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

where

– 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.

The

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 uterus.

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

time intervals.

5.1.

Fuzzy

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

given by

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.

5.2.2. Fuzzy

Reliability function for Generalized Rayleigh distribution

Consider GRD in fuzzy

environment, i.e. the fuzzy parameters

swapped with

the 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

5.2.

Application for Reliability Model

Consider

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

are

2.9246+0.7630a, 3.6876,

4.4826-0.7950a,

7.2050+0.8550a,

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.

Fuzzy

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.)

, giving

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

If T?NH

then the

density function of T is

It will often be

convenient to work with the complement of the c.d.f, the reliability or

survival function

this gives the

probability of survival of beyond time t.

The survival function of NH distribution is obtained by

An alternative

characterization of the distribution of T is given by the hazard function is

defined as

From this we get

The Hazard rate function of NH

distribution is given by

Resembling

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

decreasing for

<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.