1. Computer Simulation Henry C. Co Technology and Operations Management,
California Polytechnic and State University

2. Simulation Models (Henry C. Co)

3. Simulation Models (Henry C. Co)
Simulation: a descriptive technique that enables a decision maker to evaluate the behavior of a model under various conditions.
Simulation models complex situations
Models are simple to use and understand
Models can play “what if” experiments
Extensive software packages available
Simulation Models (Henry C. Co)

4. Simulation Models (Henry C. Co)
Analytic models: values of decision variables are the outputs.
Simulation models: values of decision variables are the inputs. Investigate the impacts on certain parameters when these values change.
Simulation Models (Henry C. Co)

5. Why Simulation?

6. Simulation Models (Henry C. Co)
Analytic models
May be difficult or impossible to obtain.
Typically predict only average or steady-state behavior.
Simulation models
Wide availability of software and more powerful PCs make implementation much easier than before.
More realistic random factors can be incorporated.
Easier to understand.
Simulation Models (Henry C. Co)

7. Simulation Process

8. Simulation Models (Henry C. Co)
Identify the problem
Develop the simulation model
Test the model
Develop the experiments
Run the simulation and evaluate results
Repeat until results are satisfactory
Simulation Models (Henry C. Co)

9. Simulation Models (Henry C. Co)
Implementation
Identify the boundaries of the system of interest.
Identify the random variables, decision variables, parameters, and the performance measure(s).
Develop an objective function for the performance measure(s) in terms of random variables, decision variables, and parameters.
Use computer to generate the simulated values of these random variables.
Compute the values of the objective function using these simulated values of random variables and values of decision variables.
Statistical analysis.
Simulation Models (Henry C. Co)

10. Monte Carlo Simulation

11. Simulation Models (Henry C. Co)
Monte Carlo method: Probabilistic simulation technique used when a process has a random component
Identify a probability distribution
Setup intervals of random numbers to match probability distribution
Obtain the random numbers
Interpret the results
Simulation Models (Henry C. Co)

12. Major Components of Models

13. Simulation Models (Henry C. Co)
Random input factors: sales, demand, stock prices, interest rates, the length of time required to perform a task.
Random performance measures:
Business profit within a time interval.
Average waiting time of a customer in a queuing system.
Random input factors  random performance measures.
Simulation Models (Henry C. Co)

14. An “Analog” Approach

15. Simulation Models (Henry C. Co)
Every point on the circumference corresponds to a number between 0 and 1.
For example, when the pointer is in the 3 O’clock position, it is pointing to the number 0.25.
“Game Spinner” for uniform random variable on the interval 0 to 1.
Simulation Models (Henry C. Co)

16. Simulating a Discrete Distribution

17. Simulation Models (Henry C. Co)
10% of the interval (0.0 to ) is mapped (assigned) to a demand d= 8.
20% of the interval (0.1 to ) is mapped to d =9.
30% of the interval (0.3 to ) is mapped to d =10.
etc., etc.
Simulation Models (Henry C. Co)

18. Excel Functions Useful in Simulation
RAND(): a volatile Excel Function
Function =RAND() generates a uniformly-distributed random number between 0 -1.
VLOOKUP
Simulation Models (Henry C. Co)

19. Simulation Models (Henry C. Co)
Use function =RAND() to generate a uniformly-distributed random number between 0 and 1.
Simulation Models (Henry C. Co)

20. Simulation Models (Henry C. Co)

21. Simulation Models (Henry C. Co)
F4=RAND() ; copy and paste F5:F13
G4=VLOOKUP(F4,$B$4:$C$10,2,1); copy and paste G5:G13
Simulation Models (Henry C. Co)

22. A Machine Breakdown Example

23. Simulation Models (Henry C. Co)
F4=RAND() ; copy and paste F5:F13
G4=VLOOKUP(F4,$B$4:$C$10,2,1); copy and paste G5:G13
Simulation Models (Henry C. Co)

24. Simulating a Continuous Distribution

25. Simulation Models (Henry C. Co)
The inverse transformation method
To transform this random number into a sample value of the random variable.
F(w) is the CDF F(x)=Prob. {W x}.
Simulation Models (Henry C. Co)

26. Inverse Transformation Method
Define F(x)=Prob. {W x} = the probability that random variable W is less than or equal to a specific value w.
Denote the 0-1 random number by u and let u = F(x).
Use =RAND() to generate a value for u, substitute it into x= F-1(u) which in turn gives a value of x.
Simulation Models (Henry C. Co)

27. Simulation Models (Henry C. Co)
EXCEL Implementation
Exponential Distribution
u = RAND()
For example, if arrival rate = 0.05, and RAND()=.75, the observation from the exponential distribution is (-1/0.05)ln(1-.75) =
Normal Dist’n: Function NORMINV
For example, NORMINV(RAND(),1000,100) returns a normally distributed random number with mean 1000 and standard deviation 100.
Simulation Models (Henry C. Co)

28. Using an EXCEL Simulation Model
Information obtained from a Simulation model:
Summary statistics about the performance measures
Downside Risk and Upside Risk
Distribution of outcomes
Based on the simulation results (Output), several alternatives (decisions) can be evaluated.
Simulation Models (Henry C. Co)

29. How Reliable is the Simulation?

30. Simulation Models (Henry C. Co)
The more trials we run, the higher the confidence we have in our results (just like any statistical analysis with real data sample).
The confidence intervals about the parameters (or any other estimated parameters) can be computed.
Given sample size and significant level  confidence intervals can be computed, or given the half width of the confidence interval and significance level  compute the minimum number of replications we have to run.
Simulation Models (Henry C. Co)

31. Advantages

32. Simulation Models (Henry C. Co)
Solves problems that are difficult or impossible to solve mathematically
Allows experimentation without risk to actual system
Compresses time to show long-term effects
Serves as training tool for decision makers
Simulation Models (Henry C. Co)

33. Limitations

34. Simulation Models (Henry C. Co)
Does not produce optimum solution
Model development may be difficult
Computer run time may be substantial
Monte Carlo simulation only applicable to random systems
Simulation Models (Henry C. Co)