A Note on Stochastic Shop Models in Which Jobs have the Same Processing Requirements on Each Machine

We consider stochastic models for flow shops, job shops and open shops in which the work required by job j is the same at each machine, being a random variable W_j. Because machines operate at different speeds, S_i, the processing time of job j at machine i is W_j/S_i,. It is the main result of this note that in a flow shop where the machine speeds increase (decrease) from the first to last machine and the workload distributions are ordered by a likelihood ratio criterion, then the makespan of the jobs is stochastically minimized by processing the jobs in the order of least to greatest (greatest to least) workload.