Application of experimental design to mixing factors and non-mixing parameters
In the field of experimental design, mixture design There are many applications in real life, such as chemical solution configuration, preparation of pharmaceutical industry preparations, etc. In the traditional mixing design, under the condition of the ingredients, the proportion of each component will be affected by each other's configuration, and even the ratio between the components is limited. The predecessors have also developed different mixing design methods for different experimental scenarios. Simplex centroid method (simplex centroid) , extreme vertex method (extreme vertices) etc.; with the increasing complexity of the industry and process, the traditional mixing design can no longer meet the existing needs, coupled with the new development of computer optimization methods such as D-optimal, I-optimal The combination of factors in experimental design is not limited to a single type of factor, so this article will use a practical case to introduce JMP custom design (Custom Design) actual use.
First, the types of mixture models that are commonly used in traditional estimation are as follows:
type of model
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Example
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Linear model
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quadratic model
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special cubic model
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full cubic model
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The common design model expressions for non-mixture factors are as follows:
In general experimental design, the admixture factor and the non-mixture factor are regarded as independent existence, but there may be interaction between the two, so we will discuss the above two cases separately.
Case 1: 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 parametric model, and c(x,z) integrates the two independent models. This model is also called a linear additive model.
The second case: there is an interaction between the two factors, the model is complex and there are many estimators
The model is as follows:
Assuming a combination of three mixing factors and two process parameters,
where f(x) is a special cubic mixture model, and g(z) is a quadratic model, so considering the interaction, the model is:
The schematic diagram of the model considering the interaction is as follows. The design of the process parameters is a cube, and the monomer (simplex) designed by the mixture is added on the vertex of the cube. , the experimental design composed is as follows:
When designing the experiment, we must pay attention, because the mixture design focuses on the predict , so when designing a model with a computer optimization method, we need to compare the designed efficiency , such as G-efficacy( Reduce the largest predicted variance in the design space ) and the mean predicted variance, which can also be compared using D-optimal and I-optimal time, to judge whose efficiency is good or bad. In the following, a few examples are introduced, and the JMP custom design (Custom Design) platform to design experiments.
case 1 : Semiconductor dry etching process
first example Note 1 As an example of a semiconductor dry etching process, this example uses a linear additive model to fit, as described below:
8 in total factor Dry Etching Experiment :
- 3 process gas : A, B, C is a mixture factor
- 5 process parameter factor : Power, Pressure, Temperature, Time, Total Flow
and given gas A and B The scale limit is shown as:
Gas C is controlled in 10% arrive 25% proportion,
The response variable is to set ER( Etch rate ) target value =100 and ER Std( Standard Deviation of Etch Rate ) The smaller the better,
The following is the response variable Y with factor X the set value table,
Response
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Goal
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ER
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Target=100
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ER Std
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Minimize
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Factor
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Low
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High
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Power
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25
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75
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Press
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100
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200
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Temp
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25
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40
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time
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30
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45
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Total Flow
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80
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120
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Gas A
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0
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1
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Gas B
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0
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1
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Gas C
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0.1
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0.25
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In Custom Design Set up separately under the platform Y and X Variable as shown 1 ,
picture 1
Gas A with gas B The proportional restriction formula can be disassembled as follows (i)(ii) , and add the restriction to the figure 2 ,
(i)
(ii)
figure 2
Considering the main effect and interaction term of the process factor and the main effect and interaction term of the mixture factor, they were added to the model, and 24 were designed. experiments, as shown in Fig. 3,
picture 3
Compare Design can be used before data collection platform to compare metrics 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 larger the value, the better ) As shown in the figure below, this example is used to compare the two designs with little 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 were screened out as shown in Figure 5 ,
Figure 5
explainable Y The ratio of the variables is shown in the figure 6:
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Image 6
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Use the Predictive Profiler with the Mixture Profiler to find the best combination of settings and workable spaces, as shown in Figure 7 with figure 8:
Figure 7
Figure 8
It can be seen from the results that ER Std R2 as well as R2 Adj Smaller and wider confidence interval of the prediction profiler, if you want to improve it later ER Std The accuracy of the prediction can be increased by using the amplification design to increase the number of experiments.
case 2 : Acrylonitrile Electromagnetic Damping Experiment
The second example references Optimum Experimental Designs. New York: Oxford University Press. Atkinson, A. C., and Donev, A. N. (1992) Example, where the response variable is Damping The damping coefficient is used to measure the electromagnetic damping of acrylonitrile, and the combination of the three mixture components is 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 categorical 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 variable, as shown in the figure 9 ,
picture 9
Add the interaction terms of all factors, in which the main effect of the wavelength factor will be automatically deleted, because there will be a collinear effect, resulting in 18 The experiment is shown in Fig. 10 ,
picture 10
Same example 1 , you can use the comparative design to compare the differences between different computer optimization methods. In this case, there are no differences, so the results are not shown.
Use Fit Model directly You can automatically bring the model items just now into the estimation as shown below 11 ,
Figure 11
Here, it should be noted that if the Fit Model is directly When you choose the effect item to be estimated, it may cause collinearity, as shown in the following figure 12 ,
picture 12
As for how to avoid 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 it is an experimental design, when selecting effect terms, JMP provides warnings and removes effect terms that may cause collinearity,
- Confirm variable correlations and handle remodeling with multivariate methods, etc.
Screening out the significant factors as shown in Figure 13 ,
Figure 13
explainable Y The ratio of the variables is shown in the figure 14 ,
Figure 14
Use the Predictive Profiler with the Mixture Profiler to find the best setting combination and workable work space, as shown in Figure 15 with figure 16
Figure 15
picture 16
summary
This article briefly introduces two different models, together with two cases, to illustrate the problems that mixture factors and non-mixture parameters may encounter when combining models. Due to space limitations, there is not much ink on the modeling process and methods. It is worth noting that when two different factor types interact, different modeling methods need to be considered. When the number of factors increases, the model will more complicated. Looking forward to familiarizing readers with JMP DOE through the description of this article Basic operation, try to use, and practically apply it to daily work.
If you are interested in mixing design, you can also refer to the following link to watch 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 Mixture Design
- https://www.statease.com/docs/v11/contents/mixture-designs/scheffe-mix-models/ Mixture Design Model Illustration
