Below is a quick walkthrough of building and analyzing a control chart using basic six sigma methodologies.
I use python for most data analysis done in for supply chain. In this case, I will be importing Numpy and Pandas libraries to add some functionality. I will also be bringing in Matplotlib in order to create the graphs we will be using to visualize the control charts.
Numpy allows us to work with arrays mathematically. so we can multiply an array [0,1,2,3,4]
times 2
and get the entire array multiplied as such [0,2,4,6,8]
. This is known as vector and matrix math and is extremely useful when working with “spreadsheet” data.
Pandas allows us to interact with CSV and excel files fairly easily. A must when using data sent to you or available from many supply chain software packages.
Matplotlib gives us the ability to create quickly and easily, excel-like charts while customizing them for our needs.
Our data consists of an excel sheet of measurements. The scenario here is that we are measuring the dimension on a part which is supposed to be 1.2 inches with a tolerance of 0.005. These are measurements of every 5 part off a swiss lathe machine. So the chart is also a run chart graphing a time series. In other words, the order of observations matters here.
We are looking to make sure the machine is “in control”. As we measure parts off the line, we use the data to make adjustments to the tooling and machine settings. The key here is that just because all the measurements are within tolerance, it doesn’t mean the process is in control.
In statistics, there is a rule known as the Empirical Rule which states that if observations are gaussian (distributed normally), then we will find
We can use this rough approximation to understand what could happen in the future. If we know the average and standard deviation of all the future parts that are going to be produced, we know it would follow the above rule… roughly. However, we don’t know this. So we use a sampling of the parts already produced in order to try to estimate the average and the standard deviation.
For a part with an acceptable range of 1.00 to 1.10 inches, and observed average measurement of 1.05 and observed standard deviation of 0.025, we can anticipate problems. Even if no actual measurements are out of tolerance from our test run, we can anticipate that about 5% of our parts will be out of tolerance given the empirical rule.
Even more than this, there are other rules which we will cover below which can help us estimate if the average has shifted or there was some change in the machine in the middle of the run. Things like trends and clusters that have a statistically low chance of occurring can clue us to this.
So let’s get to the analysis.
Let’s bring in our libraries first.
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
%matplotlib inline
I am going to set the GD&T variables manually here.
# GD&T Callout
target = 1.2
tolerance = 0.005
Now let’s calculate the upper and lower tolerance from the information above.
upper_tol = target + tolerance
lower_tol = target - tolerance
And last, we’ll import our data and take a quick look at it to make sure we understand it.
measurement_data = pd.read_excel('../data/line measurements.xlsx')
This dataset has two columns, real simple.
measurement_data.columns
#=> Index(['Measurement Number', 'Measurement'], dtype='object')
We’ll now parse each column into a separate variable to make it easier to use in the analysis.
meas_num = measurement_data['Measurement Number']
meas = measurement_data.Measurement
Because we are going to need to do some vector-wise calculations and graphing, we’ll need each measurement row to have the tolerances in it. So we’ll create those columns here (this helps us create the lines where tolerances are).
measurement_data['target'] = target
measurement_data['upper_tol'] = upper_tol
measurement_data['lower_tol'] = lower_tol
plt.plot(meas_num, meas, color='b', marker='o')
plt.plot(meas_num, measurement_data['target'], color='g', linewidth=0.75)
plt.plot(meas_num, measurement_data['upper_tol'], color='r', linewidth=0.5, linestyle='dashed')
plt.plot(meas_num, measurement_data['lower_tol'], color='r', linewidth=0.5, linestyle='dashed')
plt.xlabel('Measurement Number')
plt.ylabel('Measurement')
plt.title('Tolerance Chart')
plt.show()
We can see a few measurements are out of tolerance. I placed these out a bit for some of the zone analysis below.
Let’s move to create a control chart now.
We are going to plot each measurement as is first to get a base chart.
plt.plot(meas_num, meas, color='b')
plt.xlabel('Measurement Number')
plt.ylabel('Measurement')
plt.title('X-Bar Run Chart')
plt.show()
Now we are going to add markers and zone lines.
First, let’s find the average ad standard deviation using the Numpy functions on the Pandas Series.
# Statistics
avg = np.average(meas)
measurement_data['avg'] = avg
std = np.std(meas)
Now we’ll create the zone lines with the standard deviation calculated. We will also create columns that have those numbers so we can graph them as lines and not just points.
upper_1 = avg + std*1
upper_2 = avg + std*2
upper_3 = avg + std*3
lower_1 = avg - std*1
lower_2 = avg - std*2
lower_3 = avg - std*3
measurement_data['upper_1'] = upper_1
measurement_data['upper_2'] = upper_2
measurement_data['upper_3'] = upper_2
measurement_data['lower_1'] = lower_1
measurement_data['lower_2'] = lower_2
measurement_data['lower_3'] = lower_3
Finally, we will create the graph. You can see for the meas
series (which is our measured data) we added a marker of 'o'
which plots the actual point on the line and not just the line.
plt.plot(meas_num, meas, color='b', marker='o')
plt.plot(meas_num, measurement_data['avg'], linestyle='solid', linewidth=0.75, color='g')
plt.plot(meas_num, measurement_data['upper_1'], linestyle='dashed', linewidth=0.5, color='r')
plt.plot(meas_num, measurement_data['upper_2'], linestyle='dashed', linewidth=0.5, color='r')
plt.plot(meas_num, measurement_data['upper_3'], linestyle='dashed', linewidth=0.5, color='r')
plt.plot(meas_num, measurement_data['lower_1'], linestyle='dashed', linewidth=0.5, color='r')
plt.plot(meas_num, measurement_data['lower_2'], linestyle='dashed', linewidth=0.5, color='r')
plt.plot(meas_num, measurement_data['lower_3'], linestyle='dashed', linewidth=0.5, color='r')
plt.xlabel('Measurement Number')
plt.ylabel('Measurement')
plt.title('X-Bar Run Chart')
plt.show()
print(round(upper_3,4))
print(round(upper_2,4))
print(round(upper_1,4))
print(round(avg,4))
print(round(lower_1,4))
print(round(lower_2,4))
print(round(lower_3,4))
1.212
1.2086
1.2051
1.2017
1.1983
1.1948
1.1914
Now for the automated checks!
As mentioned above, it is import to make sure that there is not a statistically high chance of future part rejections based on the data we have so far. The basic rules you can use to check this are below.
We are going to create a set
called errors. As the checks find errors, they will add them to this set. In the end, we will have a list of all the problems with our data. Because I run this multiple times I data, I always clear the set of previous errors before doing the checks.
errors = set()
errors.clear
I am also going to create a few calculations that will make my life easier later.
differentials to target
will be the mathematical distance from the GD&T callout.differentials to the avg
will be the mathematical distance from the average.first differences
is going to be the change from the previous measurement.differentials_to_target = meas - target
differentials_to_avg = meas - avg
first_differences = np.ediff1d(meas)
Our first check will be to see if any measurements fall outside the tolerance. In this case they do.
### Tolerances
if np.max(np.absolute(differentials_to_target)) > tolerance:
errors.add("Dimensions are out of tolerance")
Next, we will see if there are outliers in our data. This will be observations outside of 3 standard deviations. Instead of looking for measurements above and below +3 and -3 std, I take the absolute value of the differentials so I only have to look towards the upper line. Also, since I am not counting how many failed points, I am just looking at the max differential.
### Outliers
absolute_differentials = np.absolute(differentials_to_avg)
if np.max(absolute_differentials) > upper_3:
errors.add("Outliers Exist")
errors
#=> {'Dimensions are out of tolerance'}
Next, we’ll check rule 2. We will look at all the measurements in blocks of 3, moving down the line. If any of the windows have 2 or more in the outer region, or above upper_2, we have a process that is not matching the empirical rule about where observations should fall. This shows us that the standard deviation (or mean) is not what we thought it was, it is most likely higher. This will result in more future measurements being farther out than we initially anticipated or have yet observed.
### Outer Regions
for index, i in enumerate(absolute_differentials):
if index < 2:
continue
count = np.count_nonzero(absolute_differentials[index-2:index+1] > (std*2))
if count >= 2:
errors.add("Outer Zone Clusters")
errors
#=> {'Dimensions are out of tolerance'}
In this case, we look good so far.
We are going to be looking for similar evidence in the other zones below using rules 3 and 4.
### Middle Regions
for index, i in enumerate(absolute_differentials):
if index < 4:
continue
count = np.count_nonzero(absolute_differentials[index-4:index+1] > (std*1))
if count >= 4:
errors.add("Middle Zone Clusters")
errors
#=> {'Dimensions are out of tolerance', 'Middle Zone Clusters'}
Here we do have something that should not occur assuming our calculated average and standard deviation. These shifts are small shifts but statistically significant. They should not occur if “nothing” changed. Changes can be occurrences such as:
### Inner Regions
for index, i in enumerate(absolute_differentials):
if index < 6:
continue
count = np.count_nonzero(differentials_to_target[index-6:index+1] > 0)
if count >= 7:
errors.add("Inner Zone Clusters")
count = np.count_nonzero(differentials_to_target[index-6:index+1] < 0)
if count >= 7:
errors.add("Inner Zone Clusters")
errors
#=> {'Dimensions are out of tolerance', 'Middle Zone Clusters'}
The next thing we are going to look for is trends. These are representative of something wrong. Trends, specifically of this magnitude should now occur in random data. Some of the causes of trends can be tooling wear or ongoing temperature changes.
### Trends
for index, i in enumerate(first_differences):
if index < 5:
continue
count = np.count_nonzero(first_differences[index-5:index+1] > 0)
if count >= 6:
errors.add("Trending Data is Present")
count = np.count_nonzero(first_differences[index-5:index+1] < 0)
if count >= 6:
errors.add("Trending Data is Present")
errors
#=> {'Dimensions are out of tolerance', 'Middle Zone Clusters'}
Then next check is for including too many processes in the control chart. The distribution is not spread out enough, so there might be 2 or more controls being performed before this measurement is taken. For example, maybe a part goes through 2 processes, and this measurement is after the second process (grinding after machining maybe. So the machining operator is already disposing of bad parts before they even get to you. This would result in a mixture or a stratification problem.
### Mixture
for index, i in enumerate(absolute_differentials):
if index < 7:
continue
count = np.count_nonzero(absolute_differentials[index-7:index+1] > upper_3)
if count == 0:
errors.add("No Mixture")
errors
#=> {'Dimensions are out of tolerance', 'Middle Zone Clusters', 'No Mixture'}
### Stratification
for index, i in enumerate(absolute_differentials):
if index < 14:
continue
count = np.count_nonzero(absolute_differentials[index-14:index+1] > upper_1)
if count == 0:
errors.add("No Stratification")
errors
#=> {'Dimensions are out of tolerance',
'Middle Zone Clusters',
'No Mixture',
'No Stratification'}
The final check is for manipulation, cherry-picking results, or tweaking a process mid-measurement. The results are “too good”.
### Over-Control
def sign_change(a, b):
if a > 0 and b > 0:
return 0
elif a < 0 and b < 0:
return 0
else:
return 1
changes = []
for index, i in enumerate(first_differences):
if index == 0:
continue
change = sign_change(first_differences[index],first_differences[index-1])
changes.append(change)
for index, i in enumerate(changes):
if index < 12:
continue
if np.array(changes[index-12:index+1]).sum() >= 13:
errors.add("Over Control")
errors
#=> {'Dimensions are out of tolerance',
'Middle Zone Clusters',
'No Mixture',
'No Stratification'}
What we end up with a control chart with a postscript that looks for all the problems and helps identify next steps.
Some future builds would be to chart where the errors are occurring by marking those markers red, something I have planned at the moment but isn’t terribly necessary since a quick visual look can often find these just as fast.
Taelynn
Thanks a lot for the post.Really thank you! Much obliged.
AmeliaroseReindlhix
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