Robust engineering is a technique for consistently producing products that meet specifications, even when unavoidable variations exist in the process.
"Stable" here refers to a state in which quality can be maintained without being affected by fluctuations in the manufacturing environment ( error factors ) such as temperature and humidity. Error factors are usually difficult to control, but many of the factors that affect product response are treated as controllable factors ( control factors ).
Therefore, in order to ensure that the product remains within the specification limits even when the error factors fluctuate, it is necessary to find and set appropriate conditions (levels) for the control factors.
In this case, JMP allows you to use the Prediction Profiler and Simulator to find the optimal conditions for control factors.
In this article, we will introduce the steps through an example.
Example: How to stabilize the hole diameter at 10mm during drilling
A material processing company has set the specification limit for hole diameters in drilling as 9.9mm to 10.1mm. It is desirable to consistently achieve the target hole diameter of 10mm.
Here, we consider the following two control factors and two noise factors (all continuous factors):
Control Factors
A: Rotation speed (rpm) [1500 – 3500]
B: Feed rate (mm/rev) [0.05 – 0.2]
Error factors (uncontrollable factors)
N1: Material temperature (℃) [15 – 30]
N2: Tool wear (mm) [0 – 0.1]
First, we used "Custom Design" to create an experimental design that included all response (hole diameter), control factors, and noise factors. The model included all main effects and two-way interactions, and the number of experiments was set to 12.
The data table shows the results of creating a plan, running an experiment, and entering the response "Hole diameter (mm)."
The response column "Hole Diameter" has the following response variable limits and specification limits:
Run "Fit Model" (or the script attached to the data table) on this data and view the report "Prediction Profiler" .
Here, the error factors "material temperature" and "tool wear amount" are both fixed to the center of the experimental range, and the conditions for the control factors that achieve the target value of 10 mm are searched for using "maximization of satisfaction."
To lock a factor to a certain value, hold down the Alt key while clicking on the factor graph and check "Lock Factor Settings ." (For checked factors, the red dotted line indicating the factor value becomes a solid line.)
The predicted profile after desirability maximization is shown below. With the noise factors fixed at the center of the factor range, the control factor settings are Speed = 1481.7 and Feed Rate = 0.17386, and the predicted response Hole Diameter is exactly 10, which matches the target response.
Note : There may not be one combination of factor settings that will achieve your target, so your analysis results may not match those presented here.
We also display a Contour Profiler to view these results in contour plots. The target response is 10, and the specification limits are 9.9 to 10.1. The red line is the Y=10 contour, and the unshaded white area indicates the area within the specification limits.
In this example, the rotation speed = 1481.7 and the feed rate = 0.17386 are points on the contour line, but they are just an example. All points on this contour line will meet the target values.
However, this setting is for the case where the noise factor is fixed at the center. In reality, the noise factor cannot be controlled and fluctuates with each experiment. Therefore, it is necessary to set a robust control factor that can achieve the target value even if the noise factor fluctuates.
Verification using a simulator
To assess the effectiveness of these factor settings against the specified specification limits and target values, use the Prediction Profiler option , Simulator .
The simulator allows you to perform Monte Carlo simulations by adding random error to the model factors and predictions. In this example, we fix the factors at their optimal settings and generate random numbers for the uncontrollable factors. The simulator allows us to see what percentage of the responses fall outside the specification limits.
Here, the error factor "Material temperature" is set to be assigned a random value from N(22.5,3), and "Tool wear amount" is set to be assigned a random value from N(0.05,0.02).
After setting the parameters, clicking the [Simulate] button will perform 10,000 random number simulations, and the percentage of those that did not fall within the specification limits will be displayed as the non-conformance rate.
In the example above, 1,132 of the 10,000 simulations were outside the specification limits [9.9 - 10.1], resulting in a non-conformance rate of 0.1132.
Notice : The value of the response variable may also be affected by random errors as well as factors in the model, but here we have not added random errors to the response variable.
This shows that when the error factors are fixed at the center, the predicted hole diameter is the target value of 10 mm when the rotation speed is 1481.7 rpm and the feed rate is 0.17386 mm/rev. However, when the error factors change, it can be seen that under these conditions, approximately 11.3% are out of specification.
For this materials company, 11.3% non-conformance is unacceptable and needs to be kept to a few percent.
Approaches to making processes robust
To make a process robust, it is effective to align the target value with the flattest part of the response surface with respect to the error factors, thereby minimizing the effect of the error factors on the process.
Mathematically, we seek the conditions under which the equation ( first derivative) obtained by first differentiating the response with respect to each error factor becomes 0. Specifically, (prediction equation Y (Equation obtained by partially differentiating with N1 ) ≒ 0 and (Prediction formula Y Add the condition that (the partial derivative of N2 ) ≒ 0. In JMP , you can set a condition by specifying a satisfaction value for the differential expression.
This approach can be implemented by specifying the columns containing the noise factors in the Profile platform found in the Graph menu.
- Select Save Columns > Prediction Formula from the red triangle in the upper left corner of the model fit report to add a prediction formula column to the end of the data table.
- Select [Graph] > [Profile] from the menu bar, specify the "Prediction formula: Hole diameter (mm)" created in step 1 as [Y, Prediction formula], check [Specify error factors] at the bottom left, and click [OK].
- In the error factor selection window, select the error factors this time: "N1: Material temperature (℃)" and "N2: Tool wear amount (mm)".
When you maximize satisfaction with the displayed profile, the system prompts you for factor settings that "bring Y closer to the target value" and "bring the partial derivative with respect to the error factor closer to 0" at the same time.
As a result of maximizing the degree of satisfaction, we can see that the graph of the prediction formula against the noise factors is flatter than before, which indicates that it is robust to fluctuations in the noise factors.
In this way, when you set an error factor in the "Profiler," a profile of the differential equation partially differentiated by the error factor is added to the prediction profiler, and the satisfaction level for this equation is automatically set so that the satisfaction level approaches 1 when the value is 0.
Furthermore, when the simulator was run with these settings, the non-conformance rate was 0.0275, a significant improvement.
summary
We have explained how to perform robust engineering using JMP using the following steps:
- Create an experimental plan including noise factors and control factors and enter response values.
- Fit the model and save the prediction formula.
- Specify the error factors in the "Profile".
- Maximize satisfaction with predictive profilers.
- The failure rate is calculated using a simulator to verify the robustness of the factor settings.
by Naohiro Masukawa (JMP Japan)
Naohiro Masukawa - JMP User Community
