Response Surface Grey Relational Analysis On The Manufacturing of High Grade Biomedical Ti-13Zr-13Nb

Optimization of the manufacturing conditions with more than one performance characteristics have been a thing of concern, especially for Response Surface Method (RSM) optimization. Hence, this study addressed this challenge by reanalyzing a data presented in a previous study using grey relational analysis (GRA) and regression analysis. Central Composite Design (CCD) of RSM with high and low values of manufacturing conditions; voltage (50, 70) V, current (8, 16) A, pulse ON time (6, 10) μs, and pulse OFF time (7, 11) μs. The manufacturing conditions for optimal biomedical Ti-13Zr-13Nb alloy were obtained to be 50V voltage, 8A current, 6 μs pulse ON time, and 11 μs pulse OFF time. It was also revealed that the mathematical model was very ecient because the modeled GRG was in consonant with the experimental one. In addition, it was also established that current was the most signicant manufacturing condition with a contribution of 47.27%. Voltage, factors interactions and residual error were insignicant on the GRG value of the titanium alloy. In conclusion, it can be deduced that the a small value of voltage within the considered settings could be used to manufacture better grade Ti-13Zr-13Nb alloy and also the small value of residual error showed the high manufacturability of the material.


Introduction
It has been shown consistently that traditional machining method has been ine cient and cumbersome in the manufacturing of a material with high strength and high temperature resistance capacity (Rizwee Despite several studies on the use of EDM in the manufacturing of materials, optimization of machining conditions for multiple performance characteristics has been a point of concern. This studies employed the use of grey relational analysis as a unique assistance for response surface methodology. Response surface method (RSM) is a collection of mathematical and statistical techniques for exploring the relationships between several inputs as design variables and one or more response variables as an outcome. RSM uses a sequence of suitably designed experiments to fnd an optimal response that is only an approximation of experimental model. This approximated model is adequate to estimate and apply, even when little is known about the process. Statistical and mathematical approaches such as RSM can be used to extract a model that present optimized operational factors (Dadrasi et  Pathak & Pandey (2020) employed pressure-less microwave sintering assisted by CCD technique to fabricate zinc-hydroxyapatite (ZHAp) biodegradable composite for load bearing orthopedic application. The CCD was used to investigate the effect and to optimize process factors, namely; wt% of hydroxyapatite, compaction pressure, and microwave sintering factors such as sintering temperature, heating rate, and soaking time on the compressive yield strength and sintered density. The regression analysis in the CCD technique brought out tthe optimum processing conditions and the conditions were validated through con rmation analysis. It was further noted that the fabricated ZHAp correlated with the human native bone required mechanical and degradation characteristics. Ebrahimi et al. (2021) employed CCD to optimize pH, temperature, and hydrothermal treatment time for high yield, size, and crystallinity. It was observed that pH is the most in uencing factor affecting the yield, size, and crystallinity of the synthesized HAp. Fern & Salimi (2021) employed CCD technique to examine the effect of processing temperatures (30-50 ˚C), stirring time (30-60 min) and stirring rates (300-500 rpm) on the crystallite size of HAp. The variance analysis from the design revealed R 2 coe cient to be 0.8736 and established processing temperature to be the most in uencing factor affecting the crystallite size of HAp.

Research Methodology
This study employed a data from Data in Brief article by . The data was only presented and analyzed using response surface methodology for individual response optimization. However, this study optimized multiple responses for better manufacturing conditions with the assistance of grey relational analysis. Table 1 shows the manufacturing conditions employed in the analysis. The analysis was done using central composite design (CCD) and it is as shown in Table 2. The breakdown of the experimental run and data are also displayed in in Table 3 and Table 4, respectively. The multiple responses considered for the optimization of the manufacturing of Ti-13Zr-13Nb alloy are electrode wear rate (EWR), surface roughness (SR), and material removal rate (MRR).

Data Analysis
The multiple responses optimization was done using grey relational analysis. First, the data was normalized using equation 1 and 2. The higher-the-better (equation 1) was chosen to normalize MRR because as high as possible metal removal rates was desired, while the smaller-the-better (equation 2) was chosen for EWR and SR because minimization was desired for both.
Note that x i (k) is the normalized of the ith experiment, and y i (k) is the initial data of the mean of each response.
Next, deviation data sequence was calculated using equation 3. The normalized data and the deviation data sequence are presented in Table 5 Δ Note that Δ oi (k), x o (k), and x i (k) are the deviation data sequence and the ideal data sequence, respectively. Note that ξ i (k) is the GRC responses calculated, which is in terms of Δ min and Δ max , the smallest and the highest deviation data. ζ denotes distinguishing coe cient (0 ∼ 1), but a coe cient of 0.5 is normally allotted.
Next, the grey relational grade (GRG) was calculated using equation 5, then the ranking was done based on the amount of GRG values. The GRC, GRG values and the ranking are presented in Table 6. The run with the highest GRG value is the optimal setting for the manufacturing of biomedical Ti-13Zr-13Nb alloy.
Note that γ i denotes GRG values for the ith experiment, n is the combined count of the responses.

Optimal settings determination
As it was mentioned that the experimental run with the highest GRG value which was ranked number 1 is the optimal setting to manufacture high grade biomedical Ti-13Zr-13Nb alloy. The 2nd run displayed the highest GRG value, and it is properly presented in Fig. 1. Based on the re ections in Table 3, the 2nd experimental run has the manufacturing conditions at 50V voltage, 8A current, 6 μs pulse ON time, 11 μs pulse OFF time. This conclusively gave the optimal settings for a sustainable manufacturing of Ti-13Zr-

Mathematical modeling and interaction of manufacturing conditions
The mathematical model of the manufacturing conditions for GRG values are presented in equation 6 using regression analysis. The modeled GRG data corresponding to its respective experimental data is presented in Fig. 2. The modeled data generated from equation 6 which is presented in Fig. 2 shows the same behavioral pattern with the experimental data. This shows the e cacy of the mathematical model.
The interaction of the manufacturing conditions relative to its corresponding GRG is also presented in Fig. 3. This shows how combinations of manufacturing conditions can give a desired GRG value. GRG = 0.892 -0.00192A -0.0217B -0.0259C + 0.0264D (6) 4.3 Signi cance of manufacturing conditions Table 7 re ects the analysis of variance (ANOVA) which highlights the contributions of each factor and their interactions on the GRG values. The factor B (Current) re ects the most contributing factor, having 47.27%, followed by pulse OFF time. This is better described in Fig. 4. The results revealed that Voltage, interactions and residual error were not signi cant on the GRG values. In other words, minimum value of voltage could produce excellent results and also the insigni cance of the residual error shows high manufacturability of Ti-13Zr-13Nb alloy.

Conclusion And Recommendation
The optimal manufacturing conditions and modeling of multiple response of clinical grade Ti-13Zr-13Nb alloy using response surface grey relational and regression analysis have been presented in this study.
The results showed that the optimal manufacturing conditions to manufacture high grade biomedical Ti-13Zr-13Nb alloy are obtained to be 50V voltage, 8A current, 6 μs pulse ON time, and 11 μs pulse OFF time. These ndings conclusively cleared up the uncertain manufacturing conditions presented in the work of . It was also shown that the mathematical model was e cacious as the modeled GRG aligned with the experimental one. Furthermore, it was also established that Current was the most signi cant manufacturing condition with a contribution of 47.27%, however, Voltage, factors interactions and residual error were insigni cant on the GRG value of the alloy. In conclusion, inference is that minimum value of voltage could be used to manufacture good grade Ti-13Zr-13Nb alloy and also the small value of residual error showed the high manufacturability of the material.

Declarations
Funding: This research did not receive any funding