Batching identical jobs

We study the problems of scheduling jobs, with different release dates and equal processing times, on two types of batching machines. All jobs of the same batch start and are completed simultaneously. On a serial batching machine, the length of a batch equals the sum of the processing times of its jobs and, when a new batch starts, a constant setup time s occurs. On a parallel batching machine, there are at most b jobs per batch and the length of a batch is the largest processing time of its jobs. We show that in both environments, for a large class of so called ‘ordered’ objective functions, the problems are polynomially solvable by dynamic programming. This allows us to derive that the problems where the objective is to minimize the weighted number of late jobs, or the weighted flow time, or the total tardiness, or the maximal tardiness are polynomial. In other words, we show that 1|p-batch, b < n, ri, pi = p|F and 1|s-batch, ri, pi = p|F, are polynomial for F ∈ {ΣwiUi, ΣwiCi, ΣTi, Tmax}. The complexity status of these problems was unknown before.