1. 1 1 Slide Simulation Professor Ahmadi

2. 2 2 Slide Simulation Chapter Outline n Computer Simulation n Simulation Modeling n Random Variables and Pseudo-Random Numbers n Time Increments n Other Simulation Issues n Validation and Statistical Considerations

3. 3 3 Slide Computer Simulation n Computer simulation is one of the most frequently employed management science techniques. n It is typically used to model random processes that are too complex to be solved by analytical methods.

4. 4 4 Slide Advantages of Computer Simulation n Among the advantages of computer simulation is the ability to gain insights into the model solution which may be impossible to attain through other techniques. n Also, once the simulation has been developed, it provides a convenient experimental laboratory to perform "what if" and sensitivity analysis.

5. 5 5 Slide Disadvantages of Computer Simulation n A large amount of time may be required to develop the simulation. n There is no guarantee that the solution obtained will actually be optimal. n Simulation is, in effect, a trial and error method of comparing different policy inputs. n It does not determine if some input which was not considered could have provided a better solution for the model.

6. 6 6 Slide Simulation Modeling n One begins a computer simulation by developing a mathematical statement of the problem. n The model should be realistic yet solvable within the speed and storage constraints of the computer system being used. n Input values for the model as well as probability estimates for the random variables must then be determined.

7. 7 7 Slide Random Variables n Random variable values are utilized in the model through a technique known as Monte Carlo simulation. n Each random variable is mapped to a set of numbers so that each time one number in that set is generated, the corresponding value of the random variable is given as an input to the model. n The mapping is done in such a way that the likelihood that a particular number is chosen is the same as the probability that the corresponding value of the random variable occurs.

8. 8 8 Slide Pseudo-Random Numbers n Because a computer program generates random numbers for the mapping according to some formula, the numbers are not truly generated in a random fashion. n However, using standard statistical tests, the numbers can be shown to appear to be drawn from a random process. n These numbers are called pseudo-random numbers.

9. 9 9 Slide Static and Dynamic Simulation Models n Static Simulation Models: In these types of models, the simulation runs are independent of each other. The state of the system at one point in time does not affect the system at future points in time. For each time period a different set of data from the input sequence is used to calculate the effects on the model. n Dynamic Simulation Models: In these types of models, the state of the system at one point in time does affect the future of the system.

10. 10 Slide Model Validation n Models which do not accurately reflect real world behavior cannot be expected to generate meaningful results. n Likewise, errors in programming can result in nonsensical results. n Validation is generally done by having an expert review the model and the computer code for errors. n If possible, the simulation should be run using actual past data. n Predictions from the simulation model should be compared with historical results.

11. 11 Slide Experimental Design n Experimental design is an important consideration in the simulation process. n Issues such as the length of time of the simulation and the treatment of initial data outputs from the model must be addressed prior to collecting and analyzing output data. n Normally one is interested in results for the steady state (long run) operation of the system being modeled. n The initial data inputs to the simulation generally represent a start-up period for the process and it may be important that the data outputs for this start-up period be neglected for predicting this long run behavior.

12. 12 Slide Experimental Design n For each policy under consideration by the decision maker, the simulation is run by considering a long sequence of input data values (given by a pseudo- random number generator). n Whenever possible, different policies should be compared by using the same sequence of input data.

13. 13 Slide Example: Probabilistic, Inc. The price change of shares of Probabilistic, Inc. has been observed over the past 50 trades. The frequency distribution is as follows: Price Change Number of Trades Price Change Number of Trades -3/8 4 -3/8 4 -1/4 2 -1/4 2 -1/8 8 -1/8 8 0 20 0 20 +1/8 10 +1/8 10 +1/4 3 +1/4 3 +3/8 2 +3/8 2 +1/2 1 +1/2 1

14. 14 Slide Example: Probabilistic, Inc. n Relative Frequency Distribution and Random Number Mapping Price Change Relative Frequency Random Numbers Price Change Relative Frequency Random Numbers -3/8.08 00 and under 07 -3/8.08 00 and under 07 -1/4.0408 and under 11 -1/4.0408 and under 11 -1/8.1612 and under 27 -1/8.1612 and under 27 0.40 28 and under 67 0.40 28 and under 67 +1/8.20 68 and under 87 +1/8.20 68 and under 87 +1/4.06 88 and under 93 +1/4.06 88 and under 93 +3/8.0494 and under 97 +3/8.0494 and under 97 +1/2.02 98 and under 99 +1/2.02 98 and under 99 TOTAL = 1.00 TOTAL = 1.00

15. 15 Slide Example: Probabilistic, Inc. If the current price per share of Probabilistic is 23, use random numbers to simulate the price per share over the next 10 trades. For random numbers, use the following: 21, 84, 07, 30, 94, 57, 57, 19, 84, 84 21, 84, 07, 30, 94, 57, 57, 19, 84, 84

16. 16 Slide Example: Probabilistic, Inc. n Simulation Worksheet Trade Random Price Stock Trade Random Price Stock Number Number Change Price Number Number Change Price 1 21 -1/8 22 7/8 1 21 -1/8 22 7/8 2 84 +1/8 23 2 84 +1/8 23 3 07 -3/8 22 5/8 3 07 -3/8 22 5/8 4 30 0 22 5/8 4 30 0 22 5/8 5 94 +3/8 23 5 94 +3/8 23 6 57 0 23 6 57 0 23 7 57 0 23 7 57 0 23 8 19 -1/8 22 7/8 8 19 -1/8 22 7/8 9 84 +1/8 23 9 84 +1/8 23 10 84 +1/8 23 1/8 10 84 +1/8 23 1/8

17. 17 Slide Example: Coin Toss n Use Excel to generate 200 random numbers and simulate a coin toss.