A new method for ranking of fuzzy numbers through using distance method

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1 From the SelectedWorks of Saeid bbasbandy 3 new method for ranking of fuzzy numbers through using distance method S. bbasbandy. Lucas. sady vailable at:

2 new method for ranking of fuzzy numbers through using distance method S. bbasbandy a*. Lucas b. sady c a Department of Mathematics Imam Khomeini International University Qazvin Iran.(saeid@abbasbandy.com) b enter of Ecellence for ontrol and Intelligent processing Tehran University and SIS IPM Tehran Iran. c Department of Mathematics rak ranch zad University rak Iran. bstract In this paper by using a new approach on distance between two fuzzy numbers we construct a new ranking system for fuzzy number which is very realistic and also matching our intuition as the crisp ranking system on R. Keywords: Fuzzy numbers; Ranking of fuzzy numbers; Distance. Introduction In many fuzzy applications defuzzification of fuzzy numbers is very important. The differences between this study and Yager [7] Yao and Wu [8] and hen [3] are as follows : Yager [7] used a weighted mean value (or centroid ( ) d µ ( ) µ d ) to define ordering. This is different from our work and results in a difference which is stated in section 3 eample set 3 (see Table ). Yao and Wu [8] used from signed distance to define ordering. It is different from our work and results in a difference which is in section 3 eample set 4 (see Table ). hen [3] first normalized fuzzy numbers and then used from maimizing set and minimizing set to define ordering. It is different from our work and results in a difference which is stated in section 3 eample set 3 (see Table ).

3 First we define a fuzzy origin for fuzzy numbers then according to the distance of fuzzy numbers with respect to this origin we rank them. The basic definitions of fuzzy number are given as follows [5]. Definition.. fuzzy number is a fuzzy set u : R I = [] which satisfies:. u is upper semicontinuous. u ( ) = outside some interval [cd] 3. There are real numbers a b such that c a b d and 3. u () is monotonic increasing on [ c a] 3. u () is monotonic decreasing on [ b d] 3.3 u ( ) = a b. The set of all the fuzzy numbers ( as given by Definition. ) is denoted by E. n equivalent parametric is also given in [5]. Definition.. fuzzy number u in parametric form is a pair ( u u) of functions u ( u ( r which satisfy the following requirements:. u ( is a bounded monotonic increasing left continuous function. u ( is a bounded monotonic decreasing left continuous function.3 u( u( r.

4 popular fuzzy number is the trapezoidal fuzzy number u y σ ) with two ( β defuzzifiers y and left fuzziness σ and right fuzziness β where the membership function is ( + σ ) σ u ( ) = ( y + β ) β y σ y y Otherwise. + β Definition.3. For arbitrary fuzzy numbers u = ( u u) and v = ( v v) the quantity d ( u v) ( u( v( ) dr + ( u( v( ) dr = is the distance between u and v [] [6].. Ranking of fuzzy number with distance method Let all of fuzzy numbers are positive or negative. Without less of generality assume that all of them are positive. The membership function of a R is u a ( ) = if = a ; and u a ( ) = if a. Hence if a = we have the following u ( = ) =. Since u ( ) E left fuzziness σ and right fuzziness β are so for each u( ) E Thus we have the following definition. d ( u u ) = ( ) ( ) u r dr + u r dr. 3

5 Definition.. For u and v d u u ) > d( v ) if and only if u f v ( u d u u ) < d( v ) if and only if u p v ( u d u u ) = d( v ) if and only if u v. ( u Property.. Suppose u and v (i) if u = v then u v (ii) if v u and E define the ranking of u and v by saying E are arbitrary then: + u( > v( v( ) for all [] u ( + r r then v p u. Remark.. The distance triangular fuzzy number u = ( σ β ) of u is defined as following [ + σ 3 + β 3 + ( β )]. d ( u u ) = σ Remark.. The distance trapezoidal fuzzy number u = ( y σ β ) of u is defined as following [ + y + σ 3 + β 3 σ y ]. d ( u u ) = + β Remark.3. If u v it is not necessary that u = v. Since if u v and + u( ) = ( v( v( ) ) then v ( u ( + r u. 4

6 3. Discussion popular fuzzy number u is the symmetric triangular fuzzy number s [ σ ] centered at with basis σ ( + σ ) σ u ( ) = ( + σ ) σ σ otherwise + σ which its parametric form is u( σ + σr u( + σ σr = = Remark 3.. The distance triangular fuzzy number s [ σ ] of u is defined as σ ] u ) = [ + ] d ( s[ σ. 3 Our work is essentially two different approaches essentially yield the same result. If for ranking fuzzy numbers s [ σ ] s [ β ] and β σ we used Yao and Wu [8] method we would have s σ ] = s[ ]. ut with our method we have s σ ] s[ ]. [ β [ β We shall now compare the methods used by other authors in [] [3] [4] [7] [8] and our method with four sets of eamples taken Yao and Wu [8]. Set : =(.5..5) =(.7.3.3) =(.9.5.). Set : =(.4.7..)(trapezoidal fuzzy numbe =(.7.4.) =(.7..). Set 3: =(.5..) =(.5.8..) =(.5..4). Set 4: =( ) =(.5.3.4) =(.6.5.). 5

7 Figure Figure 6

8 Table set Fuzzy number 3 4 hoobinech and Li < < < < < < < < Yager hen aldwin and Guild < < < < < < < < < < < < < < < < < < < < < < Wu < < < < < < < y using our method we have the following results: For set : d ( u )=.8869 d ( u )=.94 d ( u )=.65 and by Definition.: For set : d ( u )=.8756 d ( u )=.95 d ( u )=.33 and by Definition.: For set 3: d ( u )=.757 d ( u )=.946 d ( u )=.865 and by Definition.: For set 4: d ( u )=.7853 d ( u )=.7958 d ( u )=8386 and by Definition.: < < < < <. < < <. 7

9 4. onclusion In Table we have the following results: In set our method has the same result as in other five papers. In set our method has the same result as in the other four papers. However in set 3 we and Wu have p p but all the other four papers have p p. We can see from Fig. that define ordering p p is better than define ordering p p. In set 4 Fig p p our method leads to the same result as that of hoobinech and Li Yager and hen. The rest of Yao and Wu aldwin and Guild have p. References. S. bbasbandy and. sady note on " new approach for defuzzification" Fuzzy Sets and Systems 8 () J.F. aldwin and N..F. Guild omparison of fuzzy numbers on the same decision space Fuzzy Sets and Systems (979) S. hen Ranking fuzzy numbers with maimizing set and minimizing set Fuzzy Sets and Systems 7 (985) F. hoobinech and H. Li n inde for ordering fuzzy numbers Fuzzy Sets and Systems 54 (993) R. Goetschel W. Voman Elementary calculus Fuzzy Sets and Systems 8 (986) Ming Ma. Kandel and M. Friedman orrection to " New approach for defuzzification"fuzzy Sets and Systems 8 () R.R. Yager procedure for ordering fuzzy subests of the unit interval Inform. Sci. 4 (98) J. Yao and K. Wu Ranking fuzzy numbers based on decomposition principle and signed distance Fuzzy Sets and Systems 6 ()