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Optimization Approaches for Product Family DesignOptimization Approaches for Product Family Design

Timothy W. SimpsonProfessor of Mechanical & Industrial Engineering and Engineering DesignThe Pennsylvania State University

University Park, PA 16802 USA

phone: (814) 863-7136email: tws8@psu.edu

http://www.mne.psu.edu/simpson/courses/me546

ME 546 - Designing Product Families - IE 546

© T. W. SIMPSON

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Optimization in Product Family DesignOptimization in Product Family Design

• Optimization can be a helpful tool to support design decision-making

• Optimization is frequently used in product design to help determine values of design variables, x, that minimize (or maximize) one or more objectives, f(x), with satisfying a set of constraints, {g(x), h(x)}

• In product family design, optimization can be used to help balance the tradeoff between commonality and individual product performance in the family

• Let’s consider a motivating example to define key terms and introduce different optimization formulations

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Motivating ExampleMotivating Example

Objective: Design a family of ten (10) universal electric motors based on a product platform to provide a variety of power and torque outputs

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• Universal motor is most common component in power tools

• Challenge: redesign the universal motor to fit into 122 basic tools with hundreds of variations

• Result: a common platform where geometry and axial profile common stack length varied from 0.8”-1.75”

to obtain 60-650 Watts fully automated assembly process material, labor, and overhead costs

reduced from $0.51 to $0.31 labor reduced from $0.14 to $0.02

Universal Motor Platform Example

Universal Motor Platform Example

Electric motor field componentsprior to standardization

Universal motor variants0.8” 1.75”

60

650

Wat

ts

Stack length

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Scale-based Family: Rolls Royce Engines

Scale-based Family: Rolls Royce Engines

• Rolls Royce scales its aircraft engines to efficiently and effectively satisfy a variety of performance requirements

Incremental improvements and variations made to increase thrust and reduce fuel consumption

RTM322 is common to turboshaft, turboprop, and turbofan engines When scaled 1.8x, RTM322 serves as the core for RB550 series

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Example Leveraging Strategies: Boeing Aircraft

Example Leveraging Strategies: Boeing Aircraft

• Boeing 737 is divided into 3 platforms: Initial-model (100 and 200) Classic (300, 400, and 500) Next generation (600, 700,

800, and 900 models)

• The new 777 is also being designed knowing a priori that it will be stretched to carry more passengers and increase range

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Boeing 737 Interior Layouts

Boeing 737 Interior Layouts

737-300126 passengers (8 first class)

737-400147 passengers (10 first class)

737-500110 passengers (8 first class)

737-600110 passengers (8 first class)

737-700126 passengers (8 first class)

737-800162 passengers (12 first class)

737-900177 passengers (12 first class)

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Flight Ranges for 737-300, -500, -600, and -700

Flight Ranges for 737-300, -500, -600, and -700

Flight Ranges for 737-700

Flight Ranges for 737-300

Capacity: 126 Passengers Capacity: 110 Passengers

Flight Ranges for 737-600

Flight Ranges for 737-500

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Dimensions of Boeing 737-300, -400, and -500

Dimensions of Boeing 737-300, -400, and -500

Boeing 737-300 Boeing 737-400 Boeing 737-500

• All three aircraft share common height and width...

…but their fuselage lengths are different:

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Boeing 737-600

Dimensions of Boeing 737-600, -700, -800, and -900

Dimensions of Boeing 737-600, -700, -800, and -900

• The same holds true for the 737-600 through 900

Boeing 737-900

Boeing 737-700

Boeing 737-800

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Optimization for Single Product DesignOptimization for Single Product Design

Generic Form:

• Find: x

• Minimize: f(x)

• Subject to: g(x) < 0h(x) = 0

Definitions:• x = design variables• f(x) = objective function• g(x) = inequality constraints• h(x) = equality constraints

For Motor Example:• Find: r, t, AA, NA,

AF, NF, I, L

• Minimize: Mass• Maximize: Efficiency,

• Subject to: MagInt, H < 5000

Mass < 2 kg

Eff, > 70 %r > tPower = 300 WTorque = 0.5

Nm

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Optimization for Product Family Design

Optimization for Product Family Design

Generic Form:

• Find: xi

• Minimize: fi(xi)

• Subject to: gi(xi) < 0

hi(xi) = 0

Definitions:• i = 1, 2, …, p • p = number of products in

the family

For Motor Family Example:• Find: ri, ti, AA,i, NA,i,

AF,i, NF,i, Ii, Li

• Minimize: Massi

• Maximize: Efficiencyi

• Subject to: MagInt, Hi < 5000

Massi < 2 kg

Eff, i > 70 %

ri > ti

Poweri = 300 W

Torquei = Ti

where:Ti = {0.05, 0.1, 0.125, 0.2, 0.25, 0.3, 0.35,

0.4, 0.45, 0.5} Nm

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Challenges in Product Family OptimizationChallenges in Product Family Optimization

• The dimensionality and size of the optimization problem increases very quickly as the number of products in the family increases

• For motor example, p = 10: Number of design variables = 8 x p = 8 x 10 = 80 Number of objective functions = 2 x p = 2 x 10 = 20 Number of constraints = 6 x p = 6 x 10 = 60

• Using a product platform will reduce the dimensionality of the optimization problem but not the size (i.e., the number of objectives or constraints): Number of design variables = c + (n-c) x p

where: c = number of common (platform) variablesn = number of design variables for each of the p

products

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Product Platform Concept Exploration Method

Product Platform Concept Exploration Method

Step 1Create Market Segmentation Grid

Step 2Classify Factors and Ranges

Step 3Simulation Analysis/Metamodels

Step 4Aggregate Product Platform Specifications

Step 5Develop Product Platform and Family

MarketSegmentation

Grid

Robust DesignPrinciples

MetamodelingTechniques

MultiobjectiveOptimization

Product Platform and Product Family Specifications

Overall Design Requirements

The PPCEM provides a Method that facilitates the synthesis and Exploration of a common Product Platform Concept that can be scaled into an appropriate family of products to satisfy a variety of market niches

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Robust Design and Scalable Product Platforms

Robust Design and Scalable Product Platforms

• Robust design principles are used to minimize the sensitivity of a product platform (and resulting product family) to changes in one or more scale factors

Functional• torque = fcn(motor stack length)• thrust = fcn(# compressor stages)

Conceptual/configurational• # passengers on an aircraft• size of an automobile underbody

Example Scaling VariablesPlatform

High

Mid

Low

Segment A Segment B Segment C

High

Mid

Low

Segment A Segment CSegment B

Platform

Scale dow

n

Sca

le u

p

Low-End Platform Leveraging

High-End Platform Leveraging

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Compromise Decision Support ProblemCompromise Decision Support Problem

A hybrid of Goal Programming and Math Programming used to determine the values of design variables that satisfy a set of constraints and achieve as closely as possible a set of conflicting goals

Given Assumptions to model domain of interestSimulation and analyses to relate X and Y

Find Xi i = 1, …, n di-, di

+ i = 1, …,

m

SatisfySystem constraints (linear, nonlinear)

gi(X) = 0 ; i = 1, .., p

gi(X) < 0 ; i = p+1, .., p+q

System goals (linear, nonlinear)Ai(X) + di

- + di+ = Gi ; i = 1, …, m

BoundsXj

min < Xj < Xjmin; j = 1, …, n

di-, di

+ < 0 ; di- • di

+ = 0 ; i = 1, …, m

MinimizeDeviation Function

Z = { f1(di-, di

+), ..., fk(dk-, dk

+) }

Given Assumptions to model domain of interestSimulation and analyses to relate X and Y

Find Xi i = 1, …, n di-, di

+ i = 1, …,

m

SatisfySystem constraints (linear, nonlinear)

gi(X) = 0 ; i = 1, .., p

gi(X) < 0 ; i = p+1, .., p+q

System goals (linear, nonlinear)Ai(X) + di

- + di+ = Gi ; i = 1, …, m

BoundsXj

min < Xj < Xjmin; j = 1, …, n

di-, di

+ < 0 ; di- • di

+ = 0 ; i = 1, …, m

MinimizeDeviation Function

Z = { f1(di-, di

+), ..., fk(dk-, dk

+) }

FeasibleDesignSpace

Deviation Function

AspirationSpace

ConstraintsBounds

Goals

x1

x2

Reference: (Mistree, et al., 1993)

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Platform Leveraging Strategy

Platform Leveraging Strategy

PowerTools

Lawn &Garden

KitchenAppliances

High CostHigh Performance

Mid-Range

Low CostLow Performance

Ver

tical

Sca

ling

Standardizing motorinterfaces will facilitatehorizontal leveraging

to new segments

Standardizing motorinterfaces will facilitatehorizontal leveraging

to new segments

Lawn &Garden

KitchenAppliances

Universal Motor Platform(Common Design Variable Settings)

Design a single motor platform scaled by stack length

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Electric Motor Family Design Problem IElectric Motor Family Design Problem I

• Platform parameters (common to all motors): radius of motor, r on armature:

– wire x-sectional area, AA

– number of wraps, NA

• Scaling variable (1/motor): i = 1, …, 10 stack length, Li

• Constraints (6/motor) and Objectives (2/motor):

thickness of motor, t on field:

– wire x-sectional area, AF

– number of wraps, NF

Name ConstraintMagnetizing Intensity, H Hi = 5000 Amp·turns/mFeasible geometry ro,i > tiPower, P Pi = 300 WTorque, T Ti = {0.05, 0.1, 0.125, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.5} NmEfficiency,

i = 0.15 (Target = 70%)Mass, M Mi = 2.0 kg (Target = 0.5 kg)

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Two-Stage Optimization Approach in PPCEM

Two-Stage Optimization Approach in PPCEM

Stage 2: Design individual products based on platform Fix common platform parameters and instantiate each product by solving p one-dimensional optimization problems to satisfy individual constraints while trying to meet performance targets

Stage 1: Identify best platform variable settings Using robust design principles, solve one optimization problem of size n+1 to find best settings of common platform parameters, allowing one scaling variable to vary (s, s)

S

6S

Y

+3Y

-3Y

Y

S

LowerLimit

UpperLimit

Each line represents a different product architecture,i.e., a different combination of:

[ x1, x2, x3, …., xn-1, s, s ][ x1, x2, x3, …., xn-1, s, s ][ x1, x2, x3, …., xn-1, s, s ][ x1, x2, x3, …., xn-1, s, s ][ x1, x2, x3, …., xn-1, s, s ]

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Stage 1 Using robust design principles, solve one optimization problem of size 8 to find best settings of common platform parameters, allowing one scaling variable to vary (stack_length, stack_length)

T

+3T

-3T

T

L

LowerTorqueLimit

UpperTorqueLimit

L

6L

Each line represents a different product architecture,i.e., a different combination of:

[r, t, Aarmature, Narmature, Afield, Nfield][r, t, Aarmature, Narmature, Afield, Nfield][r, t, Aarmature, Narmature, Afield, Nfield][r, t, Aarmature, Narmature, Afield, Nfield][r, t, Aarmature, Narmature, Afield, Nfield]

Optimization Problem for Motor FamilyOptimization Problem for Motor Family

Stage 2 Fix common platform parameters and instantiate each product by solving 10 one-dimensional optimization problems to satisfy individual constraints while trying to meet performance targets

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L [cm] T [Nm] [%] M [kg]

2.44 0.50 47.9 0.83

2.40 0.40 53.1 0.82

2.33 0.35 55.9 0.80

2.21 0.30 58.8 0.78

2.04 0.25 61.8 0.73

1.81 0.20 65.1 0.68

1.50 0.15 68.5 0.61

1.32 0.13 70.3 0.56

1.11 0.10 72.2 0.51

0.62 0.05 76.0 0.40

Resulting Product Family Specifications

Resulting Product Family Specifications

Universal Motor Platform{Nc, Ns, Awa, Awf, r, t}

1273, 61, 0.27, 0.27, 2.67, 7.75

High

Mid

Low

Product platform obtainedusing PPCEM

Platform instantiations

Na, Nf, A f, Aa, r, t L [cm] T [Nm] [%] M [kg]

1087, 72, 0.28, 0.25, 2.71, 7.15 3.16 0.50 55.3 0.99

1082, 72, 0.27, 0.24, 2.58, 6.67 2.87 0.40 57.7 0.84

1056, 73, 0.26, 0.24, 2.51, 6.46 2.81 0.35 59.8 0.78

1030, 73, 0.25, 0.23, 2.44, 6.35 2.74 0.30 62.2 0.71

1007, 73, 0.25, 0.22, 2.35, 6.17 2.61 0.25 64.9 0.64

988, 74, 0.24, 0.22, 2.26, 5.75 2.38 0.20 67.9 0.56

785, 95, 0.21, 0.21, 2.82, 8.88 1.63 0.15 70.5 0.50

760, 89, 0.19, 0.20, 3.12, 11.20 1.41 0.13 70.0 0.50

750, 76, 0.19, 0.20, 3.31, 11.77 1.28 0.10 70.6 0.50

730, 45, 0.20, 0.21, 3.62, 9.69 0.998 0.05 71.4 0.50

Group of individually designed motors

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Comparison of Results: Individual Motors

Comparison of Results: Individual Motors

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

40% 50% 60% 70% 80%

Efficiency

Mas

s (k

g)

Benchmark GroupPPCEM (s=length)

DesiredEfficiency(> 70%)

Desired Mass(< 0.5 kg)

4

3

2

1

8

10

9

8

7

6

5

4321

6

Desired Performance Region(i.e., targets for mass and efficiency are achieved)

9

5

710

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Single-Stage Optimization ApproachOptimize product platform and product family members simultaneously by determine values of c common parameters for the product platform and s scaling variables for each product by solving one optimization problem of dimension (c + s*p)

where:

p = # products in the familyn = # design variables per product in the familys = # scaling variables per product in the familyc = # common platform variables (n = c + s)

Single-Stage Optimization Approach

Single-Stage Optimization Approach

• Use multiobjective optimization to formulate the product family optimization problem and resolve the tradeoff between commonality and individual performance

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Universal Motor Family Design Problem II

Universal Motor Family Design Problem II

• Design variables (8/motor): i = 1, …, 10 stack length, Li

radius of motor, ri

on armature:– wire x-sectional area, AA,i

– number of wraps, NA,i

• Constraints (6/motor) and Objectives (2/motor):

current, Ii

thickness of motor, ti

on field:– wire x-sectional area, AF,i

– number of wraps, NF,i

Name ConstraintMagnetizing Intensity, H Hi = 5000 Amp·turns/mFeasible geometry ro,i > tiPower, P Pi = 300 WTorque, T Ti = {0.05, 0.1, 0.125, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.5} NmEfficiency,

i = 0.15 (Target = 70%)Mass, M Mi = 2.0 kg (Target = 0.5 kg)

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10

Comparison of Results: Individual Motors

Comparison of Results: Individual Motors

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

40% 50% 60% 70% 80%

Efficiency

Mas

s (k

g)

DesiredEfficiency(> 70%)

Desired Mass(< 0.5 kg)

108

6

5

4

3

1

9

7

2

1

2

4

5

678

3

10

9

8

7

6

5

4321

Desired Performance Region(i.e., targets for mass and efficiency are achieved)

10 98

7

6

5

4

3

2

1

Benchmark Group

PhysPro (s=length)PPCEM (s=length)

PhysPro (s=radius)

9

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Comparison of ApproachesComparison of Approaches

• Single-stage approaches:+ yield performance improvements over two-stage approaches + use only a single optimization to determine best settings of

common and scaling variables

- increases dimensionality of optimization (many local optima)- assume best scaling variables are known a priori

• Two-stage approaches:+ provides flexible formulation for determining best combination

of common parameters and scaling variables within a family+ reduces dimensionality of optimization

- increases number of optimizations that must be solved- segments optimization of platform from individual products

which can lead to performance degradation within family

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Varying Platform Commonality

Varying Platform Commonality

• Ideally, an optimization algorithm would search all possible product platform combinations:

where:

the number of possible combinations of making n design variables common to platform c at a time

the null platform, i.e., no commonality within the family

and provide the designer with information about the:1) design variables that should be made common2) the values that they should take3) the values the remaining unique variables should take

n

0

n

1

n

2

n

1-n

n

n

n2esalternativ platform #

c

n

0

n

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Genetic AlgorithmsGenetic Algorithms

• Genetic algorithms (GAs) have shown great promise in many product design and optimization applications

• GAs are well suited for product family design due to the combinatorial nature of the problem, but the associated computational costs are high

• What is a Genetic Algorithm? Optimization algorithm based on evolutionary principles

(survival of the fittest) that do not require gradient information Use strings of chromosomes to represent design variables Each chromosome is evaluated for its “fitness” where those

with higher fitness reproduce to form a new population New populations of chromosomes are generated using

selection, cross-over, and mutation

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GA TerminologyGA Terminology

Chromosome

Population

Generation k Generation k+1

Selection CrossoverMutationInsertion

Geneticoperators

Individuals

gene

0 1 0 1 1 1 1 0 1 0 0 1 ….. 0 1

alleles

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Encoding - DecodingEncoding - Decoding

PhenotypeGenotype

Biology

Design

“blue eye”

UGCAACCGU(“DNA” blocks)

010010011110

expression

(chromosome)

decoding

encoding

Radius R=2.57 [m]

H

sequencing

coded domain decision domain

0 1 0 1 1 1 1 0 1 0 0 1 ….. 0 1

Radius Height Material

x1 x2 xn

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Basic Operation of a Genetic AlgorithmBasic Operation of a Genetic Algorithm

Initialize Population (initialization)

Select individual for mating (selection)

Mate individuals and produce children (crossover)

Mutate children (mutation)

Insert children into population (insertion)

Are stopping criteria satisfied?

Finish

yn

nex

t ge

ner

atio

n

Reference: Goldberg, D.E., 1989, Genetic Algorithms in Search, Optimization and Machine Learning, Addison Wesley

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Genetic Operators: Selection Genetic Operators: Selection

Roulette Wheel Selection

12

3

45

6

Probabilistically select individuals based on some measure of their performance.

Sum Sum of individual’sselection probabilities

3rd individual in currentpopulation mapped to interval[0,Sum]

• Selection: generate random number in [0,Sum]• Repeat process until desired # of individuals areselected• Basically: stochastic sampling with replacement

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Genetic Operators: SelectionGenetic Operators: Selection

2 members of currentpopulation chosen randomly

Dominant performerplaced in intermediatepopulation of survivors

PopulationFilled ?

Crossover andMutation form newpopulation

Old Population Fitness101010110111 8100100001100 4001000111110 6

Survivors Fitness101010110111 8001000111110 6101010110111 8

n

y

Tournament Selection

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Genetic Operators: Crossover and Mutation

Genetic Operators: Crossover and Mutation

Classical: single point crossover

0 1 1 0 1

1 0 0 1 1

The parents

0 1 1 1 1

1 0 0 0 1

crossoverpoint

The children(“offspring”)

P1

P2

O1

O2

• Crossover takes 2 solutions and creates 1 or 2 more

• Mutation randomly changes one or more alleles in the chromosome to increase diversity in the population

With mutation probability Pm, O2: 1 0 0 0 1 1 0 1 0 1

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Genetic Operators: InsertionGenetic Operators: Insertion

• Replacement scheme specifies how individuals from the parent generation k are chosen to be replaced by children from next generation k+1: Can replace an entire population at a time (go from generation k to k+1

with no survivors)– select N/2 pairs of parents– create N children, replace all parents– polygamy is generally allowed

Can select two parents at a time– create one child– eliminate one member of population (usually the weakest)

“Elitist” strategy– small number of fittest individuals survive unchanged

“Hall-of-fame” strategy– remember best past individuals, but do not use them for progeny

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Stopping CriteriaStopping Criteria

Generation

Globaloptimum

(unknown)

Converged toofast (mutation ratetoo small?)A

vera

ge f

itnes

s

Typical convergenceTypical convergence

• There are a variety of stopping criteria: A specific number of generations completed - typically O(100) Mean deviation in individual performance falls below a

threshold k< (i.e., genetic diversity has become small) Stagnation - no or marginal improvement from one generation

to the next: (Fn+1 - Fn)<

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Using GAs in Product Family DesignUsing GAs in Product Family Design

• Chromosomes typically represent a single product:

• For product family design, one can use multiple chromosomes to represent the products in the family:

• This requires added overhead to: make sure all products exist in equal numbers cluster products into families within each population ensure that selection and cross-over operators are performed

only on similar products

0 1 0 1 1 0 … 1 = one motor

= motor # 1

= motor # 2

= motor # 3

= motor # 8

= motor # 9

= motor # 10

0 1 0 1 0 0 … 0

0 1 1 0 1 0 … 0

1 1 0 1 1 0 … 0

1 1 1 1 1 0 … 1

0 1 1 1 1 1 … 1

1 1 1 1 1 1 … 1

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Using GAs in Product Family Design (cont.)Using GAs in Product Family Design (cont.)

• Alternatively, you can extend a single chromosome to represent the entire product family:

• Adds overhead during the decoding process, but fitness function will be evaluated for the entire family genetic operators can be applied with little to no modification

• Challenge is to determine how to represent a platform within the family of products Specify common/unique variables a priori during initialization? Or let the GA vary the levels of commonality of the platform?

…0 1 0 1 0 0 … 0 1 0 0 1 0 0 … 1 1 1 1 1 1 0 … 1

motor # 1 motor # 2 motor # 10

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Varying Platform Commonality with GA

Varying Platform Commonality with GA

• Add n commonality controlling genes to chromosomeThe length, L, of each chromosome in the GA is

determined by the number of design variables, n, and the number of products in the family, p:

L = n + np

...

Commonalitycontrolling genes

Design variablesfor Product 1

Design variablesfor Product p

0 1 0 0 ... 1 u11 c2 u31 u41… cn u1p c2 u3p u4p

… cn

• First n genes in the chromosome control the level of platform commonality: 0=unique, 1=common to family

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Product Family Penalty FunctionProduct Family Penalty Function

• Incorporate a Product Family Penalty Function (PFPF) as an additional objective function, which provides a surrogate for manufacturing cost savings

• PFPF was introduced by Martinez, Messac, & Simpson (2000) to minimize variability of design variables within a product family to promote commonality

pvarj is the percent variation of the jth design variable:

p

p1i

ijj

xx

j

j

xvarjpvar

n

1ijpvarPFPF :Min

p

p1i

2jij

j)1()xx(

varwhere:

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Step 1:Identify design

variables that could be made common

Step 2:Perform DOE to

check for possible reduction in design variables

Step 3:Identify reduced set of

design variables

Step 4:Make sample runsto determine GA

parameters

Step 7:Check constraint

violation and design feasibility

Step 8:Compute fitness values

for each designconfiguration

Step 5:Use GA to generate

design variable configurations

Step 6:Run simulation/synthesis

program for productfamily using GA

Manufacturingfeasibilityanalysis

Costanalysis

IdentifyBest

Design

Finalgen?

Yes

No

GA-Based Method for Product Family Design

GA-Based Method for Product Family Design

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Applying the GA-based Method to GAA Example

Applying the GA-based Method to GAA Example

• Step 1: Identify design variables that could be made common to the platform There are 8 design variables that define each motor:

x = (r, t, Aa, Na, Af, Nf, I, L)

• Step 2: Perform DOE to check for possible reduction in number of design variables Typically used if design variables are > 8-10 Not needed for motor example

• Step 3: Identify reduced set of design variables Not necessary for this motor example

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2.71

• Step 4: Setup GA for varying platform commonalityEach chromosome is 88 genes long (8 + 8*10)

Varying Platform Commonality in GAA Example

Varying Platform Commonality in GAA Example

Commonalitycontrolling genes

(0=unique, 1=common)

Design variablesfor 1st motor

1

These genes are treated as variables that can take values of {0,1} and are subject to

mutation and cross-over

These genes are treated as variables that can take values of {0,1} and are subject to

mutation and cross-over

These genes can take on any real value within each variable’s boundsThese genes can take on any real

value within each variable’s bounds1 1 1 1 1 0 0

7.15 750 0.28 120 0.25 3.32 0.95 2.71 7.15 750 0.28 120 0.25 4.56 3.21

Design variablesfor 10th motor

...

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Simulate Performance of GAA Families

Simulate Performance of GAA Families

• Step 5: Use GA to generate a population of solutions Create product family alternatives (chromosomes) using

selection, cross-over, and mutation We use NSGA-II algorithm from: <http://www.iitk.ac.in/kangal/>

• Step 6: Run simulation and/or analysis for each product in the family using GA generated design variables Developed a set of analytical equations to evaluate performance

of each motor: mass, efficiency, power, torque, etc.

• Step 7: Check each chromosome for constraint violation and design feasibility Each motor is checked against the set of constraints to ensure

that is feasible

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• Step 8: Compute the three “fitness” values for each motor family (chromosome) in the generation

Fitness Function 1 (to minimize) = Mi

Fitness Function 2 (to maximize) = i

Fitness Function 3 (to minimize) = pvarj

where:– Mi and i are summed over i = 1, …, 10

– pvarj is the % variation in the jth design variable, j = 1, …, 8

Compute Fitness and PFPFCompute Fitness and PFPF

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Result: Multiple Platforms and Multiple Families

Result: Multiple Platforms and Multiple Families

New challenge: which platform and family do we choose?

A: -NSGA-II families(Simpson, et al., 2005)

B: NSGA-II families(Simpson, et al., 2005)

C: Two-stage; radius scaled(Nayak, et al., 2002)

D: Single-stage; length scaled(Messac, et al., 2002)

E: Hierarchical sharing(Hernandez, et al., 2002)

F: Ant colony optimization(Kumar, et al., 2004)

G: Preference aggregation(Dai and Scott, 2004)

H: Sensitivity/cluster analysis (Dai and Scott, 2004)

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Generalizing Commonality and Scalability Issues

Generalizing Commonality and Scalability Issues

• Collaborating with Dr. Jeremy Michalek and Aida Khajavirad (CMU) to create an efficient and decomposable GA-based formulation that allows for partial commonality in a family

MOGA for 1MOGA for 1stst ProductProduct

MOGAMOGAfor Platform Selectionfor Platform Selection

Maximize Commonality

Minimize Sum of deviations from product targets received from Sub-GAs

With respect to Commonality chromosome

Minimize Deviation from 1st

product performance targets

Maximize Commonality

With respect to: 1st product design variables

Subject to: 1st productperformance constraints

MOGA for MOGA for ppthth ProductProduct

Minimize Deviation from pth

product performance targets

Maximize Commonality

With respect to: pth product design variables

Subject to: pth productperformance constraints

CommonalityCommonalityDeviationDeviation

DecomposableGA formulation

allows for parallelimplementation toimprove scalability

to large familiesof products

Source: (Khajavirad, et al., 2006)

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Chromosome Representations for Problem

Chromosome Representations for Problem

2

2

x11 x13 x14 x15

x33 x35 x41 x43 x44 x45

x23 x24 x25

Generalized commonality requires a 2D representation to define platform variable sharing and enforce design variable sharing among the variants

Product variants are represented using regular

chromosome coding

Source: (Khajavirad, et al., 2006)

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Sample ResultsSample Results

• Solutions from generalized commonality formulation dominate all of the all-or-none commonality solutions

Co

mm

on

alit

y

Performance0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.67 0.675 0.68 0.685

All-or-nonecommonality

Generalizedcommonality

Source: (Khajavirad, et al., 2006)

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A Valuable Lesson from the Motor Example

A Valuable Lesson from the Motor Example

• Optimization can provide a useful decision support tool for product family and product platform design In motor example, the resulting family should be scaled around

radius, not stack length, to achieve specified performance

• So why did B&D choose stack length? Manufacturing considerations and production costs dictated

decision: it was more economical to scale the motor along its stack length and wrap more wire around it than scale it radially

• Lesson: optimization can be useful for product family planning and strategic decision making, provided the right aspects are modeled for the individual products as well as the product family as a whole

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Ongoing and Future Research Directions

Ongoing and Future Research Directions

• Classification of product family optimization problems: Number of stages in optimization process Platform defined a priori or a posteriori Single or multiple objectives Type of optimization algorithm Number of products in the family and type of family Module and/or scale-based product family

( configuration and/or parametric variety)

• Create a product family optimization testbed (on web)

• Incorporate multiple disciplines (e.g., manufacturing, marketing) in product family optimization problems

• Approaches for designing multiple platforms in a family

• Extend to product portfolio assignment problems involving multiple families and multiple platforms

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Physical ProgrammingPhysical Programming

• Designer formulates the optimization problem in terms of physically meaningful parameters

[ ]

scn

iii

xxμPμP

1

)()( min

metrics) 4S class(for )(

metrics) 3S class(for )(

metrics) 2S class(for )(

metrics) 1S class(for )(

:Subject to

55

55

5

5

RiiLi

RiiLi

ii

ii

νxμν

νxμν

νxμ

νxμ

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© T. W. SIMPSON

Implementation of Physical Programming

Implementation of Physical Programming

• Designer enters physically meaning preferences• Numbers express desirability ranges

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• Showing all of these different objectives/ preferences gives a feel for what physical programming is capable of handling

• Number of objectives:2 motors: 12 objs.3 motors: 18 objs.5 motors: 30 objs.

10 motors: 60 objs.

Physical Programming Preferences for Motor Family

Physical Programming Preferences for Motor Family

Class

HD D T U HUith Objective

1ig 2ig 3ig 4ig 5ig

Mass - 1 1-S .20 .30 .40 .50 .60Efficiency - 1 2-S .85 .80 .75 .70 .65Mass - 2 1-S .25 .35 .45 .55 .65Efficiency - 2 2-S .80 .75 .70 .65 .60Mass - 3 1-S .30 .40 .50 .60 .70Efficiency - 3 2-S .80 .75 .70 .65 .60Mass - 4 1-S .30 .40 .50 .60 .70Efficiency - 4 2-S .80 .75 .70 .65 .60Mass - 5 1-S .30 .40 .50 .60 .70Efficiency - 5 2-S .75 .70 .65 .60 .55Mass - 6 1-S .35 .45 .55 .65 .75Efficiency - 6 2-S .75 .70 .65 .60 .55Mass - 7 1-S .45 .55 .65 .75 .85Efficiency - 7 2-S .75 .70 .65 .60 .55Mass - 8 1-S .45 .55 .65 .75 .85Efficiency - 8 2-S .70 .65 .60 .55 .50Mass - 9 1-S .55 .65 .75 .85 .95Efficiency - 9 2-S .65 .60 .55 .50 .45Mass - 10 1-S .60 .70 .80 .90 1.0Efficiency - 10 2-S .60 .55 .50 .45 .40

Unacceptable Acceptable

Mag Int. (1-10) 1-H - - - - 5000Feasibility (1-10) 2-H - - - - 1Power (1-10) 3-H - - - - 300Torque (1-10) 3-H - - - - varies

HU: Highly Undesirable, U: Undesirable, T: Tolerable, D: Desirable, HD: Highly Desirable