On Asymptotic Optimality of Permutation Schedules in Stochastic Flow Shops and Assembly Lines

Abstract

For the stochastic flow shop problem with m machines, n jobs and operation processing times being independent identically distributed random variables, we consider permutation schedules — those in which each machine processes the jobs in the same order. For fixed m and increasing n, the asymptotic behavior of an arbitrary permutation schedule length is researched. Let T be the makespan of that schedule. Considering the maximum machine load L as a lower bound of the makespan, we show that for a wide class of operation processing time distributions the average-case ratio T/L converges to 1.

Thus, an arbitrary permutation flow shop schedule is asymptotically optimal with n → ∞, and one can construct such schedule in a linear time. The same result is derived for the stochastic assembly line problem with m machines and n jobs.