Inernaional Journal of Scienific & Engineering Research, Volume 4, Issue 10, Ocober-2013 900 FUZZY MEAN RESIDUAL LIFE ORDERING OF FUZZY RANDOM VARIABLES J. EARNEST LAZARUS PIRIYAKUMAR 1, A. YAMUNA 2 1. Deparmen of Mahemaics, Tranquebar Bishop Manickam Luheran Collage, Porayar 609307, S. India.Email: jelpriyakumar@yahoo.com 2. Deparmen of Mahemaics, A. V. C College (Auonomous), Mannampandal, Mayiladuhurai - 609305, S. India. Email: yamunarun@rediff.com ABSTRACT: I is eviden ha he shape of he failure rae funcion plays an ineviable role in repair and replacemen sraegies, he mean residual life funcion is more relevan as he same summarizes he enire residual life funcions. In his paper we formulae differen fuzzy parial ordering resuls relaed o mean residual life order and proporional mean residual life model wih some characerizaion resuls. Some properies of he up mean residual life model have been obained along wih he closure of he up mean residual life order under mixure ype operaion. We consider fuzzy random variables o capure he mean residual funcion and s how ha redundancy a he componen level is no superior o ha a he sysem level. Even when he lifeimes are original and he spare componens are i.i.d., hough he resul holds for usual sochasic order. The up mean residual life order is characerized in erms of he DMRL class. We capure some of he characerizaions of he DMRL class. Keywords: Fuzzy random variables, Hazard rae order, Sochasic order, Mean residual life order, DMRL and fuzzy up mean residual order. INTRODUCTION In life esing siuaions, he mean addiional life ime given ha a componen has survived unil ime is a funcion of, called he mean residual life. We consider he special case of mean residual life inerms of fuzzy random variables. More specifically, if he fuzzy random variables X represens he life of a componen, hen he mean residual life is given by m () = E[X / X > ]. The mean residual life has been employed in life lengh by various auhors, e.g. Bryson and siddigui (1969), Hollander and Proschan (1975) and Muh (1977). Limiing properies of he mean residual life have been sudied by Meillijson (1972), Balkema and de Hann (1974), and more recenly by Bradley and Gupa [2], (2002). A smooh esimaor of he mean residual life is given by chaubey and Sen (1999). I is well known ha he failure rae funcion can be expressed quie well in erms of mean residual life and is 2013
Inernaional Journal of Scienific & Engineering Research, Volume 4, Issue 10, Ocober-2013 901 derivaive. However, he inverse problem namely ha of expressing he mean residual life in erms of he failure rae ypically involves an inegral of he complicaed expressions. 1. PRELIMINARIES Le F () = P (X > ) be he survival funcion of a random variable X having finie mean µ. Then mean residual life (mrl) m() of X is defined as m() = E[X / X > ] for < * and = 0 oherwise, where * = sup{: F () > 0}. If P(X 0) = 1 hen µ is finie and m(0) = µ. Also m() < for 0 <. If * = hen we have m() = F (x)dx F () Definiion 1.2: if m X() m Y () In his paper, we consider only non-negaive fuzzy random variables, alhough mos of he resuls can be proved for fuzzy random variables wih suppor in (0, 1]. Noe ha, alhough m() 0 for all, every nonnegaive funcion does no respec he mean residual life funcion corresponding o some random variable. In fac, a funcion m(.) represens a mean residual life funcion of some non-negaive random variable wih an absoluely coninuous disribuion funcion iff i possesses he following properies by Bhaacharjee [1], Shaked and Shanhikumar[8] and JEL Piriyakumar and N. renganahan[3] i) 0 m() < for all 0. ii) m(0) 0 iii) m() is coninuous iv) m() + is increasing on [0, ] v) when hese exiss a 0 such ha m( 0 ) = 0 hen m() = 0 for all 0. Oherwise, when here does no exis such a 0 wih m( 0 ) = 0 1 hen d = 0 m() The smaller he mean residual life funcion, he smaller he variable X should be in some sochasic sense. This saemen gives he moivaion behind he mean residual life order defined as follows: Definiion 1.1: Le X and Y be wo random variables wih mean residual life funcions m X and m Y respecively. Then X is said o be smaller han Y in he mean residual life order, denoed by X mrl Y if m X () m Y () for all. over x: X mrl Y iff F x Y x (u)du F X x (u)du F Y decreasing in x (u)du > 0 where (. F X ) denoes he survival funcion of he random variable Z. X is said o be mean residual life aging faser han Y is increasing in 0 (or) ulimaely mean residual life aging faser han Y if above holds for sufficienly large. 2. FUZZY RANDOM VARIABLES The concep of fuzzy random variable was inroduced by Kwakernaak [5,6] and Puri and Ralescu[8]. A fuzzy random variable is jus a random variable ha akes on values in a space of fuzzy ses. The oucomes of Kwakernaak s fuzzy random variables are fuzzy real subses and he exreme poins of heir -cus are classical random variables. Fuzzy random variables are mahemaical descripions for fuzzy sochasic phenomena, bu only one ime descripions. Definiion 2.1: Le (Ω, A, P) be probabiliy space. A fuzzy se valued mapping X: Ω F(R) is called a fuzzy random variable if for each B B and for each (0, 1]. X 1 (B) = {ωεω; X (ω) B φ} A. 3. FUZZY MEAN RESIDUAL LIFE ORDERING 2013
Inernaional Journal of Scienific & Engineering Research, Volume 4, Issue 10, Ocober-2013 902 In his secion he concep of fuzzy mean residual life funcions are inroduced. The concepualizaion is accomplished using resoluion ideniy. I is possible o consruc a closed fuzzy number from a family of closed inervals. Using his echnique he formulaion of fuzzy probabiliy disribuion funcions, fuzzy mean residual life funcions are inroduced. I is esablished ha under cerain assumpions he relaive fuzzy hazard rae ordering leads o he corresponding ulimae fuzzy mean residual life ordering. Le X be a non-negaive random variable wih disribuion funcion (fuzzy) F(x) and densiy funcion (fuzzy) f(x). Le F(x) denoe he failure disribuion of X, r F (x) = f(x) denoe he hazard raes and F (x) F (x) = 1 F(x) denoe he survival funcion X. Le X and Y be wo non-negaive fuzzy random variables wih survival funcions F (x) and (x) G respecively. Then X fmrl Y iff (u)du G (u)du F : F (u)du > 0 for each (0, 1]. increases in over The following resul, due o JEL Piriyakumar and A. Yamuna[4], connecs he fuzzy mean residual life order wih hazard rae order. Theorem 3.5: Le X and Y be wo non- negaive fuzzy random variables wih -level mean residual life funcions m and l respecively. Suppose ha m () l () increases in and for each (0, 1]. Then if X fmrl Y hen i follows ha X fhr Y. Definiion 3.1: 4. SOME NEW RESULTS Le X be a non-negaive fuzzy random variable wih survival funcion F and a finie mean µ. The -level mean This secion presens some new resuls concerning he fuzzy residual life of X a, for (0, 1] is defined as m () = mean residual life order and some characerizaion resuls. E[X /X > ; for < where * = sup{: F () > 0}. 0, oherwise Definiion 4.1: Definiion 3.2: Le X be anon-negaive fuzzy random variable wih an absoluely coninuous disribuion funcion F. The -level hazard rae of X is defined as Definiion 3.3: r () = f () ; R, (0, 1]. F () Le X and Y be wo non-negaive fuzzy random variables wih -level mean residual funcions m and l respecively such ha m () l () for all and for each (0, 1]. Then X is said o be smaller han Y in fuzzy mean residual life order. Symbolically, i is denoed as X fmrl Y. A non-negaive fuzzy random variable X is said o be smaller han anoher fuzzy random variable Y in he fuzzy up mean residual order, denoed by X fmrl Y if X x fmrl Y for all x 0 and for each (0, 1]. I is o be noed ha m X x () = m X (x + ) for each (0, 1]. Symbolically, X x = [X x/x > x]. Definiion 4.2: Le X be a non-negaive fuzzy random variable wih survival funcion F and a finie mean µ. The -level up mean residual life of X x a, for each (0, 1] is defined as Definiion 3.4: 2013
Inernaional Journal of Scienific & Engineering Research, Volume 4, Issue 10, Ocober-2013 903 m Xx () = E[(X x) /X x > ; for < 0, oherwise * = sup{: F () > 0}. where By (1), X x y fmrl Y y for x, y 0 Clearly m Xx () is differeniable over { : P(X x > ) > 0} fmrl Z And we ge F Xx (u)du = 1 F (x) X +x F (u)du and by (2) m Xx () = m X ( + x) for each (0, 1]. X x fmrl Z Also, X fmrl Y iff where xꞌ = x y Proposiion 4.3: ε(0,1] [X x ] fmrl ε(0,1] Z X fmrl X iff X is DMRL. X fmrl Z. Proposiion 4.5: By definiion, X fmrl X If X fmrl Y and Y fmrl X hen X = d Y. X x fmrl X for x 0 and for each (0, 1]. [X x/x > x] hmrl [X x /X > x ] I is obvious from he definiion ha up fuzzy mean residual life order implies fuzzy mean residual life order. whenever xꞌ x 0 Now, we give some condiion under which FMRL order ε(0,1] [X x X > x] hmrl [X x ε(0,1] /X > implies FMRL order. x ] whenever xꞌ x 0 X is DMRL (by definiion of DMRL) Theorem 4.6: If X fmrl Y and eiher x or Y is DMRL, hen Proposiion 4.4: If X fmrl Y and Y fmrl Z, hen X fmrl Z. X fmrl Y X x fmrl Y for all x 0, for each (0, 1]. (1) Similarly Y fmrl Z Y y fmrl Z (2) To prove X fmrl Z X fmrl Y. We know ha X fmrl Y iff, X fmrl Y iff is decreasing in over : F (u)du > 0, for all x 0 and for each (0, 1]. Now, we can wrie (u)du G (3) = F (x+u)du (u)du F (u)du F 2013
Inernaional Journal of Scienific & Engineering Research, Volume 4, Issue 10, Ocober-2013 904 We see ha he firs facor on he RHS of (3) is decreasing in if X is DMRL and he second facor is decreasing in, as X fmrl Y. Hence, for each (0, 1]. If Y is DMRL, hen proof is similar. is decreasing in for x 0 and by proposiion 4.3 and by 4.4 we see ha X fmrl Y. Now, r () = m () m () + 1 m () l () l () + 1 l () = q () i.e) r () q () for all and for each (0, 1]. Hence, ε(0,1] r () ε(0,1] q () i.e) r() q() X hr Y Remark: The non-negaive random variable X is DMRL if and only if [X / X > ] hmrl [X ꞌ / X > ꞌ]. Theorem 4.7: Le X and Y be wo non-negaive fuzzy random variables wih -level mean residual life funcions m and l respecively. Suppose ha m () is increasing in 0 and for l () each (0, 1]. Then X hr Y if X fmrl Y. he mean residual life and some oher sochasic orders of fuzzy random variables, Proceedings of iner m Xx () = E[(X x) /X x > ; for < where naional conference, CIT, Coimbaore. 0, oherwise * = sup{: F () > 0} and (0, 1]. Clearly m Xx () is differeniable over { : P(X x > ) > 0} and ha if X has he -level fuzzy hazard rae funcion r hen derivaive of m (). r () = m () + 1 m () where m ꞌ () denoe he Similarly, if Y has he -level fuzzy hazard rae funcion q, hen q () = l () + 1 l () and X fmrl Y. By sipulaion m () l (). increases in for each (0, 1] REFERENCES 1. M. C. Bhaacharjee, The class of mean residual lives and some consequences, SIAM Journal on Algebraic and discree mehods, Vol. 3, no.1, 614 619, 1995. 2. D. M. Bradley and R. C Gupa, Limiing behavior of he mean residual life, Annals of he insiue of saisical mahemaics, vol 55, 217-226, 2003. 3. J. Earnes Lazarus Piriyakumar and N. Renganahan, Sochasic orderings of fuzzy random variables, In. J. of informaion and managemen sciences, vol 12, no. 4, 29 40, 2001. 4. J. Earnes Lazarus Piriyakumar and A. Yamuna, on 5. Kwakernaak. H., Fuzzy random variables I: Definiion and heorems, Inform. Sci., 15, 1 29, 1978. 6. Kwakernaak. H., Fuzzy random variables II: Algorihms and examples for he discree case, infor. Sci., 17, 253-278, 1979. 7. Puri, M. L., and D. A. Ralescu, Fuzzy random variables, J. Mah. Anal. Appl. 114, 409-422, 1986. 8. Shaked. M and J. G. Shanhikumar, Sochasic orders, New York: Springer, 2007. This shows ha m () l () for all and for each (0, 1]. 2013