Estimating returns to scale in DEA

This paper addresses issues of returns to scale in Data Envelopment Analysis. Starting with the model developed by Banker, but avoiding Banker's conclusions on returns-to-scale, the paper shows how two close variants (inputs and outputs oriented) of the Banker-Charnes-Cooper model can be used to provide precise estimates of returns to scale. The estimation of returns to scale for each unit is done by testing the existence of solutions in four regions defined in the neighborhood of the analyzed unit. Numerical examples and graphs are used to illustrate the proposed procedures.     

A note on returns to scale in DEA

This brief note adds computational convenience and efficiency to the article by Banker and Thrall on returns to scale in DEA by modifying one of their suggestions to avoid the need for examining all alternate optima in order to reach a decision.     

Measurement of returns to scale in radial DEA models

A general approach is proposed in order to measure returns to scale and scale elasticity at projections points in the radial data envelopment analysis (DEA) models. In the first stage, a relative interior point belonging to the optimal face is found using a special, elaborated method. In previous work it was proved that any relative interior point of a face has the same returns to scale as any other interior point of this face. In the second stage, we propose to determine the returns to scale at the relative interior point found in the first stage.     

DEA cross-efficiency evaluation under variable returns to scale

Cross-efficiency evaluation in data envelopment analysis (DEA) has been developed under the assumption of constant returns to scale (CRS), and no valid attempts have been made to apply the cross-efficiency concept to the variable returns to scale (VRS) condition. This is due to the fact that negative VRS cross-efficiency arises for some decision-making units (DMUs). Since there exist many instances that require the use of the VRS DEA model, it is imperative to develop cross-efficiency measures under VRS. We show that negative VRS cross-efficiency is related to free production of outputs. We offer a geometric interpretation of the relationship between the CRS and VRS DEA models. We show that each DMU, via solving the VRS model, seeks an optimal bundle of weights with which its CRS-efficiency score, measured under a translated Cartesian coordinate system, is maximized. We propose that VRS cross-efficiency evaluation should be done via a series of CRS models under translated Cartesian coordinate systems. The current study offers a valid cross-efficiency approach under the assumption of VRS—one of the most common assumptions in performance evaluation done by DEA.     

Measurement of returns to scale using non-radial DEA models

There are some specific features of the non-radial data envelopment analysis (DEA) models which cause some problems for the returns to scale measurement. In the scientific literature on DEA, some methods were suggested to deal with the returns to scale measurement in the non-radial DEA models. These methods are based on using Strong Complementary Slackness Conditions from optimization theory. However, our investigation and computational experiments show that such methods increase computational complexity significantly and may generate as optimal, solutions contradicting optimization theory. In this paper, we propose and substantiate a direct method for the returns to scale measurement in the non-radial DEA models. Our computational experiments documented that the proposed method works reliably and efficiently on the real-life data sets.     

Measurement of returns to scale using a non-radial DEA model

Doklady Mathematics -     

On alternative optimal solutions in the estimation of returns to scale in DEA

Zhu and Shen [European Journal of Operational Research 81 (1995) 590] show that alternative optimal solutions in the estimation of returns to scale (RTS) are caused by a particular linear dependency among a set of extreme efficient DMUs when one employs the concept of most productive scale size [European Journal of Operational Research 17 (1984) 35] in data envelopment analysis (DEA). This paper demonstrates that the presence of weakly efficient DMUs may also lead to alternative optima and extends the results of Zhu and Shen to the entire frontier. Necessary and sufficient conditions for the presence of multiple optimal solutions for constant returns to scale (CRS) DMUs are established.     

Measurement of returns to scale using a non-radial DEA model: A range-adjusted measure approach

This research theoretically explores the measurement of returns to scale (RTS), using a non-radial DEA (data envelopment analysis) model. A range-adjusted measure (RAM) is used as a representative of such non-radial models. Historically, a type of RTS has been discussed within an analytical framework of radial models. The radial-based RTS measurement is replaced by the non-radial RAM/RTS measurement in this study. When discussing the non-radial RAM/RTS measurement, this study finds a problem of multiple projections that cannot be found in the radial measurement. In this research, a new linear programming approach is proposed to identify all efficient DMUs (decision making units) on a reference set. The important feature of the proposed approach is that it can deal with a simultaneous occurrence of (a) multiple reference sets, (b) multiple supporting hyperplanes and (c) multiple projections. All of the three difficulties are handled by the proposed RAM/RTS measurement. In particular, we discuss both when the three different types of multiple solutions occur on the RAM/RTS measurement and how to deal with such difficulties. Our research results make it possible to measure not only the type of RTS but also the magnitude of RTS in the RAM measurement.     

DEA congestion and returns to scale under an occurrence of multiple optimal projections

Economic implications of congestion have been recently discussed in many DEA (data envelopment analysis) studies. In addition, several previous research efforts have explored a theoretical linkage between returns to scale (RTS) and the concept of congestion, because the two economic concepts are closely connected to each other. Tone and Sahoo [Tone, K., Sahoo, B.K., 2004. Degree of scale economies and congestion: A unified DEA approach. European Journal of Operational Research 158, 755–772] have published the theoretical linkage in this journal. All of the previous studies, including their research (2004), assume a unique optimal solution in the investigation on DEA-based congestion. When multiple solutions occur in DEA-based congestion measurement, the economic implications of congestion obtained from the previous research are all problematic from both theoretical and practical perspectives. To deal with the issue, this study explores how to deal with the occurrence of multiple solutions in the DEA-based congestion measurement. This study proposes a new approach for the congestion measurement and theoretically compares the proposed approach with Tone and Sahoo (2004).     

Characterizing and finding full dimensional efficient facets in DEA: a variable returns to scale specification

The full dimensional efficient facets (FDEFs) of a production possibility set (PPS) play a key role in data envelopment analysis (DEA). Finding the FDEFs has been the subject of intensive research over the past decade. The available algorithms for finding the FDEFs in the current DEA literature either require information about the position of all the extreme efficient decision-making units on the facets of the PPS or knowledge of all extreme optimal solutions of the multiplier form of the BCC model. In this article, we develop an algorithm that does not require such crucial information that may not be easily available. To this purpose, we first carefully analyse the structure of the FDEFs of PPS with BCC technology, using basic concepts of polyhedral set theory. We then utilize this information to devise an algorithm for finding the FDEFs, using mixed integer linear programming. We illustrate our algorithm using a set of real data.     

On characterizing full dimensional weak facets in DEA with variable returns to scale technology

The frontier of the Production Possibility Set (PPS) consists of two types of full dimensional facets, efficient and weak facets. Identification of all facets of the PPS can be used in sensitivity and stability analysis, to find the closet target for inefficient Decision-Making Units (DMUs), and to determine the status of returns to scale of a DMU, among others. There has been a surge of articles on determining efficient facets in recent years. There are, however, many cases where knowledge of weak facets is required for a thorough analysis. This is the case, in particular, when the frontier of the PPS is constructed only of weak facets. The existing algorithms for finding weak facets either require knowledge of all extreme directions of the PPS or applicable only under some restrictions on the position of weak efficient DMUs. We provide a complete characterization of weak facets. Using this characterization, we then devise a different algorithm to find weak facets. We illustrate our algorithm using a numerical example.     

Constant and variable returns to scale DEA models for socially responsible investment funds

In order to evaluate the performance of socially responsible investment (SRI) funds, we propose some models which use data envelopment analysis (DEA) and can be computed in all phases of the business cycle. These models focus on the most crucial elements of an investment in mutual funds.     

An additive model approach for estimating returns to scale in imprecise data envelopment analysis

In this paper, additive model is used to provide an alternative approach for estimating returns to scale in data envelopment analysis. The proposed model is developed in both stochastic and fuzzy data envelopment analysis. Deterministic (crisp) equivalents are obtained which correspond to the stochastic and fuzzy models. Numerical examples are, also, used to illustrate the proposed approaches.     

A dual simplex-based method for determination of the right and left returns to scale in DEA

This study reviews the concept of the “right” and the “left” returns to scale (RTS) in data envelopment analysis (DEA), and a dual simplex-based method, for determining these two notions in RTS, is proposed, which has computational advantages as compared to the customary method.     

Linear programming solutions and distance functions under a constant returns to scale technology

In this paper, we investigate the various relationships among the linear programming solutions of data envelopment analysis (DEA) models under a constant returns to scale technology. We derive the analytical relationships among the efficiency measures and the activity variables for four separate models: the input-based, the output-based, the hyperbolic, and the proportional distance functions. We apply our results in order to derive a test of consistency that can be used in assessing the returns to scale among differing DEA models.     

Duality in data envelopment analysis under constant returns to scale

This paper explores duality in models of data envelopment analysis(DEA) for assessing the productive efficiencies of organizationalunits where efficient production is characterized by constantreturns to scale. The paper identifies dualityof the spacesin which efficiency is measured and discusses the practicalimplications of duality in DEA.     

Chapter 11 Simulation studies of efficiency,returns to scale and misspecification with nonlinear functions in DEA

Using statistically designed experiments, 12,500 observations are generated from a 4-pieced Cobb-Douglas function exhibiting increasing and decreasing returns to scale in its different pieces. Performances of DEA and frontier regressions represented by COLS (Corrected Ordinary Least Squares) are compared at sample sizes ofn=50, 100, 150 and 200. Statistical consistency is exhibited, with performances improving as sample sizes increase. Both DEA and COLS generally give good results at all sample sizes. In evaluating efficiency, DEA generally shows superior performance, with BCC models being best (except at corner points), followed by the CCR model and then by COLS, with log-linear regressions performing better than their translog counterparts at almost all sample sizes. Because of the need to consider locally varying behavior, only the CCR and translog models are used for returns to scale, with CCR being the better performer. An additional set of 7,500 observations were generated under conditions that made it possible to compare efficiency evaluations in the presence of collinearity and with model misspecification in the form of added and omitted variables. Results were similar to the larger experiment: the BCC model is the best performer. However, COLS exhibited surprisingly good performances — which suggests that COLS may have previously unidentified robustness properties — while the CCR model is the poorest performer when one of the variables used to generate the observations is omitted.     

Improving the identification of returns to scale in data envelopment analysis

In order to assess the efficiencies of a set of production units, it is necessary to identify the nature of returns to scale which characterise efficient production. Some methods have been developed to test the nature of the scale elasticity across the full range of scale sizes. However these tests are heavily weighted by the majority of the units and may not identify small ranges of scale size where different returns to scale hold. This paper develops a procedure based on a combination of Data Envelopment Analysis and regression analysis to identify the ranges of scale size where the returns to scale may differ from those in other ranges for the single-output, multi-input case. We also develop a measure of scale size across different input mixes.     

An additive utility mixed integer programming model for nonlinear discriminant analysis

Mathematical programming (MP) discriminant analysis models can be used to develop classification models for assigning observations of unknown class membership to one of a number of specified classes using values of a set of features associated with each observation. Since most MP discriminant analysis models generate linear discriminant functions, these MP models are generally used to develop linear classification models. Nonlinear classifiers may, however, have better classification performance than linear classifiers. In this paper, a mixed integer programming model is developed to generate nonlinear discriminant functions composed of monotone piecewise-linear marginal utility functions for each feature and the cut-off value for class membership. It is also shown that this model can be extended for feature selection. The performance of this new MP model for two-group discriminant analysis is compared with statistical discriminant analysis and other MP discriminant analysis models using a real problem and a number of simulated problem sets.