Monitoring the Weibull shape parameter by control charts for the sample range

Research Article

Published online 24 March 2010 in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/qre.1099

Monitoring the Weibull Shape Parameter byControl Charts for the Sample RangeFrancis Pascuala∗† and Huifang Zhangb

In this paper, we propose control charts for monitoring changes in the Weibull shape parameter b. These charts arebased on the range of a random sample from the smallest extreme value distribution. The control chart limits dependonly on the sample size, the desired stable average run length (ARL), and the stable value of b. We derive controllimits for both one- and two-sided control charts. They are unbiased with respect to the ARL. We discuss sample sizerequirements if the stable value of b is estimated from past data. The proposed method is applied to data on thebreaking strengths of carbon fibers. We recommend one-sided charts for detecting specific changes in b because theyare expected to signal out-of-control sooner than the two-sided charts. Copyright © 2010 John Wiley & Sons, Ltd.

Keywords: average run length; statistical process control; unbiased control charts

1. Introduction

The Weibull distribution is named after W. Weibull, who used it to characterize material breaking strengths, see1, 2. It isnow widely used to describe lifetimes of electronic components and products with increasing or decreasing failure rates.The Weibull distribution also describes failure of composite materials such as ceramics as a result of degradation or crack

growth. It adequately describes the strength distributions of steels and composite materials. For example, Basu et al.3 used theWeibull distribution to describe the strength of hot pressed ZrO2 (2.5 mol% yttria-stabilized)-30 vol% TiB2 (TZP-TiB2) composites.Padgett and Spurrier4 used the Weibull distribution to characterize the strength distribution of carbon fibers. The proposedmethod here will be applied to their data set found in Table I. Zhang and Chen5 used the Weibull distribution to describe therust-resisting properties of a process for painting metal electrical boxes. A wide array of applications of the Weibull distribution isdescribed in6--8.

There are potential problems with applying the familiar normal-based X , R, and s charts to asymmetric distributions such asWeibull. For example, Padgett and Spurrier4 remarked that these charts might not effectively detect shifts in low populationpercentiles of these distributions. Their simulations showed that the X and R charts falsely signaled out-of-control (OOC) toooften. Nelson9 stated that when the normal-based median and R charts were applied to Weibull, the probabilities of OOC signalsand, equivalently, average run lengths (ARLs) were substantially different from what were expected.

Control charts for Weibull processes found in the literature always assume that one or more Weibull parameters are knownand stable. Also, in the absence of analytical results, control limits are often obtained by maximum likelihood estimation andsimulation. For example, Zhang and Chen5 studied one-sided EWMA charts for monitoring changes in the Weibull mean assumingthat the shape parameter was known and stable. Ramalhoto and Morais10 studied schemes for monitoring the Weibull scaleparameter under the assumption that the shape and threshold parameters were known and stable. Nichols and Padgett11 useda bootstrap method with pivotal quantities to monitor Weibull percentiles. Johnson12 used the cumulative sum control chartsto monitor the Weibull mean assuming that the shape parameter is known. The control limits for the Weibull median and rangecontrol charts presented in9 required two or more parameters to be known. Note that these studies suggested ways to monitorshifts in functions of the Weibull distribution parameters, but they did not provide ways to determine which parameter specificallyhad changed when an OOC signal occurred.

We feel that methods for detecting shifts in � are valuable tools for practitioners. First, the Weibull shape parameter � playsan important role in survival analysis. It determines the nature of the hazard function. In particular, the hazard is decreasing if�<1, constant if �=1, and increasing if �>1. With the scale parameter fixed, low percentiles are lower for smaller �, whereashigh percentiles are lower with larger �. Also, the coefficient of variation (the ratio of the standard deviation to the mean) is

aDepartment of Statistics, Washington State University, Pullman, WA 99164-3144, U.S.A.bWashington State Department of Social and Health Services, 640 Woodland Square Loop SE, Lacey, WA 98503, U.S.A.∗Correspondence to: Francis Pascual, Department of Statistics, Washington State University, Pullman, WA 99164-3144, U.S.A.†E-mail: [email protected]

Copyright © 2010 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2011, 27 15--25

15

F. PASCUAL AND H. ZHANG

Table I. Stress (GPa) at breaking of carbon fibers

Sample Stress (GPa)

1 3.70 2.74 2.73 2.50 3.602 3.11 3.27 2.87 1.47 3.113 4.42 2.41 3.19 3.22 1.694 3.28 3.09 1.87 3.15 4.905 3.75 2.43 2.95 2.97 3.396 2.96 2.53 2.67 2.93 3.227 3.39 2.81 4.20 3.33 2.558 3.31 3.31 2.85 2.56 3.569 3.15 2.35 2.55 2.59 2.3810 2.81 2.77 2.17 2.83 1.9211 1.41 3.68 2.97 1.36 0.9812 2.76 4.91 3.68 1.84 1.5913 3.19 1.57 0.81 5.56 1.7314 1.59 2.00 1.22 1.12 1.7115 2.17 1.17 5.08 2.48 1.1816 3.51 2.17 1.69 1.25 4.3817 1.84 0.39 3.68 2.48 0.8518 1.61 2.79 4.70 2.03 1.8019 1.57 1.08 2.03 1.61 2.1220 1.89 2.88 2.82 2.05 3.65

a decreasing function of �. Second, control charts for � can be used to check the stable-� assumption that is made by othercharting methods for monitoring the mean or, equivalently, the Weibull scale parameter. This is analogous to the strategy ofstudying the R or S chart first before considering the X chart in the normal case.

In this paper, we propose a way to monitor � that depends on the sample size, the desired stable-process ARL, and the stablevalue of �, while the other Weibull parameters need not be specified. The control charts we propose are based on the range ofa random sample from a smallest extreme value (SEV) population. We consider one- and two-sided Shewhart-type charts withcontrol limits and study their ARL properties. We investigate sample size requirements if a stable value of � is estimated fromdata. We apply the proposed methods to the data set provided by Padgett and Spurrier4.

2. Assumptions and notation

2.1. Distribution assumption

Assume that Y has a two-parameter Weibull distribution with scale parameter � and shape parameter �. Write Y ∼WEI(�,�). It is awell-known fact that X = log(Y) (natural logarithm) has an SEV distribution with location parameter �= log(�) and scale parameter�=1 / �. X ∼SEV(�,�) has a probability density function (pdf) given by:

fX (x)= 1

�exp

[x−�

�−exp

(x−�

�

)], x ∈�.

The variance of SEV is given by �2�2 / 6. Note that monitoring the Weibull parameter � is equivalent to monitoring either theSEV parameter � or the SEV variance. However, monitoring � is not equivalent to monitoring the Weibull variance because it isa function of both � and �.

Assume that Y1,. . . , Yn are random samples from WEI(�,�). Then, X1 = log(Y1),. . . , Xn = log(Yn) are SEV random samples. Let

the range of the SEV sample be R=X(n) −X(1) where X(n) =maxi Xi and X(1) =mini Xi . Let X′i = (Xi −�) / � for i=1,. . . , n. Because

X′i ∼SEV(0, 1), then X

′1,. . . , X

′n are essentially random samples from a standard SEV distribution. Furthermore,

R′ =X

′(n) −X

′(1) =

X(n) −�

�− X(1) −�

�= X(n) −X(1)

�,

that is, R′ =R / �=�R. In the following section, we study the properties of the standard SEV range R

′. These properties can be

extended to any SEV range R through simple transformations (e.g. E(R)=�E(R′)=E(R

′) / �).

16



2.2. Distribution of the SEV range

From the results of Vogt13, we derive the pdf and cumulative distribution function (cdf) of R′

as:

fR′ (r)=nern−1∑i=1

i(−1)i+1(n−1i

)[i(er −1)+n]2

and

FR′ (r)=1+nn−1∑i=1

(−1)i(n−1i

)i(er −1)+n

, r�0. (1)

Apart from the argument r, the pdf and cdf of R′

depend only on the sample size n. The q-quantile of R′

is the inverse of thecdf function, that is,

Q′(q)=F−1

R′ (q). (2)

Using the results of Lieblein14, we can show that the mean and variance of R′

are:

�0 = n

[g1(n)−

n−1∑j=0

(−1)jg1(j+1)

(n−1

j

)](3)

�20 = n

n−1∑j=0

(−1)j(

n−1

j

)g2(j+1)−n2

[n−1∑j=0

(−1)j(

n−1

j

)g1(j+1)

]2

+ng2(n)−n2[g1(n)]2 −2n(n−1)n−2∑i=0

(−1)i(

n−2

i

)�(i+1, n− i−1)

+2n2g1(n)

[n−1∑j=0

(−1)j(

n−1

j

)g1(j+1)

](4)

where

gk(c)= (−1)k∫ +∞

0(log u)ke−cu du, �(t, u)=

∫ +∞

−∞

∫ y

−∞xyex−tex

ey−ueydx dy.

The latter integrals are computed numerically using the results of Lieblein14 and the QUADPACK Fortran algorithms in15.The pdf, cdf, q-quantile, mean, and variance of the range R of any SEV random sample are, respectively,

fR(r) = 1

�fR′

( r

�

)=�fR′ (�r), FR(r)=FR′

( r

�

)=FR′ (�r), QR(q)=�QR′ (q)= QR′ (q)

�,

�R = ��0 = �0

�, �2

R =�2�20 = �2

0

�2.

(5)

Note that R has a scale distribution with scale parameter �=1 / �, and, thus, R / �=�R is a pivotal quantity whose distributiondoes not depend on � and �. Also, the forms of QR or �R suggest that it is reasonable to use the sample range R to monitor�=1 / �. In particular, an increase in � or, equivalently, a decrease in � results in a larger R on average.

3. Control chart schemes

In this section, we present control chart schemes using the range R for monitoring the SEV � or Weibull �. The control limitsdepend on the sample size n and stable process value of �=1 / � and not on the Weibull scale �. Also, the stable and OOC ARLsdepend on only the ratio between the stable and true values of �=1 / �.

3.1. Control limits for the SEV range

Here, we consider three types of control charts for monitoring changes in �=1 / �. The objectives and priorities of the practitionerdetermine which of the charts is most appropriate for implementation. We discuss specific situations below.

• A high-side control chart has an upper control limit UCL, and an OOC signal is given if R>UCL.The high-side chart is most appropriate if an increase in � (decrease in �) is detrimental. For example, an increase in �results in an increase in the coefficient of variation or an increase in the variance of the logarithm of responses. If thescale parameter � is relatively stable, a larger � results in an increase in the hazard functions at shorter product lives, anddecreases in low percentiles such as the 1st, 5th, and 10th.

• A low-side control chart has a lower control limit LCL, and an OOC signal is given if R<LCL.


17


The low-side chart is most appropriate for detecting decreases in � (increases in �). In particular, a decrease from �>1 to�<1 (or from �<1 to �>1) represents a change from a decreasing hazard function to an increasing one.If � is relatively stable, a smaller � results in an increase in the hazard rates at longer lives and reductions in high percentilessuch as the 90, 95, and 99th.

• The two-sided control chart has both limits UCL and LCL. An OOC signal is given if R>UCL or R<LCL.The two-sided chart is appropriate if any change in � or � is deemed undesirable.

Let �S and �S be the stable-process values of � and �, respectively, pa =Pr(R>UCL)=1−FR(UCL), and pb =Pr(R<LCL)=FR(LCL).

In terms of the quantiles of R′, the control limits are given by:

UCL=�S ×QR′ (1−pa)=QR′ (1−pa) / �S, LCL=�S ×QR′ (pb)=QR′ (pb) / �S. (6)

The center line is at CL=�R =�S ×�0 =�0 / �S. Alternatively, the median �S ×QR′ (0.50)=QR′ (0.50) / �S may be used. These suggestthat a control scheme is chosen by specifying pa and/or pb. Also, the control limits do not require a specified value of the Weibullscale parameter �.

3.2. Average run length

The number of periods T till the monitoring scheme gives an OOC signal for the first time is called the run length of the chart.The expected value of T , called the ARL, is often used to evaluate and compare control charts. When the process is stable, theARL denoted by L0 should ideally be long. When the process has shifted, the chart should quickly detect the shift with an OOCalarm. Thus, the ARL should be shortened, that is, ARL<L0 when the process is OOC.

In practice, the control limits UCL and LCL are often obtained from a predetermined stable ARL, say, L0. In particular, we set

pa =pb =1 / L0 (7)

in Equation (6) for the high and low-side charts. For two-sided charts,

pa +pb =1 / L0. (8)

The above equations suggest that there are unique one-sided charts and infinitely many two-sided charts with the stable ARL of L0.

3.2.1. Out-of-control ARL. Below, assume that �S and �S are the stable-process values and �T and �T are the actual values of� and �, respectively. Let L(�T ) denote the ARL when the true SEV scale is �T . For independent samples, the run length T is ageometric random variable with success probability Pr(OOC|�T ), the probability of an OOC signal under �T . It follows that theARL is

L(�T )= 1

Pr(OOC|�T ). (9)

We refer to L0 =L(�S) as the all-OK ARL, and L(�T ) as an OOC ARL if �S �=�T .Let �=�T / �S =�S / �T represent the process shift. For a high-side chart, it can be shown using Equation (6) that

Pr(OOC|�T )=1−FR′ (�T ×UCL)=1−FR′[

1

�QR′ (1−pa)

]. (10)

For a low-side chart,

Pr(OOC|�T )=FR′ (�T ×LCL)=FR′[

1

�QR′ (pb)

]. (11)

For a two-sided chart, we have

Pr(OOC|�T ) = 1−FR′ (�T ×UCL)+FR′ (�T ×LCL)

= 1−FR′[

1

�QR′ (1−pa)

]+FR′

[1

�QR′ (pb)

]. (12)

We have the following observations:

• The OOC ARL is a function of the ratio � which represents the magnitude of the change in �=1 / �. More importantly, itsuffices to assume that �S =1 (the standard SEV case) and consider only the values of � to study ARL properties.

• Computations do not require both the stable and true values of the Weibull scale �. The ARL study done here is valideven if � varies across samples as long as it does not change within the sample. This assumption about � holds under thefundamental concept in control charting which Shewhart referred to as ‘rational subgrouping’. Under this concept, samplesare chosen so as to maximize the differences between samples and minimize differences within samples. Strategies forrational subgroups are discussed in Section 4-3.4 of Reference16.

18



3.2.2. ARL-unbiased control charts. It is possible, e.g. for a two-sided chart, that some OOC ARLs are longer than the all-OKARL L0. This is undesirable because quick detection of a process shift is ideal. To address this, Pignatiello et al.17 defined anARL-unbiased control chart which, applied to the current problem, satisfies

• L(�S)=L0 and• L(�T )<L(�S) if �T >�S for a high-side, if �T <�S for a low-side, or if �T �=�S for a two-sided chart.

The ARL-unbiased, or simply unbiased, design ensures that the ARL is maximized at the stable value of �. For these charts,the time to detect process shifts that are of interest to the practitioner are shorter than L0. Below, we consider only unbiasedcontrol charts.

3.2.3. Sample size requirements for Phase I of control charts. The standards-given and retrospective are two approaches in settingup limits for control charts. See, for example, page 64 of Reference18 and Sections 4-3.7 and 5-2.3 of Reference16. In thestandards-given approach, in-control parameter values are suggested by previous experiences with the same or a similar process.Such information may also be derived from engineering specifications. In the retrospective or as-past-data approach, no suchprior information is available, and it is customary to derive estimates of process parameters from m previous samples and thenuse them to compute control limits. This procedure is often referred to as Phase I of control charting. If the m past samples arejudged to be in control relative to these estimated limits, these limits are then used to monitor future samples (Phase II).

Control limits are truly random variables under the retrospective approach, while they are fixed quantities under the standards-given approach. Consequently, the run length distributions and ARL properties differ between these scenarios. Hence, it isimportant to determine how many previous samples m are needed so that the respective run length distributions under thesetwo approaches are similar. A literature review on the effects of parameter estimation on control chart behavior is given by19.The procedure in20 is followed for the method proposed here.

Assume that m random samples each of size n are obtained from a stable process. Let Ri be the (SEV) sample range forsample i for i=1,. . . , m. An unbiased estimator of � is Sm =R / �0, where R=∑

Ri / m and �0 is given by (4). Let U=Sm / �. By theCentral Limit Theorem with m large, U has an approximate normal distribution with mean 1 and variance �2

0 / (m�20), where �2

0 isgiven by (3). Let fU(u) be the normal pdf that approximates the pdf of U.

Suppose that � is estimated by Sm in Phase I. Recall the assumptions and notation of Section 3.2.1. Under the standards-givenscenario, run length T is a Geometric random variable with success probability given by (9) and one of (10), (11), and (12). Forthe retrospective scenario, it can be shown that an approximation of the cdf of T under a shift � is

Pr(T�t)=̇1−∫ +∞

−∞[1−Pr(OOC|U=u,�)]tfU(u) du

where Pr(OOC|U=u,�) is given by (10), (11), or (12) with the cdf argument multiplied by u. For example, Pr(OOC|U=u,�)=FR′ [uQR′ (pb) / �] for the low-side chart. The integral in the last equation is evaluated using the Fortran codes in15. The cdf of Tunder the two scenarios are compared for different values of m and �.

Suppose n=5 and L0 =100. Figure 1 plots the cdfs of T under the standards-given (� known) and retrospective (� estimated)scenarios for the low-side chart with m=75 and �=0.6, 0.8, 1. These plots suggest that estimating � using m�75 samples of sizen=5 would result in run length behavior similar to that when � is given. Figures 2 and 3 are for the high-side (�=1, 1.2, 1.4) andtwo-sided charts (�= .75, 1, 1.25), respectively. They suggest m�150 for the high-side and m�75 for the two-sided chart. Similarresults are obtained for n=4, 6,. . . , 10 and other all-OK ARL L0. For comparison, Chen20 recommended m�75 Phase I sampleswhen using R, S, and S2 charts to monitor changes in the normal distribution standard deviation.

3.3. Numerical results

3.3.1. Tables of control limits. The control limits for one-sided charts are computed using Equations (1), (2), (6), and (7). Tables IIand III give values of the standardized control limits �S ×UCL and �S ×LCL for high-side and low-side charts, respectively, andsample size n=4,. . . , 10 and all-OK ARL L0 =50, 100, 200, 300, 370, 400, 500, 1000. For example, for n=4, L0 =100, and stable shapeparameter �S, the control limit for a low-side chart is given by LCL=0.4811 / �S. The one-sided charts in these tables are allunbiased.

There are infinitely many two-sided charts that have an all-OK ARL of L0, but not all of them are unbiased. We performnumerical searches for unbiased two-sided charts by determining combinations of pa and pb that satisfy Equation (8) such thatthe ARL given by Equations (9) and (12) is equal to L0 when �=1 and below L0 when � �=1. The values of �S ×UCL and �S ×LCL ofunbiased 2-sided charts are given in Table IV. For example, for n=5 and L0 =370, we have UCL=9.4907 / �S and LCL=0.4790 / �S.

3.3.2. ARL comparisons. The effects of � and sample size n on the ARL for the low-side, high-side, and two-sided charts arestudied here. Equations (10)–(12) are used to compute the ARL.

Figure 4 is a plot of ARL versus � for the three schemes for n=5 and L0 =100. This plot shows that the one-sided charts detectthe shift significantly more quickly for relatively smaller shifts �. The differences diminish for much larger shifts, i.e. at extremevalues of �. If a specific shift in �=1 / � is crucial, it is recommended that an appropriate one-sided chart be implemented. Similarresults hold for other sample sizes and all-OK ARLs. Table V gives ARL values for the three schemes for n=5, L0 =100, 370, 500,and specific values of �. The low-side scheme is used for ��1 and the high-side for ��1.


19


0

0.2

Run Length

CD

F o

f Run

Len

gth

m=75,rho=0.6

Sigma KnownSigma Estimated

100 200 300 400

0.4

0.6

0.8

1.0

0.0

Run Length

CD

F o

f Run

Len

gth

m=75,rho=0.8

Run Length

CD

F o

f Run

Len

gth

m=75,rho=1

0 100 200 300 400 0 100 200 300 400

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Figure 1. Cdf of run length T under standards-given (� known) and retrospective (� estimated) scenarios for the low-side controlchart with m=75, n=5, L0 =100, and �=0.6, 0.8, 1

m=150,rho=1 m=150,rho=1.2 m=150,rho=1.4

0

0.2

0.0

Run Length

CD

F o

f Run

Len

gth


100 200 300 400

0.4

0.6

0.8

1.0

0.0

Run Length

CD

F o

f Run

Len

gth

Run Length

CD

F o

f Run

Len

gth

0 100 200 300 400 0 100 200 300 400

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

Figure 2. Cdf of run length T under standards-given (� known) and retrospective (� estimated) scenarios for the high-side controlchart with m=150, n=5, L0 =100, and �=1, 1.2, 1.4

Figure 5 is a plot of ARL versus � for the two-sided chart with n=5, 6, 8, 10 and L0 =100. As expected, larger samples resultin faster detection of shifts. Table VI gives ARL values corresponding to this plot.

4. Application

Padgett and Spurrier4 described monitoring a manufacturing process for carbon fibers that were used in fibrous compositematerials. Samples of n=5 specimens were taken periodically and their breaking strengths, measured in giga-Pascals (GPa), were

20



m=75,rho=0.75 m=75,rho=1 m=75,rho=1.25

0

0.2

0.0

Run Length

CD

F o

f Run

Len

gth


100 200 300 400

0.4

0.6

0.8

1.0

0.0

Run Length

CD

F o

f Run

Len

gth

Run Length

CD

F o

f Run

Len

gth

0 100 200 300 400 0 100 200 300 400

0.2

0.4

0.6

0.8

1.0

0.2

0.0

0.4

0.6

0.8

1.0

Figure 3. Cdf of run length T under standards-given (� known) and retrospective (� estimated) scenarios for the two-sided controlchart with m=75, n=5, L0 =100, and �=1, 1.2, 1.4

Table II. Standardized control limits �S ×UCL for high-side Charts

n

�S ×UCL 4 5 6 7 8 9 10

L050 5.8930 6.2440 6.5180 6.7426 6.9328 7.0977 7.2432100 6.5919 6.9429 7.2169 7.4415 7.6317 7.7966 7.9420200 7.2879 7.6389 7.9129 8.1375 8.3277 8.4926 8.6380300 7.6943 8.0453 8.3193 8.5439 8.7341 8.8990 9.0444370 7.9044 8.2554 8.5294 8.7540 8.9442 9.1090 9.2545400 7.9825 8.3335 8.6074 8.8320 9.0223 9.1871 9.3326500 8.2059 8.5569 8.8309 9.0555 9.2457 9.4106 9.55601000 8.8996 9.2506 9.5246 9.7492 9.9394 10.1043 10.2497

Table III. Standardized control limits �S ×LCL for low-side charts

n

�S ×LCL 4 5 6 7 8 9 10

L050 0.6115 0.8886 1.1276 1.3342 1.5150 1.6750 1.8182100 0.4811 0.7370 0.9642 1.1640 1.3405 1.4977 1.6391200 0.3798 0.6139 0.8286 1.0208 1.1925 1.3466 1.4858300 0.3311 0.5525 0.7596 0.9471 1.1157 1.2678 1.4056370 0.3084 0.5233 0.7265 0.9115 1.0785 1.2294 1.3664400 0.3004 0.5129 0.7146 0.8986 1.0650 1.2155 1.3523500 0.2787 0.4843 0.6817 0.8631 1.0277 1.1770 1.31281000 0.2208 0.4056 0.5897 0.7624 0.9213 1.0666 1.1997

determined. Table I presents the data for 20 samples. These were from a study conducted by the U.S. Army Materials TechnologyLaboratory in Watertown, Massachusetts. Changes in the strength distribution, modeled as Weibull, were of interest because, inparticular, decreases in the percentiles of the distribution meant a deterioration of the fiber.


21


Table IV. Standardized control limits �S ×UCL and �S ×LCL for two-sided charts

n�S ×UCL�S ×LCL 4 5 6 7 8 9 10

L050 7.1429 7.3393 7.5094 7.6593 7.7932 7.9130 8.0225

0.5432 0.7948 1.0139 1.2048 1.3730 1.5224 1.6569

100 7.9017 8.0859 8.2472 8.3901 8.5186 8.6342 8.74020.4318 0.6654 0.8747 1.0600 1.2246 1.3719 1.5050

200 8.6578 8.8311 8.9838 9.1202 9.2437 9.3552 9.45740.3438 0.5586 0.7572 0.9360 1.0966 1.2413 1.3725

300 9.0978 9.2653 9.4149 9.5474 9.6679 9.7774 9.87780.3010 0.5047 0.6967 0.8714 1.0294 1.1724 1.3025

370 9.3261 9.4907 9.6369 9.7683 9.8875 9.9957 10.09510.2810 0.4790 0.6674 0.8401 0.9967 1.1387 1.2681

400 9.4110 9.5746 9.7194 9.8505 9.9688 10.0770 10.17580.2739 0.4698 0.6569 0.8287 0.9848 1.1265 1.2556

500 9.6520 9.8135 9.9566 10.0854 10.2022 10.3091 10.40810.2546 0.4444 0.6278 0.7972 0.9517 1.0924 1.2209

1000 10.4006 10.5540 10.6917 10.8150 10.9275 11.0308 11.125910.2029 0.3741 0.5458 0.7076 0.8570 0.9943 1.1203

0.5

0

ρ = σT/σS = βS/βT

AR

L

1.0 1.5 2.0 2.5

20

40

60

80

100

Figure 4. Average run length for low-side, high-side and two-sided control charts with n=5, L0 =100 and different values of the shift �

Below, standards-given and retrospective charts are constructed for the data. The first 10 samples were believed to have comefrom a stable process with �S =4.8. The last 10 samples represented a process shift with �T =2. High-side and two-sided chartsare applied to samples 11 through 20 to monitor a decrease in � (increase in �).

4.1. Control limits

Assume that an all-OK ARL of L0 =100 is desired. For the standards-given chart, assign �S = 14.8 =0.2083. Equation (4) yields

�0 =2.8768. Thus, the center line for all charts is CL= 2.87684.8 =0.5993. Using Table II, we compute the upper control limit for

the high-side chart to be UCL= 6.94294.8 =1.4464. Using Table IV, we obtain the control limits for the two-sided chart, namely,

UCL= 8.08594.8 =1.6846 and LCL= 0.6654

4.8 =0.1386.

For the retrospective approach, the unbiased estimate Sm =R / �0 of � is obtained from the first 10 samples and is used inplace of �S. Thus, Sm = 0.5301

2.8768 =0.1843, and the center line is CL=R=0.5301. The control limits are UCL=1.2793 for the high-sidechart, whereas UCL=1.4899 and LCL=0.1226 for the two-sided chart.

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Table V. Average run lengths of low-side, two-sided and high-side control charts for n=5 and L0 =100, 370, 500

�

L0 Scheme 0.25 0.5 0.75 0.9 1 1.1 1.25 1.5 2 3 4

100 Low/high-side 2 9 35 68 100 53 25 10 4 2 1Two-sided 2 12 50 88 100 88 53 21 6 2 1



n=5n=6n=8n=10

0.5

0

AR

L

1.0 1.5 2.0 2.5

20

40

60

80

100

= T/ S = S/ T

Figure 5. Average run length for two-sided control charts with n=5, 6, 8, 10 and L0 =100

Table VI. Average run lengths of two-sided charts for n=5, 6, 8, 10, L0 =100 and different values of �

�

n 0.25 0.5 0.75 0.9 1 1.1 1.5 2 3 4

5 2 12 50 88 100 88 21 6 2 16 1 9 43 85 100 86 18 5 2 18 1 5 33 80 100 82 15 4 1 110 1 4 27 76 100 79 13 3 1 1

4.2. Results

Figure 6 gives the standards-given and retrospective high-side control charts. The former gives out OOC signals on samples 13,15, and 17 (3rd, 5th, and 7th after monitoring begins). The retrospective chart signals OOC on the same samples including the11th. An increase in � or a decrease in � is detected. Recall that �=2.0 for samples 11–20. With �= 4.8

2 =2.4, (9) and (10) give anARL of 2.3 for the standards-given chart. The OOC signals of the standards-given chart are consistent with this expected value.See Table V (row 1) and Figure 4 for comparable ARLs.

Figure 7 gives the standards-given and retrospective two-sided control charts. Both charts signal OOC on the 13th and 17thsamples. Equations (9) and (12) give an ARL of 3.4 for the standards-given chart. Comparable ARLs are found in Table V (row 2)and Figures 4 and 5.

The low-side charts, not included here, do not give out OOC signals as it should not.

5. Conclusions

The Weibull shape parameter � plays an important role in practical applications. For example, in reliability and survival analysis, �determines the nature of the hazard function. Unfortunately, there have not been methods in the literature that clearly suggest


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Sample

Sam

ple

Ran

ge

CL=0.5993

UCL=1.4464

11

1.0

Standards Given

Sample

Sam

ple

Ran

ge

CL=0.5301

UCL=1.2793

0.5

Retrospective

12 13 14 15 16 17 18 19 20 11 12 13 14 15 16 17 18 19 20

1.5

2.0

1.0

1.5

2.0

Figure 6. Standards-given and retrospective high-side control charts for monitoring carbon fiber data with n=5 and L0 =370

Sample

Sam

ple

Ran

ge

11

1.0

0.5

Standards Given

Sample

Sam

ple

Ran

ge

0.5

Retrospective

12 13 14 15 16 17 18 19 20 11 12 13 14 15 16 17 18 19 20

1.5

2.0

1.0

1.5

2.0

CL=0.5993

UCL=1.6846

LCL=0.1386

CL=0.5301

UCL=1.4899

LCL=0.1226

Figure 7. Standards-given and retrospective two-side control charts for monitoring carbon fiber data with n=5 and L0 =370

how to monitor � without estimating or specifying nuisance parameters such as the Weibull scale � or without using simulations.To address these issues, we propose a method based on the range R of the logarithm of Weibull data for monitoring �. Thecontrol limits of the proposed methods depend on the sample size, the desired all-OK ARL, and the stable value of �.

Here, we studied unbiased one- and two-sided control charts with the simple OOC criterion of one-point-outside-the-control-limits. We expect improvements if additional OOC criteria are considered in addition to the control limits. For instance, the authorsof Wetherill and Brown21 discussed the inclusion of what they called ‘warning lines’ so that an OOC signal also occurred whentwo consecutive values fall between a pair of adjacent warning line and control limit. They compared ARL performance of charts

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with and without warning lines. For a one standard deviation change in the normal mean, they showed that charts with warninglines had shorter OOC ARL by 13–16 periods than those without. The use of warning lines and other alarm rules such as thosesummarized in Tables 3.5–3.7 of Reference18 should be investigated to determine effective ways of improving chart performance.

In general, we recommend using one-sided charts if specific changes, i.e. increase or decrease in �, are of practical importanceto practitioners. Also, unbiasedness and ARL properties should be considered in selecting a practical control scheme.

References1. Weibull W. A statistical theory of the strength of material. Report No. 151, Ingeniors Vetenskaps Akademiens Handligar, Stockholm, 1939.2. Weibull W. The phenomenon of rupture in solids. Report No. 153, Ingeniors Vetenskaps Akademiens Handligar, Stockholm, 1939.3. Basu B, Tiwari D, Kundu D, Prasad R. Is Weibull distribution the most appropriate statistical strength distribution for brittle materials? Ceramics

International 2009; 35:237--246.4. Padgett WJ, Spurrier JD. Shewhart-type charts for percentiles of strength distributions. Journal of Quality Technology 1990; 22:283--288.5. Zhang L, Chen G. EWMA charts for monitoring the mean of censored Weibull lifetimes. Journal of Quality Technology 2004; 36:321--328.6. Lawless JF. Statistical Models and Methods for Lifetime Data. Wiley: New York, 1982.7. Nelson W. Applied Life Data Analysis. Wiley: New York, 1982.8. Meeker WQ, Escobar L. Statistical Methods for Reliability Data. Wiley: New York, 1998.9. Nelson PR. Control charts for Weibull processes with standards given. IEEE Transactions on Reliability 1979; 28:283--288.

10. Ramalhoto MF, Morais M. Shewhart control charts for the scale parameter of a Weibull control variable with fixed and variable samplingintervals. Journal of Applied Statistics 1999; 26:129--160.

11. Nichols MD, Padgett WJ. A bootstrap control chart for Weibull percentiles. Quality and Reliability Engineering International 2006; 22:141--151.12. Johnson NL. Cumulative sum control charts and the Weibull distribution. Technometrics 1966; 8:481--491.13. Vogt VH. Zur Parameter und Prozentpunktschätzung von Lebensdauerverteilungen bei kleinem Stichprobenumfang. Metrika 1969; 14:117--131.14. Lieblein J. On the exact valuation of the variance and covariance of order statistics in samples from the extreme-value distribution. Annals

of Mathematical Statistics 1953; 24:282--287.15. Piessens R, deDoncker-Kapenga E, Uberhuber C, Kahaner D. QUADPACK: A Subroutine Package for Automatic Integration. Springer: New York,

1983.16. Montgomery D. Introduction to Statistical Quality Control. Wiley: New York, 2005.17. Pignatiello JJ, Acosta-Mejia CA, Rao BV. The performance of control charts for monitoring process dispersion. Proceedings of the 4th Industrial

Engineering Research Conference, 1995; 320--328.18. Vardeman SB, Jobe JM. Statistical Quality Assurance Methods for Engineers. Wiley: New York, 1999.19. Jensen WA, Jones-Farmer LA, Champ CW, Woodall WH. Effects of parameter estimation on control chart properties: A literature review.

Journal of Quality Technology 2006; 25:237--247.20. Chen G. The run length distributions of the R, s and s2 control charts when � is estimated. The Canadian Journal of Statistics 1998;

26:311--322.21. Wetherill GB, Brown DW. Statistical Process Control. Chapman & Hall: London, 1991.

Authors’ biographies

Francis Pascual completed his PhD under the supervision of Professor W. Q. Meeker at Iowa State University. He is an associateprofessor at Washington State University in Pullman, Washington. He received the Jack Youden Prize for the Best ExpositoryPaper and the Frank Wilcoxon Prize for the Best Practical Application Paper in Technometrics in 2000 and the ASA’s OutstandingStatistical Application Award in 2001. His research interests include planning accelerated life tests, analysis of reliability data, andquality control.

Huifang Zhang received her PhD in Economics under the supervision of Professors Jill McCluskey and Thomas Marsh fromWashington State University. She completed her Master’s degree in Statistics in 2009 under the supervision of Professor FrancisPascual at the same university. Her research interests in Statistics include econometrics and quality control.


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Monitoring the Weibull shape parameter by control charts for the sample range

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