The problem of
approximating the reliability and failure rate values in statistical
distributions used to learning a certain occurrence is one of the significant
problems facing constantly those who are interested in life time data analysis.
The exponential distribution is often used in reliability theory and
applications. The cause for that it has a constant failure rate. The
exponential distribution is not guarantees to fit well a given set of real
data. Other distributions have been used in reliability theory. Some were taken
from the twelve different forms of distributions introduced by Burr 17 to
model data. Among those different distributions, Burr-Type X and Burr-Type XII
received the most attention. There is an exhaustive analysis of Burr-Type XII
distribution in Rodriguez 103, see also Wingo 128 for a sufficient
description of it.
Surles and Padgett 112
introduced two-parameter Burr Type X distribution, which can also be described
as generalized Rayleigh distribution (GRD). It is observed that the
two-parameter generalized Rayleigh distribution can be used quite effectively
in modelling strength and general lifetime data. Burr Type X distribution was
also discussed by Raqab, M Z and Kundu D 101.
The parameter estimation for GRD by different method was discussed by
Kundu D and Raqab, M Z 68.
Among the parametric
models, the exponential distribution is perhaps the extensively realistic
statistical distribution in several fields. One of the reasons for its
prominence is that the exponential distribution has constant hazard rate
function. The exponentiated exponential (EE) distribution was introduced by
Gupta et. al. 56. Also Generalized
exponential distribution development was discussed by Gupta and Kundu 57.
A new generalization of
the exponential distribution as an alternative to the gamma, Weibull and EE
distributions was recently proposed by Nadarajah and Haghighi in 201190. If a random variable T follows Nadarajah and
Haghighi (NH) distribution and it is denoted by T?NH
– scale parameter,
– shape parameter. It has some
inspiration properties. The NH distribution density function can be
monotonically decreasing and yet its hazard rate function can be increasing.
gamma, Weibull and EE distributions do not consent for an increasing hazard
function when their respective densities are monotonically decreasing and it is
related with the ability (or the inability) of the NH distribution to model
data that have their mode fixed at zero. The gamma, Weibull and EE
distributions are not suitable for situations of this kind. Statistical
inference for survival data analysis, refer 76.
Oxytocin is a protein
produced by the pituitary gland of mammals including man. Pitocin is a man-made
version of oxytocin used for stimulating contraction of the uterus. Oxytocin
works by increasing the concentration of calcium inside muscle cells that
control contraction of the uterus. Increased calcium increases contraction of
The FDA approved
oxytocin in November 1980, Post-delivery haemorrhage (PPH) is possibly a
serious obstacle of both vaginal and caesarean delivery. The potential risks of
oxytocin boluses in women with significant cardiovascular diseases were studied
by Camann W R 21, Mayer D82. The
prevalence of PPH is approximately 6% of
all deliveries 80.
The most frequent cause
of PPH is uterine atony; therefore, active management of the third stage of
labour rather than expectant management is recommended 22, 24. Currently
vein vaccination of 5 iu of oxytocin is suggested as the prophylactic drug of
choice to reduce the occurrence and sternness of PPH 32, 100. The
Haemodynamic effects of various types are discussed by 85, 115.
In this chapter we explore
two mathematical models using Fuzzy generalized Rayleigh distribution and Fuzzy
Nadarajah and Haghighi distribution for the effect of Oxytocin administration
to determine the reliability (survival) and hazard rate function for different
Generalized Rayleigh distribution Model for Reliability
5.2.1. Reliability function for
Generalized Rayleigh distribution
A random variable T follows the GRD has the reliability function
The failure rate function of GRD is
There are numerous
approaches and examples in classical reliability theory, which assume that all
parameters of lifetime density functions are accurate. Though, in the reality
randomness and fuzziness are often mixed up in the lifetimes of systems. But,
the parameters sometimes cannot record precisely due to machine faults, trial,
individual judgment, approximation or certain unexpected situations. When
parameter in the lifetime distribution is fuzzy, the conventional reliability
system may have trouble for handling reliability and failure rate functions.
The theory of fuzzy reliability was proposed and development by several authors
Cai and et al. 18 – 20, Karpisek, Z64, Hammer 58, Garg, H47, BalouiJamkhaneh E 37. Here we establish
a fuzzy reliability model 121 using GRD.
Reliability function for Generalized Rayleigh distribution
Consider GRD in fuzzy
environment, i.e. the fuzzy parameters
. A random
variable T follows the fuzzy generalized distribution is denoted by
. The fuzzy
reliability function of the FGRD distribution is defined as
The fuzzy hazard function of the
FGRD distribution is defined as
Application for Reliability Model
the experiment by Pinder. A.J. 96 discussed in section 4.3., and the parameters for GRD for the cardiac output after
10u dose of oxytocin are
. The equivalent fuzzy triangular numbers are
2.9246, 3.6876, and 4.4826 and
7.2050, 8.0600, 8.8440. The corresponding
8.0600, 8.8440-0.7840a. The fuzzy reliability and failure rate
values were given the Table 5.1. to Table 5.4. for different time values.5.1.
NH Distribution ModelIn our model 121, we
are investigating Nadarajah and Haghighi distribution in fuzzy environment. The
fuzzy hazard rate function and fuzzy survival function are defined for the
proposed distribution. The fuzzy hazard and survival values are calculated for
different time intervals for the maternal heart rate effects of the women after
the administration of the study medicine. 5.4.1.
NH distributionLet T be a continuous random variable with probability density function
(p.d.f.) f(t) and cumulative
distribution function (c.d.f.)
the probability that the event has occurred by duration t. The
NH distribution is modest and it is raised from the exponentiated exponential
(EE) distribution. The c.d.f. of NH distribution is given by
density function of T is
It will often be
convenient to work with the complement of the c.d.f, the reliability or
this gives the
probability of survival of beyond time t.
The survival function of NH distribution is obtained by
characterization of the distribution of T is given by the hazard function is
From this we get
The Hazard rate function of NH
distribution is given by
to Weibull as well as EE distributions, note that the NH distribution has
closed-form expressions for the survival and hazard rate functions. Moreover,
the hazard rate function can be monotonically increasing for
>1 and monotonically
<1. For =1, the hazard rate function becomes constant.5.4.2. Fuzzy NH DistributionEvery so often we face circumstances the parameters are ambiguous. Thus we consider the NH distribution with fuzzy parameters. The triangular fuzzy numbers are replaced in NH distribution. A random variable T follows Fuzzy NH distribution is denoted by T?FNHD . The fuzzy of the random variable T defined the interval c, d is as is as and compute itsa– cut as follows for all a where such that and such that . Therefore such that . The p.d.f. of a random variable T?FNHD with fuzzy parameters is defined as follows: where .The fuzzy survival function is given by where such that and such that Additional fuzzy epitomizes of the lifetime distribution is the fuzzy hazard function of NH distribution is . We propose the concept of a fuzzy hazard function based on the fuzzy probability measure and named hazard band. The fuzzy hazard rate function is given by where 5.4.3. Results and ApplicationConsider the study by 85, drug was directed as an arterial bolus (delivered in 10 seconds) by the anesthetist after the delivery of the baby. The observing and anesthetic techniques were indistinguishable for all women. For a ?uid preload, 500 ml of 6% hydroxyethyl starch (130/0.4) and 500 ml Ringer's solution were administered. After the patient had entered the operating theatre a local anesthetic (lidocaine hydrochloride) was injected in preparation for spinal anaesthesia by a single-shot technique in a sitting position. The spinal anesthetics (17 mg of ropivacaine and 20 µg of fentanyl) were injected intrathecally at L2/3. Fluid, as well as ephedrine infusion or boluses, could be given as required to achieve haemodynamic stabilization. The caesarean technique was as follows. Laparotomy was performed by a modi?ed Misgav–Ladach technique or Pfannenstiel incision, if necessary. Following uterine incision, delivery of the baby, and cord clamping, the placenta was delivered by cord traction. For uterine repair the uterus was exteriorised. The maternal heart rate (HR) after the administration of the study medication oxytocin is shown in Fig 5.4.1.The shape parameter for NH distribution is = 0.1389 and the scale parameter is = 80.0. The corresponding fuzzy triangular number is = 0.1382, 0.1389, 0.1401 and =78.65, 80.00, 81.23. The alpha cut of the shape parameter is = 0.1382+0.0007a, 0.1389, 0.1401-0.0012a. Likewise, the alpha cut of scale parameter is = 78.65+1.35a, 80.00, 81.23-1.23a. The survival rate and hazard rate after the administration of the drug oxytocin for different t values were shown in the Table 5.5. to Table 5.8.In section 5.3 the GRD and its reliability and failure rate function was successfully established in the fuzzy state. The reliability values and failure rate values were calculating for the doses of 10 u oxytocin. The results show that the reliability values are decreases for lower alpha cuts and increases for upper alpha cuts. In the meantime, the failure rate values are increases for the lower alpha cuts and decreases for upper alpha cut. In section 5.4, using the NH distribution model the fuzzy survival rate and hazard rates are calculated. The result shows that if the survival rate increases then hazard rate decreases with respect to the time intervals. We therefore conclude that oxytocin is uterotonic drug with an acceptable safety profile prophylactic use at the indicated doses are reduce maternal morbidity and mortality caused by PPH.