Abstract
The enhanced Russell graph measure, ERG, (Pastor et al., 1999) was designed as a new global efficiency measure to overcome the computational difficulties of the Russell graph measure of technical efficiency, RG (Färe et al., 1985). Historically, Farrell (1957) implemented the first measure of technical efficiency, while Färe and Lovell (1978), after suggesting some desirable properties that an ideal technical efficiency measure should satisfy, proposed the so-called Russell input measure of technical efficiency. An output version and a graph version were presented in the first book by Färe et al. (1985). These three initial Russell measures have been already presented in Chap. 5. The difficulty for solving the graph version motivated, in the last 1990s, the search for a new formulation easier to handle and to solve, and the proposed solution was the ERG measure. In the literature, two papers can be found, the first one by Pastor et al. (1999) and the second one by Tone (2001). Both proposed exactly the same linear fractional programming model for solving the Russell graph measure through the same reformulation. Hence, the ERG model was consecutively published twice with a gap of 2 years, which is something that occurs quite seldom in research. Pastor et al. (1999) published it first in the European Journal of Operational Research, and Tone (2001) published exactly the same models in the same journal under a different name, SBM, and related them also to the RG measure. It shows that two groups of researchers, without any interaction, may developed the same idea in a quasi-contemporary way without being aware of it. This is the reason why we refer to both measures, which are the same, as ERG=SBM and call it the “enhanced Russell graph slack-based measure.”
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Notes
- 1.
- 2.
With respect to the output-oriented measure, we only want to point out that, in Chap. 5 and for reasons of simplicity, instead of minimizing \( 1/\frac{1}{N}\sum \limits_{n=1}^N{\phi}_n \), we proposed directly to maximize its inverse \( \frac{1}{N}\sum \limits_{n=1}^N{\phi}_n \).
- 3.
As usual, just by modifying the equality in the convexity constraint, \( \sum \limits_{j=1}^J{\lambda}_j=1 \), we get the NIRS (≤) and the NDRS (≥). If we want to get the CRS version, as originally formulated by Pastor et al. (1999), we just need to delete the convexity constraint. Model (7.1) was the only one not considered by Tone (2001).
- 4.
Equalities (7.2) and the CRS version of model (7.3) appeared in Pastor et al. (1999) referenced as (3) and (4). Moreover, the CRS version of model (7.3) appeared in Tone (2001) referenced as (7). It should be clear that from the beginning we must assume that all input and output values are positive.
- 5.
The new variable is equal to the inverse of the denominator of the objective function; see next section.
- 6.
The inequality β ≥ 0 is a must for having a well-defined linear program as (7.5), although we know that, due to its definition, β is always a positive number.
- 7.
Geometrically, the intuition is confirmed: when using any type of slack-based model, the distance to the frontier can be measured along the L1 path defined by the corresponding slacks.
- 8.
In Aparicio et al. (2017a), the normalization procedure was not derived directly, as explained later on. Moreover, until now, the usual way of relating profit inefficiency with technical inefficiency has been obtaining a suitable Fenchel-Mahler inequality based on duality results. Here, we are proposing a more direct approach taking advantage of the final inequality obtained in the last-mentioned paper. However, developing this process has inspired us for proposing an alternative way of getting a profit inefficiency decomposition, as proposed later on in Chap. 13 based on Pastor et al. (2021b).
- 9.
The inequality is an equality if, and only if, the projection \( \left({\hat{x}}_o,{\hat{y}}_o\right) \) achieves maximum profit.
- 10.
The normalized profit inefficiency was originally called by Chambers et al. (1998) the Nerlovian profit inefficiency in honor of the economist Marc Nerlove, who was the first to suggest that profit inefficiency must be normalized in order to obtain a dimensionless number that is comparable to technical inefficiency, which, as already shown, is also a pure number.
- 11.
According to Proposition 1 in Aparicio et al. (2017a), the other decomposition was achieved realizing that the numerator of TIERG = SBM(xo, yo), \( \sum \limits_{n=1}^N\frac{1}{Ny_{on}}{s}_n^{+}+\sum \limits_{m=1}^M\frac{1}{Mx_{om}}{s}_m^{-} \), is a weighted additive function. The inequality obtained was similar to (7.9) but less tightened since the normalizing denominator was simply \( {\delta}_{\left({x}_o,{y}_o,p,w\right)} \). In the last-mentioned paper, this first decomposition was used for deriving the second decomposition (7.9) that we have derived here directly.
- 12.
Not all the firms generate multiple optimal solutions, as our next example shows.
- 13.
It is well known that the profit-maximizing point(s) must be located close to the NW corner of the sample, which intuitively means “less inputs and more outputs.” As a consequence, a non-efficient point can never be a profit maximizing point.
- 14.
An output-weak efficient point is a point whose projection on the strong efficient frontier by means of the additive model only identifies nonzero output slacks.
- 15.
It may happen that the projection identified by either of the last two models gets an optimal projection with an output value \( \sum \limits_{j=1}^J{\lambda}_j^{\ast }{y}_j\ge {y}_o \) but different from yo. But since \( \left({\hat{x}}_o,\sum \limits_{j=1}^J{\lambda}_j^{\ast }{y}_j\right)\ge \left({\hat{x}}_o,{y}_o\right) \) and we are sure that the obtained projection belongs to T, the properties of T guarantee that \( \left({\hat{x}}_o,{y}_o\right) \) also belongs to T, and consequently, it belongs to L(yo).
- 16.
Since the measure we are using is a strong efficient measure, we can use to classify the firms the—simplest—additive model or, alternatively, draw a picture in the two-input plane to see geometrically the position of the firms; see Fig. 7.2.
- 17.
Let us point out that the normalizing factor of the traditional approach does only depend on the input components of the firm being rated and on their market costs, which shows that in case of multiple projections, any of them will give rise to the same normalized cost decomposition.
- 18.
Obviously, when dealing with single-value inefficiency measures, the negative conclusion cannot be formulated.
- 19.
An input-weak efficient point is a point whose projection on the strong efficient frontier by means of the additive model only identifies nonzero input slacks.
- 20.
It may happen that the projection identified by model (7.13) gets an optimal projection with an input value \( \sum \limits_{j=1}^J{\lambda}_j^{\ast }{x}_j\le {x}_o \) but different from xo. But since \( \left(\sum \limits_{j=1}^J{\lambda}_j^{\ast }{x}_j,{\hat{y}}_o\right)\le \left({x}_o,{\hat{y}}_o\right) \) and the obtained projection belong to T, the properties of T guarantee that \( \left({x}_o,{\hat{y}}_o\right) \) also belongs to T, and consequently, it belongs to P(xo).
- 21.
Since we are assuming a VRS technology, each optimal slack is upper bounded. Being more precise, for firm (xo, yo), its vector of optimal output slack s+∗ is upper bounded by Y − yo, where Y is the vector of maximum output values over the sample of firms.
- 22.
Since the measure we are using is a strong efficient measure, we can use to classify the firms the additive model or even draw a picture, like Fig. 7.3, in the two output planes to see geometrically the location of the firms.
- 23.
Since the normalizing factor is independent of the projection, we can choose any of them for the two mentioned firms with double projections. Hence, the uniqueness of the revenue inefficiency decompositions is guaranteed in this case, being independent of the assigned projection. We have add an asterisk to the projection of each of this two units in Table 7.3 to indicate that a second projection is possible.
- 24.
This property is always satisfied by the general direct approach, as we are going to show very soon.
- 25.
We refer the reader to Sect. 2.6.1 in Chap. 2 for the installation of the “Benchmarking Economic Efficiency” Julia package. All Jupyter Notebooks implementing the different economic models in this book can be downloaded from the reference site: http://www.benchmarkingeconomicefficiency.com.
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Pastor, J.T., Aparicio, J., Zofío, J.L. (2022). The Enhanced Russell Graph Measure (ERG=SBM): Economic Inefficiency Decompositions. In: Benchmarking Economic Efficiency. International Series in Operations Research & Management Science, vol 315. Springer, Cham. https://doi.org/10.1007/978-3-030-84397-7_7
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