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Six-Sigma Quality, Process Capability,
and Statistical Process ControlSelected Slides from Jacobs et al, 9thEdition
Operations and Supply Management
Chapter 9 and 9AEdited, Annotated and Supplemented by
Peter Jurkat
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Total Quality Management (TQM)
Total qualitymanagementis definedas managing the entireorganization so that it
excels on all dimensionsof products and servicesthat are important to thecustomer
Design quality: Inherentvalue of the product inthe marketplace
Dimensions include:Performance, Features,Reliability/Durability,Serviceability, Aesthetics,and Perceived Quality.
Conformance quality:Degree to which theproduct or service designspecifications are met
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Six Sigma Quality
A philosophy and set of methods
companies use to eliminate defects
in their products and processes Seeks to reduce variation in the
processes that lead to product
defects
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McGraw-Hill/Irwin Copyright 2009 by The McGraw-Hill Companies, Inc. All rights reserved.
Gurus and their wisdom
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Consensus
Gurus had considerable differences
After 30 years get some consensus
Senior level leadership
Customer focus
Work force involvement
Process analysis
Continuous improvement
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Costs of Quality
External Failure
Costs
Appraisal Costs
Prevention Costs
Internal Failure
Costs
Costs of
Quality
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9-7
Cost of Quality Example
At 20% of sales, represents about $2M sales, at 2.5% about $73M sales
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Six Sigma Quality Measurement
Six Sigma allows managers to readily describe process
performance using a common metric: Defects Per MillionOpportunities (DPMO)
Defect associated with a critical-to-quality characteristic: a
measurable quantity used to identify failure
Statistical six sigma goal is 3.4 failures per 1,000,000 opportunities 1
failure in about 300,000 DPMO (actually 1 in 294,118)
1,000,000x
unitsofNo.x
unitpererrorfor
iesopportunitofNumber
defectsofNumber
DPMO
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9A-9Process Capability
Shows to what extent (probability) parts are produced that meet and fall
outside of specifications
Achieved when process variation (s.d.) is so small that an acceptable
proportion are defectsSix-Sigma goal is 3.4 out of one million
Bearing Diameter
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Why 3.4 DPMO?
Six sigma between mean and, say, upper
specification limit results 1 defect in100,000,000 (see SixSigmaOrigin.xls)
Experience has shown that in long term
processes have a wider variation than in shortterm studies, which results in defects with
probability beyond 4.5 sigma, i.e.3.4 DPMO
(1.5 sigma less than 6 sigma) See origin of this at http://en.wikipedia.org/wiki/Six_Sigma ( which bases
above on Tennant, Geoff (2001). SIX SIGMA: SPC and TQM in
Manufacturing and Services. Gower Publishing, Ltd., p. 25. ISBN
0566083744.)
http://en.wikipedia.org/wiki/Six_Sigmahttp://books.google.com/books?id=O6276jidG3IC&printsec=frontcoverhttp://books.google.com/books?id=O6276jidG3IC&printsec=frontcoverhttp://en.wikipedia.org/wiki/Special:BookSources/0566083744http://en.wikipedia.org/wiki/Special:BookSources/0566083744http://en.wikipedia.org/wiki/Special:BookSources/0566083744http://en.wikipedia.org/wiki/Special:BookSources/0566083744http://books.google.com/books?id=O6276jidG3IC&printsec=frontcoverhttp://books.google.com/books?id=O6276jidG3IC&printsec=frontcoverhttp://en.wikipedia.org/wiki/Six_Sigma8/10/2019 OM-09-QualityCapabilitySPC.ppt
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9A-11
Mean shift during process improvement
Still an improvement but capability is now measured against closest of LTL and UTL
LTL = lower tolerance limit UTL = upper tolerance limit
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Measure Example
Example of Defects Per MillionOpportunities (DPMO) calculation.Suppose we observe 200 letters deliveredincorrectly to the wrong addresses in asmall city during a single day when atotal of 200,000 letters were delivered.
What is the DPMO in this situation?
000,1 1,000,000x
200,000x1
200DPMO
So, for every onemillion letters
delivered thiscitys postalmanagers canexpect to have1,000 lettersincorrectly sent to
the wrongaddress.
Cost of Quality: What might that DPMO mean in terms ofover-time employment to correct the errors?
9-12
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8-13Service Blueprint, Failure Anticipation, and Poka-Yokes
Complete blueprint (p262-3)identifies 16 failure opportunities
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Toyota Dealer Service Example
Blueprint identified 16 failure opportunitiesper customer
Assume 20 customers /day => 80,000
customers/year for 250 working days per year At 3.4 failures per 1,000,000 opportunities this
would allow .272 failures/year, or 3 2/3 years
between failures What is the critical-to-quality characteristic of
the first identified failure? Second failure?
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DMAIC Cycle
GE developedmethodology
Overall focus is tounderstand andachieve what the
customer wants (Juran) Identifies defects and
variation in processesas underlying cause ofdefects (Deming)
A 6-sigma programseeks to reduce thevariation in theprocesses that lead tothese defects
Define customers andtheir priorities
Measure process and itsperformance
Analyze causes ofdefects
Improve by removingcauses
Control to maintainquality
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DMAIC in Action
We are the maker of acereal. Consumer Reportshas just published an articlethat shows that wefrequently have less than 16
ounces of cereal in a box. What should we do?
1. Define
a. What is the critical-to-quality characteristic?
b. The CTQ (critical-to-quality) characteristic inthis case is the weight ofthe cereal in the box.
2. Measurea. How would we measure to
evaluate the extent of theproblem?
b. What are acceptable limitson this measure?
c. Lets assume that thegovernment says that wemust be within 5 percentof the weight advertised onthe box.
d. Upper Tolerance Limit = 16 +.05(16) = 16.8 ouncese. Lower Tolerance Limit = 16
.05(16) = 15.2 ouncesf. Survey: 1000 boxes have
mean weight = 15.875 ozwith s.d. = .529
9 17
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Upper Tolerance
= 16.8Lower Tolerance
= 15.2
Process
Mean = 15.875
Std. Dev. = .529
What percentage of boxes are defective (i.e. less than 15.2 oz)?
Z = (xMean)/Std. Dev. = (15.215.875)/.529 = -1.276
NORMSDIST(Z) = NORMSDIST(-1.276) = .100978
Approximately, 10 percent of the boxes have less than 15.2
Ounces of cereal in themway out of six-sigma specs
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DMAIC in Action
3. Analyze - how can we
improve the
capability of our cereal
box filling process?a. Decrease Variation
b. Center Process
c. Tighten Specifications
4. ImproveHow good is
good enough?
a. Set center spec at goal
(16 oz in this case)b. Set controls so that a
deviation of 6 s.d.
occurs only 3.4 times
out of a million
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DMAIC in Action
5. Statistical ProcessControl
a. Use data from actualprocess
b. Estimate distributions
c. Calculate capabilitydobetter if not adequate(actually do better all thetime)
d. Statistically monitor the
process over time
e. Tools1) Process flow charts (e.g.,
Toyota service blueprint)
2) Run charts
3) Pareto charts
4) Check sheets5) Cause-and-effect
diagrams
6) Opportunity flowdiagrams
7) Failure mode and effect
analysis (FMEA)8) Statistical Process Control(SPC) and Control charts
9) Design of Experiments(DOE)
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9-20
9-21
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9 22
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9 23
Severity: cost of damage, rating number
Occurrence: observed relative frequency,
predicted probability
Detection: probability of detection
RPN = Occurrence X Severity X Detection
Failure Mode and Effect Analysis
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9 24
Part of Statistical Process Control
Uses statistical theory and practice to follow processes in order to determine if they are
within specification/control
Also used to predict if a process might be going out of control while still within specs
General approach is to sample a process at intervals, plot the results and compare
these to control limits
Upper
Control
Limit
Lower
ControlLimit
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Statistical Process Control
Assignable variationis causedby factors that can be clearlyidentified and possiblymanaged
Common variationis inherentin the production process
Example: A poorly trained
employee that creates
variation in finished product
output.
Example: A molding process
that always leaves burrs or
flaws on a molded item.
Based on statistical theory of variation (dispersion)
Defines process capability
Establishes process control limits
Controls process bases on periodic sampling (small samples
as opposed to inspecting/measuring everything)
Variation
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Taguchis View of Variation
Incremental
Cost of
Variability
High
Zero
Lower
Spec
Target
Spec
Upper
Spec
Traditional View
Incremental
Cost of
Variability
High
Zero
Lower
Spec
Target
Spec
Upper
Spec
Taguchis View
Traditional view is that quality within the LS and US is good and that
the cost of quality outside this range is constant, Taguchi views costs as
increasing as variability increases, so seek to achieve zero defects and
that will truly minimize quality costs.
Upper and lower specs are also called upper and lower tolerance limits (UTL and LTL)
9A-27
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Process Capability Index, Cpk
3X-UTLor
3LTLXmin=Cpk
Shifts in Process Mean
Capability Index shows
how well parts being
produced fit into design
limit specifications.
As a production process
produces items small
shifts in equipment or
systems can cause
differences inproduction
performance from
differing samples.
LTL/UTL = lower/upper tolerance limit
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The Cereal Box Example
We are the maker of this cereal. Consumer reports hasjust published an article that shows that we frequentlyhave less than 16 ounces of cereal in a box.
Lets assume that the government says that we must
be within 5 percent of the weight advertised on thebox.
Upper Tolerance Limit = 16 + .05(16) = 16.8 ounces
Lower Tolerance Limit = 16.05(16) = 15.2 ounces
We go out and buy 1,000 boxes of cereal and find thatthey weight an average of 15.875 ounces with astandard deviation of .529 ounces.
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Cereal Box Process Capability
Specification or Tolerance
Limits Upper Spec = 16.8 oz
Lower Spec = 15.2 oz
Observed Weight
Mean = 15.875 oz
Std Dev = .529 oz
3
;3
XUTLLTLXMinCpk
)529(.3
875.158.16;
)529(.3
2.15875.15MinCpk
5829.;4253.MinCpk
4253.pkC
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What does a Cpkof .4253 mean?
An index that shows how well theunits being produced fit within the
specification limits. This is a process that will produce a
relatively high number of defects.
Many companies look for a Cpkof 1.3or better 6-Sigma company wants2.0!
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Types of Statistical Sampling in SPC
Attribute (overall acceptable or not)
Defectives refers to the acceptability of product across arange of characteristics.
Defects refers to the number of defects per unit whichmay be higher than the number of defectives.
p-chart application (p for proportion)
Variable (Continuous)
Usually actual dimensions (length, weight, hardness, )
Usually measured by the mean and the standarddeviation.
X-bar and R chart applications (x-bar for mean and R forrangemuch easier to measure than s.d.)
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Control Charts
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Statistical Process Control Formulas:
Attribute Measurements (p-Chart)
p =T o t al N u m b e r o f D e fe c tiv e s
T o ta l N u m b e r o f O b s e rv a tio n s
n
s
)p-(1p=p
p
p
z-p=LCL
z+p=UCL
s
s
Given:
Compute control limits:
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1. Calculate the sample
proportions, p (these
are what can be plotted
on thep-chart) for each
sample
Sample n Defectives p
1 100 4 0.04
2 100 2 0.02
3 100 5 0.05
4 100 3 0.03
5 100 6 0.06
6 100 4 0.04
7 100 3 0.03
8 100 7 0.07
9 100 1 0.01
10 100 2 0.02
11 100 3 0.03
12 100 2 0.02
13 100 2 0.02
14 100 8 0.0815 100 3 0.03
Example of Constructing ap-chart
2. Calculate theaverage and s.d. of the
sample proportions
0.036=1500
55=p
.0188=
100
.036)-.036(1=
)p-(1p=p
n
s
E l f C t ti Ch t9A-35
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Example of Constructing ap-Chart
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Observation
p
UCL
LCL
Calculate control limits and plot the individual sample proportions, the
average of the proportions, and the control limits 3(.0188).036
UCL = 0.0924
LCL = -0.0204 (or 0)
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Example of x-bar and R charts: Steps: Calculate x-bar Chart and Plot Values
10.601
10.856
=).58(0.2204-10.728RA-x=LCL
=).58(0.2204-10.728RA+x=UCL
2
2
10.550
10.600
10.650
10.700
10.750
10.800
10.850
10.900
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
S a m p l e
Means
UCL
LCL
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Example of x-bar and R charts: Calculate R-chart and Plot Values
0
0.46504
)2204.0)(0(RD=LCL
)2204.0)(11.2(RD=UCL
3
4
0 . 0 0 0
0 . 1 0 0
0 . 2 0 0
0 . 3 0 0
0 . 4 0 0
0 . 5 0 0
0 . 6 0 0
0 . 7 0 0
0 . 8 0 0
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5
S a m p l e
RUCL
LCL
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Common criteria for concluding
process is out of control or in
danger of being so
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Advantages Economy Less handling damage Fewer inspectors Upgrading of the
inspection job Applicability to destructive
testing Entire lot rejection
(motivation forimprovement)
Disadvantages
Risks of accepting badlots and rejecting goodlots
Added planning anddocumentation
Sample provides less
information than 100-percent inspection
Acceptance Sampling
PurposesDetermine quality level of acquired goods or services(after the fact) when no sampling of productionprocess is availableEnsure quality is within predetermined level
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Risk
Acceptable Quality Level (AQL)
Max. acceptable percentage of defectives defined
by producer
The (Producers risk)
The probability of rejecting a good lot
Probability of Type I error based on consumersnull hypothesis that lot is good
Lot Tolerance Percent Defective (LTPD)
Percentage of defectives that defines consumers
rejection point
The (Consumers risk)
The probability of accepting a bad lot
Probability of Type II error based on consumers
null hypothesis that lot is good
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Operating Characteristic Curve
n = 99
c = 4
AQL LTPD
00.1
0.2
0.3
0.4
0.50.6
0.7
0.8
0.9
1
1 2 3 4 5 6 7 8 9 10 11 12
Percent defective
Probabilityofacceptinglots
withgive%o
fdefectives
=.10(consumers risk = accept bad lot)
= .05 (producers risk
= reject good lot)
The OCC brings the concepts of producers risk, consumers
risk, sample size, and maximum defects allowed together
The shape
or slope of
the curve is
dependent
on a
particular
combination
of the four
parameters
n = sample size
c = acceptance number (max
defectives allowed before lotis rejected)
H0: Lot is good
Ha: Lot is bad
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Example: Acceptance Sampling Problem
Zypercom, a manufacturer of video interfaces,
purchases printed wiring boards from an outside
vender, Procard. Procard has set an acceptable
quality level of 1% and accepts a 5% risk of rejecting
lots at or below this level. Zypercom considers lots
with 3% defectives to be unacceptable and will assume
a 10% risk of accepting a defective lot.
Develop a sampling plan for Zypercom and determine
a rule to be followed by the receiving inspection
personnel.
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Example: Step 2. Determine c
First divide LTPD by AQL
LTPDAQL
= .03.01
= 3
Then find the value for c by selecting the value in the QA-12 (on disk)
n(AQL)column that is equal to or just greater than the ratio above.
Exhibit QA-12
c LTPD/AQL n AQL c LTPD/AQL n AQL
0 44.890 0.052 5 3.549 2.6131 10.946 0.355 6 3.206 3.286
2 6.509 0.818 7 2.957 3.981
3 4.890 1.366 8 2.768 4.695
4 4.057 1.970 9 2.618 5.426
So, c = 6.
LTPD = Lot tolerant percent defective (buyers)
AQL = Acceptable quality level (seller)