A Decision System Using ANP and Fuzzy Inputs Jaroslav Ramík 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 MADM problem – AHP AHP- supermatrix AHP- limiting matrix Content

MADM problem – AHP - Criteria - Variants Content

AHP – supermatrix Supermatrix: Content

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

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

Dependent evaluation criteria – ANP Feedback Content

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

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

Uncertain evaluations 1 0 aL aM aU Triangular fuzzy number Content

Uncertain pair-wise comparisons Reciprocity 0 ¼ 1/3 ½ 1 2 3 4 Content

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

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

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

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

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

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

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

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

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

Case study - criteria Case study

Case study - variants Case study

Case study - feedback Case study

Case study - W32* and W22* Case study

Case study - synthesis Case study

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

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

Case study - comparison

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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DĚKUJI VÁM (Thank You)