[Design of a
flexible zone individuals control chart]讀後心得
Introduction
The control chart for individual units is an important statistical process control
(SPC) tool used in the following situations: (1) only one measurement or observation
is meaningful at each time, such as for humidity, voltage, results of chemical processes, accounting data (Nelson 1982), suspended air particles count, and machine
down time, (2) the production rate is so slow that it takes a day, a week, or a month
to get a single observation (Burr 1976), (3) inspection is expensive or lengthy (Juran
1974), (4) an automated process is used so that every unit manufactured is automatically inspected, (5) a direct comparison of plotted measurements to speci® cations limits is needed, and (6) the operator needs a chart that is easy to use and
understand or is customary in his ® eld as in the practice of quality control in clinical
chemistry.
The operation of a chart for individuals is like any Shewhart chart in that
samples of size one are taken and the measurements, X, are plotted on a chart
with control limit ks above and below the process mean ¹0 or its estimate X. The
value of s is usually estimated from the average of moving ranges of size two (see
Nelson 1982).
Shewhart (1931) recommends k = 3 and his original `Criterion I’ (also known as
AT&T rule 1) suggests that if a point falls outside the control limits, then an action
should be taken to ® nd the possible assignable cause.
One disadvantage of the chart for individuals is its insensitivity in detecting small
shifts. For example, it takes an average of 155.22 samples (observations) for a shift of
0.5s in the process mean to be detected. This average number of samples needed
until detection is called the `average run length’ (ARL). In this paper, the ARL value
when there is no shift in the process is called ARL 1. The ARL value when there is a
shift in the process is called ARL 2.
One remedy for this insensitivity to small shifts is the use of the traditional runs
rules, such as the AT&T rules as described in the Western Electric Handbook (1956).
For example, if AT&T rules 1, 2, and 3 are used, then the ARL 2 is reduced to 38.64
when detecting a shift of 0.5s in the process mean. This implies that the use of
additional runs rules has the desirable e€ ect of reducing the average time of
detecting
the shift. However, this speedy detection time is achieved at the cost of reducing
ARL1. This implies a larger probability of false alarm; an undesirable property. For
example, the false alarm probability increases almost threefold; from 0.0027 for an
individuals chart with no runs rules, to 0.0075 when rules 1, 2, and 3 are used.
Parkhideh and Parkhideh (1996) introduced the flexible zone methodology in
order to keep the advantage of the traditional runs rules while decreasing its disadvantage of high false alarms. This research uses the ¯ exible zone philosophy in
order to design the individuals chart based on the practitioner’s desired ARL 1 and
ARL2 values. The advantages of designing charts based on the desired ARL values
are discussed by Woodall (1985). Originally, Page (1961) designed CuSum charts to
have speci® ed ARL values for: (1) when the shift is zero, and (2) when the shift is
greater than or equal to a value considered important enough to detect. Finally, the
results of the research are summarized in the form of ARL tables such that the
¯ exible zone individuals charts with runs rules 1 and 2, or runs rules 1, 2, and 3
can be readily compared against each other as well as against the Shewhart charts
with the traditional runs rules.
Flexible zone runs rules
Figure 1 shows a flexible zone chart. Traditionally, the use of supplementary runs
rules requires that the chart be divided into six equal zones of width s (Western
Electric 1956, Roberts 1958). Starting from the upper or lower control limit and
moving toward the centre of the chart, these zones are called A, B, and C, respectively. In a ¯ exible zone chart (Parkhideh and Parkhideh 1996), the width of the
zones A, B, and C do not have to be equal. In fact, the lines ¹0 + k2s and ¹0 + k3s
define flexible zone A on the upper portion of the chart. Similarly, the lines ¹0 - k2s
and ¹0 - k3s define flexible zone A on the lower portion of the chart. Flexible zone B
on the upper portion of
¹0 + k2s , and so on.
the chart is de® ned as the area between ¹0 + k1s
Figure 1.
Flexible zones for rules 1, 2, and 3.
and
Using these zones the most popular of the runs rules are:
Rule 1. A point falling beyond zone A (above the upper control limit or below the
lower control limit) indicates lack of control.
Rule 2. Two out of three points in zone A or beyond on the same side, indicate lack
of control.
Rule 3. Four out of five successive points in zone B or beyond on the same side,
indicate lack of control.
Traditionally, the values of k1,k2, and k3 are set to 1, 2, and 3, respectively. In the
¯ exible zone methodology, the best values of k1,k2, and k3 are determined by optimization of the
is that
chart. The only restriction
placed on these
values
0 £ k1 £ k2 £ k3.
Objective function and constraints
The objective of the chart design is based on the observation (Roberts 1958, Page
1961, Woodall 1985) that a good control chart has a suitably large ARL 1 and a small
ARL 2. A small ARL 1 will lead to many false alarms which is undesirable at least
in
two respects. First, a control chart that exhibits frequent false alarms might not be
deemed reliable when it actually detects an out of control situation. Further, excessive false alarms produce many unnecessary work stoppages and an over-adjustment
of the process which results in an unstable process. A large ARL 2, on the other hand,
prevents the timely detection of a signi® cant process shift. For example, a shift that
is
greater than or equal to a multiple; d , of the process standard deviation, s , could
result in poor quality and must be detected as soon as possible.
In order to achieve a chart that provides ARL values that are close to the desired
values, the following objective function is minimized:
Objective function = ARL 1 - arl1
(
)2+ ARL 2 - arl2
(
)2,
(1)
where arl1 and arl2 are the speci® ed or desired values of ARL 1 and ARL 2,
respectively.
The ARL 1 and ARL 2 values for the combined rules 1, 2, and/or 3 are obtained
using the Markov chain approach of Brook and Evans (1972). Champ and Woodall
(1987) provide an e• cient technique for implementing this approach. The technique
is based on the realization that with a Markov-chain representation of the chart the
ARL values can be obtained by solving a matrix equation. The interested reader is
referred to Appendix A where a complete and easy-to-follow example is provided.
This Markov-chain approach implemented as a computer program is used to calculate the ARL values in the objective function.
A multi-dimensional computer search technique, as developed by Nelder and
Mead (1964), is used to optimize the objective function. For a detailed implementation of computer optimization technique and for a general discussion of optimization, the interested reader should refer to Kuester and Mize (1973) and Fletcher
(1988), respectively. As such, the best values of k1,k2, and k3 are obtained that
minimize the objective function of (1) subject to the following constraint required
for the ¯ exible zone chart:
0 £ k1 £ k2 £ k3.
(2)
Furthermore, in order to provide an easy comparison of the results, see tables 1± 6, it
was decided to design charts which will have ARL 1 values that are very close to
(withan error of 6 0.01) the desired arl1 values of 10,20,40,100,150,. . . ,400. As such,
the
following additional constraint was used:
|ARL 1 - arl1| £ 0.01.
(3)
Note that both constraints (2) and (3) were implemented using a penalty function
(Fletcher 1988.)
Comparison of the results
A significant number of flexible zone charts are designed in order to illustrate the
use of theflexible zone chart as well as properly compare the advantage of this chart
over the Shewhart individuals chart with the traditional run rules. As such, the
flexible zone charts with rules 1 and 2, as well as rules 1, 2, and 3 are designed for a
range of ARL 1 values (from 10 to 400) for each of a range of process shift values
(from 0.5s to 2s ). The flexible zone chart’s designs and the resulting ARL1 and
ARL 2
values, are summarized in tables 1± 6. Note that each table is for a particular value of
the shift in the process mean.
An obvious advantage of the flexible zones is that it provides the practitioner
with the ability to choose from a set of possible (ARL 1,ARL 2) pair values. This is in
contrast to the one pair value provided by the traditional runs rules. For example,
the practitioner does not need to accept the pair (ARL 1 = 132.89, ARL2 = 79.59)
value which is provided by the traditional rules 1, 2, and 3 for the detection of a shift
of 0.5s . Instead, the practitioner who desires a much lower false alarm rate can
choose the pair (ARL 1 = 370, ARL 2 = 70.88) from the flexible zone rules 1, 2, and
3.
This new point provides the same low false alarm rate of an ordinary Shewhart
individuals chart with the added advantage of detecting the shift on
the 71st
sample (70.88 on the average to be exact) rather than the 155th sample of the
ordinary individuals chart.
A general look at the results shows that the flexible zone chart with rules 1, 2, and
3 is always superior to the flexible zone chart with rules 1 and 2. This is represented
by the observation that for any row in tables 1± 6, the ARL 2 of the chart with
flexible
zone rules 1, 2 and 3 is always smaller (better) than the ARL 2 value of the corresponding flexible zone chart with rules 1 and 2. Similarly, it can be seen from these
tables that the flexible zone chart with rules 1 and 2 is always superior to the flexible
zone chart with rule 1. Further, it can be noted that the ¯ exible zone rules are always
better than or equal to their corresponding traditional zone (AT&T) rules including
the Shewhart chart.
Conclusion
A model for the design of a flexible zone individuals chart based on the desired
in-control and out-of-control ARL values is developed. Numerical optimization
determines the best zone widths for the rules 1 and 2 as well as the rules 1, 2, and
3 on an individuals chart. Examples of the flexible zone chart design are provided for
false alarm rates in the range of 0.0025 to 0.1 for the process shift values of 0.5, 0.75,
1.0, 1.25, 1.5, and 2.0 multiples of the process standard deviation. The results indicate that the design of flexible zone chart combined with rules 1, 2, and 3 is always
equal or superior to the Shewhart individuals chart with the traditional rules 1, or
rules 1 and 2, or rules 1, 2 and 3.
Acknowledgments
The authors would like to thank the referees for suggesting inclusion of the
Appendix and comments that improved this article.
Appendix
This appendix provides a complete example of using the Markov-chain approach
for calculating ARL values for a control chart with supplementary rules. Without
loss of generality, consider a process being monitored by an individuals chart with
traditional AT&T rules 1 and 2. Let k1 = 0, k2 = 0, and k3 = 3 in figure 1. As such
the process is deemed out-of-control when either (a) a single observation falls in any
of the regions (- ¥ ,- 3s ) or (3s ,¥ ) or (b) two out of three consecutive observations
falls on the same side of the regions (- ¥ ,- 2s ) or (2s ,¥ ). Now, observe that there
are eight states (including the absorbing state) as defined in table 7.
For example, state 4 of table 7 represents that the last two consecutive observations fell in the regions (- 2s ,2s ) and (- 3s ,- 2s ), respectively.
The following state transition matrix describes the movements between the states
of this Markov chain:
Note that the rows 1 to 8 of S are for states 1 to 8. Also, columns 1 to 5 of S represent
the regions (- ¥ ,- 3s ), (- 3s ,- 2s ), (- 2s ,2s ), (2s ,3s ), and (3s ,¥ ), respectively.
For example, the entry at row 1 and column 1 of S has a value of 8 indicating
that if the process is in state 1 and a sample point falls in the region (- ¥ ,- 3s ),
then the new state is equal to 8 (an out-of-control signal). As another example, the
entry at row 3 and column 2 of S has a value of 4 indicating that if the process is in
state 3 and a sample point falls in the region (- 3s ,- 2s ), then the new state is equal
to 4 (see table 7).
The probability transition matrix for the non-absorbing states, R, can be de® ned
as follows:
where Pi,j is the probability of going from state i to state j. Consider the case that the
shift d is zero, the P1,1 is equal to the probability of staying
in state 1 or
P(- 2 <Z <2), where Z is the standard normal random variable. Similarly, P1,2
is equal to the probability of going from state 1 to state 2 or P(2 <Z <3). However,
P1,3,P1,5,P1,6, and P1,7 are zero since these states (3, 5, 6, and 7) are not reachable
from state 1 (as can be seen from the ® rst row of matrix S.) Thus, R is equal to:
Then, from Brook and Evans (1972):
where I is the identity matrix,
1 is a vector of 1’s, and
A RL is a vector of average run lengths for all the possible initial states (see states
1 to 7 of table 7).
In other words, the vector A RL contains the ARL values corresponding to all the
possible starting states (see states 1 to 7 of table 7) for the control chart. Assuming
that, the chart starts from the in-control state 1 (see state 1 of table 7), then the ARL
of interest is equal to the value of the ® rst element of the A RL vector. Using (4), it is
now a matter of calculation to ® nd that the ARL of interest (corresponding to state 1
of table 7) is 225.36.
利用馬可夫鏈計算出區域管制圖的常數，讓區域管制圖的常數更富有彈性而不只
能運用在整數。