Planning for Package Deliveries in Risky Environments Over Multiple Epochs

Abstract

We study a risk-aware robot planning problem where a dispatcher must construct a package delivery plan that maximizes the expected reward for a robot delivering packages across multiple epochs. Each package has an associated reward for delivery and a risk of failure. If the robot fails while delivering a package, no future packages can be delivered and the cost of the robot is incurred. The package delivery plan takes place over the course of either a finite or an infinite number of epochs, denoted as the finite horizon problem and infinite horizon problem, respectively. The dispatcher has to weigh the risk and reward of delivering packages during any given epoch against the potential loss of any future epoch's reward. By using the ratio between a package's expected reward and its risk of failure, we provide an optimal solution to both the infinite and finite horizon problems. The finite horizon problem with the same set of packages for each epoch can be solved optimally in O(K + n log n) time where K is the number of epochs and n is the number of packages. If each epoch has a unique set of packages, the finite horizon problem can be solved optimally in O(Kn log n) time. For the infinite horizon problem, the optimal solution is to deliver the (single) package with the highest reward-to-risk ratio at each epoch, which can be found in O(n) time.