## 2 Harmony Search Algorithm and Selected Variants
### Improved Harmony Search (IHS)
Further, \(HMCR\) and \(PAR\) are assumed to be normally distributed where \(HMCR\sim N(HMCRm,0.01), HMCR\in[0.9,1.0]\) and \(PAR\sim N(PARm,0.05),PAR\in[0.0,1.0]\). The initial values of \(HMCRm\) and \(PARm\) are set at 0.98 and 0.9, respectively. Subsequently, SGHS begins with \(HMCR\) and \(PAR\) values being generated from the Normal distribution. During each iteration, the values of \(HMCR\) and \(PAR\) that corresponds to a replacement of new harmony in HM is recorded until a number of solutions are generated within the specified learning period \(LP\). Once \(LP\) is reached, the recorded \(HMCR\) and \(PAR\) values in previous iterations are averaged to obtain new \(HMCRm\) and \(PARm\) to be used in upcoming iterations. This process is repeated until the termination criterion is satisfied. As for \(BW\), the values are dynamically adapted, as follows:

\[BW_{t}= \begin{cases}BWmax-\frac{BWmax-BWmin}{MaxImp}&t<\frac{MaxImp}{2}\\ BWmin&t\geq\frac{MaxImp}{2}\end{cases} . \tag{8}\]

SGHS is outlined, as in Algorithm 4. ```
1:Set \(HMS\), \(HMCRm\), \(PARm\), \(BW\) using (8), \(LP\), and \(MaxImp\)
2:\(\chi^{\prime}_{i}=LB_{i}+U(0,1)\times(UB_{i}-LB_{i})\), \(\forall i=1,2,...,n\) and \(\forall j=1,2,...,HMS\)
3:Compute \(f(\mathbf{x}^{\prime}),\forall j=1,2,...,HMS\)
4:Initialize solution counter \(lp=1\)
5:Generate \(HMCR\) and \(PAR\) based on \(HMCRm\) and \(PARm\)
6:while (\(t\leq MaxImp\)) do
7:for each \(i\in[1,n]\)do
8:if\(U(0,1)\leq HMCR\)then
9:\(\chi^{\prime}_{i}=\chi^{\prime}_{i}\pm U(0,1)\times BW_{i}\) where \(j\sim U(1,HMS)\)
10:if\(U(0,1)\leq PAR\)then
11:\(\chi^{\prime}_{i}=\chi^{\prime}_{i}\)
12:endif
13:else
14:\(\chi^{\prime}_{i}=LB_{i}+U(0,1)\times(UB_{i}-LB_{i})\)
15:endif
16:endfor
17:if\(f(\mathbf{x}^{\prime})<f(\mathbf{x}^{\prime\prime\prime})\)then
18: replace \(\mathbf{x}^{\prime\prime\prime}\) in HM with \(\mathbf{x}^{\prime}\)
19: record the values of \(HMCR\) and \(PAR\)
20:endif
21:if\(lp=LP\)then
22: recompute \(HMCRm\) and \(PARm\) by averaging the recorded values of
23:\(HCMR\) and \(PAR\)
24: reset \(lp=1\)
25:else
26:\(lp=lp+1\)
27:endif
```
**Algorithm 4**************Self-Adaptive Global Best Harmony Search

### Intelligent Tuned Harmony Search (ITHS)

Based on the idea of sub-population approach for optimization (such as [56]) and decision-making theory, the Intelligent Tuned Harmony Search (ITHS) [57] attempts to intelligently control the exploration and exploitation in HS based on consciousness or previous experience. This approach begins by assigning the \(\mathbf{x}^{best}\) as the leader of the whole population. The leader divides the HM into two groups (sub-populations), say Group I and Group II, in order to achieve a good balance between exploration and exploitation. Group I consists of the harmonies with fitness less than or equal to average fitness and Group II vice-versa. In that sense, Group I will undergo both exploration and exploitation stages, while Group II will only undergo exploration stage. ITHS uses the same adaptation of dynamic \(PAR\) as the Self-Adaptive Harmony Search (SAHS) [53] that is given by:

\[PAR_{t}=PARmax-(PARmax-PARmin)\times\frac{t}{MaxImp}. \tag{9}\]

Algorithm 5 presents the computational steps of ITHS. ### Novel Self-Adaptive Harmony Search (NSHS)

The Novel Self-Adaptive Harmony Search (NSHS) is a HS variant that was developed by [51], being inspired by the defects that the creator found in SHS and other variants, namely IHS [50], GHS [52], SAHS [53], Dynamic Local Harmony Search (DLHS) [56], and SGHS [55]. In NSHS, the \(HMCR\) parameter is constructed based on the dimension of the optimization problem to be solved,

\[HMCR=1-\frac{1}{n+1}. \tag{10}\]

With reference to Equation (10), \(HMCR\) is set to be directly proportional to \(n\), in order to use the HM more frequently, and it lies in the interval \((0.5,1)\). The parameter \(PAR\) is removed.