Semi-online scheduling of two job types on a set of multipurpose machines

In the majority of studies of online scheduling on m multipurpose machines, there is complete uncertainty about the scheduling instance. In contrast, we consider a semi-online environment where there is prior knowledge about some parameters of the problem and the objective is to minimise the makespan. In our problem, there are two job types, each of which can be processed on a unique subset of an arbitrary number of machines and the processing sets are of arbitrary structure. We analyse three distinct cases, corresponding to prior knowledge of the following three values: (1) the optimal (offline) solution value; (2) the value of the total processing time; (3) the (constant) value of the largest processing time or an upper bound on the largest processing time. We provide a semi-online algorithm with a competitive ratio of 2 (Formula presented.) 1/m for the first two variations. For the last case, we show a competitive ratio as a function of the processing set parameters. In this case, we prove that the algorithm is asymptotically optimal for any structure of the multipurpose machines and that the competitive ratio in the worst case tends to 4/3.