Quality INSPECTION (Sampling Plan)

Problem – 3Problem – 3 : : (Multiple Sampling Plan)State the steps and also the cases of acceptance of the Lot , in case you are adopting Multiple Stage Sampling Plan (with 3 stages) for inspecting a lot of 20,000 items, if for 1st , 2nd and 3rd stage

sample sizes are 25, 30 & 35, the Acceptance No.s are 1, 4 & 7 and the Rejection No.s are 4, 6 & 8, respectively.

Solution – 3Solution – 3 : :

Given, N=20,000, n1= 25, n2= 30, n3= 35, CC11= 1= 1, RR11= 4= 4 CC22= 4= 4, RR22= 6= 6 CC33= 7= 7,

RR33= 8= 8

StepsSteps : : i) Inspect items of first samplefirst sample of size 25 of size 25,ii) If no. of defectives is 00 or 11, Accept the lotAccept the lotiii) If no. of defectives is 4 or more4 or more, Reject the lotReject the lotiv) If no. of defectives is 22 or 3,3, Inspect items of second second samplesample of size of size 3030 and then,

1) If no. of defectives in the combined sample (25 + 30) does not exceed 44, Accept the lotAccept the lot

2) If no. of defectives in combined sample (25 + 30) is 6 or 6 or moremore, Reject Reject the lotthe lot..

3) If no. of defectives in the combined sample (25 + 30) is 55, Inspect items of third sample of size 35third sample of size 35, and then,

(a) If no. of defectives in combined sample (25 + 30 + 35) does not exceed 77, Accept the lotAccept the lot

(b) If no. of defectives in combined sample (25 + 30 + 35) is 88 oror moremore, , Reject the lotReject the lot..

Quality INSPECTION (Sampling Plan)

Solution to Problem – 3Solution to Problem – 3 : : (Multiple Sampling Plan) [contd..]

– Given, N=20,000,

n1= 25, n2= 30, n3= 35,

CC11= 1= 1, RR11= 4= 4 CC22= 4= 4, RR22= 6= 6 CC33= 7= 7, RR33= 8= 8

Cases of Acceptance of the LotCases of Acceptance of the Lot : : The lot is accepted when no. of defectives found in,

1) Sample – 1 : d1 = 00 or 11

2) Sample – 1 : d1 = 22 and then Sample – 2 : d2 = 00 or 11 or 22

3) Sample – 1 : d1 = 33 and then Sample – 2 : d2 = 00 or 11

4) Sample – 1 : d1 = 2 2 and then Sample – 2 : d2 = 33 (d1 + d2 = 5)

and then Sample – 3 : d3 = 00 or 11 or 22

5) Sample – 1 : d1 = 33 and then Sample – 2 : d2 = 2 2 and then Sample – 3 : d3 = 00 or 11 or 22

Quality INSPECTION (Sampling Plan)

Comparison for Applicability among Comparison for Applicability among Single, Double & Multiple Sampling PlansSingle, Double & Multiple Sampling Plans

Single Sampling PlanSingle Sampling Plan : : 1.1. SimpleSimple Sampling Plan2.2. Less storage spaceLess storage space required for inspection3.3. Cost of Inspection is lowCost of Inspection is low due to short continuing period of inspection 4.4. Average no. of itemsAverage no. of items undergoing inspection per lotper lot may be the largelarge5.5. Lower flexibility ie higher risk for “acceptability of the lotLower flexibility ie higher risk for “acceptability of the lot” towards the towards the

producer/supplierproducer/supplier (more chance of rejection)

Multiple Sampling PlanMultiple Sampling Plan : : 1.1. ComplicatedComplicated Sampling Plan 2.2. More storage spaceMore storage space required for inspection3.3. Cost of Inspection is highCost of Inspection is high due to longer continuing period of

inspection4.4. Average no. of itemsAverage no. of items undergoing inspection per lotper lot may be the lowlow5.5. Higher flexibility ie lower risk for “acceptability of the lotHigher flexibility ie lower risk for “acceptability of the lot” towards

producer/supplier (more chance of acceptance)

# # Double Sampling PlanDouble Sampling Plan having the criteria in between single

and multiple ones, is most acceptableis most acceptable

Quality INSPECTION (contd.)

(C) (C) ACCEPTABLE QUALITY LEVEL (ACCEPTABLE QUALITY LEVEL (AQLAQL)) : : AQLAQL is the HIGHESTHIGHEST ““Fraction DefectivesFraction Defectives” ” in the inspection-sample sizesample size (nn)

prescribed for the lot (offered for inspection), that can allow allow acceptingaccepting the lotthe lot .

The whole lot with FRACTION DEFECTIVEFRACTION DEFECTIVE EQUALEQUAL to or LOWERLOWER than AQLAQL is quality-wise acceptedquality-wise accepted. AQL is related to ACCEPTANCE NUMBERACCEPTANCE NUMBER (CC).

((CC / / nn) x 100) x 100 % = % = AQLAQL or or CC = = nn x (% x (% AQL AQL / 100/ 100))AQL values usually lie between 0.5% and 3.0%.0.5% and 3.0%. AQL value is prefixed and mutually accepted by both Buyer and seller.

# # Rejecting a lot at AQL isRejecting a lot at AQL is “AT PRODUCER’S RISK”. “AT PRODUCER’S RISK”.

(D)(D) Rejectable Quality Level (Rejectable Quality Level (RQLRQL)) : :It is the MINIMUM “MINIMUM “Fraction DefectiveFraction Defective” in the inspection-sample size prescribed for the lot (offered for inspection), , that can allow allow rejection of the lotrejection of the lot .

The whole lot with FRACTION DEFECTIVEFRACTION DEFECTIVE EQUAL EQUAL to or MORE MORE than RQL is quality-wise REJECTEDquality-wise REJECTED. RQL is related to REJECTION NUMBERREJECTION NUMBER (RR).

((RR / / nn) x 100 ) x 100 %% = = RQLRQL or or RR = = nn x (% x (% RQLRQL / 100) / 100)

[RQL is also denoted by LOT TOLERANCE PERCENT DEFECTIVESLOT TOLERANCE PERCENT DEFECTIVES (LTPDLTPD)].

As R > C,R > C, RQL is always > AQLRQL is always > AQL . . # # Accepting a lot at RQL isAccepting a lot at RQL is “AT CONSUMER’S RISK”. “AT CONSUMER’S RISK”.

Quality INSPECTION (contd.)

OPERATING CHARACTERISTICS CURVEOPERATING CHARACTERISTICS CURVE ( ( O.C. CURVEO.C. CURVE ) showing ) showing Producer’s Risk ( Producer’s Risk ( αα ) ) &&

Consumer’s Risk (Consumer’s Risk (ββ))Pa

(Fd)(Fd) Fraction Fraction Defectives Defectives (% Defective)(% Defective)

100%89%

0.01 (1%) AQLAQL

8%0.03 (3%)

35%

0.05 (5%) RQLRQL (LTPDLTPD)

10%

Producer’s RiskProducer’s Risk ( (αα)) = 100% – 89% = 11% (Rejection probability)

Consumer’s RiskConsumer’s Risk ( (ββ)) = 10%

A lot with 3% defective has chance of 35% to be accepted

100% Rejection Probability

89% Acceptance ProbabilityThe O.C. Curve for an Attribute Sampling Plan is the graph of the “Probability of AcceptanceProbability of Acceptance” (PaPa)

against “Fraction DefectivesFraction Defectives” (FdFd) in a lot . O.C. Curve is used for Quality Acceptance or Rejection Scheme.In any Sampling Plan, the O.C. Curve shows the Probability of Acceptance (PaPa) of the lot [on Y-axis] varies depending on variation ofvaries depending on variation of Fraction Defective (FdFd) [or Percentage Defective (100.Fd)] [on X-axis].

●

●

90% Rejection Probability

Quality INSPECTION (contd.)

(E)(E) AVERAGE OUTGOING QUALITY (AOQ)AVERAGE OUTGOING QUALITY (AOQ) : : It is the average percentage defectivesaverage percentage defectives in the outgoing products

including all accepted lots and also all rejected lots which have been 100%

inspected and all defectives have been replaced with non-defective ones.

If, KK = No. of lots, N N = Lot size, n n = Sample size,

FFdd = Percent or Fraction Defectives, PPaa = Probability of Acceptance,

Total no. of items in outgoing lots = K.N (at Fraction Defective of Fd)

Total no. of defectives originally = K.N.Fd

Total no. of defectives in accepted lots = K.N.Fd x Pa

AOQAOQ = K. N.Fd .Pa = K. N. Fd .Pa = Fd . PaFd . Pa

K.N.Pa + K.N.(1 – Pa) K.N

AOQ = Fd . PaAOQ = Fd . Pa

(vi)(vi) AVERAGE OUTGOING QUALITY LIMIT(AOQL)AVERAGE OUTGOING QUALITY LIMIT(AOQL) : :It is the maximum AOQmaximum AOQ. It is the max. average Fraction defectives in out going lotsmax. average Fraction defectives in out going lots when all rejected lots are 100% inspected and all defectives are replaced by good ones.

(vii)(vii) Producer’s Risk ( Producer’s Risk ( αα ) ) ::It is the probability of rejecting a good qualityprobability of rejecting a good quality (AQL qualityAQL quality) lot in the Sampling Plan. The risk of rejecting the lots of AQL quality in a sampling plan, should be smaller than the designated Producer’s Risk (α). Rejecting a lot at AQL isRejecting a lot at AQL is “AT PRODUCER’S RISK”. “AT PRODUCER’S RISK”.

(viii)(viii) Consumer’s Risk (Consumer’s Risk (ββ)) : :It is the probability of accepting a bad qualityprobability of accepting a bad quality (RQL qualityRQL quality) lot in the Sampling Plan. The risk of accepting the lots of RQL quality in the sampling plan, should be smaller than designated Consumer Risk(β). Accepting a lot at RQL isAccepting a lot at RQL is “AT CONSUMER’S RISK”. “AT CONSUMER’S RISK”.

AOQLAOQL

AOQAOQ [= Fd x Pa]

FdFd (Fraction Defective)

Max. AOQMax. AOQ

0

AOQL occurs at a particular FdAOQL occurs at a particular Fd.

FFdd**

● ●

Quality INSPECTION (contd.)

Problem - 3Problem - 3 : : In inspection of a lot comprising of 40,000 items with an AQL of 2%2%, the sample size is 15001500. What is the Acceptance Number ? Explain the condition of acceptability of the lot. What will be your decision on acceptance of the lot of 40,000 items, if the no. of defectives found in the sample is 3030 ?

- The sample size for inspection is 1500 pieces for the lot of

40,000. In inspection of 1500 items of the sample-size, if number of defective items found does not exceed the Acceptance Number (C) the whole lot is accepted.

AQL being 2% of sample size,

The Acceptance Number,The Acceptance Number, CC = % = % AQL x (n /100) AQL x (n /100) = 2 x (1500 = 2 x (1500 /100)/100) == 3030

The whole lot of 40,000 items will be accepted, because number of defectives (found in the sample) 3030, is not greater thannot greater than the Acceptance Number 3030.

AOQL on AOQ vs p curve

Problem – 4Problem – 4 : : Find the AOQLAOQL and show it on AOQ vs PercentAOQ vs Percent Defectives Defectives curve, based on the data given below, as available from OC Curve.

Fd (%)

Pa (%)

0.4 96.3

1.0 82.7

1.4 71.8

1.6 66.3

1.8 61.0

2.0 55.8

2.2 50.9

2.5 44.1

3.0 34.3

3.5 26.2

4.0 19.9

AOQL on AOQ vs p curve

Solution – 4Solution – 4 : : Page – 1of 2

Fd(%) Pa % AOQ % AOQ (Fd x Pa) x Pa)0.4 0.963 0.3850.385

1.0 0.827 0.8270.827

1.4 0.718 1.0051.005

1.6 0.663 1.0611.061

1.8 0.610 1.0981.098

2.0 0.558 1.1161.116

2.2 0.509 1.1201.120 Max ValueMax Value

2.5 0.441 1.1021.102

3.0 0.343 1.0291.029

3.5 0.262 0.9170.917

4.0 0.199 0.7960.796

Fd (%)

Pa (%)

0.4 96.3

1.0 82.7

1.4 71.8

1.6 66.3

1.8 61.0

2.0 55.8

2.2 50.9

2.5 44.1

3.0 34.3

3.5 26.2

4.0 19.9

SolutionSolution::

AOQLAOQL = =

Given

Solution – 4Solution – 4 : :Page – 2 of 2

Average Outgoing Quality Limit (Average Outgoing Quality Limit (AOQLAOQL)) : :

AOQLAOQL

% AOQ% AOQ

Fd (Fraction Defective)Fd (Fraction Defective)

1.5

1.0

0.385

1.2

0.0220.022 0.04

●

●

●

●

AOQ – curveAOQ – curve (Drawn on given data given in the example, next page)

Max. AOQ

AOQ = Fd x Pa (with rectification)

0.004 0.01

1.121.12

0.796

CONTROLCONTROLCHARTSCHARTS

&&

CONTROL CONTROL LIMITSLIMITS

CONTROL CHARTSCONTROL CHARTS are the graphical representation of the collected information / observations / measurements plotted with respect to the Control Limits, which detects the variation (defects) in production process and warns if there is any Deviation from the specified TOLERANCE LIMIT.Purpose of Control Chart :(i) Specifies the state of statistical controlstate of statistical control of Quality of products,(ii) Indicates tolerance for inspectiontolerance for inspection to attain statistical Q. control,(iii) Judges whether the process is under controlwhether the process is under control. _ (1)Quality Characteristics ( X, R, S

charts)

CONTROL CHARTSCONTROL CHARTS (2) Fraction DefectiveFraction Defective (P – Chart) & No. of Defects per Unit (C –

Chart) (3) Non-conformitiesNon-conformities (C – chart) &

Non-conformities per unit (u – chart)

[ (1) is “Control Chart for VariablesControl Chart for Variables” (2) & (3) are “Control Charts for AttributeControl Charts for Attribute”.] _ X – Chart : Average Variable (on measurements) Chart R – Chart : RangeRange Chart (SPREAD of measurements)

SS – Chart : Standard DeviationStandard Deviation of process

Control Chart for VariablesControl Chart for VariablesLet, Variables (individual measurements/observations) = Xi , (where i = 1 to n),

Sample SizeSample Size for each sub-group (sample) = nn,

No. of Sub-group (sample)No. of Sub-group (sample) in the lot = NN. __ Sample AverageSample Average (Sample Mean), XXmean, X = ( X = ( ∑∑ XXi ) ) ÷ nn where i = 1 to n

= = – – Process AverageProcess Average (Process Mean), X = ( X = ( ∑∑ XXi ) ) ÷ NN where i = 1 to N

RangeRange (in each sample), RRi = ( X = ( X max – X – Xmin))

– – Mean RangeMean Range (of the Process), R = ( R = ( ∑∑ RRi ) ) ÷ NN where i = 1 to N

Standard DeviationStandard Deviation (in each Sample), SSj = √√ [[ ∑∑ ( X( Xi – X– Xmean ) )22 ÷ (n-1) ](n-1) ]

_ _

Mean Standard DeviationMean Standard Deviation for the lot, SS = (∑∑ Sj) ÷ NN

Standard DeviationStandard Deviation of population,

__σ = σ = R R ÷ dd22

__σ = σ = S S ÷ CC22

_ _ XX

NN

LCLLCL

UCLUCL

=XX●

● ●

●

●

●

●

●

● ●

NN

RR

__RR

UCLUCL

LCLLCL

●

●

● ●

●

●

●

● ●

_ _ X - ChartX - Chart R - ChartR - Chart

– – X – Chart and R-ChartX – Chart and R-Chart

– – X – ChartX – Chart

R-ChartR-Chart

Central Line (CL) =X

– – R

Upper Control Limit (UCL) = – X + A2.R

__ D4 . R

Lower Control Limit (LCL) = – X – A2.R

__

D3 . R

_ _ X – Chart and R-ChartX – Chart and R-Chart

_ _ X – ChartX – Chart

R-ChartR-Chart

Central Line (CL) =X

__R

Upper Control Limit (UCL) = – X + A2.R

__ D4 . R

Lower Control Limit (LCL) = – X – A2.R

__

D3 . R

_ _ X – Chart and S-X – Chart and S-ChartChart

_ _ X – ChartX – Chart

S-ChartS-Chart

Central Line (CL) =X

__S

Upper Control Limit (UCL)

= – X + A3.S

__ B4 . S

Lower Control Limit (LCL) = – X – A3.S

_ B3 . S## [ X – S charts are preferablyX – S charts are preferably applicable over X – R chart applicable over X – R chart, , when

(1) Sample size, (1) Sample size, n > 10n > 10, , (2) (2) Tight control of Process VariabilityTight control of Process Variability is needed, and is needed, and (3) Sample size (3) Sample size n variesn varies.].]

Values of Statistical ConstantsValues of Statistical Constants

Sample Size (n)

A2 D3 D4 A3 B3 B4 d2 C2

22 1.88 0 3.27 2.66 0 3.27 1.128 0.7979

33 1.02 0 2.57 1.95 0 2.57 1.693 0.8862

44 0.73 0 2.28 1.63 0 2.27 2.059 0.9213

55 0.58 0 2.11 1.43 0 2.09 2.326 0.9400

66 0.48 0 2.00 1.29 0.03 1.97 2.534 0.9515

77 0.42 0.08 1.92 1.18 0.12 1.88 2.704 0.9594

88 0.37 0.14 1.86 1.10 0.19 1.81 2.847 0.9650

99 0.34 0.18 1.82 1.03 0.24 1.76 2.970 0.9693

1010 0.31 0.22 1.78 0.98 0.28 1.72 3.078 0.9727

A Model of “Data Sheet” for X – R chartA Model of “Data Sheet” for X – R chart----------------------------------------------------------------------------------------------------------Department : Specification :Drawing No. : USL : LSL :Part Name. : Sample Size :Part No. : Sample frequency :Machine : Total No. of Sample :Operator : Inspector :-----------------------------------------------------------------------------------------------------------------------------------Sample No. X1 X2 X3 X4 X5 Xmean Range R

(= ∑ Xi ÷ n) (Xmax – Xmin) 1. 2. 3. 4. : : N (normally 20 -25)---------------------------------------------------------------------------------------------======-----------======= ∑ Xmean ∑ R = _ X = ( X = ( ∑∑ XXmean ) ) ÷ N = R = N = R = ∑∑ R R ÷ N = N =

CL = CL =UCL = UCL =LCL = LCL =

_ _ σ σ = R / d= R / d22 PCRPCR = (USL – LSL) = (USL – LSL) ÷ 6 σ 6 σ

CAUSES OF VARITIONCAUSES OF VARITIONOF MEASUREMENTSOF MEASUREMENTS

Variation in measurement (quality) can be caused in two ways,i)i) AssignableAssignable Causesii)ii) NaturalNatural or ChanceChance or RandomRandom Causes.

ASSIGNABLEASSIGNABLE CAUSES CAUSESBasic NatureBasic Nature : :1) For Assignable Causes the variations are generally greater in greater in

magnitudemagnitude (in comparison to Natural Causes) 2) The variations do not follow Normal Statisticsdo not follow Normal Statistics.3) The Assignable causes & variations are not inevitablenot inevitable.4) Assignable Causes of variation are generally easily traceable & easily traceable &

identifiable identifiable 5) Assignable Causes of variation normally can be avoidedavoided /

rectifiedrectified / eliminatedeliminated. 6) If Assignable Causes are not rectified, the process becomes

“Out of ControlOut of Control”.

CAUSES OF VARITION

Basic Assignable CausesBasic Assignable Causes : :1.1. Faulty processFaulty process 2.2. Quality of raw materialsQuality of raw materials and/or componentscomponents,3.3. Wear & TearWear & Tear of the machine and equipment4.4. Operating Condition of EquipmentOperating Condition of Equipment / machineries, like vibration,

leveling defect, defective centering, non-alignment, etc5.5. Improper fixing / fittingImproper fixing / fitting of toolstools6.6. Wrong toolsWrong tools7.7. Wrong settingWrong setting of machine 8.8. Faulty jigs and fixturesFaulty jigs and fixtures9.9. ImproperImproper Maintenance Maintenance 10. Use of uncalibrateduncalibrated or poorly calibrated instrumentinstrument,, gauges gauges,

etc.11.11. Faulty Environmental ConditionFaulty Environmental Condition of working place (improper

lightinglighting & ventilatioventilation, dustdust, smokesmoke, temperature variation, toxic or corrosive atmosphere, humidity, etc),

12. Operating Condition of InstrumentationCondition of Instrumentation & Control,13.13. Inadequate Skill of workersInadequate Skill of workers / operators,14.14. Mental conditionMental condition / Mood / Mood of workers / operators,15.15. CarelessnessCarelessness and mistake on the part of operators,

CAUSES OF VARITION

NaturalNatural or or ChanceChance or or RandomRandom Causes Causes

Basic NatureBasic Nature : :

1. Such causes are inherentinherent in the process and thus are INEVITABLEINEVITABLE,

2. The variations in measurement are generally smaller in smaller in magnitudemagnitude

3. Such causes are unpredictableunpredictable and thus non-rectifiablenon-rectifiable in nature.

4. The causes are very difficult to identify & tracedifficult to identify & trace and thus very difficult to rectify rectify even under best care & conditions.

5. Variations due to such causes followfollow Statistical Rules Statistical Rules andand Normal CurveNormal Curve (chance factor ie Probabilitychance factor ie Probability is involved).

6.6. With this type of variationsWith this type of variations, the process is said to be UNDER-UNDER-CONTROLCONTROL..

7. If the variations do not follow Normal Curvefollow Normal Curve, there must be some Assignable Causessome Assignable Causes which can be detected and rectified.

Distinguish between Distinguish between Natural Causes and Assignable CausesNatural Causes and Assignable Causes

Natural ErrorNatural Error Assignable ErrorAssignable Error1) Inherent in the process of production Exists Outside the process2) Inevitable to occur Can be avoided3) Difficult to identify / detect Easy to identify / detect4) Very difficult to eliminate / rectify can be easily eliminated / rectified 5) Errors (variations) are very small in Errors (variations) are great in magnitude. magnitude 6) Variations follow standard Statistical Variations does not follow any Laws of Variations. standard Statistical Laws of Variations. 7) Variations or Errors occur at random Variations or Errors occur steadily and continuously

Possible Possible ERRORERROR in in MEASUREMENT MEASUREMENT of Variations in a Lotof Variations in a Lot

1.1. SamplingSampling Error2.2. Testing ProcessTesting Process / Method Error3. Error in Measuring devices or InstrumentInstrument4.4. Personal ErrorPersonal Error.

Normal Distribution CurveNormal Distribution CurveFrequency of Frequency of occurrenceoccurrence

Deviation Deviation LimitsLimits

σσ 22σσ 33σσ--33σσ --22σσ -σ-σ = = X X ((μ) μ) 68.28%68.28%

95.46%95.46%

99.73%99.73%

Area under the Curve Area under the Curve indicates Populationindicates Population

Normal Distribution Curve & Six SigmaNormal Distribution Curve & Six Sigma

99.99966% ie 99.99966% ie 3.4 defects per million3.4 defects per million population come under the range population come under the range

((μ μ ± ± 4.54.5 σσ) . So, Six Sigma range covers 3.4 defects per million ) . So, Six Sigma range covers 3.4 defects per million population, even with a shift of Mean population, even with a shift of Mean μμ by 1.5 Sigma. by 1.5 Sigma.

## [ X – S charts are preferablyX – S charts are preferably applicable over X – R chart applicable over X – R chart, , when (1) Sample size, (1) Sample size, n > 10n > 10, , (2) (2) Tight control of Process VariabilityTight control of Process Variability is needed, and is needed, and (3) Sample size (3) Sample size n variesn varies.].]

SPECIFIED TOLERANCESPECIFIED TOLERANCE : : Process Control Limits :

USL = Upper specified Limit UCL = Upper Control Limit

LSL = Lower specified Limit LCL = Lower Control Limit(USL – LSLUSL – LSL) = Specified Spread Specified Spread (S) (S)Center Line of Specified Spread = Center Line of Specified Spread = (USL + LSL) / 2(USL + LSL) / 2

NATURAL TOLERANCENATURAL TOLERANCE : : ==Process Central LineProcess Central Line = XX

Upper Natural Tolerance LimitUpper Natural Tolerance Limit = UNTLUNTL == Process CL + 3 σProcess CL + 3 σ

Lower Natural Tolerance LimitLower Natural Tolerance Limit = LNTL =LNTL = Process CL – 3 σProcess CL – 3 σ

(NaturalNatural) ProcessProcess SPREAD = SPREAD = (UNTL – LNTLUNTL – LNTL) = 6 σ6 σ

PROCESS CAPABILITY RATIOPROCESS CAPABILITY RATIO (PCR) (PCR) = Specified SpreadSpecified Spread. Process SpreadProcess Spread

PCR PCR = = (USL – LSLUSL – LSL) = S S

6 σ 6 σ6 σ 6 σ

TOLERANCE TOLERANCE Tolerance is the acceptable / allowed limits of variations of the measurements(dimensions, strength, properties, finish, etc) of the items of the lot.

Tolerance Limits are the maximum variations of the measurement those canbe acceptable / allowed, above (Upper Control Limit) or below (Lower ControlLimit) the corresponding specified measurement. Example : + 0.2 + 0.8 +10Diameter 455 mm, Cr content 14 %, Strength 540 kg/cm2

– 0.3 – 0. 5 – 0

Tolerance may be : (i) Natural ToleranceNatural Tolerance and/or (ii) Specified ToleranceSpecified Tolerance

Natural ToleranceNatural Tolerance : : If the process is “Under Control” (ie the variations are small and follow statistical laws), the Upper Control Level (UCL) and theLower Control Limit (LCL) together indicate the Natural Tolerance.The gap between Natural Tolerance Limits UCLUCL & LCLLCL (ie 6 6 σσ) is BANDBAND. Specified ToleranceSpecified Tolerance : : The tolerance limits (U & L) predefined or specified by the designer or the user of the product (depending on thequality needed to fit it in its use) is termed as Specified Tolerance.Specified Tolerance is not directly related to or dependent on Natural Tolerance ie Natural Control Limits. Specified Tolerance is important andconsidered in QC of industrial Products.The gap between Specified Tolerance Limits USL & LSL is SPREADSPREAD. .

Quality-Control based on SPREAD and Process BANDQuality-Control based on SPREAD and Process BAND 1) If 6 6 σσ < S < SThe process can be easily bekept under control by adjusting =the position of X very near to CentreLine of the Spread (S).Position 2 is shifted and kept at position 1. 2) If 6 6 σσ = S = S “Strict Centering” care to be taken =to keep the Control (Centre) Line Xperfectly at the Centre Line of Spread. It is also better to relax the SpecifiedTolerance to make S> 6 σ’. Position 2 is shifted and kept at position 1.

3) If 6 6 σσ > S > S Even with Extreme care & control andPerfect Centering, the process is not under Control. The Process is said to be “Out of Control”. Quality non-conformance will be there and there by Quality Rejection Will take place. Care to be taken : (1) Fundamental change in process,(2) Revision of Specified Tolerance, (3) Perfect Centering, (4) 100% inspection

21

=X

CLCL

USLUSL

LSLLSL

6 σ

6σSS

USLUSL

LSLLSL

CLCL SS2

1

6σ = X

6σ

==XX

==X X