Risk Analysis and Probabilistic Design with Monte Carlo Simulation

Monte Carlo simulation can be used to perform simple relationship-based simulations. The new `User Defined' distribution feature allows you to specify an equation relating different random variables. You can then determine the joint $$pdf$$  for the simulated data set. This type of simulation has many applications in probabilistic design, risk analysis, quality control, etc. More advanced analysis, involving complex relationships, can be performed with advanced software specifically designed for stochastic event simulation such as ReliaSoft's RENO (see http://www.reliasoft.com/RENO).

Example 16
A hinge is made up of four components A, B, C, D. Seven units of each component were taken from the assembly line and the following measurements (in cm) were recorded:

$$\begin{matrix} \text{Dimensions for A} & \text{Dimensions for B} & \text{Dimensions for C} & \text{Dimensions for D} \\ \text{2}\text{.0187} & \text{1}\text{.9795} & \text{30}\text{.4216} & \text{33}\text{.6573} \\ \text{1}\text{.9996} & \text{2}\text{.0288} & \text{29}\text{.9818} & \text{34}\text{.5432} \\ \text{2}\text{.0167} & \text{1}\text{.9883} & \text{29}\text{.9724} & \text{34}\text{.6218} \\ \text{2}\text{.0329} & \text{2}\text{.0327} & \text{30}\text{.192} & \text{34}\text{.7538} \\ \text{2}\text{.0233} & \text{2}\text{.0119} & \text{29}\text{.9421} & \text{35}\text{.1508} \\ \text{2}\text{.0273} & \text{2}\text{.0354} & \text{30}\text{.1343} & \text{35}\text{.2666} \\ \text{1}\text{.984} & \text{1}\text{.9908} & \text{30}\text{.0423} & \text{35}\text{.7111} \\ \end{matrix}$$

Determine the number of times (A+B+C) will be greater than D.

Solution to Example 17
The parts dimensions measurements were entered into a Weibull++ standard folio as separate data sheets and were analyzed assuming normal distribution and RRX as the analysis method. The parameters are:

$$\begin{matrix} \text{A} & \text{B} & \text{C} & \text{D} \\ \hat{\mu }=2.0146 & \hat{\mu }=2.0096 & \hat{\mu }=30.0981 & \hat{\mu }=34.8149 \\ \hat{\sigma }=0.0181 & \hat{\sigma }=0.0249 & \hat{\sigma }=0.1762 & \hat{\sigma }=0.7121 \\ \end{matrix}$$

Based on the above parameters, a Monte Carlo simulation can 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. Choose User Defined under Distribution and use the Insert Data Source... to use the A, B and C measurements data sheets to generate 100 data points that represent (A+B+C). The new created sheet is then renamed to Simulated A+B+C.

Following the same steps, use the D measurements data sheets to generate 100 data points that represent D. The new created sheet is then renamed to Simulated D.

The two data sets that represent A+B+C and D are modeled with a normal distribution using RRX as the analysis method. Using the Test of Comparison tool, which is under Tools, the two data sets can be compared.

Therefore, the probability that (A+B+C) will be greater than D is $$100-83.67=16.23%$$  (note that the results could vary because of the randomness in the simulation.)