Quality Control

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 = = nai 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 = nmai 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 can

s

pressure (psi)

50 55 60 65 7045403530 50 55 60 65 7045403530

%. o

f can

s

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)(

21

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 distributionwith mean = nSn, 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

Outline