## 4 Proposed Methodology Description
### Mathematical Formulation of the Problem  
According to [PERSON] [64], robust design can be realized via a set of quality engineering activities whose goal is the development of low-cost, yet high quality, products and processes. in this direction, statistically planned experiments are used in order to identify the most influential factors in final product quality. Most importantly, the above-mentioned statistical methods are used in order to optimize the nominal levels of the most influential factors. Robust engineering includes concept design, parameter design and tolerance design [65]. [PERSON]'s robust design method uses parameter design to place the design in a position where random \"noise\" DOE does not cause failure, and to determine the proper design parameters and their levels [66]. The basic idea of parameter design in [PERSON]'s robust design is to identify appropriate settings for control factors that make the system's performance robust in relation to changes in the noise factors [67]. Thus, the control factors are assigned to an inner array in an orthogonal array, and the noise factors are assigned to an outer array. [PERSON] in [68] also describes robust engineering as a method for studying large numbers of decision variables with the bare minimum  
Figure 6: Workflow of the Proposed Methodology. number of experiments required. In order to accomplish this, orthogonal arrays are utilized. Furthermore, robust engineering is taken into consideration as an approximation for the implementation/calculation of the DOE and ANOVA method and is presented and described in Figure 7. Robust engineering/statistical DOE based on the [PERSON] approach is also used, in order to calculate the minimum number of experiments required for the validation of the digital model. Following the model validation, ANOM (ANalysis Of Means) and ANOVA methods are used for the recognition of the most affecting control factors. In the left part of Figure 7, the flowchart for the DOE analysis is presented. Similarly, in the right part of Figure 7 the corresponding pseudocode for the DOE algorithm is presented. The preliminary design of the problem and the planning consists of (i) problem definition, (ii) identification of control factors based on a cause-causality diagram, (iii) level identification, (iv) orthogonal array selection, and (v) data analysis. The data connection between the digital model and the orthogonal array is performed in step 2. The third step consists of (i) data analysis and (ii) the prediction of the optimum level for each control factor. ANOVA analysis is used to identify the control factor that affects the system the most (highest percentage). Subsequently, the key control factor with the highest percentage affecting the efficiency of the system is selected. The selection of the appropriate objective functions is very important for measuring the robustness of a system and solving engineering problems in order to identify control factors that reduce the variability in a product or process by minimizing the effects of uncontrollable factors. In a Taguchi-designed experiment, noise factors are manipulated to force variability and, from the results, to identify the optimal control factor settings that make the process or product robust or resistant to variation from the noise factors. Higher values of the signal-to-noise ratio (SNR) identify control factor settings that minimize the effects of the noise factors; types of SNR are presented in Table 5.  
Figure 7: (left) DOE flowchart; (right) DOE pseudocode. The ultimate goal in the current case study is to increase the overall throughput of the system. Therefore, and according to [47; 68; 69], the \"Larger is better\" SNR is selected, as indicated in Equation (1):  
\[\text{SNR}=-10 log_{10}\left(\frac{1}{k}{\sum\limits_{i=1}^{\text{k}}}\frac{1}{y _{i}^{2}}\right) \tag{1}\]  
where SNR is the signal-to-noise ratio, k is the number of trials and \(y_{i}\) is the response in each experiment.