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Several papers have described special structural properties for which the minimum makespan flow-shop problem can be efficiently solved. We describe a new special case of the 3 × n problem with small second-stage processing times relative to the first and third stages for which Johnson's algorithm gives an optimal schedule.

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In an epoch-setting paper on scheduling theory, Johnson determined an efficient algorithm for solving the two-stage flow-shop problem and two special cases of the three-stage problem. In his concluding remarks he suggested another special condition in which one would obtain a three-stage solution. We give an example to show that this conjecture is not quite true. A true statement is obtained, however, by only a slight strengthening of the original. The primary issue concerns the impact of the two-stage no-preference precedent rules on the three-stage problem.

an extension to one of the classes described by Szwarc.