An exact algorithm for the identical parallel machine scheduling problem

The NP-hard classical deterministic scheduling problem of minimizing the makespan on identical parallel machines is considered. The polyhedral structure of the problem is analyzed, and strong valid inequalities are identified for fixed values of the makespan. Thus, partial linear descriptions of the associated polytope are obtained and used to build an exact algorithm. Linear programming relaxations are solved iteratively until the integer solution is reached. The algorithm includes a preprocessing phase which reformulates the problem and simplifies it. Different constructive rules have been tried out in the computation of upper bound values and the incidence of this choice is reported. Computational results show that the proposed algorithm yields, in a short computational time, optimal solutions in almost all tested cases.