Distributionally Robust Budget Allocation for Earthquake Risk Mitigation in Buildings

Destructive earthquakes are rare, high-impact, uncertain events. When they occur, the physical infrastructure is damaged, resulting in major economic losses. Risk mitigation planning aims to prevent extreme adverse effects and address tail risk; however, it requires large up-front investments. The large underlying uncertainties generate high variation in the avoided losses. Disregarding these uncertainties does not eliminate their presence nor detract from their importance. Two distributionally robust optimization (DRO) models are proposed to select building groups for pre-earthquake retrofit considering the uncertainties in the (1) earthquake occurrence probabilities, and (2) within-scenario building damage costs. The models minimize the worst-case expected objective function cost given the uncertainty in the random variables, promoting informed decisions under incomplete information. The conditional value at risk (CVaR) measure is incorporated into the optimization framework to model the cognitive loss-averse bias in decision-making for low-probability, high-consequence events. CVaR is derived by taking a weighted average of the extreme damage costs in the tail of the distribution, beyond the value at risk cutoff point, refining previous research that measured risk by setting arbitrary thresholds that are hard to define in practice. Implemented for the city of San Francisco, the risk-based models guard against high damage costs at the right tail of the distribution at the expense of higher up-front costs. The objective function cost was evaluated using out-of-sample data to assess the model performance under unseen data. The DRO reformulations resulted in improved model performance in the out-of-sample testing relative to the nonrobust approach, mitigating the optimizer's curse, but may lead to overly cautious retrofit decisions if uncertainties are overestimated.