Transcript Slide 1

Quality Control
Agenda
- What is quality?
- Approaches in quality control
- Accept/Reject testing
- Sampling (statistical QC)
- Control Charts
- Robust design methods
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:
Tri-Band mobile phone vs. Dual-Band mobile phone
Notebook cursor control (IBM joystick vs. touchpad)
What is ‘Quality’
Reliability:
- A product that needs frequent repair has ‘poor quality’
Example:
Consumer Reports surveyed the owners of > 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)
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 Men’s Shoe
(no warranty)
What is ‘Quality’
Aesthetics:
- A product that is ‘better looking’ or ‘more appealing’
Examples?
or
?
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
Japan
Lower variability 
Lower failure rate
US
LSL
T arget
USL
Distribution of critical dimensions from transmissions
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 ?
QC Approaches
(1) Accept/Reject testing
(2) Sampling (statistical QC)
(3) Statistical Process Control [Shewhart]
(4) Robust design methods (Design Of Experiments) [Taguchi]
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
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
Problems with Accept/Reject testing
Solution: only measure a subset of all samples
This approach is called: Statistical Quality Control
What is statistics?
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 = (19+21+18+20+20+21+20+20) / 8 = 19.875
The variance s2 =

(a i   )
2
≈ 0.8594
n
The standard deviation = s =

(a i   )
n
2
= √( s2) ≈ 0.927.
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 = (18+19+20+21+22) / 5 = 20.
BUT: is m =  ?
Background: Statistics...
Example (continued) m = S ai / n = (18+19+20+21+22) / 5 = 20.
BUT: is m =  ?
If: sample points are selected randomly,
thermometer is accurate, …
then m is an unbiased estimator of .
- take many samples of 5 data points,
- the mean of the set of m-values will approach 
- how good is the estimate?
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.
sn =
BUT: is sn = s ?
 (a
i
 m)
2
≈ 1.4142
n
No!
The unbiased estimator of stdev of a sample = s =

(a i  m )
n 1
2
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?
Sampling: Example..
How can we know the distribution of pressure among all cans?
% . o f can s
Plot a histogram showing %-cans with pressure in different ranges
30
35
40
45
50
55
p ressu re (p si)
60
65
70
Sampling: Example…
Limit (as histogram step-size)  0: probability density function
1
s
30
35
40
45
55
50
pressure (psi)
60
65
2

e
(z  )
2s
70
pdf is (almost) the familiar bell-shaped Gaussian curve!
why?
True Gaussian curve: [-∞ , ∞]; pressure: [0, 95psi]
2
2
Why is everything normal?
pdf of many natural random variables ~ normal distribution
WHY ?
Central Limit Theorem
Let X random variable, any pdf, mean, , and variance, s2
Let Sn = sum of n randomly selected values of X;
As n  ∞
Sn approaches normal distribution
with mean = n, and variance = ns2.
Central limit theorem..
X1 =
-1, with probability 1/3
0, with probability 1/3
1, with probability 1/3
p(S1)
Example
S1
X1 + X2
-1
-1
-1
0
0
0
1
1
1
-1
0
1
-1
0
1
-1
0
1
-2
-1
0
-1
0
1
0
1
2
X1 + X2 + X3 =
-3,
-2,
-1,
0,
1,
2,
3,
X1 + X2 =
with probability 1/27
with probability 3/27
with probability 6/27
with probability 7/27
with probability 6/27
with probability 3/27
with probability 1/27
-2,
-1,
0,
1,
2,
with probability 1/9
with probability 2/9
with probability 3/9
with probability 2/9
with probability 1/9
p(S2)
X2
S2
-2
1
0
-1
2
Gaussian curve
Curve joining p(S3)
p(S3)
X1
1
0
-1
S3
-3
-2
-1
0
1
2
3
(Weaker) Central Limit Theorem...
Let Sn = X1 + X2 + … + Xn
Different pdf, same  and s
normalized Sn is ~ normally distributed
Another Weak CLT:
Under some constraints, even if Xi are from different pdf’s,
with different  and s, the normalized sum is nearly normal!
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
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 , s ?
Answer:
We will estimate these values  Samples
Background: Scaling of Normal Distribution
If x is N(, s), then z = (x – )/s is N( 0, 1)
 Standard Normal distribution tables
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 - )/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 =  + 2.14s, is 1 - 0.9838 = 0.0162
Samples taken from a Normally Distributed Variable
Central Limit Theorem
Let X random variable, any pdf, mean, , and variance, s2
Let Sn = sum of n randomly selected values of X;
As n  ∞
Sn approaches normal distribution
with mean = n, and variance = ns2.
+ Scaling 
Mean of the sample, m estimates mean of distribution
Stdev of sample = s /√n .
Estimates reliability of m as an estimate of 
 Standard error
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?
Example: QC for raw materials..
(i) Shell-C: Full tank of fuel   ~ 510 Km, s ~ 31 Km.
P( mean distance)40 is in [500 Km, 520 Km] ?
Mean distance ≈ N( 510, s/√40 ) = N( 510, 31/√40 ) ≈ 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%
Example: QC for raw materials...
(ii) Shell-C: Full tank of fuel   ~ 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
Summary/Comments on Sampling
- Statistics provides basis for reasoning;
- Sampling is economical and more efficient than accept/reject
- We may not know the population  and/or s
 more complex reasoning (not covered in this course)
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’
Process Control Charts: example
Piston rings manufacturing
Critical dimension: inside diameter
Mfg process designed for: mean diameter = 74mm, s = 0.01 mm
Measure random sample of 5 rings in each hour
Record mean value of the inside diameter
Plot
x
x
Process Control Charts example: X-bar charts
Mfg process designed for: mean diameter = 74mm, s = 0.01 mm
[source: Montgomery]
X-bar charts – UCL and LCL
s = 0.01, and n = 5;
x
is normally distributed with s x = 0.01/√5 = 0.0045
Process is in-control  We should avoid a “False rejection”
Accept the claim
 lies in
 lies in the
acceptance
interval
rejection
interval
No error
Type II error
a = P( Type I error)
Reject the claim
Type I error
No error
X-bar charts – UCL and LCL..
Process is in-control  We should avoid a “False rejection”
 lies in
 lies in the
acceptance
interval
rejection
interval
Accept the claim
No error
Type II error
Reject the claim
Type I error
No error
If we never reject the claim  never commit Type I error
a = P( Type I error)
100(1 - a)% of the sample m must lie in
[ 74 - Za/2(0.0045), 74 + Za/2(0.0045)]
x
is N( 74, 0.0045)
Typical:
P( Type I error) < 0.0027  Za/2 = 3
X-bar charts – UCL and LCL...
Avoid “False rejection”  P( Type I error) < 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]
X-bar charts: relationship between sample and x-bar
[source: Montgomery]
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+]
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
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
Deterioration/wear of equipment
Improvement/Deterioration of technique
downward
Cycling of data
Temperature or recurring changes
Operator/Operating differences
Regular rotation of machines
Difference in measuring devices used in
rotation
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), …
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!
Ina Tile Example..
Cause: Different temperature profile in different regions
O u tsid e
tile s
In s id e
tile s
O u ts id e
tile s
TARGET
In sid e
tile s
LSL
SPC approach: Eliminate cause  redesign Kiln
USL
Ina Tile Example...
Cause: Different temperature profile in different regions
SPC approach: Eliminate cause  reduce Temp variation 
How ?  redesign Kiln  Expensive!
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!
Ina Tile example: Taguchi Method..
Experiment with different values of Control Parameters 
Higher Limestone content  desensitize design to noise
In sid e
tile s
O u tsid e
tile s
after
before
TARGET
LSL
USL
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 ?
Design of Experiments
C ontrollable input
param eters
x1
x2
…
xn
O utput, y
Input
P rocess
z1
z2
…
zm
U ncontrollable
factors (noise)
Typical Objectives of DOE
(i) Determine which input variables have the most influence on the output;
(ii) Determine what value of xi’s will lead us closest to our desired value of y;
(iii) Determine where to set the most influential xi’s so as to
reduce the variability of y;
(iv) Determine where to set the most influential xi’s such that
the effects of the uncontrollable variables (zi’s) are minimized.
C ontrollable input
param eters
x1
x2
…
xn
O utput, y
Input
P rocess
z1
z2
…
zm
U ncontrollable
factors (noise)
Tool used:
ANalysis Of VAriance  ANOVA
Concluding Remarks
Statistical Tools are critical to QC
QC is critical to all productive activities
next topic: review for exam!