Finding extreme supported solutions of biobjective network flow problems: An enhanced parametric programming approach

Problem: find complete set of extreme supported efficient solutions of biobjective min cost flow.We show that only a subset of arcs needs to be considered in every parametric simplex iteration.This leads to the proposed improvement of the classical parametric network simplex.We analyse worst-case time and space complexity.Extensive computational experiments compare performance of proposed and standard approaches. We address the problem of determining a complete set of extreme supported efficient solutions of biobjective minimum cost flow (BMCF) problems. A novel method improving the classical parametric method for this biobjective problem is proposed. The algorithm runs in O(Nn(m+nlogn)) time determining all extreme supported non-dominated points in the outcome space and one extreme supported efficient solution associated with each one of them. Here n is the number of nodes, m is the number of arcs and N is the number of extreme supported non-dominated points in outcome space for the BMCF problem. The memory space required by the algorithm is O(n+m) when the extreme supported efficient solutions are not required to be stored in RAM. Otherwise, the algorithm requires O(N+m) space. Extensive computational experiments comparing the performance of the proposed method and a standard parametric network simplex method are presented.