A New Control Chart for Weibull Technological Processes

Quality Technology & Quantitative Management

Vol. 4, No. 4, pp. 553-567, 2007

QTQM © ICAQM 2007

A New Control Chart for Weibull Technological Processes Pasquale Erto and Giuliana Pallotta Department of Aerospace Engineering, University of Naples “Federico II”, Naples, Italy (Received January 2006, accepted April 2007)

______________________________________________________________________ Abstract: This work provides a new Shewhart-type control chart of the Weibull percentile. The few

alternative available charts are much less practical than the one proposed. The chart exploits existing estimators, known as Practical Bayes Estimators (PBE), able to integrate both technological and statistical information analytically, using the Bayes theorem. In this way the chart is able to improve the process control making use of any available information. The operative steps of the employed estimation procedure are fully explained. The performance of the chart is investigated carrying out a large Monte Carlo study. The use of the proposed chart is illustrated by means of a real applicative example.

Keywords: Average Run Length, bayesian estimation procedure, non-normal control charts, practical Bayes estimators, statistical process control, Weibull distribution.

______________________________________________________________________ 1. Introduction

T

oday, a technological product must ensure a specified reliability level otherwise the company image is damaged and will be restored with difficulty. Generally, the reliability targets can be achieved by means of a good design and many tests before the launch on the market. Then, experimental data must be controlled to detect a possible discrepancy between the in-service and the expected reliability, in order to adopt the necessary corrective actions immediately. Nearly always, the distributions of the reliability parameters are skewed and the normalizing effect of the X chart is not effective or impossible due the extremely small size of the available samples. Unfortunately, very few papers dealing with non-Normal populations and reliability control can be found in literature (Kanji et al. [11]; Padgett et al. [12]; Shore [14]; Zhang [20]). Moreover, reliability control based on pure classical statistical methods, carried out ignoring all the available technological knowledge, fails very often in its aim. This is mainly due to two things: (a) frequent scarcity of data (caused by the low number of the items in operation, the very high level of their reliability, etc.) makes it impossible to provide reliable “classical” (i.e. non-Bayesian) statistical estimates; (b) technicians’ lack of understanding (of the meaning of the statistical estimates) since, very often, they have not been involved in the estimation process. To overcome these difficulties we can use reliability estimators based on the application of the Bayes theorem, which has got the peculiarity of allowing one to merge technological and statistical information. In this way, the estimators provide the statistical

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estimation process with more efficiency by not wasting any information achieved by usual engineering design and process control practice. Besides, as a further consequence, they facilitate technicians’ commitment to continuous design improvement. The control chart proposed in this work uses existing Bayesian reliability estimators, known as Practical Bayes Estimators (PBE), since they were developed from an Engineers’ point of view and applied in several technological contexts during past years (Erto [1]; Erto et al. [3], [5], [6], [7]). They reduce the needed number of failure data reinforcing the statistical experimental information by incorporating the available “a priori” Engineers’ knowledge in the estimation procedure. The Bayes approach to reliability evaluation is not new for the Weibull distribution, but it is very easy only if the shape parameter is known ([16],[17]). When both shape and scale parameters are unknown, the problem is decidedly more difficult. If the Gamma and a discrete prior distribution are assumed for scale and shape parameters, respectively, the joint posterior distribution can be found in closed form [16]. The problem can also be solved if one assumes the Inverted Gamma for the scale parameter and the Uniform model for the shape parameter [17]. There are two difficulties, from a practical point of view, in these approaches: 1.

They always use priors defined on the scale parameter of the Weibull. This implies that one has a prior knowledge about it. On the contrary, an Engineer normally does not think in terms of the scale parameter, but he is usually trained to express his knowledge in terms of the less alien and more practical concept of a percentile of the Weibull distribution.

2.

Both the priors are rigid (i.e., completely specified) and so they rarely correspond exactly to the prior knowledge. Then, they often include other information that is not prior knowledge.

So, for these reasons, the PBE incorporate directly into the estimation process the prior belief about a Weibull percentile, for an arbitrary but fixed reliability level, (Section 4.3 and Section 4.5 in Section 4) and does not explicitly specify the shape parameter of the Inverse Weibull prior distribution (Section 4.4 in Section 4).

2. Practical and Modified-Practical Bayes Estimators The Practical Bayes Estimators (PBE), introduced for the Weibull model in [1], first were used in technological applications in [5], [6], then were formulated for the Inverse Weibull model in [7] and applied to accelerated tests too in [4]. The PBE incorporate Engineers’ information directly (i.e., without any unnatural manipulation and/or addition) into the reliability estimation process. Technically the PBE approach can be seen as a compromise between the Classical Bayes approach and the Empirical one. The former is based on the use of completely specified prior distributions, the latter uses prior distributions fitted to past and independent experimental observations in a context of a two-stage sampling plan. Instead the PBE approach requires only the specification of those parameters, of the prior distributions, in which the Engineers’ prior knowledge can really be converted, leaving the remaining parameters unspecified. The choice of the parameters of the priors of these estimators has to comply with the inequality (4) that produced a critical comment [9] and, consequently, an improved formulation of the priors [3]. However, since the above restriction does not seriously affect the use of the PBE, this paper refers to their original formulation in Erto [1], which is the simplest one, leaving the Modified Practical Bayes Estimators (MPBE) [3] to more

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experienced users and/or to very critical situations. Acronyms & Notation

Sf {} ⋅

survival function

pdf {} ⋅

probability density function

Γ ( ⋅)

Gamma function

R

reliability level

K

constant equal to ln (1 R )

xR

equivalent to the 1 − R percentile of the Weibull distribution such as that Sf { x R } = R

E {x R }

anticipated (mean) value given by prior knowledge for x R

x

data set

α

tail area of a pdf

δ,β

scale and shape parameters of the Weibull distribution

γ

parameter of the standard Gamma pdf

a, b

scale and shape parameters of the Inverse Weibull prior distribution of x R

N

size of the whole data set

n

size of the data subset (i.e.: subgroup)

β1 , β 2

prior numerical interval for β

∧

implies an estimate

PBE

Practical Bayes Estimators (or Estimates)

L

Likelihood

z

standard Gamma random variable

3. The Problem and the Bayes Approach The Weibull distribution is well known and widely used as a reliability model, since it possesses many convenient properties: 1.

it is characterized by two parameters (very seldom a third parameter must be considered);

2.

it assumes a wide range of shapes, depending on the values of the parameters;

3.

the meaning of its parameters is clear;

4.

the values of its shape parameter is closely related to the nature of the involved mechanism of failure;

5.

a simple likelihood function is associated to its samples.

The Weibull survival function is: β Sf { x ; δ , β } = exp ⎡ − ( x δ ) ⎤ ; ⎣ ⎦

x ≥ 0;

δ , β > 0,

(1)

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that can be immediately reparameterized in terms of the percentile x R and shape parameter, β , in which the Engineers’ knowledge can be more easily converted: β Sf { x ; x R , β } = exp ⎡ − K ( x x R ) ⎤ , x ≥ 0, x R , β > 0, K = ln (1 R ) , ⎣ ⎦

(2)

x R and β both being unknown. As an example of, we can refer to life tests where x represents the time to failure: if R = 0.90 and x R = 1,000 hours, then 90% of the items have lives greater than 1,000 hours. Generally speaking, a reliability test on n items results in a sample of n experimental data. If n is very small, say 3, but various difficulties (item cost, time, etc.) prevent one from having other applicable observations, then it is convenient to use other sources of information ([13];[15]). On the other hand, in engineering, very often some knowledge exists about the mechanism of failure under consideration, which can be converted into quantitative form about β . Moreover, an engineer presumably knows more than the simple order of magnitude of the reliability performance which the designed item has, e.g., he has a quite precise knowledge about an x R . Then, with both these pieces of information, he can formulate a numerical interval ( β1 , β 2 ) for β and an anticipated value for x R . The PBE allow combining this prior knowledge about β and x R with a few experimental data to give very good Weibull parameter estimates.

4. Technological Knowledge and Prior Distributions In order to evaluate the PBE, of the Weibull parameters, the following elementary steps must be performed. These are also useful to highlight all the assumptions on which the resulting estimates are based. 4.1. Assumption of the Prior Probability Density Function for the Shape Parameter β

The uniform prior probability density function in the interval ( β1 , β 2 ) is assumed to fit the degree of belief in the shape parameter β of the sampling distribution:

⎧1 ( β 2 − β1 ) ; β 2 ≥ β ≥ β1 > 0; pdf{β } = ⎨ ⎩0; elsewhere

β 2 > β1

,

(3)

it appears to be as non-restrictive as feasible. 4.2. Formulation of the Prior Information on β

The prior knowledge for β is converted into the values β1 and β 2 , of the prior (3), using well known relationships between the mechanism of failure and the value of β (e.g., with reference to life tests, early failures imply β < 1; chance failures imply β = 1; wear out failures imply β > 1 ). These two parameters can be effectively and easily anticipated since, as already noted in (Erto [1]), the engineer’s information, about the mechanism of failure, can always be expressed in terms of an interval ( β1 , β 2 ). This interval must be chosen wide enough in order to plausibly contain the unknown (true) value of the Weibull shape parameter. The only restriction, already mentioned in Section 2, which the anticipated values

β1 and β 2 must be subjected to, is the following one: β1 + β 2 > 2,

(4)

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since their sum will be used to set up the argument of the Gamma function in (8). 4.3. Assumption of the Prior Probability Density Function for the Percentile x R

For the selected percentile x R (corresponding to the fixed reliability level R ) the prior probability density function is assumed to be the Inverse Weibull ([2], [10]):

pdf{x R } = a b (a x R ) − ( b +1) exp ⎡⎣ −(a x R ) −b ⎤⎦ ;

x R ≥ 0;

a, b > 0,

(5)

where a and b are scale and shape parameters respectively. This distribution fits well with the prior knowledge about x R and is tractable. It has one mode and tends to be bell shaped (its variance decreasing) for increasing values of b. So, as example, for a strong prior knowledge about x R a high value of b must be chosen. 4.4. Assuming the Equality of the Prior and Sampling Shape Parameters

It is assumed b = β . Generally, the greater β is, the more peaked the Weibull probability density function is, the smaller the uncertainty in x R is and then greater b must be. Therefore, b = β is the simplest choice. As a consequence, b remains unspecified in the prior (5). This assumption overcomes many practical difficulties, freeing the engineers from the obligation to choose a fixed value, say b = 3, which often includes other information that is not prior knowledge about x R . 4.5. Expression of the Prior Belief about a Percentile x R

The prior information for x R must be converted only into the mean value E {x R } of the probability density function (5) which is: E { x R } = (1 a ) Γ(1 − 1 b ).

(6)

As a consequence of the assumptions (in Section 4.3 and Section 4.4), the probability density function of x R (5) is converted into the conditional prior:

pdf { x R β } = a β (a x R ) − ( β +1) exp ⎡⎣ −(a x R ) − β ⎤⎦ ;

a, β > 0,

(7)

with fixed (anticipated) mean E { x R }. 4.6. Evaluation of the Prior Scale Parameter a

Having assumed b = β , the parameters of the (7) to be anticipated are reduced to a only. Moreover, in order to simplify the procedure, in the Equation (6) (and only in this one) β can be substituted by its prior mean, obtaining an effective anticipated value for the prior parameter a automatically and without employing any further information: a=

Γ (1 − 1 β m ) E {x R }

;

β m = ( β1 + β 2 ) 2.

(8)

4.7. Some Comments

Using prior technological knowledge, in a statistical estimation procedure, improves

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efficiency by not wasting important information, since it immediately indicates the statistical method where reasonable estimates are located. Employing a particular probability density function for formulating the prior technological knowledge, about a statistical parameter, describes the “strength” and the “shape” of the advanced information. This improves the efficiency further, since it tells the statistical method how the belief is distributed on the range of reasonable estimates. The PBE use the prior probability density functions only for describing the prior information, without supposing necessarily that the corresponding parameters (of the sampling distributions) vary effectively from one item to another. It is supposed that true and fixed values of the unknown parameters really exist (however, the validity of the PBE holds even if the stochastic nature of the reliability parameters is assumed). According to this hypothesis, to investigate the bias and efficiency of the estimators, many Monte Carlo simulation studies were carried out ([3], [7], [1]). The effects of poor priors, (i.e., priors not centered on the true value of the corresponding unknown parameters), were fully investigated, always obtaining good results.

5. Reliability Tests and Bayes Theorem Usually, a sample array x of n experimental data is available. If the reliability (measured in terms of lifetime, tensile strength, breaking strength, etc.) of the items is characterized by the model (2), the likelihood of the sample is given by: n

⎛ β ⎞ n ⎛ K n L ( x x R , β ) ∝ ⎜ β ⎟ ∏ x iβ −1 exp ⎜ − β ∑ x iβ ⎝ x R ⎠ i =1 ⎝ x R i =1

⎞ ⎟. ⎠

(9)

Multiplying the two priors (3) and (7), the joint prior probability density function of x R and β is obtained: −1 pdf{x R , β } = ( β 2 − β1 ) a β (a x R ) − ( β +1) exp ⎡⎣ −(a x R ) − β ⎤⎦ .

(10)

So, we can use the Bayes theorem that substantially says:

⎛ joint posterior probability density ⎞ ⎜ ⎟ of unknown parameters ⎝ ⎠

∝

⎛ their joint prior ⎞ ⎜ ⎟ ⎝ probability density ⎠

⎞ × ⎛⎜⎝ likelihood ⎟, function ⎠

“Prior” and “posterior” mean before and after obtaining experimental data respectively. So, in this way, the theorem fuses the technological prior knowledge, summarized into joint prior, with all the information (data and shape of the reliability model) included into likelihood. More rigorously, from Bayes theorem this joint posterior of the two parameters to be estimated follows: pdf {x R , β } L ( x x R , β ) pdf {x R , β x } = pdf{x } =

pdf{x R , β } L ( x x R , β ) β ∞ ∫β12 ∫0 pdf{x R , β } L

(x

x R , β ) dx R d β

.

(11)

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Then, using (9), (10) and performing the first integration, the following joint posterior probability density is obtained [1]: ⎡ ⎣

n

pdf { x R , β x } =

⎤ ⎠⎦

β n +1 a − β x R − β ( n +1) −1 ∏ x iβ −1 exp ⎢ − x R − β ⎜⎛ a − β + K ∑ x iβ ⎟⎞ ⎥ i =1

n ! ∫ββ12

n −β

β a

n

β −1

∏ xi

i =1

n

⎝

i =1

⎛ a−β + K n x β ⎞ ∑ i ⎟ ⎜ i =1 ⎝ ⎠

− ( n +1)

.

(12)

dβ

We can say that this density function describes the residual uncertainty which exists about the two parameters. So we could estimate the parameters x R and β adopting their modal or median or mean values. The PBE choose the last ones, that is, the expectations of x R and β : I I E {x R x } = 3 ; E {β x } = 2 , (13) I1 I1 where n n m I j = ∫ββ12 β j a − β ∏ x iβ −1 ⎛⎜ a − β + K ∑ x iβ ⎞⎟ i =1 i =1 ⎝ ⎠

− ( n +1) + k j

(

)

Γ n +1− kj dβ

j = 1, 2, 3,

(14)

with the following values for the parameters m j and k j : m1 ≡ m3 = n; m2 = n + 1; k1 ≡ k2 = 0; k3 = 1 β .

(15)

It is interesting to note (see Appendix) that the bivariate random variable ( x R , β ) can be transformed into a standard Gamma one, using the transformation: n z = x R− β ⎛⎜ a − β + K ∑ x iβ ⎞⎟ . i =1 ⎝ ⎠

(16)

6. A Shewhart-Type Control Chart of the Weibull Percentiles Following the Shewhart approach, the Center Line (CL) of the x R control chart can be calculated on the basis of all the available data (i.e., the whole data set x ) using the first estimator (13). In this way, we can obtain a constant and robust Center Line. For each subgroup of size n, the same estimator (13) is used to calculate the current point value xˆ R . Similarly, on the basis of the whole data set x , the second estimator (13) is used to calculate a robust β estimate βˆ . This estimate is inserted in the inverse of the transformation (16):

xR = z

−

1

βˆ

1

⎛ a − βˆ + K n x β ⎞ βˆ , ∑ i ⎟ ⎜ i =1 ⎝ ⎠

(17)

in order to obtain a good estimate of the percentiles of the pdf{ x R x } as simple transformations of the percentiles of the standard Gamma (see Appendix). These percentiles are the needed control limits of the x R control chart, given a false alarm risk α. Besides, in practice, computing the control limits by inserting in the Equation (17) a ˆ ˆ more robust value Nn ⋅ ∑ iN=1 x iβ in place of ∑ in=1 x iβ was found more effective. In this way, the same control limits can be used for all the subgroups.

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In all the analysed (both real and simulated) applicative cases, the above practical choices lead to very satisfactory results. In order to illustrate the new chart operatively, the following real applicative example proposed in [12] is used. The control of the first percentile (corresponding to a reliability level R = 0.99 ) of the breaking strength distribution of a certain type of carbon fibers is considered. This kind of carbon fiber is produced to manufacture composite materials which need fibers with a breaking strength greater than 1.22 GPa (Giga-Pascal) with 99% probability. In order to control that this specification ( x R = 1.22, R = 0.99) is met, a sample of n = 5 fibers, each 50 mm long, is selected from the manufacturing process periodically, and the breaking stress of each fiber is measured. When the process was certainly in-control, ten samples of size n = 5 of breaking stress of these carbon fibers were sampled (see Table 1, [12]). On the basis of some past knowledge and experience, the Weibull distribution is considered to closely fit such breaking strength and is assumed to be the underlying distribution of the control chart. Table 1. Breaking stresses (GPa) of carbon fibers (in control state, x R = 1.22, R = 0.99). j-th sample Stress 1 3.70 2.74 2.73 2.50 3.60 2 3.11 3.27 2.87 1.47 3.11 3 4.42 2.41 3.19 3.22 1.69 4 3.28 3.09 1.87 3.15 4.90 5 3.75 2.43 2.95 2.97 3.39 6 2.96 2.53 2.67 2.93 3.22 7 3.39 2.81 4.20 3.33 2.55 8 3.31 3.31 2.85 2.56 3.56 9 3.15 2.35 2.55 2.59 2.38 10 2.81 2.77 2.17 2.83 1.92

Figure 1. Weibull control chart of the first percentile ( x R , R = 0.99) for the in-control process data (Table 1) with n = 5 .

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Figure 1 shows the control chart of the first percentile estimated by means of the PBE. The chart is drawn using the data of Table 1. On the basis of the information provided in [12], the prior interval (2.8, 6.8) for β and the anticipated (mean) value 1.22 for x R are adopted. Applying the estimator (13) to the whole data set (50 data) the estimates βˆ = 4.69 and xˆ R = 1.20 are obtained. So, the Center Line (CL) is xˆ R = 1.20 ; the Upper Control Limits (UCL) and the Lower Control Limits (LCL), for both α = 5% and α = 0.1%, are calculated transforming the corresponding percentiles of the standard Gamma on the basis of the properties in the beginning of this Section and in the Appendix. Using the same prior interval ( 2.6,6.8 ) for β and the anticipated (mean) value 1.22 for x R , the ordinates of the ten points (i.e. the estimates of the breaking strength) from all the ten samples of size n = 5 are calculated. Table 2. Breaking stresses (GPa) of carbon fibers (out-of-control state, x R = 0.26, R = 0.99). j-th sample Stress 1 1.41 3.68 2.97 1.36 0.98 2 2.76 4.91 3.68 1.84 1.59 3 3.19 1.57 0.81 5.56 1.73 4 1.59 2.00 1.22 1.12 1.71 5 2.17 1.17 5.08 2.48 1.18 6 3.51 2.17 1.69 1.25 4.38 7 1.84 0.39 3.68 2.48 0.85 8 1.61 2.79 4.70 2.03 1.80 9 1.57 1.08 2.03 1.61 2.12 10 1.89 2.88 2.82 2.05 3.65

Figure 2. Weibull control chart of the first percentile ( x R , R = 0.99 ) for the out-of-control process data (Table 2) with n = 5 .

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Differently from previous samples, the ten samples reported in Table 2 [12] are representatives of an out-of-control state, characterized by the first percentile shifted to 0.26 GPa from the original value of 1.22 GPa. Then, the above chart is applied to these data. Again, the prior interval ( 2.6, 6.8 ) for β and the anticipated (mean) value 1.22 for x R are used in order to calculate the estimates from all the ten samples of size n = 5 (Figure 2). Finally, the chart is applied to the same data (Table 2) collected in subgroups of both size n = 3 (Figure 3) and size n = 1 (Figure 4). Always, the prior interval (2.8, 6.8) for β and the anticipated (mean) value 1.22 for x R are used.

Figure 3. Weibull control chart of the first percentile ( x R , R = 0.99 ) for the out-of-control process data (Table 2) with n = 3 .

Figure 4. Weibull control chart of the first percentile ( x R , R = 0.99 ) for the out-of-control process data (Table 2) with n = 1 .

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In all cases, the diagnostic property of the chart seems to be good. Even when the subgroups are very small and the process is out-of-control (Figure 2, Figure 3 and Figure 4) the PBE can still be used as a simple tool to update the expected value for x R (i.e. the Center Line). In general, the PBE work as a filter that always adjust — in mean — the anticipated value for x R if changed, or substantially confirms if not. Table 3 provides the few “Mathematica” commands needed to calculate the whole sequence of the 10 percentile estimates, reported on the control chart. Table 3. “Mathematica” commands used to obtain the 10 percentile estimates of x R , ( N = 50; n = 5 ). (* Conversion of prior knowledge into prior parameters* ) b=(β 1+ β 2)/2; a=N[1/xR Gamma[1 − 1/b] ]; (* Posterior expectations of N/n values of X R *) n

Ig1=Table[NIntegrate[ β a n

Ig3=Table[NIntegrate[β a

−β

−β

( (

j

∏

x[ [i ] ]

β −1

i=j − n+1 j

∏

x[ [i ] ]

i=j − n+1

β −1

)( )(

a

−β

j

+ Log[1/R] ∑

x[ [i] ]

β

i=j − n+1

a

−β

j

+ Log[1/R] ∑

x[ [i] ]

i=j − n+1

β

) )

− (n+1)

Gamma[n+1],{ β ,β 1 ,β 2 }],{j,n,N,n}]; − (n+1)+1/ β

Gamma[n+1 − 1/ β ],{ β , β 1 ,β 2 }],{j,n,N,n}];

Xr2=Table[Ig3[ [i] ]/Ig1[ [i] ],{i,1,N/n}]

7. Performance of the Weibull Control Chart In order to evaluate the performance of the Weibull control chart, its ARL (Average Run Length) is investigated, using 99.73% control limits for the first percentile (equivalent to x R , R = 0.99 .). The ARL is explored by means of a large Monte Carlo study, setting a continuous shift of x R from its target value. Analytically, the ARL is computed as the reciprocal of the probability of signal, estimated as the sum of the out-of-control tail areas of the posterior probability density function of the shifted x R . A large number of Weibull samples are generated to compute the control limits at first, using the prior interval ( 2.6,6.8 ) for β and the anticipated (mean) value 1.22 for x R . Then an out-of-control state is simulated, setting a shift of x R and the probability of signal is calculated from 10,000 samples with a fixed sample size n. Following the choice of the classical approach, the variance of the Weibull distribution is left unchanged. In this way, usually a change of both scale and shape parameters follows for any percentile shift. Figure 5 presents ARL curves for sample sizes n = 1, 2, 3, 4, 5. The ARL is plotted versus the ratio Δx R /σ of the percentile shift to standard deviation. As we can see, the ARL trend highlights the conservative responsiveness of the chart: •

facing a likely moderate process improvement, (equivalent to Δx R σ > 0 ), the chart reacts by means of a (conservative) increased ARL value. Only when the percentile shift reaches a considerably higher level, the chart quickly signals the out-of-control-state. This “virtuous” ARL trend is not new since it was already found in a different context ([18], [19]);

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•

facing a likely process deterioration, (equivalent to Δx R σ < 0 ), the chart reacts by means of a prompt and significant decreased ARL value, to quickly signal that the process is getting worse.

Figure 5. Performance evaluation: ARL of the Weibull control chart (Figure 1 and Figure 2) under normalized percentile shift.

Figure 6. Performance evaluation: ARL of the Weibull control chart (Figure 1 and Figure 2) under scale parameter shift. Sometimes a process change affects the scale parameter only, being the shape parameter closely linked to the unchanged failure mechanism of the process. Consequently,

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the ARL investigation is extended to cover the case in which only the scale parameter varies, resulting in a simultaneous shift of both mean and standard deviation. The prior interval (2.8, 6.8) for β and the anticipated (mean) value 1.22 for x R are used again. Figure 6 presents the obtained ARL curves for sample sizes n = 1, 2, 3, 4, 5. The ARL is plotted versus the ratio Δx R /σ of the percentile shift to the initial standard deviation. As we can see, the ARL behaves the same as before (Figure 5). The confirmed robustness property fully exemplifies the idea imbedded in the PBE: the chart does not merely hear what data suggest but processes them in order to provide the analyst with a reliable decision tool, whenever scale and/or shape parameters actually change.

8. Concluding Remarks When the experimental data are very few (say 2, 3) the Practical Bayes Estimators (PBE) of the 2-parameter Weibull reliability model can still be used effectively. In this case the controversy about whether to use Bayesian or non-Bayesian methods appears surmounted since alternative classical estimators, like the Maximum Likelihood ones, give estimates that are very often worse than and/or in contrast with elementary technological knowledge. This work has shown that the PBE are able to provide a new approach to Shewhart-type chart for Weibull percentiles, where very few alternative methods are available. Due to the specific characteristics of the PBE, the approach can be also used for individual control charts, using data which are not collected in subgroups. On the basis of some applicative examples and preliminary simulation studies the chart seems to work satisfactorily, showing a peculiar ARL curve. Obviously, further and deeper research about its statistical properties is needed.

Acknowledgements We have to thank the Referee who gave us valuable suggestions and deep comments that induced us to change the manuscript very significantly. The work was supported by the Italian PRIN Grant (‘Statistical design of “continuous” product innovation’).”

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10. Johnson, L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd edition. Vol. 1, 662, 693, Wiley, New York. 11. Kanji, G. K. and Arif, O. H. (2001). Median rankit control chart for Weibull distribution. Total Quality Management, 12(5), 629-642. 12. Padgett, W. J. and Spurrier, J. D. (1990). Shewhart-type charts for percentiles of strength distributions. Journal of Quality Technology, 22(4), 283-288. 13. Shooman, M. L. and Sinkar, S. (1977) Generation of reliability and safety data by analysis of expert opinion. Proceeding of 1977 IEEE-ARMS, 186-193. 14. Shore, H. (2004). Non-normal populations in quality applications: a revisited perspective. Quality and Reliability Engineering International, 20, 375-382. 15. Singpurwalla, N. D. and Song, M. S. (1988). Reliability analysis using Weibull lifetime data and expert opinion. IEEE Transactions on Reliability, 37(3), 340-347. 16. Soland, R. M. (1969). Bayesian analysis of the Weibull process with unknown scale and shape parameters. IEEE Transactions on Reliability, 18(4), 181-184. 17. Tsokos, C. P. (1972). A Bayesian approach to reliability theory and simulation. Proc. 1972 IEEE-ARMS, 78-87. 18. Vermaat, M. B. and Does, R. J. M. M. (2006). A Semi-Bayesian method for Shewhart individual control charts. Quality Technology and Quantitative Management, 3(1), 111-125. 19. Vermaat, M. B., Ion, R. A., Does, R. J. M. M. and Klaassen, C. A. J. (2003). A comparison of Shewhart individuals control charts based on normal, non-parametric and extreme value theory. Quality and Reliability Engineering International, 19, 337-353. 20. Zhang, L. and Chen, G. (2004). EWMA charts for monitoring the mean of censored Weibull lifetimes. Journal of Quality Technology, 36(3), 321-328.

Appendix The Standard Gamma Transformation

In order to derive the distribution of (16), we apply the change-of-variable technique with the companion transformation β = β . In this way, the Jacobian of the transformation is: ∂x R ∂z J= ∂β ∂z

∂x R 1 ⎛1 ⎞ −⎜ +1 ⎟ n ∂β 1 ⎛ −β β ⎞β = − ⋅ ⎜ a + K ∑ xi ⎟ z ⎝ β ⎠ ∂β β ⎝ i =1 ⎠ ∂β

(18)

A New Control Chart for Weibull Technological Processes

567

and, from the joint probability density function of x R and β (12), we obtain the joint probability density function of z and β :

{

}

pdf { z , β x } = pdf x R ( z , β ) , β x ⋅ J

β n a − β ∏ x iβ −1 ⎜⎛ a − β + K ∑ x iβ ⎟⎞ n

=

n

⎝

i =1

n ! ∫ββ12

n −β

β a

n

β −1

∏ xi

i =1

⎠

i =1

−( n +1)

z n exp ( − z )

⎛ a−β + K n x β ⎞ ∑ i ⎟ ⎜ i =1 ⎝ ⎠

− ( n +1)

.

(19)

dβ

Integrating the (19) with respect to β , the standard Gamma follows: pdf { z x } = ∫ββ12 pdf {z , β x } ⋅ d β =

z n exp ( − z ) n!

=

z γ −1 exp( − z ); z ≥ 0; γ = n + 1 . Γ(γ )

(20)

When β is a constant value, e.g. an estimate βˆ , the same result still stands, being pdf z | x , βˆ equal to (20).

{

}

Authors’ Biographies: Pasquale Erto is Professor of Statistics and Probability at the University of Naples ‘Federico II’. He is member of the International Statistical Institute, the IEEE Reliability Society and fellow of the Royal Statistical Society. He was Associate Professor of Reliability and Quality Control at the University of Naples until 1990. He has worked with major industries as a statistical consultant on engineering and scientific applications involving reliability analysis, maintenance and quality. He has published two books and more than 90 papers in journals that include IEEE Transactions on Reliability, Journal of Applied Statistics, Quality and Reliability Engineering International, Reliability Engineering and System Safety and Total Quality Management. Giuliana Pallotta is a PhD student in Total Quality Management at the University of Naples ‘Federico II’. In 2005 she graduated in Industrial Engineering at the University of Naples. Her research interests are in the areas of Statistical Process Control and Bayesian inference. Her current works focus on the opportunities provided by a statistical integrated approach aimed at continuous quality improvement.