1. Outline of Chapter 9: Using Simulation to Solve Decision Problems Real world decisions are often too complex to be analyzed effectively using influence diagrams or decision trees. Monte Carlo simulation offers us a way to model these decisions that contain a large amount of data. What is a Monte Carlo Simulation? Monte Carlo simulation is a method analyze a decision by repeatedly evaluating the decision process using random values for uncertain values. The average outcome of the simulation over many runs defines the probability distribution for decision.

2. Using Random Numbers to Simulate Reality Monte Carlo simulations allow a decision model to become stochastic like the real world. Evaluating the model numerous times can give an accurate probability distribution of the various outcomes. “A random number can be any number (x) from a group of uniformly distributed numbers that falls within an established boundary, usually between 0 and 1.” –D.S. A random number can be used to choose a value from any random distribution- i.e. to decide the outcome of an event in a simulation. Computers are a good way to through random numbers.

3. The Power of Spreadsheets Spreadsheets can be used to throw random numbers. A model can be built in a spreadsheet and macros used to update the spreadsheet thousands of time and cumulate possible outcomes. Example, see figure 9-4.

4. Generating Uniform Distributions Given a random variable, x, from a uniform distribution on the interval [0,1] we can compute a uniform random variable, y, on any continuous interval [a,b]. The formula to do this is: y = a + x * ( b – a ) Using Discrete Distributions Discrete, uniform distributions can be created from the uniform distribution [0,1] by dividing the interval [1,0] into segments. i.e. if x < 0.50 then option A, else option B gives a discrete, uniform distribution with 2 outcomes. Lookup tables are useful to create distributions with more possible outcomes.

5. Using the Results of a Monte Carlo A histogram and a cumulative distribution are two graphs that are effective at displaying the results of a Monte Carlo simulation. Here is a simulation done in Excel.

6. A histogram shows the relative frequency of each of the possible outcomes. Cumulative probability distributions are useful to determine the risk profile of the possible outcomes.

7. Commercial Software Spreadsheets like Microsoft Excel are useful in doing Monte Carlo simulations. There are also software packages available such as: Crystal Ball by Decisioneering @Risk by Palisade. Both software packages can create models with many variable inputs and outcomes and create histograms and cumulative distributions over numerous runs.

8. The Role of Monte Carlo Monte Carlo simulation is not a replacement for DA, but is useful to find distributions of uncertain quantities. The thought and effort put into creating the simulations should be just as rigorous as that put into DA. “Monte Carlo simulation provides another approach to dealing with uncertain quantities. We can use this method to learn about the interactions of many uncertainties… with the use of a computer… (but) the same amount of care must be taken as when building any decision model.”