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Frontier Visualization and Estimation of Returns to Scale in Free Disposal Hull Models

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Abstract

The free disposal hull (FDH) model of production technology assumes free disposability of all inputs and outputs, but does not make any convexity assumptions. In this paper, we develop an algorithm for the reconstruction of the frontier of the FDH technology that corresponds to its sections by different two-dimensional planes. From a practical perspective, the suggested algorithm is useful for the visualization and exploration of the efficient frontier of the FDH technology. Furthermore, based on the suggested algorithm, we develop a new procedure for the evaluation of returns to scale in the FDH technology. Compared to the existing methods, our approach does not require assessing the efficiency of productive units in the reference technologies, which is known to be a computationally intensive task. Our theoretical results are confirmed by computational experiments in applications with real-life data sets from different industries.

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Notes

  1. Banker, Cooper, Seiford and Zhu [27], Zelenyuk [28], Sahoo and Tone [29] and Krivonozhko, Førsund and Lychev [30] provide reviews of recent developments of the literature on RTS. Podinovski, Chambers, Atici and Deineko [31] extend the notion of RTS based on the one-sided scale elasticity to any technology which is a polyhedral set. Podinovski [23] develops a further extension of RTS to any convex technology.

  2. It is worth noting that, without any additional assumptions or heuristic rules, the standard notion of local RTS in the FDH technology is trivial and uninteresting. Indeed, at any efficient observed DMU in the FDH technology, the right-hand scale elasticity is equal to zero and the left-hand scale elasticity is undefined (or can be formally taken equal to \( + \infty \)). Therefore, every efficient DMU in the FDH technology exhibits decreasing RTS on the right and increasing RTS on the left. Cesaroni et al. [19] suggest an alternative definition of local RTS in the FDH technology which takes into account the ray average productivity of DMUs in some small but finite (not marginal) ε-neighbourhood of the DMU under the assessment. The resulting RTS characterization depends on the value of ε chosen by the analyst.

  3. By definition, DMU \(({{X}_{o}},{{Y}_{o}})\) is (strongly) efficient if there exists no DMU \((X',Y') \in {{T}_{{{\text{FDH}}}}}\) such that \(X' \leqslant {{X}_{o}}\), \(Y' \geqslant {{Y}_{o}}\), and \(({{X}_{o}},{{Y}_{o}}) \ne (X',Y')\). The assumption that DMU \(({{X}_{o}},{{Y}_{o}})\) is strongly efficient is not necessary and made only in order to simplify the exposition. For example, for the notion of (global) RTS to be correctly defined, it suffices to assume that DMU \(({{X}_{o}},{{Y}_{o}})\) is only output radial efficient in technology \({{T}_{{{\text{FDH}}}}}\). In order to evaluate RTS using the input-oriented programs (2) and (3), we need to assume that DMU \(({{X}_{o}},{{Y}_{o}})\) is both input and output radial efficient [16, pp. 243–244]. This is clearly a weaker assumption than the assumption of strong efficiency.

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ACKNOWLEDGMENTS

This study is supported by the Russian Science Foundation (project no. 17-11-01353). The authors are grateful to V.V. Podinovski whose suggestions have led to improvements of this paper.

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Authors and Affiliations

  1. National University of Science and Technology MISiS, 119049, Moscow, Russia

    V. E. Krivonozhko & A. V. Lychev

  2. The faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, GSP-1, 119991, Moscow, Russia

    V. E. Krivonozhko

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  1. V. E. Krivonozhko
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Krivonozhko, V.E., Lychev, A.V. Frontier Visualization and Estimation of Returns to Scale in Free Disposal Hull Models. Comput. Math. and Math. Phys. 59, 501–511 (2019). https://doi.org/10.1134/S0965542519030114

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