Statistical Process Control Charts. Module 5. Goal. monitor behaviour of process using measurements in order to determine whether process operation is statistically stable stable properties not changing in time mean variance on target?. Approach.
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… essentially a graphical form of hypothesis test
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… are used for testing stability of the mean operation
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Control Limits  determined using estimate of dispersion (variability) in process
1. Using Estimated Standard Deviation
where sjis the sample standard devn. for the jth sample in the reference data set
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The sample standard dev’n provides a biased estimate of the true standard dev’n:
Thus, use the following as an estimate:
The control limits are:
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2. Using Range
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centre
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significant change?
look for
assignable cause
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UCL
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centre
line
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LCL
sample
number
(or time)
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Monitor range to determine whether variability is stable.
Range provides an indication of dispersion, and is easy to calculate.
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Monitor standard deviation to determine whether variability is stable  standard dev’n provides an indication of dispersion.
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How can we measure dispersion when we collect only one data point per sample?
Answer  using the moving range  difference between adjacent sample values:
Use this approach to 
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MR
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j
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Calculate average moving range from reference data set:
Convert AMR into an estimate for the standard deviation using the constant “d2” for n=2 sample points:
centreline  use either a target value, or the average of the samples in the reference data set
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AMR
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centre
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Use MR to monitor variability if you are only collecting one point per sample:
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The control limits and stopping rules influence:
When the number of data points per sample is fixed, there is a tradeoff between false alarm and failure to detect rates.
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Simplest stopping rule 
We can conduct numerical simulation experiments (Monte Carlo simulations) to identify 
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… average time until a shift of a specified size is detected
ARL(0)
ARL(1)
s
/
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Simple stopping rules may lead to unacceptable false alarm rates, or failure to detect modest shifts
We can modify the rules to address these shortcomings  for example, look for:
One such set of guidelines are known as the Western Electric Stopping Rules.
unacceptable ARL’s
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1) Stop if 2 out of 3 consecutive points are on the same side of the centre line, and more than 2 std. dev’ns from certain (warning lines)
upper control limit
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centre line
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2) 4 out of 5 consecutive points lie on one side of the centre line, and are more than 1 standard dev’n from the centre line
upper control limit
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centre line
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3) 8 consecutive points occurring on one side of the centre line
upper control limit
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centre line
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Stop if one of the following Trend Patterns occur:
upper control limit
7 consecutive rising points (or falling points)
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Trend Patterns 
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Cusum  “cumulative sum”
Shewhart charts are effective at detecting major shifts in process operation.
The goal of Cusum charts is more rapid detection of modest shifts in operation.
Scenario  sustained shift, which is not large enough to exceed Shewhart chart limits  is something happening in the process?
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Approach  look at cumulative departures of the measured quantity (average, standard devn) from the target value
For automated detection, keep two running totals:
Initialize these sums at 0.
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Constant “k”  typically chosen as D/2, where D is the magnitude of the shift to be detected
Chart Limit 
where
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probability of type I error  false alarm rate
b
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probability of type II error  failure to detect rate
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Shewhart Charts
Cusum Charts
Is there a compromise between these extremes?
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… Exponentially Weighted Moving Average
Use a moving average which weights recent values more heavily than older values
Exponentially Weighted Moving Average
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weighting factor
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To see exponential weighting, consider
Common values for weighting are
however the weighting factor can be any value between 0 and 1.
Large weighting factor = short memory.
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For a charted characteristic,
Mean
Variance
where are properties of the characteristic being charted. For example, if we are charting the sample average,
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Using the statistical properties of the EWMA’s, choose control chart limits as:
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Shewhart Charts assume process is
But what if it isn’t?
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normally distributed
random noise with zero mean,
constant variance
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EWMA type charts account for possible dependencies between the random components (common cause variation) in the data, and are thus more representative.
Causes of time dependencies in the common cause variation 
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… can be defined using concepts from Normal distribution
Concept  compare specification limits to statistical variation in process
Question
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Specification limits
Statistical variation
Cp :

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Interpretation
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Capability Index Cpk
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Capability Index Cpk
Example  measurements of top surface colour of 49 pancakes


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Example
Interpretation  current performance is unacceptable, and process is not capable of meeting specifications.


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