Multi Objective Optimization MOO with iSIGHTFD 2'0 Please fill out short survey

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Multi Objective Optimization (MOO) with iSIGHT-FD
2.0Please fill out short survey

• David J. Powell, PhD
• dpowell2_at_elon.edu
• Last modified March 11, 07

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Introductory Remarks

• Thanks Jeff, Liz, Marijo, Oleg, Wei-Shan
• iSIGHT classic MOO update for FD
• Designer now has a 3rd decision to make
• Parallel evaluation makes and will make more
  pragmatic.
• Survey
• Web site

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Goals

• Directly use Multiple Objective Optimization
  (MOO) in iSIGHT-FD with selective optimization
  techniques
• Use MOO in iSIGHT-FD by additional modeling with
  all optimization techniques
• Classical Numerical Approaches
• Understand iSIGHT-FD flexibility to help you
  generate multiple points or a single point on the
  pareto optimal front.
• Understand iSIGHT-FD tools for data analysis of
  pareto front.
• Understand NSGA-II Genetic Algorithm approach for
  MOOP
• Understand how to model soft constraints

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Examples of MOOP

• The traditional portfolio optimization problem
  attempts to simultaneously minimize the risk and
  maximize the return.
• A good sunroof design in a car could aim to
  minimize the noise the driver hears and maximize
  the ventilation.
• In bridge construction, a good design is
  characterized by low total mass and high
  stiffness.
• Aircraft design requires simultaneously
  optimization of fuel efficiency, payload and
  weight.
• (info from http//www-fp.mcs.anl.gov/otc/Guide/Opt
  Web/multiobj/ )

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Primary References

• Marler, R. (2005), A Study of Multi-Objective
  Optmization Methods for Engineering Applications,
  PhD Thesis, University of Iowa.
• Deb, K. (2001), Multi-Objective Optimization
  using Evolutionary Algorithms, John Wiley Sons.
• Miettinen (1999). Nonlinear Multiobjective
  Optimization. Kluwer Academic Publishers.
• Anderson, J. (2000), A Survey of Multiobjective
  Optimization in Engineering Design, Linkoping
  University, Technical Report 1097.
• Osyczka, A. (1985). Multicriteria Optimization
  for Engineering Design, In Design Optimization,
  pp 193-225.

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MOO General Form
Minimize fk(x), k
1,2,,K Subject to gj(x) lt 0,
j 1,2,.,J hm(x) 0,
m 1,2,.,M
xi(L) lt xi lt xi(U) , i 1,2, , N
where Xi Rn
continuous variables
Xi In
integer variables Xi (X1, X2,
) discrete variables
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iSIGHT-FD Formulation
Scaling is critical in all optimization single
and multiple
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Three Scaling Approaches
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Scaling Approach 2
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Scaling Approach 3
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Ideal or Utopian Solution Vector

• For each of the K objectives, there exists one
  different optimal solution.
• An objective vector constructed with these
  individual optimal objective values constitutes
  the ideal objective vector or utopian vector.
• In general, this is never obtainable
• What is its use
• Individual optimal objective values used for
  normalization
• Used by some classical techniques as solutions
  closer to ideal are better.

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Utopian Objective Vector
Nadir upperbound of eachindividualoptimizedo
bjective
Utopia lowest value of each objective
Figure from Deb p. 27
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Domination

• A solution x(1) is said to dominate the other
  solution x(2), if following 2 conditions are
  true
• The solution x(1) is no worse than x(2) in all
  objectives for j 1, 2, , K
• The solution x(1) is strictly better than x(2) in
  at least one objective

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Domination Example
1 dominates 25 dominates 1
Figure from Deb page 29
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Pareto Optimal

• Globally Pareto-optimal set. The non-dominated
  set of the entire feasible search space S is the
  globally Pareto-optimal set.

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Pareto Optimal Front
Figure from Anderson
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Pareto Optimal Fronts
Figure from Deb p. 32
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Classification of MOO Techniques

• No articulation of preference information
• Global criterion (SC)
• MinMax (SN)
• Benson (SN)
• Prior
• Weighted Sum (C,SC)
• Goal Programming (SN)
• Lexicographic (SN)

• Posterior
• Weighted Sum (SC)
• eConstraint (SN)
• Genetic Algorithm (N)
• Weighted MinMax (SN)
• Weighted Goal Programming
• Progressive
• Satisficing Tradeoff Analysis (SN)
• Guess (SN)

For clarity I will present in category order but
deviate on individual techniques
Simple programming/modeling required
S Convex objective space
C Nonconvex objective space
N
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IBeam Example
Figure from Osyczka p 196
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IBeam MOOP
Minimize Area Minimize Static
Deflection Subject to Strength lt 16
10 lt BeamHeight (x1) lt 80 10 lt
FlangeWidth (x2) lt 50 0.9 lt
WebThickness (x3) lt 5 0.9 lt
FlangeThickness (x4) lt 5 Starting design x0
75, 45, 2, 2 Area 322 Static Deflection
0.01669 Strength 5.605
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Excel Spreadsheet for IBeam
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iSIGHT-FD Optimization
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My Favorite Smooth Optimizer
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Starting Formulation for Optimal Area
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Optimal Area
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Optimal Area Nadir Static Deflection
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Excel Optimizer - GRG
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Excel GRG Options
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Excel Solution
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Starting Formulation for Optimal Static Deflection
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Optimal Static Deflection
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Optimal Deflection Nadir Area
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Start of Standard Tradeoff Curve
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How about others?

• NLPJOB by Schittkowski
• Weighted sum
• Lexicographic
• eConstraint on Tradeoff Method
• Global Criterion p 1, p 2
• MinMax

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Exercise 1 Ad Exposures
You work for Burnit advertising company and your
job is to determine the optimal number of ads to
run to maximize the exposure to men and women.
Your budget is 1.5 million. Write the calculation
and problem formulation for iSIGHT using the data
and constraints in Table 9.1 and 9.2. Note if 5
ads are placed in Sport shows then this will
achieve 15 sqrt(5) 33.541 exposures.
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Exercise 1 Ad Exposures
You need to calculate the utopia point for the
objectives.You first perform an optimization on
only ExposuresToMen andget the
values ExposuresToMen 89.5147
ExposuresToWomen 79.3875 You then perform an
optimization on only ExposuresToWomenand get the
values ExposuresToMen 84.8987 ExposuresToWom
en 89.2199 What are the utopian objective
values? What are the nadir values?
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Exercise 1 Solution
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Exercise 1 Solution
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Problem Formulation
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Exercise 1 Ad Exposures
You need to calculate the utopia point for the
objectives.You first perform an optimization on
only ExposuresToMen andget the
values ExposuresToMen 89.5147
ExposuresToWomen 79.3875 You then perform an
optimization on only ExposuresToWomenand get the
values ExposuresToMen 84.8987 ExposuresToWom
en 89.2199 What are the utopian objective
values? 89.5147, 89.2199 What are the nadir
values? 84.8987, 79.3875