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 s olving complex scheduling problems. The technique has successfully bee n used to obtain near-optimal solutions for single machine and paralle l machine scheduling problems. The approach consists of relaxing the c apacity constraints on machines by using Lagrange multipliers. The rel axed 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 multipl iers to relax the precedence constraints among operations, so that eac h 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 osci llation from iteration to iteration and, in many cases, prevent conver gence of the algorithm. Although several methods to prevent the soluti on from oscillation have been proposed, none of them is really satisfa ctory. In this paper, we propose an efficient pseudo-polynomial time d ynamic programming algorithm to solve relaxed job level subproblems. W e show that, by extending the technique to job shop scheduling problem s, the relaxation of the precedence constraints becomes unnecessary, a nd 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 opt imize ''min-max'' criteria by Lagrangean relaxation, These criteria ha ve been neglected in the Lagrangean relaxation literature due to their indecomposability. Computational results are given to demonstrate the improvements due to this algorithm.