V. Control Charts A. Overview Consider an injection molding process for a pen barrel.
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Transcript of V. Control Charts A. Overview Consider an injection molding process for a pen barrel.
V. Control Charts
A. Overview
Consider an injection molding process for a pen barrel.
The goal of this process: To produce barrels whose true mean outside diameter is μ0.
Consider a plot of the mean, μ(t), over time.
What are our options?
• Caveat emptor. “Let the buyer beware.”
Basic attitude: The customer will buy whatever we make.
Therefore, let the process mean drift!
• Let μ(t) wander and perform 100% inspection expansion.
Basic idea: Inspect out the bad barrels.
Problems:
• Very expensive in terms of inspector's time.
• Lots of wasted barrels.
• Even 100% inspection is not 100% effective!
• Use a statistical sampling plan.
Basic idea: Use statistics to suggest ways to cut down the inspection costs.
Advantage: Generally cheaper than 100% inspection.
Problems:
• Lots of wasted barrels.
• Definitely not 100% effective.
• Statistical Process Control
Basic idea: Constantly monitor the process; adjust the process once a drift is detected.
Advantage: Stops problems before they become serious.
Drawback: Does not directly deal with the sources of problems.
• Create processes which do not produce unacceptable product.
Basic idea: Identify sources of problems; set up safeguards to prevent their occurrence.
Primary statistical tool: Experimental Designs (Chapter 7)
Consider monitoring a process.
Recall, our goal is to keep μ(t) at some target value, μ0.
Note: μ0 may or may not be the “ideal” value for our characteristic of interest.
The first goal in any quality improvement program is to get the process stable.
Only after the process is stable, can we begin to move μ0 to its “ideal” value.
When μ(t) = μ0, we say that the process is in control.
The process is assumed to remain in control until acted upon by an assignable cause.
The assignable cause shifts μ(t) to some other value μ1 and the process is said to be out of control.
Our goal, then, is to develop a method for monitoring μ(t).
Problem: Do we ever see μ(t)?
No, but we do see estimates of μ(t)!
Therefore, what should we do?
Assume that we are concerned with the mean of the characteristic ofinterest, and that the characteristics variance is known.
What seems reasonable?
Step 1: Take samples at regular intervals.
Step 2: Perform hypothesis tests of the form
H0: μ = μ0 process is in control H1: μ ≠ μ0 process is out of control
Step 3: The appropriate test statistic
Step 4: The critical region
We reject if H0
What does this critical region mean?
n
yZ
/0
2/|| zZ
We conclude the process is out of control if
Equivalently, we act as if the process is in control (we have insufficient evidence to conclude the process is out of control) if
2/2/or zZzZ
nzynz
nzynz
zn
yz
zZz
//
//
/
2/02/0
2/02/
2/
0
2/
2/2/
As a result, we conclude that the process is out of control if either
or
We call
• the upper control limit (UCL) and
• the lower control limit (LCL).
What about ?
Traditionally, , which corresponds to an =.0027.
nzy /2/0
nzy /2/0
32/z
nz /2/0
nz /2/0
We summarize this sequence of tests by a graphical procedure.
Any observed sample mean above or below the control limits is considered evidence that the process is out of control.
If a “out of control” condition signals, then we must search for the assignable cause, correct it, and restart the chart.
Note: It is possible for a “false alarm” to occur.
False alarm: The chart indicates that the process is out of control when it is not.
What is P(false alarm)?
Let N be the number of samples until the process signals an out of control condition.
If the process is in control and we use , 32/z
4.3700027.0
11)(
NE
It is important to note: A control chart will signal on out of control situation at some point in time even if the process remains in control the entire time.
was chosen to insure that the process will signal very infrequently when it is in control.
On the other hand, if μ1 is significantly different from μ0, then the chart will signal quickly.
How quickly depends upon
32/z
n/01
The general form of our control charts is
θ represents the characteristic of interest we wish to monitor.
We shall consider the cases of:
• sample mean,
• sample range,
• sample proportion
• sample “counts”.
θ0 represents the target value for θ.
represents the standard error for our estimator of θ. ˆ
0
ˆ0
ˆ0
CL
3LCL
3UCL
B. - and R-Charts
Consider monitoring the ash contents for a pencil lead process.
Suppose we take a sample of 5 lots each shift.
1. The R-Chart
First, consider monitoring the within sample variability.
Thus, at each time period, we need to estimate σ.
Let yij be the jth observation (j = 1, …, n) taken at the ith interval (i = 1,2, …, ).
Let Ri be the sample range of these n observations.
X
smallest -largest )1()(
iniiyyR
We note that Ri is a measure of the “spread” or variability at the ith times interval.
Also, it is easy to calculate.
If our data follow a normal distribution, then Ri is an estimate of a multiple of σ.
Typically, we do not know the true in control value for σ.
Problem: How should we estimate it?
Solution: A “base period”.
We use an initial base period of m samples.
m may be as small as 20 intervals.
Personally, I recommend 40-60 intervals.
We then base our control limits for the sample ranges on the average sample range from this base period, , given by
The control limits for the range are:
where D3 and D4 are appropriate constants. (see the text).
Note: for n ≤ 6, D3 = 0.
For our example, the data follow.
Let the first 20 samples form the base period.
R
m
iiR
mR
1
1
RD
RD
3
4
LCL
UCL
OBS ASH1 ASH2 ASH3 ASH4 ASH5 MEAN RANGE 1 41.3 42.1 41.5 41.5 41.9 41.66 0.8 2 42.4 41.7 40.6 40.2 43.3 41.64 3.1 3 41.0 41.6 41.8 41.2 41.6 41.44 0.8 4 42.9 41.7 41.5 42.5 40.3 41.78 2.6 5 43.4 43.0 42.7 42.2 42.3 42.72 1.2 6 42.1 42.6 42.4 43.1 42.8 42.60 1.0 7 43.2 42.7 40.9 40.8 42.3 41.98 2.4 8 42.3 42.5 43.0 42.5 42.2 42.50 0.8 9 42.6 42.5 42.5 42.2 42.3 42.42 0.4 10 42.6 40.2 42.0 41.4 42.1 41.66 2.4 11 42.0 41.3 40.3 41.7 41.2 41.34 1.7 12 42.3 42.4 41.7 42.5 42.5 42.28 0.8 13 43.2 40.7 41.0 41.4 42.0 41.66 2.5 14 42.0 42.2 42.0 42.0 42.5 42.14 0.5 15 43.7 42.6 43.7 41.3 42.7 42.80 2.4 16 42.7 42.5 42.1 42.2 42.3 42.36 0.6 17 40.3 42.6 42.6 42.1 42.4 42.00 2.3 18 41.0 42.3 42.2 42.3 40.9 41.74 1.4 19 41.0 40.3 40.5 40.1 40.5 40.48 0.9 20 40.7 40.8 40.5 41.5 42.3 41.16 1.8 21 41.9 41.8 41.7 41.6 41.4 41.68 0.5 22 41.3 41.7 42.8 42.4 41.5 41.94 1.5 23 41.7 42.7 41.0 40.4 42.3 41.62 2.3 24 40.4 42.4 42.9 42.1 42.7 42.10 2.5 25 42.0 42.8 42.3 42.6 42.5 42.44 0.8 26 42.5 42.6 42.0 42.0 42.6 42.34 0.6 27 42.9 41.8 42.4 42.4 42.5 42.40 1.1 28 42.6 42.2 42.3 42.3 42.3 42.34 0.4 29 42.3 41.6 41.6 42.1 43.0 42.12 1.4 30 42.1 42.2 41.8 42.2 43.3 42.32 1.5
Example: the data are:
Let the first 20 samplesform the base period.
In this example,
n = 5 D4 = 2.115 D3 = 0
The control limits are:
The resulting chart is:
0LCL
21.3)52.1)(115.2()115.2(UCL
R
2. The -Chart
The -Chart simply refers to a control chart for sample means or averages.
Let be the sample mean of the n observations taken at the ith time interval; thus,
Often, both μ0 and σ0 are unknown.
In such situations, we use a base period of m samples to estimate it by
X
X
iy
n
jijiy
ny
1
1
m
iiy
my
10
1̂
Traditionally, people have used to estimate σ.
Actually a better estimate of σ is based on ,
where the is the sample standard deviation for the ith sample.
See the text for details.
iRmR )/1(
2s
m
iis
ms
1
22 1
2
is
It can be shown that
which is an estimate of
and
which is an estimate of
where A2 is a constant which comes from the Table 5 in the Appendix.
Note: A2 depends upon the number of observations in each sample.
RAy2
UCL
n
30
RAy2
LCL
n
30
The center line for this chart is .
The control chart thus will have the form:
________________________________
________________________________
________________________________
As an example, again consider the pencil lead ash data set.
The observations are ash contents for lots of pencil lead.
Five lots constitute a sub-group or sample.
y
RAy2
y
RAy2
Again, we shall use the first 20 samples as our base period.
Since n=5, A2 = 0.577, and .
Thus, our control limits are
877.02
RA
52.1)4.30(20
1
20
1
918.41)36.838(20
1
20
1
20
1
20
1
ii
ii
RR
yy
041.41877.0918.41LCL
794.42877.0918.41UCL
2
2
RAy
RAy
3. Comments or and R charts:
1. Typically both charts are run simultaneously.
Recall, a basis assumption of the chart is that σ2 is constant.
Thus, the chart assumes that the R chart is in control.
2. In general the R chart is more stable than the chart.
Generally, the mean tends to change more frequently then the variance.
3. In some cases, the R chart will signal a shift in the mean before the chart.
This can occur if the shift in the mean happens during a particular sample.
In such a case, the Range, which is sensitive to “outliers”, may pick up the shift while may not.
X
X
X
X
X
X
4. The R-chart assumes that the data come from a normal distribution (quite restrictive!).
5. The -chart assumes the Central Limit Theorem.
We should use graphical techniques such as the stem-and-leaf, boxplot and the normal probability plot on the observations in the base period to determine how reasonable these assumptions are.
X
Once again, consider the pencil lead ash data.
The stem-and-leaf for the base period follows.
N = 100 Median = 42.1 Quartiles = 41.3, 42.5
Decimal point is at the colon
40 : 1223333 40 : 5556778899 41 : 00002233344 41 : 555566777789 42 : 000000111112222223333333334444 42 : 555555555666666777789 43 : 0012234 43 : 77
C. The np-chart
In class, simulate a process which produces good/bad data.
A prime example is Deming's Red Bead Experiment, available from several sources (check the periodical Quality Progress published by the American Society for Quality).
The basic idea: The experiment uses a container filled with 500 red and 2500 white beads (20% red beads).
Using a special paddle with 50 holes in it, students sample from the population.
Someone records the number of red beads on each paddle drawn.
An inexpensive variant is to use red and white poker chips.
In class, construct the appropriate control chart.
• yi represents the number of red beads on the ith paddle drawn.
• p is the true proportion of red beads which should be drawn.
Note: As a first approximation, p is 0.2, but as Deming pointed out every time he conducted the experiment, the true value of p is actually something different, close but different.
The red and white beads are actually different sizes, so there is no reason to expect that the true proportion is exactly .2.
Deming's paddle actually had a true proportion of something less than 0.2, based on the many times he conducted the experiment!
• n is the number of beads drawn each time. Usually, n = 50.
The number of red beads per paddle follows a binomial distribution; thus,
• E(yi) = np, and
• var(yi) = np(1-p).
If p is unknown, then we can use a base period of m samples to estimate p by
where
The appropriate control limits are
yn
p1
m
iiy
my
1
1
pn
ppnpn
ppnpn
CL
)1(3LCL
)1(3UCL
D. The c-Chart
Consider items which we evaluated in terms of the number of defects present.
Let ci represent the “count” for the number of defects found inthe ith sample of n items inspected.
Note: n may be one.
Assume that n is fixed and constant for all samples.
What is an appropriate distribution for ci?
Poisson
Let λ be the expected number of defects per samples.
What is a reasonable procedure for monitoring ci?
A sequence of hypothesis tests of the form
H0: λ = λ0
Ha: λ ≠ λ0.
The Test Statistic:
Critical Region
Reject H0 if |Zi| > 3
This leads to the following control limits
Problem: Do we know ?
0
0
i
i
cZ
00
00
3LCL
3UCL
0
No, therefore we must estimate it.
Again use a base period of m samples.
Therefore,
m
iic
mc
10
1̂
c
cc
cc
CL
3LCL
3UCL
The following data represent the number of contaminating particles on silicon wafers after a rinsing operation.
Base Period
7 4 9 9 2 10 3 6 6 55 7 5 7 3 4 8 4 5 5
Next 40 samples
9 8 8 8 13 10 6 10 11 311 13 9 11 13 15 6 10 11 1212 2 4 7 2 4 7 6 7 4 6 4 6 6 8 5 6 9 3 6
1141
m
iic
For our data, m = 20 and ; thus,
The resulting control limits are
1141
m
i ic
7.520
11411
m
iic
mc
046.17.537.53LCL
86.127.537.53UCL
cc
cc
E. CUSUM and EWMA Charts
Standard control charts use only the information in the current subgroup. This leads to them having problems detecting small shifts. Another approach is to use information from successive groups.
Let’s start with the CUSUM (cumulative sum) chart. Consider the hypotheses
Let Si be the CUSUM statistic for the ith observation where
Si-1 is the value of the CUSUM statistic from previous observation,
Zi=(yi-μ0)/σ, d is an appropriate constant.
0
00
:
:
aH
H
)(,0max1
dzSSiii
If Si>0, the current information suggests the process might be out-of-control. How large is large? We need a threshold, so if Si>h, we will claim the process has shifted. We typically choose h=5 along with S0=0 and d=0.5.
Consider testing: then we say the process is out-of control if Si>5.
We estimate the standard deviation from our data as , then
The data and the CUSUM values are shown below:
y 9.45 7.99 9.29 11.66 12.16 10.18 8.04 11.46 9.20 10.34 9.03 11.47 10.51 9.40 10.08 Si 0 0 0 .636 1.614 1.238 0 .499 0 0 0 .506 .355 0 0
10:
10:0
aH
H
461.1
10,0max
1
i
ii
ySS
461.1ˆ2
d
RM
Day
Si
151413121110987654321
5
4
3
2
1
0
h=5
CUSUM Chart
We see from the CUSUM chart that the process is in control.
The CUSUM chart is typically used for one-sided tests, but can be used for two-sided alternatives by plotting the values we discussed and comparing them to h=5 and plotting
using h=-5 as the threshold.
An alternative time-weighted chart is the EWMA (exponentially weighted moving average) chart. This chart is typically used for two-sided alternatives. The EWMA statistic is
where Xi is ith observation, Zi-1 is value of the EWMA statistic for the previous observation and 0 < Φ ≤ 1 is weighting constant. We will let
Z0=μ0.
)(,0min1
dzSSiii
1)1(
iiiZXZ
The control limits for the EWMA are
with the center line at μ0. Using k=2.7 and Φ=0.1 is approximately equal to the CUSUM with h=5 and d=0.5.
Consider the previous example:
])1(1[)2(
2
0
ikUCL
])1(1[)2(
2
0
ikLCL
10:
10:0
aH
H