Parallel Machine Batch Scheduling

The problem of scheduling G groups of jobs on m unrelated parallel machines is considered. Each group consists of several identical jobs. Implicit in a schedule is a partition of a group into batches; all jobs of a batch are processed jointly. It is possible for diierent batches of the same group to be processed concurrently on diierent machines. However, at any time, a batch can be processed on at most one machine. A sequence independent setup time is required immediately before a batch of a group is processed. A deadline is associated with each group. The objective is to nd a schedule which is feasible with respect to deadlines. The problem is shown to be NP-hard even for the case of two identical machines, unit processing times, unit setup times and a common deadline. It is strongly NP-hard if machines are uniform and processing times, setup times and deadlines are unit. Special cases which are polynomially solvable are discussed. For the general problem, a dynamic programming algorithm and a family fDP " g of approximation algorithms is constructed. For any " > 0, algorithm DP " delivers a schedule in which the completion time of each group is at most (1+") times the value 1 of its deadline if there exists a schedule which is feasible with respect to the deadlines. The time complexity of DP " is O(G 2m+1 =" 2m).