Duality, classification and slacks in DEA

This paper presents seven theorems which expand understanding of the theoretical structure of the Charnes-Cooper-Rhodes model of Data Envelopment Analysis, especially with respect to slacks and the underlying structure of facets and faces. These theorems also serve as a basis for new algorithms which will provide optimal primal and dual solutions that satisfy the strong complementary slackness conditions for many (if not most) non-radially efficient DMUs; an improved procedure for identifying the set E of extreme efficient DMUs; and may, for many DEA domains, also settle in a single pass the existence or non-existence of input or output slacks in each of their DMUs. This paper also introduces the concept of a positive goal vector G, which is applied to characterize the set of all possible maximal optimal slack vectors. The appendix C presents an example which illustrates the need for a new concept, face regular, which focuses on the role of convexity in the intersections of radial efficient facets with the efficient frontier FR. The same example also illustrates flaws in the popular ‘sum of the slacks’ methodology.