Scheduling Job Shops with Batch Machines Using the Lagrangian Relaxation Technique

Scheduling is a key factor for manufacturing productivity. Effective scheduling can improve on-time delivery, reduce work-in-process inventory, cut lead time, and improve machine utilization. Motivated by the extensive use of batch machines in manufacturing industries, the scheduling of job shops with batch machines is studied in this paper. Unlike machines that can process one part at a time (called "standard machines" for simplicity), a batch machine is the one that can simultaneously process several parts with the same processing requirement as a batch, subject to the capacity of the batch machine. This simultaneous processing requires the "synchronization" (batch formation) of different parts for a batch operation. In view that a part may have to be processed by many standard and batch machines, multiple batch formation and batch splitting result in complicated coupling among the parts. In this paper, batches are viewed as "virtual" facilities that host and "synchronize" parts for processing, and compete for batch machine capacity. A novel integer programming formulation with a "separable" structure is then obtained. To solve this problem, a Lagrangian relaxation approach is used to decompose the problem into part and batch subproblems that can be solved by using an efficient dynamic programming algorithm. The multipliers are updated at the high level by using an “interleaved conjugate gradient method,” and a heuristic is developed to obtain feasible schedules based on subproblem solutions. Numerical testing shows that the integrated consideration of batch formation and sequencing results in high quality schedules, and the algorithm is computationally efficient to solve practical problems.