Two Solvable Cases of Three-machine Production Scheduling with Stochastic Breakdowns

The problem of scheduling a three-machine production, where machines suffer random breakdowns, is addressed with respect to the makespan objective. For our problem, obtaining an optimal schedule by implicit enumeration techniques (after dominance relations are established) is very hard, mainly due to the total idle time on machine 3. That is because it is the maximum of n terms, each of which includes the difference of two random sums of random variables, and furthermore, the terms are dependent. Two dominance relations for stochastically minimizing makespan are established in the flowshop where machines suffer random breakdowns and processing times of the jobs conform to the conditions. Furthermore, it is shown that algorithm for the deterministic problem stochastically minimizes makespan when random breakdowns are present.