Hadi s Method and It s Advantage in Ranking Fuzzy Numbers

Transcription

1 ustralian Journal of Basic and pplied Sciences, 4(1): , 1 ISSN Hadi s Method and It s dvantage in anking Fuzz Numbers S.H. Nasseri, M. Sohrabi Department of Mathematical Sciences, Mazandaran Universit, P.O.Bo , Babolsar, Iran. bstract: anking fuzz numbers plas an ver important role in linguistic decision making and some other fuzz application sstems. Man methods have been proposed to deal ith ranking fuzz numbers. The approach is illustrated b numerical eamples, shoing that it overcomes the shortcoming eisting fuzz ranking approaches, especiall hen it is difficult to use other methods to solve ranking fuzz problems. Ke ords: fuzz number, Hadi method, ranking, radius of gration. INTODUCTION anking fuzz numbers is important in decision-making, data analsis, artificial intel-ligence and socioeconomic sstems. Jain (1976,1978), Dubois and Prade (1978) introduced the relevant concepts of fuzz numbers. Bortolan and Degani (1985) revieed some methods to rank fuzz numbers, Chen and Hang (199) proposed fuzz multiple attribute decision making, Choobineh and i (199) proposed an inde for ordering fuzz numbers, Dias (199) ranked alternatives b ordering fuzz numbers, ee et al. (1994) ranked fuzz numbers ith a satisfaction function, equena et al. (1994) utilized artificial neural netorks for the automatic ranking of fuzz numbers, Fortemps and oubens (1996) presented ranking and defuzzication methods based on area compensation, and aj et al. (1999) investigated maimizing and minimizing sets to rank fuzz alternatives ith fuzz eights. Hoever, Chu and Tsao () proposed a method of ranking fuzz numbers ith an area beteen the centroid and original points. Chu and Tsaos method originated from the concepts of ee, i (1988) and Cheng (1998). ee and i proposed the comparison of fuzz numbers, for hich the considered mean and standard deviation values for fuzz numbers based on the uniform and proportional probabilit distributions. Then Cheng proposed the coefficient of variance to improve ee and i s method based Cheng also proposed a ne distance inde to improve the method proposed b Murakami et al. (198) ecentl, Wang and ee (6) presented a ne method of ranking fuzz numbers ith using radius of gration. In Section, e introduce some preliminaries. In Section, e remind other ranking method. In Section 4, e eplain Hadi method and in last Section, e compare Hadi method ith other ranking fuzz number methods.. Preliminaries: In practical use, ranking fuzz numbers is ver important in fuzz topics. For eample, the concept of optimum or the best choice is completel based on ranking or comparison. Therefore, to set the rank of fuzz numbers is one of the main problems. The concept of fuzz sets as originall introduced b Zadeh. The concept of fuzz numbers is presented b Jain (1978) and Dubois and Prade (1978). anking methods are classified into four major classed according to Chen and Hang (199) hich are listed as follos: (1) preference relation a) Degree of optimalit b) Hamming distance c) -cut d) Comparison function () Fuzz mean and spread probabilit distribution ()Fuzz scoring a) Proportion to optimal b) eft/ight scores c) Centroid inde d) rea measurement (4)inguistic epression a) Intuition b) inguistic approimation Corresponding uthor: S.H. Nasseri, M. Sohrabi, Department of Mathematical Sciences, Mazandaran Universit, P.O.Bo , Babolsar, Iran. nasseri@umz.ac.ir 46

2 ust. J. Basic & ppl. Sci., 4(1): , 1 Definition.1: et U be a universe set. fuzz set of U is defined b a member-ship function μ [, 1], here μ (), U, indicates the degree of in. Definition.: fuzz subset of universe set U is normal if and onl if sup U () = 1, here U is the universe set. Definition.: The - cut set of a fuzz set is a crisp set defined b = { X μ () } Definition.4: fuzz subset of universe set U is conve if and onl if μ ( + (1 - )) min{(μ () ^ μ ())},, U, [,1]. Definition.5: fuzz set is a fuzz number if and onl if is normal and conve on U. Definition.6: trapezoidal fuzz number is a fuzz number ith a membership function μ defined b: a1, a1 a, a a1 1, a a, ( ) a4, a a4, a4 a, otherise hich can be denoted as a quartet (a 1, a, a, a 4 ). Definition.7. The membership function f of can be epressed as f f, a a,, b c, ( ) f, c d,, otherise (1) here f :[ ab, ] [, ] and f :[ cd, ] [, ]. Since f :[ ab, ] [, ] is continuous and strictl increasing, the inverse function of f eists. Similarl, since f :[ ab, ] [, ] is continuous f and strictl decreasing, the inverse function of also eists. The inverse functions of and can be denoted b g and g, respectivel. Since f :[ ab, ] [, ] is continuous and strictl increasing, g :[, ] [ a, b] is also continuous and strictl increasing. Similarl, since f :[ ab, ] [, ] is continuous and strictl decreasing, g :[, ] [ c, d] is also continuous and strictl increasing, and g are continuous on [, ]; the are integrable on [, ]. That is, both f g and d g d f g eist. 461

3 ust. J. Basic & ppl. Sci., 4(1): , 1. Other anking:.1. the Cheng and Chu-tsao Methods: The centroid point of a fuzz number corresponds to an value on the horizontal ais and a value on the vertical ais. The centroid point (, ) for a fuzz number : ( ) ( ) f b c d a b c b c d a b c ( f ) d d ( f ) d ( f ) d d ( f ) d ( g ) d ( g ) d ( g ) d ( g ) d f here and are the left and right membership functions of fuzz number, respectivel. and g g are the inverse functions of and. f f Cheng method: The centroid point (,) for a fuzz number : ( ). Chu-Tsao method: S( )... The OG Method: adius of gration is a concept in mechanics. In this paper, e present a ne area method to rank fuzz numbers ith the radius of gration (OG) points. For better understanding our method, e briefl introduce some important definitions (Hibbeler,.C., ). The moment of inertia of the area ith respect to the ais, and the moment of inertia of the area ith respect to the ais are defined, respectivel, as I d, I d () The radius of gration of an area ith respect to the ais is defined as the quantit r, that satisfies the relation, I, here I is the moment of inertia of ith respect to the ais. Solving equation for r, e have () I (4) In a similar a, e define the radius of gration of an area ith respect to the ais is I, (5) 46

4 ust. J. Basic & ppl. Sci., 4(1): , 1 I (6) Fig..1: In the folloing of this paper, hen a generalized fuzz number is given, the radius of gration (OG) points of the generalized fuzz number is denoted as (, ) hose value can be obtained b equations () and (5). For an area made up of a number of simple shapes, the moment of inertia of the entire area is the sum of the moments of inertia of each of the individual area about the ais desired. For eample, the moment of inertia of the generalized trapezoidal fuzz number in Figure 1 can be obtained as follos, I I I I, I I I I 1 1 (7) Eample 1: Determine the moment of inertia and the radius of gration of the generalized trapezoidal fuzz number. First, the trapezoid (a, e, f, d) can be divided into three parts, (a, e, b), (b, e, f, c) and (c, f, d). The moment of inertia of the area (aeb) ith respect to ais, and the moment of inertia of the area (a, e, b) ith respect to ais can be calculated ( b a)( ) ( b a) I d d 1 ( ) 1. aeb ( I ) 1 d aeb ( ) ( ) ( ) 4 ba ba a ba (8) The moment of inertia of area befc and cfd, ith respect to ais and ais, can be obtained, respectivel, as follos, ( c b) ( I ), ( d c) ( I ), 1 46

5 ust. J. Basic & ppl. Sci., 4(1): , 1 ( c b) ( I ) ( cb) b ( cb) b, ( I ) ( d c) ( d c) c ( d c) c 1 (9) So, the (OG) point of generalized trapezoidal fuzz number can be calculated as ( I) 1( I) ( I) ((( cb) ( d a)). ) / ( I ) ( I ) ( I ) 1 ((( cb) ( d a)). ) / (1) 4. Hadi s Method for anking Trapezoidal Fuzz Numbers: In this section, e briefl eplain Hadi s method for ranking trapezoidal fuzz numbers hich is ver simple and efficient (see [16]). U For an trapezoidal fuzz number a ( a, a, a, a ), define 1 m 1 a a h, a a h m a a U a a a a am ha andha a a a a here, No assume that et ab (, ) ab, ab (, ) ab U U a ( a, a, a, a ), b ( b, b, b, b ) be to trapezoidal fuzz numbers. Definition 4.1: U U ssume a ( a, a, a, a ), b ( b, b, b, b ) be to trapezoidal fuzz numbers and ab (, ) (If ab (, ), then change role a and b ). Define the relation and on F() as follos: i) a b if and onl if ( b, a) ( a, b) ii) a b if and onl if ( b, a) ( a, b) emark 4.1: We denote a b if and onl if and. Then a b if and onl if and a b a b ab (, ) ab (, ). lso a b if and onl if b a. ab (, ) 464

6 ust. J. Basic & ppl. Sci., 4(1): , 1 emark 4.: We let (,,,) as the zero trapezoidal fuzz number. Thus an a such that, is a zero fuzz number too. emma 4.1: ssume a b, then. a b a emma 4.: ssume ab, F( ). The relation is a partial order on F(). emark 4.: We emphasis that the relation is a linear order on F() too, becaquse an to elements in F() are comparable b this relation. emma 4.: If a b and c d, then. lgorithm 4.1: Hadi s method: ac bd For to trapezoidal fuzz numbers a and b, assume that b a. Compute: ab (, ) ab and ( ba, ) ba. ba (, ) ( ab, ) a b a b b a et = -. Then if =, then, else if >, then, else. 5. Numerical Eamples: In this section, e solve some eamples to illustrate the advantage of Hadi s method in ranking fuzz numbers. Eample 5.1: Consider the folloing fuzz numbers: Fig. 5.1: 465

7 ust. J. Basic & ppl. Sci., 4(1): , 1 method Cheng Chu-Tsao OG Hadi result s e see in Fig. 5.1 the fuzz number 1 is bigger than hile OG and Cheng s methods sho that 1. Eample 5.: Consider the folloing fuzz numbers: 1 = (-16,-6,4,4), = (8,9,,) Fig. 5.: method Cheng Chu-Tsao OG Hadi result Fig. 5. shos 1 hile OG and Cheng s methods sho 1. Eample 5.: Consider the folloing fuzz numbers: 1 = (-,,1,), = (4,7,1,), = (4,7,,), 4 = (4,8,1,1) Fig. 5.: method Cheng Chu-Tsao OG Hadi result

8 ust. J. Basic & ppl. Sci., 4(1): , 1 The above tableau shos Hadi and Chu-Tsao s methods can order fuzz numbers in the good strateg. 6. Conclusion: In this paper, e focused on Hadi s method for ranking trapozoidal fuzz numbers. We eamined the advantage of Hadi s method b comparing to Chu-Tsao and Cheng s methods and recentl proposed method entitled OG method. We sa Hadi s method in all eamples obtained better results as ell as Chu-Tsao method. This method also ill be useful in solving fuzz linear programming problems. CKNOWEDGMENT The first author thanks to the esearch Center of lgebraic Hperstructures and Fuzz Mathematics, Babolsar, Iran and National Elite Foundation, Tehran, Iran for their supports. EFEENCES Bortolan, G.,. Degani, revie of some methods for ranking fuzz numbers, Fuzz Sets and Sstems, 15: Chen, S.J., C.. Hang, 199. Fuzz Multiple ttribute Decision Making, Springer, Ne York. Cheng, C.H., ne approach for ranking fuzz numbers b distance method, Fuzz Sets and Sstems, 95: Choobineh, F., H. i, 199. n inde for ordering fuzz numbers, Fuzz Sets and Sstems, 54: Chu, T.C., C.T. Tsao,. anking fuzz numbers ith an area beteen the centroid point and the original point, Computers and Mathematics ith pplications, 4: Deng, Y., Z. Zhenfu,. Qi, 6. anking fuzz numbers ith an area method using radius of gration, Computers and Mathematics ith pplications, 51: Dias, G., 199. anking alternatives using fuzz numbers: computational approach, Fuzz Sets and Sstems, 56: Dubois, D., H. Prade, Operations on fuzz numbers, The International Journal of Sstems Sciences, 9: Fortemps, P., M. oubens, anking and defuzzification methods based on area compensation, Fuzz Sets and Sstems, 8(): 19-. Hibbeler,.C.,. Mechanics of Materials, Prentice Hall, Ne Jerse. Jain,., Decision-making in the presence of fuzz variables,ieee Transactions on Sstems, Man and Cbernetics, 6: Jain,., procedure for multi-aspect decision making using fuzz sets, The International Journal of Sstems Sciences, 8: 1-7. ee, K.M., C.H. Cho, H. ee-kang, anking fuzz values ith satisfaction function, Fuzz Sets and Sstems, 64: ee, E.S.,.J. i, Comparison of fuzz numbers based on the probabilit measure of fuzz events, Computers and Mathematics ith pplications, 15: Murakami, S., S. Maeda, S. Imamura, 198. Fuzz decision analsis on the development of centralized regional energ control sstem, in: IFC Smp. on Fuzz Inform. Knoledge epresentation and Decision nal., Nasseri, S.H., anking trapezoidal fuzz numbers b using Hadi method, ustralian Journal of Basic and pplied Science, in press. aj,., D.N. Kumar, anking alternatives ith fuzz eights using maimizing set and minimizing set, Fuzz Sets and Sstems, 15: equena, I., M. Delgado, J.I. Verdaga, utomatic ranking of fuzz numbers ith the criterion of decision-maker learntb an artificial neural netork, Fuzz Sets and Sstems, 64: