An np control chart for monitoring the mean of a variable
based on an attribute inspection
Zhang Wu* (corresponding author)
School of Mechanical and Aerospace Engineering
Nanyang Technological University, Singapore 639798
Tel: (65) 67904445, Fax: (65) 67911859, Email: mzwu@ntu.edu.sg
B. C. Michael Khoo
School of Mathematical Science
University Sains Malaysia, Penang 11800, Malaysia
Lianjie Shu
Faculty of Business Administration, University of Macau, Macau
Wei Jiang
Department of Systems Engineering and Engineering Management
Stevens Institute of Technology, Hoboken, NJ 07030, USA
Abstract
This article proposes a new np control chart, called the npx chart, that employs an
attribute inspection (inspecting whether a unit is conforming or nonconforming) to monitor
the mean value of a variable x. The distinctive feature of the npx chart is using the statistical
warning limits to replace the specification limits for the classification of conforming or
nonconforming units. By optimizing the warning limits, the npx chart usually outperforms the
X chart considerably on the basis of same inspection cost. In addition, the npx chart often
works as a leading indicator of trouble and allows operators to take corrective action before
any defectives are actually produced.
Keywords:
Quality control; Statistical process control; Attribute and variable control
charts; Attribute inspection; Average Time to Signal; Loss function.
1
Zhang Wu, School of Mechanical and Aerospace Engineering, Nanyang Technological University,
Singapore 639798, Tel: (65) 67904445, Fax: (65) 67911859, Email: mzwu@ntu.edu.sg
2
1.
Introduction
In Statistical Process Control (SPC), control charts for attributes (such as the np, p, c
and u charts) detect the out-of-control conditions of a process by checking the number of
nonconforming units or nonconformities in a sample. The attribute control charts are widely
used largely owing to its simplicity in implementation. By using attribute charts, “expensive
and time-consuming measurements may be avoided by attributes inspection” (Montgomery
2005). The sample sizes of the attribute charts are usually much larger than that of the
variable charts. This also implies that the attribute inspection is much simpler and less timeconsuming than the variable inspection.
Nevertheless, it is commonly believed that attribute charts are unable or inefficient to
deal with a quality characteristic that is of a variable type. The effectiveness of a control chart
is usually measured by the Average Time to Signal (ATS) which is the average time required
to signal a process shift (e.g., a mean shift ) after it occurs. A small out-of-control ATS
means that the out-of-control conditions will be signalled promptly and the amount of
defectives produced during the out-of-control status would be reduced. On the other hand, the
in-control ATS0 should be large so that the false alarm rate is low.
The Shewhart X chart is widely applied to monitoring the mean of a variable. It
inspects the sample mean x .
nx
x
 xi
i 1
nx
,
(1)
where xi is the ith observation in a sample of size n x . The more advanced CUSUM chart
monitors the process mean by checking the cumulative sum Ct. The implementation of both
X and CUSUM charts relies on a variable inspection which measures the actual value of x
and requires the calculation of x and/or Ct.
3
Wu and Jiao (2007) proposed an attribute chart (namely the MON chart) for
monitoring the mean of a variable. This chart checks the run length between two consecutive
nonconforming samples.
Among all of the attribute control charts, the np chart may be the simplest one for
understanding and implementation. The np chart is equivalent to the p chart when the sample
size is constant, but the former is easier for non-statistically trained personnel to understand
and handle (Montgomery 2005). The np chart counts the number, d, of the nonconforming
units in a sample of size nnp. If d falls beyond the lower control limit LCLnp or upper control
limit UCLnp of the np chart, the process is thought out of control.
Whether the np chart is able to detect the mean shift of a variable with satisfactory
efficiency? Montgomery gave an example in his text (2005), in which an X chart and an np
chart are compared for detecting the mean shift of a quality characteristic x with a normal
distribution ~N(50, 22). The X chart uses the 3-sigma control limits and a sample size n x of
nine. Its power for detecting a mean shift of 2 (= 1σ) is equal to 0.50. In contrast, to detect the
same mean shift, the sample size nnp of the np chart must be at least equal to 60 in order to
achieve the same detection power and to maintain the same false alarm frequency. The ratio
between the sample sizes (i.e., nnp/ n x ) is as large as 6.667. This huge difference will scare
away many Quality Assurance (QA) practitioners from considering the np chart for
monitoring the mean of a variable.
In this article, a new type of np control chart, the npx chart, is proposed to compete the
X chart for detecting mean shift δ. It is found that even when using the same sample size and
sampling frequency, the npx chart is less effective than the X chart only by 30% to 40 %. In
fact, since the npx chart employs the simple attribute inspection and eliminates the need for
any computation, it may use a greater sample size and/or sampling frequency than the X
4
chart in many SPC applications on the basis of same inspection cost. A greater sample size
and/or sampling frequency will, in turn, make the npx chart more effective than the X chart,
measured by both Average Time to Signal (ATS) and Extra Quadratic Loss (EQL) (Reynolds
and Stoumbos 2004).
A distinctive feature of the npx chart is the use of the statistical warning limits wL and
wU to replace the specification limits for the classification of conforming and nonconforming
units. While the specification limits are fixed by design engineers, the warning limits can be
set at an optimal level by the QA engineers so that the npx chart has the highest detection
power. Moreover, since wL and wU are continuous variables, they can be adjusted so that the
in-control ATS0 of the npx chart is exactly equal to a specified value τ. It ensures that the npx
chart can meet the requirement on false alarm rate, and meanwhile has the tightest possible
control limit and the highest possible detection power.
Finally, since the warning limits are often much closer to the target value than the
specification limits, a nonconforming unit that falls beyond one warning limit of an npx chart
may still reside within the specifications. Or in other words, a nonconforming unit classified
by the npx chart may not be a defective. It means that the npx chart often provides an
indication of impending trouble and allows operators to take corrective action before any
defectives are actually produced. It is an advantage that only the variable charts have in the
past (Montgomery 2005).
The rationality of the warning limits lies in the fact that the major objective of SPC is
to quickly detect the occurrence of assignable causes of process shifts using the control chart
as an on-line process-monitoring tool (Montgomery 2005), rather than evaluating the quality
level of the units in a few samples.
As aforementioned, while the X chart adopts variable inspections, the npx chart
employs the simple attribute inspections. In an attribute inspection, the only concern is
5
whether a unit is conforming (e.g., whether x falls within the warning limits) rather than the
actual value of x. A typical example of the attribute inspection is to use a “Go/No Go” ring
gage to check whether the diameter x of a shaft exceeds a limit (Kennedy et al. 1987).
Suppose the calibrated dimension of the ring gage is made equal to the upper warning limit
wU, a shaft is deemed to be oversized if it cannot pass through the gage. Figure 1 shows (a) a
double-end gage and (b) a progressive gage. They are able to check both oversizing and
undersizing in one run (the calibrated dimension in the left hand side equals wU and that in
the right hand side equals wL). Other examples include using a snap gage to check whether
the thickness of a component passes a specification, and using a horizontal line (bar) to
decide whether a bus passenger is taller than a standard. A special spring scale can be
designed to check the weights of objects. The scale will ring or blink if an object is heavier
than a predetermined limit. Montgomery (2005) observed that “variable-type inspection is
usually much more expensive and time consuming on a per unit basis than attribute
inspection.” It thereby may be more appropriate and fair to compare the effectiveness of the
charts based on identical inspection cost per unit time (W), rather than based on the equal
sample size (n) and sampling interval (h).
W 
cn
,
h
(2)
where c is the inspection cost per unit. If the cost for instrument is neglected, c is simply
proportional to the time t spent on the inspection of a single unit. The following condition
must be satisfied for a fair comparison between an X chart and an npx chart.
c x n x cnp nnp

.
hx
hnp
(3)
In Montgomery’s example aforementioned (2005), as both the X and np charts use the same
sampling interval h, accordingly,
6
nnp
nx

cx
.
cnp
(4)
That is, the sample size should be inversely proportional to the inspection cost per unit for
identical h. On the other hand, if two charts use the same sample size but different sample
intervals, Equation (2) leads to:
hnp
hx

cnp
cx
.
(5)
Since an npx chart simply checks whether the units are conforming (within the
warning limits) or nonconforming (beyond a warning limit) and does not require any
calculation, the inspection cost cnp required by this chart must be substantially lower than the
inspection cost c x required by an X chart for many SPC applications. According to Equation
(2), the sample size nnp of the npx chart may be allowed to be larger than n x of the X chart,
or the sampling interval hnp be smaller than h x .
The actual value of the ratio ( c x /cnp) (or nnp/ n x , or h x /hnp) in a particular application
depends on the extent of simplicity of the attribute inspection with respect to the variable
inspection. For example, if the charts are implemented manually, the operator of an X chart
has to conduct the time consuming variable inspection and also calculate sample mean x .
Under such circumstances, the inspection cost c x may be much higher than cnp. Based on
same W value (inspection cost per unit time), the npx chart may use a much larger sample size
and/or a much smaller sampling interval. In example 1 in the latter section 4, the ratio of
( c x /cnp) in a typical experiment in mechanical engineering is equal to 10.082 ( c x includes the
time for measuring x and for computing x by a calculator). It means that the inspection cost
for an X chart using a sample size of n x and a sampling interval h x is nearly the same as the
inspection cost for an npx chart using (nnp = 10 n x , hnp = h x ) or (nnp = n x , hnp = h x /10).
7
On the other hand, if a computer-aided system is available for the SPC
implementation as in many modern industries, the operators are released from calculating the
sample mean x . They only have to measure the value of x and enter the reading into a
computer through the keyboard. However, the variable inspection used by the X chart is still
intrinsically more or much more difficult than the attribute inspection. In fact, in many
applications, just keying in a reading (for example, the measured diameters 74.030, 74.002,
74.019, 73.992, 74.008 … of the forged piston rings (Montgomery 2005)) through a keyboard
may take longer time than carrying out an attribute inspection. In example 1 in section 4,
when an on-site computer is in place, the ratio of ( c x /cnp) is equal to 4.482, where the cost c x
includes the time spent on measuring x and keying the reading into a computer. This suggests
that the sample size or sampling frequency of the npx chart may be 4 to 5 times higher than
that of the X chart even for computer-aided SPC.
In this article, the observations of a quality characteristic x are assumed to be
independent and have an identical normal distribution with known in-control mean μ0 and
standard deviation σ0. It is also assumed that the process variance remains unchanged. When
process shift occurs, the mean value  will change, i.e.,
  0  0 ,
(6)
where  is the mean shift in terms of 0.
The remainder of the article proceeds as follows. The implementation and design of
the npx chart is introduced in Section 2. This chart is compared with the X chart in Section 3.
Subsequently, two practical examples are illustrated in Section 4. Finally, the conclusions and
discussions are drawn in the last Section.
2.
Implementation and design of the npx chart
8
Implementation
The npx chart has five parameters: the sample size nnp, the sampling interval hnp, the
lower and upper warning limits wL and wU, and the upper control limit UCLnp. Unlike a
conventional np chart, the npx chart does not need a lower control limit. The operation of an
npx chart is as simple as that of the conventional np charts except the specification limits
being replaced by the warning limits. During the implementation, nnp units as a sample are
inspected at the end of each sampling interval hnp. If the number, d, of nonconforming units is
larger than UCLnp, the process is signalled as out of control; otherwise the process is thought
in control. The whole procedure does not need any calculation. It is noted that, with only an
upper control limit UCLnp, an npx chart is able to detect both increasing ( > 0) and
decreasing ( < 0) mean shifts, depending on whether the x values of the d nonconforming
units fall above the upper warning limit wU or below the lower warning limit wL. The upper
control limit UCLnp is used to check d and, therefore, is an integer. However, the warning
limits wL and wU are variables. They are symmetrical about the in-control mean μ0.
wL  0  k w 0
wU  0  k w 0
,
(7)
where, kw is called the warning limit coefficient.
Design specifications
To design an npx chart, the following five specifications need to be decided:
nnp
the sample size
hnp
the sampling interval

the allowed minimum in-control Average Time to Signal ATS0
μ0
the in-control process mean
9
σ0
the in-control process standard deviation
The sample size nnp and sampling interval hnp are also the charting parameters and
decided based on the available inspection resources (e.g., manpower and instrument). The
value of  is decided with regard to the tolerable false alarm rate. The values of the in-control
mean 0 and standard deviation 0 are usually estimated from the data observed during the
pilot runs in phase I operation. For a process, the control charts used in phase I and phase II
operations may not be the same. For example, Montgomery (2005) suggested using a simple
X chart in phase I operation and a more advanced EWMA or CUSUM chart in phase II
operation, as it is much simpler to use an X chart to collect the sample data for estimating
process parameters. With the same reason, one may employ an X chart in phase I operation
to estimate 0 and 0 in order to build the npx chart. Then he can use the npx chart to monitor
the process mean in phase II operation.
Constraint function
The actual in-control ATS0 should be no smaller than τ in order to satisfy the
requirement on false alarm rate. However, ATS0 cannot be too large, otherwise the out-ofcontrol ATS will also be quite large and the detection effectiveness is low. It is most ideal that
ATS0 be equal to τ.
  ATS0  hnp /  ,
(8)
where, α is the probability of type I error (the probability that the control chart produces an
out-of-control signal when the process is in fact in control). On the other hand, if the number
d of nonconforming units in a sample follows a binomial distribution, the value of α produced
by an npx chart is calculated by
10
  1
UCLnp
 Ci
i 0
n np
p0i (1  p0 )
n np  i
,
(9)
where, p0 is the probability that a unit is nonconforming (falling beyond one of the warning
limits) when the process is in control. Referring to Equation (7),
 w  0 
 w  0 
   L

p0  1   U
 0 
 0  ,
 1   k w     k w 
(10)
where, Ф() is the cumulative probability function of the standard normal distribution.
Combining Equations (8), (9) and (10), we have
hnp

1
UCLnp
i
n
 Ci 1  kw    kw  kw    kw n
np
np  i
.
(11)
i 0
When the upper control limit UCLnp is given, the warning limit coefficient kw is the only
unknown in the above equation and can be solved by any numerical method. The resultant
value of kw will ensure the satisfaction of the constraint (ATS0 = τ).
Objective function
In many control chart designs, the out-of-control ATS at a specified mean shift level is
used as the objective function to be minimized. However, since the goal of the design is to
make the control chart efficient at signalling a wide range of mean shifts, it is therefore more
desirable that the objective function measures the holistic performance of the charts cross a
process shift range rather than just at a specific point. Researchers usually compare the
performance of two charts by examining the corresponding out-of-control ATS values of the
two charts at some discrete points of process shift δ cross a range. For most of the cases, one
chart is unable to excel another chart at all the points. However, as long as one chart has
smaller ATS at more points and/or to a larger degree, this chart is thought more effective than
11
the other. Moreover, since it is usually assumed that all mean shifts within a range are equally
important (Sparks 2000), a uniform distribution for δ is implied. Such comparison scenario
may be formulated as follows:

RATS 
 max
max
1
ATS ( )
 
d ,
  min  ATSbenchmark ( )
(12)
min
where, RATS indicates the average of the ratio of the ATS values of two charts cross the range
(δmin ≤ δ ≤ δmax). In Equation (12), ATS() is produced by one chart at  and ATSbenchmark() is
generated by another chart that acts as the benchmark. Obviously, if the RATS value of a chart
is larger than one, this chart is generally less effective than the benchmark, and vice versa.
An alternative is to use the Extra Quadratic Loss (EQL) (Reynolds and Stoumbos
2004) to measure and compare the performance of the charts. For a given mean shift δ, the
incurred quadratic loss is simply equal to δ2. Moreover, the loss in quality is proportional to
ATS(δ). Consequently, the overall EQL can be calculated as follows:

EQL 
max
1
   2  ATS ( )  d .
 max   min 
(13)
min
Both EQL and RATS are integrated cross the whole shift domain rather than just being
evaluated at a particular point. The integration can be computed by any numerical method.
The out-of-control ATS is calculated under the steady-state mode. This mode assumes that the
process has reached a stationary status at the time when the process shift occurs and the
random time of process shift has a uniform distribution within the sampling interval
(Reynolds et al. 1990). As such,
12
 1

ATS ( )  ( ARL  0.5)hnp  
 0.5 hnp
1 


UCLnp
 Ci
n np
p i (1  p )
n np  i
i 0
(14)
 w  ( 0  0 ) 
 w  ( 0  0 ) 
    L

p  1    U
0
0




 1   k w       k w   ,
where, ARL is the Average Run Length (the average number of samples required to signal a
process shift after it occurs); β is the probability of type II error (the probability that the
control chart fails to produce an out-of-control signal when the process is already out of
control); and p is the probability that a unit is nonconforming when the process is out of
control.
The index EQL based on loss function has two advantages over RATS. First, the loss
function is a more comprehensive measure of the charting performance than ATS, because it
considers not only the time to signal but also the amount of loss incurred by process shifts.
Secondly, the evaluation of EQL does not require the predetermination of a benchmark chart.
In view of this, EQL will be used as the objective function for the optimization design of the
npx chart. The minimization of EQL will reduce the loss in quality (or the cost, or the damage)
incurred during the out-of-control cases. Like RATS, a ratio between the EQL values of two
control charts serves as an indication of the relative effectiveness of the two charts over a
mean shift range.
Design procedure
Most of the advanced design algorithms for control charts are to minimize the
objective function on the condition that the in-control ATS0 is equal to a specified value τ (Xie
et al. 1995, Wu and Spedding 2000, Reynolds and Stoumbos 2004). Similarly, an
13
optimization design for an npx chart aims to identify the optimal values of the upper control
limit UCLnp, the lower and upper warning limits wL and wU, so that the objective function
EQL is minimized and meanwhile the constraint (ATS0 = τ) is met. Other two charting
parameters nnp and hnp are specifications. Among the three charting parameters to be designed,
only is UCLnp an independent variable. The dependent parameters wL and wU are determined
so that the ATS0 of the npx chart is equal to τ. The optimal values of UCLnp, wL and wU are
determined as follows:
(1)
Enter the specifications of nnp, hnp,  , 0 and 0.
(2)
Search the optimal value of UCLnp by checking every possible value over the range of
(0 ≤ UCLnp < nnp).
(3)
For a given value of UCLnp determined in step (2), the warning limit coefficient kw is
determined by solving Equation (11). Then, the warning limits wL and wU can be
calculated by Equation (7).
(4)
Based on the tentatively determined values of UCLnp, wL and wU, calculate the
objective function EQL by Equations (13) and (14).
(5)
If the calculated EQL is the smallest one up to date, keep this EQL value and also
store the current values of UCLnp, wL and wU as a temporary optimal design.
(6)
At the end of the entire search of UCLnp over the range of (0 ≤ UCLnp < nnp), the npx
chart that results in the minimum EQL and satisfies the constraint of (ATS0 = τ) can be
identified. The optimal values of the charting parameters UCLnp, wL and wU are also
determined.
A computer program can be easily coded to design the npx chart. The design of an npx
chart can be completed almost in no time in a personal computer.
14
A conventional np chart using the fixed specification limits adjusts the control limits
in order to satisfy the constraint on false alarm rate (i.e., ATS0 ≥ τ). However, due to the
discrete nature of the control limits, the resultant ATS0 may be substantially larger than τ. This
will, in turn, significantly impair the detection effectiveness of the np chart (Wu et al. 2001).
On the other hand, an attempt of tightening control limits (e.g., reducing UCLnp by one) will
reduce ATS0 sharply, usually leading to a false alarm rate much higher than specification. This
problem may places the QA engineer in a no-way-out dilemma between the false alarm rate
and the detection power. For the npx chart, the ATS0 value is finalized by adjusting the
warning limit coefficient kw (in step (3) in the design procedure). Since kw is a continuous
variable, ATS0 can be made exactly equal to τ. This feature helps maximize detection
effectiveness of the npx chart, and meanwhile meet the requirement on false alarm rate.
3.
Comparative studies
The effectiveness of npx and X charts is compared in this section. Both charts are
required to produce an ATS0 value equal to the specification τ. Without losing the generality,
0 and 0 are fixed as zero and one, respectively.
Study 1: Comparison under a general condition
The comparison is first conducted under the following general conditions:
 = 370, n x = 6, h x = 1.
where, n x and h x are the sample size and sampling interval of the X chart. First, suppose an
npx chart also uses the same sample size and sampling interval ( i.e., nnp = 6, hnp = 1). With
these specifications, the two control charts are designed and the resultant charting parameters
are displayed in Table 1. The ATS values of the two charts are also enumerated in columns (2)
15
and (3), respectively, in Table 1, within a mean shift range of (0 ≤ δ ≤ 3.5). Since the ATS
values of the two charts are symmetrical for increasing and decreasing mean shifts, only the
positive δ has to be investigated. It is interesting to observe the following findings.
(1)
Firstly, both charts generate the ATS0 values very close to  when the process is in
control. This ensures a fair comparison between the two charts.
(2)
Within the whole mean shift range, the X chart is superior to the npx chart for
detecting the mean shift δ of any size. However, the difference between the ATS
values of the two charts for a given δ value is always within 100%. The maximum
difference is 84.0% that occurs when δ is equal to 1.0. Then the difference diminishes
gradually until (δ = 2.75). After that, the two charts are as effective as each other.
(3)
The values of EQL of the two charts, together with the ratio EQLnp / EQLx are
displayed below the ATS values in Table 1. These values reveal that, cross the entire
mean shift range, the X chart is, on average, more effective than the npx chart by
31.4%. It is much smaller than that illustrated in Montgomery’s example (2005).
(4)
The RATS value that indicates the average ratio between ATSnp and ATSx over the
whole mean shift range is also shown at the bottom of Table 1. It is equal to 1.355 and
confirms that the X chart outdoes the npx chart by about 30%.
It is recalled that, based on equal inspection cost per unit time W (Equation (2)), the
npx chart may use a larger sample size and/or a smaller sampling interval compared with the
X chart due to the much smaller inspection cost per unit cnp for many practical applications.
For this study with ( = 370, n x = 6, h x = 1), it is interesting to further compare the
performance of the npx and X charts when c x /cnp equal to two (in many SPC applications,
16
this ratio is often much larger than 2). Two special cases will be studied: (1) nnp = n x , hnp =
0.5 h x ; and (2) nnp = 2 n x , hnp = h x . The charting parameters of the new npx charts have to be
calculated again in order to minimize EQL over the mean shift range and meanwhile make
ATS0 equal to . The resultant charting parameters, as well as the ATS values, for the two new
npx charts are displayed in columns (4) and (5), respectively, in Table 1.
It is found from column (4) that, if the sampling interval hnp is allowed to be reduced
to half of h x , the performance of the npx chart can be substantially improved. The ATS value
of the new npx chart is slightly larger than that of the X chart only for (δ ≤ 1.5). When δ
becomes larger, the ATS value of the npx chart becomes smaller and smaller compared to that
of the X chart. Both the ratio EQLnp / EQLx and the value of RATS show that the new npx
chart is, on average, more effective than the X chart by 12.3% cross the entire shift range.
Similarly, if the sample size nnp is allowed to be the double of n x (column (5)), the
detection effectiveness of the npx chart will also be enhanced to a significant degree. The ATS
value of the new npx chart is uniformly smaller than that of the X chart over the whole mean
shift range. Both the EQLnp / EQLx ratio and RATS value are equal to about 0.85. This
indicates that the new npx chart is considerably more effective than the X chart from a
holistic viewpoint. Comparing columns (4) and (5), it is found that reducing the sampling
interval hnp can improve the performance of the npx chart more effectively for large mean
shifts; and increasing the sample size nnp is more helpful for small mean shifts.
Study 2: A factorial experiment on n x and τ
17
Next, the influence of n x and τ on the comparison between the two charts is studied in
a 22 factorial experiment. Each of these two specifications varies at two levels, resulting in
the following four runs corresponding to four different combinations of n x and τ.
( n x = 3,  = 250), ( n x = 9,  = 250), ( n x = 3,  = 500), ( n x = 9,  = 500).
The other three specifications are fixed as ( h x = 1), (0 = 0) and (0 = 1).
The ATS values of the two charts in each run are calculated and analyzed for three
cases, i.e., (nnp = n x , hnp = h x ), (nnp = n x , hnp = 0.5 h x ) and (nnp = 2 n x , hnp = h x ). All the
results, together with the charting parameters, are displayed in Tables 2 to 5. The results
reveal that the relative effectiveness of the two charts in all four runs is similar to that shown
in Table 1 for ( n x = 6,  = 370). While n x has some influence on the relative effectiveness
between the two charts,  imposes little impact on the comparison.
Even though the 22 factorial experiment is not exhaustive, it convincingly shows that
the npx chart is less effective than the X chart only by 40% or less even when identical
sample size and sampling interval are used by both charts. If the fact that variable inspection
is much more expensive and time consuming is taken in consideration, the npx chart should
be allowed to use a larger sample size and/or smaller sampling interval, and is surely superior
to the X chart to a considerable or significant degree.
From Tables 1 to 5, it is interesting to note that, in each case, the EQLnp / EQLx ratio
and RATS value are usually very close to each other. Both always lead to the similar results
for the comparison.
Furthermore, the warning limits of all the npx charts shown in these Tables are much
smaller than the specification limits (The specification limits should be no smaller than 3σ0 in
order to satisfy the minimum requirement on process capability). Due to the small warning
18
limits, a nonconforming unit classified by the npx chart is likely to reside within the
specification limits, and therefore may not be a defective. It means that the npx chart often
produces the out-of-control signal when the process has an early and minor mean shift but the
whole distribution is still well within the specification limits. As a result, like the variable
control charts, the npx chart will work as a leading indicator of trouble and will warn
operators to take corrective action before any defectives are actually produced.
4.
Examples
Two practical examples are demonstrated in this section. In the first one, only the
sample size nnp of the npx chart is increased compared with the X chart due to the smaller
cost cnp; and in the second example, both the sample size and sampling frequency of the npx
chart are increased.
Example 1: the diameter of a shaft
The diameter x of a shaft is a key dimension. An X chart and an npx chart are to be
designed and studied. One of them will be eventually selected to monitor the mean of x. The
probability distribution of x can be very well approximated by a normal one and the in-control
0 and 0 are estimated as 8.00mm and 0.012mm, respectively. Other design specifications are:
 = 740hr, n x = 4, and h x = 1hr. Since only oversizing has to be detected, the one-sided
control charts are designed. For a 3-sigma X chart, the in-control ATS0 is equal to 740.
A real field experiment is conducted to estimate the inspection costs per unit c x and
cnp. In the following two tests, each of four operators inspects the diameters x of 10 specimens.
(1)
Test one: attribute inspection for the npx chart
19
The operators use a simple MituToyo ring gage to check whether a specimen is
oversized. The average time tnp (average over the 40 readings from ten readings taken
by each of the four operators) spent on an individual inspection is 2.125 seconds.
(2)
Test two: variable inspection and computer-aided calculation for the X chart
The operators use a more delicate MituToyo digital micrometer to measure the
diameter x of each specimen and key in the reading immediately to a computer. The
average time t x spent on an individual specimen is 9.525 seconds (If no computer is
available, the operator has to calculate sample mean x by a calculator, and the
average time t x is as large as 21.425 seconds).
If the initial investment on equipment (i.e., a MituToyo ring gage for the npx chart and
a MituToyo digital micrometer plus a computer for the X chart) is neglected, the ratio
between the inspection costs ( c x /cnp) is equal to ( t x / tnp ). Then, if the npx chart opts to use
the same sampling interval but larger sample size, referring to Equation (4),
nnp
nx

cx
t
9.525
 x 
 4.482 .
cnp tnp 2.125
Since n x = 4, the matched nnp should be between 17 and 18. Here, (nnp = 17) is selected.
With all of the design specifications are available (  = 740, μ0 = 8, σ0 = 0.012, n x = 4,
h x = 1, nnp = 17, hnp = 1), the charting parameters for the two control charts can be designed
and listed below:
X chart:
n x = 4, h x = 1, UCLx = 8.0180 (= μ0 + 1.5σ0).
npx chart:
nnp = 17, hnp = 1, wU = 8.0073 (= μ0 + 0.6σ0), UCLnp = 10.
The attribute inspection for the npx chart is carried out by using a ring gage, the
calibrated dimension of which is made equal to wU. Since nnp = 17 and UCLnp = 10, if more
20
than ten out of 17 units in a sample are detected as nonconforming (i.e., the unit cannot pass
through the gage), the npx chart will produce an out-of-control signal.
The curves of the relative ATS (i.e., ATS/ ATSx ) of the two charts versus  in the shift
domain of (0    3.5) are illustrated in Figure 2. It can be seen that the npx chart is
significantly more effective than the X chart over the shift domain in this example. The
results of ( EQLnp / EQLx = 0.545) and (RATS = 0.620) also lead to the same conclusion.
Example 2: the UTS of a bar element
A process fabricates a bar element used to undertake tensile stress in a structure (Shu
et al. 2007). The element is made of Aluminum 6061 T-6 and has a cross-sectional area of
78.54mm2. Currently, a one-sided X chart with only a lower control limit LCLx is employed
to detect the possible decrease of the Ultimate Tensile Strength (UTS) of the element. The incontrol mean value μ0 and standard deviation σ0 of UTS are estimated as 240MPa and
3.8MPa, respectively. For each sampling interval ( h x = 2hr), the value of UTS of a single
specimen ( n x = 1) is measured by an Instron 5565 testing machine. It takes about 11 minutes
to increase the load gradually until UTS is reached or a pronounced necking is observed. The
in-control ATS0 is specified as  = 740 × h x = 1480hr. The charting parameters of the X
chart are listed below:
n x = 1, h x = 2, UCLx = 228.60 (= μ0 - 3σ0).
If an npx chart is to be considered, a simple attribute test can be completed in about
one third of the time required by a variable test (i.e., t x / tnp = 3). During the attribute test, a
fixed load equal to the lower warning limit wL of the npx chart will be directly exerted on the
21
specimen. If UTS is reached or necking occurs under this load, the specimen is
nonconforming.
Since c x /cnp = t x / tnp = 3, the npx chart may use a sample size nnp of two and a
sampling interval hnp of 4/3hr (or 80 minutes). It is decided based on the practical conditions.
It is noted that the X and npx charts have the identical value of W (the inspection cost per
unit time) as calculated by Equation (2).
Based on these specifications (  = 1480, μ0 = 240, σ0 = 3.8, nnp = 2, hnp = 4/3), the
charting parameters of the npx chart can be determined and listed below:
nnp = 2, hnp = 4/3, wL = 232.85 (= μ0 - 1.88σ0), UCLnp = 1.
It means that, during the attribute test for the npx chart, a fixed load equal to 232.85Mpa will
be charged to two specimens (nnp = 2) for every 80 minutes (hnp = 4/3hr). If both specimens
are nonconforming (i.e., d > UCLnp), an out-of-control signal will be produced by the npx
chart; otherwise, the process is though in control.
The curves of the relative ATS of the two charts for this example is shown in Figure 3.
Again, the npx chart overwhelmingly outperforms the X chart over the whole shift domain.
The ratio of EQLnp / EQLx is equal to 0.428; and the value of RATS equals 0.417.
5.
Conclusions
This article presents the general idea, design, operation, as well as performance
assessment, of the npx control chart. While this np chart still employs the simple attribute
inspection and eliminates the need for any computation, it is able to monitor the mean of a
variable characteristic effectively. It is attributable to the use of the warning limits to replace
the specification limits for the product classification. The results of numerical studies show
that the npx chart is usually less effective than the X chart only by 40% or less even when
22
using same sample size and sampling interval. Since the attribute inspection is much less
expensive and time consuming in most applications, the npx chart may use a larger sample
size and/or sampling frequency based on equal inspection cost, and therefore has a higher or
substantially higher detection effectiveness, measured by both ATS and EQL. It may be
somewhat surprising that an attribute chart may outperform the variable chart in detecting
mean shifts for virtually many types of applications. Moreover, the reliance on the knowledge
level of the operators and on the equipment for attribute inspections are alleviated compared
with that for variable inspections. The instruments used for attribute inspections are often
relatively simpler and need less adjustment, and therefore less expensive and more reliable
than the instruments used for variable inspections.
It has also been found that the npx chart often works as a leading indicator by
producing timely out-of-control signal before any defectives are actually produced. Moreover,
since the warning limits are continuous variables, the npx chart is able to make the in-control
ATS0 equal to any specification in order to achieve a most desirable trade-off between the
requirements on detection effectiveness and false alarm rate.
The implementation of a npx chart is exactly as easy as any ordinary np chart, and its
design is also relatively simple.
One may concern that replacing the variable inspection by the attribute inspection and
substituting the warning limits for the specification limits may disable the control chart to
estimate the value of the quality characteristic x or to diagnose the assignable causes.
However, in fact, it may be inadequate to estimate x just based on the information from a few
units in a sample. It is suggested that when the npx chart produces a signal because of an outof-control case or a false alarm, the operator may measure the actual x values of sufficient
number of units in order to accurately estimate x. This will cost little extra inspection effort,
because production processes often operate in the in-control condition for most or relatively
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long periods of time (Montgomery 2005), both the out-of-control case and false alarm are rare
events.
It is important to further study the performance of the npx control chart when some
commonly adopted assumptions cannot stand. For example, the quality characteristic x may
not follow a normal distribution and/or correlation may exist between observations. Such
sensitivity analysis of the npx chart suggests itself to be the avenue for future researches.
Furthermore, due to the widespread use of on-line measurement and distributed
computing systems in today’s SPC applications, the CUSUM control charts have been
increasingly adopted across industries. Thus, it may be also interesting to study the feasibility
of using the attribute CUSUM charts (e.g., Poisson CUSUM chart, run-length CUSUM chart)
to monitor variables.
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Montgomery, D. C. (2005). Introduction to Statistical Quality Control. John Wiley & Sons,
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