A New Approach to the Inventory Problem in Life Cycle Assessment

Environmental life cycle inventory and assessment is a method which is used to quantify the environmental load and effects associated with a product, process, or service. The scientific literature offers a significant number of case studies in which the Life Cycle Assessment (LCA) methodology is applied. However, the computational structure of LCA, despite being of paramount importance for a deep comprehension of the validity and reliability of the results obtained in the inventory analysis phase, is rarely taken into account. LCA practitioners are generally interested in guidelines on which data to collect, which choices to make, and how to report assumptions and results and thus scientific publications generally try to answer to those needs. Practitioners generally limit themselves to commercial LCA software used to accomplish the study and rarely delve into the mathematical details. The most widely used approach for the solution of the inventory problem is the matrix method. It determines the inventory vector related to a specific single or multifunctional unit process by solving a system of linear equations through the simple inversion of the so-called technology matrix A [1]. The first step for the calculation of the inventory vector of the studied process is represented by the determination of a scaling vector s, through the resolution of the system of equations As = f . The matrix A represents the flows within the economic system, and the vector f , called final demand vector or external demand vector, is an exogenously defined set of economic flows the operator requires that the system produce. After computing the scaling vector, it is possible to determine the inventory vector g (i.e. the vector containing all environmental flows associated with the reference process) by simply applying the equation g = Bs, where B (which is called intervention matrix ) represents the environmental interventions of the system of unit processes. The computation of the inventory vector related to the studied process is the goal of the Life Cycle Inventory (LCI) phase. When the technology matrix is square and invertible the solution of the problem is straightforward; however, when it is rectangular (with more rows than columns) the system is over-determined and a direct solution cannot be found. A rectangular data matrix is very common in LCI problems in different cases, such as: • Cut-off of economic flows • Multifunctional unit processes (processes delivering more than one valuable output)