Nowadays, the aging population and the little availability of informal care are two of the several factors leading to an increased need for assisted living support. Hence, home healthcare (HHC) operations including a set of nurses and patients have been developed recently by both academia and health practitioners to consider elderlies’ preferences willing to receive their cares at their homes instead of hospitals or retirement homes. Commonly, different services, e.g., nursing, physiotherapy, housekeeping and cleaning, for an HHC system are performed by nurses at patients’ homes after scheduling and routing the nurses by decision makers. Due to the difficulty of the problem, recent studies show a great deal of interest in applying various metaheuristics and heuristics to solve this problem. To alleviate the drawbacks of previous works and make HHC more practical, this paper develops not only a new mathematical formulation considering new suppositions in this research area but also a lower bound based on Lagrangian relaxation theory has been employed. As such, three new heuristics and a hybrid constructive metaheuristic are utilized in this study to solve the proposed model. Finally, the performance of the proposed algorithms is validated by the developed lower bound and also analyzed by different criteria and also the efficiency of developed formulation is probed through some sensitivity analyses.