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Time management under uncertainty is essential to large scale projects. From space exploration to industrial production, there is a need to schedule and perform activities. given complex specifications on timing. In order to generate schedules that are robust to uncertainty in the duration of activities, prior work has focused on a problem framing that uses an interval-bounded uncertainty representation. However, such approaches are unable to take advantage of known probability distributions over duration.
In this paper we concentrate on a probabilistic formulation of temporal problems with uncertain duration, called the probabilistic simple temporal problem. As distributions often have an unbounded range of outcomes, we consider chance-constrained solutions, with guarantees on the probability of meeting temporal constraints. By considering distributions over uncertain duration, we are able to use risk as a resource, reason over the relative likelihood of outcomes, and derive higher utility solutions. We first demonstrate our approach by encoding the problem as a convex program. We then develop a more efficient hybrid algorithm whose parent solver generates risk allocations and whose child solver generates schedules for a particular risk allocation. The child is made efficient by leveraging existing interval-bounded scheduling algorithms, while the parent is made efficient by extracting conflicts over risk allocations. We perform numerical experiments to show the advantages of reasoning over probabilistic uncertainty, by comparing the utility of schedules generated with risk allocation against those generated from reasoning over bounded uncertainty. We also empirically show that solution time is greatly reduced by incorporating conflict-directed risk allocation.