## 3 Mathematical Modeling  
As mentioned earlier, home visiting in a HHC system can be formulated as a VRPTW problem, which can then be written as a complete graph, \(G=(N,E)\) having \(N=\{0,1,\ldots,n\}\) being customers (or patients) where 0 is the starting point of the route. The set \(E\), the edges in the graph, represents the travelling connections between nodes. The travel times between points and service time at each point are given. Since the medical needs of patients \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Objective** & **Proposed Algorithms** & **Reference** \\ \hline \multirow{4}{*}{Total cost} & Insertion and saving heuristic & [24] \\ \cline{2-3}  & Adaptive large neighborhood search with removal and insertion operator & [25] \\ \cline{2-3}  & Adaptive genetic algorithm with 2-opt procedure & [26] \\ \cline{2-3}  & Particle swarm optimization with simulated annealing & [27] \\ \hline \multirow{4}{*}{Total distance} & Probabilistic tabu search & [18] \\ \cline{2-3}  & Branch and bound algorithm with greedy algorithm and large neighborhood search & [28] \\ \cline{2-3}  & Large neighborhood search with ruin and recreate procedure & [29] \\ \cline{2-3}  & Parallel cutting plane of branch and bound & [17] \\ \cline{2-3}  & Parallel strategy of branch and bound & [15] \\ \cline{2-3}  & Simulated annealing with ruin and recreate procedure & [30] \\ \cline{2-3}  & Lagrangian relaxation and Column generation & [16] \\ \cline{2-3}  & Genetic algorithm & [23] \\ \cline{2-3}  & Variable neighborhood search with pruning and propagation techniques & [31] \\ \cline{2-3}  & of constraint & \\ \cline{2-3}  & Simulated annealing with K-restart & [32] \\ \cline{2-3}  & Parallel hybrid genetic algorithm with evolution function & [33] \\ \cline{2-3}  & Two-phase set partitioning and genetic algorithm & [20] \\ \cline{2-3}  & Multi-parametric mutation procedure with 1 + 1 evolution strategies algorithm & [34] \\ \cline{2-3}  & Adaptive large neighborhood search & [35] \\ \cline{2-3}  & Parallel simulated annealing & [21] \\ \cline{2-3}  & Particle swarm optimization with adaptive strategies & [36] \\ \hline \multirow{4}{*}{Bi-level Minimizing the number of vehicles and total distance} & Two evolutionary strategies algorithm with representation and mutation operators & [37] \\ \cline{2-3}  & Simulated annealing (for solving primary) and large neighborhood search (for solving secondary) & [38] \\ \cline{2-3}  & Genetic algorithm with Pareto ranking technique & [19] \\ \cline{2-3}  & Arc-guided evolution strategies & [39] \\ \cline{2-3}  & Bi-level & Iterated local search and decomposition & [40] \\ \cline{2-3}  & Minimizing cost and maximizing path length & \\ \hline Multiple problems: & & \\ - & Minimize number of vehicles & Multiple ant colony system (one for number of vehicles and one for total distance) & [41] \\ - & Minimize total distance & \\ \hline Multiple problems: & & \\ - & Minimizing total waiting time & \\ - & Minimizing average waiting time & Greedy randomized adaptive search and Variable neighborhood search & [42] \\ - & Maximizing slack time & \\ \hline Multiple problems: & & \\ - & Minimizing total distance & Genetic algorithm with ruin and recreate procedure & [22] \\ - & Minimizing number of vehicles & \\ \hline The combination of total distance and lateness penalty (soft time windows) & Tabu search with multiple exchange procedures & [43] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Literature review for VRPTW with various objective functions and their proposed algorithms. or customers are unique and may be time-bound, if the service providers (or caretakers) arrive before the specified time window, they will have to wait until the appropriate time. This problem is considered a hard time windows problem. The model constructed here has the objective to minimize total travel costs with the constraints to meet all the traveling/service/matching conditions.