## 3 Proposed Methodology
### Decoding the Customer Vector
```
Input \(P\): a pool of global best solutions Output \(P\): The refined pool of solutions Procedure 1. for each solution \(s\) in \(P\)do
2. if \(s\) is the best solution in \(P\)then
3. add \(s\) to \(P\)'
4. else if \(s\) was updated in the previous iteration then
5. add \(s\) to \(P\)'
6. endif
7. endfor
```

**Algorithm 2** Refining the pool of solutions

## 4 Computational Result Analysis

We have implemented the proposed bilayer local search enhanced particle swarm optimization (BLS-PSO) in MATLAB R2017a environment on a personal computer with a processor Intel(r) Pentium(r) Dual-Core CPU T4300 @ 2.10 GHz (only single core is used), memory (RAM) of 2.0 GB, and Windows 10 Education version. We have considered seven sets of capacitated vehicle routing benchmark problems of the well-known CVRPLIB problem set which are also used for the verification of many PSO-based proposals [41], can be downloaded from [[http://vrp.atd-lab.inf.puc-rio.br/index.php/en/](http://vrp.atd-lab.inf.puc-rio.br/index.php/en/)]([http://vrp.atd-lab.inf.puc-rio.br/index.php/en/](http://vrp.atd-lab.inf.puc-rio.br/index.php/en/)). The benchmark problem sets are called Set A, Set B, Set P [55], Set E, Set F, Set M [56] and Set CMT [57]. The problems in the Sets A, B, E, F, M and P specify only capacity constraints for the vehicles, but not any constraint on the distance travelled by a vehicle. Total number of customers varies from 15 to 199 in these sets, and locations of the customers, except Sets B and M, are generated in random scattered manner or, following semi-clustered fashion [52]. The problems in Set B and M are generated in a clustered layout [52]. Though some researchers [41] pick a subset of the 92 problems to demonstrate the performances of their proposed algorithms, we have considered all the problems to show the real portrayal of the performance of our proposed BLS-PSO on a variety of problems. [PERSON] et al. [41] considered all the 14 problems of the CMT problem set. CMT1, CMT2, CMT3, CMT4, CMT5, CMT11 and CMT12 consider only the capacity constraint for the vehicles whereas CMT6, CMT7, CMT8, CMT9, CMT10, CMT13 and CMT14 have additional restrictions on the maximum travelling distance of a vehicle. In CMT, problem sets include 50 to 199 customers, whose locationsare randomly scattered in CMT1 to CMT10 problem sets while clustered locations are set in CMT11 to CMT14. ### Setting the Parameters

A fundamental characteristic of metaheuristic algorithms is that their performance is greatly parameter value intensive. Quality of solution obtained and required computational time to solve a problem critically depend on the parameter sets. As optimizing the parameter sets for any metaheuristic algorithm is itself an optimization problem, we have run all the problems for different parameter sets and selected the best result achieving parameter set for this article. In the parameter set optimization step, there are several parameter values to be decided such as population size or the number of particles \(PS_{\textit{Particle}}\), maximum number of iterations, \(T\), constants in PSO equation \(k_{1}\), \(k_{2}\) and \(k_{3}\), termination criteria and so on. We have performed a set of experiments to fix the parameter sets to obtain the best out of the proposed bilayer local search-based particle swarm optimization (BLS-PSO) algorithm. The experimental designed setup for the experiments are portrayed in Table 3.

Then, \(k_{1}\) value is predicted via Equation (12) as used in [58]. \[k_{1}=\left(0.1-\left(1-\sqrt{1-2\times t/T}\right)\times\frac{1}{2}\right) \tag{12}\]

The termination criteria used in our approach is that the program run until the maximum number of iterations, \(T\). For BLS-PSO, we have reported the average of the performances of 20 runs for the best combination of \(k_{2}\) and \(k_{3}\) for a problem, unless specified otherwise.