Robust Permutation Flowshops Under Budgeted Uncertainty

2026-02-25

AI Summary

This study addresses the permutation flow shop scheduling problem under budgeted uncertainty, where processing times of up to a specified number of jobs on each machine may deviate from their nominal values. Building upon the Bertsimas–Sim robust optimization framework, the authors employ duality techniques to reformulate the min–max objective into an equivalent nominal instance of polynomial size. The main contributions include the first proof that the two-machine case admits an exact polynomial-time algorithm, the development of a polynomial-time approximation scheme for any fixed number of machines, and a substantial improvement in computational efficiency over naive reduction approaches—yielding a logarithmic-factor speedup for the three-machine case and significantly reduced overall complexity.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We consider the robust permutation flowshop problem under the budgeted uncertainty model, where at most a given number of job processing times may deviate on each machine. We show that solutions for this problem can be determined by solving polynomially many instances of the corresponding nominal problem. As a direct consequence, our result implies that this robust flowshop problem can be solved in polynomial time for two machines, and can be approximated in polynomial time for any fixed number of machines. The reduction that is our main result follows from an analysis similar to Bertsimas and Sim (2003) except that dualization is applied to the terms of a min-max objective rather than to a linear objective function. Our result may be surprising considering that heuristic and exact integer programming based methods have been developed in the literature for solving the two-machine flowshop problem. We conclude by showing a logarithmic factor improvement in the overall running time implied by a naive reduction to nominal problems in the case of two machines and three machines.

Problem

• robust optimization
• permutation flowshop
• budgeted uncertainty
• scheduling
• processing time deviation

Innovation

• robust optimization
• permutation flowshop
• budgeted uncertainty
• polynomial-time solvability
• min-max duality