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Continuous models combining slacks-based measures of efficiency and super-efficiency

Vicente J. Bolos, Rafael Benitez, Vicente Coll-Serrano

Abstract

In the framework of data envelopment analysis (DEA), Tone (2001) introduced the slacks-based measure (SBM) of efficiency, which is a nonradial model that incorporates all the slacks of the evaluated decision-making units (DMUs) into their efficiency scores, unlike classical radial efficiency models. Next, Tone (2002) developed the SBM super-efficiency model in order to differentiate and rank efficient DMUs, whose SBM efficiency scores are always $1$. However, as pointed out by Chen (2013), some interpretation problems arise when the so-called super-efficiency projections are weakly efficient, leading to an overestimation of the SBM super-efficiency score. Moreover, this overestimation is closely related to discontinuity issues when implementing SBM super-efficiency in conjunction with SBM efficiency. Chen (2013) and Chen et al. (2019) treated these problems, but they did not arrive to a fully satisfactory solution. In this paper, we review these papers and propose a new complementary score, called composite SBM, that actually fixes the discontinuity problems by counteracting the overestimation of the SBM super-efficiency score. Moreover, we extend the composite SBM model to different orientations and variable returns to scale, and propose additive versions. Finally, we give examples and state some open problems.

Continuous models combining slacks-based measures of efficiency and super-efficiency

Abstract

In the framework of data envelopment analysis (DEA), Tone (2001) introduced the slacks-based measure (SBM) of efficiency, which is a nonradial model that incorporates all the slacks of the evaluated decision-making units (DMUs) into their efficiency scores, unlike classical radial efficiency models. Next, Tone (2002) developed the SBM super-efficiency model in order to differentiate and rank efficient DMUs, whose SBM efficiency scores are always . However, as pointed out by Chen (2013), some interpretation problems arise when the so-called super-efficiency projections are weakly efficient, leading to an overestimation of the SBM super-efficiency score. Moreover, this overestimation is closely related to discontinuity issues when implementing SBM super-efficiency in conjunction with SBM efficiency. Chen (2013) and Chen et al. (2019) treated these problems, but they did not arrive to a fully satisfactory solution. In this paper, we review these papers and propose a new complementary score, called composite SBM, that actually fixes the discontinuity problems by counteracting the overestimation of the SBM super-efficiency score. Moreover, we extend the composite SBM model to different orientations and variable returns to scale, and propose additive versions. Finally, we give examples and state some open problems.

Paper Structure

This paper contains 15 sections, 8 theorems, 37 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Score functions and efficiency scores
  5. Original SBM models
  6. Global continuous SBM models
  7. The joint SBM model
  8. The continuous SBM model
  9. The composite SBM model
  10. Definitions and properties
  11. Computational aspects
  12. Extensions
  13. Examples
  14. Concluding remarks
  15. Appendix

Key Result

Proposition 3.1

The SBM efficiency score $\rho ^*|_P$ is strongly monotonic.

Figures (3)

  • Figure 1: a) Classically, a model is applied to a DMU (called DMU$_o$) in a given set of DMUs in order to obtain its score. b) Given a model and a set of reference DMUs, we construct a score function defined on activities. The image of an activity $\left( \mathbf{x},\mathbf{y}\right)$ is the score that the model would assign to a new hypothetical DMU with activity $\left( \mathbf{x},\mathbf{y}\right)$.
  • Figure 2: Diagrams representing binary logarithms of different scores (with respect to the reference DMUs of Example \ref{['ex81']}) of activities with a normalized output equal to $1$: (a) SBM efficiency in conjunction with S-SBM, (b) J-SBM, (c) CSBM, (d) CompSBM. The solid lines separate activities with score $<1$ from activities with score $\geq 1$, and dashed lines represent the weakly efficient frontier. In plots (b), (c) and (d), the zones between solid and dashed lines are the super-inefficiency zones, fixing the discontinuity issues in (c) and (d).
  • Figure 3: Details of super-inefficiency zones in Figure \ref{['figmapas']}.

Theorems & Definitions (27)

  • Proposition 3.1
  • Proposition 3.2
  • Remark 3.1: Strong monotonicity
  • Proposition 4.1
  • Proposition 4.2
  • Example 4.1
  • Remark 5.1: Unit-invariance
  • Proposition 5.1
  • Proposition 5.2
  • Proposition 5.3
  • ...and 17 more