Ranking with Data Envelopment Analysis vs. Partial Order

There are many ranking schemes of complex entities defined by multiple attributes. Their purpose is to determine which decision-making unit (DMU) is “better”, which one is “worse”, or if one unit “dominates” another. Most of those ranking endeavours require an initial groundwork of data, which frequently introduces some subjective restrictions to the analysis. To avoid any subjectivity, the best would be to use the “raw” numerical values of attributes with no normalization, standardization, aggregation, etc. In contrast with the widely used, conventional multi-dimensional multi-criteria decision support, the authors of this paper join those who say "let the data speak first ...". This idea is realized in practice only by the partial order theory. It uses the "raw" data and undeniably has the strongest mathematical basis. For ranking purposes, a graphical representation of a partial order in the form of a Hasse diagram is especially advantageous. It is obtained from the Hasse matrix, embodying the relations between all the DMUs. The present paper provides evidence that the ranking with the Data Envelopment Analysis (DEA), which is based on a concept of efficiency does not coincide with the ranking based on a mathematical notion of the partial order. Moreover, if all the attributes belong to the class of outputs (“the bigger the better”) or to the class of inputs (“the smaller the better”), the modified  algorithm of the DEA with outputs only or inputs only could be employed. If the attributes of the system belong to both those classes, the standard DEA algorithm for DMUs with outputs only (or inputs only) could be used after changing the values of inputs on opposite and considering them as outputs (or changing the values of outputs on opposite and considering them as inputs). Also, this procedure sanctions the use of standard programs for the partial order.