On the variable common due date, minimal tardy jobs bicriteria two-machine flow shop problem with ordered machines

TL;DR

This paper improves the algorithmic efficiency for a complex two-machine flow shop scheduling problem, reducing the running time from quadratic to logarithmic scale using advanced data structures.

Contribution

The paper presents an optimized O(n log n) algorithm for a bicriteria scheduling problem, enhancing previous quadratic solutions with a novel data structure implementation.

Findings

Reduced algorithm complexity from O(n^2) to O(n log n)

Efficient implementation using modified binary trees

Applicable to scheduling problems with ordered machines

Abstract

We consider a special case of the ordinary NP-hard two-machine flow shop problem with the objective of determining simultaneously a minimal common due date and the minimal number of tardy jobs. In [S. S. Panwalkar, C. Koulamas, An O(n^2) algorithm for the variable common due date, minimal tardy jobs bicriteria two-machine flow shop problem with ordered machines, European Journal of Operational Research 221 (2012), 7-13.], the authors presented quadratic algorithm for the problem when each job has its smaller processing time on the first machine. In this note, we improve the running time of the algorithm to O(n log n) by efficient implementation using recently introduced modified binary tree data structure.