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Optimization Approaches for Product Family Design. ME 546 - Designing Product Families - IE 546. Timothy W. Simpson Professor of Mechanical & Industrial Engineering and Engineering Design The Pennsylvania State University University Park, PA 16802 USA phone: (814) 863-7136

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Optimization Approaches for Product Family Design

ME 546 - Designing Product Families - IE 546

Timothy W. Simpson

Professor of Mechanical & Industrial

Engineering and Engineering Design

The Pennsylvania State University

University Park, PA 16802 USA

phone: (814) 863-7136

email: [email protected]

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

PENNSTATE

• 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

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

• 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

Electric motor field components

prior to standardization

650

Watts

60

0.8”

Stack length

1.75”

Universal motor variants

• 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

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)

110 passengers (8 first class)

737-300

126 passengers (8 first class)

737-700

126 passengers (8 first class)

737-400

147 passengers (10 first class)

737-800

162 passengers (12 first class)

737-500

110 passengers (8 first class)

737-900

177 passengers (12 first class)

Boeing 737 Interior Layouts

Flight Ranges for 737-500

Flight Ranges for 737-700

Flight Ranges for 737-600

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

Capacity: 126 Passengers

Capacity: 110 Passengers

Boeing 737-400

Boeing 737-500

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

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

…but their fuselage lengths are different:

Boeing 737-600

Boeing 737-900

Boeing 737-700

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

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

Find: x

Minimize: f(x)

Subject to: g(x) < 0

h(x) = 0

Definitions:

x = design variables

f(x) = objective function

g(x) = inequality constraints

h(x) = equality constraints

Optimization for Single Product Design

• For Motor Example:

• Find: r, t, AA, NA,

• AF, NF, I, L

• Minimize: Mass

• Maximize: Efficiency, h

• Subject to: MagInt, H < 5000

• Mass < 2 kg

• Eff, h> 70 %

• r > t

• Power = 300 W

• Torque = 0.5 Nm

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

Optimization for Product Family Design

• 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, hi> 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

• 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) variables

n = number of design variables for each of the p products

Overall Design Requirements

Market

Segmentation

Grid

Step 1

Create Market Segmentation Grid

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

Step 2

Classify Factors and Ranges

Robust Design

Principles

Step 3

Simulation Analysis/Metamodels

Metamodeling

Techniques

Step 4

Aggregate Product Platform Specifications

Step 5

Develop Product Platform and Family

Multiobjective

Optimization

Product Platform and

Product Family Specifications

• 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

Example Scaling Variables

Platform

High

Functional

• torque = fcn(motor stack length)

• thrust = fcn(# compressor stages)

Scale up

Mid

Scale down

Low

Platform

Segment A

Segment B

Segment C

High-End Platform Leveraging

High

Conceptual/configurational

• # passengers on an aircraft

• size of an automobile underbody

Mid

Low-End Platform Leveraging

Low

Segment A

Segment B

Segment C

Space

Deviation Function

Feasible

Design

Space

x2

x1

Bounds

Constraints

Goals

Compromise Decision Support Problem

Given

Assumptions to model domain of interest

Simulation and analyses to relate X and Y

Find Xii = 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

Systemgoals (linear, nonlinear)

Ai(X) + di- + di+ = Gi ; i = 1, …, m

Bounds Xjmin< Xj< Xjmin; j = 1, …, n

di-, di+< 0 ; di- • di+ = 0 ; i = 1, …, m

Minimize

Deviation Function

Z = { f1(di-, di+), ..., fk(dk-, dk+) }

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

Reference: (Mistree, et al., 1993)

High Performance

Mid-Range

Vertical Scaling

Low Cost

Low Performance

Kitchen

Appliances

Power

Tools

Lawn &

Garden

Kitchen

Appliances

Lawn &

Garden

Universal Motor Platform

(Common Design Variable Settings)

Platform Leveraging Strategy

 Design a single motor platform scaled by stack length

Standardizing motor

interfaces will facilitate

horizontal leveraging

to new segments

>

<

Electric Motor Family Design Problem I

• Platform parameters (common to all motors):

• 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

different product architecture,

i.e., a different combination of:

[ x1, x2, x3, …., xn-1, ms, ss ]

[ x1, x2, x3, …., xn-1, ms, ss ]

[ x1, x2, x3, …., xn-1, ms, ss]

[ x1, x2, x3, …., xn-1, ms, ss]

[ x1, x2, x3, …., xn-1, ms, ss]

Y

Upper

Limit

+3Y

Y

-3Y

Lower

Limit

6S

S

S

Two-Stage Optimization Approach in PPCEM

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 (ms, ss)

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

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]

T

Upper

Torque

Limit

+3T

T

-3T

Lower

Torque

Limit

6L

L

L

Optimization Problem for Motor Family

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 (mstack_length, sstack_length)

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

Resulting Product Family Specifications

High

Mid

Low

Platform instantiations

Universal Motor Platform

Product platform obtainedusing PPCEM

{Nc, Ns, Awa, Awf, r, t}

1273, 61, 0.27, 0.27, 2.67, 7.75

Benchmark Group

PPCEM (s=length)

1

10

0.9

Desired

Efficiency

(> 70%)

9

9

8

10

0.8

7

6

8

7

5

0.7

Mass (kg)

6

4

0.6

3

5

2

0.5

4321

Desired Performance Region

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

Desired Mass

(< 0.5 kg)

0.4

1

0.3

40%

50%

60%

70%

80%

Efficiency

Single-Stage Optimization Approach

Optimize 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 family

n = # design variables per product in the family

s = # scaling variables per product in the family

c = # common platform variables (n = c + s)

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

>

<

Universal Motor Family Design Problem II

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

• stack length, Li

• 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

9

8

7

6

5

4

3

2

1

Comparison of Results: Individual Motors

Benchmark Group

PPCEM (s=length)

1

10

PhysPro (s=length)

0.9

Desired

Efficiency

(> 70%)

9

9

10

9

10

8

0.8

7

6

7

8

8

6

7

5

0.7

5

Mass (kg)

4

6

3

4

0.6

2

3

5

2

0.5

4321

Desired Performance Region

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

Desired Mass

(< 0.5 kg)

0.4

1

1

0.3

40%

50%

60%

70%

80%

Efficiency

• 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

• 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 common

2) the values that they should take

3) the values the remaining unique variables should take

• 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

Chromosome

alleles

gene

0 1 0 1 1 1 1 0 1 0 0 1 ….. 0 1

Population

Individuals

Selection

Crossover

Mutation

Insertion

Genetic

operators

Generation k

Generation k+1

Genotype

Phenotype

coded domain

decision domain

expression

UGCAACCGU

Biology

sequencing

(“DNA” blocks)

“blue eye”

decoding

Design

010010011110

H

encoding

(chromosome)

x1

x2

xn

0 1 0 1 1 1 1 0 1 0 0 1 ….. 0 1

Height

Material

Initialize Population (initialization)

Select individual for mating (selection)

Mate individuals and produce children (crossover)

next generation

Mutate children (mutation)

Insert children into population (insertion)

Are stopping criteria satisfied?

n

y

Finish

Reference:

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

Roulette Wheel Selection

Probabilistically select individuals based on some measure of their performance.

1

2

6

Sum of individual’s

selection probabilities

Sum

3

3rd individual in current

population mapped to interval

[0,Sum]

5

4

• Selection: generate random number in [0,Sum]

• Repeat process until desired # of individuals areselected

• Basically: stochastic sampling with replacement

Tournament Selection

Dominant performer

placed in intermediate

population of survivors

2 members of current

population chosen randomly

n

Population

Filled ?

y

Crossover and

Mutation form new

population

Old Population Fitness

101010110111 8

100100001100 4

001000111110 6

Survivors Fitness

101010110111 8

001000111110 6

101010110111 8

• Crossover takes 2 solutions and creates 1 or 2 more

crossover

point

Classical: single point crossover

O1

P1

0 1 1 1 1

0 1 1 0 1

O2

P2

1 0 0 1 1

1 0 0 0 1

The children

(“offspring”)

The parents

• 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

• 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

Typical convergence

Global

optimum

(unknown)

• 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 sk< e (i.e., genetic diversity has become small)

• Stagnation - no or marginal improvement from one generation to the next: (Fn+1 - Fn)< e

Average fitness

Converged too

fast (mutation rate

too small?)

Generation

• Chromosomes typically represent a single product:

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

• 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

0 1 0 1 0 0 … 0

= motor # 1

1 1 1 1 1 0 … 1

= motor # 8

0 1 1 0 1 0 … 0

= motor # 2

0 1 1 1 1 1 … 1

= motor # 9

= motor # 3

= motor # 10

1 1 0 1 1 0 … 0

1 1 1 1 1 1 … 1

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

• 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

• Add n commonality controlling genes to chromosome

• The 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

...

...

0

1

0

0

1

u11

c2

u31

u41

cn

u1p

c2

u3p

u4p

cn

Commonality

controlling genes

Design variables

for Product 1

Design variables

for Product p

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

• 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:

where:

Step 1:

Identify design

variables that could

Step 2:

Perform DOE to

check for possible reduction

in design variables

Step 3:

Identify reduced set of

design variables

Step 4:

Make sample runs

to determine GA

parameters

Step 5:

Use GA to generate

design variable

configurations

Step 6:

Run simulation/synthesis

program for product

family using GA

Manufacturing

feasibility

analysis

Cost

analysis

No

Step 8:

Compute fitness values

for each design

configuration

Step 7:

Check constraint

violation and

design feasibility

Identify

Best

Design

Final

gen?

Yes

• 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

• Step 4: Setup GA for varying platform commonality

• Each chromosome is 88 genes long (8 + 8*10)

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

Commonality

controlling genes

(0=unique, 1=common)

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

1

1

1

1

1

1

0

0

...

2.71

120

0.25

3.32

2.71

120

0.25

4.56

7.15

750

0.28

0.95

7.15

750

0.28

3.21

Design variables

for 1st motor

Design variables

for 10th motor

• 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

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

• Fitness Function 1 (to minimize) = SMi

• Fitness Function 2 (to maximize) = Shi

• Fitness Function 3 (to minimize) = Spvarj

where:

• Mi and hi are summed over i = 1, …, 10

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

A: e-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)

New challenge: which platform and family do we choose?

• 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

Decomposable

GA formulation

allows for parallel

implementation toimprove scalability

to large families

of products

Source:

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:

Sample Results

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

1.0

Generalized

commonality

0.9

0.8

0.7

Commonality

0.6

All-or-none

commonality

0.5

0.4

0.3

0.2

Performance

0.67

0.675

0.68

0.685

• 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

• 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

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

• Designer enters physically meaning preferences

• Numbers express desirability ranges

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