Linear programming relaxations of a network flow problem with applications in machinery industry

Globalization of design, manufacturing and logistics has made the production-transportation problem (PTP) an important aspect of the supply chain optimization. This problem is usually formulated with piecewise linear concave cost functions for both production and transportation costs, and then solved using a mixed-integer programming (MIP) algorithm. This paper studies the application of three different relaxations of the MIP formulation of the PTP and compares their computational efficiencies. A cutting-plane algorithm is adapted to improve the computational efficiency. Although the relaxations tend to provide similar optimal costs, they differ significantly with respect to computational complexity.