Ordering Generalized Trapezoidal Fuzzy Numbers

Int. J. Contemp. Math. Sciences, Vol. 7,, no., 555-57 Ordering Generalized Trapezoidal Fuzzy Numbers Y. L. P. Thorani, P. Phani Bushan Rao and N. Ravi Shankar Dept. of pplied Mathematics, GIS, GITM University, Visakhapatnam, India drravi68@gmail.com (N. Ravi Shankar) Dept. of Mathematics, GIT, GITM University, Visakhapatnam, India bstract This paper describes a ranking method for ordering fuzzy numbers based on rea, Mode, Spreads and Weights of generalized trapezoidal fuzzy numbers. The area used in this method is obtained from the generalized trapezoidal fuzzy number, first by splitting the generalized trapezoidal fuzzy numbers into three plane figures and then calculating the centroids of each plane figure followed by the incentre of these centroids and then finding the area of this incentre from the original point. In this paper, we also apply mode and spreads in those cases where the discrimination is not possible. This method is simple in evaluation and can rank various types of trapezoidal fuzzy numbers and also crisp numbers which are considered to be a special case of fuzzy numbers. Keywords: Ranking function; incentre ; centroid points; Generalized trapezoidal fuzzy numbers. Introduction Ranking fuzzy numbers is an important tool in decision making. In fuzzy decision analysis, fuzzy quantities are used to describe the performance of alternatives in modeling a real-world problem. Most of the ranking procedures proposed so far in literature cannot discriminate fuzzy quantities and some are counter-intuitive. s fuzzy numbers are represented by possibility distributions, they may overlap with each other and hence it is not possible to order them. It is true that fuzzy numbers are frequently partial order and cannot be compared like

556 Y. L. P. Thorani, P. Phani Bushan Rao and N. Ravi Shankar real numbers which can be linearly ordered. In order to rank fuzzy quantities, each fuzzy quantity is converted into a real number and compared by defining a ranking function from the set of fuzzy numbers to a set of real numbers which assigns a real number to each fuzzy number where a natural order exists. Usually by reducing the whole of any analysis to a single number, much of the information is lost and hence an attempt is to be made to minimize this loss. Various ranking procedures have been developed since 976 when the theory of fuzzy sets was first introduced by Zadeh []. Ranking fuzzy numbers was first proposed by Jain [] for decision making in fuzzy situations by representing the ill-defined quantity as a fuzzy set. Since then, various procedures to rank fuzzy quantities are proposed by various researchers. Bortolan and Degani [] reviewed some of these ranking methods [-] for ranking fuzzy subsets. Chen [] presented ranking fuzzy numbers with maximizing set and minimizing set. Dubois and Prade [6] presented the mean value of a fuzzy number. Lee and Li [8] presented a comparison of fuzzy numbers based on the probability measure of fuzzy events. Delgado, Verdegay and Vila [9] presented a procedure for ranking fuzzy numbers. Campos and Munoz [] presented a subjective approach for ranking fuzzy numbers. Kim and Park [] presented a method of ranking fuzzy numbers with index of optimism. Yuan [] presented a criterion for evaluating fuzzy ranking methods. Heilpern [] presented the expected value of a fuzzy number. Saade and Schwarzlander [] presented ordering fuzzy sets over the real line. Liou and Wang [5] presented ranking fuzzy numbers with integral value. Choobineh and Li [6] presented an index for ordering fuzzy numbers. Since then several methods have been proposed by various researchers which include ranking fuzzy numbers using area compensation, distance method, decomposition principle and signed distance [7, 8, 9]. Wang and Kerre [, ] classified all the above ranking procedures into three classes. The first class consists of ranking procedures based on fuzzy mean and spread [6, 7, 8, 9,, 5, 6, 7] and second class consists ranking procedures based on fuzzy scoring [,,,, 7, ] whereas the third class consists of methods based on preference relations [, 5,,, 5, 9,, ] and concluded that the ordering procedures associated with first class are relatively reasonable for the ordering of fuzzy numbers specially the ranking procedure presented by damo [7] which satisfies all the reasonable properties for the ordering of fuzzy quantities. The methods presented in the second class are not doing well and the methods [, 5,, ] which belong to class three are reasonable. Later on, ranking fuzzy numbers by preference ratio [], left and right dominance [], area between the centroid point and original point [], ], sign distance [6], distance minimization [7] came into existence. Later in 7, Garcia and Lamata [8] modified the index of Liou and Wang [5] for ranking fuzzy numbers, by stating that the index of optimism is not alone sufficient to discriminate fuzzy numbers and proposed an index of modality to rank fuzzy numbers. Most of the methods presented above cannot discriminate fuzzy numbers and some methods do not agree with human intuition whereas, some methods cannot rank crisp numbers which are a special case of fuzzy numbers.

Ordering generalized trapezoidal fuzzy numbers 557 In this paper a new method is proposed which is based on Incentre of Centroids to rank fuzzy quantities. In a trapezoidal fuzzy number, first the trapezoid is split into three parts where the first, second and third parts are a triangle, a rectangle and a triangle respectively. Then the centroids of these three parts are calculated followed by the calculation of the centroid of these centroids. Finally, a ranking function is defined which is the Euclidean distance between the centroid point and the original point to rank fuzzy numbers. Most of the ranking procedures proposed in literature use Centroid of trapezoid as reference point, as the Centroid is a balancing point of the trapezoid. But the Incentre of centroids can be considered a much more balancing point than the centroid. Further, this method uses an index of optimism to reflect the decision maker s optimistic attitude and also uses an index of modality that represents the neutrality of the decision maker. The work is organized as follows: Section briefly introduces the basic concepts and ranking of fuzzy numbers. Section presents the proposed new method. In Section the proposed method has been explained with examples which describe the advantages and the efficiency of the method which ranks generalized fuzzy numbers, images of fuzzy numbers and even crisp numbers. In Section 5, the method demonstrates its robustness by comparing with other methods like Liou and Wang, Yager and others where the methods cannot discriminate fuzzy quantities and do not agree with human intuition. Finally, the conclusions of the work are presented in Section 6.. Fuzzy concepts. Fuzzy concepts and ranking of fuzzy numbers Definition. Let U be a universe set. fuzzy set of U is defined by a membership function f : U [,], where f ( x) is the degree of x in, x U. Definition. fuzzy set of universe set U is normal if and only if sup f ( x) = x U Definition. fuzzy set of universe set U is convex if and only if f ( λx + ( λ ) y) min f ( x), f ( y), x, y U and λ [,]. Definition. fuzzy set of universe set U is a fuzzy number iff is normal and convex on U.

558 Y. L. P. Thorani, P. Phani Bushan Rao and N. Ravi Shankar Definition5. real fuzzy number is described as any fuzzy subset of the real line with membership function ( x ) possessing the following properties: (i) ( x ) is a continuous mapping from R to the closed interval f [, w ]. < w f ( x =, for all x (,a] (ii) ) (iii) f ( x) (iv) f ( x) (v) f ( x) (vi) f ( x) f is strictly increasing on [ a, b] =, for all x [ b, c] is strictly decreasing on [ c, d] =, for all x ( d, ], where a, b, c, d are real numbers Definition.6. The membership function of the real fuzzy number is given by L f, a x b, w, b x c, f ( x ) = () R f, c x d,, otherwise, where < w is a constant, a, b, c, d are real numbers and f L : [ a,b] [,w], f R : [ c,d] [,w] are two strictly monotonic and continuous functions from R to the closed interval[,w]. It is customary to write a fuzzy number as = ( a,b,c,d;w ). If w =, then = ( a,b,c,d; ) is a normalized fuzzy number, otherwise is said to be a generalized or non-normal fuzzy number If the membership function ( x ) is piecewise linear, then is said to be a f trapezoidal fuzzy number. The membership function of a trapezoidal fuzzy number is given by:

Ordering generalized trapezoidal fuzzy numbers 559 w( x a ), a x b, b a w, b x c, f ( x ) = () w( x d ), c x d, c d, otherwise, If w=, then = ( a, b, c, d;) is a normalized trapezoidal fuzzy number and is a generalized or non normal trapezoidal fuzzy number if < w <. The image of = ( a,b,c,d;w ). is given by = ( d, c, b, a;w ).. s a particular case ifb = c, the trapezoidal fuzzy number reduces to a triangular fuzzy number given by = ( a,b,d;w ). The value of b corresponds with the mode or core and [a, d] with the support. If w=, then = ( a, b, d). is a normalized triangular fuzzy number is a generalized or non normal triangular fuzzy number if < w <. s f L : [ a,b] [,w] and f R : [ c,d] [,w] are strictly monotonic and continuous functions, their inverse functions : [,w] [ a,b] : [,w] [ c,d] g are integrable on [,w] g L and g R are also continuous and strictly monotonic. Hence g L and R.. Ranking of fuzzy numbers.. Cheng s Ranking method Cheng [8] ranked fuzzy numbers with the distance method using the Euclidean distance between the Centroid point and original point. For a generalized fuzzy number = ( a,b,c,d;w ). the centroid is given by: w ( ) ( d c + b a + dc ab) + ( c b ) x, y = w d ( c + b a) + 6( c b) w, + ( b + c) ( a + d )( w) ( ) ( ) b + c a d + a + d w

56 Y. L. P. Thorani, P. Phani Bushan Rao and N. Ravi Shankar and the ranking function > associated with as ( ) i j Let i and j two fuzzy numbers, (i) If ( i ) R( j ) (ii) If R( ) R( ) < (iii) If ( ) R( ) then i > j. R i = j then i = j He further improved Lee and Li s method by proposing the index of coefficient σ of variation (CV) ascv = whereσ is standard error and μ is mean, μ μ and σ >,, the fuzzy number with smaller CV is ranked higher... Wang et al. Ranking method Wang et al. [5] found that the centroid formulae proposed by Cheng [8] are incorrect and have led to some misapplications such as by Chu and Tsao []. They presented the correct centroid formulae, for a generalized fuzzy number = ( a,b,c,d;w ). as: (, y ) x = a + b + c + d dc ab w, + c b ( d + c) ( a + b) ( d + c) ( a + b) and the ranking function associated with as ( ) Let i and (ii) If ( ) R( ) j R = x + y two fuzzy numbers, (i) If R( ) R( ) R i < j then i j < (iii) If ( ) R( ) > i then j i > j R i = j then i = j.. Liou and Wang s Ranking method Liou and Wang [5] ranked fuzzy numbers with total integral value. For a fuzzy number defined by definition 6 the total integral value is defined as I α T ( ) = αi ( ) + ( α ) I ( ) where ( I ) = R g ( y)dy and ( ) L I = g ( y )dy R L R are the right and left integral values of, respectively and α [,] is the index of optimism which represents the degree of optimism of a decision maker. Ifα = the total integral value represents a pessimistic decision maker s view L

Ordering generalized trapezoidal fuzzy numbers 56 point which is equal to left integral value. If α =, the total integral value represents an optimistic decision maker s view point and is equal to the right integral value and whenα =. 5, the total integral value represents an moderate decision maker s view point and is equal to the mean of right and left integral values. For a decision maker, the larger the value of α is, the higher is the degree of optimism... Garcia and Lamata s Ranking method Garcia and Lamata [8] modified the index of Liou and Wang [5] for ranking fuzzy numbers. This method use an index of optimism to reflect the decision maker s optimistic attitude, which is not enough to discriminate fuzzy numbers, but rather it also uses an index of modality that represents the neutrality of the decision maker. For a fuzzy number defined by definition 6, Garcia and Lamata [8] proposed an index associated with the ranking as the convex combination: ( ) ( α I S ) ( ) I ( β, α = β M + β T ), where S ( M ) is the area of the core of the fuzzy number which is equal to b for a triangular fuzzy number defined by = ( a,b,d;w ). and the average value of the plateau in case of a trapezoidal =, β [,] is the index of modality that represents the importance of central value against the extreme values, α [, ] is fuzzy number given by ( a,b,c,d;w ). α the degree of optimism of the decision maker and I T ( ) defined in ranking method... has its own meaning as. Proposed ranking Method The Centroid of a trapezoid is considered as the balancing point of the trapezoid (Fig.). Divide the trapezoid into three plane figures. These three plane figures are a triangle (PB), a rectangle (BPQC), and a triangle (CQD), respectively. Let the Centroids of the three plane figures beg, G and G respectively. The Incenter of these Centroids G, G and G is taken as the point of reference to define the ranking of generalized trapezoidal fuzzy numbers. The reason for selecting this point as a point of reference is that each Centroid point are balancing points of each individual plane figure, and the Incentre of these Centroid points is a much more balancing point for a generalized trapezoidal fuzzy number. Therefore, this point would be a better reference point than the Centroid point of the trapezoid.

56 Y. L. P. Thorani, P. Phani Bushan Rao and N. Ravi Shankar w P (b, w) Q (c, w) O (a, ) B (b, ) C(c, ) D (d, ) Fig.. Trapezoidal fuzzy number Consider a generalized trapezoidal fuzzy number = ( a,b,c,d;w ). (Fig.). The Centroids of the three plane figures a + b w b + c w c + d w areg =,, G =, andg =, respectively. Equation of the line G G is w y = and G does not lie on the line G G. Therefore, G G and G are non-collinear and they form a triangle. We define the Incentre ( x y ) I of the triangle with vertices G, G and G of, the generalized trapezoidal fuzzy number = ( abcdw,,, ; ) as I ( x, y ) α = ( a + ) + β ( b + ) + γ ( c + ) α ( w) + β ( w) + γ ( w) c d b α + β + γ, α + β + γ () where α = c b+ d + w 6 β = c+ d a b γ = c a b + w 6 s a special case, for triangular fuzzy number = ( a, b, d; w). i.e., c = b the incentre of centroids is given by

Ordering generalized trapezoidal fuzzy numbers 56 I ( x, y ) x = ( a + ) + yb + z( b + ) x( w) + y( w) + z( w) d b x + y + z, x + y + z () Where w d b + d a b a w + x = y = z = 6 6 The ranking function of the generalized trapezoidal fuzzy number = ( a,b,c,d;w ). which maps the set of all fuzzy numbers to a set of real numbers is defined as: ( ) ( ) ( ) ( ) ( ) a + + yb + z + x w + y w + z w = = b b d x R( ) x y x + y + z x + y + z (5) This is the rea between the incenter of the centroids I ( x, y ) as defined in Eq.() and the original point. The Mode (m) of the generalized trapezoidal fuzzy number = ( a,b,c,d;w ). is w = (6) defined as: m ( b + c) dx = ( b + c) w The Spread(s) of the generalized trapezoidal fuzzy number = ( a, b, c, d; w). is defined as: w ( d a) dx = w( d a) s = (7) The Left spread (ls)of the generalized trapezoidal fuzzy number = ( a,b,c,d;w ). is defined as: w ( b a) dx = w( b a) ls = (8) The Right spread (rs) of the generalized trapezoidal fuzzy number = ( a,b,c,d;w ). is defined as: w ( d c) dx = w( d c) rs = (9) Using the above definitions we now define the ranking procedure of two generalized trapezoidal fuzzy numbers.

56 Y. L. P. Thorani, P. Phani Bushan Rao and N. Ravi Shankar Let = ( a,b,c,d ;w ) and B = a, b, c, d ; w ) be two generalized ( trapezoidal fuzzy numbers. The working procedure to compare and B is as follows: Step : Find R and R B Case (i) If R > R B then > B Case (ii) If R < R B then m B then > B Case (ii) If m < m B then s B then B Case (iii) If s = s B comparison is not possible, then go to step. Step : Find ls and ls B Case (i) Ifls > ls B then > B Case (ii) Ifls < ls B then < B Case (iii) Ifls = ls B comparison is not possible, then go to step 5. Step 5: Examine w and w

Ordering generalized trapezoidal fuzzy numbers 565 Case (i) If w > w then > B Case (ii) If w < w then < B Case (iii) If w = w then B. Numerical Examples In this section, the proposed method is first explained by ranking some fuzzy numbers. Example. Let 5 = (,5 7, ;) and B =,5, ; 8 Step : Then I ( x, y ) ( 5.,.877) = and I ( x, y ) ( 5.7,.) B = Therefore, R ( ) =. 985 and R ( B ) =. 67 R < R B < Since ( ) ( ) B Example. Let 7 = (,,; ) and B =,, ; 5 Step : Then I ( x,y ) (.9998,.) = and I ( x, y ) (.69,.9) B = Therefore, R ( ) =. 9 and R ( B ) =. 69 Since R ( ) < R( B ) R( B ) > B From examples. and. we see that the proposed method can rank fuzzy numbers and their images as it is proved that B. Example. Let = (.,.,.5; ) and B = (.,.,.; ) Step :

566 Y. L. P. Thorani, P. Phani Bushan Rao and N. Ravi Shankar Then I ( x, y ) (.996,.78) = and I ( x, y ) (.995,.6) B = Therefore, R ( ) =. and ( ). 79 Since R ( ) > R( B ) > B R B = Example.5 Let = (.,.,.5;.8 ) and B = (.,.,.5; ) Step : Then I ( x, y ) (.995,.77) = and I ( x, y ) (.996,.78) B = Therefore, R ( ) =. 9and R ( B ) =. Since R ( ) < R( B ) < B From example.5 it is clear that the proposed method can rank fuzzy numbers with different height and same spreads. Example.6 Let = (.,.,.,.5; ) and B = (.,.,.5; ) Step : Then I ( x, y ) (.997,.97) = and I ( x, y ) (.996,.78) B = Therefore, R ( ) =. 9 and R ( B ) =. R > R B > Since ( ) ( ) B 5. Results and discussion In this section the advantages of the proposed method is shown by comparing with other existing methods in literature, where the methods cannot discriminate fuzzy numbers. The results are shown in Table and Table. Example 5. Consider two fuzzy numbers = (,,5 ) and B = (,,6 ) By Liou and Wang Method [5],it is clear that the two fuzzy numbers are equal for all the decision makers as α I ( T ) =.5α + ( α ). 5 and α I ( B T ) =.5α + ( α ). 5 Which is not even true by intuition. By using our method we have I ( x, y ) (.987,.5) = and I ( x, y ) (.7,.5) B = Therefore, R ( ) =. 656 and R ( B ) =. 5 > B Since R ( ) > R( B ) > B Example 5. Let (.,.,.5; ), B =,,,; = ( )

Ordering generalized trapezoidal fuzzy numbers 567 Cheng [8] ranked fuzzy numbers with the distance method using the Euclidean distance between the Centroid point and original point. Where as Chu and Tsao [] proposed a ranking function which is the area between the centroid point and original point.their centroid formulae are given by w ( ) ( d c + b a + dc ab) + ( c b ) x, y = w d ( c + b a) + 6( c b) w ( ) ( d c + b a + dc ab) + ( c b ) x, y = w d ( c + b a) + 6( c b) w, + ( b + c) ( a + d )( w) ( ) ( ) b + c a d + a + d w ( b + c) w, + a + b + c + d Both these centroid formulae cannot rank crisp numbers which are a special case of fuzzy numbers as it can be seen from the above formulae that the denominator in the first coordinate of their centroid formulae is zero, and hence centroid of crisp numbers are undefined for their formulae. By using our method, we have ( x, y ) (.996,.78) = and I ( x, y ) (,.) B = I Therefore, R ( ) =. and ( ). Since R ( ) < R( B ) < B R B = From this example it is proved that the proposed method can rank crisp numbers whereas, other methods failed to do so. Example 5. Consider four fuzzy numbers (.,.,.; ) = (.,.5,.8; ) = (.,.,.9; ) (.6,.7,.8; ) =,,, = Which were ranked earlier by Yager[8], Fortemps and Roubens[7], Liou and Wang[5], and Chen and Lu [] as shown in Table.

568 Y. L. P. Thorani, P. Phani Bushan Rao and N. Ravi Shankar Table. Comparison of various ranking methods Method Yager [8]..5.5.7 Fortemps Roubens[7] &..5.5.7 Liou & Wang[5] α =.5.65.65.75 Ranking order > > > > > > α =.5..5.5.7 > > α =.5.5.5.65 Chen [] β = -... -. β =.5 -... -. β = -... -. Proposed method.7.98.68.5 > > > > > > > > It can be seen from Table that none of the methods discriminates fuzzy numbers. Yager [8] and Fortemps and Roubens [7] methods failed to discriminate the fuzzy numbers and, Whereas the methods of Liou and Wang[], and Chen and Lu [] Cannot discriminate the fuzzy numbers, and, By using our method, we have ( x, y ) (.998,.65) =, I ( x, y ) (.999,.898) = I ( x, y ) (.7,.88) =, I ( x, y ) (.6997,.65) = I Therefore, R( ) =. 7, R( ) =. 98, R( ) =. 68, R( ) =. 5

Ordering generalized trapezoidal fuzzy numbers 569 Example 5. > > > In this we consider seven sets of fuzzy numbers available in literature and the comparative study is presented in Table..,.,.6,.8;.5 B =.,.,.,.;.7 Set : = ( ) and ( ) Set : = (.,.,.,.5; ) and B = (.,.,.,.5; ) Set : = (.,.,.,.5; ) and B = (,,,; ) Set : = (.5,.,..;), and B = (.,.,.,.5; ) Set 5: = (.,.5,.5,; ), and B = (.,.6,.6,.8; ) Set 6: = (,.,.6,.8; ), (.,.5,.5,.9; ) C = Set 7: = (.,.,.,.5; ), and B = (,,,; ) B = and (.,.6,.7,.8; ), Table. Comparison of the ranking results for different approaches Methods Set Set Set Set Set 5 Set 6 Set 7 Cheng [] B B C < < ** Chu and Tsao B C < < ** Chen and C B < < > B bbasbandy Not B B (9) ble Chen and B C < < > B Kumar et B C < < > B Proposed B B **Not Comparable

57 Y. L. P. Thorani, P. Phani Bushan Rao and N. Ravi Shankar 6. Conclusions and future work This paper proposes a method that ranks fuzzy numbers which is simple and concrete. This method ranks trapezoidal as well as triangular fuzzy numbers and their images. This method also ranks crisp numbers which are special case of fuzzy numbers whereas some methods proposed in literature cannot rank crisp numbers. This method which is simple and easier in calculation not only gives satisfactory results to well defined problems, but also gives a correct ranking order to problems. Comparative examples are used to illustrate the advantages of the proposed method. pplication of this ranking procedure in various decision making problems such as, fuzzy risk analysis and in fuzzy optimization like network analysis is left as future work. References. L.. Zadeh, Fuzzy sets, Information and control 8 (965) 8-5.. R. Jain, Decision making in the presence of fuzzy variables, IEEE Trans. on Sys., Man and Cybernetics 6 (976) 698-7.. S.M. Bass, H. Kwakernaak, Rating and ranking of multiple-aspect alternatives using Fuzzy sets, utomatica (977) 7-58.. R. Jain, procedure for multi aspect decision making using fuzzy sets. Int. J. Syst. Sci. 8 (978) -7. 5. J. F. Baldwin, N.C.F Guild, Comparison of fuzzy sets on the same decision space, Fuzzy Sets and Systems (979) -. 6. R.R. Yager, On choosing between fuzzy subsets, Kybernetes 9 (98) 5-5. 7. J.M. damo, Fuzzy decision trees, Fuzzy Sets and Systems (98) 7-9 8. R. R. Yager, procedure for ordering fuzzy subsets of the unit interval, Information Sciences (98) -6. 9. W. Chang, Ranking of fuzzy utilities with triangular membership functions, Proceedings of International Conference on Policy nalysis and Systems, 98, pp. 6-7.

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