  In robust combinatorial optimization with discrete uncertainty, two general
approximation algorithms are frequently used, which are both based on
constructing a single scenario representing the whole uncertainty set. In the
midpoint method, one optimizes for the average case scenario. In the
element-wise worst-case approach, one constructs a scenario by taking the worst
case in each component over all scenarios. Both methods are known to be
$N$-approximations, where $N$ is the number of scenarios.
  In this paper, these results are refined by reconsidering their respective
proofs as optimization problems. We present a linear program to construct a
representative scenario for the uncertainty set, which guarantees an
approximation guarantee that is at least as good as for the previous methods.
Incidentally, we show that the element-wise worst-case approach can have an
advantage over the midpoint approach if the number of scenarios is large. In
numerical experiments on the selection problem we demonstrate that our approach
can improve the approximation guarantee of the midpoint approach by around 20%.
