Using Parameterized Pareto Sets to Model Design Concepts

(1)

Systems Realization Laboratory

Woodruff School of Mechanical Engineering

to Model Design Concepts

IMECE2007-43226

Rich Malak and Chris Paredis Systems Realization Laboratory

Woodruff School of Mechanical Engineering Georgia Institute of Technology

Atlanta, GA

(2)

Systems Realization

Motivation: System-Level Modeling and Decision Making

• Which do you prefer more?

• Depends on:

– Preferences

– Implementation details

• How to model concepts for subsystems?

Hybrid Car Hybrid Car

Electrical Power Storage Electrical

Power Control Power Control

Unit

Transmission Electrical Motor/Generator

Internal Combustion

Engine

Fuel Tank Assembly

Differential Left Drive

Wheel

Right Drive Wheel

Traditional Car Traditional Car

Internal Combustion

Engine Power

Control Unit

Transmission

Fuel Tank Assembly

Wheel

Right Drive Wheel

(3)

Key Ideas

• Concepts as Sets

• Predictive Tradeoff Models

• Parameterized Pareto Sets

• Assumption: no uncertainty

(4)

Concepts as Sets of Designs

Ideally:

• Evaluate all

• Select most preferred

Design Decision

Design 1 Design 2 Design 3

Design n .. .

One-Shot Decision

In practice:

• Too many possibilities

• Decompose into

“easier” decisions

Sequential Decisions

Decision 3.1

Decision 1

Decision 3.2

Decision 3.3

Design…

Decision 3.4

Design n-1 Design n Decision

2.1

Decision 2.2

Concept 2.1

Concept 2.2

Concept 3.1

Concept 3.2

Concept 3.3

Concept 3.4

Decision 2.1

Design 1 Design 2 Design 3 Design 4

Sequential Decisions

Decision 3.1

Decision 1

Decision 3.2

Decision 3.3

Design…

Decision 3.4

Design n-1 Design n Decision

2.1

Decision 2.2

Concept 2.1

Concept 2.2

Concept 3.1

Concept 3.2

Concept 3.3

Concept 3.4

Decision 2.1

Design 1 Design 2 Design 3 Design 4

Implication:

Design Concept = Set of Design

Implementations

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Predictive Tradeoff Modeling

• Predictive: association, not causation

• Fit to data

x₁ x₂

Attribute Space Decision criteria Design Spaces

Concept

implementations

Tradeoff Model:

x

₂

= M(x

₁

)

Analysis Model 1

Analysis Model 2

Analysis Model 3

Analysis Models

Physics, engineering

relationships, etc.

(6)

Predicting Tradeoffs

• Search over

“independent”

attribute

Tradeoff Model:

x

₂

= M(x

₁

) Tradeoff

Model:

x

₂

= M(x

₁

)

Preference Model:

V = V(x

₁

,x

₂

) Preference

Model:

V = V(x

₁

,x

₂

)

V = V(x

₁

,M(x

₁

)) V = V(x

₁

,M(x

₁

))

Increasing preference

x₁ x₂

Predicted most-preferred

tradeoff

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Pareto Set

• Not all implementations are “good”

• Eliminate dominated ones

– Never would choose them

– Improves prediction accuracy

Analysis Model 1

Analysis Model 2

Analysis Model 3

x₁ x₂

Increasing Preference Pareto Set Point Dominated Point

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Limitation of Pareto Dominance

• Assumes monotonic preferences

• Non-monotonic:

– “Target seeking”

– “Target avoiding”

!Problem dependence

x Preference

“Target-avoiding”

x Preference

“Target-seeking”

• Proposed solution:

– Parameterize tradeoff

models using target

(9)

Why is problem-dependence bad?

• Subsystem-level targets depend on system- level considerations

• Plus: reuse

Traditional Car Traditional Car

Internal Combustion

Engine Power

Control Unit

Transmission

Fuel Tank Assembly

Wheel

Right Drive Wheel

What is most

preferred gear ratio?

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0.95 0.96 0.97 0.98 0.99 1

200 250 300 350 400

Reliability

Cost ($)

g y

Pareto Domination at a Fixed Gear Ratio

Parameterized Pareto Set

2.6122 2.9375

3.2500 3.5789

3.8750 0.95

0.96 0.97

0.98 0.99

1 180

200 220 240 260 280 300 320 340 360

Reliability

Gear Ratio Reliability

Reliability

Cost ($)

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Summary of Modeling Process

Parameterized

Pareto Dominance Parameterized

Pareto Dominance

Predictive Model Fitting Predictive Model Fitting

x₁ x₂

x₂ = M(x₁)

Predictive Tradeoff Model

Parameterized Pareto Set Implementations

Concept Implementations

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Example: Conceptual Design of a Gearbox

• Fixed-ratio gearbox

• Three concepts

• Multiple decision scenarios

– Physical system – Preferences

• Compare to

exhaustive search

Engine

CVT Fixed-ratio Gearbox

Rear Differential

Drive Wheels

Off-Road Vehicle

Scenario 1:

• Cost

• Reliability

• Winnings

Scenario 2:

• Cost

• Reliability

• V

_max

• A

_max

Rear Differential

(13)

Example: Tradeoff Models

2 1 4 3

6 5

0.8 0.85 0.9 0.95 1 150

200 250 300 350 400

Reliability Abstracted Model

Gear Ratio

Cost

DGB

1 2 3 4 5

0.8 0.85 0.9 0.95 1 150

200 250 300 350 400

Reliability

SGB

Cost

SGB

3 2.5 4 3.5

5 4.5

0.85

0.9

0.95

1 200

220 240 260 280 300 320 340

Reliability

PGB

Gear Ratio

Cost

PGB

All models:

RMS Error < 5%

Kriging using

DACE Toolkit

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Example: Key Results

• Consistent with exhaustive search

• Both scenarios

– Correct concept chosen

– Accurate tradeoff predictions

• Most attributes <5%

• One attribute ~ 11%

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Future Work

• Uncertainty

– In attributes: Stochastic Dominance – In tradeoff model

• Compositional Modeling

Compositional Modeling

Characteristics of most

preferred

implementation

Prediction Procedure Prediction Procedure

Decision Preferences

Tradeoff Models

System-Level Tradeoffs

Subsystem Tradeoffs System Mapping

Model of System-Level Alternative

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Summary

• Predictive tradeoff modeling

– Instead of analysis

• Parameterized Pareto sets

– Non-dominated implementations – Problem independent

• Step toward system-level decision making

• Questions?

Abstracting sets of design

implementations

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Additional Slides: Info about example

problem

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Ring (Stationary)

Sun (Input) Planet Carrier (Output)

Input pinion Output Gear

Input-side View

Input Output

Top View

Input-side View

Top View

Input Output

Input pinion Output Gear

Gearbox Configurations

(19)

Scenario 1

Using Tradeoff Models

Using Extensive Search

Percent Difference

PGB Maximum Value

682.65 681.72 0.14

Gear Ratio 4.14 4.13 0.12

Reliability 0.994 0.994 0.00

Cost ($) 262.42 262.91 0.19

SGB Maximum Value

651.20 670.87 2.93

Gear Ratio 4.12 4.27 3.51

Reliability 0.968 0.988 2.02

Cost ($) 268.20 266.55 0.62

DGB Maximum Value

613.40 606.34 1.16

Gear Ratio 4.16 4.27 2.58

Reliability 0.980 0.984 0.41

Cost ($) 319.00 327.47 2.59

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Scenario 2

Using tradeoff Models

Using Extensive Search

Percent Difference

PGB Maximum

Value

0.702 0.698 0.57

Gear Ratio 2.55 2.55 0.00

Reliability 0.99 0.987 0.30

Cost ($) 246.12 246.81 0.28

SGB Maximum

Value

0.769 0.806 4.53

Gear Ratio 1.96 1.96 0.00

Reliability 0.993 0.986 0.71

Cost ($) 237.47 213.54 11.2

DGB Maximum

Value

0.719 0.715 0.56

Gear Ratio 2.01 1.96 2.55

Reliability 0.989 0.991 0.20

Cost ($) 272.65 275.16 0.91