On the polyhedral structure of a multi-item production planning model with setup times

We present and study a mixed integer programming model that arises as a substructure in many industrial applications. This model provides a relaxation of various capacitated production planning problems, more general fixed charge network flow problems, and other structured mixed integer programs. After distinguishing the general case, which is NP–hard, from a polynomially solvable case, we analyze the polyhedral structure of the convex hull of this model, as well as of a strengthened LP relaxation. Among other results, we present valid inequalities that induce facets of the convex hull in the general case. These inequalities suffice to solve this model by linear programming in the polynomially solvable case mentioned above, and they have proven computationally effective in solving capacitated lot–sizing problems (see also Miller et al. [2000a, 2000b]).