Z charts to monitoring the process mean and variance in the second step are also investigated

Key words: Control charts; dependent process steps; optimization technique; Markov chain.

1. INTRODUCTION

Control charts are important tools in statistical quality control. They are used to effectively monitor and determine whether a process is in-control or out-of-control.

Shewhart (1931) developed the _X control chart which is easy to implement and has been widely used for industrial processes. However, even though Shewhart X control charts,

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always monitor a process by taking samples of equal size at a fixed sampling interval (FSI), they are usually slow in signaling small to moderate shifts in the process mean.

Consequently, in recent years several alternatives have been developed to improve the performance of X control charts. One of the useful approaches to improving the detection ability is to use a variable sampling interval (VSI) and/or a variable sample size (VSS) control chart instead of the traditional FSI and/or fixed sample size (FSS). Whenever there is some indication that a process parameter may have changed, the next sampling interval should be shorter and/or the next sample should be larger. On the other hand, if there is no indication of a parameter change, then the next sampling interval should be longer and/or the next sample should be smaller.

The properties of the X chart with VSIs were studied by Reynolds, et al. (1988).

Their paper has been extended by several others: Reynolds and Arnold (1989); Amin and Miller (1993); Baxley (1996); Reynolds, and Arnold, and Baik (1996). Tagaras (1998) reviewed the literature on adaptive control charts. Very little work has been down on VSI control charts for simultaneously monitoring process mean and variance. Chengular, et al.

(1989) detected process mean and variance using VSI X and R control charts.

Reynolds and Stoumbos (2001) discussed the properties of VSI X and MR control charts.

These papers show that most work on developing VSI control charts had aimed to solve the problem of monitoring process mean.

However, these articles assume that there is only a single process step whereas many products are currently produced with several dependent process steps. Consequently, it is not appropriate to monitor these process steps by utilizing a control chart for each individual process step. Zhang (1984) proposed the simple cause-selecting control chart to control the specific quality in the current process by adjusting the effect of the in-coming quality variable (X) on out-going quality variable (Y), since the in-coming quality variable on the first process step and the out-going quality variable on the second process step are dependent. The cause-selecting values (

e

) are Y minus the effect of X, and the cause-selecting control chart is constructed accordingly. Wade and Woodall (1993) reviewed and analyzed the cause-selecting control chart and examine the relationship between the cause-selecting control chart and the Hotelling T² control chart. In their opinion the cause-selecting control chart outperforms the Hotelling T² control chart, since it is easy to distinguish whether the second step of the process is out-of-control. Therefore, it seems reasonable to develop variable control schemes to control dependent process steps.

However, the properties of the variable sampling interval (VSI) control charts used to

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control mean and variance on two dependent process steps have not yet been addressed.

Therefore, studying the performance of the joint VSI control charts on two dependent process steps is reasonable. In this paper, the joint VSI

Z -

_X Z S2 and

Z -

_e 2

Se

Z control charts with variable sampling intervals are proposed for the control of mean and variance in two dependent process steps. In the next section, the performance of the VSI

Z -

_X Z S2

and

Z -

_e 2

Se

Z control charts is measured by the adjusted average time to signal (AATS), which is derived using a Markov chain approach. Finally, we illustrate the application of the proposed control charts using the example of process control for automobile braking

system on the process steps, and then compare the performance between the VSI

Z -

X Z S2 and

Z -

_e 2

Se

Z control charts and FSI

Z -

_X Z S2 and

Z -

_e 2

Se

Z control charts.

Whenever quality engineers cannot specify the values of variable sampling intervals, the optimal VSI

Z -

_X Z S2 and

Z -

_e 2

Se

Z control charts are suggested. Furthermore, the impacts of misusing

Z -

_Y 2

Sy

Z charts to monitoring the process mean and variance in the

second step are also investigated.

2. DESCRIPTION OF THE JOINT VSI Z

_X -Z S2

AND Z -

_e 2

Se

Z

CHARTS

Consider a process with two dependent process steps controlled by the joint VSI

Z

X -Z S2 and

Z -

_e 2

Se

Z control charts. Let X be the measurable in-coming quality variable on the first process step. Assume further that this process starts in a state of statistical control, that is, X follows a normal distribution with the mean at its target value,μ , and the standard deviation at its target value_X

σ

_X ; let Y be the measurable out-going quality characteristic of interest for the second process step, and follow a normal distribution conditional on X. Since the two process steps are dependent, and the second process step is affected by the first process step, then following Wade and Woodall (1993), the relationship between X and Y is generally expressed as

Y

_i

X

_i

= f ( ) X

_i

+

ε_i

, i = 1 , 2 , 3 ,... m

(2-1)

where, it is assumed that

ε

_i ~

NID

(0,

σ

²). Let Y instead of

Y | X . If the function f

(

X

_i) is known, the values of the standardized error term

ε

_i^* =

^Y

^{i −}

σ ^f

⁽

^X

ⁱ⁾ are called the cause-selecting values since they are the values of

Y adjusted for the effects of

_i

X . In

_i practice, the true function

f

(

X

_i) is usually unknown and thus must be estimated using the

m

observations obtained from the initial

_m

samples of size one, and thus the estimate for

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) (

X

_i

f

will be

Y

^{^}_i (Yang (2003)). The residuals,

e

_i =

Y

_i −

Y

^{^}_i, are generated by the model

used. Hence,

e ~NID(0,

_i

σ

_e²). Consequently, the standard residuals

e i i i

Y e Y

σ

^

*

= −

are called the cause-selecting values. The X chart is thus constructed to monitor the mean of

X on

_i the first step, and the

_e

chart is constructed to monitor the mean of

e on the second

_i step.

However, in our study the chosen sample size is not one and the rational samples of size ( n ) are taken from the two dependent process steps. Plotting the sample data to obtain a sample profile and then establish the reference line of Y and X (see Kim, Mahmoud and Woodall (2003)). To monitor the mean and variance of X on the first step the X -

S

² charts should be constructed, and to monitor the mean and variance of e on the second step the

_e

-

S charts should be constructed. The

_e²

X - S charts and

²

e

-

S charts are

_e² called cause-selecting control charts.

For engineers to use the control charts easily, the sample means and sample variances are standardized as follows.

) , n ~N(

/ σ

μ Z X

X i X

X_i = − 01 , ( 1)₂ ²

2

x i

σ S Z

n-Si = ~

χ

²(n-1)^,

) 1 , 0 (

/ ~

N

n Z e

e i

ei ⁼

σ

^{, 2}

) 2

1 (

2

e e

S

σ

S

Z n-

ⁱ

ei = ~

χ

²(n-1) , (2-2)

where ¹

n X X

n j

ij i

∑

=

= ,

n-) X (X S

n j

i ij

i 1

1

2

2

∑

=

−

= , ¹

n e e

n j

ij i

∑

=

= and

n-) e (e S

n j

i ij

e_i 1

1

2

2

∑

=

−

= .

,...., 3 , 2 , 1 , ,..., 3 , 2 ,

1 m j n

i= =

Assume that once a special cause occurs it affects the X-variable with probability v and the functional relationship (or

_e

-variable) with probability 1-v. That is, the mean of

X shifts from

ij

μ

_X ^to

μ

_X⁺

δ

₁

σ

_X ⁽

δ

₁

≠ 0

) and the variance shifts from

σ

_X ^to

δ

₂

σ

_X (

δ

₂

_{> 1}

) with probability v, and the mean of the

e shifts from 0 to

_ij

δ

₃ ⁽

δ

₃ ≠ 0 ^{) and} the variance shifts from

σ

_e to

δ

₄

σ

_e (

δ

₄

_{> 1}

) with probability 1-v. The out-of-control distribution of

X and/or

_ij

e will be adjusted to in-control state, once at least one true

_ij

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signal is obtained from the proposed control charts. Let

T be the time until the

_sc occurrence of special cause, and follow an exponential distribution of the form

f

(

t

)=

λ

exp(−

λ t

)

t

>0 (2-3) where

1 /

λ is the mean time that the process remains in a state of statistical control.

An in-control state analysis for the joint VSI

Z -

X Z S2 and

Z -

e 2

Se

Z control charts is performed since the shifts in the process mean and variance on step 1 and/or step 2 do not occur when the process is just starting, but occur at some time in the future. The standardized samples

Xi

z -

zS2 and

ei

z -

2

Se

z

are plotted on the joint VSI

Z -

X Z S2 and

Z

e

-

₂

Se

Z control charts with warning limits of the form

w

X

± , ₂

w ,

S ±

w

_e and ₂

Se

w ,

在文檔中 相依製程上平均值和變異數過度調整之適應性管制(II) (頁 60-64)