  We consider a problem of ranking and selection via simulation in the context
of personalized decision making, where the best alternative is not universal
but varies as a function of some observable covariates. The goal of ranking and
selection with covariates (R&S-C) is to use simulation samples to obtain a
selection policy that specifies the best alternative with certain statistical
guarantee for subsequent individuals upon observing their covariates. A linear
model is proposed to capture the relationship between the mean performance of
an alternative and the covariates. Under the indifference-zone formulation, we
develop two-stage procedures for both homoscedastic and heteroscedastic
simulation errors, respectively, and prove their statistical validity in terms
of average probability of correct selection. We also generalize the well-known
slippage configuration, and prove that the generalized slippage configuration
is the least favorable configuration for our procedures. Extensive numerical
experiments are conducted to investigate the performance of the proposed
procedures, the experimental design issue, and the robustness to the linearity
assumption. Finally, we demonstrate the usefulness of R&S-C via a case study of
selecting the best treatment regimen in the prevention of esophageal cancer. We
find that by leveraging disease-related personal information, R&S-C can
substantially improve patients' expected quality-adjusted life years by
providing patient-specific treatment regimen.
