Inspiring by real-world assumptions, this paper aims to present a robust optimization model for a home health care routing-scheduling problem with uncertain service and travel times. The constraints include temporal precedence and synchronization constraints, timing constraints for the transferring biological samples, and constraints associated with the multiple deployments of caregivers to provide the possibility of visiting one patient multiple times in the one-day planning horizon. Also, the existing uncertainty in service and travel times arising from an increase in customer-oriented service strategies is another crucial issue that should be appropriately addressed. Due to the complexity of this problem, three meta-heuristic algorithms (i.e., simulated annealing, genetic algorithm, and memetic algorithm) are proposed to solve the developed model. A series of experiments are implemented and shown that the memetic algorithm outperforms other proposed algorithms for large-sized problems. Furthermore, a detailed analysis of the results achieved by solving deterministic and robust models demonstrates the advantage of using the robust model. Finally, the dynamic version of this problem is proposed, and two dispatching policies are used to address this problem.