1. Introduction to Management Science
8th Edition by
Bernard W. Taylor III
Chapter 4
Simulation
Chapter 4 - Simulation

2. Chapter Topics The Monte Carlo Process
Computer Simulation with Excel Spreadsheets
Simulation of a Queuing System
Continuous Probability Distributions
Statistical Analysis of Simulation Results
Crystal Ball
Verification of the Simulation Model
Areas of Simulation Application
Chapter 4 - Simulation

3. Overview
Analogue simulation replaces a physical system with an analogous physical system that is easier to manipulate.
In computer mathematical simulation a system is replaced with a mathematical model that is analyzed with the computer.
Simulation offers a means of analyzing very complex systems that cannot be analyzed using the other management science techniques in the text.
Chapter 4 - Simulation

4. Homework Reduction
From problems 5, 9, 13, 15, 19, 21, 23: select four, and complete only those four.
Then, do:
Either one of the problems from among 25, 27, 29: (Qrystal Ball program — so, you must describe how you used the program, cite the results, and explain what the results indicate.)
Or do problem 17 (by hand or using Excel) Simulate five trials of ten random turns around the corner
Chapter 4 - Simulation

5. Monte Carlo Process
A large proportion of the applications of simulations are for probabilistic models.
The Monte Carlo technique is defined as a technique for selecting numbers randomly from a probability distribution for use in a trial (computer run) of a simulation model.
The basic principle behind the process is the same as in the operation of gambling devices in casinos (such as those in Monte Carlo, Monaco).
Gambling devices produce numbered results from well-defined populations.
Chapter 4 - Simulation

6. Probability Distribution of Demand for Laptop PC’s
Monte Carlo Process
Use of Random Numbers (1 of 10)
In the Monte Carlo process, values for a random variable are generated by sampling from a probability distribution.
Example: ComputerWorld demand data for laptops selling for $4,300 over a period of 100 weeks.
Table 4.1
Probability Distribution of Demand for Laptop PC’s
Chapter 4 - Simulation

7. Use of Random Numbers (2 of 10)
Monte Carlo Process
Use of Random Numbers (2 of 10)
The purpose of the Monte Carlo process is to generate the random variable, demand, by sampling from the probability distribution P(x).
The partitioned roulette wheel replicates the probability distribution for demand if the values of demand occur in a random manner.
The segment at which the wheel stops indicates demand for one week.
Chapter 4 - Simulation

8. A Roulette Wheel for Demand
Monte Carlo Process
Use of Random Numbers (3 of 10)
Figure 4.1
A Roulette Wheel for Demand
Chapter 4 - Simulation

9. Use of Random Numbers (4 of 10)
Monte Carlo Process
Use of Random Numbers (4 of 10)
When wheel is spun actual demand for PC’s is determined by a number at rim of the wheel.
Figure 4.2
Numbered Roulette Wheel
Chapter 4 - Simulation

10. Generating Demand from Random Numbers
Monte Carlo Process
Use of Random Numbers (5 of 10)
Process of spinning a wheel can be replicated using random numbers alone.
Transfer random numbers for each demand value from roulette wheel to a table.
Table 4.2
Generating Demand from Random Numbers
Chapter 4 - Simulation

11. Use of Random Numbers (6 of 10)
Monte Carlo Process
Use of Random Numbers (6 of 10)
Select number from a random number table:
Table 4.3
Random Number Table
Chapter 4 - Simulation

12. Use of Random Numbers (7 of 10)
Monte Carlo Process
Use of Random Numbers (7 of 10)
Repeating selection of random numbers simulates demand for a period of time.
Estimated average demand = 31/15 = 2.07 laptop PCs per week.
Estimated average revenue = $133,300/15 = $8, ($133,300 = $4,300  31).
Chapter 4 - Simulation

13. Use of Random Numbers (8 of 10)
Monte Carlo Process
Use of Random Numbers (8 of 10)
Chapter 4 - Simulation

14. Use of Random Numbers (9 of 10)
Monte Carlo Process
Use of Random Numbers (9 of 10)
Average demand could have been calculated analytically:
Chapter 4 - Simulation

15. Use of Random Numbers (10 of 10)
Monte Carlo Process
Use of Random Numbers (10 of 10)
The more periods simulated, the more accurate the results.
Simulation results will not equal analytical results unless enough trials have been conducted to reach steady state.
Often difficult to validate results of simulation - that true steady state has been reached and that simulation model truly replicates reality.
When analytical analysis is not possible, there is no analytical standard of comparison thus making validation even more difficult.
Chapter 4 - Simulation

16. Computer Simulation with Excel Spreadsheets
Generating Random Numbers (1 of 2)
As simulation models get more complex they become impossible to perform manually.
In simulation modeling, random numbers are generated by a mathematical process instead of a physical process (such as wheel spinning).
Random numbers are typically generated on the computer using a numerical technique and thus are not true random numbers but pseudorandom numbers.
Chapter 4 - Simulation

17. Computer Simulation with Excel Spreadsheets
Generating Random Numbers (2 of 2)
Artificially created random numbers must have the following characteristics:
The random numbers must be uniformly distributed.
The numerical technique for generating the numbers must be efficient.
The sequence of random numbers should reflect no (discernible) pattern.
Chapter 4 - Simulation

18. Simulation with Excel Spreadsheets (1 of 3)
Exhibit 4.1
Chapter 4 - Simulation

19. Simulation with Excel Spreadsheets (2 of 3)
“Lookup”
Exhibit 4.2
Chapter 4 - Simulation

20. Simulation with Excel Spreadsheets (3 of 3)
Exhibit 4.3
Chapter 4 - Simulation

21. Computer Simulation with Excel Spreadsheets
Decision Making with Simulation (1 of 2)
Revised ComputerWorld example; order size of one laptop each week.
Exhibit 4.4
=1+MAX(G6-H6,0)
Order one laptop each week
Chapter 4 - Simulation

22. Computer Simulation with Excel Spreadsheets
Decision Making with Simulation (2 of 2)
Order size of two laptops each week.
=2+MAX(G6-H6,0)
Order two laptops each week
Exhibit 4.5
Chapter 4 - Simulation

23. Simulation of a Queuing System Burlingham Mills Example (1 of 3)
Table 4.5
Distribution of Arrival Intervals
Table 4.6
Distribution of Service Times
Chapter 4 - Simulation

24. Simulation of a Queuing System Burlingham Mills Example (2 of 3)
Average waiting time = 12.5days/10 batches
= 1.25 days per batch
Average time in the system = 24.5 days/10 batches
= 2.45 days per batch
Chapter 4 - Simulation

25. Simulation of a Queuing System Burlingham Mills Example (3 of 3)
Caveats:
Results may be viewed with skepticism.
Ten trials do not ensure steady-state results. In fact, the statistical error for N data-points is sqrt(N), so the relative statistical error is ~1/sqrt(N).
Starting conditions can affect simulation results.
If no batches are in the system at start, simulation must run until it replicates normal operating system.
If system starts with items already in the system, simulation must begin with items in the system.
Chapter 4 - Simulation

26. Computer Simulation with Excel Burlingham Mills Example
Exhibit 4.6
Chapter 4 - Simulation

27. Continuous Probability Distributions4
1/2
Chapter 4 - Simulation

28. Machine Breakdown and Maintenance System Simulation (1 of 6)
Bigelow Manufacturing Company must decide if it should implement a machine maintenance program at a cost of $20,000 per year that would reduce the frequency of breakdowns and thus time for repair which is $2,000 per day in lost production.
A continuous probability distribution of the time between machine breakdowns:
f(x) = x/8, 0  x  4 weeks, where x = weeks between machine breakdowns
x = 4*sqrt(r1), value of x for a given value of r1.
Chapter 4 - Simulation

29. Probability Distribution of Machine Repair Time
Machine Breakdown and Maintenance System
Simulation (2 of 6)
Table 4.8
Probability Distribution of Machine Repair Time
Chapter 4 - Simulation

30. Machine Breakdown and Maintenance System Simulation (3 of 6)
Revised probability of time between machine breakdowns:
f(x) = x/18, 0  x6 weeks where x = weeks between machine breakdowns
x = 6*sqrt(r1)
Table 4.9
Revised Probability Distribution of Machine Repair Time with the Maintenance Program
Chapter 4 - Simulation

31. Machine Breakdown and Maintenance System Simulation (4 of 6)
Simulation of system without maintenance program (total annual repair cost of $84,000):
x = 4*sqrt(r1)
Table 4.10
Simulation of Machine Breakdowns and Repair Times
Chapter 4 - Simulation

32. Machine Breakdown and Maintenance System Simulation (5 of 6)
Simulation of system with maintenance program (total annual repair cost of $42,000):
x = 6*sqrt(r1)
Table 4.11
Simulation of Machine Breakdowns and Repair with the Maintenance Program
Chapter 4 - Simulation

33. Machine Breakdown and Maintenance System Simulation (6 of 6)
Results and caveats:
Implement maintenance program since cost savings appear to be $42,000 per year and maintenance program will cost $20,000 per year.
However, there are potential problems caused by simulating both systems only once.
Simulation results could exhibit significant variation since time between breakdowns and repair times are probabilistic.
To be sure of accuracy of results, simulations of each system must be run many times and average results computed.
Efficient computer simulation required to do this.
Chapter 4 - Simulation

34. Machine Breakdown and Maintenance System
Simulation with Excel (1 of 2)
Original machine breakdown example:
Exhibit 4.7
Chapter 4 - Simulation

35. Machine Breakdown and Maintenance System
Simulation with Excel (2 of 2)
Simulation with maintenance program.
Exhibit 4.8
Chapter 4 - Simulation

36. Statistical Analysis of Simulation Results (1 of 2)
Outcomes of simulation modeling are statistical measures such as averages.
Statistical results are typically subjected to additional statistical analysis to determine their degree of accuracy.
Confidence limits are developed for the analysis of the statistical validity of simulation results.
Chapter 4 - Simulation

37. Statistical Analysis of Simulation Results (2 of 2)
Formulas for 95% confidence limits:
upper confidence limit
lower confidence limit
where is the mean and  the standard deviation from a sample of size n from any population.
We can be 95% confident that the true population mean will be between the upper confidence limit and lower confidence limit.
Chapter 4 - Simulation

38. Statistical Analysis with Excel (1 of 3)
Simulation Results
Statistical Analysis with Excel (1 of 3)
Simulation with maintenance program.
Exhibit 4.9
Chapter 4 - Simulation

39. Statistical Analysis with Excel (2 of 3)
Simulation Results
Statistical Analysis with Excel (2 of 3)
Exhibit 4.10
Chapter 4 - Simulation

40. Statistical Analysis with Excel (3 of 3)
Simulation Results
Statistical Analysis with Excel (3 of 3)
Exhibit 4.11
Chapter 4 - Simulation

41. Crystal Ball
Overview
Many realistic simulation problems contain more complex probability distributions than those used in the examples.
However there are several simulation add-ins for Excel that provide a capability to perform simulation analysis with a variety of probability distributions in a spreadsheet format.
Crystal Ball, published by Decisioneering, is one of these.
Crystal Ball is a risk analysis and forecasting program that uses Monte Carlo simulation to provide a statistical range of results.
Chapter 4 - Simulation

42. Simulation of Profit Analysis Model (1 of 17)
Crystal Ball
Simulation of Profit Analysis Model (1 of 17)
Recap of Western Clothing Company break-even and profit analysis:
Price (p) for jeans is $23; variable cost (cv) is $8; fixed cost (cf ) is $10,000.
Profit Z = vp - cf - vc; break-even volume v = cf/(p - cv) = 10,000/(23-8) = pairs.
Chapter 4 - Simulation

43. Simulation of Profit Analysis Model (2 of 17)
Crystal Ball
Simulation of Profit Analysis Model (2 of 17)
Modifications to demonstrate Crystal Ball:
Assume volume is now volume demanded and is defined by a normal probability distribution with mean of 1,050 and standard deviation of 410 pairs of jeans.
Price is uncertain and defined by a uniform probability distribution from $20 to $26.
Variable cost is not constant but defined by a triangular probability distribution.
Will determine average profit and profitability with given probabilistic variables.
Chapter 4 - Simulation

44. Simulation of Profit Analysis Model (3 of 17)
Crystal Ball
Simulation of Profit Analysis Model (3 of 17)
Exhibit 4.12
Chapter 4 - Simulation

45. Simulation of Profit Analysis Model (4 of 17)
Crystal Ball
Simulation of Profit Analysis Model (4 of 17)
Exhibit 4.13
Chapter 4 - Simulation

46. Simulation of Profit Analysis Model (5 of 17)
Crystal Ball
Simulation of Profit Analysis Model (5 of 17)
Exhibit 4.14
Chapter 4 - Simulation

47. Simulation of Profit Analysis Model (6 of 17)
Crystal Ball
Simulation of Profit Analysis Model (6 of 17)
Exhibit 4.15
Chapter 4 - Simulation

48. Simulation of Profit Analysis Model (7 of 17)
Crystal Ball
Simulation of Profit Analysis Model (7 of 17)
Exhibit 4.16
Chapter 4 - Simulation

49. Simulation of Profit Analysis Model (8 of 17)
Crystal Ball
Simulation of Profit Analysis Model (8 of 17)
Exhibit 4.17
Chapter 4 - Simulation

50. Simulation of Profit Analysis Model (9 of 17)
Crystal Ball
Simulation of Profit Analysis Model (9 of 17)
Exhibit 4.18
Chapter 4 - Simulation

51. Simulation of Profit Analysis Model (10 of 17)
Crystal Ball
Simulation of Profit Analysis Model (10 of 17)
Exhibit 4.19
Chapter 4 - Simulation

52. Simulation of Profit Analysis Model (11 of 17)
Crystal Ball
Simulation of Profit Analysis Model (11 of 17)
Exhibit 4.20
Chapter 4 - Simulation

53. Simulation of Profit Analysis Model (12 of 17)
Crystal Ball
Simulation of Profit Analysis Model (12 of 17)
Exhibit 4.21
Chapter 4 - Simulation

54. Simulation of Profit Analysis Model (13 of 17)
Crystal Ball
Simulation of Profit Analysis Model (13 of 17)
Exhibit 4.22
Chapter 4 - Simulation

55. Simulation of Profit Analysis Model (14 of 17)
Crystal Ball
Simulation of Profit Analysis Model (14 of 17)
Exhibit 4.23
Chapter 4 - Simulation

56. Simulation of Profit Analysis Model (15 of 17)
Crystal Ball
Simulation of Profit Analysis Model (15 of 17)
Exhibit 4.24
Chapter 4 - Simulation

57. Simulation of Profit Analysis Model (16 of 17)
Crystal Ball
Simulation of Profit Analysis Model (16 of 17)
Exhibit 4.25
Chapter 4 - Simulation

58. Simulation of Profit Analysis Model (17 of 17)
Crystal Ball
Simulation of Profit Analysis Model (17 of 17)
Exhibit 4.26
Chapter 4 - Simulation

59. Verification of the Simulation Model (1 of 2)
Analyst wants to be certain that model is internally correct and that all operations are logical and mathematically correct.
Testing procedures for validity:
Run a small number of trials of the model and compare with manually derived solutions.
Divide the model into parts and run parts separately to reduce complexity of checking.
Simplify mathematical relationships (if possible) for easier testing.
Compare results with actual real-world data.
Chapter 4 - Simulation

60. Verification of the Simulation Model (2 of 2)
Analyst must determine if model starting conditions are correct (system empty, etc).
Must determine how long model should run to insure steady-state conditions.
A standard, fool-proof procedure for validation is not available.
Validity of the model rests ultimately on the expertise and experience of the model developer.
Chapter 4 - Simulation

61. Some Areas of Simulation Application
Queuing
Inventory Control
Production and Manufacturing
Finance
Marketing
Public Service Operations
Environmental and Resource Analysis
Chapter 4 - Simulation

62. Example Problem Solution (1 of 6)
Data
Willow Creek Emergency Rescue Squad
Minor emergency requires two-person crew, regular, a three-person crew, and major emergency, a five- person crew.
Chapter 4 - Simulation

63. Example Problem Solution (2 of 6)
Distribution of number of calls per night and emergency type:
Required: Manually simulate 10 nights of calls; determine average number of calls each night and maximum number of crew members that might be needed on any given night.
Chapter 4 - Simulation

64. Example Problem Solution (3 of 6)
Solution Step 1: Develop random number ranges for the probability distributions.
Chapter 4 - Simulation

65. Example Problem Solution (4 of 6)
Step 2: Set Up a Tabular Simulation (use second column of random numbers in Table 4.3).
Chapter 4 - Simulation

66. Example Problem Solution (5 of 6)
Step 2 continued:
Chapter 4 - Simulation

67. Example Problem Solution (6 of 6)
Step 3: Compute Results:
average number of minor emergency calls per night = 10/10 =1.0
average number of regular emergency calls per night = 14/10 = 1.4
average number of major emergency calls per night = 3/10 = 0.30
If calls of all types occurred on same night, maximum number of squad members required would be 4.
Chapter 4 - Simulation

68. Chapter 4 - Simulation