## 2 Harmony Search Algorithm and Selected Variants  
The Harmony Search (HS) algorithm [34] is a well-known population-based metaheuristic algorithm. The optimization process in HS is a mimicry of the underpinning principles of jazz music orchestra, where musicians attain a pleasant harmony through several improvisation steps. HS has been successfully applied to a wide variety of real-world optimization problems, such as system reliability, robot path planning, renewable energy systems, hyper-parameter tuning of deep neural networks, intelligent manufacturing, and credit scoring (see [35; 36]); university timetabling, structural design, water distribution, and supply chain management (see [37; 38]); and, music composition, Sudoku puzzle solving, tour planning, web page clustering, vehicle routing, dam scheduling, groundwater modeling, soil stability analysis, ecological conservation, heat exchanger design, transportation energy modeling, satellite heat pipe design, medical physics, medical imaging, RNA structure prediction, and image segmentation (see [39], among others). Besides that, the implementation of HS in various parameter estimation studies indicated the potentiality of HS as an effective parameter estimation tool. Some of the notable parameter estimation problems that applied HS include parameter estimation of the nonlinear Muskingum model [40; 41], parameter estimation in vapor-liquid equilibrium modeling [42], parameterestimation in an electrochemical lithium-ion battery model [43], parameter identification of synthetic gene networks [44], and design storm estimation from probability distribution models [45]. In addition, HS was also successfully employed in human activity pattern modeling, such as disease spread and disaster response [46]. Hereinafter, the term HS represents the family of HS variants, while the standard HS is denoted SHS. The five primary steps in SHS, as outlined in [47], are as follows:  
1. **Initialization of HS parameters and the objective function**. The control parameters are Harmony Memory Size \(HMS\), Harmony Memory Considering Rate \(HMCR\), Pitch-Adjusting Rate \(PAR\), bandwidth (now known as fret width) \(BW\), and maximum improvisations \(MaxImp\). If \(f(\cdot)\) is the objective function with \(n\) decision variables \(\mathbf{x}=(x_{1},...,x_{n})\) in the range \((LB_{i},UB_{i})\), then the continuous optimization problem can be formulated as follows: \[\text{minimize }f(\mathbf{x})\quad\text{ s.t }\quad x_{i}\in(LB_{i},UB_{i}).\] (3)
2. **Initialization of Harmony Memory (HM). ** HM is a \(HMS\times n\) dimensional matrix that consists of randomly generated harmonics (candidate solutions) from the uniform distribution \(U(0,1)\) within the ranges of the decision variables. In general, it is more convenient to represent HM as an augmented matrix of order \(HMS\times(n+1)\), as follows: \[HM= \begin{bmatrix}x_{1}^{1}&\cdots&x_{n}^{1}&f(\mathbf{x}^{1})\\ \vdots&\ddots&\ldots&\vdots\\ x_{1}^{HMS}&\cdots&x_{n}^{HMS}&f(\mathbf{x}^{HMS})\end{bmatrix} ,\] (4) where each row of the HM represents the solution vector \(\mathbf{x}^{i}\), \((j=1,2,...,HMS)\) in the first \(n-\)columns, followed by the fitness that is generated from the solution vector, \(f(\mathbf{x}^{i})\). It is also common to have HM as a sorted matrix in the ascending order of the fitness value (the last column of HM). 3. **Improvisation. ** Improvisation is performed to generate a new harmony by exploring and exploiting the search space. Thus, a new harmony is randomly selected from the HM with a probability of \(HMCR\) or it is randomly generated outside of HM with probability \(1-HMCR\). If a new harmony is obtained from HM, then the harmony may be improvised by adjusting the harmony with neighborhood values that are based on \(BW\) with probability of \(PAR\) or remain as is with probability \(1-PAR\). Note that \(HMCR\) is inversely proportional to the explorative power of different search spaces, while \(PAR\) is directly proportional to the exploitation power of local search space. 4. **Update HM.