利用MINITAB做蒙特卡洛模拟版

1、最新资料推荐Doi ng Monte Carlo Simulation in Min itab Statistical SoftwareDoing Monte Carlo simulati ons inMin itab Statistical Software isvery easy. This article illustrates how to use Min itabfor Monte Carlosimulations using both a known engineering formula and a DOE equation.by Paul Sheehy and Est on M

2、artzMonte Carlo simulation uses repeated random sampling to simulate data for a given mathematical model and evaluate the outcome. Thismethod was in itially applied back in the 1940s, whe n scie ntists worki ng on the atomic bomb used it to calculate the probabilities of onefissioning uranium atom c

3、ausing a fission reaction in another. With uranium in short supply, there was little room for experimental trial anderror . The scie ntists discovered that as long as they created eno ugh simulated data, they could compute reliable probabilitiesand reducethe amo unt of uranium n eeded for testi ng.T

4、oday, simulated data is rout in elyused in situati ons where resources are limited or gatheri ng real data would be too expe nsiveorimpractical. By using Minitab ability to easily create random data, you can use Monte Carlo simulation to: Simulate the range of possible outcomes to aid in decisi on-m

5、ak ing Forecast finan cial results or estimate project timeli nes Un dersta nd the variability in a process or system Find problems with in a process or system Man age risk by un dersta nding cost/be nefit relati on shipsSteps in the Monte Carlo ApproachDepe nding on the nu mber of factors in volved

6、, simulati ons can be very complex. But at a basic level, all Monte Carlo simulati ons have foursimple steps:1. Ide ntify the Tran sfer Equati onTo do a Monte Carlo simulati on, you n eed a qua ntitativemodel of the bus in ess activity,pla n, or process you wish to explore. Themathematical expressi

7、on of your process is called thefran sfer equati on.” This may be a known engin eeri ng or bus in ess formula, or it maybe based on a model created from a desig ned experime nt (DOE) or regressi on an alysis.2. Define the In put ParametersFor each factor in your tran sfer equati on, determ ine how i

8、ts data are distributed. Some in puts may follow the no rmal distributi on, whileothers follow a tria ngular or uniform distributi on. You the n n eed to determ ine distributi on parameters for each in put. For in sta nce, youwould n eed to specify the mea n and stan dard deviati on for in puts that

9、 follow a no rmal distributi on.3. Create Ra ndom DataTo do valid simulati on, you must create a very large, ran dom data set for each in put somethi ng on the order 100,000 in sta nces. Theseran dom data points simulate the values that would be see n over a long period for each in put. Min itab can

10、 easily create ran dom data thatfollow almost any distributi on you are likely to encoun ter.4. Simulate and An alyze Process OutputWith the simulated data in place, you can use your tran sfer equati on to calculate simulated outcomes. Running a large eno ugh qua ntity ofsimulated in put data throug

11、h your model will give you a reliable in dicati on of what the process will output over time, give n the an ticipated variati on in the in puts.Those are the steps any Monte Carlo simulation needs to follow. Here how to apply them in Minitab.Monte Carlo Using a Known Engineering FormulaA manu factur

12、i ng compa ny n eeds to evaluate the desig n of a proposed product: a small pist on pump that must pump 12 ml of fluid perminu te. You want to estimate the probable performa nce over thousa nds of pumps, give n n atural variati on in pist on diameter (D), strokelength (L), and strokes per minute (RP

13、M). Ideally, the pump flow across thousands of pumps will have a standard deviation no greater than 0.2 ml.Step 1: Identify the Transfer EquationThe first step in doing a Monte Carlo simulation is to determine the transfer equation. In this case, you can simply use an establishedengineering formula

14、that measures pump flow:2Flow (in ml) = MD/2)? L ? RPMStep 2: Define the In put ParametersNow you must defi ne the distributi on and parameters of each in put used in the tran sfer equati on. The pump pist on diameter and strokelen gth are known, but you must calculate the strokes-per-m inute (RPM)

15、n eeded to atta in the desired 12 ml/m inute flow rate. Volumepumped per stroke is give n by this equati on:XD/2)2 * LGive n D = 0.8 and L = 2.5, each stroke displaces 1.256 ml. So to achieve a flow of 12 ml/mi nute the RPM is 9.549.Based on the performa nce of other pumps your facility has manu fac

16、tured, you can say that pist on diameter is no rmally distributed with amean of 0.8 cm and a standard deviation of 0.003 cm. Stroke length is normally distributed with a mean of 2.5 cm and a standard deviationof 0.15 cm. Fin ally, strokes per mi nute is no rmally distributed with a mea n of 9.549 RP

17、M and a sta ndard deviation of 0.17 RPM.Step 3: Create Ra ndom DataNow you e ready to set up the simulation in Minitab. With Minitab you can instantaneously create 100,000 rows of simulated data. Startingwith the simulated pist on diameter data, chooseCalc Ran dom Data Normal.In the dialog box, en t

18、er 100,000 in Number of rows ofdata to gen erate, and en terD ” as the colum n in which to store the data. En ter the mean and sta ndard deviati on for pist on diameter in theappropriate fields. Press OK to populate the worksheet with 100,000 data points randomly sampled from the specified normal di

19、stribution.6The n simply repeat this process for Stroke Len gth (L) and Strokes per Minute (RPM).Step 4: Simulate and An alyze Process OutputNow create a fourth colum n in the worksheet, Flow, to hold the results of your process output calculati ons. With the ran domly gen eratedCalc Calculatorin pu

20、t data in place, you can set up Min itab calculator to calculate the output and store it in the Flow colum n. Go toand set up the flow equation like this:and select the Flow colu mn. Min itab willMin itab will quickly calculate the output for each row of simulated data.Now you re ready to look at th

21、e results. SelectStat Basic Statistics Graphical Summarygen erate a graphical summary that in eludes four graphs: a histogram of data with an overlaid no rmal curve, boxplot, and con fide neein tervals for the mea n and the media n. The graphical summary also displays An ders on-Darl ing Normality T

22、est results, descriptive statistics, and con fide nce in tervals for the mea n, media n, and sta ndard deviati on.HunEMHfciftThe graphical summary of your Monte Carlo simulation output will look like this:Summary for FlowtJ T513 314mmA-SQUrtd2M0.005MwnHOCH5tDvOJ57wise 审0.573Swaney0.043400.011133aN1D

23、OOODMlwnufflB.882lit Qwrtice11A91MedianIL996如 QuMJiensn15.59*Anderson-DttrtiHg htormality Tea95% GorrMence IrwwvWl fer11,999U.D09笳監 务denMr Median11妙L2.OO295% ConMwc* Interval for StiDwvOJSJ0.760For the ran dom data gen erated to write this article, the mea n flow rate is 12.004 based on 100,000 samp

24、les. On average, we are on target,but the smallest value was 8.882 and the largest was 15.594. That quite a ran ge. The tran smitted variati on (of all comp onen ts) resultsin a sta ndard deviati on of 0.757 ml, far exceedi ng the 0.2 ml target. Also, we see that the 0.2 ml target falls outside of t

25、he con fide ncein terval for the sta ndard deviati on.It looks like this pump design exhibits too much variation and needs to be further refined before it goes into production; Monte Carlosimulation with Minitab let us find that out without incurring the expense of manufacturing and testing thousand

26、s of prototypes.Lest you wonder whether these simulated results hold up, try it yourself! Creating different sets of simulated random data will result inminor variati ons, but the end resultan un acceptable amo unt of variati on in the flow ratewill be con siste nt every time. Thatof the Mon te Carl

27、o method.Monte Carlo Using a DOE Response EquationWhat if you donknow what equation to use, or you are trying to simulate the outcome of a unique process?the powerAn electro nics manu facturer has assig ned you to improve its electroclea ning operati on, which prepares metal parts for electroplat in

28、g.Electroplat ing lets manu facturers coat raw materials with a layer of a differe nt metal to achieve desired characteristics. Plat ing will not adhere to a dirty surface, so the compa ny has a con ti nu ous-flow electroclea ning system that conn ects to an automatic electroplat ing machi ne. A con

29、 veyer dips each part into a bath which sends voltage through the part, clea ning it. In adequate clea ning results in a high Root Mean Square Average Rough ness value, or RMS, and poor surface fini sh. Properly clea ned parts have a smooth surface and a low RMS.To optimize the process, you can adju

30、st two critical in puts: voltage (Vdc) and curre nt den sity (ASF). For your electroclea ning method, the typical engineering limits for Vdc are 3 to 12 volts. Limits for current density are 10 to 150 amps per square foot (ASF).Step 1: Ide ntify the Tran sfer Equati onYou cannot use an established t

31、extbook formula for this process, but you can set up a Resp onse Surface DOE in Min itab to determ ine thetran sfer equati on. Resp onse surface DOEs are ofte n used to optimize the resp onse by finding the best sett ings for a vital few con trollable factors.In this case, the response will be the s

32、urface quality of parts after they have been cleaned.To create a resp onse surface experime nt in Min itab, chooseStat DOE Resp onse Surface Create Resp onse Surface Desig nBecause we have two factorsvoltage (Vdc) and current density (ASF)we lselect a two-factor central composite design, which has 1

33、3runs.After Minitab creates your designed experiment, you need to perform your 13 experimental runs, collect the data, and record the surfaceroughness of the 13 finished parts. Minitab makes it easy to analyze the DOE results, reduce the model, and check assumptions usingresidual plots. Using the fi

34、nal model and Minitab response optimizer , you can find the optimum settings for your variables. In this case,you set volts to 7.74 and ASF to 77.8 to obtain a roughness value of 39.4.The resp onse surface DOE yields the follow ing tran sfer equati on for the Monte Carlo simulati on:2 2Roughness = 9

35、57.8- 189.4(Vdc)- 4.81(ASF) + 12.26(Vdc) + 0.0309(ASF)Step 2: Define the In put ParametersNow you can set the parametric defi niti ons for your Monte Carlo simulati on in puts. (The stan dard deviati ons must be known or estimatedbased on existing process knowledge.) Volts are normally distributed w

36、ith a mean of 7.74 Vdc and a standard deviation of 0.14 Vdc. Ampsper Square Foot (ASF) are no rmally distributed with a mea n of 77.8 ASF and a sta ndard deviati on of 3 ASF.Step 3: Create Ra ndom DataWith the parameters defi ned, itsimple to create 100,000 rows of simulated data for our two in puts

37、 using Min itab Calc Ran domData Normal dialog.Step 4: Simulate and An alyze Process OutputNow we can use the Calculator to en ter our formula, followed byStat Basic Statistics Graphical Summary.Summary for RMS roughness956 Corrfvdna IntarvalcAndegbiCwling Normality TestASquared P Wlu 0.005MW39劇网10,271Skewness2.01510KurtositN10000039.3M7Med on披加QuvtileMaxuTH