Robust Permutation Flowshops Under Budgeted Uncertainty

TL;DR

This paper demonstrates that the robust permutation flowshop problem under budgeted uncertainty can be solved efficiently by reducing it to multiple nominal problems, with polynomial-time solutions for two machines and approximations for fixed machine counts.

Contribution

It introduces a polynomial-time reduction method for the robust flowshop problem under budgeted uncertainty, extending to multiple machines and improving computational efficiency.

Findings

Polynomial-time solutions for two-machine flowshops.

Approximation algorithms for fixed number of machines.

Logarithmic factor improvement in running time for certain cases.

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. Next, we show 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. We conclude by noting that our reduction appears to have more general consequences for robust optimization problems under budgeted uncertainty having a similar form.