The Job Grouping Problem consists of assigning a set of jobs, each with a specific set of tool requirements, to machines with a limited tool capacity in order to minimize the number of machines needed. Traditionally, a formulation has been used that assigns jobs to machines. However, such a formulation contains a lot of symmetry since the machines are identical and they can be permuted in any feasible solution. We propose a new formulation for this problem, based on the Asymmetric Representatives Formulation (ARF) idea. This formulation eliminates the symmetry between the identical machines. We further propose various symmetry breaking constraints, including Variable Reduction and Lexicographic Ordering constraints, which can be added to the traditional formulation. These formulations are tested on a data set from the literature and newly generated data sets using a state-of-the-art commercial solver, which includes symmetry breaking features.
A first remarkable conclusion is that the order of the input data has a significant effect on the LP relaxation bound and CPU times for the ARF and some of the traditional formulations with variable reduction. The CPU time of the formulation with Lexicographic Ordering constraints on the jobs is also substantially affected, but not its LP bound. A second interesting conclusion is that the ARF is able to solve the problems of a large standard set from the literature to optimality in 2.5% of the CPU time of the traditional formulation and in 15% of the CPU time of a specialized Branch-and-Bound algorithm from the literature. For a data set with large instances, the traditional formulation extended with limited lexicographic constraints seems to be the best formulation.
Published November 2011 , 21 pages