Flow Shop Scheduling with Stochastic Reentry

2026-04-20

AI Summary

This study addresses the flow shop scheduling problem with stochastic reentry, where each job traverses all machines multiple times according to a discrete probability distribution, aiming to minimize expected performance metrics. By transforming the original problem into an equivalent parallel machine scheduling problem with machine release times—while preserving the expected objective value—the authors exploit structural properties of the auxiliary formulation to derive effective scheduling policies. The work establishes the first theoretical guarantees of optimality and approximation for this setting: under geometric and monotone hazard rate distributions, simple priority rules are shown to be optimal for minimizing expected makespan and total completion time; for weighted total completion time, an approximation ratio is provided that depends solely on the squared coefficient of variation of the underlying distribution.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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.

Problem

• Flow Shop Scheduling
• Stochastic Reentry
• Makespan
• Total Completion Time
• Approximation Guarantee

Innovation

• stochastic reentry
• flow shop scheduling
• reduction to parallel machine scheduling
• priority policies
• approximation guarantees