In this paper we propose a new control chart, the synthetic control chart, for the detection of increases in the fraction nonconforming (p). The operation, design, and performance of this control chart are described. The synthetic chart is a combination of the np chart and the conforming run length chart. It provides the user with some freedom in adjusting the control chart parameters so that the out-of-control average time to signal can be minimized. Numerical tests have indicated that the synthetic chart has a higher power for detecting process shifts in the fraction nonconforming than both the np chart and the conforming run length chart. The improvement is significant for all cases, resulting in a 50% or greater reduction of the out-of-control average time to signal. The application of the synthetic chart may therefore substantially enhance the effectiveness of statistical process control for attributes.

Keywords: Average Time to Signal, np Control Charts.

*by* **ZHANG WU, SONG HUAT YEO and TREVOR A. SPEDDING,
Nanyang Technological University, Singapore 639798 **

*INTRODUCTION*

The np control chart has been widely used in industry to
monitor the number *d* of nonconforming units in samples
of *n* inspected units. A nonconforming unit is a product
which fails to meet at least one specified requirement. The
np chart is equivalent to the fraction nonconforming *p*
chart, but it is easier to interpret when the sample sizes
are constant (Montgomery (1997)). The parameter *p* is
defined as the ratio of the number of nonconforming units
in a population to the total number of units in that population.

In this paper we designate a process shift in which the fraction
nonconforming of the process has been increased or decreased
from the in-control value of *p*_{0} as an *upward
shift* or a *downward shift*, respectively. For the
np chart, if *d* in a sample of *n* units satisfies
*LCL _{np}*

More recently, the development of the conforming run length
(CRL) control chart has been documented by many researchers
(Bourke (1991), Kaminsky et. al. (1992), Nelson (1994), Glushkovsky
(1994), Xie et. al. (1995)). It has also been highly recommended
in a recent review paper by Woodall (1997). Basically, the
random variable CRL is the number of inspected units between
two consecutive nonconforming units, inclusive of the nonconforming
unit at the end. The idea is that the distribution of the
conforming run length CRL will change when the fraction nonconforming
*p* of a process shifts. In particular the expected CRL
is shortened as *p* increases, and it is lengthened as
*p* decreases. In Figure
1, the white and black dots are used to denote conforming
units and nonconforming units respectively in a process. The
process starts at *t* = 0, and three samples of CRL are
shown with *CRL*_{1} = 4, *CRL*_{2}
= 5, and *CRL*_{3} = 3. For a CRL chart, if a
CRL value satisfies *LCL*_{CRL} __<__ *CRL*
__<__ *UCL*_{CRL}, then the process is
considered to be under control. Here, *LCL*_{CRL}
and *UCL*_{CRL} are the lower and upper control
limits of the CRL chart. However, if *CRL* < *LCL*_{CRL},
then an upward process shift is signaled, and if *CRL*
> *UCL*_{CRL}, then a downward process shift
is signaled. In contrast to the np chart, the lower control
limit in the CRL chart is used to detect the upward process
shift, and the upper control limit is used to detect the downward
process shift. The description of the CRL chart is usually
based on 100% inspection, because every unit has to be
accounted for and classified as either a conforming unit or
a nonconforming one.

The effectiveness of a control chart can be measured by the
average time to signal (ATS). For the attribute control charts,
ATS is defined as the average number of inspected units required
to signal an out-of-control case (with a certain *p*
value) after it has occurred. The smaller the ATS, the earlier
the process shift is signaled and the more effective the attribute
control chart.

In this paper a synthetic control chart is proposed by combining the operations of an np chart and a CRL chart. It is shown that the out-of-control ATS may be substantially reduced by properly designing the parameters of this new chart. As a result the effectiveness of the attribute statistical process control (SPC) can be improved. The proposed chart is closely related to the control chart for monitoring the process mean discussed in Wu and Spedding (2000a, b).

The presentation of the new chart is based on 100% inspection of all process outputs, since this is a common procedure for an attribute control chart (especially the CRL chart). However, as explained in Bourke (1991), the approach can be easily applied to non-100% inspection cases. In these cases, one uses the synthetic chart to check the number of the conforming units observed between successive observed nonconforming units, ignoring any units produced during periods of non-inspection.

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