Ole Bent Olesen

Identification and Use of Efficient Faces and Facets in DEA

This paper provides an outline of possible uses of complete information on the facial structure of a polyhedral empirical production possibility set obtained by DEA. It is argued that an identification of all facets can be used for a characterization of basic properties of the empirical production frontier. Focus is on the use of this type of information for (i) the specification of constraints on the virtual multipliers in a cone-ratio model, (ii) a characterization of the data generation process for the underlying observed data set, and (iii) the estimation of isoquants and relevant elasticities of substitution reflecting the curvature of the frontier. The relationship between the so-called FDEF approach and the cone-ratio model is explored in some detail. It is demonstrated that a decomposition of the facet generation process followed by the use of one of the available (exponential) convex hull algorithms allows for an explicit identification of the facial structure of the possibility set in fairly large DEA data sets. It is a main point to be made that the difficulties encountered for an identification of all facets in a DEA-possibility set can be circumvented in a number of empirical data sets and that this type of information can be used for a characterization of the structural properties of the frontier.

The Use of Data Envelopment Analysis with Probabilistic Assurance Regions for Measuring Hospital Efficiency

This paper uses Data Envelopment Analysis (DEA) for an estimation of the cost efficiency of 70 Danish hospitals. The analysis relates to a cost function based on 483 outputs in combination with a set of probabilistic assurance regions defined by the cost distributions for each output. It is demonstrated that the probabilistic assurance region approach allows for (i) a frontier estimation in the full output space, i.e., no fixed aggregation is required, and (ii) a controlling of the variation in heterogeneity of the output clusters, in casu Diagnosis Related Groups. The likelihood of the estimated efficiency score for a given hospital can be measured based on the sensitivity of the score w.r.t. the probability levels used in the specification of confidence intervals for the probabilistic assurance regions.

Stochastic Data Envelopment Analysis — A review

This paper provides a review of stochastic Data Envelopment Analysis (DEA). We discuss extensions of deterministic DEA in three directions: (i) deviations from the deterministic frontier are modeled as stochastic variables, (ii) random noise in terms of measurement errors, sample noise, and specification errors is made an integral part of the model, and (iii) the frontier is stochastic as is the underlying Production Possibility Set (PPS).

Efficiency analysis with ratio measures

In applications of data envelopment analysis (DEA) data about some inputs and outputs is often available only in the form of ratios such as averages and percentages. In this paper we provide a positive answer to the long-standing debate as to whether such data could be used in DEA. The problem arises from the fact that ratio measures generally do not satisfy the standard production assumptions, e.g., that the technology is a convex set. Our approach is based on the formulation of new production assumptions that explicitly account for ratio measures. This leads to the estimation of production technologies under variable and constant returns-to-scale assumptions in which both volume and ratio measures are native types of data. The resulting DEA models allow the use of ratio measures “as is”, without any transformation or use of the underlying volume measures. This provides theoretical foundations for the use of DEA in applications where important data are reported in the form of ratios.

Maintaining the Regular Ultra Passum Law in data envelopment analysis

The variable returns to scale data envelopment analysis (DEA) model is developed with a maintained hypothesis of convexity in input-output space. This hypothesis is not consistent with standard microeconomic production theory that posits an S-shape for the production frontier, i.e. for production technologies that obey the Regular Ultra Passum Law. Consequently, measures of technical efficiency assuming convexity are biased downward. In this paper, we provide a more general DEA model that allows the S-shape.

Efficiency measures and computational approaches for data envelopment analysis models with ratio inputs and outputs

In a recent paper to this journal, the authors developed a methodology that allows the incorporation of ratio inputs and outputs in the variable and constant returns-to-scale DEA models. Practical evaluation of efficiency of decision making units (DMUs) in such models generally goes beyond the application of standard linear programming techniques. In this paper we discuss how the DEA models with ratio measures can be solved. We also introduce a new type of potential ratio (PR) inefficiency. It characterizes DMUs that are strongly efficient in the model of technology with ratio measures but become inefficient if the volume data used to calculate ratio measures become available. Potential ratio inefficiency can be tested by the programming approaches developed in this paper.

Facet Analysis in Data Envelopment Analysis

Data Envelopment Analysis (DEA) employs mathematical programming to measure the relative efficiency of Decision Making Units (DMUs). One of the topics of this chapter is concerned with development of indicators to determine whether or not the specification of the input and output space is supported by data in the sense that the variation in data is sufficient for estimation of a frontier of the same dimension as the input output space. Insufficient variation in data implies that some inputs/outputs can be substituted along the efficient frontier but only in fixed proportions. Data thus locally support variation in a subspace of a lower dimension rather than in the input output space of full dimension. The proposed indicators are related to the existence of so-called Full Dimensional Efficient Facets (FDEFs). To characterize the facet structure of the CCR- or the BCC-estimators, (Charnes et al. Eur J Oper Res 2:429–444, 1978; Banker et al. Manage Sci 30(9):1078–1092, 1984) of the efficient frontier we derive a dual representation of the technologies. This dual representation is derived from polar cones. Relying on the characterization of efficient faces and facets in Steuer (Multiple criteria optimization. Theory, computation and application, 1986), we use the dual representation to define the FDEFs. We provide small examples where no FDEFs exist, both for the CCR- and the BCC estimator. Thrall (Ann Oper Res 66:109–138, 1996) introduces a distinction between interior and exterior facets. In this chapter we discuss the relationship between this classification of facets and the distinction in Olesen and Petersen (Manage Sci 42:205–219, 1996) between non-full dimensional and full dimensional efficient facets. Procedures for identification of all interior and exterior facets are discussed and a specific small example using Qhull to generate all facets is presented. In Appendix B we present the details of the input to and the output from Qhull. It is shown that the existence of well-defined marginal rates of substitution along the estimated strongly efficient frontier segments requires the existence of FDEFs. A test for the existence of FDEFs is developed, and a technology called EXFA that relies only on FDEFs and the extension of these facets is proposed, both in the context of the CCR-model and the BCC-model. This technology is related to the Cone-Ratio DEA. The EXFA technology is used to define the EXFA efficiency index providing a lower bound on the efficiency rating of the DMU under evaluation. An upper bound on the efficiency rating is provided by a technology defined as the (non-convex) union of the input output sets generated from FDEFs only. Finally, we review recent uses of efficient faces and facets in the literature.

A homothetic reference technology in data envelopment analysis

The assumption of a homothetic production function is often maintained in production economics. In this paper we explore the possibility of maintaining homotheticity within a nonparametric DEA framework. The main contribution of this paper is to use the approach suggested by Hanoch and Rothschild (1972) to define a homothetic reference technology. We focus on the largest subset of data points that is consistent with such a homothetic production function. We use the HR-approach to define a piecewise linear homothetic convex reference technology. We propose this reference technology with the purpose of adding structure to the flexible non-parametric BCC DEA estimator. Motivation for why such additional structure sometimes is warranted is provided. An estimation procedure derived from the BCC-model and from a maintained assumption of homotheticity is proposed. The performance of the estimator is analyzed using simulation.

Nonparametric Production Technologies with Multiple Component Processes

We develop a nonparametric methodology for assessing the efficiency of decision-making units operating in a production technology with several component processes. The latter is modeled by the new multiple hybrid returns-to-scale (MHRS) technology, formally derived from an explicitly stated set of production axioms. In contrast with the existing models of data envelopment analysis (DEA), the MHRS technology allows the incorporation of component-specific and shared inputs and outputs that represent several proportional (scalable) component production processes as well as nonproportional inputs and outputs. Our approach does not require information about the allocation of shared inputs and outputs to component processes or any assumptions about this allocation. We demonstrate the usefulness of the suggested approach in an application in the context of secondary education and also in a Monte Carlo study based on a simulated data generating process.

Testing over-representation of observations in subsets of a DEA technology

This paper proposes a test for whether data are over-represented in a given production zone, i.e. a subset of a production possibility set which has been estimated using the non-parametric Data Envelopment Analysis (DEA) approach. A binomial test is used that relates the number of observations inside such a zone to a discrete probability weighted relative volume of that zone. A Monte Carlo simulation illustrates the performance of the proposed test statistic and provides good estimation of both facet probabilities and the assumed common inefficiency distribution in a three dimensional input space. Potential applications include tests for whether benchmark units dominate more (or less) observations than expected.