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The Hölder Distance Functions: Economic Inefficiency Decompositions

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Part of the International Series in Operations Research & Management Science book series (ISOR,volume 315)

Abstract

Data Envelopment Analysis can determine both a technical efficiency score and benchmarking information on how to change inputs and outputs to reach the efficient frontier if the firm under evaluation is technically inefficient. All measures studied in this book resort to the determination of benchmarking information through the calculation of the farthest targets for the evaluated DMU—“farthest” in the sense that the measure corresponds to the maximum value of the sum of slacks of the corresponding model or some variation. In the case of the (weighted) additive model discussed in Chap. 6, for example, this is clearly shown in the objective function of program (6.1). The objective function of this mathematical model consists of the weighted sum of the slack in each dimension (input and output), i.e., the difference between the target located on the strongly efficient frontier and the evaluated unit. In the case of the radial measures, i.e., the Farrell measure of technical efficiency, the situation is not so evident. In DEA, it is not strange to apply a second stage when the radial model is utilized in order to determine Pareto-efficient targets from the projection point. This second stage exploits the additive model. Consequently, we are also maximizing slacks. This means that most of the traditional measures in DEA generate the farthest targets. In other words, they yield the targets that are the most difficult ones to be achieved for the firm/organization in the short run.

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Fig. 9.1
Fig. 9.2
Fig. 9.3
Fig. 9.4

Notes

  1. 1.

    For the norm ‖ . ‖h on Rg, the dual norm ‖ . ‖q on Rg is defined as \( {\left\Vert z\right\Vert}_q=\underset{{\left\Vert w\right\Vert}_h=1}{\max}\left\{\sum \limits_{j=1}^g{z}_j{w}_j\right\} \) (see, for example, Mangasarian, 1999).

  2. 2.

    A free academic license can be obtained from https://www.gurobi.com/downloads/end-user-license-agreement-academic/. Once registered, it is possible to download it from https://www.gurobi.com/downloads/gurobi-optimizer-eula/. Upon installation, it is possible to add the package “Gurobi.jl” to Julia by running the following commands: “,” “,” and “.”

  3. 3.

    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

  4. 4.

    Table 9.11 reports the optimal values of the lambda multipliers λj, obtained when solving program (9.36) with g = (gx, gy) = (0M, yο). For firm F, there are two optimal solutions with β = 0: λC= 1 and λF= 1. In the first case, shown in Table 9.11, the output constraint is satisfied as an inequality, while in the second case, it is satisfied as an equality.

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Pastor, J.T., Aparicio, J., Zofío, J.L. (2022). The Hölder Distance Functions: 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_9

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