## 2 Materials and Methods  
As mentioned earlier, managing home visits in HHC systems can be viewed as a variety of VRPTW. In addition to minimizing the arrival time, finishing time, or processing time in careaker-customer matching, the models may require further constraints to better suit the problems. Some existing works consider the travel cost, penalty cost, fixed cost, and overtime cost of the healthcare worker in their objective function. [PERSON] et al. [4] find the minimum staff numbers including minimum number per hour and per customer and the number of staff replacing/switching times per customer. [PERSON] et al. [5] minimize the weighted sum costs while maximizing service level; therefore, the costs considered comprise the total costs and penalty costs that arise when customers find it inconvenient to receive service. Putting compatibility in the objective function, [PERSON] et al. [6] minimize total travel time as well as non-preferences between careakers and patients to maximize compatibility while adding synchronization constraints, as some patients may need multiple services at the same time. [PERSON] et al. [7], also with synchronized visits to a home, construct a heuristic to solve the problem where each stop has only one task that might need multiple caretakers. Considering compatibility in the constraints, [PERSON] et al. [8] minimize maximum traveling time with the vehicle allowed to visit only compatible nodes. [PERSON] et al. [9] consider minimizing total distance traveled but with caretaker qualification requirements in the constraints. In another work, [PERSON] et al. [10] consider the weighted sum of total distance traveled, total travel cost, overtime wages and number of working staff; their notable constraints include the lunch break of the nurses. [PERSON] and [PERSON] [11] only consider the minimum total travel cost of staff but have synchronization constraints between nurses and vehicles as they consider the travel costs of the two separately. [PERSON] et al. [12] and [PERSON] et al. [13] give the most recent extended review of literature in HHC routing and scheduling problems with variety of constraints to date.  
Of the recently proposed heuristics for solving VRPTW that are more complicated than an NP-hard VRP, most are metaheuristics. The benchmark instances given in [14], known as [PERSON]'s instances, can be used to assess constructed heuristics. Table 1 showsthe objective function values studied in the most recent literature. Note that only works that consider hard time windows, i.e., where the service providers need to wait for customers to become available, are presented in the table.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multicolumn{2}{c}{**Best-Known Solution**} & \multirow{2}{*}{**References**} \\ \cline{2-3}  & **Number of Vehicles** & & **Total Distance** \\ \hline C101 & 10 & 827.30 * & [15,16] \\ C102 & 10 & 8  
There are three groups of [PERSON]'s instances. After plotting these instances, we can see that the customers in Group C instances appear to be scattered in groups or clusters, as seen in Figure 1. These instances are similar to real-world problems where customers live in geographically scattered villages. The customers in Group R instances (Figure 2) are relatively randomly placed, corresponding to customers living in a big city. The customers in Group RC (Figure 3) are randomly clustered, that is, they appear in groups, but within these groups, they are more or less randomly placed. This system can also be found in a typical real-world system as well. Since VRPTW is very applicable and useful for real-world problems, heuristics for solving VRPTW have been widely studied and still attract many researchers in the field. The relevant recent works are summarized in Table 2.