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Scalable Neighborhood Local Search for Single-Machine Scheduling with Family Setup Times

Kaja Balzereit, Niels Grüttemeier, Nils Morawietz, Dennis Reinhardt, Stefan Windmann, Petra Wolf

TL;DR

This work tackles makespan minimization for a single-machine scheduling problem with sequence-dependent family setup times, formalized as $1|ST_{sd,f},\overline{d}_j|C_{\max}$, and proposes parameterized local search across four distance notions. It establishes fixed-parameter tractability for window and multi-window distances with time $k^{\min(k,t)}\cdot |I|^{O(1)}$, while proving W[1]-hardness for insert and swap distances, supplemented by a practical $n^{2k+1}$ brute-force option. The authors design Hill-Climbing variants (Win, MW, and their Swap-enhanced versions) built on EDDS-based subroutines (Internal MM) and evaluate them against PILS1 and genetic algorithms on real-world and benchmark data, showing competitive performance, especially on larger infeasible instances. The results highlight both the practical viability of parameterized local search in this domain and the theoretical limits of certain distance-based approaches, suggesting avenues for future work on new parameterizations and objective functions. Overall, the work provides a principled framework and empirical evidence that parameterized local search can be a viable alternative to existing heuristics in complex scheduling with setups.

Abstract

In this work, we study the task of scheduling jobs on a single machine with sequence dependent family setup times under the goal of minimizing the makespan, that is, the completion time of the last job in the schedule. This notoriously NP-hard problem is highly relevant in practical productions and requires heuristics that provide good solutions quickly in order to deal with large instances. In this paper, we present a heuristic based on the approach of parameterized local search. That is, we aim to replace a given solution by a better solution having distance at most $k$ in a pre-defined distance measure. This is done multiple times in a hill-climbing manner, until a locally optimal solution is reached. We analyze the trade-off between the allowed distance $k$ and the algorithm's running time for four natural distance measures. Example of allowed operations for our considered distance measures are: swapping $k$ pairs of jobs in the sequence, or rearranging $k$ consecutive jobs. For two distance measures, we show that finding an improvement for given $k$ can be done in $f(k) \cdot n^{\mathcal{O}(1)}$ time, while such a running time for the other two distance measures is unlikely. We provide a preliminary experimental evaluation of our local search approaches.

Scalable Neighborhood Local Search for Single-Machine Scheduling with Family Setup Times

TL;DR

This work tackles makespan minimization for a single-machine scheduling problem with sequence-dependent family setup times, formalized as , and proposes parameterized local search across four distance notions. It establishes fixed-parameter tractability for window and multi-window distances with time , while proving W[1]-hardness for insert and swap distances, supplemented by a practical brute-force option. The authors design Hill-Climbing variants (Win, MW, and their Swap-enhanced versions) built on EDDS-based subroutines (Internal MM) and evaluate them against PILS1 and genetic algorithms on real-world and benchmark data, showing competitive performance, especially on larger infeasible instances. The results highlight both the practical viability of parameterized local search in this domain and the theoretical limits of certain distance-based approaches, suggesting avenues for future work on new parameterizations and objective functions. Overall, the work provides a principled framework and empirical evidence that parameterized local search can be a viable alternative to existing heuristics in complex scheduling with setups.

Abstract

In this work, we study the task of scheduling jobs on a single machine with sequence dependent family setup times under the goal of minimizing the makespan, that is, the completion time of the last job in the schedule. This notoriously NP-hard problem is highly relevant in practical productions and requires heuristics that provide good solutions quickly in order to deal with large instances. In this paper, we present a heuristic based on the approach of parameterized local search. That is, we aim to replace a given solution by a better solution having distance at most in a pre-defined distance measure. This is done multiple times in a hill-climbing manner, until a locally optimal solution is reached. We analyze the trade-off between the allowed distance and the algorithm's running time for four natural distance measures. Example of allowed operations for our considered distance measures are: swapping pairs of jobs in the sequence, or rearranging consecutive jobs. For two distance measures, we show that finding an improvement for given can be done in time, while such a running time for the other two distance measures is unlikely. We provide a preliminary experimental evaluation of our local search approaches.
Paper Structure (23 sections, 6 theorems, 17 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 23 sections, 6 theorems, 17 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contribution.
  3. Preliminaries
  4. Parameterized Local Search.
  5. Local Search in Window Neighborhoods
  6. Problem Definitions.
  7. An Observation on the Solution Structure.
  8. Subroutine Inside Windows.
  9. Fixed-Parameter Algorithms.
  10. Swap distance and insert distance
  11. Derived hill climbing algorithms
  12. The Win and MW Algorithms.
  13. The Win+Swap and MW+Swap Algorithms.
  14. Tardiness.
  15. Considered Starting Solutions.
  16. ...and 8 more sections

Key Result

Proposition 1

Let $\ell$ be a setup mapping such that $\ell(\tau,\tau)=0$ for every type $\tau$, and $\ell$ satisfies the triangle inequality. Moreover, let $\delta$ be the distance measure for the window distance or the multi-window distance. If $(J,\ell,\pi,k)$ is an instance of $\delta$ LS MM such that there e

Figures (1)

  • Figure 1: Examples of the considered distance measures: the insert distance of $\pi$ and $\pi_1$ is 1, the swap distance of $\pi$ and $\pi_2$ is 3, the window distance of $\pi$ and $\pi_3$ is 4, and the multi-window distance of $\pi$ and $\pi_4$ is 3.

Theorems & Definitions (15)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 3
  • proof : Proof of \ref{['Theorem: Window-Algo']}.
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 5 more