New upper and lower bounds for online scheduling with machine cost

This paper considers the online scheduling problem with machine cost. We are given a sequence of independent jobs with positive sizes. Jobs come one by one and it is required to schedule jobs irrevocably to a machine as soon as they are given, without any knowledge about jobs that follow later on. No machines are initially provided. When a job is revealed, the algorithm has the option to purchase new machines. The objective is to minimize the sum of the makespan and cost of purchased machines. We prove that 2 is a lower bound of the problem, which significantly improves the previous one of 4/3. We also present a new algorithm with competitive ratio (2+7)/3≈1.5486, which improves the current best algorithm with competitive ratio (26+3)/5≈1.5798. Moreover, we prove that applying only the lower bounds on the optimum objective value introduced before, no algorithm can be proven to have a competitive ratio less than 3/2.