G-OPTIMAL DESIGN OF NON-LINEAR MODEL TO INCREASE PURITY LEVELS OF SILICON DIOXIDE

ABSTRACT


INTRODUCTION
A series of statistical applications known as the experimental design is used to classify and quantify the correlation between the input and output variables in the process under research which aims to find the settings and conditions under which processes are optimized [1]. The experiment aims to obtain information or facts that follow the study's objectives by considering a minimum of time, cost, effort, and experimental materials [2]. The design of experiment (DOE) is a statistical method widely used in various scientific and industrial fields to support the design, development, and optimization of items and processes [3]. One use of the experimental design is the design for experiments on silicon dioxide.
Silica, often known as silicon dioxide (SiO2), is a mineral commonly found on Earth and obtained from mining materials. Silica is available as a naturally occurring resource that can be exhausted because of its many benefits in various fields. As a result, its selling price is extremely high. Efforts must be made to maintain the availability of silica and produce silica with a high purity level capable of providing opportunities in the business and industrial fields. Several studies have shown that it turns out that silica can be made from several materials that are around humans, such as rice husks, corn cobs, palm oil, bamboo leaves, and others for vegetable silica.
Based on previous research, the silica content in rice husk ash is 90-98% dry weight [4]. The second highest silica content is from bamboo leaf ash, which is 75.9% [5], while straw ash has a silica content of 75% [6]. The highest silica content is found in rice husks. Compared to mineral silica, silica made from rice husk has several advantages. It is more reactive, has more refined grains, is easier to produce, costs less, and is supported by the availability of abundant and renewable raw materials [7]. Complete combustion of rice husk ash produces silica as much as 90% to 98% dry weight. Silica with low purity can be increased through the purity process by adjusting the influencing factors.
The combination of these influencing factors will further increase silica purity. Research or experiments that want to know the effects of several experimental factors can be studied through a design that is processed using optimal design theory [8]. The optimal design is part of the design that estimates the parameters without bias with the smallest variance to produce accurate statistical inference [9]. An optimal design is needed to determine the points to be tested to optimize the design with the desired criteria.
Optimal design is part of the experimental design that allows parameter estimates without bias and has a minimum variance. The optimal design depends on the model used and the number of observations desired by estimating using optimal criteria. The G-optimal design minimizes the maximum mean squared prediction error in the experimental region by minimizing the maximum variance value of each predicted value [10].
A non-linear model is a relationship between the response variable and the explanatory variable that is not linear in the parameters. Taylor series is one approach that can be used to approach non-linear equations through linear equations. This research is a case study of silicon dioxide by using the temperature factor and the rate of temperature increase as factors that affect the rate of increase in the purity of silicon dioxide. The design used is the optimal design with G-optimal criteria on non-linear models. The optimal design leverages computational power and algorithms to generate a broad class of designs that can be used for many problems [12]. The VNS algorithm will be used to select the design point candidates. The VNS algorithm can be used to explore the neighborhood sequentially from the neighborhood with the fewest solutions to the neighborhood with the most solutions [13].

Model Used
The model used in this study is a non-linear model with temperature (°C) and rate of temperature rise (°C /min) as factors and silica purity levels (percent) as a response. The higher the temperature at which silicon dioxide is burned, the higher the purity of silicon dioxide. However, the increase in silicon dioxide purity follows a non-linear trend, with the increase becoming smaller as it approaches 100 percent purity [11]. Exponential decay is a non-linear model in chemical kinetics that approximates an asymptote value [8]. This is the non-linear exponential decay model that was used: where: ( , ) = expected value of the response 0 = constant 1 , 2 = parameters = temperature = rate of temperature rise The non-linear model is more complicated than the linear model, so it requires more analysis in the research process. One method for approaching non-linear equations through linear equations is the Taylor series. The multivariable Taylor method is written as follows: where ( , ) is a function of and , and are constants.

Steps of the Variable Neighborhood Search (VNS) Algorithm
The steps to choosing the optimal design point are as follows: 1. Make a list of candidate sets based on a combination of temperature points and rate of temperature rise.
The temperature was raised from 800 to 900 °C, and the rate of temperature rise from 1.67 °C/min to 5 °C/min, with a 0.5 °C/min increase. 2. Make a starting design (initial design) with the following steps: a. To create the initial design, choose points randomly from candidate sets.
b. Calculate and determine the estimated variance of the predicted value (SPV(x)). SPV can be calculated by [14]: c. In the initial design, look at the highest predicted variance ( ( )) value as the best option. 3. Explore the neighborhood that has been set in the application of the Variable Neighborhood Search algorithm with the following steps: a. Neighborhood N0 1) Add a point at random from the list of candidate sets to the initial design.
2) Using the same calculation as in step 2, calculate the expected variance of the predicted value of the current design. 3) Review the maximum predictive variance ( ( )) value as the optimal solution in the current design. 4) Compare the optimal design in the N0 neighborhood to the initial design, choosing the one with the minimum and maximum variance. If the design in the N0 neighborhood is not better than the first, the next neighborhood will be explored. b. Neighborhood N1 1) Replace two points from design N1 with two points from the candidate sets.
2) Using the same calculation as in step 2, calculate the variance of the predicted value of the current design. 3) Review the maximum predictive variance ( ( ))value as the optimal solution in the current design. 4) Compare the optimal design in the N1 neighborhood with the N0 design, and select the design with the minimum and maximum variance values. 4. Repeat steps 2 to 4 in the previous neighborhood. The third stage is repeated an infinite number of times. 5. Calculating the G-efficiency value in the chosen design. In the G-optimal design, the G-efficiency formula is [15]: where is the number of parameters in the model and is a lower bound for ∈ ( ).

Non-linear Model Approach
A Taylor series approach with a k th order polynomial will be used to approach the model employed in this study. The initial solution approach is used because it has the smallest mistake. The model's MSE (Mean Square Error) value is used to determine the order. The MSE values for the results of the SiO2 purity level given from the Taylor approximation simulation using 1 = 0,005 and 2 = 0,005 is as follows: Based on the simulation in Table 1, the Taylor polynomial selected is in the second order. The secondorder Taylor polynomial has a fairly small MSE and an uncomplicated model. Taylor's model uses Equation (2) with the second order as follows: ( , ) = 0,79413127 + 0,00037952 -0,000379 -1,8130105.10 -7 2 + 3,62602.10 -7 -1,8130105 10 -7 2 .

G-Optimal Design on Silicon Dioxide Purity Levels
n This study resulted in a design point for SiO2 purity levels in the temperature range of 800 o C to 900 o C with a different number of points, namely the design of SiO2 purity levels with seven design points, 12 design points, and 20 design points. The best design results are presented in the following table:

. Visualization of SiO2 design points with 7 points
The visualization of the design points obtained by design at a temperature of 800 o C to 900 o C with seven design points can be seen in Figure 1. The minor design point in the figure is at 800 o C, and the rate of temperature increase is 1,67 o C/minute, while the greatest is at 900 o C, when the rate of temperature increase is 3,33 o C/minute. The temperature points of 815 o C and 825 o C will then be changed to the lowest temperature point obtained, which is 800 o C. The point of temperature rise rate of 3,34 o C/minute and 2,50 o C/minute is converted into the point of the maximum temperature rise rate obtained, which is 3,50 o C/minute, to see if the design is better compared to the design obtained. The results are presented in Table 3.
The results from alternative 1, which replaced the temperature points of 815 o C and 825 o C for the lowest temperature point of 800 o C, resulted in a lower G-efficiency value of 85,90%. Alternative two, which is obtained by replacing the point of the temperature rise rate of 3,34 o C/minute and the rate of temperature increase of 2.50 o C/minute to the point of the maximum temperature rise rate obtained is 3,50 o C/minute, also obtains a smaller G-efficiency of 85,91%. That proved the G-optimal design at the level of SiO2 purity levels with 7 points is the best design with a large enough G-efficiency value of 96.41%. The design point for the level of purity of SiO2 in the temperature range of 800 o C to 900 o C with 12 design points and 20 design points can be seen in Table 4 and Table 5 below.    Table 4 shows the optimal design point for SiO2 purity using a temperature range of 800 o C to 900 o C using 12 design points with a G-efficiency value of 88,22%. The best design point at the level of SiO2 purity using a temperature range of 800 o C to 900 o C with 20 design points is presented in Table 5. Based on Table  5, the G-efficiency value obtained is 76,38%. Figure 2 shows the visualization results of the design points obtained by design at a temperature of 800 o C to 900 o C with 12 and 20 design points. The figure shows that the lowest temperature for the 12-point design is 800 o C, while the highest is 895 o C. The lowest temperature rise rate in the design with 12 points is 1,67 o C/minute, and the largest temperature rise rate is 4.50 o C/minute. The design with 20 points shows that the lowest temperature is 800 o C, and the highest is 900 o C. The lowest temperature rise rate in the design with 20 points is 1.67 oC/minute, and the largest temperature rise rate is 5 o C/minute.
The DETMAX algorithm on the SAS program will be used to display the findings of this study's search for the best design utilizing the G-optimal criteria. The results of the VNS algorithm compared to the optimal design of the DETMAX algorithm to test whether the VNS design gives a better optimal design. The point exchange process is carried out by the DETMAX algorithm, namely, adding a design point from the initial design point, then reducing the current design point. The optimal design results using the DETMAX algorithm can be seen in Tables 6 -8.  Based on Tables 6 -8, it can be concluded that the optimal design using the DETMAX algorithm has a lower G-efficiency value than the optimal design obtained using the VNS algorithm. Figure 2 and Figure 3 show the results of the G-optimal design points obtained using the DETMAX algorithm and the VNS algorithm in common; the design points obtained are very diverse and irregular. Based on the results of this study, it can be concluded that the greater the number of points used, the greater the value of the maximum predictive variance, so that the value of G-efficiency obtained will be smaller. The best G-optimal design is the design using 7 points with a G-efficiency value of 96.41%.

CONCLUSIONS
The best design point obtained from the G-optimal design on the relationship between temperature ( o C) and the rate of temperature increase ( o C/min) on the response variable of the level of purity of SiO2 with a temperature range of 800 o C to 900 o C is a pair of points 800 o C and 1,67 o C /min, 800 o C and 2,17 o C/min, 815 o C and 2,50 o C/min, 825 o C and 2,00 o C/min, 845 o C and 2,34 o C/min, 895 o C and 3,34 o C/min 900 o C and 3,50 o C/min with a G-efficiency of 96,41%.