In the field of lifetime modeling, exponential distribution (ED) has superior status to study the reliability characteristics of any lifetime phenomenon. The admiration of this model has been conferred by a number of authors. Although it became greatest widespread due to its persistent hazard rate, but in numerous real-world situation this distribution is not suited to study the phenomenon where hazard rate is not constant. In recent years, some fresh classes of models were presented based an amendment of an exponential distribution. For example, Gupta and Kundu (1999) 53 presented an extension of the exponential distribution typically called the generalized exponential (GE) distribution.
Reliability (Survival) has regularly remained a main part in the design of any kind of systems. Analyzing of reliability in life time distribution is discussed by Ezzatallah Baloui Jamkhaneh 33, 34. Gonzalez David Gonzalez et al. 48 were presented an analysis of reliability censored data. The reader should refer more detail of survival analysis for statistical model in 73.
Oxytocin is a Nano peptide synthesized in the hypothalamus and stored in and secreted from the posterior pituitary gland. Oxytocin is liable for uterine contractions and milk letdown and has been recognized more recently to performance a role as a neurotransmitter. Clinically, oxytocin infusion is used for induction and augmentation of labor and for the prevention and treatment of postpartum hemorrhage (PPH). Increasing induction rates representations more women to protracted, incessant infusion of oxytocin, which is related with dysfunctional labor and increased rate of cesarean delivery. Haemodynamic changes caused by oxytocin during caesarean conversed by 94, 112, and the postpartum haemorrahage in the third stage labour were discussed by 22, 23. The identification of a model to assess the effect of Oxytocin in diverse consequence is a too aspiring task because different individuals (due to different body conditions, emotive states, oldness, masculinity, etc.) show different biological responses (assessed by evaluating the heart rate signal) under the administration of the drug Oxytocin.
The objective of this chapter to present a mathematical model using ENH distribution in fuzzy environment and that model can be used to calculate the effect of oxytocin. Further, compute the hazard rate and survival rate for the different time interval after administration of the medicine.
6.1.1. ENH Distribution
A random variable T said to follow the GE distribution then its density function is given by
such that and .
Also Nadarajah and Haghighi (2011) 88 presented alternative extension of the exponential model. According to Nadarajah and Haghighi’s, if a random variable T follows the NH distribution then its probability density function is given by the equation (5.13).
Most recently Artur J. Lemonte (2013) 78 introduced a new three-parameter family of distribution called the exponentiated NH ENH distribution. Let a random variable T follows the ENH distribution is denoted by and the density function of ENH distribution is defined by
where are shape parameter and is the scale parameter. The cumulative function of ENH distribution is given by
The survival function of ENH distribution by using equation (5.14), (5.15) and (5.16),
The ENH hazard (failure) rate function is given by using equation (5.18) and (5.19)
6.1. Fuzzy ENH Distribution
In our model 124, we developed a fuzzy-based modeling technique using Exponentiated Nadarajah and Haghighi distribution. The proposed technique of a fuzzy model assesses the effect of oxytocin for women by evaluating the survival and hazard rate of their heart rate under the administration of oxytocin.
We may consider the ENH distributions with fuzzy parameters that is replaced in ENH distribution. A random variable T follows fuzzy ENH distributionwith fuzzy parameter is known as fuzzy ENH distribution and it is denoted by . The fuzzy probability of a random variable in the interval c, d, c?0 is as and compute its cut as follows:
for all where
The density function with fuzzy parameters of a random variable is defined as follows:
The Fuzzy survival (or fuzzy reliability) function is the fuzzy probability of an item survives beyond time t. Let the random variable T denote lifetime in a model and T ~FENHD with density function then the fuzzy cumulative distribution is
In these conditions the fuzzy survival function at time t of the fuzzy ENH distribution is defined as
One fuzzier characterizes of the lifetime distribution is the fuzzy hazard function. This function is also known as instantaneous failure rate function. We propose the theory of fuzzy hazard function based on the fuzzy probability measures and – cut hazard band. The fuzzy hazard function of is the fuzzy conditional probability of an item failing in the short time interval t to t + dt given that it has not failed at time t. We would define the fuzzy hazard function asA fuzzy mathematical model for the effect of the drug oxytocin is established successfully. The fuzzy probability logic for ENH distribution hazard rate and survival rate have been effectively assessed in this paper. The results show that the survival rate is increased and the hazard rate is decreased in the lower – cut values. Similarly the survival rate is decreased and the hazard rate is increased in the upper – cut values.