# Balancing Staff Finishing Times vs. Minimizing Total Travel Distance in Home Healthcare Scheduling
## 4 Solution Method
### Initial Solution
We analyzed a total of 56 instances. As shown in Figure 7, the values for each objective function are close to each other. For this reason, we have selected instance C102 to show the relationship between total finishing time difference and total distance for each objective function in Tables 4 and 5, respectively. Tables 4 and 5 show the calculated total finishing time difference and total distance for each objective function solved by our algorithm, with each row in the tables showing the trend when increasing the weight for total finishing time difference. Table 4 shows that, in almost every instance, the highest value is obtained for the total finishing time difference when the weight for the total distance is zero (w0_w10). Moreover, increasing the weight of total finishing time difference does not result in a decreased value. Conversely, when aiming to minimize the total distance, the distance decreases inversely in proportion to an increase in weight. In general, the solution for a multi-objective problem is not unique; in the end, it needs to be left to the administrator to decide the weights required to convert to a single objective or choose between the Pareto solutions. We can specifically consider the solutions on the Pareto front line to be the best solutions. Figure 5: Boxplot between objective functions and calculated total finishing time differences. Figure 6: Plot between total distance and total finishing time differences for 10 solutions and optimal solutions for R101. Figure 7: Plot between total distance and total finishing time differences for 10 solutions and optimal solutions for C102. objective functions, the most appropriate test to use is the Kruskal-Wallis test [37]. Under this test, data cannot be assumed to come from any specific distribution. The primary purpose of the test is to determine whether there exists a statistical difference between the medians of distinct groups. In this case, we aim to determine whether the median values of the total difference for each objective function and the total distance between each objective function are statistically significant. Each Solomon's instance is independently tested with Kruskal test. The null and alternative hypotheses are determined as follows:

\(H0\): There is no significant difference in the total distance or total difference in the objective functions with a 95% confidence interval. \(Ha\): There is a significant difference in the total distance or total difference in the objective functions with a 95% confidence interval. In this case, the \(p\)-value for the Kruskal test will accept the hypothesis when the \(p\)-value is \(<0.05\). We separated the test according to the values of the calculated total distance and total finishing time difference and tested the hypothesis grouping by the objective functions. We tested two settings: testing between all objective values, i.e., w_0_w10, w_1_w9,..., w9_w1 and w10_w0, named Setting1, and testing for partial group of objectives functions, named Setting2. After trying several groups, we chose as the best setting {w0_w10, w1_w9, w2_w8, w8_w2, w9_w1, w10_w0}. The first setting aims to test the similarity between all objective functions while the second setting aims to cut off the objective functions that are similar. The results shown in Table 6 are the \(p\)-values for each instance and each setting, ignoring the row that has all zero values. We can see that there is no significant difference in total distance and total finishing time difference in Setting1 for each instance (\(p\)-value \(<0.05\)). Moreover, most of the cases in Setting2 are significant except for two, RC103 and RC106 (\(p\)-value \(>0.05\)). The results suggest that solving only the objective functions in Setting2 is enough for use as the Pareto set. ## 6 Conclusions

This paper investigates multi-objective vehicle routing and scheduling in home healthcare systems. Modeling for this problem is based on the vehicle routing problem with time windows.