ustralian Journal of Basic and pplied Sciences, 5(): 6-69, ISSN 99-878 Ranking Fuzzy Numbers Based on mbiguity Degrees Tayebeh Hajjari Department of Mathematics, Islamic zad University, Firuz Kuh Branch, Firuz Kuh, Iran bstract: Since much of human reasoning is based on imprecise, vague and subjective values, most of decision-making processing, in reality, requires handling and evaluation of fuzzy numbers. Zadeh s (Zadeh 965) fuzzy logic has given analysts a tool to present the human behavior more precisely, especially where relatively few data exist, and where the expert knowledge about the system is vague and linguistic (Hoogerdoorn et al. 999). Therefore, ranking fuzzy numbers is one of very important research topics in fuzzy set theory because it is a base of decision-making in applications. However, fuzzy numbers may not be easily ordered into one sequence according to their magnitudes because they represent uncertain values. lthough so far, many methods for ranking of fuzzy numbers have been discussed broadly, most of them contained some shortcomings, such as requirement of complicated calculations, inconstancy with human intuition and indiscrimination. However, these methods just can apply to rank some types of fuzzy numbers (i.e. normal, non-normal, positive, and negative fuzzy numbers), and many ranking cases can just rank by their graphs intuitively. Therefore, it is important to use proper methods in the right condition. The motivation of this study is to present a model for ranking fuzzy numbers based on their ambiguities. The advantages of the new proposed are that it can be applied for most of the defuzzification and the calculation is far simple and easy than previous methods. The effectiveness of the proposed method is finally demonstrated by including a comprehensive comparing different ranking method with the present one. Key words: mbiguity, Fuzzy numbers, membeship function, Ranking INTRODUCTION Many authors have introduced various ranking approaches of fuzzy numbers, these approaches are developed in the literature for multiple attribute decision-making problems, therefore in fuzzy environments, ranking fuzzy numbers is very important decision making procedure, because a good decision is necessary for achieving desired targets. This means that the quantity of a fuzzy number is not the only consideration in the ranking process for decision-making. In other words, the quality factor is suitably incorporated into the determination of the preference order of fuzzy numbers for the decision-making purpose in the fuzzy conditions. ny ranking existing method has a shortcoming that cannot rank special cases of fuzzy numbers. number of ranking methods focus on the centroid point and its distance from origin or the area measurement for a fuzzy numbers (bbasbandy, S. and T. Hajjari, 9; Hajjari, T., ; Hajjari, T., 8; Hajjari, T., 8; Cheng, C.H., 998; Chu, T. and C. Tsao, ), some have used from dispersion and distance indexes (bbasbandy, S. and B. sady, 6; Chen, S., 985; Tran, L., L. Duckstein, ), other some applied simulation and statistics methods for ranking fuzzy numbers (Wang Liu, X. and S. Lina Han, 5; Huijun, S. and W. Jianjun, 6; Lee, E.S., R.J. Li, 988). lmost each method, however has pitfalls in some aspect or have led to some misapplications, such as inconsistency with human intuition, indiscrimination, and difficulty. Consequently seems that uniquely the best method there does not exist for comparing fuzzy numbers, and different methods may satisfy different desirable criteria. The present study has been motivated by the author interest in making decisions in fuzzy environment. In this work, we attempt to combine some commonly used defuzzification methods to define some new definition into a certain uniform. The reminder of this paper is organized as follows: Section contains some basic notation, the definitions and some existing ranking methods, which used in this work. Section includes the developed method of Mag ranking (bbasbandy, S. and T. Hajjari, 9) and a new approach that can apply for most of the defuzzification methods. Some numerical examples demonstrate this idea to compare with previous methods. Corresponding uthor: Tayebeh Hajjari, Department of Mathematics, Islamic zad University, Firuz Kuh Branch, Firuz Kuh, Iran E-mail: tayebehajjari@iaufb.ac.ir 6
ust. J. Basic & ppl. Sci., 5(): 6-69, Finally, concluding remarks are given in Section 4. Background Information: In general, a generalized fuzzy number is described as any fuzzy subset of real line R, whose membership ( x ) can be defined as (Dubios, D. and H. Prade, 987). L( x) a x b b x c ( x) R( x) c x d otherwise, () where is a constant, and L : a, b, and R : c, d, are two strictly monotonical and continuous mapping from R to closed interval [,ω]. If ω =, then is a normal fuzzy number; otherwise, it is a trapezoidal fuzzy number and is usually denoted by = (a, b, c, d, ω) or = (a, b, c, d) if ω =. In particular, when b = c, the trapezoidal fuzzy number is reduced to a triangular fuzzy number denoted by = (a, b, c, d, ω) or = (a, b, c, d) if ω =. Therefore, triangular fuzzy numbers are special cases of trapezoidal fuzzy numbers. Since L ( x ) and R( x) are both strictly monotonically and continuous functions, their inverse functions exist and should be continuous and strictly monotonically. Let L :, ab, and R :, c, d be the inverse functions of L ( x ) and R( x), respectively. Then L () r and R () r should be integrable on the close interval [. ω]. In other words, both and R () rdrshould exist. In the case L () r dr of trapezoidal fuzzy number, the inverse functions L r and R () r can be analytically expressed as () L () r a( ba) r/, () R () r d ( d c) r/, () The set of all elements that have a nonzero degree of membership in, it is called the support of, i.e. Supp ( ) xx ( x) (4) The set of elements having the largest degree of membership in, it is called the core of, i.e. Core ( ) xx ( x ) sup L ( x ). xx (5) In the following, we will always assume that is continuous and bounded support supp () = (a, d). The strong support of should be supp () = [a, d]. Definition.. function s:[.]º[,] is a reducing function if is s increasing and s() = and s() =. We say that s is a regular function if sxdx ( ). 6
ust. J. Basic & ppl. Sci., 5(): 6-69, Definition. If is a fuzzy number with r-cut representation, function, then the value of (with respect to s); it is defined by ( L( r), R ( r)) and s is a reducing ( ) ( )[ ( ) ( )] Val s r L r R r dr Definition. If is a fuzzy number with r-cut representation function then the ambiguity of (with respect to s) is defined by ( ) ( )[ ( ) ( )]. mb s r R r L r dr ( L( r), R ( r)) (6), and s is a reducing (7) Definition.4 The first of maxima (FOM) is the smallest element of core () i.e.. FOM min core( ) Definition.5 The last of maxima (LOM) is the greatest element of core() i.e. LOM max core( ). Some Existing Ranking Methods: In this part briefly, we review nine existing defuzzificationmethods: centroid point (Cheng, C.H., 998), Chu and Tsao s method ()Chu, T. and C. Tsao,, sign distance (bbasbandy, S. and B. sady, 6) and Mag Ranking method (bbasbandy, S. and T. Hajjari, 9) respectively. Centroid Point: In order to determine the centroid point ( x, y ) of a fuzzy number, Cheng (998) provided a formulae. Wang et al. (985) found from the point of view of analytical geometry and showed the corrected centroid point as follows: x y b c d xl ( x) dx xdx xr ( x) dx a b c b c d L ( x) dx dx R ( x) dx a b c yr ( y) dy yl ( y) dy. R ( y) dy L ( y) dy (8) For non-normal trapezoidal fuzzy number = (a, b, c, d, ω) formulas (8) lead to following results respectively. dc ab x a b c d ( d c) ( ab) y c d. ( d c) ( ab) (9) 64
ust. J. Basic & ppl. Sci., 5(): 6-69, Since non-normal triangular fuzzy numbers are special cases of normal trapezoidal fuzzy numbers with b = c, formulas (9) can be simplified as x abcd y. () In this case, normal triangular fuzzy numbers can be compared or ranked directly in terms of their centroid coordinates on horizontal axis. Cheng (998) formulated his idea as follows: R ( ) x( ) y( ) () Chu and Tsao s Method: Chu and Tsao ()Chu, T. and C. Tsao, computed the area between the centroid and original points to rank fuzzy numbers as: S( ) x ( ). y ( ) Sign Distance Method: Sign distance method (bbasbandy, S. and B. sady, 6) denoted by as follows d D (, ) p ( ) (, ) d ( p, ) (), which is computed () such that ( ): E {,}, (E stands for fuzzy numbers) and ( ) [ ( ) ( )] sign L x R x dx where (4) ( ) sign L x R x dx sign L x R x dx [ ( ) ( )] [ ( ) ( )] (5) also, if x ( ) if x (6) and p p p (, ) ( ( ) ( ) ). D L x R x dx (7) 65
ust. J. Basic & ppl. Sci., 5(): 6-69, Mag Ranking Method: For an arbitrary trapezoidal fuzzy number ( x, y,, ), with parametric form L x R x ( ( ), ( )), bbasbandy and Hajjari presented the magnitude of the trapezoidal fuzzy number as Mag L x R x x y s x dx ( ) ( ) ( ) ( ), (8) Where the function s(x) is a non-negative and increasing function on [, ] with s()=, s()= and sxdx ( ). Obviously, function s(x) can be considered as a weighting function. For more details, we refer the reader to (bbasbandy, S. and T. Hajjari, 9). New Method for General Fuzzy Numbers: In this part, we first develop Mag ranking method (bbasbandy, S. and T. Hajjari, 9) for all fuzzy numbers. Let be a fuzzy number with parametric form ( L ( x), R ( x)), which L (), R () are FOM and LOM respectively. Then the developed Mag ranking method can be written as L( x) R ( x) Mag( ) s( x) dx, L() R () (9) It is clear that the developed method has all properties, which the previous one has. For more detail, we refer the reader to (bbasbandy, S. and T. Hajjari, 9). Example.6. Consider two fuzzy numbers from (Wang, Z.X., 9) i.e. the triangular fuzzy number =(,,5) and fuzzy number B, shown in Fig., which the membership function of B is defined by x when x B x when x 4 when ( ), ( ) ( ),4. otherwise Fig. : 66
ust. J. Basic & ppl. Sci., 5(): 6-69, In Lious and Wang s ranking method (Wang, Z.X., 9), different ranking are produced for the same problem when applying different indices of optimism. Distance method with p, d (, ) 5, d ( B, ) 4.78 and p p p, d (, ) 5.957, d ( B, ).845, p p the ranking order is B. In Chu and Tsao s ranking method S() =.445 and S(B) =.8, therefore B From Mag ranking method Mag() = 4.5 and Mag() = 4 the ranking order is B. The results by Z-x Wang et al. s are the same. Moreover, if the approach in (Wang, Y.J. and H. Sh. Lee, 8) is used, the ranking order is B. We can concluded that B is more consistent with human intuition. s we know in most of fuzzy ranking methods, all symmetric fuzzy numbers with the same core and fuzzy numbers with the identical centroid points have the equal ranking order. This case will be more explained this by the next example. Example.7 Consider to triangular fuzzy number =(-.5,,.5) and B= (-,, ). By using Cheng s distance R() = R(), Chu and Tsao s S() = S(B) = and Mag ranking method Mag()=Mag(B)=. Hence and B have the same ranking order i.e..b. See Fig.. Fig. : Now we would like to introduce a new approach to apply our method and some other methods more precisely. In point of our view, more close to crisp number, less ambiguity because a crisp number or a certain number has no ambiguity. Consequently, the more ambiguity, the fuzzy number will be smaller. So we apply the value of ambiguity of fuzzy number as the degree of ordering in case that two fuzzy numbers are equal. The aforementioned method can apply for all defuzzification methods. In general, consider the defuzzification method DM(.) and for two fuzzy numbers and B the ranking order be based on the following situations:. If DM( ) DM( B) then B,. If DM( ) DM( B) then B,. If DM( ) DM( B) then B. In the third case, we can compare two fuzzy numbers precisely. In other words, first it should be computed the ambiguity of two fuzzy numbers then we will have one of the next conditions: 67
ust. J. Basic & ppl. Sci., 5(): 6-69,. If amb( ) amb( B) then B,. If amb( ) amb( B) then B,. If amb( ) amb( B) then B. For example consider two fuzzy number and B from Example.7 that have the same ranking order. Now we compute the ambiguity of and B by using Eq. (7) amb( ) 6 and amb( B). since amb( ) amb( B) then B. Example.8 Consider the data used in (Chen, L.H., H.W. Lu, ), i.e. two fuzzy numbers =(, 6, 9) and B=(5, 6, 7), as shown in Fig.. Fig. : Through the proposed approach in this paper, since Mag()=Mag(B)= and amb()=and amb( B), the ranking order is B. This result is the same as Z-X Wang s method (Wang, Z.X., 9). Because fuzzy numbers and B have the same mode and symmetric spread, most of existing approaches fail. For instance by bbasbandy and sady approach (bbasbandy, S. and B. sady, 6), different ranking orders are optained when different index values (p) are taken. When p= and p=, the ranking order of fuzzy numbers is.b and B respectively. Meanwhile, using the approaches in (Chu, T. and C. Tsao, ; Wang, Y.J. and H. Sh. Lee, 8; Yao, J.S., K. WU, 996), the ranking order is the same, i.e..b. Nevertheless, inconsistent results produced when distance index and CV index of cheng s approach (Cheng, C.H., 998) are respectively used. Moreover, the ranking order obtained by Wang et al. s approach (Wang, M.L., 5) is B. dditionally, by the approaches provided in (Liu, X., ; Matarazzo, B., G. Munda, ; Yao, J.S. and F.T. Lin, ) different ranking order is obtained when different indices of optimism are taken. However, decision makers prefer to result B in tuition ally. Conclusion: In spite of many ranking methods, no one can rank fuzzy numbers with human intuition consistently in all cases. Here, we first developed Mag ranking method, we then introduced an approach to ranking fuzzy numbers based on their ambiguity. The proposed approach can be utilized for most of defuzzification precisely and may be corrected them in some situations. The new method does not imply much computational effort and does not require a priori knowledge of the set of all alternatives. We also used comparative examples to illustrate the advantages of the proposed method. 68
ust. J. Basic & ppl. Sci., 5(): 6-69, CKNOWLEDGMENT This work was partly supported by Islamic zad University-Firuz-Kuh Branch. REFERENCES bbasbandy, S. and T. Hajjari, 9. new approach for ranking of trapezoidal fuzzy numbers, Comput. Math. ppl., 57(4): 4-49. bbasbandy, S. and B. sady, 6. Ranking of fuzzy numbers by sign distance, Inform. Sci., 76: 45-46. Chen, L.H., H.W. Lu,. The preference order of fuzzy numbers, Comput. Math. ppl., 44: 455-465. Chen, S., 985. Ranking fuzzy numbers with maximizing set and minimizing set, Fuzzy Sets and Systems, 7: -9. Cheng, C.H., 998. new approach for ranking fuzzy numbers by distance method, Fuzzy Sets and Systems, 95: 7-7. Chu, T. and C. Tsao,. Ranking fuzzy numbers with an area between the centroid point and orginal point, Comput. Math. ppl., 4: -7. Hajjari, T.,. Ranking of Fuzzy Numbers by Middle of Expected Interval, In: First International Conference in Mathematics and Statistics, Sharjah. Hajjari, T., 8. Comparison of Fuzzy numbers by Modified Centroid Point Method. In: Third International Conference in Mathematical Sciences Dubai, pp: 9-45. Hajjari, T., 8. Fuzzy Euclidean Distance for Fuzzy Data, In: 8th Iranian Conference on Fuzzy Systems, Iran, pp: 7-. Hajjari, T., 7. Ranking Fuzzy Numbers by Sign Length, In: 7th Iranian Conference on Fuzzy Systems, Iran, pp: 97-. Huijun, S. and W. Jianjun, 6. new approach for ranking fuzzy numbers based on fuzzy simulation analysis method, pplied Mathematics and Computation, 74: 755-767. Dubios, D. and H. Prade, 987. Operations on fuzzy numbers, Internat. J. System Sci., 9: 6-66. Lee, E.S., R.J. Li, 988. Comparestion of fuzzy numbers based on the prob- ability measure of fuzzy event, Computer and Mathematics with pplication, 5: 887-896. Matarazzo, B., G. Munda,. New approaches for the comparisonof L-R fuzzy numbers: a theoritical and operationalanalysis, Fuzzy Sets and Systems, 8: 47-48. Liu, X.,. Measuring the satisfaction of constrains in fuzzy linear programming, Fuzzy Sets and Systems, : 6-75. Tran, L., L. Duckstein,. Comparison of fuzzy numbers using a fuzzy distance measure, Fuzzy Sets and Systems, : -4. Wang, M.L., H.F. Wang and L.C. Lung, 5. Ranking fuzzy numbers based on lexicographic screening procedure, International Journal of Information Technology and Decision Making, 4: 66-678. Wang Liu, X. and S. Lina Han, 5. Ranking fuzzy numbers with preference weighting function expectations, Computer and Mathematics with ppications, 49: 7-75. Wang, Z.X., Y.J. Lio, Z.P. Fan and B. Feng, 9. Rnking LR fuzzy number based on deviation degree, Info. Sciences, 79: 7-77. Wang, Y.J. and H. Sh. Lee, 8. The revised method of ranking fuzzy numbers with an area between the centroid and original points, Comput. Math. ppl., 55: -4. Yao, J.S., K. WU, 996. Ranking fuzzy numbers based on decomposition principle and signed distance, Fuzzy Sets and Systems, 8: 67-76. Yao, J.S. and F.T. Lin,. Fuzzy critical path method based on decomposition principle and signed distance, Fuzzy Sets and Systems, 6: 67-76. 69