Minimal-cost network flow problems with variable lower bounds on arc flows

The minimal-cost network flow problem with fixed lower and upper bounds on arc flows has been well studied. This paper investigates an important extension, in which some or all arcs have variable lower bounds. In particular, an arc with a variable lower bound is allowed to be either closed (i.e., then having zero flow) or open (i.e., then having flow between the given positive lower bound and an upper bound). This distinctive feature makes the new problem NP-hard, although its formulation becomes more broadly applicable, since there are many cases where a flow distribution channel may be closed if the flow on the arc is not enough to justify its operation. This paper formulates the new model, referred to as MCNF-VLB, as a mixed integer linear programming, and shows its NP-hard complexity. Furthermore, a numerical example is used to illustrate the formulation and its applicability. This paper also shows a comprehensive computational testing on using CPLEX to solve the MCNF-VLB instances of up to medium-to-large size.