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Fuzzy Ordering

Fuzzy Ordering. C i ’ = min f(x i | x) i = 1,2,…,n C i ’ is the membership ranking for the i th variable. Example:. Computing C matrix and C’. Fuzzy Ordering. C =. The order is x 1 , x 4 , x 3 , x 2. Preference and Consensus. Crisp set approach is too restrictive.

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Fuzzy Ordering

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  1. Fuzzy Ordering Ci’ = min f(xi | x) i = 1,2,…,n Ci’ is the membership ranking for the ith variable. Example: Computing C matrix and C’

  2. Fuzzy Ordering C = The order is x1, x4, x3, x2

  3. Preference and Consensus Crisp set approach is too restrictive. Define reciprocal relation R iii = 0 rij + rji = 1 rij = 1 implies that alternative I is definitely preferred to alternative j If rij = rji = 0.5, there is equal preference. Two common measures of preference: Average fuzziness: Average certainty:

  4. Preference and Consensus C is minimum, F maximum; rij = rji = 0.5 C is maximum, F minimum; rij = 1 0   1/2 1/2   1 They are useful to determine consensus. There are different types of consensus.

  5. Antithesis of consensus M1: Complete ambivalence or maximally fuzzy M1 = M2: every pair of alternatives in definitely ranked All non-diagonal elements is 0 or 1. Alternative 1 is over alternative 2 M2 =

  6. Antithesis of consensus Three types of consensus: Type 1: one clear choice and remaining (n-1) alternatives have equal secondary preference. (rkj = 0.5 k  j) M1* = Alternative 2 has clear consensus.

  7. Antithesis of consensus Type 2: one clear choice and remaining (n-1) alternatives have definite secondary preference. (rkj = 1 k  j) M2* =

  8. Antithesis of consensus Type 3: Fuzzy consensus Mf*: a unanimous decision and remaining (n-1) alternatives have infinitely many fuzzy secondary preference. Mf* = Cardinality of a relation is the number of possible combinations of that type.

  9. (Type 1) (Type 1) (Type fuzzy) Antithesis of consensus

  10. For M1 preference relation For M2 preference relation For M1* consensus relation For M2* consensus relation Distance to consensus

  11. Example It does not have consensus properties. We compute: Notice m(M1) = 1 m(M2*) = 0 Complete ambivalence

  12. Multi-objective Decision Making A = {a1,a2,…,an}: set of alternatives O = {o1,o2,…,or}: set of objectives The degree of membership of alternative a in Oj is given below. Decision function: The optimum decision a*

  13. Multi-objective Decision Making Define a set of preferences {P} Parameter bi is contained on set {P}

  14. Multi-objective Decision Making If two alternatives x and y are tied, Since, D(a) = mini[Ci(a)], there exists some alternative k, s.t. Ck(x) = D(x) and alternative g, s.t. Cg(y) = D(y) If a tie still presents, continue the process similar to the one above.

  15. Fuzzy Bayesian Decision Method • First consider probabilistic decision analysis • S = {S1,S2,…,Sn} Set of states • P = {P(s1), P(s2),…, P(sn)} • P(si) = 1 P(si): probability of state I. It is called “prior probability”, expressing prior knowledge A = {a1, a2,…, am}, set of alternatives. For aj, we assign a utility value uji if the future state is Si

  16. Fuzzy Bayesian Decision Method Utility matrix Associated with the jth alternative

  17. Fuzzy Bayesian Decision Method Example: Decide if should drill for natural gas. a1: drill for gas a2: do not drill u11: the decision is correct and big reward +5 u12: decision wrong, costs a lot –10 u21: lost –2 u22: 4 U = 5 -10 -2 4

  18. Decision Tree utility a1 S1 0.5 u11 = 5 S2 0.5 u12 = -10 a2 S1 0.5 u11 = -2 S2 0.5 u12 = 4 E(u1) = 0.5  5 + 0.5  (-10) = 2.5 E(u2) = 0.5  (-2) + 0.5  (4) = 1 So, E(u2) is bigger, this is from the alternative a2, the decision “ not drill” should be made. Should you need more information? Fuzzy Bayesian Decision Method

  19. Fuzzy Bayesian Decision Method X = {x1,x2,…,xr} from r experiments or observations, used to update the prior probabilities. 1. New information is expressed in conditional probabilities.

  20. The value of information V(x): X = {x1,x2,…,xr} imperfect information V(x) = E(ux*) – E(u*) Perfect information is represented by posterior probabilities of 0 or 1. Perfect information Xp The value of perfect information Fuzzy Bayesian Decision Method

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