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Robust Permutation Flowshops Under Budgeted Uncertainty

Noam Goldberg, Danny Hermelin, Dvir Shabtay

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.

Robust Permutation Flowshops Under Budgeted Uncertainty

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.
Paper Structure (13 sections, 8 theorems, 14 equations, 1 figure, 3 algorithms)

This paper contains 13 sections, 8 theorems, 14 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Permutation Flowshops
  3. Robust Permutation Flowshops
  4. Robust Permutation Flowshops under Budgeted Uncertainty
  5. Our Contribution
  6. Related Work
  7. Min-Max Formulation
  8. Reduction to Nominal Instances
  9. Algorithmic Consequences
  10. Exact Solution for Two Machines
  11. Three Machines
  12. More Than Three Machines
  13. Summary and Discussion

Key Result

Theorem 1

An optimal solution to the $Fm\mid prmu\mid C^{\Gamma}_{\max}$ problem can be obtained by solving $O(n^m)$ instances of the nominal $Fm\mid prmu\mid C_{\max}$ problem. This also extends to approximate solutions: For any $\rho \geq 1$, an $\rho$-approximate solution to the nominal problem yields an $

Figures (1)

  • Figure 1: An depiction of the flowshop digraph for a flowshop with three machines and four jobs, where the node labels indicate their weights (processing times). An example of a critical path is highlighted in bold, with $k_1=2$ and $k_2=4$.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Corollary 2
  • Lemma 1
  • proof
  • Corollary 3
  • proof
  • Lemma 2
  • ...and 1 more