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Optimization under uncertainty is important in many applications,particularly to inform policy and decision making in areas such as publichealth. A key source of uncertainty arises from the incorporation ofenvironmental variables as inputs into computational models or simulators. Suchvariables represent uncontrollable features of the optimization problem andreliable decision making must account for the uncertainty they propagate to thesimulator outputs. Often, multiple, competing objectives are defined from theseoutputs such that the final optimal decision is a compromise between differentgoals.  Here, we present emulation-based optimization methodology for such problemsthat extends expected quantile improvement (EQI) to address multi-objectiveoptimization. Focusing on the practically important case of two objectives, weuse a sequential design strategy to identify the Pareto front of optimalsolutions. Uncertainty from the environmental variables is integrated out usingMonte Carlo samples from the simulator. Interrogation of the expected outputfrom the simulator is facilitated by use of (Gaussian process) emulators. Themethodology is demonstrated on an optimization problem from public healthinvolving the dispersion of anthrax spores across a spatial terrain.Environmental variables include meteorological features that impact thedispersion, and the methodology identifies the Pareto front even when there isconsiderable input uncertainty.

Multi-objective optimisation using expected quantile improvement for decision making in disease outbreaks