A probabilistic approach to rank complex fuzzy numbers

A probabilistic

approach to rank complex fuzzy numbers #. Paul Yom

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membership function gives less discriminatia~ power amang members in a fwzy set. When the .y~/ftirti distributiori is used. dwntiin and range crf the resulting distribution are reduced simuftaneoukiy. But the’ reduced Ior increased1 domain in$catesthe comptete ejection (or addition] uf some members fr’rnm ICK to) the set. Hence tht: unifmn distribution xveals more ,Gdesirabk pr&rtk Ftirthsrmore: the. oonwrsion into a pra~rtion;~I dktribution &MS .nut rquirc anq’ computation. WC therefore. suggestthe use of the piopurtional dlstribut ian whrn comp;trin~ fwq mimbers.

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order of’ random variables of ~@ru&~cts a.@. ;ciuotients without going through ‘the ifiv&on ~roc&. It is kt~even nvsary ,to-‘taLe.-derivaiiv~. Park, ,[I31 u~d,fhe,-MetIin.transformation~in prababilistic ctish iI&w mode!iing. .‘. . ‘, : ,, : .

This equation is precisely the form of the prohability density, fuoction, of the random variable 2 =’ X Y.,where X and -F’ar& continuously distributed..independent random variables with probability density ftinctions y[.~) and h( J). res@vely. The Transform of this special convolution reduces to a simpte product of rvIeIlin transforms. If we define the Slellin transform of J(z) as .52,(.$ it follows that

In !++ctia?

2. we have shown’ the conversion of

triangtilar ,arid trapezoida fuzzy m&&ership fsmnctions into corresponding probability density futici tions. Meliin transforms of these two ,density functions will now & derived. A triangular density function is defkd as f(s) =

,W(.G= 33,j.q.2f+ j. The Mellin sonvolution may be extended to the product of 11independent random variables like Y = x&. *.. A’,. The techniques fcx finding MeIIin transforms of the @fs for products. quotien~s.and pchvera! indekndenr random variables are summarized in 3’able 1.

The M&n

transform is then obtained by

sppkabilitv of the %lellin transform incompa&g ” kmplicated fuzzy ,numbers. 1 Let a fuzzy .number X be two TFN A and B. Thai is E.WqdL~

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x = .4(: )B. where their triplets ilre defmcd as .3 = (1.46) and B = (2.3.6 1.Similarly we define 1. = C-I- ,I). \vhere C = t2.3.5) and D = (3.56). NnH. a comparisot? nf two resulting fuzzy numbers X ,and I’ by wa! of the Me&n ttansfkmation is presented.The proportional probabihty distribution. which does net require lltngchange of triplets. .is used. The Mellin transform of X \vo,uId be :\!,(sj = M,,(s) MJ.4. M-here AI,,t.5) ?

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The Mellin transform for Z is given as M,(s) = M,(sj qjs). where

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mo&rate betwe+

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$evencars are available ;tt the garage which deals with wcond-hand cars. b;:eestablishesfor decision atiributc-s: age (XI ). price (X1). gas consumption IX.31. tiild maximum speed cX,) of a crir. Table 3 shows his decision matrix which is expresser’b! fuzzy and crisp data. The fuzzy numbers. which represent the linguistic terms in Tabte 3. are summtirized below: Xl :. ige of the car; new I2:, recent vi21 l&s than 3 around 5 S-10 old N,: selling price: che
This is a multiple attribute decisicti making (MADM) problem whew a decision maker is to mak;: preferencedecisionsover the avail8Wealternativeswhich are charaoterizecfby multi& usually conflic:ing. attributes. The MADM algorithms with crisp data are presentedin 17.171. Reviewson f;i~z~’ XIADM methods arc presented in [2. IO. ?I. 223.iMost fuzzy-!vMDM methods invoCve complicated mat hcmatical operations. However. if

which picks -4+ as the first .
where .Yii&the performance rating of the ith ;tltzrhive whh respect‘to the jth attribute. and wi is t.he weight assigned to the jth attribute. Because the Meltin trgwform of random variable s” is M,.(b.s - h + 1t. the Mellin transform of L’; is @v,enas

B&awe tif the exponent property. this method reqbiics that all ratings be -9reatcr than I. For instance. when an attribute has fractional ratings. all ratings in that attribute xe multiplied by 10” to meet Lis requiremeut.. Note also that a negatir;e weight’ t i.e. - 1t.j) shou!d be’ given to a c&t attribute (the more rating value. th: less preference). This method was introduced long ago by Bridgman Cl]. and has recently been advocated by Swr [f S]-

Since fuzzy numbers ~,!;endo nut yield 3 totally ordcrell set. many comparison-methods have be& proposed to rank them. These methods presume the availability of exact membership function of hwy nuthers. But the fuzzy numbers after arithmetic operations are nighly complicated and their exac; membership functions may not be available. in this paper a probabilistic approach w&sch&en Membership functions are first converted into probabilistic density functions. ihen the Mehin: transform is used to compute the mean and the, var%w of the complex fuzzy truz&ers. The fuzzy number with the hi&her mean& then ranked higher than the fuzzy number. with a ,lou-er mean. If the meansare kqual, the one v&h the smaller variance is judged hi&r rank. Numerical examples reveal the efficiency of the Mellin trsnsform approach in comparing complicated f;luy nutnbers. There ‘is orle caveat to, be raised. The resulting transform may- not bc analytic for aH valves of s. especially when quotiegs .ars involved. For example, with a iand& wriabfz 1’. !’ = !,Y wh& .Y is a random variable uniformly distributed o&r [O. I]. the Mellin transform of j’( ,r) is Y%!,(3)= .1fl2 -- s) = I.-(1 - s). To fi,nd E [ ~-1, t;ve need to evaluate Xf,.iZ), but it is not definsd. If the trans; fcjrrn is not analytic at s = 2 or s -c 3. Park 1131and, Youu; and Contreras [;SJ sugested other t,xlrniques,such as Laptar;e transform. ,’