Variable sampling interval run sum median charts with known and estimated process parameters

Computers & Industrial Engineering xxx (xxxx) xxx–xxx

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Variable sampling interval run sum median charts with known and estimated process parameters Sajal Sahaa, Michael B.C. Khooa, , P.S. Ngb, Z.L. Chongb ⁎

a b

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia Department of Physical and Mathematical Science, Faculty of Science, Universiti Tunku Abdul Rahman, 31900 Kampar, Perak, Malaysia

ARTICLE INFO

ABSTRACT

Keywords: Median control charts Known process parameters Estimated process parameters Markov chain Phase-I samples

The variable sampling interval run sum (VSI RS) median chart with known and estimated process parameters are proposed for efficiently monitoring the process mean. The construction, optimal designs, performance assessment and implementation of the VSI RS median chart, are discussed in this paper. The VSI RS median chart is compared with the Shewhart median, run sum median and exponentially weighted moving average median charts, in terms of average time to signal and standard deviation of the time to signal criteria, when process parameters are known, while the average of the average time to signal criterion is used when process parameters are estimated. For both cases of known and estimated process parameters, the VSI RS median chart is found to outperform the Shewhart median chart for all sizes of mean shifts and the exponentially weighted moving average median chart for moderate and large mean shifts. When process parameters are estimated, the standard deviation of the average time to signal criterion is used to evaluate the average of the average time to signal performance of the VSI RS median chart, when the process is in-control. Based on the standard deviation of the average time to signal criterion, the number of Phase-I samples required by the chart to have an in-control average of the average time to signal performance close to its in-control average time to signal performance is recommended.

1. Introduction The mean chart operates by monitoring the mean value of the process, while the median chart operates by monitoring the median value of the process. Even though the mean chart is more efficient than the median chart, the mean chart is not robust against outliers or unusual data points that violate the normality assumption. This phenomenon may lead to wrong conclusions, as practitioners could interpret that some process disturbances have occurred in the data but in actual fact, these process disturbances are due to outliers instead (see Khoo, 2005; Yang, Pai, & Wang, 2010). Thus, the median charts were developed by researchers to address this drawback. The advantages of the median chart are rooted in the chart’s simpler implementation and outlier resistant, hence, the median type charts are extensively explored by researchers. For instance, Janacek and Meikle (1997) proposed the median chart to overcome the problem of nonnormality in the samples. To improve the sensitivity of the median chart in the detection of out-of-control signals, Castagliola (2001) developed the exponentially weighted moving average (EWMA) median chart to monitor the process mean by using the sample range statistic to

⁎

compute the control limits. Khoo (2005) proposed the median chart to detect permanent shifts in the process mean. Park (2009) studied the performance of the median chart by using different bootstrap methods, while Graham, Human, and Chakraborti (2010) proposed a non-parametric Shewhart type chart based on the median to monitor the location of a continuous variable in the Phase-I process. However, all these median charts were proposed under the assumption of known process parameters. In practice, process parameters are rarely known and the estimation of these parameters is inaccurate if a limited number of in-control Phase-I samples is employed in the estimation. To circumvent the problem of estimation error that leads to performance deterioration, control charts with estimated process parameters are extensively studied by researchers. A comprehensive review of charts involving parameter estimation can be found in Jensen, Jones-Farmer, Champ, and Woodall (2006) and Psarakis, Vyniou, and Castagliola (2014). Nevertheless, very little attention is given to parameter estimation for median type charts. The median type chart with estimated process parameters was first presented by Castagliola and Figueiredo (2013). Castagliola, Maravelakis, and Figueiredo (2016) suggested the EWMA

Corresponding author. E-mail addresses: [email protected] (S. Saha), [email protected] (M.B.C. Khoo), [email protected] (P.S. Ng), [email protected] (Z.L. Chong).

https://doi.org/10.1016/j.cie.2018.10.049 Received 5 May 2018; Received in revised form 23 October 2018; Accepted 27 October 2018 0360-8352/ © 2018 Elsevier Ltd. All rights reserved.

Please cite this article as: SAHA, S., Computers & Industrial Engineering, https://doi.org/10.1016/j.cie.2018.10.049

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median chart with estimated process parameters. Even though the EWMA median chart is quick in detecting small shifts, the chart does not plot the sample median statistic in its original scale of measurement, hence, the EWMA median chart could be less comprehensible to practitioners in process monitoring. In this study, we propose and recommend the use of the VSI RS median chart as it has a superb performance in detecting moderate and large shifts besides enabling the sample median statistic to be plotted on its original scale of measurement. See Champ and Rigdon (1997), Lim, Khoo, Teoh, and Haq (2017) and Saha, Khoo, Teoh, and Lee (2017) for some research works on RS charts. Numerous findings have shown that control charts with the variable sampling interval (VSI) scheme are substantially more efficient than those with the fixed sampling interval (FSI) scheme. The VSI scheme is based on the idea of using two sampling intervals (i.e. long sampling interval and short sampling interval) and a fixed sample size, where the choice of the sampling interval length depends on where the previous sample is plotted on the chart. Reynolds, Amin, Arnold, and Nachlas (1988) adopted the VSI scheme on the X chart. Since then, the use of the VSI procedure on various types of charts is extensively explored, most of which assume that process parameters are known (see Amdouni, Castagliola, Taleb, and Celano (2017), Chew, Khoo, Teh, and Castagliola (2015), and Kosztyán and Katona (2018), to name a few). Nevertheless, very little research is concerned with the VSI type charts with estimated process parameters. Jensen, Bryce, and Reynolds (2008) evaluated the impact of parameter estimation on adaptive X chart by means of simulation. The results show that the run length properties of adaptive X chart are seriously affected by the estimation of process parameters and that a larger number of Phase-I samples is needed for the chart with estimated process parameters to have a similar performance to its known process parameters counterpart. Similar findings were observed by Zhang, Castagliola, Wu, and Khoo (2012). Even though the RS and VSI control charting procedures are found to be efficient and useful, no research has been made to combine these two methods. This paper takes research works on the RS and VSI methods a step further by integrating the two methods, where the variable sampling interval run sum (VSI RS) median charts with known and estimated process parameters are proposed and their performances are investigated. As it is interesting to investigate how the new VSI RS median charts fare, a comparison with existing median type charts when both process parameters are known and estimated, is conducted in this paper. Formulae to compute the average time to signal (ATS) and standard deviation of the time to signal (SDTS) for the known process parameters case; as well as the average of the ATS (AATS) and standard deviation of the ATS (SDATS) for the estimated process parameters case are derived. Optimal designs for minimizing the out-of-control ATS (ATS1) and out-of-control AATS (AATS1) values, of the VSI RS median charts with known and estimated process parameters, respectively, based on the Markov chain approach, are developed. Additionally, based on the SDATS performance metric, the recommended number of Phase-I samples in estimating process parameters so that the performance difference between the VSI RS median charts with known and estimated process parameters becomes negligible (or significantly reduced) is provided. This paper is structured as follows: Section 2 presents the properties of the VSI RS median charts with known and estimated process parameters and formulae for computing the performance measures. The optimization models of the VSI RS median chart in minimizing ATS1 and AATS1 values, for cases with known and estimated process parameters, respectively, are presented in Section 3. Section 4 compares the performance of the VSI RS median chart with that of the Shewhart (SH), EWMA and run sum (RS) median charts for both cases of known and estimated process parameters. The impact of the number of Phase-I samples on the VSI RS median chart with estimated process parameters is studied in Section 5. To demonstrate the application of the VSI RS

median chart with estimated process parameters, an illustrative example is presented in Section 6. Lastly, conclusions are drawn in Section 7. 2. Properties of the VSI RS median chart The properties of the VSI RS median charts with known and estimated process parameters are explained in Sections 2.1 and 2.2, respectively. 2.1. VSI RS median chart with known process parameters Suppose that {Yr ,1, Yr ,2, ...,Yr , n} are samples of size n, where r = 1, 2, …, is the sample number. These samples are assumed to be independent of one another and Yr , j (for j = 1, 2, …, n) follows a normal, N (µ 0 + 0, 02) distribution. Here, δ is the standardized mean shift, while µ 0 and 0 are the values of the in-control process mean and 0, the standard deviation, respectively. This indicates that when process mean has shifted, otherwise the process is statistically in-control. Let Yr be the sample median of sample r, i.e.

Yr =

Yr

if n is odd

((n + 1) 2)

Yr (n 2) + Yr (n 2 + 1) 2

if n is even

,

(1)

where (Yr (1), Yr (2), ...,Yr (n) ) is the ordered (ascending order) sample. Fig. 1 presents the K regions VSI RS median chart. The K regions above the centre line, CL (=µ 0 ) are assigned scores + S1, + S2 , …, + SK , while the K regions below the CL are assigned scores S1, S2 , …, SK . Note that S1 S2 ... SK . The regions above (i.e. Rl ) and below (i.e. Rlb ) the CL, for l = 1, 2, …, K are represented by

r th

Rl = [µ 0 + Al

1 0,

(2a)

µ 0 + Al 0 )

and

Rlb = (µ 0

Al 0, µ 0

Al

(2b)

1 0].

[3l (K 1)], for l = 1, 2, …, Here, A0 = 0 , AK = ∞ and Al = K − 1. The parameter k is a constant chosen to attain a specified ATS0 k n

Fig. 1. A graphical view of the K regions VSI RS median chart. 2

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value. The VSI RS median chart reacts based on the cumulative sums. The cumulative sums for accumulating scores when Yr falls above and below the CL, denoted by Ur and Lr , respectively, are obtained as follows:

Ur =

0 Ur

in region Rlb , for l = 1, 2, …, K − 1 is computed as

=F

Yr

0 Lr

1

if

Yr

CL

+ S (Yr ) if Yr < CL

+ Sl if

Yr

Rl

Sl if

Yr

Rlb

(3a)

,

,

Ur < + SK or

SK G

Ur < + SK G or

SK G < Lr

(

( Al

)|

n+1 n+1 , 2 2

= F Yr (µ 0 + Al 0)

µ0 0

(6)

n+1 n+1 , , 2 2

(7)

µ0 =

where ( · ) is the cdf of the standard normal random variable, while F ( · a , b) is the cdf of the beta random variable with parameters a and b. From Eqs. (6) and (7),

n+1 n+1 )| , 2 2

F

(Al

1

(9)

qT t

1

qT

2

(10)

(qT 1 ) 2 , 1 (2D t 1

t (2) ),

qT

P ),

(11)

When the process parameters µ 0 and 0 are unknown, it is necessary to estimate them from the in-control Phase-I dataset that comprises, say m samples, each of size n observations (Xr 1, Xr 2 , ...,Xr n ), for r = 1, 2, ...,m . Assume that there is independence between and within samples and Xrj , for r = 1, 2, ...,m and j = 1, 2, ...,n , is normally distributed with mean µ 0 and standard deviation 0 . The following estimators of the parameters µ 0 and 0 , suggested by Castagliola and Figueiredo (2013) are used in this study:

where l = 1, 2, …, K − 1 and F Yr ( · ) is the cumulative distribution function (cdf) of Yr . Here, F Yr (x ) is defined as (Castagliola and Figueiredo, 2013)

x

(8d)

2.2. VSI RS median chart with estimated process parameters

Rl ) 1 0 ),

n+1 n+1 , . 2 2

where P is the proportion of time for adopting the short sampling interval (d1) , while 1 – P is the proportion of time for using the long sampling interval (d2) .

Yr < µ 0 + Al 0) F Yr (µ 0 + Al

)|

1

ASI0 = d1 P + d2 (1

The probability of Yr falling in region Rl , denoted as Pl , is computed as follows:

1 0

(8c)

= (1, 0, 0, ...,0) is where 1 = (I Q) , 2 = (I Q ) the initial probability vector, Q is the transition probability matrix (TPM) for the transient states of the Markov chain model (see Appendix A), t = (t1, t2, ...,th ) is a column vector of sampling intervals, whose entry tu (for u = 1, 2, …, h) is the sampling interval (d1 or d2) adopted next when the current sample is in state u. Eqs. (5a) and (5b) provide guidelines for choosing the sampling intervals, d1 and d2 . The vector t (2) consists of the squares of the elements in vector t. Note that the dimension of Q is h × h, where the value of h depends on the scores {± S1, ± S2, ..., ± SK } of the VSI RS median chart. The matrix Dt is a diagonal matrix whose diagonal elements are taken from the vector t, while I is an identity matrix. Note that Eq. (9) computes the average time to signal starting from the first sample (Saccucci et al., 1992). In order to ensure a fair comparison with other control charts, the in-control average sampling interval (ASI 0) for all the competing charts have to be equal. The formula for computing ASI0 of the VSI RS median chart is given as follows (Saccucci et al., 1992):

Step 1. Specify the values of the parameters U0,L0 , n, K, d1, d2 and G. Step 2. Take a sample of size n and compute the sample median Yr (r being the sample number) using Eq. (1). Step 3. Compute the cumulative score Ur using Eq. (3a) if Yr Rl , and simultaneously set Lr = 0. Otherwise, compute the score Lr using Eq. (3b) when Yr Rlb , and simultaneously set Ur = 0 . Step 4. Adopt the sampling interval d1 for taking sample r + 1 if + SK G Ur < + SK or SK < Lr SK G. On the contrary, adopt d2 for taking sample r + 1 if 0 Ur < +SK G or SK G < Lr 0. SK , Step 5. Signal an out-of-control at sample r if Ur + SK or Lr where SK and SK serve as the signalling scores. Otherwise, the control flow returns to Step 2, where the next sample is taken.

Pl = Pr(Yr

n+1 n+1 , 2 2

)|

1

1t

where G is a positive integer set by the user to control the threshold for switching between the two sampling intervals, while d1 and d2 denote the short and long sampling intervals, respectively. Note that 0 < d1 < d2. The operation of the known process parameters based VSI RS median chart is given as follows:

= Pr(µ 0 + Al

).

and

(5a) (5b)

0,

(AK

( AK

ATS = qT

SDTS =

(Al

F

The ATS and SDTS values of the VSI RS median chart with known process parameters are computed by using the Markov chain approach presented in Saccucci, Amin, and Lucas (1992) and Jensen et al. (2008), respectively, as follows:

(4)

SK < Lr

F

PKb = F

and

F Yr (x ) = F

)

and

(3b)

for l = 1, 2, …, K and r = 1, 2, …. In this paper, the initial cumulative sums are set as U0 = L 0 = 0. The sampling interval between samples r and r + 1 is

d1 if + SK G

)|

1

When l = K, it is easily shown that

PK = 1

S (Yr ) =

Pl = F

( Al

(8b)

CL

for r = 1, 2, …, and the score function S (Yr ) is

d2 if 0

(

Rlb)

if Yr < CL

1 + S (Yr ) if

and

Lr =

Plb = Pr(Yr n+1 n+1 , 2 2

1 m

m

Xr

(12)

r=1

and

n+1 n+1 )| , 2 2

0

(8a)

=

1 1 d2, n m

m

Rr , r=1

(13)

where d2, n = E (Rr ) 0 is a constant tabulated for the case of normal distribution, Xr is the median of sample r and Rr = Xr (n) Xr (1) . Note

is obtained. By using a similar computation, the probability of Yr falling 3

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Table 1 Optimal parameters of the VSI RS median chart with known process parameters when ATS0 (= ) = 370, ASI0 (=t0 ) = 1, K {0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0} for n = 5.

{4, 7}, d1

{0.01, 0.1}, G

{3, 4} and

K=7

K=4 d2

S1

S2

S3

S4

k

d2

S1

S2

S3

S4

S5

S6

S7

d1 = 0.01, G = 3 0.2 1.7907 0.4 1.6785 0.6 1.6785 0.8 1.6785 1.0 1.6785 1.2 1.6785 1.4 1.6785 1.6 1.6785 1.8 1.6785 2.0 1.6785

1.2086 1.3748 1.3748 1.3748 1.3748 1.3748 1.3748 1.3748 1.3748 1.3748

1 0 0 0 0 0 0 0 0 0

2 2 2 2 2 2 2 2 2 2

4 3 3 3 3 3 3 3 3 3

10 6 6 6 6 6 6 6 6 6

1.5388 1.5448 2.1100 1.6870 1.6870 1.6870 1.6870 1.6870 1.7134 1.7134

1.2784 1.2756 1.296 1.3688 1.3688 1.3688 1.3688 1.3688 1.3514 1.3514

0 0 0 0 0 0 0 0 0 0

1 1 1 0 0 0 0 0 0 0

2 2 2 2 2 2 2 2 2 2

3 3 3 2 2 2 2 2 2 2

4 4 4 3 3 3 3 3 3 3

4 5 4 4 4 4 4 4 6 6

9 9 6 6 6 6 6 6 6 6

d1 = 0.01, G = 4 0.2 1.4325 0.4 1.4325 0.6 1.4418 0.8 1.4418 1.0 1.3526 1.2 1.3526 1.4 1.3526 1.6 1.3526 1.8 1.3526 2.0 1.3526

1.5893 1.5893 1.5794 1.5794 1.6823 1.6823 1.6823 1.6823 1.6823 1.6823

0 0 0 0 0 0 0 0 0 0

3 3 1 3 2 2 2 2 2 2

4 4 2 5 3 3 3 3 3 3

10 10 4 10 8 8 8 8 8 8

1.6861 1.6897 1.6897 2.8505 1.3739 1.3739 1.3739 1.3739 1.3739 1.3739

1.5412 1.5381 1.5381 1.5973 1.6560 1.6560 1.6560 1.6560 1.6560 1.6560

0 0 0 0 0 0 0 0 0 0

1 1 1 1 0 0 0 0 0 0

2 2 2 2 2 2 2 2 2 2

3 3 3 3 2 2 2 2 2 2

4 4 4 4 3 3 3 3 3 3

4 5 5 4 5 5 5 5 5 5

8 8 8 4 8 8 8 8 8 8

d1 = 0.1, G 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

=3 1.7907 1.6785 1.6785 1.7497 1.7497 1.7497 1.7497 1.8961 1.8961 1.8961

1.1899 1.3410 1.3410 1.2988 1.2988 1.2988 1.2988 1.2268 1.2268 1.2268

1 0 0 0 0 0 0 0 0 0

2 2 2 1 1 2 2 1 1 1

4 3 3 2 2 3 3 3 3 3

10 6 6 3 3 5 5 3 3 3

1.5448 1.5448 2.11 1.687 1.7413 1.7929 1.7929 1.2779 1.2779 1.2779

1.2509 1.2509 1.2693 1.3356 1.3033 1.2758 1.2758 1.2178 1.2178 1.2178

0 0 0 0 0 0 0 0 0 0

1 1 1 0 0 0 0 0 0 0

2 2 2 2 2 2 2 0 0 0

3 3 3 2 3 3 3 3 3 3

4 4 4 3 3 4 4 4 4 4

5 5 4 4 4 6 6 5 5 5

9 9 6 6 6 6 6 9 9 9

d1 = 0.1, G 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

=4 1.4325 1.4418 1.4418 1.4418 1.4418 1.4418 1.4418 1.4418 1.8212 1.8212

1.5362 1.5271 1.5271 1.5271 1.5271 1.5271 1.5271 1.5271 1.2613 1.2613

0 0 0 0 0 0 0 0 0 0

3 1 1 2 2 1 1 1 1 1

4 2 2 5 5 2 2 2 4 4

10 4 4 8 8 4 4 4 4 4

1.6861 1.6897 1.6897 2.8505 2.8505 1.3739 1.3739 1.5866 1.5399 1.5062

1.4924 1.4896 1.4896 1.5434 1.5434 1.5968 1.5968 1.4039 1.4404 1.4685

0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 0 0 0 0 0

2 2 2 2 2 2 2 2 1 2

3 3 3 3 3 2 2 3 1 2

4 4 4 4 4 3 3 5 2 3

4 5 5 4 4 5 5 8 4 8

8 8 8 4 4 8 8 8 4 8

k

that Xr (1) and Xr (n) denote the smallest and largest observations, respectively, in sample r. The regions above (i.e. Rl ) and below (i.e. Rlb) the CL of the estimated process parameters based VSI RS median chart are

Rl = [µ 0 + Al

1 0,

µ 0 + Al

0)

Pl = Pr(Yr = Pr(µ 0 + Al

1 0

= F Yr (µ 0 + Al

0)

(

Pl = F (14b)

F

respectively, where l = 1, 2, …, K, A0 = 0 , AK = and k Al = n [3l (K 1)]. The parameter k is a constant chosen to attain a specified value of the in-control average of the average time to signal (AATS0 ) criterion. The operation of the VSI RS median chart with estimated process parameters is similar to that of the known process parameters based chart, explained in Section 2.1. The probability of Yr falling in the region Rl , where l = 1, 2, …, K – 1, is computed as

=F

Al

0,

µ0

Al

1 0],

F Yr (µ 0 + Al

0)

(15)

1 0 ).

By adopting Eq. (7), Eq. (15) becomes

(14a)

and

Rlb = (µ 0

Rl ) Yr < µ0 + Al

µ0

µ0 0

(

µ0

+ Al µ0

0

(U + Al V

F

(

where U =

0

+ Al

(

)

0

)

0

1 0

)|

µ0 0

and V =

)| 0 0

n+1 n+1 , 2 2

n+1 n+1 , 2 2

(U + Al 1 V

µ0

n+1 n+1 , 2 2

)

n+1 n+1 , 2 2

),

(16a)

.

Similarly, the probability of Yr falling in the region Rlb , where l = 1, 2, …, K – 1, is obtained as

4

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Table 2 Optimal parameters of the VSI RS median chart with known process parameters when ATS0 (= ) = 370, ASI0 (=t0 ) = 1, K {0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0} for n = 7.

{4, 7}, d1

{0.01, 0.1}, G

{3, 4} and

K=7

K=4 d2

S1

S2

S3

S4

k

d2

S1

S2

S3

S4

S5

S6

S7

d1 = 0.01, G = 3 0.2 1.8145 0.4 1.7014 0.6 1.7014 0.8 1.7014 1.0 1.7014 1.2 1.7014 1.4 1.7014 1.6 1.7014 1.8 1.9201 2.0 1.9201

1.2087 1.3748 1.3748 1.3748 1.3748 1.3748 1.3748 1.3748 1.2500 1.2500

1 0 0 0 0 0 0 0 0 0

2 2 2 2 2 2 2 2 1 1

4 3 3 3 3 3 3 3 3 3

10 6 6 6 6 6 6 6 3 3

1.5654 2.1386 2.1386 1.7098 1.7098 1.7098 1.7098 1.7362 1.7362 1.2818

1.2755 1.2960 1.2960 1.3690 1.3690 1.3690 1.3690 1.3516 1.3516 1.2491

0 0 0 0 0 0 0 0 0 0

1 1 1 0 0 0 0 0 0 0

2 2 2 2 2 2 2 2 2 0

3 3 3 2 2 2 2 2 2 2

4 4 4 3 3 3 3 3 3 2

5 5 5 4 4 4 4 6 6 3

9 6 6 6 6 6 6 6 6 6

d1 = 0.01, G = 4 0.2 1.4513 0.4 1.4607 0.6 1.4607 0.8 1.4607 1.0 1.3702 1.2 1.3702 1.4 1.3702 1.6 1.3702 1.8 1.3592 2.0 1.3592

1.5902 1.5802 1.5802 1.5802 1.6840 1.6840 1.6840 1.6840 1.6976 1.6976

0 0 0 0 0 0 0 0 0 0

2 2 1 3 2 2 2 2 1 1

3 5 2 5 3 3 3 3 1 1

7 8 4 10 8 8 8 8 4 4

1.7088 1.7123 2.8883 1.3917 1.3917 1.3917 1.3917 1.3917 1.3596 1.5233

1.5415 1.5390 1.5980 1.6576 1.6576 1.6576 1.6576 1.6576 1.6972 1.5182

0 0 0 0 0 0 0 0 0 0

1 1 1 0 0 0 0 0 0 0

2 2 2 2 2 2 2 2 2 1

3 3 3 2 2 2 2 2 2 1

4 4 3 3 3 3 3 3 2 1

4 5 3 5 5 5 5 5 3 4

8 8 4 8 8 8 8 8 8 4

d1 = 0.1, G 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

=3 1.4513 1.7014 1.7729 1.7729 1.7729 1.7729 1.9201 1.9201 1.9201 1.9201

1.0898 1.3411 1.2992 1.2992 1.2992 1.2992 1.2276 1.2276 1.2276 1.2276

0 0 0 0 0 0 0 0 0 0

2 2 2 1 1 2 1 1 1 1

3 3 5 2 2 3 3 3 3 3

7 6 6 3 3 5 3 3 3 3

1.5654 1.5654 2.1386 1.7646 1.8165 1.8165 1.2941 1.2941 1.9337 1.2818

1.2508 1.2508 1.2695 1.3036 1.2759 1.2759 1.2191 1.2191 1.2221 1.2268

0 0 0 0 0 0 0 0 0 0

1 1 1 0 0 0 0 0 0 0

2 2 2 2 2 2 0 0 2 0

3 3 3 3 3 3 3 3 3 2

4 4 4 3 4 4 4 4 6 2

5 5 5 4 6 6 5 5 6 3

9 9 6 6 6 6 9 9 6 6

d1 = 0.1, G 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

=4 1.4513 1.4607 1.4607 1.4607 1.4607 1.4607 1.4607 1.8438 1.8438 1.8438

1.5369 1.5278 1.5278 1.5278 1.5278 1.5278 1.5278 1.2625 1.2625 1.2625

0 0 0 0 0 0 0 0 0 0

2 2 1 1 1 2 1 1 1 1

3 5 2 2 2 5 2 4 4 4

7 8 4 4 4 8 4 4 4 4

1.7088 1.7123 2.8883 2.8885 1.3917 1.3917 1.6069 1.5595 1.5252 1.2300

1.4927 1.4904 1.544 1.5441 1.5982 1.5982 1.4054 1.4415 1.4697 1.2617

0 0 0 0 0 0 0 0 0 0

1 1 1 1 0 0 0 0 0 0

2 2 2 2 2 2 2 2 2 0

3 3 3 3 2 2 3 2 2 2

4 4 3 4 3 3 5 5 3 2

4 5 4 4 5 5 8 8 8 3

8 8 4 4 8 8 8 8 8 8

k

Plb = Pr(Yr = Pr(µ 0

Al

0

= F Yr (µ 0

Al

1 0)

=F

(

(U

F

(

Al (U

< Yr 1V

Al V

where the vectors t and q, and matrix I are defined in Section 2.1, while Q is the TPM (for the transient states) of the estimated process parameters based chart. The procedure for obtaining Q is similar to that for determining Q (explained in the Appendix A), except that the probabilities Pl and Plb are replaced by Pl and Plb , respectively, for l = 1, 2, …, K. As the Phase-I samples that practitioners use differ, the ATS values obtained from Eq. (17) vary among practitioners. This variation is defined as the practitioner to practitioner variability (Saleh, Mahmoud, Jones-Farmer, Zwetsloot, & Woodall, 2015). To address this problem, the average of the ATS values (denoted as AATS) or simply the expectation of ATS , i.e. E (ATS) is usually adopted. To consider the amount of practitioner to practitioner variability for the VSI RS median chart, the standard deviation of ATS (denoted as SDATS) discussed in Jones and Steiner (2012) is used. The AATS and SDATS values are computed as follows (Hu and Castagliola, 2017):

Rlb) µ0

Al

F Yr (µ 0

1 0)

Al

n+1 n+1 , 2 2

)| )|

0)

) ).

n+1 n+1 , 2 2

(16b)

When l = K, the probabilities

PK = 1

F

(U + AK

1V

)|

n+1 n+1 , 2 2

(16c)

and

PKb = F

(U

AK

1V

)|

n+1 n+1 , 2 2

(16d)

are obtained. For the VSI RS median chart with estimated process parameters, the ATS is computed as

ATS =

qT (I

Q)

1t

qT t ,

AATS =

(17)

and 5

+

+ 0

ATSf(U , V ) (u , v m , n) dvdu

(18)

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Table 3 Optimal parameters of the VSI RS median chart with estimated process parameters when AATS0 (= ) = 370, ASI0 (=t0 ) = 1, K m {20, 50, 80} for n = 5. m

δ

{4, 7}, d1 = 0.1, G = 3 and

K=7

K=4

k

d2

S1

S2

S3

S4

k

d2

S1

S2

S3

S4

S5

S6

S7

20

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

1.4628 1.4692 1.7793 1.7793 1.7793 1.7793 1.7793 1.892 1.892 1.892

1.0751 1.0733 1.2526 1.2526 1.2526 1.2526 1.2526 1.1988 1.1988 1.1988

0 0 0 0 0 0 0 0 0 0

2 2 1 1 1 2 1 1 1 1

3 5 2 2 2 5 2 3 3 3

7 8 3 3 3 6 3 3 3 3

1.6043 1.6043 1.3353 1.3353 1.2731 1.2731 1.2731 1.2731 1.2731 1.2731

1.2121 1.2121 1.2017 1.2017 1.1902 1.1902 1.1902 1.1902 1.1902 1.1902

0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0

2 2 1 1 0 0 0 0 0 0

3 3 2 2 3 3 3 3 3 3

4 4 2 2 4 4 4 4 4 4

5 5 3 3 5 5 5 5 5 5

9 9 6 6 9 9 9 9 9 9

50

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

1.4565 1.7149 1.7149 1.7746 1.7746 1.7746 1.7746 1.902 1.902 1.902

1.081 1.31 1.31 1.2745 1.2745 1.2745 1.2745 1.2128 1.2128 1.2128

0 0 0 0 0 0 0 0 0 0

2 2 2 1 2 2 1 1 1 1

3 3 3 2 3 3 2 3 3 3

7 6 6 3 5 5 3 3 3 3

1.5866 1.5866 1.3365 1.7682 1.281 1.281 1.281 1.281 1.281 1.281

1.2288 1.2288 1.219 1.2783 1.2044 1.2044 1.2044 1.2044 1.2044 1.2044

0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0

2 2 1 2 0 0 0 0 0 0

3 3 2 3 3 3 3 3 3 3

4 4 2 3 4 4 4 4 4 4

5 5 3 4 5 5 5 5 5 5

9 9 6 6 9 9 9 9 9 9

80

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

1.4507 1.7056 1.7056 1.7688 1.7688 1.7688 1.7688 1.9017 1.9017 1.9017

1.0826 1.3187 1.3187 1.2811 1.2811 1.2811 1.2811 1.2168 1.2168 1.2168

0 0 0 0 0 0 0 0 0 0

2 2 2 2 2 1 1 1 1 1

3 3 3 3 5 2 2 3 3 3

7 6 6 5 6 3 3 3 3 3

1.5759 1.5759 1.3336 1.7619 1.2812 1.2812 1.2812 1.2812 1.2812 1.2812

1.2344 1.2344 1.2243 1.285 1.2082 1.2082 1.2082 1.2082 1.2082 1.2082

0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0

2 2 1 2 0 0 0 0 0 0

3 3 2 3 3 3 3 3 3 3

4 4 2 3 4 4 4 4 4 4

5 5 3 5 5 5 5 5 5 5

9 9 6 6 9 9 9 9 9 9

SDATS =

2

(19)

[E (ATS)]2 ,

E (ATS )

µ 2 (U )

where 2

E (ATS ) =

+

+

2

ATS f(U , V ) (u , v m , n) dvdu

0

fU (u m , n) × fV (v m , n),

2 (U )

b (u

bsinh

)2 + a2

1

(21)

u a

.

e

fV (v m , n)

ed 2,2 n v 2

2

c2

(22)

c

2

e .

(23)

1)

,

2+2 1+2

B ( 2 + 2 1 + 2B A2 )3 + A2 16

1

(28)

A 1+

1 1 + 4e 32e 2

1 , 128e3

(29)

In this section, two optimal designs of the VSI RS median chart, i.e. with known and estimated process parameters, for minimizing the outof-control ATS (ATS1) and out-of-control AATS (AATS1) , respectively, are presented, based on a specified shift size, δ. Here, δ is the size of a mean shift, where the effectiveness of a control chart, based on its speed in detecting an out-of-control condition, is measured by the aforementioned performance criteria. The two optimal design procedures mentioned above need to satisfy the ASI0 requirement. The

(24)

2 ln( 2( 2 (U ) + 2)

(27)

3. Optimal designs and performance assessment

and

b=

2( 3) . m (n + 2)

m with d2, n and d3, n being the usual conwhere A = d2, n and B = stants in Statistical Process Control (SPC), tabulated for the case of normal distribution. Also, note that f 2 ( · e ) is the pdf of the chi-square random variable with e degrees of freedom.

2µ 2 (U ) 2( 2 (U ) + 2)

(26)

d3,2 n

In Eq. (22), ( · ) is the pdf of the standard normal random variable, while a and b can be calculated as follows:

a=

)

and

and

2ed 2.2 n v f c2

2(n +

1 2)3

In Eq. (23),

where

fU (u m , n)

(

2 13 14

and

(20)

Castagliola and Figueiredo (2013) showed that the joint probability density function (pdf) of f(U , V ) (u, v m , n) in Eq. (18) can be approximated closely by the product of the marginal pdfs, fU (u m , n) of U and fv (v m , n) of V, i.e.

f(U , V ) (u, v m , n)

2 1 + + m 2(n + 2) 4(n + 2) 2

(25)

where 6

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Table 4 Optimal parameters of the VSI RS median chart with estimated process parameters when AATS0 (= ) = 370, ASI0 (=t0 ) = 1, K m {20, 50, 80} for n = 7. m

δ

{4, 7}, d1 = 0.1, G = 3 and

K=7

K=4

k

d2

S1

S2

S3

S4

k

d2

S1

S2

S3

S4

S5

S6

S7

20

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

1.4961 1.7682 1.8205 1.8205 1.8205 1.8205 1.9375 1.9375 1.9375 1.9375

1.0767 1.2900 1.2585 1.2585 1.2585 1.2585 1.2044 1.2044 1.2044 1.2044

0 0 0 0 0 0 0 0 0 0

2 2 2 2 1 1 1 1 1 1

3 3 3 5 2 2 3 3 3 3

7 6 5 6 3 3 3 3 3 3

1.6391 1.6391 1.367 1.8458 1.3041 1.3041 1.3041 1.3041 1.3041 1.4833

1.2154 1.2154 1.2075 1.2451 1.1961 1.1961 1.1961 1.1961 1.1961 1.1168

0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0

2 2 1 2 0 0 0 0 0 0

3 3 2 3 3 3 3 3 3 2

4 4 2 5 4 4 4 4 4 3

5 5 3 5 5 5 5 5 5 6

9 9 6 6 9 9 9 9 9 6

50

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

1.4812 1.7441 1.8051 1.8051 1.8051 1.8051 1.9348 1.9348 1.9348 1.9348

1.0822 1.3125 1.2771 1.2771 1.2771 1.2771 1.2159 1.2159 1.2159 1.2159

0 0 0 0 0 0 0 0 0 0

2 2 2 2 1 2 1 1 1 1

3 3 5 5 2 5 3 3 3 3

7 6 6 6 3 6 3 3 3 3

1.613 1.613 1.3596 1.7988 1.3034 1.3034 1.3034 1.3034 1.3034 1.2915

1.2305 1.2305 1.2217 1.2806 1.2072 1.2072 1.2072 1.2072 1.2072 1.2146

0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0

2 2 1 2 0 0 0 0 0 0

3 3 2 3 3 3 3 3 3 2

4 4 2 3 4 4 4 4 4 2

5 5 3 4 5 5 5 5 5 3

9 9 6 6 9 9 9 9 9 6

80

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

1.4733 1.7326 1.7967 1.7967 1.7967 1.7967 1.9313 1.9313 1.9313 1.9313

1.0835 1.3196 1.2826 1.2826 1.2826 1.2826 1.219 1.219 1.219 1.219

0 0 0 0 0 0 0 0 0 0

2 2 1 2 1 1 1 1 1 1

3 3 2 3 2 2 3 3 3 3

7 6 3 5 3 3 3 3 3 3

1.6002 1.6002 1.3545 1.7898 1.3013 1.3013 1.3013 1.3013 1.3013 1.2892

1.2356 1.2356 1.2262 1.2865 1.2103 1.2103 1.2103 1.2103 1.2103 1.2174

0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0

2 2 1 2 0 0 0 0 0 0

3 3 2 3 3 3 3 3 3 2

4 4 2 3 4 4 4 4 4 2

5 5 3 4 5 5 5 5 5 3

9 9 6 6 9 9 9 9 9 6

Table 5 ATS1 and SDTS1 of the SH median, EWMA median, RS median and VSI RS median charts with known process parameters when ATS0 (= ) = 370, ASI0 (=t0 ) = 1 and n = 5. SH median

EWMA median

RS median

VSI RS median

d1 = 0.1

d1 = 0.01 G=3

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

G=4

G=3

G=4

ATS1 SDTS1

ATS1 SDTS1

K=4 ATS1 SDTS1

K=7 ATS1 SDTS1

K=4 ATS1 SDTS1

K=7 ATS1 SDTS1

K=4 ATS1 SDTS1

K=7 ATS1 SDTS1

K=4 ATS1 SDTS1

K=7 ATS1 SDTS1

K=4 ATS1 SDTS1

K=7 ATS1 SDTS1

217.09 217.59 84.34 84.84 33.88 34.38 14.89 15.38 7.109 7.59 3.63 4.10 1.94 2.38 1.06 1.48 0.58 0.96 0.31 0.64

38.65 23.39 13.84 8.03 7.09 4.17 4.23 2.55 2.73 1.75 1.84 1.31 1.24 1.08 0.81 0.90 0.50 0.72 0.29 0.55

93.98 89.03 22.53 19.92 8.87 6.73 4.81 3.13 3.06 1.97 2.03 1.52 1.29 1.24 0.82 0.90 0.52 0.68 0.30 0.60

82.98 79.43 20.25 17.26 8.47 6.17 4.61 3.21 2.86 1.84 1.88 1.48 1.24 1.06 0.80 0.87 0.49 0.69 0.29 0.55

87.09 85.82 17.03 17.66 4.85 5.46 1.87 2.43 0.85 1.35 0.42 0.83 0.21 0.54 0.10 0.35 0.05 0.23 0.02 0.14

75.58 75.26 14.62 14.48 4.56 4.70 1.86 2.42 0.85 1.34 0.41 0.83 0.20 0.54 0.10 0.35 0.05 0.23 0.02 0.14

85.31 86.05 15.53 16.27 4.36 5.05 1.65 2.27 0.72 1.29 0.33 0.79 0.15 0.50 0.07 0.32 0.03 0.20 0.01 0.12

74.79 74.97 14.08 14.38 4.27 4.65 1.63 2.26 0.71 1.27 0.33 0.78 0.15 0.50 0.07 0.32 0.03 0.20 0.01 0.12

87.72 86.10 17.66 17.94 5.27 5.56 2.16 2.46 1.07 1.35 0.58 0.84 0.34 0.55 0.20 0.39 0.11 0.26 0.06 0.17

76.27 75.63 15.14 14.70 4.92 4.78 2.15 2.44 1.07 1.35 0.58 0.85 0.33 0.55 0.19 0.39 0.10 0.26 0.05 0.17

86.16 86.52 16.17 16.52 4.77 5.13 1.93 2.29 0.94 1.26 0.51 0.78 0.29 0.50 0.17 0.33 0.10 0.25 0.05 0.16

75.54 75.33 14.65 14.59 4.65 4.72 1.92 2.27 0.94 1.26 0.50 0.77 0.28 0.50 0.16 0.34 0.09 0.22 0.05 0.14

7

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Table 6 ATS1 and SDTS1 of the SH median, EWMA median, RS median and VSI RS median charts with known process parameters when ATS0 (= ) = 370, ASI0 (=t0 ) = 1 and n = 7. SH median

EWMA median

RS median

VSI RS median

d1 = 0.1

d1 = 0.01 G=3

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

G=4

ATS1 SDTS1

K=4 ATS1 SDTS1

K=7 ATS1 SDTS1

K=4 ATS1 SDTS1

K=7 ATS1 SDTS1

K=4 ATS1 SDTS1

K=7 ATS1 SDTS1

K=4 ATS1 SDTS1

K=7 ATS1 SDTS1

K=4 ATS1 SDTS1

K=7 ATS1 SDTS1

185.96 186.46 61.17 61.66 22.00 22.50 8.93 9.42 4.00 4.48 1.92 2.37 0.95 1.36 0.47 0.83 0.22 0.52 0.10 0.33

31.08 18.75 10.80 6.36 5.39 3.20 3.13 1.96 1.96 1.37 1.23 1.07 0.75 0.86 0.42 0.66 0.21 0.47 0.10 0.32

71.49 68.59 15.70 13.24 6.33 4.43 3.52 2.30 2.17 1.59 1.29 1.24 0.76 0.86 0.44 0.63 0.22 0.50 0.10 0.33

62.50 59.04 14.43 11.62 6.13 4.16 3.30 2.17 2.01 1.47 1.23 1.05 0.73 0.88 0.41 0.63 0.21 0.47 0.10 0.31

65.41 64.20 10.72 11.35 2.92 3.51 1.09 1.61 0.47 0.90 0.20 0.54 0.09 0.33 0.04 0.19 0.02 0.13 0.00 0.07

55.12 54.84 9.51 9.58 2.88 3.06 1.09 1.60 0.46 0.90 0.20 0.54 0.09 0.33 0.04 0.20 0.01 0.11 0.00 0.07

62.15 62.90 9.71 10.43 2.60 3.26 0.94 1.51 0.38 0.85 0.15 0.50 0.06 0.29 0.02 0.17 0.01 0.09 0.00 0.04

54.40 54.60 9.04 9.38 2.57 3.24 0.93 1.52 0.37 0.84 0.15 0.50 0.06 0.29 0.02 0.17 0.01 0.09 0.00 0.05

66.03 65.28 11.27 11.55 3.27 3.57 1.33 1.62 0.64 0.90 0.34 0.54 0.18 0.37 0.09 0.23 0.04 0.13 0.02 0.08

55.81 55.20 9.96 9.58 3.19 3.11 1.32 1.61 0.64 0.91 0.33 0.55 0.17 0.37 0.08 0.23 0.04 0.13 0.02 0.07

63.01 63.36 10.25 10.61 2.94 3.30 1.18 1.51 0.56 0.84 0.29 0.50 0.16 0.30 0.08 0.22 0.03 0.13 0.01 0.07

55.14 54.96 9.53 9.53 2.91 3.27 1.17 1.51 0.55 0.83 0.28 0.49 0.14 0.31 0.07 0.19 0.03 0.11 0.01 0.07

Step 6. Return to Step 2. Step 7. The minimum ATS1 ( ) value is given by ATSmin while the corresponding parameter combination (n, k , d1, d2, K , G, {S1, S2, ...,SK }) that produces this ATSmin value is adopted as the optimal parameter combination of the VSI RS median chart.

3.1. Optimal design of the VSI RS median chart with known process parameters

By using the above procedure, optimal parameters are computed for the 4 and 7 regions VSI RS median charts with known process parameters. Note that τ = 370, t 0 = 1, n ∈ {5, 7}, d1 {0.01, 0.1}, G {3, 4} and K {4, 7} are adopted in this paper. The optimal parameters computed are presented in Tables 1 and 2, for n = 5 and 7, respectively. For example, when n = 5, t 0 = 1, δ = 0.4, d1 = 0.01 and G = 3, the optimal parameter combination of the K = 4 regions VSI RS median chart are (k, d2 , {S1, S2, S3, S4 }) = (1.6785, 1.3748, {0, 2, 3, 6}) (see Table 1). By using this optimal parameter combination, ATS1 = 17.03 and SDTS1 = 17.66 are obtained (see Table 5). Here, ATS1 = 17.03 is the smallest ATS1 value computed for the K = 4 regions VSI RS median chart, for the shift δ = 0.4, among all the schemes having the same value of τ. Note that SDTS1 denotes the out-of-control SDTS, where a smaller value is desirable.

When process parameters are known, the optimization model for the VSI RS median chart in minimizing ATS1 ( ) is given as follows:

Minimize ATS1 ( )

k, d2,{S1, S2,..., SK }

(30a)

subject to the constraints (30b)

and

ASI0 = t 0.

G=4

ATS1 SDTS1

optimization programs for computing the optimal parameters are written in the Matlab software.

ATS0 =

G=3

(30c)

Note that ATS0 is computed using Eq. (9) by letting δ = 0, while in Eq. (30b) denotes the desired in-control ATS value. The procedure in obtaining the optimal parameter combination (k, d2, {S1, S2, ...,Sk }) , based on pre-specified values of sample size (n), short sampling interval (d1) and number of regions (K) is obtained as follows:

3.2. Optimal design of the VSI RS median chart with estimated process parameters

Step 1. Specify the desired values of τ, t 0 , n, d1, K, G and shift size (δ) where a quick detection is needed. Additionally, initialize ATSmin = . Step 2. Select a score combination {S1, S2, ...,SK } , that satisfies the constraint 0 S1 S2 ... SK 10 and proceed to Step 3. If no new combination is possible, proceed to Step 7. Step 3. Compute the values of k and d2 that satisfy Eqs. (30b) and (30c). Step 4. Compute ATS1 ( ) using Eq. (9), for the shift δ, based on the parameter values n, k, d1, d2 , τ, t 0 , K, G, {S1, S2, ...,SK } , either specified or computed prior to this step. Step 5. If ATS1 ( ) < ATSmin , let ATSmin = ATS1 ( ) . Here, ATSmin records the minimum ATS1 ( ) value.

When process parameters are estimated, the optimization model of the VSI RS median chart for minimizing AATS1 ( ) is given as follows:

Minimize

k , d2,{S1, S2,..., SK }

AATS1 ( )

(31a)

subject to constraints

AATS0 =

(31b)

and

ASI0 = t 0.

(31c)

It should be noted that AATS0 is computed using Eq. (18) by letting δ = 0. 8

9

1.0

0.8

0.6

0.4

0.2

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

90.85 88.70 17.79 16.94 5.82 5.33 2.69 2.28 1.52 1.30

K=4 k = 1.3298 d2 = 1.12 ATS1 SDTS1

G = 3, d1 = 0.1

89.12 89.40 16.44 16.66 4.88 5.11 2.05 2.25 1.06 1.23

K=7 k = 1.6401 d2 = 1.34 ATS1 SDTS1 86.42 85.96 14.97 15.63 4.14 4.92 1.49 2.12 0.64 1.17

K=4 k = 1.3298 d2 = 1.66 ATS1 SDTS1

G = 4, d1 = 0.01

88.04 88.75 15.70 16.37 4.38 5.01 1.67 2.22 0.76 1.23

K=7 k = 1.6401 d2 = 1.37 ATS1 SDTS1

78.75 77.44 11.97 11.20 3.76 3.21 1.82 1.48 1.05 0.89 0.61 0.65 0.34 0.48 0.17 0.34 0.08 0.22 0.03 0.14

219.43 219.90 85.61 86.26 34.65 35.68 15.23 15.55 7.25 7.74

ATS1 SDTS1

k = 1.6158

SH median

75.38 72.30 15.16 12.82 5.95 3.99 3.37 1.91 2.30 1.14 1.70 0.86 1.23 0.74 0.87 0.64 0.60 0.57 0.37 0.50

VSI RS median

86.82 82.37 17.56 13.56 7.65 4.22 4.86 2.11 3.59 1.40 2.79 1.15 2.18 1.00 1.70 0.84 1.36 0.63 1.16 0.43

ATS1 SDTS1

K=4 k = 1.1691 d2 = 1.09 ATS1 SDTS1

t (50)

47.59 43.51 11.05 7.65 5.08 2.84 3.08 1.51 2.10 0.98 1.54 0.71 1.19 0.54 0.97 0.44 0.80 0.45 0.62 0.50

ATS1 SDTS1

K=7 k = 1.3805

t (20)

250.92 255.87 107.30 107.75 42.33 43.28 17.10 17.68 7.07 7.50 3.02 3.52 1.35 1.78 0.61 0.99 0.27 0.58 0.11 0.36

ATS1 SDTS1

K=4 k = 1.1691

61.25 57.76 15.29 11.80 7.00 4.37 4.19 2.26 2.86 1.44

k = 0.5066 λ=0.2 ATS1 SDTS1

EWMA median

66.04 65.95 9.67 9.90 2.71 2.88 1.16 1.34 0.61 0.74 0.36 0.43 0.24 0.27 0.15 0.19 0.09 0.14 0.05 0.10

K=7 k = 1.3795 d2 = 1.30 ATS1 SDTS1

97.87 93.54 24.07 20.37 10.17 6.73 6.09 3.14 4.36 1.92

ATS1 SDTS1

K=4 k = 1.3423

RS median

71.97 72.31 8.91 9.68 2.14 2.76 0.72 1.23 0.28 0.67 0.12 0.39 0.05 0.23 0.02 0.14 0.01 0.09 0.00 0.06

K=4 k = 1.1691 d2 = 1.52 ATS1 SDTS1

G = 4, d1 = 0.01

99.06 95.19 23.05 20.46 9.01 6.88 4.97 3.19 3.26 1.79

ATS1 SDTS1

K=7 k = 1.6680

65.06 65.40 9.02 9.69 2.29 2.84 0.84 1.33 0.35 0.74 0.15 0.43 0.07 0.26 0.03 0.16 0.02 0.10 0.01 0.07

K=7 k = 1.3795 d2 = 1.33 ATS1 SDTS1

90.56 89.46 18.15 17.50 5.85 5.31 2.73 2.33 1.55 1.33

K=4 k = 1.3421 d2 = 1.11 ATS1 SDTS1

G = 3, d1 = 0.1

VSI RS median

225.27 225.11 89.75 90.61 36.14 36.68 15.99 16.40 7.38 7.79 3.67 4.11 1.90 2.36 1.03 1.47 0.56 0.94 0.29 0.62

ATS1 SDTS1

k = 1.6099

k = 0.4232 λ=0.2 ATS1 SDTS1

G = 3, d1 = 0.1

k = 1.4757

RS median

SH median

EWMA median

SH median

VSI RS median

t (20)

t (4)

91.15 90.17 16.99 17.24 5.11 5.26 2.13 2.32 1.13 1.30

K=7 k = 1.6662 d2 = 1.34 ATS1 SDTS1

59.56 54.95 15.06 11.58 6.78 4.21 4.07 2.19 2.78 1.40 2.05 0.99 1.58 0.75 1.27 0.58 1.05 0.48 0.88 0.45

k = 0.4981 λ=0.2 ATS1 SDTS1

EWMA median

87.38 87.89 15.65 16.49 4.30 5.00 1.61 2.24 0.69 1.22

90.70 90.17 16.36 17.07 4.64 5.19 1.76 2.31 0.82 1.31

K=7 k = 1.6662 d2 = 1.38 ATS1 SDTS1

96.80 94.35 22.51 19.82 8.81 6.77 4.78 3.02 3.15 1.73 2.29 1.16 1.75 0.92 1.33 0.77 1.01 0.69 0.74 0.62

ATS1 SDTS1

K=7 k = 1.6417

(continued on next page)

K=4 k = 1.3421 d2 = 1.68 ATS1 SDTS1

G = 4, d1 = 0.01

97.32 91.88 23.51 19.47 10.00 6.72 6.02 3.07 4.29 1.85 3.33 1.42 2.65 1.19 2.13 1.02 1.74 0.87 1.46 0.70

ATS1 SDTS1

K=4 k = 1.3300

RS median

Table 7 ATS1 and SDTS1 of the SH median, EWMA median, RS median and VSI RS median charts with known process parameters, for the t distribution when ATS0 (= ) = 370, ASI0 (=t0 ) = 1 and n = 5.

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0.32 0.77 0.16 0.50 0.07 0.32 0.03 0.19 0.01 0.12 0.67 0.81 0.42 0.52 0.29 0.36 0.20 0.25 0.13 0.16

0.41 0.82 0.20 0.53 0.11 0.36 0.05 0.24 0.02 0.13

K=4 k = 1.3421 d2 = 1.68 ATS1 SDTS1

K=7 k = 1.6662 d2 = 1.34 ATS1 SDTS1

2.36 1.20 1.79 0.93 1.37 0.78 1.05 0.71 0.78 0.64 3.36 1.42 2.67 1.20 2.16 1.04 1.77 0.88 1.48 0.72

0.98 0.92 0.62 0.69 0.38 0.53 0.22 0.40 0.12 0.29

4.1. Comparison of charts’ performances when process parameters are known

0.63 0.75 0.40 0.50 0.27 0.33 0.19 0.23 0.12 0.16

0.31 0.74 0.14 0.45 0.06 0.29 0.03 0.18 0.01 0.11

0.37 0.75 0.19 0.49 0.09 0.32 0.05 0.21 0.02 0.13

3.59 4.04 1.90 2.36 1.04 1.49 0.56 0.94 0.30 0.62

Tables 5 and 6 show the performances of the SH, EWMA, RS and VSI RS median charts when process parameters are known and the underlying distribution is normal, for n {5, 7}, τ = 370, t 0 = 1, d1 {0.01, 0.1}, G {3, 4} and {0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4 , 1.6, 1.8, 2.0}. The results show that the EWMA median, RS median and VSI RS median charts (with K = 4 and 7 regions) have significantly smaller ATS1 and SDTS1 values than the SH median chart, for all sizes of shifts, δ, hence the SH median chart has the worst performance. It is also noticed that the VSI RS median chart outperforms the RS median chart, in detecting all sizes of mean shifts, when the two charts have the same number of regions. The EWMA median chart has the best ATS1 performance in detecting small shifts, for example δ ≤ 0.4 when n = 5 (see Table 5) and δ = 0.2 when n = 7 (see Table 6). However, the ATS1 results show that the VSI RS median charts (with K = 4 and 7 regions) outperform the EWMA median chart, in detecting moderate and large mean shifts (see boldfaced entries in Tables 5 and 6), except for a single case, where (K , n , d1, G, )=(4, 7, 0.1, 3, 0.4) . Furthermore, in terms of the SDTS1 criterion, the VSI RS median chart prevails over the EWMA median chart when δ ≥ 0.8. It is also observed that both the K = 4 and 7 regions VSI RS median charts have almost comparable SDTS1 performances when δ ≥ 0.8 (see Tables 5 and 6). However, the 7 regions VSI RS median chart has smaller ATS1 and SDTS1 values than the 4 regions VSI RS median chart when δ ≤ 0.6. This indicates that the performance of the VSI RS median chart generally improves with an increase in the number of regions. The performances of the proposed VSI RS median and other median type charts discussed in the first and second paragraphs of this section are investigated under the assumption that the process follows a normal distribution. From here onwards, the performances of these charts are investigated with the assumption that the underlying distribution is

2.0

1.8

1.6

1.4

0.96 0.90 0.59 0.67 0.37 0.52 0.21 0.39 0.11 0.28 1.2

ATS1 SDTS1

In this section, the performances of the 4 and 7 regions VSI RS median charts are compared with that of the existing SH median (Castagliola and Figueiredo, 2013) and EWMA median (Castagliola et al., 2016) charts. In addition, the RS median chart is also considered in the performance comparison. Note that the SH, EWMA, RS and VSI RS median charts with known and estimated process parameters are adopted in the comparison. For the charts with known process parameters, comparisons are made in terms of ATS1 and SDTS1 criteria (see Tables 5–8), while for the charts with estimated process parameters, comparisons are made in terms of the AATS1 criterion (see Table 9). The boldfaced entries in Tables 5–9 show that the VSI RS median chart has the lowest ATS1, SDTS1 and AATS1 values compared with the SH, EWMA and RS median charts.

2.10 1.01 1.62 0.78 1.30 0.60 1.08 0.50 0.90 0.46

ATS1 SDTS1 ATS1 SDTS1

K=4 k = 1.3421 d2 = 1.11 ATS1 SDTS1

K=7 k = 1.6680

k = 0.5066 λ=0.2 ATS1 SDTS1

K=4 k = 1.3298 d2 = 1.66 ATS1 SDTS1 K=4 k = 1.3298 d2 = 1.12 ATS1 SDTS1

K=7 k = 1.6401 d2 = 1.34 ATS1 SDTS1

G = 4, d1 = 0.01

G = 3, d1 = 0.1

Table 7 (continued)

The step-by-step procedure explained in Section 3.1 can also be used to determine the optimal parameter combination (k , d2 , {S1, S2, ...,SK }) of the estimated process parameters based VSI RS median chart, based on pre-specified values of n, d1 and K, except that ATSmin and ATS1 ( ) are replaced by AATSmin and AATS1 ( ) , respectively. The optimal parameter combination that yields the smallest AATS1 value, for a specified shift size δ is tabulated in Tables 3 and 4, for n = 5 and 7, respectively. Note that τ = 370, t 0 = 1, d1 = 0.1, G = 3, K ∈ {4, 7} and m ∈ {20, 50, 80} are considered in these tables. The AATS1 values computed using these optimal parameter combinations are shown in Table 9. For example, when (K, n, m, d1, G, δ, t 0 ) = (4, 5, 20, 0.1, 3, 0.4, 1) is considered, the optimal parameters of the estimated process parameters based VSI RS median chart and the corresponding AATS1 (k , d2 , {S1, S2, S3, S4 }) value are obtained as = (1.4692, 1.0733, {0, 2, 5, 8}) (see Table 3) andAATS1 = 35.27 (see Table 9). 4. Comparative studies

K=4 k = 1.3423

G = 3, d1 = 0.1

k = 1.6158

SH median VSI RS median

K=7 k = 1.6401 d2 = 1.37 ATS1 SDTS1

t (50) t (20)

EWMA median

RS median

VSI RS median

G = 4, d1 = 0.01

K=7 k = 1.6662 d2 = 1.38 ATS1 SDTS1

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11

1.0

0.8

0.6

0.4

0.2

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

578.4 573.9 143.7 142.3 26.09 25.38 6.68 5.86 2.91 2.18

K=4 k = 1.5881 d2 = 1.09 ATS1 SDTS1

G = 3, d1 = 0.1

543.9 550.5 104.1 103.3 19.51 19.66 5.02 5.19 1.99 2.08

K=7 k = 1.7645 d2 = 1.30 ATS1 SDTS1 662.5 657.9 157.0 157.2 23.70 24.74 4.59 5.32 1.44 2.03

K=4 k = 1.5881 d2 = 1.55 ATS1 SDTS1

G = 4, d1 = 0.01

549.1 556.2 103.9 103.5 18.78 19.40 4.48 5.13 1.59 2.10

K=7 k = 1.7645 d2 = 1.33 ATS1 SDTS1

367.5 362.8 74.38 75.52 15.75 15.18 5.11 4.52 2.43 1.96 1.45 1.14 0.94 0.82 0.60 0.64 0.35 0.49 0.16 0.33

225.8 226.4 135.7 135.5 80.48 81.40 48.06 49.15 28.53 29.19

ATS1 SDTS1

k = 1.6660

SH median

326.7 323.4 67.70 65.91 18.97 16.40 7.95 5.58 4.49 2.62 3.01 1.54 2.22 1.02 1.74 0.81 1.38 0.71 1.09 0.66

VSI RS median

359.9 353.1 80.73 79.70 21.04 17.95 8.61 5.75 4.98 2.69 3.41 1.61 2.53 1.25 1.93 1.10 1.45 1.01 1.04 0.93 Gamma (0.5, 1)

223.2 221.8 37.41 33.24 12.39 8.98 6.33 3.75 3.92 2.02 2.74 1.28 2.05 0.92 1.59 0.70 1.27 0.56 1.04 0.46

ATS1 SDTS1

ATS1 SDTS1

K=4 k = 1.4925 d2 = 1.10 ATS1 SDTS1

Gamma (2, 1)

174.8 178.0 86.08 86.56 43.25 43.63 22.39 22.78 11.88 12.19 6.45 6.92 3.59 4.08 1.99 2.44 1.05 1.47 0.55 0.92

SDTS1

K=7 k = 1.7557

K=4 k = 1.4936

1462.8 1456.5 188.2 182.3 35.28 30.64 11.52 7.54 5.63 2.68

k = 0.5166 λ=0.2 ATS1 SDTS1

EWMA median

345.2 342.9 62.45 62.39 13.27 13.52 4.06 4.21 1.76 1.89 0.95 1.08 0.56 0.66 0.35 0.38 0.24 0.21 0.18 0.13

K=7 k = 1.7548 d2 = 1.31 ATS1 SDTS1

447.50 439.03 248.5 250.8 61.66 58.02 12.66 6.58 7.14 3.46

ATS1 SDTS1

K=4 k = 1.4555

RS median

396.8 393.3 75.85 78.43 13.59 14.54 3.45 4.17 1.19 1.80 0.49 0.96 0.21 0.56 0.08 0.31 0.02 0.13 0.01 0.03

K=4 k = 1.4925 d2 = 1.59 ATS1 SDTS1

G = 4, d1 = 0.01

401.6 397.1 167.6 168.9 36.03 32.92 9.79 5.79 5.46 3.05

ATS1 SDTS1

K=7 k = 1.4975

346.7 344.7 61.79 62.16 12.60 13.31 3.58 4.16 1.39 1.89 0.65 1.09 0.30 0.66 0.12 0.37 0.05 0.19 0.02 0.08

K=7 k = 1.7548 d2 = 1.34 ATS1 SDTS1

478.6 470.7 258.1 261.9 53.04 52.34 6.81 4.77 3.71 2.59

K=4 k = 1.4552 d2 = 1.09 ATS1 SDTS1

G = 3, d1 = 0.1

VSI RS median

191.1 189.1 100.9 100.5 53.23 53.55 28.95 29.79 15.67 16.28 8.75 9.22 4.90 5.38 2.70 3.22 1.46 1.91 0.76 1.15

ATS1 SDTS1

k = 1.6910

k = 0.5313 λ=0.2 ATS1 SDTS1

G = 3, d1 = 0.1

k = 1.6584

RS median

SH median

EWMA median

SH median

VSI RS median

Gamma (2, 1)

Gamma (4, 1)

474.0 470.2 181.9 182.6 28.57 28.54 4.43 4.24 2.15 2.21

K=7 k = 1.4975 d2 = 1.27 ATS1 SDTS1

403.7 399.2 57.93 53.65 16.36 12.54 7.56 4.59 4.51 2.28 3.04 1.39 2.22 0.95 1.72 0.72 1.37 0.58 1.11 0.47

k = 0.5397 λ=0.2 ATS1 SDTS1

EWMA median

614.5 604.5 309.1 314.1 50.02 51.03 3.99 4.57 1.81 2.39

482.5 478.7 183.8 184.7 27.80 28.27 3.81 4.31 1.73 2.25

K=7 k = 1.4975 d2 = 1.30 ATS1 SDTS1

491.7 498.0 107.8 103 26.34 23.64 9.54 6.94 5.07 2.93 3.31 1.66 2.35 1.08 1.82 0.78 1.47 0.67 1.19 0.67

ATS1 SDTS1

K=7 k = 1.7670

(continued on next page)

K=4 k = 1.4552 d2 = 1.49 ATS1 SDTS1

G = 4, d1 = 0.01

556.9 551.6 148.7 145.3 32.64 29.36 10.97 7.56 5.80 2.96 3.91 1.71 2.87 1.26 2.22 1.05 1.74 1.00 1.32 0.95

ATS1 SDTS1

K=4 k = 1.5589

RS Median

Table 8 ATS1 and SDTS1 of the SH median, EWMA median, RS median and VSI RS median charts with known process parameters, for the gamma distribution when ATS0 (= ) = 370, ASI0 (=t0 ) = 1 and n = 5.

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0.55 0.98 0.03 0.01 0.02 0.01 0.02 0.01 0.02 0.01 0.58 1.03 0.03 0.01 0.03 0.01 0.02 0.01 0.02 0.01 0.85 0.96 0.27 0.07 0.24 0.09 0.21 0.10 0.15 0.08 1.88 1.18 1.05 0.52 0.83 0.58 0.52 0.53 0.16 0.16 2.96 1.42 1.77 0.51 1.60 0.61 1.31 0.66 0.88 0.49 4.04 1.64 2.64 0.76 2.34 0.90 1.86 0.91 1.41 0.82

ATS1 SDTS1 ATS1 SDTS1

non-normal symmetrical and skewed. Borror, Montgomery, and Runger (1999) considered the t and gamma distributions to investigate the robustness of the EWMA chart to non-normality. On similar lines, this paper also adopts the t and gamma distributions for the non-normal symmetrical and skewed distributions, respectively, in studying the robustness of the charts. The simulation approach is used to compute the ATS1 and SDTS1 values of the SH median, EWMA median, RS median and VSI RS median charts, for the t and gamma distributions. Tables 7 and 8 present the ATS1 and SDTS1 values of the charts, for the t and gamma distributions, respectively. The parameters of the charts are chosen to satisfy ATS0 (= ) = 370, ASI0 (=t 0 ) = 1 and n = 5. For example, in Table 7, the parameters k = 1.1691 and d2 = 1.09 are chosen to attain ATS0 = 370, for the 4 regions VSI RS median chart when G = 3, d1 = 0.1 and the underlying distribution is t (4), i.e. the data are generated from the t distribution with 4 degrees of freedom. When the data are generated from the gamma (0.5, 1) distribution (here, 0.5 and 1 represent the shape and scale parameters, respectively), the corresponding parameters of the 4 regions VSI RS median chart with the same values of G and d1 are k = 1.4552 and d2 = 1.09 (see Table 8). In Tables 7 and 8, the ATS1 and SDTS1 results for two combinations of (G, d1) = (4, 0.01) and (3, 0.1) for the VSI RS median chart are presented. Note that (G, d1) = (4, 0.01) and (3, 0.1) represent the combinations of the charts with the best and worst performances, respectively, from four different (G, d1) ∈ {(3, 0.01), (3, 0.1), (4, 0.01), (4, 0.1)} combinations. In Tables 7 and 8, the score combinations {S1, S2, S3, S4 } = {0, 2, 3, 8} and {S1, S2, ...,S7} = {0, 0, 2, 2, 3, 4, 6} are adopted for the 4 and 7 regions VSI RS median charts, respectively. In Table 7, the ATS1 and SDTS1 values of the VSI RS median chart for the t distribution with different degrees of freedom show a similar performance to that of the normal distribution, where the EWMA median chart’s performance is the best for small mean shifts. However, for moderate and large mean shifts, the VSI RS median chart outperforms all the other median charts under consideration by having smaller ATS1 and SDTS1 values, for the same shift size (δ), where δ is the shift size in the mean, in multiples of standard deviation. Table 8 presents the ATS1 and SDTS1 values for the four median charts, for several gamma distributions. In this table, the SH median chart has the best ATS1 and SDTS1performances among all the competing charts when = 0.2 . The EWMA median chart surpasses the other charts for δ = 0.4 and 0.6, except for the heavily skewed distribution, i.e. gamma (0.5, 1), where the 7 regions VSI RS median chart prevails when δ = 0.6. For δ ≥ 0.8 and δ ≥ 1, the VSI RS median chart surpasses the other median charts, in terms of the ATS1 and SDTS1 criteria, respectively, for all the gamma distributions considered.

16.88 17.24 9.78 10.27 5.44 5.96 2.78 3.28 1.23 1.67 0.75 1.18 0.33 0.71 0.12 0.37 0.04 0.15 0.02 0.03

4.2. Comparison of charts’ performances when process parameters are estimated Table 9 presents the AATS1 values of the SH, EWMA, RS and VSI RS median charts when process parameters are estimated and the underlying distribution is normal, for n {5, 7}, τ = 370, t 0 = 1, d1 = 0.1, G = 3, m ∈ {20, 50, 80} and {0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0}. Note that Table 9 only considers G = 3 and d1 = 0.1 as this scheme has the poorest ATS1 performance among all (G, d1) combinations, for the VSI RS median chart with known process parameters. Consequently, the aim here is to investigate whether the VSI RS median chart having G = 3 and d1 = 0.1 is superior to the SH, EWMA and RS median charts when process parameters are estimated. In Table 9, it is found that the AATS1 performances of the EWMA median, RS median and VSI RS median (with K = 4 and 7 regions) charts surpass that of the SH median chart, for all sizes of mean shifts. For small shifts, say δ ≤ 0.4, the EWMA median chart outperforms the VSI RS median chart, as the former has smaller AATS1 values. However, the VSI RS median charts (with K = 4 and 7 regions) produce smaller AATS1 values than the EWMA median chart for moderate and large mean shifts (δ ≥ 0.6). Similar to the known process parameters chart,

2.0

1.8

1.6

1.4

1.2

1.75 1.24 1.14 0.85 0.75 0.65 0.47 0.54 0.26 0.41

1.06 1.17 0.60 0.70 0.36 0.38 0.24 0.17 0.19 0.10

0.60 1.08 0.24 0.61 0.08 0.29 0.02 0.09 0.01 0.01

ATS1 SDTS1

3.43 1.30 2.38 0.82 1.78 0.60 1.42 0.54 1.06 0.38

K=4 k = 1.4552 d2 = 1.49 ATS1 SDTS1 K=4 k = 1.4552 d2 = 1.09 ATS1 SDTS1 K=4 k = 1.4555

k = 0.5166 λ=0.2 ATS1 SDTS1 k = 1.6660

K=7 k = 1.7645 d2 = 1.33 ATS1 SDTS1 K=4 k = 1.5881 d2 = 1.55 ATS1 SDTS1 K=4 k = 1.5881 d2 = 1.09 ATS1 SDTS1

G = 3, d1 = 0.1

VSI RS median

Gamma (2, 1)

Table 8 (continued)

K=7 k = 1.7645 d2 = 1.30 ATS1 SDTS1

G = 4, d1 = 0.01

SH median

Gamma (0.5, 1)

EWMA median

RS median

K=7 k = 1.4975

G = 3, d1 = 0.1

VSI RS median

K=7 k = 1.4975 d2 = 1.27 ATS1 SDTS1

G = 4, d1 = 0.01

K=7 k = 1.4975 d2 = 1.30 ATS1 SDTS1

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Table 9 AATS1 of the SH median, EWMA median, RS median and VSI RS median charts with estimated process parameters when AATS0 (= ) = 370, ASI0 (=t0 ) = 1, K {4, 7},d1 = 0.1, G = 3 and m {20, 50, 80} for n {5, 7} . n=7

n=5

m

SH median

EWMA median

RS median

VSI RS median

SH median

EWMA median

RS median

AATS1

AATS1

K=4

VSI RS median

K=7

K=4

K=7

AATS1

AATS1

K=4

K=7

K=4

K=7

AATS1

AATS1

AATS1

AATS1

AATS1

AATS1

AATS1

AATS1

20

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

258.52 110.64 42.79 17.80 8.12 4.01 2.09 1.13 0.61 0.33

148.50 24.50 8.99 4.88 3.01 1.97 1.30 0.84 0.52 0.31

181.33 39.87 11.61 5.52 3.31 2.11 1.34 0.84 0.53 0.32

177.21 36.69 11.10 5.23 3.08 1.96 1.27 0.81 0.50 0.30

177.48 35.27 7.82 2.66 1.21 0.65 0.37 0.21 0.12 0.06

166.40 30.20 7.35 2.62 1.30 0.67 0.36 0.20 0.11 0.06

230.12 80.94 27.56 10.60 4.56 2.13 1.04 0.51 0.24 0.11

111.50 16.20 6.57 3.56 2.15 1.33 0.80 0.45 0.23 0.11

145.86 25.17 7.77 3.94 2.33 1.37 0.80 0.46 0.24 0.11

137.10 22.97 7.51 3.68 2.16 1.30 0.77 0.43 0.23 0.11

141.47 20.91 4.43 1.57 0.73 0.38 0.20 0.10 0.05 0.02

131.13 17.64 4.27 1.57 0.76 0.37 0.19 0.09 0.04 0.02

50

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

237.69 95.07 37.42 16.08 7.54 3.79 2.01 1.09 0.60 0.32

85.54 17.15 7.96 4.56 2.88 1.91 1.28 0.83 0.51 0.30

135.91 28.30 9.94 5.14 3.19 2.08 1.32 0.84 0.53 0.31

125.07 25.52 9.73 4.90 3.09 1.93 1.26 0.81 0.50 0.29

131.04 23.57 6.19 2.37 1.14 0.62 0.36 0.21 0.11 0.06

118.59 19.84 5.98 2.36 1.23 0.65 0.35 0.20 0.11 0.06

207.08 69.06 24.23 9.62 4.24 2.01 0.99 0.49 0.23 0.10

59.95 12.82 5.95 3.35 2.06 1.29 0.77 0.43 0.22 0.10

103.07 18.94 6.94 3.73 2.25 1.33 0.78 0.45 0.23 0.10

93.46 17.39 6.74 3.49 2.09 1.27 0.76 0.42 0.22 0.10

97.88 14.41 3.71 1.43 0.68 0.36 0.19 0.09 0.04 0.02

86.66 12.45 3.66 1.43 0.72 0.36 0.18 0.09 0.04 0.02

80

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

230.91 91.14 36.12 15.65 7.39 3.74 1.98 1.08 0.59 0.32

68.53 15.97 7.68 4.46 2.84 1.89 1.27 0.83 0.51 0.30

120.89 26.03 9.56 5.03 3.15 2.07 1.31 0.83 0.52 0.31

109.42 23.47 9.36 4.81 3.05 1.92 1.25 0.81 0.50 0.29

115.73 21.19 5.84 2.29 1.11 0.61 0.35 0.20 0.11 0.06

102.66 17.95 5.70 2.29 1.21 0.64 0.35 0.19 0.10 0.05

199.93 66.16 23.41 9.38 4.16 1.98 0.98 0.48 0.23 0.10

48.85 12.13 5.78 3.28 2.02 1.27 0.77 0.43 0.22 0.10

90.94 17.70 6.73 3.66 2.23 1.32 0.77 0.45 0.23 0.10

81.33 16.27 6.54 3.43 2.07 1.25 0.75 0.42 0.22 0.10

85.56 13.16 3.55 1.40 0.67 0.35 0.18 0.09 0.04 0.02

74.39 11.47 3.52 1.40 0.71 0.35 0.17 0.08 0.04 0.02

adding more regions and consequently scores, to the estimated process parameters based VSI RS median chart, will increase the chart’s sensitivity towards shifts.

converges to its known process parameters counterpart’s ATS0 performance when m is large. Jones and Steiner (2012), Gandy and Kvaløy (2013), and Zhang, Megahed, and Woodall (2014) suggested that the SDATS0 value should be at most 10% of the ATS0 value, in order for the estimated process parameters based chart (adopting the charting parameters of its known process parameters counterpart) to have a satisfactory AATS0 performance. Table 10 presents the AATS0 and SDATS0 values, for the K = 4 and 7 regions VSI RS median chart with estimated process parameters, computed for m ∈ {100, 200, 300, …, 1200, ∞}, using its known process parameters counterpart’s scores {S1, S2, S3, S4 } = {0, 1, 2, 4}, {S1, S2, S3, S4, S5, S6, S7} = {0, 1, 2, 3, 4, 5, 8} and parameters (k, d2 ) that give ATS0 {200, 370} when n ∈ {5, 7}. For example, by using the parameters (k, d2 ) = (1.3351, 1.1181) and scores {S1, S2, S3, S4 } = {0, 1, 2, 4} of the known process parameters based chart that gives ATS0 = 200 when K = 4 and n = 5, the corresponding estimated process parameters based chart will give (AATS0 , SDATS0) = (196.5, 26.84) when m = 500. Note that with the same (k, d2 ) and {S1, S2, S3, S4 } combinations, the SDATS0 value of the estimated process parameters based chart decreases as m increases and when m = 900, the SDATS0 value becomes less than 10% of the corresponding ATS0 value. The boldfaced entries in Table 10 show the largest SDATS0 values that do not exceed 10% of the corresponding ATS0 values, together with the minimum number of Phase-I samples, m, required.

5. Impact of the number of Phase-I samples on the VSI RS median chart with estimated process parameters The AATS1 value of a chart with estimated process parameters (see Table 9) differs significantly from its corresponding ATS1 value when process parameters are known (see Tables 5 and 6), where the disparity is more pronounced for small and moderate shifts. For instance, when n = 5, d1 = 0.1, G = 3, K = 4 and = 0.4, the AATS1 values of the VSI RS median chart with estimated process parameters, for m = 20, 50 and 80 are 35.27, 23.57 and 21.19 (see Table 9), respectively, while ATS1 = 17.66 (see Table 5) for the corresponding chart with known process parameters, where a huge difference exists. This example shows that the AATS1 value changes according to m. As m increases, the difference between the AATS1 (for chart with estimated process parameters) and ATS1 (for chart with known process parameters) values decreases. The analysis in Table 10 is to identify the number of Phase-I samples, m, that needs to be adopted so that the VSI RS median chart’s AATS0 value (when process parameters are estimated), computed using the scores and charting parameters associated with its known process parameters counterpart, is close enough to the latter’s ATS0 value, while ensuring that the in-control SDATS (SDATS0) value obtained for the former is sufficiently small. Table 10 shows that the AATS0 performance of the VSI RS median chart with estimated process parameters

6. An illustrative example The data adopted from Montgomery (2009) are considered in this 13

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Table 10 AATS0 and SDATS0 for the VSI RS median chart with estimated process parameters, computed using the scores {S1, S2, S3, S4} = {0, 1, 2, 4} when K = 4 and {S1, S2, S3, S4, S5, S6, S7} = {0, 1, 2, 3, 4, 5, 8} when K = 7, based on the parameters d1 = 0.1, G = 3, n {5, 7} and different (k, d2 ) combination which give ATS0 (= ) ∈ {200, 370}. n

m

ATS0 = 200 K=4

AATS0

5

k = 1.3351 d2 = 1.1181 186.49 192.13 194.44 195.70 196.50 197.05 197.45 197.76 198.00 198.19 198.35 198.49 200

100 200 300 400 500 600 700 800 900 1000 1100 1200

7

k = 1.3526 d2 = 1.1181 183.86 190.69 193.44 194.92 195.86 196.50 196.97 197.32 197.60 197.83 198.02 198.17 200

100 200 300 400 500 600 700 800 900 1000 1100 1200

ATS0 = 370 K=4

K=7 SDATS0

AATS0

61.36 42.97 34.88 30.09 26.84 24.45 22.60 21.11 19.88 18.85 17.96 17.18 0

k = 1.5422 d2 = 1.2497 181.12 188.97 192.19 193.95 195.06 195.83 196.40 196.82 197.16 197.44 197.66 197.85 200

51.7 36.24 29.35 25.27 22.5 20.47 18.91 17.65 16.61 15.73 14.98 14.33 0

k = 1.5628 d2 = 1.2500 179.31 187.99 191.51 193.43 194.64 195.48 196.09 196.55 196.92 197.22 197.46 197.67 200

SDATS0

AATS0

54.11 38.07 30.85 26.55 23.63 21.49 19.84 18.51 17.41 16.49 15.70 15.01 0

k = 1.4418 d2 = 1.0873 340.56 352.38 357.42 360.22 362.00 363.23 364.14 364.84 365.38 365.83 366.20 366.50 370

46.8 32.77 26.41 22.63 20.07 18.20 16.77 15.62 14.67 13.88 13.20 12.61 0

k = 1.4607 d2 = 1.0872 334.68 349.18 355.17 358.47 360.56 362.00 363.06 363.87 364.50 365.02 365.45 365.80 370

K=7 SDATS0

AATS0

SDATS0

126.53 88.53 71.84 61.94 55.23 50.29 46.47 43.40 40.86 38.73 36.89 35.29 0

k = 1.6897 d2 = 1.1917 328.28 344.93 352.00 355.94 358.45 360.2 361.49 362.47 363.26 363.89 364.41 364.85 370

111.91 79.00 64.05 55.11 49.03 44.56 41.11 38.34 36.05 34.13 32.48 31.04 0

106.43 74.76 60.56 52.12 46.39 42.19 38.94 36.33 34.18 32.37 30.82 29.47 0

k = 1.7123 d2 = 1.1920 324.22 342.72 350.48 354.77 357.51 359.41 360.81 361.88 362.73 363.41 363.98 364.46 370

96.89 68.28 55.09 47.20 41.84 37.92 34.90 32.48 30.49 28.82 27.40 26.16 0

Table 11 Phase-II data of flow width measurements (in microns) from the hard-bake process. Measurements in each sample r

Yr1

Yr2

Yr3

Yr4

Yr5

Yr

(Ur , Lr )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1.4483 1.5435 1.5175 1.5454 1.4418 1.4301 1.4981 1.3009 1.4132 1.3817 1.5765 1.4936 1.5729 1.8089 1.6236 1.412

1.5458 1.6899 1.3446 1.0931 1.5059 1.2725 1.4506 1.506 1.4603 1.3135 1.7014 1.4373 1.6738 1.5513 1.5393 1.7931

1.4538 1.583 1.4723 1.4072 1.5124 1.5945 1.6174 1.6231 1.5808 1.4953 1.4026 1.5139 1.5048 1.825 1.6738 1.7345

1.4303 1.3358 1.6657 1.5039 1.462 1.5397 1.5837 1.5831 1.7111 1.4894 1.2773 1.4808 1.5651 1.4389 1.8698 1.6391

1.6206 1.4187 1.6661 1.5264 1.6263 1.5252 1.4962 1.6454 1.7313 1.4596 1.4541 1.5293 1.7473 1.6558 1.5036 1.7791

1.4538 1.5435 1.5175 1.5039 1.5059 1.5252 1.4981 1.5831 1.5808 1.4596 1.4541 1.4936 1.5729 1.6558 1.6236 1.7345

(0, (0, (0, (0, (0, (0, (0, (0, (0, (0, (0, (0, (0, (1, (2, (4,

section. These data deal with the flow width measurements (in microns) of resist from a hard-bake process. In the preliminary analysis, the Phase-I data of 25 samples, each consisting of 5 observations, are considered. These Phase-I data are plotted on both the X and R charts, where the results show that the data are statistically in-control (see Montgomery, 2009, p. 232, Table 6.1). The implementation of the estimated process parameters based VSI RS median chart (with K = 4 regions) for the Phase-II process

0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0)*

Sampling interval d1 or d2

Total time elapsed (hours)

1.2526 1.2526 1.2526 1.2526 1.2526 1.2526 1.2526 1.2526 1.2526 1.2526 1.2526 1.2526 1.2526 0.1000 0.1000

– 1.2526 2.5052 3.7578 5.0104 6.263 7.5156 8.7682 10.0208 11.2734 12.526 13.7786 15.0312 16.2838 16.3838 16.4838

monitoring, based on AATS0 = 370 , is illustrated. Suppose that process parameters are estimated from the first m = 20 Phase-I samples of the hard-bake process, each with n = 5 observations, using Eqs. (12) and (13), which give µ 0 = 1.4818 and 0 = 0.1512 , respectively. Additionally, the short sampling interval d1 = 0.1, mean shift where a quick detection is important, δ = 0.6 and G = 3, are adopted. Consequently, the optimal parameters (k , d2 , {S1, S2, S3, S4 }) = (1.7793, 1.2526 {0, 1, 2, 3}) are obtained from Table 3. 14

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according to Step 5 of Section 2.1, the VSI RS median chart with estimated process parameters signals the first out-of-control at sample r = 16, i.e. after 16.4838 h have elapsed. The VSI RS median chart with estimated process parameters is shown in Fig. 2, where the 16 sample medians, Yr , for r = 1, 2, …, 16, from Table 11, are plotted on the chart. The value in each of the circles representing the sample medians is the cumulative score, Ur Lr . Fig. 2 shows that an out-of-control signal is detected at sample 16 as U16 (= 4) ≥ + S4 (=+3). Following this out-of-control signal, an investigation needs to be conducted to identify the assignable causes so that appropriate corrective actions can be taken. 7. Conclusions In this paper, the construction, optimal design, performance assessment and implementation of the VSI RS median chart with known and estimated process parameters are presented. Performance measures, i.e. ATS and SDTS are used to evaluate the chart’s performances when process parameters are known, while AATS is used when process parameters are estimated. The VSI RS median chart is compared with the SH, EWMA and RS median charts for both cases of known and estimated process parameters. The VSI RS median chart is found to be better than the SH and RS median charts in detecting all sizes of mean shifts, while the EWMA median chart is superior to the VSI RS median 0.4 ). However, when δ > 0.4, chart for detecting small mean shifts ( the VSI RS median chart surpasses the EWMA median chart. In addition, an analysis using SDATS shows that a large number of Phase-I samples (m) is required for the estimated process parameters based VSI RS median chart adopting the process parameters of its known process parameters counterpart, to have the same performance as the known process parameters based VSI RS median chart. Additionally, based on the SDATS value obtained, the minimum number of Phase-I samples (m), required by the estimated process parameters based VSI RS median chart to achieve a similar performance to the known process parameters based VSI RS median chart are provided. Since this work focuses on the univariate VSI RS median chart, in future, the multivariate median type charts can be investigated under the assumptions of known and estimated process parameters. Research works can also be made to improve the efficiency of median type charts by adopting the auxiliary information (AI), double sampling (DS), variable sample size (VSS), and variable sample size and sampling interval (VSSI) approaches. Moreover, the use of nonparametric techniques on median type charts can also be investigated.

Fig. 2. Four regions (K = 4) VSI RS median chart with estimated process parameters for the flow width measurements of the hard-bake process.

By employing the above values of µ 0 , 0 , k and n, the regions above the CL of the VSI RS median chart with estimated process parameters are obtained as R1 = [1.4818, 1.6021), R2 = [1.6021, 1.7224), R3 = [1.7224, 1.8427) and R 4 = [1.8427, ∞). Similarly, the regions below the CL of the chart are found to be R1b = (1.3615, 1.4818], R2b = (1.2412, 1.3615], R3b = (1.1209, 1.2412] and R 4b = (−∞, 1.1209]. From Table 11, Y1 = 1.4538 falls in R1b , hence, the score S (Y1) = S1 = 0 is assigned and L1 = L0 + S (Y1 ) = 0 is obtained according to Step 3 of Section 2.1. As Y1 < CL, U1 = 0. Note that U0 = L 0 = 0 are adopted. The VSI RS median chart with estimated process parameters adopts the long sampling interval, d2 = 1.2526 h, for taking the next sample when 0 Ur < +SK G (i.e. Ur ∈ [0, 1)) or SK G < Lr 0 (i.e. Lr ∈ (−1, 0]), while it adopts the short sampling interval, d1 = 0.1 h when + SK G Ur < + SK (i.e. Ur ∈ [1, 3)) or SK < Lr SK G (i.e. Lr ∈ (−3, −1]) (see Step 4 of Section 2.1). Hence, sample 2 is taken after the long sampling intervald2 = 1.2526 h, since U1 = 0 ∈ [0, 1) and L1 = 0 ∈ (−1, 0]. As the charting statisticY2 = 1.5435 R1, a score S (Y2) = + S1 = +0 is added to U1, so that the cumulative scores at sample r = 2 are (U2, L2 ) = (0, 0) and the long sampling interval (d2 = 1.2526) is adopted for taking sample 3, as U2 = 0 ∈ [0, 1) and L2 = 0 ∈ (−1, 0]. The procedure of assigning score, S (Yr ) and adding it to either Ur or Lr (for r = 3, 4, …, 12) continues according to Steps 2–4 of Section 2.1 until sample 13, where (U13, L13) = (0, 0) is obtained. At sample r = 14, the charting statisticY14 = 1.6558 is obtained, where Y14 R2 . Thus, a score S (Y14 ) = + S2 = +1 is added to U13 and (U14, L14) = (1, 0) is obtained. As U14 = 1 ∈ [1, 3), sample r = 15 will be taken after the short sampling interval d1 = 0.1 h. Then r = 15 givesY15 = 1.6236 R2 , hence, S (Y15) = + S2 = +1 is added to U14 and the cumulative scores become (U15, L15) = (2, 0). As U15 = 2, the short sampling interval d1 = 0.1 hours is adopted to take sample r = 16. At r = 16, Y16 = 1.7345 R3 , thus the score S (Y16) = + S3 = +2 is added to U15 . Consequently, the cumulative scores (U16, L16) = (4, 0) are obtained. As U16 = 4 ≥ + S4 = +3, then

Acknowledgements This research is supported by the Universiti Sains Malaysia (USM) Fellowship and is funded by the USM, Fundamental Research Grant Scheme, number 203.PMATHS.6711603. This research is conducted when the corresponding author (Michael B.C. Khoo) is on sabbatical leave at Universiti Tunku Abdul Rahman (UTAR), Kampar.

Appendix A The Markov chain model for the current state to the next state transition and the procedure for constructing the TPM Q of the VSI RS median chart discussed in Section 2.1 are presented here. For illustration, the K = 4 regions VSI RS median chart with scores {S1, S2, S3, S4}={0, 2, 3, 6} having initial cumulative scores (U0, L0) = (0, 0), are considered. Table A1 shows all the possible transient states of the Markov chain model, the current cumulative score (at sample r) and the next cumulative score (at sample r + 1). Note that the scores (Ur , Lr ) ∈ {(0, 0), (0, −5), (0, −4), (0, −3), (0, −2), (2, 0), (3, 0), (4, 0), (5, 0)} represent all the possible in-control cumulative score combinations that correspond to the respective transient states of the Markov chain model. For illustration, suppose that the current cumulative score at sample r is (Ur , Lr ) = (3, 0) and the median for sample r + 1, Yr + 1 falls in region R2b , then the next cumulative score at sample r + 1 becomes (Ur + 1, Lr + 1) = (0, −2) (see Table A1). Table A2 presents the current state (at sample r) to the next state (at sample r + 1) transitions, for the Markov chain model. For example, if the current state (at sample r) is state 5 which corresponds to cumulative scores (Ur , Lr ) = (0, −2) and the sample median at sample r + 1, i.e. Yr + 1, falls in region R2 , where the score + S2 = +2 is added to Ur and Lr is reset to zero, then the cumulative scores for the next sample (at sample r + 1) becomes 15

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Table A1 Current cumulative scores (at sample r) and next cumulative scores (at sample r + 1), for the 4 regions VSI RS median chart with scores {S1, S2, S3, S4} = {0, 2, 3, 6}. Current state

Current cumulative scores, (Ur , Lr )

Next cumulative scores, (Ur + 1, Lr + 1) , if Yr + 1

R3b

R 4b 1 2 3 4 5 6 7 8 9

(0, (0, (0, (0, (0, (2, (3, (4, (5,

0) −5) −4) −3) −2) 0) 0) 0) 0)

(0, (0, (0, (0, (0, (0, (0, (0, (0,

−6) −11) −10) −9) −8) −6) −6) −6) −6)

(0, (0, (0, (0, (0, (0, (0, (0, (0,

R2b −3) −8) −7) −6) −5) −3) −3) −3) −3)

(0, (0, (0, (0, (0, (0, (0, (0, (0,

−2) −7) −6) −5) −4) −2) −2) −2) −2)

(0, (0, (0, (0, (0, (0, (0, (0, (0,

R2

R1

R1b 0) −5) −4) −3) −2) 0) 0) 0) 0)

(0, (0, (0, (0, (0, (2, (3, (4, (5,

0) 0) 0) 0) 0) 0) 0) 0) 0)

(2, (2, (2, (2, (2, (4, (5, (6, (7,

R4

R3 0) 0) 0) 0) 0) 0) 0) 0) 0)

(3, (3, (3, (3, (3, (5, (6, (7, (8,

0) 0) 0) 0) 0) 0) 0) 0) 0)

(6, 0) (6, 0) (6, 0) (6, 0) (6, 0) (8, 0) (9, 0) (10, 0) (11, 0)

Table A2 Current state (at sample r) to next state (at sample r + 1) transitions, for the 4 regions VSI RS median chart with scores {S1, S2, S3, S4} = {0, 2, 3, 6}. Current state

1 2 3 4 5 6 7 8 9

Current cumulative scores, (Ur , Lr )

(0, (0, (0, (0, (0, (2, (3, (4, (5,

Next state, if Yr + 1

0) −5) −4) −3) −2) 0) 0) 0) 0)

R 4b

R3b

R2b

R1b

R1

R2

R3

R4

Ab Ab Ab Ab Ab Ab Ab Ab Ab

4 Ab Ab Ab 2 4 4 4 4

5 Ab Ab 2 3 5 5 5 5

1 2 3 4 5 1 1 1 1

1 1 1 1 1 6 7 8 9

6 6 6 6 6 8 9 Ab Ab

7 7 7 7 7 9 Ab Ab Ab

Ab Ab Ab Ab Ab Ab Ab Ab Ab

Table A3 Complete transition probabilities (from the current state at sample r to the next state at sample r + 1) of the TPM, for the 4 regions VSI RS median chart with scores {S1, S2, S3, S4} = {0, 2, 3, 6}. Current state

1 2 3 4 5 6 7 8 9 Ab

Current cumulative scores, (Ur , Lr )

(0, 0) (0, −5) (0, −4) (0, −3) (0, −2) (2, 0) (3, 0) (4, 0) (5, 0) 6

Next state 1

2

3

4

5

6

7

8

9

Ab

P1 + P1b P1 P1 P1 P1 P1b P1b P1b P1b 0

0 P1b 0 P2b P3b 0 0 0 0 0

0 0 P1b 0 P2b 0 0 0 0 0

P3b 0 0 P1b 0 P3b P3b P3b P3b 0

P2b 0 0 0 P1b P2b P2b P2b P2b 0

P2 P2 P2 P2 P2 P1 0 0 0 0

P3 P3 P3 P3 P3 0 P1 0 0 0

0 0 0 0 0 P2 0 P1 0 0

0 0 0 0 0 P3 P2 0 P1

P4 P4 P4 P4 P4 P4 P4 P4 P4 1

+ + + + + + + + +

P4b P2b + P3b + P4b P2b + P3b + P4b P3b + P4b P4b P4b P3 + P4b P3 + P2 + P4b P3 + P2 + P4b

(Ur + 1, Lr + 1) = (2, 0), which is associated with state 6 (see Table A2). Consequently, the transition probability from the current state 5 (at sample r) to the next state 6 (at sample r + 1) is obtained as Pr(Yr + 1 R2) = P2 , which is shown in Table A3. Note that Table A3 presents the complete TPM for the 4 regions VSI RS median chart with scores {S1, S2, S3, S4 } = {0, 2, 3, 6}, where Ab denotes the absorbing state.

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