Decision support by interval SMART/SWING Methods to incorporate uncertainty into multiattribute analysis

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Decision support by interval SMART/SWING
Methods to incorporate uncertainty into
multiattribute analysis

• Ahti Salo
• Jyri Mustajoki
• Raimo P. Hämäläinen
• Systems Analysis Laboratory
• Helsinki University of Technology
• www.sal.hut.fi

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Multiattribute value tree analysis

• Value tree
• Value of an alternative x
• wi is the weight of attribute i
• vi(xi) is the component value of an alternative
  x with respect to attribute i

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Ratio methods in weight elicitation

• SWING
• 100 points to the attribute for which the swing
  from the lowest level to the highest is most
  preferred
• Fewer points to attributes for which the swings
  are less important
• Weights by normalizing the sum to one
• SMART
• 10 points to the least important attribute
• Otherwise similar

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Questions of interest

• Role of the reference attribute
• What if this is not the most or the least
  important as in SMART/SWING?
• How to incorporate preferential uncertainty?
• Uncertainties can be modeled as intervals of
  ratios instead of pointwise estimates
• Are there behavioral or procedural benefits?

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Generalized SMART and SWING

• Extensions
• 1. The reference attribute can be any of the
  attributes
• 2. The DM may reply with intervals instead of
  exact point estimates
• 3. The reference attribute, too, can be assigned
  an interval
• ? A family of Interval SMART/SWING methods
• Mustajoki, Hämäläinen and Salo, 2001

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Generalized SMART and SWING
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Some interval methods

• Preference Programming (Interval AHP)
• (Arbel, 1989 Salo and Hämäläinen, 1995)
• PAIRS (Preference Assessment by Imprecise Ratio
  Statements)(Salo and Hämäläinen, 1992)
• PRIME (Preference Ratios In Multiattribute
  Evaluation) (Salo and Hämäläinen, 2001)
• Robust Portfolio Modeling (Liesiö, Mild and
  Salo, 2007,2008)

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Classification of ratio methods
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Interval SMART/SWING Simple PAIRS

• PAIRS
• Constraints on any weight ratios
• ? Feasible region S
• Interval SMART/SWING
• Constraints from the ratios of the points

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1. Relaxing the reference attribute

• Any attribute can be selected as the reference
  attribute
• Weight ratios calculated from ratios of point
  assignments
• ? Technically no difference to SMART and SWING
• Possibility of behavioral biases
• How to guide the DM?
• Experimental research needed

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2. Interval judgments about ratio estimates

• Interval SMART/SWING
• The reference attribute given any (exact) number
  of points
• Points to non-reference attributes given as
  intervals

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Interval judgments about ratio estimates

• Max/min ratios of points constrain the feasible
  region of weights
• Can be calculated with PAIRS
• Pairwise dominance
• A dominates B pairwisely, if the value of A is
  greater than the value of B for every feasible
  weight combination

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Choice of the reference attribute

• Only the weight ratio constraints including the
  reference attribute are given
• ? Feasible region depends on the choice of the
  reference attribute
• Example
• Three attributes A, B, C
• 1) A as reference attribute
• 2) B as reference attribute

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Example A as reference

• A given 100 points
• Point intervals given to the other attributes
• 50-200 points to attribute B
• 100-300 points to attribute C
• Weight ratio between B and C not yet given by the
  DM

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Feasible region S
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Example B as reference

• A given 50-200 points
• Ratio between A and B as before
• The DM gives a pointwise ratio between B and C
  200 points for C
• Less uncertainty in results ? smaller feasible
  region

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Feasible region S'
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Which attribute to select as the reference
attribute?

• An attribute against which one can readily
  compare the other ones
• Possibly directly measurable (e.g. money)
• Elimination of remaining uncertainties through
  narrower intervals leads to more conclusive
  results

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3. Using an interval on the reference attribute

• Interpretations of intervals
• Preferences of multiple stakeholders
• Ambiguous interpretations of the attribute
• Degree of confidence about ones preferences
• Feasible region from the max/min ratios

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Interval reference

• A 50-100 points
• B 50-100 points
• C 100-150 points

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Implies additional constraints

• Feasible region S

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Using an interval on the reference attribute

• Are DMs able to compare against intervals?
• Two helpful procedures
• 1. First give points with
  pointwise
  reference
  attribute and then
  extend these to
  intervals
• 2. Use of external anchoring attribute, e.g.
  money

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WINPRE software

• Weighting methods
• Preference programming
• PAIRS
• Interval SMART/SWING
• Interactive graphical user interface
• Instantaneous identification of dominance
• ? Interval sensitivity analysis
• Available free for academic use
• www.decisionarium.hut.fi

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Vincent Sahid's job selection example

• (Hammond, Keeney and Raiffa, 1999)

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Consequences table
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Imprecise rating of the alternatives
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Interval SMART/SWING weighting
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Value intervals

• Jobs C and E
  dominated
• ? Can be eliminated
• Process continues by narrowing the ratio
  intervals of attribute weights
• Easier as Jobs C and E are eliminated

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Conclusions

• Interval SMART/SWING
• An easy method to model uncertainty by intervals
• Linear programming algorithms involved
• Computational support needed
• WINPRE software available for free
• How do the DMs use the intervals?
• Procedural and behavioral aspects should be
  addressed

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References

• Arbel, A., 1989. Approximate articulation of
  preference and priority derivation, European
  Journal of Operational Research 43, 317-326.
• Hammond, J.S., Keeney, R.L., Raiffa, H., 1999.
  Smart Choices. A Practical Guide to Making Better
  Decisions, Harvard Business School Press, Boston,
  MA.
• Mustajoki, J., Hämäläinen, R.P., Salo, A., 2005.
  Decision support by interval SMART/SWING
  Incorporating imprecision in the SMART and SWING
  methods, Decision Sciences, 36(2), 317-339.
• Salo, A., Hämäläinen, R.P., 1992. Preference
  assessment by imprecise ratio statements,
  Operations Research 40 (6), 1053-1061.
• Salo, A., Hämäläinen, R.P., 1995. Preference
  programming through approximate ratio
  comparisons, European Journal of Operational
  Research 82, 458-475.
• Salo, A., Hämäläinen, R.P., 2001. Preference
  ratios in multiattribute evaluation (PRIME) -
  elicitation and decision procedures under
  incomplete information. IEEE Trans. on SMC 31
  (6), 533-545.
• Downloadable publications at www.sal.hut.fi/Public
  ations