1. Fuzzy Integrals in Multi- Criteria Decision Making Dec. 2011 Jiliang University China

2.  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

3. Multi-Criteria Decision Making Problem

4. Aggregation in MCDM

5.  Mathematical Properties Properties of extreme values Idempotency Continuity Non-decreasing w.r.t. each argument Stability under the same positive linear transform Requirements of Aggregation Operator

6.  Behavioral Properties Expressing the weights of unequal importance on criteria Expressing the behavior of decision maker from perfect tolerance (disjunctive behavior) to total intolerance (conjunctive behavior) 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 important jointly Easy semantic interpretation of aggregation operator Requirements of Aggregation Operator

7.  Quasi-arithmetic Mean Example: Mean and Generalized Mean Common Aggregation Operator

8.  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 importance in the aggregation process When all weights are 0, then weighted maximum becomes the max-operator Common Aggregation Operator

9.  Ordered weighted averaging (OWA) Weighted average of ordered input Note: Common Aggregation Operator

10.  Fuzzy measure  Additivity, Super-additivity, Sub-additivity Fuzzy Measures

11.  Sugeno’s g-lamda measure Fuzzy Measure

13. Note: We need only n numbers of fuzzy density instead of 2 n.

14. Fuzzy Measures and Integrals

16.  Sugeno and Choquet integral are idempotent, continuous, and monotonically non-decreasing operators.  Choquet integral with additive measure coincide with a weighted arithmetic mean.  Choquet integral is stable under positive linear transforms.  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

17.  Any OWA operator is a Choquet integral.  Sugeno and Choquet integral contains all order statistics, thus in particular, min, max, and the median.  Weighted minimum and weighted maximum are special case of Sugeno integral Properties of Fuzzy Integrals

18. Example: Importance of Criteria and Interaction Rank Order: A > C > B

19. Importance of Criteria and Interaction Rank Order: C > A > B

20. Index for Importance Multiplied by n

21. Index of Interaction Note: Redundancy and synergy

22.  Identification Based on Semantics Importance of criteria Interaction between criteria Symmetric criteria {math, physics} Veto effect Pass effect Identification of Fuzzy Measure

23.  Identification Based on Learning Data Identification of Fuzzy Measure M. Grabish, H. T. Nguyen, and E. A. Walker, Fundamentals of Uncertainty Calculi, with Applications to Fuzzy Inference, Kluwer Academic, 1995

24. Decision Making in Pattern recognition Feature level simple classifier Aggregation of class memberships Input pattern Class label

25. Decision Making in Pattern recognition Input pattern Complex classifiers Aggregation of class memberships Class label

26.  Multi-Criteria Decision Making Problem and Aggregation Operators  Fuzzy Integrals have useful properties required for aggregation operator in multi-criteria decision making Not only degree of importance foe a separate criterion but also redundancy 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