Issue 56, October 2005

Probabilistic design is a methodology aimed at producing robust and reliable products. A common mistake made by design engineers is not considering variations in the factors that affect the reliability or performance of their product and instead using only the nominal or mean values of these factors. By understanding the probability distributions of the design parameters, environmental usage profile and manufacturing variations, the design engineer can design for a specific reliability or quality level by producing designs that are robust to variations.

Generally, the probabilistic design approach follows these steps:

1. Identify the product's failure modes.

2. Identify factors, causes (internal or external to the product) and design properties that produce these failure modes.

3. Determine the relationship between these factors and a design characteristic that describes the reliability of the product.

4. Construct distribution models that capture the variability of each of these factors.

5. Derive a distribution model that results from the distributions of these factors and the relationship that relates them to the product design characteristic.

6. Study the resulting distribution and estimate deviations from the goal or specifications.

This process leads to determining ways to create a more robust design that takes into consideration manufacturing, environmental and use variations.

A simple example that demonstrates the process of considering the variation of different design parameters and obtaining a combined effect (variation) is the area of a rectangle. If the height and length of a rectangle are distributed, the area of the item is distributed as well. In order to find the distribution of the area, we can generate random height and length values based on their corresponding distributions and then apply the equation A = H x L. A distribution can then be fitted to the resulting set of area values. After fitting a distribution model to the area A, the proportion of products that falls outside an acceptable specification interval can be estimated.

Monte Carlo simulation is a great tool for performing probabilistic design analysis, as it can be used to perform simple relationship-based simulations. The new User Defined distribution feature in Weibull++ 7 allows you to specify an equation relating different random variables. You can then determine the joint pdf for the simulated data set. In addition to probabilistic design, this type of simulation has many applications in risk analysis, quality control, etc.

Example

A company manufactures hinges which are made up of four components A, B, C and D. The next figure shows a schematic of the hinge's assembly.

The manufacturer wants to determine the percentage of hinges that would fall out of specifications. Specifically, the manufacturer wants to estimate the probability that (A+B+C) will be greater than D. For that purpose, the manufacturer collected data about the dimensions of each of the manufactured components. Of each of the components (A, B, C, D), seven units were taken from the assembly line and the following measurements (in cm) were recorded.

The first step in this analysis is to model the dimensions of each of the parts using the collected data. The part dimension measurements are entered into a Weibull++ standard folio as separate data sheets and are analyzed assuming a normal distribution and RRX as the analysis method. The parameters are:

Component A: μ_A = 2.0146, σ_A = 0.0181

Component B: μ_S = 2.0096, σ_S = 0.0249

Component C: μ_C = 30.0981, σ_C = 0.1762

Component D: μ_D = 34.8149, σ_D = 0.7121

A Monte Carlo simulation can now be performed to estimate the number of times (A+B+C) will be greater than D. Select Generate Monte Carlo Data from the Tools menu. Under Distribution, select User Defined and use the Insert Data Source button to select the A, B and C part measurements data sheets to generate 1000 data points that represent (A+B+C-D).

Weibull++ creates a Free-Form (Probit) data sheet that is made up of X data and Y position data in %. Assuming a normal distribution and RRX as the analysis method, the estimated parameters are  μ_A = -0.6807 and σ  = 0.7338.

The following is a pdf plot of (A+B+C-D).

The probability that (A+B+C) > D is equivalent to the probability that (A+B+C-D) > 0. So the area under the curve of the pdf from time = 0 to infinity—where “time” is actually the value of (A+B+C-D)—will tell us how likely it is that the product will fall out of specifications (i.e., the likelihood that (A+B+C) will be greater than D). We will use the QCP to calculate this value, as shown next. (Note that we will use the Results as Reliability option in the QCP, even though we are finding the likelihood of falling out of specifications, since the calculation we wish to perform is equivalent to finding the reliability of a randomly generated data set at time = 0.)

Therefore, (A+B+C) will be greater than D 17.68% of the time. Note that the result might vary depending on the seed value chosen by the software (which can be specified by the user).

Comparison to Analytical Results

For the simple problem described above, an analytical solution is possible to obtain. For complex relationships that involve a variety of distributions types, however, analytical solutions could become more complex or impossible and simulation remains the preferred analysis choice.

The mean of A+B+C can be derived as follows:

The standard deviation of A+B+C can be derived as follows:

To calculate the probability that (A+B+C) > D:

We calculate the test statistic:

Alternatively, we can find the P_value as follows:

Therefore, 17.25% of the times A+B+C will be greater then D, which agrees with the simulation results.

Note:

RENO, which is an advanced software specifically designed for stochastic event simulation, can be used for more advanced simulation analysis involving complex relationships, complex diagrams or flowcharts that can not be expressed using the User Defined distribution feature in Weibull++ 7. It can also be used to optimize the setting of the controllable factors to produce robust designs.
 

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