Computer Simulation

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

2. Simulation Models (Henry C. Co)

3. Simulation Model • 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. 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. 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. 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. 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. 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. 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. “Game Spinner” for uniform random variable on the interval 0 to 1. • 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. Simulation Models (Henry C. Co)

16. Simulating a Discrete Distribution

17. 10% of the interval (0.0 to 0.09999) is mapped (assigned) to a demand d= 8. • 20% of the interval (0.1 to 0.29999) is mapped to d =9. • 30% of the interval (0.3 to 0.59999) 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. 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. 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. 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. 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. 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) = 23.73. • 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. 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. 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. 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)