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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
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Content Problem -AHP Dependent criteria – ANP Solution Case study
Conclusion
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Problem- AHP MADM problem – AHP AHP- supermatrix AHP- limiting matrix
Content
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MADM problem – AHP - Criteria - Variants Content
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AHP – supermatrix Supermatrix: Content
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AHP- limiting matrix Limiting matrix:
- vector of evaluations of variants (weights) Content
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Dependent criteria – ANP
Dependent evaluation criteria – ANP Dependent criteria – supermatrix Dependent criteria – limiting matrix Uncertain evaluations Uncertain pair-wise comparisons Content
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Dependent evaluation criteria – ANP
Feedback Content
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Dependent criteria – supermatrix
- matrix of feedback between the criteria Content
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Dependent criteria – limiting matrix
- vector of evaluations of variants (weights) Content
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Uncertain evaluations
1 aL aM aU Triangular fuzzy number Content
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Uncertain pair-wise comparisons
Reciprocity 0 ¼ 1/3 ½ Content
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Solution Fuzzy evaluations Fuzzy arithmetic Fuzzy weights and values
Defuzzyfication Algorithm Content
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Fuzzy evaluations Fuzzy values (of criteria/variants):
Triangular fuzzy numbers: , k = 1,2,...,r Normalized fuzzy values: Content
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Fuzzy arithmetic aL > 0, bL > 0 Addition: Subtraction:
Multiplication: Division: Particularly: aL > 0, bL > 0 Content
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Fuzzy weights and values
Triangular fuzzy pair-wise comparison matrix (reciprocal): approximation of the matrix: Content
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Fuzzy weights and values
Solve the optimization problem: subject to Solution: i = 1,2,...,r Logarithmic method Content
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Defuzzyfication Result of synthesis: Triangular fuzzy vector, i.e.
Corresponding crisp (nonfuzzy) vector: where 1/3 zL zM xg zU Content
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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
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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
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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
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Case study - criteria Case study
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Case study - variants Case study
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Case study - feedback Case study
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Case study - W32* and W22* Case study
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Case study - synthesis Case study
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Case study - crisp case with fedback
Crisp case: aL = aM = aU Case study
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Case study - crisp case NO fedback
Crisp case: aL = aM = aU, W22 = 0 Case study
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Case study - comparison
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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
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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
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DĚKUJI VÁM (Thank You)
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