## 2 Harmony Search Algorithm and Selected Variants
### Improved Harmony Search (IHS)  
The Improved Harmony Search (IHS) [50] is the prototypical HS variant. While still requiring to fine-tune the \(HMCR\) parameter, the parameters \(PAR\) and \(BW\) are made dynamic in this variant with the introduction of the \(PARmax\) (\(BWmax\)) and \(PARmin\) (\(BWmin\)) iterative parameters. Although IHS has shown better performance than SHS, this variant increased the burdensome process of setting suitable values for four parameters instead of just two in SHS [51]. Nevertheless, IHS is considered to be a breakthrough that paved the way for the development of various HS variants to date. \(PAR\) and \(BW\) are adjusted at each iteration while using:  
\[PAR_{t}=PARmin+\frac{PARmax-PARmin}{MaxImp}\times t, \tag{5}\]  
\[BW_{t}=BWmax\times(\frac{BWmin}{BWmax})^{\frac{t}{MaxImp}}. \tag{6}\]  
The computational procedure of IHS is provided, as in Algorithm 2. ```
1: Set \(HMS\), \(HMCR\), \(PAR\) using (5), \(BW\) using (6), and \(MaxImp\)
2:\(x_{i}^{\prime}=LB_{i}+U(0,1)\times(UB_{i}-LB_{i}),\forall i=1,2,...,n\) and \(\forall j=1,2,...,HMS\)
3: Compute \(f(\mathbf{x}^{\prime}),\forall j=1,2,...,HMS\)
4:while\((t\leq MaxImp)\)do
5:for each \(i\in[1,n]\)do
6:if\(U(0,1)\leq HMCR\)then
7:\(x_{i}^{\prime}=x_{i}^{\prime}\) where \(j\sim U(1,HMS)\)
8:if\(U(0,1)\leq PAR_{t}\)then
9:\(x_{i}^{\prime}=x_{i}^{\prime}+(2\times U(0,1)-1)\times BW



replaced with a random selection of best harmony of any decision variable from the HM, as follows:

\[x_{i}^{\prime}=x_{k}^{best},\quad i\in[1,n],\quad k\sim U(1,n). \tag{7}\]

In general, GHS is claimed to perform better than SHS and IHS, especially in high-dimensional optimization problems. However, [53] asserted that GHS has flaws that will cause premature convergence and the name of this variant is also said to be misleading. The most serious flaw, as noted by [51], is the frequent generation of infeasible new harmonies, whenever the upper and lower bounds of each decision variable are not identical in the given optimization problem. Hence, GHS is not considered in this paper. ### Novel Global Harmony Search (NGHS)

The Novel Global Harmony Search (NGHS) [54] adapts the swarm intelligence concept of the PSO algorithm in the improvisation step of the SHS. This approach enables the new harmony to mimic the global-best harmony in the HM. Thus, the \(HMCR\) and \(PAR\) parameters are removed and the improvisation is only dependent on the best and worst harmonies in HM. The random generation of harmony is, in fact, analogous to SHS, and the only difference is that, instead of randomly selecting a harmony with the probability of \(1-HMCR\), NGHS performs genetic mutation with the probability of \(Pm\), based on the ideas from evolutionary algorithms. NGHS is presented, as in Algorithm 3. ```
1: Set \(HMS\), \(Pm\), and \(MaxImp\)
2:\(x_{i}^{\prime}=LB_{i}+U(0,1)\times(UB_{i}-LB_{i}),\forall i=1,2,...,n\) and \(\forall j=1,2,...,HMS\)
3: Compute \(\{\mathbf{f}(y)\},i=1,2,...,HMS\)
4:while\(\{i\in\mathit{MaxImp}\}\)do
5:for each \(i\in[1,n]\)do
6:\(x_{B}=2\times x^{\prime}\) - \(x_{i}^{\prime}\) - \(x_{i}^{\prime}\) - \(x_{i}^{\prime}=min(max_{i}^{\prime},LB_{i}),UB_{i})\)
7:\(x_{i}^{\prime}=x_{i}^{\prime}=x_{i}^{\prime}+U(0,1)\times(x_{R}-x_{i}^{ \prime})\) % position updating
9:if\((U(0,1)\leq Pm)\)then % genetic mutation
10:\(x_{i}^{\prime}=LB_{i}+U(0,1)\times(UB_{i}-LB_{i})\)
11:endif
12:endfor
13:if\(f(\mathbf{x}^{\prime})<f(\mathbf{x}^{\prime})\)then
14: replace \(\mathbf{x}^{\prime}\) in HM with \(\mathbf{x}^{\prime}\)
15:endif
16:endwhile
```
**Algorithm 3**Novel Global Harmony Search

### Self-Adaptive Global Best Harmony Search (SGHS)

The Self-Adaptive Global Best Harmony Search (SGHS) [55] aims to improve the GHS [52] in terms of avoiding getting trapped at local optima. In this approach, \(HMCR\) and \(PAR\) are dynamically adjusted to a suitable range after a number of iterations by tracking their previous values that allowed for the replacement of new harmony in HM.