  We want to select the best systems out of a given set of systems (or rank
them) with respect to their expected performance. The systems allow random
observations only and we assume that the joint observation of the systems has a
multivariate normal distribution with unknown mean and covariance. We allow
dependent marginal observations as they occur when common random numbers are
used for the simulation of the systems. In particular, we focus on positively
dependent observations as they might be expected in heuristic optimization
where `systems' are different solutions to an optimization problem with common
random inputs. In each iteration, we allocate a fixed budget of simulation runs
to the solutions. We use a Bayesian setup and allocate the simulation effort
according to the posterior covariances of the solutions until the ranking and
selection decision is correct with a given high probability. Here, the complex
posterior distributions are approximated only but we give extensive empirical
evidence that the observed error probabilities are well below the given bounds
in most cases. We also use a generalized scheme for the target of the ranking
and selection that allows to bound the error probabilities with a Bonferroni
approach. Our test results show that our procedure uses less simulations than
comparable procedures from literature even in most of the cases where the
observations are not positively correlated.
