# 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 T1 & 100 & 10 & 150 & 1467.68 & 11 & 1479.50 & 11 & 0.81\% \\ T2 & 100 & 20 & 150 & 1449.20 & 11 & 1456.03 & 11 & 0.47\% \\ T3 & 100 & 10 & 150 & 1394.80 & 10 & 1394.93 & 10 & 0.01\% \\ T4 & 100 & 20 & 150 & 1432.29 & 11 & 1432.96 & 11 & 0.05\% \\ T5 & 100 & 10 & 150 & 1167.16 & 11 & 1170.25 & 11 & 0.26\% \\ T6 & 100 & 20 & 150 & 1102.24 & 10 & 1104.37 & 10 & 0.19\% \\ T7 & 100 & 10 & 150 & 791.66 & 11 & 794.23 & 11 & 0.32\% \\ T8 & 100 & 20 & 150 & 728.3 & 11 & **728.3** & **11** & **0.00\%** \\ T9 & 100 & 10 & 150 & 1238.49 & 11 & 1249.32 & 11 & 0.87\% \\ T10 & 100 & 20 & 150 & 1245.31 & 11 & 1247.86 & 11 & 0.20\% \\ T11 & 100 & 10 & 150 & 902.26 & 12 & **902.26** & **12** & **0.00\%** \\ T12 & 100 & 20 & 150 & 1018.29 & 11 & 1020.73 & 11 & 0.24\% \\ T13 & 150 & 10 & 150 & 1866.75 & 16 & 1951.25 & 16 & 4.53\% \\ T14 & 150 & 20 & 150 & 1833.95 & 16 & 1882.15 & 16 & 2.63\% \\ T15 & 150 & 10 & 150 & 1978.27 & 17 & 2020.79 & 17 & 2.15\% \\ T16 & 150 & 20 & 150 & 1801.39 & 16 & 1828.45 & 16 & 1.50\% \\ T17 & 150 & 10 & 150 & 1447.66 & 16 & 1456.56 & 16 & 0.61\% \\ T18 & 150 & 20 & 150 & 1441.98 & 15 & 1460.56 & 16 & 1.29\% \\ T19 & 150 & 10 & 150 & 1205.09 & 17 & 1207.05 & 17 & 0.16\% \\ T20 & 150 & 20 & 150 & 930.99 & 17 & 933.89 & 17 & 0.31\% \\ T21 & 150 & 10 & 150 & 1699.92 & 17 & 1737.36 & 17 & 2.20\% \\ T22 & 150 & 20 & 150 & 1400.01 & 16 & 1405.58 & 16 & 0.40\% \\ T23 & 150 & 10 & 150 & 1199.51 & 16 & 1214.89 & 16 & 1.28\% \\ T24 & 150 & 20 & 150 & 1152.18 & 16 & 1154.92 & 16 & 0.24\% \\ T25 & 200 & 10 & 150 & 2279.92 & 21 & 2329.67 & 21 & 2.18\% \\ T26 & 200 & 20 & 150 & 2185.41 & 22 & 2247.71 & 22 & 2.85\% \\ T27 & 200 & 10 & 150 & 2234.78 & 21 & 2280.79 & 21 & 2.06\% \\ T28 & 200 & 20 & 150 & 2241.04 & 21 & 2263.51 & 21 & 1.00\% \\ T29 & 200 & 10 & 150 & 2089.77 & 21 & 2129.20 & 21 & 1.89\% \\ T30 & 200 & 20 & 150 & 1709.56 & 21 & 1759.26 & 20 & 2.91\% \\ T31 & 200 & 10 & 150 & 1466.62 & 20 & 1482.62 & 21 & 1.09\% \\ T32 & 200 & 20 & 150 & 1084.78 & 22 & 1091.28 & 22 & 0.60\% \\ T33 & 200 & 10 & 150 & 1970.44 & 21 & 2000.46 & 21 & 1.52\% \\ T34 & 200 & 20 & 150 & 1918.93 & 21 & 1972.85 & 21 & 2.81\% \\ T35 & 200 & 10 & 150 & 1771.06 & 21 & 1790.42 & 22 & 1.09\% \\ T36 & 200 & 20 & 150 & 1393.16 & 22 & 1400.70 & 22 & 0.54\% \\ \hline \multicolumn{7}{c}{Average} & 19.3684 & 19.4737 & 1.15\% \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Computational results LRP with time windows for Tuzun and Burke’s instances. algorithm can solve 36 instances with an average relative gap of 1.15%, and varies from 0.00% to 4.53%. The number of vehicles used is slightly more than the number reported in the result of BKS. VNSPR has an average vehicle use of 19.47 while BKS has an average vehicle use of 19.36. Overall, the proposed algorithm is efficient and effective for finding good solutions after adding time window constraints. Compared to CLRP's best-known solutions, the average gap is relatively smaller, and the number of vehicles (\(NV\)) used is marginally higher. Considering the additional constraint added to the instances, this should be a reasonable result.