# A Location Routing Problem with Time Windows Consideration: A Metaheuristics Approach
## 3 Results
### Algorithm Verification  
The VNSPR algorithm is initially implemented on all three sets of CLRP benchmark instances so that verification can be performed. The proposed algorithm is then evaluated in comparison to four of the most recently developed heuristics that have been proposed to solve LRP. These heuristics are SALRP [3], GRASP+ ELS [57], modified ant colony algorithm (MACO) [58], and two-phase hybrid heuristic (HH) [59]. [PERSON] et al. [3] reported that these four algorithms produce among the best solutions for LRP. The notations in Table 1 are used for Tables 2-4. The result formatted in bold indicates that it either achieve the same value as the Best Known Solution (BKS) or represent the best result among the results generated by other algorithms for the particular instance. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Notation** & **Description** \\ \hline _Prob. ID_ & problem number \\ \(n\) & quantity of customers \\ \(m\) & quantity of depots \\ \(Q\) & vehicle capacity \\ _BKS_ & Best Know Solution \\ _Gap_ \% & percentage gap of the solution cost found with respect to BKS \\ _CPU_ & running time in seconds on the CPU used by each algorithm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notation for CLRP instance. \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 B1 & 21 & 5 & 6000 & **424.9** & **0.00\%** & 18.3 & **0.00\%** & 0 & **0.00\%** & 5.64 & **0.00\%** & 6 & **0.00\%** & 2.2 \\ B2 & 22 & 5 & 4500 & **585.1** & **0.00\%** & 16.6 & **0.00\%** & 15 & **0.00\%** & 4.69 & **0.00\%** & 9 & **0.00\%** & 3.3 \\ B3 & 29 & 5 & 4500 & **512.1** & **0.00\%** & 23.9 & **0.00\%** & 9 & **0.00\%** & 8.59 & **0.00\%** & 11 & **0.00\%** & 5.2 \\ B4 & 32 & 5 & 8000 & **562.2** & **0.00\%** & 27 & **0.00\%** & 18 & **0.00\%** & 13.05 & **0.00\%** & 40 & **0.00\%** & 9.1 \\ B5 & 32 & 5 & 11,000 & **504.3** & **0.00\%** & 25.1 & **0.00\%** & 34 & **0.00\%** & 9.56 & **0.00\%** & 22 & **0.00\%** & 10.2 \\ B6 & 36 & 5 & 250 & **460.4** & **0.00\%** & 31.7 & **0.00\%** & 0 & **0.00\%** & 13.14 & **0.00\%** & 39 & **0.00\%** & 10.8 \\ B7 & 50 & 5 & 160 & **565.6** & **0.00\%** & 52.8 & **0.00\%** & 8 & **0.00\%** & 29.16 & 2.62\% & 45 & **0.00\%** & 15 \\ B8 & 75 & 10 & 140 & **844.4** & 0.43\% & 126.8 & **0.76\%** & 86 & 0.06\% & 58.72 & 0.53\% & 94 & **0.00\%** & 110.3 \\ B9 & 100 & 10 & 200 & **833.4** & 0.59\% & 330.8 & **0.00\%** & 127 & 0.40\% & 83.92 & 0.62\% & 234 & **0.00\%** & 184 \\ B10 & 12 & 2 & 140 & **204.00\%** & 6.8 & - & **0.00\%** & 2.09 & - & - & **0.00\%** & 0.8 \\ B11 & 55 & 15 & 120 & **1112.1** & 0.06\% & 112.4 & - & - & 0.04\% & 28.81 & - & - & **0.00\%** & 37.4 \\ B12 & 85 & 7 & 160 & **1625.5** & **0.00\%** & 213.1 & - & - & 0.04\% & 77.86 & - & - & 0.63\% & 120.5 \\ B13 & 318 & 4 & 2500 & 557275 & 1.12\% & 2806.5 & - & - & 0.53\% & 831.34 & - & - & 0.35\% & 2189.07 \\ B14 & 318 & 4 & 8000 & **67019** & 2.10\% & 3352.5 & - & - & **0.00\%** & 1986.33 & - & - & 1.28\% & 2614.95 \\ B15 & 27 & 5 & 2500 & **3062** & **0.00\%** & 23.3 & **0.00\%** & 35 & **0.00\%** & 8.59 & **0.00\%** & 11 & **0.00\%** & 4.8 \\ B16 & 134 & 8 & 850 & **5709** & **0.00\%** & 522.4 & **0.18\%** & 280 & **0.00\%** & 136.63 & 3.18\% & 252 & **0.00\%** & 407.472 \\ B17 & 88 & 8 & 9,000,00 & 3558.8 & **0.00\%** & 226.9 & **0.00\%** & 130 & **0.00\%** & 99.53 & 1.74\% & 148 & **0.00\%** & 148.4 \\ B18 & 150 & 10 & 8,000,00 & **43919.9** & 2.71\% & 577 & **0.10\%** & 1697 & 0.48\% & 166.95 & 1.50\% & 456 & 0.90\% & 450.06 \\ B19 & 117 & 14 & 150 & **12,290.3** & 1.71\% & 323.4 & - & - & 0.53\% & 77.45 & - & - & 2.00\% & 252.252 \\ \hline \multicolumn{2}{c}{Mean gap (\%)} & \multicolumn{2}{c}{0.43\%} & \multicolumn{2}{c}{0.08\%} & \multicolumn{2}{c}{0.11\%} & \multicolumn{2}{c}{0.78\%} & \multicolumn{2}{c}{0.27\%} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of CLRP for the Barreto’s [41] instances.