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UIUC ECE 307 - Simulation

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ECE 307 – Techniques for Engineering Decisions 14. SimulationSIMULATIONSIMULATION EXAMPLESIMULATION EXAMPLESIMULATION EXAMPLESIMULATION EXAMPLESIMULATION EXAMPLEBASIC ALGORITHMSIMULATION EXAMPLESIMULATION EXAMPLEGENERATION OF RANDOM DRAWSGENERATION OF RANDOM DRAWSGENERATION OF RANDOM DRAWSSOFT PRETZEL EXAMPLESOFT PRETZEL EXAMPLESOFT PRETZEL EXAMPLEMANUFACTURING CASE STUDYSlide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27SIMULATION RESULTSSIMULATION RESULTSc.d.f.s OF THE TWO PROCESSES© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 1ECE 307 – Techniques for Engineering Decisions14. SimulationGeorge GrossDepartment of Electrical and Computer EngineeringUniversity of Illinois at Urbana–Champaign© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 2SIMULATION Simulation provides a systematic approach to deal with uncertainty by “flipping a coin” or “ rolling a die” to represent the outcome or realization of each uncertain event In many real-world situations, simulation may be the only viable means to quantitatively deal with a problem under uncertainty Effective simulation requires implementation of appropriate approximations at many and, some-times, at possibly every stage of the problem© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 3SIMULATION EXAMPLE The problem is concerned with the purchase of fabric by a fashion designer The two choices offered by textile suppliers are:supplier 1: fixed price – constant 2 $/ydsupplier 2: variable price dependent on quantity at2.10 $/yd for the first 20,000 yd;1.90 $/yd for the next 10,000 yd;1.70 $/yd for the next 10,000 yd; 1.50 $/yd thereafter© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 4SIMULATION EXAMPLE The purchaser is uncertain about the demand but determines an appropriate model is: The decision may be represented in form of the following decision branches:© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 5SIMULATION EXAMPLE© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 6SIMULATION EXAMPLE Supplier 1 has a simple linear cost function Supplier 2 has a more complicated scheme to evaluate costs: in effect, the range of the demand and the corresponding probability for to be in a particular segment of the range must be known, as well as the expected value of for each range© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 7SIMULATION EXAMPLE We simulate the situation in the decision tree by “drawing multiple samples from the appropriatepopulation” We systematically tabulate the results and evaluate the required statistics The algorithm for the simulation consists of a few simple steps which are repeated until we collect an appropriatly sized sample© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 8BASIC ALGORITHMStep 0 : store the distribution ; determine , the maximum number of draws; setStep 1 : if , stop; else setStep 2 : draw a random sample from the normal distributionStep 3 : evaluate the outcomes on both branches; enter each outcome into the data base and return to Step 1© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 9SIMULATION EXAMPLE Application of the algorithm allows the determi–nation of the histogram of the cost figures and then the evaluation of the expected costs For the assumed demand, for supplier 1, we have the straightforward case ofand© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 10SIMULATION EXAMPLEand the use of the algorithm may be bypassed For the supplier 2, the algorithm is applied for the random draws The actual simulation is an exercise left to the reader© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 11GENERATION OF RANDOM DRAWS A key issue is the generation of random draws for which we need a random number generator There are various random number generator algorithms One intuitive scheme is based on the use of a uniformly distributed r.v. between 0 and 1© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 12GENERATION OF RANDOM DRAWS011.0probability© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 13GENERATION OF RANDOM DRAWS We draw a random value of , say and work through the c.d.f. to get the value of the r.v. 1.00yprobabilityx *y *© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 14SOFT PRETZEL EXAMPLE The market size is unknown, but we assume that the market size is a normally distributed r.v. with We are interested to determine the fraction of the market the new company is able to capture We model the distribution of using the discrete distribution tabulated below:© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 1516 0.1519 0.3525 0.3528 0.15SOFT PRETZEL EXAMPLE© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 16SOFT PRETZEL EXAMPLE Sales price of a pretzel is $ 0.50 Variable costs are represented by a uniformly distributed r.v. in the range [0.08 , 0.12] $/pretzel Fixed costs are also random The contributions to profits are given byand may be evaluated via simulation We can use simulation to approximate© 2006 – 2021 George Gross, University of Illinois at


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