The Budget-Constrained Min Cost Flow Problem
نویسندگان
چکیده
In this paper we describe a problem that we de ne as the Budget-Constrained Minimum Cost Flow (BCMCF) problem. The BCMCF problem is a natural extension of the well-known Minimum Cost Flow (MCF) [2] problem, with a xed cost related to the use of arcs and a budget constraint. Network ow problems often become hard when extra constraints are added. Ahuja and Orlin [1] discuss the constrained maximum ow problem with a budget constraint related to the cost of ow. Beasley and Christo des [3] study the resource constrained shortest path problem. Demgensky et al. [4] deal with a ow cost lowering problem with an upgrade budget to buy upgrade units on arcs. The BCMCF has, to the best of our knowledge, not been described in literature before. We show that the Accessibility Arc Upgrading Problem (AAUP) [5] is a special case of the BCMCF problem. We propose an exact solution method based on Lagrange relaxation and a heuristic approach based on variable neighborhood search. Let us start by describing the BCMCF problem in detail. In this problem a ow has to be sent from a set of supply nodes or sources, through the arcs of a network, to a set of demand nodes or sinks. For each arc in the network, there is a cost per unit of ow over the arc, and a xed cost associated to the use of the arc. Note that there might be several arcs connecting any speci c pair of nodes. The problem is to nd a minimal ow cost such that the sum of the xed costs incurred by using the arcs to transport the ow does not exceed a given budget. We consider the uncapacitated as well as the capacitated case. Basically, this problem is a standard minimum cost ow problem in which an additional decision indicates which arcs are actually used to pass the ow, such that the cost for using these arcs does not exceed a xed budget. A relevant application of the BCMCF problem comes from the accessibility arc upgrading problem (AAUP) [5]. AAUP is a network upgrading problem in which resources have to be allocated in order to improve the accessibility to a set of vertices in a network. In the domain of rural road network planning, this problem
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