New indices for analysing partial ranking diagrams

Interest is growing in decision making strategies and several techniques are now available. The assessment of priorities is a typical premise before final decisions are taken. Total and partial order ranking (POR) strategies, which from a mathematical point of view are based on elementary methods of discrete mathematics, appear as an attractive and simple tool to asses priorities. Despite the well-known total ranking strategies, which are scalar methods combining the different criteria values into a global index which always ranks elements in an ordered sequence, the partial order ranking is a vectorial approach which recognises that not all the elements can be directly compared with all the others. In fact when many criteria are considered, contradictions in the ranking are bound to exist and the higher the number of criteria, the higher the probability that contradictions in the ranking occur. The Hasse diagram technique (HDT) is a very useful tool to perform partial order ranking. The results of the partial order ranking are visualised in a diagram, called Hasse diagram. Incomparable elements are located at the same geometrical height and as high as possible in the diagram, thus incomparable elements are arranged in levels. The quality of a ranking procedure has to be evaluated by a deep analysis and by several indices, i.e. scalar functions that describe features of an ordered set and allow comparison among different rankings. For this purpose, new indices for ranking analysis are proposed here, compared with the ones found in literature and tested on theoretical examples and on real data.

[1]  A. Hartmann,et al.  Use of hasse diagram technique for evaluation of phospholipid fatty acids distribution as biomarkers in selected soils , 1995 .

[2]  Rainer Brüggemann,et al.  An algebraic/graphical tool to compare ecosystems with respect to their pollution. The German River Elbe as an example. I : Hasse-diagrams , 1994 .

[3]  B. Münzer,et al.  A graph-theoretical tool for priority setting of chemicals , 1993 .

[4]  Efraim Halfon,et al.  Comparison of an index function and a vectorial approach method for ranking waste disposal sites , 1989 .

[5]  Rainer Brüggemann,et al.  A Theoretical Concept To Rank Environmentally Significant Chemicals , 1999, J. Chem. Inf. Comput. Sci..

[6]  B. Mogensen,et al.  Pesticide leaching assessment method for ranking both single substances and scenarios of multible substance use , 1998 .

[7]  Rainer Brüggemann,et al.  Application of the Concept of Partial Order on Comparative Evaluation of Environmental Chemicals , 1999 .

[8]  Rainer Brüggemann,et al.  An algebraic/graphical tool to compare ecosystems with respect to their pollution IV: Comparative regional analysis by boolean arithmetics , 1999 .

[9]  S. Pudenz,et al.  An algebraic/graphical tool to compare ecosystems with respect to their pollution by PB/CD III: Comparative regional analysis by applying a similarity index , 1998 .

[10]  Rainer Brüggemann,et al.  An algebraic/graphical tool to compare ecosystems with respect to their pollution II: Comparative regional analysis , 1994 .

[11]  Johann Gasteiger,et al.  Comparative Evaluation of Chemical and Environmental Online and CD-ROM Databases , 2000, J. Chem. Inf. Comput. Sci..

[12]  P. Lewi,et al.  Multicriteria decision making using Pareto optimality and PROMETHEE preference ranking , 1992 .

[13]  R. Brüggemann,et al.  An algebraic/graphical tool to compare ecosystems with respect to their pollution V: cluster analysis and Hasse diagrams. , 2000, Chemosphere.

[14]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[15]  Marcello G. Reggiani,et al.  On ranking chemicals for environmental hazard , 1986 .

[16]  Jean Pierre Brans,et al.  Multicriteria decision making: A case study , 1991 .

[17]  R. Brüggemann,et al.  Applying Hasse diagram technique for the evaluation of toxicological fish tests , 1995 .

[18]  R. Brüggemann,et al.  Application of formal concept analysis to evaluate environmental databases , 1997 .

[19]  R. Brüggemann,et al.  Ranking of aquatic effect tests using hasse diagrams , 1997 .

[20]  A. Smilde,et al.  Multicriteria decision making , 1992 .