# A Swarm Intelligence Solution for the Multi-Vehicle Profitable Pickup and Delivery Problem
## 4 Proposed Method
### ABC without Clustering (ABC)
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Dataset**} & \multicolumn{4}{c}{**Population Size**} \\ \cline{2-5}  & **30** & **40** & **50** & **60** \\ \hline
13-100-F-S & 39,438.58 & 39,438.58 & 41,472.86 & 41,472.86 \\
14-100-F-L & 63,455.13 & 66,453.66 & 66,453.66 & 66,453.66 \\
15-100-P-S & 72,032.22 & 72,032.22 & 72,170.54 & 72,170.54 \\
16-100-P-L & 131,404.9 & 131,404.9 & 131,404.9 & 131,404.9 \\
17-100-R-S & 74,330.95 & 77,484.06 & 77,484.06 & 77,484.06 \\
18-100-R-L & 119,991.9 & 120,249.7 & 126,504.3 & 126,504.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of tuning the population size of the ABC. limit value to more than 100 did not contribute to an improvement in the objective function value. Thus, the limit value was taken to be 100. ### Parameter Tuning of ABC(C)

Since we used the same ABC algorithm, with little modification, in the ABC with the clustering approach, we adopted the same parameter values that have been used in the ABC for our ABC(C). ### Experimental Results

As previously mentioned, we proposed two versions of the ABC to improve the initial solutions: the ABC used with GRASP, and the ABC(C) used with the clustering methods, i.e., K-means-GRASP(C), modified K-means-GRASP(C), and ACO-GRASP(C). We also have two separate strategies for the scout be phase. S1 constructs the solution by randomly selecting one of these strategies: random insertion, greedy, and randomised insertion or greedy insertion. After testing this strategy, we found the random insertion heuristic did not improve the results compared to others, so we eliminated this choice and kept the other two. The second strategy, S2, works like the GRASP in [7]. Therefore, we have eight overall proposed methods named as follows: (1) ABC(S1), (2) ABC(S2), (3) K-means-GRASP(C)-ABC(C)S1, (4) K-means-GRASP(C)-ABC(C)S2, (5) modified K-means-GRASP(C)-ABC(C)S1, (6) modified K-means-GRASP(C)-ABC(C)S2, (7) ACO-GRASP(C)-ABC(C)S1, and (8) ACO-GRASP(C)-ABC(C)S2. All proposed algorithms were run five times, as performed with the previous algorithms in [6] and [7]. We compared our proposed algorithms and the state-of-the-art algorithms presented in [10; 11], which are C1-Seq VNS, C2-Seq VNS, C1-Self Adaptive VNS, C2-Self Adaptive VNS and IPPD, for solving the MVPPDP in terms of the processing time and profits gained. First, with respect to the processing time, all previous algorithms have been run for 5 min for each instance. We note, though, that this time is not considered acceptable for the m-commerce apps (e.g., Hungerstation, MRSOOL, etc.) that are used daily and rely on the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Dataset** & \multicolumn{4}{c}{**Demon Value**} \\ \cline{2-5}
**Instances** & **1000** & **2500** & **5000** & **7500** \\ \hline
13-100-F-S & 41,472.86 & 41,596.33 & 41,954.86 & 41,954.86 \\
14-100-F-L & 66,453.66 & 66,683.84 & 66,683.84 & 66,683.84 \\
15-100-P-S & 72,170.54 & 72,449.56 & 72,449.56 & 72,449.56 \\
16-100-P-L & 131,404.9 & 131,404.9 & 131,404.9 & 131,404.9 \\
17-100-R-S & 77,484.06 & 77,484.06 & 77,484.06 & 77,484.06 \\
18-100-R-L & 126,504.3 & 126,504.3 & 126,504.3 & 126,504.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of tuning the demon value of the ABC. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Dataset** & **Limit** & \\
**Instances** & **100** & **200** \\ \hline
13-100-F-S & 41,472.86 & 41,472.86 \\
14-100-F-L & 66,453.66 & 66,453.66 \\
15-100-P-S & 72,170.54 & 72,170.54 \\
16-100-P-L & 131,404.9 & 131,404.9 \\
17-100-R-S & 77,484.06 & 77,484.06 \\
18-100-R-L & 126,504.3 & 126,504.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of tuning the limit value of the ABC. speed of responding, so in our case, we used the stopping condition of 500 iterations to measure the processing time. Table 5 presents the comparison results between our proposed algorithms and state-of-the-art algorithms in terms of time (in seconds).