TQM control charts for variables and attributes

control chart in process capability

1. production process.
With the help of a control chart, one can find out the natural capcbility of a
production troubles.
Control chart makes possible the diagnosis and correction of the many
statically or not.
The control chart is used to check whether the process is controlled
in a process.
V It is a tool to distinguish between chance and assignable causes of variations
time and the
variations of this data.
A con trol chart is a graph that displays data taken ove
G12.9.1. What is it?
12.9. CONTROL CHART

2. Control Charts & Process Copability
/ The general form of the control chart is illustrated in Fig.12.8.
Quality characteristic value
The control chart is used to evaluate process stability and to decide when to
adjust the process.
+36
4 10
Sample values
12 14
Sample number, S
G12.9.2. Types of Control Charts
16
The two basic types of control charts used are:
Fig. 12.8. General form of control chart
1. Control charts for variables, and
2. Control charts for attributes.
18
Out-of-control
20
G12.9.3. Analysis of Patterns in Control Charts
UCL
12.25
Centre line
LCL
Control charts for variables require a measurement of the quality
characteristic of interest.
Control charts for attributes require a determination of whether a part is
defective or how many defects are there in the sample.
The main objective of using control charts is to determine when a process is
Out-of-control so that necessary actions may be taken.
V Criteria other than a plotted point falling outside the control limits are also
used to determine whether a process is out-of-control.
Plot patterns often indicate whether the process is in control or not.
Analyzing plot patterns is more difficult than plotting the control charts.

3. 12.28
12.10. CONTROL CHARTS FOR VARIABLES
The quality characteristics which can be measured and expressed in specific
units of measurements are called variables.
Control charts based upon measurements of quality characteristics are called
as control charts for variables.
Types of variable control charts: The most commonly used variable control
charts are:
() X-or average-charts,
() R-or range-charts, and
(üi) s-or standard deviation-charts.
V The X-chart is used to monitor the centering of the process to control its
accuracy.
The R-chart monitors the dispersion or precision of the process.
V The s-chart shows the variation of the process.
12.10.1. Construction of X-and R-Charts
Total Quality Management
Step 1:Select the characteristics for applying a control chart.
Step 2: Select the appropriate type of control chart.
Step 3: Collect the data.
Step 4: Choose the rational sub-group i.e., sample.
Step 5: Calculate the average (X) and range (R) for each sample.
For example, if a sub-group contains 5 items whose dimensions (say diameter length or weight or etc.) are x, x2, X3, X and x5, then
Sub-group average, X
5
and subgroup range, R
Maximum value -
Minimum value
t selection of sample size:
Subgroups (i.e., sample size) must be large enougn
to
detect
points or patterns
indicating a lack of control when a lack of
control exists. The
larger
the
sample size, the better the
chances of
detecting the
pattern.

4. 12.29
Control
Charts
&
Process
Capability
5tep
6:
Calculate
the
average
of
the
ayerages
(}
)
and
average
of
range
(R).
Let
N
=
Number
of
sub-groups
Then,
average
of
averages
(or
grand
average)
is
given
by
2X
N
Similarly
average
of
range
(R)
is
given
by
2R
N
R
Step
7:
Calculate
the
control
limits
for
X
and
R-charts.
(a)
Control
limits
of
X-chart
Control
limit
or
centre
line,
CLy
:
X
Upper
control
limit,
UCLy
=
X
+
A
R
Lower
control
limit,
LCL
=
X
-
A,
R
and
Where
A
=
Factor
or
constant
for
X-chart,
taken
from
the
Table
A.2.
(see Apperndix).
(b)
Control
limits
for
R-chart
R
Control
limit
or
centre
line,
CLp
=
D4 R
Upper
control
limit,
UCLR
:
Lower
control
limit,
LCRR
and
where
Da
and
D4
are
statistical
factors,
taken
from
Table
A,2.
(see Appendix)
Step
8:
Plot
CL,
UCL
and
LCL
on
the
chart.
Step.9:
Plot
individual
X
and
R
values
on
the
chart.
Step
10:
Check
whether
the
process
is
in
control
or
not.
Step
11:
Revise
the
control
limits
if
the
points
are
out-of-control,
by
removing
the out-of-control points.
=

5. 3
6.34
6.40
6.34
6.36
2
6.46
6.37
6.36
6.41
1
6.35
6.40
6.32
6.37
Sample Number
X1
X3
Measurements
now
under
statistical
control.
revised
mean
and
range
charts
and
plot
the
values.
(vi)
State
whether
the
process
is
which
are
rectified
now.
Revise
the
central
line
and
control
limits.
()
Draw
the
control.
(iv)
If
not,
assume
that
the
deviation
occurred
due
to
assignable
causes
range
charts
and
plot
the
values.
(iii)
State
whether
the
process
is
under
statistical
()
Determine
the
trial
central
line
and
control
limits.
ii)
Draw
the
mean
and
[A.U, Nov/Dec 2O03]
process. molding
process.
Mean
and
range
charts
are
required
to
be
established
for this
Example 12.9
plastic
component
in
grams.
This
component
is
manufactured
using
a
plastic
injection
Following
table
contains
the
data
on
the
weight
ofa

6. and
R=
N
10
XR
1,11
: 0,111
N EX 64.22
= 6.422
EX= 64.22
10
6.38
6.40
SR = 1,11
6.45
6.37
6.40
9
6.56
6.55
0.08
6.45
6.48
6.51
8
6.35
0.11
6.41
6.37
6.38
6.37
7
6.38
0.06
6.44
6.28
6.58
6.42
6.41
0.30
6.40
6.29
6.34
6.36
5
6.38
0.12
6.34
6.44
6.40
6.39
0.10
4
6.69
6.64
6.68
6.59
6.65
0.10
3
6.34
6.40
6.34
6.36
6.36
0.06
2
6.46
6.37
6.36
6.41
6.40
0.10
6.35
6.40
6.32 6.37
6.36
0.08
Number
X1
X3
4
Sample
X1
+
X2
+
X3
+
X4
Xmin
Measurements
X
R = Xmax
are
tabulated
as
follows:
0Solution:
In
order
to
calculate
X
and
R,
the
results
of
the
given
observations
To
find:
Centre
line
and
control
limits
for
X-
and
R-charts
Given
Data:
Subgroup
size,
n
=
4;
N=
10
10
6.38
6.40
6.45
6.37
6.56
6.55
6.45
6.48
6.35
6.41
6.37
6.38
7
6.38
6.44
6.28
6.58
6
6.41
6.40
6.29
6.34
5
6.38
6.34
6.44
6.40
4
6.69
6.64
6.68
6.59
Control
Charts
&
Process
Capability
12.31|
=

7. Fig. 12.14. X-chart
Subgroup No.
1
2
10
4
5
6
7
625 6.30 6
LCLZ = 6341
6.40 -
CL = 6.422
6.45 .
6
UCL = 6.503
6.55 6.60 6.65 670 -
and 12.15 respectively.
Using
the
calculated
control
limits,
the
X-
and
R-charts
are
shown
in
Figs.12.14
(i)
To
draw
the
and
R-charts:
and
Lower
control
limit,
LCLe
:
Da.
R
:
0
(0.111)
=
0
Ans.
Upper
control
limit,
UCL
=
D4. R
:0.253
Ans.
= 2.282 (0.111)
Control
limit
or
Central
line,
CL
:
R
=0.111
Ans.
o
For R-chart:
:
6.341
Ans.
D
and
Lower
control
limit,
Lcl
=
X-
Az
R=
6.422
-0.729
(0.111)
Upper
control
limit,
UCL,
:6.503 Ans :
X+
Ag
R=
6.422
+0.729
(0.111)
X
Control
limit
or
Central
line,
CL-
=
X=
6.422
Ans.
For X-chart: ()
To
determine
trail
centre
line
and
control
limits:
For
a
subgroup
size,
n
=
4,
Table
A.2
gives
the
following
statistical
factors:
Ag
=
0.729:
Da
=
0:
and
D
=
2.282
12,32
Total Quality Management
35
50

8. Appendix
Observations
in Sample, n
3
4
5
6
7
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
A2. FACTORS FOR COMPUTING CENTRE LINE AND THREE-SIGMA CONTROL LIMITS
25
Factors for Control Limits
A
2.121 1.880
1.732
1.500
1.342
1.134
1.061
1.000
1.225 0.483
0.949
0.905
0.866
0.832
0.802
0.775
0.750
0.728
0.707
0.688
0.671
0.655
0.640
0.626
K-charts
0.612
0.600
For n> 25,
1.023
0.729
0.577
0.419
0.373
0.337
0.308
0.285
0.249
0.235
0.223
0.212
0.203
0.194
0.187
0.180
0.173
0.167
0.162
0.157
0.153
Source: O ASTM.
A :
A3
2.659
1.954
1.628
0.266 0.886 0.9776
0.850 0.9794
0.9810
.182
1.099
1.032
1.287 0.9515
0.975
0.927
1.427 0.9400 1.0638
0.817
0.789
0.763
0.739
0.718
0.698
0.680
0.663
0.647
0.633
0.619
0.606
B4 = C4
Factors for
Center Line
1/lc4 CA
0.7979
3
0.8862
0.9213
0.9594
0.9650
0.9693
0.9727
0.9754
olollL
0.9845
0.9854
0.9862
0.9869
0.9876
0.9882
0.9887
0.9892
0.9896
1.2533
3
1.1284
V2(n-1)
1.0854
0.9823 1.0180
0.9835
1.0510
1.0423
1.0363
1.0317
1.0281
1.0252
1.0229
1.0210
1.0194
1.0148
1.0140
1.0133
1.0126
1.0119
1.0114
1.0109
1.0105
3
S-Charts
c4 Vn
Factors for Control Limits
0.428
1.0168 0.448
and
B3
1.0157 0.466
0.030
0.118
0.185
0.239
0.284
0.321
0.354
0.382
0,406
0.482
0.497
0.510
0.523
0.534
0.545
0.555
0.565
B4
3.267
2.568
2.266
2.089
1.970
1.882
1.815
1.761
1.716
1.646
1.618
1.594
1.572
1.679 0.313
1.552
1.534
1.518
1.503
1.490
1.477
1.466
1.455
1,445
1.435
Bs
0.029
0.113
0.179
B, = C4
0.232
0.276
0.346
0.374
0.399
0.421
0.440
0.458
0,475
0.490
0.504
0.516
0.528
0.539
0.549
4 (n-1)
0.559
3
B6 1/dg dz
2,606 1.128 0.8865 0.853
2.276 1.693 0.5907 0.888
2.088 2.059 0.4857 0.880
1.964 2.326 0.4299 0.864
1.874 2.534 0.3946 0.848
1.806 2.704 9-3698 0.833
1.751
1.707
1.669
1.637
1.610
1.585
1.563
1.526
1.511
1.496
1.483
3.407
1.544 3.472
1.470
1.459
1.448
1.438
1.429.
C4 4 n-3 B 1-
Factors for
Center Line
1.420
2.970
3.078
3.173
3.258
3.532
3.588
3.640
0.3152 0.787
0.3069 0.778
3.336 0.2998 0.770
0.2935 0.763
0.2880
3.689
3.735
3.778
3.819
3.858
3.895
3.931
0.3512
3
0.3367
0.3249 0.797
0.2831
0.2787
0.2747
0.2711
0.2677
0.2647
02618
02592
02567
0.820
02544
0.808
c4V2 (n-1)
0.756
0.750
0.744
0.739
0.734
0.729
0.724
0.720
0.716
0.712
0.708
R-Charts
Factors for Control Limits
DË
0.204
0.388
0.547
0.687
0.811
0.922
1.025
1.118
1.203
1.282
1.356
1.424
1.487
1.549
1.605
1.659
1.710
1.759
1.806
B = 1+
D2
3.686
4.358
4.698
4.918
5.078
5.204
5.306
5.393
5.469
5.535
5.594
S.647
5.696
S.741
S.782
5.820
5856
5.891
5.921
S.951
S.979
6.006
6.031
6.056
3
D3
0
0.076
0.136
0.184
0.223
0.256
0.283
0.307
0.328
0.347
0.363
0.378
0.391
0.403
0.415
0.425
0.434
0.443
0.451
0.459
C4 V 2 (n -1)
D4
3.267
2.574
2.284
2.114
2.004
1.924
1.864
1.816
1.777
1.744
1.717
1.693
1.672
1.653
1.637
1.622
1.608
1.597
1.583
1.575
1.566
A.3
1.557
1.548
1.541
2.847

9. and
Lower
control
limit,
LCLe
3
D3.R
=0
(0.073)
:
0
Ans.
Upper
: 0.166 Ans.
control limit, UCLR
D4.
R=
2.282
(0.783)
Revised
R =0.073 Ans,
and
Revised :
N-3
2X-
(6.65
+6.42
+
6.51)
10-3
64.22 -19.58
6.377
Revised R =
N-3
ER
-
(0.10
+
0.30
+
0.11)
With
samples
4,
7
and
9
deleted,
we
get
10-3
1,11 -0.51
= 0.073
removing
the
out-of-control
points.
ine
and
control
limits
can
be
revised.
Revised
control
limits
can
be
determined
by
identified
and
they
are
eliminated
through
appropriate
remedial
actions,
the
centre
Assuming
that
the
assignable
causes
for
these
three
samples
(4,
7
and
9)
are
(i)
To
revise
the
centre
line
and
con::
ol
limits:
limits,
therefore
the
process
is
statistically
out
of
control.
Ans.
X-chart
(refer
Fig.12.14).
Since
those
three
points
are
outside
the
upper
control
chart
(refer
Fig.12.15)
and
samples
4
and
9
are
above
the
upper
control
limits
of
the
It
may
be
observed
that
sample
7
is
above
the
upper
control
limits
on
the
R
(i)
To
check
the
process
for
statistical
control:
Fig. 12.15. R-chart
Subgroup No.
2
4
5
6
7
9
10
0.00
LCLR =0
0.05 0.10
Range
CLR =0.111
0.15
4 0.20
0.25
UCLR = C.253
0.30
Control Charts.
12.33|
for R-chart:
control
limits
Control
limit
or
Central
line,
CLg
Process
apability

10. 12.34
Revised control limits for X-chart:
() To draw the revised X -and R-charts:
(x) ueen
Now using the revised control limits, the X-and R-charts are drawn as shown in
Figs.12.16 and 12.17 respectively.
6.70
6.65
6.60
6.55
6.50
R 6.45
6.40
6.35
6.30
6.25
0
0.30
0.25
0.20
Control limit or Central = X-A, R = 6.377 -0.729 (0.073)
: 6.324 Ans.
O 0.15
0.10
0.05
0.00
1
1 2
3 4
3
5
Subgroup No.
4 5
6
Fig. 12. 16. Revised X-chart
Subgroup No.
7
6
Total Quality Management
8
7
Fig. 12.17. Revised R-chart
9
UCLy = 6.430
CLz = 6.377
LCLy = 6324
10
UCLR = 0.166
CLR =0.073
10
LCL = 0