Formulación y Aprobación

= 0.045 x

The variables and being zero, are omitted here. Variable x, has a somewhat odd position in this overview. For variables to the coefficients represent process characteristics and therefore are valid at all times. For variable x, this is not so: does not really represent a flow but is the net result of all in- and outflows of node 2. Its magnitude therefore is coincidental, and its coefficient is only valid for the one year, 1990.

In this case, only the inflows of the system are fixed, the other stocks and flows being determined by the specification of the relations between the stocks and flows. In a stationary state model, the missing equations (there are seven equations, but ten variables) are provided by balancing formulae:

- - = 0

This set of equations can be represented as a matrix equation, provided that, as in this case, all equations are linear:

o o o "

0

1

0

0

0

0

0 0 0 0

0

-0.045 0 0

-0.4

0 -0.4 0

0 0 -0.7

-0.8

0

0

0

0

0

0

1 0 0.2 0

0 0 0

0 0 0

0 0 0

"xf

"20"

or more compactly:

The description of the system as such a matrix equation opens possibilities for various types of analysis: the existence of solutions, the solution space, and the robustness of the solution can be studied by means of standard algebraic techniques, as is shown by Heijungs (1994) for the related product Life Cycle Assessment.

As stated above, in this case the interdependency of flows and stocks must be specified rather than the magnitude. This implies that other data are required: not so much data on products and contents, but data on the processes and the way substance flows are redirected by them, or in other words their distribution characteristics. Process data can be obtained from various sources: - Many environmental processes have been modelled far enough to serve as a basis for the

distribution characteristics on the environmental side. This is also true of processes related to agriculture: the efficiency of the minerals taken up by crop, and of the conversion from fodder to animal food products by cattle, for example, are quite well known (Veen et al. (1993) for example contains many of such data).

- For a number of production processes in the economy (for example, the ongoing series of documentation of industrial processes by the Dutch Ministry of the Environment), process data are available as mass balances or efficiency and emission factors, and can be translated into the right format.

In other cases, and this is true especially for economic processes "downstream", data are sorely lacking. Distribution factors then have to be estimated by other means:

- The bookkeeping results for a given year could be used, similarly to the economic or sectoral Input-Output Analysis modelling technique, to derive distribution factors by determining the contribution of the various input flows to the output (Leontief, 1966). In IOA such factors, directly calculated from the Input-Output table for a given year, are called technical coefficients. In SFA practice it is often more useful to determine the division of the input over the various output flows, which could be described as "reversed technical coefficients" or output coefficients. (1994) introduces the concept of "performance variables" which could be derived as such output coefficients, but could also be composed out of other information. These PVs can be used to comment on the efficiency of the process they refer to. The mass balancing principle can be used to estimate a missing flow by appointing it as a

balancing item. This has been applied in case studies, especially for estimating unknown but large quantities of generated solid waste and flows from producers to consumers.

A limited but adequate number of stocks or flows must be fixed, however, to be able to solve the set of equations. These given flows together then determine the magnitude of all the others. It depends on the purpose of the analysis which variables should be selected as the fixed variables. SFINX can be used as a static model. As such, it has been applied in the case studies referred to before. It has been used for two specific purposes: to analyze the origins of certain pollution problems, and to estimate the effectiveness of certain policy measures.

Origins analysis

For a pollutants policy, insight into the origins of pollution problems is essential. The origins of one specific problematical flow can be traced at several levels. In the conducted case studies, three levels are distinguished:

- direct causes, derived directly from the nodes balance (for example, one of the direct causes of the cadmium soil load is atmospheric deposition);

- the economic sectors, or environmental policy target groups, directly responsible for the problem, identified by following the path back from node to node to the point of emission (for example, waste incineration is one of the economic sectors responsible for the cadmium soil load);