The Comprehensive Product Platform Planning (CP3) framework presents a flexible mathematical model of the platform planning process, which allows (i) the formation of sub-families of products, and (ii) the simultaneous identification and quantification of plat- form/scaling design variables. The CP3 model is founded on a generalized commonality matrix that represents the product platform plan, and yields a mixed binary-integer non- linear programming problem. In this paper, we develop a methodology to reduce the high dimensional binary integer problem to a more tractable integer problem, where the com- monality matrix is represented by a set of integer variables. Subsequently, we determine the feasible set of values for the integer variables in the case of families with 3 − 7 kinds of products. The cardinality of the feasible set is found to be orders of magnitude smaller than the total number of unique combinations of the commonality variables. In addition, we also present the development of a generalized approach to Mixed-Discrete Non-Linear Optimization (MDNLO) that can be implemented through standard non-gradient based op- timization algorithms. This MDNLO technique is expected to provide a robust and compu- tationally inexpensive optimization framework for the reduced CP3 model. The generalized approach to MDNLO uses continuous optimization as the primary search strategy, how- ever, evaluates the system model only at the feasible locations in the discrete variable space.

1. Developing a Non-gradient Based Mixed-Discrete
Optimization Approach for Comprehensive Product
Platform Planning (CP3)
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
13th AIAA/ISSMO Multidisciplinary Analysis Optimization (MAO) Conference
September 13-15, 2010
Fort Worth, Texas

2. Presentation Outline
 Motivation and technical background
 Objectives of this paper
 Generalized Mixed-Discrete Non-Linear Optimization (MDNLO)
 Comprehensive Product Platform Planning (CP3) Model
 Simplification of the CP3 Model
 Concluding Remarks
2

3. Motivation
• Development of a family of products that satisfies different market niches
introduces significant challenges to today’s manufacturing industries.
• A comprehensive product family design methodology can offer a
powerful solution to these daunting challenges.
• Product platform planning is generally combinatorial in nature, and
typical engineering systems involve highly non-linear criterion functions.
• Hence, product family design methodologies generally demand solution
of a complex mixed-integer non-linear programming (MINLP) problem.
• To this end, a generalized technique that can be implemented through
robust optimization, will be very useful.
3

4. 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 4

5. Existing Product Family Design Methods (Scalable)
5
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
Likely to introduce a significant
source of sub-optimality
Exhaustive approach
Computationally prohibitive for
large scale systems
Selection Integrated Optimization*
Provides a computationally inexpensive single stage approach
Genetic Algorithm based Approach
Provides a robust optimization approach. Allows formation of sub-families**
* Khire and Messac, 2008; ** Khajavirad et al., 2009

6. Mixed-Discrete Non-Linear Optimization (MDNLO)
MDNLO
Criterion
Functions
Non-linear
Objectives
Non-linear
constraints
Design
Variables
Continuous
Variables
Product family design model
• Generally combinatorial in nature
• The CP3 model yields a mixed
binary-integer programming
(BIP) problem
• Mixed-BIP is a subset of
MDNLO
Discrete
Variables
Uniformly
Distributed
e.g. Integers
Non-uniformly
Distributed
6

7. Existing algorithms for MDNLO
Gradient based algorithms: Branch and Bound, Cutting Plane and Outer
Approximation algorithms*
• Provides a proof of optima
• Do not readily apply to highly non-linear and multi-modal problems
• Computationally prohibitive for a large number of feasible discrete
combinations
7
Binary Genetic** and Binary Swarm based*** Algorithms:
• Provides a robust optimization approach
• Cannot readily avoid combinations of binary variables known to be
infeasible apriori
* MINLP World, 2010
** Deb , 2002
*** Kennedy and Eberhart, 1997

8. Specific Research Objectives
• Develop a generalized approach to solve complex MDNLO problems,
such as that presented by the CP3 model.
• Develop a strategy to avoid redundancy in the commonality matrix that
represents the platform planning process in CP3 model.
• Reduce the high dimensional mixed binary-integer problem (BIP),
presented by the CP3 model, into a more tractable mixed integer problem.
8

9. 9
Part 1 • Generalized Approach to MDNLO
• Comprehensive Product Platform
Planning (CP3) Model Part 2
Part 3 • Simplification of the CP3 Model

10. Generalized Approach to MDNLO - Process
Requires specification of
the feasible set of values
for each discrete variable
10
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
Non-gradient
based
optimization
Criterion for
selecting local
discrete points

11. Vertex Approximation Techniques
In the discrete variable domain, the
location of a candidate solution can be
defined by a local hypercube
Nearest Vertex Approach (NVA)
Approximates to the nearest discrete
location based on Euclidean distance.
Shortest Normal Approach (SNA)
Approximates to the discrete location with
shortest normal to the connecting vector.
11

12. Reduction of High Dimensional BIP* Problem
• A BIP with m binary variables yields a single hypercube with 2m vertices.
• The binary variables can be aggregated into binary strings, e.g.
• Binary strings are then converted into integer variables.
0 1 1 0
• Benefit: Infeasible combinations of binary variables (known apriori) can
be easily eliminated by modifying the feasible set of values for the integer
variable z, without applying additional constraints.
• Application: Product family design problem
*BIP: Binary Integer Programming 12

13. 13
Part 1 • Generalized Approach to MDNLO
• Comprehensive Product Platform
Planning (CP3) Model Part 2
Part 3 • Simplification of the CP3 Model

14. Basic Components of the CP3 Framework
CP3 Model
 Formulates an integrated mathematical model, yielding a MINLP* problem
 Allows the formation of sub-families of products
 Allows simultaneous selection and optimization of platform/scaling design
variables
 Seeks to eliminate distinctions between modular and scalable families
CP3 Optimization (In Original)
this paper)
 Solves the actual MINLP problem using the generalized approach to MDNLO that
Converts the MINLP into a continuous optimization problem using a set of Gaussian
can pdfs be collectively implemented called through the Platform a non-gradient Segregating based Mapping algorithm
function (PSMF)
*MINLP: Mixed Integer Non-Linear Programming 14

15. 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.
 
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
15

16. 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

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)
16

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

18. 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
18

19. 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 19

20. Commonality Matrix Redundancy
20
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?

21. 21
Part 1 • Generalized Approach to MDNLO
• Comprehensive Product Platform
Planning (CP3) Model Part 2
Part 3 • Simplification of the CP3 Model

22. Converting CP3 Model from BIP to IP* Problem
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 
 
 
  
 
 
 
1 0 0 0 0 1
33 j z 
For a family of N kinds of products and a set of n design variables:
 No. of integer variables = No. of Binary Strings = n
 String length =
N N 1 2
 Range of integer variables:
 1 2 1 0,2N N z    
 
*BIP: Binary Integer Programming; IP: Integer Programming 22

23. Reduction of the CP3 Model
 Identify the infeasible combinations of binary commonality variables
 2
2 0 kl kl kl
j j j     
 Eliminate the corresponding integer values from the set of values for each
integer variable, , to form the feasible set Zfeas.
23
 1 2 1 0,1,2, ,2 N N  
 

24. Modified MINLP Problem
Performance objective
 
 
Max
Min
s.t. 0
 
 
 
 0, 
1,2,....,
 0, 
1,2,....,
 

 
 


 
Cost 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
24
Feasible set of values for
each integer variable

25. Interesting Observations
 Number of combinations of platform/scaling variables (n-variable system):
25
2 n  n
Z M
MZ – size of the feasible set of values (Zfeas)
Grows exponentially with the number of product kinds
 This difference is attributed to the allowance of sub-families of products.
 E.g. A 7-product family yields a feasible set with size MZ = 877
 The feasible set Zfeas applies to each integer variable for any scale-based
product family
 For modular product families, additional restrictions owing to the product
architecture might further reduce the feasible set

26. Further Interesting Observations
26
For product families with 2-7 kinds of products
Number of platform/scaling
combinations for each design variable
Feasible values of the integer variables

27. Concluding Remarks
 The proposed generalized approach to mixed-discrete non-linear optimization
can be readily applied to optimizing the CP3 model.
 An additional constraint reduces the number of feasible combinations of
platform/scaling design variables to several orders of magnitude less than the
number of unique commonality matrices.
 The binary-integer programming problem is successfully reduced to a more
tractable integer programming problem
 We found that the number of possible combinations of platform/scaling
variables is exponentially higher than 2n
 The feasible set of values for the integer variables, once determined, can be
used for any scalable product family, which is uniquely helpful.
27

28. Future Work
 Current research is focused on implementation of the mixed-discrete non-linear
optimization (MDNLO) approach through evolutionary and swarm
based algorithms.
 Subsequent application of the new CP3 optimization technique to design a
product family of universal electric motors would establish the true
potential of this approach.
28

29. Selected References
1. Chowdhury, S., Messac, A., and Khire, R., “Comprehensive Product Platform Planning (CP3)
Framework: Presenting a Generalized Product Family,” 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural
Dynamics, and Materials Conference, Orlando, Fl, April 2010.
2. http://www.chevrolet.com/, GM (Chevrolet) official website.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
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, pp. 182-197.
10. Simpson, T. W., Maier, J. R. A. and Mistree, F., “Product Platform Design: Method and Application,” Research in
Engineering Design, Vol. 13, No. 1, 2001, pp. 2–22.
11. “MINLP World,” http://www.gamsworld.org/minlp/, 2010.
29

30. Acknowledgement
30
Thank you

31. Questions
or
Comments
31

32. 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 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
32
, , , ,
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