fuzzy reliability estimation for exponential distribution ...€¦ · 04.01.2017 · the...
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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