Online and semi-online scheduling on two hierarchical machines with a common due date to maximize the total early work

TL;DR

This paper develops optimal online and semi-online algorithms for scheduling jobs on two hierarchical machines with a common due date, aiming to maximize early work, and provides competitive ratio guarantees.

Contribution

It introduces new optimal algorithms for various online and semi-online scheduling scenarios with hierarchical machines and common due date.

Findings

Optimal online algorithm with a competitive ratio of √2.

Semi-online algorithm with a competitive ratio of 4/3.

Algorithms for specific cases with known largest processing time.

Abstract

In this study, we investigated several online and semi-online scheduling problems on two hierarchical machines with a common due date to maximize the total early work. For the pure online case, we designed an optimal online algorithm with a competitive ratio of $\sqrt 2$. For the case when the total processing time is known, we proposed an optimal semi-online algorithm with a competitive ratio of $\frac{4}{3}$. Additionally, for the cases when the largest processing time is known, we gave optimal algorithms with a competitive ratio of $\frac{6}{5}$ if the largest job is a lower hierarchy one, and of $\sqrt 5-1$ if the largest job is a higher hierarchy one, respectively.