## 3 Proposed Methodology
### Decoding the Customer Vector
The experiments show that performance of the proposed algorithm get improved with increasing both population size, \(PS_{\textit{Particle}}\) and maximum number of iteration, \(T\) portrayed in Figure 5. It is observed that even if the \(PS_{\textit{Particle}}\) and \(T\) is increased up to certain limit, the proposed algorithm can find results equal to best known solutions at most for 85 benchmark instances. When the setting \(PS_{\textit{Particle}}\) is equal to the number of nodes in respective problem statement and \(T\) is 50 or more it can find the results equal to the best-known solutions for maximum 85 benchmark problems. Not finding the outputs equal to the best-known solutions for the remaining problem instances are likely due to having two reasons either the small population size or the number of maximum iteration which is set as low as 100. However, as an approximate algorithm is expected to find a near optimal or, a \"good\" solution within a small computational time, we have chosen the \(PS_{\textit{Particle}}\) as equal to \(n\) and \(T\) as 50. For setting the value of \(k_{2}\) and \(k_{3}\), we have run the program again with the parameter setting of \(PS_{\textit{Particle}}\) and \(T\) as mentioned above. In these experimental setups, for a particular \(k_{2}\) we have run the program for all \(k_{3}\) values. For any setup, the number of best known results found is marked on the pie chart in Figure 6. Close observation portrays that there is no significant converging point to fix the value of \(k_{2}\) and \(k_{3}\) to obtain the best performance of the proposed algorithm. However, the computational time required for one complete run of the proposed algorithm for an instance is not significant. Thus, experiments for all the combinations of \(k_{2}\) and \(k_{3}\) for each instance is performed thoroughly and the best result achieved by the proposed BLS-PSO is reported in later subsections. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(PS_{\textit{Particle}}\) & \(T\) & \(k_{2}\) & \(k_{3}\) \\ \hline \{\(\ln\)/101, \(\ln\)/81, \(\ln\)/61, & \multirow{2}{*}{\(\{10, 20, 50, 60, 80, 100\}\)} & \{between 0 and 1 & \{between 0 and 1 \\ \(\ln\)/41, \(\ln\)/31, \(\ln\)/21, \(\ln\)} & \{with 0.1 interval\}\} & \{with 0.1 interval\} \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Different values of parameters used in experimental setup. ### Result Analysis

Tables 4-10 present the computational results obtained by BLS-PSO for several benchmark problem sets, where \(m\) represents the required number of vehicles for the corresponding problem

Figure 5: Variation in convergence of BLS-PSO depending on different \(PS_{\text{\emph{Particle}}}\) & \(T\). Figure 6: Scattered variation of convergence of the proposed BLS-PSO depending on different values of parameters \(k_{2}\) and \(k_{3}\). instance, _Cost_ reports the best fitness obtained by an algorithm along with the corresponding _gap_ in percentage, representing the quality of a solution, calculated as

\[gap=\ \frac{Cost_{BLS-PSO}-BKS}{BKS}\times 100\% \tag{13}\]

where \(Cost_{BLS-PSO}\) is the best solution obtained by the proposed method and \(BKS\) is the best-known solution for the same instance reported in the literature. \(AvgT\) is average of CPU times required by 20 individual runs for particular \(k_{2}\) and \(k_{3}\) value to reach the corresponding best solution. As we have mentioned, \(k_{2}\) and \(k_{3}\) values are not same for all the problems, we have mentioned the specific \(k_{2}\) and \(k_{3}\) values for each instance. Although the values of \(k_{2}\) and \(k_{3}\) tabulated in the tables are not mandatorily the only combination to find the results rather there may many other combinations giving the same results. We have mentioned only one combination here for the consistency in the presentation of the tables.