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 à   such that:

      •   Ã is convex and normal

      •   μà 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 person asking.
                                          (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


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