Introduction to partial order theory exemplified by the Evaluation of Sampling Sites

The first part of this chapter gives a detailed introduction to partial order ranking and Hasse Diagram Technique (HDT). Thus, the construction of Hasse diagrams is elucidated as is the different concepts associated with the diagrams. The analysis of Hasse diagrams is disclosed including structural analysis, dimension analysis and sensitivity analysis. Further the concept of linear extensions is introduced including ranking probability and averaged rank. The evaluation of sampling sites is, in the second part of the chapter, used as an illustrative example of the advantageous use of partial order ranking and Hasse Diagram Technique.

When a ranking of some objects (chemicals, geographical sites, river sections etc.) by a multicriteria analysis is of concern, it is often difficult to find a common scale among the criteria and therefore even the simple sorting process is performed by applying additional constraints, just to get a ranking index. However, such additional constraints, often arising from normative considerations are controversial. The theory of partially ordered sets and its graphical representation (Hasse diagrams) does not need such additional information just to sort the objects.

Here, the approach of using partially ordered sets is described by applying it to a battery of tests on sediments of the Lake Ontario. In our analysis we found: (1) the dimension analysis of partially ordered sets suggests that there is a considerable redundancy with respect to ranking. The partial ranking of the sediment sites can be visualized within a two-dimensional grid. (2) Information, obtained from the structure of the Hasse diagram: For example six classes of sediment sites have high priority, each class exhibits a different pattern of results. (3) The sensitivity analysis identifies one test as most important, namely the test for Fecal Coliforms/

Escherichia coli

. This means that the ranking of samples is heavily influenced by the results of this specific test.