PF_MAO2010 Souma

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

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

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

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*
4* GM (Chevrolet) official website

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 2Step 1
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

Mixed-Discrete Non-Linear Optimization (MDNLO)
MDNLO
Criterion
Functions
Non-linear
Objectives
Non-linear
constraints
Design
Variables
Continuous
Variables
Discrete
Variables
Uniformly
Distributed
e.g. Integers
Non-uniformly
Distributed
6
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

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

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
• Generalized Approach to MDNLOPart 1
• Comprehensive Product Platform
Planning (CP3) ModelPart 2
• Simplification of the CP3 ModelPart 3

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
XC
i
Discrete variable
space location
XD
i
Approximate to
nearby feasible
discrete location
XD-feas
i
Non-gradient
based
optimization
Criterion for
selecting local
discrete points

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

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.
• 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
12*BIP: Binary Integer Programming
0 1 1 0

13

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 this paper)
 Solves the actual MINLP problem using the generalized approach to MDNLO that
can be implemented through a non-gradient based algorithm
*MINLP: Mixed Integer Non-Linear Programming 14
CP3 Optimization (Original)
 Converts the MINLP into a continuous optimization problem using a set of Gaussian
pdfs collectively called the Platform Segregating Mapping function (PSMF)

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.
 
 
     
 
 
2 2 212 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,....,
, , , , ,
PERFORMANCE
COST
i
i
f Y
f Y
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
1 1 1 2 2 2
1 2 3 1 2 3
1 2 3
, , ,
, , , , ,
, , : 0, 1
x
X x x x x x x
B B
  
  

 
1 2 12
12 1 2
if , then 0
if 1, then
j j j
j j j
x x
x x


 
 
1
1x
1
2x
1
3x
2
1x
2
2x
2
3x
12
1
12
2
12
3
0
1
15

Commonality Constraint
12 12 1
1 1 1
12 12 2
1 1 1
12 12 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
0 0 0 0
0 0 0 0
0 0 0 0
x
x
x
x x x x x x
x
x
x
 
 
 
 
 
 
   
   
   
   
          
   
   
      
     
2 2 212 1 2 12 1 2 12 1 2
1 1 1 2 2 2 3 3 3 0x x x x x x       
Commonality Constraint Matrix (Λ)
16

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

Generalized Commonality Constraint Matrix
1 1
1 1
1
1
1 1
1 1
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
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
k N
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

Generalized Commonality Matrix
11 1
1 1
1
1 1
11 1
1
11 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
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
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
1 , iff variable is included in product-
0 , iff variable is NOT included in product-
kk ll l k
j j j jl
k l
th
kk
j th
x x
j k
j k
 


  
 


 

Corresponds to the jth design variable*
19* Chowdhury et al., 2010

Commonality Matrix Redundancy
20
 
2
2 0kl kl kl
j j j     
The value of should
never be equal to 2
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?
ID - Indeterminate

21

1 1 0 0
1 1 0 0
0 0 1 1
0 0 1 1
j
 
 
 
 
 
 
Converting CP3 Model from BIP to IP* Problem
22*BIP: Binary Integer Programming; IP: Integer Programming
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 0 0 0 0 1
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 =
 Range of integer variables:
 1 2N N 
 1 2 1
0,2
N N
z
 
  
33jz 

Reduction of the CP3 Model
 Identify the infeasible combinations of binary commonality variables
 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 
 
 
 
2
2 0kl kl kl
j j j     

Modified MINLP Problem
 
 
 
 
 
 
 
1 2 1 2 1
1 1 1
Max
Min
s.t. 0
0, 1,2,....,
0, 1,2,....,
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


 
 
 
 


 2
1 2 feas,
, 1,2, , ; 1,2, ,
TN
n n
j n j
x
Z z z z z z Z
k l N j n
  
   
   
Performance objective
Cost objective
24
Feasible set of values for
each integer variable

Interesting Observations
 Number of combinations of platform/scaling variables (n-variable system):
25
2n
 
n
ZM
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

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

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

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

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

Platform: Definition and Demo.
32
“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 2 3 4 5
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 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
    
         
         
             
         
         
         

PF_MAO2010 Souma

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