Revisiting Johnson's rule for minimizing makespan in the Two-Machine Flow Shop scheduling problem

TL;DR

This paper improves the efficiency of Johnson's rule for two-machine flow shop scheduling by identifying conditions for linear-time computation, supported by probabilistic analysis and experiments.

Contribution

It demonstrates that Johnson's rule can be computed in linear time under certain conditions, bypassing full sorting in typical cases.

Findings

Linear-time detection of optimal scheduling possible in many cases

Probabilistic analysis shows high likelihood of linear-time performance

Experimental results support theoretical findings

Abstract

We consider Johnson's rule for minimizing the makespan in the two-machine flow shop problem. Although its worst-case time complexity is O(n log n), we show that it is possible to detect in linear time whether a full sorting of jobs can be avoided and an optimal solution can be computed in O(n) time. A probabilistic analysis indicates that linear time complexity holds with high probability under uniformly distributed processing times, a result further supported by extensive computational experimentation.