Control Charts for Variables with Specified Process Capability Indices

Abstract

Control charts, also known as Shewhart charts in statistical process control are the statistical tools used to determine whether a manufacturing process is in a state of statistical control or not. If analysis of the control chart indicates that the process is currently under statistical control (i.e. is stable, with variation only co ming fro m sources common to the process) then no corrections or changes to process control parameters are needed or desirable. In addition, data fro m the controlled process can be used to predict the future performance of the process with the help of process capability indices. That is, the suitability and the performance of the manufacturing process are assessed in two stage processes namely control charts and process capability indices. Th is paper deals with the techniques namely, capability based control charts which combines the two stage processes into a single stage process in industrial applicat ions for on line process control. The relative performance of the capability based control charts for variables is assessed with that of existing usual control charts for variables, namely Charts X − and Charts R −. It is observed that the proposed method is simp le to apply and does not warrant any tedious computations both for control charts and for co mputing process capability indices. Fo r the sake o f convenience we have presented the control charts constants for constructing capability based Charts R − and Charts X −. The proposed method is also illustrated with the help of a numerical examp le.

Full text

International Journal of Probability and Statistics 2012, 1(4): 101-110
DOI: 10.5923/j.ijps.20120104.03
Control Charts for Variables with Specified Process
Capability Indices
J. Subramani1,*, S. B alamurali2
1Department of Stat istics, Pondicherry University, R V Nagar, Kalapet, Puducherry, 605 014, India
2Department of Comp uter Applications, Kalasalingam University, Krishnankoil, 6426 190, Srivilliputhur,Tamilnadu, India
Abs trac t Control charts, also known as Shewhart charts in s tatistical process control are the statistical tools used to
determine whether a manufacturing process is in a state of statistical control or not. If analysis of the control chart indicates
that the process is currently under statistical control (i.e. is stable, with variation only coming fro m sources common to the
process) then no corrections or changes to process control parameters are needed or desirable. In addition, data from the
controlled process can be used to predict the future performance of the process with the help of process capability indices.
That is, the suitability and the performance of the manufacturing process are assessed in two stage processes namely control
charts and process capability indices. This paper deals with the techniques namely, capability based control charts which
combines the two stage processes into a single stage process in industrial applicat ions for on line process control. The relative
performance of the capability based control charts for variables is assessed with that of existing usual control charts for
variables, namely
ChartsX −
and
ChartsR −
. It is observed that the proposed method is simple to apply and does not warrant
any tedious computations both for control charts and for computing process capability indices . For the sake of convenience
we have presented the control charts constants for constructing capability based
ChartsR −
and
ChartsX −
. The proposed
method is also illustrated with the help of a numerical examp le.
Ke y wo rd s Control Charts, Industrial Applications, Process Capability Index, Quality Characteristics
1. Introduction
The present scenario of total quality management
demands the effective use of statistical tools for analyzing
quality problems and improving the performance of
production process. Measuring, monitoring and ma intaining
several quality characteristics to achieve fu rth er
improvements are required in every production process.
Many of the quality characteristics are measurable in nature
and can be expressed in terms of numerical measurement by
measuring them using micrometer, vernier calipers etc. For
example, length, diameter, weight, pocket width, distance
and groove diameter of a shaft are some of the quantifiab le
quality characteristics encountered during the production of
shafts. When dealing with measurable quality characteristics,
it is usually neces sary to monitor the behaviour of the mean
value of the quality characteristics and its variability. To
study the behaviour of the production process and to take a
necessary action on the process, control charts for variables
are very much used which requires to co llec t random
samples from the production process and to compute the
* Corresponding author:
drjsubramani@yahoo. co.in (J. S ubramani)
Published online at http://journal.sapub.org/ijps
Copyright © 2012 Scientific & Academic Publ ishing. All Rights Reserved
estimate of the quality characteristics. Further one has to
show that the process is under statistical control before
computing the process capability indices. Nowadays process
capability analysis is a mandatory requirement and to have a
minimum prescribed value for the process capability indices
p
C
and
pk
C
to be a supplier for original manufacturing
companies, particularly in the field of automobile industries.
Process capability indices (P CI) have proliferated in both use
and variety during the last three decades. Their widespread
and often uncritical use led to improvements in quality.
Process capability indices are considered as a practical
tool by many advocates of the statistical process control
industry. They are used to determine whether the
manufacturing process is capable of producing units within
the product or process specifications. A capability index is a
dimensionless measure based on the process parameters and
the process specifications, designed to quantify, in a simple
and easily understood way, the performance of the process.
Several process capability indices have been developed
among which the basic and widely used indices are Cp and
Cpk. (see Kan e[8]). A detailed review of the process
capability indices can be seen in Kotz and Johnson[9].
The process capability indices are unit less measures
showing the capability of a manufacturing process whet her it
is capable of operating within its specifications limits. If the
value of the PCI exceeds one, then the process is considered

102 J. Subraman i et a l.: Control Charts for Variables with Specified Process Capability Indices
to be satis factory or capable. Howeve r this approach ignores
the fact that these PCIs are rando m variables with
distributions. Confidence limits play a major role for the
correct interpretation of PCIs. Recently, techniques and
tables were developed to construct confidence limits for the
process capability ind ices. These techniques are based on the
assumption that the underlying process is normal. Moreover,
since many processes can frequently have skewed or heavily
tailed distributions. In practice, the interval estimation
technique that is free from the assumption of distribution is
desirable. The various other properties and the behaviour of
process capability indices at different type of distributions
were studied by many authors. For exa mple, the 95%
confidence limits for the process capability indices Cp and
Cpk were constructed by Chou, Owen and Bo rrego [5]. As
their limits on Cpk can produce 97% or 98% lower confidence
limits (instead of 95%) ma king them conservative, an
approximation presented by Bis s e ll[4] is recommended.
Franklin and Wass erman[6] presented an initial study of the
properties of these three bootstrap confidence intervals for
Cpk. Franklin and Wasserman[7] constructed the confidence
limits for s ome b asic capability indices and examined their
behaviors when the underlying process is normal, skewed or
heavily ta iled distributions. Balamurali andKalyanasundara
m[3] constructed bootstrap confidence limits for the
ca pab ilit y indices Cp, Cpk and Cpm based on lognormal and
chi-squared distributions. Balamurali[2] considered the
above capability indices under short run production
processes and constructed the confidence limits.
However the problems one has to face are the following:
There is a considerable time taken for establishing and
monitoring the process through control charts for variables;
inadequate technical personnel to correctly interpret the
outcome of the control charts; unnecessary inventory and
delay due to the computation of process capability indices
and to take a decision on the process based on the resulting
values of process capability indices. Hence it is felt to have
an alternative method to achieve the benefit of both control
charts for variables and the process capability analysis. As a
result we have proposed new type of control charts for
variables based on the required process capability indices
p
C
and
pk
C
, called as “control charts for variables with
specified value for process capability indices”. Further we
have presented the control charts constants to construct the
proposed control charts and are expla ined with the help of a
numerical example. For a more detailed discussion on the
control charts for variables and process capability analysis,
the readers are refe rred to Montgomery[10], Spiring[11],
Sarka r and Pa l[1], Subraman i[12, 13 ], Subramani and
Balamurali[14] and the references cited therein. In this paper,
we propose a capability based control chart for on line
process control. The relative performance of this control
charts for variables is assessed with that of existing usual
control charts for variables. It is observed that the proposed
method is simple to apply and does not warrant any tedious
computations both for control charts and for computing
process capability indices. We are also presenting the control
charts constants for constructing capability based control
charts. The proposed method is also illustrated with a
numerical example.
2. Control Charts and Process
Capability Analysis
2.1 . Usual Method
S et the
m anufactur ing pr ocess
Im pl em ent contr ol char ts
for var iables
Is
R-Cha r t
U nder C ontr ol?
Is
X-bar -Char t
U nder C ontr ol?
Com pute Pr ocess capability
Indi ces C p and C pk
Is
C p V alue OK ?
Is
C pk V alue OK?
No
No
No
No
Y es
Y es
Y es
Y es
E stablish the
m anufactur ing pr ocess
C ontinue the m anufac tur ing
D o you
want to i m pr ove
the pr ocess?
Y es No
Fi gure 1 . P r esent M an uf acturin g Syst em
For the sake of convenience to the readers, we have
presented the control charts for variables for checking
whether the process is under statistical control or not. That is ,
to take a decision whether the process can be allowed further
without making any adjustment or to take corrective actions
if any, to bring back the process under statistical control.
Once the process has been brought under the state of
statistical control by both
ChartR −
and
ChartX −
then
the process cap ability ind ices
pkp
CandC
can be co mpu ted.
If the values of the process capability indices are at the
satisfactory level, the process can be allowed further without
making any change or adjustment in the process. Otherwise
suitable adjustments have to be made so that the process shall
satisfy both the requirements of control charts and the
process capability indices. The present manufacturing

International Journal of Probability and Statistics 2012, 1(4): 101-110 103
process can be explained in the flo w chart as shown in Fig ure
1.
The following are the various computational formulae
involved in constructing the control charts and for the
computation of process capability indices.
1
1
n
ii
i
Xx
n
=
=∑
: Mean of ith sample
1
1N
i
i
Xx
N
=
=
∑
: Grand Mean
() ()
iii
R Max x Min x= −
: Range of ith sample
1
1
k
i
i
RR
k=
=
∑
: Mean of Range values
2x
UCL X A R= +
: Upper Control Limit for -Cha rt
2x
LCL X A R= −
: Lower Control Limit for -Ch art
4R
UCL D R=
: Upper Control Limit for R-Ch ar t
3R
LCL D R=
: Lower Control Limit for R-Chart
2
6/
p
USL LSL
CRd
−
=
: Proce s s Capab ility
2
3/
pu
USL X
CRd
−
=
: Upper Process Capability Index
: Lower Process Capability Index
: Process Capability Index
LSL: Lower Specification Limit of the Quality
Characteristics
USL: Upper Specification Limit of the Quality
Characteristics
When the sample size n=5, the values for the control chart
constants obtained from the table are as given below:
D4 = 2. 114, D3 = 0, d2 = 2.326 and A2 = 0.577
The Control limits for constructing and
are as given below:
Control limits for :
R
CL R=
3R
LCL D R=
4R
UCL D R=
Control limits for :
x
CL X
=
2x
LCL X A R= −
2x
UCL X A R= +
2.2 . Propos ed Method- Co ntr ol C har ts for Varia bles wi th
Specifie d
p
C
Val u e
Suppose that a customer wants to have products produced
in a process with a specified value for the process capability
index
p
C
. To achieve this, the manufacturer has to establish
a process under statistical control and then the process
capability inde x has to be computed. If the computed value
of
p
C
is more than the specified value given by the
customer, the process can be run without any adjustment.
Otherwis e suitable actions have to be taken to meet the
required process capability values. The drawback of this
method is that the manufacturer has to keep all the produced
items in the inventory until the computation of the process
capability inde x and to take a decision on the process based
on the process capability index. It is a time consuming
procedure as well as requires addit ional skilled personnel. To
avoid this, we have proposed control charts for variables
with s p e cified p rocess capability indices. In the proposed
method, the process is under statistical control means that it
satisfies the required process capability index. Further it is
not necessary to compute the process capability indices
separately. The proposed manufacturing process can be
explained in the flow chart shown in F ig ure 2.
Se t the
manufacturing proc es s
Implement c ontrol char ts for
variable s w ith Cp and Cpk
Is
R-Char t
Under Contr ol?
Is
X-bar -Char t
Under Contr ol?
No
No
Yes
Yes
Establish the
manufacturing proc es s
Continue the manufa ctur ing
Do you
want to impr ove
the process?
Yes No
Spe cify Cp and Cpk
Fi gure 2 . P r esent M an uf acturin g Syst em
The following are the derivation of various computational
formulae involved in constructing the proposed control
charts with specified process capabilit y inde x
p
C
.
X
X
2
/3 dR
LSLX
Cpl
−
=
),
(plpu
pk CC
MinC =
LSLUSLTTolerance −==
2
arg LSLUSL
MetT +
==
ChartsR −
ChartsX −
ChartR −
ChartX −

104 J. Subraman i et a l.: Control Charts for Variables with Specified Process Capability Indices
Consider
6
p
USL LSL
C
σ
−
=
2
6/
p
T
CRd
⇒=
If value is given then value can be obtained from
the above formula as
*
p
T
RD
C
⇒=
Where
*2
6
d
D=
Similarly one can obtain the lower control limit (LCL) and
upper control limit (UCL) of the as given
below:
Center Line=
*
R
p
T
CL R D C
= =
3R
LCL D R=
32
6
R
p
DdT
LCL C
⇒=
*
3R
p
T
LCL D C
⇒=
Where
*32
3
6
Dd
D=
4R
UCL D R=
42
6
R
p
DdT
UCL C
⇒=
In the similar manner one can compute the control limits
to construct with s pecif ied value as given
below.
Center line=
Lower control limit for the proposed is
obtained as given below:
2x
LCL X A R= −
*
2x
p
T
LCL X A D C
⇒=−
*
2x
p
T
LCL X A C
⇒=−
Where
**
22
A AD=
*
4R
p
T
UCL D C
⇒=
Where
*42
4
6
Dd
D=
Ta b l e 1 . Contro l Cha rt s Constant s for specif ied
p
C
va lue
n
2
d
3
D
4
D
2
A
2
1.128
0
3.267
1.880
0.1 880
0.0 000
0.6 142
0.3 534
3 1.693 0 2.574 1.023 0 .2 822 0.0 000 0 .7263 0 .2887
4
2.059
0
2.282
0.729
0.3 432
0.0 000
0.7 831
0.2 502
5 2.326 0 2.115 0.557 0 .3 877 0.0 000 0 .8199 0 .2159
6
2.534
0
2.004
0.483
0.4 223
0.0 000
0.8 464
0.2 040
7
2.704
0.076
1.924
0.419
0.4 507
0.0 343
0.8 671
0.1 888
8
2.847
0.136
1.864
0.373
0.4 745
0.0 645
0.8 845
0.1 770
9
2.970
0.184
1.816
0.337
0.4 950
0.0 911
0.8 989
0.1 668
10 3.078 0.223 1.777 0.308 0 .5 130 0.1 144 0 .9116 0 .1580
11
3.173
0.256
1.744
0.285
0.529
0.135
0.922
0.151
12
3.258
0.283
1.717
0.266
0.543
0.154
0.932
0.144
13 3.336 0.307 1.693 0.249 0.556 0.171 0.941 0.138
14
3.407
0.328
1.672
0.235
0.568
0.186
0.949
0.133
15 3.472 0.347 1.653 0.223 0.579 0.201 0.957 0.129
16
3.532
0.363
1.637
0.212
0.589
0.214
0.964
0.125
17 3.588 0.378 1.622 0.203 0.598 0.226 0.970 0.121
18
3.640
0.391
1.608
0.194
0.607
0.237
0.976
0.118
19
3.689
0.403
1.597
0.187
0.615
0.248
0.982
0.115
20
3.735
0.415
1.585
0.180
0.623
0.258
0.987
0.112
21
3.778
0.425
1.575
0.173
0.630
0.268
0.992
0.109
22 3.819 0.434 1.566 0.167 0.637 0.276 0.997 0.106
23
3.858
0.443
1.557
0.162
0.643
0.285
1.001
0.104
24
3.895
0.451
1.548
0.157
0.649
0.293
1.005
0.102
25
3.931
0.459
1.541
0.153
0.655
0.301
1.010
0.100
p
C
R
p
C
Td
R6
2
=
ChartR −
ChartX −
p
C
XCL
X
=
ChartX −
*
D
*
3
D
*
4
D
*
2
A

International Journal of Probability and Statistics 2012, 1(4): 101-110 105
One can use the above control limits to construct
with s pecif ied value.
Similarly one may obtain the upper control limit for the
proposed as
2x
UCL X A R= +
*
2x
p
T
UCL X A D C
⇒=+
*
2x
p
T
UCL X A C
⇒=+
Where
**
22
A AD=
Thus the control limits for the control charts with s pecified
p
C
value are as given below:
Control limits for proposed
Chart
R−
:
*
R
p
T
CL D C
=
*
3R
p
T
LCL D C
=
*
4R
p
T
UCL D C
=
Control limits for proposed :
x
CL X
=
*
2x
p
T
LCL X A C
= −
*
2x
p
T
UCL X A C
= +
One can use the above control limits to construct the
ChartR −
and
ChartX −
with s pecified
p
C
value. For
the sample s ize
n
( )
252 ≤≤ n
, we have presented the
values of the control charts constants
*** *
23 2
,,D D D and A
in the Table 1 given belo w.
The advantageous of the proposed control charts is that the
observed range values are not required for computing the
control limits. Further if the process is under the state of
statistical control means that it automatically satisfies the
required conditions regarding the values of process
capability inde x
p
C
.
2.3 . Proposed Method- Co ntr ol Char ts for Vari ables with
Specifie d
pk
C
valu e
In the section 2.2, we have introduced a method for
constructing control charts for variables with specified value
for the process capability index
p
C
. In the simila r man ner
we have developed a method for constructing control charts
for v ariab les with specified value for the process capability
index
pk
C
. The follo wing are the derivation of various
computational formulae involved in constructing control
charts for variables with specified process capability index
pk
C
Consider
{ }
,
pk pl pu
C Min C C=
Where
2
3/
pu
USL X
CRd
−
=
and
2
3/
pl
X LSL
CRd
−
=
After a little algebra, one can rewrite the as
2
2
3/
pk
TXM
CRd
−−
=
If value is given then value can be obtained fro m
the above formula as
2
2
2
22
2
3/ 3 3
pk pk pk
TT
Td XM XM
XM d
RCd C C

−− −−
−− 

= = =
*2
k
pk
TXM
RD C

−−


⇒=
Where
*2
3
k
d
D=
Similarly one can obtain the lower control limit (LCL) and
upper control limit (UCL) of the as given
below:
*
2
Rk
pk
TXM
CenterLine CL R D C

−−


= = =
3R
LCL D R=
*
3
2
Rk
pk
TXM
LCL D D C

−−


⇒=
*
3
2
Rk
pk
TXM
LCL D C

−−


⇒=
Where
**
32
33
3
kk
Dd
D DD= =
4R
UCL D R=
*
4
2
Rk
pk
TXM
UCL D D C

−−


⇒=
*
4
2
Rk
pk
TXM
UCL D C

−−


⇒=
where
**
42
44
3
kk
Dd
D DD= =
One can use the above control limits to construct
ChartR −
p
C
ChartX −
Chart
X−
pk
C
pk
C
R
ChartR −

106 J. Subraman i et a l.: Control Charts for Variables with Specified Process Capability Indices
with s pecif ied value.
In the similar manner one can compute the control limits
to construct with s pec ifie d value as given
below.
Center line=
Lower control limit for the proposed is
obtained as given below:
*
2
k
pk
TXM
RD C

−−


⇒=
2x
LCL X A R= −
*
2
2
xk
pk
TXM
LCL X A D C

−−


⇒=−
2
*
2
k
pk
x
TXM
LCL X A C

−−


⇒=−
Where
**
22kk
A AD=
Similarly one may obtain the upper control limit for the
proposed as
2x
UCL X A R= +
2
*2
xk
pk
TXM
UCL X A D C

−−


⇒=+
2
*
2
xk
pk
TXM
UCL X A C

−−


⇒=+
Where
22
**
kk
A AD=
Thus the control limits for the proposed control charts
with s pecif ied value are as given below:
Control limits for with s pecifie d value
*
2
Rk
pk
TXM
CL R D C

−−


= =
*
3
2
Rk
pk
TXM
LCL D C

−−


=
*
4
2
Rk
pk
TXM
UCL D C

−−


=
Control limits for the proposed with
specified value:
x
CL X=
*
2
2
xk
pk
TXM
LCL X A C

−−


= −
*
2
2
xk
pk
TXM
UCL X A C

−−


= +
The above control limits can be further simp lified by
substituting the values of and , which lead to the
following two cases:
Cas e 1 : When
Control limits for w ith s pecified value
*
Rk
pk
USL X
CL R D C

−
= = 


*
3Rk
pk
USL X
LCL D C

−
=


*
4Rk
pk
USL X
UCL D C

−
=


Control limits for the proposed with
specified value:
x
CL X=
*
2xk
pk
USL X
LCL X A C

−
= − 


*
2xk
pk
USL X
UCL X A C

−
= + 


Cas e 2 : When
Control limits for with s pecifie d value
*
Rk
pk
X LSL
CL R D C

−
= = 


*
3Rk
pk
X LSL
LCL D C

−
=


*
4Rk
pk
X LSL
UCL D C

−
=


Control limits for the proposed
ChartX −
with
specified value:
x
CL X=
*
2xk
pk
X LSL
LCL X A C

−
= − 


*
2xk
pk
X LSL
UCL X A C

−
= + 


ChartR −
pk
C
Chart
X−
pk
C
XCL
X
=
ChartX −
ChartX −
pk
C
ChartR −
pk
C
ChartX −
pk
C
T
M
MX >
ChartR −
pk
C
Chart
X−
pk
C
MX <
ChartR −
pk
C
pk
C

International Journal of Probability and Statistics 2012, 1(4): 101-110 107
Ta b l e 2 . Control Chart s Constants for specified
pk
C
va lue
2
1.128
0
3.267
1.880
0.376
0.000
1.228
0.707
3
1.693
0
2.574
1.023
0.564
0.000
1.453
0.577
4
2.059
0
2.282
0.729
0.686
0.000
1.566
0.500
5
2.326
0
2.115
0.557
0.775
0.000
1.640
0.432
6
2.534
0
2.004
0.483
0.845
0.000
1.693
0.408
7
2.704
0.076
1.924
0.419
0.901
0.069
1.734
0.378
8
2.847
0.136
1.864
0.373
0.949
0.129
1.769
0.354
9
2.970
0.184
1.816
0.337
0.990
0.182
1.798
0.334
10
3.078
0.223
1.777
0.308
1.026
0.229
1.823
0.316
11
3.173
0.256
1.744
0.285
1.058
0.271
1.845
0.301
12
3.258
0.283
1.717
0.266
1.086
0.307
1.865
0.289
13
3.336
0.307
1.693
0.249
1.112
0.341
1.883
0.277
14
3.407
0.328
1.672
0.235
1.136
0.372
1.899
0.267
15
3.472
0.347
1.653
0.223
1.157
0.402
1.913
0.258
16
3.532
0.363
1.637
0.212
1.177
0.427
1.927
0.250
17
3.588
0.378
1.622
0.203
1.196
0.452
1.940
0.243
18
3.640
0.391
1.608
0.194
1.213
0.474
1.951
0.235
19
3.689
0.403
1.597
0.187
1.230
0.496
1.964
0.230
20
3.735
0.415
1.585
0.180
1.245
0.517
1.973
0.224
21
3.778
0.425
1.575
0.173
1.259
0.535
1.983
0.218
22
3.819
0.434
1.566
0.167
1.273
0.552
1.994
0.213
23
3.858
0.443
1.557
0.162
1.286
0.570
2.002
0.208
24
3.895
0.451
1.548
0.157
1.298
0.586
2.010
0.204
25
3.931
0.459
1.541
0.153
1.310
0.601
2.019
0.200
One can use the above control limits to construct the
and with s pe cified value. For
the given sample size , we have presented the
values of the control charts constants
** * *
23 2
,,
kkk k
D D D and A
in the Table 2 given below.
The advantageous of the proposed control charts with
specified and values, is that the observed range
values are not required for computing the control limits.
Further if the process is under the state of statistical control
means that it automatically satisfies the required conditions
regarding the values of process capability indices a nd
.
3. Numerical Example
Consider the data given below in Table 3 is the Exa mple
5.1 (Montgomery[10], page 213). The data is pertaining to
the manufacturing of Piston Rings for an auto motive engine
produced by a forging process. Twenty five samples, each of
size five have been taken and the inside d ia meter is measured.
The resulting data together with the sample means and
sample range values are given below in Table 3.
Fro m the above values we have obtained the follo wing:
0011.74=X
,
0232.0R =
and
0099.0/dR
2
==
σ
Cas e 1 : Usual Method of Control Charts and the
Co mputation of Process Capability Indices
Control Limits for R-Chart are obtained as:
0.02324
R
CL R= =
3* 0* 0.02324 0
R
LCL D R= = =
4
* 2.115* 0.02324 0.0492
R
UCL D R= = =
Control Limits for -Chart are obtained as:
74.00118
x
CL X= =
By plotting the control limits, sample ranges and sample
means, one may get and as
given in Figure 3 and Figure 4. From the Chart, we
observe that all the plotted sample range values are falling
within the control limits. Hence we may conclude that the
variations are under control. Similarly fro m the ,
we observe that all the plotted sample means are falling
inside the control limits. Hence we may conclude that the
averages are also under the statistical control.
n
2
d
3
D
4
D
2
A
*
k
D
*
3k
D
*
4k
D
*
2k
A
ChartR −
ChartX −
pk
C
n
( )
252 ≤≤ n
p
C
pk
C
p
C
pk
C
X
98777.7302324.0*577.000118.74
*
2
=−=
−= RAX
x
LCL
01459.7402324.
0*577.000118.74
*
2
=−=
+= RAXUCLx
ChartR −
ChartX −
−R
ChartX −

108 J. Subraman i et a l.: Control Charts for Variables with Specified Process Capability Indices
Ta b l e 3 . Data of Inside Diameter of Piston Rings (Spec: 74.000±0.05 mm)
Sam ple Num ber
x1
x2
x3
x4
x5
Sam ple Mean
Range R
1
74. 030
74. 002
74. 019
73. 992
74. 008
74. 010 2
0.038
2
73. 995
73. 992
74. 001
74. 011
74. 004
74. 000 6
0.019
3
73. 988
74. 024
74. 021
74. 005
74. 002
74. 008 0
0.036
4
74. 002
73. 996
73. 993
74. 015
74. 009
74. 003 0
0.022
5
73. 992
74. 007
74. 015
73. 989
74. 014
74. 003 4
0.026
6
74. 009
73. 994
73. 997
73. 985
73. 993
73. 995 6
0.024
7
73. 995
74. 006
73. 994
74. 000
74. 005
74. 000 0
0.012
8
73. 985
74. 003
73. 993
74. 015
73. 988
73. 996 8
0.030
9
74. 008
73. 995
74. 009
74. 005
74. 004
74. 004 2
0.014
10
73. 998
74. 000
73. 990
74. 007
73. 995
73. 998 0
0.017
11
73. 994
73. 998
73. 994
73. 995
73. 990
73. 994 2
0.008
12
74. 004
74. 000
74. 007
74. 000
73. 996
74. 001 4
0.011
13
73. 983
74. 002
73. 998
73. 997
74. 012
73. 998 4
0.029
14
74. 006
73. 967
73. 994
74. 000
73. 984
73. 990 2
0.039
15
74. 012
74. 014
73. 998
73. 999
74. 007
74. 006 0
0.016
16
74. 000
73. 984
74. 005
73. 998
73. 996
73. 996 6
0.021
17
73. 994
74. 012
73. 986
74. 005
74. 007
74. 000 8
0.026
18
74. 006
74. 010
74. 018
74. 003
74. 000
74. 007 4
0.018
19
73. 984
74. 002
74. 003
74. 005
73. 997
73. 998 2
0.021
20
74. 000
74. 010
74. 013
74. 020
74. 003
74. 009 2
0.020
21
73. 982
74. 001
74. 015
74. 005
73. 996
73. 999 8
0.033
22
74. 004
73. 999
73. 990
74. 006
74. 009
74. 001 6
0.019
23
74. 010
73. 989
73. 990
74. 009
74. 014
74. 002 4
0.025
24
74. 015
74. 008
73. 993
74. 000
74. 010
74. 005 2
0.022
25
73. 982
73. 984
73. 995
74. 017
74. 013
73. 998 2
0.035
Av er a ge
74. 001 18
0.02324
Fi gure 3. R-C h art - Usual Method
Fi gure 4. X-Bar Chart- Usua l Metho d
The manufacturing capability of a process can normally be
evaluated in terms of process capability indices
p
C
and
pk
C
. The process capability indices obtained from the above
values are given below:
2
74.05 73.95 1.6681
6 / 6 *0.2324 / 2.326
p
USL LSL
CRd
−−
= = =
2
74.05 74.00118 1.6287
3 / 3* 0.2324 / 2.326
pu
USL X
CRd
−−
= = =
2
74.00118 73.95 1.7075
3 / 3* 0.2324 / 2.326
pl
X LSL
CRd
−−
= = =
( , ) (1.6287,1.7075) 1.6287
pk pu pl
C Min C C Min= = =
If the customer’s requirement of the process capability
index
p
C
is 1.5, then one may conclude from the above
capability indices that the process is an efficient one.
However If the customer’s require ment of the process
capability inde x
p
C
is 2.0, then one may conclude from the
above capability indices that the process is not an efficient
one and hence the process has to be adjusted so as the
resulting process capability index
p
C
is at leas t 2.0.
Cas e 2 : Co ntrol Charts with specified
5.1=
p
C
In this case we construct the control limits for the given
value of process capability index
p
C
and hence separate
computation of the process capability index is not required.
Further the proposed control charts states that the process is
under the statistical control means that the process satisfies
0.00
0.01
0.02
0.03
0.04
0.05
0.06
1
3
5
7
9
11
13
15
17
19
21
23
25
Range Values
LCL-R R-Bar UCL-R Range
73.98
73.99
74.00
74.01
74.02
74.03
1
3
5
7
9
11
13
15
17
19
21
23
25
Average Values
S_ M ean
LCL
X-Bar
UCL

International Journal of Probability and Statistics 2012, 1(4): 101-110 109
the requirement of the process capability index. Hence one
can construct several control limits for different values of
process capability index Cp. If any plotted points falls
o uts id e th e con t ro l limit s wit h s p ecifie d Cp= 1.5 means that
the process does not satisfy the required process capability
index Cp= 1. 5.
Let the given process capability index Cp= 1.5. Then the
Control limits for
ChartR −
with s p e cified Cp= 1. 5 are
obtained as given below:
*
0.1
. 0.3877 0.2585
1.5
R
p
T
CL D C

= = =


*
3
0.1
00
1.5
R
p
T
LCL D C

= = =


*
4
0.1
. 0.8199 0.0547
1.5
R
p
T
UCL D C

= = =


Control limits for with s pecifie d
1.5
p
C=
74.00118
x
CL X= =
2
*
0.1
. 74.00118 0.2159 73.9868
1.5
x
p
T
LCL A C
X

=−= − =


2
*
0.1
. 74.00118 0.2159 74.0156
1.5
x
p
T
UCL X A C

=+= + =


Fi gure 5. R- Chart with Specified Cp=1.5
Fi gure 6. X-Bar Chart with Specified Cp=1 .5
By plotting the control limits, sample ranges and sample
means, one may get and wit h
specified as given in Figure 5 and Figure 6. From
the , we observe that all the plotted sample range
values are fa lling within the control limits. Hence we may
conclude that the variations are under control. Similar ly from
the , we observe that all the plotted sample
means are falling inside the control limits. Hence we may
conclude that the averages are als o under the statistical
control. Further we conclude that the process capability
index for the given process is at least 1.5 and satisfies the
customer’s requirements.
Cas e 3 : Control Charts with specified Cp=1.5
In this case we construct the control limits for the given
value of process capability index Cpk
and hence separate
computation of the process capability index is not required.
Further the proposed control charts states that the process is
under the statistical control means, the process satisfies the
requirement of the process capability index. Hence one can
construct several control limits for different values of
process capability index. If any p lotted points falls outside
the control limits with specified Cpk=1.5 means that the
process does not satisfy the required process capability index.
Cpk=1.5.
Let the given process capability index Cp = 1. 5. Furthe r
00.7400118.74 =>= MX
the contro l li mit s with s p ecified
process capability index Cpk=1.5 are obtained as given
below:
Control limits for
ChartR −
with s pecifie d Cpk value
*74.05 74.00118
0.775 0.025224
1.5
Rk
pk
USL
CL R D C
X

−−

= = = =





*
3
74.05 74.00118
00
1.5
Rk
pk
USL X
LCL D C

−−

= = =





4
*74.05 74.00118
1.64 0.053377
1.5
Rk
pk
USL
UCL D C
X

−−

= = =





Control limits for the proposed with s p e cified
value are obtained as given below:
74.00118
x
CL X
= =
2
*
74.05 74.00118
74.00118 0.432 1.5
73.98712
xk
pk
USL X
LCL X A C

−
= − 


−

= − 

=
2
*
74.05 74.00118
74.00118 0.432 1.5
74.01524
k
pk
x
USL X
UCL X A C

−
= + 


−

= + 

=
By plotting the control limits, sample ranges and sample
means, one may get and w it h s pec ified
Cpk=1.5 as given in Figure 7 and Figure 8. Fro m the ,
we observe that all the plotted sample range values are
falling within the control limits. Hence we may conclude that
the variations are under control. Similarly from the ,
ChartX −
ChartR −
Chart
X−
5.1=
p
C
ChartR −
ChartX −
ChartX −
5.1
=
pk
C
ChartR −
ChartX −
ChartR −
ChartX −

110 J. Subraman i et a l.: Control Charts for Variables with Specified Process Capability Indices
we observe that all the plotted sample means are falling
inside the control limits. Hence we may conclude that the
averages are also under the s tatistical control. Further we
conclude that the process capability index Cpk for the given
process is at least 1.5 and satisfies the customer’s
requirements.
Fi gure 7. R-Chart with Sp ecified Cpk=1.5
Fi gure 8. X-Bar Chart with Specified Cpk =1.5
4. Conclusions
In this paper, we have proposed a control chart which is
based on the process capability indices Cp and Cpk for on line
process control, which have the benefit of the usual control
charts and the process capability indices. That is, the
proposed control charts combines the two stage controlling
mechanis m namely, Control Charts and the process
capability indices into a single stage controlling mechanism
to monitor the process online and to assess the suitabilit y of
the manufacturing process. It has been shown that the
relative performance of the proposed control charts for
variables can be assessed with that of usual var ia b le control
charts. It has als o been shown that the proposed control chart
is simple to apply and does not warrant any tedious
computations both for control charts and for computing
process capability indices . Moreover, we have also presented
tables fo r the control charts constants for computing the
control limits of the capability based control charts. The
proposed method has a lso been illustrated with a numerical
example. This study can also be extended to develop control
charts based on other process capability indices.
ACKNOWLEDGEMENTS
The authors are thankful to the editor and the refe rees for
their useful co mments, which have helped to improve the
presentation of the paper.
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