1 Scheduling of Manufacturing Systems Using the Lagrangian Relaxation Technique

Scheduling is one of the most basic but the most difficult problems encountered in the manufacturing industry. Generally, some degree of time-consuming and impractical enumeration is required to obtain optimal solutions. Industry has thus relied on a combination of heuristics and simulation to solve the problem, resulting in unreliable and often infeasible schedules. Yet, there is a great need for an improvement in scheduling operations in complex and turbulent manufacturing environments. The logical strategy is to find scheduling methods which consistently generate good schedules efficiently. However, it is often difficult to measure the quality of a schedule without knowing the optimum. In this paper, the practical scheduling of three manufacturing environments are examined in the increasing order of complexity. The first problem considers scheduling singleoperation jobs on parallel, identical machines; the second one is concerned with scheduling multiple-operation jobs with simple fork/join precedence constraints on parallel, identical machines; and lastly, the job shop scheduling problem is considered, where multipleoperation jobs with general precedence constraints are to be scheduled on multiple machine types. The Lagrangian relaxation technique is used to decompose each problem into job-level or operation-level subproblems, which are much easier to solve than the original problem and have intuitive appeals. This technique then results in algorithms which generate near-optimal schedules efficiently, while giving a lower bound on the optimal cost. In addition, the Lagrange multipliers from the previous schedule can be used to initialize the algorithms in the day-to-day scheduling operations, further reducing the computation time. All algorithms are demonstrated with examples using actual factory data. The successful application of the 1 This research was supported in part by the Department of Higher Education, State of Connecticut and Pratt & Whitney, East Hartford, CT, under the Cooperative High Technology Research and Development Grant Program, and by the National Science Foundation under Grant ECS-8717167. The authors would like to thank Mr. Scott Bailey, Mr. Curtis Cook and Mr. Jack Gibbs of Pratt & Whitney for their invaluable suggestions and support. An earlier version of this work has been presented as a plenary paper at the 1991 IFAC Workshop on Discrete Event System Theory and Applications in Manufacturing and Social Phenomena, Shenyang, P. R. China. 2 Peter B. Luh is with the Department of Electrical & Systems Engineering, University of Connecticut, Storrs, CT, 06269-3157, and Debra J. Hoitomt is with Pratt & Whitney, East Hartford, CT, 06108.