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

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, 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

   1 1 Mahadik (2013c).

   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

REFERENCES

1. Reynolds, M. R., Jr., Amin, R. W., Arnold, J. C., and Nachlas, J. A.

   (1988). X charts with variable sampling interval. Technometrics

   30:181-192.

2. Prabhu, S. S., Runger, G. C., and Keats, J. B. (1993). An adaptive

   sample size X chart. International Journal of Production Research 31:2895-2909.

3. Costa, A. F. B. (1994). X charts with variable sample size.

   Journal of Quality Technology 26:155-163.

4. 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 26:164-176.

5. Costa, A. F. B. (1999). X charts with variable parameters. Journal of Quality Technology 31:408-416.

6. 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. 1284-1299.

7. Amin, R. W. and Letsinger, W. C. (1991). Improved switching rules in control procedures using variable sampling intervals. Communications in Statistics Computations and Simulations 20:205-230.

8. 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 22:2081-2102.

9. 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. 4453-4469.

   the ith trial when Z

   i1

   I2

10. 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. 535-544.

11. Chen, Y-K, Liao, H-C, and Chang, H-H. (2011). Re-evaluation of

    adaptive X control charts: a cost-effectiveness perspective. International Journal of Innovative Computing, Information and Control 7:1229-1242.

12. 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 27:877-884.

13. 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 52:799-811.

14. Nenes, G. (2011). A new approach for the economic design of ully adaptive control charts. International Journal of Production Economics 131:631-642.

15. Kooli, I., Limam, M. (2011). Economic design of an attribute np control chart using a variable sample size. Sequential Analysis 30:145-159.

16. Lee, P-H. (2011). Adaptive R charts with variable parameters.

    Computational Statistics and Data Analysis 55:2003-2010.

17. 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: 1031-1038.

18. Lee P-H. and Lin C-S. (2012) Adaptive Max charts for monitoring process mean and variability, Journal of the Chinese Institute of Industrial Engineers 29: 193-205.

19. 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

20. 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. 63-76.

21. 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

22. Kuo, T-I. and Lee, P-H. (2013) Design of adaptive s control charts, Journal of Statistical Computation and Simulation 83: 2002- 2014.

23. 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: NSGA-II approach, Journal of Statistical Computation and Simulation, DOI:10.1080/00949655.2014.931404

24. Faraz, A. Heuchenne, C., Saniga E., and Costa A. F. B. (2014) Double-objective economic statistical design of the VP T2 control chart: Wald's identity approach, Journal of Statistical Computation and Simulation 84: 2123-2137.

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

26. Mahadik, S. B. (2013d) X charts with variable sampling interval and warning limits, Journal of Academia and Industrial Research, Vol 2, No. 2, pp. 103-110.