# A Hybrid Grasshopper Optimization Algorithm Applied to the Open Vehicle Routing Problem
## 4 Computational Experiments
### Results with the Hierarchical Objective Function
The HGOALS performs the best in instances with route length constraints, improving the best-known solutions for six of the seven instances. The HGOALS obtains new best solutions for C2, C6, C7, C9, C10, C13 and C14. For instances C6, C13 and C14 the algorithm found solutions using one less vehicle. Finally, it is relevant to mention that the algorithm struggles with C11 instance obtaining the largest gap (1.18). The latter instance has a cluster like distribution. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Instance** & \(n\) & **Q** & **\(BK\)S** & **\(NV\)** & **HGOALS** & **BBMO**[37] & **BLSA**[38] & **HBMO**[44] & **ORTR**[35] \\ \hline C1 & 50 & 160 & 416.06 & 5 & **416.06** & **416.06** & **416.06** & **416.06** & **416.06** \\ C2 & 75 & 140 & 567.14 & 10 & **560.19** & 567.14 & 567.14 & 567.14 & 567.14 \\ C3 & 100 & 200 & 639.74 & 8 & **639.74** & **639.74** & **639.74** & 640.25 & **639.74** \\ C4 & 150 & 200 & 733.13 & 12 & 733.64 & 735.18 & 733.13 & 738.49 & 733.13 \\ C5 & 199 & 200 & 893.39 & 16 & 895.49 & 895.19 & **893.39** & 902.17 & 924.96 \\ C6 & 50 & 160 & 412.96 & 6 & **422.25 (5)** & 412.96 & - & 412.96 & 412.96 \\ C7 & 75 & 140 & 583.19 & 10 & **567.77** & 583.19 & - & 583.19 & 568.49 \\ C8 & 100 & 200 & 644.63 & 9 & 646.75 & **644.63** & - & 644.63 & **644.63** \\ C9 & 150 & 200 & 757.69 & 13 & **742.35** & 759.35 & - & 765.95 & **756.38** \\ C10 & 199 & 200 & 874.71 & 17 & **873.70** & 879.38 & - & 884.28 & 876.02 \\ C11 & 120 & 200 & 682.12 & 7 & 690.194 & **682.12** & **682.12** & 683.15 & 682.54 \\ C12 & 100 & 200 & 534.24 & 10 & **534.24** & 535.28 & **534.24** & 536.37 & **534.24** \\ C13 & 120 & 200 & 899.16 & 11 & **865.80 (10)** & 904.04 & - & 905.18 & 896.50 \\ C14 & 100 & 200 & 591.87 & 11 & **556.32 (10)** & 592.16 & - & 593.95 & 591.87 \\ \hline
**Instance** & \(n\) & **Q** & **\(BK\)S** & **MVNS**[34] & **HES**[36] & **TSB**[32] & **ALNS**[45] & **ACO**[46] & **HPSO**[47] \\ \hline C1 & 50 & 160 & 416.06 & 5 & **416.06** & **416.06** & **416.06** & **416.06** & **416.06** \\ C2 & 75 & 140 & 567.14 & 10 & 567.14 & 567.14 & 574.05 & 567.14 & 567.14 \\ C3 & 100 & 200 & 639.74 & 8 & **639.74** & **639.74** & 641.60 & 641.76 & 639.74 \\ C4 & 150 & 200 & 733.13 & 12 & **733.13** & **733.13** & 740.80 & **733.13** & **733.13** \\ C5 & 199 & 200 & 893.39 & 16 & 905.96 & 894.11 & 953.40 & 896.08 & 977.95 \\ C6 & 50 & 160 & 412.96 & 6 & 412.96 & 412.96 & 412.96 & 412.96 & 412.96 \\ C7 & 75 & 140 & 583.19 & 10 & 596.47 & 584.15 & 634.54 & 583.19 & 568.49 \\ C8 & 100 & 200 & 644.63 & 9 & **644.63** & **644.63** & **644.63** & 645.16 & **644.63** \\ C9 & 150 & 200 & 757.69 & 13 & 760.06 & 764.56 & 785.20 & 757.84 & 758.97 \\ C10 & 199 & 200 & 874.71 & 17 & **875.67** & 888.46 & 884.63 & **875.67** & 886.08 \\ C11 & 120 & 200 & 682.12 & 7 & 682.12 & 682.12 & 683.40 & 682.12 & 683.11 \\ C12 & 100 & 200 & 534.24 & 10 & **534.24** & **534.24** & 535.10 & **534.24** & **534.24** \\ C13 & 120 & 200 & 899.16 & 11 & 904.04 & 910.26 & 943.66 & 909.80 & 904.01 \\ C14 & 100 & 200 & 591.87 & 11 & 591.87 & 591.87 & 597.30 & 591.87 & 591.87 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison between the HGOALS results and previous research for instances in group C (hierarchical objective function). The comparison of the results with previous research for instances in group O is made in Table 4. In this set of instances, the BLSA performs the best obtaining the best-known solutions for six of the eight benchmark problems. On the other hand, the HGOALS performs better than the other algorithms (except BLSA), and finds a new best solution for instance O4 using nine vehicles instead of ten. Notice that test problems in this set do not consider route length constraints. Finally, Table 5 compares results for instances in group K. The comparison is made with upper bounds obtained by the GRASP algorithm proposed in [42] and the BRKGA in [48]. The results show that the HGOALS outperforms the other two metaheuristics by obtaining six new best solutions for the eight instances in the set.