## 3 Proposed Methodology
### Decoding the Customer Vector
Hence, although the gap values obtained by the proposed algorithm are higher than the other methods for some instances, BLS-PSO is found to reach to the best known results for most of the cases in significantly lower computational times as compared to the other algorithms. This advocates for its efficiency. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & & \multicolumn{2}{c}{820.62} & \multicolumn{2}{c}{**819.56**} & \multicolumn{2}{c}{**819.56**} & \multicolumn{2}{c}{**819.56**} & \multicolumn{2}{c}{**819.56**} & \multicolumn{2}{c}{**819.56**} \\ CMT12 & 819.56 & 60 (0.13) & 88 (0.00) & 149 (0.00) & 23 (0.00) & 10.00 (0.00) & 10.00 \\ \hline  & & 1566.32 & 1546.2 & 1546.2 & 1544.18 & 1546.36 & 1558.35 \\ CMT13 & 1541.14 & 86 (1.63) & 160 (0.33) & 121 (1.4) & 25 (0.197) & 15.54 (0.34) & 1.12 \\ \hline  & & 867.13 & 866.37 & 866.37 & 866.37 & 866.37 & 866.37 & 866.37 \\ CMT14 & 866.37 & 64 (0.09) & 99 (0.00) & 173 (0.00) & 122 (0.00) & 11.07 (0.00) & 10.00 (0.00) \\ \hline  & & – & – & – & – & – & – \\ Average & 84.1 & 163.1 & 302.58 & 48.13 & 13.33 & \\  & (1.788) & (0.874) & (0.913) & (0.084) & (0.64) & 1.52 (1.52) \\ \hline \end{tabular} _Inf_ indicates when an algorithm could not achieve any feasible solution. a indicates the results compared to those BLS-PSO has obtained better results.

\end{table} 
Table 12: _Cont._

Figure 8: Performances of different algorithms in solving Set CMT instances. (**a**) gap (%); (**b**) Computational Time (s). ## 5 Conclusions

In this article, we have proposed a bilayer local search-based Particle Swarm Optimization (BLS-PSO) incorporated with a simple chromosome structure to apply PSO operations easily. We have also proposed a decoding technique that depends on the number of vehicles needed to serve the customers which is specified in the problem statement of benchmark instances considered in this article, and then build the routes in a comprehensible solution on which we apply the proposed local searches. The first layer of the proposed bilayer local search approach searches the neighborhood of all particles to find better solution whereas the second layer is applied on the pool of the best particle of different generations. The first layer is important for exploring the neighborhoods of the particles before modifying the chromosomes by PSO operations, and the second layer gives more chances to the best selected particles to exploit more of their neighborhoods. Such a contribution helps the search strategy to obtain optimal or near optimal solutions with the expense of very short computational time, which is also evident from the experimental results obtained by applying the method on different benchmark problem sets. The proposed decoding method can also be easily integrated with other population-based metaheuristics to solve CVRP where a solution is encoded as a permutation of customer nodes. The bilayer search strategy may also enhance the performances of other nature-inspired searching strategies. Moreover, introduction of a probabilistic approach to the decoding algorithm may also contribute to generate different potential solutions. The estimation procedure of the required number of vehicles may also be enhanced by using a stochastic technique other than taking as input from the problem statement to solve generalized vehicle routing problems. We leave all these possibilities to be investigated in our future endeavors. This work was supported by Hankuk University of Foreign Studies Research Fund. [PERSON] and [PERSON] conceived and designed the experiments; [PERSON]. [PERSON] performed the experiments; [PERSON] and [PERSON] analyzed the data; [PERSON]. [PERSON] contributed analysis tools; [PERSON] wrote the paper. The authors declare no conflict of interest. ## References

* (1) [PERSON]; [PERSON]; [PERSON] Green logistic vehicle routing problem: Routing light delivery vehicles in urban areas using a neuro-fuzzy model. _Expert Syst. Appl._**2014**, _41_, 4245-4258.