# A Hybrid Grasshopper Optimization Algorithm Applied to the Open Vehicle Routing Problem
## 4 Computational Experiments
### Results with the Hierarchical Objective Function
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Instance** & \(n\) & \(Q\) & \(NV\) & \(BKS\) & \(Avg\) & \(Best\) & \(CPU\) & \%\(gap\) \\ \hline C1 & 50 & 160 & 5 & 416.06 & 416.95 & **416.06** & 3.96 & 0.21 \\ C2 & 75 & 140 & 10 & 567.14 & 563.40 & **560.19** & 20.31 & \(-\)1.23 \\ C3 & 100 & 200 & 8 & 639.74 & 643.65 & **639.74** & 23.28 & 0.00 \\ C4 & 150 & 200 & 12 & 733.13 & 734.16 & 733.64 & 94.81 & 0.07 \\ C5 & 199 & 200 & 16 & 893.39 & 885.25 & 895.49 & 158.32 & 0.24 \\ C6 & 50 & 160 & 5 (6) & 412.96 & 417.75 & **422.25** & 5.70 & 2.25 \\ C7 & 75 & 140 & 10 & 583.19 & 567.96 & **567.77** & 9.45 & \(-\)2.64 \\ C8 & 100 & 200 & 9 & 644.63 & 647.74 & 646.75 & 24.12 & 0.33 \\ C9 & 150 & 200 & 13 & 757.69 & 743.19 & **742.35** & 135.98 & \(-\)2.02 \\ C10 & 199 & 200 & 17 & 874.71 & 878.73 & **873.70** & 243.11 & \(-\)0.12 \\ C11 & 120 & 200 & 7 & 682.12 & 695.83 & 600.194 & 62.00 & 1.18 \\ C12 & 100 & 200 & 10 & 534.24 & 535.99 & **534.24** & 30.37 & 0.00 \\ C13 & 120 & 200 & 10 (11) & 899.16 & 877.21 & **865.80** & 122.00 & \(-\)3.71 \\ C14 & 100 & 200 & 10 (11) & 591.87 & 557.74 & **556.32** & 39.06 & \(-\)6.01 \\ \hline \multicolumn{2}{c}{\(Avg\)} & & & & & & 69.46 & -0.82 \\ \hline O1 & 200 & 900 & 5 & 6018.52 & 6035.89 & **6018.52** & 519.37 & 0.00 \\ O2 & 240 & 550 & 9 & 4557.38 & 4571.43 & 4560.29 & 820.07 & 0.06 \\ O3 & 280 & 900 & 7 & 7731.00 & 7735.45 & 7734.70 & 825.82 & 0.05 \\ O4 & 320 & 700 & 9(10) & 7251.74 & 7274.23 & **7367.91** & 740.80 & 1.60 \\ O5 & 360 & 900 & 8 & 9193.15 & 9198.73 & **9193.15** & 919.40 & 0.00 \\ O6 & 400 & 900 & 9 & 9793.72 & 9807.46 & **9793.72** & 1000.00 & 0.00 \\ O7 & 440 & 900 & 10 & 10,344.37 & 10,387.18 & 10,363.80 & 1000.00 & 0.19 \\ O8 & 400 & 1000 & 10 & 12,415.36 & 12,465.65 & 12,420.81 & 943.79 & 0.04 \\ \hline \multicolumn{2}{c}{\(Avg\)} & & & & & & 846.16 & 0.24 \\ \hline K01 & 240 & 550 & 9 & 4629.21 & 4642.75 & **4622.58** & 760.49 & \(-\)0.14 \\ K02 & 320 & 700 & 9 & 7285.98 & 7303.10 & 7293.50 & 1000.00 & 0.10 \\ K03 & 400 & 900 & 10 & 9837.79 & 9879.08 & **9831.07** & 1000.00 & \(-\)0.07 \\ K04 & 480 & 1000 & 10 & 12,431.50 & 12,475.34 & **12,422.20** & 1000.00 & \(-\)0.07 \\ K05 & 200 & 900 & 5 & 6342.66 & 6052.56 & **6041.33** & 595.51 & \(-\)4.75 \\ K06 & 280 & 900 & 7 & 7731.21 & 7734.46 & **7731.21** & 756.28 & 0.00 \\ K07 & 360 & 900 & 9 & 9188.11 & 9172.43 & **9162.29** & 1000.00 & \(-\)0.28 \\ K08 & 440 & 900 & 10 & 10,488.50 & 10,472.46 & **10,457.60** & 1000.00 & \(-\)0.29 \\ \hline \multicolumn{2}{c}{\(Avg\)} & & & & & 889.04 & \(-\)0.69 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results obtained by the HGOALS for all the test instances (hierarchical objective function). (5, 10, 10 and 9, respectively) than the solutions found in previous research (6, 11, 11 and 10). Finally, the algorithm struggles with the C11 instance. The average percentage deviation from the best-known solution (gap) is the worst for all benchmark problems (1.18%). The reason is the cluster distribution of the clients and the tendency of the feasibility recovery procedure to create more routes than the minimum required. Note that instance C6 has a higher gap, but it uses fewer vehicles than the best-known solution. The CPU times increase significantly as the benchmark problem size increases. The algorithm reached the 1000 s limit for seven of the benchmark instances, two from set O and five from set K. The algorithm performs the worst in the set O with an average deviation from the best-known solution of 0.24. For group C, the algorithm performs the best having an average deviation from the best-known solutions of \(-0.82\), while for group K this metric is \(-0.69\). Table 3 compares the proposed algorithm with the results from previous research. Considering all instances in group C, the HGOALS outperforms the other algorithms. The Broad Local Search Algorithm (BLSA) performs better than the HGOALS in instances without route length constraints in this group.