ch05a process capability[1]

8/7/2019 Ch05a Process Capability[1]

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Chapter 5a Process Capability

This chapter introduces the topic of processcapability studies. The theory behind

process capability and the calculation of Cpand Cpk is presented


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Specification Limit &Process Limit

Look at indv. values and avg. values of xsIndv xs values n = 84 - considered as populationAvgs n = 21 - sample taken

= same (in this case)

Normally distributed individual xs and avg. valueshaving same mean, only the spread is different W > R elationship = popu. Std. dev. of avgs

If n = 5 = 0.45 W W = popn std dev of indiv. xsSP RE AD OF AVGS IS HALF OF SP RE AD FO R INDV.VALU E S

xXandX

xW

nx !

xW


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R elationship betweenpopulation and sample values

Assume Normal Dist.E stimate popu. std. dev.

c4 } ; n = 84

@ (c4 = 0.99699)

= 4.17

= 2.09417.4

nx!

W!W

4c

!W

3n4)1n(4

0.996994.16 !


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C entral Limit Theorem

If the population from which samples aretaken is NOT normal, the distribution of

SAMPLE AVERAGES will tend towardnormality provided that sample size, n, is atleast 4.Tendency gets better as n oStandardized normal for distribution of averages Z =

n

x


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Central Limit Theoremis one reason whycontrol chart worksNo need to worry aboutdistribution of s is notnormal, i.e. indv.values.Averages distributionwill tend to ND


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C ontrol Limits &Specifications

C ontrol limits - limits for avgs, and establishedas a func. of avgsSpecification limits -allowable variation insize as per designdocuments e.g. drawing@ for individual valuesestimated by designengineers


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C ontrol limits, Process spread, Dist of averages, & distribution of individual valuesare interdependent. determined by theprocessC . C harts C ANNOT determine process meetsspec.


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Process C apability &Tolerance

When spec. established without knowingwhether process capable of meeting it or not serious situations can resultProcess capable or not actually lookingat process spread, which is called processcapability (6 W)Lets define specification limit as tolerance(T) : T = USL -LSL

3 types of situation can resultthe value of 6 W < USL-LSLthe value of 6 W = USL - LSLthe value of 6 W > USL - LSL


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C ase I and C ase II situations


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C ase 3 situation


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Process C apabilityProcedure (s method)

1. Take subgroup size 4 for 20 subgroups2. Calculate sample s.d., s, for each subgroup3. Calculate avg. sample s.d. Ds = s/g4. Calculate est. population s.d.5. Calculate Process Capability =

R - method1. Same as 1. above2. Calculate R for each subgroup3. Calculate avg. Range, = R/g4. Calculate5. Calculate Calculate 6

4o cs !WW

R

oW2o dR !


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Process C apability (6 W) AndTolerance

Cp - Capability IndeT = U-LCp = 1 Case II 6W = TCp > 1 Case I 6W < TCp < 1 Case III 6W > TUsually Cp = 1.33 (de facto

std.)Measure of processperformanceShortfall of Cp - measurenot in terms of nominal or target value >>> must useCpk

Formulas

Cp = (T)/6 W

Cpk =3

Z(min)

Z (USL) =xUSL

WLSLx


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E xampleDetermine Cp and Cpk for aprocess with average 6.45,W = 0.030, having USL =

6.50 , LSL = 6.30 -- T = 0.2

SolutionCp= T/6 W= 0.2/6(0.03)=1.11Cpk = Z(min)/3Z(U) = (USL - Dx )/ W =

6.50-6.45)/0.03 = 1.67Z(L) = ( Dx LSL)/ W = 6.45-

6.30)/0.03 = 5.00@Cpk = 1.67/3 = 0.56

Process NOT capable sincenot centered. Cp > 1 doesntmean capable. Have tocheck Cpk

UL T

6.30 6.506.45=x


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C omments On C p, C pkCp does not change when process center (avg.) changesCp = Cpk when process is centred

Cpk e Cp always this situationCpk = 1.00 de facto standardCpk < 1.00 p process producing rejectsCp < 1.00 p process not capableCpk = 0 process center is at one of spec.limit (U or L)Cpk < 0 i.e. ve value, avg outside of limits


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E xercise

1. Find Cp, CpkDx = 129.7 (Length of radiator hose)W = 2.35Spec. 130.0 s 3.0What is the %defective?

2. Find Cp, Cpk

Spec.U = 58 mmL = 42 mmW = 2 mmWhen = 50When = 54

3. Find Cp, CpkU = 56L = 44W = 2

When = 50When = 56

ch05a process capability[1]

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