1. A Decision System Using ANP and Fuzzy Inputs Jaroslav Ramík Silesian University Opava School of Business Administration Karviná Czech Republic FUR XII, Rome, June 2006

2. Content Problem -AHP Dependent criteria – ANP Solution Case study
Conclusion

3. Problem- AHP MADM problem – AHP AHP- supermatrix AHP- limiting matrix
Content

4. MADM problem – AHP
- Criteria
- Variants
Content

5. AHP – supermatrix
Supermatrix:
Content

6. AHP- limiting matrix Limiting matrix:
- vector of evaluations of variants (weights)
Content

7. Dependent criteria – ANP
Dependent evaluation criteria – ANP
Dependent criteria – supermatrix
Dependent criteria – limiting matrix
Uncertain evaluations
Uncertain pair-wise comparisons
Content

8. Dependent evaluation criteria – ANP
Feedback
Content

9. Dependent criteria – supermatrix
- matrix of feedback between the criteria
Content

10. Dependent criteria – limiting matrix
- vector of evaluations of variants (weights)
Content

11. Uncertain evaluations1
aL aM aU
Triangular fuzzy number
Content

12. Uncertain pair-wise comparisons
Reciprocity
0 ¼ 1/3 ½
Content

13. Solution Fuzzy evaluations Fuzzy arithmetic Fuzzy weights and values
Defuzzyfication
Algorithm
Content

14. Fuzzy evaluations Fuzzy values (of criteria/variants):
Triangular fuzzy numbers: , k = 1,2,...,r
Normalized fuzzy values:
Content

15. Fuzzy arithmetic aL > 0, bL > 0 Addition: Subtraction:
Multiplication:
Division:
Particularly:
aL > 0, bL > 0
Content

16. Fuzzy weights and values
Triangular fuzzy pair-wise comparison matrix (reciprocal):
approximation
of the matrix:
Content

17. Fuzzy weights and values
Solve the optimization problem:
subject to
Solution:
i = 1,2,...,r
Logarithmic method
Content

18. Defuzzyfication Result of synthesis: Triangular fuzzy vector, i.e.
Corresponding crisp (nonfuzzy) vector:
where
1/3
zL zM xg zU
Content

19. Algorithm Step 1: Calculate triangular fuzzy weights
(of criteria, feedback and variants):
Step 2: Calculate the aggregating triangular fuzzy evaluations of the variants: or
Step 3: Find the „best“ variant using a ranking method (e.g. Center Gravity)
Content

20. Case study Case study - outline Case study - criteria
Case study - variants
Case study - feedback
Case study - W32* and W22*
Case study - synthesis
Case study - crisp case with fedback
Case study - crisp case NO fedback
Case study - comparison
Content

21. Case study - outline
Problem: Buy the best product (a car) 3 criteria 4 variants
Data: triangular fuzzy pair-wise comparisons  fuzzy weights
Calculations: 1. with feedback
2. without feedback
Crisp case: „middle values of triangles“
Case study

22. Case study - criteria
Case study

23. Case study - variants
Case study

24. Case study - feedback
Case study

25. Case study - W32* and W22*
Case study

26. Case study - synthesis
Case study

27. Case study - crisp case with fedback
Crisp case: aL = aM = aU
Case study

28. Case study - crisp case NO fedback
Crisp case: aL = aM = aU, W22 = 0
Case study

29. Case study - comparison

30. Conclusion
Fuzzy evaluation of pair-wise comparisons may be more comfortable and appropriate for DM
Occurance of dependences among criteria is realistic and frequent
Dependences among criteria may influence the final rank of variants
Presence of fuzziness in evaluations may change the final rank of variants
Case study

31. References
Buckley, J.J., Fuzzy hierarchical analysis. Fuzzy Sets and Systems 17, 1985, 1, p , ISSN
Chen, S.J., Hwang, C.L. and Hwang, F.P., Fuzzy multiple attribute decision making. Lecture Notes in Economics and Math. Syst., Vol. 375, Springer-Verlag, Berlin – Heidelberg 1992, ISBN
Horn, R. A., Johnson, C. R., Matrix Analysis, Cambridge University Press, 1990, ISBN
Ramik, J., Duality in fuzzy linear programming with possibility and necessity relations. Fuzzy Sets and Systems 157, 2006, 1, p , ISSN
Saaty, T.L., Exploring the interface between hierarchies, multiple objectives and fuzzy sets. Fuzzy Sets and Systems 1, 1978, p , ISSN
Saaty, T.L., Multicriteria decision making - the Analytical Hierarchy Process. Vol. I., RWS Publications, Pittsburgh, 1991, ISBN .
Saaty, T.L., Decision Making with Dependence and Feedback – The Analytic Network Process. RWS Publications, Pittsburgh, 2001, ISBN
Van Laarhoven, P.J.M. and Pedrycz, W., A fuzzy extension of Saaty's priority theory. Fuzzy Sets and Systems 11, 1983, 4, p , ISSN

32. DĚKUJI VÁM (Thank You)