[Paper Review] Recoverable Robust Single Machine Scheduling with Polyhedral Uncertainty

Matthew Bold, Marc Goerigk|arXiv (Cornell University)|Nov 12, 2020

Scheduling and Optimization Algorithms4 citations

TL;DR

This paper proposes a recoverable robust approach for single-machine scheduling under polyhedral uncertainty, where a first-stage schedule can be adjusted via up to Δ disjoint job swaps after processing times are realized. It introduces compact mixed-integer programming formulations using matching and assignment structures, with computational results showing that even a single swap captures most of the benefit of recourse, and the assignment-based model is the strongest among the three proposed formulations.

ABSTRACT

This paper considers a recoverable robust single-machine scheduling problem under polyhedral uncertainty with the objective of minimising the total flow time. In this setting, a decision-maker must determine a first-stage schedule subject to the uncertain job processing times. Then following the realisation of these processing times, they have the option to swap the positions of up to Delta disjoint pairs of jobs to obtain a second-stage schedule. We first formulate this scheduling problem using a general recoverable robust framework, before we examine the incremental subproblem in further detail. We prove a general result for max-weight matching problems, showing that for edge weights of a specific form, the matching polytope can be fully characterised by polynomially many constraints. We use this result to derive a matching-based compact formulation for the full problem. Further analysis of the incremental problem leads to an additional assignment-based compact formulation. Computational results on budgeted uncertainty sets compare the relative strengths of the three compact models we propose.

Motivation & Objective

• To address the robust single-machine scheduling problem with uncertain job processing times under a recoverable robustness framework.
• To develop compact mixed-integer programming formulations that can efficiently model the recoverable robust scheduling problem with limited recourse actions.
• To analyze the incremental subproblem of job swapping to derive tailored formulations based on matching and assignment structures.
• To compare the computational performance of multiple compact formulations under budgeted uncertainty sets.
• To investigate the impact of increasing the number of allowed swaps (Δ) and uncertainty budget (Γ) on solution quality and robustness.

Proposed method

• Formulates the recoverable robust scheduling problem using a general framework where a first-stage schedule is adjusted via up to Δ disjoint job swaps after uncertainty realization.
• Proves that for edge weights of a specific form, the matching polytope can be fully described by polynomially many constraints, eliminating the need for odd-cycle inequalities.
• Derives a matching-based compact formulation for the full problem using this structural result.
• Develops a symmetric assignment-based compact formulation, transferring the integral matching result to this alternative model.
• Proposes three exact models: matching-based, assignment-based, and a restricted version with Δ=0 for comparison.
• Employs computational experiments on instances with n ≤ 20 to compare the relative strength of the three formulations under budgeted uncertainty.

Experimental results

Research questions

• RQ1 How can a recoverable robust single-machine scheduling problem with limited recourse swaps be formulated as a compact mixed-integer program?
• RQ2 What structural properties of the incremental subproblem (job swapping) allow for compact formulation without odd-cycle constraints?
• RQ3 How does increasing the number of allowed swaps (Δ) affect the objective value and robustness of the solution?
• RQ4 How does increasing the uncertainty budget (Γ) impact the worst-case total flow time under different recourse levels?
• RQ5 Which of the proposed compact formulations—matching-based, assignment-based, or restricted—provides the strongest bound and best computational performance?

Key findings

• The assignment-based formulation is the strongest among the three compact models proposed, providing the tightest bounds and best computational performance.
• Allowing just one swap captures the vast majority of the benefit of recourse, as increasing Δ beyond 1 yields diminishing returns in objective improvement.
• For a fixed Δ, increasing the uncertainty budget (Γ) leads to a monotonic increase in the worst-case total flow time, with average objective values rising by up to 22.1% when Γ increases from 3 to 7 for n=10.
• When Δ increases from 0 to 3, the average objective value decreases by 1.0% on average across n=15 and n=20 instances, indicating a consistent but modest benefit from additional swaps.
• The matching-based formulation is valid and compact due to the integrality of solutions under the specific weight structure, eliminating the need for odd-cycle constraints.
• The computational results are limited to instances with n ≤ 20 due to the high computational cost of exact models, suggesting a need for heuristics in larger-scale applications.