# A Phase Angle-Modulated Bat Algorithm with Application to Antenna Topology Optimization
## 4 Experimental Results and Discussions
### Zero-One Knapsack Problems
Moreover, the average value of the iterative data of the examplesk7-k12 in 30 optimization tests was selected, and the iterative convergence of the five algorithms was compared, as shown in Figure 5.  
\begin{table}
\begin{tabular}{c c l l} \hline
**No. ** & **D** & **Parameter (\(\boldsymbol{u,p,C}\))** & **Opt** \\ \hline \(k_{1}\) & 10 & \(w\) = (95, 4, 60, 32, 23, 72, 80, 62, 65, 46); & 295 \\  & & \(p\) = (55, 10, 47, 5, 4, 50, 8, 61, 85, 87); & \\  & & \(C\) = 269 \\ \hline \(k_{2}\) & 20 & \(w\) = (92, 4, 43, 83, 84, 68, 92, 82, 6, 44, 32, 18, 56, 83, 25, 96, 70, 48, 14, 58); & 1024 \\  & & \(p\) = (44, 46, 90, 72, 91, 40, 75, 35, 8, 54, 78, 40, 77, 15, 61, 17, 75, 29, 75, 63); & \\  & & \(C\) = 878 \\ \hline \(k_{3}\) & 50 & \(w\) = (80, 82, 85, 70, 72, 70, 66, 50, 55, 25, 50, 55, 40, 48, 59, 32, 22, 60, 30, 32, 40, 38, 35, 32, 25, 28, 30, 22, 50, 30, 45, 30, 60, 50, 20, 65, 20, 25, 30, 10, 20, 25, 15, 10, 10, 10, 4, 4, 2, 1); & 3103 \\  & & \(p\) = (220, 208, 198, 192, 180, 180, 165, 162, 160, 158, 155, 130, 125, 122, 120, 118, 115, 110, 105, 101, 100, 98, 96, 95, 90, 88, 82, 80, 77, 75, 73, 72, 70, 69, 66, 65, 63, 60, 58, 56, 50, 30, 20, 15, 10, 8, 5, 3, 1); & 3103 \\  & & \(C\) = 1000 \\ \hline \(k_{4}\) & 80 & \(w\) = (40, 27, 5, 21, 51, 16, 42, 18, 52, 28, 57, 34, 44, 43, 52, 55, 53, 42, 47, 56, 57, 44, 16, 2, 12, 9, 40, 23, 56, 3, 39, 16, 54, 36, 52, 5, 53, 48, 23, 47, 41, 49, 22, 42, 10, 16, 53, 58, 40, 1, 43, 56, 40, 32, 44, 35, 37, 45, 52, 56, 40, 2, 23,49, 50, 26, 11, 35, 32, 34, 58, 6, 52, 26, 31, 23, 4, 52, 53, 19); & \\  & & \(p\) = (199, 194, 193, 191, 189, 178, 174, 169, 164, 161, 158, 157, 154, 152, 152, 149, 142, 131, 125, 124, 124, 122, 119, 116, 114, 113, 111, 110, 109, 100, 97, 94, 94, 91, 82, 82, 81, 80, 80, 79, 77, 76, 74, 72, 71, 70, 69,68, 65, 65, 61, 56, 55, 54, 53, 47, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Parameters** & **Value** \\ \hline \multirow{6}{*}{BPSO} & Population size & 30 \\  & \(C_{1}\), \(C_{2}\) & 2 \\  & \(W\) & Decrease linearly from 0.4 to 1.2 \\  & Max iteration & 500 \\  & Max velocity & 2 \\  & Stopping criterion & Max iteration \\ \hline \multirow{6}{*}{BBA} & Population size & 30 \\  & \(F_{min}\) & 0 \\  & \(F_{max}\) & 2 \\  & \(A\) & 0.25 \\  & \(r\) & 0.5 \\  & \(\varepsilon\) & \([-1,1]\) \\  & \(\alpha\) & 0.9 \\  & \(\gamma\) & 0.9 \\  & Max iteration & 500 \\  & Stopping criterion & Max iteration \\ \hline \multirow{6}{*}{AMBA} & Population size & 30 \\  & Artificial bat (_a,b,c,d,e_) & \([-1,1]\) \\  & \(F_{min}\) & 0 \\  & \(r\) & 0.5 \\  & \(\varepsilon\) & \([-1,1]\) \\  & \(\alpha\) & 0.9 \\  & \(\gamma\) & 0.9 \\  & Max iteration & 500 \\  & Stopping criterion & Max iteration \\ \hline \multirow{6}{*}{A-AMBA} & Population size & 30 \\  & Artificial bat (_a,b,c,d,e_) & \([-1,1]\) \\  & \(F_{min}\) & 0 \\  & \(F_{max}\) & 2 \\  & \(A\) & 0.25 \\  & \(r\) & 0.5 \\  & \(\varepsilon\) & \([-1,1]\) \\  & \(\alpha\) & 0.9 \\  & \(\gamma\) & 0.9 \\  & Max iteration & 500 \\  & Stopping criterion & Max iteration \\ \hline \multirow{6}{*}{P-AMBA} & Population size & 30 \\  & Artificial bat (_a,b,c,d,e,g_) & \([-1,1]\) \\  & \(F_{min}\) & 0 \\  & \(F_{max}\) & 2 \\  & \(A\) & 0.25 \\  & \(r\) & 0.5 \\  & \(\varepsilon\) & \([-1,1]\) \\  & \(\alpha\) & 0.9 \\  & \(\gamma\) & 0.9 \\  & Max iteration & 500 \\  & Stopping criterion & Max iteration \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Initial parameters for phase angle-modulated bat algorithm (P-AMBA), amplitude angle-modulated bat algorithm (A-AMBA), angle-modulated bat algorithm (AMBA), binary bat algorithm (BBA), and binary partical swarm algorithm (BPSO) in the knapsack problem. \begin{table}
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**No. ** & **Alg.