Six-Sigma

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Six-Sigma Training Book
Six-Sigma
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Continuous Distributions
Normal Exponential Weibull Lognormal t c2 f

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Sampling Distributions
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The most widely used model for the distribution
of continuous random variable. Arises in the
study of numerous physical phenomena, such as the
velocity of molecules.
Plot is known as Probability Density Function of X
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• Many natural phenomena and man-made processes are
  observed to have normal distributions, or can be
  closely represented as normally distributed.
• For example, the length of a machined part is
  observed to vary about its mean due to
• temperature drift, humidity change, vibrations,
  cutting angle variations, cutting tool wear,
  bearing wear, rotational speed variations,
  fixturing variations, raw material changes and
  contamination level changes
• If these sources of variation are small,
  independent and equally likely to be positive or
  negative, the length will closely approximate a
  normal distribution.

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???? - ???

• First introduced by French mathematician Abraham
  DeMoivre in 1733.
• Made famous in 1809 by German mathematician K.F.
  Gauss when he also developed a normal
  distribution independently and used it in his
  study of astronomy.
• As a result, it is also known as the Gaussian
  distribution.
• During mid to late nineteenth century, many
  statisticians believed that it was normal for
  most well-behaved data to follow this curve.

Karl Friedrich Gauss
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• ????????, ????, ???????????????????????????????.
• ??, ????????????????????? (???) ??????????????.
• ???????????????.

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• A normal distribution can be completely described
  by knowing only the
• Mean (m)
• Variance (s2)

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X N(m, s2)
What is the difference between the 3 normal
distributions?
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?????????
What is the difference between process A B
for each case?
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• The mean, median and mode all coincide at the
  same value - m. There is perfect symmetry.

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• If a set of observations is arranged in an
  increasing order of magnitude (ranked data), the
  middle value is called the median.
• If the number of observations is odd, the median
  is the value of the middle number.
• If the number of observations is even, there are
  2 middle numbers, and the median is the average
  of the 2 values.

The mean represents the arithmetic average of all
observations in a data set.
The mode is the observation that occurs most
frequently in the sample.
Mean Median Mode
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?????????

• The area under sections of the curve can be used
  to estimate the cumulative probability of a
  certain event occurring

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µ
Point of Inflection
m /- 3s is often referred to as the width of a
normal distribution
1s

-
68.27
95.45
99.73
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?????????

• Lets compute the cumulative probabilities of the
  following distributions

m 16.6 s 2.8
m 3.5 s 0.6
20.0
1.8
m -1.5 s 0.9
-2.8
0.5
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?????????

• MiniTab
• Calc ð Probability Distributions ð Normal...

Enter m value
Enter s value
Enter x value
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?
?
??? 6s
?
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6s ??
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The Focus of Six Sigma

• Identifying critical aspects of the business with
  problems or opportunities for improvement.
• Targeting those critical areas and designating
  improvement efforts as Six Sigma Black Belt
  projects.
• Selecting top people to work on the
  projects--full time.
• Ensuring these people have the time, tools, and
  resources they need to succeed.

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Customer Focus A Model For Success
What purpose is Six-sigma ?
People
Processes
Processes
People
Technology
Organization
Technology
Organization
Capability
Capability

• ??????????????????????????.
• ??????????, ??, ??????.
• ??,??, ?????????????.

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Six Sigma Vision
What purpose is Six-sigma ?

• The Vision of Six Sigma is to delight customers
  by delivering world-class quality products
  through the achievement of Six Sigma levels of
  performance in everything we do.

Six Sigma Philosophy
The philosophy of Six Sigma is to apply a
structured, systematic approach to achieve
breakthrough improvement across all areas of our
business.
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Six Sigma - Aggressive Goal
What purpose is Six-sigma ?
?
PPM
Process Capability
Defects per Million Opp.
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Statistical Definition of n-Sigma
s
- n
s
n
This is the so-called n-sigma
m
o
scale
Process Width
Design Width
Sigma is a statistical unit of measure that
reflects process capability. The sigma scale of
measure is perfectly correlated to such
characteristics as defects-per-unit, parts-per
million defective, and the probability of a
failure/error.
scale
LSL
USL
T
LSL
USL
T
scale
LSL
USL
T
LSL
USL
T
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Statistical Definition of 6s
s
6
st
This is the six- sigma we said
m
o
s
3
st
scale
Process Width
Design Width
scale
LSL
USL
T
LSL
USL
T
.001
ppm
.001
ppm
lt LSL
gt USL
scale
LSL
USL
T
LSL
USL
T
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6s - Performance Target
Sigma
Long-Term Yield
Standard
3 Sigma
93.32
Historical
4 Sigma
99.379
Current
5 Sigma
99.9767
Intermediate
6 Sigma
99.99966
Long-Run
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The Strategy
USL
LSL

• Characterize
• Optimize
• Breakthrough

T
USL
LSL
T
USL
LSL
T
USL
LSL
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The Breakthrough Phases
Phase 1
Measurement
Characterization
Phase 2
Analysis
Phase 3
Improvement
Optimization
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The Breakthrough Phases Analysis tool and method
Dot plot , Box plot,
Histogram chart, Pareto chart
Capability study ( Cpk)
Phase 2
Analysis
GR R study
Causeeffect analysis Fishbone and CE matrix
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The Breakthrough Phases Improvement tool and
method
Set up process Map
Improvement
Set up FMEA
and control
Analysis rolled throughput yield
.
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? ? ? ? ? ?
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? ? ? ?

• ?????????
• ????
• ??, ????????
• ?????????
• ?????????
• 6s ??

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• 1. ????
• ?????????????? , ???,?????????????.
• ?? ?????,??????????????? ????????????.
• ????????????????????????????,???,
  ??????????????????? .
• ????????? ?????????????? .

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• 2. ????
• ?? ?????????
• ?? ????? ? ????? , ????????
• ???????????(???)?????????????? .

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• ?????????
• ???????????????????
• ??????????????????????

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• ????????
• ???? a) ??????
• b) ?????
• c) ???????
• ?????????????????? .

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• ???? ???
• ??????????
• ???????????
• ??????????????

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a) ????? b)???????? c)???????
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??????

• ???? (LSL and USL)
• created by design engineering in response to
  customer requirements to specify the tolerance
  for a products characteristic
• ???? (LPL and UPL)
• measures the variation of a process
• the natural 6? limits of the measured
  characteristic
• ???? (LCL and UCL)
• measures the variation of a sample statistic
  (mean, variance, proportion, etc)

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??????

• ?????????
• ????
• Cp
• ????
• Cpu
• Cpl
• Cpk

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• Cp ???????? (6?)??????.

??
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• ???, Cp ? 1.0 ?????????????????.
• ???????????, ???0.27 ??????.
• Cp ???
• 1.00 0.270
• 1.33 0.007
• 1.50 6.8 ppm
• 2.00 2.0 ppb

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a) ????? (Cpgt2) b) ????? (Cp1 to 2) c) ?????
(Cplt1)
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• The Cp index compares the allowable spread
  (USL-LSL) against the process spread (6?). It
  fails to take into account if the process is
  centered between the specification limits.

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• The Cpk index relates the scaled distance between
  the process mean and the nearest specification
  limit.

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• Cpk ???
• 1.0 0.13 0.27
• 1.1 0.05 0.10
• 1.2 0.02 0.03
• 1.3 48.1 96.2 ppm
• 1.4 13.4 26.7 ppm
• 1.5 3.4 6.8 ppm
• 1.6 794 1589 ppb
• 1.7 170 340 ppb
• 1.8 33 67 ppb
• 1.9 6 12 ppb
• 2.0 1 2 ppb

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????
a)?????(Cpkgt1.5) b)?????(Cpk1 to
1.5) c)?????(Cpklt1)
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• (a) Poor Process Potential (b) Poor Process
  Performance

• Experimental Design
• to reduce variation

• Experimental Design
• to center mean
• to reduce variation

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• A process is stable if the distribution of
  measurements made on the given feature is
  consistent over time.

Stable Process
Time
Unstable Process
Time
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• ??????(previously called short-term capability)
  shows the inherent variability of a
  machine/process operating within a brief period
  of time.
• ??????(previously called long-term capability)
  shows the variability of a machine/process
  operating over a period of time. It includes
  sources of variation in addition to the
  short-term variability.

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• Within Overall
• Sample Size 30 50 units ? 100 units
• Number of Lots single lot several lots
• Period of Time hours or days weeks or months
• Number of Operators single operator different
  operators
• Process Potential Cp Pp
• Process Performance Cpk Ppk

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• The length of a camshaft for an automobile engine
  is specified at 600 2 mm. Control of the length
  of the camshaft is critical to avoid
  scrap/rework.
• The camshaft is provided by external suppliers.
  Assess the process capability for this supplier.
• The data is available in Camshaft.MTW. Data are
  collected in subgroups of 5 each.

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Example 4

• Minitab
• Stat ? Quality Tools ? Capability Analysis
  (Normal)

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Example 4
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Example 5

• Histogram of the camshaft length suggests mixed
  populations. Further investigation revealed that
  there are two suppliers for the camshaft. Data
  were now collected on camshafts from each source
  without combining both. Subgroup size is 5 for
  each supplier.
• Are the two suppliers similar in performance?
• If not, what are your recommendations?

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Example 5

• MiniTab
• Stat ? Quality Tools ? Capability Sixpack(Normal)

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Example 5
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Example 5
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?Box-Cox ??????????

• When the process data are not normal, the Cpk or
  Ppk indices are not accurate or reliable, because
  these indices are computed on the basis that the
  data are normally distributed.
• Dppm values associated with the indices will not
  be near to the actual performance when the normal
  curve does not model the actual data well.

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?Box-Cox ??????????

• If the process data are somewhat bell-shaped but
  skewed, Box-Cox transformation can be used to
  make the data normal before we assess the process
  capability.
• Remember to transform the specification limits
  too before we compute Cpk or Ppk!

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?Box-Cox ??????????

• Minitab
• Stat ? Quality Tools ? Capability Analysis
  (Normal)

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?Box-Cox ??????????
Example 6
Open the file named Dimension.MTW in the Day-2
folder again. Compute the process capability
with the specification limits LSL 0.1 USL
10 Are the data normally distributed? Compute
the process capability again with Box-Cox
transformation.
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?Box-Cox ??????????
Example 6
Cpk of 0.41 is reported in the SSAT package. This
value is not reliable or accurate if the data are
not normal.
Data is not normal
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?Box-Cox ??????????
Example 6
Cpk has increased from 0.41 to 0.81
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Whats 6? Quality Then
Sigma Level Capability

• Original Definition by Motorola
• In the short term, the specification limits are
  at least 6? away from the process mean ?, i.e.
  Cp ? 2,
• In the long run, the process will shift by less
  than 1.5?, i.e. Ppk ? 1.5,
• The process will yield less than 3.4 dppm
  rejected parts.

Shift 1.5?
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Whats Six Sigma Quality Now
Sigma Level Capability

• Mikel J Harry claims that the process mean
  between lots will vary, with an average process
  shift of 1.5?.

Note Sigma Capability ƒ(dpmo) ? ƒ(dppm)
Shift 1.5?
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Types of Variation

• 1. Positional Variation
• Same process, variation at differing locations
  simultaneously
• Temperature variations inside a thermal chamber
• Cavity-to-cavity variations in an injection mold
• 2. Cyclical Variation
• Sequential repetitions of a process over fairly
  short time, say, less than 15 mins
• Variations between consecutive batches of a
  process
• Differences from lot to lot of raw materials

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Types of Variation

• 3. Temporal Variation
• Variations over longer periods of time, such a
  several hours, days or weeks.

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Measurement System Analysis Approach

• There are two types of measurements possible
• Variable
• Data can be described on a continuous scale
• Attribute
• Data cannot be adequately described on a
  continuous scale
• Pass / Fail, very low counts
• Each must be approached differently.