A hybrid algorithm for solving network flow problems with side constraints

Abstract

Many planning problems can be formulated as minimum cost network flow models. There exists a multitude of applications modeling a wide range of practical situations, e.g. production planning, distribution, project planning, scheduling, manpower planning, and many others. Such model formulations can be solved very efficiently using specialized network optimization algorithms. Large network flow problems containing thousands of variables and constraints can be solved to optimality on a desktop computer in the order of seconds. Solving the same problem with a general linear programming optimizer would take at least 100 times longer. Unfortunately, many real world problems contain additional linear constraints, termed side constraints, which destroy the network structure of the coefficient matrix making it necessary to solve the problem as a general linear program. Several methods have been proposed, therefore, to handle such side constraints efficiently within the framework of a network algorithm. In this paper three of the more successful methods are described and tested in detail and a hybrid approach is suggested which combines the advantages of two of these methods. We consider network flow problems with few additional linear side constraints. Three approaches for solving such problems are presented. The first method is a specialized Simplex Algorithm with primal partitioning of the basis. Secondly, Lagrangean relaxation is used, solving the dual problem by subgradient optimization. Finally, good starting solutions are generated by relaxing the side constraints, solving the resulting pure network problem, and using the solution of the pure network problem as an advanced start for a general LP-optimizer. Numerical tests show that a hybrid combination of Lagrangean relaxation and subsequent optimization by primal partitioning is most efficient.