In the field of lifetime modeling, exponential distribution(ED) has superior status to study the reliability characteristics of anylifetime phenomenon. The admiration of this model has been conferred by anumber of authors.
Although it became greatest widespread due to its persistenthazard rate, but in numerous real-world situation this distribution is notsuited to study the phenomenon where hazard rate is not constant. In recentyears, some fresh classes of models were presented based an amendment of anexponential distribution. For example, Gupta and Kundu (1999) 56 presented anextension of the exponential distribution typically called the generalizedexponential (GE) distribution. Reliability(Survival) has regularly remained a main part in the design of any kind ofsystems.
Analyzing of reliability in life time distribution is discussed byEzzatallah Baloui Jamkhaneh 36, 37. Gonzalez. D-Gonzalez David et al. 51were presented an analysis of reliability censored data. The reader shouldrefer more detail of survival analysis for statistical model in 76. Oxytocin is a Nano peptide synthesized in the hypothalamus andstored in and secreted from the posterior pituitary gland. Oxytocin is liablefor uterine contractions and milk letdown and has been recognized more recentlyto performance a role as a neurotransmitter. Clinically, oxytocin infusion isused for induction and augmentation of labor and for the prevention andtreatment of postpartum hemorrhage (PPH).
Increasing induction ratesrepresentations more women to protracted, incessant infusion of oxytocin, whichis related with dysfunctional labor and increased rate of cesarean delivery.Haemodynamic changes caused by oxytocin during caesarean conversed by 96, 115,and the postpartum haemorrahage in the third stage labour were discussed by 25-27.The identification of a model to assess the effect of Oxytocin in diverseconsequence is a too aspiring task because different individuals (due todifferent body conditions, emotive states, oldness, masculinity, etc.) showdifferent 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 modelusing ENH distribution in fuzzy environment and that model can be used tocalculate the effect of oxytocin. Further, compute the hazard rate and survivalrate for the different time interval after administration of the medicine.6.1.
1. ENHDistributionA random variable T said to follow the GEdistribution then its density function is given by suchthat and andthe development of GE was discussed in 2007 by Gupta and Kundu 57. AlsoNadarajah and Haghighi (2011) 95 presented alternative extension of theexponential model. According to Nadarajah and Haghighi’s, if a random variableT follows the exponential distribution (NHE) then its probability densityfunction is given by Mostrecently Artur J. Lemonte (2013) 78 introduced a new three-parameter familyof distribution called the exponentiated NH ENH distribution. Let a randomvariable T follows the ENH distribution is denoted by and the densityfunction of ENH distribution is defined by where are shapeparameter and is the scale parameter. The cumulativefunction of ENH distribution is given by Thesurvival function of ENH distribution by using equation (5.
14), (5.15) and(5.16), TheENH hazard (failure) rate function is given by using equation (5.18) and (5.19) 6.1. Fuzzy ENH DistributionIn our model 122, we developed a fuzzy-basedmodeling technique using Exponentiated Nadarajah and Haghighi distribution. Theproposed technique of a fuzzy model assesses the effect of oxytocin for womenby evaluating the survival and hazard rate of their heart rate under theadministration of oxytocin.
We may consider the ENH distributions with fuzzyparameters that is replaced in ENH distribution. A random variable T follows fuzzyENH distributionwith fuzzy parameter is known as fuzzy ENH distribution and itis denoted by . The fuzzy probability of a random variable in the interval c, d, c?0 is as and compute its cutas follows:The Fuzzy survival (or fuzzy reliability) function is the fuzzy probability of an item survives beyondtime t. Let the random variable T denote lifetime in a model and T ~FENHD withdensity function then the fuzzy cumulative distribution is In these conditions the fuzzy survival function at time t of thefuzzy ENH distribution is defined asOne fuzzier characterizes of the lifetime distribution is thefuzzy hazard function.
This function isalso known as instantaneous failure rate function. We propose the theory offuzzy hazard function based on the fuzzy probability measures and –cut hazard band. The fuzzy hazard function of is the fuzzy conditional probability of an itemfailing in the short time interval tto t + dt given that it has not failed at time t. We would define the fuzzyhazard function asA fuzzy mathematical model for the effect of thedrug oxytocin is established successfully. The fuzzy probability logic for ENHdistribution hazard rate and survival rate have been effectively assessed inthis paper.
The results shows that the survivalrate is increased and the hazard rate is decreased in the lower –cut values. Similarly the survival rate is decreased and thehazard rate is increased in the upper –cut values.