X Charts with Variable Sampling Interval, Control limits, and Warning Limits

DOI : 10.17577/IJERTV3IS080722

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X Charts with Variable Sampling Interval, Control limits, and Warning Limits

X Charts with Variable Sampling Interval, Control limits, and Warning Limits

Department of Statistics, Solapur University, Solapur, INDIA 413255

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

Key words: Adaptive control chart, average number of samples to signal, average number of switches to signal, steady-state average time to signal.

1. INTRODUCTION

Shewhart control chart is an effective on-line 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.

2. 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

w1 ) or ( t2 , L2 , w2 ), where t1 , t2 , L1 , L2 , w1 , and w2

At start-up 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 out-of-control 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 out-of-control 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

3. PERFORMANCE MEASURES

The appropriate measures of statistical

performance of a VSICWL X chart are the steady-state 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,

4. 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 in-control 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 out-of-control ANSW values of VSICWL charts are smaller than that of VSIWL charts although the in-control 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
5. 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

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