Ordering Generalized Hexagonal Fuzzy Numbers Using Rank, Mode, Divergence and Spread

Transcription

1 IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn: x. Volume 1, Issue 3 Ver. II (May-Jun. 14), PP 15-.iosrjournals.org Ordering Generalized Hexagonal Fuzzy Numbers Using Rank, Mode, Divergence and Spread P. Rajarajesari 1,.Sahaya Sudha 1 (Department of Mathematics, Chikkanna Government rts College, Tirupur, India.) (Department of Mathematics, Nirmala College for omen, Coimbatore, India.) bstract: The Fuzzy set theory has been applied in many fields such as management, engineering and almost in every business enterprise as ell as day to day activities. Ordering fuzzy numbers plays an important role in approximate reasoning, optimization, forecasting, decision making, risk analysis, controlling, scheduling and various other usages in our day to day activities. This paper describes a ranking method for ordering fuzzy numbers based on rea, Mode, divergence, Spreads and Weights of generalized (non-normal) hexagonal fuzzy numbers. Keyords: Ranking function, Hexagonal fuzzy numbers, Centroid points, rea I. Introduction Ranking fuzzy number is used mainly in data analysis, artificial intelligence and various other fields of operations research. In fuzzy environment ranking fuzzy numbers is a very important in decision making procedure. Ranking fuzzy numbers ere first proposed by Jain [1] for decision making in fuzzy situations by representing the ill- defined quantity as a fuzzy set. Some of these ranking methods have been compared and revieed by Bortolan and Degani [], and more recently by Chen and Hang [3].Lee and Li [4] proposed the comparison of fuzzy numbers. Liou and Wang [5] presented ranking fuzzy numbers ith interval values. The centroids of fuzzy numbers have been examined recently. One of the most commonly used methods under the class of fuzzy scoring is the centroid point method.cheng [6] used a centroid based distance method to rank fuzzy numbers in 1998.Then Chu and Tsao[7] utilized the area beteen the centroid point and the origin to rank fuzzy numbers in.bbasbandi and sady[8] suggested a sign distance method for ranking fuzzy numbers in 6. Wang Y.J and Lee.H.S[9] proposed the revised method of ranking fuzzy numbers ith an area beteen the centroid and original points in8.since then several methods have been proposed by various researchers hich includes ranking fuzzy numbers using maximizing and minimizing set[1] decomposition principle and signed distance [11],different heights and spreads[1], rank, mode,divergence and,spread [13],area compensation distance method[14],ordering of trapezoidal fuzzy numbers[15].1 DEFINITION: [16] II. The Characteristic function (x ) Preliminaries of a crisp set X assigns a value of either 1 or to each individual in the universal setx.this function can be generalized to a function such that the value assigned to the element of the universal set X fall ithin a specified range (i.e) : X [,1 ]. The assigned value indicates the membership grade of the element in the set. The function is called the membership function and the set à = {( (x))/xєx} defind by for each x X. DEFINITION: [16] is called a fuzzy set. fuzzy set ~ defined on the universal set of real numbers R is said to be a fuzzy number of its membership function has the folloing characteristics (i) ~ : R [,1 ] is continuous for all x, a ] U[ a, ] (ii) ~ ( x) [ 1 4 (iii) ~ ( x) is strictly increasing on [ a ] and strictly decreasing on [ a,a 3 4 ] for all a, a ] (iv) ~ ( x) 1 1,a x [ 3, here a1 a a3 a4.iosrjournals.org 15 Page

2 .3 DEFINITION: [16] fuzzy set ~ defined on the universal set of real numbers R is said to be generalized fuzzy number of its membership function has the folloing characteristics (i) ~ : R [,1 ] is continuous for all x, a ] U[ a, ] (ii) ~ ( x) [ 1 4 (iii) ~ ( x) is strictly increasing on [ a a 1, ] and strictly decreasing on [ a,a 3 4 ] (iv) ~ ( x) for all x a, a ], here 1 [ 3.4 DEFINITION: [16] fuzzy number is a trapezoidal fuzzy number denoted by (a 1, a, a 3,a 4 ) and its membership function is given belo, here a 1 a a 3 a 4 (x) =.5 DEFINITION: [16] generalized fuzzy number ~ =( a 1, a, a3, a4 number if its membership function is given by (x) = ~ x a1 ( ), a1 x a a a1, a x a3 a4 x ( ), a3 x a a4 a3 4 ;) is said to be a generalized trapezoidal fuzzy III. Hexagonal Fuzzy Numbers 3.1 DEFINITION: [17] fuzzy number is a hexagonal fuzzy number denoted by (a 1, a, a 3, a 4, a 5, a 6 ) here a 1, a, a 3, a 4, a 5, a 6 are real numbers and its membership function (x) is given belo. (x) =.iosrjournals.org 16 Page

3 3. DEFINITION: [17] generalized fuzzy number is a hexagonal fuzzy number denoted by =(a 1, a, a 3, a 4, a 5, a 6, ) here a 1, a, a 3, a 4, a 5, a 6 are real numbers and its membership function (x) is given belo. (x) = 3.3 ORDERING OF HEXGONL FUZZY NUMBER: [18] Let = (a 1, a, a 3, a 4, a 5, a 6 ) and = (b 1, b, b 3, b 4, b 5, b 6 ) be in F(R) be the set of all real hexagonal fuzzy numbers i) if and only if a i = b i, i=1,,3,4,5,6 ii) if and only if a i b i, i=1,,3,4,5,6 iii) if and only if a i b i, i=1,,3,4,5,6 3.4 RNKING OF HEXGONL FUZZY NUMBERS: [18] n efficient approach for comparing the fuzzy numbers is by the use of a ranking function R: F (R) R, here F (R) is a set of fuzzy numbers defined on set of real numbers, hich maps each fuzzy number into a real number, here a natural order exists. For any to hexagonal fuzzy numbers =(a 1,a,a 3,a 4,a 5,a 6 )and =(b 1,b,b 3,b 4,b 5,b 6 ) e have the folloing comparison i) R( ) ii) R( ) R( ) iii) R( ) R( ) III. Proposed ranking Method Fig.1 Generalized hexagonal fuzzy number The centroid of a hexagonal fuzzy number is considered to be the balancing point of the hexagon (Fig.1).Divide the hexagonal into three plane figures.these three plane figures are a Triangle BQ, Hexagon CDERQB and again a triangle REF respectively. The circumcenter of the centroids of these three plane figures.iosrjournals.org 17 Page

4 is taken as the point of reference to define the ranking of generalized Hexagonal fuzzy numbers. Let the centroid of the three plane figures be G 1, G, G 3, respectively. The centroid of the three plane figures is G 1 = ; G = G 3 = respectively. Equation of the line is y= and G does not lie on the line G 3. Therefore G 1,G and G 3 are non- collinear and they form a triangle. We define the centroid of the triangle ith vertices G 1,G and G 3 of the generalized hexagonal fuzzy number =(a 1, a,a 3, a 4,a 5, a 6 ;) as The ranking function of the generalized hexagonal fuzzy number maps the set of all fuzzy numbers to a set of real numbers is defined as: (1) =(a 1, a,a 3, a 4,a 5, a 6 ;),hich R( = () This is the area beteen the centroid of the centroids original point. The mode of the generalized hexagonal fuzzy number Mode 1 ( a3 a4 ) dx (3) The divergence of the generalized hexagonal fuzzy number Diverence ( a a ) dx...(4 6 1 ) The left spread of the generalized hexagonal fuzzy number Leftspread ls ( a a1) dx ( a3 a ) dx ls ( a a1) dx (5) 3 The right spread of the generalized hexagonal fuzzy number Rightspread rs rs ( a a ) 4 dx 6 ( a6 a5 ) dx ( a5 a4 ) dx (6) =(a 1, a,a 3, a 4,a 5, a 6 ;) is defined as: as defined in (1) and () the =(a 1, a,a 3, a 4,a 5, a 6 ;) is defined as =(a 1, a,a 3, a 4,a 5, a 6 ;) is defined as =(a 1, a,a 3, a 4,a 5, a 6 ;) is defined as 4.1. PROPOSITION: If =(a 1, a,a 3, a 4,a 5, a 6 ; 1 ) and =(b 1,b,b 3,b 4,b 5,b 6 ; ) are to generalized Hexagonal fuzzy numbers then (i) R( = R ( (ii) Mode( = Mode( (iii) Divergence( Divergence( then,.iosrjournals.org 18 Page

5 (a) Left spread( > Left spread( ) if 1 a 3 > b 3 (b) Left spread( < Left spread( ) if 1 a 3 < b 3 (c) Left spread( = Left spread( ) if 1 a 3= b 3 Proof: From the assumptions (i) R( =R( (i.e.) (ii) Mode( = Mode( (i.e.) a a ) ( b b ) (8) 1( (iii) Divergence( Divergence( (i.e.) a a ) ( b ) (9) 1( b1 Solving (7),(8) and (9) 1 a 1= b 1 1 a 6= b 6 a a ) ( b ) 1( b4 1( a a5) ( b b5 (a) Left spread( > Left spread( ) a a ) ( b ) 1( b1 1 a 3 > b 3 ( 1 a 1= b 1 ) Hence Left spread( > Left spread( ) iff 1 a 3 > b 3 ) = (b) Left spread( < Left spread( ) a a ) ( b ) 1( b1 1 a 3 < b 3 ( 1 a 1= b 1 ) Hence Left spread( < Left spread( ) iff 1 a 3 < b 3 (c) Left spread( = Left spread( ) a a ) ( b ) 1( b1 1 a 3= b 3 (( 1 a 1= b ) Hence Left spread( = Left spread( ) iff 1 a 3 = b COROLLRY: ll the results of proposition 4.1 also hold for right spread. 4. PROPOSED PPROCH FOR RNKING GENERLIZED HEXGONL FUZZY NUMBER: If =(a 1, a,a 3, a 4,a 5, a 6 ; 1 ) and =(b 1,b,b 3,b 4,b 5,b 6 ; ) are to generalized hexagonal fuzzy numbers then Find R( and R ( Case (i) If R( > R ( then > Case (ii) If R( < R ( then <.iosrjournals.org 19 Page

6 Case (iii) If R( = R ( then = then go to Step Step Find mode ( and mode ( Case (i) If mode ( > mode ( then > Case (ii) If mode ( < mode ( then < Case (iii) If mode ( = mode ( then = then go to Step 3 Step 3 Find divergence ( and mode ( Case (i) If divergence ( > divergence ( then > Case (ii) If divergence ( < divergence ( then < Case (iii) If divergence ( = divergence ( then = then go to Step 4 Step 4 Find Left spread ( and Left spread ( Case (i) Left spread( > Left spread( ) i.e. 1 a 3 > b 3 then > (from proposition 4.1.) Case (ii) Left spread( < Left spread( ) ie 1 a 3 < b 3 then < (from proposition 4.1 ) Case (iii) Left spread( = Left spread( ) Step 5 Find 1 and i.e 1 a 3= b 3 then = (from proposition 4.1 ) Case (i) If 1 > then > Case (ii) If 1 < then < Case (iii) If 1 = then = IV. Result and Discussions Example: 1 Let =(.,.3,.5,.6,.8,.9;.7) and =(.4,.6,.1,.1,.16,.18;.35) R( =.693, R ( =.693, Since R( = R ( go to Step Step Mode( =.77, Mode ( =.77, Since Mode( = Mode ( go to Step 3 Step3 Divergence( =.49, Divergence ( =.49, Since Divergence( = Divergence ( go to Step 4 Step4 Left spread( =.1, Left spread ( =.1, Since Left spread( = Left spread ( go to Step 5 Step5.iosrjournals.org Page

7 1 =.7, =.35 Since 1 >, > Example: Let =(.17,.,.5,.38,.4,.45;.4) and =(.84,.4,.5,.76,.8,.9;.8) R( =. R ( =8.896, Since R( < R (, < Example:3 Let =(.3,.5,.5,.7,.8,.9;1) and =(.1,.5,.6,.7,.8,.9;1) R( =11.1, R ( = 11.1, Since R( = R (, go to Step Step Mode( =1., Mode ( =1.3, Since Mode( < Mode (, < Example:4 Let =(.1,.,.3,.6,.7,.8;1) and =(.,.,.4,.5,.7,.8;1) R( =8.3 R ( =8.3, Since R( = R (, go to Step Step Mode( =.9, Mode ( =.9, Since Mode( = Mode (, go to Step3 Step3 Divergence( =.7 Divergence ( =.6, Since Divergence( > Divergence ( > V. Conclusion In this paper, a hexagonal fuzzy ranking is used for centroid of a triangle from an earlier version. [17, 18] In this ranking method e have used rank, mode, divergence and spreads are used and e infer the results found ere satisfactory and the conditions are satisfied and this has been proved ith illustrative examples and if more number of parameters also taken e can make the decision making process simple in orking on risk and uncertainty ith optimized ranking order. This ranking procedure can be applied in various decision making problems such as, fuzzy optimization, Fuzzy HP. References [1] Jain.R Decision making in the presence of fuzzy variables, IEEE Transactions on systems, Man and Cybernetics 6 (1976) [] Bortolan, G and Degani R. revie of some methods for ranking fuzzy subsets, fuzzy sets and systems 15(1985) [3] Chen S.J and Hang C.L Fuzzy multiple attribute decision making, Springer, Berlin (199) [4] Lee E.S and Li.R.J, Comparison of fuzzy numbers based on the probability number of fuzzy events computers and mathematics ith applications 15 (1988) [5] Liou, T-S., Wang, M.J., (199), Ranking fuzzy numbers ith integral value. Fuzzy Sets and Systems, 5, pp [6] Cheng C.H, ne approach for ranking fuzzy numbers by distance method,fuzzy sets and systems Vol 95 (1998), [7] Chu T.C. and Tsao.C T. Ranking fuzzy numbers ith an area beteen the centroid points and the original point. Computers and mathematics ith applications 43 () [8] bbasbandy S and sady B Ranking of fuzzy numbers by sign distance, Information sciences 176 (6) [9] Wang Y.J and Lee.H.S The revised method of ranking fuzzy numbers ith an area beteen the centroid and original points,computers and mathematics ith applications,vol 55 No (8) [1] Chen S.H, Rank in fuzzy numbers ith maximizing set and minimizing set, Fuzzy sets and systems 17 (1985) [11] Yao, J., Wu, K., (), Ranking fuzzy numbers based on decomposition principle and signed distance, Fuzzy sets and Systems, 116, pp [1] Chen SM., Chen JH., Fuzzy risk analysis based on ranking generalized fuzzy numbers ith different heights and different spreads. Expert Systems ith pplications, 36(3), (9) [13] Kumar,., Singh P., Kaur., and Kaur, P., Ranking of generalized trapezoidal fuzzy numbers based on rank, mode, divergence and spread. Turkish journal of Fuzzy Systems, (1) vol.1, No, iosrjournals.org 1 Page

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