A warm-start dual simplex solution algorithm for the minimum flow networks with postoptimality analyses

This paper presents an algorithm for finding the minimum flow in general (s, t) networks with m directed arcs. The minimum flow problem (MFP) arises in many transportation and communication systems. Here, we construct a linear programming (LP) formulation of MFP and develop a simplex-type algorithm to find a minflow. Unlike other simplex-like algorithms, the algorithm developed here starts with an incomplete Basic Variable Set (BVS) initially and then fills-up the BVS completely while pushing toward an optimal vertex. If this results in pushing too far into infeasibility, the next step pulls the solution back to feasibility. Both phases use the Gauss-Jordan pivoting transformation used in the standard simplex and dual simplex algorithms.

The proposed approach has some common features with Dantzig's self-dual simplex algorithm. We have avoided, however, the need for extra variables (slack and surplus) for equality constraints, as well as an artificial constraint for the self-dual algorithm for initial phase and the dual simplex, respectively. The proposed Push phase takes at most 2m − 1 iterations to complete a tree (this augmentation may not be feasible). An infeasible path to the optimal solution contains at most 2m − 1 iterations; therefore in theory, the algorithm needs a sequence of at most 4m − 2 iterations.

Furthermore, the algorithm developed here makes available the full power of LP perturbation analysis and facilitates introducing network structural changes and side constraints also. It can also detect clerical errors in data entry which may make the problem infeasible or unbounded. It is assumed that the reader is familiar with LP terminology.