  Optimization under uncertainty is important in many applications,
particularly to inform policy and decision making in areas such as public
health. A key source of uncertainty arises from the incorporation of
environmental variables as inputs into computational models or simulators. Such
variables represent uncontrollable features of the optimization problem and
reliable decision making must account for the uncertainty they propagate to the
simulator outputs. Often, multiple, competing objectives are defined from these
outputs such that the final optimal decision is a compromise between different
goals.
  Here, we present emulation-based optimization methodology for such problems
that extends expected quantile improvement (EQI) to address multi-objective
optimization. Focusing on the practically important case of two objectives, we
use a sequential design strategy to identify the Pareto front of optimal
solutions. Uncertainty from the environmental variables is integrated out using
Monte Carlo samples from the simulator. Interrogation of the expected output
from the simulator is facilitated by use of (Gaussian process) emulators. The
methodology is demonstrated on an optimization problem from public health
involving the dispersion of anthrax spores across a spatial terrain.
Environmental variables include meteorological features that impact the
dispersion, and the methodology identifies the Pareto front even when there is
considerable input uncertainty.
