Semi-on-line scheduling problem for maximizing the minimum machine completion time on three special uniform machines

In this paper, we investigate a semi-on-line version for a special case of three machines M-1, M-2, M-3 where the processing time of the largest job is known in advance. A speed s(i)(s(1) = s(2) = 1, 1 <= s(3) = s) is associated with machine M-i. Our goal is to maximize the C-min - the minimum workload of three machines. We give a C-min3 algorithm and prove its competitive ratio is max {2, 3s+2/2+s} and the algorithm is the best possible for 1 <= s <= 2. We also claim the competitive ratio of algorithm C-min3 is tight.