Rigorous proof of fuzzy error propagation with matrix-based LCI

Abstract

Propagation of parametric uncertainty in life cycle inventory (LCI) models is usually performed based on probabilistic Monte Carlo techniques. However, alternative approaches using interval or fuzzy numbers have been proposed based on the argument that these provide a better reflection of epistemological uncertainties inherent in some process data. Recent progress has been made to integrate fuzzy arithmetic into matrix-based LCI using decomposition into α-cut intervals. However, the proposed technique implicitly assumes that the lower bounds of the technology matrix elements give the highest inventory results, and vice versa, without providing rigorous proof. This paper provides formal proof of the validity of the assumptions made in that paper using a formula derived in 1950. It is shown that an increase in the numerical value of an element of the technology matrix A results in a decrease of the corresponding element of the inverted matrix A –1, provided that the latter is non-negative. It thus follows that the assumption used in fuzzy uncertainty propagation using matrix-based LCI is valid when A –1 does not contain negative elements. In practice, this condition is satisfied by feasible life cycle systems whose component processes have positive scaling factors. However, when avoided processes are used in order to account for the presence of multifunctional processes, this condition will be violated. We then provide some guidelines to ensure that the necessary conditions for fuzzy propagation are met by an LCI model. The arguments presented here thus provide rigorous proof that the algorithm developed for fuzzy matrix-based LCI is valid under specified conditions, namely when the inverse of the technology matrix is non-negative. This paper thus gives the conditions for which computationally efficient propagation of uncertainties in fuzzy LCI models is strictly valid.