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To Accompany Russell and Taylor, Operations Management, 4th Edition, 2003 Prentice-Hall, Inc. All rights reserved.
Chapter 15Chapter 15
Statistical Statistical Quality ControlQuality Control
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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The Normal DistributionThe Normal Distribution
=0=0 11 22 33-1-1-2-2-3-3
95%
99.74%
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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
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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
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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
p =
= 200 / 20(100)= 0.10
total defectives
total sample observations
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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
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
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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
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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
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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
Example 15.2Example 15.2
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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
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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
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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
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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
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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
Example 15.3Example 15.3
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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
Example 15.3Example 15.3
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 –
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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==
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x-Chart Examplex-Chart Example
Example 15.4Example 15.4
OBSERVATIONS (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
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
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x-Chart Examplex-Chart Example
Example 15.4Example 15.4
OBSERVATIONS (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
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=
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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
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Control Chart PatternsControl Chart Patterns
Figure 15.3Figure 15.3
UCLUCL
LCLLCL
Sample observationsSample observationsconsistently above theconsistently above thecenter linecenter line
LCLLCL
UCLUCL
Sample observationsSample observationsconsistently below theconsistently below thecenter linecenter line
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Control Chart PatternsControl Chart Patterns
Figure 15.3Figure 15.3
LCLLCL
UCLUCL
Sample observationsSample observationsconsistently increasingconsistently increasing
UCLUCL
LCLLCL
Sample observationsSample observationsconsistently decreasingconsistently decreasing
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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
Figure 15.4Figure 15.4
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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.
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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
Example 15.5Example 15.5
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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
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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
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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.
Design Design SpecificationsSpecifications
ProcessProcess
Figure 15.5Figure 15.5
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Process CapabilityProcess Capability
Figure 15.5Figure 15.5
(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.
Design Design SpecificationsSpecifications
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.
Design Design SpecificationsSpecifications
ProcessProcess
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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
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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)
Example 15.6Example 15.6
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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
==
,,
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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
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