Multiperson Decision Making Based on Multiplicative Preference Relations

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Multiperson Decision Making Based on
Multiplicative Preference Relations

• CIS6660.1 Seminar
• Stelian Coros
• University of Guelph

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• F. Chiclana, F. Herrera, E. Herrera-Viedma
• Multiperson Decision Making Based on
  Multiplicative Preference Relations.
• European Journal of Operational Research, vol.
  129, pp. 372-385, 2001

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Contents

• Introduction to the Problem
• Preference Representations
• Making the Information Uniform
• Decision Model
• Conclusion

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The Multiperson Decision Making Problem an
introduction
Lets ask the experts!!
Dynamic Programming
Genetic Algorithms
Brute Force
Which algorithm should we use to solve a certain
optimization problem?
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The Multiperson Decision Making Problem an
introduction

• Two main problems
• The experts will have different opinions
• The experts will express their opinions using
  different preference representations

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The Multiperson Decision Making Problem an
introduction
Record Expert Preferences
Present the information in a uniform manner
Run decision model
Selection Set of Alternative(s)
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Contents

• Introduction to the Problem
• Preference Representations
• Making the Information Uniform
• Decision Model
• Conclusion

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Preference Representations

• Let X x1,x2,,xn be a set of alternatives
• n 2
• Let E e1,e2,,em be a set of experts
• m 2

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Preference Representations

• 1. A preference ordering of alternatives
• alternatives are ordered from best to worst
• For ek? E
• Ok ok(1),,ok(n) a permutation of 1,..,n
• xi is rated ok(i)
• the lower the position, the better it suits
  the expert

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Preference RepresentationsPreference Ordering
of Alternatives

• Example
• Let X x1,x2,x3,x4
• O1 3,1,4,2 - preference indicated by expert 1
• o1(1) 3
• o1(2) 1 ? Alternative most preferred by
  expert 1
• o1(3) 4 ? Alternative least preferred by
  expert 1
• o1(4) 2

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Preference Representations

• 2. Utility functions
• associate each alternative to a real number
• For ek? E
• Uk uk1,, ukn set of n utility values
• uki1..n ? 01
• the higher the value, the better it suits the
  expert

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Preference RepresentationsUtility Functions

• Example
• Let X x1,x2,x3,x4
• U3 0.5,0.7,1,0.1 preference indicated by
  expert 3
• x3 ? alternative highly preferred by expert 3

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Preference Representations

• 3. Multiplicative preference relations
• indicate the indifference between every pair of
  alternatives
• For ek? E
• Ak ? X x X positive preference relation
• akij intensity of preference (1 to 9 scale)
• akij akji 1 (multiplicatively reciprocal)
• the higher the value of akij, the more expert k
  prefers alternative xi to xj

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Preference Representations Multiplicative
preference relations

• Example
• Let X x1,x2,x3,x4
• ? x1 is absolutely preferred to x4 according to
  expert 6
• ? aijlt1 j is preferred to i
• ? aijgt1 i is preferred to j

- preference indicated by expert 6
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Contents

• Introduction to the Problem
• Preference Representations
• Making the Information Uniform
• Decision Model
• Conclusion

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Making the Information Uniform

• Use multiplicative preference relations as base
  of uniform preference representation
• utility functions ?
• preference ordering ?

• multiplicative preference
• relations

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Utility Functions ? Multiplicative Preference
Relations

• Recall
• Uk uk1,, ukn , uki1..n ? 01
• Transformation function
• akij h(uki,ukj) uki/ukj

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Utility Functions ? Multiplicative Preference
Relations

• Properties
• 1. h(uki,ukj) h(ukj,uki) 1, ? i, j ?
  1,..,n
• 2. h(uki,uki) 1, ? i ? 1,..,n
• 3. h(uki,ukj) gt 1 if ukigtukj, ? i, j ? 1,..,n
  

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Preference Ordering ? Multiplicative Preference
Relations

• Recall
• Ok ok(1),,ok(n)
• Transformation function
• akij g(ok(i), ok(j)) 9 ,
• (n-ok(i))
• (n-1)

uki-ukj
uki
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Preference Ordering ? Multiplicative Preference
Relations

• Properties
• 1. g(ok(i), ok(j)) ? 1/9,9, ? i, j ? 1,..,n
• 2. g(ok(i), ok(j)) g(ok(j), ok(i)) 1, ? i, j
  ? 1,..,n
• 3. g(ok(i), ok(j)) gt 1 if ok(i) lt ok(j), ? i,
  j ? 1,..,n

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Contents

• Introduction to the Problem
• Preference Representations
• Making the Information Uniform
• Decision Model
• Fuzzy Majority and OWG operator
• Collective Multiplicative Preference Relation
• Choosing the alternatives
• Conclusion

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Fuzzy Majority

• soft majority concept ? fuzzy linguistic
  quantifier
• fuzzy linguistic quantifier ? most, at least
  half, as many as possible
• fuzzy quantifier semantic ? fuzzy sets

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Fuzzy Linguistic Quantifiers
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The Ordered Weighted Geometric Operator

• Let a1,a2,,am list of values to aggregate
• OWG operator
• FG Rm ? R
• ck kth largest value in a1,a2,,am
• - order value vector
• wk Q(k/m) Q((k-1)/m), k 1,,m
• - exponential weighting vector

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The Ordered Weighted Geometric Operator

• OWG - fuzzy majority guided aggregation operator

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Collective Multiplicative Preference Relation

• aggregate existing multiplicative preference
  relations ? Ac
• use the OWG operator at least half

• indicates how much xi is preferred to xj
• according to at least half the experts

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Choosing the Alternative(s)

• Use two quantifier guided choice degrees of
  alternatives
• Quantifier Guided Dominance Degree
• Quantifier Guided Non Dominance Degree
• defined over Ac, based on fuzzy majority (most)

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Quantifier Guided Dominance Degree
for any xi
aggregate
- quantifies the degree to which alternative xi
dominates over most alternatives according
to at least half of the experts
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Quantifier Guided Dominance Degree

• Maximum dominance elements
• alternatives that most dominate the other ones

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Quantifier Guided Non Dominance Degree
for any xi
- quantifies the degree to which alternative xi
is not dominated by most alternatives
according to at least half of the experts
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Quantifier Guided Non Dominance Degree

• Maximal non-dominance elements
• alternatives that are least dominated by the
  other ones

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Choosing the Alternative(s)
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Choosing the Alternative(s)

• What if the intersection is empty?
• dominance based sequential selection
• select those elements from XMQGDD that have the
  highest MQGNDD values
• non dominance based sequential selection
• select those elements from XMQGNDD that have the
  highest MQGDD values

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Contents

• Introduction to the Problem
• Preference Representations
• Making the Information Uniform
• Decision Model
• Conclusion

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Conclusion
Record Expert Preferences preference
ordering utility functions multiplicative
relations
Present the information in a uniform manner by
means of multiplicative relations
Run decision model collective multiplicative
preference relation OWG operator dominance
degree non dominance degree
Selection Set of Alternative(s)
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Questions