Effects of Measurement Error on Controlling Two ... - CiteSeerX

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ISSN 0940-5151

Economic Quality Control Vol 22 (2007), No. 1, 127 – 139

Effects of Measurement Error on Controlling Two Dependent Process Steps Su-Fen Yang, Han-Wei Ho and M. A. Rahim

Abstract: Imprecise measurements may seriously affect the efficiency of process control and production cost. For two dependent processes, the in-coming quality variable in the first process may influence the out-going quality variable in the second process. To effectively monitor and distinguish the state of the two dependent process steps with imprecise measurements, the EWMA control chart and cause-selecting control chart are constructed by taking account of the measurement errors. The EWMA control chart for imprecise measurements is constructed to monitor small shifts in the process mean on the first step, and the cause-selecting control chart for imprecise measurements, is constructed to monitor small shifts in the process mean in the second step. The effects of imprecise measurements on the performance of the two proposed control charts are examined for the case, where the mean of each process may be changed by the occurrence of a special cause. The performance of the proposed control charts is measured by the average run length. A sensitivity analysis indicates that a larger variation of imprecise measurements leads to a larger average run length for the out-of-control process and a larger average loss. Keywords: Imprecise measurement, dependent process steps, average run length.

1

Introduction

Any control chart for process control is essentially based on measurements. The performance of control charts and other statistical process control tools might be seriously affected by errors due to the measuring instrument. The effect of measurement errors on the operating characteristics of an X chart in cases where only the process mean shifts, is discussed by Bennett [2], Mizuno [12], Abraham [1], and Mittag and Stemann [11]. Kanazuka [6] and Mittag [9] investigate the power of the X-R control charts, where both the process mean and process spread change. Mittag and Stemann [11] extend the results of Mittag [9], referring to the X-S control chart. Rahim [15] analyzes the effects of imprecise measurement devices on the design parameters of economic control charts. Linna and Woodall [7] address the effect of measurement errors on the performance of charts using a linear covariate. Yang [19] investigates the effect of measurement errors on the design parameters of the economic asymmetric and S control charts. However, all the above papers discuss only a single process. In reality, there are generally several different processes to transform the raw materials into products. If the processes are mutually independent or uncorrelated, then separate control charts to monitor the processes may be used. However, the processes are often dependent. The former process will have influence on the latter one. If the control charts to the observations at each process are applied without removing the effect of the former process, then the state of the latter process might be misjudged. Zhang

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[20] proposes the cause-selecting control chart to monitor the state of the second process. The cause-selecting control chart effectively distinguishes, whether or not the second process is out of control. Wade and Woodall [17] review the concepts of the cause-selecting approach and propose using prediction limits instead of Zhang’s control limits of the cause-selecting control chart. Yang and Chen [19] propose economic X and cause-selecting control charts to monitor two dependent processes under two failure mechanisms. However, the performance of two dependent process controls when there are measurement errors, has not been evaluated so far. In this article, a process model is derived to take account of measurement errors on two dependent processes. We propose an EWMA control chart based on observations to monitor small shifts in the mean of the first process and a cause-selecting control chart based on residuals to monitor small shifts in the mean of the second process. The effect of measurement errors on the resulting EWMA and cause-selecting control charts is investigated. The performance of the proposed control charts is measured by the average run length (ARL). The investigations indicate that imprecise measurements will considerably improve the detection ability of the proposed control charts compared with proposed control charts neglecting measurement errors. This advises a manager to propose some appropriate strategies for measurement devices. The applications of the proposed control charts is illustrated by a numerical example.

2

Modelling the Two Dependent Processes

In this paper, it is assumed that the process means are shifted due to the occurrence of special causes. For modelling the two dependent production processes the following assumptions are made and notations are used.

2.1

Assumptions

1.

There are two dependent serial production processes, the first process and the second process. The outcome of the first process effects the result of the second one.

2.

Let XT represent the quality characteristic of interest for the first process and it is assumed that XT is normally distributed with mean µx and variance σx2 when the process is in statistical control. Similarly, let YT stand for the quality characteristic of the second process. The relationship between XT and YT is expressed by the folloing regression model: YT |xT = F (xT ) + T

(1)

The unconditional YT is also assumed to be normally distributed. When the second process is in statistical control, the mean is denoted by µy and the variance by σy2 . 3.

There are measurement errors due to the measurement devices employed. The measurement error for the first process, x and the second process, y are assumed to increase the process variance by δx2 σx2 and δy2 σy2 , respectively, where δx = 1 and δy = 1. Again, the normal distribution is assumed:

Effects of Measurement Error on Controlling Two Dependent Process Steps

x ∼ N (0, δx2 σx2 ) y ∼ N (0, δy2 σy2 )

129

(2)

Moreover, it is assumed that the two errors x and y are statistically independent. In addition, it is assumed that YT and x , XT and y are also independent. The observed in-coming quality from the first process and out-going quality from the second process are denoted by Xo and Yo , respectively, where Xo = XT + x ∼ N (µx , (1 + δx2 )σx2 ) Yo = YT + y ∼ N (µy , (1 + δy2 )σy2 )

(3)

4.

Let AC1 denote a special cause, which may only occur in the first process and shift the process mean from µx to µx + δ(10) σx . Another special cause, say AC2 , may only occur in the second process and shifts the process mean from µy to µy + δ(01) σy .

5.

At the end of the second process, a pair of observations (Xo(i,j,k) , Yo(i,j,k) ) are randomly drawn at a fixed time interval with sample size n = 1, where i is the indicator variable of the occurrence of AC1 in the first process, j is the indicator variable of the occurrence of AC2 in the second process, and k denotes the number of samples taken, k = 1, 2, . . ..

6.

EWMA control charts are effective in detecting small shifts in the process mean when the sample size is one (Montgomery [13]), therefore, EWMA charts are used for controlling the first process. To effectively detect the state of the second process, the effects of Xo given by errors or residuals should be removed from Yo . Thus, the cause-selecting control chart based on taking account of errors or residuals is selected for controlling the second process.

2.2

Notations

αe1 : probability that the EWMA chart based on observations trigger a signal given the first process is in control βe1 : probability that the EWMA chart based on observations does not trigger a signal given the first process is out of control αe2 : probability that the cause-selecting control chart based on residuals trigger a signal given the second process is in control βe2 : probability that the cause-selecting control chart based on residuals does not trigger a signal given the second process is out of control

2.3

The Relationships Between Xo and Yo

The relations between Xo and Yo can take many forms. Let i represent the state of the first process, where i = 1 stands for the out-of-control state of the first process, and i = 0 for the in-control state. Let j stand for the state of the second process, where j = 1 represents the the out-of-control state of the second process, and j = 0 the in-control state. It is assumed here that a linear regression model appropriately describes the relationships between Xo and Yo : Yo(i,j,k) |Xo(i,j,k) = β0 + β1 Xo(i,j,k) + k + y

(4)

The errors, k + y , are the effects of Xo on Yo , which must be removed. If the regression model is unknown, the residuals e = Yo − Yˆo are used to replace k + y .

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The Distributions of Xo and Yo

3

The desired EWMA chart for the in-coming quality and the desired cause-selecting control chart for the residuals are based on the distributions of the plotted statistics and residuals, respectively. There are four possible distributions for the in-coming quality and out-going quality based on the possible states of the two dependent processes.

3.1

First Process and the Second Process are In-Control

The distributions of the in-coming quality and the measurement error in case of an in-control first process are assumed as follows: XT(0,0,k) ∼ N (µx , σx2 ) x ∼ N (0, δx2 σx2 )

(5)

Therefore, the in-control distribution of the observed in-coming quality is given by: Xo(0,0,k) = XT(0,0,k) + x ∼ N (µx , (1 + δx2 )σx2 )

(6)

From the equation (4), the in-control distribution of the observed out-going quality, given that the in-control observed in-coming quality, is: 2 Yo(0,0,k) |Xo(0,0,k) ∼ N (β0 + β1 Xo(0,0,k) , σy|x + δy2 σy2 )

(7)

The in-control expectation and variance of the observed out-going quality can be derived as follows: 



(8) E Yo(0,0,k) = E E Yo(0,0,k) |Xo(0,0,k) = E β0 + β1 Xo(0,0,k) = β0 + β1 µx




 V Yo(0,0,k) = V E Yo(0,0,k) |Xo(0,0,k) + E V Yo(0,0,k) |Xo(0,0,k) 

2 + δy2 σy2 = V β0 + β1 Xo(0,0,k) + E σy|x 2 + δy2 σy2 = β12 (1 + δx2 )σx2 + σy|x

Hence, the distribution of the in-control observed out-going quality is given by:   2 + δy2 σy2 Yo(0,0,k) ∼ N β0 + β1 µx , β12 (1 + δx2 )σx2 + σy|x

3.2

(9)

(10)

First Process is Out-of-Control, Second Process is In-Control

After the occurrence of AC1 , the out-of-control distribution of the observed in-coming quality is given by:   X(1,0,k) ∼ N µx + δ(1,0) σx , (1 + δx2 )σx2 (11) The out-of-control in-coming quality does affect the out-going quality, hence the distribution of the observed out-going quality is   2 (12) + δy2 σy2 Yo(1,0,k) ∼ N β0 + β1 µx + δ(1,0) σx , β12 (1 + δx2 )σx2 + σy|x

Effects of Measurement Error on Controlling Two Dependent Process Steps

3.3

131

First Process is In-Control, Second Process is Out-of-Control

When the second process is out-of-control by the occurrence of AC2 , the distribution of the observed in-coming quality is unchanged since AC2 has no effect on the first process. Thus, distribution of the in-going quality is given by: Xo(0,1,k) ∼ N (µx , (1 + δx2 )σx2 )

(13)

After occurrence of AC2 , the mean of the observed out-going quality, given that the in-control observed in-coming quality, shifts to δ(0,1) σy . Hence, the out-of-control distribution of the outgoing quality is given by:   2 (14) + δy2 σy2 Yo(0,1,k) ∼ N β0 + β1 µx + δ(0,1) σy , β12 (1 + δx2 )σx2 + σy|x

3.4

First Process and Second Process are Out-of-Control

After occurrence of AC1 , the out-of-control distribution of the observed in-coming quality is   Xo(1,1,k) ∼ N µx + δ(1,0) σx , (1 + δx2 )σx2 (15) After occurrence of AC2 , the mean of the observed out-going quality, given that the out-ofcontrol observed in-coming quality, shifts to β1 δ(1,0) σx + δ(0,1) σy , implying the out-of-control distribution of the out-going quality:    2 + δ 2 σ 2 , β 2 (1 + δ 2 )σ 2 + σ 2 + δ 2 σ 2 Yo(1,1,k) ∼ N β0 + β1 (µx + δ(1,0) σx ) + δ(0,1) σy|x (16) 1 x x y y y y y|x

4

The EWMA Control Chart and the Cause-Selecting Control Chart

For distinguishing the occurrences of AC2 in the second process and AC1 in the first one, the EWMA control chart is developed for the distribution of the observed in-coming quality and the cause-selecting control chart for the distribution of the residuals to diagnose the states of the first process and the second process.

4.1

EWMA Control Chart for the First Process

When the first process is in control, the distribution (6) of Xo is used to construct the EWMA control chart. The statistic Q(0,0,k) of the EWMA control chart is defined as: Q(0,0,k) = λ1 Xo(0,0,k) + (1 − λ1 Q(0,0,k−1) )

for k = 1, 2, . . .

where λ1 is a constant with 0 < λ < 1. Let the starting value Q(0,0,0) be the first process in-control mean µx . Hence:

(17)

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Su-Fen Yang, Han-Wei Ho and M. A. Rahim

E[Q(0,0,k) ] = E[λ1 Xo(0,0,k) + (1 − λ1 Q(0,0,k−1) )]

 k−1 (1 − λ1 )ν Xo(0,0,k−ν) + (1 − λ1 )k Q(0,0,0) = E λ1  =

ν=0

  1 − (1 − λ1 )k + (1 − λ1 )k E[Q(0,0,0) ] = µx

(18)

V [Q(0,0,k) ] = V [λ1 Xo(0,0,k) + (1 − λ1 Q(0,0,k−1) )]

 k−1 (1 − λ1 )ν Xo(0,0,k−ν) + (1 − λ1 )k Q(0,0,0) = V λ1 ν=0

= λ21 =

k−1 ν=0

(1 − λ1 )2ν V [Xo(0,0,k−ν) ]

    λ1  1 − δx2 σx2 1 − (1 − λ1 )2k 2 − λ1

Thus, the distribution of the EWMA statistic is obtained:     2 λ1  2k 2 1 − (1 − λ1 ) 1 − δx σx Q(0,0,k) ∼ N µx , 2 − λ1 The control limits of the EWMA control chart are readily obtained by From (20):  λ1 (1 − (1 − λ1 )2k ) ((1 − δx2 ) σx2 ) UCL = µx + k1 2−λ 1  λ1 (1 − (1 − λ1 )2k ) ((1 − δx2 ) σx2 ) LCL = µx − k1 2−λ 1 Note that for k → ∞, the control limits converges to

λ1 (1 − δx2 ) σx2 µx ± k1 2 − λ1

4.2

(19)

(20)

(21)

(22)

Cause-Selecting Control Chart for the Second Process

Since the out-going quality Yo depends on in-coming quality Xo , the actual state of the second process cannot be identified until the effect of the in-coming quality has been removed. Therefore, the actual state of the second process may be specified by the cause-selecting values or residuals. The cause-selecting control chart is constructed by the in-control distribution of the causeselecting values. Suppose that there is a statistically significant fitted regression model between Xo and Yo : = βˆ0 + βˆ1 Xo (23) Yˆo (i,j,k)

(i,j,k)

with cause-selecting values given by: − Yˆo e(i,j,k) = Yo (i,j,k)

(i,j,k)

(24)

During the in-control process, the distribution of the cause-selecting values is given as follows:   (25) e(0,0,k) = Yo(0,0,k) − Yˆo(0,0,k) ∼ N 0, σe2 where σe2 is estimated as follows:

Effects of Measurement Error on Controlling Two Dependent Process Steps 

2   Xo(0,0,k) − X o(0,0,k)    2 1  σ + δy2 σy2  + σe2 =  1 −   y|x m  m 2   Xo(00ν) − X o(00ν) 

133



(26)

ν=1

The cause-selecting control chart is thus constructed based on (25). The statistic of the causeselecting control chart is denoted by Z(0,0,k) and defined as follows: Z(0,0,k) = λ2 e(0,0,k) + (1 − λ2 )Z(0,0,k−1)

for k = 1, 2, . . .

(27)

where λ2 is a constant with 0 < λ2 < 1. The initial value is set to Z(0,0,0) = 0. For the in-control second process, the expectation and variance of Z(0,0,k) are derived as follows:     E Z(0,0,k) = E λ2 e(0,0,k) + (1 − λ2 )Z(0,0,k−1)

 k−1 (1 − λ2 )ν e(0,0,k−ν) + (1 − λ2 )k Z(0,0,0) = E λ2  =

ν=0

  ! − (1 − λ2 )k + (1 − λ2 )k E[Z(0,0,0) ] = 0

(28)

V [Z(0,0,k) ] = V [λ2 e(0,0,k) + (1 − λ2 Z(0,0,k−1) ]

 k−1 (1 − λ2 )ν e(0,0,k−ν) + (1 − λ2 )k Z(0,0,0) = V λ2 ν=0

= λ22 =

k−1

(1 − λ2 )2ν V [e(0,0,k−ν) ]

ν=0

 λ2  1 − (1 − λ2 )2k σe2 2 − λ2

Thus, with (28) and (29) the following distribution of the statistic Z(0,0,k) is obtained.   λ2  2k 1 − (1 − λ2 ) σe2 Z(0,0,k) ∼ N 0, 2 − λ2 With (30) the control limits of the cause-selecting control chart are as follows:  λ2 UCL = k2 2−λ (1 − (1 − λ2 )2k ) σe2  2 λ2 LCL = −k2 2−λ (1 − (1 − λ2 )2k ) σe2 2 For k → ∞ the control limits converge to

λ2 ±k2 σ2 2 − λ2 e

(29)

(30)

(31)

(32)

The proposed EWMA control chart and cause-selecting control charts are used to monitor and to distinguish if the first process and/or the second process is/are out-of-control. A signal of the EWMA control chart, but no signal of the cause-selecting control chart, indicates that the first process is out-of-control, but the second process is in-control. No signal of the EWMA control chart, but a signal of the cause-selecting control chart, indicates that the first process is in-control, but the second process is out-of-control. If the EWMA control chart and the cause-selecting control chart give simultaneously a signal, then both processes seem to be outof-control.

134

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Su-Fen Yang, Han-Wei Ho and M. A. Rahim

Average Run Length Calculation

The effects of imprecise measurements on the performance of the proposed control charts are evaluated by the average run length (ARL). When the processes are all in-control, it is desirable to have a large ARL corresponding to a low false alarm rate. When at least one of the processes is out-of-control, it is desirable to have a small ARL corresponding to a high true alarm rate. Next the average run lengths for the four possible process states are derived.

5.1

First Process and the Second Processes are In-Control

Let W1 denote the number of samples until the first signal is triggered either by the EWMA chart or by the cause-selecting control chart on the condition that the first and the second processes are in-control. The probability distribution of W1 is given as follows: P r(W1 = k) = (1 − αe1 )k−1 (1 − αe2 )k−1 (1 − (1 − αe1 )(1 − αe2 ))

for k = 1, 2, . . .

(33)

With (33), the average run length or the expectation of W1 is easily obtained: E[W1 ] =

5.2

1 1 − (1 − αe1 )(1 − αe2 )

(34)

First Process is Out-of-Control, Second Process is In-Control

Let W2 denote the number of samples until the first true alarm is triggered by the EWMA chart on the condition that the first process is out-of-control while the second one is in-control. The probability distribution of W2 is given as follows: (1 − βe1 ) P r(W2 = k) = βek−1 1

(35)

From (35) the average run length is readily obtained: E[W2 ] =

5.3

1 1 − βe1

(36)

First Process is In-Control, Second Process is Out-of-Control

Let W3 denote the number of samples until the first true alarm is triggered by the cause selecting control chart on the condition that the second process is out-of-control while the first one is incontrol. The probability distribution of W2 is given as follows: (1 − βe2 ) P r(W3 = k) = βek−1 2

(37)

From (37) the average run length is readily obtained: E[W3 ] =

1 1 − βe2

(38)

Effects of Measurement Error on Controlling Two Dependent Process Steps

5.4

135

First Process and Second Process are Out-of-Control

Let W4 denote the number of samples until the first true alarm is triggered either by the EWMA chart or by the cause-selecting control chart on the condition that the first and the second processes are out-of-control. The probability distribution of W4 is given as follows: P r(W4 = k) = (βe1 βe2 )k−1 (1 − βe1 βe2 )

(39)

From (39) the average run length of W4 is obtained: E[W4 ] =

1 1 − βe1 βe2

(40)

The calculation of the ARLs given in (33) to (40) is complicated. Crowder [3, 4, 5] and Lucas and Saccucci [8] propose approximations and a Fortran program to calculate the ARL for a single EWMA control chart. We have modified the Fortran program proposed by Crowder [4] to calculate approximately the values of αe1 , βe1 and αe2 , βe2 With these values the various ARL of the proposed control charts for a sensitivity study were obtained.

6

Sensitivity Analysis

The sensitivity analysis illustrates the effects of variation of the measurement errors on the performance of the proposed control charts. Let Ψ1 and Ψ2 be the ratios of measurement error variation and the total variation of the first process and the second process, respectively: Ψ1 = Ψ2 =

δx2 1+δx2 δy2 σy2 σy|x +δy2 σy2

(41)

Usually it is assumed that Ψ1 and Ψ2 are less than 0.5, since the variations from the measurement errors are always smaller than or equal to all the variations from the processes. The values of the average run length under various combinations of Ψ1 , Ψ2 , δ(1,0) and δ(0,1) are shown in Table 1, which is organized into 36 separate panels, corresponding to the following values: δ(1,0) δ(0,1) Ψ1 Ψ2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5

From Table 1, we find that when the first process is out of control (δ(1,0) = 0), but the second process is in control (δ(0,1) = 0),an increase in Ψ1 results in an increase of the ARL; while increases in Ψ2 leave the ARL unchanged. When the second process is out of control (δ01) = 0), but the first process is in control (δ(1,0) = 0), an increase in Ψ1 leaves the ARL unchanged; while an increase in Ψ2 results in an increase of the ARL. When the first and the second processes are both out of control (δ(1,0) = 0,δ01) = 0), then any increase of either Ψ1 or Ψ2 leads to an increase of the ARL; while increases in Ψ2 lead to increases in ARL.

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Effects of Measurement Error on Controlling Two Dependent Process Steps

137

We conclude, that imprecise measurement devices may seriously affect the performance of control charts. When Ψ1 and/or Ψ2 increase(s), the ability to detect shifts of the processes gets poor. The results of sensitivity analysis are summarized in the Table 2.

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Su-Fen Yang, Han-Wei Ho and M. A. Rahim

Conclusions

An EWMA control chart and a cause-selecting control chart are proposed for monitoring and distinguishing the states of two dependent production processes with imprecise measurement devices. The EWMA control chart controls the in-coming quality for detecting small shifts in the mean of the first process. The cause-selecting control chart controls the residuals of the second process for detecting small shifts in the process mean of the second process. The performance of the proposed EWMA and cause-selecting control charts under different measurement errors is evaluated by means of the average run length. It is shown that the presence of measurement errors may seriously affect the ability of the proposed control charts to detect process disturbances quickly. Increasing variations of the measurement errors lead to decreasing probabilities of detecting the occurrence of special cause. These finding highlight the importance of proper calibration or regular maintenance of the measurement devices. The adaptive statistical process control approach may provide an improvement of the proposed control charts under various variations of measurement errors.

Acknowledgments. Support for this research was provided in part by the National Science Council of the Republic of China, grant No. NSC-91-2118-M-004-008. Financial assistance of NSERC of Canada to the third author is greatly acknowledged as well.

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Su-Fen Yang, Han-Wei Ho Department of Statistics National Chengchi University Taipei, 116 Taiwan

M. A. Rahim Faculty of Business Administration University of New Brunswick Fredericton Canada E3B 5A3