Volume 6, Issue 2

Reliability Edge Home

The invention of the Monte Carlo simulation method is credited to Stanislaw Ulam, a Polish-born mathematician who worked for John von Neumann on the United States' Manhattan Project during World War II. He devised the method, which involves the use of repeated experiments to estimate a solution, in 1946 while pondering the probabilities of winning a card game of solitaire.

To begin to understand how the Monte Carlo simulation method can be used to analyze probabilistic problems, consider a simple example. To estimate the probability of rolling a given number with a single die, one option is to use a computer to simulate a large number of dice rolls (randomly generating a number from 1 to 6 during each simulation) and count the number of times that the number appears. The percentage of times that the number appears across all simulations provides an estimate of the probability.

This method can be applied to solve much more complex problems in a variety of areas, including risk analysis, safety analysis, reliability modeling, maintenance planning, optimization, operational research, financial analysis, etc. The following examples demonstrate three possible applications. ReliaSoft's new RENO software is used to perform each analysis.

Example 1 - Warranty Prediction Based on Variable Usage
Reliability analysis for a certain snow blower yielded a Weibull life distribution with Beta = 2 and Eta = 400 hours. Although the reliability of the snow blower is known, the usage in various regions of the country is variable. For this example, suppose that the snow blower will be deployed in a part of the country where the probability that it will snow on any given winter day is 1 in 10. If it snows, then the snow depth is normally distributed with Mean = 6 inches and Standard Deviation = 2. Furthermore, the area of the driveways in that region is also normally distributed with Mean = 300 square feet and Standard Deviation = 40.

If the snow blower clears 30 cubic feet an hour, what would the reliability of the blower be over 100 winter days?

Simulation Solution
It is possible to create a flowchart that models the reliability and usage scenario described for the snow blower. This model will rely on a collection of defined variables that represent the probabilities and other characteristics for the given scenario. There are likely to be many different ways to model any given scenario. One approach to solving this problem would require the definition of the following variables:

Probabilities

• The probability that it will snow on any given winter day: 1 in 10 or 0.10

Random Variables

• The amount of snowfall on the days that it snows: Normal (6, 2)
• The area of the driveways in the region: Normal (300, 40)

Equations

• The snow blower usage for 100 winter days, which can be calculated based on the amount of snowfall, the area of the driveways and rate at which the blower clears the snow: (((Snowfall_inches/12)*Area_to_clean_ft2)/30)

A flowchart model can then be constructed as shown in Figure 1.

Figure 1: Flowchart Model for Snow Blower Example

In this model, the simulation determines whether it snowed on a given day by randomly generating a number from 1 to 10 (uniform probability) and then comparing it to the fixed probability of snow on any given winter day (0.10). If the randomly generated number is less than or equal to the fixed probability, this indicates that it snowed and the simulation uses the equation described above to calculate the snow blower usage hours for that day. Figure 2 shows the Conditional Block construct that is used in the RENO software to instruct the simulation to perform this check.

Figure 2: The Conditional Block from the RENO Flowchart passes the calculated snow blower usage hours on days when it snows

The Weibull Reliability function returns the reliability for a given time based on a Weibull distribution with specified parameters. The average hours of snow blower usage that are estimated by the simulation can then be used as the Time input to this function, as shown in Figure 3. The reserved keyword IN inserts the usage value from the previous construct in the flowchart.

Figure 3: The Weibull Reliability function estimates reliability for the given usage hours

When this flowchart model has been simulated 1,000 times, the reliability for the snow blower under the specified usage conditions is estimated to be 98.5975%.

Example 2 - Risk Analysis: Manufacturing Variability
A certain chemical is added during the production of a suntan lotion and the manufacturing process contains some minor variability. The concentration of chemical X is normally distributed with Mean = 0.005 and Standard Deviation = 0.0001 ml of chemical X per ml of lotion. The amount of lotion applied by a person also varies and is known to be normally distributed with Mean = 100 ml and Standard Deviation = 20 ml. It is suspected that 10% of the population may, depending on the amount used, have an allergic reaction to chemical X. From the 10% who are susceptible, the probability that someone will have an allergic reaction is a function of the amount of the chemical applied. This probability is given by a Weibull distribution with Beta = 2.5 and Eta = 1.8.

Determine the percentage of the population that is expected to experience an allergic reaction.

Simulation Solution
One approach to solving this problem via simulation would require the definition of the following variables:

Random Variables

• The concentration of the chemical in the lotion: Normal (.005, .0001)
• The amount of lotion applied by a user: Normal (100, 20)

Equations

• The amount of the chemical applied by a user, which can be calculated by multiplying the concentration of the chemical by the amount of lotion applied

Probabilities

• The probability that the user will have a reaction, which is based on the amount of chemical applied and Weibull (2.5, 1.8)

A flowchart model can then be constructed, as shown next.

In this model, the simulation randomly generates values for the concentration of the chemical and the amount of lotion applied based on the specified distributions and parameters. It then calculates the amount of chemical applied by the user based on the specified equation. Finally, it determines the probability of a reaction based on the specified distribution/parameters and the amount of chemical applied. Figure 4 shows the construct used within the RENO flowchart to generate the desired result as a percentage.

Figure 4: RENO construct to estimate the percentage of users who experience an allergic reaction

When this flowchart model has been simulated 2,000 times, the percentage of the population that will experience an allergic reaction to the suntan lotion is estimated to be .4203%.

Example 3 - Financial Analysis of Investment Strategies
In planning for your retirement, let's assume that you are choosing to invest some of your money in the bank and some in the stock market (NASDAQ and tied to the NASDAQ Composite Index). After researching the two investment vehicles, you determine the following:

• The interest from the bank investment follows a generalized gamma distribution with Mean = 1.7406, Standard Deviation = 0.1141 and Lambda = 4.3106.
• The profit from the NASDAQ investment follows a normal distribution with Mean = 13 and Standard Deviation = 28.

Conclusion
As this article demonstrates, simulation can be a powerful tool for analyzing complex problems in a variety of areas. The RENO software provides a visual, intuitive and flexible way to construct flowcharts to model any probabilistic scenario and then use simulation to estimate results of interest. More information is available on the Web at http://RENO.ReliaSoft.com.