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## Chapter 4 Simulation

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**Introduction to Management Science**8th Edition by Bernard W. Taylor III Chapter 4 Simulation Chapter 4 - Simulation**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**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**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**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**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**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**Monte Carlo Process**Use of Random Numbers (3 of 10) Figure 4.1 A Roulette Wheel for Demand Chapter 4 - Simulation**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**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**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**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($133,300 = $4,300 31). Chapter 4 - Simulation**Monte Carlo Process**Use of Random Numbers (8 of 10) Chapter 4 - Simulation**Monte Carlo Process**Use of Random Numbers (9 of 10) • Average demand could have been calculated analytically: Chapter 4 - Simulation**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**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**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**Simulation with Excel Spreadsheets (1 of 3)**Exhibit 4.1 Chapter 4 - Simulation**Simulation with Excel Spreadsheets (2 of 3)**Exhibit 4.2 “Lookup” Chapter 4 - Simulation**Simulation with Excel Spreadsheets (3 of 3)**Exhibit 4.3 Chapter 4 - Simulation**=1+MAX(G6-H6,0)**Order one laptopeach week 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 Chapter 4 - Simulation**=2+MAX(G6-H6,0)**Order two laptopseach week Computer Simulation with Excel Spreadsheets Decision Making with Simulation (2 of 2) • Order size of two laptops each week. Exhibit 4.5 Chapter 4 - Simulation**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**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**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 statisticalerror 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**Computer Simulation with Excel**Burlingham Mills Example Exhibit 4.6 Chapter 4 - Simulation**1/2**0 4 Continuous Probability Distributions Chapter 4 - 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. • 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**Machine Breakdown and Maintenance System**Simulation (2 of 6) Table 4.8 Probability Distribution of Machine Repair Time Chapter 4 - Simulation**Machine Breakdown and Maintenance System**Simulation (3 of 6) • Revised probability of time between machine breakdowns: f(x) = x/18, 0 x6 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**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**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**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**Machine Breakdown and Maintenance System**Simulation with Excel (1 of 2) • Original machine breakdown example: Exhibit 4.7 Chapter 4 - Simulation**Machine Breakdown and Maintenance System**Simulation with Excel (2 of 2) • Simulation with maintenance program. Exhibit 4.8 Chapter 4 - Simulation**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**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**Simulation Results**Statistical Analysis with Excel (1 of 3) • Simulation with maintenance program. Exhibit 4.9 Chapter 4 - Simulation**Simulation Results**Statistical Analysis with Excel (2 of 3) Exhibit 4.10 Chapter 4 - Simulation**Simulation Results**Statistical Analysis with Excel (3 of 3) Exhibit 4.11 Chapter 4 - Simulation**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**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) = 666.7 pairs. Chapter 4 - Simulation**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 probabilitydistribution 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**Crystal Ball**Simulation of Profit Analysis Model (3 of 17) Exhibit 4.12 Chapter 4 - Simulation**Crystal Ball**Simulation of Profit Analysis Model (4 of 17) Exhibit 4.13 Chapter 4 - Simulation**Crystal Ball**Simulation of Profit Analysis Model (5 of 17) Exhibit 4.14 Chapter 4 - Simulation**Crystal Ball**Simulation of Profit Analysis Model (6 of 17) Exhibit 4.15 Chapter 4 - Simulation**Crystal Ball**Simulation of Profit Analysis Model (7 of 17) Exhibit 4.16 Chapter 4 - Simulation**Crystal Ball**Simulation of Profit Analysis Model (8 of 17) Exhibit 4.17 Chapter 4 - Simulation**Crystal Ball**Simulation of Profit Analysis Model (9 of 17) Exhibit 4.18 Chapter 4 - Simulation