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!