MIT 2.810 Fall 2016 Homework 3 Solutions 1 MIT 2.810 Manufacturing Processes and Systems Homework 3 Solutions - Process Control - 2016 Problem 1. Control Charts The data shown in Table 1 are x ¯ and R values for 24 samples of size n = 5 taken from a process producing bearings. The measurements are made on the inside diameter of the bearing, with only the last three decimals recorded (i.e. 34.5 should be 0.50345). (a) Set up x ¯ and R charts on this process. Does the process seem to be in statistical control? (b) If specifications on the diameter are 0.503 ± 0.0010, find the percentage of non-conforming bearings produced by this process. What assumption(s) do you have to make to determine this number? Sample Number x ¯ R Sample Number x ¯ R 1 34.5 3 13 35.4 8 2 34.2 4 14 34.0 6 3 31.6 4 15 37.1 5 4 31.5 4 16 34.9 7 5 35.0 5 17 33.5 4 6 34.1 6 18 31.7 3 7 32.6 4 19 34.0 8 8 33.8 3 20 35.1 4 9 34.8 7 21 33.7 2 10 33.6 8 22 32.8 1 11 31.9 3 23 33.5 3 12 38.6 9 24 34.2 2 Table 1: Bearing diameter data Answer: The center line and limits for the x ¯ and R charts are given below: Chart Center Line Control Limits x ¯ ± ! R UCL = D 4 , LCL = D 3
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From the x̄ chart, we can conclude that the process is out of control based on themeasurements of the 12th and 15th samples. Out of 24 samples, we have 2 samples fallingoutside the3σ limitsoraprobabilityof8.3%.Weknow that for thenormaldistribution, theprobabilitythatapointwillfalloutsidethe3σlimitsis0.27%.Sothisoccurrenceisveryunusualandtheprocessneedstobeevaluated.NotethattheRchartdoesnotrevealanyirregularitiesintheprocess.ButthechartwouldstillbeusefulifweweretoapplytheWesternElectricrules(seeProblem3)heretogiveawarningsignalofaprocesslikelytogooutofcontrol.
Problem2.ControlChartsThefillvolumeofsoft-drinkbeveragebottlesisanimportantqualitycharacteristic.Thevolumeismeasured (approximately)byplacingagaugeover thecrownandcomparing theheightofthe liquid in the neck of the bottle against a coded scale. On this scale, a reading of zerocorrespondstothecorrect fillheight.Fifteensamplesofsizen=10havebeenanalyzed,andthefillheightsareshowninTable2.
a. Setupx̄andscontrolchartsonthisprocess.Doestheprocessexhibitstatisticalcontrol?Ifnecessary,constructrevisedcontrollimits.
b. SetupanRchart,andcompareitwiththeschartinparta.SampleNumber
Thoserulesapplytoonesideofthecenterlineatatime.Therefore,apointabovetheupperwarninglimitfollowedimmediatelybyapointbelowthelowerwarninglimitwouldnotsignalanout-of-controlalarm.Calculate the probability of the first two patterns occurring assuming the data points areindependentbutidenticallydistributedwithanormaldistribution.
calculate theprobabilityof apoint fallingoutside thewarning limit.Wehave r points fallingbeyondawarninglimitand(n-r)pointsfallingbeforeit.Sincethepointsareindependent,theprobabilityofapatternofpointsistheproductoftheirindividualprobabilities.Answer:FromProf.Hardt’slecturenotes,weknowthattheareaunderthenormaldistributioncurveforvariousspreads,zσ,isapproximatelywrittenas:
Within1σ 0.68Within2σ 0.95Within3σ 0.997
Therefore, by symmetry, theareabetween1σand2σ limitsoneither side is (0.95-0.68)/2=0.135.Theareabetween2σand3σlimitsoneithersideis(0.997-0.95)/2=0.0235.1. Theprobabilitythatonepointplotsoutsidethe3σlimitsis(1–0.997)=0.003or0.3%.2. Theprobabilitythatapointwillfallbeyondthe2σlimitononesideofthecenterlineis(1-