Operation of the simulation on the basis of the 'Monte Carlo' method

Introduction

Simulation can be loosely translated to 'pretending', or in relation to our situation 'calculating an accurate reflection of reality'. Simulation is often times used in situations where the complexity and multiplicity of calculations make it impossible to do the calculations analytically.

Complex situations such as described above are found in Maintenance as well. Estimating the expected costs of a single component of which it is certain that it is broken only once every 5 years, is not particularly difficult. However, suppose that the failure behavior is not as clear. How can you estimate the costs when the expected life span with a 70% certainty is between 4 and 6 years? How about when each quarter of the year condition based measurements are done and based on the extreme limit (Profile left on tires from a car for example), you have to decide when to do preventive maintenance. Furthermore to add to the complexity, what if the time of replacement depends on other parts as well? If an organization wants to know what the optimal time to replace a certain part is in relation to risks, costs and safety is it still possible to calculate this with a set of formulas in an analytical way?

The maintenance model will become really complex really fast that the analytical calculation methods are no longer sufficient. It is too complicated and time consuming to try and solve by using analytical calculations. Using simulation in Asset Strategy Optimization, we are can determine the possible solutions. It is not without reason that simulation is called the art of solving those problems that seem impossible or very difficult by using an analytical approach.

The 'Monte Carlo' method

The simulation method in Asset Strategy Optimization is based on the 'Monte Carlo' method. It would require too much time and details to explain this method here. We would like to refer you to the Asset Strategy Optimization training for that and will only provide an example here.

The "Monte Carlo' method can best be characterized by the generation of random values of know statistical distributions. These statistical distributions represent the service life, replacement and repair times for preventive maintenance in Asset Strategy Optimization, the lead time and MTTCC (Mean Time to Critical Condition) of the state-dependent maintenance, which are inherently stochastic of nature.

It is common that the maintenance concept is based on the failure behavior and the corresponding durability of failure behavior. The model-based description of failure behavior is done by using a risk analysis in which the failure modes, failure causes and failure conditions including durability, type of distributions are described with characteristic parameters. The description of the failure behavior by means of distribution and types of characteristic parameters is one of the most important relationships to the simulation. On the basis of these data, namely, through simulation, we are able to calculate some indicators that are necessary to make informed decisions with respect to maintenance.

Assume that we are dealing with one component of which the failure behavior is described by one failure mode, one failure cause and one failure condition. The 'Monte Carlo' method will generate a number between 0 and 1, 0.35 for example (the chance for any other number is the same). Subsequently we are interested in the translation of this chance in to lifespan. The lifespan is called Time To Failure (TTF). In order to determine the TTF, we have to determine at which TTF the area is lower than the probability distribution function 0.35.

A useful tool for determining this is the cumulative probability density function. The cumulative density probability function f (t) is displayed below, based on a normal distribution with an average life of 1 year and a standard deviation of 3 months, as used in the example.

Cumulative density probability function F(t)

The vertical axis shows the result of the "Monte Carlo" simulation, the random number between 0 and 1. On this axis, we have indicated the point at 0.35. In the same figure, we have plotted the cumulative probability density function. If we then draw a horizontal line from the y-axis (follow the arrows in Figure 2), starting at 0.35, then this line intersects the cumulative probability density function. At this point of intersection we then draw a vertical line downward which intersects the horizontal x-axis. This point is the time to failure (TTF) that is associated with the number 0.35 on the y-axis. In other words, the result of the integral is derived below, so the above-TTF (see figure below) .

Intersection 0.35 with cumulative probability density function F (t) for determination TTF

Asset Strategy Optimization shows the user how much time should be covered by the simulation calculations. We call this the simulation period. A simulation period of, for example, 10 years is not unusual. Asset Strategy Optimization then uses a process that we can visualize as follows. Imagine an imaginary timeline for the simulation period. Let us call the time on this timeline t, where t = 0 (year): the beginning of the simulation period, and t = 10 (years of age): the end of the simulation period. In this imaginary timeline we then draw the first TTF. At t = TTF1 the component will fail, assuming that there is no preventive maintenance for this part (see figure below).

In Asset Strategy Optimization we can indicate which follow-up action must take place. Assume the part is replaced, requiring 12 hours of maintenance time.

Then Asset Strategy Optimization will calculate the next time of failure. How does it work?

The process is similar as before. Once again, a random number will be generated between 0 and 1; then, based on the cumulative probability density function, a TTF will be derived. Suppose that the random number is 0.78, in the figure below the process is indicated (again with arrows).

Next, the part will suffer another failure at time t = TTF1 + 12 hours + TTF2. This is under the assumption that no preventive maintenance or other relationships have been incorporated in the model. This process will repeat itself until the end of the set simulation period (10 years). In Asset Strategy Optimization running through a timeline is called a 'run'. The process to arrive at a Time To Failure is called a 'draw' in Statistics.

Time-independent and time-dependent opportunities

In Asset Strategy Optimization we know time-independent and time-dependent opportunities. In this context, these concepts define whether or not there is a relationship between failing and the age of the component. An example of this is a nail puncturing a tire. The chance of a puncture caused by this nail is independent of the age of the tire and hence of the moment in time. In other words: chances are equal that the tire will encounter the nail on Tuesday or on Saturday, independent whether the tire used is new or worn. This is called a time-independent probability. In case of time-dependent probability, however, then indeed the probability of failure is dependent on the age of the part.

Typical examples are those parts where wear and aging processes play a significant role. This means that the probability of failure increases as the component ages. The failure behavior is often expressed by use of normal and Weibull distributions. The exponential distribution, by contrast, is often used for the modeling of time-independent processes.

Reliability Test in Asset Strategy Optimization

The reliability of the simulation results is largely determined by the accuracy. Accuracy in Asset Strategy Optimization is defined as the degree to which the (calculated) average of the simulation results in Asset Strategy Optimization is in accordance with the real average simulation results of the simulation model.

Since the central limit theory applies, this means that the simulation results in a large number of runs converge to the final outcome.

To test the reliability we use a renown test algorithm, the Students-T test. In this test, the user has to indicate the confidence interval. Valid values in Asset Strategy Optimization are 90%, 95% and 99%.

A confidence interval of 95% in Asset Strategy Optimization means that we accept an 'error margin' of 5% (i.e. 100% minus the confidence interval). This implies that 5% of simulation results (from which we calculate the reliability) is located outside of the test area. The simulation result is defined as the number of operations at system level. If 95% lies within the test area, then the number of maintenance actions meets the specified confidence interval. Asset Strategy Optimization will stop simulating.

For a confidence interval of 90% we will accept an 'error margin' of 10%. This means that if there is less than 10% outside of the test area, the simulation results (number of operations) meet the specified confidence interval.

The number of runs required for a 99% confidence interval is generally a factor 5 to 10 higher than in the situation of a 90% confidence interval; this also depends on various aspects such as the entered simulation model. In Asset Strategy Optimization, the test algorithm is applied starting at 11 runs. It is possible that a simulation meets the specified confidence interval from 11 runs.

Should the user pursue maximum reliability of simulation results (on the basis of the number of maintenance actions), then the user needs to choose the 99% confidence interval. If the user wants to get insight in the simulation results, then the user would best choose the 90% confidence interval.