Exact algorithm for batch scheduling on unrelated machine

In this paper, we propose a new linear algorithm to tackle a specific class of unrelated machine scheduling problem, considered as an important real-life situation, which we called Batch Scheduling on Unrelated Machine (BSUM), where we have to schedule a batch of identical and non-preemptive jobs on unrelated parallel machines. The objective is to minimize the makespan (Cmax) of the whole schedule. For this, a mathematical formulation is made and a lower bound is computed based on the potential properties of the problem in order to reduce the search space size and thus accelerate the algorithm. Another property is also deducted to design our algorithm that solves this problem. The latter is considered as a particular case of RmCmax family problems known as strongly NP-hard, therefore, a polynomial reduction should realize a significant efficiency to treat them. As we will show, Batch BSUM is omnipresent in several kind of applications as manufacturing, transportation, logistic and routing. It is of major importance in several company activities. The problem complexity and the optimality of the algorithm are reported, proven and discussed.