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Title: Six-Sigma


1
Six-Sigma Training Book
Six-Sigma
2
????


3
????
Continuous Distributions
Normal Exponential Weibull Lognormal t c2 f


Sampling Distributions
4
????
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
5
????
  • 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.

6
???? - ???
  • 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
7
????
  • ????????, ????, ???????????????????????????????.
  • ??, ????????????????????? (???) ??????????????.
  • ???????????????.

8
?????????
  • A normal distribution can be completely described
    by knowing only the
  • Mean (m)
  • Variance (s2)

1
X N(m, s2)
What is the difference between the 3 normal
distributions?
9
?????????
What is the difference between process A B
for each case?
10
?????????
  • The mean, median and mode all coincide at the
    same value - m. There is perfect symmetry.

2
  • 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
11
?????????
  • The area under sections of the curve can be used
    to estimate the cumulative probability of a
    certain event occurring

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

-
68.27
95.45
99.73
12
?????????
  • 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
13
?????????
  • MiniTab
  • Calc ð Probability Distributions ð Normal...

Enter m value
Enter s value
Enter x value
14
?
?
??? 6s
?
15
6s ??
16
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.

17
Customer Focus A Model For Success
What purpose is Six-sigma ?
People
Processes
Processes
People
Technology
Organization
Technology
Organization
Capability
Capability
  • ??????????????????????????.
  • ??????????, ??, ??????.
  • ??,??, ?????????????.

18
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.
19
Six Sigma - Aggressive Goal
What purpose is Six-sigma ?
?
PPM
Process Capability
Defects per Million Opp.
20
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
21
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
22
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
23
The Strategy
USL
LSL
  • Characterize
  • Optimize
  • Breakthrough

T
USL
LSL
T
USL
LSL
T
USL
LSL
24
The Breakthrough Phases
Phase 1
Measurement
Characterization
Phase 2
Analysis
Phase 3
Improvement
Optimization
25
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
26
The Breakthrough Phases Improvement tool and
method
Set up process Map
Improvement
Set up FMEA
and control
Analysis rolled throughput yield
.
27
? ? ? ? ? ?
28
? ? ? ?
  • ?????????
  • ????
  • ??, ????????
  • ?????????
  • ?????????
  • 6s ??

29
?????????
  • 1. ????
  • ?????????????? , ???,?????????????.
  • ?? ?????,??????????????? ????????????.
  • ????????????????????????????,???,
    ??????????????????? .
  • ????????? ?????????????? .

30
?????????
  • 2. ????
  • ?? ?????????
  • ?? ????? ? ????? , ????????
  • ???????????(???)?????????????? .

31
?????
  • ?????????
  • ???????????????????
  • ??????????????????????

32
?????
  • ????????
  • ???? a) ??????
  • b) ?????
  • c) ???????
  • ?????????????????? .

33
????
  • ???? ???
  • ??????????
  • ???????????
  • ??????????????

34
?????????
a) ????? b)???????? c)???????
35
??????
  • ???? (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)

36
??????
  • ?????????
  • ????
  • Cp
  • ????
  • Cpu
  • Cpl
  • Cpk

37
????
  • Cp ???????? (6?)??????.

??
38
????
  • ???, Cp ? 1.0 ?????????????????.
  • ???????????, ???0.27 ??????.
  • Cp ???
  • 1.00 0.270
  • 1.33 0.007
  • 1.50 6.8 ppm
  • 2.00 2.0 ppb

39
????
a) ????? (Cpgt2) b) ????? (Cp1 to 2) c) ?????
(Cplt1)
40
????
  • 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.

41
????
  • The Cpk index relates the scaled distance between
    the process mean and the nearest specification
    limit.

42
????
  • 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

43
????
a)?????(Cpkgt1.5) b)?????(Cpk1 to
1.5) c)?????(Cpklt1)
44
?????????
  • (a) Poor Process Potential (b) Poor Process
    Performance
  • Experimental Design
  • to reduce variation
  • Experimental Design
  • to center mean
  • to reduce variation

45
?????????
46
?????
  • A process is stable if the distribution of
    measurements made on the given feature is
    consistent over time.

Stable Process
Time
Unstable Process
Time
47
?????????????
  • ??????(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.

48
?????????????
  • 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

49
????
  • 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.

50
Example 4
  • Minitab
  • Stat ? Quality Tools ? Capability Analysis
    (Normal)

51
Example 4
52
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?

53
Example 5
  • MiniTab
  • Stat ? Quality Tools ? Capability Sixpack(Normal)

54
Example 5
55
Example 5
56
?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.

57
?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!

58
?Box-Cox ??????????
  • Minitab
  • Stat ? Quality Tools ? Capability Analysis
    (Normal)

59
?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.
60
?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
61
?Box-Cox ??????????
Example 6
Cpk has increased from 0.41 to 0.81
62
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?
63
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?
64
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

65
Types of Variation
  • 3. Temporal Variation
  • Variations over longer periods of time, such a
    several hours, days or weeks.

66
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.
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