Interfaces with Other DisciplinesA generalized multiplicative directional distance function for efficiency measurement in DEA
Introduction
Data envelopment analysis (DEA), introduced by Charnes et al., 1978, Charnes et al., 1979 and extended by Banker, Charnes, and Cooper (1984), is a linear programming (LP) based non-parametric methodology for measuring the relative efficiency of a set of homogenous decision making units (DMUs) with multiple inputs and outputs. DEA is used to construct a reference technology relative to which the efficiency of individual DMUs can be estimated. The frontier of the reference technology provides a conservative inner (empirical) approximation of a true but unobserved production function.
Based on the axiomatic approach and the types of convexity postulate, two different characterizations of a technology structure – piece-wise linear and piece-wise log-linear – are suggested for estimating the relative efficiency in the DEA literature. The piece-wise linear technology is constructed based on normal convexity postulate (Banker et al., 1984), which requires the marginal products of factor inputs to be non-increasing. The piece-wise log-linear technology is, however, constructed based on geometric convexity (log-convexity) postulate (Banker & Maindiratta, 1986). The motivation for introducing the log-linear technology is that, unlike the linear one, it allows for increasing, constant and decreasing marginal productivities along the production frontier. More precisely, this technology structure is flexible enough to simultaneously capture the three production features – convexity, linearity, and concavity – of a production function.
The piece-wise log-linear technology pertains to a class of DEA models, known as the multiplicative models, which were developed by Charnes, Cooper, Seiford, and Stutz (1982) based on the log-linear envelopment principle introduced by Banker, Charnes, Cooper, and Schinnar (1981) (For more details about the multiplicative models, see e.g., Chang and Guh, 1994, Seiford and Zhu, 1998, Sueyoshi and Chang, 1989, among others).
The directional distance function (DDF) of Chambers et al., 1996, Chambers et al., 1998 is a useful generalization of the Shephard’s (1970) distance functions. Using the concept of DDF, Peyrache and Coelli (2009) recently introduced the concept of multiplicative directional distance function (MDDF), which completely characterizes the technology, and serves as a measure of technical efficiency (TE). The MDDF also encompasses the hyperbolic, the modified hyperbolic (Cuesta, Lovell, & Zofio, 2009), and the Shephard’s input and output distance functions as special cases. However, if slacks are considered to be additional sources of inefficiency, then the MDDF is unable to yield a comprehensive measure of TE in the sense of Pareto–Koopmans (Koopmans, 1951); and hence, as a result, it is not strongly monotonic.
In this paper, first, we demonstrate the capability of the MDDF for evaluating the TE of DMUs operating on the non-concave and non-convex frontier regions of the technology by redefining the MDDF relative to the log-linear technology. We also propose a DEA formulation for the MDDF, and a computational method for its estimation. Considering slacks as the additional sources of inefficiency, we then extend the MDDF to a generalized multiplicative directional distance function (GMDDF), which provides a complete characterization of the technology. This GMDDF yields a comprehensive measure of TE as it satisfies several desirable properties of an ideal efficiency measure such as strong monotonicity, unit invariance, translation invariance, and positive affine transformation invariance.
The GMDDF-based TE measure yields a straightforward interpretation – it is the product of the geometric means of the input and output efficiencies. As regards the practical advantages of our proposed TE measure, first, due to the flexibility in computer programming, the computations of TE scores and the projections of inefficient observations towards different facets of the efficient frontier can be easily provided in any standard DEA software. Second, it allows for the incorporation of decision makers’ preferences into efficiency assessment, which will generate a more realistic TE measure by assigning weights for individual inputs and outputs that are more consistent with the underlying objectives of decision makers.
While the GMDDF-based TE measure is non-radial in nature, the MDDF-based measure is radial. On comparison between the two measures, one can argue that the former is to be preferred to the latter, on the ground that the MDDF does not encapsulate all sources of inefficiency as it suffers from the problem of slacks, and the GMDDF has the flexibility to enable itself to better reflect the trade-offs between the inputs and/or outputs in its efficiency measures. However, purely on the axiomatic ground of efficiency measurement, the MDDF-based measure can be favored, given its economic interpretation for the class of log-convex monotonic technologies. Furthermore, from a purely practical point of view, the proportional interpretation of the radial MDDF measure facilitates the use of its benchmarking results in managerial contexts. For more details on the debate as to the choice of radial vis-a-vis non-radial measures, see Mahlberg and Sahoo, 2011, Sahoo and Acharya, 2010, Sahoo and Acharya, 2012, Sahoo et al., 2011, Sahoo and Tone, 2009a, Sahoo and Tone, 2009b, among others.
The remainder of the paper unfolds as follows. Section 2 deals in detail with the description of technology in an analytical framework, followed by an introduction of the MDDF-based TE measure. Section 3, first, develops the GMDDF-based measure of TE that accounts for all types of slacks; second, it describes a computational method for the practical execution of this measure, and finally, it presents a discussion of its properties. Section 4 deals with an illustrative empirical application on real-life data set of hardware computer companies in India for the period 2001–2010. Section 5 presents the conclusion with some observations and remarks.
Section snippets
Technology
Let us consider a technology involving n observed DMUs; each uses m inputs to produce s outputs. Let and be, respectively, the input and output vectors of DMUj, j ∈ J = {1, … , n}. The superscript T stands for a vector transpose. We further consider o as the index of the DMU under evaluation.
A production technology transforming an input vector into an output vector can be characterized by the technology set , defined as
Generalized multiplicative directional distance function
Consider the following definition. Definition 3.1.1 Let be an input–output vector and be a user-specified direction vector. Then, the functionis called the generalized multiplicative directional distance function (GMDDF).
The GMDDF possesses the following properties:
P.1 (Representation). completely characterizes T2, i.e., if and only if (x, y) ∈ T2.
P.2 (
An illustrative empirical application
For an illustrative empirical illustration, we analyze sample data of 20 hardware computer companies operating in India over the period 2001–2010, which was originally used and analyzed in an earlier study by Sahoo, Kerstens, and Tone (2012). The data consists of one output, i.e., gross sales, and three inputs – manufacturing cost, overhead cost and maintenance cost.
The computer hardware market comprises Indian branded players, MNC players, and assembly players. To introduce efficiency and
Concluding remarks
The analysis of efficiency relies extensively on the types of convexity postulate used in the construction of non-parametric technology. Two different characterizations of the non-parametric technology – piece-wise linear and piece-wise log-linear – are generally used to estimate the TE. While the linear technology requires normal convexity, the log-linear one uses geometric convexity. The latter technology is flexible enough to accommodate the three production structures – convexity,
Acknowledgements
We are grateful to Robert G. Dyson (Editor) and two anonymous referees of the journal ever for their constructive comments and valuable suggestions; and B.P. Mishra for his careful proofread of the manuscript. The usual disclaimer applies.
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