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Introduction to Management Science 9 th Edition by Bernard W. Taylor III

Introduction to Management Science 9 th Edition by Bernard W. Taylor III. Chapter 14 Simulation. © 2007 Pearson Education . Chapter Topics. The Monte Carlo Process Computer Simulation with Excel Spreadsheets Simulation of a Queuing System Continuous Probability Distributions

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Introduction to Management Science 9 th Edition by Bernard W. Taylor III

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  1. Introduction to Management Science 9th Edition by Bernard W. Taylor III Chapter 14 Simulation © 2007 Pearson Education Chapter 14 - 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 • Verification of the Simulation Model • Areas of Simulation Application Chapter 14 - 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 14 - Simulation

  4. Monte Carlo Process • A large proportion of the applications of simulations are for probabilistic models. • The Monte Carlotechnique 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 14 - Simulation

  5. 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 14.1 Probability Distribution of Demand for Laptop PC’s Chapter 14 - Simulation

  6. 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 14 - Simulation

  7. Monte Carlo Process Use of Random Numbers (3 of 10) Figure 14.1 A Roulette Wheel for Demand Chapter 14 - Simulation

  8. 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 14.2 Numbered Roulette Wheel Chapter 14 - Simulation

  9. 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 14.2 Generating Demand from Random Numbers Chapter 14 - Simulation

  10. Monte Carlo Process Use of Random Numbers (6 of 10) • Select number from a random number table: Table 14.3 Random Number Table Chapter 14 - Simulation

  11. 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 = $8,886.67. Chapter 14 - Simulation

  12. Monte Carlo Process Use of Random Numbers (8 of 10) Chapter 14 - Simulation

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

  14. 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 14 - Simulation

  15. 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 14 - Simulation

  16. 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 pattern. Chapter 14 - Simulation

  17. Simulation with Excel Spreadsheets (1 of 3) Exhibit 14.1 Chapter 14 - Simulation

  18. Simulation with Excel Spreadsheets (2 of 3) Exhibit 14.2 Chapter 14 - Simulation

  19. Simulation with Excel Spreadsheets (3 of 3) Exhibit 14.3 Chapter 14 - Simulation

  20. Computer Simulation with Excel Spreadsheets Decision Making with Simulation (1 of 2) • Revised ComputerWorld example; order size of one laptop each week. Exhibit 14.4 Chapter 14 - Simulation

  21. Computer Simulation with Excel Spreadsheets Decision Making with Simulation (2 of 2) • Order size of two laptops each week. Exhibit 14.5 Chapter 14 - Simulation

  22. Simulation of a Queuing System Burlingham Mills Example (1 of 3) Table 14.5 Distribution of Arrival Intervals Table 14.6 Distribution of Service Times Chapter 14 - Simulation

  23. 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 14 - Simulation

  24. 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. • 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 14 - Simulation

  25. Computer Simulation with Excel Burlingham Mills Example Exhibit 14.6 Chapter 14 - Simulation

  26. Continuous Probability Distributions Chapter 14 - Simulation

  27. 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(ri), value of x for a given value of ri. Chapter 14 - Simulation

  28. Machine Breakdown and Maintenance System Simulation (2 of 6) Table 14.8 Probability Distribution of Machine Repair Time Chapter 14 - Simulation

  29. 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(ri) Table 14.9 Revised Probability Distribution of Machine Repair Time with the Maintenance Program Chapter 14 - Simulation

  30. Machine Breakdown and Maintenance System Simulation (4 of 6) • Simulation of system without maintenance program (total annual repair cost of $84,000): Table 14.10 Simulation of Machine Breakdowns and Repair Times Chapter 14 - Simulation

  31. Machine Breakdown and Maintenance System Simulation (5 of 6) • Simulation of system with maintenance program (total annual repair cost of $42,000): Table 14.11 Simulation of Machine Breakdowns and Repair with the Maintenance Program Chapter 14 - Simulation

  32. 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 14 - Simulation

  33. Machine Breakdown and Maintenance System Simulation with Excel (1 of 2) • Original machine breakdown example: Exhibit 14.7 Chapter 14 - Simulation

  34. Machine Breakdown and Maintenance System Simulation with Excel (2 of 2) • Simulation with maintenance program. Exhibit 14.8 Chapter 14 - Simulation

  35. 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 14 - Simulation

  36. Statistical Analysis of Simulation Results (2 of 2) • Formulas for 95% confidence limits: upper confidence limit lower confidence limit where is the mean and s 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 14 - Simulation

  37. Simulation Results Statistical Analysis with Excel (1 of 3) • Simulation with maintenance program. Exhibit 14.9 Chapter 14 - Simulation

  38. Simulation Results Statistical Analysis with Excel (2 of 3) Exhibit 14.10 Chapter 14 - Simulation

  39. Simulation Results Statistical Analysis with Excel (3 of 3) Exhibit 14.11 Chapter 14 - Simulation

  40. 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 14 - Simulation

  41. 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 14 - Simulation

  42. Some Areas of Simulation Application • Queuing • Inventory Control • Production and Manufacturing • Finance • Marketing • Public Service Operations • Environmental and Resource Analysis Chapter 14 - Simulation

  43. 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 14 - Simulation

  44. 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 14 - Simulation

  45. Example Problem Solution (3 of 6) • Solution Step 1: Develop random number ranges for the probability distributions. Chapter 14 - Simulation

  46. Example Problem Solution (4 of 6) • Step 2: Set Up a Tabular Simulation (use second column of random numbers in Table 14.3). Chapter 14 - Simulation

  47. Example Problem Solution (5 of 6) • Step 2 continued: Chapter 14 - Simulation

  48. 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 14. Chapter 14 - Simulation

  49. End of chapter The rest of the transparencies are given as a brief overview of Crystall Ball software; which not included in the exam. Chapter 14 - Simulation

  50. 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 14 - Simulation

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