On the "rigorous proof of fuzzy error propagation with matrix-based LCI"

This short note re-examines the proof of fuzzy error propagation with matrix-based LCI in Heijungs and Tan (Int J Life Cycle Assess 15:1014-1019, 2010) paper (referred to as HT hereafter), published in this journal. We provide counter examples to the claims made therein, point out the key error in their proof and identify correct sufficient conditions under which the largest (smallest) values of the given alpha-cuts of fuzzy numbers in the technology matrix yield the smallest (largest) scaling factors. HT uses iterative perturbations of a matrix to seemingly provide a rigorous proof of this result. Flaws in their arguments are identified and demonstrated by way of a counterexample. A classical result on monotonic property of the inverse of M-matrices leads to the correct sufficient conditions under which HT result holds. Since counter examples can be found, the result stated in HT is not, in general, guaranteed. As claimed in the HT paper, checking the upper and lower bounds of alpha-cuts may not be sufficient to describe the uncertainty (the full range of values) in the final inventory. However, slightly stronger conditions on the fuzzy technology matrix provide these inventory bounds.