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1Chapter 14 - Simulation Introduction to Management Science 9 th Edition by Bernard W. Taylor III Chapter 14 Simulation © 2007 Pearson Education

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2Chapter 14 - Simulation 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 Topics

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3Chapter 14 - Simulation 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. Overview

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4Chapter 14 - Simulation 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. Monte Carlo Process

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5Chapter 14 - Simulation Table 14.1 Probability Distribution of Demand for Laptop PCs In the Monte Carlo process, values for a random variable are generated by sampling from a probability distribution. 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. Example: ComputerWorld demand data for laptops selling for $4,300 over a period of 100 weeks. Monte Carlo Process Use of Random Numbers (1 of 10)

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6Chapter 14 - Simulation The purpose of the Monte Carlo process is to generate the random variable, demand, by sampling from the probability distribution P(x). 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 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. The segment at which the wheel stops indicates demand for one week. Monte Carlo Process Use of Random Numbers (2 of 10)

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

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8Chapter 14 - Simulation Figure 14.2 Numbered Roulette Wheel Monte Carlo Process Use of Random Numbers (4 of 10) When wheel is spun actual demand for PCs is determined by a number at rim of the wheel. When wheel is spun actual demand for PCs is determined by a number at rim of the wheel.

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9Chapter 14 - Simulation Table 14.2 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. 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. Transfer random numbers for each demand value from roulette wheel to a table.

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

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11Chapter 14 - Simulation Repeating selection of random numbers simulates demand for a period of time. 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 demand = 31/15 = 2.07 laptop PCs per week. Estimated average revenue = $133,300/15 = $8,886.67. Estimated average revenue = $133,300/15 = $8,886.67. Monte Carlo Process Use of Random Numbers (7 of 10)

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

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

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14Chapter 14 - Simulation The more periods simulated, the more accurate the results. 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. 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. 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. When analytical analysis is not possible, there is no analytical standard of comparison thus making validation even more difficult. Monte Carlo Process Use of Random Numbers (10 of 10)

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15Chapter 14 - Simulation As simulation models get more complex they become impossible to perform manually. 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). 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. Random numbers are typically generated on the computer using a numerical technique and thus are not true random numbers but pseudorandom numbers. Computer Simulation with Excel Spreadsheets Generating Random Numbers (1 of 2)

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16Chapter 14 - Simulation Artificially created random numbers must have the following characteristics: The random numbers must be uniformly distributed. The random numbers must be uniformly distributed. The numerical technique for generating the numbers must be efficient. The numerical technique for generating the numbers must be efficient. The sequence of random numbers should reflect no pattern. The sequence of random numbers should reflect no pattern. Computer Simulation with Excel Spreadsheets Generating Random Numbers (2 of 2)

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

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

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

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

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

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

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23Chapter 14 - Simulation 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 Simulation of a Queuing System Burlingham Mills Example (2 of 3)

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24Chapter 14 - Simulation Simulation of a Queuing System Burlingham Mills Example (3 of 3)Caveats: Results may be viewed with skepticism. Results may be viewed with skepticism. Ten trials do not ensure steady-state results. Ten trials do not ensure steady-state results. Starting conditions can affect simulation 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.

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

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26Chapter 14 - Simulation Continuous Probability Distributions

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27Chapter 14 - Simulation 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. 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: 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(r i ), value of x for a given value of r i.

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

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29Chapter 14 - Simulation Table 14.9 Revised Probability Distribution of Machine Repair Time with the Maintenance Program Machine Breakdown and Maintenance System Simulation (3 of 6) Revised probability of time between machine breakdowns: Revised probability of time between machine breakdowns: f(x) = x/18, 0 x 6 weeks where x = weeks between machine breakdowns x = 6*sqrt(r i ) x = 6*sqrt(r i )

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

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

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32Chapter 14 - Simulation 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. 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. 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. 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. 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. Efficient computer simulation required to do this.

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

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

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35Chapter 14 - Simulation Outcomes of simulation modeling are statistical measures such as averages. 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. 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. Confidence limits are developed for the analysis of the statistical validity of simulation results. Statistical Analysis of Simulation Results (1 of 2)

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36Chapter 14 - Simulation Formulas for 95% confidence limits: 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. We can be 95% confident that the true population mean will be between the upper confidence limit and lower confidence limit. Statistical Analysis of Simulation Results (2 of 2)

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

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

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

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40Chapter 14 - Simulation 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. Verification of the Simulation Model (1 of 2)

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41Chapter 14 - Simulation 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. Verification of the Simulation Model (2 of 2)

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

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43Chapter 14 - Simulation Data Willow Creek Emergency Rescue Squad Minor emergency requires two-person crew, regular, a three-person crew, and major emergency, a five- person crew. Example Problem Solution (1 of 6)

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44Chapter 14 - Simulation 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. Example Problem Solution (2 of 6)

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

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

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

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48Chapter 14 - Simulation 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. Example Problem Solution (6 of 6)

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

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50Chapter 14 - Simulation Crystal Ball Overview Many realistic simulation problems contain more complex probability distributions than those used in the examples. 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. 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, 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. Crystal Ball is a risk analysis and forecasting program that uses Monte Carlo simulation to provide a statistical range of results.

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51Chapter 14 - Simulation Recap of Western Clothing Company break-even and profit analysis: Recap of Western Clothing Company break-even and profit analysis: Price (p) for jeans is $23; variable cost (c v ) is $8; fixed cost (c f ) is $10,000. Price (p) for jeans is $23; variable cost (c v ) is $8; fixed cost (c f ) is $10,000. Profit Z = vp - c f - v c ; break-even volume v = c f /(p - c v ) = 10,000/(23-8) = 666.7 pairs. Profit Z = vp - c f - v c ; break-even volume v = c f /(p - c v ) = 10,000/(23-8) = 666.7 pairs. Crystal Ball Simulation of Profit Analysis Model (1 of 17)

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52Chapter 14 - Simulation Modifications to demonstrate Crystal Ball: 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. 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. 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. Variable cost is not constant but defined by a triangular probability distribution. Will determine average profit and profitability with given probabilistic variables. Will determine average profit and profitability with given probabilistic variables. Crystal Ball Simulation of Profit Analysis Model (2 of 17)

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53Chapter 14 - Simulation Exhibit 14.12 Crystal Ball Simulation of Profit Analysis Model (3 of 17)

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54Chapter 14 - Simulation Exhibit 14.13 Crystal Ball Simulation of Profit Analysis Model (4 of 17)

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55Chapter 14 - Simulation Exhibit 14.14 Crystal Ball Simulation of Profit Analysis Model (5 of 17)

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56Chapter 14 - Simulation Exhibit 14.15 Crystal Ball Simulation of Profit Analysis Model (6 of 17)

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57Chapter 14 - Simulation Exhibit 14.16 Crystal Ball Simulation of Profit Analysis Model (7 of 17)

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58Chapter 14 - Simulation Exhibit 14.17 Crystal Ball Simulation of Profit Analysis Model (8 of 17)

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59Chapter 14 - Simulation Exhibit 14.18 Crystal Ball Simulation of Profit Analysis Model (9 of 17)

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60Chapter 14 - Simulation Exhibit 14.19 Crystal Ball Simulation of Profit Analysis Model (10 of 17)

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61Chapter 14 - Simulation Exhibit 14.20 Crystal Ball Simulation of Profit Analysis Model (11 of 17)

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62Chapter 14 - Simulation Exhibit 14.21 Crystal Ball Simulation of Profit Analysis Model (12 of 17)

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63Chapter 14 - Simulation Exhibit 14.22 Crystal Ball Simulation of Profit Analysis Model (13 of 17)

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64Chapter 14 - Simulation Exhibit 14.23 Crystal Ball Simulation of Profit Analysis Model (14 of 17)

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65Chapter 14 - Simulation Exhibit 14.24 Crystal Ball Simulation of Profit Analysis Model (15 of 17)

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66Chapter 14 - Simulation Exhibit 14.25 Crystal Ball Simulation of Profit Analysis Model (16 of 17)

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67Chapter 14 - Simulation Exhibit 14.26 Crystal Ball Simulation of Profit Analysis Model (17 of 17)

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