Allocation in life cycle inventory: partial set of solutions to an ill-posed problem

• Published in: The international journal of life cycle assessment 2014-11, Vol.19 (11), p.1854-1865
• Main Authors: Cruze, Nathan B., Goel, Prem K., Bakshi, Bhavik R.
• Format: Article
• Language: English
• Subjects: Allocations; Earth and Environmental Science; Environment; Environmental Chemistry; Environmental Economics; Environmental Engineering/Biotechnology; Ill posed problems; Inventories; Lci Methodology and Databases; Least squares method; Life cycle analysis; Life cycle engineering; Life cycles; Mathematical models; Matrix; Partitioning; Stockpiling
• ISSN: 0948-3349; 1614-7502
• DOI: 10.1007/s11367-014-0785-3

Background, aim, and scope The primary aim of this paper is to indicate that partitioning allocation methods yields only a small subset of solutions to an ill-posed problem that has potentially infinitely many exact solutions. It will be shown that each of the existing partitioning methods arrives at just one particular solution from among infinitely many solutions of an underdetermined system of linear equations . Materials and methods Some life cycle inventories fall into a class of functions called estimable functions in linear model framework, in which case they are invariant to allocation assumptions. This class of functions unites results described by Heijungs and Frischknecht (Int J Life Cycle Assess 3:321–332, 1998 ) and Heijungs and Suh ( 2002 , Conjecture 1, p. 91). The inventories for non-estimable functions obtained through allocation are, in fact, derived under a set of additional implicit equality constraints called side conditions, often resulting in inventory results which differ greatly from one allocation to the next. Results and discussions This paper explicates (1) identification of all estimable functions from any given technology matrix and (2) recovery of side conditions imposed on non-estimable functions through partitioning. These methods are illustrated in a simple example, and their relation to least squares techniques for allocation explored by Marvuglia et al. (Int J Life Cycle Assess 15:1020–1040, 2010 ) ;(Int J Agr Environ Inf Syst 3:51–71, 2012 ) are discussed. Conclusions and outlook Recommendations are made that may lead to more meaningful ways to obtain additional data or include additional information in life cycle inventories in the future.