## 3 Materials and Methods
### Proposed Adaptive Large Neighborhood Search
#### 3.2.1 Initial Solution
In the first scenario, the total tardiness is 2652.56 s, while in the second scenario, it was 0 s.

### Evaluation Perspective from Fleet Management

A business may not always have as many vehicles as the number of routes required to ensure customer satisfaction. Ensuring minimum tardiness with fewer vehicles is important to decrease costs and satisfy customer time windows. Total tardiness according to fleet size scenarios is analyzed for the problems with 40 and 60 customers and given in Tables 6 and 7. Figure 6: Scenario 1, cost-oriented solution for R05 problem, representation of routes on OSM map. (Blue icon indicates the depot, red icons represent delivery locations, and blue lines show the vehicle routes). Figure 7: Scenario 2, customer-oriented solution for R05 problem, representation of routes on OSM map. (Blue icon indicates the depot, red icons represent delivery locations, and blue lines show the vehicle routes). The results show that total tardiness increases rapidly as the number of vehicles decreases. Additional analyses were conducted to assess the statistical significance of this trend. The following hypotheses were formulated:

**H\({}_{0}\):**: _There is no relationship between fleet size and total tardiness._
**H\({}_{a}\):**: _There is a relationship between fleet size and total tardiness._

In response to these hypotheses, the regression statistics and coefficient values for the relationship between fleet size and total tardiness are presented in Tables 8 and 9, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Test** & **\# of** & **\# of** & **\# of Tardy** & **Total Tardiness (s)** \\
**Problems** & **Routes** & **Vehicles** & **Deliveries** & **Total Tardiness (s)** \\ \hline C40 & 11 & 11 & 0 & 0 \\ C40 & 11 & 10 & 2 & 1095.15 \\ C40 & 11 & 9 & 7 & 2874.56 \\ C40 & 11 & 8 & 10 & 5712.73 \\ C40 & 11 & 7 & 13 & 9236.65 \\ C40 & 11 & 6 & 17 & 14,240.29 \\ \hline R40 & 11 & 11 & 0 & 0 \\ R40 & 11 & 10 & 2 & 804.75 \\ R40 & 11 & 9 & 5 & 3424.78 \\ R40 & 11 & 8 & 9 & 7390.73 \\ R40 & 11 & 7 & 13 & 12,823.69 \\ R40 & 11 & 6 & 17 & 19,573.68 \\ \hline RC40 & 10 & 10 & 0 & 0 \\ RC40 & 10 & 9 & 2 & 1213.66 \\ RC40 & 10 & 8 & 7 & 3883.94 \\ RC40 & 10 & 7 & 9 & 8276.91 \\ RC40 & 10 & 6 & 16 & 13,543.13 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Number of tardy deliveries in problems with 40 customers. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Test** & **\# of** & **\# of** & **Tardy Deliveries** & **Total Tardiness (s)** \\
**Problems** & **Routes** & **Vehicles** & **Tardy Deliveries** & **Total Tardiness (s)** \\ \hline C60 & 12 & 12 & 0 & 0 \\ C60 & 12 & 11 & 1 & 225.94 \\ C60 & 12 & 10 & 4 & 1272.41 \\ C60 & 12 & 9 & 8 & 3080.85 \\ C60 & 12 & 8 & 14 & 6265.13 \\ C60 & 12 & 7 & 23 & 10,587. Table 8 shows the basic statistics of the regression analysis for the problem sets with 60 customers (R60, C60, and RC60). The high R\({}^{2}\) value (0.9616) indicates that the model explains a high proportion of the variability in the dependent variable (number of tardy deliveries). This means that 96.16% of the variation in the number of tardy deliveries can be explained by fleet size. The \(p\)-value \(<\) 0.05 confirms that there is a statistically significant relationship between fleet size and tardiness. Table 9 shows the coefficients of the regression model that examines the effect of fleet size on the number of tardy deliveries. The coefficient for the number of vehicles is \(-\)4.7381, which implies that each additional vehicle reduces the number of tardy deliveries by 4.74 units on average. The \(p\)-value is quite low, which implies that the effect is statistically significant. Table 10 presents the ANOVA results that assess the significance of the regression model. In Table 10, the F-statistic is quite high at 475.31, indicating that the model explains a large and significant portion of the variance in the number of tardy deliveries. Moreover, the significance value (Significance F = 6.6 \(\times\) 10\({}^{-15}\)) is well below the 0.05 significance level.