Fuzzy Integrals in Multi- Criteria Decision Making Dec. 2011 Jiliang University China.
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Transcript of Fuzzy Integrals in Multi- Criteria Decision Making Dec. 2011 Jiliang University China.
![Page 1: Fuzzy Integrals in Multi- Criteria Decision Making Dec. 2011 Jiliang University China.](https://reader035.fdocuments.in/reader035/viewer/2022062809/5697bfe31a28abf838cb5086/html5/thumbnails/1.jpg)
Fuzzy Integrals in Multi-Crite-ria Decision Making
Dec. 2011 Jiliang University China
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Multi-Criteria Decision Making Problem Aggregation• Requirements of aggregation operators
• Common aggregation operators Fuzzy Measure and Integrals Properties of Fuzzy Integral Importance and Interaction of Criteria Decision Making in Pattern Recognition Summary
Contents
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Multi-Criteria Decision Making Problem
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Aggregation in MCDM
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Criteria
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Result Final
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Mathematical Properties• Properties of extreme values
• Idempotency
• Continuity
• Non-decreasing w.r.t. each argument
• Stability under the same positive linear trans-form
Requirements of Aggregation Operator
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Behavioral Properties• Expressing the weights of unequal importance on criteria
• Expressing the behavior of decision maker from perfect toler-ance (disjunctive behavior) to total intolerance (conjunctive be-havior)• Accept when some criteria are met
• Demand all criteria have to be equally met
• Expressing compensatory effect: • Redundancy when two criteria express the same things
• Synergy of two criteria: little importance separately but impor-tant jointly
• Easy semantic interpretation of aggregation operator
Requirements of Aggregation Operator
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Quasi-arithmetic Mean
Example: Mean and Generalized Mean
Common Aggregation Operator
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Median: mid-ordered data after sorting Weighted minimum and maximum
• When all weights are 1, then weighted minimum becomes the min-operator
• The larger weight value represents the more degree of impor-tance in the aggregation process
• When all weights are 0, then weighted maximum becomes the max-operator
Common Aggregation Operator
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Ordered weighted averaging (OWA)• Weighted average of ordered input
Note:
Common Aggregation Operator
)()2()1(
11)(1,...,
....
1 ),...,(OWA1
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mean Trimmed )0,2
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Fuzzy measure
Additivity, Super-additivity, Sub-additivity
Fuzzy Measures
)()( implies )2(
1)( ,0)( (1)
axioms. following satisfying ]1,0[)(:
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additivity sub :)()()(
additivitysuper :)()()(
measurey probabilitin additivity :)()()(
BABA
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Sugeno’s g-lamda measure
Fuzzy Measure
.or measure Sugeno called is gThen
.1 somefor g(B)g(A)g(B)g(A))(
, with )( allFor
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Fuzzy Measure
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Fuzzy Measure
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Note: We need only n numbers of fuzzy density instead of 2n.
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Fuzzy Measures and Integrals
)()1()()()()2()1(
)()(11
,...,, and )(...)()(0 where
)()(),...,(
w.r.t.]1,0[:function a of integral Sugeno
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Fuzzy Measures and Integrals
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3.0,6.0,9.0
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1.0,4.0,3.0
: w.r.t.function a of integral Sugeno
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Sugeno and Choquet integral are idempotent, contin-uous, and monotonically non-decreasing operators.
Choquet integral with additive measure coincide with a weighted arithmetic mean.
Choquet integral is stable under positive linear trans-forms.
Choquet integral is suitable for cardinal aggregation where numbers have a real meaning.
Sugeno integral is suitable for ordinal aggregation where only order makes sense.
Properties of Fuzzy Integrals
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Any OWA operator is a Choquet integral. Sugeno and Choquet integral contains all order sta-
tistics, thus in particular, min, max, and the median. Weighted minimum and weighted maximum are
special case of Sugeno integral
Properties of Fuzzy Integrals
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Example:
Importance of Criteria and Interaction
2 3 literphysicsmath www
25.15 8
102163 183_
AStudent
Rank Order: A > C > B
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Importance of Criteria and Interaction
1,, 0
synergy of because 3.045.09.0,
synergy of because 3.045.09.0,
redundancy of because 45.045.05.0,
3.0 45.0
literaturephysicsmath
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Rank Order: C > A > B
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Index for Importance
1
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indexShapley : of Importance Global
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jXj
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Multiplied by n
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Index of Interaction
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and !
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[-1,1] |,,
: and between Index n Interactio Average
,
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X
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xx
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xxXAjiXji
ji
ji
Note: Redundancy and synergy
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Identification Based on Semantics• Importance of criteria
• Interaction between criteria
• Symmetric criteria {math, physics}
• Veto effect
• Pass effect
Identification of Fuzzy Measure
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),...,,,...,(),...,,...,( 1111
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Identification Based on Learning Data
Identification of Fuzzy Measure
lkyzyzzCE kk
l
kkknk ,...,1 ),,( data learningfor ),...,(
error theminimize that measurefuzzy heIdentify t
1
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2
M. Grabish, H. T. Nguyen, and E. A. Walker, Fundamentals of Uncertainty Calculi, with Applications to Fuzzy Inference, Kluwer Academic, 1995
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Decision Making in Pattern recognition
x
1C
2C
3C
4C
1x
2x
3x
4x
)(1 H
Decision
Class
)(2 H
Feature level simple classi-fier
Aggregation of class member-ships
Input pattern Class
label
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Decision Making in Pattern recognition
1C
2C
3C
4C
)(HDecision
Class
x
Input pattern
Complexclassi-fiers
Aggregation of class memberships
Classlabel
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Multi-Criteria Decision Making Problem and Aggregation Op-erators
Fuzzy Integrals have useful properties required for aggrega-tion operator in multi-criteria decision making• Not only degree of importance foe a separate criterion but also redun-
dancy and synergy effects between criteria Identification of Fuzzy measure based on• Semantic involved in the decision making problem
• Learning data
• Semantics and learning data Application are diverse • Pattern Recognition
• Multi-sensor Fusion
Summary