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We consider the two-stage proportionate blocking flow shop scheduling problem with the objective of simultaneously minimising the mean and the variation of job waiting times. We show that the problem can be solved intime by evaluating at most two candidate optimal sequences. In some cases, the shortest processing time (SPT) sequence is optimal for any set of processing times. In all other cases, the optimal sequence is either the SPT sequence or a V-shaped sequence. We show that the optimal solution value for the corresponding no-wait flow shop scheduling problem is no worse than the optimal solution value for the blocking flow shop problem. We then analyse the blocking flow shop problem with the option of rejecting jobs from the sequence by paying their job-specific rejection penalties. We show that the problem with the job rejection option can be solved intime by dynamic programming (DP). Finally, we show that our results also apply to the corresponding flow shop scheduling problem with synchronous job transfers.

Simultaneous minimisation of mean and variation of waiting times in a two-stage proportionate blocking flow shop