Scheduling unrelated parallel machines with optional machines and jobs selection

In this paper we study two generalizations of the well known unrelated parallel machines scheduling problem under makespan minimization. First, a situation in which not every available parallel machine should be used and it is desirable to employ only a subset of the parallel machines. This is referred to as ``Not all Machines'' or NAM in short. This environment applies frequently in production shops where capacity exceeds demand or when production capacity can be lent to third companies. Also, NAM can be used to increase production capacity and it is not clear how many additional machines should be acquired. The second studied generalization has been referred to as ``Not All Jobs'' or NAJ. Here, there is no obligation to process all available jobs. Profit maximization or cost minimization are two clear examples. Despite the many potential applications of these generalizations, we are not aware of comprehensive studies. We propose Mixed Integer Programming mathematical formulations for both NAM and NAJ, and it shown that the later can be effectively solved with modern commercial solvers. We also present three algorithms to solve the NAM problem. These algorithms are compared with the proposed MIP formulation when solved with IBM ILOG CPLEX 12.1. Comprehensive computational and statistical experiments prove that our proposed algorithms significantly improve the results given by the solver.