RICH Rank Inclusion in Criteria Hierarchies

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RICH Rank Inclusion in Criteria Hierarchies
A. Salo, A. Punkka (2005) Rank Inclusion in
Criteria Hierarchies, EJOR 163/2, 338-356.
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Multi-attribute weighting
Subcontractor
Schedule(a1)
Overall cost (a3)
Quality of work (a2)
References (a4)
Possibility of changes (a5)
Large firm (x1)
Small entrepreneur (x2)
Medium-sized firm (x3)
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Ordinal preference information

• Approaches to the elicitation of ordinal
  information
• Ask the DM to rank the attributes in terms of
  importance
• Derive a representative weight vector from the
  ranking
• e.g., SMARTER (Edwards and Barron 1994)
• ranks reciprocal weights
• centroid weights
• Incomplete ordinal preference information
• The DM(s) may be unable to rank the attributes
• contentious issuses which is more important -
  economy or environmental impacts
• Equal weights sometimes used as an approximation
  
• Here Associate a set of possible rankings with a
  given set of attributes

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Modelling incomplete ordinal information

• Rank-orderings correspond to complete ordinal
  information
• Known ranking for each alternative
• A bijection from attributes to rankings2nd
  attribute is the most important one, followed by
  the 1st and then the 3rd
• Rank Inclusion in Criteria Hierarchies (Salo and
  Punkka, 2005)
• Admits incomplete ordinal statements
• Cost or Quality is the most important attribute
• Environmental concerns are among the three most
  important attributes
• Location is not the least important attribute
• These are compatible with several rank orderings
• May lead to a non-convex feasible weight region

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Preference elicitation example 1

• The most important attribute is either a1 or a2

• Compatible rank orders are (a1,a2,a3),
  (a1,a3,a2), (a2,a1,a3), (a2,a3,a1)
• Feasible region not convex

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Analysis of incomplete ordinal information

• I is a set of attributes, J a set of rank numbers
• r a rank-ordering is a mapping from attributes to
  
• r(ai) is the rank of attribute i
• Compatible rank orders
• Feasible region for a given rank order r
• Feasible region for rank orders compatible with
  sets I and J

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Preference elicitation - example 1

• The most important attribute is either a1 or a2
• This leads to attribute set Ia1,a2 and rank
  set J1

• Compatible rank orders are (a1,a2,a3),
  (a1,a3,a2), (a2,a1,a3), (a2,a3,a1)
• Feasible region not convex

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Preference elicitation example 2

• Attributes a1 and a2 are the two most important
  attributes
• This leads to attribute set Ia1,a2 and rank
  set J1,2

• Compatible rank orders are (a1,a2,a3) and
  (a2,a1,a3)
• Sp(I)S(I,1,,p)

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Theorems on feasible regions

• Feasible region associated with any given I and J
  is equal to that of complement of I and
  complement of J
• If there are more ranks in J than attributes in
  I, the feasible region gets smaller when
  attributes are added to I
• If there are less ranks in J than attributes in
  I, the feasible region gets smaller when
  attributes are removed from to I

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Theorems on feasible regions

• When there are less ranks in J than attributes in
  I, the feasible region gets smaller when ranks
  are added to J
• When there are more ranks in J than attributes in
  I, the feasible region gets smaller when ranks
  are removed from J

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Special cases of feasible regions

• The subset of p most important attributes
• Lemma 1.
  is convex only if
• Lemma 2. If ,
  then

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Measuring the size of the feasible region

• Measure of completeness
• Compares the number of compatible rank-orderings
  to the total number of rank-orderings

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Effectiveness of incomplete ordinal information

• Questions
• How effective is incomplete ordinal information?
• Which decision rules are best?
• Randomly generated problems
• n5,7,10 attributes m5,10,15 alternatives
• 3 different preference statements
• A. DM knows the most important attribute
• B. DM knows two most important attributes
• C. DM knows the set of 3 attributes which
  contains the 2 most important
• Statements compared to equal weights and complete
  rank orders
• Efficiency studied using central values (appeared
  to be best)
• 5000 problem instances
• Values computed at extreme points

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Percentage of correct choices
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Expected loss of value
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Results

• Statements improve performance in relation to
  equal weights
• Rank order is better than the studied statements
• Statement B gives the best results
• The feasible region is smallest

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Computational shortcuts

• Theorem. If is a rank-ordering, then
  the extreme points of the feasible region
  are given by
• Use of this result
• All extreme points for rank-orderings can be
  computedin advance (when the number of
  attributes is not large)
• Extreme points of feasible region obtained by
  pruning these points
• Very (and increasingly) fast computations

See E. Carrizosa, E. Conde, F.R. Fernández, J.
Puerto (1995). Multi-criteria analysis with
partial information about the weighting
coefficients, EJOR 81, 291-301.
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Decision criteria (1/2)

• Provide decision guidance when dominance does not
  hold
• Associated loss of value must be examined,
  however!
• Max-max - Choose the alternatives with the
  highest possible value
• Max-min Choose the alternative with the
  highest minimum value

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Decision criteria (2/2)

• Minimax regret Choose the alternative for which
  the value difference to any other alternative is
  minimized (loss of value)
• Central values mid point of overall value
  intervals
• Central weights same with normalized central
  weights, assuming known scores

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Application of RICH to the risk management
planning
O. Ojanen, S. Makkonen, A. Salo (2005). A
Multi-Criteria Framework for the Selection of
Risk Analysis Methods at Energy Utilities, Int.
Journal of Risk Assessment and Mgmt 5/1, 16-35.
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Assessment of risk mgmt methods
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Group interaction

• Supplier group
• w.r.t total utility,
• information utility gt usability gt costs
• w.r.t information utility
• total normal risk meast most important
• extreme risk meast 2nd most important
• sensitivity meast 3rd most imporant
• attribution to risk factors least important
• w.r.t usability
• intuitiveness gt flexibility gt authority

• User group
• w.r.t total utility
• cost is most important
• usability and info utility 2nd and 3rd most
  important
• w.r.t information utilility
• extreme risk meast and total normal risk 1st and
  2nd most important
• attribution to portfolio components 3rd
• attribution to risk factors and sensivity meast
  4th and 5th most important
• w.r.t usability
• intuitiveness gt flexibility gt authority