A rigourous mathematical framework for computing a sustainability ratio: The emergy

TL;DR: The Track Summing method of Tennenbaum as mentioned in this paper was developed precisely for interconnected networks with feedbacks and splits of emergy and it is partially parallelizable, but it is not well suited to account for the co-product problem which induces the idempotent operator max.

Abstract: The computational problem of emergy within a general system of interconnected processes at steady state is a subject of interest in literature today. When there is no co-product the proposed method coincides with the Track Summing Method of Tennenbaum which was developed precisely for interconnected networks with feedbacks and splits of emergy. As the underlying algebraic structure of the Tennenbaum's method is the linear algebra, it is not well-suited to account for the co-product problem which induces the idempotent operator max. Thus, authors have chosen another underlying algebraic structure which is the idempotent semiring structure (i.e. a semiring equipped with an idempotent addition). This method is divided into two parts. The first part is the emergy flow enumeration, where paths from an emergy source to the input of a given process are enumerated avoiding double counting of emergy assignation. This part is a path-finding problem which is a slight modification of gerbier of null square approach to find elementary/simple paths in a graph. The second part evaluates the emergy flowing between two components of the system. It is a quantitative part in which the problem of avoiding double counting split and co-product flows are dealt with by introducing a way to mark splits and co-products flows. The method is partially parallelizable. However, the method enumerates paths on a graph thus, in worst cases, its complexity is not polynomial. This paper provides a rigorous framework based on an axiomatic basis to conduct the emergy evaluation of an emergy graph. © 2012 ISEIS All rights reserved.