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The Traditional Engineering Methods Finance Essay

Risk analysis addresses how safe or how risky a real world situation is in a clear-cut way when traditional engineering methods do not explicitly express this. Projects/problems that require risk analysis are generally of high cost or unique or strategically important and multidisciplinary in nature. Risk analysis is routinely used in decision-making concerning weapons systems, space systems, transportation and communication systems, insurance problems, safety of structures and process plants exposed to natural or other hazards. Risk can be quantified for both hazards, which are (i) beyond human control (e.g. natural hazards), and (ii) highly controlled systems (e.g. process plants) where human intervention is p rogrammed and controlled.

Risk assessment of structures in nuclear, chemical, offshore industries and other engineered facilities exposed to natural hazards is of considerable importance to structural engineers. Recently for example, hurricane Katrina was an extraordinary act of nature, which resulted in the most destructive natural disaster in American history (Ayyub 2009). In this regard, it is necessary to understand what risks are taken into account as part of designing the facilities. Sometimes tolerance in risk is preferred to acceptance in risk to express the willingness of individuals and groups to live with a risk to secure certain benefits and in the confidence that it is being properly controlled (Gebhard 2005). In addition, as the value-system of a nation changes (Erich 2002) and as the natural boundary conditions are modified by human actions or global changes, an existing system will be found not meeting the demands of the present society. The decisions for change (Erich 2002) also depend on the changes in options available, perhaps in the case of handling a flood (Renata 2003) situation, as well as on the changes in risk perception and attitudes towards risk.

The assessment has to consider hazards arising from internal or external sources and should also consider structural performance in the process of estimating the safety of structures besides fatalities/injury consequences of the event. Firb e m, explosions, external impacts and seismic events could be among the events to be considered. Perhaps due to fire, progressive collapse (Brian and Halil 2009) has been of an increasing concern in the structural engineering community, especially since the collapse of the World Trade Center towers in 2001.

Risk is generally defined as chance of damage or loss, or probability of loss or as impact of hazards. As stated by Ayyub (2003), in the engineering community, risk is generally defined as the potential of losses for a system resulting from an uncertain exposure to a hazard or as a result of an uncertain event. And, risk is quantified as the rate (measured in identified risk-events per unit time, such as a year) that lives, economic, environmental, and social and cultural losses will occur due to the nonperformance of an engineered system or component. Risk is therefore often viewed as a function of probabilities and adverse events, that can be quantified objectively through risk assessment (Therese and Wibecke 2004). Risk analysis for natural and man-made hazards (or risk assessment) is the process of quantifying the probabilities of potential consequences in various hazard scenarios (Paolo and Colleen 2009), and of evaluating that information to decide whether and how to act, under conditions of uncertainty (Vose 2000; Bedford and Cooke 2001). Overall, risk analysis is the process of producing overall summary measures of the level of safety, cost, or health being assessed.

Since risk analysis involves forecasting of undesirable events (hazardous events), it is not strictly a scientific discipline (Lind 1987). Nevertheless, it is not an arbitrary procedure or a free art form; it employs logical analysis and operates on data obtained by scientific methods that are useful for the context. Moreover, risk analyses are scrutinized by critiques from within the profession, by the clients, and by engineers and scientists in contiguous disciplines. This process is analogous to the way in which new knowledge is accepted in science, and high standards of skepticism are applied (Pidgeon and O'Leary 2000). The assessment of risk, by analysis or judgment, is an appropriate prerequisite for all prudent actions, whether it is a matter of a pedestrian crossing the street or the setting of a hazardous facility.

Risk analysis need not always be quantitative (Kleindorfer and Kunreuther 1987). Although quantification may give an additional dimension, getting the logic right is more important. People often argue that it is not possible to perform a risk analysis when there are not enough data available. In fact, when less information is available, one is more in need of a good risk analysis (Kleindorfer and Kunreuther 1987). Even without detailed information, domain experts can gain insight into the problem. Risk analysis helps to disclose the structure of a problem and its interconnections in a clear and logical way.

Since aim of risk analysis is to facilitate decision making, not only a qualitative but also a quantitative result may be important. For quantification of risk, information/data has to be collected; tests and experiments may have to be conducted in order to obtain confirmed data. However, where such information is lacking it is legitimate and meaningful to base risk estimates mainly on subjective consensus data.

One must be aware that carrying out a risk analysis does not result in reducing risk. In addition, it will not change anything concerning the knowledge of the physical and the background information of a problem. However, it is an attempt to make one-step in the direction of getting a better grasp of the problem of safety. The only thing we are able to achieve by analyzing risks is the way one understands the problem for taking suitable steps to face the risks. The overall objective of risk analysis is to provide a basis for the decision making process concerning risks by comparison of the risks with risk criteria.

Risk estimation can assign a scale for the hazard and thus help both insurers and regulators, where enforcement and regulation requirements are fixed based on appropriate decisions on the acceptability of risk (individual or societal). In addition, outcome of risk analysis helps in reducing risk by adopting suitable measures — e.g. including scrubbers at coal-fired power stations or by adding additional levels of containment at nuclear installation (Bernard 2005) but it is not possible to avoid risks entirely. As an example, estimating the risks associated with a plant gives the owner an indication of the extent of his potential financial liability due to possible damage resulting from a fault within plant. This can be useful in calculating the amount of insurance cover that is required. The process of estimating risks also shows where the system can be modified to improve its reliability and efficiency. Estimation of risk gives the regulator a useful basis for assessing acceptability. The difficulty faced by safety assessors’ lies in convincing regulators and decision-makers that at some point the system is safe enough.

2.1 A BRIEF HISTORICAL PERSPECTIVE

Norman C. Rasmussen of the Massachusetts Institute of Technology was among the first (Morgan 1993) to use rational methods such as graphical trees (event/fault tree) for risk analysis on a large scale when he directed a study of nuclear reactor safety in 1975. Boeing Company carries out risk analysis of design of large aircraft and also to predict the frequency of catastrophic events. Alcoa Company also applied risk analysis and consequently reduced the likelihood of explosions by a factor of 20. Major chemical companies such as Du Pont, Monsanto and Union Carbide have also employed risk analysis techniques in designing processes for chemical plants, in deciding where to build plants and to evaluate the risks of transporting chemicals. In site-specific contamination problems, effective cost reduction is achieved by using risk analysis.

Following the Piper Alpha (Cullen 1990) disaster of an offshore structure in North Sea, UK in 1988, it became clear that the complexity of large and critical engineering systems required new approaches to risk analysis and management. This disaster was the trigger for the introduction of a risk-based goal-setting regulatory regime in the offshore oil and gas sector, and similarly the Seveso incident ultimately led to the formation of Control of Industrial Major Accident Hazards regulations (CIMAH 1984). More recently, a safety case regime has been introduced into the rail sector, and risk-based approach is currently being scrutinized by the International Maritime Organization. There is another recent introduction of Construction Design and Management Regulation (CDM Regulations 1995) with new additional requirements for structural stability. The regulations for structural integrity of new constructions already came under the Building Act and the Building Regulations (The Building Act 1991), which are the responsibility of the department of the Environment (DoE), USA. The SAFESA (SAFE Structural Analysis 1985) process was developed by a consortium to minimize errors in certain avenues with the support of finite element analysis.

In spite of the advances brought about by science and technology in the areas of health, longevity and safety of constructions, society continues to face new hazards. Although certain degree of risk-taking is essential to the development of human society (Zeckhauser and Viscusi 1990), it is also of increasing concern to governments, organizations, and the general public that the risk from new technologies be minimized as far as practicable (Royal Society 1992; Health and Safety Executive 1996). New problems of hazards are being faced because of day by day increasing complexity of problems due to technological development (risk due to technology!). Moreover, the risk analyst would face difficulties in analysis due to the unfamiliarity of the problem. Therefore, better understanding and improved techniques are required for carrying out risk analysis.

From the past history of software systems for supporting application of risk analysis, one can notice that a number of systems are available today for analyzing risk. Some of such systems are given below:

IRAS (Insurance/Investment Risk Analysis System), Stanford University, California — is a system to assess seismic risk of buildings and structures.

SPERIL (Structural PERIL) — is a system for damage assessment of existing structures subjected to earthquake excitations.

Demos, Lumina Decision Systems in Palo Alto, California — for estimating number of deaths caused every year by chemical pollutants.

WIND WRITE, Texas Tech University — for classification of buildings based on vulnerability of buildings against cyclones.

A number of such supporting systems for risk analysis are being developed for meeting the increasing needs. Steps involved for development of systems by using modeling of risk analysis problem is described in the following section.

2.3 STEPS INVOLVED IN MODELING OF RISK ANALYSIS PROBLEM

Model of a real world problem or a complex problem is constructed with a set of inter-related events/parameters and their relationships (Durgaprasad 1997) describing the intrinsic structure and characteristics of that problem. Towards this, knowledge describing various aspects of the problem has to be abstracted and organized in the form of a model. An overview of various problems involved in modeling a real world problem is given in the following.

Abstracted knowledge for different problems may be ranging from rich (complete) evidence base to poor (incomplete) evidence base. If the evidence base of knowledge is rich, then the relationships between the parameters involved can be well defined. Models based on such relationships may be referred as certain type; e.g., problems of engineering mechanics modeled by using classical deterministic methods. If the evidence base of knowledge is poor (incomplete), then the relationships between parameters will be ill-defined. Models based on such relationships may be referred as uncertain type; e.g., models of risk analysis problems based on probabilistic or fuzzy set approach. If the evidence base of information is between rich and poor, then the relationships between parameters may be defined empirically. Models based on such relationships may be classified as empirical type; e.g., clauses in engineering codes of practice.

Formulation of a model of a complex problem like risk analysis problem is often much harder and associated with uncertainties. Model uncertainty seems likely to be more serious, where careful enumeration and control of variables as in laboratory-based experiments, is not possible (Chatfield 1995). There are typically three main sources of uncertainty in any problem (Hodges 1987):

Uncertainty about the structure of the model, which can arise in different ways such as model misspecification (e.g. omitting a variable by mistake)

Uncertainty about estimation of the model parameters, assuming that the structure of the model is known

Unexplained random/fuzzy variation in observed variables even when the structure of the model and the values of the model parameters are known.

The above aspects of modeling can be classified under two broad categories such as qualitative and quantitative. As pointed out by Fishwick (1991), the qualitative aspects may deal with modeling of the problem, methodology, and conceptual framework, while content or the quantitative data is utilized in the quantitative aspects. The qualitative and the quantitative aspects of modeling are the pyramid and inverted pyramid respectively as pictorially illustrated in Fig. 1, which may be elaborated as in the following:

Qualitative aspects of modeling are more related to abstraction of various aspects of problem in terms of ideas and concepts (acquired knowledge/information), and much less to data (numbers) as indicated by the broken arrow.

As one travels towards quantitative aspects of modeling, data (numbers) become more prominent (continuous arrows), and their accuracy increases.

Quantitative

Qualitative

Knowledge

Data

Concepts

Ideas

Accuracy

FIG. 1. Qualitative and quantitative aspects of modeling

Thus, ideas and concepts (knowledge) are mostly on the qualitative side, and numerical accuracy increases towards quantitative side.

The role of qualitative and quantitative aspects may vary in modeling of different problems. Little theory is available to guide us, and the biases (Miller 1990) which result when different models are formulated and fitted to the same data are not well understood. This is because, as stated by Chatfield (1995) that the unexplained random/fuzzy variation will depend not only on unknown variations in sampling units and nuisance variables but also on all the ignored variables and factors. In connection with this, Chatfield (1995) and Box (1976) described that model building in general and of any risk analysis problem can be done in an iterative and interactive way [e.g., Box (1994) on the continuing search for quality improvement] based on the steps given below:

Scope definition

Hazard identification

Knowledge integration

Risk estimation

Computation of risk and understanding

A brief discussion about these steps is given in the following.

2.3.1 Scope Definition

Scope definition involves describing the problem requiring risk analysis, formulating the objective of risk analysis, identifying the concern of interest for risk analysis (e.g. undesirable outcome), selection of the methods/models to be used along with appropriate assumptions and constraints in the process and also identification of potential analysts.

Categorization of risk in different contexts may be of interest and useful. Risks can be categorized by the nature of hazards (e.g. natural hazards, technical hazards, social hazards, and lifestyle hazards) leading to the consequences of interest. Risks can also be categorized by the nature of consequences due to (i) structural safety (impact of damage of structures on the contents), (ii) public safety (economic loss or damage to public health), and (iii) occupational safety (damage to health of workers). In addition, risk may also be classified (Gheorghe 1996) as catastrophic/minor, controllable/non-­controllable, direct/indirect, equitable/inequitable, expected/unexpected, familiar/new, fatal/non-fatal, general/specific, internal/external, instantaneous/gradual, reversible/irreversible, and temporary/permanent.

2.3.2 Hazard Identification

Hazard identification deals with identification of range of possible undesirable events, which have the potential to cause damage, harm, injury or loss. Potential analysts that are identified in the scope definition step may also be involved in hazard identification process. Hazard identification step is vital and most important along with systematic investigation of all the undesirable consequences of events.

It is vital to remember the cautionary words namely that the analysts and decision-makers must understand the problem of risk analysis and should have relevant knowledge on which to base their assessment. If these cautionary words are not carefully thought about, the methodologies can easily lead to very doubtful and misleading results. Relevant knowledge/information must be provided about all the technical, environmental, organizational and human circumstances to the activity/problem being investigated, with special emphasis on safety implications.

Blockly (1996) defined hazard as a group of different undesirable events (characteristics), external or internal to the system, that individually or together represent a set of developing current events which could result in an undesirable future event (hazardous trigger event). For the safe management of hazardous activities, hazard identification has to be carried out thoroughly. This may start from identifying the group (set) of events that may contribute to the hazard, alternative groups of the events that may follow, and the different consequences of each of the event group.

The size of the trigger event (e.g. a high wind, an earthquake or a simple human error) is not the only important cause of the damage/accident, rather one of the main tasks is to identify the preconditions. These preconditions represent the developing potential for failures and damages/hazards/accidents. In view of this, identification of triggering conditions, which could transfer the hazard potential into one or more undesirable events, is important to consider in hazard identification (Sanchez-slva 1994). In this context, Kletz (1992) describes HAZOP (HAZards and OPerability) for chemical process plants in which every component and link is brainstormed for possible undesirable events, consequences and prevention. The brainstorming team might include a design engineer, process engineer, commissioning manager, instrument engineer, research-chemist and an independent chairperson. Results from HOZOP frequently suggest recommendations for modifications to improve the safety and operability of the plant. Hambly (1992) describes HAZAN (HAZard ANalysis) for systematic analysis of hazard and their potential consequences.

In most situations, there will be number of undesirable events to be considered in the hazard identification process using expertise from domain experts, which finally will form as acquired-knowledge. In a constructive approach to knowledge acquisition, domain experts and knowledge engineers cooperatively build models for risk analysis. However, it may be difficult to predict all possible future events. Since many failures are the unintended and unforeseen consequences of hazards (Blockly 1992) a risk prediction may not include the events that actually cause failure. Thus, risk analysis may be an incomplete exercise (Blockly 1996). Moreover, as mentioned earlier, acquired knowledge may be uncertain due to vagueness (Ayyub 1998), which is in the form of relationships between the identified events/parameters.

To facilitate processing of above elicited knowledge, each piece/fragment of knowledge has to be defined as a set of events/parameters and their relationships (Durgaprasad 1997). These identified sets of events/parameters acquired from domain experts may be in verbal and raw form, which may have some meaning of hazard. For example, for the problem of risk analysis of a roof structure against damage subjected to cyclone; intensity of wind speed is a major hazardous factor, since a higher wind speed increases debris potential that results in higher intensity of debris hazard. This relationship is represented symbolically as set of undesirable events to represent a piece of knowledge. pk1= {intensity of wind speed, debris potential, debris hazard}, where pk1 is a set of undesirable events.

Like this, different pieces of knowledge will form the knowledge base and it can be used in risk assessment. Uncertainties inherent in the elicited knowledge can be handled by processing knowledge while integrating knowledge.

2.3.4 Knowledge Integration

Integration of knowledge includes processing of knowledge (Durgaprasad 2001) to make the acquired knowledge more consistent and in a readily useable form. Success of risk analysis depends upon how the possible undesirable events are identified and interrelated.

Risk analysis may give misleading results due to either lack of consistency and completeness in the knowledge or due to failure in taking into account all the relevant undesirable events. As emphasized elsewhere (J. C. Consultants 1986), concern about these aspects has been voiced in studies made by the nuclear and the process industries.

Processing of acquired knowledge is essential in-order to identify the conflicting pieces of information, gaps, and redundancies. Further, it is necessary to eliminate conflicts and represent the knowledge in a consistent and complete manner suitable for carrying out risk analysis. Towards this, rational techniques such as graph theoretic techniques, which can have iterative interaction with domain expert with the help of a computer, are needed for processing of knowledge. The domain expert can use this approach to develop a knowledge base on his own or to consolidate the pieces of knowledge automatically. Processing (Durgaprasad 2001) mainly consists of two main stages, (i) inspection of pieces of knowledge for consistency and completeness, and (ii) synthesis of pieces of knowledge for streamlining and finally to make the knowledge base healthier.

Use of graphs in modeling is, perhaps, the most commonly used method for qualitative aspects of modeling (Fishwick 1991). Graphs provide (Harary 1981; Roberts 1976; Roberts 1981) a buffer between the analyst/domain-expert and the knowledge/information that are used to model the system. A form of directed acyclic graph (causal network/tree form) is used to provide a systematic way of relating the events without circularity (loops). Acquired knowledge is processed, made effective and represented as a directed acyclic graph listing all the events/parameters of a system to display their relationships graphically. This directed acyclic graph is oriented in a particular manner, such that only one branch enters each node. Each edge (branch) represents a possible state or trajectory in time, with time flowing in the direction of the orientation. In addition, an uncertainty (likelihood/probability) is associated with each edge, conditional on the preceding edge. Each vertex (node) represents a possible transition to alternative states or trajectories and marks the introduction of a new state event/parameter.

2.3.5 Risk Estimation

Risk estimation deals with structured assemblies of an estimate, which includes quantification of uncertainty of undesirable events identified at the hazard identification stage, by assessment of their likelihoods and by assessment of the respective consequences. Estimation of risk associated with an event depend on how likely the event is and how serious its occurrence will be, risk must be regarded as a two-dimensional entity comprising the likelihood (probability) and magnitude of adverse consequences. Risk estimation or expected loss for each event can be defined quantitatively as the product of the consequence (C) of a specific undesirable event and its likelihood/probability (P):

R = C X P (1)

Estimation of total risk may be done by associating the risk/product (R) with every branch of directed acyclic graph. The product (R) of these for each branch is then a component of the total risk, and the total risk is obtained by summing over all the possible outcomes. Risk analysis thus amounts to the calculation of top event uncertainties (e.g. probabilities) based on assigned uncertainties.

However, assessments of respective consequences based on their significance’s’ are important to consider. For example, an apparently less significant and relatively small component in a system may, in some circumstances, lead to a high consequence result if failure occurs. An example of this could be in a rail bridge structure where the main body of the structure was found adequate but failure of a connection to the rail could cause a derailment with subsequent personal injuries. The main body of the structure would not have been prone to failure – unless a consequential impact on the main structure then resulted. This illustrates the need for sequential consequences to be identified and assessed.

It may be important to note that the total risk can be reduced by reducing C or by reducing P at each event level.

Deterministic and non-deterministic approaches used for risk estimation are briefly explained below.

2.3.5.1 Deterministic and Non-deterministic Estimation

Deterministic estimation deals with non-consideration of uncertainties of identified events in estimating risk. Simplest form of risk estimation is deterministic estimation (Peacock and Whyte 1992) based on point estimates (usually mean or median). All uncertainties in the values of the input variables (identified events) are treated independently and no attempt is made to quantify their uncertainties. In fact, there is no way to judge the conservatism of these point estimates, especially for problems that can be subjected to the extreme values of external loads (e.g. severe storm runoffs for a dam).

The plausible upper bound is generally used as a cut-off point for policy decisions and the residual risk is unknown. This approach does not allow a meaningful comparison of risk. The ranking of risks according to plausible upper bounds has no theoretical reason to be the same as the ranking of the means. In fact, the ratio of the plausible upper bounds to the mean of the outcome distribution can vary widely, from factors of two or three to several orders of magnitude (Pate-Cornell 1996). Therefore, they can lead to the wrong priorities in allocation of resources. Furthermore, in practice, it is often quickly forgotten that these upper bounds can be far from central values and it is usual for policy makers to subsequently use them as if they represented some kind of a mean.

Although, there is no reason to believe that priorities set based on plausible upper bounds will ensure maximum risk reduction for the money spent, and it is on this ground that the overall conservatism can be challenged. In addition, absolute safety is impossibility and world resources are limited. Excessive expenditure in protection against one risk implies less protecting elsewhere (Hambly 1994). Although, many engineers are familiar with risk estimation based on deterministic functions, but estimations must be done probabilistically to be realistic. To make a meaningful comparison of costs and benefits, one needs to consider uncertainties of undesirable events (constituent parameters). The engineer should consider the range of possible hazards and associated losses and their respective likelihood so as not to over- or under-predict the real damage or loss (Robin 1991).

As explained earlier, non-deterministic estimation deals with consideration of uncertainties of identified events in estimating risk. Uncertainty can be considered for both components of an event (likelihood and consequence) or for a single component (likelihood) of an event. This permits representation of risk either not by a point estimate or by a partially point estimate. Considering uncertainties (e.g. probability distributions) including over consequences is generally regarded as providing a more complete characterization of risk. Handling and assessment of different uncertainties for estimating risk is explained in the following.

2.3.5.2 Uncertainty Analysis

Uncertainty analysis deals with handling of uncertainties involved in parameters including assessment of their uncertainties with the aid of statistics, experiments, analytical methods, experience, and judgment.

Uncertainty can be classified as randomness, fuzziness and incompleteness (Blockly 1996). Uncertainty in structural engineering was classified into objective type that corresponds to randomness, and subjective type that corresponds to fuzziness. The objective types include physical, statistical and modeling sources of uncertainty. The subjective types are based on lack of knowledge and expert-based judgment (e.g. assessment of structural performance).

Uncertainties in structural engineering systems can be mainly attributed to ambiguity and vagueness in assigning the values to different variables/parameters and their inter-relationships. The ambiguity component is generally attributed to non-cognitive source, which includes that relating to randomness. The vagueness-related uncertainty is due to cognitive sources such as the assessment of structural performance, quality, deterioration, and condition of existing structures, and also in defining-the inter-relationships among the variables/parameters of the problem. These sources include fuzziness.

Above uncertainties may be quantified and represented by using probability and fuzzy set principles. Binomial, Poisson, Extreme Value, LogNormal, Negative Exponential, and Weibull Distributions are used to represent uncertainty using statistical methods in probability theory. Extreme value theory (Vesely 1984) has been used to predict frequencies of catastrophic floods, tornadoes and hurricanes. For example, Venkateswarlu et al. (1985) used lognormal distributions to propose a theoretical model for predicting the severe cyclonic wind speeds at a given risk level.

Analysis used to handle the uncertainties is known as Uncertainty Analysis (UA). Methods for UA involve (i) processing of relevant historical/statistical data, and (ii) use of expert judgment techniques. Generally, a combination of these methods is necessary. Uncertainty analysis in the first place is more complex than the upper-bound approach (deterministic). Probability theory as a measure of randomness is commonly used in UA.

Randomness (due to ambiguity) uncertainties can be treated by classical frequentist methods and propagated through the analysis, for example by Monte Carlo simulation. The greater challenge is to treat epistemic (due to vagueness) uncertainties that come from incomplete (or lack of) knowledge about fundamental phenomena. These can be approached only through Bayesian (subjective) probability and expert opinions. Expert opinions can be aggregated into full (composite) probability distributions that can be combined through Bayesian computations with the other variables of the model.

Risk estimation methods for quantifying and propagating uncertainty through models differ according to the degree to which the methods reflect an objective or judgmental perspective of risk. The objective perspective of classical statistics views risk as a measurable property of the physical world. A method with a strong objective perspective will adopt a definition of probability related to the frequency with which events occur, will rely heavily on empirical data and empirically validated models, and will base conclusions on statistical inference. The outputs of risk estimation adopting an objective perspective include both computed outcome values and probabilities, and measures of the uncertainties in these estimates.

The judgmental perspective associated with the eighteenth-century mathematician, the Reverend Thomas Bayes, regards risks as a product of perceptions. The estimation of consequences of hazard is based on perception of seriousness of the hazard. A method with a strong judgmental perspective will adapt the Bayesian definition of probability: i.e. uncertainty is a degree of belief, and probabilities depend on the information, experience, theories, etc. of the individual. Bayesian risk estimation methods tend to make explicit use of expert judgment and theoretical models. The “quality” of the outputs produced by a Bayesian analysis can, however, be reflected by a variety of computed measures, such as the sensitivity of those values to the range of judgments provided by different experts and measures that reflect the extent to which judgmental estimates might change in the face of new evidence.

In spite of the great significance and practical success of probabilistic information theory, it has increasingly been recognized that probability theory captures only one type of uncertainty (Klir 1991). Uncertainty that is not of a statistical nature (the phenomena are not sufficiently random to yield meaningful statistical averages), often may find in natural language and human reasoning of risk analysis.

Since the mid-1960s, several alternative mathematical theories became available for characterizing situations under uncertainty. They are comprised under two broad theories, a theory of fuzzy sets and a theory of fuzzy measures (Zadeh 1965; Sugeno 1969; Klir and Folger 1988). The former is a generalization of classical set theory; the latter is a generalization of probability theory. The concept of a fuzzy set provides a basic mathematical framework for dealing with vagueness (or fuzziness), an important type of uncertainty, which is closely connected with natural language and commonsense reasoning. In contrast to fuzzy sets, which are capable of capturing vagueness of concepts embedded in natural language, fuzzy measures capture uncertainty (incompleteness) caused by lack of knowledge. Well-developed theory of fuzzy measures includes Belief and Plausibility measures. Application of fuzzy measures for risk analysis is not covered in this paper.

2.3.6 Computation of Risk and Understanding

Computational techniques to assess risk will help in propagating the parametric uncertainties along a directed acyclic graph (tree-like structure), where the flow of information is predefined. The directed acyclic graph helps inference mechanism to traverse along each node (parameter/event) to estimate risk. This predefined flow of uncertain information may be termed as solution process. This process deals with the assembly of a composite representation of risk, entails the sequential consideration of uncertainties of parameters one after the other resulting finally in the estimation of total risk.

A more sophisticated synthesis may be in the form of an automated system for computer application. Such systems express in a quantified way the flow of uncertainties of parameters. Towards implementing this process, search techniques are useful, where search is a structured way for propagating parametric uncertainties to arrive at the final risk. Suitable action sequence for tree search and its representation based on probabilistic analysis, bayesian analysis and fuzzy set analysis can be generated. Search techniques such as breadth first and backtracking can be used to pickup uncertainties at each level of parameters and to propagate to higher levels. The process of propagating uncertainties from one level to another higher level will ultimately lead to the final goal of estimating the total risk. This will also help in understanding the estimated risk.

Understanding risk helps in taking decisions and also for introducing necessary changes in the project for reducing the risk. It also may help in identifying human errors, which contribute significantly in effecting the actual risk (Kletz 1990; Humbly 1994). Risk understanding can be categorized into the following two areas:

Information that helps a domain expert/assessor to understand the events that are involved and their sequence, etc. This can be called as risk visualization. A common example of a risk visualization mechanism is showing the ordering of different events.

Information that helps a domain expert to understand why some event is a part of the system/model for risk analysis, or how the system is used to estimate a certain amount of risk, etc. This helps in risk explanation.

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