## 2 Literature Review  
HHC-related models started with [PERSON] et al. [15] in 1997. Currently, HHC problems are an extension of the VRP augmented by many side constraints that are specific to the HHC context linked to a scheduling problem. This new problem's formulation is called the Home Health Care Routing and Scheduling Problem (HHCRSP). To establish an HHCRSP, an extended definition of HHC is used, which includes various health-related parameters.  
According to [PERSON] et al. [16], HHC deals with various challenges such as partitioning a territory, dimensioning human resources, and assigning and scheduling caregivers to patients or tasks. Mostly, the objective of a scheduling problem is to find an ideal pairing of caregivers and patients, whereas the routing problem aims to design caregivers' tours while minimizing the total travel time. The HHCRSP is an NP-hard problem [17], and due to this, it has received the most attention. The proposed models are evaluated on homemade datasets or derived from protected data [13]. Knowing that different sets of constraints and objectives can be considered, it is impossible to make a comparison between different solving methods. Indeed, in HHC, there is no commonly accepted benchmark due to medical confidentiality.  
[PERSON] et al. [17] proposed an HHCRSP model formulated by taking into account workload balance for caregivers and introducing time windows. A time window represents a time interval in which something has to be started or can describe the working time or availability of someone. In 2013, [PERSON] et al. [18] formulated an HHCRP with time window constraints. They also proposed a model with simultaneous delivery and pickup of drugs and biological samples. [PERSON] et al. [12] proposed an HHCRSP as being multi-period and multi-service. The model takes into account the personal qualifications of the staff, possible interdependencies among services, and the individual requirements of the patients. [PERSON] et al. [9] proposed a multimodal HHCRSP from an Austrian HHC provider based on multiple modes of transportation. Its aim was to determine efficient multimodal tours while considering staff and patient satisfaction. [PERSON] et al. [19] studied scheduling and routing problems related to HHC with the consideration of lunch break requirements and working legal requirements. [PERSON] et al. [20] presented a model based on preferred time windows for patient and synchronization constraints. Among all these models, no comparison has so far been presented in the literature.  
Each HHCRSP is distinguished regarding the planning horizon [14] and its composition in terms of strength (number of caregivers, patients) [21]. Some HC providers pursue continuity of care. Patients are assigned to only one caregiver named the reference caregiver, who follows the entire patient care pathway. A reference caregiver provides all the visits that are pertinent [22]. On another note, some HHCRSP definitions include patients' and caregivers' preferences [23; 24], a minimal and maximal amount of working time for a caregiver, and time windows for patients. In the HHCRSP, some implicit parameters are not mentioned in the model such as the share of time when a caregiver is on the road or the capacity of the HC provider. The current literature suggests a rate between 15% and 20% for the share of traveling time [13]. The capacity of an HC provider is related to some constraints such as working legal requirements or the option for lunch breaks [10; 13].