Download shewhart and time weight control charts for continuous variables in

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Bootstrapping (statistics) wikipedia, lookup

History of statistics wikipedia, lookup

Resampling (statistics) wikipedia, lookup

Time series wikipedia, lookup

Transcript
SHEWHART AND TIME WEIGHT CONTROL CHARTS FOR
CONTINUOUS VARIABLES IN QUALITY CONTROL - THEORY
Amhammed Almhdi Abdallah1 , Nuri Omar Ali Al-Hammali2
1- Faculty of Education Brack–Sebha University–Libya– [email protected]
2- Faculty of Sciences – Sebha University–Libya – [email protected]
Abstract
Statistical Process Control (SPC) involves using statistical techniques to measure and
analyze the variation in processes. SPC is very important and most often used for
manufacturing processes, the intent of SPC is to monitor product quality and maintain
processes to fixed targets. Statistical quality control refers to using statistical
techniques for measuring and improving the quality of processes and includes SPC in
addition to other techniques, such as sampling plans, experimental design, variation
reduction, process capability analysis, and process improvement plans.
We present shewhart and time weight control charts for continuous variables.
Introduction
Statistical process control is not a magic formula for curing all production ills; it is a
very useful tool to be used in promoting and maintaining the health of a commercial
or industrial enterprise.
The first step in SPC is to define the process from the point of the view of the
financial manager. Then, the characteristics of the process are observed and measured
over time. The numbers obtained from these observations are used to monitor the
process by calculating the average and examining the natural variation around the
average (mean) over time. This method of studying the variation from the mean is
known as control charting, as it pinpoints if the process has encountered any special
variation that needs special attention. Control charts that depict no special problems
indicate that the process is in control and predictable. However, if the control charts
show unusual variations (by points outside the acceptable range) it may indicate some
problem within the process. Usually the problem is caused by a temporary
circumstance and thus, can be resolved by a localized solution rather than changing
the general policy. it is worth noting here that SPC is used:
1- To improve quality.
2- To increase yield (or maintain yield at reduced cost).
Parts of SPC:
The SPC methods can be described as two parts :
1- ON-LINE SPC methods.
2- OFF-LINE SPC methods.
ON-LINE SPC methods divide into two types, screening or preventative.
In screening, we inspect the output, and if the quality is not satisfactory, we screen out
the substandard items for reworking, for selling at a reduced price or for scrap. This is
usually done by a system of sampling inspection. Screening for quality is usually very
expensive, and not recommended.
In preventative SPC methods we inspect the process, and try to use process control to
avoid defective items being produced.
Some people contrast control charts, as preventative methods, with sampling
inspection as a screening procedure. This distinction is not correct. Control charts can
be used as a screening mechanism, and sampling inspection can be used in a
preventative manner. But the importance of on-line SPC methods lies in their use as
preventative procedures.
SPC methods concentrate on trying to control process average level and process
spread. In particular, process spread or variability is a special enemy of quality, and
needs to be tackled with some vigour. Indeed the vast majority of discussions on
quality between manufacturers, their customers and suppliers is centered around the
consistency of feed stocks and products.
OFF- LINE SPC This is often the next stage on from on-line SPC, although ideally it
should be built into designing and setting up a product and its production process
from the start.
The aim is to reduce or remove the effect of potential causes of variability by
modifying the process, or the product, so making it less sensitive to these causes. This
generally requires skill and ingenuity from a team of people with different expertise.
Some Basic Definitions
Suppose we take samples. We let the successive results be denoted x1 , x2 ,..... then for
the first n we can calculate the sample mean
2
x   xi / n
and the sample variance
s 2   ( xi  x) 2 /( n  1)
or the standard deviation s, which is' the square root of the variance.
We regard any data set, such as a collection of 100 successive results, as a random
sample from a population. For the example we are considering the population is
summarized by the relative frequency table for a very large set of results taken under
the same conditions. Models for populations are probability distributions. For
countable data, as in our example above, a suitable probability distribution is a set of
values P(r), for r = 0, 1, 2, . . ., such that p(r) ≥ 0 and

p(r )  1
For continuous data, a suitable probability distribution is defined by a probability
density function f(x) such that f(x) ≥ 0,


f ( x)dx 1

and the probability of getting a result between L and U is
U

L
f ( x)dx
The mean of the population, rather than the sample, is called the expectation, and is
defined by
E ( x)   xf ( x)dx
for continuous data . the variance of the population is given by
V ( x)   ( x   ) 2 f ( x)dx
These quantities can be regarded as the sample mean and variance, but calculated for
the whole population rather than for a sample.
Basic Shewhart control charts for continuous variables
Introduction
The idea of the control chart is to operate a simple mechanism for controlling the
average level and spread of a process. A minimum of two charts is required. one to
control process average level and one to control process spread
In control - Out of control
In any production process, some variation in quality is unavoidable, and the theory
behind the control chart originated by Dr W. A. Shewhart is that this variation can be
divided into two categories.
random variation. and variation due to special or assignable causes. Variations in
quality which are due to causes over which we have some degree of control. Such as a
different quality of raw material, or new and unskilled workers are called special
causes of variation.
The random variation is the variation in quality which is the result of many complex
causes, the result of each cause being slight. By, and large nothing can be done about
this source of variation except to modify the process.
3
If data from a process are such that they might have come from a single distribution,
having certain desired properties such as a mean in a specified range, the process is
said to be in control.
If, on the other hand, variation due to one or more special causes is present, the
process is said to be out of control. The Shewhart control chart is a simple device
which enables us to define this stale of statistical control more precisely, and which
also enables us to judge when it has been attained.
Sampling Risks
When operating SPC we take small samples from the process at regular intervals and
plot, Say, the mean and the range on a chart. As a result, we conclude that the process
is either in control or out of control. If the process is out of control. this may be due to
a change in process average level, or process spread. or due to a particular problem at
a specified time point. Because of the variation inherent in sampling. the average
levels and the spreads as indicated by the samples will vary from sample to sample
even if the true process average and spread are constant. This gives rise to two
dangers when sample observations are plotted on a control chart. These are as
follows:
Type I risk: The risk that a legitimately extreme sample will give a spurious 'action'
decision when no change has occurred in the process.
Type II risk: The risk that a sample will fall within the control limits although there
has been a real change in the process: the change is not signalled. (The size of this risk
will get smaller as the size of the change increases.)
Control Charts for Average Level

x -Charts: Chart Construction

^
The construction of an x -chart relies on having good estimates  of the process
^
average level and  e , of the standard error of the group means.
These estimates are derived either from data from process capability studies. or from
fresh data.
Method summary

Construction of x -Chart
^
Step 1 Obtain estimates of the process average level,  , and the process variability,
^
and also obtain the estimated standard error of group means,  e .
^
Step 2 Choose the scale of the chart so that  is near the centre. and so that the scale
^
^
covers approximately  4  e , from  . where n is the sample size at each sampling
point.
^
^
^
^
Step 3 Mark the action lines at   3.09 e (probability);   3 e (popular)
^
^
^
^
Step 4 Mark the warning lines at   1.96 e (probability)   2 e (popular)
Charts for Control of (Within-Group) Process Spread
Range Charts – Construction
4
There are two ways of setting up the range chart - the 'range' method and the 'σ'
method. If process capability data are used to get an estimate of σ w , then this is fed
into the appropriate step of the a method. These methods are set out below.
Method Summary
Construction of A Range Chart by The Range Method
Step 1 Obtain the average range R either from process capability studies data. or
from at least 20 groups of fresh data.
Step 2 Choose the scale of the range chart so that the range goes down to zero, and up
to about twice the largest range observed in the trial data sets.
Step 3 Mark the action and warning limits on the chart:
Lower action limit: D1 R
Upper action limit:
D2 R
Lower warning limit: D3 R
Upper warning limit: D4 R
where the D1 , D2 , D3 and D4 values are given in Table (factors for Construction
range charts from an average. The factors are obtained from the distribution of the
range in Normal samples, and from the conversion factors from range to estimates of
σ ).
Method Summary
Construction of A Range Chart by The 'σ' Method
Step 1 Obtain an estimate of σw either from process capability studies data, or from at
least 20 groups of fresh data.
Step 2 Use the factors from Table (factors for Construction range charts from an
^
estimate of σ). and multiply by  w to find where to plot the action and warning
limits. The scale should be chosen to extend to about 50% greater than the upper
action limit.
Standard Deviation Charts
These are very similar to the range charts, except that standard deviations are
calculated and plotted for each group.
Method Summary
Step 1 Obtain an estimate of σw either from process capability studies data, or from at
least 20 groups of fresh data.
Step 2 Use the factors from Table (factors for Construction deviation charts from an
estimate of standard deviation).
^
Lower action limit:
D9  W
Upper action limit:
D10  W
^
^
Lower warning limit: D11  W
^
Upper warning limit: D12  W
Control of Between-Group Spread
Control of between-group variation can be affected by calculating a moving-range
chart or a moving-standard-deviation chart, based on the group means. The commonly
used method is to operate an X and R or S chart.
5
The Average Run Length
In order to design our charts in a more precise way, and in order to compare properties
of alternative charts we need to introduce the concept of run length. The run length is
the number of observations plotted on the charts until an 'out of control' signal is
given. Run lengths are usually calculated assuming that observations are sampled
independently from a specific population, and here we assume that there is simple
random variation only.
Special Problems
Non-normality
Basically, non-normality does not matter much for controlling process average level.
provided the group size is at least four. but it can affect charts for control of spread
markedly. Frequently investigations will show that Non-normality is due to some way
in which the process is operated, such as due to tile merging of streams. and this can
be corrected. Failing this, it is often possible to transform data to normality by using a
simple transformation such as logarithm, square root. etc. If there is non-normality
and this cannot be corrected or transformations used, then the methods Liven are
readily transferred to another distribution. The procedure involved would be to collect
sufficient data to fit a distribution, and then fit action and warning lines at the
appropriate percentage points.
Unequal numbers in groups
If the numbers in the groups change, it is a simple matter to change the limits.
Correlation
It is clear that if successive group means are correlated, the Shewhart chart could be
affected markedly. Vasilopoulos and Stamboulis (1978) have shown how to alter the
decision limits for a simple autoregressive model.
Some Theoretical Results for Shewhart Charts
Shewhart chart with action lines only
we assume simple random variation only, first we deal with a Shewhart chart for
means with action lines only. Suppose we take observations in groups of n at a time.
n
which we denote Xij, j=1,2,…,n, . . .. n. i=1,2,…. The group means x i   j 1 X ij / n
are plotted on a chart with upper and lower action limits at xUA , xLA . If all observations
are independently and identically distributed with an N(  , 2 ) distribution, then the
probability of being outside the action limits is
xLA

n
n
p
exp{n( x   ) 2 / 2 2 }dx  
exp{n( x   ) 2 / 2 2 }dx

xUA
(2 )
(2 )
 x  
   xUA 
  LA
  

/ n 
/ n 
It is clear that the distribution of run length R is geometric with parameter p. This
means that the ARL is
ARL = E(R) = 1/p
and the variance of the run length distribution is V(R)=(1- p)/p2
If the chart is set up in the standard way using the exact μ and σ, then p = 0.002 , so
that
E(R) = 500
and
V(R) = 249 500
and the standard deviation of R is 499.5. The highly skewed nature of the geometric
distribution needs to be taken note of. If charts are being designed using ARL as a
6
basis, this point about the shape of the rum-length distribution needs to be watched
carefully.
Shewhart Chart With Warning Lines
We now consider the modified Shewhart chart with warning lines. That is, action is
taken when one point is outside the action line or two successive points outside the
warning lines. Let p0, p1 and p2 be the probability of sample means falling in the
regions, and let the run lengths from points within these regions be L0, L1 and L2.
By taking one observation. we easily generate the following equations:
L0  1  p0 L0  p1 L1  p2 L2
L1  1  p0 L0  p2 L2
L2  1  p0 L0  p1 L1
We then easily obtain
L0  (1  p1  p2  p1 p2 ) /(1  p0  p1 p2  p0 p1  p0 p2  p0 p1 p2 )
In this expression. it is relatively easy to work out the ARL for any distribution, so
long as the Xs are i.i.d (independent and identically distributed). Some Values of the
ARLs for the standard chart with boundaries at  3.09 / n and  196 / n
('probability' limits), and  3 / n and  2 / n ('popular' limits) .
The Position of The Warning Lines
By using L0  (1  p1  p2  p1 p2 ) /(1  p0  p1 p2  p0 p1  p0 p2  p0 p1 p2 ) we can
vary the values of p0 and p1 to get the same ARL when the process is on target.
There are two limiting cases:
(1) No 'warning' region. Here we have
p0  ( L0  1) / L0
where L0 is the ARL when the process is on target.
(2) No 'action' region, but simply a warning region. Here we have
p1  {1  (1  8L0 )} / 4 L0
A Modified Scheme
The following rule is suggested: Take action if one point is outside the action limits,
or two out of three points outside the same warning limit. The ARL function for this
rule can be derived in a similar way to the Shewhart chart with warning lines method,
and we have the formula
1  (1  p0 )( p1  p2 )  2 p1 p2 (1  p0 ) 2
L0 
{1  p0  p1 p2 (1  p0 ) 2  p0 p1 p2 (1  p02 )  p02 ( p1  p2 )}
Extensions to Shewhart Charts for One-At-A-Time Data
One-At-A-Time Sampling
Circumstances Giving Rise to One-At-A-Time Sampling
One-at-a-time data arises frequently in practice. especially in the process industries. In
the component manufacturing field, many individual components are produced, and
each stands on its own merits. It is then easy to sample a group of n at a time to get
information about the process average level and process spread. In continuous
processes typically a rather different situation applies. The instantaneous variation of
7
the chemical, fluid. etc.. may be quite small, but the process varies gently in time for
many reasons often incompletely understood. For a process of this type. selecting a
group of observations close together in time will only represent measurement or
sampling error. which may be negligible. In order to be able to get close control of
such a process over time. it is necessary to use rational blocking or moving averages.
Charts of individual values will not detect the kind of slow drifts experienced. In this
chapter the construction of 'moving' type charts is described. but there are some
preliminary points to settle first.
i  1,2,..., k
It is convenient to think first in terms of Model xij     Bi   W  ij
j  1,2,..., n
. where there is both between- and within-group variation. with variances  B2 and  W2
respectively. Here  B2 represents variation in the process average level over time, and
 W2 the local or sampling variation. The variance of a group of n observations taken at
one point in time is
V ( xi. )   B2   W2 / n
whereas the variance of a mean of n observations sampled singly over a period is
V ( x i. )  ( B2   W2 ) / n
showing that one-at-a-time sampling is more efficient. However, two points should be
made. Firstly local variation,  W2 , may well be crucially important, rather than
variation of the process average; level over time since in some cases the latter night be
subject to some rough control. At least two observations need to be sampled at a time
even to estimate  W2 ,. and rather more than two observations are really needed.
Secondly, the cost of sampling a group of observations is often much less than the
cost of obtaining the same number of single observations. The sampling cost is made
up of staff costs, machinery costs, cost of disturbance to the production line etc.. and
the actual cost of using these facilities to obtain. say. five observations rather than
one, is often much less than five times the cost of a single observation.
The key reason why observations are often sampled in' groups for SPC is that control
of process spread is critical, and grouped observations are necessary to estimate
process spread. In the process industries, or in other applications where there is
substantial between -group variability, one-at-a-time sampling is often more
appropriate. Control of process spread can then be achieved through a moving-range
or moving-standard-deviation chart.
Some Alternative Charting Methods
If the group size is small, or perhaps one, the charting methods cannot be used
effectively. For example. although we can construct a Shewhart chart for group size
n=1, the limits are very wide. so that the chart has little power. Some alternative
approaches when the group size is small or one are as follows.
1- Rational blocking
In this method we artificially block the data by shift, batch. day etc. We hope to have
some rational method of doing this, so that special causes of variation occur between
rather than within groups. Although this is a very simple way of getting round the
problem, an unfortunate choice of blocking can mask chances, and inflated estimates
of spread can also result. leading to wide limits.
2- Moving-averages, moving-average
8
The alternative to rational blocking is to use moving averages or moving ranges. The
disadvantage of any moving-average chart is that action is necessarily delayed by the
averaging.
A key problem with one-at-a-time data is the estimation of the standard error σe of the
points being plotted.
Details of Further Control Charts for Control of Process Average Level
The construction methods given below apply for groups of size n. but for one-at-atime data put n=1 throughout. We shall assume that estimates of the process
variability are available. We shall denote the group means by X 1 , X 2 ,... and the
^
estimate of standard error of the X i is denoted
 e . If we have one-at-a-time
^
sampling,  e is simply an estimate of σ . The following subsections assume that this
^
estimate  e is available.
Moving-average charts
In the moving-average chart. we plot the averages of the last k groups of size n.
The points on the time axis represent times at which groups of size n are taken. For a
moving average of 3s. these averages are represented at the middle of the three points.
For a moving average of 4s. the averages fall half-way between two time points at
which groups are sampled.
Points are lost at the start and finish of a moving-average chart. because there is not
enough data to complete the chart. If desired. these missing points could be filled in
with the averages of the points available at the ends, but if this is done. the action
limits calculated below should bo disregarded for these points. In a moving-average
chart, only action lines are used because successive moving averages are highly
correlated. The usual method of constructing charts, ignores the dependence between
successive moving-averages.
Construction of A Moving-Average Chart
Method Summary
Construction of A Moving-Average Chart
^
Step 1(Grouped or blocked data) Obtain estimates of the process average level,  ,
and process variability. and also obtain the estimated standard error of group means
^
e.
^
Step 1 (One-at-a-time data) Obtain an estimate of the process average level,  . Also
^
obtain an estimate of the process variability. and choose a suitable  e for charting.
^
Step 2 Choose the scale of the chart so that  is near the centre. and so that the scale
^
covers approximately  4 e / (k ) .
^
^
Step 3 Mark the action lines at ia   3.09 e / (k ) .
Note:
This assumes that all groups are of the same size.
An alternative to disregarding the first few points is to plot the moving-averages of
2,3,… points, etc., until k points are available. If this method is used the
corresponding action lines have to be recalculated at each point until k points are
available.
9
Exponentially Weighted Moving-Average Charts (EWMA charts)
The exponentially weighted moving-average chart is particularly useful either when
we have one-at-a-time data, or else when great precision is needed to detect small
changes. Basically the method is to form a new moving-average at each sampling
point by calculating a weighted average of the new value and the previous movingaverage. Therefore in the exponentially weighted moving-average chart all of the past
data has some effect on the current value, but it rapidly lose influence.
Method Summary
Construction of Exponentially Weighted Moving-Average Charts
^
Step 1 (Grouped or blocked data) Obtain estimates of the process average level,  ,
and process variability, and also obtain the estimated standard error of group means
^
e.
^
Step 2 (One-at-a-time data) Obtain an estimate of the process average level,  . Also
^
obtain an estimate of the process variability, and choose a suitable  e for charting.
Step 2 Choose a value for p in the range 0.1-0.5. This is the amount of weight put on
the current value.
Step 3 Choose a starting value, k0 , as either the overall mean or a target value.
Step 4 Calculate and plot the moving average ki using the formula
ki  p X i  (1  p)ki1
where the X i are group means, or for one-at-a-time data, single values.
^
Step 5 Use action limits only, and place them at   A1  e
where the A1 values are given in Table (factors for construction exponentially
weighted moving-average charts from an estimates of σ.)
Exponentially Weighted Moving-Averages: Starting-Up Problems
The factors given in Table ( factors for construction exponentially weighted movingaverage charts from an estimates of σ.( A1 )) are based on the asvmptotic variance of
mi. If we write the asymptotic variance V(k), then we have
V (k )  p 2 ( X i )  (1  p) 2 V (k )
leading to
p
p 2
V (k ) 
V (X i ) 
( 2  p)
n(2  p)
However. if we start with a fixed k0, then we have
V (k i )  p 2 2 / n
V (k 2 )  [ p 2  p 2 (1  p) 2 ] 2 / n
V (k 3 )  [ p 2  p 2 (1  p ) 2  p 2 (1  p ) 4 ] 2 / n, etc.
The convergence of these to the asymptotic formula given above can sometimes be
rather slow, especially if p is rather small.
Control of Process Spread
Moving-Range Charts
For one-at-a-time data, a moving-range chart or a moving-standard-deviation chart is
the only way of controlling process spread. However, one of these charts may be
10
useful with grouped data, as an extra chart, if there is a substantial between-group
variation. In this case we plot the moving range of the group means.
Method Summary
Construction of Moving-Range Chart
Step 1 Determine k, the number of sampling points the range is taken over. This is
often determined by practical considerations.
^
Step 2 (One-at-a-time data) Obtain an estimate  of the process spread by using
moving ranges of k points.
^
Step 2 (Grouped or blocked data) Obtain an estimate  of the standard error of group
means.
Step 3 Choose the scale of the moving-range chart so that the range goes down to
zero. and up to about twice the largest range observed.
Step 4 Mark the action limits on the chart:
^
Lower action limit
D5 
Upper action limit
D6 
^
D5 and D6
where D5 and D6 are given in Table (fators for construction range charts from an
estimates of σ.(D5 , D6, D7 , D8)).
There are no warning limits on a moving-range chart.
Choice of Charting Method
Choice Between Charts for Control of Average Level
Here we summarize some of the advantages and disadvantages of the alternative
Shewhart charts for controlling the average level of a process.
X Charts
advantages
(1) Good at detecting sudden jumps in average level
(2) Well used and reliable
(3) Easy to understand
Disadvantages (1) Rather slow to detect drifts in average level
(2) Not very good at picking out small changes in average level.
Moving-Average Charts
Advantages (1) Better than X charts at detectino slow drifts
(2) Can be used when group size n is small, and when n=1
Disadvantages (1) Delays in responding to sudden jumps in average level.
Exponentially Weighted Moving-Average Charts
Advantages (1) Good at detecting slow drifts
(2) Can be used with small group sizes, and when n=1
Disadvantages (1) Delays in responding to sudden jumps.
For one-at-a-time. individual charts are sometimes used, for example when data is
infrequent. and moving averages would cause undue delay. A moving-average chart
or EWMA chart should then be used as well.
Charts for Control of Process Spread
If the data is one-at-a-time, then the moving-range chart is the only one usable. If the
data is grouped or blocked, ordinary range or standard deviation charts should be
used. However, even in this case, a moving-range chart based on group averages will
help to keep control of between-sample variation. if there is any.
11
Practical use of Shewhart and Moving-Average Charts
Relation Between 'Control' and Process Capability
A process is said to be `in control' if there is no evidence from either the chart for
control of average level or the chart for control of process spread that assignable
causes of variation may be present. It is important to note that we can have a process
in control, but producing defective quality material because of low relative capability.
The reverse can be true when the process is of high capability.
Out of Control
When the charts show an `out of control' point which is not due to miscalculation, we
may have evidence from either or both of the charts for average level and the chart for
process spread. Apart from the specific rules given in previous sections, there may be
other obviously non-random patterns in the plots. For example, We may have too
much clustering of the data in the centre of the chart, or too much clustering towards
the extremes. Alternatively. we may observe `cycles' in the charts. The great value of
the charts is that they show when to set up procedures to look for special causes of
variation (which may involve stopping the process) and when to leave the process
alone. By seeing whether there is evidence of a shift in level, or spread, or both, and
by seeing whether this is a sudden change or a steady drift, we can get some clues as
to the possible cause. The step of 'problem-hunting' for special causes of variation is
expensive and time-consuming - but not so expensive as bad product. In some
industries, indications of when to leave a process alone are as valuable as indications
of when to search for a special cause.
Sequential Tests for Persistent Shift of The Mean
CUSUMs do frequently give us a device for early detection that the shift has occurred.
Usually the standard Shewhart control chart will detect the change in level easily,
provided the change is not too small. Let us assume that the mean μ0 of the production
process has the required value but a shift to some other value μ1 is possible. Let us
assume also that the variance, σ2, is known and constant regardless of whether the
shift of the mean occurs or not. As in the previous sections, we denote the running
number of lots taken, each of size n, by N. Let
x N  0
zN 
/ n
be the standardized sample mean of the Nth lot. consists in calculating the following
two cumulative sums after each Nth lot has been obtained:
SH N  max [0, ( z N  k )  SH N 1
and
SLN  max [0, ( z N  k )  SH N 1
where SH0 = SL0 = 0 and k is a suitable constant. Note that the upper cumulative
sum, SHN, is always nonnegative and increases whenever zN is greater than k.
Analogously, the lower CUSUM, SLN, is nonnegative as well and increases whenever
zN is smaller than -k. The value of k is usually selected to be half the assumed value of
the (absolute) mean shift in standardized units. That is, if the magnitude of the mean
shift is believed to be equal to 1   0 , we get the shift n 1   0 /  in
standardized units and we usually set k  n 1   0 /( 2 ) . Now, if either SHN or
SLN becomes greater than some prescribed constant h, the process is considered to be
out of control. More precisely, an upward shift can be believed to have happened in
the former and a downward shift in the latter case. The value of h is usually selected
12
to be equal to 4 or, less frequently, to 5. We shall see later that there is some rationale
behind the given choices of k and h.
We define the concept of the average run length (ARL) of a test as the expected value
of the number of lots until the first out of-control lot is detected, given the magnitude
of the mean shift that occurred. The ARL's can easily be calculated for the mean
control chart. Indeed, the ARL corresponds then to the expected value of the number
of trials until the first success occurs in a sequence of Bernoulli experiments, where
the success consists in producing an out-of-control signal by a single trial, that is, by a
single lot. The ARL is, thus, equal to the mean of the geometric distribution, 1/p,
where p is the probability of producing an out-of-control signal by a single lot. Now, p
can be immediately calculated for any given problem. For example, if, in standardized
units, the mean shift of absolute value 3 occurred, we get that the new mean lies on
one of the control limits (we assume in our analysis that both the mean before the shift
and the constant variance of the production process under scrutiny are known). Hence,
under the usual normality assumption,
p  P(z N  3  3 or z N  3  3 )
 P(z N  0 )  P(z N   6 )  1/ 2
and the ARL is equal to 2. Analogously, one obtains that the ARL's for the
standardized shifts of values 4, 2, 1 and 0 are 1/.8413=1.189, 1/.1587=6.301,
1/.023=43.956 and 1/.0027=370.37, respectively. We note that the last number is the
ARL until a false alarm occurs. Computing the ARL's for the Page CUSUMs is a
rather difficult task. The results for the Page CUSUMs with k = .5 (again for
standardized mean shifts) in following table.
Shift
0
1
2
3
4
h=4
168
8.38
3.34
2.19
1.71
h=5
465
10.4
4.01
2.57
21.01
Comparing the ARL's for the standard mean control chart with those for the Page
CUSUMs, we find that it may indeed be advantageous to use the latter on data with
small, single mean shift. We note also that the Page test with h = 4 is likely to ring a
false alarm much earlier than the corresponding test with h = 5.
A thorough analysis of the AR.L's for the Page tests reveals that it is indeed
reasonable to use k equal to one-half of the absolute value of the mean shift and to
choose h equal to 4 or 5. The given values of h have been selected to give the largest
in-control ARL consistent with an adequately small out-of-control ARL.
The EWMA charts are based on calculating Exponentially Weighted Moving
Averages of sample means. Let us assume that a certain number of lots has been
observed and that the sample mean across all the lots is x .The exponentially weighted
moving average for the Nth lot is computed as
^
^
x N  r x N  (1  r ) x N 1
13
^
where x 0 is most often set equal to x . If the means of first lots can be safely assumed
^
to be equal to some μ0, we can let x 0 = μ0. The weighting factor r is positive and not
greater than 1.
^
We note that no averaging takes place , x N  x N , if r=1. Otherwise, past sample
^
^
means x N i are included in the moving average x N with the exponential weight
r (1  r ) i . Clearly, if there is a shift in the lot means, it will likely be reflected in the
values of the moving average. Careful analysis shows that, in order that small mean
shifts be handily detected, the factor r should lie somewhere between .2 and .5. The
most often choices are .25 and .333. Control limits for the EWMA chart may be
written as
a ( n) s
r
UCL  x  3
2r
n
and
a ( n) s
r
UCL  x  3
n 2r
where a(n) is a constant with values given by
^
  a ( n) s
and folloing table
Sample size
2
3
4
5
6
7
8
9
10
15
20
an
1.253
1.1284
1.0854
1.0638
1.0510
1.0423
1.0363
1.0317
1.0281
1.0180
1.0133
bn
0.8865
0.5907
0.4857
0.4300
0.3946
0.3698
0.3512
0.3367
0.3249
0.2880
0.2677
cn
1.1829
1.0646
1.0374
1.0260
1.0201
1.0161
1.0136
1.0116
1.0103
1.0063
1.0046
The second term on the right hand sides of the above expressions is an approximation
^
of the exact value of three standard deviations of the moving average x N . In fact,
^
standard deviation of x N is equal to
  r (1  (1  r ) 2 N 
1/ 2

 ,
2r
n

where, as usual, σ is the population standard deviation.
14
CUSUM Performance on Data with Upward Variance Drift
The performance of CUSUM approaches should improve on data where there is a
stochastic drift upward in the variance of the output variable. We assume that if lots of
observations of size n are drawn from a normal distribution with mean μ and variance
σ2 then letting sample variance be given by
n
1 j
2

( x j ,i  x j )2
s j nj 
i 1
we have the fact that
(n  1) 2
zj  j 2 sj

has the Chi-square distribution with nj -1 degrees of freedom. That is
z
j
n j 1
1
1  2
2
f (z j ) 
z
e
n 1 j
n j  1 j2
(
)2
2
Then, assuming that each lot is of the same size, n, following the argument, we have
that, after the N'th lot the likelihood ratio interval corresponding to a continuation of
sampling is given by
ln( k0 )  R3  ln( k1 )
where
 1 N n 2 3
n 1 N 2 
(
)
exp(

s )
 2
2  j 

2

j

1
1
1

R3  ln 
 1 N n 2 3
n 1 N 2 
exp(  2  s j ) 
( 2 )
2 0 j 1 
 0
A straightforward Shewhart CUSUM chart can easily be constructed. We recall that
for each sample of size nj ,
1 n
s 2j 
 ( x j ,i  x j )2
n j  1 i 1
is an unbiased estimator for the underlying population variance, σ2. Thus, the average
of N sample variances is also an unbiased estimator for σ2 , i.e.,
1 n
E (  s 2j )   2
n j 1
Recalling the fact that if the xj,i are independently drawn from a normal distribution
with mean μ and variance σ2, then
(n  1) s 2
  2 (n  1)
2

For normal data, then, we have,
Var ( s 2 )  E ( s 2   2 ) 2
1  4 4! n  3 4
 (

 )
n 4 2! n  1
2 4

n 1
So, then, a natural Shewhart CUSUM statistic is given by
15
N
R4 
s
j 1
2
j
 N 02
 02
2
N n 1
Clearly, statistics R3 and R4 can be used for detecting a persistent shift in the variance
from one value to another. As in the case of using statistic R1 to pick up a mean shift,
we can replace a single test based on R3 by a sequence of such tests. Starting the test
anew whenever a decision is taken enables us to get rid of the inertia of the single test.
  
  2 
R1  N 1 2 0  x N  0
2 


n
x N  0
R2 
0
nN
Acceptance-Rejection CUSUMS
For Quality Assurance, Suppose that we have a production system where the
proportion of defective goods at the end of the process is given by p. A manufacturer
believes it appropriate to aim for a target of p0 as the proportion of defective goods.
When that proportion rises to p1 he proposes to intervene and see what is going
wrong.
Let us suppose that the size of lot j is nj. Then the likelihood ratio test is given by
N
n j!
x
n x
p1 j (1  p1 ) j j

j 1 x j !( n j  x j )!
R5  ln( N
)
n j!
xj
n j x j
p0 (1  p0 )

j 1 x j !( n j  x j )!
16
Conclusion
The primary purpose of SPC techniques is to encourage continuous improvement and
to simplify the administrative and production processes. Control charts help managers
in eliminating waste by allowing them to evaluate the stability of a particular process,
determine the mean and the variability within the process, and monitor the effects to
improve the process.
17
References
12345-
Hanter, J.S. (1986). " The exponentially weighted moving average" . Journal of
Quality Technology, vol 18, pp. 203-210.
Thompson, J.R., Koronacki, J., (1993) ." Statistical Process Control for Quality
Improvement ".Chapman and Hall Press
Walton, M.(1986)."The Deming Management Method –The Complete Guide to
Quality Management ". foreword by Deming, W.Edwards.
Wetherill, G.B., Brown, D.W. (1993). "Statistical Process Control Theory and
Practice". Chapman and Hall Press.
Andrews, D.F., Bickel, P.J., Hampel, F.R., Huber, P.J, Rogers, W.H., And
Tukey, J.W. (1972). Robust Estimates of Location. Princeton: Princeton
University Press.
18