| Abstract |
The purpose of this paper is to enhance the mathematical and physical understanding of practitioners of uncertainty analysis of life cycle inventory (LCI), on the application of possibility theory. The main questions dealt with are (1) clear definition of the terms—“necessity–possibility,” “probability,” “belief–plausibility,” and of their mutual relationships; (2) what justifies the substitution of classical probability for possibility; (3) mutual comparison of, and transformations in both senses between probability and possibility uncertainty measures; (4) how to construct meaningful input possibility measures from available probabilistic/statistic information; and (5) comparative analysis of the solutions of the problem of data uncertainty propagation in LCI, afforded, respectively, by probabilistic Monte Carlo simulation and possibilistic fuzzy interval arithmetic.The questions above are addressed from the rigorous mathematical formulations of the theories of probability and statistics, of possibility, and of random sets and belief/plausibility functions, although directed to LCI uncertainty analysis practitioners. On this respect, the paper allows two different levels of reading: a basic level (main text) and a deeper level (Electronic supplementary material). Particular tools used are (a) various transformations between possibility and probability distributions, in both senses, for the continuous case, proposed by Dubois et al. (e.g., Reliable Comp 10:273-297, 2004); (b) Monte Carlo simulation for either independent or dependent input random variables; (c) fuzzy interval arithmetic; and (d) Heijungs and Suh’s (2002) matrix formulation of LCI problems.The links among uncertainty measures, uncertain variables, and uncertainty analysis are cleared up. It is recalled how a probability measure can be constructed and attached to an input variable, and its probability distribution and unknown “correct value” be related, in a physically meaningful way. It is justified that, usually, a dual necessity–possibility measure has much less uncertainty information than a comparable probability measure. Although the specialists are not unanimous, it is opined that the theoretical framework developed by Dubois et al. (e.g., Reliable Comp 10:273-297, 2004) is the most convenient one to use in uncertainty analysis, to compare and mutually transform probability and possibility data. This is exemplified in (a) the transformation of the very common triangular possibility and normal standard probability distributions; (b) the general construction of possibility measures from different probability data previously available; and, above all, (c) the comparison of the output information of possibilistic and probabilistic uncertainty analyses of an LCI problem proposed by Tan (Int J Life Cycle Assess 13:585–592, 2008). The general problem of data uncertainty propagation through deterministic models (e.g., of LCI) is tackled with (1) classical probabilistic Monte Carlo simulation (for either independent or dependent input random variables); (2) possibilistic fuzzy interval arithmetic; and (3) hybrid methods (only mentioned).(1) The practical conditions in which an analysis of uncertainty should switch from a probability to a possibility basis are still ill-defined, but that seems to be the case when the input information is based on states of large ignorance. (2) A dual necessity–possibility uncertainty measure can be viewed both as an imprecise probability measure that substitutes a definite probability for an interval, and as a belief–plausibility measure. (3) A possibility distribution can be mathematically and physically interpreted as a random set of nested prediction/confidence intervals for the “correct value” of the variable, with confidence levels ranging from 0 to 1. (4) There exist mathematically and physically sound rules to compare/transform probability and possibility uncertainty information under different applicable paradigms (e.g., based on the reliability of the input information). (5) Sometimes, Geer and Klir’s (Int J Gen Syst 20:143–176, 1992) confidence index has a physically counterintuitive behavior in uncertainty analysis. (6) The probabilistic Monte Carlo simulation can be used also for dependent random input variables, only requiring more exigent input information (conditional probability distributions) than in the case of independency. (7) The possibilistic fuzzy interval arithmetic uncertainty analysis, although computationally cheap, generates output information quite poor—a good point estimate but a set of (roughly) confidence intervals with very large amplitudes for the “correct value” of each output variable.(1) In probability uncertainty analysis, pay attention to the relation between the “correct value” of a random variable and the parameters of its probability distribution (e.g., mean or mode). (2) Do not precipitate in changing from probabilistic to possibilistic uncertainty analysis: it may be theoretically unjustified and much output uncertainty information is lost. (3) Respect the well-established applicable rules in going from probability to possibility uncertainty information, or vice versa. (4) Be attentive to possible counterintuitive physical meaning of Geer and Klir’s (Int J Gen Syst 20:143–176, 1992) confidence index in possibilistic uncertainty analysis. |
