Convex hull results for generalizations of the constant capacity single node flow set

Abstract

For single node flow sets with fixed costs and constant capacities on the inflow and outflow arcs, a family of constant capacity flow covers are known to provide the convex hull in different special cases and are conjectured to provide it in the general case. Here we study more general mixed integer sets for which such single node flow cover inequalities suffice to give the convex hull. In particular we consider the case of a path in which each node has one (or several) incoming and outgoing arcs with constant capacities and fixed costs. This can be seen as a lot-sizing set with production and sales decisions driven by costs and prices and by the lower and upper bounds on stocks instead of being driven by demands as in the standard lot-sizing model. The approach we take is classical: we characterize the extreme points, derive tight extended formulations and project out the additional variables. Specifically we show that Fourier–Motzkin elimination, though far from elegant, can be used to carry out the non-trivial projections. The validity of the conjecture for the single node flow set follows from our results.