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: tws8@psu.edu
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
PENNSTATE
© T. W. SIMPSON
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
PENNSTATE
© T. W. SIMPSON
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

Watts
650
60
0.8”
Stack length
1.75”
Universal motor variants
PENNSTATE
© T. W. SIMPSON
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
PENNSTATE
© T. W. SIMPSON
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
PENNSTATE
© T. W. SIMPSON
Boeing 737 Interior Layouts
737-600
737-300
110 passengers (8 first class)
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)
PENNSTATE
737-900
177 passengers (12 first class)
© T. W. SIMPSON
Flight Ranges for 737-300, -500, -600, and -700
Capacity: 126 Passengers
Capacity: 110 Passengers
Flight Ranges for 737-300
Flight Ranges for 737-500
Flight Ranges for 737-700
Flight Ranges for 737-600
PENNSTATE
© T. W. SIMPSON
Dimensions of Boeing 737-300, -400, and -500
• All three aircraft share common height and width...
…but their fuselage lengths are different:
Boeing 737-300
PENNSTATE
Boeing 737-400
Boeing 737-500
© T. W. SIMPSON
Dimensions of Boeing 737-600, -700, -800, and -900
• The same holds true for the 737-600 through 900
Boeing 737-800
Boeing 737-600
Boeing 737-900
Boeing 737-700
PENNSTATE
© T. W. SIMPSON
Optimization for Single Product Design
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
PENNSTATE
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
© T. W. SIMPSON
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
PENNSTATE
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
© T. W. SIMPSON
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:
PENNSTATE
c = number of common (platform) variables
n = number of design variables for each of the p products
© T. W. SIMPSON
Product Platform Concept Exploration Method
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
Step 1
Create Market Segmentation Grid
Market
Segmentation
Grid
Step 2
Classify Factors and Ranges
Robust Design
Principles
Step 3
Simulation Analysis/Metamodels
Step 4
Metamodeling
Techniques
Aggregate Product Platform Specifications
Step 5
Develop Product Platform and Family
Multiobjective
Optimization
Product Platform and
Product Family Specifications
PENNSTATE
© T. W. SIMPSON
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
Scale down
Mid
Low
Scale up
High
Functional
• torque = fcn(motor stack length)
• thrust = fcn(# compressor stages)
Platform
Segment A Segment B Segment C
High
High-End Platform Leveraging
Conceptual/configurational
Mid
Low-End Platform Leveraging
Low
• # passengers on an aircraft
• size of an automobile underbody
Segment A Segment B Segment C
PENNSTATE
© T. W. SIMPSON
Compromise 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
Deviation Function
x2
Find
Xi
i = 1, …, n
di-, di+
Bounds
Constraints
Goals
i = 1, …, m
Satisfy
System 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
Bounds
Xjmin < Xj < Xjmin; j = 1, …, n
di-, di+ < 0 ; di- • di+ = 0 ; i
Feasible
Design
Space
x1
PENNSTATE
Aspiration
Space
Given
Assumptions to model domain of interest
Simulation and analyses to relate X and Y
= 1, …, m
Minimize
Deviation Function
Z = { f1(di-, di+), ..., fk(dk-, dk+) }
Reference: (Mistree, et al., 1993)
© T. W. SIMPSON
Platform Leveraging Strategy
Standardizing motor
interfaces will facilitate
horizontal leveraging
to new segments
Vertical Scaling
High Cost
High Performance
Mid-Range
Low Cost
Low Performance
Kitchen
Appliances
Power
Tools
Lawn &
Garden
Universal Motor Platform
(Common Design Variable Settings)
 Design a single motor platform scaled by stack length
PENNSTATE
© T. W. SIMPSON
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

thickness of motor, t
 on field:
– wire x-sectional area, AF
– number of wraps, NF

• Scaling variable (1/motor):

i = 1, …, 10
stack length, Li
• Constraints (6/motor) and Objectives (2/motor):
Name
Magnetizing Intensity, H
Feasible geometry
Power, P
Torque, T
Efficiency, h
Mass, M
PENNSTATE
Constraint
Hi =
 5000 Amp·turns/m
ro,i > ti
Pi = 300 W
Ti = {0.05, 0.1, 0.125, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.5} Nm
hi >= 0.15 (Target = 70%)
< 2.0 kg (Target = 0.5 kg)
Mi =
© T. W. SIMPSON
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)
Upper
Limit
Y
mY
Lower
Limit
Each line represents a
different product architecture,
i.e., a different combination of:
+3sY
-3sY
6sS
mS
S
[ 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 ]
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
PENNSTATE
© T. W. SIMPSON
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)
Upper
Torque
Limit
T
mT
Lower
Torque
Limit
Each line represents a
different product architecture,
i.e., a different combination of:
+3sT
-3sT
6sL
mL
L
[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]
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
PENNSTATE
© T. W. SIMPSON
Resulting Product Family Specifications
L [cm] T [Nm] h [%] M [kg]
High
0.50
47.9
0.83
1087, 72, 0.28, 0.25, 2.71, 7.15
3.16
0.50
55.3
0.99
2.40
0.40
53.1
0.82
1082, 72, 0.27, 0.24, 2.58, 6.67
2.87
0.40
57.7
0.84
2.33
0.35
55.9
0.80
1056, 73, 0.26, 0.24, 2.51, 6.46
2.81
0.35
59.8
0.78
2.21
0.30
58.8
0.78
1030, 73, 0.25, 0.23, 2.44, 6.35
2.74
0.30
62.2
0.71
2.04
0.25
61.8
0.73
1007, 73, 0.25, 0.22, 2.35, 6.17
2.61
0.25
64.9
0.64
1.81
0.20
65.1
0.68
988, 74, 0.24, 0.22, 2.26, 5.75
2.38
0.20
67.9
0.56
1.50
0.15
68.5
0.61
785, 95, 0.21, 0.21, 2.82, 8.88
1.63
0.15
70.5
0.50
1.32
0.13
70.3
0.56
760, 89, 0.19, 0.20, 3.12, 11.20
1.41
0.13
70.0
0.50
1.11
0.10
72.2
0.51
750, 76, 0.19, 0.20, 3.31, 11.77
1.28
0.10
70.6
0.50
0.62
0.05
76.0
0.40
730, 45, 0.20, 0.21, 3.62, 9.69
0.998
0.05
71.4
0.50
Platform instantiations
Universal Motor Platform
{Nc, Ns, Awa, Awf, r, t}
1273, 61, 0.27, 0.27, 2.67, 7.75
PENNSTATE
L [cm] T [Nm] h M [kg]
2.44
Mid
Low
Na , Nf, Af, Aa , r, t
Group of individually designed motors
Product platform obtained
using PPCEM
© T. W. SIMPSON
Comparison of Results: Individual Motors
Benchmark Group
PPCEM (s=length)
1
10
0.9
10
Mass (kg)
0.8
9
Desired
Efficiency
(> 70%)
9
8
7
6
8
0.7
7
5
6
0.6
4
3
5
0.5
0.4
4321
Desired Performance Region
(i.e., targets for mass and
efficiency are achieved)
Desired Mass
(< 0.5 kg)
0.3
40%
50%
60%
2
1
70%
80%
Efficiency
PENNSTATE
© T. W. SIMPSON
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
PENNSTATE
© T. W. SIMPSON
Universal Motor Family Design Problem II
• Design variables (8/motor):
stack length, Li
 radius of motor, ri
 on armature:
– wire x-sectional area, AA,i
– number of wraps, NA,i

i = 1, …, 10
current, Ii
 thickness of motor, ti
 on field:
– wire x-sectional area, AF,i
– number of wraps, NF,i

• Constraints (6/motor) and Objectives (2/motor):
Name
Magnetizing Intensity, H
Feasible geometry
Power, P
Torque, T
Efficiency, h
Mass, M
PENNSTATE
Constraint
Hi =
 5000 Amp·turns/m
ro,i > ti
Pi = 300 W
Ti = {0.05, 0.1, 0.125, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.5} Nm
hi >= 0.15 (Target = 70%)
< 2.0 kg (Target = 0.5 kg)
Mi =
© T. W. SIMPSON
Comparison of Results: Individual Motors
1
Benchmark Group
PPCEM (s=length)
PhysPro (s=length)
PhysPro (s=radius)
10
0.9
10
Mass (kg)
0.8
10
9
8 7
0.7
6
7 8
8
10
Desired
Efficiency
(> 70%)
9
9
9
8
6
7
7
5
0.6
5
6
6
4
5
5
3
4
3
2
4 2
4321
0.5
3
0.4
Desired Performance Region
(i.e., targets for mass and
efficiency are achieved)
Desired Mass
(< 0.5 kg)
0.3
40%
50%
60%
2
1
1
1
70%
80%
Efficiency
PENNSTATE
© T. W. SIMPSON
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
PENNSTATE
© T. W. SIMPSON
Varying Platform Commonality
• Ideally, an optimization algorithm would search all
possible product platform combinations:
 n  n 
 n  n  n
# platform alternatives     
             2n
 n   n - 1
 2  1 0 
where:
 n  the number of possible combinations of making n design
 
 c  variables common to platform c at a time
 n  the null platform, i.e., no commonality within the family
 
0
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
PENNSTATE
© T. W. SIMPSON
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

PENNSTATE
© T. W. SIMPSON
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
PENNSTATE
Generation k+1
© T. W. SIMPSON
Encoding - Decoding
Genotype
coded domain
Biology
UGCAACCGU
(“DNA” blocks)
Phenotype
decision domain
expression
sequencing
“blue eye”
Design
010010011110
(chromosome)
x1
x2
decoding
encoding
xn
H
Radius R=2.57 [m]
0 1 0 1 1 1 1 0 1 0 0 1 ….. 0 1
Radius
PENNSTATE
Height
Material
© T. W. SIMPSON
Basic Operation of a Genetic Algorithm
Initialize Population (initialization)
next generation
Select individual for mating (selection)
Mate individuals and produce children (crossover)
Mutate children (mutation)
Insert children into population (insertion)
n
Are stopping criteria satisfied?
y
Finish
PENNSTATE
Reference:
Goldberg, D.E., 1989, Genetic Algorithms in Search,
Optimization and Machine Learning, Addison Wesley
© T. W. SIMPSON
Genetic Operators: Selection
Roulette Wheel Selection
1
2
6
3
5
4
Probabilistically select
individuals based on some
measure of their
performance.
Sum
Sum of individual’s
selection probabilities
3rd individual in current
population mapped to interval
[0,Sum]
• Selection: generate random number in [0,Sum]
• Repeat process until desired # of individuals areselected
• Basically: stochastic sampling with replacement
PENNSTATE
© T. W. SIMPSON
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
PENNSTATE
Survivors
101010110111
001000111110
101010110111
Fitness
8
6
8
© T. W. SIMPSON
Genetic Operators: Crossover and Mutation
• Crossover takes 2 solutions and creates 1 or 2 more
crossover
point
Classical: single point crossover
P1
0 1 1 0 1
0 1 1 1 1
O1
P2
1 0 0 1 1
1 0 0 0 1
O2
The parents
The children
(“offspring”)
• 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
PENNSTATE
 1 0 1 0 1
© T. W. SIMPSON
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
PENNSTATE
© T. W. SIMPSON
Stopping Criteria
Average fitness
Typical convergence
Global
optimum
(unknown)
Converged too
fast (mutation rate
too small?)
Generation
• 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

PENNSTATE
© T. W. SIMPSON
Using GAs in Product Family Design
• Chromosomes typically represent a single product:
0 1 0 1 1 0
…
1
= one motor
• For product family design, one can use multiple
chromosomes to represent the products in the family:
0 1 0 1 0 0
…
0 1 1 0 1 0
0 = motor # 1
…
1 1 0 1 1 0
0 = motor # 2
…
0 = motor # 3
1 1 1 1 1 0
…
0 1 1 1 1 1
1 = motor # 8
…
1 1 1 1 1 1
1 = motor # 9
…
1 = motor # 10
• 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

PENNSTATE
© T. W. SIMPSON
Using GAs in Product Family Design (cont.)
• Alternatively, you can extend a single chromosome to
represent the entire product family:
0 1 0 1 0 0
motor # 1
…
0 1 0 0 1 0 0
motor # 2
…
1
…
1 1 1 1 1 0
…
1
motor # 10
• 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?

PENNSTATE
© T. W. SIMPSON
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
Commonality
controlling genes
Design variables
for Product 1
... u
1p
c2 u3p u4p … cn
Design variables
for Product p
• First n genes in the chromosome control the level of
platform commonality: 0=unique, 1=common to family
PENNSTATE
© T. W. SIMPSON
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
n
Min : PFPF   pvar j
i 1
pvarj is the percent variation of the jth design variable:
varj
pvar j 
xj
PENNSTATE
p
xij
where: xj  
p
i 1
( xij  xj) 2
varj  
( p  1)
i 1
p
© T. W. SIMPSON
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 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
Step 7:
Check constraint
violation and
design feasibility
PENNSTATE
Step 3:
Identify reduced set of
design variables
Cost
analysis
Step 8:
Compute fitness values
for each design
configuration
No
Final
gen?
Yes
Identify
Best
Design
© T. W. SIMPSON
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
PENNSTATE
© T. W. SIMPSON
Varying Platform Commonality in GAA Example
• Step 4: Setup GA for varying platform commonality
 Each chromosome is 88 genes long (8 + 8*10)
Commonality
controlling genes
(0=unique, 1=common)
1
1
1
1
1
1
0
These genes are treated as variables that
can take values of {0,1} and are subject to
mutation and cross-over
0
These genes can take on any real
value within each variable’s bounds
2.71 7.15 750 0.28 120 0.25 3.32 0.95
Design variables
for 1st motor
PENNSTATE
...
2.71 7.15 750 0.28 120 0.25 4.56 3.21
Design variables
for 10th motor
© T. W. SIMPSON
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
PENNSTATE
© T. W. SIMPSON
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
PENNSTATE
© T. W. SIMPSON
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?
PENNSTATE
© T. W. SIMPSON
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 Platform Selection
Maximize Commonality
Minimize Sum of deviations
from product targets received from
Sub-GAs
With respect to Commonality
chromosome
Commonality
Deviation
Deviation
Commonality
MOGA for 1st Product
MOGA for pth Product
Minimize Deviation from 1st
product performance targets
Minimize Deviation from pth
product performance targets
Maximize Commonality
Maximize Commonality
With respect to:
1st product design
variables
With respect to:
pth product design
variables
Subject to: 1st product
performance constraints
Subject to: pth product
performance constraints
PENNSTATE
Decomposable
GA formulation
allows for parallel
implementation to
improve scalability
to large families
of products
Source:
(Khajavirad, et al., 2006)
© T. W. SIMPSON
Chromosome Representations for Problem
Generalized commonality requires a
2D representation to define platform
variable sharing and enforce design
variable sharing among the variants
2
Product variants are
represented using regular
chromosome coding
2
x11
x13 x14 x15
x33
PENNSTATE
x23 x24 x25
x35
x41
x43 x44 x45
Source:
(Khajavirad, et al., 2006)
© T. W. SIMPSON
Sample Results
Source:
(Khajavirad, et al., 2006)
• Solutions from generalized commonality formulation
dominate all of the all-or-none commonality solutions
1.0
Generalized
commonality
0.9
Commonality
0.8
0.7
0.6
All-or-none
commonality
0.5
0.4
0.3
0.2
Performance
0.67
PENNSTATE
0.675
0.68
0.685
© T. W. SIMPSON
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
PENNSTATE
© T. W. SIMPSON
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
PENNSTATE
© T. W. SIMPSON
PENNSTATE
© T. W. SIMPSON
Physical Programming
• Designer formulates the
optimization problem in
terms of physically
meaningful parameters
nsc
min P( μ)   Pi μi ( x)
x
i 1
Subject to :
μi ( x)  νi 5
(for class 1S metrics)
μi ( x)  νi 5
(for class 2S metrics)
νi 5 L  μi ( x)  νi 5 R
(for class 3S metrics)
νi 5 L  μi ( x)  νi 5 R
(for class 4S metrics)
PENNSTATE
© T. W. SIMPSON
Implementation of Physical Programming
• Designer enters physically meaning preferences
• Numbers express desirability ranges
PENNSTATE
© T. W. SIMPSON
Physical Programming Preferences for Motor Family
ith Objective
Class
HD


D


gi1
Mass - 1
Efficiency - 1
Mass - 2
Efficiency - 2
Mass - 3
Efficiency - 3
Mass - 4
Efficiency - 4
Mass - 5
Efficiency - 5
Mass - 6
Efficiency - 6
Mass - 7
Efficiency - 7
Mass - 8
Efficiency - 8
Mass - 9
Efficiency - 9
Mass - 10
Efficiency - 10
Mag Int. (1-10)
Feasibility (1-10)
Power (1-10)
Torque (1-10)
1-S
2-S
1-S
2-S
1-S
2-S
1-S
2-S
1-S
2-S
1-S
2-S
1-S
2-S
1-S
2-S
1-S
2-S
1-S
2-S
1-H
2-H
3-H
3-H
T


gi 2
.20
.85
.25
.80
.30
.80
.30
.80
.30
.75
.35
.75
.45
.75
.45
.70
.55
.65
.60
.60
-
.30
.80
.35
.75
.40
.75
.40
.75
.40
.70
.45
.70
.55
.70
.55
.65
.65
.60
.70
.55
Unacceptable
-
U


gi 3
HU


gi 4
gi 5
.40
.75
.45
.70
.50
.70
.50
.70
.50
.65
.55
.65
.65
.65
.65
.60
.75
.55
.80
.50
.50
.70
.55
.65
.60
.65
.60
.65
.60
.60
.65
.60
.75
.60
.75
.55
.85
.50
.90
.45
-
-
.60
.65
.65
.60
.70
.60
.70
.60
.70
.55
.75
.55
.85
.55
.85
.50
.95
.45
1.0
.40
Acceptable
5000
1
300
varies
HU: Highly Undesirable, U: Undesirable, T: Tolerable, D: Desirable, HD: Highly Desirable
PENNSTATE
• 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.
© T. W. SIMPSON