Quality Control - What is quality? - Approaches in quality control - Accept/Reject testing - Sampling (statistical QC) - Control Charts - Robust design methods Agenda
Quality Control
- What is quality?
- Approaches in quality control
- Accept/Reject testing
- Sampling (statistical QC)
- Control Charts
- Robust design methods
Agenda
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 CoupesHonda S2000Mazda MX-5 Miata (2005)Lexus SC430Chevrolet 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
TargetLSL USL
Japan
US
TargetLSL USL
Japan
US
Distribution of critical dimensions from transmissions
Lower variability Lower failure rate
Definitions
Quality is inversely proportional to variability
Quality improvement is the reduction in variabilityof 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
target
Lower Specified Limit Upper Specified Limit
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 = ai / n = (19+21+18+20+20+21+20+20) / 8 = 19.875
The variance s2 = ≈ 0.8594
n
ai 2)(
The standard deviation = = n
ai 2)( = √(2) ≈ 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 = ai / n = (18+19+20+21+22) / 5 = 20.
BUT: is m = ?
Background: Statistics...
Example (continued) m = ai / n = (18+19+20+21+22) / 5 = 20.
BUT: is m = ?
then m is an unbiased estimator of .
If: sample points are selected randomly, thermometer is accurate, …
- 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.
BUT: is sn = ? No!
sn = n
mai 2)( ≈ 1.4142
The unbiased estimator of stdev of a sample = s = 1
)( 2
n
mai
Sampling: Example
Soda can production:Design spec: pressure of a sealed can 50PSI at 10C
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..
50 55 60 65 7045403530
%. o
f ca
ns
pressure (psi)
50 55 60 65 7045403530 50 55 60 65 7045403530
%. o
f ca
ns
pressure (psi)
How can we know the distribution of pressure among all cans?
Plot a histogram showing %-cans with pressure in different ranges
Sampling: Example…
Limit (as histogram step-size) 0: probability density function
50 55 60 65 7045403530pressure (psi)
pdf is (almost) the familiar bell-shaped Gaussian curve!why?
True Gaussian curve: [-∞ , ∞]; pressure: [0, 95psi]
2
2
2
)(
2
1
z
e
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, 2
Let Sn = sum of n randomly selected values of X;
As n ∞ Sn approaches normal distribution
with mean = n, and variance = n2.
Central limit theorem..
Example X1 =-1, with probability 1/3 0, with probability 1/3 1, with probability 1/3
-1 0 1S1
p(S
1)
X1 + X2 + X3 =
-3, with probability 1/27-2, with probability 3/27-1, with probability 6/27 0, with probability 7/27 1, with probability 6/27 2, with probability 3/27 3, with probability 1/27
-1 0 1-2 2-3 3S3
p(S
3)
Gaussian curveCurve joining p(S3)
X1 X2 X1 + X2
-1 -1 -2-1 0 -1-1 1 0 0 -1 -1 0 0 0 0 1 1 1 -1 0 1 0 1 1 1 2
X1 + X2 =
-2, with probability 1/9-1, with probability 2/9 0, with probability 3/9 1, with probability 2/9 2, with probability 1/9
-1 0 1-2 2S2
p(S
2)
(Weaker) Central Limit Theorem...
Let Sn = X1 + X2 + … + Xn
Different pdf, same and
normalized Sn is ~ normally distributed
Another Weak CLT:Under some constraints, even if Xi are from different pdf’s,with different and , 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 thephysical 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 , ?
Answer: We will estimate these values Samples
Background: Scaling of Normal Distribution
If x is N(, ), then z = (x – )/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 - )/ = (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 = + 2.14, is 1 - 0.9838 = 0.0162
Samples taken from a Normally Distributed Variable
Standard error
Central Limit Theorem
Let X random variable, any pdf, mean, , and variance, 2
Let Sn = sum of n randomly selected values of X;
+ Scaling Mean of the sample, m estimates mean of distributionStdev of sample = /√n.
As n ∞ Sn approaches normal distribution
with mean = n, and variance = n2.
Estimates reliability of m as an estimate of
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, ~ 31 Km.
P( mean distance)40 is in [500 Km, 520 Km] ?
Mean distance ≈ N( 510, /√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%
(ii) Shell-C: Full tank of fuel ~ 510 Km, ~ 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?
Example: QC for raw materials...
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
more complex reasoning (not covered in this course)
Control Charts in QC
1. Use sampling of product/process2. Repeat sampling at regular intervals3. Plot the time series data4. Look for any ‘patterns’ that may indicate ‘out-of-control’ process
4.1. Look for problem4.2. Solve problem bring process back to ‘under-control’
Process Control Charts: example
Piston rings manufacturingCritical dimension: inside diameter
Mfg process designed for: mean diameter = 74mm, = 0.01 mm
Measure random sample of 5 rings in each hour
Record mean value of the inside diameter x
Plot x
Process Control Charts example: X-bar charts
[source: Montgomery]
Mfg process designed for: mean diameter = 74mm, = 0.01 mm
X-bar charts – UCL and LCL
= 0.01, and n = 5;
x is normally distributed with = 0.01/√5 = 0.0045 x
Process is in-control We should avoid a “False rejection”
Accept the claim
Reject the claim
lies inacceptance
interval
lies in therejectioninterval
No error Type II error
No errorType I error
= P( Type I error)
X-bar charts – UCL and LCL..
Process is in-control We should avoid a “False rejection”
Accept the claim
Reject the claim
lies inacceptance
interval
lies in therejectioninterval
No error Type II error
No errorType I error
= P( Type I error)
If we never reject the claim never commit Type I error
x is N( 74, 0.0045)
100(1 - )% of the sample m must lie in
[ 74 - Z/2(0.0045), 74 + Z/2(0.0045)]
Typical: P( Type I error) < 0.0027 Z/2 = 3
X-bar charts – UCL and LCL...
Avoid “False rejection” P( Type I error) < 0.0027 Z/2 = 3
Piston Rings:Control limits = 74 ± 3(0.0045) UCL = 74.0135, LCL = 73.9865
3-sigma control limits
[source: Montgomery]
X-bar charts: relationship between sample and x-bar
[source: Montgomery]
Points of interest
[source: Montgomery+]
-- larger sample size control limit lines move close together
-- Larger sample size control chart can identify smaller shifts in the process
-- ±2 warning lines
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 outsidethe 2-sigma warning limits butstill inside the control limits
Large shift in the process in the equipment, methods, materials, or operatorShift 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
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
Outsidetiles
Insidetiles
Outsidetiles
Insidetiles
SPC approach: Eliminate cause redesign Kiln
Insidetiles
Outsidetiles
LSL USL
TARGET
Insidetiles
Outsidetiles
LSL USL
TARGET
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 LimestoneFineness of additiveAmount of AgalmatoliteType of AgalmatoliteRaw material Charging QuantityAmount of Waste ReturnAmount 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
Insidetiles
Outsidetiles
LSL USL
TARGET
before
after
Insidetiles
Outsidetiles
LSL USL
TARGET
before
after
Robust Design definition
A method of designing a process or product aimed atreducing the variability (deviations from target performance)by lowering sensitivity to noise.
HOW ?
Design of Experiments
Process
x1
Input Output, y
x2 xn…
z1 z2 zm…
Controllable inputparameters
Uncontrollablefactors (noise)
Process
x1
Input Output, y
x2 xn…
z1 z2 zm…
Controllable inputparameters
Uncontrollablefactors (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.
Process
x1
Input Output, y
x2 xn…
z1 z2 zm…
Controllable inputparameters
Uncontrollablefactors (noise)
Process
x1
Input Output, y
x2 xn…
z1 z2 zm…
Controllable inputparameters
Uncontrollablefactors (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!