Six Sigma Nitish Nagar
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Transcript of Six Sigma Nitish Nagar
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Module-7Control Charts
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Control Charts – Learning Objectives
At the end of this section, delegates will:
• Understand how control charts can show if a
process is stable
• Generate and interpret control charts for variable
and attribute data
• Understand the role of control charts within the
DMAIC improvement process
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Control Charts – Agenda1. Introduction to Statistical Process Control, SPC
2. Control Limits
3. Individual and Moving Range Chart
4. Workshop on Control Charts
5. Defective (Binomial) p-Chart
6. Defects (Poisson) u-Chart
7. Workshop on Attribute Control Charts
8. Uses of Control Charts
9. Summary
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What is Statistical Process Control?
• Statistical Process Control is a method of monitoring and detecting changes in processes.
• SPC uses an advanced form of Time Series plots.
• SPC provides an easy method of deciding if a process has changed (in other words, is the process “in-control”?).
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We Need Ways of Interpreting Data
• Everyday we are flooded by data and we are
forced to make decisions:
• Calls handled decreases by 4%
• UK trade deficit rises by £5 billion
• Company X’s earnings are $240Million less
than the previous quarter
• Should we take action ?
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Leave it alone -it ain’t broke
Pain &suffering
Pain &suffering
Lower “Customer”Requirement
Upper “Customer”Requirement
How do we manage data historically?
This Method
• Tells you where you are in relation to customer’s needs
• It will NOT tell you how you got there or what to do next
• Means that pressure to achieve customer requirements will cause you to:
• Actually Fix The Process
• Sabotage The Process
• Sabotage The Data (Integrity)
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What do Control Charts detect?
• Control Charts detect changes in a process.
• All processes change slightly, but process control
aims to detect ‘statistically significant’ changes that
are not just random variation.
• Processes can change in several different ways…
• the process average can change
• the process variation can change
• the process may contain one-off events
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Process Control
• Process control refers to the evaluation of process stability over time
• Process Capability refers to the evaluation of how well a process meets specifications
LSL USL
0 5 10 15 20 25
Time
UCL
LCL
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Why would a Process be Incapable?There are a number of reasons why a process may not be capable of meeting specification:
1. The specification is incorrect!
2. Excessive variation
3. The process is not on target
4. A combination of the above
5. Errors are being made
6. The process is not stable
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The specification is incorrect
• This issue was discussed during the Customer Focus section of this course
• If specifications are not clearly related to customer requirements, then it is always a good idea to challenge the specification before attempting to improve the process
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Excessive variationUpper
Specification
Limit
Lower
Specification
LimitTarget
• Excessive variation means that we have a variation reduction issue
• We will need to understand which process inputs are causing the variation in the process output
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The process is not on targetUpper
Specification
Limit
Lower
Specification
LimitTarget
• In this situation we have a process targeting issue
• We will need to understand which process inputs are causing the process to be off-target
• This situation is sometimes simple to solve!
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Excessive variation and not on targetUpper
Specification
Limit
Lower
Specification
LimitTarget
• In this situation we have both excessive variation and a process targeting issue
• We will need to understand which process inputs are causing the excessive variation and which are causing the process to be off-target
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Errors are being madeUpper
Specification
Limit
Lower
Specification
LimitTarget
• A situation such as this might indicate that errors are being made which result in occasional excursions outside of the specification
• This is often an indication of a mistake proofing issue
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The process is not stable
• A situation such as this is an indication that the process is unstable
• Whenever this situation is encountered in a DMAIC activity, then the reason(s) for the instability must be found and removed before assessing process capability
Upper
Specification
Limit
Lower
Specification
Limit Target
Last week
This week
Next week?
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The process is not stableUpper
Specification
Limit
Lower
Specification
LimitTarget
Last week
This week
Next
week?
• Causes of process instability are sometimes referred
to as “special causes”
• Removing these special causes may result in the
process becoming capable of consistently meeting the
target
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Unstable Process
“Special”causes of variation are present
Tim
e
TotalVariation
Target
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Stable Process
Tim
eTarget
TotalVariation
Only “Common”causes of variation are present
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Capable Process
Tim
e
Spec Limits
CAPABLE
NOT
CAPABLE Process is Stable but Process is not Capable
Process is Stable and Process is Capable
Management Action
(DMAIC) to reducecommon cause variation
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Control Charts test for Stability(Control Chart of Average)
0 5 10 15 20 25
Time
1.26
1.27
1.28
1.29
1.3
Pro
cess A
vera
ge
Upper Control Limit
Lower Control Limit
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Transactional Improvement Process
� Select Project
� Define Project
Objective
� Form the Team
� Map the Process
� Identify Customer Requirements
� Identify Priorities
� Update Project File
� Select Project
� Define Project
Objective
� Form the Team
� Map the Process
� Identify Customer Requirements
� Identify Priorities
� Update Project File
� Control Critical x ’s
� Monitor y’s
� Validate Control Plan
� Identify further opportunities
� Close Project
15 20 25 30 35
LSL USL
Phase Review
1 5 10 15 20
10.2
10.0
9.8
9.6
Upper Control Limit
Lower Control Limit
y
Phase Review
� Develop Detailed Process Maps
� Identify Critical Process Steps (x ’s) by looking for:
– Process Bottlenecks
– Rework / Repetition
– Non-value Added Steps
– Sources of Error / Mistake
� Map the Ideal Process
� Identify gaps between current and ideal
START
PROCESSSTEPS
DECISION
STOP
Phase Review
� Brainstorm Potential Improvement Strategies
� Select Improvement Strategy
� Plan and Implement Pilot
� Verify Improvement
� Implement Countermeasures
Criteria A B C D
Time + s - +
Cost + - + s
Service - + - +
Etc s s - +
15 20 25 30 35
LSL USL
Phase Review
Analyse Improve ControlMeasureDefine
Phase Review
Define
� Define Measures (y ’s)
� Check Data Integrity
� Determine Process Stability
� Determine Process Capability
� Set Targets for Measures
15 20 25 30 35
LSL USL
15 20 25 30 35
LSL USL
Phase Review
1 5 10 15 20
10.2
10.0
9.8
9.6
Upper Control Limit
Lower Control Limit
y
Phase Review
� Develop Detailed Process Maps
� Identify Critical Process Steps (x ’s) by looking for:
– Process Bottlenecks
– Rework / Repetition
– Non-value Added Steps
– Sources of Error / Mistake
� Map the Ideal Process
� Identify gaps between current and ideal
START
PROCESSSTEPS
DECISION
STOP
Phase Review
� Brainstorm Potential Improvement Strategies
� Select Improvement Strategy
� Plan and Implement Pilot
� Verify Improvement
� Implement Countermeasures
Criteria A B C D
Time + s - +
Cost + - + s
Service - + - +
Etc s s - +
15 20 25 30 35
LSL USL
Phase Review
Analyse Improve ControlMeasureDefine
Phase Review
Define
� Define Measures (y ’s)
� Check Data Integrity
� Determine Process Stability
� Determine Process Capability
� Set Targets for Measures
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Role of Control ChartsMeasure Phase:
• used during capability studies to assess process stability
Improve Phase:
• used to establish if the modified, improved process is stable
Control Phase
• used to control critical process input variables (x’s) in order to
reduce variability in process outputs (y’s)
• used to monitor process outputs (y’s) on an ongoing basis to
ensure that the process remains in control
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Control Charts
VariableData
NoSubgroups
Subgroupsn = 2-9
Subgroupsn > 9
Individuals &Moving Range
Chart
X Bar & RChart
X Bar & sChart
AttributeData
Defect Data(Poisson)
Defective Data(Binomial)
VaryingSubgroup
Size
ConstantSubgroup
Size
VaryingSubgroup
Size
ConstantSubgroup
Size
u Chart c Chart p Chart np Chart
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What do Control Charts tell us?
• Is the process stable? • Should we be taking action?• Are there any special causes?• What is the average process output?• What is the variability?
0 5 10 15 20 25
Subgroup
1.26
1.27
1.28
1.29
1.3
X-b
ar
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Control Limits
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Total Variation
TotalVariation
Within SubgroupVariation
Between SubgroupVariation
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Control Limits Use Within Subgroup Variation
• The total variation and Within Subgroup variation are the same only if the process is stable
• The Within Subgroup variation is an estimate of what the total variation would be if the process were stable
• The Within Subgroup variation is used to calculate the control limits since these limits represent the range of values expected for a stable process
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Controls Limits
• Controls limits are always:
Average ± 3 Standard DeviationsWhere the average and standard deviation are the
average and standard deviation of whatever data
is plotted:
99.7%99.7%99.7%99.7%
−−−−4σ4σ4σ4σ −−−−3σ3σ3σ3σ −−−−2σ2σ2σ2σ −−−−1σ1σ1σ1σ +1σ+1σ+1σ+1σ +2σ+2σ+2σ+2σ +3σ+3σ+3σ+3σ +4σ+4σ+4σ+4σ0
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Control LimitsUpper Control Limit
Lower Control Limit
• Control Limits are statistical boundaries which tell us whether or not the process is stable
• Based on the normal distribution, 99.7% of the points plot within the control limits if the process is stable
• The chance of a point outside the control limits, falsely indicating the process is unstable, is only 0.3% or 1 in 370
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Individuals Control Chart
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Individuals Control Chart• Used when only a single observation per time period
(subgroup):� Monthly reporting data:
• On-time shipments, In-process Inventory, Complaints, etc.
� Rare events
� Sales
� Stock Price
� Inventory Levels
� Customer Response Time
� Lost Time Accidents
� Complaints
� Anything that can be measured and varies
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Within Subgroup Variation• The best estimate is obtained by taking the differences between
consecutive samples i.e. the Moving Range (MR).
• We can use the the average MR, R, or the median MR, R
• When using R the Short Term standard deviation is estimated by:
1.128
R
d
Rσ
2
Within ==
˜
• When using R the standard deviation is estimated by:˜
0.954
R~
d
R~
σ
4
Within==
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Table of Constants for ImR chartsSample size d 2 d 3 d 4 D 3 D 4 D 5 D 6 E 2 E 5
2
3
4
56
7
8
9
10
0.853
0.888
0.880
0.8640.848
0.833
0.820
0.808
0.797
0.954
1.588
1.978
2.2572.472
2.645
2.791
2.915
3.024
3.267
2.574
2.282
2.1142.004
1.924
1.864
1.816
1.777
2.970
3.078
3.865
2.744
2.376
0
0
0
0 2.179
0.209
1.075
1.029
0.9921.809 0.975
0
0.055
0.119
0.168
2.054
1.967
1.901
1.850
1.128
1.693
2.059
2.3262.534
2.704
2.847
0
0
0
00
0.076
0.136
0.184
0.223
2.660
1.772
1.457
1.2901.184
1.109
1.054
1.010
3.145
1.889
1.517
1.3291.214
1.134
We would generally calculate the differences between consecutive samples, which corresponds to a “sample size” of 2 in this table.Minitab will calculate the control chart limits for us!
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Control Limits
The controls limits for the average, based on R are:
R2.66XREX1.128
R3X
d
R3X3σX 2
2
Within ±=±=±=±=±
The controls limits for the range, based on R are:
R3.267RDUCL
0R0RDLCL
4Range
3Range
==
=×==
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Control Limits
R~
3.145XR~
EX0.953
R~
3Xd
R~
3X3σX 5
4
Within ±=±=±=±=±
~The controls limits for the average, based on R are:
The controls limits for the range, based on R are:~
R~
3.865R~
DUCL
0R~
0R~
DLCL
6Range
5Range
==
=×==
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R Versus R• Some of the differences may be contaminated by
shifts in the mean (special Causes).
• R, the median MR, is more robust to this contamination so is generally preferred.
• When many of the differences are zero, it might be necessary to use R instead.
• A conversion factor can be developed:
182.1R~
954.0
128.1R~
R
954.0
R~
128.1
Rwithin
××××====××××====
====σσσσ====
~
¯ ˜
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Call Out TimeSample Number Call Out Time
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
5.35
3.28
1.07
1.06
4.29
3.23
5.40
6.42
3.25
8.55
4.26
7.48
5.35
2.14
4.24
6.44
3.21
9.66
4.28
5.33
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Call Out TimeSample Number Call out Time
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Difference
5.35
3.28
1.07
1.06
4.29
3.23
5.40
6.42
3.25
8.55
4.26
7.48
5.35
2.14
4.24
6.44
3.21
9.66
4.28
5.33
2.07
2.21
0.01
3.23
1.06
2.07
1.02
5.30
4.29
3.22
2.13
3.21
2.10
2.20
3.23
6.45
5.38
1.05
2.21
4.71 2.82
Average Average Difference
Ordered
0.01
1.02
1.05
1.06
2.07
2.07
2.10
2.13
2.20
2.21
2.21
3.21
3.22
3.23
3.23
4.29
5.30
5.38
6.45
Median
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Individuals chart - MinitabOpen Worksheet: Call Out Time
Stat>Control Charts>Variables Charts for Individuals>I-MR
Select Variable: Call Out Time
Click “I-MR” Options
Click “Estimate” – select “Median moving range”
Click “Tests” - select “1 point > 3 standard deviations from center line”
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IMR Chart – Minitab (using R)
Observation
Indiv
idual Valu
e
2018161412108642
12
8
4
0
_X=4.71
UC L=11.66
LC L=-2.24
Observation
Movin
g R
ange
2018161412108642
8
6
4
2
0
__MR=2.613
UC L=8.538
LC L=0
I-MR Chart of Call out Time
˜
613.2182.1R~
R
21.2R~
Median
====××××====
====
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Lognormal and other non –normal data
• When dealing with lognormal and other non-normal data
we need to be cautious.
• X-bar and Range charts will be acceptable for most non-
normal data with a sub-group size of 5 or larger.
• I-MR charts may give false indications of instability.
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Workshop – Individuals Control Chart
Open Minitab worksheet: PAYMENT TIMES.MTW
Use Minitab to create:
• Individuals and Moving Range chart
• Use the Median with a moving range of 2
• Assess the stability of the process
• What would you want to do next?
• Prepare a short report of your findings
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Attribute Control Charts
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Control Charts for Defective Items (Binomial)
• A p-Chart is used to track the proportion defective
• The p-Chart is constructed using data on the number of defectives from varying (or fixed) subgroup sizes
• The data opposite shows the number of defective orders from random samples taken over 10 working days
• The subgroups should be large enough to contain 5 or more defective items
Sample
Number
Defectives
(np)
Subgroup
Size
1 8 96
2 12 104
3 13 99
4 8 100
5 7 103
6 13 110
7 6 97
8 7 88
9 10 111
10 8 105
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Control Charts for Defective Items (Binomial)
• The p-chart must satisfy the requirements for the Binomial
Distribution. The particular requirements affecting the p-chart are:
1. Each unit (e.g., transaction, invoice, …) can only be classified
as pass or fail
2. If one unit (e.g., transaction, invoice, …) fails, then the chance
of the next unit failing is not affected
• If the Binomial distribution is not appropriate then it may be
possible to use the Individuals control chart already discussed
• Since we are charting defective items this chart should not be
used when the number of defectives is zero or there are a large
number of zeros (80-90%)
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P Chart Construction
Sample
Number
Defectives
(np)
Subgroup
Size (n)
Proportion
Defective
(p)
Average
Proportion
Defective
(pbar) 3 Sigma UCL(p) LCL(p)
1 8 96 0.083 0.091 0.088062 0.179062 0.002938
2 12 104 0.115 0.091 0.084608 0.175608 0.006392
3 13 99 0.131 0.091 0.086718 0.177718 0.004282
4 8 100 0.08 0.091 0.086283 0.177283 0.004717
5 7 103 0.068 0.091 0.085017 0.176017 0.005983
6 13 110 0.118 0.091 0.082268 0.173268 0.008732
7 6 97 0.062 0.091 0.087607 0.178607 0.003393
8 7 88 0.08 0.091 0.091978 0.182978 -0.00098
9 10 111 0.091 0.091 0.081896 0.172896 0.009104
10 8 105 0.076 0.091 0.084204 0.175204 0.006796
Total 92 1013
n
)p-(1p3-p3σpLCL,
n
)p-(1p3p3σpUCL
n
)p-(1pσ0.091,
1013
92
Σn
Σnpp
pp =−=+=+=
====
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P Chart - Minitab
Open Worksheet P Chart
Stat>Control Charts>Attributes Charts>P
Variable: Defectives
Subgroups in: “Subgroup Size” Click “P Chart – Options”
Click “Tests” – select “1 point > 3 standard deviations from center line”
Sample
Pro
port
ion
10987654321
0.20
0.15
0.10
0.05
0.00
_P=0.0908
UCL=0.1749
LCL=0.0067
P Chart of Defectives
Tests performed with unequal sample sizes
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48
Control Charts for Defects (Poisson)
• A u Chart is used to track the number of defects per unit (e.g., transaction, invoice, …).
• The u chart is constructed using data on the number of defects from varying subgroup sizes (number of units).
• The data opposite shows the number of defects in the given number of invoices sampled randomly from 10 weeks of invoicing.
Sample
Number
Defects
c
Invoices
n
1 7 40
2 4 45
3 8 33
4 5 40
5 3 39
6 8 46
7 5 27
8 7 45
9 9 38
10 4 39
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49
Control Charts for Defects (Poisson)
• Since the u-chart is based on the Poisson Distribution, the data should be tested to see if it fits the Poisson distribution (e.g., some “count data”such as complaints and late shipments may not fit the Poisson distribution)
• If the Poisson distribution does not fit then it may be possible to use the Individuals control chart already discussed
• Since we are charting defects, this chart should not be used when the number of defects is zero or there are a large number of zeros (80-90%)
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U Chart - ConstructionSample
Number
Defects
c
Invoices
n
DPU
u u bar LCL(u) UCL(u)
1 7 40 0.175 0.153 0 0.34
2 4 45 0.089 0.153 0 0.328
3 8 33 0.242 0.153 0 0.307
4 5 40 0.125 0.153 0 0.339
5 3 39 0.077 0.153 0 0.341
6 8 46 0.174 0.153 0 0.326
7 5 27 0.185 0.153 0 0.378
8 7 45 0.156 0.153 0 0.327
9 9 38 0.237 0.153 0 0.343
10 4 39 0.103 0.153 0 0.341
Total 60 392 0.153
n
u3u3uLCL,
n
u3u3uUCL
uσ 0.153,392
60
Σn
Σcu
uu −−−−====−−−−====++++====++++====
================
σσσσσσσσ
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U Chart - Minitab
Open Worksheet: U Chart
Stat>Control Charts>Attributes Charts>U
Variable: Defects Subgroups in: “Units”
Click “U Chart Options”
Click “Tests” – select “1 point > 3 standard deviations from center line”
Sample
Sample C
ount Per Unit
10987654321
0.4
0.3
0.2
0.1
0.0
_U=0.1531
UCL=0.3410
LCL=0
U Chart of Defects
Tests performed with unequal sample sizes
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Workshop - Attributes Control Chart
• Using the packets of sweets provided (assume that each packet has been taken from a different batch of production over the last few days):
� Randomly select 20 sweets from each packet
� Inspect the sweets for two types of defect-
• Badly mis-shaped/damaged sweet
• Missing or poorly printed logo
• Using Minitab, assess the stability of the process
• Prepare a short report of your findings
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Uses of Control Charts
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Control Charts
Variable
Data
No
Subgroups
Subgroups
n = 2-9
Subgroups
n > 9
Individuals &
Moving Range
Chart
X Bar & R
Chart
X Bar & s
Chart
Attribute
Data
Defect Data
(Poisson)
Defective Data
(Binomial)
Varying
Subgroup
Size
Constant
Subgroup
Size
Varying
Subgroup
Size
Constant
Subgroup
Size
u Chart c Chart p Chart np Chart
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Improvement
• Control charts are one of many variation reduction
tools
• Controls charts detect change of the output variable
(y)
• The output changes because a critical input variable
(x) has changed
• Control charts provide clues that can help to identify
these critical inputs (x’s)
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Clues to Discovering Critical x’s
• When did the change occur?
• What patterns are emerging?
� Shifts
• Gradual or Sudden?
� Trends
• Increasing or Decreasing?
� Unusual patterns or cycles?
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Identification of Critical x’s
• To determine the critical x, i.e., the input causing
the shift, we need to consider:
� Delayed detection
� Multiple inputs causing shifts
� Lack of information on inputs
• We can also use screening experiments, scatter
diagrams, … to determine critical x’s
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Transmission of Variation, y = f(x)
• Control charts can help to discover critical x’s that
are causing the process to shift
• Tighter control of these critical x’s will make the
process more stable
OUTPUT
INPUT
Relationship BetweenInput and Output
Variation of Input
TransmittedVariation
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Transactional Improvement Process
� Select Project
� Define Project
Objective
� Form the Team
� Map the Process
� Identify Customer Requirements
� Identify Priorities
� Update Project File
� Select Project
� Define Project
Objective
� Form the Team
� Map the Process
� Identify Customer Requirements
� Identify Priorities
� Update Project File
� Control Critical x ’s
� Monitor y’s
� Validate Control Plan
� Identify further opportunities
� Close Project
15 20 25 30 35
LSL USL
Phase Review
1 5 10 15 20
10.2
10.0
9.8
9.6
Upper Control Limit
Lower Control Limit
y
Phase Review
� Develop Detailed Process Maps
� Identify Critical Process Steps (x ’s) by looking for:
– Process Bottlenecks
– Rework / Repetition
– Non-value Added Steps
– Sources of Error / Mistake
� Map the Ideal Process
� Identify gaps between current and ideal
START
PROCESSSTEPS
DECISION
STOP
Phase Review
� Brainstorm Potential Improvement Strategies
� Select Improvement Strategy
� Plan and Implement Pilot
� Verify Improvement
� Implement Countermeasures
Criteria A B C D
Time + s - +
Cost + - + s
Service - + - +
Etc s s - +
15 20 25 30 35
LSL USL
Phase Review
Analyse Improve ControlMeasureDefine
Phase Review
Define
� Define Measures (y ’s)
� Check Data Integrity
� Determine Process Stability
� Determine Process Capability
� Set Targets for Measures
15 20 25 30 35
LSL USL
15 20 25 30 35
LSL USL
Phase Review
1 5 10 15 20
10.2
10.0
9.8
9.6
Upper Control Limit
Lower Control Limit
y
Phase Review
� Develop Detailed Process Maps
� Identify Critical Process Steps (x ’s) by looking for:
– Process Bottlenecks
– Rework / Repetition
– Non-value Added Steps
– Sources of Error / Mistake
� Map the Ideal Process
� Identify gaps between current and ideal
START
PROCESSSTEPS
DECISION
STOP
Phase Review
� Brainstorm Potential Improvement Strategies
� Select Improvement Strategy
� Plan and Implement Pilot
� Verify Improvement
� Implement Countermeasures
Criteria A B C D
Time + s - +
Cost + - + s
Service - + - +
Etc s s - +
15 20 25 30 35
LSL USL
Phase Review
Analyse Improve ControlMeasureDefine
Phase Review
Define
� Define Measures (y ’s)
� Check Data Integrity
� Determine Process Stability
� Determine Process Capability
� Set Targets for Measures
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Control Charts - Summary
• Charts can be constructed for variable or attribute data
• I-MR Charts should always be considered for attribute data
• Control Charts are used during capability studies to
determine process stability
• Real-Time control charts are used to detect shifts so that
causes of shifts can be identified and eliminated
• Should be used to control critical x’s (process input
variables) in order to reduce variability in process outputs
(y’s).
• Used to monitor y’s on an ongoing basis to ensure that the
process remains in control.