An online concierge system displays an interface to a user identifying an estimated time of arrival for an order. To generate the estimated time of arrival for the order, the online concierge system trains a prediction engine to predict delivery time based on a predicted selection time for a shopper to select the order for fulfillment and predicted travel time for the shopper to deliver items of the order to a location identified by the order. The online concierge system generates a policy optimization model that computes an adjustment for the predicted delivery time. The adjustment is determined by solving a stochastic optimization problem with a constraint on a probability of the order being delivered after the estimated time of arrival. The predicted delivery time combined with the adjustment determines the estimated time of delivery displayed to the user to balance between minimizing late deliveries and wait times.