Article Preview
TopIntroduction
It is most important to know the nature of all available information, data or model parameters in risk assessment. More often, it is seen that available information/data is interpreted in probabilistic sense because probability theory is a very strong and well established mathematical tool to deal with aleatory uncertainty arises due to inherent variability, natural stochasticity, environmental or structural variation across space or time, due to heterogeneity or the random character of natural processes. However, it is clear that not all available information, data or model parameters are affected by aleatory uncertainty and can be handled by traditional probability theory. Uncertainty may also occur due to scarce or incomplete information or data, measurement error or data obtain from expert judgment or subjective interpretation of available data or information. Thus, model parameters, data may be affected by such kind of (epistemic) uncertainty. Traditional probability theory is inappropriate to represent epistemic uncertainty. To overcome the limitation of probabilistic method, L.A Zadeh in 1965 introduced a new concept called fuzzy set theory and Dempster in 1967 put forward another theory which is known as evidence theory or Dempster- Shafer theory (1976). Fuzzy set theory is more appropriate in situation where uncertainty arises due to vagueness, imprecision, lack of data, small sample sizes etc. On the other hand, the use of Dempster-Shafer theory in risk analysis has many advantages over the conventional probabilistic approach. It provides convenient and comprehensive way to handle engineering problems like imprecisely specified distributions, poorly known and unknown correlation between different variables. It is especially helpful for modelling uncertainty when no data are available and one has to depend on expert opinion. The fundamental objects of DST are called focal elements and the primitive function associated with it is called basic probability assignment (BPA). A DSS can be described by focal elements and its corresponding BPA.
Experts’ opinions are needed when encountering uncertainty, ignorance and complexity. It is needed to deal with the situation where cost of technical difficulties involved or uniqueness of the situation under study make it difficult/impossible to make enough observations to quantify the models with real data. Sometimes, these are also used to refine the estimate obtained from real data as well. Generally, in Dempster-Shafer theory of evidence, experts provide basic probability assignments (bpa) for interval focal elements as data are imprecise or incomplete due to insufficient information (Dutta et al 2011b). Due to the presence of uncertainty, data can be treated as generalized triangular/normal triangular fuzzy number (TFN) because TFN encodes only most likely value (mode) and the spread (confidence interval). Thus we get a comprehensive form of Dempster-Shafer theory of evidence. Straszecka (2003, 2006a, 2006b) studied the basic framework of DST with fuzzy focal elements and used in medical diagnosis. Every disease is associated with a set of symptoms. The symptoms are usually of fuzzy nature (e.g., low blood pressure, high body temperature etc.) and so, use of fuzzy focal elements in DST is justifiable. Membership functions for these symptoms can be defined in consultation with an expert (a physician) or during training data investigation. Then bpas are assigned to the focal elements. In the calculation of belief and plausibility for the disease only those focal elements (symptoms) will take part for which the membership value corresponding to the observed value (laboratory test), exceeds some given threshold value. [Bel(D), Pl(D)] determines the credibility of the diagnosis. Dutta et al (2011b) studied DST with fuzzy focal elements and proposed arithmetic operations on DSTs. Dutta et al (2011c) also discussed DST with fuzzy focal elements and devised a method for obtaining belief and plausibility measure from bpa’s assigned to fuzzy focal elements. Further, Dutta, (2015) studied DST with generalized/normal fuzzy focal elements and an approach was devised to combine DSTs using possibilistic sampling technique. Bauer, (1997), Beynon et al., (2000), Beynon et al., (2001), Beynon, (2002), Beynon, (2005), Mercier et al., (2007), Srivastava & Liu, (2003), Wu, (2009), Yager, (2008), Yang & Sen, (1997), Yang & Xu, (2002), Deng & Chan, (2011) applied DST in decision making.