Quality Control

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Quality Control
Agenda
- What is quality? - Approaches in quality
control - Accept/Reject testing - Sampling
(statistical QC) - Control Charts - Robust
design methods
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What is Quality
Performance
- A product that performs better than others at
same function Example Sound quality of Apple
iPod vs. iRiver
- Number of features, user interface Examples T
ri-Band mobile phone vs. Dual-Band mobile
phone Notebook cursor control (IBM joystick vs.
touchpad)
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What is Quality
Reliability
- A product that needs frequent repair has poor
quality
Example Consumer Reports surveyed the owners of
gt 1 million vehicles. To calculate predicted
reliability for 2006 model-year vehicles, the
magazine averaged overall reliability scores for
the last three model years (two years for newer
models) Best predicted reliability Sporty
cars/Convertibles Coupes Honda S2000 Mazda MX-5
Miata (2005) Lexus SC430 Chevrolet Monte Carlo
(2005)
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What is Quality
Durability
- A product that has longer expected service life
Adidas Barricade 3 Men's Shoe (6-Month outsole
warranty)
Nike Air Resolve Plus Mid Mens Shoe (no
warranty)
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What is Quality
Aesthetics
- A product that is better looking or more
appealing Examples?
?
or
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Defining quality for producers..
Example Montgomery - Real case study
performed in 1980 for a US car manufacturer -
Two suppliers of transmissions (gear-box) for
same car model Supplier 1 Japanese Supplier
2 USA - USA transmissions has 4x
service/repair costs than Japan transmissions
Lower variability ? Lower failure rate
Distribution of critical dimensions from
transmissions
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Definitions
Quality is inversely proportional to variability
Quality improvement is the reduction in
variability of products/services.
How to reduce in variability of products/services
?
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QC Approaches
(1) Accept/Reject testing (2) Sampling
(statistical QC) (3) Statistical Process Control
Shewhart (4) Robust design methods (Design Of
Experiments) Taguchi
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Accept/Reject testing
- Find the characteristic that defines
quality - Find a reliable, accurate method to
measure it - Measure each item - All items
outside the acceptance limits are scrapped
Lower Specified Limit
Upper Specified Limit
target
Measured characteristic
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Problem with Accept/Reject testing
(1) May not be possible to measure all
data Examples Performance of
Air-conditioning system, measure temperature of
room Pressure in soda can at 10 (2) May
be too expensive to measure each
sample Examples Service time for customers
at McDonalds Defective surface on small
metal screw-heads
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Problems with Accept/Reject testing
Solution only measure a subset of all samples
This approach is called Statistical Quality
Control
What is statistics?
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Background Statistics
Average value (mean) and spread (standard
deviation) Given a list of n numbers, e.g. 19,
21, 18, 20, 20, 21, 20, 20. Mean m S ai / n
(1921182020212020) / 8 19.875 The
variance s2 0.8594
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Background Statistics..
Example. Air-conditioning system cools the living
room and bedroom to 20? Suppose now I want to
know the average temperature in a room
- Measure the temperature at 5 different
locations in each room. Living Room 18, 19,
20, 21, 22. Bedroom 19, 20, 20, 20, 19.
What is the average temperature in the living
room?
m S ai / n (1819202122) / 5 20.
BUT is m m ?
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Background Statistics...
Example (continued)
m S ai / n (1819202122) / 5 20.
BUT is m m ?
If sample points are selected randomly,
thermometer is accurate,
then m is an unbiased estimator of m.
- take many samples of 5 data points, - the mean
of the set of m-values will approach m
- how good is the estimate?
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Background Statistics....
Example. Air-conditioning system cools the living
room and bedroom to 20? Suppose now I want to
know the variation of temperature in a room
- Measure the temperature at 5 different
locations in each room. Living Room 18, 19,
20, 21, 22.
BUT is sn s ? No!
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Sampling Example
Soda can production Design spec pressure of a
sealed can 50PSI at 10?C Testing sample few
randomly selected cans each hour Questions
How many should we test? Which cans should we
select?
To Answer We need to know the distribution of
pressure among all cans
Problem How can we know the distribution of
pressure among all cans?
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Sampling Example..
How can we know the distribution of pressure
among all cans?
Plot a histogram showing -cans with pressure in
different ranges
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Sampling Example
Limit (as histogram step-size) ? 0 probability
density function
why?
pdf is (almost) the familiar bell-shaped Gaussian
curve!
True Gaussian curve -8 , 8 pressure 0,
95psi
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Why is everything normal?
pdf of many natural random variables normal
distribution
WHY ?
Central Limit Theorem
Let X random variable, any pdf, mean, m, and
variance, s2
Let Sn sum of n randomly selected values of X
As n ? 8 Sn approaches normal distribution with
mean nm, and variance ns2.
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Central limit theorem..
Example
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(Weaker) Central Limit Theorem...
Let Sn X1 X2 Xn
Different pdf, same m and s
normalized Sn is normally distributed
Another Weak CLT Under some constraints, even if
Xi are from different pdfs, with different m and
s, the normalized sum is nearly normal!
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Central Limit Therem....
Observation For many physical processes/objects
variation is f( many independent
factors) effect of each individual factor is
relatively small
Observation CLT ? The variation of
parameter(s) measuring the physical phenomenon
will follow Gaussian pdf
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Sampling for QC
Soda Can Problem, recalled How can we know the
distribution of pressure among all cans?
Answer We can assume it is normally distributed
Problem But what is the m, s ?
Answer We will estimate these values ? Samples
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Background Scaling of Normal Distribution
If x is N(m, s), then z (x m)/s is N( 0, 1)
? Standard Normal distribution tables
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Normal Distribution scaling example
A manufacturer of long life milk estimates that
the life of a carton of milk (i.e. before it goes
bad) is normally distributed with a mean 150
days, with a stdev 14 days. What fraction of
milk cartons would be expected to still be ok
after 180 days?
Z 180 days (Z - m)/s (180 - 150)/14 2.14
Use tables Z 2.14 ? area 0.9838 Fraction
of milk cartons that are ok Z 180 days or Z
m 2.14s, is 1 - 0.9838 0.0162
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Samples taken from a Normally Distributed Variable
Central Limit Theorem
Let X random variable, any pdf, mean, m, and
variance, s2
Let Sn sum of n randomly selected values of X
As n ? 8 Sn approaches normal distribution with
mean nm, and variance ns2.
Scaling ? Mean of the sample, m estimates
mean of distribution Stdev of sample s /vn .
Estimates reliability of m as an estimate of m
? Standard error
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Example QC for raw materials
A logistics company buys Shell-C brand diesel for
its trucks. Full tank of fuel ? average truck
travel 510 Km, stdev 31 Km. New seller
provides a cheaper fuel, Caltex-B, Claim that
it will give similar mileage as the Shell-C. (i)
What is the probability that the mean distance
traveled over 40 full-tank journeys of
Shell-C is between 500 Km and 520 Km? (ii) Mean
distance covered by 40 full-tank journeys using
Caltex-B 495 Km. What is the
probability that Caltex-B is equivalent to
Shell-C?
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Example QC for raw materials..
(i) Shell-C Full tank of fuel ? m 510 Km, s
31 Km. P( mean distance)40 is in 500 Km, 520
Km ?
Mean distance N( 510, s/v40 ) N( 510, 31/v40
) N( 510, 4.9)
Use tables, Area between z (500 -510)/4.9
-2.04 and z (520 - 510)/4.9
2.04
Area 1 - (( 1 - 0.9793) (1 - 0.9793))
0.9586
P( mean distance)40 ? 500 Km, 520 Km 95.86
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Example QC for raw materials...
(ii) Shell-C Full tank of fuel ? m 510 Km, s
31 Km. Mean distance covered by 40 full-tank
journeys using Caltex-B 495 Km. What is
the probability that Caltex-B is equivalent to
Shell-C?
P(mean distance over 40 journeys) 495 ?
m 495 ? z (495 - 510)/4.9 -3.06 ? P(
m40 using Shell-C or similar 495) 0.9989
? P(Caltex-B is equivalent to Shell-C) (1 -
0.9989) 0.0011
This method of reasoning is related to Hypothesis
Testing
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Summary/Comments on Sampling
- Statistics provides basis for reasoning -
Sampling is economical and more efficient than
accept/reject - We may not know the population
m and/or s ? more complex reasoning (not
covered in this course)
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Control Charts in QC
1. Use sampling of product/process 2. Repeat
sampling at regular intervals 3. Plot the time
series data 4. Look for any patterns that may
indicate out-of-control process 4.1. Look for
problem 4.2. Solve problem ? bring process back
to under-control
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Process Control Charts example
Piston rings manufacturing Critical dimension
inside diameter
Mfg process designed for mean diameter 74mm, s
0.01 mm
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Process Control Charts example X-bar charts
Mfg process designed for mean diameter 74mm, s
0.01 mm
source Montgomery
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X-bar charts UCL and LCL
s 0.01, and n 5
Process is in-control ? We should avoid a False
rejection
a P( Type I error)
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X-bar charts UCL and LCL..
Process is in-control ? We should avoid a False
rejection
If we never reject the claim ? never commit Type
I error
100(1 - a) of the sample m must lie in 74 -
Za/2(0.0045), 74 Za/2(0.0045)
a P( Type I error)
Typical P( Type I error) lt 0.0027 ?
Za/2 3
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X-bar charts UCL and LCL...
Avoid False rejection ? P( Type I error) lt
0.0027 ? Za/2 3
3-sigma control limits
Piston Rings Control limits 74 3(0.0045) ?
UCL 74.0135, LCL 73.9865
source Montgomery
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X-bar charts relationship between sample and
x-bar
source Montgomery
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Points of interest
-- larger sample size ? control limit lines move
close together
-- Larger sample size ? control chart can
identify smaller shifts in the process
-- 2s warning lines
source Montgomery
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Using Control Charts
Observation Possible Cause
One or more points outside of the control limits A special cause of variance due to material, equipment, method or measurement system change Error in measurement of part(s) Error in plotting (or calculating point) Error in plotting/calculating limits
Run of eight points on one side of the center line Shift in the process output due to changes in the equipment, methods, or materials Shift in the measurement system
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Using Control Charts..
Observation Possible Cause
Two of three consecutive points outside the 2-sigma warning limits but still inside the control limits Large shift in the process in the equipment, methods, materials, or operator Shift in the measurement system
Four of five consecutive points beyond the 1-sigma limits -same-
Trend of seven points in a row upward or downward Deterioration/wear of equipment Improvement/Deterioration of technique
Cycling of data Temperature or recurring changes Operator/Operating differences Regular rotation of machines Difference in measuring devices used in rotation
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Process Control Charts
- Great practical use in factories - First
introduced by Walter A. Shewhart - Help to
reduce variability - Monitor performance over
time - Trends and out-of-control are immediately
detected - Other common control charts
Range-charts (R-charts),
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Robust Design and Taguchi Methods
Example The INA Tile Company
- Tiles made in Kiln - Variability in size too
high - Variation due to baking process -
Accept/Reject is expensive!
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Ina Tile Example..
Cause Different temperature profile in different
regions
SPC approach Eliminate cause ? redesign Kiln
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Ina Tile Example...
Cause Different temperature profile in different
regions
SPC approach Eliminate cause ? reduce Temp
variation ?
How ? ? redesign Kiln ? Expensive!
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Ina Tile example Taguchi Method
Response Tile dimension Control Parameters
(tile design) Amount of Limestone Fineness of
additive Amount of Agalmatolite Type of
Agalmatolite Raw material Charging
Quantity Amount of Waste Return Amount of
Feldspar Noise parameter was the temperature
gradient.
Taguchi Experiment with different values of
Control Parameters!
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Ina Tile example Taguchi Method..
Experiment with different values of Control
Parameters ?
Higher Limestone content ? desensitize design to
noise
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Robust Design definition
A method of designing a process or product aimed
at reducing the variability (deviations from
target performance) by lowering sensitivity to
noise.
HOW ?
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Design of Experiments
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Typical Objectives of DOE
(i) Determine which input variables have the most
influence on the output (ii) Determine what
value of xis will lead us closest to our desired
value of y (iii) Determine where to set the
most influential xis so as to reduce the
variability of y (iv) Determine where to set
the most influential xis such that the
effects of the uncontrollable variables (zis)
are minimized.
Tool used ANalysis Of VAriance ? ANOVA
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Concluding Remarks
Statistical Tools are critical to QC QC is
critical to all productive activities
next topic review for exam!