Abstract
Abstract We first consider the ordinary NP-hard three-machine proportionate open shop minimum makespan O 3| prpt | C max problem and show that it is solvable in O ( n log n ) time when certain conditions on the total machine load are met. When these conditions are not met, we derive an approximate solution by implementing the optimal solution for the O 3| prpt | C max problem when the two longest jobs are of equal length. In that case, both absolute and ratio worst-case bounds are derived. We also consider the more general mixed shop problem M 3| prpt | C max in which a given job subset must be processed as in a flow shop while the remaining jobs can be processed as in the O 3| prpt | C max problem. We show that our open shop solution techniques can be implemented to derive exact and approximate solutions for the M 3| prpt | C max problem. Finally, we discuss the applicability of our open shop results to the proportionate open shop problem with unequal machine speeds.
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