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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

© T. W. SIMPSON

Optimization 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

Motivating 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

Universal Motor Platform Example

- 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

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

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

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 LayoutsFlight Ranges for 737-500

Flight Ranges for 737-700

Flight Ranges for 737-600

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

Generic Form:

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

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

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

Challenges 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) variables

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

Product Platform Concept Exploration Method

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 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

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 ProblemGiven

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):
- 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

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

+3Y

Y

-3Y

Lower

Limit

6S

S

S

Two-Stage Optimization Approach in PPCEMStage 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)

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

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

+3T

T

-3T

Lower

Torque

Limit

6L

L

L

Optimization Problem for Motor FamilyStage 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

Group of individually designed motors

Resulting Product Family SpecificationsHigh

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

Comparison of Results: Individual Motors

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

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
- 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

9

8

7

6

5

4

3

2

1

Comparison of Results: Individual MotorsBenchmark Group

PPCEM (s=length)

1

10

PhysPro (s=length)

PhysPro (s=radius)

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

Comparison 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

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 common

2) the values that they should take

3) the values the remaining unique variables should take

Genetic 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

GA Terminology

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

Encoding - Decoding

Genotype

Phenotype

coded domain

decision domain

expression

UGCAACCGU

Biology

sequencing

(“DNA” blocks)

“blue eye”

decoding

Design

010010011110

H

encoding

(chromosome)

Radius R=2.57 [m]

x1

x2

xn

0 1 0 1 1 1 1 0 1 0 0 1 ….. 0 1

Radius

Height

Material

Basic Operation of a Genetic Algorithm

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

Genetic Operators: Selection

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

Genetic Operators: Selection

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

Genetic Operators: Crossover and Mutation

- 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

Genetic 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

Stopping Criteria

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

Using 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

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

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

Varying Platform Commonality with GA

- 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

Product 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:

where:

GA-Based Method for Product Family Design

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 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

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

Varying Platform Commonality in GAA 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

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

Compute Fitness and PFPF

- 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

Result: Multiple Platforms and Multiple Families

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?

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

Decomposable

GA formulation

allows for parallel

implementation toimprove scalability

to large families

of products

Source:

(Khajavirad, et al., 2006)

Chromosome Representations for Problem

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)

(Khajavirad, et al., 2006)

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

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

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

Physical Programming

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

Implementation of Physical Programming

- 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
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