1. Simulation ensembles: Monte-Carlo sampling
In this video, we will learn how Monte Carlo simulation ensembles can help study the response of systems to different scenarios.
2. System response to different scenarios
Non-deterministic processes can lead to variations in a system's response and performance; and now that we know how to represent such non-determinism in models, we can use this to characterize how uncertainty propagates through a model.
This can be helpful to test our model's response to different scenarios and to such randomness.
It can assist in planning future developments and business expansions, performing stress tests, and preparing for extreme situations.
3. Monte Carlo sampling
Monte-Carlo sampling is a class of algorithms that relies on repeated random sampling.
The idea is to run a model with small changes to examine the system's response.
As the number of samples increases, the parameter space increases as well; in other words, the more samples we have, the better the picture of the model response is. However, running thousands of scenarios has a computational cost, and a compromise is often required.
The plot below shows 100 Monte Carlo samples. As we can see, the number of samples is insufficient to reveal patterns in the model results.
The following plot shows 1500 samples. Patterns start to emerge, but still provide a limited picture.
The last plot has 5000 samples, and grid patterns can now be clearly seen.
As you can see, in this example, we couldn't understand the system's behavior with only 100 samples, but that picture became much clearer as the number of samples increased.
4. Monte Carlo sampling: Process investigation
How can Monte Carlo sampling be used to investigate processes?
Here we show a generator that uses the random-dot-gauss method to generate numbers based on a normal distribution with an average of 25 and a standard deviation of five.
The scatterplot shown displays a range of outputs that can be expected by sampling from this normal distribution.
As we can see, sampling results of our Monte Carlo simulation are very close to the average value of 25. The further a value is from the average value, the fewer samples there are.
5. Monte Carlo sampling: Discrete-event models
The underlying objective of using Monte Carlo sampling in discrete-event models is to explore model uncertainty arising from non-deterministic processes.
In other words, this can help characterize uncertainty to support concrete management decisions.
The diagram on the right shows an activity that involves four states, which are affected by processes one to three. "Process 1" can change model "State 1" in six ways, producing six "State 2" scenarios. "Process 2" can only change "State 2" in one way, but the result is still determined the by various scenarios of "State 2". Finally, "Process 3" can change "State 3" in three ways, resulting in a compounded effect because the six possible scenarios of "State 3" can now have three scenarios each, producing a total of 18 scenarios by the end of the simulation.
6. Monte Carlo sampling: Discrete-event models
As we have seen, uncertainty in each process duration propagates through a system, producing different model trajectories, often referred to as the response envelope.
In this example, we have four processes, and we show here dictionaries for two of these processes. Process 1 is named "Raw_material" and has an average duration of 20 hours and a maximum delay of ten percent. Process 2 is named "Unloading" and has an average duration of 15 hours and a maximum delay of five percent.
The plot shows the different trajectories of this activity and how variations in the process performances become compounded. The model response envelope shows that the entire activity can last between 150 and 500 hours.
7. Let's practice!
Let's put these new concepts into practice.