1. Simulation is the process of studying the behavior of a real system by using a model that replicates the system under different scenarios. A simulation model is constructed by identifying the mathematical expressions and logical relationships that describe how the system operates.

2. Advantages of Computer Simulation It offers the ability to gain insights into the model solution which may be impossible to attain through other techniques. It provides a convenient experimental laboratory to perform "what if" and risk analysis.

3. Disadvantages of Computer Simulation A large amount of time may be required to develop the simulation model. Simulation is, in effect, a trial and error method of comparing different policy inputs. It does not determine if some input which was not considered could have provided a better solution for the model.

4. Building a Simulation Model 1. Identify the decision variables, random variables and objective in the problem. 2. Model the logic of the problem: Flowchart Formulas to describe relationships Probability distributions for random variables Program code 3. Validate the model 4. Experimental Design 5. Perform simulation runs and analyze output results

5. Random Variables Random variable values are utilized in the model through a technique known as Monte Carlo simulation. Each random variable is mapped to a set of numbers N so that each time one number in N is generated, the corresponding value of the random variable is given as an input to the model. The mapping is done in such a way that the long run percentage of time that a particular number is simulated in the model occurs according to the probability of that value for the random variable.

6. Excel’s Random Number Generator (RNG) =rand() Randomly simulates a value between 0 and 1 in the cell where the function is entered In PC’s press [F9] to recalculate the function manually Function value is recalculated whenever a number or formula is entered in another cell unless Calculation Options in Formula ribbon is set to Manual

7. Continuous Distributions The values generated for a random variable are specified from a set of uninterrupted values over a range; an infinite number of values is possible For example, the interest rate next year could be modeled as a continuous random variable between 0% to 8%.

8. Common Continuous Distributions Normal Distribution: A symmetrical bell shaped curve that is centered around a specified mean μ with a spread described by the standard deviation σ Uniform Distribution: A rectangular curve where it is assumed that all values between a specified minimum and a specified maximum are equally likely to occur

9. Modeling Continuous Distributions In Excel, for the Normal distribution: =norminv(random #, μ, σ) Values will be simulated from a symmetrical bell-shaped curve where the most likely value is μ and 64% of the values have a chance of lying within 1 σ (in either direction) of μ

10. Discrete Distributions The values generated for a random variable must be from a finite distinct set of individual values. For example, the number of passengers who may try to buy airline tickets is a discrete random variable that is limited to positive integer values in a certain range.

11. =Randbetween(a,b) function Simulates an integer value between a and b Assumes that every number between a and b is equally likely to occur in the system Maps numbers generated between 0 and 1 using rand() function to the interval (a,b)

12. Modeling Discrete Distributions In Excel, when every random variable value is not equally likely and there are limited choices, use the Vlookup function: =vlookup(value to look up in column 1, table to look in, column to report result from) See Vlookup function Excel snippit in MyLMUConnect.

13. Modeling Discrete Distributions In Excel, when every random variable value is not equally likely and there are many choices, use a continuous distribution with the Int function: =Int(norminv(random #, μ, σ) Replace the continuous distribution with appropriate shape for likelihood as appropriate.

14. Model Validation Models based on assumptions which do not accurately reflect real world behavior cannot be expected to generate meaningful results. Errors in programming can result in nonsensical results. Validation is generally done by having an expert review the model and the computer code for errors. If possible, the simulation should be run using actual past data. Predictions from the simulation model should be compared with historical results.

15. Experimental Design Policies under consideration for implementation in the real system must be identified. For each policy under consideration by the decision maker, the simulation requires performing many runs. Whenever possible, different policies should be compared by using the same sequence of random numbers.

16. “Trials”, “Runs” and “Iterations” Every time a set of input values are simulated, output results should be collected. The outputs associated with a trial represent one snapshot of what could occur in the real system and under what conditions Many trials (e.g. runs, iterations) should be performed so that a distribution describing the key outputs can be created and the mean outcomes and risk can be viewed

17. Excel Simulation Add-ins Risk Solver Crystal Ball @Risk Provides built-in functions for probability distributions Performs simulation trials, captures outputs and summarizes results with histograms and statistics