A flow shop scheduling problem with machine-dependent speeds: An ensemble approach with worst-case analysis

We consider a flow shop scheduling problem to minimize the total weighted completion time with any given job having the same processing requirement at all stages and each machine operating at its own speed. This problem is known to be NP-hard even for just two machines. In this paper, we propose a simple yet efficient approximation algorithm that leverages the complementary strengths of existing approaches. We first establish that our algorithm achieves a strictly better approximation ratio than previously known methods for the two-machine case. We then extend our analysis to settings with more than two machines, focusing on scenarios where the number of distinct machine speeds is fixed. Finally, we use these insights to derive an approximation ratio for the general case of arbitrary machine speeds.