CPK与抽样检查

91力Cpk理路e-mail:[e-mail]力Cpk數量Cpk良率不料數力CpkCpk來量良率Cpk率度數(PDF)累數(CDF)率來Cpk力良率CpkMatlab了力例立理論力力Cp,Cpk,Cpm,Cpmk,((Kane(1986),Chanetal.(1988)),數量:Cp=σ6LSLUSL−,92Cpk=⎩⎨⎧⎭⎬⎫−−σµσµ3,3minLSLUSL,Cpm=22)(6TLSLUSL−+−µσ,Cpmk=⎪⎩⎪⎨⎧⎪⎭⎪⎬⎫−+−−+−2222)(3,)(3minTLSLTUSLµσµµσµ,USLLSLµ數σTUSLLSL不µσ料數X=nXnii/1∑=S=[∑−=niiX1(2)X/(n-1)]1/2來CpkXSpkCˆ數力Cpk理pkCˆ來量良率力良率了pkCˆ率度數(PDF)累數(CDF)pkCˆt(bivariatenon-centralt-distributed)數聯率Pearnetal.(1992)folded-normal)chi-square)兩數聯率PDFCDF不易pkCˆ不利pkCˆ[1]Zhangetal.(1990)100γ%pkCˆ)(ZhLpkC=⎥⎥⎦⎤⎢⎢⎣⎡⎭⎬⎫⎩⎨⎧−Γ−Γ−−−−−∧22))2/)1(((2))2/)2(()(1(31)(1nnnnnZCpkγ,93Z(γ)γ數lowerpercentileofthestandardnormaldistribution).[2]Bissell(1990)100γ%pkCˆ:)(BiLpkC=()22912−+−∧∧nCnZCpkpkγ(1)[3)NagataandNagahata(1994)(1)100γ%pkCˆ)(NNLpkC=)1(2ˆ91)(ˆ)1(5212−+−−−nCnZCnpkpkγ.[4]Chouetal.(1990)andLevinson(1997)不t(bivariatenon-centralt-distributed)數聯率k1=k2100γ%pkCˆPr(Cpk≥)(ChLpkC)=Pr(Cpl≥)(ChLpkC,Cpu≥)(ChLpkC)=γPrγσµσµ=⎥⎦⎤⎢⎣⎡≥−−−≤−−)(2)(13,3ChLpkChLpkCSkXCSkX,Pr[111)(tTn≤−δ221)(tTn≤−δ]=γ.()~1δ−nTtnon-centraltdistribution)度1−n參數noncentralityparameter)δnkt11−=nkt22=)(13ChLpkCn−=δ)(23ChLpkCn=δk1=k2=3Cpl=3Cpu了SXUSLCpu3)(ˆ−=SLSLXCpl3)(ˆ−=不FranklinandWasserman(1992)KushlerandHurley(1992)Rodridguez(1992)NagataandNagahata(1994)Pr(plpuCCˆˆ=)率0Cpk良率2Φ(3Cpk)–1≤yield≤Φ(3Cpk)力Cpk1.0094論不降異不良率2700(ppm)不力1.00≤Cpk1.33論RS降異不良率66-2700(ppm)不力1.33≤Cpk1.67論不良率0.54-66(ppm)不力1.67≤Cpk2.00論不良率0.002-0.54(ppm)不Cpk≥2.00論不良率0.002(ppm)不1.論力不數1.力不數ConditionCpkvaluesppmInadequateCpk1.00NC2700Capable1.00≤Cpk1.33NC2700Satisfactory1.33≤Cpk1.67NC66Excellent1.67≤Cpk2.00NC0.54Super2.00≤CpkNC0.002(Montgomery(2001))力22.力良率類Cpk良率ExistingProcesses1.3399.9933896%NewProcesses1.5099.9993198%ExistingProcessesonSafety,Strength,orCriticalParameters(suchasmanufacturingsoftdrinksorchemicalsolutionbottledwithglasscontainers)1.5099.9993198%NewProcessesonSafety,Strength,orCriticalParameters1.6799.9999455%95CpkPDFCDF率Matlab來Cpk例來pkCˆ率了pkCˆ率(1)D=(n−1)1/2d/σ(2)a=[(n–1)/n]1/2(3)K=(n–1)S2/σ2K~21−nχ,(4)Z=n1/2(X−T)/σZ~N(δ,1)δ=n1/2(µ−T)/σ,(5)Y=Z2Y率度數:fY(y)=()()()yfyfyZZ+−21y0(2)pkCˆ:KYaDCpk3ˆ−=(x)FpkCˆ=Pr(pkCˆ≤x)=P(KYaD3−≤x)=1–∫=−≤∞0)()|3(dyyfyYxYaDKPYK~21−nχPr(xyaDK3−≤)=0y(D/a)2x0.(x)FpkCˆ=1–∫−≤203)a/D(Ydy)y(f)xyaDKPr(=1–∫⎟⎟⎠⎞⎜⎜⎝⎛−2)/(022)(9)(aDYKdyyfxyaDF(2)不率累度數96)(ˆxFpkC=()∫⎟⎟⎠⎞⎜⎜⎝⎛−−2202291aDKxyaDF()()()yfyfyZZ+−21dy(3))(ˆxfpkC=()()()()()yfyfyxyaDxyaDfZZaDK+−−⎟⎟⎠⎞⎜⎜⎝⎛−∫322209922dy(4)(3)數y=t2pkCˆ累度數Chi-squaredistribution)normaldistribution):(x)FpkCˆ=1-[],)()(9))(1(22dtntntnxtnbnGnb0ξφξφ−++∫⎟⎟⎠⎞⎜⎜⎝⎛−−x0b=d/σFK(.)ordinarycentralChi-squaredistribution)21−nχ累度數,φ(.)standardnormaldistribution)N(0,1)率度數.1(a)-1(d)pkCˆPDFCDFξ=0.01b=3d=2不數n=1020501(a).pkCˆPDFξ=0.0,b=3,d=2n=10,20,50().1(b).pkCˆPDFξ=1.0,b=3,d=2,n=10,20,50().1(c).pkCˆCDFξ=0.5,b=3,d=2,n=10,20,50().1(d).pkCˆCDFξ=1.0,b=3,d=2,n=10,20,50().97CpkCpk=Cb=d/σb=3C+|ξ|.Cpkξ數.Cpk=σµ3||md−−=3||ξσ−d/,whereξ=(µ–T)/σ.數nγpkCˆ參數ξ列(5)數參數ξ=(µ–T)/σξˆ=(X–T)/S(5)ξ數ξˆ=ξˆ0ξˆ=ξˆ0都dtntntCntnbnGnb0pk⎥⎦⎤⎢⎣⎡−++∫⎟⎟⎠⎞⎜⎜⎝⎛−−∧∧∧)()(9))(1(22ξφξφ=1γ.(5)(LowerConfidenceBounds;LCB)了)(PSLpkC率LCB數[i]21−nχFK(⋅)[ii]N(0,1)φ(⋅)[iii]recursiveadaptiveSimpsonquardrature—“quad”數數t0t1(1t0t10).LCBMatlab(availableuponrequest力Step1.讀料(X1,X2,…,Xn)LSLUSLTandγStep2.料XSξˆpkCˆStep3.利pkCˆrecursiveadaptiveSimpsonquardrature—“quad”(t0)Step4.t1pkCˆ)(PSLpkCrecursiveadaptiveSimpsonquardrature—“quad”Tol10-4Step5.論“ThetruevalueofthemanufacturingcapabilityCpkisnolessthanthe)(PSLpkCwith100γ%levelofconfidence”98)(PSLpkC參數ξ參數µσξ=(µ–T)/σ若數µσ更不行量數ξ=0(0.05)3.00n=10(5)200pkCˆ=0.7(0.1)3.0γ=0.95)(PSLpkC參數ξ[i])(PSLpkCξn[ii]Cξ=1.00ξ≥1.00(5)pkCˆnγξˆ=ξ=1.00來)(PSLpkC論數數例2-5)(PSLpkCv.s.|ξ|pkCˆ0.91.22.02.5γ0.95n=30,50,70,100,150,2003-4pkCˆ=0.7(0.1)3.0n=10(5)200γ=0.95了力例pkCˆ=1.5n=1003-4力Cpk不1.315;Cpk1.3152.)(PSLpkCv.s.|ξ|pkCˆ=0.9,γ=0.95n=30,50,70,100,150,200().3.)(PSLpkCv.s.|ξ|pkCˆ=1.2,γ=0.95n=30,50,70,100,150,200().994.)(PSLpkCv.s.|ξ|pkCˆ=2.0,γ=0.95n=30,50,70,100,150,200().5.)(PSLpkCv.s.|ξ|pkCˆ=2.5,γ=0.95n=30,50,70,100,150,200().PVCHigh-ImpactPVCPVCPVC良度易流力不廉便料PVC''211例6USL48.4mmLSL47.6mmT48mm6.PVC''2115.100.(mm)48.07048.08748.12148.07247.98848.02948.10048.14048.14448.12747.98348.15148.00748.14848.11848.14348.07848.08248.15648.10648.10948.05948.15048.15748.02648.13648.17748.07448.16648.05648.12048.25348.21448.15048.19948.21947.96948.07948.03148.061484.0±mm10048.02048.09148.05248.19048.04448.14148.13047.99748.11548.13148.18348.10848.16048.14748.13748.05548.16348.08448.11748.03448.18348.17548.18848.18348.11548.12748.15148.10848.03048.15548.09748.10447.98848.01648.03648.02948.14148.12248.04848.14048.12348.09347.99948.09947.94848.09948.10348.20148.17648.04348.11248.04248.14048.08948.09648.09748.14748.07548.09148.081100PVC料料5料料Histogram)normalprobabilityplot)料更Shapiro-Wilk量W=0.989p-value=0.6557料論料了行Matlab讀料(X1,X2,…,Xn)LSL=47.6mmUSL=48.4mmT=48mmγ=0.95ξ=(µ–T)/σ=1.0數=X48.103S=0.0608pkCˆ=1.628)(PSLpkC=1.428行論95%力Cpk不1.428;Cpk1.428不良率不19(ppm)不MatlabExecutionInput&Output-------------------------------------------------------------------------------------------------Output:TheSampleMeanis48.10