# A Method for Ranking Fuzzy Numbers and Its Application to Decision by rt3463df

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```									 A Method for Ranking Fuzzy
Numbers and Its Application to
Decision-Making

H. Lee-Kwang
Jee-Hyong Lee
Outline

The Problem - "Decision Making with Uncertainty"
Concept          - Fuzzy Numbers

The Solution - "Method to Rank Fuzzy Numbers"
Phase 1           - Evaluation
Concept           - Viewpoint
Concept           - Satisfaction Function
Concept           - Evaluation Value
Phase 2           - Ordering
Concept           - Relative Index

The Conclusion
2         Review, Examples, and Summary
THE PROBLEM
DECISION MAKING UNDER UNCERTAINTY

Identifying the Problem

Faced with a collection of possible options,
we would like to know which possibility is
better than all the others.

In order to determine which is better, we must
be able to assign a value to how good each
option is and then rank them in order.

But what if the values for how 'good' a
3   possibility is are fuzzy values?
THE PROBLEM
DECISION MAKING UNDER UNCERTAINTY

What Are Fuzzy Numbers?

"a QUANTITY with an IMPRECISE VALUE"

A fuzzy number is a fuzzy set Ã   such that:

•   Ã is convex and normal

•   μÃ is (at least) segmentally continuous

•   μÃ(x) = 1 for one and only one value in Ã
4
THE PROBLEM
DECISION MAKING UNDER UNCERTAINTY

A Motivational Example

Which number is larger?
1
F1                F2
It depends on the needs of
(the viewpoint of the user)
0           5              10

Optimistic Viewpoint:              Pessimistic Viewpoint:
"since F1 could be larger"          "since F1 could be smaller"

5                F1 > F2                         F2 > F1
THE SOLUTION
RANKING FUZZY NUMBERS - OVERVIEW

Proposed Method For Ranking

1.    EVALUATION

a)   fuzzy numbers are compared with a viewpoint
provided by the user

b)   the degree to which each fuzzy number is
larger than the viewpoint is evaluated

2.    ORDERING

a)   fuzzy numbers are put into order according to
the degrees evaluated
6
THE SOLUTION
RANKING FUZZY NUMBERS - EVALUATION

Evaluation - What Is A Viewpoint

For set of fuzzy numbers X, a fuzzy set V which:

i) ÃX, Support(A)  Support(V)

ii)      ∞
-∞∫    μv(x) dx exists and is non-zero

is a viewpoint.

7
THE SOLUTION
RANKING FUZZY NUMBERS - EVALUATION

Evaluation - Example Viewpoints

"INTERVAL"
uniform distribution of membership

"OPTIMISTIC"            "PESSIMISTIC"

membership increases    membership decreases
8   as numbers increase     as numbers increase
THE SOLUTION
RANKING FUZZY NUMBERS - EVALUATION

Evaluation - A Satisfaction Function

To compare a fuzzy number to a viewpoint, a
satisfaction function is used.

A   B             Think of the satisfaction
x         function as the area on
A                         the graph where the value
A>B   of A is greater than the
B                         value of B, divided by
y                     the whole area.
y=x
9
THE SOLUTION
RANKING FUZZY NUMBERS - EVALUATION

Evaluation - A Satisfaction Function

S(A > B) =      ∞   y
-∞∫ -∞∫  μA(x) μB(y) dx dy

∞   ∞
-∞∫ -∞∫ μA(x) μB(y) dx dy

S(A > B) represents the possibility that fuzzy
value A is greater than fuzzy value B.

This formula uses a t-norm , which is a fuzzy
intersection. For simplicity, we will use
multiplication, which is a valid t-norm.
10
THE SOLUTION
RANKING FUZZY NUMBERS - EVALUATION

Evaluation - Evaluation Values

The evaluation value of A compared to a
viewpoint V is denoted EV(A) and is defined as:

EV(A) = S (A > V)

It is the ratio of integration of μA(x)μV(y)
of the area where A > V, over the entire
area.
11
THE SOLUTION
RANKING FUZZY NUMBERS - ORDERING

Ordering - Relative Indices

Fuzzy numbers evaluated against a viewpoint
can be ordered with a standard ranking
algorithm based on their EV(A) values.

The relative index for a fuzzy number A, from
set X, for a viewpoint V, is defined as:

RV(A) =       EV(A)       RV(A) shows how close
    A is to the best member
12             maxFX{EV(F)}   of X for viewpoint V.
THE SOLUTION
RANKING FUZZY NUMBERS - REVIEW

Review of the Proposed Method

For ranking a set of fuzzy numbers, X:

1) Define a fuzzy set V to act as viewpoint

2) Evaluate EV(A) for each A  X

3) Rank the fuzzy numbers according to EV
(using a standard sorting algorithm)

13   4) Evaluate RV(A) for each A  X
THE CONCLUSION
EXAMPLE

Sample (With Interval Viewpoint)

1        F2   F4 F3   F1      1
V

0 0.2 0.4 0.6 0.8 1.0         0 0.2 0.4 0.6 0.8 1.0

F1      F2          F3      F4
EV   0.900   0.500       0.650   0.400
14            RV   1.000   0.556       0.722   0.444
THE CONCLUSION
EXAMPLE

Sample (With Optimistic Viewpoint)

1        F2   F4 F3   F1      1     OPTIMISTIC
V

0 0.2 0.4 0.6 0.8 1.0         0 0.2 0.4 0.6 0.8 1.0

F1      F2          F3      F4
EV   0.810   0.271       0.420   0.172
15            RV   1.000   0.334       0.518   0.212
THE CONCLUSION
SUMMARY

Summary

 Compared with a number of existing methods

 Applied to a game of fuzzy profit and loss

 Positive, intuitive results

Questions?