In a recent Opinion, Schaubroeck (2022) argues that the methodology for life cycle assessment (LCA) should be reconstructed, allowing for “non-linear, dynamic, and integrated effects”, moving away from “linear inverse modeling,” as it “impede[s] more accurate quantification”. Schaubroeck (2022) extensively refers to our book The Computational Structure of Life Cycle Assessment (Heijungs and Suh, 2002) and suggests that our book presents the “linear inverse modeling” as the only conceivable approach to LCA computation.

In this response, we address three topics: (1) we clarify that our aforementioned book does not preclude other approaches to LCA computation than the linear approach, (2) we highlight other non-linear approaches to LCA in the existing literature, and (3) we comment on the proposed set-up by Schaubroeck (2022).

First, our book did not claim that “linear inverse modeling” is “thé mathematical framework.” Schaubroeck (2022) writes that the equation like h = QBA⁻¹f in Heijungs and Suh (2002) “should never have been introduced as thé mathematical framework” (italics and accent in original). We believe that this rendering is a simple misunderstanding.

We therefore disagree with the view by Schaubroeck (2022) that our book impeded the advancement of the field by presenting the matrix-based approach as the only conceivable approach to LCA.

Let us also add that the term “linear inverse modeling”, which Schaubroeck (2022) uses to describe our approach does not occur in Heijungs and Suh (2002). As far as we know, its first occurrence is by the author himself in Schaubroeck et al. (2013), citing Suh and Huppes (2005), who in turn do not use the term.

Second, non-linear LCA is not at all new. As a matter of fact, we contributed to the development of non-linear LCA. One of us (Suh) co-authored a recent paper on “Non-linearity in marginal LCA” (Qin et al., 2021) and the other (Heijungs) published another recent paper with an explicit section on “Nonlinear LCA” (Heijungs, 2020). Further, this journal (of which we are associate editor and chief editor) has an ongoing research topic “Non-linearity in Life Cycle Assessment”, which has so-far featured several interesting articles that discuss approaches for developing non-linear LCA (e.g., Li et al., 2020; Pizzol et al., 2021). We believe that it would be important to understand the contribution by Schaubroeck (2022) in the context of these ongoing contributions to non-linearity in LCA rather than as a lone Copernican Revolution as the author seems to suggest.

Third, the proposed approach by Schaubroeck is unclear. Schaubroeck (2022) presents a “basic framework”. The “basic general equation of a process” in Schaubroeck et al. (2021) is reproduced here:

{F_x, F_v, …} = p_p({F_y, F_z, …}, t, OC)

in which the F-terms are flow amounts, t is time and OC are “other condition parameters that in this case might influence the process”. Further, p_p is “a function that represents a certain process p (which may also be abstract),” according to Schaubroeck et al. (2021). First about the mathematical conventions in this notation: brackets, like {F_x, F_v, …} and {F_y, F_z, …}, indicate sets, and the idea of having a function that takes a set as an argument and returns a set as a function value is completely unknown to us. It would help if just a small example would have been given. We are afraid that the expression is incomprehensible due partly to the ‘unconventional' use of mathematical notations. Second, the function presented above is too abstract to be meaningful in our view. It was the British writer, Clive Staples Lewis, who famously said “You can make anything by writing.” Likewise, we suppose that one may well define an abstract function as the function that governs how the entire universe operates given the set of all relevant inputs. How useful it is to merely define such a function, however, is the core of the issue.

We hope that this commentary helps clarify some of the misunderstandings and confusions that readers of Schaubroeck (2022) may have encountered.