# A Location Routing Problem with Time Windows Consideration: A Metaheuristics Approach
## 3 Results
### Algorithm Verification
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline
**Prob. ID** & **n** & **m** & **Q** & **BKS** & **Gap** & **CPU** & **Gap** & **CPU** & **Gap** & **CPU** & **Gap** & **CPU** & **Gap** & **CPU** \\ \hline P1 & 20 & 5 & 70 & **54.793** & **0.00\%** & 19.8 & **0.00\%** & 0 & **0.00\%** & 4.08 & **0.00\%** & 3 & **0.00\%** & 3.7 \\ P2 & 20 & 5 & 150 & **39.104** & **0.00\%** & 15 & **0.00\%** & 0 & **0.00\%** & 4.78 & **0.00\%** & 4 & **0.00\%** & 4.8 \\ P3 & 20 & 5 & 70 & **48.00\%** & **0.00\%** & 19.3 & **0.00\%** & 0 & **0.00\%** & 3.92 & 0.08\% & 3 & **0.00\%** & 4.4 \\ P4 & 20 & 5 & 150 & **37.542** & **0.00\%** & 15 & **0.00\%** & 0 & **0.00\%** & 5.45 & **0.00\%** & 4 & **0.00\%** & 3.3 \\ P5 & 50 & 5 & 70 & **90.111** & **0.00\%** & 74.7 & **0.00\%** & 3 & **0.00\%** & 24.47 & 0.32\% & 27 & **0.00\%** & 34.3 \\ P6 & 50 & 5 & 150 & **63.242** & **0.00\%** & 57.7 & **0.00\%** & 0 & **0.00\%** & 21.31 & 1.31\% & 27 & **0.00\%** & 26 \\ P7 & 50 & 5 & 70 & **88,298** & **0.00\%** & 95 & 0.39\% & 11 & **0.00\%** & 24.19 & 1.18\% & 23 & **0.00\%** & 37.4 \\ P8 & 50 & 5 & 150 & **67.308** & 0.05\% & 58.6 & **0.00\%** & 16 & **0.00\%** & 20 The performance of the VNSPR is evaluated based on the cost of the solution as well as the percentage difference in the solution that exists between the BKS, the results produced by the VNSPR, and the outcomes produced by other algorithms. The gap is computed for each instance by dividing the difference in value that exists between the proposed algorithm's solution values and the BKS value. Based on the quality of the results, VNSPR has the potential to replace SALRP, GRASP+ ELS, and the two-phase hybrid heuristic (HH). This is measured in terms of the mean gap percentage. The best result for MACO was found while the parameters were being set, whereas the best result for VNSPR is based on a single run, despite the solution quality being a little bit lower. The computational findings demonstrate that VNSPR is capable of successfully resolving CLRP. It is important to note that VNSPR has the smallest average percentage gap out of all of the algorithms in the comparisons; this indicates that the algorithm is  
For Barreto's instances, the suggested VNSPR can unravel 14 problems of all of the 19 datasets in these instances. The algorithm, however, is unable to find an optimal solution, particularly for instances B12, B13, B14, B18, and B19, in which aside from instance B12, are large-sized problems with more than 100 customer nodes. The percentage difference over BKS varies between 0.00 and 2.00%, with a mean percentage difference of 0.27%. In terms of mean percentage difference, VNSPR are able to outperform the SALRP algorithm. VNSPR performs better than SALRP and MACO algorithms in terms of quantity of BKSs discovered. For [PERSON] et al. instances, among 30 instances of this dataset, the proposed VNSPR obtains 11 best solutions when compared to the best-known solutions. The percentage difference between BKS and results of the proposed VNSPR spans between 0.00 and 1.53%, with a mean percentage difference is 0.34%. VNSPR's mean percentage difference is smaller than those of SALRP, GRASP + ELS, and HH, although MACO has a slightly lower mean percentage gap of 0.33%. Although VNSPR could not find an optimal solution for a large-sized problem (nodes > 100), the proposed algorithm has a 1.53% smaller maximum gap compared to MACO and still outperforms MACO in large instances. In the case of [PERSON] and [PERSON]'s instances, VNSPR achieves five best-known solutions, with average gaps between them ranging from 0.0% to 3.18%. VNSPR is competitive in terms of solution quality with other algorithms (SALRP, GRASP+ ELS, MACO, and HH), despite the fact that it is unable to discover the best new solutions. This is due to the smallest average percentage gap. In addition, the results demonstrate that the proposed VNSPR methodology has the smallest difference (3.18%) between the smallest and largest gaps.