Throughput maximization of real-time scheduling with batching

We consider the following scheduling with batching problem that has many applications, for example, in multimedia-on-demand and manufacturing of integrated circuits. The input to the problem consists of n jobs and k parallel machines. Each job is associated with a set of time intervals in which it can be scheduled (given either explicitly or nonexplicitly), a weight, and a family. Each family is associated with a processing time. Jobs that belong to the same family can be batched and executed together on the same machine. The processing time of each batch is the processing time of the family of jobs it contains. The goal is to find a nonpreemptive schedule with batching that maximizes the weight of the scheduled jobs. We give constant factor (4 or 4 + ε) approximation algorithms for two variants of the problem, depending on the precise representation of the input. When the batch size is unbounded and each job is associated with a time window in which it can be processed, these approximation ratios reduce to 2 and 2 + ε, respectively. We also give approximation algorithms for two special cases when all release times are the same.