# Calibration of GA Parameters for Layout Design Optimization Problems Using Design of Experiments
## 5 Case Study on Layout Design Optimization  
The procedure we decided to apply to analyzing the influence of genetic parameters on the final solution quality for reorganization of machines and parts into cells is an experimental design method developed by [PERSON]. Here we established a P-diagram (see Figure 2) that identifies the inputs and outputs of the system together with control and noise factors. The [PERSON] experimental method consists of performing selected experiments to study the influence of several operating factors on output-parameter values. [PERSON] separates the factors into two domains: control and noise factors. The difference between these two factor groups is that we cannot control them directly. The elimination of the noise factors is often impractical and impossible so the [PERSON] method seeks to minimize the effect of noise, using optimal levels of important control factors based on the concept of robustness [52]. These three fundamental control factors are the (A), probability of mutation (B), and balance weight factor (C). The levels of each set of control factors are shown in Table 1.  
Based on the principles of the genetic algorithm, these control-factor values are decided taking a reference from the optimization of control parameters for a genetic algorithms using an image registration problem [53]. In the evaluation of a final feasible solution, there also exist the so-called noise factors that have an impact on the results. These factors cause deviation in the search space size and the noise factors, such as the size of the machine-part matrix (D). The offered factors are composed of individual levels, wherein each level is completely independent. This particular case is under consideration with three levels of control and two levels of noise factors. It is proposed to apply Taguchi quality concept for the assessment of the final solution, and, in this context, the L9 orthogonal array has been chosen. The results of the experiments are subsequently transformed into a signal-to-noise ratio. For reducing the variability of solutions around a target, the smaller-is-better S/N ratio (SNR) calculation is applied [54]:  
\[SNR=-10\log[\frac{i}{n}\sum_{i=1}^{n}Y_{i}^{2}] \tag{15}\]  
A signal-to-noise ratio is a measure of robustness that can be used to identify the control factor settings that minimize the effect of noise on the response. In the parameter design, there are two factors: the signal factor that can be controlled and the noise factor that is too expensive to control. In this research work, the signal factors are A, B, and C, wherein the noise factor is the exceptional element. \(Yi^{2}\) is obtained from the mean spare deviation (MSD) of signal factors and of exceptional elements. Here the '\(n\)' represents the number of experiments that are performed using [PERSON]'s experiment design. Higher values of SNR will control the noise factors that lead to the minimization of the objective function. Since, in this research work, the objective is to minimize the combination of two functions, it is recommenced from [PERSON]'s SNR principle that the'smaller is better' is suitable for this approach. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Control Factors**} & \multirow{2}{*}{**Description**} & \multicolumn{3}{c}{**Levels**} \\ \cline{3-5}  & & – & **0** & **+** \\ \hline A & Probability of crossover (\(P_{c}\)) & 0.3 & 0.6 & 0.8 \\ B & Probability of mutation (\(P_{m}\)) & 0.01 & 0.05 & 0.1 \\ C & Balance weight factor (\(\alpha\)) & 0.4 & 0.6 & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Input signal factors and levels. Figure 2: Parameter diagram for a genetic algorithm. The stopping criterion used to test algorithms was set at a number of generations [55], fixed to min 120. In order to conduct the experiment, we used a set of four instances and implemented the algorithms in PHP script. As a common performance measure, the number of exceptional elements (EE) has been used.