In JMP's Prediction Profiler, you can use "Maximize Desirability" to find the optimal combination of factors (explanatory variables) that simultaneously satisfies the objectives of each response (larger is better, smaller is better, or match the target value) as much as possible.
However, the optimal value of the factor found is only one point, and in actual processes, it is difficult to control temperature, humidity, etc. to the exact optimal value, and it is natural to think that the factor will fluctuate around the optimal value due to environmental factors, etc.
If the factor varies, the response will vary accordingly, and so the response values will also form a distribution. If the response has a specification range (LSL: lower limit, USL: upper limit), some of the distribution may exceed the specification limits, resulting in nonconformance.
The figure below shows an example of the distribution of the response (Error1) when factor variability is assumed. Looking at the histogram, we can see that there is a certain amount of data that falls outside the specification range. In such a case, even if the factor is set to the optimal value, there is a possibility that a certain percentage of the data will fall outside the specification limits, i.e., non-conformance will occur.
Therefore, even if the optimal value obtained by "optimizing satisfaction" is a good condition on average, it may not be optimal from the perspective of staying within the specification range as much as possible .
So how can we find factor values that are as close to within the specification range as possible?
JMP has a feature called " simulation experiments " that allows you to perform multiple simulations assuming variability in factors and consider conditions that minimize the non-conformance rate (the rate that does not fall within the specification range) .
Example of optimization
Here, the following settings are made using experimental design.
The response has two errors (Error1 and Error2), and since both are errors, the target value is set to 0. However, the specification limits are set to a range of -1.5 to 1.5.
An experiment was conducted based on a response surface design, and the data table showing the experimental points and response values (Error1, Error2) obtained is shown below. Using this data, a response surface model is fitted to Error1 and Error2, respectively.
The following is a report from the Prediction Profiler using Maximize Desirability. The response Error1 and Error2 are both close to zero, which satisfies the desire for zero error. The optimal factor values were Temp1 = 73.2, Temp2 = 102.4, and Humidity = 15.6.
Evaluation of variability using a simulator
To represent the distribution of factors around their optimum values, we use the Simulator option in the Prediction Profiler, which assumes that the optimum value for each factor follows a normal distribution with a mean and a constant standard deviation.
In the simulation, 10,000 pairs of random numbers are generated from these distributions, and response values are calculated based on the response surface prediction equation. The proportion of the 10,000 response values that fall outside the specification range (-1.5 to 1.5) is calculated as the non-conformance rate .
The figure below shows an example of the simulation results. The non-conformance rate for "Error1" is 0.4051, and the non-conformance rate for "Error2" is 0.286. This means that even if the optimal values are set, there is still a possibility of non-conformance occurring at rates of around 40% and 29%, respectively.
*Since this is a simulation using random numbers, the numbers will fluctuate slightly each time you run it.
Conditions for minimizing non-conformance through "simulation experiments"
Therefore, we will conduct a simulation experiment to find factor settings that will further reduce the non-conformance rate , even if it means some deviation from the target response value (Error1=0, Error2=0).
The simulation experiment window allows you to specify various experiment settings, but note that here we have set the number of experiments to 128 and the number of simulations per experiment to 10,000.
Click [OK] and run the program. A data table with 128 rows will be output. Each row corresponds to one experiment, and each row contains the results of 10,000 simulations.
The simulation experiment here involves examining how the non-conformance rate changes when the location (mean) of the factor distribution is changed .
For example, in the first line, 10,000 random numbers are generated from a normal distribution (more precisely, a weighted normal distribution) with the mean set to Temp1 = 73.21, Temp2 = 107.9, and Humidity = 14.00 and the same standard deviation as the simulator, and the response value is calculated. The percentage of values outside the specification limits is then calculated.
As a result, the non-conformance rate for Error1 was 0.202 and the non-conformance rate for Error2 was 0.654, indicating that the "overall non-conformance rate" (where at least one of the results is non-conforming) was 0.670. The above data table shows the results of 128 similar experiments.
These 128 experimental points were selected based on a space-filling design that covers the entire experimental space widely and evenly.
Creating a defect rate model
Using the simulation results, we create a model that predicts the failure rate using three factors (Temp1, Temp2, and Humidity). In JMP, we use a Gaussian process model, which is suitable for modeling simulation experiments. Using the fitted model, we determine the factor settings that minimize the failure rate.
The objective variable is not the non-conformance rate itself, but the common logarithm of the "Log10 non-conformance rate." Using this model, we find the factor settings that minimize the "Log10 non-conformance rate" by "optimizing satisfaction."
This shows the state after fitting a Gaussian process model and using "optimization of satisfaction" to find the factor settings that minimize the "Log 10 nonconformance rate." The factor values in this case are Temp1 = 71.7, Temp2 = 106.9, and Humidity = 15.3, which are slightly different from the optimal values found at the beginning.
If we run the simulator again using these factor settings as the average, the predicted values for Error1 and Error2 will both be slightly off from 0, but the non-conformance rates will be 0.1179 and 0.119, respectively, which is a significant improvement over the initial optimization results (0.4051 and 0.286).
summary
As this example shows, when staying within specification limits is important, the conditions that best meet response targets are not necessarily the conditions that minimize nonconformance rates.
By conducting simulation experiments assuming that factors will vary, it is possible to consider factor settings that will keep the non-conformance rate low.
If we consider the perspective that "it is important to create products that meet specifications properly," the approach introduced here will be an effective method in many workplaces.
by Naohiro Masukawa (JMP Japan)
Naohiro Masukawa - JMP User Community
