cause-selecting control charts with variable parameters

CAUSE-SELECTING CONTROL CHARTS WITH VARIABLE PARAMETERS

Rassoul Noorossana1 and Maryam Shekary A.2

Industrial Engineering Department, Iran University of Science and Technology, Tehran, Iran, (+9821) 77240540-50 Ext. 3017, [email protected], [email protected]

Abstract: Cause-selecting control charts are effective tools to monitor and detect shifts in a process with two dependent steps. In this paper, we extend the study of by xZ and eZ control charts with variable parameters in order to monitor quality characteristics, which are produced by the first and the second step of a two-stage process, respectively. We utilize three different sample sizes, sampling intervals, and 2 different control limit coefficients. Choosing any couple of oncoming parameter is based on the previous sample statistics. Finally, the method of constructing the proposed control chart and determining its parameters is illustrated. Keywords: control charts, adaptive, cause-selecting, two steps, variable parameters. Introduction Control charts are effective tools in statistical process control which was introduced by Shewhart at the first time. A control chart has three parameters: the sample size (n), the sampling interval (h) and the control limit coefficient (k). Control charts with fixed parameters (traditional Shewhart charts) have been widely used in industrial process control, Because of their ease of construction and operation. This kind of control charts can detect large shifts in the process mean, but they are weak in detecting small and moderate shifts. Consequently several alternatives have been extended to overcome this weakness of Shewhart control charts. One of the most useful one is using adaptive control chart which has at least one variable parameter. In adaptive control charts, switching among different parameters depends on the current sample statistics plotted on the control chart. If the current plotted point is close to the center line, there is no evidence that the process parameter has changed, so the next sample will be taken from the process with a smaller sample size and/or a bigger sampling interval, and/or a larger control limit coefficient will be used. If the current sample statistic is close to the control limit but still within them, there is an indication of process parameter changes, therefore the next sample will be taken from the process with bigger sample size and/or smaller sampling interval, and/or smaller control limit coefficient will be utilized. Adaptive control charts was first proposed by Reynolds et al. (1988). They investigated the X control chart with variable sampling intervals (VSI). They utilize average time to signal and average number of sample to signal as performance measure and illustrated that their control chart works more efficient than Shewhart control chart. Several authors such as Cui and Reynolds (1988), Reynolds and Arnold (1989), Runger and Pignatiello (1991), and Amin and Miller (2007) have extended this concept. Prabhu et al. (1993) and Costa (1994) independently investigated the X control chart with variable sample sizes (VSS). Other researchers such as Zimmer et al. (1998) ,and Epprecht & Costa (2001) contributed on this concept. Prabhu et al. (1994) introduced the X control chart with variable sample sizes variable sampling intervals (VSSI) at the first time. Costa (1997), Costa (1999a), and Mahadik & Shirke (2009) expanded these kind of control charts. X control chart with variable parameters (VP) was first proposed by Costa (1999b). Other authors such Costa (2000) and Magalhães et al. (2002) extended the VP control charts. Tagaras (1998) review the literature on adaptive charts.

1- Corresponding to: Dr. Rassoul Noorossana, Professor of Applied Statistics. He is a senior member of ASQ. 2 - Maryam Shekary A. is a graduate student at the Industrial Engineering Department.

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Proceedings of the 41st International Conference on Computers & Industrial Engineering

696

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The idea of cause-selecting control charts was first proposed by Zhang (1984, 1985a) to monitor a multi stage process. When you want to monitor a process consisting of several dependent steps, and your product passes these steps respectively, it is not appropriate to utilize a separate control chart for each step. Zhang considered a two dependent process steps and adjusted the effect of in-coming quality variable (X) on out-going quality variable (Y). Wade and Woodall (1993) reviewed and analyzed the cause-selecting control chart and examined the relationship between the cause-selecting control chart and Hotelling T2 control chart for a process with two dependent steps. In their opinions, the cause-selecting control chart outperforms the Hotelling T2 control chart because it is easier to distinguish which process step is out of control, and unlike Hotelling T2 control chart, it has the ability of handling complicated relationships between variables. Yang and Su (2006) proposed VSS cause-selecting control charts using three variable sample sizes. Yang and Su (2007a) investigated VSI cause-selecting control charts with three different sampling intervals. Yang and Su (2007b) proposed VSSI cause-selecting control charts with three variable sample sizes and sampling intervals. Yang and Chen (2009) and Yang (2010) extended VSI cause-selecting control charts to detect shifts in the means and variances of two dependent process steps. In this paper, a set of VP cause-selecting control charts using three different sample sizes and sampling intervals and control limit coefficients are proposed to monitor a process with two dependent steps. The performance of the proposed chart is evaluated by average time to signal (ATS) criterion. Performance of the cause-selecting control chart with fixed parameters is compared to the proposed chart numerically. Design of the xZ and eZ Control Charts with variable parameters

Through this paper, a process with two dependent steps is considered. Let X and Y represent the quality characteristic of interest for the first and second step respectively, where X is assumed to have a normal distribution with in-control mean x and standard deviation x and Y is assumed to have a

normal distribution conditioned on X. Moreover, it is assumed that that the process starts in an out of control state and values of X cannot be observed in the first step but can be measured at the end of the second step. When a sample of size ns is taken after sampling interval hs, the paired observations (Xi, Yi) are measured and the corresponding statistic is compared with ks. Since the two process steps are dependent and Y is affected by X, the relationship between X and Y is expressed as

( ) , 1, 2,3,..., , (1)i i iiY X f X i m

where the random error 2(0, )i NID , when the second process step is in control. Since the exact

function ( )if x is usually unknown, it should be estimated using the initial m samples of size one taken

from the process. Let �iY be an estimate for ( )if x . Consequently, one can compute the residuals of the

model using �i i ie Y Y , where the random variable 2(0, )ie NID . The standard residuals

* ( )i i ie Y Y

are called the cause-selecting values because they are the values of Yi adjusted for the effect of Xi. In order to monitor the mentioned process, two separate control charts will be constructed. One control chart for monitoring the mean of the quality characteristic X denoted by xZ

and another for monitoring the mean of standardized residuals denoted by eZ . When the first step is

in an out of control state, the mean changes to 1x x , and when the second step shifts to an out of

control state, the mean of ie changes from 0 to 2 ,where 2 0 .If at least one control chart produces a signal, the process will be stopped to search for assignable cause. The standardized sample statistics xZ and eZ are defined as

/i

i xx

x s

xz

n

and (2)

/i

ie

s

ez

n


697

where 1sn

i ij sjX X n , 1sn

i ij sje e n , and �ij ij ije Y Y for 1,2,...,i m and 1,..., sj n .

The position of the current sample statistic plotted on each control chart determines the next sample size and sampling interval and control limits. Three different sample sizes and sampling intervals and two different control limit coefficients are utilized, 1 2 30 n n n , 3 2 10 h h h and

2 10 k k . Corresponding to each control limit, a warning limit will be calculated, where

0 s sw k . These warning and control limits divide each X adaptive control chart into the following three zones:

11 1 1( , )xI w w (Central region of wide xZ )

21 1 1 1 1, ,xI k w w k

(Warning region wide xZ )

31 1 1( , )xI k k (Control region of wide xZ ) (3)

12 2 2( , )xI w w (Central region of narrow xZ )

22 2 2 2 2, ,xI k w w k (Warning region narrow xZ )

32 2 2( , )xI k k (Control region of narrow xZ )

The different zones of each e control chart (wide or narrow) are as same as above. If both current sample points xZ and eZ of wide control charts fall within central region, the next sample size and

sampling interval would be 1 1( , )n h , and the next statistic will be compared with w1 and k1. If one of the current sample points falls within the central region and the other falls within the warning region of wide control charts, the next sample will be taken with medium sample size and sampling interval

2 2( , )n h , and the calculated statistic will be compared with w1 and k1. If both current sample points xZ

and eZ of wide control charts fall within the warning region, it is reasonable to use the biggest sample

size and smaller sampling interval 3 3( , )n h , and w2 and k2 will be used as control and warning limits. If

both current sample statistics xZ and eZ of narrow control charts falls within the central region, the

small sample size and large sampling interval 1 1( , )n h will be used for the next sampling and w1 and k1

will be used as control and warning lines. If one of the current sample statistics xZ and eZ of narrow

control charts falls within the central region, the medium sample size and sampling interval 2 2( , )n h

will be used for the next sampling, and the plotted point will be compared with w1 and k1. If both current sample statistics xZ and eZ of narrow control charts falls within the warning region, the next

sample will be taken from the process with biggest sample size and smaller sampling interval 3 3( , )n h and w2 and k2 will be utilized as control and warning limits. If at least one of the sample points falls outside the related control region, the process should be stopped to search for the assignable cause. The relationship for selecting the next sample size and sampling interval and control limits using the current sample statistics ( , )

i ix eZ Z are shown in equations 4.

1 11 1

2 11 2

2 22 2

12 1

1 1 1 1 21 2

2 2 1 1 11 1

3 3 2 1 21 2

2 2 1 22 2

( , , )

( , , )( , , )

( , , )

( , , )

i ii i

i ii i

i ii i

i ii i

e e e eX X X X

e e e eX X X Xi i i

e e e eX X X X

e e eX X X X

n h k if Z I Z I or Z I Z I

n h k if Z I Z I or Z I Z In h k



22

2,3, 4,... (4)

e

i

It is assumed that the biggest sample size, smallest sampling interval and narrow control limits are used at the start of the process to indicate initial process changes. If the cause-selecting control scheme with variable parameters is to be compared to the corresponding chart with fixed parameters, both charts should have the same average sample size and sampling interval during in control period, Hence


698

3 3 1 2 0( , , 0, 0) (5)s x x e eE n Z I Z I n

3 3 1 2 0( , , 0, 0) (6)s x x e eE h Z I Z I h

where 0n , 0h ,and 0k are the parameters of the cause selecting scheme with fixed parameters. Based

on equation (5), the following equation can be obtained:

1 , 1 1 1 1 2 , 1 1 1 1 2

1 , 1 2 1 1 2 , 1 2 1 1 2

2 , 1 1 1 1 2 , 1 1 2 1 2

2 , 1 1 2 1 2 , 1 1 1 1 2

( 0, 0) ( 0, 0) (7)

( 0, 0) ( 0, 0)

( 0, 0) ( 0, 0)

( 0, 0) ( 0, 0)

x i x e i e

x i x e i e

x i x e i e

x i x e i e

n P Z I P Z I

n P Z I P Z I

n P Z I P Z I

n P Z I P Z I

n

2 , 1 2 1 1 2 , 1 2 2 1 2

2 , 1 2 2 1 2 , 1 2 1 1 2

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

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

0

( 0, 0) ( 0, 0)

( 0, 0) ( 0, 0)

( 0, 0) ( 0, 0)

( 0, 0) ( 0, 0)

(

x i x e i e

x i x e i e

x i x e i e

x i x e i e

P Z I P Z I

n P Z I P Z I

n P Z I P Z I

n P Z I P Z I

n P

3 1 2 3 1 20, 0) ( 0, 0)x x e eZ I P Z I

The same equation can be obtained for the expected value of sampling interval by replacing h instead of n in the above equation. In addition, the probability for a point falls within the central region of narrow and wide xZ control chart should be equal. Because the control and warning limits of xZ and

eZ are the same, this condition will be established for eZ chart too spontaneously. Moreover, the cause selecting control chart with fixed parameters and the proposed chart should have the same false alarm rate. Therefore:

1 1 1 3 1 2 2 1 2 3 1 2( , 0, 0) ( , 0, 0) (8)x x x xX X X XP Z I Z I P Z I Z I

21 , 1 1 1 1 2 , 1 1 1 1 2

21 , 1 2 1 1 2 , 1 2 1 1 2

21 , 1 2 2 1 2 , 1 2 1 1 2

1 2 ( ) 1 ( 0, 0) ( 0, 0) (9)

1 2 ( ) 1 ( 0, 0) ( 0, 0)

1 2 ( ) 1 ( 0, 0) ( 0,

x i x e i e

x i x e i e

x i x e i e

k P Z I P Z I

k P Z I P Z I

k P Z I P Z I

21 , 1 2 1 1 2 , 1 2 2 1 2

22 , 1 1 2 1 2 , 1 1 2 1 2

22 , 1 2 2 1 2 , 1 2 2 1

0)

1 2 ( ) 1 ( 0, 0) ( 0, 0)

1 2 ( ) 1 ( 0, 0) ( 0, 0)

1 2 ( ) 1 ( 0, 0) ( 0

x i x e i e

x i x e i e

x i x e i e

k P Z I P Z I

k P Z I P Z I

k P Z I P Z I

2

20

, 0)

1 (2 ( ) 1)k

The proposed control charts have ten parameters 1 2 3 1 2 3 1 2 1 2( , , , , , , , , , )n n n h h h k k w w and we have 4

equations (5, 6, 8, 9) which should be satisfied. We suggest first determine 1 2 3 1 2 3 1( , , , , , , )n n n h h h k as

known parameters and then calculate 2 1 2( , , )k w w from above equations to construct and utilize the proposed cause-selecting control charts. Conclusions In this article, adaptive xZ and eZ cause-selecting control charts with three variable sample sizes and sampling intervals and two variable control limit coefficients were proposed to control two dependent


699

process steps. The method of switching among different parameters was explained obviously. Finally, the method of constructing cause-selecting control chart with variable parameters and calculating its warning limits was demonstrated. References Amin, R. W. and Miller, R. W. (2007). A robustness study of X charts with variable sampling intervals. Journal of Quality Technology 25: 36-44.

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cause-selecting control charts with variable parameters

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