The Monte Carlo method

What is a simulation?

Let’s roll a lot of dice. Every time we roll, that’s a simulation. This allows us to find experimental probabilities:

Law of Large Numbers (or law of averages)

As the total number of simulations gets larger, the experimental probability gets closer to the theoretical probability:

\[P(\textrm{rolling a 6})\rightarrow \frac{1}{6}\]

Can computers find random numbers?

Not exactly! But what they can do is generate large amounts of pseudorandom numbers. That is, a computer will use complex algorithms to find something that is close enough to random to satisfy most mathematicians.

EXCEL:

Our major tools will be the Excel functions: RAND() and RANDBETWEEN(a,b). To simulate a large lumber of dice rolls, we use the function RANDBETWEEN(1,6), which randomly chooses a number between 1 and 6.

R:

R is easier: we will use “r” commands (rnorm(), rpois(), rbinom(), ect) to generate strings of random numbers.

Examples of simulations in Business

Simulations can do more than let us roll pretend dice. They can be used to model various real-world situations such as:

• Product demand
• Lead time for orders to arrive
• Time between customers, machine breakdowns, or any other event
• Service Time
• Tim to complete a project or activity
• Number of employees absent on a certain day
• Stock market performance¹

Advantages to using simulation instead of other methods

• Simulations rely less on theoretical knowledge (and fun Statistics theorems like the Central Limit theorem), making them easier to understand.
• Can analyze complex real world problems like State economies and world food systems that can’t be solved using pure math.
• We don’t have to actually try out potentially disruptive experiments to see what will happen. For example, we can simulate our production with a new machine before actually purchasing the machine.
• We can look at the effects of different variables, all from the comfort of our desk chairs.
• We can look at things over time without a time machine.

Disadvantages to Simulation

• It can be very expensive and time consuming to produce
• Our model rests on any underlying assumptions we have made. In the dice rolling experiment, we assumed that we were working with a fair six-sided die. What if we weren’t?

E

Why not just use the Expected Value formula?

Good question! The mean, or expected value of this distribution is:

\[E(X)=\Sigma X P(X)=\$5.50 \times 0.5 +\$4.25\times 0.4+\$4.75\times 0.1=\$4.93\]

But this is the long range average. On any given conference, we won’t have exactly $4.93 per person (or $172.38 total revenue) When we are forecasting demand over a short time period, things can be much more volatile.