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8/06/2007 ENGN8101 Modelling and Optimization 1
To provide direction in understanding the process and application of Parameter Design in the
development of robust products and processes
PurposePurpose
8/06/2007 ENGN8101 Modelling and Optimization 2
Parameter Design Course Objectives
Describe how Parameter Design relates to quality engineering.Apply the Engineered System model to the product/process of study.Identify the ideal function of the system.Identify control and noise factors affecting the system.
8/06/2007 ENGN8101 Modelling and Optimization 3
Parameter Design Course Objectives (continued)
Develop an experimental plan for both static and dynamic Parameter Design experiments.Use statistical concepts and tools from Experimental Design to analyze data from Parameter Design experiments.Make predictions and confirm improved robustness.Apply Parameter Design principles to teamwork to optimize project outcomes.
Introduction to Robustness
Introduction
to Robustness
8/06/2007 ENGN8101 Modelling and Optimization 5
Traditional Philosophy of Quality
No Good No GoodGood
No Loss Loss
LowerLimit
yTargetValue
Loss
ResponseUpperLimit
8/06/2007 ENGN8101 Modelling and Optimization 6
Either Side ofthe Specification Limit
Lower Limit
Target Value
Upper Limit
Just “in Spec”
Just“out of Spec”
8/06/2007 ENGN8101 Modelling and Optimization 7
Lower Limit
Target Value
Upper Limit
The Difference Between Targetand Specification Limit
8/06/2007 ENGN8101 Modelling and Optimization 8
Meeting the Target
The further away from target value, the lower the quality and the less satisfied the customer
Meet the TargetMeet the Target
The goal is to meet the target, not the specification
Lower Limit
Target Value
Upper Limit
8/06/2007 ENGN8101 Modelling and Optimization 9
Loss Function Philosophy
TargetPoor
ym
Loss
Poor
BestGood Good
FairFair
$
Response
8/06/2007 ENGN8101 Modelling and Optimization 10
How the Distribution Affects Quality Loss
Producing this distribution, we would experience some loss
Lower Limit
Target Value
Upper Limit
Loss Function
8/06/2007 ENGN8101 Modelling and Optimization 11
How the Distribution Affects Quality Loss
Producing this distribution, that has less variance from target, the loss would be lower
Lower Limit
Target Value
Upper Limit
Loss Function
8/06/2007 ENGN8101 Modelling and Optimization 12
Definition of Noise
NOISE FACTORS:Parameters which affect system response variability and are difficult, impossible or expensive to control.
8/06/2007 ENGN8101 Modelling and Optimization 13
Robustness
ROBUSTNESS: Low system response variability in the presence of noise.
8/06/2007 ENGN8101 Modelling and Optimization 14
For Example...
As consumers, we expect our automobiles to start every time, on the first turn of the key, regardless of:
Ambient temperatureAltitudeAge of the carHot or cold start, etc.
8/06/2007 ENGN8101 Modelling and Optimization 15
The Robust Product/Process
The robust product/process can operate within a wide range of conditions and still perform its intended function at a high quality level as perceived by the customer.
8/06/2007 ENGN8101 Modelling and Optimization 16
The Non-Robust Product/Process
The non-robust product/process will operate at a high quality level only within a limited range of conditions.
Parameter Design Introduction
Parameter
Design Intro
8/06/2007 ENGN8101 Modelling and Optimization 18
Dealing With Noise
Traditional– Eliminate or reduce– Compensate– Overlook
Parameter Design– Minimize the effect of noise
8/06/2007 ENGN8101 Modelling and Optimization 19
Parameter Designat the INA Tile Company
Kiln Wall
BurnersTiles
Initial Distribution
TargetAcceptable Deviation
TileDimension
8/06/2007 ENGN8101 Modelling and Optimization 20
The Traditional Approach
Upgrade technology– Build a new kiln– Add more burners
Feedback control– Control the temperature variation of the oven
8/06/2007 ENGN8101 Modelling and Optimization 21
Alternative Solutions
1. Redesign and build a new kiln» Cost = $1 million
2. Adjust the parameters that could be changed easily and inexpensively
» Cost = Unknown
8/06/2007 ENGN8101 Modelling and Optimization 22
Actions
$$ $$ $
INA Tile Company AlternativesCost Comparisons
Inspection and Scrap
Redesigned Kiln Increased Lime Content
8/06/2007 ENGN8101 Modelling and Optimization 23
Parameter Design at the INA Tile Company Summary
TargetAcceptable Deviation
Improved Distribution
Initial Distribution
Tile Dimension
8/06/2007 ENGN8101 Modelling and Optimization 24
What is the intent and how do we maximize it?
Shift from:
What is wrong and how do we fix it?
to:
Dr. Genichi Taguchi’s Approach
8/06/2007 ENGN8101 Modelling and Optimization 25
Making the system insensitive to usage variations
Shift from:
Controlling, compensating or eliminating usage variationsto:
Dr. Genichi Taguchi’s Approach
8/06/2007 ENGN8101 Modelling and Optimization 26
Taguchi’s 3-Step Robust Engineering Process
1. Concept Design
2. Parameter Design
3. Tolerance Design
8/06/2007 ENGN8101 Modelling and Optimization 27
The Impact ofIgnoring Parameter Design
Concept DesignOften requires longer lead timesMay introduce unnecessarily high levels of technology
Tolerance DesignMakes products more expensive to manufacture
8/06/2007 ENGN8101 Modelling and Optimization 28
Parameter Design
2.Selecting optimum values for parameters that will reduce the variability of the response to noise
The best design is determined at the leastcost by:1. Running a designed experiment with
low cost components and processsettings
8/06/2007 ENGN8101 Modelling and Optimization 29
Parameter Design
4.Selecting the lowest cost setting for factors that have minimal effect onresponse
The best design is determined at the leastcost by:3. Selecting optimum values for control
factors that will shift the mean of theresponse toward target
8/06/2007 ENGN8101 Modelling and Optimization 30
Steps in Parameter Design
1. Identify Project and Team
2. Formulate Engineered System: Ideal Function
3. Formulate Engineered System: Parameters
4. Assign Control Factors to Inner Array
5. Assign Noise Factors to Outer Array
6. Conduct Experiment and Collect Data
7. Analyze Data and Select Optimal Design
8. Predict and Confirm
Parameter Design: Static
Parameter
Design: Static
8/06/2007 ENGN8101 Modelling and Optimization 32
Steps in Parameter Design: Static
1. Identify Project and Team
2. Formulate Engineered System: Ideal Function
3. Formulate Engineered System: Parameters
4. Assign Control Factors to Inner Array
5. Assign Noise Factors to Outer Array
6. Conduct Experiment and Collect Data
7. Analyze Data and Select Optimal Design
8. Predict and Confirm
8/06/2007 ENGN8101 Modelling and Optimization 33
Project Selection
Project selection should be based on the potential to:– Increase customer satisfaction– Increase reliability– Incorporate new technology– Reduce cost– Reduce warranty– Achieve Best-in-Class
8/06/2007 ENGN8101 Modelling and Optimization 34
Successful Project Requirements
A clear objective Cross-functional team Thorough planning Management support
8/06/2007 ENGN8101 Modelling and Optimization 35
Steps in Parameter Design: Static
1. Identify Project and Team
2. Formulate Engineered System: Ideal Function
3. Formulate Engineered System: Parameters
4. Assign Control Factors to Inner Array
5. Assign Noise Factors to Outer Array
6. Conduct Experiment and Collect Data
7. Analyze Data and Select Optimal Design
8. Predict and Confirm
8/06/2007 ENGN8101 Modelling and Optimization 36
Customer’s World
“Voice of the Customer”
INTENT:What the customer expects
PERCEIVED RESULT:What the
customer gets
Customer’s World
8/06/2007 ENGN8101 Modelling and Optimization 37
SYSTEM:Transforms intent into perceived
result
SIGNAL:Causes the system to fulfill the intent
RESPONSE (y):System output that determines the perceived result
Engineer’s World
Engineer’s World
8/06/2007 ENGN8101 Modelling and Optimization 38
Signal Energy
Transformations
Engineer’s World
Response (y)
Intended Result
Unintended Result
Maximizing energy which produces the intended result will minimize the unintended results (error states)
System:
Energy Transfer
8/06/2007 ENGN8101 Modelling and Optimization 39
The ideal function is the transfer of energy to the intended result
Ideal Function
Engineer’s World
Response (y)
Intended Result
SystemSignal
8/06/2007 ENGN8101 Modelling and Optimization 40
Measuring the transformation of energy
Shift from:
Measuring symptoms of poor quality
to:
Dr. Genichi Taguchi’s Approach
8/06/2007 ENGN8101 Modelling and Optimization 41
Response
A response should:Be related to the Perceived ResultBe an engineering metricBe related to the Ideal Function of the systemQuantify the energy transfer
8/06/2007 ENGN8101 Modelling and Optimization 42
Composite View
“Voice of the Customer”
System Response (y)Signal
Customer Intent
PerceivedResult
8/06/2007 ENGN8101 Modelling and Optimization 43
Steps in Parameter Design: Static
1. Identify Project and Team
2. Formulate Engineered System: Ideal Function
3. Formulate Engineered System: Parameters
4. Assign Control Factors to Inner Array
5. Assign Noise Factors to Outer Array
6. Conduct Experiment and Collect Data
7. Analyze Data and Select Optimal Design
8. Predict and Confirm
8/06/2007 ENGN8101 Modelling and Optimization 44
Completing the Engineered System
Signal ResponseSystem
CONTROL FACTORS: parameters that are controllable by the engineer
NOISE FACTORS: parameters that affect the system and are difficult, impossible or expensive to control
8/06/2007 ENGN8101 Modelling and Optimization 45
Where to Look for Noise
environment
customer usage
neighboring systems
deterioration
manufacturing
8/06/2007 ENGN8101 Modelling and Optimization 46
Noise or Control?
Is material hardness (Rockwell) a noise or control factor?
ControlNoise
8/06/2007 ENGN8101 Modelling and Optimization 47
Steps in Parameter Design: Static
1. Identify Project and Team
2. Formulate Engineered System: Ideal Function
3. Formulate Engineered System: Parameters
4. Assign Control Factors to Inner Array
5. Assign Noise Factors to Outer Array
6. Conduct Experiment and Collect Data
7. Analyze Data and Select Optimal Design
8. Predict and Confirm
8/06/2007 ENGN8101 Modelling and Optimization 48
Experimental PlanSelect Control Factors
Filter control factor list based on:– Improvement potential– Change feasibility– Cost– Testing resources – Prioritize list
Control FactorsA B C …
8/06/2007 ENGN8101 Modelling and Optimization 49
Experimental PlanIdentify Levels
Keep in mind:– 2 or 3 levels is most
common– Level range should be
as wide as possible– One level may
represent baseline– More levels require
more testing
A1
A2
Leve
ls…
Control FactorsA B C …
8/06/2007 ENGN8101 Modelling and Optimization 50
Experimental PlanSelect Orthogonal Array
Select based on:– Number of factors– Number of levels– Testing resources
Control FactorsA B C …
Leve
ls
Orthogonal array L?This array is called the Inner Array
8/06/2007 ENGN8101 Modelling and Optimization 51
Steps in Parameter Design: Static
1. Identify Project and Team
2. Formulate Engineered System: Ideal Function
3. Formulate Engineered System: Parameters
4. Assign Control Factors to Inner Array
5. Assign Noise Factors to Outer Array
6. Conduct Experiment and Collect Data
7. Analyze Data and Select Optimal Design
8. Predict and Confirm
8/06/2007 ENGN8101 Modelling and Optimization 52
Inner and Outer Arrays
A B CControl Factors
...
Noise Factors
Outer Orthogonal Array
Response Data
Inner Orthogonal
Array
8/06/2007 ENGN8101 Modelling and Optimization 53
Noise
Noise Factors
Outer Array – Effect on energy transfer– Effect on response
variability– Engineering knowledge– Screening experiments– Customer usage profiles
• Prioritize noise factor list based on:
8/06/2007 ENGN8101 Modelling and Optimization 54
Steps in Parameter Design: Static
1. Identify Project and Team
2. Formulate Engineered System: Ideal Function
3. Formulate Engineered System: Parameters
4. Assign Control Factors to Inner Array
5. Assign Noise Factors to Outer Array
6. Conduct Experiment and Collect Data
7. Analyze Data and Select Optimal Design
8. Predict and Confirm
8/06/2007 ENGN8101 Modelling and Optimization 55
Preliminary Run
Dot plot
ResponseN1 N2
8/06/2007 ENGN8101 Modelling and Optimization 56
Experimental Layout
An experiment can be run in any order
Outer Array
Inner Array
A B C …...y11 y12
y21 y22 ...
............
123
Run N1 N2…
8/06/2007 ENGN8101 Modelling and Optimization 57
Steps in Parameter Design: Static
1. Identify Project and Team
2. Formulate Engineered System: Ideal Function
3. Formulate Engineered System: Parameters
4. Assign Control Factors to Inner Array
5. Assign Noise Factors to Outer Array
6. Conduct Experiment and Collect Data
7. Analyze Data and Select Optimal Design
8. Predict and Confirm
8/06/2007 ENGN8101 Modelling and Optimization 58
1900305000.0On
15
Level2
600456000.5Off
9
Ball-and-Ramp Analysis: Introduction
Response:Swirl Time (secs)
Control Factors (Inner Array):L : Run Length (mm)A : Ramp-to-Funnel Angle (deg)H : Run End Height (mm)C : Clamping (turns unscrewed)O : Operator Hat
Noise Factor (Outer Array)Ball Size (mm)
8/06/2007 ENGN8101 Modelling and Optimization 59
Experimental Plan
25
11
14
267 28 2
Run No. A
123 1
1
1
2
122
H
12
2
1
2
211
O
12
C
1
1
2
212
21
L
2
1
1
112
22
1
1
1
221
22
c6
2
1
2
121
21
c5NOISE
N1 N2
8/06/2007 ENGN8101 Modelling and Optimization 60
Calculating Averages
13.0 13.0 13.5
7.6 7.8 7.6 11.5 11.6 12.02 1 2 1 25
1 1 1 1 1 8.3 8.4 8.11
8.5 8.3 8.2 11.4 11.5 11.41 2 2 2 14
9.1 9.4 9.3 11.7 11.9 11.72 1 2 2 167 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 18 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2
Run No. A H O C L N1
NOISEN2
5.4 5.1 5.4 6.7 6.8 6.21 1 1 2 22
N1 N2 y
3 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2
Avg 1Avg 2Effect Effect
Avg N2Avg N1
8/06/2007 ENGN8101 Modelling and Optimization 61
13.0 13.0 13.5
9.77.6 7.8 7.6 11.5 11.6 12.02 1 2 1 25
10.71 1 1 1 1 8.3 8.4 8.11
9.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 14
10.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 167 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 12.68 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 9.2
Run No. A H O C L N1
NOISEN2
5.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 22
N1 N2 y
8.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2
Avg 1Avg 2Effect
Avg N2Avg N1Effect
Calculating Averages
8/06/2007 ENGN8101 Modelling and Optimization 62
13.0 13.0 13.5
9.77.6 7.8 7.6 11.5 11.6 12.02 1 2 1 2 7.75
10.71 1 1 1 1 8.3 8.4 8.1 8.31
9.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 1 8.34
10.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 1 9.367 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 9.9 12.68 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 7.7 9.2
Run No. A H O C L N1
NOISEN2
5.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 2 5.32
N1 N2 y
8.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2 6.5
Avg 1Avg 2Effect Effect
Avg N2Avg N1
Calculating Averages
8/06/2007 ENGN8101 Modelling and Optimization 63
9.77.6 7.8 7.6 11.5 11.6 12.02 1 2 1 2 7.7 11.75
10.71 1 1 1 1 8.3 8.4 8.1 13.0 13.0 13.5 8.3 13.21
9.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 1 8.3 11.44
10.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 1 9.3 11.867 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 9.9 15.3 12.68 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 7.7 9.210.7
Run No. A H O C L N1
NOISEN2
5.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 2 5.3 6.62
N1 N2 y
8.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2 6.5 11.1
Avg 1Avg 2Effect Effect
Avg N2Avg N1
Calculating Averages
8/06/2007 ENGN8101 Modelling and Optimization 64
Calculating Control Factor Effects
9.77.6 7.8 7.6 11.5 11.6 12.02 1 2 1 2 7.7 11.75
10.71 1 1 1 1 8.3 8.4 8.1 13.0 13.0 13.5 8.3 13.21
9.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 1 8.3 11.44
10.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 1 9.3 11.867 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 9.9 15.3 12.68 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 7.7 9.210.7
Run No. A H O C L N1
NOISEN2
5.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 2 5.3 6.62
N1 N2 y
8.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2 6.5 11.1
Avg 1Avg 2Effect Effect
Avg N2Avg N1
8/06/2007 ENGN8101 Modelling and Optimization 65
10.7
9.710.512.6
9.2
Calculating Control Factor Effects
7.6 7.8 7.6 11.5 11.6 12.02 1 2 1 2 7.7 11.75
1 1 1 1 1 8.3 8.4 8.1 13.0 13.0 13.5 8.3 13.21
9.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 1 8.3 11.44
9.1 9.4 9.3 11.7 11.9 11.72 1 2 2 1 9.3 11.867 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 9.9 15.38 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 7.7 10.7
Run No. A H O C L N1
NOISEN2
5.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 2 5.3 6.62
N1 N2 y
8.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2 6.5 11.1
8.8Avg 1Avg 2Effect Effect
Avg N2Avg N1
8/06/2007 ENGN8101 Modelling and Optimization 66
Calculating Control Factor Effects
9.77.6 7.8 7.6 11.5 11.6 12.02 1 2 1 2 7.7 11.75
10.71 1 1 1 1 8.3 8.4 8.1 13.0 13.0 13.5 8.3 13.21
9.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 1 8.3 11.44
10.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 1 9.3 11.867 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 9.9 15.3 12.68 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 7.7 9.210.7
Run No. A H O C L N1
NOISEN2
5.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 2 5.3 6.62
N1 N2 y
8.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2 6.5 11.1
10.58.8Avg 1
Avg 2Effect Effect
Avg N2Avg N1
8/06/2007 ENGN8101 Modelling and Optimization 67
8.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2 6.5 11.1
9.7
Calculating Control Factor Effects
7.6 7.8 7.6 11.5 11.6 12.02 1 2 1 2 7.7 11.75
10.71 1 1 1 1 8.3 8.4 8.1 13.0 13.0 13.5 8.3 13.21
9.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 1 8.3 11.44
10.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 1 9.3 11.867 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 9.9 15.3 12.68 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 7.7 9.210.7
Run No. A H O C L N1
NOISEN2
5.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 2 5.3 6.62
N1 N2 y
-1.710.5
8.8Avg 1Avg 2Effect Effect
Avg N2Avg N1
8/06/2007 ENGN8101 Modelling and Optimization 68
Calculating Control Factor Effects
9.77.6 7.8 7.6 11.5 11.6 12.02 1 2 1 2 7.7 11.75
10.71 1 1 1 1 8.3 8.4 8.1 13.0 13.0 13.5 8.3 13.21
9.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 1 8.3 11.44
10.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 1 9.3 11.867 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 9.9 15.3 12.68 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 7.7 9.210.7
Run No. A H O C L N1
NOISEN2
5.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 2 5.3 6.62
N1 N2 y
8.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2 6.5 11.1
Avg 1Avg 2Effect Effect
9.2 9.6 10.5 10.910.1 9.7 8.9 8.4-0.9 -0.1 1.6 2.5
Avg N2Avg N1
-1.710.5
8.8
8/06/2007 ENGN8101 Modelling and Optimization 69
9.77.6 7.8 7.6 11.5 11.6 12.02 1 2 1 2 7.7 11.75
10.71 1 1 1 1 8.3 8.4 8.1 13.0 13.0 13.5 8.3 13.21
9.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 1 8.3 11.44
10.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 1 9.3 11.867 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 9.9 15.3 12.68 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 7.7 9.210.7
Run No. A H O C L N1
NOISEN2
5.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 2 5.3 6.62
N1 N2 y
8.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2 6.5 11.1
-1.710.5
8.8Avg 1Avg 2Effect Effect
9.2 9.6 10.5 10.910.1 9.7 8.9 8.4-0.9 -0.1 1.6 2.5
Avg N2Avg N1
Calculating the Noise Effect
8/06/2007 ENGN8101 Modelling and Optimization 70
10.7
11.5 11.6 12.0
13.0 13.0 13.5
11.4 11.5 11.4
11.7 11.9 11.715.5 15.4 15.110.8 10.6
6.7 6.8 6.211.0 11.2 11.2
11.5
9.77.6 7.8 7.62 1 2 1 2 7.7 11.75
10.71 1 1 1 1 8.3 8.4 8.1 8.3 13.21
9.98.5 8.3 8.21 2 2 2 1 8.3 11.44
10.59.1 9.4 9.32 1 2 2 1 9.3 11.867 9.8 10.0 10.02 2 1 1 1 9.9 15.3 12.68 2 7.4 7.9 7.72 1 2 2 7.7 9.210.7
Run No. A H O C L N1
NOISEN2
5.95.4 5.1 5.41 1 1 2 2 5.3 6.62
N1 N2 y
8.83 6.5 6.5 6.51 2 2 1 2 6.5 11.1
-1.710.5
8.8Avg 1Avg 2Effect Effect
9.2 9.6 10.5 10.910.1 9.7 8.9 8.4-0.9 -0.1 1.6 2.5
Avg N2Avg N1
Calculating the Noise Effect
8/06/2007 ENGN8101 Modelling and Optimization 71
Calculating the Noise Effect
13.0
11.5
9.77.6 7.8 7.6 11.5 11.6 12.02 1 2 1 2 7.7 11.75
10.71 1 1 1 1 8.3 8.4 8.1 13.0 13.5 8.3 13.21
9.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 1 8.3 11.44
10.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 1 9.3 11.867 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 9.9 15.3 12.68 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 7.7 9.210.7
Run No. A H O C L N1
NOISEN2
5.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 2 5.3 6.62
N1 N2 y
8.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2 6.5 11.1
-1.710.5
8.8Avg 1Avg 2Effect Effect
9.2 9.6 10.5 10.910.1 9.7 8.9 8.4-0.9 -0.1 1.6 2.5
7.9Avg N2Avg N1
8/06/2007 ENGN8101 Modelling and Optimization 72
Calculating the Noise Effect
3.6
13.0
11.5
9.77.6 7.8 7.6 11.5 11.6 12.02 1 2 1 2 7.7 11.75
10.71 1 1 1 1 8.3 8.4 8.1 13.0 13.5 8.3 13.21
9.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 1 8.3 11.44
10.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 1 9.3 11.867 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 9.9 15.3 12.68 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 7.7 9.210.7
Run No. A H O C L N1
NOISEN2
5.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 2 5.3 6.62
N1 N2 y
8.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2 6.5 11.1
-1.710.5
8.8Avg 1Avg 2Effect Effect
9.2 9.6 10.5 10.910.1 9.7 8.9 8.4-0.9 -0.1 1.6 2.5
7.9Avg N2Avg N1
8/06/2007 ENGN8101 Modelling and Optimization 73
Avg 1Avg 2Effect
A H O C L N8.8 9.2 9.6 10.5 10.9 7.9
10.5 10.1 9.7 8.9 8.4 11.5− 1.6 − 0.9 − 0.1 1.6 2.5 3.6
Main Effects Plot
6789
10111213
A1 A2 H1 H2 O1 O2 C1 C2 L1 L2 N1 N2
Factors
Ave
rage
Tim
e (s
ecs)
8/06/2007 ENGN8101 Modelling and Optimization 74
1
2
11.48.3
7.7 10.7
7.7
8.3
9.39.9
5.36.5
11.7
13.2
11.815.3
6.611.1
Calculating Noise x Control Interaction Effects
N1 N2 N1 N2 N1 N2 N1 N2 N1 N2CF 1CF 2
Effect
N1 N2 yRun No. A H O C L N1 N212345678
DATA2 1 2 1 2
1 1 1 1 1
1 2 2 2 1
1 2 2 12 2 1 1 1
22 1 2 2
1 1 1 2 22 2 1 2
9.7
10.7
9.9
10.512.6
9.2
5.98.8
8/06/2007 ENGN8101 Modelling and Optimization 75
7.7
11.4
10.7
7.7 11.79.3 11.89.9 15.3
2222
Calculating Noise x Control Interaction Effects
8.3
7.1N1 N2 N1 N2 N1 N2 N1 N2 N1 N2
CF 1CF 2
Effect
N1 N2 yRun No. A H O C L N1 N212345678
DATA1 2 1 2
1 1 1 1 1
1 2 2 2 1
1 2 2 12 1 1 12 1 2 2
1 1 1 2 21 2 2 1 2
9.7
10.78.3 13.2
9.9
10.512.6
9.2
5.95.3 6.68.86.5 11.1
8/06/2007 ENGN8101 Modelling and Optimization 76
15.3
11.4
15.3
11.7
13.2
11.8
10.7
6.611.1
8.3
Calculating Noise x Control Interaction Effects
7.18.6
N1 N2 N1 N2 N1 N2 N1 N2 N1 N2CF 1CF 2
Effect
N1 N2 yRun No. A H O C L N1 N212345678
DATA2 1 2 1 2
1 1 1 1 1
1 2 2 2 1
2 1 2 2 12 2 1 1 1
22 1 2 2
1 1 1 2 22 2 1 2
9.77.7
10.78.3
9.9
10.59.39.9 12.67.7 9.2
5.98.86.5
8/06/2007 ENGN8101 Modelling and Optimization 77
11.411.16.5
7.715.310.7
8.3
10.67.18.6
N1 N2 N1 N2 N1 N2 N1 N2 N1 N2CF 1CF 2
Effect
Calculating Noise x Control Interaction Effects
N1 N2 yRun No. A H O C L N1 N212345678
DATA2 1 2 1 2
1 1 1 1 1
1 2 2 2 1
2 1 2 2 12 2 1 1 1
22 1 2 2
1 1 1 2 21 2 2 1 2
9.77.7 11.7
10.78.3 13.2
9.9
10.59.3 11.89.9 12.6
9.2
5.95.3 6.68.8
8/06/2007 ENGN8101 Modelling and Optimization 78
10.7
6.5
7.7
12.410.67.1
8.6
Calculating Noise x Control Interaction Effects
8.3
N1 N2 N1 N2 N1 N2 N1 N2 N1 N2CF 1CF 2
Effect
N1 N2 yRun No. A H O C L N1 N212345678
DATA2 1 2 1 2
1 1 1 1 1
1 2 2 2 1
2 1 2 2 12 2 1 1 1
22 1 2 2
1 1 1 2 21 2 2 1 2
9.77.7 11.7
10.78.3 13.2
9.911.4
10.59.3 11.89.9 15.3 12.6
9.2
5.95.3 6.68.811.1
8/06/2007 ENGN8101 Modelling and Optimization 79
Calculating Interaction Effects
+−
++
−− + −−+
A
B
(− −) = B(− +) −Interactioneffect
( − ) =2
A B
(+ +) − (+ −) = A
8/06/2007 ENGN8101 Modelling and Optimization 80
12.410.6
Calculating Noise x Control Interaction Effects
8.3
7.18.6
N1 N2 N1 N2 N1 N2 N1 N2 N1 N2CF 1CF 2
Effect
N1 N2 yRun No. A H O C L N1 N212345678
DATA2 1 2 1 2
1 1 1 1 1
1 2 2 2 1
2 1 2 2 12 2 1 1 1
22 1 2 2
1 1 1 2 21 2 2 1 2
9.77.7 11.7
10.78.3 13.2
9.911.4
10.59.3 11.89.9 15.3 12.67.7 9.210.7
5.95.3 6.68.86.5 11.1
0.1
Effect( ) ( )
2
+ + + − − + − −− −−=
8/06/2007 ENGN8101 Modelling and Optimization 81
Calculating Noise x Control Interaction Effects
8.3
N1 N2 N1 N2 N1 N2 N1 N2 N1 N2CF 1CF 2
Effect
N1 N2 yRun No. A H O C L N1 N212345678
DATA2 1 2 1 2
1 1 1 1 1
1 2 2 2 1
2 1 2 2 12 2 1 1 1
22 1 2 2
1 1 1 2 21 2 2 1 2
9.77.7 11.7
10.78.3 13.2
9.911.4
10.59.3 11.89.9 15.3 12.67.7 9.210.7
5.95.3 6.68.86.5 11.1
0.1 0.4 0.0 -1.1 -0.4
7.6 10.8 7.8 11.4 8.1 12.8 9.0 12.98.1 12.2 7.9 11.5 7.6 10.1 6.8 10.012.4
10.67.18.6 Effect
( ) ( )2
+ + + − − + − −− −−=
8/06/2007 ENGN8101 Modelling and Optimization 82
Noise x Control Interaction Plot
4
6
8
10
12
14
N1 N2
Ave
rage
Tim
e
CF Level 1CF Level 2
A H O C LN1 N2
CF 1 7.1 10.6 7.6 10.8 7.8 11.4 8.1 12.8 9.0 12.9CF 2 8.6 12.4 8.1 12.2 7.9 11.5 7.6 10.1 6.8 10.0Effect 0.1 0.4 0.0 -1.1 -0.4
N1 N2 N1 N2N1 N2 N1 N2
N1 N2N1 N2 N1 N2 N1 N2
8/06/2007 ENGN8101 Modelling and Optimization 83
Average Effects Daniel PlotN
xO
O NxA
Nxc
5
Nxc
6
c5 NxL
c6 NxH
H NxC
C A L N
Absolute Effect
Half-NormalScores
NxC
N
L
AC
H
0.0
1.0
2.0
3.0
4.0
0.00 0.50 1.00 1.50 2.00
Half-Normal Scores
Abs
olut
e Ef
fect
s
0.0 0.1 0.1 0.2 0.2 0.3 0.4 0.4 0.4 0.9 1.1 1.6 1.7 2.5 3.6.04 .12 .21 .29 .38 .47 .57 .67 .77 .89 1.02 1.17 1.36 1.61 2.10
8/06/2007 ENGN8101 Modelling and Optimization 84
1. Reduce variabilityIdentify factors which interact with noise -- use these to reduce variability.
2. Adjust to mean to targetIdentify factors with location effects, that do not interact with noise -- use these to adjust the mean.
2-Step Optimization
8/06/2007 ENGN8101 Modelling and Optimization 85
=Energy producing the intended resultEnergy wasted on unintended results
SN
Signal-to-Noise Ratio
8/06/2007 ENGN8101 Modelling and Optimization 86
y
Intended vs. Unintended Results
measured in σ’s
8/06/2007 ENGN8101 Modelling and Optimization 87
SN = 10 log10( )
2yσ
Robustness Metric
Always maximize S/N to improve robustness
8/06/2007 ENGN8101 Modelling and Optimization 88
Calculating Signal-to-Noise Ratio
Run No. N1 N2 σ S/Ny
Avg 1Avg 2Effect
9.77.6 7.8 7.6 11.5 11.6 12.02 1 2 1 25
13.0 13.0 13.5 10.71 1 1 1 1 8.3 8.4 8.11
9.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 14
10.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 16
8 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 9.2
5.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 228.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2
7 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 12.6
A H O C L
8/06/2007 ENGN8101 Modelling and Optimization 89
13.5 2.7
Run No. N1 N2 σ S/N
Calculating Signal-to-Noise Ratio
y
Effect
2.29.77.6 7.8 7.6 11.5 11.6 12.02 1 2 1 25
13.0 13.0 10.71 1 1 1 1 8.3 8.4 8.11
1.79.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 14
1.410.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 16
1.78 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 9.2
0.75.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 222.58.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2
7 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 12.6 3.0
Avg 1Avg 2
A H O C L
8/06/2007 ENGN8101 Modelling and Optimization 90
12.0
S
Calculating Signal-to-Noise Ratio
N = 10 log10 ( )2y
σ
Run No. N1 N2 σ S/Ny
Effect
2.2 12.89.77.6 7.8 7.6 11.5 11.6 12.02 1 2 1 25
2.713.0 13.0 13.5 10.71 1 1 1 1 8.3 8.4 8.11
1.7 15.39.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 14
1.4 17.710.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 16
1.7 14.88 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 9.2
0.7 18.25.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 222.5 10.88.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2
7 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 12.6 12.63.0
Avg 1Avg 2
A H O C L
8/06/2007 ENGN8101 Modelling and Optimization 91
Calculating Signal-to-Noise Effects
Run No. N1 N2 σ S/Ny
Effect
2.2 12.89.77.6 7.8 7.6 11.5 11.6 12.02 1 2 1 25
2.7 12.013.0 13.0 13.5 10.71 1 1 1 1 8.3 8.4 8.11
1.7 15.39.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 14
1.4 17.710.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 16
1.7 14.88 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 9.2
0.7 18.25.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 222.5 10.88.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2
7 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 12.6 12.63.0
Avg 1Avg 2
A H O C L
8/06/2007 ENGN8101 Modelling and Optimization 92
14.1
Calculating Signal-to-Noise Effects
Run No. A H O C L N1 N2 σ S/Ny
Effect
2.2 12.89.77.6 7.8 7.6 11.5 11.6 12.02 1 2 1 25
2.7 12.013.0 13.0 13.5 10.71 1 1 1 1 8.3 8.4 8.11
1.7 15.39.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 14
1.4 17.710.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 16
1.7 14.88 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 9.2
0.7 18.25.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 222.5 10.88.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2
7 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 12.6 12.63.0
Avg 1Avg 2
8/06/2007 ENGN8101 Modelling and Optimization 93
Calculating Signal-to-Noise Effects
15.3
14.1
Run No. A H O C L N1 N2 σ S/Ny
Effect
2.2 12.89.77.6 7.8 7.6 11.5 11.6 12.02 1 2 1 25
2.7 12.013.0 13.0 13.5 10.71 1 1 1 1 8.3 8.4 8.11
1.79.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 14
1.4 17.710.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 16
1.7 14.88 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 9.2
0.7 18.25.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 222.5 10.88.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2
7 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 12.6 12.63.0
Avg 1Avg 2 14.5
8/06/2007 ENGN8101 Modelling and Optimization 94
Calculating Signal-to-Noise Effects
15.3
14.1
Run No. A H O C L N1 N2 σ S/Ny
Effect
2.2 12.89.77.6 7.8 7.6 11.5 11.6 12.02 1 2 1 25
2.7 12.013.0 13.0 13.5 10.71 1 1 1 1 8.3 8.4 8.11
1.79.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 14
1.4 17.710.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 16
1.7 14.88 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 9.2
0.7 18.25.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 222.5 10.88.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2
7 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 12.6 12.63.0
Avg 1Avg 2 14.5
-0.4
8/06/2007 ENGN8101 Modelling and Optimization 95
15.3
14.1
Run No. A H O C L N1 N2 σ S/N
Calculating Signal-to-Noise Effects
y
Effect
2.2 12.89.77.6 7.8 7.6 11.5 11.6 12.02 1 2 1 25
2.7 12.013.0 13.0 13.5 10.71 1 1 1 1 8.3 8.4 8.11
1.79.98.5 8.3 8.2 11.4 11.5 11.41 2 2 2 14
1.4 17.710.59.1 9.4 9.3 11.7 11.9 11.72 1 2 2 16
1.7 14.88 2 7.4 7.9 7.7 10.8 10.6 10.72 1 2 2 9.2
0.7 18.25.95.4 5.1 5.4 6.7 6.8 6.21 1 1 2 222.5 10.88.83 6.5 6.5 6.5 11.0 11.2 11.21 2 2 1 2
7 9.8 10.0 10.0 15.5 15.4 15.12 2 1 1 1 12.6 12.63.0
Avg 1Avg 2 14.5
-0.4 1.8
15.213.4
0.3
14.414.1
-4.4
12.116.5
0.2
14.414.2
8/06/2007 ENGN8101 Modelling and Optimization 96
S/N Ratio Effects Plot
A H O C L14.1 15.2 14.4 12.1 14.414.5 13.4 14.1 16.5 14.2- 0.4 1.8 0.3 - 4.4 0.2
101112131415161718
A1 A2 H1 H2 O1 O2 C1 C2 L1 L2Factors
Ave
rage
S/N
Avg 1Avg 2Effect
8/06/2007 ENGN8101 Modelling and Optimization 97
S/N Daniel Plot
L O A c5 c6 H CAbsolute Effect 0.2 0.3 0.4 0.9 1.1 1.8 4.4
Half-Normal Score 0.09 0.27 0.45 0.66 0.90 1.20 1.71
C
0.0
1.0
2.0
3.0
4.0
5.0
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80Half-Normal Score
Abs
olut
e Ef
fect
8/06/2007 ENGN8101 Modelling and Optimization 98
2-Step Optimization
1. Reduce variabilityIdentify factors with large S/N effects--use these to reduce variability.
2. Adjust mean to targetIdentify factors with location effects--use these to adjust the mean.
8/06/2007 ENGN8101 Modelling and Optimization 99
2-Step Optimization Example
A1 A2 H1 H2 O1 O2 C1 C2 L1 L210
12
14
16
18
Ave
rage
S/N
6
8
10
12
Ave
rage
Tim
e
A1 A2 H1 H2 O1 O2 C1 C2 L1 L2
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