Separating Product Design Optimization
Peyman Karimian, Jeffrey W. Herrmann
Decomposition
Optimization
Designers use optimization techniques to
determine values for design variables to
optimize an objective function while satisfying
performance requirements and other
constraints.
Separation
The difficulty of solving large scale
optimization problems has motivated various
decomposition approaches, but many of
these methods require multiple iterations to
converge to a feasible, optimal solution,
which leads to extensive effort in some
cases.
P*
Our approach, which we call separation,
replaces a large design optimization
problem with a set of subproblems, solves
each subproblem once, and produces a
feasible solution without iterative cycles.
An exact separation finds an optimal
solution to the original problem.
Approximate separations find near-optimal
solutions.
P1
P1
P2
P2
P3
P3
A typical decomposition scheme uses a second-level
problem to coordinate the subproblems.
Separation yields a set of subproblems.
Solving one subproblem provides the input to the next.
Profit Maximization without Competition
Profit Maximization under Competition
Product Family Design under Competition
We considered different separations of the
profit maximization problem for an universal
electric motor and compared to the solution
found from an all-at-once formulation. We
found that maximizing customer satisfaction
leads to better solutions that optimizing mass
or efficiency.
We modeled the problem of designing an
universal electric motor as a mixed motive
game in which each manufacturer makes
independent product design and pricing
decisions. We found an exact separation with
three subproblems that yield the optimal
product design and price.
We modeled the problem of designing a
family of universal electric motors (with
ten different torques) as a zero sum game.
We compared different separations and
found that a separation that maximized
total utility yielded a near-optimal product
family.
A, p
P2:
X, p
P3:
max Π
max Π
(a)
min ∆
P2:
max u1 ( X1 )
X
P1:
max ∏
X
P5:
p
max Π
P4:
min
X 1 , X 2 , p1 , p2
(b)
P4:
min M
X1
P3:
max u2 ( X 2 )
(a)
p1 , p2
2
 ∂ ∏1   ∂ ∏ 2 

 +

 ∂R1   ∂R2 
X2
(b)
2
Mass (kg)
P1:
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Player 2
Family A
Family B
0.6
(c)
(a) The all-at-once formulations (A1) and (A2) maximize profit.
(b) Separation S1 finds the most profitable attribute values and
price and then sets the design variables to satisfy them. (c)
Separation S2 finds the best design and then sets the price to
maximize profit.
0.7
0.8
0.9
1
Efficiency (% )
(a) The all-at-once formulations to maximize profit.
(b) Separation P1 and P2 finds the best utility values and
design variables for both player and then sets price for the best design.
(a)
(a) The Mass and Efficiency of Each Player’s Products.