Probability-credibility health risk assessment under uncertain environment

The ability to describe variables in a health risk model through probability theory enables us to estimate human health risk. These types of risk assessment are interpreted as probabilistic risk assessment (PRA). Generally, PRA requires specific estimate of the parameters of the probability density of the input variables. In all circumstances, such estimates of the parameters may not be available due to the lack of knowledge or information. Such types of variables are treated as uncertain variables. These types of information are often termed uncertainty which are interpreted through fuzzy theory. The ability to describe uncertainty through fuzzy set theory enables us to process both random variable and fuzzy variable in a single framework. The method of processing aleatory and epistemic uncertainties into a same framework is coined as hybrid method. In this paper, we are going to talk about such type of hybrid methodology for human health risk assessment. Risk assessment on human health through different pathways of exposure has been attempted many a times combining Monte Carlo analysis and extension principle of fuzzy set theory. The emergence of credibility theory enables transforming fuzzy variable into credibility distribution function which can be used in those hybrid analyses. Hence, an attempt, for the first time, has been made to combine probability theory and credibility theory to estimate risk in human health exposure. This method of risk assessment in the presence of credibility theory and probability theory is identified as probabilistic-credibility method (PCM). The results obtained are then interpreted through probability theory, unlike the other hybrid methodology where the results are interpreted in terms of possibility theory. The results obtained are then compared with probability-fuzzy risk assessment (PFRA) method. Generally, decision under hybrid methodology is made on the index of optimism. An optimistic decision maker estimates from the \(\alpha\)-cut at 1, whereas a pessimistic decision maker estimates from the \(\alpha\)-cut at 0. The PCM is an optimistic approach as the decision is always made at \(\alpha\)=1.     

Risk tolerance measure for decision-making in fuzzy analysis: a health risk assessment perspective

In risk assessment studies it is important to determine how uncertain and imprecise knowledge should be included into the simulation and assessment models. Thus, proper evaluation of uncertainties has become a major concern in environmental and health risk assessment studies. Previously, researchers have used probability theory, more commonly Monte Carlo analysis, to incorporate uncertainty analysis in health risk assessment studies. However, in conducting probabilistic health risk assessment, risk analyst often suffers from lack of data or the presence of imperfect or incomplete knowledge about the process modeled and also the process parameters. Fuzzy set theory is a tool that has been used in propagating imperfect and incomplete information in health risk assessment studies. Such analysis result in fuzzy risks which are associated with membership functions. Since possibilistic health risk assessment studies are relatively new, standard procedures for decision-making about the acceptability of the resulting fuzzy risk with respect to a crisp standard set by the regulatory agency are not fully established. In this paper, we are providing a review of several available approaches which may be used in decision-making. These approaches involve defuzzification techniques, the possibility and the necessity measures. In this study, we also propose a new measure, the risk tolerance measure, which can be used in decision making. The risk tolerance measure provides an effective metric for evaluating the acceptability of a fuzzy risk with respect to a crisp compliance criterion. Fuzzy risks with different membership functions are evaluated with respect to a crisp compliance criterion by using the possibility, the necessity, and the risk tolerance measures and the results are discussed comparatively.     

A naive uncertainty model for measuring operational risks faced by financial institutions

When insufficient data are available for measuring operational risk faced by a financial institute, most of the models depending on the probability theory are failure. Differing from that we use a probability distribution to depict random uncertainty, in this paper we use a number to represent the naive uncertainty in a phase serving for operational risk identification. The simplest form of the naive uncertainty model for measuring operational risk with multiple phases is the weighted mean with the uncertainties. It is also valid when we have a rough judgment for the uncertainties with intervals or fuzzy values. In this paper, we give a calculation case in lending operational risk to demonstrate the model validity.     

Partitioned multiobjective risk modeling of carcinogenic compounds in groundwater

Quantifying human cancer risk arising from exposure to contaminated groundwater is complicated by the many hydrogeological, environmental, and toxicological uncertainties involved. In this study, we used Monte Carlo simulation to estimate cancer risk associated with tetrachloroethene (PCE) dissolved in groundwater by linking three separate models for: (1) reactive contaminant transport; (2) human exposure pathways; and (3) the PCE cancer potency factor. The hydrogeologic model incorporates an analytical solution for a one-dimensional advective–dispersive–reactive transport equation to determine the PCE concentration in a water supply well located at a fixed distance from a continuous source. The pathway model incorporates PCE exposure through ingestion, inhalation, and dermal contact. The toxicological model combines epidemiological data from eight rodent bioassays of PCE exposure in the form of a composite cumulative distribution frequency curve for the human PCE cancer potency factor. We assessed the relative importance of individual model variables through their correlation with expected cancer risk calculated in an ensemble of Monte Carlo simulations with 20,000 trials. For the scenarios evaluated, three factors were most highly correlated with cancer risk: (1) the microbiological decay constant for PCE in groundwater, (2) the linear groundwater pore velocity, and (3) the cancer potency factor. We then extended our analysis beyond conventional expected value risk assessment using the partitioned multiobjective risk method (PMRM) to generate expected-value functions conditional to a 1 in 100,000 increased cancer risk threshold. This approach accounts for low probability/high impact outcomes separately from the conventional unconditional expected values. Thus, information on potential worst-case outcomes can be quantified for decision makers. Using PMRM, we evaluated the cost-benefit relationship of implementing several postulated risk management alternatives intended to mitigate the expected and conditional cancer risk. Our results emphasize the importance of hydrogeologic models in risk assessment, but also illustrate the importance of integrating environmental and toxicological uncertainty. When coupled with the PMRM, models integrating uncertainty in transport, exposure, and potency constitute an effective risk assessment tool for use within a risk-based corrective action (RBCA) framework.     

Grey fuzzy optimization model for water quality management of a river system

A grey fuzzy optimization model is developed for water quality management of river system to address uncertainty involved in fixing the membership functions for different goals of Pollution Control Agency (PCA) and dischargers. The present model, Grey Fuzzy Waste Load Allocation Model (GFWLAM), has the capability to incorporate the conflicting goals of PCA and dischargers in a deterministic framework. The imprecision associated with specifying the water quality criteria and fractional removal levels are modeled in a fuzzy mathematical framework. To address the imprecision in fixing the lower and upper bounds of membership functions, the membership functions themselves are treated as fuzzy in the model and the membership parameters are expressed as interval grey numbers, a closed and bounded interval with known lower and upper bounds but unknown distribution information. The model provides flexibility for PCA and dischargers to specify their aspirations independently, as the membership parameters for different membership functions, specified for different imprecise goals are interval grey numbers in place of a deterministic real number. In the final solution optimal fractional removal levels of the pollutants are obtained in the form of interval grey numbers. This enhances the flexibility and applicability in decision-making, as the decision-maker gets a range of optimal solutions for fixing the final decision scheme considering technical and economic feasibility of the pollutant treatment levels. Application of the GFWLAM is illustrated with case study of the Tunga–Bhadra river system in India.     

An integrated fuzzy-stochastic modeling approach for assessing health-impact risk from air pollution

High concentrations of air pollutants in the ambient environment can result in breathing problems with human communities. Effective assessment of health-impact risk from air pollution is important for supporting decisions of the related detection, prevention, and correction efforts. However, the quality of information available for environmental/health risk assessment is often not satisfactory enough to be presented as deterministic numbers. Stochastic method is one of the methods for tackling those uncertainties, by which uncertain information can be presented as probability distributions. However, if the uncertainties can not be presented as probabilities, they can then be handled through fuzzy membership functions. In this study, an integrated fuzzy-stochastic modeling (IFSM) approach is developed for assessing air pollution impacts towards asthma susceptibility. This development is based on Monte Carlo simulation for the fate of SO₂ in the ambient environment, examination of SO₂ concentrations based on the simulation results, quantification of evaluation criteria using fuzzy membership functions, and risk assessment based on the combined fuzzy-stochastic information. The IFSM entails (a) simulation for the fate of pollutants in ambient environment, with the consideration of source/medium uncertainties, (b) formulation of fuzzy air quality management criteria under uncertain human-exposure pathways, exposure dynamics, and SPG-response variations, and (c) integrated risk assessment under complexities of the combined fuzzy/stochastic inputs of contamination level and health effect (i.e., asthma susceptibility). The developed IFSM is applied to a study of regional air quality management. Reasonable results have been generated, which are useful for evaluating health risks from air pollution. They also provide support for regional environmental management and urban planning.     

2D Monte Carlo versus 2D Fuzzy Monte Carlo health risk assessment

Risk estimates can be calculated using crisp estimates of the exposure variables (i.e., contaminant concentration, contact rate, exposure frequency and duration, body weight, and averaging time). However, aggregate and cumulative exposure studies require a better understanding of exposure variables and uncertainty and variability associated with them. Probabilistic risk assessment (PRA) studies use probability distributions for one or more variables of the risk equation in order to quantitatively characterize variability and uncertainty. Two-dimensional Monte Carlo Analysis (2D MCA) is one of the advanced modeling approaches that may be used to conduct PRA studies. In this analysis the variables of the risk equation along with the parameters of these variables (for example mean and standard deviation for a normal distribution) are described in terms of probability density functions (PDFs). A variable described in this way is called a second order random variable. Significant data or considerable insight to uncertainty associated with these variables is necessary to develop the appropriate PDFs for these random parameters. Typically, available data and accuracy and reliability of such data are not sufficient for conducting a reliable 2D MCA. Thus, other theories and computational methods that propagate uncertainty and variability in exposure and health risk assessment are needed. One such theory is possibility analysis based on fuzzy set theory, which allows the utilization of incomplete information (incomplete information includes vague and imprecise information that is not sufficient to generate probability distributions for the parameters of the random variables of the risk equation) together with expert judgment. In this paper, as an alternative to 2D MCA, we are proposing a 2D Fuzzy Monte Carlo Analysis (2D FMCA) to overcome this difficulty. In this approach, instead of describing the parameters of PDFs used in defining the variables of the risk equation as random variables, we describe them as fuzzy numbers. This approach introduces new concepts and risk characterization methods. In this paper we provide a comparison of these two approaches relative to their computational requirements, data requirements and availability. For a hypothetical case, we also provide a comperative interpretation of the results generated.     

Assessment and management of risk in subsurface hydrology: A review and perspective

Uncertainty plagues every effort to model subsurface processes and every decision made on the basis of such models. Given this pervasive uncertainty, virtually all practical problems in hydrogeology can be formulated in terms of (ecologic, monetary, health, regulatory, etc.) risk. This review deals with hydrogeologic applications of recent advances in uncertainty quantification, probabilistic risk assessment (PRA), and decision-making under uncertainty. The subjects discussed include probabilistic analyses of exposure pathways, PRAs based on fault tree analyses and other systems-based approaches, PDF (probability density functions) methods for propagating parametric uncertainty through a modeling process, computational tools (e.g., random domain decompositions and transition probability based approaches) for quantification of geologic uncertainty, Bayesian algorithms for quantification of model (structural) uncertainty, and computational methods for decision-making under uncertainty (stochastic optimization and decision theory). The review is concluded with a brief discussion of ways to communicate results of uncertainty quantification and risk assessment.     

Uncertainty Analysis for a Dynamic Phosphorus Model with Fuzzy Parameters

A simplified method based on fuzzy set theory is presented to incorporate uncertainty of parameters into a dynamic total phosphorus model. Uncertainty may arise from difference between calibrated conditions and projected condition as well as from inconsistency of available data in the literature. The uncertainty in parameters was represented by fuzzy numbers that can be generated through various ways such as model calibration process, soft interpretation of literature data, and subjective opinions of experts. The proposed fuzzy approach decomposed fuzzy parameters into interval numbers at different level cuts, and solved for interval solutions through very simple calculation instead of solving nonlinear programming models. The interval solutions at each level cut were could be combined to obtain fuzzy solutions. This method has been applied to the phosphorus load-response model of the Triadelphia Reservoir near the Washington, DC area. Two pollution control scenarios have been simulated with fuzzy parameters. The measures of necessity and possibility have been used to analyze the potential risk of the two scenarios. The research results indicated that uncertainty is a very important factor in water quality modeling. By incorporating uncertainty into model framework, the fuzzy model identified the highly risky scenario that was considered preferable based on solutions of the deterministic model.     

Modelling uncertainties in short-term reservoir operation using fuzzy sets and a genetic algorithm / Modélisation d’incertitudes dans la gestion de barrage à court terme grâce à des ensembles flous et à un algorithme génétique

Abstract

Abstract Various uncertainties are inherent in modelling any reservoir operation problem. Two of these are addressed in this study: uncertainty involved in the expression of reservoir penalty functions, and uncertainty in determining the target release value. Fuzzy set theory was used to model these uncertainties where the preferences of the decision maker for the fuzzified parameters are expressed as membership functions. Nonlinear penalty functions are used to determine the penalties due to deviations from targets. The optimization was performed using a genetic algorithm with the objectives to minimize the total penalty and to maximize the level of satisfaction of the decision maker with fuzzified input parameters. The proposed formulation was applied to the problem of finding the optimal release and storage values, taking Green reservoir in Kentucky, USA as a case study. The approach offers more flexibility to reservoir decision-making by demonstrating an efficient way to represent subjective uncertainties, and to deal with non-commensurate objectives under a fuzzy multi-objective environment.     

Fuzzy rule based seismic risk assessment of one-story precast industrial buildings

Efficient tools capable of using uncertain data to produce fast and approximate results are more practical in rapid decision-making applications when compared to conventional methods. From this point of view, this study introduces a risk assessment model for one-story precast industrial buildings by fuzzy logic which builds a bridge between uncertainty and precision. The input, output and relations of the fuzzy based risk assessment model(FBRAM) were determined by reference buildings. The Monte Carlo simulation method was used to handle uncertainties associated with the structural characteristics of the reference buildings. Section dimension, longitudinal reinforcement ratio, column height related to building elevation, confinement ratio and seismic hazard are regarded as input and the plastic demand ratio is considered as the output parameter by the mathematical formulation of strength and deformation capacity of the buildings. The supervised learning method was used to determine the membership function of fuzzy sets. Fuzzy rules of FBRAM were constructed from Monte Carlo simulation by mapping of inputs and output. FBRAM was evaluated by a group of simulated buildings and two existing precast industrial buildings. Comparisons have shown significant agreement with analytical model results in both cases. Consequently, it is anticipated that the proposed model can be used for the seismic risk mitigation of precast buildings.     

Non-linear fuzzy-set based uncertainty propagation for improved DO prediction using multiple-linear regression

In this paper, a new non-linear fuzzy-set based methodology is proposed to characterize and propagate uncertainty through a multiple linear regression (MLR) model to predict DO using flow and water temperature as the regressors. The output is depicted as probabilistic rather than deterministic and is used to calculate the risk of low DO concentration. To demonstrate the new method, data from the Bow River in Calgary, Alberta from 2006 to 2008 are used. Low DO concentration has been occasionally observed in the river and correctly predicting, and quantifying the associated uncertainty and variability of DO is of interest to the City of Calgary. Flow, temperature and DO data were used to construct five MLR models, using different combinations of linear and non-linear fuzzy membership functions. The results show that non-linear representation of variance is superior to the linear approach based on model performance. Normal and Gumbel based membership functions produced the best results. The outputs from two non-linear fuzzy membership models were used to calculate risk of low DO. The predicted risk was between 3.9 and 4.9 %. This is an improvement over the traditional method, which can not indicate a risk of low DO for the same time period. This study demonstrates that water resource managers can adequately use MLR models to predict the risk of low DO using abiotic factors.     

Global sensitivity analysis for stochastic ground motion modeling in seismic-risk assessment

Seismic risk assessment requires adoption of appropriate models for the earthquake hazard, the structural system and for its performance, and quantification of the uncertainties involved in these models through appropriate probability distributions. Characterization of the seismic hazard comprises undoubtedly the most critical component of this process, the one associated with the largest amount of uncertainty. For applications involving dynamic analysis this hazard is frequently characterized through stochastic ground motion models. This paper discusses a novel, global sensitivity analysis for the seismic risk with emphasis on such a stochastic ground motion modeling. This analysis aims to identify the overall (i.e. global) importance of each of the uncertain model parameters, or of groups of them, towards the total risk. The methodology is based on definition of an auxiliary density (distribution) function, proportional to the integrand of the integral quantifying seismic risk, and on comparison of this density to the initial probability distribution for the model parameters of interest. Uncertainty in the rest of the model parameters is explicitly addressed through integration of their joint auxiliary distribution to calculate the corresponding marginal distributions. The relative information entropy is used to quantify the difference between the compared density functions and an efficient approach based on stochastic sampling is introduced for estimating this entropy for all quantities of interest. The framework is illustrated in an example that adopts a source-based stochastic ground motion model, and valuable insight is provided for its implementation within structural engineering applications.     

Bayesian flood frequency analysis in the light of model and parameter uncertainties

The specific objective of the paper is to propose a new flood frequency analysis method considering uncertainty of both probability distribution selection (model uncertainty) and uncertainty of parameter estimation (parameter uncertainty). Based on Bayesian theory sampling distribution of quantiles or design floods coupling these two kinds of uncertainties is derived, not only point estimator but also confidence interval of the quantiles can be provided. Markov Chain Monte Carlo is adopted in order to overcome difficulties to compute the integrals in estimating the sampling distribution. As an example, the proposed method is applied for flood frequency analysis at a gauge in Huai River, China. It has been shown that the approach considering only model uncertainty or parameter uncertainty could not fully account for uncertainties in quantile estimations, instead, method coupling these two uncertainties should be employed. Furthermore, the proposed Bayesian-based method provides not only various quantile estimators, but also quantitative assessment on uncertainties of flood frequency analysis.     

Environmental decision-making under uncertainty using intuitionistic fuzzy analytic hierarchy process (IF-AHP)

Analytic hierarchy process (AHP) is a utility theory based decision-making technique, which works on a premise that the decision-making of complex problems can be handled by structuring them into simple and comprehensible hierarchical structures. However, AHP involves human subjective evaluation, which introduces vagueness that necessitates the use of decision-making under uncertainty. The vagueness is commonly handled through fuzzy sets theory, by assigning degree of membership. But, the environmental decision-making problem becomes more involved if there is an uncertainty in assigning the membership function (or degree of belief) to fuzzy pairwise comparisons, which is referred to as ambiguity (non-specificity). In this paper, the concept of intuitionistic fuzzy set is applied to AHP, called IF-AHP to handle both vagueness and ambiguity related uncertainties in the environmental decision-making process. The proposed IF-AHP methodology is demonstrated with an illustrative example to select best drilling fluid (mud) for drilling operations under multiple environmental criteria.     

Model and parameter uncertainty in IDF relationships under climate change

Quantifying distributional behavior of extreme events is crucial in hydrologic designs. Intensity Duration Frequency (IDF) relationships are used extensively in engineering especially in urban hydrology, to obtain return level of extreme rainfall event for a specified return period and duration. Major sources of uncertainty in the IDF relationships are due to insufficient quantity and quality of data leading to parameter uncertainty due to the distribution fitted to the data and uncertainty as a result of using multiple GCMs. It is important to study these uncertainties and propagate them to future for accurate assessment of return levels for future. The objective of this study is to quantify the uncertainties arising from parameters of the distribution fitted to data and the multiple GCM models using Bayesian approach. Posterior distribution of parameters is obtained from Bayes rule and the parameters are transformed to obtain return levels for a specified return period. Markov Chain Monte Carlo (MCMC) method using Metropolis Hastings algorithm is used to obtain the posterior distribution of parameters. Twenty six CMIP5 GCMs along with four RCP scenarios are considered for studying the effects of climate change and to obtain projected IDF relationships for the case study of Bangalore city in India. GCM uncertainty due to the use of multiple GCMs is treated using Reliability Ensemble Averaging (REA) technique along with the parameter uncertainty. Scale invariance theory is employed for obtaining short duration return levels from daily data. It is observed that the uncertainty in short duration rainfall return levels is high when compared to the longer durations. Further it is observed that parameter uncertainty is large compared to the model uncertainty.     

Transmissivity and storage coefficient estimation by coupling the Cooper–Jacob method and modified fuzzy least-squares regression

Traditionally the Cooper–Jacob equation is used to determine the transmissivity and the storage coefficient for an aquifer using pump test results. This model, however, is a simplified version of the actual subsurface and does not allow for analysis of the uncertainty that comes from a lack of knowledge about the heterogeneity of the environment under investigation. In this paper, a modified fuzzy least-squares regression (MFLSR) method is developed that uses imprecise pump test data to obtain fuzzy intercept and slope values which are then used in the Cooper–Jacob method. Fuzzy membership functions for the transmissivity and the storage coefficient are then calculated using the extension principle. The supports of the fuzzy membership functions incorporate the transmissivity and storage coefficient values that would be obtained using ordinary least-squares regression and the Cooper–Jacob method. The MFLSR coupled with the Cooper–Jacob method allows the analyst to ascertain the uncertainty that is inherent in the estimated parameters obtained using the simplified Cooper–Jacob method and data that are uncertain due to lack of knowledge regarding the heterogeneity of the aquifer.     

Joint Monte Carlo and possibilistic simulation for flood damage assessment

A joint Monte Carlo and fuzzy possibilistic simulation (MC-FPS) approach was proposed for flood risk assessment. Monte Carlo simulation was used to evaluate parameter uncertainties associated with inundation modeling, and fuzzy vertex analysis was applied for promulgating human-induced uncertainty in flood damage estimation. A study case was selected to show how to apply the proposed method. The results indicate that the outputs from MC-FPS would present as fuzzy flood damage estimate and probabilistic-possibilistic damage contour maps. The stochastic uncertainty in the flood inundation model and fuzziness in the depth-damage functions derivation would cause similar levels of influence on the final flood damage estimate. Under the worst scenario (i.e. a combined probabilistic and possibilistic uncertainty), the estimated flood damage could be 2.4 times higher than that computed from conventional deterministic approach; considering only the pure stochastic effect, the flood loss would be 1.4 times higher. It was also indicated that uncertainty in the flood inundation modeling has a major influence on the standard deviation of the simulated damage, and that in the damage-depth function has more notable impact on the mean of the fitted distributions. Through applying MC-FPS, rich information could be derived under various α-cut levels and cumulative probabilities, and it forms an important basis for supporting rational decision making for flood risk management under complex uncertainties.     

Sensitivity of seismic hazard evaluations to uncertainties determined from seismic source characterization

The sensitivity and overall uncertainty in peak ground acceleration (PGA)estimates have been calculated for the city of Tabriz, northwestern Iran byusing a specific randomized blocks design. Eight seismic hazard models andparameters with randomly selected uncertainties at two levels have beenconsidered and then a linear model between predicted PGA at a givenprobability level and the uncertainties has been performed. The inputmodels and parameters are those related to the attenuation, magnituderupture-length and recurrence relationships with their uncertainties.Application of this procedure to the studied area indicates that effects ofthe simultaneous variation of all eight input models and parameters on thesensitivity of the seismic hazard can be investigated with a decreasingnumber of computations for all possible combinations at a fixed annualprobability. The results show that the choice of a mathematical model ofthe source mechanism, attenuation relationships and the definition ofseismic parameters are most critical in estimating the sensitivity of seismichazard evaluation, in particular at low levels of probability of exceedance.The overall uncertainty in the expected PGA for an annual probability of0.0021 (10% exceedence in 50 yr) is expressed by a coefficient ofvariation (CV) of about 34% at 68% confidence level for a distance ofabout 5km from the field of the major faults. The CV will decrease withincreasing site-source distance and remains constant, CV = 15%, fordistances larger than 15 km. Finally, treating alternative models on theoverall uncertainty are investigated by additional outliers in input decision.     

Two-stage fuzzy chance-constrained programming: application to water resources management under dual uncertainties

In this study, a two-stage fuzzy chance-constrained programming (TFCCP) approach is developed for water resources management under dual uncertainties. The concept of distribution with fuzzy probability (DFP) is presented as an extended form for expressing uncertainties. It is expressed as dual uncertainties with both stochastic and fuzzy characteristics. As an improvement upon the conventional inexact linear programming for handling uncertainties in the objective function and constraints, TFCCP has advantages in uncertainty reflection and policy analysis, especially when the input parameters are provided as fuzzy sets, probability distributions and DFPs. TFCCP integrates the two-stage stochastic programming (TSP) and fuzzy chance-constrained programming within a general optimization framework. TFCCP incorporates the pre-regulated water resources management policies directly into its optimization process to analyze various policy scenarios; each scenario has different economic penalty when the promised amounts are not delivered. TFCCP is applied to a water resources management system with three users. Solutions from TFCCP provide desired water allocation patterns, which maximize both the system’s benefits and feasibility. The results indicate that reasonable solutions were generated for objective function values and decision variables, thus a number of decision alternatives can be generated under different levels of stream flows, α-cut levels and fuzzy dominance indices.