Inequalities for stochastic flow shops and job shops

Abstract

AbstractConsider an m‐machine flow shop with n jobs. The processing time of job j, j = 1,…, n, on each one of the m machines is equal to the random variable Xj and is distributed according to Fj. We show that, under certain conditions, more homogeneous distributions F1,…, Fn result in a smaller expected makespan. We also study the effect of the variability of distribution Fj on the expected waiting costs of the n jobs and on the job sequencing which minimizes this total expected waiting cost. We show that the smallest (largest) variance first rule minimizes the total expected waiting cost on a single machine when the waiting cost function is increasing convex (concave). We also show that the smallest variance first rule minimizes, under given conditions, the total expected waiting cost in an m machine flow shop when the waiting cost function is increasing convex. Similar results are also obtained for the two‐machine job shop. Similar results cannot be obtained when the processing times of job j on the various machines are i.i.d. and distributed according to Fj.