# An Improved Particle-Swarm-Optimization Algorithm for a Prediction Model of Steel Slab Temperature
## 4 Simulations and Discussion
### Validation of XPSO by Benchmark Test Functions  
The XPSO algorithm is compared with some popular variant PSO algorithms (PSO [32], IPSO [33], IPSO2 [30], HPSO [34], CPSO [35]) on a series of widely used optimization benchmark functions. A set of benchmark functions were selected from papers ([36; 37]). The benchmark set consisted of three main groups of benchmark functions: 4 unimodal (UM) functions, 2 multimodal (MM) functions and 3 composition (CM) functions. The UM functions (f1-f4) with a unique global best can expose the intensification capacities of different algorithms. The MM functions (f5-f6) can expose the diversification of algorithms. The CM functions (f7-f9) were selected from the IEEE CEC 2005 competition [37], which are also utilized in many papers to test the performances (balancing the exploration and exploitation inclinations and escaping from local optima) of algorithms. The mathematical formulation and characteristics of UM and MM functions are shown in Table 1. Details of the CM functions are shown in Table 2. The parameters of both the PSO algorithms and the optimization problem (2) were as follows. The specific parameter combinations for the inertia weight were \([w_{s1},w_{e1}]=[0.9,0.4]\), \([w_{s2},w_{e2}]=[0.65,0]\) and \([w_{s3},w_{e3}]=[0.55,0.05]\). The maximum and minimum velocities of the particle were \(V_{max}=0.1\) and \(V_{min}=-0.1\), respectively. In addition, the acceleration coefficients \(c_{1}\) and \(c_{2}\) were selected to be 2.5 and 1.5, respectively. The inertia weight of the HPSO varied randomly in the range (0, 1). The parameters related to jumping out of local optimal were \(s=270\) and \(\varepsilon=0.001\). The values of the regularization parameters in the optimization problem were \(\lambda_{1}=1.2,\lambda_{2}=1.0\). The maximum number of iterations for all benchmark functions (f1-f9) was selected as 8000. The dimensions of these benchmark functions (f1-f9) were selected as 10, 30 and 50. Thus, the performances of the six variant PSO algorithms can be compared in different dimensions. Each algorithm was run individually 10 times, and the average statistical error was calculated. The mean of objective values (Mean) and standard deviation of its solving error (S.D.) were chosen as the performance measures for each algorithm. The simulation results are shown in Table 3 and Figures 2-10. Table 3 shows best values in bold. \begin{table}
\begin{tabular}{c c c c} \hline
**Function** & **Name** & **Function’s Expressions** & **Search Range** & **Global opt.**1  \\ \hline f1 & Sphere & \(f_{1}=\sum_{i=1}^{n}x_{i}^{2}\) & \([-100,100]^{n}\) & 0 \\ f2 & Schwefel’s 1.2 & \(f_{2}=\sum_{i=1}^{n}\sum_{j=1}^{n}x_{j}^{2}\) & \([-100,100]^{n}\) & 0 \\ f3 & Schwefel’s 2.21 & \(f_{3}=\max\{|x_{i}|,1\leq i\leq n\}\) & \([-100,100]^{n}\) & 0 \\ f4 & Quartic Noise & \(f_{4}=\sum_{i=1}^{n}x_{i}^{4}+random[0.1)\) & \([-1.28,1.28]^{n}\) & 0 \\ f5 & Generalized & \(f_{5}=\sum_{i=1}^{n}\big{[}x_{i}^{2}-10\cos(2\pi x_{i})+10\big{]}\) & \([-5.12,5.12]^{n}\) & 0 \\  & Rastrigin & \(f_{6}=0.1\sin^{2}(3\pi x_{1})+0.1\sum_{i=1}^{n}(x_{i}-1)^{2}\big{[}1+\sin^{2}( 3\pi x_{i}+1)\big{]}\) & \([-50,50]^{n}\) & 0 \\ f6 & Penalized & \(+0.1(x_{n}-1)^{2}\big{[}1+\sin^{2}(2\pi x_{n})\big{]}+\sum_{i=1}^{n}u(x_{i},5,100,4)\) & \([-50,50]^{n}\) & 0 \\ \hline \end{tabular}
\end{table} 
Table 1: Descriptions of unimodal and multimodal benchmark functions. According to Table 3, the XPSO algorithm is superior to other PSO algorithms in terms of solution accuracy. Considering the UM benchmark functions, the results of (f1-f3) solved by XPSO are better than those of the other algorithms. For f4, the best value of Mean was obtained by the XPSO algorithm for the (10, 30, 50)-dimensional cases. The best values of S.D for the (10, 30, 50)-dimensional cases were obtained by XPSO, HPSO and IPSO2. Except f5 in the 10-dimensional case and f6 in the 50-dimensional case, the results for the MM benchmark functions by XPSO are superior to those of the other algorithms.