity network flow problem with piecewise linear cost structures by assuming the less-than-truckload transportation mode and non-linear production cost structure. We use polymatroid inequalities to strengthen its linear programming relaxation and present a cutting plane approach to tackle it. The computational results show that the strong formulation gives a tight lower bound and the cutting plane approach can solve the problem efficiently.