Scheduling unrelated machines with two types of jobs

In this paper, we consider the problem of scheduling a set of jobs having only two possible processing times on a set of unrelated parallel machines. This problem is a generalisation of the much more common problem of scheduling equal-length jobs on identical machines. Such a situation may occur in the production of two different types of products. First, we show that an earlier open problem of scheduling jobs with two possible processing times p and 2p on unrelated machines with the objective to minimise the makespan can be polynomially solved by an algorithm consisting of two phases. A slight modification of this algorithm yields an absolute worst-case error of q for the case of two arbitrary processing times p and q, p < q. Thus, for practical problems of a large size with two types of products and two possible processing times, the approximation algorithm generates schedules very close to an optimal one.