ORIGINAL PAPER E. Kentel  M. M. Aral Probabilistic-fuzzy health risk modeling Abstract Health risk analysis of multi-pathway expo- sure to contaminated water involves the use of mecha- nistic models that include many uncertain and highly variable parameters. Currently, the uncertainties in these models are treated using statistical approaches. How- ever, not all uncertainties in data or model parameters are due to randomness. Other sources of imprecision that may lead to uncertainty include scarce or incom- plete data, measurement error, data obtained from ex- pert judgment, or subjective interpretation of available information. These kinds of uncertainties and also the non-random uncertainty cannot be treated solely by statistical methods. In this paper we propose the use of fuzzy set theory together with probability theory to incorporate uncertainties into the health risk analysis. We identify this approach as probabilistic-fuzzy risk assessment (PFRA). Based on the form of available information, fuzzy set theory, probability theory, or a combination of both can be used to incorporate parameter uncertainty and vari- ability into mechanistic risk assessment models. In this study, tap water concentration is used as the source of contamination in the human exposure model. Ingestion, inhalation and dermal contact are considered as multiple exposure pathways. The tap water concentration of the contaminant and cancer potency factors for ingestion, inhalation and dermal contact are treated as fuzzy variables while the remaining model parameters are treated using probability density functions. Combined utilization of fuzzy and random variables produces membership functions of risk to individuals at different fractiles of risk as well as probability distributions of risk for various alpha-cut levels of the membership function. The proposed method provides a robust approach in evaluating human health risk to exposure when there is both uncertainty and variability in model parameters. PFRA allows utilization of certain types of information which have not been used directly in existing risk assessment methods. Keywords Human health risk  Probability theory  Fuzzy set theory  Exposure  Uncertainty  Variability 1 Introduction Risk assessment has been used to quantify the human health impacts due to exposure to contaminated water via multiple exposure routes such as ingestion (drink- ing), inhalation (breathing volatilized contaminants during showering) and dermal contact (contact of con- taminated water with skin, for example while shower- ing). The goal of risk assessment is to estimate the severity and likelihood of harm to human health from exposure to a substance or activity that under plausible circumstances can cause harm to human health. Quan- titative risk characterization involves evaluating expo- sure estimates against a benchmark of toxicity, such as a cancer slope factor. Risk is calculated by multiplying the cancer potency factor (slope factor) of the toxic sub- stance by the dose an individual receives. Uncertainties in risk estimates may arise from many different sources such as measurement or estimation of parameters, natural variability in individual’s response, variability in environmental concentration of toxicants over time and space and unverifiable assumptions in dose-response models or extrapolations of the results of these models. For example, variability in individuals is not only due to response. Air intake or amount of water consumed per day may vary from person to person. Variability in environment is not only associated with concentration of toxicant in the medium, but also vola- tilization of contaminant from tap water to air as it varies as a function of temperature. In risk assessment studies, Stoch Envir Res and Risk Ass (2004) 18: 324–338 Ó Springer-Verlag 2004 DOI 10.1007/s00477-004-0187-3 E. Kentel Æ M. M. Aral (&) Multimedia Environmental Simulations Laboratory, School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, Georgia, USA e-mail: mustafa.aral@ce.gatech.edu Tel.: +/) 404–8942243 Fax: +/) 404–894-5111