# New Shewhart-EWMA and Shewhart-CUSUM control charts for

## Transcript Of New Shewhart-EWMA and Shewhart-CUSUM control charts for

Scientia Iranica E (2019) 26(6), 3796{3818

Sharif University of Technology

Scientia Iranica

Transactions E: Industrial Engineering http://scientiairanica.sharif.edu

New Shewhart-EWMA and Shewhart-CUSUM control charts for monitoring process mean

M. Awais and A. Haq

Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan. Received 7 August 2017; received in revised form 19 February 2018; accepted 2 July 2018

KEYWORDS

Average run length; CUSUM; Control chart; EWMA; Perfect and imperfect rankings; Statistical process control.

Abstract. This study proposes new Shewhart-EWMA (SEWMA) and Shewhart-CUSUM

(SCUSUM) control charts using Varied L Ranked Set Sampling (VLRSS) for monitoring the process mean, namely the SEWMA-VLRSS and SCUSUM-VLRSS charts. The run length characteristics of the proposed charts are computed using extensive Monte Carlo simulations. The proposed charts are compared with their existing counterparts in terms of the average and standard deviations of run lengths. It is found that, with perfect and imperfect rankings, the SEWMA-VLRSS and SCUSUM-VLRSS charts are more sensitive than their analogous charts based on simple random sampling, Ranked Set Sampling (RSS), and median RSS schemes. A real dataset is also used to explain the implementation of the proposed control charts.

© 2019 Sharif University of Technology. All rights reserved.

1. Introduction

The Statistical Process Control (SPC) is a collection of tools that help distinguish between two types of variation, namely the natural- and special-cause variations. A process is said to be in statistical control when only natural-caused variations are present, while the process with the special-cause variations is said to be out of control. There are seven major tools in the SPC, including the histogram, check sheet, Pareto chart, cause-and-e ect diagram, defect concentration diagram, scatter diagram, and control charts. Statistical quality control charts are very e ective SPC tools that are frequently used to monitor special-cause variations in a production/manufacturing process.

The control charts are divided into two categories, memory-less and memory-type control charts. The

*. Corresponding author. E-mail address: [email protected] (A. Haq)

doi: 10.24200/sci.2018.4962.1011

Shewhart-type charts fall in the memory-less category because they completely rely on the present information. A major limitation of the Shewhart control chart is that it is less sensitive against small and moderate shifts in the process parameter(s). On the other hand, the Exponentially Weighted Moving Average (EWMA) and the CUmulative SUM (CUSUM) control charts fall in the memory-type category. The reason is that both of these control charts take into account past and current information to maintain their plotting statistics. This feature of the memory-type control charts helps them swiftly react against small to moderate shifts in the process parameter(s).

The CUSUM chart was rst developed by Page [1]. Lucas and Crosier [2] associated a Fast Initial Response (FIR) feature with the CUSUM chart to further enhance its sensitivity by giving head-starts to the plotting CUSUMs at the beginning of a process. Lucas [3] used both the Shewhart and CUSUM charts simultaneously for monitoring small and large shifts in a process, named the Shewhart-CUSUM (SCUSUM) chart. For monitoring changes in the process mean,

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Roberts [4] was the rst to introduce the EWMA chart. Lucas and Saccucci [5] attached an FIR feature with the EWMA chart to increase its sensitivity to the start-up/initial problems. Moreover, they coupled the Shewhart chart with the EWMA chart, named the Shewhart-EWMA (SEWMA) chart, for detecting small and large shifts simultaneously. There are many new advancements and improvements in the control charting structures of the EWMA and CUSUM charts. For some related works on these control charts, we refer to Knoth [6], Lucas and Crosier [2], Chiu [7], Abbas et al. [8], Haq [9], Haq et al. [10], and the references cited therein.

Ranked-Set Sampling (RSS) was rst introduced by McIntyre [11] for estimating mean pasture and forage yields. RSS is a cost-e ective alternative to Simple Random Sampling (SRS) in situations where the units to be sampled could be ranked with respect to each other prior to the formal measurements. Ranking may be done visually based on personal judgment or using the ranks of an auxiliary variable, provided that it is highly correlated with the study variable. For example, if the quanti cation of the underlying quality characteristic(s) is laborious, costly and timeconsuming, it may involve breaking the product{which is expensive and might be hard to construct, etc.{but using some experts' knowledge, it might be possible to rank the quality characteristic according to its quality level or using any less expensive method. There also exist the situations where the error is inevitable while ranking the units, particularly when ranking the units in large set sizes. Dell and Clutter [12] showed that, despite the presence of ranking errors, the mean estimator with RSS is not only unbiased but also more precise than the mean estimator with SRS. A simple imperfect ranking model was designed by Stokes [13], whereby the study variable could be ranked using the ranks of an auxiliary variable. For precisely estimating the mean of a symmetric population, Extreme RSS (ERSS) and Median RSS (MRSS) schemes were suggested by Samawi et al. [14] and Muttlak [15], respectively. The mean estimator with MRSS is more precise than those with the SRS and RSS, when sampling from a symmetric population. Muttlak [16] suggested using the Quartile RSS (QRSS) as a better alternative to SRS, RSS, and MRSS schemes for estimating the population mean when sampling from an asymmetric population. Utilizing the idea of L moments, Al-Nasser [17] proposed a generalized sampling scheme, named L RSS (LRSS), for estimating the population mean. The LRSS scheme encompasses existing RSS schemes, such as the RSS, QRSS, ERSS, and MRSS. Haq et al. [18] further extended the work of Al-Nasser [17] and generalized the LRSS scheme for e ciently estimating the population mean, named the Varied LRSS (VLRSS) scheme. For a symmetric

population, the VLRSS scheme{with both perfect and imperfect rankings{not only is a cost-e ective alternative to the exiting ranking schemes, but it also encompasses them, i.e., the mean estimator with the VLRSS scheme is better than the mean estimator based on SRS, RSS, ERSS, MRSS, and QRSS. For more details, we refer to Haq et al. [18].

As the mean estimators with the RSS schemes are more precise than the mean estimator based on SRS scheme, this fact has led many researchers to construct more sensitive quality control charts. Salazar and Sinha [19] were the rst to propose a Shewhart chart using RSS for monitoring the process mean. Their work, later on, extended by Muttlak and AlSabah [20], who suggested several Shewhart-type mean charts using the RSS, ERSS, and MRSS schemes under both perfect and imperfect rankings. Abujiya and Muttlak [21] and Al-Omari and Haq [22] used the double RSS schemes to construct the Shewhart charts for monitoring the process mean. Al-Sabah [23] suggested new CUSUM mean charts using RSS and MRSS. He showed that the CUSUM charts with RSS and MRSS schemes were more sensitive than that using SRS. Recently, Abujiya et al. [24,25] proposed SEWMA and SCUSUM mean charts using RSS and MRSS schemes. In another work, Awais and Haq [26,27] have suggested improved EWMA and CUSUM charts for monitoring the process mean, respectively. For more related works on the RSS-based control charts, we refer to Haq [28], Mehmood et al. [29,30], Haq et al. [31-36], Abbasi and Riaz [37], Abid et al. [38,39], Munir and Haq [40], and the references cited therein.

Since the VLRSS mean estimator is more precise than the mean estimators based on SRS, RSS, and MRSS schemes when sampling from a symmetric population, we believe that the control charts with the VLRSS would be more sensitive than those based on SRS, RSS, and MRSS schemes. This paper proposes new SEWMA and SCUSUM mean charts using VLRSS, named SEWMA-VLRSS and SCUSUMVLRSS charts, respectively. Monte Carlo simulations are used to compute the run length characteristics of the proposed control charts, including the Average Run Length (ARL) and the Standard Deviation of the Run Length (SDRL). The proposed charts are compared with their counterparts based on SRS, RSS, and MRSS schemes. It turns out that the proposed charts are more sensitive than the existing charts.

The rest of the paper is structured in the following order: Sections 2 and 3 brie y review SEWMA and SCUSUM charts with SRS, respectively. In Section 4, the VLRSS scheme is discussed with both perfect and imperfect rankings. The proposed charts are presented in Section 5. A comparative study is conducted in Section 6. An illustrative example is presented in Section 7, and Section 8 summarizes the main ndings.

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2. The SEWMA chart

The classical SEWMA chart is a mixture of two control

charts. The Shewhart and EWMA charts provide

protection against the large and small-to-moderate

shifts in the process mean, respectively [cf., 5].

Let Y denote the study variable and let fYtg,

for t = 1; 2; , be a sequence of Independent and

Identically Distributed (IID) random variables. Here,

it is assumed that Yt is a normally distributed random

variable with the in-control mean Y and the in-control

variance Y2 , i.e., Yt N(Y ; Y2 ) for t 1. Let

fY SRS;tg be a sequence of IPIDn random variables with

SRS, where Y SRS;t = (1=n) i=1 Yi;t. Here, Yi;t is the

ith observation in the tth simple random sample of size

n, for i = 1; 2; ; n. Note that Y SRS;t is also a normal

random variable with the mean Y and the variance

Y2 =n, i.e., Let

Y=SRSp;tnjNY (Y;Y;1Y2j==nY).

be

the

amount

of

standardized shift to be detected in the in-control

process mean Y , where Y;1 is the out-of-control

process mean.

By using Y SRS;t, an EWMA statistic, say Zt, is

given by:

Zt = Y SRS;t + (1 )Zt 1;

(1)

where Zt and Zt 1 are the current and past infor-

mation, respectively, and 0 < 1 is a smoothing

constant. The starting value of Zt is set equal to the incontrol process mean Y , i.e., Z0 = Y . The variance of Zt is:

Var(Zt) = Y2 1 (1 )2t :

(2)

n (2 )

Here, if the time t gets large, the term [1 (1 )2t] approaches unity. The asymptotic variance of the EWMA statistic Zt is given by:

Var(Zt) = Y2 :

(3)

n (2 )

The Upper Control Limit (UCL) and the Lower Control Limit (LCL) of the EWMA chart based on the asymptotic variance of Zt are given by:

s

UCL = Y + LpYn (2 );

and: s

LCL = Y LpYn (2 ): (4)

Similarly, the UCL and LCL of the Shewhart chart based on Y SRS;t are given by:

UCL = Y + d1 pYn ;

and:

LCL = Y d1 pYn: (5)

The Central Limits (CLs) of both the EWMA and Shewhart charts are set equal to the in-control process mean, i.e., CL = Y . Here, L and d1 are the design parameters of the SEWMA chart, respectively, and their values depend on the choices of and the desired in-control ARL. The SEWMA chart triggers an out-ofcontrol signal when either Zt falls outside the EWMA control limits or Y SRS;t falls outside the Shewhart control limits. When working with the SEWMA chart, the range of d1 should be 3:0 < d1 < 4:5 [cf., 5].

As mentioned earlier, the EWMA chart is very e ective in detecting small-to-moderate process shifts. However, there might exist a situation where the process{initially or in the startup{may make tracks in a di erent direction from the process target or after the process is recouped from an out-of-control state. In such situations, giving a head-start to the EWMA chart may help earlier detection of shifts in the process target. The FIR feature in the EWMA charting structure was rst suggested by Lucas and Saccucci [5] to overcome such situations. They suggested using two one-sided EWMA charts, each with a head-start. Their work was further extended by Rhoads et al. [41] who used two one-sided EWMA charts with head-starts and the time-varying control limits. To further reduce the time-varying control limits of the EWMA chart for the rst few samples, say ten or twenty, Steiner [42] used an exponentially decreasing adjustment factor. The new control limits of the EWMA chart with the FIRadjustment factor are as follows:

s

UCL=Y +L pYn 1 (1 f)1+a(t 1) (2 ) ; (6)

s

LCL=Y L pYn 1 (1 f)1+a(t 1) (2 ) ; (7)

where f and a are known constants. The choice of a, suggested by Steiner [42], for which the FIR adjustment has little e ect after the 20th observation is a = (1=19)(2=log(1 f) + 1). For instance, with f = 0:5, we get a = 0:3. For more details regarding the FIR feature with the EWMA chart, we refer to Rhoads et al. [41], Steiner [42], Knoth [6], and Haq et al. [10].

3. The Shewhart-CUSUM chart

SCUSUM chart was rst suggested by Lucas [3], who integrated the Shewhart chart with the CUSUM chart; it is useful to detect small and large shifts in the process target simultaneously. In the SCUSUM charting structure, the CUSUM chart quickly detects small shifts, whereas the Shewhart chart swiftly detects large shifts in the process target.

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SCUSUM chart works similarly to the classical CUSUM chart. The CUSUM chart works with the two CUSUMs, upward and downward, say Ct+ and Ct , respectively, given by:

Ct+ = Max[0; +(Y SRS;t Y ) K + Ct+ 1]; (8) Ct = Max[0; (Y SRS;t Y ) K + Ct 1]; (9)

where K = kY =pn is the reference of slack value of

the CUSUM chart. Here, k is usually taken as half of the magnitude of the shift to be detected in Y , i.e., k = =2. The CUSUM chart declares the out-of-control

process if either Ct+ or Cpt exceeds the predetermined

decision value H = hY = n, where h is selected to get the desired in-control ARL of the CUSUM chart. In the SCUSUM chart, the above CUSUM is integrated with the Shewhart chart, as explained in the previous section. The SCUSUM chart triggers an out-of-control signal when Ct+ or Ct exceeds H or if YSRS;t exceeds the control limits given in Eq. (5) in either direction. It is customary to take d1 = 3:5 [cf., 3].

The FIR feature with the CUSUM chart was rst suggested by Lucas and Crosier [2], which enables the CUSUM chart to react quickly against the startup/initial problems. The FIR feature in the CUSUM chart works by resetting the starting values of both plotting-CUSUMs to some non-zero constants. They recommended using H=2 for a 50% head-start, i.e., by setting C0+ = C0 = H=2 . For more details, see Lucas and Crosier [2] and Haq et al. [10].

4. The VLRSS scheme

In this section, we brie y review the mean estimator using VLRSS under both perfect and imperfect rankings. The VLRSS scheme is a cost-e ective alternative to the SRS and RSS schemes. This scheme provides not only an unbiased and precise mean estimator when sampling from a symmetric population, but also plenty of options to the experimenter in selecting di erent representative samples with the less number of identi ed units compared to that using the RSS scheme, i.e., the ranking costs with VLRSS could be more or less than that with the RSS. It is worth mentioning that the VLRSS scheme provides a more e cient mean estimator than the mean estimators based on RSS and MRSS schemes when ranking costs are negligible. However, when ranking costs are high, it is still bene cial to use VLRSS scheme with the less ranking cost than that with the RSS schemes [cf., 18].

The main steps involved in selecting a varied L ranked set sample of size n are presented as follows:

Step 1: Select the value of the VLRSS coe cient,

say w = [al], where 0 a < 0:5 . Here, [ ] is the

largest possible integer value;

Step 2: Select 2wl units from the target population.

Divide these units into 2w sets, with each set consisting of l units;

Step 3: Rank the units within each set by any cheap

or inexpensive method with respect to the study variable or using ranks of an auxiliary variable;

Step 4: Select the vth and (l v + 1)th smallest

ranked units from the rst and last w sets, respectively, where v = 1; 2; ; [l=2];

Step 5: Identify m(m 2w) units from the target

population and, then, divide these units into m 2w sets, with each set comprising m units;

Step 6: Select the ith smallest ranked unit from the

(i+w)th set of m units, for i = w+1; w+2; ; m w;

Step 7: This completes one cycle of a varied L ranked

set sample of size m. Steps 1-6 could be repeated, if necessary, r number of times to get a total sample of size n = mr units.

Symbolically, let (Yi1j; Yi2j; ; Yilj), i = 1; 2; ; 2w, be 2w samples, each of size l, for the jth cycle, where j = 1; 2; ; r. Let Yi(v:l)j denote the vth order statistic of (Yi1j; Yi2j; ; Yilj) for i = 1; 2; ; w, and let Yi(l v+1:l)j be the (l v + 1)th order statistic of (Yi1j; Yi2j; ; Yilj) for i = w + 1; w + 2; ; 2w. Let (Y(i+w)1j; Y(i+w)2j; ; Y(i+w)mj), i = w+1; 2; ; m w, denote m 2w samples, each of size m, for the jth cycle. Let Yi+w(i:m)j denote the ith order statistic of (Y(i+w)1j; Y(i+w)2j; ; Y(i+w)mj) for i = 1; 2; ; m w.

The sample mean based on a varied L ranked set sample of size n, denoted by Y VLRSS, and its variance, respectively, are given by:

Y VLRSS = 1 X r

X w

Yi(v:l)j +

X 2w

Yi(l

v+1:l)j

n j=1 i=1

i=w+1

mXw

!

+

Yi+w(i:m)j ;

(10)

i=w+1

Var(Y VLRSS) = n1m w(Y2 (v:l) + Y2 (l v+1:l))

mXw

!

+

Y2 (i:m) ;

(11)

i=w+1

where: Y2 (v:l) = Var(Yi(v:l)j);

Y2 (l v+1:l) = Var(Yi(l v+1:l)j); and

Y2 (i:m) = Var(Yi(i:m)j): For more details regarding the computation of the

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variances of order statistics, we refer to David and Nagaraja [43].

For a symmetric population, Haq et al. [18] showed that Y VLRSS is an unbiased estimator of Y . They have also shown that, with some suitable choices of v, l, and w, the existing RSS schemes are special cases of VLRSS. For instance, for w = 0, VLRSS becomes RSS; for w = [(m 1)=2], l = m, and v = w+1, VLRSS becomes MRSS, etc. While selecting a varied L ranked set sample of size n, the experimenter needs to identify nm 2w(m l)r units, while the classical RSS and MRSS require identifying nm units when selecting a sample of size n. It is to be noted that when m > l, VLRSS requires fewer identi ed units than that when using the RSS or MRSS [cf., 18].

4.1. The imperfect VLRSS scheme

There may exist a situation where it is not possible to rank the study variable visually, or it is costly and time consuming. This issue can be solved by ranking the study variable (Y ) using the ranks of a highly correlated variable, say X, given that it is readily available. Stokes [13] suggested a simple model for the imperfect rankings, given by

Yi[i:u]j = Y + XY Xi(i:u)j X + ij;

i = 1; 2; ; u; j = 1; 2; ; r;

(12)

where u = l; m; X and X are the population mean and standard deviation of X, respectively, and is the

correlation between Y and X. Here, ij N(0; Y2 (1

2)), and Xi(i:u)j and ij are mutually independent. Yi[i:u]j is the ith concomitant or induced order statistic corresponding to the ith order statistic Xi(i:u)j, i = 1; 2; ; u. The values of X are perfectly ranked; however, those of Y are ranked with error. On the lines of Stokes [13], the sample mean under Imperfect VLRSS (IVLRSS), say Y IVLRSS, and its variance, are, respectively, given by:

Y IVLRSS = 1 X r

X w

Yi[v:l]j +

X 2w

Yi[l

v+1:l]j

n j=1 i=1

i=w+1

mXw

!

+

Yi+w[i:m]j ;

(13)

i=w+1

and:

Var(Y VLRSS) = n1m w(Y2 [v:l] + Y2 [l v+1:l])

mXw

!

+

Y2 [i:m]

(14)

i=w+1

(

= n1m mY2 (1 2)+2 XY22

mXw

!)

+

X2 (i:m) ;

i=w+1

2wX2 (v:l) (15)

where:

Y2 [v:l] = Var(Yi[v:l]j);

Y2 [l v+1:l] = Var(Yi[l v+1:l]j); and

Y2 [i:m] = Var(Yi[i:m]j):

For more details on the computation of these variances, we refer to see David and Nagaraja [43]. When sampling from a symmetric bivariate population, Y IVLRSS is unbiased, is characterized by reasonable assumptions, and is more precise than the mean estimators based on imperfect RSS and MRSS schemes [cf., 18].

5. The proposed control charts

In this section, new SEWMA and SCUSUM control charts are proposed for e ciently monitoring the process mean Y under both perfect and imperfect VLRSS schemes. The run length characteristics of these control charts are also computed through Monte Carlo simulations.

5.1. The SEWMA chart

Suppose that a sample of size n is selected from the

target population with S scheme at the time t( 1), where S = VLRSS and IVLRSS. Let fY S;tg be a

sequence of IID random variables for t = 1; 2; .

By considering fY S;tg, it is possible to construct an

SEWMA chart for monitoring Y . The plottingstatistic of the SEWMA chart with S scheme is given by:

Qt = Y S;t + (1 )Qt 1;

(16)

where is a smoothing constant. The asymptotic variance of Qt is:

Var(Qt) = Var(Y S;t) (2 ); (17)

where Var(Y S;t) denotes the variance of fY S;tg at the

time t. The control limits of the SEWMA chart with S scheme are:

q

s

UCL = Y + L Var(Y S;t) (2 );

and:

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q

s

LCL = Y L Var(Y S;t) (2 ) :

(18)

Similarly, the control limits of the Shewhart chart,

based on fY S;tg, are given by:

q

UCL = Y + d1 Var(Y S;t);

and:

q

LCL = Y d1 Var(Y S;t);

(19)

where L and d1 are positive control charting multipliers that are selected to ensure that the in-control ARL of the SEWMA chart has reached a certain level. The SEWMA chart triggers an out-of-control signal

whenever Qt or fY S;tg falls outside their respective

control limits' intervals, i.e., [LCL, UCL]. As mentioned earlier, the sensitivity of the SEWMA chart can be enhanced by giving a head-start to the SEWMA chart with the FIR-adjustment [cf., 42]; on the similar lines, an FIR feature is associated with the proposed SEWMA chart by setting f = 0:5 and a = 0:3, as recommended by Steiner [42].

5.2. The SCUSUM chart

To construct an SCUSUM chart for monitoring Y ,

consider the sequence fY S;tg for t = 1; 2; . The

plotting-statistics (upper and lower CUSUMs) of the proposed SCUSUM chart using S scheme are, respectively, de ned by:

Ct+ = Max[0; +(Y S;t Y ) K + Ct+ 1];

(20)

Ct = Max[0; (Y S;t Y ) K + Ct 1];

(21)

where C0+ = C0 = 0. The reference value, K, and the decision interval, H, of the SCUSUM chart are respectively as follows:

q

K = k Var(Y S;t);

(22)

q

H = h Var(Y S;t);

(23)

where the values of k and h are the same as explained in the previous section. Similarly, the control limits of

the Shewhart chart based on fY S;tg are given by:

q

UCL = Y + d1 Var(Y S;t);

and:

q

LCL = Y d1 Var(Y S;t):

(24)

The SCUSUM triggers an out-of-control signal if Ct+ or Ct exceeds H or if Y S;t is less than LCL or greater

than UCL of the Shewhart chart. The sensitivity of the CUSUM chart for the start-up problems, as suggested by Lucas and Crosier [2], could increase with a headstart feature. They recommended resetting the starting values of Ct+ and Ct to non-zero constants, like C0+ = C0 = H=2 for an 50% head-start [cf., 2]. On the same lines, an FIR feature is attached to the SCUSUM chart with 50% head-start.

5.3. Run length evaluation

Generally, the run length performance of a control chart is evaluated in terms of its run length characteristics including the ARL and the SDRL. For an in-control process, the in-control ARL should be large enough to avoid false alarms, while, for an out-of-control process, it should be as small as possible so that the control chart can swiftly trigger an out-of-control signal. In the literature, there exist some methods that could be used to compute the run length characteristics of a control chart, including the integral equations, Markov chain, and the Monte Carlo simulations. The Monte Carlo simulation method is broadly used to compute the run length characteristics of the control charts, and thus it is used here.

In order to evaluate the run length performances of the proposed control charts, we generate samples under VLRSS from a normal distribution. The incontrol ARL is set equal to 500 a choice recommended by the SPC practitioners. Here, each simulation run comprises 50,000 iterations of the run length. In Tables 1 and 2, the values of (; L) and (k; h) are reported for the SEWMA and SCUSUM charts, respectively, with di erent possible values of (m; l; v) with r = 1 when the in-control ARL is matched as 500. These constants could be used when using the proposed charts with di erent choices of (m; l; v) when the in-control ARL is xed to 500.

For brevity of discussion, without loss of generality, with n = 5 and r = 1, using di erent pairs of (l; v) with w = 2, we compute the ARLs and SDRLs of the proposed control charts in Tables 3-6 (with and without FIR features). It is to be noted that, for a given sample size n, we consider those choices of w and (l; v) with the VLRSS scheme for which the mean estimator is precise [cf., 18]. Di erent values of are considered, i.e., = 0(0:25)4. For both the SEWMA and SCUSUM charts, di erent values of and k have been considered. The values of d1 for the SEWMA and SCUSUM charts are set equal to 3.31 and 3.50, respectively. Moreover, the results are computed when sampling from a normal distribution. Here, under each simulation run, 50,000 replications of the run length are considered. It is observed that the out-of-control ARLs tend to decrease as the value of increases, and vice versa. A similar trend is observed when a control chart is constructed with the FIR feature.

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Table 1. The values of L with di erent choices of (l; v) when the in-control ARL of the SEWMA-VLRSS chart is 500.

m=2

(0; 0)

m=4 (2; 1) (3; 2) (4; 1) (4; 2) (5; 3) (6; 3)

0.05 2.2582

2.8510 2.8523 2.8800 2.8490 2.8492 2.8485

0.10 2.4260

3.0270 3.0164 3.0463 3.0257 3.0225 3.0230

0.25 2.5789

3.1700 3.1666 3.1856 3.1650 3.1659 3.1648

0.50 2.6350

3.1755 3.1700 3.1889 3.1747 3.1695 3.1755

0.75 2.6470

3.1230 3.1186 3.1390 3.1190 3.1155 3.1161

1.00 2.6465

3.1050 3.1000 3.1158 3.0990 3.1000 3.1000

m = 3; w = 1 (2; 1) (3; 1) (3; 2) (4; 1) (4; 2) (5; 1) (5; 2) (5; 3)

0.05 2.8600 2.8800 2.8519 2.8890 2.8510 2.8990 2.8585 2.8522

0.10 3.0307 3.0469 3.0285 3.0599 3.0285 3.0690 3.0308 3.0290

0.25 3.1730 3.1860 3.1710 3.2004 3.1710 3.2100 3.1683 3.1660

0.50 3.1763 3.1921 3.1758 3.2056 3.1759 3.2105 3.1800 3.1759

0.75 3.1240 3.1377 3.1200 3.1499 3.1207 3.1545 3.1230 3.1203

1.00 3.1070 3.1157 3.1025 3.1260 3.1028 3.1313 3.1070 3.1013

m=5

w=1

w=2

(2; 1) (3; 2) (4; 2) (5; 1)

(5; 3) (6; 3) (7; 4) (8; 4)

0.05 2.8494 2.8490 2.8480 2.8740

2.8450 2.8460 2.8444 2.8448

0.10 3.0270 3.0190 3.0250 3.0450

3.0170 3.0179 3.0164 3.0164

0.25 3.1695 3.1654 3.1645 3.1850

3.1570 3.1579 3.1584 3.1590

0.50 3.1747 3.1690 3.1755 3.1920

3.1660 3.1669 3.1670 3.1673

0.75 3.1195 3.1150 3.1150 3.1370

3.1150 3.1150 3.1134 3.1155

1.00 3.1000 3.1000 3.1000 3.1147

3.0980 3.0968 3.0945 3.0970

Table 2. The values of h with di erent choices of (l; v) when the in-control ARL of the SCUSUM-VLRSS chart is 500.

m=2

k

(0; 0)

m = 4; w = 1 (2; 1) (3; 2) (4; 1) (4; 2) (5; 3) (6; 3)

0.25 9.1100

9.0950 9.0740 9.1100 9.0970 9.0760 9.0875

0.50 5.3280

5.3166 5.3105 5.3360 5.3050 5.3050 5.3077

0.75 3.6980

3.6785 3.6765 3.6950 3.6753 3.6758 3.6760

1.00 2.7450

2.7282 2.7255 2.7440 2.7245 2.7240 2.7252

m = 3; w = 1 k (2; 1) (3; 1) (3; 2) (4; 1) (4; 2) (5; 1) (5; 2) (5; 3)

0.25 9.0953 9.1109 9.0700 9.1230 9.0950 9.1560 9.0950 9.0747

0.50 5.3164 5.3340 5.3133 5.3430 5.3085 5.3780 5.3143 5.3060

0.75 3.6791 3.6980 3.6767 3.7122 3.6790 3.7181 3.6800 3.6767

1.00 2.7320 2.7450 2.7280 2.7547 2.7284 2.7670 2.7290 2.7287

m=5

w=1

w=2

k (2; 1) (3; 2) (4; 2) (5; 1)

(5; 3) (6; 3) (7; 4) (8; 4)

0.25 9.0959 9.0739 9.0755 9.1110

9.0490 9.0499 9.0508 9.0510

0.50 5.3109 5.3080 5.3080 5.3270

5.2930 5.2935 5.2938 5.2938

0.75 3.6760 3.6755 3.6746 3.6960

3.6670 3.6682 3.6666 3.6682

1.00 2.7250 2.7245 2.7260 2.7460

2.7170 2.7193 2.7185 2.7186

5.4. When the process parameters are unknown

If the underlying process parameters are not known in advance{phase-I monitoring, then it is customary to estimate them using a large historical dataset, provided

that it has been obtained from an in-control process. Suppose that, from an in-control process, q subgroups and each of size m are available with S scheme.

In the perfect ranking case, Y and Var(Y VLRSS) could be estimated by using their respective unbiased

M. Awais and A. Haq/Scientia Iranica, Transactions E: Industrial Engineering 26 (2019) 3796{3818

3803

Table 3. The run length pro les of SEWMA-VLRSS chart when the in-control ARL is 500.

= 0:05

(l; v) (5; 3) (6; 3) (7; 4) (8; 4)

L 2.8450 2.8460 2.8444 2.8448

0.00

ARL SDRL

500.25 494.42

499.43 490.67

500.87 491.50

499.53 490.78

0.25

ARL SDRL

36.42 22.74

32.97 19.85

30.09 17.63

28.20 15.99

0.50

ARL SDRL

13.33 5.62

12.29 5.08

11.37 4.58

10.77 4.28

0.75

ARL SDRL

7.89 3.01

7.29 2.80

6.72 2.62

6.35 2.52

1.00

ARL SDRL

5.35 2.24

4.93 2.13

4.48 2.03

4.19 1.96

1.25

ARL SDRL

3.81 1.86

3.41 1.76

3.04 1.64

2.81 1.56

1.50

ARL SDRL

2.69 1.52

2.39 1.38

2.10 1.23

1.92 1.13

1.75

ARL SDRL

1.95 1.15

1.72 1.00

1.53 0.84

1.41 0.73

2.00

ARL SDRL

1.49 0.80

1.34 0.65

1.23 0.52

1.16 0.44

2.25

ARL SDRL

1.23 0.52

1.14 0.40

1.08 0.30

1.06 0.24

2.50

ARL SDRL

1.10 0.33

1.05 0.24

1.03 0.16

1.01 0.12

2.75

ARL SDRL

1.03 0.19

1.02 0.13

1.01 0.08

1.00 0.05

3.00

ARL SDRL

1.01 0.10

1.00 0.07

1.00 0.04

1.00 0.02

3.25

ARL SDRL

1.00 0.05

1.00 0.03

1.00 0.01

1.00 0.00

3.50

ARL SDRL

1.00 0.02

1.00 0.01

1.00 0.00

1.00 0.00

3.75

ARL SDRL

1.00 0.01

1.00 0.00

1.00 0.00

1.00 0.00

4.00

ARL SDRL

1.00 0.00

1.00 0.00

1.00 0.00

1.00 0.00

= 0:25

(l; v) (5; 3) (6; 3) (7; 4) (8; 4)

L 3.1570 3.1579 3.1584 3.1590

0.00

ARL SDRL

500.20 500.00

500.80 497.87

502.48 501.34

500.74 498.89

0.25

ARL SDRL

69.36 64.36

61.38 57.35

53.18 48.42

48.84 44.58

0.50

ARL SDRL

14.56 10.63

12.78 9.03

11.35 7.69

10.30 6.74

0.75

ARL SDRL

6.52 3.59

5.88 3.12

5.30 2.74

4.94 2.47

1.00

ARL SDRL

4.04 1.89

3.68 1.69

3.37 1.53

3.16 1.43

1.25

ARL SDRL

2.88 1.29

2.63 1.19

2.40 1.09

2.25 1.03

1.50

ARL SDRL

2.18 1.00

1.99 0.92

1.80 0.84

1.69 0.79

1.75

ARL SDRL

1.71 0.79

1.55 0.71

1.43 0.63

1.34 0.56

2.00

ARL SDRL

1.39 0.60

1.29 0.51

1.20 0.43

1.15 0.37

2.25

ARL SDRL

1.20 0.43

1.13 0.35

1.08 0.28

1.05 0.23

2.50

ARL SDRL

1.09 0.29

1.05 0.22

1.03 0.16

1.01 0.12

2.75

ARL SDRL

1.03 0.18

1.02 0.12

1.01 0.08

1.00 0.05

3.00

ARL SDRL

1.01 0.11

1.00 0.06

1.00 0.04

1.00 0.02

3.25

ARL SDRL

1.00 0.06

1.00 0.03

1.00 0.01

1.00 0.01

3.50

ARL SDRL

1.00 0.03

1.00 0.01

1.00 0.01

1.00 0.01

3.75

ARL SDRL

1.00 0.01

1.00 0.00

1.00 0.00

1.00 0.00

4.00

ARL SDRL

1.00 0.00

1.00 0.00

1.00 0.00

1.00 0.00

(5; 3) 3.0170 502.58 494.28 42.10 32.78 12.37

6.33 6.89 2.89 4.64 1.90 3.34 1.52 2.47 1.25 1.87 0.99 1.46 0.72 1.22 0.50 1.10 0.32 1.04 0.20 1.01 0.11 1.00 0.06 1.00 0.02 1.00 0.01 1.00 0.00 (5; 3) 3.1660 501.10 508.07 118.70 117.02 23.68 21.49 8.30 6.38 4.30 2.70 2.82 1.49 2.06 0.98 1.64 0.71 1.37 0.54 1.19 0.41 1.09 0.29 1.03 0.18 1.01 0.10 1.00 0.05 1.00 0.03 1.00 0.01 1.00 0.00

= (6; 3) 3.0179 502.48 500.68 37.61 28.23 11.34 5.57 6.34 2.61 4.27 1.79 3.04 1.43 2.21 1.15 1.66 0.87 1.33 0.61 1.14 0.39 1.05 0.24 1.02 0.13 1.00 0.07 1.00 0.03 1.00 0.01 1.00 0.00 1.00 0.00

= (6; 3) 3.1669 500.45 504.69 106.57 103.89 20.41 18.15 7.19 5.31 3.79 2.26 2.52 1.29 1.88 0.87 1.50 0.63 1.27 0.47 1.13 0.34 1.05 0.22 1.02 0.13 1.00 0.07 1.00 0.03 1.00 0.01 1.00 0.00 1.00 0.00

0:10 (7; 4) 3.0164 500.45 500.09 33.53 24.51 10.34 4.93 5.84 2.40 3.90 1.68 2.77 1.34 1.99 1.05 1.50 0.76 1.22 0.49 1.08 0.30 1.02 0.16 1.01 0.08 1.00 0.03 1.00 0.01 1.00 0.00 1.00 0.00 1.00 0.00

0:50 (7; 4) 3.1670 501.94 500.30 93.83 91.67 17.45 15.24 6.24 4.39 3.40 1.95 2.28 1.12 1.72 0.76 1.39 0.56 1.19 0.41 1.08 0.27 1.03 0.16 1.01 0.08 1.00 0.04 1.00 0.01 1.00 0.00 1.00 0.00 1.00 0.00

(8; 4) 3.0164 502.76 498.02 30.84 22.16

9.73 4.53 5.53 2.26 3.67 1.61 2.56 1.28 1.83 0.97 1.40 0.67 1.16 0.42 1.05 0.24 1.01 0.12 1.00 0.06 1.00 0.03 1.00 0.00 1.00 0.00 1.00 0.00 1.00 0.00 (8; 4) 3.1673 500.08 502.16 85.87 83.88 15.45 13.37 5.67 3.89 3.13 1.73 2.14 1.02 1.62 0.70 1.31 0.51 1.14 0.35 1.05 0.22 1.01 0.12 1.00 0.05 1.00 0.02 1.00 0.01 1.00 0.00 1.00 0.00 1.00 0.00

3804

M. Awais and A. Haq/Scientia Iranica, Transactions E: Industrial Engineering 26 (2019) 3796{3818

Table 4. The run length pro les of SCUSUM-VLRSS chart when the in-control ARL is 500.

k = 0:25

(l; v) (5; 3) (6; 3) (7; 4) (8; 4)

h 9.0490 9.0499 9.0508 9.0510

0.00

ARL SDRL

500.80 486.65

500.62 490.59

500.72 492.92

500.26 487.82

0.25

ARL SDRL

36.00 21.62

32.87 18.97

30.02 16.75

28.32 15.31

0.50

ARL SDRL

13.64 5.36

12.66 4.82

11.76 4.36

11.17 4.09

0.75

ARL SDRL

8.25 2.88

7.65 2.66

7.11 2.49

6.73 2.38

1.00

ARL SDRL

5.71 2.11

5.27 2.02

4.85 1.94

4.54 1.88

1.25

ARL SDRL

4.15 1.81

3.76 1.73

3.38 1.65

3.12 1.58

1.50

ARL SDRL

3.02 1.56

2.69 1.45

2.37 1.33

2.16 1.24

1.75

ARL SDRL

2.18 1.25

1.92 1.11

1.69 0.96

1.55 0.85

2.00

ARL SDRL

1.65 0.92

1.46 0.77

1.31 0.62

1.23 0.51

2.25

ARL SDRL

1.31 0.62

1.20 0.49

1.12 0.36

1.08 0.30

2.50

ARL SDRL

1.14 0.39

1.08 0.29

1.04 0.20

1.02 0.16

2.75

ARL SDRL

1.05 0.24

1.03 0.16

1.01 0.11

1.01 0.07

3.00

ARL SDRL

1.02 0.13

1.01 0.09

1.00 0.04

1.00 0.04

3.25

ARL SDRL

1.01 0.07

1.00 0.04

1.00 0.02

1.00 0.01

3.50

ARL SDRL

1.00 0.03

1.00 0.02

1.00 0.01

1.00 0.00

3.75

ARL SDRL

1.00 0.02

1.00 0.01

1.00 0.00

1.00 0.00

4.00

ARL SDRL

1.00 0.00

1.00 0.00

1.00 0.00

1.00 0.00

k = 0:75

(l; v) (5; 3) (6; 3) (7; 4) (8; 4)

h 3.6670 3.6682 3.6666 3.6682

0.00

ARL SDRL

502.85 499.40

502.68 497.36

502.63 501.27

500.05 495.78

0.25

ARL SDRL

70.37 66.21

61.78 57.04

53.89 49.40

48.85 44.55

0.50

ARL SDRL

13.86 10.04

12.16 8.45

10.70 7.10

9.81 6.29

0.00

ARL SDRL

502.85 499.40

502.68 497.36

502.63 501.27

500.05 495.78

0.25

ARL SDRL

70.37 66.21

61.78 57.04

53.89 49.40

48.85 44.55

0.50

ARL SDRL

13.86 10.04

12.16 8.45

10.70 7.10

9.81 6.29

0.75

ARL SDRL

6.23 3.32

5.60 2.85

5.11 2.51

4.73 2.25

1.00

ARL SDRL

3.92 1.75

3.59 1.55

3.28 1.38

3.09 1.30

1.25

ARL SDRL

2.83 1.17

2.61 1.07

2.40 0.99

2.25 0.93

1.50

ARL SDRL

2.19 0.90

2.01 0.84

1.84 0.77

1.74 0.73

1.75

ARL SDRL

1.76 0.73

1.61 0.67

1.48 0.60

1.40 0.56

2.00

ARL SDRL

1.45 0.58

1.34 0.52

1.25 0.46

1.19 0.40

2.25

ARL SDRL

1.25 0.45

1.17 0.39

1.11 0.32

1.07 0.26

2.50

ARL SDRL

1.12 0.33

1.07 0.26

1.04 0.20

1.02 0.15

2.75

ARL SDRL

1.05 0.22

1.02 0.15

1.01 0.10

1.01 0.08

3.00

ARL SDRL

1.02 0.13

1.01 0.08

1.00 0.05

1.00 0.03

3.25

ARL SDRL

1.00 0.07

1.00 0.04

1.00 0.02

1.00 0.01

3.50

ARL SDRL

1.00 0.04

1.00 0.02

1.00 0.00

1.00 0.00

3.75

ARL SDRL

1.00 0.02

1.00 0.00

1.00 0.00

1.00 0.00

4.00

ARL SDRL

1.00 0.00

1.00 0.00

1.00 0.00

1.00 0.00

(5; 3) 5.2930 502.65 497.32 47.16 39.74 12.17

6.73 6.48 2.76 4.37 1.71 3.23 1.28 2.48 1.08 1.94 0.91 1.54 0.73 1.28 0.54 1.13 0.36 1.05 0.23 1.02 0.13 1.01 0.07 1.00 0.04 1.00 0.01 1.00 0.00 (5; 3) 2.7170 500.36 497.91 96.77 94.02 17.85 15.22 500.36 497.91 96.77 94.02 17.85 15.22 6.69 4.378 3.84 1.99 2.68 1.18 2.08 0.84 1.69 0.66 1.44 0.55 1.24 0.44 1.12 0.33 1.05 0.22 1.02 0.13 1.01 0.08 1.00 0.04 1.00 0.01 1.00 0.01

k= (6; 3) 5.2935 500.08 493.26 41.55 34.25 11.00 5.83 5.95 2.44 4.05 1.56 2.97 1.21 2.26 1.01 1.75 0.84 1.40 0.64 1.19 0.43 1.08 0.28 1.03 0.16 1.01 0.09 1.00 0.04 1.00 0.02 1.00 0.01 1.00 0.01

k= (6; 3) 2.7193 499.33 496.82 86.37 84.06 15.30 12.52 499.33 496.82 86.37 84.06 15.30 12.52 5.91 3.68 3.48 1.71 2.46 1.05 1.92 0.76 1.57 0.61 1.33 0.49 1.17 0.38 1.07 0.26 1.03 0.16 1.01 0.09 1.00 0.04 1.00 0.01 1.00 0.00 1.00 0.00

0:50 (7; 4) 5.2938 501.11 494.90 36.91 29.62 9.96 5.06 5.50 2.22 3.73 1.44 2.72 1.14 2.05 0.95 1.58 0.75 1.28 0.53 1.12 0.34 1.04 0.20 1.01 0.11 1.00 0.05 1.00 0.02 1.00 0.01 1.00 0.00 1.00 0.00

1:00 (7; 4) 2.7185 501.55 500.44 75.50 72.87 13.16 10.48 501.55 500.44 75.50 72.87 13.16 10.48 5.26 3.13 3.16 1.49 2.27 0.95 1.77 0.69 1.46 0.56 1.24 0.44 1.11 0.31 1.04 0.20 1.01 0.10 1.00 0.04 1.00 0.02 1.00 0.01 1.00 0.00 1.00 0.00

(8; 4) 5.2938 499.16 491.04 33.68 26.48

9.32 4.60 5.17 2.06 3.52 1.37 2.55 1.09 1.91 0.90 1.48 0.69 1.21 0.46 1.08 0.28 1.02 0.15 1.01 0.08 1.00 0.04 1.00 0.01 1.00 0.00 1.00 0.00 1.00 0.00 (8; 4) 2.7186 500.13 498.09 68.85 66.29 11.72 9.08 500.13 498.09 68.85 66.29 11.72 9.08 4.81 2.76 2.96 1.36 2.13 0.87 1.67 0.65 1.38 0.52 1.19 0.40 1.07 0.26 1.02 0.15 1.01 0.07 1.00 0.03 1.00 0.01 1.00 0.00 1.00 0.00 1.00 0.00

M. Awais and A. Haq/Scientia Iranica, Transactions E: Industrial Engineering 26 (2019) 3796{3818

3805

Table 5. The run length pro les of SEWMA-VLRSS chart with the FIR feature for the time when the in-control ARL is

500.

= 0:05

= 0:10

(l; v) (5; 3) (6; 3) (7; 4) (8; 4)

(5; 3) (6; 3) (7; 4) (8; 4)

L 2.8479 2.8479 2.8475 2.8472

3.0297 3.0294 3.0294 3.0283

0.00

ARL SDRL

499.65 493.15

501.08 496.13

502.93 494.98

500.20 492.83

501.52 503.53

501.09 510.07

502.12 504.01

500.81 507.82

0.25

ARL SDRL

35.57 23.10

32.18 20.41

29.27 17.88

27.25 16.22

39.47

34.99

30.98

28.24

33.06

28.62

24.90

22.39

0.50

ARL SDRL

12.07 5.93

11.06 5.39

10.06 4.87

9.46 4.51

10.28

9.22

8.29

7.70

6.59

5.79

5.08

4.64

0.75

ARL SDRL

6.53 3.02

5.96 2.72

5.46 2.46

5.10 2.32

5.12

4.64

4.21

3.96

2.80

2.45

2.19

2.00

1.00

ARL SDRL

4.20 1.89

3.83 1.72

3.50 1.57

3.28 1.48

3.27

3.01

2.76

2.60

1.54

1.37

1.22

1.13

1.25

ARL SDRL

2.98 1.35

2.71 1.24

2.46 1.14

2.30 1.08

2.40

2.22

2.06

1.95

1.02

0.92

0.83

0.78

1.50

ARL SDRL

2.23 1.05

2.01 0.96

1.83 0.87

1.70 0.80

1.90

1.76

1.64

1.56

0.76

0.69

0.63

0.60

1.75

ARL SDRL

1.73 0.82

1.57 0.72

1.43 0.64

1.35 0.57

1.58

1.47

1.37

1.30

0.61

0.56

0.52

0.48

2.00

ARL SDRL

1.40 0.62

1.29 0.53

1.20 0.44

1.15 0.38

1.35

1.26

1.19

1.14

0.50

0.45

0.39

0.35

2.25

ARL SDRL

1.20 0.44

1.13 0.36

1.08 0.28

1.05 0.23

1.19

1.13

1.08

1.05

0.40

0.33

0.27

0.22

2.50

ARL SDRL

1.09 0.30

1.05 0.22

1.03 0.16

1.01 0.12

1.09

1.05

1.03

1.01

0.28

0.22

0.16

0.12

2.75

ARL SDRL

1.04 0.19

1.02 0.12

1.01 0.08

1.00 0.06

1.03

1.02

1.01

1.00

0.18

0.13

0.08

0.06

3.00

ARL SDRL

1.01 0.10

1.00 0.07

1.00 0.04

1.00 0.02

1.01

1.00

1.00

1.00

0.11

0.06

0.04

0.02

3.25

ARL SDRL

1.00 0.06

1.00 0.03

1.00 0.01

1.00 0.01

1.00

1.00

1.00

1.00

0.05

0.03

0.02

0.01

3.50

ARL SDRL

1.00 0.03

1.00 0.01

1.00 0.01

1.00 0.01

1.00

1.00

1.00

1.00

0.02

0.01

0.01

0.00

3.75

ARL SDRL

1.00 0.01

1.00 0.01

1.00 0.00

1.00 0.00

1.00

1.00

1.00

1.00

0.01

0.01

0.00

0.00

4.00

ARL SDRL

1.00 0.00

1.00 0.00

1.00 0.00

1.00 0.00

1.00

1.00

1.00

1.00

0.00

0.00

0.00

0.00

= 0:25

= 0:50

(l; v) (5; 3) (6; 3) (7; 4) (8; 4)

(5; 3) (6; 3) (7; 4) (8; 4)

L 3.2014 3.1999 3.1990 3.1995

3.2490 3.2490 3.2486 3.2480

0.00

ARL SDRL

500.07 535.91

500.87 537.36

501.93 535.19

502.28 534.75

499.56 579.34

499.95 580.42

500.56 584.53

501.07 583.79

0.25

ARL SDRL

61.29 69.10

52.22 59.21

45.27 51.49

40.63 45.93

103.34 89.54

76.76

69.54

134.54 117.61 102.91 94.93

0.00

ARL SDRL

500.07 535.91

500.87 537.36

501.93 535.19

502.28 534.75

499.56 579.34

499.95 580.42

500.56 584.53

501.07 583.79

0.25

ARL SDRL

61.29 69.10

52.22 59.21

45.27 51.49

40.63 45.93

103.34 89.54

76.76

69.54

134.54 117.61 102.91 94.93

0.50

ARL SDRL

9.50 9.74

8.22 8.15

7.08 6.77

6.38 5.94

13.13

10.53

8.48

7.42

20.25

15.89

12.67

10.89

0.75

ARL SDRL

3.83 2.91

3.42 2.47

3.06 2.11

2.84 1.87

3.50

3.02

2.66

2.41

3.97

3.19

2.52

2.12

1.00

ARL SDRL

2.34 1.38

2.14 1.18

1.98 1.06

1.86 0.95

1.93

1.77

1.63

1.54

1.40

1.17

0.99

0.88

1.25

ARL SDRL

1.73 0.85

1.60 0.75

1.50 0.67

1.42 0.61

1.43

1.34

1.27

1.22

0.74

0.64

0.55

0.49

1.50

ARL SDRL

1.40 0.59

1.32 0.52

1.24 0.46

1.20 0.42

1.20

1.15

1.10

1.08

0.46

0.39

0.33

0.28

1.75

ARL SDRL

1.21 0.43

1.15 0.37

1.11 0.31

1.08 0.27

1.09

1.06

1.04

1.02

0.30

0.24

0.19

0.15

2.00

ARL SDRL

1.10 0.30

1.06 0.24

1.04 0.19

1.02 0.15

1.03

1.02

1.01

1.01

0.18

0.14

0.10

0.08

2.25

ARL SDRL

1.04 0.19

1.02 0.14

1.01 0.10

1.01 0.08

1.01

1.01

1.00

1.00

0.10

0.07

0.05

0.03

2.50

ARL SDRL

1.01 0.11

1.01 0.08

1.00 0.05

1.00 0.03

1.00

1.00

1.00

1.00

0.05

0.04

0.02

0.01

2.75

ARL SDRL

1.00 0.06

1.00 0.03

1.00 0.02

1.00 0.02

1.00

1.00

1.00

1.00

0.02

0.01

0.01

0.01

3.00

ARL SDRL

1.00 0.03

1.00 0.01

1.00 0.00

1.00 0.00

1.00

1.00

1.00

1.00

0.01

0.00

0.00

0.00

3.25

ARL SDRL

1.00 0.01

1.00 0.01

1.00 0.00

1.00 0.00

1.00

1.00

1.00

1.00

0.00

0.00

0.00

0.00

3.50

ARL SDRL

1.00 0.00

1.00 0.00

1.00 0.00

1.00 0.00

1.00

1.00

1.00

1.00

0.00

0.00

0.00

0.00

3.75

ARL SDRL

1.00 0.00

1.00 0.00

1.00 0.00

1.00 0.00

1.00

1.00

1.00

1.00

0.00

0.00

0.00

0.00

4.00

ARL SDRL

1.00 0.00

1.00 0.00

1.00 0.00

1.00 0.00

1.00

1.00

1.00

1.00

0.00

0.00

0.00

0.00

Sharif University of Technology

Scientia Iranica

Transactions E: Industrial Engineering http://scientiairanica.sharif.edu

New Shewhart-EWMA and Shewhart-CUSUM control charts for monitoring process mean

M. Awais and A. Haq

Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan. Received 7 August 2017; received in revised form 19 February 2018; accepted 2 July 2018

KEYWORDS

Average run length; CUSUM; Control chart; EWMA; Perfect and imperfect rankings; Statistical process control.

Abstract. This study proposes new Shewhart-EWMA (SEWMA) and Shewhart-CUSUM

(SCUSUM) control charts using Varied L Ranked Set Sampling (VLRSS) for monitoring the process mean, namely the SEWMA-VLRSS and SCUSUM-VLRSS charts. The run length characteristics of the proposed charts are computed using extensive Monte Carlo simulations. The proposed charts are compared with their existing counterparts in terms of the average and standard deviations of run lengths. It is found that, with perfect and imperfect rankings, the SEWMA-VLRSS and SCUSUM-VLRSS charts are more sensitive than their analogous charts based on simple random sampling, Ranked Set Sampling (RSS), and median RSS schemes. A real dataset is also used to explain the implementation of the proposed control charts.

© 2019 Sharif University of Technology. All rights reserved.

1. Introduction

The Statistical Process Control (SPC) is a collection of tools that help distinguish between two types of variation, namely the natural- and special-cause variations. A process is said to be in statistical control when only natural-caused variations are present, while the process with the special-cause variations is said to be out of control. There are seven major tools in the SPC, including the histogram, check sheet, Pareto chart, cause-and-e ect diagram, defect concentration diagram, scatter diagram, and control charts. Statistical quality control charts are very e ective SPC tools that are frequently used to monitor special-cause variations in a production/manufacturing process.

The control charts are divided into two categories, memory-less and memory-type control charts. The

*. Corresponding author. E-mail address: [email protected] (A. Haq)

doi: 10.24200/sci.2018.4962.1011

Shewhart-type charts fall in the memory-less category because they completely rely on the present information. A major limitation of the Shewhart control chart is that it is less sensitive against small and moderate shifts in the process parameter(s). On the other hand, the Exponentially Weighted Moving Average (EWMA) and the CUmulative SUM (CUSUM) control charts fall in the memory-type category. The reason is that both of these control charts take into account past and current information to maintain their plotting statistics. This feature of the memory-type control charts helps them swiftly react against small to moderate shifts in the process parameter(s).

The CUSUM chart was rst developed by Page [1]. Lucas and Crosier [2] associated a Fast Initial Response (FIR) feature with the CUSUM chart to further enhance its sensitivity by giving head-starts to the plotting CUSUMs at the beginning of a process. Lucas [3] used both the Shewhart and CUSUM charts simultaneously for monitoring small and large shifts in a process, named the Shewhart-CUSUM (SCUSUM) chart. For monitoring changes in the process mean,

M. Awais and A. Haq/Scientia Iranica, Transactions E: Industrial Engineering 26 (2019) 3796{3818

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Roberts [4] was the rst to introduce the EWMA chart. Lucas and Saccucci [5] attached an FIR feature with the EWMA chart to increase its sensitivity to the start-up/initial problems. Moreover, they coupled the Shewhart chart with the EWMA chart, named the Shewhart-EWMA (SEWMA) chart, for detecting small and large shifts simultaneously. There are many new advancements and improvements in the control charting structures of the EWMA and CUSUM charts. For some related works on these control charts, we refer to Knoth [6], Lucas and Crosier [2], Chiu [7], Abbas et al. [8], Haq [9], Haq et al. [10], and the references cited therein.

Ranked-Set Sampling (RSS) was rst introduced by McIntyre [11] for estimating mean pasture and forage yields. RSS is a cost-e ective alternative to Simple Random Sampling (SRS) in situations where the units to be sampled could be ranked with respect to each other prior to the formal measurements. Ranking may be done visually based on personal judgment or using the ranks of an auxiliary variable, provided that it is highly correlated with the study variable. For example, if the quanti cation of the underlying quality characteristic(s) is laborious, costly and timeconsuming, it may involve breaking the product{which is expensive and might be hard to construct, etc.{but using some experts' knowledge, it might be possible to rank the quality characteristic according to its quality level or using any less expensive method. There also exist the situations where the error is inevitable while ranking the units, particularly when ranking the units in large set sizes. Dell and Clutter [12] showed that, despite the presence of ranking errors, the mean estimator with RSS is not only unbiased but also more precise than the mean estimator with SRS. A simple imperfect ranking model was designed by Stokes [13], whereby the study variable could be ranked using the ranks of an auxiliary variable. For precisely estimating the mean of a symmetric population, Extreme RSS (ERSS) and Median RSS (MRSS) schemes were suggested by Samawi et al. [14] and Muttlak [15], respectively. The mean estimator with MRSS is more precise than those with the SRS and RSS, when sampling from a symmetric population. Muttlak [16] suggested using the Quartile RSS (QRSS) as a better alternative to SRS, RSS, and MRSS schemes for estimating the population mean when sampling from an asymmetric population. Utilizing the idea of L moments, Al-Nasser [17] proposed a generalized sampling scheme, named L RSS (LRSS), for estimating the population mean. The LRSS scheme encompasses existing RSS schemes, such as the RSS, QRSS, ERSS, and MRSS. Haq et al. [18] further extended the work of Al-Nasser [17] and generalized the LRSS scheme for e ciently estimating the population mean, named the Varied LRSS (VLRSS) scheme. For a symmetric

population, the VLRSS scheme{with both perfect and imperfect rankings{not only is a cost-e ective alternative to the exiting ranking schemes, but it also encompasses them, i.e., the mean estimator with the VLRSS scheme is better than the mean estimator based on SRS, RSS, ERSS, MRSS, and QRSS. For more details, we refer to Haq et al. [18].

As the mean estimators with the RSS schemes are more precise than the mean estimator based on SRS scheme, this fact has led many researchers to construct more sensitive quality control charts. Salazar and Sinha [19] were the rst to propose a Shewhart chart using RSS for monitoring the process mean. Their work, later on, extended by Muttlak and AlSabah [20], who suggested several Shewhart-type mean charts using the RSS, ERSS, and MRSS schemes under both perfect and imperfect rankings. Abujiya and Muttlak [21] and Al-Omari and Haq [22] used the double RSS schemes to construct the Shewhart charts for monitoring the process mean. Al-Sabah [23] suggested new CUSUM mean charts using RSS and MRSS. He showed that the CUSUM charts with RSS and MRSS schemes were more sensitive than that using SRS. Recently, Abujiya et al. [24,25] proposed SEWMA and SCUSUM mean charts using RSS and MRSS schemes. In another work, Awais and Haq [26,27] have suggested improved EWMA and CUSUM charts for monitoring the process mean, respectively. For more related works on the RSS-based control charts, we refer to Haq [28], Mehmood et al. [29,30], Haq et al. [31-36], Abbasi and Riaz [37], Abid et al. [38,39], Munir and Haq [40], and the references cited therein.

Since the VLRSS mean estimator is more precise than the mean estimators based on SRS, RSS, and MRSS schemes when sampling from a symmetric population, we believe that the control charts with the VLRSS would be more sensitive than those based on SRS, RSS, and MRSS schemes. This paper proposes new SEWMA and SCUSUM mean charts using VLRSS, named SEWMA-VLRSS and SCUSUMVLRSS charts, respectively. Monte Carlo simulations are used to compute the run length characteristics of the proposed control charts, including the Average Run Length (ARL) and the Standard Deviation of the Run Length (SDRL). The proposed charts are compared with their counterparts based on SRS, RSS, and MRSS schemes. It turns out that the proposed charts are more sensitive than the existing charts.

The rest of the paper is structured in the following order: Sections 2 and 3 brie y review SEWMA and SCUSUM charts with SRS, respectively. In Section 4, the VLRSS scheme is discussed with both perfect and imperfect rankings. The proposed charts are presented in Section 5. A comparative study is conducted in Section 6. An illustrative example is presented in Section 7, and Section 8 summarizes the main ndings.

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2. The SEWMA chart

The classical SEWMA chart is a mixture of two control

charts. The Shewhart and EWMA charts provide

protection against the large and small-to-moderate

shifts in the process mean, respectively [cf., 5].

Let Y denote the study variable and let fYtg,

for t = 1; 2; , be a sequence of Independent and

Identically Distributed (IID) random variables. Here,

it is assumed that Yt is a normally distributed random

variable with the in-control mean Y and the in-control

variance Y2 , i.e., Yt N(Y ; Y2 ) for t 1. Let

fY SRS;tg be a sequence of IPIDn random variables with

SRS, where Y SRS;t = (1=n) i=1 Yi;t. Here, Yi;t is the

ith observation in the tth simple random sample of size

n, for i = 1; 2; ; n. Note that Y SRS;t is also a normal

random variable with the mean Y and the variance

Y2 =n, i.e., Let

Y=SRSp;tnjNY (Y;Y;1Y2j==nY).

be

the

amount

of

standardized shift to be detected in the in-control

process mean Y , where Y;1 is the out-of-control

process mean.

By using Y SRS;t, an EWMA statistic, say Zt, is

given by:

Zt = Y SRS;t + (1 )Zt 1;

(1)

where Zt and Zt 1 are the current and past infor-

mation, respectively, and 0 < 1 is a smoothing

constant. The starting value of Zt is set equal to the incontrol process mean Y , i.e., Z0 = Y . The variance of Zt is:

Var(Zt) = Y2 1 (1 )2t :

(2)

n (2 )

Here, if the time t gets large, the term [1 (1 )2t] approaches unity. The asymptotic variance of the EWMA statistic Zt is given by:

Var(Zt) = Y2 :

(3)

n (2 )

The Upper Control Limit (UCL) and the Lower Control Limit (LCL) of the EWMA chart based on the asymptotic variance of Zt are given by:

s

UCL = Y + LpYn (2 );

and: s

LCL = Y LpYn (2 ): (4)

Similarly, the UCL and LCL of the Shewhart chart based on Y SRS;t are given by:

UCL = Y + d1 pYn ;

and:

LCL = Y d1 pYn: (5)

The Central Limits (CLs) of both the EWMA and Shewhart charts are set equal to the in-control process mean, i.e., CL = Y . Here, L and d1 are the design parameters of the SEWMA chart, respectively, and their values depend on the choices of and the desired in-control ARL. The SEWMA chart triggers an out-ofcontrol signal when either Zt falls outside the EWMA control limits or Y SRS;t falls outside the Shewhart control limits. When working with the SEWMA chart, the range of d1 should be 3:0 < d1 < 4:5 [cf., 5].

As mentioned earlier, the EWMA chart is very e ective in detecting small-to-moderate process shifts. However, there might exist a situation where the process{initially or in the startup{may make tracks in a di erent direction from the process target or after the process is recouped from an out-of-control state. In such situations, giving a head-start to the EWMA chart may help earlier detection of shifts in the process target. The FIR feature in the EWMA charting structure was rst suggested by Lucas and Saccucci [5] to overcome such situations. They suggested using two one-sided EWMA charts, each with a head-start. Their work was further extended by Rhoads et al. [41] who used two one-sided EWMA charts with head-starts and the time-varying control limits. To further reduce the time-varying control limits of the EWMA chart for the rst few samples, say ten or twenty, Steiner [42] used an exponentially decreasing adjustment factor. The new control limits of the EWMA chart with the FIRadjustment factor are as follows:

s

UCL=Y +L pYn 1 (1 f)1+a(t 1) (2 ) ; (6)

s

LCL=Y L pYn 1 (1 f)1+a(t 1) (2 ) ; (7)

where f and a are known constants. The choice of a, suggested by Steiner [42], for which the FIR adjustment has little e ect after the 20th observation is a = (1=19)(2=log(1 f) + 1). For instance, with f = 0:5, we get a = 0:3. For more details regarding the FIR feature with the EWMA chart, we refer to Rhoads et al. [41], Steiner [42], Knoth [6], and Haq et al. [10].

3. The Shewhart-CUSUM chart

SCUSUM chart was rst suggested by Lucas [3], who integrated the Shewhart chart with the CUSUM chart; it is useful to detect small and large shifts in the process target simultaneously. In the SCUSUM charting structure, the CUSUM chart quickly detects small shifts, whereas the Shewhart chart swiftly detects large shifts in the process target.

M. Awais and A. Haq/Scientia Iranica, Transactions E: Industrial Engineering 26 (2019) 3796{3818

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SCUSUM chart works similarly to the classical CUSUM chart. The CUSUM chart works with the two CUSUMs, upward and downward, say Ct+ and Ct , respectively, given by:

Ct+ = Max[0; +(Y SRS;t Y ) K + Ct+ 1]; (8) Ct = Max[0; (Y SRS;t Y ) K + Ct 1]; (9)

where K = kY =pn is the reference of slack value of

the CUSUM chart. Here, k is usually taken as half of the magnitude of the shift to be detected in Y , i.e., k = =2. The CUSUM chart declares the out-of-control

process if either Ct+ or Cpt exceeds the predetermined

decision value H = hY = n, where h is selected to get the desired in-control ARL of the CUSUM chart. In the SCUSUM chart, the above CUSUM is integrated with the Shewhart chart, as explained in the previous section. The SCUSUM chart triggers an out-of-control signal when Ct+ or Ct exceeds H or if YSRS;t exceeds the control limits given in Eq. (5) in either direction. It is customary to take d1 = 3:5 [cf., 3].

The FIR feature with the CUSUM chart was rst suggested by Lucas and Crosier [2], which enables the CUSUM chart to react quickly against the startup/initial problems. The FIR feature in the CUSUM chart works by resetting the starting values of both plotting-CUSUMs to some non-zero constants. They recommended using H=2 for a 50% head-start, i.e., by setting C0+ = C0 = H=2 . For more details, see Lucas and Crosier [2] and Haq et al. [10].

4. The VLRSS scheme

In this section, we brie y review the mean estimator using VLRSS under both perfect and imperfect rankings. The VLRSS scheme is a cost-e ective alternative to the SRS and RSS schemes. This scheme provides not only an unbiased and precise mean estimator when sampling from a symmetric population, but also plenty of options to the experimenter in selecting di erent representative samples with the less number of identi ed units compared to that using the RSS scheme, i.e., the ranking costs with VLRSS could be more or less than that with the RSS. It is worth mentioning that the VLRSS scheme provides a more e cient mean estimator than the mean estimators based on RSS and MRSS schemes when ranking costs are negligible. However, when ranking costs are high, it is still bene cial to use VLRSS scheme with the less ranking cost than that with the RSS schemes [cf., 18].

The main steps involved in selecting a varied L ranked set sample of size n are presented as follows:

Step 1: Select the value of the VLRSS coe cient,

say w = [al], where 0 a < 0:5 . Here, [ ] is the

largest possible integer value;

Step 2: Select 2wl units from the target population.

Divide these units into 2w sets, with each set consisting of l units;

Step 3: Rank the units within each set by any cheap

or inexpensive method with respect to the study variable or using ranks of an auxiliary variable;

Step 4: Select the vth and (l v + 1)th smallest

ranked units from the rst and last w sets, respectively, where v = 1; 2; ; [l=2];

Step 5: Identify m(m 2w) units from the target

population and, then, divide these units into m 2w sets, with each set comprising m units;

Step 6: Select the ith smallest ranked unit from the

(i+w)th set of m units, for i = w+1; w+2; ; m w;

Step 7: This completes one cycle of a varied L ranked

set sample of size m. Steps 1-6 could be repeated, if necessary, r number of times to get a total sample of size n = mr units.

Symbolically, let (Yi1j; Yi2j; ; Yilj), i = 1; 2; ; 2w, be 2w samples, each of size l, for the jth cycle, where j = 1; 2; ; r. Let Yi(v:l)j denote the vth order statistic of (Yi1j; Yi2j; ; Yilj) for i = 1; 2; ; w, and let Yi(l v+1:l)j be the (l v + 1)th order statistic of (Yi1j; Yi2j; ; Yilj) for i = w + 1; w + 2; ; 2w. Let (Y(i+w)1j; Y(i+w)2j; ; Y(i+w)mj), i = w+1; 2; ; m w, denote m 2w samples, each of size m, for the jth cycle. Let Yi+w(i:m)j denote the ith order statistic of (Y(i+w)1j; Y(i+w)2j; ; Y(i+w)mj) for i = 1; 2; ; m w.

The sample mean based on a varied L ranked set sample of size n, denoted by Y VLRSS, and its variance, respectively, are given by:

Y VLRSS = 1 X r

X w

Yi(v:l)j +

X 2w

Yi(l

v+1:l)j

n j=1 i=1

i=w+1

mXw

!

+

Yi+w(i:m)j ;

(10)

i=w+1

Var(Y VLRSS) = n1m w(Y2 (v:l) + Y2 (l v+1:l))

mXw

!

+

Y2 (i:m) ;

(11)

i=w+1

where: Y2 (v:l) = Var(Yi(v:l)j);

Y2 (l v+1:l) = Var(Yi(l v+1:l)j); and

Y2 (i:m) = Var(Yi(i:m)j): For more details regarding the computation of the

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M. Awais and A. Haq/Scientia Iranica, Transactions E: Industrial Engineering 26 (2019) 3796{3818

variances of order statistics, we refer to David and Nagaraja [43].

For a symmetric population, Haq et al. [18] showed that Y VLRSS is an unbiased estimator of Y . They have also shown that, with some suitable choices of v, l, and w, the existing RSS schemes are special cases of VLRSS. For instance, for w = 0, VLRSS becomes RSS; for w = [(m 1)=2], l = m, and v = w+1, VLRSS becomes MRSS, etc. While selecting a varied L ranked set sample of size n, the experimenter needs to identify nm 2w(m l)r units, while the classical RSS and MRSS require identifying nm units when selecting a sample of size n. It is to be noted that when m > l, VLRSS requires fewer identi ed units than that when using the RSS or MRSS [cf., 18].

4.1. The imperfect VLRSS scheme

There may exist a situation where it is not possible to rank the study variable visually, or it is costly and time consuming. This issue can be solved by ranking the study variable (Y ) using the ranks of a highly correlated variable, say X, given that it is readily available. Stokes [13] suggested a simple model for the imperfect rankings, given by

Yi[i:u]j = Y + XY Xi(i:u)j X + ij;

i = 1; 2; ; u; j = 1; 2; ; r;

(12)

where u = l; m; X and X are the population mean and standard deviation of X, respectively, and is the

correlation between Y and X. Here, ij N(0; Y2 (1

2)), and Xi(i:u)j and ij are mutually independent. Yi[i:u]j is the ith concomitant or induced order statistic corresponding to the ith order statistic Xi(i:u)j, i = 1; 2; ; u. The values of X are perfectly ranked; however, those of Y are ranked with error. On the lines of Stokes [13], the sample mean under Imperfect VLRSS (IVLRSS), say Y IVLRSS, and its variance, are, respectively, given by:

Y IVLRSS = 1 X r

X w

Yi[v:l]j +

X 2w

Yi[l

v+1:l]j

n j=1 i=1

i=w+1

mXw

!

+

Yi+w[i:m]j ;

(13)

i=w+1

and:

Var(Y VLRSS) = n1m w(Y2 [v:l] + Y2 [l v+1:l])

mXw

!

+

Y2 [i:m]

(14)

i=w+1

(

= n1m mY2 (1 2)+2 XY22

mXw

!)

+

X2 (i:m) ;

i=w+1

2wX2 (v:l) (15)

where:

Y2 [v:l] = Var(Yi[v:l]j);

Y2 [l v+1:l] = Var(Yi[l v+1:l]j); and

Y2 [i:m] = Var(Yi[i:m]j):

For more details on the computation of these variances, we refer to see David and Nagaraja [43]. When sampling from a symmetric bivariate population, Y IVLRSS is unbiased, is characterized by reasonable assumptions, and is more precise than the mean estimators based on imperfect RSS and MRSS schemes [cf., 18].

5. The proposed control charts

In this section, new SEWMA and SCUSUM control charts are proposed for e ciently monitoring the process mean Y under both perfect and imperfect VLRSS schemes. The run length characteristics of these control charts are also computed through Monte Carlo simulations.

5.1. The SEWMA chart

Suppose that a sample of size n is selected from the

target population with S scheme at the time t( 1), where S = VLRSS and IVLRSS. Let fY S;tg be a

sequence of IID random variables for t = 1; 2; .

By considering fY S;tg, it is possible to construct an

SEWMA chart for monitoring Y . The plottingstatistic of the SEWMA chart with S scheme is given by:

Qt = Y S;t + (1 )Qt 1;

(16)

where is a smoothing constant. The asymptotic variance of Qt is:

Var(Qt) = Var(Y S;t) (2 ); (17)

where Var(Y S;t) denotes the variance of fY S;tg at the

time t. The control limits of the SEWMA chart with S scheme are:

q

s

UCL = Y + L Var(Y S;t) (2 );

and:

M. Awais and A. Haq/Scientia Iranica, Transactions E: Industrial Engineering 26 (2019) 3796{3818

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q

s

LCL = Y L Var(Y S;t) (2 ) :

(18)

Similarly, the control limits of the Shewhart chart,

based on fY S;tg, are given by:

q

UCL = Y + d1 Var(Y S;t);

and:

q

LCL = Y d1 Var(Y S;t);

(19)

where L and d1 are positive control charting multipliers that are selected to ensure that the in-control ARL of the SEWMA chart has reached a certain level. The SEWMA chart triggers an out-of-control signal

whenever Qt or fY S;tg falls outside their respective

control limits' intervals, i.e., [LCL, UCL]. As mentioned earlier, the sensitivity of the SEWMA chart can be enhanced by giving a head-start to the SEWMA chart with the FIR-adjustment [cf., 42]; on the similar lines, an FIR feature is associated with the proposed SEWMA chart by setting f = 0:5 and a = 0:3, as recommended by Steiner [42].

5.2. The SCUSUM chart

To construct an SCUSUM chart for monitoring Y ,

consider the sequence fY S;tg for t = 1; 2; . The

plotting-statistics (upper and lower CUSUMs) of the proposed SCUSUM chart using S scheme are, respectively, de ned by:

Ct+ = Max[0; +(Y S;t Y ) K + Ct+ 1];

(20)

Ct = Max[0; (Y S;t Y ) K + Ct 1];

(21)

where C0+ = C0 = 0. The reference value, K, and the decision interval, H, of the SCUSUM chart are respectively as follows:

q

K = k Var(Y S;t);

(22)

q

H = h Var(Y S;t);

(23)

where the values of k and h are the same as explained in the previous section. Similarly, the control limits of

the Shewhart chart based on fY S;tg are given by:

q

UCL = Y + d1 Var(Y S;t);

and:

q

LCL = Y d1 Var(Y S;t):

(24)

The SCUSUM triggers an out-of-control signal if Ct+ or Ct exceeds H or if Y S;t is less than LCL or greater

than UCL of the Shewhart chart. The sensitivity of the CUSUM chart for the start-up problems, as suggested by Lucas and Crosier [2], could increase with a headstart feature. They recommended resetting the starting values of Ct+ and Ct to non-zero constants, like C0+ = C0 = H=2 for an 50% head-start [cf., 2]. On the same lines, an FIR feature is attached to the SCUSUM chart with 50% head-start.

5.3. Run length evaluation

Generally, the run length performance of a control chart is evaluated in terms of its run length characteristics including the ARL and the SDRL. For an in-control process, the in-control ARL should be large enough to avoid false alarms, while, for an out-of-control process, it should be as small as possible so that the control chart can swiftly trigger an out-of-control signal. In the literature, there exist some methods that could be used to compute the run length characteristics of a control chart, including the integral equations, Markov chain, and the Monte Carlo simulations. The Monte Carlo simulation method is broadly used to compute the run length characteristics of the control charts, and thus it is used here.

In order to evaluate the run length performances of the proposed control charts, we generate samples under VLRSS from a normal distribution. The incontrol ARL is set equal to 500 a choice recommended by the SPC practitioners. Here, each simulation run comprises 50,000 iterations of the run length. In Tables 1 and 2, the values of (; L) and (k; h) are reported for the SEWMA and SCUSUM charts, respectively, with di erent possible values of (m; l; v) with r = 1 when the in-control ARL is matched as 500. These constants could be used when using the proposed charts with di erent choices of (m; l; v) when the in-control ARL is xed to 500.

For brevity of discussion, without loss of generality, with n = 5 and r = 1, using di erent pairs of (l; v) with w = 2, we compute the ARLs and SDRLs of the proposed control charts in Tables 3-6 (with and without FIR features). It is to be noted that, for a given sample size n, we consider those choices of w and (l; v) with the VLRSS scheme for which the mean estimator is precise [cf., 18]. Di erent values of are considered, i.e., = 0(0:25)4. For both the SEWMA and SCUSUM charts, di erent values of and k have been considered. The values of d1 for the SEWMA and SCUSUM charts are set equal to 3.31 and 3.50, respectively. Moreover, the results are computed when sampling from a normal distribution. Here, under each simulation run, 50,000 replications of the run length are considered. It is observed that the out-of-control ARLs tend to decrease as the value of increases, and vice versa. A similar trend is observed when a control chart is constructed with the FIR feature.

3802

M. Awais and A. Haq/Scientia Iranica, Transactions E: Industrial Engineering 26 (2019) 3796{3818

Table 1. The values of L with di erent choices of (l; v) when the in-control ARL of the SEWMA-VLRSS chart is 500.

m=2

(0; 0)

m=4 (2; 1) (3; 2) (4; 1) (4; 2) (5; 3) (6; 3)

0.05 2.2582

2.8510 2.8523 2.8800 2.8490 2.8492 2.8485

0.10 2.4260

3.0270 3.0164 3.0463 3.0257 3.0225 3.0230

0.25 2.5789

3.1700 3.1666 3.1856 3.1650 3.1659 3.1648

0.50 2.6350

3.1755 3.1700 3.1889 3.1747 3.1695 3.1755

0.75 2.6470

3.1230 3.1186 3.1390 3.1190 3.1155 3.1161

1.00 2.6465

3.1050 3.1000 3.1158 3.0990 3.1000 3.1000

m = 3; w = 1 (2; 1) (3; 1) (3; 2) (4; 1) (4; 2) (5; 1) (5; 2) (5; 3)

0.05 2.8600 2.8800 2.8519 2.8890 2.8510 2.8990 2.8585 2.8522

0.10 3.0307 3.0469 3.0285 3.0599 3.0285 3.0690 3.0308 3.0290

0.25 3.1730 3.1860 3.1710 3.2004 3.1710 3.2100 3.1683 3.1660

0.50 3.1763 3.1921 3.1758 3.2056 3.1759 3.2105 3.1800 3.1759

0.75 3.1240 3.1377 3.1200 3.1499 3.1207 3.1545 3.1230 3.1203

1.00 3.1070 3.1157 3.1025 3.1260 3.1028 3.1313 3.1070 3.1013

m=5

w=1

w=2

(2; 1) (3; 2) (4; 2) (5; 1)

(5; 3) (6; 3) (7; 4) (8; 4)

0.05 2.8494 2.8490 2.8480 2.8740

2.8450 2.8460 2.8444 2.8448

0.10 3.0270 3.0190 3.0250 3.0450

3.0170 3.0179 3.0164 3.0164

0.25 3.1695 3.1654 3.1645 3.1850

3.1570 3.1579 3.1584 3.1590

0.50 3.1747 3.1690 3.1755 3.1920

3.1660 3.1669 3.1670 3.1673

0.75 3.1195 3.1150 3.1150 3.1370

3.1150 3.1150 3.1134 3.1155

1.00 3.1000 3.1000 3.1000 3.1147

3.0980 3.0968 3.0945 3.0970

Table 2. The values of h with di erent choices of (l; v) when the in-control ARL of the SCUSUM-VLRSS chart is 500.

m=2

k

(0; 0)

m = 4; w = 1 (2; 1) (3; 2) (4; 1) (4; 2) (5; 3) (6; 3)

0.25 9.1100

9.0950 9.0740 9.1100 9.0970 9.0760 9.0875

0.50 5.3280

5.3166 5.3105 5.3360 5.3050 5.3050 5.3077

0.75 3.6980

3.6785 3.6765 3.6950 3.6753 3.6758 3.6760

1.00 2.7450

2.7282 2.7255 2.7440 2.7245 2.7240 2.7252

m = 3; w = 1 k (2; 1) (3; 1) (3; 2) (4; 1) (4; 2) (5; 1) (5; 2) (5; 3)

0.25 9.0953 9.1109 9.0700 9.1230 9.0950 9.1560 9.0950 9.0747

0.50 5.3164 5.3340 5.3133 5.3430 5.3085 5.3780 5.3143 5.3060

0.75 3.6791 3.6980 3.6767 3.7122 3.6790 3.7181 3.6800 3.6767

1.00 2.7320 2.7450 2.7280 2.7547 2.7284 2.7670 2.7290 2.7287

m=5

w=1

w=2

k (2; 1) (3; 2) (4; 2) (5; 1)

(5; 3) (6; 3) (7; 4) (8; 4)

0.25 9.0959 9.0739 9.0755 9.1110

9.0490 9.0499 9.0508 9.0510

0.50 5.3109 5.3080 5.3080 5.3270

5.2930 5.2935 5.2938 5.2938

0.75 3.6760 3.6755 3.6746 3.6960

3.6670 3.6682 3.6666 3.6682

1.00 2.7250 2.7245 2.7260 2.7460

2.7170 2.7193 2.7185 2.7186

5.4. When the process parameters are unknown

If the underlying process parameters are not known in advance{phase-I monitoring, then it is customary to estimate them using a large historical dataset, provided

that it has been obtained from an in-control process. Suppose that, from an in-control process, q subgroups and each of size m are available with S scheme.

In the perfect ranking case, Y and Var(Y VLRSS) could be estimated by using their respective unbiased

M. Awais and A. Haq/Scientia Iranica, Transactions E: Industrial Engineering 26 (2019) 3796{3818

3803

Table 3. The run length pro les of SEWMA-VLRSS chart when the in-control ARL is 500.

= 0:05

(l; v) (5; 3) (6; 3) (7; 4) (8; 4)

L 2.8450 2.8460 2.8444 2.8448

0.00

ARL SDRL

500.25 494.42

499.43 490.67

500.87 491.50

499.53 490.78

0.25

ARL SDRL

36.42 22.74

32.97 19.85

30.09 17.63

28.20 15.99

0.50

ARL SDRL

13.33 5.62

12.29 5.08

11.37 4.58

10.77 4.28

0.75

ARL SDRL

7.89 3.01

7.29 2.80

6.72 2.62

6.35 2.52

1.00

ARL SDRL

5.35 2.24

4.93 2.13

4.48 2.03

4.19 1.96

1.25

ARL SDRL

3.81 1.86

3.41 1.76

3.04 1.64

2.81 1.56

1.50

ARL SDRL

2.69 1.52

2.39 1.38

2.10 1.23

1.92 1.13

1.75

ARL SDRL

1.95 1.15

1.72 1.00

1.53 0.84

1.41 0.73

2.00

ARL SDRL

1.49 0.80

1.34 0.65

1.23 0.52

1.16 0.44

2.25

ARL SDRL

1.23 0.52

1.14 0.40

1.08 0.30

1.06 0.24

2.50

ARL SDRL

1.10 0.33

1.05 0.24

1.03 0.16

1.01 0.12

2.75

ARL SDRL

1.03 0.19

1.02 0.13

1.01 0.08

1.00 0.05

3.00

ARL SDRL

1.01 0.10

1.00 0.07

1.00 0.04

1.00 0.02

3.25

ARL SDRL

1.00 0.05

1.00 0.03

1.00 0.01

1.00 0.00

3.50

ARL SDRL

1.00 0.02

1.00 0.01

1.00 0.00

1.00 0.00

3.75

ARL SDRL

1.00 0.01

1.00 0.00

1.00 0.00

1.00 0.00

4.00

ARL SDRL

1.00 0.00

1.00 0.00

1.00 0.00

1.00 0.00

= 0:25

(l; v) (5; 3) (6; 3) (7; 4) (8; 4)

L 3.1570 3.1579 3.1584 3.1590

0.00

ARL SDRL

500.20 500.00

500.80 497.87

502.48 501.34

500.74 498.89

0.25

ARL SDRL

69.36 64.36

61.38 57.35

53.18 48.42

48.84 44.58

0.50

ARL SDRL

14.56 10.63

12.78 9.03

11.35 7.69

10.30 6.74

0.75

ARL SDRL

6.52 3.59

5.88 3.12

5.30 2.74

4.94 2.47

1.00

ARL SDRL

4.04 1.89

3.68 1.69

3.37 1.53

3.16 1.43

1.25

ARL SDRL

2.88 1.29

2.63 1.19

2.40 1.09

2.25 1.03

1.50

ARL SDRL

2.18 1.00

1.99 0.92

1.80 0.84

1.69 0.79

1.75

ARL SDRL

1.71 0.79

1.55 0.71

1.43 0.63

1.34 0.56

2.00

ARL SDRL

1.39 0.60

1.29 0.51

1.20 0.43

1.15 0.37

2.25

ARL SDRL

1.20 0.43

1.13 0.35

1.08 0.28

1.05 0.23

2.50

ARL SDRL

1.09 0.29

1.05 0.22

1.03 0.16

1.01 0.12

2.75

ARL SDRL

1.03 0.18

1.02 0.12

1.01 0.08

1.00 0.05

3.00

ARL SDRL

1.01 0.11

1.00 0.06

1.00 0.04

1.00 0.02

3.25

ARL SDRL

1.00 0.06

1.00 0.03

1.00 0.01

1.00 0.01

3.50

ARL SDRL

1.00 0.03

1.00 0.01

1.00 0.01

1.00 0.01

3.75

ARL SDRL

1.00 0.01

1.00 0.00

1.00 0.00

1.00 0.00

4.00

ARL SDRL

1.00 0.00

1.00 0.00

1.00 0.00

1.00 0.00

(5; 3) 3.0170 502.58 494.28 42.10 32.78 12.37

6.33 6.89 2.89 4.64 1.90 3.34 1.52 2.47 1.25 1.87 0.99 1.46 0.72 1.22 0.50 1.10 0.32 1.04 0.20 1.01 0.11 1.00 0.06 1.00 0.02 1.00 0.01 1.00 0.00 (5; 3) 3.1660 501.10 508.07 118.70 117.02 23.68 21.49 8.30 6.38 4.30 2.70 2.82 1.49 2.06 0.98 1.64 0.71 1.37 0.54 1.19 0.41 1.09 0.29 1.03 0.18 1.01 0.10 1.00 0.05 1.00 0.03 1.00 0.01 1.00 0.00

= (6; 3) 3.0179 502.48 500.68 37.61 28.23 11.34 5.57 6.34 2.61 4.27 1.79 3.04 1.43 2.21 1.15 1.66 0.87 1.33 0.61 1.14 0.39 1.05 0.24 1.02 0.13 1.00 0.07 1.00 0.03 1.00 0.01 1.00 0.00 1.00 0.00

= (6; 3) 3.1669 500.45 504.69 106.57 103.89 20.41 18.15 7.19 5.31 3.79 2.26 2.52 1.29 1.88 0.87 1.50 0.63 1.27 0.47 1.13 0.34 1.05 0.22 1.02 0.13 1.00 0.07 1.00 0.03 1.00 0.01 1.00 0.00 1.00 0.00

0:10 (7; 4) 3.0164 500.45 500.09 33.53 24.51 10.34 4.93 5.84 2.40 3.90 1.68 2.77 1.34 1.99 1.05 1.50 0.76 1.22 0.49 1.08 0.30 1.02 0.16 1.01 0.08 1.00 0.03 1.00 0.01 1.00 0.00 1.00 0.00 1.00 0.00

0:50 (7; 4) 3.1670 501.94 500.30 93.83 91.67 17.45 15.24 6.24 4.39 3.40 1.95 2.28 1.12 1.72 0.76 1.39 0.56 1.19 0.41 1.08 0.27 1.03 0.16 1.01 0.08 1.00 0.04 1.00 0.01 1.00 0.00 1.00 0.00 1.00 0.00

(8; 4) 3.0164 502.76 498.02 30.84 22.16

9.73 4.53 5.53 2.26 3.67 1.61 2.56 1.28 1.83 0.97 1.40 0.67 1.16 0.42 1.05 0.24 1.01 0.12 1.00 0.06 1.00 0.03 1.00 0.00 1.00 0.00 1.00 0.00 1.00 0.00 (8; 4) 3.1673 500.08 502.16 85.87 83.88 15.45 13.37 5.67 3.89 3.13 1.73 2.14 1.02 1.62 0.70 1.31 0.51 1.14 0.35 1.05 0.22 1.01 0.12 1.00 0.05 1.00 0.02 1.00 0.01 1.00 0.00 1.00 0.00 1.00 0.00

3804

M. Awais and A. Haq/Scientia Iranica, Transactions E: Industrial Engineering 26 (2019) 3796{3818

Table 4. The run length pro les of SCUSUM-VLRSS chart when the in-control ARL is 500.

k = 0:25

(l; v) (5; 3) (6; 3) (7; 4) (8; 4)

h 9.0490 9.0499 9.0508 9.0510

0.00

ARL SDRL

500.80 486.65

500.62 490.59

500.72 492.92

500.26 487.82

0.25

ARL SDRL

36.00 21.62

32.87 18.97

30.02 16.75

28.32 15.31

0.50

ARL SDRL

13.64 5.36

12.66 4.82

11.76 4.36

11.17 4.09

0.75

ARL SDRL

8.25 2.88

7.65 2.66

7.11 2.49

6.73 2.38

1.00

ARL SDRL

5.71 2.11

5.27 2.02

4.85 1.94

4.54 1.88

1.25

ARL SDRL

4.15 1.81

3.76 1.73

3.38 1.65

3.12 1.58

1.50

ARL SDRL

3.02 1.56

2.69 1.45

2.37 1.33

2.16 1.24

1.75

ARL SDRL

2.18 1.25

1.92 1.11

1.69 0.96

1.55 0.85

2.00

ARL SDRL

1.65 0.92

1.46 0.77

1.31 0.62

1.23 0.51

2.25

ARL SDRL

1.31 0.62

1.20 0.49

1.12 0.36

1.08 0.30

2.50

ARL SDRL

1.14 0.39

1.08 0.29

1.04 0.20

1.02 0.16

2.75

ARL SDRL

1.05 0.24

1.03 0.16

1.01 0.11

1.01 0.07

3.00

ARL SDRL

1.02 0.13

1.01 0.09

1.00 0.04

1.00 0.04

3.25

ARL SDRL

1.01 0.07

1.00 0.04

1.00 0.02

1.00 0.01

3.50

ARL SDRL

1.00 0.03

1.00 0.02

1.00 0.01

1.00 0.00

3.75

ARL SDRL

1.00 0.02

1.00 0.01

1.00 0.00

1.00 0.00

4.00

ARL SDRL

1.00 0.00

1.00 0.00

1.00 0.00

1.00 0.00

k = 0:75

(l; v) (5; 3) (6; 3) (7; 4) (8; 4)

h 3.6670 3.6682 3.6666 3.6682

0.00

ARL SDRL

502.85 499.40

502.68 497.36

502.63 501.27

500.05 495.78

0.25

ARL SDRL

70.37 66.21

61.78 57.04

53.89 49.40

48.85 44.55

0.50

ARL SDRL

13.86 10.04

12.16 8.45

10.70 7.10

9.81 6.29

0.00

ARL SDRL

502.85 499.40

502.68 497.36

502.63 501.27

500.05 495.78

0.25

ARL SDRL

70.37 66.21

61.78 57.04

53.89 49.40

48.85 44.55

0.50

ARL SDRL

13.86 10.04

12.16 8.45

10.70 7.10

9.81 6.29

0.75

ARL SDRL

6.23 3.32

5.60 2.85

5.11 2.51

4.73 2.25

1.00

ARL SDRL

3.92 1.75

3.59 1.55

3.28 1.38

3.09 1.30

1.25

ARL SDRL

2.83 1.17

2.61 1.07

2.40 0.99

2.25 0.93

1.50

ARL SDRL

2.19 0.90

2.01 0.84

1.84 0.77

1.74 0.73

1.75

ARL SDRL

1.76 0.73

1.61 0.67

1.48 0.60

1.40 0.56

2.00

ARL SDRL

1.45 0.58

1.34 0.52

1.25 0.46

1.19 0.40

2.25

ARL SDRL

1.25 0.45

1.17 0.39

1.11 0.32

1.07 0.26

2.50

ARL SDRL

1.12 0.33

1.07 0.26

1.04 0.20

1.02 0.15

2.75

ARL SDRL

1.05 0.22

1.02 0.15

1.01 0.10

1.01 0.08

3.00

ARL SDRL

1.02 0.13

1.01 0.08

1.00 0.05

1.00 0.03

3.25

ARL SDRL

1.00 0.07

1.00 0.04

1.00 0.02

1.00 0.01

3.50

ARL SDRL

1.00 0.04

1.00 0.02

1.00 0.00

1.00 0.00

3.75

ARL SDRL

1.00 0.02

1.00 0.00

1.00 0.00

1.00 0.00

4.00

ARL SDRL

1.00 0.00

1.00 0.00

1.00 0.00

1.00 0.00

(5; 3) 5.2930 502.65 497.32 47.16 39.74 12.17

6.73 6.48 2.76 4.37 1.71 3.23 1.28 2.48 1.08 1.94 0.91 1.54 0.73 1.28 0.54 1.13 0.36 1.05 0.23 1.02 0.13 1.01 0.07 1.00 0.04 1.00 0.01 1.00 0.00 (5; 3) 2.7170 500.36 497.91 96.77 94.02 17.85 15.22 500.36 497.91 96.77 94.02 17.85 15.22 6.69 4.378 3.84 1.99 2.68 1.18 2.08 0.84 1.69 0.66 1.44 0.55 1.24 0.44 1.12 0.33 1.05 0.22 1.02 0.13 1.01 0.08 1.00 0.04 1.00 0.01 1.00 0.01

k= (6; 3) 5.2935 500.08 493.26 41.55 34.25 11.00 5.83 5.95 2.44 4.05 1.56 2.97 1.21 2.26 1.01 1.75 0.84 1.40 0.64 1.19 0.43 1.08 0.28 1.03 0.16 1.01 0.09 1.00 0.04 1.00 0.02 1.00 0.01 1.00 0.01

k= (6; 3) 2.7193 499.33 496.82 86.37 84.06 15.30 12.52 499.33 496.82 86.37 84.06 15.30 12.52 5.91 3.68 3.48 1.71 2.46 1.05 1.92 0.76 1.57 0.61 1.33 0.49 1.17 0.38 1.07 0.26 1.03 0.16 1.01 0.09 1.00 0.04 1.00 0.01 1.00 0.00 1.00 0.00

0:50 (7; 4) 5.2938 501.11 494.90 36.91 29.62 9.96 5.06 5.50 2.22 3.73 1.44 2.72 1.14 2.05 0.95 1.58 0.75 1.28 0.53 1.12 0.34 1.04 0.20 1.01 0.11 1.00 0.05 1.00 0.02 1.00 0.01 1.00 0.00 1.00 0.00

1:00 (7; 4) 2.7185 501.55 500.44 75.50 72.87 13.16 10.48 501.55 500.44 75.50 72.87 13.16 10.48 5.26 3.13 3.16 1.49 2.27 0.95 1.77 0.69 1.46 0.56 1.24 0.44 1.11 0.31 1.04 0.20 1.01 0.10 1.00 0.04 1.00 0.02 1.00 0.01 1.00 0.00 1.00 0.00

(8; 4) 5.2938 499.16 491.04 33.68 26.48

9.32 4.60 5.17 2.06 3.52 1.37 2.55 1.09 1.91 0.90 1.48 0.69 1.21 0.46 1.08 0.28 1.02 0.15 1.01 0.08 1.00 0.04 1.00 0.01 1.00 0.00 1.00 0.00 1.00 0.00 (8; 4) 2.7186 500.13 498.09 68.85 66.29 11.72 9.08 500.13 498.09 68.85 66.29 11.72 9.08 4.81 2.76 2.96 1.36 2.13 0.87 1.67 0.65 1.38 0.52 1.19 0.40 1.07 0.26 1.02 0.15 1.01 0.07 1.00 0.03 1.00 0.01 1.00 0.00 1.00 0.00 1.00 0.00

M. Awais and A. Haq/Scientia Iranica, Transactions E: Industrial Engineering 26 (2019) 3796{3818

3805

Table 5. The run length pro les of SEWMA-VLRSS chart with the FIR feature for the time when the in-control ARL is

500.

= 0:05

= 0:10

(l; v) (5; 3) (6; 3) (7; 4) (8; 4)

(5; 3) (6; 3) (7; 4) (8; 4)

L 2.8479 2.8479 2.8475 2.8472

3.0297 3.0294 3.0294 3.0283

0.00

ARL SDRL

499.65 493.15

501.08 496.13

502.93 494.98

500.20 492.83

501.52 503.53

501.09 510.07

502.12 504.01

500.81 507.82

0.25

ARL SDRL

35.57 23.10

32.18 20.41

29.27 17.88

27.25 16.22

39.47

34.99

30.98

28.24

33.06

28.62

24.90

22.39

0.50

ARL SDRL

12.07 5.93

11.06 5.39

10.06 4.87

9.46 4.51

10.28

9.22

8.29

7.70

6.59

5.79

5.08

4.64

0.75

ARL SDRL

6.53 3.02

5.96 2.72

5.46 2.46

5.10 2.32

5.12

4.64

4.21

3.96

2.80

2.45

2.19

2.00

1.00

ARL SDRL

4.20 1.89

3.83 1.72

3.50 1.57

3.28 1.48

3.27

3.01

2.76

2.60

1.54

1.37

1.22

1.13

1.25

ARL SDRL

2.98 1.35

2.71 1.24

2.46 1.14

2.30 1.08

2.40

2.22

2.06

1.95

1.02

0.92

0.83

0.78

1.50

ARL SDRL

2.23 1.05

2.01 0.96

1.83 0.87

1.70 0.80

1.90

1.76

1.64

1.56

0.76

0.69

0.63

0.60

1.75

ARL SDRL

1.73 0.82

1.57 0.72

1.43 0.64

1.35 0.57

1.58

1.47

1.37

1.30

0.61

0.56

0.52

0.48

2.00

ARL SDRL

1.40 0.62

1.29 0.53

1.20 0.44

1.15 0.38

1.35

1.26

1.19

1.14

0.50

0.45

0.39

0.35

2.25

ARL SDRL

1.20 0.44

1.13 0.36

1.08 0.28

1.05 0.23

1.19

1.13

1.08

1.05

0.40

0.33

0.27

0.22

2.50

ARL SDRL

1.09 0.30

1.05 0.22

1.03 0.16

1.01 0.12

1.09

1.05

1.03

1.01

0.28

0.22

0.16

0.12

2.75

ARL SDRL

1.04 0.19

1.02 0.12

1.01 0.08

1.00 0.06

1.03

1.02

1.01

1.00

0.18

0.13

0.08

0.06

3.00

ARL SDRL

1.01 0.10

1.00 0.07

1.00 0.04

1.00 0.02

1.01

1.00

1.00

1.00

0.11

0.06

0.04

0.02

3.25

ARL SDRL

1.00 0.06

1.00 0.03

1.00 0.01

1.00 0.01

1.00

1.00

1.00

1.00

0.05

0.03

0.02

0.01

3.50

ARL SDRL

1.00 0.03

1.00 0.01

1.00 0.01

1.00 0.01

1.00

1.00

1.00

1.00

0.02

0.01

0.01

0.00

3.75

ARL SDRL

1.00 0.01

1.00 0.01

1.00 0.00

1.00 0.00

1.00

1.00

1.00

1.00

0.01

0.01

0.00

0.00

4.00

ARL SDRL

1.00 0.00

1.00 0.00

1.00 0.00

1.00 0.00

1.00

1.00

1.00

1.00

0.00

0.00

0.00

0.00

= 0:25

= 0:50

(l; v) (5; 3) (6; 3) (7; 4) (8; 4)

(5; 3) (6; 3) (7; 4) (8; 4)

L 3.2014 3.1999 3.1990 3.1995

3.2490 3.2490 3.2486 3.2480

0.00

ARL SDRL

500.07 535.91

500.87 537.36

501.93 535.19

502.28 534.75

499.56 579.34

499.95 580.42

500.56 584.53

501.07 583.79

0.25

ARL SDRL

61.29 69.10

52.22 59.21

45.27 51.49

40.63 45.93

103.34 89.54

76.76

69.54

134.54 117.61 102.91 94.93

0.00

ARL SDRL

500.07 535.91

500.87 537.36

501.93 535.19

502.28 534.75

499.56 579.34

499.95 580.42

500.56 584.53

501.07 583.79

0.25

ARL SDRL

61.29 69.10

52.22 59.21

45.27 51.49

40.63 45.93

103.34 89.54

76.76

69.54

134.54 117.61 102.91 94.93

0.50

ARL SDRL

9.50 9.74

8.22 8.15

7.08 6.77

6.38 5.94

13.13

10.53

8.48

7.42

20.25

15.89

12.67

10.89

0.75

ARL SDRL

3.83 2.91

3.42 2.47

3.06 2.11

2.84 1.87

3.50

3.02

2.66

2.41

3.97

3.19

2.52

2.12

1.00

ARL SDRL

2.34 1.38

2.14 1.18

1.98 1.06

1.86 0.95

1.93

1.77

1.63

1.54

1.40

1.17

0.99

0.88

1.25

ARL SDRL

1.73 0.85

1.60 0.75

1.50 0.67

1.42 0.61

1.43

1.34

1.27

1.22

0.74

0.64

0.55

0.49

1.50

ARL SDRL

1.40 0.59

1.32 0.52

1.24 0.46

1.20 0.42

1.20

1.15

1.10

1.08

0.46

0.39

0.33

0.28

1.75

ARL SDRL

1.21 0.43

1.15 0.37

1.11 0.31

1.08 0.27

1.09

1.06

1.04

1.02

0.30

0.24

0.19

0.15

2.00

ARL SDRL

1.10 0.30

1.06 0.24

1.04 0.19

1.02 0.15

1.03

1.02

1.01

1.01

0.18

0.14

0.10

0.08

2.25

ARL SDRL

1.04 0.19

1.02 0.14

1.01 0.10

1.01 0.08

1.01

1.01

1.00

1.00

0.10

0.07

0.05

0.03

2.50

ARL SDRL

1.01 0.11

1.01 0.08

1.00 0.05

1.00 0.03

1.00

1.00

1.00

1.00

0.05

0.04

0.02

0.01

2.75

ARL SDRL

1.00 0.06

1.00 0.03

1.00 0.02

1.00 0.02

1.00

1.00

1.00

1.00

0.02

0.01

0.01

0.01

3.00

ARL SDRL

1.00 0.03

1.00 0.01

1.00 0.00

1.00 0.00

1.00

1.00

1.00

1.00

0.01

0.00

0.00

0.00

3.25

ARL SDRL

1.00 0.01

1.00 0.01

1.00 0.00

1.00 0.00

1.00

1.00

1.00

1.00

0.00

0.00

0.00

0.00

3.50

ARL SDRL

1.00 0.00

1.00 0.00

1.00 0.00

1.00 0.00

1.00

1.00

1.00

1.00

0.00

0.00

0.00

0.00

3.75

ARL SDRL

1.00 0.00

1.00 0.00

1.00 0.00

1.00 0.00

1.00

1.00

1.00

1.00

0.00

0.00

0.00

0.00

4.00

ARL SDRL

1.00 0.00

1.00 0.00

1.00 0.00

1.00 0.00

1.00

1.00

1.00

1.00

0.00

0.00

0.00

0.00