To Accompany Russell and Taylor, Operations Management, 4th Edition, 2003 Prentice-Hall, Inc. All rights reserved. Chapter 15 Statistical Quality Control

To Accompany Russell and Taylor, Operations Management, 4th Edition, 2003 Prentice-Hall, Inc. All rights reserved.

Chapter 15Chapter 15

Statistical Statistical Quality ControlQuality Control



Statistical Process Statistical Process ControlControl

• Take periodic samples from processTake periodic samples from process

• Plot sample points on control chartPlot sample points on control chart

• Determine if process Determine if process is within limitsis within limits

• Prevent quality Prevent quality problemsproblems

UCLUCL

LCLLCL


VariationVariation Common CausesCommon Causes

Variation inherent in a processVariation inherent in a processCan be eliminated only through Can be eliminated only through

improvements in the systemimprovements in the system

Special CausesSpecial CausesVariation due to identifiable factorsVariation due to identifiable factorsCan be modified through operator or Can be modified through operator or

management actionmanagement action


Types of DataTypes of DataAttribute dataAttribute data

Product characteristic Product characteristic evaluated with a discrete choiceevaluated with a discrete choice

Good/bad, yes/noGood/bad, yes/no

Variable dataVariable data Product characteristic that Product characteristic that

can be measuredcan be measured Length, size, weight, height, Length, size, weight, height,

time, velocitytime, velocity


SPC Applied to SPC Applied to ServicesServices

Nature of defect is different in Nature of defect is different in servicesservices

Service defect is a failure to meet Service defect is a failure to meet customer requirementscustomer requirements

Monitor times, customer Monitor times, customer satisfactionsatisfaction


Service Quality ExamplesService Quality ExamplesHospitals Hospitals

Timeliness, responsiveness, Timeliness, responsiveness, accuracy of lab testsaccuracy of lab tests

Grocery StoresGrocery StoresCheck-out time, stocking, cleanlinessCheck-out time, stocking, cleanliness

AirlinesAirlinesLuggage handling, waiting times, Luggage handling, waiting times,

courtesycourtesyFast food restaurantsFast food restaurants

Waiting times, food quality, Waiting times, food quality, cleanliness, employee courtesycleanliness, employee courtesy


Service Quality ExamplesService Quality ExamplesCatalog-order companiesCatalog-order companies

Order accuracy, operator Order accuracy, operator knowledge and courtesy, knowledge and courtesy, packaging, delivery time, packaging, delivery time, phone order waiting timephone order waiting time

Insurance companiesInsurance companiesBilling accuracy, timeliness of claims Billing accuracy, timeliness of claims

processing, agent availability and processing, agent availability and response timeresponse time


Control ChartsControl Charts

Graph establishing process control Graph establishing process control limitslimits

Charts for variablesCharts for variablesMean (x-bar), Range (R)Mean (x-bar), Range (R)

Charts for attributesCharts for attributesp and cp and c


Process Control ChartProcess Control Chart

11 22 33 44 55 66 77 88 99 1010

Sample numberSample number

UpperUppercontrolcontrol

limitlimit

ProcessProcessaverageaverage

LowerLowercontrolcontrol

limitlimit

Out of controlOut of control

Figure 15.1Figure 15.1


A Process is In Control ifA Process is In Control if

1.1. No sample points outside limitsNo sample points outside limits

2.2. Most points near process averageMost points near process average

3.3. About equal number of points About equal number of points above & below centerlineabove & below centerline

4.4. Points appear randomly Points appear randomly distributeddistributed


Development of Development of Control ChartControl Chart

Based on in-control dataBased on in-control data

If non-random causes present If non-random causes present discard datadiscard data

Correct control chart limitsCorrect control chart limits


Control Charts for Control Charts for AttributesAttributes

p Chartsp ChartsCalculate percent defectives in sampleCalculate percent defectives in sample

c Chartsc ChartsCount number of Count number of

defects in itemdefects in item


p-Chartp-ChartUCL = UCL = pp + + zzpp

LCL = LCL = pp - - zzppwherewhere

zz == the number of standard deviations the number of standard deviations from the process averagefrom the process averagepp == the sample proportion defective; an the sample proportion defective; an estimate of the process averageestimate of the process averagepp == the standard deviation of the the standard deviation of the

sample proportionsample proportion

pp = = pp(1 - (1 - pp))

nn


The Normal DistributionThe Normal Distribution

=0=0 11 22 33-1-1-2-2-3-3

95%

99.74%


Control Chart Z ValuesControl Chart Z Values

Smaller Z values make more Smaller Z values make more sensitive chartssensitive charts

Z = 3.00 is standardZ = 3.00 is standardCompromise between sensitivity Compromise between sensitivity

and errorsand errors


p-Chart Examplep-Chart Example20 samples of 100 pairs of jeans20 samples of 100 pairs of jeans

NUMBER OFNUMBER OF PROPORTIONPROPORTIONSAMPLESAMPLE DEFECTIVESDEFECTIVES DEFECTIVEDEFECTIVE

11 66 .06.06

22 00 .00.00

33 44 .04.04

:: :: ::

:: :: ::

2020 1818 .18.18

200200

Example 15.1Example 15.1




11 66 .06.06

22 00 .00.00

33 44 .04.04

:: :: ::

:: :: ::

2020 1818 .18.18

200200


p =

= 200 / 20(100)= 0.10

total defectives

total sample observations




11 66 .06.06

22 00 .00.00

33 44 .04.04

:: :: ::

:: :: ::

2020 1818 .18.18

200200


pp = 0.10 = 0.10

UCL = p + z = 0.10 + 3p(1 - p)

n

0.10(1 - 0.10)

100

UCL = 0.190

LCL = 0.010

LCL = p - z = 0.10 - 3p(1 - p)

n

0.10(1 - 0.10)

100


0.020.02

0.040.04

0.060.06

0.080.08

0.100.10

0.120.12

0.140.14

0.160.16

0.180.18

0.200.20

Pro

po

rtio

n d

efec

tive

Pro

po

rtio

n d

efec

tive

Sample numberSample number22 44 66 88 1010 1212 1414 1616 1818 2020

UCL = 0.190

LCL = 0.010

p = 0.10

p-Chartp-Chart


c-Chartc-Chart

UCL = UCL = cc + + zzcc

LCL = LCL = cc - - zzcc

cc = = cc

wherewhere

cc = number of defects per sample = number of defects per sample


c-Chartc-ChartThe number of defects in 15 sample roomsThe number of defects in 15 sample rooms

11 121222 8833 1616

:: :::: ::

1515 1515

190190

SAMPLESAMPLE NUMBER OF DEFECTSNUMBER OF DEFECTS

cc = = 12.67 = = 12.67190190

1515

UCLUCL = = cc + + zzcc

= 12.67 + 3 12.67= 12.67 + 3 12.67= 23.35= 23.35

LCLLCL = = cc + + zzcc

= 12.67 - 3 12.67= 12.67 - 3 12.67= 1.99= 1.99



c-Chartc-Chart

33

66

99

1212

1515

1818

2121

2424

Nu

mb

er o

f d

efec

tsN

um

ber

of

def

ects

Sample numberSample number

22 44 66 88 1010 1212 1414 1616

UCL = 23.35

LCL = 1.99

c = 12.67


Control Charts for Control Charts for VariablesVariables

Mean chart ( x -Chart )Mean chart ( x -Chart )Uses average of a sampleUses average of a sample

Range chart ( R-Chart )Range chart ( R-Chart )Uses amount of dispersion in Uses amount of dispersion in

a samplea sample


Range ( R- ) ChartRange ( R- ) Chart

UCL = UCL = DD44RR LCL = LCL = DD33RR

RR = = RRkk

wherewhere

RR = range of each sample= range of each samplekk = number of samples= number of samples


Range ( R- ) ChartRange ( R- ) Chartn A2 D3 D4

SAMPLE SIZE FACTOR FOR x-CHART FACTORS FOR R-CHART

2 1.88 0.00 3.273 1.02 0.00 2.574 0.73 0.00 2.285 0.58 0.00 2.116 0.48 0.00 2.007 0.42 0.08 1.928 0.37 0.14 1.869 0.44 0.18 1.82

10 0.11 0.22 1.7811 0.99 0.26 1.7412 0.77 0.28 1.7213 0.55 0.31 1.6914 0.44 0.33 1.6715 0.22 0.35 1.6516 0.11 0.36 1.6417 0.00 0.38 1.6218 0.99 0.39 1.6119 0.99 0.40 1.6120 0.88 0.41 1.59

Table 15.1Table 15.1


R-Chart ExampleR-Chart ExampleOBSERVATIONS (SLIP-RING DIAMETER, CM)OBSERVATIONS (SLIP-RING DIAMETER, CM)

SAMPLE SAMPLE kk 11 22 33 44 55 xx RR

11 5.025.02 5.015.01 4.944.94 4.994.99 4.964.96 4.984.98 0.080.0822 5.015.01 5.035.03 5.075.07 4.954.95 4.964.96 5.005.00 0.120.1233 4.994.99 5.005.00 4.934.93 4.924.92 4.994.99 4.974.97 0.080.0844 5.035.03 4.914.91 5.015.01 4.984.98 4.894.89 4.964.96 0.140.1455 4.954.95 4.924.92 5.035.03 5.055.05 5.015.01 4.994.99 0.130.1366 4.974.97 5.065.06 5.065.06 4.964.96 5.035.03 5.015.01 0.100.1077 5.055.05 5.015.01 5.105.10 4.964.96 4.994.99 5.025.02 0.140.1488 5.095.09 5.105.10 5.005.00 4.994.99 5.085.08 5.055.05 0.110.1199 5.145.14 5.105.10 4.994.99 5.085.08 5.095.09 5.085.08 0.150.15

1010 5.015.01 4.984.98 5.085.08 5.075.07 4.994.99 5.035.03 0.100.10

50.0950.09 1.151.15



R-Chart ExampleR-Chart ExampleOBSERVATIONS (SLIP-RING DIAMETER, CM)OBSERVATIONS (SLIP-RING DIAMETER, CM)


11 5.025.02 5.015.01 4.944.94 4.994.99 4.964.96 4.984.98 0.080.0822 5.015.01 5.035.03 5.075.07 4.954.95 4.964.96 5.005.00 0.120.1233 4.994.99 5.005.00 4.934.93 4.924.92 4.994.99 4.974.97 0.080.0844 5.035.03 4.914.91 5.015.01 4.984.98 4.894.89 4.964.96 0.140.1455 4.954.95 4.924.92 5.035.03 5.055.05 5.015.01 4.994.99 0.130.1366 4.974.97 5.065.06 5.065.06 4.964.96 5.035.03 5.015.01 0.100.1077 5.055.05 5.015.01 5.105.10 4.964.96 4.994.99 5.025.02 0.140.1488 5.095.09 5.105.10 5.005.00 4.994.99 5.085.08 5.055.05 0.110.1199 5.145.14 5.105.10 4.994.99 5.085.08 5.095.09 5.085.08 0.150.15

1010 5.015.01 4.984.98 5.085.08 5.075.07 4.994.99 5.035.03 0.100.10

50.0950.09 1.151.15


Rk

R = = = 0.115 1.1510

UCL = D4R = 2.11(0.115) = 0.243

LCL = D3R = 0(0.115) = 0

UCL = 0.243

LCL = 0

Ra

ng

e

Sample number

R = 0.115

|1

|2

|3

|4

|5

|6

|7

|8

|9

|10

0.28 –

0.24 –

0.20 –

0.16 –

0.12 –

0.08 –

0.04 –

0 –


x-Chart Calculationsx-Chart Calculations

xx = = xx11 + + xx22 + ... + ... xxkk

kk==

UCL = UCL = xx + + AA22RR LCL = LCL = xx - - AA22RR== ==

wherewhere

xx = the average of the sample means= the average of the sample means==


x-Chart Examplex-Chart Example


OBSERVATIONS (SLIP-RING DIAMETER, CM)OBSERVATIONS (SLIP-RING DIAMETER, CM)


11 5.025.02 5.015.01 4.944.94 4.994.99 4.964.96 4.984.98 0.080.0822 5.015.01 5.035.03 5.075.07 4.954.95 4.964.96 5.005.00 0.120.1233 4.994.99 5.005.00 4.934.93 4.924.92 4.994.99 4.974.97 0.080.0844 5.035.03 4.914.91 5.015.01 4.984.98 4.894.89 4.964.96 0.140.1455 4.954.95 4.924.92 5.035.03 5.055.05 5.015.01 4.994.99 0.130.1366 4.974.97 5.065.06 5.065.06 4.964.96 5.035.03 5.015.01 0.100.1077 5.055.05 5.015.01 5.105.10 4.964.96 4.994.99 5.025.02 0.140.1488 5.095.09 5.105.10 5.005.00 4.994.99 5.085.08 5.055.05 0.110.1199 5.145.14 5.105.10 4.994.99 5.085.08 5.095.09 5.085.08 0.150.15

1010 5.015.01 4.984.98 5.085.08 5.075.07 4.994.99 5.035.03 0.100.10

50.0950.09 1.151.15

UCL = x + A2R = 5.01 + (0.58)(0.115) = 5.08

LCL = x - A2R = 5.01 - (0.58)(0.115) = 4.94

=

=

x = = = 5.01 cm= x

k50.09

10


x-Chart Examplex-Chart Example


OBSERVATIONS (SLIP-RING DIAMETER, CM)OBSERVATIONS (SLIP-RING DIAMETER, CM)


11 5.025.02 5.015.01 4.944.94 4.994.99 4.964.96 4.984.98 0.080.0822 5.015.01 5.035.03 5.075.07 4.954.95 4.964.96 5.005.00 0.120.1233 4.994.99 5.005.00 4.934.93 4.924.92 4.994.99 4.974.97 0.080.0844 5.035.03 4.914.91 5.015.01 4.984.98 4.894.89 4.964.96 0.140.1455 4.954.95 4.924.92 5.035.03 5.055.05 5.015.01 4.994.99 0.130.1366 4.974.97 5.065.06 5.065.06 4.964.96 5.035.03 5.015.01 0.100.1077 5.055.05 5.015.01 5.105.10 4.964.96 4.994.99 5.025.02 0.140.1488 5.095.09 5.105.10 5.005.00 4.994.99 5.085.08 5.055.05 0.110.1199 5.145.14 5.105.10 4.994.99 5.085.08 5.095.09 5.085.08 0.150.15

1010 5.015.01 4.984.98 5.085.08 5.075.07 4.994.99 5.035.03 0.100.10

50.0950.09 1.151.15

UCL = x + A2R = 5.01 + (0.58)(0.115) = 5.08

LCL = x - A2R = 5.01 - (0.58)(0.115) = 4.94

=

=

x = = = 5.01 cm= x

k50.09

10

UCL = 5.08

LCL = 4.94

Mea

n

Sample number

|1

|2

|3

|4

|5

|6

|7

|8

|9

|10

5.10 –

5.08 –

5.06 –

5.04 –

5.02 –

5.00 –

4.98 –

4.96 –

4.94 –

4.92 –

x = 5.01=


Using x- and R-Charts Using x- and R-Charts TogetherTogether

Each measures the process Each measures the process differently differently

Both process average and variability Both process average and variability must be in controlmust be in control


Control Chart PatternsControl Chart Patterns


UCLUCL

LCLLCL

Sample observationsSample observationsconsistently above theconsistently above thecenter linecenter line

LCLLCL

UCLUCL

Sample observationsSample observationsconsistently below theconsistently below thecenter linecenter line


Control Chart PatternsControl Chart Patterns


LCLLCL

UCLUCL

Sample observationsSample observationsconsistently increasingconsistently increasing

UCLUCL

LCLLCL

Sample observationsSample observationsconsistently decreasingconsistently decreasing


Zones for Pattern TestsZones for Pattern TestsUCL

LCL

Zone A

Zone B

Zone C

Zone C

Zone B

Zone A

Process average

3 sigma = x + A2R=

3 sigma = x - A2R=

2 sigma = x + (A2R)= 23

2 sigma = x - (A2R)= 23

1 sigma = x + (A2R)= 13

1 sigma = x - (A2R)= 13

x=

Sample number

|1

|2

|3

|4

|5

|6

|7

|8

|9

|10

|11

|12

|13



Control Chart PatternsControl Chart Patterns1.1. 8 consecutive points on one side of the 8 consecutive points on one side of the

center line.center line.

2.2. 8 consecutive points up or down across 8 consecutive points up or down across Zones.Zones.

3.3. 14 points alternating up or down.14 points alternating up or down.

4.4. 2 out of 3 consecutive points in Zone A 2 out of 3 consecutive points in Zone A but still inside the control limits.but still inside the control limits.

5.5. 4 out of 5 consecutive points in Zone A 4 out of 5 consecutive points in Zone A or B.or B.


Performing a Pattern TestPerforming a Pattern Test

11 4.984.98 BB —— BB

22 5.005.00 BB UU CC

33 4.954.95 BB DD AA

44 4.964.96 BB DD AA

55 4.994.99 BB UU CC

66 5.015.01 —— UU CC

77 5.025.02 AA UU CC

88 5.055.05 AA UU BB

99 5.085.08 AA UU AA

1010 5.035.03 AA DD BB

SAMPLESAMPLE xx ABOVE/BELOWABOVE/BELOW UP/DOWNUP/DOWN ZONEZONE



Sample Size DeterminationSample Size Determination

Attribute control chartsAttribute control charts50 to 100 parts in a sample50 to 100 parts in a sample

Variable control chartsVariable control charts2 to 10 parts in a sample2 to 10 parts in a sample


Process CapabilityProcess Capability Range of natural variability in processRange of natural variability in process

Measured with control charts.Measured with control charts.

Process cannot meet specifications if Process cannot meet specifications if natural variability exceeds tolerancesnatural variability exceeds tolerances

3-sigma quality3-sigma quality Specifications equal the process control Specifications equal the process control

limits. limits.

6-sigma quality6-sigma quality Specifications twice as large as control Specifications twice as large as control

limitslimits


Process CapabilityProcess Capability

(b) Design specifications (b) Design specifications and natural variation the and natural variation the same; process is capable same; process is capable of meeting specifications of meeting specifications most the time.most the time.

Design Design SpecificationsSpecifications

ProcessProcess

(a) Natural variation (a) Natural variation exceeds design exceeds design specifications; process specifications; process is not capable of is not capable of meeting specifications meeting specifications all the time.all the time.


ProcessProcess



Process CapabilityProcess Capability


(c) Design specifications (c) Design specifications greater than natural greater than natural variation; process is variation; process is capable of always capable of always conforming to conforming to specifications.specifications.


ProcessProcess

(d) Specifications greater (d) Specifications greater than natural variation, but than natural variation, but process off center; process off center; capable but some output capable but some output will not meet upper will not meet upper specification.specification.


ProcessProcess


Process Capability Process Capability MeasuresMeasures

Process Capability RatioProcess Capability Ratio

CCpp ==

==

tolerance rangetolerance range

process rangeprocess range

upper specification limit - upper specification limit - lower specification limitlower specification limit

66


Computing CComputing Cpp

Net weight specification = 9.0 oz Net weight specification = 9.0 oz 0.5 oz 0.5 ozProcess mean = 8.80 ozProcess mean = 8.80 ozProcess standard deviation = 0.12 ozProcess standard deviation = 0.12 oz

CCpp ==

= = 1.39= = 1.39

upper specification limit - upper specification limit - lower specification limitlower specification limit

66

9.5 - 8.59.5 - 8.5

6(0.12)6(0.12)



Process Capability Process Capability MeasuresMeasures

Process Capability IndexProcess Capability Index

CCpkpk = minimum = minimum

xx - lower specification limit - lower specification limit

33

==

upper specification limit -upper specification limit - x x

33

==

,,


Computing CComputing Cpkpk

Net weight specification = 9.0 oz Net weight specification = 9.0 oz 0.5 oz 0.5 ozProcess mean = 8.80 ozProcess mean = 8.80 ozProcess standard deviation = 0.12 ozProcess standard deviation = 0.12 oz

CCpkpk = minimum= minimum

= minimum , = 0.83= minimum , = 0.83

xx - lower specification limit - lower specification limit

33

==

upper specification limit -upper specification limit - x x

33

==,,

8.80 - 8.508.80 - 8.50

3(0.12)3(0.12)

9.50 - 8.809.50 - 8.80

3(0.12)3(0.12)Example 15.7Example 15.7

Documents

To Accompany Russell and Taylor, Operations Management, 4th Edition, 2003 Prentice-Hall, Inc. All rights reserved. Chapter 15 Statistical Quality Control