Generalized lexicographically optimal flows in networks with multiple sources and sinks

Abstract

Abstract Lexicographically optimal flows with multiple sources and sinks in networks dealt with by Megiddo ( 1974) and Fujishigo ( 1980) are extended to treat more important and practical network flow problems in systems science. First, we consider the problem of finding a flow which has the minimum cost among all lexicographically optimal flows with respect to a given weight vector. Here, the lexicographically optimal flow with respect to a given weight vector is a flow which not only attains the maximum flow value but also distributes supplies in sources and demands in sinks as proportional to the given weights as possible. Secondly, we consider the problem of finding a flow which has the minimum cost among all the flows which are lexicographically optimal with respect to weight vectors belonging to an admissible region. Algorithms for solving these newly formulated network flow problems are proposed. In the light of matroid theory, the proposed algorithms are discussed. An illustrative example is also ...