Abstract

McNaughton's theorem (1959) states that preemptions in scheduling arbitrary processing time jobs on identical parallel machines to minimize the total weighted completion time are redundant. Du et al. (1991) proved that this remains true even though the jobs have precedence constraints in the form of chains. There are known simple counterexamples showing that other extensions of McNaughton's theorem to other criteria or more general precedence constraints such as intrees or outtrees, or different release dates of jobs, or different speeds of machines, are not true even for equal weights of jobs. In this paper we show that in the case of two machines and unit processing times, preemptions are still advantageous for intrees or machines with different speeds even for equal weights, or outtrees for different weights, but become redundant for outtrees and equal weights even for different release dates. We also conjecture that the latter statement is actually true for any number of machines.

Article Outline

1. Introduction

1.1. Definitions and denotations

1.2. Earlier results on preemption redundancy

1.3. Cases with two machines where preemptions of unit processing time jobs are advantageous

1.4. Cases with two machines where preemptions of unit processing time jobs are redundant

2. Main result

2.1. Passage to integer preemptions

2.2. Expanded problem and theorem formulation

2.3. Theorem proof

3. Concluding remark

References