International Journal of Contemporary Mathematical Sciences

Vol. 12, 2017, no. 1, 31 - 42

HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/ijcms.2017.612158

Fuzzy Reliability Estimation for Exponential

Distribution Using Ranked Set Sampling

M. A. Hussian and Essam A. Amin

Department of Mathematical Statistics, Institute of Statistical Studies and

Research (ISSR), Cairo University, Egypt

Copyright © 2016 M. A. Hussian and Essam A. Amin. This article is distributed under the

Creative Commons Attribution License, which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, the estimation of stress strength models in the presence of fuzziness

is discussed when X and Y are independent exponential random variables with

different parameters. A fuzzy membership function is defined as a function of the

difference between stress and strength values and is increasing whenever yx .

The estimation is made using the maximum likelihood estimator (MLE) of FR

and under two sampling schemes, the simple random sampling (SRS) and the

ranked set sampling (RSS) schemes. Monte Carlo simulations are performed to

compare the estimators obtained using both approaches. The comparison is based

on biases, mean squared errors (MSEs) and the efficiency of the estimators of FR

based on RSS with respect to those based on SRS.

Keywords: Fuzzy; Exponential distribution; Reliability; Stress-strength; Ranked

set sampling; Simple random sampling; Maximum likelihood estimators,

Membership function

1. Introduction

The problem of estimating the reliability measure has received a great

attention and interest. The most widely used approach for reliability estimation is

the well-known stress-strength model. This model is used in many applications of

physics and engineering such as strength failure and the system collapse.

)( XYPR is a measure of component reliability when it is subjected to random

stress Y and has strength X. In this context, R can be considered as a measure of

32 M. A. Hussian and Essam A. Amin

system performance and it is naturally arise in electrical and electronic systems.

The reliability of the system can be also considered as the probability that the

system is strong enough to overcome the stress imposed on it. In addition, it may be

mentioned that R equals the area under the receiver operating characteristic (ROC) curve

for diagnostic test or biomarkers with continuous outcome, see; Bamber (1975).

Let X and Y be two independent continuous random variables having

cumulative distribution function [cdf ( )(xF )] and probability distribution function

[pdf ( )(xf )]. The traditional stress-strength model for 0x is given by

,)()()(

xy

YX ydFxdFXYPR (1)

In fuzzy environment, the membership function plays an important role in

converting fuzzy numbers into real numbers. On this perspective, Eryilmaz and

Tütüncüb (2015) proposed the fuzzy stress-strength as follows

),()()()(0

)( ydFxdFxXYPR YX

y

yAF

(2)

where xyyyA :)( is a fuzzy set and )()( xyA is the appropriate membership

function on )(yA attached to the difference between stress and strength values

where

,

),(

,0

)()(

xyfyxP

xyif

xyA (3)

for an increasing function (.)P .

According to fuzzy stress–strength interference, for xX and yY , with

an increase in the values of yx , the system becomes more reliable and

therefore, such a consideration may provide a more sensitive analysis to the

reliability of the system.

Most of the studies of the traditional stress-strength model focus on the

computation and estimation of the reliability for various stress and strength

distributions such as exponential, Weibull, normal, and gamma. For example,

Raqab et. al., (2008) estimated the reliability model for a 3-parameter generalized

exponential distribution, Wong (2012) estimated confidence intervals of )( XYP

for the generalized Pareto distribution and Asgharzadeh et. al., (2013) studied the

stress-strength model for the generalized logistic distribution. A comprehensive

review of the topic is presented in Kotz et. al., (2003).

Recently, many authors have been interested in estimating the traditional R

using RSS. For example, Sengupta and Mukhuti (2008), considered an unbiased

estimation of traditional R using RSS for exponential populations. Muttlak et al.,

(2010), proposed three estimators of R using RSS when X and Y are independent

Fuzzy reliability estimation for exponential distribution… 33

one-parameter exponential populations. Furthermore, Dong et. al., (2012)

considered nonparametric estimation of reliable life based on ranked set sampling

and proposed a reliable life estimator under a modified ranked set sampling

protocol.

In a RSS procedure, m independent sets of SRS each of size m are drawn

from the distribution under consideration. these samples are ranked by some

auxiliary criterion that does not require actual measurements and only the ith

smallest observation is quantified from the ith set, i = 1,2,…,m. This completes a

cycle of the sampling. Then, the cycle is repeated k times to obtain a ranked set

sample of size n = m k ; see for example Wolfe (2004), Chen and Sinha (2010)

and Bocci and Rocco (2010).

In this article, we consider the estimation of fuzzy stress-strength )( XYPRF

, when X and Y are independent but not identically distributed exponential random

variables.

According to fuzzy stress–strength interference, for xX and yY , with an

increase in the values of yx , the system becomes more reliable. Therefore, such

a consideration may provide a more sensitive analysis. The rest of paper is

organized as follows: the derivation of the traditional and fuzzy stress-strength

models for the exponential distribution is discussed in Section 2. In Sections 3 and

4, the MLEs of R using SRS and RSS approaches for both traditional and fuzzy

models are derived. In Section 5 simulation studies and comparison between

different estimators are discussed. Concluding remarks are given in Section 6.

2. The Stress Strength Model

Let X and Y be two independent exponential random variables with scale

parameters and respectively. The exponential distribution ( )(Exp ) has the

following cdf and pdf for 0x : xexF 1)( and

xexf )( (4)

respectively. The traditional stress-strength model for the exponential distribution

had been extensively studied and calculated to be )/( R ; (see for example

Kotz et. al., (2003)). The membership function defined in Eq. (3) can then be

defined as

,

,1

,0

)()(

)(

xyfe

xyif

xyxc

yA (5)

for some constant 0c . Therefore the fuzzy stress–strength reliability

)( XYPRF is given by

34 M. A. Hussian and Essam A. Amin

.11)(0

)( Rc

c

adydxeeeXYPR x

y

yxa

F

(6)

It is clear that RRF and tends to the traditional reliability R as a .

3. Likelihood Estimation Using SRS:

Let ),...,,( 21 nXXX and ),...,,( 21 mYYY be two independent random samples from

)(Exp and )(Exp respectively. The likelihood function of and for the observed

samples is

]exp[]exp[),;(11

m

jj

nn

ii

n

s yxdataL (7)

Therefore, the log-likelihood function of and is given by

m

j

j

n

i

iss ymxnL11

logloglog

The MLE’s s and s of the parameters and respectively can be

obtained as

1

1

)(ˆ

n

i

is xn , and 1

1

)(ˆ

m

j

js ym

Using the invariance property of the MLE’s, one can get the MLE’s of R

and FR as follows

,ˆˆ

ˆˆ

ss

ssR

and ,ˆ

ˆˆ

s

s

Fs Rc

cR

(8)

4. Likelihood Estimation Using RSS:

Assume that xxjmi ,...,rjmiXx

1,,...,1,):( is a ranked set sample with

sample size xxx rmn drawn from )(Exp where xm and xr are the set size and

the number of cycles of the RSS sample. Let yylmk ,...,rlmkYy

1,,...,1,):( be an

independent ranked set sample with sample size yyy rmn drawn from )(Exp

where ym and yr are the set size and the number of cycles of the RSS sample.

For simplification purposes, jmiX ):( and lmkY ):( will be denoted as ijX and klY

respectively. The pdf of the random variables ijX and klY are given by

,)]exp(1[)][exp()()!1(

!)( 11

1

i

ij

im

ij

x

xij xx

imi

mxg x

,)]exp(1[)][exp()()!1(

!)( 11

2

k

kl

km

kl

y

y

kl yykmk

myg y

Fuzzy reliability estimation for exponential distribution… 35

The likelihood function of and for the observed samples is given by

x x

x

r

j

m

i

i

ij

im

ijr xxKdataL1 1

11)]exp(1[)][exp(),;(

y y

y

r

l

m

k

k

kl

km

kl yy1 1

11)]exp(1[)][exp( (9)

Therefore, the log-likelihood function of and will be

y yx xr

l

m

kkl

r

j

m

iij ykxi

1 11 1

)]exp(1log[)1()]exp(1log[)1(

where *K is constant. This implies that

x xx x

r

j

m

i ijr

ijrijr

j

m

iijx

r

xx

x

xxixim

rm

1 11 1

0)]ˆexp(1[

)ˆexp()1()1(

ˆ

,

and

y yy y

r

l

m

k klr

klrklr

l

m

kkly

r

yy

y

yyjykm

rm

1 11 1

0)]ˆexp(1[

)ˆexp()1()1(

ˆ

,

where r r are the MLEs of the parameters and under the RSS scheme. By

solving the nonlinear equations (15) and (16) and by the invariance property of

the MLEs, the MLE of R and FR is given by

,ˆˆ

ˆˆ

rr

rrR

and ,ˆ

ˆˆ

r

r

Fr Rc

cR

(10)

5. Asymptotic Distribution

In case of the SRS scheme and based on the asymptotic properties under

general conditions of the MLEs s and s , the asymptotic distribution of the

MLEs immediately follows from the Fisher information matrix of and

(Lehmann (1999)). That is, as mn , , FsR is asymptotically normal with

mean FR and asymptotic variance (see Rao (1965)),

1

22

2

1

12

1

11

2

2 2

I

RI

RRI

R FFFFRFs

,

where

22

22

)()(

)(

a

aaRF

,

)()( 2 a

aRF

,

y yx xr

l

m

kkly

r

j

m

iijxyyxxrr ykmximrmrmKL

1 11 1

* )1()1(logloglog

36 M. A. Hussian and Essam A. Amin

and 1

ijI is the (i,j)th element of the inverse of the observed Fisher information

matrix ),( I . According to Eq. (5), we get

m

nII

III

s

s

ss

ss

s 2

2

1

22)(21)(

12)(11)(1

0

0),(

,

where

211)(

nI s , 021)(12)( ss II and

222)(

mI s

The asymptotic %100)1( confidence interval of FsR , is then given by

FsFs RFsRFs zRzR

ˆˆ,ˆˆ2/12/1 (11)

where 2ˆFsR is the estimator of 2

FsR which can be obtained by replacing and

involved in 2

FsR by their corresponding MLEs and z is the quantile of the

standard normal distribution. Similar results can be derived in case of the RSS

scheme with different Fisher information matrix given by

22)(21)(

12)(11)(

)( ),(rr

rr

r II

III ,

where

x xr

j

m

i ij

ijijijijxxr

x

xxxxi

rmI

1 12211)(

)]exp(1[

]1)exp()1)[(exp()1(

,

021)(12)( rr II ,

and

y yr

l

m

k kl

klklklklyy

ry

yyyyj

rmI

1 1222)( .

)]exp(1[

]1)exp()1)[(exp()1(

6. Simulation Study

In this Section, we present a numerical comparison between the MLEs of

traditional and fuzzy stress strength measure R using both SRS and RSS

approaches. The estimation is made through biases and mean squared errors

(MSE) of the MLEs FsR and FrR , and the efficiency of the estimator FrR with

respect the estimator FsR where the efficiency of a parameter 2 with respect to

the parameter 1 is given by )ˆ(/)ˆ()ˆ,ˆ( 2121 MSEMSEeff . We also compute the

expected length (EL) and coverage probability (CP) for both asymptotic

confidence intervals obtained for FR and based on SRS and RSS schemes.

Fuzzy reliability estimation for exponential distribution… 37

The simulations are made using MATHEMATICA for several combinations

of the parameters n , m , xm , xr , ym , yr and R , for 5 yx rr , the random

samples of )(E and )(E are generated using the forms

,)1log(

ux

and ,

)1log(

vy

where 0 , 0 , 10 u and 10 v are uniform random variables. Different

simulations are based on 10000 replications. The results are shown in Tables 1-6.

Table 1: MLE of FR for different values of when 1.0 , 1a and 5 yx rr

FR (n,m) mx,my Bias MSE

)ˆ( FrReff FsR

FrR FsR FrR

0.25 (10,10) (2,2) 0.0764 0.0566 1.8788 1.6493 1.1392 (10,15) (2,3) 0.0686 0.0509 1.3739 1.0999 1.2491 (10,25) (2,5) 0.0639 0.0495 1.0973 0.8497 1.2913 (15,25) (3,5) 0.0608 0.0452 0.8451 0.6278 1.3461 (25,25) (5,5) 0.0559 0.0404 0.6979 0.4686 1.4894 0.50 (10,10) (2,2) 0.0482 0.0351 1.8234 1.5075 1.2096 (10,15) (2,3) 0.0404 0.0294 1.3334 1.0049 1.3269 (10,25) (2,5) 0.0341 0.0259 1.0649 0.7761 1.3721 (15,25) (3,5) 0.0312 0.0231 0.8202 0.5738 1.4294 (25,25) (5,5) 0.0277 0.0221 0.6773 0.4282 1.5817 0.75 (10,10) (2,2) 0.0446 0.0293 1.7936 1.5639 1.1469 (10,15) (2,3) 0.0347 0.0238 1.2719 1.0120 1.2568 (10,25) (2,5) 0.0255 0.0222 0.9987 0.7753 1.2882 (15,25) (3,5) 0.0241 0.0193 0.7536 0.5667 1.3298 (25,25) (5,5) 0.0209 0.0156 0.6080 0.4184 1.4533 0.95 (10,10) (2,2) 0.0332 0.0218 1.6907 1.2323 1.3720 (10,15) (2,3) 0.0260 0.0179 1.2120 0.8216 1.4751 (10,25) (2,5) 0.0251 0.0157 0.9807 0.6346 1.5454 (15,25) (3,5) 0.0216 0.0133 0.7542 0.4689 1.6084 (25,25) (5,5) 0.0190 0.0101 0.6247 0.3504 1.7828

Table 2: MLE of FR for different values of when 1 , 1a and 5 yx rr .

FR (n,m) mx,my Bias MSE

)ˆ( FrReff FsR

FrR FsR

FrR

0.255 (10,10) (2,2) 0.0804 0.0590 1.9777 1.7180 1.1511 (10,15) (2,3) 0.0730 0.0536 1.4616 1.1578 1.2624 (10,25) (2,5) 0.0673 0.0515 1.1550 0.8851 1.3049 (15,25) (3,5) 0.0634 0.0476 0.8803 0.6608 1.3321 (25,25) (5,5) 0.0589 0.0430 0.7346 0.4985 1.4738 0.500 (10,10) (2,2) 0.0508 0.0366 1.9193 1.5703 1.2223 (10,15) (2,3) 0.0430 0.0310 1.4186 1.0578 1.3410 (10,25) (2,5) 0.0359 0.0270 1.1210 0.8085 1.3866 (15,25) (3,5) 0.0325 0.0244 0.8544 0.6040 1.4146 (25,25) (5,5) 0.0292 0.0235 0.7129 0.4555 1.5651 0.667 (10,10) (2,2) 0.0469 0.0305 1.8880 1.6290 1.1589 (10,15) (2,3) 0.0370 0.0251 1.3531 1.0653 1.2702 (10,25) (2,5) 0.0269 0.0232 1.0512 0.8076 1.3017

38 M. A. Hussian and Essam A. Amin

Table 2: (Continued): MLE of FR for different values of when 1 , 1a and

5 yx rr .

(15,25) (3,5) 0.0251 0.0203 0.7850 0.5966 1.3159 (25,25) (5,5) 0.0220 0.0166 0.6400 0.4451 1.4380 0.951 (10,10) (2,2) 0.0350 0.0228 1.7797 1.2836 1.3865 (10,15) (2,3) 0.0277 0.0189 1.2893 0.8649 1.4908 (10,25) (2,5) 0.0264 0.0164 1.0323 0.6610 1.5617 (15,25) (3,5) 0.0225 0.0140 0.7857 0.4936 1.5916 (25,25) (5,5) 0.0200 0.0108 0.6576 0.3728 1.7640

Table 3: MLE of FR for different values of when 1.0 , 10a and

5 yx rr .

FR (n,m) mx,my Bias MSE

)ˆ( FrReff FsR

FrR FsR FrR

0.255 (10,10) (2,2) 0.0458 0.0289 1.1273 0.8428 1.3375 (10,15) (2,3) 0.0400 0.0275 0.8013 0.5939 1.3491 (10,25) (2,5) 0.0378 0.0260 0.6496 0.4466 1.4544 (15,25) (3,5) 0.0359 0.0247 0.4990 0.3432 1.4538 (25,25) (5,5) 0.0322 0.0218 0.4020 0.2531 1.5883 0.500 (10,10) (2,2) 0.0290 0.0179 1.0721 0.7704 1.3918 (10,15) (2,3) 0.0235 0.0159 0.7777 0.5427 1.4331 (10,25) (2,5) 0.0202 0.0136 0.6304 0.4079 1.5454 (15,25) (3,5) 0.0184 0.0126 0.4843 0.3137 1.5439 (25,25) (5,5) 0.0160 0.0120 0.3901 0.2313 1.6868 0.667 (10,10) (2,2) 0.0268 0.0150 1.1192 0.7991 1.4006 (10,15) (2,3) 0.0202 0.0129 0.7417 0.5465 1.3573 (10,25) (2,5) 0.0151 0.0117 0.5913 0.4075 1.4510 (15,25) (3,5) 0.0143 0.0106 0.4697 0.3098 1.5159 (25,25) (5,5) 0.0120 0.0084 0.3794 0.2259 1.6792 0.951 (10,10) (2,2) 0.0200 0.0111 1.0009 0.6297 1.5895 (10,15) (2,3) 0.0152 0.0096 0.7265 0.4437 1.6374 (10,25) (2,5) 0.0149 0.0082 0.5806 0.3335 1.7407 (15,25) (3,5) 0.0127 0.0072 0.4576 0.2564 1.7849 (25,25) (5,5) 0.0109 0.0055 0.3699 0.1892 1.9547

Table 4: MLE of FR for different values of when 1 , 10a and 5 yx rr .

FR (n,m) mx,my Bias MSE

)ˆ( FrReff FsR

FrR FsR FrR

0.255 (10,10) (2,2) 0.0611 0.0413 1.5030 1.2040 1.2484 (10,15) (2,3) 0.0556 0.0382 1.1129 0.8249 1.3490 (10,25) (2,5) 0.0511 0.0356 0.8778 0.6118 1.4349 (15,25) (3,5) 0.0499 0.0348 0.6930 0.4834 1.4335 (25,25) (5,5) 0.0447 0.0303 0.5583 0.3515 1.5886 0.500 (10,10) (2,2) 0.0386 0.0256 1.4587 1.1005 1.3255 (10,15) (2,3) 0.0327 0.0221 1.0801 0.7537 1.4331 (10,25) (2,5) 0.0273 0.0186 0.8519 0.5588 1.5246 (15,25) (3,5) 0.0256 0.0178 0.6726 0.4418 1.5222 (25,25) (5,5) 0.0222 0.0166 0.5418 0.3212 1.6872 0.667 (10,10) (2,2) 0.0357 0.0214 1.4349 1.1416 1.2569 (10,15) (2,3) 0.0281 0.0179 1.0302 0.7590 1.3574

Fuzzy reliability estimation for exponential distribution… 39

Table 4: (Continued): MLE of FR for different values of when 1 , 10a

and 5 yx rr .

(10,25) (2,5) 0.0204 0.0160 0.7990 0.5582 1.4313 (15,25) (3,5) 0.0198 0.0149 0.6180 0.4364 1.4162 (25,25) (5,5) 0.0167 0.0117 0.4864 0.3138 1.5500 0.951 (10,10) (2,2) 0.0266 0.0159 1.3526 0.8996 1.5035 (10,15) (2,3) 0.0211 0.0134 0.9817 0.6162 1.5932 (10,25) (2,5) 0.0201 0.0113 0.7846 0.4569 1.7171 (15,25) (3,5) 0.0177 0.0102 0.6184 0.3611 1.7129 (25,25) (5,5) 0.0152 0.0076 0.4998 0.2628 1.9017

Based on Tables 1 - 4, it is clear that the biases and MSEs of both FsR and

FrR

decreases as the sample size increases. Also, it can be observed that as the

reliability parameter R increases, the biases and MSEs for both FsR and

FrR

decreases. Furthermore, as increases biases and MSEs increase as well for fixed

a. Comparing the two schemes, the biases and MSEs of FrR is always better than

those of FsR which can be noted from the efficiency of

FrR with respect to FsR .

The efficiency of FrR is always greater than one and increases as the sample sizes

increase.

Finally, we observe that the asymptotic confidence interval based on both

schemes works well in terms of coverage probability and is wider when increases for the fixed a. For fixed and as a increases the intervals is better in

the sense of expected length. When comparing both schemes the asymptotic

intervals based on SRS scheme is always wider than those produced under the

RSS scheme.

Table 5: Expected lengths (EL) and coverage probability (CP) of the asymptotic

confidence intervals with 95.0)1( , 1a and 5 yx rr .

FR (n,m) mx,m

y

SRS RSS

1.0EL

1.0CP

1EL

1CP

1.0EL

1.0CP

1EL

1CP

0.25 (10,10) (2,2) 0.5844 0.976 0.615

2

0.956 0.5260 0.976 0.553

7

0.954 (10,15) (2,3) 0.5585 0.981 0.587

9

0.960 0.5027 0.985 0.529

1

0.958 (10,25) (2,5) 0.4707 0.969 0.495

5

0.954 0.4236 0.976 0.446

0

0.955 (15,25) (3,5) 0.4492 0.984 0.472

8

0.975 0.4043 0.985 0.425

5

0.963 (25,25) (5,5) 0.4329 0.980 0.455

7

0.969 0.3896 0.984 0.410

1

0.966 0.50 (10,10) (2,2) 0.6344 0.981 0.667

8

0.962 0.5710 0.971 0.601

0

0.974 (10,15) (2,3) 0.6129 0.986 0.645

2

0.971 0.5516 0.966 0.580

7

0.965 (10,25) (2,5) 0.5818 0.970 0.612

4

0.958 0.5236 0.980 0.551

2

0.955 (15,25) (3,5) 0.5199 0.979 0.547

3

0.963 0.4679 0.969 0.492

6

0.965 (25,25) (5,5) 0.4751 0.985 0.500

1

0.970 0.4276 0.984 0.450

1

0.973 0.75 (10,10) (2,2) 0.5185 0.984 0.545

8

0.968 0.4667 0.979 0.491

2

0.970 (10,15) (2,3) 0.5125 0.986 0.539

5

0.972 0.4613 0.985 0.485

6

0.974 (10,25) (2,5) 0.4965 0.972 0.522

6

0.959 0.4469 0.982 0.470

3

0.962 (15,25) (3,5) 0.4778 0.983 0.502

9

0.976 0.4300 0.989 0.452

6

0.980 (25,25) (5,5) 0.3998 0.989 0.420

8

0.981 0.3598 0.988 0.378

7

0.979 0.95 (10,10) (2,2) 0.3951 0.985 0.415

9

0.979 0.3556 0.987 0.374

3

0.978 (10,15) (2,3) 0.3494 0.979 0.367

8

0.965 0.3145 0.989 0.331

0

0.969 (10,25) (2,5) 0.3184 0.980 0.335

2

0.973 0.2866 0.988 0.301

7

0.978 (15,25) (3,5) 0.2810 0.982 0.295

8

0.980 0.2529 0.979 0.266

2

0.981 (25,25) (5,5) 0.2419 0.985 0.254

6

0.981 0.2177 0.990 0.229

1

0.988

40 M. A. Hussian and Essam A. Amin

Table 6: Expected lengths (EL) and coverage probability (CP) of the asymptotic

confidence intervals with 95.0)1( , 10a and 5 yx rr .

FR (n,m) mx,m

y

SRS RSS

1.0EL

1.0CP

5.0EL

5.0CP

1.0EL

1.0CP

5.0EL

5.0CP

0.2

5

(10,10

)

(2,2) 0.5610 0.977 0.5906 0.975 0.4892 0.979 0.5149 0.976 (10,15

)

(2,3) 0.5362 0.982 0.5644 0.980 0.4675 0.984 0.4921 0.981 (10,25

)

(2,5) 0.4519 0.970 0.4757 0.968 0.3939 0.972 0.4148 0.969 (15,25

)

(3,5) 0.4312 0.985 0.4539 0.983 0.3760 0.987 0.3957 0.984 (25,25

)

(5,5) 0.4156 0.981 0.4375 0.979 0.3623 0.983 0.3814 0.980 0.5

0

(10,10

)

(2,2) 0.6090 0.982 0.6411 0.980 0.5310 0.984 0.5589 0.981 (10,15

)

(2,3) 0.5884 0.987 0.6194 0.985 0.5130 0.989 0.5401 0.986 (10,25

)

(2,5) 0.5585 0.971 0.5879 0.969 0.4869 0.973 0.5126 0.970 (15,25

)

(3,5) 0.4991 0.980 0.5254 0.978 0.4351 0.982 0.4581 0.979 (25,25

)

(5,5) 0.4561 0.986 0.4801 0.984 0.3977 0.988 0.4186 0.985 0.7

5

(10,10

)

(2,2) 0.4978 0.985 0.5240 0.983 0.4340 0.987 0.4568 0.984 (10,15

)

(2,3) 0.4920 0.987 0.5179 0.985 0.4290 0.989 0.4516 0.986 (10,25

)

(2,5) 0.4766 0.973 0.5017 0.971 0.4156 0.975 0.4374 0.972 (15,25

)

(3,5) 0.4587 0.984 0.4828 0.982 0.3999 0.986 0.4209 0.983 (25,25

)

(5,5) 0.3838 0.990 0.4040 0.988 0.3346 0.992 0.3522 0.989 0.9

5

(10,10

)

(2,2) 0.3793 0.986 0.3993 0.984 0.3307 0.988 0.3481 0.985 (10,15

)

(2,3) 0.3354 0.980 0.3531 0.978 0.2925 0.982 0.3078 0.979 (10,25

)

(2,5) 0.3057 0.981 0.3218 0.979 0.2665 0.983 0.2806 0.980 (15,25

)

(3,5) 0.2698 0.983 0.2840 0.981 0.2352 0.985 0.2476 0.982 (25,25

)

(5,5) 0.2322 0.986 0.2444 0.984 0.2025 0.991 0.2131 0.988

7. Conclusions

In this paper, we have addressed the problem of estimating the fuzzy stress-

strength reliability )( XYPRF for the exponential distribution which makes the

analysis more sensitive and more reliable. The estimation is made using both

simple random sampling SRS and ranked set sampling RSS schemes. Both

estimators are based on a new approach in literature and may cause a new point of

view to stress-strength models over different lifetime distributions. Also it is clear

that different membership function will provide different measures of the fuzzy

stress-strength model.

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Received: September 12, 2016; Published: February 15, 2017