10-1 Quality Control

CHAPTER 10

Quality Control

Homework Problems: # 1,2,4,5,7,9,13,14,22,24 on pp. 448-453.

10-2 Quality Control

Phases of Quality Assurance

Acceptance sampling

Process control

Continuous improvement

Inspec'onbefore/a/erproduc'on

Inspec'onandcorrec'veac'onduringproduc'on/

service

Qualitybuiltintoproduct/process

Theleastprogressive

Themostprogressive

Chapter 10

Chapter 9

10-3 Quality Control

Inspection •  Inspec3on

• Anappraisalac'vitythatcomparesgoodsorservicestoastandard.

•  Inspec'onissues:1.  Howmuchtoinspectandhowo/en2.  Atwhatpointsintheprocesstoinspect3.  Whethertoinspectinacentralizedoron-site

loca'on4.  WhethertoinspectaGributesorvariables

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How Much to Inspect?

TotalCost

10-5 Quality Control

Where to Inspect in the Process

• Rawmaterialsandpurchasedparts

•  Finishedproducts

• Beforeacostlyopera'on

• Beforeanirreversibleprocess

• Beforeacoveringprocess

10-6 Quality Control

Examples of Inspection Points

10-7 Quality Control

Centralized vs. On-Site Inspection

•  Effectsoncostandlevelofdisrup'onareamajorissueinselec'ngcentralizedvs.on-siteinspec'on•  Centralized

•  Specializedteststhatmaybestbecompletedinalab•  Morespecializedtes'ngequipmente.g.computerchips•  Morefavorabletes'ngenvironment

•  On-Site•  Quickerdecisionsarerendered•  Avoidintroduc'onofextraneousfactors•  Qualityatthesource

10-8 Quality Control

Statistical Process Control (SPC)

• Qualitycontrolseeks• QualityofConformance

•  Aproductorserviceconformstospecifica'ons

• Atoolusedtohelpinthisprocess:•  SPC

•  Sta's'calevalua'onoftheoutputofaprocessduringproduc'onorservicedelivery

•  Helpstodecideifaprocessis“incontrol”orifcorrec'veac'onisneeded

10-9 Quality Control

Process Variability

• Twobasicques'onsconcerningvariability:1.  Arethevaria'onsrandom?

•  Processcontrole.g.,

2.  Givenastableprocess,istheinherentvariabilityoftheprocesswithinarangethatconformstoperformancecriteria?•  Processcapabilitye.g.,

10-10 Quality Control

Variation and Control • Varia3on• Random/Common(cause)varia3on:

•  Naturalvaria'onintheoutputofaprocess,createdbycountlessminorfactor,

• Assignable/Special(cause)varia3on:•  Avaria'onwhosecausecanusuallybeiden'fied

10-11 Quality Control

Distribution of a Variable: weight in a box of cereal

Note:thepurple(lightblue)linesshowachangeindistribu'on

Whathaschanged?

10-12 Quality Control

Effects of “Assignable Causes” on Process Control

10-13 Quality Control

Statistical Process Control (SPC)

•  Theessenceofsta's'calprocesscontrolistoassurethattheoutputofaprocessisrandomsothatfutureoutputwillberandomaswell.

10-14 Quality Control

Sampling and Sampling Distribution

•  SamplingDistribu3on•  Atheore'caldistribu'onthatdescribestherandomvariabilityofsamplesta's'cs

•  Thenormaldistribu'oniscommonlyusedforthispurpose

•  CentralLimitTheorem•  Thedistribu'onofsampleaverages(i.e.,samplemeans)tendstobenormalregardlessoftheshapeoftheprocessdistribu'on

10-15 Quality Control

Sampling and Sample Distribution

•  SPCinvolvesperiodicallytakingsamplesofprocessoutputandcompu'ngsamplesta's'cs:•  Samplemeans•  Thenumberofoccurrencesofsomeoutcome

•  Samplesta's'csareusedtojudgetherandomnessofprocessvaria'on

10-16 Quality Control

Sampling Distribution

10-17 Quality Control

Normal Distribution Figure 10.5

10-18 Quality Control

Sampling and Process Distribution: Examples

Anindustrialprocessthatmakes3-footsec'onofplas'cpipeproducespipewithanaverageinsidediameterof1(one)inchandastandarddevia'onof0.05inch.A).Ifyourandomlyselectonepieceofpipe,whatistheprobabilitythatitsinside

diameterwillexceed1.02inches,assumingthepopula'onisnormal?B).Ifyouselectarandomsampleof25piecesofpipe,whatistheprobabilitythat

thesamplemeanwillexceed1.02inches?

10-19 Quality Control 10-20 Quality Control

Table B (p.869)

Areasunderthestandardnormalcurve,from-∞to+z-∞ 0

10-21 Quality Control

Control Chart: The Voice of the Process

• ControlChart•  Purpose:tomonitorprocessoutputtoseeifitisrandom

• A'meorderedplotofrepresenta'vesamplesta's'csobtainedfromanongoingprocess(e.g.samplemeans),usedtodis'nguishbetweenrandomandnonrandomvariability

• Upperandlowercontrollimitsdefinetherangeofacceptablevaria'on

10-22 Quality Control

Statistical Control Process

•  TheControlProcess• Define• Measure• Compare• Evaluate• Correct• Monitorresults

10-23 Quality Control

Control Chart Examples 10-24 Quality Control

Control Limits Fig. 10.7

Random

variation

10-25 Quality Control

Observations from Sample Distribution

Sample number

UCL

LCL

1 2 3 4

10-26 Quality Control

Control Chart Examples

Conclusion?

10-27 Quality Control

Control Chart Examples

Conclusion?

10-28 Quality Control

Control Chart Examples

Conclusion?

10-29 Quality Control

Control Chart Examples

Conclusion?

10-30 Quality Control

Control Chart Examples

Conclusion?

10-31 Quality Control

SPC Errors

•  TypeIerror•  Concludingaprocessisnotincontrolwhenitactuallyis.

•  Theprobabilityofrejec'ngthenullhypothesiswhenthenullhypothesisistrue.

•  Concludingnon-randomvaria'onispresentwhenonlyrandomnessispresent

•  Manufacturer’sRisk

•  TypeIIerror•  Concludingaprocessisincontrolwhenitisnot.

•  Theprobabilityoffailingtorejectthenullhypothesiswhenthenullhypothesisisfalse.

•  Consumer’sRisk

10-32 Quality Control

Type I Error

Q.HowcanweadjustLCLandUCLsoastoreduceTypeIerror?

10-33 Quality Control

Q.FindtheZvaluefor5% 𝛼risk?risk?

Example of Finding Z value 10-34 Quality Control

Control Charts for Variables

• Meancharts

• Usedtomonitorthecentraltendencyofaprocess.

• Xbarcharts

• Rangecharts

• Usedtomonitortheprocessdispersion

• Rcharts

Variables are continuous, generate data that are measured.

10-35 Quality Control

Control Charts for Variables

)min()max( ii xxR −=

Example:whatisthemeanandrangeof10,8,11,12,9

10-36 Quality Control

X bar (mean) charts

• Centerlineisthegrandmean(Xdoublebar)• PointsareXbars

xzXUCL σ+=

xzXLCL σ−=

RAXUCL 2+=

RAXLCL 2−=or

A2=acontrolchartfactorbasedonsamplesize,n(seeTable10.3)

_

=

10-37 Quality Control

Range (R) Charts

• CenterlineistheaverageRange(Rbar)• PointsareR• D3andD4valuesaretabledaccordington(samplesize,seeTable10.3)

RDUCL 4= RDLCL 3=

10-38 Quality Control

Examples for Mean and Range charts

Sample Sample Number 1 2 3 4 R x

1 0.5014 0.5022 0.5009 0.5027 2 0.5021 0.5041 0.5024 0.5020 3 0.5018 0.5026 0.5035 0.5023 4 0.5008 0.5034 0.5024 0.5015 5 0.5041 0.5056 0.5034 0.5047

Special Metal Screw

_

10-39 Quality Control

Examples for Mean and Range charts

Sample Sample Number 1 2 3 4 R x

1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018 2 0.5021 0.5041 0.5024 0.5020 0.0021 0.5027 3 0.5018 0.5026 0.5035 0.5023 0.0017 0.5026 4 0.5008 0.5034 0.5024 0.5015 0.0026 0.5020 5 0.5041 0.5056 0.5034 0.5047 0.0022 0.5045

Special Metal Screw

𝑅 =𝑿 =

10-40 Quality Control

Examples for Mean and Range charts

•  SpecialmetalScrew

Control Charts x-Charts

UCLx = x + A2R LCLx = x - A2R

= =

R = 0.0021 A2 = 0.73 x = 0.5027 =

UCLx = LCLx =

10-41 Quality Control

Table 10.3 Factors for 3-sigma control limits for Mean and Range charts

10-42 Quality Control

Mean Charts Plots

Sample plots

Conclusion?

10-43 Quality Control

Range Charts Results

•  SpecialMetalScrew

UCLR = LCLR =

Control Charts R-Charts R = 0.0021 D4 = 2.282

D3 = 0 UCLR = D4R LCLR = D3R

10-44 Quality Control

Range charts Plots

Sample plots

Conclusion?

10-45 Quality Control

Mean and Range Charts

UCL

LCL

x-Chart

UCL

LCL

R-chart

Sampling Distribution

10-46 Quality Control

Mean and Range Charts

UCL

LCL

R-chart

Sampling Distribution

x-Chart

UCL

LCL

10-47 Quality Control

Control Charts for Variables Using Sigma

• MeanCharts

UCLx =

LCLx =

UCLx = LCL x =

=

Sunny Dale Bank

x = 5.0 minutes σ = 1.5 minutes n = 6 customers z = 1.96

=

10-48 Quality Control

Using Mean and Range Charts •  Todetermineini'alcontrollimits:

•  Obtain20to25samples•  Computeappropriatesamplesta's'cse.g.,mean,range..

•  Establishpreliminarycontrollimits•  Determineifanypointsfalloutsidethecontrollimits

•  Ifyoufindnoout-of-controlsignals,assumetheprocessisincontrol.(youcanusethesecontrollimits)

•  Ifyoufindanout-of-controlsignal,searchforandcorrecttheassignablecauseofvaria'on.Thenresumetheprocessandcollectanothersetofobserva'onsonwhichcontrollimitscanbebased.

•  Plotthedataonthecontrolchartandcheckforout-of-controlsignals

10-49 Quality Control

Control Chart for Attributes

• P-Chart:Controlchartusedtomonitorthepropor'onofdefec'vesinaprocess

• C-Chart:Controlchartusedtomonitorthenumberofdefectsperunit

Attributes are discrete, generate data that are counted.

10-50 Quality Control

Use of p-Charts

• Whenobserva'onscanbeplacedintotwocategories.• Goodorbad• Passorfail• Operateordon’toperate

• Whenthedataconsistsofmul'plesamplesofseveralobserva'onseach

10-51 Quality Control

P- Charts

)ˆ(LCL)ˆ(UCL

pp

pp

zpzpσ

σ

−=

+=

σσ ppreplcesˆIf p is unknown, it can be estimated from samples. replaces p,

npp

p)1(ˆ −

=σ

10-52 Quality Control

P-Charts Example

• HometownBank(usingZ=3) Sample Wrong Number Account Number

1 15 2 12 3 19 4 2 5 19 6 4 7 24 8 7 9 10 10 17 11 15 12 3

Total 147

p =

Given n = 2500

10-53 Quality Control

P-Charts Plots

Conclusion?

10-54 Quality Control

C- Charts

•  C-charts:usedtocountdefectsinaconstantsamplesize

czcUCL +=

czcLCL −= −

10-55 Quality Control

C-Charts Example •  AHighwayintersec'onaverages3accidentspermonth.Lastmonth7accidentsoccurred.Hassomethingchangedattheintersec'on?UseZ=3.Q. Does this appear to be a quality control problem?

10-56 Quality Control

Use of c-Charts

• Useonlywhenthenumberofoccurrencesperunitofmeasurecanbecounted;non-occurrencescannotbecounted.• Scratches,chips,dents,orerrorsperitem• Cracksorfaultsperunitofdistance• BreaksorTearsperunitofarea• Bacteriaorpollutantsperunitofvolume• Calls,complaints,failuresperunitof'me

10-57 Quality Control

Run Tests

•  Evenifaprocessappearstobeincontrol,thedatamays'llnotreflectarandomprocess

•  Analystso/ensupplementcontrolchartswitharuntest•  Runtest

•  AtestforpaGernsinasequence•  Run

•  Sequenceofobserva'onswithacertaincharacteris'c

10-58 Quality Control

Run Tests

•  Controlchartstestforpointsthataretooextreme(e.g.,3σfromthemean)tobeconsideredrandom.But,evenifallpointsarewithinthecontrollimits,theprocessmaynotberandom.

•  AnysortofpaGern(e.g.,trend,cycle,bias,meanshi/,toomuchdispersion)inthedatawouldsuggestanon-randomprocess.

•  Arunisdefinedasasequenceofobserva'onswithacertaincharacteris'c,followedbyoneormoreobserva'onswithadifferentcharacteris'c.

10-59 Quality Control

Nonrandom Patterns in Control charts

•  Trend•  Cycles•  Bias• Meanshi/•  Toomuchdispersion

10-60 Quality Control

Counting Above/Below Median Runs (7 runs)

B A A B A B B B A A B

Counting Runs

Q.Howtofindthemedianforanoddorevennumberofobserva'ons?

median

10-61 Quality Control

Counting Above/Below Median Runs (7 runs)

Counting Up/Down Runs (8 runs)

U U D U D U D U U D

B A A B A B B B A A B

Figure10.12Mediantests

Figure10.13Up/downtests

Counting Runs

Firstobserva'on

10-62 Quality Control

Run Tests Formula (Table 10.6)

N:the#ofsamplesobserva'ons?

10-63 Quality Control

Run Tests Example •  Thenumberofdefec'veitemspersamplefor11samplesisshownbelow.DetermineifrandompaGernsarepresentinthesequenceusingbothmedianandUp-downtests.UseZ=2.

1234567891011Numberof 2217192518202117232324defec'ves

Sample

10-64 Quality Control

•  Specifica'onsortolerancesn Rangeofacceptablevaluesestablishedbyengineeringdesignorcustomerrequirements

•  Processvariabilityn Naturalvariabilityinaprocess

•  Processcapabilityn Processvariabilityrela3vetospecifica'onseeexamples

•  Oneoftheuniqueproper'esofanormaldistribu'onisthat99.74%oftheobserva'onswillfallwithin3standarddevia'on(σ)fromthemean(μ).Thus,weexpectaprocessthatisincontroltoproducethemajorityoftheoutput(99.74%)betweenμ-3σ andμ+3σ, whereμistheprocessmean.Thatis,thenaturaltolerancelimitsoftheprocessareμ±3σ, andthusweuse6σasameasureofprocesscapability.

Process Capability

10-65 Quality Control

Process Capability 10-66 Quality Control

Process Capability Ratio

Processcapabilityra'o,Cp= specifica'onwidthprocesswidth

Upperspecifica'on–lowerspecifica'on6σ

Cp=

l NotethataCpvaluelessthan1.0indicatesthattheprocessisnotcapableofmee'ngspecifica'ons;avalue≥1.0correspondstoaprocessthatiscapableofmee'ngspecifica'on.

l ACpvalueof1.0impliesthatthefirmisproducing3σquality(0.26%defects)andthattheprocessisconsistentlyproducingoutputswithinspecifica'onseventhoughsomedefectsaregenerated.PreDygood?

10-67 Quality Control

Process Capability Ratio Ø ACpvalueof1.0impliesthatthefirmisproducing3σquality(0.26%defects).Fairlydecentindeed,but

l ?lostpiecesofmaileveryday.l Atleast?wrongdrugprescrip'onseachyearl ?incorrectsurgicalopera'onseachyear.

Cp=0.66=2σquality=Cp=1.0=3σquality=Cp=1.33=4σquality=Cp=2.0=6σquality=Notethatunlesstheprocessmeaniscenteredbetweentheupperandlowerspecifica'ons,theCpra'ocanbemisleading.

10-68 Quality Control

Process Capability Index

Process Capability Index, Cpk = Min (Cpu , Cpl)

Where,

μ–Lowerspecifica'on3σ

Cpl =

Upperspecifica'on–μ3σCpu =

µ = the process average/mean

10-69 Quality Control

Process Capability Ratio & Index Example

•  Theintensivecareunitlabprocesshasanaverageturnaround'meof26.1minutesandastandarddevia'onof1.2minutes.Thetarget(nominal)valueforthisserviceis25minuteswithanupperspecifica'onlimitof30minutesandalowerspecifica'onlimitof20minutes.Managementwantstohavefour-sigmaperformanceforthelab.Isthelabcapableofthislevelofperformance?

Cp =

Cpk =

10-70 Quality Control

Product/service Reliability •  SeriesSystem

component # 1 component # 2 component # n

P1 P2 Pn

P1=probabilitythatcomponent#1willfunc'on

Reliabilityoftheproduct/system=P1xP2x●●●Pn

●●●

Example:

10-71 Quality Control

Product/service Reliability • ParallelSystem

component # 1

component # 2

component # n

P1

P2

Pn

Reliability=1–(1-P1)x(1-P2)x●●●(1-Pn)

●●●

Example: