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# PRODUCTIONS/OPERATIONS MANAGEMENT

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```Chapter 6 Part 3
X-bar and R Control Charts
Attribute Data
Data that is discrete
п‚· Discrete data is based on вЂњcounts.вЂќ
п‚· Assumes integer values
п‚·
пЃ‡ Number
of defective units
пЃ‡ Number of customers who are вЂњvery satisfiedвЂќ
пЃ‡ Number of defects
Variables Data
X-bar and R chart is used to monitor mean
and variance of a process when quality
characteristic is continuous.
п‚· Continuous values (variables data) can
theoretically assume an infinite number of
values in some interval.
п‚·
Time
п‚· Weight
п‚· Ounces
п‚· Diameter
п‚·
X-bar and R Chart
X-bar chart monitors the process mean by
using the means of small samples taken
frequently
п‚· R chart monitors the process variation by
using the sample ranges as the measure of
variability
п‚·
п‚·
Range = Maximum value вЂ“ Minimum value
Notation
X пЂЅ quality
characteri stic
n пЂЅ sample size
k пЂЅ number
of samples
X пЂЅ sample mean ( X - bar)
пЂЅ
пѓҐ
n
X
Notation
X пЂЅ sample mean of X ( X - bar)
X пЂЅ mean of the X s
X is also called
" X double bar, "
the estimated
the overall
process
mean
mean, or
Notation
R пЂЅ Sample Range
пЂЅ Maximum
value - Minimum
R пЂЅ Mean of ranges
value
R
пѓҐ
пЂЅ
k
пЃіЛ† пЂЅ estimated
пЃіЛ†
n
пЂЅ estimated
standard
deviation
of X
standard error of X
Example of Notation
A company monitors the time (in minutes) it
takes to assemble a product.
п‚· The company decides to sample 3 units of the
product at three different times tomorrow:
п‚·
9 AM
п‚· 12 Noon
п‚· 3 PM
п‚·
What is the sample size, n?
п‚· What is k, the number of samples?
п‚·
п‚·
Suppose the following data are obtained.
Hour
п‚·
Assembly Time
(minutes)
X1
X2
X3
9:00 AM
5
12
4
12 Noon
3:00 PM
6
9
8
4
10
2
How would you compute
п‚· X-bar?
п‚· R
п‚· R-bar
п‚· X double bar
Example of Notation
Hour
Assembly Time
(minutes)
X1
X2
X3
Sample
Mean, X
Sample
Range, R
9:00 AM
5
12
4
21/3 = 7
12 вЂ“ 4 = 8
12 Noon
6
8
10
24/3 = 8
10 вЂ“ 6 = 4
3:00 PM
9
4
2
15/3 = 5
9вЂ“2=7
X = 20/3 = 6.7 R =19/3 =6.3
Sample Means
X пЂЅ
пѓҐ
X
n
First Sample
X пЂЅ
пѓҐ
n
X
пЂЅ
5 пЂ« 12 пЂ« 4
3
пЂЅ
21
3
пЂЅ 7
Sample Means
Second Sample
X пЂЅ
пѓҐX
пЂЅ
n
6 пЂ« 8 пЂ« 10
пЂЅ
3
24
пЂЅ8
3
Third Sample
X пЂЅ
пѓҐX
n
пЂЅ
9пЂ«4пЂ«2
3
пЂЅ
15
3
пЂЅ5
Estimated Process Mean
X пЂЅ
пѓҐ
X
k
пЂЅ
7пЂ«8пЂ«5
3
пЂЅ
20
3
пЂЅ 6 .7
Sample Ranges
R пЂЅ Maximum
First Sample :
R пЂЅ 12 пЂ­ 4 пЂЅ 8
Second Sample :
R пЂЅ 10 - 6 пЂЅ 4
Third Sample :
R пЂЅ 9-2 пЂЅ 7
Value - Minimum
Value
Mean of R
R пЂЅ
пѓҐR
k
пЂЅ
8пЂ«4пЂ«7
3
пЂЅ
19
3
пЂЅ 6 .3
Underlying Distributions
п‚·
When constructing an X-bar chart, we
actually have two distributions to consider:
п‚·
The distribution of the sample means X , and
п‚·
The process distribution, the distribution of the
quality characteristic itself, X.
п‚·
The distribution of
averages.
п‚·
The distribution of X is a distribution of ???
X
is a distribution of
Underlying Distributions
п‚·
These distributions have the same mean
Mean of X пЂЅ Mean of X
Their variances (or standard deviations) are
different.
п‚· Which distribution has the bigger variance?
п‚·
п‚·
п‚·
Would you expect more variability among
averages or among individual values?
The variability among the individual values is
???
Underlying Distributions
п‚·
The standard deviation among the sample
means is smaller by a factor of 1
n
п‚·
Therefore,
Std of X пЂЅ пЃі
пѓ¦ 1 пѓ¶
пЃіЛ†
Std of X пЂЅ пЃіЛ† пѓ§пѓ§
пѓ·пѓ· пЂЅ
n
пѓЁ n пѓё
Underlying Distributions
Sampling
distribution
of X
Distribution
of X
X
Distribution of X
пЃіЛ†
M
M
m
LCL
X
M UCL
M
M
M X
m
m
The distribution of X is assumed to be normal.
This assumption needs to be tested in practice.
Distribution of X-bar
пЃіЛ†
n
M
LCL
M
X
m
M UCL
M
M
M X
m
m
If the distribution of X is normal, the distribution of
X-bar will be normal for any sample size.
Control Limits for X-bar Chart
п‚·
Since we are plotting sample means on the Xbar chart, the control limits are based on the
distribution of the sample means.
п‚·
The control limits are therefore
UCL пЂЅ X пЂ« 3
пЃіЛ†
n
LCL пЂЅ X пЂ­ 3
пЃіЛ†
n
Control Limits for X-bar Chart
Distribution of X
LCL
X пЂ­3
UCL
пЃіЛ†
n
X
X пЂ«3
пЃіЛ†
n
Control Limits for X-bar Chart
LCL пЂЅ X пЂ­ 3
пЃіЛ†
n
UCL пЂЅ X пЂ« 3
пЃіЛ†
n
LCL пЂЅ X пЂ­ A 2 R
UCL пЂЅ X пЂ« A 2 R
Control Limits for X-bar Chart
3
пЃіЛ†
n
пЂЅ A2 R
A2 is a factor that depends on the n, the sample size,
and will be given in a table.
Example of X-bar Chart
п‚·
п‚·
п‚·
п‚·
п‚·
A company that makes soft drinks wants to
monitor the sugar content of its drinks.
The sugar content (X) is normally
distributed, but the means and variances
are unknown.
The target sugar level for one of its drinks
is 15 grams.
The lower spec limit is 10 grams.
The upper spec limit is 20 grams.
Example of X-bar Chart
The company wants to know how much sugar
on average is being put into this soft drink and
how much variability there is in the sugar
content in each bottle.
п‚· The company also wants to know if the mean
sugar content is on target.
п‚· Lastly, the company wants to know the
percentage of drinks that are too sweet and
the percentage that are not sweet enough.
(Next section)
п‚·
Example of X-bar Chart
п‚·
To obtain this information, the company
decides to sample 3 bottles of the soft drink at
3 different time each day:
10 A.M,
п‚· 1:00 P.M. and
п‚· 4:00 P.M.
п‚·
The company will use this data to construct an
X-bar and R chart. (In practice, you need 2530 samples to construct the control limits.)
п‚· For the past two days, the following data were
collected:
п‚·
Example of X-bar Chart
Day
Hour
X1
X2
X3
1
10 am
17
13
6
2
1 pm
15
12
24
4 pm
12
21
15
10 am
13
12
17
1 pm
18
21
15
4 pm
10
18
17
What is n?
What is the k?
What is the next
step?
Example of X-bar Chart
Day Hour
1
10 am
2
X1
17
X2
13
X
X3
6 36/3 =12
R
11
1 pm
15
12
24 51/3 =17
12
4 pm
12
21
15 48/3 =16
9
10 am
13
12
17 42/3 =14
5
1 pm
18
21
15 54/3 =18
6
4 pm
10
18
17 45/3 =15
8
X = 92/6 R = 51/6
= 15.33
= 8.5
X-bar Chart Control Limits
LCL пЂЅ X пЂ­ A 2 R
UCL пЂЅ X пЂ« A 2 R
Table A: X-bar Chart Factor, A2
n
A2
2
1.88
3
1.02
4
0.73
5
0.58
X-bar Chart Control Limits
X пЂЅ 15 . 33
R пЂЅ 8 .5
From Table A in notes or Table 6 - 1, p. 182 of text
n пЂЅ3
A 2 пЂЅ 1 . 02
X-bar Chart Control Limits
LCL пЂЅ X пЂ­ A 2 R
пЂЅ 15 . 33 пЂ­ 1 . 02 ( 8 . 5 )
пЂЅ 6 . 66
UCL пЂЅ X пЂ« A 2 R
пЂЅ 15 . 33 пЂ« 1 . 02 ( 8 . 5 )
пЂЅ 24 . 0
X -b a r C h a rt fo r S u g a r C o n te n t
30.00
25.00
20.00
15.00
10.00
5.00
0.00
10
1
4
10
1
H our
H our
1
2
D ay
4
Interpretation of X-bar Chart
п‚·
п‚·
п‚·
The X-bar chart is in control because ????
This means that the only source of
variation among the sample mean is due
to random causes.
The process mean is therefore stable and
predictable and, consequently, we can
estimate it.
Interpretation of X-bar Chart
п‚·
п‚·
Our best estimate of the mean is the
center line on the control chart, which is
the overall mean (X-double bar) of 15.33
grams.
If the process remains in control, the
company can predict that all bottles of this
soft drink produced in the future will have
a sugar content of, on average, 15.33
grams.
Interpretation of X-bar Chart
This prediction, however, indicates that there
is a problem with the location of the mean.
п‚· The process mean is off target by 0.33 grams
(15.33 -15.00).
п‚· The process mean, although stable and
predictable, is at the wrong level and should
be corrected to the target.
п‚·
Interpretation of X-bar Chart
Since the process mean is in control, there
are no special causes of variation that may be
responsible for the mean being off target.
п‚· Since the operators are responsible for
correcting problems due to special causes
and management is responsible for correcting
problems due to random causes of variation,
management action is required to fix this
problem.
п‚·
Interpretation of X-bar Chart
The reason is that, because the process is in
control, the filling machines require more than
a simple adjustments (typically due to special
causes) which can be made by the operators.
п‚· The machines may require
п‚· new parts,
п‚· a complete overhaul, or
п‚· they may simply not be capable of operating
on target, in which case a new machine is
required.
п‚·
Interpretation of X-bar Chart
Expecting the operators to adjust the mean to
the target when the process is in control is
analogous to requiring that you produce zero
to toss a fair coin 100 times each day.
п‚· Why?
п‚·
R Chart
Monitors the process variability (the variability
of X)
п‚· Tells us when the process variability has
changed or is about to change.
п‚· R chart must be in control before we can use
the X-bar chart.
п‚·
R Chart
п‚·
Rules for detecting changes in variance:
If at least one sample range falls above the upper
control limit, or there is an upward trend within the
control limits, process variability has increased.
п‚· If at least one sample range falls on or below the
lower control limit, or there is a downward trend
within the control limits, process variability has
decreased.
п‚·
R Chart Control Limits
LCL пЂЅ D 3 R
UCL пЂЅ D 4 R
Table B: Factors for R Chart
n
D3
D4
2
3
4
5
0
0
0
0
3.27
2.57
2.28
2.11
R Chart Control Limits
n пЂЅ3
D3 пЂЅ 0
D 4 пЂЅ 2 . 57
LCL пЂЅ 0 ( 8 . 5 )
пЂЅ 0
UCL пЂЅ 2 . 57 ( 8 . 5 )
пЂЅ 21 . 85
R C h a rt fo r W e ig h t
25
20
R
15
LCL
10
UCL
R -b a r
5
0
10
1
4
10
1
Hour
Hour
1
2
Day
4
Interpretation of R Chart
Since all of the sample ranges fall within the
control limits, the R chart is in control.
п‚· The standard deviation is stable and
predictable and can be estimatedвЂ”done in
next section.
п‚· This does not necessarily mean that the
amount of variation in the process is
acceptable.
п‚·
Interpretation of R Chart
Continuous improvement means the company
should continuously reduce the variance.
п‚· Since the process variation is in control,
management action is required to reduce the
variation.
п‚·
Expected Pattern in a Stable Process
X-bar Chart
UCL
LCL
Time
Expected pattern is a normal distribution
How Non-Random Patterns Show Up
Sampling
Distribution
(process variability is increasing)
UCL
x-Chart
LCL
Does not
reveal increase
UCL
R-chart
Reveals increase
LCL
How Non-Random Patterns Show Up
(process mean is
shifting upward)
Sampling
Distribution
UCL
Detects shift
x-Chart
LCL
UCL
R-chart
LCL
Does not
detect shift
Is a Stable Process a Good Process?
п‚·
вЂњIn controlвЂќ indicates that the process mean is
stable and hence predictable.
п‚·
A stable process, however, is not necessary a
вЂњgoodвЂќ (defect free) process.
The process mean, although stable, may be
far off target, resulting in the production of
defective product.
п‚· In this case, we have, as Deming puts it, вЂњA
stable process for the production of defective
product.вЂќ
п‚·
Control Limits vs. Spec. Limits
п‚·
Control limits apply to sample means, not
individual values.
п‚·
п‚·
Mean diameter of sample of 5 parts, X-bar
Spec limits apply to individual values
п‚·
Diameter of an individual part, X
Control Limits vs. Spec. Limits
Sampling
distribution,
X-bar
Process
distribution,
X
Mean=
Target
LSL
Lower
control
limit
Upper
control
limit
USL
Underlying Distributions
Sampling
distribution
X
of X
Distribution
of X
LSL
USL
X
LCL
UCL
Control limits are put on distribution of X-bar
Spec limits apply to the distribution of X
Responsibility for Corrective Action
Special Causes
(Process out of
control)
Operators
(workers)
Random Variation
(Process in Control)
Management
Benefits of Control Charts
п‚·
Control charts prevent unnecessary
п‚· If process is in control, do not adjust it.
п‚· Adjustments will increase the variance.
п‚· Management action is required to improve
process.
special causes occur.
Benefits of Control Charts
Control charts assign responsibility for
corrective action.
п‚· Control charts are the only statistical valid way
to estimate the mean and variance of a
process or product.
п‚· Control charts make it possible to predict
future performance of a process and thereby
take early corrective action.
п‚·
```
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