1. 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’

2. Fuzzy Ordering x1x1 x2x2 x3x3 x4x4 x1x1 1111 x2x2 0.7110.380.11 x3x3 0.6110.43 x4x4 0.111 C = C’ min f(x i | x) 1 0.11 0.43 0.67 The order is x 1, x 4, x 3, x 2

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

4. Preference and Consensus C is minimum, F maximum; r ij = r ji = 0.5 C is maximum, F minimum; r ij = 1 0   1/21/2   1 They are useful to determine consensus. There are different types of consensus.

5. Antithesis of consensus M 1 : Complete ambivalence or maximally fuzzy M 1 = M 2 : every pair of alternatives in definitely ranked All non-diagonal elements is 0 or 1. Alternative 1 is over alternative 2 M 2 = 00.5 0 0 0 0101 0010 1001 0100

6. Antithesis of consensus Three types of consensus: Type 1: one clear choice and remaining (n-1) alternatives have equal secondary preference. (r kj = 0.5k  j) M 1 * = Alternative 2 has clear consensus. 0 00.5 1011 00 0 0

7. Antithesis of consensus Type 2: one clear choice and remaining (n-1) alternatives have definite secondary preference. (r kj = 1k  j) M 2 * = 0 011 1011 1001 1010

8. Antithesis of consensus Type 3: Fuzzy consensus M f * : a unanimous decision and remaining (n-1) alternatives have infinitely many fuzzy secondary preference. M f * = Cardinality of a relation is the number of possible combinations of that type. 0 00.50.6 1011 0.5000.3 0.400.70

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

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

11. Example 010.50.2 000.30.9 0.50.700.6 0.80.10.40 It does not have consensus properties. We compute: Notice m(M 1 ) = 1 m(M 2 * ) = 0 Complete ambivalence

12. Multi-objective Decision Making A = {a 1,a 2,…,a n }: set of alternatives O = {o 1,o 2,…,o r }: set of objectives The degree of membership of alternative a in O j is given below. Decision function: The optimum decision a *

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

14. Multi-objective Decision Making If two alternatives x and y are tied, Since, D(a) = min i [C i (a)], there exists some alternative k, s.t. C k (x) = D(x) and alternative g, s.t. C g (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 = {S 1,S 2,…,S n }Set of states P = {P(s 1 ), P(s 2 ),…, P(s n )}  P(s i ) = 1 P(s i ): probability of state I. It is called “prior probability”, expressing prior knowledge A = {a 1, a 2,…, a m }, set of alternatives. For a j, we assign a utility value u ji if the future state is S i

16. Fuzzy Bayesian Decision Method Utility matrix s1s1 s2s2 … snsn a1a1 u 11 u 12 … u 1n ::::: amam u m1 u m2 … u mn Associated with the j th alternative

17. Fuzzy Bayesian Decision Method Example: Decide if should drill for natural gas. a 1 : drill for gas a 2 : do not drill u 11 : the decision is correct and big reward +5 u 12 : decision wrong, costs a lot –10 u 21 : lost –2 u 22 : 4U = 5 -10 -2 4

18. Fuzzy Bayesian Decision Method Decision Tree utility a 1 S 1 0.5u 11 = 5 S 2 0.5u 12 = -10 a 2 S 1 0.5u 11 = -2 S 2 0.5u 12 = 4 E(u 1 ) = 0.5  5 + 0.5  (-10) = 2.5 E(u 2 ) = 0.5  (-2) + 0.5  (4) = 1 So, E(u 2 ) is bigger, this is from the alternative a 2, the decision “ not drill” should be made. Should you need more information?

19. Fuzzy Bayesian Decision Method X = {x 1,x 2,…,x r } from r experiments or observations, used to update the prior probabilities. 1. New information is expressed in conditional probabilities.

20. Fuzzy Bayesian Decision Method The value of information V(x): X = {x 1,x 2,…,x r } imperfect information V(x) = E(u x * ) – E(u * ) Perfect information is represented by posterior probabilities of 0 or 1. Perfect information X p The value of perfect information