The comparison of stochastic and deterministic DEA models

The idea of deterministic data envelopment analysis (DEA) models is well-known. If we start to study the stochastic DEA models, it seems at first glance, that the ideas of these two models are totally different. The aim of this paper is to explain that there is a really nice connection between the ideas of stochastic and deterministic models. The idea of the piecewise-linear convex hull as a frontier estimation for effective units goes back to Farrell (1957) and it was consider by a few authors in following years. The main attention received this topic when Charnes, Cooper and Rhodes (1978) presented their paper, where the Data Envelopment Analysis (DEA) was introduced. Since then there were published a large number of papers and books which extended these results and DEA methodology - see for example Coelli (2009). The basic idea of DEA is the comparison of some units which are characterized by several inputs and outputs. The main aim of this analysis is to identify so-called efficient units. More precisely, the aim is to estimate frontier function and to measure the efficiencies of units relative to this estimated frontier. The other possible point of view at this problem comes from the decision making theory. In fact, we have some units and we know there evaluation according to several criteria, some of them are cost type, some of them are benefit type. Our aim is to find such units (alternatives in the terms of multiple attribute decision making), which are not dominated and which are Pareto efficient. The solving of DEA problems leads to the problem of mathematical programming, in case of deterministic models to linear programming, in case of stochastic model to non-linear programming problems. The paper is organized as follows. In the section 2 we introduce the basic notation necessary for the setting of these problems. Section 3 introduces the deterministic DEA models and the stochastic DEA models are given in section 4. To show the connection between deterministic and stochastic models, we present the deterministic models in non- classical way. We do not use a formulation based on the idea of optimization of efficiency - which is widely used for deterministic models, but we use a dual approach - an approach of dominated alternatives. This allows us to show a close connection between deterministic and stochastic formulation of the problem. 2 Methods