# A Location Routing Problem with Time Windows Consideration: A Metaheuristics Approach
## 4 Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Prob. ID** & **n** & **m** & **Q** & **BKS** & **NV** & **VNSPR** & **NV** & **Gap (\%)** \\ \hline P1 & 20 & 5 & 70 & 54,793 & 5 & 54,793 & 5 & **0.00\%** \\ P2 & 20 & 5 & 150 & 39,104 & 3 & 39,104 & 3 & **0.00\%** \\ P3 & 20 & 5 & 70 & 48,908 & 5 & 48,908 & 5 & **0.00\%** \\ P4 & 20 & 5 & 150 & 37,542 & 3 & 37,542 & 3 & **0.00\%** \\ P5 & 50 & 5 & 70 & 90,111 & 12 & 90,111 & 12 & **0.00\%** \\ P6 & 50 & 5 & 150 & 63,242 & 6 & 63,242 & 6 & **0.00\%** \\ P7 & 50 & 5 & 70 & 88,298 & 12 & 88,298 & 12 & **0.00\%** \\ P8 & 50 & 5 & 150 & 67,308 & 6 & 67,340 & 6 & 0.05\% \\ P9 & 50 & 5 & 70 & 84,055 & 12 & 84,055 & 12 & **0.00\%** \\ P10 & 50 & 5 & 150 & 51,822 & 6 & 51,822 & 6 & **0.00\%** \\ P11 & 50 & 5 & 70 & 86,203 & 12 & 86,496 & 12 & 0.34\% \\ P12 & 50 & 5 & 150 & 61,830 & 6 & 61,830 & 6 & **0.00\%** \\ P13 & 100 & 5 & 70 & 275,419 & 24 & 277,348 & 24 & 0.70\% \\ P14 & 100 & 5 & 150 & 213,615 & 11 & 217,336 & 11 & 1.74\% \\ P15 & 100 & 5 & 70 & 193,671 & 23 & 193,993 & 23 & 0.17\% \\ P16 & 100 & 5 & 150 & 157,150 & 11 & 160,776 & 11 & 2.31\% \\ P17 & 100 & 5 & 70 & 200,079 & 23 & 201,497 & 24 & 0.71\% \\ P18 & 100 & 5 & 150 & 152,441 & 11 & 155,638 & 11 & 2.10\% \\ P19 & 100 & 10 & 70 & 287,983 & 26 & 294,244 & 26 & 2.17\% \\ P20 & 100 & 10 & 150 & 231,763 & 12 & 239,026 & 12 & 3.13\% \\ P21 & 100 & 10 & 70 & 243,590 & 24 & 246,464 & 24 & 1.18\% \\ P22 & 100 & 10 & 150 & 203,988 & 11 & 208,617 & 11 & 2.27\% \\ P23 & 100 & 10 & 70 & 250,882 & 24 & 253,070 & 24 & 0.87\% \\ P24 & 100 & 10 & 150 & 204,317 & 11 & 211,926 & 11 & 3.72\% \\ P25 & 200 & 10 & 70 & 477,248 & 47 & 494,705 & 47 & 3.66\% \\ P26 & 200 & 10 & 150 & 378,351 & 22 & 391,889 & 22 & 3.58\% \\ P27 & 200 & 10 & 70 & 449,849 & 48 & 454,117 & 48 & 0.95\% \\ P28 & 200 & 10 & 150 & 374,330 & 22 & 394,890 & 23 & 5.49\% \\ P29 & 200 & 10 & 70 & 472,472 & 47 & 487,694 & 47 & 3.22\% \\ P30 & 200 & 10 & 150 & 362,817 & 22 & 377,792 & 22 & 4.13\% \\ \hline \multicolumn{8}{c}{Average} & \multicolumn{8}{c}{22.3684} & \multicolumn{8}{c}{22.47368} \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Computational results LRP with time windows for [PERSON] et al.’s instances. Table 6 displays the results that were obtained from [PERSON]'s datasets. The results demonstrate that the proposed VNSPR is capable of solving all extensive instances of [PERSON]'s datasets. With differences over BKS ranging from 0.00% to 3.58% and an average relative differences of 1.00%, the proposed VNSPR is capable of obtaining the best solution to 9 out of the 19 problems. In instances \(117\times 4\) and \(150\times 5\), VNSPR has a higher number of vehicles used. It can happen because these two instances have narrow time windows, which makes it more difficult to solve. The same goes for [PERSON] et al. 's datasets. When contrasted to the best solutions that are currently known for all 30 instances of this dataset, the proposed VNSPR obtains 10 of the best solutions. Table 7 presents the information of BKS for [PERSON] et al. 's instances and the computational results of the VNSPR. The gap between the solutions varies from 0.00-5.49% with an average gap of 1.42%. The gap happens due to problem complexity as the consequence of the time window constraint. In order to obtain a feasible solution, a larger number of vehicles used were needed. The additional numbers of vehicles increase vehicle cost and the total cost. Table 8 presents an additional comparison of the results obtained from the use of various algorithms and proposes VNSPR with regard to the third dataset. The proposed