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Cross-Efficiency in Fuzzy Data Envelopment Analysis (FDEA): Some Proposals

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Performance Measurement with Fuzzy Data Envelopment Analysis

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 309))

Abstract

Different techniques have been proposed in the literature to rank decision making units (DMUs) in the context of Fuzzy Data Envelopment Analysis. In our opinion, those that result from using a ranking method to order the fuzzy efficiencies obtained are susceptible to a serious criticism: they are not based on objective criteria. Cross-efficiency evaluation was introduced as an extension of DEA aimed at ranking the DMUs. This methodology has found a significant number of applications and has been extensively investigated. In this chapter, we discuss some difficulties that arise with the definition of fuzzy cross-efficiencies and we propose a fuzzy cross-efficiency evaluation based on the FDEA model by Guo and Tanaka. Such model relies on the dual multiplier formulation of the CCR model and the fuzzy efficiency of a given DMU is defined in a ratio form in terms of the input and output weights obtained. This allows us to define the cross-efficiencies in an analogous manner to that of the fuzzy efficiency. The resulting cross-efficiencies are consistent in the sense that the cross-efficiency of a given DMU, calculated with its own input and output weights, is equal to the relative efficiency of this unit. We illustrate our methodology with an example.

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Notes

  1. 1.

    A triangular fuzzy number is, in particular, convex and its height is equal to 1. Therefore, according to Remarks 2.1 and 2.2 in Wang and Kerre [20], Y 2 is equal both to an index proposed by Campos and Muñoz [21] and the index of Liou and Wang [22] in the particular case of considering an optimism index equal to 1/2. Besides, it is straightforward to check that Y 2 also coincides with Fortemps and Roubens’ index [23]. Finally, Proposition 2.1 states also an equivalence of Y 2 with a third index proposed in Choobineh [24].

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Acknowledgments

This work has been partially supported by the Ministerio de Ciencia e Innovación of Spain, Research Projects TIN2009-14392-C02-01 and MTM2009-10479.

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Authors and Affiliations

  1. Centro de Investigación Operativa, Universidad Miguel Hernández, Alicante, Spain

    Inmaculada Sirvent

  2. Departament de Estadística e Investigación Operativa, Universitat de Valencia, Valencia, Spain

    Teresa León

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  1. Inmaculada Sirvent
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  2. Teresa León
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Correspondence to Teresa León .

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Editors and Affiliations

  1. Operations and Information Management Group, Aston Business School, Aston University, Birmingham, United Kingdom

    Ali Emrouznejad

  2. Business Systems and Analytics Department, La Salle University, Philadelphia, Pennsylvania, USA

    Madjid Tavana

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Sirvent, I., León, T. (2014). Cross-Efficiency in Fuzzy Data Envelopment Analysis (FDEA): Some Proposals. In: Emrouznejad, A., Tavana, M. (eds) Performance Measurement with Fuzzy Data Envelopment Analysis. Studies in Fuzziness and Soft Computing, vol 309. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41372-8_5

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  • DOI: https://doi.org/10.1007/978-3-642-41372-8_5

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