Statistical Process Control
Introduction
There are 2 general approaches to improving quality: 1)
Inspection and 2) Detection.
Inspection
With inspection, products are produced and at the completion
of the production process, someone inspects the product to determine if it
conforms to specifications. If it does
not conform, it is either discarded or repaired.
Inspection
has several drawbacks:
It is expensive to produce substandard products regardless
of whether they are discarded or fixed.
Detection
Rather than inspecting the product after it has been
produced, we inspect the process to determine when the process starts producing
units that do no conform to specifications.
This approach allows us to correct the production process
before it creates a large number of defective products.
Process
Variation
We have 2 types of variation: Common Cause (Chance Cause)
and Special Cause (Assignable Cause).
Common Cause (Chance Cause) = Caused by a number of randomly occurring events
that are part of the production process and that in general cannot be
eliminated without changing he process.
Special Cause (Assignable Cause) = Caused
by specific events or factors that are frequently temporary and that can
usually be identified and eliminated.
Say that we are producing breakfast cereal in 16 ounce
boxes. Each box will vary in weight by
some small amount. To model this
variation, we could assign the weight of the box to be a random variable.
If the only sources of variation are caused by chance, then
each box's weight is drawn from the same distribution. That is, each distribution has the same
shape, mean, and standard deviation.
Show figure 23.1
In this figure, the process is said to be under
control.
Most companies realize that a process will have common cause
variation and assign specification limits to the product. For us, we must produce boxes of cereal that
weigh 16 ounces +/- .02 ounces. If the
process is in control, we will produce boxes within this range.
If the boxes fall consistently outside this range, we have a
special cause variation.
Special Cause variation usually stem from the following
sources:
SIX M’s
•
differences among machines
and tools - MACHINES
•
differences among workers
and supervisors - MAN
•
differences among
materials - MATERIAL
•
differences among work
methods - METHODS
•
differences among
measurement methods - MEASUREMENTS
•
differences in the
environment - MOTHER NATURE
A special cause variation will be noticeable through the
following graphs:
·
Level shift
·
Instability
·
Trend
·
Cycle
Level
shift - this is a change in the mean of the process
distribution.
Show figure 23.2
Instability - this is
a change in the standard deviation of the process distribution
Show figure 23.3
Trend - there is
a slow, steady shift (either up or down) in the process distribution mean
Show figure 23.4
Cycle - a
repeated series of small observations followed by large observations
Show figure 23.5
Control
Charts
A Control Chart is a plot of statistics over time.
Each control chart contains a centerline, a lower control
limit and an upper control limit.
Show figure 23.6
If, when we plot the sample statistics, all points are
randomly distributed between the control limits, we conclude that the process
is under control.
If the points are not randomly distributed between the
control limits, we conclude that the process is out of control.
What if we
wanted to know whether the central location of the distribution has changed
from one period to another?
We use a x-bar chart.
For now, let us assume that we know the mean, m and the
standard deviation s of the
process when it is under control. With
this information we can construct an x-bar chart.
The vertical-axis represents the values of x-bar and the
horizontal-axis tracks the samples in the order in which they were taken.
The centerline is the value of m.
The control limits are 3 standard deviations from the
centerline. Since the standard
deviation of x-bar is s/n 1/2:
Lower control limit = 
Upper Control limit = 
Show figure 23.7
Say that we take a sample of four boxes every 30 minutes, so
n = 4. We know that the mean weight of
the box when the process is under control, m = 16.01 and that the standard deviation, s =
.02. What does the x-bar chart look
like?
Show figure 23.8 - special cause and common
cause variation
Control
Charts for Variables: x-bar and S charts
Charts for variables are used when we measure the product in
some way, such as its length, wide, weight, or variable that can be
measured.
We typically measure the variables of the product that are
critical to the design and manufacture of the product or critical to the
customer.
To determine if the distribution mean has changed, we use
the x-bar chart
To determine if the distribution standard deviation has
changed, we use the S chart (Standard
deviation) or R chart (Range)
In industry, people often use the range chart instead of the
sample standard deviation because it is easier to compute.
When we examine a process in industry, we will probably not know
the mean or standard deviation of the process distribution. Thus, in order to construct the x-bar chart,
we need to estimate the parameters from the data.
We begin by drawing samples when the process is under
control. For each sample, we compute the
mean and standard deviation.
The estimator of the mean of the distribution is the mean of
the sample means.
Where x-barj is the mean of the jth sample and
there are k samples.
The estimator of the standard deviation of the distribution
is S.
Centerline = 
Lower control limit for x-bar = 
Upper control limit for x-bar = 
Example:
Lear Seating manufacturers seats for Ford, Chrysler and
General Motors. One of the components
of a front-seat cushion is a wire spring, produced from 4 mm steel wire. A machine is employed to bend the wire so
that that the spring's length is 500 mm.
If the springs are longer than 500 mm, they will loosen and eventually
fall out. If they are too short, they
will not easily fit into position (in fact short springs have led to injuries
in the past to workers attempting to install them). In order to determine if the process is under control, random
samples of four springs are taken every 2 hours. The last 25 samples are shown below. Construct an x-bar chart from these data.
|
Sample |
|
|||
|
1 |
501.02 |
501.65 |
504.34 |
501.10 |
|
2 |
499.80 |
498.89 |
499.47 |
497.90 |
|
3 |
497.12 |
498.35 |
500.34 |
499.33 |
|
… |
…. |
… |
… |
… |
|
25 |
502.03 |
501.44 |
502.76 |
503.79 |
Menu
commands:
1.
Import the data
2.
Click Stat, Control
Charts, and Xbar
3.
Type the variable name
4.
Use the cursor to select Subgroup size, (4) hit tab,
and type the sample size
5.
Use the cursor to select Pooled std. Dev.
6.
Click O.K.
Pattern
Tests to Determine When the Process is out of control
To determine if a process is out of control, we need to
examine the pattern made by the samples when they are plotted on a control
chart.
To examine patterns, we need to divide the control chart
into zones.
Show Figure 23.9
Zone C = 1 standard deviation from the centerline (MEAN)
Zone B = 2 standard deviations from the centerline (MEAN)
Zone A = 3 standard deviations from the centerline (MEAN)
Now that we have the zones defined, we can apply 8 rules to
the control chart to determine if the process is in control.
Test 1: one point beyond zone A. We conclude that the process is out o control that the process is
out o control if any point is outside the control limits.
Test 2: nine points in a row in Zone C or beyond (on the
same side of the centerline)
Test 3: six increasing or decreasing points in a row
Test 4: Fourteen points in a row in alternating up and down
Test 5: two out of three points in a row in Zone A o beyond
(on the same side of he centerline)
Test 6: four out of five points in a row in Zone B o beyond
(on the same side of the centerline)
Test 7: fifteen points in a row in Zone C (on both sides of
the centerline)
Test 8: eight points in row beyond Zone C (on both sides of
the centerline)
When any of these patterns is recognized, we have reason to
believe that the process is out of control.
Show Figure 23.11
In Minitab, there are 8 pattern tests for x-bar charts, but
no tests for S and R charts. Minitab
has four pattern tests for P charts.
Menu
commands:
Click Tests for
Special Causes and All eight
Use the cursor to select the tests you want
To draw the zones, click S limits and type 1, 2, 3
in the Sigma Limit positions box
Click O.K. twice
S Charts
The S Chart graphs sample standard deviations to determine
if the process distribution standard deviation has changed.
The format for an S Chart and other controls charts are the
same: a centerline and control limits, but the formulas are different.
S for each subgroup:
Centerline (s) = 
Lower control limit for S = 
If B3 = 
Upper control limit for S = 
If B4 = 
Where B3 , B4 and c4 is a
constant that depends on the sample size.
Minitab
Commands:
1.
Type or import the data into one column
2.
Click Stat, Control
Charts, and S
3.
Type the variable name
4.
Use the cursor to select Subgroup Size, hit tab,
and type the sample size
5.
Use the cursor to select Pooled std. Dev.
6.
Click O.K.
In interpreting the results:
Points do no fall outside the control limits and we have not
applied the pattern tests.
When examining an S chart, we look for increases and
decreases in the standard deviation.
Increases are obviously bad and we want to determine the reasons for
their occurrence. Decreases in S.D. are
good, but we also need to investigate their cause. If the decrease has an assignable cause, we want to make that
part of the operating procedures. Often
though, decreases in S.D. are due to poor sampling techniques.
Using the x-bar and S charts
The x-bar and S charts are used together. Why?
To construct the x-bar chart, we use the value S.
Thus, if the S chart indicates that the process is out of
control, then the value of S will not lead to an accurate estimate of the
process standard deviation for use in the x-bar chart.
In practice, we draw the S chart first. If it indicates that the process is under
control then we construct the x-bar chart.
If the x-bar chart also indicates that the process is in control, then
we can use both charts to maintain control over the process.
If either chart indicates that the process is out of
control, then we stop the process, fix the problem, collect more samples, and
re-draw the charts.
x-Bar
and R Charts
Calculating the S chart can be time-consuming if done by
hand. Many companies use the sample
range to estimate the process standard deviation. This change affects the creation of the x-bar chart and how we
test to see if the process standard deviation has changed.
The sample range = 
Lower control limit for x-bar = 
Upper control limit for x-bar = 
Where d2 is a constant that depends on the sample
size.
If we let:
The control limits can also be calculated by:
Lower control limit for x-bar = 
Upper control limit for x-bar = 
Where A2 is a constant that depends on the sample
size.
Building the range chart:
The centerline is the mean of the ranges, R-bar.
Lower control limit for R-bar = 
Upper control limit for R-bar = 
Where d2 and d3 is a
constant that depends on the sample size.
If we let:
The control limits can also be calculated by:
Lower control limit for R-bar = 
Upper control limit for R-bar = 
Where D3 and D4 are constants that
depends on the sample size.
Minitab
Commands:
1.
Type or import the data into one column
2.
Click Stat, Control
Charts, and R
3.
Type the variable name
4.
Use the cursor to select Subgroup Size, hit tab,
and type the sample size
5.
Use the cursor to select Rbar estimate
6.
Click O.K.
Menu
commands:
1.
Import the data
2.
Click Stat, Control
Charts, and Xbar
3.
Type the variable name
4.
Use the cursor to select Subgroup size, hit tab,
and type the sample size
5.
Use the cursor to select Rbar estimate
6.
Click O.K.
Getting
stated: A 13-step process for R charts
1. Choose
what to measure
·
Determine those product characteristics that are important
to the customer and to production requirements.
·
You will not be able to measure every product characteristic
that you consider important. Why? Cost is too high. You must narrow your list of characteristics to measure down to a
critical few.
·
You may not be able to directly measure everything that is
important. In these cases, you must
find something measurable that will allow the important characteristic to be
controlled. For example, you may want
to measure the hardness of rubber. You
can control the amount of time for curing, which in turn controls the
hardness.
2. Take
the Samples
When taking samples to set up the range chart, make sure
that you are only measuring variation inherent to the process. Thus you are attempting to measure the
process without including assignable sources of variation. How do you do this? 1) Take each sample over a very short period
of time. 2) Take each sample for a
single source of data, such as one machine, one operator, one batch of
material, etc..
3. Set up
forms for data and graphs
4. Collect
the samples and record measurements
5. Make
sure that you have recorded the measurements in the order of production
Calculate the averages
6.
Calculate the overall average
7.
Determine the ranges for the samples
8.
Calculate the average range
9.
Determine scales for the graphs and plot the data
Find the largest and smallest x-bar and range values.
Pick a scale that will allow these values to easily fit into
the chart area with room for points beyond these values to account for out of
control conditions.
10.
Determine the control limits for ranges
We must determine if the range is within control before
looking at the x-bar.
11.
Determine if the ranges are in statistical control
There are three possible answers to this question:
a) All the ranges fall inside the control limits.
If all the ranges fall within the control limits, the range
is within statistical control and you can construct the x-bar chart. If a point falls on the upper or lower
control limit, the point is still within the limits and is in control.
b) One or two ranges fall outside the limits
When setting up control charts, it is common to throw out
one or points that fall outside the limits and recalculate the overall mean and
average range and the control limits.
After you refigure these numbers, one of two things will happen. One, you will find that all the remaining
points fall within the new limits and you can proceed with the x-bar chart or
two, one or two points will be outside the new limits. In this case, you have an out of control
condition and you will have to find and remove the special cause variation and
start the entire process over again.
c) Three or more ranges fall outside the limits
The process is out of control. You will have to find and remove the special cause variation and
start the entire process over again.
12.
Determine the control limits for the averages
13.
Determine if the averages are in statistical control
Use the same approach you used for the ranges: three choices
a, b, or c.
Control Chart for Attributes: p Chart
What if we have a product that has quality characteristics
that are not easily measured and given a numerical result?
For example, how do we measure the beauty of a rose?
What if the quality characteristics are too
expensive o too difficult to measure?
What if a product has so many small characteristics
that to measure all of them would be too expensive?
In many cases, a single attribute chart can be used in place
of a numerous x-bar charts.
A p Chart is a control chart used to monitor a process whose
results are categorized as either defective or non-defective. We use the p chart to track the proportion
of defective units in a series of samples.
Thus an attribute chart will tell us if the part is good or
bad. An attribute chart is not as
efficient as an x-bar chart.
Classifying a part as simply good or bad does no provide as much
information as actual measurements that permit comparison o engineering
specifications.
We overcome this comparative lack of efficiency by using
appreciably larger sample sizes in attribute charts.
We construct a p chart the same way as we did for
x-bar. We draw a sample of size n from
the process at a minimum of 25 periods.
We then calculate the proportion of defective units.
Centerline = 
Lower control limit for p chart = 
Upper control limit for p chart = 
If the lower control limit is negative, set it equal to
zero.
Pattern Tests for p charts
Test 1: one point beyond zone A
Test 2: nine points in a row in Zone C or beyond (on same
side of the centerline)
Test 3: six increasing points or six decreasing points in a
row
Test 4: fourteen points in a row alternating up and down
Example:
A company that process 3.5 inch computer disks has been
receiving complaints from its customers about the large number of disks that
will not store data properly. Company
management has decided to institute statistical process control in order to
remedy the problem. Every hour, a
random sample of 200 disks is taken, and each disk is tested to determine
whether it is defective. The results of
the first 40 hours are shown below.
Using the data, draw a p chart to monitor the production process. Was
the process out of control when the sample was taken?
|
Sample |
Number of Defectives |
|
1 |
19 |
|
2 |
5 |
|
3 |
16 |
|
4 |
20 |
|
5 |
6 |
|
6 |
12 |
|
7 |
18 |
|
8 |
6 |
|
9 |
13 |
|
10 |
15 |