International Journal of Computer Applications (0975 – 8887) Volume 35– No.11, December 2011 47 Comparison of a Case Study of Uncertainty Propagation using Possibility-Probability Transformation Tazid Ali Dept. Of Mathematics Dibrugarh University Dibrugarh,-786004, India Palash Dutta Dept. Of Mathematics Dibrugarh University Dibrugarh,-786004, India Hrishikesh Boruah Dept. Of Mathematics Dibrugarh University Dibrugarh,-786004, India ABSTRACT Uncertainty parameters in risk assessment can be modeled by different ways viz. probability distribution, possibility distribution, belief measure, depending upon the nature and availability of the data. Different transformations exist for converting expression of one form of uncertainty to another form. They differ from one another substantially, ranging from simple ratio scaling to more sophisticated transformation based upon various principles. These transformations should satisfy certain consistency principles. Several researchers viz., Zadeh, Klir, Dubois & Prade have given such type of consistency laws. The weakest consistency rule that any probability-possibility transformation should satisfy is pro(A) pos(A) i.e., probability of any event is less than or equal to possibility of that event. The strongest among such transformation law Pro(A) > 0  Pos(A) =1. Though possibility and probability capture different types of uncertainty, still transformations are used because it is essential in solving many practical problems. In this paper, we reviewed the consistency principles as given by the above authors. Then we have made a comparative case study of uncertainty propagation by three different methods using probability- possibility transformation satisfying consistency conditions. Keywords Uncertainty, Risk Assessment, Hybrid method, Probability- possibility transformation 1. INTRODUCTION Risk assessment methods have become more and more popular support tools in decision making process. The goal of risk assessment is to estimate the severity and likelihood of harm to humans’ health from exposure to a substance or activity that under plausible circumstances can cause harm to human health. Uncertainty in risk assessment may arise from many different sources such as scarce or incomplete information or data, measurement error or data obtain from expert judgment or subjective interpretation of available data or information. Here we will consider four different types of uncertainties: firstly, random variable observed with total precision which can be represented by a classical probability measure. Secondly, deterministic parameters whose value is imprecisely known, which can be modeled in a natural way by possibility distribution. Thirdly, imprecisely known observed random variables which can be represented by a p-box. Fourthly, there may be a case in which we do not know the representation of the parameters. i.e., it is random variable or deterministic but we know only the range of the values of the parameter and the most likely value. That kind of uncertainty can be either modeled by a possibility distribution or a fuzzy number. As in the last type, where we do not have the proper idea about the representation of the parameter, so we can perform probability/possibility transformation. Human being is always exposed to radiation either from natural or anthropogenic sources in the environment. While there have been natural nuclides since the beginning the earth’s existence, manmade nuclides have been released from nuclear installations and fallouts from the nuclear test and nuclear accident. Also produced water is the most significant source of waste generated in the production phase of oil and gas operations. Once discharged into the ocean, a number of heavy metals and poly aromatic hydrocarbon in produced water may introduce toxicity and bioaccumulation in marine organisms. These compounds are harmful to fish and therefore human can be affected through intake of such fishes. Consequently, we can say that human health can also be indirectly (or directly) affected through different pathways such as inhalation, ingestion, submersion and dermal contact. For this purpose, risk assessment is performed to quantify the potential detriment to human and evaluate the effectiveness of proposed remediation measures. To demonstrate and make use of the transformations a hypothetical case study for non-cancer human health risk assessment is presented here by considering three scenarios and each scenario contains three cases. In the first case, the representation of the some parameters are taken to be possibilistic (fuzzy number) while some are taken to be probabilistic and some are considered as constants. In the second case, we transform the possibilistic distribution (fuzzy number) to triangular probability distribution. In the third case, we will consider the triangular fuzzy numbers as uniform probability distribution with the same support. All the calculations have been performed using Risk calc 4 [7]. 2. PROBABILITY THEORY Probability theory frequently used in uncertainty analysis. If parameters used in prescribed models are random in nature and followed well define distribution, then probabilistic methods are most suitable and well accepted approach for risk assessment. A random variable is a variable in a study in which subjects are randomly selected. Let X be a discrete random variable. A probability mass function is a function such that (i) f(x i )  0, (ii) 1 n i f   (x i ) = 1, (iii) f(x i ) = p(x = x i )