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FUZZY MCDM

FUZZY MCDM. Fuzzy number. DEFINITION is defined to be a fuzzy triangular number if its membership function. Operations of fuzzy numbers. The basic operations on fuzzy triangular numbers. RANKING OF FUZZY NUMBERS.

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FUZZY MCDM

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  1. FUZZY MCDM Borgulya I

  2. Fuzzy number • DEFINITION is definedto be a fuzzy triangularnumberifitsmembershipfunction Borgulya I

  3. Operations of fuzzy numbers • The basicoperationson fuzzy triangularnumbers Borgulya I

  4. RANKING OF FUZZY NUMBERS • Let μi(x) denote the membership function for the fuzzy number ni. • nidominates (or outranks) nj, written as ni> nj, if and only ifeij= 1 and eji < Q, where Q is some fixed positive fraction less than one. Borgulya I

  5. RANKING OF FUZZY NUMBERS • EXAMPLE Suppose that the importances of two alternatives A1 andA2 are represented by the two fuzzy triangular numbers nl= (0.2, 0.4, 0.6)and n2= (0.4,0.7,0.9), respectively. Next, observe thate21= 1 and e12 = 0.4 < Q = 0.9. Therefore, the best alternative is A1. Borgulya I

  6. RANKING OF FUZZY NUMBERS Borgulya I

  7. Borgulya I

  8. The Fuzzy Weighted-SumModel • Crispmodel: • Fuzzy model: • the jth criterion is a fuzzy triangular number denoted asaij= (aijl, aijm, aiju).Weights are denoted as wj= (Wyl, Wjm , Wju). Borgulya I

  9. The Fuzzy Weighted-SumModel Borgulya I

  10. The Fuzzy Weighted-SumModel Borgulya I

  11. The Fuzzy Weighted-ProductModel • Crispmodel: • Fuzzy model: • where aKj, aLj, and wjare fuzzy triangular numbers. Alternative AKdominates alternative AL if and only if thenumerator inR is greaterthanthedenominator (R greaterthanorequaltoone) Borgulya I

  12. The Fuzzy Weighted-ProductModel Borgulya I

  13. The Fuzzy Weighted-ProductModel e31 = e32 = e12 = 1, and e13 , e23 , e21 are less than Q (= 0.9). Borgulya I

  14. The Fuzzy TOPSIS Method Crispmodel: • The basicconcept of thismethod is thattheselectedbestalternativeshouldhavetheshortestdistancefromtheidealsolutionand thefarthestdistancefromthenegative-idealsolutionin a geometrical (i.e., Euclidean) sense. Borgulya I

  15. TOPSIS Method Technique for Order Preference by Similarity to Ideal Solution • Step1: Constructthenormalizeddecisionmatrix • Step2: ConstructtheweightednormalizeddecisionmatrixV = (rijWj). • Step3: Determinetheideal and thenegative-idealsolutions. The ideal (A*) and thenegative-ideal (A-) solutionsaredefinedasfollows: Borgulya I

  16. TOPSIS Method Borgulya I

  17. TOPSIS Method • Step 4: Calculate the separation measure. • Step 5: Calculate the relative closeness to the ideal solution Step 6: Rank the preference order (Ci*) Borgulya I

  18. Fuzzy TOPSIS Method • Fuzzy model, EXAMPLE • Step 1: Construct the normalized decision matrix • Step 2: Construct the weighted normalized decision matrix Borgulya I

  19. Fuzzy TOPSIS Method • Step 3: Determine the idealand negative-ideal solutions. • Step 4: Calculate the separation measure Borgulya I

  20. Fuzzy TOPSIS Method • E.g. • Step 5: Calculate the relative closeness to the ideal solution Borgulya I

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