Flow Shop Scheduling with Stochastic Reentry

TL;DR

This paper introduces a reduction approach for flow shop scheduling with stochastic reentry, enabling the transfer of optimality and approximation guarantees from classical problems to this complex setting.

Contribution

It provides the first reduction-based framework that extends structural results and guarantees to stochastic reentrant flow shop scheduling problems.

Findings

Proves the optimality of simple priority policies for makespan and total completion time.

Derives an approximation guarantee for total weighted completion time based on distribution variability.

Establishes the first known guarantees for flow shops with stochastic reentry.

Abstract

We study flow shop scheduling with stochastic reentry, where jobs must complete multiple passes through the entire shop, and the number of passes that a job requires for completion is drawn from a discrete probability distribution. The goal is to find policies that minimize performance measures in expectation. Our main contribution is a reduction to a classical parallel machine scheduling problem augmented with machine arrivals. This reduction preserves expected objective values and enables transferring structural results and performance guarantees from the auxiliary problems to the reentrant flow shop setting. We demonstrate the usefulness of this reduction by proving the optimality of simple priority policies for minimizing the makespan and the total completion time in expectation under geometric and, more generally, monotone hazard rate distributions. For minimizing the total weighted completion time, we derive an approximation guarantee that depends only on the squared coefficient of variation of the underlying distributions for a simple priority policy. Our results constitute the first optimality and approximation guarantees for flow shops with stochastic reentry and demonstrate that established scheduling policies naturally extend to this setting through the proposed reduction.