 Open Access
 Total Downloads : 148
 Authors : Shashibhushan B. Mahadik
 Paper ID : IJERTV3IS080722
 Volume & Issue : Volume 03, Issue 08 (August 2014)
 Published (First Online): 21082014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
X Charts with Variable Sampling Interval, Control limits, and Warning Limits
X Charts with Variable Sampling Interval, Control limits, and Warning Limits
Shashibhushan B. Mahadik
Department of Statistics, Solapur University, Solapur, INDIA 413255
Abstract – The idea of variable sampling interval, control limits, and warning limits (VSICWL) is proposed for X
charts. Expressions for performance measures for
VSICWL X charts are developed using a Markov chain approach. The performances of these charts are compared numerically with that of variable sampling interval and warning limits (VSIWL) and variable
control and warning limits (VCWL) X charts. It is observed that the statistical performance of a VSICWL
X char is better than that of VSIWL and VCWL X
charts while its administrative performance is better than that of VSIWL X chart.
Key words: Adaptive control chart, average number of samples to signal, average number of switches to signal, steadystate average time to signal.

INTRODUCTION
Shewhart control chart is an effective online process control technique for detecting the occurrence of an assignable cause variability in manufacturing and other processes. It has three design parameters, viz, sampling interval length, sample size, and warning limit(s). The original control chart is static in the sense that its design parameters are kept fixed throughout the period of its implementation. A control chart is termed to be adaptive if at least one of its design parameter is a variable and takes a value for the next sample according to the status of the process indicated by the current sample.
It has been proved in the literature that the adaptive control charts monitor processes more efficiently than the static ones. Reynolds et al. (1988) proposed the
first adaptive control chart. It is the X chart with variable sampling interval. Then, Prabhu, et al. (1993) and Costa (1994) independently proposed variable sample size X
charts. Prabhu, et al. (1994) proposed variable sample size
and sampling interval X charts. Costa (1999) proposed the adaptive X charts in which all the three design parameters are variable. Mahadik and Shirke (2009) proposed a special variable sample size and sampling
interval X chart.
The weakness of an adaptive control chart is the inconvenience in its administration due to frequent switches between the values of its adaptive design
parameters. Some modifications have been suggested in the literature in order to lessen this inconvenience. See, for example, Amin and Letsinger (1991), Amin and Hemasinha (1993), and Mahadik (2012a, b).
Some recent references on adaptive control charts include Chen et al. (2011), Dai et al. (2011), Faraz and Saniga (2011), Nenes (2011), Kooli and Limam (2011), and Lee (2011), Zhang, et al. (2011), Lee and Lin (2012), Huang (2013), Mahadik, S. B. (2013a, b), Kuo and Lee (2013), Seif, et al. (2014), and Faraz, et al. (2014).
In the present paper, the idea of variable sampling interval, control limits, and warning limits (VSICWL) is
proposed for X charts. The performances of these charts are compared numerically with that of variable sampling interval and warning limits (VSIWL) and variable control
and warning limits (VCWL) X charts. It is observed that the statistical performance of the proposed chart is better
than that of VSIWL and VCWL X charts while its administrative performance is better than that of VSIWL
X chart.
The remainder of the paper is organized as follows. The subsequent sections describe the design principle of a VSICWL X chart. Expressions for
performance measures for this chart are derived. Its
statistical and administrative performances are compared
numerically with that of VSIWL and VCWL X charts. This is followed by the Conclusions.

A VSICWL X CHART
Let the quality characteristic X to be monitored follows a normal distribution with mean and a known and constant standard deviation . Suppose 0 is the target value of An occurrence of an assignable cause results in a shift of size in , where is expressed in
units. It is assumed that remains constant following the occurrence of a shift until it is detected. A VSICWL
X chart to monitor is as described below.
The chart statistic is the standardized sample mean
i
Zi nX 0 , where X i , i = 1, 2, , is the
mean of ith sample of size is n drawn on X. Note that when
= 0 , Z i N (0, 1), and when = 0 + Z i N
( n , 1). Let t(i) be the length of sampling interval between the (i 1)st and ith trials, i = 1, 2, . Let L(i) be
the distance of each control limit and w(i) be the distance of each warning limit of the chart from its centerline for the
ith trial. The values of (t(i), L(i), w(i)) can be either ( t , L ,
shows a typical VSICWL X chart. We note that its appearance is same as the VCWL X chart proposed by
1 1 Mahadik (2013c).
w1 ) or ( t2 , L2 , w2 ), where t1 , t2 , L1 , L2 , w1 , and w2
At startup the values of (t(1), L(1), w(1)) can be
are such that
tmax
t1
t2
tmin ,
tmax and
tmin
chosen using an arbitrary probability distribution. In
practice, it is recommended to use the triplet ( t , L , w )
being the longest and shortest possible sampling intervals, respectively, > L1 L2 > 0, 0 < w1 < L1 , 0 < w2 < L2 , and w1 w2 . When Zi1 falls within (L(i 1), L(i
1)), the triplet of values of (t(i), L(i), w(i)), i = 2, 3, ,
between ( t , L , w ) and ( t , L , w ) is chosen
2 2 2
for the first trial to provide additional protection against the problems that may exist initially. The trial following an outofcontrol signal is again treated to be the first trial and the mechanism of choosing (t(i), L(i), w(i)) is restarted from that.
1 1 1
2 2 2
Note that when L = L , a VSICWL X chart is
according to the following rule
1 2
(t1 , L1 ,w1 ),
if Zi1 I1
a VSIWL X chart proposed by Mahadik (2013d) and
(t(i), L(i), w(i)) (t
, L ,w ),
if Z
I ,
when
t1 =
t2 , it is a VCWL X chart proposed by
where
2 2 2
I1 = [w(i 1), w(i 1)] and
i1 2
I 2 = (L(i 1), w(i
Mahadik (2013c). In the next section, expressions for performance measures for a VSICWL X chart are
1)) (w(i 1), L(i 1)).
The chart signals an outofcontrol state at the ith
derived.
trial, i = 1, 2, , if Z i
falls beyond (L(i), L(i)). Figure 1
L1 L2
w1 w2
Zi 0
w2
w1
L2
L1
Sample Number
Figure 1: A VSICWL X chart

PERFORMANCE MEASURES
The appropriate measures of statistical
performance of a VSICWL X chart are the steadystate average time to signal (SSATS) and the average number of samples to signal (ANSS). SSATS is the expected value of the time between a shift that occurs at some random time after the process starts and the time the chart signals while ANSS is the expected value of the number of samples taken from a shift to the time the chart signals. The administrative performance can be measured through average number of switches to signal (ANSW). ANSW is
the expected value of the number of switches between two sampling interval lengths from a shift to the signal.
Let SSATS, ANSS, and ANSW be the SSATS, ANSS, and ANW, respectively of a control chart when the process mean has shifted from 0 to 1 = 0 + The expressions for SSATS and ANSS are derived below using a Markov chain approach.
Henceforth, the ith trial refers to the ith trial after a shift when i > 0 and the last trial before the (i + 1)st trial when i 0. Also, Z i refers to the sample point
corresponding to the ith trial.
Define the three states 1, 2, and 3 of the Markov Chain corresponding to whether a sample point for the ith
trial is plotted in I1 , I 2 , and I 3 = ( , L(i)] [L(i),
1, if (Zi1 I1 , Zi I2 )
2 , if (Zi1 I2 , Zi I1 )
), respectively i = 1, 2, .. State 3 is the absorbing state,
Yi 3 , if (Zi1 I1 , Zi I1 )
, i = 1, 2,
as the process of taking samples is restarted when a sample
point falls in region I . The transition probability matrix is
4 , if (Z
i1
I2
, Zi
I2 )
3
given by
5 , if Zi
L(i)
p p
11 12
P p p
p
13
p ,
It is easy to see that { Yi , i = 1, 2, } is a Markov
Chain with transition probability matrix
21 22
23
0 p 0 p p
0 0
1
p
21
0 p
22 23
0 p
where p
is the transition probability that j is the prior
12 11
13
jk Q p
0 p
0 p .
i
2
i 1
1
state and k is the current state, when the process mean has shifted by . For example,
12 11
p
0 21 0
13
p
p
22 23
12
p =
Pr [ Z I  Z
I ]
0
0 0 0
1
Pr [ Z i I 2  L(i) = L1 , w(i) = w1 ]
= Pr [ L1 < Z i < w1 ]
Then, the expression for ANSW is given by
1
1
ANSW a(I1 Q ) e
where, I is the identity matrix of order 4, Q is the sub
+ P[ w1 < Z i < L1 ]
w1 n ) L1
n ) + L1
1 1
matrix of Q that contains the probabilities associated with the transient states only, e = (1,1, 0, 0) , and a =
n ) w1 n ),
(a , a , a , a ) , a being the initial probability of
1 2 3 4 j
where is the cumulative distribution function of standard normal variate.
state j, j = 1, 2, 3, 4, given by
b p , j 1
Then, SSATS and ANSS are given by
SSATS b(I P )1 t E(U) (1)
1 12
b p , j 2
1 a Pr [Y
j] 2 21 .
and
j 1
b p , j 3
ANSS
b(I P )1 1,
1 11
b p ,
j 4
1
2 22
where I is the identity matrix of order 2, P1
is the sub
The following section evaluates the performances
matrix of P that contains the probabilities associated with the transient states only, t = ( t1 , t2 ), 1 = (1, 1), and
of VSICWL X charts in comparison with the VSIWL and VCWL X charts.
b = ( b1 ,
b2 ), b j being the conditional probability that
Z0 falls in
I j given that it falls within the control limits,

PERFORMANCE EVALUATION OF VSICWL X
CHARTS
j = 1, 2. We note that b2 = 1 b1 . The Expression for b1
is derived by Mahadik (2013d) and is
2( w2 ) 1
b 2( L2 ) 1 .
1 2( w ) 1 2( w ) 1
1 1 2
2( L1 ) 1 2( L2 ) 1
In this section, the performances of VSICWL X
charts are evaluated by comparing that with that of VSIWL
and VCWL X charts. The three charts are designed such that their incontrol statistical performances are matched. Such charts are called matched charts. The matching of the charts is achieved by choosing the values of design parameters of the charts such that E[t(1)] as well as
P[ Z > L(i)] are the same for all the charts.
E(U) in equation (1) is the expected value of the 1
time U between the 0th trial and the shift. Assuming that an
Obviously, the ANSS values of the matched
assignable cause of a process shift occurs according to a VSIWL and VSICWL charts are the same. Further, we note
Poisson process, it can be shown that E(U) = 2 . Hence,
1
SSATS b(I P )1 t 2 .
The expression for ANSW is also derived using a Markov Chain approach. For, let
that the VCWL charts are free from the problem of switches between the sampling interval lengths and thus have better administrative performance than that of VSICWL charts.
Table 1 shows the design parameters of two sets
of the matched VCWL, VSIWL, and VSICWL X charts while tables 2, 3 and 4, respectively, show the ANSS,
SSATS, and ANSW values of these charts for the shifts in mean of various sizes. These tables clearly indicate that the SSATS values of VSICWL charts are uniformly smaller than that of VCWL and VSIWL charts for a wide range of shift size. The ANSS values of VSICWL charts are smaller than that of VSIWL charts for small to moderate shifts and are similar to that of VSIWL charts for large shifts. Also,
the outofcontrol ANSW values of VSICWL charts are smaller than that of VSIWL charts although the incontrol ANSW values of the two charts are almost the same. Thus, the statistical performance of a VSICWL chart is superior to that of VSIWL and VCWL charts while its administrative performance is superior to that of a VSIWL chart.
Table 1: Design parameters of the matched charts
Chart
n
t1
t2
L1
L2
w1
w2
Set 1
VCWL
4
1.00
1.00
3.20
2.26
2.00
1.00
VSIWL
4
1.05
0.20
3.00
3.00
2.00
1.00
VSICWL
4
1.05
0.20
3.20
2.26
2.00
1.00
Set 2
VCWL
3
1.00
1.00
3.20
2.15
2.00
1.75
VSIWL
3
1.04
0.10
3.00
3.00
2.00
1.75
VSICWL
3
1.04
0.10
3.20
2.15
2.00
1.75
Table 2: ANSS values of the matched charts
Chart
ANSS values for the shift in mean of size
0
0.25
0.5
0.75
1
1.5
Set 1
VCWL/ VSICWL
370.40
138.25
30.93
9.44
4.26
1.81
1.21
1.03
1.00
VSIWL
370.40
155.22
43.89
14.97
6.30
2.00
1.19
1.02
1.00
Set 2
VCWL/
VSICWL
370.43
173.1
48.81
16.09
6.85
2.41
1.45
1.13
1.02
VSIWL
370.40
184.24
60.69
22.48
9.76
2.91
1.47
1.10
1.01
Table 3: SSATS values of the matched charts
Chart
SSATS values for the shift in mean of size
Set 1
VCWL
369.90
137.75
30.43
8.94
3.76
1.31
0.71
0.53
0.50
VSIWL
369.90
151.62
39.90
11.94
4.19
0.97
0.56
0.51
0.50
VSICWL
370.03
133.57
26.65
6.67
2.43
0.83
0.56
0.51
0.50
Set 2
VCWL
369.93
172.61
48.31
15.59
6.35
1.91
0.95
0.63
0.52
VSIWL
369.90
180.42
55.74
18.30
6.65
1.35
0.64
0.52
0.50
VSICWL
369.93
169.56
44.91
13.24
4.81
1.20
0.63
0.52
0.50
Table 4: ANSW values of the matched charts
Chart
ANSW values for the shift in mean of size
Set 1
VSIWL
29.84
19.26
10.29
5.27
2.50
0.55
0.14
0.02
0.00
VSICWL
30.30
16.88
6.60
2.62
1.23
0.49
0.18
0.03
0.00
Set 2
VSIWL
30.30
20.90
12.07
6.77
3.54
0.88
0.28
0.08
0.01
VSICWL
30.77
19.98
9.85
4.91
2.57
0.87
0.36
0.12
0.02

CONCLUSIONS
L2 : distance of each control limit from the centerline for
The proposed chart is the fusion of VSIWL and
VCWL X charts. The expressions for performance
the ith trial when Z
i1
I2
measures, viz, SSATS, ANSS, and ANSW for this chart
p : Pr [ Z I  Z I ]
jk i k i1 j
are developed using a Markov chain approach. This chart exhibit better statistical performance than that of VSIWL and VCWL X charts. Also, its administrative performance is better than that of VSIWL X chart.
APPENDIX: NOTATION
X : quality characteristic to be monitored
: mean of X
standard deviation of X
0 : target value of
size of shift in in units Z i : standardized sample mean n : sample size
t(i) : length of the sampling interval between the (i 1)st
and ith trials
w(i) : distance of each warning limit from the centerline for the ith trial
L(i) : distance of each control limit from the centerline for the ith trial
I1 : [w(i), w(i)] for ith trial
I 2 : (L(i), w(i)) (w(i), L(i)) for ith trial
I 3 : ( , L(i)] [L(i), )
t1 : long sampling interval
: cumulative distribution function of standard normal variate
b j : conditional probability that Z0 falls in I j given that it falls within its control limits
U : time between the 0th trial and the shift
REFERENCES

Reynolds, M. R., Jr., Amin, R. W., Arnold, J. C., and Nachlas, J. A.
(1988). X charts with variable sampling interval. Technometrics
30:181192.

Prabhu, S. S., Runger, G. C., and Keats, J. B. (1993). An adaptive
sample size X chart. International Journal of Production Research 31:28952909.

Costa, A. F. B. (1994). X charts with variable sample size.
Journal of Quality Technology 26:155163.

Prabhu, S. S., Montgomery, D. C., and Runger, G. C. (1994). A
combined adaptive sample size and sampling interval X control scheme. Journal of Quality Technology 26:164176.

Costa, A. F. B. (1999). X charts with variable parameters. Journal of Quality Technology 31:408416.

Mahadik, S. B. and Shirke, D. T. (2009) A special variable sample
size and sampling interval X chart, Communications in Statistics
Theory and Methods, Vol. 38 No. 8, pp. 12841299.

Amin, R. W. and Letsinger, W. C. (1991). Improved switching rules in control procedures using variable sampling intervals. Communications in Statistics Computations and Simulations 20:205230.

Amin, R. W. and Hemasinha, R. (1993). The switching behavior of
t2 : short sampling interval
w1 : distance of each warning limit from the centerline for
X charts with variable sampling intervals. Communications in Statistics Theory and Methods 22:20812102.

Mahadik, S. B. (2012a) Exact results for variable sampling interval
the ith trial when Z
i1
I1
Shewhart control charts with runs rules for switching between sampling interval lengths, Communications in statistics Theory
w2 : distance of each warning limit from the centerline for
and Methods, Vol. 41 No. 24, pp. 44534469.
the ith trial when Z
i1
I2

Mahadik, S. B. (2012b) X charts with variable sample size, sampling interval, and warning limits,Quality and Reliability
L1 : distance of each control limit from the centerline for the ith trial when Zi1 I1
Engineering International, Vol 29, No. 4, pp. 535544.

Chen, YK, Liao, HC, and Chang, HH. (2011). Reevaluation of
adaptive X control charts: a costeffectiveness perspective. International Journal of Innovative Computing, Information and Control 7:12291242.

Dai, Y., Luo, Y., Li, Z., and Wang, Z. (2011). A new adaptive CUSUM control chart for detecting the multivariate process mean. Quality and Reliability Engineering International 27:877884.

Faraz, A. and Saniga, E. (2011). A unification and some corrections to Markov chain approaches to develop variable ratio sampling scheme control charts. Statistical Papers 52:799811.

Nenes, G. (2011). A new approach for the economic design of ully adaptive control charts. International Journal of Production Economics 131:631642.

Kooli, I., Limam, M. (2011). Economic design of an attribute np control chart using a variable sample size. Sequential Analysis 30:145159.

Lee, PH. (2011). Adaptive R charts with variable parameters.
Computational Statistics and Data Analysis 55:20032010.

Zhang, J., Li, Z., and Wang, Z. (2011) A new adaptive control chart for monitoring process mean and variability, The International Journal of Advanced Manufacturing Technology 60: 10311038.

Lee PH. and Lin CS. (2012) Adaptive Max charts for monitoring process mean and variability, Journal of the Chinese Institute of Industrial Engineers 29: 193205.

Huang, C. C. (2013) Max control chart with adaptive sample sizes for jointly monitoring process mean and standard deviation, Journal of the Operational Research Society, doi:10.1057/jors.2013.155

Mahadik, S. B. (2013a) Variable sample size and sampling interval
X charts with runs rules for switching between sample sizes and
sampling interval lengths, Quality and Reliability Engineering International, Vol 29, No.1, pp. 6376.

Mahadik, S. B. (2013b) Variable sample size and sampling interval
Hotellings T 2 charts with runs rules for switching between sample sizes and sampling interval lengths, International Journal of Reliability, Quality and Safety Engineering, Vol. 20, No. 4, DOI: 10.1142/S0218539313500150

Kuo, TI. and Lee, PH. (2013) Design of adaptive s control charts, Journal of Statistical Computation and Simulation 83: 2002 2014.

Seif, A., Faraz, A., and Sadeghifar, M. (2014) Evaluation of the economic statistical design of the multivariate T2 control chart with multiple variable sampling intervals scheme: NSGAII approach, Journal of Statistical Computation and Simulation, DOI:10.1080/00949655.2014.931404

Faraz, A. Heuchenne, C., Saniga E., and Costa A. F. B. (2014) Doubleobjective economic statistical design of the VP T2 control chart: Wald's identity approach, Journal of Statistical Computation and Simulation 84: 21232137.

Mahadik, S. B. (2013c) X charts with variable control and warning limits, Economic Quality Control, Vol28, No. 2, pp. 117 124.

Mahadik, S. B. (2013d) X charts with variable sampling interval and warning limits, Journal of Academia and Industrial Research, Vol 2, No. 2, pp. 103110.