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Chapter 14 Simulation Chapter Topics • Monte Carlo Process • Statistical Analysis of Simulation Results • Verification of the Simulation Model • Computer Simulation with Excel Spreadsheets • Areas of Simulation Application Overview • Simulation replaces physical systems • 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 with other OR techniques Monte Carlo Process • Many applications of simulations are for probabilistic models • Monte Carlo technique: a technique for selecting numbers randomly from a probability distribution • Generate the random variable, demand, by sampling from the probability distribution P (x) • Example: Demand data for an item selling for $100 over a period of 100 weeks Demand/ Week Freq. of Demand Probability 0 1 2 3 4 20 40 20 10 10 0.2 0.4 0.2 0.1 0.1 Cumulative Corresponding Probability RN 0.2 0.6 0.8 0.9 1.00 0-19 20-59 60-79 80-89 90-99 Monte Carlo Process Use of Random Numbers • Select number from a random number table: Monte Carlo Process Use of Random Numbers • Repeat selection of random numbers to simulate demand (say for 15 week) • Calculate average demand = 31/15 = 2.07 units per week • Estimated average revenue = $3100/15 =206 • Expected average demand (analytically): n E( x) P( x ) x i1 E( x) (.20)(0) (.40)(1) (.20)(2) (.10)(3) (.10)(4) 1.5 units/wk i i Monte Carlo Process Use of Random Numbers • More periods simulated, the more accurate the results • Have to have enough trials in order to have identical results (reach steady state) • Often difficult to validate results of simulation • When reaches the steady state, simulation model truly replicates reality • When analytical analysis is not possible, there is no comparison; validation even more difficult Computer Simulation with Excel Spreadsheets Generating Random Numbers (1 of 2) • Random numbers are typically generated using a numerical technique • Thus are not true random numbers but pseudorandom numbers • Random numbers must have the following characteristics: • Must be uniformly distributed • Numerical technique used for generating the numbers must be efficient • Sequence of random numbers should not reflect any pattern Simulation with Excel Spreadsheets (1 of 3)