statistical basis of the control charts - eskisehir.edu.tr
Post on 03-Oct-2021
6 Views
Preview:
TRANSCRIPT
Chapter 5 1 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
• A process is operating with only chance (common) causes of variation
present is said to be in statistical control.
• A process that is operating in the presence of assignable (special) causes is
said to be out of control.
Chance and Assignable Causes of Variation
Chapter 5 2 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
• A process that is operating with only chance causes of variation is
said to be in statistical control.
– Variation that is random in nature. This type of variation cannot be
completely eliminated unless there is a major change in the
equipment or material used in the process.
– Internal machine friction,
– slight variations in material or process conditions (such as the
temperature of the mold being used to make glass bottles),
– atmospheric conditions (such as temperature, humidity, and the
dust content of the air), and vibrations transmitted to a machine
from a passing forklift
Chance and Assignable Causes of Variation
Chapter 5 3 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
• A process that is operating in the presence of assignable causes is
said to be out of control.
– Variation that is not random. It can be eliminated or reduced by
investigating the problem and finding the cause.
– Improperly adjusted or controlled machines, operator errors,
defective raw material.
– If the hole drilled in a piece of steel is too large due to a dull
drill, the drill may be sharpened or a new drill inserted.
– An operator who continually sets up the machine incorrectly can
be replaced or retained.
– If the roll of steel to be used in the process does not have the
correct tensile strength, it can be rejected.
Chance and Assignable Causes of Variation
Chapter 5 4 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
• As long as the points plot within the control limits in a random manner, the process is assumed to be in statistical control.
• A point that plots outside the control limits is evidence that the process is out of control
– Investigation and corrective action are required to find and eliminate assignable cause(s)
Statistical Basis of the Control Chart
• Even if all the points plot inside the control limits, if they behave in a systematic or non-random manner, then this could be an indication that the process is out of control.
Chapter 5 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
Photolithography Example
• Important quality characteristic in hard bake is resist flow width
• Process is monitored by average flow width
– Process mean is 1.5 microns
– Process standard deviation is 0.15 microns
– Sample of 5 wafers
• Note that all plotted points fall inside the control limits
– Process is considered to be in statistical control
How the Shewhart Control Chart Works
Chapter 5 6 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
Determination of the Control Limits
Chapter 5 7 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
Shewhart Control Chart Model
Chapter 5 8 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
Chapter 5 9 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
• There is a close connection between control charts and hypothesis
testing
– Probability of type I error of the control chart: probability of
concluding the process is out of control when it is really in
control
– Probability of type II error of the control chart: probability of
concluding the process is in control when it is really out of
control
• Operating-characteristic curve of a control chart displays its
probability of type II error; an indication of the ability of the control
chart to detect process shifts of different magnitudes.
Statistical Basis of the Control Chart
Chapter 5 10 Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012 John Wiley & Sons, Inc.
Chapter 5 11 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
• 3-Sigma Control Limits
– Probability of type I error is 0.0027
• Probability Limits
– Type I error probability is chosen directly
– For example, 0.001 probability limits (type I error probability is 0.002) gives 3.09-sigma control limits
• Warning Limits
– Typically selected as 2-sigma limits
Choice of Control Limits
Chapter 5 12 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
Sample Size and Sampling Frequency
• Average run length (ARL) of a control chart: Average number of points
that must be plotted before a point indicates an out-of-control condition.
• If the process observations are uncorrelated, then for any Shewhart control
chart, the ARL can be calculated as
ARL =1
𝑝
𝑝: the probability that any point exceeds the control limits.
• The average run length of the 𝑥 chart with three-sigma limits when the
process in in control is
𝐴𝑅𝐿0 =1
0.0027= 370
Even if the process remains in control, an out-of-control signal will
be generated every 370 samples, on the average.
Chapter 5 13 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
Sample Size and Sampling Frequency
• Average time to signal (ATS): If samples are taken at fixed intervals of time
that are ℎ hours apart, then
ATS = ARLℎ
• Hard-bake process example: Suppose we are sampling every hour. Then, we
will have a false alarm about every 370 hours on the average.
• The mean shifts to 1.725 microns. The out-of-control ARL (called ARL1) is
ARL1 =1
𝑝=1
0.65= 1.54
The control chart will require 1.54 samples to detect the process
shift, on the average.
• The average time required to detect this shift is
ATS = ARL1ℎ = 1.54 1 = 1.54 hours
Chapter 5 14 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
Example: An 𝑥 chart is in use with the following parameters:
UCL = 363
Center line = 360
LCL = 357
The sample size is n =9. The 𝑥 chart exhibits control. The quality
characteristic is normally distributed with standard deviation 3.
a) What is the 𝛼-risk associated with the 𝑥 chart?
b) Specifications on this quality characteristic are 358 ± 6. What are your
conclusions regarding the ability of the process to produce items within
specifications?
c) Suppose the mean shifts to 357. What is the probability that the shift will
not be detected on the first sample following the shift?
d) What would be the appropriate control limits for the 𝑥 chart if the type I
error probability were to be 0.01?
Chapter 5 15 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
Example: An 𝑥 chart with three-sigma limits has parameters as follows:
UCL = 104
Center line = 100
LCL = 96
n =5
a) Suppose the process quality characteristics being controlled is
normally distributed with a true mean of 98 and a standard deviation of
8. What is the probability that the control chart would exhibit lack of
control by at least the third point plotted?
b) Find the ARL for the chart.
Chapter 5 16 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
Out-Of-Control-Action Plans
Chapter 5 17
Chapter 5 18 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
Reasons for Popularity of Control Charts
1. Control charts are a proven technique for improving
productivity.
2. Control charts are effective in defect prevention.
3. Control charts prevent unnecessary process
adjustment.
4. Control charts provide diagnostic information.
5. Control charts provide information about process
capability.
Chapter 5 19 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
• Pattern is very nonrandom in appearance
• 19 of 25 points plot below the center line, while only 6 plot
above
• Following 4th point, 5 points in a row increase in
magnitude, a run up
• There is also an unusually long run down beginning with
18th point
Patterns on Control Charts
Chapter 5 20 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
The Cyclic Pattern
• They all fall within the control limits.
• Such a pattern may indicate a problem with the
process such as;
• Operator fatigue,
• Raw material deliveries,
• Heat or stress buildup etc.
Chapter 5 21 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
Chapter 5 22 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
Discussion of the Sensitizing Rules
Chapter 5 23 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
See Champ and Woodall (1987)
In general, care should be exercised when using several decision rules
simultaneously. Suppose that the analyst uses 𝑘 decision rules and that
criterion 𝑖 has type I error probability 𝛼𝑖. Then the overall type I error or
false alarm probability for the decision based on all 𝑘 tests is
𝛼 = 1 − 1− 𝛼𝑖
𝑘
𝑖=1
provided that all 𝑘 decision rules are independent.
Chapter 5 24 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
Phase I and Phase II of Control Chart Application
Standard control chart usage involves phase I and phase II applications.
• In Phase I, a set of process data is gathered and analyzed all at once in a
retrospective analysis, constructing trial control limits to determine if the
process has been in control over the period of time where the data were
collected.
– Charts are effective at detecting large, sustained shifts in process
parameters, outliers, measurement errors, data entry errors, etc.
– Facilitates identification and removal of assignable causes
• In phase II, the control chart is used to monitor the process
– Process is assumed to be reasonably stable
– Emphasis is on process monitoring, not on bringing an unruly process
into control
top related