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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

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

Nike Air Resolve Plus Mid Men’s Shoe

(no warranty)

Adidas Barricade 3 Men's Shoe

(6-Month outsole 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

Lower variability

Lower failure rate

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?

The standard deviation =s=

=√( s2) ≈ 0.927.

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 = ≈ 0.8594

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 =Sai / n = (18+19+20+21+22) / 5 = 20.

BUT: is m = m ?

Background: Statistics...

Example (continued)

m =Sai / n = (18+19+20+21+22) / 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?

≈ 1.4142

sn=

The unbiased estimator of stdevof a sample = s =

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!

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..

How can we know the distribution of pressure among all cans?

Plot a histogram showing %-cans with pressure in different ranges

30

40

35

45

55

70

60

65

50

pressure (psi)

Sampling: Example…

Limit (as histogram step-size) 0: probability density function

why?

pdf is (almost) the familiar bell-shaped Gaussian curve!

True Gaussian curve: [-∞ , ∞]; pressure: [0, 95psi]

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 ∞Sn approaches normal distribution

with mean = nSn, and variance = ns2.

-1, with probability 1/3

0, with probability 1/3

1, with probability 1/3

p(S1)

X1 =

S1

1

0

-1

-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

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 =

p(S2)

S2

1

2

0

-2

-1

-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

Gaussian curve

Curve joining p(S3)

X1 + X2 + X3 =

p(S3)

3

1

2

S3

0

-2

-1

-3

Central limit theorem..

Example

(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 pdf’s,

with different m 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 m, s ?

Answer:

We will estimate these values

Outline