ON THE PERFORMANCE OF COMBINED EWMA ...

ON THE PERFORMANCE OF COMBINED EWMA SCHEMES FOR µ AND σ : A MARKOVIAN APPROACH Manuel Cabral Morais

António Pacheco

Departamento de Matemática and Centro de Matemática Aplicada, Instituto Superior Técnico, Av. Rovisco Pais, 1096 Lisboa Codex, PORTUGAL E-mails: [email protected]

[email protected]

Key Words and Phrases: two-dimensional Markov chain, average run length, run length percentage points, probability of a misleading signal.

ABSTRACT Changes in the process mean (µ ) or in the process standard deviation (σ ) ought to be regarded as an indication that a production process is out of control. This paper considers the problem of the joint monitoring of these two parameters — when the quality characteristic follows a normal distribution —, using a combined Exponentially Weighted Moving Average (CEWMA) scheme. Three performance measures of this joint control scheme are investigated under shifts in the process mean or inflations of the process standard deviation, and under the adoption of head starts: the average run length, the run length percentage points and the probability of a misleading signal. Approximations to these three performance indicators will be obtained considering a two-dimensional Markov chain. The independence between the horizontal and vertical transitions of this approximating two-dimensional Markov chain plays an important role in providing simple expressions to those performance measures which avoid the computation of a probability transition matrix with unusual dimensions. A numerical comparison between these three performance measures and the corresponding ones of the matched combined Shewhart (CShewhart ) scheme ( X, S 2 ) will be also presented, leading to the conclusion that the substituition of this combined scheme by the CEWMA scheme can improve the joint monitoring of the process mean and standard deviation.

1

1. Introduction Control charts are widely used as process monitoring tools, primarily to detect changes in the process mean or in its standard deviation which can indicate a deterioration in quality. The joint monitoring of these two parameters can be achieved by running what is called a combined or joint scheme. Combined schemes for µ and σ — when the quality characteristic has a normal distribution — have received a great deal of attention in the quality control literature. The several combined schemes that have been proposed and studied can be divided in two broad and distinct categories: the combined schemes which make use of one control chart for an univariate summary statistic (Chengalur et al. (1989), Domangue and Patch (1991)) or a bivariate summary statistic (Takahashi (1989)); and the popular combined schemes that result from running simultaneously two control charts — a chart for µ and another one for σ (Crowder (1987), Saniga (1989), Gan (1989, 1995), St. John and Bragg (1991), Morais (1998)). The combined schemes in this last category trigger an out-of-control signal at the t th sampling period if either individual chart signals at that sampling period. The average run length ( ARL) — which is the average number of sampling periods before an out-of-control signal is given by the control scheme — has been used for some time to describe the likely performance of a control procedure: the ARL is supposed to be large, when the production process is stable or in control, and small, otherwise. Nevertheless some authors have argued that the percentage points of the run length ( RL ) provide a second and more appropriate performance measure when Shewhart control charts are not used. In addition, St. John and Bragg (1991) pointed out a serious problem in the joint monitoring of µ and σ : the existence of misleading signals that can send the user of the combined scheme in the wrong direction in attempt to diagnose and correct the cause of the out-ofcontrol signal. This suggests the use of a third performance measure, the probability of a misleading signal ( PMS ) . Primary interest is usually in detecting changes in the process mean and only inflations in the process standard deviation, therefore only the joint use of standard EWMA ( Shewhart ) charts for µ and upper one-sided EWMA ( Shewhart ) charts for σ is considered in this paper. The performances of such combined schemes are studied — with special emphasis on the combined EWMA scheme — and compared in this paper for various settings of the shifts in the nominal level of the

2

process mean and standard deviation, and for several head starts given to the combined EWMA scheme. Three performance measures — ARL's, RL percentage points, and PMS — are considered and approximated, in the case of the combined EWMA scheme, using the Markovian approach instead of the integral equation approach (used by Gan (1995) to provide an approximation to the ARL). Simple guidelines, based on all three performance measures, are provided for the design of combined EWMA schemes. All the numerical results were obtained running several programs for the package Mathematica (Wolfram (1996)). Larger tables with values of all the performance measures considered here are available from the authors on request.

2. Description and an Example of the Combined EWMA Scheme Suppose one wishes to control the process mean and standard deviation — respectively, at known nominal values µ 0 and σ 0 — when the quality characteristic, denoted here by the random variable X , has a distribution belonging to the parametric model Normal(µ , σ 2 ), − ∞ < µ < +∞, σ 2 > 0 . It is important to notice that, since µ 0 and σ 0 are assumed to be known, there is no need to collect k

{

}

samples of size n to estimate the nominal values for µ and σ and the control limits. A change in the process production can obviously be represented by a change in the parameters of the quality characteristic distribution: the change in µ will be represented in terms of the nominal value of the sample mean standard deviation

δ = n (µ − µ 0 ) σ 0 ( −∞ < δ < +∞) ,

(1)

and the inflation of the process standard deviation will be measured by

θ = σ σ 0 (θ ≥ 1) .

(2)

It is obvious that in control (δ , θ ) = (0,1) and out-of-control (δ , θ ) takes a constant value (assumed to be known) in ( −∞, +∞) × [1, +∞) \ (0,1) .

The combined EWMA (CEWMA) control scheme considered in this paper uses two separate charts: a standard EWMA control chart for µ and an upper onesided EWMA control chart for σ . The process is deemed out-of-control at time t if the observation of the EWMA statistic for µ ,

3

wµ , 0 , t = 0 Wµ , t =   1 − λ µ × Wµ , t −1 + λ µ × Xt , t = 1,2,..., is beyond the region CE − µ = LCLE − µ ,UCLE − µ = µ 0 ± γ E − µ × σ 0 λ µ 

(

)

[

]

(3)

[(2 − λ )n]  , µ

(4)

or the observed value of the EWMA statistic for σ , wσ , 0 , t = 0 Wσ , t =  max ln(σ 02 ), (1 − λ σ ) × Wσ , t −1 + λ σ × ln( St2 ) , t = 1,2,...,  does not belong to

{

}

 λ σ ψ ′[(n − 1) 2]  CE − σ = [ LCLE − σ ,UCLE − σ ] = ln(σ 02 ),ln(σ 02 ) + γ E − σ ×  2 − λσ  

(5)

(6)

where n is the sample size, Xt = ∑ i =1 Xit n, St2 = ∑ i =1 ( Xit − Xt ) (n − 1) and ψ ′ represent the t th random sample mean and variance and the trigamma function, n

n

2

respectively. It will be shown later (in particular in section 7) that the constants in the expressions of the summary statistics and of the control limits are chosen by the user to produce a desired performance for the combined control scheme. Example 1: The temperature of a chemical reactant is a crucial factor in obtaining satisfactory yield from a chemical process. The nominal value for the mean and the standard deviation of the chemical reactant temperature are µ 0 = 100 o C and

σ 0 = 1o C , respectively. Suppose groups of five temperatures of the reactant are recorded every hour for ten consecutive hours with the process with a mean (standard deviation) level that was initially off target and equal to µ = 100.5o C (σ = 1.1o C). The simulated temperatures are shown in Table I, along with the observed values of: the random sample mean and variance, Xt and St2 (the Shewhart summary statistics); and of the EWMA summary statistics Wµ , t and Wσ , t , considering λ µ = λ σ = 0.05, wµ , 0 = µ 0 and wσ , 0 = ln(σ 02 ). The CEWMA scheme in Figure 1 is run considering γ E − µ = 2.624 , CE − µ = [99.812,100.188], γ E − σ = 1.27 and CE − σ = [0,0.163]. This combined scheme produced an out-of-control signal by the 5th observation since the UCL line of the EWMA control chart for µ was crossed. Note that the upper one-sided EWMA control chart for σ did not give out-of-control signal within the first 10 observations.

4

2

TABLE I. Temperatures and values of the xt , st , wµ , t and wσ , t N 1 2 3 4 5 6 7 8 9 10

x1N 100.6 101.0 101.5 101.7 103.6 101.0 101.8 99.9 103.6 103.0

x2 N 101.0 100.3 103.6 101.1 101.2 100.9 100.5 100.9 99.3 98.6

x3 N 99.7 101.1 98.7 100.5 101.3 100.9 98.8 103.4 101.0 99.7

x4 N 102.1 99.6 99.5 100.9 100.7 101.7 100.8 100.1 100.5 98.4

x5 N 101.4 102.3 100.7 99.0 102.4 101.7 100.5 101.5 101.2 101.4

xt 100.96 100.86 100.80 100.64 101.84 101.24 100.48 101.16 101.12 100.22

2

st 0.803 1.013 3.610 1.028 1.353 0.178 1.167 1.978 2.467 3.832

wµ , t

wσ , t

100.048 100.089 100.124 100.150 100.234 100.285 100.295 100.338 100.377 100.369

0 0.001 0.065 0.063 0.075 0 0.008 0.041 0.085 0.147

Before making any considerations about the performance of the CEWMA scheme, it should be added that the individual charts for µ and σ of the combined Shewhart (CShewhart ) scheme use the summary statistics Xt and St2 (respectively), and the control limits LCLS − µ = µ 0 − γ S − µ × σ 0 n , UCLS − µ = µ 0 + γ S − µ × σ 0 n , LCLS − µ = 0 and UCLS − σ = γ S − σ × σ 02 (n − 1) (see for instance Chengalur et al., 1989).

3. The Run Length Survival Function A change in the process production must be detected quickly so that a corrective action can be taken. Thus, it comes as no surprise that the run length ( RL ) — which is the number of sampling periods before an out-of-control signal is given by the control procedure — is usually used to describe the performance of a control scheme. An approximation to this random variable distribution, when a CEWMA scheme is used, can be obtained using two Markov chains with h and v states in the Markovian approach as described in Appendix A. Let RLiE − µ (δ , θ ) , RLEj − σ (θ ) and RLi,CEj (δ , θ ) denote the run length of the standard EWMA chart for µ , the upper one-sided EWMA for σ and the CEWMA scheme, respectively — conditional to (δ , θ ) , and to the fact that the initial value of the summary statistics wµ , 0 and wσ , 0 are associated to the states Eµ , i and Eσ , j of the Markov chains. Note that, if i ≠ [h 2] ( j > 1), where [ x ] denotes the integer part of x , a head start (Lucas and Crosier, 1982, Lucas and Saccucci, 1990) has

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been given to the EWMA chart for µ (the upper one-sided EWMA for σ ). A signal is given by the CEWMA scheme whenever an out-of-control signal is observed on either individual chart. Following Gan (1995) (who, as mentioned before, proposed the use of a CEWMA scheme which comprises individual charts for µ and σ without head starts) and Woodall and Ncube (1985) (who considered the simultaneous use of m univariate CUSUM charts for m expected values), the run length of the CEWMA scheme studied here is given by

{

}

RLi,CEj (δ , θ ) = min RLiE − µ (δ , θ ), RLEj − σ (θ ) .

(7)

Then, since Wµ , t and Wσ , t are independent, given (δ , θ ) , the survival function of RLi,CEj (δ , θ ) is given by

[

]

FRLi, j ( δ , θ ) ( s ) = P RLi,CEj (δ , θ ) > s = FRLi CE

E− µ

( δ ,θ )

(s) × FRL

j E− σ

(θ )

(s)

(8)

for − ∞ < s < +∞, i = 1,...,h − 1 and j = 1,...,v − 1, where the approximations to FRLi ( δ , θ ) ( s ) and FRL j ( θ ) ( s ) are given in Appendix A. E− µ

E− σ

Now let RLS − µ (δ , θ ) , RLS − σ (θ ) and RLCS (δ , θ ) be the run length of the standard Shewhart chart for µ , the upper one-sided Shewhart for σ and the CShewhart scheme, respectively. Recall that RLS − µ (δ , θ ) and RLS − σ (θ ) are independent geometric random variables with parameters 1 − Φ γ S−µ − δ θ − Φ −γ S−µ − δ θ and 1 − Fχ 2 ( γ S − σ θ 2 ) , ( n−1) respectively. Analogously RLCS (δ , θ ) = min RLS − µ (δ , θ ), RLS − σ (θ ) , and the survival function of this random variable equals the product of the survival functions of RLS − µ (δ , θ ) and RLS − σ (θ ) . Thus, RLCS (δ , θ ) is also geometric with parameter α (δ , θ ) = 1 − Φ γ S − µ − δ θ − Φ − γ S − µ − δ θ × 1 − Fχ 2 ( γ S − σ θ 2 ) . ( n−1)

( { [(

) ] [(

{ [(

) ]})

{

{

}

}

) ]}

) ] [(

4. ARL: Approximations and Tables

[

]

[[

]]

Let U(δ , θ ) = ARLi,CEj (δ , θ ) i =1,...,h = E RLi,CEj (δ , θ ) i =1,...,h be the ARL's matrix of j =1,...,v j =1,...,v the CEWMA scheme, considering that the initial values wµ , 0 and wσ , 0 are associated to the, in control or out of control, states Eµ , i and Eσ , j . An approximation to this matrix can be obtained adapting the iterative procedure proposed by Prabhu and Runger (1996) to this particular control scheme

6

[

]

U ( k +1) (δ , θ ) = T • T + P µ (δ , θ ) × U ( k ) (δ , θ ) × P σ (θ ) , k = 0,1,...

(9)

where: U ( k ) (δ , θ ) represents the approximation to U(δ , θ ) in the k th iteration; U ( 0 ) (δ , θ ) = 0 / µ × 0 σ ; P µ (δ , θ ) and P σ (θ ) are probability transition matrices defined by expressions (A.1) and (A.2), respectively, in Appendix A; 1µ × 1 / σ T= / 0 σ

0µ   0 

(10)

is a h × v matrix which indicates the in control states of the two-dimensional Markov chain described in Appendix A; and the symbol • indicates elementwise multiplication of the matrices. This procedure was used by Runger and Prabhu (1996) to obtain approximate values to the ARL of a Multivariate EWMA chart for the control of a multivariate normal mean vector.

[

]

A second approximation for ARL(δ , θ ) = ARLi,CEj (δ , θ ) i =1,...,h −1 — the j =1,...,v −1 ARL's matrix of the CEWMA scheme when wµ , 0 and wσ , 0 are associated only to the in control states Eµ , i and Eσ , j — involves the survival function of RLi,CEj (δ , θ ) which is the product of RLiE − µ (δ , θ ) and RLEj − σ (θ ) survival functions: ARL(δ , θ ) =

[∑

+∞ s=0

FRLi

E− µ

( δ ,θ )

(s) × FRL

j E− σ

(θ )

]

(s) i =1,...,h −1 .

(11)

j =1,...,v −1

Approximate values to the entries ARLi,CEj (δ , θ ) of the matrix ARL(δ , θ ) can be obtained using the approximations (A.13) and (A.14) to the two survival functions in (11) and truncating the series. The second approximation procedure was used to compute the ARL's values of the CEWMA scheme, assuming that the convergence of the series (11) is attained as soon as the relative error is less than 10 −6 , and considering the parameters of Example 1 ( n = 5; λ µ = λ σ = 0.05, γ E − µ = 2.624 , γ E − σ = 1.27) and h − 1 = v − 1 = 51 in control states for the individual EWMA charts. These parameters yield to a CEWMA scheme (without head starts) with ARL at the h 2 ],1 nominal values µ 0 and σ 0 equal to ARL[CE (0,1) = 255.46; the in-control ARL's of the individual EWMA charts of this combined scheme are ARL[Eh−2µ] (0,1) = 500.13 and ARL1E − σ (1) = 500.86 . It is important to add that the in-control ARL's of the CShewhart scheme individual charts were matched to those of the combined CEWMA scheme, i.e. ARLS − µ (0,1) = ARL[Eh−2µ] (0,1) and ARLS − σ (1) = ARL1E − σ (1) ; thus,

7

{

]} and γ

[

γ S − µ = Φ −1 1 − 1 2 × ARLS − µ (0,1)

S−σ

= Fχ−12

( n−1)

[1 − 1 ARLS − σ (1)] .

The first iterative procedure, which was used by Morais (1998) to h 2 ],1 approximate ARL[CE (δ , θ ) , requires a large number of iterations until convergence

h 2 ],1 is attained in the case of approximating ARL[CE (0,1) and a steadily decreasing number of iterations when δ or θ increases. The results obtained here lead to the same conclusion and made one realize that the number of iterations is virtually independent of the head starts given to the individual EWMA control charts.

Table II has ARL's of the CShewhart and CEWMA schemes for various settings of the shifts in the nominal level of the process mean and standard deviation; several head starts ( HS ) for the individual charts of the CEWMA scheme were also considered: HSµ = 0%, ±50%, ±75%, and HSσ = 0%, 50%, 75%. It is important to note that, due to the symmetry of Wµ , t distribution, RLi,CEj (δ , θ ) and ′, j RLiCE ( −δ , θ ) are identically distributed if i and i ′ are associated to symmetric head starts; thus one omits the ARL values (percentage points values and probabilities of certain misleading signals, in the next sections) for negative values of δ . The numerical results in Table II — in particular the values of the ARL reduction that follows from the substitution of the CShewhart scheme by the

[

]

CEWMA scheme, 1 − ARLi,CEj (δ , θ ) ARLCS (δ , θ ) × 100% — suggest that the adoption of the CEWMA scheme can substantially reduce the ARL when moderate shifts occur in the process mean and standard deviation. It is interesting to notice that the adoption of positive head starts HSµ leads to a CEWMA scheme more sensitive to inflations in both parameters at the cost of smaller ARL's when the process is in control. In the CEWMA scheme, the use of a negative HSµ head start leads to a smaller average detection speed of inflations in µ or in σ ; nevertheless this combined scheme is still more sensitive to changes in the values of µ or σ than the CShewhart scheme, in most cases.

8

TABLE II. Values of ARLCS (δ , θ ) , ARLCE (δ , θ ) and the ARL reduction — 1 − ARLi,CEj (δ , θ ) ARLCS (δ , θ ) × 100% (listed in order corresponding to HSσ = 0%, 50%, 75%)

[

i, j

]

(1 − ARLCE

ARLCE HSµ

θ δ ARLCS 1.0 0.0 250.5 0.1

244.1

0.5

144.0

1.0

49.3

2.0

7.2

1.1 0.0

81.6

0.1

80.4

0.5

58.6

1.0

27.8

2.0

6.0

1.5 0.0

6.9

0.1

6.8

0.5

6.4

1.0

5.4

2.0

3.1

-75% 215.7 207.0 182.2 173.9 167.0 147.0 36.2 35 31 16.5 16.0 14.4 8.1 7.9 7.3 49.8 43.5 33.9 51.6 45.1 35.1 28.9 25.5 20.2 15.6 14.2 11.5 8.0 7.5 6.4 6.1 4.1 2.7 6.3 4.2 2.8 6.8 4.5 2.9 6.9 4.6 3.0 6.0 4.2 2.8

-50% 243.5 233.8 205.7 185.9 178.5 157.2 34.3 33.2 29.5 15.0 14.6 13.2 7.2 7.1 6.6 57.3 50.0 38.9 56.5 49.3 38.4 27.9 24.7 19.5 14.4 13.1 10.6 7.2 6.8 5.8 6.9 4.6 3.0 7.0 4.6 3.0 7.1 4.7 3.0 6.9 4.6 3.0 5.7 4.1 2.7

ARLCS ) × 100% HSµ

0% +50% +75% 255.5 243.5 215.7 245.2 233.8 207.0 215.8 205.7 182.2 182.7 161.6 134.3 175.4 155.2 129.0 154.4 136.6 113.7 28.2 19.0 12.9 27.2 18.5 12.6 24.3 16.5 11.3 11.4 6.9 4.4 11.1 6.8 4.3 10.1 6.2 4.1 5.2 3.1 1.9 5.2 3.0 1.9 4.9 2.9 1.9 60.7 57.3 49.8 53.0 50.0 43.5 41.1 38.9 33.9 56.4 49.8 41.0 49.2 43.5 36.0 38.3 33.9 28.1 23.5 16.4 11.3 20.9 14.7 10.3 16.6 11.9 8.5 11.1 6.8 4.4 10.2 6.4 4.2 8.4 5.5 3.7 5.3 3.1 2.0 5.1 3.1 1.9 4.4 2.8 1.9 7.2 6.9 6.1 4.7 4.6 4.1 3.0 3.0 2.7 7.2 6.7 5.8 4.7 4.5 4.0 3.0 2.9 2.7 7.0 6.0 4.8 4.6 4.2 3.5 3.0 2.8 2.4 6.3 4.9 3.6 4.4 3.6 2.8 2.9 2.5 2.1 4.7 3.0 2.0 3.6 2.6 1.9 2.1 2.0 1.6

9

-75% 13.9 17.3 27.3 28.8 31.6 39.8 74.8 75.7 78.4 66.5 67.5 70.7 -12.4 -10.5 -1.5 38.9 46.6 58.4 35.8 43.9 56.3 50.7 56.4 65.6 43.8 49.1 58.8 -33.7 -26.0 -6.6 11.7 39.7 60.1 8.6 38.1 59.3 -5.2 30.0 54.6 -27.7 14.9 44.9 -91 -34.3 10.8

-50% 2.8 6.7 17.9 23.8 26.9 35.6 76.1 77.0 79.5 69.5 70.3 73.2 -0.6 0.9 8.5 29.8 38.7 52.4 29.7 38.6 52.3 52.4 57.8 66.6 48.3 53.0 61.8 -20.1 -13.9 2.8 -0.1 33.4 56.9 -1.7 32.5 56.4 -10.5 27.1 53.1 -27.8 14.6 44.7 -81.4 -30.0 12.8

0% +50% +75% -2.0 2.8 13.9 2.1 6.7 17.3 13.9 17.9 27.3 25.2 33.8 45.0 28.1 36.4 47.2 36.7 44.0 53.4 80.4 86.8 91.0 81.1 87.2 91.3 83.1 88.5 92.1 76.9 86.1 91.2 77.4 86.3 91.2 79.5 87.3 91.8 26.9 57.4 73.1 27.5 57.6 73.1 32.2 59.3 73.7 25.6 29.8 38.9 35.1 38.7 46.6 49.6 52.4 58.4 29.9 38.1 49.0 38.8 45.9 55.3 52.4 57.8 65.0 59.9 72.1 80.7 64.3 74.8 82.4 71.6 79.7 85.6 60.2 75.6 84.3 63.3 76.9 84.9 69.7 80.3 86.7 12.3 48.6 67.3 15.2 48.4 67.5 25.7 53.3 69.0 -4.7 -0.1 11.7 31.1 33.4 39.7 55.8 56.9 60.1 -4.9 1.4 14.6 31.0 34.0 41.1 55.7 57.1 60.8 -8.3 6.1 25.1 28.0 34.9 45.5 53.5 56.8 62.3 -17.9 9.2 33.8 18.9 32.5 47.3 46.7 53.3 61.3 -49.2 2.9 34.9 -14.5 16.2 40.1 34.5 36 50.2

5. RL Percentage Points: Tables of Approximate Values The average run length ( ARL) has been used for some time as a sole performance measure of a control chart: the ARL is supposed to be large, when the production process is stable or in control, and small, otherwise. Nevertheless, some authors have argued that in some cases the percentage points (or the survival function) of the run length provide a more appropriate performance measure. −1 The p × 100% (0 < p < 1) percentage point of RLi,CEj (δ , θ ) , FRL ( p) , is i, j ( δ ,θ ) CE

defined by the smallest integer m satisfying FRLi, j ( δ , θ ) ( m ) ≤ 1 − p.

(12)

CE

Approximations to this percentage point can be easily obtained doing the approximations for the run length survival function given by (A.12)—(A.14). −1 As for the p × 100% percentage point of RLCS (δ , θ ) , FRL ( p), is can be CS ( δ , θ ) defined as the smallest integer m satisfying FRLCS ( δ , θ ) ( m ) ≤ 1 − p, or as equal to the

integer part of ln(1 − p) ln[1 − α (δ , θ )].

TABLE III. ARLCS (δ , θ ) and RLCS (δ , θ ) percentage points p

θ

δ ARLCS

1.0

0.0 250.5 ±0.1 244.1 ±0.5 144.0 ±1.0 49.3 ±2.0 7.2 1.1 0.0 81.6 ±0.1 80.4 ±0.5 58.6 ±1.0 27.8 ±2.0 6.0 1.5 0.0 6.9 ±0.1 6.8 ±0.5 6.4 ±1.0 5.4 ±2.0 3.1

.1% 1% 1 3 1 3 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

5% 10% 20% 30% 40% 50% 60% 1 3 2 7 5 6 9 0 128 174 230 1 3 2 6 5 5 8 7 125 169 224 8 1 6 3 3 5 2 7 4 100 132 3 6 11 18 25 34 45 1 1 2 3 4 5 7 5 9 19 29 42 57 75 5 9 18 29 41 56 74 3 7 13 21 30 41 54 2 3 7 10 14 19 26 1 1 2 2 3 4 6 1 1 2 3 4 5 6 1 1 2 3 4 5 6 1 1 2 3 4 5 6 1 1 2 2 3 4 5 1 1 1 1 2 2 3

70% 301 294 173 59 9 98 97 70 33 7 8 8 8 6 4

80% 403 393 231 79 11 131 129 94 45 9 11 11 10 8 5

90% 576 561 331 113 16 187 185 134 63 13 15 15 14 12 6

95% 99% 749 1152 730 1122 430 661 147 225 20 31 243 374 240 369 174 268 8 2 126 17 26 20 30 19 30 18 28 15 23 8 12

Table III has several percentage points of RLCS (δ , θ ) for different pairs of (δ , θ ) . In Table IV one can find FRL−1CS ( δ ,θ ) ( p) − FRL−1i, j ( δ ,θ ) ( p) — the difference

between those percentage points and the ones of RLi,CEj (δ , θ ) — for the following head starts: HSµ , HSσ = ( −50%,50%), (0%,0%), (50%,50%) . CE

(

)

10

TABLE IV. ARLCE (δ , θ ) and FRL ( δ , θ ) ( p) − FRLi, j δ , θ ( p) ) CS CE ( −1

i, j

−1

listed in order corresponding to ( HSµ , HSσ ) = (-50%,50%), (0%,0%), (50%, 50%) p

θ

δ

1.0

0.0

0.1

0.5

1.0

2.0

1.1

0.0

0.1

0.5

1.0

2.0

1.5

0.0

0.1

0.5

1.0

2.0

ARLCE .1% 233.8 255.5 233.8 178.5 182.7 155.2 33.2 28.2 18.5 14.6 11.4 6.8 7.1 5.2 3.0 50.0 60.7 50.0 49.3 56.4 43.5 24.7 23.5 14.7 13.1 11.1 6.4 6.8 5.3 3.1 4.6 7.2 4.6 4.6 7.2 4.5 4.7 7.0 4.2 4.6 6.3 3.6 4.1 4.7 2.6

-1 -5 -1 -1 -5 -1 -1 -4 -1 -1 -3 -1 -1 -2 0 -1 -3 -1 -1 -3 -1 -1 -3 0 -1 -2 0 -1 -1 0 0 -1 0 0 -1 0 0 -1 0 0 -1 0 0 -1 0

1% 0 -8 0 0 -7 0 -1 -5 0 -2 -4 -1 -2 -2 -1 -1 -5 -1 -1 -5 -1 -1 -4 -1 -1 -3 -1 -1 -2 0 0 -2 0 0 -2 0 0 -2 0 0 -2 0 0 -1 0

5% 10% 20% 30% 40% 50% 60% 70% 6 11 13 14 15 16 17 18 -10 -9 -9 -8 -8 -6 -5 -5 6 11 13 14 15 16 17 18 5 4 8 16 28 41 58 80 -7 -5 4 13 25 38 55 76 7 1 5 3 0 4 2 5 5 6 8 8 5 106 -2 1 1 3 2 9 4 8 7 0 9 8 134 -2 4 1 8 3 4 5 3 7 6 104 140 4 1 1 2 6 4 3 6 3 8 6 115 151 -5 -3 0 6 12 20 30 42 -3 -1 3 9 15 23 33 46 0 3 7 14 20 28 39 51 -4 -4 -4 -3 -3 -2 0 1 -2 -3 -2 -1 -1 0 2 3 -1 -1 0 1 1 2 4 6 2 4 10 14 19 24 30 38 -4 -4 0 2 7 12 19 26 2 4 10 14 19 24 30 38 2 4 8 12 16 21 28 37 -4 -4 -1 3 8 14 21 30 2 4 10 16 22 28 36 45 -1 2 2 6 11 19 28 40 -5 -2 1 6 13 21 31 43 0 3 8 14 21 30 40 53 -2 -3 -2 0 2 6 12 17 -3 -3 0 2 5 9 15 20 0 0 3 6 9 13 20 26 -3 -4 -3 -4 -3 -3 -1 -1 -2 -3 -2 -2 -2 -1 1 1 -1 -1 0 0 0 1 3 4 -1 -1 0 0 1 1 2 3 -2 -3 -2 -2 -2 -1 -1 0 -1 -1 0 0 1 1 2 3 -1 -1 0 0 1 1 2 3 -2 -3 -2 -2 -2 -1 -1 0 -1 -1 0 0 1 1 2 3 -1 -1 0 0 1 1 2 3 -2 -3 -2 -2 -2 -1 -1 0 -1 -1 0 0 1 2 2 3 -1 -1 0 0 0 0 1 1 -2 -3 -2 -3 -2 -2 -1 -1 0 -1 0 0 0 1 1 2 -1 -1 -1 -2 -1 -2 -1 -1 -2 -2 -2 -3 -2 -2 -2 -1 0 -1 -1 -1 0 0 0 1

11

80% 21 -2 21 110 107 137 186 192 203 61 65 70 3 5 7 49 38 49 49 43 58 58 61 72 28 31 36 1 3 5 5 1 5 5 1 5 3 1 4 1 0 3 -1 -1 2

90% 24 1 24 161 158 188 275 282 293 92 96 102 7 9 12 69 57 69 72 66 81 89 92 103 43 46 52 4 6 9 6 3 6 6 3 7 5 3 6 4 2 6 -1 -1 2

95% 27 5 27 212 209 239 364 370 382 124 128 133 10 12 15 88 76 88 94 87 102 120 123 134 59 63 68 7 9 12 9 6 9 8 5 8 7 5 8 5 4 7 0 1 3

99% 36 13 36 332 329 359 571 577 589 196 200 206 20 22 25 133 121 133 145 139 154 193 196 208 97 101 107 15 17 20 14 10 14 14 11 14 12 10 14 9 8 12 2 3 6

The results in Table IV suggest that RLCS (δ , θ ) (which has a geometric distribution) has, for moderate shifts in both parameters, heavier right tails than RLi,CEj (δ , θ ) even when a negative HSµ is adopted; therefore the CEWMA scheme tends to produce short run lengths more frequently which means faster detection of changes in µ or in σ .

6. The Probability of a Misleading Signal The diagnostic procedures that follow an out-of-control signal can differ depending on whether the signal was given by the chart for µ or the chart for σ , or whether the signal given by the chart for µ is on positive or negative side. Thus a misleading signal could possibly send the user in the wrong direction in the attempt to diagnose the cause of the out-of-control signal. In fact, misleading signals can be a serious problem for the user of combined charts for multiple parameters. St. John and Bragg (1991) identified the following types of misleading signals arising in combined charts for µ and σ : I. the process mean increases but the out-of-control signal is given by the

II.

chart for σ , or the out-of-control signal is observed on the negative side of the chart for µ (that is the observed value of the summary statistic is below the lower control limit); µ shifts down but the out-of-control signal is observed on the chart for

σ , or the chart for µ gives an out-of-control signal on the positive side; III. an inflation of the process standard deviation occurs but the out-ofcontrol signal is given by the chart for µ . Only type III correspond to what can be called a "pure misleading signal", i.e., a change in the value of one of the parameters is followed by an out-of-control signal by the chart for the other parameter. However, there is a situation that also leads to a "pure misleading signal" and is related to both misleading signals of types I and II: IV. a shift ocurs in µ but the out-of-control signal is observed on the chart for σ . This will be called a mislealing signal of type IV (although it is a sub-type of types I or II). The misleading signals I-IV are graphically described in Figure 2. Only the

12

types III and IV are going to studied in this paper since they correspond to what can be called a "pure misleading signal". The expressions for the types III and IV misleading signals probabilities are:

[

]

i, j PMSCE ( III; θ ) = P RLEj − σ (θ ) > RLiE − µ (0, θ )

= ∑ s =1 FRL j +∞

E− σ

(θ )

[

(s) × FRL

[

i E− µ

( 0, θ )

(s − 1) − FRL

i E− µ

(s) , θ > 1,

]

(13)

(s) , δ ≠ 0,

]

(14)

( 0, θ )

]

i, j PMSCE ( IV; δ ) = P RLiE − µ (δ ,1) > RLEj − σ (1)

= ∑ s =1 FRLi +∞

E− µ

( δ ,1)

[

(s) × FRL

j E− σ

(1 )

(s − 1) − FRL

(1 )

j E− σ

respectively. Approximate values to these probabilities can be obtained by using the approximations to the survival functions (A.13) and (A.14) and truncating the series, as in the evaluation of the ARL. The corresponding types III and IV misleading signal probabilities for the CShewhart scheme are:

[

]

PMSCS ( III; θ ) = P RLS − σ (θ ) > RLS − µ (0, θ ) =1−

[ (

1 − Fχ 2

)

(

( n−1)

(γ

S−σ

θ2)

)]

1 − Φ γ S − µ θ − Φ − γ S − µ θ × Fχ 2

[ 1 − [ Φ( γ

( n−1)

]

(γ

S−σ

θ2)

, θ > 1,

(15)

PMSCS ( IV; δ ) = P RLS − µ (δ ,1) > RLS − σ (1) =1−

[ (

S−µ

)

)

(

− δ − Φ −γ S−µ − δ

(

)]

)]

1 − Φ γ S − µ − δ − Φ − γ S − µ − δ × Fχ 2

( n−1)

(γ S −σ )

, δ ≠ 0.

(16)

As an illustration, these probabilities were obtained for the CShewhart and CEWMA schemes described in the previous sections, considering a relative error of 10 −6 in the truncation of both series. Note that Table V has PMS 's of type III only ′, j for positive values of HSµ because RLi,CEj (0, θ ) and RLiCE (0, θ ) are identically distributed if i and i ′ are associated to symmetric head starts; thus the i, j PMSCE ( III; θ ) values, when HSµ = −50%, −75%, equal those obtained for HSµ = 50%,75%.

13

TABLE V. PMS of Type III for the CShewhart and CEWMA schemes (listed in order corresponding to HSσ =0%, 50%, 75%) PMSCE HSµ

θ 1.05

PMSCS 0.446

1.1

0.402

1.5

0.240

0% 0.293558 0.269358 0.221725 0.177123 0.152522 0.116706 0.023177 0.010230 0.004722

+50% 0.332817 0.306229 0.252613 0.228097 0.198159 0.152511 0.092820 0.049574 0.024782

+75% 0.418719 0.390112 0.326151 0.335398 0.300055 0.237205 0.243236 0.161741 0.091477

TABLE VI. PMS of Type IV for the CShewhart and CEWMA schemes (listed in order corresponding to HSσ =0%, 50%, 75%) PMSCE HSµ

δ 0.05

PMSCS 0.496

0.1

0.486

0.5

0.286

1.0

0.097

2.0

0.012

-75% 0.408590 0.430208 0.495589 0.337871 0.362838 0.437056 0.058515 0.096576 0.205692 0.018685 0.057897 0.171733 0.003396 0.036092 0.147359

-50% 0.449246 0.471412 0.535307 0.361849 0.387676 0.461834 0.054637 0.093142 0.203256 0.015857 0.054676 0.168592 0.002357 0.032442 0.141890

0% 0.455727 0.477927 0.541324 0.355191 0.381492 0.456608 0.042274 0.080810 0.192090 0.009247 0.045233 0.157926 0.000686 0.021489 0.121960

+50% 0.417909 0.441082 0.508410 0.312608 0.339885 0.419333 0.024840 0.059770 0.169422 0.003045 0.027544 0.128575 0.000055 0.006485 0.073885

+75% 0.357561 0.380094 0.449945 0.258302 0.284167 0.364686 0.014723 0.041351 0.138381 0.001124 0.014358 0.089574 0.000007 0.001528 0.034520

According to Table V, the adoption of the CEWMA scheme lead to a i, j substantially decrease in PMSCE ( III; θ ) , the probability of having the EWMA chart for µ responsible for a misleading signal (with one exception: θ = 1.5, HSσ = 0% and HSµ = 75%). The numerical results in Table VI suggest that the adoption of the CEWMA i, j scheme also reduces PMSCE ( IV; δ ) , at least if either δ or HSσ is not large.

14

The monotonic results for these two tables are due to the fact that the adoption of the head start HSµ ( HSσ ) is usually followed by a decrease of run length of the EWMA control chart for µ (σ ) . Therefore there is: an increase (decrease) in i, j PMSCE ( III; θ ) with HSµ considering a fixed HSσ (with HSσ for a fixed HSµ ); i, j and a decrease (increase) in PMSCE ( IV; δ ) with HSµ for a constant HSσ (with HSσ considering a fixed HSµ ).

7. Design Strategy for CEWMA schemes It is very dificult to design a CEWMA control scheme that is jointly optimal in detecting shifts in the process mean and inflations of the process standard deviation, mainly because of the complexity of this scheme performance indicators and the several parameters involved in the design. Gan (1995) suggested a simpler and reasonable approach: the separate design of the individual control charts involved in the combined scheme. This design approach is adopted here considering a criteria that involves two performance measures — the ARL and the RL percentage points —, and fixing the sample size (n) , the number of in control states for both individual charts ( h − 1 and v − 1) and their initial in control states ( i and j ). Let ARL*E − µ and ARL*E − σ denote the smallest economically acceptable in control ARL's for the individual charts; consider (δ * , θ * ) , the magnitude of a change in both parameters one wants to detect quickly; and set pE* − µ ( pE* − σ ) as the lower limit for the probability of not taking more than RL*CE samples before the EWMA control chart for µ (σ ) detects the change (δ * , θ * ) (θ * ) . The proposed design strategy has the following steps: • Step 1 — choose λ µ and γ E − µ that verify the following conditions ARLiE − µ (0,1) ≥ ARL*E − µ ,

[

(17)

]

P RLiE − µ (δ * , θ * ) ≤ RL*CE > pE* − µ ; •

Step 2 — obtain λ σ and γ E − σ such that ARLEj − σ (1) ≥ ARL*E − σ ,

[

(19)

]

P RLEj − σ (θ * ) ≤ RL*CE > pE* − σ ; •

(18)

(20)

Step 3 — repeat steps 1 and 2 a couple of times, and pick the pairs

15

(λ

)

, γ E − µ and (λ σ , γ E − σ ) which yield the larger values of the probabilities in (18) and (20); •

µ

Step 4 — investigate the overall behaviour of the designed combined scheme, in terms of in control and out-of-control ARL's, RL percentage points and probabilities of misleading signals.

A few notes on this design procedure. Step 4 can be eventually followed by the adoption of head starts for the individual charts; but recall that these new charts have to verify conditions (17) to (20). Since RLiE − µ (δ * , θ * ) and RLiE − σ (θ * ) are independent, given (δ * , θ * ) , one concludes that the combined scheme designed here verifies

[

]

[

] [

]

P RLi,CEj (δ * , θ * ) ≤ RL*CE > 1 − 1 − pE* − µ × 1 − pE* − σ .

(21)

To simplify Steps 1 and 2 of the design procedure, one suggests: choosing λ first; then, since the in control ARL's of the individual EWMA charts are monotonous functions of γ , adopting of a binary search procedure to find the values of γ that satisfies (17) and (19); and finally checking if the two pairs (λ , γ ) satisfy inequalites (18) and (20). It should also be noted that the probabilities P RLiE − µ (δ , θ ) ≤ ARLiE − µ (δ , θ ) and P RLEj − σ (θ ) ≤ ARLEj − σ (θ ) usually exceed 50%. Suppose one desires a CEWMA control scheme with the following

[

]

[

]

characteristics: n = 5 and 51 in control states for each EWMA chart; ARL[Eh−2µ] (0,1) ≥ 500 , P RL[Eh−2µ] (0.5,1.2) ≤ 25 > 50%; and ARL1E − σ (1) ≥ 500, P RL1E − σ (1.2) ≤ 25 > 50%. The CEWMA control scheme with no head starts, presented in the previous sections (i.e. λ µ = λ σ = 0.05, γ E − µ = 2.624 and γ E − σ = 1.27), has these characteristics. However, a change to a second scheme with

[

]

[

]

λ µ = λ σ = 0.06 , γ E − µ = 2.68, γ E − σ = 1.35, HSσ = 0% or HSσ = 50% ( j = [v 2]) slightly improves the properties of the first combined scheme (see Table VII).

16

TABLE VII. Properties of three CEWMA schemes Performance measures ARLE − µ ( 0,1) i

() ( ) i ARLE − µ ( 0.5,1. 2 ) j ARLE − σ (1. 2 ) i, j ARLCE ( 0. 5,1. 2 ) i P[ RLE − µ ( 0.5,1. 2 ) ≤ 25] j P[ RLE − σ (1. 2 ) ≤ 25] i, j P[ RLCE ( 0.5,1. 2 ) ≤ 25] i, j PMSCE ( III;1. 2 ) i, j PMSCE ( IV; 0. 5) j ARLE − σ 1 i, j ARLCE 0,1

1st CEWMA scheme 2nd CEWMA scheme 2nd CEWMA scheme HSσ = 0% HSσ = 50% 500.13

500.06

500.06

500.86

519.71

500.52

255.46

255.17

245.653

27.37

26.89

26.89

25.32

25.81

20.30

16.77

16.52

13.37

0.568

0.583

0.583

0.632

0.624

0.720

0.841

0.844

0.884

0.079262

0.089597

0.066968

0.042274

0.041632

0.076813

8. Concluding Remarks Since charts for the process mean and standard deviation are often used jointly it is recommendable to discuss the performance of combined control schemes like the CEWMA scheme. In this paper attention was given not only to the average run length, which provides a unidimensional snapshot of the performance of the CEWMA scheme, but also to two other performance indicators: the run length percentage points and the probability of a misleading signal. The Markovian approach and the independence between the horizontal and vertical transitions of the approximating two-dimensional Markov chain played an important role in providing tractable expressions for these three performance measures. The adoption of a CEWMA scheme as an alternative to a CShewhart scheme usually leads to a decrease in the probability of originating a misleading signal. However, this adoption must be done with some care since the numerical results suggest a substantial run length reduction — in terms of average and percentage points — only for moderate shifts in µ and σ . When convenient head starts (e.g. positive HSµ for δ > 0) have been given to the EWMA charts for µ and σ , the

17

combined EWMA scheme tends to produce short run lengths more frequently, yielding to a more sensitive control scheme if the process is out-of-control and to a slight decrease of the in control run length. Finally, a design procedure based in all three performance measures, and which provides simple guidelines for a convenient choice of the combined scheme parameters, was described.

Appendix A — Approximating the Run Length Distribution of the CEWMA Scheme Using the Markovian Approach An approximation to the run length distribution of the CEWMA scheme can be obtained approximating the properties of the continuous-state two-dimensional Markov chain Wµ , t ,Wσ , t ,t = 0,1,... by a two-dimensional Markov chain with

{(

}

)

discrete state space. Due to the fact that Wµ , t and Wσ , t are independent, given (δ , θ ) , the horizontal and vertical transitions of each one of those two chains are also independent. Therefore, the behaviour of the approximating chain can be described by two one-dimensional chains — one for each individual chart — whose definition involves, following Brook and Evans (1972), Woodall (1984), Lucas and Sacucci (1990) and others: • dividing CE − µ (given in (4)) in h − 1 intervals of equal range ∆ µ that are

•

•

associated to the ordered transient states Eµ , i , i = 1,...,h − 1 ( h should be an even number); taking the lower limit of the first interval, eµ , 1 , equal to LCLE − µ and eµ , i = eµ , 1 + (i − 1)∆ µ , i = 2,...,h; associating the set ln(σ 02 ) to the transient state Eσ , 1 , and dividing

{

{

}

}

CE − σ \ ln(σ ) in v − 2 intervals of range ∆ σ associated to the ordered transient states Eσ , j , j = 2,...,v − 1; considering the lower limit of the first interval, eσ , 2 , equal to LCLE − σ and eσ , j = eσ , 2 + ( j − 2)∆ σ , j = 2,...,v − 1; and associating the absorbing state of the first (second) chain, Eµ , h 2 0

( E ) , to (−∞, LCL ) ∪ (UCL

)

, +∞ ((UCLE − σ , +∞)) . The approximations to the probability transition matrices of these two independent Markov chains, represented in partitioned form, are as follows σ, v

E −µ

E −µ

[

]

 Q (δ , θ ) I µ − Q µ (δ , θ ) × 1µ  P µ (δ , θ ) =  / µ  1 0 µ 

18

(A.1)

Q (θ ) [ I σ − Q σ (θ )] × 1σ  P σ ( δ , θ ) = P σ (θ ) =  / σ , 0 1 σ  

(A.2)

where: 1µ (1σ ) is a column vector of h − 1 (v − 1) ones, 0 µ (0 σ ) is a column vector of h − 1 (v − 1) zeros; I µ ( I σ ) is the identity matrix with rank h − 1 (v − 1) ; and the matrices Q µ (δ , θ ) and Q σ (θ ) have entries given by

e +e   qµ , ij (δ , θ ) = P Wµ , t ∈Eµ , j Wµ , t −1 = µ , i µ , i +1 , δ , θ , i, j = 1,...,h − 1 (A.3) 2   and

(

)

qσ , 1 j (θ ) = P Wσ , t ∈Eσ , j Wσ , t −1 ∈Eσ ,1 , θ , j = 1,...,v − 1,

(A.4)

e +e   qσ , ij (θ ) = P Wσ , t ∈Eσ , j Wσ , t −1 = σ , i σ , i +1 , θ , 2   i = 2,...,v − 1, j = 1,...,v − 1,

(A.5)

respectively. These entries can be written in an alternative way: qµ , ij (δ , θ ) = aµ , ij (δ , θ ) − aµ , i j −1 (δ , θ ), i, j = 1,...,h − 1

(A.6)

qσ , ij (θ ) = aσ , ij (θ ) − aσ , i j −1 (θ ), i, j = 1,...,v − 1

(A.7)

where

[ (

and

)

]

 2γ   E − µ × j − 1 − λ µ (i − 1 2) − λ µ (h − 1) 2 δ  aµ , ij (δ , θ ) = Φ  − , θ θ (h − 1) λ µ 2 − λ µ   i = 1,...,h − 1, j = 0,...,h − 1

(A.8)

aσ , i 0 (θ ) = 0, i = 1,...,v − 1,

(A.9)

(

)

n −1 aσ , 1 j (θ ) = Fχ 2  2 × exp[( j − 1)∆ σ λ σ ], j = 1,...,v − 1, ( n−1)  θ  n −1 aσ , ij (θ ) = Fχ 2  2 × exp ( j − 1) − (1 − λ σ )(i − 3 2) ∆ σ λ σ  ,  ( n−1)  θ i = 2,...,v − 1, j = 1,...,v − 1.

{[

]

(A.10)

}

(A.11)

Note that entries (A.9) to (A.11) do not depend on δ leading to conclusion that the run length of the upper one-sided EWMA chart for σ (or any other EWMA chart for σ ) does not depend on the actual value of µ , therefore the omition of δ in the representation of this chart run length. Recall that, given (δ , θ ) , RLi,CEj (δ , θ ) , RLiE − µ (δ , θ ) and RLEj − σ (θ ) denote the

19

run length of the combined EWMA scheme, the standard EWMA chart for µ and the upper one-sided EWMA chart for σ , conditional to the fact the two summary statistics initial values — wµ , 0 and wσ , 0 — belong to the transient states Eµ , i (i = 1,...,h − 1) and Eσ , j ( j = 1,...v − 1) , respectively. In this case RLi,CEj (δ , θ ) has survival function given by FRLi, j ( δ , θ ) ( s ) = FRLi

E− µ

CE

( δ ,θ )

(s) × FRL

j E− σ

(θ )

(s)

(A.12)

for − ∞ < s < +∞, i = 1,...,h − 1 and j = 1,...,v − 1, where FRLi ( δ , θ ) ( s ) and E− µ FRL j ( θ ) ( s ) can be approximated by E− σ 1, s < 1 * s FRL = (A.13) ( ) i e ′ × Q ( δ , θ ) [ s ] × 1 , s ≥ 1 E− µ ( δ , θ ) µ µ  µ, i

[

* FRL i

E− σ

]

1, s < 1 [s] e′ σ , i × [Q σ (θ )] × 1σ , s ≥ 1,

(s) =  (θ )

(A.14)

( )

respectively, with e µ , i e σ , j denoting the i th ( j th ) vector of the orthonormal basis for Rh −1 (Rv −1 ) and [ s] representing the integer part of s . Finally, note that the vectors of the ARL's for the individual charts of the CEWMA scheme are:

[

]

ARL E − µ (δ , θ ) = ARLiE − µ (δ , θ )

[

]

ARL E − σ (θ ) = ARLEj − σ (θ )

i =1,...,h −1

[

]

= I µ − Q µ (δ , θ )

= [ I σ − Q σ (θ )] × 1σ . −1

j =1,...,v −1

−1

× 1µ ;

(A.15) (A.16)

ACKNOWLEDGEMENTS We thank the referee for some thoughtful comments. This paper was written with the partial support of grants Praxis PCEX/P/MAT/41/96 and Praxis PCEX/P/MAT/10002/98.

BIBLIOGRAPHY Brook, D., and Evans, D.A. (1972). "An approach to the probability distribution of CUSUM run length," Biometrika, 59, 539-549. Chengalur, I.N., Arnold, J.C. and Reynolds Jr., M.R. (1989). "Variable sampling

20

intervals for multiparameter Shewhart charts," Communications in Statistics, Part A — Theory and Methods, 18, 1769-1792. Crowder, S.V. (1987). "Computation of ARL for combined individual measurements and movinge range charts," Journal of Quality Technology, 19, 98-102. Crowder, S.V. and Hamilton, M.D. (1992). "An EWMA for monitoring a process standard deviation," Journal of Quality Technology, 24, 12-21. Domangue, R. and Patch, S.C. (1991). "Some Omnibus Exponentially Weighted Moving Average statistical process monitoring schemes," Technometrics, 32, 299-314. Gan, F.F. (1989). "Combined mean and variance control charts," ASQC Quality Congress Transactions — Toronto, 129-139. Gan, F.F. (1991). "Computing the percentage points of the run length distribution of an Exponentially Weighted Moving Average control chart," Journal of Quality Technology, 23, 359-365. Gan, F.F. (1995). "Joint monitoring of process mean and variance using Exponentially Weighted Moving Average control charts," Technometrics, 37, 446-453. Lucas, J.M. and Crosier, R.B. (1982). "Fast initial response for CUSUM qualitycontrol schemes: give your CUSUM a head start," Technometrics, 24, 199205. Lucas, J.M. and Saccucci, M.S. (1990). "Exponentially Weighted Moving Average control schemes: Properties and enhancements," Technometrics, 32, 1-12. Morais, M.C. (1998). "Combined EWMA schemes for the simultaneous control of µ and σ 2 — A Markovian approach" (in portuguese), Estatística: a Diversidade na Unidade (M. Souto de Miranda and I. Pereira, Eds.). Edições Salamandra, Lisboa, 207-219. Prabhu, S.S. and Runger, G.C. (1996). "Analysis of a two-dimensional Markov chain," Communications in Statistics Part B — Simulation and Computation, 25, 75-79.

21

Runger, G.C and Prabhu, S.S. (1996). "A Markov chain model for the multivariate Exponentially Weighted Moving Average control chart," Journal of the American Statistical Association, 91, 1701-6. Saniga, E.M. (1989). "Economic statistical control chart design with an application to X and R charts," Technometrics, 31, 313-320. St. John, R.C. and Bragg, D.J. (1991). "Joint X-bar & R charts under shift in mu or sigma," ASQC Quality Congress Transactions — Milwaukee, 547-550. Takahashi, T. (1989). "Simultaneous control charts based on ( x, s ) and ( x,r ) for multiple decision on process change," Reports of Statistical Application Research 36, 1-20. Wolfram, S. (1996). The Mathematica Book — 3rd edition (Mathematica Version 3.0), Wolfram Media, Cambridge University Press. Woodall, W.H. (1984). "On the Markov chain approach to the two-sided CUSUM procedure," Technometrics, 26, 41-46. Woodall, W.H. and Ncube, M.M. (1985). "Multivariate CUSUM quality-control procedures," Technometrics, 27, 285-292.

22

FIGURE 1 — Control charting using the EWMA summary statistics W µ , t and W µ , t

UCLE − µ

100.4 100.3 100.2

wµ , t LCLE − µ

100.1 100 99.9 99.8 99.7 99.6 1

2

3

4

5

6

7

8

9

10

5

6

7

8

9

10

t

0.2

UCLE − σ wσ , t

0.15 0.1 0.05

LCLE − σ

0 1

2

3

4

-0.05

23

t

FIGURE 2 — Types of misleading signals Type I — δ > 0, θ = 1

Type II — δ < 0, θ = 1

UCLσ

UCLσ

LCLσ

LCLσ LCLµ

LCLµ

UCLµ

Type III — δ = 0, θ > 1

UCLµ

Type IV — δ ≠ 0, θ = 1

UCLσ

UCLσ

LCLσ

LCLσ LCLµ

LCLµ

UCLµ

24

UCLµ