PAIRS Preference Assessment by Imprecise Ratio Statements

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PAIRS Preference Assessment by Imprecise Ratio
Statements
A. Salo, A.A. and R.P. Hämäläinen (1992).
Preference Assessment by Imprecise Ratio
Statements, Operations Research 40/6, 1053-1061.
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Hierarchical value trees
Higher level attributes
Subcontractor
Collaboration
Proposal content
Twig-level attributes
Schedule(a1)
Overall cost (a3)
Quality of work (a2)
Reputation (a4)
Possibility of changes (a5)
Large firm (x1)
Small entrepreneur (x2)
Medium-sized firm (x3)
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Hierarchical value trees
Subcontractor
Collaboration
Proposal content
Schedule(a1)
Overall cost (a3)
Quality of work (a2)
Reputation (a4)
Possibility of changes (a5)
Large firm (x1)
Small entrepreneur (x2)
Medium-sized firm (x3)
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Ratio comparisons in weight elicitation

• Pairwise comparisons for the lower-atttibutes
• Can be elicited through ratio comparisons
• Which attribute is more important, proposed
  content or collaboration?
• How much more important is this attribute?
• Cf. Analytical Hierarchy Process (AHP Saaty,
  1980)
• Observations
• Need to be interpreted in terms of value
  differences
• However, the mode of questioning does not
  necessarily ensure this
• Incomplete information
• PAIRS admits incomplete information through
  interval-value ratio statements
• These correspond to linear constraints on the
  feasible weights

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Feasible weight regions

• Characterized by interval-valued ratio statements
  
• The first attribute is more important than the
  second, but at most twice as important
• The first attribute is more important than the
  third, but at most three times as important
• These statements results in linear constraints

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Dealing with inconsistencies

• Without support, the elicitation of
  interval-valued ratio statements could result in
  inconsistencies
• For instance, attribute 3 can be no more than 2
  times more important than attribute 2, because
  the preceding inequalities give
• For each ratio, consistency bounds are defined
  so that
• Consistency is ensured by showing these bounds
  when eliciting new statements

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Concerns with verbal statements in ratio
elicitation

• Verbal expressions are often employed in ratio
  comparisons
• eg, AHP (Saaty, 1980) makes use of statements
• 1 equally important
• 3 somewhat more importat
• 5 strongly more important
• 7 very strongly more important
• 9 extremely more important
• Weights are relatively insensitive in the upper
  end

A. Salo and R.P. Hämäläinen (1997). On the
Measurement of Preferences in the Analytic
Hierarchy Process, J of MCDA 6/6, 309-319.
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Alternative ratio scales can be employed

• Discrete weight distributions can be converted
  into corresponding ratios
• Care is needed when attaching verbal expressions
  to these
• Use of numerical values may be more warranted

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Decision recommendations with incomplete
information

• Each alternative has a unique overall value for
  any point estimate combination of scores and
  weights
• When scores and weights vary over their
  respective intervals and feasible weight regions,
  value intervals are associated with alternatives
  
• These intervals may overlap ? decision guidance
  is needed
• Is further preference information needed?
• Can some alternatives can be eliminated?
• What dominance relationships are there?

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Dominance structures

• Absolute dominance
• Lowest value of x exceeds the highest value of
  x
• Pairwise dominance
• Value of x exceeds that of x for all feasible
  parameters
• Properties
• Absolute dominance implies pairwise dominance
• Dominance relations become more conclusive with
  more information
• if the feasible sets are large, there is no
  single non-dominated alternative

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Computational issues

• Problems of nonlinearity
• Weights are elicited separately for each higher
  attribute
• Overall value of an alternative is a
  multiplicative expression of weights on the
  higher levels of the value tree
• Selections of feasible attribute weights on the
  higher levels are not independent
• How to determine weight intervals and dominance
  relationships?

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Principle

• Hierarchical propagation of bounds
• Start with score intervals at the lowest level
  of the value tree
• Propagate bounds for ? value intervals and ?
  pairwise dominance by using results from the
  lower levels as coefficients in a set of
  hierarchically structured LP programs
• Let
• the levels of the value tree
  and
• the twig-level attributes on the lowest
  level
• the sub-attributes of attribute i (with
  attribute 0 as the topmost one)

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Value intervals

• Theorem. Starting from level and moving
  towards the higher levels of the value tree,
  level by level, let Then
• Value intervals can be computed efficiently from
  a series of LP problems.

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Pairwise dominance

• Theorem. For attributes on level of the
  value tree, letThen,moving from level
  towards the higher levels of the value tree,
  level by level, let Then
• Pairwise dominance can be computed from LP
  problems
• A check is needed only if

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Size of feasible weight regions

• How complete is the weight information for
  higher-level attributes?
• This can be measured by the ambiguity index,
  defined as
• Has intuitively appearling properties
• Can be quickly computed from pairwise bounds

A. Salo and R.P. Hämäläinen (1995). Preference
Programming through Approximate Ratio
Comparisons, EJOR 82, 458-475.
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Case study Countermeasures for a nuclear accident

• Temporal combinations of countermeasures
• An exercise in decision workshop with relevant
  authorities
• Actions during weeks 2-5 and 6-12 after the
  accident
• - - - Do nothing
• Fod Provide clean fodder to cattle
• Prod Production change from milk to e.g.
  cheese
• Ban Ban the milk
• CombinationsFodFod clean fodder for both
  weeks 2-5 and 6-12

J. Mustajoki and R.P. Hämäläinen (2005). Using
Intervals for Global Sensitivity and Worst-Case
Analyses in Multiattribute Value Trees, EJOR.
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Conventional analysis

• Complete information
• FodFod is the most preferred alternative

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Incomplete information in weight assessment

• Error ratio 2 on each weight ratio
• E.g., Initial Health/Socio-psychological ratio 2
  modified to ½,221,4
• FodFod still dominates all the other alternatives

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Imprecision in score estimation

• 10 of the score intervals
• Independent changes under all Socio-psychological
  attribute
• point estimate replaced by
• FodFod dominates all the other alternatives
  except ProdFod

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Incompleteness in weight and score estimation

• Combining the two previous
• ------ is now the only dominated alternative

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Summary

• PAIRS
• Admits incomplete information through
  interval-valued ratio statements
• Is computationally efficient enough for the
  interactive analysis of alternatives value
  intervals and dominance structures
• Preserves the consistency of DMs statements
  throught consistency bounds
• Considerations
• Ratio statements do not criteria weights
  explicitly to the alternatives ranges
• Does not offer clear-cut guidance for setting
  where dominance rules do not hold
• Software support
• Freely available decision support tool available
  at http//www.decisionarium.hut.fi/