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Six-Sigma Six-Sigma Training BookTraining Book Dec 18,20011BPTL Confidential4 數据分布2BPTL Confidential4 NormalExponentialWeibullLognormaltc2fContinuous DistributionsSampling Distributions數据分布3BPTL ConfidentialThe most widely used model for the distribution of continuous random variable.Arises in the study of numerous physical phenomena,such as the velocity of molecules.正態分布正態分布Plot is known as Probability Density Functionof X4BPTL Confidential4Many natural phenomena and man-made processes are observed to have normal distributions,or can be closely represented as normally distributed.4For example,the length of a machined part is observed to vary about its mean due to:temperature drift,humidity change,vibrations,cutting angle variations,cutting tool wear,bearing wear,rotational speed variations,fixturing variations,raw material changes and contamination level changes4If these sources of variation are small,independent and equally likely to be positive or negative,the length will closely approximate a normal distribution.正態分布正態分布5BPTL Confidential4First introduced by French mathematician Abraham DeMoivre in 1733.4Made famous in 1809 by German mathematician K.F.Gauss when he also developed a normal distribution independently and used it in his study of astronomy.4As a result,it is also known as the Gaussian distribution.4During mid to late nineteenth century,many statisticians believed that it was“normal”for most well-behaved data to follow this curve.正態分布正態分布-歷程表歷程表Karl Friedrich Gauss6BPTL Confidential4正態分布易于理解,具有特性,統計學提供了許多基于正態分布的強有力的分析方法來幫助人們做決定.4因此,我們通常會試圖用正態分布去近似模擬其它分布(如可能)或轉化數据以“使”它遵從正態分布.4它是分析過程能力的首選分布形式.正態分布正態分布7BPTL Confidential4A normal distribution can be completely described by knowing only the:Mean()Variance(2)正態分布的一些特性正態分布的一些特性Distribution OneDistribution TwoDistribution ThreeWhat is the difference between the 3 normal distributions?X N(,2)18BPTL ConfidentialANormal(A,A)BNormal(B,B)ANormal(A,A)BNormal(B,B)ANormal(A,A)BNormal(B,B)What is the difference between process A&B for each case?正態分布的一些特性正態分布的一些特性9BPTL Confidential4The mean,median and mode all coincide at the same value-.There is perfect symmetry.+-MeanMedianMode2The mean represents the arithmetic average of all observations in a data set.If a set of observations is arranged in an increasing order of magnitude(ranked data),the middle value is called the median.lIf the number of observations is odd,the median is the value of the middle number.lIf the number of observations is even,there are 2 middle numbers,and the median is the average of the 2 values.The mode is the observation that occurs most frequently in the sample.正態分布的一些特性正態分布的一些特性10BPTL Confidential4The area under sections of the curve can be used to estimate the cumulative probability of a certain“event”occurring:Point of Inflection1+-68.27%95.45%99.73%+/-3 is often referred to as the width of a normal distribution3正態分布的一些特性正態分布的一些特性11BPTL ConfidentialLets compute the cumulative probabilities of the following distributions:+-m=3.5s=0.61.8+-20.0=16.6=2.8+-m=-1.5s=0.9-2.80.5正態分布的一些特性正態分布的一些特性12BPTL ConfidentialMiniTab:Calc Probability Distributions Normal.Enter valueEnter s valueEnter x value正態分布的一些特性正態分布的一些特性13BPTL Confidential什么是什么是 6?14BPTL Confidential6 簡介15BPTL ConfidentialThe Focus of Six Sigma4Identifying critical aspects of the business with problems or opportunities for improvement.4Targeting those critical areas and designating improvement efforts as Six Sigma Black Belt projects.4Selecting top people to work on the projects-full time.4Ensuring these people have the time,tools,and resources they need to succeed.16BPTL ConfidentialCustomer Focus:A Model For SuccessTechnologyTechnologyCapabilityCapabilityOrganizationOrganizationPeoplePeopleProcessesProcesses商務上的生存競爭有賴于我們多大程度上讓我們的客戶滿意商務上的生存競爭有賴于我們多大程度上讓我們的客戶滿意.客戶滿意才能体現品質客戶滿意才能体現品質,价格价格,和貨期的意義和貨期的意義.品質品質,成本成本,准時走貨無不依耐于工序能力准時走貨無不依耐于工序能力.What purpose is Six-sigma?17BPTL ConfidentialSix Sigma VisionThe Vision of Six Sigma is to delight customers by delivering world-class quality products through the achievement of Six Sigma levels of performance in everything we do.What purpose is Six-sigma?Six Sigma PhilosophyThe philosophy of Six Sigma is to apply a structured,systematic approach to achieve breakthrough improvement across all areas of our business.18BPTL Confidential PPMProcessCapabilityDefects per Million Opp.Six Sigma-Aggressive GoalWhat purpose is Six-sigma?19BPTL ConfidentialStatistical Definition of n-SigmaLSLLSLUSLUSLProcess WidthoDesign WidthTTscaleLSLLSLUSLUSLscaleTT+n scale-n This is the so-called n-sigma Sigma is a statistical unit of measure that reflects process capability.The sigma scale of measure is perfectly correlated to such characteristics as defects-per-unit,parts-per million defective,and the probability of a failure/error.20BPTL ConfidentialStatistical Definition of 6This is the six-sigma we said LSLLSLUSLUSLProcess WidthoDesign Width-3 st+3 stTT.001 ppm USL.001 ppm 2)b)工序能力可工序能力可(Cp=1 to 2)c)工序能力差工序能力差(Cp1.5)b)工序能力可工序能力可(Cpk=1 to 1.5)c)工序能力差工序能力差(Cpk1)a)Cp=2Cpk=2b)Cp=2Cpk=1c)Cp=2Cpk 143BPTL Confidential工序潛力与工序表現工序潛力与工序表現(a)Poor Process Potential(b)Poor Process PerformanceLSLUSLLSLUSLExperimental Design to reduce variationExperimental Design to center mean to reduce variation44BPTL Confidential工序潛力与工序表現工序潛力与工序表現a)Cp=2Cpk=2b)Cp=2Cpk=1c)Cp=2Cpk 145BPTL Confidential工序穩定性工序穩定性A process is stable if the distribution of measurements made on the given feature is consistent over time.TimeStable ProcessTimeUnstable Processucllclucllcl46BPTL Confidential短期工序能力与長期工序能力短期工序能力与長期工序能力短期工序能力短期工序能力(previously called short-term capability)shows the inherent variability of a machine/process operating within a brief period of time.長期工序能力長期工序能力(previously called long-term capability)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.47BPTL ConfidentialWithinOverallSample Size30 50 units 100 unitsNumber of Lotssingle lotseveral lotsPeriod of Timehours or daysweeks or monthsNumber of Operatorssingle operatordifferent operatorsProcess Potential Cp PpProcess Performance Cpk Ppk短期工序能力与長期工序能力短期工序能力与長期工序能力48BPTL Confidential實戰演練實戰演練The length of a camshaft for an automobile engine is specified at 600 2 mm.Control of the length of the camshaft is critical to avoid scrap/rework.The camshaft is provided by external suppliers.Assess the process capability for this supplier.The data is available in Camshaft.MTW.Data are collected in subgroups of 5 each.49BPTL ConfidentialExample 4Minitab:Stat Quality Tools Capability Analysis(Normal)50BPTL ConfidentialExample 451BPTL ConfidentialExample 5Histogram of the camshaft length suggests mixed populations.Further investigation revealed that there are two suppliers for the camshaft.Data were now collected on camshafts from each source without combining both.Subgroup size is 5 for each supplier.Are the two suppliers similar in performance?If not,what are your recommendations?52BPTL ConfidentialExample 5MiniTab:Stat Quality Tools Capability Sixpack(Normal)53BPTL ConfidentialExample 554BPTL ConfidentialExample 555BPTL Confidential用用Box-Cox 轉化后的工序能力分析轉化后的工序能力分析4When the process data are not normal,the Cpk or Ppk indices are not accurate or reliable,because these indices are computed on the basis that the data are normally distributed.4Dppm values associated with the indices will not be near to the actual performance when the normal curve does not model the actual data well.56BPTL Confidential4If the process data are somewhat bell-shaped but skewed,Box-Cox transformation can be used to make the data normal before we assess the process capability.4Remember to transform the specification limits too before we compute Cpk or Ppk!用用Box-Cox 轉化后的工序能力分析轉化后的工序能力分析57BPTL ConfidentialMinitab:Stat Quality Tools Capability Analysis(Normal)用用Box-Cox 轉化后的工序能力分析轉化后的工序能力分析58BPTL ConfidentialExample 6Open the file named Dimension.MTW in the Day-2 folder again.Compute the process capability with the specification limits:LSL:0.1USL:10Are the data normally distributed?Compute the process capability again with Box-Cox transformation.用用Box-Cox 轉化后的工序能力分析轉化后的工序能力分析59BPTL ConfidentialCpk of 0.41 is reported in the SSAT package.This value is not reliable or accurate if the data are not normal.Data is not normalExample 6用用Box-Cox 轉化后的工序能力分析轉化后的工序能力分析60BPTL ConfidentialExample 6Cpk has increased from 0.41 to 0.81用用Box-Cox 轉化后的工序能力分析轉化后的工序能力分析61BPTL ConfidentialWhats“6”Quality Then4Original Definition by Motorola:4In the short term,the specification limits are at least 6 away from the process mean,i.e.Cp 2,4In the long run,the process will shift by less than 1.5,i.e.Ppk 1.5,4The process will yield less than 3.4 dppm rejected parts.66Shift1.54.5“Sigma Level”Capability62BPTL ConfidentialWhats Six Sigma Quality NowMikel J Harry claims 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)“Sigma Level”Capability63BPTL ConfidentialTypes of Variation1.Positional VariationSame process,variation at differing locations simultaneously:Temperature variations inside a thermal chamberCavity-to-cavity variations in an injection mold2.Cyclical VariationSequential repetitions of a process over fairly short time,say,less than 15 mins:Variations between consecutive batches of a processDifferences from lot to lot of raw materials64BPTL ConfidentialTypes of Variation3.Temporal VariationVariations over longer periods of time,such a several hours,days or weeks.65BPTL ConfidentialMeasurement System Analysis ApproachThere are two types of measurements possible:4VariableData can be described on a continuous scale4AttributeData cannot be adequately described on a continuous scalePass/Fail,very low counts Each must be approached differently.66BPTL Confidential
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