Open Journal of Mathematical Optimization (May 2021)
Exact makespan minimization of unrelated parallel machines
Abstract
We study methods for the exact solution of the unrelated parallel machine problem with makespan minimization, generally denoted as $R||C_\text{max}$. Our original application arises from the automotive assembly process where tasks needs to be distributed among several robots. This involves the solutions of several $R||C_\text{max}$ instances, which proved hard for a MILP solver since the makespan objective induces weak LP relaxation bounds. To improve these bounds and to enable the solution of larger instances, we propose a branch–and–bound method based on a Lagrangian relaxation of the assignment constraints. For this relaxation we derive a criterion for variable fixing and prove the zero duality gap property for the case of two parallel machines. Our computational studies indicate that the proposed algorithm is competitive with state-of-the-art methods on different types of instances. Moreover, the impact of each proposed feature is analysed.
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