## 3 Object System by Using HS Theory
### HS Searching Procedure  
The HS algorithm was conceptualized using the musical process of searching for a perfect state of harmony. Musical performances seek pleasing harmony (a perfect state), as determined by an aesthetic standard, just as the optimization process seeks to find a global solution (a perfect state) as determined by an objective function. The pitch of each musical instrument determines the aesthetic quality, just as the objective function value is determined by the set of values assigned to each decision variable [11]. The searching procedure using HS theory for \(\beta_{\mathrm{opt}}\) and \(\gamma_{\mathrm{opt}}\)[13] is as follows:  
Step 1: Initialize the optimization problem and algorithm parameters. First, the optimization problem is specified as Equations (12)-(14). The HS algorithm parameters required to solve the optimization problem (i.e., Equations (12)-(14) are also set in this step: harmony memory size (number of solution vectors in harmony memory, HMS) = 6, harmony memory considering rate (HMCR) = 0.9, pitch adjusting rate (PAR) = (0.4, 0.9), and termination criterion (maximum number of searches) = 2000. Step 2: Initialize the harmony memory (HM). The solution vectors in the HM matrix are generated randomly and sorted by the values of the objective function. The HM is given by (15):  
\[HM=\left[ \begin{array}{cccc}\beta^{1}&\beta^{2}&\ldots&\beta^{HMS}\\ \gamma^{1}&\gamma^{2}&\ldots&\gamma^{HMS}\end{array} \right]^{T}. \tag{15}\]  
Step 3: Improvise a new harmony from the HM. A new harmony vector, (\(\beta^{\prime}\), \(\gamma^{\prime}\)) is generated from the HM based on memory considerations, pitch adjustments, and randomization. For example, the value of the first design variable (\(\beta^{\prime}\)) for the new vector can be chosen from any value in the specified HM range (\(\beta^{\prime}-\beta^{\mathrm{HMS}}\)). Values of the other design variables (\(\gamma^{\prime}\)) can be chosen in the same way. Here, the algorithm chooses the new value with HMCR = 0.9:  
\[\beta^{\prime}\leftarrow\left\{ \begin{array}{c}\beta^{\prime}\in\left\{ \beta^{1},\beta^{2},\ldots,\beta^{HMS}\right\},with\ probability\ HMCR\\ \beta^{\prime}\in\left(\beta_{\min},\beta_{\min}\right),with\ probability\ (1-HMCR) \end{array} \right., \tag{16}\]  
\[\gamma^{\prime}\leftarrow\left\{ \begin{array}{c}\gamma^{\prime}\in\left\{ \gamma^{1},\gamma^{2},\ldots,\gamma^{HMS}\right\},with\ probability\ HMCR\\ \gamma^{\prime}\in\left(\gamma_{\min},\gamma_{\min}\right),with\ probability\ (1-HMCR) \end{array} \right.. \tag{17}\]  
The pitch adjusting process is performed until a value is chosen from the HM. PAR of 0.4 indicates that it is possible to choose a neighboring value using 40% \(\times\) HMCR. The value (1-PAR) sets the rate of doing nothing. _Pitch adjusting decision for_  
\[\beta^{\prime},\gamma^{\prime}\leftarrow\left\{ \begin{array}{c}\text{ \emph{Yes with probability PAR}}\\ \text{\emph{No with probability }}(1-PAR)\end{array} \right.. \tag{18}\]  
If the pitch adjustment decision for \(\beta^{\prime}\), \(\gamma^{\prime}\) is \"Yes\", and \(\beta^{\prime}\), \(\gamma^{\prime}\) are assumed to be \(\beta^{\prime}\) (\(k\)), \(\gamma^{\prime}\) (\(k\)), the \(k\)th element in \(\beta\) and \(\gamma\), the pitch-adjusted values of \(\beta\) (\(k\)), (\(k\)) are:  
\[\beta^{\prime}\leftarrow\beta^{\prime}+\alpha, \tag{19}\]  
\[\gamma^{\prime}\leftarrow\gamma^{\prime}+\alpha, \tag{20}\]  
where \(\alpha=bw\times u(-1,1)\), \(bw\in\) (0.0001,1), which is an arbitrary distance bandwidth for the continuous design variable, and \(u(-1,1)\) is a uniform distribution between \(-1\) and \(1\). Step 4: Update the HM. If the objective function value of new harmony vector is larger than the worst harmony in the HM, include the new harmony in the HM and exclude the existing worst harmony from the HM. Then, sort the HM by the objective function value. Step 5: Repeat Steps 3 and 4 until the termination criterion is met.