MonteCarloMethod

• The Essay topic: Monte Carlo Method in Simulations

• Class:4IT496 Simulation of Systems (WS 2017/2018)

• Author: Bc. Yauheniya Andreyuk

1. Introduction

Monte Carlo method is an algorithm that rely on repeated random sampling to obtain numerical results. Essential idea of Monte Carlo is using randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration, and generating draws from a probability distribution. Monte Carlo simulation is a technique that allows to see all the possible outcomes of the decisions and assess the impact of risk, allowing better decision making under uncertainty. The technique is used by professionals in such fields as finance, project management, energy, manufacturing, engineering, research and development, insurance, oil & gas, transportation, and the environment. The simulation furnishes the decision-maker with a range of possible outcomes and the probabilities they will occur for any choice of action. It shows the extreme possibilities — the outcomes of going for broke and for the most conservative decision — along with all possible consequences for middle-of-the-road decisions.

2. History

Principles of Monte Carlo Simulations the way we know it today were developed at the beginning of the 18th century. The history of Monte Carlo started with the work of a French scientist Georges Louis LeClerc, Comte de Buffon (1707-1788), who created “Buffon’s Needle.” In his work he used a method of dropping random needles on a lined background to estimate π. He proved that for a needle the same length as the distance between the lines the probability that each time a needle would intersect a line was 2/ π. He tested his theory by throwing baguettes over his shoulder on to a tile floor.

His formula was:

                π = 2N/X

Where N is the number of tosses and X is the number of times a needle intersected a line.

The idea of using randomness in a determinative manner was revolutionary due to the fact that simulations beforehand were using a deterministic problem with statistical sampling to estimate the uncertainties in simulations. In the 19th and early 20th centuries, simulation was increasingly used as an experimental means of confirming theory, analyzing data, or supplementing intuition in mathematical statistics.

These were seminal studies, but there is a significant difference between them and typical modern Monte Carlo simulations studying problems that are otherwise intractable, e.g. galaxy formation modeling. The early simulations dealt with previously understood deterministic problems. Modern Monte Carlo simulations invert this approach, solving deterministic problems using a probabilistic analog and solving the problem probabilistically.

These simulations were used in investigating radiation shielding and the distance that neutrons would likely travel through various materials. This form of simulation was used during the Manhattan Project, the American World War II effort to develop nuclear weapons. During their work on the Manhattan Project John von Neumann and Stanislaw Ulam named the method after the Monte Carlo Casino in Monaco.

Since then Monte Carlo simulations have proven to be extremely useful in other fields such as physics, physical chemistry, operations research, business, and medical fields. Monte Carlo simulation is now a much-used scientific tool for problems that are analytically intractable and for which experimentation is too time-consuming, costly, or impractical. Researchers explore complex systems, examine quantities that are hidden in experiments, and easily repeat or modify experiments. But the simulation also has some disadvantages: it doesn't give exact solutions and results are only as good as the model and inputs used; it can require huge computing resources and simulation software, like any software, is prone to bugs. Nowadays there are some disputes that simulations are over used because they remove the need for analytical or experimental approaches and that people take it as a lazy way out.

3. How Monte Carlo simulation works.

Monte Carlo Simulation is used for obtaining a distribution of possible numerical outcome values for a given problem. It performs analysis by building models of possible results by substituting a range of values — a probability distribution — for any factor that has inherent uncertainty. It then calculates results over and over, each time using a different set of random values from the probability functions. Depending upon the number of uncertainties and the ranges specified for them, a Monte Carlo simulation could involve thousands of recalculations before it is complete. Monte Carlo simulation produces distributions of possible outcome values.

Monte Carlo Simulation has 3 basic elements:

1. A mathematical formula that represents how inputs turn into an output

2. A reasonable estimation of the variation of each input.

3. An idea of what output performance is acceptable.

The above model illustrates a visual representation of the three elements. The inputs are probability distributions and can vary in type. These are the possible range of values that the input values will take in the randomization process. These various types of distributions are listed below. The model is like the equation of how inputs turn into outputs found in our example in the previous sub-page, and finally the output is of course our results. This too will take the form of a distribution.

Input Probability Distributions Normal – Or “bell curve.” Values in the middle near the mean are most likely to occur. It is symmetric.

Log-normal – Values are positively skewed, not symmetric like a normal distribution. It is used to represent values that don’t go below zero but have unlimited positive potential.

Uniform – All values have an equal chance of occurring, and the user simply defines the minimum and maximum.

Triangular – The user defines the minimum, most likely, and maximum values. Values around the most likely are more likely to occur.

PERT- The user defines the minimum, most likely, and maximum values, just like the triangular distribution. Values around the most likely are more likely to occur. However values between the most likely and extremes are more likely to occur than the triangular; that is, the extremes are not as emphasized.

Discrete – The user defines specific values that may occur and the likelihood of each.