Approximation algorithms for mixed batch scheduling on parallel machines

In this paper, we consider two mixed batch scheduling problems on parallel machines, in which multiple jobs can be processed on the same machine simultaneously as a batch, and the processing time of each batch is equal to the weighted sum of the maximum processing time and the total processing time of jobs in this batch. In the first problem, we assume that each job has an inclusive processing restriction, and all the machines have the same capacity. In the second problem, we assume that machines can have different capacities, and there is no inclusive processing restriction for each job. The objectives of both problems are to minimize their makespan. For the first problem, we propose an approximation algorithm whose performance ratio is (2+alpha) with 0 <=alpha <= 1. For the second problem, we derive an approximation algorithm with a performance ratio of (1+rho-(1-alpha)/m-alpha/(mKmin)), where rho is the ratio of the maximum machine capacity to the minimum machine capacity and Kmin is the minimum machine capacity.