A SYNTHETIC CONTROL CHART FOR ...

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International Journal of Reliability, Quality and Safety Engineering Vol. 15, No. 3 (2008) 217–245 c World Scientific Publishing Company

A SYNTHETIC CONTROL CHART FOR MONITORING THE PROCESS MEAN OF SKEWED POPULATIONS BASED ON THE WEIGHTED VARIANCE METHOD

MICHAEL B. C. KHOO∗,‡ , ZHANG WU†,§ and ABDU M. A. ATTA∗,¶ ∗School

of Mathematical Sciences, Univertiti Sains Malaysia 11800 Minden, Penang, Malaysia

†School of Mechanical and Aerospace Engineering Nanyang Technological University, Singapore 639798 ‡[email protected] §[email protected] ¶[email protected]

Received 26 September 2007 Revised 8 April 2008 A synthetic control chart for detecting shifts in the process mean integrates the Shewhart X chart and the conforming run length chart. It is known to outperform the Shewhart X chart for all magnitudes of shifts and is also superior to the exponentially weighted moving average chart and the joint X-exponentially weighted moving average charts for shifts of greater than 0.8σ in the mean. A synthetic chart for the mean assumes that the underlying process follows a normal distribution. In many real situations, the normality assumption may not hold. This paper proposes a synthetic control chart to monitor the process mean of skewed populations. The proposed synthetic chart uses a method based on a weighted variance approach of setting up the control limits of the X/S sub-chart for skewed populations when process parameters are known and unknown. For symmetric populations, however, the limits of the new X/S sub-chart are equivalent to that of the existing X/S sub-chart which assumes a normal underlying distribution. The proposed synthetic chart based on the weighted variance method is compared by Monte Carlo simulation with many existing control charts for skewed populations when the underlying populations are Weibull, lognormal, gamma and normal and it is generally shown to give the most favourable results in terms of false alarm and mean shift detection rates. Keywords: Synthetic control chart; skewed populations; weighted variance; weighted standard deviation; skewness correction; conforming run length.

1. Introduction A control chart is used to determine whether a process is in a state of statistical control, to bring out-of-control process into in-control and to monitor a process to ensure that it remains in-control.

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M. B. C. Khoo, Z. Wu & A. M. A. Atta

A synthetic control chart for the process mean which integrates the X chart and the conforming run length (CRL) chart provides significant improvement in terms of detection power over the basic X chart for all levels of mean shifts. It also surpasses the exponentially weighted moving average (EWMA) and the joint X– EWMA charts in the detection power for a mean shift of greater than 0.8σ.24 Other works on univariate synthetic charts include that of Wu and Spedding,25 Wu et al.,27 Wu and Yeo,26 Calzada and Scariano,2 Davis and Woodall,13 Scariano and Calzada,21 Huang and Chen,16 Chen and Huang8 and Costa and Rahim.11 Among the research on using synthetic control charts in multivariate process monitoring are those by Ghute and Shirke15 and Costa and Machado.10 Synthetic control charts for variables data and the other commonly used control charting approaches such as the X, EWMA and cumulative sum (CUSUM) charts are all based on the assumption that the distribution of the quality characteristic is normal or approximately normal. For nonnormal underlying distributions, three different approaches are currently employed to deal with this problem.22 The first approach is to transform the original data so that the transformed data have an approximate normal distribution, followed by plotting the transformed data on the standard control charts. The second approach involves increasing the sample size until the sample average has a distribution that is well modeled by the normal distribution. The third approach is designing control charts using heuristic methods, such as the X and R charts based on the weighted variance (WV) method proposed by Bai and Choi,1 the X, CUSUM and EWMA charts based on the weighted standard deviation (WSD) method suggested by Chang and Bai6 and the X and R charts based on the skewness correction (SC) method presented by Chan and Cui.4 Other research papers that deal with skewed univariate control charts include that of Yourstone and Zimmer,28 Wu,23 Castagliola,3 Dou and Sa,14 Chen,9 Nichols and Padgett20 and Tsai.22 Among the works on multivariate charts for skewed populations are those by Chang and Bai7 and Chang.5 In this research, a new approach is designed to construct a synthetic control chart for detecting shifts in the process mean of skewed populations using the weighted variance (WV) method, where no assumptions of the underlying process is required. The new synthetic chart will be called the synthetic WV–X chart hereafter. The synthetic WV–X chart is found to provide a huge improvement over all the existing control charting approaches for skewed populations. In the rest of this paper, Sec. 2 reviews some literatures on the Shewhart X chart, the synthetic X chart and the heuristic charts for monitoring the mean of skewed populations. The synthetic WV–X chart is developed in Sec. 3.1. A simulation study is conducted to evaluate the performance of the synthetic WV–X chart in Sec. 4. An example is provided to illustrate the construction of the synthetic WV–X chart in Sec. 5. Finally, conclusions are drawn in Sec. 6.


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2. Literature Review 2.1. Shewhart X control chart Assume that a process follows a normal distribution with in-control mean, µX , and standard deviation, σX . If the target values of µX and σX are unknown, they are estimated from an in-control historical data set consisting of m samples, each of size n. Let Xij denotes the jth observation in the ith sample of a certain quality characteristic of interest, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. Also, let X i and Ri represent the sample mean and the sample range of the ith sample, respectively. If the sample grand mean and the average sample range are denoted as m m 1 1 X = m i=1 X i and R = m i=1 Ri , respectively, then the control limits of the Shewhart X chart are defined as19 UCLX = X + 3

R √ , d2 n

CLX = X

(1a) (1b)

and LCLX = X − 3

R √ . d2 n

(1c)

The value of the control chart constant, d2 which depends on the sample size, n, is given in most statistical quality control text books. 2.2. Synthetic X control chart The synthetic X control chart which makes a Shewhart X chart and a CRL chart work together is shown to be effective in detecting shifts in the process mean. The synthetic X chart consists of a X/S sub-chart and a CRL/S sub-chart.24 Figure 1 illustrates how the determination of a CRL value, i.e. the number of inspected units between two consecutive nonconforming units (inclusive of the ending nonconforming unit) is made, assuming that a process starts at t = 0. For this CRL 1

CRL 2

CRL 3

t=0

Conforming unit Nonconforming unit

Fig. 1.

Conforming run length.


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case, CRL1 = 4, CRL2 = 3 and CRL3 = 5. Note that an X sample in a synthetic chart corresponds to a unit in the CRL chart. The control flow of the synthetic X chart is outlined as follows24 : Step 1: Determine the lower control limit, L, of the CRL/S sub-chart and compute the control limits, UCLX/S and LCLX/S of the X/S sub-chart as follows: UCLX/S = µX + kσX

(2a)

LCLX/S = µX − kσX ,

(2b)

and

Step 2: Step 3:

Step 4: Step 5:

Step 6: Step 7:

where µX denotes the in-control mean and σX is the standard deviation of the sample mean. A procedure and formulae are provided by Wu and Spedding24 to determine the optimal design parameters k and L which give the minimum out-of-control ARL for a desired shift, δd , based on an in-control ARL of interest. Take a random sample of size, n, at each inspection point and compute the sample mean, X. If LCLX/S < X < UCLX/S , the sample is classified as a conforming sample and the control flow returns to Step 2. Otherwise, the sample is considered as a nonconforming sample and the control flow proceeds to Step 4. Check the number of X samples between the present and the last nonconforming samples. This number is the CRL sample of the CRL chart. If CRL ≥ L, the process is in-control and the control flow returns to Step 2. Otherwise, the process is out-of-control and the control flow proceeds to Step 6. Signal the out-of-control status. Find and remove assignable cause(s). Then return to Step 2.

2.3. Heuristic control charts for monitoring the mean of skewed populations Heuristic X charts, namely the WV–X chart by Bai and Choi,1 the WSD–X, WSD–CUSUM and WSD–EWMA charts by Chang and Bai6 and the SC–X chart by Chan and Cui4 are developed to monitor the mean of skewed populations. These charts overcome the problem of a high false alarm rate faced by the Shewhart X chart when the underlying process distribution is not symmetrical. The WV and WSD methods are based on the idea that a skewed distribution can be splitted into two segments at its mean and each segment is used to create a new symmetric distribution. The two new distributions created from the original skewed distribution have the same mean but different standard deviations. The WV and WSD methods use the two created symmetric distributions to set up the limits of the chart.22 The WV method decomposes the process variance, σX2 , into two parts while the WSD method decomposes the process standard deviation, σX , into two parts.


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In the WV method, (2PX )σX2 and [2(1 − PX )]σX2 are used in place of σX2 for the computation of UCLWV–X and LCLWV–X , respectively.1 On the contrary, in the WSD method, (2PX )σX and [2(1 − PX )]σX are used in place of σX for computing UCLWSD–X and LCLWSD−X , respectively.6 This explains why for the limits of the WV–X chart [see Eqs. (3a) and (3b)], there is a square root for the terms 2PX and 2(1 − PX ), while for the WSD–X chart [see Eqs. (6a) and (6b)], there is no square root for these two terms. The control limits of the WV–X chart are1 σ UCLWV−X = µX + 3 √X 2PX (3a) n and

σ LCLWV−X = µX − 3 √X 2(1 − PX ), n

(3b)

where µX is the in-control mean of X and σX is the standard deviation of X while PX = P (X ≤ µX ). If the process parameters are unknown, the limits are computed as follows: 3R 2PˆX = X + WU R (4a) UCLWV−X = X + √ d2 n and LCLWV−X

3R = X − √ d2 n

2(1 − PˆX ) = X − WL R.

(4b)

Here, X and R denote the sample grand mean and the average sample range, respectively, d2 is the constant for the given skewed population corresponding to d2 for the normal distribution and m n i=1 j=1 I(X − Xij ) ˆ PX = , (5) m×n where m and n are the number of samples and the number of observations in a sample, and I(X − Xij ) = 1 if X ≥ Xij or I(X − Xij ) = 0 otherwise. WU and WL are the chart’s constants whose values are given in Bai and Choi.1 The limits of the WSD–X chart are6 σ UCLWSD−X = µX + 3 √X (2PX ) (6a) n and σ LCLWSD−X = µX − 3 √X [2(1 − PX )] n

(6b)

when parameters are known and UCLWSD−X = X + 3

R √ (2PˆX ) dWSD n 2

(7a)


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and LCLWSD−X = X − 3

R dWSD 2

√ [2(1 − PˆX )] n

(7b)

when parameters are unknown. Here, ≡ PX d2 (2n(1 − PX )) + (1 − PX )d2 (2nPX ), dWSD 2

(8)

where d2 (n) is d2 for the normal distribution when the sample size is n. Note that 2nPX and 2n(1 − PX ) are not integers in general. Here, linear interpolation is used to obtain d2 (2nPX ) and d2 (2n(1 − PX )). For example, d2 (3.4) can be obtained as d2 (3) + [d2 (4) − d2 (3)](0.4). The WSD–CUSUM control chart statistics are6 WSD WSD WSD CU,i = max{0, CU,i−1 + (ZU,i − γUWSD )},

for i = 1, 2, . . .

(9a)

WSD WSD WSD = min{0, CL,i−1 + (ZL,i + γLWSD )}, CL,i

for i = 1, 2, . . .

(9b)

and

WSD WSD where CU,0 = CL,0 = 0, while the standardized statistics are

X i − µX √ 2PX σX / n

(10a)

X i − µX √ . 2(1 − PX )σX / n

(10b)

δd 4PX

(11a)

δd . 4(1 − PX )

(11b)

WSD ZU,i =

and WSD = ZL,i

The reference values are γUWSD = and γLWSD =

Here, δd is the magnitude of a shift in the mean where a quick detection is required. WSD WSD > h or CL,i < −h, where h An out-of-control signal is given at time, i, if CU,i is selected to give a desired in-control ARL. The WSD–EWMA chart statistics are6 Ei = λ X i + (1 − λ )Ei−1 ,

for i = 1, 2, . . . ,

(12)

where λ (0 < λ ≤ 1) is the smoothing constant and E0 = µX . The limits of the WSD–EWMA chart are λ σ (2PX ) (13a) UCLWSD−EWMA = µX + K √X n 2 − λ


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and LCLWSD−EWMA

σ = µX − K √X n

λ [2(1 − PX )], 2 − λ

223

(13b)

where the selection of λ and K is based on the approach discussed by Crowder12 and Lucas and Saccucci.18 Note that in Eqs. (10a), (10b), (11a), (11b), (13a) and (13b), R if the process parameters are unknown, µX , σX and PX are estimated by X, dWSD 2 and PˆX , respectively.

Chan and Cui4 proposed the SC–X chart based on the Cornish-Fisher expansion. The limits of the SC–X chart are √ 4α3 /(3 n) σX √ (14a) UCLSC−X = µX + 3 + 1 + 0.2α23 /n n and

√ 4α3 /(3 n) σX √ . LCLSC−X = µX + −3 + 1 + 0.2α23 /n n

(14b)

Here, α3 is the skewness of X. When the exact values of the process parameters are unknown, Chan and Cui4 define the estimators of µX , σX m n Xij −X 3 1 and α3 as X, dR∗ and nm−3 , respectively, where SX = i=1 j=1 SX 2 m n 1 2 i=1 j=1 (Xij − X) , m is the number of samples and n is the numnm−1 ber of observations in a sample. Note that the estimator of α3 is consistent and it is essentially the third moment estimator. A table of the values of d∗2 is given in Chan and Cui.4 3. The Proposed Synthetic WV–X Chart 3.1. Methodology The WSD–X chart is shown to provide considerable improvement over the WV–X chart, in terms of false alarm rates by Chang and Bai.6 However, from a simulation study conducted, the synthetic WSD–X chart is found to give higher false alarm rates and lower mean shift detection rates than the synthetic WV–X chart for most cases involving skewed populations when process parameters are known or unknown. Thus, the synthetic WV–X chart is proposed instead of the synthetic WSD–X chart. The synthetic WV–X chart is based on the idea of integrating the WV method of Bai and Choi1 and the synthetic X chart approach of Wu and Spedding.24 The synthetic WV–X chart gives a higher out-of-control detection rate (smaller Type-II error) and a lower false alarm rate (smaller Type-I error) than the existing heuristic control charts. The synthetic WV–X chart consists of a WV–X/S sub-chart and a CRL/S sub-chart.


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The operation of the synthetic WV–X chart is similar to that of the synthetic X chart described in Sec. 2.2, except that the limits in Eqs. (2a) and (2b) are replaced with that of the WV–X/S sub-chart given as follows: σ UCLWV−X/S = µX + k √X 2PX n

(15a)

σ LCLWV−X/S = µX − k √X 2(1 − PX ), n

(15b)

and

where PX = P (X ≤ µX ). Similarly, in Step 3 (see Sec. 2.2), the classification of a sample as either conforming or nonconforming is made based on the limits in Eqs. (15a) and (15b). The determination of k and L is made using the method discussed in Wu and Spedding,24 based on PX = 0.5, so that the synthetic WV–X chart becomes the basic synthetic X chart when the underlying distribution is symmetric. When PX = 0.5, the design parameters, k and L are chosen using the following Eqs. (16) and (17) given by Wu and Spedding24 : ARL(0) =

1 2Φ(−k){1 − [1 − 2Φ(−k)]L }

(16)

1 . P [1 − (1 − P )L ]

(17)

and ARL(δd ) =

√ √ Here, P = 1 − Φ(k − δd n) + Φ(−k − δd n), where δd is the magnitude of a shift in the mean, in multiples of standard deviation, that needs a quick detection. The procedure works by first setting the desired in-control ARL, i.e., ARL(0). Eq. (16) is then used to determine the design parameters k and L for the synthetic WV–X chart based on the value of ARL(0) set. This is made by increasing the value of L from 1 to 10 and obtaining the corresponding k by solving Eq. (16) numerically. From the results of intensive numerical tests, the optimal value of L is always much smaller than 10. Once the pairs of values (L, k), for L = 1, 2, . . . , 10, that correspond to ARL(0) are obtained, Eq. (17) is used to compute the corresponding ARL(δd ) values for the (L, k) combinations. Next, the pair (L, k) which produces the smallest ARL(δd ) value is identified. Table 1 gives the values of ARL(δd ) for different combinations of (L, k, n) when δd = 1 and identifies the pair (L, k) that gives the smallest ARL(δd ) for sample sizes, n = 4, 5, 7 and 10. Here, the optimal pairs are (L, k) = (5, 2.260), (4, 2.219), (3, 2.164) and (2, 2.085) for n = 4, 5, 7 and 10, respectively, where the values in bold refer to the smallest ARL(δd ) values corresponding to n.


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Table 1. Values of ARL(δd ) for different (L, k, n) combinations when δd = 1, where the optimal (L, k) pair for each n corresponds to the boldfaced ARL(δd ) value. L

k

1 2 3 4 5 6 7 8 9 10

n

1.944 2.085 2.164 2.219 2.260 2.294 2.322 2.346 2.367 2.385

4

5

7

10

3.66475 3.00280 2.80587 2.74442 2.73319 2.75129 2.78077 2.81691 2.85637 2.89503

2.64481 2.21413 2.11242 2.09719 2.11144 2.14071 2.17402 2.20874 2.24325 2.27538

1.73777 1.52992 1.50693 1.52238 1.54619 1.57223 1.59666 1.61934 1.64027 1.65893

1.26690 1.18720 1.19397 1.20993 1.22493 1.23864 1.25060 1.26131 1.27101 1.27959

When process parameters are unknown, the limits in Eqs. (15a) and (15b) are computed as follows: R 2PˆX = X + kRSU , (18a) UCLWV−X/S = X + k √ d2 n and LCLWV−X/S = X − k Pm

X

Pm

R √ d2 n

2(1 − PˆX ) = X − kRSL ,

(18b)

R

i i where X = i=1 and R = i=1 are the sample grand mean and the average m m sample range, respectively, estimated from a preliminary data set of m in-control subgroups. The formula, PˆX , is given in Eq. (5), while the control chart constant, d2 , is the same as that defined in Sec. 2.3 for the WV–X chart. The optimal pair, √ (L, k)

is determined using the approach described earlier. Also, note that SL = √ ˆ and SU = d2√PnX .

2(1−PˆX ) √ d2 n

2

The WV–X/S sub-chart constant, d2 , for nonnormal distributions, such as Weibull, lognormal, Burr’s, etc., is the mean of the relative range, σR . In other X

1 words, d2 = E(R) σX . Similar to Bai and Choi for the WV–X chart, we consider here a heuristic method based on the Weibull, lognormal and Burr’s distributions in the computation of d2 . The cumulative distribution functions (cdfs) of the Weibull, lognormal and Burr’s distributions are1

F (x) = 1 − exp[−(λx)β ], x > 0, log x − θ F (x) = Φ , x>0 σ

(19) (20)

and F (x) = 1 − (1 + xc )−b ,

x > 0,

(21)


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respectively, where λ and β are the scale and shape parameters, respectively, θ and σ are the location and scale parameters, respectively and b and c are parameters (b, c ≥ 1). Letting λ = 1 and θ = 0, we have PX = 1 − exp[−(Γ(1 + 1/β))β ]

(22)

PX = Φ(σ/2)

(23)

and for the Weibull and lognormal distributions, respectively.1 It can also be easily shown that

c −b Γ(1/c)Γ(b − 1/c) (24) PX = 1 − 1 + cΓ(b) for the Burr’s distribution. The values of β and σ satisfying a given PX in Eqs. (22) and (23), respectively, can be obtained uniquely using a numerical method. Also, the skewness coefficient, α3 , is unique for a given β or σ. Unlike the Weibull or lognormal distributions, PX for the Burr’s distribution depends on two parameters, b and c. Therefore, (c, b) cannot be obtained uniquely for a given value of PX . A procedure similar to that in Bai and Choi1 is employed to compute the constant, d2 , for the Weibull, lognormal and Burr’s distributions. The numerical method is used for the Weibull and Burr’s distributions while the simulation approach is considered for the lognormal distribution. Table 2 gives some of the values of d2 for n = 5 and PX = 0.50(0.04)0.98. Here, α3 represents the skewness coefficient while β denotes the shape parameter of the Weibull distribution, σ the scale parameter of the lognormal distribution, and b and c the parameters of the Burr’s distribution. For each PX , the average value of d2 is computed. For example, if PX = 0.58 and n = 5, d2 = (2.23512 + 2.21799 + 2.19357 + 2.20891 + 2.22570)/5 = 2.21626. Using a similar procedure, the average values of d2 are computed for the Weibull, lognormal and three of Burr’s distributions for n = 2(1)10, 15, 20, 25 and PX = 0.50(0.02)0.98. If the underlying population is skewed to the left, then PX is less than 0.5. When PX < 0.5, a d2 value corresponding to 1−PX can be used. For example, if PX = 0.3 and n = 5, then we use d2 = 1.65803, corresponding to PX = 1 − 0.3 = 0.7. After the average √ values of d2 for all√(n, PX ) combinations are obtained, the 2(1−Pˆ )

ˆ

2PX √ X and SU = √ , of the WV–X/S sub-chart when constants SL = d2 n d2 n parameters are unknown are computed and are given in Table 3. The values, SL for the case of PX ≤ 0.5 are similar to that of SU for 1 − PX , and vice-versa.

3.2. Discussion Based on a sample size, n, a desired ARL(0) value and the magnitude of a shift, δd , where a quick detection is desired, Eqs. (16) and (17) are used in determining

0.74

0.70

0.66

0.62

0.58

0.54

α3 d2

α3 d2

α3 d2

α3 d2

α3 d2

α3 d2

(λ, β) = (1, 0.5448) 5.335 1.57890

(λ, β) = (1, 0.672) 3.686 1.79477

(λ, β) = (1, 0.842) 2.581 1.97720

(λ, β) = (1, 1.083) 1.774 2.12486

(λ, β) = (1, 1.45) 1.138 2.23512

(λ, β) = (1, 2.085) 0.582 2.30601

(λ, β) = (1, 3.6286) 0.000 2.33229

0.50

α3 d2

Weibull

(θ, σ) = (0, 1.287) 10.759 1.37624

(θ, σ) = (0, 1.049) 6.186 1.65552

(θ, σ) = (0, 0.825) 3.710 1.89415

(θ, σ) = (0, 0.611) 2.267 2.08285

(θ, σ) = (0, 0.404) 1.322 2.21799

(θ, σ) = (0, 0.201) 0.612 2.29911

(θ, σ) = (0, 0.001) 0.000 2.32628

Lognormal

(c, b) = (0.85, 2.96) 144.697 1.12153

(c, b) = (1.02, 2.99) 43.802 1.52120

(c, b) = (1.21, 3.32) 8.481 1.84143

(c, b) = (1.62, 2.96) 3.618 2.03965

(c, b) = (2.19, 3.00) 1.668 2.19357

(c, b) = (3.33, 2.95) 0.767 2.28538

(c, b) = (6.29, 3.03) 0.068 2.31947

(c, b) = (0.78, 3.69) 84.229 1.25032

(c, b) = (0.95, 3.61) 24.989 1.58579

(c, b) = (1.11, 4.37) 5.287 1.88314

(c, b) = (1.45, 4.08) 2.701 2.06797

(c, b) = (1.92, 4.37) 1.441 2.20891

(c, b) = (2.81, 4.53) 0.690 2.29386

(c, b) = (4.85, 5.02) 0.056 2.32530

Burr’s

(c, b) = (0.63, 9.17) 12.144 1.48219

(c, b) = (0.77, 9.23) 5.572 1.73288

(c, b) = (0.96, 9.21) 3.223 1.93981

(c, b) = (1.22, 9.94) 2.010 2.10576

(c, b) = (1.64, 9.87) 1.240 2.22570

(c, b) = (2.38, 9.99) 0.620 2.30159

(c, b) = (4.09, 9.6) 0.045 2.32905

1.36184

1.65803

1.90713

2.08422

2.21626

2.29719

2.32648

Average (d2 )

The values of d2 under some selected values of PX for Weibull, lognormal and Burr’s distributions when n = 5.

PX

Table 2.

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A Synthetic Control Chart for Monitoring the Process Mean of Skewed Populations 227

0.98

0.94

0.90

0.86

0.82

α3 d2

α3 d2

α3 d2

α3 d2

α3 d2

(λ, β) = (1, 0.1193) 152.378 0.04682

(λ, β) = (1, 0.1807) 73.686 0.22216

(λ, β) = (1, 0.2366) 39.100 0.47988

(λ, β) = (1, 0.2964) 21.629 0.76928

(λ, β) = (1, 0.3644) 12.734 1.05875

(λ, β) = (1, 0.4452) 8.016 1.33174

0.78

α3 d2

Weibull

PX

(θ, σ) = (0, 4.108) 202.746 0.05833

(θ, σ) = (0, 3.1095) 139.112 0.15081

(θ, σ) = (0, 2.5631) 93.617 0.29805

(θ, σ) = (0, 2.1606) 58.874 0.50872

(θ, σ) = (0, 1.8307) 34.496 0.77512

(θ, σ) = (0, 1.5444) 19.303 1.07430

Lognormal

(c, b) = (0.33, 3.20) 402.757 0.01485

(c, b) = (0.39, 3.09) 391.139 0.02832

(c, b) = (0.46, 2.98) 378.660 0.06267

(c, b) = (0.54, 2.88) 363.309 0.14268

(c, b) = (0.61, 3.01) 329.457 0.35040

(c, b) = (0.72, 2.95) 264.358 0.68531

Burr’s

(c, b) = (0.21, 5.89) 395.447 0.01581

(c, b) = (0.27, 6.03) 370.563 0.05471

(c, b) = (0.33, 6.05) 342.092 0.16467

(c, b) = (0.40, 5.86) 291.009 0.38183

(c, b) = (0.49, 5.31) 215.245 0.66941

(c, b) = (0.62, 4.32) 153.208 0.94303

(Continued )

(c, b) = (0.17, 9.40) 386.038 0.01924

(c, b) = (0.24, 8.54) 357.426 0.07752

(c, b) = (0.29, 9.93) 299.982 0.26257

(c, b) = (0.36, 9.05) 216.170 0.52411

(c, b) = (0.43, 9.53) 94.979 0.87207

(c, b) = (0.52, 9.27) 33.780 1.19215

0.03137

0.10670

0.22063

0.47132

0.74515

1.04531

Average (d2 )

228

Table 2.



0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98

PX

0.62579 0.61500 0.60605 0.60452 0.60467 0.60769 0.61697 0.62318 0.64505 0.67506 0.71106 0.76287 0.83156 0.91673 1.03309 1.16809 1.35112 1.58913 1.96227 2.40146 3.02466 3.86028 5.00304 6.19195 7.54247

2

0.34104 0.33519 0.33141 0.32954 0.32965 0.33208 0.33624 0.34354 0.35177 0.36775 0.38672 0.41343 0.44838 0.49179 0.55143 0.62133 0.71584 0.83829 1.03023 1.25702 1.59925 2.07717 2.78474 3.64670 5.57344

3

0.24286 0.23873 0.23603 0.23445 0.23424 0.23505 0.23796 0.24242 0.24747 0.25753 0.26948 0.28628 0.30843 0.33567 0.37349 0.41829 0.48000 0.56008 0.68721 0.83887 1.08547 1.44928 2.06197 2.88615 4.16667

4 0.19223 0.18899 0.18673 0.18537 0.18494 0.18526 0.18706 0.19001 0.19337 0.20048 0.20893 0.22098 0.23680 0.25650 0.28379 0.31604 0.36010 0.41746 0.50209 0.61660 0.90650 1.04124 1.45192 2.02386 2.85122

5

6

8 0.12426 0.12222 0.12062 0.11942 0.11733 0.11819 0.11838 0.11914 0.12007 0.12285 0.12623 0.13133 0.13818 0.14098 0.15908 0.17369 0.19392 0.22023 0.26266 0.31215 0.39339 0.51374 0.71451 0.99870 1.46947

7 SL 0.13978 0.13746 0.13571 0.13447 0.13375 0.13349 0.13402 0.13525 0.13671 0.14039 0.14485 0.15141 0.16015 0.17113 0.18661 0.20503 0.23045 0.26360 0.31697 0.37992 0.48293 0.63638 0.89299 1.25874 1.85595

n 9 0.11230 0.11040 0.10898 0.10781 0.10698 0.10640 0.10632 0.10674 0.10729 0.10943 0.11208 0.11620 0.12181 0.12890 0.13904 0.15107 0.16778 0.18947 0.22443 0.26473 0.33051 0.42698 0.58662 0.81284 1.19303

WV–X/S sub-chart constants.

0.16107 0.15838 0.15642 0.15513 0.15453 0.15450 0.15554 0.15745 0.15967 0.16472 0.17076 0.17949 0.19101 0.20538 0.22543 0.24911 0.28154 0.32364 0.39069 0.48796 0.59527 0.77880 1.07602 1.48706 2.12298

Table 3.

0.10284 1.10119 0.09971 0.09862 0.09772 0.09716 0.09680 0.09694 0.09719 0.09886 0.10093 0.10425 0.10890 0.11477 0.12324 0.13324 0.14731 0.16548 0.19480 0.22816 0.28284 0.36348 0.49568 0.68277 1.00390

10 0.07447 0.07329 0.07217 0.07111 0.07011 0.06923 0.06841 0.06640 0.06734 0.06758 0.06807 0.06929 0.07121 0.07379 0.07779 0.08177 0.08953 0.09859 0.11355 0.13020 0.15507 0.19765 0.26327 0.36220 0.50612

15 0.05997 0.05903 0.05805 0.05720 0.05602 0.05505 0.05404 0.05320 0.05357 0.05217 0.05215 0.05276 0.05400 0.05593 0.05929 0.06363 0.07031 0.07960 0.09557 0.11553 0.14533 0.19257 0.27605 0.39571 0.62477

20 0.05100 0.05021 0.04932 0.04836 0.04734 0.04635 0.04526 0.04429 0.04340 0.04288 0.04256 0.04274 0.04338 0.04454 0.04684 0.04960 0.05416 0.06058 0.07184 0.08584 0.10695 0.14035 0.20039 0.28956 0.46849

25



0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98

PX

0.62579 0.64011 0.65664 0.68199 0.71057 0.74426 0.78807 0.83091 0.89873 0.98406 1.08616 1.22332 1.40288 1.63132 1.94524 2.33617 2.88379 3.64116 4.86345 6.50318 9.07397 13.0908 19.8026 30.3342 52.7973

2

0.34104 0.34888 0.35907 0.37177 0.38739 0.40671 0.42949 0.45805 0.49011 0.53608 0.59072 0.66296 0.75644 0.87515 1.03830 1.24266 1.52787 1.92077 2.55341 3.40403 4.79775 7.04404 11.0223 17.8651 38.0140

3

0.24286 0.24848 0.25573 0.26450 0.27526 0.28788 0.30395 0.32323 0.34479 0.37542 0.41164 0.45907 0.52034 0.59733 0.70327 0.83658 1.02450 1.28331 1.70323 2.27166 3.25641 4.91473 8.16149 14.1392 29.1667

4 0.19223 0.19671 0.20232 0.20913 0.21733 0.22690 0.23894 0.25335 0.26941 0.29224 0.31914 0.35435 0.39950 0.45644 0.53436 0.63209 0.76859 0.95651 1.24441 1.66975 2.71949 3.53102 5.74685 9.91484 19.9585

5 0.16107 0.16485 0.16948 0.17501 0.18159 0.18922 0.19867 0.20994 0.22246 0.24012 0.26084 0.28782 0.32224 0.36548 0.42446 0.49822 0.60091 0.74156 0.96832 1.32140 1.78580 2.64105 4.25901 7.28507 14.8608

6

8 0.12426 0.12721 0.13069 0.13472 0.13788 0.14475 0.15121 0.15885 0.16729 0.17908 0.19281 0.21059 0.23311 0.25807 0.29954 0.34738 0.41389 0.50461 0.65099 0.84532 1.18016 1.74216 2.82812 4.89262 10.2863

7 SU 0.13978 0.14308 0.14704 0.15170 0.15718 0.16349 0.17119 0.18033 0.19047 0.20466 0.22126 0.24279 0.27018 0.30453 0.35138 0.41005 0.49186 0.60398 0.78560 1.02884 1.44879 2.15808 3.53457 6.16652 12.9917

n

(Continued )

0.11230 0.11491 0.11808 0.12162 0.12572 0.13031 0.13581 0.14232 0.14949 0.15952 0.17120 0.18633 0.20549 0.22937 0.26180 0.30215 0.35811 0.43413 0.55625 0.71689 0.99152 1.44796 2.32190 3.98207 8.35123

9 0.10284 0.10532 0.10803 0.11126 0.11483 0.11900 0.12364 0.12925 0.13541 0.14411 0.15417 0.16717 0.18372 0.20423 0.23206 0.26649 0.31442 0.37917 0.48280 0.61785 0.84853 1.23262 1.96194 3.34487 7.02728

10 0.07447 0.07628 0.07819 0.08222 0.08238 0.08479 0.08738 0.08853 0.09382 0.09851 0.10399 0.11112 0.12013 0.13131 0.14647 0.16354 0.19110 0.22590 0.28142 0.35257 0.46522 0.67027 1.04205 1.77439 3.54286

15 0.05997 0.06144 0.06290 0.06453 0.06583 0.06743 0.06903 0.07094 0.07464 0.07605 0.07966 0.08460 0.09110 0.09953 0.11165 0.12725 0.15006 0.18238 0.23686 0.31286 0.43599 0.65302 1.09264 1.93855 4.37342

20 0.05100 0.05226 0.05344 0.05456 0.05564 0.05677 0.05781 0.05906 0.06047 0.06251 0.06501 0.06853 0.07319 0.07926 0.08820 0.09920 0.11560 0.13881 0.17805 0.23246 0.32086 0.47596 0.79316 1.41855 3.27946

25

230

Table 3.




SPI-J072 00305


231

the optimal design parameters, k and L for the synthetic WV–X chart, assuming that the underlying process follows a normal distribution. The following discussion explains why the optimal (L, k) combination obtained for the normal distribution is still employed for the case of skewed populations: (i) Control charts for skewed populations such as the WV–X and the WSD–X charts are constructed based on the 3-sigma approach of the Shewhart chart for a normally distributed process, except that the standard deviation of X is multiplied by two different factors, one for the UCLWV−X (or UCLWSD−X ) while the other for the LCLWV−X (or LCLWSD−X ) (see Eqs. (3a), (3b), (4a), (4b), (6a), (6b), (7a) and (7b)). For skewed populations, i.e., PX = 0.5, the WV and WSD methods provide asymmetric control limits in accordance with the direction and degree of skewness estimated from the sample data by using different variances in computing upper and lower control limits. These asymmetric limits are made possible because of the two different factors used for the UCLWV−X (or UCLWSD−X ) and for the LCLWV−X (or LCLWSD−X ), even though the constant “3” denoting the 3-sigma width remains unchanged. The construction of the WV–X/S sub-chart for the synthetic WV–X chart is based on the same procedure, i.e., in computing the limits of the WV–X/S sub-chart, the standard deviation of X is multiplied by two different factors, one for the UCLWV−X/S [see Eq. (15a)] and the other for the LCLWV−X/S [see Eq. (15b)]. The main idea is to use the heuristic method proposed by Bai and Choi.1 For the WV–X chart to construct the WV–X/S sub-chart and then merge the WV–X/S sub-chart with the CRL/S sub-chart to form the proposed synthetic WV–X chart so that the proposed chart becomes more sensitive in detecting moderate and large process mean shifts. Thus, the optimal k (representing the k-sigma width of the WV–X/S sub-chart) and L values computed for the normal distribution using Eqs. (16) and (17) remain unchanged for any direction and degree of skewness as long as the sample size, n, remains constant. For this case, the asymmetric limits of the WV–X/S subchart are adjusted in accordance with the degree of skewness of the underlying process distribution by only the factors used for the limits UCLWV−X/S and LCLWV−X/S in Eqs. (15a) and (15b), respectively. Hence, the (L, k) values obtained for the synthetic WV–X chart based on the normality assumption is still reasonably optimal for skewed populations. (ii) Unlike the normal distribution, the distribution of the sample mean, X, for skewed distributions such as lognormal, Weibull and Burr’s is intractable. Thus, Eqs. (16) and (17) cannot be obtained and solved numerically for nonnormal distributions. (iii) The false alarm rates of the synthetic WV–X chart for any degree of skewness are all lower than that of the normal theory value, when the process parameters are known (see Table 4). When process parameters are unknown, Table 5 shows that the false alarm rates of the synthetic WV–X chart for skewed

1.5

2.0

2.5

3.0

0.9987

0.8598

0.7637

1.0

1.5688

1.2123

0.5

2.0

2.5

3.0

0.5593

0.6525

0.7315

2.0

2.5

3.0

0.983

0.648

0.442

3.913

1.5

1.0

15.4

1.788

0.0

0.5

38000

Gamma

1.5

1.0

0.3170

0.4484

0.5

0.1656

0.0023

0.0022

0.0019

0.0019

0.0023

0.0026

0.0027

0.0024

0.0023

0.0023

0.0023

0.0025

0.0026

0.0027

0.0024

0.0022

0.0020

0.0019

0.0021

0.0023

0.0025

0.0027

-

WV X

Syn

0.0029

0.0023

0.0020

0.0025

0.0026

0.0027

0.0027

0.0061

0.0059

0.0051

0.0041

0.0033

0.0028

0.0027

0.0030

0.0025

0.0020

0.0019

0.0020

0.0021

0.0022

0.0027

0.0037

0.0034

0.0032

0.0028

0.0026

0.0027

0.0027

0.0056

0.0051

0.0044

0.0037

0.0031

0.0028

0.0027

0.0043

0.0037

0.0031

0.0026

0.0022

0.0021

0.0022

0.0027

0.0078

0.0069

0.0060

0.0049

0.0038

0.0030

0.0027

0.0089

0.0081

0.0069

0.0055

0.0041

0.0030

0.0027

0.0082

0.0071

0.0059

0.0045

0.0033

0.0024

0.0022

0.0027

SC − X WSD − X WV − X

0.0026

0.0024

0.0025

0.0025

0.0026

0.0027

0.0027

0.0027

0.0027

0.0027

0.0028

0.0027

0.0027

0.0027

0.0027

0.0024

0.0025

0.0025

0.0026

0.0026

0.0026

0.0085

0.0063

0.0049

0.0037

0.0031

0.0028

0.0027

0.0049

0.0045

0.0040

0.0036

0.0031

0.0028

0.0027

0.0069

0.0058

0.0048

0.0040

0.0032

0.0028

0.0027

0.0027

EWMA

0.0027

WSD−

WSD−

CUSUM

The boldfaced values denote the lowest false alarm rates corresponding to α 3

α

σ

0.0

0.0010

Lognormal

β

0.0

2.2266

0.0

3.6286

Weibull

Normal

Distribution

α3

4

0.0146

0.0126

0.0105

0.0078

0.0055

0.0035

0.0027

0.0138

0.0124

0.0106

0.0082

0.0057

0.0035

0.0027

0.0146

0.0126

0.0102

0.0077

0.0051

0.0029

0.0022

0.0027

Std X

0.0014

0.0016

0.0019

0.0022

0.0025

0.0027

0.0027

0.0019

0.0021

0.0023

0.0024

0.0026

0.0027

0.0027

0.0016

0.0017

0.0019

0.0022

0.0024

0.0025

0.0026

0.0027

-

WV X

Syn

0.0021

0.0021

0.0024

0.0027

0.0027

0.0028

0.0027

0.0063

0.0053

0.0044

0.0036

0.0031

0.0028

0.0027

0.0026

0.0025

0.0024

0.0024

0.0023

0.0024

0.0024

0.0027

0.0023

0.0022

0.0022

0.0024

0.0025

0.0027

0.0027

0.0042

0.0037

0.0034

0.0030

0.0028

0.0027

0.0027

0.0028

0.0024

0.0022

0.0022

0.0022

0.0023

0.0024

0.0055

0.0049

0.0042

0.0036

0.0031

0.0028

0.0027

0.0070

0.0061

0.0051

0.0042

0.0033

0.0028

0.0027

0.0061

0.0051

0.0042

0.0033

0.0027

0.0024

0.0024

0.0027

X −WV − X

0.0027

SC − X WSD

7

n

WSD−

0.0032

0.0031

0.0029

0.0028

0.0028

0.0027

0.0027

0.0028

0.0028

0.0028

0.0028

0.0027

0.0027

0.0027

0.0029

0.0029

0.0029

0.0028

0.0027

0.0027

0.0027

0.0027

CUSUM

0.0099

0.0072

0.0054

0.0041

0.0033

0.0028

0.0027

0.0056

0.0050

0.0044

0.0038

0.0032

0.0029

0.0027

0.0080

0.0068

0.0054

0.0043

0.0034

0.0028

0.0026

0.0027

EWMA

WSD−

0.0116

0.0098

0.0080

0.0062

0.0044

0.0032

0.0027

0.0114

0.0101

0.0083

0.0064

0.0045

0.0032

0.0027

0.0116

0.0097

0.0078

0.0060

0.0041

0.0028

0.0024

0.0027

Std X

0.0019

0.0022

0.0024

0.0025

0.0026

0.0027

0.0027

0.0021

0.0023

0.0024

0.0026

0.0027

0.0027

0.0027

0.0020

0.0022

0.0024

0.0025

0.0025

0.0026

0.0027

0.0027

-

WV X

Syn

10

0.0021

0.0024

0.0026

0.0027

0.0027

0.0027

0.0027

0.0056

0.0047

0.0039

0.0033

0.0029

0.0028

0.0027

0.0028

0.0027

0.0026

0.0025

0.0024

0.0024

0.0025

0.0027

0.0026

0.0027

0.0027

0.0026

0.0027

0.0027

0.0027

0.0038

0.0035

0.0033

0.0031

0.0029

0.0028

0.0027

0.0028

0.0027

0.0026

0.0025

0.0025

0.0024

0.0025

0.0027

0.0044

0.0039

0.0034

0.0030

0.0029

0.0028

0.0027

0.0059

0.0051

0.0043

0.0036

0.0031

0.0028

0.0027

0.0048

0.0041

0.0034

0.0029

0.0026

0.0025

0.0025

0.0027

SC − X WSD − X WV − X

False alarm rates of the various charts based on known parameters.

WSD−

0.0039

0.0034

0.0032

0.0030

0.0028

0.0027

0.0027

0.0030

0.0030

0.0029

0.0028

0.0028

0.0027

0.0027

0.0034

0.0033

0.0031

0.0030

0.0028

0.0027

0.0027

0.0027

CUSUM

0.0106

0.0077

0.0057

0.0043

0.0033

0.0029

0.0027

0.0059

0.0053

0.0046

0.0040

0.0033

0.0029

0.0027

0.0087

0.0072

0.0058

0.0045

0.0035

0.0028

0.0027

0.0027

EWMA

WSD−

0.0098

0.0083

0.0067

0.0051

0.0039

0.0031

0.0027

0.0101

0.0087

0.0072

0.0055

0.0040

0.0030

0.0027

0.0100

0.0084

0.0067

0.0051

0.0037

0.0027

0.0025

0.0027

Std X

232

Table 4.



1.5

2.0

2.5

3.0

1.2123

0.9987

0.8598

0.7637

1.0

1.5688

2.0

2.5

3.0

0.5593

0.6525

0.7315

1.5

2.0

2.5

3.0

0.648

0.442

3.913

0.983

1.0

15.4

1.788

0.0

0.5

38000

Gamma

1.5

0.4484

1.0

0.3170

0.0039

0.0039

0.0038

0.0040

0.0042

0.0043

0.0045

0.0055

0.0052

0.0050

0.0047

0.0045

0.0045

0.0045

0.0045

0.0042

0.0039

0.0036

0.0036

0.0038

0.0040

0.0045

-

WV X

0.0080

0.0060

0.0050

0.0048

0.0045

0.0038

0.0042

0.0091

0.0083

0.0074

0.0064

0.0052

0.0040

0.0042

0.0078

0.0063

0.0050

0.0041

0.0035

0.0028

0.0035

0.0042

SC − X

0.0082

0.0078

0.0063

0.0051

0.0046

0.0044

0.0045

0.0111

0.0095

0.0078

0.0063

0.0052

0.0046

0.0044

0.0098

0.0083

0.0063

0.0046

0.0037

0.0035

0.0036

0.0044

WSD − X

0.0087

0.0083

0.0077

0.0065

0.0051

0.0042

0.0040

0.0123

0.0110

0.0094

0.0075

0.0056

0.0043

0.0039

0.0102

0.0091

0.0076

0.0060

0.0044

0.0034

0.0032

0.0039

WV − X

0.0061

0.0093

0.0098

0.0219

0.0072

0.0066

0.0064

0.0064

0.0105

0.0094

0.0082

0.0073

0.0068

0.0065

0.0064

0.0103

0.0097

0.0083

0.0069

0.0063

0.0061

0.0164

0.0149

0.0118

0.0093

0.0079

0.0073

0.0071

0.0148

0.0129

0.0109

0.0091

0.0080

0.0074

0.0071

0.0164

0.0146

0.0118

0.0091

0.0075

0.0069

0.0067

0.0071

EWMA

0.0064

WSD−

WSD−

CUSUM

0.0368

0.0280

0.0205

0.0137

0.0085

0.0051

0.0041

0.0273

0.0231

0.0185

0.0135

0.0087

0.0052

0.0040

0.0337

0.0272

0.0203

0.0136

0.0080

0.0043

0.0033

0.0040

Std X

0.0022

0.0026

0.0032

0.0034

0.0036

0.0037

0.0038

0.0040

0.0040

0.0039

0.0038

0.0038

0.0038

0.0038

0.0027

0.0029

0.0032

0.0034

0.0035

0.0036

0.0036

0.0038

-

WV X

Syn

0.0060

0.0053

0.0048

0.0042

0.0039

0.0037

0.0036

0.0080

0.0071

0.0060

0.0051

0.0043

0.0038

0.0036

0.0064

0.0055

0.0047

0.0040

0.0036

0.0034

0.0033

0.0036

SC − X

0.0065

0.0061

0.0054

0.0047

0.0042

0.0038

0.0038

0.0090

0.0078

0.0066

0.0055

0.0046

0.0040

0.0038

0.0074

0.0063

0.0052

0.0045

0.0039

0.0035

0.0034

0.0038

WSD − X

0.0059

0.0059

0.0056

0.0047

0.0040

0.0036

0.0035

0.0094

0.0083

0.0069

0.0055

0.0045

0.0038

0.0035

0.0070

0.0063

0.0054

0.0044

0.0037

0.0033

0.0032

0.0035

WV − X

7

n

WSD−

0.0111

0.0102

0.0087

0.0075

0.0066

0.0061

0.0060

0.0101

0.0092

0.0083

0.0075

0.0067

0.0062

0.0061

0.0109

0.0100

0.0087

0.0077

0.0069

0.0062

0.0060

0.0061

CUSUM

0.0194

0.0158

0.0123

0.0097

0.0080

0.0070

0.0067

0.0146

0.0127

0.0110

0.0095

0.0081

0.0071

0.0068

0.0179

0.0153

0.0125

0.0102

0.0083

0.0071

0.0067

0.0068

EWMA

WSD−

0.0279

0.0211

0.0157

0.0102

0.0065

0.0043

0.0036

0.0218

0.0181

0.0141

0.0102

0.0067

0.0044

0.0036

0.0256

0.0203

0.0151

0.0102

0.0064

0.0040

0.0033

0.0036

Std X

0.0021

0.0029

0.0034

0.0035

0.0035

0.0028

0.0025

0.0035

0.0036

0.0036

0.0036

0.0034

0.0028

0.0025

0.0026

0.0030

0.0033

0.0036

0.0037

0.0030

0.0026

0.0025

-

WV X

Syn

0.0056

0.0051

0.0047

0.0043

0.0037

0.0035

0.0034

0.0068

0.0061

0.0053

0.0047

0.0039

0.0035

0.0034

0.0059

0.0051

0.0046

0.0044

0.0037

0.0034

0.0033

0.0034

SC − X

False alarm rates of the various charts based on unknown parameters.

The boldfaced values denote the lowest false alarm rates corresponding to α 3

α

σ

0.0

0.5

0.0010

0.1656

Lognormal

β

0.0

0.5

3.6286

0.0

α3

2.2266

Weibull

Normal

Distribution

Syn

4

Table 5.

0.0092

0.0074

0.0060

0.0049

0.0042

0.0038

0.0036

0.0080

0.0070

0.0060

0.0050

0.0043

0.0038

0.0036

0.0082

0.0071

0.0059

0.0050

0.0042

0.0037

0.0035

0.0036

WSD − X

0.0047

0.0047

0.0050

0.0041

0.0037

0.0035

0.0034

0.0077

0.0067

0.0056

0.0046

0.0039

0.0035

0.0034

0.0056

0.0050

0.0044

0.0039

0.0036

0.0034

0.0033

0.0034

WV − X

10 WSD−

0.0111

0.0100

0.0088

0.0075

0.0066

0.0062

0.0060

0.0105

0.0094

0.0082

0.0073

0.0068

0.0065

0.0064

0.0105

0.0097

0.0087

0.0079

0.0071

0.0064

0.0061

0.0060

CUSUM

0.0111

0.0100

0.0088

0.0075

0.0066

0.0062

0.0060

0.0105

0.0094

0.0082

0.0073

0.0068

0.0065

0.0064

0.0105

0.0097

0.0087

0.0079

0.0071

0.0064

0.0061

0.0067

EWMA

WSD−

0.0222

0.0170

0.0127

0.0085

0.0058

0.0041

0.0035

0.0180

0.0148

0.0115

0.0084

0.0058

0.0041

0.0035

0.0205

0.0164

0.0123

0.0087

0.0058

0.0041

0.0034

0.0035

Std X




234

SPI-J072 00305


populations are reasonably close to that of the synthetic WV–X chart for a normal distribution. These results show the robustness of the synthetic WV–X chart to violations of the normality assumption. These results, combined with the fact that the synthetic WV–X chart outperforms most of the other existing control charts for skewed populations in detecting moderate and large shifts in the mean (see Tables 6 and 7) make the synthetic WV–X chart appealing to practitioners. Calzada and Scariano2 found that the standard synthetic X chart is moderately robust to processes from skewed populations for sample sizes, n ≥ 6. The robustness property of the standard synthetic X chart, together with the fact that the limits of the WV–X/S sub-chart are computed based on a multiplication of the standard deviation of X by two different factors, one for the UCLWV−X/S while the other for the LCLWV−X/S make the proposed synthetic WV–X chart even more robust to skewed populations. 4. Performance of the Synthetic WV−X Chart The synthetic WV–X chart is compared with the existing heuristic charts for skewed data in terms of the false alarm rates when the process is in-control and mean shift detection rates when the process is out-of-control, via a Monte Carlo simulation using SAS version 9. The false alarm rate of a chart is defined as the proportion of subgroup points plotting beyond the limits of the chart, given that the process is actually in-control. The mean shift detection rate measures the ability of a chart in responding to a shift in the process mean and it represents the proportion of subgroup points plotting beyond the limits of a chart when the mean of a process has shifted. The existing heuristic charts considered are the WV–X, SC−X, WSD–X, WSD–CUSUM and WSD–EWMA charts. The standard Shewhart X chart is also considered here in the comparison. The skewed distributions considered are Weibull, lognormal and gamma because they can represent a wide variety of shapes from nearly symmetric to highly skewed. Note that the gamma distribution and not the Burr’s distribution is considered here although the latter is one of the three distributions used in Sec. 3.1 to compute the constant, d2 . The use of the gamma distribution in this section shows that the synthetic WV–X chart performs equally well for any skewed distribution, irrespective of whether it is one of the three distributions considered in Sec. 3.1 for computing d2 . For convenience, a scale parameter of one is chosen for the Weibull and gamma distributions while a location parameter of zero is selected for the lognormal distribution since the skewness does not depend on the parameters of these distributions. For a gamma distribution with a scale parameter of one and location parameter of zero, its cdf is given by Johnson and Kotz.17 Γx (α) , x ≥ 0, α > 0, (25) F (x) = Γ(α)

x

∞ where Γx (α) = 0 mα−1 e−m dm and Γ(α) = 0 mα−1 e−m dm. Then for this case PX = F (α)

(26)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.6286

2.2266

1.5688

1.2123

0.9987

0.8598

0.7637

0.25 0.50 1.00 1.50 2.00

0.25 0.50 1.00 1.50 2.00

0.25 0.50 1.00 1.50 2.00

0.25 0.50 1.00 1.50 2.00

0.25 0.50 1.00 1.50 2.00

0.25 0.50 1.00 1.50 2.00

0.25 0.50 1.00 1.50 2.00

δ

-

0.0073 0.0214 0.1494 0.5622 0.9959

0.0074 0.0233 0.1713 0.5903 0.9859

0.0074 0.0258 0.1972 0.6209 0.9748

0.0074 0.0284 0.2304 0.6578 0.9671

0.0074 0.0327 0.2707 0.6965 0.9622

0.0077 0.0387 0.3178 0.7349 0.9602

0.0081 0.0463 0.3731 0.7695 0.9593

WV X

0.0053 0.0093 0.0298 0.0930 0.2754

0.0046 0.0088 0.0319 0.1079 0.3201

0.0041 0.0087 0.0366 0.1340 0.3930

0.0038 0.0091 0.0456 0.1771 0.4934

0.0038 0.0106 0.0633 0.2475 0.6126

0.0044 0.0142 0.0965 0.3591 0.7378

0.0059 0.0222 0.1598 0.5020 0.8342

SC − X

0.0076 0.0136 0.0430 0.1342 0.3796

0.0071 0.0136 0.0479 0.1588 0.4417

0.0066 0.0139 0.0556 0.1928 0.5208

0.0062 0.0145 0.0676 0.2422 0.6051

0.0060 0.0157 0.0866 0.3131 0.6950

0.0060 0.0188 0.1185 0.4072 0.7795

0.0062 0.0233 0.1644 0.5086 0.8402

0.0147 0.0263 0.0820 0.2475 0.6361

0.0137 0.0260 0.0884 0.2718 0.6680

0.0124 0.0251 0.0965 0.3015 0.7009

0.0109 0.0244 0.1062 0.3399 0.7371

0.0092 0.0239 0.1198 0.3910 0.7743

0.0077 0.0234 0.1388 0.4447 0.8081

0.0060 0.0228 0.1621 0.5051 0.8377

WSD − X WV − X

4

0.0163 0.0708 0.1986 0.3161 0.4177

0.0174 0.0736 0.2037 0.3237 0.4303

0.0188 0.0769 0.2108 0.3335 0.4447

0.0206 0.0810 0.2193 0.3454 0.4610

0.0226 0.0866 0.2300 0.3618 0.4778

0.0254 0.0933 0.2436 0.3814 0.4949

0.0112 0.0532 0.1776 0.2935 0.3935

0.0127 0.0582 0.1850 0.3036 0.4060

0.0144 0.0644 0.1943 0.3155 0.4217

0.0174 0.0716 0.2065 0.3307 0.4424

0.0211 0.0807 0.2218 0.3484 0.4636

0.0264 0.0924 0.2394 0.3726 0.4837

0.0330 0.1053 0.2601 0.3978 0.5041

EWMA

CUSUM

0.0285 0.1010 0.2598 0.4028 0.5134

WSD−

WSD−

0.0261 0.0460 0.1433 0.4038 0.8850

0.0239 0.0448 0.1479 0.4196 0.8690

0.0211 0.0425 0.1526 0.4338 0.8547

0.0176 0.0391 0.1563 0.4517 0.8447

0.0137 0.0345 0.1591 0.4676 0.8387

0.0097 0.0287 0.1600 0.4847 0.8356

0.0059 0.0222 0.1598 0.5020 0.8342

X

Std

0.0077 0.0366 0.4152 0.9813 1.0000

0.0080 0.0410 0.4453 0.9743 1.0000

0.0083 0.0468 0.4801 0.9699 1.0000

0.0088 0.0545 0.5211 0.9661 1.0000

0.0099 0.0652 0.5651 0.9639 1.0000

0.0115 0.0801 0.6128 0.9626 0.9999

0.0136 0.0990 0.6620 0.9634 0.9996

-

WV X

Syn

0.0057 0.0136 0.0751 0.3415 0.8962

0.0054 0.0142 0.0882 0.4019 0.9225

0.0052 0.0157 0.1094 0.4760 0.9482

0.0054 0.0185 0.1420 0.5628 0.9655

0.0060 0.0232 0.1931 0.6597 0.9779

0.0072 0.0318 0.2669 0.7580 0.9856

0.0094 0.0464 0.3612 0.8288 0.9895

SC − X

0.0068 0.0163 0.0889 0.3925 0.9389

0.0066 0.0173 0.1043 0.4522 0.9541

0.0064 0.0189 0.1263 0.5182 0.9656

0.0064 0.0216 0.1596 0.5964 0.9740

0.0069 0.0261 0.2101 0.6879 0.9816

0.0080 0.0351 0.2818 0.7717 0.9878

0.0099 0.0485 0.3688 0.8337 0.9900

WSD − X

0.0145 0.0351 0.1748 0.6338 0.9999

0.0136 0.0357 0.1920 0.6638 0.9986

0.0127 0.0363 0.2130 0.6951 0.9966

0.0118 0.0373 0.2411 0.7315 0.9944

0.0110 0.0393 0.2745 0.7682 0.9928

0.0103 0.0429 0.3171 0.8012 0.9915

0.0097 0.0475 0.3655 0.8310 0.9897

WV − X

n 7

0.0315 0.1130 0.2764 0.4164 0.5172

0.0330 0.1164 0.2824 0.4290 0.5214

0.0350 0.1212 0.2916 0.4423 0.5284

0.0375 0.1267 0.3030 0.4575 0.5420

0.0406 0.1338 0.3176 0.4729 0.5685

0.0444 0.1428 0.3348 0.4892 0.6107

0.0490 0.1528 0.3556 0.5091 0.6712

CUSUM

WSD−

0.0208 0.0932 0.2553 0.3917 0.5086

0.0233 0.0993 0.2639 0.4041 0.5121

0.0266 0.1068 0.2751 0.4206 0.5167

0.0314 0.1159 0.2883 0.4395 0.5266

0.0375 0.1270 0.3058 0.4600 0.5453

0.0451 0.1409 0.3273 0.4803 0.5816

0.0540 0.1562 0.3521 0.5002 0.6418

EWMA

WSD−

0.0281 0.0647 0.3013 0.8519 1.0000

0.0258 0.0638 0.3118 0.8436 1.0000

0.0232 0.0620 0.3226 0.8385 0.9999

0.0202 0.0595 0.3333 0.8340 0.9992

0.0166 0.0560 0.3427 0.8289 0.9966

0.0130 0.0515 0.3529 0.8284 0.9935

0.0094 0.0464 0.3612 0.8288 0.9895

X

Std

0.0089 0.0593 0.7100 1.0000 1.0000

0.0095 0.0674 0.7293 1.0000 1.0000

0.0102 0.0774 0.7487 0.9999 1.0000

0.0114 0.0903 0.7736 0.9993 1.0000

0.0131 0.1086 0.7963 0.9987 1.0000

0.0157 0.1323 0.8177 0.9980 1.0000

0.0194 0.1618 0.8394 0.9972 1.0000

-

WV X

Syn

0.0066 0.0205 0.1666 0.7260 1.0000

0.0067 0.0229 0.2009 0.7833 1.0000

0.0068 0.0265 0.2460 0.8389 0.9999

0.0073 0.0323 0.3074 0.8816 0.9999

0.0085 0.0416 0.3832 0.9151 0.9999

0.0104 0.0562 0.4699 0.9422 0.9998

0.0136 0.0781 0.5644 0.9577 0.9997

SC − X

0.0067 0.0208 0.1687 0.7309 1.0000

0.0066 0.0228 0.2001 0.7824 1.0000

0.0067 0.0259 0.2420 0.8344 0.9999

0.0071 0.0315 0.3025 0.8768 0.9999

0.0083 0.0410 0.3786 0.9133 0.9999

0.0106 0.0568 0.4725 0.9437 0.9998

0.0142 0.0810 0.5729 0.9590 0.9997

WSD − X

0.0153 0.0462 0.3221 0.9295 1.0000

0.0146 0.0482 0.3479 0.9321 1.0000

0.0139 0.0506 0.3766 0.9360 1.0000

0.0134 0.0548 0.4159 0.9411 1.0000

0.0133 0.0602 0.4603 0.9487 0.9999

0.0135 0.0686 0.5151 0.9530 0.9999

0.0139 0.0795 0.4318 0.9583 0.9997

WV − X

10

0.0464 0.1460 0.3336 0.4943 0.5653

0.0485 0.1505 0.3429 0.4998 0.5855

0.0510 0.1562 0.3532 0.5067 0.6146

0.0538 0.1632 0.3667 0.5142 0.6604

0.0575 0.1716 0.3829 0.5283 0.7285

0.0625 0.1828 0.4017 0.5519 0.8051

0.0682 0.1950 0.4235 0.5890 0.8741

CUSUM

WSD−

0.0317 0.1265 0.3120 0.4733 0.5387

0.0352 0.1333 0.3222 0.4826 0.5532

0.0399 0.1417 0.3341 0.4926 0.5751

0.0459 0.1514 0.3506 0.5019 0.6118

0.0532 0.1640 0.3704 0.5152 0.6737

0.0625 0.1796 0.3923 0.5348 0.7584

0.0732 0.1972 0.4183 0.5689 0.8423

EWMA

WSD−

Mean shift detection rates of the various charts when the underlying distribution is Weibull based on known parameters.

The boldfaced values denote the highest mean shift detection rates corresponding to α 3

α3

β

Syn

Table 6.

0.0307 0.0886 0.5041 0.9948 1.0000

0.0286 0.0885 0.5126 0.9897 1.0000

0.0260 0.0877 0.5247 0.9827 1.0000

0.0234 0.0868 0.5354 0.9753 1.0000

0.0202 0.0843 0.5453 0.9690 1.0000

0.0171 0.0815 0.5549 0.9625 0.9999

0.0136 0.0781 0.5644 0.9577 0.9997

X

Std



0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.6286

2.2266

1.5688

1.2123

0.9987

0.8598

0.7637

0.25 0.50 1.00 1.50 2.00

0.25 0.50 1.00 1.50 2.00

0.25 0.50 1.00 1.50 2.00

0.25 0.50 1.00 1.50 2.00

0.25 0.50 1.00 1.50 2.00

0.25 0.50 1.00 1.50 2.00

0.25 0.50 1.00 1.50 2.00

δ

-

0.0128 0.0339 0.1726 0.5048 0.8389

0.0128 0.0357 0.1882 0.5375 0.8707

0.0126 0.0377 0.2086 0.5764 0.8995

0.0121 0.0399 0.2350 0.6216 0.9229

0.0115 0.0425 0.2684 0.6683 0.9378

0.0114 0.0470 0.3121 0.7150 0.9471

0.0115 0.0531 0.3627 0.7560 0.9525

WV X

0.0136 0.0241 0.0739 0.2109 0.4894

0.0111 0.0209 0.0696 0.2058 0.4835

0.0090 0.0183 0.0679 0.2107 0.4974

0.0173 0.0305 0.0927 0.2567 0.5484

0.0156 0.0290 0.0946 0.2694 0.5840

0.0128 0.0258 0.0929 0.2749 0.6010

0.0104 0.0229 0.0935 0.2887 0.6237

0.0091 0.0226 0.0158 0.3329 0.6798

0.0063 0.0161 0.0812 0.2775 0.6167

0.0074 0.0165 0.0707 0.2318 0.5401

0.0086 0.0246 0.1326 0.4102 0.7610

0.0086 0.0283 0.1728 0.5019 0.8290

0.0181 0.0319 0.0968 0.2662 0.5649

0.0170 0.0315 0.1021 0.2850 0.6000

0.0156 0.0309 0.1088 0.3096 0.6410

0.0137 0.0298 0.1705 0.3421 0.6885

0.0116 0.0284 0.1277 0.3825 0.7368

0.0096 0.0273 0.1440 0.4345 0.7845

0.0076 0.0259 0.1652 0.4926 0.8249

WSD-/ X WV − X

0.0057 0.0170 0.1018 0.3491 0.7069

0.0061 0.0209 0.1433 0.4557 0.7991

SC − X

4

0.0309 0.0877 0.2550 0.3492 0.4541

0.0308 0.0889 0.2280 0.3546 0.4619

0.0295 0.0879 0.2274 0.3551 0.4640

0.0281 0.0869 0.2271 0.3560 0.4661

0.0282 0.0891 0.2331 0.3653 0.4766

0.0299 0.0947 0.2454 0.3828 0.4952

0.0250 0.0723 0.2037 0.3273 0.4316

0.0253 0.0752 0.2091 0.3343 0.4407

0.0248 0.0763 0.2113 0.3368 0.4443

0.0246 0.0780 0.2144 0.3404 0.4486

0.0265 0.0838 0.2242 0.3528 0.4619

0.0305 0.0939 0.2410 0.3740 0.4833

0.0358 0.1056 0.2602 0.3979 0.5060

EWMA

0.0321 0.1014 0.2599 0.4027 0.5162

WSD−

WSD−

CUSUM

X

Std

0.0593 0.1036 0.2966 0.6766 0.9567

0.0502 0.0908 0.2694 0.6265 0.9375

0.0401 0.0767 0.2423 0.5789 0.9102

0.0298 0.0617 0.2159 0.5377 0.8777

0.0203 0.0473 0.1918 0.5061 0.8479

0.0127 0.0351 0.1739 0.4898 0.8295

0.0077 0.0262 0.1667 0.4957 0.8275

The boldfaced values denote the highest mean shift detection rates corresponding to α 3

α3

β

Syn

0.0118 0.0488 0.3817 0.8798 0.9957

0.0124 0.0541 0.4188 0.9063 0.9983

0.0130 0.0607 0.4630 0.9285 0.9994

0.0135 0.0681 0.5092 0.9431 0.9997

0.0143 0.0780 0.5577 0.9516 0.9997

0.0157 0.0924 0.6106 0.9580 0.9995

0.0172 0.1086 0.6559 0.9609 0.9990

-

WV X

Syn

0.0127 0.0298 0.1440 0.4844 0.8734

0.0109 0.0278 0.1467 0.4990 0.8880

0.0095 0.0267 0.1559 0.5312 0.9121

0.0087 0.0273 0.1768 0.5860 0.9398

0.0088 0.0312 0.2195 0.6688 0.9654

0.0094 0.0381 0.2802 0.7202 0.9800

0.0097 0.0452 0.3427 0.8388 0.9849

SC − X

0.0152 0.0354 0.1679 0.5385 0.9062

0.0136 0.0343 0.1755 0.5642 0.9267

0.0120 0.0330 0.1852 0.5922 0.9442

0.0106 0.0339 0.2077 0.6392 0.9588

0.0105 0.0365 0.2432 0.7009 0.9732

0.0108 0.0425 0.2968 0.7661 0.9818

0.0123 0.0544 0.3737 0.8293 0.9874

WSD − X

0.0169 0.0394 0.1830 0.5636 0.9120

0.0164 0.0408 0.2011 0.6065 0.9393

0.0157 0.0423 0.2238 0.6552 0.9617

0.0146 0.0435 0.2494 0.7023 0.9756

0.0135 0.0454 0.2810 0.7472 0.9824

0.0127 0.0486 0.3229 0.7921 0.9862

0.0114 0.0515 0.3660 0.8256 0.9872

WV − X

n 7

0.0450 0.1303 0.3045 0.4502 0.5618

0.0448 0.1310 0.3068 0.4544 0.5635

0.0443 0.1316 0.3092 0.4586 0.5669

0.0448 0.1342 0.3153 0.4668 0.5779

0.0459 0.1384 0.3247 0.4785 0.5961

0.0478 0.1444 0.3376 0.4930 0.6282

0.0514 0.1534 0.3556 0.5126 0.6804

CUSUM

WSD−

0.0351 0.1116 0.2831 0.4275 0.5358

0.0359 0.1144 0.2872 0.4332 0.5389

0.0367 0.1176 0.2918 0.4390 0.5434

0.0391 0.1235 0.3008 0.4498 0.5382

0.0428 0.1317 0.3134 0.4643 0.5702

0.0482 0.1425 0.3303 0.4816 0.6003

0.0558 0.1567 0.3525 0.5029 0.6520

EWMA

WSD−

0.0593 0.1309 0.4900 0.9334 0.9996

0.0506 0.1173 0.4599 0.9134 0.9994

0.0413 0.1027 0.4310 0.8898 0.9989

0.0320 0.0876 0.4042 0.8652 0.9976

0.0235 0.0729 0.3814 0.8435 0.9949

0.0164 0.0604 0.3668 0.8292 0.9911

0.0113 0.0513 0.3655 0.8255 0.9872

X

Std

0.0124 0.0712 0.6296 0.9893 1.0000

0.0134 0.0803 0.6676 0.9935 1.0000

0.0149 0.0928 0.7107 0.9958 1.0000

0.0164 0.1077 0.7513 0.9968 1.0000

0.0182 0.1252 0.7852 0.9968 1.0000

0.0206 0.1462 0.8129 0.9963 1.0000

0.0238 0.1722 0.8367 0.9956 1.0000

-

WV X

Syn

0.0131 0.0387 0.2502 0.7571 0.9874

0.0119 0.0385 0.2651 0.7813 0.9913

0.0115 0.0406 0.2965 0.8213 0.9955

0.0115 0.0452 0.3450 0.8697 0.9983

0.0116 0.0513 0.4018 0.9065 0.9992

0.0126 0.0625 0.4738 0.9342 0.9994

0.0138 0.0765 0.5450 0.9508 0.9993

SC − X

0.0139 0.0411 0.2630 0.7800 0.9909

0.0129 0.0415 0.2827 0.8101 0.9948

0.0122 0.0430 0.3107 0.8414 0.9974

0.0120 0.0469 0.3539 0.8785 0.9987

0.0125 0.0545 0.4133 0.9113 0.9993

0.0140 0.0676 0.4891 0.9387 0.9995

0.0170 0.0891 0.5753 0.9578 0.9950

WSD − X

0.0175 0.0510 0.3086 0.8263 0.9936

0.0172 0.0540 0.3389 0.8594 0.9972

0.0172 0.0585 0.3788 0.8925 0.9989

0.0170 0.0635 0.4232 0.9190 0.9996

0.0167 0.0693 0.4700 0.9369 0.9997

0.0163 0.0763 0.5181 0.9485 0.9996

0.0160 0.0856 0.5684 0.9566 0.9995

WV − X

10

0.0575 0.1616 0.3580 0.5087 0.6566

0.0580 0.1636 0.3627 0.5136 0.6672

0.0587 0.1663 0.3690 0.5201 0.6850

0.0603 0.1709 0.3785 0.5294 0.7161

0.0624 0.1771 0.3912 0.5432 0.7608

0.0656 0.1853 0.4708 0.5644 0.8272

0.0699 0.1958 0.4259 0.5983 0.8750

CUSUM

WSD−

0.0446 0.1421 0.3360 0.4876 0.6120

0.0464 0.1462 0.3422 0.4942 0.6209

0.0489 0.1516 0.3503 0.5021 0.6372

0.0528 0.1594 0.3623 0.5125 0.6660

0.0582 0.1696 0.3780 0.5261 0.7117

0.0654 0.1824 0.3976 0.5459 0.7744

0.0744 0.1979 0.4204 0.5781 0.8449

EWMA

WSD−

Mean shift detection rates of the various charts when the underlying distribution is Weibull based on unknown parameters.

0.0601 0.1596 0.9954 0.9920 1.0000

0.0524 0.1472 0.6408 0.9889 1.0000

0.0442 0.1340 0.6180 0.9842 1.0000

0.0361 0.1203 0.5977 0.9777 1.0000

0.0282 0.1068 0.5810 0.9701 1.0000

0.0214 0.0947 0.5702 0.9627 0.9998

0.0160 0.0856 0.5688 0.9568 0.9995

X

Std

236

Table 7.




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237

since µX = α. Here, α denotes the shape parameter. For the sake of comparison, the normal distribution is also considered since the standard X chart is based on this distribution. The false alarm and mean shift detection rates are computed for all the charts when process parameters are known and unknown using the formulae described in the previous sections. The mean shift detection rates are computed under the zerostate mode, where a mean shift is assumed to occur immediately after inspecting an entire sample or at the beginning of a new sample. The WL and WU constants of the WV–X chart are computed for n = 4, 7, 10 and PX = 0.50(0.02)0.98 so that the computation of the limits of the chart can be made when parameters are unknown. The following sample sizes are considered in the simulation for all the charts: n = 4, 7 and 10. Note that the individual measurements (n = 1) case is excluded because the formulae for the computation of the control limits of all the charts considered when process parameters are unknown require that n ≥ 2 so that the charts’ constants can be determined. The shape parameters, β and α, of the Weibull and gamma distributions, respectively and the scale parameter, σ, of the lognormal distribution are determined in such a way so that the coefficient of skewness, α3 = 0.5(0.5)3. All the charts considered in this study are designed based on an in-control ARL of 370. The synthetic WV–X chart is designed for a quick detection of a shift of one standard deviation in the mean, i.e., δd = 1. Similarly, the WSD–CUSUM and WSD–EWMA charts are also designed so that they are optimal in detecting a shift of one standard deviation in the mean. For this case, h = 4.777 for the WSD–CUSUM chart and (λ , K) = (0.14, 2.785) for the WSD–EWMA chart. A shift in the process mean is represented by µX ,1 = µX ,0 + δσX ,0 , where δ > 0 is the magnitude of a shift, in terms of the number of standard deviation units while µX ,0 and σX ,0 represent the in-control mean and the in-control standard deviation, respectively. For a random variable, X, from the Weibull, lognormal and gamma distributions considered in this study, their in-control means are 1 µX ,0 = Γ 1 + , (27) β 1 (28) µX ,0 = exp σ 2 2 and µX ,0 = α, respectively, while their in-control standard deviations are 2 2 1 σX ,0 = Γ 1 + , − Γ 1+ β β σX ,0 = exp(σ 2 )[exp(σ 2 ) − 1]

(29)

(30) (31)


238

SPI-J072 00305


and σX ,0 =

√ α,

(32)

respectively. Note that when δ = 0, the process is in-control. For the out-of-control process, δ = 0.25, 0.5(0.5)2 are considered. For the cases when the process parameters are known, the false alarm rates and the mean shift detection rates are computed for the various charts based on 1000 in-control (δ = 0) subgroups and 1000 out-of-control (δ > 0) subgroups, each of size n, respectively. For this case, the limits of the various charts are calculated using their respective in-control process parameters, µX ,0 and σX ,0 . For the case involving unknown process parameters, the false alarm rates and the mean shift detection rates are obtained as follows: First, 30 in-control subgroups, each of size n, are generated from a particular distribution and the limits of the various charts are estimated. Next, another 1000 in-control subgroups (or 1000 outof-control subgroups), each having the same size, n, are generated from the same distribution and the false alarm rates (or mean shift detection rates) are computed for the various charts. This process of generating 30 initial subgroups followed by 1000 subsequent subgroups is repeated 10,000 times and the average false alarm rates (or the average mean shift detection rates) for the various charts are recorded. Note that we only consider 1000 subgroups instead of say, 10000 subgroups, for the in-control and out-of-control cases when the process parameters are known and unknown because both subgroup sizes are found to give almost similar false alarm rates and mean shift detection rates. Tables 4 and 5 give the false alarm rates of the various charts for known and unknown process parameters, respectively. The mean shift detection rates of the various charts involving known and unknown process parameters are given in Tables 6 and 7 respectively. The normal, Weibull, lognormal and gamma distributions are considered in the computation of the false alarm rates of the various charts. However, due to space constraint, only the Weibull distribution is considered in computing the mean shift detection rates of the charts. Tables 4 and 5 show that the synthetic WV–X chart gives the lowest false alarm rates in most of the cases, when the underlying population is skewed. The synthetic WV–X chart provides a remarkable improvement, in terms of reduced false alarm rates, compared to its WV–X chart counterpart and the other heuristic charts for skewed data, especially when the skewness, α3 , increases. It is found that the existing heuristic charts, i.e., the SC−X, WSD−X, WV–X, WSD–CUSUM and WSD–EWMA charts have large false alarm rates when the population is skewed, where the false alarm rates become more pronounced for large α3 and when the process parameters are unknown. As expected the standard X chart has the highest false alarm rates when α3 > 0. Note also that for the normal population, the false alarm rates of all the charts are the same, i.e., 0.0027, when process parameters are known (see Table 4). However, for the normal population, when process parameters


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239

are unknown, the synthetic WV–X chart has among the lowest false alarm rates, in comparison with the other charts (see Table 5). We will now discuss the mean shift detection abilities of the charts. We notice from Tables 6 and 7 when process parameters are known or unknown, respectively, that the synthetic WV–X chart provides the highest mean shift detection rates when δ ≥ 1 and n = 4, 7, 10, except for some cases in Table 7 when α3 ≥ 2 and n = 4, 7, where the standard X chart gives the highest detection rates. Even for these few cases, the synthetic WV–X chart is not much less sensitive than the standard X chart. For example, when α3 = 2, n = 4 and δ = 1, 1.5, 2 (see Table 7), the mean shift detection rates of the synthetic WV–X chart are {0.2086, 0.5764, 0.8995} while that of the standard X chart are {0.2423, 0.5789, 0.9102}, where the former is only slightly less sensitive than the latter. Furthermore, the standard X chart has a very high false alarm rate for skewed populations which render it less useful in such situations. For small shifts, say δ = 0.25 and 0.5, Tables 6 and 7 show that usually either the WSD–CUSUM chart or the WSD–EWMA chart gives the highest mean shift detection rates, across all values of α3 . However, the false alarm rates of the WSD– CUSUM and WSD–EWMA charts are quite high, especially when process parameters are unknown (see Table 5). Tables 6 and 7 reveal that the synthetic WV–X chart significantly outperforms its WV–X chart counterpart as well as the WSD−X and SC−X charts, in terms of the mean shift detection rates for both small and large shifts. In general, the synthetic WV–X chart is found to have the most favorable performance among all the charts considered for both cases of known and unknown process parameters, in terms of false alarm and mean shift detection rates. The synthetic WV–X chart gives the lowest false alarm rates in most cases of the skewed populations considered, irrespective of the skewness, α3 and the sample size, n. The synthetic WV–X chart also provides the highest mean shift detection rates when δ ≥ 1 in the majority of the cases for the various skewness, α3 and sample sizes, n. Even though the WSD–CUSUM and the WSD–EWMA charts have higher mean shift detection rates than the synthetic WV–X chart for small shifts, the false alarm rates of the former charts are greatly larger than that of the latter, hence making the two memory charts less desirable in practical situations. 5. An Illustrative Example The data in Table 8, generated from a Weibull distribution for the purpose of illustration, consist of 200 skewed observations and are grouped into 40 subgroups of size, n = 5 each. The shape parameter, β, is chosen to be 0.7637 so that the skewness, α3 , is 3. The in-control mean is given by 1 µX ,0 = λΓ 1 + , (33) β



240

Table 8. Sample number, i

Ri

Observed values Xi,1

Xi,2

Xi,3

Xi,4

1

5.5181

3.6795

14.6212

66.3914

8.7067 377.761

47.9608 70.8019

31.8635 6.1323

0.1024 1.849

4

68.7161

39.3142

0.1571

15.8283

173.63 177.058 127.239 10.2096 3.0277 68.6576

22.3091 38.2975

7

46.4027

8

23.9032

9 10

34.4272 37.5365

11

69.4955 235.59

12 13

9.3264 46.3876

22.4358 156.509 5.3193

7.6337

108.607

Xi

CRLSynX CRLSynWV−X

Xi,5 0.9339

65.4575

18.2288

9.2668 47.8584 19.58 80.9042 375.912 107.4897∗ 54.0056

68.559

42.4676 154.7489 108.5407∗ 25.3201 65.6299 29.1025 81.7607 134.0732

42.9689

45.0472 11.6264

17.0272 79.7482 28.545 250.6586

35.9092 73.1517

18.3365

16.7741

50.3702 218.8159

78.1133

0.9532 28.6258 0.0263 100.437

20.7178 18.3208

9.9281 27.6726 18.9897 100.4107

13.9103 36.8323

14

41.2945

50.7075

11.0628

4.6931

2.3729

48.3346

22.0262

49.2082

86.3268

26.7127

32.5357

54.9649

59.6141

49.9497

16 17

164.803 51.2249

11.7028 44.9619

5.4833 0.1457

1.7302 22.1518

44.2832 163.0728 60.9098 60.7641

45.6005 35.8788

18

1.0078

19

18.3748

2.5108 102.865

20 21

7.2352 100.073 70.2105 107.898

22

69.1715

23 24

78.3785 120.157

33.3378

101.8572

27.8757

8.1903

4.3708

13.2459

28.967

15.5039

5.3346 66.5964

48.1447 96.1047

7.4998 87.8226

94.7384 41.3016

33.6575 85.7264

69.6173 183.968

16.317

87.3395 112.244

71.4867 67.0292

67.1451 183.817 115.235 96.0451 99.0554 104.946 74.3424

25

66.4697

70.1768

26

87.8118

67.8211 180.46

27 28

71.1532 69.3772 81.6418 178.381 127.711

16.678

106.552

116.6719 103.2125∗ 53.1278 97.4465∗

68.66

78.6414

84.1118 112.6389

73.8687 104.378 74.2189 104.618 118.926

114.7965 104.4681∗

73.1718

69.9913

7.8727

57.7197 102.6814∗

30

79.7264

89.0195 261.517

167.347

76.1676 185.3494 134.7555∗+

31

67.109

69.857

116.386

85.0796 194.063

32

70.5192

66.7438 378.88

33

66.302

96.5758

34

77.3884 256.341

35

70.6055

36 37 38 39 40

72.2236

261.172 67.662 166.196 77.7691

182.304 137.676

215.117

91.1394 267.737

94.1074 80.9611 269.036 604.381 266.405 103.152 111.328 74.5677 91.1999 110.289 82.3005 166.52 75.178

119.9207∗+

90.7405 312.1362 157.8375∗+

96.6789 353.037

136.0511∗+

178.9526 170.5437∗+ 197.1315 115.8949∗

523.4199 262.9781∗+ 36.7603 98.1073∗ ∗

79.7572 133.222

132.527

61.9844

99.8089

91.6874 106.81

143.033

178.275

86.5876 137.2651∗+

69.227

118.235

49.008

88.8468

107.145

74.4488

71.2376

286.735

17 1 1

70.5641

99.7692∗

483.256 413.8788 160.4066∗+ 83.5923 104.1621 104.4904∗ 90.2269

2

83.143

175.6075

1.6481 81.3963 25.7656 262.285

1.1905 176.798

3

35.6043

15

29

+

Simulated data for illustration.

2 3 5 6

∗

SPI-J072 00305

2 1 1

27

1 1

3

1

1

1

1

1

1

1

1

1 1 1

2

1 1

3

denotes nonconforming sample on the X/S sub-chart. denotes nonconforming sample on the WV−X/S sub-chart.

where λ is the scale parameter. λ = 42.61 is chosen so that µX ,0 = 50. The in-control standard deviation is computed using Eq. (34) to be σX ,0 = 66.3. 2 2 1 (34) σX ,0 = λ Γ 1 + − Γ 1+ β β Subgroups 1 to 20 correspond to an in-control process while subgroups 21 to 40 correspond to an out-of-control process. The out-of-control mean is represented by µX ,1 = µX ,0 + δσX ,0 , where δ = 1.


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241

From the data in subgroups 1 to 20, we obtain µ ˆX ,0 = X = 45.653, R = 116.125 and PˆX = 0.67. Using Table 3, we have 0.68 − 0.67 0.67 − 0.66 SL = 0.19337 × + 0.20048 × = 0.1969 0.02 0.02 and

SU =

0.68 − 0.67 0.26941 × 0.02

0.67 − 0.66 + 0.29224 × = 0.2808, 0.02

by linear interpolation. Assuming a Type-I error of 0.0027, the WV–X/S subchart’s limits are computed using Eqs. (18a) and (18b) to be UCLWV−X/S = 45.653 + 2.219(116.125)(0.2808) = 118.01 and LCLWV−X/S = 45.653 − 2.219(116.125)(0.1969) = −5.08. For this case, the optimal (L, k) combination (see Table 1) obtained using the procedure discussed in Sec. 3.1 is (4, 2.219). Note that k = 2.219 is a constant that controls the width of the limits of the WV–X/S sub-chart [see Eqs. (18a) and (18b)], while L = 4 is the lower limit of the CRL/S sub-chart. For the sake of comparison, we also consider the construction of the basic synthetic X chart by Wu and Spedding.24 The optimal (L, k) combination is also found to be (4, 2.219). The X/S sub-chart’s limits are computed using Eqs. (2a) and (2b) ˆX = X and σ ˆX = d2R√n , where to be UCLX/S = 95.20 and LCLX/S = −3.89. Here, µ d2 = 2.326 since n = 5. The conforming run length (CRL) values of the basic synthetic X and the synthetic WV–X charts, denoted as CRLSynX and CRLSynWV−X , respectively, are computed and shown in Table 8. The WV–X/S and X/S sub-charts are plotted in Fig. 2 while Fig. 3 shows the corresponding CRL/S sub-charts. From the CRL/S sub-charts in Fig. 3, we notice that the first out-of-control signal is detected by the synthetic WV–X chart at the 30th sample and by the basic synthetic X chart at the 3rd sample. A signal at the 3rd sample corresponds to a false out-of-control signal because the process is actually in-control. The basic synthetic X chart is too sensitive when the underlying process comes from a skewed population. For this case the synthetic WV–X chart provides a more reliable conclusion about the state of a process. 6. Conclusions In this paper, we have suggested a new synthetic WV–X chart based on the weighted variance approach. The simulations have proven that the synthetic WV–X chart gives the most favourable results, in terms of false alarms and mean shift detection rates, and for cases with known or unknown parameters. The performance of this new chart is particularly good when the skewness of the distribution


242

SPI-J072 00305


Xi

300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0 -20 1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

39

Sample no., i

UCL X / S / LCL X / S = 95.2 / −3.89 UCL WV- X / S / LCLWV- X / S = 118.01 / −5.08 Fig. 2.

CRL

WV–X/S and X/S sub-charts.

30

20

10

L=4 0 1

3

5

7

9

11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Sample no., i CRL of the synthetic WV− X chart CRL of the basic synthetic X chart

Fig. 3.

CRL/S sub-charts.


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243

is considerable or serious. In comparison to the basic synthetic X chart, the synthetic WV–X chart also provides a more reliable conclusion about the state of a process when the underlying distribution is skewed, as illustrated by the example in Sec. 5. Consequently, the synthetic WV–X chart can favourably replace the existing heuristic charts for skewed populations discussed in Sec. 2.3. Three future works on this topic that are worthy of pursuing are as follows: (i) A study on the performance of the synthetic WV–X chart under a steady-state mode, for which the mean shift occurs randomly at any time. (ii) Construction of the synthetic X chart incorporating the weighted standard deviation (WSD) or the skewness correction (SC) method. (iii) Construction of a synthetic control chart for process dispersion, such as the synthetic S chart (see Huang and Chen,16 ), incorporating the weighted variance (WV), weighted standard deviation (WSD) or skewness correction (SC) method. Acknowledgment This research is supported by the Universiti Sains Malaysia, Short Term Research Grant, no. 304/PMATHS/637011. References 1. D. S. Bai and I. S. Choi, X and R control charts for skewed populations, Journal of Quality Technology 27 (1995) 120–131. 2. M. E. Calzada and S. M. Scariano, The robustness of the synthetic control chart to non-normality, Communications in Statistics — Simulation and Computation 30 (2001) 311–326. 3. P. Castagliola, X control chart for skewed populations using a scaled weighted variance method, International Journal of Reliability, Quality and Safety Engineering 7 (2000) 237–252. 4. L. K. Chan and H. J. Cui, Skewness correction X and R charts for skewed distributions, Naval Research Logistics 50 (2003) 555–573. 5. Y. S. Chang, Multivariate CUSUM and EWMA control charts for skewed populations using weighted standard deviations, Communications in Statistics — Simulation and Computation 36 (2007) 921–936. 6. Y. S. Chang and D. S. Bai, Control charts for positively-skewed populations with weighted standard deviations, Quality and Reliability Engineering International 17 (2001) 397–406. 7. Y. S. Chang and D. S. Bai, A multivariate T 2 control chart for skewed populations using weighted standard deviations, Quality and Reliability Engineering International 20 (2004) 31–46. 8. F. L. Chen and H. J. Huang, A synthetic control chart for monitoring process dispersion with sample range, International Journal of Advanced Manufacturing Technology 26 (2005) 842–851. 9. Y. K. Chen, Economic design of X control charts for non-normal data using variable sampling policy, International Journal of Production Economics 92 (2004) 61–74.


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10. A. F. B. Costa and M. A. G. Machado, Synthetic control charts with two-stage sampling for monitoring bivariate processes, Pesquisa Operacional 27 (2007) 117–130. 11. A. F. B. Costa and M. A. Rahim, A synthetic control chart for monitoring the process mean and variance, Journal of Quality in Maintenance Engineering 12 (2006) 81–88. 12. S. V. Crowder, Design of exponentially weighted moving average schemes, Journal of Quality Technology 21 (1989) 155–162. 13. R. B. Davis and W. H. Woodall, Evaluating and improving the synthetic control chart, Journal of Quality Technology 34 (2002) 200–208. 14. Y. Dou and P. Sa, One-sided control charts for the mean of positively skewed distributions, Total Quality Management 13 (2002) 1021–1033. 15. V. B. Ghute and D. T. Shirke, Joint monitoring of multivariate process using synthetic control charts, International Journal of Statistics and Management System 1 (2007) 129–141. 16. H. J. Huang and F. L. Chen, A synthetic control chart for monitoring process dispersion with sample standard deviation, Computers & Industrial Engineering 49 (2005) 221–240. 17. N. L. Johnson and S. Kotz, Distributions in Statistics: Continuous Univariate Distributions-1 (Houghton Mifflin Company, Boston, 1970). 18. J. M. Lucas and M. S. Saccucci, Exponentially-weighted moving average control schemes: properties and enhancements, Technometrics 32 (1990) 1–12. 19. D. C. Montgomery, Introduction to Statistical Quality Control, 5th ed. (John Wiley & Sons, New York, 2005). 20. M. D. Nichols and W. J. Padgett, A bootstrap control chart for Weibull percentiles, Quality and Reliability Engineering International 22 (2005) 141–151. 21. S. M. Scariano and M. E. Calzada, A note on the lower-sided synthetic chart for exponentials, Quality Engineering 15 (2003) 677–680. 22. T. R. Tsai, Skew normal distribution and the design of control charts for averages, International Journal of Reliability, Quality and Safety Engineering 14 (2007) 49–63. 23. Z. Wu, Asymmetric control limits of the X chart for skewed process distributions, International Journal of Quality and Reliability Management 13 (1996) 49–60. 24. Z. Wu and T. A. Spedding, A synthetic control chart for detecting small shifts in the process mean, Journal of Quality Technology 32 (2000a) 32–38. 25. Z. Wu and T. A. Spedding, Implementing synthetic control charts, Journal of Quality Technology 32 (2000b) 75–78. 26. Z. Wu and S. H. Yeo, Implementing synthetic control charts for attributes, Journal of Quality Technology 33 (2001) 112–114. 27. Z. Wu, S. H. Yeo and T. A. Spedding, A synthetic control chart for detecting fraction nonconforming increases, Journal of Quality Technology 33 (2001) 104–111. 28. S. A. Yourstone and W. J. Zimmer, Non-normality and the design of control charts for averages, Decision Sciences 23 (1992) 1099–1113.

About the Authors Michael B. C. Khoo is an Associate Professor in the School of Mathematical Sciences, Universiti Sains Malaysia (USM). His research interests are statistical process control and reliability analysis. He is a member of the American Society for Quality. He also serves as a member of the editorial boards of Quality Engineering, International Journal of Statistics and Management System and Journal of Modern Applied Statistical Methods.


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Zhang Wu is an Associate Professor in the School of Mechanical and Aerospace Engineering of Nanyang Technological University, Singapore. His current research interests include quality control, reliability, nonlinear optimization and geometrical tolerance. Dr. Wu is a senior member of the American Society for Quality. Abdu M. A. Atta is a Ph.D. student in the School of Mathematical Sciences, Universiti Sains Malaysia (USM). He obtained his B.Sc. and M.Sc. in statistics from University of Baghdad and University of Almustansiria, both in Iraq, respectively.