# Estimating the Tour Length for the Close Enough Traveling Salesman Problem
## 3 Regression Results
### Model Validation  
We generated 234 new CETSP instances, similar to the 780 instances we used to develop the regression models, to test the Best BIC model with eight variables. The node locations (depot and the customers) are generated randomly and all customers in an instance have the same radius for the service regions. The instances have 6, 8, 10, 12, 14, 16, 18, 20, 25, or 30 customers. The radii for the customer service regions are 0.25, 0.50, or 1.00. There are nine new instances for each combination of the radius of the customer service regions and the number of customers up to 20. There are three new instances for each combination of the radius of the customer service regions and the number of customers greater than 20. We estimated the SZVNS tour lengths for the 234 new instances using the Best BIC model. The out-of-sample MPE and MAPE values are \(-0.344\%\) and \(4.233\%\), respectively, which indicate that the Best BIC model performs well in estimating the SZVNS tour lengths on the new CETSP instances. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Coefficient** & **Best Adjusted R\({}^{2}\)** & **Best Mallows’s \(C_{p}\)** & **Best BIC** \\ \hline Intercept (\(\beta_{1}\)) & 15.320 **** & 15.485 **** & 16.334 **** \\ n (\(\beta_{2}\)) & 0.188 & & \\ A (\(\beta_{3}\)) & 0.048 **** & 0.046 **** & 0.044 **** \\ MinP (\(\beta_{4}\)) & \(-0.668\) **** & \(-0.734\) **** & \(-0.674\) **** \\ MaxP (\(\beta_{5}\)) & 0.224 **** & 0.230 **** & \\ VarP (\(\beta_{6}\)) & 0.101 * & & \\ SumMinP (\(\beta_{7}\)) & 0.359 **** & 0.362 **** & 0.362 **** \\ SumMaxP (\(\beta_{8}\)) & 0.036 **** & 0.025 **** & 0.027 **** \\ MinM (\(\beta_{9}\)) & 0.455 **** & 0.508 **** & 0.498 **** \\ MaxM (\(\beta_{10}\)) & \(-0.410\) **** & \(-0.273\) **** & \\ SumM (\(\beta_{11}\)) & \(-0.064\) **** & & \\ VarM (\(\beta_{12}\)) & 0.685 **** & 0.805 **** & 0.797 **** \\ VarX\(\times\)VarY (\(\beta_{13}\)) & 0.014 **** & 0.011 **** & 0.012 **** \\ AvgR (\(\beta_{14}\)) & \(-9.059\) **** & \(-9.059\) **** & \(-9.059\) **** \\ SZ (\(\beta_{15}\)) & & & \\ \hline Number of variables & 13 & 10 & 8 \\ Adjusted R\({}^{2}\) & 0.921 & 0.921 & 0.920 \\ Mallows’s C\({}_{p}\) & & 13.0 & \\ BIC & & \(-1921\) \\ MPE & \(-0.192\%\) & \(-0.194\%\) & \(-0.202\%\) \\ MAPE & 3.983\% & 3.995\% & 4.008\% \\ \hline \hline \end{tabular} \({}^{*}p<0.1\); ** \(p<0.05\); **** \(p<0.001\); **** \(p<0.001\).  
\end{table} 
Table 5: Regression results on the three selected best subset models. We generated the optimal tour lengths for the 234 new instances using a branch-and-bound algorithm [14] in order to test the usefulness of the eight independent variables selected in the Best BIC model in estimating the optimal tour lengths. SZ is the only independent variable specific to the SZVNS algorithm and is not a part of any of the three models shown in Table 5. Therefore, the eight independent variables in the Best BIC model should have more general usage in estimating tour lengths. The regression model built using the eight independent variables in the Best BIC model and trained on the optimal tour lengths of the 234 new instances is given by: \(optimal_{i}=13.394+0.054\times\mathrm{A}_{i}-0.226\times\mathrm{MinP}_{i}+0.316 \times\mathrm{SumMinP}_{i}+0.033\times\mathrm{SumMaxP}_{i}+0.640\times\mathrm{ MinM}_{i}+0.852\times\mathrm{VarM}_{i}+0.018\times(\mathrm{VarX}\times\mathrm{VarY} )_{i}-8.774\times\mathrm{AvgR}_{i}\), where \(optimal_{i}\) denotes the optimal tour length for instance \(i\). The signs and the orders of magnitude of the coefficients are the same as the coefficients in the Best BIC model. The regression model for estimating the optimal tour lengths with eight independent variables has an adjusted R\({}^{2}\) value of 0.937, indicating a very good model fit. The MPE value of \(-0.161\%\), which is close to zero, indicates that there is almost an equal distribution of instances with positive and negative residuals. The MAPE value of 3.735% also indicates the usefulness of this model in estimating optimal tour lengths.