# Advanced Parameter-Setting-Free Harmony Search Algorithm
## 3 Advanced Parameter-Setting-Free HS
### Pitch Adjusting Bandwidth Dependent on the Object  
In addition, the pitch adjusting bandwidth was used to determine the maximum range of this adjustment level when using the existing value as the probability of HMCR and adjusting the value with the probability of PAR. However, this method was possible only when using an object, not when using iteration when defining an HMCR. In a typical HS, 0.1% of the full range was used when setting the adjustment range, \(\mathbf{b}\). However, depending on the problem given, the appropriate range was different, and in the case of a function, if the entire range is a different size, the appropriate adjustment range is also different. Due to this, there are various methods of determining pitch adjusting bandwidth [15], but most of them are determined by specifying the maximum and minimum values of the adjustment range in advance. Therefore, Equation (9) was proposed to obtain an appropriate adjustment range without specifying these values.  
\(\mathbf{b}_{\mathbf{i}}\) is the pitch adjusting bandwidth used until the HS is iterated i times the HMS and then i + 1 times the HMS. Set \(\mathbf{b}\) to any value within the entire input range until the first time HS is repeated the same number of times as HMS. U and L are the maximum and minimum values of the entire problem input range; \(Loss_{i}\) is the average of the HM results when HS is repeated i times the HMS; \(Loss_{obj}\) is the object value, and \(Loss_{start}\) is the mean of the function result stored in the HM immediately after performing the same number of times as HMS. In the case of repeating the same number of times as the first HMS, \(Loss_{0}\) uses \(Loss_{start}\).  
\[\left\{\begin{array}{l}\mathbf{b}_{\mathbf{i}}=(\mathrm{U-L})\times\frac{ Loss_{i-1}-Loss_{i}}{Loss_{start}-Loss_{obj}},\ \ \mathbf{b}_{\mathbf{i}}\geq(\mathrm{U-L})\times 0.0001\\ \mathbf{b}_{\mathbf{i}}=(\mathrm{U-L})\times\left(1-\frac{Loss_{start}-Loss_{ i}}{Loss_{start}-Loss_{obj}}\right)\times 0.1,\ \ \ \ \ \ \ \ \ \ else\end{array}\right. \tag{9}\]  
Figure 7 shows the results of the advanced PSF HS produced by HMCR using objects in the Griewank function. Even though the same function is shown in this figure, the optimum pitch adjusting bandwidth was different when the input range was different. The advanced \(\mathbf{b}\) obtained by Equation (9) proposed in this study did not produce the best value but showed good results.  
Figure 6: Advanced PSF-PAR using the HMCR value from Figure 5a.  
In the newly designed method, changing the HMCR and the PAR requires the maximum number of iterations or the object value. Therefore, the case of using the maximum number of iterations is the iteration PSF scheme and the case requiring the object value is denoted the object PSF scheme. Applying this method to HS is equivalent to changing the \"Calculate and Compare HM\" box in Figures 1-8.  
Figure 8: Expansion of the “Calculate and Compare HM” box for advanced PSF-HS in Figure 1: (**a**) iteration PSF and (**b**) object PSF schemes.  
Figure 7: Calculation of Griewank function in 32 dimensions using HS Equation (9): (**a**) results for the range of [\(-\)10, 10] and (**b**) results for the range of [\(-\)600, 600].