We consider a scheduling problem introduced by Ahmadi et at., batching and scheduling jobs on batch and discrete processors, in which each job has to be prepared before it can be processed. The preparation is performed by a batching machine; it can prepare at most c jobs in each run, which takes an amount of time that is independent of the number and identity of the jobs under preparation. We present a very strong Lagrangian lower bound by using the new concept of positional completion times. This bound can be computed in O(n log n) time only, where n is the number of jobs. We further present two classes of O(n log n) heuristics that transform an optimal schedule for the Lagrangian dual problem into a feasible schedule. Any heuristic in one class has performance guarantee of 3/2. We further show that even the most naive heuristic in this class has a compelling empirical performance.
