fuzzy ordering

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


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


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


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


*

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*


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


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


Utility matrixsn…s2s1

u1n…u12u11a1

:::::

umn…um2um1am

n

iijij sPuuE

1

Associated with the jth alternative

jj

uEuE max*


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


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?


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|

||

|

||

|


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

fuzzy ordering

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