# Pilot Sequence Allocation Schemes in Massive MIMO Systems Using Heuristic Approaches
## 4 Proposed Solutions
### Variable Neighbourhood Search
The argument here is that increasing the minimal SE may result in a reduction in terms of outage probability. Thus, Table 4 presents the minimal SE for different combinations of maximum number of iterations and for each algorithm. It is worth noting that only the optimal heuristic parameters were used in this simulation. Furthermore, it is clear that the tested heuristics are divided into two groups when we compare their performance: the low performing group includes RA, BPSO and GA while the high performing group is composed of SPV-PSO, SPV-VNS-PSO, SPVG-PSO and SPVG-VNS-PSO. Besides the aforementioned fact, one can also imply that the average variation in performance among all algorithms in the high performance bracket when the number of iterations is increased from 50 to 100 is less than 2%. Additionally, even though Table 4 shows that RA, BPSO and GA had the same performance despite the change in the number \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{3}{c}{**Max. \# of Iterations (\(I\))**} \\ \cline{2-4}  & **50** & **100** & **200** \\ \hline RA & \(7.56\times 10^{-5}\) & \(7.56\times 10^{-5}\) & \(7.56\times 10^{-5}\) \\ BPSO & \(7.56\times 10^{-5}\) & \(7.56\times 10^{-5}\) & \(7.56\times 10^{-5}\) \\ GA & \(7.56\times 10^{-5}\) & \(7.56\times 10^{-5}\) & \(7.56\times 10^{-5}\) \\ SPV PSO & 1.4143 & 1.4329 & 1.4246 \\ SPV VNS PSO & 1.3762 & 1.3364 & 1.3239 \\ SPVG PSO & 1.3897 & 1.3794 & 1.3956 \\ SPVG VNS PSO & 1.3191 & 1.3419 & 1.2982 \\ \hline \end{tabular}
\end{table} 
Table 4: Minimal spectral efficiency (in \(\text{bps}/\text{Hz}\)) using the optimal parameters for each heuristic (\(\omega=0.25\), \(v_{\max}=2\), \(VNS=0.30\), \(T_{m}=0.50\) and \(K=20\)). \begin{table}
\begin{tabular}{c c} \hline \hline
**BPSO Parameters** & **Tested Value(s)** \\ \hline Inertia (\(\omega\)) & \(\{0.1,0.25,\mathbf{0.5}\}\) \\ Max. Velocity (\(v_{\max}\)) & \(\{1,2,3\}\) \\ Coefficient (\(c_{1},c_{2}\)) & \(\{(\mathbf{1},\mathbf{1}),(2),(2),(3,3)\}\) \\ Max. \# of Iterations (\(I\)) & \(\{50,100,\mathbf{200}\}\) \\ Population Size (\(N\)) & \(2\times K\) \\ \hline
**SPV(G)-PSO Parameters** & **Tested Value(s)** \\ \hline Variable neighborhood search (VNS) & \(\{0.15,0.30,0.60\}\) \\ Max. \# of Iterations (\(I\)) & \(\{50,100,\mathbf{200}\}\) \\ Population Size (\(N\)) & \(2\times K\) \\ \hline
**GA Parameters** & **Tested Value(s)** \\ \hline Mutation Rate (\(T_{m}\)) & \(\{0.25,\,0.50,\mathbf{0.75}\}\) \\ Max. \# of Generations (G_{\max}\) & \(\{50,100,\mathbf{200}\}\) \\ Population Size (\(N\)) & \(2\times K\) \\ \hline \end{tabular}
\end{table} 
Table 3: Set of tested heuristic parameters. The parameters in bold are the ones which resulted in the best system spectral efficiency. of iterations, the average difference in performance was in the order of 0.0013% when the number of iterations rise from 50 to 100. Finally, as it is commonplace, increasing the number of iterations directly impacts the computational time of all the algorithms but RA. Hence, we present Figure 3 which shows the average time needed to compute the solution. At this point, it is clear when the performance results are combined with the execution time results that the best approach is to use as few iterations as possible in each of the algorithms due to the high cost and low increase in performance. From now on, all the simulations use the optimal parameters for each heuristic, and the number of iterations is set to 50. As one may observe, each algorithm has a set of parameters that need to be adapted to the discussed problem. In order to find the best parameters, we considered a macro-cell scenario scenario with \(K=20\) users per cell of \(R=1000\) m in radius and a maximum number of iterations \(I=50\) to all heuristics. For the PSO-based algorithms, we first analyzed the inertia coefficient \(\omega\). Figure 4 shows the results which demonstrate that \(\omega=0.25\) is the best configuration for both BPSO and SPVG-PSO, while \(\omega=0.1\) is the best option for SPV-PSO.