Control chart for multivariate attribute processes - Taylor & Francis ...

int. j. prod. res., 1998, vol. 36, no. 12, 3477± 3489

Control chart for multivariate attribute processes X. S. LU² , M. XIE² *, T. N. GOH² and C. D. LAI³ Many industrial processes are multivariate in nature since the quality of a product depends on more than one variable. Multivariate control procedures can be used to capture the relationship between the variables and to provide more sensitive control than that provided by the application of univariate control procedures on each variable. Much has been done on the multivariate variable processes, such as embodied in control procedures based on Hotelling’ s T 2 statistic. However, little work has been done to deal with the control of multivariate attribute processes, which is very important in practical production processes. In this paper, we develop a Shewhart-type control chart to deal with multivariate attribute processes, which is called the multivariate np chart (MNP chart). The control chart uses the weighted sum of the counts of nonconforming units with respect to all the quality characteristics as the plotted statistics. It enhances the e ciency of identifying the critical assignable cause when an out-of-control signal appears. It is also simple to interpret for out-of-control signals. The practical application of the MNP chart is also discussed in this paper with an example presented to demonstrate the approach of the MNP chart and to compare with the univariate np control charts which are commonly used in industry.

1.

Introduction

With modern data-acquisition equipment and on-line computers it is now common practice to monitor several quality characteristics simultaneously rather than a single quality characteristic during production. For example, a chemical process may be a function of temperature, pressure, and ¯ owrates, all of which need to be monitored carefully. Another illustration is a box that can be regarded as nonconforming if its width, length, or weight do not all meet the speci® cations. Montgomery (1996) names the former process a multivariate variable process and we similarly call the latter a multivariate attribute process. The most common type of control chart used in a production process is the Shewhart control chart. A multivariate quality characteristics process could be monitored by applying an univariate Shewhart chart to each quality characteristic. If these quality characteristics are independent of each other, this would be an adequate procedure. However, multivariate quality characteristics in many production processes tend to be correlated and therefore the results could be misleading and di cult to interpret. More speci® cally, Montgomery and Mastrangelo (1991) have shown that the use of univariate Shewhart charts in a multivariate quality process has distorted the simultaneous monitoring of the characteristics, in that the type I error and the probability of a point correctly plotting in control are not equal to their advertised levels for the individual control charts. Thus it would be necessary to use a Revision received November 1997.

² Department of Industrial and Systems Engineering, The National University of

Singapore, Singapore. ³ Massey University, Palmerston North, New Zealand. * To whom correspondence should be addressed. 0020± 7543/98 $12. 00

Ñ

1998 Taylor & Francis Ltd.

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X. S. L u et al.

multivariate control procedure which will take into account the internal relationship of the correlated quality characteristics. Furthermore, it will be more practical and economical to use a single multivariate control scheme rather than several univariate Shewhart control charts. Lowery and Montgomery (1995) have shown that normally a multivariate control scheme has a better sensitivity than one based on the univariate control charts in monitoring multivariate quality processes. The use of multivariate control procedures for production processes monitoring is increasingly popular. This is a result of recent advances in multivariate quality control (MQC), such as in multivariate cumulative sum control charts (e.g. Woodall and Ncube (1985), Crosier (1988), Healy (1987), Pignatiello and Runger (1990), and Hawkins (1993)), multivariate exponentially weighted moving average control charts 2 (e.g. Lowery et al. 1992), and Hotelling’s T charts, as well as the improved e€ ectiveness of these techniques to identify the cause of an out-of-control signal. Other perspectives about MQC are available in Nomikos and MacGregor (1995), Margavia and Conerly (1995), Kourti and MacGregor (1996), Runger (1996), and Aparisi (1996) etc. Jackson (1985, 1991), Ryan (1989), Wierda (1994) and Lowery and Montgomery (1995) provided excellent summaries and discussions of multivariate control charts. However, almost all papers on multivariate control charts deal with multivariate variable processes although it is easy to collect count related data as exact measurements are not needed (see Robinson and Miller 1989, Bourke 1991, Xie and Goh 1992, Kaminsky et al. 1992, Rowlands 1992, Glushkovsky 1994 and Montgomery 2 1996). In an early paper, Patel (1973) proposed a Hotelling’s type c chart to monitor the time-dependent observations from multivariate binomial or multivariate Poisson populations. Because of its complexity, the scheme has not been widely implemented in practice. Moreover, similar to any Hotelling’ s multivariate type 2 control procedure, Patel’s c chart only has the upper control limit, which cannot be used to detect both process deterioration and process improvement. As to the basis of many existing multivariate control charts, Hotelling’s T 2 statistic is based on the assumption that the multivariate quality characteristics conform to normal distributions. In case of multivariate attribute processes, the count of nonconforming units of each quality characteristic usually conforms to a Binomial distribution. In this paper, an attempt is made to establish the multivariate np control chart (MNP control chart) to deal with multivariate attribute processes. A new statistic X, which is the weighted sum of the counts of nonconforming units of each quality characteristic in a sample, is introduced and the control limits are derived. Based on the MNP chart we can easily judge the performance of a multivariate attribute process on the basis whether the statistic X is out of the control limits. The practical operation of the MNP control chart is illustrated and the condition for the choice of sample size is also given. An approach is described to easily and e ciently identify the critical quality characteristic which is the major contributor to an out-of-control signal. A numerical example is also given to show the approach of applying the MNP chart. It has been found that our MNP chart which can be easily implemented is more sensitive than the univariate np chart in detecting process shifts of a multivariate attribute process. 2.

Framework for the M NP chart

For the process being monitored, assume that there are m quality characteristics. Denote by pi the probability that an item is nonconforming with respect to quality

Multivariate attribute processes: a control chart

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characteristic i. The quality characteristics may not be independent and we denote the correlation coe cient between characteristic i and characteristic j by d ij . Note that: d ij

=d

ji

|d ij | £ 1 d ij = 1, i = j . P = (p1 , p2 , . . . , pn )

( 1)

Using a matrix notation, let be the fraction nonconforming vector and S = d ij m ´ m be the correlation coe cient matrix. Denote by C = (C1 , C2 , . . . , Cm ) the vector of counts of nonconforming units, where Ci is the count of nonconforming units with respect to quality characteristic i in a sample. We now introduce a new statistic X, which is the weighted sum of the nonconforming units of all the quality characteristics in a sample:

[]

m

X= i= 1

Ci /Ï pi .

( 2)

The quality characteristics of a multivariate attribute process a€ ect the process di€ erently even if these characteristics change by the same amount. More speci® cally the smaller the fraction nonconforming pi , the more the count Ci contributes to the statistic X of the multivariate attribute process. In other words, the change of the statistic X is more sensitive to the change of the count Ci with a smaller fraction nonconforming. Therefore, it is obvious that the weight of count Ci must be in inverse proportion to the fraction nonconforming pi . The reasons for the choice of 1 /Ï pi as the weight of quality characteristic i’s contribution to test statistic X are as below: 2 First of all, unlike Hotelling’s T type control charts for multivariate variable processes, test statistic X of the MNP chart is a straightforward extension of that of the univariate np chart. Second, the expected value of the test statistic m E (X) = n i= 1 Ï pi is also an increasing function of pj , which is related to the internal properties of the process being monitored and agrees with its de® nition as the weighted sum of nonconforming units with respect to all quality characteristics. Finally, this particular choice makes the derivation of the control limits and sample size requirement easy and presents a clearly de® ned MNP chart, the practical merits enhancing the chances of being adopted by industry. It should be noted that the nonconformance of the quality characteristics may carry di€ erent physical severity, that is, the nonconformance in one dimension may be physically more serious than that in another dimension. This reality is commonly found in such multivariate attribute processes as produce complex products, for example, automobiles, computers, major appliances, etc. Montgomery (1996) suggested a reasonable remedy to classif y the nonconformity according to its severity by including a demerit system. Denote di the number of demerits which indicates the severity of the nonconformance in quality characteristic i, the above test statistic X can be extended to be more applicable for the real situations in the following manner: m XD =

i =1

di Ci /Ï pi .

Accordingly, the nonconformance vector D is introduced as

( 3)

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X. S. L u et al. D = (d1 , d2 , . . . , dm ) .

(4)

The mean and variance of the test statistic X can be calculated as follows. m

E (XD ) = E

j =1

dj Cj /Ï pj

m

=

j= 1

dj E (Cj )

j =1

Ï pj = n

dj npj

m

= m

j= 1

Ï pj

dj Ï pj ,

(5)

and m

Var ( XD ) = Var

dj Cj j= 1

m

=

2

dj Var (Cj )

Ï pj . pj + 2

j =1 m

= j =1

di dj d

Var (Ci ) Var (Cj )

ij

i