# Balancing Staff Finishing Times vs. Minimizing Total Travel Distance in Home Healthcare Scheduling
## 4 Solution Method
### Initial Solution
The objectives are to minimize total finishing time difference and minimize the total travel distance, combining typical objective functions for routing and services scheduling problems. Home healthcare has been a crucial support system for the overall health system as it reduces the burden on individual patients who would otherwise have to travel to the hospital regularly to receive care. In real-life situations, minimizing costs alone may not adequately answer the needs of a business, especially those in the healthcare sector. The best way to allocate resources (costs and caretakers) is to consider both aspects simultaneously. For this reason, we have tried to fill the gap in the literature by proposing new objective functions. We chose the weighted-sum approach to solve multiple-objective optimization problems; and given the NP-hard nature of these problems, an algorithm based on simulated annealing (SA) has been constructed. The problem was converted into a single-objective problem with different choices of weights (a total of 11 choices), then the SA was deployed to solve the problem according to those weights and the results are combined together to obtain the Pareto set. The initial solution was generated using the inserting procedure. Then, \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{3}{c}{**Total Distance**} & \multicolumn{3}{c}{**Total Finishing Time Difference**} \\ \cline{2-5}  & **Setting1** & **Setting2** & **Setting1** & **Setting2** \\ \hline R110 & 0.00 & 0.00 & 0.03 & 0.01 \\ R112 & 0.00 & 0.00 & 0.01 & 0.00 \\ R209 & 0.00 & 0.00 & 0.00 & 0.01 \\ RC103 & 0.00 & 0.00 & 0.02 & 0.12 \\ RC106 & 0.00 & 0.00 & 0.00 & 0.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The \(p\)-value for each instance of testing for total distance and total finishing time difference. the improvement operators, swapping, moving, path-exchange, and ruin and recreate, are applied to find the improved solution. To assess the performance of our algorithm, we used it to solve the well-known benchmark instances called Solomon's instances. The results, when compared to the optimal solutions, show that in most cases our algorithm is able to find the optimal solution with the objective function to minimize total finishing time difference. Moreover, the results of testing the median between the objective functions converted from the weighted sum approach suggest that the Pareto set can be obtained by solving only a partial set of objective functions. In terms of future work, methods should be investigated to solve multiple objective functions based on the idea of transforming the objectives into a single objective by adding weights. Choosing the weight is an important decision to make as the objective values may not be on the same scale. Moreover, to make the problem more realistic, additional constraints may have to be considered, for instance, multiple depots (multiple-health centers), simultaneous visits, and preferences for patients or caretakers. The model can be extended by considering the combination of new objective functions such as finishing time, total traveling time and total number of workloads, etc. Conceptualization, P.S. and C.L. ; Data curation, P.S. ; Formal analysis, C.L. ; Investigation, P.S. ; Methodology, P.S. and C.L. ; Software, P.S. ; Supervision, C.L. ; Validation, P.S. and C.L. ; Visualization, C.L. ; Writing--original draft, P.S. ; Writing--review and editing, C.L. All authors have read and agreed to the published version of the manuscript. This research was supported by Fundamental Fund 2023 [Grant No. FF66/024], Chiang Mai University. Not applicable. The original data presented in the study are openly available at [[http://web.cba.neu.edu/](http://web.cba.neu.edu/)]([http://web.cba.neu.edu/](http://web.cba.neu.edu/))\(\sim\)msolomon/problems.htm (accessed on 1 July 2024). This research was supported by Chiang Mai University. The authors appreciate [PERSON]'s proof-reading help. The authors declare no conflicts of interest.