# Analysis and Improvement of Fireworks Algorithm
## 3 Analysis and Improvement of Fireworks Algorithm
### Opposition-Based Learning Population Initialization
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Functions** & & **FWA** & **FWA-I** & **FWA-II** & **FWA-III** & **IFWA** \\ \hline f1 & Mean & \(-1396.7\) & \(-1396.81\) & \(-1400\) & \(-1400\) & \(-1400\) \\ f2 & Mean & \(2.3\times 10^{7}\) & \(2.21\times 10^{7}\) & \(\mathbf{3.62\times 10^{5}}\) & \(3.39\times 10^{6}\) & \(4.03\times 10^{5}\) \\ f3 & Mean & \(7.2\times 10^{9}\) & \(7.0\times 10^{9}\) & \(6.76\times 10^{8}\) & \(4.92\times 10^{8}\) & \(\mathbf{1.21\times 10^{8}}\) \\ f4 & Mean & \(2.18\times 10^{4}\) & \(2.13\times 10^{4}\) & \(1.33\times 10^{4}\) & \(1.47\times 10^{4}\) & \(-1099.89\) \\ f5 & Mean & \(-997.58\) & \(-997.71\) & \(-1000\) & \(-1000\) & \(-1000\) \\ f6 & Mean & \(-815\) & \(-820.71\) & \(-856.34\) & \(-859\) & \(-872\) \\ f7 & Mean & \(-639\) & \(-646.13\) & \(-614.85\) & \(-528\) & \(-709\) \\ f8 & Mean & \(-679.06\) & \(-679.07\) & \(-679.07\) & \(-679.08\) & \(-679.13\) \\ f9 & Mean & \(-565.52\) & \(-566.13\) & \(-566.46\) & \(-566.78\) & \(-576.12\) \\ f10 & Mean & \(-464.8\) & \(-468.12\) & \(-499.63\) & \(-499.70\) & \(-499.978\) \\ f11 & Mean & \(-384.10\) & \(-385.32\) & \(-368.52\) & \(-348.19\) & \(-304.89\) \\ f12 & Mean & \(114.19\) & \(121.19\) & \(8.43\) & \(6.78\) & \(-158.02\) \\ f13 & Mean & \(191.23\) & \(185.04\) & \(165.74\) & \(159.10\) & \(-1.124\) \\ f14 & Mean & \(647.11\) & \(723.65\) & \(609.74\) & \(797.02\) & \(2644.91\) \\ f15 & Mean & \(5014.04\) & \(5012.96\) & \(4473.48\) & \(4601.22\) & \(3930.46\) \\ f16 & Mean & \(201.73\) & \(2 Figure 2: The FWA, EFWA, dynFWA, SPSO2011 and IFWA searching curves. (**a**) f2 function; (**b**) f12 function; (**c**) f22 function; (**d**) f23 function; (**e**) f25 function; (**f**) f26 function; (**g**) f27 function; (**h**) f28 function. Figure 2 shows that IFWA have faster convergence for eight functions than FWA, EFWA, dynFWA. However, in f2 and f12, the SPSO2011 is better, and in other functions, IFWA still has faster convergence. Thus, IFWA is the best one in terms of solution accuracy on most functions. #### 5.2.3 Comparison of Average Fitness Value and Average Rank

Table 3 shows comparison of average fitness value and total number of rank \(1\) for FWA, EFWA, dynFWA, SPSO2011 and IFWA. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Functions** & & **SPSO2011** & **FWA** & **EFWA** & **dynFWA** & **IFWA** \\ \hline f1 & Mean & \(\mathbf{-1400}\) & \(-1396.7\) & \(-1399\) & \(\mathbf{-1400}\) & \(\mathbf{-1400}\) \\  & Rank & \(1\) & \(3\) & \(2\) &



The results from Table 3 indicate that the total number of rank 1 of IFWA (17) is the best in the five algorithms. #### 5.2.4 Comparison of Average Run-Time Cost

Figures 3 and 4 show comparison of average run-time cost in 28 functions for FWA, EFWA, dynFWA, SPSO2011 and IFWA. The results from Figures 3 and 4 indicate that the average run-time cost of SPSO2011 is the most expensive among the five algorithm. The time cost of IFWA is the least. #### 5.2.5 Comparison of Statistical Test

To evaluate whether the IFWA results were significantly different from those of the FWA, EFWA, dynFWA and SPSO2011, the IFWA mean results during iteration for each test function were compared with those of the FWA, EFWA, dynFWA and SPSO2011. The \(T\) test [24], which is a safe and robust, was utilized at the 5% level to detect significant differences between these pairwise samples for each test function. The test2 function in Matlab R2014a was used to run the T test, as shown in Table 4. The null hypothesis is that the results of IFWA, SPSO2011, FWA, EFWA and dynFWA are derived from distributions of equal mean, and in order to avoid to increase type I error, we correct the \(p\)-values using the Holm's method, and order the \(p\)-value for the four hypotheses being tested from smallest to largest, and here we have four t tests. Thus, the \(p\)-value 0.05 is changed to 0.0125, 0.0167, 0.025, and 0.05, and then the corrected \(p\)-values were used to compare with p-values respectively. Here \(p\)-value is the result of the T test.