Several semi-online scheduling problems on two identical machines with combined information

In this paper we consider several semi-online scheduling problems on two identical machines with combined information. The objective of each problem is to minimize the makespan. The first problem is semi-online scheduling with known optimal solution value and maximum job size. We obtain a lower bound 6/5 and design an optimal algorithm with a competitive ratio 6/5. The second problem is semi-online scheduling with a buffer of size k, where k(k >= 1) is a finite positive integer, and known maximum job size. We obtain a lower bound 6/5 and design an algorithm with a competitive ratio 5/4. The third problem is semi-online scheduling with a buffer of size 1 and jobs arriving in decreasing order of their processing times. We obtain a lower bound 7/6, which matches an upper bound in the literature. The last problem is semi-online scheduling with a buffer of size 1 and all the job processing times being bounded in the interval [1, t](t >= 1). We obtain a lower bound max {min{4/3, t+2/6}, min{5/4, t+1/4}, min{7/6, t+2/3}}, where the lower bound 4/3 for t >= 6 matches an upper bound in the literature, and design an algorithm with a competitive ratio max{t+2/3, 8/7} for 1 <= t <= 3/2, which is optimal for 10/7 <= t <= 3/2. (C) 2012 Elsevier B.V. All rights reserved.