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This paper presents seven theorems which expand understanding of the theoretical structure of the Charnes­Cooper­Rhodes (CCR) 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 (SCSC) 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. [ABSTRACT FROM AUTHOR]

This paper considers the problem of finding minimal length tree networks on the unit sphere Φ of a given point set (V) where distance is measured along great circular arcs. The related problems of finding a Steiner Minimal Tree SMT (V) and of finding a Minimum Spanning Tree MST(V) are treated through a simplicial decomposition technique based on computing the Delaunay Triangulation DT (V) and the Voronoi Diagram VD(V) of the given point set. O (N log N) algorithms for computing DT(V), VD(V), and WST(V) as well as an O(N log N) heuristic for finding a sub-optimal SMT(V) solution are presented, together with experimental results for randomly distributed points on Φ. [ABSTRACT FROM AUTHOR]

The development of mathematical programming algorithms and heuristics has traditionally been hampered by the separation of modeller and optimiser. Developers either have to sacrifice performance by switching between a modeller and optimiser and communicating through cumbersome matrix files; or they can work solely with the in-core matrix in the optimiser, where knowledge about the structured algebraic model is missing and thus difficult to exploit. We present EMOSL, a new combined modelling and optimisation subroutine library that overcomes these drawbacks. It allows the problem to be manipulated using the notation of the model language, which is of course close to the algebraic formulation of the problem. Access is provided directly to the in-core matrix of the optimiser, eliminating the need to communicate via files. Thus the scope and ease of algorithm development is greatly improved, without any degradation of performance. We substantiate these claims by describing four applications: a block structure extraction tool, and the implementations of three MIP heuristics and cutting planes techniques. [ABSTRACT FROM AUTHOR]