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

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

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

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

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

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

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

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 = $8,886.67 ($133,300 = $4,300  31). Chapter 4 - Simulation

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

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

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

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) “Lookup” Exhibit 4.2 Chapter 4 - Simulation

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

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

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

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 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

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

Continuous Probability Distributions 4 1/2 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

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

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

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

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

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

Statistical Analysis with Excel (3 of 3) 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

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) = 666.7 pairs. Chapter 4 - Simulation

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Chapter 4 - Simulation