Abstract-Random variability and imprecision are two distinct facets of the uncertainty affecting parameters that influence the assessment of risk. While random variability can be represented by probability distribution functions, imprecision (or partial ignorance) is better accounted for by possibility distributions (or families of probability distributions). Because practical situations of risk computation often involve both types of uncertainty, methods are needed to combine these two modes of uncertainty representation in the propagation step. A hybrid method is presented here, which jointly propagates probabilistic and possibilistic uncertainty. It produces results in the form of a random fuzzy interval. This paper focuses on how to properly summarize this kind of information; and how to address questions pertaining to the potential violation of some tolerance threshold. While exploitation procedures proposed previously entertain a confusion between variability and imprecision, thus yielding overly conservative results, a new approach is proposed, based on the theory of evidence, and is illustrated using synthetic examples.

Uncertainty is a major aspect of the estimation, using models, of the risk of human exposure to pollutants. The Monte Carlo method, which applies probability theory to address model parameter uncertainty, relies on a statistical representation of available information. In recent years, the theory of possibilities has been proposed as an alternative approach to address model parameter uncertainty in situations where available information are insufficient to identify statistically representative probability distributions, due in particular to data scarcity. In practice, it may occur that certain model parameters can be reasonably represented by probability distributions, because there is sufficient data available to substantiate such distributions by statistical analysis, while others are better represented by fuzzy numbers (due to data scarcity). The question then arises as to how these two modes of representation of model parameter uncertainty can be combined for the purpose of estimating the risk of exposure. In this paper an approach (termed a hybrid approach) for achieving such a combination is proposed, and applied to the estimation of human exposure, via vegetable consumption, to cadmium present in the surficial soils of an industrial site located in the north of France. The application illustrates the potential of the proposed approach, which allows the uncertainty affecting model parameters to be represented in a fashion which is consistent with the information at hand.

26 27Uncertainty analysis in LCA studies has been subject to major progress over the last years. In the context of waste 28 management, various methods have been implemented but a systematic method for uncertainty analysis of waste-29 LCA studies is lacking. The objective of this paper is (1) to present the sources of uncertainty specifically inherent to 30 waste-LCA studies, (2) to select and apply several methods for uncertainty analysis and (3) to develop a general 31 framework for quantitative uncertainty assessment of LCA of waste management systems. The suggested method is a 32 sequence of four steps combining the selected methods: (Step 1) a sensitivity analysis evaluating the sensitivities of 33 the results with respect to the input uncertainties, (Step 2) an uncertainty propagation providing appropriate tools for 34 representing uncertainties and calculating the overall uncertainty of the model results, (Step 3) an uncertainty 35 contribution analysis quantifying the contribution of each parameter uncertainty to the final uncertainty and (Step 4) 36 as a new approach, a combined sensitivity analysis providing a visualization of the shift in the ranking of different 37 options due to variations of selected key parameters. This tiered approach optimizes the resources available to LCA 38 practitioners by only propagating the most influential uncertainties. Waste management has during the last decade been subject to a range of life cycle assessment (LCA; described in 48 ISO, 2006) studies e.g. Damgaard et al. (2011, Lazarevic et al. (2010) and Pires et al. 49 (2011). The purposes of these studies have been to help quantifying, for example, where in the waste management 50 system the environmental loads and savings are taking place, which technologies are preferable under specific 51 conditions, or the balance between material and energy recovery. LCA-models specifically focusing on waste 52 management systems are available; see Gentil et al. (2010) for a review of the models. 53As for any LCA study, results are subject to uncertainty due to the combined effects of data variability, 54 erroneous measurements, wrong estimations, unrepresentative or missing data and modelling assumptions. 55Uncertainty is of two different natures: while epistemic uncertainty relates to an incomplete state of knowledge 56 (Hoffman and Hammonds, 1994), stochastic uncertainty originates from the inherent variability of the natural world. 57Such uncertainty can be spatial (e.g. when the farming practice of land receiving compost varies spatially) or 58 temporal (e.g. when the performance of a process varies with time). These two different natures of the uncertainty are 59 usually treated together and referred to by the term "uncertainty". 60 They found that stochastic modelling was the most frequently-used method to propagate uncertainties in LCA. This 75 method propagates probability distributions using random sampling like the Monte Carlo analysis. However, they 76 noted that many of the studies using such modelling...

This paper explores flows and stocks, at the scale of the European Union, of certain rare earth elements (REEs; Pr, Nd, Eu, Tb, Dy and Y) which are associated with products that are important for the decarbonisation of the energy sector and that also have strong recycling potential. Material flow analyses were performed considering the various steps along the value chain (separation of rare earth oxides, manufacture of products, etc.) and including the lithosphere as a potential stock (potential geological resources). Results provide estimates of flows of rare earths into use, in-use stocks and waste streams. Flows into use of, e.g., Tb in fluorescent lamp phosphors, Nd and Dy in permanent magnets and Nd in battery applications were estimated, for selected reference year 2010, as 35, 1230, 230 and 120 tons respectively. The proposed Sankey diagrams illustrate the strong imbalance of flows of permanent magnet REEs along the value chain, with Europe relying largely on the import of finished products (magnets and applications). It is estimated that around 2020, the amounts of Tb in fluorescent lamps and Nd in permanent magnets recycled each year in Europe, could be on the order of 10 tons for Tb and between 170 and 230 tons for Nd.

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