An improvement of the Lagrangean relaxation approach for job shop scheduling: A dynamic programming method

Lagrangean relaxation has recently emerged as a promising method for solving complex scheduling problems. The technique has successfully been used to obtain near-optimal solutions for single machine and parallel machine scheduling problems. The approach consists of relaxing the capacity constraints on machines by using Lagrange multipliers. The relaxed problem can be decomposed into independent job level subproblems, Peter B, Luh and his colleagues extended the technique to general job shop scheduling problems by introducing additional Lagrangean multipliers to relax the precedence constraints among operations, so that each job level relaxed subproblem can be further decomposed into a set of operation level subproblems which can easily be solved by enumeration, Unfortunately, the operation level subproblems exhibit solution oscillation from iteration to iteration and, in many cases, prevent convergence of the algorithm. Although several methods to prevent the solution from oscillation have been proposed, none of them is really satisfactory. In this paper, we propose an efficient pseudo-polynomial time dynamic programming algorithm to solve relaxed job level subproblems. We show that, by extending the technique to job shop scheduling problems, the relaxation of the precedence constraints becomes unnecessary, and thus the oscillation problem vanishes. This algorithm significantly improves the efficiency of the Lagrangean relaxation approach to job-shop scheduling problems. This algorithm also makes it possible to optimize "min-max" criteria by Lagrangean relaxation, These criteria have been neglected in the Lagrangean relaxation literature due to their indecomposability. Computational results are given to demonstrate the improvements due to this algorithm.