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Ranking with a Euclidean common set of weights in data envelopment analysis: with application to the Eurozone banking sector

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Abstract

This paper provides a new method to define a Euclidean common set of weights (ECSW) in data development analysis (DEA) that (1) allows ranking both efficient and inefficient firms, (2) is more realistic in terms of determination of weights, and (3) generates rankings for banks consistent with their credit ratings. We first use DEA to determine the efficient frontier and then estimate a common set of weights that can minimize the Euclidean distance between the firms and that frontier. This process is illustrated by a simple numerical example and is extended to a real-life situation using the Eurozone banking sector. Our ECSW approach outperforms other common set of weights approaches in both numerical and real-life examples, and in terms of providing rankings consistent with banks’ credit ratings.

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Notes

  1. Doyle (1995) used a cross-efficiency technique to define the CSW as the average of all optimal weights of all DMUs. This approach, however, is unable to fully account for the information contained in outputs and inputs (Kao and Hung 2005).

  2. The first two methods deal with weight restriction (because the weights for individual inputs or outputs for DMUs can be very low, allowing those inputs or outputs to be effectively ignored) and thus require additional information. The last two do not and thus are more appropriate within the general DEA context.

  3. Although for the case of L1 norm, the distance measured by Kao and Hung (2005) becomes identical to the model of maximizing the average efficiency scores of Roll and Golany (1993), or the model of maximizing individual efficiency scores of Jahanshaloo et al. (2005).

  4. Mathematically, a norm is a total size or length of all vectors in a vector space or matrices. Hence, the L1 norm is defined as the sum of the absolute values of its components, and the L2 norm is the square root of the sum of the squares of the absolute values of its components. Meanwhile, the L∞ norm is defined as the maximum of the absolute values of its components (Horn and Johnson 2012).

  5. Additionally, the L1 norm, also known as “city block” or Manhattan distance, is not the shortest distance between two points, in comparison to the Euclidean geometric distance of the L2 norm.

  6. This approach was also used in Wang et al. (2011), although they referred to it as a regression analysis.

  7. Although there is a constraint requiring the efficiency scores of other DMUs using that set of weights to also have to be less than or equal to one—see Eq. (1).

  8. In DEA, there must be at least one efficient DMU that lies on the frontier with efficiency score equal to one.

  9. If \( p = 1 \), we have D1 as the “city block” or Manhattan measure of distance (L1 norm). If \( p = 2 \), we have D2 as the Euclidean distance measure (L2 norm). And if \( p = \infty \), we have D∞ as the Chebyshev distance (L∞ norm).

  10. Wang et al. (2011) also used the L2 norm distance in calculating their CSW, although they called it the regression analysis method (RAM).

  11. DEA can only rank inefficient DMUs whilst super-efficiency DEA (Andersen and Petersen 1993) can rank efficient DMUs; they thus both belong to the DSW approach. The LP2008 approach is therefore only focused on ranking efficient DMUs.

  12. \( \Delta_{j}^{O} \) and \( \Delta_{j}^{I} \) are identical to ∆Y and ∆X in Fig. 1 (Panel B), respectively. Notice that this (artificial) virtual position lies on the slope 1.0 frontier, and thus has the characteristic of equal value in both coordinates, i.e. x = y. Consequently, the (absolute value) of the gaps should be equal as well, i.e. \( \left| {\Delta_{j}^{O} } \right| = \left| {\Delta_{j}^{I} } \right| \). This position, therefore, may not identical with the (optimal) virtual position using the DSW.

  13. Ramezani-Tarkhorani et al. (2014) argued that the second criteria should be achieved by comparing the virtual inputs, i.e. DMU uses less virtual input will be better, rather than the virtual gaps reduction.

  14. Although some weights may not be unique, the virtual inputs and outputs are. Since we focus on the virtual value, e.g. \( U_{j}^{*} y_{j} \), we therefore can ignore this uniqueness issue.

  15. The advantages of the ECSW approach shown in this study are consistent with an earlier version (Ngo and Tripe, 2014), although that earlier study used data for 2013 only for New Zealand banks.

  16. Readers are encouraged to look for more discussion regarding those variables, which are very popular in the banking (efficiency) literature, used in previous studies such as Altunbas et al. (2000), Chang et al. (2012), Kao and Liu (2013), and Ouenniche and Carrales (2018). There is no single agreed list of preferred variables, however, and other variables have been selected in other studies. As we discuss in Sect. 4.2 below, however, it is not obvious that a different set of variables would have any major impact on our results.

  17. The Spearman’s ranking correlations between those measurements are all statistically significant at 1% level but were not reported to save space.

  18. The t test for those differences or reductions are all statistically significant at 1% level but were not reported to save space.

  19. Similar argument, although not using credit rating scores, was used in Barr et al. (1994) and Bauer et al. (1998).

  20. Please note that banks with no credit rating scores are excluded and thus, the numbers of observations differ across the measurements (e.g. TR, SP or beta) and across the years.

  21. One can also argue that DEA rankings are close to those of Standard & Poor’s as the relationship between SP and all DEA efficiency scores, no matter which weighting approach is used, is always found to be significantly positive (see rows 8–11 of Table 8).

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Authors and Affiliations

  1. Rennes School of Business, Rennes, France

    Helmi Hammami

  2. School of Aviation, Massey University, Palmerston North, New Zealand

    Thanh Ngo

  3. VNU University of Economics and Business, Hanoi, Vietnam

    Thanh Ngo

  4. School of Economics and Finance, Massey University, Palmerston North, New Zealand

    David Tripe

  5. IPAG Business School, Paris, France

    Dinh-Tri Vo

  6. University of Economics Ho Chi Minh City, Ho Chi Minh City, Vietnam

    Dinh-Tri Vo

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  1. Helmi Hammami
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Correspondence to Thanh Ngo.

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Hammami, H., Ngo, T., Tripe, D. et al. Ranking with a Euclidean common set of weights in data envelopment analysis: with application to the Eurozone banking sector. Ann Oper Res 311, 675–694 (2022). https://doi.org/10.1007/s10479-020-03759-6

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