A relational input-output-model consists of an input space Ø≠V⊆R
, an output space Ø≠X⊆R
, and a total relation Q⊆VxX called the input-output relation. This input-output relation includes all tuples of input and output vectors which represent technologically feasible activities, and which in that sense form a technology set. In a given activity the output vector is interpreted as a list of gross production and gross disposal, while the input vector is interpreted as a list of gross consumption and gross emission. The term “technology” includes semantically not only classical production processes, but also contemporary disposal processes. The result of an activity is defined as the difference between the output vector and the input vector, and is the list of net production/disposal and net consumption/emission. The result of an activity is a list of expenditures and returns, and can be conceived as quantities of production factors and products.
Having defined the above, it is then possible to consider all activities through which a desired minimum result can be achieved. The goal here is to yield at least a minimum quantity of products while at the same time respecting a limit of the quantities of production factors. This restriction permits the definition of dual planning tasks in the direction of either activity minimization or activity maximization. Necessary and sufficient conditions are then given for their unique solvability. Finally models are dealt with in which either an input or an output function exists which models a technological process. These models ensure the unique solvability of one of the dual planning tasks.