## 1 Introduction  
In the year 2020, the proportion of the Thai population aged over 65 was 19.8%, more than double the global average of 9.3%. This figure is expected to increase to 29.6% over the next 30 years [1]. This means that in a short period of time Thailand will become a Super Aged country, defined as a country where more than 28% of the population is over 60 years of age. As projected by the Office of the National Economic and Social Development Board, Office of the Prime Minister, Bangkok, Thailand [2], the number of elderly people living alone with health problems will also increase. Home healthcare (HHC), which plays a major role in a proactive healthcare system, is essential for this population group, especially those with chronic diseases and bedridden patients requiring long-term care [3]. Caring for these people may involve injections, wound dressing, physical therapy, and other rehabilitation procedures, as well as health-related advice-giving to ensure the sustainability and improved life quality of the coming Aged Society. For the elderly, home visits represent more efficient management of human resources than having them visit the hospital individually.  
In attempting to manage home visits in an HHC system, the following need to be considered: (1) compatibility matching of patients/customers and physicians, experts, or healthcare staff; (2) best routing for healthcare personnel to visit their customers; (3) meeting applicable service conditions, and (4) minimizing the cost of the whole system.  
The second consideration together with the fourth bear similarities to a regular VRP, and if the third is also considered, the problem becomes a vehicle routing problem with time windows (VRPTW). In this work, we will also integrate the first aspect into the VRPTW to better capture the nature of the HHC system. When all these aspects are taken into consideration, the problem becomes even more complicated and requires a custom-made heuristic to find a solution.  
Most of the VRPTWs showcased in the literature already consider minimizing distance traveled as the main objective. The model we constructed can also deal with minimizing the total cost, with main objectives ranging from total distance traveled to completion time. This, along with the new general set of compatibility constraints, will make it more relevant for real-world applications, as patient-careaker compatibility matching is sometimes necessary and minimizing waiting time and balancing job numbers per vehicle are a more important issue in VRPTWs that are similar to HHC systems.  
The work process in this study goes as follows: First, a VRPTW with compatibility matching constraints is formulated. Subsequently, a hybrid heuristic combining local search (LS), Ruin and recreate procedure (R&R), and particle swarm optimization (PSO) is proposed for solving the problem, which is NP-hard in nature. Finally, simulation results tested on best known benchmark instances, namely [PERSON]'s instances, are presented to show our heuristic's efficiency in solving VRPTW problems with and without compatibility matching constraints.  
This paper is organized as follows: Section 2 provides a review of literature on VRPTW. Section 3 presents a mathematical formulation of the problem. Section 4 describes the solution approach. Computational experiments on [PERSON]'s instances and conclusions are presented in Sections 5 and 6, respectively.