Weighting intervals and ranking, exemplified by leaching potential of pesticides

Often a ranking based on a multi-indicator system is performed by construction of a composite indicator, which is generally computed as a weighted average of the indicators. The set of weighttuples is introduced: the g-space. Each point of this space represents a tuple of weight values, which lead together with the indicators of an object to a certain value of the composite indicator. The composite indicator induces a weak or linear order, when the associated space of an object set together with a data matrix is available. Each hypercube in the g-space, corresponding to intervals of the weights, can be represented by a partial order which is not necessarily a weak or even a linear order. Changing from one point of the hypercube to another will often not change the partial order. We have a freedom of changing weights according to the dimensionality of the g-space -1 (because of normalization). When other hypercubes are selected, then other partial orders can be found. The boundary between two hypercubes with different partial orders is a lower dimensional sphere, with fewer degrees of freedom. In the current paper we treat these two points: (i) how to use weight intervals to determine the resulting partial orders and (ii) if the number of indicators is not too large and if the focus is on a pair of objects instead on the whole set, then equations are given which can be helpful.