A Decision System Using ANP and Fuzzy Inputs

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 Goal: find the best variant C 1 V 1 C 2 … Cn - Criteria V 2 … Vm - 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 GOAL CRITERIA Feedback VARIANTS Content

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

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

Uncertain evaluations 1 0 a. L a. M a. U 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 a. L > 0, b. L > 0 • Addition: • Subtraction: • Multiplication: • Division: • Particularly: 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 i = 1, 2, . . . , r Solution: Logarithmic method Content

Defuzzyfication • Result of synthesis: Triangular fuzzy vector, i. e. • Corresponding crisp (nonfuzzy) vector: where 1/3 z. L z. M x g z. U 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 - W 32* and W 22* 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 - W 32* and W 22* Case study

Case study - synthesis Case study

Case study - crisp case with fedback Crisp case: a. L = a. M = a. U Case study

Case study - crisp case NO fedback Crisp case: a. L = a. M = a. U, W 22 = 0 Case study

Case study - comparison Case study

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)