The Profiler has identified conditions that predict a high average response. However, if the actual settings are not exactly the same each time, and the temperature, concentration, and time fluctuate slightly, knowing only the average of the nominal optimum point will not reveal the risk of failure.

Expand the prediction of a single point to an input distribution.

In U25, x and y were entered into the Profiler as fixed values. In robustness simulation, each input is represented as a distribution.

• Input mean: The set point you want to run.
• Input SD: The actual fluctuation of the equipment, process, or preparation.
• Distribution shape: Normal, uniform, empirical distribution, etc.
• Input correlation: The degree to which the two settings move together.
• Output noise: Measurement or process variation not explained by the model.

Monte Carlo repeatedly generates input combinations from this distribution, passes them through the fitted model, and approximates the output distribution. The simulation does not create new, real data, but rather propagates the results of the current model and input assumptions.

Even with the same mean, different tails result in different pass rates.

Even if the output mean is at the center of the specification, a large SD will cause the tails to extend outside the criteria. Conversely, even if the predicted mean is slightly lower, if the input sensitivity is small and the distribution is narrow, the pass rate may be high.

Before converting to a normal distribution based only on the mean and SD, check the histogram, skewness, and multimodality of the simulation table.

Input distributions need justification, not just convenience

Setting the temperature SD to 0.2°C versus 1°C significantly changes the results. Obtain justification from equipment logs, process history, measurement systems, and engineering tolerances. The rule of arbitrarily converting 1/6 of the range to the SD hides the additional assumptions that the distribution is normal and the range is ±3SD.

If the inputs are correlated, do not sample independently. For example, if the total volume is fixed, the concentrations of two components may move in opposite directions. Incorrect correlation can create many combinations that are actually impossible or miss dangerous combinations.

A Monte Carlo pass-rate estimate also has simulation error

If the estimated pass rate in N trials is p̂, the simple approximate standard error for independent simulations is:

MC SE ≈ √[p̂(1−p̂)/N]

To observe rare failures, such as a 99.9% pass rate, 500 trials do not provide sufficient tail precision. Fixing the seed allows for the reproduction of the same run, but verify that the conclusion is stable with different seeds and more iterations. Fixing the seed does not eliminate real-world uncertainty.

Capability is followed by stability and distribution assumptions

Capability indices such as Cp and Cpk summarize the process location and dispersion relative to the specification. However, the SD from a short-term synthetic simulation should not be used as the long-term process sigma. Before interpreting actual capability, check for time-series stability, independence, distribution fit, and the measurement system.

This section teaches the difference between the output distribution and the pass rate. It does not declare a design space, proven acceptable range, or long-term process capability for regulatory purposes.

In-Silico Lab: Place the tail next to the mean

1. Increase the input SD from 0.05 to 0.30 and observe the output SD and pass rate.
2. Change the LSL from 87 to 91 to check the specification dependence.
3. Change the correlation to -0.7, 0, and +0.7 and observe the changes in the output distribution.
4. Increase the number of simulations from 500 to 10,000 to check if the MC SE decreases.
5. Change the seed to observe the repeatability stability of the tail estimation.

Each row in the Lab represents a Monte Carlo draw with a synthetic input x·y and one output. The input is a correlated normal teaching model, and the output has a small amount of independent noise added. This is not the actual process distribution.

In JMP, the settings and results table are preserved together.

The input distribution, mean, SD, correlation, response noise, and number of runs of the Profiler Simulation are saved in the receipt. Simulate to Table shows each row, so you can trace back the failure conditions. After checking the specification and stability assumptions, interpret the Distribution and Capability output.

Example of a result statement

Using a prefitted quadratic model, we performed 10,000 simulations with seed 26026 and specified means, SDs, and a correlation of 0.3 for x and y. The conditional pass rate satisfying the LSL of 89 was 98.7%, and the MC SE was 0.11%p. This value is an educational estimate conditional on the specified input distribution and model and does not imply long-term capability or regulatory operating range.

Concluding the section

• The nominal optimum and the robust region are different questions.
• Monte Carlo transfers the input distribution to the output distribution through a model.
• The input SD, distribution, and correlation require empirical justification.
• The pass rate includes both the specification and the Monte Carlo error.
• Capability is not a measure of long-term performance without process stability and distributional assumptions.

In the next section, we will look at how to select candidate points in a constrained region where a standard CCD or factorial design cannot be directly implemented.

Supplementary Materials

• JMP Help · Simulator
• NIST/SEMATECH · Process Capability
• NIST/SEMATECH · Monte Carlo Methods

This article and the Lab are synthetic materials for educational purposes and should not be used as evidence for actual research, process, quality, or regulatory decisions.