Optimal algorithm for semi-online scheduling on two machines under GoS levels

Recently, Liu et al. (J Combin Optim 21:138–149, 2011) studied the semi-online scheduling problem on two machines under a grade of service provision. As the sum of jobs’ processing times $$\varSigma $$Σ is known in advance and the processing times are bounded by an interval $$[1,\alpha ]$$[1,α] where $$1< \alpha <2$$1<α<2, they presented an algorithm which is $$\frac{1+\alpha }{2}$$1+α2-competitive when $$\varSigma \ge \frac{2\alpha }{\alpha -1}$$Σ≥2αα-1. In this paper, we give a modified algorithm which is shown to be optimal for arbitrary $$\alpha $$α and $$\varSigma $$Σ.