1、Learning ObjectivesBrief Revision on Process Potential vs Process PerformanceWithin vs Overall Process CapabilityIntroduction to Z-scoreProcess Capability for Non-Normal DataCycle-Time(Exponential Distribution)Reject Rate(Binomial Distribution)Defect Rate(Poisson Distribution)RevisionProcess Capabil
2、ityProcess Capability is the inherent reproducibility of a processs output.It measures how well the process is currently behaving with respect to the output specifications.It refers to the uniformity of the process.Capability is often thought of in terms of the proportion of output that will be with
3、in product specification tolerances.The frequency of defectives produced may be measured ina)percentage(%)b)parts per million(ppm)c)parts per billion(ppb)Process CapabilityProcess Capability studies can indicate the consistency of the process output indicate the degree to which the output meets spec
4、ifications be used for comparison with another process or competitorProcess Capability Indices Two measures of process capability Process PotentialCp Process PerformanceCpuCplCpkProcess PotentialThe Cp index assesses whether the natural tolerance(6)of a process is within the specification limits.6LS
5、LUSLToleranceNaturalTolerancegEngineerinCpProcess PerformanceThe Cpk index relates the scaled distance between the process mean and the nearest specification limit.3USLCpu3LSLCplplpupkCCMinimumC,Process Potential vs Process Performancea)Cp=2Cpk=2b)Cp=2Cpk=1c)Cp=2Cpk 1Cp Cpk Missed OpportunityWithin
6、vs Overall CapabilityWithin Capability(previously called short-term capability)shows the inherent variability of a machine/process operating within a brief period of time.Overall Capability(previously called long-term capability)shows the variability of a machine/process operating over a period of t
7、ime.It includes sources of variation in addition to the short-term variability.Within vs Overall CapabilityWithinOverallSample Size30 50 units 100 unitsNumber of Lotssingle lotseveral lotsPeriod of Timehours or daysweeks or monthsNumber of Operatorssingle operatordifferent operatorsProcess Potential
8、 Cp PpProcess Performance Cpk PpkWithin CapabilityOverall CapabilityThe key difference between the two sets of indices lies in the estimates for Within and Overall.Within vs Overall CapabilityWithinp6LSLUSLCWithinpl3LSLCWithinpu3USLCWithinpk3NSLCOverallp6LSLUSLPOverallpl3LSLPOverallpu3USLPOverallpk3
9、NSLPIntroduction toZ-SCORES13Assuming Normality.sxXZUSLpsxUSLZsxLSLPsxUSLsxXsxLSLPUSLXLSLPpzZPUSLXPZ is Normally distributed with Mean=0 and SD=1LSLLSLXPZ score14USLpsxUSLZsxLSLPsxUSLsxXsxLSLPUSLXLSLPUSLXPLSLLSLXPZLSLZUSLUSLsxUSLscoreZZ-Score interpretation:How many standard deviations,s or s-hats,i
10、s the mean,x-bar,from some specified value,x.xLets assume there is only an USL?s0.001ppmUSLTA Six Sigma Process25,000ppmUSLTA Two Sigma ProcessBasic Instructions for MinitabComputing Standard Normal Probabilities Select“Calc”,“Probability Distributions”and“Normal”.Select“Cumulative Probability”,ente
11、r the“Mean”and“Standard Deviation”,click on“Input constant”,enter the value and click on“OK”.Cumulative Distribution FunctionNormal with mean=11.0000 and standard deviation=1.00000 x P(X=x)12.0000 0.84131587.08413.011USLXPUSLXPUSL=12T 11A One Sigma ProcessDPPM=(1-0.8413)x 1,000,000 =15870011 11-12 s
12、xUSLsxxZscoreZUSLCumulative Distribution FunctionNormal with mean=11.0000 and standard deviation=1.00000 x P(X=x)12.0000 0.8413 Select“Inverse Cumulative probability”,set the“Mean”=0 and“Standard Deviation”=1,click on“Input constant”,enter the total area associated with fallout and click on“OK”.p=0.
13、84139998.08413.09998.08413.0scoreZZPzZPInverse Cumulative Distribution FunctionNormal with mean=0 and standard deviation=1.00000 P(X=x)x 0.8413 0.9998 1LSL=9USL=12T 11Determine the DPPMZLSL,ZUSL and the Z scoreExercise Select“Calc”,“Probability Distributions”and“Normal”.Select“Cumulative probability
14、”,enter the“Mean”and“Standard Deviation”,click on“Input constant”,enter the value and click on“OK”.Cumulative Distribution FunctionNormal with mean=11.0000 and standard deviation=1.00000 x P(X=x)12.0000 0.841311 11-12 sxUSLsxxZUSLSolution:Minitab Select“Cumulative probability”,enter the“Mean”and“Sta
15、ndard Deviation”,click on“Input constant”,enter the value and click on“OK”.Cumulative Distribution FunctionNormal with mean=11.0000 and standard deviation=1.00000 x P(X=x)9.0000 0.0228DPPM=(0.0228)+(1-0.8413)x 1,000,000 =181500Solution:MinitabZLSL=(9-11)/1 Select“Inverse Cumulative probability”,set
16、the“Mean”=0 and“Standard Deviation”=1,click on“Input constant”,enter the total area associated with fallout and click on“OK”.p=1-(0.0228)+(1-0.8413)=1-0.1815=0.81859097.08185.09097.08185.0scoreZZPzZPInverse Cumulative Distribution FunctionNormal with mean=0 and standard deviation=1.00000 P(X USLPPM
17、USLPPM LSLPpkPPLPPUPpScaleShapeSample NMeanLSLTargetUSL122970.80122970.80 *75000.00 75000.00 *0.39 *0.39 *3.341.004003.34 *7.00Expected LT PerformanceObserved LT PerformanceOverall(LT)CapabilityProcess DataExample of Process Capability Study on Cycle TimeProcess Capability for Reject RateFor a Norma
18、l Distribution,the proportion of parts produced beyond a specification limit is)Z(F1USLZPr1USLZPrUSLXPrReject RateProcess Capability for Reject RateThus,for every reject rate there is an accompanying Z-Score,whereRecall thatHence3,3USLLSLMinPpkLimitSpecScoreZ3ScoreZPpkProcess Capability for Reject R
19、ateEstimation of Ppk for Reject Rate Determine the long-term reject rate(p)Determine the inverse cumulative probability for p,using Calc Probability Distribution Normal Z-Score is the magnitude of the returned value Ppk is one-third of the Z-ScoreExample of Process Capability Study for Reject RateA
20、sales manager plans to assess the process capability of his telephone sales departments handling of incoming calls.The following data was collected over a period of 20 days:number of incoming calls per daynumber of unanswered calls per daysStat Quality Tools Capability Analysis(Binomial)Example of P
21、rocess Capability Study for Reject Rate201000.260.250.240.230.220.210.200.19Sample NumberProportionP=0.22643.0SL=0.2555-3.0SL=0.1973201023.522.521.5Sample Number%Defective2624222020501950185026252423222120%DefectiveSample SizeProcess Capability of Telephones SalesSummary StatsCumulative%DefectiveDis
22、t of%DefectiveP ChartRate of Defectives(denotes 95%C.I.)Average P:%Defective:Target:PPM Def.:Process Z:0.22642722.64302264270.751(0.2222,0.2307)(22.22,23.07)(222241,230654)(0.737,0.765)Ppk=0.25Example of Process Capability Study for Reject RateProcess Capability for Defect RateOther applications,app
23、roximating a Poisson Distribution:error ratesparticle countchemical concentrationProcess Capability for Defect RateEstimation of Ppk for Defect Rate Determine the long-term defects per opportunity(d)d =defects per unit opportunities per unit Determine the inverse cumulative probability for d,using C
24、alc Probability Distribution Normal Z-Score is the magnitude of the returned value Ppk is one-third of the Z-ScoreExample of Process Capability Study for Defect Rate The process manager for a wire manufacturer is concerned about the effectiveness of the wire insulation process.Random lengths of elec
25、trical wiring are taken and tested for weak spots in their insulation by means of a test voltage.The number of weak spots and the length of each piece of wire are recorded.Stat Quality Tools Capability Analysis(Poisson)Example of Process Capability Study for Defect Rate 10090807060504030201000.080.0
26、70.060.050.040.030.020.010.00Sample NumberSample CountU=0.026523.0SL=0.06904-3.0SL=0.0001009080706050403020100.0300.0250.0200.015Sample NumberDPU0.0750.0500.0250.000Target1501401301201101000.080.070.060.050.040.030.020.010.00DPUSample SizeProcess Capability for Wire InsulationSummary StatsCumulative
27、 DPUDist of DPUU ChartDefect Rate(denotes 95%C.I.)Mean DPU:Min DPU:Max DPU:Targ DPU:0.026519400.07534250(0.0237309,0.0295455)Opportunities per Unit =1Defects per Opportunity =0.0265194Z-Score =Abs(-1.935)=1.935Ppk =1/3(Z-Score)=0.645Example of Process Capability Study for Defect Rate End of TopicAny questionReading ReferenceIntroduction to Statistical Quality Control,Douglas C.Montgomery,John Wiley&Sons,ISBN 0-471-30353-4
