Research on approximation algorithms of two uniform machines in parallel machines scheduling

We consider a parallel machine scheduling problem: the number of machines is two and the machines are uniform. We have n jobs, the processing times of which are pi p2…, pn if processed on the first machine, and ap1, ap2…apn if processed on the second machine. 0 < a ≤ 1. The number of jobs on each machine is unlimited. The weight of each job is w1, w2,…wn respectively. Our goal is to assign the jobs on two machines and determine the order of jobs on each machine, and aim to minimize the total weighted completion time. We derive a 1.1755-approximation algorithm for this problem.