In home health care (HHC) operations, one of the major aims of HHC centers is to timely meet patients' demands. According to the feedback from HHC centers, their decision-makers deal with some complexity in scheduling and routing of their health workers. Inspired by this point, the present research introduces a new HHC routing and scheduling problem considering different skill levels of health workers and different levels of patients’ needs. So, in such a condition, a highly qualified health worker can visit those patients who need lower-skilled demands while a low-qualified health worker cannot visit those who request higher skills. In this way, the total cost of the system will be lower compared to the situation in which the patients' needs exactly match the health workers' skills. Moreover, we consider that the maximum number of homes each health worker is tasked to visit during the day is specified and if more patients than this specified limit are assigned to each health worker, an additional cost will be imposed on the center in proportion to the excess number of patients. Since patient satisfaction, which is obtained with timely visits, is important for each HHC center, a hard time window is considered for each patient. The presented model is solved using the GAMS software with the CPLEX solver. Along with the MIP approach, a metaheuristic algorithm based on a simulated annealing algorithm is adopted to solve the problem. The results give the managers insight into this method of cost management in comparison with manual and traditional traditional planning. This study may help the decision-makers of HHC centers make more accurate decisions which, in turn, result in timelier service provision, increase the patients' satisfaction level, and improve the overall efficiency of HHC centers.