Development of a family of products that satisfies different sectors of the market introduces significant challenges to today’s manufacturing industries – from development time to aftermarket services. A product family with a common platform paradigm offers a powerful solution to these daunting challenges. The Comprehensive Product Platform Planning (CP3) framework formulates a flexible product family model that (i) seeks to eliminate traditional boundaries between modular and scalable families, (ii) allows the formation of sub-families of products, and (iii) yield the optimal depth and number of platforms. In this paper, the CP3 framework introduces a solution strategy that obviates common assumptions; namely (i) the identification of platform/non-platform design variables and the determination of variable values are separate processes, and (ii) the cost reduction of creating product platforms is independent of the total number of each product manufactured. A new Cost Decay Function (CDF) is developed to approximate the reduction in cost with increasing commonalities among products, for a specified capacity of production. The Mixed Integer Non-Liner Programming (MINLP) problem, presented by the CP3 model, is solved using a novel Platform Segregating Mapping Function (PSMF). The proposed CP3 framework is implemented on a family of universal electric motors.

1. Comprehensive Product Platform
Planning (CP3)
Framework: Presenting a Generalized
Product Family
Souma Chowdhury, Achille Messac,
Rensselaer Polytechnic Institute
Department of Mechanical, Aerospace, and Nuclear Engineering
Multidisciplinary Design and Optimization Laboratory
and
Ritesh Khire
United Technologies Research Center

2. A guide to the next 20 minutes
 Brief overview of product family design methodologies
 Introduction to the Comprehensive Product Platform Planning (CP3) framework
 Mathematical representation of the CP3 model
 Key aspects of the CP3 optimization strategy
 Application of the CP3 framework to a family of Universal Electric Motors
2

3. Product Family
A typical product family consists of multiple products that share common features
embodied in a, so-called, platform, defined in terms of platform design variables.
3
Product Family Structure
GM Chevrolet Product Line
 Efficient product platform planning
generally leads to reduced overhead
that results in lower per product cost.
 Product family design relies on
quantitative optimization
methodologies.

4. Types of Product Families
In scale based product families two critical decisions are typically made:
• the selection of platform and scaling design variables (combinatorial)
• the determination of the values of these design variables (continuous)
The design process of module-based product family is conceptually divided
into the following three levels:
• Architectural level
• Configuration level
• Instantiation level
4

5. Comprehensive Product Platform Planning (CP3)
Objectives
• To develop an integrated mathematical model of the product platform
planning process.
• To avoid the typical design barriers between scalable and modular product
families.
• To develop a robust solution strategy that optimizes the product platform
model.
5

6. Earlier Product Platform Planning Methods
Scale based product families
6
Combinatorial
in nature
Continuous/Discrete
in nature
Select platform and
scaling design
variables
Determine optimal
values of platform and
scaling design variables
Step 1 Step 2
Platform/Scaling
Combination #1
(optimized)
Platform/Scaling
Combination #2n
(optimized)
Compare
all 2n
optimal
designs and
select
overall
optimal
Two-Step approach
This method is likely to introduce a
significant source of sub-optimality
Exhaustive approach
This method is expected to be
computationally prohibitive for
large scale systems

7. Earlier Product Platform Planning Methods…
Modular product families
7
Instantiation Level
Fixed module
combination
Predefined
module
candidates
Simultaneous optimization
of module attribute and
module combination
Do not readily apply to scalable
product families

8. Recent Product Platform Planning Method
Recent methods in scalable product family design such as Genetic Algorithm
based approaches, Selection Integrated Optimization approaches effectively
address the typical limitations of the earlier methods. However these
methods
• Assume that a platform is formed only when a design variable is common to
all products (the “all common/all distinct” restriction),
• Do not readily apply to both modular and scale-based product families,
• Assume that the cost reduction resulting from platform planning is
independent of the total number of each product manufactured, and
• Assume that the cost reduction resulting from platform planning is equally
sensitive to each design variable comprising the product.
8

9. Basic Components of the CP3 Framework
CP3 Model
• Formulates an integrated mathematical model yielding a MINLP* problem
• Seeks to eliminate distinctions between modular and scalable families
• Allows the formation of sub-families of products
CP3 Optimization
• Provides a robust solution to the MINLP problem
• Uses the Particle Swarm Optimization (PSO) algorithm
• Accounts for the effect of the number of each product manufactured on the
cost objective (cost of product family to be minimized)
9
*MINLP: Mixed Integer Non-Linear Programming

10. Physical Design Variable Product-1 Product-2 Integer Variables
1st variable
2nd variable
3rd variable
CP3 Model
The generalized CP3 model develops a MINLP problem. This is illustrated by a
2-product/3-variable product family.
10
 
f Y
PERFORMANCE
f Y
 
     
 
 
2 2 2
12 1 2 12 1 2 12 1 2
1 1 1 2 2 2 3 3 3
      
  
  
  1 1 1 2 2

1 2 3 1 2
 
   
Max
Min
s.t. 0
0, 1,2,....,
Design Constraints
0, 1,2,....,
, , , , ,
COST
i
i
x x x x x x
g X i p
h X i q
Y x x x x x
2
3 1 2 3
x
1 1 1 2 2 2
1 2 3 1 2 3
X 
x x x x x x
1 2 3
, , ,
, , , , ,
B B
, , : 0, 1
  
  
 
1 2 12
x x
if , then 0
if 1, then
 

 
j j j
12 1 2
x x

j j j
1
1x
1
2x
1
3x
2
1x
2
2x
2
3x
12
1
12
2
12
3
0
1

11. Commonality Constraint
11
      2 2 2
12 1 2 12 1 2 12 1 2
1 1 1 2 2 2 3 3 3  x  x  x  x  x  x  0
12 12 1
1 1 1
12 12 2
1 1 1
  0 0 0 0
  
   
 0 0 0 0
  
 0 0 12  12 0 0
  1

1 2 1 2 1 2 2 2 2
1 1 2 2 3 3 12 12 2
         2 2   2

 12  12   1

 3 3   3

 12 12 2
 3 3   3

0
0 0 0 0
0 0 0 0
0 0 0 0
x
x
x
x x x x x x
x
x
x
 
 
 
 
 
 
Commonality Constraint Matrix (Λ)

12. Generalized Commonality Constraint Matrix
k N
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
12
1 1
1 1
1
1
1 1
0 0 0 0 0 0
0 0 0 1 1
0 0 0
1
0 0 0 0 0 0
0 0 0 1
0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0 0
1 1
1
1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
k
N Nk
k N
k N
j j
k
N Nk
j j
k N
k N
n n
k
N Nk
n n
k N
 
 
 
 
 
 






 



 




 










number of products
number of design variables
N
n








 
 
 
 
 
 
 
 
 
 
 
 
 


 Corresponds to the jth design variable

13. Generalized Commonality Matrix
13
11 1
1 1
1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
1 1
0 0 0 0 0 0
0 0 0 11 1
0 0 0
0 0 0 0 0 0
0 0 0 1
0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0 0
11 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
1
N
N NN
N
j j
N NN
j j
N
n n
N NN
n n
k
j
 
 
 

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1 , iff =1 and
0 , otherwise
 product-kk

ll l k
1 , iff variable is included in product-
0 , iff variable is NOT included in l j j j j
k l
th

kk
j th
x x
j k
j k

  
 

 
Corresponds to the jth design variable

14. Platform: Definition and Demo.
“A product platform is said to be created when more than one product in a
family have the same magnitude of a particular design variable”
CP3 classifies design variables into: (1) platform, (2) sub-platform, and (3) non-platform
14
variables
1 1 1 1 1 0 0 0 1 1 0 0 1 0 0 0 1 0 0 1
1 1 1 1 0 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0
         
         
              
         
         
         
 ,  ,  ,  ,

1 2 3 4 5
1 1 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0
1 1 1 1 0 0 0 1 0 0 1 1 0 0 0 1 1 0 0 1

15. Product Family Cost Analysis
15
CF  CFD CFO
Net Product Family Cost Direct Cost Auxiliary Cost
    
 
C f X m diag f X m
 , , 
, ,

,
F FD FO
f X
c
 

m: Number of products manufactured (Capacity vector)

16. Nature of Cost Variation
16
Direct Cost
2
 
   
f  0 & f  0  k 
1, 2, ...,
N
m k FD ( m
k ) 2 FD  
  0
& : Auxiliary Cost per product
FO
FO
f



M
M 
 m f
Auxiliary Cost
Number of similar products manufactured

17. Generalized MINLP Problem
Performance objective
Cost objective
Commonality Constraint
  
17
 
 
Max
Min
s.t. 0
 
 
 
 0, 
1,2,....,
 0, 
1,2,....,
 


 
 
1 2 1 2 1 2
1 1 1
where
,
p
c
T
i
i
M
T
N N N
j j j n n n
f Y
f Y
X X
g X i p
h X i q
C
Y 
X
X x x x x x x x x x

18. CP3 Optimization: Cost Objective
Cost Decay Function (CDF)
• An increase in (i) the specified capacity of production m and/or (ii)
1
commonalities λ in the product family tend to reduce the cost of
manufacturing per product.
0.9
k)
j
k (CDF
j
• Hence the Cost Decay Function (CDF) that represents the variation of the
18
cost of manufacturing per product is defined as
  c

 2
    1
 1
 
  1

3 2
3
1
1
c
k k c
j c j
CDF M c c
c
M  m




0.8
0.7
 c1: coefficient that controls the rate of cost decrease per product
 c2: coefficient that provides the practical extent of this cost decrease
4 0.5
 c3: coefficient 0
that provides the maximum possible capacity of production
10
1
10
2
10
3
10
10
0.6
k (M
Number of products that share design variable x
j
k)
j
Cost Decay Function for variable x
c
1
= 0.1
c
1
= 0.2
c
1
= 0.3
c
1
= 0.4
c
1
= 0.5
c
1
= 0.6
c
1
= 0.7
c
1
= 0.8
c
1
= 0.9
c
1
= 1.0
c
2
= 0.5
c
3
= 104

19. CP3 Optimization: Commonality Constraint
Platform Segregating Mapping Function (PSMF)
• The commonality constraint can be reformulated as
• A continuous approximation of this expression is achieved using a set of
Gaussian probability distribution function for each design variable
19
XT X 
 2
 x k  x
l

1.0
0.8
kl j j
j
  
2 exp
 2


 j

)
 a
  PSMFX 
0.6
 1,


0.4
 
x
10
p b x
  
10

2 2ln10
1
10
a
0.2
• The full width at one-tenth maximum for each design variable is given by
 10  j 10 j
0.0
x  x x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Magnitude of jth design variable, x
j
Commonality variable (kl
j
product 1
product 2
product 3
product 4
product 5

20. Overall CP3 Optimization Strategy
N
usly optimize products using PSO (solve Eq. 30)
   
Npop istage istage
1.0
0.8
0.6
0.4
0.2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
20
Approximated MINLP problem
Pseudo-code
     
w f X w f X w
X X
g X i p
h X i q
1 1 1 Max 1 , 0.5
s.t.
p s
 
 
0, 1,2,....,
0, 1,2,....,
 
 
where
PSMF
T
i
i
M
C
X



   
 
 
 
1. Optimize each product using PSO (maximizing performance)
2. Determine the range for implementing PSO on each
3. Initiate a random population of size
 
 max
)
4. Set 10 10
& 1
5. Simultaneo
j x
Npop
x  x istage 
1
min 1
1 10
10 10 10 10
max
10
6. Set , where
7. Choose the optimal configuration as one of the starting point
Nstage
istage istage frac frac
x
x x x x
x

       
    
s
8. Initiate a random population of size -1, & set 1
9. If istage Nstage
go to step 5, else terminate solution
 

0.0
Magnitude of jth design variable, x
j
Commonality variable (kl
j
delx = 10.0
delx = 8.0
delx = 6.0
delx = 4.0
delx = 2.0
delx = 1.0
delx = 0.5
delx = 0.1

21. Constrained Particle Swarm Optimization (PSO)
21
Swarm Motion
t  1 t t

1
i i i
t t t t
i i l i i g g i
x x v
v  v  r p x  r p x
   
1
1 2

 
    
Constraint Dominance Principle
Solution-i is said to dominate solution-j if,
• solution-i is feasible and solution-j is infeasible or,
• both solutions are infeasible and solution-i has a smaller constraint violation
than solution-j or,
• both solutions are feasible and solution-i weakly dominates solution-j.

22. Test Problem: Universal Electric Motor
In this example, the objective is to develop a scale-based product family of
five universal electric motors that are required to satisfy different torque
requirements (Trq)
22
Motor 1 2 3 4 5
Torque N/m 0.1 0.2 0.3 0.4 0.5
Design Variable Lower Limit Upper Limit
Number of turns on the armature (Nc) 100 1500
Number of turns on each field pole (Ns) 1 500
Cross-sectional area of the armature wire (Awa) 0.01 mm2 1.00 mm2
Cross-sectional area of the field pole wire (Awf) 0.01 mm2 1.00 mm2
Radius of the motor (ro) 10.00 mm 100.00 mm
Thickness of the stator (t) 0.50 mm 10.00 mm
Stack length of the motor (L) 1.00 mm 100.00 mm
Current drawn by the motor (I) 0.1 Amp 6.0 Amp

23. Test Problem Optimization
Performance obj. Cost obj.
N N n
1 1
23
      1 1 Max 1
  
f 
f m
N Nn
T T k 1, 2, ...,
N
P k N
M k
w f X w f X
300 N/m 1, 2, ...,
2 kg 1, 2,
s.t.
p c
k k
rq
k
out
k
total
  
  
  
   ...,
5000 Amp.turns/m 1, 2, ..., Physi
0.15 1, 2, ...,
1 1, 2, ...,
k
k

k
o
k
N
H k N
k N
r
k N
t





  

   

   

  
X X

 
 
  
1 1 1
cal design constraints
where
Commonality constraint

PSMF
T
M
T
C s wa wf o
C
X

X N N A A r t L I

  
 
  
CDF
5, 7
k
p k c j
k k j
N n
 
    
 
 

24. CP3 Optimization Results
Three different cases are analyzed: classified by the number of each product
manufactured (capacity vector m)
Case 1: m10
Case 2: m100
Case 3: m10000
24
0.25
29
27
0.2
25
23
0.15
21
19
4 15
0.05
0
3
4 10
1
10
2
10
3
10
10
17
1
2
Capacity of production (mk)
Number of adaptive variables
0
10
10
10
10
0.1
Capacity of production (mkExtent of commonality (EC)

25. Concluding Remarks
 The CP3 technique provides a comprehensive mathematical model of the
platform planning process which is unique in the literature.
 The CP3 model accounts for certain aspects the instantiation level of modular
product families.
 The CP3 technique performs simultaneous selection of platform design
variables and optimization of design variable values
 The “all common/all distinct” restriction is avoided.
 The set of product platforms obtained is not necessarily independent
“specified number of products manufactured”.
25

26. Concluding Remarks
 The CP3 model formulates a generic MINLP problem.
 The Platform Segregating Mapping Function (PSMF) approximates the
MINLP problem into a continuous problem.
 A Cost Decay Function (CDF) approximates the cost per product attributed
to the total number of products that share a particular design variable.
FutureWork
 The solution of the exact MINLP problem, instead of a continuous
approximation is being pursued.
 A multi-objective scenario will also be investigated, to explore the trade-offs
between product performances and net cost reduction resulting from
platform planning.
 Further exploration of module-based product family applications will be
performed to establish the true potential of this new method.
26

27. References
1. http://www.chevrolet.com/, GM (Chevrolet) official website.
2. Simpson, T. W., and D'Souza, B. “Assessing variable levels of platform commonality within a
product family using a multiobjective genetic algorithm,” Concurrent Engineering: Research
and Applications, Vol. 12, No. 2, 2004, pp. 119-130.
3. Stone, R. B., Wood, K. L., and Crawford, R. H., “A heuristic method to identify modules from a
functional description of a product,” Design Studies, Vol. 21, No. 1, 2000, pp. 5-31.
4. Messac, A., Martinez, M. P., and Simpson, T. W., “Introduction of a Product Family Penalty
Function Using Physical Programming,” ASME Journal of Mechanical Design, Vol. 124, No. 2,
2002, pp. 164-172.
5. Khire, R. A., Messac, A., and Simpson, T. W., “Optimal design of product families using
Selection-Integrated Optimization (SIO) Methodology,” In: 11th AIAA/ISSMO Symposium on
Multidisciplinary Analysis and Optimization, Portsmouth, VA September 2006.
6. Khajavirad, A., Michalek, J. J., and Simpson, T. W., “An Efficient Decomposed Multiobjective
Genetic Algorithm for Solving the Joint Product Platform Selection and Product Family Design
Problem with Generalized Commonality,” Structural and Multidisciplinary Optimization, Vol.
39, No. 2, 2009, pp. 187-201.
7. Chen, C., and Wang, L. A., “Modified Genetic Algorithm for Product Family Optimization with
Platform Specified by Information Theoretical Approach,” J. Shanghai Jiaotong University
(Science), Vol. 13, No. 3, 2008, pp. 304–311.
27

28. References
8. Kennedy, J., and Eberhart, R. C., “Particle Swarm Optimization,” In Proceedings of the 1995
IEEE International Conference on Neural Networks, 1995, pp. 1942-1948.
9. Deb, K., Pratap, A., Agarwal, S., and Meyarivan, “T. A Fast and Elitist Multi-objective Genetic
Algorithm: NSGA-II,” IEEE Transactions on Evolutionary Computation, Vol 6, No. 2, April 2002,
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29. Thank you
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30. Questions
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