# A Novel Multi-Objective Harmony Search Algorithm with Pitch Adjustment by Genotype
## 2 Non-Disruptive Multi-Objective HS Algorithms
### Pitch Adjustment Variants  
Several pitch adjustment techniques have been proposed for HS [20]. In the original version, the pitch adjustment for a new variable \(x_{i}\) that belongs to a solution vector \(\vec{x}=(x_{1}x_{2}x_{3}...x_{M})\) with \(M\) design variables is given by expression (1):  
\[x_{i}=\left\{ \begin{array}{ll}HM(r,i)+rand(-1,1)\times bw,&\text{probability }(PAR)\\ HM(r,i),&\text{probability }(1-PAR)\end{array} \right. \tag{1}\]  
where \(N\) is the number of harmonies; \(r\) is a random integer from \(1\) to \(N\); \(rand(-1,1)\) is a random number generated between \(-1\) and \(1\); \(bw\) is the bandwidth; and \(PAR\) is the pitch adjustment rate. \(PAR\) and \(bw\) are preset to fixed values. This scheme is used by [PERSON] et al. [14] in the MOHS2 algorithm, where diverse problems are solved with fixed values for \(PAR\) and \(bw\) that are proportional to the range of each variable. [PERSON] et al. [26] proposed one of the most used pitch adjustment methods with fixed parameters, with a fixed value of \(PAR\), and a value of \(bw\) for each variable given by Equation (2), where \(L_{b}\) and \(U_{b}\) are the lower and upper limits of the specific variable, respectively:  
\[bw=\frac{(U_{b}-L_{b})}{1000} \tag{2}\]  
[PERSON] et al. [16] presented an improvement to HS through the linear increase of \(PAR\) and the exponential decrease of \(bw\) in every iteration, as shown in Equations (3) and (4) where \(PAR_{min}\) and \(PAR_{max}\) are the minimum and maximum values of \(PAR\), respectively; \(g\) is the current iteration; \(NI\) is the total number of iterations; and \(bw_{max}\) and \(bw_{min}\) are the maximum and minimum bandwidth, respectively. This encourages exploration in the early iterations of the algorithm and the subsequent intensification of exploitation. This scheme was applied by [PERSON] et al. [15] to the development of the MOHS3. \[PAR(g)=PAR_{min}+(PAR_{max}-PAR_{min})\times\frac{g}{NI} \tag{3}\]  
\[bw(g)=bw_{max}\times exp\left[ln\left(\frac{bw_{min}}{bw_{max}}\right)\times \frac{g}{NI}\right] \tag{4}\]  
[PERSON] et al. [7] proposed a fixed \(PAR\), and a \(bw\) that decreases exponentially depending on the range of each variable, as shown in Equation (5), where \(a\) is a positive constant in the range from \(0\) to \(1\). It generates a \(bw\) value for each variable, eliminating the parameters required in [16]. \[bw=\frac{(U_{b}-L_{b})}{g^{a}} \tag{5}\]  
In [27], [PERSON] and [PERSON] presented a pitch adjustment with decreasing \(PAR\), \(PAR_{min}\) set to \(0\) and \(PAR_{max}\) set to \(1\), as calculated in Equation (6). The pitch adjustment for \(x_{i}\) is given by expression Equation (7). The self-adaptive bandwidth \(bw\) is based on harmonic memory awareness, that is, depends on the extreme values of \(HM\), where \(min(HM)^{i}\) and \(max(HM)^{i}\) are the minimum and maximum values of the \(i\)th variable in \(HM\), respectively. Thus, the selection of \(PAR\) and \(bw\) values is eliminated. \[PAR(g)=PAR_{max}-(PAR_{max}-PAR_{min})\times\frac{g}{NI} \tag{6}\]  
\[x_{i}=\left\{ \begin{array}{ll}HM(r,i)+(max(HM)^{i}-HM(r,i))\times rand(0,1),& \text{probability }(0.5\times PAR)\\ HM(r,i)+(min(HM)^{i}-HM(r,i))\times rand(0,1),&\text{probability }(0.5\times PAR)\\ HM(r,i),&\text{probability }(1-PAR)\end{array} \right. \tag{7}\]  
[PERSON] and [PERSON], inspired by PSO, proposed the random selection of variables from the best \(HM\) vector for the pitch adjustment, eliminating the \(bw\) parameter and adding a social dimension to the algorithm [19]. This is indicated in Equation (8), where\(x_{k}^{best}\) is a random variable of the best solution vector. \(PAR\) grows linearly, as shown in Equation (3). \[x_{i}=\left\{ \begin{array}{ll}x_{k}^{best},&probability\ (PAR)\\ HM(r,i),&probability\ (1-PAR)\end{array} \right. \tag{8}\]  
As can be seen, only two proposals for pitch adjustment have been brought to the multi-objective extensions of HS.