Heuristics for Online Scheduling on Identical Parallel Machines with Two GoS Levels

This paper considers the online scheduling problem on m (m ≥ 3) parallel machines (the first k machines with grade 1 and the remaining m − k machines with grade 2) with two GoS levels and makespan as the objective function. The jobs arrive over time with grade 1 or 2 and an arrival job can be assigned to a machine only when the grade of the job is no less than the grade of the machine. Three cases are considered: (i) For k = 1, the authors present an online algorithm with competitive ratio of 9/5. (ii) For 1 < k < m − 1, an online algorithm with competitive ratio of 2.280 is proposed. (iii) For k = m − 1, an online algorithm is presented with competitive ratio of 2. All the three algorithms are based on greedy algorithm with a similar structure. At last, numerical instances are given and the average competitive ratios of the instances show good performance of the proposed algorithms.