Simulation

results

aka scenarios. contributes a little bit to our understanding of a problem and helps inform our decision

Uniform Distribution

everything is equally probable

Best Case/Worst Case Analysis

- Best Case - The most optimistic values for uncertain values - Worst Case - Use the most pessimistic values for uncertain values - can be done easily - but does not provide a distribution of possible outcomes within bounds.

What if Analysis

- Enter uncertain values - Check output values - Problem - values entered may be biased - tedious work with one-at-a-time - results may be too narrow

Normal Distribution

- Familiar bell-shapes curve. - Useful in simulation modeling as a continuous input distribution. - not always an appropriate distribution because it is symmetric -Do not use if: *a skewed distribution is a better representation of the data. *negative values are invalid.

Simulation Analysis

- Is similar to an automated what-if analysis - Value selected is unbiased - Computer generates hundreds (thousands) of scenarios. -Analyze the results to better understand the behavior under uncertainty. -Decision making is based on empirical evidence.

Methods of Risk Analysis

-Best case/Worst case analysis -What if analysis -Simulation

Characteristics of Probability Distribution

-Discrete (whole units) (computer) vs continuous (any value) (the world of nature) *Digitized - has been turned into discrete values -symmetric vs skewed -bounded vs unbounded

Binomial Distribution

-Discrete distribution -applies when of independent and identical trials occur. ex. world series games -each trial results in success or a failure -the number of successes in these trials - n - the number of trials - p - the probability of success for each trial.

Simulation

-Producing models results using one or more computers. -Uses a computational model as input -Provides computational results as output *simulated results imitate some real-world system.

Probability Distribution for Input Variables

- are the sine qua non of simulation models. - the difference between the other spreadsheet models and the simulation models is that at least one of the input variable cells in a simulation contains random numbers. -each time the spreadsheet recalculates, random numbers change, and the new random values of the inputs produce new values of the outputs.

Continuous

A probability distribution is continuous if its possible values are essentially a continuum. *ex: amount of rain that falls during a month. -characterized by a density function, a smooth curve -the height of a density function above any value indicates the relative likelihood of that value. -probabilities can be calculated as areas under the curve.

Discrete

A probability distribution is discrete if it has finite numbers of possible values. ex: sum of the faces of two dice -Discrete distribution is a sequence of spikes *the height of each spike is the probability of the corresponding value.

Non-Negative vs Unrestricted

-Special case of bounded distributions is when the only possible values are nonnegative. In such cases, model the randomness with a probability distribution that is bounded below 0. Rules out non-negative numbers that make no sense.

Why are simulations models useful?

-allow "what-ifs" evaluation without implementing a system. -Show system's sensitivity to changes in variables.

Use discrete distribution when ...

...the uncertain quantity is not really continuous or you want a discrete approximation to a continuous variable. specify: -the possible values -their probabilities. make sure the probabilities sum to 1.

Bounded vs Unbounded

Probability distributions are bounded if there are values such as A and B such that no possible value can be less than or greater than B. Unbounded - no restrictions

What is a simulation model?

Simulation models express a system with equations and algorithms -explicitly uses uncertainty -uses randomnization to model uncertainty. -Produces a distribution of results instead of a single optimal value. -Each result set varies -probability distribution (key to this) for certain inputs -> simulation model -> probability distribution for important outputs.

Random

something that comes about thats a consequence of a probability distribution