Six-Sigma Quality, Process Capability, and Statistical Process
ControlSelected Slides from Jacobs et al, 9th Edition
Operations and Supply Management Chapter 9 and 9A
Edited, Annotated and Supplemented byPeter Jurkat
Total Quality Management (TQM)• Total quality
management is defined as managing the entire organization so that it excels on all dimensions of products and services that are important to the customer
• Design quality: Inherent value of the product in the marketplace
– Dimensions include: Performance, Features, Reliability/Durability, Serviceability, Aesthetics, and Perceived Quality.
• Conformance quality: Degree to which the product or service design specifications are met
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
9-3
McGraw-Hill/Irwin
Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.
Gurus and their wisdom
Consensus
• Gurus had considerable differences• After 30 years get some consensus
– Senior level leadership– Customer focus– Work force involvement– Process analysis– Continuous improvement
Costs of Quality
External Failure Costs
Appraisal Costs
Prevention Costs
Internal FailureCosts
Costs ofQuality
9-6
9-7Cost of Quality Example
At 20% of sales, represents about $2M sales, at 2.5% about $73M sales
Six Sigma Quality Measurement
• Six Sigma allows managers to readily describe process performance using a common metric: Defects Per Million Opportunities (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)
1,000,000 x
units of No. x
unit per error for
iesopportunit ofNumber
defects ofNumber
DPMO 1,000,000 x
units of No. x
unit per error for
iesopportunit ofNumber
defects ofNumber
DPMO
9-8
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 defects – Six-Sigma goal is 3.4 out of one million
Bearing Diameter
Why 3.4 DPMO?• Six sigma between mean and, say, upper
specification limit results 1 defect in 100,000,000 (see SixSigmaOrigin.xls)
• Experience has shown that in long term processes have a wider variation than in short term 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.)
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
Measure Example
Example of Defects Per Million Opportunities (DPMO) calculation. Suppose we observe 200 letters delivered incorrectly to the wrong addresses in a small city during a single day when a total of 200,000 letters were delivered. What is the DPMO in this situation?
Example of Defects Per Million Opportunities (DPMO) calculation. Suppose we observe 200 letters delivered incorrectly to the wrong addresses in a small city during a single day when a total of 200,000 letters were delivered. What is the DPMO in this situation?
000,1 1,000,000 x
200,000 x 1
200DPMO
000,1 1,000,000 x
200,000 x 1
200DPMO
So, for every one million letters delivered this city’s postal managers can expect to have 1,000 letters incorrectly sent to the wrong address.
So, for every one million letters delivered this city’s postal managers can expect to have 1,000 letters incorrectly sent to the wrong address.
Cost of Quality: What might that DPMO mean in terms of over-time employment to correct the errors?
Cost of Quality: What might that DPMO mean in terms of over-time employment to correct the errors?
9-12
8-13Service Blueprint, Failure Anticipation, and Poka-Yokes
Complete blueprint (p262-3) identifies 16 failure opportunities
Toyota Dealer Service Example
• Blueprint identified 16 failure opportunities per 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?
DMAIC Cycle• GE developed
methodology• Overall focus is to
understand and achieve what the customer wants (Juran)
• Identifies defects and variation in processes as underlying cause of defects (Deming)
• A 6-sigma program seeks to reduce the variation in the processes that lead to these defects
• Define customers and their priorities
• Measure process and its performance
• Analyze causes of defects
• Improve by removing causes
• Control to maintain quality
DMAIC in Action• We are the maker of a
cereal. Consumer Reports has just published an article that shows that we frequently 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 in this case is the weight of the cereal in the box.
2. Measurea. How would we measure to
evaluate the extent of the problem?
b. What are acceptable limits on this measure?
c. Let’s assume that the government says that we must be within ± 5 percent of the weight advertised on the box.
d. Upper Tolerance Limit = 16 + .05(16) = 16.8 ounces
e. Lower Tolerance Limit = 16 – .05(16) = 15.2 ounces
f. Survey: 1000 boxes have mean weight = 15.875 oz with s.d. = .529
Upper Tolerance = 16.8
Lower Tolerance = 15.2
ProcessMean = 15.875Std. Dev. = .529
What percentage of boxes are defective (i.e. less than 15.2 oz)?
Z = (x – Mean)/Std. Dev. = (15.2 – 15.875)/.529 = -1.276
NORMSDIST(Z) = NORMSDIST(-1.276) = .100978
Approximately, 10 percent of the boxes have less than 15.2 Ounces of cereal in them – way out of six-sigma specs
9-17
DMAIC in Action
3. Analyze - how can we improve the capability of our cereal box filling process?a. Decrease Variationb. Center Processc. Tighten Specifications
4. Improve – How 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
DMAIC in Action5. Statistical Process
Controla. Use data from actual
processb. Estimate distributionsc. Calculate capability – do
better if not adequate (actually do better all the time)
d. Statistically monitor the process over time
e. Tools1) Process flow charts (e.g.,
Toyota service blueprint)2) Run charts3) Pareto charts4) Check sheets5) Cause-and-effect
diagrams6) Opportunity flow
diagrams7) Failure mode and effect
analysis (FMEA)8) Statistical Process Control
(SPC) and Control charts9) Design of Experiments
(DOE)
9-20
9-21
9-22
9-23
Severity: cost of damage, rating numberOccurrence: observed relative frequency, predicted probabilityDetection: probability of detectionRPN = Occurrence X Severity X Detection
Failure Mode and Effect Analysis
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 Control Limit
Statistical Process Control
• Assignable variation is caused by factors that can be clearly identified and possibly managed
• Common variation is inherent in the production process
Example: A poorly trained employee that creates variation in finished product output.
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.
Example: A molding process that always leaves “burrs” or flaws on a molded item.
9A-25
•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
Taguchi’s View of Variation
IncrementalCost of Variability
High
Zero
LowerSpec
TargetSpec
UpperSpec
Traditional View
IncrementalCost of Variability
High
Zero
LowerSpec
TargetSpec
UpperSpec
Taguchi’s 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.
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.
9A-26
Upper and lower specs are also called upper and lower tolerance limits (UTL and LTL)
Process Capability Index, Cpk
3
X-UTLor
3
LTLXmin=C pk
Shifts in Process Mean
Capability Index shows how well parts being produced fit into design limit specifications.
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 in production performance from differing samples.
As a production process produces items small shifts in equipment or systems can cause differences in production performance from differing samples.
9A-27
LTL/UTL = lower/upper tolerance limit
The Cereal Box Example
• We are the maker of this cereal. Consumer reports has just published an article that shows that we frequently have less than 16 ounces of cereal in a box.
• Let’s assume that the government says that we must be within ± 5 percent of the weight advertised on the box.
• 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 that
they weight an average of 15.875 ounces with a standard deviation of .529 ounces.
9A-28
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
XUTLLTLXMinC pk
)529(.3
875.158.16;
)529(.3
2.15875.15MinC pk
5829.;4253.MinC pk
4253.pkC
9A-29
What does a Cpk of .4253 mean?
• An index that shows how well the units 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 Cpk of 1.3 or better… 6-Sigma company wants 2.0!
9A-30
Types of Statistical Sampling in SPC
• Attribute (overall acceptable or not)– Defectives refers to the acceptability of product across a
range of characteristics.– Defects refers to the number of defects per unit which
may 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 standard
deviation.– X-bar and R chart applications (x-bar for mean and R for
range – much easier to measure than s.d.)
9A-31
9A-32
Control Charts
Statistical Process Control Formulas:Attribute Measurements (p-Chart)
p =Total Number of Defectives
Total Number of Observationsp =
Total Number of Defectives
Total Number of Observations
ns
)p-(1 p = p n
s)p-(1 p
= p
p
p
z - p = LCL
z + p = UCL
s
s
p
p
z - p = LCL
z + p = UCL
s
s
Given:
Compute control limits:
9A-33
1. Calculate the sample proportions, p (these are what can be plotted on the p-chart) for each sample
1. Calculate the sample proportions, p (these are what can be plotted on the p-chart) for each sample
Sample n Defectives p1 100 4 0.042 100 2 0.023 100 5 0.054 100 3 0.035 100 6 0.066 100 4 0.047 100 3 0.038 100 7 0.079 100 1 0.01
10 100 2 0.0211 100 3 0.0312 100 2 0.0213 100 2 0.0214 100 8 0.0815 100 3 0.03
Example of Constructing a p-chart
9A-34
2. Calculate the average and s.d. of the sample proportions
2. Calculate the average and s.d. of the sample proportions
0.036=1500
55 = p 0.036=1500
55 = p
.0188= 100
.036)-.036(1=
)p-(1 p = p n
s .0188= 100
.036)-.036(1=
)p-(1 p = p n
s
Example of Constructing a p-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
Calculate control limits and plot the individual sample proportions, the average of the proportions, and the control limits
9A-35
3(.0188) .036 3(.0188) .036 UCL = 0.0924LCL = -0.0204 (or 0)
UCL = 0.0924LCL = -0.0204 (or 0)
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
10.601
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
Sample
Mea
ns
UCL
LCL
9A-36
Example of x-bar and R charts: Calculate R-chart and Plot Values
0
0.46504
)2204.0)(0(R D= LCL
)2204.0)(11.2(R D= UCL
3
4
0
0.46504
)2204.0)(0(R D= LCL
)2204.0)(11.2(R D= UCL
3
4
0.000
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Sample
RUCL
LCL
9A-37
9A-38
Common criteria for concluding process is out of control or in
danger of being so
• Advantages– Economy– Less handling damage– Fewer inspectors– Upgrading of the
inspection job– Applicability to destructive
testing– Entire lot rejection
(motivation for improvement)
• Disadvantages– Risks of accepting “bad”
lots and rejecting “good” lots
– Added planning and documentation
– Sample provides less information than 100-percent inspection
Acceptance Sampling
•Purposes•Determine quality level of acquired goods or services (“after the fact”) when no sampling of production process is available•Ensure quality is within predetermined level
Risk
• Acceptable Quality Level (AQL)– Max. acceptable percentage of defectives defined
by producer
• The(Producer’s risk)– The probability of rejecting a good lot– Probability of Type I error based on consumer’s
null hypothesis that lot is good
• Lot Tolerance Percent Defective (LTPD)– Percentage of defectives that defines consumer’s
rejection point
• The (Consumer’s risk)– The probability of accepting a bad lot– Probability of Type II error based on consumer’s
null hypothesis that lot is good
9A-40
Operating Characteristic Curve
n = 99c = 4
AQL LTPD
00.10.20.30.40.50.60.70.80.9
1
1 2 3 4 5 6 7 8 9 10 11 12
Percent defective
Pro
bab
ilit
y of
acc
epti
ng
lots
w
ith
giv
e %
of
def
ecti
ves
=.10(consumer’s risk = accept bad lot)
= .05 (producer’s risk = reject good lot)
The OCC brings the concepts of producer’s risk, consumer’s risk, sample size, and maximum defects allowed together
The OCC brings the concepts of producer’s risk, consumer’s 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
The shape or slope of the curve is dependent on a particular combination of the four parameters
9A-41
n = sample sizec = acceptance number (max defectives allowed before lot is rejected)
H0: Lot is goodHa: Lot is bad
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.
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.
9A-42
Example: What is given and what is not?
In this problem, AQL is given to be 0.01 and LTDP is given to be 0.03. We are also given an alpha of 0.05 and a beta of 0.10.
In this problem, AQL is given to be 0.01 and LTDP is given to be 0.03. We are also given an alpha of 0.05 and a beta of 0.10.
What you need to determine is your sampling plan is “c” and “n.”
What you need to determine is your sampling plan is “c” and “n.”
9A-43
LTPD = Lot tolerant percent defective (buyers)AQL = Acceptable quality level (seller)
For a give allowed sampling error SE and confidence C = 1 - α the sample size is determined by:
2
22/α
SE
pqzn =
where p = probability of a defective and q = 1 - p
Example: Step 2. Determine “c”
First divide LTPD by AQLFirst divide LTPD by AQL
LTPD
AQL =
.03
.01 = 3
LTPD
AQL =
.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.
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-12Exhibit QA-12
c LTPD/AQL n AQL c LTPD/AQL n AQL0 44.890 0.052 5 3.549 2.6131 10.946 0.355 6 3.206 3.2862 6.509 0.818 7 2.957 3.9813 4.890 1.366 8 2.768 4.6954 4.057 1.970 9 2.618 5.426
So, c = 6.So, c = 6.
9A-44
LTPD = Lot tolerant percent defective (buyers)AQL = Acceptable quality level (seller)
Recommended