Revisiting Johnson's rule for minimizing makespan in the Two-Machine Flow Shop scheduling problem
Federico Della Croce, Quentin Schau
TL;DR
The paper revisits Johnson's rule for the two-machine flow shop problem $F2 \;||\; C_{\max}$ and shows that, although Johnson’s algorithm runs in $O(n \log n)$ time due to sorting, one can detect in linear time whether a full sort can be avoided and an optimal solution can be obtained in $O(n)$ time. It introduces linear-time criteria based on partitioning jobs into sets $A$ and $B$, along with conditions that ensure optimality with only small prefixes/suffixes needing sorting, and proves high-probability linear-time solvability under standard distributions; when the criteria fail, worst-case instances require full sorting. The authors provide probabilistic bounds via $P^* = P_1^* P_2^*$ and dynamic-programming arguments to bound success probabilities, and validate the approach with extensive computational experiments showing linear-time performance and large sets of equivalent optimal sequences. This work offers a practically efficient framework for solving and enumerating optimal two-machine flow-shop schedules in realistic settings.
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.