## 3 Results and Discussion
### Multi-Objective Optimization Based on Grey Relational Analysis  
While ANOVA on the Taguchi orthogonal experimental design is suitable for single objective optimization, but it is ineffective in multi-objective optimization [41]. [PERSON] developed the grey relational analysis to complete the multi-objective optimization [42,43]. By combining the Taguchi orthogonal experimental design with grey relational analysis, analysis on the influence of multiple factors (all the processing parameters) and multi-objective optimization (maximize micro-hardness and minimize wear volume) are effectively able to accomplish at the same time [44]. The process of completing the multi-objective optimization in this study is summarized in three steps [45,46]. Step One is data normalization on the S/N of micro-hardness and S/N of wear volume. The data after normalization will range from 0 to 1, regardless of its original unit and data range. Equations (3) and (4) below were used to conduct the data normalization for the data set of S/N of micro-hardness and S/N of wear volume,  
\[X_{i}(SNMH)=\frac{Y_{i}(SNMH)-minY_{i}(SNMH)}{maxY_{i}(SNMH)-minY_{i}(SNMH)} \tag{3}\]  
\[X_{i}(SNWV)=\frac{Y_{i}(SNWV)-minY_{i}(SNWV)}{maxY_{i}(SNWV)-minY_{i}(SNWV)} \tag{4}\]  
where \(X_{i}\) represents the corresponding value after data normalization; \(Y_{i}\) stands for the corresponding original results; is the ith experimental run (i = 1, 2, 3,..., 16); SNMH stands for S/N of micro-hardness, and SNWV stands for S/N of wear volume; \(minY_{i}\) and \(maxY_{i}\) represent the minimum and maximum value of corresponding \(Y_{i}\). Step Two is grey relational coefficient calculation using Equations (5) and (6) for the data set of S/N of micro-hardness and S/N of wear volume,  
\[\text{GRC}_{i}(SNMH)=\frac{\min\Delta_{i}(SNMH)+\xi\max\Delta_{i}(SNMH)}{ \Delta_{i}(SNMH)+\xi\max\Delta_{i}(SNMH)} \tag{5}\]  
\[\text{GRC}_{i}(SNWV)=\frac{\min\Delta_{i}(SNWV)+\xi\max\Delta_{i}(SNWV)}{ \Delta_{i}(SNWV)+\xi\max\Delta_{i}(SNWV)} \tag{6}\]  
where \(\text{GRC}_{i}\) stands for the grey relational coefficient (GRC) for the corresponding objectives in the \(i\)th run; \(\Delta_{i}\) is the corresponding deviation between the normalized value and the reference value 1, which is obtained by equation \(\Delta_{i}(SNMH)=\left|1-X_{i}(SNMH)\right|\) and \(\Delta_{i}(SNWV)=\left|1-X_{i}(SNWV)\right|\); \(\min\Delta_{i}\) and \(\max\Delta_{i}\) represent the corresponding minimum and maximum value of \(\Delta_{i}\); \(\xi\) is the distinguishing coefficient with the range \(\xi\in[0,1]\). Because of the moderate distinguishing effect and stability, usually  
Figure 4: Main effects plot for the S/N of wear volume. 0.5 is selected for the value of \(\xi\), [47]. The result of normalization and grey relational coefficient for each run are shown in Table 9.  
The last step is calculating the grey relational grade (GRG). The goal of this study is improving the micro-hardness of the clad layer, while reducing the wear volume at the same time, so that the high performance clad layer could be obtained. Therefore, equivalent weight factor was assigned to both objectives and the GRG calculation was conducted using Equation (7),  
\[\text{GRG}_{\text{i}}=\frac{\text{GRG}_{\text{i}}(\text{SNMH})+\text{GRG}_{ \text{i}}(\text{SNWV})}{2} \tag{7}\]  
where \(\text{GRG}_{\text{i}}\) is the grey relational grade for the ith run. After all the GRGs for the 16 runs were derived, S/N conversion was conducted using Equation (1) and the converted values were sorted from largest to smallest as the rank in last column in Table 10. A larger GRG and its corresponding S/N GRG indicate attaining the preferable micro-hardness and wear volume concurrently. Thus, the best experiment setup in the L16 Taguchi orthogonal design was the 6 th run, which consisted a laser power of 1400 W, scanning speed of 7 mm/s, gas flow of 1500 L/h, and SiC powder ratio of 30%.