## 2 Materials and Methods
### Multi-Response Optimization through Grey Relational Analysis (GRA)  
The Taguchi approach is only capable of optimizing a single response. However, when two or more responses of distinct features are involved, the Taguchi technique is limited. Thus, an optimization method called grey relational analysis (GRA) becomes a panacea. Taguchi OA was used to obtain the optimal levels of the tribological parameters. Data normalization is categorized as \"the smaller the better\" and \"the larger the better\". Let the actual sequence and the comparison sequences be \(X_{i}^{*}(k)\) and \(\varphi_{i}(k)\), respectively. \(i\) = 1, 2, 3... ; \(m\) = 1, 2, 3... and \(n\) and \(m\) represent the total number of experiments and experimental values, respectively. Data preprocessing is used to transform the actual sequence into an identical sequence. Many data preprocessing techniques can be utilized in GRA, depending upon the features of the actual sequence. Generally, the series is normalized between 0 and 1 [27].  
For this study, the target value is \"the smaller the better\". Therefore, the actual sequence is normalized using Equation (5):  
\[X_{i}^{*}(k)=\frac{max\varphi_{i}(k)-\varphi_{i}(k)}{max\varphi_{i}(k)-min \varphi_{i}(k)} \tag{5}\]  
where \(X_{i}^{*}(k)\) = normalized for the ith experiment and \(\varphi_{i}(k)\) = initial sequence of the average responses.  
After data normalization, the succeeding phase is the computation of the deviation sequence of the normalized data using the mathematical Equation (6), shown below:  
\[\Delta_{oi}(k)=|X_{0}^{*}(k)-X_{i}^{*}(k)| \tag{6}\]  
where \(\Delta_{oi}(k)\) stands for deviation, \(X_{0}^{*}(k)\) denotes the normalized data, and \(X_{i}^{*}(k)\) refers to the comparability sequence. GRC is thus estimated through Equation (7):  
\[\xi_{i}(k)=\frac{\Delta_{min}+\xi\Delta_{max}}{\Delta_{oi}(k)+\xi\Delta_{max}} \tag{7}\]  
where \(\xi_{i}(k)\) = GRC of each response, \(\Delta_{min}\) and \(\Delta_{max}\) = the lowest and the highest deviations of the individual target factor, respectively. The differentiating or identification coefficient is symbolized by \(\zeta\) and is demarcated within the range of \(\xi\epsilon[0,1]\). This is usually set at \(\forall 2\)  
Figure 3: Taguchi L\({}_{27}\)(3\({}^{4}\)) OA.  
to assign equivalent weights to every variable. As indicated in Equation (8), the GRG is then determined by taking the mean of the GRG of each output parameter:  
\[\gamma_{i}=\frac{1}{n}\sum_{i=1}^{n}\xi_{i}(k) \tag{8}\]  
where \(\gamma_{i}\) = GRG obtained for ith test run, and \(n\) = summation count of performance attributes. Following the determination of the optimal levels of the parameter, the last phase is to predict and validate the result using Equation (9):  
\[\gamma_{predicted}=\gamma_{m}+\sum_{i=1}^{q}\gamma_{0}-\gamma_{m} \tag{9}\]  
where \(\gamma_{0}\) represents the highest of the mean GRG at optimum levels of the variables and \(\gamma_{m}\) defines the average GRG. \(q\) = parameter that signifies the factors influencing the target values.  
To find out the contribution of each parameter on the wear rate, an ANOVA is applied. A large percentage contribution (%C) signifies a parameter exerting a significant influence on the responses.