Determining the optimal sequences and the distributional properties of their completion times in stochastic flow shops

This paper deals with a stochastic flow shop problem consisting of M machines and N jobs. The processing time of job j on machine i is an independent random variable with a given probability distribution function. It is assumed that preemption is not allowed and the existence of unlimited intermediate storage between the machines. The objective here, as it is in most flow shop investigations, is to determine a permutation (sequence) of the N jobs with the minimum makespan, which is the completion time of the last job on the last machine. Calculating the minimum makespan in the stochastic flow shop, if compared with the deterministic case, is harder, on the one hand; and on the other, the problem involves issues not existent in the deterministic case. It is shown in this paper that the minimum makespan (MM) in the stochastic case is a random variable (r.v.) not always connected to a particular sequence. Different realizations of the r.v. may correspond to different sequences. Consequently, a new concept for the optimal sequence is introduced. The distributional properties of the MM and their relations to those of the makespan of any sequence are analyzed. A lower bound on the expected value of the makespan of any sequence is derived. A practical heuristic procedure for determining or approximating the optimal sequence is developed. It is practtical in the sense that it is less restrictive than what has been developed so far and easier to implement. Computational experience is also provided.