quality and operations management process control and capability analysis
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Quality and OperationsManagement
Process Control and Capability Analysis
Process Control
• Recognizes that variance exists in all processes• Sources of variation
– systematic
– assignable
• Purpose – to detect and eliminate ‘out-of-control’ conditions
– to return a process to an ‘in-control’ state
• Basic tool -- the SPC chart(s)
Measuring A Process
• Types of measurements– variables data
• length, weight, speed, output, etc• discrete values
– attributes data• good vs bad, pass vs fail, etc• binary values
• Types of charts– variables -- X-R chart– attributes -- p, np, c and u
• Basic assumption -- sample means are normally distributed
Getting Started with SPCX-R Charts
• Determine sample size and frequency of data collection• Collect sufficient historical data• Ensure normality of distribution• Calculate factors for control charts
• Construct control chart• Plot data points• Determine outliers and eliminate assignable causes• Recalculate control limits with reduced data set• Implement new process control chart
X
R
UCLx LCLx
UCLrLCLr
Basic Properties
• x = std dev of sample mean = /n (where = process standard deviation)
• conventional approach uses 3 /n• limitations of control charts
– Type I Error: probability that an in-control value would appear as out-of-control
– Type II Error: probability that a shift causing an out-of-control situation would be mis-reported as in-control
– delays due to sampling interval– charting without taking action on assignable causes– over control actions
Type 1 and Type 2 Error
Type 1error
Type 2error
No error
No error
Alarm No Alarm
In Control
Out of Control
Suppose 1 > , thenType 2 Error = Z [( + 3 x - 1) / x ]
Type 1 Error = 0.0027 for 3 charts
Type 2 Error Example
Suppose: = 101= 10.2 = 4/3n = 9thus,x = 4/9
Then, Type 2 Error = Z [( + 3 x - 1) / x ]= Z [(10 + 12/9 - 10.2) / (4/9)]= Z [2.55] = 0.9946
if 1= 11.0, then Type 2 Error = Z[0.75] = .7734if 1= 12.0, then Type 2 Error = Z[-1.50] = .0668
Prob.{shift will be detected in 3rd sample after shift occurs}= 0.0668*0.0668*(1-0.0668) = 0.0042Average number of samples taken before shift is detected= 1/(1-0.0668) = 1.0716Prob.{no false alarms first 32 runs, but false alarm on 33rd}= (0.9973)32*(0.0027) = .0025Average number of samples taken before a false alarm= 1/0.0027 = 370
Tests for Unnatural Patterns
• Probability that “odd” patterns observed are not “natural” variability are calculated by using the probabilities associated with each zone of the control chart
• Use the assumption that the population is normally distributed
• Probabilities for X-chart are shown on next slide
Normal Distribution Applied to X-R Control Charts
A
A
B
B
C
C
+3
+2
+1
-1
-2
-3
Probability = .00135
Probability = .1360
Probability = .3413
Probability = .3413
Probability = .1360
Probability = .02135
Probability = .00135
Probability = .02135UCLx
LCLx
X
Outer 3rd
Outer 3rd
Middle 3rd
Middle 3rd
Inner 3rd
Inner 3rd
A Few Standard Tests
• 1 point outside Zone A
• 2 out of 3 in Zone A or above (below)
• 4 out of 5 in Zone B or above (below)
• 8 in a row in Zone C or above (below)
• 10 out of 11 on one side of center
Tests for Unnatural Patterns
• 2 out of 3 in A or beyond– .0227 x .0227 x (1-.0227) x 3 = .0015
• 4 out of 5 in B or beyond– .15874 x (1-.1587) x 5 = .0027
• 8 in a row on one side of center– .508 = .0039
Other Charts
• P-chart– based on fraction (percentage) of defective units in a
varying sample size
• np-chart– based on number of defective units in a fixed sample size
• u-chart– based on the counts of defects in a varying sample size
• c-chart– based on the count of defects found in a fixed sample size
SPC Quick Reference Card
• P-chart– based on fraction (percentage) of defective units in a varying sample size– UCL/LCLp = p 3(p)(1-p)/n
• np-chart– based on number of defective units in a fixed sample size– UCL/LCLnp = np 3(np)(1-p)
• u-chart– based on the counts of defects in a varying sample size– UCL/LCLu = u 3u/n
• c-chart– based on the count of defects found in a fixed sample size– UCL/LCLc = c 3c
• X-R chart– variables data
– UCL/LCLX = X 3 x = X 3 / n = X A2R where R/d2
– UCLR = D4R and A2 = 3/d2 n– LCLR = D3R
• for p, np, u, c and R chart the LCL can not be less than zero.
Process Capability
• Cp: process capability ratio– a measure of how the distribution compares to the width of
the specification– not a measure of conformance– a measure of capability, if distribution center were to match
center of specification range
• Cpk: process capability index– a measure of conformance (capability) to specification – biased towards “worst case”– compares sample mean to nearest spec. against distribution
width
How Good is Good Enough?
• Cp = 1.0 => 3 => 99.73% (in acceptance) => .9973 => 2700 ppm out of tolerance– PG&E operates non-stop
• 23.65 hours per year without electricity
– average car driven 15,000 miles per year• 41 breakdowns or problems per year
• Cp = 2.0 => 6 => 99.99983% (in acceptance) => .9999983 => 3.4 ppm out of tolerance– PG&E operates non-stop
• 1.78 minutes per year without electricity
– average car driven 15,000 miles per year• 0.051 breakdowns or problems per year or one every 20 years
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