Multiple Criteria Optimization and Interactive Procedures

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Multiple Criteria Optimization and Interactive
Procedures Part 1 Basic Concepts
Ralph E. Steuer (University of Georgia) Yue
York Qi (University of Georgia) Markus
Hirschberger (University of Eichstätt-Ingolstadt)
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rsteuer_at_uga.edu
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Tchebycheff contour
probing direction
Z
feasible region in criterion space
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Production planning min cost min fuel
consumption min production in a given
geographical area
River basin management achieve BOD standards
min nitrate standards min pollution
removal costs achieve municipal water
demands min groundwater pumping
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Oil refining min cost min imported crude
min environmental pollution min
deviations from demand slate
Sausage blending min cost max protein min
fat min deviations from moisture target
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Portfolio selection in finance max capital
appreciation max dividends max liquidity
max social responsibility min number of
securities in portfolio
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Discrete Alternative Methods
Multiple Criteria Optimization
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• Decision Space vs. Criterion Space
• Contenders for Optimality
• Criterion and Semi-Positive Polar Cones
• Graphical Detection of the Efficient Set
• Graphical Detection of the Nondominated Set
• Nondominated Set Detection with Min and Max
  Objectives
• Image/Inverse Image Relationship and Collapsing
• Unsupported Nondominated Criterion Vectors
• Ideal way?
• Contours, Upper Level Sets and Quasiconcavity
• More-Is-Always-Better-Than-Less vs.
  Quasiconcavity

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In the general case, we have
If all objectives are linear, we write
If all objectives and all constraints are linear,
we have a multiple objective linear program
(MOLP).
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1. Decision Space versus Criterion Space
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Morphing of S into Z as we change coordinate
system
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2. Contenders for Optimality
Points in decision space are either efficient or
inefficient. Criterion vectors (points) in
criterion space are either nondominated or
dominated. We are interested in efficient points
and nondominated criterion vectors because only
they are contenders for optimality.
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3. Criterion and Nonnegative Polar Cones
Criterion cone is the convex cone generated by
the gradients of the objective functions.
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Criterion cone is the convex cone generated by
the gradients of the objective functions. The
larger the criterion cone (that is, the more
conflict there is in the problem), the bigger the
efficient set.
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Nonnegative polar of the criterion cone is the
set of vectors that make a ? 90o angle or less
with all objective function gradients. In the
case of an MOLP, it is given by
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4. Graphical Detection of the Efficient Set
Example 1
Observe the criterion cone.
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Construct nonnegative polar cone.
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Move nonnegative polar cone around.
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Only when there are no points in the translated
nonnegative polar cone other than at the vertex
is the point at the vertex efficient.
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Graphical Detection of the Efficient Set Example 2
Observe criterion cone.
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Construct nonnegative polar cone.
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(observe that x2 and x4 are not efficient)
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Graphical Detection of the Efficient Set Example 3
Note small size of criterion cone and that S
consists of only 6 points.
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Small criterion cone results in a large
nonnegative polar cone. (this makes it harder for
points to be efficient).
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Moving nonnegative polar cone around.
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5. Graphical Detection of the Nondominated Set
To determine if a criterion vector in Z is
nondominated, translate nonnegative orthant in Rk
to the point.
Now, move nonnegative orthant around.
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Now, try to identify the entire nondominated set.
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Now, move nonnegative orthant around.
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6. Nondominated Set Detection with Min and Max
Objectives
maxmax
Z
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maxmax
Z
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maxmax
Z
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maxmax
Z
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maxmax
Z
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minmax
Z
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minmax
Z
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minmax
Z
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minmax
Z
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minmin
Z
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minmin
Z
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maxmin
Z
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maxmin
Z
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In other words, the image of an efficient point
is a nondominated criterion vector. And an
inverse image of a nondominated criterion vector
is an efficient point.
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7. Image/Inverse Image Relationship and
Collapsing
Upper bound on dimensionality of S is n, but
upper bound on dimensionality of Z is k.
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8. Unsupported Nondominated Criterion Vectors
A nondominated criterion vectors is either
supported or unsupported. A nondominated
criterion vector is unsupported if and only if it
is dominated by a convex combination of other
feasible criterion vectors.
Unsupported nondominated criterion vectors are
typically hard to compute.
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9. Ideal Way?
Assess a decision makers utility function
UUUUUUaand solve
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Maybe not such a good idea for four reasons.

• Difficulty in assessing U
• U is almost certainly nonlinear
• Generates only one solution
• Does not allow for learning

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10. Contours, Upper Level Sets and Quasiconcavity
A U is quasiconcave if and upper level sets are
convex.
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Quasiconcave functions have at most one top.
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11. More-Is-Always-Better-Than-Less vs.
Quasiconcavity
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More-is-always-better-than-less does not imply
that all local optima are global optima.
z1 is a local optimum, but z2 is the global
optimum.
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More-is-always-better-than-less does not imply
quasiconcavity
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Nondominated set is important because in it is
the decision makers optimal criterion vector.
Efficient set is important because in it are all
of the decision makers optimal solutions.
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Multiple Criteria Optimization and Interactive
Procedures Part 2 Techniques
Ralph E. Steuer Terry College of Business
University of Georgia Athens, Georgia 30602-6253
USA
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• ADBASE
• Size of the Nondominated Set
• Criterion Value Ranges over Nondominated Set
• Nadir Criterion Values
• Payoff Tables
• Filtering
• Benchmark Computer Times
• Stamp/Coin Example
• Weighted-Sums Method
• e-Constraint Method

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12. ADBASE
ADBASE is a computer code for computing all
efficient extreme points of an MOLP.
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13. Size of the Nondominated Set
Can provide a finely-grained discrete covering of
the nondominated set.
Although nondominated set is a portion of the
surface of Z, it is a non-trivial task to find
the best point in the nondominated set.
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14. Criterion Value Ranges over the Nondominated
Set
If know nondominated set ahead of time, can warm
up decision maker with following information.
Upper bounds are easy. But lower bounds on the
ranges, called nadir criterion values, are
difficult to obtain.
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15. Nadir Criterion Values
Nadir criterion vector is vector of minimal
criterion vales over the nondominated set.
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16. Payoff Tables
Often over-estimate minimum criterion values over
nondominated set.
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In this problem E S. True nadir value for Obj1
is 0 not 8.
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The larger the problem, the greater the
likelihood that the payoff table column minimum
values will be wrong. After about 5 x 20 x 30,
most will be wrong.
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17. Filtering
Reducing 8 vectors down to a dispersed subset of
size 5
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First point (z1) always retained by filter.
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z2 retained by filter, but z3 discarded.
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z4 retained by filter, but z5 discarded.
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z6 retained by filter, but z7 and z8 discarded.
Wanted 5 but got 4. Reduce neighborhood, then do
again. After a number of iterations, will
converge to desired size.
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18. Benchmark Computer Times (in seconds)
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19. Stamp/Coin Example
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20. Weighted-Sums Method
But how to pick the weights because they are a
function of

• decision-makers preferences.
• scale in weight the objectives are measures
  (e.g., cubic feet versus board feet of lumber).
• shape of the feasible region

May also get flip-flopping behavior.
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Purpose of weighted-sums approach is to obtain
information from the DM so as to create a
l-vector that causes the composite gradient lTC
of the weighted-sums program
to point in the same direction as the utility
function gradient.
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2.
Boss says to go with 50/50 weights.
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Boss likes resulting solution and is proud his
50/50 weights. Then asks that second objective
be changed from cubic feet to board feet of
timber production.
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With 50/50 weights, that causes composite
gradient to point in a different direction
resulting in a completely different solution.
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3.
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Utility function is quasiconcave. Assuming we
get perfect information from DM, weighted-sums
method will iterate forever!
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21. e-Constraint Method
Basically trial-and-error
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Multiple Criteria Optimization and Interactive
Procedures Part 3 Interactive Principles
Ralph E. Steuer Terry College of Business
University of Georgia Athens, Georgia 30602-6253
USA
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• Overall Interactive Algorithmic Structure
• Vector-Maximum/Filtering
• Goal Programming
• Lp-Metrics
• Weighted Lp-Metrics
• Reference Criterion Vector
• Wierzbickis Aspiration Criterion Vector Method
• Lexicographic Tchebycheff Sampling Program
• Tchebycheff Procedure (overview)
• Tchebycheff Procedure (in more detail)
• Tchebycheff Vertex l-Vector
• How to Compute Dispersed Probing Rays
• Projected Line Search Method
• List of Interactive Procedures

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22. Overall Interactive Algorithmic Structure
Controlling Parameters
weighting vector ei RHS values
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23. Vector-Maximum/Filtering

• Let number of solutions shown be 8, convergence
  rate be 1/6.
• Solve an MOLP for, say 66,000, nondominated
  extreme points.
• Filter to obtain the 8 most different among the
  66,000.
• Decision maker selects z(1), the most
  preferred of the 8.
• Filter to obtain the 8 most different among the
  11,000 closest to z(1).
• Decision maker selects z(2) , the most
  preferred of the new 8.
• Filter to obtain the 8 most different among the
  1,833 closets to z(2).
• Decision maker selects z(3) , the most
  preferred of the new 8.
• And so forth.

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24. Goal Programming
Must choose a target vector and then select
deviational variable weights. Goal programming
uses weighted L1-metric.
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25. Lp-Metrics
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26. Weighted Lp-Metrics
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27. Reference Criterion Vector
Constructed so as to dominate every point in the
nondominated set
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usually good enough to round to next largest
integer
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28. Wierzbickis Reference Point Procedure
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First iteration
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Second iteration
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Third iteration
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29. Lexicographic Tchebycheff Sampling Program
Geometry carried out by lexicographic Tchebycheff
sampling program
Minimizing a causes non-negative orthant contour
to slide up the probing ray until it last touches
the feasible region Z.
Perturbation term
is there to break ties.
Direction of the probing ray emanating from zref
is given by
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Two lexicographic minimum solutions, but both
nondominated
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30. Tchebycheff Method (Overview)
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First iteration
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Second iteration
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Third iteration
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Controlling Parameters
target vector, weights q(i) aspiration vectors
li multipliers
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31. Tchebycheff Method (in more detail)
Let P number of solutions to be presented to
the DM at each iteration 4 Let r reduction
factor 0.5 Let t number of iterations 4
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Now, form reference criterion vector zref.
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Now, form L(1) and obtain 4 dispersed l-vectors
from it.
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Now, solve four lexicographic Tchebycheff
sampling programs (one for each probing ray).
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Now, select most preferred, designating it z(1).
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Now, form L(2) and obtain 4 dispersed l-vectors
from it.
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32. Tchebycheff Vertex l-Vector
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33. How to Compute Dispersed Probing Rays
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Now, solve four lexicographic Tchebycheff
sampling programs.
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Now, select most preferred, designating it z(2).
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Now, form L(3) and obtain 4 dispersed l-vectors
from it.
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Now, solve four lexicographic Tchebycheff
sampling programs.
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Now, select most preferred, designating it z(3)
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Now, form L(4) and obtain 4 dispersed l-vectors
from it.
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And so forth.
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34. Projected Line Search Method
Like driving across surface of moon.
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Drive straight awhile, turn, drive straight
awhile, turn, drive straight awhile, and so forth.
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35. List of Interactive Procedures

• Weighted-sums (traditional)
• e-constraint method (traditional)
• Goal programming (mostly US, 1960s)
• STEM (France Russia, 1971)
• Geoffrion, Dyer, Feinberg procedure (US, 1972)
• Vector-maximum/filtering (US, 1976)
• Zionts-Wallenius Procedure (US Finland, 1976)
• Wierzbickis reference point method (Poland,
  1980)
• Tchebycheff method (US Canada, 1983)
• Satisficing tradeoff method (Japan, 1984)
• Pareto Race (Finland, 1986)
• AIM (US South Africa, 1995)
• NIMBUS (Finland, 1998)

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The End
www.terry.uga.edu/rsteuer/install