Home health care (HHC) services represent a set of medical services given to patients at their homes. The patients require a set of care that must be coordinated and treated by skilled caregivers corresponding to their needs. This study proposes an HHC routing and assignment approach based on a mixed-integer linear programming model that aims to minimize total route cost. The HHC approach takes into account a set of HHC specific constraints and criteria. Secondly, we propose a new robust counterpart HHC model under uncertainty based on the well-known budgeted uncertainty set. The robust counterpart HHC model deals with travel and service times uncertainty. The computational results compare the deterministic model with its robust counterpart model. The small and medium instances have been solved using TSP benchmarks with specific data concerning HHC problems. The models have been implemented using ILOG CPLEX Optimization Studio. The computational results of small and medium instances indicated the efficiency of the proposed approach. Robustness analysis of the obtained results was conducted using a Monte Carlo simulation and indicated the price of robustness. The increase of route cost in comparison with the risk of infeasibility shows the importance of the designed robust routes for HHC routing and scheduling problems.