part 4: statistical process control control charts
TRANSCRIPT
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Part 4: Statistical Process Control
Control Charts
Henriqueta Nóvoa | José Sarsfield Cabral
Faculdade de Engenharia da Universidade do Porto
Mestrado em Engenharia Mecânica
Mestrado em Engenharia Industrial e Gestão
Mestrado em Engenharia Electrotécnica e de Computadores
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Deming
4
“Why are we here?”
“We are here to make
another world.”
Another quote :
“If I had to reduce my message for management
to just a few words, I’d say it all had to do
with reducing variation” (Neave 1990).
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Timeline
1938 W. E. Deming invites Shewhart to present seminars on control charts at
the U.S. Department of Agriculture Graduate School.
1946 The American Society for Quality Control (ASQC) is formed as the
merger of various quality societies.
The International Standards Organization (ISO) is founded.
Deming is invited to Japan by the Economic and Scientific Services Section of the
U.S. War Department to help occupation forces in rebuilding Japanese industry.
The Japanese Union of Scientists and Engineers (JUSE) is formed.
1946–1949 Deming is invited to give statistical quality control seminars to Japanese
industry.
1951 A. V. Feigenbaum publishes the first edition of his book, Total Quality Control.
JUSE establishes the Deming Prize for significant achievement in quality
control and quality methodology.
W. Edwards Deming 1900- 1993
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Major contributions:“have one aim: to make it possible for people to work with joy”
1. Create a constancy of purpose focused on the improvement of products
and services. Constantly try to improve product design and
performance. Investment in research, development, and innovation will have long-term payback to the organization.
2. Adopt a new philosophy of rejecting poor workmanship, defective products,
or bad service.
3. Do not rely on mass inspection to “control” quality.
4. Do not award business to suppliers on the basis of price alone, but also
consider quality.
W. Edwards Deming 1900- 1993
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Major contributions:
5. Focus on continuous improvement. Constantly try to improve the
production and service system. Involve the workforce in these activities and
make use of statistical methods.
6. Practice modern training methods and invest in training for all
employees.
7. Improve leadership, and practice modern supervision methods.
8. Drive out fear. Many workers are afraid to ask questions, report problems,
or point out conditions that are barriers to quality and effective production.
9. Break down the barriers between functional areas of the business.
10. Eliminate targets, slogans, and numerical goals for the workforce.
W. Edwards Deming 1900- 1993
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Major contributions:
11. Eliminate numerical quotas and work standards. These standards have
historically been set without regard to quality.
12. Remove the barriers that discourage employees from doing their jobs.
Management must listen to employee suggestions, comments, and
complaints.
13. Institute an ongoing program of training and education for all
employees. Education in simple, powerful statistical techniques should be
mandatory for all employees.
14. Create a structure in top management that will vigorously advocate the first
13 points.
W. Edwards Deming 1900- 1993
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Statistical Process Control
Statistical process control is a collection of
tools that when used together can result in
process stability and variability reduction
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Statistical Process Control
The seven major tools are
1) Histogram or Stem and Leaf plot
2) Check Sheet
3) Pareto Chart
4) Cause and Effect Diagram
5) Defect Concentration Diagram
6) Scatter Diagram
7) Control Chart
Douglas C. Mongomery “Introduction to Statistical Quality Control”
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The rest of the “Magnificent Seven”
The control chart is most effective when
integrated into a comprehensive SPC program.
The seven major SPC problem-solving tools should
be used routinely to identify improvement
opportunities.
The seven major SPC problem-solving tools should
be used to assist in reducing variability and
eliminating waste.
Douglas C. Montgomery “Introduction to Statistical Quality Control”, Wiley
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The first control chart
In May 1924, Shewhart (1891-1967) proposes a method
for evaluating the stability overtime of the proportion
of defective appliances (p).
Upper Control Limit
Lower Control Limit
P
“this point indicates trouble”
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The first control chart
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…That all changed on May 16, 1924. Dr.
Shewhart's boss, George D. Edwards, recalled:
"Dr. Shewhart prepared a little memorandum only
about a page in length. About a third of that
page was given over to a simple diagram which
we would all recognize today as a
schematic control chart.”
That diagram, and the short text which preceded
and followed it, set forth all of the essential
principles and considerations which are involved
in what we know today as process quality
control.
Shewhart's work pointed out the importance of
reducing variation in a manufacturing process
and the understanding that continual process-
adjustment in reaction to non-conformance
actually increased variation and degraded
quality. Font: Wikipedia
Western Electric and Bell
Telephone Laboratories
1918-24
1891-1967
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Tampering
https://www.youtube.com/watch?v=2VogtYRc9dA
Lessons from the Funnel Experiment
• Despite good intentions to fix a system and move it closer to the target of
the process, manipulations only made outcomes worse overtime;
• All systems have a certain level of inherent variability;
• Attempting to adjust a stable process will make things worse.
Font: Marco Santos
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?
”A phenomenon will be said to be controlled when, through the use of past experience, we can
predict, at least within limits, how the phenomenon may be expected to vary in the future”
(W.A.Shewhart, 1931)
Management is prediction!
Font: Marco Santos
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Chance causes vs. assignable causes of variation
The value of any quality charateristic varies from part to
part, from day to day…
The dispersion of the results is due to a variety of reasons acting randomly, that could be either known or unknown, whose
effect on the individual variable is small
Common causes are inherent to the processes (typically, their
elimination is expensive or difficult).
Together, their action determines that the quality characteristic
will behave in accordance with a stable probability distribution.
chance causes (or common)
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Chance causes vs. assignable causes of variation
Definition of a Normal random variable:
“ A continuous random variable X follows (approximately) a Normal
distribution with parameters m and s if the sum of a large number of
effects caused by independent causes, in which the effect of each cause
is very small when compared to the sum of the effects of all the other
causes.”
Chance (or common) causes of variation
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The variation may also reflect a situation of instability
This will happen if, in addition to common causes of variation,
other type of causes occur, possibly avoidable, which effects
acting together can cause a change in the parameters/shape of the
underlying distribution.
assignable (special) causes of variation
Chance causes vs. assignable causes of variation
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Chance causes vs. assignable causes of variation
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Shewhart Control Charts
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Chance causes vs. assignable causes of variation
Appropriately reacting to the source of variation in a process provides the correct
economic balance between overreacting and under-reacting to variation from a process.
chance causes assignable causes
chance
causes
assignable
causes
Type I error
Change
(increases variability)
Focus on how to
fundamentally
change the process
Type II error
Sub-react
(lost prevention)
Focus in the research
of assignable causes
The
variability
is really
caused by…
How do you deal with variability...
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Control Charts Objectives
A process that is operating with only common (chance)
causes of variation present is said to be in statistical
control.
A process that is operating in the presence of
assignable causes of variation is said to be out of
control.
O objectivo fundamental das Cartas de Controlo é o
de identificar Causas Assinaláveis de Variação,
balizando os momentos em que tais fenómenos
ocorrem.
The fundamental objective of Control Charts is to
identify assignable causes of variation, marking
the moments when they occur.
Douglas C. Mongomery “Introduction to Statistical Quality Control”
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Control Charts Objectives
The control chart is the tool used to
distinguish between common and
assignable causes of variation;
The control limits represent the
expected variation due to common
causes.
Control Limits are often called “the voice of the process”
and used to identify assignable causes of variation
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Common vs. Assignable causes of variation
Eliminate ACV Reduce CCV
A process with
assignable causes
A stable and
more capable
processA stable process
Out of control
Unpredictable
In statistical control
Predictable - stable
TUM, Slides on Quality Management
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A brief introduction to process capability
Assume that a certain
quality characteristic follows
a normal distribution with a
mean m and a standard
deviation, s.The natural tolerance limits
of the process are:
UNTL = m + 3s
LNTL = m - 3s
O
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A brief introduction to process capability
Whatever the distribution of the quality charateristic is, it can be
proved that
For k = 3, it turns out that the probability of having an
observation within the interval average ± 3s is at least 88.9%.
O
1k with
2
11
kkXkP msms
What if the distribution of the quality characteristic is non-normal?
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A brief introduction to process capability
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Specification Limits:
Often called “voice of the customer” and used to determine if the product
meets a customer requirement.
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A brief introduction to process capability
A simple and quantitative way to express process capability is
through the Process Capability Ratio Cp:
USL: Upper Specification Limit
LSL: Lower Specification Limit
s: Standard deviation (s)
Meaning, the ability to produce products or provide services that
meet specifications defined by the customers' needs.
If Cp > 1, then a low number of nonconforming items will be produced;
If Cp = 1, assuming a normal distribution of QC, then we are producing
0,27% nonconforming;
If Cp < 1, then a large number of nonconforming items will be produced.
In practice, we look for a Cp > 1.3
s6
LSL-USL
pC
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A brief introduction to process capability
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Shewhart Control Charts
A machine produces cylindrical shafts at a rate of 4000 parts/hour.
Over a period of eight hours of uninterrupted work, a piece is
removed every four minutes (equivalent to 15 shafts per hour)
constituting a sample of 120 shafts.
The nominal (target) shaft diameter is 10.00 mm, and shafts whose
diameters do not deviate from the target value by more than
0.02mm (s = 0.01) are accepted by the client.
USL= 10.02
LSL = 9.98
N = 120
S = 0.01
Chance causes vs. assignable causes of variation
Exemplo 1:
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Shewhart Control Charts
66.001.06
04.0
ˆ6
s
LSLUSLC
p
9.98
9.99
10.00
10.01
10.02 Limite Sup. Tolerância
1ª hora 2ª hora 3ª hora 4ª hora 5ª hora 6ª hora 7ª hora 8ª hora
(ii)
Limite Inf. Tolerância
10.00x
01.0s
10.029.98
0%
5%
10%
15%
20%
25%
30%
Fre
quên
cia
rela
tiva
LIT LST
66.0
01.06
04.0
6
s
Tpc
(i)
Chance causes vs. assignable causes of variation
s =0.01
USLLSL
Upper specification limit
Lower specification limit
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Shewhart Control Charts
Conclusions:
The process is stable and (apparently) is operating with only
chance (or common) causes of variation;
The shaft variability is caused by a high number of potential
causes, such as: fluctuations in the temperature of the plant,
alterations in raw material, variations in the power supply of the
machine, vibrations in the support of the machine, etc.
Chance causes vs. assignable causes of variation
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Shewhart Control Charts
Example 2:
assignable cause
10.00x
01.0s
10.029.98
0%
5%
10%
15%
20%
25%
30%
Fre
quên
cia
rela
tiva
LIT LST
66.0
01.06
04.0
6
s
Tpc
(i)
9.98
9.99
10.00
10.01
10.02 Limite Superior da Tolerância
1ª hora 2ª hora 3ª hora 4ª hora 5ª hora 6ª hora 7ª hora 8ª hora
(ii)
Limite Inferior da Tolerância
66.0ˆ6
s
LSLUSLC
p
Chance causes vs. assignable causes of variation
USLLSL
Upper specification limit
Lower specification limit
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Shewhart Control Charts
Conclusions:
The process is not stabilised, suggesting the influence of
assignable causes of variation;
An analysis a posteriori identified the source of the problem: the
operator “adjusted”/fine-tuned the machine hourly, believing
that this procedure was effective to control the quality of the
final product.
Chance causes vs. assignable causes of variation
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Shewhart Control Charts
Chance causes vs. assignable causes of variation
s = 0,01
10.029.98
x = 10,00
0.67
0.016
0.04
σ6 ˆ
USL-LSLC
p
LIT LST
9.98
9.99
10.00
10.01
10.02
1ªhora 2ªhora 3ªhora 4ªhora 5ªhora 6ªhora 7ªhora 8ªhora
LST
LIT
“In control”
9.98
9.99
10.00
10.01
10.02
1ª hora 2ª hora 3ª hora 4ª hora 5ª hora 6ª hora 7ª hora 8ª hora
LST
LIT
“Out-of-control”
USL
LSL
USL
LSL
USLLSL
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Shewhart Control Charts
Variação associada a
Causa Especiais
Variação associada a
Causa Comuns
Fora do normal Normal
Perturbada Natural
Instável Estável
Não homogénea Homogénea
Mista Uma única distribuição
Errática Sem mudança
Aos “saltos” Constante
Imprevisível Previsível
Inconsistente Consistente
Pouco comum Habitual
Diferente Semelhante
Importante Pouco importante
Significativa Não significativa
Chance causes vs. assignable causes of variation
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Shewhart Control Charts
In a process in statistical control with unknown parameters of the
distribution of the quality charateristic, the probability of having values
within this interval
deviation) standard valueexpected ()( sm k
is constant
Control Charts Basic Principles
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Shewhart Control Charts
In a normal distribution:
Control Charts Basic Principles
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Shewhart Control Charts
Control Charts Basic Principles
sample of size n
H0: m = m0
H1: m ≠ m0
xm = m0
x N(m0,sx)
xm = m0
ET = x N(m0,sx)
Don´t Reject H0 Reject H0Reject H0
a/2a/2
m0 + ksxm0 - ksx
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Shewhart Control Charts
Control Charts Basic Principles
x3
x4
x1
x2
m0 + ksx
m0 - ksx
m0
H0 Rejection region
H0 Rejection region
1ª sample 2ª sample 3ª sample 4ª sample
H0: m = m0
H1: m ≠ m0
xm = m0
Don´t Reject H0 Reject H0Reject H0
a/2a/2
Shewhart Control Charts
Control Limits
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Shewhart Control Charts
Control limits
Usual significance levels (typically a = 5%) would lead, in
most industrial environments, to an excessive number of
false alarms.
Therefore, Shewhart has proposed k = 3, which leads
(assuming normal distributions) to an a = 0,27%.
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Shewhart Control Charts
Generic assumptions
Lower control limit
Upper control limit
+ 3s
- 3s
m0 → Central line
1 2 3 4 5 6 7 8 9 10 11 12 13
h
l
Usual scale lh 6LCL-UCL
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Shewhart Control Charts
Statistical Basis of the control chart
Relationship between hypothesis testing and control charts:
Example:
o We have a process that we assume the true process mean
is m = 1.5 and the process standard deviation is s = 0.15.
o Samples of size 5 are taken giving a standard deviation of the
sample average, , as x
0671.05
15.0
nx
ss
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Shewhart Control Charts
Statistical Basis of the control chart
Example:
Control limits are set at 3 standard deviations from the
mean;
The 3-Sigma Control Limits are:
UCL = 1.5 + 3(0.0671) = 1.7013
CL = 1.5
LCL = 1.5 - 3(0.0671) = 1.2987
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Shewhart Control Charts
Statistical Basis of the control chart
Choosing the control limits is equivalent to setting up the
critical region for testing hypothesis:
H0: m = 1.5
H1: m 1.5
Douglas C. Mongomery “Introduction to Statistical Quality Control”
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Shewhart Control Charts
Patterns for identifying assignable causes
If the control charts consist of a time series of
sample statistics, will the analysis of sequences of
observations enhance their diagnostic ability?
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Shewhart Control Charts
(ii) Abnormal sequence of points
Patterns for identifying assignable causes
Tempo
A sequence of 8 points
bellow the central line
One point beyond control limits
Tempo
Linha +2s
Linha -2s
Linha -1s
Linha +1s
Four out of five points
bellow the line -1sigma
Two out of three points
above the line +2sigma
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Shewhart Control Charts
(iii) - a) Trends – seven or more points going up or down
Tempo
Linha +2sigma
Linha -2sigma
(iii) - b) Big oscillations – between three consecutive points, one is
between the UCL and the line +2sigma and the other is between
the LCL and the line -2sigma
Tempo
Tendência ascendente
não-linear
Tendência descendente (sete
pontos sucessivos a descer)
Patterns for identifying assignable causes
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Shewhart Control Charts
(iii) - c) Proximity to the center line – almost all points within the limits 1.5sigma
(indicating a probable mixture of populations with different expected values);
(iii) - b) Cyclic patterns: seasonality
Tempo
Linha +1.5sigma
Linha -1.5sigma
Tempo
Periodicidade
Patterns for identifying assignable causes
Line
Line
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Shewhart Control Charts
Patterns for identifying assignable causes
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Shewhart Control Charts
Patterns for identifying assignable causes
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Shewhart Control Charts
Patterns for identifying assignable causes
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Shewhart Control Charts
Patterns for identifying assignable causes
Although these pattern tests allow the Shewhart control charts
to be more sensitive, if some tests are used together, then
the probability of a type I error (a may become too large,
thus accentuating the risk of false alarm.
These effect will be even bigger when more rules
are considered simultaneously (it is common to find
recommendations of 6 or 8!)
E
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Shewhart Control Charts
Minitab: Patterns for identifying assignable causes
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Shewhart Control Charts
(i) process out-of-control and producing defective units;
(ii) process out-of-control and not producing defects units;
(iii) Process in control and producing defective units;
(iv) process in control and not producing defects units.
Limite Superior de Controlo
Limite Inferior de Controlo
Limite Superior da Tolerância
Limite Inferior da Tolerância
1x
Defeituosa
Amostra 1
Amostra 2
2x
Specification Limits are set up taking into account customers’s requests and the
characteristics of the productive process.
Control Limits are calculated taking into account the variability of the process.
Difference between specification limits and control limits
Upper Specification Limit
Lower Specification Limit
Upper Control Limit
Lower Control Limit
sample 1
sample 2
defect
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Relationship between Natural Tolerance Limits, Control
Limits ans Specification Limits
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Shewhart Control Charts
The Warning Limits are specified at two-sigma;
If one or more points fall between the warning
limits and the control limits, or very close to
the warning limit, we should be suspicious that
the process may not be operating properly;
Advantage: the use of warning limits can
increase the sensitivity of the control chart;
Disadvantage: the use of warning limits can
also result in an increased risk of false alarms.
Warning Limits
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Shewhart Control Charts
Guidelines:
In designing a control chart, both the sample size to be
selected and the frequency of selection must be specified;
Larger samples make it easier to detect small shifts in the
process;
Current practice tends to favor smaller, more frequent
samples.
Sample Size and Sampling Frequency
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Shewhart Control Charts
Guidelines:
The collection of initial data must be preceded by a careful
study of the process in order to identify potential assignable
causes;
Pilot samples (K samples of size N) for the initial set-up of the
control limits must be collected during time periods when it is
assured that assignable potential causes are absent.
Guidelines for the design of the control chart
It is essential to update the control limits regularly.
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Shewhart Control Charts
Example:Frozen orange juice concentrate is packed in 6-oz cardboard cans. These cans are
formed on a machine by spinning them from cardboard stock and attaching a metal
bottom panel. By inspection of a can, we may determine whether, when filled, it
could possibly leak either on the side seam or around the bottom joint. Such a
nonconforming can has an improper seal on either the side seam or the bottom
panel.
Set up a control chart to improve the fraction of nonconforming cans produced by
this machine.
Guidelines for the design of the control chart
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Shewhart Control Charts
The power of the test is reduced, thus reducing the sensitivity of
the control chart.
Sensitivity of the control chart
By doing k = 3, it decreases a but increases up b !
What are the practical consequences of such a rule?
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Shewhart Control Charts
Sensitivity of the control chart
By doing k = 3, it
decreases a but
increases up b !
b 1b
a
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84
Shewhart Control Charts
How to evaluate the ability of Shewhart control charts
to detect out-of-control situations?
The capacity of a control chart for detecting an out-of-control
situation can be measured by the power of the test (1-b).
Sensitivity of the control chart
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Shewhart Control Charts
Sensitivity of the control chart – Operating Characteristic Curves (OCC)
H0: m= 1200
H1: m≠ 1200
s =300
N=100
a= 0,05
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86
Shewhart Control Charts
Example: Control Charts for Averages
.2
34
333 0000
sm
sm
smsm
NX
Assuming X Normal (m0, s), and N = 4
LC = 0mm X
s 5.0
H0: m = m0
H1: m m0
Sensitivity of the control chart – Operating Characteristic Curves (OCC)
If the expected value suffers a deviation of between the time elapsed
between the first and second sample, what is the probability of having an out-of-control
signal in the sample following the shift?
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87
Shewhart Control Charts
2)()3()( 000
x
xx
x
x
x
x LSCxz
s
smsm
s
sm
s
m
0228.02zP )-(1 Pd b
Sensitivity of the control chart – Operating Characteristic Curves (OCC)
Pd = (1- b)
m1 = m0 +
LIC
LSC
a/2
m0
H0: m = m0
H1: m m0
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89
Shewhart Control Charts
In Statitical Process Control it is common to talk about the Operating
Characteristic Curves (OCC), a curve that reflects the ability (or sensitivity)
of the chart to detect deviations from H0.
Sensitivity of the control chart – Operating Characteristic Curves (OCC)
H0: m = m0
(H1: m = m0 ± sx)
0.2
0.4
0.6
0.8
1.0
b
%
a = 0.27%
0.0-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0
(50.0%)
(84.1%)
(97.7%)
OCC for a control chart for averages with N = 4
The OCC represents the
probability that a point is
within the control limits, as a
function of each possible
value of the parameter of the
underlying distribution.
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90
Shewhart Control Charts
Sensitivity of the control chart – Operating Characteristic Curves (OCC)
H0: m= 1200
H1: m≠ 1200
s =300
N=100
a= 0,0027
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Shewhart Control Charts
ARL- Average Run Length
For a certain deviation of the value of the parameter in relation to H0,
the average run length is the expected value of the number of samples that we
need to gather until having an out-of-control signal.
Pd = probability of detecting the deviation on the 1º sample
(1- Pd). Pd = probability of detecting the deviation on the 2º sample …
= probability of detecting the deviation on the K sample
d
d
1k
d1k
d
1k
dP
1PP1kPP1E
dP
1ARL
Sensitivity of the control chart
1 − 𝑃𝑑𝑘−1. 𝑃𝑑
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Shewhart Control Charts
What is the ARL corresponding to a deviation of ?Xs
Pd = 0.0228
86.430228.0
1
P
1ARL
d
meaning that, if the expected value of the x variable suffers a shift
of 0.5s, you need 44 samples, in average, in order to detect this
shift (N=4).
Sensitivity of the control chart
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Shewhart Control Charts
ARL for a control chart for averages with N = 4
= 0
ARL = 6.3s 0.1
37.3700027.0
1110
ad
HP
ARL
Sensitivity of the control chart
AR
L
0
5
10
15
20
25
30
35
40
45
50
-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0
H0: m = m0
(H1: m = m0 ± sx)
(6.3)
(43.9)
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Shewhart Control Charts
Sensitivity of the control chart
if the process is in statistical control, then:
control-of-out point 1 P
1ARL
a
1ARL0
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Shewhart Control Charts
Measures of sensitivity of the control chart
if the process is out-of-control, then:
b
1
1ARL1
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Control Charts for Attributes
Measures of sensitivity of the control chart
If ARL is considered unsatisfactory, then action must be taken to
reduce the out-of-control ARL1.
Possible alternatives:
1. Increasing the sample size would result in a smaller value of b and a
shorter out-of-control ARL1;
2. Reduce the interval between samples (e.g. if we are currently
sampling every hour, it will take about seven hours, on the average, to
detect the shift. If we take the sample every half hour, it will require
only three and a half hours, on the average, to detect the shift);
3. Use a control chart that is more responsive to small shifts, such as the
cumulative sum charts (CUSUM).
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97
Types of Control Charts
Mean and variation (moving range)
( 𝒙; Rmoving)
Number nonconforming (Np)(Número de defeituosas)
Number of nonconformities (c)(Número de defeitos)
Number of nonconformities per unit (u)(average number of non-conformities per unit |
número de defeitos por unidade)
Fraction nonconforming (p)(Proporção de defeituosas)
Mean and variation (std deviation)
( 𝒙; s)N < 8
Mean and variation (range)
( 𝒙; R)
Control
Charts
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Types of Control Charts
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99
Types of Control Charts
Np and p control charts
Control charts for counting one random variable that only has two
possible outcomes:
defective (non conforming) or nondefective (conforming)
Sample size (N) constant Np Chart (number of defective items )
Sample size (N) variable p Chart (fraction nonconforming)
Y (number of defectives in a subgroup ) Binomial (N, p)
H0: p = p0
H1: p p0
Control Charts for Attributes
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100
Types of Control Charts
c and u control charts
Control charts for variables dealing with the number of defects or
nonconformities for a given “opportunity area” (piece, component,
time or space unit)
“opportunity area” constant c control chart (nº of defects)
“opportunity area” variable u control chart (nº of defects/unity)
Y (number of occurrences per opportunity area) Poisson (l)
H0: l = l0
H1: l l0
Control Charts for Attributes
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101
Types of Control Charts
Control charts and
Sample size N >8 control chart best estimator of s is
the sample standard deviation
Sample size N 8 control chart best estimator of s is
the sample range
H0: s = s0
H1: s s0
)( s;x )( R;x
)( s;x
H0: m = m0
H1: m m0
Assumptions of the and control charts: )( s;x
x ),( 0 xN sm
),( 0 smN
Control Charts for Variables
x
)( R;x
)( R;x
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Types of Control Charts
Control chart for individual measurements
Moving range: a statistic consisting in the absolute difference between
successive values of the variable under analysis
H0: s = s0
H1: s s0
H0: m = m0
H1: m m0.
)(moving
R;x
k
iii xx
kAM
21
1
1
Individual values control chart (x)
Moving range control chart (MR)
Control Charts for Variables
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Types of Control Charts
Mean and variation (moving range)
( 𝒙; Rmoving)
Number nonconforming (Np)(Número de defeituosas)
Number of nonconformities (c)(Número de defeitos)
Number of nonconformities per unit (u)(average number of non-conformities per unit |
número de defeitos por unidade)
Fraction nonconforming (p)(Proporção de defeituosas)
Mean and variation (std deviation)
( 𝒙; s)N < 8
Mean and variation (range)
( 𝒙; R)
Control
Charts
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Control Charts for Attributes
Control Chart for number nonconforming (Np) and
fraction nonconforming (p)
If Y represent the number of time a certain event occurs over N identical
experiments, and if
(i) correspond to each experience only two possible outcomes
(“sucess” and “failure”),
(ii) if the probability of each outcome is constant for all the N experiences and if,
(iii) the results of each experiment are independent
then Y (number of “sucesses”) B(N, p), with parameters
pN m )1( ppN s
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Control Charts for Attributes
Number nonconforming control chart – Np Chart
Standard Given
For standard given
)1(3 ppNpN
pN
)1(3 ppNpN
UCL =
CL =
LCL =
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Control Charts for Attributes
Fraction Nonconforming Control Chart– p Chart
Standard Given
For standard given
Assuming that the proportion of nonconforming (defectives) in a
subgroup = Y/N has the parameters:
UCL =
CL =
LCL =
ppNN
1
mN
ppppN
N
)1()1(
12
2 s
then,Nppp )1(3
p
Nppp )1(3
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Control Charts for Attributes
In a boiler assembly line, 200 boilers
are produced per day. After assembly,
the boilers undergo through a large set
of tests and verifications. When a
faulty boiler is detected, it is diverted
from the assembly line, entering a
particular station for repairs.
The responsible of the process wants
to evaluate if some recent changes in
the process had any impact on the
fraction nonconfoming (or the number
of defects) of the process.
The number of nonconforming boilers
was steady for a long time, being of 5
boilers a day.
Np and p chart (standard given – N constant sample size) : Example
Daily boiler rejections in the final tests (N = 200)
Day Number of
Defects
Percentage of
defects
Day Number of
Defects
Percentage of
defects
1 7 0.035 16 4 0.020
2 7 0.035 17 6 0.030
3 6 0.030 18 7 0.010
4 2 0.010 19 5 0.025
5 11 0.055 20 1 0.005
6 2 0.010 21 2 0.010
7 4 0.020 22 5 0.025
8 3 0.015 23 10 0.050
9 4 0.020 24 5 0.025
10 1 0.005 25 2 0.010
11 3 0.015 26 4 0.020
12 7 0.035 27 2 0.010
13 7 0.035 28 3 0.015
14 4 0.020 29 8 0.040
15 5 0.025 30 4 0.020
Daily boiler rejections 30 days AFTER process changes
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Control Charts for Attributes
Y (number of rejected boilers) B(N, p) assuming that Np= 5
Np and p chart (standard given – N constant sample size) : Example
208.2975.0025.0200)1( ppNs
624.11208.235)1(3 ppNpN
5 pN
0624.1208.235)1(3 ppNpN
UCL =
CL =
LCL =
Dia
0
2
4
6
8
10
12
5 10 15 20 25 30
LC = 5
LSC = 11.624
LSC = 0
UCL
LCL
CL
Day
There is no evidence that the changes performed in the process had any impact in the boiler’s defect rate
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Control Charts for Attributes
Y (number of rejected boilers) B(N, p) assuming that Np= 5
UCL = 11,624
CL = 5
LCL = 0
Np and p chart (standard given – N constant sample size) : Example
There is no evidence that the changes performed in the process had any impact in the boiler’s defect rate
28252219161310741
12
10
8
6
4
2
0
Sample
Sam
ple
Co
un
t
__NP=5
UCL=11,62
LCL=0
Np Chart Number Nonconforming Boilers
An estimated historical parameter is used in the calculations.
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Control Charts for Attributes
p control chart - fraction nonconforming (proporção de defeituosas)
Np and p chart (standard given – N constant sample size) : Example
LSC =
LC =
LIC =
0581.0200975.0025.03025.0)1(3 Nppp
025.0p
00081.0200975.0025.0305.0)1(3 Nppp
Dia
0.00
0.01
0.02
0.03
0.04
0.05
0.06
5 10 15 20 25 30
LC = 0.025
LSC = 0.0581
LIC = 0
UCL
CL
LCL
Day
There is no evidence that the changes performed in the process had any impact in the boiler’s defect rate
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Control Charts for Attributes
p control chart - fraction nonconforming (proporção de defeituosas)
Np and p chart (standard given – N constant sample size) : Example
UCL = 0.058
CL = 0.025
LCL = 0
There is no evidence that the changes perormed in the process had any impact in the boiler’s defect rate
28252219161310741
0,06
0,05
0,04
0,03
0,02
0,01
0,00
Sample
Pro
po
rtio
n
_P=0,025
UCL=0,05812
LCL=0
p Chart for Fraction Nonconforming Boilers
An estimated historical parameter is used in the calculations.
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Control Charts for Attributes
Number nonconforming control chart – Np Chart
No Standard Given
No Standard Given
items ofnumber total
items defective ofnumber totalˆ
1
1
K
k
k
K
k
k
N
y
p
)ˆ1(ˆ3ˆ ppNpN
pN ˆ
)ˆ1(ˆ3ˆ ppNpN
UCL =
CL =
LCL =
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Control Charts for Attributes
Fraction Nonconforming Control Chart – standard given vs. no standard given
28252219161310741
0,06
0,05
0,04
0,03
0,02
0,01
0,00
Sample
Pro
port
ion
_P=0,025
UCL=0,05812
LCL=0
28252219161310741
0,06
0,05
0,04
0,03
0,02
0,01
0,00
Sample
Pro
port
ion
_P=0,0235
UCL=0,05563
LCL=0
p Chart for Fraction Nonconforming Boilers
An estimated historical parameter is used in the calculations.
p Chart for Fraction Nonconforming Boilers
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117
Control Charts for Attributes
Np and p chart: Control Limits
Guidelines for the definition of control limits:
Number of samples
N ( sample size) should be such that, preferably and, at
least ;
In situations where p is very low (or very high), the previous rule
might lead to impossible sample sizes
(ex. if p = 1 ppm the size of the sample would be N= 1 million).
But, although you might follow all the rules, you can never be sure that the
process was in control while the samples were gathered.
3025 k
1ˆ pN
5ˆ pN
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118
Control Charts for Attributes
Np and p chart: Control Limits
Basic rules for the elimination of “outliers”:
If control limits for current or future production are to be meaningful, they
must be based on data from a process that is in control. Therefore, when
the hypothesis of past control is rejected, it is necessary to revise the trial
control limits, examining each of the out-of-control points, looking for an
assignable cause;
Any points that exceed the trial control limits should be investigated. If
assignable causes for these points are discovered, they should be discarded
and new trial control limits determined;
Whenever possible, the number of discarded samples should not exceed
more than 10% of the total number of samples.
The computed control limits based on the preliminar samples are known as
Trial Control Limits – Phase I of control chart usage.
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Types of Control Charts
Mean and variation (moving range)
( 𝒙; Rmoving)
Number nonconforming (Np)(Número de defeituosas)
Number of nonconformities (c)(Número de defeitos)
Number of nonconformities per unit (u)(average number of non-conformities per unit |
número de defeitos por unidade)
Fraction nonconforming (p)(Proporção de defeituosas)
Mean and variation (std deviation)
( 𝒙; s)N < 8
Mean and variation (range)
( 𝒙; R)
Control
Charts
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120
Control Charts for Attributes
Number of Nonconformities control chart – c chart
There are many instances where an item will contain nonconformities
but the item itself is not classified as nonconforming (or defective)
(nonconformities vs nonconforming ).
It is often important to construct control charts for the total number
of nonconformities or the average number of nonconformities for a
given “area of opportunity”. The inspection unit must be the same for
each unit.
The number of nonconformities in a given area of opportunity can be
modeled by the Poisson distribution.
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121
Control Charts for Attributes
Control Charts for Nonconformities – c and u charts
Nonconformities vs Nonconforming
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122
Control Charts for Attributes
C: number of occurences that a given phenomenon occurs in a continuous
interval – “area of opportunity” or inspection unit (volume, area, lenght or
space; time intervals or any other entity considered a continuum)
The Poisson distribution has the following characteristics:
It is a discrete distribution.
It describes rare events.
Each occurrence is independent of the other occurrences.
It describes discrete occurrences over a continuum or interval.
The occurrences in each interval can range from zero to infinity.
The expected number of occurrences must hold constant throughout the experiment.
C (number of occurences in an area of opportunity) Poisson (l)
l: average number of nonconformities by inspection unit
Control Charts for Nonconformities – c and u charts
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123
Control Charts for Attributes
Properties of the Poisson distribution:
Any linear combination of independent poisson variables still follows a poisson
distribution (ex. total number of painting defects, i.e. non-conformities);
Certain types of binomial distribution problems can be approximated by using
the Poisson distribution. Binomial problems with large sample sizes and small
values of p, which then generate rare events, are potential candidates for use
of the Poisson distribution. As a rule of thumb:
Control Charts for Nonconformities – c and u charts
20N 7 pN
UCL = CC mm 3
CL = Cm
LCL = CC mm 3
cXVar
cXE
l
l
)(
)(
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124
Control Charts for Attributes
Control Charts for Variables
Standard Given No Standard Given
Number of Nonconformities control chart | constant sample size –> c chart
c3cLCL
cCL
c3cUCL
c3cLCL
cCL
c3cUCL
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125
Control Chart for Nonconformities: no standard given
The table bellow represents the number of nonconformities observed in 26
successive samples of 100 printed circuit boards. For reasons of convenience, the
inspection unit is defined as 100 boards.
Douglas C. Montgomery “Introduction to Statistical Quality Control”, Wiley
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126
Control Chart for Nonconformities: no standard given
The table bellow represents the number of nonconformities observed in 26
successive samples of 100 printed circuit boards. For reasons of convenience, the
inspection unit is defined as 100 boards.
C (number of nonconformities by inspection unit) Poisson (c)
c: average number of nonconformities by inspection unit
Douglas C. Montgomery “Introduction to Statistical Quality Control”, Wiley
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127
Control Chart for Nonconformities: no standard given
48.619.85385.193
85.19 lineCenter
33.2219.85319.85c3
85.1926
516
ccLCL
c
cUCL
c
Douglas C. Montgomery “Introduction to Statistical Quality Control”, Wiley
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128
Control Chart for Nonconformities: no standard given
36.619.67367.193
67.19 lineCenter
32.9719.67319.67c3
67.1924
472
ccLCL
c
cUCL
c
Twenty new samples, each consisting in one inspection unit (i.e. 100 boards) are
subsequently collected.
Revised Control Limits
Douglas C. Montgomery “Introduction to Statistical Quality Control”, Wiley
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129
Control Chart for Nonconformities: no standard given
By analysing the nonconformities by type, we can often gain insight into their cause.
Douglas C. Montgomery “Introduction to Statistical Quality Control”, Wiley
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Control Chart for Nonconformities: no standard given
Further Analysis of Nonconformities (defect data for 500 boards)
60% of the total number of defects is due to two defect types: solder insufficiency
and solder cold joints Problems with the wave soldering process!
As the process manufactures several different types of printed circuit boards, it may be
helpful to examine the occurrence of defect type by type of printed circuit board.
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Control Chart for Nonconformities: no standard given
Further Analysis of Nonconformities (defect data for 500 boards)
As the process manufactures several different types of printed circuit boards, it may be
helpful to examine the occurrence of defect type by type of printed circuit board.
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Control Chart for Nonconformities: no standard given
Further Analysis of Nonconformities (defect data for 500 boards)
As the process manufactures several different types of printed circuit boards, it may be
helpful to examine the occurrence of defect type by type of printed circuit board.
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Control Chart for Nonconformities: no standard given
Further Analysis of Nonconformities
This tool is useful in focusing the attention of operators, manufacturing
engineers and managers on quality problems. Developing a good cause-and-effect
diagram usually advances the level of technological understanding of
the process.Douglas C. Montgomery “Introduction to Statistical Quality Control”, Wiley
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134
Types of Control Charts
Mean and variation (moving range)
( 𝒙; Rmoving)
Number nonconforming (Np)(Número de defeituosas)
Number of nonconformities (c)(Número de defeitos)
Number of nonconformities per unit (u)(average number of non-conformities per unit |
número de defeitos por unidade)
Fraction nonconforming (p)(Proporção de defeituosas)
Mean and variation (std deviation)
( 𝒙; s)N < 8
Mean and variation (range)
( 𝒙; R)
Control
Charts
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135
Control Charts for Attributes
UCL =
CL =
LCL =
uu sm 3
um
uu sm 3
Number of Nonconformities control chart | variable sample size –> u chart
Nk : number of “areas of opportunity” or inspection units in a given sample k (or
the size of the sample k in multiples of the same unit). For example, number
of square meters, of equal pieces, production hours;
yk : number of nonconformities (defects) present in the Nk “areas of opportunity”
of sample K;
uk : average number of nonconformities (defects) per unit, in the Nk “areas of
opportunity” of sample K ( );
mu : expected value of the average number of nonconformities (defects) per
inspection unit
su : standard deviation of the average number of nonconformities (defects) per
inspection unit
kkk Nyu
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136
If we assume that follows a Poisson distribution, thenkkk uNy
ukyy N msm 2
kkk Nyu
2
2
2 1)( y
k
ukN
uVar ss k
uuk
k
uN
NN
mms
2
2 1
UCL =
CL =
LCL =
um
kuu Nmm 3
kuu Nmm 3
K
k
K
k
k
u
kN
y
1
1m̂
kuu Nmm ˆ3ˆ
um̂
kuu Nmm ˆ3ˆ
UCL =
CL =
LCL=
No standard given
k
uu
N
ms
Number of Nonconformities control chart | variable sample size –> u chart
Control Charts for Attributes
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138
Numa fábrica têxtil que fabrica tecidos de várias texturas e padrões, foi instituído um
processo de amostragem que analisa a ocorrência de defeitos por 50 m2 de tecido. Os
dados de 10 rolos encontram-se na tabela seguinte. Com estes dados pretende-se construir
uma carta de controlo do número de não-conformidades por unidade de inspeção.
Number of Nonconformities control chart | variable sample size –> u chart
Control Charts for Attributes
K
k
K
k
k
u
kN
y
1
1m̂Número
do rolo Nº de m2
Nº Total de não-
conformidades
1 500 14
2 400 12
3 650 20
4 500 11
5 475 7
6 500 10
7 600 21
8 525 16
9 600 19
10 625 23
kuu Nmm ˆ3ˆ
um̂
kuu Nmm ˆ3ˆ
LSC =
LC =
LIC =
153
Nº de unidades
de inspecção
por rolo
10.0
8.0
13.0
10.0
9.5
10.0
12.0
10.5
12.0
12.5
107.5
42.15.107
153ˆ um
UCL LCL
2.555 0.291
2.689 0.158
2.416 0.431
2.555 0.291
2.584 0.262
2.555 0.291
2.456 0.390
2.528 0.319
2.456 0.390
2.436 0.411
k
kk
N
yu
Nº de não-conformidades
por unidade de inspecção
1.40
1.50
1.54
1.10
0.74
1.00
1.75
1.52
1.58
1.84
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Number of Nonconformities control chart | variable sample size –> u chart
Control Charts for Attributes
191715131197531
1,4
1,2
1,0
0,8
0,6
0,4
0,2
0,0
Sample
Sam
ple
Co
un
t P
er
Un
it
_U=0,701
UCL=1,249
LCL=0,153
U Chart of Total Number of Imperfections
Tests performed with unequal sample sizes
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140
Control Charts for Attributes
Measures of sensitivity of the control chart
The sensitivity of the control charts is associated with its ability to
detect shifts in the process parameters;
The power of the tests with the binomial and Poisson distribution is
low;
Control Charts are not the best tool to control processes with low
process capability.
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141
Control Charts for Attributes
Measures of sensitivity of the control chart
The Operating-Characteristic Function
It is a graphical display of the probability of incorrectly accepting the
hypothesis of statistical control (i.e., a type II or b-error) against the process
fraction nonconforming.
The OCC provides a measure of the sensitivity of the control chart.
i.e. for N=50 and =0.20, UCL=0.3697 and LCL=0.0303
𝑝 =𝐷
𝑛D is a binomial random variable with parameters n and p
Note that when the LCL is negative, the second term on the right-hand side of
equation should be dropped.
p
pp
pp
LCLDPUCLDP
LCLpPUCLpP
ˆˆb
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142
Control Charts for Attributes
Measures of sensitivity of the control chart
The Operating-Characteristic Function
pp
LCLDPUCLDP b
Introduction to Statistical Quality Control, Douglas C. Montgomery.
485.180.369750UCLN
515.10.030350LCLN
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143
Control Charts for Attributes
Measures of sensitivity of the control chart
The Operating-Characteristic Curve
For N=50 and =0.20, UCL=0.3697 and LCL=0.0303p
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144
Control Charts for Attributes
Measures of sensitivity of the control chart
Calculate the Average Run Lenght (ARL) for the fraction nonconforming
control chart (p chart)
if the process is in control, then:
controlof out plots point sample P
1ARL
a
1ARL0
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145
Control Charts for Attributes
Measures of sensitivity of the control chart
if the process is in control, then:
a
1ARL0
UCLLCL
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146
Control Charts for Attributes
Measures of sensitivity of the control chart
Calculate the Average Run Lenght (ARL) for the fraction
nonconforming control chart (p chart)
if it is out of control, then:
b
1
1ARL1
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147
Control Charts for Attributes
Measures of sensitivity of the control chart
Example:
Suppose that the process shifts out of control to p = 0.3
(in control p= 𝒑 =0,2)
From the table b= 0.8594
b
1
1ARL
1
7
0.8594-1
1ARL1
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148
Control Charts for Attributes
Measures of sensitivity of the control chart
Example:
Suppose that the process shifts out of control to p = 0.3
(in control p= 𝒑 =0,2)|From the table b= 0.8594
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149
Control Charts for Attributes
Measures of sensitivity of the control chart
Suppose that the process shifts out of control to p = 0.3
b
1
1ARL
1 7
0.8594-1
1ARL1
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150
Control Charts for Attributes
Measures of sensitivity of the control chart
If ARL is considered unsatisfactory, then action must be taken to
reduce the out-of-control ARL1.
Possible alternatives:
1. Increasing the sample size would result in a smaller value of b and a
shorter out-of-control ARL1;
2. Reduce the interval between samples (e.g. if we are currently
sampling every hour, it will take about seven hours, on the average, to
detect the shift. If we take the sample every half hour, it will require
only three and a half hours, on the average, to detect the shift);
3. Use a control chart that is more responsive to small shifts, such as the
cumulative sum charts (CUSUM).
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151
Control Charts for Attributes
Measures of sensitivity of the control chart
1. Increasing the sample size would result in a smaller value of b and a
shorter out-of-control ARL1;
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152
Types of Control Charts
Mean and variation (moving range)
( 𝒙; Rmoving)
Number nonconforming (Np)(Número de defeituosas)
Number of nonconformities (c)(Número de defeitos)
Number of nonconformities per unit (u)(average number of non-conformities per unit |
número de defeitos por unidade)
Fraction nonconforming (p)(Proporção de defeituosas)
Mean and variation (std deviation)
( 𝒙; s)N < 8
Mean and variation (range)
( 𝒙; R)
Control
Charts
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153
Control Charts for Variables
Control charts for variables are built based on quantitative data –
Quality characteristics are measured on a numerical scale.
Average control chart - evaluates process center
Range control chart - evaluates process variation
X
x
x
x LSC
LIC
LC
0
0
ss
mm
0
0
ss
mm
0
0
ss
mm
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154
Shewhart Control Charts
Control Charts for Variables and R
Notation for variables control charts:
n - size of the sample (sometimes called a subgroup) chosen at a
given point in time
k - number of samples selected
m is the true process mean
s is the true process standard deviation
x
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155
Shewhart Control Charts
Control Charts for Variables
Notation for variables control charts:
= average of the observations in the ith sample (where i = 1, 2, ..., k)
= grand average or “average of the averages”
(this value is used as the center line of the control chart)
Ri = range of the values in the ith sample
Ri = xmax - xmin
= average range for all k samplesR
Control Charts for Variables and R x
ix
x
Shewhart Control Charts
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Standard Given No Standard Given
What do you know by now?
Control Charts for VariablesControl Charts for Variables and R x
Shewhart Control Charts
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Control Charts for Variables
mm xNx ss X ( , )
UCL = Nsm 3
CL = m
LCL = Nsm 3
Control Charts for Variables : Standard Given and No Standard Givenx
Control Chart: standard given X
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Standard Given No Standard Given
What do you know by now?
Control Charts for VariablesControl Charts for Variables and R x
Shewhart Control Charts
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159
Control Charts for Variables
R Control Chart: standard given | Sample size N < 8
UCL = RR
sm 3
CL = R
m
LCL = RR
sm 3
For Normal variables with (m, s) and sample sizes N
UCL = ssss 23232 )3(3 Ddddd
CL = s2d
LCL = ssss 13232 )3(3 Ddddd .
sm 2
dR
ss 3
dR
Control Charts for VariablesControl Charts for Variables R: Standard Given
UCL = 𝑫𝟐.s
CL = 𝒅𝟐.s
𝐋𝐂𝐋 = 𝑫𝟏.s
Control Limits for the R chart: standard given
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160
Control Charts for Variables
R Control Chart: no standard given | Sample size N 8
If s is unknown, then:
Control Charts for VariablesControl Charts for Variables R: No Standard Given
Control Limits for the R chart: no standard given
2
ˆd
Rs
K
kk
RK
R1
1where
LSC = RDd
RD
4
2
2
LC = Rd
Rd
2
2
LIC = RDd
RD
3
2
1.
N D4 D3
2 3,267 0
3 2,575 0
4 2,282 0
values that only depend on the
sample size N
<
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Standard Given No Standard Given
What do you know by now?
Control Charts for VariablesControl Charts for Variables and R x
Shewhart Control Charts
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162
Control Charts for Variables
If the parameters of X are unknown, then
mm xNx ss X ( , )
K
k
kxK
x1
1m̂
and if X follows a Normal distribution
2
ˆd
Rs
UCL = Nsm 3
CL = m
LCL = Nsm 3
UCL = RAxNd
Rx
2
2
3
CL = x
LCL = RAxNd
Rx
2
2
3
Control Charts for Variables : Standard Given and No Standard Givenx
Control Chart: standard given X
Control Chart: no standard given X
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163
Shewhart Control Charts
Control Charts for Variables
Control Limits for the chartx
UCL = 𝒙 + 𝑨𝟐. 𝑅
CL = 𝑥
𝑳𝑪𝑳 = 𝒙 − 𝑨𝟐. 𝑅
UCL = 𝑫𝟒. 𝑅
CL = 𝑅
𝑳𝑪𝑳 = 𝑫𝟑. 𝑅
Control Limits for the R chart
Control Charts for Variables and R: No Standard Givenx
Small sample sizes N < 8
No standard given
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164
Shewhart Control Charts
Control Charts for Variables
Control Limits for the chartx
UCL = 𝒙 + 𝑨𝟐. 𝑅
CL = 𝑥
𝑳𝑪𝑳 = 𝒙 − 𝑨𝟐. 𝑅
UCL = 𝑫𝟒. 𝑅
CL = 𝑅
𝑳𝑪𝑳 = 𝑫𝟑. 𝑅
Control Limits for the R chart
Control Charts for Variables and R: No Standard Givenx
Small sample sizes N < 8
No standard given
K
k
kxK
x1
1m̂
2
ˆd
Rs
If x follows a Normal distribution
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165
Control Charts for Variables
Its is recomended that and estimates are calculated having a
minimum of 100 values (K: number of samples and typically n is
between 3 and 5);
The control limits of the control chart should only be considered definite
when all points on the plot in-control.
x R
20K
x
Control Limits for the chartx
UCL = 𝒙 + 𝑨𝟐. 𝑅
CL = 𝑥
𝑳𝑪𝑳 = 𝒙 − 𝑨𝟐. 𝑅
UCL = 𝑫𝟒. 𝑅
CL = 𝑅
𝑳𝑪𝑳 = 𝑫𝟑. 𝑅
Control Limits for the R chart
R
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Shewhart Control Charts
167
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168
Control Charts for Variables
Samples should be selected in such a way that maximizes the chances for
shifts in the process average to occur between samples, and thus to
show up as out-of-control points on the averages chart;
The R chart, on the other hand, measures the variability within a
sample. Therefore, samples should be selected so that variability within
samples measures only chance or random causes;
Another way of saying this:
X bar chart monitors the between sample variability
the variability of the process over time
R chart monitors the within sample variability
the “instantaneous” variability in a given moment
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171
Control Charts for Variables
In a casting process, two samples of molten steel are taken from
the ladle per day (the first is taken early in the morning and the
second one in the afternoon).
For each sample of molten steel, two dog-bone specimens
(provetes) are casted. These specimens are used to perform tensile
tests and to obtain their corresponding mechanical properties,
mainly the yield stress (tensão de cedência).
There are reasons to suspect that, from the point of view of the
mechanical properties of steel, significant alterations have occurred
between morning and the afternoon.
control charts: exampleRx,
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172
Control Charts for Variables
Numa determinado fundição, são retiradas da colher de vazamento
duas amostras de aço por dia (uma ao princípio da manhã e outra ao
fim da tarde). A partir de cada amostra são vazados dois provetes
que, posteriormente são ensaiados à tracção.
Há razões para suspeitar que, do ponto de vista das propriedades
mecânicas do aço, se verificam alterações significativas de manhã
para a tarde.
Na Tabela seguinte registam-se os valores da tensão de cedência
(em MPa) obtidos nos ensaios efectuados sobre os provetes
recolhidos nos últimos 25 dias úteis.
Cartas de Médias e Amplitudes (Cartas ) - ExemploAx,
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173
Yield Stress (MPa) in two specimen (provetes) of molten steel
Morning Afternoon Morning Afternoon
Day Spec. 1 Spec. 2 Spec. 1 Spec. 2 Day Spec. 1 Spec. 2 Spec. 1 Spec. 2
19/04 197.6 193.7 199.7 203.5 06/05 197.6 198.7 204.8 203.8
20/04 195.7 199.8 199.5 198.5 07/05 198.1 198.9 203.0 196.0
21/04 192.5 190.3 209.8 210.3 10/05 197.6 200.1 196.4 197.5
22/04 198.5 199.1 201.1 196.5 11/05 206.8 205.7 195.5 192.7
23/04 201.7 199.2 201.8 201.4 12/05 199.0 205.4 194.5 196.6
26/04 204.7 199.4 196.5 197.8 13/05 196.9 198.7 199.2 194.6
27/04 197.8 195.4 199.6 196.1 14/05 197.9 198.6 200.9 201.8
28/04 200.4 200.6 197.5 198.1 17/05 200.5 201.3 209.4 208.6
29/04 204.3 202.2 197.7 198.4 18/05 199.5 202.5 193.6 197.7
30/04 200.6 206.9 205.5 197.8 19/05 202.3 202.0 199.5 196.6
03/05 198.8 202.7 194.9 200.3 20/05 200.4 196.3 199.4 199.6
04/05 195.1 199.2 202.7 200.9 21/05 212.8 211.1 191.7 193.5
05/05 203.1 200.9 193.8 192.8
Control Charts for Variables
control charts: exampleRx,
The following table record the values of yield stress (MPa) obtained in tests carried out
on samples collected in the last 25 days.
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174
Control Charts for Variables
Taking into considerations that each group of observations comprehends two
specimens (provetes) of the same ladle (colher de vazamento) (N=2)
51.250
1 50
1
k
RR
Control Limits for the R chart
Control Limits for the chart
UCL = 42.20451.2880.170.1992
RAx
CL = 70.199x
LCL = 98.19451.2880.170.1992
RAx
UCL = 20.851.2267.34
RD
CL = 51.2R
LCL = 0
x
control charts: exampleRx,
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Shewhart Control Charts
175
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176
Control Charts for Variables
Process is
out-of-
control
control charts: exampleRx,
464136312621161161
210
205
200
195
Sample
Sa
mp
le M
ea
n
__
X=199,69
UCL=204,40
LCL=194,98
464136312621161161
7,5
5,0
2,5
0,0
Sample
Sa
mp
le R
an
ge
_R=2,504
UCL=8,181
LCL=0
1
1
1
1
1
1
1
1
subgroup size 2
Xbar-R Chart of Ordenado
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177
Control Charts for Variables
The standard deviation of the yield stress of the specimen can be calculated
according to:
22.2128.1
51.2ˆ
2
d
Rs << 31.4
1100
1ˆ
100
1
2
i
iT xxss
control charts: exampleRx,
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Control Charts for Variables
If it was not considered the information that there are suspicions considering
potential alterations of the yield stress between morning and afternoon,
then, with N = 4:
control charts: exampleRx,
Yield Stress (MPa) in two specimen (provetes) of molten steel
Morning Afternoon Morning Afternoon
Day Prov. 1 Prov. 2 Prov. 1 Prov. 2 Day Prov. 1 Prov. 2 Prov. 1 Prov. 2
19/04 197.6 193.7 199.7 203.5 06/05 197.6 198.7 204.8 203.8
20/04 195.7 199.8 199.5 198.5 07/05 198.1 198.9 203.0 196.0
21/04 192.5 190.3 209.8 210.3 10/05 197.6 200.1 196.4 197.5
22/04 198.5 199.1 201.1 196.5 11/05 206.8 205.7 195.5 192.7
23/04 201.7 199.2 201.8 201.4 12/05 199.0 205.4 194.5 196.6
26/04 204.7 199.4 196.5 197.8 13/05 196.9 198.7 199.2 194.6
27/04 197.8 195.4 199.6 196.1 14/05 197.9 198.6 200.9 201.8
28/04 200.4 200.6 197.5 198.1 17/05 200.5 201.3 209.4 208.6
29/04 204.3 202.2 197.7 198.4 18/05 199.5 202.5 193.6 197.7
30/04 200.6 206.9 205.5 197.8 19/05 202.3 202.0 199.5 196.6
03/05 198.8 202.7 194.9 200.3 20/05 200.4 196.3 199.4 199.6
04/05 195.1 199.2 202.7 200.9 21/05 212.8 211.1 191.7 193.5
05/05 203.1 200.9 193.8 192.8
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179
Control Charts for Variables
If it was not considered the information that there are suspicions considering
potential alterations of the yield stress between morning and afternoon,
then, with N = 4:
92.725
1 25
1
k
RR >> 51.2R (N = 2)
UCL = 47.20592.7729.070.1992
RAx
CL = 70.199x
LCL = 93.19392.7729.070.1992
RAx
UCL = 07.1892.7282.24
RD
CL = 92.7R
LCL = 0
control charts: exampleRx,
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252321191715131197531
205
200
195
Sample
Sa
mp
le M
ea
n
__X=199,69
UCL=205,45
LCL=193,93
252321191715131197531
20
10
0
Sample
Sa
mp
le R
an
ge
_R=7,90
UCL=18,03
LCL=0
1
1
subgroup size 4
Xbar-R Chart of Ordenado
method for estimating within subgroup variation - R bar
180
Control Charts for Variables
Previous limit: 8.2
control charts: exampleRx,
Process is out-of-
control (assignable
causes are present
between morning
and night)
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181
Control Charts for Variables
Selection of the sample size
The ability of the and R charts to detect shifts in process quality
is described by their operating characteristic(OC) curves
OCC of the control chart (ii) – CCO da Carta AX
control charts: exampleRx,
OCC of the R control chart
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182
Types of Control Charts
Mean and variation (moving range)
( 𝒙; Rmoving)
Number nonconforming (Np)(Número de defeituosas)
Number of nonconformities (c)(Número de defeitos)
Number of nonconformities per unit (u)(average number of non-conformities per unit |
número de defeitos por unidade)
Fraction nonconforming (p)(Proporção de defeituosas)
Mean and variation (std deviation)
( 𝒙; s)N < 8
Mean and variation (range)
( 𝒙; R)
Control
Charts
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OCAP- Out of Control Action Plan
When a control chart is
introduced, an initial OCAP
should accompany
it. Control charts without an
OCAP are not likely to be
useful as a process
improvement tool.
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184
Shewhart Control Charts
Control Charts for Variables
Notation for variables control charts:
si = standard deviation of the values in the ith sample
= average standard deviation for all k sampless
Control Charts for Variables and R x
k
i
is sk
s1
1m̂
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185
Shewhart Control Charts
Control Charts for Variables
Control Limits for the chartx
UCL = 𝒙 + 𝑨𝟑. 𝑠
CL = 𝑥
𝑳𝑪𝑳 = 𝒙 − 𝑨𝟑. 𝑠
UCL = 𝑩𝟒. 𝑠
CL = 𝑠
𝑳𝑪𝑳 = 𝑩𝟑. 𝑠
Control Limits for the s chart
Control Charts for Variables and sx
Small sample sizes N ≥8
No standard given
K
k
kxK
x1
1m̂
4
ˆc
ss
If x follows a Normal distribution
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186
Shewhart Control Charts
Control Charts for Variables
Control Limits for the chartx
UCL = 𝒙 + 𝑨𝟑. 𝑠
CL = 𝑥
𝑳𝑪𝑳 = 𝒙 − 𝑨𝟑. 𝑠
UCL = 𝑩𝟒. 𝑠
CL = 𝑠
𝑳𝑪𝑳 = 𝑩𝟑. 𝑠
Control Limits for the s chart
Control Charts for Variables and s x
Small sample sizes N > 8
No standard given
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187
Types of Control Charts
Mean and variation (moving range)
( 𝒙; Rmoving)
Number nonconforming (Np)(Número de defeituosas)
Number of nonconformities (c)(Número de defeitos)
Number of nonconformities per unit (u)(average number of non-conformities per unit |
número de defeitos por unidade)
Fraction nonconforming (p)(Proporção de defeituosas)
Mean and variation (std deviation)
( 𝒙; s)N < 8
Mean and variation (range)
( 𝒙; R)
Control
Charts
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Shewhart Control Charts
188
There are many situations in which the sample size used for process monitoring is n = 1; that
is, the sample consists of an individual unit. Examples:
1. Automated inspection and measurement technology is used, and every unit manufactured
is analyzed so there is no basis for rational subgrouping;
2. Data comes available relatively slowly, and it is inconvenient to allow sample sizes of
n > 1 to accumulate before analysis. The long interval between observations will cause
problems with rational subgrouping. This occurs frequently in both manufacturing and
nonmanufacturing situations;
3. Repeat measurements on the process differ only because of laboratory or analysis error,
as in many chemical processes;
4. In process plants, such as papermaking, measurements on some parameter such as coating
thickness across the roll will differ very little and produce a standard deviation that
is much too small if the objective is to control coating thickness along the roll.
Control Charts for Individual Measurements
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190
Shewhart Control Charts
Control Charts for VariablesControl Charts for Individual Measurements
No Standard Given
Control Limits for Individual Units (x chart)
UCL = 𝒙 + 3.𝑴𝑹
𝑑2CL = 𝒙
𝑳𝑪𝑳 = 𝒙 − 3.𝑴𝑹
𝑑2
But, how can we estimate process variability? A good estimator is based on the moving range (MR):
iiixxMR
1i = 2, 3, ..., k
k
iii
xxk
MR2
1
1
1
UCL = 𝑫𝟒.𝑀𝑅
CL = 𝑀𝑅
𝑳𝑪𝑳 = 𝑫𝟑.𝑀𝑅
Control Limits for the Moving Range chart
A good estimator of s is then
with N=2, d2=1.128
2
ˆd
MRs
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Shewhart Control Charts
191
Control Charts for Individual Measurements
The mortgage loan processing unit of a bank monitors the
costs of processing loan applications. The quantity tracked is
the average weekly processing costs, obtained by dividing
total weekly costs by the number of loans processed during
the week. The processing costs for the most recent 20 weeks
are shown in the table.
Set up individual and moving range control charts for these
data.
UCL = 𝒙 + 3.𝑴𝑹
𝑑2CL = 𝒙
𝑳𝑪𝑳 = 𝒙 − 3.𝑴𝑹
𝑑2
UCL = 𝒙 + 3.𝑴𝑹
𝑑2= 300.5 + 3
7.79
1.1.28= 321.22
CL = 𝑥 = 300.5
𝑳𝑪𝑳 = 𝒙 − 3.𝑴𝑹
𝑑2= 300.5 − 3
7.79
1.128= 279.78
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Shewhart Control Charts
192
Control Charts for Individual Measurements
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Shewhart Control Charts
193
Control Charts for Individual Measurements
Disadvantages of the (x, MR) control charts
Less sensitive than the xbar-R charts;
One of the assumptions is that the observations follow a Normal distribution;
The successive values of the ranges could be highly correlated;
Advantages of the (x, MR) control charts
Simple interpretation;
Immediate update;
Direct comparison with the specifications limits.
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204
Shewhart Control Charts
Control charts are used in two diferent ways:
1) As a monitoring and surveillance online process tool,
functioning as an early warning system for the occurrence of
uncontrollable situations;
2) As a retrospective process analysis tool, with the aim of
identifying the presence (or absence) of special causes of
variation and provide their elimination.
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205
Process Analysis with control charts
The analysis of processes through control charts has two main
purposes:
(1) To estimate the real and the potential capability of
processes;
(2) To reduce the process’s variability.
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206
Process Analysis with control charts
(1) Process Capability Estimation
The head of the packing division was having
problems with the detergent powder
packing machine, and thought that the
machine didn´t have sufficient accuracy to
meet customers' requirements.
The specifications for the detergent weight
are 4 grams around the nominal value of 504
grams.
504g ± 4𝑔
There is a suspicion that, besides the equipment, the supplier and the origin of the raw
materials is a factor that might influence the weight of the packages.
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207
2Process Analysis with control charts
(1) Process Capability Estimation
In order to evaluate the problem, the net
weights (in grams) observed in samples of
size 4 taken randomly from 25 different lots
were gathered (each lot has 5000 packets).
The origin of the raw materials might
change from lot to lot.
1 lot = 5000 packetsRaw materials of the same
supplier
25 lots
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208
Tabela– Pesos (em gramas) de detergente em pó
Lote no. x1 x2 x3 x4x A
1 504.9 506.0 505.2 504.7 505.2 1.32 501.2 499.5 501.1 500.8 500.7 1.73 506.6 503.4 505.6 503.7 504.8 3.24 503.2 502.5 503.2 503.5 503.1 1.05 498.5 500.2 499.1 500.7 499.6 2.2
6 504.0 503.1 502.9 504.4 503.6 1.57 503.7 502.6 502.9 502.9 503.0 1.18 502.7 503.7 503.7 504.5 503.7 1.89 505.0 504.2 506.7 504.9 505.2 2.510 504.9 504.4 505.2 504.0 504.6 1.211 506.3 505.9 504.7 507.0 506.0 2.312 504.2 503.5 504.2 502.5 503.6 1.713 500.3 501.7 500.3 502.6 501.2 2.3
14 501.6 504.0 501.3 502.7 502.4 2.715 502.5 503.5 502.8 503.3 503.0 1.016 504.0 503.3 504.4 506.5 504.6 3.217 502.5 498.9 501.8 501.3 501.1 3.618 502.3 500.2 503.7 503.6 502.5 3.519 503.8 505.2 504.6 504.7 504.6 1.420 506.4 505.8 508.0 505.2 506.4 2.821 505.2 506.6 504.8 504.1 505.2 2.5
22 505.3 504.6 505.9 506.4 505.6 1.823 506.9 503.7 504.0 505.9 505.1 3.224 504.9 506.1 505.7 504.4 505.3 1.725 509.1 506.8 507.8 507.0 507.7 2.3
503.9x 14.2A
Process Analysis with control charts
(1) Process Capability Estimation
Weight (grams) of the 25 samples
R
𝑅
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209
Equipment Raw materials
In order to ensure that we meet specifications, we must understand:
Process Analysis with control charts
(1) Process Capability Estimation
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210
Peso
499 500 501 502 503 504 505 506 507 508 509
Lim
ite
Infe
rior
da
Tol
erân
cia
Lim
ite
Sup
erio
r da
Tol
erân
cia
0
5
10
15
20
Fre
quên
cia
x = 503.9
s = 2.09
The weight distribution is centred, unimodal and not too asymmetrical
Specifications: 504g ± 4𝑔
Process Analysis with control charts
(1) Process Capability Estimation
LSL USL
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211
Process Analysis with control charts
(1) Process Capability Estimation
An advantage of using the histogram to estimate process capability is
that it gives an immediate, visual impression of process performance.
But, they do not necessarily display the potential capability of the
process because they do not address the issue of statistical control.
8
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Knowing that , the process capability index is (capacidade do processo): 09.2 ss
64.009.26
8
6
s
LSLUSLC
p
Conclusion: The head of the packing division is right, the equipment is not capable
of meeting specifications.
When estimating global variability through the standard deviation s, what factors of
variation are included in this value?
09.2s
Equipment ? Raw materials ?
Question
Process Analysis with control charts
(1) Process Capability Estimation
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213
2Process Analysis with control charts
(1) Process Capability Estimation
S = 2.09
Includes the variation:
-> within subgroups;
-> between subgroups, due to changes in
raw materials.
The origin of the raw materials might
change from lot to lot.
So, we need to compare this estimate of s with another estimate that only takes into account
the within subgroup variation.
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214
Tabela– Pesos (em gramas) de detergente em pó
Lot no. x1 x2 x3 x4 x R
1 504.9 506.0 505.2 504.7 505.2 1.3
2 501.2 499.5 501.1 500.8 500.7 1.7
3 506.6 503.4 505.6 503.7 504.8 3.2
4 503.2 502.5 503.2 503.5 503.1 1.0
5 498.5 500.2 499.1 500.7 499.6 2.2
6 504.0 503.1 502.9 504.4 503.6 1.5
7 503.7 502.6 502.9 502.9 503.0 1.1
8 502.7 503.7 503.7 504.5 503.7 1.8
9 505.0 504.2 506.7 504.9 505.2 2.5
10 504.9 504.4 505.2 504.0 504.6 1.2
11 506.3 505.9 504.7 507.0 506.0 2.3
12 504.2 503.5 504.2 502.5 503.6 1.7
13 500.3 501.7 500.3 502.6 501.2 2.3
14 501.6 504.0 501.3 502.7 502.4 2.7
15 502.5 503.5 502.8 503.3 503.0 1.0
16 504.0 503.3 504.4 506.5 504.6 3.2
17 502.5 498.9 501.8 501.3 501.1 3.6
18 502.3 500.2 503.7 503.6 502.5 3.5
19 503.8 505.2 504.6 504.7 504.6 1.4
20 506.4 505.8 508.0 505.2 506.4 2.8
21 505.2 506.6 504.8 504.1 505.2 2.5
22 505.3 504.6 505.9 506.4 505.6 1.8
23 506.9 503.7 504.0 505.9 505.1 3.2
24 504.9 506.1 505.7 504.4 505.3 1.7
25 509.1 506.8 507.8 507.0 507.7 2.3
503.9x 14.2R
Raw material constant =
same supplier
14.2R
The average range only
takes into account
within subgroup
variation
Process Analysis with control charts
(1) Process Capability Estimation
Looking at the available information, this means:
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215
Process Analysis with control charts
(1) Process Capability Estimation
252321191715131197531
508
506
504
502
500
Sample
Sa
mp
le M
ea
n
__X=503,903
UCL=505,462
LCL=502,344
252321191715131197531
4,8
3,6
2,4
1,2
0,0
Sample
Sa
mp
le R
an
ge
_R=2,14
UCL=4,882
LCL=0
1
1
1
11
1
1
1
Detergent powder packing machine
Raw material might change between subgroups
But, is the process stable?
Process variability
is in-control
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216
Re-estimating the standard deviation with the Average Range (amplitude média):
04.1059.2
14.2ˆ
2
d
Rd
s where sd only accounts for within subgroup variation
28.104.16
8
ˆ6
d
p
TC
s
The equipment is capable for producing packages according
to specifications.
Nevertheless, instability due the changes in raw materials,
or due to any other assignable cause that might occur
between subgroups, should be eliminated.
Cp represents the process capability that can be achieved if all assignable
causes are removed between subgroups, i.e. the potential process
capability
Process Analysis with control charts
(1) Process Capability Estimation
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21728.1
04.16
8
ˆ6
d
p
TC
s
Process Analysis with control charts
(1) Process Capability Estimation
64.009.26
8
6
s
TPp
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218
28.104.16
8
ˆ6
d
p
TC
s
Process Analysis with control charts
(1) Process Capability Estimation
64.009.26
8
6
s
TPp
Cp: is the best indicator of potential process capability because it assumes a
stable and on-target process.
Pp: is an indicator of actual process performance if the process is on-target.
It does take into consideration the long-term variability.
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220
Procedure for Process Improvement with control charts:
(i) Identification of possible and plausible assignable causes of variation
(i.e. factors) and collect samples in rational subgroups;
(ii) Confirmation or rejection of the hypothesis considered in (i);
(iii) Create an hierarchy of assignable causes of variation/ factors according
to the magnitude of its effects;
(iv) Analyze and eliminate the causes that are at the root of such factors of
variation;
(v) Confirm the results.
Process Analysis with control charts
Process Improvement with control charts
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221
ijE ),( 2
eoN sm ),( 2
doN sm
tD
Variation within groups Variation between groups
Distribution in
t = 1 (m1, sd)Distribution in
t = 2 (m2, sd)
mk
Distribution in
t = k (mk, sd)
Dk
xkj
Ek
m
Global Distribution
(m, s)
t = 1
Time
t = 2 t = k
Assuming that the process is only unstable in what concerns location.
Process Analysis with control charts
(2) Process Improvement with control charts
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222
tjttjttj EDEx mm
t: index related to the instante when the sample was taken t (t = 1, 2, ..., K)
j: index related to the position of one observation within each sample (j = 1, 2, ..., N)
xtj: j observation of the sample taken at time t
mt: expected value of the distribution at time t
Etj: error (deviation in relation to mt) associated to the j-ésima observation of the sample
taken at time t
m: global expected value
Dt: deviation in relation to m of the expected value of the distribution at time t
ANOVA Model (1 factor, fixed effects)
Process Analysis with control charts
(2) ) Process Improvement with control charts
Assuming that the process is only unstable in what concerns location.
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223
)1(
)1(1222
NK
Ned sss
)( 22 sEs : represents the expected value of the global sampling variance
(variância amostral global) (for all data)
)ˆ( 22dd E ss : represents the expected value of the variance within groups, that,
in the case of R charts is given by 2
2dR
)ˆ( 22ee E ss : represents the expected value of the variance between groups
that, in the case of xbar charts is given by )1/()(ˆ 2
1
2
kxxK
t
txs
K>> N 222
ed sss
Process Analysis with control charts
(2) Process Improvement with control charts
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225
222
ed sss
is related with the variation that occurs within groups (assuming that
it is homogeneous), due to common causes of variation
results from a change in the process location, possibly due to
assignable (or special) causes of variation
2
ds
2
es
In a process completely in control, = 0 2
es 22
dss
N
dex
222 ˆ
ˆˆs
ss The variability in the xbar chart depends on both
components:
the variability between and within groups
Process Analysis with control charts
(2) Process Improvement with control charts
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226
222
ed sss
estimate based on orR s
Ndxe222
ˆˆˆ sss estimate based on
(variance of the observations on the
xbar chart)
Process Analysis with control charts
(2) Process Improvement with control charts
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227
Tabela– Pesos (em gramas) de detergente em pó
Lote no. x1 x2 x3 x4x A
1 504.9 506.0 505.2 504.7 505.2 1.32 501.2 499.5 501.1 500.8 500.7 1.73 506.6 503.4 505.6 503.7 504.8 3.24 503.2 502.5 503.2 503.5 503.1 1.05 498.5 500.2 499.1 500.7 499.6 2.2
6 504.0 503.1 502.9 504.4 503.6 1.57 503.7 502.6 502.9 502.9 503.0 1.18 502.7 503.7 503.7 504.5 503.7 1.89 505.0 504.2 506.7 504.9 505.2 2.510 504.9 504.4 505.2 504.0 504.6 1.211 506.3 505.9 504.7 507.0 506.0 2.312 504.2 503.5 504.2 502.5 503.6 1.713 500.3 501.7 500.3 502.6 501.2 2.3
14 501.6 504.0 501.3 502.7 502.4 2.715 502.5 503.5 502.8 503.3 503.0 1.016 504.0 503.3 504.4 506.5 504.6 3.217 502.5 498.9 501.8 501.3 501.1 3.618 502.3 500.2 503.7 503.6 502.5 3.519 503.8 505.2 504.6 504.7 504.6 1.420 506.4 505.8 508.0 505.2 506.4 2.821 505.2 506.6 504.8 504.1 505.2 2.5
22 505.3 504.6 505.9 506.4 505.6 1.823 506.9 503.7 504.0 505.9 505.1 3.224 504.9 506.1 505.7 504.4 505.3 1.725 509.1 506.8 507.8 507.0 507.7 2.3
503.9x 14.2A
(iii) Create an hierarchy of assignable causes of variation/ factors according
to the magnitude of its effects;
Process Analysis with control charts
(2) Process Improvement with control charts
Weights (grams) of powder detergent
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228
N = 4
With the 100 observations, we can calculate:
For each sample, we can calculate:
x - estimate of the expected value m
s - estimate of the standard deviation s
- estimate of the expected value mt of the weight distribution at time t
- estimate of the standard deviation sd
x
2d
R
Process Analysis with control charts
(2) Process Improvement with control charts
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229
You now suspect that the shift (turno) might have a decisive influence on
the final weight of the packages
Process Analysis with control charts
(2) Process Improvement with control charts
Turno A Turno B
Lote no. x1 x2 x3 x4
1 504.9 506.0 505.2 504.7
2 501.2 499.5 501.1 500.8
3 506.6 503.4 505.6 503.7
4 503.2 502.5 503.2 503.5
. . .
Turno A Turno B
Lote no. x1 x2 x3 x4
1 504.9 506.0 505.2 504.7
2 501.2 499.5 501.1 500.8
3 506.6 503.4 505.6 503.74 503.2 502.5 503.2 503.5
. . .
N = 2 N = 4
Shift A Shift B
Lot #
Shift A Shift B
Lot #
N = 2 N = 4
Turno A Turno B
Lote no. x1 x2 x3 x4
1 504.9 506.0 505.2 504.72 501.2 499.5 501.1 500.83 506.6 503.4 505.6 503.74 503.2 502.5 503.2 503.5. . . . .. . . . .. . . . .
Shift A Shift B
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230
Components of variation within and between subgroups:
Process Analysis with control charts
(2) Process Improvement with control charts
Grouping by lot and shift
(N = 2)
Grouping by lot
(N = 4)
Causes for
variation within
subgroups
Causes for
variation between
subgroups
• Measurement error
• Sampling error
• Other factors of variation
acting within subgroups
• Measurement error
• Sampling error
• Factor “change of shift”
• Factor “change of raw material”
• Measurement error
• Sampling error
• Factor “change of shift”
• Other factors of variation
acting within subgroups
• Measurement error
• Sampling error
• Factor “change of raw
material”
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231
Control charts Average-Range with N = 2
144.ˆ2
11.128
1.29
d
Rd
s
77.1
2
144.1801.3
ˆˆˆ
2
22
N
dxe
sss
Process Analysis with control charts
(2) Process Improvement with control charts
464136312621161161
508
506
504
502
500
Sample
Sa
mp
le M
ea
n
__X=503,903
UCL=506,337
LCL=501,469
464136312621161161
4
3
2
1
0
Sample
Sa
mp
le R
an
ge
_R=1,294
UCL=4,228
LCL=0
1
1
1
1
1
1
11
1
1
1
Detergent powder packing machine
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232
847.
04.683.
ˆˆˆ
2
2
1
4
13
2
N
d
xe
sss
04.1ˆ2
d
Rd
s
1,919+1,04Process Analysis with control charts
(2) Process Improvement with control charts
Control charts Average-Range with N = 4
252321191715131197531
508
506
504
502
500
Sample
Sa
mp
le M
ea
n
__
X=503,903
UCL=505,462
LCL=502,344
252321191715131197531
4,8
3,6
2,4
1,2
0,0
Sample
Sa
mp
le R
an
ge
_
R=2,14
UCL=4,882
LCL=0
1
1
1
11
1
1
1
Detergent powder packing machine
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233
R control charts with N = 4
286.3ˆ
082.1ˆ
2
2
e
d
s
s
Process Analysis with control charts
(2) Process Improvement with control chartsThink Break
R control charts with N = 2
147.3ˆ
309.1ˆ
2
2
e
d
s
s
Grouping by lot and shift
(N = 2)
Grouping by lot
(N = 4)
Causes for
variation within
subgroups
Causes for
variation between
subgroups
• Measurement error
• Sampling error
• Other factors of variation
acting within subgroups
• Measurement error
• Sampling error
• Factor “change of shift”
• Factor “change of raw material”
• Measurement error
• Sampling error
• Factor “change of shift”
• Other factors of variation
acting within subgroups
• Measurement error
• Sampling error
• Factor “change of raw
material”
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234
%6.70309.1147.3
147.3
ˆ
ˆ2
2
total
e
s
s
70.6% of the variation is explained by the
factors operating between groups: “Change of
Shift”; “Change of Raw Material”;
N = 2
N = 4 %2.751.0823.286
3.286
ˆ
ˆ2
2
total
e
s
s
75.2% of the variation is explained by the
factors operating between groups: “Change of
Raw Material”;
Conclusion: A better process capability index does not depend on particular working
methods of any of the two shifts, but seems to be related with problems related to
raw materials (and different suppliers) used in detergent manufacturing.
Process Analysis with control charts
(2) Process Improvement with control charts
What is the relative importance of each factor?
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What is the relative importance of each factor?
Process Analysis with control charts
(2) Process Improvement with control charts
247.0090.2
039.1
ˆ
ˆ2
2
2
2
t
d
OverallStDev
WithinStDev
s
s
3011.0090.2
147.12
2
2
2
t
d
all)StDev(Over
in)StDev(With
s
s
0541.0
N=4 Between Groups: Raw Materials N=2 Between groups: Raw Materials and
Change of Shift (Turno)
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248
Shewhart Control Charts
Control charts are used in two diferent ways:
1) As a retrospective process analysis tool, with the aim of
identifying the presence (or absence) of special causes of
variation and provide their elimination;
2) As a monitoring and surveillance online process tool,
functioning as an early warning system for the occurrence of
uncontrollable situations.
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249
Guidelines for implementing control charts
Process Analysis.
Characterization of variation.
Process improvement.
Fixing the definitive control limits.
Phase 1(historical data, no standard given)
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250
Guidelines for implementing control charts
Determine which process characteristics to control and evaluation of the measurement process;
Identify possible assignable causes of variation and collect the data accordingly. Verify/ensure the independence of observations.
Calculate the trial control limits; confirm or reject the previous hypotheses and rank the assignable causes of variation (i.e. factors) according to the magnitude of its effects.
Further analize and eliminate the root causes of the relevant factors of variation (process improvement).
Confirm the improvement and calculate the control limits for future production.
Phase 1: initial setup
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251
Guidelines for implementing control charts
Before you can install a control chart on the line and expect
positive results, you must first understand process variation, how
to sample to quantify that variation and how to use SPC to guide the
reduction of variation.
Quoting Lynne Hare (Quality Progress, August 2001)
SPC is not a tool to steer the process; it is a means of gaining
process understanding to aid in process improvement.
Phase 1
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252
Guidelines for implementing control charts
Phase 2(stable process, standard given)
Control and process monitoring.
Periodic re-estimation of the
control limits.
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253
Guidelines for implementing control charts
Initially, use the control limits previously calculated in Phase 1.
Keep the rational subgroup approach to sampling, in order to
better identify assignable causes.
The sampling frequency should be set taking into account the
advantages of maximizing the chance of occurring assignable
causes of variation between samples and the need to obtain
independent observations within the samples.
Control limits should be recalculated periodically.
Phase 2 – process control and monitoring
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Please note that in Phase 2, even with reasonable
sized samples, the Shewhart control charts should
only be used in processes with a good or even really
good process capability (Cp = Cpk > 1,3).
Always remembre that there are better alternatives!
E
The CUSUM and EWMA control charts!
Guidelines for implementing control charts
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Unfortunately, in practice, what normally occurs is that:
Ignore its usefulness in Phase 1
Make a bad use of them in Phase 2
Conclusion
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Strategy for process improvement (ISO/FDIS 7870-2)
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Systems approach to the construction of variables control charts
(ISO/FDIS 7870-2)
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