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 -     

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.     

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.     

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 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.     

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.     

Calculating scale elasticity in DEA models

In the data envelopment analysis (DEA) efficiency literature, qualitative characterizations of returns to scale (increasing, constant, or decreasing) are most common. In economics it is standard to use the scale elasticity as a quantification of scale properties for a production function representing efficient operations. Our contributions are to review DEA practices, apply the concept of scale elasticity from economic multi-output production theory to DEA piecewise linear frontier production functions, and develop formulas for scale elasticity for radial projections of inefficient observations in the relative interior of fully dimensional facets. The formulas are applied to both constructed and real data and show the differences between scale elasticities for the two valid projections (input and output orientations). Instead of getting qualitative measures of returns to scale only as was done earlier in the DEA literature, we now get a quantitative range of scale elasticity values providing more information to policy-makers.     

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).     

Stability of the classification of returns to scale in FDH models

This paper deals with the estimation of returns to scale (RTS) in free disposal hull (FDH) models and provides some stability intervals for preserving the RTS classification. It has been shown that the proposed stability intervals can be obtained via a polynomial-time algorithm based on the calculation of certain ratios of inputs and outputs, without solving any mathematical programming problem. The results of the study have been proved via some lemmas and theorems and have been illustrated by numerical examples and a real application.     

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.     

The utility of returns to scale in DEA programming: An analysis of Michigan rural hospitals

In 1984, Banker, Charnes, and Cooper introduced the capability of using data envelopment analysis to assess increasing, decreasing, or constant returns to scale. This analysis would appear to make an important contribution to the health care field because of the regulatory environment within which the industry exists and the competition among hospitals for additional services and capacity. In many states, hospitals must submit a “certificate of need” to prove eligibility to add capacity or services. Agency administrators at the state level should analyze each hospital's production performance to determine the effectiveness of resource utilization. Residents of a state where hospitals are regulated need to know the effectiveness of agencies in allowing resources to be properly allocated to hospitals. Returns to scale analysis can help provide answers to these concerns. We examine Michigan rural hospitals and propose a simple, yet logical procedure for evaluating returns to scale for technically inefficient hospitals.     

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.     

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.     

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.     

Returns to scale in dynamic DEA

Two different types of inputs (variable inputs and quasi-fixed inputs) are incorporated into an analytical framework of dynamic data envelopment analysis (DEA). A unique feature of the quasi-inputs is that those are considered as outputs at the current period, while being treated as inputs at the next period. The dynamic DEA can measure interdependency among consecutive periods. This study incorporates the concept of returns to scale into the dynamic DEA.     

Measuring returns to scale in DEA models when the firm is regulated

In the standard framework of data envelopment analysis (DEA) models, the returns to scale are fully characterized using the multiplier on the convexity constraint of inefficient decision making units (DMU) using the projection of the input–output vector on the frontier. In this note, we investigate how the returns to scale measurements in DEA models are affected by the presence of regulatory constraints. These additional constraints change the role played by the convexity constraint. In order to avoid biased estimation of the returns to scale, we show that the interaction between the regulatory and the convexity constraints has to be taken into account.     

Non-parametric tests of returns to scale

This paper discusses various statistics for testing hypotheses regarding returns to scale in the context of non-parametric models of technical efficiency. In addition, the paper presents bootstrap estimation procedures which yield appropriate critical values for the test statistics. Evidence on the true sizes and power of the various proposed tests is obtained from Monte-Carlo experiments. This paper is an extension of earlier work in [Manage. Sci. 44 (1998) 49; J. Appl. Statist. 27 (2000b) 779].