fuzzy ordering
Post on 01-Jan-2016
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Fuzzy Ordering
Ci’ = min f(xi | x) i = 1,2,…,nCi’ is the membership ranking for the ith variable.
Example:
0.13.01.03.0
7.00.13.05.0
9.08.00.17.0
2.03.05.00.1
4321
4321
4321
4321
4444
3333
2222
1111
xfxfxfxf
xfxfxfxf
xfxfxfxf
xfxfxfxf
xxxx
xxxx
xxxx
xxxx
Computing C matrix and C’
Fuzzy Ordering
x1 x2 x3 x4
x1 1 1 1 1
x2 0.71 1 0.38 0.11
x3 0.6 1 1 0.43
x4 0. 1 1 1
C =
C’
min f(xi | x)
1
0.11
0.43
0.67
The order is x1, x4, x3, x2
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:
2/1
2/1
~~
~
2
~
~
nn
RRtRC
nn
RtRF
Tr
r
Preference and Consensus
1~~
RCRF
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.
~RF
~RC
Antithesis of consensus
M1: Complete ambivalence or maximally fuzzy
M1 =
M2: every pair of alternatives in definitely rankedAll non-diagonal elements is 0 or 1.Alternative 1 is over alternative 2
M2 =
0 0.5 0.5 0.5
0.5 0 0.5 0.5
0.5 0.5 0 0.5
0.5 0.5 0.5 0
0 1 0 1
0 0 1 0
1 0 0 1
0 1 0 0
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.
0 0 0.5 0.5
1 0 1 1
0.5 0 0 0.5
0.5 0 0.5 0
Antithesis of consensus
Type 2: one clear choice and remaining (n-1) alternatives have definite secondary preference.
(rkj = 1 k j)
M2* =
0 0 1 1
1 0 1 1
1 0 0 1
1 0 1 0
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 possiblecombinations of that type.
0 0 0.5
0.6
1 0 1 1
0.5
0 0 0.3
0.4
0 0.7
0
Antithesis of consensus
*
2
21
2
23*2
*1
2
1
2
2
2
1
2
f
nn
nn
M
n
nM
nM
M
M
(Type 1)
(Type 1)
(Type fuzzy)
Distance to consensus
0
21
0
1
121
~
21
~
~
~
21
~~
RM
nRM
RM
RM
RCRM
For M1 preference relation
For M2 preference relation
For M1* consensus relation
For M2* consensus relation
Example
0 1 0.5 0.2
0 0 0.3 0.9
0.5 0.7 0 0.6
0.8 0.1 0.4 0
~R
It does not have consensus properties.
We compute:
Notice m(M1) = 1 m(M2*) = 0
Complete ambivalence
293.0
395.0683.0
*1
~~
Mm
RmRC
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*
aaa
OOOD
rOOO
r
,...,min
...
1
21
aa DAa
D
max*
Multi-objective Decision Making
Define a set of preferences {P}Parameter bi is contained on set {P}
aaaa
Hence
aaa
ObCLet
aObD
aOb
aObbaOM
bOM
bOMbOMbOMD
r
iii
CCCAa
D
ObC
iii
ii
ii
iiii
ii
rr
,...,,minmax
,
,max
,
,
,...,,
21
'
*
'
'
'
2211
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)
yxa
aDyDxDeiAa
max..
.,ˆˆ
ˆˆ
minˆ
minˆ
xselectweyDxDIf
yDxDcompareThen
yCyD
xCxDLet
igl
iki
If a tie still presents, continue the process similar to the one above.
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
Fuzzy Bayesian Decision Method
Utility matrixsn…s2s1
u1n…u12u11a1
:::::
umn…um2um1am
n
iijij sPuuE
1
Associated with the jth alternative
jj
uEuE max*
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
Fuzzy Bayesian Decision Method
Decision Tree utility
a1 S1 0.5 u11 = 5S2 0.5 u12 = -10
a2 S1 0.5 u11 = -2S2 0.5 u12 = 4
E(u1) = 0.5 5 + 0.5 (-10) = 2.5E(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
X = {x1,x2,…,xr} from r experiments or observations, used to update the prior probabilities.
1. New information is expressed in conditional probabilities.
k
r
kkx
kjj
k
kijikj
iikk
k
iikki
ik
xPxuEuE
xuExuE
xSPuxuE
SPSxPxP
xP
SPSxPxSP
SxP
1
**
*
|
|max|
||
|
||
|
Fuzzy Bayesian Decision Method
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
0
1| ki xsP
**
1
** |
uEuExV
xPxuEuE
xpp
k
r
kkxpxp
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