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On scheduling coupled tasks with exact delays to minimize maximum lateness

Wiesław Kubiak

TL;DR

The paper shows polynomial time algorithms for \emph{agreeable} ($d_i\geq d_j$ implies $b_i\geq b_j$) and \emph{disagreeable} ($d_i\geq d_j$ implies $b_i\leq b_j$) cases.

Abstract

This paper studies scheduling coupled tasks with exact delays to minimize maximum lateness. The first task has processing time $p>0$ and the second $b_i\geq 0$, also the second needs to start exactly $p$ units of time after the completion of the first. The couple has due date $d_i$. The tasks are scheduled on a single machine to minimize maximum lateness. The problem has been left open in the literature which offer hardly any results on scheduling coupled tasks with exact delays to minimize maximum lateness. The paper shows polynomial time algorithms for \emph{agreeable} ($d_i\geq d_j$ implies $b_i\geq b_j$) and \emph{disagreeable} ($d_i\geq d_j$ implies $b_i\leq b_j$) cases. The complexity of the general problem remains open.

On scheduling coupled tasks with exact delays to minimize maximum lateness

TL;DR

The paper shows polynomial time algorithms for \emph{agreeable} ( implies ) and \emph{disagreeable} ( implies ) cases.

Abstract

This paper studies scheduling coupled tasks with exact delays to minimize maximum lateness. The first task has processing time and the second , also the second needs to start exactly units of time after the completion of the first. The couple has due date . The tasks are scheduled on a single machine to minimize maximum lateness. The problem has been left open in the literature which offer hardly any results on scheduling coupled tasks with exact delays to minimize maximum lateness. The paper shows polynomial time algorithms for \emph{agreeable} ( implies ) and \emph{disagreeable} ( implies ) cases. The complexity of the general problem remains open.
Paper Structure (19 sections, 25 theorems, 124 equations, 6 figures)

This paper contains 19 sections, 25 theorems, 124 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Agreeable case
  3. Pairs without long jobs
  4. Sequences without long jobs
  5. Order of jobs in the interlaced pairs
  6. Sequences with long jobs
  7. From semi-feasible to feasible sequences
  8. The graphs for the agreeable case
  9. The algorithm for the agreeable case
  10. Disagreeable case
  11. Sequence for instance without long jobs
  12. Sequences for instance with long jobs
  13. Trimming algorithm for instances without long jobs
  14. Pivotal jobs for $C_{\max}$ and $L_{\max}$
  15. Possible ends
  16. ...and 4 more sections

Key Result

Lemma 2.1

For pairs $(i, j)(k,l)$ in an optimal schedule we have $\min\{d_i,d_j\} \leq \min\{d_k,d_l\}$ and $\max\{d_i,d_j\} \leq \max\{d_k,d_l\}$.

Figures (6)

  • Figure 1: The conditions for the arc between $(\{i, j\}, c)$ and $(\{k, l\}, c-2)$ to exist.
  • Figure 2: The conditions for the arc between $((i, j), c)$ and $((k, l), c-2)$ to exist.
  • Figure 3: The two nodes $((i, j), c)$ and $((j, i), c)$ (and their respective schedules) to replace $(\{i<j\}, c)$, $i, j\in\mathcal{I}\setminus T_{\mathcal{I}}$.
  • Figure 4: The construction of graph $G'$.
  • Figure 5: The construction of graph $G'$.
  • ...and 1 more figures

Theorems & Definitions (48)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 38 more