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Increasing e-commerce activity, competition for shorter delivery times, and innovations in transportation technologies have pushed the industry toward instant delivery logistics. This paper studies a facility location and online demand allocation problem applicable to a logistics company expanding to offer instant delivery service using unmanned aerial vehicles or drones. The problem is decomposed into two stages. During the planning stage, the facilities are located, and product and battery capacity are allocated. During the operational stage, customers place orders dynamically and real-time demand allocation decisions are made. The paper explores a multi-armed bandit framework for maximizing the cumulative reward realized by the logistics company subject to various capacity constraints and compares it with other strategies. The multi-armed bandit framework provides about 7% more rewards than the second-best strategy when tested on standard test instances. A case study based in Portland Metro Area showed that multi-armed bandits can outperform the second-best strategy by more than 20%.