Coupled tasks belong to the class of multi-operation tasks, where two consecutive operations are separated by a certain time interval of fixed duration. The objective is to schedule the tasks on one machine in order to minimize the makespan. Such problems occur especially in the case of the management of radar systems. The complexity status of this problem has been settled for all particular cases, except the case of the identical coupled task problem, where one has multiple copies of a single coupled task. This scheduling problem seems to be simple at first glance. However, its complexity status (for both, the cyclic and the finite case) has remained open for many years. Recently, Vassilissa Lehoux-Lebacque, Gerd Finke and Nadia Brauner proved that the cyclic case is polynomial. This report aims at studying the finite case by using some of the results and structures known from the cyclic case. We have been able to see that the finite case strongly differs from the cyclic case. Using integer programming and the commercial solver CPLEX, we have obtained a series of optimal solutions for small instances. The structures underlying these solutions are much more complicated than those of the cyclic case. Eventually, we have obtained results that let us think that this problem is not in NP.
