The single machine semi-online scheduling problem with the objective of
minimizing total completion time is investigated with the assumption that the
ratio of the longest to the shortest processing time is not greater than a
constant $\gamma$. A semi-online algorithm is designed and its competitive
ratio is proven to be $1+ \frac{\gamma - 1}{1 + \sqrt {1 + \gamma (\gamma -
1)}}$. The competitive analysis method is as following: it starts from an arbitrary instance and
modifies the instance towards the possible structure of the worst-case instance
with respect to the given online algorithm. The modification guarantees that
the performance ratio does not decrease. Eventually, it comes up with a
relatively simple instance with a special structure, whose performance ratio
can be directly analyzed and serves as an upper bound on the competitive ratio.