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A Decision System Using ANP and Fuzzy Inputs Jaroslav Ram ík Silesian University Opava School of Business Administration Karviná Czech Republic e-mail: ramik@opf.slu.cz FUR XII, Rome, June 2006. Content. Problem -AHP Dependent criteria – ANP Solution Case study Conclusion. Problem- AHP.

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  1. A Decision System Using ANP and Fuzzy InputsJaroslav RamíkSilesian University Opava School of Business Administration Karviná Czech Republice-mail: ramik@opf.slu.czFUR 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 Supermatrix: - matrix of feedback between the criteria Content

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

  11. Uncertain evaluations 1 0 aL aM aU Triangular fuzzy number Content

  12. Uncertain pair-wise comparisons Reciprocity 0 ¼ 1/3 ½ 1 2 3 4 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: 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 Case study

  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. 233-247, ISSN 0165-0114. • 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 3-540-54998-6. • Horn, R. A., Johnson, C. R.,Matrix Analysis, Cambridge University Press, 1990, ISBN 0521305861. • Ramik, J., Duality in fuzzy linear programming with possibility and necessity relations. Fuzzy Sets and Systems 157, 2006, 1, p. 1283-1302, ISSN 0165-0114. • Saaty, T.L., Exploring the interface between hierarchies, multiple objectives and fuzzy sets. Fuzzy Sets and Systems 1, 1978, p. 57-68, ISSN 0165-0114. • 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 0-9620317-9-8. • Van Laarhoven, P.J.M. and Pedrycz, W., A fuzzy extension of Saaty's priority theory. Fuzzy Sets and Systems 11, 1983, 4, p. 229-241, ISSN 0165-0114.

  32. DĚKUJI VÁM(Thank You)

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