A New Approach on Proportional Fuzzy Likelihood Ratio orderings of Triangular Fuzzy Random Variables

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International Journal of Research in Engineering and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Print): 2320-9356 www.ijres.org Volume 5 Issue 7 Ç July. 2017 Ç PP. 06-13

A New Approach on Proportional Fuzzy Likelihood Ratio orderings of Triangular Fuzzy Random Variables 1

D.Rajan,2C.Senthilmurugan

1

Associate Professor Of Mathematics, T.B.M.L.College, Porayar South India. Assistant Professor Of Mathematics, Swami Dayananda College Of Arts And Science, Manjakkudi , South India.

2

ABSTRACT: In this chapter, we introduce a new approach on the concept of proportional fuzzy likelihood ratio orderings, increasing and decreasing proportional fuzzy likelihood ratio orderings of triangular fuzzy random variables are presented. Based on these orderings, some theorems are also established.

I.

INTRODUCTION

Likelihood ratio ordering established in stochastic processes is very useful in developing bounds and approximation for performance measures of stochastic systems. Two triangular fuzzy random variables R and T with parameters means đ?œ‡1 , đ?œ‡2 and standard deviationsđ?œŽ1 , đ?œŽ2 respectively, are ordered in the sense of likelihood ratio ordering, when the ratio

P{(R LÎą (ď Ąď€­1) Ďƒ2 Ο2 ) ď‚ł 0 ďƒš(R U Îą + (ď Ąď€­1)Ďƒ2 Ο2 ) ď‚Ł 0} P{(đ?‘‡ÎąL  ď Ąď€­1 Ďƒ2 Ο2 )ď‚ł 0 ďƒš (đ?‘‡ÎąU + ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0}

of their probability density functions is

also an increasing function. In this chapter, we discuss about the Proportional fuzzy likelihood ratio orderings of triangular fuzzy random variables based on Kwakernaak’s [2] fuzzy random variables. The association of the paper is as follows. In section 2, briefly mention the ď Ą-cut of the triangular fuzzy random variables with parameters mean ď ­ and standard deviation ď ł and some new definitions of the proportional, increasing and decreasing proportional fuzzy likelihood ratio orderings are presented. In section 3, we prove some theorems of proportional, increasing and decreasing proportional fuzzy likelihood ratio orderings are proved. 1.1Preliminaries In this section some well - known basic definitions and notions will be discussed. Definition:1.2.1 A fuzzy set A on the universal set X is defined as the set ordered pairs A = {(x, ÂľA(x)) : x ∈X, ÂľA(x) ∈ [0,1]}, where y= ÂľA(x) is its membership function. Definition:1.2.2 The support of fuzzy set A is the set of all points x in X such thatÂľA(x)>0. That is, Support (A) = {x ∈X / ÂľA(x) > 0}. Definition: 1.2.3 The ď Ą-cut of ď Ą-level set of fuzzy set A is a set consisting of those elements of the universe X whose membership values exceed the threshold level ď Ą. (i.e.,) Aď Ą = {X / ď ­A(x) ď‚łď Ą} Definition: 3.2.4 A fuzzy set A on R must possess at least the following three properties to qualify as a fuzzy number. i. A must be a normal fuzzy set ii. Aď Ą must be closed interval for every ď ĄďƒŽ [0,1] iii. The support of A, 0+A must be bounded. Among the various shapes of fuzzy numbers, triangular fuzzy number (TFN) is the most popular one. A TFN is defined as follows:

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A New Approach On Proportional Fuzzy Likelihood Ratio Orderings Of …. Definition: 1.2.5 Atriangular fuzzy number A = (a1, a2, a3) with a1 ,a2, a3 ∈R, a1  a2  a3, is a fuzzy number with membership function for x  a1 0  xa 1 A(x) =  for a1  x  a 2  a 2  a1    a3  x for a  x  a 2 3  a3  a 2   for x  a3 0

It is easy to check that the -cut of a TFN A = (a1, a2, a3) is of the form A = [a1 , a 3 ] with

a 1 = (a2 a1)  + a1, a 3 =  (a3 a2) + a3 Definition: 3.2.6 If R and T are triangular fuzzy random variables with means 1, 2 and standard deviations 1, 2 respectively. Then R is said to be fuzzy likelihood ratio order of triangular fuzzy number (LRO) which is less than (or) equal to T, if whenever s  t and u  v. Here,s = 11,t = 22, u = 1 + 1andv = 2 + 2. P {(RLα  (1)2 2)  0  (RUα + (1)2 2)  0} P {(TαL  (1)1 1)  0  (TαU + (1)1 1)  0}  P {(RLα  (1)1 1)  0  (RUα + (1)1 1)  0} P {(TαL  (1)2 2)  0  (TαU + (1)2 2)  0} It can also be written as P TαL  1 σ1 μ1  0  TαU + 1 σ1 μ1  0 P RLα  1 σ1 μ1  0  RUα + 1 σ1 μ1  0 ≤ We will write R 

LRO

P{(TαL  1 σ2 μ2 ) 0  (TαU + 1 σ2 μ2 )  0}

P{(RLα (1) σ2 μ2 )  0 (RUα + (1)σ2 μ2 )  0}

T.

Definition: 1.2.7 If R and T are triangular fuzzy random variables with means 1, 2 and standard deviations 1, 2 respectively, and T has a log – concave function of the triangular fuzzy random variable. Then R is said to be log – concave function of fuzzy likelihood ratio order of triangular fuzzy number (LCFLR), which is less than (or) equal to T. if whenever s  t andu v. Here, s = 11, t = 22,u = 1 + 1 and v = 2 + 2. Since T is log – concave function of triangular fuzzy random variable. ThenP {bTαL + (1-b)TαU } ≥ P { TαL b } P {{ TαU 1−b } P {(bTαL (1)bσ1 bμ1 )  0  ((1 − b)TαU + (1)(1 − b)σ1 (1 − b)μ1 )  0} b

1−b

≥P { (TαL (1)σ1 μ1 )  0 } P { (TαU + (1)σ1 μ1 )  0 }, 0 ≤ b ≤ 1. Now, (LCFLR) can be written as P{(bTαL  1 bσ1  bμ1 )  0  ((1 − b)TαU + (1)(1 − b)σ1 (1 − b)μ1 ))  0}

. P{(RLα  1 σ1 μ1 )  0 (RUα + 1 σ1 μ1 )  0} P{(bTαL  1 bσ2  bμ2 )  0  ( 1 − b TαU + 1 1 − b σ2  (1 − b)μ2 ))  0} ≤ . P{(RLα  1 σ2 μ2 )  0 (RUα + 1 σ2 μ2 )  0} (3.2.1) And it can also be written as P{ (TαL  1 σ1 μ1 )  0

b

} P{ (TαU + 1 σ1 μ1 )  0

1−b

}

P{ RLα  1 σ1 μ1  0  (RUα + (1)σ1 μ1 )  0} ≤

P{ (TαL  1 σ2 μ2 )  0

b

} P{ (TαU + 1 σ2 μ2 )  0

1−b

P{(RLα  1 σ2 μ2 )  0  (RUα + 1 σ2 μ2 )  0}

(3.2.2)

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}

A New Approach On Proportional Fuzzy Likelihood Ratio Orderings Of ‌. Here, LCFLR of the equation (3.2.1) ≼ equation (3.2.2), otherwise, is called LCONFLROTFN. It will write R ď‚ŁLCFLR T andR ď‚ŁLCONFLR T respectively. Here, the constants b and (1-b) are the left and right part of the triangular fuzzy number.In this chapter, we use the equation (3.2.1) only. 3.3. Proportional Fuzzy Likelihood Ratio Order of Triangular Fuzzy Random Variables 1.3.1 Proportional Fuzzy Likelihood Ratio Order Definition: 1.3.1.1 If R and T are triangular fuzzy random variables with means đ?œ‡1 , đ?œ‡2 and standard deviations đ?œŽ1 ,đ?œŽ2 respectively. Then R is said to be proportional fuzzy likelihood ratio order of the triangular fuzzy number (PFLR) which is less than (or)equal to T. if whenever s ≤ t and u ≤ v. Here, s = ď ­1ď€­ď ł1, t = ď ­2ď€­ď ł2, u = ď ­1 + ď ł1, v = ď ­2 + ď ł2and đ?œ† < 1. P {(RLÎą  (ď Ąď€­1) ď ł2ď€­ď ­2) ď‚ł 0 ďƒš (RUÎą + (ď Ąď€­1)ď ł2ď€­ď ­2) ď‚Ł 0} L P{(ÎťTÎą  (ď Ąď€­1) Îťď ł1ď€­Îťď ­1) ď‚ł 0 ďƒš (ÎťTÎąU + (ď Ąď€­1) Îťď ł1 ď€­Îťď ­1) ď‚Ł 0} ď‚Ł P {(RLÎą  (ď Ąď€­1) ď ł1 ď€­ď ­1) ď‚ł 0 ďƒš (RUÎą + (ď Ąď€­1)ď ł1 ď€­ď ­1) ď‚Ł 0} P{( ÎťTÎąL  (ď Ąď€­1)Îťď ł2 ď€­Îťď ­2) ď‚ł 0 ďƒš (ÎťTÎąU + (ď Ąď€­1) Îťď ł2ď€­Îťď ­2) ď‚Ł 0} It can be written as P {(ÎťTÎąL  (ď Ąď€­1)ÎťĎƒ1 ΝΟ1 ) ď‚ł 0 ďƒš (ÎťTÎąU + (ď Ąď€­1)ÎťĎƒ1 ΝΟ1 ) ď‚Ł 0} P {(RLÎą  (ď Ąď€­1)Ďƒ1  Îź1 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0} ≤

P {(ÎťT LÎą  ď Ąď€­1 ÎťĎƒ2  ΝΟ2 ) ď‚ł 0 ďƒš (ÎťT U Îą + ď Ąď€­1 ÎťĎƒ2  ΝΟ2 ) ď‚Ł 0} P {(R LÎą  ď Ąď€­1 Ďƒ2  Îź2 ) ď‚ł 0 ďƒš (R U Îą + ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0}

(3.3.1.1)

Here, the RHS is increasing in triangular fuzzy random variable for all đ?œ† in (0, 1). We will write R ď‚ŁPFLR T. Theorem: 1.3.1.2 If R and T are triangular fuzzy random variables with means đ?œ‡1 , đ?œ‡2 and standard deviations đ?œŽ1 ,đ?œŽ2 respectively. If R ď‚ŁPFLROTFN T. Thenđ?œ‡đ?‘… ≤ đ?œ‡ đ?‘‡ . Proof: If R and T are triangular fuzzy random variables with means đ?œ‡1 , đ?œ‡2 and standard deviations đ?œŽ1 ,đ?œŽ2 respectively. Whenever, P{đ?œ‡1 − đ?œŽ1 ≤ R ≤ đ?œ‡1 + đ?œŽ1 } = P {(RLÎą  (ď Ąď€­1) đ?œŽ1  đ?œ‡1 ) ď‚ł 0 ďƒš (Rđ?‘ˆÎą + ď Ąď€­1 đ?œŽ1 đ?œ‡1 ) ď‚Ł 0} = P {(RLÎą ď€­ď Ąđ?œŽ1  đ?œ‡1 − đ?œŽ1 ) ď‚ł 0 ďƒš (RUÎą + Îą đ?œŽ1  đ?œ‡1 + đ?œŽ1 ) ď‚Ł 0} (3.3.1.2) P {Îź1 − Ďƒ1 ≤ T ≤ Îź1 + Ďƒ1 } = P{(TÎąL  (ď Ąď€­1)Ďƒ1 Ο1 ) ď‚ł 0 ďƒš (TÎąU + ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0} = P {(TÎąL ď€­ď ĄĎƒ1  Îź1 − Ďƒ1 ) ď‚ł 0 ďƒš (TÎąU + ď ĄĎƒ1  (Îź1 + Ďƒ1 )) ď‚Ł 0} (3.3.1.3) P{Îź2 − Ďƒ2 ≤ R ≤ Îź2 + Ďƒ2 } = P {(RLÎą  ď Ąď€­1 Ďƒ2  Îź2 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0} = P{(RLÎą ď€­ď ĄĎƒ2  Îź2 − Ďƒ2 ) ď‚ł 0 ďƒš (RUÎą + ď ĄĎƒ2  Îź2 + Ďƒ2 ) ď‚Ł 0}(3.3.1.4) P {Îź2 − Ďƒ2 ≤T ≤ Îź2 + Ďƒ2 } = P {(TÎąL  (ď Ąď€­1)ď ł2 ď€­ď ­2) ď‚ł 0 ďƒš (TÎąU + (ď Ąď€­1)ď ł2 ď€­ď ­2) ď‚Ł 0} = P{(TÎąL ď€­ď ĄĎƒ2 (Îź2 − Ďƒ2 )) ď‚ł 0 ďƒš (TÎąU + ď ĄĎƒ2 (Îź2 + Ďƒ2 )) ď‚Ł 0}(3.3.1.5) P {Îź2 − Ďƒ2 ≤ ÎťT ≤ Îź2 + Ďƒ2 } = P {( ÎťTÎąL  (ď Ąď€­1)Îťď ł2 ď€­Îťď ­2) ď‚ł0 ďƒš (ÎťTÎąU + (ď Ąď€­1)Îťď ł2 Îťď ­2) ď‚Ł 0} = P{(ÎťTÎąL ď€­ď ĄÎťĎƒ2  Îť(Îź2 − Ďƒ2 )) ď‚ł 0 ďƒš (ÎťTÎąU + ď ĄÎťĎƒ2  Îť(Îź2 + Ďƒ2 )) ď‚Ł 0}(3.3.1.6) P {Îź2 − Ďƒ2 ≤

T Îť

≤ Îź 2 + Ďƒ2 } = P T LÎą

=P

a

−

ÎąĎƒ2 Îť

T LÎą a

(Îąâˆ’1)Ďƒ2

−

Îť

Îź2 âˆ’Ďƒ2

−

−

Îź2 Îť

≼ 0 ∨

Ν �

TU Îą

≼ 0 ∨ TU Îą a

+

a ÎąĎƒ2 Îť

+ −

(Îąâˆ’1)Ďƒ2 Îť Îź2 âˆ’Ďƒ2

−

Îź2 Îť

≤ 0

≤ 0

Îť

(3.3.1.7)

Let đ?‘‡đ?œ† be the triangular fuzzy number of . Suppose, by contradiction, that đ?œ‡đ?‘… > đ?œ‡ đ?‘‡ . đ?œ†

1

�

1

Since, P {đ?œ‡ − Ďƒ ≤ đ?œ†T ≤ đ?œ‡ + Ďƒ} = P{đ?œ‡ − Ďƒ ≤ ≤ đ?œ‡ + Ďƒ}, Where đ?œ† = <1, a>1. đ?œ† đ?œ† đ?‘Ž In triangular fuzzy random variable, this equation can be written as follows: (1) If T is a triangular fuzzy random variable. Then P {( ÎťTÎąL  (ď Ąď€­1)Îťď ł2 Ν ď ­2) ď‚ł 0 ďƒš (ÎťTÎąU + (ď Ąď€­1)Îťď ł2  Îťď ­2) ď‚Ł 0} TL

1

(Îąâˆ’1)Ďƒ

TU

Îź

(Îąâˆ’1)Ďƒ

Îź

2 2 = P Îąâˆ’ − 2 ≼ 0 ∨ Îą + − 2 ≤ 0 Îť a Îť Îť a Îť Îť (2) If R is a triangular fuzzy random variable. Then P {( ÎťRLι (ď Ąď€­1) Îťď ł2Ν ď ­2) ď‚ł 0 ďƒš(ÎťRUÎą + (ď Ąď€­1)Îť ď ł2 Îťď ­2) ď‚Ł 0}

=

1 Îť

P

R LÎą a

−

(Îąâˆ’1)Ďƒ2 Îť

−

Îź2 Îť

≼ 0 ∨

RU Îą a

+

(Îąâˆ’1)Ďƒ2 Îť

−

Îź2 Îť

≤ 0 (3.3.1.8)

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A New Approach On Proportional Fuzzy Likelihood Ratio Orderings Of ‌. 1

P{đ?œ†đ?œ‡ −đ?œ†Ďƒ ≤đ?œ†T ≤ đ?œ†đ?œ‡ +đ?œ†Ďƒ}

đ?‘Ž

P{đ?œ‡ âˆ’Ďƒ ≤ đ?‘… ≤ đ?œ‡ +Ďƒ}

Where đ?œ† = <1,a>1.It follows from the assumption that

is increasing in triangular fuzzy

random variable for all đ?œ† in (0, 1). Hence, S (đ?‘‡đ?œ† - R) = 1 for each đ?œ† in (0, 1). Here, S (đ?‘‡đ?œ† - R) means that the number of sign changes of the functionsđ?‘‡đ?œ† and R. (i.e.,) R and đ?‘‡đ?œ† are stochastically ordered for eachđ?œ† in (0, 1). In đ?œ‡ đ?œ‡ đ?‘‡ particular, by taking đ?œ† = đ?‘‡ <1. It follows that the triangular fuzzy random variables R and đ?‘… are stochastically ordered. Since R and

đ?œ‡đ?‘… đ?œ‡đ?‘… đ?‘‡ đ?œ‡đ?‘‡

đ?œ‡đ?‘‡

have the same mean, ordinary stochastic order is possible. If they have the same

distribution. This contradicts (3.3.8) and hence đ?œ‡đ?‘… ≤ đ?œ‡ đ?‘‡ holds. Theorem: 1.3.1.3: If R ≤PFLR T, then đ?œ‡1 − đ?œŽ1 ≤ đ?œ‡2 − đ?œŽ2 andđ?œ‡1 + đ?œŽ1 ≤ đ?œ‡2 + đ?œŽ2 . Proof: Supposeđ?œ‡1 − đ?œŽ1 > đ?œ‡2 − đ?œŽ2 . Let ∈1 and ∈2 be such that đ?œ‡2 − đ?œŽ2 <∈1 < đ?œ‡1 − đ?œŽ1 < ∈2 <min{đ?œ‡1 + đ?œŽ1 , đ?œ‡2 + đ?œŽ2 } and letđ?œ† ∈ (0,1) such that đ?œ‡2 − đ?œŽ2 < đ?œ† ∈1 < đ?œ‡1 − đ?œŽ1 < đ?œ† ∈2 < min{đ?œ‡1 + đ?œŽ1 , đ?œ‡2 + đ?œŽ2 }. By definition of PFLR under thegiven condition, we get, P

ÎťTÎąL ď€­ď ĄÎťĎƒ2 Ν Îź2 − Ďƒ2 P

RLÎą ď€­ď ĄĎƒ2  Îź2 − Ďƒ2

ď‚ł 0 ďƒš ÎťTÎąU + ď Ą ÎťĎƒ2 Ν Îź2 + Ďƒ2 ď‚ł 0 ďƒš RUÎą + Îą Ďƒ2  Îź2 + Ďƒ2 ≤

P

∴

P {(ÎťTÎąL ď€­ď ĄÎťĎƒ2 Ν ∈1 ) ď‚ł P {(RLÎą ď€­ď ĄĎƒ2  Îź2 − Ďƒ2 ≤

ď‚Ł0

ÎťTÎąL ď€­ď ĄÎťĎƒ1  Îť Îź1 − Ďƒ1 P

2

ď‚Ł0

ď‚ł 0 ďƒš ÎťTÎąU + ď Ą ÎťĎƒ1 Ν Îź1 + Ďƒ1

RLÎą ď€­ď ĄĎƒ1  Îź1 − Ďƒ1 0 ďƒš

(ÎťTÎąU

ď‚ł 0 ďƒš RUÎą + Îą Ďƒ1  Îź1 + Ďƒ1

ď‚Ł0

ď‚Ł0

+ ď Ą ÎťĎƒ2 Ν Îź2 + Ďƒ2 ) ď‚Ł 0}

) ď‚ł 0 ďƒš (RUÎą + Îą Ďƒ2  Îź2 + Ďƒ2 ) ď‚Ł 0} P {(ÎťTÎąL ď€­ď ĄÎťĎƒ1 Ν2 ∈2 ) ď‚ł 0 ďƒš (ÎťTÎąU + ď Ą ÎťĎƒ1 Ν (Îź1 + Ďƒ1 )) ď‚Ł 0}

P {(RLÎą ď€­ď ĄĎƒ1  Îź1 − Ďƒ1 ) ď‚ł 0 ďƒš (RUÎą + Îą Ďƒ1  Îź1 + Ďƒ1 ) ď‚Ł 0} Which is a contradiction to the definition of PFLR. Therefore, we must have đ?œ‡1 − đ?œŽ1 ≤ đ?œ‡2 − đ?œŽ2 . Similarly, it can be shown that đ?œ‡1 + đ?œŽ1 ≤ đ?œ‡2 + đ?œŽ2 . Theorem: 1.3.1.4 If R and T are triangular fuzzy random variables with means đ?œ‡1 , đ?œ‡2 and standard deviationsđ?œŽ1 ,đ?œŽ2 respectively. Satisfy Rď‚ŁPFLR T if and only if R ď‚ŁFLR aT,a >1. Proof: Since R ď‚ŁPFLR T P {(ÎťTÎąL  ď Ąď€­1 ÎťĎƒ1  ΝΟ1 ) ď‚ł 0 ďƒš (ÎťTÎąU + (ď Ąď€­1)ÎťĎƒ1 ΝΟ1 ) ď‚Ł 0} â&#x;ş P {(RLÎą  (ď Ąď€­1)Ďƒ1  Îź1 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0} P {(ÎťTÎąL  (ď Ąď€­1)ÎťĎƒ2 ΝΟ2 ) ď‚ł 0 ďƒš (ÎťTÎąU + (ď Ąď€­1)ÎťĎƒ2 ΝΟ2 ) ď‚Ł 0} ≤ P {(RLÎą  (ď Ąď€­1)Ďƒ2  Îź2 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0} Where the RHS is increasing in triangular fuzzy random variable for all đ?œ† in (0, 1). 1 Here, đ?œ† = <1, a>1. We get, đ?‘Ž

â&#x;ş

P{(

T LÎą a

 ď Ąď€­1

Ďƒ1 a

Îź

TU Îą

a

a

 1) ď‚ł 0 ∨ (

+ (ď Ąď€­1)

Ďƒ1 a

Îź

 1 ) ≤ 0} a

P {(RLÎą  (ď Ąď€­1)Ďƒ1  Îź1 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0} ≤

TL Ďƒ Îź TU Ďƒ Îź P{( Îą  (ď Ąď€­1) 2  2 ) ď‚ł 0 ∨ ( Îą + ď Ąď€­1 2  2 ) ≤ 0} a

a

a

a

a

a

P {(R LÎą  ď Ąď€­1 Ďƒ2  Îź2 ) ď‚ł 0 ďƒš (R U Îą + ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0}

By using equation (3.3.1.8), we get P {(aTÎąL  (ď Ąď€­1) aĎƒ1 aÎź1 ) ď‚ł 0 ďƒš (aTÎąU + (ď Ąď€­1) aĎƒ1 aÎź1 ) ď‚Ł 0} â&#x;ş P {(RLÎą (ď Ąď€­1) Ďƒ1  Îź1 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0} P {(aTÎąL (ď Ąď€­1) aĎƒ2 aÎź2 ) ď‚ł 0 ďƒš (aTÎąU + (ď Ąď€­1) aĎƒ2 aÎź2 ) ď‚Ł 0} ≤ P {(RLÎą  (ď Ąď€­1) Ďƒ2  Îź2 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0} Here, the RHS is increasing in triangular fuzzy random variable for all đ?œ† in (0, 1). Which implies that R ď‚ŁFLR aT. Theorem: 1.3.1.5

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A New Approach On Proportional Fuzzy Likelihood Ratio Orderings Of ‌. Let T be a triangular fuzzy random variable with mean đ?œ‡1 and standard deviation đ?œŽ1 and T is log – concave. Then T ď‚ŁFLR aT. Proof: Since T is log – concave function of triangular fuzzy random variable with mean đ?œ‡1 and standard deviation đ?œŽ1 .Then P{cTÎąL + (1-c)TÎąU } ≼ P { TÎąL c } P { TÎąU 1−c } P {(cTÎąL (ď Ąď€­1)cĎƒ1 cÎź1 ) ď‚ł 0 ďƒš ((1 − c)TÎąU + (ď Ąď€­1)(1 − c)Ďƒ1 (1 − c)Îź1 ) ď‚Ł 0} c

1−c

≼ P { (TÎąL  (ď Ąď€­1) Ďƒ1 Ο1 ) ď‚ł 0 } P { (TÎąU + (ď Ąď€­1)Ďƒ1 Ο1 ) ď‚Ł 0 } (3.3.1.9) Here, the constants c and (1-c) are the left and right part of the triangular fuzzy number.andaT is also a log – concave function of triangular fuzzy random variable with mean ađ?œ‡1 = đ?œ‡2 and Standard deviation ađ?œŽ1 = đ?œŽ2 . We get P {(abTÎąL  ď Ąď€­1 bĎƒ2  bÎź2 ) ď‚ł 0 ďƒš (a(1 − b)TÎąU + (ď Ąď€­1)(1 − b)Ďƒ2 (1 − b)Îź2 ) ď‚Ł 0} ≼ b

1−b

P { (aTÎąL  ď Ąď€­1 Ďƒ2 Ο2 ) ď‚ł 0 } P { (aTÎąU + ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0 } (3.3.1.10) Here, the constants b and (1-b) are the left and right part of the triangular fuzzy number. By using the equation (3.3.1.1) in the equation (3.3.1.9) and (3.10). We get, P {(abTÎąL  (ď Ąď€­1)bĎƒ1 bÎź1 ) ď‚ł 0 ďƒš (a(1 − b)TÎąU + (ď Ąď€­1)(1 − b)Ďƒ1 (1 − b)Îź1 ) ď‚Ł 0}

P {(cTÎąL  (ď Ąď€­1) cĎƒ1 cÎź1 ) ď‚ł 0 ďƒš ((1 − c)TÎąU + (ď Ąď€­1)(1 − c)Ďƒ1 (1 − c)Îź1 ) ď‚Ł 0} P {(abTÎąL (ď Ąď€­1)bĎƒ2 bÎź2 ) ď‚ł 0 ďƒš (a(1 − b)TÎąU + (ď Ąď€­1)(1 − b)Ďƒ2 (1 − b)Îź2 ) ď‚Ł 0} ≤ P {(cTÎąL  ď Ąď€­1 cĎƒ2  cÎź2 ) ď‚ł 0 ďƒš ((1 − c)TÎąU + (ď Ąď€­1)(1 − c)Ďƒ2 (1 − c)Îź2 ) ď‚Ł 0} FLR Which is implies that T ď‚Ł aT. Theorem: 1.3.1.6 If R and T are triangular fuzzy random variables with means đ?œ‡1 , đ?œ‡2 and standard deviations đ?œŽ1 ,đ?œŽ2 respectively. Suppose that T has a log – concave function. Then R ď‚ŁFLR T ⇒ R ď‚ŁPFLR T. Proof: Since R ď‚ŁFLR T and T has a log – concave function. Then P {(bTÎąL  (ď Ąď€­1)bĎƒ1 bÎź1 ) ď‚ł 0 ďƒš ((1 − b)TÎąU + (ď Ąď€­1)(1 − b)Ďƒ1 (1 − b)Îź1 ) ď‚Ł 0}

P {(RLÎą (ď Ąď€­1)Ďƒ1  Îź1 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0} P{(bTÎąL  ď Ąď€­1 bĎƒ2  bÎź2 ) ď‚ł 0 ďƒš ((1 − b)TÎąU + (ď Ąď€­1)(1 − b)Ďƒ2 (1 − b)Îź2 ) ď‚Ł 0} ≤ P {(RLÎą  (ď Ąď€­1)Ďƒ2  Îź2 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0} Here, b and (1 – b) are the left and right part of the triangular fuzzy number and 0 ≤ b ≤ 1. Now, we use the relationship between đ?œ† and b (đ?œ† < b, đ?œ† ≠0). We get P {(ÎťTÎąL  ď Ąď€­1 ÎťĎƒ1  ΝΟ1 ) ď‚ł 0 ďƒš ((1 − Îť)TÎąU + (ď Ąď€­1)(1 − Îť)Ďƒ1  (1 − Îť)Îź1 ) ď‚Ł 0} P {(RLÎą  (ď Ąď€­1)Ďƒ1  Îź1 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0} P {(ÎťTÎąL  ď Ąď€­1 ÎťĎƒ2  ΝΟ2 ) ď‚ł 0 ďƒš ( 1 − Îť TÎąU + ď Ąď€­1 1 − Îť Ďƒ2  (1 − Îť)Îź2 ) ď‚Ł 0} ≤ P {(RLÎą  (ď Ąď€­1) Ďƒ2  Îź2 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0} PFLR Which implies that R ď‚Ł T. 3.3.2 Increasing and Decreasing Proportional Fuzzy Likelihood Ratio Order Definition: 1.3.2.1 Let R be a triangular fuzzy random variable with mean Âľ and standard deviation Ďƒ. Then R is said to be increasing proportional fuzzy likelihood ratio order of triangular fuzzy number (IPFLROTFN) R ď‚ŁIPFLROTFNR. If P {(ÎťRLÎą  ď Ąď€­1 ÎťĎƒ1  ΝΟ1 ) ď‚ł 0 ďƒš (ÎťRUÎą + ď Ąď€­1 ÎťĎƒ1  ΝΟ1 ) ď‚Ł 0}

P {(RLÎą  (ď Ąď€­1)Ďƒ1  Îź1 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0} P {(ÎťRLÎą  ď Ąď€­1 ÎťĎƒ2  ΝΟ2 ) ď‚ł 0 ďƒš (ÎťRUÎą + (ď Ąď€­1) ÎťĎƒ2 ΝΟ2 ) ď‚Ł 0} ≤ P {(RLÎą  ď Ąď€­1 Ďƒ2  Îź2 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0} the RHS is increasing in triangular fuzzy random variable for all đ?œ† in (0,1). By theorem (3.3.1.1), we have R ď‚ŁIPFLRR with means đ?œ‡1 = đ?œ‡2 and đ?œŽ1 =đ?œŽ2 respectively. It will be said that R is decreasing proportional fuzzy likelihood ratio order of the triangular fuzzy number (DPFLR) R ď‚ŁDPFLR R. If P {(ÎťRLÎą  ď Ąď€­1 ÎťĎƒ1  ΝΟ1 ) ď‚ł 0 ďƒš (ÎťRUÎą + ď Ąď€­1 ÎťĎƒ1  ΝΟ1 ) ď‚Ł 0} P {(RLÎą  (ď Ąď€­1)Ďƒ1  Îź1 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0} P {(ÎťRLÎą  ď Ąď€­1 ÎťĎƒ2  ΝΟ2 ) ď‚ł 0 ďƒš (ÎťRUÎą + ď Ąď€­1 ÎťĎƒ2  ΝΟ2 ) ď‚Ł 0} ≤ P {(RLÎą  (ď Ąď€­1) Ďƒ2  Îź2 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0} www.ijres.org

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A New Approach On Proportional Fuzzy Likelihood Ratio Orderings Of ‌. the RHS is decreasing in triangular fuzzy random variable for all đ?œ† in (0,1). Theorem: 1.3.2.2: Let R be a triangular fuzzy random variable with mean đ?œ‡1 and standard deviation đ?œŽ1 . The following conditions are equivalent:1) R ď‚ŁIPFLRR 2) R ď‚ŁFLR aR, ∀ a > 1. 3) R ď‚ŁPFLR R Proof: Since Rď‚ŁIPFLR R P ÎťRLÎą  ď Ąď€­1 ÎťĎƒ1  ΝΟ1 ď‚ł 0 ďƒš ÎťRUÎą + ď Ąď€­1 ÎťĎƒ1  ΝΟ1 ď‚Ł 0 RLÎą  ď Ąď€­1 Ďƒ1  Îź1 ď‚ł 0 ďƒš RUÎą + ď Ąď€­1 Ďƒ1 Ο1 ď‚Ł 0 P {(ÎťRLÎą (ď Ąď€­1)ÎťĎƒ2 ΝΟ2 ) ď‚ł 0 ďƒš (ÎťRLÎą + (ď Ąď€­1)ÎťĎƒ2 ΝΟ2 ) ď‚Ł 0} ≤ P {(RLÎą (ď Ąď€­1)Ďƒ2  Îź2 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0} the RHS is increasing in triangular fuzzy random variable for all đ?œ† in (0,1). By using equation (3.3.1.8), we get P

P{(

R LÎą Îť

 − ď Ąď€­1

Ďƒ1 Îť

Îź

 1) ď‚ł 0 ∨ ( Îť

P {(RLÎą  (ď Ąď€­1)Ďƒ1  Îź1 ) ď‚ł 0

RU Îą

Îť ďƒš (RUÎą

+ ď Ąď€­1

Ďƒ1 Îť

Îź

 1 ) ≤ 0} Ν

+ ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0} ≤

1

RL Ďƒ Îź RU Ďƒ Îź P{( Îą  ď Ąď€­1 2  2 ) ď‚ł 0 ∨ ( Îą + ď Ąď€­1 2  2 ) ≤ 0} Îť

Îť

Îť

Îť

Îť

Îť

P {(R LÎą  ď Ąď€­1 Ďƒ2  Îź2 ) ď‚ł 0 ďƒš (R U Îą + ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0}

Put đ?‘Ž= , a > 1, we get Îť P {(aRLÎą  ď Ąď€­1 aĎƒ1  aÎź1 ) ď‚ł 0 ďƒš (aRUÎą + ď Ąď€­1 aĎƒ1  aÎź1 ) ď‚Ł 0} P {(RLÎą  (ď Ąď€­1)Ďƒ1  Îź1 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0}

aR LÎą  ď Ąď€­1 aĎƒ2  aÎź2 ď‚ł 0 ďƒš aR U Îą + ď Ąď€­1 aĎƒ2  aÎź2 ď‚Ł 0

P

≤

P

R LÎą  ď Ąď€­1 Ďƒ2  Îź2 ď‚ł 0 ďƒš R U Îą + ď Ąď€­1 Ďƒ2 Ο2 ď‚Ł 0

Which implies that R ď‚Ł aR, ∀ a > 1. Hence, (1) ⇒ (2) 2) Since R ď‚ŁFLR aR, ∀ a > 1. P {(aRLÎą  (ď Ąď€­1) aĎƒ1 aÎź1 ) ď‚ł 0 ďƒš (aRUÎą + (ď Ąď€­1) aĎƒ1 aÎź1 ) ď‚Ł 0} P {(RLÎą  (ď Ąď€­1)Ďƒ1  Îź1 ) ď‚ł 0 ďƒš (RUÎą + ď Ąď€­1 Ďƒ1  Îź1 ) ď‚Ł 0} P aRLÎą  ď Ąď€­1 aĎƒ2  aÎź2 ď‚ł 0 ďƒš aRUÎą + ď Ąď€­1 aĎƒ2  aÎź2 ď‚Ł 0 ≤ P RLÎą  ď Ąď€­1 Ďƒ2  Îź2 ď‚ł 0 ďƒš RUÎą + ď Ąď€­1 Ďƒ2 Ο2 ď‚Ł 0 Here, the RHS is increasing triangular fuzzy random variable for all a >1. 1 Put đ?‘Ž= , a > 1, we get FLRO

P{(

R LÎą Îť

Îť

 − ď Ąď€­1

Ďƒ1 Îť

Îź

 1) ď‚ł 0 ∨ ( Îť

P {(RLÎą  (ď Ąď€­1)Ďƒ1  Îź1 ) ď‚ł 0

RU Îą

Îť ďƒš (RUÎą

+ ď Ąď€­1

Ďƒ1 Îť

Îź

 1 ) ≤ 0} Ν

+ ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0} ≤ P

R LÎą Îť

−

{(RLι 

Îąâˆ’1 Ďƒ2 Îť

−

Îź2 Îť

≼ 0 ∨

P (ď Ąď€­1) Ďƒ2  Îź2 ) ď‚ł 0 ďƒš By equation (3.3.1.8), P {(đ?œ† RLÎą  ď Ąď€­1 đ?œ† đ?œŽ 1 đ?œ† đ?œ‡ 1 ) ď‚ł 0 ďƒš (đ?œ† Rđ?‘ˆÎą + (ď Ąď€­1) đ?œ† đ?œŽ 1 đ?œ† đ?œ‡ 1 ) ď‚Ł 0}

RU Îą Îť (RUÎą

+

(Îąâˆ’1)Ďƒ2 Îť

−

Îź2 Îť

≤ 0

+ ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0}

P {(đ?‘… đ??żđ?›ź  (ď Ąď€­1)đ?œŽ 1  đ?œ‡ 1 ) ď‚ł 0 ďƒš (đ?‘… đ?‘ˆđ?›ź + ď Ąď€­1 đ?œŽ 1 đ?œ‡ 1 ) ď‚Ł 0} P {(đ?œ† đ?‘… đ??żđ?›ź  ď Ąď€­1 đ?œ† đ?œŽ 2 đ?œ† đ?œ‡ 2 ) ď‚ł 0 ďƒš (đ?œ† đ?‘… đ?‘ˆđ?›ź + ď Ąď€­1 đ?œ† đ?œŽ 2 đ?œ† đ?œ‡ 2 ) ď‚Ł 0} ≤ P {(RLÎą  (ď Ąď€­1)đ?œŽ 2  đ?œ‡ 2 ) ď‚ł 0 ďƒš (RU Îą + ď Ąď€­1 đ?œŽ 2 đ?œ‡ 2 ) ď‚Ł 0} Where the RHS is increasing in triangular fuzzy random variable for all đ?œ† < 1. Which implies that R ď‚ŁPFLR R. Hence, (2) ⇒ (3). 3) Since R ď‚ŁPFLR R. P {(ÎťRLÎą  ď Ąď€­1 ÎťĎƒ1  ΝΟ1 ) ď‚ł 0 ďƒš (ÎťRU Îą + ď Ąď€­1 ÎťĎƒ1  ΝΟ1 ) ď‚Ł 0} P {(RLÎą  ď Ąď€­1 Ďƒ1  Îź1 ) ď‚ł 0 ďƒš (RU Îą + ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0} P {(ÎťRLÎą  ď Ąď€­1 ÎťĎƒ2  ΝΟ2 ) ď‚ł 0 ďƒš (ÎťRU Îą + (ď Ąď€­1)ÎťĎƒ2 ΝΟ2 ) ď‚Ł 0} ≤ L U P {(RÎą  ď Ąď€­1 Ďƒ2  Îź2 ) ď‚ł 0 ďƒš (RÎą + ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0} 1

Here, the RHS is increasing in triangular fuzzy random variable for all đ?œ† < 1 and đ?œ† = , a > 1. Which implies that (IPFLR).Hence, R ď‚ŁPFLR R

⇒R ≤IPFLR R.Hence, (3) ⇒ (1). www.ijres.org

đ?‘Ž

11 | Page

A New Approach On Proportional Fuzzy Likelihood Ratio Orderings Of ‌.

Theorem: 1.3.2.3 If R and T are triangular fuzzy random variables with means đ?œ‡ 1 , đ?œ‡ 2 and standard deviations đ?œŽ 1 ,đ?œŽ 2 respectively. If R ď‚ŁFLR T and T is IPFLR. Then R ď‚ŁPFLR T. 3.16. Proof: Since R ď‚ŁFLR T P TLÎą  ď Ąď€­1 Ďƒ1 Ο1 ď‚ł 0 ďƒš TU Îą + ď Ąď€­1 Ďƒ1 Ο1 ď‚Ł 0 P

RLÎą  ď Ąď€­1 Ďƒ1  Îź1 ď‚ł 0 ďƒš RU Îą + ď Ąď€­1 Ďƒ1 Ο1 ď‚Ł 0 ≤

P

TLÎą  ď Ąď€­1 Ďƒ2 Ο2 ď‚ł 0 ďƒš TU Îą + ď Ąď€­1 Ďƒ2 Ο2 ď‚Ł 0

P

RLÎą  ď Ąď€­1 Ďƒ2  Îź2 ď‚ł 0 ďƒš RU Îą + ď Ąď€­1 Ďƒ2 Ο2 ď‚Ł 0

And if T is IPFLR. Then P {(ÎťTLÎą (ď Ąď€­1)ÎťĎƒ1 ΝΟ1 ) ď‚ł 0 ďƒš (ÎťTU Îą + (ď Ąď€­1)ÎťĎƒ1 ΝΟ1 ) ď‚Ł 0} P {(TLÎą  ď Ąď€­1 Ďƒ1 Ο1 ) ď‚ł 0 ďƒš (TU Îą + ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0} P ÎťTLÎą  ď Ąď€­1 ÎťĎƒ2  ΝΟ2 ď‚ł 0 ďƒš ÎťTU Îą + ď Ąď€­1 ÎťĎƒ2  ΝΟ2 ď‚Ł 0 ≤ L U P TÎą  ď Ąď€­1 Ďƒ2  Îź2 ď‚ł 0 ďƒš TÎą + ď Ąď€­1 Ďƒ2  Îź2 ď‚Ł 0 Multiply the above two inequalities, we get P {(ÎťTLÎą  ď Ąď€­1 ÎťĎƒ1 ΝΟ1 ) ď‚ł 0 ďƒš (ÎťTU Îą + (ď Ąď€­1)ÎťĎƒ1 ΝΟ1 ) ď‚Ł 0} P {(RLÎą  (ď Ąď€­1)Ďƒ1  Îź1 ) ď‚ł 0 ďƒš (RU Îą + ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0} ≤ Which implies that R ď‚ŁPFLR T.

U P {(ÎťTL Îą (ď Ąď€­1)ÎťĎƒ2 ΝΟ2 ) ď‚ł 0 ďƒš (ÎťTÎą + ď Ąď€­1 ÎťĎƒ2  ΝΟ2 ) ď‚Ł 0} U P {(RL  ď Ąď€­1 Ďƒ Ο ) ď‚ł 0 ďƒš (R Îą 2 2 Îą + ď Ąď€­1 Ďƒ2 Ο2) ď‚Ł 0}

Theorem: 1.3.2.4 Let R be a triangular fuzzy random variable with mean Âľ and standard deviation Ďƒ. Then R is IPFLR (DPFLR) if and only if R is log – concave (log – convex). Proof: We will prove the result for the DPFLR case; the IPFLR case can be proven in the same way.Suppose that the triangular fuzzy random variable T is log – convex. Then b

1−b

L U P {bTLÎą + (1-b)TU }, (0 ≤ b ≤ 1) Îą } ≤ P { TÎą } P {{ TÎą P {(bTLÎą (ď Ąď€­1)bĎƒ1 bÎź1 ) ď‚ł 0 ďƒš ((1 − b)TU Îą + (ď Ąď€­1)(1 − b)Ďƒ1 (1 − b)Îź1 ) ď‚Ł 0}

≤ P { (TLÎą  (ď Ąď€­1)Ďƒ1 Ο1 ) ď‚ł 0

b

} P { (TU Îą + (ď Ąď€­1)Ďƒ1 Ο1 ) ď‚Ł 0

1−b

}

Here, the triangular fuzzy number (a1, a2, a3) have the equal distance from the mean (a2 = Âľ). Therefore, the middle point đ?‘? +(1−đ?‘? ) of the triangular fuzzy number is = 0.5 = đ?œ† . By using definition (3.2.7) and definition (3.3.2.1), we get 2

P {(ÎťRLÎą  ď Ąď€­1 ÎťĎƒ1  ΝΟ1 ) ď‚ł 0 ďƒš (ÎťRU Îą + ď Ąď€­1 ÎťĎƒ1  ΝΟ1 ) ď‚Ł 0} P {(RLÎą  (ď Ąď€­1)Ďƒ1  Îź1 ) ď‚ł 0 ďƒš (RU Îą + ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0} P {(đ?œ† RLÎą  ď Ąď€­1 đ?œ† đ?œŽ 2 đ?œ† đ?œ‡ 2 ) ď‚ł 0 ďƒš (đ?œ† RU Îą + ď Ąď€­1 đ?œ† đ?œŽ 2 đ?œ† đ?œ‡ 2 ) ď‚Ł 0} ≤ L U P {(RÎą  ď Ąď€­1 đ?œŽ 2  đ?œ‡ 2 ) ď‚ł 0 ďƒš (RÎą + ď Ąď€­1 đ?œŽ 2 đ?œ‡ 2 ) ď‚Ł 0} Here, the RHS is decreasing in triangular fuzzy random variable for all đ?œ† in (0, 1). This implies that R is DPFLR. 1 Conversely, assume that R is DPFLR and Let a >1, đ?œ† = < 1. đ?‘Ž

P {(đ?œ† RLÎą  ď Ąď€­1 đ?œ† đ?œŽ 1 đ?œ† đ?œ‡ 1 ) ď‚ł 0 ďƒš (đ?œ† RU Îą + ď Ąď€­1 đ?œ† đ?œŽ 1 đ?œ† đ?œ‡ 1 ) ď‚Ł 0} P {(RLÎą (ď Ąď€­1) đ?œŽ 1  đ?œ‡ 1 ) ď‚ł 0 ďƒš (RU Îą + ď Ąď€­1 đ?œŽ 1  đ?œ‡ 1 ) ď‚Ł 0} P {(đ?œ† RLÎą  ď Ąď€­1 đ?œ† đ?œŽ 2 đ?œ† đ?œ‡ 2 ) ď‚ł 0 ďƒš (đ?œ† RU Îą + ď Ąď€­1 đ?œ† đ?œŽ 2 đ?œ† đ?œ‡ 2 ) ď‚Ł 0} ≤ L U P {(RÎą  ď Ąď€­1 đ?œŽ 2  đ?œ‡ 2 ) ď‚ł 0 ďƒš (RÎą + ď Ąď€­1 đ?œŽ 2 đ?œ‡ 2 ) ď‚Ł 0} Here, the RHS is decreasing in triangular fuzzy random variable for all đ?œ† in (0, 1). By equation (3.8),

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A New Approach On Proportional Fuzzy Likelihood Ratio Orderings Of ‌. U RL Îą − Îąâˆ’1 Ďƒ1 − Îź1 ≼ 0 ∨ RÎą +(Îąâˆ’1)Ďƒ1 − Îź1 ≤ 0 Îť Îť Îť Îť Îť Îť

P

⇒

U P {(RL Îą  (ď Ąď€­1)Ďƒ1  Îź1 ) ď‚ł 0 ďƒš (RÎą + ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0}

P

≤

U RL Îą − Îąâˆ’1 Ďƒ2 âˆ’Îź2 ≼ 0 ∨ RÎą +(Îąâˆ’1)Ďƒ2 âˆ’Îź2 ≤ 0 Îť Îť Îť Îť Îť Îť

U P {(RL Îą (ď Ąď€­1) Ďƒ2  Îź2 ) ď‚ł 0 ďƒš (RÎą + ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0}

1

Then put a = , a> 1, we get đ?œ†

P {(aRLÎą (ď Ąď€­1)aĎƒ1 aÎź1 ) ď‚ł 0 ďƒš (aRU Îą + (ď Ąď€­1)aĎƒ1 aÎź1 ) ď‚Ł 0} P {(RLÎą (ď Ąď€­1) Ďƒ1  Îź1 ) ď‚ł 0 ďƒš (RU Îą + ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0} P {(aRLÎą (ď Ąď€­1)aĎƒ2 aÎź2 ) ď‚ł 0 ďƒš (aRU Îą + (ď Ąď€­1)aĎƒ2 aÎź2 ) ď‚Ł 0} ≤ L U P {(RÎą (ď Ąď€­1)Ďƒ2  Îź2 ) ď‚ł 0 ďƒš (RÎą + ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0} đ?‘? +(1−đ?‘? )

Here, a = 0.5 = is the middle part of the triangular fuzzy random variable, b and (1-b) are the left and 2 right part of the log – convex triangular fuzzy random variable. Hence, P{(bRLÎą (ď Ąď€­1)bĎƒ1 bÎź1 ) ď‚ł 0 ďƒš ((1 − b)RU Îą + (ď Ąď€­1)(1 − b)Ďƒ1 (1 − b)Îź1 )) ď‚Ł 0} P{(RLÎą  ď Ąď€­1 Ďƒ1 Ο1 ) ď‚ł 0 ďƒš(RU Îą + ď Ąď€­1 Ďƒ1 Ο1 ) ď‚Ł 0} P{(bRLÎą (ď Ąď€­1)bĎƒ2 bÎź2 ) ď‚ł 0 ďƒš ((1 − b)RU Îą + (ď Ąď€­1)(1 − b)Ďƒ2 (1 − b)Îź2 )) ď‚Ł 0} ≤ L . P{(RÎą  ď Ąď€­1 Ďƒ2 Ο2 ) ď‚ł 0 ďƒš(RU Îą + ď Ąď€­1 Ďƒ2 Ο2 ) ď‚Ł 0} LCONFLR Which implies that R ď‚Ł R so that R is log – convex. Hence, the proof.

REFERENCES [1]. [2]. [3]. [4]. [5]. [6].

George Shanthikumar.J, David D.Yao. “The Preservation of likelihood Ratio ordering under Convolution�, Stochastic Processes and their Applications Vol.23, 259-267, 1986. Kwakernaak. H, Fuzzy random variables – I, Definitions and theorems, Information Science,Vol.15, pp.1-29, 1978. NagoorGani. A, and Mohamed Assarudeen. S.N, “A New operation on triangular fuzzy Number for solving fuzzy linear programming problem�, Applied Mathematical Science,Vol.6, No.11, 525-532, 2012. Ramos Romero. H.M,Sordodiaz. M.A, “The Proportional likelihood ratio order and application�, Vol.25, 2, P.211-223, 2001. Stochastic ordering-from wikipedia, the free encyclopedia.The likelihood ratio ordering-wikipedia.

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