Better online algorithms for scheduling with machine cost

For most scheduling problems the set of machines is fixed initially and remains unchanged for the duration of the problem. Recently Imreh and Noga proposed adding the concept of machine cost to scheduling problems and considered the so-called list model problem. For this problem, we are given a sequence of independent jobs with positive sizes, which must be processed nonpreemptively on a machine. No machines are initially provided, and 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 machines. In this paper, we first present an online algorithm with a competitive ratio at most 1.5798, which improves the known upper bound 1.618. Then for a special case where every job size is no greater than the machine cost, we present an optimal online algorithm with a competitive ratio 4/3. Last, we present an algorithm with a competitive ratio at most 3/2 for the semionline problem with known largest size, which improves the known upper bound 1.5309.