# A Swarm Intelligence Solution for the Multi-Vehicle Profitable Pickup and Delivery Problem
## 4 Proposed Method
### ABC without Clustering (ABC)
For example, the first algorithm refers to generating the initial solution using the GRASP after clustering the search space by using the K-means algorithm and then improving it using the ABC with reconstruction strategy S1. ## 5 Computational Experiments

All algorithms have been implemented in MATLAB (R2017b) on a laptop powered by an Intel Core i7-4510U processor operating at 2.00 GHz (2601 MHz, two cores, four logical processors). The following subsections provide descriptions of the datasets utilised and the specifics of parameter tuning prior to presenting the experimental results. ### Test Instances

To test the performance of our proposed methods, we used the same instances used by [4]. The dataset comprises 36 instances, which have been categorised into three distinct groups: small size (consisting of 20 and 50 customers, respectively, served by two and three vehicles), medium size (comprising 100 and 250 customers, served by 4 and 5 vehicles, respectively), and large size (500 and 1000 customers served by 6 and 8 vehicles, respectively). Moreover, twelve instances comprise each group. These occurrences exhibit variability in terms of both time and revenue. To generate short and long routes, the total time limit is either small or large, with a range of 2500 to 15,000 s.

Furthermore, the revenue amounts are established in one of three ways: at random, proportional to demand, or equal for all customers. Revenue is acquired from the delivery customer subsequent to the shipment of merchandise originating from the pickup customer. The quantity of goods is specified as a positive integer between 1 and 50. ### Parameter Tuning

#### 5.2.1 Parameter Tuning of ABC without Clustering

Each data instance of 100 customers was tested with different numbers of iterations: 10, 50, 100, 500, and 750. As seen in Table 1, the results of testing showed that there was no enhancement in the objective function values after 500 iterations. Thus, the maximum number of iterations was taken to be 500. The same dataset instances were tested again to select the appropriate number of solutions in the population. At the beginning, the population size was set to 30, because it provided the best results given in [7]. Then, we increased the population to different sizes: 40, 50, and 60 solutions. Table 2 presents the values of the OF after setting the number of iterations to 500. The results showed that increasing the population size to more than 50 did not lead to an improvement in the objective function value. Thus, the population size was taken to be 50. The limit value of the non-improvement counter is another important parameter of the ABC algorithm because it affects the conversion of the search process. Thus, the value selected for the limit should be suitable for the number of iterations. Therefore, we tested the limit using two values: 100 and 200. The results in Table 3 show that increasing the \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Dataset**} & \multicolumn{4}{c}{**Number of Iterations**} \\ \cline{2-5}  & **10** & **50** & **100** & **500** & **750** \\ \hline
13-100-F-S & 32,274,04 & 35,432.98 & 39,437.98 & 39,438.58 & 39,438.58 \\
14-100-F-L & 63,311.86 & 63,311.86 & 63,311.86 & 63,455.13 & 63,455.13 \\
15-100-P-S & 68,625.48 & 68,625.48 & 71,185.28 & 72,032.22 & 72,032.22 \\
16-100-P-L & 115,792.8 & 115,792.8 & 124,608.2 & 131,404.9 & 131,404.9 \\
17-100-R-S & 67,974.56 & 74,246.27 & 74,246.27 & 74,330.95 & 74,330.95 \\
18-100-R-L & 112,986.9 & 112,986.9 & 116,775.2 & 119,991.9 & 119,991.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of tuning the number of iterations of the ABC.