A product family with a common platform paradigm can increase the flexibility and responsiveness of the product- manufacturing process and help take away market share from competitors that develop one product at a time. The recently developed Comprehensive Product Platform Planning (CP3 ) method allows (i) the formation of sub-families of products, and (ii) the simultaneous identification and quantification of platform/scaling design variables. The CP3 model is founded on a generalized commonality matrix representation of the product-platform-plan. In this paper, a new commonality index is developed and introduced in CP3 to simultaneously account for the degree of inter-product commonalities and for the overlap between groups of products sharing different platform variables. To maximize both the performance of the product family and the new commonality measure, we develop and apply an advanced mixed-discrete Particle Swarm Optimization (MDPSO) algorithm. In the MDPSO algo- rithm, the discrete variables are updated using a deterministic nearest-feasible-vertex criterion after each iteration of the conventional PSO. Such an approach is expected to avoid the undesirable discrepancy in the rate of evolution of discrete and continuous variables. To prevent a premature stagnation of solutions (likely in conventional PSO), while solving the high dimensional MINLP problem presented by CP3, we introduce a new adaptive diversity-preservation technique. This technique first characterizes the population diversity and then applies a stochastic update of the discrete variables based on the estimated diversity measure. The potential of the new CP3 optimization methodology is illustrated through its application to design a family of universal electric motors. The optimized platform plans provide helpful insights into the importance of accounting for the overlap between different product platforms, when quantifying the effective commonality in the product family.

1. Comprehensive Product Platform Planning (CP3)
Using Mixed-Discrete Particle Swarm Optimization
and a New Commonality Index
Souma Chowdhury*, Achille Messac#, and Ritesh Khire**
* Rensselaer Polytechnic Institute, Department of Mechanical, Aerospace, and Nuclear Engineering
# Syracuse University, Department of Mechanical and Aerospace Engineering
** United Technologies Research Center
ASME 2012 International Design Engineering Technical Conference
August 12-15, 2012
Chicago, IL

2. Product Family Design
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.
 Efficient product platform planning generally leads to
reduced overhead that results in lower per product cost.
 Product family design relies on quantitative
optimization methodologies.
GM Chevrolet Product Line*
* GM (Chevrolet) official website 2

3. Product Family Design Methods
 A significant number of product family design methods have appeared in
the literature, which can be classified according to*:
• Suitable for module-based and/or scale based families
• One-stage or two-stage methods (specifies platform a priori)
• Single or multi-objective
• Models market demand and/or manufacturing costs
• Considers uncertainty or not
• Type of optimization (gradient-based, heuristic population based)
 Desirable Attributes:
1. Adequately quantifies and considers important objectives
2. Addresses a wide range of product types/classes
3. Formulates a tractable optimization problem and effectively solve it
*Product Platform and Product Family Design, Springer, 2006 3

4. Diverse Commonality Objectives
Existing Commonality Objectives
 Penalty Function: Accounts for the difference in the values of each part
(design variable) between different products#
 Commonality Index: Accounts for the ratio of the actual number of
unique parts to the maximum number of unique parts possible*
 Martin and Ishii (1996) pointed out that delaying the branching and the
sub-branching of the product variation graph is preferable from a
commonality perspective.
 “The sharing of multiple parts in the same group of products (sub-family)”
is thereby more helpful than “the sharing of one part in one
group of products and another part in a different group of products”.
*Martin and Ishii 1996, Khajavirad and Michalek 2008; 4 # Khire et al. 2006

5. Presentation Topics
Comprehensive Product Platform Planning
Novel Commonality Index
Mixed-Discrete Particle Swarm Optimization
Case Study: Family of Electric Motors
5

6. Presentation Outline
Comprehensive Product Platform Planning
Novel Commonality Index
Mixed-Discrete Particle Swarm Optimization
Case Study: Family of Electric Motors
6

7. Comprehensive Product Platform Planning (CP3) Model
The generalized CP3 model develops a MINLP problem. This is illustrated by
a 2-product/3-variable product family.
Physical Design Variable Product-1 Product-2 Binary Variables
1st variable
2nd variable
3rd variable
 
 
f X
f X
Max
performance
Max ,
commonality
2 2 2
     
 
 
12 1 2 12 1 2 12 1 2
1 1 1 2 2 2 3 3 3
s.t. 0
0, 1,2,....,
1 1 1
1 2 3
Design Constraints
0, 1,2,....,
, , ,
i
i
x x x x x x
g X i p
h X i q
X x x x

      
  
  
  
 
2 2 2
1 2 3
x x x
 
12 12 12
1 2 3
 
  
12 12 12
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
7

8. Generalized Commonality Matrix
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*
* Chowdhury et al. 2010, Khajavirad and Michalek 2008 8

9. Converting the Combinatorial Problem into a Tractable
The upper off-diagonal elements of each block of the commonality matrix are aggregated
into a binary string, which yields an integer variable – e.g., in a 4-product family:
1 1 0 0
1 1 0 0
0 0 1 1
0 0 1 1
j 
Integer Problem
 
 
  
 
 
 
1 0 0 0 0 1
33 j z 
For a family of N kinds of products and a set of n design variables:
 String length = ; Set of allowed of integer variables:
N N 1 2  1 2 1 0,2N N z    
 
 Now, some combinations can be known to be infeasible/undesirable and the
corresponding integer values can be removed from the set of allowed values
*BIP: Binary Integer Programming; IP: Integer Programming 9

10. MINLP Problem Definition
Performance objective
 
 
Max
Max
s.t. 0
 
 
 
 0, 
1,2,....,
 0, 
1,2,....,
 

 
 


 
Commonality Objective
Commonality Constraint
 2
1 2 1 2 1
1 1 1
where
, Z
p
c
T
i
i
con

j bi j
N N
 j j j n

f Y
f Y
X X
g X i p
h X i q
f
f z
Y X
X x x x x x x x x
   ,

1 2 feas    
, 1,2, , ; 1,2, ,
T
N
n n
j n j
x
Z z z z z z Z
k l N j n
Allowed set of values for
each integer variable
10

11. Presentation Topics
Comprehensive Product Platform Planning
Novel Commonality Index
Mixed-Discrete Particle Swarm Optimization
Case Study: Family of Electric Motors
11

12. Attributes of Commonality (Family of Glassware)
12
body diameter (D)
body height (H)
body
base
base diameter (B)
D2
D1 D2
H2
H1 H1
B1 B1 B2
LARGE MEDIUM SMALL
It would have been more beneficial if the base (B) was shared
between the small and the medium glassware, while the
commonality index (CI) won’t change.

13. Commonality Index (CI)
Standard Commonality Index (CI):
In terms of the commonality matrix (for scaling families):
N: no. of product types; n: number of parts in each scaling product
u: actual number of unique parts in the family
nk: Number of parts in the kth product
R: Rank of the commonality matrix
13

14. Cross-Commonality Index
 The degree of similarity between two commonality matrix blocks, i and
j, provides an effective representation of the overlap between the
platforms corresponding to the ith and the jth parts.
Ri: Rank of the ith block (i) of the commonality matrix
Ri j: Rank of the element-by-element product of i and j
 Since , we can simplify:
Platform Variation
Products A, B, C shares part 1, and products C, D, E shares part 2 – Maximum platform variation
Products A, B, C shares part 1, and products A, B, C shares part 2 – No/Minimum platform variation
14

15. Cross-Commonality Index (CCI)
 CCI is weighted combination where
 Both the 2nd and the 3rd term promotes product variation to occur further
upstream in the product differentiation chain.
 In the case of scale-based product family (all products comprise n parts):
15
Product variation with respect to
the product-parts/components
Platform variation with respect to
their product memberships
 0,1

16. Comparing CI and CCI
Family of 3 products, each comprising of 2 parts
16
Commonality Index (CI) Cross-Commonality Index (CCI)
z1 and z2 are the integer commonality variables representing the
platform-plans for parts 1 and 2, respectively.
Except for z1 = z2 platform combinations, the CCI values are
lower than the CI values for any given platform combination.

17. Presentation Topics
Comprehensive Product Platform Planning
Novel Commonality Index
Mixed-Discrete Particle Swarm Optimization
Case Study: Family of Electric Motors
17

18. Basic PSO Dynamics
Location Update:
Velocity Update:
Inertia Personal/Exploitive
Behavior
Social/Explorative
Behavior
Diversity Preserving
Behavior
18

19. Diversity Preservation (Discrete Variables)
 Performed through the modification of the otherwise deterministic update
process (i.e. updating the discrete component of the design vector to the
nearest allowed discrete point).
 A stochastic update process is introduced to help particles (discrete
component) jump out of the local hypercube.
r bound of the local cell
th
r
r
If 

, use the nearest neighboring vertex update
4 d
,
i
If 

, update randomly to the upper or lowe
d i
4 ,
r : random number between 0 and 1;

: diversity coefficient for i
d
d
i
iscrete variable
4 ,
 Separate diversity coefficients for each discrete variable-i
 Owing to different numbers of available feasible values
 Owing to different distribution of the feasible values
푥푖 ∈ 1,10,100
푥푗 ∈ 1,2, … , 100
e.g.
19

20. Measure of Diversity (Discrete Variables)
 Diversity Metric: Separate diversity metric for each discrete variable-i
 Diversity Coefficient: Defined as
a monotonically decreasing
function of the discrete variable
diversity metric
Mi represents the size of the set of allowed values for the ith discrete variable
20

21. Presentation Topics
Comprehensive Product Platform Planning
Novel Commonality Index
Mixed-Discrete Particle Swarm Optimization
Case Study: Family of Electric Motors
21

22. Test Application: Universal Electric Motor
In this example, the objective is to develop a scale-based product families of
2, 4 and 6 universal electric motors that are required to satisfy different
torque requirements
Motor 1 2 3 4 5 6
Torque
N/m
0.05 0.1 0.125 0.15 0.20 0.25
Design Variable Lower
22
Limit
Upper
Limit
Number of turns on the armature (Nc) 100 1500
Number of turns on each field pole (Ns) 1 500
Cross-sect. area of the armature wire (Awa) 0.01 mm2 1.00 mm2
Cross-sect. 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. Case Study Formulation
1. Design families of 2, 4, and 6 motors
by simultaneously maximizing the
aggregate performance of the family
and the CI.
2. Design families of 4 and 6 motors by
simultaneously maximizing the
aggregate performance of the family
and the CCI.
23

24. Case Study Results: 4-Motor Family
Maximizing Commonality Index (CI) Maximizing Cross-Commonality Index (CCI)
 Maximizing CCI facilitated more commonality among similar groups of
products – motors 2 and 3, and motor 1 and 3.
 However, knowledge of the actual manufacturing process chain to
conclusively compare the actual benefits of the platform plans.
24

25. Case Study Results: 6-Motor Family
Maximizing Commonality Index (CI) Maximizing Cross-Commonality Index (CCI)
 Maximizing CCI facilitated more commonality among similar groups of
products .
25

26. Concluding Remarks
 The Comprehensive Product Platform (CP3) method provides a tractable
model of the complex combinatorial process of product platform planning.
 Optimal product platform planning is performed using a novel PSO algorithm
capable of addressing (and preserving diversity in) discrete variables.
 A more advanced measure of commonality (CCI) is formulated.
• Considers the overlap among product platforms – facilitates more commonality
among similar groups of products
• Effectiveness of CCI is illustrated using the family of motor example
 Consideration of the actual product differentiation chain (manufacturing
process) would allow more realistic quantification of commonality.
26

27. Acknowledgement
• I would like to acknowledge my research adviser Prof.
Achille Messac, and Dr. Ritesh Khire for their contributions
to this paper.
• Support from NSF awards CMMI-1100948 and CMMI-
0946765 is also gratefully acknowledged
27

28. Thank you
Questions
and
Comments
28

29. Research Objectives
 Formulate a more comprehensive measure of inter-product
commonality (in a family) that accounts for the membership-overlap
among the parts-based product platforms.
 Develop and apply an optimization strategy to solve the reduced
MINLP problem yielded by the CP3 model, using a new mixed-discrete
Particle Swarm Optimization algorithm.
29

30. Commonality Constraint
      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

30
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)

31. Generalized Commonality Constraint Matrix
k N
1 1
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 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
31

32. Commonality Matrix Redundancy
32
ID - Indeterminate
The value of should
never be equal to 2
 2
2 0 kl kl kl
j j j  
Hence, the constraint should be applied
for all combinations of i, j, and k
Can we avoid the evaluation of this likely expensive constraint during the
course of optimization?

33. START
i = 0
i = i+1
Is i ≤ mn
No
Yes Yes
Is λj
kl = 1
Platform
Variable Check
Is
xj
k = xj
l
or
|xj
k – xj
l| ≤ e
Scaling Variable
Check
Is
xj
k ≠ xj
l
or
|xj
k – xj
l| > e
1, x2
1, x2
Infeasible Product
Family Design
Feasible Product
Family Design
Yes
No
Product Designs
Product-1
X = {x1
1,…, xn
1}
Product-2
X = {x1
2, x2
2,…, xn
2}
Product-N
X = {x1
N, x2
N,…, xn
N}
Platform Plan
λ12 , i=1
1
λ13 , i=2
1
1N , i=N
λ1
λj
kl , i=mp
N-1N , i=mn
λn
Product-1
X = {x1
1,…, xn
1}
Product-2
X = {x1
2, x2
2,…, xn
2}
Product-N
X = {x1
N, x2
N,…, xn
N}
Yes
No No

34. Discrete Variables in PSO
Iteration: t = t + 1
Apply continuous
Optimization
ith candidate
solution
Xi
Evaluate system
model
Fi (Xc
i, XD-feas
i)
Cont. variable
space location
i
XC
Discrete variable
space location
i
XD
Approximate to
nearby feasible
discrete location
i
XD-feas
Neighboring
discrete-point
selection
criterion
Enclosing Cell
Nearest Vertex Approach
The allowed values of each discrete variable is known a priori 34

35. Optimized Product Family Metrics
Maximizing Commonality Index (CI) Maximizing Cross-Commonality Index (CCI)
 Maximizing CCI allowed more commonality.
 Further refinement of solutions with local search might be helpful.
35