Autonomous Mobile Robots (AMRs) play a crucial role in transportation and service tasks at hospitals, contributing to enhanced efficiency and meeting medical demands. This paper investigates the optimization problem of scheduling strategies for AMRs at smart hospitals, where the service and travel times of AMRs are stochastic. We formulate a stochastic mixed integer programming model to minimize the total cost of the hospital by reducing the number of AMRs and travel distance while satisfying constraints such as AMR battery state of charge, AMR capacity, and time windows for medical requests. To address this objective, we identify several properties for generating high-quality solutions efficiently. We improve the Variable Neighborhood Search (VNS) algorithm by incorporating the properties of the AMR scheduling problem to solve the model. Experimental results demonstrate that VNS generates higher-quality solutions compared to existing methods. By intelligently arranging the driving routes of AMRs for both charging and service requests, we achieve substantial cost reductions for hospitals, enhancing the utilization of medical resources.

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