In the field of experimental design, mixture design It has many applications in real life, such as chemical solution preparation and preparation of pharmaceutical industry preparations. In traditional mixture design, under the constraints of the ingredients, the proportions of each ingredient will be affected by the configuration of each other, and even the proportions of each ingredient have restrictions. Predecessors have also developed different mixture design methods for different experimental scenarios, such as simplex centroid method (simplex centroid) , extreme vertex method (extreme vertices) wait.
As the complexity of industries and processes increases, traditional mixing design no longer meets existing needs. In addition, newly developed computer optimization methods such as D-optimal, I-optimal With the development of other countries, the factor combination of experimental design is not limited to a single type of factors, so this article will use several practical cases to JMP custom design (Custom Design) Design experiments with mixing factors and non-mixing parameters to select effect items. Letâs take a look at how to start a mixing experiment in two scenarios: when mixing factors and non-mixing parameters are independent of each other and when there is interaction. and analyse.
First, the types of mixture models that are traditionally commonly used for estimation are as follows:
Common design model expressions for non-mixture factors are as follows:
In general experimental design, mixing factors and non-mixing factors are regarded as independent entities, but there may be interactions between the two, so next we discuss the above two situations separately.
The first situation: the two factors are independent of each other
The model is as follows:
f(x) is a mixture model, g(z) is a non-mixture parameter model, and c(x,z) integrates the two independent models. This model is also called a linear additive model.
The second situation: there is an interaction between the two factors, the model is complex and there are many estimation items
The model is as follows:
Assuming a combination of three mixing factors and two process parameters,
Among them, f(x) is a special cubic mixture model, and g(z) is a quadratic model. Therefore, considering the interaction, the model is:
The schematic diagram of the model considering the interaction is as follows. The process parameters are designed as a cube, and the monomer (simplex) of the mixture design is added to the vertices of the cube. , the experimental design composed of is as shown below:
When designing experiments, we must pay attention to the fact that mixture design focuses on predict , so when designing a model using computer optimization methods, we need to compare the designed efficiency , such as G-efficacy( Reduce the maximum number of predicted variations in the design space ) and the average predicted variation, which can also be used for comparison at this time D-optimal and I-optimal time to judge whose efficiency is good or bad.
The following is introduced through several examples and based on JMP custom design (Custom Design) platform to design experiments.
Case 1 : The two factors are independent of each other
first example Note 1 This is an example of a semiconductor dry etching process. This example uses a linear addition model to fit it. The description is as follows:
total 8 factorial Dry etching experiment :
- 3 process gas : A, B, C is a mixture factor
- 5 process parameter factors : Power, Pressure, Temperature, Time, Total Flow
And given gas A and B The proportion limit is as follows:
Gas C It is controlled in 10% arrive 25% proportion,
The reaction variable is set ER( Etching rate ) target value =100 and ER Std( Standard deviation of etching rate ) The smaller the better,
The following is the reaction variable Y with factors X set value table,
Response
|
Goal
|
ER
|
Target=100
|
ER Std
|
Minimize
|
Factor
|
Low
|
High
|
Power
|
25
|
75
|
Press
|
100
|
200
|
Temp
|
25
|
40
|
time
|
30
|
45
|
Total Flow
|
80
|
120
|
Gas A
|
0
|
1
|
Gas B
|
0
|
1
|
Gas C
|
0.1
|
0.25
|
In Custom Design Set up separately under the platform Y and X Variables as shown in the figure 1 ,
picture 1
Gas A with gas B The proportional restriction formula of can be disassembled as follows (i)(ii) , and add the restrictive expression as shown in the figure 2 ,
(i)
(ii)
figure 2
Taking into account the main effect and interaction term of the process factor and the main effect and interaction term of the mixture factor, respectively, were added to the model to design 24 Experiment, as shown in the figure 3.
picture 3
You can use Compare Design before collecting data Platform to compare indicators, such as Power( The higher the better ), Prediction Variance( The smaller the better ), Design Space down Prediction Variance Variety ( The smaller the better ) and Design Efficiency( by G-efficiency and I-efficiency Mainly, the bigger the value, the better ) As shown below, this is an example to compare the two designs. There is not much difference.
Power
Prediction Variance
Design Space Prediction Variance
Design Efficiency
Then use the least squares method to fit the model as shown in Figure 4
picture 4
The significant factors are screened out as shown in Figure 5 ,
Figure 5
Can explain Y The proportion of variables is shown in the figure 6. According to the selected significant effect items, the R-square of explained ER is 0.99, the adjustable R-square is 0.98, the R-square of explained ER Std is 0.87, and the adjustable R-square is 0.64, which means that the model established by these effect items can Make the next prediction. As for ER Std, the adjustable R-square is small and the confidence interval of the prediction profiler is wide. If you want to improve the accuracy of ER Std prediction in the future, you can use the expanded design to increase the number of experiments.
|
|
Figure 6
|
Use the prediction profiler with the mixture profiler to find the best setting combination and operable working space, as shown in Figure 7 with diagram 8:
Figure 7
Figure 8
As can be seen from the figure, the optimal ER is 91.3, and the ER Std is 0.84. The setting values of Power, Press, Temp, Total Flow, Gas A, Gas B, and Gas C are: 60.61, 100, 40, 45, 80, 0.9, 0, 0.1, while the blank area of the ternary diagram represents the operable operating space.
If you want to improve in the future ER Std To improve the accuracy of prediction, amplification design can be used to increase the number of experiments.
Case 2 : There is an interaction between the two factors
The second example quotes Optimum Experimental Designs. New York: Oxford University Press. Atkinson, A. C., and Donev, A. N.(1992) Example, where the reaction variable is Damping The damping coefficient is used to measure the electromagnetic damping of acrylonitrile, and the three mixture composition combinations are as follows :
CuSO4 , the ratio is between 0.2 arrive 0.8 ,
Na2S2O3 , the ratio is between 0.2 arrive 0.8 ,
Glyoxal , the ratio is between 0 arrive 0.6 ,
Contains a non-mixture category factor :
Wavelength There are three levels of wavelength : L1, L2, L3 , because the experimenter is only interested in these three levels,
Also in custom design (Custom Design) Enter under the platform Y and X Variables, as shown in the figure 9 ,
picture 9
Add the interaction terms of all factors, and the main effect of the wavelength factor will be automatically deleted because there will be collinearity effects, generating 18 The experiment is shown in the figure 10 ,
picture 10
Same example 1 , you can use comparative design to compare the differences between different computer optimization methods. There is no difference in this example, so the results are not displayed.
Use Fit Model directly You can automatically bring in the model terms just now for estimation as shown below 11 ,
Figure 11
Here, one thing to note is that if you directly use the Fit Model When you choose the effect terms to be estimated, collinearity may occur, as shown in the figure below 12 ,
picture 12
As for how to avoid the occurrence of collinearity, due to the limited space of the article, here are several reference methods:
- use stepwise Method selection model
- use JMP Pro of Generalized Regression deal with
- If you are doing an experimental design, when selecting effect items, JMP Warnings are provided and effects terms that may cause collinearity are removed.
- Confirm the correlation of variables and use multi-variable methods to process and re-model, etc.
The significant factors are screened out as shown in Figure 13 ,
Figure 13
Can explain Y The proportion of variables is shown in the figure 14 , According to the selected significant effect terms, the R-square of Damping that can be explained is 0.88, and the adjustable R-square is 0.82, which means that the model established by these effect terms can be used for the next step of prediction.
Figure 14
Using the prediction profiler together with the mixture profiler, you can find the best setting combination and operable working space, as shown in Figure 15 with diagram 16,
The optimal Damping is 9.36. The setting values of CuSO4, Na2S2O3, Glyoxal and Wavelength are 0.629, 0.37, 0 and L2 respectively. The workable space is the blank area of the ternary diagram.
Figure 15
picture 16
summary
This article briefly introduces two different models, and also uses two cases to illustrate the problems that may be encountered when mixing factors and non-mixing parameters are combined with the model. Due to space limitations, not much is discussed on the modeling process and methods. It is worth noting that when two different factor types interact, different modeling techniques need to be considered. When the number of factors increases, the model will more complicated. We hope that through the explanation in this article, readers can become familiar with JMP DOE Basic operations, try to use and apply them to daily work.
If you are interested in hybrid design, you can also refer to the following link to view more JMP DOE resource:
References:
- Note 1 for reference https://community.jmp.com/t5/Discovery-Summit-Korea-2020/Design-and-Analysis-of-a-Multiple-Response-Mixture-Experiment/ta-p/333014 Case
- https://community.jmp.com/kvoqx44227/attachments/kvoqx44227/sample-data/69/1/Mixture_with_Process.pdf Introduction to Mix Design
