Scheduling a hybrid flowshop with batch production at the last stage

In this paper, we address the problem of scheduling n jobs in an s-stage hybrid flowshop with batch production at the last stage with the objective of minimizing a given criterion with respect to the completion time. The batch production at stage s is referred to as serial batches by Hopp and Spearman where the processing time of a batch is equal to the sum of the processing times of all jobs included in it. This paper establishes an integer programming model and proposes a batch decoupling based Lagrangian relaxation algorithm for this problem. In this algorithm, after capacity constraints are relaxed by Lagrangian multipliers, the relaxed problem is decomposed based on a batch, unlike the commonly used job decoupling, so that it can be decomposed into batch-level subproblems, each for a specific batch. An improved forward dynamic programming algorithm is then designed for solving these subproblems where all operations within a batch form an in-tree structure and the precedence relations exist not only between the operations of a job but between the jobs in this batch at the last stage. A computational comparison is provided for the developed algorithm and the commonly used Lagrangian relaxation algorithm which, after capacity constraints and precedence relations within a batch are relaxed, decomposes the relaxed problem into job-level subproblems and solves the subproblems by using dynamic programming. Numerical results show that the designed Lagrangian relaxation method provides much better schedules and converges faster for small to medium sized problems, especially for larger sized problems.