Priori Aggregation of Preference Information

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Priori Aggregation of Preference Information

• Weighted Sum (No programming required)
• Lexicographic (No programming required)
• Goal Programming
• Introduces idea of soft constraints
• Weighted Min Max (previously demonstrated)

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Weighted Sum Approach

• Minimize f(x) S wmfm(x)
• Subject to gj(x) lt 0, j 1,2,.,J
• hk(x) 0, k 1,2,.,K
• M
• S m1 wm 1
• xi(L) lt xi lt xi(U) , i 1,2, , n
• Need to normalize each objective for weights to
  be meaningful.

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Pareto Optimal Theorems (Deb)

• The solution to the problem represented by
  Weighted Sum is Pareto optimal if the weight is
  positive for all objectives.
• If x is a pareto-optimal solution of a convex
  multi-objective optimization problem then there
  exists a non-zero positive weight vector w such
  that x is a solution to the problem.

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iSIGHT Formulation 1

• iSIGHT Directly supports through GUI a weighted
  formulation for a single set of weights.
• User can add weights and normalization scale
  factors and easily obtain a Pareto optimal point.
• User can then adjust weighting and rerun

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Formulation with even weighting and scaling
between 0 - 10
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iSIGHT Formulation
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Optimization Results with GRG
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Standard Tradeoff Curve
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Advantages and Disadvantages

• Advantage
• Simplicity
• Ease of implementation
• Disadvantages
• Uniformly distributed weight vectors need not
  find a uniformly distributed set of
  Pareto-optimal solutions
• Cannot find certain solutions in a non convex
  space.

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Lexicographic

• Deal with one objective at a time in priority
  order.
• Step 1 Optimize on Cross Section Area
• Step 2 Take optimal Cross Section Area and set
  an upper bound constraint on Cross Section Area
  with( 1 e / 100) (Optimal CrossSectionArea)
  where e is a percentage if e 0 then no
  compromise on gain.
• Step 3 Optimize on Static Deflection
• Possible iSIGHT implementations
• Manual No Programming Required
• Task Plan
• Rules

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Optimize on CrossSectionArea
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Set Upper Bound Constraint on Cross Section Area

• User must decide on how much to allow (if any) of
  CrossSectionArea optimum to give up while
  minimizing Static Deflection.
• Lets assume 30 so new bound for CrossSectionArea
  is(1 30/100) Optimal CrossSectionArea1.3
  127.427 165.655
• Note The only objective now in StaticDeflection

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New Problem Formulation
Note Upper bound on CrossSectionArea and now new
objective to minimize StaticDeflection
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Optimize with GRG
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Standard Tradeoff Curve
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Goal Programming
M Minimize Sj1 (pj
nj) Subject to fj(x) pj nj tj, j1,2,,M
nj,pj gt 0,
j1,2,,M
Define targets for each objective. Targets may or
maynot be reachable Could add weights to p and n
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Goal Programming
Figure from Deb
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iSIGHT Formulation

• Single level task requires a calculation to
  calculate new objective. (Alternatively, could
  minimize weighted iSIGHT objective)
• Add two variables for each objective to track
  the positive and negative deviations
• Add constraint for each objective
• Add constraint to insure each deviation is gt 0.

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Calculation for Goal Programming
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Problem Formulation
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Optimization with GRG
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Standard Tradeoff Curve
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Summary of Priori Aggregation of Preference
Information

• Weighted Sum (No programming required)
• Lexicographic (No programming required)
• Goal Programming
• Could easily extend to Weighted Goal Programming
• Could be considered to be a satisficing method
  depending on attainability of targets
• Weighted Min Max (previously demonstrated)