Direct methods for sparse matrices
Direct methods for sparse matrices
Network flow problems with one side constraint: a comparison of three solution methods
Computers and Operations Research
A dual approach to primal degeneracy
Mathematical Programming: Series A and B - Mathematical Models and Their Solutions
Linear programming and network flows (2nd ed.)
Linear programming and network flows (2nd ed.)
A practical anti-cycling procedure for linearly constrained optimization
Mathematical Programming: Series A and B
New crash procedures for large systems of linear constraints
Mathematical Programming: Series A and B
COMO '86 Lectures given at the third session of the Centro Internazionale Matematico Estivo (C.I.M.E.) on Combinatorial optimization
Network flows: theory, algorithms, and applications
Network flows: theory, algorithms, and applications
On the computational behavior of a polynomial-time network flow algorithm
Mathematical Programming: Series A and B
Mathematical Programming: Series A and B - Special issue on interior point methods for linear programming: theory and practice
Optimization Software Guide
A tabular simplex-type algorithm as a teaching aid for general LP models
Mathematical and Computer Modelling: An International Journal
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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.