Quality Design and Control

Prof. Pradip Kumar Ray

Department of Industrial and Systems Engineering
Indian Institute of Technology, Kharagpur

Lecture – 24
Statistical Process Control – II (Contd.)

Continuing our discussion on the special purpose control charts.

(Refer Slide Time: 00:17)

We have already refer to modified control chart if you recollect.

(Refer Slide Time: 00:22)

And what we have mentioned that the modified control chart is to be used along with the
acceptance control charts. Like, we say that r control chart, and x bar control chart must
be use simultaneously. And similarly, we also say that the modified control chart, and
acceptance control charts. These 2 chart must be used simultaneously. There is a specific
reason for this one. And we will come to know what is the, that reason when I discuss the
acceptance control charts.

Now, there are if you just mention that there is there are ah, there are 2 conditions to be
satisfied. The first condition is you need to determine the upper limit as well as the lower
limit of the process plane in such a way that the proportion and confirm is restricted to
delta; that is, the first condition we have to satisfy. And for that you need to use the
modified control chart. Now the second condition that you frequent may frequently come
across; that is, these the proportion and conforming which we have set as a standard. Say,
1 percent, 0.5 percent, is it ok? So, that is the set; that is, and the value of delta. There is
no guarantee that in an actual process at that point in time there could be take many
reasons, that this actual proportion nonconforming may be greater than, substantially
greater than the delta, which is to be specified.

So, what do you do? Obviously, you need to use another control chart, which will help
you in detecting this you know unacceptable or the proportional conforming this amount
as quickly as possible. So, initially you design the modified control chart with respect to
the proportional conforming delta; which is acceptable to you. Next what you will do?
That means, if the delta changes to gamma; that means, say 1 percent, it was it is delta is
1 percent. Suppose because of some reason, you are getting a value of say 3 percent
proportion non-conforming.

So, whether you know you are able to detect this change in the proportional conforming
as quickly as possible. So, for this you need to use acceptance control charts. So, I think
it is clear that you know, what is the specification, that is why the acceptance for these
acceptance control chart is used. Now this so, this point I will elaborated, when I say that
this chart is used when the proportion non-conform proportion of non-conforming output
is not restricted to specified value delta.

If you remember, that when we design when we design the modified control chart the
modified control chart is designed with respect to a specified value delta, is it ok? But
because of some reason suppose the delta is changed to some higher value, say gamma.
Gamma is substantially greater than delta. So, the delta is 1, say 1 percent and the
gamma maybe 3 percent. Now these chart is designed in such a way that these higher
value of proportion nonconforming is detected with a probability of 1 minus beta. Beta
as we know these are probability of making type 2 error; that means, essentially
probability of non-detection.

Obviously 1 minus beta is probability of detection. So, that is to be assured; that means, a
high value of probability of detection is to be assured, is it? These chart is to be used
jointly with a modified control chart. These points already are elaborated.
(Refer Slide Time: 04:53)

So, what is the problem statement? Problem statement is very simple. We use to detect
the level of proportion nonconforming gamma with probability probability of 1 minus
beta, is it ok? So, if the detection probability and make sure that whenever the
professional nonconforming becomes gamma, ok? It should be detected as quickly as
possible; that means, make sure that the average (Refer Time: 05:26) if you recollect we
have use the term we have defined what is a r l; that means, on an average how many
sample points you are going to plot before and out of control condition is reached, is it
ok is detected.

Now; obviously, if the process has already gone out of control so, that error should be as
minimum as possible, say 3 to 4 or say 2 to 3 sample points as in quikly as possible.
Whether of suppose you know the process remains in control. So, the (Refer Time:
06:04) should be you know, as early as possible (Refer Time: 06:10). So, here in this case
has you are defining the out of control condition with respect to the proportion of non-
conforming. And with a specific value of proportion of non-conforming gamma, we say
that the process has gone out of control. So obviously, the a r l value should be as
minimum as possible?
(Refer Slide Time: 06:34)

Now, again we referring to the say distribution of individual values, that is x. So, again
what we are assuming now that already the mu l, and mu u. These 2 values are specified
against the process mean, is it ok? And your this is the upper value. This is the lower
value, is it ok? Now you know this is the distribution of x and so, the corresponding
value of the standard deviation, suppose standard deviation is sigma and if the proportion
of non-conforming gamma, which is not acceptable to your. So, this (Refer Time: 07:19)
distance is z gamma into sigma, is it ok?

So, this is the distance from mu l to USL. USL is already specified for the given quality
characteristics for which already you have drawn, you have been using the modified
control chart. And similarly, on the other side this is the area under the curve of beyond
LSL that is gamma, is it ok? And this is a standardized distance; that is, z gamma into
sigma ok. Distance of mu l from LSL is, it is ok. So, we are referring to the distribution
of individual values, and we are assuming that the central limit theorem holds, that is
why the distribution of x is assumed to be normal.

When you refer to the distribution of the sample mean that is x bar; that means, around
mu l around mu u; obviously, you know as per central limit theorem. So, this is the
standard deviation, we will be sigma upon root n that man’s sigma z bar is equals to
sigma upon root n. And this we refer to the distribution of x bar. So now, you have the
say l LCL is here. And UCL is here. So, even if you know the value is at mu l, what do
you find that there is an area, which is within this LCL within the UCL this zone.

So, obviously, this is a proportion of say this is the proportion indicative of the
probability of making type 2 error; that means, the process has gone out of control, but
actually it is shown as in control, is it ok? So, this is a type 2 error. Similarly, when the
upper value is mu u ok, and the process mean, now the area which is within the upper
control limit that is beta, is it ok? So, again this is when you reach at this at this point.
So, you have this area that is this indicative of probability of making type 2 error.

So; obviously, the standard size distance from UCL to mu u is z 2 beta z beta into sigma
upon root n. And similarly, the standardized distance of LCL from mu l is z beta z beta
into sigma upon root n, is it clear? So, we are referring to distribution of the individual
values x. And then we are referring to the distribution of the sample mean x bar, is it
clear?

So, we are defined what is USL, what is LSL. We have defined what is gamma. We have
defined what is beta; we have you know defined what is UCL and what is LCL. Now
what you try to do; that means, ultimately in terms of all this parameters ok, you need to
have an expression for UCL in terms of USL. And you must have an expression of LCL
in terms of LSL; that means what you are trying to do, we are trying to explore the
relationship between the control limits as and the specification limits, is it ok?

(Refer Slide Time: 10:44)

So, and then you know; obviously, you know whenever you are referring to say the delta
or gamma, the proportion of non-conforming. So, these are you know this proportion
nonconforming is always defined with respect to in reference to other specification
limits. And so, we are going one step further; that means, till now we have not refer to
the specification limits, is it ok in the control chart.

Now, here we are determining the control charts in such a way that that the relationship
between the control limits and the specification limits are so, given the problem. Another
important aspect we must look into; that is the specification limit is always you know
applicable to individual values. Whereas, the control limits not necessarily accepting,
when you draw the control chart for the individual units, that is x control chart. And
along with the x control chart if you remember, we have the m r or moving range control
chart, is it ok?

So, accepting you know those that that particular case, otherwise always we will find that
the sample size is greater than 1, and in the sample is represented by it is statistics. So, x
bar is it ok? So, so, whenever you refer to the distribution of say the process the
parameter, say the process mean; that means, the sample size is greater than one.

So, we refer to the distribution of x bar, is it clear? Whereas, whenever you refer to the
specification limit, we are referring to the distribution of x of the individual values. Now
what are the set of equations, we if we refer to this figure, what you will find that the mu
l is equals to LSL plus z gamma into sigma. We refer to the first figure; that means the
institution of x for the individual values. And similarly, you get mu u is equals to USL
minus z gamma into sigma, is it ok?

Now, we must have an expression for mu l, and mu u from the second figure; that means,
you refer to the next figure, we will find the upper control limit is mu u minus z beta,
what is beta? Beta is the probability of making type 2 errors into sigma divided by root n.
What is n? N is sample size. Similarly, LCL you refer to the same figure you will find
that the LCL is equals to mu l plus z beta into sigma divided by root over n. How do you
get this sigma upon root n is basically sigma x bar? And as for the central limit theorem,
you can say that sigma x bar is nothing but sigma upon root n.

So, these 2 equations, we are referring to the distribution of individual values. Whereas,
these 2 equations, for this equation, we are referring to the distribution of x bar is it?
(Refer Slide Time: 14:35)

So, whenever. So, related to the acceptance control charts. So, what is the upper control
limit? is equals to upper specification limit minus these expression into; that means, z
gamma plus z beta upon root n into sigma, is it ok?

So, sigma is essentially the process standard deviation. Similarly, the LCL is equals to
LSL plus z gamma plus z beta divided by root over n into sigma. Now again how do you
determine sigma? So, that would be good estimate, acceptable estimate; that is, you
referred to r bar control r control chart or s control chart. And you say that the sigma, if
you refer to the r r control chart, is it ok? So, you can have an estimate of sigma, and that
is sigma hat is equals to r bar by d 2. And similarly, if you instead of constructing that
instead of using r chart suppose you use s chart.

So, similarly. So, related to say the s chart, you have you know the value of c 4 and as
well as the s bar. So, another you know the expression for sigma, it is an estimate of
sigma is say s bar divided by c 4, is it ok? So, so, whenever you declare a process to be
in control; that means we say that the first I try to control the process with respect to it is
variability. So, if the variability is under control, then I use the control chart, another
control chart in such a way, that I say that the mean or the process mean is under control,
is it ok?

So, so, the so; obviously, you know when you assume that the process is in statistical
control, as in all likelihood the value of sigma; that is, estimate of sigma is made
available. Now when has a we have been saying that are modified control chart must be
used along with the acceptance control chart. So, when the principles of both the charts
are applicable simultaneously, reading my point? So, in the first case, we know that,
what is the mu u, what is the value of mu l; we know what is the distribution of x.

We also know what is the distribution of x bar. We know given the quality characteristics
USL and LSL. And then we say that the given control chart there is a probability of
making type one error, is it ok? And then there is a probability of making type 2 error;
when the proportion of non-conforming changes say any value from say the delta, is it
ok? So, then we say that under these 4 verses 6 conditions or 4 or 5 conditions, we have
to meet all these conditions simultaneously.

So; obviously, the modified control chart must be used along with the acceptance
contract chart. So, that is the norm. So, when so, under in this context we say the
principles of both the charts are applicable simultaneously, is it clear? I think it is clear.
And simple in sample size n it given by this one is it clear? That means what you try to
do; that means, UCL for the acceptance control chart the expression for UCL is equated
weight the expression for UCL for the modified control chart, is it ok?

So, either you equate both these expressions. And get the expressions for n, is it ok?
Sample size or you can say that I will equate LCL expression of acceptance control
charts with the LCL expressions for the modified control chart. So, again you will be
getting an expression for n. So, what is the expression for say sample size when you
combine both these charts? That is z alpha plus z beta, divided by; this is z 1, this is not z
alpha by a beta, there is some problem.

So, this is z alpha plus z beta into say you have z delta z delta gamma minus z delta. So,
it is z delta z gamma minus z delta, is it clear ok? So, so we will we will make this
changes later one.
(Refer Slide Time: 19:54)

Now so, this is the special purpose say we are discussing the special purpose control
chart, right. Now as I have already told you, there are there are variety of special purpose
control charts So, so the 3 specific control charts we are used till now.

First one is the regression control chart, or the trend control chart that is the first one. The
second one that we have discussed that is the modified control chart and the third one is
the acceptance control charts. Now the next important control chart. We are going to use
that is the cumulative sum or cusum control chart, is it ok? Now I will before I discuss in
detail, the cumulative sum of control chart. let me say highlight some of the important
points related to control charting.

As you are aware that are the basic theory behind control charting was introduced many
years back in 1920's, and by Walter (Refer Time: 21:14). So, that is why I know many of
these control charts are named after him. And their effect has Shewhart control chart. So,
the Shewhart control charts are very less sensitive to small shift in the process. These
small shift maybe, in the order of 1 to 1 0.5 sigma or less, is it ok? It is not one one to it
is 1.5 sigma or less, is it ok? That means, when the sigma of the process is wrong, that is
the first exercise you carry out before you start using a control chart; that means, given a
quality characteristics given a process, is it ok? And it is running process. So, you must
know with respect to the quality characteristics, what is the value of sigma? And this
sigma with respect to the quality characteristics may be referred to as the process
standard deviation, and estimate of the process standard deviation.

Now, what we are trying to do; that means, we are using control chart that is the first
objective, what extends we are able to control the variability. So, the variability; that
means, the sigma. That is also random variable. So, and so, if the process mean changes;
suppose you need to control the process mean. So, the change in the process mean; that
means, shift magnitude many a time, it is it is expressed in terms of sigma; that means,
what you are assuming that in a given process, both mu and sigma may change. And
their essentially, they are changing simultaneously ok.

Now, we need to define the small shift as well as the large shift, is it ok? Large shift are
basically you know more than 1.5 sigma. That is the norm, or say more than 2 sigma, is it
ok? That is the norms. So, what the researchers have found, that the Shewhart control
chart is very, very sensitive to a small to large shift, is it ok? Whereas, it may not be that
sensitive ok, when the shift magnitude is very, very small; that means, less than 1.5
sigma.

In many sophisticated processes, providing high quality products with very intricate
shapes, and very tight tolerances like say you know automobile parts or the aircraft parts,
is it ok? So, small shift in the process may make the process out of control, is it ok? So, if
there is a small shift in the process parameter in the process; I mean process parameters
ah, you say that the process has gone out of control. Because very tight tolerances you
not be able to achieve.

So, the control chart is special purpose control charts need to be designed in such a case;
that means, you design the control chart, you re design the control chart existing control
chart in such a way. That even we have the this the small shift in the process parameter
value can be detected quickly ok. So, a control chart needs to be designed in such a way
that the small shifts can be detected as quickly as possible ok.

So, the cusum control chart is designed and used to make this purposes, is it ok? Now so,
these are very special purpose control chart. And I will just explain. What are the steps to
be followed ok. And what is the rational behind following the steps ok, in developing a
cusum control chart, is it ok?
(Refer Slide Time: 25:30)

Now what do you try to do in the in the previous in previous cases? Like, when you use
the Shewhart control charts, what do you assume that all the samples they are
independent with one other; that means the sample values are not dependent, is it ok?
And at any point in time at a particular when you draw a sample and at a particular point
in time you are you know you are using a sample statistics. And you plot the value of the
sample statistic, is it ok? That means, you know the value you plot is indicative of a
condition at that point in time, at that during that period of time, is it ok?.

So, as got a no relationship with the previous conditions, getting my points now, what we
are trying to do; that means, you try to incorporate all the information in the sequence of
sample values by plotting the cumulative sum of the deviations of the sample values
from a target value; that means, here is here is a process I have designed. Now with
respect to the process parameter ok, given process parameter, I must know what is this
target value?

Now, the actual value when you get at a particular sample. So, you know that, what is
this you know the deviation from the actual value? So, that is a particular sample, what
you need to do when you arrive at a particular sample, now you need to consider for all
the previous samples the deviations from the targets. And you have to you know the add
them together at a particular point in time. So, the cusum charts incorporates, all the
information in the sequence of sample values; that means, (Refer Time: 27:57) is to be
maintained that I have already told you, that when you construct a control chart make
sure that the there the samples are drawn ok, as per you know the time sequence.

By plotting the cumulative sum of the deviations of the sample values from a target
value; that means, at a particular sample ok, when I when I define the deviation, now this
deviation incorporates all the deviations that that have already occurred in the previous
samples, is it ok? From the first sample, just I will explain it means how do you arrive
this community value. Let x j bar be the average value average value of the jth sample
with n equals to n greater than 1.

So, this is the average value, not values ok. So, just one value; that means, you have n
number of values in a particular sample jth sample you compute the average value. Mu 0
be the target for the process mean as already mentioned. Now how do you compute the
cusum? So, the cusum at the ith sample; that means, how do you arrive at the ith sample;
that means, you have started from the first sample second sample third sample like this
you proceed, and ultimately you reach the ith sample, is it clear?

So, the I could be 1, I could be 2, I could be 3, is it ok? So, I is the variable. So, at the ith
sample, what is how do you compute the cumulative the sum; that is, from j equals to 1
to i up to i, including i you compute the x j bar minus mu 0; that means, j equals to 1 to i,
is it ok? That means, up to I including I th point. Some up to I mean including ith sample,
is it clear? I think it is clear, when is this is you are the cumulative sum of the deviations
sum of the deviations from the target value up to the ith sample point and including the
ith sample, is it ok?

So, the cusum chart is very effective when n equals to 1, as you when you will be
referring to that the control chart for the individual units; that means, x chart ok, you
know what are the for under which conditions you have no other alternative, but to go for
you know, using x chart is it ok. So, the sample size is very, very less ok. So, . So, that is
one. So, the so, in many cases we will find, like if you deal with say the process
industries, is it ok? The case of n equals to 1 is very, very common.

So, in this case average sample values changes to the individual value; that means, there
will be just x j not x j bar, is it ok? So, what you know this cumulative sum control chart
maybe recommended to be used for say the process industries, and whenever say is a
chemical plant or the chemical industries ok, as an example. So, you will find that you
need to assume that n equals to 1, is it ok? And accordingly you define the cumulative
sum, is it ok?

So, the first thing you have to do that is cumulative say the sum you have to defined with
respect to the given the process parameter, is it ok? And for which target value is to be
specified.

(Refer Slide Time: 31:53)

Now so, just what I am telling you certain decision rules, let me just highlight, and then
in the next sections, I will be discussing the other aspects of cusum chart.

So, what is the decision rules that is to be known. That is if the process remains at mu 0;
that means, again control states that is the target value. C i has random fluctuations with
mean at 0. Always you know the x i values is there will be randomness; that means, there
is randomness only when the process is in control. So, obviously, the mean at 0, is it ok?
If the process mean shifts upward to say mu 1 which is greater than mu 0 substantially
greater than mu 0 c I shows an upward positive drift and similarly if the process mean
shifts downward to mu 2, which is substantially less than mu 0, c i shows a downward a
downward or negative drift, is it ?

A methodology or a scheme is to be developed. So, that from the trend pattern of c i as

being observed we may be able to conclude that there is a significant change in the
process mean determine the changed value of the process mean, is it ok? That you have
to do and identify the sample, where the change in the process mean taking place due to
assignable causes; that means, at what point in time for which sample the process has
gone out of control. So, that particular the sample you have to detect . And you refer
back to that particular sample, is it ok? And you just like check the process conditions,
and whether you are able to identify you know the assignable causes related to out of
control conditions.

Now, here whenever you talk about cusum control charts, the researchers have developed
2 approaches, is it ok? In fact, there could be other approaches, but this 2 approaches are
always refer to. So, you must be aware of in. In fact, on the cusum chart till lots of
research going on, is it ok? So, the from. So, in perfect the systems you may be going to
a perfect system, is it ok? But the 2 important schemes they refer to the scheme or the
approach for the cusum control chart, you must be you must be discussing. And that is
first one is the tabular or algorithmic cusum, this is the scheme you need to use. And the
next one is the v mask template. So, in the next sections, I am going to discuss these 2
schemes.