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.