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1、Process Capability Analysis(Measure Phase)Scope of ModuleProcess VariationProcess CapabilitySpecification, Process and Control LimitsProcess Potential vs Process PerformanceShort-Term vs Long-Term Process CapabilityProcess Capability for Non-Normal DataCycle-Time(Exponential Distribution)Reject Rate
2、(Binomial Distribution)Defect Rate(Poisson Distribution)Process VariationProcess Variation is the inevitable differences among individual measurements or units produced by a process.Sources of Variationwithin unit(positional variation)between units(unit-unit variation)between lots(lot-lot variation)
3、between lines(line-line variation)across time(time-time variation)measurement error(repeatability & reproducibility)Types of VariationInherent or Natural VariationDue to the cumulative effect of many small unavoidable causesA process operating with only chance causes of variation present is said to
4、be “in statistical control” Types of VariationSpecial or Assignable VariationMay be due to a) improperly adjusted machine b) operator error c) defective raw materialA process operating in the presence of assignable causes of variation is said to be “out-of-control”Process CapabilityProcess Capabilit
5、y 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 within product specifi
6、cation 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 outputindicate the degree to which the output meets specificationsbe
7、 used for comparison with another process or competitorProcess Capability vs Specification Limitsa)b)c)a) Process is highly capableb) Process is marginally capablec) Process is not capableThree Types of LimitsSpecification Limits (LSL and USL) created by design engineering in response to customer re
8、quirements to specify the tolerance for a products characteristicProcess Limits (LPL and UPL)measures the variation of a processthe natural 6 limits of the measured characteristicControl Limits (LCL and UCL)measures the variation of a sample statistic (mean, variance, proportion, etc)Three Types of
9、LimitsDistribution of Individual ValuesDistribution of Sample AveragesProcess Capability IndicesTwo measures of process capabilityProcess PotentialCpProcess PerformanceCpuCplCpkProcess PotentialThe Cp index assesses whether the natural tolerance (6) of a process is within the specification limits.Pr
10、ocess PotentialA Cp of 1.0 indicates that a process is judged to be “capable”, i.e. if the process is centered within its engineering tolerance, 0.27% of parts produced will be beyond specification limits. Cp Reject Rate1.000.270 %1.330.007 %1.506.8 ppm2.002.0 ppbProcess Potentiala)b)c)a) Process is
11、 highly capable (Cp2)b) Process is capable (Cp=1 to 2)c) Process is not capable (Cp1.5)b) Process is capable (Cpk=1 to 1.5)c) Process is not capable (Cpk1)a)Cp = 2Cpk = 2b)Cp = 2Cpk = 1c)Cp = 2Cpk 1Example 1Specification Limits:4 to 16 gMachineMeanStd Dev(a) 10 4(b) 10 2(c) 7 2(d) 13 1Determine the
12、corresponding Cp and Cpk for each machine.Example 1AExample 1BExample 1CExample 1DProcess CapabilityFor a normally distributed characteristic, the defective rate F(x) may be estimated via the following:For characteristics with only one specification limit:a) LSL onlyb) USL onlyLSLUSLExample 2Specifi
13、cation Limits:4 to 16 gMachineMeanStd Dev(a) 10 4(b) 10 2(c) 7 2(d) 13 1Determine the defective rate for each machine.Example 2Mean Std Dev ZLSL ZUSL F(xUSL) F(x) 10 4 -1.51.5 66,807 66,807133,614 10 2 -3.03.0 1,350 1,350 2,700 7 2 -1.54.5 66,807 3 66,811 13 1 -9.03.0 0 1,350 1,350Lower Spec Limit =
14、 4 gUpper Spec Limit = 16 gProcess Potential vs Process Performance(a) Poor Process Potential(b) Poor Process PerformanceLSLUSLLSLUSLExperimental Design to reduce variationExperimental Design to center mean to reduce variationProcess Potential vs Process Performance Process Potential Index (Cp) Cpk
15、1.0 1.2 1.4 1.6 1.8 2.0 1.02,699.9 1,363.3 1,350.0 1,350.0 1,350.0 1,350.0 1.2 318.3 159.9 159.1 159.1 159.1 1.4 26.7 13.4 13.4 13.4 1.6 1.6 0.8 0.8 1.8 0.1 0.0 2.0 0.0Defective Rate (measured in dppm) is dependent on the actual combination of Cp and Cpk.Process Potential vs Process Performancea)Cp
16、= 2Cpk = 2b)Cp = 2Cpk = 1c)Cp = 2Cpk 1Cp Cpk Missed OpportunityAlternative Process Performance IndexProcess capability statistics measure process variation relative to specification limits. The Cp statistic compares the engineering tolerance against the processs natural variation.The Cpk statistic t
17、akes into account the location of the process relative to the midpoint between specifications. If the process target is not centered between specifications, the Cpm statistic is preferred.Process StabilityA process is stable if the distribution of measurements made on the given feature is consistent
18、 over time.TimeStable ProcessTimeUnstable ProcessucllclucllclWithin 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)
19、 shows the variability of a machine/process operating over a period of time. 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 mon
20、thsNumber of Operatorssingle operatordifferent operatorsProcess Potential Cp PpProcess Performance Cpk PpkWithin vs Overall CapabilityWithin CapabilityOverall CapabilityThe key difference between the two sets of indices lies in the estimates for Within and Overall .Estimating Within and OverallConsi
21、der the following observations from a Control Chart: S/NX1X2 XkMeanRangeStd Dev1x1,1x2,1 xk,1 X1 R1 S12x1,2x2,2 xk,2 X2 R2 S2: : : : : : :mx1,mx2,m xk,m Xm Rm SmThe overall variation Overall is estimated byEstimating Within and OverallThe within variation Within may be estimated by one of the follow
22、ing:(a) R-bar Methodwhered2 is a Shewhart constant = (k)(b)S-bar Methodwherec4 is a Shewhart constant = (k)(c) Pooled Standard Deviation MethodIn MiniTab, the Pooled Standard Deviation is the default method.Estimating Within and OverallIn cases where there is only 1 observation per sub-group (i.e. k
23、=1), the Moving Range Method is used, where .The within variation Within is then estimated using eithera) the Average Moving Range :b) the Median Moving Range :Example 3The length of a camshaft for an automobile engine is specified at 600 2 mm. Control of the length of the camshaft is critical to av
24、oid scrap/rework.The camshaft is provided by an external supplier. Assess the process capability for this supplier.The data is available in Process Capability Analysis.MTW.Example 3Stat Quality Tools Capability Analysis (Normal)Example 3Example 3AHistogram of camshaft length suggests mixed populatio
25、ns. Further investigation revealed that there are two suppliers for the camshaft. Data was collected over camshafts from both sources.Are the two suppliers similar in performance?If not, what are your recommendations?Example 3AStat Quality Tools Capability Sixpack(Normal)Example 3AExample 3AWhats Si
26、x Sigma Quality ThenOriginal Definition by Motorola:if the specification limits are at least 6 away from the process mean , i.e. Cp 2,and the process shifts by less than 1.5, i.e. Cpk 1.5,then the process will yield less than 3.4 dppm rejects.66Shift1.54.5Whats Six Sigma Quality NowMikel J Harry cla
27、ims that the process mean between lots will vary, with an average process shift of 1.5.k = z + 1.5 k = z + 1.5 Shift1.5zNote:Sigma Capability = (dpmo) (dppm)Process Capability for Non-Normal DataNot every measured characteristic is normally distributed.CharacteristicDistributionCycle TimeExponential
28、 Reject RateBinomialDefect RatePoissonProcess Capability for Cycle TimeThe Weibull Distribution is a general family of distribution withwherescale parameter is the value at which CDF=68.17%,andshape parameter determines the shape of the PDF.Process Capability for Cycle TimeAt =1,the Weibull Distribu
29、tion is reduced toFor an Exponential Distribution,The Exponential Distribution is thus a Weibull Distribution with =1.Weibull (x; =1, )Exponential (x; )Example 4A customer service manager wants to determine the process capability for his department. A primary performance index is the time taken to c
30、lose a customer complaint. The goal for this index is to close a complaint within one calendar week.Performance over the last 400 complaints was reviewed.Example 4Stat Quality Tools Capability Analysis (Weibull)Example 4Example 4AStat Quality Tools Capability Sixpack (Weibull)Example 4AProcess Capab
31、ility for Reject RateFor a Normal Distribution, the proportion of parts produced beyond a specification limit is Reject RateProcess Capability for Reject RateThus, for every reject rate there is an accompanying Z-Score, whereRecall thatHenceProcess Capability for Reject RateEstimation of Ppk for Rej
32、ect RateDetermine the long-term reject rate (p)Determine the inverse cumulative probability for p,using Calc Probability Distribution NormalZ-Score is the magnitude of the returned valuePpk is one-third of the Z-ScoreExample 5A sales manager plans to assess the process capability of his telephone sa
33、les 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 daysExample 5Stat Quality Tools Capability Analysis (Binomial)Example 5Ppk = 0.25Process Capability for Defect RateOther applications,
34、approximating a Poisson Distribution :error ratesparticle countchemical concentrationProcess Capability for Defect RateEstimation of Ytp for Defect RateDefine size of an inspection unitDetermine the long-term defects per unit (DPU)DPU= Total Defects Total UnitsDetermine the throughput yield (Ytp)Ytp
35、= expDPUProcess Capability for Defect RateEstimation of Sigma-Capability for Defect RateDetermine the opportunities per unitDetermine the long-term defects per opportunity (d)d= defects per unit opportunities per unitDetermine the inverse cumulative probability for d,using Calc Probability Distribut
36、ion NormalZ-Score is the magnitude of the returned valueSigma-Capability = Z-Score + 1.5Example 6The process manager for a wire manufacturer is concerned about the effectiveness of the wire insulation process. Random lengths of electrical wiring are taken and tested for weak spots in their insulatio
37、n by means of a test voltage. The number of weak spots and the length of each piece of wire are recorded. Example 6Stat Quality Tools Capability Analysis (Poisson)Example 6Defects per Unit = 0.0265194Throughput Yield = expDPU = exp0.0265194 = 0.9738c.f. First-Time Yield = 2 / 100 = 0.02Example 6Defi
38、ne1 Inspection Unit= 125 unit length of wirei.e.Units= Length 125Example 6AStat Quality Tools Capability Analysis (Poisson)Example 6ADefects per Unit = 3.31493Throughput Yield = expDPU = exp3.31493 = 0.0363c.f. First-Time Yield = 2 / 100 = 0.02Example 6BDefects per Unit = 3.31493Opportunities per Unit = 1Defects per Opportunity = 3.31493Z-Score = ?Example 6B1 inspection unit = 1 unit length of wireOpportunities per Unit = 1 Defects per Opportunity = 329 12,406 = 0.0265Z-Score = Abs1(0.0265) = 1.935Sigma-Capability = Z-Score + 1.5 = 3.435Choice of Six Sigma Metric
