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Theoretical Lower Bounds for the Oven Scheduling Problem

Francesca Da Ros, Marie-Louise Lackner, Nysret Musliu

TL;DR

These problem-specific lower bounds enable us to assess the solution quality for large instances for which exact methods often fail to provide tight lower bounds, and enable us to assess the solution quality for large instances for which exact methods often fail to provide tight lower bounds.

Abstract

The Oven Scheduling Problem (OSP) is an NP-hard real-world parallel batch scheduling problem arising in the semiconductor industry. The objective of the problem is to schedule a set of jobs on ovens while minimizing several factors, namely total oven runtime, job tardiness, and setup costs. At the same time, it must adhere to various constraints such as oven eligibility and availability, job release dates, setup times between batches, and oven capacity limitations. The key to obtaining efficient schedules is to process compatible jobs simultaneously in batches. In this paper, we develop theoretical, problem-specific lower bounds for the OSP that can be computed very quickly. We thoroughly examine these lower bounds, evaluating their quality and exploring their integration into existing solution methods. Specifically, we investigate their contribution to exact methods and a metaheuristic local search approach using simulated annealing. Moreover, these problem-specific lower bounds enable us to assess the solution quality for large instances for which exact methods often fail to provide tight lower bounds.

Theoretical Lower Bounds for the Oven Scheduling Problem

TL;DR

These problem-specific lower bounds enable us to assess the solution quality for large instances for which exact methods often fail to provide tight lower bounds, and enable us to assess the solution quality for large instances for which exact methods often fail to provide tight lower bounds.

Abstract

The Oven Scheduling Problem (OSP) is an NP-hard real-world parallel batch scheduling problem arising in the semiconductor industry. The objective of the problem is to schedule a set of jobs on ovens while minimizing several factors, namely total oven runtime, job tardiness, and setup costs. At the same time, it must adhere to various constraints such as oven eligibility and availability, job release dates, setup times between batches, and oven capacity limitations. The key to obtaining efficient schedules is to process compatible jobs simultaneously in batches. In this paper, we develop theoretical, problem-specific lower bounds for the OSP that can be computed very quickly. We thoroughly examine these lower bounds, evaluating their quality and exploring their integration into existing solution methods. Specifically, we investigate their contribution to exact methods and a metaheuristic local search approach using simulated annealing. Moreover, these problem-specific lower bounds enable us to assess the solution quality for large instances for which exact methods often fail to provide tight lower bounds.
Paper Structure (28 sections, 1 theorem, 20 equations, 6 figures, 5 tables)

This paper contains 28 sections, 1 theorem, 20 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The Oven Scheduling Problem
  3. Solution methods for the
  4. Lower bounds on the optimal solution cost
  5. Minimum number of batches required and minimal cumulative batch processing time
  6. Bound based on machine capacities and job sizes.
  7. Refinement of the bound for small jobs based on machine eligibility.
  8. Alternative refinement of the bound for small jobs based on compatible job processing times.
  9. Overall bound on the number of batches and the minimal cumulative processing time.
  10. Bounds on the other components of the objective function
  11. Setup costs.
  12. Number of tardy jobs.
  13. Including lower bounds in solution methods
  14. Experimental evaluation
  15. Benchmark instances
  16. ...and 13 more sections

Key Result

theorem thmcountertheorem

For any given set of unit size jobs $\mathcal{I}$ and for any given constant $c \in \mathbb{N}$, Algorithm GAC+ solves the problem OSP*, i.e., $GACb(\mathcal{I},c)$ is the minimum number of batches required under the condition that a batch may not contain more than $c$ jobs. Moreover, the cumulative

Figures (6)

  • Figure 1: Overview of the goals targeted by this work.
  • Figure 2: Gap[%] between the known optimum and the calculated lower bounds.
  • Figure 3: Gap[%] between the best solution found and the best lower bounds.
  • Figure 4: Minimum time required by to reach a given gap[%] w.r.t. $obj$.
  • Figure 5: Gantt chart of a solution of the for an instance with 10 jobs. The label of each bar represents the jobs processed in the batch. Unavailabilities ("unavail.") are reported in gray. Batches with attribute 1 are colored in green, whereas those referring to attribute 2 are colored in magenta.
  • ...and 1 more figures

Theorems & Definitions (2)

  • theorem thmcountertheorem
  • proof : of Theorem \ref{['thm:algo-GAC+']}