Branch-and-Price-and-Cut for Single-Commodity Fixed-Charge Network Flow Problems

We consider the single-commodity single-origin fixed-charge network flow problem (SSFNP) defined on a directed network G = (V,A), where V is the node set and A is the arc set, with each arc a represented by the ordered pair of head and tail nodes (h(a), t(a)). The node set includes a single origin (with no incoming arc), multiple destinations (with no outgoing arc) and transshipment nodes (with both incoming and outgoing arcs). The destinations and the transshipment nodes are grouped in sets D and T , respectively. Each destination k ∈ D has an associated demand dk > 0. On each arc a ∈ A, there is a capacity ua (with 0 < ua ≤ ∑ k∈D d k), a per unit transportation cost ca ≥ 0 and a fixed cost fa ≥ 0 that is incurred whenever some flow is sent through arc a. The problem consists in satisfying the demands at minimum cost, while respecting capacity and flow conservation constraints. By introducing flow variables xa on the amount of flow on each arc and binary variables ya that specifies whether arc a is used or not, the SSFNP can be modeled as follows: