# A Hybrid Grasshopper Optimization Algorithm Applied to the Open Vehicle Routing Problem
## 4 Computational Experiments
### Results with the Traveling Cost Minimization Objective Function  
The obtained results by the HGOALS for the benchmark problems with the second objective function of minimizing the total traveling cost are presented in Table 6. The results show that the algorithm was able to improve three of the 14 instances of group C, one of group O and five on group K. The HGOALS also reaches the best-known solutions for five instances in group C and one for group K. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Instance** & \(n\) & \(Q\) & \(NV\) & \(BKS\) & **HGOALS** & **ORTR [35]** & **BLSA [38]** \\ \hline O1 & 200 & 900 & 5 & 6018.52 & **6018.52** & **6018.52** & **6018.52** \\ O2 & 240 & 550 & 9 & 4557.38 & 4560.29 & 4584.55 & **4557.38** \\ O3 & 280 & 900 & 7 & 7731.00 & 7734.70 & 7732.85 & **7731.00** \\ O4 & 320 & 700 & 10 & 7251.74 & **7367.91 \({}^{**}\)(9)** & 7291.89 & 7253.20 \\ O5 & 360 & 900 & 8 & 9193.15 & **9193.15** & 9197.61 & **9193.15** \\ O6 & 400 & 900 & 9 & 9793.72 & **9793.72** & 9803.80 & **9793.72** \\ O7 & 440 & 900 & 10 & 10,344.37 & 10,363.80 & 10,374.97 & 10347.70 \\ O8 & 400 & 1000 & 10 & 12,415.36 & 12,420.81 & 12,429.56 & **12,415.36** \\ \hline
**Instance** & \(n\) & \(Q\) & \(NV\) & \(BKS\) & **HES [36]** & **HBMO [44]** & **BBMO [37]** \\ \hline O1 & 200 & 900 & 5 & 6018.52 & **6018.52** & 6023.48 & 6021.11 \\ O2 & 240 & 550 & 9 & 4557.38 & 4583.70 & 4561.18 & **4557.38** \\ O3 & 280 & 900 & 7 & 7731.00 & 7733.77 & 7745.16 & 7735.14 \\ O4 & 320 & 700 & 10 & 7251.74 & 7271.24 & 7287.49 & 7267.18 \\ O5 & 360 & 900 & 8 & 9193.15 & 9254.15 & 9201.25 & 9198.25 \\ O6 & 400 & 900 & 9 & 9793.72 & 9821.09 & 9809.48 & 9798.19 \\ O7 & 440 & 900 & 10 & 10,344.37 & 10,363.40 & 10,401.24 & 10,351.18 \\ O8 & 400 & 1000 & 10 & 12,415.36 & 12,428.20 & 12,429.57 & 12,418.57 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison between route decoder and previous research for instances in group O (hierarchical objective function). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Instance** & \(n\) & \(Q\) & \(NV\) & \(BKS\) & **HGOALS** & **GRASP [42]** & **BRKGA [43]** \\ \hline K01 & 240 & 550 & 9 & 4629.21 & **4622.58** & 4790.31 & 4629.21 \\ K02 & 320 & 700 & 10 & 7285.98 & 7293.50 & 8407.07 & **7285.98** \\ K03 & 400 & 900 & 10 & 9837.79 & **9831.07** & 11,286.40 & 9837.78 \\ K04 & 480 & 1000 & 10 & 12,431.50 & **12,422.20** & 14,325.87 & 12,431.50 \\ K05 & 200 & 900 & 5 & 6342.66 & **6041.33** & 6342.66 & 6004.65 \({}^{*}\)(12) \\ K06 & 280 & 900 & 7 & 7731.21 & **7731.21** & 8557.82 & **7731.21** \\ K07 & 360 & 900 & 9 & 9188.11 & **9162.29** & 10,624.48 & 9188.11 \\ K08 & 440 & 900 & 10 & 10,488.50 & **10,457.60** & 12,129.07 & 10,488.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison between route decoder results and previous research for instances in group K (hierarchical objective function). The results show that the proposed algorithm is capable of improving the best-known solution in nine of the 30 instances used to test the algorithm. Notice that eight of these instances consider route length constraints. It is also capable of finding the best-known solutions in other nine test instances. In brief, the HGOALS matches or improves the best-known solutions for 18 of the 30 test instances. On the other hand, the CPU times required by the algorithm reach 1000 s for six instances in group O. It is also in this group where the algorithm performs the worst with an average deviation from the best solution of 0.02. For group K, the algorithm performs the best having an average deviation from the best solution of \(-0.08\), while for group C this metric is of \(-0.07\). Table 7 compares the proposed algorithm with previous research for benchmark problems in group C. For instances with no route length constraints, the HGOALS performs worse than the BLSA and BRKGA but better than the other algorithms.