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Approximation algorithms for scheduling with rejection in green manufacturing

Mingyang Gong, Brendan Mumey

TL;DR

This paper addresses scheduling with rejection under a total energy bound in green manufacturing, where machines have non-uniform energy rates and each job may be rejected at a penalty or accepted and scheduled incurring energy costs. The authors develop a suite of approximation schemes: a $(2+\epsilon)$-approximation for the general case with variable $m$, a QPTAS for general $m$, a PTAS for uniform energy rates, and an FPTAS when the number of machines $m$ is fixed. The key techniques combine a $\gamma$-critical/$\gamma$-tiny decomposition, LP-based rejection decisions for tiny jobs, and leveraging the LO24 PTAS as a subroutine to schedule accepted jobs. These results advance energy-aware scheduling with rejection and have practical implications for energy-constrained production settings.

Abstract

Motivated by green manufacturing, this paper investigates a scheduling with rejection problem subject to an energy consumption constraint. Machines are associated with non-uniform energy consumption rates, defined as the energy consumed per unit time. Each job is either rejected with a rejection penalty or accepted and scheduled on some machine for processing, which incurs energy consumption. The problem aims to minimize the makespan of the accepted jobs plus the total penalty of the rejected jobs while the total energy consumption is bounded by a given threshold. In this paper, when the number of machines is part of the input, we develop the first $(2+ε)$-approximation algorithm for any fixed constant $ε$ and a simple QPTAS as well as a PTAS for uniform energy consumption rates. Moreover, we present an FPTAS when the number of machines is a fixed constant.

Approximation algorithms for scheduling with rejection in green manufacturing

TL;DR

This paper addresses scheduling with rejection under a total energy bound in green manufacturing, where machines have non-uniform energy rates and each job may be rejected at a penalty or accepted and scheduled incurring energy costs. The authors develop a suite of approximation schemes: a -approximation for the general case with variable , a QPTAS for general , a PTAS for uniform energy rates, and an FPTAS when the number of machines is fixed. The key techniques combine a -critical/-tiny decomposition, LP-based rejection decisions for tiny jobs, and leveraging the LO24 PTAS as a subroutine to schedule accepted jobs. These results advance energy-aware scheduling with rejection and have practical implications for energy-constrained production settings.

Abstract

Motivated by green manufacturing, this paper investigates a scheduling with rejection problem subject to an energy consumption constraint. Machines are associated with non-uniform energy consumption rates, defined as the energy consumed per unit time. Each job is either rejected with a rejection penalty or accepted and scheduled on some machine for processing, which incurs energy consumption. The problem aims to minimize the makespan of the accepted jobs plus the total penalty of the rejected jobs while the total energy consumption is bounded by a given threshold. In this paper, when the number of machines is part of the input, we develop the first -approximation algorithm for any fixed constant and a simple QPTAS as well as a PTAS for uniform energy consumption rates. Moreover, we present an FPTAS when the number of machines is a fixed constant.

Paper Structure

This paper contains 13 sections, 17 theorems, 18 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Model
  3. Literature review for related problems
  4. Our results
  5. Preliminaries
  6. Determining rejection decisions for $\gamma$-critical jobs
  7. A $(2+\epsilon)$-approximation for $P \mid rej, U, r_i \mid C_{\max}(A) + \sum_{J_j \in R} e_j$
  8. Determining rejection decisions for $\gamma$-tiny jobs
  9. The complete $(2+\epsilon)$-approximation algorithm
  10. A QPTAS for $P \mid rej, U, r_i \mid C_{\max}(A) + \sum_{J_j \in R} e_j$
  11. A PTAS for $P \mid rej, U, r_i=r \mid C_{\max}(A) + \sum_{J_j \in R} e_j$
  12. An FPTAS for $Pm \mid rej, U, r_i \mid C_{\max}(A) + \sum_{J_j \in R} e_j$
  13. Conclusions

Key Result

Lemma 1

Given $x_1, x_2, \ldots, x_m \ge 0$ and $y_1, y_2, \ldots, y_m \ge 0$ with $\sum_{i=1}^m x_i \ge \sum_{i=1}^m y_i$, if there exists a non-negative integer $\ell$ such that $y_i > x_i$ if and only if $i \le \ell$, then we have

Theorems & Definitions (26)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Definition 2
  • Lemma 3
  • Definition 3
  • Definition 4
  • Example 1
  • Lemma 4
  • Lemma 5
  • ...and 16 more